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

Newform orbit 57.8.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 9q^{2} + 189q^{3} + 589q^{4} - 27q^{5} + 243q^{6} + 1889q^{7} + 3933q^{8} + 5103q^{9} + 8232q^{10} + 1071q^{11} + 15903q^{12} + 14216q^{13} + 5094q^{14} - 729q^{15} + 16009q^{16} - 22743q^{17} + 6561q^{18} + 48013q^{19} - 43146q^{20} + 51003q^{21} + 138486q^{22} + 87930q^{23} + 106191q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} + 18x^{6} - 440184x^{5} - 12400776x^{4} + 54737490963x^{3} + 3479081791590x^{2} - 1975610049808500x - 184101405051708360$

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	189	−27	1889	1071	14216	−22743	48013	87930	131166	555464	445922	389232	−168079	219915	−1306710	−1737072	−524563	967724	−8722548	5447675	−1429066	−20709144	3404898	34668350

$\operatorname{tr} a_n$ 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)[3]   # 57.8.a.d

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

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