# Properties

 Label 448.4 Level 448 Weight 4 Dimension 9826 Nonzero newspaces 16 Sturm bound 49152 Trace bound 25

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## Defining parameters

 Level: $$N$$ = $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$16$$ Sturm bound: $$49152$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(448))$$.

Total New Old
Modular forms 18864 10046 8818
Cusp forms 18000 9826 8174
Eisenstein series 864 220 644

## Trace form

 $$9826q - 32q^{2} - 24q^{3} - 32q^{4} - 32q^{5} - 32q^{6} - 28q^{7} - 80q^{8} + 14q^{9} + O(q^{10})$$ $$9826q - 32q^{2} - 24q^{3} - 32q^{4} - 32q^{5} - 32q^{6} - 28q^{7} - 80q^{8} + 14q^{9} - 32q^{10} + 16q^{11} - 32q^{12} - 176q^{13} - 40q^{14} - 304q^{15} - 32q^{16} - 264q^{17} - 32q^{18} - 72q^{19} - 32q^{20} - 28q^{21} - 1024q^{22} - 20q^{23} - 2032q^{24} - 142q^{25} + 48q^{26} + 348q^{27} + 720q^{28} + 720q^{29} + 4608q^{30} + 700q^{31} + 2448q^{32} + 1940q^{33} + 1968q^{34} + 448q^{35} + 1680q^{36} + 1008q^{37} - 912q^{38} - 20q^{39} - 3312q^{40} - 2088q^{41} - 3200q^{42} - 1732q^{43} - 2032q^{44} - 3000q^{45} - 32q^{46} - 1908q^{47} - 32q^{48} - 790q^{49} + 5632q^{50} - 8972q^{51} + 6592q^{52} - 848q^{53} + 3424q^{54} - 1184q^{55} - 432q^{56} + 2056q^{57} - 4784q^{58} + 8896q^{59} - 9824q^{60} + 2128q^{61} - 6016q^{62} + 7648q^{63} - 12176q^{64} + 4460q^{65} - 11104q^{66} + 12024q^{67} - 4160q^{68} + 1240q^{69} - 2056q^{70} + 840q^{71} + 1264q^{72} - 2088q^{73} + 5232q^{74} - 14760q^{75} + 11872q^{76} - 3164q^{77} + 3904q^{78} - 20180q^{79} - 8560q^{80} - 14018q^{81} - 13952q^{82} - 5144q^{83} - 4184q^{84} - 3392q^{85} + 1008q^{86} - 20q^{87} + 6208q^{88} + 7096q^{89} + 18688q^{90} - 812q^{91} + 25152q^{92} + 5632q^{93} + 17824q^{94} + 2172q^{95} + 25808q^{96} + 9176q^{97} + 12064q^{98} + 4028q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(448))$$

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
448.4.a $$\chi_{448}(1, \cdot)$$ 448.4.a.a 1 1
448.4.a.b 1
448.4.a.c 1
448.4.a.d 1
448.4.a.e 1
448.4.a.f 1
448.4.a.g 1
448.4.a.h 1
448.4.a.i 1
448.4.a.j 1
448.4.a.k 1
448.4.a.l 1
448.4.a.m 1
448.4.a.n 1
448.4.a.o 1
448.4.a.p 1
448.4.a.q 2
448.4.a.r 2
448.4.a.s 2
448.4.a.t 2
448.4.a.u 3
448.4.a.v 3
448.4.a.w 3
448.4.a.x 3
448.4.b $$\chi_{448}(225, \cdot)$$ 448.4.b.a 4 1
448.4.b.b 4
448.4.b.c 6
448.4.b.d 6
448.4.b.e 8
448.4.b.f 8
448.4.e $$\chi_{448}(223, \cdot)$$ 448.4.e.a 8 1
448.4.e.b 8
448.4.e.c 8
448.4.e.d 24
448.4.f $$\chi_{448}(447, \cdot)$$ 448.4.f.a 2 1
448.4.f.b 4
448.4.f.c 8
448.4.f.d 8
448.4.f.e 24
448.4.i $$\chi_{448}(65, \cdot)$$ 448.4.i.a 2 2
448.4.i.b 2
448.4.i.c 2
448.4.i.d 2
448.4.i.e 2
448.4.i.f 2
448.4.i.g 4
448.4.i.h 4
448.4.i.i 4
448.4.i.j 6
448.4.i.k 6
448.4.i.l 6
448.4.i.m 6
448.4.i.n 8
448.4.i.o 12
448.4.i.p 12
448.4.i.q 12
448.4.j $$\chi_{448}(111, \cdot)$$ 448.4.j.a 4 2
448.4.j.b 88
448.4.m $$\chi_{448}(113, \cdot)$$ 448.4.m.a 34 2
448.4.m.b 38
448.4.p $$\chi_{448}(255, \cdot)$$ 448.4.p.a 2 2
448.4.p.b 2
448.4.p.c 2
448.4.p.d 2
448.4.p.e 4
448.4.p.f 6
448.4.p.g 6
448.4.p.h 20
448.4.p.i 48
448.4.q $$\chi_{448}(31, \cdot)$$ 448.4.q.a 32 2
448.4.q.b 32
448.4.q.c 32
448.4.t $$\chi_{448}(289, \cdot)$$ 448.4.t.a 32 2
448.4.t.b 32
448.4.t.c 32
448.4.u $$\chi_{448}(57, \cdot)$$ None 0 4
448.4.x $$\chi_{448}(55, \cdot)$$ None 0 4
448.4.z $$\chi_{448}(47, \cdot)$$ n/a 184 4
448.4.ba $$\chi_{448}(81, \cdot)$$ n/a 184 4
448.4.bc $$\chi_{448}(29, \cdot)$$ n/a 1152 8
448.4.bd $$\chi_{448}(27, \cdot)$$ n/a 1520 8
448.4.bh $$\chi_{448}(9, \cdot)$$ None 0 8
448.4.bi $$\chi_{448}(87, \cdot)$$ None 0 8
448.4.bm $$\chi_{448}(3, \cdot)$$ n/a 3040 16
448.4.bn $$\chi_{448}(37, \cdot)$$ n/a 3040 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(448))$$ into lower level spaces

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