Space of modular forms of level 660, weight 2, and trivial character

• Label: 660.2.a
• Level: $660$
• Weight: $2$
• Character orbit: 660.a
• Rep. character: $\chi_{660}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $6$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	660.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(660))\).

	Total	New	Old
Modular forms	156	8	148
Cusp forms	133	8	125
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(5\)	\(11\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(2\)
Plus space				\(+\)	\(2\)
Minus space				\(-\)	\(6\)

Trace form

\( 8 q + 8 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(660))\) 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	5	11
660.2.a.a	$660$	$2$	660.a	1.a	$1$	$1$	$1$	$5.270$	\(\Q\)	None	✓	✓	✓	✓	660.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
660.2.a.b	$660$	$2$	660.a	1.a	$1$	$1$	$1$	$5.270$	\(\Q\)	None	✓	✓	✓	✓	660.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}+q^{11}-4q^{13}+q^{15}+\cdots\)
660.2.a.c	$660$	$2$	660.a	1.a	$1$	$1$	$1$	$5.270$	\(\Q\)	None	✓	✓	✓	✓	660.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}+q^{9}-q^{11}-4q^{13}+\cdots\)
660.2.a.d	$660$	$2$	660.a	1.a	$1$	$1$	$1$	$5.270$	\(\Q\)	None	✓	✓	✓	✓	660.2.a.d	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+2q^{7}+q^{9}+q^{11}+2q^{13}+\cdots\)
660.2.a.e	$660$	$2$	660.a	1.a	$1$	$2$	$2$	$5.270$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	660.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(1+\beta )q^{7}+q^{9}+q^{11}+\cdots\)
660.2.a.f	$660$	$2$	660.a	1.a	$1$	$2$	$2$	$5.270$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	660.2.a.f	$1$	$0$	\(0\)	\(2\)	\(2\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(1+\beta )q^{7}+q^{9}-q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(660))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(660)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(330))\)\(^{\oplus 2}\)