Atlas › Modular forms › Weight 6 › Level 51 › 51.6.a › 51.6.a.c

Newform orbit 51.6.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 5q^{2} + 36q^{3} + 113q^{4} + 146q^{5} + 45q^{6} + 60q^{7} - 153q^{8} + 324q^{9} - 536q^{10} + 1114q^{11} + 1017q^{12} - 166q^{13} + 1738q^{14} + 1314q^{15} + 2745q^{16} - 1156q^{17} + 405q^{18} + 3454q^{19} + 5748q^{20} + 540q^{21} - 6368q^{22} + 998q^{23} - 1377q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} - 151x^{3} + 3991x^{2} + 123177x + 672282$

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
$\operatorname{tr} a_p$	5	36	146	60	1114	−166	−1156	3454	998	3048	−7808	−1700	−1386	−27654	38156	52648	50028	13076	7880	106836	52488	37688	9812	20876	−1996

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 8, 32] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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, 6)[2]   # 51.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 51:6 1000