Space of modular forms of level 2008, weight 2, and trivial character

• Label: 2008.2.a
• Level: $2008$
• Weight: $2$
• Character orbit: 2008.a
• Rep. character: $\chi_{2008}(1,\cdot)$
• Character field: $\Q$
• Dimension: $63$
• Newform subspaces: $4$
• Sturm bound: $504$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(504\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2008))\).

	Total	New	Old
Modular forms	256	63	193
Cusp forms	249	63	186
Eisenstein series	7	0	7

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

\(2\)	\(251\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(19\)
\(+\)	\(-\)	$-$	\(12\)
\(-\)	\(+\)	$-$	\(23\)
\(-\)	\(-\)	$+$	\(9\)
Plus space		\(+\)	\(28\)
Minus space		\(-\)	\(35\)

Trace form

\( 63 q - 2 q^{3} - 4 q^{7} + 69 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2008))\) 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	251
2008.2.a.a	$2008$	$2$	2008.a	1.a	$1$	$9$	$9$	$16.034$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-5\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{5})q^{5}+(\beta _{6}-\beta _{7}+\cdots)q^{7}+\cdots\)
2008.2.a.b	$2008$	$2$	2008.a	1.a	$1$	$12$	$12$	$16.034$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(5\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{8}q^{5}+\beta _{5}q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
2008.2.a.c	$2008$	$2$	2008.a	1.a	$1$	$19$	$19$	$16.034$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-8\)	\(-11\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{4}q^{5}+(-1-\beta _{13})q^{7}+\cdots\)
2008.2.a.d	$2008$	$2$	2008.a	1.a	$1$	$23$	$23$	$16.034$		None	✓	$1$	$0$	\(0\)	\(2\)	\(8\)	\(2\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2008))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2008)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(251))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(502))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1004))\)\(^{\oplus 2}\)