# Properties

 Label 540.2 Level 540 Weight 2 Dimension 3190 Nonzero newspaces 18 Newform subspaces 41 Sturm bound 31104 Trace bound 5

## Defining parameters

 Level: $$N$$ = $$540 = 2^{2} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$41$$ Sturm bound: $$31104$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(540))$$.

Total New Old
Modular forms 8376 3382 4994
Cusp forms 7177 3190 3987
Eisenstein series 1199 192 1007

## Trace form

 $$3190 q - 10 q^{2} - 18 q^{4} - 32 q^{5} - 36 q^{6} - 12 q^{7} + 2 q^{8} - 36 q^{9} + O(q^{10})$$ $$3190 q - 10 q^{2} - 18 q^{4} - 32 q^{5} - 36 q^{6} - 12 q^{7} + 2 q^{8} - 36 q^{9} - 15 q^{10} - 20 q^{11} + 6 q^{12} - 32 q^{13} + 54 q^{14} + 9 q^{15} + 2 q^{16} + 40 q^{17} + 30 q^{18} + 10 q^{19} + 45 q^{20} - 12 q^{21} + 30 q^{22} + 78 q^{23} - 30 q^{25} + 54 q^{27} - 20 q^{28} + 102 q^{29} - 33 q^{30} + 16 q^{31} - 90 q^{32} + 84 q^{33} - 42 q^{34} + 99 q^{35} - 96 q^{36} + 46 q^{37} - 86 q^{38} + 54 q^{39} - 23 q^{40} + 108 q^{41} - 132 q^{42} + 64 q^{43} - 162 q^{44} - 9 q^{45} - 82 q^{46} + 36 q^{47} - 162 q^{48} + 96 q^{49} - 92 q^{50} - 78 q^{51} - 18 q^{52} - 32 q^{53} - 180 q^{54} + 22 q^{55} - 250 q^{56} - 72 q^{57} - 42 q^{58} - 74 q^{59} - 126 q^{60} - 56 q^{61} - 248 q^{62} - 114 q^{63} - 102 q^{64} - 76 q^{65} - 222 q^{66} + 18 q^{67} - 280 q^{68} - 54 q^{69} - 135 q^{70} - 48 q^{71} - 144 q^{72} - 120 q^{73} - 282 q^{74} - 69 q^{75} - 182 q^{76} - 96 q^{77} - 132 q^{78} + 16 q^{79} - 336 q^{80} - 108 q^{81} - 140 q^{82} - 120 q^{83} - 228 q^{84} - 72 q^{85} - 290 q^{86} + 6 q^{87} - 162 q^{88} - 232 q^{89} - 174 q^{90} - 112 q^{91} - 118 q^{92} - 246 q^{93} - 138 q^{94} - 226 q^{95} + 12 q^{96} - 26 q^{97} - 56 q^{98} - 174 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(540))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
540.2.a $$\chi_{540}(1, \cdot)$$ 540.2.a.a 1 1
540.2.a.b 1
540.2.a.c 1
540.2.a.d 1
540.2.a.e 1
540.2.a.f 1
540.2.d $$\chi_{540}(109, \cdot)$$ 540.2.d.a 2 1
540.2.d.b 2
540.2.d.c 4
540.2.e $$\chi_{540}(431, \cdot)$$ 540.2.e.a 16 1
540.2.e.b 16
540.2.h $$\chi_{540}(539, \cdot)$$ 540.2.h.a 4 1
540.2.h.b 4
540.2.h.c 16
540.2.h.d 24
540.2.i $$\chi_{540}(181, \cdot)$$ 540.2.i.a 2 2
540.2.i.b 6
540.2.j $$\chi_{540}(53, \cdot)$$ 540.2.j.a 8 2
540.2.j.b 8
540.2.k $$\chi_{540}(163, \cdot)$$ 540.2.k.a 4 2
540.2.k.b 4
540.2.k.c 4
540.2.k.d 4
540.2.k.e 32
540.2.k.f 48
540.2.n $$\chi_{540}(179, \cdot)$$ 540.2.n.a 4 2
540.2.n.b 4
540.2.n.c 8
540.2.n.d 48
540.2.q $$\chi_{540}(71, \cdot)$$ 540.2.q.a 48 2
540.2.r $$\chi_{540}(289, \cdot)$$ 540.2.r.a 12 2
540.2.u $$\chi_{540}(61, \cdot)$$ 540.2.u.a 30 6
540.2.u.b 42
540.2.x $$\chi_{540}(17, \cdot)$$ 540.2.x.a 24 4
540.2.y $$\chi_{540}(127, \cdot)$$ 540.2.y.a 128 4
540.2.bb $$\chi_{540}(59, \cdot)$$ 540.2.bb.a 24 6
540.2.bb.b 600
540.2.bd $$\chi_{540}(49, \cdot)$$ 540.2.bd.a 108 6
540.2.be $$\chi_{540}(11, \cdot)$$ 540.2.be.a 432 6
540.2.bh $$\chi_{540}(7, \cdot)$$ 540.2.bh.a 1248 12
540.2.bj $$\chi_{540}(77, \cdot)$$ 540.2.bj.a 216 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(540))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(540)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(540))$$$$^{\oplus 1}$$