## Defining parameters

 Level: $$N$$ = $$539 = 7^{2} \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newform subspaces: $$70$$ Sturm bound: $$47040$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(539))$$.

Total New Old
Modular forms 12360 11534 826
Cusp forms 11161 10662 499
Eisenstein series 1199 872 327

## Trace form

 $$10662q - 119q^{2} - 121q^{3} - 127q^{4} - 125q^{5} - 142q^{6} - 152q^{7} - 237q^{8} - 149q^{9} + O(q^{10})$$ $$10662q - 119q^{2} - 121q^{3} - 127q^{4} - 125q^{5} - 142q^{6} - 152q^{7} - 237q^{8} - 149q^{9} - 164q^{10} - 151q^{11} - 344q^{12} - 146q^{13} - 180q^{14} - 265q^{15} - 205q^{16} - 164q^{17} - 221q^{18} - 168q^{19} - 232q^{20} - 194q^{21} - 302q^{22} - 331q^{23} - 288q^{24} - 205q^{25} - 242q^{26} - 223q^{27} - 236q^{28} - 282q^{29} - 322q^{30} - 197q^{31} - 289q^{32} - 222q^{33} - 416q^{34} - 222q^{35} - 361q^{36} - 163q^{37} - 204q^{38} - 182q^{39} - 126q^{40} - 158q^{41} - 138q^{42} - 246q^{43} - 162q^{44} - 284q^{45} - 70q^{46} - 160q^{47} - 126q^{48} - 68q^{49} - 463q^{50} - 154q^{51} - 86q^{52} - 192q^{53} - 226q^{54} - 161q^{55} - 276q^{56} - 348q^{57} - 210q^{58} - 199q^{59} - 2q^{60} - 124q^{61} - 206q^{62} - 174q^{63} - 197q^{64} - 176q^{65} + 77q^{66} - 349q^{67} - 130q^{68} - 95q^{69} - 210q^{70} - 171q^{71} + 27q^{72} - 146q^{73} - 216q^{74} + 4q^{75} + 132q^{76} - 153q^{77} - 262q^{78} - 178q^{79} + 56q^{80} + 178q^{81} - 20q^{82} - 18q^{83} + 292q^{84} - 110q^{85} + 162q^{86} + 168q^{87} + 117q^{88} - 225q^{89} + 472q^{90} - 76q^{91} - 164q^{92} + 185q^{93} - 58q^{94} - 72q^{95} + 406q^{96} - 199q^{97} + 156q^{98} - 553q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(539))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
539.2.a $$\chi_{539}(1, \cdot)$$ 539.2.a.a 1 1
539.2.a.b 1
539.2.a.c 1
539.2.a.d 1
539.2.a.e 2
539.2.a.f 2
539.2.a.g 3
539.2.a.h 3
539.2.a.i 3
539.2.a.j 3
539.2.a.k 4
539.2.a.l 10
539.2.b $$\chi_{539}(538, \cdot)$$ 539.2.b.a 8 1
539.2.b.b 12
539.2.b.c 16
539.2.e $$\chi_{539}(67, \cdot)$$ 539.2.e.a 2 2
539.2.e.b 2
539.2.e.c 2
539.2.e.d 2
539.2.e.e 2
539.2.e.f 2
539.2.e.g 2
539.2.e.h 2
539.2.e.i 4
539.2.e.j 4
539.2.e.k 4
539.2.e.l 6
539.2.e.m 6
539.2.e.n 8
539.2.e.o 20
539.2.f $$\chi_{539}(148, \cdot)$$ 539.2.f.a 4 4
539.2.f.b 4
539.2.f.c 8
539.2.f.d 8
539.2.f.e 16
539.2.f.f 16
539.2.f.g 20
539.2.f.h 20
539.2.f.i 48
539.2.i $$\chi_{539}(362, \cdot)$$ 539.2.i.a 4 2
539.2.i.b 8
539.2.i.c 12
539.2.i.d 16
539.2.i.e 32
539.2.j $$\chi_{539}(78, \cdot)$$ 539.2.j.a 144 6
539.2.j.b 144
539.2.m $$\chi_{539}(195, \cdot)$$ 539.2.m.a 48 4
539.2.m.b 96
539.2.p $$\chi_{539}(76, \cdot)$$ 539.2.p.a 324 6
539.2.q $$\chi_{539}(214, \cdot)$$ 539.2.q.a 8 8
539.2.q.b 16
539.2.q.c 16
539.2.q.d 16
539.2.q.e 32
539.2.q.f 32
539.2.q.g 32
539.2.q.h 40
539.2.q.i 96
539.2.r $$\chi_{539}(23, \cdot)$$ 539.2.r.a 276 12
539.2.r.b 276
539.2.s $$\chi_{539}(19, \cdot)$$ 539.2.s.a 16 8
539.2.s.b 16
539.2.s.c 16
539.2.s.d 48
539.2.s.e 192
539.2.v $$\chi_{539}(15, \cdot)$$ 539.2.v.a 1296 24
539.2.w $$\chi_{539}(10, \cdot)$$ 539.2.w.a 648 12
539.2.z $$\chi_{539}(6, \cdot)$$ 539.2.z.a 1296 24
539.2.bc $$\chi_{539}(4, \cdot)$$ 539.2.bc.a 2592 48
539.2.bf $$\chi_{539}(17, \cdot)$$ 539.2.bf.a 2592 48

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(539))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(539)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 2}$$