# Properties

 Label 3600.3 Level 3600 Weight 3 Dimension 282859 Nonzero newspaces 56 Sturm bound 2073600 Trace bound 45

## Defining parameters

 Level: $$N$$ = $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$56$$ Sturm bound: $$2073600$$ Trace bound: $$45$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(3600))$$.

Total New Old
Modular forms 697472 284519 412953
Cusp forms 684928 282859 402069
Eisenstein series 12544 1660 10884

## Trace form

 $$282859 q - 78 q^{2} - 78 q^{3} - 84 q^{4} - 120 q^{5} - 168 q^{6} - 77 q^{7} - 72 q^{8} - 34 q^{9} + O(q^{10})$$ $$282859 q - 78 q^{2} - 78 q^{3} - 84 q^{4} - 120 q^{5} - 168 q^{6} - 77 q^{7} - 72 q^{8} - 34 q^{9} - 288 q^{10} - 109 q^{11} - 104 q^{12} - 119 q^{13} - 32 q^{14} - 96 q^{15} - 52 q^{16} - 92 q^{17} - 4 q^{18} - 76 q^{19} - 96 q^{20} - 169 q^{21} - 80 q^{22} + 109 q^{23} - 136 q^{24} + 24 q^{25} - 340 q^{26} - 6 q^{27} - 160 q^{28} + 35 q^{29} - 128 q^{30} - 69 q^{31} + 132 q^{32} - 291 q^{33} + 312 q^{34} - 24 q^{35} - 364 q^{36} - 58 q^{37} - 428 q^{38} + 57 q^{39} - 176 q^{40} - 143 q^{41} - 304 q^{42} + 101 q^{43} - 880 q^{44} - 60 q^{45} - 1276 q^{46} + 393 q^{47} - 52 q^{48} + 47 q^{49} - 536 q^{50} + 208 q^{51} - 488 q^{52} + 426 q^{53} + 204 q^{54} + 22 q^{55} + 232 q^{56} - 84 q^{57} + 660 q^{58} - 653 q^{59} - 128 q^{60} - 259 q^{61} - 288 q^{62} - 987 q^{63} - 384 q^{64} - 736 q^{65} - 768 q^{66} + 165 q^{67} - 1224 q^{68} - 1183 q^{69} - 744 q^{70} - 806 q^{71} + 8 q^{72} - 1592 q^{73} - 880 q^{74} - 480 q^{75} - 1056 q^{76} - 409 q^{77} + 308 q^{78} - 629 q^{79} - 16 q^{80} + 286 q^{81} - 1344 q^{82} - 561 q^{83} + 424 q^{84} + 856 q^{85} + 704 q^{86} - 213 q^{87} + 668 q^{88} + 956 q^{89} - 128 q^{90} - 1856 q^{91} + 532 q^{92} + 375 q^{93} + 1524 q^{94} - 798 q^{95} - 100 q^{96} + 315 q^{97} - 674 q^{98} + 27 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(3600))$$

