Space of modular forms of level 196, weight 3, and Character 196.g

• Label: 196.3.g
• Level: $196$
• Weight: $3$
• Character orbit: 196.g
• Rep. character: $\chi_{196}(67,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $72$
• Newform subspaces: $11$
• Sturm bound: $84$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	196.g (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(84\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(196, [\chi])\).

	Total	New	Old
Modular forms	128	88	40
Cusp forms	96	72	24
Eisenstein series	32	16	16

Trace form

\( 72 q + 3 q^{2} + 7 q^{4} + 2 q^{5} + 12 q^{6} + 6 q^{8} + 86 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(196, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
196.3.g.a	$196$	$3$	196.g	28.g	$6$	$4$	$2$	$5.341$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+2\beta _{2}q^{2}+(-4-4\beta _{2})q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
196.3.g.b	$196$	$3$	196.g	28.g	$6$	$4$	$2$	$5.341$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+2\beta _{2}q^{2}+(\beta _{1}-\beta _{3})q^{3}+(-4-4\beta _{2}+\cdots)q^{4}+\cdots\)
196.3.g.c	$196$	$3$	196.g	28.g	$6$	$4$	$2$	$5.341$	\(\Q(\sqrt{-3}, \sqrt{-5})\)	None		$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+2\beta _{2})q^{2}-\beta _{1}q^{3}-4\beta _{2}q^{4}+\cdots\)
196.3.g.d	$196$	$3$	196.g	28.g	$6$	$4$	$2$	$5.341$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	\(\Q(\sqrt{-7}) \)		$8$	$0$	\(-3\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-2\beta _{2}+\beta _{3})q^{2}+(1-3\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
196.3.g.e	$196$	$3$	196.g	28.g	$6$	$4$	$2$	$5.341$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q+(2+2\beta _{2})q^{2}+4\beta _{2}q^{4}+\beta _{1}q^{5}-8q^{8}+\cdots\)
196.3.g.f	$196$	$3$	196.g	28.g	$6$	$4$	$2$	$5.341$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(8\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+2q^{2}+(\beta _{1}-\beta _{3})q^{3}+4q^{4}+4\beta _{1}q^{5}+\cdots\)
196.3.g.g	$196$	$3$	196.g	28.g	$6$	$4$	$2$	$5.341$	\(\Q(\sqrt{-3}, \sqrt{-5})\)	None		$4$	$0$	\(8\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+2q^{2}+\beta _{1}q^{3}+4q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
196.3.g.h	$196$	$3$	196.g	28.g	$6$	$8$	$4$	$5.341$	8.0.207360000.1	None		$8$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{2}-\beta _{3})q^{2}+(\beta _{1}-2\beta _{4}-2\beta _{5}-2\beta _{7})q^{3}+\cdots\)
196.3.g.i	$196$	$3$	196.g	28.g	$6$	$12$	$6$	$5.341$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(-2\)	\(0\)	\(2\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{6}q^{2}-\beta _{4}q^{3}+(-\beta _{1}+\beta _{5}+\beta _{9}+\cdots)q^{4}+\cdots\)
196.3.g.j	$196$	$3$	196.g	28.g	$6$	$12$	$6$	$5.341$	12.0.\(\cdots\).1	None		$4$	$0$	\(1\)	\(0\)	\(-4\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}-\beta _{4}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
196.3.g.k	$196$	$3$	196.g	28.g	$6$	$12$	$6$	$5.341$	12.0.\(\cdots\).1	None		$4$	$0$	\(1\)	\(0\)	\(4\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}-\beta _{4}q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(196, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(196, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)