## Defining parameters

 Level: $$N$$ = $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$21$$ 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

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

## 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

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