# Properties

 Label 392.2.i Level $392$ Weight $2$ Character orbit 392.i Rep. character $\chi_{392}(177,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $20$ Newform subspaces $8$ Sturm bound $112$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 392.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$8$$ Sturm bound: $$112$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(392, [\chi])$$.

Total New Old
Modular forms 144 20 124
Cusp forms 80 20 60
Eisenstein series 64 0 64

## Trace form

 $$20 q + 2 q^{3} - 2 q^{5} - 8 q^{9} + O(q^{10})$$ $$20 q + 2 q^{3} - 2 q^{5} - 8 q^{9} + 2 q^{11} + 8 q^{13} + 12 q^{15} - 2 q^{17} + 6 q^{19} - 14 q^{23} - 24 q^{25} - 28 q^{27} + 8 q^{29} - 6 q^{31} - 6 q^{33} - 6 q^{37} + 20 q^{39} + 24 q^{41} + 24 q^{43} - 4 q^{45} + 6 q^{47} + 22 q^{51} + 10 q^{53} + 4 q^{55} - 44 q^{57} + 18 q^{59} - 2 q^{61} + 36 q^{65} - 2 q^{67} + 4 q^{69} - 48 q^{71} + 14 q^{73} - 8 q^{75} - 14 q^{79} + 2 q^{81} - 16 q^{83} - 60 q^{85} - 20 q^{87} - 2 q^{89} + 6 q^{93} - 2 q^{95} - 8 q^{97} - 80 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(392, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
392.2.i.a $2$ $3.130$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$4$$ $$0$$ $$q+(-2+2\zeta_{6})q^{3}+4\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots$$
392.2.i.b $2$ $3.130$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-1$$ $$0$$ $$q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots$$
392.2.i.c $2$ $3.130$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots$$
392.2.i.d $2$ $3.130$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots$$
392.2.i.e $2$ $3.130$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$-4$$ $$0$$ $$q+(2-2\zeta_{6})q^{3}-4\zeta_{6}q^{5}-\zeta_{6}q^{9}-8q^{15}+\cdots$$
392.2.i.f $2$ $3.130$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-1$$ $$0$$ $$q+(3-3\zeta_{6})q^{3}-\zeta_{6}q^{5}-6\zeta_{6}q^{9}+(1+\cdots)q^{11}+\cdots$$
392.2.i.g $4$ $3.130$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+5\beta _{2}q^{9}+\cdots$$
392.2.i.h $4$ $3.130$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+(-2\beta _{1}-2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(392, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(392, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(98, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(196, [\chi])$$$$^{\oplus 2}$$