Introduction

The space $S_2(\Gamma_0(163))$ of weight-2 cusp forms for the congruence subgroup $\Gamma_0(163)$ has dimension $g=13$, where $g$ is the genus of the modular curve $X_0(163)$. The prime $p=163$ is distinguished as the largest prime for which $\mathbb{Q}(\sqrt{-p})$ has class number one, a list completed by Heegner , Stark , and Baker .

For each prime $d$ in the set $$S = \{3,\, 7,\, 11,\, 19,\, 43,\, 67,\, 163\},$$ the Hecke operator $T_d$ acts on $S_2(\Gamma_0(163))$. (At $d=163$, the operator is $T_{163}=U_{163}$, since $163$ divides the level.) The Hecke trace Gram matrix $$\label{eq:G} G(d,d') = \mathop{\mathrm{Tr}}(T_d\, T_{d'} \mid S_2(\Gamma_0(163)))$$ is a $7\times 7$ symmetric positive definite matrix with integer entries, computable via the Eichler–Selberg trace formula or directly in a computer algebra system supporting modular forms (we used SageMath ).

The present note has two layers.

First, we record the exact distance-geometric identity $$B := JGJ = -\tfrac{1}{2}JDJ,$$ where $D(d,d')=G(d,d)+G(d',d')-2G(d,d')$ and $J=I_7-\frac17\mathbf{1}\mathbf{1}^T$ is the centering matrix. This is the classical double-centering identity of Schoenberg , but here it identifies the centered Hecke operator with the canonical centered Gram matrix recovered from Hecke trace distances.

Second, we compare $B$ to the cotangent Laplacian $L_{\mathrm{cot}}$ built from the same distance data. The complete graph $K_7$ endowed with edge lengths $\ell(d,d')=\sqrt{D(d,d')}$ carries a canonical formal cotangent operator. Although this graph is not a triangulated surface in the usual Regge sense, the resulting symmetric Laplacian is numerically almost diagonal in a $B$-eigenbasis. This suggests that the Hecke trace geometry and the discrete cotangent geometry are much closer than one would expect a priori. The level 163 example is the main case study; a later scan is included only to show that the phenomenon is broader than a single isolated example.

The Hecke trace Gram matrix

All computations were performed in SageMath using ModularSymbols(163, 2) and verified by independent evaluation of traces.

[prop:G] The matrix $G = \bigl(\mathop{\mathrm{Tr}}(T_d\,T_{d'}\mid S_2(\Gamma_0(163)))\bigr)_{d,d'\in S}$, with rows and columns ordered as $(3,7,11,19,43,67,163)$, is $$G = \begin{pmatrix} 46 & -18 & -12 & 20 & -44 & -70 & 6\\ -18 & 92 & -10 & -44 & -4 & -60 & 4\\ -12 & -10 & 136 & -38 & -88 & 120 & 6\\ 20 & -44 & -38 & 282 & -102 & 6 & -6\\ -44 & -4 & -88 & -102 & 583 & -164 & -13\\ -70 & -60 & 120 & 6 & -164 & 640 & -14\\ 6 & 4 & 6 & -6 & -13 & -14 & 13 \end{pmatrix}.$$ Its eigenvalues, in increasing order, are approximately $$10.37,\; 19.44,\; 76.78,\; 98.09,\; 274.14,\; 491.62,\; 821.55.$$ In particular, $G$ is positive definite and $G(163,163)=13=g$.

Direct computation in SageMath. The identity $G(163,163)=\mathop{\mathrm{Tr}}(T_{163}^2\mid S_2(\Gamma_0(163))) = \mathop{\mathrm{Tr}}(U_{163}^2) = \dim S_2(\Gamma_0(163)) = 13$ follows from the fact that $U_{163} = -w_{163}$ (the Atkin–Lehner involution) at prime level, so $U_{163}^2 = w_{163}^2 = \mathrm{Id}$ on the $13$-dimensional space.

The double-centering identity

Given the Gram matrix $G$, define the Hecke distance matrix by $$D(d,d') = G(d,d) + G(d',d') - 2\,G(d,d'),$$ and the centering matrix $$J = I_7 - \tfrac{1}{7}\,\mathbf{1}\mathbf{1}^T,$$ where $\mathbf{1}=(1,\ldots,1)^T\in\mathbb{R}^7$. The centered Hecke operator is $B = JGJ$.

[thm:centering] With $G$, $D$, $J$, and $B$ as above, $$\label{eq:centering} B = JGJ = -\tfrac{1}{2}\,JDJ.$$ Both equalities hold exactly over $\mathbb{Q}$.

