## Defining parameters

 Level: $$N$$ = $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newforms: $$21$$ Sturm bound: $$3136$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(147))$$.

Total New Old
Modular forms 904 613 291
Cusp forms 665 517 148
Eisenstein series 239 96 143

## Trace form

 $$517q + 3q^{2} - 16q^{3} - 39q^{4} - 6q^{5} - 30q^{6} - 44q^{7} - 21q^{8} - 28q^{9} + O(q^{10})$$ $$517q + 3q^{2} - 16q^{3} - 39q^{4} - 6q^{5} - 30q^{6} - 44q^{7} - 21q^{8} - 28q^{9} - 72q^{10} - 24q^{11} - 34q^{12} - 56q^{13} - 24q^{14} - 33q^{15} - 71q^{16} - 6q^{17} - 18q^{18} - 50q^{19} - 30q^{20} - 25q^{21} - 90q^{22} - 24q^{23} - 42q^{24} - 83q^{25} - 42q^{26} - 28q^{27} - 108q^{28} - 42q^{29} - 51q^{30} - 86q^{31} - 69q^{32} - 45q^{33} - 108q^{34} - 42q^{35} - 28q^{36} - 28q^{37} + 36q^{38} + 14q^{39} + 132q^{40} + 78q^{41} + 51q^{42} - 30q^{43} + 168q^{44} + 15q^{45} + 138q^{46} + 24q^{47} + 141q^{48} + 78q^{49} + 81q^{50} + 57q^{51} + 180q^{52} + 42q^{53} - 18q^{54} + 96q^{55} + 150q^{56} - 65q^{57} + 48q^{58} - 24q^{59} + 51q^{60} - 6q^{61} - 84q^{62} - 36q^{63} - 75q^{64} - 96q^{65} - 57q^{66} - 106q^{67} - 138q^{68} - 45q^{69} - 150q^{70} - 48q^{71} - 42q^{72} - 56q^{73} - 90q^{74} - 22q^{75} - 182q^{76} - 72q^{77} + 15q^{78} - 118q^{79} - 48q^{80} + 80q^{81} + 6q^{82} + 84q^{83} + 164q^{84} - 42q^{85} + 174q^{86} + 129q^{87} + 216q^{88} + 138q^{89} + 372q^{90} + 22q^{91} + 126q^{92} + 247q^{93} + 168q^{94} + 228q^{95} + 225q^{96} + 58q^{97} + 438q^{98} + 96q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(147))$$

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
147.2.a $$\chi_{147}(1, \cdot)$$ 147.2.a.a 1 1
147.2.a.b 1
147.2.a.c 1
147.2.a.d 2
147.2.a.e 2
147.2.c $$\chi_{147}(146, \cdot)$$ 147.2.c.a 2 1
147.2.c.b 8
147.2.e $$\chi_{147}(67, \cdot)$$ 147.2.e.a 2 2
147.2.e.b 2
147.2.e.c 2
147.2.e.d 4
147.2.e.e 4
147.2.g $$\chi_{147}(68, \cdot)$$ 147.2.g.a 2 2
147.2.g.b 16
147.2.i $$\chi_{147}(22, \cdot)$$ 147.2.i.a 24 6
147.2.i.b 36
147.2.k $$\chi_{147}(20, \cdot)$$ 147.2.k.a 96 6
147.2.m $$\chi_{147}(4, \cdot)$$ 147.2.m.a 48 12
147.2.m.b 60
147.2.o $$\chi_{147}(5, \cdot)$$ 147.2.o.a 12 12
147.2.o.b 192

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(147))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(147)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 2}$$