# Properties

 Label 693.2.i Level $693$ Weight $2$ Character orbit 693.i Rep. character $\chi_{693}(100,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $68$ Newform subspaces $12$ Sturm bound $192$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$693 = 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 693.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$12$$ Sturm bound: $$192$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(693, [\chi])$$.

Total New Old
Modular forms 208 68 140
Cusp forms 176 68 108
Eisenstein series 32 0 32

## Trace form

 $$68 q - 36 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8} + O(q^{10})$$ $$68 q - 36 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8} + 6 q^{10} + 8 q^{13} + 4 q^{14} - 36 q^{16} - 6 q^{17} - 10 q^{19} - 8 q^{20} + 4 q^{23} - 42 q^{25} + 14 q^{26} - 64 q^{28} + 24 q^{29} + 6 q^{31} + 32 q^{32} + 48 q^{34} - 42 q^{35} - 4 q^{38} + 24 q^{40} + 24 q^{41} - 52 q^{43} + 4 q^{44} + 10 q^{46} + 22 q^{47} - 4 q^{49} - 88 q^{50} + 54 q^{52} - 10 q^{53} + 16 q^{55} - 30 q^{56} - 2 q^{58} + 40 q^{59} + 28 q^{61} + 68 q^{62} + 44 q^{64} - 10 q^{65} + 28 q^{67} - 52 q^{68} - 6 q^{70} - 12 q^{71} - 46 q^{73} - 8 q^{74} + 32 q^{76} - 2 q^{77} - 20 q^{79} - 2 q^{80} - 30 q^{82} + 84 q^{83} + 80 q^{85} + 6 q^{86} + 12 q^{88} + 6 q^{89} - 26 q^{91} - 124 q^{92} - 22 q^{94} - 24 q^{95} - 84 q^{97} + 18 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(693, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
693.2.i.a $2$ $5.534$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$0$$ $$0$$ $$5$$ $$q-2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+(2+\zeta_{6})q^{7}+\cdots$$
693.2.i.b $2$ $5.534$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$4$$ $$-4$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+4\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots$$
693.2.i.c $2$ $5.534$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$-5$$ $$q+(2-2\zeta_{6})q^{4}-4\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots$$
693.2.i.d $2$ $5.534$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$-5$$ $$q+(2-2\zeta_{6})q^{4}+4\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots$$
693.2.i.e $2$ $5.534$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-4$$ $$q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+(-1-2\zeta_{6})q^{7}+\cdots$$
693.2.i.f $4$ $5.534$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-2$$ $$0$$ $$-4$$ $$4$$ $$q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
693.2.i.g $6$ $5.534$ 6.0.1783323.2 None $$0$$ $$0$$ $$-2$$ $$2$$ $$q+(\beta _{3}-\beta _{5})q^{2}+(-1-\beta _{1}+\beta _{4}+\beta _{5})q^{4}+\cdots$$
693.2.i.h $6$ $5.534$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$q+(\zeta_{18}-\zeta_{18}^{2}-\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots$$
693.2.i.i $8$ $5.534$ 8.0.$$\cdots$$.5 None $$2$$ $$0$$ $$4$$ $$2$$ $$q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{6})q^{4}+\cdots$$
693.2.i.j $10$ $5.534$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$2$$ $$0$$ $$-4$$ $$-1$$ $$q+\beta _{3}q^{2}+(-2-2\beta _{2}-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots$$
693.2.i.k $12$ $5.534$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$-4$$ $$6$$ $$q+\beta _{6}q^{2}+(-2+2\beta _{4}-\beta _{8})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots$$
693.2.i.l $12$ $5.534$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$4$$ $$6$$ $$q-\beta _{6}q^{2}+(-2+2\beta _{4}-\beta _{8})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(693, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(693, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(77, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(231, [\chi])$$$$^{\oplus 2}$$