Properties

Label 192.9.e
Level $192$
Weight $9$
Character orbit 192.e
Rep. character $\chi_{192}(65,\cdot)$
Character field $\Q$
Dimension $62$
Newform subspaces $12$
Sturm bound $288$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(288\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

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

Total New Old
Modular forms 268 66 202
Cusp forms 244 62 182
Eisenstein series 24 4 20

Trace form

\( 62 q - 2 q^{9} + O(q^{10}) \) \( 62 q - 2 q^{9} + 51396 q^{13} + 13124 q^{21} - 4218754 q^{25} + 1665792 q^{33} - 4288956 q^{37} - 14593536 q^{45} + 41177146 q^{49} + 7045116 q^{57} - 8181820 q^{61} - 73684480 q^{69} - 4 q^{73} + 105439678 q^{81} - 108522496 q^{85} + 80483780 q^{93} - 105074564 q^{97} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
192.9.e.a 192.e 3.b $1$ $78.217$ \(\Q\) \(\Q(\sqrt{-3}) \) 12.9.c.a \(0\) \(-81\) \(0\) \(4034\) $\mathrm{U}(1)[D_{2}]$ \(q-3^{4}q^{3}+4034q^{7}+3^{8}q^{9}+35806q^{13}+\cdots\)
192.9.e.b 192.e 3.b $1$ $78.217$ \(\Q\) \(\Q(\sqrt{-3}) \) 12.9.c.a \(0\) \(81\) \(0\) \(-4034\) $\mathrm{U}(1)[D_{2}]$ \(q+3^{4}q^{3}-4034q^{7}+3^{8}q^{9}+35806q^{13}+\cdots\)
192.9.e.c 192.e 3.b $2$ $78.217$ \(\Q(\sqrt{-2}) \) None 6.9.b.a \(0\) \(-126\) \(0\) \(-5572\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-63-3\beta )q^{3}+34\beta q^{5}-2786q^{7}+\cdots\)
192.9.e.d 192.e 3.b $2$ $78.217$ \(\Q(\sqrt{-110}) \) None 12.9.c.b \(0\) \(-102\) \(0\) \(6188\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-51+\beta )q^{3}+18\beta q^{5}+3094q^{7}+\cdots\)
192.9.e.e 192.e 3.b $2$ $78.217$ \(\Q(\sqrt{-14}) \) None 3.9.b.a \(0\) \(-90\) \(0\) \(-3500\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-45+3\beta )q^{3}+10\beta q^{5}-1750q^{7}+\cdots\)
192.9.e.f 192.e 3.b $2$ $78.217$ \(\Q(\sqrt{-14}) \) None 3.9.b.a \(0\) \(90\) \(0\) \(3500\) $\mathrm{SU}(2)[C_{2}]$ \(q+(45-3\beta )q^{3}+10\beta q^{5}+1750q^{7}+\cdots\)
192.9.e.g 192.e 3.b $2$ $78.217$ \(\Q(\sqrt{-110}) \) None 12.9.c.b \(0\) \(102\) \(0\) \(-6188\) $\mathrm{SU}(2)[C_{2}]$ \(q+(51-\beta )q^{3}+18\beta q^{5}-3094q^{7}+\cdots\)
192.9.e.h 192.e 3.b $2$ $78.217$ \(\Q(\sqrt{-2}) \) None 6.9.b.a \(0\) \(126\) \(0\) \(5572\) $\mathrm{SU}(2)[C_{2}]$ \(q+(63+3\beta )q^{3}+34\beta q^{5}+2786q^{7}+\cdots\)
192.9.e.i 192.e 3.b $8$ $78.217$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 24.9.e.a \(0\) \(-56\) \(0\) \(1584\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-7+\beta _{2})q^{3}+(-\beta _{2}+\beta _{3})q^{5}+(198+\cdots)q^{7}+\cdots\)
192.9.e.j 192.e 3.b $8$ $78.217$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 24.9.e.a \(0\) \(56\) \(0\) \(-1584\) $\mathrm{SU}(2)[C_{2}]$ \(q+(7-\beta _{2})q^{3}+(-\beta _{2}+\beta _{3})q^{5}+(-198+\cdots)q^{7}+\cdots\)
192.9.e.k 192.e 3.b $16$ $78.217$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 96.9.e.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(-3\beta _{1}+\beta _{2})q^{7}+\cdots\)
192.9.e.l 192.e 3.b $16$ $78.217$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 96.9.e.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)

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

\( S_{9}^{\mathrm{old}}(192, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(6, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 2}\)