Atlas › Modular forms › Weight 2 › Level 781 › 781.2.a › 781.2.a.g

Newform orbit 781.2.a.g

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 3q^{2} + 9q^{3} + 25q^{4} + 9q^{5} - 2q^{6} - 2q^{7} + 3q^{8} + 33q^{9} + 2q^{10} + 20q^{11} + 19q^{12} - 12q^{13} + 16q^{14} + 10q^{15} + 31q^{16} + 9q^{17} - 11q^{18} + 12q^{19} + 21q^{20} - 6q^{21} + 3q^{22} + 17q^{23} - q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 2T_{3} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{5} + 7T_{19} + 8T_{23}$ , whose minimal polynomial is

$x^{20} - 286x^{19} - 841x^{18} + 8268674x^{17} - 858344114x^{16} - 21535270027x^{15} + 7932331480382x^{14} - 276012214986854x^{13} - 22433541322644245x^{12} + 1653624868075358377x^{11} + 10271743984685063612x^{10} - 3477118764652482941162x^{9} + 52126473809360611813307x^{8} + 3266970533935251974735960x^{7} - 92723778363307267628068485x^{6} - 1181770569739347831832309813x^{5} + 59098309356910258790112115442x^{4} - 50837096490116755330623684696x^{3} - 14061647363371277044144983106464x^{2} + 79014950725796244201690710991792x + 639296340536105681830117724166048$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
71	$+1$

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
$\operatorname{tr} a_p$	3	9	9	−2	20	−12	9	12	17	11	14	−4	14	7	23	8	77	−34	−5	−20	−22	−10	58	11	21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.407 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, 11, 71, 781]: [0, 2, 7, 74] (certified at ell = 2147483171)
T = [(2, 1), (3, 2), (7, 3), (13, 4), (17, 5), (5, 6), (19, 7), (23, 8)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 16 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(781, 2)[6]   # 781.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 781:2 1000