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

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

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 167
668.2.a.a 668.a 1.a $2$ $5.334$ \(\Q(\sqrt{13}) \) None \(0\) \(1\) \(-6\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-3q^{5}+(1+\beta )q^{7}+\beta q^{9}+(4+\cdots)q^{13}+\cdots\)
668.2.a.b 668.a 1.a $5$ $5.334$ 5.5.826865.1 None \(0\) \(3\) \(10\) \(9\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{2})q^{3}+2q^{5}+(2-\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots\)
668.2.a.c 668.a 1.a $7$ $5.334$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-4\) \(-2\) \(-12\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{3})q^{3}+(\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)

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

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