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Properties

Label 192.9.b
Level $192$
Weight $9$
Character orbit 192.b
Rep. character $\chi_{192}(31,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $3$
Sturm bound $288$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(288\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(192, [\chi])\).

Total New Old
Modular forms 268 32 236
Cusp forms 244 32 212
Eisenstein series 24 0 24

Trace form

\( 32 q + 69984 q^{9} + O(q^{10}) \) \( 32 q + 69984 q^{9} - 463680 q^{17} - 375136 q^{25} + 13124160 q^{41} - 38487648 q^{49} + 3100032 q^{57} + 135700992 q^{65} - 40027840 q^{73} + 153055008 q^{81} + 182669760 q^{89} + 215937472 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(192, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
192.9.b.a 192.b 8.d $8$ $78.217$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-3^{3}\beta _{1}q^{3}+(34\beta _{2}+\beta _{3})q^{5}+(-91\beta _{4}+\cdots)q^{7}+\cdots\)
192.9.b.b 192.b 8.d $12$ $78.217$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{3}-\beta _{2}q^{5}+(20\beta _{7}-\beta _{9})q^{7}+\cdots\)
192.9.b.c 192.b 8.d $12$ $78.217$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-3^{3}\beta _{6}q^{3}+(-3\beta _{1}+\beta _{4})q^{5}+(-3\beta _{7}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(192, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(192, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)