# Properties

 Label 3600.2 Level 3600 Weight 2 Dimension 141014 Nonzero newspaces 56 Sturm bound 1382400 Trace bound 45

## Defining parameters

 Level: $$N$$ = $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$56$$ Sturm bound: $$1382400$$ Trace bound: $$45$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3600))$$.

Total New Old
Modular forms 351872 142675 209197
Cusp forms 339329 141014 198315
Eisenstein series 12543 1661 10882

## Trace form

 $$141014 q - 78 q^{2} - 78 q^{3} - 76 q^{4} - 120 q^{5} - 168 q^{6} - 53 q^{7} - 72 q^{8} - 22 q^{9} + O(q^{10})$$ $$141014 q - 78 q^{2} - 78 q^{3} - 76 q^{4} - 120 q^{5} - 168 q^{6} - 53 q^{7} - 72 q^{8} - 22 q^{9} - 288 q^{10} - 81 q^{11} - 104 q^{12} - 81 q^{13} - 96 q^{14} - 96 q^{15} - 148 q^{16} - 168 q^{17} - 124 q^{18} - 152 q^{19} - 96 q^{20} - 223 q^{21} - 124 q^{22} - 55 q^{23} - 136 q^{24} - 16 q^{25} - 304 q^{26} - 96 q^{27} - 336 q^{28} - 147 q^{29} - 128 q^{30} - 123 q^{31} - 188 q^{32} - 255 q^{33} - 216 q^{34} - 84 q^{35} - 172 q^{36} - 364 q^{37} - 144 q^{38} - 135 q^{39} - 176 q^{40} - 109 q^{41} - 64 q^{42} - 149 q^{43} - 104 q^{44} - 180 q^{45} - 444 q^{46} - 225 q^{47} - 52 q^{48} - 308 q^{49} - 136 q^{50} - 374 q^{51} - 80 q^{52} - 266 q^{53} - 60 q^{54} - 298 q^{55} - 24 q^{56} - 132 q^{57} - 40 q^{58} - 287 q^{59} - 128 q^{60} - 277 q^{61} + 24 q^{62} - 219 q^{63} - 256 q^{64} - 316 q^{65} - 48 q^{66} - 311 q^{67} + 120 q^{68} - 151 q^{69} - 24 q^{70} - 330 q^{71} + 8 q^{72} - 162 q^{73} + 156 q^{74} - 120 q^{75} - 152 q^{76} - 185 q^{77} + 68 q^{78} - 201 q^{79} - 16 q^{80} - 302 q^{81} - 96 q^{82} - 191 q^{83} + 40 q^{84} - 224 q^{85} + 148 q^{86} + 15 q^{87} + 172 q^{88} - 28 q^{89} - 128 q^{90} - 384 q^{91} + 356 q^{92} - 195 q^{93} + 132 q^{94} - 198 q^{95} - 100 q^{96} - 291 q^{97} + 398 q^{98} + 9 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3600))$$

