Space of modular forms of level 100, weight 9, and Character 100.b

• Label: 100.9.b
• Level: $100$
• Weight: $9$
• Character orbit: 100.b
• Rep. character: $\chi_{100}(51,\cdot)$
• Character field: $\Q$
• Dimension: $73$
• Newform subspaces: $7$
• Sturm bound: $135$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	100.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(135\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(100, [\chi])\).

	Total	New	Old
Modular forms	126	79	47
Cusp forms	114	73	41
Eisenstein series	12	6	6

Trace form

\( 73 q - 2 q^{2} - 330 q^{4} - 742 q^{6} + 2728 q^{8} - 144623 q^{9} + O(q^{10}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(100, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
100.9.b.a	$100$	$9$	100.b	4.b	$2$	$1$	$1$	$40.738$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓		4.9.b.a	$2$	$0$	\(-16\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2^{4}q^{2}+2^{8}q^{4}-2^{12}q^{8}+3^{8}q^{9}+\cdots\)
100.9.b.b	$100$	$9$	100.b	4.b	$2$	$2$	$2$	$40.738$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-5}) \)			20.9.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+8iq^{2}+79iq^{3}-2^{8}q^{4}-2528q^{6}+\cdots\)
100.9.b.c	$100$	$9$	100.b	4.b	$2$	$2$	$2$	$40.738$	\(\Q(\sqrt{-39}) \)	None			4.9.b.b	$2$	$0$	\(20\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(10-\beta )q^{2}-8\beta q^{3}+(-56-20\beta )q^{4}+\cdots\)
100.9.b.d	$100$	$9$	100.b	4.b	$2$	$16$	$16$	$40.738$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None			20.9.b.a	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$2^{61}\cdot 5^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+(-3-\beta _{3}+\cdots)q^{4}+\cdots\)
100.9.b.e	$100$	$9$	100.b	4.b	$2$	$16$	$16$	$40.738$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	100.9.b.e	$2$	$0$	\(-3\)	\(0\)	\(0\)	\(0\)	$2^{56}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-21-\beta _{4})q^{4}+\cdots\)
100.9.b.f	$100$	$9$	100.b	4.b	$2$	$16$	$16$	$40.738$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	100.9.b.e	$2$	$0$	\(3\)	\(0\)	\(0\)	\(0\)	$2^{56}\cdot 5^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-21+\beta _{4})q^{4}+\cdots\)
100.9.b.g	$100$	$9$	100.b	4.b	$2$	$20$	$20$	$40.738$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None			20.9.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{75}\cdot 3^{4}\cdot 5^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{8})q^{3}+(38+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(100, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(100, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)