Atlas › Modular forms › Weight 2 › Level 707 › 707.2.a › 707.2.a.e

Newform orbit 707.2.a.e

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 2q^{2} - 9q^{3} + 6q^{4} - 7q^{5} - 5q^{6} + 10q^{7} - 3q^{8} + 9q^{9} - 13q^{10} - 2q^{11} - 12q^{12} - 32q^{13} - 2q^{14} + 6q^{15} + 6q^{16} - 6q^{17} + 4q^{18} - 27q^{19} + 4q^{20} - 9q^{21} - 6q^{22} + q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 11x^{9} + 33x^{8} - 18x^{7} - 188x^{6} - 76x^{5} + 301x^{4} + 92x^{3} - 128x^{2} - 9x + 4$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
101	$-1$

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
$\operatorname{tr} a_p$	−2	−9	−7	10	−2	−32	−6	−27	1	0	−26	−39	−4	−13	−6	−7	7	−43	3	16	−34	16	−3	14	−19

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.319 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, 7, 101, 707]: [0, 1, 9, 70] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 5 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(707, 2)[4]   # 707.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 707:2 1000