Atlas › Modular forms › Weight 8 › Level 57 › 57.8.a › 57.8.a.c

Newform orbit 57.8.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 15q^{2} - 162q^{3} + 681q^{4} - 921q^{5} + 405q^{6} - 855q^{7} + 1167q^{8} + 4374q^{9} - 1020q^{10} + 3639q^{11} - 18387q^{12} + 1602q^{13} - 17562q^{14} + 24867q^{15} + 83433q^{16} - 27333q^{17} - 10935q^{18} + 41154q^{19} - 331662q^{20} + 23085q^{21} - 240198q^{22} - 130962q^{23} - 31509q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} + 936x^{5} + 157002x^{4} - 61780950x^{3} - 15822464559x^{2} - 105612333246x + 11168440783056$

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
$\operatorname{tr} a_p$	−15	−162	−921	−855	3639	1602	−27333	41154	−130962	−207024	−5808	−717708	−29370	427677	−2183541	−2003292	−1562412	5614815	−83040	−7194780	3744873	−5961186	−15553068	−40110192	−22111596

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 12, 48] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 7 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, 8)[2]   # 57.8.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 57:8 1000