Atlas › Modular forms › Weight 6 › Level 3 › 3.6.a › 3.6.a.a

Newform orbit 3.6.a.a

$q$ -expansion

$q - 6q^{2} + 9q^{3} + 4q^{4} + 6q^{5} - 54q^{6} - 40q^{7} + 168q^{8} + 81q^{9} - 36q^{10} - 564q^{11} + 36q^{12} + 638q^{13} + 240q^{14} + 54q^{15} - 1136q^{16} + 882q^{17} - 486q^{18} - 556q^{19} + 24q^{20} - 360q^{21} + 3384q^{22} - 840q^{23} + 1512q^{24} + O(q^{25})$

Coefficient field

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

$x$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$

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
$\operatorname{tr} a_p$	−6	9	6	−40	−564	638	882	−556	−840	4638	4400	−2410	−6870	9644	−18672	33750	−18084	39758	−23068	−4248	−41110	21920	82452	−94086	49442

$\operatorname{tr} a_n$ 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 3, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.093 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]: [1, 3] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 1 = 1 x 1 (phi(ord eps) x dim S_k^new); 1 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(3, 6)[0]   # 3.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 3:6 1000