# Properties

 Label 1638.2 Level 1638 Weight 2 Dimension 18134 Nonzero newspaces 84 Sturm bound 290304 Trace bound 27

## Defining parameters

 Level: $$N$$ = $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$84$$ Sturm bound: $$290304$$ Trace bound: $$27$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1638))$$.

Total New Old
Modular forms 74880 18134 56746
Cusp forms 70273 18134 52139
Eisenstein series 4607 0 4607

## Trace form

 $$18134 q - 4 q^{2} - 12 q^{3} - 8 q^{4} - 12 q^{5} + 12 q^{6} - 24 q^{7} + 2 q^{8} + 12 q^{9} + O(q^{10})$$ $$18134 q - 4 q^{2} - 12 q^{3} - 8 q^{4} - 12 q^{5} + 12 q^{6} - 24 q^{7} + 2 q^{8} + 12 q^{9} - 42 q^{10} - 36 q^{11} - 54 q^{13} + 20 q^{14} + 48 q^{15} + 42 q^{17} + 24 q^{18} + 4 q^{19} + 30 q^{20} + 84 q^{21} + 84 q^{22} + 96 q^{23} + 12 q^{24} + 50 q^{25} + 38 q^{26} + 72 q^{27} + 20 q^{28} + 138 q^{29} + 120 q^{30} + 168 q^{31} - 4 q^{32} + 156 q^{33} + 132 q^{34} + 228 q^{35} + 108 q^{36} + 146 q^{37} + 184 q^{38} + 228 q^{39} - 12 q^{40} + 342 q^{41} + 72 q^{42} + 132 q^{43} + 96 q^{44} + 336 q^{45} + 48 q^{46} + 288 q^{47} + 12 q^{48} + 136 q^{49} + 38 q^{50} + 60 q^{51} + 24 q^{52} - 120 q^{53} - 108 q^{54} - 120 q^{55} - 28 q^{56} - 180 q^{57} - 30 q^{58} - 336 q^{59} - 144 q^{60} - 66 q^{61} - 272 q^{62} - 312 q^{63} - 2 q^{64} - 54 q^{65} - 192 q^{66} + 92 q^{67} - 114 q^{68} - 72 q^{69} + 12 q^{72} + 280 q^{73} - 134 q^{74} + 12 q^{75} + 40 q^{76} + 156 q^{77} - 60 q^{78} + 280 q^{79} - 18 q^{80} + 180 q^{81} + 186 q^{82} + 444 q^{83} - 12 q^{84} + 450 q^{85} + 76 q^{86} + 408 q^{87} + 36 q^{88} + 552 q^{89} + 96 q^{90} + 398 q^{91} + 168 q^{92} + 360 q^{93} + 336 q^{94} + 624 q^{95} + 24 q^{96} + 420 q^{97} + 200 q^{98} + 24 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1638))$$

