Atlas › Modular forms › Weight 12 › Level 20 › 20.12.a › 20.12.a.b

Newform orbit 20.12.a.b

Trace form

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

$2q + 220q^{3} + 6250q^{5} + 3340q^{7} + 43738q^{9} + 298320q^{11} + 1209820q^{13} + 687500q^{15} + 9056340q^{17} + 19439368q^{19} + 41862752q^{21} + 55926420q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{46729})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} - 3560x - 2341505904$

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	220	6250	3340	298320	1209820	9056340	19439368	55926420	41841708	62230792	−729235940	−608419068	−1129440740	−1653072900	5724887340	3756433896	4923703564	−5843244140	−43352162664	33647099620	−54799425296	−9303032100	3688968372	65892157780

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.353 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]: [2, 4, 7, 7, 19, 36] (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, 12)[1]   # 20.12.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 20:12 1000