# Properties

 Label 735.2.q Level $735$ Weight $2$ Character orbit 735.q Rep. character $\chi_{735}(79,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $80$ Newform subspaces $8$ 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.q (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$35$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$8$$ Sturm bound: $$224$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$19$$, $$73$$

## Dimensions

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

Total New Old
Modular forms 256 80 176
Cusp forms 192 80 112
Eisenstein series 64 0 64

## Trace form

 $$80 q + 40 q^{4} - 2 q^{5} + 8 q^{6} + 40 q^{9} + O(q^{10})$$ $$80 q + 40 q^{4} - 2 q^{5} + 8 q^{6} + 40 q^{9} + 4 q^{10} + 4 q^{15} - 48 q^{16} + 24 q^{19} + 8 q^{20} + 12 q^{24} + 4 q^{25} + 12 q^{26} - 40 q^{29} + 16 q^{30} - 16 q^{31} - 16 q^{34} + 80 q^{36} + 4 q^{39} - 32 q^{40} - 16 q^{41} - 44 q^{44} + 2 q^{45} + 48 q^{46} - 40 q^{50} - 4 q^{51} + 4 q^{54} - 8 q^{55} - 4 q^{59} - 28 q^{60} - 16 q^{61} - 112 q^{64} + 14 q^{65} - 28 q^{66} - 40 q^{69} + 72 q^{71} - 152 q^{74} - 8 q^{75} + 64 q^{76} + 16 q^{79} - 52 q^{80} - 40 q^{81} - 16 q^{85} + 80 q^{86} - 16 q^{89} + 8 q^{90} + 32 q^{94} + 42 q^{95} - 8 q^{96} + O(q^{100})$$

## 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.q.a $4$ $5.869$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots$$
735.2.q.b $4$ $5.869$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{3})q^{3}-\zeta_{12}^{2}q^{4}+\cdots$$
735.2.q.c $8$ $5.869$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$-12$$ $$0$$ $$q+(\zeta_{24}-\zeta_{24}^{5}+\zeta_{24}^{7})q^{2}+(-\zeta_{24}^{2}+\cdots)q^{3}+\cdots$$
735.2.q.d $8$ $5.869$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$12$$ $$0$$ $$q+(\zeta_{24}-\zeta_{24}^{5}+\zeta_{24}^{7})q^{2}+(\zeta_{24}^{2}+\cdots)q^{3}+\cdots$$
735.2.q.e $12$ $5.869$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+(-\beta _{3}-\beta _{4})q^{2}+(-\beta _{5}-\beta _{7})q^{3}+\cdots$$
735.2.q.f $12$ $5.869$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$2$$ $$0$$ $$q+(-\beta _{3}-\beta _{4})q^{2}+(\beta _{5}+\beta _{7})q^{3}+(1+\cdots)q^{4}+\cdots$$
735.2.q.g $16$ $5.869$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q+(-\beta _{5}+\beta _{6}-\beta _{15})q^{2}-\beta _{3}q^{3}+(1+\cdots)q^{4}+\cdots$$
735.2.q.h $16$ $5.869$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{3}+\beta _{8})q^{2}-\beta _{4}q^{3}+(1-\beta _{6}+\cdots)q^{4}+\cdots$$

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

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