Atlas › Modular forms › Weight 2 › Level 356 › 356.2.a › 356.2.a.b

Newform orbit 356.2.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + q^{3} + 3q^{5} + 16q^{9} + 8q^{11} + 10q^{13} + q^{15} + 7q^{17} + 13q^{19} + 20q^{21} - 3q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{7} - 183x^{6} + 2162x^{5} + 1028640x^{4} - 46258905x^{3} + 146798311x^{2} + 9322528422x - 56115796062$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 356$ )
89	$+1$

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	1	3	0	8	10	7	13	−3	4	−9	2	8	13	−30	−7	0	18	−2	−30	19	−18	−20	−7	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.212 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, 2, 4, 89, 178, 356]: [0, 1, 2, 8, 24, 48] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(356, 2)[1]   # 356.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 356:2 1000