Atlas › Modular forms › Weight 6 › Level 80 › 80.6.a › 80.6.a.e

Newform orbit 80.6.a.e

$q$ -expansion

$q + 4q^{3} + 25q^{5} - 192q^{7} - 227q^{9} + 148q^{11} + 286q^{13} + 100q^{15} - 1678q^{17} - 1060q^{19} - 768q^{21} - 2976q^{23} + O(q^{25})$

Coefficient field

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

$x + 188$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 80$ )
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	4	25	−192	148	286	−1678	−1060	−2976	−3410	2448	182	−9398	1244	12088	23846	20020	32302	−60972	32648	−38774	33360	−16716	101370	−119038

$\operatorname{tr} a_n$ 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 80, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.281 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, 16, 20, 40, 80]: [1, 2, 4, 3, 7, 9, 12, 18, 34, 64] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 10 = 1 x 10 (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(80, 6)[4]   # 80.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 80:6 1000