# Properties

 Label 1323.4 Level 1323 Weight 4 Dimension 143672 Nonzero newspaces 32 Sturm bound 508032 Trace bound 9

## 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

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

## 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 available 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

$$S_{4}^{\mathrm{old}}(\Gamma_1(1323)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(189))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(441))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1323))$$$$^{\oplus 1}$$