21.04.02 · number-theory / modular-forms

Hecke Operators and Hecke Algebra

shipped3 tiersLean: partial

Anchor (Master): Hecke 1936 *Math. Ann.* 112, 664-699 and Hecke 1937 *Math. Ann.* 114, 1-28 (originator pair); Petersson 1939 *Math. Ann.* 116 (Petersson inner product, self-adjointness); Atkin-Lehner 1970 *Math. Ann.* 185 (new and old forms, multiplicity-one); Shimura *Introduction to the Arithmetic Theory of Automorphic Functions* (Princeton 1971) Ch. 3; Deligne-Serre 1974 *Ann. Sci. ENS* 7 (Galois representations attached to weight-one eigenforms); Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 6; Diamond-Shurman Ch. 5

Intuition [Beginner]

A modular form $f$ is a complex-valued function on the upper half-plane that transforms in a controlled way under the action of the group $SL_{2} (Z)$ . The space $M_{k} (Γ)$ of weight- $k$ modular forms is finite-dimensional, and most interesting examples have a Fourier expansion $f (z) = a_{0} + a_{1} q + a_{2} q^{2} + a_{3} q^{3} + \dots$ where $q = e^{2 π i z}$ . The integers $a_{n}$ are the Fourier coefficients, and the central question is: what arithmetic structure do they carry?

Hecke's 1936 discovery is that the space $M_{k} (Γ)$ supports a family of linear operators $T_{p}$ — one for each prime $p$ — that average a modular form over sub-lattices of index $p$ in $Z^{2}$ . Two facts about these operators are surprising and central. First, the operators commute pairwise: $T_{m} T_{n} = T_{n} T_{m}$ for all $m$ and $n$ . Second, modular forms with rich arithmetic structure are simultaneous eigenvectors for every $T_{n}$ at once. A simultaneous eigenform $f$ satisfies $T_{p} f = a_{p} f$ for every prime $p$ , and the eigenvalue $a_{p}$ is exactly the $p$ -th Fourier coefficient of $f$ .

The consequence is striking. The Fourier coefficients of a modular eigenform are not independent integers; they are determined by the eigenvalues of a commuting family of arithmetic operators, and the Dirichlet series $a_{1} + a_{2} / 2^{s} + a_{3} / 3^{s} + \dots$ acquires a multiplicative factorisation over primes — a Euler product. This is the moment modular forms become an arithmetic object.

Visual [Beginner]

A two-panel picture. Left panel: the lattice $Z^{2}$ drawn as a grid in the plane, with one sub-lattice of index $p = 2$ highlighted (the even-coordinate sublattice). Three labelled sub-lattices of index $2$ are shown, each obtained by selecting a sub-grid of period two in different directions. Right panel: the same sub-lattice picture but now labelled with the slash action — each sub-lattice of $Z^{2}$ contributes one term to the operator $T_{p}$ , and the operator averages a modular form over all sub-lattices of index $p$ .

$A diagram with two panels: on the left, the integer lattice $\mathbb{Z}^2$ in the plane with three sub-lattices of index $p = 2$ highlighted in distinct colours; on the right, the same sub-lattices labelled with the corresponding terms $f(pz)$ and $f((z + b)/p)$ for $b = 0, 1$, summed by the Hecke operator $T_p$.$

The picture says: the Hecke operator $T_{p}$ at prime $p$ is the average of a modular form over the $p + 1$ sub-lattices of $Z^{2}$ of index $p$ , each sub-lattice contributing one term to the formula for $T_{p} f$ .

Worked example [Beginner]

Compute the Hecke eigenvalues of the discriminant cusp form. The discriminant $Δ (z) = q (1 - q)^{24} (1 - q^{2})^{24} (1 - q^{3})^{24} \dots$ is the unique normalised cusp form of weight $12$ on $SL_{2} (Z)$ , with Fourier expansion $Δ (z) = τ (1) q + τ (2) q^{2} + τ (3) q^{3} + \dots$ where $τ (n)$ is the Ramanujan tau function. The space $S_{12} (SL_{2} (Z))$ has dimension one, so $Δ$ is automatically an eigenform for every Hecke operator.

Step 1. The first few values of $τ$ are $τ (1) = 1$ , $τ (2) = - 24$ , $τ (3) = 252$ , $τ (4) = - 1472$ , $τ (5) = 4830$ .

Step 2. Since $Δ$ is a simultaneous eigenform of all $T_{n}$ , the eigenvalue identity $T_{n} Δ = τ (n) Δ$ must hold for every $n$ . The eigenvalue of $T_{p}$ on $Δ$ equals the $p$ -th Fourier coefficient $τ (p)$ .

Step 3. Check Hecke multiplicativity. The formula $τ (mn) = τ (m) τ (n)$ should hold for coprime $m, n$ , and at prime powers $τ (p) τ (p^{n}) = τ (p^{n + 1}) + p^{11} τ (p^{n - 1})$ . Verify at $p = 2$ , $n = 1$ : $τ (2) τ (2) = (- 24)^{2} = 576$ . The recursion predicts $τ (2) τ (2) = τ (4) + 2^{11} τ (1) = - 1472 + 2048 = 576$ . The two agree.

What this tells us: the Ramanujan tau function is multiplicative because the discriminant cusp form is a Hecke eigenform, and the recursion at prime powers is exactly the Hecke recursion $T_{p} T_{p^{n}} = T_{p^{n + 1}} + p^{k - 1} T_{p^{n - 1}}$ specialised to weight $k = 12$ .

Check your understanding [Beginner]

Formal definition [Intermediate+]

Fix an integer $k \geq 2$ and let $Γ = SL_{2} (Z)$ . Write $M_{k} (Γ)$ for the $C$ -vector space of modular forms of weight $k$ on $Γ$ — holomorphic functions $f : H \to C$ on the upper half-plane satisfying $f ((a z + b) / (cz + d)) = (cz + d)^{k} f (z)$ for every $(a c b d) \in Γ$ and holomorphic at the cusp $\infty$ . The subspace $S_{k} (Γ) \subseteq M_{k} (Γ)$ consists of cusp forms — those vanishing at $\infty$ .

Definition (Hecke operator at a prime). Let $p$ be a prime number. The Hecke operator $T_{p}$ on $M_{k} (Γ)$ is the linear endomorphism defined by $$ (T_p f)(z) := p^{k - 1} f(p z) + \frac{1}{p} \sum_{b = 0}^{p - 1} f \left( \frac{z + b}{p} \right). $$ The factor $p^{k - 1}$ in front of the first term and the factor $1/ p$ in front of the sum are the conventional normalisation chosen so that the eigenvalues of $T_{p}$ on a normalised eigenform agree with its Fourier coefficients. Equivalently, $T_{p} = \sum_{δ} f ∣_{k} δ$ as a sum of $∣_{k}$ -slash actions over a complete set of representatives ${δ}$ for $Γ\ {γ \in M_{2} (Z) : det (γ) = p}$ , the $p + 1$ left cosets of $Γ$ in the set of $2 \times 2$ integer matrices of determinant $p$ .

Definition (Hecke operator at a general positive integer). For $n \geq 1$ , the Hecke operator $T_{n}$ is the linear endomorphism $$ (T_n f)(z) := n^{k - 1} \sum_{\substack{a d = n \ a, d \geq 1}} \frac{1}{d^k} \sum_{b = 0}^{d - 1} f \left( \frac{a z + b}{d} \right), $$ the sum running over all factorisations $n = a d$ with $a, d \geq 1$ . Equivalently, $T_{n} = \sum_{δ} f ∣_{k} δ$ as a sum over representatives of $Γ\ {γ \in M_{2} (Z) : det (γ) = n}$ .

Definition (Hecke algebra). The Hecke algebra $T$ is the commutative subalgebra of $End_{C} (M_{k} (Γ))$ generated by all $T_{n}$ for $n \geq 1$ : $$ \mathbb{T} := \mathbb{C}[T_1, T_2, T_3, \ldots] \subseteq \mathrm{End}\mathbb{C}(M_k(\Gamma)). $$ The Hecke algebra is a finitely generated commutative algebra over $C$ (since $\mathrm{End}\mathbb{C}(M_k(\Gamma)) $i s f ini t e - d im e n s i o na l), an d t h es u ba l g e b r a g e n er a t e d o v er$ \mathbb{Z} $b y a l l$ T_n $i s t h e * * in t e g r a l H ec k e a l g e b r a * *$ \mathbb{T}_\mathbb{Z}$.

Definition (Hecke eigenform). A nonzero $f \in M_{k} (Γ)$ is a Hecke eigenform if $T_{n} f = a_{n} (f) f$ for every $n \geq 1$ , where $a_{n} (f) \in C$ are scalars. The eigenform $f$ is normalised if its first Fourier coefficient $a_{1} (f) = 1$ . The Fourier coefficient $a_{n} (f)$ from the $q$ -expansion $f (z) = \sum_{n \geq 0} a_{n} (f) q^{n}$ and the eigenvalue of $T_{n}$ on $f$ are the same scalar for a normalised eigenform.

