# Properties

 Label 728.2.k Level $728$ Weight $2$ Character orbit 728.k Rep. character $\chi_{728}(337,\cdot)$ Character field $\Q$ Dimension $20$ Newform subspaces $2$ Sturm bound $224$ Trace bound $1$

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## Defining parameters

 Level: $$N$$ $$=$$ $$728 = 2^{3} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 728.k (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$224$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(728, [\chi])$$.

Total New Old
Modular forms 120 20 100
Cusp forms 104 20 84
Eisenstein series 16 0 16

## Trace form

 $$20 q + 16 q^{9} + O(q^{10})$$ $$20 q + 16 q^{9} + 8 q^{13} + 8 q^{17} - 32 q^{25} + 4 q^{29} + 12 q^{35} - 16 q^{39} - 20 q^{43} - 20 q^{49} + 8 q^{51} - 20 q^{53} - 64 q^{55} - 40 q^{61} + 24 q^{65} + 8 q^{69} + 40 q^{75} + 8 q^{77} + 60 q^{81} + 56 q^{87} + 4 q^{91} + 20 q^{95} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(728, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
728.2.k.a $8$ $5.813$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(-\beta _{1}+\beta _{5})q^{5}-\beta _{5}q^{7}+\cdots$$
728.2.k.b $12$ $5.813$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+(-\beta _{1}+\beta _{5}+\beta _{7})q^{5}-\beta _{4}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(728, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(728, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(91, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(104, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(182, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(364, [\chi])$$$$^{\oplus 2}$$