Atlas › Modular forms › Weight 12 › Level 7 › 7.12.a › 7.12.a.a

Newform orbit 7.12.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 54q^{2} + 120q^{3} + 4100q^{4} - 13500q^{5} - 43668q^{6} + 33614q^{7} - 363960q^{8} - 104526q^{9} + 431880q^{10} - 750816q^{11} + 2429112q^{12} - 9548q^{13} - 907578q^{14} - 1214280q^{15} + 11267600q^{16} + 4160052q^{17} - 2029158q^{18} - 17998712q^{19} - 31313520q^{20} + 2016840q^{21} + 48302112q^{22} - 66161016q^{23} - 80862480q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3369})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{2} - 66x - 83136$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$

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
$\operatorname{tr} a_p$	−54	120	−13500	33614	−750816	−9548	4160052	−17998712	−66161016	61515612	−15281552	−527218340	−178276140	1826745232	568240704	−4185816372	3111345000	15042595060	9856523968	−24312011328	−30890001932	1992804256	5277014568	−101541312828	−192228621116

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.311 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]: [2, 9] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 2 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(7, 12)[0]   # 7.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 7:12 1000