# Properties

 Label 1050.3.l Level $1050$ Weight $3$ Character orbit 1050.l Rep. character $\chi_{1050}(43,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $72$ Newform subspaces $8$ Sturm bound $720$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.l (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$8$$ Sturm bound: $$720$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1050, [\chi])$$.

Total New Old
Modular forms 1008 72 936
Cusp forms 912 72 840
Eisenstein series 96 0 96

## Trace form

 $$72 q + 8 q^{2} - 16 q^{8} + O(q^{10})$$ $$72 q + 8 q^{2} - 16 q^{8} + 24 q^{13} - 288 q^{16} - 24 q^{17} + 24 q^{18} + 16 q^{22} + 16 q^{23} + 80 q^{26} - 64 q^{31} - 32 q^{32} - 144 q^{33} + 432 q^{36} - 8 q^{37} - 320 q^{41} + 32 q^{43} + 256 q^{46} + 64 q^{47} + 48 q^{52} + 152 q^{53} - 32 q^{58} + 672 q^{61} - 256 q^{62} - 384 q^{66} + 48 q^{68} - 512 q^{71} + 48 q^{72} - 312 q^{73} + 224 q^{77} - 648 q^{81} + 320 q^{82} + 576 q^{83} + 128 q^{86} + 192 q^{87} - 32 q^{88} + 32 q^{92} - 240 q^{93} - 664 q^{97} - 56 q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1050, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1050.3.l.a $8$ $28.610$ 8.0.$$\cdots$$.1 None $$-8$$ $$0$$ $$0$$ $$0$$ $$q+(-1-\beta _{3})q^{2}+\beta _{6}q^{3}+2\beta _{3}q^{4}+\cdots$$
1050.3.l.b $8$ $28.610$ 8.0.$$\cdots$$.1 None $$-8$$ $$0$$ $$0$$ $$0$$ $$q+(-1-\beta _{3})q^{2}-\beta _{6}q^{3}+2\beta _{3}q^{4}+\cdots$$
1050.3.l.c $8$ $28.610$ 8.0.$$\cdots$$.1 None $$-8$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\beta _{3})q^{2}+\beta _{1}q^{3}-2\beta _{3}q^{4}+\cdots$$
1050.3.l.d $8$ $28.610$ 8.0.$$\cdots$$.1 None $$-8$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\beta _{3})q^{2}+\beta _{1}q^{3}-2\beta _{3}q^{4}+\cdots$$
1050.3.l.e $8$ $28.610$ 8.0.$$\cdots$$.1 None $$8$$ $$0$$ $$0$$ $$0$$ $$q+(1+\beta _{3})q^{2}-\beta _{6}q^{3}+2\beta _{3}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots$$
1050.3.l.f $8$ $28.610$ 8.0.$$\cdots$$.1 None $$8$$ $$0$$ $$0$$ $$0$$ $$q+(1-\beta _{3})q^{2}-\beta _{1}q^{3}-2\beta _{3}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots$$
1050.3.l.g $8$ $28.610$ 8.0.$$\cdots$$.1 None $$8$$ $$0$$ $$0$$ $$0$$ $$q+(1-\beta _{3})q^{2}-\beta _{1}q^{3}-2\beta _{3}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots$$
1050.3.l.h $16$ $28.610$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$16$$ $$0$$ $$0$$ $$0$$ $$q+(1+\beta _{2})q^{2}+\beta _{4}q^{3}+2\beta _{2}q^{4}+(\beta _{3}+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1050, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1050, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(350, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(525, [\chi])$$$$^{\oplus 2}$$