## Defining parameters

 Level: $$N$$ = $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$10$$ Sturm bound: $$2112$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(138))$$.

Total New Old
Modular forms 616 133 483
Cusp forms 441 133 308
Eisenstein series 175 0 175

## Trace form

 $$133q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$133q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} + 8q^{7} + q^{8} + q^{9} + 6q^{10} + 12q^{11} + q^{12} + 14q^{13} + 8q^{14} - 16q^{15} + q^{16} - 26q^{17} - 43q^{18} - 24q^{19} - 38q^{20} - 58q^{21} - 32q^{22} - 65q^{23} + q^{24} - 57q^{25} - 30q^{26} - 65q^{27} - 36q^{28} - 14q^{29} - 38q^{30} - 12q^{31} + q^{32} - 10q^{33} + 18q^{34} + 4q^{35} + q^{36} - 50q^{37} + 20q^{38} - 30q^{39} + 6q^{40} - 2q^{41} + 8q^{42} - 44q^{43} + 12q^{44} + 6q^{45} + 23q^{46} - 40q^{47} + q^{48} - 75q^{49} + 31q^{50} - 26q^{51} + 14q^{52} + 10q^{53} + 23q^{54} - 16q^{55} + 8q^{56} + 42q^{57} + 30q^{58} + 16q^{59} + 28q^{60} + 62q^{61} + 32q^{62} + 118q^{63} + q^{64} + 84q^{65} + 100q^{66} + 68q^{67} + 18q^{68} + 133q^{69} + 48q^{70} + 72q^{71} + 45q^{72} + 74q^{73} + 38q^{74} + 163q^{75} + 20q^{76} + 96q^{77} + 80q^{78} + 36q^{79} + 6q^{80} + 45q^{81} + 42q^{82} + 40q^{83} + 30q^{84} + 64q^{85} + 44q^{86} - 36q^{87} + 12q^{88} + 46q^{89} + 6q^{90} + 24q^{91} + 23q^{92} + 32q^{93} + 48q^{94} - 12q^{95} + q^{96} - 78q^{97} - 31q^{98} - 98q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(138))$$

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
138.2.a $$\chi_{138}(1, \cdot)$$ 138.2.a.a 1 1
138.2.a.b 1
138.2.a.c 1
138.2.a.d 2
138.2.d $$\chi_{138}(137, \cdot)$$ 138.2.d.a 8 1
138.2.e $$\chi_{138}(13, \cdot)$$ 138.2.e.a 10 10
138.2.e.b 10
138.2.e.c 10
138.2.e.d 10
138.2.f $$\chi_{138}(5, \cdot)$$ 138.2.f.a 80 10

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(138))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(138)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(46))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(69))$$$$^{\oplus 2}$$