Setup

Let $H=\{2,3,7,11,19,43,67,163\}$ be the eight rational primes $p$ for which $\mathbb{Q}(\sqrt{-p})$ has class number one . Set $$H' \;=\; \{N\in H : \dim S_2(\Gamma_0(N))>0\} \;=\; \{11,19,43,67,163\}.$$ At each level $N\in H'$ we form the nested gate set $$S_N \;=\; \{p\in H\setminus\{2\}:p\leq N\},$$ i.e. $S_{11}=\{3,7,11\}$, $S_{19}=\{3,7,11,19\}$, $S_{43}=\{3,7,11,19,43\}$, $S_{67}=\{3,7,11,19,43,67\}$, and $S_{163}=\{3,7,11,19,43,67,163\}$. The gate $T_N$ is interpreted as the Atkin–Lehner-twisted operator $U_N=-w_N$ at the prime level.

The Hecke trace Gram matrix is $$G_N(d,d') \;=\; \mathop{\mathrm{Tr}}\!\bigl(T_dT_{d'}\mid S_2(\Gamma_0(N))\bigr).$$ At $N=163$ the inequality $$\label{eq:ineq} |G_N(d,d')| \;\leq\; 2\min\bigl(G_N(d,d),G_N(d',d')\bigr)$$ holds for all $21$ pairs in $S_{163}$ . The present note tests [eq:ineq] at the four other Heegner-prime levels.

Computation

[prop:gram-tables] The trace Gram matrices on $S_N$ at the five levels of $H'$, computed in SageMath via modular symbols, are:

$N=11$, $S_{11}=\{3,7,11\}$: $$G_{11}=\begin{pmatrix}1 & 2 & -1\\ 2 & 4 & -2\\ -1 & -2 & 1\end{pmatrix}$$

$N=19$, $S_{19}=\{3,7,11,19\}$: $$G_{19}=\begin{pmatrix}4 & 2 & -6 & -2\\ 2 & 1 & -3 & -1\\ -6 & -3 & 9 & 3\\ -2 & -1 & 3 & 1\end{pmatrix}$$

$N=43$, $S_{43}=\{3,7,11,19,43\}$: $$G_{43}=\begin{pmatrix} 8 & -4 & -14 & 12 & 2\\ -4 & 12 & 12 & 0 & -4\\ -14 & 12 & 27 & -18 & -5\\ 12 & 0 & -18 & 28 & -2\\ 2 & -4 & -5 & -2 & 3 \end{pmatrix}$$

$N=67$, $S_{67}=\{3,7,11,19,43,67\}$: $$G_{67}=\begin{pmatrix} 14 & -4 & 14 & -26 & -10 & 2\\ -4 & 30 & -6 & 0 & -12 & 0\\ 14 & -6 & 28 & -54 & 14 & -2\\ -26 & 0 & -54 & 135 & 2 & -5\\ -10 & -12 & 14 & 2 & 134 & -14\\ 2 & 0 & -2 & -5 & -14 & 5 \end{pmatrix}$$

$N=163$, $S_{163}=\{3,7,11,19,43,67,163\}$: as in .

Direct computation in SageMath using ModularSymbols(N,2,sign=1).cuspidal_subspace() and hecke_matrix. Reproducible by the script in the appendix.

Status of the inequality at each level

[thm:status] The trace inequality [eq:ineq] holds at $N\in\{11,43,163\}$ and fails at $N\in\{19,67\}$. Specifically:

1. $N=11$: holds for all $\binom{3}{2}=3$ pairs.

2. $N=19$: fails for the two pairs $(7,11)$ and $(11,19)$, where $|G_{19}|=3$ exceeds the bound $2$.

3. $N=43$: holds for all $\binom{5}{2}=10$ pairs.

4. $N=67$: fails for the single pair $(43,67)$, where $|G_{67}(43,67)|=14$ exceeds the bound $2\min(134,5)=10$.

5. $N=163$: holds for all $\binom{7}{2}=21$ pairs .

Read off the Gram matrices of Proposition [prop:gram-tables] and check $|G_N(d,d')|$ against $2\min(G_N(d,d),G_N(d',d'))$ for each pair. The non-trivial line is at $N=67$: $$|G_{67}(43,67)|=14,\qquad 2\min(G_{67}(43,43),G_{67}(67,67))=2\min(134,5)=10.$$

The dimension-one reduction

At $N\in\{11,19\}$ the cuspidal space is one-dimensional, so $G_N$ has rank one and decomposes as $G_N(d,d')=a_d(f^N)\,a_{d'}(f^N)$ where $f^N$ is the unique normalised newform. In this case [eq:ineq] is equivalent to a statement about the spread of the $a_p$ values.

