## Defining parameters

 Level: $$N$$ = $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$20$$ Sturm bound: $$153600$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(800))$$.

Total New Old
Modular forms 58496 30067 28429
Cusp forms 56704 29657 27047
Eisenstein series 1792 410 1382

## Trace form

 $$29657 q - 52 q^{2} - 40 q^{3} - 52 q^{4} - 64 q^{5} - 84 q^{6} - 24 q^{7} - 52 q^{8} - 31 q^{9} + O(q^{10})$$ $$29657 q - 52 q^{2} - 40 q^{3} - 52 q^{4} - 64 q^{5} - 84 q^{6} - 24 q^{7} - 52 q^{8} - 31 q^{9} - 64 q^{10} - 64 q^{11} - 100 q^{12} - 170 q^{13} - 260 q^{14} - 48 q^{15} - 384 q^{16} - 206 q^{17} - 232 q^{18} - 32 q^{19} - 64 q^{20} + 172 q^{21} + 144 q^{22} + 592 q^{23} - 96 q^{24} - 96 q^{25} - 144 q^{26} - 88 q^{27} + 328 q^{28} + 886 q^{29} - 64 q^{30} - 204 q^{31} + 568 q^{32} - 408 q^{33} + 480 q^{34} - 504 q^{35} + 1376 q^{36} - 1858 q^{37} + 932 q^{38} - 1384 q^{39} - 64 q^{40} - 2042 q^{41} - 2312 q^{42} - 960 q^{43} - 2092 q^{44} - 424 q^{45} - 1524 q^{46} + 108 q^{47} - 2496 q^{48} + 2165 q^{49} - 64 q^{50} + 1472 q^{51} - 2572 q^{52} + 2686 q^{53} - 1136 q^{54} + 452 q^{55} + 1112 q^{56} + 524 q^{57} + 3152 q^{58} + 1344 q^{59} - 1088 q^{60} - 1018 q^{61} + 3336 q^{62} - 68 q^{63} - 10552 q^{64} - 4352 q^{65} - 20452 q^{66} - 1696 q^{67} - 9152 q^{68} - 6292 q^{69} - 1648 q^{70} + 752 q^{71} + 9284 q^{72} + 3910 q^{73} + 13724 q^{74} + 3256 q^{75} + 11868 q^{76} + 14588 q^{77} + 34852 q^{78} + 7172 q^{79} + 12576 q^{80} + 18173 q^{81} + 16908 q^{82} + 7760 q^{83} + 29264 q^{84} + 4640 q^{85} + 8464 q^{86} - 216 q^{87} + 3664 q^{88} - 3274 q^{89} - 7504 q^{90} - 7056 q^{91} - 12072 q^{92} - 14816 q^{93} - 16552 q^{94} - 3412 q^{95} - 31696 q^{96} - 12294 q^{97} - 14072 q^{98} - 5472 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(800))$$

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
800.4.a $$\chi_{800}(1, \cdot)$$ 800.4.a.a 1 1
800.4.a.b 1
800.4.a.c 1
800.4.a.d 1
800.4.a.e 1
800.4.a.f 1
800.4.a.g 1
800.4.a.h 1
800.4.a.i 1
800.4.a.j 1
800.4.a.k 1
800.4.a.l 2
800.4.a.m 2
800.4.a.n 2
800.4.a.o 2
800.4.a.p 2
800.4.a.q 2
800.4.a.r 2
800.4.a.s 2
800.4.a.t 2
800.4.a.u 3
800.4.a.v 3
800.4.a.w 3
800.4.a.x 3
800.4.a.y 4
800.4.a.z 4
800.4.a.ba 4
800.4.a.bb 4
800.4.c $$\chi_{800}(449, \cdot)$$ 800.4.c.a 2 1
800.4.c.b 2
800.4.c.c 2
800.4.c.d 2
800.4.c.e 2
800.4.c.f 2
800.4.c.g 2
800.4.c.h 4
800.4.c.i 4
800.4.c.j 4
800.4.c.k 4
800.4.c.l 4
800.4.c.m 6
800.4.c.n 6
800.4.c.o 8
800.4.d $$\chi_{800}(401, \cdot)$$ 800.4.d.a 2 1
800.4.d.b 12
800.4.d.c 12
800.4.d.d 12
800.4.d.e 16
800.4.f $$\chi_{800}(49, \cdot)$$ 800.4.f.a 4 1
800.4.f.b 12
800.4.f.c 12
800.4.f.d 24
800.4.j $$\chi_{800}(407, \cdot)$$ None 0 2
800.4.l $$\chi_{800}(201, \cdot)$$ None 0 2
800.4.n $$\chi_{800}(543, \cdot)$$ n/a 108 2
800.4.o $$\chi_{800}(143, \cdot)$$ n/a 104 2
800.4.q $$\chi_{800}(249, \cdot)$$ None 0 2
800.4.s $$\chi_{800}(7, \cdot)$$ None 0 2
800.4.u $$\chi_{800}(161, \cdot)$$ n/a 360 4
800.4.v $$\chi_{800}(43, \cdot)$$ n/a 856 4
800.4.y $$\chi_{800}(101, \cdot)$$ n/a 900 4
800.4.ba $$\chi_{800}(149, \cdot)$$ n/a 856 4
800.4.bb $$\chi_{800}(107, \cdot)$$ n/a 856 4
800.4.be $$\chi_{800}(209, \cdot)$$ n/a 352 4
800.4.bg $$\chi_{800}(129, \cdot)$$ n/a 360 4
800.4.bj $$\chi_{800}(81, \cdot)$$ n/a 352 4
800.4.bl $$\chi_{800}(23, \cdot)$$ None 0 8
800.4.bm $$\chi_{800}(41, \cdot)$$ None 0 8
800.4.bp $$\chi_{800}(47, \cdot)$$ n/a 704 8
800.4.bq $$\chi_{800}(63, \cdot)$$ n/a 720 8
800.4.bt $$\chi_{800}(9, \cdot)$$ None 0 8
800.4.bu $$\chi_{800}(87, \cdot)$$ None 0 8
800.4.bx $$\chi_{800}(67, \cdot)$$ n/a 5728 16
800.4.by $$\chi_{800}(29, \cdot)$$ n/a 5728 16
800.4.ca $$\chi_{800}(21, \cdot)$$ n/a 5728 16
800.4.cd $$\chi_{800}(3, \cdot)$$ n/a 5728 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(800))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(800)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 2}$$