# Properties

 Label 3840.2.d Level $3840$ Weight $2$ Character orbit 3840.d Rep. character $\chi_{3840}(2689,\cdot)$ Character field $\Q$ Dimension $96$ Newform subspaces $38$ Sturm bound $1536$ Trace bound $35$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3840 = 2^{8} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3840.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$40$$ Character field: $$\Q$$ Newform subspaces: $$38$$ Sturm bound: $$1536$$ Trace bound: $$35$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$, $$31$$, $$37$$, $$43$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3840, [\chi])$$.

Total New Old
Modular forms 816 96 720
Cusp forms 720 96 624
Eisenstein series 96 0 96

## Trace form

 $$96 q + 96 q^{9} + O(q^{10})$$ $$96 q + 96 q^{9} - 96 q^{49} + 32 q^{65} + 96 q^{81} + 64 q^{89} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3840, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3840.2.d.a $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-4$$ $$0$$ $$q-q^{3}+(-2+i)q^{5}+4iq^{7}+q^{9}+\cdots$$
3840.2.d.b $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-4$$ $$0$$ $$q-q^{3}+(-2-i)q^{5}+4iq^{7}+q^{9}+\cdots$$
3840.2.d.c $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-4$$ $$0$$ $$q-q^{3}+(-2+i)q^{5}+q^{9}-2iq^{11}+\cdots$$
3840.2.d.d $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q-q^{3}+(-1+i)q^{5}+iq^{7}+q^{9}+iq^{11}+\cdots$$
3840.2.d.e $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q-q^{3}+(-1+i)q^{5}+iq^{7}+q^{9}-3iq^{11}+\cdots$$
3840.2.d.f $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q-q^{3}+(-1+i)q^{5}+q^{9}+2iq^{11}+\cdots$$
3840.2.d.g $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q-q^{3}+(-1-i)q^{5}+iq^{7}+q^{9}+iq^{11}+\cdots$$
3840.2.d.h $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q-q^{3}+(-1+i)q^{5}+2iq^{7}+q^{9}+\cdots$$
3840.2.d.i $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q-q^{3}+(1+i)q^{5}+2iq^{7}+q^{9}-6q^{13}+\cdots$$
3840.2.d.j $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q-q^{3}+(1-i)q^{5}+iq^{7}+q^{9}-iq^{11}+\cdots$$
3840.2.d.k $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q-q^{3}+(1+i)q^{5}+iq^{7}+q^{9}+3iq^{11}+\cdots$$
3840.2.d.l $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q-q^{3}+(1-i)q^{5}+q^{9}+2iq^{11}+2q^{13}+\cdots$$
3840.2.d.m $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$2$$ $$0$$ $$q-q^{3}+(1+i)q^{5}+iq^{7}+q^{9}-iq^{11}+\cdots$$
3840.2.d.n $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$4$$ $$0$$ $$q-q^{3}+(2+i)q^{5}+q^{9}+2iq^{11}-2q^{13}+\cdots$$
3840.2.d.o $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$4$$ $$0$$ $$q-q^{3}+(2-i)q^{5}+4iq^{7}+q^{9}+4iq^{11}+\cdots$$
3840.2.d.p $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$4$$ $$0$$ $$q-q^{3}+(2+i)q^{5}+4iq^{7}+q^{9}+4q^{13}+\cdots$$
3840.2.d.q $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-4$$ $$0$$ $$q+q^{3}+(-2-i)q^{5}+4iq^{7}+q^{9}+\cdots$$
3840.2.d.r $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-4$$ $$0$$ $$q+q^{3}+(-2+i)q^{5}+4iq^{7}+q^{9}+\cdots$$
3840.2.d.s $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-4$$ $$0$$ $$q+q^{3}+(-2+i)q^{5}+q^{9}+2iq^{11}+\cdots$$
3840.2.d.t $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q+q^{3}+(-1+i)q^{5}+q^{9}-2iq^{11}+\cdots$$
3840.2.d.u $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q+q^{3}+(-1-i)q^{5}+iq^{7}+q^{9}-3iq^{11}+\cdots$$
3840.2.d.v $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q+q^{3}+(-1-i)q^{5}+iq^{7}+q^{9}+iq^{11}+\cdots$$
3840.2.d.w $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q+q^{3}+(-1-i)q^{5}+2iq^{7}+q^{9}+\cdots$$
3840.2.d.x $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q+q^{3}+(-1+i)q^{5}+iq^{7}+q^{9}+iq^{11}+\cdots$$
3840.2.d.y $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+q^{3}+(1+i)q^{5}+iq^{7}+q^{9}-iq^{11}+\cdots$$
3840.2.d.z $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+q^{3}+(1-i)q^{5}+2iq^{7}+q^{9}-6q^{13}+\cdots$$
3840.2.d.ba $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+q^{3}+(1-i)q^{5}+iq^{7}+q^{9}-iq^{11}+\cdots$$
3840.2.d.bb $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+q^{3}+(1-i)q^{5}+q^{9}-2iq^{11}+2q^{13}+\cdots$$
3840.2.d.bc $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+q^{3}+(1-i)q^{5}+iq^{7}+q^{9}+3iq^{11}+\cdots$$
3840.2.d.bd $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$4$$ $$0$$ $$q+q^{3}+(2-i)q^{5}+q^{9}+2iq^{11}-2q^{13}+\cdots$$
3840.2.d.be $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$4$$ $$0$$ $$q+q^{3}+(2+i)q^{5}+4iq^{7}+q^{9}+4iq^{11}+\cdots$$
3840.2.d.bf $2$ $30.663$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$4$$ $$0$$ $$q+q^{3}+(2-i)q^{5}+4iq^{7}+q^{9}+4q^{13}+\cdots$$
3840.2.d.bg $4$ $30.663$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q-q^{3}-\beta _{3}q^{5}-\beta _{1}q^{7}+q^{9}-\beta _{2}q^{11}+\cdots$$
3840.2.d.bh $4$ $30.663$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+q^{3}-\beta _{3}q^{5}-\beta _{1}q^{7}+q^{9}-\beta _{2}q^{11}+\cdots$$
3840.2.d.bi $6$ $30.663$ 6.0.350464.1 None $$0$$ $$-6$$ $$0$$ $$0$$ $$q-q^{3}+\beta _{1}q^{5}+(-\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots$$
3840.2.d.bj $6$ $30.663$ 6.0.350464.1 None $$0$$ $$-6$$ $$0$$ $$0$$ $$q-q^{3}+\beta _{3}q^{5}+(-\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots$$
3840.2.d.bk $6$ $30.663$ 6.0.350464.1 None $$0$$ $$6$$ $$0$$ $$0$$ $$q+q^{3}-\beta _{3}q^{5}+(-\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots$$
3840.2.d.bl $6$ $30.663$ 6.0.350464.1 None $$0$$ $$6$$ $$0$$ $$0$$ $$q+q^{3}-\beta _{1}q^{5}+(-\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3840, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3840, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(320, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(480, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(640, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(960, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1280, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1920, [\chi])$$$$^{\oplus 2}$$