Atlas › Modular forms › Weight 10 › Level 10 › 10.10.a › 10.10.a.c

Newform orbit 10.10.a.c

$q$ -expansion

$q + 16q^{2} + 174q^{3} + 256q^{4} - 625q^{5} + 2784q^{6} + 4658q^{7} + 4096q^{8} + 10593q^{9} - 10000q^{10} + 28992q^{11} + 44544q^{12} - 164446q^{13} + 74528q^{14} - 108750q^{15} + 65536q^{16} - 594822q^{17} + 169488q^{18} - 295780q^{19} - 160000q^{20} + 810492q^{21} + 463872q^{22} + 2544534q^{23} + 712704q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{7}$ is a root of

$x - 4832$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$+1$

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	174	−625	4658	28992	−164446	−594822	−295780	2544534	−3722970	2335772	10840418	21593862	10832294	5172138	98179674	16162860	−43928158	−81557422	161307732	−247147966	−583345720	−14571786	470133690	−117838462

$\operatorname{tr} a_n$ 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 10, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 5, 10]: [1, 3, 5, 15] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 3 = 1 x 3 (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(10, 10)[2]   # 10.10.a.c

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

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