Atlas › Modular forms › Weight 6 › Level 31 › 31.6.a › 31.6.a.b

Newform orbit 31.6.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 7q^{2} - 2q^{3} + 149q^{4} + 128q^{5} + 72q^{6} + 88q^{7} + 924q^{8} + 1512q^{9} + 1581q^{10} + 574q^{11} - 46q^{12} - 122q^{13} - 309q^{14} - 524q^{15} + 833q^{16} + 1932q^{17} - 6845q^{18} - 1796q^{19} - 37q^{20} + 996q^{21} - 9000q^{22} - 4136q^{23} - 17468q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 5x^{7} - 1990x^{6} + 15987x^{5} + 1159441x^{4} - 9889532x^{3} - 211195384x^{2} + 1094532384x + 13261728528$

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
$\operatorname{tr} a_p$	7	−2	128	88	574	−122	1932	−1796	−4136	5050	7688	23674	44828	21058	5348	4926	16944	−73682	−134768	123724	70792	−26036	21882	164392	−131948

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.217 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, 15] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 3 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, 6)[1]   # 31.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 31:6 1000