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

Newform orbit 199.4.a.b

Trace form

The 29 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$29q + 9q^{2} + 10q^{3} + 137q^{4} + 39q^{5} + 33q^{6} + 21q^{7} + 111q^{8} + 397q^{9} + 45q^{10} + 172q^{11} + 104q^{12} + 105q^{13} + 295q^{14} + 84q^{15} + 621q^{16} + 432q^{17} + 409q^{18} + 134q^{19} + 384q^{20} + 376q^{21} + 217q^{22} + 415q^{23} + 581q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 29. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{29} - 19x^{28} - 627x^{27} + 13025x^{26} + 162063x^{25} - 3753648x^{24} - 23676366x^{23} + 612616475x^{22} + 2271818024x^{21} - 63670952297x^{20} - 159224442389x^{19} + 4457485677316x^{18} + 8767698254632x^{17} - 215580762996142x^{16} - 384893480335733x^{15} + 7241699773476434x^{14} + 12889853272525812x^{13} - 167236022610880566x^{12} - 307637218451360090x^{11} + 2589045565729072916x^{10} + 4873514246024378627x^{9} - 25767918988547285321x^{8} - 47148125437660891483x^{7} + 155014579681611110263x^{6} + 244556935935120404330x^{5} - 516434274323054048009x^{4} - 510446992822672391656x^{3} + 787293864860812930884x^{2} + 26215861224697139160x - 31965347185408255104$

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 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
$\operatorname{tr} a_p$	9	10	39	21	172	105	432	134	415	1315	163	−20	2306	961	143	1055	1370	−559	−1114	2416	−698	−1033	50	5895	70

$\operatorname{tr} a_n$ 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)[1]   # 199.4.a.b

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

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