Space of modular forms of level 324, weight 8, and trivial character

• Label: 324.8.a
• Level: $324$
• Weight: $8$
• Character orbit: 324.a
• Rep. character: $\chi_{324}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $5$
• Sturm bound: $432$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(432\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(324))\).

	Total	New	Old
Modular forms	396	28	368
Cusp forms	360	28	332
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
\(+\)	\(+\)	\(+\)		\(102\)	\(0\)	\(102\)		\(90\)	\(0\)	\(90\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(99\)	\(0\)	\(99\)		\(87\)	\(0\)	\(87\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(96\)	\(13\)	\(83\)		\(90\)	\(13\)	\(77\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(99\)	\(15\)	\(84\)		\(93\)	\(15\)	\(78\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(201\)	\(15\)	\(186\)		\(183\)	\(15\)	\(168\)		\(18\)	\(0\)	\(18\)
Minus space		\(-\)		\(195\)	\(13\)	\(182\)		\(177\)	\(13\)	\(164\)		\(18\)	\(0\)	\(18\)

Trace form

28 * q - 166 * q^7 + 5204 * q^13 - 24436 * q^19 + 212620 * q^25 + 91130 * q^31 + 40058 * q^37 + 1496114 * q^43 + 1676142 * q^49 - 2336490 * q^55 - 793660 * q^61 - 2559490 * q^67 + 2439674 * q^73 - 6139318 * q^79 + 4039110 * q^85 - 4395122 * q^91 + 17505014 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(324))\) 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
324.8.a.a	$324$	$8$	324.a	1.a	$1$	$3$	$3$	$101.213$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			324.8.a.a	$1$	$1$	\(0\)	\(0\)	\(-57\)	\(114\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-19+\beta _{1})q^{5}+(38-\beta _{2})q^{7}+(146+\cdots)q^{11}+\cdots\)
324.8.a.b	$324$	$8$	324.a	1.a	$1$	$3$	$3$	$101.213$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			324.8.a.a	$1$	$1$	\(0\)	\(0\)	\(57\)	\(114\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(19-\beta _{1})q^{5}+(38-\beta _{2})q^{7}+(-146+\cdots)q^{11}+\cdots\)
324.8.a.c	$324$	$8$	324.a	1.a	$1$	$7$	$7$	$101.213$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		36.8.e.a	$1$	$1$	\(0\)	\(0\)	\(-321\)	\(83\)		$-$	$+$	$-$	$2^{10}\cdot 3^{15}$	$\mathrm{SU}(2)$	\(q+(-46-\beta _{1})q^{5}+(12-\beta _{2})q^{7}+(2^{4}+\cdots)q^{11}+\cdots\)
324.8.a.d	$324$	$8$	324.a	1.a	$1$	$7$	$7$	$101.213$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				36.8.e.a	$1$	$0$	\(0\)	\(0\)	\(321\)	\(83\)		$-$	$-$	$+$	$2^{10}\cdot 3^{15}$	$\mathrm{SU}(2)$	\(q+(46+\beta _{1})q^{5}+(12-\beta _{2})q^{7}+(-2^{4}+\cdots)q^{11}+\cdots\)
324.8.a.e	$324$	$8$	324.a	1.a	$1$	$8$	$8$	$101.213$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		324.8.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-560\)		$-$	$-$	$+$	$2^{16}\cdot 3^{20}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{5}+(-70-\beta _{1})q^{7}+(\beta _{4}+\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(324)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)