Atlas › Modular forms › Weight 8 › Level 51 › 51.8.a › 51.8.a.a

Newform orbit 51.8.a.a

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 7q^{2} + 54q^{3} + 357q^{4} + 320q^{5} + 189q^{6} - 1500q^{7} + 4473q^{8} + 1458q^{9} + 12890q^{10} - 8832q^{11} + 9639q^{12} - 21804q^{13} - 21728q^{14} + 8640q^{15} + 14097q^{16} + 9826q^{17} + 5103q^{18} + 58552q^{19} + 139510q^{20} - 40500q^{21} - 129780q^{22} + 128420q^{23} + 120771q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1177})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 327x - 103032$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
17	$-1$

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
$\operatorname{tr} a_p$	7	54	320	−1500	−8832	−21804	9826	58552	128420	39600	185724	531432	527652	301008	−286392	1807068	−1073928	−2077688	1981080	−34772	1469876	−115252	−3331592	−6022252	−17925524

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.294 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 engine's checks

dim M^+ at levels [1, 3, 17, 51]: [1, 3, 12, 44] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (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(51, 8)[0]   # 51.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 51:8 1000