## Defining parameters

 Level: $$N$$ = $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$14$$ Newform subspaces: $$45$$ Sturm bound: $$10752$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(208))$$.

Total New Old
Modular forms 4200 2290 1910
Cusp forms 3864 2192 1672
Eisenstein series 336 98 238

## Trace form

 $$2192 q - 20 q^{2} - 22 q^{3} - 40 q^{4} - 22 q^{5} + 40 q^{6} + 30 q^{7} + 64 q^{8} + 16 q^{9} + O(q^{10})$$ $$2192 q - 20 q^{2} - 22 q^{3} - 40 q^{4} - 22 q^{5} + 40 q^{6} + 30 q^{7} + 64 q^{8} + 16 q^{9} + 112 q^{10} - 142 q^{11} - 224 q^{12} - 50 q^{13} - 424 q^{14} + 246 q^{15} - 584 q^{16} - 146 q^{17} - 372 q^{18} + 122 q^{19} + 368 q^{20} + 58 q^{21} + 1152 q^{22} - 130 q^{23} + 1672 q^{24} + 230 q^{25} + 240 q^{26} - 100 q^{27} - 584 q^{28} - 22 q^{29} - 2496 q^{30} - 1074 q^{31} - 1960 q^{32} - 398 q^{33} - 896 q^{34} - 1066 q^{35} + 1168 q^{36} + 314 q^{37} + 2440 q^{38} - 1366 q^{39} + 2624 q^{40} - 1026 q^{41} + 1336 q^{42} + 754 q^{43} - 1760 q^{44} + 78 q^{45} - 2288 q^{46} + 4042 q^{47} - 3560 q^{48} + 1452 q^{49} - 1476 q^{50} + 6260 q^{51} + 212 q^{52} + 3532 q^{53} + 2728 q^{54} + 2966 q^{55} + 952 q^{56} + 1354 q^{57} - 40 q^{58} - 4946 q^{59} + 360 q^{60} - 5290 q^{61} + 136 q^{62} - 9690 q^{63} - 1048 q^{64} - 2662 q^{65} + 808 q^{66} - 5794 q^{67} + 1736 q^{68} - 1638 q^{69} - 344 q^{70} + 1438 q^{71} - 2208 q^{72} - 314 q^{73} + 880 q^{74} + 7020 q^{75} + 2432 q^{76} + 2612 q^{77} + 748 q^{78} + 7484 q^{79} + 5272 q^{80} - 558 q^{81} + 1384 q^{82} + 5578 q^{83} + 4288 q^{84} + 5930 q^{85} + 11936 q^{86} + 4482 q^{87} + 13360 q^{88} + 8838 q^{89} + 20832 q^{90} - 278 q^{91} + 2864 q^{92} + 4418 q^{93} + 3976 q^{94} - 15854 q^{95} - 11464 q^{96} - 4946 q^{97} - 16420 q^{98} - 16934 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(208))$$

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
208.4.a $$\chi_{208}(1, \cdot)$$ 208.4.a.a 1 1
208.4.a.b 1
208.4.a.c 1
208.4.a.d 1
208.4.a.e 1
208.4.a.f 1
208.4.a.g 1
208.4.a.h 2
208.4.a.i 2
208.4.a.j 2
208.4.a.k 2
208.4.a.l 3
208.4.b $$\chi_{208}(105, \cdot)$$ None 0 1
208.4.e $$\chi_{208}(25, \cdot)$$ None 0 1
208.4.f $$\chi_{208}(129, \cdot)$$ 208.4.f.a 2 1
208.4.f.b 2
208.4.f.c 2
208.4.f.d 4
208.4.f.e 10
208.4.i $$\chi_{208}(81, \cdot)$$ 208.4.i.a 2 2
208.4.i.b 2
208.4.i.c 2
208.4.i.d 4
208.4.i.e 4
208.4.i.f 6
208.4.i.g 8
208.4.i.h 12
208.4.k $$\chi_{208}(31, \cdot)$$ 208.4.k.a 2 2
208.4.k.b 12
208.4.k.c 28
208.4.l $$\chi_{208}(83, \cdot)$$ 208.4.l.a 164 2
208.4.n $$\chi_{208}(53, \cdot)$$ 208.4.n.a 144 2
208.4.p $$\chi_{208}(77, \cdot)$$ 208.4.p.a 164 2
208.4.s $$\chi_{208}(99, \cdot)$$ 208.4.s.a 164 2
208.4.u $$\chi_{208}(135, \cdot)$$ None 0 2
208.4.w $$\chi_{208}(17, \cdot)$$ 208.4.w.a 2 2
208.4.w.b 2
208.4.w.c 8
208.4.w.d 8
208.4.w.e 20
208.4.z $$\chi_{208}(9, \cdot)$$ None 0 2
208.4.ba $$\chi_{208}(121, \cdot)$$ None 0 2
208.4.bc $$\chi_{208}(7, \cdot)$$ None 0 4
208.4.bf $$\chi_{208}(11, \cdot)$$ 208.4.bf.a 328 4
208.4.bh $$\chi_{208}(69, \cdot)$$ 208.4.bh.a 328 4
208.4.bj $$\chi_{208}(29, \cdot)$$ 208.4.bj.a 328 4
208.4.bk $$\chi_{208}(115, \cdot)$$ 208.4.bk.a 328 4
208.4.bm $$\chi_{208}(15, \cdot)$$ 208.4.bm.a 4 4
208.4.bm.b 24
208.4.bm.c 28
208.4.bm.d 28

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(208))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(208)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(104))$$$$^{\oplus 2}$$