# Properties

 Label 3420.2 Level 3420 Weight 2 Dimension 130024 Nonzero newspaces 96 Sturm bound 1244160 Trace bound 26

## Defining parameters

 Level: $$N$$ = $$3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$96$$ Sturm bound: $$1244160$$ Trace bound: $$26$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3420))$$.

Total New Old
Modular forms 316800 131856 184944
Cusp forms 305281 130024 175257
Eisenstein series 11519 1832 9687

## Trace form

 $$130024 q - 54 q^{2} - 66 q^{4} - 154 q^{5} - 196 q^{6} - 16 q^{7} - 54 q^{8} - 168 q^{9} + O(q^{10})$$ $$130024 q - 54 q^{2} - 66 q^{4} - 154 q^{5} - 196 q^{6} - 16 q^{7} - 54 q^{8} - 168 q^{9} - 231 q^{10} - 44 q^{11} - 32 q^{12} - 128 q^{13} + 18 q^{14} - 6 q^{15} - 102 q^{16} - 74 q^{17} + 32 q^{18} + 30 q^{19} - 46 q^{20} - 380 q^{21} + 30 q^{22} + 66 q^{23} + 12 q^{24} - 116 q^{25} - 26 q^{26} + 48 q^{27} - 104 q^{28} - 48 q^{30} - 124 q^{32} + 12 q^{33} - 60 q^{34} + 76 q^{35} - 252 q^{36} - 332 q^{37} - 190 q^{38} + 100 q^{39} - 60 q^{40} - 160 q^{41} - 104 q^{42} - 66 q^{43} - 144 q^{44} - 48 q^{45} - 496 q^{46} - 78 q^{47} - 4 q^{48} - 108 q^{49} - 160 q^{50} - 84 q^{51} - 70 q^{52} - 116 q^{53} - 4 q^{54} - 56 q^{55} - 244 q^{56} - 224 q^{57} - 148 q^{58} - 194 q^{59} - 200 q^{60} - 432 q^{61} - 164 q^{62} - 148 q^{63} - 138 q^{64} - 277 q^{65} - 392 q^{66} - 238 q^{67} - 196 q^{68} - 320 q^{69} - 57 q^{70} - 366 q^{71} - 348 q^{72} - 584 q^{73} - 66 q^{74} - 138 q^{75} + 44 q^{76} - 546 q^{77} - 376 q^{78} - 180 q^{79} - 233 q^{80} - 624 q^{81} + 260 q^{82} - 258 q^{83} - 272 q^{84} - 306 q^{85} - 80 q^{86} - 260 q^{87} + 174 q^{88} - 466 q^{89} - 294 q^{90} - 216 q^{91} - 76 q^{92} - 140 q^{93} + 204 q^{94} - 69 q^{95} - 352 q^{96} + 122 q^{97} + 288 q^{98} + 64 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3420))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
3420.2.a $$\chi_{3420}(1, \cdot)$$ 3420.2.a.a 1 1
3420.2.a.b 1
3420.2.a.c 1
3420.2.a.d 1
3420.2.a.e 1
3420.2.a.f 1
3420.2.a.g 2
3420.2.a.h 2
3420.2.a.i 2
3420.2.a.j 3
3420.2.a.k 3
3420.2.a.l 3
3420.2.a.m 3
3420.2.a.n 3
3420.2.a.o 3
3420.2.b $$\chi_{3420}(2431, \cdot)$$ n/a 200 1
3420.2.e $$\chi_{3420}(1709, \cdot)$$ 3420.2.e.a 16 1
3420.2.e.b 24
3420.2.f $$\chi_{3420}(1369, \cdot)$$ 3420.2.f.a 4 1
3420.2.f.b 6
3420.2.f.c 6
3420.2.f.d 8
3420.2.f.e 10
3420.2.f.f 12
3420.2.i $$\chi_{3420}(1331, \cdot)$$ n/a 144 1
3420.2.j $$\chi_{3420}(341, \cdot)$$ 3420.2.j.a 24 1
3420.2.m $$\chi_{3420}(379, \cdot)$$ n/a 296 1
3420.2.n $$\chi_{3420}(2699, \cdot)$$ n/a 216 1
3420.2.q $$\chi_{3420}(121, \cdot)$$ n/a 160 2
3420.2.r $$\chi_{3420}(1141, \cdot)$$ n/a 144 2
3420.2.s $$\chi_{3420}(2101, \cdot)$$ n/a 160 2
3420.2.t $$\chi_{3420}(1261, \cdot)$$ 3420.2.t.a 2 2
3420.2.t.b 2
3420.2.t.c 2
