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

Newform orbit 785.2.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - q^{2} - 2q^{3} + 11q^{4} - 9q^{5} - 2q^{6} - 22q^{7} + 3q^{8} + 9q^{9} + q^{10} - 2q^{11} - 38q^{12} - q^{13} - 3q^{14} + 2q^{15} - 9q^{16} - 17q^{17} + 4q^{18} - 5q^{19} - 11q^{20} + 7q^{21} - 17q^{22} - 3q^{23} + q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 3x^{8} - 26x^{7} - 35x^{6} + 228x^{5} - 65x^{4} - 359x^{3} + 136x^{2} + 174x - 29$

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 odd 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
$\operatorname{tr} a_p$	−1	−2	−9	−22	−2	−1	−17	−5	−3	20	−25	−17	3	−24	−3	−11	13	−7	−34	12	17	−61	−33	−15	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 0.239 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)[1]   # 785.2.a.b

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

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