## Defining parameters

 Level: $$N$$ = $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$301056$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1568))$$.

Total New Old
Modular forms 77184 41111 36073
Cusp forms 73345 40141 33204
Eisenstein series 3839 970 2869

## Trace form

 $$40141 q - 124 q^{2} - 94 q^{3} - 124 q^{4} - 126 q^{5} - 124 q^{6} - 108 q^{7} - 220 q^{8} - 187 q^{9} + O(q^{10})$$ $$40141 q - 124 q^{2} - 94 q^{3} - 124 q^{4} - 126 q^{5} - 124 q^{6} - 108 q^{7} - 220 q^{8} - 187 q^{9} - 116 q^{10} - 94 q^{11} - 108 q^{12} - 118 q^{13} - 144 q^{14} - 162 q^{15} - 104 q^{16} - 58 q^{17} - 104 q^{18} - 94 q^{19} - 108 q^{20} - 144 q^{21} - 208 q^{22} - 86 q^{23} - 136 q^{24} - 185 q^{25} - 144 q^{26} - 58 q^{27} - 144 q^{28} - 230 q^{29} - 156 q^{30} - 74 q^{31} - 144 q^{32} - 308 q^{33} - 136 q^{34} - 108 q^{35} - 248 q^{36} - 126 q^{37} - 116 q^{38} - 34 q^{39} - 112 q^{40} - 174 q^{41} - 144 q^{42} - 134 q^{43} - 84 q^{44} - 10 q^{45} - 92 q^{46} - 18 q^{47} - 72 q^{48} - 24 q^{49} - 348 q^{50} + 22 q^{51} - 116 q^{52} + 2 q^{53} - 112 q^{54} - 42 q^{55} - 144 q^{56} - 136 q^{57} - 128 q^{58} - 30 q^{59} - 128 q^{60} - 6 q^{61} - 144 q^{62} - 48 q^{63} - 256 q^{64} - 224 q^{65} - 148 q^{66} - 62 q^{67} - 104 q^{68} + 4 q^{69} - 144 q^{70} - 174 q^{71} - 100 q^{72} - 190 q^{73} - 108 q^{74} - 46 q^{75} - 124 q^{76} - 144 q^{77} - 180 q^{78} - 90 q^{79} - 128 q^{80} - 123 q^{81} - 124 q^{82} - 42 q^{83} - 144 q^{84} - 204 q^{85} - 144 q^{86} - 2 q^{87} - 120 q^{88} - 174 q^{89} - 400 q^{90} - 150 q^{91} - 304 q^{92} - 232 q^{93} - 400 q^{94} - 106 q^{95} - 600 q^{96} - 482 q^{97} - 312 q^{98} - 360 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1568))$$

