Atlas › Modular forms › Weight 4 › Level 91 › 91.4.a › 91.4.a.b

Newform orbit 91.4.a.b

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 4q^{2} - 5q^{3} + 26q^{4} - 36q^{5} - 45q^{6} + 28q^{7} - 30q^{8} + 21q^{9} - 44q^{10} - 95q^{11} - 17q^{12} - 52q^{13} - 28q^{14} - 16q^{15} + 58q^{16} - 146q^{17} + 65q^{18} - 48q^{19} - 474q^{20} - 35q^{21} - 143q^{22} - 121q^{23} - 469q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{4} + 9x^{3} - 8x^{2} - 66x + 72$

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
$\operatorname{tr} a_p$	−4	−5	−36	28	−95	−52	−146	−48	−121	−440	−283	−209	−93	526	−783	−340	−922	−141	−523	1468	−47	1025	−1190	−2962	2715

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.104 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, 3, 5, 30] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 18 = 1 x 18 (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(91, 4)[1]   # 91.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 91:4 1000