Atlas › Modular forms › Weight 8 › Level 40 › 40.8.a › 40.8.a.a

Newform orbit 40.8.a.a

$q$ -expansion

$q - 36q^{3} + 125q^{5} + 776q^{7} - 891q^{9} - 124q^{11} - 13082q^{13} - 4500q^{15} - 15950q^{17} - 20516q^{19} - 27936q^{21} - 29224q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{7}$ is a root of

$x - 740$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 40$ )
5	$-1$

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	−36	125	776	−124	−13082	−15950	−20516	−29224	−221482	−109760	73422	12762	290548	1269152	−395778	421492	−2122250	−3132868	−5376552	4985466	3867504	−6190196	1124394	9968098

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ for the primes $p < 1000$ not dividing 40, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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, 5, 8, 10, 20, 40]: [1, 3, 5, 5, 9, 13, 24, 46] (certified at ell = 2147483629)
T = [(3, 1), (7, 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(40, 8)[0]   # 40.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 40:8 1000