Space of modular forms of level 2004, weight 2, and trivial character

• Label: 2004.2.a
• Level: $2004$
• Weight: $2$
• Character orbit: 2004.a
• Rep. character: $\chi_{2004}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $4$
• Sturm bound: $672$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2004.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(672\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2004))\).

	Total	New	Old
Modular forms	342	28	314
Cusp forms	331	28	303
Eisenstein series	11	0	11

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

\(2\)	\(3\)	\(167\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(9\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(5\)
\(-\)	\(-\)	\(-\)	$-$	\(9\)
Plus space			\(+\)	\(10\)
Minus space			\(-\)	\(18\)

Trace form

\( 28 q + 28 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2004))\) 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	167
2004.2.a.a	$2004$	$2$	2004.a	1.a	$1$	$5$	$5$	$16.002$	5.5.149169.1	None	✓	$1$	$1$	\(0\)	\(-5\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-\beta _{1}+\beta _{3}-\beta _{4})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
2004.2.a.b	$2004$	$2$	2004.a	1.a	$1$	$5$	$5$	$16.002$	5.5.161121.1	None	✓	$1$	$1$	\(0\)	\(5\)	\(-7\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta _{3}-\beta _{4})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
2004.2.a.c	$2004$	$2$	2004.a	1.a	$1$	$9$	$9$	$16.002$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-9\)	\(1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{1}q^{5}+(-\beta _{3}-\beta _{6})q^{7}+q^{9}+\cdots\)
2004.2.a.d	$2004$	$2$	2004.a	1.a	$1$	$9$	$9$	$16.002$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(9\)	\(9\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{5}-\beta _{6}q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2004))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2004)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(334))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(501))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(668))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\)\(^{\oplus 2}\)