# Properties

 Label 1600.4 Level 1600 Weight 4 Dimension 118997 Nonzero newspaces 28 Sturm bound 614400 Trace bound 12

## Defining parameters

 Level: $$N$$ = $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$28$$ Sturm bound: $$614400$$ Trace bound: $$12$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1600))$$.

Total New Old
Modular forms 232416 119899 112517
Cusp forms 228384 118997 109387
Eisenstein series 4032 902 3130

## Trace form

 $$118997 q - 104 q^{2} - 78 q^{3} - 104 q^{4} - 128 q^{5} - 168 q^{6} - 80 q^{7} - 104 q^{8} - 157 q^{9} + O(q^{10})$$ $$118997 q - 104 q^{2} - 78 q^{3} - 104 q^{4} - 128 q^{5} - 168 q^{6} - 80 q^{7} - 104 q^{8} - 157 q^{9} - 128 q^{10} - 146 q^{11} - 104 q^{12} - 32 q^{13} - 104 q^{14} - 96 q^{15} - 168 q^{16} - 78 q^{17} - 104 q^{18} - 54 q^{19} - 128 q^{20} - 180 q^{21} + 368 q^{22} - 80 q^{23} + 896 q^{24} - 160 q^{25} - 368 q^{26} - 264 q^{27} - 864 q^{28} - 504 q^{29} - 128 q^{30} - 488 q^{31} - 1344 q^{32} - 1060 q^{33} - 1104 q^{34} - 96 q^{35} - 1048 q^{36} - 624 q^{37} + 336 q^{38} - 64 q^{39} - 128 q^{40} + 814 q^{41} + 3056 q^{42} + 758 q^{43} + 896 q^{44} - 128 q^{45} - 168 q^{46} + 864 q^{47} - 104 q^{48} + 881 q^{49} - 128 q^{50} + 4228 q^{51} - 3416 q^{52} + 304 q^{53} - 1832 q^{54} - 96 q^{55} + 224 q^{56} - 776 q^{57} + 2272 q^{58} - 4538 q^{59} - 128 q^{60} - 1248 q^{61} + 2888 q^{62} - 2268 q^{63} + 5944 q^{64} + 3840 q^{65} + 5368 q^{66} + 1290 q^{67} + 1960 q^{68} + 5212 q^{69} - 128 q^{70} - 1472 q^{71} - 752 q^{72} - 3842 q^{73} - 2736 q^{74} - 6704 q^{75} - 6280 q^{76} - 10996 q^{77} - 2096 q^{78} - 2656 q^{79} - 128 q^{80} - 13187 q^{81} + 6856 q^{82} - 8238 q^{83} + 4040 q^{84} - 4832 q^{85} - 688 q^{86} - 5232 q^{87} - 3224 q^{88} - 2290 q^{89} - 128 q^{90} + 3324 q^{91} - 12720 q^{92} + 11296 q^{93} - 9032 q^{94} + 7632 q^{95} - 13088 q^{96} + 5098 q^{97} - 12208 q^{98} + 6842 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1600))$$

