# Properties

 Label 1368.1 Level 1368 Weight 1 Dimension 110 Nonzero newspaces 15 Newform subspaces 28 Sturm bound 103680 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$15$$ Newform subspaces: $$28$$ Sturm bound: $$103680$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1368))$$.

Total New Old
Modular forms 2032 416 1616
Cusp forms 304 110 194
Eisenstein series 1728 306 1422

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 94 8 8 0

## Trace form

 $$110 q + 3 q^{2} - 9 q^{4} + 2 q^{6} - 10 q^{7} - 3 q^{8} + 4 q^{9} + O(q^{10})$$ $$110 q + 3 q^{2} - 9 q^{4} + 2 q^{6} - 10 q^{7} - 3 q^{8} + 4 q^{9} - 4 q^{10} + 4 q^{11} + 4 q^{12} - 2 q^{14} + 7 q^{16} + 4 q^{17} - 4 q^{18} + 13 q^{19} + 9 q^{22} + 2 q^{23} + 2 q^{24} - 3 q^{25} + 2 q^{26} - 12 q^{27} - 2 q^{28} + 4 q^{30} + 8 q^{31} + 3 q^{32} - 8 q^{34} - 4 q^{35} - 8 q^{38} - 6 q^{39} - 2 q^{40} - 2 q^{41} - 10 q^{42} - 4 q^{43} - 25 q^{44} - 8 q^{46} + 2 q^{47} - 18 q^{48} - q^{49} + 3 q^{50} - 4 q^{51} + 4 q^{52} + 14 q^{54} - 4 q^{56} - 3 q^{57} - 14 q^{58} + 2 q^{59} + 4 q^{62} + 12 q^{63} - 3 q^{64} + 4 q^{66} - 21 q^{68} - 16 q^{72} + 9 q^{73} - 2 q^{74} + 2 q^{75} - 4 q^{76} - 4 q^{79} + 4 q^{81} + 2 q^{82} + 10 q^{83} - 6 q^{87} + 4 q^{88} - 10 q^{89} + 2 q^{90} + 2 q^{92} - 12 q^{94} - 4 q^{96} - 12 q^{97} - 3 q^{98} - 10 q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1368))$$

