## Defining parameters

 Level: $$N$$ = $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$24$$ Sturm bound: $$259200$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(900))$$.

Total New Old
Modular forms 109120 46151 62969
Cusp forms 106880 45783 61097
Eisenstein series 2240 368 1872

## Trace form

 $$45783 q - 21 q^{2} + 12 q^{3} - 43 q^{4} - 67 q^{5} - 67 q^{6} + 93 q^{7} - 510 q^{8} - 266 q^{9} + O(q^{10})$$ $$45783 q - 21 q^{2} + 12 q^{3} - 43 q^{4} - 67 q^{5} - 67 q^{6} + 93 q^{7} - 510 q^{8} - 266 q^{9} - 640 q^{10} + 1877 q^{11} - 510 q^{12} - 1187 q^{13} - 1524 q^{14} - 2172 q^{15} - 6159 q^{16} - 756 q^{17} + 2224 q^{18} + 11488 q^{19} + 9530 q^{20} + 12043 q^{21} - 7431 q^{22} - 48121 q^{23} - 31333 q^{24} - 10137 q^{25} - 6700 q^{26} + 13080 q^{27} + 52546 q^{28} + 37427 q^{29} + 32408 q^{30} + 32151 q^{31} - 3911 q^{32} + 75139 q^{33} - 44457 q^{34} - 76828 q^{35} - 95721 q^{36} + 13949 q^{37} - 40009 q^{38} - 62519 q^{39} + 8540 q^{40} + 59183 q^{41} + 78804 q^{42} - 79875 q^{43} - 51760 q^{44} + 100576 q^{45} - 179390 q^{46} + 140825 q^{47} + 22813 q^{48} - 21847 q^{49} + 344498 q^{50} - 168108 q^{51} + 397792 q^{52} - 333529 q^{53} + 90499 q^{54} - 31232 q^{55} - 350504 q^{56} + 323604 q^{57} - 457930 q^{58} + 554785 q^{59} - 205292 q^{60} + 522073 q^{61} - 272400 q^{62} - 396147 q^{63} + 174254 q^{64} - 602361 q^{65} + 281962 q^{66} - 503541 q^{67} + 505735 q^{68} - 434921 q^{69} + 130254 q^{70} - 69120 q^{71} - 434533 q^{72} + 471560 q^{73} - 1053810 q^{74} + 802648 q^{75} - 420739 q^{76} + 1768023 q^{77} + 1419340 q^{78} - 127363 q^{79} + 831538 q^{80} + 207518 q^{81} - 393404 q^{82} - 913121 q^{83} - 286410 q^{84} - 1035021 q^{85} - 1095253 q^{86} - 1503841 q^{87} + 1190373 q^{88} - 1723121 q^{89} - 1190344 q^{90} + 1009926 q^{91} - 1389574 q^{92} + 518323 q^{93} - 223774 q^{94} + 613532 q^{95} + 681044 q^{96} + 1797689 q^{97} + 1663538 q^{98} + 694537 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(900))$$

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
900.6.a $$\chi_{900}(1, \cdot)$$ 900.6.a.a 1 1
900.6.a.b 1
900.6.a.c 1
900.6.a.d 1
900.6.a.e 1
900.6.a.f 1
900.6.a.g 1
900.6.a.h 1
900.6.a.i 1
900.6.a.j 1
900.6.a.k 1
900.6.a.l 2
900.6.a.m 2
900.6.a.n 2
900.6.a.o 2
900.6.a.p 2
900.6.a.q 2
900.6.a.r 2
900.6.a.s 2
900.6.a.t 2
900.6.a.u 2
900.6.a.v 2
900.6.a.w 3
900.6.a.x 3
900.6.d $$\chi_{900}(649, \cdot)$$ 900.6.d.a 2 1
900.6.d.b 2
900.6.d.c 2
900.6.d.d 2
900.6.d.e 2
900.6.d.f 2
900.6.d.g 2
900.6.d.h 2
900.6.d.i 2
900.6.d.j 4
900.6.d.k 4
900.6.d.l 4
900.6.d.m 4
900.6.d.n 4
900.6.e $$\chi_{900}(251, \cdot)$$ n/a 190 1
900.6.h $$\chi_{900}(899, \cdot)$$ n/a 180 1
900.6.i $$\chi_{900}(301, \cdot)$$ n/a 190 2
900.6.j $$\chi_{900}(557, \cdot)$$ 900.6.j.a 16 2
900.6.j.b 20
900.6.j.c 24
900.6.k $$\chi_{900}(307, \cdot)$$ n/a 446 2
900.6.n $$\chi_{900}(181, \cdot)$$ n/a 252 4
900.6.o $$\chi_{900}(299, \cdot)$$ n/a 1072 2
900.6.r $$\chi_{900}(551, \cdot)$$ n/a 1128 2
900.6.s $$\chi_{900}(49, \cdot)$$ n/a 180 2
900.6.v $$\chi_{900}(71, \cdot)$$ n/a 1200 4
900.6.w $$\chi_{900}(109, \cdot)$$ n/a 248 4
900.6.z $$\chi_{900}(179, \cdot)$$ n/a 1200 4
900.6.be $$\chi_{900}(257, \cdot)$$ n/a 360 4
900.6.bf $$\chi_{900}(7, \cdot)$$ n/a 2144 4
900.6.bg $$\chi_{900}(61, \cdot)$$ n/a 1200 8
900.6.bj $$\chi_{900}(127, \cdot)$$ n/a 2984 8
900.6.bk $$\chi_{900}(17, \cdot)$$ n/a 400 8
900.6.bn $$\chi_{900}(59, \cdot)$$ n/a 7168 8
900.6.bq $$\chi_{900}(169, \cdot)$$ n/a 1200 8
900.6.br $$\chi_{900}(11, \cdot)$$ n/a 7168 8
900.6.bs $$\chi_{900}(67, \cdot)$$ n/a 14336 16
900.6.bt $$\chi_{900}(77, \cdot)$$ n/a 2400 16

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

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(900))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(900)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 18}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 9}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 18}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 9}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 9}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(450))$$$$^{\oplus 2}$$