Atlas › Modular forms › Weight 10 › Level 21 › 21.10.a › 21.10.a.a

Newform orbit 21.10.a.a

$q$ -expansion

$q - 24q^{2} + 81q^{3} + 64q^{4} - 144q^{5} - 1944q^{6} + 2401q^{7} + 10752q^{8} + 6561q^{9} + 3456q^{10} - 15030q^{11} + 5184q^{12} - 151486q^{13} - 57624q^{14} - 11664q^{15} - 290816q^{16} - 350448q^{17} - 157464q^{18} - 691108q^{19} - 9216q^{20} + 194481q^{21} + 360720q^{22} + 892458q^{23} + 870912q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + T_{5}$ is a root of

$x + 168$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
7	$-1$

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
$\operatorname{tr} a_p$	−24	81	−144	2401	−15030	−151486	−350448	−691108	892458	1648518	−3734296	−11471902	13985724	16794524	−14012052	−97439910	110798304	−93816682	−122446456	206197398	250337558	−38314852	−514086924	−1061294916	−73841578

$\operatorname{tr} a_n$ 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 21, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 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, 3, 7, 21]: [1, 4, 7, 26] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 4 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(21, 10)[0]   # 21.10.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 21:10 1000