Atlas › Modular forms › Weight 2 › Level 958 › 958.2.a › 958.2.a.f

Newform orbit 958.2.a.f

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{11} - 7x^{10} - 31x^{9} + 311x^{8} - 122x^{7} - 3348x^{6} + 5879x^{5} + 7054x^{4} - 19863x^{3} + 2694x^{2} + 8524x + 1000$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
479	$+1$

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
$\operatorname{tr} a_p$	11	4	3	6	11	5	13	10	21	12	10	3	−9	10	7	9	6	−4	1	17	−28	5	−4	−27	−38

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.379 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, 2, 479, 958]: [0, 1, 41, 122] (certified at ell = 2147482877)
T = [(3, 1), (5, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 5 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(958, 2)[5]   # 958.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 958:2 1000