Atlas › Modular forms › Weight 4 › Level 202 › 202.4.a › 202.4.a.b

Newform orbit 202.4.a.b

$q$ -expansion

$q + 2q^{2} - 2q^{3} + 4q^{4} + 3q^{5} - 4q^{6} - 30q^{7} + 8q^{8} - 23q^{9} + 6q^{10} + 22q^{11} - 8q^{12} - 51q^{13} - 60q^{14} - 6q^{15} + 16q^{16} - 13q^{17} - 46q^{18} - 71q^{19} + 12q^{20} + 60q^{21} + 44q^{22} - 41q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

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

$x - 1$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
101	$+1$

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 odd 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
$\operatorname{tr} a_p$	2	−2	3	−30	22	−51	−13	−71	−41	204	97	−434	−240	−440	497	122	590	−728	862	627	280	−335	−328	994	674

$\operatorname{tr} a_n$ 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 202, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 101, 202]: [1, 2, 27, 79] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 5 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(202, 4)[1]   # 202.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 202:4 1000