## Defining parameters

 Level: $$N$$ = $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$8$$ Newform subspaces: $$27$$ Sturm bound: $$8192$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(192))$$.

Total New Old
Modular forms 3216 1318 1898
Cusp forms 2928 1274 1654
Eisenstein series 288 44 244

## Trace form

 $$1274q - 6q^{3} - 16q^{4} - 8q^{6} - 8q^{7} - 10q^{9} + O(q^{10})$$ $$1274q - 6q^{3} - 16q^{4} - 8q^{6} - 8q^{7} - 10q^{9} - 16q^{10} + 40q^{11} - 8q^{12} - 160q^{13} - 128q^{15} - 16q^{16} - 208q^{17} - 8q^{18} - 60q^{19} + 4q^{21} - 960q^{22} - 1008q^{24} - 122q^{25} + 80q^{26} + 126q^{27} + 1504q^{28} + 800q^{29} + 2312q^{30} + 712q^{31} + 2480q^{32} + 868q^{33} + 1984q^{34} + 456q^{35} + 872q^{36} + 16q^{37} - 880q^{38} - 604q^{39} - 3296q^{40} - 1888q^{41} - 3168q^{42} - 820q^{43} - 2000q^{44} + 476q^{45} - 16q^{46} - 8q^{48} - 714q^{49} + 5712q^{50} - 3080q^{51} + 6608q^{52} + 424q^{54} - 1448q^{55} - 784q^{56} - 308q^{57} - 4768q^{58} + 6880q^{59} - 4904q^{60} - 16q^{61} - 5856q^{62} + 2520q^{63} - 12112q^{64} + 4048q^{65} - 5544q^{66} + 12036q^{67} - 4128q^{68} + 844q^{69} - 4048q^{70} + 896q^{71} - 8q^{72} - 2068q^{73} + 5264q^{74} - 6982q^{75} + 11888q^{76} - 3504q^{77} + 12112q^{78} - 17336q^{79} + 10032q^{80} + 1202q^{81} - 16q^{82} - 2680q^{83} + 4136q^{84} - 2368q^{85} - 1292q^{87} - 16q^{88} + 352q^{89} - 9368q^{90} + 3056q^{91} - 176q^{93} - 16q^{94} + 7728q^{95} - 12928q^{96} + 6788q^{97} - 844q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(192))$$

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
192.4.a $$\chi_{192}(1, \cdot)$$ 192.4.a.a 1 1
192.4.a.b 1
192.4.a.c 1
192.4.a.d 1
192.4.a.e 1
192.4.a.f 1
192.4.a.g 1
192.4.a.h 1
192.4.a.i 1
192.4.a.j 1
192.4.a.k 1
192.4.a.l 1
192.4.c $$\chi_{192}(191, \cdot)$$ 192.4.c.a 2 1
192.4.c.b 4
192.4.c.c 4
192.4.c.d 12
192.4.d $$\chi_{192}(97, \cdot)$$ 192.4.d.a 4 1
192.4.d.b 4
192.4.d.c 4
192.4.f $$\chi_{192}(95, \cdot)$$ 192.4.f.a 4 1
192.4.f.b 4
192.4.f.c 8
192.4.f.d 8
192.4.j $$\chi_{192}(49, \cdot)$$ 192.4.j.a 24 2
192.4.k $$\chi_{192}(47, \cdot)$$ 192.4.k.a 44 2
192.4.n $$\chi_{192}(25, \cdot)$$ None 0 4
192.4.o $$\chi_{192}(23, \cdot)$$ None 0 4
192.4.r $$\chi_{192}(13, \cdot)$$ 192.4.r.a 384 8
192.4.s $$\chi_{192}(11, \cdot)$$ 192.4.s.a 752 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(192))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(192)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 2}$$