# Properties

 Label 504.2 Level 504 Weight 2 Dimension 2860 Nonzero newspaces 30 Newform subspaces 81 Sturm bound 27648 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$30$$ Newform subspaces: $$81$$ Sturm bound: $$27648$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(504))$$.

Total New Old
Modular forms 7488 3040 4448
Cusp forms 6337 2860 3477
Eisenstein series 1151 180 971

## Trace form

 $$2860 q - 14 q^{2} - 18 q^{3} - 14 q^{4} - 8 q^{5} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 30 q^{9} + O(q^{10})$$ $$2860 q - 14 q^{2} - 18 q^{3} - 14 q^{4} - 8 q^{5} - 8 q^{6} - 16 q^{7} - 8 q^{8} - 30 q^{9} - 6 q^{10} + 10 q^{11} + 4 q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 2 q^{17} - 24 q^{18} - 10 q^{19} - 20 q^{20} + 12 q^{21} - 34 q^{22} + 36 q^{23} - 72 q^{24} - 2 q^{25} - 44 q^{26} - 24 q^{27} - 14 q^{28} - 12 q^{29} - 116 q^{30} + 56 q^{31} - 64 q^{32} - 10 q^{33} + 46 q^{34} + 18 q^{35} - 148 q^{36} + 30 q^{37} - 76 q^{38} - 60 q^{39} - 8 q^{40} + 46 q^{41} - 74 q^{42} - 38 q^{43} - 126 q^{44} + 44 q^{45} - 116 q^{46} - 12 q^{47} - 64 q^{48} + 56 q^{49} - 134 q^{50} - 6 q^{51} - 20 q^{52} + 82 q^{53} - 24 q^{54} + 32 q^{55} - 28 q^{56} - 38 q^{57} - 28 q^{58} + 16 q^{59} + 72 q^{60} + 16 q^{61} + 56 q^{62} - 56 q^{63} - 110 q^{64} + 32 q^{65} + 112 q^{66} + 26 q^{67} + 30 q^{68} - 80 q^{69} - 140 q^{70} - 160 q^{71} + 48 q^{72} - 102 q^{73} - 50 q^{74} - 230 q^{75} - 166 q^{76} - 138 q^{77} - 68 q^{78} - 160 q^{79} - 168 q^{80} - 106 q^{81} - 310 q^{82} - 286 q^{83} - 176 q^{84} - 160 q^{85} - 242 q^{86} - 168 q^{87} - 262 q^{88} - 146 q^{89} - 224 q^{90} - 258 q^{91} - 402 q^{92} - 52 q^{93} - 318 q^{94} - 368 q^{95} - 288 q^{96} - 114 q^{97} - 326 q^{98} - 180 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(504))$$

