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

Newform orbit 60.6.a.c

$q$ -expansion

$q + 9q^{3} - 25q^{5} - 244q^{7} + 81q^{9} - 144q^{11} + 50q^{13} - 225q^{15} - 1914q^{17} + 140q^{19} - 2196q^{21} - 624q^{23} + O(q^{25})$

Coefficient field

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

$x + 388$

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	−244	−144	50	−1914	140	−624	−3126	−5176	15698	12570	11516	−26736	−19158	27984	22022	−12676	−59520	−67102	11048	−115284	73650	35522

$\operatorname{tr} a_n$ 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)[2]   # 60.6.a.c

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

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