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

Newform orbit 787.2.a.a

Trace form

The 28 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$28q - 12q^{2} - 8q^{3} + 24q^{4} - 29q^{5} - 4q^{6} - 9q^{7} - 30q^{8} + 10q^{9} + 8q^{10} - 16q^{11} - 11q^{12} - 15q^{13} - 10q^{14} - 5q^{15} + 24q^{16} - 24q^{17} - 25q^{18} + 2q^{19} - 50q^{20} - 28q^{21} - 8q^{22} - 44q^{23} - 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 28. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{28} + 20x^{27} + 117x^{26} - 192x^{25} - 4385x^{24} - 9436x^{23} + 51988x^{22} + 226749x^{21} - 192839x^{20} - 2160977x^{19} - 943448x^{18} + 11107653x^{17} + 11907183x^{16} - 33807354x^{15} - 50188115x^{14} + 63609358x^{13} + 113342457x^{12} - 76665345x^{11} - 147487918x^{10} + 62290982x^{9} + 108318027x^{8} - 34850024x^{7} - 40340027x^{6} + 11661431x^{5} + 5906869x^{4} - 1684562x^{3} - 105311x^{2} + 16146x + 299$

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 odd 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	−8	−29	−9	−16	−15	−24	2	−44	−45	−5	−50	−23	2	−37	−105	−24	−18	2	−52	−17	9	−31	−25	−13

$\operatorname{tr} a_n$ 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)[0]   # 787.2.a.a

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

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