# Properties

 Label 224.2.a Level $224$ Weight $2$ Character orbit 224.a Rep. character $\chi_{224}(1,\cdot)$ Character field $\Q$ Dimension $6$ Newform subspaces $4$ Sturm bound $64$ Trace bound $3$

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## Defining parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$64$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(224))$$.

Total New Old
Modular forms 40 6 34
Cusp forms 25 6 19
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$7$$FrickeDim
$$+$$$$+$$$+$$$1$$
$$+$$$$-$$$-$$$3$$
$$-$$$$+$$$-$$$2$$
Plus space$$+$$$$1$$
Minus space$$-$$$$5$$

## Trace form

 $$6 q + 4 q^{5} + 14 q^{9} + O(q^{10})$$ $$6 q + 4 q^{5} + 14 q^{9} + 4 q^{13} - 4 q^{17} - 6 q^{25} + 4 q^{29} - 16 q^{33} + 20 q^{37} - 36 q^{41} - 28 q^{45} + 6 q^{49} - 44 q^{53} + 20 q^{61} - 8 q^{65} - 48 q^{69} + 12 q^{73} + 22 q^{81} + 40 q^{85} + 12 q^{89} - 32 q^{93} + 28 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(224))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 7
224.2.a.a $1$ $1.789$ $$\Q$$ None $$0$$ $$-2$$ $$0$$ $$-1$$ $+$ $+$ $$q-2q^{3}-q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots$$
224.2.a.b $1$ $1.789$ $$\Q$$ None $$0$$ $$2$$ $$0$$ $$1$$ $+$ $-$ $$q+2q^{3}+q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots$$
224.2.a.c $2$ $1.789$ $$\Q(\sqrt{5})$$ None $$0$$ $$-2$$ $$2$$ $$2$$ $+$ $-$ $$q+(-1-\beta )q^{3}+(1-\beta )q^{5}+q^{7}+(3+\cdots)q^{9}+\cdots$$
224.2.a.d $2$ $1.789$ $$\Q(\sqrt{5})$$ None $$0$$ $$2$$ $$2$$ $$-2$$ $-$ $+$ $$q+(1+\beta )q^{3}+(1-\beta )q^{5}-q^{7}+(3+2\beta )q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(224))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(224)) \simeq$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(14))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(56))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(112))$$$$^{\oplus 2}$$