## Defining parameters

 Level: $$N$$ = $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$16$$ Sturm bound: $$2400$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(100))$$.

Total New Old
Modular forms 970 503 467
Cusp forms 830 463 367
Eisenstein series 140 40 100

## Trace form

 $$463 q - 6 q^{2} - 8 q^{3} - 10 q^{4} - 35 q^{5} - 34 q^{6} + 32 q^{7} + 78 q^{8} + 198 q^{9} + O(q^{10})$$ $$463 q - 6 q^{2} - 8 q^{3} - 10 q^{4} - 35 q^{5} - 34 q^{6} + 32 q^{7} + 78 q^{8} + 198 q^{9} + 64 q^{10} + 40 q^{11} + 150 q^{12} - 276 q^{13} - 10 q^{14} - 172 q^{15} - 386 q^{16} - 108 q^{17} - 622 q^{18} + 128 q^{19} - 326 q^{20} + 316 q^{21} - 730 q^{22} + 344 q^{23} + 623 q^{25} + 892 q^{26} + 844 q^{27} + 1750 q^{28} + 408 q^{29} + 1230 q^{30} + 184 q^{31} + 1254 q^{32} - 660 q^{33} - 10 q^{34} - 628 q^{35} - 1290 q^{36} - 645 q^{37} - 720 q^{38} - 72 q^{39} + 204 q^{40} - 1588 q^{41} - 1630 q^{42} - 80 q^{43} - 1340 q^{44} - 2185 q^{45} + 626 q^{46} - 2056 q^{47} - 680 q^{48} - 3099 q^{49} - 2206 q^{50} - 532 q^{51} - 832 q^{52} + 707 q^{53} - 3780 q^{54} - 740 q^{55} - 4534 q^{56} + 2108 q^{57} - 3824 q^{58} + 596 q^{59} + 2810 q^{60} + 2516 q^{61} + 460 q^{62} + 2068 q^{63} + 4820 q^{64} + 2395 q^{65} + 5070 q^{66} + 2792 q^{67} + 4846 q^{68} + 4804 q^{69} + 3030 q^{70} + 512 q^{71} + 4064 q^{72} - 2036 q^{73} + 3692 q^{75} - 2140 q^{76} - 2740 q^{77} - 7180 q^{78} + 1424 q^{79} - 2106 q^{80} - 2758 q^{81} - 9362 q^{82} + 1704 q^{83} - 11290 q^{84} + 3277 q^{85} + 106 q^{86} - 332 q^{87} + 830 q^{88} + 2797 q^{89} + 3494 q^{90} - 464 q^{91} + 7970 q^{92} - 4388 q^{93} + 11030 q^{94} - 2204 q^{95} + 6506 q^{96} - 6960 q^{97} + 12998 q^{98} - 3340 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(100))$$

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
100.4.a $$\chi_{100}(1, \cdot)$$ 100.4.a.a 1 1
100.4.a.b 1
100.4.a.c 1
100.4.a.d 2
100.4.c $$\chi_{100}(49, \cdot)$$ 100.4.c.a 2 1
100.4.c.b 2
100.4.e $$\chi_{100}(7, \cdot)$$ 100.4.e.a 2 2
100.4.e.b 2
100.4.e.c 2
100.4.e.d 8
100.4.e.e 12
100.4.e.f 24
100.4.g $$\chi_{100}(21, \cdot)$$ 100.4.g.a 28 4
100.4.i $$\chi_{100}(9, \cdot)$$ 100.4.i.a 32 4
100.4.l $$\chi_{100}(3, \cdot)$$ 100.4.l.a 8 8
100.4.l.b 336

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(100))$$ into lower level spaces

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