Atlas › Modular forms › Weight 2 › Level 768 › 768.2.a › 768.2.a.l

Newform orbit 768.2.a.l

Trace form

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

$2q + 2q^{3} + 2q^{9} + 8q^{11} - 4q^{17} + 8q^{19} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = 1904T_{5} + 4595T_{7} + 5697T_{13} + 9850T_{17} + 5931T_{19} + 8769T_{11} + 1705T_{23} + 9694T_{29}$ , whose minimal polynomial is

$x^{2} - 78200x - 1062902008$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 768$ )
3	$-1$

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	0	0	8	0	−4	8	0	0	0	0	−20	24	0	0	−8	0	8	0	4	0	−8	−12	28

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.327 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, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 768]: [0, 1, 1, 2, 3, 3, 5, 4, 8, 6, 12, 10, 20, 18, 36, 34, 68, 132] (certified at ell = 2147483137)
T = [(5, 1904), (7, 4595), (13, 5697), (17, 9850), (19, 5931), (11, 8769), (23, 1705), (29, 9694)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 10 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(768, 2)[11]   # 768.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 768:2 1000