Atlas › Modular forms › Weight 4 › Level 53 › 53.4.a › 53.4.a.c

Newform orbit 53.4.a.c

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q + 2q^{2} + 16q^{3} + 44q^{4} + q^{5} + 9q^{6} + 68q^{7} + 6q^{8} + 82q^{9} + 31q^{10} + 79q^{11} + 171q^{12} + 104q^{13} - 170q^{14} - 128q^{15} + 132q^{16} + 103q^{17} - 405q^{18} + 160q^{19} - 729q^{20} - 96q^{21} - 431q^{22} - 105q^{23} - 727q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{8} - 18x^{7} - 50x^{6} + 2556x^{5} - 10869x^{4} - 60400x^{3} + 562168x^{2} - 1419120x + 1156176$

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 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
$\operatorname{tr} a_p$	2	16	1	68	79	104	103	160	−105	−6	45	422	188	1149	736	−424	−73	−112	2196	−1666	−634	−651	−246	−2649	1363

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 15] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 13 = 1 x 13 (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(53, 4)[2]   # 53.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 53:4 1000