Atlas › Modular forms › Weight 4 › Level 31 › 31.4.a › 31.4.a.b

Newform orbit 31.4.a.b

Trace form

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

$5q + 3q^{2} + 4q^{3} + 29q^{4} + 15q^{5} + 24q^{6} + 9q^{7} + 12q^{8} + 29q^{9} - 187q^{10} + 88q^{11} - 190q^{12} - 28q^{13} + 3q^{14} - 130q^{15} + 9q^{16} + 138q^{17} - 225q^{18} - 43q^{19} - 21q^{20} + 170q^{21} - 40q^{22} + 206q^{23} - 36q^{24} + O(q^{25})$

Coefficient field

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

$x^{5} - 7x^{4} - 116x^{3} + 1079x^{2} - 2997x + 2738$

Atkin–Lehner signs

$p$	$w_p$
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
$\operatorname{tr} a_p$	3	4	15	9	88	−28	138	−43	206	474	−155	−508	473	−82	−644	374	−541	440	−1884	491	302	−1244	1544	3056	583

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.091 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 engine's checks

dim M^+ at levels [1, 31]: [1, 9] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 1 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(31, 4)[1]   # 31.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 31:4 1000