## Defining parameters

 Level: $$N$$ = $$21 = 3 \cdot 7$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$7$$ Sturm bound: $$96$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(21))$$.

Total New Old
Modular forms 44 26 18
Cusp forms 20 18 2
Eisenstein series 24 8 16

## Trace form

 $$18q - 6q^{3} - 22q^{4} - 6q^{5} - 6q^{6} + 2q^{7} - 6q^{8} - 24q^{9} + O(q^{10})$$ $$18q - 6q^{3} - 22q^{4} - 6q^{5} - 6q^{6} + 2q^{7} - 6q^{8} - 24q^{9} - 12q^{10} + 6q^{11} + 18q^{12} + 2q^{13} + 24q^{14} + 48q^{15} + 90q^{16} + 48q^{17} + 96q^{18} + 2q^{19} - 24q^{21} - 108q^{22} - 96q^{23} - 150q^{24} - 132q^{25} - 126q^{26} - 132q^{27} - 102q^{28} - 12q^{29} + 24q^{30} + 86q^{31} + 78q^{32} + 186q^{33} + 240q^{34} + 210q^{35} + 198q^{36} + 160q^{37} + 174q^{38} + 66q^{39} + 96q^{40} - 72q^{42} - 188q^{43} - 168q^{44} - 198q^{45} - 204q^{46} - 222q^{47} - 210q^{48} - 120q^{49} - 114q^{50} - 168q^{51} + 152q^{52} - 12q^{53} - 84q^{54} + 156q^{55} + 18q^{56} + 108q^{57} + 84q^{59} + 96q^{60} + 38q^{61} + 78q^{63} - 34q^{64} + 18q^{65} - 36q^{66} + 102q^{67} + 36q^{68} + 288q^{69} - 60q^{70} + 120q^{71} + 174q^{72} - 100q^{73} + 90q^{74} + 180q^{75} - 292q^{76} + 72q^{77} + 24q^{78} + 162q^{79} + 48q^{80} + 108q^{82} + 102q^{84} - 168q^{85} + 210q^{86} + 12q^{87} - 24q^{88} - 60q^{89} - 132q^{90} - 148q^{91} - 84q^{92} - 264q^{93} - 324q^{94} - 438q^{95} - 402q^{96} - 376q^{97} - 270q^{98} - 432q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(21))$$

We only show spaces with odd 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
21.3.b $$\chi_{21}(8, \cdot)$$ 21.3.b.a 4 1
21.3.d $$\chi_{21}(13, \cdot)$$ 21.3.d.a 2 1
21.3.f $$\chi_{21}(10, \cdot)$$ 21.3.f.a 2 2
21.3.f.b 2
21.3.f.c 2
21.3.h $$\chi_{21}(2, \cdot)$$ 21.3.h.a 2 2
21.3.h.b 4

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(21))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(21)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 2}$$