# Properties

 Label 90.2.l Level $90$ Weight $2$ Character orbit 90.l Rep. character $\chi_{90}(23,\cdot)$ Character field $\Q(\zeta_{12})$ Dimension $24$ Newform subspaces $2$ Sturm bound $36$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 90.l (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$45$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$2$$ Sturm bound: $$36$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(90, [\chi])$$.

Total New Old
Modular forms 88 24 64
Cusp forms 56 24 32
Eisenstein series 32 0 32

## Trace form

 $$24q + 4q^{3} - 8q^{6} + O(q^{10})$$ $$24q + 4q^{3} - 8q^{6} - 24q^{11} - 4q^{12} - 8q^{15} + 12q^{16} - 8q^{18} - 12q^{20} - 8q^{21} - 24q^{23} - 12q^{25} - 8q^{27} - 24q^{30} + 16q^{33} + 16q^{36} - 24q^{37} + 36q^{38} + 36q^{41} + 44q^{42} + 68q^{45} - 24q^{46} + 48q^{47} + 8q^{48} + 48q^{50} - 16q^{51} - 24q^{55} + 12q^{56} + 52q^{57} + 12q^{58} + 4q^{60} - 12q^{61} - 80q^{63} - 24q^{65} - 8q^{66} - 12q^{67} - 36q^{68} - 16q^{72} + 8q^{75} - 48q^{77} - 24q^{78} - 4q^{81} - 48q^{82} + 60q^{83} - 24q^{85} - 72q^{86} - 56q^{87} - 8q^{90} + 48q^{91} - 24q^{92} + 52q^{93} + 60q^{95} - 4q^{96} - 36q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(90, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
90.2.l.a $$8$$ $$0.719$$ $$\Q(\zeta_{24})$$ None $$0$$ $$4$$ $$12$$ $$-8$$ $$q+\zeta_{24}^{7}q^{2}+(1+\zeta_{24}^{2}-\zeta_{24}^{4}+\zeta_{24}^{5}+\cdots)q^{3}+\cdots$$
90.2.l.b $$16$$ $$0.719$$ 16.0.$$\cdots$$.9 None $$0$$ $$0$$ $$-12$$ $$8$$ $$q-\beta _{11}q^{2}+(-\beta _{3}-\beta _{4}-\beta _{5}+\beta _{8}+\cdots)q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(90, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(90, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$