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

Newform orbit 57.4.a.c

Trace form

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

$3q - 3q^{2} - 9q^{3} + 9q^{4} - 12q^{5} + 9q^{6} - 18q^{7} - 105q^{8} + 27q^{9} - 48q^{10} - 54q^{11} - 27q^{12} - 66q^{13} - 42q^{14} + 36q^{15} + 273q^{16} - 48q^{17} - 27q^{18} + 57q^{19} + 114q^{20} + 54q^{21} + 234q^{22} - 456q^{23} + 315q^{24} + O(q^{25})$

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

$x^{3} + 15x^{2} - 75x - 285$

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 odd 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
$\operatorname{tr} a_p$	−3	−9	−12	−18	−54	−66	−48	57	−456	30	−144	162	−354	−246	−258	−906	420	996	492	−624	564	−300	504	930	−618

$\operatorname{tr} a_n$ 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)[2]   # 57.4.a.c

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

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