Space of modular forms of level 189, weight 2, and trivial character

• Label: 189.2.a
• Level: $189$
• Weight: $2$
• Character orbit: 189.a
• Rep. character: $\chi_{189}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $6$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(48\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(189))\).

	Total	New	Old
Modular forms	30	8	22
Cusp forms	19	8	11
Eisenstein series	11	0	11

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(8\)	\(3\)	\(5\)		\(5\)	\(3\)	\(2\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(10\)	\(3\)	\(7\)		\(7\)	\(3\)	\(4\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(7\)	\(1\)	\(6\)		\(4\)	\(1\)	\(3\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(12\)	\(2\)	\(10\)		\(7\)	\(2\)	\(5\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(18\)	\(6\)	\(12\)		\(12\)	\(6\)	\(6\)		\(6\)	\(0\)	\(6\)

Trace form

8 * q + 12 * q^4 - 4 * q^10 - 12 * q^13 + 12 * q^16 + 12 * q^19 - 4 * q^22 - 16 * q^28 + 20 * q^31 - 36 * q^34 - 40 * q^37 - 48 * q^40 + 28 * q^43 - 12 * q^46 + 8 * q^49 - 8 * q^52 + 52 * q^55 + 8 * q^58 - 12 * q^61 - 4 * q^64 + 16 * q^70 + 8 * q^73 + 40 * q^76 + 4 * q^79 + 28 * q^82 - 12 * q^85 - 36 * q^88 + 4 * q^91 - 12 * q^94 - 28 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(189))\) 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	7
189.2.a.a	$189$	$2$	189.a	1.a	$1$	$1$	$1$	$1.509$	\(\Q\)	None	✓	✓	✓	✓	189.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{5}-q^{7}+2q^{10}+\cdots\)
189.2.a.b	$189$	$2$	189.a	1.a	$1$	$1$	$1$	$1.509$	\(\Q\)	None	✓	✓	✓	✓	189.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}+q^{7}-6q^{11}-4q^{13}+\cdots\)
189.2.a.c	$189$	$2$	189.a	1.a	$1$	$1$	$1$	$1.509$	\(\Q\)	None	✓	✓			189.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+3q^{5}+q^{7}+6q^{11}-4q^{13}+\cdots\)
189.2.a.d	$189$	$2$	189.a	1.a	$1$	$1$	$1$	$1.509$	\(\Q\)	None	✓	✓			189.2.a.a	$1$	$0$	\(2\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{5}-q^{7}+2q^{10}+\cdots\)
189.2.a.e	$189$	$2$	189.a	1.a	$1$	$2$	$2$	$1.509$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		189.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+\beta q^{5}+q^{7}-\beta q^{8}+3q^{10}+\cdots\)
189.2.a.f	$189$	$2$	189.a	1.a	$1$	$2$	$2$	$1.509$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓		189.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+5q^{4}-\beta q^{5}-q^{7}+3\beta q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(189))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(189)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)