## Defining parameters

 Level: $$N$$ = $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$36$$ Sturm bound: $$230400$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1400))$$.

Total New Old
Modular forms 59616 29028 30588
Cusp forms 55585 28208 27377
Eisenstein series 4031 820 3211

## Trace form

 $$28208 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 2 q^{5} - 86 q^{6} - 74 q^{7} - 132 q^{8} - 134 q^{9} + O(q^{10})$$ $$28208 q - 54 q^{2} - 54 q^{3} - 54 q^{4} - 2 q^{5} - 86 q^{6} - 74 q^{7} - 132 q^{8} - 134 q^{9} - 64 q^{10} - 108 q^{11} - 6 q^{12} - 14 q^{13} - 42 q^{14} - 144 q^{15} - 22 q^{16} - 106 q^{17} + 76 q^{18} + 10 q^{19} - 24 q^{20} + 16 q^{21} - 22 q^{22} - 20 q^{23} + 94 q^{24} - 138 q^{25} - 72 q^{26} + 24 q^{27} - 26 q^{28} - 24 q^{29} - 56 q^{30} - 36 q^{31} - 104 q^{32} - 102 q^{33} - 146 q^{34} - 68 q^{35} - 310 q^{36} + 28 q^{37} - 116 q^{38} + 80 q^{39} - 144 q^{40} - 164 q^{41} - 142 q^{42} + 44 q^{43} - 126 q^{44} + 158 q^{45} - 212 q^{46} + 156 q^{47} - 130 q^{48} - 72 q^{49} - 120 q^{50} + 60 q^{51} - 12 q^{52} + 120 q^{53} - 14 q^{54} + 104 q^{55} - 36 q^{56} + 112 q^{57} + 108 q^{58} + 254 q^{59} + 24 q^{60} + 96 q^{61} + 132 q^{62} + 194 q^{63} - 30 q^{64} - 2 q^{65} - 14 q^{66} + 92 q^{67} + 42 q^{68} + 236 q^{69} - 76 q^{70} - 144 q^{71} - 188 q^{72} + 202 q^{73} - 34 q^{74} - 112 q^{75} - 190 q^{76} + 130 q^{77} - 288 q^{78} - 148 q^{79} - 104 q^{80} + 72 q^{81} - 270 q^{82} - 266 q^{83} - 174 q^{84} - 98 q^{85} - 154 q^{86} - 380 q^{87} - 446 q^{88} - 44 q^{89} - 664 q^{90} - 254 q^{91} - 502 q^{92} - 234 q^{93} - 514 q^{94} - 320 q^{95} - 238 q^{96} - 204 q^{97} - 234 q^{98} - 604 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1400))$$