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
1638.2.a $$\chi_{1638}(1, \cdot)$$ 1638.2.a.a 1 1
1638.2.a.b 1
1638.2.a.c 1
1638.2.a.d 1
1638.2.a.e 1
1638.2.a.f 1
1638.2.a.g 1
1638.2.a.h 1
1638.2.a.i 1
1638.2.a.j 1
1638.2.a.k 1
1638.2.a.l 1
1638.2.a.m 1
1638.2.a.n 1
1638.2.a.o 1
1638.2.a.p 1
1638.2.a.q 1
1638.2.a.r 1
1638.2.a.s 1
1638.2.a.t 1
1638.2.a.u 2
1638.2.a.v 2
1638.2.a.w 2
1638.2.a.x 2
1638.2.a.y 2
1638.2.c $$\chi_{1638}(883, \cdot)$$ 1638.2.c.a 2 1
1638.2.c.b 2
1638.2.c.c 2
1638.2.c.d 2
1638.2.c.e 2
1638.2.c.f 2
1638.2.c.g 2
1638.2.c.h 4
1638.2.c.i 6
1638.2.c.j 6
1638.2.c.k 6
1638.2.e $$\chi_{1638}(1637, \cdot)$$ 1638.2.e.a 16 1
1638.2.e.b 16
1638.2.g $$\chi_{1638}(755, \cdot)$$ 1638.2.g.a 8 1
1638.2.g.b 8
1638.2.g.c 8
1638.2.g.d 8
1638.2.i $$\chi_{1638}(625, \cdot)$$ n/a 192 2
1638.2.j $$\chi_{1638}(235, \cdot)$$ 1638.2.j.a 2 2
1638.2.j.b 2
1638.2.j.c 2
1638.2.j.d 2
1638.2.j.e 2
1638.2.j.f 2
1638.2.j.g 2
1638.2.j.h 2
1638.2.j.i 2
1638.2.j.j 2
1638.2.j.k 4
1638.2.j.l 4
1638.2.j.m 4
1638.2.j.n 4
1638.2.j.o 4
1638.2.j.p 6
1638.2.j.q 6
1638.2.j.r 8
1638.2.j.s 10
1638.2.j.t 10
1638.2.k $$\chi_{1638}(211, \cdot)$$ n/a 168 2
1638.2.l $$\chi_{1638}(373, \cdot)$$ n/a 224 2
1638.2.m $$\chi_{1638}(289, \cdot)$$ 1638.2.m.a 2 2
1638.2.m.b 2
1638.2.m.c 2
1638.2.m.d 2
1638.2.m.e 2
1638.2.m.f 6
1638.2.m.g 8
1638.2.m.h 8
1638.2.m.i 8
1638.2.m.j 10
1638.2.m.k 10
1638.2.m.l 16
1638.2.m.m 16
1638.2.n $$\chi_{1638}(547, \cdot)$$ n/a 144 2
1638.2.o $$\chi_{1638}(841, \cdot)$$ n/a 168 2
1638.2.p $$\chi_{1638}(919, \cdot)$$ 1638.2.p.a 2 2
1638.2.p.b 2
1638.2.p.c 2
1638.2.p.d 2
1638.2.p.e 2
1638.2.p.f 6
1638.2.p.g 8
1638.2.p.h 8
1638.2.p.i 8
1638.2.p.j 10
1638.2.p.k 10
1638.2.p.l 16
1638.2.p.m 16
1638.2.q $$\chi_{1638}(529, \cdot)$$ n/a 224 2
1638.2.r $$\chi_{1638}(757, \cdot)$$ 1638.2.r.a 2 2
1638.2.r.b 2
1638.2.r.c 2
1638.2.r.d 2
1638.2.r.e 2
1638.2.r.f 2
1638.2.r.g 2
1638.2.r.h 2
1638.2.r.i 2
1638.2.r.j 2
1638.2.r.k 2
1638.2.r.l 2
1638.2.r.m 2
1638.2.r.n 2
1638.2.r.o 2
1638.2.r.p 2
1638.2.r.q 2
1638.2.r.r 2
1638.2.r.s 2
1638.2.r.t 2
1638.2.r.u 4
1638.2.r.v 4
1638.2.r.w 4
1638.2.r.x 4
1638.2.r.y 4
1638.2.r.z 4
1638.2.r.ba 4
1638.2.s $$\chi_{1638}(445, \cdot)$$ n/a 224 2
1638.2.t $$\chi_{1638}(835, \cdot)$$ n/a 224 2
1638.2.u $$\chi_{1638}(79, \cdot)$$ n/a 192 2
1638.2.x $$\chi_{1638}(307, \cdot)$$ 1638.2.x.a 8 2
1638.2.x.b 8
1638.2.x.c 8
1638.2.x.d 8
1638.2.x.e 24
1638.2.x.f 32
1638.2.y $$\chi_{1638}(827, \cdot)$$ 1638.2.y.a 12 2
1638.2.y.b 12
1638.2.y.c 16
1638.2.y.d 16
1638.2.z $$\chi_{1638}(571, \cdot)$$ n/a 224 2
1638.2.bc $$\chi_{1638}(251, \cdot)$$ 1638.2.bc.a 40 2
1638.2.bc.b 40
1638.2.bd $$\chi_{1638}(173, \cdot)$$ n/a 224 2
1638.2.bg $$\chi_{1638}(563, \cdot)$$ n/a 224 2
1638.2.bi $$\chi_{1638}(1213, \cdot)$$ n/a 224 2
1638.2.bj $$\chi_{1638}(127, \cdot)$$ 1638.2.bj.a 4 2
1638.2.bj.b 4
1638.2.bj.c 4
1638.2.bj.d 4
1638.2.bj.e 4
1638.2.bj.f 8
1638.2.bj.g 12
1638.2.bj.h 16
1638.2.bj.i 16
1638.2.bm $$\chi_{1638}(205, \cdot)$$ n/a 224 2
1638.2.bn $$\chi_{1638}(311, \cdot)$$ n/a 224 2
1638.2.bq $$\chi_{1638}(971, \cdot)$$ 1638.2.bq.a 72 2
1638.2.br $$\chi_{1638}(965, \cdot)$$ n/a 224 2
1638.2.bu $$\chi_{1638}(887, \cdot)$$ n/a 224 2
1638.2.cd $$\chi_{1638}(731, \cdot)$$ n/a 224 2
1638.2.ce $$\chi_{1638}(419, \cdot)$$ n/a 224 2
1638.2.cf $$\chi_{1638}(521, \cdot)$$ 1638.2.cf.a 8 2
1638.2.cf.b 8
1638.2.cf.c 24
1638.2.cf.d 24
1638.2.cg $$\chi_{1638}(131, \cdot)$$ n/a 192 2
