# Properties

 Label 224.4.a Level $224$ Weight $4$ Character orbit 224.a Rep. character $\chi_{224}(1,\cdot)$ Character field $\Q$ Dimension $18$ Newform subspaces $8$ Sturm bound $128$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 224.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$128$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(224))$$.

Total New Old
Modular forms 104 18 86
Cusp forms 88 18 70
Eisenstein series 16 0 16

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.
$$+$$$$+$$$$+$$$$5$$
$$+$$$$-$$$$-$$$$3$$
$$-$$$$+$$$$-$$$$4$$
$$-$$$$-$$$$+$$$$6$$
Plus space$$+$$$$11$$
Minus space$$-$$$$7$$

## Trace form

 $$18q - 4q^{5} + 122q^{9} + O(q^{10})$$ $$18q - 4q^{5} + 122q^{9} + 92q^{13} + 308q^{17} + 318q^{25} + 284q^{29} + 1232q^{33} - 660q^{37} - 588q^{41} + 1788q^{45} + 882q^{49} + 1388q^{53} - 768q^{57} + 780q^{61} - 1304q^{65} - 1488q^{69} - 1212q^{73} - 2078q^{81} + 1880q^{85} + 388q^{89} - 5856q^{93} - 3628q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(224))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 7
224.4.a.a $$1$$ $$13.216$$ $$\Q$$ None $$0$$ $$-2$$ $$0$$ $$7$$ $$+$$ $$-$$ $$q-2q^{3}+7q^{7}-23q^{9}+20q^{11}-20q^{13}+\cdots$$
224.4.a.b $$1$$ $$13.216$$ $$\Q$$ None $$0$$ $$2$$ $$0$$ $$-7$$ $$-$$ $$+$$ $$q+2q^{3}-7q^{7}-23q^{9}-20q^{11}-20q^{13}+\cdots$$
224.4.a.c $$2$$ $$13.216$$ $$\Q(\sqrt{37})$$ None $$0$$ $$-6$$ $$-6$$ $$14$$ $$+$$ $$-$$ $$q+(-3-\beta )q^{3}+(-3+\beta )q^{5}+7q^{7}+\cdots$$
224.4.a.d $$2$$ $$13.216$$ $$\Q(\sqrt{37})$$ None $$0$$ $$6$$ $$-6$$ $$-14$$ $$+$$ $$+$$ $$q+(3+\beta )q^{3}+(-3+\beta )q^{5}-7q^{7}+(19+\cdots)q^{9}+\cdots$$
224.4.a.e $$3$$ $$13.216$$ 3.3.2981.1 None $$0$$ $$-8$$ $$10$$ $$-21$$ $$+$$ $$+$$ $$q+(-3-\beta _{1})q^{3}+(3+\beta _{2})q^{5}-7q^{7}+\cdots$$
224.4.a.f $$3$$ $$13.216$$ 3.3.621.1 None $$0$$ $$0$$ $$-6$$ $$-21$$ $$-$$ $$+$$ $$q+\beta _{1}q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}-7q^{7}+\cdots$$
224.4.a.g $$3$$ $$13.216$$ 3.3.621.1 None $$0$$ $$0$$ $$-6$$ $$21$$ $$-$$ $$-$$ $$q-\beta _{1}q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+7q^{7}+\cdots$$
224.4.a.h $$3$$ $$13.216$$ 3.3.2981.1 None $$0$$ $$8$$ $$10$$ $$21$$ $$-$$ $$-$$ $$q+(3+\beta _{1})q^{3}+(3+\beta _{2})q^{5}+7q^{7}+(11+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(224))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(224)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(14))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(56))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(112))$$$$^{\oplus 2}$$