Space of modular forms of level 147, weight 7, and Character 147.b

• Label: 147.7.b
• Level: $147$
• Weight: $7$
• Character orbit: 147.b
• Rep. character: $\chi_{147}(50,\cdot)$
• Character field: $\Q$
• Dimension: $77$
• Newform subspaces: $6$
• Sturm bound: $130$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	147.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(130\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(147, [\chi])\).

	Total	New	Old
Modular forms	120	87	33
Cusp forms	104	77	27
Eisenstein series	16	10	6

Trace form

\( 77 q - 25 q^{3} - 2236 q^{4} - 350 q^{6} + 331 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(147, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
147.7.b.a	$147$	$7$	147.b	3.b	$2$	$1$	$1$	$33.818$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(27\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{3}q^{3}+2^{6}q^{4}+3^{6}q^{9}+12^{3}q^{12}+\cdots\)
147.7.b.b	$147$	$7$	147.b	3.b	$2$	$12$	$12$	$33.818$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(-52\)	\(0\)	\(0\)	$2^{3}\cdot 3^{8}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-4-\beta _{3})q^{3}+(-43+\beta _{2}+\cdots)q^{4}+\cdots\)
147.7.b.c	$147$	$7$	147.b	3.b	$2$	$12$	$12$	$33.818$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{19}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-21+\beta _{4})q^{4}+\cdots\)
147.7.b.d	$147$	$7$	147.b	3.b	$2$	$14$	$14$	$33.818$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		$2$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)	$2^{5}\cdot 3^{11}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(-3^{3}+\beta _{2})q^{4}+\cdots\)
147.7.b.e	$147$	$7$	147.b	3.b	$2$	$14$	$14$	$33.818$	\(\mathbb{Q}[x]/(x^{14} + \cdots)\)	None		$2$	$0$	\(0\)	\(1\)	\(0\)	\(0\)	$2^{5}\cdot 3^{11}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(-3^{3}+\beta _{2})q^{4}+\cdots\)
147.7.b.f	$147$	$7$	147.b	3.b	$2$	$24$	$24$	$33.818$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(147, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(147, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)