Space of modular forms of level 192, weight 7, and Character 192.e

• Label: 192.7.e
• Level: $192$
• Weight: $7$
• Character orbit: 192.e
• Rep. character: $\chi_{192}(65,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $10$
• Sturm bound: $224$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	192.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(224\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(192, [\chi])\).

	Total	New	Old
Modular forms	204	50	154
Cusp forms	180	46	134
Eisenstein series	24	4	20

Trace form

\( 46 q - 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(192, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
192.7.e.a	$192$	$7$	192.e	3.b	$2$	$1$	$1$	$44.170$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-27\)	\(0\)	\(286\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{3}q^{3}+286q^{7}+3^{6}q^{9}-506q^{13}+\cdots\)
192.7.e.b	$192$	$7$	192.e	3.b	$2$	$1$	$1$	$44.170$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(27\)	\(0\)	\(-286\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{3}q^{3}-286q^{7}+3^{6}q^{9}-506q^{13}+\cdots\)
192.7.e.c	$192$	$7$	192.e	3.b	$2$	$2$	$2$	$44.170$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(-42\)	\(0\)	\(4\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-21-\beta )q^{3}+10\beta q^{5}+2q^{7}+(153+\cdots)q^{9}+\cdots\)
192.7.e.d	$192$	$7$	192.e	3.b	$2$	$2$	$2$	$44.170$	\(\Q(\sqrt{-5}) \)	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(-484\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3-\beta )q^{3}+6\beta q^{5}-242q^{7}+(-711+\cdots)q^{9}+\cdots\)
192.7.e.e	$192$	$7$	192.e	3.b	$2$	$2$	$2$	$44.170$	\(\Q(\sqrt{-5}) \)	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(484\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3+\beta )q^{3}+6\beta q^{5}+242q^{7}+(-711+\cdots)q^{9}+\cdots\)
192.7.e.f	$192$	$7$	192.e	3.b	$2$	$2$	$2$	$44.170$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(42\)	\(0\)	\(-4\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(21+\beta )q^{3}+10\beta q^{5}-2q^{7}+(153+\cdots)q^{9}+\cdots\)
192.7.e.g	$192$	$7$	192.e	3.b	$2$	$6$	$6$	$44.170$	6.0.1173604352.2	None		$2$	$0$	\(0\)	\(-10\)	\(0\)	\(-156\)	$2^{22}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-3^{3}+\cdots)q^{7}+\cdots\)
192.7.e.h	$192$	$7$	192.e	3.b	$2$	$6$	$6$	$44.170$	6.0.1173604352.2	None		$2$	$0$	\(0\)	\(10\)	\(0\)	\(156\)	$2^{22}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(3^{3}-3\beta _{1}+\cdots)q^{7}+\cdots\)
192.7.e.i	$192$	$7$	192.e	3.b	$2$	$12$	$12$	$44.170$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{56}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-\beta _{3}q^{5}-\beta _{1}q^{7}+(-2^{6}+\beta _{3}+\cdots)q^{9}+\cdots\)
192.7.e.j	$192$	$7$	192.e	3.b	$2$	$12$	$12$	$44.170$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{56}\cdot 3^{19}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+\beta _{2}q^{5}+(-2\beta _{3}-\beta _{6})q^{7}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(192, [\chi])\) into lower level spaces

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