# Properties

 Label 252.9 Level 252 Weight 9 Dimension 6072 Nonzero newspaces 20 Sturm bound 31104 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$9$$ Nonzero newspaces: $$20$$ Sturm bound: $$31104$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(\Gamma_1(252))$$.

Total New Old
Modular forms 14064 6160 7904
Cusp forms 13584 6072 7512
Eisenstein series 480 88 392

## Trace form

 $$6072 q - 3 q^{2} - 42 q^{3} + 657 q^{4} - 2073 q^{5} - 2730 q^{6} + 2153 q^{7} + 23235 q^{8} - 14430 q^{9} + O(q^{10})$$ $$6072 q - 3 q^{2} - 42 q^{3} + 657 q^{4} - 2073 q^{5} - 2730 q^{6} + 2153 q^{7} + 23235 q^{8} - 14430 q^{9} - 50322 q^{10} + 39177 q^{11} + 51324 q^{12} - 280198 q^{13} - 82065 q^{14} - 195450 q^{15} - 268323 q^{16} - 70959 q^{17} + 1116324 q^{18} - 193693 q^{19} - 872970 q^{20} - 815799 q^{21} + 423270 q^{22} - 1803969 q^{23} - 73278 q^{24} + 3710310 q^{25} + 4544946 q^{26} - 550026 q^{27} + 1806273 q^{28} - 4848942 q^{29} - 5476782 q^{30} - 2475427 q^{31} - 9181173 q^{32} - 5995152 q^{33} + 6015606 q^{34} + 16385067 q^{35} + 1210542 q^{36} - 2980025 q^{37} - 26869104 q^{38} - 14359794 q^{39} - 9123402 q^{40} + 3713016 q^{41} + 19665096 q^{42} + 24365200 q^{43} + 20755086 q^{44} + 1792506 q^{45} + 20488830 q^{46} - 20157579 q^{47} - 34534998 q^{48} + 96667353 q^{49} - 14139657 q^{50} - 51710634 q^{51} + 53024298 q^{52} + 102502281 q^{53} + 47327220 q^{54} - 26343324 q^{55} - 78138369 q^{56} - 40055970 q^{57} + 14040828 q^{58} + 1202697 q^{59} + 38221470 q^{60} - 23761027 q^{61} - 23445948 q^{62} + 61425531 q^{63} + 137108613 q^{64} + 122874276 q^{65} + 114911850 q^{66} + 49208979 q^{67} - 185343078 q^{68} - 194799990 q^{69} + 57932058 q^{70} - 189429480 q^{71} + 25652520 q^{72} + 246847001 q^{73} - 128905734 q^{74} + 291745866 q^{75} + 171601794 q^{76} + 170123106 q^{77} + 391344264 q^{78} - 115684563 q^{79} + 165269430 q^{80} - 478457790 q^{81} - 118932690 q^{82} - 104964174 q^{83} - 297452502 q^{84} + 218034894 q^{85} - 446636652 q^{86} - 8406828 q^{87} - 187924332 q^{88} + 1006244853 q^{89} + 137844354 q^{90} + 13333214 q^{91} + 302879424 q^{92} - 805966476 q^{93} - 134024832 q^{94} + 20983023 q^{95} + 751803840 q^{96} + 1192323188 q^{97} - 84413661 q^{98} + 629532330 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(\Gamma_1(252))$$

We only show spaces with odd 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
252.9.c $$\chi_{252}(197, \cdot)$$ 252.9.c.a 16 1
252.9.d $$\chi_{252}(181, \cdot)$$ 252.9.d.a 2 1
252.9.d.b 6
252.9.d.c 8
252.9.d.d 10
252.9.g $$\chi_{252}(127, \cdot)$$ n/a 120 1
252.9.h $$\chi_{252}(251, \cdot)$$ n/a 128 1
252.9.m $$\chi_{252}(65, \cdot)$$ n/a 128 2
252.9.p $$\chi_{252}(61, \cdot)$$ n/a 128 2
252.9.q $$\chi_{252}(143, \cdot)$$ n/a 256 2
252.9.r $$\chi_{252}(131, \cdot)$$ n/a 760 2
252.9.s $$\chi_{252}(83, \cdot)$$ n/a 760 2
252.9.u $$\chi_{252}(151, \cdot)$$ n/a 760 2
252.9.v $$\chi_{252}(43, \cdot)$$ n/a 576 2
252.9.y $$\chi_{252}(163, \cdot)$$ n/a 316 2
252.9.z $$\chi_{252}(73, \cdot)$$ 252.9.z.a 2 2
252.9.z.b 10
252.9.z.c 10
252.9.z.d 12
252.9.z.e 20
252.9.bc $$\chi_{252}(13, \cdot)$$ n/a 128 2
252.9.bd $$\chi_{252}(229, \cdot)$$ n/a 128 2
252.9.bg $$\chi_{252}(29, \cdot)$$ 252.9.bg.a 96 2
252.9.bh $$\chi_{252}(137, \cdot)$$ n/a 128 2
252.9.bk $$\chi_{252}(53, \cdot)$$ 252.9.bk.a 44 2
252.9.bl $$\chi_{252}(67, \cdot)$$ n/a 760 2
252.9.bn $$\chi_{252}(47, \cdot)$$ n/a 760 2

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

## Decomposition of $$S_{9}^{\mathrm{old}}(\Gamma_1(252))$$ into lower level spaces

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