# Properties

 Label 3330.2 Level 3330 Weight 2 Dimension 71372 Nonzero newspaces 80 Sturm bound 1181952 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$80$$ Sturm bound: $$1181952$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3330))$$.

Total New Old
Modular forms 300096 71372 228724
Cusp forms 290881 71372 219509
Eisenstein series 9215 0 9215

## Trace form

 $$71372 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 10 q^{5} + 12 q^{6} - 24 q^{7} + 6 q^{8} + 28 q^{9} + O(q^{10})$$ $$71372 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 10 q^{5} + 12 q^{6} - 24 q^{7} + 6 q^{8} + 28 q^{9} + 22 q^{10} + 52 q^{11} + 16 q^{12} + 28 q^{13} + 40 q^{14} + 48 q^{15} - 6 q^{16} + 36 q^{17} + 8 q^{18} + 16 q^{19} + 22 q^{20} + 72 q^{21} + 20 q^{22} + 72 q^{23} - 12 q^{24} + 34 q^{25} - 30 q^{26} + 48 q^{27} - 24 q^{28} - 60 q^{29} - 200 q^{31} - 6 q^{32} - 20 q^{33} - 152 q^{34} - 140 q^{35} - 20 q^{36} - 142 q^{37} - 156 q^{38} - 128 q^{39} - 39 q^{40} - 368 q^{41} - 128 q^{42} - 164 q^{43} - 64 q^{44} - 160 q^{45} - 144 q^{46} - 192 q^{47} - 20 q^{48} - 102 q^{49} - 103 q^{50} - 60 q^{51} - 20 q^{52} - 60 q^{53} - 60 q^{54} + 40 q^{55} - 40 q^{56} - 52 q^{57} - 68 q^{58} - 92 q^{59} + 16 q^{60} - 118 q^{61} + 80 q^{63} + 6 q^{64} + 3 q^{65} + 96 q^{66} - 36 q^{67} + 72 q^{68} + 128 q^{69} + 48 q^{70} + 12 q^{72} - 28 q^{73} + 50 q^{74} - 12 q^{75} + 92 q^{76} - 120 q^{77} + 56 q^{78} - 80 q^{79} - 10 q^{80} - 4 q^{81} + 116 q^{82} - 144 q^{83} + 24 q^{84} - 21 q^{85} + 140 q^{86} + 40 q^{87} + 20 q^{88} + 2 q^{89} + 112 q^{90} + 128 q^{91} + 288 q^{92} + 368 q^{93} + 488 q^{94} + 408 q^{95} + 16 q^{96} + 952 q^{97} + 918 q^{98} + 760 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3330))$$

