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

• Label: 3200.2.c
• Level: $3200$
• Weight: $2$
• Character orbit: 3200.c
• Rep. character: $\chi_{3200}(2049,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $28$
• Sturm bound: $960$
• Trace bound: $29$

Defining parameters

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(960\)
Trace bound:			\(29\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(29\)

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

\( 72 q - 72 q^{9} - 16 q^{41} - 72 q^{49} + 136 q^{81} - 112 q^{89}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3200.2.c.a	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		✓			3200.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+4 i q^{7}-6 q^{9}-3 q^{11}+\cdots\)
3200.2.c.b	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		✓			3200.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}-4 i q^{7}-6 q^{9}-3 q^{11}+\cdots\)
3200.2.c.c	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		✓			3200.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+4 i q^{7}-6 q^{9}+3 q^{11}+\cdots\)
3200.2.c.d	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		✓			3200.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}-4 i q^{7}-6 q^{9}+3 q^{11}+\cdots\)
3200.2.c.e	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					128.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+2\beta q^{7}-q^{9}-2 q^{11}+\beta q^{13}+\cdots\)
3200.2.c.f	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					128.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-2\beta q^{7}-q^{9}-2 q^{11}-\beta q^{13}+\cdots\)
3200.2.c.g	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					640.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-q^{9}-2 q^{11}-\beta q^{13}+3\beta q^{17}+\cdots\)
3200.2.c.h	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					640.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-q^{9}-2 q^{11}+\beta q^{13}+3\beta q^{17}+\cdots\)
3200.2.c.i	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					640.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-q^{9}+2 q^{11}+\beta q^{13}-3\beta q^{17}+\cdots\)
3200.2.c.j	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					640.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-q^{9}+2 q^{11}-\beta q^{13}-3\beta q^{17}+\cdots\)
3200.2.c.k	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					128.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+2\beta q^{7}-q^{9}+2 q^{11}-\beta q^{13}+\cdots\)
3200.2.c.l	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					128.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-2\beta q^{7}-q^{9}+2 q^{11}+\beta q^{13}+\cdots\)
3200.2.c.m	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		✓			3200.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}-q^{11}+2 i q^{13}-3 i q^{17}+\cdots\)
3200.2.c.n	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		✓			3200.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}-q^{11}-2 i q^{13}-3 i q^{17}+\cdots\)
3200.2.c.o	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		✓			3200.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}+q^{11}-2 i q^{13}+3 i q^{17}+\cdots\)
3200.2.c.p	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None		✓			3200.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 q^{9}+q^{11}+2 i q^{13}+3 i q^{17}+\cdots\)
3200.2.c.q	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					640.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{7}+3 q^{9}-6 q^{11}+\beta q^{13}+3\beta q^{17}+\cdots\)
3200.2.c.r	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					640.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+3 q^{9}-6 q^{11}-\beta q^{13}+3\beta q^{17}+\cdots\)
3200.2.c.s	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					640.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+3 q^{9}+6 q^{11}+\beta q^{13}+3\beta q^{17}+\cdots\)
3200.2.c.t	$3200$	$2$	3200.c	5.b	$2$	$2$	$2$	$25.552$	\(\Q(\sqrt{-1}) \)	None					640.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{7}+3 q^{9}+6 q^{11}+\beta q^{13}-3\beta q^{17}+\cdots\)
3200.2.c.u	$3200$	$2$	3200.c	5.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{5})\)	None					640.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(-3+\beta _{3})q^{9}+\cdots\)
3200.2.c.v	$3200$	$2$	3200.c	5.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{5})\)	None					640.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(-3+\beta _{3})q^{9}+\cdots\)
3200.2.c.w	$3200$	$2$	3200.c	5.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{5})\)	None					640.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(-3+\beta _{3})q^{9}+\cdots\)
3200.2.c.x	$3200$	$2$	3200.c	5.b	$2$	$4$	$4$	$25.552$	\(\Q(i, \sqrt{5})\)	None					640.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(-3+\beta _{3})q^{9}+\cdots\)
3200.2.c.y	$3200$	$2$	3200.c	5.b	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		✓			3200.2.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{3}+(2\beta_{2}-2\beta_1)q^{7}+\cdots\)
3200.2.c.z	$3200$	$2$	3200.c	5.b	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		✓			3200.2.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{3}+(-2\beta_{2}+2\beta_1)q^{7}+\cdots\)
3200.2.c.ba	$3200$	$2$	3200.c	5.b	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		✓			3200.2.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{3}+(2\beta_{2}-2\beta_1)q^{7}+\cdots\)
3200.2.c.bb	$3200$	$2$	3200.c	5.b	$2$	$4$	$4$	$25.552$	\(\Q(\zeta_{8})\)	None		✓			3200.2.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta_{2}+\beta_1)q^{3}+(-2\beta_{2}+2\beta_1)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(3200, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(320, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(640, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1600, [\chi])\)\(^{\oplus 2}\)