# Properties

 Label 975.2 Level 975 Weight 2 Dimension 22496 Nonzero newspaces 40 Newform subspaces 243 Sturm bound 134400 Trace bound 6

## Defining parameters

 Level: $$N$$ = $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$40$$ Newform subspaces: $$243$$ Sturm bound: $$134400$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(975))$$.

Total New Old
Modular forms 34944 23400 11544
Cusp forms 32257 22496 9761
Eisenstein series 2687 904 1783

## Trace form

 $$22496 q + 2 q^{2} - 60 q^{3} - 106 q^{4} + 12 q^{5} - 80 q^{6} - 96 q^{7} + 60 q^{8} - 50 q^{9} + O(q^{10})$$ $$22496 q + 2 q^{2} - 60 q^{3} - 106 q^{4} + 12 q^{5} - 80 q^{6} - 96 q^{7} + 60 q^{8} - 50 q^{9} - 124 q^{10} + 20 q^{11} - 14 q^{12} - 102 q^{13} + 72 q^{14} - 68 q^{15} - 146 q^{16} + 22 q^{17} - 52 q^{18} - 100 q^{19} - 72 q^{20} - 88 q^{21} - 128 q^{22} - 8 q^{23} - 144 q^{24} - 220 q^{25} + 66 q^{26} - 102 q^{27} - 160 q^{28} + 2 q^{29} - 124 q^{30} - 172 q^{31} + 26 q^{32} - 52 q^{33} - 68 q^{34} + 40 q^{35} - 174 q^{36} - 94 q^{37} - 16 q^{38} - 106 q^{39} - 388 q^{40} + 86 q^{41} - 192 q^{42} - 108 q^{43} - 36 q^{44} - 224 q^{45} - 156 q^{46} + 16 q^{47} - 368 q^{48} - 162 q^{49} - 236 q^{50} - 324 q^{51} - 440 q^{52} - 208 q^{53} - 548 q^{54} - 344 q^{55} - 396 q^{56} - 360 q^{57} - 490 q^{58} - 260 q^{59} - 412 q^{60} - 222 q^{61} - 452 q^{62} - 332 q^{63} - 596 q^{64} - 134 q^{65} - 508 q^{66} - 252 q^{67} - 386 q^{68} - 200 q^{69} - 200 q^{70} - 56 q^{71} - 318 q^{72} - 176 q^{73} - 86 q^{74} + 12 q^{75} - 708 q^{76} - 24 q^{77} - 302 q^{78} - 232 q^{79} + 140 q^{80} - 134 q^{81} - 10 q^{82} + 92 q^{83} - 104 q^{84} - 268 q^{85} + 104 q^{86} - 26 q^{87} - 80 q^{88} - 24 q^{89} - 48 q^{90} - 116 q^{91} + 104 q^{92} - 152 q^{93} - 284 q^{94} - 112 q^{95} - 78 q^{96} - 340 q^{97} - 34 q^{98} + 8 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(975))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
975.2.a $$\chi_{975}(1, \cdot)$$ 975.2.a.a 1 1
975.2.a.b 1
975.2.a.c 1
975.2.a.d 1
975.2.a.e 1
975.2.a.f 1
975.2.a.g 1
975.2.a.h 1
975.2.a.i 1
975.2.a.j 1
975.2.a.k 1
975.2.a.l 2
975.2.a.m 3
975.2.a.n 3
975.2.a.o 3
975.2.a.p 3
975.2.a.q 3
975.2.a.r 5
975.2.a.s 5
975.2.b $$\chi_{975}(376, \cdot)$$ 975.2.b.a 2 1
975.2.b.b 2
975.2.b.c 2
975.2.b.d 2
975.2.b.e 2
