## Defining parameters

 Level: $$N$$ = $$804 = 2^{2} \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newform subspaces: $$39$$ Sturm bound: $$71808$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(804))$$.

Total New Old
Modular forms 18612 7978 10634
Cusp forms 17293 7722 9571
Eisenstein series 1319 256 1063

## Trace form

 $$7722q - 66q^{4} - 33q^{6} - 66q^{9} + O(q^{10})$$ $$7722q - 66q^{4} - 33q^{6} - 66q^{9} - 66q^{10} - 33q^{12} - 132q^{13} - 66q^{16} - 33q^{18} - 66q^{21} - 66q^{22} - 33q^{24} - 132q^{25} - 66q^{28} - 33q^{30} - 66q^{33} - 66q^{34} - 33q^{36} - 132q^{37} - 66q^{40} - 33q^{42} - 66q^{45} - 66q^{46} - 33q^{48} - 132q^{49} - 66q^{52} + 66q^{53} - 33q^{54} + 198q^{55} + 11q^{57} - 66q^{58} + 132q^{59} - 33q^{60} + 132q^{61} + 11q^{63} - 66q^{64} + 264q^{65} - 66q^{66} + 132q^{67} - 66q^{70} + 264q^{71} - 33q^{72} + 154q^{73} + 132q^{75} - 66q^{76} + 132q^{77} - 33q^{78} + 154q^{79} - 66q^{81} - 66q^{82} + 66q^{83} - 33q^{84} - 132q^{85} - 66q^{88} - 33q^{90} - 66q^{93} - 66q^{94} - 33q^{96} - 132q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(804))$$

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
804.2.a $$\chi_{804}(1, \cdot)$$ 804.2.a.a 1 1
804.2.a.b 1
804.2.a.c 1
804.2.a.d 1
804.2.a.e 3
804.2.a.f 5
804.2.c $$\chi_{804}(671, \cdot)$$ 804.2.c.a 4 1
804.2.c.b 128
804.2.e $$\chi_{804}(535, \cdot)$$ 804.2.e.a 34 1
804.2.e.b 34
804.2.g $$\chi_{804}(401, \cdot)$$ 804.2.g.a 2 1
804.2.g.b 4
804.2.g.c 16
804.2.i $$\chi_{804}(37, \cdot)$$ 804.2.i.a 2 2
804.2.i.b 2
804.2.i.c 2
804.2.i.d 8
804.2.i.e 8
804.2.j $$\chi_{804}(499, \cdot)$$ 804.2.j.a 68 2
804.2.j.b 68
804.2.l $$\chi_{804}(431, \cdot)$$ 804.2.l.a 264 2
804.2.o $$\chi_{804}(365, \cdot)$$ 804.2.o.a 2 2
804.2.o.b 4
804.2.o.c 4
804.2.o.d 36
804.2.q $$\chi_{804}(25, \cdot)$$ 804.2.q.a 60 10
804.2.q.b 60
804.2.s $$\chi_{804}(5, \cdot)$$ 804.2.s.a 20 10
804.2.s.b 200
804.2.u $$\chi_{804}(43, \cdot)$$ 804.2.u.a 340 10
804.2.u.b 340
804.2.w $$\chi_{804}(59, \cdot)$$ 804.2.w.a 1320 10
804.2.y $$\chi_{804}(49, \cdot)$$ 804.2.y.a 100 20
804.2.y.b 120
804.2.ba $$\chi_{804}(41, \cdot)$$ 804.2.ba.a 20 20
804.2.ba.b 440
804.2.bd $$\chi_{804}(23, \cdot)$$ 804.2.bd.a 2640 20
804.2.bf $$\chi_{804}(7, \cdot)$$ 804.2.bf.a 680 20
804.2.bf.b 680

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(804))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(804)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(67))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(134))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(201))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(268))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(402))$$$$^{\oplus 2}$$