Space of modular forms of level 324, weight 6, and trivial character

• Label: 324.6.a
• Level: $324$
• Weight: $6$
• Character orbit: 324.a
• Rep. character: $\chi_{324}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $5$
• Sturm bound: $324$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(324\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(324))\).

	Total	New	Old
Modular forms	288	20	268
Cusp forms	252	20	232
Eisenstein series	36	0	36

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(72\)	\(0\)	\(72\)		\(60\)	\(0\)	\(60\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(75\)	\(0\)	\(75\)		\(63\)	\(0\)	\(63\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(72\)	\(11\)	\(61\)		\(66\)	\(11\)	\(55\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(69\)	\(9\)	\(60\)		\(63\)	\(9\)	\(54\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(141\)	\(9\)	\(132\)		\(123\)	\(9\)	\(114\)		\(18\)	\(0\)	\(18\)
Minus space		\(-\)		\(147\)	\(11\)	\(136\)		\(129\)	\(11\)	\(118\)		\(18\)	\(0\)	\(18\)

Trace form

20 * q + 58 * q^7 - 1556 * q^13 - 956 * q^19 + 15332 * q^25 - 3350 * q^31 + 790 * q^37 - 5102 * q^43 + 64386 * q^49 - 10458 * q^55 - 16124 * q^61 - 27458 * q^67 + 170350 * q^73 + 204106 * q^79 + 120186 * q^85 + 225974 * q^91 + 9634 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(324))\) 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
324.6.a.a	$324$	$6$	324.a	1.a	$1$	$3$	$3$	$51.964$	3.3.513129.1	None	✓	✓			324.6.a.a	$1$	$0$	\(0\)	\(0\)	\(-33\)	\(-30\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{1})q^{5}+(-10-\beta _{2})q^{7}+\cdots\)
324.6.a.b	$324$	$6$	324.a	1.a	$1$	$3$	$3$	$51.964$	3.3.513129.1	None	✓	✓			324.6.a.a	$1$	$0$	\(0\)	\(0\)	\(33\)	\(-30\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(11+\beta _{1})q^{5}+(-10-\beta _{2})q^{7}+(-10+\cdots)q^{11}+\cdots\)
324.6.a.c	$324$	$6$	324.a	1.a	$1$	$4$	$4$	$51.964$	\(\Q(\sqrt{3}, \sqrt{91})\)	None	✓	✓			324.6.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(176\)		$-$	$-$	$+$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(2\beta _{1}+\beta _{2})q^{5}+(44+\beta _{3})q^{7}+(-11\beta _{1}+\cdots)q^{11}+\cdots\)
324.6.a.d	$324$	$6$	324.a	1.a	$1$	$5$	$5$	$51.964$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		36.6.e.a	$1$	$1$	\(0\)	\(0\)	\(-21\)	\(-29\)		$-$	$-$	$+$	$2^{4}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{2})q^{5}+(-6-\beta _{1}+\beta _{2})q^{7}+\cdots\)
324.6.a.e	$324$	$6$	324.a	1.a	$1$	$5$	$5$	$51.964$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		36.6.e.a	$1$	$0$	\(0\)	\(0\)	\(21\)	\(-29\)		$-$	$+$	$-$	$2^{4}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{2})q^{5}+(-6-\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(324)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)