Atlas › Modular forms › Weight 8 › Level 50 › 50.8.a › 50.8.a.d

Newform orbit 50.8.a.d

$q$ -expansion

$q - 8q^{2} + 87q^{3} + 64q^{4} - 696q^{6} + 1366q^{7} - 512q^{8} + 5382q^{9} - 1083q^{11} + 5568q^{12} - 5468q^{13} - 10928q^{14} + 4096q^{16} - 25269q^{17} - 43056q^{18} + 33485q^{19} + 118842q^{21} + 8664q^{22} - 5838q^{23} - 44544q^{24} + O(q^{25})$

Coefficient field

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

$x - 1453$

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
$\operatorname{tr} a_p$	−8	87	0	1366	−1083	−5468	−25269	33485	−5838	125280	−73798	395926	−22683	−100148	−1145244	354882	1098360	−422998	−2558579	−2287428	−6372443	−2019250	−7972983	2185935	5823646

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.315 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, 3, 5, 13, 19, 55] (certified at ell = 2147482921)
T = [(3, 1), (7, 1)]; h of degree 12 = 1 x 12 (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(50, 8)[3]   # 50.8.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 50:8 1000