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

Newform orbit 713.2.a.g

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 5q^{2} + 2q^{3} + 25q^{4} + q^{5} + 4q^{6} + 23q^{7} + 15q^{8} + 20q^{9} + q^{10} + 3q^{11} - 5q^{12} + 2q^{13} + 5q^{14} + 15q^{15} + 35q^{16} - 9q^{17} + 29q^{18} + 12q^{19} - 8q^{20} - 6q^{21} - 12q^{22} + 18q^{23} - 40q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{18} - 11x^{17} - 315x^{16} + 3351x^{15} + 39329x^{14} - 387049x^{13} - 2609631x^{12} + 21796294x^{11} + 102548464x^{10} - 633932929x^{9} - 2402253608x^{8} + 9277641200x^{7} + 30472997400x^{6} - 64249268704x^{5} - 183601428336x^{4} + 190476212576x^{3} + 454883377792x^{2} - 182408157696x - 331599840512$

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 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	2	1	23	3	2	−9	12	18	12	−18	21	10	43	25	−27	−6	−10	55	−27	9	43	1	−14	−20

$\operatorname{tr} a_n$ 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)[6]   # 713.2.a.g

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

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