[prop:dim1] For $N$ with $\dim S_2(\Gamma_0(N))=1$, the inequality [eq:ineq] on the gate set $S_N$ is equivalent to $$\max_{p\in S_N}|a_p(f^N)| \;\leq\; 2\min_{p\in S_N}|a_p(f^N)|.$$

$|G_N(d,d')|=|a_d a_{d'}|=|a_d|\cdot|a_{d'}|$, and $2\min(G_N(d,d),G_N(d',d'))=2\min(a_d^2,a_{d'}^2) =2\min(|a_d|,|a_{d'}|)^2$. The inequality $|a_d||a_{d'}|\leq 2\min(|a_d|,|a_{d'}|)^2$ rearranges to $\max(|a_d|,|a_{d'}|)\leq 2\min(|a_d|,|a_{d'}|)$. Quantifying over all pairs gives the global ratio bound.

[cor:dim1-eval] For $N=11$, $f^N=11a1$ has $(a_3,a_7,a_{11})=(-1,-2,1)$, so the absolute values are $(1,2,1)$ with ratio $2$ — the inequality holds (at the boundary). For $N=19$, $f^N=19a1$ has $(a_3,a_7,a_{11},a_{19})=(-2,-1,3,1)$, so the absolute values are $(2,1,3,1)$ with ratio $3>2$ — the inequality fails on the $(7,11)$ and $(11,19)$ pairs.

Direct from Proposition [prop:dim1] and the $a_p$ data computed in SageMath.

The dimension-one cases show that whether [eq:ineq] holds at a small level is essentially a question about whether the eigenvalues of the unique newform happen to lie within a factor of $2$ of one another on the chosen gate set. This is a property of $f^N$, not of any Heegner structure. At $N=11$ it holds (just barely); at $N=19$ it fails (just barely).

What the data shows

[thm:negative] The level-$163$ trace inequality  is not a uniform property of the class-number-one Heegner-prime family $H'$. It holds at three of five levels and fails at the other two.

Theorem [thm:status].

The two failures are mild:

• At $N=19$, the violating ratio is $|G_{19}|/\text{bound}=3/2$.

• At $N=67$, the violating ratio is $|G_{67}(43,67)|/\text{bound}=14/10=7/5$.

In particular, the constant $2$ in [eq:ineq] cannot be strengthened uniformly, but the family-wide constant could plausibly be replaced by something around $7/5\cdot 2=14/5$ if a uniform statement is desired. We do not pursue this.

The levels where the inequality holds are $N\in\{11,43,163\}$ and where it fails are $N\in\{19,67\}$. We have no conceptual explanation for this partition. The dimensions of the cuspidal spaces are $(1,1,3,5,13)$ at $(11,19,43,67,163)$, so the holding set has dimensions $(1,3,13)$ and the failing set has dimensions $(1,5)$. The successive ratios in the holding set $(1\to 3\to 13)$ are $(3,13/3)$ and the failing-set ratio $(1\to 5)$ is $5$, but we record this without interpretation.

What is not proved here

• No conceptual reason is given for the partition $\{11,43,163\}\sqcup\{19,67\}$.

• No uniform constant $C\geq 2$ is computed for which $|G_N(d,d')|\leq C\min(G_N(d,d),G_N(d',d'))$ holds across all of $H'$. The data of Theorem [thm:status] show $C\geq 7/5\cdot 2$ is necessary; we do not show it is sufficient at all levels (in particular, larger gate sets at higher levels could produce worse ratios).

• No statement is made about non-prime $N$, about non-Heegner levels, or about the same construction at higher weight.

• The negative result of Theorem [thm:negative] should be read as a statement about the specific inequality $|G|\leq 2\min(G_{dd},G_{d'd'})$. Variants with different constants or different gate sets are not addressed.

Reproducibility

The complete calculation of all Gram matrices and the inequality check is the following SageMath script:

heegner_primes = [3, 7, 11, 19, 43, 67, 163]
levels = [11, 19, 43, 67, 163]

def gate_set(N):
    return [p for p in heegner_primes if p <= N]

def trace_gram(N, S):
    M = ModularSymbols(N, 2, sign=1).cuspidal_subspace()
    n = len(S)
    G = matrix(ZZ, n, n)
    for i, d in enumerate(S):
        Td = M.hecke_matrix(d)
        for j, dp in enumerate(S):
            G[i,j] = (Td * M.hecke_matrix(dp)).trace()
    return G

for N in levels:
    S = gate_set(N)
    G = trace_gram(N, S)
    n = len(S)
    violations = [(S[i], S[j], abs(G[i,j]), 2*min(G[i,i], G[j,j]))
                  for i in range(n) for j in range(i+1, n)
                  if abs(G[i,j]) > 2*min(G[i,i], G[j,j])]
    print(N, "violations:", violations)

9

R. Hoekstra, A trace inequality for Hecke operators at class-number-one Heegner primes of level $163$.

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