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
3600.2.a $$\chi_{3600}(1, \cdot)$$ 3600.2.a.a 1 1
3600.2.a.b 1
3600.2.a.c 1
3600.2.a.d 1
3600.2.a.e 1
3600.2.a.f 1
3600.2.a.g 1
3600.2.a.h 1
3600.2.a.i 1
3600.2.a.j 1
3600.2.a.k 1
3600.2.a.l 1
3600.2.a.m 1
3600.2.a.n 1
3600.2.a.o 1
3600.2.a.p 1
3600.2.a.q 1
3600.2.a.r 1
3600.2.a.s 1
3600.2.a.t 1
3600.2.a.u 1
3600.2.a.v 1
3600.2.a.w 1
3600.2.a.x 1
3600.2.a.y 1
3600.2.a.z 1
3600.2.a.ba 1
3600.2.a.bb 1
3600.2.a.bc 1
3600.2.a.bd 1
3600.2.a.be 1
3600.2.a.bf 1
3600.2.a.bg 1
3600.2.a.bh 1
3600.2.a.bi 1
3600.2.a.bj 1
3600.2.a.bk 1
3600.2.a.bl 1
3600.2.a.bm 1
3600.2.a.bn 1
3600.2.a.bo 1
3600.2.a.bp 1
3600.2.a.bq 1
3600.2.a.br 1
3600.2.a.bs 2
3600.2.b $$\chi_{3600}(2951, \cdot)$$ None 0 1
3600.2.d $$\chi_{3600}(649, \cdot)$$ None 0 1
3600.2.f $$\chi_{3600}(2449, \cdot)$$ 3600.2.f.a 2 1
3600.2.f.b 2
3600.2.f.c 2
3600.2.f.d 2
3600.2.f.e 2
3600.2.f.f 2
3600.2.f.g 2
3600.2.f.h 2
3600.2.f.i 2
3600.2.f.j 2
3600.2.f.k 2
3600.2.f.l 2
3600.2.f.m 2
3600.2.f.n 2
3600.2.f.o 2
3600.2.f.p 2
3600.2.f.q 2
3600.2.f.r 2
3600.2.f.s 2
3600.2.f.t 2
3600.2.f.u 2
3600.2.f.v 2
3600.2.h $$\chi_{3600}(1151, \cdot)$$ 3600.2.h.a 2 1
3600.2.h.b 2
3600.2.h.c 2
3600.2.h.d 4
3600.2.h.e 4
3600.2.h.f 4
3600.2.h.g 4
3600.2.h.h 4
3600.2.h.i 4
3600.2.h.j 8
3600.2.k $$\chi_{3600}(1801, \cdot)$$ None 0 1
3600.2.m $$\chi_{3600}(1799, \cdot)$$ None 0 1
3600.2.o $$\chi_{3600}(3599, \cdot)$$ 3600.2.o.a 4 1
3600.2.o.b 8
3600.2.o.c 8
3600.2.o.d 8
3600.2.o.e 8
3600.2.q $$\chi_{3600}(1201, \cdot)$$ n/a 222 2
3600.2.t $$\chi_{3600}(901, \cdot)$$ n/a 374 2
3600.2.u $$\chi_{3600}(899, \cdot)$$ n/a 288 2
3600.2.w $$\chi_{3600}(593, \cdot)$$ 3600.2.w.a 4 2
3600.2.w.b 4
3600.2.w.c 4
3600.2.w.d 4
3600.2.w.e 4
3600.2.w.f 4
3600.2.w.g 4
3600.2.w.h 4
3600.2.w.i 8
3600.2.w.j 8
3600.2.w.k 8
3600.2.w.l 8
3600.2.w.m 8
3600.2.x $$\chi_{3600}(2143, \cdot)$$ 3600.2.x.a 2 2
3600.2.x.b 2
3600.2.x.c 2
3600.2.x.d 4
3600.2.x.e 4
3600.2.x.f 4
3600.2.x.g 8
3600.2.x.h 8
3600.2.x.i 8
3600.2.x.j 8
3600.2.x.k 8
3600.2.x.l 8
3600.2.x.m 8
3600.2.x.n 8
3600.2.x.o 8
3600.2.z $$\chi_{3600}(2107, \cdot)$$ n/a 356 2
3600.2.bc $$\chi_{3600}(2357, \cdot)$$ n/a 288 2
3600.2.bd $$\chi_{3600}(307, \cdot)$$ n/a 356 2
3600.2.bg $$\chi_{3600}(557, \cdot)$$ n/a 288 2
3600.2.bi $$\chi_{3600}(343, \cdot)$$ None 0 2
3600.2.bj $$\chi_{3600}(2393, \cdot)$$ None 0 2
3600.2.bl $$\chi_{3600}(251, \cdot)$$ n/a 304 2
3600.2.bm $$\chi_{3600}(1549, \cdot)$$ n/a 356 2
3600.2.bp $$\chi_{3600}(721, \cdot)$$ n/a 296 4
3600.2.bs $$\chi_{3600}(1199, \cdot)$$ n/a 216 2
3600.2.bu $$\chi_{3600}(599, \cdot)$$ None 0 2
3600.2.bw $$\chi_{3600}(601, \cdot)$$ None 0 2
3600.2.bx $$\chi_{3600}(2351, \cdot)$$ n/a 228 2
3600.2.bz $$\chi_{3600}(49, \cdot)$$ n/a 212 2
