Atlas › Modular forms › Weight 4 › Level 229 › 229.4.a

Newspace 229.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(229))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
229.4.a.a	26	degree 26	$26q - 15q^{2} - 17q^{3} + 89q^{4} - 30q^{5} - 53q^{6} - 55q^{7} + O(q^{8})$	−	$-1$
229.4.a.b	31	degree 31	$31q + 11q^{2} + 19q^{3} + 149q^{4} + 30q^{5} + 31q^{6} + 29q^{7} + O(q^{8})$	+	$+1$

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 $T$
229.4.a.a	$x^{26} + 32x^{25} + 9x^{24} - 9031x^{23} - 56608x^{22} + 1055910x^{21} + 9552425x^{20} - 67744462x^{19} - 762511015x^{18} + 2702035176x^{17} + 35594895865x^{16} - 73791244712x^{15} - 1042378665363x^{14} + 1510192793863x^{13} + 19523098258117x^{12} - 23995557195654x^{11} - 230672022482177x^{10} + 276272453127444x^{9} + 1643346858747103x^{8} - 2016228341466699x^{7} - 6493387171681120x^{6} + 8062436773093723x^{5} + 12195194822001421x^{4} - 14877244708978490x^{3} - 7194929158026650x^{2} + 9585169902402500x - 1700449901125000$
229.4.a.b	$x^{31} - 30x^{30} - 367x^{29} + 18311x^{28} + 3126x^{27} - 4634392x^{26} + 17410725x^{25} + 651824178x^{24} - 3702862859x^{23} - 57454286416x^{22} + 388749609093x^{21} + 3384370938808x^{20} - 24819849617343x^{19} - 138058438425777x^{18} + 1030894208563515x^{17} + 3949657643576784x^{16} - 28505814688491465x^{15} - 78122675881146596x^{14} + 525342919106882527x^{13} + 1015371447468136145x^{12} - 6381828149885594116x^{11} - 7777664063494704445x^{10} + 50066780534268035823x^{9} + 26603354807267517422x^{8} - 241288790204262957948x^{7} + 24563552556077848792x^{6} + 628159922567349794424x^{5} - 402723332826731389824x^{4} - 614151207945499813984x^{3} + 702006389344174588416x^{2} - 87632935406700121728x - 71796309641843151360$

Orbit

Characteristic polynomial of

T

229.4.a.a

x^{26} + 32x^{25} + 9x^{24} - 9031x^{23} - 56608x^{22} + 1055910x^{21} + 9552425x^{20} - 67744462x^{19} - 762511015x^{18} + 2702035176x^{17} + 35594895865x^{16} - 73791244712x^{15} - 1042378665363x^{14} + 1510192793863x^{13} + 19523098258117x^{12} - 23995557195654x^{11} - 230672022482177x^{10} + 276272453127444x^{9} + 1643346858747103x^{8} - 2016228341466699x^{7} - 6493387171681120x^{6} + 8062436773093723x^{5} + 12195194822001421x^{4} - 14877244708978490x^{3} - 7194929158026650x^{2} + 9585169902402500x - 1700449901125000

229.4.a.b

x^{31} - 30x^{30} - 367x^{29} + 18311x^{28} + 3126x^{27} - 4634392x^{26} + 17410725x^{25} + 651824178x^{24} - 3702862859x^{23} - 57454286416x^{22} + 388749609093x^{21} + 3384370938808x^{20} - 24819849617343x^{19} - 138058438425777x^{18} + 1030894208563515x^{17} + 3949657643576784x^{16} - 28505814688491465x^{15} - 78122675881146596x^{14} + 525342919106882527x^{13} + 1015371447468136145x^{12} - 6381828149885594116x^{11} - 7777664063494704445x^{10} + 50066780534268035823x^{9} + 26603354807267517422x^{8} - 241288790204262957948x^{7} + 24563552556077848792x^{6} + 628159922567349794424x^{5} - 402723332826731389824x^{4} - 614151207945499813984x^{3} + 702006389344174588416x^{2} - 87632935406700121728x - 71796309641843151360

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(229))$	57
Old forms	0
Cusp forms $S_{4}(\Gamma_0(229))$	57
Eisenstein series $E_{4}(\Gamma_0(229))$	2
Modular forms $M_{4}(\Gamma_0(229))$	59

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.157 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, 229]: [1, 59] (certified at ell = 2147478649)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 8 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(229, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 229:4 1000

← 228.4.a · 230.4.a →