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

Newform orbit 219.4.a.b

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - 24q^{3} + 24q^{4} - 42q^{5} + 28q^{7} + 72q^{9} + 44q^{10} - 36q^{11} - 72q^{12} - 64q^{13} - 156q^{14} + 126q^{15} + 12q^{16} - 90q^{17} - 122q^{19} - 720q^{20} - 84q^{21} - 200q^{22} - 204q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{8} + 42x^{7} + 225x^{6} - 7740x^{5} - 53685x^{4} + 539298x^{3} + 2132811x^{2} - 14545008x + 12719592$

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
$\operatorname{tr} a_p$	0	−24	−42	28	−36	−64	−90	−122	−204	−576	−144	26	−264	−36	−798	−930	−1518	270	2326	−1272	584	1342	−2694	−2496	1330

$\operatorname{tr} a_n$ 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)[1]   # 219.4.a.b

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

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