# Properties

 Label 378.4.g Level $378$ Weight $4$ Character orbit 378.g Rep. character $\chi_{378}(109,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $64$ Newform subspaces $8$ Sturm bound $288$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 378.g (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$8$$ Sturm bound: $$288$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(378, [\chi])$$.

Total New Old
Modular forms 456 64 392
Cusp forms 408 64 344
Eisenstein series 48 0 48

## Trace form

 $$64 q - 128 q^{4} + 52 q^{7} + O(q^{10})$$ $$64 q - 128 q^{4} + 52 q^{7} + 12 q^{10} + 236 q^{13} - 512 q^{16} + 266 q^{19} - 312 q^{22} - 806 q^{25} - 104 q^{28} + 110 q^{31} + 240 q^{34} - 262 q^{37} + 48 q^{40} - 3280 q^{43} - 96 q^{46} - 3104 q^{49} - 472 q^{52} + 744 q^{55} + 816 q^{58} + 836 q^{61} + 4096 q^{64} - 790 q^{67} + 780 q^{70} - 2578 q^{73} - 2128 q^{76} - 2254 q^{79} - 1056 q^{82} + 8448 q^{85} + 624 q^{88} - 2692 q^{91} + 2544 q^{94} + 1124 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(378, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
378.4.g.a $6$ $22.303$ 6.0.$$\cdots$$.2 None $$-6$$ $$0$$ $$-5$$ $$8$$ $$q-2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots$$
378.4.g.b $6$ $22.303$ 6.0.$$\cdots$$.2 None $$6$$ $$0$$ $$5$$ $$8$$ $$q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(2\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots$$
378.4.g.c $8$ $22.303$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-8$$ $$0$$ $$2$$ $$6$$ $$q+(-2+2\beta _{1})q^{2}-4\beta _{1}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
378.4.g.d $8$ $22.303$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-8$$ $$0$$ $$4$$ $$25$$ $$q+2\beta _{1}q^{2}+(-4-4\beta _{1})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
378.4.g.e $8$ $22.303$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$8$$ $$0$$ $$-4$$ $$25$$ $$q-2\beta _{1}q^{2}+(-4-4\beta _{1})q^{4}+(\beta _{1}+\beta _{6}+\cdots)q^{5}+\cdots$$
378.4.g.f $8$ $22.303$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$8$$ $$0$$ $$-2$$ $$6$$ $$q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
378.4.g.g $10$ $22.303$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-10$$ $$0$$ $$2$$ $$-13$$ $$q+(-2-2\beta _{2})q^{2}+4\beta _{2}q^{4}+\beta _{5}q^{5}+\cdots$$
378.4.g.h $10$ $22.303$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$10$$ $$0$$ $$-2$$ $$-13$$ $$q+(2+2\beta _{2})q^{2}+4\beta _{2}q^{4}-\beta _{5}q^{5}+(3\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(378, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(378, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(189, [\chi])$$$$^{\oplus 2}$$