Atlas › Modular forms › Weight 2 › Level 831 › 831.2.a › 831.2.a.d

Newform orbit 831.2.a.d

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 4q^{2} - 17q^{3} + 20q^{4} + 9q^{5} - 4q^{6} - 5q^{7} + 9q^{8} + 17q^{9} - 4q^{10} + 16q^{11} - 20q^{12} - 4q^{13} + 19q^{14} - 9q^{15} + 34q^{16} + 24q^{17} + 4q^{18} - 3q^{19} + 33q^{20} + 5q^{21} + 2q^{22} + 22q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{17} - 31x^{16} - 79x^{15} + 10769x^{14} - 57519x^{13} - 1303771x^{12} + 11569500x^{11} + 62285178x^{10} - 811381342x^{9} - 794475550x^{8} + 23798762769x^{7} - 8960806659x^{6} - 290409176787x^{5} + 53592720177x^{4} + 1486760130450x^{3} + 986997713456x^{2} - 1110086276833x - 859513119073$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
277	$-1$

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
$\operatorname{tr} a_p$	4	−17	9	−5	16	−4	24	−3	22	29	−11	8	21	−5	10	43	30	14	−43	26	14	−7	14	53	−26

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 3, 277, 831]: [0, 1, 23, 94] (certified at ell = 2147483137)
T = [(2, 1), (5, 3)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 7 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(831, 2)[3]   # 831.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 831:2 1000