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
3330.2.a $$\chi_{3330}(1, \cdot)$$ 3330.2.a.a 1 1
3330.2.a.b 1
3330.2.a.c 1
3330.2.a.d 1
3330.2.a.e 1
3330.2.a.f 1
3330.2.a.g 1
3330.2.a.h 1
3330.2.a.i 1
3330.2.a.j 1
3330.2.a.k 1
3330.2.a.l 1
3330.2.a.m 1
3330.2.a.n 1
3330.2.a.o 1
3330.2.a.p 1
3330.2.a.q 1
3330.2.a.r 1
3330.2.a.s 1
3330.2.a.t 1
3330.2.a.u 1
3330.2.a.v 1
3330.2.a.w 1
3330.2.a.x 1
3330.2.a.y 1
3330.2.a.z 1
3330.2.a.ba 1
3330.2.a.bb 2
3330.2.a.bc 2
3330.2.a.bd 2
3330.2.a.be 2
3330.2.a.bf 2
3330.2.a.bg 3
3330.2.a.bh 3
3330.2.a.bi 3
3330.2.a.bj 4
3330.2.a.bk 5
3330.2.a.bl 5
3330.2.d $$\chi_{3330}(1999, \cdot)$$ 3330.2.d.a 2 1
3330.2.d.b 2
3330.2.d.c 2
3330.2.d.d 2
3330.2.d.e 2
3330.2.d.f 2
3330.2.d.g 2
3330.2.d.h 4
3330.2.d.i 4
3330.2.d.j 4
3330.2.d.k 4
3330.2.d.l 4
3330.2.d.m 4
3330.2.d.n 6
3330.2.d.o 8
3330.2.d.p 10
3330.2.d.q 14
3330.2.d.r 14
3330.2.e $$\chi_{3330}(739, \cdot)$$ 3330.2.e.a 2 1
3330.2.e.b 2
3330.2.e.c 10
3330.2.e.d 10
3330.2.e.e 16
3330.2.e.f 16
3330.2.e.g 20
3330.2.e.h 20
3330.2.h $$\chi_{3330}(2071, \cdot)$$ 3330.2.h.a 2 1
3330.2.h.b 2
3330.2.h.c 2
3330.2.h.d 2
3330.2.h.e 2
3330.2.h.f 2
3330.2.h.g 2
3330.2.h.h 2
3330.2.h.i 2
3330.2.h.j 4
3330.2.h.k 4
3330.2.h.l 4
3330.2.h.m 6
3330.2.h.n 6
3330.2.h.o 8
3330.2.h.p 16
3330.2.i $$\chi_{3330}(1111, \cdot)$$ n/a 288 2
3330.2.j $$\chi_{3330}(121, \cdot)$$ n/a 304 2
3330.2.k $$\chi_{3330}(2341, \cdot)$$ n/a 132 2
3330.2.l $$\chi_{3330}(211, \cdot)$$ n/a 304 2
3330.2.n $$\chi_{3330}(179, \cdot)$$ n/a 152 2
3330.2.o $$\chi_{3330}(1153, \cdot)$$ n/a 190 2
3330.2.p $$\chi_{3330}(593, \cdot)$$ n/a 144 2
3330.2.q $$\chi_{3330}(1997, \cdot)$$ n/a 152 2
3330.2.r $$\chi_{3330}(253, \cdot)$$ n/a 190 2
3330.2.x $$\chi_{3330}(1511, \cdot)$$ n/a 112 2
3330.2.y $$\chi_{3330}(619, \cdot)$$ n/a 456 2
3330.2.z $$\chi_{3330}(2209, \cdot)$$ n/a 456 2
3330.2.be $$\chi_{3330}(841, \cdot)$$ n/a 304 2
3330.2.bf $$\chi_{3330}(961, \cdot)$$ n/a 304 2
3330.2.bk $$\chi_{3330}(2971, \cdot)$$ n/a 132 2
3330.2.bn $$\chi_{3330}(2119, \cdot)$$ n/a 456 2
3330.2.bo $$\chi_{3330}(1849, \cdot)$$ n/a 456 2
3330.2.bp $$\chi_{3330}(889, \cdot)$$ n/a 432 2
3330.2.bq $$\chi_{3330}(529, \cdot)$$ n/a 456 2
3330.2.bv $$\chi_{3330}(1639, \cdot)$$ n/a 192 2
3330.2.bw $$\chi_{3330}(1009, \cdot)$$ n/a 188 2
3330.2.bx $$\chi_{3330}(751, \cdot)$$ n/a 304 2
3330.2.ca $$\chi_{3330}(571, \cdot)$$ n/a 912 6
3330.2.cb $$\chi_{3330}(181, \cdot)$$ n/a 372 6
3330.2.cc $$\chi_{3330}(601, \cdot)$$ n/a 912 6
3330.2.cd $$\chi_{3330}(569, \cdot)$$ n/a 912 4
3330.2.cg $$\chi_{3330}(191, \cdot)$$ n/a 608 4
3330.2.ch $$\chi_{3330}(1361, \cdot)$$ n/a 608 4
3330.2.ck $$\chi_{3330}(251, \cdot)$$ n/a 224 4
3330.2.cl $$\chi_{3330}(1207, \cdot)$$ n/a 380 4
3330.2.cm $$\chi_{3330}(233, \cdot)$$ n/a 304 4
3330.2.cn $$\chi_{3330}(2267, \cdot)$$ n/a 304 4
3330.2.co $$\chi_{3330}(1657, \cdot)$$ n/a 380 4
