Space of modular forms of level 324, weight 3, and Character 324.g

• Label: 324.3.g
• Level: $324$
• Weight: $3$
• Character orbit: 324.g
• Rep. character: $\chi_{324}(53,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $16$
• Newform subspaces: $4$
• Sturm bound: $162$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(162\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(324, [\chi])\).

	Total	New	Old
Modular forms	252	16	236
Cusp forms	180	16	164
Eisenstein series	72	0	72

Trace form

\( 16 q + 5 q^{7} - 25 q^{13} - 46 q^{19} + 34 q^{25} - 10 q^{31} + 194 q^{37} + 140 q^{43} + 57 q^{49} - 252 q^{55} - 97 q^{61} - 157 q^{67} + 38 q^{73} - 397 q^{79} + 36 q^{85} + 466 q^{91} + 305 q^{97}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(324, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
324.3.g.a	$324$	$3$	324.g	9.d	$6$	$2$	$1$	$8.828$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					108.3.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-11\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-11+11\zeta_{6})q^{7}-23\zeta_{6}q^{13}-37q^{19}+\cdots\)
324.3.g.b	$324$	$3$	324.g	9.d	$6$	$2$	$1$	$8.828$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					12.3.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-2+2\zeta_{6})q^{7}+22\zeta_{6}q^{13}+26q^{19}+\cdots\)
324.3.g.c	$324$	$3$	324.g	9.d	$6$	$4$	$2$	$8.828$	\(\Q(\zeta_{12})\)	None					108.3.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_1 q^{5}+7\beta_{2} q^{7}+(\beta_{3}-\beta_1)q^{11}+\cdots\)
324.3.g.d	$324$	$3$	324.g	9.d	$6$	$8$	$4$	$8.828$	\(\Q(\zeta_{24})\)	None		✓	✓		324.3.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$3^{10}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta_{5} q^{5}+(-\beta_{2}+\beta_1)q^{7}+\beta_{7} q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(324, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)