# Properties

 Label 722.2.e Level $722$ Weight $2$ Character orbit 722.e Rep. character $\chi_{722}(99,\cdot)$ Character field $\Q(\zeta_{9})$ Dimension $174$ Newform subspaces $19$ Sturm bound $190$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$722 = 2 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 722.e (of order $$9$$ and degree $$6$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{9})$$ Newform subspaces: $$19$$ Sturm bound: $$190$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$7$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(722, [\chi])$$.

Total New Old
Modular forms 690 174 516
Cusp forms 450 174 276
Eisenstein series 240 0 240

## Trace form

 $$174 q + 3 q^{3} + 3 q^{6} + 12 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10})$$ $$174 q + 3 q^{3} + 3 q^{6} + 12 q^{7} + 3 q^{8} + 3 q^{9} + 12 q^{11} - 12 q^{13} - 12 q^{14} + 6 q^{15} + 12 q^{17} + 6 q^{18} - 24 q^{20} - 24 q^{21} + 12 q^{23} + 3 q^{24} - 6 q^{26} - 3 q^{27} + 6 q^{28} + 18 q^{29} - 6 q^{31} - 3 q^{33} + 12 q^{34} + 12 q^{35} + 3 q^{36} + 12 q^{37} - 24 q^{39} - 3 q^{41} + 12 q^{42} + 6 q^{43} + 24 q^{45} - 30 q^{47} - 6 q^{48} - 45 q^{49} - 3 q^{50} - 21 q^{51} + 6 q^{52} - 24 q^{53} - 9 q^{54} - 18 q^{55} - 12 q^{56} - 48 q^{58} + 3 q^{59} + 6 q^{60} - 6 q^{61} - 18 q^{62} - 12 q^{63} - 87 q^{64} - 12 q^{65} - 3 q^{66} + 9 q^{67} + 9 q^{68} + 6 q^{69} + 12 q^{70} + 18 q^{71} - 6 q^{72} + 30 q^{73} + 18 q^{74} - 12 q^{77} + 18 q^{78} - 6 q^{79} + 33 q^{81} - 3 q^{82} + 36 q^{83} - 6 q^{84} + 24 q^{85} - 12 q^{86} + 24 q^{87} + 6 q^{88} - 12 q^{90} + 12 q^{91} - 6 q^{92} - 6 q^{93} + 12 q^{94} - 3 q^{97} - 21 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(722, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
722.2.e.a $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$-6$$ $$0$$ $$-6$$ $$q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(-1-\zeta_{18}^{2}+\cdots)q^{3}+\cdots$$
722.2.e.b $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$-3$$ $$0$$ $$-6$$ $$q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+(-1+\zeta_{18}^{3}-\zeta_{18}^{5})q^{3}+\cdots$$
722.2.e.c $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$-9$$ $$q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots$$
722.2.e.d $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$-9$$ $$q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots$$
722.2.e.e $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$3$$ $$q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots$$
722.2.e.f $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$3$$ $$q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots$$
722.2.e.g $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$9$$ $$q+(\zeta_{18}-\zeta_{18}^{4})q^{2}-3\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots$$
722.2.e.h $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$9$$ $$q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+3\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots$$
722.2.e.i $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots$$
722.2.e.j $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}-\zeta_{18}q^{3}-\zeta_{18}^{5}q^{4}+\cdots$$
722.2.e.k $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$3$$ $$0$$ $$-6$$ $$q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(\zeta_{18}^{2}+\zeta_{18}^{3}+\cdots)q^{3}+\cdots$$
722.2.e.l $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$3$$ $$0$$ $$-6$$ $$q+(-\zeta_{18}+\zeta_{18}^{4})q^{2}+(1-\zeta_{18}^{3}+\zeta_{18}^{5})q^{3}+\cdots$$
722.2.e.m $6$ $5.765$ $$\Q(\zeta_{18})$$ None $$0$$ $$6$$ $$0$$ $$-6$$ $$q+(\zeta_{18}-\zeta_{18}^{4})q^{2}+(1+\zeta_{18}^{2})q^{3}+\cdots$$
722.2.e.n $12$ $5.765$ 12.0.$$\cdots$$.3 None $$0$$ $$0$$ $$0$$ $$-6$$ $$q+\beta _{10}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+\beta _{2}q^{4}+\cdots$$
722.2.e.o $12$ $5.765$ 12.0.$$\cdots$$.3 None $$0$$ $$0$$ $$0$$ $$-6$$ $$q-\beta _{10}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+\beta _{2}q^{4}+\cdots$$
722.2.e.p $12$ $5.765$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$6$$ $$q-\beta _{9}q^{2}-2\beta _{7}q^{3}-\beta _{11}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
722.2.e.q $12$ $5.765$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$6$$ $$q+(\beta _{3}-\beta _{9})q^{2}-2\beta _{1}q^{3}+(-\beta _{5}+\beta _{11})q^{4}+\cdots$$
722.2.e.r $24$ $5.765$ None $$0$$ $$0$$ $$0$$ $$6$$
722.2.e.s $24$ $5.765$ None $$0$$ $$0$$ $$0$$ $$6$$

## Decomposition of $$S_{2}^{\mathrm{old}}(722, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(722, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(19, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(361, [\chi])$$$$^{\oplus 2}$$