# Properties

 Label 1024.2.e Level $1024$ Weight $2$ Character orbit 1024.e Rep. character $\chi_{1024}(257,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $56$ Newform subspaces $16$ Sturm bound $256$ Trace bound $19$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.e (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$16$$ Sturm bound: $$256$$ Trace bound: $$19$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$47$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1024, [\chi])$$.

Total New Old
Modular forms 304 72 232
Cusp forms 208 56 152
Eisenstein series 96 16 80

## Trace form

 $$56 q + O(q^{10})$$ $$56 q + 16 q^{17} - 16 q^{33} + 8 q^{49} - 16 q^{65} + 24 q^{81} - 16 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1024, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1024.2.e.a $2$ $8.177$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-2+2i)q^{3}-5iq^{9}+(2+2i)q^{11}+\cdots$$
1024.2.e.b $2$ $8.177$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$-4$$ $$0$$ $$q+(-1+i)q^{3}+(-2-2i)q^{5}-4iq^{7}+\cdots$$
1024.2.e.c $2$ $8.177$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$4$$ $$0$$ $$q+(-1+i)q^{3}+(2+2i)q^{5}+4iq^{7}+\cdots$$
1024.2.e.d $2$ $8.177$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$-4$$ $$0$$ $$q+(1-i)q^{3}+(-2-2i)q^{5}+4iq^{7}+\cdots$$
1024.2.e.e $2$ $8.177$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$4$$ $$0$$ $$q+(1-i)q^{3}+(2+2i)q^{5}-4iq^{7}+iq^{9}+\cdots$$
1024.2.e.f $2$ $8.177$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$4$$ $$0$$ $$0$$ $$q+(2-2i)q^{3}-5iq^{9}+(-2-2i)q^{11}+\cdots$$
1024.2.e.g $4$ $8.177$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1+\zeta_{8}-\zeta_{8}^{2})q^{3}+(-2\zeta_{8}+3\zeta_{8}^{2}+\cdots)q^{9}+\cdots$$
1024.2.e.h $4$ $8.177$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1+\zeta_{8}^{2})q^{3}-\zeta_{8}q^{5}+(-\zeta_{8}-\zeta_{8}^{3})q^{7}+\cdots$$
1024.2.e.i $4$ $8.177$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}q^{3}+\zeta_{8}^{3}q^{5}+4\zeta_{8}^{2}q^{7}+\zeta_{8}^{2}q^{9}+\cdots$$
1024.2.e.j $4$ $8.177$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}q^{5}+3\zeta_{8}^{2}q^{9}-3\zeta_{8}^{3}q^{13}-2q^{17}+\cdots$$
1024.2.e.k $4$ $8.177$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}q^{5}+3\zeta_{8}^{2}q^{9}-\zeta_{8}^{3}q^{13}+2q^{17}+\cdots$$
1024.2.e.l $4$ $8.177$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}q^{3}+\zeta_{8}^{2}q^{9}-3\zeta_{8}^{3}q^{11}+6q^{17}+\cdots$$
1024.2.e.m $4$ $8.177$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}q^{3}-\zeta_{8}^{3}q^{5}-4\zeta_{8}^{2}q^{7}+\zeta_{8}^{2}q^{9}+\cdots$$
1024.2.e.n $4$ $8.177$ $$\Q(\zeta_{8})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1-\zeta_{8}^{2})q^{3}-\zeta_{8}q^{5}+(\zeta_{8}+\zeta_{8}^{3})q^{7}+\cdots$$
1024.2.e.o $4$ $8.177$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1+\zeta_{8}+\zeta_{8}^{2})q^{3}+(2\zeta_{8}+3\zeta_{8}^{2}+\cdots)q^{9}+\cdots$$
1024.2.e.p $8$ $8.177$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}^{7}q^{3}+\zeta_{24}q^{5}-\zeta_{24}^{6}q^{7}-3\zeta_{24}^{2}q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1024, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1024, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 7}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(128, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(256, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(512, [\chi])$$$$^{\oplus 2}$$