Atlas › Modular forms › Weight 6 › Level 7 › 7.6.a › 7.6.a.a

Newform orbit 7.6.a.a

$q$ -expansion

$q - 10q^{2} - 14q^{3} + 68q^{4} - 56q^{5} + 140q^{6} - 49q^{7} - 360q^{8} - 47q^{9} + 560q^{10} + 232q^{11} - 952q^{12} - 140q^{13} + 490q^{14} + 784q^{15} + 1424q^{16} - 1722q^{17} + 470q^{18} - 98q^{19} - 3808q^{20} + 686q^{21} - 2320q^{22} + 1824q^{23} + 5040q^{24} + O(q^{25})$

Coefficient field

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

$x + 24$

Atkin–Lehner signs

$p$	$w_p$
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 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
$\operatorname{tr} a_p$	−10	−14	−56	−49	232	−140	−1722	−98	1824	3418	−7644	−10398	−17962	10880	9324	2262	−2730	25648	−48404	−58560	68082	31784	−20538	−50582	−58506

$\operatorname{tr} a_n$ 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 7, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.198 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, 7]: [1, 5] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 3 = 1 x 3 (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(7, 6)[0]   # 7.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 7:6 1000