# Properties

 Label 819.2.o Level $819$ Weight $2$ Character orbit 819.o Rep. character $\chi_{819}(568,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $72$ Newform subspaces $10$ Sturm bound $224$ Trace bound $11$

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## Defining parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.o (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$10$$ Sturm bound: $$224$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$2$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(819, [\chi])$$.

Total New Old
Modular forms 240 72 168
Cusp forms 208 72 136
Eisenstein series 32 0 32

## Trace form

 $$72 q - 2 q^{2} - 38 q^{4} - 8 q^{5} + 24 q^{8} + O(q^{10})$$ $$72 q - 2 q^{2} - 38 q^{4} - 8 q^{5} + 24 q^{8} + 10 q^{10} + 6 q^{11} - 14 q^{13} - 8 q^{14} - 34 q^{16} + 10 q^{17} + 8 q^{19} - 10 q^{20} + 14 q^{22} + 12 q^{23} + 76 q^{25} + 32 q^{26} + 10 q^{29} + 28 q^{31} - 6 q^{32} - 64 q^{34} + 6 q^{35} + 4 q^{37} - 4 q^{38} - 80 q^{40} - 2 q^{43} + 24 q^{44} + 20 q^{46} + 20 q^{47} - 36 q^{49} + 50 q^{50} + 22 q^{52} - 32 q^{53} + 6 q^{55} + 12 q^{56} - 50 q^{58} + 22 q^{59} + 12 q^{61} - 40 q^{62} + 80 q^{64} + 10 q^{65} - 20 q^{67} - 38 q^{68} - 48 q^{70} - 8 q^{71} - 8 q^{73} + 12 q^{74} - 26 q^{76} - 24 q^{77} + 8 q^{79} + 30 q^{80} + 16 q^{82} - 124 q^{83} + 14 q^{85} - 128 q^{86} + 18 q^{88} - 16 q^{89} - 14 q^{91} - 16 q^{92} - 36 q^{94} + 22 q^{95} - 26 q^{97} - 2 q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(819, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
819.2.o.a $4$ $6.540$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+\zeta_{12}^{3}q^{5}+\cdots$$
819.2.o.b $4$ $6.540$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+\zeta_{12}^{3}q^{5}+\cdots$$
819.2.o.c $4$ $6.540$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$3$$ $$0$$ $$-6$$ $$-2$$ $$q+(1+\beta _{1}+\beta _{3})q^{2}+(3\beta _{1}+3\beta _{2})q^{4}+\cdots$$
819.2.o.d $6$ $6.540$ 6.0.771147.1 None $$-2$$ $$0$$ $$0$$ $$3$$ $$q+(1-\beta _{2}-\beta _{3}+\beta _{4}+\beta _{5})q^{2}+(-2+\cdots)q^{4}+\cdots$$
819.2.o.e $6$ $6.540$ 6.0.64827.1 None $$-2$$ $$0$$ $$0$$ $$-3$$ $$q+(-1+\beta _{4}+\beta _{5})q^{2}+(-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
819.2.o.f $6$ $6.540$ 6.0.6040683.1 None $$0$$ $$0$$ $$-4$$ $$-3$$ $$q+(-\beta _{1}-\beta _{2})q^{2}+(\beta _{3}-\beta _{4}-\beta _{5})q^{4}+\cdots$$
819.2.o.g $6$ $6.540$ $$\Q(\zeta_{18})$$ None $$0$$ $$0$$ $$-12$$ $$3$$ $$q+(-\zeta_{18}+\zeta_{18}^{2}+\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+\cdots$$
819.2.o.h $8$ $6.540$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-1$$ $$0$$ $$14$$ $$4$$ $$q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots$$
819.2.o.i $12$ $6.540$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-6$$ $$q+\beta _{7}q^{2}+(-1-\beta _{3}+\beta _{4})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots$$
819.2.o.j $16$ $6.540$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+(\beta _{1}+\beta _{9})q^{2}+(-1-\beta _{3}-\beta _{8}-\beta _{11}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(819, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(819, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(91, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(117, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(273, [\chi])$$$$^{\oplus 2}$$