Space of modular forms of level 200, weight 2, and Character 200.k

• Label: 200.2.k
• Level: $200$
• Weight: $2$
• Character orbit: 200.k
• Rep. character: $\chi_{200}(43,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $32$
• Newform subspaces: $8$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	200.k (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 40 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(200, [\chi])\).

	Total	New	Old
Modular forms	72	40	32
Cusp forms	48	32	16
Eisenstein series	24	8	16

Trace form

32 * q + 2 * q^2 + 4 * q^3 + 4 * q^6 - 4 * q^8 - 8 * q^11 - 12 * q^12 - 28 * q^16 + 8 * q^17 - 10 * q^18 - 12 * q^22 - 52 * q^26 - 8 * q^27 + 20 * q^28 + 32 * q^32 + 16 * q^33 + 76 * q^36 + 4 * q^38 - 8 * q^41 + 20 * q^42 - 28 * q^43 + 8 * q^46 - 16 * q^48 - 40 * q^51 - 20 * q^52 - 32 * q^56 - 8 * q^57 - 20 * q^58 - 40 * q^62 - 100 * q^66 + 28 * q^67 + 4 * q^68 + 20 * q^72 - 16 * q^73 + 52 * q^76 + 40 * q^78 - 40 * q^81 + 28 * q^82 + 44 * q^83 + 96 * q^86 - 16 * q^88 + 56 * q^91 - 20 * q^92 + 44 * q^96 - 16 * q^97 + 6 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(200, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
200.2.k.a	$200$	$2$	200.k	40.k	$4$	$2$	$1$	$1.597$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		✓			200.2.k.a	$4$	$1$	\(-2\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-i-1)q^{2}+(2 i-2)q^{3}+2 i q^{4}+\cdots\)
200.2.k.b	$200$	$2$	200.k	40.k	$4$	$2$	$1$	$1.597$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-10}) \)		✓			200.2.k.b	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(i-1)q^{2}-2 i q^{4}+(2 i-2)q^{7}+\cdots\)
200.2.k.c	$200$	$2$	200.k	40.k	$4$	$2$	$1$	$1.597$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-10}) \)		✓			200.2.k.b	$4$	$0$	\(2\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-i+1)q^{2}-2 i q^{4}+(-2 i+2)q^{7}+\cdots\)
200.2.k.d	$200$	$2$	200.k	40.k	$4$	$2$	$1$	$1.597$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		✓			200.2.k.a	$4$	$0$	\(2\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(i+1)q^{2}+(-2 i+2)q^{3}+2 i q^{4}+\cdots\)
200.2.k.e	$200$	$2$	200.k	40.k	$4$	$4$	$2$	$1.597$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-2}) \)		✓			200.2.k.e	$4$	$0$	\(-4\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(-1-\beta _{2})q^{2}+(1-\beta _{2}+\beta _{3})q^{3}+\cdots\)
200.2.k.f	$200$	$2$	200.k	40.k	$4$	$4$	$2$	$1.597$	\(\Q(i, \sqrt{6})\)	\(\Q(\sqrt{-2}) \)		✓			200.2.k.e	$4$	$0$	\(4\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q+(1+\beta _{2})q^{2}+(-1+\beta _{2}-\beta _{3})q^{3}+\cdots\)
200.2.k.g	$200$	$2$	200.k	40.k	$4$	$8$	$4$	$1.597$	8.0.3317760000.5	None		✓			200.2.k.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+(\beta _{4}+\beta _{6})q^{3}+\beta _{2}q^{4}+(2+\cdots)q^{6}+\cdots\)
200.2.k.h	$200$	$2$	200.k	40.k	$4$	$8$	$4$	$1.597$	\(\Q(\zeta_{20})\)	None					40.2.k.a	$4$	$0$	\(2\)	\(4\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_{7} q^{2}+(\beta_{6}-\beta_{5}-\beta_{3}+1)q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(200, [\chi])\) into lower level spaces

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