## Defining parameters

 Level: $$N$$ = $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newforms: $$13$$ 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

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

## 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 the 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

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