# Properties

 Label 3900.2 Level 3900 Weight 2 Dimension 159746 Nonzero newspaces 80 Sturm bound 1612800 Trace bound 24

## Defining parameters

 Level: $$N$$ = $$3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$80$$ Sturm bound: $$1612800$$ Trace bound: $$24$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3900))$$.

Total New Old
Modular forms 409920 161546 248374
Cusp forms 396481 159746 236735
Eisenstein series 13439 1800 11639

## Trace form

 $$159746q + 8q^{3} - 148q^{4} + 4q^{5} - 122q^{6} - 12q^{7} - 48q^{8} - 154q^{9} + O(q^{10})$$ $$159746q + 8q^{3} - 148q^{4} + 4q^{5} - 122q^{6} - 12q^{7} - 48q^{8} - 154q^{9} - 192q^{10} - 20q^{11} - 80q^{12} - 296q^{13} - 28q^{15} - 180q^{16} - 114q^{17} - 12q^{18} - 88q^{19} + 40q^{20} - 290q^{21} - 144q^{22} - 104q^{23} + 10q^{24} - 452q^{25} - 28q^{26} + 20q^{27} - 80q^{28} - 62q^{29} - 4q^{30} - 8q^{31} + 20q^{32} - 150q^{33} - 68q^{34} - 8q^{35} - 44q^{36} - 218q^{37} + 184q^{38} + 12q^{39} - 112q^{40} - 2q^{41} + 192q^{42} + 80q^{43} + 340q^{44} + 4q^{45} + 84q^{46} + 136q^{47} + 170q^{48} - 196q^{49} + 328q^{50} - 48q^{51} + 72q^{52} - 44q^{53} - 34q^{54} + 92q^{56} - 162q^{57} + 36q^{58} - 148q^{59} - 4q^{60} - 682q^{61} + 36q^{62} - 62q^{63} - 64q^{64} - 162q^{65} - 292q^{66} - 360q^{67} - 64q^{68} - 106q^{69} - 168q^{70} - 168q^{71} + 68q^{72} - 288q^{73} + 44q^{75} - 380q^{76} - 24q^{77} + 108q^{78} - 24q^{79} + 8q^{80} - 30q^{81} - 52q^{82} + 116q^{83} + 132q^{84} + 108q^{85} + 64q^{86} + 344q^{87} - 120q^{88} + 692q^{89} - 76q^{90} + 180q^{91} - 104q^{92} + 326q^{93} - 244q^{94} + 160q^{95} + 92q^{96} + 604q^{97} - 356q^{98} + 14q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3900))$$

