Atlas › Modular forms › Weight 2 › Level 905 › 905.2.a › 905.2.a.e

Newform orbit 905.2.a.e

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q - 6q^{2} - 8q^{3} + 12q^{4} + 13q^{5} - 6q^{6} - 26q^{7} - 15q^{8} + 13q^{9} - 6q^{10} - 3q^{11} - 9q^{12} - 13q^{13} + 5q^{14} - 8q^{15} + 14q^{16} - 13q^{17} - 9q^{18} - 24q^{19} + 12q^{20} + 10q^{21} - 5q^{22} - 29q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

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

$x^{13} + 14x^{12} + 59x^{11} + 3x^{10} - 511x^{9} - 769x^{8} + 942x^{7} + 2011x^{6} - 483x^{5} - 1407x^{4} - 142x^{3} + 205x^{2} + 34x + 1$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
181	$-1$

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
$\operatorname{tr} a_p$	−6	−8	13	−26	−3	−13	−13	−24	−29	5	−23	−5	5	−32	−29	−5	2	−7	−34	−6	−16	−20	−50	16	−40

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 5, 181, 905]: [0, 1, 15, 92] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (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(905, 2)[4]   # 905.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 905:2 1000