# Properties

 Label 350.2.e Level $350$ Weight $2$ Character orbit 350.e Rep. character $\chi_{350}(51,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $24$ Newform subspaces $12$ Sturm bound $120$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$12$$ Sturm bound: $$120$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$11$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(350, [\chi])$$.

Total New Old
Modular forms 144 24 120
Cusp forms 96 24 72
Eisenstein series 48 0 48

## Trace form

 $$24 q - 12 q^{4} + 8 q^{7} - 8 q^{9} + O(q^{10})$$ $$24 q - 12 q^{4} + 8 q^{7} - 8 q^{9} - 4 q^{14} - 12 q^{16} - 4 q^{17} + 8 q^{18} + 4 q^{19} + 4 q^{21} - 8 q^{22} + 12 q^{23} + 8 q^{26} - 24 q^{27} - 4 q^{28} - 8 q^{29} - 20 q^{31} - 24 q^{33} - 16 q^{34} + 16 q^{36} - 4 q^{37} - 8 q^{38} + 8 q^{39} - 24 q^{41} - 4 q^{42} + 40 q^{43} - 8 q^{46} + 20 q^{47} - 12 q^{51} + 4 q^{53} + 12 q^{54} - 4 q^{56} + 24 q^{57} + 4 q^{59} + 32 q^{61} + 48 q^{62} - 36 q^{63} + 24 q^{64} - 4 q^{68} + 48 q^{69} - 24 q^{71} + 8 q^{72} - 28 q^{73} + 20 q^{74} - 8 q^{76} + 8 q^{77} + 16 q^{78} + 12 q^{79} - 20 q^{81} - 16 q^{82} + 32 q^{83} + 28 q^{84} - 8 q^{86} + 24 q^{87} + 4 q^{88} - 8 q^{89} + 28 q^{91} - 24 q^{92} - 12 q^{93} + 4 q^{94} - 64 q^{97} - 24 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(350, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.2.e.a $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-2$$ $$0$$ $$-5$$ $$q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
350.2.e.b $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-2$$ $$0$$ $$4$$ $$q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
350.2.e.c $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots$$
350.2.e.d $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$1$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(2-3\zeta_{6})q^{7}+\cdots$$
350.2.e.e $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$0$$ $$1$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
350.2.e.f $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$3$$ $$0$$ $$5$$ $$q-\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
350.2.e.g $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-3$$ $$0$$ $$-5$$ $$q+\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
350.2.e.h $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-2$$ $$0$$ $$4$$ $$q+\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
350.2.e.i $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-1$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-2+3\zeta_{6})q^{7}+\cdots$$
350.2.e.j $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$4$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(3-2\zeta_{6})q^{7}+\cdots$$
350.2.e.k $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$1$$ $$2$$ $$0$$ $$5$$ $$q+\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
350.2.e.l $2$ $2.795$ $$\Q(\sqrt{-3})$$ None $$1$$ $$3$$ $$0$$ $$-1$$ $$q+\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(350, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(350, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$