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

Newform orbit 51.8.a.e

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - 15q^{2} - 135q^{3} + 553q^{4} + 173q^{5} + 405q^{6} + 400q^{7} - 3537q^{8} + 3645q^{9} - 4728q^{10} + 395q^{11} - 14931q^{12} + 393q^{13} - 13598q^{14} - 4671q^{15} + 88313q^{16} - 24565q^{17} - 10935q^{18} + 135897q^{19} + 183164q^{20} - 10800q^{21} + 198976q^{22} + 74963q^{23} + 95499q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{5} - 158x^{4} - 172640x^{3} + 9979260x^{2} + 6607989171x - 80591922378$

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
$\operatorname{tr} a_p$	−15	−135	173	400	395	393	−24565	135897	74963	68334	140834	388712	788355	−816809	−406850	−1560550	−1839342	6972412	5431884	3328472	8135658	19025102	7064622	22446956	25321108

$\operatorname{tr} a_n$ 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)[4]   # 51.8.a.e

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

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