# Properties

 Label 1008.2.r Level $1008$ Weight $2$ Character orbit 1008.r Rep. character $\chi_{1008}(337,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $72$ Newform subspaces $14$ Sturm bound $384$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1008.r (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$14$$ Sturm bound: $$384$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1008, [\chi])$$.

Total New Old
Modular forms 408 72 336
Cusp forms 360 72 288
Eisenstein series 48 0 48

## Trace form

 $$72 q + 4 q^{9} + O(q^{10})$$ $$72 q + 4 q^{9} - 12 q^{11} - 12 q^{15} + 8 q^{17} + 12 q^{23} - 36 q^{25} + 36 q^{27} + 12 q^{33} + 24 q^{35} + 36 q^{39} + 4 q^{41} + 12 q^{47} - 36 q^{49} - 12 q^{51} - 4 q^{57} - 12 q^{59} - 8 q^{65} - 80 q^{71} + 24 q^{73} - 68 q^{75} - 20 q^{81} + 4 q^{87} - 16 q^{89} - 24 q^{93} + 32 q^{95} - 12 q^{97} + 20 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1008, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1008.2.r.a $2$ $8.049$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-2$$ $$-1$$ $$q+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots$$
1008.2.r.b $2$ $8.049$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-2$$ $$1$$ $$q+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots$$
1008.2.r.c $2$ $8.049$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$1$$ $$q+(-1+2\zeta_{6})q^{3}+\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots$$
1008.2.r.d $2$ $8.049$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$3$$ $$1$$ $$q+(1-2\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots$$
1008.2.r.e $4$ $8.049$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$-4$$ $$-2$$ $$-2$$ $$q+(-1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots$$
1008.2.r.f $4$ $8.049$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$1$$ $$-3$$ $$2$$ $$q+(\beta _{2}+\beta _{3})q^{3}+(1-2\beta _{1}-2\beta _{2}+\beta _{3})q^{5}+\cdots$$
1008.2.r.g $6$ $8.049$ 6.0.309123.1 None $$0$$ $$-2$$ $$3$$ $$3$$ $$q+(-1+\beta _{2}+\beta _{3}+\beta _{4})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
1008.2.r.h $6$ $8.049$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$-3$$ $$3$$ $$q+(\zeta_{18}^{2}+\zeta_{18}^{4})q^{3}+(-1+\zeta_{18}+\zeta_{18}^{5})q^{5}+\cdots$$
1008.2.r.i $6$ $8.049$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$3$$ $$3$$ $$q+(\zeta_{18}^{4}-\zeta_{18}^{5})q^{3}+(1-\zeta_{18}-\zeta_{18}^{2}+\cdots)q^{5}+\cdots$$
1008.2.r.j $6$ $8.049$ 6.0.309123.1 None $$0$$ $$2$$ $$-1$$ $$-3$$ $$q+(1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots$$
1008.2.r.k $6$ $8.049$ 6.0.309123.1 None $$0$$ $$4$$ $$5$$ $$-3$$ $$q+(1-\beta _{5})q^{3}+(2+\beta _{2}+2\beta _{4}-\beta _{5})q^{5}+\cdots$$
1008.2.r.l $8$ $8.049$ 8.0.2091141441.1 None $$0$$ $$1$$ $$-3$$ $$-4$$ $$q+(-\beta _{1}+\beta _{6})q^{3}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$
1008.2.r.m $8$ $8.049$ 8.0.508277025.1 None $$0$$ $$4$$ $$4$$ $$4$$ $$q+(1-\beta _{2})q^{3}+(\beta _{2}-\beta _{3}+\beta _{4}+2\beta _{5}+\cdots)q^{5}+\cdots$$
1008.2.r.n $10$ $8.049$ 10.0.$$\cdots$$.1 None $$0$$ $$0$$ $$-3$$ $$-5$$ $$q-\beta _{6}q^{3}+(-\beta _{2}-\beta _{7})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1008, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1008, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(144, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 2}$$