Atlas › Modular forms › Weight 6 › Level 51 › 51.6.a › 51.6.a.a

Newform orbit 51.6.a.a

Trace form

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

$2q - 7q^{2} + 18q^{3} + 33q^{4} - 113q^{5} - 63q^{6} - 399q^{8} + 162q^{9} + 758q^{10} + 297q^{11} + 297q^{12} - 1479q^{13} - 2320q^{14} - 1017q^{15} + 993q^{16} + 578q^{17} - 567q^{18} - 2147q^{19} - 4402q^{20} - 1692q^{22} - 6395q^{23} - 3591q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{145})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 120x + 2295$

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 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
$\operatorname{tr} a_p$	−7	18	−113	0	297	−1479	578	−2147	−6395	−4584	−1110	14790	−15435	3723	5286	−21120	10002	−56654	−25776	−145996	28760	−28990	−82606	37558	85714

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 3, 8, 32] (certified at ell = 2147483489)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 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, 6)[0]   # 51.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 51:6 1000