975.2.b.f 4
975.2.b.g 4
975.2.b.h 4
975.2.b.i 6
975.2.b.j 6
975.2.b.k 6
975.2.b.l 6
975.2.c $$\chi_{975}(274, \cdot)$$ 975.2.c.a 2 1
975.2.c.b 2
975.2.c.c 2
975.2.c.d 2
975.2.c.e 2
975.2.c.f 2
975.2.c.g 2
975.2.c.h 4
975.2.c.i 6
975.2.c.j 6
975.2.c.k 6
975.2.h $$\chi_{975}(649, \cdot)$$ 975.2.h.a 2 1
975.2.h.b 2
975.2.h.c 2
975.2.h.d 2
975.2.h.e 4
975.2.h.f 4
975.2.h.g 4
975.2.h.h 8
975.2.h.i 12
975.2.i $$\chi_{975}(451, \cdot)$$ 975.2.i.a 2 2
975.2.i.b 2
975.2.i.c 2
975.2.i.d 2
975.2.i.e 2
975.2.i.f 2
975.2.i.g 2
975.2.i.h 2
975.2.i.i 2
975.2.i.j 4
975.2.i.k 4
975.2.i.l 6
975.2.i.m 6
975.2.i.n 12
975.2.i.o 12
975.2.i.p 12
975.2.i.q 12
975.2.k $$\chi_{975}(307, \cdot)$$ 975.2.k.a 4 2
975.2.k.b 4
975.2.k.c 8
975.2.k.d 28
975.2.k.e 40
975.2.m $$\chi_{975}(443, \cdot)$$ 975.2.m.a 32 2
975.2.m.b 32
975.2.m.c 32
975.2.m.d 48
975.2.n $$\chi_{975}(749, \cdot)$$ 975.2.n.a 4 2
975.2.n.b 4
975.2.n.c 4
975.2.n.d 4
975.2.n.e 4
975.2.n.f 4
975.2.n.g 4
975.2.n.h 4
975.2.n.i 4
975.2.n.j 4
975.2.n.k 4
975.2.n.l 4
975.2.n.m 8
975.2.n.n 8
975.2.n.o 8
975.2.n.p 8
975.2.n.q 40
975.2.n.r 40
975.2.o $$\chi_{975}(476, \cdot)$$ 975.2.o.a 4 2
975.2.o.b 4
975.2.o.c 4
975.2.o.d 4
975.2.o.e 4
975.2.o.f 4
975.2.o.g 4
975.2.o.h 4
975.2.o.i 4
975.2.o.j 4
975.2.o.k 4
975.2.o.l 8
975.2.o.m 8
975.2.o.n 8
975.2.o.o 8
975.2.o.p 40
975.2.o.q 48
975.2.s $$\chi_{975}(818, \cdot)$$ 975.2.s.a 8 2
975.2.s.b 8
975.2.s.c 16
975.2.s.d 32
975.2.s.e 32
975.2.s.f 64
975.2.t $$\chi_{975}(268, \cdot)$$ 975.2.t.a 4 2
975.2.t.b 4
975.2.t.c 8
975.2.t.d 28
975.2.t.e 40
975.2.v $$\chi_{975}(196, \cdot)$$ 975.2.v.a 52 4
975.2.v.b 52
975.2.v.c 68
975.2.v.d 68
975.2.w $$\chi_{975}(49, \cdot)$$ 975.2.w.a 4 2
975.2.w.b 4
975.2.w.c 4
975.2.w.d 4
975.2.w.e 4
975.2.w.f 4
975.2.w.g 8
975.2.w.h 8
975.2.w.i 8
975.2.w.j 8
975.2.w.k 24
975.2.bb $$\chi_{975}(724, \cdot)$$ 975.2.bb.a 4 2
975.2.bb.b 4
975.2.bb.c 4
975.2.bb.d 4
975.2.bb.e 4
975.2.bb.f 4
975.2.bb.g 4
975.2.bb.h 4
975.2.bb.i 8
975.2.bb.j 12
975.2.bb.k 12
975.2.bb.l 24
975.2.bc $$\chi_{975}(751, \cdot)$$ 975.2.bc.a 2 2
975.2.bc.b 2
975.2.bc.c 2
975.2.bc.d 2
975.2.bc.e 2
975.2.bc.f 2
975.2.bc.g 2
975.2.bc.h 4
975.2.bc.i 8
975.2.bc.j 8
975.2.bc.k 12
975.2.bc.l 12
975.2.bc.m 16
975.2.bc.n 16
975.2.bf $$\chi_{975}(64, \cdot)$$ 975.2.bf.a 288 4
