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

Newform orbit 91.4.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + q^{2} + q^{3} - 3q^{4} - 22q^{5} - 23q^{6} - 21q^{7} - 27q^{8} - 32q^{9} - 20q^{10} - 31q^{11} - 3q^{12} + 39q^{13} - 7q^{14} - 60q^{15} - 127q^{16} - 140q^{17} + 18q^{18} + 68q^{19} + 100q^{20} - 7q^{21} - 19q^{22} - 221q^{23} + 129q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 2x^{2} - 10x + 4$

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
$\operatorname{tr} a_p$	1	1	−22	−21	−31	39	−140	68	−221	−114	167	129	−505	−290	−53	−214	−130	375	333	−1392	569	−2219	354	−268	−1729

$\operatorname{tr} a_n$ 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)[0]   # 91.4.a.a

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

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