Atlas › Modular forms › Weight 4 › Level 51 › 51.4.a › 51.4.a.e

Newform orbit 51.4.a.e

Trace form

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

$3q + 5q^{2} + 9q^{3} + 13q^{4} + 8q^{5} + 15q^{6} - 8q^{7} + 33q^{8} + 27q^{9} - 38q^{10} + 34q^{11} + 39q^{12} + 36q^{13} - 104q^{14} + 24q^{15} - 79q^{16} + 51q^{17} + 45q^{18} - 142q^{19} - 126q^{20} - 24q^{21} - 248q^{22} + 110q^{23} + 99q^{24} + O(q^{25})$

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

$x^{3} - 13x^{2} + 13x + 171$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
17	$-1$

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
$\operatorname{tr} a_p$	5	9	8	−8	34	36	51	−142	110	90	−148	110	720	−146	500	610	−216	−18	−1404	−960	−794	−276	−1552	1394	402

$\operatorname{tr} a_n$ 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, 17, 51]: [1, 2, 6, 20] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(51, 4)[4]   # 51.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 51:4 1000