Atlas › Modular forms › Weight 6 › Level 42 › 42.6.a › 42.6.a.e

Newform orbit 42.6.a.e

$q$ -expansion

$q + 4q^{2} - 9q^{3} + 16q^{4} + 76q^{5} - 36q^{6} - 49q^{7} + 64q^{8} + 81q^{9} + 304q^{10} + 650q^{11} - 144q^{12} + 762q^{13} - 196q^{14} - 684q^{15} + 256q^{16} - 556q^{17} + 324q^{18} - 2452q^{19} + 1216q^{20} + 441q^{21} + 2600q^{22} - 2950q^{23} - 576q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + T_{11}$ is a root of

$x - 726$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$+1$
7	$+1$

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
$\operatorname{tr} a_p$	4	−9	76	−49	650	762	−556	−2452	−2950	−674	−3024	7730	−17016	21836	−23940	15594	5608	150	−43784	−39178	−23570	−17892	38972	6024	108430

$\operatorname{tr} a_n$ 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 42, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.131 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, 2, 3, 6, 7, 14, 21, 42]: [1, 2, 3, 7, 5, 12, 16, 44] (certified at ell = 2147483647)
T = [(5, 1), (11, 1)]; 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(42, 6)[4]   # 42.6.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 42:6 1000