Atlas › Modular forms › Weight 10 › Level 20 › 20.10.a › 20.10.a.b

Newform orbit 20.10.a.b

Trace form

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

$2q - 260q^{3} - 1250q^{5} - 380q^{7} + 34882q^{9} + 102720q^{11} + 179140q^{13} + 162500q^{15} + 316020q^{17} + 137272q^{19} + 2840312q^{21} - 665460q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{79})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{7}$ , whose minimal polynomial is

$x^{2} + 640x - 98995200$

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	−260	−1250	−380	102720	179140	316020	137272	−665460	−6893748	291832	11261380	29773452	−11708180	62493300	9417780	−92930856	195673924	−219767420	311207016	−99224060	542261776	−1256915700	−462291852	1671716740

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.305 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, 6, 5, 15, 30] (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, 10)[1]   # 20.10.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 20:10 1000