Atlas › Modular forms › Weight 4 › Level 39 › 39.4.a › 39.4.a.b

Newform orbit 39.4.a.b

Trace form

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

$2q + 2q^{2} - 6q^{3} + 14q^{4} + 24q^{5} - 6q^{6} + 54q^{8} + 18q^{9} - 32q^{10} - 44q^{11} - 42q^{12} - 26q^{13} - 56q^{14} - 72q^{15} - 30q^{16} + 164q^{17} + 18q^{18} + 48q^{19} + 56q^{20} - 380q^{22} + 8q^{23} - 162q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{14})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{2} - 74x + 1019$

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 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	−6	24	0	−44	−26	164	48	8	404	40	−100	200	−616	−324	−164	140	628	−472	428	−900	−432	−1388	960	−532

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.099 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, 2, 5, 16] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 6 = 1 x 6 (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(39, 4)[1]   # 39.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 39:4 1000