Atlas › Modular forms › Weight 2 › Level 765 › 765.2.a › 765.2.a.g

Newform orbit 765.2.a.g

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{3})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{7} + 5T_{13} + 7T_{19} + 9T_{23}$ , whose minimal polynomial is

$x^{2} - 36x - 39$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 765$ )
5	$+1$
17	$-1$

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	−2	−2	−6	−8	2	4	6	0	10	−8	0	−8	−12	−12	−12	4	−20	−6	−8	−2	−24	12	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.572 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, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765]: [0, 1, 1, 2, 4, 2, 8, 8, 10, 20, 40, 112] (certified at ell = 2147483137)
T = [(2, 1), (7, 3), (13, 5), (19, 7), (23, 9)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 8 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(765, 2)[6]   # 765.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 765:2 1000