# Properties

 Label 1728.3.q Level $1728$ Weight $3$ Character orbit 1728.q Rep. character $\chi_{1728}(449,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $92$ Newform subspaces $12$ Sturm bound $864$ Trace bound $41$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.q (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$12$$ Sturm bound: $$864$$ Trace bound: $$41$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1728, [\chi])$$.

Total New Old
Modular forms 1224 100 1124
Cusp forms 1080 92 988
Eisenstein series 144 8 136

## Trace form

 $$92 q - 6 q^{5} + O(q^{10})$$ $$92 q - 6 q^{5} + 2 q^{13} + 188 q^{25} - 6 q^{29} + 8 q^{37} - 138 q^{41} - 240 q^{49} + 2 q^{61} + 6 q^{65} - 8 q^{73} - 6 q^{77} - 48 q^{85} - 2 q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1728, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1728.3.q.a $2$ $47.085$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$-2$$ $$q+(4-2\zeta_{6})q^{5}+(-2+2\zeta_{6})q^{7}+(-1+\cdots)q^{11}+\cdots$$
1728.3.q.b $2$ $47.085$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$2$$ $$q+(4-2\zeta_{6})q^{5}+(2-2\zeta_{6})q^{7}+(1+\zeta_{6})q^{11}+\cdots$$
1728.3.q.c $4$ $47.085$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$-18$$ $$-2$$ $$q+(-3-3\beta _{2})q^{5}+(\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots$$
1728.3.q.d $4$ $47.085$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$-18$$ $$2$$ $$q+(-3-3\beta _{2})q^{5}+(-\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots$$
1728.3.q.e $4$ $47.085$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$-6$$ $$q+(1-\beta _{1}+\beta _{2})q^{5}+(-\beta _{1}-3\beta _{2}-\beta _{3})q^{7}+\cdots$$
1728.3.q.f $4$ $47.085$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$6$$ $$q+(1-\beta _{1}+\beta _{2})q^{5}+(\beta _{1}+3\beta _{2}+\beta _{3})q^{7}+\cdots$$
1728.3.q.g $4$ $47.085$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$0$$ $$9$$ $$-1$$ $$q+(3-\beta _{1}+\beta _{2})q^{5}+(-1+2\beta _{1}-\beta _{3})q^{7}+\cdots$$
1728.3.q.h $4$ $47.085$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$0$$ $$9$$ $$1$$ $$q+(3-\beta _{1}+\beta _{2})q^{5}+(1-2\beta _{1}+\beta _{3})q^{7}+\cdots$$
1728.3.q.i $8$ $47.085$ 8.0.$$\cdots$$.9 None $$0$$ $$0$$ $$-6$$ $$-6$$ $$q+(-1+\beta _{1}+\beta _{2}-\beta _{4})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots$$
1728.3.q.j $8$ $47.085$ 8.0.$$\cdots$$.9 None $$0$$ $$0$$ $$-6$$ $$6$$ $$q+(-1+\beta _{1}+\beta _{2}-\beta _{4})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots$$
1728.3.q.k $24$ $47.085$ None $$0$$ $$0$$ $$0$$ $$0$$
1728.3.q.l $24$ $47.085$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1728, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1728, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 14}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 7}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(144, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(216, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(288, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(432, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(576, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(864, [\chi])$$$$^{\oplus 2}$$