Atlas › Modular forms › Weight 4 › Level 50 › 50.4.a › 50.4.a.c

Newform orbit 50.4.a.c

$q$ -expansion

$q - 2q^{2} + 8q^{3} + 4q^{4} - 16q^{6} + 4q^{7} - 8q^{8} + 37q^{9} + 12q^{11} + 32q^{12} + 58q^{13} - 8q^{14} + 16q^{16} - 66q^{17} - 74q^{18} - 100q^{19} + 32q^{21} - 24q^{22} - 132q^{23} - 64q^{24} + O(q^{25})$

Coefficient field

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

$x - 12$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	— ( $5^2 \mid 50$ )

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
$\operatorname{tr} a_p$	−2	8	0	4	12	58	−66	−100	−132	−90	152	34	−438	−32	204	−222	420	902	1024	432	−362	−160	−72	810	−1106

$\operatorname{tr} a_n$ 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 50, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 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, 5, 10, 25, 50]: [1, 2, 3, 7, 9, 25] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 1 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(50, 4)[2]   # 50.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 50:4 1000