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

Newform orbit 713.2.a.f

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{15} - 13x^{14} - 207x^{13} + 3242x^{12} + 11778x^{11} - 284982x^{10} - 30449x^{9} + 11431542x^{8} - 13937952x^{7} - 226250688x^{6} + 337235440x^{5} + 2394440880x^{4} - 2824262304x^{3} - 13042354112x^{2} + 7382578560x + 27053777664$

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
$\operatorname{tr} a_p$	1	4	1	23	7	18	5	18	−15	−8	15	19	−30	29	13	17	−6	16	21	17	19	19	11	−12	18

$\operatorname{tr} a_n$ 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)[5]   # 713.2.a.f

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

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