Space of modular forms of level 84, weight 6, and trivial character

• Label: 84.6.a
• Level: $84$
• Weight: $6$
• Character orbit: 84.a
• Rep. character: $\chi_{84}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(84))\).

	Total	New	Old
Modular forms	86	6	80
Cusp forms	74	6	68
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
\(+\)	\(+\)	\(+\)	\(+\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(11\)	\(0\)	\(11\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(12\)	\(0\)	\(12\)		\(10\)	\(0\)	\(10\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(11\)	\(0\)	\(11\)		\(9\)	\(0\)	\(9\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(11\)	\(2\)	\(9\)		\(10\)	\(2\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(11\)	\(2\)	\(9\)		\(10\)	\(2\)	\(8\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(41\)	\(2\)	\(39\)		\(35\)	\(2\)	\(33\)		\(6\)	\(0\)	\(6\)
Minus space			\(-\)		\(45\)	\(4\)	\(41\)		\(39\)	\(4\)	\(35\)		\(6\)	\(0\)	\(6\)

Trace form

6 * q + 44 * q^5 + 486 * q^9 - 356 * q^11 - 396 * q^13 + 396 * q^15 + 2932 * q^17 + 2040 * q^19 + 882 * q^21 + 4396 * q^23 + 5730 * q^25 + 8252 * q^29 - 7320 * q^31 + 360 * q^33 + 6076 * q^35 + 20412 * q^37 - 11160 * q^39 - 34428 * q^41 + 22632 * q^43 + 3564 * q^45 + 16728 * q^47 + 14406 * q^49 + 16884 * q^51 - 37332 * q^53 - 105720 * q^55 - 3888 * q^57 - 38824 * q^59 - 1836 * q^61 + 50904 * q^65 - 105456 * q^67 - 47160 * q^69 + 7876 * q^71 - 40260 * q^73 + 18864 * q^75 + 23912 * q^77 + 4296 * q^79 + 39366 * q^81 - 127472 * q^83 + 173856 * q^85 - 48312 * q^87 + 149316 * q^89 + 5880 * q^91 + 6984 * q^93 - 74384 * q^95 + 109020 * q^97 - 28836 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(84))\) 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
84.6.a.a	$84$	$6$	84.a	1.a	$1$	$1$	$1$	$13.472$	\(\Q\)	None	✓	✓	✓	✓	84.6.a.a	$1$	$1$	\(0\)	\(-9\)	\(6\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+6q^{5}+7^{2}q^{7}+3^{4}q^{9}-108q^{11}+\cdots\)
84.6.a.b	$84$	$6$	84.a	1.a	$1$	$1$	$1$	$13.472$	\(\Q\)	None	✓	✓	✓	✓	84.6.a.b	$1$	$1$	\(0\)	\(9\)	\(-34\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-34q^{5}-7^{2}q^{7}+3^{4}q^{9}-332q^{11}+\cdots\)
84.6.a.c	$84$	$6$	84.a	1.a	$1$	$2$	$2$	$13.472$	\(\Q(\sqrt{5569}) \)	None	✓	✓	✓	✓	84.6.a.c	$1$	$0$	\(0\)	\(-18\)	\(-6\)	\(-98\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-9q^{3}+(-3-\beta )q^{5}-7^{2}q^{7}+3^{4}q^{9}+\cdots\)
84.6.a.d	$84$	$6$	84.a	1.a	$1$	$2$	$2$	$13.472$	\(\Q(\sqrt{505}) \)	None	✓	✓	✓	✓	84.6.a.d	$1$	$0$	\(0\)	\(18\)	\(78\)	\(98\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+9q^{3}+(39-\beta )q^{5}+7^{2}q^{7}+3^{4}q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(84))\) into lower level spaces

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