# Properties

 Label 735.2.i Level $735$ Weight $2$ Character orbit 735.i Rep. character $\chi_{735}(226,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $52$ Newform subspaces $14$ Sturm bound $224$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 735.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$14$$ Sturm bound: $$224$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$, $$13$$, $$17$$

## Dimensions

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

Total New Old
Modular forms 256 52 204
Cusp forms 192 52 140
Eisenstein series 64 0 64

## Trace form

 $$52 q - 8 q^{2} - 2 q^{3} - 32 q^{4} + 48 q^{8} - 26 q^{9} + O(q^{10})$$ $$52 q - 8 q^{2} - 2 q^{3} - 32 q^{4} + 48 q^{8} - 26 q^{9} - 4 q^{10} - 12 q^{11} - 4 q^{12} + 4 q^{13} - 36 q^{16} + 8 q^{17} - 8 q^{18} + 6 q^{19} - 16 q^{20} - 8 q^{22} - 16 q^{23} + 12 q^{24} - 26 q^{25} + 8 q^{26} + 4 q^{27} + 40 q^{29} + 6 q^{31} - 36 q^{32} - 8 q^{33} + 16 q^{34} + 64 q^{36} + 2 q^{37} - 32 q^{38} + 22 q^{39} - 12 q^{40} - 8 q^{41} + 20 q^{43} - 28 q^{44} - 8 q^{46} + 8 q^{47} + 16 q^{48} + 16 q^{50} - 12 q^{51} + 24 q^{55} - 68 q^{57} + 4 q^{58} - 4 q^{59} + 4 q^{61} - 24 q^{62} + 56 q^{64} + 12 q^{65} + 12 q^{66} - 2 q^{67} - 12 q^{68} - 24 q^{69} - 24 q^{72} + 10 q^{73} - 12 q^{74} - 2 q^{75} - 40 q^{76} + 8 q^{78} + 6 q^{79} + 16 q^{80} - 26 q^{81} + 28 q^{82} - 8 q^{83} - 8 q^{85} + 132 q^{86} - 24 q^{87} + 136 q^{88} - 16 q^{89} + 8 q^{90} - 200 q^{92} + 34 q^{93} + 24 q^{94} - 8 q^{95} + 24 q^{96} + 32 q^{97} + 24 q^{99} + O(q^{100})$$

## 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.i.a $2$ $5.869$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-1$$ $$-1$$ $$0$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots$$
735.2.i.b $2$ $5.869$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$1$$ $$0$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots$$
735.2.i.c $2$ $5.869$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-1$$ $$0$$ $$q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+\cdots$$
735.2.i.d $2$ $5.869$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots$$
735.2.i.e $2$ $5.869$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$-1$$ $$0$$ $$q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots$$
735.2.i.f $2$ $5.869$ $$\Q(\sqrt{-3})$$ None $$2$$ $$1$$ $$1$$ $$0$$ $$q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots$$
735.2.i.g $4$ $5.869$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-2$$ $$-2$$ $$-2$$ $$0$$ $$q+(-1+\beta _{1}-\beta _{2})q^{2}+\beta _{2}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots$$
735.2.i.h $4$ $5.869$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-2$$ $$2$$ $$2$$ $$0$$ $$q+(-1+\beta _{1}-\beta _{2})q^{2}-\beta _{2}q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots$$
735.2.i.i $4$ $5.869$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q-\beta _{2}q^{2}+\beta _{1}q^{3}+3\beta _{1}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots$$
735.2.i.j $4$ $5.869$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-1-\beta _{2})q^{5}+\beta _{3}q^{6}+\cdots$$
735.2.i.k $4$ $5.869$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q-\beta _{2}q^{2}-\beta _{1}q^{3}+3\beta _{1}q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots$$
735.2.i.l $4$ $5.869$ $$\Q(\zeta_{12})$$ None $$2$$ $$-2$$ $$2$$ $$0$$ $$q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(-1+\zeta_{12})q^{3}+\cdots$$
735.2.i.m $8$ $5.869$ 8.0.$$\cdots$$.10 None $$-4$$ $$-4$$ $$4$$ $$0$$ $$q+(-1-\beta _{1}-\beta _{5})q^{2}+\beta _{5}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots$$
735.2.i.n $8$ $5.869$ 8.0.$$\cdots$$.10 None $$-4$$ $$4$$ $$-4$$ $$0$$ $$q+(-1-\beta _{1}-\beta _{5})q^{2}-\beta _{5}q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(735, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(245, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(147, [\chi])$$$$^{\oplus 2}$$