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

Newform orbit 219.4.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 4q^{2} - 30q^{3} + 40q^{4} + 28q^{5} - 12q^{6} - 28q^{7} + 48q^{8} + 90q^{9} + 44q^{10} + 74q^{11} - 120q^{12} + 66q^{13} - 16q^{14} - 84q^{15} + 140q^{16} + 114q^{17} + 36q^{18} + 138q^{19} + 582q^{20} + 84q^{21} + 382q^{22} + 180q^{23} - 144q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} - 32x^{9} - 409x^{8} + 17264x^{7} + 36326x^{6} - 3053836x^{5} + 811770x^{4} + 211567808x^{3} - 131877759x^{2} - 4766372324x + 3250268695$

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
$\operatorname{tr} a_p$	4	−30	28	−28	74	66	114	138	180	502	−16	386	1104	−466	454	860	−288	934	−1946	−1312	−730	−1638	452	5188	10

$\operatorname{tr} a_n$ 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)[2]   # 219.4.a.c

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

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