Atlas › Modular forms › Weight 2 › Level 791 › 791.2.a › 791.2.a.f

Newform orbit 791.2.a.f

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q - 2q^{2} + 9q^{3} + 26q^{4} + 12q^{5} + 14q^{6} - 18q^{7} + 3q^{8} + 31q^{9} + 16q^{10} + 4q^{11} + 5q^{12} + 12q^{13} + 2q^{14} - 7q^{15} + 30q^{16} + 7q^{17} - 2q^{18} + 23q^{19} + 16q^{20} - 9q^{21} - 13q^{22} + 4q^{23} + 26q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{18} - 341x^{17} + 27625x^{16} + 3031323x^{15} - 579332950x^{14} + 11963583945x^{13} + 2924921156924x^{12} - 185830560248457x^{11} - 2687688378995646x^{10} + 553581362812294739x^{9} - 11249604020098197800x^{8} - 450348321872986721019x^{7} + 21356050344542396829970x^{6} - 166087803168872775588425x^{5} - 5549655927551330651323304x^{4} + 129363953098817091274064513x^{3} - 992255898942148508372782559x^{2} + 2358503616042503709826174138x + 2087568437849369020875416155$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
113	$-1$

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	9	12	−18	4	12	7	23	4	−3	27	4	4	−18	30	5	−10	69	7	−4	49	−31	−17	68	43

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.367 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 7, 113, 791]: [0, 1, 10, 78] (certified at ell = 2147480161)
T = [(2, 1), (3, 3), (13, 5), (17, 7), (19, 9)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 15 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(791, 2)[5]   # 791.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 791:2 1000