## Defining parameters

 Level: $$N$$ = $$3456 = 2^{7} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$30$$ Sturm bound: $$1327104$$ Trace bound: $$55$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(3456))$$.

Total New Old
Modular forms 336576 148224 188352
Cusp forms 326977 146688 180289
Eisenstein series 9599 1536 8063

## Trace form

 $$146688 q - 128 q^{2} - 144 q^{3} - 224 q^{4} - 128 q^{5} - 192 q^{6} - 168 q^{7} - 128 q^{8} - 240 q^{9} + O(q^{10})$$ $$146688 q - 128 q^{2} - 144 q^{3} - 224 q^{4} - 128 q^{5} - 192 q^{6} - 168 q^{7} - 128 q^{8} - 240 q^{9} - 224 q^{10} - 96 q^{11} - 192 q^{12} - 224 q^{13} - 128 q^{14} - 144 q^{15} - 224 q^{16} - 192 q^{17} - 192 q^{18} - 168 q^{19} - 128 q^{20} - 192 q^{21} - 224 q^{22} - 96 q^{23} - 192 q^{24} - 280 q^{25} - 128 q^{26} - 144 q^{27} - 512 q^{28} - 128 q^{29} - 192 q^{30} - 168 q^{31} - 128 q^{32} - 384 q^{33} - 224 q^{34} - 96 q^{35} - 192 q^{36} - 224 q^{37} - 128 q^{38} - 144 q^{39} - 224 q^{40} - 160 q^{41} - 192 q^{42} - 168 q^{43} - 128 q^{44} - 192 q^{45} - 224 q^{46} - 104 q^{47} - 192 q^{48} - 336 q^{49} - 128 q^{50} - 144 q^{51} - 224 q^{52} - 128 q^{53} - 192 q^{54} - 384 q^{55} - 128 q^{56} - 240 q^{57} - 224 q^{58} - 96 q^{59} - 192 q^{60} - 224 q^{61} - 128 q^{62} - 144 q^{63} - 224 q^{64} - 128 q^{65} - 192 q^{66} - 168 q^{67} - 128 q^{68} - 192 q^{69} - 224 q^{70} - 96 q^{71} - 192 q^{72} - 280 q^{73} - 128 q^{74} - 144 q^{75} - 224 q^{76} - 184 q^{77} - 192 q^{78} - 200 q^{79} - 128 q^{80} - 288 q^{81} - 512 q^{82} - 176 q^{83} - 192 q^{84} - 288 q^{85} - 128 q^{86} - 144 q^{87} - 224 q^{88} - 288 q^{89} - 192 q^{90} - 264 q^{91} - 128 q^{92} - 192 q^{93} - 224 q^{94} - 216 q^{95} - 192 q^{96} - 576 q^{97} - 128 q^{98} - 144 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(3456))$$

