Atlas › Modular forms › Weight 4 › Level 125 › 125.4.a › 125.4.a.d

Newform orbit 125.4.a.d

Trace form

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

$8q + 44q^{4} + 46q^{6} + 96q^{9} + 76q^{11} + 98q^{14} + 468q^{16} + 280q^{19} + 236q^{21} + 870q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 256x^{6} + 24291x^{4} - 1011046x^{2} + 15550931$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 125$ )

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	0	0	0	76	0	0	280	0	300	996	0	−764	0	0	0	660	756	0	1896	0	940	0	2300	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.109 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, 5, 25, 125]: [1, 3, 9, 39] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 4 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(125, 4)[3]   # 125.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 125:4 1000