Space of modular forms of level 648, weight 4, and trivial character

• Label: 648.4.a
• Level: $648$
• Weight: $4$
• Character orbit: 648.a
• Rep. character: $\chi_{648}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $12$
• Sturm bound: $432$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(432\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(648))\).

	Total	New	Old
Modular forms	348	36	312
Cusp forms	300	36	264
Eisenstein series	48	0	48

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
\(+\)	\(+\)	\(+\)		\(89\)	\(10\)	\(79\)		\(77\)	\(10\)	\(67\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(86\)	\(8\)	\(78\)		\(74\)	\(8\)	\(66\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(85\)	\(9\)	\(76\)		\(73\)	\(9\)	\(64\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(88\)	\(9\)	\(79\)		\(76\)	\(9\)	\(67\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(177\)	\(19\)	\(158\)		\(153\)	\(19\)	\(134\)		\(24\)	\(0\)	\(24\)
Minus space		\(-\)		\(171\)	\(17\)	\(154\)		\(147\)	\(17\)	\(130\)		\(24\)	\(0\)	\(24\)

Trace form

\( 36 q - 18 q^{13} - 90 q^{19} + 666 q^{25} + 180 q^{31} - 594 q^{37} + 90 q^{43} + 1368 q^{49} - 612 q^{55} + 738 q^{61} + 774 q^{67} + 144 q^{73} + 252 q^{79} + 2826 q^{85} + 108 q^{91} + 342 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(648))\) 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
648.4.a.a	$648$	$4$	648.a	1.a	$1$	$1$	$1$	$38.233$	\(\Q\)	None	✓	✓			648.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(36\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{5}+6^{2}q^{7}+2^{6}q^{11}-65q^{13}+\cdots\)
648.4.a.b	$648$	$4$	648.a	1.a	$1$	$1$	$1$	$38.233$	\(\Q\)	None	✓	✓			648.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(36\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{5}+6^{2}q^{7}-2^{6}q^{11}-65q^{13}+\cdots\)
648.4.a.c	$648$	$4$	648.a	1.a	$1$	$2$	$2$	$38.233$	\(\Q(\sqrt{201}) \)	None	✓	✓			648.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-30\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{5}+(-15+\beta )q^{7}+(23+\cdots)q^{11}+\cdots\)
648.4.a.d	$648$	$4$	648.a	1.a	$1$	$2$	$2$	$38.233$	\(\Q(\sqrt{129}) \)	None	✓	✓			648.4.a.d	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(-6\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{5}+(-3-\beta )q^{7}+(-5+\cdots)q^{11}+\cdots\)
648.4.a.e	$648$	$4$	648.a	1.a	$1$	$2$	$2$	$38.233$	\(\Q(\sqrt{129}) \)	None	✓	✓			648.4.a.d	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-6\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(-3-\beta )q^{7}+(5+3\beta )q^{11}+\cdots\)
648.4.a.f	$648$	$4$	648.a	1.a	$1$	$2$	$2$	$38.233$	\(\Q(\sqrt{201}) \)	None	✓	✓			648.4.a.c	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-30\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{5}+(-15+\beta )q^{7}+(-23+\cdots)q^{11}+\cdots\)
648.4.a.g	$648$	$4$	648.a	1.a	$1$	$4$	$4$	$38.233$	4.4.29952.1	None	✓	✓			648.4.a.g	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-2+\cdots)q^{11}+\cdots\)
648.4.a.h	$648$	$4$	648.a	1.a	$1$	$4$	$4$	$38.233$	4.4.72153.1	None	✓				72.4.i.a	$1$	$1$	\(0\)	\(0\)	\(-5\)	\(-3\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{5}+(-1-\beta _{3})q^{7}+(4+\cdots)q^{11}+\cdots\)
648.4.a.i	$648$	$4$	648.a	1.a	$1$	$4$	$4$	$38.233$	4.4.72153.1	None	✓		✓		72.4.i.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-3\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{5}+(-1-\beta _{3})q^{7}+(-4+\cdots)q^{11}+\cdots\)
648.4.a.j	$648$	$4$	648.a	1.a	$1$	$4$	$4$	$38.233$	4.4.29952.1	None	✓	✓			648.4.a.g	$1$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$-$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(2-\beta _{1}+\cdots)q^{11}+\cdots\)
648.4.a.k	$648$	$4$	648.a	1.a	$1$	$5$	$5$	$38.233$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		72.4.i.b	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(3\)		$-$	$-$	$+$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{5}+(1+\beta _{1})q^{7}+(5-2\beta _{2}+\cdots)q^{11}+\cdots\)
648.4.a.l	$648$	$4$	648.a	1.a	$1$	$5$	$5$	$38.233$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		72.4.i.b	$1$	$0$	\(0\)	\(0\)	\(5\)	\(3\)		$+$	$+$	$+$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{5}+(1+\beta _{1})q^{7}+(-5+2\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(648))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(648)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 2}\)