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	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(3\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(7\)

Trace form

\( 12 q + O(q^{10}) \)

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

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

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