# Properties

 Label 1728.2.s Level $1728$ Weight $2$ Character orbit 1728.s Rep. character $\chi_{1728}(575,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $44$ Newform subspaces $7$ Sturm bound $576$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.s (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$36$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$7$$ Sturm bound: $$576$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1728, [\chi])$$.

Total New Old
Modular forms 648 52 596
Cusp forms 504 44 460
Eisenstein series 144 8 136

## Trace form

 $$44 q - 6 q^{5} + O(q^{10})$$ $$44 q - 6 q^{5} + 2 q^{13} + 12 q^{25} - 6 q^{29} + 8 q^{37} + 30 q^{41} + 8 q^{49} + 2 q^{61} + 6 q^{65} - 8 q^{73} - 6 q^{77} - 8 q^{85} - 2 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1728, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1728.2.s.a $2$ $13.798$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-6$$ $$-6$$ $$q+(-4+2\zeta_{6})q^{5}+(-2-2\zeta_{6})q^{7}+\cdots$$
1728.2.s.b $2$ $13.798$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-6$$ $$6$$ $$q+(-4+2\zeta_{6})q^{5}+(2+2\zeta_{6})q^{7}+(3+\cdots)q^{11}+\cdots$$
1728.2.s.c $2$ $13.798$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$3$$ $$-3$$ $$q+(2-\zeta_{6})q^{5}+(-1-\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots$$
1728.2.s.d $2$ $13.798$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$3$$ $$3$$ $$q+(2-\zeta_{6})q^{5}+(1+\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots$$
1728.2.s.e $4$ $13.798$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q+(2-\zeta_{12}^{2})q^{5}-\zeta_{12}q^{7}+(-\zeta_{12}+\cdots)q^{11}+\cdots$$
1728.2.s.f $8$ $13.798$ 8.0.170772624.1 None $$0$$ $$0$$ $$-6$$ $$0$$ $$q+(-1-\beta _{2})q^{5}+(\beta _{1}-\beta _{6})q^{7}+(-\beta _{3}+\cdots)q^{11}+\cdots$$
1728.2.s.g $24$ $13.798$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1728, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1728, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(144, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(288, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(432, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(576, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(864, [\chi])$$$$^{\oplus 2}$$