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

Newform orbit 16.10.a.d

$q$ -expansion

$q + 156q^{3} + 870q^{5} + 952q^{7} + 4653q^{9} + 56148q^{11} + 178094q^{13} + 135720q^{15} - 247662q^{17} - 315380q^{19} + 148512q^{21} - 204504q^{23} + O(q^{25})$

Coefficient field

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

$x - 1026$

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	156	870	952	56148	178094	−247662	−315380	−204504	−3840450	1309408	4307078	1512042	−33670604	10581072	16616214	−112235100	−33197218	121372252	387172728	255240074	−492101840	457420236	−31809510	−673532062

$\operatorname{tr} a_n$ 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)[3]   # 16.10.a.d

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

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