Properties

Label 168.2.a
Level $168$
Weight $2$
Character orbit 168.a
Rep. character $\chi_{168}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $64$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(64\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(168))\).

Total New Old
Modular forms 40 2 38
Cusp forms 25 2 23
Eisenstein series 15 0 15

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2 q + 4 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{5} + 2 q^{9} + 4 q^{13} + 4 q^{17} - 2 q^{21} - 8 q^{23} - 2 q^{25} - 4 q^{29} - 16 q^{31} - 4 q^{37} - 8 q^{39} - 12 q^{41} + 8 q^{43} + 4 q^{45} + 2 q^{49} + 8 q^{51} - 4 q^{53} - 8 q^{57} + 16 q^{59} - 12 q^{61} + 8 q^{65} + 24 q^{67} - 8 q^{71} - 12 q^{73} + 2 q^{81} + 16 q^{83} + 8 q^{85} + 16 q^{87} + 4 q^{89} + 8 q^{91} + 20 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(168))\) 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 3 7
168.2.a.a 168.a 1.a $1$ $1.341$ \(\Q\) None \(0\) \(-1\) \(2\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+q^{7}+q^{9}+6q^{13}-2q^{15}+\cdots\)
168.2.a.b 168.a 1.a $1$ $1.341$ \(\Q\) None \(0\) \(1\) \(2\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}-q^{7}+q^{9}-2q^{13}+2q^{15}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(168))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(168)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)