Atlas › Modular forms › Weight 2 › Level 710 › 710.2.a › 710.2.a.j

Newform orbit 710.2.a.j

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13}$ , whose minimal polynomial is

$x^{5} - 77x^{4} + 206x^{3} + 46256x^{2} + 393820x + 749292$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$+1$
71	$+1$

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
$\operatorname{tr} a_p$	5	3	−5	3	6	5	8	7	11	−12	1	0	−4	9	−5	2	−20	−4	−16	−5	17	−2	−8	−27	16

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.346 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, 5, 10, 71, 142, 355, 710]: [0, 1, 1, 3, 7, 20, 38, 112] (certified at ell = 2147482681)
T = [(3, 1), (7, 3), (11, 5), (13, 7)]; h of degree 25 = 1 x 25 (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(710, 2)[9]   # 710.2.a.j

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

cargo run --release -p sagebrush-web --example atlas -- 710:2 1000