Atlas › Modular forms › Weight 2 › Level 713 › 713.2.a › 713.2.a.c

Newform orbit 713.2.a.c

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{5} + 4x^{4} - 119x^{3} - 477x^{2} + 1363x + 2521$

Atkin–Lehner signs

$p$	$w_p$
23	$+1$
31	$+1$

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
$\operatorname{tr} a_p$	−4	0	−6	−12	11	−3	−6	−8	−5	1	−5	0	1	−31	16	−16	−19	4	4	19	4	−37	−32	−14	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.239 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, 23, 31, 713]: [0, 3, 3, 66] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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(713, 2)[2]   # 713.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 713:2 1000