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

Newform orbit 39.4.a.c

Trace form

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

$3q + 2q^{2} + 9q^{3} + 10q^{4} + 4q^{5} + 6q^{6} + 30q^{7} - 6q^{8} + 27q^{9} - 4q^{10} - 16q^{11} + 30q^{12} + 39q^{13} - 176q^{14} + 12q^{15} - 110q^{16} - 146q^{17} + 18q^{18} + 94q^{19} - 244q^{20} + 90q^{21} - 56q^{22} - 48q^{23} - 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{3} - 14x^{2} - 2247x - 27696$

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	9	4	30	−16	39	−146	94	−48	−2	302	374	480	−260	−24	−678	−1788	230	74	−948	−222	−24	−796	1436	3242

$\operatorname{tr} a_n$ 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)[2]   # 39.4.a.c

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

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