# Properties

 Label 1300.2 Level 1300 Weight 2 Dimension 26323 Nonzero newspaces 40 Sturm bound 201600 Trace bound 13

## Defining parameters

 Level: $$N$$ = $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$40$$ Sturm bound: $$201600$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1300))$$.

Total New Old
Modular forms 52080 27227 24853
Cusp forms 48721 26323 22398
Eisenstein series 3359 904 2455

## Trace form

 $$26323 q - 66 q^{2} - 8 q^{3} - 58 q^{4} - 162 q^{5} - 90 q^{6} + 6 q^{7} - 42 q^{8} - 120 q^{9} + O(q^{10})$$ $$26323 q - 66 q^{2} - 8 q^{3} - 58 q^{4} - 162 q^{5} - 90 q^{6} + 6 q^{7} - 42 q^{8} - 120 q^{9} - 64 q^{10} - 6 q^{11} - 52 q^{12} - 148 q^{13} - 136 q^{14} + 4 q^{15} - 122 q^{16} - 99 q^{17} - 64 q^{18} + 28 q^{19} - 84 q^{20} - 134 q^{21} - 28 q^{22} + 76 q^{23} - 30 q^{24} - 78 q^{25} - 180 q^{26} + 112 q^{27} - 28 q^{28} - 49 q^{29} - 76 q^{30} + 22 q^{31} + 4 q^{32} - 70 q^{33} - 34 q^{34} + 16 q^{35} - 106 q^{36} - 185 q^{37} - 112 q^{38} - 68 q^{39} - 280 q^{40} - 251 q^{41} - 276 q^{42} - 118 q^{43} - 228 q^{44} - 302 q^{45} - 270 q^{46} - 110 q^{47} - 366 q^{48} - 196 q^{49} - 284 q^{50} + 24 q^{51} - 220 q^{52} - 194 q^{53} - 360 q^{54} - 8 q^{55} - 264 q^{56} + 42 q^{57} - 214 q^{58} + 126 q^{59} - 276 q^{60} + 53 q^{61} - 170 q^{62} + 362 q^{63} - 160 q^{64} - 31 q^{65} - 276 q^{66} + 284 q^{67} - 28 q^{68} + 306 q^{69} - 76 q^{70} + 164 q^{71} + 32 q^{72} + 56 q^{73} + 6 q^{74} + 276 q^{75} - 66 q^{76} + 212 q^{77} + 16 q^{78} + 196 q^{79} - 44 q^{80} - 218 q^{81} + 146 q^{82} + 166 q^{83} + 144 q^{84} - 246 q^{85} + 84 q^{86} - 16 q^{87} + 224 q^{88} - 310 q^{89} + 236 q^{90} - 14 q^{91} + 88 q^{92} - 318 q^{93} + 250 q^{94} - 72 q^{95} + 20 q^{96} - 500 q^{97} + 226 q^{98} - 174 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1300))$$

