# Properties

 Label 361.2.e Level $361$ Weight $2$ Character orbit 361.e Rep. character $\chi_{361}(28,\cdot)$ Character field $\Q(\zeta_{9})$ Dimension $120$ Newform subspaces $13$ Sturm bound $63$ Trace bound $12$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 361.e (of order $$9$$ and degree $$6$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{9})$$ Newform subspaces: $$13$$ Sturm bound: $$63$$ Trace bound: $$12$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(361, [\chi])$$.

Total New Old
Modular forms 252 216 36
Cusp forms 132 120 12
Eisenstein series 120 96 24

## Trace form

 $$120 q + 6 q^{2} + 3 q^{3} + 6 q^{5} - 3 q^{6} - 3 q^{7} - 6 q^{8} - 3 q^{9} + O(q^{10})$$ $$120 q + 6 q^{2} + 3 q^{3} + 6 q^{5} - 3 q^{6} - 3 q^{7} - 6 q^{8} - 3 q^{9} - 9 q^{10} - 3 q^{11} - 3 q^{12} + 3 q^{13} + 3 q^{14} - 3 q^{15} + 18 q^{16} - 3 q^{17} + 6 q^{18} - 78 q^{20} - 6 q^{23} - 15 q^{24} - 39 q^{26} - 6 q^{27} - 6 q^{28} + 3 q^{29} - 24 q^{30} - 9 q^{31} - 9 q^{32} + 9 q^{33} - 6 q^{35} + 24 q^{36} - 48 q^{39} - 21 q^{41} + 3 q^{42} + 3 q^{43} - 9 q^{44} + 18 q^{46} + 3 q^{47} + 3 q^{48} + 69 q^{49} + 15 q^{50} - 3 q^{51} - 15 q^{52} + 3 q^{53} - 30 q^{54} - 18 q^{55} + 6 q^{56} - 120 q^{58} - 12 q^{59} + 6 q^{60} + 12 q^{61} + 12 q^{62} - 12 q^{63} + 120 q^{64} + 12 q^{65} + 9 q^{66} + 30 q^{67} - 15 q^{68} + 12 q^{69} + 9 q^{70} + 6 q^{71} + 12 q^{72} + 12 q^{73} - 15 q^{74} - 30 q^{75} - 24 q^{77} - 15 q^{78} + 39 q^{79} - 3 q^{80} - 6 q^{81} + 54 q^{82} - 24 q^{83} - 3 q^{84} - 24 q^{86} - 9 q^{87} - 9 q^{88} + 12 q^{89} - 18 q^{90} - 15 q^{91} - 42 q^{92} - 9 q^{93} - 18 q^{94} + 18 q^{96} - 18 q^{97} + 9 q^{98} - 9 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(361, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
361.2.e.a $6$ $2.883$ $$\Q(\zeta_{18})$$ None $$-3$$ $$-6$$ $$3$$ $$0$$ $$q+(\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}-\zeta_{18}^{4})q^{2}+\cdots$$
361.2.e.b $6$ $2.883$ $$\Q(\zeta_{18})$$ None $$-3$$ $$3$$ $$3$$ $$0$$ $$q+(-1+\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{4}+\cdots)q^{2}+\cdots$$
361.2.e.c $6$ $2.883$ $$\Q(\zeta_{18})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$0$$ $$-9$$ $$q+2\zeta_{18}^{5}q^{4}-\zeta_{18}^{4}q^{5}+(-3+3\zeta_{18}^{3}+\cdots)q^{7}+\cdots$$
361.2.e.d $6$ $2.883$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$3$$ $$q+2\zeta_{18}q^{3}+2\zeta_{18}^{5}q^{4}+3\zeta_{18}^{4}q^{5}+\cdots$$
361.2.e.e $6$ $2.883$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$0$$ $$3$$ $$q-2\zeta_{18}q^{3}+2\zeta_{18}^{5}q^{4}+3\zeta_{18}^{4}q^{5}+\cdots$$
361.2.e.f $6$ $2.883$ $$\Q(\zeta_{18})$$ None $$3$$ $$-3$$ $$3$$ $$0$$ $$q+(1-\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots$$
361.2.e.g $6$ $2.883$ $$\Q(\zeta_{18})$$ None $$3$$ $$6$$ $$3$$ $$0$$ $$q+(-\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots$$
361.2.e.h $6$ $2.883$ $$\Q(\zeta_{18})$$ None $$6$$ $$3$$ $$-6$$ $$0$$ $$q+(1+\zeta_{18}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+(1+\cdots)q^{3}+\cdots$$
361.2.e.i $12$ $2.883$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$-18$$ $$q-\beta _{7}q^{2}+(-\beta _{1}-2\beta _{3})q^{3}+(\beta _{5}+\beta _{10}+\cdots)q^{4}+\cdots$$
361.2.e.j $12$ $2.883$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$-18$$ $$q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{7}-2\beta _{9})q^{3}+(\beta _{4}+\cdots)q^{4}+\cdots$$
361.2.e.k $12$ $2.883$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$6$$ $$q+(-\beta _{5}-2\beta _{10}+\beta _{11})q^{2}+2\beta _{5}q^{3}+\cdots$$
361.2.e.l $12$ $2.883$ 12.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$6$$ $$q+(-\beta _{5}-2\beta _{10}+\beta _{11})q^{2}-2\beta _{5}q^{3}+\cdots$$
361.2.e.m $24$ $2.883$ None $$0$$ $$0$$ $$0$$ $$24$$

## Decomposition of $$S_{2}^{\mathrm{old}}(361, [\chi])$$ into lower level spaces

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