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

• Label: 1568.2.i
• Level: $1568$
• Weight: $2$
• Character orbit: 1568.i
• Rep. character: $\chi_{1568}(961,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $80$
• Newform subspaces: $25$
• Sturm bound: $448$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(448\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	512	80	432
Cusp forms	384	80	304
Eisenstein series	128	0	128

Trace form

\( 80 q - 40 q^{9} + O(q^{10}) \)

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.i.a	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
1568.2.i.b	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
1568.2.i.c	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
1568.2.i.d	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(-2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
1568.2.i.e	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-4\zeta_{6}q^{5}+3\zeta_{6}q^{9}-4q^{13}+(-8+\cdots)q^{17}+\cdots\)
1568.2.i.f	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)		$4$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}-6q^{13}+(2-2\zeta_{6})q^{17}+\cdots\)
1568.2.i.g	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+6q^{13}+(-2+\cdots)q^{17}+\cdots\)
1568.2.i.h	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+4\zeta_{6}q^{5}+3\zeta_{6}q^{9}+4q^{13}+(8-8\zeta_{6})q^{17}+\cdots\)
1568.2.i.i	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(4+\cdots)q^{11}+\cdots\)
1568.2.i.j	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
1568.2.i.k	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+\cdots\)
1568.2.i.l	$1568$	$2$	1568.i	7.c	$3$	$2$	$1$	$12.521$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(-4+\cdots)q^{11}+\cdots\)
1568.2.i.m	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
1568.2.i.n	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1568.2.i.o	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1}-\beta _{2})q^{3}+(2\beta _{1}-\beta _{2}+2\beta _{3})q^{5}+\cdots\)
1568.2.i.p	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+3\beta _{2}q^{5}+4\beta _{2}q^{9}+\beta _{1}q^{11}+\cdots\)
1568.2.i.q	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{9}+(-2-2\beta _{2})q^{11}+\cdots\)
1568.2.i.r	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{9}+(2+2\beta _{2})q^{11}-2\beta _{3}q^{13}+\cdots\)
1568.2.i.s	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+3\beta _{1}q^{5}+(3+3\beta _{2})q^{9}+5\beta _{3}q^{13}+\cdots\)
1568.2.i.t	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-1}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-\beta _{1}q^{5}+(3+3\beta _{2})q^{9}+\beta _{3}q^{13}+(-5\beta _{1}+\cdots)q^{17}+\cdots\)
1568.2.i.u	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}^{2}q^{3}+(1-\zeta_{12})q^{5}+3\zeta_{12}^{2}q^{11}+\cdots\)
1568.2.i.v	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
1568.2.i.w	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(2\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\cdots\)
1568.2.i.x	$1568$	$2$	1568.i	7.c	$3$	$4$	$2$	$12.521$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{3}+(-2\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+\cdots\)
1568.2.i.y	$1568$	$2$	1568.i	7.c	$3$	$8$	$4$	$12.521$	8.0.207360000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{3}+(-2\beta _{1}+2\beta _{4})q^{5}-7\beta _{3}q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1568, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(784, [\chi])\)\(^{\oplus 2}\)