Atlas › Modular forms › Weight 2 › Level 787 › 787.2.a › 787.2.a.b

Newform orbit 787.2.a.b

Trace form

The 37 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$37q + 12q^{2} + 6q^{3} + 42q^{4} + 31q^{5} + 4q^{6} + 9q^{7} + 36q^{8} + 47q^{9} + 4q^{10} + 18q^{11} + 15q^{12} + 13q^{13} + 8q^{14} + 3q^{15} + 48q^{16} + 18q^{17} + 17q^{18} + 40q^{20} - 4q^{21} - 4q^{22} + 42q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 37. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{37} - 18x^{36} + 21x^{35} + 1440x^{34} - 7400x^{33} - 41468x^{32} + 375833x^{31} + 348513x^{30} - 9430734x^{29} + 8483157x^{28} + 140222126x^{27} - 284566626x^{26} - 1308853874x^{25} + 3975455229x^{24} + 7661017857x^{23} - 33438147279x^{22} - 25944763728x^{21} + 185194975755x^{20} + 31762724681x^{19} - 700232362524x^{18} + 111761792272x^{17} + 1840743391053x^{16} - 591980804109x^{15} - 3391167606467x^{14} + 1227784558422x^{13} + 4356954220228x^{12} - 1357087007000x^{11} - 3800384395595x^{10} + 774251462436x^{9} + 2115849927956x^{8} - 178461872279x^{7} - 678787952054x^{6} + 3551214415x^{5} + 115757873788x^{4} + 3163463288x^{3} - 9415076120x^{2} - 210843664x + 280444288$

Atkin–Lehner signs

$p$	$w_p$
787	$-1$

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
$\operatorname{tr} a_p$	12	6	31	9	18	13	18	0	42	45	−3	52	23	4	45	103	2	−6	24	32	5	−27	23	9	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 787]: [0, 66] (certified at ell = 2147472259)
T = [(2, 1), (3, 1)]; h of degree 65 = 1 x 65 (phi(ord eps) x dim S_k^new); 7 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(787, 2)[1]   # 787.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 787:2 1000