Space of modular forms of level 3200, weight 2, and Character 3200.f

• Label: 3200.2.f
• Level: $3200$
• Weight: $2$
• Character orbit: 3200.f
• Rep. character: $\chi_{3200}(449,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $19$
• Sturm bound: $960$
• Trace bound: $49$

Defining parameters

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(960\)
Trace bound:			\(49\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3200, [\chi])\).

	Total	New	Old
Modular forms	528	72	456
Cusp forms	432	72	360
Eisenstein series	96	0	96

Trace form

\( 72 q + 72 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(3200, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3200.2.f.a	$3200$	$2$	3200.f	40.f	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+4iq^{7}-2q^{9}+3iq^{11}-iq^{17}+\cdots\)
3200.2.f.b	$3200$	$2$	3200.f	40.f	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-4iq^{7}-2q^{9}+3iq^{11}-iq^{17}+\cdots\)
3200.2.f.c	$3200$	$2$	3200.f	40.f	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{9}-4q^{13}+iq^{17}-2iq^{29}+\cdots\)
3200.2.f.d	$3200$	$2$	3200.f	40.f	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{9}+4q^{13}+iq^{17}+2iq^{29}+\cdots\)
3200.2.f.e	$3200$	$2$	3200.f	40.f	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+4iq^{7}-2q^{9}-3iq^{11}-iq^{17}+\cdots\)
3200.2.f.f	$3200$	$2$	3200.f	40.f	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+4iq^{7}-2q^{9}+3iq^{11}+iq^{17}+\cdots\)
3200.2.f.g	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}+2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.h	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}^{3}q^{3}+\zeta_{8}^{2}q^{7}-q^{9}+2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.i	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.j	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}^{3}q^{3}+3\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.k	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}^{3}q^{3}+\zeta_{8}^{2}q^{7}-q^{9}-2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.l	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{8}^{3}q^{3}-3\zeta_{8}^{2}q^{7}-q^{9}-2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.f.m	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+2q^{9}+\beta _{2}q^{11}-4q^{13}+\cdots\)
3200.2.f.n	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+2q^{9}-\beta _{2}q^{11}+4q^{13}+\cdots\)
3200.2.f.o	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{8}^{3}q^{3}+5q^{9}-\zeta_{8}^{2}q^{11}-3\zeta_{8}q^{17}+\cdots\)
3200.2.f.p	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-\beta _{2}q^{7}+7q^{9}-6q^{13}-\beta _{1}q^{17}+\cdots\)
3200.2.f.q	$3200$	$2$	3200.f	40.f	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+\beta _{2}q^{7}+7q^{9}+6q^{13}-\beta _{1}q^{17}+\cdots\)
3200.2.f.r	$3200$	$2$	3200.f	40.f	$2$	$8$	$8$	$25.552$	8.0.40960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{2}q^{7}+2q^{9}+\beta _{3}q^{11}+\cdots\)
3200.2.f.s	$3200$	$2$	3200.f	40.f	$2$	$8$	$8$	$25.552$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 5^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\zeta_{24}q^{3}+(2-\zeta_{24}^{3})q^{9}+\zeta_{24}^{6}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3200, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3200, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 10}\)