Space of modular forms of level 10, weight 9, and Character 10.c

• Label: 10.9.c
• Level: $10$
• Weight: $9$
• Character orbit: 10.c
• Rep. character: $\chi_{10}(3,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $8$
• Newform subspaces: $2$
• Sturm bound: $13$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	10.c (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(13\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(10, [\chi])\).

	Total	New	Old
Modular forms	28	8	20
Cusp forms	20	8	12
Eisenstein series	8	0	8

Trace form

\( 8 q + 140 q^{3} - 780 q^{5} + 512 q^{6} + 4540 q^{7} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(10, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
10.9.c.a	$10$	$9$	10.c	5.c	$4$	$4$	$2$	$4.074$	\(\Q(i, \sqrt{249})\)	None		$2$	$0$	\(-32\)	\(54\)	\(90\)	\(-1186\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-8+8\beta _{1})q^{2}+(13+13\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
10.9.c.b	$10$	$9$	10.c	5.c	$4$	$4$	$2$	$4.074$	\(\Q(i, \sqrt{601})\)	None		$2$	$0$	\(32\)	\(86\)	\(-870\)	\(5726\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(8+8\beta _{1})q^{2}+(22-21\beta _{1}+\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(10, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(10, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 2}\)