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

Newform orbit 50.6.a.d

$q$ -expansion

$q + 4q^{2} - 24q^{3} + 16q^{4} - 96q^{6} + 172q^{7} + 64q^{8} + 333q^{9} + 132q^{11} - 384q^{12} + 946q^{13} + 688q^{14} + 256q^{16} + 222q^{17} + 1332q^{18} + 500q^{19} - 4128q^{21} + 528q^{22} - 3564q^{23} - 1536q^{24} + O(q^{25})$

Coefficient field

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

$x - 148$

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	−24	0	172	132	946	222	500	−3564	2190	2312	11242	1242	−20624	−6588	21066	7980	16622	−1808	−24528	−20474	−46240	51576	−110310	78382

$\operatorname{tr} a_n$ 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)[3]   # 50.6.a.d

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

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