Space of modular forms of level 198, weight 4, and trivial character

• Label: 198.4.a
• Level: $198$
• Weight: $4$
• Character orbit: 198.a
• Rep. character: $\chi_{198}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $10$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	198.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(198))\).

	Total	New	Old
Modular forms	116	13	103
Cusp forms	100	13	87
Eisenstein series	16	0	16

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	Dim
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(1\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(5\)

Trace form

\( 13 q - 2 q^{2} + 52 q^{4} - 12 q^{5} + 52 q^{7} - 8 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(198))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	11
198.4.a.a	$198$	$4$	198.a	1.a	$1$	$1$	$1$	$11.682$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-10\)	\(16\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-10q^{5}+2^{4}q^{7}-8q^{8}+\cdots\)
198.4.a.b	$198$	$4$	198.a	1.a	$1$	$1$	$1$	$11.682$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(3\)	\(-10\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+3q^{5}-10q^{7}-8q^{8}+\cdots\)
198.4.a.c	$198$	$4$	198.a	1.a	$1$	$1$	$1$	$11.682$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(8\)	\(-22\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+8q^{5}-22q^{7}-8q^{8}+\cdots\)
198.4.a.d	$198$	$4$	198.a	1.a	$1$	$1$	$1$	$11.682$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-14\)	\(-8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-14q^{5}-8q^{7}+8q^{8}+\cdots\)
198.4.a.e	$198$	$4$	198.a	1.a	$1$	$1$	$1$	$11.682$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-8\)	\(-22\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-8q^{5}-22q^{7}+8q^{8}+\cdots\)
198.4.a.f	$198$	$4$	198.a	1.a	$1$	$1$	$1$	$11.682$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+14q^{7}+8q^{8}-11q^{11}+\cdots\)
198.4.a.g	$198$	$4$	198.a	1.a	$1$	$1$	$1$	$11.682$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(19\)	\(14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+19q^{5}+14q^{7}+8q^{8}+\cdots\)
198.4.a.h	$198$	$4$	198.a	1.a	$1$	$2$	$2$	$11.682$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(-10\)	\(-2\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-5-\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
198.4.a.i	$198$	$4$	198.a	1.a	$1$	$2$	$2$	$11.682$	\(\Q(\sqrt{70}) \)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(8\)	\(36\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(4+\beta )q^{5}+(18+\beta )q^{7}+\cdots\)
198.4.a.j	$198$	$4$	198.a	1.a	$1$	$2$	$2$	$11.682$	\(\Q(\sqrt{70}) \)	None	✓	$1$	$0$	\(4\)	\(0\)	\(-8\)	\(36\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-4+\beta )q^{5}+(18-\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(198))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(198)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)