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
1368.1.b $$\chi_{1368}(1367, \cdot)$$ None 0 1
1368.1.c $$\chi_{1368}(1027, \cdot)$$ None 0 1
1368.1.h $$\chi_{1368}(305, \cdot)$$ None 0 1
1368.1.i $$\chi_{1368}(37, \cdot)$$ 1368.1.i.a 1 1
1368.1.i.b 1
1368.1.l $$\chi_{1368}(683, \cdot)$$ 1368.1.l.a 4 1
1368.1.m $$\chi_{1368}(343, \cdot)$$ None 0 1
1368.1.n $$\chi_{1368}(989, \cdot)$$ None 0 1
1368.1.o $$\chi_{1368}(721, \cdot)$$ None 0 1
1368.1.u $$\chi_{1368}(373, \cdot)$$ None 0 2
1368.1.v $$\chi_{1368}(353, \cdot)$$ None 0 2
1368.1.ba $$\chi_{1368}(619, \cdot)$$ 1368.1.ba.a 2 2
1368.1.ba.b 2
1368.1.ba.c 4
1368.1.bb $$\chi_{1368}(407, \cdot)$$ None 0 2
1368.1.bc $$\chi_{1368}(847, \cdot)$$ None 0 2
1368.1.bd $$\chi_{1368}(107, \cdot)$$ None 0 2
1368.1.bg $$\chi_{1368}(601, \cdot)$$ None 0 2
1368.1.bh $$\chi_{1368}(581, \cdot)$$ None 0 2
1368.1.bj $$\chi_{1368}(77, \cdot)$$ None 0 2
1368.1.bk $$\chi_{1368}(265, \cdot)$$ None 0 2
1368.1.bo $$\chi_{1368}(227, \cdot)$$ 1368.1.bo.a 2 2
1368.1.bo.b 2
1368.1.bp $$\chi_{1368}(799, \cdot)$$ None 0 2
1368.1.bs $$\chi_{1368}(7, \cdot)$$ None 0 2
1368.1.bt $$\chi_{1368}(635, \cdot)$$ 1368.1.bt.a 2 2
1368.1.bt.b 2
1368.1.bv $$\chi_{1368}(145, \cdot)$$ None 0 2
1368.1.bw $$\chi_{1368}(125, \cdot)$$ 1368.1.bw.a 8 2
1368.1.bz $$\chi_{1368}(163, \cdot)$$ 1368.1.bz.a 2 2
1368.1.ca $$\chi_{1368}(791, \cdot)$$ None 0 2
1368.1.cd $$\chi_{1368}(761, \cdot)$$ None 0 2
1368.1.ce $$\chi_{1368}(493, \cdot)$$ 1368.1.ce.a 6 2
1368.1.ce.b 6
1368.1.ch $$\chi_{1368}(445, \cdot)$$ None 0 2
1368.1.ci $$\chi_{1368}(425, \cdot)$$ None 0 2
1368.1.cj $$\chi_{1368}(691, \cdot)$$ 1368.1.cj.a 2 2
1368.1.cj.b 2
1368.1.cj.c 4
1368.1.ck $$\chi_{1368}(335, \cdot)$$ None 0 2
1368.1.cn $$\chi_{1368}(455, \cdot)$$ None 0 2
1368.1.co $$\chi_{1368}(115, \cdot)$$ None 0 2
1368.1.cr $$\chi_{1368}(829, \cdot)$$ None 0 2
1368.1.cs $$\chi_{1368}(809, \cdot)$$ None 0 2
1368.1.cw $$\chi_{1368}(673, \cdot)$$ None 0 2
1368.1.cx $$\chi_{1368}(653, \cdot)$$ None 0 2
1368.1.cy $$\chi_{1368}(463, \cdot)$$ None 0 2
1368.1.cz $$\chi_{1368}(563, \cdot)$$ 1368.1.cz.a 2 2
1368.1.cz.b 2
1368.1.df $$\chi_{1368}(167, \cdot)$$ None 0 6
1368.1.dh $$\chi_{1368}(43, \cdot)$$ 1368.1.dh.a 6 6
1368.1.dh.b 6
1368.1.dk $$\chi_{1368}(155, \cdot)$$ 1368.1.dk.a 6 6
1368.1.dk.b 6
1368.1.dm $$\chi_{1368}(175, \cdot)$$ None 0 6
1368.1.do $$\chi_{1368}(205, \cdot)$$ None 0 6
1368.1.dq $$\chi_{1368}(137, \cdot)$$ None 0 6
1368.1.dr $$\chi_{1368}(557, \cdot)$$ None 0 6
1368.1.ds $$\chi_{1368}(433, \cdot)$$ None 0 6
1368.1.dv $$\chi_{1368}(17, \cdot)$$ None 0 6
1368.1.dx $$\chi_{1368}(109, \cdot)$$ None 0 6
1368.1.dy $$\chi_{1368}(193, \cdot)$$ None 0 6
1368.1.ea $$\chi_{1368}(5, \cdot)$$ None 0 6
1368.1.ec $$\chi_{1368}(367, \cdot)$$ None 0 6
1368.1.ee $$\chi_{1368}(59, \cdot)$$ 1368.1.ee.a 6 6
1368.1.ee.b 6
1368.1.ef $$\chi_{1368}(71, \cdot)$$ None 0 6
1368.1.eh $$\chi_{1368}(595, \cdot)$$ 1368.1.eh.a 6 6
1368.1.ek $$\chi_{1368}(395, \cdot)$$ None 0 6
1368.1.em $$\chi_{1368}(55, \cdot)$$ None 0 6
1368.1.en $$\chi_{1368}(283, \cdot)$$ 1368.1.en.a 6 6
1368.1.en.b 6
1368.1.ep $$\chi_{1368}(383, \cdot)$$ None 0 6
1368.1.er $$\chi_{1368}(245, \cdot)$$ None 0 6
1368.1.es $$\chi_{1368}(97, \cdot)$$ None 0 6
1368.1.ev $$\chi_{1368}(329, \cdot)$$ None 0 6
1368.1.ex $$\chi_{1368}(13, \cdot)$$ None 0 6

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1368))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1368)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(171))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(228))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(456))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(684))$$$$^{\oplus 2}$$