Properties

Label 162.9.d
Level $162$
Weight $9$
Character orbit 162.d
Rep. character $\chi_{162}(53,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $64$
Newform subspaces $8$
Sturm bound $243$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 8 \)
Sturm bound: \(243\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(162, [\chi])\).

Total New Old
Modular forms 456 64 392
Cusp forms 408 64 344
Eisenstein series 48 0 48

Trace form

\( 64 q + 4096 q^{4} + 9230 q^{7} + O(q^{10}) \) \( 64 q + 4096 q^{4} + 9230 q^{7} + 16850 q^{13} - 524288 q^{16} - 1279684 q^{19} + 123648 q^{22} + 1823212 q^{25} + 2362880 q^{28} - 870700 q^{31} - 440832 q^{34} - 3404740 q^{37} + 2714960 q^{43} - 14834688 q^{46} - 36124338 q^{49} - 2156800 q^{52} + 48748104 q^{55} + 39264000 q^{58} - 24448462 q^{61} - 134217728 q^{64} - 12429238 q^{67} + 112757760 q^{70} - 35788732 q^{73} - 81899776 q^{76} - 100297126 q^{79} + 134701056 q^{82} + 148652136 q^{85} - 15826944 q^{88} - 177687860 q^{91} + 23665152 q^{94} - 29767690 q^{97} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(162, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
162.9.d.a 162.d 9.d $4$ $65.995$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-5572\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{3})q^{2}+(2^{7}-2^{7}\beta _{2})q^{4}+\cdots\)
162.9.d.b 162.d 9.d $4$ $65.995$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-3304\) $\mathrm{SU}(2)[C_{6}]$ \(q+(8\beta _{1}-8\beta _{3})q^{2}+(2^{7}-2^{7}\beta _{2})q^{4}+\cdots\)
162.9.d.c 162.d 9.d $4$ $65.995$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(4130\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{3})q^{2}+(2^{7}-2^{7}\beta _{2})q^{4}+\cdots\)
162.9.d.d 162.d 9.d $4$ $65.995$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(7064\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-8\beta _{1}+8\beta _{3})q^{2}+(2^{7}-2^{7}\beta _{2})q^{4}+\cdots\)
162.9.d.e 162.d 9.d $8$ $65.995$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(-308\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2\zeta_{24}^{4}-2\zeta_{24}^{5})q^{2}+(2^{7}-2^{7}\zeta_{24}^{2}+\cdots)q^{4}+\cdots\)
162.9.d.f 162.d 9.d $8$ $65.995$ 8.0.\(\cdots\).28 None \(0\) \(0\) \(0\) \(-164\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}-2^{7}\beta _{1}q^{4}+(21\beta _{2}-\beta _{4})q^{5}+\cdots\)
162.9.d.g 162.d 9.d $16$ $65.995$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-1492\) $\mathrm{SU}(2)[C_{6}]$ \(q+8\beta _{8}q^{2}+(2^{7}-2^{7}\beta _{1})q^{4}+(\beta _{8}+151\beta _{9}+\cdots)q^{5}+\cdots\)
162.9.d.h 162.d 9.d $16$ $65.995$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(8876\) $\mathrm{SU}(2)[C_{6}]$ \(q+8\beta _{9}q^{2}+2^{7}\beta _{1}q^{4}+(67\beta _{8}-\beta _{12}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(162, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(162, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)