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

• Label: 147.7.f
• Level: $147$
• Weight: $7$
• Character orbit: 147.f
• Rep. character: $\chi_{147}(19,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $80$
• Newform subspaces: $6$
• Sturm bound: $130$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	147.f (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(130\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

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

	Total	New	Old
Modular forms	240	80	160
Cusp forms	208	80	128
Eisenstein series	32	0	32

Trace form

\( 80 q + 20 q^{2} - 1100 q^{4} + 336 q^{5} - 3812 q^{8} + 9720 q^{9} + O(q^{10}) \)

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.f.a	$147$	$7$	147.f	7.d	$6$	$8$	$4$	$33.818$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(-5\)	\(-108\)	\(42\)	\(0\)	$3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{1}-\beta _{2})q^{2}+(-9-9\beta _{2})q^{3}+\cdots\)
147.7.f.b	$147$	$7$	147.f	7.d	$6$	$8$	$4$	$33.818$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(-5\)	\(-108\)	\(252\)	\(0\)	$2^{2}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{1}-\beta _{2})q^{2}+(-18-9\beta _{2}+\cdots)q^{3}+\cdots\)
147.7.f.c	$147$	$7$	147.f	7.d	$6$	$8$	$4$	$33.818$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(-5\)	\(108\)	\(-252\)	\(0\)	$2^{2}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{1}-\beta _{2})q^{2}+(18+9\beta _{2})q^{3}+\cdots\)
147.7.f.d	$147$	$7$	147.f	7.d	$6$	$8$	$4$	$33.818$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(-5\)	\(108\)	\(294\)	\(0\)	$2^{2}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{1}-\beta _{2})q^{2}+(18+9\beta _{2})q^{3}+\cdots\)
147.7.f.e	$147$	$7$	147.f	7.d	$6$	$24$	$12$	$33.818$		None		$2$	$0$	\(20\)	\(-324\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
147.7.f.f	$147$	$7$	147.f	7.d	$6$	$24$	$12$	$33.818$		None		$2$	$0$	\(20\)	\(324\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

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

\( S_{7}^{\mathrm{old}}(147, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)