# Properties

 Label 3360.1 Level 3360 Weight 1 Dimension 80 Nonzero newspaces 5 Newform subspaces 14 Sturm bound 589824 Trace bound 21

## Defining parameters

 Level: $$N$$ = $$3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$5$$ Newform subspaces: $$14$$ Sturm bound: $$589824$$ Trace bound: $$21$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(3360))$$.

Total New Old
Modular forms 6856 680 6176
Cusp forms 712 80 632
Eisenstein series 6144 600 5544

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 80 0 0 0

## Trace form

 $$80q + 4q^{9} + O(q^{10})$$ $$80q + 4q^{9} + 4q^{15} - 4q^{21} + 6q^{25} + 4q^{31} - 12q^{33} - 4q^{45} - 4q^{49} - 12q^{55} + 8q^{57} - 4q^{63} - 16q^{69} - 4q^{79} - 24q^{81} - 12q^{87} - 16q^{93} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(3360))$$

We only show spaces with odd 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
3360.1.b $$\chi_{3360}(1679, \cdot)$$ None 0 1
3360.1.c $$\chi_{3360}(799, \cdot)$$ None 0 1
3360.1.h $$\chi_{3360}(2449, \cdot)$$ None 0 1
3360.1.i $$\chi_{3360}(449, \cdot)$$ None 0 1
3360.1.l $$\chi_{3360}(1121, \cdot)$$ None 0 1
3360.1.m $$\chi_{3360}(3121, \cdot)$$ None 0 1
3360.1.n $$\chi_{3360}(1471, \cdot)$$ None 0 1
3360.1.o $$\chi_{3360}(2351, \cdot)$$ None 0 1
3360.1.r $$\chi_{3360}(2801, \cdot)$$ None 0 1
3360.1.s $$\chi_{3360}(1441, \cdot)$$ None 0 1
3360.1.x $$\chi_{3360}(3151, \cdot)$$ None 0 1
3360.1.y $$\chi_{3360}(671, \cdot)$$ None 0 1
3360.1.bb $$\chi_{3360}(3359, \cdot)$$ None 0 1
3360.1.bc $$\chi_{3360}(2479, \cdot)$$ None 0 1
3360.1.bd $$\chi_{3360}(769, \cdot)$$ None 0 1
3360.1.be $$\chi_{3360}(2129, \cdot)$$ None 0 1
3360.1.bh $$\chi_{3360}(1583, \cdot)$$ None 0 2
3360.1.bi $$\chi_{3360}(223, \cdot)$$ None 0 2
3360.1.bn $$\chi_{3360}(1217, \cdot)$$ 3360.1.bn.a 4 2
3360.1.bn.b 4
3360.1.bn.c 4
3360.1.bn.d 4
3360.1.bo $$\chi_{3360}(337, \cdot)$$ None 0 2
3360.1.bq $$\chi_{3360}(2087, \cdot)$$ None 0 2
3360.1.br $$\chi_{3360}(2057, \cdot)$$ None 0 2
3360.1.bt $$\chi_{3360}(1177, \cdot)$$ None 0 2
3360.1.bw $$\chi_{3360}(727, \cdot)$$ None 0 2
3360.1.by $$\chi_{3360}(1639, \cdot)$$ None 0 2
3360.1.bz $$\chi_{3360}(601, \cdot)$$ None 0 2
3360.1.cc $$\chi_{3360}(839, \cdot)$$ None 0 2
3360.1.cd $$\chi_{3360}(281, \cdot)$$ None 0 2
3360.1.cf $$\chi_{3360}(1511, \cdot)$$ None 0 2
3360.1.ci $$\chi_{3360}(1289, \cdot)$$ None 0 2
3360.1.cj $$\chi_{3360}(631, \cdot)$$ None 0 2
3360.1.cm $$\chi_{3360}(1609, \cdot)$$ None 0 2
3360.1.cn $$\chi_{3360}(377, \cdot)$$ None 0 2
3360.1.cq $$\chi_{3360}(407, \cdot)$$ None 0 2
3360.1.cs $$\chi_{3360}(2407, \cdot)$$ None 0 2
3360.1.ct $$\chi_{3360}(2857, \cdot)$$ None 0 2
3360.1.cv $$\chi_{3360}(1553, \cdot)$$ 3360.1.cv.a 4 2
3360.1.cv.b 4
3360.1.cv.c 8
3360.1.cv.d 8
3360.1.cw $$\chi_{3360}(673, \cdot)$$ None 0 2
3360.1.db $$\chi_{3360}(1247, \cdot)$$ None 0 2
3360.1.dc $$\chi_{3360}(1903, \cdot)$$ None 0 2
3360.1.dd $$\chi_{3360}(1151, \cdot)$$ None 0 2
3360.1.de $$\chi_{3360}(751, \cdot)$$ None 0 2
3360.1.dj $$\chi_{3360}(481, \cdot)$$ None 0 2
3360.1.dk $$\chi_{3360}(401, \cdot)$$ None 0 2
3360.1.dm $$\chi_{3360}(1649, \cdot)$$ 3360.1.dm.a 4 2
3360.1.dm.b 4
3360.1.dn $$\chi_{3360}(1249, \cdot)$$ None 0 2
3360.1.do $$\chi_{3360}(79, \cdot)$$ None 0 2
3360.1.dp $$\chi_{3360}(479, \cdot)$$ None 0 2
3360.1.ds $$\chi_{3360}(1409, \cdot)$$ 3360.1.ds.a 8 2
3360.1.ds.b 8
3360.1.dt $$\chi_{3360}(1489, \cdot)$$ None 0 2
3360.1.dy $$\chi_{3360}(319, \cdot)$$ None 0 2
