# Properties

 Label 336.4.q Level $336$ Weight $4$ Character orbit 336.q Rep. character $\chi_{336}(193,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $48$ Newform subspaces $13$ Sturm bound $256$ Trace bound $5$

# Related objects

## Defining parameters

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

## Dimensions

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

Total New Old
Modular forms 408 48 360
Cusp forms 360 48 312
Eisenstein series 48 0 48

## Trace form

 $$48 q + 6 q^{3} - 18 q^{7} - 216 q^{9} + O(q^{10})$$ $$48 q + 6 q^{3} - 18 q^{7} - 216 q^{9} + 20 q^{11} + 318 q^{19} + 80 q^{23} - 516 q^{25} - 108 q^{27} - 400 q^{29} - 402 q^{31} - 12 q^{33} + 468 q^{35} + 8 q^{37} + 210 q^{39} - 592 q^{41} - 892 q^{43} - 204 q^{47} + 592 q^{49} + 16 q^{53} - 376 q^{55} + 168 q^{57} + 688 q^{59} + 200 q^{61} - 162 q^{63} - 280 q^{65} - 750 q^{67} - 1704 q^{71} + 1092 q^{73} + 1050 q^{75} + 184 q^{77} - 942 q^{79} - 1944 q^{81} - 4152 q^{83} + 1296 q^{85} - 1044 q^{87} + 1712 q^{89} - 1138 q^{91} - 456 q^{93} - 36 q^{95} + 2328 q^{97} - 360 q^{99} + O(q^{100})$$

## 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.q.a $2$ $19.825$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-7$$ $$35$$ $$q+(-3+3\zeta_{6})q^{3}-7\zeta_{6}q^{5}+(21-7\zeta_{6})q^{7}+\cdots$$
336.4.q.b $2$ $19.825$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-2$$ $$-35$$ $$q+(-3+3\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-14+\cdots)q^{7}+\cdots$$
336.4.q.c $2$ $19.825$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$11$$ $$-7$$ $$q+(-3+3\zeta_{6})q^{3}+11\zeta_{6}q^{5}+(7-21\zeta_{6})q^{7}+\cdots$$
336.4.q.d $2$ $19.825$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$15$$ $$-35$$ $$q+(-3+3\zeta_{6})q^{3}+15\zeta_{6}q^{5}+(-21+\cdots)q^{7}+\cdots$$
336.4.q.e $2$ $19.825$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$3$$ $$7$$ $$q+(3-3\zeta_{6})q^{3}+3\zeta_{6}q^{5}+(-7+21\zeta_{6})q^{7}+\cdots$$
336.4.q.f $2$ $19.825$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$6$$ $$7$$ $$q+(3-3\zeta_{6})q^{3}+6\zeta_{6}q^{5}+(14-21\zeta_{6})q^{7}+\cdots$$
336.4.q.g $4$ $19.825$ $$\Q(\sqrt{-3}, \sqrt{505})$$ None $$0$$ $$-6$$ $$-9$$ $$0$$ $$q+(-3+3\beta _{2})q^{3}+(-\beta _{1}-4\beta _{2})q^{5}+\cdots$$
336.4.q.h $4$ $19.825$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$-6$$ $$3$$ $$20$$ $$q+3\beta _{2}q^{3}+(1+2\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+\cdots$$
336.4.q.i $4$ $19.825$ $$\Q(\sqrt{-3}, \sqrt{193})$$ None $$0$$ $$6$$ $$-11$$ $$-6$$ $$q+(3-3\beta _{2})q^{3}+(\beta _{1}-6\beta _{2})q^{5}+(1-2\beta _{1}+\cdots)q^{7}+\cdots$$
336.4.q.j $4$ $19.825$ $$\Q(\sqrt{-3}, \sqrt{1345})$$ None $$0$$ $$6$$ $$-5$$ $$0$$ $$q+(3-3\beta _{2})q^{3}+(-\beta _{1}-2\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots$$
336.4.q.k $6$ $19.825$ 6.0.9924270768.1 None $$0$$ $$-9$$ $$-11$$ $$13$$ $$q+3\beta _{3}q^{3}+(-4+\beta _{2}-4\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots$$
336.4.q.l $6$ $19.825$ 6.0.$$\cdots$$.1 None $$0$$ $$9$$ $$11$$ $$1$$ $$q+(3-3\beta _{3})q^{3}+(4\beta _{3}-\beta _{4}-\beta _{5})q^{5}+\cdots$$
336.4.q.m $8$ $19.825$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$12$$ $$-4$$ $$-18$$ $$q-3\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{6})q^{5}+(-2+\cdots)q^{7}+\cdots$$

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

$$S_{4}^{\mathrm{old}}(336, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$