# Properties

 Label 1008.3.cg Level $1008$ Weight $3$ Character orbit 1008.cg Rep. character $\chi_{1008}(145,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $78$ Newform subspaces $17$ Sturm bound $576$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1008.cg (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$17$$ Sturm bound: $$576$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1008, [\chi])$$.

Total New Old
Modular forms 816 82 734
Cusp forms 720 78 642
Eisenstein series 96 4 92

## Trace form

 $$78 q + 3 q^{5} - 6 q^{7} + O(q^{10})$$ $$78 q + 3 q^{5} - 6 q^{7} - 9 q^{11} + 3 q^{17} - 45 q^{19} + 7 q^{23} + 182 q^{25} + 36 q^{29} - 45 q^{31} - 147 q^{35} - 25 q^{37} - 108 q^{43} - 147 q^{47} - 98 q^{49} - 47 q^{53} + 93 q^{59} + 69 q^{61} - 32 q^{65} + 33 q^{67} - 100 q^{71} - 123 q^{73} + 89 q^{77} - 71 q^{79} - 86 q^{85} + 75 q^{89} + 336 q^{91} + 117 q^{95} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1008, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1008.3.cg.a $2$ $27.466$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-9$$ $$-13$$ $$q+(-6+3\zeta_{6})q^{5}+(-5-3\zeta_{6})q^{7}+\cdots$$
1008.3.cg.b $2$ $27.466$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-3$$ $$-13$$ $$q+(-2+\zeta_{6})q^{5}+(-5-3\zeta_{6})q^{7}+(3+\cdots)q^{11}+\cdots$$
1008.3.cg.c $2$ $27.466$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-3$$ $$14$$ $$q+(-2+\zeta_{6})q^{5}+7q^{7}+(-15+15\zeta_{6})q^{11}+\cdots$$
1008.3.cg.d $2$ $27.466$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-13$$ $$q+(-8+3\zeta_{6})q^{7}+(-7+14\zeta_{6})q^{13}+\cdots$$
1008.3.cg.e $2$ $27.466$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$11$$ $$q+(8-5\zeta_{6})q^{7}+(-15+30\zeta_{6})q^{13}+\cdots$$
1008.3.cg.f $2$ $27.466$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$7$$ $$q+(4-2\zeta_{6})q^{5}+7\zeta_{6}q^{7}+(-10+10\zeta_{6})q^{11}+\cdots$$
1008.3.cg.g $2$ $27.466$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$9$$ $$7$$ $$q+(6-3\zeta_{6})q^{5}+(7-7\zeta_{6})q^{7}+(11-11\zeta_{6})q^{11}+\cdots$$
1008.3.cg.h $4$ $27.466$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$-12$$ $$10$$ $$q+(-4-2\beta _{1}+\beta _{3})q^{5}+(-5\beta _{1}-\beta _{3})q^{7}+\cdots$$
1008.3.cg.i $4$ $27.466$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-14$$ $$q+\beta _{3}q^{5}+7\beta _{1}q^{7}+(-2\beta _{2}-2\beta _{3})q^{11}+\cdots$$
1008.3.cg.j $4$ $27.466$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ None $$0$$ $$0$$ $$0$$ $$-14$$ $$q+\beta _{2}q^{5}+(-7-7\beta _{1})q^{7}+(2\beta _{2}-\beta _{3})q^{11}+\cdots$$
1008.3.cg.k $4$ $27.466$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$26$$ $$q+(-\beta _{2}+\beta _{3})q^{5}+(8-3\beta _{1})q^{7}+\beta _{2}q^{11}+\cdots$$
1008.3.cg.l $4$ $27.466$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$-8$$ $$q+(2+\beta _{1}+2\beta _{3})q^{5}+(-1+2\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
1008.3.cg.m $4$ $27.466$ $$\Q(\sqrt{-3}, \sqrt{65})$$ None $$0$$ $$0$$ $$9$$ $$-2$$ $$q+(2+\beta _{1}-\beta _{3})q^{5}+(-1+\beta _{2})q^{7}+\cdots$$
1008.3.cg.n $8$ $27.466$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$-6$$ $$4$$ $$q+(-1-\beta _{4})q^{5}+(-\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots$$
1008.3.cg.o $8$ $27.466$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{5}q^{5}+(2-\beta _{1}-3\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots$$
1008.3.cg.p $8$ $27.466$ 8.0.$$\cdots$$.9 None $$0$$ $$0$$ $$6$$ $$-8$$ $$q+(1-\beta _{1}+\beta _{6})q^{5}+(-1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
1008.3.cg.q $16$ $27.466$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{7}q^{5}+(-1+\beta _{2}-\beta _{5})q^{7}+(\beta _{4}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1008, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1008, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(336, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(504, [\chi])$$$$^{\oplus 2}$$