## Defining parameters

 Level: $$N$$ = $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$48$$ Sturm bound: $$311040$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(1368))$$.

Total New Old
Modular forms 105408 46220 59188
Cusp forms 101952 45608 56344
Eisenstein series 3456 612 2844

## Trace form

 $$45608q - 46q^{2} - 60q^{3} - 58q^{4} - 80q^{6} - 98q^{7} - 106q^{8} - 156q^{9} + O(q^{10})$$ $$45608q - 46q^{2} - 60q^{3} - 58q^{4} - 80q^{6} - 98q^{7} - 106q^{8} - 156q^{9} - 178q^{10} - 122q^{11} - 44q^{12} + 32q^{13} + 102q^{14} - 84q^{15} + 190q^{16} - 104q^{17} + 120q^{18} - 164q^{19} + 192q^{20} + 72q^{21} + 22q^{22} + 102q^{23} - 24q^{24} - 110q^{25} - 102q^{26} + 288q^{27} - 498q^{28} + 360q^{29} + 76q^{30} + 282q^{31} - 226q^{32} - 100q^{33} - 258q^{34} + 426q^{35} - 76q^{36} - 120q^{37} + 56q^{38} - 156q^{39} + 598q^{40} - 524q^{41} - 460q^{42} + 30q^{43} - 314q^{44} - 520q^{45} - 82q^{46} - 870q^{47} - 736q^{48} - 630q^{49} - 1114q^{50} - 492q^{51} - 802q^{52} - 648q^{54} - 970q^{55} - 1302q^{56} + 146q^{57} - 696q^{58} + 70q^{59} - 1284q^{60} - 100q^{61} - 1584q^{62} + 908q^{63} - 874q^{64} + 624q^{65} - 416q^{66} - 730q^{67} - 896q^{68} + 208q^{69} - 1018q^{70} - 270q^{71} + 84q^{72} + 222q^{73} - 210q^{74} - 980q^{75} - 56q^{76} - 900q^{77} + 820q^{78} + 942q^{79} + 1914q^{80} - 940q^{81} + 1712q^{82} + 178q^{83} + 1832q^{84} + 376q^{85} + 3328q^{86} - 348q^{87} + 3070q^{88} + 748q^{89} + 2428q^{90} + 2550q^{91} + 3672q^{92} + 704q^{93} + 2286q^{94} - 1314q^{95} + 1632q^{96} - 1332q^{97} + 2102q^{98} - 816q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(1368))$$