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
3456.2.a $$\chi_{3456}(1, \cdot)$$ 3456.2.a.a 1 1
3456.2.a.b 1
3456.2.a.c 1
3456.2.a.d 1
3456.2.a.e 1
3456.2.a.f 1
3456.2.a.g 1
3456.2.a.h 1
3456.2.a.i 1
3456.2.a.j 1
3456.2.a.k 1
3456.2.a.l 1
3456.2.a.m 1
3456.2.a.n 1
3456.2.a.o 1
3456.2.a.p 1
3456.2.a.q 2
3456.2.a.r 2
3456.2.a.s 2
3456.2.a.t 2
3456.2.a.u 2
3456.2.a.v 2
3456.2.a.w 2
3456.2.a.x 2
3456.2.a.y 2
3456.2.a.z 2
3456.2.a.ba 2
3456.2.a.bb 2
3456.2.a.bc 2
3456.2.a.bd 2
3456.2.a.be 2
3456.2.a.bf 2
3456.2.a.bg 2
3456.2.a.bh 2
3456.2.a.bi 2
3456.2.a.bj 2
3456.2.a.bk 2
3456.2.a.bl 2
3456.2.a.bm 2
3456.2.a.bn 2
3456.2.c $$\chi_{3456}(3455, \cdot)$$ 3456.2.c.a 8 1
3456.2.c.b 8
3456.2.c.c 8
3456.2.c.d 8
3456.2.c.e 8
3456.2.c.f 8
3456.2.c.g 8
3456.2.c.h 8
3456.2.d $$\chi_{3456}(1729, \cdot)$$ 3456.2.d.a 2 1
3456.2.d.b 2
3456.2.d.c 2
3456.2.d.d 2
3456.2.d.e 2
3456.2.d.f 2
3456.2.d.g 2
3456.2.d.h 2
3456.2.d.i 4
3456.2.d.j 4
3456.2.d.k 4
3456.2.d.l 4
3456.2.d.m 8
3456.2.d.n 8
3456.2.d.o 8
3456.2.d.p 8
3456.2.f $$\chi_{3456}(1727, \cdot)$$ 3456.2.f.a 4 1
3456.2.f.b 4
3456.2.f.c 8
3456.2.f.d 8
3456.2.f.e 8
3456.2.f.f 8
3456.2.f.g 8
3456.2.f.h 8
3456.2.f.i 8
3456.2.i $$\chi_{3456}(1153, \cdot)$$ 3456.2.i.a 2 2
3456.2.i.b 2
3456.2.i.c 2
3456.2.i.d 2
3456.2.i.e 10
3456.2.i.f 10
3456.2.i.g 10
3456.2.i.h 10
3456.2.i.i 12
3456.2.i.j 12
3456.2.i.k 12
3456.2.i.l 12
3456.2.k $$\chi_{3456}(865, \cdot)$$ n/a 128 2
3456.2.l $$\chi_{3456}(863, \cdot)$$ n/a 128 2
3456.2.p $$\chi_{3456}(575, \cdot)$$ 3456.2.p.a 4 2
3456.2.p.b 4
3456.2.p.c 8
3456.2.p.d 16
3456.2.p.e 16
3456.2.p.f 24
3456.2.p.g 24
3456.2.r $$\chi_{3456}(577, \cdot)$$ 3456.2.r.a 4 2
3456.2.r.b 4
3456.2.r.c 4
3456.2.r.d 4
3456.2.r.e 16
3456.2.r.f 16
3456.2.r.g 24
3456.2.r.h 24
3456.2.s $$\chi_{3456}(1151, \cdot)$$ 3456.2.s.a 24 2
3456.2.s.b 24
3456.2.s.c 24
3456.2.s.d 24
3456.2.v $$\chi_{3456}(433, \cdot)$$ n/a 256 4
3456.2.w $$\chi_{3456}(431, \cdot)$$ n/a 256 4
3456.2.y $$\chi_{3456}(385, \cdot)$$ n/a 864 6
3456.2.z $$\chi_{3456}(287, \cdot)$$ n/a 176 4
3456.2.bc $$\chi_{3456}(289, \cdot)$$ n/a 176 4
3456.2.be $$\chi_{3456}(217, \cdot)$$ None 0 8
3456.2.bf $$\chi_{3456}(215, \cdot)$$ None 0 8
3456.2.bj $$\chi_{3456}(193, \cdot)$$ n/a 864 6
3456.2.bl $$\chi_{3456}(191, \cdot)$$ n/a 864 6
3456.2.bm $$\chi_{3456}(383, \cdot)$$ n/a 864 6
3456.2.bo $$\chi_{3456}(145, \cdot)$$ n/a 368 8
3456.2.br $$\chi_{3456}(143, \cdot)$$ n/a 368 8
3456.2.bt $$\chi_{3456}(109, \cdot)$$ n/a 4096 16
3456.2.bu $$\chi_{3456}(107, \cdot)$$ n/a 4096 16
3456.2.bw $$\chi_{3456}(97, \cdot)$$ n/a 1680 12
3456.2.bz $$\chi_{3456}(95, \cdot)$$ n/a 1680 12
3456.2.cb $$\chi_{3456}(71, \cdot)$$ None 0 16
3456.2.cc $$\chi_{3456}(73, \cdot)$$ None 0 16
3456.2.cf $$\chi_{3456}(47, \cdot)$$ n/a 3408 24
3456.2.cg $$\chi_{3456}(49, \cdot)$$ n/a 3408 24
3456.2.ci $$\chi_{3456}(35, \cdot)$$ n/a 6080 32
3456.2.cl $$\chi_{3456}(37, \cdot)$$ n/a 6080 32
3456.2.cm $$\chi_{3456}(25, \cdot)$$ None 0 48
3456.2.cp $$\chi_{3456}(23, \cdot)$$ None 0 48
3456.2.cr $$\chi_{3456}(13, \cdot)$$ n/a 55104 96
3456.2.cs $$\chi_{3456}(11, \cdot)$$ n/a 55104 96

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(3456))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(3456)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 14}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 15}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 7}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(384))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(432))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(576))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(864))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1152))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1728))$$$$^{\oplus 2}$$