3600.2.cb $$\chi_{3600}(1849, \cdot)$$ None 0 2
3600.2.cd $$\chi_{3600}(551, \cdot)$$ None 0 2
3600.2.cg $$\chi_{3600}(431, \cdot)$$ n/a 240 4
3600.2.ci $$\chi_{3600}(289, \cdot)$$ n/a 296 4
3600.2.ck $$\chi_{3600}(1369, \cdot)$$ None 0 4
3600.2.cm $$\chi_{3600}(71, \cdot)$$ None 0 4
3600.2.co $$\chi_{3600}(719, \cdot)$$ n/a 240 4
3600.2.cq $$\chi_{3600}(359, \cdot)$$ None 0 4
3600.2.cs $$\chi_{3600}(361, \cdot)$$ None 0 4
3600.2.cu $$\chi_{3600}(349, \cdot)$$ n/a 1712 4
3600.2.cv $$\chi_{3600}(851, \cdot)$$ n/a 1800 4
3600.2.cy $$\chi_{3600}(7, \cdot)$$ None 0 4
3600.2.db $$\chi_{3600}(857, \cdot)$$ None 0 4
3600.2.dc $$\chi_{3600}(293, \cdot)$$ n/a 1712 4
3600.2.df $$\chi_{3600}(43, \cdot)$$ n/a 1712 4
3600.2.dg $$\chi_{3600}(893, \cdot)$$ n/a 1712 4
3600.2.dj $$\chi_{3600}(643, \cdot)$$ n/a 1712 4
3600.2.dk $$\chi_{3600}(257, \cdot)$$ n/a 424 4
3600.2.dn $$\chi_{3600}(607, \cdot)$$ n/a 432 4
3600.2.dq $$\chi_{3600}(299, \cdot)$$ n/a 1712 4
3600.2.dr $$\chi_{3600}(301, \cdot)$$ n/a 1800 4
3600.2.ds $$\chi_{3600}(241, \cdot)$$ n/a 1424 8
3600.2.dt $$\chi_{3600}(179, \cdot)$$ n/a 1920 8
3600.2.du $$\chi_{3600}(181, \cdot)$$ n/a 2384 8
3600.2.dy $$\chi_{3600}(233, \cdot)$$ None 0 8
3600.2.dz $$\chi_{3600}(487, \cdot)$$ None 0 8
3600.2.eb $$\chi_{3600}(53, \cdot)$$ n/a 1920 8
3600.2.ee $$\chi_{3600}(523, \cdot)$$ n/a 2384 8
3600.2.ef $$\chi_{3600}(197, \cdot)$$ n/a 1920 8
3600.2.ei $$\chi_{3600}(163, \cdot)$$ n/a 2384 8
3600.2.ek $$\chi_{3600}(127, \cdot)$$ n/a 600 8
3600.2.el $$\chi_{3600}(17, \cdot)$$ n/a 480 8
3600.2.ep $$\chi_{3600}(109, \cdot)$$ n/a 2384 8
3600.2.eq $$\chi_{3600}(611, \cdot)$$ n/a 1920 8
3600.2.er $$\chi_{3600}(121, \cdot)$$ None 0 8
3600.2.et $$\chi_{3600}(119, \cdot)$$ None 0 8
3600.2.ev $$\chi_{3600}(239, \cdot)$$ n/a 1440 8
3600.2.ez $$\chi_{3600}(311, \cdot)$$ None 0 8
3600.2.fb $$\chi_{3600}(169, \cdot)$$ None 0 8
3600.2.fd $$\chi_{3600}(529, \cdot)$$ n/a 1424 8
3600.2.ff $$\chi_{3600}(191, \cdot)$$ n/a 1440 8
3600.2.fi $$\chi_{3600}(11, \cdot)$$ n/a 11456 16
3600.2.fj $$\chi_{3600}(229, \cdot)$$ n/a 11456 16
3600.2.fk $$\chi_{3600}(223, \cdot)$$ n/a 2880 16
3600.2.fn $$\chi_{3600}(113, \cdot)$$ n/a 2848 16
3600.2.fo $$\chi_{3600}(187, \cdot)$$ n/a 11456 16
3600.2.fr $$\chi_{3600}(173, \cdot)$$ n/a 11456 16
3600.2.fs $$\chi_{3600}(67, \cdot)$$ n/a 11456 16
3600.2.fv $$\chi_{3600}(77, \cdot)$$ n/a 11456 16
3600.2.fw $$\chi_{3600}(137, \cdot)$$ None 0 16
3600.2.fz $$\chi_{3600}(103, \cdot)$$ None 0 16
3600.2.ga $$\chi_{3600}(61, \cdot)$$ n/a 11456 16
3600.2.gb $$\chi_{3600}(59, \cdot)$$ n/a 11456 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3600))$$ into lower level spaces

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