Atlas › Modular forms › Weight 2 › Level 917 › 917.2.a › 917.2.a.b

Newform orbit 917.2.a.b

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 3q^{2} - 12q^{3} + 9q^{4} - 9q^{5} + 14q^{7} - 6q^{8} + 16q^{9} - 19q^{10} - 7q^{11} - 23q^{12} - 31q^{13} - 3q^{14} - 6q^{15} + 7q^{16} - 4q^{17} - 5q^{18} - 36q^{19} + 3q^{20} - 12q^{21} - 10q^{22} + 6q^{23} + 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 15x^{13} + 65x^{12} - 45x^{11} - 1003x^{10} - 1843x^{9} + 2506x^{8} + 9635x^{7} + 2096x^{6} - 14377x^{5} - 8229x^{4} + 7924x^{3} + 5581x^{2} - 1374x - 1051$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
131	$-1$

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 odd 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	−12	−9	14	−7	−31	−4	−36	6	2	−30	−15	−10	−62	−5	−3	−9	−43	−40	7	−30	5	−38	14	−27

$\operatorname{tr} a_n$ 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, 7, 131, 917]: [0, 1, 12, 90] (certified at ell = 2147481571)
T = [(2, 1), (3, 1)]; h of degree 65 = 1 x 65 (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(917, 2)[1]   # 917.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 917:2 1000