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

Newform orbit 10.12.a.b

$q$ -expansion

$q - 32q^{2} + 738q^{3} + 1024q^{4} - 3125q^{5} - 23616q^{6} + 25574q^{7} - 32768q^{8} + 367497q^{9} + 100000q^{10} + 769152q^{11} + 755712q^{12} - 918982q^{13} - 818368q^{14} - 2306250q^{15} + 1048576q^{16} + 10312794q^{17} - 11759904q^{18} - 5521660q^{19} - 3200000q^{20} + 18873612q^{21} - 24612864q^{22} - 39973422q^{23} - 24182784q^{24} + O(q^{25})$

Coefficient field

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

$x - 26312$

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
$\operatorname{tr} a_p$	−32	738	−3125	25574	769152	−918982	10312794	−5521660	−39973422	−15269010	−241583788	−25751446	−1217700138	−683436262	1537395294	3572891298	−1069039020	−2091535078	−1462369186	9660178332	−5603447662	5026936280	−38405955462	35558583210	10572232514

$\operatorname{tr} a_n$ 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)[1]   # 10.12.a.b

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

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