Space of modular forms of level 1176, weight 3, and Character 1176.d

• Label: 1176.3.d
• Level: $1176$
• Weight: $3$
• Character orbit: 1176.d
• Rep. character: $\chi_{1176}(785,\cdot)$
• Character field: $\Q$
• Dimension: $82$
• Newform subspaces: $8$
• Sturm bound: $672$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1176.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(672\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1176, [\chi])\).

	Total	New	Old
Modular forms	480	82	398
Cusp forms	416	82	334
Eisenstein series	64	0	64

Trace form

\( 82 q + 2 q^{3} + 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1176, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1176.3.d.a	$1176$	$3$	1176.d	3.b	$2$	$2$	$2$	$32.044$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta )q^{3}+2\beta q^{5}+(-7+2\beta )q^{9}+\cdots\)
1176.3.d.b	$1176$	$3$	1176.d	3.b	$2$	$2$	$2$	$32.044$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+2\beta )q^{3}+3\beta q^{5}+(-7-4\beta )q^{9}+\cdots\)
1176.3.d.c	$1176$	$3$	1176.d	3.b	$2$	$2$	$2$	$32.044$	\(\Q(\sqrt{-2}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+2\beta )q^{3}+3\beta q^{5}+(-7+4\beta )q^{9}+\cdots\)
1176.3.d.d	$1176$	$3$	1176.d	3.b	$2$	$8$	$8$	$32.044$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-2\beta _{1}-\beta _{3}-\beta _{4})q^{5}+(2+\cdots)q^{9}+\cdots\)
1176.3.d.e	$1176$	$3$	1176.d	3.b	$2$	$12$	$12$	$32.044$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(-\beta _{3}-\beta _{10})q^{5}+(\beta _{1}-\beta _{4}+\cdots)q^{9}+\cdots\)
1176.3.d.f	$1176$	$3$	1176.d	3.b	$2$	$16$	$16$	$32.044$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+\beta _{2}q^{9}-\beta _{6}q^{11}+\cdots\)
1176.3.d.g	$1176$	$3$	1176.d	3.b	$2$	$16$	$16$	$32.044$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{3}-\beta _{2}q^{5}+(1-\beta _{3})q^{9}+\beta _{8}q^{11}+\cdots\)
1176.3.d.h	$1176$	$3$	1176.d	3.b	$2$	$24$	$24$	$32.044$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(1176, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1176, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)