## Defining parameters

 Level: $$N$$ = $$540 = 2^{2} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$18$$ Sturm bound: $$46656$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(540))$$.

Total New Old
Modular forms 16152 6666 9486
Cusp forms 14952 6474 8478
Eisenstein series 1200 192 1008

## Trace form

 $$6474 q - 6 q^{2} - 10 q^{4} - 4 q^{5} - 36 q^{6} + 8 q^{7} - 54 q^{8} - 36 q^{9} + O(q^{10})$$ $$6474 q - 6 q^{2} - 10 q^{4} - 4 q^{5} - 36 q^{6} + 8 q^{7} - 54 q^{8} - 36 q^{9} - 43 q^{10} - 120 q^{11} - 102 q^{12} - 148 q^{13} - 146 q^{14} - 45 q^{15} - 62 q^{16} - 120 q^{17} + 30 q^{18} - 2 q^{19} + 41 q^{20} - 156 q^{21} + 94 q^{22} + 102 q^{23} + 252 q^{24} + 40 q^{25} + 540 q^{26} + 300 q^{28} - 302 q^{29} + 129 q^{30} - 92 q^{31} + 534 q^{32} + 30 q^{33} + 222 q^{34} - 153 q^{35} - 96 q^{36} + 14 q^{37} - 198 q^{38} - 54 q^{39} - 95 q^{40} + 652 q^{41} + 228 q^{42} - 20 q^{43} - 210 q^{44} + 477 q^{45} - 498 q^{46} + 528 q^{47} + 414 q^{48} + 246 q^{49} + 372 q^{50} + 534 q^{51} + 78 q^{52} + 852 q^{53} - 180 q^{54} + 272 q^{55} + 478 q^{56} + 234 q^{57} + 910 q^{58} + 108 q^{59} - 234 q^{60} + 652 q^{61} + 360 q^{62} - 762 q^{63} + 386 q^{64} + 162 q^{65} - 834 q^{66} - 238 q^{67} - 252 q^{68} - 1098 q^{69} - 127 q^{70} - 336 q^{71} + 288 q^{72} + 228 q^{73} - 42 q^{74} + 561 q^{75} - 278 q^{76} - 120 q^{77} + 192 q^{78} + 100 q^{79} + 938 q^{80} + 324 q^{81} - 292 q^{82} + 840 q^{83} - 408 q^{84} + 88 q^{85} - 1058 q^{86} + 1122 q^{87} - 290 q^{88} + 586 q^{89} - 1146 q^{90} - 28 q^{91} - 1686 q^{92} + 2814 q^{93} + 310 q^{94} - 12 q^{95} - 2508 q^{96} + 122 q^{97} - 2232 q^{98} + 690 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(540))$$

We only show spaces with odd 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
540.3.b $$\chi_{540}(269, \cdot)$$ 540.3.b.a 4 1
540.3.b.b 4
540.3.b.c 8
540.3.c $$\chi_{540}(271, \cdot)$$ 540.3.c.a 32 1
540.3.c.b 32
540.3.f $$\chi_{540}(379, \cdot)$$ 540.3.f.a 8 1
540.3.f.b 40
540.3.f.c 48
540.3.g $$\chi_{540}(161, \cdot)$$ 540.3.g.a 2 1
540.3.g.b 2
540.3.g.c 2
540.3.g.d 4
540.3.l $$\chi_{540}(217, \cdot)$$ 540.3.l.a 8 2
540.3.l.b 8
540.3.l.c 16
540.3.m $$\chi_{540}(107, \cdot)$$ n/a 192 2
540.3.o $$\chi_{540}(341, \cdot)$$ 540.3.o.a 4 2
540.3.o.b 12
540.3.p $$\chi_{540}(19, \cdot)$$ n/a 136 2
540.3.s $$\chi_{540}(91, \cdot)$$ 540.3.s.a 96 2
540.3.t $$\chi_{540}(89, \cdot)$$ 540.3.t.a 24 2
540.3.v $$\chi_{540}(37, \cdot)$$ 540.3.v.a 48 4
540.3.w $$\chi_{540}(143, \cdot)$$ n/a 272 4
540.3.z $$\chi_{540}(29, \cdot)$$ n/a 216 6
540.3.ba $$\chi_{540}(31, \cdot)$$ n/a 864 6
540.3.bc $$\chi_{540}(41, \cdot)$$ n/a 144 6
540.3.bf $$\chi_{540}(79, \cdot)$$ n/a 1272 6
540.3.bg $$\chi_{540}(23, \cdot)$$ n/a 2544 12
540.3.bi $$\chi_{540}(13, \cdot)$$ n/a 432 12

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(540))$$ into lower level spaces

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