Atlas › Modular forms › Weight 4 › Level 57 › 57.4.a › 57.4.a.b

Newform orbit 57.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{2} - 6q^{3} + q^{4} + 22q^{5} - 3q^{6} + 36q^{7} + 9q^{8} + 18q^{9} + 44q^{10} + 30q^{11} - 3q^{12} - 54q^{13} + 84q^{14} - 66q^{15} - 119q^{16} - 12q^{17} + 9q^{18} - 38q^{19} + 44q^{20} - 108q^{21} - 216q^{22} + 244q^{23} - 27q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{33})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 23x + 58$

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
$\operatorname{tr} a_p$	1	−6	22	36	30	−54	−12	−38	244	50	118	−218	−50	196	792	−486	588	−64	−44	−684	−172	−918	838	58	1340

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 2, 6, 22] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 2 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, 4)[1]   # 57.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 57:4 1000