# Properties

 Label 180.2.k Level $180$ Weight $2$ Character orbit 180.k Rep. character $\chi_{180}(127,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $26$ Newform subspaces $5$ Sturm bound $72$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 180.k (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q(i)$$ Newform subspaces: $$5$$ Sturm bound: $$72$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$7$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(180, [\chi])$$.

Total New Old
Modular forms 88 34 54
Cusp forms 56 26 30
Eisenstein series 32 8 24

## Trace form

 $$26 q + 2 q^{2} + 4 q^{5} + 8 q^{8} + O(q^{10})$$ $$26 q + 2 q^{2} + 4 q^{5} + 8 q^{8} - 10 q^{10} - 2 q^{13} - 16 q^{16} + 14 q^{17} - 16 q^{20} - 16 q^{22} + 6 q^{25} - 20 q^{26} - 32 q^{28} - 28 q^{32} - 6 q^{37} - 16 q^{38} + 20 q^{40} + 16 q^{46} + 30 q^{50} + 52 q^{52} - 22 q^{53} + 64 q^{56} + 24 q^{58} - 40 q^{61} + 56 q^{62} - 22 q^{65} + 28 q^{68} + 16 q^{70} - 38 q^{73} - 48 q^{76} - 48 q^{77} - 20 q^{80} - 32 q^{82} - 26 q^{85} - 64 q^{86} + 32 q^{88} - 56 q^{92} - 54 q^{97} - 38 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(180, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
180.2.k.a $2$ $1.437$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$-2$$ $$0$$ $$2$$ $$0$$ $$q+(-1-i)q^{2}+2iq^{4}+(1-2i)q^{5}+\cdots$$
180.2.k.b $2$ $1.437$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$2$$ $$0$$ $$-2$$ $$0$$ $$q+(1+i)q^{2}+2iq^{4}+(-1+2i)q^{5}+\cdots$$
180.2.k.c $2$ $1.437$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$2$$ $$0$$ $$4$$ $$0$$ $$q+(1+i)q^{2}+2iq^{4}+(2-i)q^{5}+(-2+\cdots)q^{8}+\cdots$$
180.2.k.d $8$ $1.437$ 8.0.157351936.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{2}+\beta _{4})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(2\beta _{4}+\cdots)q^{5}+\cdots$$
180.2.k.e $12$ $1.437$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{1}+\beta _{7}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(180, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(180, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 2}$$