# Properties

 Label 336.4.bc Level $336$ Weight $4$ Character orbit 336.bc Rep. character $\chi_{336}(17,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $92$ Newform subspaces $6$ Sturm bound $256$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.bc (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ Sturm bound: $$256$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(336, [\chi])$$.

Total New Old
Modular forms 408 100 308
Cusp forms 360 92 268
Eisenstein series 48 8 40

## Trace form

 $$92 q + 3 q^{3} - 14 q^{7} - q^{9} + O(q^{10})$$ $$92 q + 3 q^{3} - 14 q^{7} - q^{9} - 50 q^{15} + 276 q^{19} - 97 q^{21} - 868 q^{25} - 624 q^{31} - 3 q^{33} - 2 q^{37} - 302 q^{39} - 580 q^{43} - 663 q^{45} - 388 q^{49} + 141 q^{51} + 790 q^{57} - 6 q^{61} - 1207 q^{63} - 1192 q^{67} - 330 q^{73} + 222 q^{75} + 1124 q^{79} + 459 q^{81} - 804 q^{85} + 3756 q^{87} - 3822 q^{91} + q^{93} - 3566 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(336, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
336.4.bc.a $2$ $19.825$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-9$$ $$0$$ $$-37$$ $$q+(-3-3\zeta_{6})q^{3}+(-18-\zeta_{6})q^{7}+\cdots$$
336.4.bc.b $2$ $19.825$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$9$$ $$0$$ $$17$$ $$q+(3+3\zeta_{6})q^{3}+(18-19\zeta_{6})q^{7}+3^{3}\zeta_{6}q^{9}+\cdots$$
336.4.bc.c $12$ $19.825$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$42$$ $$q+(-\beta _{3}+\beta _{4})q^{3}-\beta _{11}q^{5}+(2-2\beta _{1}+\cdots)q^{7}+\cdots$$
336.4.bc.d $12$ $19.825$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$3$$ $$0$$ $$56$$ $$q+\beta _{1}q^{3}+\beta _{4}q^{5}+(4+2\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots$$
336.4.bc.e $16$ $19.825$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-80$$ $$q-\beta _{4}q^{3}-\beta _{9}q^{5}+(-6-\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots$$
336.4.bc.f $48$ $19.825$ None $$0$$ $$0$$ $$0$$ $$-12$$

## Decomposition of $$S_{4}^{\mathrm{old}}(336, [\chi])$$ into lower level spaces

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