# Properties

 Label 288.2.p Level $288$ Weight $2$ Character orbit 288.p Rep. character $\chi_{288}(47,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $20$ Newform subspaces $2$ Sturm bound $96$ Trace bound $1$

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## Defining parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.p (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$72$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(288, [\chi])$$.

Total New Old
Modular forms 112 28 84
Cusp forms 80 20 60
Eisenstein series 32 8 24

## Trace form

 $$20q + 4q^{3} - 4q^{9} + O(q^{10})$$ $$20q + 4q^{3} - 4q^{9} + 6q^{11} + 8q^{19} - 4q^{25} + 16q^{27} - 2q^{33} - 18q^{41} + 2q^{43} - 4q^{49} - 28q^{51} - 20q^{57} - 30q^{59} - 6q^{65} + 2q^{67} - 8q^{73} - 68q^{75} + 8q^{81} - 54q^{83} + 36q^{91} - 2q^{97} - 10q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(288, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
288.2.p.a $$4$$ $$2.300$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-2$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+(1-2\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots$$
288.2.p.b $$16$$ $$2.300$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$6$$ $$0$$ $$0$$ $$q+(-\beta _{1}+\beta _{3}-\beta _{8})q^{3}-\beta _{9}q^{5}-\beta _{4}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(288, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(288, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 3}$$