Atlas › Modular forms › Weight 6 › Level 64 › 64.6.a › 64.6.a.h

Newform orbit 64.6.a.h

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 92q^{5} + 1050q^{9} + 84q^{13} + 1924q^{17} + 9216q^{21} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 92x + 1348$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 64$ )

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	0	−92	0	0	84	1924	0	0	5108	0	−23900	−10156	0	0	39428	0	−58636	0	0	75828	0	0	27860	327204

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.25 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, 64]: [1, 2, 4, 7, 12, 22, 42] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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(64, 6)[7]   # 64.6.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 64:6 1000