Atlas › Modular forms › Weight 2 › Level 782 › 782.2.a › 782.2.a.i

Newform orbit 782.2.a.i

Trace form

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

$2q + 2q^{2} + 2q^{3} + 2q^{4} + 6q^{5} + 2q^{6} + 3q^{7} + 2q^{8} - 4q^{9} + 6q^{10} + q^{11} + 2q^{12} - 2q^{13} + 3q^{14} + 6q^{15} + 2q^{16} - 2q^{17} - 4q^{18} + 6q^{20} + 3q^{21} + q^{22} - 2q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{37})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{29} + 8T_{31}$ , whose minimal polynomial is

$x^{2} + 83x - 1617$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
17	$+1$
23	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	2	2	6	3	1	−2	−2	0	−2	−4	−9	−1	−6	−4	−3	10	3	1	3	19	3	−6	29	−6	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.777 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, 17, 23, 34, 46, 391, 782]: [0, 1, 2, 3, 6, 8, 38, 112] (certified at ell = 2147482481)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (29, 7), (31, 8)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 9 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(782, 2)[8]   # 782.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 782:2 1000