Space of modular forms of level 84, weight 4, and trivial character

• Label: 84.4.a
• Level: $84$
• Weight: $4$
• Character orbit: 84.a
• Rep. character: $\chi_{84}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $64$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(64\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(84))\).

	Total	New	Old
Modular forms	54	2	52
Cusp forms	42	2	40
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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(8\)	\(0\)	\(8\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(29\)	\(2\)	\(27\)		\(23\)	\(2\)	\(21\)		\(6\)	\(0\)	\(6\)
Minus space			\(-\)		\(25\)	\(0\)	\(25\)		\(19\)	\(0\)	\(19\)		\(6\)	\(0\)	\(6\)

Trace form

2 * q + 20 * q^5 + 18 * q^9 + 40 * q^11 + 116 * q^13 + 24 * q^15 + 100 * q^17 + 16 * q^19 - 42 * q^21 - 176 * q^23 - 18 * q^25 - 52 * q^29 - 256 * q^31 - 96 * q^33 - 56 * q^35 - 20 * q^37 - 24 * q^39 - 12 * q^41 - 680 * q^43 + 180 * q^45 - 720 * q^47 + 98 * q^49 - 384 * q^51 + 204 * q^53 + 272 * q^55 + 504 * q^57 + 32 * q^59 + 932 * q^61 + 1128 * q^65 - 168 * q^67 - 384 * q^69 + 832 * q^71 + 468 * q^73 + 480 * q^75 + 224 * q^77 + 464 * q^79 + 162 * q^81 - 656 * q^83 + 488 * q^85 - 480 * q^87 - 444 * q^89 + 56 * q^91 - 96 * q^93 + 832 * q^95 - 492 * q^97 + 360 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(84))\) 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
84.4.a.a	$84$	$4$	84.a	1.a	$1$	$1$	$1$	$4.956$	\(\Q\)	None	✓	✓	✓	✓	84.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(6\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+6q^{5}+7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
84.4.a.b	$84$	$4$	84.a	1.a	$1$	$1$	$1$	$4.956$	\(\Q\)	None	✓	✓	✓	✓	84.4.a.b	$1$	$0$	\(0\)	\(3\)	\(14\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+14q^{5}-7q^{7}+9q^{9}+4q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(84))\) into lower level spaces

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