## Defining parameters

 Level: $$N$$ = $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newforms: $$60$$ Sturm bound: $$24576$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(448))$$.

Total New Old
Modular forms 6576 3422 3154
Cusp forms 5713 3202 2511
Eisenstein series 863 220 643

## Trace form

 $$3202q - 32q^{2} - 24q^{3} - 32q^{4} - 32q^{5} - 32q^{6} - 28q^{7} - 80q^{8} - 34q^{9} + O(q^{10})$$ $$3202q - 32q^{2} - 24q^{3} - 32q^{4} - 32q^{5} - 32q^{6} - 28q^{7} - 80q^{8} - 34q^{9} - 32q^{10} - 16q^{11} - 32q^{12} - 16q^{13} - 40q^{14} - 48q^{15} - 32q^{16} - 40q^{17} - 32q^{18} - 8q^{19} - 32q^{20} - 44q^{21} - 96q^{22} - 20q^{23} - 112q^{24} - 62q^{25} - 112q^{26} - 36q^{27} - 80q^{28} - 112q^{29} - 192q^{30} - 68q^{31} - 112q^{32} - 76q^{33} - 112q^{34} - 32q^{35} - 240q^{36} - 48q^{37} - 112q^{38} - 20q^{39} - 112q^{40} - 40q^{41} - 80q^{42} - 36q^{43} - 48q^{44} - 24q^{45} - 32q^{46} + 12q^{47} - 32q^{48} - 54q^{49} - 32q^{50} - 76q^{51} + 64q^{52} + 16q^{53} + 96q^{54} - 160q^{55} + 16q^{56} - 88q^{57} + 112q^{58} - 160q^{59} + 160q^{60} - 16q^{61} + 32q^{62} - 96q^{63} + 112q^{64} - 84q^{65} + 128q^{66} - 200q^{67} + 64q^{68} - 8q^{69} + 56q^{70} - 184q^{71} + 112q^{72} - 40q^{73} + 80q^{74} - 136q^{75} + 96q^{76} - 44q^{77} - 32q^{78} - 84q^{79} - 112q^{80} - 98q^{81} - 192q^{82} - 24q^{83} - 152q^{84} - 96q^{85} - 240q^{86} - 20q^{87} - 192q^{88} - 136q^{89} - 320q^{90} - 108q^{91} - 384q^{92} - 160q^{93} - 224q^{94} - 132q^{95} - 304q^{96} - 200q^{97} - 176q^{98} - 228q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(448))$$

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
448.2.a $$\chi_{448}(1, \cdot)$$ 448.2.a.a 1 1
448.2.a.b 1
448.2.a.c 1
448.2.a.d 1
448.2.a.e 1
448.2.a.f 1
448.2.a.g 1
448.2.a.h 1
448.2.a.i 2
448.2.a.j 2
448.2.b $$\chi_{448}(225, \cdot)$$ 448.2.b.a 2 1
448.2.b.b 2
448.2.b.c 4
448.2.b.d 4
448.2.e $$\chi_{448}(223, \cdot)$$ 448.2.e.a 8 1
448.2.e.b 8
448.2.f $$\chi_{448}(447, \cdot)$$ 448.2.f.a 2 1
448.2.f.b 2
448.2.f.c 2
448.2.f.d 8
448.2.i $$\chi_{448}(65, \cdot)$$ 448.2.i.a 2 2
448.2.i.b 2
448.2.i.c 2
448.2.i.d 2
448.2.i.e 2
448.2.i.f 2
448.2.i.g 4
448.2.i.h 4
448.2.i.i 4
448.2.i.j 4
448.2.j $$\chi_{448}(111, \cdot)$$ 448.2.j.a 4 2
448.2.j.b 4
448.2.j.c 4
448.2.j.d 16
448.2.m $$\chi_{448}(113, \cdot)$$ 448.2.m.a 2 2
448.2.m.b 2
448.2.m.c 8
448.2.m.d 12
448.2.p $$\chi_{448}(255, \cdot)$$ 448.2.p.a 2 2
448.2.p.b 2
448.2.p.c 4
448.2.p.d 4
448.2.p.e 16
448.2.q $$\chi_{448}(31, \cdot)$$ 448.2.q.a 8 2
448.2.q.b 12
448.2.q.c 12
448.2.t $$\chi_{448}(289, \cdot)$$ 448.2.t.a 8 2
448.2.t.b 12
448.2.t.c 12
448.2.u $$\chi_{448}(57, \cdot)$$ None 0 4
448.2.x $$\chi_{448}(55, \cdot)$$ None 0 4
448.2.z $$\chi_{448}(47, \cdot)$$ 448.2.z.a 56 4
448.2.ba $$\chi_{448}(81, \cdot)$$ 448.2.ba.a 4 4
448.2.ba.b 4
448.2.ba.c 48
448.2.bc $$\chi_{448}(29, \cdot)$$ 448.2.bc.a 8 8
448.2.bc.b 184
448.2.bc.c 192
448.2.bd $$\chi_{448}(27, \cdot)$$ 448.2.bd.a 16 8
448.2.bd.b 480
448.2.bh $$\chi_{448}(9, \cdot)$$ None 0 8
448.2.bi $$\chi_{448}(87, \cdot)$$ None 0 8
448.2.bm $$\chi_{448}(3, \cdot)$$ 448.2.bm.a 992 16
448.2.bn $$\chi_{448}(37, \cdot)$$ 448.2.bn.a 992 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(448))$$ into lower level spaces

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