Hecke Kernel Transformer at Level 163

Setup

• level N = 163
• genus g = 13
• discriminants [3, 7, 11, 19, 43, 67, 163]
• Hecke module: cuspidal modular symbols for Gamma_0(163), dimension 13

Define

G(d,d') = Tr(T_d T_d' | S_2(Gamma_0(163)))

for d,d' in [3, 7, 11, 19, 43, 67, 163].

Full 7x7 matrix

Rows/columns ordered as [3, 7, 11, 19, 43, 67, 163]:

[  46  -18  -12   20  -44  -70    6]
[ -18   92  -10  -44   -4  -60    4]
[ -12  -10  136  -38  -88  120    6]
[  20  -44  -38  282 -102    6   -6]
[ -44   -4  -88 -102  583 -164  -13]
[ -70  -60  120    6 -164  640  -14]
[   6    4    6   -6  -13  -14   13]

Requested checks

• G(163,163) = 13.
• G(163,d) row: {'3': 6, '7': 4, '11': 6, '19': -6, '43': -13, '67': -14, '163': 13}.
• |G(163,d)| <= 26 for all d: True.
• maximum off-diagonal |G(d,d')|: {'abs_value': 164, 'pair': [43, 67], 'value': -164}.
• transformer-ratio inequality |G(d,d')| <= 2 min(G(d,d), G(d',d')) for every pair: True.
• violations: [].

Diagonal self-energies

{'3': 46, '7': 92, '11': 136, '19': 282, '43': 583, '67': 640, '163': 13}

Eigenvalues

7x7 matrix eigenvalues:

['10.37188113408177?', '19.43971877724339?', '76.78154057670428?', '98.0936915437378?', '274.1373051897647?', '491.6242100246581?', '821.551652753810?']

Numerically:

[10.371881134081772, 19.439718777243385, 76.78154057670429, 98.09369154373778, 274.1373051897647, 491.62421002465805, 821.55165275381]

Positive semidefinite: True.

This is expected structurally, because G is a Gram matrix:

G(d,d') = sum_f a_d(f) a_d'(f)

over the 13 Galois-conjugate newforms.

163 - 2 G(163,d)

{'3': 151, '7': 155, '11': 151, '19': 175, '43': 189, '67': 191, '163': 137}

The value 137 = 163 - 2*13 appears only at d = 163 itself. The other entries are 151, 155, 151, 175, 189, 191, so there is no nontrivial second appearance of 137 in this matrix.

6x6 matter submatrix

Excluding d = 163:

[  46  -18  -12   20  -44  -70]
[ -18   92  -10  -44   -4  -60]
[ -12  -10  136  -38  -88  120]
[  20  -44  -38  282 -102    6]
[ -44   -4  -88 -102  583 -164]
[ -70  -60  120    6 -164  640]

Eigenvalues:

['18.32286919004750?', '76.76556178874808?', '97.8930888116512?', '273.4009044696592?', '491.0714441558965?', '821.5461315839976?']

Numerically:

[18.3228691900475, 76.76556178874807, 97.89308881165121, 273.40090446965917, 491.07144415589653, 821.5461315839975]

Honest read

• The matrix is completely explicit and integral.
• G(163,163)=13 holds exactly because T_163 has eigenvalues ±1, so squaring and tracing returns the full dimension.
• The requested pairwise inequality happens to hold for all 49 pairs in this data set.
• The matrix is positive semidefinite for the simplest possible reason: it is the Gram matrix of the Hecke-eigenvalue vectors indexed by the seven discriminants.
• The 137 numerology does not propagate through the off-diagonal 163 row; it only appears in the tautological diagonal slot.