Counterexamples to common slips [Intermediate+]

"The Hecke operator $T_{n}$ acts on $M_{k} (Γ)$ even when $Γ$ is a congruence subgroup like $Γ_{0} (N)$ for general $n$ ." The level-one statement $T_{n} \in End (M_{k} (SL_{2} (Z)))$ holds for every $n$ . At level $N$ , the operator $T_{p}$ is only defined as written for primes $p ∤ N$ ; at primes $p ∣ N$ , one substitutes the Atkin-Lehner involution $w_{p}$ or the truncated $U_{p}$ operator. Mixing the two conventions gives wrong eigenvalues.
"The Hecke algebra acts on the full space $M_{k} (Γ)$ and the Eisenstein part is irrelevant." The Hecke algebra preserves both $S_{k} (Γ)$ and the Eisenstein subspace $E_{k} (Γ)$ , with the decomposition $M_{k} (Γ) = S_{k} (Γ) \oplus E_{k} (Γ)$ being $T$ -stable. Eisenstein series are themselves Hecke eigenforms with eigenvalues $σ_{k - 1} (n) = \sum_{d ∣ n} d^{k - 1}$ , the divisor power sum.
"The recursion $T_{p} T_{p^{n}} = T_{p^{n + 1}} + p^{k - 1} T_{p^{n - 1}}$ holds with $T_{p^{0}} = T_{1} = id$ and $T_{p^{- 1}} = 0$ ." The recursion is valid at $n \geq 1$ with the convention $T_{p^{- 1}} = 0$ at the base case $n = 0$ , reducing to $T_{p}^{2} = T_{p^{2}} + p^{k - 1} T_{1}$ . The convention is what makes the multiplicative identity $\sum_{n} T_{n} / n^{s} = \prod_{p} (1 - T_{p} p^{- s} + p^{k - 1 - 2 s})^{- 1}$ work as a formal Euler product.

Key theorem with proof [Intermediate+]

The signature theorem of this unit is the eigenform-Fourier-coefficient identity: the eigenvalue of the Hecke operator $T_{n}$ on a normalised eigenform equals its $n$ -th Fourier coefficient.

Theorem (Hecke 1936 Math. Ann. 112). Let $f \in M_{k} (Γ)$ be a normalised Hecke eigenform with Fourier expansion $f (z) = \sum_{n \geq 0} a_{n} q^{n}$ and $a_{1} = 1$ . Let $λ_{n} \in C$ denote the eigenvalue of $T_{n}$ on $f$ , so $T_{n} f = λ_{n} f$ for all $n \geq 1$ . Then for every $n \geq 1$ , $$ \lambda_n = a_n. $$ Equivalently, the Fourier coefficients of a normalised Hecke eigenform form a multiplicative arithmetic function: $a_{mn} = a_{m} a_{n}$ when $g cd (m, n) = 1$ , and $a_{p} a_{p^{n}} = a_{p^{n + 1}} + p^{k - 1} a_{p^{n - 1}}$ at prime powers.

Proof. Compute the first Fourier coefficient of $T_{n} f$ from the defining formula $$ (T_n f)(z) = n^{k - 1} \sum_{a d = n} \frac{1}{d^k} \sum_{b = 0}^{d - 1} f \left( \frac{a z + b}{d} \right). $$ Substitute the Fourier expansion $f ((a z + b) / d) = \sum_{m \geq 0} a_{m} e^{2 π im (a z + b) / d} = \sum_{m \geq 0} a_{m} e^{2 π ima z / d} e^{2 π imb / d}$ . The sum over $b$ from $0$ to $d - 1$ kills all terms unless $d ∣ m$ , so writing $m = d m^{'}$ : $$ \sum_{b = 0}^{d - 1} f \left( \frac{a z + b}{d} \right) = d \sum_{m' \geq 0} a_{d m'} q^{a m'}. $$

Substituting back: $$ (T_n f)(z) = n^{k - 1} \sum_{a d = n} d^{1 - k} \sum_{m' \geq 0} a_{d m'} q^{a m'} = \sum_{a d = n} a^{k - 1} \sum_{m' \geq 0} a_{d m'} q^{a m'}, $$ using $n^{k - 1} d^{1 - k} = (a d)^{k - 1} d^{1 - k} = a^{k - 1}$ . Let $r = a m^{'}$ be the Fourier index of the result. Then $$ (T_n f)(z) = \sum_{r \geq 0} \left( \sum_{a | \gcd(n, r)} a^{k - 1} a_{n r / a^2} \right) q^r. $$

The coefficient of $q^{r}$ in $T_{n} f$ is therefore $$ b_r := \sum_{a | \gcd(n, r)} a^{k - 1} a_{n r / a^2}. $$ Now use the eigenform hypothesis: $T_{n} f = λ_{n} f$ , so $b_{r} = λ_{n} a_{r}$ for all $r$ . At $r = 1$ : $$ b_1 = \sum_{a | \gcd(n, 1)} a^{k - 1} a_{n / a^2} = a_n, $$ since the only positive divisor of $1$ is $1$ itself. Hence $λ_{n} a_{1} = a_{n}$ , and using $a_{1} = 1$ , we get $λ_{n} = a_{n}$ .

The multiplicative identities follow from substituting back: at $r = m$ with $g cd (m, n) = 1$ , $$ \lambda_n a_m = \sum_{a | \gcd(n, m)} a^{k - 1} a_{n m / a^2} = a_{n m}, $$ since the gcd is $1$ . Combined with $λ_{n} = a_{n}$ , this gives $a_{n} a_{m} = a_{nm}$ for coprime $(m, n)$ .

At $r = p^{j}$ for a prime $p ∣ n$ , write $n = p^{i} n_{0}$ with $g cd (p, n_{0}) = 1$ . The divisor structure of $g cd (n, p^{j}) = p^{m i n (i, j)}$ produces the recursion $a_{p} a_{p^{n}} = a_{p^{n + 1}} + p^{k - 1} a_{p^{n - 1}}$ by setting $n = p$ and $r = p^{j}$ and unpacking the resulting double sum. The details mirror the multiplicativity derivation and use the Hecke algebra relation $T_{p} T_{p^{j}} = T_{p^{j + 1}} + p^{k - 1} T_{p^{j - 1}}$ . $□$

Bridge. This identity builds toward the Euler product of the modular $L$ -function attached to $f$ , since $L (f, s) = \sum_{n} a_{n} n^{- s} = \prod_{p} (1 - a_{p} p^{- s} + p^{k - 1 - 2 s})^{- 1}$ factors over primes by the multiplicativity of ${a_{n}}$ . It appears again in the Eichler-Shimura correspondence between weight- $2$ cusp eigenforms and $2$ -dimensional Galois representations on Tate modules of modular Jacobians. The central insight is that the Hecke operators generate a commutative algebra acting on the finite-dimensional space $M_{k} (Γ)$ , so simultaneous diagonalisation produces a finite basis of eigenforms whose eigenvalues are the Fourier coefficients. This is exactly the foundational reason that Dirichlet series attached to modular eigenforms admit Euler products: the Hecke recursion at primes is the Euler factor expanded as a geometric series, and the eigenvalue $a_{p}$ is the trace of Frobenius on the associated Galois representation. The bridge is from a holomorphic-analytic object (the modular form) to an arithmetic-multiplicative object (its Fourier coefficients viewed as a Dirichlet series), and the bridge runs through the commutative algebra $T$ generated by Hecke operators.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Show that the Hecke operator $T_{p}$ at prime $p$ on the Eisenstein series $E_{k} (z) = 1 - (2 k / B_{k}) \sum_{n \geq 1} σ_{k - 1} (n) q^{n}$ has eigenvalue $σ_{k - 1} (p) = 1 + p^{k - 1}$ , where $σ_{k - 1} (n) = \sum_{d ∣ n} d^{k - 1}$ is the divisor power sum.

Hint

The Eisenstein series $E_{k}$ is the unique (up to scalar) Hecke eigenform in the Eisenstein subspace of $M_{k} (SL_{2} (Z))$ . Compute $T_{p} E_{k}$ via the Fourier-coefficient formula derived in the Key theorem proof, applied to the divisor power sum.

Answer

The $n$ -th Fourier coefficient of $E_{k}$ (after normalisation so $a_{1} = 1$ ) is $σ_{k - 1} (n) = \sum_{d ∣ n} d^{k - 1}$ . By the Hecke eigenform identity, $T_{p} E_{k} = a_{p} (E_{k}) E_{k} = σ_{k - 1} (p) E_{k}$ . Compute $σ_{k - 1} (p) = 1^{k - 1} + p^{k - 1} = 1 + p^{k - 1}$ since the only divisors of a prime are $1$ and $p$ .

To verify Hecke multiplicativity directly: for $g cd (m, n) = 1$ , $σ_{k - 1} (mn) = σ_{k - 1} (m) σ_{k - 1} (n)$ since the divisors of $mn$ factor uniquely as products of coprime divisor-pairs. At prime powers: $σ_{k - 1} (p) σ_{k - 1} (p^{n}) - p^{k - 1} σ_{k - 1} (p^{n - 1}) = σ_{k - 1} (p^{n + 1})$ . Both identities follow from the explicit divisor structure of $p$ -power integers.

Exercise 4 (medium, symbolic).

State the Euler product for the Hecke $L$ -function $L (f, s) = \sum_{n \geq 1} a_{n} (f) n^{- s}$ attached to a normalised cusp eigenform $f$ of weight $k$ on $SL_{2} (Z)$ , and derive it from the multiplicativity of the Fourier coefficients.

Hint

The multiplicativity $a_{mn} = a_{m} a_{n}$ for $g cd (m, n) = 1$ factorises the Dirichlet series over primes. The recursion at prime powers determines the local Euler factor.

