## 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

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

## 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

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