Space of modular forms of level 1006, weight 2, and trivial character

• Label: 1006.2.a
• Level: $1006$
• Weight: $2$
• Character orbit: 1006.a
• Rep. character: $\chi_{1006}(1,\cdot)$
• Character field: $\Q$
• Dimension: $41$
• Newform subspaces: $10$
• Sturm bound: $252$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1006 = 2 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1006.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(252\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1006))\).

	Total	New	Old
Modular forms	128	41	87
Cusp forms	125	41	84
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.

\(2\)	\(503\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(8\)
\(+\)	\(-\)	$-$	\(13\)
\(-\)	\(+\)	$-$	\(12\)
\(-\)	\(-\)	$+$	\(8\)
Plus space		\(+\)	\(16\)
Minus space		\(-\)	\(25\)

Trace form

\( 41 q - q^{2} - 4 q^{3} + 41 q^{4} - 4 q^{7} - q^{8} + 41 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1006))\) 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	503
1006.2.a.a	$1006$	$2$	1006.a	1.a	$1$	$1$	$1$	$8.033$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{8}-3q^{9}+4q^{11}+2q^{13}+\cdots\)
1006.2.a.b	$1006$	$2$	1006.a	1.a	$1$	$1$	$1$	$8.033$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-2\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-2q^{5}-q^{6}+q^{7}+\cdots\)
1006.2.a.c	$1006$	$2$	1006.a	1.a	$1$	$1$	$1$	$8.033$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-3\)	\(0\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}-3q^{6}-3q^{7}+q^{8}+\cdots\)
1006.2.a.d	$1006$	$2$	1006.a	1.a	$1$	$1$	$1$	$8.033$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
1006.2.a.e	$1006$	$2$	1006.a	1.a	$1$	$1$	$1$	$8.033$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(-4\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-4q^{5}+q^{6}+q^{7}+\cdots\)
1006.2.a.f	$1006$	$2$	1006.a	1.a	$1$	$2$	$2$	$8.033$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta q^{3}+q^{4}+(-1-\beta )q^{5}+\beta q^{6}+\cdots\)
1006.2.a.g	$1006$	$2$	1006.a	1.a	$1$	$5$	$5$	$8.033$	5.5.36497.1	None	✓	$1$	$1$	\(-5\)	\(0\)	\(-1\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(\beta _{1}-\beta _{3}+\beta _{4})q^{3}+q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
1006.2.a.h	$1006$	$2$	1006.a	1.a	$1$	$5$	$5$	$8.033$	5.5.205225.1	None	✓	$1$	$1$	\(5\)	\(-4\)	\(-3\)	\(-9\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1006.2.a.i	$1006$	$2$	1006.a	1.a	$1$	$12$	$12$	$8.033$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(-12\)	\(-3\)	\(7\)	\(-2\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{4}q^{3}+q^{4}+(\beta _{7}-\beta _{10})q^{5}+\cdots\)
1006.2.a.j	$1006$	$2$	1006.a	1.a	$1$	$12$	$12$	$8.033$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(12\)	\(5\)	\(5\)	\(8\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{9}q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1006))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1006)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\)\(^{\oplus 2}\)