Space of modular forms of level 171, weight 4, and trivial character

• Label: 171.4.a
• Level: $171$
• Weight: $4$
• Character orbit: 171.a
• Rep. character: $\chi_{171}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $9$
• Sturm bound: $80$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(80\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(171))\).

	Total	New	Old
Modular forms	64	22	42
Cusp forms	56	22	34
Eisenstein series	8	0	8

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
\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(6\)
\(-\)	\(-\)	$+$	\(8\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(8\)

Trace form

\( 22 q + 94 q^{4} + 6 q^{5} + 36 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(171))\) 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
171.4.a.a	$171$	$4$	171.a	1.a	$1$	$1$	$1$	$10.089$	\(\Q\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(6\)	\(-16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+6q^{5}-2^{4}q^{7}+21q^{8}+\cdots\)
171.4.a.b	$171$	$4$	171.a	1.a	$1$	$1$	$1$	$10.089$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(12\)	\(-20\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}+12q^{5}-20q^{7}-15q^{8}+\cdots\)
171.4.a.c	$171$	$4$	171.a	1.a	$1$	$1$	$1$	$10.089$	\(\Q\)	None	✓	$1$	$1$	\(3\)	\(0\)	\(-6\)	\(-16\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-6q^{5}-2^{4}q^{7}-21q^{8}+\cdots\)
171.4.a.d	$171$	$4$	171.a	1.a	$1$	$1$	$1$	$10.089$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(0\)	\(12\)	\(11\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+12q^{5}+11q^{7}-21q^{8}+\cdots\)
171.4.a.e	$171$	$4$	171.a	1.a	$1$	$2$	$2$	$10.089$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-22\)	\(36\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+\beta q^{4}+(-10-2\beta )q^{5}+(2^{4}+\cdots)q^{7}+\cdots\)
171.4.a.f	$171$	$4$	171.a	1.a	$1$	$3$	$3$	$10.089$	3.3.3144.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-14\)	\(-35\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}+\beta _{2})q^{2}+(8-\beta _{1}-2\beta _{2})q^{4}+\cdots\)
171.4.a.g	$171$	$4$	171.a	1.a	$1$	$3$	$3$	$10.089$	3.3.2700.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(12\)	\(-18\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(3+4\beta _{1}+\beta _{2})q^{4}+(4+\cdots)q^{5}+\cdots\)
171.4.a.h	$171$	$4$	171.a	1.a	$1$	$4$	$4$	$10.089$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-3\)	\(0\)	\(6\)	\(38\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(6+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)
171.4.a.i	$171$	$4$	171.a	1.a	$1$	$6$	$6$	$10.089$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(20\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(7+\beta _{3})q^{4}+(2\beta _{1}+\beta _{5})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(171))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(171)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 2}\)