Atlas › Modular forms › Weight 2 › Level 780 › 780.2.a › 780.2.a.f

Newform orbit 780.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{7} + 3T_{11} + 5T_{19} + 7T_{23} + 9T_{29} + 11T_{31}$ , whose minimal polynomial is

$x^{2} + 11x + 26$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 780$ )
3	$-1$
5	$-1$
13	$+1$

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	2	3	3	−2	5	0	3	0	−4	−3	5	−4	20	−11	14	−9	6	−9	−12	−5	−6	−5	−9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.892 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, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780]: [0, 1, 1, 2, 1, 3, 3, 5, 1, 4, 6, 5, 10, 6, 10, 18, 8, 18, 24, 34, 32, 48, 92, 180] (certified at ell = 2147483629)
T = [(7, 1), (11, 3), (19, 5), (23, 7), (29, 9), (31, 11)]; h of degree 8 = 1 x 8 (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(780, 2)[5]   # 780.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 780:2 1000