Space of modular forms of level 1800, weight 2, and trivial character

• Label: 1800.2.a
• Level: $1800$
• Weight: $2$
• Character orbit: 1800.a
• Rep. character: $\chi_{1800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $24$
• Sturm bound: $720$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(720\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1800))\).

	Total	New	Old
Modular forms	408	24	384
Cusp forms	313	24	289
Eisenstein series	95	0	95

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
\(+\)	\(+\)	\(+\)	\(+\)		\(48\)	\(2\)	\(46\)		\(37\)	\(2\)	\(35\)		\(11\)	\(0\)	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)		\(54\)	\(3\)	\(51\)		\(42\)	\(3\)	\(39\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(51\)	\(4\)	\(47\)		\(39\)	\(4\)	\(35\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(51\)	\(3\)	\(48\)		\(39\)	\(3\)	\(36\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(54\)	\(2\)	\(52\)		\(42\)	\(2\)	\(40\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(48\)	\(3\)	\(45\)		\(36\)	\(3\)	\(33\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(51\)	\(3\)	\(48\)		\(39\)	\(3\)	\(36\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(51\)	\(4\)	\(47\)		\(39\)	\(4\)	\(35\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(198\)	\(11\)	\(187\)		\(151\)	\(11\)	\(140\)		\(47\)	\(0\)	\(47\)
Minus space			\(-\)		\(210\)	\(13\)	\(197\)		\(162\)	\(13\)	\(149\)		\(48\)	\(0\)	\(48\)

Trace form

24 * q - 4 * q^7 + 6 * q^11 - 4 * q^13 - 4 * q^17 - 6 * q^19 - 12 * q^23 - 8 * q^29 - 4 * q^31 - 4 * q^37 - 26 * q^41 + 12 * q^43 + 20 * q^47 + 32 * q^49 + 20 * q^53 + 32 * q^59 + 8 * q^61 + 4 * q^67 + 32 * q^71 + 4 * q^73 - 16 * q^77 - 12 * q^79 - 20 * q^83 - 6 * q^89 - 32 * q^91 + 36 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1800))\) 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
1800.2.a.a	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				600.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{7}+6q^{11}-3q^{13}+2q^{17}+q^{19}+\cdots\)
1800.2.a.b	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				360.2.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-4q^{11}+4q^{13}+6q^{17}-4q^{19}+\cdots\)
1800.2.a.c	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				120.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+6q^{13}-2q^{17}+4q^{19}-8q^{23}+\cdots\)
1800.2.a.d	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				360.2.f.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+4q^{11}+4q^{13}-6q^{17}-4q^{19}+\cdots\)
1800.2.a.e	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				600.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{7}-2q^{11}+3q^{13}+6q^{17}-7q^{19}+\cdots\)
1800.2.a.f	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				360.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-2q^{11}-4q^{13}-2q^{17}+4q^{19}+\cdots\)
1800.2.a.g	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				120.2.f.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-2q^{11}+2q^{13}-6q^{17}+8q^{19}+\cdots\)
1800.2.a.h	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				200.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-q^{11}-4q^{13}+5q^{17}+q^{19}+\cdots\)
1800.2.a.i	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				360.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+2q^{11}-4q^{13}+2q^{17}+4q^{19}+\cdots\)
1800.2.a.j	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				40.2.c.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+4q^{11}-4q^{13}-4q^{19}-2q^{23}+\cdots\)
1800.2.a.k	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓	✓			1800.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}-4q^{11}+q^{13}+4q^{17}+q^{19}+\cdots\)
1800.2.a.l	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓	✓			1800.2.a.k	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{7}+4q^{11}+q^{13}-4q^{17}+q^{19}+\cdots\)
1800.2.a.m	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				24.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{11}+2q^{13}+2q^{17}-4q^{19}+\cdots\)
1800.2.a.n	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				120.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{11}-6q^{13}-6q^{17}-4q^{19}+\cdots\)
1800.2.a.o	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓	✓			1800.2.a.k	$1$	$1$	\(0\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}-4q^{11}-q^{13}-4q^{17}+q^{19}+\cdots\)
1800.2.a.p	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓	✓			1800.2.a.k	$1$	$0$	\(0\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{7}+4q^{11}-q^{13}+4q^{17}+q^{19}+\cdots\)
1800.2.a.q	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				120.2.f.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-2q^{11}-2q^{13}+6q^{17}+8q^{19}+\cdots\)
1800.2.a.r	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				200.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-q^{11}+4q^{13}-5q^{17}+q^{19}+\cdots\)
1800.2.a.s	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				40.2.c.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+4q^{11}+4q^{13}-4q^{19}+2q^{23}+\cdots\)
1800.2.a.t	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				600.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}-2q^{11}-3q^{13}-6q^{17}-7q^{19}+\cdots\)
1800.2.a.u	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				360.2.f.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-4q^{11}-4q^{13}-6q^{17}-4q^{19}+\cdots\)
1800.2.a.v	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				40.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-4q^{11}+2q^{13}+2q^{17}+4q^{19}+\cdots\)
1800.2.a.w	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				360.2.f.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}+4q^{11}-4q^{13}+6q^{17}-4q^{19}+\cdots\)
1800.2.a.x	$1800$	$2$	1800.a	1.a	$1$	$1$	$1$	$14.373$	\(\Q\)	None	✓				600.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{7}+6q^{11}+3q^{13}-2q^{17}+q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1800))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1800)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(300))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(600))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(900))\)\(^{\oplus 2}\)