Atlas › Modular forms › Weight 8 › Level 31 › 31.8.a › 31.8.a.b

Newform orbit 31.8.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 15q^{2} + 40q^{3} + 613q^{4} + 570q^{5} + 768q^{6} + 540q^{7} - 2220q^{8} + 9518q^{9} + 8653q^{10} + 14526q^{11} + 36050q^{12} + 14440q^{13} + 51531q^{14} + 42530q^{15} + 56673q^{16} + 51930q^{17} + 102795q^{18} + 24124q^{19} + 107979q^{20} + 40910q^{21} - 16040q^{22} + 72510q^{23} - 71412q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 55x^{9} - 15896x^{8} + 695035x^{7} + 86166363x^{6} - 2886863810x^{5} - 202197440780x^{4} + 4611097028760x^{3} + 220580468861120x^{2} - 2496232761718400x - 93064067996201984$

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
$\operatorname{tr} a_p$	15	40	570	540	14526	14440	51930	24124	72510	331608	−297910	120250	1188306	−318500	2363760	2717430	4115088	6429700	5202840	−2846208	−138320	4272482	−182550	18325602	16774310

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.266 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, 19] (certified at ell = 2147482951)
T = [(2, 1), (3, 1)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 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, 8)[1]   # 31.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 31:8 1000