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 - 10 * q^10 - q^11 + 6 * q^13 - 4 * q^14 + 17 * q^16 + 7 * q^17 + 2 * q^19 + 8 * q^20 - 20 * q^22 - 4 * q^23 + 13 * q^25 - 14 * q^26 - 30 * q^28 + 4 * q^29 - 4 * q^31 - 21 * q^32 - 22 * q^34 - 15 * q^35 + 20 * q^37 - 3 * q^38 - 66 * q^40 + 16 * q^41 - 17 * q^43 + 10 * q^44 - 8 * q^46 - 3 * q^47 + 13 * q^49 + q^50 + 14 * q^52 - 10 * q^53 - 3 * q^55 + 14 * q^58 + 26 * q^59 + 17 * q^61 - 16 * q^62 + 53 * q^64 + 36 * q^65 - 8 * q^67 - 16 * q^68 + 24 * q^70 - 6 * q^71 - 5 * q^73 + 26 * q^74 + 5 * q^76 + 17 * q^77 + 8 * q^79 - 22 * q^80 - 22 * q^82 - 44 * q^83 + 39 * q^85 - 12 * q^88 - 14 * q^89 - 20 * q^91 + 4 * q^92 + 24 * q^94 - 7 * q^95 - 10 * q^97 + 47 * q^98

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	Twist minimal	Largest	Maximal	Minimal twist	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	✓				57.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{5}+3q^{8}-2q^{10}+\cdots\)
171.2.a.b	$171$	$2$	171.a	1.a	$1$	$1$	$1$	$1.365$	\(\Q\)	None	✓		✓	✓	19.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}-q^{7}-3q^{11}-4q^{13}+\cdots\)
171.2.a.c	$171$	$2$	171.a	1.a	$1$	$1$	$1$	$1.365$	\(\Q\)	None	✓				57.2.a.b	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{5}+3q^{7}-2q^{10}+\cdots\)
171.2.a.d	$171$	$2$	171.a	1.a	$1$	$1$	$1$	$1.365$	\(\Q\)	None	✓				57.2.a.a	$1$	$0$	\(2\)	\(0\)	\(3\)	\(-5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+3q^{5}-5q^{7}+6q^{10}+\cdots\)
171.2.a.e	$171$	$2$	171.a	1.a	$1$	$4$	$4$	$1.365$	4.4.13068.1	None	✓	✓	✓	✓	171.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{3})q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)

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

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