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

Newform orbit 917.2.a.a

Trace form

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

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

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

$x^{14} + 11x^{13} + 21x^{12} - 141x^{11} - 459x^{10} + 687x^{9} + 2910x^{8} - 2097x^{7} - 8732x^{6} + 5557x^{5} + 12811x^{4} - 10502x^{3} - 6373x^{2} + 8592x - 2279$

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 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
$\operatorname{tr} a_p$	−5	−6	3	−14	−11	−7	−6	−20	−10	−14	−6	−37	16	−50	−27	−15	1	23	−68	1	−24	−39	−12	−4	−11

$\operatorname{tr} a_n$ 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)[0]   # 917.2.a.a

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

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