## Defining parameters

 Level: $$N$$ = $$21 = 3 \cdot 7$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$224$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(21))$$.

Total New Old
Modular forms 108 72 36
Cusp forms 84 64 20
Eisenstein series 24 8 16

## Trace form

 $$64 q + 51 q^{3} - 134 q^{4} - 336 q^{5} - 6 q^{6} + 1240 q^{7} + 1410 q^{8} + 387 q^{9} + O(q^{10})$$ $$64 q + 51 q^{3} - 134 q^{4} - 336 q^{5} - 6 q^{6} + 1240 q^{7} + 1410 q^{8} + 387 q^{9} - 5172 q^{10} - 8232 q^{11} - 2130 q^{12} + 4016 q^{13} + 12792 q^{14} + 4818 q^{15} + 11842 q^{16} + 1680 q^{17} + 1572 q^{18} - 24706 q^{19} + 7479 q^{21} + 66228 q^{22} - 35616 q^{23} - 93390 q^{24} - 156752 q^{25} - 6678 q^{26} + 128064 q^{27} + 181162 q^{28} + 84720 q^{29} + 52944 q^{30} + 10742 q^{31} - 110922 q^{32} - 171615 q^{33} - 200352 q^{34} - 164640 q^{35} - 48858 q^{36} + 120110 q^{37} + 426174 q^{38} + 495450 q^{39} + 553776 q^{40} - 544392 q^{42} - 900016 q^{43} - 674184 q^{44} - 171783 q^{45} + 218100 q^{46} + 566160 q^{47} + 541014 q^{48} + 215920 q^{49} - 11514 q^{50} - 111975 q^{51} - 871432 q^{52} - 321720 q^{53} - 656292 q^{54} - 500484 q^{55} + 110898 q^{56} + 412254 q^{57} + 2991744 q^{58} + 1628592 q^{59} + 1606956 q^{60} + 1105262 q^{61} + 120519 q^{63} - 2502074 q^{64} - 1639512 q^{65} - 2497176 q^{66} - 4138858 q^{67} - 2437596 q^{68} - 1367694 q^{69} + 604740 q^{70} + 1161744 q^{71} - 837330 q^{72} + 637046 q^{73} + 1264914 q^{74} + 3090750 q^{75} + 6800876 q^{76} + 2274624 q^{77} + 2817960 q^{78} + 62486 q^{79} - 1247232 q^{80} + 1511127 q^{81} - 3740436 q^{82} - 482094 q^{84} + 1504212 q^{85} + 2360274 q^{86} - 1369284 q^{87} - 5260632 q^{88} - 2759232 q^{89} - 12501852 q^{90} - 3538384 q^{91} - 9135924 q^{92} - 2268483 q^{93} + 2471484 q^{94} + 4723152 q^{95} + 13330758 q^{96} + 8833592 q^{97} + 8734554 q^{98} + 6639210 q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(21))$$

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
21.7.b $$\chi_{21}(8, \cdot)$$ 21.7.b.a 12 1
21.7.d $$\chi_{21}(13, \cdot)$$ 21.7.d.a 8 1
21.7.f $$\chi_{21}(10, \cdot)$$ 21.7.f.a 8 2
21.7.f.b 8
21.7.h $$\chi_{21}(2, \cdot)$$ 21.7.h.a 28 2

## Decomposition of $$S_{7}^{\mathrm{old}}(\Gamma_1(21))$$ into lower level spaces

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