3330.2.db $$\chi_{3330}(547, \cdot)$$ n/a 912 4
3330.2.dc $$\chi_{3330}(1363, \cdot)$$ n/a 912 4
3330.2.dd $$\chi_{3330}(97, \cdot)$$ n/a 912 4
3330.2.de $$\chi_{3330}(1037, \cdot)$$ n/a 864 4
3330.2.df $$\chi_{3330}(677, \cdot)$$ n/a 912 4
3330.2.dg $$\chi_{3330}(767, \cdot)$$ n/a 912 4
3330.2.dh $$\chi_{3330}(137, \cdot)$$ n/a 912 4
3330.2.di $$\chi_{3330}(47, \cdot)$$ n/a 912 4
3330.2.dj $$\chi_{3330}(443, \cdot)$$ n/a 912 4
3330.2.dk $$\chi_{3330}(193, \cdot)$$ n/a 912 4
3330.2.dl $$\chi_{3330}(43, \cdot)$$ n/a 912 4
3330.2.dm $$\chi_{3330}(637, \cdot)$$ n/a 912 4
3330.2.ds $$\chi_{3330}(1289, \cdot)$$ n/a 912 4
3330.2.dt $$\chi_{3330}(29, \cdot)$$ n/a 912 4
3330.2.dw $$\chi_{3330}(1799, \cdot)$$ n/a 304 4
3330.2.dx $$\chi_{3330}(911, \cdot)$$ n/a 608 4
3330.2.ed $$\chi_{3330}(289, \cdot)$$ n/a 576 6
3330.2.ee $$\chi_{3330}(139, \cdot)$$ n/a 1368 6
3330.2.ef $$\chi_{3330}(49, \cdot)$$ n/a 1368 6
3330.2.eg $$\chi_{3330}(379, \cdot)$$ n/a 564 6
3330.2.eh $$\chi_{3330}(361, \cdot)$$ n/a 372 6
3330.2.ei $$\chi_{3330}(151, \cdot)$$ n/a 912 6
3330.2.er $$\chi_{3330}(691, \cdot)$$ n/a 912 6
3330.2.es $$\chi_{3330}(229, \cdot)$$ n/a 1368 6
3330.2.et $$\chi_{3330}(169, \cdot)$$ n/a 1368 6
3330.2.eu $$\chi_{3330}(13, \cdot)$$ n/a 2736 12
3330.2.ev $$\chi_{3330}(77, \cdot)$$ n/a 2736 12
3330.2.ey $$\chi_{3330}(83, \cdot)$$ n/a 2736 12
3330.2.ez $$\chi_{3330}(283, \cdot)$$ n/a 2736 12
3330.2.fe $$\chi_{3330}(131, \cdot)$$ n/a 1824 12
3330.2.ff $$\chi_{3330}(89, \cdot)$$ n/a 912 12
3330.2.fg $$\chi_{3330}(161, \cdot)$$ n/a 576 12
3330.2.fh $$\chi_{3330}(59, \cdot)$$ n/a 2736 12
3330.2.fm $$\chi_{3330}(457, \cdot)$$ n/a 2736 12
3330.2.fn $$\chi_{3330}(163, \cdot)$$ n/a 1140 12
3330.2.fo $$\chi_{3330}(527, \cdot)$$ n/a 2736 12
3330.2.fp $$\chi_{3330}(53, \cdot)$$ n/a 912 12
3330.2.fu $$\chi_{3330}(287, \cdot)$$ n/a 912 12
3330.2.fv $$\chi_{3330}(263, \cdot)$$ n/a 2736 12
3330.2.fw $$\chi_{3330}(217, \cdot)$$ n/a 1140 12
3330.2.fx $$\chi_{3330}(277, \cdot)$$ n/a 2736 12
3330.2.ga $$\chi_{3330}(479, \cdot)$$ n/a 2736 12
3330.2.gb $$\chi_{3330}(281, \cdot)$$ n/a 1824 12

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3330))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(3330)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(37))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(74))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(111))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(185))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(222))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(333))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(370))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(555))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(666))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1110))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1665))$$$$^{\oplus 2}$$