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

Newform orbit 710.2.a.i

Trace form

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

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

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

$x^{5} - 51x^{4} - 402x^{3} + 66920x^{2} - 1356380x + 8522388$

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	−1	5	−1	4	5	−4	5	−1	10	13	6	2	−3	−17	10	16	14	22	−5	23	30	−20	9	10

$\operatorname{tr} a_n$ 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)[8]   # 710.2.a.i

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

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