## Defining parameters

 Level: $$N$$ = $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$8$$ Newform subspaces: $$14$$ Sturm bound: $$1536$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(84))$$.

Total New Old
Modular forms 636 246 390
Cusp forms 516 230 286
Eisenstein series 120 16 104

## Trace form

 $$230q + 10q^{4} + 12q^{5} + 42q^{6} - 56q^{7} - 102q^{8} - 42q^{9} + O(q^{10})$$ $$230q + 10q^{4} + 12q^{5} + 42q^{6} - 56q^{7} - 102q^{8} - 42q^{9} - 148q^{10} + 96q^{11} - 78q^{12} + 304q^{13} + 312q^{14} + 180q^{15} + 610q^{16} - 24q^{17} + 372q^{18} - 184q^{19} - 438q^{21} - 756q^{22} - 60q^{23} - 930q^{24} + 322q^{25} - 750q^{26} - 1062q^{28} + 1620q^{29} + 324q^{30} + 8q^{31} - 210q^{32} + 576q^{33} - 112q^{34} - 816q^{35} - 114q^{36} - 116q^{37} + 1494q^{38} - 696q^{39} + 104q^{40} + 12q^{41} - 948q^{42} - 952q^{43} + 1848q^{44} - 1500q^{45} + 2388q^{46} - 684q^{47} + 1014q^{48} - 3922q^{49} - 1050q^{50} - 360q^{51} - 2872q^{52} - 1752q^{53} - 2112q^{54} - 468q^{55} - 5022q^{56} + 1788q^{57} - 2320q^{58} + 1128q^{59} - 72q^{60} + 6208q^{61} + 2508q^{63} + 694q^{64} + 2832q^{65} + 732q^{66} + 3536q^{67} + 7452q^{68} + 1572q^{69} + 9444q^{70} + 1224q^{71} + 2970q^{72} - 2192q^{73} + 3234q^{74} - 2592q^{75} - 2388q^{76} + 468q^{77} + 336q^{78} - 3544q^{79} - 5976q^{80} - 3006q^{81} - 7228q^{82} - 4272q^{83} - 1470q^{84} - 6008q^{85} - 6342q^{86} - 4320q^{87} + 3144q^{88} + 708q^{89} + 5148q^{90} + 1472q^{91} + 7644q^{92} - 1452q^{93} + 14292q^{94} + 684q^{95} + 4878q^{96} + 3256q^{97} + 11130q^{98} + 9144q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(84))$$

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
84.4.a $$\chi_{84}(1, \cdot)$$ 84.4.a.a 1 1
84.4.a.b 1
84.4.b $$\chi_{84}(55, \cdot)$$ 84.4.b.a 12 1
84.4.b.b 12
84.4.e $$\chi_{84}(71, \cdot)$$ 84.4.e.a 36 1
84.4.f $$\chi_{84}(41, \cdot)$$ 84.4.f.a 8 1
84.4.i $$\chi_{84}(25, \cdot)$$ 84.4.i.a 4 2
84.4.i.b 4
84.4.k $$\chi_{84}(5, \cdot)$$ 84.4.k.a 2 2
84.4.k.b 2
84.4.k.c 12
84.4.n $$\chi_{84}(11, \cdot)$$ 84.4.n.a 88 2
84.4.o $$\chi_{84}(19, \cdot)$$ 84.4.o.a 24 2
84.4.o.b 24

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(84))$$ into lower level spaces

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