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

Newform orbit 244.4.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 3q^{3} + 24q^{5} + 13q^{7} + 140q^{9} - 14q^{11} - 2q^{13} + 16q^{15} + 112q^{17} + 181q^{19} + 536q^{21} + 393q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 21x^{8} - 892x^{7} + 23543x^{6} + 78008x^{5} - 5401249x^{4} + 24726784x^{3} + 271213889x^{2} - 2313081501x + 4093495326$

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	−3	24	13	−14	−2	112	181	393	739	263	375	743	248	408	717	204	549	−1164	265	553	−2892	−2083	476	−1897

$\operatorname{tr} a_n$ 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)[2]   # 244.4.a.c

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

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