Atlas › Modular forms › Weight 2 › Level 727 › 727.2.a

Newspace 727.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(727))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
727.2.a.a	24	degree 24	$24q - 13q^{2} - 8q^{3} + 21q^{4} - 8q^{5} - 3q^{6} - 7q^{7} + O(q^{8})$	+	$-1$
727.2.a.b	36	degree 36	$36q + 14q^{2} + 6q^{3} + 40q^{4} + 6q^{5} + 3q^{6} + q^{7} + O(q^{8})$	−	$+1$

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 $T$
727.2.a.a	$x^{24} + 21x^{23} + 152x^{22} + 240x^{21} - 2346x^{20} - 11214x^{19} + 540x^{18} + 102005x^{17} + 153597x^{16} - 360545x^{15} - 1040209x^{14} + 264958x^{13} + 2955777x^{12} + 1554402x^{11} - 3742820x^{10} - 4195891x^{9} + 1144272x^{8} + 3617149x^{7} + 1316949x^{6} - 662407x^{5} - 639589x^{4} - 180440x^{3} - 18165x^{2} - 98x + 4$
727.2.a.b	$x^{36} - 20x^{35} + 60x^{34} + 1316x^{33} - 9714x^{32} - 23294x^{31} + 401970x^{30} - 317203x^{29} - 8325606x^{28} + 19649023x^{27} + 99800890x^{26} - 371471804x^{25} - 709760653x^{24} + 4044364450x^{23} + 2647919271x^{22} - 28882234242x^{21} - 346001238x^{20} + 142608889677x^{19} - 48711964474x^{18} - 499141043025x^{17} + 258762725221x^{16} + 1251121171888x^{15} - 703058274250x^{14} - 2243311897247x^{13} + 1102242869087x^{12} + 2833426822282x^{11} - 922056603112x^{10} - 2417127378592x^{9} + 235793570629x^{8} + 1257427593087x^{7} + 185545354316x^{6} - 302120342853x^{5} - 116814392666x^{4} + 1797487715x^{3} + 5545435697x^{2} + 270113574x - 71144036$

Orbit

Characteristic polynomial of

T

727.2.a.a

x^{24} + 21x^{23} + 152x^{22} + 240x^{21} - 2346x^{20} - 11214x^{19} + 540x^{18} + 102005x^{17} + 153597x^{16} - 360545x^{15} - 1040209x^{14} + 264958x^{13} + 2955777x^{12} + 1554402x^{11} - 3742820x^{10} - 4195891x^{9} + 1144272x^{8} + 3617149x^{7} + 1316949x^{6} - 662407x^{5} - 639589x^{4} - 180440x^{3} - 18165x^{2} - 98x + 4

727.2.a.b

x^{36} - 20x^{35} + 60x^{34} + 1316x^{33} - 9714x^{32} - 23294x^{31} + 401970x^{30} - 317203x^{29} - 8325606x^{28} + 19649023x^{27} + 99800890x^{26} - 371471804x^{25} - 709760653x^{24} + 4044364450x^{23} + 2647919271x^{22} - 28882234242x^{21} - 346001238x^{20} + 142608889677x^{19} - 48711964474x^{18} - 499141043025x^{17} + 258762725221x^{16} + 1251121171888x^{15} - 703058274250x^{14} - 2243311897247x^{13} + 1102242869087x^{12} + 2833426822282x^{11} - 922056603112x^{10} - 2417127378592x^{9} + 235793570629x^{8} + 1257427593087x^{7} + 185545354316x^{6} - 302120342853x^{5} - 116814392666x^{4} + 1797487715x^{3} + 5545435697x^{2} + 270113574x - 71144036

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(727))$	60
Old forms	0
Cusp forms $S_{2}(\Gamma_0(727))$	60
Eisenstein series $E_{2}(\Gamma_0(727))$	1
Modular forms $M_{2}(\Gamma_0(727))$	61

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.239 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, 727]: [0, 61] (certified at ell = 2147482591)
T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (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(727, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 727:2 1000

← 726.2.a · 728.2.a →