Atlas › Modular forms › Weight 4 › Level 248 › 248.4.a › 248.4.a.a

Newform orbit 248.4.a.a

Trace form

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

$4q - 2q^{3} + 9q^{5} - 45q^{7} + 24q^{9} - 54q^{11} - 16q^{13} - 98q^{15} - 76q^{17} - 141q^{19} - 66q^{21} - 158q^{23} + O(q^{25})$

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

$x^{4} - 7x^{3} - 155x^{2} - 57x + 2322$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 248$ )
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	−2	9	−45	−54	−16	−76	−141	−158	−94	124	92	7	−816	−904	−198	−487	236	−216	−859	524	480	−194	910	21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.156 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, 8, 31, 62, 124, 248]: [1, 2, 3, 5, 9, 26, 51, 100] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 22 = 1 x 22 (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(248, 4)[0]   # 248.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 248:4 1000