Space of modular forms of level 650, weight 6, and Character 650.b

• Label: 650.6.b
• Level: $650$
• Weight: $6$
• Character orbit: 650.b
• Rep. character: $\chi_{650}(599,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $15$
• Sturm bound: $630$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(630\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(650, [\chi])\).

	Total	New	Old
Modular forms	538	90	448
Cusp forms	514	90	424
Eisenstein series	24	0	24

Trace form

\( 90 q - 1440 q^{4} - 304 q^{6} - 6282 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(650, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
650.6.b.a	$650$	$6$	650.b	5.b	$2$	$2$	$2$	$104.249$	\(\Q(\sqrt{-1}) \)	None					26.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4iq^{2}-2^{4}q^{4}+170iq^{7}-2^{6}iq^{8}+\cdots\)
650.6.b.b	$650$	$6$	650.b	5.b	$2$	$4$	$4$	$104.249$	\(\Q(i, \sqrt{235})\)	None					130.6.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+(-13\beta _{1}+\beta _{3})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.c	$650$	$6$	650.b	5.b	$2$	$4$	$4$	$104.249$	\(\Q(i, \sqrt{14})\)	None					130.6.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}+(8\beta _{1}+3\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.d	$650$	$6$	650.b	5.b	$2$	$4$	$4$	$104.249$	\(\Q(i, \sqrt{2785})\)	None					26.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{2}q^{2}+(\beta _{1}-4\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.e	$650$	$6$	650.b	5.b	$2$	$4$	$4$	$104.249$	\(\Q(i, \sqrt{19})\)	None					130.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+(-3\beta _{1}+3\beta _{3})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.f	$650$	$6$	650.b	5.b	$2$	$4$	$4$	$104.249$	\(\Q(i, \sqrt{145})\)	None					130.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+(-3\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.g	$650$	$6$	650.b	5.b	$2$	$4$	$4$	$104.249$	\(\Q(i, \sqrt{10})\)	None					130.6.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+(2\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.h	$650$	$6$	650.b	5.b	$2$	$4$	$4$	$104.249$	\(\Q(i, \sqrt{849})\)	None					26.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{2}q^{2}+(\beta _{1}-4\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.i	$650$	$6$	650.b	5.b	$2$	$6$	$6$	$104.249$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					130.6.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{2}q^{2}+(\beta _{1}+3\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.j	$650$	$6$	650.b	5.b	$2$	$6$	$6$	$104.249$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					130.6.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{1}q^{2}+(7\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.k	$650$	$6$	650.b	5.b	$2$	$8$	$8$	$104.249$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			650.6.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{2}q^{2}+(\beta _{1}+5\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.l	$650$	$6$	650.b	5.b	$2$	$8$	$8$	$104.249$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					130.6.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}+(-5\beta _{1}-\beta _{2})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.m	$650$	$6$	650.b	5.b	$2$	$10$	$10$	$104.249$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			650.6.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{5}q^{2}+(\beta _{1}-2\beta _{5})q^{3}-2^{4}q^{4}+\cdots\)
650.6.b.n	$650$	$6$	650.b	5.b	$2$	$10$	$10$	$104.249$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			650.6.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{6}q^{2}+\beta _{1}q^{3}-2^{4}q^{4}-4\beta _{2}q^{6}+\cdots\)
650.6.b.o	$650$	$6$	650.b	5.b	$2$	$12$	$12$	$104.249$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		650.6.a.s	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta _{7}q^{2}+(\beta _{1}+2\beta _{7})q^{3}-2^{4}q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(650, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(650, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 2}\)