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

Newform orbit 21.8.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{2} + 81q^{3} + 225q^{4} - 114q^{5} - 81q^{6} + 1029q^{7} + 417q^{8} + 2187q^{9} + 9294q^{10} + 8736q^{11} + 6075q^{12} + 12762q^{13} - 1029q^{14} - 3078q^{15} - 3327q^{16} + 5490q^{17} - 2187q^{18} + 8004q^{19} - 119922q^{20} + 27783q^{21} - 108276q^{22} - 101868q^{23} + 11259q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} + 117x^{2} - 189732x + 14892120$

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
$\operatorname{tr} a_p$	−3	81	−114	1029	8736	12762	5490	8004	−101868	−255342	80592	800322	1198002	119052	1019256	−1174878	−692556	4507314	−2951364	−2009844	−3938874	6406008	4575444	−11158974	−2153394

$\operatorname{tr} a_n$ 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)[3]   # 21.8.a.d

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

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