Space of modular forms of level 324, weight 2, and Character 324.h

• Label: 324.2.h
• Level: $324$
• Weight: $2$
• Character orbit: 324.h
• Rep. character: $\chi_{324}(107,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $44$
• Newform subspaces: $6$
• Sturm bound: $108$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	324.h (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 36 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(108\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(324, [\chi])\).

	Total	New	Old
Modular forms	132	52	80
Cusp forms	84	44	40
Eisenstein series	48	8	40

Trace form

44 * q + 2 * q^4 + 4 * q^10 + 4 * q^13 + 2 * q^16 + 6 * q^22 + 18 * q^25 + 12 * q^28 - 32 * q^34 - 8 * q^37 - 38 * q^40 - 72 * q^46 + 14 * q^49 - 8 * q^52 - 20 * q^58 + 4 * q^61 - 100 * q^64 - 18 * q^70 - 32 * q^73 + 24 * q^76 - 32 * q^82 - 16 * q^85 + 78 * q^88 + 84 * q^94 - 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(324, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
324.2.h.a	$324$	$2$	324.h	36.h	$6$	$4$	$2$	$2.587$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					108.2.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{3}q^{2}-2q^{4}+2\beta _{1}q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots\)
324.2.h.b	$324$	$2$	324.h	36.h	$6$	$4$	$2$	$2.587$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None					108.2.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+2\beta _{1}q^{5}+(-2+\cdots)q^{7}+\cdots\)
324.2.h.c	$324$	$2$	324.h	36.h	$6$	$4$	$2$	$2.587$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-1}) \)					36.2.b.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(2-2\beta _{2})q^{4}+\beta _{1}q^{5}+\cdots\)
324.2.h.d	$324$	$2$	324.h	36.h	$6$	$8$	$4$	$2.587$	8.0.12960000.1	None					108.2.b.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\beta _{3}+\beta _{7})q^{2}+(-\beta _{1}-\beta _{2})q^{4}+\cdots\)
324.2.h.e	$324$	$2$	324.h	36.h	$6$	$8$	$4$	$2.587$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-1}) \)		✓			324.2.b.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{6}$	$\mathrm{U}(1)[D_{6}]$	\(q-\beta_{3} q^{2}+(-2\beta_1+2)q^{4}+(-\beta_{6}+\beta_{5})q^{5}+\cdots\)
324.2.h.f	$324$	$2$	324.h	36.h	$6$	$16$	$8$	$2.587$	16.0.\(\cdots\).1	None		✓	✓		324.2.b.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{15}q^{2}+(-1+\beta _{8}-\beta _{9}-\beta _{11}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(324, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)