# Properties

 Label 735.2.d Level $735$ Weight $2$ Character orbit 735.d Rep. character $\chi_{735}(589,\cdot)$ Character field $\Q$ Dimension $40$ Newform subspaces $6$ Sturm bound $224$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$224$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(735, [\chi])$$.

Total New Old
Modular forms 128 40 88
Cusp forms 96 40 56
Eisenstein series 32 0 32

## Trace form

 $$40 q - 40 q^{4} - 4 q^{5} + 4 q^{6} - 40 q^{9} + O(q^{10})$$ $$40 q - 40 q^{4} - 4 q^{5} + 4 q^{6} - 40 q^{9} + 8 q^{10} - 4 q^{15} + 24 q^{16} + 28 q^{20} - 12 q^{24} + 8 q^{25} - 24 q^{26} - 8 q^{29} + 8 q^{30} + 16 q^{31} - 32 q^{34} + 40 q^{36} + 8 q^{39} - 16 q^{40} - 8 q^{41} + 56 q^{44} + 4 q^{45} - 32 q^{50} - 8 q^{51} - 4 q^{54} + 8 q^{55} + 16 q^{59} + 16 q^{60} + 16 q^{61} - 8 q^{64} - 68 q^{65} - 8 q^{66} - 8 q^{69} - 24 q^{71} + 32 q^{74} - 16 q^{75} - 16 q^{76} + 32 q^{79} - 44 q^{80} + 40 q^{81} - 56 q^{85} - 32 q^{86} + 40 q^{89} - 8 q^{90} - 32 q^{94} - 36 q^{95} + 68 q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(735, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
735.2.d.a $2$ $5.869$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+iq^{2}-iq^{3}+q^{4}+(-1+2i)q^{5}+\cdots$$
735.2.d.b $6$ $5.869$ 6.0.350464.1 None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-\beta _{1}q^{2}-\beta _{4}q^{3}+(-1+\beta _{3}+\beta _{5})q^{4}+\cdots$$
735.2.d.c $8$ $5.869$ 8.0.309760000.3 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{5}q^{2}-\beta _{4}q^{3}+(-2+\beta _{6}-\beta _{7})q^{4}+\cdots$$
735.2.d.d $8$ $5.869$ 8.0.2058981376.2 None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+\beta _{6}q^{2}-\beta _{4}q^{3}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots$$
735.2.d.e $8$ $5.869$ 8.0.2058981376.2 None $$0$$ $$0$$ $$2$$ $$0$$ $$q+\beta _{6}q^{2}+\beta _{4}q^{3}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots$$
735.2.d.f $8$ $5.869$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{4}q^{2}+\zeta_{24}q^{3}+\zeta_{24}^{3}q^{4}+(-\zeta_{24}^{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(735, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(735, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(245, [\chi])$$$$^{\oplus 2}$$