## Defining parameters

 Level: $$N$$ = $$21 = 3 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$8$$ Sturm bound: $$128$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(21))$$.

Total New Old
Modular forms 60 42 18
Cusp forms 36 30 6
Eisenstein series 24 12 12

## Trace form

 $$30q + 3q^{3} - 6q^{4} - 24q^{5} - 36q^{6} - 54q^{7} + 54q^{8} + 39q^{9} + O(q^{10})$$ $$30q + 3q^{3} - 6q^{4} - 24q^{5} - 36q^{6} - 54q^{7} + 54q^{8} + 39q^{9} + 48q^{10} - 96q^{12} - 84q^{13} - 312q^{14} - 198q^{15} - 126q^{16} - 12q^{17} + 270q^{18} + 558q^{19} + 1044q^{20} + 585q^{21} + 432q^{22} - 168q^{23} + 144q^{24} - 426q^{25} - 294q^{26} - 108q^{27} - 1110q^{28} - 648q^{29} - 1530q^{30} - 750q^{31} - 1218q^{32} - 801q^{33} - 276q^{34} + 84q^{35} + 258q^{36} + 1422q^{37} + 1710q^{38} + 1212q^{39} + 1416q^{40} + 192q^{41} + 2322q^{42} + 1152q^{43} + 2184q^{44} + 2151q^{45} + 708q^{46} + 792q^{47} - 1056q^{48} - 2346q^{49} - 3258q^{50} - 3069q^{51} - 4680q^{52} - 2100q^{53} - 4158q^{54} - 1020q^{55} + 690q^{56} - 150q^{57} + 3528q^{58} - 336q^{59} + 3060q^{60} + 3138q^{61} + 1812q^{62} + 2283q^{63} + 4278q^{64} + 504q^{65} + 882q^{66} + 918q^{67} - 1536q^{68} - 1656q^{69} - 3504q^{70} - 492q^{71} - 4050q^{72} - 4326q^{73} - 2730q^{74} - 2304q^{75} - 5544q^{76} - 180q^{77} + 504q^{78} - 510q^{79} + 3096q^{80} + 3987q^{81} + 7488q^{82} + 4200q^{83} + 7956q^{84} + 4668q^{85} + 5166q^{86} + 1656q^{87} - 1812q^{88} - 1752q^{89} - 1080q^{90} + 804q^{91} - 4704q^{92} - 633q^{93} + 504q^{94} + 1344q^{95} - 2700q^{96} + 300q^{97} - 5982q^{98} - 4446q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$

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
21.4.a $$\chi_{21}(1, \cdot)$$ 21.4.a.a 1 1
21.4.a.b 1
21.4.a.c 2
21.4.c $$\chi_{21}(20, \cdot)$$ 21.4.c.a 2 1
21.4.c.b 4
21.4.e $$\chi_{21}(4, \cdot)$$ 21.4.e.a 2 2
21.4.e.b 6
21.4.g $$\chi_{21}(5, \cdot)$$ 21.4.g.a 12 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(21))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(21)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 2}$$