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

Newform orbit 21.8.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 12q^{2} - 54q^{3} + 352q^{4} - 24q^{5} - 324q^{6} - 686q^{7} + 7008q^{8} + 1458q^{9} + 4144q^{10} + 2124q^{11} - 9504q^{12} - 1084q^{13} - 4116q^{14} + 648q^{15} + 61312q^{16} - 29256q^{17} + 8748q^{18} - 25816q^{19} + 47232q^{20} + 18522q^{21} - 137336q^{22} + 68316q^{23} - 189216q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{67})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 12x - 21672$

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
$\operatorname{tr} a_p$	12	−54	−24	−686	2124	−1084	−29256	−25816	68316	211308	435840	−28428	749760	397096	840168	−246684	2199504	−1951108	1532048	2024004	−1709028	1048168	−4894296	−60864	−26046852

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

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)[2]   # 21.8.a.c

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

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