Atlas › Modular forms › Weight 4 › Level 60 › 60.4.a › 60.4.a.b

Newform orbit 60.4.a.b

$q$ -expansion

$q - 3q^{3} + 5q^{5} + 32q^{7} + 9q^{9} + 36q^{11} - 10q^{13} - 15q^{15} - 78q^{17} + 140q^{19} - 96q^{21} - 192q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{7} + 3T_{11}$ is a root of

$x - 140$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 60$ )
3	$+1$
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	−3	5	32	36	−10	−78	140	−192	6	−16	−34	−390	−52	408	−114	516	−58	−892	−120	−646	−1168	−732	−1590	194

$\operatorname{tr} a_n$ 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 60, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.123 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, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]: [1, 2, 2, 3, 3, 5, 7, 9, 8, 12, 22, 42] (certified at ell = 2147483629)
T = [(7, 1), (11, 3)]; h of degree 2 = 1 x 2 (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(60, 4)[1]   # 60.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 60:4 1000