Atlas › Modular forms › Weight 2 › Level 808 › 808.2.a › 808.2.a.f

Newform orbit 808.2.a.f

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 2q^{3} + 5q^{5} - q^{7} + 15q^{9} + 8q^{11} + 9q^{13} - 6q^{15} + 12q^{17} + 6q^{19} + 12q^{21} + 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{9} - 7x^{8} - 24x^{7} + 236x^{6} - 203x^{5} - 835x^{4} + 1061x^{3} - 147x^{2} - 68x - 4$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 808$ )
101	$-1$

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	5	−1	8	9	12	6	9	15	−15	18	12	15	11	25	0	3	2	−23	8	−41	18	20	11

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 101, 202, 404, 808]: [0, 1, 2, 3, 9, 27, 54, 106] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 25 = 1 x 25 (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(808, 2)[5]   # 808.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 808:2 1000