## Defining parameters

 Level: $$N$$ = $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Newform subspaces: $$26$$ Sturm bound: $$7680$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(165))$$.

Total New Old
Modular forms 3040 1996 1044
Cusp forms 2720 1892 828
Eisenstein series 320 104 216

## Trace form

 $$1892 q - 8 q^{2} + 2 q^{3} + 28 q^{4} - 12 q^{5} - 130 q^{6} - 20 q^{7} + 232 q^{8} + 186 q^{9} + O(q^{10})$$ $$1892 q - 8 q^{2} + 2 q^{3} + 28 q^{4} - 12 q^{5} - 130 q^{6} - 20 q^{7} + 232 q^{8} + 186 q^{9} + 352 q^{10} + 256 q^{11} + 84 q^{12} - 268 q^{13} - 1308 q^{14} - 673 q^{15} - 2188 q^{16} - 680 q^{17} + 118 q^{18} + 1392 q^{19} + 1822 q^{20} + 1976 q^{21} + 3020 q^{22} + 1336 q^{23} + 2586 q^{24} + 158 q^{25} - 364 q^{26} - 1474 q^{27} - 3512 q^{28} - 2976 q^{29} - 4182 q^{30} - 4084 q^{31} - 3792 q^{32} - 3608 q^{33} - 2664 q^{34} - 120 q^{35} - 1078 q^{36} + 2580 q^{37} + 2084 q^{38} + 3378 q^{39} + 4364 q^{40} + 5168 q^{41} + 8364 q^{42} + 5616 q^{43} + 9772 q^{44} + 2388 q^{45} + 5616 q^{46} + 2136 q^{47} + 3760 q^{48} - 1160 q^{49} - 2818 q^{50} - 412 q^{51} - 10888 q^{52} - 7104 q^{53} - 648 q^{54} - 10046 q^{55} - 17360 q^{56} - 7532 q^{57} - 10192 q^{58} - 3032 q^{59} - 8202 q^{60} - 6532 q^{61} - 5824 q^{62} - 10334 q^{63} - 2948 q^{64} + 1984 q^{65} - 9376 q^{66} + 5272 q^{67} + 10416 q^{68} - 688 q^{69} + 12860 q^{70} - 2344 q^{71} + 4448 q^{72} - 4980 q^{73} + 2852 q^{74} + 7407 q^{75} + 15760 q^{76} + 3528 q^{77} + 16560 q^{78} + 10220 q^{79} + 7802 q^{80} + 14062 q^{81} + 17676 q^{82} + 12704 q^{83} + 19428 q^{84} + 4186 q^{85} + 3152 q^{86} + 11932 q^{87} + 20356 q^{88} + 8752 q^{89} + 6192 q^{90} + 10996 q^{91} + 15188 q^{92} + 3442 q^{93} + 3728 q^{94} + 5952 q^{95} - 13736 q^{96} - 5648 q^{97} - 808 q^{98} - 10558 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(165))$$

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
165.4.a $$\chi_{165}(1, \cdot)$$ 165.4.a.a 1 1
165.4.a.b 1
165.4.a.c 2
165.4.a.d 3
165.4.a.e 3
165.4.a.f 3
165.4.a.g 3
165.4.a.h 4
165.4.c $$\chi_{165}(34, \cdot)$$ 165.4.c.a 14 1
165.4.c.b 14
165.4.d $$\chi_{165}(164, \cdot)$$ 165.4.d.a 2 1
165.4.d.b 2
165.4.d.c 64
165.4.f $$\chi_{165}(131, \cdot)$$ 165.4.f.a 48 1
165.4.j $$\chi_{165}(43, \cdot)$$ 165.4.j.a 72 2
165.4.k $$\chi_{165}(23, \cdot)$$ 165.4.k.a 60 2
165.4.k.b 60
165.4.m $$\chi_{165}(16, \cdot)$$ 165.4.m.a 24 4
165.4.m.b 24
165.4.m.c 24
165.4.m.d 24
165.4.p $$\chi_{165}(41, \cdot)$$ 165.4.p.a 192 4
165.4.r $$\chi_{165}(29, \cdot)$$ 165.4.r.a 272 4
165.4.s $$\chi_{165}(4, \cdot)$$ 165.4.s.a 144 4
165.4.v $$\chi_{165}(38, \cdot)$$ 165.4.v.a 544 8
165.4.w $$\chi_{165}(7, \cdot)$$ 165.4.w.a 288 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(165))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(165)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 2}$$