Atlas › Modular forms › Weight 6 › Level 60 › 60.6.a › 60.6.a.a

Newform orbit 60.6.a.a

$q$ -expansion

$q - 9q^{3} - 25q^{5} + 44q^{7} + 81q^{9} + 216q^{11} + 770q^{13} + 225q^{15} + 534q^{17} + 1580q^{19} - 396q^{21} + 2904q^{23} + O(q^{25})$

Coefficient field

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

$x - 260$

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	−9	−25	44	216	770	534	1580	2904	−4566	2744	1442	−13350	17204	−10824	−9942	−15576	39302	55796	57120	50402	−10552	108564	−116430	−2782

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.148 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, 3, 4, 3, 7, 9, 13, 12, 18, 34, 66] (certified at ell = 2147483629)
T = [(7, 1), (11, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 2 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, 6)[0]   # 60.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 60:6 1000