Answer

Euler product. For a normalised cusp eigenform $f$ of weight $k$ on $SL_{2} (Z)$ , $$ L(f, s) = \sum_{n \geq 1} \frac{a_n}{n^s} = \prod_p \frac{1}{1 - a_p p^{-s} + p^{k - 1 - 2 s}}. $$

Derivation. By multiplicativity $a_{mn} = a_{m} a_{n}$ for $g cd (m, n) = 1$ , the Dirichlet series factors as $L (f, s) = \prod_{p} L_{p} (f, s)$ where $L_{p} (f, s) = \sum_{n \geq 0} a_{p^{n}} p^{- n s}$ is the local factor at $p$ .

Compute the local factor using the Hecke recursion $a_{p^{n + 1}} = a_{p} a_{p^{n}} - p^{k - 1} a_{p^{n - 1}}$ : $$ L_p(f, s) = \sum_{n \geq 0} a_{p^n} p^{-n s}. $$ Define $g (X) := \sum_{n \geq 0} a_{p^{n}} X^{n}$ . The Hecke recursion translates to $$ g(X) - a_p X g(X) + p^{k - 1} X^2 g(X) = a_0 + (a_1 - a_p a_0) X = 1, $$ using $a_{p^{0}} = 1$ and $a_{p^{1}} = a_{p}$ . Hence $g (X) = (1 - a_{p} X + p^{k - 1} X^{2})^{- 1}$ . Substituting $X = p^{- s}$ gives $$ L_p(f, s) = \frac{1}{1 - a_p p^{-s} + p^{k - 1 - 2 s}}. $$ The Euler product follows.

This is Hecke 1936's central application of the Hecke-operator theory: modular cusp eigenforms have $L$ -functions with Euler products, and the local factor at $p$ is a quadratic in $p^{- s}$ whose coefficients are the eigenvalue $a_{p}$ and the weight-dependent constant $p^{k - 1}$ .

Exercise 5 (medium, symbolic).

Prove that the Hecke operators $T_{m}$ and $T_{n}$ commute on $M_{k} (SL_{2} (Z))$ for every pair of positive integers $(m, n)$ . Sketch the argument using the sub-lattice / coset interpretation.

Hint

The Hecke operator $T_{n}$ is the average over the $σ_{1} (n) = \sum_{d ∣ n} d$ sub-lattices of index $n$ in $Z^{2}$ . Composition $T_{m} T_{n}$ averages over pairs of sub-lattices, and the corresponding double coset is symmetric in $m$ and $n$ .

Answer

Setup. A sub-lattice of $Z^{2}$ of index $n$ corresponds to a matrix in $M_{2} (Z)$ of determinant $n$ , modulo $Γ = SL_{2} (Z)$ acting on the left (changing basis of the sub-lattice). There are $σ_{1} (n)$ such cosets, exactly the index- $n$ sub-lattices.

Composition. The product $T_{m} T_{n}$ averages a modular form first over index- $n$ sub-lattices, then over index- $m$ sub-lattices of each. The result is an average over pairs $(L_{n}, L_{m})$ where $L_{n} \supseteq L_{m}$ has $Z^{2} / L_{n} ≅ Z / n$ and $L_{n} / L_{m} ≅ Z / m$ . By the chain rule for indices, $Z^{2} / L_{m}$ has order $mn$ , and the pair $(L_{n}, L_{m})$ corresponds to a flag of sub-lattices.

Counting. The number of pairs $(L_{n}, L_{m})$ as above equals the number of ways to choose a sub-lattice of $Z^{2}$ of index $mn$ with a marked intermediate sub-lattice of index $n$ . By the divisor structure of $Z^{2}$ as a $Z$ -module, the count is symmetric in $m, n$ for $g cd (m, n) = 1$ (each is determined by an intermediate lattice up to the symmetric Smith normal form), and the count is given by the recursive Hecke relation at prime powers.

Commutativity. The double coset $Γ\ M_{2} (Z)_{= mn} /Γ$ has cardinality independent of whether we write $mn$ as $m \cdot n$ or $n \cdot m$ , so the operators $T_{m} T_{n}$ and $T_{n} T_{m}$ both average over the same set of double cosets with the same weighting. Hence $T_{m} T_{n} = T_{n} T_{m}$ on $M_{k} (Γ)$ .

This is the geometric content of the commutativity: Hecke operators are double cosets, and double cosets in $Γ\ GL_{2} (Q_{+})^{\times} /Γ$ commute under convolution since the Hecke double cosets form a commutative algebra (the spherical Hecke algebra of $GL_{2}$ ).

Exercise 6 (hard, symbolic).

State and prove the Petersson self-adjointness theorem: the Hecke operators $T_{n}$ on $S_{k} (SL_{2} (Z))$ are self-adjoint with respect to the Petersson inner product $$ \langle f, g \rangle_{\mathrm{Pet}} = \int_{\Gamma \backslash \mathbb{H}} f(z) \overline{g(z)} y^{k - 2} dx , dy. $$

Hint

The proof uses (a) the $Γ$ -invariance of the integrand for $f, g \in S_{k} (Γ)$ , (b) the change-of-variable invariance under the sub-lattice averaging, and (c) the convergence on the cuspidal subspace via the rapid decay of cusp forms near $\infty$ .

Answer

Theorem (Petersson 1939 Math. Ann. 116). For all $n \geq 1$ and $f, g \in S_{k} (SL_{2} (Z))$ , $$ \langle T_n f, g \rangle_{\mathrm{Pet}} = \langle f, T_n g \rangle_{\mathrm{Pet}}. $$

Proof. Write $T_{n}$ as the sum over coset representatives $T_{n} f = \sum_{δ} f ∣_{k} δ$ with $δ$ ranging over $Γ\ {γ \in M_{2} (Z) : det (γ) = n}$ , equivalently the index- $n$ sub-lattices of $Z^{2}$ . Each $δ = (a 0 b d)$ with $a d = n$ , $a, d \geq 1$ , $0 \leq b < d$ , acts by $f ∣_{k} δ (z) = d^{- k} f ((a z + b) / d) \cdot a^{k}$ — equivalently $a^{k - 1} d^{- 1} f ((a z + b) / d)$ matching the Hecke-operator formula.

Compute $$ \langle T_n f, g \rangle = \sum_\delta \int_{\Gamma \backslash \mathbb{H}} (f |k \delta)(z) \overline{g(z)} y^{k - 2} dx , dy. $$ By the $Γ$ -invariance of $\overline{g (z)} y^{k - 2} d x d y$ and the fact that $δ$ has determinant $n$ , change variables $z \mapsto δ z$ on the upper half-plane in each summand. The Jacobian compensates exactly for the $∣_{k}$ slash factor, and the integration is now against $g (δ z)$ pulled back. The sum over $δ$ ranges over a complete set of representatives $Γ\ M_{2} (Z)_{= n}$ , and by the inversion symmetry $δ \mapsto δ^{*}$ (where $δ^{*}$ is the adjoint matrix, with the same coset structure since both are determinant- $n$ matrices), the sum equals $$ \sum{\delta^} \int_{\Gamma \backslash \mathbb{H}} f(z) \overline{(g |_k \delta^)(z)} y^{k - 2} dx , dy = \langle f, T_n g \rangle. $$

The cleaner symmetry: the Hecke operator $T_{n}$ is the convolution against the double coset $Γ (10 0 n) Γ$ in the Hecke algebra of $GL_{2}$ . The double coset is invariant under the involution $γ \mapsto γ^{*}$ (transpose-and-determinant-divide), so the Hecke operator is self-adjoint with respect to any $Γ$ -invariant inner product induced from a Haar measure on $GL_{2} (R)$ . The Petersson measure $y^{k - 2} d x d y$ is the natural such measure on the cuspidal sector, and convergence on $S_{k} (Γ)$ follows from the cusp condition (rapid decay near $i \infty$ ).

Consequence. Self-adjoint commuting operators on a finite-dimensional inner-product space $S_{k} (Γ)$ admit a simultaneous orthonormal basis of eigenvectors. Hence $S_{k} (Γ)$ has an orthonormal basis of Hecke eigenforms with respect to the Petersson product, all eigenvalues real. The basis is unique up to permutation by the multiplicity-one theorem (Atkin-Lehner 1970 at higher level; Hecke at level one).

Exercise 7 (hard, symbolic).

Compute the dimension of $S_{12} (SL_{2} (Z))$ and prove that the discriminant $Δ$ is, up to scalar, the unique Hecke eigenform in this space. Conclude that the Ramanujan tau function $τ (n)$ is multiplicative.

Hint

Use the dimension formula $dim M_{k} (SL_{2} (Z))$ for weight $k$ on the full modular group. For $k = 12$ , the dimension of the cusp form space is one.

Answer

Dimension. The dimension formula for $M_{k} (SL_{2} (Z))$ for $k \geq 4$ even is $$ \dim M_k = \begin{cases} \lfloor k / 12 \rfloor + 1 & k \not\equiv 2 \pmod{12} \ \lfloor k / 12 \rfloor & k \equiv 2 \pmod{12} \end{cases} $$ with $dim S_{k} = dim M_{k} - 1$ for $k \geq 4$ .

For $k = 12$ : $⌊ 12/12 ⌋ + 1 = 2$ , so $dim M_{12} = 2$ and $dim S_{12} = 1$ .

Uniqueness of $Δ$ . Since $dim S_{12} (SL_{2} (Z)) = 1$ , the space is spanned by any nonzero cusp form, which is the discriminant $Δ (z) = q \prod_{n \geq 1} (1 - q^{n})^{24}$ . Every Hecke operator $T_{n}$ acts on $S_{12}$ — a one-dimensional space — by a scalar, so $Δ$ is automatically a simultaneous Hecke eigenform. The eigenvalue of $T_{n}$ on $Δ$ equals the $n$ -th Fourier coefficient $τ (n)$ by the Key Theorem.

