# Properties

 Label 168.4.q Level $168$ Weight $4$ Character orbit 168.q Rep. character $\chi_{168}(25,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $24$ Newform subspaces $6$ Sturm bound $128$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 168.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$6$$ Sturm bound: $$128$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(168, [\chi])$$.

Total New Old
Modular forms 208 24 184
Cusp forms 176 24 152
Eisenstein series 32 0 32

## Trace form

 $$24q - 6q^{3} + 24q^{7} - 108q^{9} + O(q^{10})$$ $$24q - 6q^{3} + 24q^{7} - 108q^{9} - 28q^{11} - 140q^{13} - 84q^{15} - 4q^{17} - 250q^{19} - 60q^{21} + 84q^{23} - 138q^{25} + 108q^{27} - 280q^{29} + 168q^{31} + 114q^{33} - 708q^{35} - 318q^{37} - 210q^{39} - 888q^{41} - 116q^{43} + 268q^{49} + 60q^{51} + 1180q^{53} + 3468q^{55} + 612q^{57} - 768q^{59} + 1572q^{61} + 54q^{63} - 992q^{65} - 1014q^{67} - 1704q^{69} - 1072q^{71} + 1550q^{73} - 1362q^{75} - 1140q^{77} + 1268q^{79} - 972q^{81} + 9032q^{83} + 4360q^{85} + 474q^{87} + 2368q^{89} - 450q^{91} - 1146q^{93} - 3392q^{95} - 5860q^{97} + 504q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(168, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
168.4.q.a $$2$$ $$9.912$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-7$$ $$-35$$ $$q+3\zeta_{6}q^{3}+(-7+7\zeta_{6})q^{5}+(-14+\cdots)q^{7}+\cdots$$
168.4.q.b $$2$$ $$9.912$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-2$$ $$35$$ $$q+3\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{5}+(21-7\zeta_{6})q^{7}+\cdots$$
168.4.q.c $$2$$ $$9.912$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$11$$ $$7$$ $$q+3\zeta_{6}q^{3}+(11-11\zeta_{6})q^{5}+(14-21\zeta_{6})q^{7}+\cdots$$
168.4.q.d $$4$$ $$9.912$$ $$\Q(\sqrt{-3}, \sqrt{505})$$ None $$0$$ $$6$$ $$-9$$ $$0$$ $$q+(3-3\beta _{2})q^{3}+(\beta _{1}-5\beta _{2})q^{5}+(8-17\beta _{2}+\cdots)q^{7}+\cdots$$
168.4.q.e $$6$$ $$9.912$$ 6.0.$$\cdots$$.1 None $$0$$ $$-9$$ $$11$$ $$-1$$ $$q+(-3+3\beta _{3})q^{3}+(4\beta _{3}-\beta _{4}-\beta _{5})q^{5}+\cdots$$
168.4.q.f $$8$$ $$9.912$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ 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 $$S_{4}^{\mathrm{old}}(168, [\chi])$$ into lower level spaces

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