Atlas › Modular forms › Weight 6 › Level 57 › 57.6.a › 57.6.a.d

Newform orbit 57.6.a.d

Trace form

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

$3q - 4q^{2} + 27q^{3} + 84q^{4} + 206q^{5} - 36q^{6} + 186q^{7} - 708q^{8} + 243q^{9} - 786q^{10} + 654q^{11} + 756q^{12} + 720q^{13} - 486q^{14} + 1854q^{15} + 4296q^{16} - 36q^{17} - 324q^{18} - 1083q^{19} + 5536q^{20} + 1674q^{21} + 2262q^{22} + 2174q^{23} - 6372q^{24} + O(q^{25})$

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

$x^{3} - 202x^{2} + 12689x - 237650$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
19	$+1$

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
$\operatorname{tr} a_p$	−4	27	206	186	654	720	−36	−1083	2174	−3914	−5478	−10668	−17254	−3174	23880	−12570	6036	10740	74952	−31656	65928	9594	9560	154322	−115686

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.242 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, 19, 57]: [1, 3, 10, 36] (certified at ell = 2147483647)
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(57, 6)[3]   # 57.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 57:6 1000