# Properties

 Label 32.8 Level 32 Weight 8 Dimension 121 Nonzero newspaces 3 Newform subspaces 6 Sturm bound 512 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$32 = 2^{5}$$ Weight: $$k$$ = $$8$$ Nonzero newspaces: $$3$$ Newform subspaces: $$6$$ Sturm bound: $$512$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_1(32))$$.

Total New Old
Modular forms 240 131 109
Cusp forms 208 121 87
Eisenstein series 32 10 22

## Trace form

 $$121 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 282 q^{5} - 4 q^{6} + 684 q^{7} - 4 q^{8} - 55 q^{9} + O(q^{10})$$ $$121 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 282 q^{5} - 4 q^{6} + 684 q^{7} - 4 q^{8} - 55 q^{9} - 13004 q^{10} - 4 q^{11} + 30668 q^{12} + 13614 q^{13} - 26196 q^{14} - 17872 q^{15} - 52784 q^{16} + 29786 q^{17} + 102056 q^{18} - 4 q^{19} + 81996 q^{20} - 25092 q^{21} - 374240 q^{22} - 142124 q^{23} + 380880 q^{24} + 114731 q^{25} - 363984 q^{26} + 476084 q^{27} - 195464 q^{28} - 188450 q^{29} + 870468 q^{30} - 625712 q^{31} + 535656 q^{32} - 221328 q^{33} - 201072 q^{34} + 816500 q^{35} - 1669712 q^{36} - 427098 q^{37} - 2533100 q^{38} + 44260 q^{39} + 2746344 q^{40} + 1015470 q^{41} + 4924936 q^{42} + 366180 q^{43} - 1585404 q^{44} - 1101030 q^{45} - 2715812 q^{46} - 1566432 q^{47} - 5044368 q^{48} - 222987 q^{49} - 1653804 q^{50} + 3002072 q^{51} + 8882212 q^{52} + 1727846 q^{53} + 6815776 q^{54} - 920756 q^{55} - 3655000 q^{56} - 447340 q^{57} - 12448736 q^{58} - 1835940 q^{59} - 6364808 q^{60} - 6248450 q^{61} + 12948600 q^{62} + 4225056 q^{63} + 16915544 q^{64} - 5521724 q^{65} - 11246068 q^{66} - 1940684 q^{67} - 8895536 q^{68} + 18598748 q^{69} - 8149480 q^{70} - 4751604 q^{71} + 12884564 q^{72} + 13488702 q^{73} + 10931436 q^{74} + 11573008 q^{75} + 14443516 q^{76} - 27558612 q^{77} - 38252364 q^{78} - 16015904 q^{79} - 11333192 q^{80} - 26840867 q^{81} + 15827196 q^{82} - 9565884 q^{83} + 8101984 q^{84} + 21578308 q^{85} - 14812992 q^{86} + 78242884 q^{87} + 5716448 q^{88} + 21030126 q^{89} + 5335400 q^{90} - 3406996 q^{91} + 27562664 q^{92} - 33442352 q^{93} - 14227032 q^{94} - 103409944 q^{95} - 25080448 q^{96} - 37388030 q^{97} + 57598104 q^{98} - 9738480 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_1(32))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
32.8.a $$\chi_{32}(1, \cdot)$$ 32.8.a.a 1 1
32.8.a.b 2
32.8.a.c 2
32.8.a.d 2
32.8.b $$\chi_{32}(17, \cdot)$$ 32.8.b.a 6 1
32.8.e $$\chi_{32}(9, \cdot)$$ None 0 2
32.8.g $$\chi_{32}(5, \cdot)$$ 32.8.g.a 108 4

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_1(32))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_1(32)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 5}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 1}$$