# Properties

 Label 128.9.d Level $128$ Weight $9$ Character orbit 128.d Rep. character $\chi_{128}(63,\cdot)$ Character field $\Q$ Dimension $32$ Newform subspaces $6$ Sturm bound $144$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 128.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$144$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(128, [\chi])$$.

Total New Old
Modular forms 136 32 104
Cusp forms 120 32 88
Eisenstein series 16 0 16

## Trace form

 $$32 q + 69984 q^{9} + O(q^{10})$$ $$32 q + 69984 q^{9} - 154560 q^{17} - 1791712 q^{25} + 3305344 q^{33} - 1127616 q^{41} - 2084832 q^{49} + 7916416 q^{57} - 14649600 q^{65} + 102896960 q^{73} - 57824352 q^{81} - 300504768 q^{89} + 157386816 q^{97} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(128, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
128.9.d.a $2$ $52.144$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{5}-3^{8}q^{9}-85iq^{13}+63358q^{17}+\cdots$$
128.9.d.b $2$ $52.144$ $$\Q(\sqrt{2})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta q^{3}+18527q^{9}+69\beta q^{11}-162434q^{17}+\cdots$$
128.9.d.c $4$ $52.144$ $$\Q(i, \sqrt{14})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}-71\beta _{1}q^{5}+9\beta _{3}q^{7}-4545q^{9}+\cdots$$
128.9.d.d $4$ $52.144$ $$\Q(\sqrt{-70}, \sqrt{102})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}+\beta _{3}q^{5}-\beta _{2}q^{7}-33q^{9}+\cdots$$
128.9.d.e $4$ $52.144$ $$\Q(i, \sqrt{30})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-5\beta _{2}q^{3}-85\beta _{1}q^{5}-7\beta _{3}q^{7}+5439q^{9}+\cdots$$
128.9.d.f $16$ $52.144$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{6}q^{3}-\beta _{4}q^{5}-\beta _{9}q^{7}+(2663+\beta _{1}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{9}^{\mathrm{old}}(128, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(128, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(8, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(64, [\chi])$$$$^{\oplus 2}$$