# Properties

 Label 378.2.h Level $378$ Weight $2$ Character orbit 378.h Rep. character $\chi_{378}(289,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $16$ Newform subspaces $4$ Sturm bound $144$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.h (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$63$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$4$$ Sturm bound: $$144$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(378, [\chi])$$.

Total New Old
Modular forms 168 16 152
Cusp forms 120 16 104
Eisenstein series 48 0 48

## Trace form

 $$16 q - 8 q^{4} - 8 q^{5} - 2 q^{7} + O(q^{10})$$ $$16 q - 8 q^{4} - 8 q^{5} - 2 q^{7} + 8 q^{11} + 2 q^{13} - 2 q^{14} - 8 q^{16} + 14 q^{17} - 4 q^{19} + 4 q^{20} - 4 q^{23} + 16 q^{25} + 16 q^{26} - 2 q^{28} + 10 q^{29} + 2 q^{31} + 14 q^{35} + 2 q^{37} - 24 q^{38} + 6 q^{41} + 2 q^{43} - 4 q^{44} - 6 q^{46} + 6 q^{47} - 14 q^{49} + 4 q^{50} - 4 q^{52} - 24 q^{53} - 12 q^{55} + 4 q^{56} - 12 q^{58} + 22 q^{59} + 8 q^{61} - 44 q^{62} + 16 q^{64} + 6 q^{65} + 14 q^{67} - 28 q^{68} - 52 q^{71} - 28 q^{73} - 12 q^{74} - 4 q^{76} + 34 q^{77} + 20 q^{79} + 4 q^{80} - 16 q^{83} + 12 q^{85} + 24 q^{86} + 36 q^{89} - 16 q^{91} + 2 q^{92} + 12 q^{94} - 34 q^{95} + 2 q^{97} + 24 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(378, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
378.2.h.a $2$ $3.018$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-6$$ $$5$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-3q^{5}+(3+\cdots)q^{7}+\cdots$$
378.2.h.b $2$ $3.018$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$6$$ $$-1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+3q^{5}+(1-3\zeta_{6})q^{7}+\cdots$$
378.2.h.c $6$ $3.018$ 6.0.309123.1 None $$-3$$ $$0$$ $$2$$ $$-4$$ $$q+(-1+\beta _{4})q^{2}-\beta _{4}q^{4}-\beta _{3}q^{5}+(-1+\cdots)q^{7}+\cdots$$
378.2.h.d $6$ $3.018$ 6.0.309123.1 None $$3$$ $$0$$ $$-10$$ $$-2$$ $$q+\beta _{4}q^{2}+(-1+\beta _{4})q^{4}+(-2-\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(378, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(378, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(189, [\chi])$$$$^{\oplus 2}$$