## Defining parameters

 Level: $$N$$ = $$28 = 2^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$3$$ Newforms: $$3$$ Sturm bound: $$96$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(28))$$.

Total New Old
Modular forms 39 20 19
Cusp forms 10 8 2
Eisenstein series 29 12 17

## Trace form

 $$8q - 3q^{2} - q^{3} - 3q^{4} - 9q^{5} - 4q^{7} - 3q^{8} - 4q^{9} + O(q^{10})$$ $$8q - 3q^{2} - q^{3} - 3q^{4} - 9q^{5} - 4q^{7} - 3q^{8} - 4q^{9} + 6q^{10} + 3q^{11} + 12q^{12} + 4q^{13} + 9q^{14} + 6q^{15} + 9q^{16} - 9q^{17} + 3q^{18} + q^{19} + 5q^{21} - 18q^{22} - 3q^{23} - 12q^{24} + 2q^{25} - 12q^{26} - 10q^{27} - 27q^{28} - 6q^{30} + 7q^{31} - 3q^{32} + 9q^{33} + 15q^{35} + 9q^{36} + 7q^{37} + 18q^{38} - 2q^{39} + 12q^{40} + 12q^{41} + 12q^{42} - 8q^{43} + 18q^{44} + 6q^{45} + 12q^{46} + 9q^{47} - 16q^{49} + 3q^{50} - 3q^{51} - 21q^{53} - 18q^{54} - 18q^{55} + 9q^{56} - 38q^{57} - 6q^{58} - 9q^{59} - 12q^{60} - 17q^{61} - 10q^{63} + 9q^{64} + 6q^{65} - 6q^{66} + 7q^{67} + 6q^{69} + 6q^{70} - 15q^{72} + 31q^{73} - 12q^{74} - 4q^{75} + 21q^{77} + 24q^{78} + 13q^{79} - 24q^{80} + 35q^{81} + 12q^{82} + 24q^{83} + 30q^{85} - 18q^{86} + 6q^{87} + 18q^{88} + 39q^{89} - 8q^{91} - 6q^{92} + 13q^{93} - 30q^{94} + 3q^{95} - 24q^{96} - 20q^{97} - 15q^{98} + 12q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$

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
28.2.a $$\chi_{28}(1, \cdot)$$ None 0 1
28.2.d $$\chi_{28}(27, \cdot)$$ 28.2.d.a 2 1
28.2.e $$\chi_{28}(9, \cdot)$$ 28.2.e.a 2 2
28.2.f $$\chi_{28}(3, \cdot)$$ 28.2.f.a 4 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(28))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(28)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 2}$$