Atlas › Modular forms › Weight 2 › Level 913 › 913.2.a › 913.2.a.e

Newform orbit 913.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$15q - 7q^{2} - 7q^{3} + 15q^{4} - 4q^{5} + 3q^{6} - 17q^{7} - 27q^{8} + 14q^{9} - 7q^{10} + 15q^{11} - 13q^{12} - 21q^{13} + 4q^{14} - 13q^{15} + 27q^{16} - 14q^{17} - 19q^{18} - 2q^{19} - 4q^{20} - 11q^{21} - 7q^{22} - 43q^{23} + 19q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{15} + 14x^{14} + 43x^{13} - 203x^{12} - 1298x^{11} - 673x^{10} + 7363x^{9} + 9917x^{8} - 16343x^{7} - 28829x^{6} + 15729x^{5} + 31632x^{4} - 5587x^{3} - 11875x^{2} - 241x + 681$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
83	$-1$

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
$\operatorname{tr} a_p$	−7	−7	−4	−17	15	−21	−14	−2	−43	−17	−10	−26	−8	−9	−41	−22	−12	−18	16	−16	−22	−31	15	27	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 0.268 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, 83, 913]: [0, 2, 8, 86] (certified at ell = 2147478731)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (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(913, 2)[4]   # 913.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 913:2 1000