# Properties

 Label 3360.2 Level 3360 Weight 2 Dimension 108696 Nonzero newspaces 80 Sturm bound 1179648 Trace bound 71

## Defining parameters

 Level: $$N$$ = $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$80$$ Sturm bound: $$1179648$$ Trace bound: $$71$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3360))$$.

Total New Old
Modular forms 301056 109896 191160
Cusp forms 288769 108696 180073
Eisenstein series 12287 1200 11087

## Trace form

 $$108696 q - 20 q^{3} - 64 q^{4} + 8 q^{5} - 96 q^{6} - 56 q^{7} - 32 q^{9} + O(q^{10})$$ $$108696 q - 20 q^{3} - 64 q^{4} + 8 q^{5} - 96 q^{6} - 56 q^{7} - 32 q^{9} - 128 q^{10} - 96 q^{12} - 112 q^{13} - 64 q^{14} - 108 q^{15} - 352 q^{16} - 64 q^{17} - 64 q^{18} - 88 q^{19} - 64 q^{20} - 152 q^{21} - 256 q^{22} - 96 q^{23} + 16 q^{24} - 160 q^{25} + 160 q^{26} - 176 q^{27} - 16 q^{29} + 48 q^{30} - 392 q^{31} + 160 q^{32} - 136 q^{33} + 32 q^{34} - 96 q^{35} - 32 q^{36} - 144 q^{37} + 160 q^{38} - 184 q^{39} - 16 q^{40} - 96 q^{41} + 80 q^{42} - 288 q^{43} + 32 q^{44} - 56 q^{45} - 192 q^{46} - 144 q^{47} + 176 q^{48} - 184 q^{49} + 48 q^{50} - 148 q^{51} + 128 q^{52} - 272 q^{53} + 176 q^{54} - 112 q^{55} + 112 q^{56} - 176 q^{57} + 224 q^{58} + 64 q^{59} + 200 q^{60} - 624 q^{61} + 192 q^{62} - 92 q^{63} + 320 q^{64} + 48 q^{65} + 256 q^{66} + 104 q^{67} + 416 q^{68} - 192 q^{69} + 192 q^{70} + 240 q^{71} + 208 q^{72} + 576 q^{74} + 98 q^{75} + 320 q^{76} + 64 q^{77} + 272 q^{78} + 200 q^{79} + 528 q^{80} + 64 q^{81} + 256 q^{82} + 160 q^{83} + 16 q^{85} + 256 q^{86} + 344 q^{87} + 288 q^{88} + 256 q^{89} + 168 q^{90} + 48 q^{91} + 352 q^{92} + 304 q^{93} + 608 q^{94} + 200 q^{95} + 80 q^{96} + 320 q^{97} + 608 q^{98} + 440 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3360))$$

