Space of modular forms of level 1600, weight 3, and Character 1600.h

• Label: 1600.3.h
• Level: $1600$
• Weight: $3$
• Character orbit: 1600.h
• Rep. character: $\chi_{1600}(1599,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $16$
• Sturm bound: $720$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(720\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1600, [\chi])\).

	Total	New	Old
Modular forms	516	74	442
Cusp forms	444	70	374
Eisenstein series	72	4	68

Trace form

\( 70 q + 202 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1600, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1600.3.h.a	$1600$	$3$	1600.h	20.d	$2$	$2$	$2$	$43.597$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(16\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4q^{3}+8q^{7}+7q^{9}+2iq^{11}-7iq^{13}+\cdots\)
1600.3.h.b	$1600$	$3$	1600.h	20.d	$2$	$2$	$2$	$43.597$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q-9q^{9}-iq^{13}+3iq^{17}+42q^{29}+\cdots\)
1600.3.h.c	$1600$	$3$	1600.h	20.d	$2$	$2$	$2$	$43.597$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(8\)	\(0\)	\(-16\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4q^{3}-8q^{7}+7q^{9}-2iq^{11}-7iq^{13}+\cdots\)
1600.3.h.d	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3+\beta _{3})q^{3}+(1+5\beta _{3})q^{7}+(5-6\beta _{3})q^{9}+\cdots\)
1600.3.h.e	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{2})q^{3}+(4-2\beta _{2})q^{7}+(6-4\beta _{2}+\cdots)q^{9}+\cdots\)
1600.3.h.f	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{3})q^{3}+(-1+\beta _{3})q^{7}+(-3+\cdots)q^{9}+\cdots\)
1600.3.h.g	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-4-2\beta _{2})q^{7}+2q^{9}+(-8\beta _{1}+\cdots)q^{11}+\cdots\)
1600.3.h.h	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{3}q^{3}+2\zeta_{12}^{3}q^{7}-6q^{9}+7\zeta_{12}^{2}q^{11}+\cdots\)
1600.3.h.i	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{15})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-2\beta _{1}q^{7}+6q^{9}+\beta _{3}q^{11}+\cdots\)
1600.3.h.j	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(4+2\beta _{2})q^{7}+2q^{9}+(-8\beta _{1}+\cdots)q^{11}+\cdots\)
1600.3.h.k	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{3})q^{3}+(1-\beta _{3})q^{7}+(-3-2\beta _{3})q^{9}+\cdots\)
1600.3.h.l	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{11})\)	None		$2$	$0$	\(0\)	\(8\)	\(0\)	\(-16\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\beta _{2})q^{3}+(-4+2\beta _{2})q^{7}+(6-4\beta _{2}+\cdots)q^{9}+\cdots\)
1600.3.h.m	$1600$	$3$	1600.h	20.d	$2$	$4$	$4$	$43.597$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(12\)	\(0\)	\(-4\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3-\beta _{3})q^{3}+(-1-5\beta _{3})q^{7}+(5-6\beta _{3})q^{9}+\cdots\)
1600.3.h.n	$1600$	$3$	1600.h	20.d	$2$	$8$	$8$	$43.597$	\(\Q(\zeta_{20})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{17}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{20}^{3}q^{3}+(\zeta_{20}^{3}+\zeta_{20}^{6})q^{7}+(1+\cdots)q^{9}+\cdots\)
1600.3.h.o	$1600$	$3$	1600.h	20.d	$2$	$8$	$8$	$43.597$	8.0.\(\cdots\).3	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+(\beta _{3}+\beta _{7})q^{7}+(4+\beta _{2})q^{9}+\cdots\)
1600.3.h.p	$1600$	$3$	1600.h	20.d	$2$	$8$	$8$	$43.597$	8.0.12960000.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{17}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{5}q^{7}+(9-\beta _{3})q^{9}-\beta _{2}q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1600, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1600, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 2}\)