# Properties

 Label 735.1.f Level $735$ Weight $1$ Character orbit 735.f Rep. character $\chi_{735}(344,\cdot)$ Character field $\Q$ Dimension $6$ Newform subspaces $4$ Sturm bound $112$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 735.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$112$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(735, [\chi])$$.

Total New Old
Modular forms 22 16 6
Cusp forms 6 6 0
Eisenstein series 16 10 6

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 6 0 0 0

## Trace form

 $$6 q + 2 q^{4} + 6 q^{9} + O(q^{10})$$ $$6 q + 2 q^{4} + 6 q^{9} - 2 q^{15} - 2 q^{16} + 6 q^{25} + 2 q^{36} - 8 q^{46} - 4 q^{51} - 6 q^{60} - 6 q^{64} - 4 q^{79} + 6 q^{81} - 4 q^{85} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(735, [\chi])$$ into newform subspaces

Label Dim $A$ Field Image CM RM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
735.1.f.a $1$ $0.367$ $$\Q$$ $D_{2}$ $$\Q(\sqrt{-15})$$, $$\Q(\sqrt{-35})$$ $$\Q(\sqrt{21})$$ $$0$$ $$-1$$ $$-1$$ $$0$$ $$q-q^{3}-q^{4}-q^{5}+q^{9}+q^{12}+q^{15}+\cdots$$
735.1.f.b $1$ $0.367$ $$\Q$$ $D_{2}$ $$\Q(\sqrt{-15})$$, $$\Q(\sqrt{-35})$$ $$\Q(\sqrt{21})$$ $$0$$ $$1$$ $$1$$ $$0$$ $$q+q^{3}-q^{4}+q^{5}+q^{9}-q^{12}+q^{15}+\cdots$$
735.1.f.c $2$ $0.367$ $$\Q(\sqrt{2})$$ $D_{4}$ $$\Q(\sqrt{-15})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q-\beta q^{2}-q^{3}+q^{4}+q^{5}+\beta q^{6}+q^{9}+\cdots$$
735.1.f.d $2$ $0.367$ $$\Q(\sqrt{2})$$ $D_{4}$ $$\Q(\sqrt{-15})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q-\beta q^{2}+q^{3}+q^{4}-q^{5}-\beta q^{6}+q^{9}+\cdots$$