## Defining parameters

 Level: $$N$$ = $$712 = 2^{3} \cdot 89$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$5$$ Newform subspaces: $$6$$ Sturm bound: $$31680$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(712))$$.

Total New Old
Modular forms 623 219 404
Cusp forms 95 45 50
Eisenstein series 528 174 354

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 45 0 0 0

## Trace form

 $$45q - 3q^{2} - 2q^{3} + q^{4} - 2q^{6} - 3q^{8} - q^{9} + O(q^{10})$$ $$45q - 3q^{2} - 2q^{3} + q^{4} - 2q^{6} - 3q^{8} - q^{9} - 2q^{11} - 2q^{12} + q^{16} - 2q^{17} - 5q^{18} - 2q^{19} - 2q^{22} - 2q^{24} + q^{25} - 4q^{27} - 3q^{32} - 4q^{33} - 2q^{34} - q^{36} - 2q^{38} - 2q^{41} - 2q^{43} - 2q^{44} - 2q^{48} + q^{49} - 3q^{50} - 4q^{51} - 4q^{54} - 4q^{57} - 2q^{59} + q^{64} - 4q^{66} - 2q^{67} - 2q^{68} + 39q^{72} - 2q^{73} - 2q^{75} - 2q^{76} - 3q^{81} - 2q^{82} - 2q^{83} - 2q^{86} - 2q^{88} - 3q^{89} - 4q^{91} - 2q^{96} - 6q^{97} - 3q^{98} - 6q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(712))$$

We only show spaces with odd 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
712.1.c $$\chi_{712}(355, \cdot)$$ 712.1.c.a 1 1
712.1.c.b 2
712.1.d $$\chi_{712}(535, \cdot)$$ None 0 1
712.1.g $$\chi_{712}(179, \cdot)$$ None 0 1
712.1.h $$\chi_{712}(711, \cdot)$$ None 0 1
712.1.i $$\chi_{712}(55, \cdot)$$ None 0 2
712.1.l $$\chi_{712}(123, \cdot)$$ 712.1.l.a 2 2
712.1.n $$\chi_{712}(393, \cdot)$$ None 0 4
712.1.p $$\chi_{712}(37, \cdot)$$ None 0 4
712.1.r $$\chi_{712}(87, \cdot)$$ None 0 10
712.1.s $$\chi_{712}(67, \cdot)$$ 712.1.s.a 10 10
712.1.v $$\chi_{712}(39, \cdot)$$ None 0 10
712.1.w $$\chi_{712}(11, \cdot)$$ 712.1.w.a 10 10
712.1.y $$\chi_{712}(99, \cdot)$$ 712.1.y.a 20 20
712.1.bb $$\chi_{712}(47, \cdot)$$ None 0 20
712.1.bc $$\chi_{712}(13, \cdot)$$ None 0 40
712.1.be $$\chi_{712}(33, \cdot)$$ None 0 40

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(712))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(712)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(356))$$$$^{\oplus 2}$$