# Properties

 Label 338.8 Level 338 Weight 8 Dimension 8184 Nonzero newspaces 8 Sturm bound 56784 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ = $$8$$ Nonzero newspaces: $$8$$ Sturm bound: $$56784$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_1(338))$$.

Total New Old
Modular forms 25071 8184 16887
Cusp forms 24615 8184 16431
Eisenstein series 456 0 456

## Trace form

 $$8184 q - 8 q^{2} + 12 q^{3} + 64 q^{4} - 210 q^{5} - 96 q^{6} - 5040 q^{7} + 2560 q^{8} - 25371 q^{9} + O(q^{10})$$ $$8184 q - 8 q^{2} + 12 q^{3} + 64 q^{4} - 210 q^{5} - 96 q^{6} - 5040 q^{7} + 2560 q^{8} - 25371 q^{9} - 18240 q^{10} + 16908 q^{11} + 14592 q^{12} + 29928 q^{13} - 29824 q^{14} - 167112 q^{15} - 28672 q^{16} - 128064 q^{17} + 191304 q^{18} + 445860 q^{19} - 2304 q^{20} - 862584 q^{21} - 8736 q^{22} + 98640 q^{23} - 6144 q^{24} + 434725 q^{25} + 257616 q^{27} - 322560 q^{28} + 904956 q^{29} + 1635648 q^{30} - 239184 q^{31} - 32768 q^{32} - 1350696 q^{33} - 1665552 q^{34} - 3840 q^{35} + 750912 q^{36} + 855492 q^{37} + 4270304 q^{38} + 1629420 q^{39} + 107520 q^{40} + 98976 q^{41} - 3380160 q^{42} - 5444372 q^{43} - 4467456 q^{44} - 12408780 q^{45} - 2242752 q^{46} + 2146704 q^{47} + 49152 q^{48} + 11360153 q^{49} + 3653752 q^{50} + 1886088 q^{51} - 1693504 q^{52} + 11017926 q^{53} + 8172288 q^{54} - 5227680 q^{55} - 3973120 q^{56} - 26435616 q^{57} - 10105152 q^{58} - 12460428 q^{59} - 2892288 q^{60} + 4560236 q^{61} + 9049280 q^{62} + 55110528 q^{63} + 1835008 q^{64} + 26353695 q^{65} + 17959296 q^{66} - 5371308 q^{67} + 662016 q^{68} - 55472352 q^{69} - 20501952 q^{70} - 50589720 q^{71} - 18393600 q^{72} - 46543926 q^{73} - 23859904 q^{74} + 65186700 q^{75} + 28535040 q^{76} + 133309872 q^{77} + 35465760 q^{78} + 43034672 q^{79} - 147456 q^{80} - 106829775 q^{81} - 79929120 q^{82} - 86720004 q^{83} - 23508480 q^{84} - 117121878 q^{85} + 6583136 q^{86} - 11452200 q^{87} - 559104 q^{88} + 81535722 q^{89} + 84047760 q^{90} + 59782244 q^{91} + 32374272 q^{92} + 93959472 q^{93} + 8044992 q^{94} - 52385352 q^{95} - 393216 q^{96} - 226231734 q^{97} - 64300872 q^{98} - 72270756 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_1(338))$$

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
338.8.a $$\chi_{338}(1, \cdot)$$ 338.8.a.a 1 1
338.8.a.b 1
338.8.a.c 1
338.8.a.d 1
338.8.a.e 2
338.8.a.f 2
338.8.a.g 3
338.8.a.h 3
338.8.a.i 4
338.8.a.j 4
338.8.a.k 5
338.8.a.l 5
338.8.a.m 8
338.8.a.n 8
338.8.a.o 9
338.8.a.p 9
338.8.a.q 12
338.8.a.r 12
338.8.b $$\chi_{338}(337, \cdot)$$ 338.8.b.a 2 1
338.8.b.b 2
338.8.b.c 2
338.8.b.d 2
338.8.b.e 4
338.8.b.f 4
338.8.b.g 6
338.8.b.h 8
338.8.b.i 16
338.8.b.j 18
338.8.b.k 24
338.8.c $$\chi_{338}(191, \cdot)$$ n/a 182 2
338.8.e $$\chi_{338}(23, \cdot)$$ n/a 180 2
338.8.g $$\chi_{338}(27, \cdot)$$ n/a 1284 12
338.8.h $$\chi_{338}(25, \cdot)$$ n/a 1296 12
338.8.i $$\chi_{338}(3, \cdot)$$ n/a 2520 24
338.8.k $$\chi_{338}(17, \cdot)$$ n/a 2544 24

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_1(338))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_1(338)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(169))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(338))$$$$^{\oplus 1}$$