# Properties

 Label 336.3.bh Level $336$ Weight $3$ Character orbit 336.bh Rep. character $\chi_{336}(145,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $32$ Newform subspaces $8$ Sturm bound $192$ Trace bound $5$

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## Defining parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 336.bh (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: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(336, [\chi])$$.

Total New Old
Modular forms 280 32 248
Cusp forms 232 32 200
Eisenstein series 48 0 48

## Trace form

 $$32 q - 8 q^{7} + 48 q^{9} + O(q^{10})$$ $$32 q - 8 q^{7} + 48 q^{9} + 16 q^{11} - 48 q^{19} - 16 q^{23} + 88 q^{25} + 32 q^{29} + 24 q^{31} - 72 q^{33} + 240 q^{35} + 48 q^{37} + 24 q^{39} + 128 q^{43} + 288 q^{47} + 48 q^{49} + 96 q^{53} - 48 q^{57} - 192 q^{59} - 144 q^{61} + 24 q^{63} - 80 q^{65} - 160 q^{67} - 384 q^{71} - 120 q^{73} - 80 q^{77} - 24 q^{79} - 144 q^{81} - 32 q^{85} + 48 q^{91} - 48 q^{93} + 96 q^{95} + 96 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(336, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
336.3.bh.a $2$ $9.155$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-9$$ $$7$$ $$q+(-1-\zeta_{6})q^{3}+(-6+3\zeta_{6})q^{5}+(7+\cdots)q^{7}+\cdots$$
336.3.bh.b $2$ $9.155$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$3$$ $$-13$$ $$q+(-1-\zeta_{6})q^{3}+(2-\zeta_{6})q^{5}+(-5+\cdots)q^{7}+\cdots$$
336.3.bh.c $2$ $9.155$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-6$$ $$7$$ $$q+(1+\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+7\zeta_{6}q^{7}+\cdots$$
336.3.bh.d $2$ $9.155$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$9$$ $$-13$$ $$q+(1+\zeta_{6})q^{3}+(6-3\zeta_{6})q^{5}+(-5-3\zeta_{6})q^{7}+\cdots$$
336.3.bh.e $4$ $9.155$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$-6$$ $$12$$ $$10$$ $$q+(-1+\beta _{1})q^{3}+(4+2\beta _{1}+\beta _{3})q^{5}+\cdots$$
336.3.bh.f $4$ $9.155$ $$\Q(\sqrt{-3}, \sqrt{65})$$ None $$0$$ $$6$$ $$-9$$ $$-2$$ $$q+(2-\beta _{1})q^{3}+(-1-2\beta _{1}-\beta _{3})q^{5}+\cdots$$
336.3.bh.g $8$ $9.155$ 8.0.$$\cdots$$.9 None $$0$$ $$-12$$ $$-6$$ $$-8$$ $$q+(-1-\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{6})q^{5}+\cdots$$
336.3.bh.h $8$ $9.155$ 8.0.$$\cdots$$.2 None $$0$$ $$12$$ $$6$$ $$4$$ $$q+(1-\beta _{3})q^{3}+(1+\beta _{4})q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(336, [\chi])$$ into lower level spaces

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