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
1568.2.a $$\chi_{1568}(1, \cdot)$$ 1568.2.a.a 1 1
1568.2.a.b 1
1568.2.a.c 1
1568.2.a.d 1
1568.2.a.e 1
1568.2.a.f 1
1568.2.a.g 1
1568.2.a.h 1
1568.2.a.i 1
1568.2.a.j 2
1568.2.a.k 2
1568.2.a.l 2
1568.2.a.m 2
1568.2.a.n 2
1568.2.a.o 2
1568.2.a.p 2
1568.2.a.q 2
1568.2.a.r 2
1568.2.a.s 2
1568.2.a.t 2
1568.2.a.u 2
1568.2.a.v 2
1568.2.a.w 2
1568.2.a.x 4
1568.2.b $$\chi_{1568}(785, \cdot)$$ 1568.2.b.a 2 1
1568.2.b.b 2
1568.2.b.c 4
1568.2.b.d 4
1568.2.b.e 6
1568.2.b.f 6
1568.2.b.g 12
1568.2.e $$\chi_{1568}(783, \cdot)$$ 1568.2.e.a 4 1
1568.2.e.b 4
1568.2.e.c 8
1568.2.e.d 8
1568.2.e.e 12
1568.2.f $$\chi_{1568}(1567, \cdot)$$ 1568.2.f.a 8 1
1568.2.f.b 16
1568.2.f.c 16
1568.2.i $$\chi_{1568}(961, \cdot)$$ 1568.2.i.a 2 2
1568.2.i.b 2
1568.2.i.c 2
1568.2.i.d 2
1568.2.i.e 2
1568.2.i.f 2
1568.2.i.g 2
1568.2.i.h 2
1568.2.i.i 2
1568.2.i.j 2
1568.2.i.k 2
1568.2.i.l 2
1568.2.i.m 4
1568.2.i.n 4
1568.2.i.o 4
1568.2.i.p 4
1568.2.i.q 4
1568.2.i.r 4
1568.2.i.s 4
1568.2.i.t 4
1568.2.i.u 4
1568.2.i.v 4
1568.2.i.w 4
1568.2.i.x 4
1568.2.i.y 8
1568.2.j $$\chi_{1568}(391, \cdot)$$ None 0 2
1568.2.m $$\chi_{1568}(393, \cdot)$$ None 0 2
1568.2.p $$\chi_{1568}(31, \cdot)$$ 1568.2.p.a 16 2
1568.2.p.b 16
1568.2.p.c 16
1568.2.p.d 32
1568.2.q $$\chi_{1568}(815, \cdot)$$ 1568.2.q.a 4 2
1568.2.q.b 8
1568.2.q.c 8
1568.2.q.d 8
1568.2.q.e 8
1568.2.q.f 8
1568.2.q.g 12
1568.2.q.h 16
1568.2.t $$\chi_{1568}(177, \cdot)$$ 1568.2.t.a 4 2
1568.2.t.b 4
1568.2.t.c 4
1568.2.t.d 8
1568.2.t.e 8
1568.2.t.f 8
1568.2.t.g 12
1568.2.t.h 24
1568.2.u $$\chi_{1568}(225, \cdot)$$ n/a 336 6
1568.2.v $$\chi_{1568}(197, \cdot)$$ n/a 636 4
1568.2.y $$\chi_{1568}(195, \cdot)$$ n/a 624 4
1568.2.ba $$\chi_{1568}(215, \cdot)$$ None 0 4
1568.2.bb $$\chi_{1568}(361, \cdot)$$ None 0 4
1568.2.bf $$\chi_{1568}(223, \cdot)$$ n/a 336 6
1568.2.bg $$\chi_{1568}(111, \cdot)$$ n/a 324 6
1568.2.bj $$\chi_{1568}(113, \cdot)$$ n/a 324 6
1568.2.bk $$\chi_{1568}(65, \cdot)$$ n/a 672 12
1568.2.bm $$\chi_{1568}(165, \cdot)$$ n/a 1248 8
1568.2.bn $$\chi_{1568}(19, \cdot)$$ n/a 1248 8
1568.2.bp $$\chi_{1568}(57, \cdot)$$ None 0 12
1568.2.bs $$\chi_{1568}(55, \cdot)$$ None 0 12
1568.2.bt $$\chi_{1568}(81, \cdot)$$ n/a 648 12
1568.2.bw $$\chi_{1568}(47, \cdot)$$ n/a 648 12
1568.2.bx $$\chi_{1568}(159, \cdot)$$ n/a 672 12
1568.2.ca $$\chi_{1568}(27, \cdot)$$ n/a 5328 24
1568.2.cd $$\chi_{1568}(29, \cdot)$$ n/a 5328 24
1568.2.cf $$\chi_{1568}(9, \cdot)$$ None 0 24
1568.2.cg $$\chi_{1568}(87, \cdot)$$ None 0 24
1568.2.cj $$\chi_{1568}(3, \cdot)$$ n/a 10656 48
1568.2.ck $$\chi_{1568}(37, \cdot)$$ n/a 10656 48

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1568))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1568)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(224))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(392))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(784))$$$$^{\oplus 2}$$