# Properties

 Label 84.2 Level 84 Weight 2 Dimension 72 Nonzero newspaces 8 Newform subspaces 13 Sturm bound 768 Trace bound 5

## Defining parameters

 Level: $$N$$ = $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$13$$ Sturm bound: $$768$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(84))$$.

Total New Old
Modular forms 252 88 164
Cusp forms 133 72 61
Eisenstein series 119 16 103

## Trace form

 $$72 q + q^{3} - 6 q^{4} + 6 q^{5} - 6 q^{6} + 8 q^{7} - 6 q^{8} - 11 q^{9} + O(q^{10})$$ $$72 q + q^{3} - 6 q^{4} + 6 q^{5} - 6 q^{6} + 8 q^{7} - 6 q^{8} - 11 q^{9} - 24 q^{10} - 6 q^{11} - 18 q^{12} - 34 q^{13} - 24 q^{14} - 18 q^{15} - 30 q^{16} - 12 q^{17} - 18 q^{18} - 16 q^{19} - 35 q^{21} + 12 q^{22} - 12 q^{23} + 6 q^{24} - 42 q^{25} + 30 q^{26} - 2 q^{27} + 42 q^{28} - 36 q^{29} + 30 q^{30} - 4 q^{31} + 30 q^{32} + 3 q^{33} + 24 q^{34} + 6 q^{35} + 30 q^{36} - 14 q^{37} + 18 q^{38} + 38 q^{39} + 24 q^{40} + 24 q^{41} + 48 q^{42} + 40 q^{43} + 24 q^{44} + 39 q^{45} + 24 q^{46} + 18 q^{47} + 54 q^{48} + 18 q^{49} + 42 q^{50} + 21 q^{51} + 24 q^{52} + 24 q^{53} + 48 q^{54} + 18 q^{56} - 10 q^{57} - 12 q^{59} + 12 q^{60} - 40 q^{61} - 17 q^{63} - 42 q^{64} - 6 q^{65} - 18 q^{66} - 60 q^{67} - 36 q^{68} - 12 q^{69} - 72 q^{70} - 30 q^{72} - 58 q^{73} - 66 q^{74} + 10 q^{75} - 84 q^{76} - 120 q^{78} - 12 q^{79} - 72 q^{80} + 73 q^{81} - 84 q^{82} - 12 q^{83} - 114 q^{84} - 36 q^{85} - 66 q^{86} + 12 q^{87} - 120 q^{88} + 12 q^{89} - 120 q^{90} + 20 q^{91} - 84 q^{92} + 5 q^{93} - 24 q^{94} - 18 q^{95} - 66 q^{96} + 56 q^{97} - 42 q^{98} - 54 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
84.2.a $$\chi_{84}(1, \cdot)$$ 84.2.a.a 1 1
84.2.a.b 1
84.2.b $$\chi_{84}(55, \cdot)$$ 84.2.b.a 4 1
84.2.b.b 4
84.2.e $$\chi_{84}(71, \cdot)$$ 84.2.e.a 12 1
84.2.f $$\chi_{84}(41, \cdot)$$ 84.2.f.a 2 1
84.2.i $$\chi_{84}(25, \cdot)$$ 84.2.i.a 2 2
84.2.k $$\chi_{84}(5, \cdot)$$ 84.2.k.a 2 2
84.2.k.b 2
84.2.k.c 2
84.2.n $$\chi_{84}(11, \cdot)$$ 84.2.n.a 24 2
84.2.o $$\chi_{84}(19, \cdot)$$ 84.2.o.a 8 2
84.2.o.b 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(84))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(84)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 1}$$