Space of modular forms of level 1200, weight 2, and trivial character

• Label: 1200.2.a
• Level: $1200$
• Weight: $2$
• Character orbit: 1200.a
• Rep. character: $\chi_{1200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $19$
• Sturm bound: $480$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(480\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1200))\).

	Total	New	Old
Modular forms	276	19	257
Cusp forms	205	19	186
Eisenstein series	71	0	71

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(33\)	\(1\)	\(32\)		\(25\)	\(1\)	\(24\)		\(8\)	\(0\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)		\(36\)	\(3\)	\(33\)		\(27\)	\(3\)	\(24\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)		\(36\)	\(4\)	\(32\)		\(27\)	\(4\)	\(23\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)		\(33\)	\(1\)	\(32\)		\(24\)	\(1\)	\(23\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)		\(36\)	\(2\)	\(34\)		\(27\)	\(2\)	\(25\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)		\(33\)	\(3\)	\(30\)		\(24\)	\(3\)	\(21\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)		\(33\)	\(2\)	\(31\)		\(24\)	\(2\)	\(22\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)		\(36\)	\(3\)	\(33\)		\(27\)	\(3\)	\(24\)		\(9\)	\(0\)	\(9\)
Plus space			\(+\)		\(132\)	\(7\)	\(125\)		\(97\)	\(7\)	\(90\)		\(35\)	\(0\)	\(35\)
Minus space			\(-\)		\(144\)	\(12\)	\(132\)		\(108\)	\(12\)	\(96\)		\(36\)	\(0\)	\(36\)

Trace form

19 * q + q^3 + 19 * q^9 - 4 * q^11 + 2 * q^13 - 2 * q^17 - 16 * q^23 + q^27 + 10 * q^29 + 12 * q^31 + 4 * q^33 + 10 * q^37 + 6 * q^39 + 6 * q^41 + 12 * q^43 + 24 * q^47 + 35 * q^49 - 2 * q^51 + 2 * q^53 + 4 * q^57 + 28 * q^59 - 14 * q^61 + 4 * q^67 - 8 * q^69 + 24 * q^71 - 2 * q^73 + 48 * q^79 + 19 * q^81 + 20 * q^83 - 18 * q^87 - 10 * q^89 + 20 * q^91 + 8 * q^93 - 10 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1200))\) 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	3	5
1200.2.a.a	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				60.2.d.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{17}+\cdots\)
1200.2.a.b	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				600.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}+q^{9}-2q^{11}-3q^{13}+\cdots\)
1200.2.a.c	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				75.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
1200.2.a.d	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓		✓	✓	24.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
1200.2.a.e	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				15.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{9}+4q^{11}+2q^{13}-2q^{17}+\cdots\)
1200.2.a.f	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				300.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}+q^{9}-6q^{11}+5q^{13}+\cdots\)
1200.2.a.g	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				30.2.c.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}-2q^{11}-6q^{13}+\cdots\)
1200.2.a.h	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				120.2.f.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}-2q^{11}+2q^{13}+\cdots\)
1200.2.a.i	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				600.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5q^{7}+q^{9}+6q^{11}-3q^{13}+\cdots\)
1200.2.a.j	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				600.2.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-5q^{7}+q^{9}+6q^{11}+3q^{13}+\cdots\)
1200.2.a.k	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				30.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{7}+q^{9}-2q^{13}-6q^{17}+\cdots\)
1200.2.a.l	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓		✓	✓	120.2.f.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
1200.2.a.m	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				30.2.c.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}-2q^{11}+6q^{13}+\cdots\)
1200.2.a.n	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				300.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}+q^{9}-6q^{11}-5q^{13}+\cdots\)
1200.2.a.o	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				120.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+4q^{11}-6q^{13}+6q^{17}+\cdots\)
1200.2.a.p	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				75.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}+q^{9}-2q^{11}+q^{13}+\cdots\)
1200.2.a.q	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				600.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}+q^{9}-2q^{11}+3q^{13}+\cdots\)
1200.2.a.r	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				120.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
1200.2.a.s	$1200$	$2$	1200.a	1.a	$1$	$1$	$1$	$9.582$	\(\Q\)	None	✓				60.2.d.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1200)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)