Atlas › Modular forms › Weight 4 › Level 220 › 220.4.a › 220.4.a.b

Newform orbit 220.4.a.b

$q$ -expansion

$q + 5q^{3} - 5q^{5} - 19q^{7} - 2q^{9} - 11q^{11} - 62q^{13} - 25q^{15} + 19q^{17} - 131q^{19} - 95q^{21} + 138q^{23} + O(q^{25})$

Coefficient field

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

$x + 52$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 220$ )
5	$+1$
11	$+1$

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	5	−5	−19	−11	−62	19	−131	138	−79	217	−91	158	120	−546	−439	290	−373	728	−709	850	−1194	58	753	1228

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 220, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 11, 20, 22, 44, 55, 110, 220]: [1, 2, 3, 3, 7, 4, 12, 11, 21, 20, 58, 114] (certified at ell = 2147482921)
T = [(3, 1), (7, 3)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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(220, 4)[1]   # 220.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 220:4 1000