# Properties

 Label 448.3.s Level $448$ Weight $3$ Character orbit 448.s Rep. character $\chi_{448}(129,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $60$ Newform subspaces $8$ Sturm bound $192$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 448.s (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$8$$ Sturm bound: $$192$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(448, [\chi])$$.

Total New Old
Modular forms 280 68 212
Cusp forms 232 60 172
Eisenstein series 48 8 40

## Trace form

 $$60 q + 6 q^{5} + 76 q^{9} + O(q^{10})$$ $$60 q + 6 q^{5} + 76 q^{9} - 6 q^{17} + 118 q^{21} + 108 q^{25} + 40 q^{29} - 6 q^{33} - 30 q^{37} + 60 q^{45} - 4 q^{49} + 82 q^{53} - 44 q^{57} + 6 q^{61} + 48 q^{65} - 6 q^{73} - 46 q^{77} - 142 q^{81} + 396 q^{85} + 282 q^{89} + 182 q^{93} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(448, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
448.3.s.a $2$ $12.207$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-3$$ $$-14$$ $$q+(-1-\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}-7q^{7}+\cdots$$
448.3.s.b $2$ $12.207$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-3$$ $$14$$ $$q+(1+\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+7q^{7}+\cdots$$
448.3.s.c $4$ $12.207$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$-6$$ $$6$$ $$-8$$ $$q+(-2-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots$$
448.3.s.d $4$ $12.207$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$6$$ $$6$$ $$8$$ $$q+(2+\beta _{1}+\beta _{3})q^{3}+(1-\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots$$
448.3.s.e $8$ $12.207$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{2}q^{3}+\beta _{7}q^{5}+(-2+3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
448.3.s.f $8$ $12.207$ 8.0.$$\cdots$$.2 None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{5}q^{3}-\beta _{4}q^{5}+(-1+3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
448.3.s.g $16$ $12.207$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}-\beta _{10}q^{5}+\beta _{13}q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots$$
448.3.s.h $16$ $12.207$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+\beta _{5}q^{5}+(-\beta _{2}-\beta _{12})q^{7}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(448, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(448, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(224, [\chi])$$$$^{\oplus 2}$$