Space of modular forms of level 84, weight 12, and trivial character

• Label: 84.12.a
• Level: $84$
• Weight: $12$
• Character orbit: 84.a
• Rep. character: $\chi_{84}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $5$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(84))\).

	Total	New	Old
Modular forms	182	10	172
Cusp forms	170	10	160
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
\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(6\)
Minus space			\(-\)	\(4\)

Trace form

\( 10 q - 15580 q^{5} + 590490 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(84))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
84.12.a.a	$84$	$12$	84.a	1.a	$1$	$1$	$1$	$64.541$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-243\)	\(-2130\)	\(16807\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}-2130q^{5}+7^{5}q^{7}+3^{10}q^{9}+\cdots\)
84.12.a.b	$84$	$12$	84.a	1.a	$1$	$2$	$2$	$64.541$	\(\Q(\sqrt{1000465}) \)	None	✓	$1$	$1$	\(0\)	\(-486\)	\(-8910\)	\(-33614\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+(-4455-5\beta )q^{5}-7^{5}q^{7}+\cdots\)
84.12.a.c	$84$	$12$	84.a	1.a	$1$	$2$	$2$	$64.541$	\(\Q(\sqrt{1435009}) \)	None	✓	$1$	$0$	\(0\)	\(-486\)	\(5496\)	\(33614\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{5}q^{3}+(2748-\beta )q^{5}+7^{5}q^{7}+3^{10}q^{9}+\cdots\)
84.12.a.d	$84$	$12$	84.a	1.a	$1$	$2$	$2$	$64.541$	\(\Q(\sqrt{37321}) \)	None	✓	$1$	$1$	\(0\)	\(486\)	\(-5130\)	\(33614\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+(-2565-5\beta )q^{5}+7^{5}q^{7}+\cdots\)
84.12.a.e	$84$	$12$	84.a	1.a	$1$	$3$	$3$	$64.541$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(729\)	\(-4906\)	\(-50421\)		$-$	$-$	$+$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{5}q^{3}+(-1635-\beta _{1})q^{5}-7^{5}q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(84))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(84)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)