Space of modular forms of level 5239, weight 2, and trivial character

• Label: 5239.2.a
• Level: $5239$
• Weight: $2$
• Character orbit: 5239.a
• Rep. character: $\chi_{5239}(1,\cdot)$
• Character field: $\Q$
• Dimension: $387$
• Newform subspaces: $22$
• Sturm bound: $970$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(970\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5239))\).

	Total	New	Old
Modular forms	498	387	111
Cusp forms	471	387	84
Eisenstein series	27	0	27

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

\(13\)	\(31\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(87\)
\(+\)	\(-\)	$-$	\(108\)
\(-\)	\(+\)	$-$	\(105\)
\(-\)	\(-\)	$+$	\(87\)
Plus space		\(+\)	\(174\)
Minus space		\(-\)	\(213\)

Trace form

\( 387 q + 2 q^{2} + 2 q^{3} + 388 q^{4} + 6 q^{6} + 4 q^{7} + 9 q^{8} + 391 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5239))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13	31
5239.2.a.a	$5239$	$2$	5239.a	1.a	$1$	$1$	$1$	$41.834$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{5}-2q^{7}+3q^{8}-3q^{9}+\cdots\)
5239.2.a.b	$5239$	$2$	5239.a	1.a	$1$	$1$	$1$	$41.834$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}-4q^{5}-2q^{7}+q^{9}+\cdots\)
5239.2.a.c	$5239$	$2$	5239.a	1.a	$1$	$1$	$1$	$41.834$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(4\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}+4q^{5}+2q^{7}+q^{9}+\cdots\)
5239.2.a.d	$5239$	$2$	5239.a	1.a	$1$	$1$	$1$	$41.834$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-3\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{5}+2q^{7}-3q^{8}-3q^{9}+\cdots\)
5239.2.a.e	$5239$	$2$	5239.a	1.a	$1$	$2$	$2$	$41.834$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-3\)	\(-4\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-2q^{3}+3\beta q^{4}+(1-2\beta )q^{5}+\cdots\)
5239.2.a.f	$5239$	$2$	5239.a	1.a	$1$	$2$	$2$	$41.834$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(-2\)	\(-2\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-2\beta q^{3}+(-1+\beta )q^{4}-q^{5}+\cdots\)
5239.2.a.g	$5239$	$2$	5239.a	1.a	$1$	$6$	$6$	$41.834$	6.6.5748973.1	None	✓	$1$	$1$	\(2\)	\(-5\)	\(9\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+(-1+\beta _{1})q^{3}+(1-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
5239.2.a.h	$5239$	$2$	5239.a	1.a	$1$	$7$	$7$	$41.834$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(5\)	\(-11\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+(1-\beta _{1})q^{3}+(\beta _{1}+\beta _{4}+\beta _{6})q^{4}+\cdots\)
5239.2.a.i	$5239$	$2$	5239.a	1.a	$1$	$8$	$8$	$41.834$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(7\)	\(-11\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
5239.2.a.j	$5239$	$2$	5239.a	1.a	$1$	$8$	$8$	$41.834$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(-3\)	\(15\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{6}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
5239.2.a.k	$5239$	$2$	5239.a	1.a	$1$	$16$	$16$	$41.834$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-2\)	\(-4\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+\beta _{9}q^{5}+\cdots\)
5239.2.a.l	$5239$	$2$	5239.a	1.a	$1$	$16$	$16$	$41.834$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(-2\)	\(4\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)
5239.2.a.m	$5239$	$2$	5239.a	1.a	$1$	$17$	$17$	$41.834$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(-7\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{9}q^{3}+(1+\beta _{2})q^{4}-\beta _{16}q^{5}+\cdots\)
5239.2.a.n	$5239$	$2$	5239.a	1.a	$1$	$17$	$17$	$41.834$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(7\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{9}q^{3}+(1+\beta _{2})q^{4}+\beta _{16}q^{5}+\cdots\)
5239.2.a.o	$5239$	$2$	5239.a	1.a	$1$	$18$	$18$	$41.834$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(-6\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{11}q^{3}+(1+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots\)
5239.2.a.p	$5239$	$2$	5239.a	1.a	$1$	$18$	$18$	$41.834$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(0\)	\(6\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{11}q^{3}+(1+\beta _{2})q^{4}-\beta _{7}q^{5}+\cdots\)
5239.2.a.q	$5239$	$2$	5239.a	1.a	$1$	$34$	$34$	$41.834$		None	✓	$1$	$1$	\(-8\)	\(0\)	\(-16\)	\(-8\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
5239.2.a.r	$5239$	$2$	5239.a	1.a	$1$	$34$	$34$	$41.834$		None	✓	$1$	$0$	\(8\)	\(0\)	\(16\)	\(8\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
5239.2.a.s	$5239$	$2$	5239.a	1.a	$1$	$36$	$36$	$41.834$		None	✓	$1$	$1$	\(-2\)	\(-5\)	\(-5\)	\(-5\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
5239.2.a.t	$5239$	$2$	5239.a	1.a	$1$	$36$	$36$	$41.834$		None	✓	$1$	$1$	\(2\)	\(-5\)	\(5\)	\(5\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
5239.2.a.u	$5239$	$2$	5239.a	1.a	$1$	$54$	$54$	$41.834$		None	✓	$1$	$0$	\(-2\)	\(7\)	\(-5\)	\(-5\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
5239.2.a.v	$5239$	$2$	5239.a	1.a	$1$	$54$	$54$	$41.834$		None	✓	$1$	$0$	\(2\)	\(7\)	\(5\)	\(5\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5239))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5239)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(403))\)\(^{\oplus 2}\)