# Properties

 Label 1053.2.e Level $1053$ Weight $2$ Character orbit 1053.e Rep. character $\chi_{1053}(352,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $96$ Newform subspaces $19$ Sturm bound $252$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1053 = 3^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1053.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$19$$ Sturm bound: $$252$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1053, [\chi])$$.

Total New Old
Modular forms 276 96 180
Cusp forms 228 96 132
Eisenstein series 48 0 48

## Trace form

 $$96 q - 48 q^{4} + O(q^{10})$$ $$96 q - 48 q^{4} - 48 q^{16} + 24 q^{19} - 12 q^{22} - 60 q^{25} - 12 q^{31} + 48 q^{34} + 24 q^{37} + 60 q^{40} - 12 q^{43} - 72 q^{46} - 72 q^{49} - 96 q^{55} + 60 q^{58} - 24 q^{61} - 24 q^{64} - 24 q^{67} + 36 q^{70} + 72 q^{73} + 24 q^{76} + 60 q^{79} + 24 q^{82} + 24 q^{88} + 36 q^{94} - 48 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1053, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1053.2.e.a $2$ $8.408$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-2$$ $$-5$$ $$q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}-2\zeta_{6}q^{5}+\cdots$$
1053.2.e.b $2$ $8.408$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-2$$ $$4$$ $$q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}-2\zeta_{6}q^{5}+\cdots$$
1053.2.e.c $2$ $8.408$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$2$$ $$-5$$ $$q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+2\zeta_{6}q^{5}+(-5+\cdots)q^{7}+\cdots$$
1053.2.e.d $2$ $8.408$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$2$$ $$4$$ $$q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+2\zeta_{6}q^{5}+(4+\cdots)q^{7}+\cdots$$
1053.2.e.e $4$ $8.408$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-2$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$$
1053.2.e.f $4$ $8.408$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$-1$$ $$0$$ $$-5$$ $$2$$ $$q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots$$
1053.2.e.g $4$ $8.408$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$-1$$ $$0$$ $$-3$$ $$0$$ $$q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3})q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
1053.2.e.h $4$ $8.408$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$-1$$ $$0$$ $$3$$ $$3$$ $$q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3})q^{4}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
1053.2.e.i $4$ $8.408$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}-2\zeta_{12}q^{7}+\cdots$$
1053.2.e.j $4$ $8.408$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$1$$ $$0$$ $$5$$ $$2$$ $$q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots$$
1053.2.e.k $4$ $8.408$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$1$$ $$0$$ $$-3$$ $$3$$ $$q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
1053.2.e.l $4$ $8.408$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$1$$ $$0$$ $$3$$ $$0$$ $$q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
1053.2.e.m $4$ $8.408$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$0$$ $$q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots$$
1053.2.e.n $6$ $8.408$ 6.0.1156923.1 None $$0$$ $$0$$ $$-3$$ $$0$$ $$q+\beta _{4}q^{2}+(-\beta _{1}+\beta _{2}-2\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots$$
1053.2.e.o $6$ $8.408$ 6.0.1156923.1 None $$0$$ $$0$$ $$3$$ $$0$$ $$q-\beta _{4}q^{2}+(-\beta _{1}+\beta _{2}-2\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots$$
1053.2.e.p $8$ $8.408$ 8.0.4678560000.4 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}+(-2-2\beta _{3}-\beta _{4})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
1053.2.e.q $8$ $8.408$ 8.0.$$\cdots$$.8 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q-\beta _{7}q^{2}+(-1+\beta _{2}-\beta _{4})q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots$$
1053.2.e.r $8$ $8.408$ 8.0.49787136.1 None $$0$$ $$0$$ $$0$$ $$10$$ $$q+\beta _{3}q^{2}+\beta _{4}q^{4}+\beta _{2}q^{5}+(2+\beta _{4}+\cdots)q^{7}+\cdots$$
1053.2.e.s $16$ $8.408$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-6$$ $$q-\beta _{13}q^{2}+(-\beta _{1}-\beta _{3}+\beta _{4}-2\beta _{10}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1053, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1053, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(117, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(351, [\chi])$$$$^{\oplus 2}$$