## 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

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

## 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

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