Space of modular forms of level 4400, weight 2, and trivial character

• Label: 4400.2.a
• Level: $4400$
• Weight: $2$
• Character orbit: 4400.a
• Rep. character: $\chi_{4400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $95$
• Newform subspaces: $57$
• Sturm bound: $1440$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 57 \)
Sturm bound:			\(1440\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4400))\).

	Total	New	Old
Modular forms	756	95	661
Cusp forms	685	95	590
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.

\(2\)	\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(10\)
\(+\)	\(+\)	\(-\)	\(-\)	\(13\)
\(+\)	\(-\)	\(+\)	\(-\)	\(15\)
\(+\)	\(-\)	\(-\)	\(+\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)	\(11\)
\(-\)	\(+\)	\(-\)	\(+\)	\(11\)
\(-\)	\(-\)	\(+\)	\(+\)	\(13\)
\(-\)	\(-\)	\(-\)	\(-\)	\(13\)
Plus space			\(+\)	\(43\)
Minus space			\(-\)	\(52\)

Trace form

\( 95 q + 2 q^{3} + 4 q^{7} + 95 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\) 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
4400.2.a.a	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				88.2.a.a	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{7}+6q^{9}+q^{11}+6q^{17}+\cdots\)
4400.2.a.b	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.b.c	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{7}+6q^{9}+q^{11}+5q^{17}+\cdots\)
4400.2.a.c	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.b.b	$1$	$2$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}-q^{11}-6q^{13}+\cdots\)
4400.2.a.d	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				550.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}+q^{11}-5q^{13}+\cdots\)
4400.2.a.e	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				220.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
4400.2.a.f	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.b.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-q^{11}-2q^{13}+6q^{17}+\cdots\)
4400.2.a.g	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				550.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-q^{11}+3q^{13}-4q^{17}+\cdots\)
4400.2.a.h	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.b.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}+q^{11}+4q^{17}+\cdots\)
4400.2.a.i	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				11.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}-q^{11}-4q^{13}+\cdots\)
4400.2.a.j	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.b.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}+q^{11}-q^{17}-q^{19}+\cdots\)
4400.2.a.k	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.b.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}-2q^{9}-q^{11}-4q^{13}+\cdots\)
4400.2.a.l	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{7}-2q^{9}-q^{11}+6q^{13}+\cdots\)
4400.2.a.m	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}-q^{11}+4q^{13}+4q^{17}+\cdots\)
4400.2.a.n	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				220.2.b.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}+q^{11}-6q^{13}-2q^{17}+\cdots\)
4400.2.a.o	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.a.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-3q^{9}+q^{11}+8q^{19}-8q^{23}+\cdots\)
4400.2.a.p	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				55.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}+q^{11}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
4400.2.a.q	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				220.2.b.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-3q^{9}+q^{11}+6q^{13}+2q^{17}+\cdots\)
4400.2.a.r	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{7}-3q^{9}+q^{11}-6q^{13}+6q^{17}+\cdots\)
4400.2.a.s	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.b.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{7}-2q^{9}-q^{11}+4q^{13}+\cdots\)
4400.2.a.t	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.a.c	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
4400.2.a.u	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.b.c	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+q^{11}+q^{17}-q^{19}+\cdots\)
4400.2.a.v	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				44.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}+q^{11}+4q^{13}+\cdots\)
4400.2.a.w	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+5q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
4400.2.a.x	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.b.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}+q^{11}-4q^{17}+\cdots\)
4400.2.a.y	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				550.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}-3q^{13}+4q^{17}+\cdots\)
4400.2.a.z	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				220.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}+4q^{17}+4q^{19}+\cdots\)
4400.2.a.ba	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.b.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}+2q^{13}-6q^{17}+\cdots\)
4400.2.a.bb	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.b.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{7}+q^{9}-q^{11}+6q^{13}+\cdots\)
4400.2.a.bc	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				550.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{7}+q^{9}+q^{11}+5q^{13}+\cdots\)
4400.2.a.bd	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				110.2.b.c	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{7}+6q^{9}+q^{11}-5q^{17}+\cdots\)
