Atlas › Modular forms › Weight 4 › Level 219 › 219.4.a › 219.4.a.a

Newform orbit 219.4.a.a

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - 2q^{2} + 18q^{3} + 4q^{4} - 28q^{5} - 6q^{6} - 56q^{7} - 24q^{8} + 54q^{9} - 72q^{10} - 46q^{11} + 12q^{12} - 86q^{13} - 88q^{14} - 84q^{15} - 180q^{16} - 126q^{17} - 18q^{18} - 250q^{19} - 14q^{20} - 168q^{21} - 22q^{22} - 120q^{23} - 72q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} + 30x^{5} + 221x^{4} - 360x^{3} - 4673x^{2} + 2586x + 14643$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
73	$+1$

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 odd 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
$\operatorname{tr} a_p$	−2	18	−28	−56	−46	−86	−126	−250	−120	110	−284	−762	−292	−286	−278	−672	−236	−830	−678	−20	−438	−1282	1180	1276	350

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.141 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, 73, 219]: [1, 2, 20, 76] (certified at ell = 2147482873)
T = [(2, 1), (5, 1)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 6 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(219, 4)[0]   # 219.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 219:4 1000