## Defining parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$20$$ Sturm bound: $$13824$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(252))$$.

Total New Old
Modular forms 5424 2342 3082
Cusp forms 4944 2250 2694
Eisenstein series 480 92 388

## Trace form

 $$2250 q - 3 q^{2} + 6 q^{3} - 31 q^{4} - 42 q^{5} - 54 q^{6} + 22 q^{7} + 33 q^{8} - 114 q^{9} + O(q^{10})$$ $$2250 q - 3 q^{2} + 6 q^{3} - 31 q^{4} - 42 q^{5} - 54 q^{6} + 22 q^{7} + 33 q^{8} - 114 q^{9} + 268 q^{10} - 114 q^{11} - 162 q^{13} - 87 q^{14} + 480 q^{15} - 619 q^{16} + 240 q^{17} + 228 q^{18} - 46 q^{19} + 450 q^{20} - 138 q^{21} + 1044 q^{22} - 696 q^{23} - 18 q^{24} - 716 q^{25} + 294 q^{26} - 900 q^{27} + 273 q^{28} - 1212 q^{29} - 1638 q^{30} - 1024 q^{31} - 3483 q^{32} + 1230 q^{33} - 1064 q^{34} + 1386 q^{35} - 1410 q^{36} + 888 q^{37} + 72 q^{38} + 1332 q^{39} + 2626 q^{40} + 1518 q^{41} + 1962 q^{42} + 266 q^{43} + 2532 q^{44} - 1008 q^{45} + 282 q^{46} + 324 q^{47} + 2670 q^{48} + 5652 q^{49} + 1671 q^{50} - 990 q^{51} + 2794 q^{52} + 1152 q^{53} + 2628 q^{54} + 72 q^{55} + 2421 q^{56} + 1974 q^{57} + 2680 q^{58} + 96 q^{59} + 2370 q^{60} - 9198 q^{61} + 3312 q^{63} - 811 q^{64} - 3576 q^{65} - 5106 q^{66} - 4354 q^{67} - 6360 q^{68} - 3420 q^{69} - 1290 q^{70} - 4200 q^{71} - 7728 q^{72} - 324 q^{73} - 12492 q^{74} - 7086 q^{75} - 2904 q^{76} - 1782 q^{77} - 9444 q^{78} + 2408 q^{79} - 9162 q^{80} + 6474 q^{81} - 4808 q^{82} + 6654 q^{83} - 3126 q^{84} + 6644 q^{85} + 7278 q^{86} + 8964 q^{87} - 810 q^{88} + 17460 q^{89} + 17718 q^{90} + 3590 q^{91} + 12222 q^{92} + 4200 q^{93} + 3486 q^{94} + 2172 q^{95} + 13920 q^{96} - 3474 q^{97} + 6999 q^{98} - 3612 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(252))$$

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
252.4.a $$\chi_{252}(1, \cdot)$$ 252.4.a.a 1 1
252.4.a.b 1
252.4.a.c 1
252.4.a.d 1
252.4.a.e 2
252.4.a.f 2
252.4.b $$\chi_{252}(55, \cdot)$$ 252.4.b.a 2 1
252.4.b.b 4
252.4.b.c 4
252.4.b.d 8
252.4.b.e 12
252.4.b.f 12
252.4.b.g 16
252.4.e $$\chi_{252}(71, \cdot)$$ 252.4.e.a 36 1
252.4.f $$\chi_{252}(125, \cdot)$$ 252.4.f.a 8 1
252.4.i $$\chi_{252}(25, \cdot)$$ 252.4.i.a 48 2
252.4.j $$\chi_{252}(85, \cdot)$$ 252.4.j.a 18 2
252.4.j.b 18
252.4.k $$\chi_{252}(37, \cdot)$$ 252.4.k.a 2 2
252.4.k.b 2
252.4.k.c 4
252.4.k.d 4
252.4.k.e 4
252.4.k.f 4
252.4.l $$\chi_{252}(193, \cdot)$$ 252.4.l.a 48 2
252.4.n $$\chi_{252}(31, \cdot)$$ n/a 280 2
252.4.o $$\chi_{252}(95, \cdot)$$ n/a 280 2
252.4.t $$\chi_{252}(17, \cdot)$$ 252.4.t.a 16 2
252.4.w $$\chi_{252}(5, \cdot)$$ 252.4.w.a 48 2
252.4.x $$\chi_{252}(41, \cdot)$$ 252.4.x.a 48 2
252.4.ba $$\chi_{252}(155, \cdot)$$ n/a 216 2
252.4.bb $$\chi_{252}(11, \cdot)$$ n/a 280 2
252.4.be $$\chi_{252}(107, \cdot)$$ 252.4.be.a 96 2
252.4.bf $$\chi_{252}(19, \cdot)$$ n/a 116 2
252.4.bi $$\chi_{252}(139, \cdot)$$ n/a 280 2
252.4.bj $$\chi_{252}(103, \cdot)$$ n/a 280 2
252.4.bm $$\chi_{252}(173, \cdot)$$ 252.4.bm.a 48 2

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(252))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(252)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 2}$$