Space of modular forms of level 200, weight 4, and trivial character

• Label: 200.4.a
• Level: $200$
• Weight: $4$
• Character orbit: 200.a
• Rep. character: $\chi_{200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $12$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(200))\).

	Total	New	Old
Modular forms	102	14	88
Cusp forms	78	14	64
Eisenstein series	24	0	24

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
\(+\)	\(+\)	\(+\)		\(27\)	\(4\)	\(23\)		\(21\)	\(4\)	\(17\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)		\(24\)	\(3\)	\(21\)		\(18\)	\(3\)	\(15\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(24\)	\(3\)	\(21\)		\(18\)	\(3\)	\(15\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(27\)	\(4\)	\(23\)		\(21\)	\(4\)	\(17\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(54\)	\(8\)	\(46\)		\(42\)	\(8\)	\(34\)		\(12\)	\(0\)	\(12\)
Minus space		\(-\)		\(48\)	\(6\)	\(42\)		\(36\)	\(6\)	\(30\)		\(12\)	\(0\)	\(12\)

Trace form

14 * q - 4 * q^3 + 12 * q^7 + 116 * q^9 - 6 * q^11 - 32 * q^13 + 136 * q^17 + 226 * q^19 + 196 * q^21 + 132 * q^23 - 568 * q^27 - 208 * q^29 - 100 * q^31 - 64 * q^33 + 656 * q^37 + 928 * q^39 + 38 * q^41 - 4 * q^43 - 980 * q^47 - 178 * q^49 + 970 * q^51 - 800 * q^53 + 1904 * q^57 + 1408 * q^59 - 780 * q^61 + 2060 * q^63 - 748 * q^67 - 2324 * q^69 - 1328 * q^71 - 1208 * q^73 + 736 * q^77 - 484 * q^79 + 1046 * q^81 - 1876 * q^83 - 2872 * q^87 - 2454 * q^89 + 632 * q^91 + 880 * q^93 - 40 * q^97 - 3324 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(200))\) 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
200.4.a.a	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓				40.4.a.c	$1$	$1$	\(0\)	\(-10\)	\(0\)	\(18\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}+18q^{7}+73q^{9}-2^{4}q^{11}+\cdots\)
200.4.a.b	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓	✓			200.4.a.b	$1$	$0$	\(0\)	\(-9\)	\(0\)	\(-26\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-26q^{7}+54q^{9}-59q^{11}+\cdots\)
200.4.a.c	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓	✓			200.4.a.c	$1$	$0$	\(0\)	\(-5\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-2q^{7}-2q^{9}+39q^{11}-84q^{13}+\cdots\)
200.4.a.d	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓				40.4.a.b	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-2^{4}q^{7}-11q^{9}+6^{2}q^{11}+\cdots\)
200.4.a.e	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓	✓			200.4.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+6q^{7}-26q^{9}-19q^{11}-12q^{13}+\cdots\)
200.4.a.f	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓	✓			200.4.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-6q^{7}-26q^{9}-19q^{11}+12q^{13}+\cdots\)
200.4.a.g	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓				8.4.a.a	$1$	$1$	\(0\)	\(4\)	\(0\)	\(-24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-24q^{7}-11q^{9}-44q^{11}+\cdots\)
200.4.a.h	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓	✓			200.4.a.c	$1$	$0$	\(0\)	\(5\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+2q^{7}-2q^{9}+39q^{11}+84q^{13}+\cdots\)
200.4.a.i	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓				40.4.a.a	$1$	$0$	\(0\)	\(6\)	\(0\)	\(34\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+34q^{7}+9q^{9}+2^{4}q^{11}-58q^{13}+\cdots\)
200.4.a.j	$200$	$4$	200.a	1.a	$1$	$1$	$1$	$11.800$	\(\Q\)	None	✓	✓			200.4.a.b	$1$	$0$	\(0\)	\(9\)	\(0\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+26q^{7}+54q^{9}-59q^{11}+\cdots\)
200.4.a.k	$200$	$4$	200.a	1.a	$1$	$2$	$2$	$11.800$	\(\Q(\sqrt{6}) \)	None	✓		✓		40.4.c.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(-2-3\beta )q^{7}+(1-4\beta )q^{9}+\cdots\)
200.4.a.l	$200$	$4$	200.a	1.a	$1$	$2$	$2$	$11.800$	\(\Q(\sqrt{6}) \)	None	✓		✓		40.4.c.a	$1$	$0$	\(0\)	\(4\)	\(0\)	\(4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(2-3\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(200))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(200)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)