Properties

Label 68.2.a
Level $68$
Weight $2$
Character orbit 68.a
Rep. character $\chi_{68}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $1$
Sturm bound $18$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(68))\).

Total New Old
Modular forms 12 2 10
Cusp forms 7 2 5
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.

\(2\)\(17\)FrickeDim.
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 4 q^{13} - 12 q^{15} - 2 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 14 q^{25} + 8 q^{27} - 2 q^{31} + 12 q^{35} + 16 q^{37} + 16 q^{39} - 12 q^{41} + 4 q^{43} - 24 q^{45} - 6 q^{49} - 2 q^{51} + 12 q^{53} - 12 q^{55} - 8 q^{57} + 12 q^{59} - 8 q^{61} - 14 q^{63} - 24 q^{65} + 16 q^{67} - 6 q^{71} + 4 q^{73} + 14 q^{75} - 14 q^{79} + 2 q^{81} - 12 q^{83} + 12 q^{87} + 12 q^{89} - 16 q^{91} + 16 q^{93} + 24 q^{95} + 4 q^{97} + 6 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(68))\) 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 17
68.2.a.a 68.a 1.a $2$ $0.543$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-2\beta q^{5}+(-1-\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(68))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(68)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)