# Properties

 Label 252.2.b Level $252$ Weight $2$ Character orbit 252.b Rep. character $\chi_{252}(55,\cdot)$ Character field $\Q$ Dimension $18$ Newform subspaces $5$ 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.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$28$$ Character field: $$\Q$$ Newform subspaces: $$5$$ 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 56 22 34
Cusp forms 40 18 22
Eisenstein series 16 4 12

## Trace form

 $$18 q + 3 q^{2} - 3 q^{4} + 9 q^{8} + O(q^{10})$$ $$18 q + 3 q^{2} - 3 q^{4} + 9 q^{8} - 5 q^{14} + 5 q^{16} + 14 q^{22} - 22 q^{25} - 3 q^{28} + 20 q^{29} - 7 q^{32} - 4 q^{37} - 42 q^{44} - 22 q^{46} + 2 q^{49} - 33 q^{50} - 12 q^{53} + 29 q^{56} - 50 q^{58} - 39 q^{64} - 16 q^{65} + 48 q^{70} + 34 q^{74} - 12 q^{77} - 16 q^{85} + 74 q^{86} + 34 q^{88} + 42 q^{92} - 37 q^{98} + O(q^{100})$$

## 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.b.a $$2$$ $$2.012$$ $$\Q(\sqrt{-7})$$ $$\Q(\sqrt{-7})$$ $$1$$ $$0$$ $$0$$ $$0$$ $$q+\beta q^{2}+(-2+\beta )q^{4}+(-1+2\beta )q^{7}+\cdots$$
252.2.b.b $$4$$ $$2.012$$ $$\Q(\sqrt{-2}, \sqrt{7})$$ $$\Q(\sqrt{-21})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-2q^{4}-\beta _{2}q^{5}+\beta _{3}q^{7}-2\beta _{1}q^{8}+\cdots$$
252.2.b.c $$4$$ $$2.012$$ $$\Q(i, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+2\beta _{2}+\cdots)q^{7}+\cdots$$
252.2.b.d $$4$$ $$2.012$$ 4.0.2312.1 None $$1$$ $$0$$ $$0$$ $$-2$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots$$
252.2.b.e $$4$$ $$2.012$$ 4.0.2312.1 None $$1$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots$$

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

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