Atlas › Modular forms › Weight 4 › Level 244 › 244.4.a › 244.4.a.b

Newform orbit 244.4.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 5q^{3} - 21q^{5} + 6q^{7} + 28q^{9} + 47q^{11} - 7q^{13} - 114q^{15} - 162q^{17} - 267q^{19} - 262q^{21} - 276q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{5} + 26x^{4} + 141x^{3} - 1207x^{2} - 13202x - 31367$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 244$ )
61	$+1$

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	−5	−21	6	47	−7	−162	−267	−276	−347	−291	−425	−698	104	−464	−917	511	−305	419	−53	976	305	1957	−2506	3263

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 61, 122, 244]: [1, 2, 3, 17, 49, 96] (certified at ell = 2147482921)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 3 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(244, 4)[1]   # 244.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 244:4 1000