Let S₂(Γ₀(163)) be the space of weight-2 cusp forms at level 163. This space has dimension 13 — the genus of the modular curve X₀(163). The prime 163 is the largest class-number-one Heegner prime: the imaginary quadratic field Q(√−163) has class number one.
The seven Heegner discriminants {3, 7, 11, 19, 43, 67, 163} define Hecke operators T_d on this space. For d = 163 we use the Atkin–Lehner operator U₁₆₃. The trace Gram matrix is
This is a 7×7 symmetric positive-definite integer matrix. Its entries encode the inner products of Hecke operators in the trace pairing.
The proof splits the 21 unordered pairs into four categories:
Nine pairs by Cauchy–Schwarz. When the ratio of the two diagonal entries is at most 4, the standard Cauchy–Schwarz bound on the Gram matrix gives the result directly.
Five vacuum-row pairs by Eichler–Selberg. For pairs involving d = 163, the prime-level Eichler–Selberg trace formula shows that most class-number terms vanish because (−t²/163) = −1 whenever 163 does not divide t. Only the t = 0 term (and t = 163 for the largest primes) survives. This rigid local cancellation is the most conceptual part of the proof.
Four pairs by the 1+5+7 orbit decomposition. The 13-dimensional space decomposes into three Galois orbits of newforms: V₁ (the rational elliptic curve 163a1, dimension 1), V₅ (degree 5), and V₇ (degree 7). Applying Cauchy–Schwarz within each orbit gives tighter bounds.
Three residual pairs by Hurwitz class-number sums. The remaining pairs are verified by exact evaluation of the Eichler–Selberg formula at composite indices n = 129, 201, 469.
The inequality is not generic. Adjoining any of the primes 17, 53, 71, 89, or 97 to the set produces violations. For example, |G(53,163)| = 43 > 26 = 2·G(163,163). The Heegner set is special — the bound holds for these seven primes and fails for larger sets at the same level.
The explicit Gram matrix, the surviving t-values for the residual Eichler–Selberg sums, and the orbit decomposition data are recorded in the paper. All computations were carried out in SageMath using modular symbols.
The Gram matrix has eigenvalues approximately 10.37, 19.44, 76.78, 98.09, 274.14, 491.62, 821.55 — confirming positive definiteness.
View PDFView HTML