Space of modular forms of level 1002, weight 2, and trivial character

• Label: 1002.2.a
• Level: $1002$
• Weight: $2$
• Character orbit: 1002.a
• Rep. character: $\chi_{1002}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $11$
• Sturm bound: $336$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1002 = 2 \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1002.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(336\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1002))\).

	Total	New	Old
Modular forms	172	29	143
Cusp forms	165	29	136
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.

\(2\)	\(3\)	\(167\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(7\)
Plus space			\(+\)	\(11\)
Minus space			\(-\)	\(18\)

Trace form

\( 29 q + q^{2} + q^{3} + 29 q^{4} + 6 q^{5} + q^{6} + 8 q^{7} + q^{8} + 29 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\) 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
1002.2.a.a	$1002$	$2$	1002.a	1.a	$1$	$1$	$1$	$8.001$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{8}+q^{9}+\cdots\)
1002.2.a.b	$1002$	$2$	1002.a	1.a	$1$	$1$	$1$	$8.001$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(3\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+3q^{5}+q^{6}+3q^{7}+\cdots\)
1002.2.a.c	$1002$	$2$	1002.a	1.a	$1$	$1$	$1$	$8.001$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-1\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{7}+\cdots\)
1002.2.a.d	$1002$	$2$	1002.a	1.a	$1$	$1$	$1$	$8.001$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-4q^{7}+\cdots\)
1002.2.a.e	$1002$	$2$	1002.a	1.a	$1$	$1$	$1$	$8.001$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-3q^{7}+\cdots\)
1002.2.a.f	$1002$	$2$	1002.a	1.a	$1$	$2$	$2$	$8.001$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(-2+\beta )q^{5}+q^{6}+\cdots\)
1002.2.a.g	$1002$	$2$	1002.a	1.a	$1$	$3$	$3$	$8.001$	3.3.1300.1	None	✓	$1$	$1$	\(-3\)	\(-3\)	\(3\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)
1002.2.a.h	$1002$	$2$	1002.a	1.a	$1$	$3$	$3$	$8.001$	3.3.148.1	None	✓	$1$	$1$	\(3\)	\(-3\)	\(-3\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(-1+\beta _{2})q^{5}-q^{6}+\cdots\)
1002.2.a.i	$1002$	$2$	1002.a	1.a	$1$	$4$	$4$	$8.001$	4.4.2777.1	None	✓	$1$	$0$	\(4\)	\(-4\)	\(5\)	\(1\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(1-\beta _{1})q^{5}-q^{6}+\cdots\)
1002.2.a.j	$1002$	$2$	1002.a	1.a	$1$	$5$	$5$	$8.001$	5.5.11256624.1	None	✓	$1$	$0$	\(-5\)	\(5\)	\(-1\)	\(9\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+\beta _{2}q^{5}-q^{6}+(2+\cdots)q^{7}+\cdots\)
1002.2.a.k	$1002$	$2$	1002.a	1.a	$1$	$7$	$7$	$8.001$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(7\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(1-\beta _{1})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1002))\) into lower level spaces

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