Multiplicativity of $τ$ . By the Key Theorem, $τ (mn) = τ (m) τ (n)$ for $g cd (m, n) = 1$ , and $τ (p^{n + 1}) = τ (p) τ (p^{n}) - p^{11} τ (p^{n - 1})$ at prime powers. These two identities are Ramanujan's 1916 conjectures, proved by Mordell 1917 Proc. Cambridge Phil. Soc. via a direct $L$ -function argument, and reinterpreted via Hecke operators in Hecke 1936. The full $L$ -function $L (Δ, s) = \sum_{n \geq 1} τ (n) n^{- s}$ has the Euler product $$ L(\Delta, s) = \prod_p \frac{1}{1 - \tau(p) p^{-s} + p^{11 - 2 s}}. $$ Deligne 1974 proved $∣ τ (p) ∣ \leq 2 p^{11/2}$ (the Ramanujan-Petersson conjecture, now a theorem), confirming that the local Euler factor has roots of absolute value $p^{- 11/2}$ .

Exercise 8 (hard, symbolic).

State the multiplicity-one theorem for newforms (Atkin-Lehner 1970 Math. Ann. 185) and explain how it generalises the level-one statement that $S_{12} (SL_{2} (Z))$ is one-dimensional.

Hint

At higher level $Γ_{0} (N)$ , the space $S_{k} (Γ_{0} (N))$ decomposes into a new part and an old part. The new part has a basis of Hecke eigenforms uniquely determined by their eigenvalues; the old part is built up from forms of lower level.

Answer

Setup. Let $N \geq 1$ be a positive integer, called the level, and $Γ_{0} (N) = {(a c b d) \in SL_{2} (Z) : N ∣ c}$ the level- $N$ congruence subgroup. The space $S_{k} (Γ_{0} (N))$ of weight- $k$ cusp forms on $Γ_{0} (N)$ decomposes as $$ S_k(\Gamma_0(N)) = S_k^{\mathrm{new}}(\Gamma_0(N)) \oplus S_k^{\mathrm{old}}(\Gamma_0(N)), $$ where the new part consists of forms not arising from forms of strictly smaller level $M ∣ N$ via the natural lifts $f (z) \mapsto f (d z)$ for $d ∣ (N / M)$ , and the old part is the span of all such lifted forms.

Atkin-Lehner multiplicity-one theorem. The new part $S_{k}^{new} (Γ_{0} (N))$ has a basis of simultaneous Hecke eigenforms for all $T_{p}$ with $p ∤ N$ and all Atkin-Lehner involutions $w_{q}$ for $q ∣ N$ , with each eigenform uniquely determined (up to scalar) by its sequence of Hecke eigenvalues ${a_{p} (f) : p prime}$ and Atkin-Lehner signs.

Level-one specialisation. At level $N = 1$ , the new part equals the entire space $S_{k} = S_{k}^{new} (SL_{2} (Z))$ , and the multiplicity-one statement says that Hecke eigenforms in $S_{k} (SL_{2} (Z))$ are uniquely determined by their eigenvalues. At $k = 12$ , $dim S_{12} = 1$ , so there is one Hecke eigenform (the discriminant $Δ$ , up to scalar) and the multiplicity-one statement is vacuous in this one-dimensional setting.

Higher-dimensional level-one example. At $k = 24$ , $dim S_{24} (SL_{2} (Z)) = 2$ , and the two Hecke eigenforms are explicit (with eigenvalues $a_{2} = 1080 \pm 12144169$ , both real algebraic integers in the totally real number field $Q (144169)$ ). Multiplicity-one at level $1$ says these two eigenforms are uniquely determined by their eigenvalue sequences.

Significance. The multiplicity-one theorem is the input to Deligne's construction of $ℓ$ -adic Galois representations attached to weight- $\geq 2$ cusp eigenforms: each eigenform corresponds to a unique two-dimensional Galois representation, with trace of Frobenius at $p ∤ N ℓ$ equal to the eigenvalue $a_{p} (f)$ . Multiplicity-one is what allows the bridge to be a function rather than a multi-valued correspondence.

Lean formalization [Intermediate+]

The companion file lean/Codex/NumberTheory/ModularForms/HeckeOperators.lean records the Hecke operator definition, multiplicativity, the eigenform-Fourier-coefficient identity, and the Petersson self-adjointness statement as sorry-stubbed declarations on Mathlib's ModularForm type. The formalisable kernel has four components.

First, a def heckeOperator (p : ℕ) [Fact (Nat.Prime p)] : ModularForm k → ModularForm k declaring the operator $T_{p}$ as the sub-lattice average $(T_{p} f) (z) = p^{k - 1} f (p z) + p^{- 1} \sum_{b = 0}^{p - 1} f ((z + b) / p)$ . The Lean declaration is type-correct because Mathlib provides the ModularForm type, the slash action of $SL_{2} (Z)$ , and the Möbius transformation of the upper half-plane; the proof of well-definedness (that the sub-lattice average is again $Γ$ -invariant) is sorry-stubbed.

Second, the multiplicativity theorem theorem hecke_multiplicative (m n : ℕ) (h : Nat.Coprime m n) : heckeOperator (m * n) = heckeOperator m ∘ heckeOperator n as a sorry-stubbed declaration. Combined with the prime-power recursion (a separate theorem statement), this encodes the full Hecke algebra structure on $M_{k} (Γ)$ .

Third, the eigenform-Fourier-coefficient identity theorem hecke_eigenform_fourier_coeff_eq_eigenvalue as a sorry-stubbed declaration: if $f$ is a normalised eigenform with $T_{n} f = λ_{n} f$ for all $n$ and $a_{1} (f) = 1$ , then $λ_{n} = a_{n} (f)$ . This is the unit's signature theorem, formalised as a statement on Mathlib's ModularForm.coeff API.

Fourth, the Petersson self-adjointness statement as a sorry-stubbed declaration: $⟨ T_{n} f, g ⟩_{Pet} = ⟨ f, T_{n} g ⟩_{Pet}$ . The Petersson inner product itself requires the integration-on-fundamental-domain machinery missing from Mathlib; the file declares the inner product as an axiom placeholder pending upstream development.

The full proofs require the sub-lattice / coset decomposition API for $SL_{2} (Z)$ acting on $M_{2} (Z)_{= n}$ (the index- $n$ sub-lattice space), plus an integration framework on the fundamental domain $Γ\ H$ . These are the items named in the lean_mathlib_gap frontmatter field.

Advanced results [Master]

The Hecke algebra and the Euler product

Theorem 1 (Hecke algebra structure; Hecke 1936 Math. Ann. 112). The Hecke algebra $T_{k} (Γ) = C [T_{n} : n \geq 1] \subseteq End_{C} (M_{k} (Γ))$ is a commutative finite-dimensional $C$ -algebra. Its generators ${T_{n}}$ satisfy the multiplicative relations $$ T_{m n} = T_m T_n \text{ for } \gcd(m, n) = 1, \qquad T_p T_{p^n} = T_{p^{n + 1}} + p^{k - 1} T_{p^{n - 1}}, \quad p \text{ prime}, $$ generating the algebra freely over $Z$ as a polynomial ring in the prime-indexed operators ${T_{p}}_{p prime}$ modulo the prime-power recursion.

Proof sketch. The two relations are derived from the sub-lattice description of Hecke operators: for $g cd (m, n) = 1$ , the index- $mn$ sub-lattices of $Z^{2}$ are products of index- $m$ and index- $n$ sub-lattices in a unique way, giving $T_{m} T_{n} = T_{mn}$ . At prime powers, the chain-of-sub-lattices structure $Z^{2} \supseteq L_{p} \supseteq L_{p^{2}} \supseteq \dots$ yields the recursion via inclusion-exclusion over flags. Commutativity follows because the index- $mn$ sub-lattice count is symmetric in $m, n$ . The Hecke algebra is finite-dimensional because $End_{C} (M_{k} (Γ))$ is finite-dimensional. $□$

Theorem 2 (Hecke 1936 Euler product). For a normalised cusp eigenform $f \in S_{k} (SL_{2} (Z))$ with Fourier coefficients ${a_{n}}_{n \geq 1}$ , $$ L(f, s) := \sum_{n \geq 1} \frac{a_n}{n^s} = \prod_p \frac{1}{1 - a_p p^{-s} + p^{k - 1 - 2 s}}. $$ The Euler product converges absolutely for $Re (s) > (k + 1) /2$ and admits analytic continuation to $C$ via the integral representation $L (f, s) = (2 π)^{- s} Γ (s) \int_{0}^{\infty} f (i y) y^{s - 1} d y$ , satisfying the functional equation $Λ (f, s) = (- 1)^{k /2} Λ (f, k - s)$ with $Λ (f, s) := (2 π)^{- s} Γ (s) L (f, s)$ .

Significance. The Euler product is the central application of the Hecke operator theory: modular forms become arithmetic objects through their attached $L$ -functions, which factor over primes by the Hecke recursion. The functional equation is the modular-form-side analogue of the Riemann functional equation for $ζ (s)$ , with the modular $L$ -function playing the role of a refined zeta function attached to the cusp form $f$ .

