Atlas › Modular forms › Weight 6 › Level 39 › 39.6.a › 39.6.a.b

Newform orbit 39.6.a.b

Trace form

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

$2q - 4q^{2} - 18q^{3} + 56q^{4} - 48q^{5} + 36q^{6} + 72q^{7} - 432q^{8} + 162q^{9} - 464q^{10} - 596q^{11} - 504q^{12} - 338q^{13} - 1376q^{14} + 432q^{15} - 480q^{16} - 268q^{17} - 324q^{18} + 1128q^{19} + 896q^{20} - 648q^{21} + 5896q^{22} - 1768q^{23} + 3888q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{14})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 52x - 220$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
13	$+1$

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
$\operatorname{tr} a_p$	−4	−18	−48	72	−596	−338	−268	1128	−1768	−7612	−4160	17468	−28000	−24328	−18108	1420	6788	−37148	106112	−30460	−37620	−2160	207004	−74136	−121156

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.232 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, 13, 39]: [1, 3, 7, 26] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 10 = 1 x 10 (phi(ord eps) x dim S_k^new); 3 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(39, 6)[1]   # 39.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 39:6 1000