## 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

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

## 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

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