## Defining parameters

 Level: $$N$$ = $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$331776$$ Trace bound: $$52$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1728))$$.

Total New Old
Modular forms 85104 37216 47888
Cusp forms 80785 36512 44273
Eisenstein series 4319 704 3615

## Trace form

 $$36512 q - 64 q^{2} - 72 q^{3} - 112 q^{4} - 64 q^{5} - 96 q^{6} - 84 q^{7} - 64 q^{8} - 120 q^{9} + O(q^{10})$$ $$36512 q - 64 q^{2} - 72 q^{3} - 112 q^{4} - 64 q^{5} - 96 q^{6} - 84 q^{7} - 64 q^{8} - 120 q^{9} - 112 q^{10} - 48 q^{11} - 96 q^{12} - 112 q^{13} - 64 q^{14} - 72 q^{15} - 112 q^{16} - 112 q^{17} - 96 q^{18} - 84 q^{19} - 64 q^{20} - 96 q^{21} - 112 q^{22} - 52 q^{23} - 96 q^{24} - 140 q^{25} - 64 q^{26} - 72 q^{27} - 256 q^{28} - 64 q^{29} - 96 q^{30} - 84 q^{31} - 64 q^{32} - 72 q^{33} - 112 q^{34} - 68 q^{35} - 96 q^{36} - 128 q^{37} - 64 q^{38} - 72 q^{39} - 112 q^{40} - 112 q^{41} - 96 q^{42} - 116 q^{43} - 64 q^{44} - 96 q^{45} - 112 q^{46} - 84 q^{47} - 96 q^{48} - 228 q^{49} - 64 q^{50} - 72 q^{51} - 112 q^{52} - 112 q^{53} - 96 q^{54} - 224 q^{55} - 64 q^{56} - 120 q^{57} - 112 q^{58} - 72 q^{59} - 96 q^{60} - 128 q^{61} - 32 q^{62} - 72 q^{63} - 112 q^{64} - 196 q^{65} - 96 q^{66} - 84 q^{67} - 64 q^{68} - 96 q^{69} - 112 q^{70} - 28 q^{71} - 96 q^{72} - 140 q^{73} - 64 q^{74} - 72 q^{75} - 112 q^{76} - 8 q^{77} - 96 q^{78} - 68 q^{79} + 64 q^{80} - 168 q^{81} - 96 q^{82} - 8 q^{83} - 96 q^{84} - 48 q^{85} + 144 q^{86} - 72 q^{87} + 48 q^{88} + 48 q^{89} - 96 q^{90} - 36 q^{91} + 240 q^{92} - 96 q^{93} + 80 q^{94} + 36 q^{95} - 96 q^{96} + 44 q^{97} + 208 q^{98} - 72 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1728))$$