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
1600.4.a $$\chi_{1600}(1, \cdot)$$ 1600.4.a.a 1 1
1600.4.a.b 1
1600.4.a.c 1
1600.4.a.d 1
1600.4.a.e 1
1600.4.a.f 1
1600.4.a.g 1
1600.4.a.h 1
1600.4.a.i 1
1600.4.a.j 1
1600.4.a.k 1
1600.4.a.l 1
1600.4.a.m 1
1600.4.a.n 1
1600.4.a.o 1
1600.4.a.p 1
1600.4.a.q 1
1600.4.a.r 1
1600.4.a.s 1
1600.4.a.t 1
1600.4.a.u 1
1600.4.a.v 1
1600.4.a.w 1
1600.4.a.x 1
1600.4.a.y 1
1600.4.a.z 1
1600.4.a.ba 1
1600.4.a.bb 1
1600.4.a.bc 1
1600.4.a.bd 1
1600.4.a.be 1
1600.4.a.bf 1
1600.4.a.bg 1
1600.4.a.bh 1
1600.4.a.bi 1
1600.4.a.bj 1
1600.4.a.bk 1
1600.4.a.bl 1
1600.4.a.bm 1
1600.4.a.bn 1
1600.4.a.bo 1
1600.4.a.bp 1
1600.4.a.bq 1
1600.4.a.br 1
1600.4.a.bs 1
1600.4.a.bt 1
1600.4.a.bu 1
1600.4.a.bv 1
1600.4.a.bw 1
1600.4.a.bx 1
1600.4.a.by 1
1600.4.a.bz 1
1600.4.a.ca 1
1600.4.a.cb 2
1600.4.a.cc 2
1600.4.a.cd 2
1600.4.a.ce 2
1600.4.a.cf 2
1600.4.a.cg 2
1600.4.a.ch 2
1600.4.a.ci 2
1600.4.a.cj 2
1600.4.a.ck 2
1600.4.a.cl 2
1600.4.a.cm 2
1600.4.a.cn 2
1600.4.a.co 2
1600.4.a.cp 2
1600.4.a.cq 3
1600.4.a.cr 3
1600.4.a.cs 3
1600.4.a.ct 3
1600.4.a.cu 4
1600.4.a.cv 4
1600.4.a.cw 4
1600.4.a.cx 4
1600.4.c $$\chi_{1600}(449, \cdot)$$ n/a 106 1
1600.4.d $$\chi_{1600}(801, \cdot)$$ n/a 114 1
1600.4.f $$\chi_{1600}(1249, \cdot)$$ n/a 108 1
1600.4.j $$\chi_{1600}(143, \cdot)$$ n/a 212 2
1600.4.l $$\chi_{1600}(401, \cdot)$$ n/a 222 2
1600.4.n $$\chi_{1600}(1343, \cdot)$$ n/a 212 2
1600.4.o $$\chi_{1600}(543, \cdot)$$ n/a 216 2
1600.4.q $$\chi_{1600}(49, \cdot)$$ n/a 212 2
1600.4.s $$\chi_{1600}(207, \cdot)$$ n/a 212 2
1600.4.u $$\chi_{1600}(321, \cdot)$$ n/a 712 4
1600.4.v $$\chi_{1600}(407, \cdot)$$ None 0 4
1600.4.y $$\chi_{1600}(201, \cdot)$$ None 0 4
1600.4.ba $$\chi_{1600}(249, \cdot)$$ None 0 4
1600.4.bb $$\chi_{1600}(7, \cdot)$$ None 0 4
1600.4.be $$\chi_{1600}(289, \cdot)$$ n/a 720 4
1600.4.bg $$\chi_{1600}(129, \cdot)$$ n/a 712 4
1600.4.bj $$\chi_{1600}(161, \cdot)$$ n/a 720 4
1600.4.bl $$\chi_{1600}(43, \cdot)$$ n/a 3440 8
1600.4.bm $$\chi_{1600}(101, \cdot)$$ n/a 3624 8
1600.4.bn $$\chi_{1600}(149, \cdot)$$ n/a 3440 8
1600.4.br $$\chi_{1600}(107, \cdot)$$ n/a 3440 8
1600.4.bt $$\chi_{1600}(303, \cdot)$$ n/a 1424 8
1600.4.bu $$\chi_{1600}(81, \cdot)$$ n/a 1424 8
1600.4.bx $$\chi_{1600}(223, \cdot)$$ n/a 1440 8
1600.4.by $$\chi_{1600}(63, \cdot)$$ n/a 1424 8
1600.4.cb $$\chi_{1600}(209, \cdot)$$ n/a 1424 8
1600.4.cc $$\chi_{1600}(47, \cdot)$$ n/a 1424 8
1600.4.cf $$\chi_{1600}(87, \cdot)$$ None 0 16
1600.4.cg $$\chi_{1600}(9, \cdot)$$ None 0 16
1600.4.ci $$\chi_{1600}(41, \cdot)$$ None 0 16
1600.4.cl $$\chi_{1600}(23, \cdot)$$ None 0 16
1600.4.cm $$\chi_{1600}(3, \cdot)$$ n/a 22976 32
1600.4.cq $$\chi_{1600}(29, \cdot)$$ n/a 22976 32
1600.4.cr $$\chi_{1600}(21, \cdot)$$ n/a 22976 32
1600.4.cs $$\chi_{1600}(67, \cdot)$$ n/a 22976 32

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1600))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1600)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 21}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 15}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 14}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 7}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(320))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(800))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1600))$$$$^{\oplus 1}$$