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

Newform orbit 664.2.a.f

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 2q^{3} + 7q^{5} + q^{7} + 16q^{9} + 10q^{11} + 7q^{13} - 14q^{15} + 15q^{17} + 7q^{19} - 5q^{21} + 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{8} - 23x^{7} - 15x^{6} + 3687x^{5} - 24444x^{4} - 9157x^{3} + 327774x^{2} - 142349x - 1236370$

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	2	7	1	10	7	15	7	4	16	3	8	24	−1	−2	7	−2	16	−25	8	14	−4	−8	20	2

$\operatorname{tr} a_n$ 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)[5]   # 664.2.a.f

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

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