Petersson inner product and self-adjointness

Theorem 3 (Petersson 1939 Math. Ann. 116). The Hecke operators ${T_{n}}_{n \geq 1}$ acting on $S_{k} (Γ)$ are self-adjoint with respect to the Petersson inner product $$ \langle f, g \rangle_{\mathrm{Pet}} := \int_{\Gamma \backslash \mathbb{H}} f(z) \overline{g(z)} y^{k - 2} dx , dy. $$ The Petersson product converges absolutely on $S_{k} (Γ) \times S_{k} (Γ)$ (cusp forms have rapid decay at $\infty$ ), is $Γ$ -invariant by the modular transformation rule, and yields the equality $⟨ T_{n} f, g ⟩ = ⟨ f, T_{n} g ⟩$ for every $n \geq 1$ and every $f, g \in S_{k} (Γ)$ .

Corollary (spectral decomposition). $S_{k} (Γ)$ admits an orthonormal basis ${f_{1}, \dots, f_{d}}$ of simultaneous Hecke eigenforms with $T_{n} f_{i} = a_{n} (f_{i}) f_{i}$ for all $n$ and $i$ . Each $f_{i}$ is normalised (after rescaling) so that $a_{1} (f_{i}) = 1$ , and the eigenvalue sequences ${a_{n} (f_{i})}_{n \geq 1}$ are uniquely determined by $f_{i}$ .

New and old forms; multiplicity one (Atkin-Lehner 1970)

Theorem 4 (Atkin-Lehner 1970 Math. Ann. 185). For $Γ = Γ_{0} (N)$ with $N \geq 1$ , the space $S_{k} (Γ_{0} (N))$ decomposes as $S_{k} (Γ_{0} (N)) = S_{k}^{new} (Γ_{0} (N)) \oplus S_{k}^{old} (Γ_{0} (N))$ , where the new part is the orthogonal complement (in the Petersson inner product) of the old part, the latter being the sum of subspaces $$ S_k^{\mathrm{old}}(\Gamma_0(N)) = \sum_{M | N, M < N} \sum_{d | (N/M)} \alpha_d S_k(\Gamma_0(M)), $$ with $α_{d} : S_{k} (Γ_{0} (M)) \to S_{k} (Γ_{0} (N))$ the lift $α_{d} (f) (z) = f (d z)$ . The new part has a basis of newforms — normalised cusp eigenforms for all $T_{p}$ with $p ∤ N$ — uniquely determined (multiplicity one) by their eigenvalue sequences.

Significance. The multiplicity-one theorem is the precise statement that distinguishes "the modular form attached to an arithmetic object" from "a modular form with the same eigenvalues but spurious differences." It allows the Hecke-eigenvalue invariants to be a complete classifying invariant for cusp eigenforms.

Galois representations attached to cusp eigenforms

Theorem 5 (Deligne 1971 Sém. Bourbaki 355, Deligne-Serre 1974 Ann. Sci. ENS 7). Let $f$ be a normalised newform of weight $k \geq 1$ on $Γ_{0} (N)$ , with eigenvalue sequence ${a_{p} (f) \in \overline{Q}}$ . For each prime $ℓ$ and each embedding $\overline{Q} ↪ \overline{Q_{ℓ}}$ , there is a continuous semisimple Galois representation $$ \rho_{f, \ell} : \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}2(\overline{\mathbb{Q}\ell}) $$ unramified outside $N ℓ$ , with $tr ρ_{f, ℓ} (Frob_{p}) = a_{p} (f)$ and $det ρ_{f, ℓ} (Frob_{p}) = p^{k - 1}$ for every prime $p ∤ N ℓ$ .

Significance. This theorem encodes the deepest content of the Hecke operator theory: the Fourier coefficients of a modular eigenform are traces of Frobenius on a $2$ -dimensional Galois representation. The Hecke algebra is in this sense a piece of the absolute Galois group $Gal (\overline{Q} / Q)$ realised geometrically through the modular curves $X_{0} (N)$ . For weight $k = 2$ , the representation $ρ_{f, ℓ}$ is realised on the $ℓ$ -adic Tate module of the abelian variety $A_{f}$ attached to $f$ by Shimura's construction — the modular abelian variety quotient of the Jacobian $Jac (X_{0} (N))$ .

Ramanujan-Petersson conjecture and the Weil bound

Theorem 6 (Deligne 1971 + 1974 Publ. Math. IHES 43; Ramanujan-Petersson conjecture, now a theorem). Let $f$ be a normalised cusp newform of weight $k$ . Then for every prime $p ∤ N$ , $$ |a_p(f)| \leq 2 p^{(k - 1)/2}. $$ Equivalently, the roots $α_{p}, β_{p}$ of the local Euler factor $1 - a_{p} X + p^{k - 1} X^{2}$ satisfy $∣ α_{p} ∣ = ∣ β_{p} ∣ = p^{(k - 1) /2}$ , lying on the circle of radius $p^{(k - 1) /2}$ in $C$ .

Significance. The Ramanujan-Petersson conjecture, raised by Ramanujan 1916 for $Δ$ (the bound $∣ τ (p) ∣ \leq 2 p^{11/2}$ ), proved by Deligne 1971 for general weight $k \geq 2$ as a consequence of the Weil conjectures, is the modular-form analogue of the Riemann hypothesis. It bounds the Fourier coefficients optimally and confirms that the Hecke eigenvalues are "as small as possible" given the local Euler factor structure — precisely the size where the Euler product converges on the critical line.

Eichler-Shimura, modular Jacobians, and the modularity theorem

Theorem 7 (Eichler 1954 Arch. Math. 5; Shimura 1958 Tohoku Math. J. 10). For a weight- $2$ cusp newform $f$ on $Γ_{0} (N)$ , the Hecke algebra acts on the first cohomology $H^{1} (X_{0} (N), C)$ of the modular curve, and the simultaneous Hecke eigenform $f$ corresponds to a $2$ -dimensional summand $V_{f} \subseteq H^{1} (X_{0} (N), Q)$ . The associated abelian variety $A_{f}$ is a quotient of $Jac (X_{0} (N))$ by an ideal of the Hecke algebra, with the property that $L (A_{f}, s) = \prod_{σ} L (f^{σ}, s)$ ranging over the Galois conjugates of $f$ .

Significance. This is the geometric realisation of weight- $2$ newforms as cohomological objects on modular curves, building toward the modularity theorem (Wiles 1995, Breuil-Conrad-Diamond-Taylor 2001): every elliptic curve $E$ over $Q$ is isogenous to $A_{f}$ for some weight- $2$ cusp newform $f$ on $Γ_{0} (N_{E})$ where $N_{E}$ is the conductor of $E$ . The Hecke operator on $f$ corresponds to the Frobenius endomorphism of $A_{f}$ , and the eigenvalue $a_{p} (f)$ corresponds to the trace of Frobenius on the $ℓ$ -adic Tate module of $A_{f}$ at $p$ .

Synthesis. The Hecke algebra is the foundational reason that modular forms become arithmetic objects. The central insight is that a commutative finitely-generated algebra of operators acts on the finite-dimensional space $M_{k} (Γ)$ , and simultaneous diagonalisation produces a basis of eigenforms whose Fourier coefficients are the eigenvalues. This is exactly the structure that identifies modular forms with $L$ -functions admitting Euler products: the prime-power Hecke recursion is the quadratic local Euler factor expanded as a geometric series, and the eigenvalue $a_{p}$ is the trace of Frobenius on the attached Galois representation. Putting these together with the Petersson self-adjointness, the spectral decomposition of $S_{k} (Γ)$ is unique, orthogonal, and uniquely determined by eigenvalue data — the multiplicity-one theorem at newform level.

The bridge from Hecke 1936-37 to the modern theory runs through three intermediate results. Petersson 1939 supplies the self-adjointness that yields the orthogonal spectral basis. Atkin-Lehner 1970 supplies the new/old decomposition at higher level $Γ_{0} (N)$ , generalising the unique-eigenform structure at level one. Eichler-Shimura 1954-58 supplies the geometric realisation on modular curves, identifying Hecke eigenvalues with Frobenius traces. Deligne 1971-74 closes the loop by attaching $2$ -dimensional Galois representations to weight- $\geq 2$ newforms, with the trace of Frobenius equal to the Hecke eigenvalue.

The pattern recurs throughout the Langlands programme: a commutative algebra of operators on an automorphic representation space produces eigenvectors whose eigenvalues match Frobenius traces on Galois representations. Hecke's 1936-37 papers are the surface case of this paradigm, generalising to $GL_{n}$ over arbitrary global fields, and the Hecke algebra of this unit is the local-global avatar of the spherical Hecke algebra $C_{c}^{\infty} (K \ G (Q_{p}) / K)$ of representation theory.

Full proof set [Master]

Proposition 8 (Hecke multiplicativity at coprime indices). Let $m, n \geq 1$ with $g cd (m, n) = 1$ . Then $T_{mn} = T_{m} T_{n}$ as operators on $M_{k} (SL_{2} (Z))$ .

Proof. Use the sub-lattice description: $T_{n} f$ is the sum of $f$ over a complete set of representatives for the $σ_{1} (n)$ sub-lattices of $Z^{2}$ of index $n$ , weighted by the slash factor $n^{k - 1}$ . Composition $T_{m} T_{n}$ sums $f$ over pairs $(L_{n}, L_{m})$ where $L_{n}$ is index $n$ in $Z^{2}$ and $L_{m}$ is index $m$ in $L_{n}$ .

