Space of modular forms of level 200, weight 2, and trivial character

• Label: 200.2.a
• Level: $200$
• Weight: $2$
• Character orbit: 200.a
• Rep. character: $\chi_{200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(200))\).

	Total	New	Old
Modular forms	42	5	37
Cusp forms	19	5	14
Eisenstein series	23	0	23

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(0\)	\(9\)		\(4\)	\(0\)	\(4\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(12\)	\(2\)	\(10\)		\(6\)	\(2\)	\(4\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(12\)	\(2\)	\(10\)		\(6\)	\(2\)	\(4\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(9\)	\(1\)	\(8\)		\(3\)	\(1\)	\(2\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(18\)	\(1\)	\(17\)		\(7\)	\(1\)	\(6\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(24\)	\(4\)	\(20\)		\(12\)	\(4\)	\(8\)		\(12\)	\(0\)	\(12\)

Trace form

5 * q + 4 * q^7 + 11 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^17 - 2 * q^19 - 4 * q^21 - 4 * q^23 - 14 * q^29 + 12 * q^31 - 6 * q^37 - 8 * q^39 - 8 * q^41 + 8 * q^43 - 4 * q^47 - 3 * q^49 - 30 * q^51 - 6 * q^53 - 12 * q^59 - 2 * q^61 - 12 * q^63 - 8 * q^67 + 4 * q^69 - 8 * q^71 + 6 * q^73 + 16 * q^77 + 44 * q^79 + 5 * q^81 + 16 * q^83 - 12 * q^89 + 40 * q^91 + 14 * q^97 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(200))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5
200.2.a.a	$200$	$2$	200.a	1.a	$1$	$1$	$1$	$1.597$	\(\Q\)	None	✓	✓			200.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{7}+6q^{9}+q^{11}+4q^{13}+\cdots\)
200.2.a.b	$200$	$2$	200.a	1.a	$1$	$1$	$1$	$1.597$	\(\Q\)	None	✓		✓	✓	40.2.c.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
200.2.a.c	$200$	$2$	200.a	1.a	$1$	$1$	$1$	$1.597$	\(\Q\)	None	✓				40.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-3q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
200.2.a.d	$200$	$2$	200.a	1.a	$1$	$1$	$1$	$1.597$	\(\Q\)	None	✓				40.2.c.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
200.2.a.e	$200$	$2$	200.a	1.a	$1$	$1$	$1$	$1.597$	\(\Q\)	None	✓	✓			200.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{7}+6q^{9}+q^{11}-4q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(200)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)