3420.2.t.d 2
3420.2.t.e 2
3420.2.t.f 2
3420.2.t.g 2
3420.2.t.h 2
3420.2.t.i 2
3420.2.t.j 2
3420.2.t.k 2
3420.2.t.l 2
3420.2.t.m 2
3420.2.t.n 2
3420.2.t.o 2
3420.2.t.p 2
3420.2.t.q 2
3420.2.t.r 4
3420.2.t.s 4
3420.2.t.t 4
3420.2.t.u 4
3420.2.t.v 6
3420.2.t.w 8
3420.2.v $$\chi_{3420}(343, \cdot)$$ n/a 540 2
3420.2.x $$\chi_{3420}(1673, \cdot)$$ 3420.2.x.a 4 2
3420.2.x.b 8
3420.2.x.c 24
3420.2.x.d 36
3420.2.z $$\chi_{3420}(683, \cdot)$$ n/a 480 2
3420.2.bb $$\chi_{3420}(37, \cdot)$$ 3420.2.bb.a 8 2
3420.2.bb.b 8
3420.2.bb.c 8
3420.2.bb.d 12
3420.2.bb.e 24
3420.2.bb.f 40
3420.2.bc $$\chi_{3420}(449, \cdot)$$ 3420.2.bc.a 8 2
3420.2.bc.b 8
3420.2.bc.c 16
3420.2.bc.d 48
3420.2.bf $$\chi_{3420}(1171, \cdot)$$ n/a 400 2
3420.2.bg $$\chi_{3420}(1151, \cdot)$$ n/a 320 2
3420.2.bj $$\chi_{3420}(1189, \cdot)$$ 3420.2.bj.a 4 2
3420.2.bj.b 4
3420.2.bj.c 20
3420.2.bj.d 32
3420.2.bj.e 40
3420.2.bk $$\chi_{3420}(239, \cdot)$$ n/a 1424 2
3420.2.bp $$\chi_{3420}(1379, \cdot)$$ n/a 1424 2
3420.2.br $$\chi_{3420}(419, \cdot)$$ n/a 1296 2
3420.2.bt $$\chi_{3420}(1361, \cdot)$$ n/a 160 2
3420.2.bv $$\chi_{3420}(1519, \cdot)$$ n/a 1424 2
3420.2.bx $$\chi_{3420}(1399, \cdot)$$ n/a 1424 2
3420.2.ca $$\chi_{3420}(1481, \cdot)$$ n/a 160 2
3420.2.cc $$\chi_{3420}(221, \cdot)$$ n/a 160 2
3420.2.ce $$\chi_{3420}(259, \cdot)$$ n/a 1424 2
3420.2.cf $$\chi_{3420}(49, \cdot)$$ n/a 240 2
3420.2.ch $$\chi_{3420}(191, \cdot)$$ n/a 864 2
3420.2.cj $$\chi_{3420}(11, \cdot)$$ n/a 960 2
3420.2.cm $$\chi_{3420}(229, \cdot)$$ n/a 216 2
3420.2.co $$\chi_{3420}(1489, \cdot)$$ n/a 240 2
3420.2.cq $$\chi_{3420}(1451, \cdot)$$ n/a 960 2
3420.2.cr $$\chi_{3420}(1471, \cdot)$$ n/a 960 2
3420.2.ct $$\chi_{3420}(749, \cdot)$$ n/a 240 2
3420.2.cv $$\chi_{3420}(569, \cdot)$$ n/a 240 2
3420.2.cy $$\chi_{3420}(31, \cdot)$$ n/a 960 2
3420.2.da $$\chi_{3420}(151, \cdot)$$ n/a 960 2
3420.2.dc $$\chi_{3420}(2729, \cdot)$$ n/a 240 2
3420.2.dd $$\chi_{3420}(559, \cdot)$$ n/a 592 2
3420.2.dg $$\chi_{3420}(521, \cdot)$$ 3420.2.dg.a 8 2
3420.2.dg.b 8
3420.2.dg.c 32
3420.2.dj $$\chi_{3420}(539, \cdot)$$ n/a 480 2
3420.2.dk $$\chi_{3420}(541, \cdot)$$ n/a 204 6
3420.2.dl $$\chi_{3420}(61, \cdot)$$ n/a 480 6
3420.2.dm $$\chi_{3420}(481, \cdot)$$ n/a 480 6
3420.2.dn $$\chi_{3420}(197, \cdot)$$ n/a 160 4
3420.2.dp $$\chi_{3420}(163, \cdot)$$ n/a 1184 4
3420.2.ds $$\chi_{3420}(407, \cdot)$$ n/a 2848 4
3420.2.dt $$\chi_{3420}(493, \cdot)$$ n/a 480 4
3420.2.du $$\chi_{3420}(1813, \cdot)$$ n/a 480 4
3420.2.dx $$\chi_{3420}(227, \cdot)$$ n/a 2848 4
3420.2.dy $$\chi_{3420}(1703, \cdot)$$ n/a 2848 4
3420.2.ec $$\chi_{3420}(373, \cdot)$$ n/a 480 4
3420.2.ee $$\chi_{3420}(463, \cdot)$$ n/a 2848 4
3420.2.ef $$\chi_{3420}(77, \cdot)$$ n/a 432 4
3420.2.eg $$\chi_{3420}(1793, \cdot)$$ n/a 480 4
