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

Newform orbit 713.2.a.e

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{9} + 20x^{8} + 53x^{7} - 941x^{6} - 4255x^{5} + 10963x^{4} + 55218x^{3} + 2932x^{2} - 52616x - 12048$

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
$\operatorname{tr} a_p$	−5	−5	1	−12	1	−18	−10	0	9	−8	9	−25	−11	−45	−29	−10	24	−1	−24	−25	−63	−9	−15	12	8

$\operatorname{tr} a_n$ 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)[4]   # 713.2.a.e

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

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