Atlas › Modular forms › Weight 2 › Level 649 › 649.2.a › 649.2.a.e

Newform orbit 649.2.a.e

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{12} + 187x^{11} + 5849x^{10} - 768978x^{9} - 54364538x^{8} - 427210528x^{7} + 42372373311x^{6} + 980392745421x^{5} + 2900673407330x^{4} - 56929210685166x^{3} - 198032612555975x^{2} - 13224954150616x + 764489292368$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
59	$+1$

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
$\operatorname{tr} a_p$	−4	−4	−5	−10	−12	−7	−8	−14	−26	−33	−12	12	−24	3	−21	−4	−12	−20	15	−43	−16	−32	6	−31	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 59, 649]: [0, 2, 6, 62] (certified at ell = 2147482481)
T = [(2, 1), (3, 3), (7, 5), (13, 7), (17, 9)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 12 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(649, 2)[4]   # 649.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 649:2 1000