Atlas › Modular forms › Weight 2 › Level 785 › 785.2.a › 785.2.a.d

Newform orbit 785.2.a.d

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 2q^{2} + 6q^{3} + 16q^{4} - 14q^{5} + 4q^{6} + 20q^{7} + 6q^{8} + 14q^{9} - 2q^{10} - 10q^{11} + 12q^{12} + 17q^{13} + 5q^{14} - 6q^{15} + 16q^{16} + 11q^{17} + 19q^{18} + 23q^{19} - 16q^{20} + 7q^{21} + 3q^{22} + 3q^{23} + 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 8x^{13} - 22x^{12} + 277x^{11} + 21x^{10} - 3300x^{9} + 1694x^{8} + 17021x^{7} - 8535x^{6} - 38913x^{5} + 11519x^{4} + 31468x^{3} - 7953x^{2} - 7107x + 1731$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
157	$-1$

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	6	−14	20	−10	17	11	23	3	−14	39	25	−7	22	19	11	−35	35	26	8	33	13	15	−11	21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.239 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, 5, 157, 785]: [0, 1, 13, 80] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (phi(ord eps) x dim S_k^new); 6 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(785, 2)[3]   # 785.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 785:2 1000