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

Newform orbit 250.4.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 4q^{2} + q^{3} + 8q^{4} - 2q^{6} + 28q^{7} - 16q^{8} - 51q^{9} - 96q^{11} + 4q^{12} + 51q^{13} - 56q^{14} + 32q^{16} + 8q^{17} + 102q^{18} + 25q^{19} + 9q^{21} + 192q^{22} - 94q^{23} - 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 29x + 209$

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
$\operatorname{tr} a_p$	−4	1	0	28	−96	51	8	25	−94	−320	−201	−142	−526	−24	−307	−519	−365	−781	−707	164	−849	120	−364	920	2943

$\operatorname{tr} a_n$ 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)[0]   # 250.4.a.a

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

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