Write $D = \mathbf{g}\,\mathbf{1}^T + \mathbf{1}\,\mathbf{g}^T - 2G$, where $\mathbf{g} = \mathop{\mathrm{diag}}(G) = (G(d,d))_{d\in S}$. Then $$-\tfrac{1}{2}\,JDJ = -\tfrac{1}{2}\,J\bigl(\mathbf{g}\,\mathbf{1}^T + \mathbf{1}\,\mathbf{g}^T - 2G\bigr)J = JGJ,$$ since $J\mathbf{1} = \mathbf{0}$ and $\mathbf{1}^T J = \mathbf{0}^T$. This is Schoenberg’s classical identity . Since all entries of $G$ and $D$ are rational integers, the identity holds exactly over $\mathbb{Q}$.

The matrix $B$ annihilates $\mathbf{1}$, hence $\operatorname{rank}(B)\leq 6$. In the present case $B$ has exactly six nonzero eigenvalues, namely $$10.632739,\; 60.562594,\; 96.738276,\; 251.545366,\; 455.849771,\; 810.671253,$$ and one zero eigenvalue in the direction of $\mathbf{1}$.

The cotangent operator

We equip the complete graph $K_7$ on vertex set $S$ with edge lengths $$\ell(d,d') = \sqrt{D(d,d')}.$$

For each triangle $(d,d',d'')$ in $K_7$, define the angle at vertex $d$ by the Euclidean law of cosines: $$\cos\alpha_d(d',d'') = \frac{\ell(d,d')^2 + \ell(d,d'')^2 - \ell(d',d'')^2}{2\,\ell(d,d')\,\ell(d,d'')}\,.$$ The cotangent weight of edge $(d,d')$ is $$w(d,d') = \frac{1}{2}\sum_{\substack{d''\in S\\d''\neq d,d'}} \cot\alpha_{d''}(d,d'),$$ where the sum runs over the five triangles containing the edge $(d,d')$.

The cotangent Laplacian is the $7\times 7$ matrix $$L_{\mathrm{cot}}(d,d') = \begin{cases} -w(d,d') & d\neq d',\\[4pt] \displaystyle\sum_{d''\neq d} w(d,d'') & d = d'. \end{cases}$$

The complete $2$-skeleton of $K_7$ is not a $2$-manifold triangulation: each edge lies in five triangles. Thus $L_{\mathrm{cot}}$ should be read as the formal cotangent operator attached to the Hecke length data, rather than as the Laplace–Beltrami operator of a genuine piecewise-flat surface.

The matrix $L_{\mathrm{cot}}$ is symmetric and satisfies $L_{\mathrm{cot}}\,\mathbf{1}=\mathbf{0}$. In the present computation its spectrum is nonnegative: $$0,\; 5.250892,\; 7.378571,\; 10.591287,\; 15.303978,\; 17.814893,\; 26.837106.$$ One cotangent edge weight is slightly negative, so we do not claim positive semidefiniteness by abstract construction alone.

Near-simultaneous diagonalizability

The exact theorem concerns $B$, not $G$. Indeed, an identity $L_{\mathrm{cot}}=G$ is impossible, because $$L_{\mathrm{cot}}\,\mathbf{1}=\mathbf{0}, \qquad G\,\mathbf{1}=(-72,\,-40,\,114,\,118,\,168,\,458,\,-4)^T.$$ Thus the correct comparison is between $L_{\mathrm{cot}}$ and the centered Hecke operator $B$ on $\mathbf{1}^\perp$.

[prop:bridge] Let $B$ and $L_{\mathrm{cot}}$ be as above, restricted to the six-dimensional subspace $\mathbf{1}^\perp$. Then:

1. The sorted nonzero eigenvalue sequences of $B$ and $L_{\mathrm{cot}}$ have Pearson correlation coefficient $$\rho = 0.982148.$$

2. Under the overlap-maximizing matching of eigenvectors, the pairing is order-reversing: the largest eigenvalue of $B$ is matched with the smallest nonzero eigenvalue of $L_{\mathrm{cot}}$, and vice versa.

3. Each eigenvector $v_i$ of $B$ on $\mathbf{1}^\perp$ has a matched eigenvector $u_{\sigma(i)}$ of $L_{\mathrm{cot}}$ such that $$|\langle v_i,\, u_{\sigma(i)}\rangle| \geq 0.930724,$$ with the maximum overlap attaining $0.996905$ and the mean overlap equal to $0.973626$.

