# Properties

 Label 588.2.b Level $588$ Weight $2$ Character orbit 588.b Rep. character $\chi_{588}(391,\cdot)$ Character field $\Q$ Dimension $40$ Newform subspaces $4$ Sturm bound $224$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 588.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$28$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$224$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(588, [\chi])$$.

Total New Old
Modular forms 128 40 88
Cusp forms 96 40 56
Eisenstein series 32 0 32

## Trace form

 $$40q + 4q^{2} - 4q^{4} + 16q^{8} + 40q^{9} + O(q^{10})$$ $$40q + 4q^{2} - 4q^{4} + 16q^{8} + 40q^{9} + 12q^{16} + 4q^{18} - 12q^{22} - 32q^{25} + 32q^{29} - 16q^{30} - 16q^{32} - 4q^{36} + 40q^{37} - 40q^{44} + 24q^{46} - 52q^{50} - 48q^{53} + 24q^{57} + 4q^{58} - 4q^{60} - 4q^{64} - 16q^{65} + 16q^{72} + 12q^{74} + 36q^{78} + 40q^{81} - 32q^{85} - 36q^{86} + 100q^{88} - 56q^{92} + 8q^{93} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(588, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
588.2.b.a $$8$$ $$4.695$$ 8.0.562828176.1 None $$-2$$ $$-8$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+(\beta _{2}-\beta _{5})q^{5}+\cdots$$
588.2.b.b $$8$$ $$4.695$$ 8.0.562828176.1 None $$-2$$ $$8$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-\beta _{2}+\beta _{5}+\cdots)q^{5}+\cdots$$
588.2.b.c $$12$$ $$4.695$$ 12.0.$$\cdots$$.1 None $$4$$ $$-12$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-q^{3}+(\beta _{5}+\beta _{6})q^{4}+(-1+\cdots)q^{5}+\cdots$$
588.2.b.d $$12$$ $$4.695$$ 12.0.$$\cdots$$.1 None $$4$$ $$12$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+q^{3}+(\beta _{5}+\beta _{6})q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(588, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(588, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(196, [\chi])$$$$^{\oplus 2}$$