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

Newform orbit 16.10.a.b

$q$ -expansion

$q - 68q^{3} + 1510q^{5} - 10248q^{7} - 15059q^{9} - 3916q^{11} - 176594q^{13} - 102680q^{15} + 148370q^{17} - 499796q^{19} + 696864q^{21} + 1889768q^{23} + O(q^{25})$

Coefficient field

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

$x - 1442$

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	−68	1510	−10248	−3916	−176594	148370	−499796	1889768	−920898	−1379360	5064966	−24100758	−25785196	60790224	29496214	−51819388	33426910	−144856196	−68397128	168216202	−235398736	64639852	−78782694	−24113566

$\operatorname{tr} a_n$ 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)[1]   # 16.10.a.b

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

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