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
1728.2.a $$\chi_{1728}(1, \cdot)$$ 1728.2.a.a 1 1
1728.2.a.b 1
1728.2.a.c 1
1728.2.a.d 1
1728.2.a.e 1
1728.2.a.f 1
1728.2.a.g 1
1728.2.a.h 1
1728.2.a.i 1
1728.2.a.j 1
1728.2.a.k 1
1728.2.a.l 1
1728.2.a.m 1
1728.2.a.n 1
1728.2.a.o 1
1728.2.a.p 1
1728.2.a.q 1
1728.2.a.r 1
1728.2.a.s 1
1728.2.a.t 1
1728.2.a.u 1
1728.2.a.v 1
1728.2.a.w 1
1728.2.a.x 1
1728.2.a.y 1
1728.2.a.z 1
1728.2.a.ba 1
1728.2.a.bb 1
1728.2.a.bc 2
1728.2.a.bd 2
1728.2.c $$\chi_{1728}(1727, \cdot)$$ 1728.2.c.a 2 1
1728.2.c.b 2
1728.2.c.c 4
1728.2.c.d 4
1728.2.c.e 4
1728.2.c.f 8
1728.2.c.g 8
1728.2.d $$\chi_{1728}(865, \cdot)$$ 1728.2.d.a 2 1
1728.2.d.b 2
1728.2.d.c 2
1728.2.d.d 2
1728.2.d.e 4
1728.2.d.f 4
1728.2.d.g 4
1728.2.d.h 4
1728.2.d.i 4
1728.2.d.j 4
1728.2.f $$\chi_{1728}(863, \cdot)$$ 1728.2.f.a 4 1
1728.2.f.b 4
1728.2.f.c 4
1728.2.f.d 4
1728.2.f.e 4
1728.2.f.f 4
1728.2.f.g 8
1728.2.i $$\chi_{1728}(577, \cdot)$$ 1728.2.i.a 2 2
1728.2.i.b 2
1728.2.i.c 2
1728.2.i.d 2
1728.2.i.e 2
1728.2.i.f 2
1728.2.i.g 2
1728.2.i.h 2
1728.2.i.i 4
1728.2.i.j 4
1728.2.i.k 4
1728.2.i.l 4
1728.2.i.m 4
1728.2.i.n 8
1728.2.k $$\chi_{1728}(433, \cdot)$$ 1728.2.k.a 4 2
1728.2.k.b 4
1728.2.k.c 24
1728.2.k.d 32
1728.2.l $$\chi_{1728}(431, \cdot)$$ 1728.2.l.a 32 2
1728.2.l.b 32
1728.2.p $$\chi_{1728}(287, \cdot)$$ 1728.2.p.a 16 2
1728.2.p.b 16
1728.2.p.c 16
1728.2.r $$\chi_{1728}(289, \cdot)$$ 1728.2.r.a 4 2
1728.2.r.b 4
1728.2.r.c 8
1728.2.r.d 8
1728.2.r.e 12
1728.2.r.f 12
1728.2.s $$\chi_{1728}(575, \cdot)$$ 1728.2.s.a 2 2
1728.2.s.b 2
1728.2.s.c 2
1728.2.s.d 2
1728.2.s.e 4
1728.2.s.f 8
1728.2.s.g 24
1728.2.v $$\chi_{1728}(217, \cdot)$$ None 0 4
1728.2.w $$\chi_{1728}(215, \cdot)$$ None 0 4
1728.2.y $$\chi_{1728}(193, \cdot)$$ n/a 420 6
1728.2.z $$\chi_{1728}(143, \cdot)$$ 1728.2.z.a 88 4
1728.2.bc $$\chi_{1728}(145, \cdot)$$ 1728.2.bc.a 4 4
1728.2.bc.b 4
1728.2.bc.c 4
1728.2.bc.d 4
1728.2.bc.e 72
1728.2.be $$\chi_{1728}(109, \cdot)$$ n/a 1024 8
1728.2.bf $$\chi_{1728}(107, \cdot)$$ n/a 1024 8
1728.2.bj $$\chi_{1728}(97, \cdot)$$ n/a 432 6
1728.2.bl $$\chi_{1728}(95, \cdot)$$ n/a 432 6
1728.2.bm $$\chi_{1728}(191, \cdot)$$ n/a 420 6
1728.2.bo $$\chi_{1728}(73, \cdot)$$ None 0 8
1728.2.br $$\chi_{1728}(71, \cdot)$$ None 0 8
1728.2.bs $$\chi_{1728}(49, \cdot)$$ n/a 840 12
1728.2.bv $$\chi_{1728}(47, \cdot)$$ n/a 840 12
1728.2.bx $$\chi_{1728}(35, \cdot)$$ n/a 1504 16
1728.2.by $$\chi_{1728}(37, \cdot)$$ n/a 1504 16
1728.2.cb $$\chi_{1728}(23, \cdot)$$ None 0 24
1728.2.cc $$\chi_{1728}(25, \cdot)$$ None 0 24
1728.2.ce $$\chi_{1728}(13, \cdot)$$ n/a 13728 48
1728.2.ch $$\chi_{1728}(11, \cdot)$$ n/a 13728 48

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1728))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1728)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 7}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(432))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(576))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(864))$$$$^{\oplus 2}$$