3360.1.dz $$\chi_{3360}(719, \cdot)$$ None 0 2
3360.1.ec $$\chi_{3360}(1391, \cdot)$$ None 0 2
3360.1.ed $$\chi_{3360}(991, \cdot)$$ None 0 2
3360.1.ee $$\chi_{3360}(241, \cdot)$$ None 0 2
3360.1.ef $$\chi_{3360}(641, \cdot)$$ None 0 2
3360.1.ej $$\chi_{3360}(349, \cdot)$$ None 0 4
3360.1.el $$\chi_{3360}(211, \cdot)$$ None 0 4
3360.1.em $$\chi_{3360}(29, \cdot)$$ None 0 4
3360.1.eo $$\chi_{3360}(251, \cdot)$$ None 0 4
3360.1.eq $$\chi_{3360}(323, \cdot)$$ None 0 4
3360.1.er $$\chi_{3360}(293, \cdot)$$ None 0 4
3360.1.eu $$\chi_{3360}(307, \cdot)$$ None 0 4
3360.1.ev $$\chi_{3360}(253, \cdot)$$ None 0 4
3360.1.fa $$\chi_{3360}(1163, \cdot)$$ None 0 4
3360.1.fb $$\chi_{3360}(1133, \cdot)$$ None 0 4
3360.1.fe $$\chi_{3360}(643, \cdot)$$ None 0 4
3360.1.ff $$\chi_{3360}(1093, \cdot)$$ None 0 4
3360.1.fg $$\chi_{3360}(379, \cdot)$$ None 0 4
3360.1.fi $$\chi_{3360}(181, \cdot)$$ None 0 4
3360.1.fl $$\chi_{3360}(419, \cdot)$$ None 0 4
3360.1.fn $$\chi_{3360}(701, \cdot)$$ None 0 4
3360.1.fq $$\chi_{3360}(367, \cdot)$$ None 0 4
3360.1.fr $$\chi_{3360}(767, \cdot)$$ None 0 4
3360.1.fs $$\chi_{3360}(193, \cdot)$$ None 0 4
3360.1.ft $$\chi_{3360}(17, \cdot)$$ 3360.1.ft.a 8 4
3360.1.ft.b 8
3360.1.fx $$\chi_{3360}(103, \cdot)$$ None 0 4
3360.1.fy $$\chi_{3360}(697, \cdot)$$ None 0 4
3360.1.ga $$\chi_{3360}(1097, \cdot)$$ None 0 4
3360.1.gd $$\chi_{3360}(23, \cdot)$$ None 0 4
3360.1.gf $$\chi_{3360}(151, \cdot)$$ None 0 4
3360.1.gg $$\chi_{3360}(409, \cdot)$$ None 0 4
3360.1.gj $$\chi_{3360}(311, \cdot)$$ None 0 4
3360.1.gk $$\chi_{3360}(569, \cdot)$$ None 0 4
3360.1.gm $$\chi_{3360}(1319, \cdot)$$ None 0 4
3360.1.gp $$\chi_{3360}(1241, \cdot)$$ None 0 4
3360.1.gq $$\chi_{3360}(919, \cdot)$$ None 0 4
3360.1.gt $$\chi_{3360}(1081, \cdot)$$ None 0 4
3360.1.gu $$\chi_{3360}(457, \cdot)$$ None 0 4
3360.1.gx $$\chi_{3360}(1447, \cdot)$$ None 0 4
3360.1.gz $$\chi_{3360}(263, \cdot)$$ None 0 4
3360.1.ha $$\chi_{3360}(857, \cdot)$$ None 0 4
3360.1.he $$\chi_{3360}(1297, \cdot)$$ None 0 4
3360.1.hf $$\chi_{3360}(257, \cdot)$$ None 0 4
3360.1.hg $$\chi_{3360}(607, \cdot)$$ None 0 4
3360.1.hh $$\chi_{3360}(527, \cdot)$$ None 0 4
3360.1.hl $$\chi_{3360}(221, \cdot)$$ None 0 8
3360.1.hn $$\chi_{3360}(59, \cdot)$$ None 0 8
3360.1.ho $$\chi_{3360}(61, \cdot)$$ None 0 8
3360.1.hq $$\chi_{3360}(499, \cdot)$$ None 0 8
3360.1.hs $$\chi_{3360}(373, \cdot)$$ None 0 8
3360.1.ht $$\chi_{3360}(187, \cdot)$$ None 0 8
3360.1.hw $$\chi_{3360}(173, \cdot)$$ None 0 8
3360.1.hx $$\chi_{3360}(443, \cdot)$$ None 0 8
3360.1.ic $$\chi_{3360}(37, \cdot)$$ None 0 8
3360.1.id $$\chi_{3360}(283, \cdot)$$ None 0 8
3360.1.ig $$\chi_{3360}(773, \cdot)$$ None 0 8
3360.1.ih $$\chi_{3360}(107, \cdot)$$ None 0 8
3360.1.ii $$\chi_{3360}(131, \cdot)$$ None 0 8
3360.1.ik $$\chi_{3360}(149, \cdot)$$ None 0 8
3360.1.in $$\chi_{3360}(331, \cdot)$$ None 0 8
3360.1.ip $$\chi_{3360}(229, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(3360))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(3360)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(224))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(420))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(480))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(560))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(672))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(840))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1680))$$$$^{\oplus 2}$$