## Defining parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$5$$ Newform subspaces: $$10$$ Sturm bound: $$5120$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(128))$$.

Total New Old
Modular forms 2128 1176 952
Cusp forms 1968 1128 840
Eisenstein series 160 48 112

## Trace form

 $$1128q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} + O(q^{10})$$ $$1128q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} - 16q^{10} - 12q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 8q^{15} - 16q^{16} - 24q^{17} - 16q^{18} - 12q^{19} - 16q^{20} + 308q^{21} - 16q^{22} + 1140q^{23} - 16q^{24} - 1364q^{25} - 16q^{26} - 3660q^{27} - 16q^{28} - 1744q^{29} - 16q^{30} - 16q^{31} - 16q^{32} + 3936q^{33} - 16q^{34} + 5172q^{35} - 16q^{36} + 3632q^{37} - 16q^{38} + 2676q^{39} - 16q^{40} - 2900q^{41} - 16q^{42} - 5580q^{43} - 16q^{44} - 3164q^{45} - 16q^{46} - 8q^{47} - 16q^{48} - 9628q^{49} - 43072q^{50} - 8400q^{51} - 17968q^{52} + 1904q^{53} + 31088q^{54} + 11764q^{55} + 49376q^{56} + 29932q^{57} + 65504q^{58} + 13044q^{59} + 63920q^{60} + 15088q^{61} + 11792q^{62} - 32q^{63} - 24400q^{64} - 16144q^{65} - 70864q^{66} - 18892q^{67} - 53296q^{68} - 38860q^{69} - 122320q^{70} - 19980q^{71} - 81664q^{72} - 29460q^{73} - 33280q^{74} + 184q^{75} + 28272q^{76} + 28404q^{77} + 99344q^{78} + 50168q^{79} + 105248q^{80} + 41452q^{81} - 16q^{82} - 10572q^{83} - 16q^{84} - 17416q^{85} - 16q^{86} - 49292q^{87} - 16q^{88} - 54548q^{89} - 16q^{90} - 31884q^{91} - 16q^{92} - 17488q^{93} - 16q^{94} - 16q^{95} - 16q^{96} + 24288q^{97} - 16q^{98} + 46904q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(128))$$

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
128.5.c $$\chi_{128}(127, \cdot)$$ 128.5.c.a 8 1
128.5.c.b 8
128.5.d $$\chi_{128}(63, \cdot)$$ 128.5.d.a 2 1
128.5.d.b 2
128.5.d.c 4
128.5.d.d 8
128.5.f $$\chi_{128}(31, \cdot)$$ 128.5.f.a 14 2
128.5.f.b 14
128.5.h $$\chi_{128}(15, \cdot)$$ 128.5.h.a 60 4
128.5.j $$\chi_{128}(7, \cdot)$$ None 0 8
128.5.l $$\chi_{128}(3, \cdot)$$ 128.5.l.a 1008 16

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(128))$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(\Gamma_1(128)) \cong$$ $$S_{5}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 5}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$