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
3600.3.c $$\chi_{3600}(449, \cdot)$$ 3600.3.c.a 4 1
3600.3.c.b 4
3600.3.c.c 4
3600.3.c.d 4
3600.3.c.e 4
3600.3.c.f 4
3600.3.c.g 8
3600.3.c.h 8
3600.3.c.i 8
3600.3.c.j 8
3600.3.c.k 8
3600.3.c.l 8
3600.3.e $$\chi_{3600}(3151, \cdot)$$ 3600.3.e.a 1 1
3600.3.e.b 1
3600.3.e.c 1
3600.3.e.d 1
3600.3.e.e 1
3600.3.e.f 2
3600.3.e.g 2
3600.3.e.h 2
3600.3.e.i 2
3600.3.e.j 2
3600.3.e.k 2
3600.3.e.l 2
3600.3.e.m 2
3600.3.e.n 2
3600.3.e.o 2
3600.3.e.p 2
3600.3.e.q 2
3600.3.e.r 2
3600.3.e.s 2
3600.3.e.t 2
3600.3.e.u 2
3600.3.e.v 2
3600.3.e.w 2
3600.3.e.x 2
3600.3.e.y 4
3600.3.e.z 4
3600.3.e.ba 4
3600.3.e.bb 4
3600.3.e.bc 4
3600.3.e.bd 4
3600.3.e.be 4
3600.3.e.bf 4
3600.3.e.bg 4
3600.3.e.bh 4
3600.3.e.bi 4
3600.3.e.bj 8
3600.3.g $$\chi_{3600}(1351, \cdot)$$ None 0 1
3600.3.i $$\chi_{3600}(2249, \cdot)$$ None 0 1
3600.3.j $$\chi_{3600}(1999, \cdot)$$ 3600.3.j.a 2 1
3600.3.j.b 4
3600.3.j.c 4
3600.3.j.d 4
3600.3.j.e 4
3600.3.j.f 4
3600.3.j.g 4
3600.3.j.h 4
3600.3.j.i 4
3600.3.j.j 8
3600.3.j.k 8
3600.3.j.l 8
3600.3.j.m 8
3600.3.j.n 8
3600.3.j.o 8
3600.3.j.p 8
3600.3.l $$\chi_{3600}(1601, \cdot)$$ 3600.3.l.a 2 1
3600.3.l.b 2
3600.3.l.c 2
3600.3.l.d 2
3600.3.l.e 2
3600.3.l.f 2
3600.3.l.g 2
3600.3.l.h 2
3600.3.l.i 2
3600.3.l.j 2
3600.3.l.k 2
3600.3.l.l 2
3600.3.l.m 4
3600.3.l.n 4
3600.3.l.o 4
3600.3.l.p 4
3600.3.l.q 4
3600.3.l.r 4
3600.3.l.s 4
3600.3.l.t 4
3600.3.l.u 4
3600.3.l.v 4
3600.3.l.w 6
3600.3.l.x 6
3600.3.n $$\chi_{3600}(3401, \cdot)$$ None 0 1
3600.3.p $$\chi_{3600}(199, \cdot)$$ None 0 1
3600.3.r $$\chi_{3600}(1099, \cdot)$$ n/a 716 2
3600.3.s $$\chi_{3600}(701, \cdot)$$ n/a 608 2
3600.3.v $$\chi_{3600}(1943, \cdot)$$ None 0 2
3600.3.y $$\chi_{3600}(793, \cdot)$$ None 0 2
3600.3.ba $$\chi_{3600}(107, \cdot)$$ n/a 576 2
3600.3.bb $$\chi_{3600}(757, \cdot)$$ n/a 716 2
3600.3.be $$\chi_{3600}(1907, \cdot)$$ n/a 576 2
3600.3.bf $$\chi_{3600}(2557, \cdot)$$ n/a 716 2
3600.3.bh $$\chi_{3600}(2593, \cdot)$$ n/a 178 2
3600.3.bk $$\chi_{3600}(143, \cdot)$$ n/a 144 2
3600.3.bn $$\chi_{3600}(1349, \cdot)$$ n/a 576 2
3600.3.bo $$\chi_{3600}(451, \cdot)$$ n/a 754 2
3600.3.bq $$\chi_{3600}(1399, \cdot)$$ None 0 2
3600.3.br $$\chi_{3600}(1001, \cdot)$$ None 0 2
3600.3.bt $$\chi_{3600}(401, \cdot)$$ n/a 450 2
3600.3.bv $$\chi_{3600}(799, \cdot)$$ n/a 432 2
3600.3.by $$\chi_{3600}(1049, \cdot)$$ None 0 2
3600.3.ca $$\chi_{3600}(151, \cdot)$$ None 0 2
3600.3.cc $$\chi_{3600}(751, \cdot)$$ n/a 456 2
3600.3.ce $$\chi_{3600}(1649, \cdot)$$ n/a 428 2
3600.3.cf $$\chi_{3600}(89, \cdot)$$ None 0 4
3600.3.ch $$\chi_{3600}(631, \cdot)$$ None 0 4
3600.3.cj $$\chi_{3600}(271, \cdot)$$ n/a 600 4
3600.3.cl $$\chi_{3600}(1169, \cdot)$$ n/a 480 4
3600.3.cn $$\chi_{3600}(919, \cdot)$$ None 0 4
3600.3.cp $$\chi_{3600}(521, \cdot)$$ None 0 4
3600.3.cr $$\chi_{3600}(161, \cdot)$$ n/a 480 4
3600.3.ct $$\chi_{3600}(559, \cdot)$$ n/a 600 4
3600.3.cw $$\chi_{3600}(1051, \cdot)$$ n/a 3624 4
