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Properties

Label 168.2.q
Level $168$
Weight $2$
Character orbit 168.q
Rep. character $\chi_{168}(25,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $3$
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.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(64\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(168, [\chi])\).

Total New Old
Modular forms 80 8 72
Cusp forms 48 8 40
Eisenstein series 32 0 32

Trace form

\( 8 q + 2 q^{3} - 4 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{3} - 4 q^{9} + 4 q^{11} + 12 q^{13} - 4 q^{15} + 4 q^{17} + 14 q^{19} - 4 q^{21} - 12 q^{23} - 14 q^{25} - 4 q^{27} - 8 q^{29} - 10 q^{33} - 36 q^{35} - 2 q^{37} - 2 q^{39} + 24 q^{41} - 4 q^{43} - 12 q^{49} - 4 q^{51} - 28 q^{53} + 28 q^{55} + 12 q^{57} - 20 q^{61} + 6 q^{63} - 16 q^{65} + 2 q^{67} + 8 q^{69} + 16 q^{71} + 10 q^{73} + 22 q^{75} + 36 q^{77} - 20 q^{79} - 4 q^{81} + 40 q^{83} + 24 q^{85} + 2 q^{87} + 32 q^{89} - 26 q^{91} + 10 q^{93} + 32 q^{95} - 12 q^{97} - 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(168, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
168.2.q.a 168.q 7.c $2$ $1.341$ \(\Q(\sqrt{-3}) \) None 168.2.q.a \(0\) \(-1\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
168.2.q.b 168.q 7.c $2$ $1.341$ \(\Q(\sqrt{-3}) \) None 168.2.q.b \(0\) \(1\) \(-2\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}+\cdots\)
168.2.q.c 168.q 7.c $4$ $1.341$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None 168.2.q.c \(0\) \(2\) \(1\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{2})q^{3}+(-1+2\beta _{1}+\beta _{2}-\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(168, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(168, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)