# Properties

 Label 180.3.f Level $180$ Weight $3$ Character orbit 180.f Rep. character $\chi_{180}(19,\cdot)$ Character field $\Q$ Dimension $28$ Newform subspaces $8$ Sturm bound $108$ Trace bound $4$

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## Defining parameters

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 180.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$108$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$7$$, $$13$$, $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(180, [\chi])$$.

Total New Old
Modular forms 80 32 48
Cusp forms 64 28 36
Eisenstein series 16 4 12

## Trace form

 $$28q - 4q^{4} + O(q^{10})$$ $$28q - 4q^{4} + 8q^{10} + 36q^{14} - 44q^{16} + 12q^{20} + 20q^{25} - 60q^{26} - 136q^{34} - 20q^{40} + 96q^{41} - 204q^{44} + 164q^{49} - 168q^{50} + 180q^{56} - 56q^{61} + 260q^{64} + 168q^{70} + 564q^{74} - 168q^{76} + 372q^{80} - 40q^{85} - 168q^{86} - 192q^{89} - 360q^{94} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(180, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
180.3.f.a $$1$$ $$4.905$$ $$\Q$$ $$\Q(\sqrt{-5})$$ $$-2$$ $$0$$ $$5$$ $$4$$ $$q-2q^{2}+4q^{4}+5q^{5}+4q^{7}-8q^{8}+\cdots$$
180.3.f.b $$1$$ $$4.905$$ $$\Q$$ $$\Q(\sqrt{-5})$$ $$2$$ $$0$$ $$5$$ $$-4$$ $$q+2q^{2}+4q^{4}+5q^{5}-4q^{7}+8q^{8}+\cdots$$
180.3.f.c $$2$$ $$4.905$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-6$$ $$0$$ $$q+iq^{2}-4q^{4}+(-3-2i)q^{5}-4iq^{8}+\cdots$$
180.3.f.d $$4$$ $$4.905$$ $$\Q(\sqrt{-3}, \sqrt{22})$$ None $$-4$$ $$0$$ $$0$$ $$0$$ $$q+(-1-\beta _{2})q^{2}+(-2+2\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$
180.3.f.e $$4$$ $$4.905$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{3})q^{2}+(2+2\zeta_{12}^{2})q^{4}+\cdots$$
180.3.f.f $$4$$ $$4.905$$ $$\Q(i, \sqrt{15})$$ $$\Q(\sqrt{-15})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(4+\beta _{3})q^{4}+5\beta _{2}q^{5}+(3\beta _{1}+\cdots)q^{8}+\cdots$$
180.3.f.g $$4$$ $$4.905$$ $$\Q(\sqrt{-3}, \sqrt{22})$$ None $$4$$ $$0$$ $$0$$ $$0$$ $$q+(1+\beta _{2})q^{2}+(-2+2\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
180.3.f.h $$8$$ $$4.905$$ 8.0.$$\cdots$$.4 None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(180, [\chi])$$ into lower level spaces

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