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

Newform orbit 219.4.a.d

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 2q^{2} + 36q^{3} + 68q^{4} + 42q^{5} + 6q^{6} + 84q^{7} + 24q^{8} + 108q^{9} + 48q^{10} + 64q^{11} + 204q^{12} + 200q^{13} + 52q^{14} + 126q^{15} + 396q^{16} + 78q^{17} + 18q^{18} + 98q^{19} - 196q^{20} + 252q^{21} - 252q^{22} - 136q^{23} + 72q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{12} - 44x^{11} - 189x^{10} + 31472x^{9} - 240374x^{8} - 6982668x^{7} + 91817334x^{6} + 415058256x^{5} - 10110679587x^{4} + 18033718176x^{3} + 320222277207x^{2} - 1671048657864x + 2292185172456$

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
$\operatorname{tr} a_p$	2	36	42	84	64	200	78	98	−136	308	704	654	308	1004	350	718	154	1202	266	−124	876	−254	−370	−1640	558

$\operatorname{tr} a_n$ 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)[3]   # 219.4.a.d

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

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