Space of modular forms of level 6039, weight 2, and trivial character

• Label: 6039.2.a
• Level: $6039$
• Weight: $2$
• Character orbit: 6039.a
• Rep. character: $\chi_{6039}(1,\cdot)$
• Character field: $\Q$
• Dimension: $250$
• Newform subspaces: $16$
• Sturm bound: $1488$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(1488\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6039))\).

	Total	New	Old
Modular forms	752	250	502
Cusp forms	737	250	487
Eisenstein series	15	0	15

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
\(+\)	\(+\)	\(+\)	$+$	\(25\)
\(+\)	\(+\)	\(-\)	$-$	\(25\)
\(+\)	\(-\)	\(+\)	$-$	\(25\)
\(+\)	\(-\)	\(-\)	$+$	\(25\)
\(-\)	\(+\)	\(+\)	$-$	\(44\)
\(-\)	\(+\)	\(-\)	$+$	\(29\)
\(-\)	\(-\)	\(+\)	$+$	\(31\)
\(-\)	\(-\)	\(-\)	$-$	\(46\)
Plus space			\(+\)	\(110\)
Minus space			\(-\)	\(140\)

Trace form

\( 250 q + 250 q^{4} + 2 q^{5} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\) 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
6039.2.a.a	$6039$	$2$	6039.a	1.a	$1$	$5$	$5$	$48.222$	5.5.24217.1	None	✓	$1$	$1$	\(2\)	\(0\)	\(2\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2\beta _{1}-\beta _{2}+\beta _{4})q^{4}+\beta _{2}q^{5}+\cdots\)
6039.2.a.b	$6039$	$2$	6039.a	1.a	$1$	$6$	$6$	$48.222$	6.6.2661761.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}-\beta _{5}q^{4}+(-\beta _{1}+\beta _{3}-\beta _{5})q^{5}+\cdots\)
6039.2.a.c	$6039$	$2$	6039.a	1.a	$1$	$11$	$11$	$48.222$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(1\)	\(-11\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{8}q^{5}+(-1+\cdots)q^{7}+\cdots\)
6039.2.a.d	$6039$	$2$	6039.a	1.a	$1$	$11$	$11$	$48.222$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(13\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{5}+\beta _{6})q^{4}+(1-\beta _{10})q^{5}+\cdots\)
6039.2.a.e	$6039$	$2$	6039.a	1.a	$1$	$12$	$12$	$48.222$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(3\)	\(-9\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}+(-1+\cdots)q^{7}+\cdots\)
6039.2.a.f	$6039$	$2$	6039.a	1.a	$1$	$12$	$12$	$48.222$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(0\)	\(7\)	\(-15\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
6039.2.a.g	$6039$	$2$	6039.a	1.a	$1$	$13$	$13$	$48.222$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(-7\)	\(5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-1+\beta _{6}+\cdots)q^{5}+\cdots\)
6039.2.a.h	$6039$	$2$	6039.a	1.a	$1$	$13$	$13$	$48.222$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(-7\)	\(7\)		$-$	$+$	$-$	$+$	$5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{5})q^{5}+\cdots\)
6039.2.a.i	$6039$	$2$	6039.a	1.a	$1$	$13$	$13$	$48.222$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(-3\)	\(11\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+(1+\beta _{11}+\cdots)q^{7}+\cdots\)
6039.2.a.j	$6039$	$2$	6039.a	1.a	$1$	$14$	$14$	$48.222$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(9\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{10}q^{5}+(1+\cdots)q^{7}+\cdots\)
6039.2.a.k	$6039$	$2$	6039.a	1.a	$1$	$19$	$19$	$48.222$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$0$	\(-5\)	\(0\)	\(0\)	\(9\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}-\beta _{16}q^{7}+\cdots\)
6039.2.a.l	$6039$	$2$	6039.a	1.a	$1$	$21$	$21$	$48.222$		None	✓	$1$	$0$	\(0\)	\(0\)	\(-7\)	\(5\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$
6039.2.a.m	$6039$	$2$	6039.a	1.a	$1$	$25$	$25$	$48.222$		None	✓	$1$	$1$	\(-5\)	\(0\)	\(-12\)	\(-4\)		$+$	$-$	$-$	$+$		$\mathrm{SU}(2)$
6039.2.a.n	$6039$	$2$	6039.a	1.a	$1$	$25$	$25$	$48.222$		None	✓	$1$	$1$	\(-5\)	\(0\)	\(-4\)	\(4\)		$+$	$+$	$+$	$+$		$\mathrm{SU}(2)$
6039.2.a.o	$6039$	$2$	6039.a	1.a	$1$	$25$	$25$	$48.222$		None	✓	$1$	$0$	\(5\)	\(0\)	\(4\)	\(4\)		$+$	$-$	$+$	$-$		$\mathrm{SU}(2)$
6039.2.a.p	$6039$	$2$	6039.a	1.a	$1$	$25$	$25$	$48.222$		None	✓	$1$	$0$	\(5\)	\(0\)	\(12\)	\(-4\)		$+$	$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6039))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6039)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(61))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(183))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(549))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(671))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2013))\)\(^{\oplus 2}\)