# Properties

 Label 668.2.a Level $668$ Weight $2$ Character orbit 668.a Rep. character $\chi_{668}(1,\cdot)$ Character field $\Q$ Dimension $14$ Newform subspaces $3$ Sturm bound $168$ Trace bound $1$

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## Defining parameters

 Level: $$N$$ $$=$$ $$668 = 2^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 668.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$168$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(668))$$.

Total New Old
Modular forms 87 14 73
Cusp forms 82 14 68
Eisenstein series 5 0 5

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

$$2$$$$167$$FrickeDim.
$$-$$$$+$$$$-$$$$7$$
$$-$$$$-$$$$+$$$$7$$
Plus space$$+$$$$7$$
Minus space$$-$$$$7$$

## Trace form

 $$14q + 2q^{5} + 14q^{9} + O(q^{10})$$ $$14q + 2q^{5} + 14q^{9} - 2q^{11} - 4q^{13} - 14q^{15} - 2q^{17} - 2q^{19} - 14q^{23} + 6q^{25} - 2q^{29} + 2q^{31} - 16q^{33} + 2q^{35} - 10q^{37} - 6q^{39} - 4q^{41} - 4q^{43} + 2q^{45} - 2q^{47} + 30q^{49} + 2q^{51} + 6q^{55} + 4q^{57} - 6q^{59} - 2q^{61} - 14q^{63} - 22q^{65} - 12q^{67} + 14q^{69} + 8q^{71} + 18q^{73} + 26q^{75} + 2q^{79} + 6q^{81} + 22q^{83} - 34q^{85} - 2q^{87} - 14q^{89} + 6q^{91} - 32q^{93} + 8q^{95} - 44q^{97} - 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(668))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 167
668.2.a.a $$2$$ $$5.334$$ $$\Q(\sqrt{13})$$ None $$0$$ $$1$$ $$-6$$ $$3$$ $$-$$ $$+$$ $$q+\beta q^{3}-3q^{5}+(1+\beta )q^{7}+\beta q^{9}+(4+\cdots)q^{13}+\cdots$$
668.2.a.b $$5$$ $$5.334$$ 5.5.826865.1 None $$0$$ $$3$$ $$10$$ $$9$$ $$-$$ $$+$$ $$q+(1-\beta _{2})q^{3}+2q^{5}+(2-\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots$$
668.2.a.c $$7$$ $$5.334$$ $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ None $$0$$ $$-4$$ $$-2$$ $$-12$$ $$-$$ $$-$$ $$q+(-1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(668))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(668)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(167))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(334))$$$$^{\oplus 2}$$