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

Newform orbit 91.4.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 7q^{2} - 5q^{3} + 53q^{4} + 16q^{5} + q^{6} - 35q^{7} + 75q^{8} + 58q^{9} + 48q^{10} + 83q^{11} - 17q^{12} - 65q^{13} - 49q^{14} - 20q^{15} + 337q^{16} - 88q^{17} + 342q^{18} + 170q^{19} + 388q^{20} + 35q^{21} - 111q^{22} + 57q^{23} - 627q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} - 2x^{4} - 142x^{3} + 36x^{2} + 4844x + 11304$

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
$\operatorname{tr} a_p$	7	−5	16	−35	83	−65	−88	170	57	376	−7	407	967	−792	−823	232	98	697	−225	2052	879	2587	284	462	−1259

$\operatorname{tr} a_n$ 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)[2]   # 91.4.a.c

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

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