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

Newform orbit 57.6.a.e

Trace form

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

$3q + 9q^{2} - 27q^{3} + 21q^{4} - 9q^{5} - 81q^{6} + 141q^{7} + 423q^{8} + 243q^{9} + 576q^{10} + 495q^{11} - 189q^{12} + 348q^{13} + 1368q^{14} + 81q^{15} + 2289q^{16} + 1539q^{17} + 729q^{18} + 1083q^{19} + 2448q^{20} - 1269q^{21} + 2160q^{22} - 1278q^{23} - 3807q^{24} + O(q^{25})$

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

$x^{3} - 4041x + 98856$

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
$\operatorname{tr} a_p$	9	−27	−9	141	495	348	1539	1083	−1278	90	−4656	1482	8496	−24879	30627	32670	58824	22047	35988	132948	1869	85470	87948	74358	−66234

$\operatorname{tr} a_n$ 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)[4]   # 57.6.a.e

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

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