## Defining parameters

 Level: $$N$$ = $$896 = 2^{7} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$20$$ Newform subspaces: $$88$$ Sturm bound: $$98304$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(896))$$.

Total New Old
Modular forms 25536 13488 12048
Cusp forms 23617 13008 10609
Eisenstein series 1919 480 1439

## Trace form

 $$13008q - 64q^{2} - 48q^{3} - 64q^{4} - 64q^{5} - 64q^{6} - 60q^{7} - 160q^{8} - 80q^{9} + O(q^{10})$$ $$13008q - 64q^{2} - 48q^{3} - 64q^{4} - 64q^{5} - 64q^{6} - 60q^{7} - 160q^{8} - 80q^{9} - 64q^{10} - 48q^{11} - 64q^{12} - 64q^{13} - 80q^{14} - 112q^{15} - 64q^{16} - 96q^{17} - 64q^{18} - 48q^{19} - 64q^{20} - 68q^{21} - 160q^{22} - 32q^{23} - 64q^{24} - 48q^{25} - 64q^{26} - 80q^{28} - 128q^{29} - 64q^{30} - 8q^{31} - 64q^{32} - 64q^{33} - 64q^{34} - 36q^{35} - 160q^{36} - 32q^{37} - 64q^{38} - 64q^{40} - 48q^{41} - 80q^{42} - 104q^{43} - 64q^{44} - 72q^{45} - 64q^{46} - 40q^{47} - 64q^{48} - 120q^{49} - 256q^{50} - 56q^{51} - 256q^{52} - 128q^{53} - 320q^{54} - 112q^{55} - 192q^{56} - 328q^{57} - 352q^{58} - 112q^{59} - 448q^{60} - 192q^{61} - 256q^{62} - 120q^{63} - 544q^{64} - 192q^{65} - 448q^{66} - 128q^{67} - 256q^{68} - 168q^{69} - 272q^{70} - 184q^{71} - 352q^{72} - 208q^{73} - 288q^{74} - 72q^{75} - 320q^{76} - 84q^{77} - 352q^{78} - 40q^{79} - 160q^{80} - 104q^{81} - 64q^{82} + 32q^{83} - 80q^{84} - 176q^{85} - 64q^{86} + 64q^{87} - 64q^{88} + 48q^{89} - 64q^{90} - 12q^{91} - 160q^{92} + 32q^{93} - 64q^{94} + 72q^{95} - 64q^{96} - 80q^{98} - 16q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(896))$$

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
896.2.a $$\chi_{896}(1, \cdot)$$ 896.2.a.a 1 1
896.2.a.b 1
896.2.a.c 1
896.2.a.d 1
896.2.a.e 2
896.2.a.f 2
896.2.a.g 2
896.2.a.h 2
896.2.a.i 3
896.2.a.j 3
896.2.a.k 3
896.2.a.l 3
896.2.b $$\chi_{896}(449, \cdot)$$ 896.2.b.a 2 1
896.2.b.b 2
896.2.b.c 2
896.2.b.d 2
896.2.b.e 4
896.2.b.f 4
896.2.b.g 4
896.2.b.h 4
896.2.e $$\chi_{896}(447, \cdot)$$ 896.2.e.a 4 1
896.2.e.b 4
896.2.e.c 4
896.2.e.d 4
896.2.e.e 4
896.2.e.f 4
896.2.e.g 8
896.2.f $$\chi_{896}(895, \cdot)$$ 896.2.f.a 8 1
896.2.f.b 8
896.2.f.c 8
896.2.f.d 8
896.2.i $$\chi_{896}(513, \cdot)$$ 896.2.i.a 8 2
896.2.i.b 8
896.2.i.c 8
896.2.i.d 8
896.2.i.e 8
896.2.i.f 8
896.2.i.g 8
896.2.i.h 8
896.2.j $$\chi_{896}(223, \cdot)$$ 896.2.j.a 4 2
896.2.j.b 4
896.2.j.c 4
896.2.j.d 4
896.2.j.e 4
896.2.j.f 4
896.2.j.g 16
896.2.j.h 16
896.2.m $$\chi_{896}(225, \cdot)$$ 896.2.m.a 2 2
896.2.m.b 2
896.2.m.c 2
896.2.m.d 2
896.2.m.e 8
896.2.m.f 8
896.2.m.g 12
896.2.m.h 12
896.2.p $$\chi_{896}(255, \cdot)$$ 896.2.p.a 16 2
896.2.p.b 16
896.2.p.c 16
896.2.p.d 16
896.2.q $$\chi_{896}(703, \cdot)$$ 896.2.q.a 16 2
896.2.q.b 16
896.2.q.c 16
896.2.q.d 16
896.2.t $$\chi_{896}(65, \cdot)$$ 896.2.t.a 16 2
896.2.t.b 16
896.2.t.c 16
896.2.t.d 16
896.2.u $$\chi_{896}(113, \cdot)$$ 896.2.u.a 4 4
896.2.u.b 40
896.2.u.c 52
896.2.x $$\chi_{896}(111, \cdot)$$ 896.2.x.a 8 4
896.2.x.b 112
896.2.z $$\chi_{896}(31, \cdot)$$ 896.2.z.a 56 4
896.2.z.b 56
896.2.ba $$\chi_{896}(289, \cdot)$$ 896.2.ba.a 4 4
896.2.ba.b 4
896.2.ba.c 4
896.2.ba.d 4
896.2.ba.e 48
896.2.ba.f 48
896.2.bc $$\chi_{896}(57, \cdot)$$ None 0 8
896.2.bd $$\chi_{896}(55, \cdot)$$ None 0 8
896.2.bh $$\chi_{896}(81, \cdot)$$ 896.2.bh.a 240 8
896.2.bi $$\chi_{896}(47, \cdot)$$ 896.2.bi.a 240 8
896.2.bk $$\chi_{896}(27, \cdot)$$ 896.2.bk.a 32 16
896.2.bk.b 1984
896.2.bn $$\chi_{896}(29, \cdot)$$ 896.2.bn.a 752 16
896.2.bn.b 784
896.2.bq $$\chi_{896}(87, \cdot)$$ None 0 16
896.2.br $$\chi_{896}(9, \cdot)$$ None 0 16
896.2.bt $$\chi_{896}(3, \cdot)$$ 896.2.bt.a 4032 32
896.2.bu $$\chi_{896}(37, \cdot)$$ 896.2.bu.a 4032 32

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(896))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(896)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 7}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(224))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(448))$$$$^{\oplus 2}$$