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
3900.2.a $$\chi_{3900}(1, \cdot)$$ 3900.2.a.a 1 1
3900.2.a.b 1
3900.2.a.c 1
3900.2.a.d 1
3900.2.a.e 1
3900.2.a.f 1
3900.2.a.g 1
3900.2.a.h 1
3900.2.a.i 1
3900.2.a.j 1
3900.2.a.k 1
3900.2.a.l 1
3900.2.a.m 1
3900.2.a.n 1
3900.2.a.o 2
3900.2.a.p 2
3900.2.a.q 2
3900.2.a.r 2
3900.2.a.s 2
3900.2.a.t 2
3900.2.a.u 3
3900.2.a.v 3
3900.2.a.w 3
3900.2.a.x 3
3900.2.c $$\chi_{3900}(3301, \cdot)$$ 3900.2.c.a 2 1
3900.2.c.b 2
3900.2.c.c 2
3900.2.c.d 2
3900.2.c.e 4
3900.2.c.f 4
3900.2.c.g 4
3900.2.c.h 6
3900.2.c.i 6
3900.2.c.j 6
3900.2.c.k 6
3900.2.d $$\chi_{3900}(3899, \cdot)$$ n/a 496 1
3900.2.g $$\chi_{3900}(3251, \cdot)$$ n/a 456 1
3900.2.h $$\chi_{3900}(1249, \cdot)$$ 3900.2.h.a 2 1
3900.2.h.b 2
3900.2.h.c 2
3900.2.h.d 2
3900.2.h.e 2
3900.2.h.f 2
3900.2.h.g 2
3900.2.h.h 2
3900.2.h.i 4
3900.2.h.j 4
3900.2.h.k 6
3900.2.h.l 6
3900.2.j $$\chi_{3900}(649, \cdot)$$ 3900.2.j.a 2 1
3900.2.j.b 2
3900.2.j.c 2
3900.2.j.d 2
3900.2.j.e 2
3900.2.j.f 2
3900.2.j.g 4
3900.2.j.h 4
3900.2.j.i 4
3900.2.j.j 8
3900.2.j.k 12
3900.2.m $$\chi_{3900}(2651, \cdot)$$ n/a 520 1
3900.2.n $$\chi_{3900}(599, \cdot)$$ n/a 432 1
3900.2.q $$\chi_{3900}(601, \cdot)$$ 3900.2.q.a 2 2
3900.2.q.b 2
3900.2.q.c 2
3900.2.q.d 2
3900.2.q.e 2
3900.2.q.f 2
3900.2.q.g 2
3900.2.q.h 2
3900.2.q.i 2
3900.2.q.j 4
3900.2.q.k 4
3900.2.q.l 4
3900.2.q.m 6
3900.2.q.n 6
3900.2.q.o 8
3900.2.q.p 8
3900.2.q.q 16
3900.2.q.r 16
3900.2.r $$\chi_{3900}(1357, \cdot)$$ 3900.2.r.a 16 2
3900.2.r.b 28
3900.2.r.c 40
3900.2.u $$\chi_{3900}(1607, \cdot)$$ n/a 992 2
3900.2.v $$\chi_{3900}(1457, \cdot)$$ n/a 144 2
3900.2.y $$\chi_{3900}(2107, \cdot)$$ n/a 432 2
3900.2.z $$\chi_{3900}(749, \cdot)$$ n/a 168 2
3900.2.bc $$\chi_{3900}(151, \cdot)$$ n/a 532 2
3900.2.be $$\chi_{3900}(499, \cdot)$$ n/a 504 2
3900.2.bf $$\chi_{3900}(2501, \cdot)$$ n/a 176 2
3900.2.bh $$\chi_{3900}(1507, \cdot)$$ n/a 504 2
3900.2.bk $$\chi_{3900}(857, \cdot)$$ n/a 168 2
3900.2.bm $$\chi_{3900}(2257, \cdot)$$ 3900.2.bm.a 16 2
3900.2.bm.b 28
3900.2.bm.c 40
3900.2.bn $$\chi_{3900}(707, \cdot)$$ n/a 992 2
3900.2.bp $$\chi_{3900}(781, \cdot)$$ n/a 240 4
3900.2.bs $$\chi_{3900}(1199, \cdot)$$ n/a 992 2
3900.2.bt $$\chi_{3900}(251, \cdot)$$ n/a 1040 2
3900.2.bw $$\chi_{3900}(49, \cdot)$$ 3900.2.bw.a 4 2
3900.2.bw.b 4
3900.2.bw.c 4
3900.2.bw.d 4
3900.2.bw.e 4
3900.2.bw.f 4
3900.2.bw.g 8
3900.2.bw.h 8
3900.2.bw.i 8
3900.2.bw.j 8
3900.2.bw.k 8
3900.2.bw.l 8
3900.2.bw.m 16
3900.2.by $$\chi_{3900}(1849, \cdot)$$ 3900.2.by.a 4 2
3900.2.by.b 4
3900.2.by.c 4
3900.2.by.d 4
3900.2.by.e 4
3900.2.by.f 4
3900.2.by.g 8
3900.2.by.h 8
3900.2.by.i 12
3900.2.by.j 12
3900.2.by.k 16
3900.2.bz $$\chi_{3900}(1751, \cdot)$$ n/a 1040 2
3900.2.cc $$\chi_{3900}(1499, \cdot)$$ n/a 992 2
3900.2.cd $$\chi_{3900}(901, \cdot)$$ 3900.2.cd.a 2 2
3900.2.cd.b 2
3900.2.cd.c 2
3900.2.cd.d 4
3900.2.cd.e 4
3900.2.cd.f 4
3900.2.cd.g 4
3900.2.cd.h 4
3900.2.cd.i 4
3900.2.cd.j 8
3900.2.cd.k 8
3900.2.cd.l 8
3900.2.cd.m 8
3900.2.cd.n 12
3900.2.cd.o 12
3900.2.cg $$\chi_{3900}(469, \cdot)$$ n/a 240 4
3900.2.ch $$\chi_{3900}(131, \cdot)$$ n/a 2880 4
3900.2.ck $$\chi_{3900}(779, \cdot)$$ n/a 3328 4
