# Properties

 Label 3240.2.q Level $3240$ Weight $2$ Character orbit 3240.q Rep. character $\chi_{3240}(1081,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $96$ Newform subspaces $34$ Sturm bound $1296$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$34$$ Sturm bound: $$1296$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3240, [\chi])$$.

Total New Old
Modular forms 1392 96 1296
Cusp forms 1200 96 1104
Eisenstein series 192 0 192

## Trace form

 $$96 q + O(q^{10})$$ $$96 q + 24 q^{19} - 48 q^{25} - 60 q^{31} - 72 q^{43} - 60 q^{49} - 12 q^{61} - 12 q^{67} - 72 q^{73} + 120 q^{91} + 36 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3240, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
3240.2.q.a $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-4$$ $$q-\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{7}+6\zeta_{6}q^{13}+\cdots$$
3240.2.q.b $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-2$$ $$q-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots$$
3240.2.q.c $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-2$$ $$q-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-1+\zeta_{6})q^{11}+\cdots$$
3240.2.q.d $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-2$$ $$q-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(2-2\zeta_{6})q^{11}+\cdots$$
3240.2.q.e $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$0$$ $$q-\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{11}+3q^{17}+\cdots$$
3240.2.q.f $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$0$$ $$q-\zeta_{6}q^{5}+(1-\zeta_{6})q^{11}-7q^{19}+6\zeta_{6}q^{23}+\cdots$$
3240.2.q.g $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$0$$ $$q-\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}-6\zeta_{6}q^{13}+\cdots$$
3240.2.q.h $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$1$$ $$q-\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(2-2\zeta_{6})q^{11}+\cdots$$
3240.2.q.i $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$2$$ $$q-\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+6\zeta_{6}q^{13}+\cdots$$
3240.2.q.j $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$2$$ $$q-\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots$$
3240.2.q.k $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$4$$ $$q-\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots$$
3240.2.q.l $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$4$$ $$q-\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(2-2\zeta_{6})q^{11}+\cdots$$
3240.2.q.m $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-4$$ $$q+\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{7}+6\zeta_{6}q^{13}+\cdots$$
3240.2.q.n $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-2$$ $$q+\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-2+2\zeta_{6})q^{11}+\cdots$$
3240.2.q.o $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-2$$ $$q+\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(1-\zeta_{6})q^{11}+\cdots$$
3240.2.q.p $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-2$$ $$q+\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots$$
3240.2.q.q $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}-6\zeta_{6}q^{13}+\cdots$$
3240.2.q.r $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{5}+(-1+\zeta_{6})q^{11}-7q^{19}+\cdots$$
3240.2.q.s $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{5}+(2-2\zeta_{6})q^{11}-3q^{17}-q^{19}+\cdots$$
3240.2.q.t $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$1$$ $$q+\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(-2+2\zeta_{6})q^{11}+\cdots$$
3240.2.q.u $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$2$$ $$q+\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots$$
3240.2.q.v $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$2$$ $$q+\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+6\zeta_{6}q^{13}+\cdots$$
3240.2.q.w $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$4$$ $$q+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(-2+2\zeta_{6})q^{11}+\cdots$$
3240.2.q.x $2$ $25.872$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$4$$ $$q+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots$$
3240.2.q.y $4$ $25.872$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$0$$ $$-2$$ $$-1$$ $$q+\beta _{1}q^{5}+(-1-\beta _{1}+\beta _{3})q^{7}+(-2+\cdots)q^{11}+\cdots$$
3240.2.q.z $4$ $25.872$ $$\Q(\sqrt{-3}, \sqrt{73})$$ None $$0$$ $$0$$ $$-2$$ $$-1$$ $$q+(-1+\beta _{2})q^{5}-\beta _{1}q^{7}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots$$
3240.2.q.ba $4$ $25.872$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$0$$ $$-2$$ $$-1$$ $$q+(-1+\beta _{1})q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\beta _{3}q^{11}+\cdots$$
3240.2.q.bb $4$ $25.872$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-2$$ $$2$$ $$q+(-1+\zeta_{12})q^{5}+(\zeta_{12}-\zeta_{12}^{2})q^{7}+\cdots$$
3240.2.q.bc $4$ $25.872$ $$\Q(\sqrt{-3}, \sqrt{73})$$ None $$0$$ $$0$$ $$2$$ $$-1$$ $$q+(1-\beta _{2})q^{5}-\beta _{1}q^{7}+(\beta _{1}-\beta _{2})q^{11}+\cdots$$
3240.2.q.bd $4$ $25.872$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$0$$ $$2$$ $$-1$$ $$q+(1-\beta _{1})q^{5}+(-\beta _{1}+\beta _{3})q^{7}-\beta _{3}q^{11}+\cdots$$
3240.2.q.be $4$ $25.872$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$0$$ $$2$$ $$-1$$ $$q-\beta _{1}q^{5}+(-1-\beta _{1}+\beta _{3})q^{7}+(2+2\beta _{1}+\cdots)q^{11}+\cdots$$
3240.2.q.bf $4$ $25.872$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$2$$ $$2$$ $$q+(1-\zeta_{12})q^{5}+(\zeta_{12}-\zeta_{12}^{2})q^{7}+(-2\zeta_{12}+\cdots)q^{11}+\cdots$$
3240.2.q.bg $8$ $25.872$ 8.0.3887771904.9 None $$0$$ $$0$$ $$-4$$ $$-2$$ $$q-\beta _{1}q^{5}+(\beta _{1}-\beta _{2}-\beta _{4}+\beta _{5}-\beta _{7})q^{7}+\cdots$$
3240.2.q.bh $8$ $25.872$ 8.0.3887771904.9 None $$0$$ $$0$$ $$4$$ $$-2$$ $$q+\beta _{1}q^{5}+(\beta _{1}-\beta _{2}-\beta _{4}+\beta _{5}-\beta _{7})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3240, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3240, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(162, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(216, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(270, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(324, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(360, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(405, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(540, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(648, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(810, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1080, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1620, [\chi])$$$$^{\oplus 2}$$