Atlas › Modular forms › Weight 2 › Level 788 › 788.2.a › 788.2.a.b

Newform orbit 788.2.a.b

Trace form

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

$6q - 6q^{5} - 7q^{7} + 2q^{9} - 16q^{13} - 9q^{15} - 11q^{17} + q^{19} - 14q^{21} - 6q^{23} + O(q^{25})$

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

$x^{6} + 18x^{5} + 71x^{4} - 263x^{3} - 1361x^{2} + 1567x + 1864$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 788$ )
197	$-1$

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	0	−6	−7	0	−16	−11	1	−6	−3	−12	−22	−8	1	−15	−4	11	−6	4	7	−35	11	18	6	−44

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 197, 394, 788]: [0, 1, 2, 17, 51, 102] (certified at ell = 2147479489)
T = [(3, 1), (5, 3)]; h of degree 17 = 1 x 17 (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(788, 2)[1]   # 788.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 788:2 1000