Atlas › Modular forms › Weight 2 › Level 626 › 626.2.a › 626.2.a.e

Newform orbit 626.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{9} - 67x^{8} - 2712x^{7} + 210478x^{6} + 2542307x^{5} - 213376795x^{4} - 1339954117x^{3} + 69456620819x^{2} + 552326684881x - 987872773531$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
313	$-1$

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
$\operatorname{tr} a_p$	−9	1	−2	6	6	−3	−12	10	13	3	37	11	1	3	17	−30	−1	37	34	43	5	39	−35	−12	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 0.35 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, 2, 313, 626]: [0, 1, 26, 80] (certified at ell = 2147482273)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 6 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(626, 2)[4]   # 626.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 626:2 1000