## Defining parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$20$$ Newform subspaces: $$40$$ Sturm bound: $$6912$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(252))$$.

Total New Old
Modular forms 1968 810 1158
Cusp forms 1489 718 771
Eisenstein series 479 92 387

## Trace form

 $$718 q - 3 q^{2} + q^{4} - 3 q^{5} - 6 q^{6} - q^{7} - 15 q^{8} + O(q^{10})$$ $$718 q - 3 q^{2} + q^{4} - 3 q^{5} - 6 q^{6} - q^{7} - 15 q^{8} - 10 q^{10} + 9 q^{11} - 24 q^{12} + 10 q^{13} - 9 q^{14} - 6 q^{15} - 11 q^{16} - 9 q^{17} - 48 q^{18} + 29 q^{19} - 54 q^{20} - 3 q^{21} - 36 q^{22} + 27 q^{23} - 18 q^{24} + 36 q^{25} - 42 q^{26} + 18 q^{27} - 39 q^{28} - 6 q^{29} - 6 q^{30} + 5 q^{31} - 3 q^{32} - 78 q^{33} - 52 q^{34} - 51 q^{35} - 18 q^{36} - 69 q^{37} - 54 q^{38} - 30 q^{39} - 118 q^{40} - 162 q^{41} - 60 q^{42} - 74 q^{43} - 120 q^{44} - 102 q^{45} - 156 q^{46} - 81 q^{47} - 114 q^{48} - 71 q^{49} - 171 q^{50} - 72 q^{51} - 142 q^{52} - 87 q^{53} - 108 q^{54} - 54 q^{55} - 99 q^{56} - 108 q^{57} - 76 q^{58} - 45 q^{59} - 30 q^{60} + 7 q^{61} - 81 q^{63} - 59 q^{64} - 114 q^{65} + 54 q^{66} + 63 q^{67} + 84 q^{68} - 42 q^{69} + 102 q^{70} - 24 q^{71} + 96 q^{72} - 29 q^{73} + 210 q^{74} - 72 q^{75} + 144 q^{76} - 126 q^{77} + 84 q^{78} - 3 q^{79} + 198 q^{80} - 132 q^{81} + 80 q^{82} - 66 q^{83} + 66 q^{84} - 158 q^{85} + 126 q^{86} + 198 q^{88} - 201 q^{89} + 90 q^{90} - 10 q^{91} + 102 q^{92} - 144 q^{93} + 144 q^{94} + 3 q^{95} + 48 q^{96} - 158 q^{97} + 111 q^{98} + 6 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(252))$$

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
252.2.a $$\chi_{252}(1, \cdot)$$ 252.2.a.a 1 1
252.2.a.b 1
252.2.b $$\chi_{252}(55, \cdot)$$ 252.2.b.a 2 1
252.2.b.b 4
252.2.b.c 4
252.2.b.d 4
252.2.b.e 4
252.2.e $$\chi_{252}(71, \cdot)$$ 252.2.e.a 12 1
252.2.f $$\chi_{252}(125, \cdot)$$ 252.2.f.a 4 1
252.2.i $$\chi_{252}(25, \cdot)$$ 252.2.i.a 2 2
252.2.i.b 14
252.2.j $$\chi_{252}(85, \cdot)$$ 252.2.j.a 6 2
252.2.j.b 6
252.2.k $$\chi_{252}(37, \cdot)$$ 252.2.k.a 2 2
252.2.k.b 2
252.2.k.c 2
252.2.l $$\chi_{252}(193, \cdot)$$ 252.2.l.a 2 2
252.2.l.b 14
252.2.n $$\chi_{252}(31, \cdot)$$ 252.2.n.a 4 2
252.2.n.b 84
252.2.o $$\chi_{252}(95, \cdot)$$ 252.2.o.a 88 2
252.2.t $$\chi_{252}(17, \cdot)$$ 252.2.t.a 4 2
252.2.w $$\chi_{252}(5, \cdot)$$ 252.2.w.a 16 2
252.2.x $$\chi_{252}(41, \cdot)$$ 252.2.x.a 16 2
252.2.ba $$\chi_{252}(155, \cdot)$$ 252.2.ba.a 72 2
252.2.bb $$\chi_{252}(11, \cdot)$$ 252.2.bb.a 88 2
252.2.be $$\chi_{252}(107, \cdot)$$ 252.2.be.a 32 2
252.2.bf $$\chi_{252}(19, \cdot)$$ 252.2.bf.a 4 2
252.2.bf.b 4
252.2.bf.c 4
252.2.bf.d 4
252.2.bf.e 4
252.2.bf.f 8
252.2.bf.g 8
252.2.bi $$\chi_{252}(139, \cdot)$$ 252.2.bi.a 4 2
252.2.bi.b 4
252.2.bi.c 80
252.2.bj $$\chi_{252}(103, \cdot)$$ 252.2.bj.a 4 2
252.2.bj.b 84
252.2.bm $$\chi_{252}(173, \cdot)$$ 252.2.bm.a 16 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(252))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(252)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 2}$$