# Properties

 Label 1323.2 Level 1323 Weight 2 Dimension 47373 Nonzero newspaces 32 Sturm bound 254016 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$32$$ Sturm bound: $$254016$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1323))$$.

Total New Old
Modular forms 65304 48925 16379
Cusp forms 61705 47373 14332
Eisenstein series 3599 1552 2047

## Trace form

 $$47373 q - 126 q^{2} - 186 q^{3} - 218 q^{4} - 123 q^{5} - 180 q^{6} - 252 q^{7} - 210 q^{8} - 180 q^{9} + O(q^{10})$$ $$47373 q - 126 q^{2} - 186 q^{3} - 218 q^{4} - 123 q^{5} - 180 q^{6} - 252 q^{7} - 210 q^{8} - 180 q^{9} - 213 q^{10} - 117 q^{11} - 168 q^{12} - 207 q^{13} - 132 q^{14} - 315 q^{15} - 166 q^{16} - 75 q^{17} - 171 q^{18} - 192 q^{19} - 39 q^{20} - 216 q^{21} - 327 q^{22} - 96 q^{23} - 198 q^{24} - 184 q^{25} - 78 q^{26} - 189 q^{27} - 556 q^{28} - 198 q^{29} - 189 q^{30} - 183 q^{31} - 12 q^{32} - 180 q^{33} - 129 q^{34} - 108 q^{35} - 306 q^{36} - 154 q^{37} + 42 q^{38} - 147 q^{39} - 117 q^{40} + 3 q^{41} - 216 q^{42} - 319 q^{43} + 39 q^{44} - 189 q^{45} - 93 q^{46} - 63 q^{47} - 411 q^{48} - 250 q^{49} - 537 q^{50} - 306 q^{51} - 315 q^{52} - 390 q^{53} - 450 q^{54} - 660 q^{55} - 402 q^{56} - 471 q^{57} - 453 q^{58} - 432 q^{59} - 612 q^{60} - 315 q^{61} - 672 q^{62} - 342 q^{63} - 758 q^{64} - 477 q^{65} - 477 q^{66} - 320 q^{67} - 747 q^{68} - 351 q^{69} - 336 q^{70} - 333 q^{71} - 522 q^{72} - 315 q^{73} - 279 q^{74} - 321 q^{75} - 276 q^{76} - 162 q^{77} - 450 q^{78} - 167 q^{79} - 282 q^{80} - 144 q^{81} - 396 q^{82} + 75 q^{83} - 216 q^{84} - 261 q^{85} + 195 q^{86} - 171 q^{87} - 120 q^{88} + 117 q^{89} - 162 q^{90} - 172 q^{91} + 57 q^{92} - 219 q^{93} + 15 q^{94} + 87 q^{95} - 180 q^{96} - 66 q^{97} - 6 q^{98} - 603 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1323))$$

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
1323.2.a $$\chi_{1323}(1, \cdot)$$ 1323.2.a.a 1 1
1323.2.a.b 1
1323.2.a.c 1
1323.2.a.d 1
1323.2.a.e 1
1323.2.a.f 1
1323.2.a.g 1
1323.2.a.h 1
1323.2.a.i 1
1323.2.a.j 1
1323.2.a.k 1
1323.2.a.l 1
1323.2.a.m 1
1323.2.a.n 1
1323.2.a.o 1
1323.2.a.p 1
1323.2.a.q 1
1323.2.a.r 1
1323.2.a.s 1
1323.2.a.t 2
1323.2.a.u 2
1323.2.a.v 2
1323.2.a.w 2
1323.2.a.x 3
1323.2.a.y 3
1323.2.a.z 3
1323.2.a.ba 3
1323.2.a.bb 4
1323.2.a.bc 4
1323.2.a.bd 4
1323.2.a.be 4
1323.2.c $$\chi_{1323}(1322, \cdot)$$ 1323.2.c.a 2 1
1323.2.c.b 4
1323.2.c.c 4
1323.2.c.d 12
1323.2.c.e 16
1323.2.c.f 16
1323.2.e $$\chi_{1323}(1108, \cdot)$$ n/a 106 2
1323.2.f $$\chi_{1323}(442, \cdot)$$ 1323.2.f.a 2 2
1323.2.f.b 2
1323.2.f.c 6
1323.2.f.d 6
1323.2.f.e 10
1323.2.f.f 10
1323.2.f.g 12
1323.2.f.h 24
1323.2.g $$\chi_{1323}(361, \cdot)$$ 1323.2.g.a 2 2
1323.2.g.b 6
1323.2.g.c 6
1323.2.g.d 6
1323.2.g.e 6
1323.2.g.f 10
1323.2.g.g 12
1323.2.g.h 24
1323.2.h $$\chi_{1323}(226, \cdot)$$ 1323.2.h.a 2 2
1323.2.h.b 6
1323.2.h.c 6
1323.2.h.d 6
1323.2.h.e 6
1323.2.h.f 10
1323.2.h.g 12
1323.2.h.h 24
1323.2.i $$\chi_{1323}(521, \cdot)$$ 1323.2.i.a 2 2
1323.2.i.b 10
1323.2.i.c 12
1323.2.i.d 48
1323.2.o $$\chi_{1323}(440, \cdot)$$ 1323.2.o.a 2 2
1323.2.o.b 2
1323.2.o.c 10
1323.2.o.d 10
1323.2.o.e 48
1323.2.p $$\chi_{1323}(80, \cdot)$$ n/a 106 2
1323.2.s $$\chi_{1323}(656, \cdot)$$ 1323.2.s.a 2 2
1323.2.s.b 10
1323.2.s.c 12
1323.2.s.d 48
1323.2.u $$\chi_{1323}(190, \cdot)$$ n/a 444 6
1323.2.v $$\chi_{1323}(67, \cdot)$$ n/a 696 6
1323.2.w $$\chi_{1323}(148, \cdot)$$ n/a 708 6
1323.2.x $$\chi_{1323}(214, \cdot)$$ n/a 696 6
1323.2.z $$\chi_{1323}(188, \cdot)$$ n/a 444 6
1323.2.be $$\chi_{1323}(68, \cdot)$$ n/a 696 6
1323.2.bh $$\chi_{1323}(362, \cdot)$$ n/a 696 6
1323.2.bi $$\chi_{1323}(146, \cdot)$$ n/a 696 6
1323.2.bk $$\chi_{1323}(37, \cdot)$$ n/a 648 12
1323.2.bl $$\chi_{1323}(100, \cdot)$$ n/a 648 12
1323.2.bm $$\chi_{1323}(64, \cdot)$$ n/a 648 12
1323.2.bn $$\chi_{1323}(109, \cdot)$$ n/a 900 12
1323.2.bp $$\chi_{1323}(17, \cdot)$$ n/a 648 12
1323.2.bs $$\chi_{1323}(26, \cdot)$$ n/a 900 12
1323.2.bt $$\chi_{1323}(62, \cdot)$$ n/a 648 12
1323.2.bz $$\chi_{1323}(143, \cdot)$$ n/a 648 12
1323.2.ca $$\chi_{1323}(25, \cdot)$$ n/a 5976 36
1323.2.cb $$\chi_{1323}(22, \cdot)$$ n/a 5976 36
1323.2.cc $$\chi_{1323}(4, \cdot)$$ n/a 5976 36
1323.2.ce $$\chi_{1323}(20, \cdot)$$ n/a 5976 36
1323.2.cf $$\chi_{1323}(47, \cdot)$$ n/a 5976 36
1323.2.ci $$\chi_{1323}(5, \cdot)$$ n/a 5976 36

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1323))$$ into lower level spaces

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