# Properties

 Label 648.1.b Level $648$ Weight $1$ Character orbit 648.b Rep. character $\chi_{648}(163,\cdot)$ Character field $\Q$ Dimension $2$ Newform subspaces $2$ Sturm bound $108$ Trace bound $2$

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## Defining parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 648.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$108$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(648, [\chi])$$.

Total New Old
Modular forms 14 6 8
Cusp forms 2 2 0
Eisenstein series 12 4 8

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 2 0 0 0

## Trace form

 $$2q + 2q^{4} + O(q^{10})$$ $$2q + 2q^{4} + 2q^{16} - 2q^{19} - 2q^{22} + 2q^{25} - 2q^{34} - 2q^{43} + 2q^{49} + 2q^{64} - 2q^{67} - 2q^{73} - 2q^{76} - 2q^{82} - 2q^{88} - 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(648, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
648.1.b.a $$1$$ $$0.323$$ $$\Q$$ $$D_{3}$$ $$\Q(\sqrt{-2})$$ None $$-1$$ $$0$$ $$0$$ $$0$$ $$q-q^{2}+q^{4}-q^{8}+q^{11}+q^{16}+q^{17}+\cdots$$
648.1.b.b $$1$$ $$0.323$$ $$\Q$$ $$D_{3}$$ $$\Q(\sqrt{-2})$$ None $$1$$ $$0$$ $$0$$ $$0$$ $$q+q^{2}+q^{4}+q^{8}-q^{11}+q^{16}-q^{17}+\cdots$$