Space of modular forms of level 84, weight 6, and Character 84.i

• Label: 84.6.i
• Level: $84$
• Weight: $6$
• Character orbit: 84.i
• Rep. character: $\chi_{84}(25,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $14$
• Newform subspaces: $3$
• Sturm bound: $96$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(96\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(84, [\chi])\).

	Total	New	Old
Modular forms	172	14	158
Cusp forms	148	14	134
Eisenstein series	24	0	24

Trace form

14 * q + 9 * q^3 + 22 * q^5 + 29 * q^7 - 567 * q^9 - 178 * q^11 - 314 * q^13 - 396 * q^15 + 320 * q^17 + 3217 * q^19 + 180 * q^21 - 568 * q^23 - 5485 * q^25 - 1458 * q^27 - 16760 * q^29 + 4705 * q^31 + 6714 * q^33 + 30890 * q^35 - 8441 * q^37 - 9423 * q^39 - 12252 * q^41 - 73802 * q^43 + 1782 * q^45 + 49746 * q^47 + 72845 * q^49 - 1224 * q^51 + 4572 * q^53 - 35400 * q^55 - 45018 * q^57 + 100540 * q^59 + 112738 * q^61 + 21303 * q^63 - 121722 * q^65 - 83189 * q^67 - 67104 * q^69 - 100612 * q^71 + 51745 * q^73 + 48807 * q^75 + 168112 * q^77 + 5719 * q^79 - 45927 * q^81 + 190628 * q^83 - 82704 * q^85 - 24156 * q^87 + 52416 * q^89 + 18709 * q^91 + 27405 * q^93 - 25498 * q^95 - 386156 * q^97 + 28836 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(84, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
84.6.i.a	$84$	$6$	84.i	7.c	$3$	$2$	$1$	$13.472$	\(\Q(\sqrt{-3}) \)	None		✓			84.6.i.a	$2$	$0$	\(0\)	\(-9\)	\(69\)	\(245\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-9\zeta_{6}q^{3}+(69-69\zeta_{6})q^{5}+(98+7^{2}\zeta_{6})q^{7}+\cdots\)
84.6.i.b	$84$	$6$	84.i	7.c	$3$	$4$	$2$	$13.472$	\(\Q(\sqrt{-3}, \sqrt{7081})\)	None		✓			84.6.i.b	$2$	$0$	\(0\)	\(-18\)	\(-47\)	\(-174\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-9+9\beta _{2})q^{3}+(\beta _{1}-24\beta _{2})q^{5}+\cdots\)
84.6.i.c	$84$	$6$	84.i	7.c	$3$	$8$	$4$	$13.472$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓	✓		84.6.i.c	$2$	$0$	\(0\)	\(36\)	\(0\)	\(-42\)	$2^{6}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q-9\beta _{1}q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(10+30\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

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