Space of modular forms of level 147, weight 5, and Character 147.b

• Label: 147.5.b
• Level: $147$
• Weight: $5$
• Character orbit: 147.b
• Rep. character: $\chi_{147}(50,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $7$
• Sturm bound: $93$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	147.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(93\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(147, [\chi])\).

	Total	New	Old
Modular forms	82	60	22
Cusp forms	66	50	16
Eisenstein series	16	10	6

Trace form

\( 50 q + 2 q^{3} - 380 q^{4} + 34 q^{6} - 98 q^{9} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(147, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
147.5.b.a	$147$	$5$	147.b	3.b	$2$	$1$	$1$	$15.195$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(-9\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-9q^{3}+2^{4}q^{4}+3^{4}q^{9}-12^{2}q^{12}+\cdots\)
147.5.b.b	$147$	$5$	147.b	3.b	$2$	$1$	$1$	$15.195$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(9\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+9q^{3}+2^{4}q^{4}+3^{4}q^{9}+12^{2}q^{12}+\cdots\)
147.5.b.c	$147$	$5$	147.b	3.b	$2$	$8$	$8$	$15.195$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$3^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{3}+(-10-\beta _{2}+\cdots)q^{4}+\cdots\)
147.5.b.d	$147$	$5$	147.b	3.b	$2$	$8$	$8$	$15.195$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+\beta _{3}q^{3}+(-12+\beta _{1})q^{4}+\cdots\)
147.5.b.e	$147$	$5$	147.b	3.b	$2$	$8$	$8$	$15.195$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(-5+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots\)
147.5.b.f	$147$	$5$	147.b	3.b	$2$	$8$	$8$	$15.195$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$3^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(-10-\beta _{2}+\cdots)q^{4}+\cdots\)
147.5.b.g	$147$	$5$	147.b	3.b	$2$	$16$	$16$	$15.195$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{4}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-\beta _{4}q^{3}+(-8+\beta _{1})q^{4}+\beta _{2}q^{5}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(147, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(147, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)