975.2.bg $$\chi_{975}(79, \cdot)$$ 975.2.bg.a 104 4
975.2.bg.b 136
975.2.bh $$\chi_{975}(181, \cdot)$$ 975.2.bh.a 8 4
975.2.bh.b 128
975.2.bh.c 136
975.2.bl $$\chi_{975}(193, \cdot)$$ 975.2.bl.a 8 4
975.2.bl.b 8
975.2.bl.c 8
975.2.bl.d 8
975.2.bl.e 16
975.2.bl.f 16
975.2.bl.g 16
975.2.bl.h 32
975.2.bl.i 56
975.2.bn $$\chi_{975}(218, \cdot)$$ 975.2.bn.a 8 4
975.2.bn.b 8
975.2.bn.c 64
975.2.bn.d 96
975.2.bn.e 144
975.2.bo $$\chi_{975}(176, \cdot)$$ 975.2.bo.a 4 4
975.2.bo.b 4
975.2.bo.c 4
975.2.bo.d 8
975.2.bo.e 72
975.2.bo.f 72
975.2.bo.g 72
975.2.bo.h 96
975.2.bp $$\chi_{975}(149, \cdot)$$ 975.2.bp.a 4 4
975.2.bp.b 4
975.2.bp.c 4
975.2.bp.d 4
975.2.bp.e 8
975.2.bp.f 8
975.2.bp.g 72
975.2.bp.h 72
975.2.bp.i 72
975.2.bp.j 72
975.2.bt $$\chi_{975}(68, \cdot)$$ 975.2.bt.a 8 4
975.2.bt.b 8
975.2.bt.c 8
975.2.bt.d 8
975.2.bt.e 8
975.2.bt.f 8
975.2.bt.g 8
975.2.bt.h 8
975.2.bt.i 8
975.2.bt.j 8
975.2.bt.k 48
975.2.bt.l 96
975.2.bt.m 96
975.2.bu $$\chi_{975}(7, \cdot)$$ 975.2.bu.a 8 4
975.2.bu.b 8
975.2.bu.c 8
975.2.bu.d 8
975.2.bu.e 16
975.2.bu.f 16
975.2.bu.g 16
975.2.bu.h 32
975.2.bu.i 56
975.2.bw $$\chi_{975}(16, \cdot)$$ 975.2.bw.a 288 8
975.2.bw.b 288
975.2.bx $$\chi_{975}(73, \cdot)$$ 975.2.bx.a 560 8
975.2.bz $$\chi_{975}(38, \cdot)$$ 975.2.bz.a 64 8
975.2.bz.b 1024
975.2.cd $$\chi_{975}(86, \cdot)$$ 975.2.cd.a 1088 8
975.2.ce $$\chi_{975}(44, \cdot)$$ 975.2.ce.a 1088 8
975.2.cf $$\chi_{975}(53, \cdot)$$ 975.2.cf.a 960 8
975.2.ci $$\chi_{975}(112, \cdot)$$ 975.2.ci.a 560 8
975.2.cl $$\chi_{975}(121, \cdot)$$ 975.2.cl.a 272 8
975.2.cl.b 272
975.2.cm $$\chi_{975}(94, \cdot)$$ 975.2.cm.a 544 8
975.2.cn $$\chi_{975}(4, \cdot)$$ 975.2.cn.a 576 8
975.2.cq $$\chi_{975}(28, \cdot)$$ 975.2.cq.a 1120 16
975.2.cs $$\chi_{975}(113, \cdot)$$ 975.2.cs.a 2176 16
975.2.cw $$\chi_{975}(59, \cdot)$$ 975.2.cw.a 2176 16
975.2.cx $$\chi_{975}(11, \cdot)$$ 975.2.cx.a 2176 16
975.2.cy $$\chi_{975}(17, \cdot)$$ 975.2.cy.a 2176 16
975.2.db $$\chi_{975}(67, \cdot)$$ 975.2.db.a 1120 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(975))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(975)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(195))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(325))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(975))$$$$^{\oplus 1}$$