Atlas › Modular forms › Weight 6 › Level 50 › 50.6.a › 50.6.a.e

Newform orbit 50.6.a.e

$q$ -expansion

$q + 4q^{2} - 14q^{3} + 16q^{4} - 56q^{6} - 158q^{7} + 64q^{8} - 47q^{9} - 148q^{11} - 224q^{12} - 684q^{13} - 632q^{14} + 256q^{16} - 2048q^{17} - 188q^{18} + 2220q^{19} + 2212q^{21} - 592q^{22} + 1246q^{23} - 896q^{24} + O(q^{25})$

Coefficient field

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

$x + 172$

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
$\operatorname{tr} a_p$	4	−14	0	−158	−148	−684	−2048	2220	1246	−270	−2048	4372	−2398	−2294	10682	−2964	−39740	−42298	−32098	−4248	−30104	35280	27826	−85210	97232

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.128 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, 9, 13, 39] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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, 6)[4]   # 50.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 50:6 1000