Atlas › Modular forms › Weight 4 › Level 199 › 199.4.a › 199.4.a.a

Newform orbit 199.4.a.a

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q - 11q^{2} - 8q^{3} + 65q^{4} - 41q^{5} - 51q^{6} - 35q^{7} - 129q^{8} - 8q^{9} - 75q^{10} - 158q^{11} - 112q^{12} - 103q^{13} - 209q^{14} - 156q^{15} + 45q^{16} - 350q^{17} - 131q^{18} - 132q^{19} - 416q^{20} - 296q^{21} - 179q^{22} - 321q^{23} - 427q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} + 19x^{19} - 147x^{18} - 4087x^{17} + 5667x^{16} + 346616x^{15} + 77554x^{14} - 14628307x^{13} - 8499246x^{12} + 318870810x^{11} + 140060846x^{10} - 3395378753x^{9} - 413717113x^{8} + 16039693947x^{7} - 2876165665x^{6} - 24829115864x^{5} + 4528267229x^{4} + 13589200892x^{3} - 1847088196x^{2} - 1956622072x + 112384928$

Atkin–Lehner signs

$p$	$w_p$
199	$-1$

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
$\operatorname{tr} a_p$	−11	−8	−41	−35	−158	−103	−350	−132	−321	−1237	−457	−760	−2040	101	−1173	−1169	−652	313	840	−988	−448	−657	−320	−2669	1360

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 199]: [1, 51] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 7 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(199, 4)[0]   # 199.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 199:4 1000