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Properties

Label 168.3.z
Level $168$
Weight $3$
Character orbit 168.z
Rep. character $\chi_{168}(73,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $2$
Sturm bound $96$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 168.z (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(168, [\chi])\).

Total New Old
Modular forms 144 16 128
Cusp forms 112 16 96
Eisenstein series 32 0 32

Trace form

\( 16 q + 4 q^{7} + 24 q^{9} + O(q^{10}) \) \( 16 q + 4 q^{7} + 24 q^{9} - 8 q^{11} - 24 q^{15} + 48 q^{17} + 120 q^{19} + 24 q^{21} + 48 q^{23} + 36 q^{25} + 64 q^{29} - 84 q^{31} - 108 q^{33} - 168 q^{35} - 68 q^{37} - 24 q^{39} - 112 q^{43} + 200 q^{49} + 24 q^{51} - 40 q^{53} - 72 q^{57} - 240 q^{59} - 216 q^{61} - 48 q^{63} - 296 q^{65} + 56 q^{67} + 32 q^{71} + 156 q^{73} + 144 q^{75} + 124 q^{79} - 72 q^{81} - 32 q^{85} + 108 q^{87} - 192 q^{89} - 288 q^{91} - 36 q^{93} + 144 q^{95} - 48 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(168, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
168.3.z.a 168.z 7.d $8$ $4.578$ 8.0.\(\cdots\).2 None \(0\) \(-12\) \(6\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+\beta _{4})q^{3}+(1+\beta _{6})q^{5}+(1-3\beta _{4}+\cdots)q^{7}+\cdots\)
168.3.z.b 168.z 7.d $8$ $4.578$ 8.0.\(\cdots\).9 None \(0\) \(12\) \(-6\) \(8\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{2})q^{3}+(-1+\beta _{7})q^{5}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(168, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(168, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)