Space of modular forms of level 3456, weight 2, and Character 3456.r

• Label: 3456.2.r
• Level: $3456$
• Weight: $2$
• Character orbit: 3456.r
• Rep. character: $\chi_{3456}(577,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $96$
• Newform subspaces: $8$
• Sturm bound: $1152$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.r (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 72 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(1152\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3456, [\chi])\).

	Total	New	Old
Modular forms	1248	96	1152
Cusp forms	1056	96	960
Eisenstein series	192	0	192

Trace form

\( 96 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3456, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3456.2.r.a	$3456$	$2$	3456.r	72.n	$6$	$4$	$2$	$27.596$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-4\zeta_{12}^{2}q^{7}+(-\zeta_{12}+\zeta_{12}^{3})q^{11}+\cdots\)
3456.2.r.b	$3456$	$2$	3456.r	72.n	$6$	$4$	$2$	$27.596$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-6+\beta _{1}+3\beta _{2}-\beta _{3})q^{11}+(3-4\beta _{1}+\cdots)q^{17}+\cdots\)
3456.2.r.c	$3456$	$2$	3456.r	72.n	$6$	$4$	$2$	$27.596$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(6+\beta _{1}-3\beta _{2}-\beta _{3})q^{11}+(3+4\beta _{1}+\cdots)q^{17}+\cdots\)
3456.2.r.d	$3456$	$2$	3456.r	72.n	$6$	$4$	$2$	$27.596$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+4\zeta_{12}^{2}q^{7}+(-\zeta_{12}+\zeta_{12}^{3})q^{11}+\cdots\)
3456.2.r.e	$3456$	$2$	3456.r	72.n	$6$	$16$	$8$	$27.596$	16.0.\(\cdots\).9	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{5}-\beta _{7})q^{5}+\beta _{13}q^{7}+\beta _{10}q^{11}+\cdots\)
3456.2.r.f	$3456$	$2$	3456.r	72.n	$6$	$16$	$8$	$27.596$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{10}q^{5}+(-\beta _{3}-\beta _{15})q^{7}-\beta _{4}q^{11}+\cdots\)
3456.2.r.g	$3456$	$2$	3456.r	72.n	$6$	$24$	$12$	$27.596$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$
3456.2.r.h	$3456$	$2$	3456.r	72.n	$6$	$24$	$12$	$27.596$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(3456, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3456, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1152, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1728, [\chi])\)\(^{\oplus 2}\)