# Properties

 Label 6724.2.a Level $6724$ Weight $2$ Character orbit 6724.a Rep. character $\chi_{6724}(1,\cdot)$ Character field $\Q$ Dimension $136$ Newform subspaces $13$ Sturm bound $1722$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$6724 = 2^{2} \cdot 41^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6724.a (trivial) Character field: $$\Q$$ Newform subspaces: $$13$$ Sturm bound: $$1722$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(6724))$$.

Total New Old
Modular forms 924 136 788
Cusp forms 799 136 663
Eisenstein series 125 0 125

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

$$2$$$$41$$FrickeDim.
$$-$$$$+$$$$-$$$$73$$
$$-$$$$-$$$$+$$$$63$$
Plus space$$+$$$$63$$
Minus space$$-$$$$73$$

## Trace form

 $$136q - 2q^{3} - 4q^{5} + 128q^{9} + O(q^{10})$$ $$136q - 2q^{3} - 4q^{5} + 128q^{9} - 4q^{11} + 10q^{15} + 4q^{17} - 6q^{19} + 12q^{23} + 128q^{25} + 10q^{27} + 4q^{29} + 10q^{31} + 22q^{33} + 26q^{35} - 14q^{37} + 24q^{39} - 2q^{43} - 12q^{45} + 6q^{47} + 124q^{49} + 6q^{51} + 16q^{53} + 2q^{55} - 12q^{57} - 12q^{59} - 22q^{61} + 10q^{63} - 4q^{65} - 28q^{67} + 28q^{69} + 2q^{71} - 6q^{73} - 30q^{75} - 16q^{77} + 18q^{79} + 112q^{81} + 8q^{83} - 32q^{85} - 42q^{87} - 4q^{89} - 40q^{91} + 28q^{93} - 14q^{95} - 16q^{97} - 58q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(6724))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 41
6724.2.a.a $$3$$ $$53.691$$ 3.3.785.1 None $$0$$ $$-2$$ $$-2$$ $$7$$ $$-$$ $$-$$ $$q+(-1+\beta _{1})q^{3}+(-1+\beta _{1})q^{5}+(2+\cdots)q^{7}+\cdots$$
6724.2.a.b $$3$$ $$53.691$$ 3.3.785.1 None $$0$$ $$2$$ $$-2$$ $$-7$$ $$-$$ $$+$$ $$q+(1-\beta _{1})q^{3}+(-1+\beta _{1})q^{5}+(-2+\cdots)q^{7}+\cdots$$
6724.2.a.c $$4$$ $$53.691$$ 4.4.25808.1 None $$0$$ $$-2$$ $$4$$ $$0$$ $$-$$ $$+$$ $$q+(\beta _{1}-\beta _{2})q^{3}+(2-\beta _{2}+\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots$$
6724.2.a.d $$4$$ $$53.691$$ 4.4.25088.1 None $$0$$ $$0$$ $$4$$ $$0$$ $$-$$ $$+$$ $$q+\beta _{1}q^{3}+(1+\beta _{2})q^{5}+\beta _{3}q^{7}+\beta _{2}q^{9}+\cdots$$
6724.2.a.e $$6$$ $$53.691$$ 6.6.40716288.1 None $$0$$ $$0$$ $$-4$$ $$0$$ $$-$$ $$-$$ $$q+\beta _{1}q^{3}+(-1+\beta _{2})q^{5}+(\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots$$
6724.2.a.f $$8$$ $$53.691$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-1$$ $$3$$ $$0$$ $$-$$ $$+$$ $$q-\beta _{1}q^{3}-\beta _{6}q^{5}+(\beta _{2}+\beta _{5}+\beta _{6}+\beta _{7})q^{7}+\cdots$$
6724.2.a.g $$8$$ $$53.691$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$1$$ $$3$$ $$0$$ $$-$$ $$+$$ $$q+\beta _{1}q^{3}-\beta _{6}q^{5}+(-\beta _{2}-\beta _{5}-\beta _{6}+\cdots)q^{7}+\cdots$$
6724.2.a.h $$12$$ $$53.691$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-1$$ $$-2$$ $$0$$ $$-$$ $$-$$ $$q-\beta _{1}q^{3}-\beta _{3}q^{5}+(\beta _{4}+\beta _{10})q^{7}+(1+\cdots)q^{9}+\cdots$$
6724.2.a.i $$12$$ $$53.691$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$1$$ $$-2$$ $$0$$ $$-$$ $$+$$ $$q+\beta _{1}q^{3}-\beta _{3}q^{5}+(-\beta _{4}-\beta _{10})q^{7}+\cdots$$
6724.2.a.j $$16$$ $$53.691$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$6$$ $$0$$ $$-$$ $$+$$ $$q+\beta _{1}q^{3}-\beta _{4}q^{5}+(\beta _{1}-\beta _{11})q^{7}+(2+\cdots)q^{9}+\cdots$$
6724.2.a.k $$18$$ $$53.691$$ $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ None $$0$$ $$-5$$ $$2$$ $$-7$$ $$-$$ $$-$$ $$q-\beta _{1}q^{3}+(1+\beta _{4}+\beta _{7}+\beta _{12}-\beta _{17})q^{5}+\cdots$$
6724.2.a.l $$18$$ $$53.691$$ $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ None $$0$$ $$5$$ $$2$$ $$7$$ $$-$$ $$+$$ $$q+\beta _{1}q^{3}+(1+\beta _{4}+\beta _{7}+\beta _{12}-\beta _{17})q^{5}+\cdots$$
6724.2.a.m $$24$$ $$53.691$$ None $$0$$ $$0$$ $$-16$$ $$0$$ $$-$$ $$-$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(6724))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(6724)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(41))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(82))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(164))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(1681))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(3362))$$$$^{\oplus 2}$$