Atlas › Modular forms › Weight 6 › Level 91 › 91.6.a › 91.6.a.c

Newform orbit 91.6.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - q^{2} + 28q^{3} + 245q^{4} + 219q^{5} + 435q^{6} + 392q^{7} - 57q^{8} + 930q^{9} - 948q^{10} + 184q^{11} + 463q^{12} - 1352q^{13} - 49q^{14} - 256q^{15} + 4953q^{16} + 2278q^{17} - 5308q^{18} + 4959q^{19} + 6684q^{20} + 1372q^{21} - 2009q^{22} + 1889q^{23} + 26021q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 27x^{7} - 1758x^{6} + 50718x^{5} + 559812x^{4} - 17131764x^{3} - 53952320x^{2} + 1441170768x - 1529786880$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
13	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	−1	28	219	392	184	−1352	2278	4959	1889	15265	6805	5534	31176	−391	1539	19553	90596	−58218	−89884	2308	−69187	−167799	101449	51833	295025

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.257 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 engine's checks

dim M^+ at levels [1, 7, 13, 91]: [1, 5, 7, 50] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 6 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(91, 6)[2]   # 91.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 91:6 1000