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

Newform orbit 91.4.a.d

Trace form

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

$6q + 2q^{2} + 13q^{3} + 10q^{4} + 26q^{5} + 3q^{6} + 42q^{7} + 12q^{8} + 111q^{9} - 32q^{10} + 11q^{11} + 141q^{12} + 78q^{13} + 14q^{14} + 64q^{15} - 54q^{16} + 210q^{17} + 65q^{18} + 78q^{19} + 38q^{20} + 91q^{21} + 125q^{22} - 75q^{23} - 189q^{24} + O(q^{25})$

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

$x^{6} - 15x^{5} - 56x^{4} + 1342x^{3} - 2656x^{2} - 16508x + 47232$

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
$\operatorname{tr} a_p$	2	13	26	42	11	78	210	78	−75	306	−93	−611	131	−88	587	770	934	1629	263	1032	819	−2169	1580	1244	−1643

$\operatorname{tr} a_n$ 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)[3]   # 91.4.a.d

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

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