Atlas › Modular forms › Weight 2 › Level 219 › 219.2.a › 219.2.a.e

Newform orbit 219.2.a.e

Trace form

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

$6q - q^{2} + 6q^{3} + 7q^{4} + 5q^{5} - q^{6} + 8q^{7} - 9q^{8} + 6q^{9} - 6q^{10} - 2q^{11} + 7q^{12} + 4q^{13} - 14q^{14} + 5q^{15} + 9q^{16} - 3q^{17} - q^{18} + 5q^{19} + 6q^{20} + 8q^{21} - 6q^{23} - 9q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5}$ , whose minimal polynomial is

$x^{6} - 14x^{5} - 69x^{4} + 1306x^{3} + 125x^{2} - 20684x - 21961$

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 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
$\operatorname{tr} a_p$	−1	6	5	8	−2	4	−3	5	−6	−4	4	13	−6	−12	−19	−5	−3	11	5	−8	−6	5	−5	20	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.371 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 73, 219]: [0, 1, 6, 26] (certified at ell = 2147482873)
T = [(2, 1), (5, 3)]; h of degree 13 = 1 x 13 (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(219, 2)[4]   # 219.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 219:2 1000