Space of modular forms of level 1568, weight 3, and Character 1568.g

• Label: 1568.3.g
• Level: $1568$
• Weight: $3$
• Character orbit: 1568.g
• Rep. character: $\chi_{1568}(687,\cdot)$
• Character field: $\Q$
• Dimension: $77$
• Newform subspaces: $15$
• Sturm bound: $672$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(672\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1568, [\chi])\).

	Total	New	Old
Modular forms	480	87	393
Cusp forms	416	77	339
Eisenstein series	64	10	54

Trace form

\( 77 q - 2 q^{3} + 203 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1568, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1568.3.g.a	$1568$	$3$	1568.g	8.d	$2$	$1$	$1$	$42.725$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{3}-5q^{9}-14q^{11}-2q^{17}-34q^{19}+\cdots\)
1568.3.g.b	$1568$	$3$	1568.g	8.d	$2$	$2$	$2$	$42.725$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-4+\beta )q^{3}+(9-8\beta )q^{9}-12\beta q^{11}+\cdots\)
1568.3.g.c	$1568$	$3$	1568.g	8.d	$2$	$2$	$2$	$42.725$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-3\zeta_{6}q^{5}-8q^{9}-17q^{11}-8\zeta_{6}q^{13}+\cdots\)
1568.3.g.d	$1568$	$3$	1568.g	8.d	$2$	$2$	$2$	$42.725$	\(\Q(\sqrt{-7}) \)	\(\Q(\sqrt{-7}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-9q^{9}-6q^{11}+2\beta q^{23}+5^{2}q^{25}+\cdots\)
1568.3.g.e	$1568$	$3$	1568.g	8.d	$2$	$2$	$2$	$42.725$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{3}+23q^{9}+14q^{11}-6\beta q^{17}+\cdots\)
1568.3.g.f	$1568$	$3$	1568.g	8.d	$2$	$2$	$2$	$42.725$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+3\zeta_{6}q^{5}-8q^{9}-17q^{11}+8\zeta_{6}q^{13}+\cdots\)
1568.3.g.g	$1568$	$3$	1568.g	8.d	$2$	$2$	$2$	$42.725$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓	$2$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(4+\beta )q^{3}+(9+8\beta )q^{9}+12\beta q^{11}+\cdots\)
1568.3.g.h	$1568$	$3$	1568.g	8.d	$2$	$4$	$4$	$42.725$	\(\Q(\sqrt{2}, \sqrt{-7})\)	None		$2$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\beta _{1})q^{3}+(\beta _{2}-\beta _{3})q^{5}+(-3-4\beta _{1}+\cdots)q^{9}+\cdots\)
1568.3.g.i	$1568$	$3$	1568.g	8.d	$2$	$6$	$6$	$42.725$	6.0.700560112.1	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{3})q^{3}+(-2\beta _{1}-\beta _{5})q^{5}+\cdots\)
1568.3.g.j	$1568$	$3$	1568.g	8.d	$2$	$6$	$6$	$42.725$	6.0.15582448.1	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{1}+\beta _{2})q^{3}+(-\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
1568.3.g.k	$1568$	$3$	1568.g	8.d	$2$	$6$	$6$	$42.725$	6.0.700560112.1	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{3}+(-2\beta _{1}-\beta _{5})q^{5}+(1+\cdots)q^{9}+\cdots\)
1568.3.g.l	$1568$	$3$	1568.g	8.d	$2$	$6$	$6$	$42.725$	6.0.15582448.1	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{1}-\beta _{2})q^{3}+(-\beta _{3}+\beta _{4}-\beta _{5})q^{5}+\cdots\)
1568.3.g.m	$1568$	$3$	1568.g	8.d	$2$	$8$	$8$	$42.725$	8.0.\(\cdots\).3	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2})q^{3}+\beta _{1}q^{5}+(6+\beta _{2}-\beta _{6}+\cdots)q^{9}+\cdots\)
1568.3.g.n	$1568$	$3$	1568.g	8.d	$2$	$8$	$8$	$42.725$	8.0.\(\cdots\).5	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+\beta _{2}q^{5}+(1-\beta _{1})q^{9}+4\beta _{1}q^{11}+\cdots\)
1568.3.g.o	$1568$	$3$	1568.g	8.d	$2$	$20$	$20$	$42.725$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{38}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+\beta _{5}q^{5}+(2-\beta _{2}-\beta _{9}+\beta _{11}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1568, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1568, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(784, [\chi])\)\(^{\oplus 2}\)