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

Newform orbit 39.6.a.a

$q$ -expansion

$q + 2q^{2} + 9q^{3} - 28q^{4} - 74q^{5} + 18q^{6} - 112q^{7} - 120q^{8} + 81q^{9} - 148q^{10} + 164q^{11} - 252q^{12} + 169q^{13} - 224q^{14} - 666q^{15} + 656q^{16} - 1646q^{17} + 162q^{18} - 2052q^{19} + 2072q^{20} - 1008q^{21} + 328q^{22} + 4152q^{23} - 1080q^{24} + O(q^{25})$

Coefficient field

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

$x + 72$

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
$\operatorname{tr} a_p$	2	9	−74	−112	164	169	−1646	−2052	4152	2638	−8936	1846	8010	−19236	−12840	−1434	1428	−25202	−22868	17280	54410	65312	−70372	−76390	−174398

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.232 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)[0]   # 39.6.a.a

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

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