Space of modular forms of level 196, weight 2, and Character 196.f

• Label: 196.2.f
• Level: $196$
• Weight: $2$
• Character orbit: 196.f
• Rep. character: $\chi_{196}(19,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $32$
• Newform subspaces: $4$
• Sturm bound: $56$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.f (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(56\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(196, [\chi])\).

	Total	New	Old
Modular forms	72	48	24
Cusp forms	40	32	8
Eisenstein series	32	16	16

Trace form

32 * q + 3 * q^2 - q^4 + 6 * q^5 - 6 * q^8 - 6 * q^9 - 6 * q^10 - 12 * q^12 - 13 * q^16 + 6 * q^17 + 17 * q^18 + 12 * q^24 - 2 * q^25 + 12 * q^26 - 56 * q^29 + 34 * q^30 - 17 * q^32 - 6 * q^33 - 22 * q^36 + 14 * q^37 - 18 * q^38 - 12 * q^40 - 34 * q^44 - 8 * q^46 - 26 * q^50 - 10 * q^53 + 18 * q^54 + 28 * q^57 + 14 * q^58 - 20 * q^60 + 18 * q^61 - 22 * q^64 - 20 * q^65 + 6 * q^66 + 19 * q^72 - 30 * q^73 + 24 * q^74 + 40 * q^78 + 24 * q^80 + 20 * q^81 - 12 * q^82 - 52 * q^85 - 54 * q^86 + 54 * q^88 - 54 * q^89 + 92 * q^92 + 38 * q^93 + 30 * q^94 + 24 * q^96

Decomposition of \(S_{2}^{\mathrm{new}}(196, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
196.2.f.a	$196$	$2$	196.f	28.f	$6$	$4$	$2$	$1.565$	\(\Q(\zeta_{12})\)	None					28.2.f.a	$4$	$0$	\(-2\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-\zeta_{12}+\cdots)q^{3}+\cdots\)
196.2.f.b	$196$	$2$	196.f	28.f	$6$	$4$	$2$	$1.565$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	\(\Q(\sqrt{-7}) \)					28.2.d.a	$8$	$0$	\(1\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(3-\beta _{1}+\cdots)q^{8}+\cdots\)
196.2.f.c	$196$	$2$	196.f	28.f	$6$	$8$	$4$	$1.565$	8.0.339738624.1	\(\Q(\sqrt{-1}) \)		✓			196.2.d.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q+(-\beta _{2}-\beta _{5})q^{2}+2\beta _{4}q^{4}-\beta _{3}q^{5}+\cdots\)
196.2.f.d	$196$	$2$	196.f	28.f	$6$	$16$	$8$	$1.565$	16.0.\(\cdots\).2	None		✓	✓		196.2.d.c	$8$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{6}-\beta _{11})q^{2}+(2\beta _{1}+\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(196, [\chi])\) into lower level spaces

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