Atlas › Modular forms › Weight 8 › Level 44 › 44.8.a › 44.8.a.b

Newform orbit 44.8.a.b

Trace form

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

$4q + 75q^{3} - 219q^{5} + 1150q^{7} + 6575q^{9} + 5324q^{11} + 3410q^{13} + 14731q^{15} + 47260q^{17} + 34604q^{19} + 178878q^{21} + 83875q^{23} + O(q^{25})$

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

$x^{4} + 144x^{3} - 332904x^{2} - 55729600x + 14396461200$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 44$ )
11	$-1$

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	75	−219	1150	5324	3410	47260	34604	83875	−25066	248987	−732685	−373166	−359250	−1918600	−1671980	2314353	−3441174	125285	9506041	1846390	11096302	9080310	2687007	6773325

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 11, 22, 44]: [1, 3, 5, 8, 23, 45] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(44, 8)[1]   # 44.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 44:8 1000