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

Newform orbit 39.4.a.a

$q$ -expansion

$q - 3q^{3} - 8q^{4} - 12q^{5} + 2q^{7} + 9q^{9} - 36q^{11} + 24q^{12} + 13q^{13} + 36q^{15} + 64q^{16} - 78q^{17} + 74q^{19} + 96q^{20} - 6q^{21} - 96q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{5}$ is a root of

$x + 36$

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
$\operatorname{tr} a_p$	0	−3	−12	2	−36	13	−78	74	−96	18	−214	−286	−384	524	300	558	576	74	38	−456	−682	704	−888	−1020	110

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

Sato–Tate

The normalized $a_p / 2p^{3/2}$ for the primes $p < 1000$ not dividing 39, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

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)[0]   # 39.4.a.a

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

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