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

• Label: 3200.2.d
• Level: $3200$
• Weight: $2$
• Character orbit: 3200.d
• Rep. character: $\chi_{3200}(1601,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $23$
• Sturm bound: $960$
• Trace bound: $33$

Defining parameters

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(960\)
Trace bound:			\(33\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	528	76	452
Cusp forms	432	76	356
Eisenstein series	96	0	96

Trace form

\( 76 q - 76 q^{9} + O(q^{10}) \)

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.d.a	$3200$	$2$	3200.d	8.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-4q^{7}+2q^{9}-3iq^{11}-q^{17}+\cdots\)
3200.2.d.b	$3200$	$2$	3200.d	8.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-4q^{7}+2q^{9}+3iq^{11}+q^{17}+\cdots\)
3200.2.d.c	$3200$	$2$	3200.d	8.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{3}-5q^{9}-\beta q^{11}-6q^{17}+3\beta q^{19}+\cdots\)
3200.2.d.d	$3200$	$2$	3200.d	8.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{9}-3iq^{13}-8q^{17}-2iq^{29}+\cdots\)
3200.2.d.e	$3200$	$2$	3200.d	8.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{9}-iq^{13}+2q^{17}+iq^{29}-3iq^{37}+\cdots\)
3200.2.d.f	$3200$	$2$	3200.d	8.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{9}+3iq^{13}+8q^{17}-2iq^{29}+\cdots\)
3200.2.d.g	$3200$	$2$	3200.d	8.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+4q^{7}+2q^{9}-3iq^{11}-q^{17}+\cdots\)
3200.2.d.h	$3200$	$2$	3200.d	8.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+4q^{7}+2q^{9}+3iq^{11}+q^{17}+\cdots\)
3200.2.d.i	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\sqrt{2}, \sqrt{-5})\)	\(\Q(\sqrt{-5}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}-3\beta _{1}q^{7}-7q^{9}+3\beta _{3}q^{21}+\cdots\)
3200.2.d.j	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-\beta _{3}q^{7}-7q^{9}+3\beta _{1}q^{13}+\cdots\)
3200.2.d.k	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}-\beta _{3}q^{7}-3q^{9}-2\beta _{2}q^{11}+\cdots\)
3200.2.d.l	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+\beta _{3}q^{7}-3q^{9}+2\beta _{2}q^{11}+\cdots\)
3200.2.d.m	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\sqrt{2}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+\beta _{1}q^{7}-2q^{9}+\beta _{3}q^{11}+\cdots\)
3200.2.d.n	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}+(-2+\beta _{3})q^{9}+(\beta _{1}+2\beta _{2}+\cdots)q^{11}+\cdots\)
3200.2.d.o	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-2q^{9}+\beta _{1}q^{11}-\beta _{2}q^{13}+\cdots\)
3200.2.d.p	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-2q^{9}+\beta _{1}q^{11}-\beta _{2}q^{13}+\cdots\)
3200.2.d.q	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}+(-2+\beta _{3})q^{9}+(-\beta _{1}-2\beta _{2}+\cdots)q^{11}+\cdots\)
3200.2.d.r	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\sqrt{2}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-\beta _{1}q^{7}-2q^{9}-\beta _{3}q^{11}+\cdots\)
3200.2.d.s	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}^{2}q^{3}+3\zeta_{8}^{3}q^{7}+q^{9}+4\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.d.t	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}^{2}q^{3}-\zeta_{8}^{3}q^{7}+q^{9}-2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.d.u	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	\(\Q(\sqrt{-5}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{3}+\beta _{2}q^{7}+q^{9}+\beta _{3}q^{21}-3\beta _{2}q^{23}+\cdots\)
3200.2.d.v	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}^{2}q^{3}-\zeta_{8}^{3}q^{7}+q^{9}+2\zeta_{8}^{2}q^{11}+\cdots\)
3200.2.d.w	$3200$	$2$	3200.d	8.b	$2$	$4$	$4$	$25.552$	\(\Q(\sqrt{2}, \sqrt{-5})\)	\(\Q(\sqrt{-10}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta _{1}q^{7}+3q^{9}-\beta _{2}q^{11}+\beta _{3}q^{13}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(3200, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(640, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1600, [\chi])\)\(^{\oplus 2}\)