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
3360.2.a $$\chi_{3360}(1, \cdot)$$ 3360.2.a.a 1 1
3360.2.a.b 1
3360.2.a.c 1
3360.2.a.d 1
3360.2.a.e 1
3360.2.a.f 1
3360.2.a.g 1
3360.2.a.h 1
3360.2.a.i 1
3360.2.a.j 1
3360.2.a.k 1
3360.2.a.l 1
3360.2.a.m 1
3360.2.a.n 1
3360.2.a.o 1
3360.2.a.p 1
3360.2.a.q 1
3360.2.a.r 1
3360.2.a.s 1
3360.2.a.t 1
3360.2.a.u 1
3360.2.a.v 1
3360.2.a.w 1
3360.2.a.x 1
3360.2.a.y 1
3360.2.a.z 1
3360.2.a.ba 2
3360.2.a.bb 2
3360.2.a.bc 2
3360.2.a.bd 2
3360.2.a.be 2
3360.2.a.bf 2
3360.2.a.bg 2
3360.2.a.bh 2
3360.2.a.bi 3
3360.2.a.bj 3
3360.2.d $$\chi_{3360}(2911, \cdot)$$ 3360.2.d.a 16 1
3360.2.d.b 16
3360.2.d.c 16
3360.2.d.d 16
3360.2.e $$\chi_{3360}(911, \cdot)$$ 3360.2.e.a 4 1
3360.2.e.b 4
3360.2.e.c 44
3360.2.e.d 44
3360.2.f $$\chi_{3360}(2561, \cdot)$$ n/a 128 1
3360.2.g $$\chi_{3360}(1681, \cdot)$$ 3360.2.g.a 8 1
3360.2.g.b 12
3360.2.g.c 12
3360.2.g.d 16
3360.2.j $$\chi_{3360}(1009, \cdot)$$ 3360.2.j.a 2 1
3360.2.j.b 2
3360.2.j.c 2
3360.2.j.d 2
3360.2.j.e 32
3360.2.j.f 32
3360.2.k $$\chi_{3360}(1889, \cdot)$$ n/a 192 1
3360.2.p $$\chi_{3360}(239, \cdot)$$ n/a 144 1
3360.2.q $$\chi_{3360}(2239, \cdot)$$ 3360.2.q.a 48 1
3360.2.q.b 48
3360.2.t $$\chi_{3360}(2689, \cdot)$$ 3360.2.t.a 2 1
3360.2.t.b 2
3360.2.t.c 2
3360.2.t.d 2
3360.2.t.e 2
3360.2.t.f 2
3360.2.t.g 4
3360.2.t.h 6
3360.2.t.i 6
3360.2.t.j 8
3360.2.t.k 8
3360.2.t.l 8
3360.2.t.m 10
3360.2.t.n 10
3360.2.u $$\chi_{3360}(209, \cdot)$$ n/a 184 1
3360.2.v $$\chi_{3360}(1919, \cdot)$$ n/a 144 1
3360.2.w $$\chi_{3360}(559, \cdot)$$ 3360.2.w.a 48 1
3360.2.w.b 48
3360.2.z $$\chi_{3360}(1231, \cdot)$$ 3360.2.z.a 4 1
3360.2.z.b 4
3360.2.z.c 28
3360.2.z.d 28
3360.2.ba $$\chi_{3360}(2591, \cdot)$$ 3360.2.ba.a 4 1
3360.2.ba.b 4
3360.2.ba.c 20
3360.2.ba.d 20
3360.2.ba.e 24
3360.2.ba.f 24
3360.2.bf $$\chi_{3360}(881, \cdot)$$ n/a 128 1
3360.2.bg $$\chi_{3360}(961, \cdot)$$ n/a 128 2
3360.2.bj $$\chi_{3360}(433, \cdot)$$ n/a 192 2
3360.2.bk $$\chi_{3360}(1793, \cdot)$$ n/a 288 2
3360.2.bl $$\chi_{3360}(127, \cdot)$$ n/a 144 2
3360.2.bm $$\chi_{3360}(1007, \cdot)$$ n/a 368 2
3360.2.bp $$\chi_{3360}(967, \cdot)$$ None 0 2
3360.2.bs $$\chi_{3360}(937, \cdot)$$ None 0 2
3360.2.bu $$\chi_{3360}(2297, \cdot)$$ None 0 2
3360.2.bv $$\chi_{3360}(1847, \cdot)$$ None 0 2
3360.2.bx $$\chi_{3360}(1049, \cdot)$$ None 0 2
3360.2.ca $$\chi_{3360}(71, \cdot)$$ None 0 2
3360.2.cb $$\chi_{3360}(169, \cdot)$$ None 0 2
3360.2.ce $$\chi_{3360}(391, \cdot)$$ None 0 2
3360.2.cg $$\chi_{3360}(841, \cdot)$$ None 0 2
3360.2.ch $$\chi_{3360}(1399, \cdot)$$ None 0 2
3360.2.ck $$\chi_{3360}(41, \cdot)$$ None 0 2
3360.2.cl $$\chi_{3360}(1079, \cdot)$$ None 0 2
3360.2.co $$\chi_{3360}(2617, \cdot)$$ None 0 2
3360.2.cp $$\chi_{3360}(2647, \cdot)$$ None 0 2
3360.2.cr $$\chi_{3360}(167, \cdot)$$ None 0 2
3360.2.cu $$\chi_{3360}(617, \cdot)$$ None 0 2
3360.2.cx $$\chi_{3360}(463, \cdot)$$ n/a 144 2
3360.2.cy $$\chi_{3360}(1343, \cdot)$$ n/a 384 2
3360.2.cz $$\chi_{3360}(97, \cdot)$$ n/a 192 2
3360.2.da $$\chi_{3360}(113, \cdot)$$ n/a 288 2
3360.2.df $$\chi_{3360}(1039, \cdot)$$ n/a 192 2
3360.2.dg $$\chi_{3360}(1439, \cdot)$$ n/a 384 2
3360.2.dh $$\chi_{3360}(689, \cdot)$$ n/a 368 2
3360.2.di $$\chi_{3360}(289, \cdot)$$ n/a 192 2
3360.2.dl $$\chi_{3360}(1361, \cdot)$$ n/a 256 2
3360.2.dq $$\chi_{3360}(191, \cdot)$$ n/a 256 2
3360.2.dr $$\chi_{3360}(271, \cdot)$$ n/a 128 2
3360.2.du $$\chi_{3360}(1201, \cdot)$$ n/a 128 2
3360.2.dv $$\chi_{3360}(1601, \cdot)$$ n/a 256 2
3360.2.dw $$\chi_{3360}(431, \cdot)$$ n/a 256 2
3360.2.dx $$\chi_{3360}(31, \cdot)$$ n/a 128 2
3360.2.ea $$\chi_{3360}(1279, \cdot)$$ n/a 192 2
3360.2.eb $$\chi_{3360}(1199, \cdot)$$ n/a 368 2
3360.2.eg $$\chi_{3360}(929, \cdot)$$ n/a 384 2