3900.2.cl $$\chi_{3900}(181, \cdot)$$ n/a 288 4
3900.2.cp $$\chi_{3900}(1379, \cdot)$$ n/a 2880 4
3900.2.cq $$\chi_{3900}(311, \cdot)$$ n/a 3328 4
3900.2.ct $$\chi_{3900}(1429, \cdot)$$ n/a 272 4
3900.2.cv $$\chi_{3900}(1007, \cdot)$$ n/a 1984 4
3900.2.cw $$\chi_{3900}(1393, \cdot)$$ n/a 168 4
3900.2.cy $$\chi_{3900}(257, \cdot)$$ n/a 336 4
3900.2.db $$\chi_{3900}(43, \cdot)$$ n/a 1008 4
3900.2.dd $$\chi_{3900}(401, \cdot)$$ n/a 356 4
3900.2.de $$\chi_{3900}(799, \cdot)$$ n/a 1008 4
3900.2.dg $$\chi_{3900}(1051, \cdot)$$ n/a 1064 4
3900.2.dj $$\chi_{3900}(149, \cdot)$$ n/a 336 4
3900.2.dk $$\chi_{3900}(607, \cdot)$$ n/a 1008 4
3900.2.dn $$\chi_{3900}(893, \cdot)$$ n/a 336 4
3900.2.do $$\chi_{3900}(743, \cdot)$$ n/a 1984 4
3900.2.dr $$\chi_{3900}(193, \cdot)$$ n/a 168 4
3900.2.ds $$\chi_{3900}(61, \cdot)$$ n/a 544 8
3900.2.dt $$\chi_{3900}(47, \cdot)$$ n/a 6656 8
3900.2.dw $$\chi_{3900}(73, \cdot)$$ n/a 560 8
3900.2.dx $$\chi_{3900}(77, \cdot)$$ n/a 1120 8
3900.2.ea $$\chi_{3900}(103, \cdot)$$ n/a 3360 8
3900.2.ec $$\chi_{3900}(161, \cdot)$$ n/a 1120 8
3900.2.ed $$\chi_{3900}(619, \cdot)$$ n/a 3360 8
3900.2.ef $$\chi_{3900}(31, \cdot)$$ n/a 3360 8
3900.2.ei $$\chi_{3900}(629, \cdot)$$ n/a 1120 8
3900.2.ej $$\chi_{3900}(547, \cdot)$$ n/a 2880 8
3900.2.em $$\chi_{3900}(53, \cdot)$$ n/a 960 8
3900.2.eo $$\chi_{3900}(203, \cdot)$$ n/a 6656 8
3900.2.ep $$\chi_{3900}(697, \cdot)$$ n/a 560 8
3900.2.er $$\chi_{3900}(589, \cdot)$$ n/a 544 8
3900.2.eu $$\chi_{3900}(491, \cdot)$$ n/a 6656 8
3900.2.ev $$\chi_{3900}(419, \cdot)$$ n/a 6656 8
3900.2.ez $$\chi_{3900}(121, \cdot)$$ n/a 576 8
3900.2.fa $$\chi_{3900}(179, \cdot)$$ n/a 6656 8
3900.2.fd $$\chi_{3900}(191, \cdot)$$ n/a 6656 8
3900.2.fe $$\chi_{3900}(289, \cdot)$$ n/a 576 8
3900.2.fh $$\chi_{3900}(37, \cdot)$$ n/a 1120 16
3900.2.fi $$\chi_{3900}(227, \cdot)$$ n/a 13312 16
3900.2.fk $$\chi_{3900}(113, \cdot)$$ n/a 2240 16
3900.2.fn $$\chi_{3900}(367, \cdot)$$ n/a 6720 16
3900.2.fo $$\chi_{3900}(89, \cdot)$$ n/a 2240 16
3900.2.fr $$\chi_{3900}(271, \cdot)$$ n/a 6720 16
3900.2.ft $$\chi_{3900}(19, \cdot)$$ n/a 6720 16
3900.2.fu $$\chi_{3900}(41, \cdot)$$ n/a 2240 16
3900.2.fw $$\chi_{3900}(127, \cdot)$$ n/a 6720 16
3900.2.fz $$\chi_{3900}(17, \cdot)$$ n/a 2240 16
3900.2.ga $$\chi_{3900}(97, \cdot)$$ n/a 1120 16
3900.2.gd $$\chi_{3900}(167, \cdot)$$ n/a 13312 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3900))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(3900)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(78))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(130))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(156))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(195))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(260))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(325))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(390))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(650))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(780))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(975))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1300))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1950))$$$$^{\oplus 2}$$