Space of modular forms of level 196, weight 8, and trivial character

• Label: 196.8.a
• Level: $196$
• Weight: $8$
• Character orbit: 196.a
• Rep. character: $\chi_{196}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $6$
• Sturm bound: $224$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	196.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(224\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(196))\).

	Total	New	Old
Modular forms	208	24	184
Cusp forms	184	24	160
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(54\)	\(0\)	\(54\)		\(46\)	\(0\)	\(46\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)		\(52\)	\(0\)	\(52\)		\(44\)	\(0\)	\(44\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(50\)	\(11\)	\(39\)		\(46\)	\(11\)	\(35\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(52\)	\(13\)	\(39\)		\(48\)	\(13\)	\(35\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(106\)	\(13\)	\(93\)		\(94\)	\(13\)	\(81\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(102\)	\(11\)	\(91\)		\(90\)	\(11\)	\(79\)		\(12\)	\(0\)	\(12\)

Trace form

24 * q + 252 * q^5 + 19314 * q^9 - 982 * q^11 + 4340 * q^13 - 1930 * q^15 + 13440 * q^17 - 23408 * q^19 - 40346 * q^23 + 349774 * q^25 + 105840 * q^27 - 44300 * q^29 + 15680 * q^31 + 5600 * q^33 + 438694 * q^37 + 575980 * q^39 + 438480 * q^41 - 1802848 * q^43 + 216860 * q^45 - 651840 * q^47 - 2447114 * q^51 + 2440538 * q^53 + 4260032 * q^55 - 971946 * q^57 + 1027152 * q^59 - 1919428 * q^61 - 3914256 * q^65 + 4304762 * q^67 + 12886832 * q^69 - 1233120 * q^71 + 6865320 * q^73 - 8976464 * q^75 - 7392130 * q^79 + 26902648 * q^81 + 28388640 * q^83 - 15775530 * q^85 + 4449760 * q^87 - 10770984 * q^89 - 27577942 * q^93 + 16261754 * q^95 + 44079840 * q^97 - 33267244 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(196))\) 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
196.8.a.a	$196$	$8$	196.a	1.a	$1$	$2$	$2$	$61.227$	\(\Q(\sqrt{1009}) \)	None	✓				28.8.a.b	$1$	$0$	\(0\)	\(-14\)	\(294\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(147+11\beta )q^{5}+(-1129+\cdots)q^{9}+\cdots\)
196.8.a.b	$196$	$8$	196.a	1.a	$1$	$2$	$2$	$61.227$	\(\Q(\sqrt{3529}) \)	None	✓				28.8.a.a	$1$	$0$	\(0\)	\(14\)	\(-42\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(-21-3\beta )q^{5}+(1391+\cdots)q^{9}+\cdots\)
196.8.a.c	$196$	$8$	196.a	1.a	$1$	$4$	$4$	$61.227$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			196.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{7}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(1769-\beta _{3})q^{9}+\cdots\)
196.8.a.d	$196$	$8$	196.a	1.a	$1$	$5$	$5$	$61.227$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				28.8.e.a	$1$	$1$	\(0\)	\(-27\)	\(-249\)	\(0\)		$-$	$+$	$-$	$2^{9}\cdot 3^{2}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(-5-\beta _{1})q^{3}+(-7^{2}-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
196.8.a.e	$196$	$8$	196.a	1.a	$1$	$5$	$5$	$61.227$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		28.8.e.a	$1$	$0$	\(0\)	\(27\)	\(249\)	\(0\)		$-$	$-$	$+$	$2^{9}\cdot 3^{2}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(5+\beta _{1})q^{3}+(7^{2}+\beta _{1}-\beta _{2})q^{5}+(1140+\cdots)q^{9}+\cdots\)
196.8.a.f	$196$	$8$	196.a	1.a	$1$	$6$	$6$	$61.227$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		196.8.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 7^{4}$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{3})q^{3}+(\beta _{2}+\beta _{3})q^{5}+(50+2\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(196))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(196)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 2}\)