Atlas › Modular forms › Weight 10 › Level 8 › 8.10.a › 8.10.a.b

Newform orbit 8.10.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

$x - 1578$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 8$ )

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 8, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.278 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]: [1, 3, 6, 11] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 2 = 1 x 2 (phi(ord eps) x dim S_k^new); 1 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(8, 10)[1]   # 8.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 8:10 1000