# Properties

 Label 378.2.u Level $378$ Weight $2$ Character orbit 378.u Rep. character $\chi_{378}(43,\cdot)$ Character field $\Q(\zeta_{9})$ Dimension $108$ Newform subspaces $5$ Sturm bound $144$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.u (of order $$9$$ and degree $$6$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$27$$ Character field: $$\Q(\zeta_{9})$$ Newform subspaces: $$5$$ Sturm bound: $$144$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

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

Total New Old
Modular forms 456 108 348
Cusp forms 408 108 300
Eisenstein series 48 0 48

## Trace form

 $$108 q + 12 q^{5} + 6 q^{8} + 12 q^{9} + O(q^{10})$$ $$108 q + 12 q^{5} + 6 q^{8} + 12 q^{9} + 18 q^{11} + 6 q^{12} + 24 q^{15} - 12 q^{18} - 24 q^{20} + 18 q^{22} + 12 q^{23} + 36 q^{25} - 54 q^{27} - 72 q^{29} + 36 q^{31} + 18 q^{33} + 18 q^{34} + 12 q^{35} - 24 q^{36} + 24 q^{38} + 12 q^{39} + 54 q^{41} + 18 q^{43} + 12 q^{44} - 12 q^{45} + 72 q^{47} + 12 q^{48} + 12 q^{50} - 36 q^{51} - 72 q^{53} + 36 q^{54} - 42 q^{57} - 72 q^{59} - 54 q^{64} - 84 q^{65} - 54 q^{67} - 12 q^{68} - 96 q^{69} - 24 q^{71} - 24 q^{72} - 96 q^{74} - 120 q^{75} - 36 q^{76} - 36 q^{78} - 72 q^{79} - 24 q^{81} + 12 q^{84} - 72 q^{85} - 6 q^{86} + 60 q^{87} - 36 q^{88} - 66 q^{89} - 36 q^{90} - 24 q^{92} - 12 q^{93} + 12 q^{95} - 12 q^{96} - 54 q^{97} + 6 q^{98} + 72 q^{99} + O(q^{100})$$

## 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.u.a $6$ $3.018$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$3$$ $$0$$ $$q-\zeta_{18}q^{2}+(\zeta_{18}-2\zeta_{18}^{4})q^{3}+\zeta_{18}^{2}q^{4}+\cdots$$
378.2.u.b $12$ $3.018$ 12.0.$$\cdots$$.1 None $$0$$ $$3$$ $$0$$ $$0$$ $$q+\beta _{7}q^{2}+(-\beta _{2}+\beta _{4}+\beta _{5}-\beta _{10}+\cdots)q^{3}+\cdots$$
378.2.u.c $24$ $3.018$ None $$0$$ $$-3$$ $$3$$ $$0$$
378.2.u.d $30$ $3.018$ None $$0$$ $$-3$$ $$3$$ $$0$$
378.2.u.e $36$ $3.018$ None $$0$$ $$3$$ $$3$$ $$0$$

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

$$S_{2}^{\mathrm{old}}(378, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(189, [\chi])$$$$^{\oplus 2}$$