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

Newform orbit 7.12.a.b

Trace form

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

$3q + 77q^{2} - 140q^{3} + 5493q^{4} + 5026q^{5} + 64274q^{6} - 50421q^{7} + 125103q^{8} + 658519q^{9} + 610764q^{10} - 1039052q^{11} - 609154q^{12} - 1881222q^{13} - 1294139q^{14} - 8220752q^{15} - 9117711q^{16} + 15797334q^{17} + 38909q^{18} + 12340356q^{19} + 37982168q^{20} + 2352980q^{21} + 31890144q^{22} + 7276752q^{23} - 117296046q^{24} + O(q^{25})$

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

$x^{3} + 63x^{2} - 663088x - 106812564$

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 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
$\operatorname{tr} a_p$	77	−140	5026	−50421	−1039052	−1881222	15797334	12340356	7276752	−126853406	162184512	567121338	893682734	460197828	2723825664	93426522	5899174428	2106807738	−27027285348	16564049928	6036803934	−54895016736	123561203892	−91648411106	−71237114634

$\operatorname{tr} a_n$ 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)[1]   # 7.12.a.b

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

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