## Defining parameters

 Level: $$N$$ = $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$14$$ Sturm bound: $$38400$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(400))$$.

Total New Old
Modular forms 14792 7558 7234
Cusp forms 14008 7373 6635
Eisenstein series 784 185 599

## Trace form

 $$7373 q - 26 q^{2} - 16 q^{3} - 16 q^{4} - 40 q^{5} - 72 q^{6} - 42 q^{7} - 68 q^{8} - 13 q^{9} + O(q^{10})$$ $$7373 q - 26 q^{2} - 16 q^{3} - 16 q^{4} - 40 q^{5} - 72 q^{6} - 42 q^{7} - 68 q^{8} - 13 q^{9} - 32 q^{10} + 32 q^{11} + 76 q^{12} + 174 q^{13} + 164 q^{14} - 12 q^{15} + 240 q^{16} - 216 q^{17} + 150 q^{18} - 548 q^{19} - 32 q^{20} - 638 q^{21} - 612 q^{22} - 130 q^{23} - 872 q^{24} + 80 q^{25} - 344 q^{26} + 794 q^{27} - 768 q^{28} - 598 q^{29} - 1664 q^{30} - 30 q^{31} - 416 q^{32} + 34 q^{33} + 1564 q^{34} + 204 q^{35} + 2820 q^{36} + 2438 q^{37} + 2888 q^{38} + 1298 q^{39} + 3088 q^{40} + 2160 q^{41} + 4576 q^{42} - 44 q^{43} + 2972 q^{44} + 570 q^{45} + 708 q^{46} + 430 q^{47} - 2480 q^{48} - 1365 q^{49} - 2872 q^{50} - 2224 q^{51} - 7748 q^{52} - 3754 q^{53} - 4856 q^{54} - 1210 q^{55} - 528 q^{56} - 3466 q^{57} - 16 q^{58} - 1896 q^{59} + 480 q^{60} - 414 q^{61} - 104 q^{62} - 7478 q^{63} + 8216 q^{64} - 1524 q^{65} + 12716 q^{66} - 7648 q^{67} + 5032 q^{68} + 778 q^{69} + 760 q^{70} + 3866 q^{71} - 4452 q^{72} + 3788 q^{73} - 9100 q^{74} + 10408 q^{75} - 9884 q^{76} + 3522 q^{77} - 24332 q^{78} + 12034 q^{79} - 6352 q^{80} + 2545 q^{81} - 12536 q^{82} + 4356 q^{83} - 16000 q^{84} - 7744 q^{85} - 1012 q^{86} + 3002 q^{87} + 272 q^{88} + 272 q^{89} + 3688 q^{90} - 5058 q^{91} + 10672 q^{92} - 946 q^{93} + 11400 q^{94} - 2350 q^{95} + 20232 q^{96} + 6576 q^{97} + 7426 q^{98} - 2342 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(400))$$

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
400.4.a $$\chi_{400}(1, \cdot)$$ 400.4.a.a 1 1
400.4.a.b 1
400.4.a.c 1
400.4.a.d 1
400.4.a.e 1
400.4.a.f 1
400.4.a.g 1
400.4.a.h 1
400.4.a.i 1
400.4.a.j 1
400.4.a.k 1
400.4.a.l 1
400.4.a.m 1
400.4.a.n 1
400.4.a.o 1
400.4.a.p 1
400.4.a.q 1
400.4.a.r 1
400.4.a.s 1
400.4.a.t 1
400.4.a.u 1
400.4.a.v 2
400.4.a.w 2
400.4.a.x 2
400.4.c $$\chi_{400}(49, \cdot)$$ 400.4.c.a 2 1
400.4.c.b 2
400.4.c.c 2
400.4.c.d 2
400.4.c.e 2
400.4.c.f 2
400.4.c.g 2
400.4.c.h 2
400.4.c.i 2
400.4.c.j 2
400.4.c.k 2
400.4.c.l 2
400.4.c.m 2
400.4.d $$\chi_{400}(201, \cdot)$$ None 0 1
400.4.f $$\chi_{400}(249, \cdot)$$ None 0 1
400.4.j $$\chi_{400}(43, \cdot)$$ n/a 212 2
400.4.l $$\chi_{400}(101, \cdot)$$ n/a 222 2
400.4.n $$\chi_{400}(143, \cdot)$$ 400.4.n.a 2 2
400.4.n.b 4
400.4.n.c 4
400.4.n.d 8
400.4.n.e 8
400.4.n.f 12
400.4.n.g 16
400.4.o $$\chi_{400}(7, \cdot)$$ None 0 2
400.4.q $$\chi_{400}(149, \cdot)$$ n/a 212 2
400.4.s $$\chi_{400}(107, \cdot)$$ n/a 212 2
400.4.u $$\chi_{400}(81, \cdot)$$ n/a 176 4
400.4.w $$\chi_{400}(9, \cdot)$$ None 0 4
400.4.y $$\chi_{400}(129, \cdot)$$ n/a 176 4
400.4.bb $$\chi_{400}(41, \cdot)$$ None 0 4
400.4.bd $$\chi_{400}(3, \cdot)$$ n/a 1424 8
400.4.be $$\chi_{400}(21, \cdot)$$ n/a 1424 8
400.4.bh $$\chi_{400}(23, \cdot)$$ None 0 8
400.4.bi $$\chi_{400}(47, \cdot)$$ n/a 360 8
400.4.bl $$\chi_{400}(29, \cdot)$$ n/a 1424 8
400.4.bm $$\chi_{400}(67, \cdot)$$ n/a 1424 8

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(400))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(400)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$