Atlas › Modular forms › Weight 6 › Level 14 › 14.6.a › 14.6.a.a

Newform orbit 14.6.a.a

$q$ -expansion

$q - 4q^{2} + 10q^{3} + 16q^{4} + 84q^{5} - 40q^{6} + 49q^{7} - 64q^{8} - 143q^{9} - 336q^{10} - 336q^{11} + 160q^{12} + 584q^{13} - 196q^{14} + 840q^{15} + 256q^{16} - 1458q^{17} + 572q^{18} + 470q^{19} + 1344q^{20} + 490q^{21} + 1344q^{22} - 4200q^{23} - 640q^{24} + O(q^{25})$

Coefficient field

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

$x - 94$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
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	10	84	49	−336	584	−1458	470	−4200	4866	−7372	14330	6222	3704	−1812	−37242	34302	24476	−17452	28224	3602	42872	−35202	26730	−16978

$\operatorname{tr} a_n$ 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 14, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.108 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, 7, 14]: [1, 2, 5, 12] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 2 = 1 x 2 (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(14, 6)[0]   # 14.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 14:6 1000