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

Newform orbit 785.2.a.e

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 4x^{15} - 49x^{14} + 211x^{13} + 769x^{12} - 3719x^{11} - 4229x^{10} + 26171x^{9} + 4581x^{8} - 67822x^{7} + 5394x^{6} + 48243x^{5} - 1042x^{4} - 9157x^{3} + 632x^{2} + 509x - 61$

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
$\operatorname{tr} a_p$	0	4	16	20	18	11	1	23	−3	12	39	9	5	10	−5	−11	25	9	6	8	21	69	−1	−9	23

$\operatorname{tr} a_n$ 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)[4]   # 785.2.a.e

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

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