The space S₂(Γ₀(163)) of weight-2 cusp forms has dimension g = 13 (the genus of X₀(163)). For each prime d in the Heegner set S = {3, 7, 11, 19, 43, 67, 163}, the Hecke operator Td acts on this space (with T₁₆₃ = U₁₆₃, the Atkin-Lehner involution at the level). The trace Gram matrix G(d,d') = Tr(Td Td' | S₂(Γ₀(163))) is a 7×7 symmetric positive definite integer matrix, computed in SageMath.
The diagonal entry G(163,163) = 13 = g, since U₁₆₃² = w₁₆₃² = Id on the 13-dimensional space.
Eigenvalues of G: 10.37, 19.44, 76.78, 98.09, 274.14, 491.62, 821.55. Positive definite.
This is the classical Schoenberg identity, but here it identifies the centered Hecke operator with the canonical centered Gram matrix recovered from Hecke trace distances. The Hecke distance D(d,d') is a squared Euclidean distance in the eigenform coefficient space.
The complete graph K₇ is equipped with edge lengths ℓ(d,d') = √D(d,d'). Although this graph is not a triangulated surface in the Regge sense, the resulting cotangent Laplacian Lcot is a well-defined symmetric operator.
| Metric | Value |
|---|---|
| Relative commutator norm ‖[Lcot, B]‖ / (‖Lcot‖·‖B‖) | 0.030 |
| Eigenvector overlaps (min – max) | 0.931 – 0.997 |
| Off-diagonal Frobenius fraction in B-eigenbasis | 0.071 (7.1%) |
The cotangent geometry and the Hecke trace geometry are much closer than one would expect a priori. The off-diagonal fraction of 7.1% means Lcot is 92.9% diagonal in the B-eigenbasis.
The near-diagonalizability metric δ (off-diagonal Frobenius fraction) is computed at three Heegner levels with cumulative gate sets:
| Level N | δ(N) |
|---|---|
| 43 | 0.100 |
| 67 | 0.139 |
| 163 | 0.071 |
Level 163 has the smallest δ (strongest near-diagonalizability) among the three Heegner cases.