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

Newform orbit 21.6.a.d

$q$ -expansion

$q + 10q^{2} + 9q^{3} + 68q^{4} - 106q^{5} + 90q^{6} - 49q^{7} + 360q^{8} + 81q^{9} - 1060q^{10} + 92q^{11} + 612q^{12} + 670q^{13} - 490q^{14} - 954q^{15} + 1424q^{16} - 222q^{17} + 810q^{18} - 908q^{19} - 7208q^{20} - 441q^{21} + 920q^{22} - 1176q^{23} + 3240q^{24} + O(q^{25})$

Coefficient field

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

$x + 96$

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
$\operatorname{tr} a_p$	10	9	−106	−49	92	670	−222	−908	−1176	1118	3696	4182	−6662	−3700	−7056	−37578	32700	−10802	64996	−61320	38922	−88096	71892	111818	−150846

$\operatorname{tr} a_n$ 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)[3]   # 21.6.a.d

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

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