By the chain rule for lattice indices, $L_{m}$ has index $mn$ in $Z^{2}$ , and the pair $(L_{n}, L_{m})$ is a flag of sub-lattices. When $g cd (m, n) = 1$ , the flag is uniquely determined by the bottom lattice $L_{m}$ (since $L_{n}$ is the unique index- $n$ sub-lattice of $Z^{2}$ containing $L_{m}$ , equivalently $L_{n} = Z^{2} \cap (1/ n) L_{m}$ ). Therefore the sum over pairs $(L_{n}, L_{m})$ collapses to the sum over $L_{mn}$ , and we have $T_{m} T_{n} = T_{mn}$ .

Equivalently, the matrix double coset $Γ (0 1 n 0) Γ \cdot Γ (0 1 m 0) Γ$ equals $Γ (0 1 mn 0) Γ$ when $g cd (m, n) = 1$ , by direct computation in the Hecke ring of $GL_{2} (Q)$ . $□$

Proposition 9 (Hecke recursion at prime powers). Let $p$ be a prime and $n \geq 1$ . Then $T_{p} T_{p^{n}} = T_{p^{n + 1}} + p^{k - 1} T_{p^{n - 1}}$ as operators on $M_{k} (SL_{2} (Z))$ .

Proof. Compose $T_{p}$ and $T_{p^{n}}$ on the sub-lattice level. Pairs $(L_{p^{n}}, L_{p})$ with $L_{p}$ index $p$ in $L_{p^{n}}$ split into two cases.

Case A: $L_{p}$ has index $p^{n + 1}$ in $Z^{2}$ — the "deepest" sub-lattices. These pairs match index- $p^{n + 1}$ sub-lattices of $Z^{2}$ with an intermediate index- $p^{n}$ lattice. Counting: each index- $p^{n + 1}$ sub-lattice $L$ admits a unique intermediate index- $p^{n}$ sub-lattice $L_{p^{n}}$ (the unique sub-lattice between $L$ and $Z^{2}$ at that depth), contributing one term to the Case A sum, totalling $T_{p^{n + 1}}$ .

Case B: $L_{p}$ has index $p^{n}$ in $Z^{2}$ — the "shallow" sub-lattices where $L_{p} = L_{p^{n}}$ . These correspond to choosing an index- $p^{n}$ sub-lattice of $Z^{2}$ and degenerating the chain $L_{p^{n}} = L_{p}$ . Each such configuration contributes the slash factor $p^{k - 1}$ (the determinant power) and a sum over $L_{p^{n - 1}}$ (the further intermediate sub-lattice). Counting: $p^{k - 1}$ contributions per $L_{p^{n - 1}}$ , totalling $p^{k - 1} T_{p^{n - 1}}$ .

Adding the two cases: $T_{p} T_{p^{n}} = T_{p^{n + 1}} + p^{k - 1} T_{p^{n - 1}}$ . $□$

Proposition 10 (Petersson self-adjointness; geometric proof). For every $n \geq 1$ and every $f, g \in S_{k} (SL_{2} (Z))$ , $⟨ T_{n} f, g ⟩_{Pet} = ⟨ f, T_{n} g ⟩_{Pet}$ .

Proof. Write $T_{n}$ as the average over coset representatives $T_{n} = \sum_{δ} ∣_{k} δ$ where $δ$ runs over $Γ\ M_{2} (Z)_{= n}$ . For each $δ$ , the slash-action $f ∣_{k} δ$ satisfies $(f ∣_{k} δ) (z) = (det δ)^{k - 1} (cz + d)^{- k} f (δ z)$ where $δ = (a c b d)$ with $a d - b c = n$ .

The Petersson inner product is built from the $SL_{2} (R)$ -invariant measure $d μ = y^{- 2} d x d y$ on $H$ together with the height- $k - 2$ weight $y^{k - 2}$ . The full integrand $f \overset{g}{ˉ} y^{k - 2} d x d y = f \overset{g}{ˉ} \cdot y^{k} \cdot y^{- 2} d x d y$ is $Γ$ -invariant by direct computation: the modular transformation rule $f (γ z) = (cz + d)^{k} f (z)$ produces a phase $∣ cz + d ∣^{2 k}$ in $f \overset{g}{ˉ}$ , and the measure $y^{k}$ transforms by $∣ cz + d ∣^{- 2 k}$ since $Im (γ z) = y /∣ cz + d ∣^{2}$ . These cancel, leaving the integrand $Γ$ -invariant.

To prove self-adjointness, change variables $z \mapsto δ z$ in the integral $\int (f ∣_{k} δ) \overset{g}{ˉ} y^{k - 2} d x d y$ . The Jacobian of $δ$ on $H$ is $n /∣ cz + d ∣^{2}$ (sending the hyperbolic measure $d x d y / y^{2}$ to itself by the determinant- $n$ scaling), and the weight factor $y^{k - 2}$ transforms by $∣ cz + d ∣^{- 2 (k - 2)}$ . The combined integrand under $z \mapsto δ z$ becomes $$ (f \cdot \overline{g(\delta^{-1} z)} y^{k - 2}) \cdot (\text{compensating factors}) = f(z) \cdot \overline{(g |_k \delta^)(z)} y^{k - 2}, $$ where $\delta^ = n \delta^{-1} $i s t h e a d j o in t ma t r i x o f$ \delta $(t r an s p ose - t h e n - d i v i d e - b y - d e t er minan t), s t i l l in$ M_2(\mathbb{Z}) $w i t h$ \det \delta^* = n$.

Summing over $δ$ : the involution $δ \mapsto δ^{*}$ permutes the cosets $Γ\ M_{2} (Z)_{= n}$ , since the set of determinant- $n$ matrices is symmetric under adjoint. The result is $$ \sum_\delta \int (f |k \delta) \bar g y^{k - 2} dx dy = \sum{\delta^} \int f \overline{(g |_k \delta^)} y^{k - 2} dx dy, $$ which is $⟨ f, T_{n} g ⟩_{Pet}$ . $□$

Proposition 11 (Hecke eigenforms span $S_{k} (Γ)$ ). For $Γ = SL_{2} (Z)$ and $k \geq 4$ even, the space $S_{k} (Γ)$ has an orthonormal basis (with respect to the Petersson inner product) of normalised Hecke eigenforms.

Proof. The Hecke algebra $T_{k} (Γ)$ is a commutative algebra acting on the finite-dimensional space $S_{k} (Γ)$ . By Petersson self-adjointness (Proposition 10), every operator $T_{n} \in T_{k} (Γ)$ is self-adjoint with respect to the Petersson product. A commuting family of self-adjoint operators on a finite-dimensional complex inner-product space admits a simultaneous orthonormal basis of eigenvectors (the spectral theorem).

Hence $S_{k} (Γ)$ has an orthonormal basis ${f_{1}, \dots, f_{d}}$ of simultaneous Hecke eigenforms with $T_{n} f_{i} = λ_{n}^{(i)} f_{i}$ and $λ_{n}^{(i)} \in R$ (real because self-adjoint). After normalising each $f_{i}$ so that $a_{1} (f_{i}) = 1$ (which is possible because eigenforms have non-vanishing first Fourier coefficient at level one — proved by Atkin-Lehner via the new/old decomposition or directly for $SL_{2} (Z)$ from the dimension formula), the eigenvalue $λ_{n}^{(i)}$ equals the Fourier coefficient $a_{n} (f_{i})$ .

The eigenvalues ${a_{n} (f_{i})}$ are algebraic integers (lying in the totally real number field generated by the Hecke eigenvalues at level one), and they uniquely determine $f_{i}$ by the multiplicity-one theorem. $□$

Connections [Master]

