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

Newform orbit 202.4.a.a

$q$ -expansion

$q + 2q^{2} - 8q^{3} + 4q^{4} + 18q^{5} - 16q^{6} - 13q^{7} + 8q^{8} + 37q^{9} + 36q^{10} - 12q^{11} - 32q^{12} - 16q^{13} - 26q^{14} - 144q^{15} + 16q^{16} + 117q^{17} + 74q^{18} + 143q^{19} + 72q^{20} + 104q^{21} - 24q^{22} + 42q^{23} - 64q^{24} + O(q^{25})$

Coefficient field

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

$x - 10$

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 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
$\operatorname{tr} a_p$	2	−8	18	−13	−12	−16	117	143	42	−9	−16	440	−84	−283	354	−273	−612	83	974	−1062	272	50	−1056	720	−1531

$\operatorname{tr} a_n$ 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)[0]   # 202.4.a.a

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

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