Space of modular forms of level 147, weight 2, and trivial character

• Label: 147.2.a
• Level: $147$
• Weight: $2$
• Character orbit: 147.a
• Rep. character: $\chi_{147}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $5$
• Sturm bound: $37$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(37\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(147))\).

	Total	New	Old
Modular forms	26	7	19
Cusp forms	11	7	4
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(3\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(5\)

Trace form

\( 7 q - q^{2} - q^{3} + 7 q^{4} + 2 q^{5} + q^{6} - 9 q^{8} + 7 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(147))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7
147.2.a.a	$147$	$2$	147.a	1.a	$1$	$1$	$1$	$1.174$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+3q^{8}+\cdots\)
147.2.a.b	$147$	$2$	147.a	1.a	$1$	$1$	$1$	$1.174$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(-1\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}+2q^{5}-2q^{6}+\cdots\)
147.2.a.c	$147$	$2$	147.a	1.a	$1$	$1$	$1$	$1.174$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}-2q^{5}+2q^{6}+\cdots\)
147.2.a.d	$147$	$2$	147.a	1.a	$1$	$2$	$2$	$1.174$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
147.2.a.e	$147$	$2$	147.a	1.a	$1$	$2$	$2$	$1.174$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(2\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(147))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(147)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)