4400.2.a.be	$4400$	$2$	4400.a	1.a	$1$	$1$	$1$	$35.134$	\(\Q\)	None	✓				440.2.a.d	$1$	$0$	\(0\)	\(3\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{7}+6q^{9}+q^{11}+6q^{13}+\cdots\)
4400.2.a.bf	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{13}) \)	None	✓				1100.2.a.f	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-2-\beta )q^{7}+(1+3\beta )q^{9}+\cdots\)
4400.2.a.bg	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{5}) \)	None	✓				275.2.a.d	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-2+3\beta )q^{7}+(-1+\cdots)q^{9}+\cdots\)
4400.2.a.bh	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{13}) \)	None	✓				275.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-3+\beta )q^{7}+\beta q^{9}+q^{11}+\cdots\)
4400.2.a.bi	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{21}) \)	None	✓				1100.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-3+\beta )q^{7}+(2+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bj	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{17}) \)	None	✓				440.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-\beta q^{7}+(1+\beta )q^{9}-q^{11}-2q^{13}+\cdots\)
4400.2.a.bk	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{5}) \)	None	✓				2200.2.a.n	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-\beta )q^{7}+(-2+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bl	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{33}) \)	None	✓				110.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+\beta q^{7}+(5+\beta )q^{9}+q^{11}-2q^{13}+\cdots\)
4400.2.a.bm	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{5}) \)	None	✓				2200.2.a.p	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1+\beta )q^{7}+(-2+\beta )q^{9}-q^{11}+\cdots\)
4400.2.a.bn	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{2}) \)	None	✓				55.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2q^{7}+5q^{9}-q^{11}+(4+\beta )q^{13}+\cdots\)
4400.2.a.bo	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{5}) \)	None	✓				2200.2.a.p	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1-\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
4400.2.a.bp	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{17}) \)	None	✓				88.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{7}+(1+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bq	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{5}) \)	None	✓				2200.2.a.n	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
4400.2.a.br	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{17}) \)	None	✓				440.2.a.f	$1$	$1$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{7}+(1+\beta )q^{9}-q^{11}+\cdots\)
4400.2.a.bs	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{13}) \)	None	✓				275.2.a.e	$1$	$1$	\(0\)	\(1\)	\(0\)	\(5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(3-\beta )q^{7}+\beta q^{9}+q^{11}-5q^{13}+\cdots\)
4400.2.a.bt	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{17}) \)	None	✓				440.2.a.g	$1$	$0$	\(0\)	\(1\)	\(0\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2+\beta )q^{7}+(1+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bu	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{21}) \)	None	✓				1100.2.a.g	$1$	$0$	\(0\)	\(1\)	\(0\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(3-\beta )q^{7}+(2+\beta )q^{9}+q^{11}+\cdots\)
4400.2.a.bv	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{5}) \)	None	✓				275.2.a.d	$1$	$1$	\(0\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(2-3\beta )q^{7}+(-1+3\beta )q^{9}+\cdots\)
4400.2.a.bw	$4400$	$2$	4400.a	1.a	$1$	$2$	$2$	$35.134$	\(\Q(\sqrt{13}) \)	None	✓				1100.2.a.f	$1$	$0$	\(0\)	\(3\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(2+\beta )q^{7}+(1+3\beta )q^{9}+\cdots\)
4400.2.a.bx	$4400$	$2$	4400.a	1.a	$1$	$3$	$3$	$35.134$	3.3.837.1	None	✓		✓		2200.2.a.t	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2})q^{7}+(2+\cdots)q^{9}+\cdots\)
4400.2.a.by	$4400$	$2$	4400.a	1.a	$1$	$3$	$3$	$35.134$	3.3.1229.1	None	✓		✓		2200.2.a.u	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
4400.2.a.bz	$4400$	$2$	4400.a	1.a	$1$	$3$	$3$	$35.134$	3.3.1229.1	None	✓				2200.2.a.u	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{2})q^{7}+(2+\beta _{2})q^{9}+\cdots\)
4400.2.a.ca	$4400$	$2$	4400.a	1.a	$1$	$3$	$3$	$35.134$	3.3.837.1	None	✓		✓		2200.2.a.t	$1$	$0$	\(0\)	\(3\)	\(0\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{2})q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
4400.2.a.cb	$4400$	$2$	4400.a	1.a	$1$	$4$	$4$	$35.134$	4.4.54764.1	None	✓				440.2.b.d	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2}+\beta _{3})q^{9}+\cdots\)
4400.2.a.cc	$4400$	$2$	4400.a	1.a	$1$	$4$	$4$	$35.134$	\(\Q(\sqrt{3}, \sqrt{11})\)	None	✓		✓		55.2.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
4400.2.a.cd	$4400$	$2$	4400.a	1.a	$1$	$4$	$4$	$35.134$	\(\Q(\sqrt{3}, \sqrt{19})\)	None	✓		✓		220.2.b.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(3+\beta _{3})q^{9}+\cdots\)
4400.2.a.ce	$4400$	$2$	4400.a	1.a	$1$	$4$	$4$	$35.134$	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 \(S_{2}^{\mathrm{old}}(\Gamma_0(4400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4400)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(880))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2200))\)\(^{\oplus 2}\)