Atlas › Modular forms › Weight 4 › Level 124 › 124.4.a › 124.4.a.a

Newform orbit 124.4.a.a

Trace form

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

$4q - 4q^{3} - 14q^{5} - 12q^{7} - 24q^{9} - 98q^{11} - 128q^{13} - 162q^{15} - 86q^{17} - 116q^{19} - 118q^{21} - 214q^{23} + O(q^{25})$

Coefficient field

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

$x^{4} + 18x^{3} - 19x^{2} - 518x + 842$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 124$ )
31	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	−4	−14	−12	−98	−128	−86	−116	−214	−168	−124	598	218	−192	32	290	−376	196	40	−536	1436	−1010	−1174	−518	−466

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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 engine's checks

dim M^+ at levels [1, 2, 4, 31, 62, 124]: [1, 2, 3, 9, 26, 51] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(124, 4)[0]   # 124.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 124:4 1000