## Defining parameters

 Level: $$N$$ = $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$32$$ Sturm bound: $$508032$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1323))$$.

Total New Old
Modular forms 192312 145224 47088
Cusp forms 188712 143672 45040
Eisenstein series 3600 1552 2048

## Trace form

 $$143672q - 123q^{2} - 186q^{3} - 199q^{4} - 99q^{5} - 198q^{6} - 252q^{7} - 357q^{8} - 234q^{9} + O(q^{10})$$ $$143672q - 123q^{2} - 186q^{3} - 199q^{4} - 99q^{5} - 198q^{6} - 252q^{7} - 357q^{8} - 234q^{9} - 255q^{10} + 3q^{11} - 33q^{12} + 37q^{13} + 120q^{14} - 360q^{15} - 415q^{16} - 813q^{17} - 819q^{18} - 809q^{19} - 2157q^{20} - 216q^{21} - 1215q^{22} - 189q^{23} + 990q^{24} + 215q^{25} + 2898q^{26} + 945q^{27} + 144q^{28} + 1461q^{29} + 279q^{30} + 1369q^{31} + 2517q^{32} - 819q^{33} + 861q^{34} - 108q^{35} - 2412q^{36} - 2015q^{37} - 5685q^{38} - 948q^{39} - 5703q^{40} - 5643q^{41} - 216q^{42} - 2075q^{43} - 6393q^{44} - 540q^{45} - 1857q^{46} + 471q^{47} - 5667q^{48} - 2532q^{49} - 15294q^{50} - 5085q^{51} - 3731q^{52} - 1524q^{53} + 3114q^{54} + 4272q^{55} + 7914q^{56} + 7710q^{57} + 12525q^{58} + 17916q^{59} + 25326q^{60} + 7141q^{61} + 27234q^{62} + 7596q^{63} + 20423q^{64} + 18075q^{65} + 3105q^{66} + 4495q^{67} + 21960q^{68} - 2718q^{69} + 5292q^{70} - 1449q^{71} - 10296q^{72} - 6335q^{73} - 19785q^{74} - 14496q^{75} - 19679q^{76} - 16416q^{77} - 19890q^{78} - 12875q^{79} - 36582q^{80} + 3258q^{81} - 11202q^{82} - 7779q^{83} - 216q^{84} - 6219q^{85} - 7287q^{86} + 2700q^{87} + 3825q^{88} - 5274q^{89} - 12690q^{90} + 108q^{91} - 13887q^{92} - 11982q^{93} - 4791q^{94} - 4653q^{95} - 14274q^{96} + 361q^{97} + 3774q^{98} - 1818q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1323))$$

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
1323.4.a $$\chi_{1323}(1, \cdot)$$ 1323.4.a.a 1 1
1323.4.a.b 1
1323.4.a.c 1
1323.4.a.d 1
1323.4.a.e 1
1323.4.a.f 1
1323.4.a.g 1
1323.4.a.h 1
1323.4.a.i 1
1323.4.a.j 1
1323.4.a.k 1
1323.4.a.l 1
1323.4.a.m 1
1323.4.a.n 1
1323.4.a.o 2
1323.4.a.p 2
1323.4.a.q 2
1323.4.a.r 2
1323.4.a.s 2
1323.4.a.t 2
1323.4.a.u 2
1323.4.a.v 2
1323.4.a.w 2
1323.4.a.x 2
1323.4.a.y 3
1323.4.a.z 3
1323.4.a.ba 4
1323.4.a.bb 6
1323.4.a.bc 6
1323.4.a.bd 6
1323.4.a.be 6
1323.4.a.bf 6
1323.4.a.bg 6
1323.4.a.bh 7
1323.4.a.bi 7
1323.4.a.bj 7
1323.4.a.bk 7
1323.4.a.bl 8
1323.4.a.bm 8
1323.4.a.bn 8
1323.4.a.bo 8
1323.4.a.bp 12
1323.4.a.bq 12
1323.4.c $$\chi_{1323}(1322, \cdot)$$ n/a 160 1
1323.4.e $$\chi_{1323}(1108, \cdot)$$ n/a 320 2
1323.4.f $$\chi_{1323}(442, \cdot)$$ n/a 236 2
1323.4.g $$\chi_{1323}(361, \cdot)$$ n/a 232 2
1323.4.h $$\chi_{1323}(226, \cdot)$$ n/a 232 2
1323.4.i $$\chi_{1323}(521, \cdot)$$ n/a 232 2
1323.4.o $$\chi_{1323}(440, \cdot)$$ n/a 232 2
1323.4.p $$\chi_{1323}(80, \cdot)$$ n/a 320 2
1323.4.s $$\chi_{1323}(656, \cdot)$$ n/a 232 2
1323.4.u $$\chi_{1323}(190, \cdot)$$ n/a 1344 6
1323.4.v $$\chi_{1323}(67, \cdot)$$ n/a 2136 6
1323.4.w $$\chi_{1323}(148, \cdot)$$ n/a 2184 6
1323.4.x $$\chi_{1323}(214, \cdot)$$ n/a 2136 6
1323.4.z $$\chi_{1323}(188, \cdot)$$ n/a 1344 6
1323.4.be $$\chi_{1323}(68, \cdot)$$ n/a 2136 6
1323.4.bh $$\chi_{1323}(362, \cdot)$$ n/a 2136 6
1323.4.bi $$\chi_{1323}(146, \cdot)$$ n/a 2136 6
1323.4.bk $$\chi_{1323}(37, \cdot)$$ n/a 1992 12
1323.4.bl $$\chi_{1323}(100, \cdot)$$ n/a 1992 12
1323.4.bm $$\chi_{1323}(64, \cdot)$$ n/a 1992 12
1323.4.bn $$\chi_{1323}(109, \cdot)$$ n/a 2688 12
1323.4.bp $$\chi_{1323}(17, \cdot)$$ n/a 1992 12
1323.4.bs $$\chi_{1323}(26, \cdot)$$ n/a 2688 12
1323.4.bt $$\chi_{1323}(62, \cdot)$$ n/a 1992 12
1323.4.bz $$\chi_{1323}(143, \cdot)$$ n/a 1992 12
1323.4.ca $$\chi_{1323}(25, \cdot)$$ n/a 18072 36
1323.4.cb $$\chi_{1323}(22, \cdot)$$ n/a 18072 36
1323.4.cc $$\chi_{1323}(4, \cdot)$$ n/a 18072 36
1323.4.ce $$\chi_{1323}(20, \cdot)$$ n/a 18072 36
1323.4.cf $$\chi_{1323}(47, \cdot)$$ n/a 18072 36
1323.4.ci $$\chi_{1323}(5, \cdot)$$ n/a 18072 36

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1323))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1323)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(189))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(441))$$$$^{\oplus 2}$$