## Defining parameters

 Level: $$N$$ = $$1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$42$$ Sturm bound: $$288000$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1650))$$.

Total New Old
Modular forms 74240 16316 57924
Cusp forms 69761 16316 53445
Eisenstein series 4479 0 4479

## Trace form

 $$16316 q - 6 q^{2} - 14 q^{3} - 6 q^{4} - 20 q^{5} - 19 q^{6} - 68 q^{7} - 6 q^{8} - 26 q^{9} + O(q^{10})$$ $$16316 q - 6 q^{2} - 14 q^{3} - 6 q^{4} - 20 q^{5} - 19 q^{6} - 68 q^{7} - 6 q^{8} - 26 q^{9} - 4 q^{10} - 24 q^{11} + 8 q^{12} - 40 q^{13} - 4 q^{14} + 8 q^{15} - 6 q^{16} + 28 q^{17} + 41 q^{18} + 74 q^{19} + 16 q^{20} + 44 q^{21} + 60 q^{22} + 92 q^{23} + 31 q^{24} + 172 q^{25} - 4 q^{26} + 16 q^{27} + 32 q^{28} + 56 q^{29} + 40 q^{30} + 12 q^{31} + 14 q^{32} + 65 q^{33} + 56 q^{34} + 16 q^{35} - q^{36} + 48 q^{37} - 24 q^{38} + 118 q^{39} + 12 q^{40} - 32 q^{41} + 106 q^{42} + 412 q^{43} + 120 q^{44} + 140 q^{45} + 328 q^{46} + 464 q^{47} + 18 q^{48} + 806 q^{49} + 124 q^{50} + 325 q^{51} + 260 q^{52} + 468 q^{53} + 56 q^{54} + 344 q^{55} - 16 q^{56} + 355 q^{57} + 444 q^{58} + 380 q^{59} + 16 q^{60} + 376 q^{61} + 164 q^{62} - 18 q^{63} - 6 q^{64} + 92 q^{65} + 52 q^{66} + 164 q^{67} + 108 q^{68} - 230 q^{69} - 16 q^{70} - 128 q^{71} - 58 q^{72} - 300 q^{73} - 108 q^{74} - 328 q^{75} - 12 q^{76} - 176 q^{77} - 276 q^{78} - 188 q^{79} - 20 q^{80} - 78 q^{81} - 134 q^{82} - 100 q^{83} - 136 q^{84} - 68 q^{85} - 100 q^{86} - 124 q^{87} - 40 q^{88} - 68 q^{89} - 108 q^{90} + 200 q^{91} - 48 q^{92} + 250 q^{93} - 120 q^{94} + 128 q^{95} + 18 q^{96} + 390 q^{97} - 150 q^{98} + 410 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1650))$$

