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

Newspace 199.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(199))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
199.4.a.a	20	degree 20	$20q - 11q^{2} - 8q^{3} + 65q^{4} - 41q^{5} - 51q^{6} - 35q^{7} + O(q^{8})$	−	$-1$
199.4.a.b	29	degree 29	$29q + 9q^{2} + 10q^{3} + 137q^{4} + 39q^{5} + 33q^{6} + 21q^{7} + O(q^{8})$	+	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
199.4.a.a	$x^{20} + 19x^{19} - 147x^{18} - 4087x^{17} + 5667x^{16} + 346616x^{15} + 77554x^{14} - 14628307x^{13} - 8499246x^{12} + 318870810x^{11} + 140060846x^{10} - 3395378753x^{9} - 413717113x^{8} + 16039693947x^{7} - 2876165665x^{6} - 24829115864x^{5} + 4528267229x^{4} + 13589200892x^{3} - 1847088196x^{2} - 1956622072x + 112384928$
199.4.a.b	$x^{29} - 19x^{28} - 627x^{27} + 13025x^{26} + 162063x^{25} - 3753648x^{24} - 23676366x^{23} + 612616475x^{22} + 2271818024x^{21} - 63670952297x^{20} - 159224442389x^{19} + 4457485677316x^{18} + 8767698254632x^{17} - 215580762996142x^{16} - 384893480335733x^{15} + 7241699773476434x^{14} + 12889853272525812x^{13} - 167236022610880566x^{12} - 307637218451360090x^{11} + 2589045565729072916x^{10} + 4873514246024378627x^{9} - 25767918988547285321x^{8} - 47148125437660891483x^{7} + 155014579681611110263x^{6} + 244556935935120404330x^{5} - 516434274323054048009x^{4} - 510446992822672391656x^{3} + 787293864860812930884x^{2} + 26215861224697139160x - 31965347185408255104$

Orbit

Characteristic polynomial of

T

199.4.a.a

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

199.4.a.b

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

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(199))$	49
Old forms	0
Cusp forms $S_{4}(\Gamma_0(199))$	49
Eisenstein series $E_{4}(\Gamma_0(199))$	2
Modular forms $M_{4}(\Gamma_0(199))$	51

Decomposition of the old subspace

The old subspace is 0.

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)

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

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

← 198.4.a · 200.4.a →