Atlas › Modular forms › Weight 2 › Level 664 › 664.2.a › 664.2.a.e

Newform orbit 664.2.a.e

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 3q^{3} - 4q^{5} - 5q^{7} + 3q^{9} - 11q^{11} - 4q^{13} - 12q^{15} - 5q^{17} - 10q^{19} - 6q^{21} - 11q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{6} + 15x^{5} - 42x^{4} - 1199x^{3} - 1148x^{2} + 21660x + 38569$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 664$ )
83	$+1$

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	−3	−4	−5	−11	−4	−5	−10	−11	−17	−13	−5	−9	−20	−6	2	−21	−5	−16	−10	44	−2	−6	−2	46

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 8, 83, 166, 332, 664]: [0, 1, 2, 3, 8, 23, 45, 88] (certified at ell = 2147483323)
T = [(3, 1), (5, 3)]; h of degree 21 = 1 x 21 (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(664, 2)[4]   # 664.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 664:2 1000