Space of modular forms of level 650, weight 2, and Character 650.e

• Label: 650.2.e
• Level: $650$
• Weight: $2$
• Character orbit: 650.e
• Rep. character: $\chi_{650}(451,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $42$
• Newform subspaces: $11$
• Sturm bound: $210$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	650.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(210\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(650, [\chi])\).

	Total	New	Old
Modular forms	232	42	190
Cusp forms	184	42	142
Eisenstein series	48	0	48

Trace form

\( 42 q + q^{2} - 21 q^{4} - 4 q^{7} - 2 q^{8} - 21 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(650, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
650.2.e.a	$650$	$2$	650.e	13.c	$3$	$2$	$1$	$5.190$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(-1\)	\(-2\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
650.2.e.b	$650$	$2$	650.e	13.c	$3$	$2$	$1$	$5.190$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(0\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-3\zeta_{6}q^{7}-q^{8}+\cdots\)
650.2.e.c	$650$	$2$	650.e	13.c	$3$	$2$	$1$	$5.190$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+4\zeta_{6}q^{7}-q^{8}+\cdots\)
650.2.e.d	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$2$	$0$	\(-2\)	\(-1\)	\(0\)	\(1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1})q^{2}-\beta _{3}q^{3}+\beta _{1}q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
650.2.e.e	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		$2$	$0$	\(-2\)	\(1\)	\(0\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2})q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+\cdots\)
650.2.e.f	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(-2\)	\(2\)	\(0\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{2})q^{3}+(-1+\cdots)q^{4}+\cdots\)
650.2.e.g	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		$2$	$0$	\(2\)	\(-1\)	\(0\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
650.2.e.h	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{10})\)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
650.2.e.i	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$2$	$0$	\(2\)	\(1\)	\(0\)	\(-1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1})q^{2}+\beta _{3}q^{3}+\beta _{1}q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
650.2.e.j	$650$	$2$	650.e	13.c	$3$	$6$	$3$	$5.190$	6.0.591408.1	None		$2$	$0$	\(-3\)	\(0\)	\(0\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{4})q^{2}-\beta _{5}q^{3}-\beta _{4}q^{4}+(-\beta _{3}+\cdots)q^{6}+\cdots\)
650.2.e.k	$650$	$2$	650.e	13.c	$3$	$6$	$3$	$5.190$	6.0.591408.1	None		$2$	$0$	\(3\)	\(0\)	\(0\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{2}+(\beta _{3}-\beta _{5})q^{3}+(-1+\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(650, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(650, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 3}\)