Space of modular forms of level 800, weight 4, and Character 800.c

• Label: 800.4.c
• Level: $800$
• Weight: $4$
• Character orbit: 800.c
• Rep. character: $\chi_{800}(449,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $15$
• Sturm bound: $480$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(480\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(800, [\chi])\).

	Total	New	Old
Modular forms	384	54	330
Cusp forms	336	54	282
Eisenstein series	48	0	48

Trace form

\( 54 q - 486 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(800, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.4.c.a	$800$	$4$	800.c	5.b	$2$	$2$	$2$	$47.202$	\(\Q(\sqrt{-1}) \)	None					32.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}-8iq^{7}-37q^{9}-40q^{11}+\cdots\)
800.4.c.b	$800$	$4$	800.c	5.b	$2$	$2$	$2$	$47.202$	\(\Q(\sqrt{-1}) \)	None					32.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}-8iq^{7}-37q^{9}+40q^{11}+\cdots\)
800.4.c.c	$800$	$4$	800.c	5.b	$2$	$2$	$2$	$47.202$	\(\Q(\sqrt{-1}) \)	None		✓			800.4.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5iq^{3}+10iq^{7}+2q^{9}-15q^{11}+\cdots\)
800.4.c.d	$800$	$4$	800.c	5.b	$2$	$2$	$2$	$47.202$	\(\Q(\sqrt{-1}) \)	None		✓			800.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+5iq^{3}+10iq^{7}+2q^{9}+15q^{11}+\cdots\)
800.4.c.e	$800$	$4$	800.c	5.b	$2$	$2$	$2$	$47.202$	\(\Q(\sqrt{-1}) \)	None					160.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+3iq^{7}+23q^{9}-60q^{11}+\cdots\)
800.4.c.f	$800$	$4$	800.c	5.b	$2$	$2$	$2$	$47.202$	\(\Q(\sqrt{-1}) \)	None					160.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+3iq^{7}+23q^{9}+60q^{11}+\cdots\)
800.4.c.g	$800$	$4$	800.c	5.b	$2$	$2$	$2$	$47.202$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					32.4.a.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{3}q^{9}-9iq^{13}+47iq^{17}+130q^{29}+\cdots\)
800.4.c.h	$800$	$4$	800.c	5.b	$2$	$4$	$4$	$47.202$	\(\Q(i, \sqrt{13})\)	None					160.4.a.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{2}q^{7}-5^{2}q^{9}+3\beta _{3}q^{11}+\cdots\)
800.4.c.i	$800$	$4$	800.c	5.b	$2$	$4$	$4$	$47.202$	\(\Q(i, \sqrt{6})\)	None					160.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}-5\beta _{2})q^{7}+\cdots\)
800.4.c.j	$800$	$4$	800.c	5.b	$2$	$4$	$4$	$47.202$	\(\Q(i, \sqrt{10})\)	None					160.4.a.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+3\beta _{2}q^{7}-13q^{9}+\beta _{3}q^{11}+\cdots\)
800.4.c.k	$800$	$4$	800.c	5.b	$2$	$4$	$4$	$47.202$	\(\Q(i, \sqrt{6})\)	None					160.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}-5\beta _{2})q^{7}+\cdots\)
800.4.c.l	$800$	$4$	800.c	5.b	$2$	$4$	$4$	$47.202$	\(\Q(i, \sqrt{5})\)	None					160.4.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+7\beta _{2}q^{7}+7q^{9}-\beta _{3}q^{11}+\cdots\)
800.4.c.m	$800$	$4$	800.c	5.b	$2$	$6$	$6$	$47.202$	6.0.2068430400.1	None		✓			800.4.a.u	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+2\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots\)
800.4.c.n	$800$	$4$	800.c	5.b	$2$	$6$	$6$	$47.202$	6.0.2068430400.1	None		✓			800.4.a.u	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+2\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots\)
800.4.c.o	$800$	$4$	800.c	5.b	$2$	$8$	$8$	$47.202$	8.0.\(\cdots\).12	None		✓	✓		800.4.a.ba	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(\beta _{3}+\beta _{7})q^{7}+(-24+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(800, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)