Space of modular forms of level 171, weight 2, and trivial character

• Label: 171.2.a
• Level: $171$
• Weight: $2$
• Character orbit: 171.a
• Rep. character: $\chi_{171}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $5$
• Sturm bound: $40$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(40\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(171))\).

	Total	New	Old
Modular forms	24	8	16
Cusp forms	17	8	9
Eisenstein series	7	0	7

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

\(3\)	\(19\)	Fricke	Dim
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(3\)
\(-\)	\(-\)	$+$	\(1\)
Plus space		\(+\)	\(1\)
Minus space		\(-\)	\(7\)

Trace form

\( 8 q + 3 q^{2} + 11 q^{4} + q^{5} - q^{7} + 3 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(171))\) 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	19
171.2.a.a	$171$	$2$	171.a	1.a	$1$	$1$	$1$	$1.365$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{5}+3q^{8}-2q^{10}+\cdots\)
171.2.a.b	$171$	$2$	171.a	1.a	$1$	$1$	$1$	$1.365$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}-q^{7}-3q^{11}-4q^{13}+\cdots\)
171.2.a.c	$171$	$2$	171.a	1.a	$1$	$1$	$1$	$1.365$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{5}+3q^{7}-2q^{10}+\cdots\)
171.2.a.d	$171$	$2$	171.a	1.a	$1$	$1$	$1$	$1.365$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(3\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{5}-5q^{7}+6q^{10}+\cdots\)
171.2.a.e	$171$	$2$	171.a	1.a	$1$	$4$	$4$	$1.365$	4.4.13068.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{3})q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(171))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(171)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 2}\)