Atlas › Modular forms › Weight 2 › Level 961 › 961.2.a › 961.2.a.g

Newform orbit 961.2.a.g

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 6q^{4} + 3q^{8} - 9q^{9} - 9q^{10} + 15q^{14} + 12q^{16} + 21q^{20} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{3} - 6x - 1$

Complex multiplication by $\mathbb{Q}(\sqrt{-31})$ : $a_p = 0$ for every conjugate at the 6 primes $p \le 31$ inert in it (exactly, from $\operatorname{tr} a_{p^2} = -\dim \cdot p^{k-1}$ ), and $\operatorname{tr} a_p = 0$ at all 91 inert primes $p < 1000$ . This is evidence from the stored coefficients, not a proof.

Atkin–Lehner signs

$p$	$w_p$
31	— ( $31^2 \mid 961$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	0	0	0	0	0	0	0	0	0	0	0	0	0	24	0	0	0	−36	0	0	0	0	0	0

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 31, 961]: [0, 3, 83] (certified at ell = 2147477881)
T = [(2, 1), (3, 3)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 9 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(961, 2)[6]   # 961.2.a.g

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 961:2 1000