Atlas › Modular forms › Weight 10 › Level 22 › 22.10.a › 22.10.a.a

Newform orbit 22.10.a.a

$q$ -expansion

$q + 16q^{2} - 41q^{3} + 256q^{4} - 1039q^{5} - 656q^{6} - 3482q^{7} + 4096q^{8} - 18002q^{9} - 16624q^{10} + 14641q^{11} - 10496q^{12} - 199796q^{13} - 55712q^{14} + 42599q^{15} + 65536q^{16} + 164038q^{17} - 288032q^{18} - 277560q^{19} - 265984q^{20} + 142762q^{21} + 234256q^{22} - 1211721q^{23} - 167936q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{3} + T_{5}$ is a root of

$x + 1080$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
11	$-1$

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
$\operatorname{tr} a_p$	16	−41	−1039	−3482	14641	−199796	164038	−277560	−1211721	4248880	9112927	10500403	−844768	1083514	−45843752	5568394	−106773315	−98810468	−168277647	67984277	−65392116	85785910	−103589846	−809499425	859612633

$\operatorname{tr} a_n$ 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 22, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.31 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, 11, 22]: [1, 3, 10, 29] (certified at ell = 2147483171)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (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(22, 10)[0]   # 22.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 22:10 1000