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

Newform orbit 91.6.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 10q^{2} + 10q^{3} + 66q^{4} + 89q^{5} - 47q^{6} - 441q^{7} + 384q^{8} + 1099q^{9} + 1164q^{10} + 386q^{11} + 1725q^{12} + 1521q^{13} - 490q^{14} + 1326q^{15} + 426q^{16} + 2218q^{17} + 1935q^{18} - 1185q^{19} + 5854q^{20} - 490q^{21} + 7299q^{22} + 3193q^{23} + 15939q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} - 20x^{8} - 1573x^{7} + 22714x^{6} + 889358x^{5} - 7575628x^{4} - 204924468x^{3} + 717167488x^{2} + 16308058000x + 7436000000$

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
$\operatorname{tr} a_p$	10	10	89	−441	386	1521	2218	−1185	3193	18387	12117	28950	25826	−9123	−21689	62879	−70420	40812	14100	−23730	−33477	22077	−167391	−85093	294351

$\operatorname{tr} a_n$ 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)[3]   # 91.6.a.d

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

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