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

Newform orbit 958.2.a.d

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{8} + 5x^{7} - 17x^{6} - 61x^{5} + 76x^{4} + 158x^{3} - 71x^{2} - 110x - 25$

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 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
$\operatorname{tr} a_p$	−8	−2	−3	−2	−5	−3	−15	2	−17	−16	8	−7	−11	0	−15	−23	−4	−6	−3	−5	−18	7	−30	−33	−28

$\operatorname{tr} a_n$ 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)[3]   # 958.2.a.d

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

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