Space of modular forms of level 4006, weight 2, and trivial character

• Label: 4006.2.a
• Level: $4006$
• Weight: $2$
• Character orbit: 4006.a
• Rep. character: $\chi_{4006}(1,\cdot)$
• Character field: $\Q$
• Dimension: $166$
• Newform subspaces: $9$
• Sturm bound: $1002$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(1002\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4006))\).

	Total	New	Old
Modular forms	503	166	337
Cusp forms	500	166	334
Eisenstein series	3	0	3

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

\(2\)	\(2003\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(40\)
\(+\)	\(-\)	$-$	\(43\)
\(-\)	\(+\)	$-$	\(47\)
\(-\)	\(-\)	$+$	\(36\)
Plus space		\(+\)	\(76\)
Minus space		\(-\)	\(90\)

Trace form

\( 166 q + 2 q^{3} + 166 q^{4} + 4 q^{6} - 4 q^{7} + 164 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\) 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	2003
4006.2.a.a	$4006$	$2$	4006.a	1.a	$1$	$1$	$1$	$31.988$	\(\Q\)	None	✓	$1$	$2$	\(-1\)	\(0\)	\(-3\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-3q^{5}-2q^{7}-q^{8}-3q^{9}+\cdots\)
4006.2.a.b	$4006$	$2$	4006.a	1.a	$1$	$1$	$1$	$31.988$	\(\Q\)	None	✓	$1$	$2$	\(1\)	\(-2\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-3q^{5}-2q^{6}-2q^{7}+\cdots\)
4006.2.a.c	$4006$	$2$	4006.a	1.a	$1$	$1$	$1$	$31.988$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(2\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+2q^{5}+q^{6}-2q^{7}+\cdots\)
4006.2.a.d	$4006$	$2$	4006.a	1.a	$1$	$2$	$2$	$31.988$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(2\)	\(-4\)	\(2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-2+\beta )q^{3}+q^{4}+q^{5}+(-2+\cdots)q^{6}+\cdots\)
4006.2.a.e	$4006$	$2$	4006.a	1.a	$1$	$2$	$2$	$31.988$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+(-2+\cdots)q^{7}+\cdots\)
4006.2.a.f	$4006$	$2$	4006.a	1.a	$1$	$31$	$31$	$31.988$		None	✓	$1$	$1$	\(31\)	\(-13\)	\(-23\)	\(-18\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
4006.2.a.g	$4006$	$2$	4006.a	1.a	$1$	$40$	$40$	$31.988$		None	✓	$1$	$1$	\(-40\)	\(-1\)	\(-23\)	\(12\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
4006.2.a.h	$4006$	$2$	4006.a	1.a	$1$	$42$	$42$	$31.988$		None	✓	$1$	$0$	\(-42\)	\(0\)	\(27\)	\(-10\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
4006.2.a.i	$4006$	$2$	4006.a	1.a	$1$	$46$	$46$	$31.988$		None	✓	$1$	$0$	\(46\)	\(21\)	\(23\)	\(26\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4006))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4006)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(2003))\)\(^{\oplus 2}\)