## Defining parameters

 Level: $$N$$ = $$32 = 2^{5}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$256$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(32))$$.

Total New Old
Modular forms 112 59 53
Cusp forms 80 49 31
Eisenstein series 32 10 22

## Trace form

 $$49q - 4q^{2} - 4q^{3} - 4q^{4} - 2q^{5} - 4q^{6} + 12q^{7} - 4q^{8} + 41q^{9} + O(q^{10})$$ $$49q - 4q^{2} - 4q^{3} - 4q^{4} - 2q^{5} - 4q^{6} + 12q^{7} - 4q^{8} + 41q^{9} + 116q^{10} - 4q^{11} - 52q^{12} - 122q^{13} - 212q^{14} - 112q^{15} - 304q^{16} - 182q^{17} - 184q^{18} - 4q^{19} + 76q^{20} + 252q^{21} + 192q^{22} + 628q^{23} - 48q^{24} + 331q^{25} + 16q^{26} - 268q^{27} + 376q^{28} - 202q^{29} + 1188q^{30} - 1200q^{31} + 616q^{32} - 480q^{33} + 528q^{34} - 460q^{35} + 1456q^{36} + 830q^{37} + 980q^{38} + 1156q^{39} - 536q^{40} + 446q^{41} - 2264q^{42} + 804q^{43} - 2044q^{44} - 1230q^{45} - 1444q^{46} - 672q^{47} - 2448q^{48} - 1075q^{49} - 3564q^{50} - 1384q^{51} - 2524q^{52} + 446q^{53} - 1088q^{54} + 44q^{55} + 1192q^{56} + 1028q^{57} + 3200q^{58} + 1372q^{59} + 5752q^{60} - 938q^{61} + 3384q^{62} + 2496q^{63} + 4952q^{64} + 1716q^{65} + 5996q^{66} + 2036q^{67} + 2768q^{68} - 292q^{69} + 1400q^{70} + 364q^{71} - 1708q^{72} - 754q^{73} - 3476q^{74} - 1712q^{75} - 5124q^{76} + 620q^{77} - 11916q^{78} + 928q^{79} - 10312q^{80} - 1499q^{81} - 6404q^{82} + 2436q^{83} - 6560q^{84} - 972q^{85} - 928q^{86} - 2972q^{87} + 1248q^{88} - 1794q^{89} + 7400q^{90} - 3604q^{91} + 10152q^{92} + 5008q^{93} + 12840q^{94} - 5304q^{95} + 17792q^{96} + 2354q^{97} + 11224q^{98} - 5424q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$

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
32.4.a $$\chi_{32}(1, \cdot)$$ 32.4.a.a 1 1
32.4.a.b 1
32.4.a.c 1
32.4.b $$\chi_{32}(17, \cdot)$$ 32.4.b.a 2 1
32.4.e $$\chi_{32}(9, \cdot)$$ None 0 2
32.4.g $$\chi_{32}(5, \cdot)$$ 32.4.g.a 44 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(32))$$ into lower level spaces

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