Atlas › Modular forms › Weight 2 › Level 659 › 659.2.a › 659.2.a.d

Newform orbit 659.2.a.d

Trace form

The 37 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$37q - 2q^{2} + 6q^{3} + 50q^{4} + 13q^{5} + 5q^{6} + 6q^{7} - 3q^{8} + 55q^{9} + 6q^{10} + q^{11} + 11q^{12} + 35q^{13} + 4q^{15} + 80q^{16} + 4q^{17} - 9q^{18} + 37q^{19} + 11q^{20} + 31q^{21} - 5q^{22} - q^{23} + 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 37. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{37} - 16x^{36} - 696x^{35} + 12031x^{34} + 211097x^{33} - 4023875x^{32} - 36825451x^{31} + 795226127x^{30} + 4103295892x^{29} - 104039379128x^{28} - 306092843132x^{27} + 9562694867589x^{26} + 15547917030430x^{25} - 639100160361504x^{24} - 536807545082192x^{23} + 31682552142268125x^{22} + 12636120700201193x^{21} - 1176637299922349712x^{20} - 232065810744634873x^{19} + 32788659486195473586x^{18} + 5470966147797074895x^{17} - 680996412566936964756x^{16} - 174938589700252184827x^{15} + 10370747432214903941973x^{14} + 4403048825496625796846x^{13} - 112507316374664557977116x^{12} - 70977720906289598593693x^{11} + 829691847299743899669485x^{10} + 695283073396755435701797x^{9} - 3859431642828552834153877x^{8} - 3877555790426439552468916x^{7} + 10042617746321277632384433x^{6} + 10748829939430553931546634x^{5} - 12320357703923855355449723x^{4} - 11219221090794304360003384x^{3} + 7026971358981196439757964x^{2} + 3219979591494491808522298x - 1073425846578906609736581$

Atkin–Lehner signs

$p$	$w_p$
659	$-1$

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
$\operatorname{tr} a_p$	−2	6	13	6	1	35	4	37	−1	16	39	37	2	11	9	27	9	136	−10	−11	47	10	−24	22	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 659]: [0, 56] (certified at ell = 2147471831)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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(659, 2)[3]   # 659.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 659:2 1000