4. The relative commutator norm satisfies $$\frac{\|[B,\, L_{\mathrm{cot}}]\|_F}{\|B\|_F\,\|L_{\mathrm{cot}}\|_F} = 0.030399,$$ where $\|\cdot\|_F$ denotes the Frobenius norm.

5. In the eigenbasis of $B$, the matrix $L_{\mathrm{cot}}$ has off-diagonal Frobenius norm equal to $7.135\%$ of its total Frobenius norm.

All values are computed from the exact integer matrix $G$ of Proposition [prop:G], with cotangent weights evaluated in double-precision floating point.

The numerical evidence therefore supports not the false identity $L_{\mathrm{cot}}=G$, but the weaker and more plausible bridge principle that $L_{\mathrm{cot}}$ is nearly diagonal in the centered Hecke eigenbasis.

A local cumulative comparison

To compare levels with increasing arithmetic complexity, define the cumulative class-number-one sets $$\begin{aligned} S^{(43)} &= \{3,\,7,\,11,\,19,\,43\},\\ S^{(67)} &= \{3,\,7,\,11,\,19,\,43,\,67\},\\ S^{(163)} &= \{3,\,7,\,11,\,19,\,43,\,67,\,163\}.\end{aligned}$$ For each level $p\in\{43,67,163\}$ we form the Hecke trace matrix $G^{(p)}$ on the corresponding set $S^{(p)}$, then define $D^{(p)}$, $B^{(p)}$, and $L_{\mathrm{cot}}^{(p)}$ exactly as above.

[prop:family] The off-diagonal fraction $\delta(p)$—defined as the ratio of the off-diagonal Frobenius norm of $L_{\mathrm{cot}}^{(p)}$ in the eigenbasis of $B^{(p)}$ to the total Frobenius norm of that restricted matrix—takes the values

Among these three cumulative levels, the near-diagonalizability is strongest at level $163$.

This comparison is cumulative rather than intrinsic: the sets $S^{(43)}$, $S^{(67)}$, and $S^{(163)}$ are nested prefixes of the class-number-one primes, not the split-Heegner sets at the corresponding levels. We make this explicit because the numerical trend depends on that choice of family.

A broader scan over prime levels $N\leq 199$ with the same cumulative family choice shows that level 163 is not globally minimal for the off-diagonal fraction. Among the full seven-operator cumulative cases in that scan, levels $173$, $197$, and $199$ all give smaller values than the level-163 value $0.071352$. Thus Proposition [prop:family] should be read as a local comparison among the three displayed cumulative levels, not as a global extremality claim for 163.

Discussion

The double-centering identity (Theorem [thm:centering]) is classical, but its application to the Hecke Gram matrix at Heegner primes yields an exact arithmetic consequence: the pairwise Hecke distances determine the centered Hecke operator without loss.

The cotangent operator attached to the same distance data is not equal to the Hecke operator. Nevertheless, Proposition [prop:bridge] shows that the two operators almost share an eigenbasis. This is the mathematically robust bridge: not equality of matrices, but near-simultaneous diagonalizability on the centered subspace.

The exact content of this note is therefore asymmetric: the double-centering identity is a theorem, while the Hecke–cotangent bridge is a numerical phenomenon anchored at level 163. The local comparison of Proposition [prop:family] shows that the bridge improves from $43$ and $67$ to $163$ inside one natural cumulative family, but the broader scan shows that this does not extend to a global minimality statement. Any claim that 163 is geometrically extremal must therefore depend on a more refined arithmetic selection principle than the cumulative family alone.

One further numerical feature merits mention. If the nonzero eigenspaces are paired by maximum overlap, then the cotangent eigenvalues are fit substantially better by a decreasing rational function of the centered Hecke eigenvalues than by an affine rescaling. This suggests that the correct next conjecture is not $L_{\mathrm{cot}}\approx aB+bI$, but rather that $L_{\mathrm{cot}}|_{\mathbf{1}^\perp}$ is approximately of the form $f(B|_{\mathbf{1}^\perp})$ for a monotone decreasing scalar function $f$.

Acknowledgements

Computations were performed in SageMath , NumPy, and SciPy.

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A. Baker, Linear forms in the logarithms of algebraic numbers, Mathematika 13 (1966), 204–216.

K. Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253.

T. Miyake, Modular Forms, Springer-Verlag, Berlin, 1989.

U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2 (1993), 15–36.

T. Regge, General relativity without coordinates, Nuovo Cimento 19 (1961), 558–571.

The Sage Developers, SageMath, the Sage Mathematics Software System, https://www.sagemath.org.

I. J. Schoenberg, Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert”, Ann. of Math. 36 (1935), 724–732.

H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27.