## Defining parameters

 Level: $$N$$ = $$810 = 2 \cdot 3^{4} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$139968$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(810))$$.

Total New Old
Modular forms 53352 12672 40680
Cusp forms 51624 12672 38952
Eisenstein series 1728 0 1728

## Trace form

 $$12672 q - 24 q^{5} + 72 q^{7} + 48 q^{8} + O(q^{10})$$ $$12672 q - 24 q^{5} + 72 q^{7} + 48 q^{8} + 36 q^{10} - 102 q^{11} - 180 q^{13} - 264 q^{14} + 408 q^{17} + 828 q^{18} + 1620 q^{19} + 480 q^{20} + 432 q^{21} - 252 q^{22} - 2832 q^{23} - 1242 q^{25} - 4680 q^{26} - 2808 q^{27} - 1152 q^{28} - 3684 q^{29} - 756 q^{30} - 324 q^{31} + 1728 q^{33} + 2052 q^{34} + 7044 q^{35} + 2304 q^{36} + 4716 q^{37} + 3108 q^{38} - 432 q^{40} + 4674 q^{41} + 126 q^{43} - 528 q^{44} + 1026 q^{45} - 1008 q^{46} - 5424 q^{47} - 5364 q^{49} - 2136 q^{50} - 5922 q^{51} - 432 q^{52} - 9540 q^{53} - 1152 q^{55} + 1056 q^{56} - 4428 q^{57} + 3168 q^{58} + 162 q^{59} + 1944 q^{61} + 2976 q^{62} + 3996 q^{63} - 1152 q^{64} + 2130 q^{65} - 18720 q^{66} + 15498 q^{67} - 672 q^{68} - 8244 q^{69} + 1368 q^{70} - 2400 q^{71} + 1152 q^{72} + 1008 q^{73} + 4776 q^{74} - 3960 q^{76} + 24984 q^{77} + 21312 q^{78} - 14868 q^{79} + 1920 q^{80} + 23040 q^{81} - 2088 q^{82} + 6276 q^{83} + 4896 q^{84} - 11448 q^{85} + 7164 q^{86} + 21888 q^{87} - 1008 q^{88} + 1632 q^{89} - 1440 q^{90} + 5400 q^{91} - 2976 q^{92} - 18180 q^{93} + 12312 q^{94} - 9252 q^{95} - 5760 q^{96} + 16506 q^{97} - 21828 q^{98} - 41508 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(810))$$

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
810.4.a $$\chi_{810}(1, \cdot)$$ 810.4.a.a 1 1
810.4.a.b 1
810.4.a.c 1
810.4.a.d 1
810.4.a.e 1
810.4.a.f 1
810.4.a.g 2
810.4.a.h 2
810.4.a.i 2
810.4.a.j 2
810.4.a.k 2
810.4.a.l 2
810.4.a.m 2
810.4.a.n 2
810.4.a.o 2
810.4.a.p 2
810.4.a.q 3
810.4.a.r 3
810.4.a.s 4
810.4.a.t 4
810.4.a.u 4
810.4.a.v 4
810.4.c $$\chi_{810}(649, \cdot)$$ 810.4.c.a 10 1
810.4.c.b 10
810.4.c.c 16
810.4.c.d 18
810.4.c.e 18
810.4.e $$\chi_{810}(271, \cdot)$$ 810.4.e.a 2 2
810.4.e.b 2
810.4.e.c 2
810.4.e.d 2
810.4.e.e 2
810.4.e.f 2
810.4.e.g 2
810.4.e.h 2
810.4.e.i 2
810.4.e.j 2
810.4.e.k 2
810.4.e.l 2
810.4.e.m 2
810.4.e.n 2
810.4.e.o 2
810.4.e.p 2
810.4.e.q 2
810.4.e.r 2
810.4.e.s 2
810.4.e.t 2
810.4.e.u 2
810.4.e.v 2
810.4.e.w 2
810.4.e.x 2
810.4.e.y 4
810.4.e.z 4
810.4.e.ba 4
810.4.e.bb 4
810.4.e.bc 4
810.4.e.bd 4
810.4.e.be 4
810.4.e.bf 4
810.4.e.bg 8
810.4.e.bh 8
810.4.f $$\chi_{810}(323, \cdot)$$ n/a 144 2
810.4.i $$\chi_{810}(109, \cdot)$$ n/a 144 2
810.4.k $$\chi_{810}(91, \cdot)$$ n/a 216 6
810.4.m $$\chi_{810}(53, \cdot)$$ n/a 288 4
810.4.p $$\chi_{810}(19, \cdot)$$ n/a 324 6
810.4.q $$\chi_{810}(31, \cdot)$$ n/a 1944 18
810.4.s $$\chi_{810}(17, \cdot)$$ n/a 648 12
810.4.v $$\chi_{810}(49, \cdot)$$ n/a 2916 18
810.4.w $$\chi_{810}(23, \cdot)$$ n/a 5832 36

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(810))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(810)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(405))$$$$^{\oplus 2}$$