# Properties

 Label 400.8 Level 400 Weight 8 Dimension 17327 Nonzero newspaces 14 Sturm bound 76800 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$8$$ Nonzero newspaces: $$14$$ Sturm bound: $$76800$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_1(400))$$.

Total New Old
Modular forms 33992 17512 16480
Cusp forms 33208 17327 15881
Eisenstein series 784 185 599

## Trace form

 $$17327 q - 26 q^{2} + 8 q^{3} - 208 q^{4} - 40 q^{5} - 216 q^{6} + 646 q^{7} - 1028 q^{8} + 2573 q^{9} + O(q^{10})$$ $$17327 q - 26 q^{2} + 8 q^{3} - 208 q^{4} - 40 q^{5} - 216 q^{6} + 646 q^{7} - 1028 q^{8} + 2573 q^{9} - 32 q^{10} + 5768 q^{11} - 27380 q^{12} - 29526 q^{13} + 22244 q^{14} + 37428 q^{15} + 13296 q^{16} - 1140 q^{17} - 29370 q^{18} - 62764 q^{19} - 32 q^{20} - 8414 q^{21} - 44372 q^{22} + 166542 q^{23} - 59240 q^{24} + 32240 q^{25} - 36040 q^{26} - 592630 q^{27} - 41856 q^{28} - 173338 q^{29} + 1353856 q^{30} + 455074 q^{31} - 390816 q^{32} - 1959998 q^{33} - 4639652 q^{34} - 408276 q^{35} - 2228604 q^{36} + 1759730 q^{37} + 5190968 q^{38} + 707042 q^{39} + 4367888 q^{40} + 1867588 q^{41} - 657344 q^{42} - 325572 q^{43} - 10092836 q^{44} - 3897030 q^{45} - 9208316 q^{46} - 5180978 q^{47} + 8499280 q^{48} + 2686157 q^{49} + 7947528 q^{50} + 9227840 q^{51} + 269692 q^{52} + 1194082 q^{53} - 11883512 q^{54} + 1229030 q^{55} - 4093200 q^{56} - 8207530 q^{57} - 1860736 q^{58} - 20429536 q^{59} + 131040 q^{60} - 449378 q^{61} + 6274072 q^{62} - 276614 q^{63} + 32440088 q^{64} + 9476476 q^{65} - 49072852 q^{66} + 47900984 q^{67} - 25747928 q^{68} - 10755350 q^{69} + 19691320 q^{70} - 73203382 q^{71} + 71698908 q^{72} - 23346944 q^{73} + 22119908 q^{74} + 1361128 q^{75} - 2297564 q^{76} + 32407586 q^{77} - 95293004 q^{78} + 93661794 q^{79} - 37919952 q^{80} + 14821651 q^{81} - 12464312 q^{82} + 23103260 q^{83} + 36361856 q^{84} + 62567136 q^{85} + 25689628 q^{86} - 188724790 q^{87} + 68685072 q^{88} - 30723804 q^{89} + 40165288 q^{90} - 8408674 q^{91} - 51523024 q^{92} + 360206 q^{93} - 68976120 q^{94} + 91099250 q^{95} + 7282440 q^{96} + 76406740 q^{97} + 155030978 q^{98} + 22542178 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_1(400))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
400.8.a $$\chi_{400}(1, \cdot)$$ 400.8.a.a 1 1
400.8.a.b 1
400.8.a.c 1
400.8.a.d 1
400.8.a.e 1
400.8.a.f 1
400.8.a.g 1
400.8.a.h 1
400.8.a.i 1
400.8.a.j 1
400.8.a.k 1
400.8.a.l 1
400.8.a.m 1
400.8.a.n 1
400.8.a.o 1
400.8.a.p 1
400.8.a.q 1
400.8.a.r 1
400.8.a.s 1
400.8.a.t 2
400.8.a.u 2
400.8.a.v 2
400.8.a.w 2
400.8.a.x 2
400.8.a.y 2
400.8.a.z 2
400.8.a.ba 2
400.8.a.bb 2
400.8.a.bc 2
400.8.a.bd 2
400.8.a.be 2
400.8.a.bf 2
400.8.a.bg 2
400.8.a.bh 3
400.8.a.bi 3
400.8.a.bj 4
400.8.a.bk 4
400.8.a.bl 4
400.8.c $$\chi_{400}(49, \cdot)$$ 400.8.c.a 2 1
400.8.c.b 2
400.8.c.c 2
400.8.c.d 2
400.8.c.e 2
400.8.c.f 2
400.8.c.g 2
400.8.c.h 2
400.8.c.i 2
400.8.c.j 2
400.8.c.k 2
400.8.c.l 2
400.8.c.m 4
400.8.c.n 4
400.8.c.o 4
400.8.c.p 4
400.8.c.q 4
400.8.c.r 4
400.8.c.s 4
400.8.c.t 4
400.8.c.u 6
400.8.d $$\chi_{400}(201, \cdot)$$ None 0 1
400.8.f $$\chi_{400}(249, \cdot)$$ None 0 1
400.8.j $$\chi_{400}(43, \cdot)$$ n/a 500 2
400.8.l $$\chi_{400}(101, \cdot)$$ n/a 526 2
400.8.n $$\chi_{400}(143, \cdot)$$ n/a 126 2
400.8.o $$\chi_{400}(7, \cdot)$$ None 0 2
400.8.q $$\chi_{400}(149, \cdot)$$ n/a 500 2
400.8.s $$\chi_{400}(107, \cdot)$$ n/a 500 2
400.8.u $$\chi_{400}(81, \cdot)$$ n/a 416 4
400.8.w $$\chi_{400}(9, \cdot)$$ None 0 4
400.8.y $$\chi_{400}(129, \cdot)$$ n/a 416 4
400.8.bb $$\chi_{400}(41, \cdot)$$ None 0 4
400.8.bd $$\chi_{400}(3, \cdot)$$ n/a 3344 8
400.8.be $$\chi_{400}(21, \cdot)$$ n/a 3344 8
400.8.bh $$\chi_{400}(23, \cdot)$$ None 0 8
400.8.bi $$\chi_{400}(47, \cdot)$$ n/a 840 8
400.8.bl $$\chi_{400}(29, \cdot)$$ n/a 3344 8
400.8.bm $$\chi_{400}(67, \cdot)$$ n/a 3344 8

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

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_1(400))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_1(400)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 15}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 9}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 10}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 8}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 5}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 1}$$