# Properties

 Label 338.10 Level 338 Weight 10 Dimension 10522 Nonzero newspaces 8 Sturm bound 70980 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ = $$10$$ Nonzero newspaces: $$8$$ Sturm bound: $$70980$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_1(338))$$.

Total New Old
Modular forms 32169 10522 21647
Cusp forms 31713 10522 21191
Eisenstein series 456 0 456

## Trace form

 $$10522 q + 16 q^{2} - 156 q^{3} + 256 q^{4} + 870 q^{5} - 2496 q^{6} - 9160 q^{7} - 20480 q^{8} + 4653 q^{9} + O(q^{10})$$ $$10522 q + 16 q^{2} - 156 q^{3} + 256 q^{4} + 870 q^{5} - 2496 q^{6} - 9160 q^{7} - 20480 q^{8} + 4653 q^{9} + 320640 q^{10} - 129828 q^{11} - 205824 q^{12} - 216864 q^{13} + 712448 q^{14} + 2251512 q^{15} + 65536 q^{16} - 4213812 q^{17} - 3074832 q^{18} + 210524 q^{19} + 2064384 q^{20} - 1743408 q^{21} - 898368 q^{22} + 2362752 q^{23} - 638976 q^{24} - 12914975 q^{25} + 9023328 q^{27} - 2344960 q^{28} + 71700288 q^{29} - 28714368 q^{30} - 77363752 q^{31} + 1048576 q^{32} + 3728832 q^{33} + 64321056 q^{34} + 187172040 q^{35} - 4018944 q^{36} - 141572416 q^{37} - 145083712 q^{38} - 102830052 q^{39} + 3563520 q^{40} + 154771572 q^{41} + 226102656 q^{42} + 97810652 q^{43} + 80317440 q^{44} - 55503840 q^{45} - 215430528 q^{46} - 361503480 q^{47} - 10223616 q^{48} + 92204913 q^{49} + 267661456 q^{50} + 726481560 q^{51} - 66558720 q^{52} - 129814146 q^{53} - 714097152 q^{54} - 453799800 q^{55} - 103088128 q^{56} + 552271176 q^{57} + 521233536 q^{58} + 950011668 q^{59} + 417933312 q^{60} - 460066960 q^{61} - 462904960 q^{62} - 813493368 q^{63} - 83886080 q^{64} - 904349247 q^{65} - 674992896 q^{66} - 1804260820 q^{67} + 14932992 q^{68} + 4013955768 q^{69} + 1815247488 q^{70} + 3213554760 q^{71} + 443535360 q^{72} - 704773078 q^{73} - 974495872 q^{74} - 3094985340 q^{75} + 53894144 q^{76} - 5879467584 q^{77} - 282276288 q^{78} - 1849791232 q^{79} + 528482304 q^{80} + 571559769 q^{81} + 673058880 q^{82} + 8927768268 q^{83} + 2959638528 q^{84} + 8042361078 q^{85} + 1549982144 q^{86} - 2790832200 q^{87} - 229982208 q^{88} - 11952040710 q^{89} - 7808430240 q^{90} - 4209919920 q^{91} - 1986791424 q^{92} + 5953378536 q^{93} + 2652855168 q^{94} + 9876985032 q^{95} - 163577856 q^{96} + 7710514826 q^{97} + 10404891024 q^{98} + 20039913900 q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_1(338))$$

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
338.10.a $$\chi_{338}(1, \cdot)$$ 338.10.a.a 1 1
338.10.a.b 1
338.10.a.c 1
338.10.a.d 1
338.10.a.e 3
338.10.a.f 3
338.10.a.g 5
338.10.a.h 5
338.10.a.i 5
338.10.a.j 5
338.10.a.k 6
338.10.a.l 6
338.10.a.m 10
338.10.a.n 10
338.10.a.o 12
338.10.a.p 12
338.10.a.q 15
338.10.a.r 15
338.10.b $$\chi_{338}(337, \cdot)$$ n/a 116 1
338.10.c $$\chi_{338}(191, \cdot)$$ n/a 230 2
338.10.e $$\chi_{338}(23, \cdot)$$ n/a 232 2
338.10.g $$\chi_{338}(27, \cdot)$$ n/a 1644 12
338.10.h $$\chi_{338}(25, \cdot)$$ n/a 1632 12
338.10.i $$\chi_{338}(3, \cdot)$$ n/a 3288 24
338.10.k $$\chi_{338}(17, \cdot)$$ n/a 3264 24

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

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_1(338))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_1(338)) \cong$$ $$S_{10}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 6}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 3}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(169))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(338))$$$$^{\oplus 1}$$