## Defining parameters

 Level: $$N$$ = $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$24$$ Sturm bound: $$129600$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(900))$$.

Total New Old
Modular forms 44320 18572 25748
Cusp forms 42080 18202 23878
Eisenstein series 2240 370 1870

## Trace form

 $$18202q - 17q^{2} + 3q^{3} - 11q^{4} - 42q^{5} - 49q^{6} + 27q^{7} - 44q^{8} - 23q^{9} + O(q^{10})$$ $$18202q - 17q^{2} + 3q^{3} - 11q^{4} - 42q^{5} - 49q^{6} + 27q^{7} - 44q^{8} - 23q^{9} - 80q^{10} + 20q^{11} - 60q^{12} - 29q^{13} - 150q^{14} - 12q^{15} - 119q^{16} - 126q^{17} - 44q^{18} - 58q^{19} - 110q^{20} - 251q^{21} - 295q^{22} - 317q^{23} - 175q^{24} - 252q^{25} - 72q^{26} - 96q^{27} - 94q^{28} - 277q^{29} + 8q^{30} + 37q^{31} + 353q^{32} - 110q^{33} + 339q^{34} - 28q^{35} + 363q^{36} + 348q^{37} + 335q^{38} - 23q^{39} - 100q^{40} + 80q^{41} - 18q^{42} - 194q^{43} - 34q^{44} - 224q^{45} + 218q^{46} - 269q^{47} + 331q^{48} - 815q^{49} + 698q^{50} + 165q^{51} + 1190q^{52} - 184q^{53} + 553q^{54} - 92q^{55} + 1036q^{56} + 639q^{57} + 1072q^{58} + 436q^{59} - 92q^{60} + 495q^{61} + 672q^{62} - 327q^{63} - 182q^{64} - 636q^{65} - 314q^{66} + 264q^{67} - 1123q^{68} + 697q^{69} - 1026q^{70} + 496q^{71} - 229q^{72} + 422q^{73} - 792q^{74} + 928q^{75} - 1575q^{76} + 1807q^{77} + 1318q^{78} - 437q^{79} + 458q^{80} + 617q^{81} - 516q^{82} + 1213q^{83} + 726q^{84} - 56q^{85} + 447q^{86} + 1445q^{87} + 893q^{88} + 1372q^{89} - 184q^{90} + 774q^{91} + 952q^{92} - 473q^{93} + 2074q^{94} + 112q^{95} - 1156q^{96} + 966q^{97} + 410q^{98} - 1109q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(900))$$

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
900.3.b $$\chi_{900}(449, \cdot)$$ 900.3.b.a 4 1
900.3.b.b 8
900.3.c $$\chi_{900}(451, \cdot)$$ 900.3.c.a 1 1
900.3.c.b 1
900.3.c.c 1
900.3.c.d 1
900.3.c.e 2
900.3.c.f 2
900.3.c.g 2
900.3.c.h 2
900.3.c.i 2
900.3.c.j 2
900.3.c.k 4
900.3.c.l 4
900.3.c.m 4
900.3.c.n 8
900.3.c.o 8
900.3.c.p 8
900.3.c.q 8
900.3.c.r 8
900.3.c.s 8
900.3.c.t 8
900.3.c.u 8
900.3.f $$\chi_{900}(199, \cdot)$$ 900.3.f.a 2 1
900.3.f.b 2
900.3.f.c 4
900.3.f.d 8
900.3.f.e 8
900.3.f.f 16
900.3.f.g 16
900.3.f.h 16
900.3.f.i 16
900.3.g $$\chi_{900}(701, \cdot)$$ 900.3.g.a 2 1
900.3.g.b 2
900.3.g.c 4
900.3.g.d 4
900.3.l $$\chi_{900}(757, \cdot)$$ 900.3.l.a 2 2
900.3.l.b 4
900.3.l.c 4
900.3.l.d 4
900.3.l.e 4
900.3.l.f 4
900.3.l.g 4
900.3.l.h 4
900.3.m $$\chi_{900}(107, \cdot)$$ n/a 144 2
900.3.p $$\chi_{900}(101, \cdot)$$ 900.3.p.a 4 2
900.3.p.b 4
900.3.p.c 12
900.3.p.d 16
900.3.p.e 16
900.3.p.f 24
900.3.q $$\chi_{900}(499, \cdot)$$ n/a 424 2
900.3.t $$\chi_{900}(151, \cdot)$$ n/a 444 2
900.3.u $$\chi_{900}(149, \cdot)$$ 900.3.u.a 8 2
900.3.u.b 8
900.3.u.c 24
900.3.u.d 32
900.3.x $$\chi_{900}(91, \cdot)$$ n/a 592 4
900.3.y $$\chi_{900}(89, \cdot)$$ 900.3.y.a 80 4
900.3.ba $$\chi_{900}(161, \cdot)$$ 900.3.ba.a 80 4
900.3.bb $$\chi_{900}(19, \cdot)$$ n/a 592 4
900.3.bc $$\chi_{900}(157, \cdot)$$ n/a 144 4
900.3.bd $$\chi_{900}(407, \cdot)$$ n/a 848 4
900.3.bh $$\chi_{900}(287, \cdot)$$ n/a 960 8
900.3.bi $$\chi_{900}(37, \cdot)$$ n/a 200 8
900.3.bl $$\chi_{900}(79, \cdot)$$ n/a 2848 8
900.3.bm $$\chi_{900}(41, \cdot)$$ n/a 480 8
900.3.bo $$\chi_{900}(29, \cdot)$$ n/a 480 8
900.3.bp $$\chi_{900}(31, \cdot)$$ n/a 2848 8
900.3.bu $$\chi_{900}(23, \cdot)$$ n/a 5696 16
900.3.bv $$\chi_{900}(13, \cdot)$$ n/a 960 16

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(900))$$ into lower level spaces

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