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

Newform orbit 21.8.a.b

Trace form

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

$2q - 9q^{2} - 54q^{3} + 317q^{4} - 360q^{5} + 243q^{6} + 686q^{7} - 5067q^{8} + 1458q^{9} - 9030q^{10} - 4932q^{11} - 8559q^{12} + 7708q^{13} - 3087q^{14} + 9720q^{15} + 20033q^{16} - 28584q^{17} - 6561q^{18} - 63728q^{19} + 38790q^{20} - 18522q^{21} + 186204q^{22} + 82260q^{23} + 136809q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1065})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} + 369x - 62076$

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
$\operatorname{tr} a_p$	−9	−54	−360	686	−4932	7708	−28584	−63728	82260	−435996	−29240	−709556	−25056	496216	−1575000	2057436	−1101024	28996	−4480784	54540	666604	2322952	−7384392	1784448	16266412

$\operatorname{tr} a_n$ 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)[1]   # 21.8.a.b

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

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