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

Newform orbit 10.10.a.a

$q$ -expansion

$q - 16q^{2} - 204q^{3} + 256q^{4} + 625q^{5} + 3264q^{6} + 5432q^{7} - 4096q^{8} + 21933q^{9} - 10000q^{10} + 73932q^{11} - 52224q^{12} - 114514q^{13} - 86912q^{14} - 127500q^{15} + 65536q^{16} + 41682q^{17} - 350928q^{18} + 1057460q^{19} + 160000q^{20} - 1108128q^{21} - 1182912q^{22} + 1599336q^{23} + 835584q^{24} + O(q^{25})$

Coefficient field

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

$x - 5228$

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	−204	625	5432	73932	−114514	41682	1057460	1599336	2184510	−9619648	4799942	9531882	−13464484	11441952	53615766	81862620	−104691298	140571092	97098792	171848906	−117380080	323637636	−894379110	232678562

$\operatorname{tr} a_n$ 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)[0]   # 10.10.a.a

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

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