## Defining parameters

 Level: $$N$$ = $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$9$$ Nonzero newspaces: $$8$$ Newform subspaces: $$15$$ Sturm bound: $$3456$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(\Gamma_1(84))$$.

Total New Old
Modular forms 1596 662 934
Cusp forms 1476 638 838
Eisenstein series 120 24 96

## Trace form

 $$638 q - 12 q^{2} + 123 q^{3} + 98 q^{4} + 2346 q^{5} - 2268 q^{6} - 5230 q^{7} + 13230 q^{8} + 37071 q^{9} + O(q^{10})$$ $$638 q - 12 q^{2} + 123 q^{3} + 98 q^{4} + 2346 q^{5} - 2268 q^{6} - 5230 q^{7} + 13230 q^{8} + 37071 q^{9} - 26336 q^{10} - 56394 q^{11} - 60468 q^{12} + 168018 q^{13} + 165648 q^{14} - 80346 q^{15} - 170734 q^{16} + 166944 q^{17} - 408534 q^{18} + 696692 q^{19} + 959508 q^{20} + 377331 q^{21} - 1417224 q^{22} + 227520 q^{23} + 314196 q^{24} + 1562088 q^{25} + 1867842 q^{26} - 3858588 q^{27} - 3251814 q^{28} + 1095972 q^{29} - 4207878 q^{30} + 4138688 q^{31} + 10288518 q^{32} - 1007109 q^{33} + 6204436 q^{34} - 4260726 q^{35} - 4147950 q^{36} - 8605018 q^{37} - 15008130 q^{38} + 10214808 q^{39} - 1154624 q^{40} + 17730912 q^{41} - 8956302 q^{42} - 8765612 q^{43} + 19453656 q^{44} + 14033163 q^{45} + 15090036 q^{46} + 11782746 q^{47} - 21832416 q^{48} - 41702572 q^{49} - 19319934 q^{50} - 3948957 q^{51} + 18873064 q^{52} - 10280196 q^{53} + 5093334 q^{54} + 72314352 q^{55} - 36395838 q^{56} + 6063906 q^{57} - 63710024 q^{58} + 25922772 q^{59} - 33088704 q^{60} - 83361492 q^{61} - 44595852 q^{62} + 18177243 q^{63} + 14133878 q^{64} - 76947510 q^{65} + 53411646 q^{66} + 26733312 q^{67} + 49585128 q^{68} + 22833630 q^{69} + 100685808 q^{70} + 110006568 q^{71} + 133067946 q^{72} - 2357622 q^{73} - 69552426 q^{74} - 140598114 q^{75} - 338195736 q^{76} - 31415592 q^{77} - 5013480 q^{78} - 211737108 q^{79} + 218046096 q^{80} + 66794259 q^{81} + 162108928 q^{82} + 141383820 q^{84} + 871880204 q^{85} - 140757618 q^{86} - 95141376 q^{87} - 48603252 q^{88} - 227774076 q^{89} + 160210872 q^{90} - 247692388 q^{91} - 112924560 q^{92} + 464857005 q^{93} + 122624928 q^{94} - 258953022 q^{95} - 635282148 q^{96} - 230255952 q^{97} - 689865594 q^{98} - 159728226 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(\Gamma_1(84))$$

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
84.9.c $$\chi_{84}(29, \cdot)$$ 84.9.c.a 16 1
84.9.d $$\chi_{84}(13, \cdot)$$ 84.9.d.a 10 1
84.9.g $$\chi_{84}(43, \cdot)$$ 84.9.g.a 48 1
84.9.h $$\chi_{84}(83, \cdot)$$ 84.9.h.a 1 1
84.9.h.b 1
84.9.h.c 1
84.9.h.d 1
84.9.h.e 120
84.9.j $$\chi_{84}(47, \cdot)$$ 84.9.j.a 248 2
84.9.l $$\chi_{84}(67, \cdot)$$ 84.9.l.a 64 2
84.9.l.b 64
84.9.m $$\chi_{84}(61, \cdot)$$ 84.9.m.a 10 2
84.9.m.b 12
84.9.p $$\chi_{84}(53, \cdot)$$ 84.9.p.a 2 2
84.9.p.b 40

## Decomposition of $$S_{9}^{\mathrm{old}}(\Gamma_1(84))$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(\Gamma_1(84)) \cong$$ $$S_{9}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 2}$$