Atlas › Modular forms › Weight 4 › Level 250 › 250.4.a › 250.4.a.f

Newform orbit 250.4.a.f

Trace form

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

$4q - 8q^{2} - 3q^{3} + 16q^{4} + 6q^{6} - 9q^{7} - 32q^{8} + 43q^{9} + 63q^{11} - 12q^{12} - 108q^{13} + 18q^{14} + 64q^{16} - 44q^{17} - 86q^{18} + 170q^{19} + 203q^{21} - 126q^{22} - 143q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

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

$x^{4} + 12x^{3} - 1261x^{2} - 27582x - 107129$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	— ( $5^2 \mid 250$ )

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
$\operatorname{tr} a_p$	−8	−3	0	−9	63	−108	−44	170	−143	450	33	−19	513	812	1696	502	900	−317	−59	998	−933	−1640	1772	−35	1911

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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, 5, 10, 25, 50, 125, 250]: [1, 2, 3, 7, 9, 25, 39, 115] (certified at ell = 2147482801)
T = [(3, 1), (7, 1)]; h of degree 24 = 1 x 24 (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(250, 4)[5]   # 250.4.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 250:4 1000