Space of modular forms of level 800, weight 6, and Character 800.c

• Label: 800.6.c
• Level: $800$
• Weight: $6$
• Character orbit: 800.c
• Rep. character: $\chi_{800}(449,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $16$
• Sturm bound: $720$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(720\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(800, [\chi])\).

	Total	New	Old
Modular forms	624	90	534
Cusp forms	576	90	486
Eisenstein series	48	0	48

Trace form

\( 90 q - 7290 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(800, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
800.6.c.a	$800$	$6$	800.c	5.b	$2$	$2$	$2$	$128.307$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}-104iq^{7}+179q^{9}-536q^{11}+\cdots\)
800.6.c.b	$800$	$6$	800.c	5.b	$2$	$2$	$2$	$128.307$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{3}-104iq^{7}+179q^{9}+536q^{11}+\cdots\)
800.6.c.c	$800$	$6$	800.c	5.b	$2$	$2$	$2$	$128.307$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{5}q^{9}+597iq^{13}+1121iq^{17}+\cdots\)
800.6.c.d	$800$	$6$	800.c	5.b	$2$	$4$	$4$	$128.307$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}^{3}q^{3}-6\zeta_{12}^{3}q^{7}-525q^{9}+\cdots\)
800.6.c.e	$800$	$6$	800.c	5.b	$2$	$4$	$4$	$128.307$	\(\Q(i, \sqrt{85})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+3\beta _{2}q^{7}-97q^{9}+13\beta _{3}q^{11}+\cdots\)
800.6.c.f	$800$	$6$	800.c	5.b	$2$	$4$	$4$	$128.307$	\(\Q(i, \sqrt{70})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}-\beta _{2})q^{3}+(-26\beta _{1}-\beta _{2})q^{7}+\cdots\)
800.6.c.g	$800$	$6$	800.c	5.b	$2$	$4$	$4$	$128.307$	\(\Q(i, \sqrt{70})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}-\beta _{2})q^{3}+(-26\beta _{1}-\beta _{2})q^{7}+\cdots\)
800.6.c.h	$800$	$6$	800.c	5.b	$2$	$4$	$4$	$128.307$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{2}q^{3}+31\beta _{2}q^{7}+63q^{9}-29\beta _{3}q^{11}+\cdots\)
800.6.c.i	$800$	$6$	800.c	5.b	$2$	$4$	$4$	$128.307$	\(\Q(i, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+7\beta _{2}q^{7}+203q^{9}-57\beta _{3}q^{11}+\cdots\)
800.6.c.j	$800$	$6$	800.c	5.b	$2$	$6$	$6$	$128.307$	6.0.6140289600.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2\beta _{1}+\beta _{3})q^{3}+(-\beta _{1}+\beta _{3}+\beta _{5})q^{7}+\cdots\)
800.6.c.k	$800$	$6$	800.c	5.b	$2$	$6$	$6$	$128.307$	6.0.6140289600.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2\beta _{1}+\beta _{3})q^{3}+(-\beta _{1}+\beta _{3}+\beta _{5})q^{7}+\cdots\)
800.6.c.l	$800$	$6$	800.c	5.b	$2$	$8$	$8$	$128.307$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(6\beta _{1}-\beta _{4})q^{7}+(-181+\beta _{3}+\cdots)q^{9}+\cdots\)
800.6.c.m	$800$	$6$	800.c	5.b	$2$	$8$	$8$	$128.307$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}+(2\beta _{5}+\beta _{6})q^{7}+(-56-\beta _{3}+\cdots)q^{9}+\cdots\)
800.6.c.n	$800$	$6$	800.c	5.b	$2$	$10$	$10$	$128.307$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(-22\beta _{5}+\beta _{7})q^{7}+(-11^{2}+\cdots)q^{9}+\cdots\)
800.6.c.o	$800$	$6$	800.c	5.b	$2$	$10$	$10$	$128.307$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(-22\beta _{5}+\beta _{7})q^{7}+(-11^{2}+\cdots)q^{9}+\cdots\)
800.6.c.p	$800$	$6$	800.c	5.b	$2$	$12$	$12$	$128.307$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{34}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{7}q^{3}+(-\beta _{7}-\beta _{8})q^{7}+(-38+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(800, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(800, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 2}\)