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
1400.2.a $$\chi_{1400}(1, \cdot)$$ 1400.2.a.a 1 1
1400.2.a.b 1
1400.2.a.c 1
1400.2.a.d 1
1400.2.a.e 1
1400.2.a.f 1
1400.2.a.g 1
1400.2.a.h 1
1400.2.a.i 1
1400.2.a.j 1
1400.2.a.k 1
1400.2.a.l 1
1400.2.a.m 1
1400.2.a.n 1
1400.2.a.o 2
1400.2.a.p 2
1400.2.a.q 2
1400.2.a.r 2
1400.2.a.s 3
1400.2.a.t 3
1400.2.b $$\chi_{1400}(701, \cdot)$$ n/a 114 1
1400.2.e $$\chi_{1400}(1399, \cdot)$$ None 0 1
1400.2.g $$\chi_{1400}(449, \cdot)$$ 1400.2.g.a 2 1
1400.2.g.b 2
1400.2.g.c 2
1400.2.g.d 2
1400.2.g.e 2
1400.2.g.f 2
1400.2.g.g 2
1400.2.g.h 2
1400.2.g.i 4
1400.2.g.j 4
1400.2.g.k 4
1400.2.h $$\chi_{1400}(251, \cdot)$$ n/a 146 1
1400.2.k $$\chi_{1400}(951, \cdot)$$ None 0 1
1400.2.l $$\chi_{1400}(1149, \cdot)$$ n/a 108 1
1400.2.n $$\chi_{1400}(699, \cdot)$$ n/a 140 1
1400.2.q $$\chi_{1400}(401, \cdot)$$ 1400.2.q.a 2 2
1400.2.q.b 2
1400.2.q.c 2
1400.2.q.d 2
1400.2.q.e 2
1400.2.q.f 2
1400.2.q.g 2
1400.2.q.h 4
1400.2.q.i 6
1400.2.q.j 6
1400.2.q.k 6
1400.2.q.l 8
1400.2.q.m 8
1400.2.q.n 12
1400.2.q.o 12
1400.2.s $$\chi_{1400}(293, \cdot)$$ n/a 280 2
1400.2.t $$\chi_{1400}(407, \cdot)$$ None 0 2
1400.2.w $$\chi_{1400}(43, \cdot)$$ n/a 216 2
1400.2.x $$\chi_{1400}(657, \cdot)$$ 1400.2.x.a 16 2
1400.2.x.b 24
1400.2.x.c 32
1400.2.z $$\chi_{1400}(281, \cdot)$$ n/a 184 4
1400.2.bb $$\chi_{1400}(299, \cdot)$$ n/a 280 2
1400.2.bd $$\chi_{1400}(551, \cdot)$$ None 0 2
1400.2.bg $$\chi_{1400}(149, \cdot)$$ n/a 280 2
1400.2.bh $$\chi_{1400}(249, \cdot)$$ 1400.2.bh.a 4 2
1400.2.bh.b 4
1400.2.bh.c 4
1400.2.bh.d 4
1400.2.bh.e 4
1400.2.bh.f 4
1400.2.bh.g 8
1400.2.bh.h 12
1400.2.bh.i 12
1400.2.bh.j 16
1400.2.bk $$\chi_{1400}(451, \cdot)$$ n/a 292 2
1400.2.bm $$\chi_{1400}(501, \cdot)$$ n/a 292 2
1400.2.bn $$\chi_{1400}(199, \cdot)$$ None 0 2
1400.2.br $$\chi_{1400}(139, \cdot)$$ n/a 944 4
1400.2.bt $$\chi_{1400}(29, \cdot)$$ n/a 720 4
1400.2.bu $$\chi_{1400}(111, \cdot)$$ None 0 4
1400.2.bx $$\chi_{1400}(531, \cdot)$$ n/a 944 4
1400.2.by $$\chi_{1400}(169, \cdot)$$ n/a 176 4
1400.2.ca $$\chi_{1400}(279, \cdot)$$ None 0 4
1400.2.cd $$\chi_{1400}(141, \cdot)$$ n/a 720 4
1400.2.ce $$\chi_{1400}(257, \cdot)$$ n/a 144 4
1400.2.ch $$\chi_{1400}(107, \cdot)$$ n/a 560 4
1400.2.ci $$\chi_{1400}(207, \cdot)$$ None 0 4
1400.2.cl $$\chi_{1400}(157, \cdot)$$ n/a 560 4
1400.2.cm $$\chi_{1400}(81, \cdot)$$ n/a 480 8
1400.2.co $$\chi_{1400}(97, \cdot)$$ n/a 480 8
1400.2.cp $$\chi_{1400}(267, \cdot)$$ n/a 1440 8
1400.2.cs $$\chi_{1400}(127, \cdot)$$ None 0 8
1400.2.ct $$\chi_{1400}(13, \cdot)$$ n/a 1888 8
1400.2.cw $$\chi_{1400}(159, \cdot)$$ None 0 8
1400.2.cx $$\chi_{1400}(221, \cdot)$$ n/a 1888 8
1400.2.cz $$\chi_{1400}(131, \cdot)$$ n/a 1888 8
1400.2.dc $$\chi_{1400}(9, \cdot)$$ n/a 480 8
1400.2.dd $$\chi_{1400}(109, \cdot)$$ n/a 1888 8
1400.2.dg $$\chi_{1400}(31, \cdot)$$ None 0 8
1400.2.di $$\chi_{1400}(19, \cdot)$$ n/a 1888 8
1400.2.dk $$\chi_{1400}(117, \cdot)$$ n/a 3776 16
1400.2.dn $$\chi_{1400}(23, \cdot)$$ None 0 16
1400.2.do $$\chi_{1400}(67, \cdot)$$ n/a 3776 16
1400.2.dr $$\chi_{1400}(17, \cdot)$$ n/a 960 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1400))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1400)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(280))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(350))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(700))$$$$^{\oplus 2}$$