Atlas › Modular forms › Weight 2 › Level 783 › 783.2.a › 783.2.a.i

Newform orbit 783.2.a.i

Trace form

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

$8q - q^{2} + 11q^{4} + 2q^{5} + 5q^{7} - 3q^{8} + 7q^{10} + 2q^{11} + 11q^{13} - 8q^{14} + 21q^{16} - 6q^{17} + 5q^{19} + 27q^{20} + 17q^{22} + 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} - x^{7} - 55x^{6} + 13x^{5} + 809x^{4} - 321x^{3} - 3815x^{2} + 2827x + 2205$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 783$ )
29	$-1$

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
$\operatorname{tr} a_p$	−1	0	2	5	2	11	−6	5	4	8	−1	26	−19	16	18	19	−13	2	25	7	11	16	−19	16	31

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.251 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, 3, 9, 27, 29, 87, 261, 783]: [0, 1, 2, 4, 3, 12, 32, 92] (certified at ell = 2147483269)
T = [(2, 1), (5, 1)]; h of degree 38 = 1 x 38 (phi(ord eps) x dim S_k^new); 4 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(783, 2)[8]   # 783.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 783:2 1000