# Properties

 Label 3450.2.d Level $3450$ Weight $2$ Character orbit 3450.d Rep. character $\chi_{3450}(2899,\cdot)$ Character field $\Q$ Dimension $68$ Newform subspaces $27$ Sturm bound $1440$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$27$$ Sturm bound: $$1440$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3450, [\chi])$$.

Total New Old
Modular forms 744 68 676
Cusp forms 696 68 628
Eisenstein series 48 0 48

## Trace form

 $$68 q - 68 q^{4} + 4 q^{6} - 68 q^{9} + O(q^{10})$$ $$68 q - 68 q^{4} + 4 q^{6} - 68 q^{9} + 68 q^{16} - 8 q^{19} + 8 q^{21} - 4 q^{24} + 24 q^{26} - 8 q^{29} - 24 q^{34} + 68 q^{36} - 8 q^{39} + 56 q^{41} + 8 q^{46} - 84 q^{49} - 4 q^{54} - 32 q^{59} + 64 q^{61} - 68 q^{64} - 8 q^{69} + 32 q^{71} - 24 q^{74} + 8 q^{76} + 16 q^{79} + 68 q^{81} - 8 q^{84} - 64 q^{86} - 40 q^{89} + 64 q^{91} - 16 q^{94} + 4 q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3450, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3450.2.d.a $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+iq^{8}-q^{9}+\cdots$$
3450.2.d.b $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}+3iq^{7}+\cdots$$
3450.2.d.c $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}+iq^{7}-iq^{8}+\cdots$$
3450.2.d.d $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+2iq^{7}+\cdots$$
3450.2.d.e $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+iq^{8}-q^{9}+\cdots$$
3450.2.d.f $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}-iq^{8}-q^{9}+\cdots$$
3450.2.d.g $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+2iq^{7}+\cdots$$
3450.2.d.h $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+5iq^{7}+\cdots$$
3450.2.d.i $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}-iq^{3}-q^{4}-q^{6}+iq^{8}-q^{9}+\cdots$$
3450.2.d.j $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
3450.2.d.k $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+iq^{7}+iq^{8}+\cdots$$
3450.2.d.l $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+5iq^{7}+\cdots$$
3450.2.d.m $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}-iq^{8}-q^{9}+\cdots$$
3450.2.d.n $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+4iq^{7}+\cdots$$
3450.2.d.o $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+3iq^{7}+\cdots$$
3450.2.d.p $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+iq^{7}+iq^{8}+\cdots$$
3450.2.d.q $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
3450.2.d.r $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+iq^{8}-q^{9}+\cdots$$
3450.2.d.s $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+4iq^{7}+\cdots$$
3450.2.d.t $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-iq^{2}+iq^{3}-q^{4}+q^{6}+iq^{8}-q^{9}+\cdots$$
3450.2.d.u $2$ $27.548$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-iq^{3}-q^{4}+q^{6}+2iq^{7}+\cdots$$
3450.2.d.v $4$ $27.548$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}-q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$$
3450.2.d.w $4$ $27.548$ $$\Q(i, \sqrt{73})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}-\beta _{2}q^{3}-q^{4}-q^{6}-3\beta _{2}q^{7}+\cdots$$
3450.2.d.x $4$ $27.548$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-\beta _{1}q^{3}-q^{4}+q^{6}+2\beta _{2}q^{7}+\cdots$$
3450.2.d.y $4$ $27.548$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}+q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
3450.2.d.z $4$ $27.548$ $$\Q(i, \sqrt{57})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+\beta _{2}q^{3}-q^{4}+q^{6}-3\beta _{2}q^{7}+\cdots$$
3450.2.d.ba $6$ $27.548$ 6.0.181494784.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{4}q^{2}+\beta _{4}q^{3}-q^{4}-q^{6}-\beta _{3}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3450, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3450, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(115, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(230, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(345, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(575, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(690, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1150, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1725, [\chi])$$$$^{\oplus 2}$$