Atlas › Modular forms › Weight 8 › Level 53 › 53.8.a › 53.8.a.a

Newform orbit 53.8.a.a

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q - 16q^{2} - 149q^{3} + 768q^{4} - 374q^{5} - 615q^{6} - 3076q^{7} - 2274q^{8} + 9097q^{9} - 5559q^{10} - 4120q^{11} - 18309q^{12} - 20959q^{13} + 28280q^{14} + 20672q^{15} + 54268q^{16} - 62901q^{17} - 170973q^{18} - 152601q^{19} - 331149q^{20} - 176518q^{21} - 379857q^{22} - 98093q^{23} - 498423q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} + 165x^{13} - 6910x^{12} - 2028628x^{11} + 519086x^{10} + 9358400554x^{9} + 70576358220x^{8} - 20608246011552x^{7} - 125086856832391x^{6} + 23125397027566277x^{5} + 37222431383262342x^{4} - 12547592145144646792x^{3} + 41197958629672114836x^{2} + 2551728827283081067160x - 19272378287851051652800$

Atkin–Lehner signs

$p$	$w_p$
53	$-1$

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
$\operatorname{tr} a_p$	−16	−149	−374	−3076	−4120	−20959	−62901	−152601	−98093	−264851	−212914	−650035	467884	−1805618	−174602	2084278	2390638	946836	−5060238	−6541385	−13324340	−12751459	−13144961	−8449166	−14511129

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 53]: [1, 33] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 31 = 1 x 31 (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(53, 8)[0]   # 53.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 53:8 1000