## Defining parameters

 Level: $$N$$ = $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$21$$ Sturm bound: $$5760$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(165))$$.

Total New Old
Modular forms 2080 1356 724
Cusp forms 1760 1244 516
Eisenstein series 320 112 208

## Trace form

 $$1244 q + 8 q^{2} - 2 q^{3} - 4 q^{4} + 8 q^{5} + 26 q^{6} + 56 q^{7} + 56 q^{8} - 22 q^{9} + O(q^{10})$$ $$1244 q + 8 q^{2} - 2 q^{3} - 4 q^{4} + 8 q^{5} + 26 q^{6} + 56 q^{7} + 56 q^{8} - 22 q^{9} - 68 q^{10} - 36 q^{11} - 156 q^{12} - 80 q^{13} - 20 q^{14} + 27 q^{15} + 196 q^{16} + 100 q^{17} + 74 q^{18} + 196 q^{19} + 122 q^{20} - 20 q^{21} - 180 q^{22} - 192 q^{23} - 614 q^{24} - 302 q^{25} - 676 q^{26} - 248 q^{27} - 832 q^{28} - 320 q^{29} - 122 q^{30} - 364 q^{31} + 152 q^{32} + 396 q^{33} + 696 q^{34} + 400 q^{35} + 890 q^{36} + 396 q^{37} + 692 q^{38} + 648 q^{39} - 316 q^{40} + 352 q^{41} + 316 q^{42} - 488 q^{43} - 940 q^{44} - 217 q^{45} - 1184 q^{46} - 996 q^{47} - 1128 q^{48} - 1384 q^{49} - 1298 q^{50} - 1202 q^{51} - 1016 q^{52} - 972 q^{53} - 1812 q^{54} - 246 q^{55} - 720 q^{56} - 822 q^{57} + 624 q^{58} + 300 q^{59} + 378 q^{60} + 544 q^{61} + 1024 q^{62} + 352 q^{63} + 2572 q^{64} + 1544 q^{65} + 1396 q^{66} + 1404 q^{67} + 2288 q^{68} + 1398 q^{69} + 2020 q^{70} + 1584 q^{71} + 2132 q^{72} + 1984 q^{73} + 1460 q^{74} + 517 q^{75} + 824 q^{76} + 1052 q^{77} + 1640 q^{78} + 456 q^{79} + 42 q^{80} - 66 q^{81} - 1276 q^{82} + 412 q^{83} - 1352 q^{84} - 684 q^{85} - 592 q^{86} - 1212 q^{87} - 3052 q^{88} - 760 q^{89} - 2238 q^{90} - 3064 q^{91} - 3988 q^{92} - 2450 q^{93} - 4504 q^{94} - 2958 q^{95} - 3096 q^{96} - 2616 q^{97} - 4464 q^{98} - 1490 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(165))$$

We only show spaces with odd 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
165.3.b $$\chi_{165}(76, \cdot)$$ 165.3.b.a 16 1
165.3.e $$\chi_{165}(56, \cdot)$$ 165.3.e.a 28 1
165.3.g $$\chi_{165}(89, \cdot)$$ 165.3.g.a 2 1
165.3.g.b 2
165.3.g.c 36
165.3.h $$\chi_{165}(109, \cdot)$$ 165.3.h.a 24 1
165.3.i $$\chi_{165}(67, \cdot)$$ 165.3.i.a 40 2
165.3.l $$\chi_{165}(32, \cdot)$$ 165.3.l.a 4 2
165.3.l.b 4
165.3.l.c 4
165.3.l.d 4
165.3.l.e 4
165.3.l.f 4
165.3.l.g 64
165.3.n $$\chi_{165}(19, \cdot)$$ 165.3.n.a 96 4
165.3.o $$\chi_{165}(14, \cdot)$$ 165.3.o.a 176 4
165.3.q $$\chi_{165}(26, \cdot)$$ 165.3.q.a 128 4
165.3.t $$\chi_{165}(46, \cdot)$$ 165.3.t.a 32 4
165.3.t.b 32
165.3.u $$\chi_{165}(2, \cdot)$$ 165.3.u.a 352 8
165.3.x $$\chi_{165}(37, \cdot)$$ 165.3.x.a 192 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(165))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(165)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 2}$$