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Properties

Label 168.5.z
Level $168$
Weight $5$
Character orbit 168.z
Rep. character $\chi_{168}(73,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $32$
Newform subspaces $2$
Sturm bound $160$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(160\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(168, [\chi])\).

Total New Old
Modular forms 272 32 240
Cusp forms 240 32 208
Eisenstein series 32 0 32

Trace form

\( 32 q + 56 q^{7} + 432 q^{9} + O(q^{10}) \) \( 32 q + 56 q^{7} + 432 q^{9} + 144 q^{11} - 144 q^{15} - 864 q^{17} + 1104 q^{19} - 144 q^{21} - 1440 q^{23} + 1992 q^{25} + 384 q^{29} - 1176 q^{31} + 3240 q^{33} + 3408 q^{35} + 728 q^{37} - 1584 q^{39} + 4192 q^{43} - 9392 q^{49} - 1584 q^{51} + 3312 q^{53} - 4752 q^{57} + 4320 q^{59} + 18576 q^{61} + 1728 q^{63} + 17520 q^{65} + 4624 q^{67} - 4416 q^{71} - 15240 q^{73} - 16416 q^{75} - 16512 q^{77} - 14968 q^{79} - 11664 q^{81} - 3904 q^{85} + 16200 q^{87} + 21888 q^{89} + 27456 q^{91} + 15768 q^{93} - 17952 q^{95} + 7776 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{5}^{\mathrm{new}}(168, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
168.5.z.a 168.z 7.d $16$ $17.366$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 168.5.z.a \(0\) \(-72\) \(12\) \(40\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-6+3\beta _{1})q^{3}+(\beta _{1}+\beta _{5}+\beta _{7})q^{5}+\cdots\)
168.5.z.b 168.z 7.d $16$ $17.366$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 168.5.z.b \(0\) \(72\) \(-12\) \(16\) $\mathrm{SU}(2)[C_{6}]$ \(q+(6-3\beta _{2})q^{3}+(-1-\beta _{3})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(168, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(168, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 2}\)