## 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

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

## 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

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