Atlas › Modular forms › Weight 6 › Level 44 › 44.6.a › 44.6.a.a

Newform orbit 44.6.a.a

$q$ -expansion

$q + 7q^{3} - 79q^{5} - 50q^{7} - 194q^{9} + 121q^{11} - 380q^{13} - 553q^{15} - 1154q^{17} - 1824q^{19} - 350q^{21} + 3591q^{23} + O(q^{25})$

Coefficient field

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

$x + 72$

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	7	−79	−50	121	−380	−1154	−1824	3591	8032	−2945	6979	−520	−2486	−6920	−13718	−31779	34156	−61503	−14971	−36444	−28538	77482	36271	−49799

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.23 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, 4, 6, 17, 33] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; 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, 6)[0]   # 44.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 44:6 1000