Modular forms on $SL_{2} (Z)$ 21.04.01. Sibling unit defining the ambient space $M_{k} (SL_{2} (Z))$ on which the Hecke operators act. The sibling unit covers holomorphy on the upper half-plane, weight- $k$ transformation under $Γ$ , holomorphy at the cusp $\infty$ , the dimension formula for $M_{k}$ , and the Eisenstein-series / cusp-form decomposition. The present unit is the operator-theoretic refinement: $M_{k} (Γ)$ is not just a finite-dimensional vector space, it is a $T$ -module, and the Hecke algebra is what makes the dimension formula compatible with the multiplicative structure of the Fourier coefficients.
Eichler-Shimura correspondence 21.04.03. Successor unit (to be produced) treating the weight- $2$ Hecke eigenform / Galois-representation bridge. Weight- $2$ cusp eigenforms on $Γ_{0} (N)$ correspond to $2$ -dimensional $ℓ$ -adic Galois representations on the Tate module of the modular Jacobian $Jac (X_{0} (N))$ , with the Hecke eigenvalue $a_{p}$ equal to the trace of Frobenius. The bridge from the present unit's Hecke algebra to Eichler-Shimura's Galois-representation framework is the geometric realisation of the Hecke operators as Frobenius-correspondences on the modular curve $X_{0} (N)$ .
Riemann zeta function $ζ (s)$ 21.03.01. Sibling unit on the prototypical Dirichlet series. The modular $L$ -function $L (f, s) = \sum_{n} a_{n} n^{- s}$ generalises $ζ (s)$ in two senses: $ζ (s)$ is the level- $1$ weight- $0$ Eisenstein-series $L$ -function (with $a_{n} = 1$ for all $n$ , no cuspidal content), and $L (f, s)$ has an Euler product over primes plus a functional equation, exactly mirroring the Riemann zeta functional equation. The Hecke recursion at primes is the modular-form analogue of the Euler factor $(1 - p^{- s})^{- 1}$ in $ζ$ , generalising the multiplicative structure.
Dirichlet $L$ -functions $L (s, χ)$ 21.03.02. Sibling unit on the next-simplest Dirichlet series. The Dirichlet $L$ -function $L (s, χ) = \sum_{n} χ (n) n^{- s}$ is the prototype of an $L$ -function attached to a $1$ -dimensional Galois character $χ : Gal (\overline{Q} / Q) \to C^{\times}$ , the simplest-conductor case where the Hecke / Galois correspondence is direct and was Dirichlet's 1837 starting point. The modular $L$ -function $L (f, s)$ of the present unit is the corresponding $2$ -dimensional case: a cusp eigenform $f$ corresponds to a $2$ -dimensional Galois representation, and the Dirichlet series $L (f, s)$ generalises Dirichlet's $L (s, χ)$ from $GL_{1}$ to $GL_{2}$ .
$ℓ$ -adic Galois representations 21.05.01. Successor unit (to be produced) on the Deligne-Serre construction of $2$ -dimensional Galois representations attached to weight- $\geq 2$ cusp eigenforms. The Hecke eigenvalue $a_{p} (f)$ equals the trace of $ρ_{f, ℓ} (Frob_{p})$ on the $ℓ$ -adic representation $ρ_{f, ℓ}$ , and the local Euler factor $1 - a_{p} p^{- s} + p^{k - 1 - 2 s}$ is the characteristic polynomial of Frobenius on $ρ_{f, ℓ}$ . The present unit's Hecke algebra is, after Deligne-Serre, a piece of the absolute Galois group realised on the cohomology of modular curves.
Modularity theorem and BSD 21.06.01. Future successor unit on the Wiles-Taylor-Breuil-Conrad-Diamond modularity theorem and the Birch-Swinnerton-Dyer conjecture. Every elliptic curve $E$ over $Q$ is modular: there exists a weight- $2$ cusp eigenform $f_{E}$ on $Γ_{0} (N_{E})$ such that $L (E, s) = L (f_{E}, s)$ . The Hecke eigenvalues of $f_{E}$ are the traces of Frobenius on the $ℓ$ -adic Tate module $T_{ℓ} E$ , and the modularity theorem identifies the elliptic-curve $L$ -function with the modular-form $L$ -function attached to $f_{E}$ . The Hecke algebra of the present unit is the operator-theoretic substrate on which modularity is stated.
Dedekind zeta function, Hecke $L$ , Artin $L$ 21.03.03. Sibling unit treating the higher-rank analogues of Dirichlet's $L (s, χ)$ . The Dedekind zeta function $ζ_{K} (s)$ of a number field $K$ generalises $ζ (s)$ via the Euler product over prime ideals; the Hecke $L$ -functions $L (s, χ)$ attached to Hecke characters $χ$ of the idèle class group of $K$ generalise Dirichlet's $L$ . The modular $L$ -function $L (f, s)$ of the present unit is the $GL_{2}$ -side counterpart, with the Hecke algebra of operators replacing the Hecke characters of $GL_{1}$ .
Iwasawa $Z_{p}$ -extensions 21.07.01. Successor unit on the algebraic foundation of Iwasawa theory. The Hecke algebra of the present unit, when deformed over the Iwasawa algebra $Λ = Z_{p} [[T]]$ , becomes Hida's universal $Λ$ -adic Hecke algebra $T_{Λ}^{ord}$ for ordinary cusp forms (Hida 1985 Ann. Sci. ENS 19). The fibrewise specialisation of $T_{Λ}^{ord}$ at each integer weight $k \geq 2$ recovers the present unit's Hecke algebra acting on $S_{k} (Γ_{0} (N p^{r}))$ , packaging the entire weight-tower of ordinary newforms into a single $Λ$ -algebra.
$p$ -adic $L$ -functions and Iwasawa Main Conjecture 21.07.02. Successor unit on the analytic side of Iwasawa theory. The Mazur-Wiles 1984 proof of the Iwasawa Main Conjecture for $Q$ runs through $Λ$ -adic Eisenstein series in the universal Hida-Hecke algebra and the Eisenstein-ideal congruences between cuspidal and Eisenstein Hecke eigenvalues; the depth of these congruences encodes the $λ$ -invariant of the cyclotomic class group on the algebraic side. The Hecke algebra of the present unit is the operator-theoretic substrate on which the Eisenstein-ideal / Iwasawa-Main-Conjecture machinery rests.

Historical & philosophical context [Master]

Erich Hecke introduced the operators bearing his name in the 1936-37 pair of papers Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung ^[Hecke1936] ^[Hecke1937] in Mathematische Annalen 112 and 114. Hecke's setting was Ramanujan's 1916 conjectures on the multiplicativity of the tau function $τ (n)$ — multiplicativity at coprime indices and the prime-power recursion $τ (p) τ (p^{n}) = τ (p^{n + 1}) + p^{11} τ (p^{n - 1})$ . Ramanujan had observed these identities empirically from the Fourier expansion of $Δ$ ; Mordell 1917 Proc. Cambridge Phil. Soc. gave a direct proof via the $L$ -function. Hecke realised the deeper structural reason: the Euler products of modular $L$ -functions were not coincidental but came from a system of commuting operators ${T_{n}}$ on the modular form space, whose simultaneous eigenforms had multiplicative Fourier coefficients by construction. The 1936 paper introduced the operators and proved the multiplicative recursions; the 1937 continuation extended the theory to higher level $Γ_{0} (N)$ and proved the functional equation for the attached $L$ -function.

Petersson 1939 Math. Ann. 116 ^{[Petersson1939]} introduced the inner product on cusp forms that bears his name, and proved the Hecke operators are self-adjoint with respect to it. This single result transformed the Hecke-eigenform theory: self-adjointness yielded the spectral decomposition of $S_{k} (Γ)$ into an orthonormal basis of eigenforms, with eigenvalues now guaranteed real algebraic integers. Atkin-Lehner 1970 Math. Ann. 185 ^{[AtkinLehner1970]} introduced the new/old decomposition at higher level $Γ_{0} (N)$ , defined the Atkin-Lehner involutions $w_{q}$ for $q ∣ N$ , and proved the multiplicity-one theorem for newforms — the precise statement that uniquely determines a newform by its Hecke eigenvalue sequence.

The bridge from Hecke's combinatorial-analytic theory to the modern arithmetic-geometric framework was constructed by Eichler 1954 Arch. Math. 5 and Shimura 1958 Tohoku Math. J. 10, who realised the Hecke operators as cohomological correspondences on the modular curve $X_{0} (N)$ , identifying Hecke eigenvalues with traces of Frobenius. Deligne 1971 Sém. Bourbaki 355 + Deligne-Serre 1974 Ann. Sci. ENS 7 ^{[DeligneSerre1974]} completed the picture by constructing $2$ -dimensional $ℓ$ -adic Galois representations attached to weight- $\geq 1$ cusp eigenforms, with the trace of Frobenius equal to the Hecke eigenvalue. Deligne 1974 Publ. Math. IHES 43 proved the Ramanujan-Petersson conjecture — the bound $∣ a_{p} (f) ∣ \leq 2 p^{(k - 1) /2}$ for normalised newform Hecke eigenvalues — as a consequence of the Weil conjectures applied to the étale cohomology of modular varieties.

The Manin-Panchishkin Introduction to Modern Number Theory (Springer EMS 49, 2nd ed. 2005) Ch. 6 ^{[ManinPanchishkin2005]} codifies this synthesis for the modern student, while Diamond-Shurman 2005 A First Course in Modular Forms (GTM 228) ^{[DiamondShurman2005]} provides the canonical introductory textbook. The Hecke algebra of this unit is the operator-theoretic substrate on which the modular $\to$ Galois bridge is built, and through that bridge the Hecke algebra connects to the Langlands programme: a commutative algebra of operators on automorphic representations whose eigenvalues match traces of Frobenius on Galois representations, with the Hecke operators of $GL_{2}$ over $Q$ being the first substantive case of the general spherical-Hecke-algebra framework.

Bibliography [Master]

@article{Hecke1936,
  author  = {Hecke, Erich},
  title   = {{\"U}ber {M}odulfunktionen und die {D}irichletschen {R}eihen mit {E}ulerscher {P}roduktentwicklung. {I}},
  journal = {Mathematische Annalen},
  volume  = {112},
  year    = {1936},
  pages   = {664--699}
}

@article{Hecke1937,
  author  = {Hecke, Erich},
  title   = {{\"U}ber {M}odulfunktionen und die {D}irichletschen {R}eihen mit {E}ulerscher {P}roduktentwicklung. {II}},
  journal = {Mathematische Annalen},
  volume  = {114},
  year    = {1937},
  pages   = {1--28}
}

@article{Petersson1939,
  author  = {Petersson, Hans},
  title   = {Konstruktion der {M}odulformen und der zu gewissen {G}renzkreisgruppen geh{\"o}rigen automorphen {F}ormen von positiver reeller {D}imension und die vollst{\"a}ndige {B}estimmung ihrer {F}ourierkoeffizienten},
  journal = {Mathematische Annalen},
  volume  = {116},
  year    = {1939},
  pages   = {401--412}
}

@article{AtkinLehner1970,
  author  = {Atkin, A. O. L. and Lehner, Joseph},
  title   = {Hecke operators on {$\Gamma_0(m)$}},
  journal = {Mathematische Annalen},
  volume  = {185},
  year    = {1970},
  pages   = {134--160}
}

@article{DeligneSerre1974,
  author  = {Deligne, Pierre and Serre, Jean-Pierre},
  title   = {Formes modulaires de poids 1},
  journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
  volume  = {7},
  year    = {1974},
  pages   = {507--530}
}

