## Defining parameters

 Level: $$N$$ = $$799 = 17 \cdot 47$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$10$$ Newform subspaces: $$24$$ Sturm bound: $$105984$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(799))$$.

Total New Old
Modular forms 27232 26963 269
Cusp forms 25761 25611 150
Eisenstein series 1471 1352 119

## Trace form

 $$25611 q - 315 q^{2} - 318 q^{3} - 327 q^{4} - 324 q^{5} - 342 q^{6} - 330 q^{7} - 351 q^{8} - 345 q^{9} + O(q^{10})$$ $$25611 q - 315 q^{2} - 318 q^{3} - 327 q^{4} - 324 q^{5} - 342 q^{6} - 330 q^{7} - 351 q^{8} - 345 q^{9} - 352 q^{10} - 326 q^{11} - 342 q^{12} - 332 q^{13} - 346 q^{14} - 330 q^{15} - 327 q^{16} - 348 q^{17} - 711 q^{18} - 350 q^{19} - 376 q^{20} - 354 q^{21} - 382 q^{22} - 362 q^{23} - 406 q^{24} - 343 q^{25} - 360 q^{26} - 378 q^{27} - 378 q^{28} - 356 q^{29} - 394 q^{30} - 338 q^{31} - 383 q^{32} - 370 q^{33} - 306 q^{34} - 692 q^{35} - 235 q^{36} - 310 q^{37} - 298 q^{38} - 240 q^{39} - 108 q^{40} - 206 q^{41} - 126 q^{42} - 312 q^{43} - 182 q^{44} - 206 q^{45} - 196 q^{46} - 277 q^{47} - 332 q^{48} - 321 q^{49} - 273 q^{50} - 290 q^{51} - 640 q^{52} - 334 q^{53} - 150 q^{54} - 256 q^{55} - 150 q^{56} - 216 q^{57} - 308 q^{58} - 312 q^{59} - 290 q^{60} - 318 q^{61} - 386 q^{62} - 378 q^{63} - 463 q^{64} - 406 q^{65} - 482 q^{66} - 430 q^{67} - 374 q^{68} - 722 q^{69} - 466 q^{70} - 426 q^{71} - 499 q^{72} - 328 q^{73} - 480 q^{74} - 422 q^{75} - 394 q^{76} - 342 q^{77} - 198 q^{78} - 234 q^{79} - 190 q^{80} - 185 q^{81} - 54 q^{82} - 338 q^{83} - 58 q^{84} - 171 q^{85} - 462 q^{86} - 290 q^{87} + 26 q^{88} - 248 q^{89} + 184 q^{90} + 42 q^{91} + 80 q^{92} - 268 q^{93} + 113 q^{94} - 396 q^{95} + 110 q^{96} - 288 q^{97} - 61 q^{98} - 10 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(799))$$

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
799.2.a $$\chi_{799}(1, \cdot)$$ 799.2.a.a 1 1
799.2.a.b 1
799.2.a.c 2
799.2.a.d 8
799.2.a.e 12
799.2.a.f 17
799.2.a.g 20
799.2.b $$\chi_{799}(424, \cdot)$$ 799.2.b.a 2 1
799.2.b.b 28
799.2.b.c 40
799.2.f $$\chi_{799}(565, \cdot)$$ 799.2.f.a 60 2
799.2.f.b 80
799.2.g $$\chi_{799}(189, \cdot)$$ 799.2.g.a 4 4
799.2.g.b 4
799.2.g.c 112
799.2.g.d 152
799.2.j $$\chi_{799}(46, \cdot)$$ 799.2.j.a 80 8
799.2.j.b 480
799.2.k $$\chi_{799}(18, \cdot)$$ 799.2.k.a 704 22
799.2.k.b 704
799.2.n $$\chi_{799}(16, \cdot)$$ 799.2.n.a 1540 22
799.2.o $$\chi_{799}(4, \cdot)$$ 799.2.o.a 3080 44
799.2.r $$\chi_{799}(2, \cdot)$$ 799.2.r.a 6160 88
799.2.s $$\chi_{799}(5, \cdot)$$ 799.2.s.a 12320 176

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(799))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(799)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(47))$$$$^{\oplus 2}$$