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
504.2.a $$\chi_{504}(1, \cdot)$$ 504.2.a.a 1 1
504.2.a.b 1
504.2.a.c 1
504.2.a.d 1
504.2.a.e 1
504.2.a.f 1
504.2.a.g 1
504.2.a.h 1
504.2.b $$\chi_{504}(55, \cdot)$$ None 0 1
504.2.c $$\chi_{504}(253, \cdot)$$ 504.2.c.a 2 1
504.2.c.b 4
504.2.c.c 4
504.2.c.d 4
504.2.c.e 8
504.2.c.f 8
504.2.h $$\chi_{504}(71, \cdot)$$ None 0 1
504.2.i $$\chi_{504}(125, \cdot)$$ 504.2.i.a 8 1
504.2.i.b 24
504.2.j $$\chi_{504}(323, \cdot)$$ 504.2.j.a 24 1
504.2.k $$\chi_{504}(377, \cdot)$$ 504.2.k.a 8 1
504.2.p $$\chi_{504}(307, \cdot)$$ 504.2.p.a 2 1
504.2.p.b 4
504.2.p.c 4
504.2.p.d 4
504.2.p.e 4
504.2.p.f 4
504.2.p.g 16
504.2.q $$\chi_{504}(25, \cdot)$$ 504.2.q.a 2 2
504.2.q.b 2
504.2.q.c 22
504.2.q.d 22
504.2.r $$\chi_{504}(169, \cdot)$$ 504.2.r.a 2 2
504.2.r.b 2
504.2.r.c 6
504.2.r.d 8
504.2.r.e 8
504.2.r.f 10
504.2.s $$\chi_{504}(289, \cdot)$$ 504.2.s.a 2 2
504.2.s.b 2
504.2.s.c 2
504.2.s.d 2
504.2.s.e 2
504.2.s.f 2
504.2.s.g 2
504.2.s.h 2
504.2.s.i 4
504.2.t $$\chi_{504}(193, \cdot)$$ 504.2.t.a 2 2
504.2.t.b 2
504.2.t.c 22
504.2.t.d 22
504.2.w $$\chi_{504}(205, \cdot)$$ 504.2.w.a 184 2
504.2.x $$\chi_{504}(31, \cdot)$$ None 0 2
504.2.y $$\chi_{504}(173, \cdot)$$ 504.2.y.a 184 2
504.2.z $$\chi_{504}(95, \cdot)$$ None 0 2
504.2.be $$\chi_{504}(139, \cdot)$$ 504.2.be.a 184 2
504.2.bf $$\chi_{504}(115, \cdot)$$ 504.2.bf.a 4 2
504.2.bf.b 180
504.2.bk $$\chi_{504}(19, \cdot)$$ 504.2.bk.a 12 2
504.2.bk.b 32
504.2.bk.c 32
504.2.bl $$\chi_{504}(17, \cdot)$$ 504.2.bl.a 16 2
504.2.bm $$\chi_{504}(107, \cdot)$$ 504.2.bm.a 8 2
504.2.bm.b 8
504.2.bm.c 48
504.2.br $$\chi_{504}(155, \cdot)$$ 504.2.br.a 144 2
504.2.bs $$\chi_{504}(257, \cdot)$$ 504.2.bs.a 48 2
504.2.bt $$\chi_{504}(11, \cdot)$$ 504.2.bt.a 184 2
504.2.bu $$\chi_{504}(41, \cdot)$$ 504.2.bu.a 48 2
504.2.bz $$\chi_{504}(239, \cdot)$$ None 0 2
504.2.ca $$\chi_{504}(5, \cdot)$$ 504.2.ca.a 184 2
504.2.cb $$\chi_{504}(23, \cdot)$$ None 0 2
504.2.cc $$\chi_{504}(293, \cdot)$$ 504.2.cc.a 16 2
504.2.cc.b 168
504.2.ch $$\chi_{504}(269, \cdot)$$ 504.2.ch.a 8 2
504.2.ch.b 56
504.2.ci $$\chi_{504}(359, \cdot)$$ None 0 2
504.2.cj $$\chi_{504}(37, \cdot)$$ 504.2.cj.a 8 2
504.2.cj.b 8
504.2.cj.c 12
504.2.cj.d 16
504.2.cj.e 32
504.2.ck $$\chi_{504}(199, \cdot)$$ None 0 2
504.2.cp $$\chi_{504}(223, \cdot)$$ None 0 2
504.2.cq $$\chi_{504}(277, \cdot)$$ 504.2.cq.a 184 2
504.2.cr $$\chi_{504}(103, \cdot)$$ None 0 2
504.2.cs $$\chi_{504}(85, \cdot)$$ 504.2.cs.a 72 2
504.2.cs.b 72
504.2.cx $$\chi_{504}(185, \cdot)$$ 504.2.cx.a 48 2
504.2.cy $$\chi_{504}(347, \cdot)$$ 504.2.cy.a 184 2
504.2.cz $$\chi_{504}(187, \cdot)$$ 504.2.cz.a 4 2
504.2.cz.b 180

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(504))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(504)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(252))$$$$^{\oplus 2}$$