Space of modular forms of level 10, weight 7, and Character 10.c

• Label: 10.7.c
• Level: $10$
• Weight: $7$
• Character orbit: 10.c
• Rep. character: $\chi_{10}(3,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $10$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
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:			\(10\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(10, [\chi])\).

	Total	New	Old
Modular forms	22	6	16
Cusp forms	14	6	8
Eisenstein series	8	0	8

Trace form

\( 6 q + 8 q^{2} - 64 q^{3} + 180 q^{5} + 224 q^{6} - 696 q^{7} - 256 q^{8} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(10, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
10.7.c.a	$10$	$7$	10.c	5.c	$4$	$2$	$1$	$2.301$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(-8\)	\(-46\)	\(-150\)	\(-494\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4-4i)q^{2}+(-23+23i)q^{3}+\cdots\)
10.7.c.b	$10$	$7$	10.c	5.c	$4$	$4$	$2$	$2.301$	\(\Q(i, \sqrt{129})\)	None		$2$	$0$	\(16\)	\(-18\)	\(330\)	\(-202\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4+4\beta _{1})q^{2}+(-5+4\beta _{1}-\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(10, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(10, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 2}\)