Space of modular forms of level 2013, weight 2, and trivial character

• Label: 2013.2.a
• Level: $2013$
• Weight: $2$
• Character orbit: 2013.a
• Rep. character: $\chi_{2013}(1,\cdot)$
• Character field: $\Q$
• Dimension: $99$
• Newform subspaces: $8$
• Sturm bound: $496$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(496\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2013))\).

	Total	New	Old
Modular forms	252	99	153
Cusp forms	245	99	146
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.

\(3\)	\(11\)	\(61\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(12\)
\(+\)	\(+\)	\(-\)	$-$	\(14\)
\(+\)	\(-\)	\(+\)	$-$	\(13\)
\(+\)	\(-\)	\(-\)	$+$	\(11\)
\(-\)	\(+\)	\(+\)	$-$	\(13\)
\(-\)	\(+\)	\(-\)	$+$	\(11\)
\(-\)	\(-\)	\(+\)	$+$	\(12\)
\(-\)	\(-\)	\(-\)	$-$	\(13\)
Plus space			\(+\)	\(46\)
Minus space			\(-\)	\(53\)

Trace form

\( 99 q - 3 q^{2} - q^{3} + 93 q^{4} - 6 q^{5} - 3 q^{6} - 8 q^{7} + 9 q^{8} + 99 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	11	61
2013.2.a.a	$2013$	$2$	2013.a	1.a	$1$	$11$	$11$	$16.074$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(11\)	\(-13\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{5}+\beta _{6})q^{4}+(-1+\cdots)q^{5}+\cdots\)
2013.2.a.b	$2013$	$2$	2013.a	1.a	$1$	$11$	$11$	$16.074$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(-11\)	\(-1\)	\(-11\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)
2013.2.a.c	$2013$	$2$	2013.a	1.a	$1$	$12$	$12$	$16.074$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(-7\)	\(12\)	\(-7\)	\(-15\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
2013.2.a.d	$2013$	$2$	2013.a	1.a	$1$	$12$	$12$	$16.074$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(-12\)	\(-3\)	\(-9\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+\cdots\)
2013.2.a.e	$2013$	$2$	2013.a	1.a	$1$	$13$	$13$	$16.074$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(-13\)	\(3\)	\(11\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{8}q^{5}+\cdots\)
2013.2.a.f	$2013$	$2$	2013.a	1.a	$1$	$13$	$13$	$16.074$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(13\)	\(7\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
2013.2.a.g	$2013$	$2$	2013.a	1.a	$1$	$13$	$13$	$16.074$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(13\)	\(7\)	\(7\)		$-$	$-$	$-$	$-$	$5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{5}+\cdots)q^{5}+\cdots\)
2013.2.a.h	$2013$	$2$	2013.a	1.a	$1$	$14$	$14$	$16.074$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(-14\)	\(1\)	\(9\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+\beta _{10}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2013))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2013)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(61))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(183))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(671))\)\(^{\oplus 2}\)