# Properties

 Label 1050.2.bc Level $1050$ Weight $2$ Character orbit 1050.bc Rep. character $\chi_{1050}(157,\cdot)$ Character field $\Q(\zeta_{12})$ Dimension $96$ Newform subspaces $8$ Sturm bound $480$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.bc (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$35$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$8$$ Sturm bound: $$480$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$17$$

## Dimensions

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

Total New Old
Modular forms 1056 96 960
Cusp forms 864 96 768
Eisenstein series 192 0 192

## Trace form

 $$96q + 12q^{7} + O(q^{10})$$ $$96q + 12q^{7} - 16q^{11} + 48q^{16} + 8q^{21} - 8q^{22} + 8q^{23} + 48q^{26} + 12q^{28} + 96q^{31} - 12q^{33} - 96q^{36} + 16q^{37} + 48q^{38} - 4q^{42} + 48q^{43} + 16q^{46} + 24q^{47} - 32q^{51} - 16q^{53} - 32q^{56} + 16q^{57} - 36q^{58} + 24q^{61} + 48q^{67} + 128q^{71} + 24q^{73} + 32q^{78} + 48q^{81} - 48q^{82} + 32q^{86} - 60q^{87} - 4q^{88} + 8q^{91} - 16q^{92} - 8q^{93} - 16q^{98} + O(q^{100})$$

## 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.bc.a $$8$$ $$8.384$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{7}q^{2}+(-\zeta_{24}^{3}+\zeta_{24}^{7})q^{3}+\cdots$$
1050.2.bc.b $$8$$ $$8.384$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{24}^{3}-\zeta_{24}^{7})q^{2}+\zeta_{24}^{7}q^{3}+(\zeta_{24}^{2}+\cdots)q^{4}+\cdots$$
1050.2.bc.c $$8$$ $$8.384$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{7}q^{2}+(\zeta_{24}^{3}-\zeta_{24}^{7})q^{3}-\zeta_{24}^{2}q^{4}+\cdots$$
1050.2.bc.d $$8$$ $$8.384$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}q^{2}-\zeta_{24}^{5}q^{3}+\zeta_{24}^{2}q^{4}+\zeta_{24}^{6}q^{6}+\cdots$$
1050.2.bc.e $$16$$ $$8.384$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}-\beta _{11}q^{3}+(\beta _{8}-\beta _{9})q^{4}-\beta _{9}q^{6}+\cdots$$
1050.2.bc.f $$16$$ $$8.384$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{11}q^{2}+\beta _{2}q^{3}-\beta _{8}q^{4}+\beta _{9}q^{6}+\cdots$$
1050.2.bc.g $$16$$ $$8.384$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{6}q^{2}-\beta _{12}q^{3}+(-\beta _{5}+\beta _{13})q^{4}+\cdots$$
1050.2.bc.h $$16$$ $$8.384$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{15}q^{2}+\beta _{2}q^{3}+(\beta _{5}-\beta _{13})q^{4}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1050, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$