Atlas › Modular forms › Weight 2 › Level 731 › 731.2.a › 731.2.a.e

Newform orbit 731.2.a.e

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 2q^{2} + 5q^{3} + 26q^{4} + 11q^{5} + 3q^{6} + 7q^{7} - 6q^{8} + 28q^{9} - 2q^{10} + 4q^{11} + 9q^{12} + 14q^{13} + 5q^{14} - 7q^{15} + 32q^{16} + 19q^{17} + 12q^{18} + 12q^{19} + 23q^{20} + 16q^{21} + 36q^{22} - q^{23} - 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 7x^{18} - 53x^{17} + 435x^{16} + 894x^{15} - 10047x^{14} - 4548x^{13} + 113803x^{12} - 25108x^{11} - 683939x^{10} + 362811x^{9} + 2172759x^{8} - 1383186x^{7} - 3412797x^{6} + 1968634x^{5} + 2310386x^{4} - 872528x^{3} - 438020x^{2} + 97184x + 18664$

Atkin–Lehner signs

$p$	$w_p$
17	$-1$
43	$+1$

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
$\operatorname{tr} a_p$	2	5	11	7	4	14	19	12	−1	41	−8	50	6	−19	−21	−9	−4	68	0	23	−1	16	−32	11	36

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.236 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, 17, 43, 731]: [0, 2, 4, 68] (certified at ell = 2147480161)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(731, 2)[4]   # 731.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 731:2 1000