## Defining parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$5$$ Newform subspaces: $$17$$ Sturm bound: $$4096$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(128))$$.

Total New Old
Modular forms 1616 888 728
Cusp forms 1456 840 616
Eisenstein series 160 48 112

## Trace form

 $$840q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} + O(q^{10})$$ $$840q - 16q^{2} - 12q^{3} - 16q^{4} - 16q^{5} - 16q^{6} - 12q^{7} - 16q^{8} - 20q^{9} - 16q^{10} - 12q^{11} - 16q^{12} - 16q^{13} - 16q^{14} - 16q^{15} - 16q^{16} - 24q^{17} - 16q^{18} - 12q^{19} - 16q^{20} - 124q^{21} - 16q^{22} - 340q^{23} - 16q^{24} - 196q^{25} - 16q^{26} + 252q^{27} - 16q^{28} + 384q^{29} - 16q^{30} + 736q^{31} - 16q^{32} + 896q^{33} - 16q^{34} + 444q^{35} - 16q^{36} - 16q^{38} - 612q^{39} - 16q^{40} - 964q^{41} - 16q^{42} - 820q^{43} - 16q^{44} - 300q^{45} - 16q^{46} - 16q^{47} - 16q^{48} + 1348q^{49} + 5696q^{50} + 1368q^{51} + 6608q^{52} + 1488q^{53} + 3440q^{54} + 276q^{55} - 800q^{56} - 1364q^{57} - 4768q^{58} - 1388q^{59} - 9808q^{60} - 3664q^{61} - 5872q^{62} - 2512q^{63} - 12112q^{64} - 3920q^{65} - 10960q^{66} - 2052q^{67} - 4144q^{68} - 2236q^{69} - 4048q^{70} - 236q^{71} + 1280q^{72} + 1708q^{73} + 5248q^{74} + 1696q^{75} + 11888q^{76} + 2420q^{77} + 14096q^{78} - 16q^{79} + 10016q^{80} - 788q^{81} - 16q^{82} - 2452q^{83} - 16q^{84} + 504q^{85} - 16q^{86} + 1276q^{87} - 16q^{88} + 3372q^{89} - 16q^{90} + 3588q^{91} - 16q^{92} + 4256q^{93} - 16q^{94} + 6072q^{95} - 16q^{96} + 5920q^{97} - 16q^{98} + 5408q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(128))$$

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
128.4.a $$\chi_{128}(1, \cdot)$$ 128.4.a.a 1 1
128.4.a.b 1
128.4.a.c 1
128.4.a.d 1
128.4.a.e 2
128.4.a.f 2
128.4.a.g 2
128.4.a.h 2
128.4.b $$\chi_{128}(65, \cdot)$$ 128.4.b.a 2 1
128.4.b.b 2
128.4.b.c 2
128.4.b.d 2
128.4.b.e 4
128.4.e $$\chi_{128}(33, \cdot)$$ 128.4.e.a 10 2
128.4.e.b 10
128.4.g $$\chi_{128}(17, \cdot)$$ 128.4.g.a 44 4
128.4.i $$\chi_{128}(9, \cdot)$$ None 0 8
128.4.k $$\chi_{128}(5, \cdot)$$ 128.4.k.a 752 16

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(128))$$ into lower level spaces

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