Space of modular forms of level 198, weight 2, and trivial character

• Label: 198.2.a
• Level: $198$
• Weight: $2$
• Character orbit: 198.a
• Rep. character: $\chi_{198}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $72$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	198.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(72\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(198))\).

	Total	New	Old
Modular forms	44	5	39
Cusp forms	29	5	24
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.

\(2\)	\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(20\)	\(1\)	\(19\)		\(13\)	\(1\)	\(12\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(24\)	\(4\)	\(20\)		\(16\)	\(4\)	\(12\)		\(8\)	\(0\)	\(8\)

Trace form

5 * q - q^2 + 5 * q^4 + 2 * q^5 - q^8 - 2 * q^10 + q^11 - 2 * q^13 + 8 * q^14 + 5 * q^16 + 6 * q^17 + 4 * q^19 + 2 * q^20 - q^22 - 4 * q^23 - 5 * q^25 - 2 * q^26 - 22 * q^29 - 8 * q^31 - q^32 - 6 * q^34 - 2 * q^37 - 8 * q^38 - 2 * q^40 - 2 * q^41 - 4 * q^43 + q^44 - 8 * q^46 + 20 * q^47 - 3 * q^49 - 15 * q^50 - 2 * q^52 - 6 * q^53 - 6 * q^55 + 8 * q^56 - 2 * q^58 - 12 * q^59 - 34 * q^61 + 16 * q^62 + 5 * q^64 + 28 * q^65 + 4 * q^67 + 6 * q^68 + 4 * q^71 + 18 * q^73 - 14 * q^74 + 4 * q^76 + 24 * q^79 + 2 * q^80 + 2 * q^82 + 4 * q^83 + 12 * q^85 - q^88 - 14 * q^89 + 16 * q^91 - 4 * q^92 + 16 * q^94 - 8 * q^95 + 2 * q^97 - 9 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(198))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	11
198.2.a.a	$198$	$2$	198.a	1.a	$1$	$1$	$1$	$1.581$	\(\Q\)	None	✓		✓	✓	66.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{5}-4q^{7}-q^{8}+2q^{10}+\cdots\)
198.2.a.b	$198$	$2$	198.a	1.a	$1$	$1$	$1$	$1.581$	\(\Q\)	None	✓	✓	✓	✓	198.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}+q^{11}+2q^{13}+\cdots\)
198.2.a.c	$198$	$2$	198.a	1.a	$1$	$1$	$1$	$1.581$	\(\Q\)	None	✓		✓	✓	66.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(4\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{5}-2q^{7}-q^{8}-4q^{10}+\cdots\)
198.2.a.d	$198$	$2$	198.a	1.a	$1$	$1$	$1$	$1.581$	\(\Q\)	None	✓	✓	✓	✓	198.2.a.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}-q^{11}+2q^{13}+\cdots\)
198.2.a.e	$198$	$2$	198.a	1.a	$1$	$1$	$1$	$1.581$	\(\Q\)	None	✓		✓	✓	66.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}+q^{11}-4q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(198))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(198)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)