@article{Deligne1974,
  author  = {Deligne, Pierre},
  title   = {La conjecture de {W}eil. {I}},
  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume  = {43},
  year    = {1974},
  pages   = {273--307}
}

@book{Shimura1971,
  author    = {Shimura, Goro},
  title     = {Introduction to the Arithmetic Theory of Automorphic Functions},
  series    = {Publications of the Mathematical Society of Japan},
  volume    = {11},
  publisher = {Princeton University Press},
  year      = {1971}
}

@book{Serre1973,
  author    = {Serre, Jean-Pierre},
  title     = {A Course in Arithmetic},
  series    = {Graduate Texts in Mathematics},
  volume    = {7},
  publisher = {Springer},
  year      = {1973}
}

@book{Lang1976,
  author    = {Lang, Serge},
  title     = {Introduction to Modular Forms},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {222},
  publisher = {Springer},
  year      = {1976}
}

@book{DiamondShurman2005,
  author    = {Diamond, Fred and Shurman, Jerry},
  title     = {A First Course in Modular Forms},
  series    = {Graduate Texts in Mathematics},
  volume    = {228},
  publisher = {Springer},
  year      = {2005}
}

@book{ManinPanchishkin2005,
  author    = {Manin, Yuri I. and Panchishkin, Alexei A.},
  title     = {Introduction to Modern Number Theory},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {49},
  edition   = {2nd},
  publisher = {Springer},
  year      = {2005}
}

@article{Mordell1917,
  author  = {Mordell, Louis J.},
  title   = {On {M}r {R}amanujan's empirical expansions of modular functions},
  journal = {Proceedings of the Cambridge Philosophical Society},
  volume  = {19},
  year    = {1917},
  pages   = {117--124}
}

@article{Eichler1954,
  author  = {Eichler, Martin},
  title   = {Quatern{\"a}re quadratische {F}ormen und die {R}iemannsche {V}ermutung f{\"u}r die {K}ongruenzzetafunktion},
  journal = {Archiv der Mathematik},
  volume  = {5},
  year    = {1954},
  pages   = {355--366}
}

Prerequisites

none — this is a leaf unit

Tier anchors

beginner: Diamond-Shurman *A First Course in Modular Forms* (GTM 228, 2005) §5.1 informal — Hecke operators as sub-lattice averages and the eigenform-Fourier-coefficient identity
intermediate: Diamond-Shurman *A First Course in Modular Forms* (GTM 228, 2005) Ch. 5; Serre *A Course in Arithmetic* (GTM 7, 1973) Ch. VII §5
master: Hecke 1936 *Math. Ann.* 112, 664-699 and Hecke 1937 *Math. Ann.* 114, 1-28 (originator pair); Petersson 1939 *Math. Ann.* 116 (Petersson inner product, self-adjointness); Atkin-Lehner 1970 *Math. Ann.* 185 (new and old forms, multiplicity-one); Shimura *Introduction to the Arithmetic Theory of Automorphic Functions* (Princeton 1971) Ch. 3; Deligne-Serre 1974 *Ann. Sci. ENS* 7 (Galois representations attached to weight-one eigenforms); Manin-Panchishkin *Introduction to Modern Number Theory* (Springer EMS 49, 2nd ed. 2005) Ch. 6; Diamond-Shurman Ch. 5

References

TODO_REF
Hecke 1936 — Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I · *Mathematische Annalen* 112, 664-699; the originator paper introducing Hecke operators $T_n$ on $M_k(\mathrm{SL}_2(\mathbb{Z}))$, the multiplicativity relation $T_{mn} = T_m T_n$ for $(m, n) = 1$ and $T_p T_{p^n} = T_{p^{n+1}} + p^{k-1} T_{p^{n-1}}$, the recognition that Fourier coefficients of normalised eigenforms are multiplicative and yield Euler products for the attached Dirichlet series
TODO_REF
Hecke 1937 — Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II · *Mathematische Annalen* 114, 1-28; continuation of the 1936 paper, treating modular forms of higher level $\Gamma_0(N)$ and proving the functional equation of the attached $L$-function $L(f, s) = \sum a_n n^{-s}$ for a normalised cusp eigenform $f$
TODO_REF
Petersson 1939 — Konstruktion der Modulformen und der zu gewissen Grenzkreisgruppen gehörigen automorphen Formen · *Mathematische Annalen* 116, 401-412; introduces the Petersson inner product on cusp forms $\langle f, g \rangle = \int_{\Gamma \backslash \mathbb{H}} f \bar g y^{k-2} dx dy$ and proves the Hecke operators $T_n$ are self-adjoint with respect to it, yielding orthonormal bases of eigenforms
TODO_REF
Atkin-Lehner 1970 — Hecke operators on $\Gamma_0(m)$ · *Mathematische Annalen* 185, 134-160; the theory of new and old forms at level $\Gamma_0(N)$, the Atkin-Lehner involutions $w_q$ for $q | N$, and the multiplicity-one theorem for newforms
TODO_REF
Deligne-Serre 1974 — Formes modulaires de poids 1 · *Annales scientifiques de l'École Normale Supérieure* 7 (4), 507-530; construction of two-dimensional Galois representations attached to weight-one cusp eigenforms, Artin's conjecture in the odd icosahedral case
TODO_REF
Shimura 1971 — Introduction to the Arithmetic Theory of Automorphic Functions · Publications of the Mathematical Society of Japan 11, Princeton University Press; Ch. 3 (Hecke operators), Ch. 7 (Eichler-Shimura)
TODO_REF
Serre 1973 — A Course in Arithmetic · Graduate Texts in Mathematics 7, Springer; Ch. VII §5 (Hecke operators on $M_k(\mathrm{SL}_2(\mathbb{Z}))$), the cleanest textbook account of the original level-one theory
TODO_REF
Lang 1976 — Introduction to Modular Forms · Grundlehren 222, Springer; Hecke operators, Petersson inner product, newforms
TODO_REF
Diamond-Shurman 2005 — A First Course in Modular Forms · Graduate Texts in Mathematics 228, Springer; Ch. 5 (Hecke operators), Ch. 6 (Eisenstein series); canonical modern textbook
TODO_REF
Manin-Panchishkin 2005 — Introduction to Modern Number Theory · Springer EMS 49, 2nd edition; Ch. 6 §3 (Hecke operators and Hecke algebras), §4 (Petersson inner product and the spectral decomposition of $S_k(\Gamma)$)

Lean module

Codex.NumberTheory.ModularForms.HeckeOperators

Mathlib gap

Mathlib at present has substantial support for modular forms via
`Mathlib.NumberTheory.ModularForms.Basic` (the type `ModularForm Γ k`
with weight-$k$ slash action of $\mathrm{SL}_2(\mathbb{Z})$,
holomorphy on the upper half-plane, and cusp condition at $\infty$),
and basic results on the Eisenstein-series / cusp-form decomposition.
The full Hecke-operator development is absent: Mathlib lacks (i) the
operator `heckeOperator (p : ℕ) [Fact (Nat.Prime p)] :
ModularForm Γ k →ₗ[ℂ] ModularForm Γ k` as a sub-lattice average
$T_p f(z) = p^{k - 1} f(p z) + p^{-1} \sum_{b = 0}^{p - 1}
f((z + b)/p)$ on $\mathrm{SL}_2(\mathbb{Z})$-modular forms, with the
proof of well-definedness (the slash action commutes with the
sub-lattice sum modulo $\Gamma$); (ii) the multiplicativity theorem
`hecke_multiplicative : T_{m * n} = T_m * T_n` for $\gcd(m, n) = 1$
and the recursion `T_p * T_{p^n} = T_{p^{n+1}} + p^{k - 1} * T_{p^{n-1}}$,
giving the Hecke algebra $\mathbb{T}$ as the commutative
$\mathbb{Z}$-subalgebra of $\mathrm{End}_{\mathbb{C}}(M_k(\Gamma))$
generated by all $T_n$; (iii) the eigenform-Fourier-coefficient
identity `hecke_eigenform_fourier_coeff_eq_eigenvalue`: if $T_n f =
a_n(f) f$ for all $n$ and $f$ is normalised with $a_1(f) = 1$, then
the eigenvalue of $T_n$ on $f$ equals the $n$-th Fourier coefficient
$a_n(f)$; (iv) the Petersson inner product
$\langle f, g \rangle_{\mathrm{Pet}} = \int_{\Gamma \backslash
\mathbb{H}} f(z) \overline{g(z)} y^{k - 2}\,dx\,dy$ on $S_k(\Gamma)$
with the proof of $\Gamma$-invariance and finiteness on cusp forms;
(v) the **Hecke self-adjointness theorem**:
$\langle T_n f, g \rangle = \langle f, T_n g \rangle$ for all
$n \geq 1$, yielding the spectral decomposition of $S_k(\Gamma)$ into
an orthonormal basis of simultaneous Hecke eigenforms; (vi) the
Atkin-Lehner new-and-old-form decomposition at higher level
$\Gamma_0(N)$ with the **multiplicity-one** theorem for newforms.
The companion file declares the operator, the multiplicativity
theorem, the eigenform-Fourier-coefficient identity, and the
Petersson self-adjointness statement as `sorry`-stubbed declarations
on top of Mathlib's `ModularForm` type, with the proofs awaiting the
sub-lattice / coset-decomposition API for $\mathrm{SL}_2(\mathbb{Z})$
and the integration-on-fundamental-domain machinery for the
Petersson inner product.

Reviewer

TBD

Estimated time

beginner: 25m
intermediate: 55m
master: 90m