Atlas › Modular forms › Weight 6 › Level 32 › 32.6.a › 32.6.a.a

Newform orbit 32.6.a.a

$q$ -expansion

$q - 8q^{3} + 14q^{5} - 208q^{7} - 179q^{9} - 536q^{11} + 694q^{13} - 112q^{15} - 1278q^{17} + 1112q^{19} + 1664q^{21} + 3216q^{23} + O(q^{25})$

Coefficient field

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

$x - 6$

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	14	−208	−536	694	−1278	1112	3216	2918	−2624	−9458	170	−19928	32	−22178	41480	15462	−20744	28592	−53670	−69152	−37800	−126806	62290

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.218 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, 4, 7, 12, 22] (certified at ell = 2147483497)
T = [(3, 1), (5, 1)]; h of degree 5 = 1 x 5 (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(32, 6)[0]   # 32.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 32:6 1000