3600.3.cx $$\chi_{3600}(149, \cdot)$$ n/a 3440 4
3600.3.cz $$\chi_{3600}(193, \cdot)$$ n/a 856 4
3600.3.da $$\chi_{3600}(1343, \cdot)$$ n/a 864 4
3600.3.dd $$\chi_{3600}(157, \cdot)$$ n/a 3440 4
3600.3.de $$\chi_{3600}(443, \cdot)$$ n/a 3440 4
3600.3.dh $$\chi_{3600}(493, \cdot)$$ n/a 3440 4
3600.3.di $$\chi_{3600}(1307, \cdot)$$ n/a 3440 4
3600.3.dl $$\chi_{3600}(407, \cdot)$$ None 0 4
3600.3.dm $$\chi_{3600}(457, \cdot)$$ None 0 4
3600.3.do $$\chi_{3600}(101, \cdot)$$ n/a 3624 4
3600.3.dp $$\chi_{3600}(499, \cdot)$$ n/a 3440 4
3600.3.dv $$\chi_{3600}(341, \cdot)$$ n/a 3840 8
3600.3.dw $$\chi_{3600}(19, \cdot)$$ n/a 4784 8
3600.3.dx $$\chi_{3600}(287, \cdot)$$ n/a 960 8
3600.3.ea $$\chi_{3600}(433, \cdot)$$ n/a 1192 8
3600.3.ec $$\chi_{3600}(397, \cdot)$$ n/a 4784 8
3600.3.ed $$\chi_{3600}(467, \cdot)$$ n/a 3840 8
3600.3.eg $$\chi_{3600}(37, \cdot)$$ n/a 4784 8
3600.3.eh $$\chi_{3600}(323, \cdot)$$ n/a 3840 8
3600.3.ej $$\chi_{3600}(73, \cdot)$$ None 0 8
3600.3.em $$\chi_{3600}(503, \cdot)$$ None 0 8
3600.3.en $$\chi_{3600}(91, \cdot)$$ n/a 4784 8
3600.3.eo $$\chi_{3600}(269, \cdot)$$ n/a 3840 8
3600.3.es $$\chi_{3600}(79, \cdot)$$ n/a 2880 8
3600.3.eu $$\chi_{3600}(641, \cdot)$$ n/a 2864 8
3600.3.ew $$\chi_{3600}(41, \cdot)$$ None 0 8
3600.3.ex $$\chi_{3600}(439, \cdot)$$ None 0 8
3600.3.ey $$\chi_{3600}(209, \cdot)$$ n/a 2864 8
3600.3.fa $$\chi_{3600}(31, \cdot)$$ n/a 2880 8
3600.3.fc $$\chi_{3600}(391, \cdot)$$ None 0 8
3600.3.fe $$\chi_{3600}(329, \cdot)$$ None 0 8
3600.3.fg $$\chi_{3600}(29, \cdot)$$ n/a 22976 16
3600.3.fh $$\chi_{3600}(211, \cdot)$$ n/a 22976 16
3600.3.fl $$\chi_{3600}(313, \cdot)$$ None 0 16
3600.3.fm $$\chi_{3600}(23, \cdot)$$ None 0 16
3600.3.fp $$\chi_{3600}(83, \cdot)$$ n/a 22976 16
3600.3.fq $$\chi_{3600}(13, \cdot)$$ n/a 22976 16
3600.3.ft $$\chi_{3600}(203, \cdot)$$ n/a 22976 16
3600.3.fu $$\chi_{3600}(133, \cdot)$$ n/a 22976 16
3600.3.fx $$\chi_{3600}(47, \cdot)$$ n/a 5760 16
3600.3.fy $$\chi_{3600}(97, \cdot)$$ n/a 5728 16
3600.3.gc $$\chi_{3600}(139, \cdot)$$ n/a 22976 16
3600.3.gd $$\chi_{3600}(221, \cdot)$$ n/a 22976 16

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(3600))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(3600)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 18}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 15}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 24}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 18}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 20}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 18}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 15}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 16}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(450))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(720))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(900))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(1200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(1800))$$$$^{\oplus 2}$$