Atlas › Modular forms › Weight 10 › Level 16 › 16.10.a › 16.10.a.c

Newform orbit 16.10.a.c

$q$ -expansion

$q + 60q^{3} - 2074q^{5} + 4344q^{7} - 16083q^{9} - 93644q^{11} - 12242q^{13} - 124440q^{15} - 319598q^{17} + 553516q^{19} + 260640q^{21} + 712936q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 2014$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 16$ )

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	60	−2074	4344	−93644	−12242	−319598	553516	712936	2075838	6420448	−18197754	9033834	−19594732	18484176	10255766	−121666556	−45948962	−50535428	−267044680	−176213366	269685680	227032556	72141594	228776546

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

Sato–Tate

The normalized $a_p / 2p^{9/2}$ for the primes $p < 1000$ not dividing 16, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.296 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, 8, 16]: [1, 3, 6, 11, 20] (certified at ell = 2147483629)
T = [(3, 1), (5, 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(16, 10)[2]   # 16.10.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 16:10 1000