# Properties

 Label 1024.2.b Level $1024$ Weight $2$ Character orbit 1024.b Rep. character $\chi_{1024}(513,\cdot)$ Character field $\Q$ Dimension $28$ Newform subspaces $8$ Sturm bound $256$ Trace bound $33$

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## Defining parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$256$$ Trace bound: $$33$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$7$$

## Dimensions

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

Total New Old
Modular forms 152 36 116
Cusp forms 104 28 76
Eisenstein series 48 8 40

## Trace form

 $$28 q - 20 q^{9} + O(q^{10})$$ $$28 q - 20 q^{9} + 8 q^{17} - 12 q^{25} - 8 q^{33} + 12 q^{49} - 24 q^{57} - 8 q^{65} + 4 q^{81} - 8 q^{97} + O(q^{100})$$

## 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.b.a $2$ $8.177$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+2\beta q^{3}+\beta q^{5}-4q^{7}-5q^{9}-2\beta q^{11}+\cdots$$
1024.2.b.b $2$ $8.177$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\beta q^{3}-\beta q^{5}-2q^{7}+q^{9}+\beta q^{11}+\cdots$$
1024.2.b.c $2$ $8.177$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta q^{5}+3q^{9}-5\beta q^{13}-8q^{17}+3q^{25}+\cdots$$
1024.2.b.d $2$ $8.177$ $$\Q(\sqrt{-2})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-3\beta q^{5}+3q^{9}+\beta q^{13}+8q^{17}-13q^{25}+\cdots$$
1024.2.b.e $2$ $8.177$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta q^{3}+\beta q^{5}+2q^{7}+q^{9}+\beta q^{11}+\cdots$$
1024.2.b.f $2$ $8.177$ $$\Q(\sqrt{-2})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+2\beta q^{3}-\beta q^{5}+4q^{7}-5q^{9}-2\beta q^{11}+\cdots$$
1024.2.b.g $8$ $8.177$ $$\Q(\zeta_{16})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{16}^{2}q^{3}+\zeta_{16}q^{5}-\zeta_{16}^{3}q^{7}+(-1+\cdots)q^{9}+\cdots$$
1024.2.b.h $8$ $8.177$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{24}q^{3}+\zeta_{24}^{5}q^{5}+\zeta_{24}^{4}q^{7}+(-1+\cdots)q^{9}+\cdots$$

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

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