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

Newform orbit 791.2.a.g

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + 4q^{2} + q^{3} + 36q^{4} + 4q^{5} - 2q^{6} + 24q^{7} + 9q^{8} + 41q^{9} - 6q^{10} + 8q^{11} - 5q^{12} + 6q^{13} + 4q^{14} + 21q^{15} + 56q^{16} + 3q^{17} + 13q^{19} - 14q^{20} + q^{21} + 15q^{22} + 10q^{23} - 22q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{13} + 7T_{17} + 9T_{19}$ , whose minimal polynomial is

$x^{24} - 175x^{23} - 25674x^{22} + 5445670x^{21} + 251808924x^{20} - 70636834607x^{19} - 1159664606634x^{18} + 498379136553540x^{17} + 2398544049849903x^{16} - 2091589580304252854x^{15} - 1756794161961520844x^{14} + 5363204960593568651100x^{13} + 4092117637778718092740x^{12} - 8310942286232253588933722x^{11} - 22599875118512434578711128x^{10} + 7401897397780104326951316796x^{9} + 39929216656840860355361133815x^{8} - 3380190549953127616529673081235x^{7} - 24960265592288210055345848948234x^{6} + 637884323502465015031047092108942x^{5} + 3847266408950287706754192177560320x^{4} - 59860245023943809659867846716173103x^{3} - 142201845673299615906033274206728542x^{2} + 2448958630194483238362485853591205856x - 4623763302763408325486066852876059847$

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
$\operatorname{tr} a_p$	4	1	4	24	8	6	3	13	10	7	15	36	14	34	−28	21	6	19	23	−10	11	55	−35	28	49

$\operatorname{tr} a_n$ 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)[6]   # 791.2.a.g

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

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