1638.2.cl $$\chi_{1638}(209, \cdot)$$ n/a 192 2
1638.2.cm $$\chi_{1638}(269, \cdot)$$ 1638.2.cm.a 72 2
1638.2.cr $$\chi_{1638}(361, \cdot)$$ 1638.2.cr.a 16 2
1638.2.cr.b 20
1638.2.cr.c 20
1638.2.cr.d 36
1638.2.cs $$\chi_{1638}(589, \cdot)$$ n/a 168 2
1638.2.cv $$\chi_{1638}(277, \cdot)$$ n/a 224 2
1638.2.cy $$\chi_{1638}(1343, \cdot)$$ n/a 224 2
1638.2.cz $$\chi_{1638}(467, \cdot)$$ 1638.2.cz.a 8 2
1638.2.cz.b 8
1638.2.cz.c 32
1638.2.cz.d 32
1638.2.da $$\chi_{1638}(857, \cdot)$$ n/a 224 2
1638.2.db $$\chi_{1638}(101, \cdot)$$ n/a 224 2
1638.2.dg $$\chi_{1638}(17, \cdot)$$ 1638.2.dg.a 36 2
1638.2.dg.b 36
1638.2.dh $$\chi_{1638}(545, \cdot)$$ n/a 224 2
1638.2.dk $$\chi_{1638}(121, \cdot)$$ n/a 224 2
1638.2.dl $$\chi_{1638}(25, \cdot)$$ n/a 224 2
1638.2.dm $$\chi_{1638}(415, \cdot)$$ 1638.2.dm.a 4 2
1638.2.dm.b 8
1638.2.dm.c 12
1638.2.dm.d 12
1638.2.dm.e 20
1638.2.dm.f 40
1638.2.dn $$\chi_{1638}(43, \cdot)$$ n/a 168 2
1638.2.ds $$\chi_{1638}(337, \cdot)$$ n/a 168 2
1638.2.dt $$\chi_{1638}(1297, \cdot)$$ 1638.2.dt.a 16 2
1638.2.dt.b 20
1638.2.dt.c 20
1638.2.dt.d 36
1638.2.dv $$\chi_{1638}(335, \cdot)$$ n/a 224 2
1638.2.dw $$\chi_{1638}(647, \cdot)$$ 1638.2.dw.a 36 2
1638.2.dw.b 36
1638.2.dz $$\chi_{1638}(257, \cdot)$$ n/a 224 2
1638.2.ea $$\chi_{1638}(677, \cdot)$$ n/a 192 2
1638.2.eg $$\chi_{1638}(185, \cdot)$$ n/a 224 2
1638.2.eh $$\chi_{1638}(503, \cdot)$$ 1638.2.eh.a 80 2
1638.2.ek $$\chi_{1638}(815, \cdot)$$ n/a 224 2
1638.2.eu $$\chi_{1638}(31, \cdot)$$ n/a 448 4
1638.2.ev $$\chi_{1638}(97, \cdot)$$ n/a 448 4
1638.2.ew $$\chi_{1638}(115, \cdot)$$ n/a 448 4
1638.2.ex $$\chi_{1638}(695, \cdot)$$ n/a 448 4
1638.2.ey $$\chi_{1638}(71, \cdot)$$ n/a 112 4
1638.2.ez $$\chi_{1638}(359, \cdot)$$ n/a 160 4
1638.2.fa $$\chi_{1638}(617, \cdot)$$ n/a 336 4
1638.2.fb $$\chi_{1638}(431, \cdot)$$ n/a 144 4
1638.2.fc $$\chi_{1638}(515, \cdot)$$ n/a 448 4
1638.2.fd $$\chi_{1638}(535, \cdot)$$ n/a 448 4
1638.2.fe $$\chi_{1638}(1189, \cdot)$$ n/a 192 4
1638.2.ff $$\chi_{1638}(229, \cdot)$$ n/a 448 4
1638.2.fg $$\chi_{1638}(19, \cdot)$$ n/a 184 4
1638.2.fh $$\chi_{1638}(73, \cdot)$$ n/a 192 4
1638.2.fi $$\chi_{1638}(223, \cdot)$$ n/a 448 4
1638.2.fj $$\chi_{1638}(317, \cdot)$$ n/a 448 4
1638.2.fk $$\chi_{1638}(11, \cdot)$$ n/a 448 4
1638.2.fl $$\chi_{1638}(743, \cdot)$$ n/a 336 4
1638.2.ge $$\chi_{1638}(145, \cdot)$$ n/a 184 4
1638.2.gf $$\chi_{1638}(821, \cdot)$$ n/a 448 4
1638.2.gg $$\chi_{1638}(137, \cdot)$$ n/a 448 4
1638.2.gh $$\chi_{1638}(239, \cdot)$$ n/a 336 4
1638.2.gi $$\chi_{1638}(409, \cdot)$$ n/a 448 4
1638.2.gj $$\chi_{1638}(241, \cdot)$$ n/a 448 4
1638.2.gk $$\chi_{1638}(265, \cdot)$$ n/a 448 4
1638.2.gl $$\chi_{1638}(305, \cdot)$$ n/a 144 4

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1638))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1638)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(78))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(91))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(117))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(126))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(182))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(234))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(273))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(546))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(819))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1638))$$$$^{\oplus 1}$$