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

Newform orbit 51.4.a.d

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q - 6q^{3} + 20q^{4} + 6q^{5} - 8q^{7} + 18q^{9} + 144q^{10} + 66q^{11} - 60q^{12} - 2q^{13} - 144q^{14} - 18q^{15} - 88q^{16} - 34q^{17} - 26q^{19} + 60q^{20} + 24q^{21} - 144q^{22} - 198q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 6x - 441$

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
$\operatorname{tr} a_p$	0	−6	6	−8	66	−2	−34	−26	−198	444	532	88	570	−182	420	−300	444	400	−968	−1848	868	508	996	−288	1024

$\operatorname{tr} a_n$ 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)[3]   # 51.4.a.d

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

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