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
1650.2.a $$\chi_{1650}(1, \cdot)$$ 1650.2.a.a 1 1
1650.2.a.b 1
1650.2.a.c 1
1650.2.a.d 1
1650.2.a.e 1
1650.2.a.f 1
1650.2.a.g 1
1650.2.a.h 1
1650.2.a.i 1
1650.2.a.j 1
1650.2.a.k 1
1650.2.a.l 1
1650.2.a.m 1
1650.2.a.n 1
1650.2.a.o 1
1650.2.a.p 1
1650.2.a.q 1
1650.2.a.r 1
1650.2.a.s 1
1650.2.a.t 1
1650.2.a.u 2
1650.2.a.v 2
1650.2.a.w 2
1650.2.a.x 2
1650.2.a.y 2
1650.2.a.z 2
1650.2.c $$\chi_{1650}(199, \cdot)$$ 1650.2.c.a 2 1
1650.2.c.b 2
1650.2.c.c 2
1650.2.c.d 2
1650.2.c.e 2
1650.2.c.f 2
1650.2.c.g 2
1650.2.c.h 2
1650.2.c.i 2
1650.2.c.j 2
1650.2.c.k 2
1650.2.c.l 2
1650.2.c.m 2
1650.2.c.n 2
1650.2.c.o 4
1650.2.d $$\chi_{1650}(1451, \cdot)$$ 1650.2.d.a 2 1
1650.2.d.b 2
1650.2.d.c 8
1650.2.d.d 8
1650.2.d.e 8
1650.2.d.f 8
1650.2.d.g 8
1650.2.d.h 8
1650.2.d.i 12
1650.2.d.j 12
1650.2.f $$\chi_{1650}(1649, \cdot)$$ 1650.2.f.a 4 1
1650.2.f.b 4
1650.2.f.c 8
1650.2.f.d 8
1650.2.f.e 8
1650.2.f.f 8
1650.2.f.g 16
1650.2.f.h 16
1650.2.j $$\chi_{1650}(1343, \cdot)$$ n/a 120 2
1650.2.l $$\chi_{1650}(43, \cdot)$$ 1650.2.l.a 4 2
1650.2.l.b 4
1650.2.l.c 8
1650.2.l.d 8
1650.2.l.e 8
1650.2.l.f 8
1650.2.l.g 16
1650.2.l.h 16
1650.2.m $$\chi_{1650}(361, \cdot)$$ n/a 240 4
1650.2.n $$\chi_{1650}(301, \cdot)$$ n/a 152 4
1650.2.o $$\chi_{1650}(31, \cdot)$$ n/a 240 4
1650.2.p $$\chi_{1650}(421, \cdot)$$ n/a 240 4
1650.2.q $$\chi_{1650}(181, \cdot)$$ n/a 240 4
1650.2.r $$\chi_{1650}(331, \cdot)$$ n/a 208 4
1650.2.t $$\chi_{1650}(131, \cdot)$$ n/a 480 4
1650.2.u $$\chi_{1650}(529, \cdot)$$ n/a 192 4
1650.2.x $$\chi_{1650}(959, \cdot)$$ n/a 480 4
1650.2.bb $$\chi_{1650}(479, \cdot)$$ n/a 480 4
1650.2.be $$\chi_{1650}(29, \cdot)$$ n/a 480 4
1650.2.bf $$\chi_{1650}(149, \cdot)$$ n/a 288 4
1650.2.bk $$\chi_{1650}(239, \cdot)$$ n/a 480 4
1650.2.bm $$\chi_{1650}(281, \cdot)$$ n/a 480 4
1650.2.bn $$\chi_{1650}(229, \cdot)$$ n/a 240 4
1650.2.bp $$\chi_{1650}(169, \cdot)$$ n/a 240 4
1650.2.br $$\chi_{1650}(41, \cdot)$$ n/a 480 4
1650.2.bu $$\chi_{1650}(101, \cdot)$$ n/a 304 4
1650.2.bv $$\chi_{1650}(761, \cdot)$$ n/a 480 4
1650.2.by $$\chi_{1650}(49, \cdot)$$ n/a 144 4
1650.2.bz $$\chi_{1650}(619, \cdot)$$ n/a 240 4
1650.2.cc $$\chi_{1650}(379, \cdot)$$ n/a 240 4
1650.2.ce $$\chi_{1650}(371, \cdot)$$ n/a 480 4
1650.2.ch $$\chi_{1650}(329, \cdot)$$ n/a 480 4
1650.2.cj $$\chi_{1650}(113, \cdot)$$ n/a 960 8
1650.2.ck $$\chi_{1650}(373, \cdot)$$ n/a 480 8
1650.2.cl $$\chi_{1650}(13, \cdot)$$ n/a 480 8
1650.2.cm $$\chi_{1650}(337, \cdot)$$ n/a 480 8
1650.2.cn $$\chi_{1650}(7, \cdot)$$ n/a 288 8
1650.2.co $$\chi_{1650}(217, \cdot)$$ n/a 480 8
1650.2.cu $$\chi_{1650}(23, \cdot)$$ n/a 800 8
1650.2.cv $$\chi_{1650}(257, \cdot)$$ n/a 576 8
1650.2.cw $$\chi_{1650}(53, \cdot)$$ n/a 960 8
1650.2.cx $$\chi_{1650}(47, \cdot)$$ n/a 960 8
1650.2.cy $$\chi_{1650}(137, \cdot)$$ n/a 960 8
1650.2.df $$\chi_{1650}(73, \cdot)$$ n/a 480 8

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1650))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1650)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(66))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(110))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(165))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(275))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(330))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(550))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(825))$$$$^{\oplus 2}$$