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

• Label: 100.9.f
• Level: $100$
• Weight: $9$
• Character orbit: 100.f
• Rep. character: $\chi_{100}(57,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $24$
• Newform subspaces: $3$
• Sturm bound: $135$
• Trace bound: $1$

Defining parameters

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

Dimensions

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

	Total	New	Old
Modular forms	258	24	234
Cusp forms	222	24	198
Eisenstein series	36	0	36

Trace form

24 * q + 70 * q^3 + 2030 * q^7 + 840 * q^11 - 33180 * q^13 - 43620 * q^17 - 627196 * q^21 + 663270 * q^23 - 1576040 * q^27 + 3166524 * q^31 + 944020 * q^33 - 5344080 * q^37 + 7246344 * q^41 + 10342710 * q^43 - 19232250 * q^47 + 42485848 * q^51 + 24320640 * q^53 - 88218320 * q^57 + 45976748 * q^61 + 77441350 * q^63 - 100675930 * q^67 + 16468572 * q^71 + 93528520 * q^73 - 134199660 * q^77 + 115768396 * q^81 + 10450350 * q^83 - 164801600 * q^87 + 347762196 * q^91 + 50183620 * q^93 + 179570760 * q^97

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	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
100.9.f.a	$100$	$9$	100.f	5.c	$4$	$4$	$2$	$40.738$	\(\Q(i, \sqrt{3309})\)	None		✓			100.9.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+7\beta _{2}q^{7}-57\beta _{1}q^{9}+420q^{11}+\cdots\)
100.9.f.b	$100$	$9$	100.f	5.c	$4$	$8$	$4$	$40.738$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					20.9.f.a	$2$	$0$	\(0\)	\(70\)	\(0\)	\(2030\)	$2^{14}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(9+9\beta _{1}+\beta _{2})q^{3}+(254-254\beta _{1}+\cdots)q^{7}+\cdots\)
100.9.f.c	$100$	$9$	100.f	5.c	$4$	$12$	$6$	$40.738$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		100.9.f.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{6}\cdot 5^{12}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{3}+(7\beta _{7}-20\beta _{8}-\beta _{11})q^{7}+\cdots\)

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

\( S_{9}^{\mathrm{old}}(100, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)