# Properties

 Label 1050.2.d Level $1050$ Weight $2$ Character orbit 1050.d Rep. character $\chi_{1050}(1049,\cdot)$ Character field $\Q$ Dimension $48$ Newform subspaces $8$ Sturm bound $480$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$105$$ Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$480$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1050, [\chi])$$.

Total New Old
Modular forms 264 48 216
Cusp forms 216 48 168
Eisenstein series 48 0 48

## Trace form

 $$48q + 48q^{4} + O(q^{10})$$ $$48q + 48q^{4} + 48q^{16} + 20q^{21} + 48q^{39} + 56q^{46} - 16q^{49} + 40q^{51} + 48q^{64} - 32q^{79} - 88q^{81} + 20q^{84} - 56q^{91} - 96q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1050, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1050.2.d.a $$4$$ $$8.384$$ $$\Q(i, \sqrt{5})$$ None $$-4$$ $$-2$$ $$0$$ $$6$$ $$q-q^{2}+(-\beta _{2}-\beta _{3})q^{3}+q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots$$
1050.2.d.b $$4$$ $$8.384$$ $$\Q(i, \sqrt{6})$$ None $$-4$$ $$0$$ $$0$$ $$0$$ $$q-q^{2}-\beta _{3}q^{3}+q^{4}+\beta _{3}q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
1050.2.d.c $$4$$ $$8.384$$ $$\Q(i, \sqrt{5})$$ None $$-4$$ $$2$$ $$0$$ $$-6$$ $$q-q^{2}+(\beta _{2}+\beta _{3})q^{3}+q^{4}+(-\beta _{2}-\beta _{3})q^{6}+\cdots$$
1050.2.d.d $$4$$ $$8.384$$ $$\Q(i, \sqrt{5})$$ None $$4$$ $$-2$$ $$0$$ $$6$$ $$q+q^{2}+(-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots$$
1050.2.d.e $$4$$ $$8.384$$ $$\Q(i, \sqrt{6})$$ None $$4$$ $$0$$ $$0$$ $$0$$ $$q+q^{2}+\beta _{3}q^{3}+q^{4}+\beta _{3}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$$
1050.2.d.f $$4$$ $$8.384$$ $$\Q(i, \sqrt{5})$$ None $$4$$ $$2$$ $$0$$ $$-6$$ $$q+q^{2}+(\beta _{2}+\beta _{3})q^{3}+q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots$$
1050.2.d.g $$12$$ $$8.384$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$-12$$ $$0$$ $$0$$ $$0$$ $$q-q^{2}+\beta _{10}q^{3}+q^{4}-\beta _{10}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$$
1050.2.d.h $$12$$ $$8.384$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$12$$ $$0$$ $$0$$ $$0$$ $$q+q^{2}-\beta _{10}q^{3}+q^{4}-\beta _{10}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1050, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1050, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$