Space of modular forms of level 7056, weight 2, and Character 7056.k

• Label: 7056.2.k
• Level: $7056$
• Weight: $2$
• Character orbit: 7056.k
• Rep. character: $\chi_{7056}(881,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $9$
• Sturm bound: $2688$
• Trace bound: $121$

Defining parameters

Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(2688\)
Trace bound:			\(121\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(7056, [\chi])\).

	Total	New	Old
Modular forms	1440	80	1360
Cusp forms	1248	80	1168
Eisenstein series	192	0	192

Trace form

\( 80 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(7056, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
7056.2.k.a	$7056$	$2$	7056.k	21.c	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}-\beta _{3}q^{11}-\beta _{1}q^{13}+2\beta _{2}q^{17}+\cdots\)
7056.2.k.b	$7056$	$2$	7056.k	21.c	$2$	$4$	$4$	$56.342$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}-\beta _{1}q^{11}-3\beta _{2}q^{13}+2\beta _{3}q^{17}+\cdots\)
7056.2.k.c	$7056$	$2$	7056.k	21.c	$2$	$8$	$8$	$56.342$	\(\Q(\zeta_{16})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{16}^{5}q^{5}-\zeta_{16}q^{11}-3\zeta_{16}^{2}q^{13}+\cdots\)
7056.2.k.d	$7056$	$2$	7056.k	21.c	$2$	$8$	$8$	$56.342$	\(\Q(\zeta_{16})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{16}^{5}q^{5}-2\zeta_{16}q^{11}+(-4\zeta_{16}^{2}+\cdots)q^{13}+\cdots\)
7056.2.k.e	$7056$	$2$	7056.k	21.c	$2$	$8$	$8$	$56.342$	\(\Q(\zeta_{16})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{16}^{5}q^{5}+\zeta_{16}q^{11}+(2\zeta_{16}^{2}+\zeta_{16}^{4}+\cdots)q^{13}+\cdots\)
7056.2.k.f	$7056$	$2$	7056.k	21.c	$2$	$8$	$8$	$56.342$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{24}^{5}q^{5}+\zeta_{24}^{7}q^{11}-\zeta_{24}^{3}q^{13}+\cdots\)
7056.2.k.g	$7056$	$2$	7056.k	21.c	$2$	$8$	$8$	$56.342$	\(\Q(\zeta_{16})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{16}^{4}q^{5}+(\zeta_{16}+2\zeta_{16}^{2})q^{11}-\zeta_{16}^{5}q^{13}+\cdots\)
7056.2.k.h	$7056$	$2$	7056.k	21.c	$2$	$16$	$16$	$56.342$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{13}q^{5}-\beta _{6}q^{11}+(-2\beta _{3}+2\beta _{5}+\cdots)q^{13}+\cdots\)
7056.2.k.i	$7056$	$2$	7056.k	21.c	$2$	$16$	$16$	$56.342$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{13}q^{5}+(\beta _{5}-\beta _{6})q^{11}+\beta _{7}q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(7056, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(7056, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1008, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1176, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1764, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2352, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(3528, [\chi])\)\(^{\oplus 2}\)