Atlas › Modular forms › Weight 8 › Level 21 › 21.8.a › 21.8.a.a

Newform orbit 21.8.a.a

$q$ -expansion

$q + 2q^{2} + 27q^{3} - 124q^{4} - 278q^{5} + 54q^{6} - 343q^{7} - 504q^{8} + 729q^{9} - 556q^{10} - 4496q^{11} - 3348q^{12} - 7274q^{13} - 686q^{14} - 7506q^{15} + 14864q^{16} + 11382q^{17} + 1458q^{18} - 15884q^{19} + 34472q^{20} - 9261q^{21} - 8992q^{22} + 86100q^{23} - 13608q^{24} + O(q^{25})$

Coefficient field

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

$x + 276$

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
$\operatorname{tr} a_p$	2	27	−278	−343	−4496	−7274	11382	−15884	86100	40702	−44760	−580962	−171658	−741148	1071720	−1721778	−1557012	2597998	−963548	−4063380	−5370222	4094936	−1343124	9081574	6487914

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

Sato–Tate

The normalized $a_p / 2p^{7/2}$ 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 0.262 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, 20] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(21, 8)[0]   # 21.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 21:8 1000