Atlas › Modular forms › Weight 10 › Level 20 › 20.10.a › 20.10.a.a

Newform orbit 20.10.a.a

$q$ -expansion

$q - 48q^{3} + 625q^{5} - 532q^{7} - 17379q^{9} - 33180q^{11} - 99682q^{13} - 30000q^{15} - 443454q^{17} - 357244q^{19} + 25536q^{21} - 142956q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{7}$ is a root of

$x + 580$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 20$ )
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	−48	625	−532	−33180	−99682	−443454	−357244	−142956	1527966	7323416	−2666842	−7939014	−21174520	16059636	−87822234	120625212	93576542	193621688	417763488	−450372742	−91425472	−652637376	−170059206	−10947022

$\operatorname{tr} a_n$ 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 20, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.305 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, 5, 10, 20]: [1, 3, 6, 5, 15, 30] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 3 = 1 x 3 (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(20, 10)[0]   # 20.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 20:10 1000