Space of modular forms of level 200, weight 3, and Character 200.g

• Label: 200.3.g
• Level: $200$
• Weight: $3$
• Character orbit: 200.g
• Rep. character: $\chi_{200}(51,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $8$
• Sturm bound: $90$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	200.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(90\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(200, [\chi])\).

	Total	New	Old
Modular forms	66	41	25
Cusp forms	54	35	19
Eisenstein series	12	6	6

Trace form

35 * q + 2 * q^3 + 2 * q^4 - 2 * q^6 + 89 * q^9 + 10 * q^11 - 12 * q^12 + 32 * q^14 - 30 * q^16 - 2 * q^17 + 52 * q^18 + 34 * q^19 + 8 * q^22 - 18 * q^24 - 68 * q^26 + 68 * q^27 - 100 * q^28 - 40 * q^32 + 12 * q^33 + 42 * q^34 + 128 * q^36 - 128 * q^38 - 10 * q^41 - 140 * q^42 - 110 * q^43 - 66 * q^44 + 272 * q^46 + 312 * q^48 - 101 * q^49 - 164 * q^51 + 160 * q^52 - 226 * q^54 - 168 * q^56 + 108 * q^57 + 320 * q^58 - 206 * q^59 + 160 * q^62 + 2 * q^64 - 594 * q^66 - 222 * q^67 - 368 * q^68 - 192 * q^72 - 18 * q^73 - 108 * q^74 + 390 * q^76 - 600 * q^78 + 151 * q^81 - 172 * q^82 + 322 * q^83 - 80 * q^84 + 168 * q^86 + 512 * q^88 - 146 * q^89 + 184 * q^91 + 60 * q^92 - 88 * q^94 - 802 * q^96 - 130 * q^97 + 540 * q^98 + 478 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(200, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
200.3.g.a	$200$	$3$	200.g	8.d	$2$	$1$	$1$	$5.450$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				8.3.d.a	$2$	$0$	\(2\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+2q^{3}+4q^{4}+4q^{6}+8q^{8}+\cdots\)
200.3.g.b	$200$	$3$	200.g	8.d	$2$	$2$	$2$	$5.450$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-2}) \)	✓	✓			200.3.g.b	$2$	$0$	\(-4\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+(1+\beta )q^{3}+4q^{4}+(-2-2\beta )q^{6}+\cdots\)
200.3.g.c	$200$	$3$	200.g	8.d	$2$	$2$	$2$	$5.450$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-10}) \)					40.3.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}-4 q^{4}-3\beta q^{7}-4\beta q^{8}+\cdots\)
200.3.g.d	$200$	$3$	200.g	8.d	$2$	$2$	$2$	$5.450$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-2}) \)	✓	✓			200.3.g.b	$2$	$0$	\(4\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+(-1+\beta )q^{3}+4q^{4}+(-2+\cdots)q^{6}+\cdots\)
200.3.g.e	$200$	$3$	200.g	8.d	$2$	$6$	$6$	$5.450$	6.0.189974000.1	None		✓			200.3.g.e	$2$	$0$	\(-3\)	\(2\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
200.3.g.f	$200$	$3$	200.g	8.d	$2$	$6$	$6$	$5.450$	6.0.189974000.1	None		✓			200.3.g.e	$2$	$0$	\(3\)	\(-2\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{1})q^{2}-\beta _{2}q^{3}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
200.3.g.g	$200$	$3$	200.g	8.d	$2$	$8$	$8$	$5.450$	8.0.\(\cdots\).1	None					40.3.g.a	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+\beta _{6}q^{3}+(-1+\beta _{1})q^{4}+(-1+\cdots)q^{6}+\cdots\)
200.3.g.h	$200$	$3$	200.g	8.d	$2$	$8$	$8$	$5.450$	8.0.\(\cdots\).1	None					40.3.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{2}+\beta _{6}q^{3}+(2+\beta _{5})q^{4}+(2-\beta _{2}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(200, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(200, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 2}\)