Atlas › Modular forms › Weight 6 › Level 21 › 21.6.a › 21.6.a.a

Newform orbit 21.6.a.a

$q$ -expansion

$q - 6q^{2} - 9q^{3} + 4q^{4} + 78q^{5} + 54q^{6} + 49q^{7} + 168q^{8} + 81q^{9} - 468q^{10} + 444q^{11} - 36q^{12} - 442q^{13} - 294q^{14} - 702q^{15} - 1136q^{16} - 126q^{17} - 486q^{18} + 2684q^{19} + 312q^{20} - 441q^{21} - 2664q^{22} + 4200q^{23} - 1512q^{24} + O(q^{25})$

Coefficient field

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

$x - 72$

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 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
$\operatorname{tr} a_p$	−6	−9	78	49	444	−442	−126	2684	4200	−5442	80	−5434	7962	−11524	−13920	−9594	27492	49478	−59356	32040	−61846	−65776	40188	−7974	−143662

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

Sato–Tate

The normalized $a_p / 2p^{5/2}$ 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 0.111 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, 3, 5, 16] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 6)[0]   # 21.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 21:6 1000