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

Newform orbit 91.6.a.b

Trace form

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

$7q - 2q^{2} + 28q^{3} + 102q^{4} - 62q^{5} - 191q^{6} - 343q^{7} - 666q^{8} + 289q^{9} - 972q^{10} - 1102q^{11} + 463q^{12} - 1183q^{13} + 98q^{14} - 1484q^{15} + 1322q^{16} + 242q^{17} - 3819q^{18} - 1836q^{19} + 2590q^{20} - 1372q^{21} + 1263q^{22} - 6048q^{23} - 20517q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - 26x^{6} - 629x^{5} + 20738x^{4} - 138084x^{3} - 500004x^{2} + 8195044x - 22478352$

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 odd 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
$\operatorname{tr} a_p$	−2	28	−62	−343	−1102	−1183	242	−1836	−6048	−15518	8294	2622	−14090	11644	−6598	13618	−3688	−50200	17634	−108120	41302	1060	−33388	−118902	−203730

$\operatorname{tr} a_n$ 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)[1]   # 91.6.a.b

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

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