3360.2.eh $$\chi_{3360}(529, \cdot)$$ n/a 192 2
3360.2.ei $$\chi_{3360}(461, \cdot)$$ n/a 2048 4
3360.2.ek $$\chi_{3360}(659, \cdot)$$ n/a 2304 4
3360.2.en $$\chi_{3360}(421, \cdot)$$ n/a 768 4
3360.2.ep $$\chi_{3360}(139, \cdot)$$ n/a 1536 4
3360.2.es $$\chi_{3360}(43, \cdot)$$ n/a 1152 4
3360.2.et $$\chi_{3360}(13, \cdot)$$ n/a 1536 4
3360.2.ew $$\chi_{3360}(83, \cdot)$$ n/a 3040 4
3360.2.ex $$\chi_{3360}(533, \cdot)$$ n/a 2304 4
3360.2.ey $$\chi_{3360}(883, \cdot)$$ n/a 1152 4
3360.2.ez $$\chi_{3360}(853, \cdot)$$ n/a 1536 4
3360.2.fc $$\chi_{3360}(923, \cdot)$$ n/a 3040 4
3360.2.fd $$\chi_{3360}(197, \cdot)$$ n/a 2304 4
3360.2.fh $$\chi_{3360}(491, \cdot)$$ n/a 1536 4
3360.2.fj $$\chi_{3360}(629, \cdot)$$ n/a 3040 4
3360.2.fk $$\chi_{3360}(811, \cdot)$$ n/a 1024 4
3360.2.fm $$\chi_{3360}(589, \cdot)$$ n/a 1152 4
3360.2.fo $$\chi_{3360}(977, \cdot)$$ n/a 736 4
3360.2.fp $$\chi_{3360}(577, \cdot)$$ n/a 384 4
3360.2.fu $$\chi_{3360}(383, \cdot)$$ n/a 768 4
3360.2.fv $$\chi_{3360}(1327, \cdot)$$ n/a 384 4
3360.2.fw $$\chi_{3360}(887, \cdot)$$ None 0 4
3360.2.fz $$\chi_{3360}(233, \cdot)$$ None 0 4
3360.2.gb $$\chi_{3360}(313, \cdot)$$ None 0 4
3360.2.gc $$\chi_{3360}(487, \cdot)$$ None 0 4
3360.2.ge $$\chi_{3360}(521, \cdot)$$ None 0 4
3360.2.gh $$\chi_{3360}(359, \cdot)$$ None 0 4
3360.2.gi $$\chi_{3360}(121, \cdot)$$ None 0 4
3360.2.gl $$\chi_{3360}(199, \cdot)$$ None 0 4
3360.2.gn $$\chi_{3360}(1129, \cdot)$$ None 0 4
3360.2.go $$\chi_{3360}(871, \cdot)$$ None 0 4
3360.2.gr $$\chi_{3360}(89, \cdot)$$ None 0 4
3360.2.gs $$\chi_{3360}(1031, \cdot)$$ None 0 4
3360.2.gv $$\chi_{3360}(137, \cdot)$$ None 0 4
3360.2.gw $$\chi_{3360}(647, \cdot)$$ None 0 4
3360.2.gy $$\chi_{3360}(247, \cdot)$$ None 0 4
3360.2.hb $$\chi_{3360}(73, \cdot)$$ None 0 4
3360.2.hc $$\chi_{3360}(47, \cdot)$$ n/a 736 4
3360.2.hd $$\chi_{3360}(1087, \cdot)$$ n/a 384 4
3360.2.hi $$\chi_{3360}(737, \cdot)$$ n/a 768 4
3360.2.hj $$\chi_{3360}(817, \cdot)$$ n/a 384 4
3360.2.hk $$\chi_{3360}(109, \cdot)$$ n/a 3072 8
3360.2.hm $$\chi_{3360}(451, \cdot)$$ n/a 2048 8
3360.2.hp $$\chi_{3360}(269, \cdot)$$ n/a 6080 8
3360.2.hr $$\chi_{3360}(11, \cdot)$$ n/a 4096 8
3360.2.hu $$\chi_{3360}(653, \cdot)$$ n/a 6080 8
3360.2.hv $$\chi_{3360}(227, \cdot)$$ n/a 6080 8
3360.2.hy $$\chi_{3360}(157, \cdot)$$ n/a 3072 8
3360.2.hz $$\chi_{3360}(163, \cdot)$$ n/a 3072 8
3360.2.ia $$\chi_{3360}(53, \cdot)$$ n/a 6080 8
3360.2.ib $$\chi_{3360}(563, \cdot)$$ n/a 6080 8
3360.2.ie $$\chi_{3360}(493, \cdot)$$ n/a 3072 8
3360.2.if $$\chi_{3360}(67, \cdot)$$ n/a 3072 8
3360.2.ij $$\chi_{3360}(19, \cdot)$$ n/a 3072 8
3360.2.il $$\chi_{3360}(541, \cdot)$$ n/a 2048 8
3360.2.im $$\chi_{3360}(179, \cdot)$$ n/a 6080 8
3360.2.io $$\chi_{3360}(101, \cdot)$$ n/a 4096 8

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3360))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(3360)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 48}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 40}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 32}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 20}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 20}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 20}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(224))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(420))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(480))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(560))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(672))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(840))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1680))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3360))$$$$^{\oplus 1}$$