Space of modular forms of level 648, weight 2, and trivial character

• Label: 648.2.a
• Level: $648$
• Weight: $2$
• Character orbit: 648.a
• Rep. character: $\chi_{648}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $8$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(216\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(648))\).

	Total	New	Old
Modular forms	132	12	120
Cusp forms	85	12	73
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)		\(31\)	\(2\)	\(29\)		\(20\)	\(2\)	\(18\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(34\)	\(4\)	\(30\)		\(22\)	\(4\)	\(18\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(35\)	\(3\)	\(32\)		\(23\)	\(3\)	\(20\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(32\)	\(3\)	\(29\)		\(20\)	\(3\)	\(17\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(63\)	\(5\)	\(58\)		\(40\)	\(5\)	\(35\)		\(23\)	\(0\)	\(23\)
Minus space		\(-\)		\(69\)	\(7\)	\(62\)		\(45\)	\(7\)	\(38\)		\(24\)	\(0\)	\(24\)

Trace form

\( 12 q - 6 q^{13} + 6 q^{19} + 6 q^{25} - 12 q^{31} + 18 q^{37} - 6 q^{43} + 24 q^{49} + 12 q^{55} + 30 q^{61} + 6 q^{67} - 24 q^{73} + 12 q^{79} + 6 q^{85} - 36 q^{91} - 30 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(648))\) 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
648.2.a.a	$648$	$2$	648.a	1.a	$1$	$1$	$1$	$5.174$	\(\Q\)	None	✓				72.2.i.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}+5q^{11}-5q^{13}-2q^{17}+\cdots\)
648.2.a.b	$648$	$2$	648.a	1.a	$1$	$1$	$1$	$5.174$	\(\Q\)	None	✓	✓			648.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-4q^{11}-5q^{13}-5q^{17}+8q^{19}+\cdots\)
648.2.a.c	$648$	$2$	648.a	1.a	$1$	$1$	$1$	$5.174$	\(\Q\)	None	✓				72.2.i.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{7}-5q^{11}-5q^{13}+2q^{17}+\cdots\)
648.2.a.d	$648$	$2$	648.a	1.a	$1$	$1$	$1$	$5.174$	\(\Q\)	None	✓	✓			648.2.a.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+4q^{11}-5q^{13}+5q^{17}+8q^{19}+\cdots\)
648.2.a.e	$648$	$2$	648.a	1.a	$1$	$2$	$2$	$5.174$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		648.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}-2\beta q^{7}-2q^{11}+(1+\cdots)q^{13}+\cdots\)
648.2.a.f	$648$	$2$	648.a	1.a	$1$	$2$	$2$	$5.174$	\(\Q(\sqrt{33}) \)	None	✓		✓		72.2.i.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+(2-\beta )q^{7}+q^{11}+(2+\beta )q^{13}+\cdots\)
648.2.a.g	$648$	$2$	648.a	1.a	$1$	$2$	$2$	$5.174$	\(\Q(\sqrt{33}) \)	None	✓				72.2.i.b	$1$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(2-\beta )q^{7}-q^{11}+(2+\beta )q^{13}+\cdots\)
648.2.a.h	$648$	$2$	648.a	1.a	$1$	$2$	$2$	$5.174$	\(\Q(\sqrt{3}) \)	None	✓	✓			648.2.a.e	$1$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+2\beta q^{7}+2q^{11}+(1-2\beta )q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(648))\) into lower level spaces

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