Atlas › Modular forms › Weight 4 › Level 32 › 32.4.a › 32.4.a.c

Newform orbit 32.4.a.c

$q$ -expansion

$q + 8q^{3} - 10q^{5} + 16q^{7} + 37q^{9} - 40q^{11} - 50q^{13} - 80q^{15} - 30q^{17} + 40q^{19} + 128q^{21} + 48q^{23} + O(q^{25})$

Coefficient field

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

$x + 2$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 32$ )

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
$\operatorname{tr} a_p$	0	8	−10	16	−40	−50	−30	40	48	−34	320	310	410	152	−416	−410	−200	30	776	400	−630	−1120	552	−326	−110

$\operatorname{tr} a_n$ 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 32, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.1 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, 4, 8, 16, 32]: [1, 2, 3, 5, 8, 14] (certified at ell = 2147483497)
T = [(3, 1), (5, 1)]; h of degree 3 = 1 x 3 (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(32, 4)[2]   # 32.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 32:4 1000