## Defining parameters

 Level: $$N$$ = $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$24$$ Sturm bound: $$497664$$ Trace bound: $$52$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(1728))$$.

Total New Old
Modular forms 168048 74080 93968
Cusp forms 163728 73376 90352
Eisenstein series 4320 704 3616

## Trace form

 $$73376 q - 64 q^{2} - 72 q^{3} - 112 q^{4} - 64 q^{5} - 96 q^{6} - 84 q^{7} - 64 q^{8} - 120 q^{9} + O(q^{10})$$ $$73376 q - 64 q^{2} - 72 q^{3} - 112 q^{4} - 64 q^{5} - 96 q^{6} - 84 q^{7} - 64 q^{8} - 120 q^{9} - 112 q^{10} - 48 q^{11} - 96 q^{12} - 112 q^{13} - 64 q^{14} - 72 q^{15} - 112 q^{16} - 112 q^{17} - 96 q^{18} - 84 q^{19} - 64 q^{20} - 96 q^{21} - 112 q^{22} - 44 q^{23} - 96 q^{24} - 140 q^{25} - 64 q^{26} - 72 q^{27} - 256 q^{28} - 64 q^{29} - 96 q^{30} - 84 q^{31} - 64 q^{32} - 72 q^{33} - 112 q^{34} + 52 q^{35} - 96 q^{36} + 48 q^{37} - 64 q^{38} - 72 q^{39} - 112 q^{40} + 112 q^{41} - 96 q^{42} + 44 q^{43} - 64 q^{44} - 96 q^{45} - 112 q^{46} - 52 q^{47} - 96 q^{48} - 260 q^{49} - 64 q^{50} - 72 q^{51} - 112 q^{52} - 352 q^{53} - 96 q^{54} - 448 q^{55} - 64 q^{56} - 120 q^{57} - 112 q^{58} - 336 q^{59} - 96 q^{60} - 336 q^{61} - 128 q^{62} - 72 q^{63} - 112 q^{64} - 276 q^{65} - 96 q^{66} - 84 q^{67} - 64 q^{68} - 96 q^{69} - 112 q^{70} - 68 q^{71} - 96 q^{72} - 140 q^{73} - 64 q^{74} - 72 q^{75} - 112 q^{76} - 456 q^{77} - 96 q^{78} - 340 q^{79} - 1984 q^{80} - 168 q^{81} - 2336 q^{82} - 528 q^{83} - 96 q^{84} - 752 q^{85} - 1936 q^{86} - 72 q^{87} - 1232 q^{88} - 848 q^{89} - 96 q^{90} - 276 q^{91} - 976 q^{92} - 96 q^{93} - 304 q^{94} - 68 q^{95} - 96 q^{96} + 172 q^{97} + 752 q^{98} - 72 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(1728))$$

