# Properties

 Label 224.1.h Level $224$ Weight $1$ Character orbit 224.h Rep. character $\chi_{224}(209,\cdot)$ Character field $\Q$ Dimension $1$ Newform subspaces $1$ Sturm bound $32$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 224.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$56$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$32$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(224, [\chi])$$.

Total New Old
Modular forms 12 3 9
Cusp forms 4 1 3
Eisenstein series 8 2 6

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 1 0 0 0

## Trace form

 $$q + q^{7} - q^{9} + O(q^{10})$$ $$q + q^{7} - q^{9} - 2 q^{23} - q^{25} + q^{49} - q^{63} + 2 q^{71} + 2 q^{79} + q^{81} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(224, [\chi])$$ into newform subspaces

Label Dim $A$ Field Image CM RM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
224.1.h.a $1$ $0.112$ $$\Q$$ $D_{2}$ $$\Q(\sqrt{-7})$$, $$\Q(\sqrt{-14})$$ $$\Q(\sqrt{2})$$ $$0$$ $$0$$ $$0$$ $$1$$ $$q+q^{7}-q^{9}-2q^{23}-q^{25}+q^{49}+\cdots$$

## Decomposition of $$S_{1}^{\mathrm{old}}(224, [\chi])$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(224, [\chi]) \cong$$ $$S_{1}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 3}$$