Atlas › Modular forms › Weight 4 › Level 44 › 44.4.a › 44.4.a.a

Newform orbit 44.4.a.a

$q$ -expansion

$q - 5q^{3} - 7q^{5} - 26q^{7} - 2q^{9} - 11q^{11} + 52q^{13} + 35q^{15} + 46q^{17} - 96q^{19} + 130q^{21} + 27q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + 3T_{5}$ is a root of

$x + 26$

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	−5	−7	−26	−11	52	46	−96	27	16	−293	−29	−472	−110	−224	754	825	−548	−123	1001	−1020	526	−158	−1217	−263

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 44, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.107 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, 2, 3, 4, 11, 21] (certified at ell = 2147483171)
T = [(3, 1), (5, 3)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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, 4)[0]   # 44.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 44:4 1000