# Properties

 Label 663.2 Level 663 Weight 2 Dimension 11543 Nonzero newspaces 36 Newform subspaces 78 Sturm bound 64512 Trace bound 17

## Defining parameters

 Level: $$N$$ = $$663 = 3 \cdot 13 \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$36$$ Newform subspaces: $$78$$ Sturm bound: $$64512$$ Trace bound: $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(663))$$.

Total New Old
Modular forms 16896 12199 4697
Cusp forms 15361 11543 3818
Eisenstein series 1535 656 879

## Trace form

 $$11543q + 9q^{2} - 65q^{3} - 115q^{4} + 18q^{5} - 59q^{6} - 120q^{7} + 9q^{8} - 69q^{9} + O(q^{10})$$ $$11543q + 9q^{2} - 65q^{3} - 115q^{4} + 18q^{5} - 59q^{6} - 120q^{7} + 9q^{8} - 69q^{9} - 158q^{10} - 20q^{11} - 147q^{12} - 189q^{13} - 40q^{14} - 122q^{15} - 267q^{16} - 15q^{17} - 251q^{18} - 188q^{19} - 118q^{20} - 144q^{21} - 212q^{22} - 8q^{23} - 175q^{24} - 239q^{25} - 111q^{26} - 221q^{27} - 328q^{28} - 50q^{29} - 190q^{30} - 216q^{31} - 91q^{32} - 124q^{33} - 347q^{34} - 32q^{35} - 51q^{36} - 162q^{37} + 20q^{38} - 33q^{39} - 386q^{40} - 38q^{41} - 32q^{42} - 228q^{43} + 4q^{44} + 22q^{45} - 224q^{46} + 48q^{47} + 65q^{48} - 141q^{49} + 51q^{50} - 29q^{51} - 439q^{52} - 78q^{53} - 123q^{54} - 312q^{55} - 160q^{56} - 188q^{57} - 390q^{58} - 116q^{59} - 226q^{60} - 370q^{61} - 200q^{62} - 240q^{63} - 423q^{64} - 138q^{65} - 228q^{66} - 220q^{67} - 245q^{68} - 312q^{69} - 512q^{70} - 24q^{71} - 203q^{72} - 394q^{73} - 150q^{74} - 235q^{75} - 420q^{76} - 176q^{77} - 55q^{78} - 464q^{79} - 262q^{80} - 109q^{81} - 394q^{82} - 44q^{83} - 208q^{84} - 268q^{85} + 76q^{86} - 26q^{87} - 388q^{88} - 66q^{89} + 70q^{90} - 328q^{91} - 152q^{92} - 32q^{93} - 392q^{94} - 104q^{95} - 27q^{96} - 234q^{97} + 57q^{98} - 108q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(663))$$

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
663.2.a $$\chi_{663}(1, \cdot)$$ 663.2.a.a 1 1
663.2.a.b 1
663.2.a.c 1
663.2.a.d 3
663.2.a.e 3
663.2.a.f 5
663.2.a.g 5
663.2.a.h 6
663.2.a.i 6
663.2.b $$\chi_{663}(103, \cdot)$$ 663.2.b.a 2 1
663.2.b.b 2
663.2.b.c 6
663.2.b.d 8
663.2.b.e 8
663.2.b.f 10
663.2.e $$\chi_{663}(220, \cdot)$$ 663.2.e.a 40 1
663.2.f $$\chi_{663}(118, \cdot)$$ 663.2.f.a 2 1
663.2.f.b 16
663.2.f.c 18
663.2.i $$\chi_{663}(256, \cdot)$$ 663.2.i.a 2 2
663.2.i.b 2
663.2.i.c 2
663.2.i.d 2
663.2.i.e 12
663.2.i.f 16
663.2.i.g 18
663.2.i.h 22
663.2.j $$\chi_{663}(157, \cdot)$$ 663.2.j.a 32 2
663.2.j.b 40
663.2.m $$\chi_{663}(200, \cdot)$$ 663.2.m.a 160 2
663.2.n $$\chi_{663}(86, \cdot)$$ 663.2.n.a 76 2
663.2.n.b 76
663.2.q $$\chi_{663}(203, \cdot)$$ 663.2.q.a 8 2
663.2.q.b 8
663.2.q.c 144
663.2.r $$\chi_{663}(47, \cdot)$$ 663.2.r.a 160 2
663.2.u $$\chi_{663}(64, \cdot)$$ 663.2.u.a 80 2
663.2.w $$\chi_{663}(16, \cdot)$$ 663.2.w.a 4 2
663.2.w.b 8
663.2.w.c 76
663.2.z $$\chi_{663}(205, \cdot)$$ 663.2.z.a 2 2
663.2.z.b 4
663.2.z.c 12
663.2.z.d 16
663.2.z.e 20
663.2.z.f 22
663.2.ba $$\chi_{663}(322, \cdot)$$ 663.2.ba.a 80 2
663.2.bd $$\chi_{663}(8, \cdot)$$ 663.2.bd.a 4 4
663.2.bd.b 4
663.2.bd.c 312
663.2.bg $$\chi_{663}(196, \cdot)$$ 663.2.bg.a 64 4
663.2.bg.b 80
663.2.bh $$\chi_{663}(25, \cdot)$$ 663.2.bh.a 176 4
663.2.bi $$\chi_{663}(161, \cdot)$$ 663.2.bi.a 4 4
663.2.bi.b 4
663.2.bi.c 312
663.2.bk $$\chi_{663}(4, \cdot)$$ 663.2.bk.a 160 4
663.2.bm $$\chi_{663}(89, \cdot)$$ 663.2.bm.a 320 4
663.2.bp $$\chi_{663}(50, \cdot)$$ 663.2.bp.a 8 4
663.2.bp.b 8
663.2.bp.c 304
663.2.bq $$\chi_{663}(137, \cdot)$$ 663.2.bq.a 148 4
663.2.bq.b 148
663.2.bt $$\chi_{663}(98, \cdot)$$ 663.2.bt.a 320 4
663.2.bv $$\chi_{663}(55, \cdot)$$ 663.2.bv.a 176 4
663.2.bx $$\chi_{663}(116, \cdot)$$ 663.2.bx.a 640 8
663.2.by $$\chi_{663}(14, \cdot)$$ 663.2.by.a 288 8
663.2.by.b 288
663.2.ca $$\chi_{663}(31, \cdot)$$ 663.2.ca.a 336 8
663.2.cd $$\chi_{663}(73, \cdot)$$ 663.2.cd.a 336 8
663.2.cf $$\chi_{663}(110, \cdot)$$ 663.2.cf.a 640 8
663.2.cg $$\chi_{663}(94, \cdot)$$ 663.2.cg.a 320 8
663.2.ch $$\chi_{663}(43, \cdot)$$ 663.2.ch.a 352 8
663.2.ck $$\chi_{663}(2, \cdot)$$ 663.2.ck.a 640 8
663.2.cm $$\chi_{663}(7, \cdot)$$ 663.2.cm.a 672 16
663.2.cp $$\chi_{663}(28, \cdot)$$ 663.2.cp.a 672 16
663.2.cr $$\chi_{663}(29, \cdot)$$ 663.2.cr.a 1280 16
663.2.cs $$\chi_{663}(23, \cdot)$$ 663.2.cs.a 1280 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(663))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(663)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(221))$$$$^{\oplus 2}$$