Atlas › Modular forms › Weight 4 › Level 221 › 221.4.a › 221.4.a.a

Newform orbit 221.4.a.a

$q$ -expansion

$q + 5q^{2} + 4q^{3} + 17q^{4} + 6q^{5} + 20q^{6} - 20q^{7} + 45q^{8} - 11q^{9} + 30q^{10} + 70q^{11} + 68q^{12} - 13q^{13} - 100q^{14} + 24q^{15} + 89q^{16} - 17q^{17} - 55q^{18} + 96q^{19} + 102q^{20} - 80q^{21} + 350q^{22} - 98q^{23} + 180q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{3}$ is a root of

$x - 9$

Atkin–Lehner signs

$p$	$w_p$
13	$+1$
17	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	5	4	6	−20	70	−13	−17	96	−98	−48	−312	262	360	−460	−168	666	780	−392	−24	−320	−748	90	1280	782	−524

$\operatorname{tr} a_n$ 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 221, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 13, 17, 221]: [1, 5, 6, 66] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 48 = 1 x 48 (phi(ord eps) x dim S_k^new); 7 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(221, 4)[0]   # 221.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 221:4 1000