Space of modular forms of level 2200, weight 2, and trivial character

• Label: 2200.2.a
• Level: $2200$
• Weight: $2$
• Character orbit: 2200.a
• Rep. character: $\chi_{2200}(1,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $25$
• Sturm bound: $720$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2200.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(720\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2200))\).

	Total	New	Old
Modular forms	384	47	337
Cusp forms	337	47	290
Eisenstein series	47	0	47

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

\(2\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(45\)	\(6\)	\(39\)		\(40\)	\(6\)	\(34\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(48\)	\(6\)	\(42\)		\(42\)	\(6\)	\(36\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(51\)	\(5\)	\(46\)		\(45\)	\(5\)	\(40\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(48\)	\(7\)	\(41\)		\(42\)	\(7\)	\(35\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(51\)	\(7\)	\(44\)		\(45\)	\(7\)	\(38\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(48\)	\(4\)	\(44\)		\(42\)	\(4\)	\(38\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(45\)	\(4\)	\(41\)		\(39\)	\(4\)	\(35\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(48\)	\(8\)	\(40\)		\(42\)	\(8\)	\(34\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(186\)	\(21\)	\(165\)		\(163\)	\(21\)	\(142\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(198\)	\(26\)	\(172\)		\(174\)	\(26\)	\(148\)		\(24\)	\(0\)	\(24\)

Trace form

47 * q - 2 * q^3 - 4 * q^7 + 41 * q^9 + 3 * q^11 - 6 * q^13 + 2 * q^17 - 4 * q^19 + 4 * q^21 - 6 * q^23 - 14 * q^27 + 30 * q^29 - 14 * q^31 + 2 * q^33 + 16 * q^37 + 18 * q^41 - 16 * q^43 + 20 * q^47 + 31 * q^49 - 4 * q^51 + 10 * q^53 + 32 * q^57 + 14 * q^59 + 14 * q^61 + 24 * q^63 - 6 * q^67 + 22 * q^69 - 18 * q^71 - 18 * q^73 + 4 * q^77 - 52 * q^79 + 71 * q^81 - 8 * q^83 + 24 * q^89 - 48 * q^91 - 10 * q^93 - 8 * q^97 + 9 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2200))\) 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	11
2200.2.a.a	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.a.d	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+6q^{9}-q^{11}+6q^{13}+\cdots\)
2200.2.a.b	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.b.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}+q^{11}+6q^{13}+\cdots\)
2200.2.a.c	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.b.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}-q^{11}-4q^{17}+\cdots\)
2200.2.a.d	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.b.c	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}-q^{11}+q^{17}+q^{19}+\cdots\)
2200.2.a.e	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-3q^{9}-q^{11}-6q^{13}+6q^{17}+\cdots\)
2200.2.a.f	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}-q^{11}-8q^{19}+8q^{23}+\cdots\)
2200.2.a.g	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}+q^{11}+4q^{13}+4q^{17}+\cdots\)
2200.2.a.h	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.b.c	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}-q^{11}-q^{17}+q^{19}+\cdots\)
2200.2.a.i	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.b.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}-q^{11}+4q^{17}+\cdots\)
2200.2.a.j	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				440.2.b.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{7}+q^{9}+q^{11}-6q^{13}+\cdots\)
2200.2.a.k	$2200$	$2$	2200.a	1.a	$1$	$1$	$1$	$17.567$	\(\Q\)	None	✓				88.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2q^{7}+6q^{9}-q^{11}+6q^{17}+\cdots\)
2200.2.a.l	$2200$	$2$	2200.a	1.a	$1$	$2$	$2$	$17.567$	\(\Q(\sqrt{17}) \)	None	✓				440.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2-\beta )q^{7}+(1+\beta )q^{9}-q^{11}+\cdots\)
2200.2.a.m	$2200$	$2$	2200.a	1.a	$1$	$2$	$2$	$17.567$	\(\Q(\sqrt{17}) \)	None	✓				440.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+\beta )q^{7}+(1+\beta )q^{9}+q^{11}+\cdots\)
2200.2.a.n	$2200$	$2$	2200.a	1.a	$1$	$2$	$2$	$17.567$	\(\Q(\sqrt{5}) \)	None	✓	✓			2200.2.a.n	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-\beta )q^{7}+(-2+\beta )q^{9}-q^{11}+\cdots\)
2200.2.a.o	$2200$	$2$	2200.a	1.a	$1$	$2$	$2$	$17.567$	\(\Q(\sqrt{17}) \)	None	✓				88.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2\beta q^{7}+(1+\beta )q^{9}-q^{11}+\cdots\)
2200.2.a.p	$2200$	$2$	2200.a	1.a	$1$	$2$	$2$	$17.567$	\(\Q(\sqrt{5}) \)	None	✓	✓			2200.2.a.p	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1+\beta )q^{7}+(-2+\beta )q^{9}+q^{11}+\cdots\)
2200.2.a.q	$2200$	$2$	2200.a	1.a	$1$	$2$	$2$	$17.567$	\(\Q(\sqrt{5}) \)	None	✓	✓			2200.2.a.p	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1-\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
2200.2.a.r	$2200$	$2$	2200.a	1.a	$1$	$2$	$2$	$17.567$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		2200.2.a.n	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
2200.2.a.s	$2200$	$2$	2200.a	1.a	$1$	$2$	$2$	$17.567$	\(\Q(\sqrt{17}) \)	None	✓				440.2.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{7}+(1+\beta )q^{9}+q^{11}-2q^{13}+\cdots\)
2200.2.a.t	$2200$	$2$	2200.a	1.a	$1$	$3$	$3$	$17.567$	3.3.837.1	None	✓	✓	✓		2200.2.a.t	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2})q^{7}+(2+\cdots)q^{9}+\cdots\)
2200.2.a.u	$2200$	$2$	2200.a	1.a	$1$	$3$	$3$	$17.567$	3.3.1229.1	None	✓	✓			2200.2.a.u	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
2200.2.a.v	$2200$	$2$	2200.a	1.a	$1$	$3$	$3$	$17.567$	3.3.1229.1	None	✓	✓	✓		2200.2.a.u	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
2200.2.a.w	$2200$	$2$	2200.a	1.a	$1$	$3$	$3$	$17.567$	3.3.837.1	None	✓	✓	✓		2200.2.a.t	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{2})q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
2200.2.a.x	$2200$	$2$	2200.a	1.a	$1$	$4$	$4$	$17.567$	4.4.54764.1	None	✓		✓		440.2.b.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2}+\beta _{3})q^{9}+\cdots\)
2200.2.a.y	$2200$	$2$	2200.a	1.a	$1$	$4$	$4$	$17.567$	4.4.54764.1	None	✓		✓		440.2.b.d	$1$	$0$	\(0\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2}+\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2200))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2200)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1100))\)\(^{\oplus 2}\)