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

Newform orbit 91.6.a.a

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 13q^{2} - 26q^{3} + 49q^{4} - 130q^{5} + 147q^{6} + 294q^{7} - 159q^{8} + 120q^{9} + 388q^{10} - 748q^{11} - 3q^{12} + 1014q^{13} - 637q^{14} + 754q^{15} - 1575q^{16} - 2682q^{17} - 5386q^{18} + 1278q^{19} - 6052q^{20} - 1274q^{21} + 155q^{22} - 4512q^{23} - 4359q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 39x^{5} - 296x^{4} - 12848x^{3} + 40016x^{2} + 602452x + 984696$

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
$\operatorname{tr} a_p$	−13	−26	−130	294	−748	1014	−2682	1278	−4512	−16704	−10980	11698	−8272	11360	−38476	21836	−59780	6738	−14782	−34050	−69654	30720	−10954	−163654	−14942

$\operatorname{tr} a_n$ 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)[0]   # 91.6.a.a

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

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