# Properties

 Label 648.4.i Level $648$ Weight $4$ Character orbit 648.i Rep. character $\chi_{648}(217,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $72$ Newform subspaces $22$ Sturm bound $432$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 648.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$22$$ Sturm bound: $$432$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(648, [\chi])$$.

Total New Old
Modular forms 696 72 624
Cusp forms 600 72 528
Eisenstein series 96 0 96

## Trace form

 $$72 q + O(q^{10})$$ $$72 q + 180 q^{19} - 900 q^{25} - 450 q^{31} + 972 q^{43} - 2160 q^{49} - 3060 q^{55} + 36 q^{61} + 774 q^{67} - 2484 q^{73} - 630 q^{79} + 414 q^{85} - 9900 q^{91} + 144 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(648, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
648.4.i.a $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-16$$ $$12$$ $$q-2^{4}\zeta_{6}q^{5}+(12-12\zeta_{6})q^{7}+(-2^{6}+\cdots)q^{11}+\cdots$$
648.4.i.b $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-14$$ $$24$$ $$q-14\zeta_{6}q^{5}+(24-24\zeta_{6})q^{7}+(28-28\zeta_{6})q^{11}+\cdots$$
648.4.i.c $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-5$$ $$-36$$ $$q-5\zeta_{6}q^{5}+(-6^{2}+6^{2}\zeta_{6})q^{7}+(2^{6}+\cdots)q^{11}+\cdots$$
648.4.i.d $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$-3$$ $$q-4\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(-28+\cdots)q^{11}+\cdots$$
648.4.i.e $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$-24$$ $$q-2\zeta_{6}q^{5}+(-24+24\zeta_{6})q^{7}+(-44+\cdots)q^{11}+\cdots$$
648.4.i.f $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$9$$ $$q-\zeta_{6}q^{5}+(9-9\zeta_{6})q^{7}+(17-17\zeta_{6})q^{11}+\cdots$$
648.4.i.g $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$9$$ $$q+\zeta_{6}q^{5}+(9-9\zeta_{6})q^{7}+(-17+17\zeta_{6})q^{11}+\cdots$$
648.4.i.h $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$-24$$ $$q+2\zeta_{6}q^{5}+(-24+24\zeta_{6})q^{7}+(44+\cdots)q^{11}+\cdots$$
648.4.i.i $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$-3$$ $$q+4\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(28-28\zeta_{6})q^{11}+\cdots$$
648.4.i.j $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$5$$ $$-36$$ $$q+5\zeta_{6}q^{5}+(-6^{2}+6^{2}\zeta_{6})q^{7}+(-2^{6}+\cdots)q^{11}+\cdots$$
648.4.i.k $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$14$$ $$24$$ $$q+14\zeta_{6}q^{5}+(24-24\zeta_{6})q^{7}+(-28+\cdots)q^{11}+\cdots$$
648.4.i.l $2$ $38.233$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$16$$ $$12$$ $$q+2^{4}\zeta_{6}q^{5}+(12-12\zeta_{6})q^{7}+(2^{6}-2^{6}\zeta_{6})q^{11}+\cdots$$
648.4.i.m $4$ $38.233$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$0$$ $$-8$$ $$-24$$ $$q+(-4-4\beta _{1}-\beta _{3})q^{5}+(12\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
648.4.i.n $4$ $38.233$ $$\Q(\sqrt{-3}, \sqrt{-67})$$ None $$0$$ $$0$$ $$-8$$ $$30$$ $$q+(-4-4\beta _{1}+\beta _{3})q^{5}+(-15\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
648.4.i.o $4$ $38.233$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$0$$ $$-4$$ $$6$$ $$q+(-2\beta _{1}-\beta _{2})q^{5}+(3-3\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots$$
648.4.i.p $4$ $38.233$ $$\Q(\sqrt{-3}, \sqrt{-43})$$ None $$0$$ $$0$$ $$-4$$ $$6$$ $$q+(-2-2\beta _{1}+\beta _{3})q^{5}+(-3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
648.4.i.q $4$ $38.233$ $$\Q(\sqrt{-3}, \sqrt{-43})$$ None $$0$$ $$0$$ $$4$$ $$6$$ $$q+(2+2\beta _{1}+\beta _{3})q^{5}+(-3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
648.4.i.r $4$ $38.233$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$0$$ $$0$$ $$4$$ $$6$$ $$q+(2\beta _{1}+\beta _{2})q^{5}+(3-3\beta _{1}+2\beta _{2}+2\beta _{3})q^{7}+\cdots$$
648.4.i.s $4$ $38.233$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$0$$ $$8$$ $$-24$$ $$q+(4+4\beta _{1}+\beta _{3})q^{5}+(12\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots$$
648.4.i.t $4$ $38.233$ $$\Q(\sqrt{-3}, \sqrt{-67})$$ None $$0$$ $$0$$ $$8$$ $$30$$ $$q+(4+4\beta _{1}-\beta _{3})q^{5}+(-15\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
648.4.i.u $8$ $38.233$ 8.0.897122304.10 None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+(2\beta _{1}+\beta _{7})q^{5}+(\beta _{2}+\beta _{3}-\beta _{5}-\beta _{7})q^{7}+\cdots$$
648.4.i.v $8$ $38.233$ 8.0.897122304.10 None $$0$$ $$0$$ $$8$$ $$0$$ $$q+(2\beta _{1}+\beta _{7})q^{5}+(-\beta _{2}-\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(648, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(648, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(162, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(216, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(324, [\chi])$$$$^{\oplus 2}$$