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Properties

Label 162.9.b
Level $162$
Weight $9$
Character orbit 162.b
Rep. character $\chi_{162}(161,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $3$
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.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
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 228 32 196
Cusp forms 204 32 172
Eisenstein series 24 0 24

Trace form

\( 32 q - 4096 q^{4} - 3692 q^{7} + O(q^{10}) \) \( 32 q - 4096 q^{4} - 3692 q^{7} - 10752 q^{10} - 63860 q^{13} + 524288 q^{16} - 8036 q^{19} + 123648 q^{22} - 3885076 q^{25} + 472576 q^{28} - 1370444 q^{31} + 2032128 q^{34} + 3715192 q^{37} + 1376256 q^{40} + 1990744 q^{43} + 7417344 q^{46} + 15054252 q^{49} + 8174080 q^{52} - 2065500 q^{55} - 17909760 q^{58} - 14469284 q^{61} - 67108864 q^{64} - 47236208 q^{67} + 10000896 q^{70} + 50227084 q^{73} + 1028608 q^{76} - 24064508 q^{79} - 90682368 q^{82} + 148308768 q^{85} - 15826944 q^{88} - 34561132 q^{91} + 220356096 q^{94} + 470524984 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.b.a 162.b 3.b $8$ $65.995$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(-8876\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{4}q^{2}-2^{7}q^{4}+(-66\beta _{4}-\beta _{6}+\cdots)q^{5}+\cdots\)
162.9.b.b 162.b 3.b $8$ $65.995$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(1492\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{4}q^{2}-2^{7}q^{4}+(150\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{5}+\cdots\)
162.9.b.c 162.b 3.b $16$ $65.995$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(3692\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{8}q^{2}-2^{7}q^{4}-\beta _{10}q^{5}+(231-\beta _{1}+\cdots)q^{7}+\cdots\)

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

\( S_{9}^{\mathrm{old}}(162, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 2}\)