Atlas › Modular forms › Weight 4 › Level 21 › 21.4.a › 21.4.a.c

Newform orbit 21.4.a.c

Trace form

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

$2q - 3q^{2} + 6q^{3} + 17q^{4} + 6q^{5} - 9q^{6} + 14q^{7} - 87q^{8} + 18q^{9} - 66q^{10} - 6q^{11} + 51q^{12} + 16q^{13} - 21q^{14} + 18q^{15} + 137q^{16} - 6q^{17} - 27q^{18} + 64q^{19} + 222q^{20} + 42q^{21} - 276q^{22} + 6q^{23} - 261q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{57})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - 3x - 12$

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	6	6	14	−6	16	−6	64	6	−252	40	−248	−450	376	−12	−1104	804	−428	148	954	1072	−572	1944	366	808

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.096 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, 2, 3, 10] (certified at ell = 2147483647)
T = [(2, 1), (5, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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, 4)[2]   # 21.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 21:4 1000