Atlas › Modular forms › Weight 10 › Level 6 › 6.10.a › 6.10.a.a

Newform orbit 6.10.a.a

$q$ -expansion

$q - 16q^{2} + 81q^{3} + 256q^{4} + 2694q^{5} - 1296q^{6} - 3544q^{7} - 4096q^{8} + 6561q^{9} - 43104q^{10} + 29580q^{11} + 20736q^{12} - 44818q^{13} + 56704q^{14} + 218214q^{15} + 65536q^{16} - 101934q^{17} - 104976q^{18} - 895084q^{19} + 689664q^{20} - 287064q^{21} - 473280q^{22} - 1113000q^{23} - 331776q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{5} + T_{7}$ is a root of

$x + 850$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
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
$\operatorname{tr} a_p$	−16	81	2694	−3544	29580	−44818	−101934	−895084	−1113000	−2357346	175808	−2919418	26218794	−18762964	−20966160	57251574	33587580	82260830	−188455804	80924040	−236140918	526909808	18346452	690643098	−438251038

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

Sato–Tate

The normalized $a_p / 2p^{9/2}$ for the primes $p < 1000$ not dividing 6, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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]: [1, 3, 4, 11] (certified at ell = 2147483647)
T = [(5, 1), (7, 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(6, 10)[0]   # 6.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 6:10 1000