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
1368.3.b $$\chi_{1368}(1367, \cdot)$$ None 0 1
1368.3.c $$\chi_{1368}(1027, \cdot)$$ n/a 180 1
1368.3.h $$\chi_{1368}(305, \cdot)$$ 1368.3.h.a 16 1
1368.3.h.b 20
1368.3.i $$\chi_{1368}(37, \cdot)$$ n/a 198 1
1368.3.l $$\chi_{1368}(683, \cdot)$$ n/a 160 1
1368.3.m $$\chi_{1368}(343, \cdot)$$ None 0 1
1368.3.n $$\chi_{1368}(989, \cdot)$$ n/a 144 1
1368.3.o $$\chi_{1368}(721, \cdot)$$ 1368.3.o.a 2 1
1368.3.o.b 8
1368.3.o.c 20
1368.3.o.d 20
1368.3.u $$\chi_{1368}(373, \cdot)$$ n/a 952 2
1368.3.v $$\chi_{1368}(353, \cdot)$$ n/a 240 2
1368.3.ba $$\chi_{1368}(619, \cdot)$$ n/a 952 2
1368.3.bb $$\chi_{1368}(407, \cdot)$$ None 0 2
1368.3.bc $$\chi_{1368}(847, \cdot)$$ None 0 2
1368.3.bd $$\chi_{1368}(107, \cdot)$$ n/a 320 2
1368.3.bg $$\chi_{1368}(601, \cdot)$$ n/a 240 2
1368.3.bh $$\chi_{1368}(581, \cdot)$$ n/a 952 2
1368.3.bj $$\chi_{1368}(77, \cdot)$$ n/a 864 2
1368.3.bk $$\chi_{1368}(265, \cdot)$$ n/a 240 2
1368.3.bo $$\chi_{1368}(227, \cdot)$$ n/a 952 2
1368.3.bp $$\chi_{1368}(799, \cdot)$$ None 0 2
1368.3.bs $$\chi_{1368}(7, \cdot)$$ None 0 2
1368.3.bt $$\chi_{1368}(635, \cdot)$$ n/a 952 2
1368.3.bv $$\chi_{1368}(145, \cdot)$$ 1368.3.bv.a 20 2
1368.3.bv.b 20
1368.3.bv.c 20
1368.3.bv.d 40
1368.3.bw $$\chi_{1368}(125, \cdot)$$ n/a 320 2
1368.3.bz $$\chi_{1368}(163, \cdot)$$ n/a 396 2
1368.3.ca $$\chi_{1368}(791, \cdot)$$ None 0 2
1368.3.cd $$\chi_{1368}(761, \cdot)$$ n/a 216 2
1368.3.ce $$\chi_{1368}(493, \cdot)$$ n/a 952 2
1368.3.ch $$\chi_{1368}(445, \cdot)$$ n/a 952 2
1368.3.ci $$\chi_{1368}(425, \cdot)$$ n/a 240 2
1368.3.cj $$\chi_{1368}(691, \cdot)$$ n/a 952 2
1368.3.ck $$\chi_{1368}(335, \cdot)$$ None 0 2
1368.3.cn $$\chi_{1368}(455, \cdot)$$ None 0 2
1368.3.co $$\chi_{1368}(115, \cdot)$$ n/a 864 2
1368.3.cr $$\chi_{1368}(829, \cdot)$$ n/a 396 2
1368.3.cs $$\chi_{1368}(809, \cdot)$$ 1368.3.cs.a 40 2
1368.3.cs.b 40
1368.3.cw $$\chi_{1368}(673, \cdot)$$ n/a 240 2
1368.3.cx $$\chi_{1368}(653, \cdot)$$ n/a 952 2
1368.3.cy $$\chi_{1368}(463, \cdot)$$ None 0 2
1368.3.cz $$\chi_{1368}(563, \cdot)$$ n/a 952 2
1368.3.df $$\chi_{1368}(167, \cdot)$$ None 0 6
1368.3.dh $$\chi_{1368}(43, \cdot)$$ n/a 2856 6
1368.3.dk $$\chi_{1368}(155, \cdot)$$ n/a 2856 6
1368.3.dm $$\chi_{1368}(175, \cdot)$$ None 0 6
1368.3.do $$\chi_{1368}(205, \cdot)$$ n/a 2856 6
1368.3.dq $$\chi_{1368}(137, \cdot)$$ n/a 720 6
1368.3.dr $$\chi_{1368}(557, \cdot)$$ n/a 960 6
1368.3.ds $$\chi_{1368}(433, \cdot)$$ n/a 300 6
1368.3.dv $$\chi_{1368}(17, \cdot)$$ n/a 240 6
1368.3.dx $$\chi_{1368}(109, \cdot)$$ n/a 1188 6
1368.3.dy $$\chi_{1368}(193, \cdot)$$ n/a 720 6
1368.3.ea $$\chi_{1368}(5, \cdot)$$ n/a 2856 6
1368.3.ec $$\chi_{1368}(367, \cdot)$$ None 0 6
1368.3.ee $$\chi_{1368}(59, \cdot)$$ n/a 2856 6
1368.3.ef $$\chi_{1368}(71, \cdot)$$ None 0 6
1368.3.eh $$\chi_{1368}(595, \cdot)$$ n/a 1188 6
1368.3.ek $$\chi_{1368}(395, \cdot)$$ n/a 960 6
1368.3.em $$\chi_{1368}(55, \cdot)$$ None 0 6
1368.3.en $$\chi_{1368}(283, \cdot)$$ n/a 2856 6
1368.3.ep $$\chi_{1368}(383, \cdot)$$ None 0 6
1368.3.er $$\chi_{1368}(245, \cdot)$$ n/a 2856 6
1368.3.es $$\chi_{1368}(97, \cdot)$$ n/a 720 6
1368.3.ev $$\chi_{1368}(329, \cdot)$$ n/a 720 6
1368.3.ex $$\chi_{1368}(13, \cdot)$$ n/a 2856 6

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(1368))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(1368)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(114))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(171))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(228))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(342))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(456))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(684))$$$$^{\oplus 2}$$