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

• Label: 1568.3.c
• Level: $1568$
• Weight: $3$
• Character orbit: 1568.c
• Rep. character: $\chi_{1568}(97,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $8$
• Sturm bound: $672$
• Trace bound: $53$

Defining parameters

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(672\)
Trace bound:			\(53\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

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

	Total	New	Old
Modular forms	480	80	400
Cusp forms	416	80	336
Eisenstein series	64	0	64

Trace form

\( 80 q - 240 q^{9} - 432 q^{25} + 96 q^{29} - 160 q^{37} - 32 q^{53} - 32 q^{57} - 32 q^{65} + 496 q^{81} - 224 q^{85} + 416 q^{93}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1568.3.c.a	$1568$	$3$	1568.c	7.b	$2$	$4$	$4$	$42.725$	4.0.2048.2	\(\Q(\sqrt{-1}) \)		✓			1568.3.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(4\beta _{1}+3\beta _{3})q^{5}+9q^{9}+(-5\beta _{1}-12\beta _{3})q^{13}+\cdots\)
1568.3.c.b	$1568$	$3$	1568.c	7.b	$2$	$4$	$4$	$42.725$	4.0.2048.2	\(\Q(\sqrt{-1}) \)		✓			1568.3.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(3\beta _{1}+4\beta _{3})q^{5}+9q^{9}+(12\beta _{1}+5\beta _{3})q^{13}+\cdots\)
1568.3.c.c	$1568$	$3$	1568.c	7.b	$2$	$8$	$8$	$42.725$	8.0.\(\cdots\).26	None		✓			1568.3.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-\beta _{1}-\beta _{4})q^{5}+(-13-5\beta _{3}+\cdots)q^{9}+\cdots\)
1568.3.c.d	$1568$	$3$	1568.c	7.b	$2$	$8$	$8$	$42.725$	8.0.\(\cdots\).16	None		✓			1568.3.c.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-\beta _{1}-3\beta _{3})q^{5}+(-5+3\beta _{5}+\cdots)q^{9}+\cdots\)
1568.3.c.e	$1568$	$3$	1568.c	7.b	$2$	$12$	$12$	$42.725$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1568.3.c.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{3}+\beta _{11}q^{5}+(-3+\beta _{1}+\beta _{3}+\cdots)q^{9}+\cdots\)
1568.3.c.f	$1568$	$3$	1568.c	7.b	$2$	$12$	$12$	$42.725$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			1568.3.c.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{3}-\beta _{11}q^{5}+(-3+\beta _{1}+\beta _{3}+\cdots)q^{9}+\cdots\)
1568.3.c.g	$1568$	$3$	1568.c	7.b	$2$	$16$	$16$	$42.725$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					224.3.s.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{3}+\beta _{1}q^{5}+(-5-\beta _{4})q^{9}+(\beta _{5}+\cdots)q^{11}+\cdots\)
1568.3.c.h	$1568$	$3$	1568.c	7.b	$2$	$16$	$16$	$42.725$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					224.3.s.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{12}q^{5}+(-1+\beta _{3})q^{9}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1568, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(784, [\chi])\)\(^{\oplus 2}\)