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 + 14120 * q^12 - 39974 * q^13 + 3892 * q^14 + 192178 * q^16 - 110374 * q^17 + 443558 * q^18 + 35896 * q^21 - 72640 * q^22 - 212978 * q^24 - 1538972 * q^26 - 1987400 * q^28 - 468070 * q^29 + 2844928 * q^32 + 1827520 * q^33 + 1429838 * q^34 + 2568656 * q^36 - 2991334 * q^37 + 2600640 * q^38 - 993774 * q^41 + 1271880 * q^42 + 4071210 * q^44 + 16298188 * q^46 + 8287680 * q^48 - 38003503 * q^49 - 7169624 * q^52 + 5520026 * q^53 - 9023426 * q^54 - 1019932 * q^56 - 15851520 * q^57 - 32195604 * q^58 + 24874386 * q^61 - 2987280 * q^62 - 52515150 * q^64 + 38970930 * q^66 + 8310536 * q^68 - 83516536 * q^69 + 40562088 * q^72 + 4261146 * q^73 - 42725672 * q^74 + 43505610 * q^76 + 105280320 * q^77 + 90379280 * q^78 + 136079489 * q^81 - 98230124 * q^82 - 128420836 * q^84 - 133189792 * q^86 - 111221920 * q^88 + 179555930 * q^89 + 132386280 * q^92 - 119142080 * q^93 + 161554592 * q^94 - 72309682 * q^96 - 239709414 * q^97 + 269309638 * q^98

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.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	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+8\beta q^{2}+79\beta q^{3}-256 q^{4}-2528 q^{6}+\cdots\)
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

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