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
1300.2.a $$\chi_{1300}(1, \cdot)$$ 1300.2.a.a 1 1
1300.2.a.b 1
1300.2.a.c 1
1300.2.a.d 1
1300.2.a.e 1
1300.2.a.f 1
1300.2.a.g 2
1300.2.a.h 2
1300.2.a.i 3
1300.2.a.j 3
1300.2.a.k 3
1300.2.c $$\chi_{1300}(1249, \cdot)$$ 1300.2.c.a 2 1
1300.2.c.b 2
1300.2.c.c 2
1300.2.c.d 2
1300.2.c.e 4
1300.2.c.f 6
1300.2.d $$\chi_{1300}(649, \cdot)$$ 1300.2.d.a 4 1
1300.2.d.b 4
1300.2.d.c 6
1300.2.d.d 6
1300.2.f $$\chi_{1300}(701, \cdot)$$ 1300.2.f.a 2 1
1300.2.f.b 2
1300.2.f.c 2
1300.2.f.d 2
1300.2.f.e 6
1300.2.f.f 8
1300.2.i $$\chi_{1300}(601, \cdot)$$ 1300.2.i.a 2 2
1300.2.i.b 2
1300.2.i.c 2
1300.2.i.d 2
1300.2.i.e 2
1300.2.i.f 2
1300.2.i.g 10
1300.2.i.h 10
1300.2.i.i 12
1300.2.j $$\chi_{1300}(151, \cdot)$$ n/a 254 2
1300.2.m $$\chi_{1300}(57, \cdot)$$ 1300.2.m.a 2 2
1300.2.m.b 4
1300.2.m.c 8
1300.2.m.d 8
1300.2.m.e 8
1300.2.m.f 12
1300.2.o $$\chi_{1300}(443, \cdot)$$ n/a 216 2
1300.2.p $$\chi_{1300}(207, \cdot)$$ n/a 244 2
1300.2.r $$\chi_{1300}(957, \cdot)$$ 1300.2.r.a 2 2
1300.2.r.b 4
1300.2.r.c 8
1300.2.r.d 8
1300.2.r.e 8
1300.2.r.f 12
1300.2.u $$\chi_{1300}(99, \cdot)$$ n/a 244 2
1300.2.v $$\chi_{1300}(261, \cdot)$$ n/a 120 4
1300.2.y $$\chi_{1300}(101, \cdot)$$ 1300.2.y.a 2 2
1300.2.y.b 8
1300.2.y.c 10
1300.2.y.d 10
1300.2.y.e 16
1300.2.ba $$\chi_{1300}(49, \cdot)$$ 1300.2.ba.a 4 2
1300.2.ba.b 8
1300.2.ba.c 8
1300.2.ba.d 20
1300.2.bb $$\chi_{1300}(549, \cdot)$$ 1300.2.bb.a 4 2
1300.2.bb.b 4
1300.2.bb.c 4
1300.2.bb.d 4
1300.2.bb.e 4
1300.2.bb.f 4
1300.2.bb.g 20
1300.2.be $$\chi_{1300}(181, \cdot)$$ n/a 136 4
1300.2.bg $$\chi_{1300}(209, \cdot)$$ n/a 120 4
1300.2.bj $$\chi_{1300}(129, \cdot)$$ n/a 144 4
1300.2.bk $$\chi_{1300}(799, \cdot)$$ n/a 488 4
1300.2.bn $$\chi_{1300}(93, \cdot)$$ 1300.2.bn.a 4 4
1300.2.bn.b 4
1300.2.bn.c 16
1300.2.bn.d 20
1300.2.bn.e 40
1300.2.bo $$\chi_{1300}(43, \cdot)$$ n/a 488 4
1300.2.br $$\chi_{1300}(107, \cdot)$$ n/a 488 4
1300.2.bs $$\chi_{1300}(193, \cdot)$$ 1300.2.bs.a 4 4
1300.2.bs.b 4
1300.2.bs.c 16
1300.2.bs.d 20
1300.2.bs.e 40
1300.2.bv $$\chi_{1300}(851, \cdot)$$ n/a 508 4
1300.2.bw $$\chi_{1300}(61, \cdot)$$ n/a 288 8
1300.2.bx $$\chi_{1300}(239, \cdot)$$ n/a 1648 8
1300.2.bz $$\chi_{1300}(73, \cdot)$$ n/a 280 8
1300.2.cc $$\chi_{1300}(103, \cdot)$$ n/a 1648 8
1300.2.cd $$\chi_{1300}(27, \cdot)$$ n/a 1440 8
1300.2.cg $$\chi_{1300}(177, \cdot)$$ n/a 280 8
1300.2.ci $$\chi_{1300}(31, \cdot)$$ n/a 1648 8
1300.2.cj $$\chi_{1300}(69, \cdot)$$ n/a 288 8
1300.2.cm $$\chi_{1300}(9, \cdot)$$ n/a 272 8
1300.2.co $$\chi_{1300}(121, \cdot)$$ n/a 272 8
1300.2.cq $$\chi_{1300}(11, \cdot)$$ n/a 3296 16
1300.2.cs $$\chi_{1300}(37, \cdot)$$ n/a 560 16
1300.2.cu $$\chi_{1300}(3, \cdot)$$ n/a 3296 16
1300.2.cx $$\chi_{1300}(23, \cdot)$$ n/a 3296 16
1300.2.cz $$\chi_{1300}(33, \cdot)$$ n/a 560 16
1300.2.db $$\chi_{1300}(19, \cdot)$$ n/a 3296 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1300))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1300)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(130))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(260))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(325))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(650))$$$$^{\oplus 2}$$