Space of modular forms of level 6724, weight 2, and trivial character

• Label: 6724.2.a
• Level: $6724$
• Weight: $2$
• Character orbit: 6724.a
• Rep. character: $\chi_{6724}(1,\cdot)$
• Character field: $\Q$
• Dimension: $136$
• Newform subspaces: $13$
• Sturm bound: $1722$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 6724 = 2^{2} \cdot 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6724.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(1722\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6724))\).

	Total	New	Old
Modular forms	924	136	788
Cusp forms	799	136	663
Eisenstein series	125	0	125

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

\(2\)	\(41\)	Fricke	Dim
\(-\)	\(+\)	$-$	\(73\)
\(-\)	\(-\)	$+$	\(63\)
Plus space		\(+\)	\(63\)
Minus space		\(-\)	\(73\)

Trace form

\( 136 q - 2 q^{3} - 4 q^{5} + 128 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6724))\) 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	41
6724.2.a.a	$6724$	$2$	6724.a	1.a	$1$	$3$	$3$	$53.691$	3.3.785.1	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1})q^{5}+(2+\cdots)q^{7}+\cdots\)
6724.2.a.b	$6724$	$2$	6724.a	1.a	$1$	$3$	$3$	$53.691$	3.3.785.1	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{1})q^{5}+(-2+\cdots)q^{7}+\cdots\)
6724.2.a.c	$6724$	$2$	6724.a	1.a	$1$	$4$	$4$	$53.691$	4.4.25808.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{2})q^{3}+(2-\beta _{2}+\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots\)
6724.2.a.d	$6724$	$2$	6724.a	1.a	$1$	$4$	$4$	$53.691$	4.4.25088.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{5}+\beta _{3}q^{7}+\beta _{2}q^{9}+\cdots\)
6724.2.a.e	$6724$	$2$	6724.a	1.a	$1$	$6$	$6$	$53.691$	6.6.40716288.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{2})q^{5}+(\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
6724.2.a.f	$6724$	$2$	6724.a	1.a	$1$	$8$	$8$	$53.691$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{6}q^{5}+(\beta _{2}+\beta _{5}+\beta _{6}+\beta _{7})q^{7}+\cdots\)
6724.2.a.g	$6724$	$2$	6724.a	1.a	$1$	$8$	$8$	$53.691$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{6}q^{5}+(-\beta _{2}-\beta _{5}-\beta _{6}+\cdots)q^{7}+\cdots\)
6724.2.a.h	$6724$	$2$	6724.a	1.a	$1$	$12$	$12$	$53.691$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(\beta _{4}+\beta _{10})q^{7}+(1+\cdots)q^{9}+\cdots\)
6724.2.a.i	$6724$	$2$	6724.a	1.a	$1$	$12$	$12$	$53.691$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{5}+(-\beta _{4}-\beta _{10})q^{7}+\cdots\)
6724.2.a.j	$6724$	$2$	6724.a	1.a	$1$	$16$	$16$	$53.691$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{4}q^{5}+(\beta _{1}-\beta _{11})q^{7}+(2+\cdots)q^{9}+\cdots\)
6724.2.a.k	$6724$	$2$	6724.a	1.a	$1$	$18$	$18$	$53.691$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-5\)	\(2\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{4}+\beta _{7}+\beta _{12}-\beta _{17})q^{5}+\cdots\)
6724.2.a.l	$6724$	$2$	6724.a	1.a	$1$	$18$	$18$	$53.691$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(5\)	\(2\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{4}+\beta _{7}+\beta _{12}-\beta _{17})q^{5}+\cdots\)
6724.2.a.m	$6724$	$2$	6724.a	1.a	$1$	$24$	$24$	$53.691$		None	✓	$2$	$1$	\(0\)	\(0\)	\(-16\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6724))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6724)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(82))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(164))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1681))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3362))\)\(^{\oplus 2}\)