Space of modular forms of level 640, weight 2, and trivial character

• Label: 640.2.a
• Level: $640$
• Weight: $2$
• Character orbit: 640.a
• Rep. character: $\chi_{640}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $12$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	640.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(640))\).

	Total	New	Old
Modular forms	112	16	96
Cusp forms	81	16	65
Eisenstein series	31	0	31

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

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(2\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(10\)

Trace form

\( 16 q + 16 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(640))\) 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	5
640.2.a.a	$640$	$2$	640.a	1.a	$1$	$1$	$1$	$5.110$	\(\Q\)	None	✓	✓			640.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+q^{9}-2q^{11}+2q^{13}+\cdots\)
640.2.a.b	$640$	$2$	640.a	1.a	$1$	$1$	$1$	$5.110$	\(\Q\)	None	✓	✓			640.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-2q^{11}-2q^{13}+\cdots\)
640.2.a.c	$640$	$2$	640.a	1.a	$1$	$1$	$1$	$5.110$	\(\Q\)	None	✓	✓			640.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{9}+6q^{11}-2q^{13}+\cdots\)
640.2.a.d	$640$	$2$	640.a	1.a	$1$	$1$	$1$	$5.110$	\(\Q\)	None	✓	✓			640.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-3q^{9}-6q^{11}-2q^{13}+\cdots\)
640.2.a.e	$640$	$2$	640.a	1.a	$1$	$1$	$1$	$5.110$	\(\Q\)	None	✓	✓			640.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-3q^{9}-6q^{11}+2q^{13}+\cdots\)
640.2.a.f	$640$	$2$	640.a	1.a	$1$	$1$	$1$	$5.110$	\(\Q\)	None	✓	✓			640.2.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{7}-3q^{9}+6q^{11}+2q^{13}+\cdots\)
640.2.a.g	$640$	$2$	640.a	1.a	$1$	$1$	$1$	$5.110$	\(\Q\)	None	✓	✓			640.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+q^{9}+2q^{11}+2q^{13}+\cdots\)
640.2.a.h	$640$	$2$	640.a	1.a	$1$	$1$	$1$	$5.110$	\(\Q\)	None	✓	✓			640.2.a.a	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}+2q^{11}-2q^{13}+\cdots\)
640.2.a.i	$640$	$2$	640.a	1.a	$1$	$2$	$2$	$5.110$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		640.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-q^{5}+(-1+\beta )q^{7}+\cdots\)
640.2.a.j	$640$	$2$	640.a	1.a	$1$	$2$	$2$	$5.110$	\(\Q(\sqrt{5}) \)	None	✓	✓			640.2.a.i	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+q^{5}+(1-\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
640.2.a.k	$640$	$2$	640.a	1.a	$1$	$2$	$2$	$5.110$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		640.2.a.i	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+(1-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
640.2.a.l	$640$	$2$	640.a	1.a	$1$	$2$	$2$	$5.110$	\(\Q(\sqrt{5}) \)	None	✓	✓			640.2.a.i	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+(-1+\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(640))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(640)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 2}\)