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

Newform orbit 22.10.a.c

$q$ -expansion

$q + 16q^{2} + 201q^{3} + 256q^{4} + 2349q^{5} + 3216q^{6} - 8806q^{7} + 4096q^{8} + 20718q^{9} + 37584q^{10} - 14641q^{11} + 51456q^{12} - 131068q^{13} - 140896q^{14} + 472149q^{15} + 65536q^{16} - 55698q^{17} + 331488q^{18} + 1041824q^{19} + 601344q^{20} - 1770006q^{21} - 234256q^{22} - 662139q^{23} + 823296q^{24} + O(q^{25})$

Coefficient field

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

$x - 2550$

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 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	201	2349	−8806	−14641	−131068	−55698	1041824	−662139	−4819344	−180115	−7803025	−5927736	−5929162	61576176	7349514	−113901909	−13814260	309980903	42752631	142018340	−325376446	253502934	−994227705	−352091047

$\operatorname{tr} a_n$ 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)[2]   # 22.10.a.c

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

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