Atlas › Modular forms › Weight 2 › Level 658 › 658.2.a › 658.2.a.h

Newform orbit 658.2.a.h

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13}$ , whose minimal polynomial is

$x^{4} + 53x^{3} + 375x^{2} + 599x + 268$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
7	$+1$
47	$+1$

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
$\operatorname{tr} a_p$	−4	−1	−1	−4	−7	−2	−2	0	−14	−10	−10	−1	−11	−11	−4	−19	−5	20	16	−38	−1	−10	−29	34	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.421 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, 7, 14, 47, 94, 329, 658]: [0, 1, 1, 4, 5, 14, 34, 100] (certified at ell = 2147483137)
T = [(3, 1), (5, 3), (11, 5), (13, 7)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 6 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(658, 2)[7]   # 658.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 658:2 1000