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

Newform orbit 250.4.a.e

Trace form

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

$4q - 8q^{2} - 8q^{3} + 16q^{4} + 16q^{6} - 34q^{7} - 32q^{8} + 78q^{9} + 8q^{11} - 32q^{12} - 28q^{13} + 68q^{14} + 64q^{16} - 84q^{17} - 156q^{18} + 80q^{19} + 128q^{21} - 16q^{22} - 138q^{23} + 64q^{24} + O(q^{25})$

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

$x^{4} + 42x^{3} + 414x^{2} + 858x - 1199$

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	−8	0	−34	8	−28	−84	80	−138	130	8	−184	−542	−368	−1364	−728	−640	−642	376	−792	2212	1560	−198	−1110	1316

$\operatorname{tr} a_n$ 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)[4]   # 250.4.a.e

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

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