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

Newform orbit 51.8.a.b

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 7q^{2} - 81q^{3} + 233q^{4} - 440q^{5} + 189q^{6} + 808q^{7} + 1785q^{8} + 2187q^{9} - 4010q^{10} + 366q^{11} - 6291q^{12} - 8756q^{13} + 12344q^{14} + 11880q^{15} - 27199q^{16} + 14739q^{17} - 5103q^{18} - 51450q^{19} - 76450q^{20} - 21816q^{21} - 187284q^{22} + 31738q^{23} - 48195q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} + 447x^{2} + 35091x - 1252395$

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 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
$\operatorname{tr} a_p$	−7	−81	−440	808	366	−8756	14739	−51450	31738	151362	−158012	155254	−342000	−1220582	765084	1996194	1077744	−1027962	−5818028	−6680080	−11465770	−858604	−2737624	1947394	−1785102

$\operatorname{tr} a_n$ 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)[1]   # 51.8.a.b

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

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