Space of modular forms of level 84, weight 8, and trivial character

• Label: 84.8.a
• Level: $84$
• Weight: $8$
• Character orbit: 84.a
• Rep. character: $\chi_{84}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $128$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(128\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(84))\).

	Total	New	Old
Modular forms	118	6	112
Cusp forms	106	6	100
Eisenstein series	12	0	12

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

\(2\)	\(3\)	\(7\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(1\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(2\)

Trace form

\( 6 q + 220 q^{5} + 4374 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(84))\) 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	3	7
84.8.a.a	$84$	$8$	84.a	1.a	$1$	$1$	$1$	$26.240$	\(\Q\)	None	✓	✓	✓	✓	84.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(100\)	\(-343\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+10^{2}q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
84.8.a.b	$84$	$8$	84.a	1.a	$1$	$1$	$1$	$26.240$	\(\Q\)	None	✓	✓	✓	✓	84.8.a.b	$1$	$1$	\(0\)	\(27\)	\(-240\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-240q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
84.8.a.c	$84$	$8$	84.a	1.a	$1$	$2$	$2$	$26.240$	\(\Q(\sqrt{3649}) \)	None	✓	✓	✓	✓	84.8.a.c	$1$	$0$	\(0\)	\(-54\)	\(264\)	\(686\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(132-\beta )q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
84.8.a.d	$84$	$8$	84.a	1.a	$1$	$2$	$2$	$26.240$	\(\Q(\sqrt{21961}) \)	None	✓	✓	✓	✓	84.8.a.d	$1$	$0$	\(0\)	\(54\)	\(96\)	\(-686\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(48-\beta )q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(84))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(84)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)