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

Newform orbit 51.6.a.d

Trace form

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

$5q + q^{2} - 45q^{3} + 117q^{4} + 4q^{5} - 9q^{6} - 40q^{7} - 15q^{8} + 405q^{9} + 1234q^{10} - 46q^{11} - 1053q^{12} + 1912q^{13} + 4544q^{14} - 36q^{15} + 7465q^{16} + 1445q^{17} + 81q^{18} - 774q^{19} - 4350q^{20} + 360q^{21} + 3184q^{22} - 3914q^{23} + 135q^{24} + O(q^{25})$

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

$x^{5} - 5x^{4} - 10079x^{3} + 135327x^{2} + 24597432x - 623316384$

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
$\operatorname{tr} a_p$	1	−45	4	−40	−46	1912	1445	−774	−3914	3510	7308	5754	−3228	12662	3172	13574	−23304	78346	156628	−37488	52442	−85220	12784	92230	7606

$\operatorname{tr} a_n$ 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)[3]   # 51.6.a.d

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

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