Atlas › Modular forms › Weight 2 › Level 917 › 917.2.a › 917.2.a.c

Newform orbit 917.2.a.c

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 3q^{2} + 10q^{3} + 17q^{4} + 3q^{5} + 8q^{6} + 18q^{7} + 6q^{8} + 18q^{9} + 3q^{10} + q^{11} + 7q^{12} + 25q^{13} + 3q^{14} + 2q^{15} + 11q^{16} + 8q^{17} + q^{18} + 34q^{19} + 3q^{20} + 10q^{21} - 4q^{22} + 2q^{23} + 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{18} - 13x^{17} + 14x^{16} + 480x^{15} - 1955x^{14} - 3837x^{13} + 33844x^{12} - 24767x^{11} - 189091x^{10} + 367703x^{9} + 201463x^{8} - 1001582x^{7} + 532406x^{6} + 479977x^{5} - 497439x^{4} + 107538x^{3} + 3840x^{2} - 460x - 13$

Atkin–Lehner signs

$p$	$w_p$
7	$-1$
131	$+1$

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
$\operatorname{tr} a_p$	3	10	3	18	1	25	8	34	2	−8	26	9	2	34	−1	−29	17	37	42	1	36	29	24	2	51

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 7, 131, 917]: [0, 1, 12, 90] (certified at ell = 2147481571)
T = [(2, 1), (3, 1)]; h of degree 65 = 1 x 65 (phi(ord eps) x dim S_k^new); 7 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(917, 2)[2]   # 917.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 917:2 1000