Atlas › Modular forms › Weight 8 › Level 20 › 20.8.a › 20.8.a.b

Newform orbit 20.8.a.b

Trace form

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

$2q - 20q^{3} + 250q^{5} + 1660q^{7} + 4858q^{9} + 3600q^{11} + 13180q^{13} - 2500q^{15} + 5460q^{17} - 40472q^{19} - 97888q^{21} - 41820q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{1129})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 1640x + 383376$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 20$ )
5	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	−20	250	1660	3600	13180	5460	−40472	−41820	118668	−115928	306940	−353148	1215340	−2068500	1400460	−1992504	−1678676	3663940	1794936	5062180	10178224	−7214100	−15330828	14024020

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.261 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, 2, 4, 5, 10, 20]: [1, 3, 5, 5, 13, 24] (certified at ell = 2147483629)
T = [(3, 1), (7, 1)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 2 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(20, 8)[1]   # 20.8.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 20:8 1000