# Properties

 Label 1024.2.g Level $1024$ Weight $2$ Character orbit 1024.g Rep. character $\chi_{1024}(129,\cdot)$ Character field $\Q(\zeta_{8})$ Dimension $128$ Newform subspaces $8$ Sturm bound $256$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.g (of order $$8$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$32$$ Character field: $$\Q(\zeta_{8})$$ Newform subspaces: $$8$$ Sturm bound: $$256$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

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

Total New Old
Modular forms 608 128 480
Cusp forms 416 128 288
Eisenstein series 192 0 192

## Trace form

 $$128 q + O(q^{10})$$ $$128 q + O(q^{100})$$

## 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.g.a $16$ $8.177$ $$\Q(\zeta_{48})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+\zeta_{48}^{15}q^{3}+(-1+\zeta_{48}^{3}-\zeta_{48}^{8}+\cdots)q^{5}+\cdots$$
1024.2.g.b $16$ $8.177$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+\beta _{2}q^{3}+(\beta _{3}+\beta _{10})q^{5}+(-\beta _{4}-\beta _{9}+\cdots)q^{7}+\cdots$$
1024.2.g.c $16$ $8.177$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+\beta _{13}q^{3}+(-1-\beta _{3}+\beta _{5}+\beta _{8})q^{5}+\cdots$$
1024.2.g.d $16$ $8.177$ $$\Q(\zeta_{48})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+(\zeta_{48}^{11}+\zeta_{48}^{14})q^{3}+(-\zeta_{48}-\zeta_{48}^{2}+\cdots)q^{5}+\cdots$$
1024.2.g.e $16$ $8.177$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+\beta _{2}q^{3}+(-\beta _{3}-\beta _{10})q^{5}+(\beta _{4}+\beta _{9}+\cdots)q^{7}+\cdots$$
1024.2.g.f $16$ $8.177$ $$\Q(\zeta_{48})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+\zeta_{48}^{15}q^{3}+(1+\zeta_{48}^{8}+\zeta_{48}^{13}+\cdots)q^{5}+\cdots$$
1024.2.g.g $16$ $8.177$ $$\Q(\zeta_{48})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+(\zeta_{48}^{11}+\zeta_{48}^{14})q^{3}+(\zeta_{48}+\zeta_{48}^{2}+\cdots)q^{5}+\cdots$$
1024.2.g.h $16$ $8.177$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+\beta _{13}q^{3}+(1+\beta _{3}-\beta _{5}-\beta _{8})q^{5}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1024, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(128, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(256, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(512, [\chi])$$$$^{\oplus 2}$$