## Defining parameters

 Level: $$N$$ = $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$14$$ Newform subspaces: $$62$$ Sturm bound: $$19200$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(400))$$.

Total New Old
Modular forms 5192 2581 2611
Cusp forms 4409 2396 2013
Eisenstein series 783 185 598

## Trace form

 $$2396 q - 26 q^{2} - 20 q^{3} - 24 q^{4} - 40 q^{5} - 36 q^{6} - 18 q^{7} - 20 q^{8} - 2 q^{9} + O(q^{10})$$ $$2396 q - 26 q^{2} - 20 q^{3} - 24 q^{4} - 40 q^{5} - 36 q^{6} - 18 q^{7} - 20 q^{8} - 2 q^{9} - 32 q^{10} - 28 q^{11} - 28 q^{12} - 24 q^{13} - 28 q^{14} - 12 q^{15} - 48 q^{16} - 42 q^{17} - 22 q^{18} + 8 q^{19} - 32 q^{20} - 14 q^{21} - 24 q^{22} + 6 q^{23} - 24 q^{24} - 76 q^{26} - 14 q^{27} - 80 q^{28} - 28 q^{29} - 64 q^{30} - 62 q^{31} - 96 q^{32} - 110 q^{33} - 148 q^{34} - 36 q^{35} - 172 q^{36} - 104 q^{37} - 148 q^{38} - 70 q^{39} - 112 q^{40} - 74 q^{41} - 184 q^{42} - 40 q^{43} - 140 q^{44} - 70 q^{45} - 156 q^{46} - 2 q^{47} - 144 q^{48} - 86 q^{49} - 72 q^{50} - 24 q^{51} - 92 q^{52} - 8 q^{53} - 120 q^{54} - 10 q^{55} - 48 q^{56} + 70 q^{57} - 36 q^{58} + 44 q^{59} - 36 q^{61} - 8 q^{62} - 94 q^{63} + 24 q^{64} - 44 q^{65} + 92 q^{66} - 100 q^{67} + 88 q^{68} - 38 q^{69} + 40 q^{70} - 142 q^{71} + 220 q^{72} + 34 q^{73} + 152 q^{74} - 152 q^{75} + 60 q^{76} - 46 q^{77} + 244 q^{78} - 206 q^{79} + 48 q^{80} - 112 q^{81} + 104 q^{82} - 256 q^{83} + 272 q^{84} - 144 q^{85} + 88 q^{86} - 382 q^{87} + 160 q^{88} - 42 q^{89} + 88 q^{90} - 210 q^{91} + 32 q^{92} - 146 q^{93} + 72 q^{94} - 190 q^{95} + 72 q^{96} - 162 q^{97} - 94 q^{98} - 338 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(400))$$

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
400.2.a $$\chi_{400}(1, \cdot)$$ 400.2.a.a 1 1
400.2.a.b 1
400.2.a.c 1
400.2.a.d 1
400.2.a.e 1
400.2.a.f 1
400.2.a.g 1
400.2.a.h 1
400.2.c $$\chi_{400}(49, \cdot)$$ 400.2.c.a 2 1
400.2.c.b 2
400.2.c.c 2
400.2.c.d 2
400.2.d $$\chi_{400}(201, \cdot)$$ None 0 1
400.2.f $$\chi_{400}(249, \cdot)$$ None 0 1
400.2.j $$\chi_{400}(43, \cdot)$$ 400.2.j.a 2 2
400.2.j.b 8
400.2.j.c 16
400.2.j.d 18
400.2.j.e 24
400.2.l $$\chi_{400}(101, \cdot)$$ 400.2.l.a 2 2
400.2.l.b 2
400.2.l.c 2
400.2.l.d 4
400.2.l.e 4
400.2.l.f 12
400.2.l.g 12
400.2.l.h 16
400.2.l.i 16
400.2.n $$\chi_{400}(143, \cdot)$$ 400.2.n.a 2 2
400.2.n.b 4
400.2.n.c 4
400.2.n.d 8
400.2.o $$\chi_{400}(7, \cdot)$$ None 0 2
400.2.q $$\chi_{400}(149, \cdot)$$ 400.2.q.a 2 2
400.2.q.b 2
400.2.q.c 4
400.2.q.d 4
400.2.q.e 12
400.2.q.f 12
400.2.q.g 16
400.2.q.h 16
400.2.s $$\chi_{400}(107, \cdot)$$ 400.2.s.a 2 2
400.2.s.b 8
400.2.s.c 16
400.2.s.d 18
400.2.s.e 24
400.2.u $$\chi_{400}(81, \cdot)$$ 400.2.u.a 4 4
400.2.u.b 4
400.2.u.c 4
400.2.u.d 8
400.2.u.e 8
400.2.u.f 12
400.2.u.g 16
400.2.w $$\chi_{400}(9, \cdot)$$ None 0 4
400.2.y $$\chi_{400}(129, \cdot)$$ 400.2.y.a 8 4
400.2.y.b 8
400.2.y.c 8
400.2.y.d 32
400.2.bb $$\chi_{400}(41, \cdot)$$ None 0 4
400.2.bd $$\chi_{400}(3, \cdot)$$ 400.2.bd.a 464 8
400.2.be $$\chi_{400}(21, \cdot)$$ 400.2.be.a 464 8
400.2.bh $$\chi_{400}(23, \cdot)$$ None 0 8
400.2.bi $$\chi_{400}(47, \cdot)$$ 400.2.bi.a 8 8
400.2.bi.b 16
400.2.bi.c 16
400.2.bi.d 80
400.2.bl $$\chi_{400}(29, \cdot)$$ 400.2.bl.a 464 8
400.2.bm $$\chi_{400}(67, \cdot)$$ 400.2.bm.a 464 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(400))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(400)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$