3420.2.ej $$\chi_{3420}(1483, \cdot)$$ n/a 2592 4
3420.2.ek $$\chi_{3420}(7, \cdot)$$ n/a 2848 4
3420.2.eo $$\chi_{3420}(353, \cdot)$$ n/a 480 4
3420.2.ep $$\chi_{3420}(217, \cdot)$$ n/a 200 4
3420.2.er $$\chi_{3420}(107, \cdot)$$ n/a 960 4
3420.2.et $$\chi_{3420}(169, \cdot)$$ n/a 720 6
3420.2.ew $$\chi_{3420}(671, \cdot)$$ n/a 2880 6
3420.2.ex $$\chi_{3420}(751, \cdot)$$ n/a 2880 6
3420.2.fa $$\chi_{3420}(509, \cdot)$$ n/a 720 6
3420.2.fc $$\chi_{3420}(41, \cdot)$$ n/a 480 6
3420.2.fd $$\chi_{3420}(679, \cdot)$$ n/a 4272 6
3420.2.fh $$\chi_{3420}(359, \cdot)$$ n/a 1440 6
3420.2.fi $$\chi_{3420}(1459, \cdot)$$ n/a 1776 6
3420.2.fl $$\chi_{3420}(1421, \cdot)$$ n/a 168 6
3420.2.fn $$\chi_{3420}(479, \cdot)$$ n/a 4272 6
3420.2.fp $$\chi_{3420}(29, \cdot)$$ n/a 720 6
3420.2.fs $$\chi_{3420}(211, \cdot)$$ n/a 2880 6
3420.2.fu $$\chi_{3420}(289, \cdot)$$ n/a 300 6
3420.2.fv $$\chi_{3420}(251, \cdot)$$ n/a 960 6
3420.2.fy $$\chi_{3420}(91, \cdot)$$ n/a 1200 6
3420.2.fz $$\chi_{3420}(89, \cdot)$$ n/a 240 6
3420.2.gb $$\chi_{3420}(131, \cdot)$$ n/a 2880 6
3420.2.ge $$\chi_{3420}(709, \cdot)$$ n/a 720 6
3420.2.gg $$\chi_{3420}(119, \cdot)$$ n/a 4272 6
3420.2.gj $$\chi_{3420}(79, \cdot)$$ n/a 4272 6
3420.2.gk $$\chi_{3420}(641, \cdot)$$ n/a 480 6
3420.2.gm $$\chi_{3420}(283, \cdot)$$ n/a 8544 12
3420.2.go $$\chi_{3420}(137, \cdot)$$ n/a 1440 12
3420.2.gq $$\chi_{3420}(167, \cdot)$$ n/a 8544 12
3420.2.gs $$\chi_{3420}(143, \cdot)$$ n/a 2880 12
3420.2.gu $$\chi_{3420}(13, \cdot)$$ n/a 1440 12
3420.2.gw $$\chi_{3420}(433, \cdot)$$ n/a 600 12
3420.2.gy $$\chi_{3420}(617, \cdot)$$ n/a 1440 12
3420.2.ha $$\chi_{3420}(17, \cdot)$$ n/a 480 12
3420.2.hc $$\chi_{3420}(43, \cdot)$$ n/a 8544 12
3420.2.he $$\chi_{3420}(883, \cdot)$$ n/a 3552 12
3420.2.hg $$\chi_{3420}(193, \cdot)$$ n/a 1440 12
3420.2.hi $$\chi_{3420}(383, \cdot)$$ n/a 8544 12

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3420))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(3420)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(95))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(114))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(171))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(190))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(228))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(285))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(342))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(380))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(570))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(684))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(855))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1140))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1710))$$$$^{\oplus 2}$$