# Properties

 Label 700.2 Level 700 Weight 2 Dimension 7214 Nonzero newspaces 24 Newform subspaces 88 Sturm bound 57600 Trace bound 7

## Defining parameters

 Level: $$N$$ = $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Newform subspaces: $$88$$ Sturm bound: $$57600$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(700))$$.

Total New Old
Modular forms 15240 7626 7614
Cusp forms 13561 7214 6347
Eisenstein series 1679 412 1267

## Trace form

 $$7214 q - 27 q^{2} - 9 q^{3} - 19 q^{4} - 66 q^{5} - 24 q^{6} - 39 q^{8} - 32 q^{9} + O(q^{10})$$ $$7214 q - 27 q^{2} - 9 q^{3} - 19 q^{4} - 66 q^{5} - 24 q^{6} - 39 q^{8} - 32 q^{9} - 16 q^{10} + 3 q^{11} - 4 q^{12} - 28 q^{13} - 17 q^{14} + 4 q^{15} - 47 q^{16} - 21 q^{17} - 37 q^{18} + 25 q^{19} - 36 q^{20} - 67 q^{21} - 70 q^{22} + 61 q^{23} - 48 q^{24} + 30 q^{25} - 60 q^{26} + 138 q^{27} - 53 q^{28} + 8 q^{29} - 28 q^{30} + 79 q^{31} + 13 q^{32} + 137 q^{33} - 40 q^{34} + 44 q^{35} - 91 q^{36} - 11 q^{37} - 58 q^{38} + 22 q^{39} - 136 q^{40} - 68 q^{41} - 114 q^{42} - 80 q^{43} - 138 q^{44} - 170 q^{45} - 132 q^{46} - 71 q^{47} - 276 q^{48} - 118 q^{49} - 284 q^{50} - 75 q^{51} - 308 q^{52} - 95 q^{53} - 438 q^{54} - 104 q^{55} - 189 q^{56} - 366 q^{57} - 322 q^{58} - 117 q^{59} - 348 q^{60} - 105 q^{61} - 332 q^{62} - 188 q^{63} - 391 q^{64} - 38 q^{65} - 414 q^{66} - 97 q^{67} - 328 q^{68} - 174 q^{69} - 218 q^{70} - 64 q^{71} - 379 q^{72} - 193 q^{73} - 312 q^{74} + 60 q^{75} - 288 q^{76} - 95 q^{77} - 376 q^{78} - 67 q^{79} - 104 q^{80} - 337 q^{81} - 100 q^{82} + 52 q^{83} - 138 q^{84} - 294 q^{85} - 18 q^{86} + 2 q^{87} + 50 q^{88} - 67 q^{89} + 212 q^{90} + 76 q^{91} + 142 q^{92} - 39 q^{93} + 274 q^{94} + 24 q^{95} + 152 q^{96} - 236 q^{97} + 135 q^{98} + 84 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(700))$$

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
700.2.a $$\chi_{700}(1, \cdot)$$ 700.2.a.a 1 1
700.2.a.b 1
700.2.a.c 1
700.2.a.d 1
700.2.a.e 1
700.2.a.f 1
700.2.a.g 1
700.2.a.h 1
700.2.a.i 1
700.2.a.j 1
700.2.c $$\chi_{700}(699, \cdot)$$ 700.2.c.a 4 1
700.2.c.b 4
700.2.c.c 4
700.2.c.d 4
700.2.c.e 4
700.2.c.f 4
700.2.c.g 4
700.2.c.h 8
700.2.c.i 8
700.2.c.j 8
700.2.c.k 16
700.2.e $$\chi_{700}(449, \cdot)$$ 700.2.e.a 2 1
700.2.e.b 2
700.2.e.c 2
700.2.e.d 2
700.2.g $$\chi_{700}(251, \cdot)$$ 700.2.g.a 2 1
700.2.g.b 4
700.2.g.c 4
700.2.g.d 4
700.2.g.e 4
700.2.g.f 4
700.2.g.g 4
700.2.g.h 4
700.2.g.i 8
700.2.g.j 8
700.2.g.k 8
700.2.g.l 16
700.2.i $$\chi_{700}(401, \cdot)$$ 700.2.i.a 2 2
700.2.i.b 2
700.2.i.c 2
700.2.i.d 6
700.2.i.e 6
700.2.i.f 8
700.2.k $$\chi_{700}(43, \cdot)$$ 700.2.k.a 24 2
700.2.k.b 36
700.2.k.c 48
700.2.m $$\chi_{700}(293, \cdot)$$ 700.2.m.a 8 2
700.2.m.b 8
700.2.m.c 8
700.2.n $$\chi_{700}(141, \cdot)$$ 700.2.n.a 4 4
700.2.n.b 4
700.2.n.c 20
700.2.n.d 28
700.2.p $$\chi_{700}(451, \cdot)$$ 700.2.p.a 4 2
700.2.p.b 8
700.2.p.c 32
700.2.p.d 32
700.2.p.e 32
700.2.p.f 32
700.2.r $$\chi_{700}(149, \cdot)$$ 700.2.r.a 4 2
700.2.r.b 4
700.2.r.c 4
700.2.r.d 12
700.2.t $$\chi_{700}(199, \cdot)$$ 700.2.t.a 4 2
700.2.t.b 4
700.2.t.c 32
700.2.t.d 32
700.2.t.e 64
700.2.w $$\chi_{700}(111, \cdot)$$ 700.2.w.a 464 4
700.2.y $$\chi_{700}(29, \cdot)$$ 700.2.y.a 64 4
700.2.ba $$\chi_{700}(139, \cdot)$$ 700.2.ba.a 464 4
700.2.bc $$\chi_{700}(157, \cdot)$$ 700.2.bc.a 8 4
700.2.bc.b 16
700.2.bc.c 24
700.2.be $$\chi_{700}(107, \cdot)$$ 700.2.be.a 8 4
700.2.be.b 8
700.2.be.c 8
700.2.be.d 48
700.2.be.e 72
700.2.be.f 128
700.2.bg $$\chi_{700}(81, \cdot)$$ 700.2.bg.a 160 8
700.2.bh $$\chi_{700}(13, \cdot)$$ 700.2.bh.a 160 8
700.2.bj $$\chi_{700}(127, \cdot)$$ 700.2.bj.a 720 8
700.2.bm $$\chi_{700}(19, \cdot)$$ 700.2.bm.a 928 8
700.2.bo $$\chi_{700}(9, \cdot)$$ 700.2.bo.a 160 8
700.2.bq $$\chi_{700}(31, \cdot)$$ 700.2.bq.a 928 8
700.2.bt $$\chi_{700}(23, \cdot)$$ 700.2.bt.a 1856 16
700.2.bv $$\chi_{700}(17, \cdot)$$ 700.2.bv.a 320 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(700))$$ into lower level spaces

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