Atlas › Modular forms › Weight 12 › Level 10 › 10.12.a › 10.12.a.c

Newform orbit 10.12.a.c

$q$ -expansion

$q + 32q^{2} - 318q^{3} + 1024q^{4} - 3125q^{5} - 10176q^{6} - 70714q^{7} + 32768q^{8} - 76023q^{9} - 100000q^{10} + 238272q^{11} - 325632q^{12} - 2097478q^{13} - 2262848q^{14} + 993750q^{15} + 1048576q^{16} + 5955546q^{17} - 2432736q^{18} + 10210820q^{19} - 3200000q^{20} + 22487052q^{21} + 7624704q^{22} - 3535758q^{23} - 10420224q^{24} + O(q^{25})$

Coefficient field

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

$x + 71032$

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 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
$\operatorname{tr} a_p$	32	−318	−3125	−70714	238272	−2097478	5955546	10210820	−3535758	−139304850	−101002348	−524913814	284590422	−1253635078	−216106434	−4881275358	8692473300	3296491802	18275027966	−13287447588	−32505250798	9297455960	−22741484838	−93378882390	−5811134014

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

Sato–Tate

The normalized $a_p / 2p^{11/2}$ 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 0.332 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]: [2, 4, 7, 19] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 5 = 1 x 5 (phi(ord eps) x dim S_k^new); 3 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, 12)[2]   # 10.12.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 10:12 1000