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Hecke trace Gram matrix at level 163

Name: Hecke Trace Gram Matrix at Level 163
Creator: Richard Hoekstra
Published: 2026-04

Computed in SageMath via ModularSymbols(163, 2). Verified by independent trace evaluation.

Setup

V = S₂(Γ₀(163)), dimension 13 (genus of X₀(163)). The prime 163 is the largest class-number-one Heegner prime. The Heegner operator set is S = {3, 7, 11, 19, 43, 67, 163}, where T₁₆₃ = U₁₆₃ = −w₁₆₃ (the Atkin-Lehner involution).

The Gram matrix G(d, d') = Tr(T_d T_d' | V)

	3	7	11	19	43	67	163
3	46	−18	−12	20	−44	−70	6
7	−18	92	−10	−44	−4	−60	4
11	−12	−10	136	−38	−88	120	6
19	20	−44	−38	282	−102	6	−6
43	−44	−4	−88	−102	583	−164	−13
67	−70	−60	120	6	−164	640	−14
163	6	4	6	−6	−13	−14	13

Symmetric, integer-valued, positive definite. The entry G(163, 163) = 13 = dim V, since U₁₆₃² = Id on the 13-dimensional space.

Eigenvalues of G

#	Eigenvalue
1	10.37
2	19.44
3	76.78
4	98.09
5	274.14
6	491.62
7	821.55

The Hecke eigenvalues a_p(E) for E = 163a1

The elliptic curve E: y² + y = x³ − 2x + 1 (Cremona label 163a1, conductor 163, rank 1).

p	a_p	a_p²	v₃(a_p)
3	0	0	∞
7	2	4	0
11	−6	36	1
19	−6	36	1
43	7	49	0
67	−2	4	0

Sum: Σ a_p² = 0 + 4 + 36 + 36 + 49 + 4 = 129 = 3 · 43.

The Hecke distance matrix D(d, d')

D(d, d') = G(d, d) + G(d', d') − 2G(d, d'). This is a squared Euclidean distance in eigenform coefficient space.

	3	7	11	19	43	67	163
3	0	174	206	288	717	826	47
7	174	0	248	462	683	852	97
11	206	248	0	494	895	536	137
19	288	462	494	0	1069	904	313
43	717	683	895	1069	0	1551	635
67	826	852	536	904	1551	0	681
163	47	97	137	313	635	681	0

The double-centering identity

B = JGJ = −(1/2)JDJ, where J = I₇ − (1/7)11^T. Both equalities hold exactly over Q. This is the classical Schoenberg identity identifying the centered Hecke operator with the canonical centered Gram matrix recovered from trace distances.

Cite this data

@misc{hoekstra2026gram,
  author = {Hoekstra, Richard},
  title = {Hecke Trace Gram Matrix at Level 163},
  year = {2026},
  url = {https://richardhoekstra.nl/data/hecke-gram-matrix-163.html}
}