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

Newform orbit 917.2.a.d

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 4q^{2} + 8q^{3} + 24q^{4} - 3q^{5} - 19q^{7} + 15q^{8} + 23q^{9} + q^{10} + 9q^{11} + 17q^{12} + 7q^{13} - 4q^{14} + 14q^{15} + 26q^{16} + 34q^{18} + 26q^{19} - 3q^{20} - 8q^{21} + 28q^{22} + 10q^{23} + 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} - 12x^{18} + x^{17} + 446x^{16} - 919x^{15} - 6478x^{14} + 16889x^{13} + 50879x^{12} - 128088x^{11} - 253260x^{10} + 473350x^{9} + 802829x^{8} - 811034x^{7} - 1405689x^{6} + 529150x^{5} + 1178045x^{4} + 32772x^{3} - 346388x^{2} - 103127x - 8221$

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
$\operatorname{tr} a_p$	4	8	−3	−19	9	7	0	26	10	6	2	13	6	66	1	13	11	−35	82	7	24	9	18	−6	−15

$\operatorname{tr} a_n$ 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)[3]   # 917.2.a.d

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

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