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

Newform orbit 788.2.a.c

Trace form

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

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

Coefficient field

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

$x^{9} - 9x^{8} - 320x^{7} + 3058x^{6} + 29701x^{5} - 347161x^{4} - 355130x^{3} + 12716705x^{2} - 41087976x + 35024256$

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	−3	4	9	−3	4	7	−3	8	0	15	41	15	−15	17	5	−11	−13	−11	16	45	30	−34	8	59

$\operatorname{tr} a_n$ 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)[2]   # 788.2.a.c

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

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