Space of modular forms of level 3200, weight 1, and Character 3200.g

• Label: 3200.1.g
• Level: $3200$
• Weight: $1$
• Character orbit: 3200.g
• Rep. character: $\chi_{3200}(2751,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $5$
• Sturm bound: $480$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3200.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(480\)
Trace bound:			\(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(3200, [\chi])\).

	Total	New	Old
Modular forms	78	11	67
Cusp forms	30	11	19
Eisenstein series	48	0	48

The following table gives the dimensions of subspaces with specified projective image type.

	\(D_n\)	\(A_4\)	\(S_4\)	\(A_5\)
Dimension	11	0	0	0

Trace form

\( 11 q + 9 q^{9} + O(q^{10}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(3200, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3200.1.g.a	$3200$	$1$	3200.g	8.d	$2$	$1$	$1$	$1.597$	\(\Q\)	$D_{2}$	\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{2}) \)	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-q^{9}+2q^{17}+2q^{41}+q^{49}+2q^{73}+\cdots\)
3200.1.g.b	$3200$	$1$	3200.g	8.d	$2$	$2$	$2$	$1.597$	\(\Q(\sqrt{-1}) \)	$D_{2}$	\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \)	\(\Q(\sqrt{10}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$		\(q-q^{9}-iq^{13}-iq^{37}-q^{41}+q^{49}+\cdots\)
3200.1.g.c	$3200$	$1$	3200.g	8.d	$2$	$2$	$2$	$1.597$	\(\Q(\sqrt{3}) \)	$D_{6}$	\(\Q(\sqrt{-2}) \)	None	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q-\beta q^{3}+2q^{9}-\beta q^{11}-q^{17}+\beta q^{19}+\cdots\)
3200.1.g.d	$3200$	$1$	3200.g	8.d	$2$	$2$	$2$	$1.597$	\(\Q(\sqrt{3}) \)	$D_{6}$	\(\Q(\sqrt{-2}) \)	None	✓	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$		\(q+\beta q^{3}+2q^{9}-\beta q^{11}+q^{17}+\beta q^{19}+\cdots\)
3200.1.g.e	$3200$	$1$	3200.g	8.d	$2$	$4$	$4$	$1.597$	\(\Q(\zeta_{8})\)	$D_{4}$	\(\Q(\sqrt{-5}) \)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$		\(q+(-\zeta_{8}+\zeta_{8}^{3})q^{3}+(-\zeta_{8}-\zeta_{8}^{3})q^{7}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(3200, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(3200, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(640, [\chi])\)\(^{\oplus 2}\)