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
1728.3.b $$\chi_{1728}(1567, \cdot)$$ 1728.3.b.a 4 1
1728.3.b.b 4
1728.3.b.c 4
1728.3.b.d 4
1728.3.b.e 4
1728.3.b.f 4
1728.3.b.g 8
1728.3.b.h 8
1728.3.b.i 12
1728.3.b.j 12
1728.3.e $$\chi_{1728}(1025, \cdot)$$ 1728.3.e.a 1 1
1728.3.e.b 1
1728.3.e.c 1
1728.3.e.d 1
1728.3.e.e 2
1728.3.e.f 2
1728.3.e.g 2
1728.3.e.h 2
1728.3.e.i 2
1728.3.e.j 2
1728.3.e.k 2
1728.3.e.l 2
1728.3.e.m 2
1728.3.e.n 2
1728.3.e.o 4
1728.3.e.p 4
1728.3.e.q 4
1728.3.e.r 4
1728.3.e.s 4
1728.3.e.t 4
1728.3.e.u 8
1728.3.e.v 8
1728.3.g $$\chi_{1728}(703, \cdot)$$ 1728.3.g.a 2 1
1728.3.g.b 2
1728.3.g.c 2
1728.3.g.d 2
1728.3.g.e 2
1728.3.g.f 2
1728.3.g.g 4
1728.3.g.h 4
1728.3.g.i 4
1728.3.g.j 8
1728.3.g.k 8
1728.3.g.l 8
1728.3.g.m 8
1728.3.g.n 8
1728.3.h $$\chi_{1728}(161, \cdot)$$ 1728.3.h.a 4 1
1728.3.h.b 4
1728.3.h.c 4
1728.3.h.d 4
1728.3.h.e 4
1728.3.h.f 4
1728.3.h.g 8
1728.3.h.h 8
1728.3.h.i 12
1728.3.h.j 12
1728.3.j $$\chi_{1728}(593, \cdot)$$ n/a 128 2
1728.3.m $$\chi_{1728}(271, \cdot)$$ n/a 128 2
1728.3.n $$\chi_{1728}(737, \cdot)$$ 1728.3.n.a 8 2
1728.3.n.b 24
1728.3.n.c 32
1728.3.n.d 32
1728.3.o $$\chi_{1728}(127, \cdot)$$ 1728.3.o.a 2 2
1728.3.o.b 2
1728.3.o.c 4
1728.3.o.d 8
1728.3.o.e 8
1728.3.o.f 8
1728.3.o.g 16
1728.3.o.h 20
1728.3.o.i 24
1728.3.q $$\chi_{1728}(449, \cdot)$$ 1728.3.q.a 2 2
1728.3.q.b 2
1728.3.q.c 4
1728.3.q.d 4
1728.3.q.e 4
1728.3.q.f 4
1728.3.q.g 4
1728.3.q.h 4
1728.3.q.i 8
1728.3.q.j 8
1728.3.q.k 24
1728.3.q.l 24
1728.3.t $$\chi_{1728}(415, \cdot)$$ 1728.3.t.a 32 2
1728.3.t.b 32
1728.3.t.c 32
1728.3.u $$\chi_{1728}(55, \cdot)$$ None 0 4
1728.3.x $$\chi_{1728}(377, \cdot)$$ None 0 4
1728.3.ba $$\chi_{1728}(559, \cdot)$$ n/a 184 4
1728.3.bb $$\chi_{1728}(17, \cdot)$$ n/a 184 4
1728.3.bd $$\chi_{1728}(53, \cdot)$$ n/a 2048 8
1728.3.bg $$\chi_{1728}(163, \cdot)$$ n/a 2048 8
1728.3.bh $$\chi_{1728}(31, \cdot)$$ n/a 864 6
1728.3.bi $$\chi_{1728}(319, \cdot)$$ n/a 852 6
1728.3.bk $$\chi_{1728}(65, \cdot)$$ n/a 852 6
1728.3.bn $$\chi_{1728}(353, \cdot)$$ n/a 864 6
1728.3.bp $$\chi_{1728}(199, \cdot)$$ None 0 8
1728.3.bq $$\chi_{1728}(89, \cdot)$$ None 0 8
1728.3.bt $$\chi_{1728}(79, \cdot)$$ n/a 1704 12
1728.3.bu $$\chi_{1728}(113, \cdot)$$ n/a 1704 12
1728.3.bw $$\chi_{1728}(19, \cdot)$$ n/a 3040 16
1728.3.bz $$\chi_{1728}(125, \cdot)$$ n/a 3040 16
1728.3.ca $$\chi_{1728}(41, \cdot)$$ None 0 24
1728.3.cd $$\chi_{1728}(7, \cdot)$$ None 0 24
1728.3.cf $$\chi_{1728}(5, \cdot)$$ n/a 27552 48
1728.3.cg $$\chi_{1728}(43, \cdot)$$ n/a 27552 48

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(1728))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(1728)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 16}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 14}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 15}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 7}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(432))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(576))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(864))$$$$^{\oplus 2}$$