Space of modular forms of level 196, weight 4, and trivial character

• Label: 196.4.a
• Level: $196$
• Weight: $4$
• Character orbit: 196.a
• Rep. character: $\chi_{196}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $7$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	196.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(196))\).

	Total	New	Old
Modular forms	96	10	86
Cusp forms	72	10	62
Eisenstein series	24	0	24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(7\)	Fricke	Dim
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(6\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(4\)

Trace form

\( 10 q + 6 q^{3} + 2 q^{5} + 30 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(196))\) 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	7
196.4.a.a	$196$	$4$	196.a	1.a	$1$	$1$	$1$	$11.564$	\(\Q\)	None	✓	✓			196.4.a.a	$1$	$0$	\(0\)	\(-4\)	\(-20\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-20q^{5}-11q^{9}+44q^{11}+\cdots\)
196.4.a.b	$196$	$4$	196.a	1.a	$1$	$1$	$1$	$11.564$	\(\Q\)	None	✓				28.4.a.b	$1$	$0$	\(0\)	\(-4\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-6q^{5}-11q^{9}-12q^{11}+82q^{13}+\cdots\)
196.4.a.c	$196$	$4$	196.a	1.a	$1$	$1$	$1$	$11.564$	\(\Q\)	None	✓	✓			196.4.a.a	$1$	$0$	\(0\)	\(4\)	\(20\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+20q^{5}-11q^{9}+44q^{11}+\cdots\)
196.4.a.d	$196$	$4$	196.a	1.a	$1$	$1$	$1$	$11.564$	\(\Q\)	None	✓				28.4.a.a	$1$	$0$	\(0\)	\(10\)	\(8\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}+8q^{5}+73q^{9}-40q^{11}+\cdots\)
196.4.a.e	$196$	$4$	196.a	1.a	$1$	$2$	$2$	$11.564$	\(\Q(\sqrt{37}) \)	None	✓				28.4.e.a	$1$	$1$	\(0\)	\(0\)	\(-14\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-7+2\beta )q^{5}+10q^{9}+(2^{4}+\cdots)q^{11}+\cdots\)
196.4.a.f	$196$	$4$	196.a	1.a	$1$	$2$	$2$	$11.564$	\(\Q(\sqrt{2}) \)	None	✓	✓			196.4.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{5}-5^{2}q^{9}-26q^{11}+\cdots\)
196.4.a.g	$196$	$4$	196.a	1.a	$1$	$2$	$2$	$11.564$	\(\Q(\sqrt{37}) \)	None	✓		✓		28.4.e.a	$1$	$0$	\(0\)	\(0\)	\(14\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(7+2\beta )q^{5}+10q^{9}+(2^{4}-7\beta )q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(196))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(196)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)