Space of modular forms of level 192, weight 9, and Character 192.e

• 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: $9$

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

\( 62 q - 2 q^{9} + O(q^{10}) \)

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

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
192.9.e.a	$192$	$9$	192.e	3.b	$2$	$1$	$1$	$78.217$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-81\)	\(0\)	\(4034\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{4}q^{3}+4034q^{7}+3^{8}q^{9}+35806q^{13}+\cdots\)
192.9.e.b	$192$	$9$	192.e	3.b	$2$	$1$	$1$	$78.217$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(81\)	\(0\)	\(-4034\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{4}q^{3}-4034q^{7}+3^{8}q^{9}+35806q^{13}+\cdots\)
192.9.e.c	$192$	$9$	192.e	3.b	$2$	$2$	$2$	$78.217$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(-126\)	\(0\)	\(-5572\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-63-3\beta )q^{3}+34\beta q^{5}-2786q^{7}+\cdots\)
192.9.e.d	$192$	$9$	192.e	3.b	$2$	$2$	$2$	$78.217$	\(\Q(\sqrt{-110}) \)	None		$2$	$0$	\(0\)	\(-102\)	\(0\)	\(6188\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-51+\beta )q^{3}+18\beta q^{5}+3094q^{7}+\cdots\)
192.9.e.e	$192$	$9$	192.e	3.b	$2$	$2$	$2$	$78.217$	\(\Q(\sqrt{-14}) \)	None		$2$	$0$	\(0\)	\(-90\)	\(0\)	\(-3500\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-45+3\beta )q^{3}+10\beta q^{5}-1750q^{7}+\cdots\)
192.9.e.f	$192$	$9$	192.e	3.b	$2$	$2$	$2$	$78.217$	\(\Q(\sqrt{-14}) \)	None		$2$	$0$	\(0\)	\(90\)	\(0\)	\(3500\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(45-3\beta )q^{3}+10\beta q^{5}+1750q^{7}+\cdots\)
192.9.e.g	$192$	$9$	192.e	3.b	$2$	$2$	$2$	$78.217$	\(\Q(\sqrt{-110}) \)	None		$2$	$0$	\(0\)	\(102\)	\(0\)	\(-6188\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(51-\beta )q^{3}+18\beta q^{5}-3094q^{7}+\cdots\)
192.9.e.h	$192$	$9$	192.e	3.b	$2$	$2$	$2$	$78.217$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(126\)	\(0\)	\(5572\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(63+3\beta )q^{3}+34\beta q^{5}+2786q^{7}+\cdots\)
192.9.e.i	$192$	$9$	192.e	3.b	$2$	$8$	$8$	$78.217$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(-56\)	\(0\)	\(1584\)	$2^{42}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-7+\beta _{2})q^{3}+(-\beta _{2}+\beta _{3})q^{5}+(198+\cdots)q^{7}+\cdots\)
192.9.e.j	$192$	$9$	192.e	3.b	$2$	$8$	$8$	$78.217$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(56\)	\(0\)	\(-1584\)	$2^{42}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(7-\beta _{2})q^{3}+(-\beta _{2}+\beta _{3})q^{5}+(-198+\cdots)q^{7}+\cdots\)
192.9.e.k	$192$	$9$	192.e	3.b	$2$	$16$	$16$	$78.217$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{110}\cdot 3^{30}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(-3\beta _{1}+\beta _{2})q^{7}+\cdots\)
192.9.e.l	$192$	$9$	192.e	3.b	$2$	$16$	$16$	$78.217$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{110}\cdot 3^{24}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+\cdots\)

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

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