Space of modular forms of level 147, weight 3, and Character 147.f

• Label: 147.3.f
• Level: $147$
• Weight: $3$
• Character orbit: 147.f
• Rep. character: $\chi_{147}(19,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $26$
• Newform subspaces: $7$
• Sturm bound: $56$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.f (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(56\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(147, [\chi])\).

	Total	New	Old
Modular forms	90	26	64
Cusp forms	58	26	32
Eisenstein series	32	0	32

Trace form

\( 26 q + 4 q^{2} + 3 q^{3} - 36 q^{4} + 6 q^{5} - 4 q^{8} + 39 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(147, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
147.3.f.a	$147$	$3$	147.f	7.d	$6$	$2$	$1$	$4.005$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-3\)	\(3\)	\(-9\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-3\zeta_{6}q^{2}+(1+\zeta_{6})q^{3}+(-5+5\zeta_{6})q^{4}+\cdots\)
147.3.f.b	$147$	$3$	147.f	7.d	$6$	$2$	$1$	$4.005$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(-3\)	\(-12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(3-3\zeta_{6})q^{4}+\cdots\)
147.3.f.c	$147$	$3$	147.f	7.d	$6$	$2$	$1$	$4.005$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(-3\)	\(9\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(3-3\zeta_{6})q^{4}+\cdots\)
147.3.f.d	$147$	$3$	147.f	7.d	$6$	$2$	$1$	$4.005$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(3\)	\(12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{6}q^{2}+(1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{4}+\cdots\)
147.3.f.e	$147$	$3$	147.f	7.d	$6$	$2$	$1$	$4.005$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(2\)	\(3\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+2\zeta_{6}q^{2}+(1+\zeta_{6})q^{3}+(4-2\zeta_{6})q^{5}+\cdots\)
147.3.f.f	$147$	$3$	147.f	7.d	$6$	$8$	$4$	$4.005$	8.0.339738624.1	None		$2$	$0$	\(4\)	\(-12\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}-\beta _{7})q^{2}+(-1+\beta _{3})q^{3}+(-5+\cdots)q^{4}+\cdots\)
147.3.f.g	$147$	$3$	147.f	7.d	$6$	$8$	$4$	$4.005$	8.0.339738624.1	None		$2$	$0$	\(4\)	\(12\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}-\beta _{7})q^{2}+(1-\beta _{3})q^{3}+(-5+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(147, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(147, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)