# Properties

 Label 252.2.bf Level $252$ Weight $2$ Character orbit 252.bf Rep. character $\chi_{252}(19,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $36$ Newform subspaces $7$ Sturm bound $96$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 252.bf (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$28$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$7$$ Sturm bound: $$96$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(252, [\chi])$$.

Total New Old
Modular forms 112 44 68
Cusp forms 80 36 44
Eisenstein series 32 8 24

## Trace form

 $$36q + 6q^{5} + O(q^{10})$$ $$36q + 6q^{5} + 6q^{10} + 20q^{14} + 4q^{16} + 6q^{17} - 20q^{22} + 4q^{25} - 18q^{26} + 24q^{28} + 16q^{29} - 20q^{32} - 14q^{37} - 36q^{38} - 48q^{40} - 26q^{46} - 20q^{49} - 12q^{50} - 36q^{52} + 6q^{53} - 56q^{56} - 4q^{58} - 18q^{61} - 24q^{64} - 20q^{65} + 36q^{68} - 30q^{70} - 42q^{73} + 44q^{74} - 6q^{77} + 96q^{80} + 24q^{82} - 20q^{85} + 10q^{86} + 56q^{88} - 54q^{89} + 48q^{92} + 78q^{94} + 94q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(252, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
252.2.bf.a $$4$$ $$2.012$$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ None $$-1$$ $$0$$ $$6$$ $$0$$ $$q-\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(2-\beta _{2}+\cdots)q^{5}+\cdots$$
252.2.bf.b $$4$$ $$2.012$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+\beta _{3}q^{2}-2q^{4}-2\beta _{1}q^{5}+(1-3\beta _{2}+\cdots)q^{7}+\cdots$$
252.2.bf.c $$4$$ $$2.012$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(2\beta _{1}-2\beta _{3})q^{5}+\cdots$$
252.2.bf.d $$4$$ $$2.012$$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ None $$1$$ $$0$$ $$-6$$ $$0$$ $$q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(-2+\beta _{2}+\cdots)q^{5}+\cdots$$
252.2.bf.e $$4$$ $$2.012$$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$6$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$$
252.2.bf.f $$8$$ $$2.012$$ 8.0.562828176.1 None $$-1$$ $$0$$ $$0$$ $$-2$$ $$q+\beta _{5}q^{2}+(-\beta _{4}+\beta _{6}-\beta _{7})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$
252.2.bf.g $$8$$ $$2.012$$ 8.0.562828176.1 None $$-1$$ $$0$$ $$0$$ $$2$$ $$q+(-\beta _{1}-\beta _{5})q^{2}+(-\beta _{2}-\beta _{6}+\beta _{7})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(252, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(252, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 2}$$