Space of modular forms of level 1568, weight 2, and Character 1568.q

• Label: 1568.2.q
• Level: $1568$
• Weight: $2$
• Character orbit: 1568.q
• Rep. character: $\chi_{1568}(815,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $72$
• Newform subspaces: $8$
• Sturm bound: $448$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.q (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(448\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(3\), \(5\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1568, [\chi])\).

	Total	New	Old
Modular forms	512	88	424
Cusp forms	384	72	312
Eisenstein series	128	16	112

Trace form

\( 72 q - 6 q^{3} + 30 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1568, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1568.2.q.a	$1568$	$2$	1568.q	56.m	$6$	$4$	$2$	$12.521$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	\(\Q(\sqrt{-7}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{6}]$	\(q+(-3-3\beta _{1})q^{9}+4\beta _{1}q^{11}-\beta _{3}q^{23}+\cdots\)
1568.2.q.b	$1568$	$2$	1568.q	56.m	$6$	$8$	$4$	$12.521$	8.0.339738624.1	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-\beta _{3}-\beta _{7})q^{3}+(-1+\beta _{2}-\beta _{4}+\cdots)q^{9}+\cdots\)
1568.2.q.c	$1568$	$2$	1568.q	56.m	$6$	$8$	$4$	$12.521$	8.0.339738624.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{3})q^{3}+(-\beta _{3}+\beta _{7})q^{5}+(-\beta _{4}+\cdots)q^{9}+\cdots\)
1568.2.q.d	$1568$	$2$	1568.q	56.m	$6$	$8$	$4$	$12.521$	8.0.339738624.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{3})q^{3}+(\beta _{3}-\beta _{7})q^{5}+(-\beta _{4}+\cdots)q^{9}+\cdots\)
1568.2.q.e	$1568$	$2$	1568.q	56.m	$6$	$8$	$4$	$12.521$	\(\Q(\zeta_{24})\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{24}^{4}-\zeta_{24}^{7})q^{3}+(\zeta_{24}^{5}-\zeta_{24}^{6}+\cdots)q^{5}+\cdots\)
1568.2.q.f	$1568$	$2$	1568.q	56.m	$6$	$8$	$4$	$12.521$	8.0.339738624.1	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{U}(1)[D_{6}]$	\(q+\beta _{1}q^{3}+(-7\beta _{4}-\beta _{5})q^{9}+(-2+3\beta _{2}+\cdots)q^{11}+\cdots\)
1568.2.q.g	$1568$	$2$	1568.q	56.m	$6$	$12$	$6$	$12.521$	12.0.\(\cdots\).2	None		$4$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+\beta _{3})q^{3}+(-\beta _{7}+\beta _{11})q^{5}+\cdots\)
1568.2.q.h	$1568$	$2$	1568.q	56.m	$6$	$16$	$8$	$12.521$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{6}q^{3}+\beta _{15}q^{5}+(-1+\beta _{3}+\beta _{8}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1568, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1568, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(784, [\chi])\)\(^{\oplus 2}\)