Space of modular forms of level 4050, weight 2, and trivial character

• Label: 4050.2.a
• Level: $4050$
• Weight: $2$
• Character orbit: 4050.a
• Rep. character: $\chi_{4050}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $54$
• Sturm bound: $1620$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 54 \)
Sturm bound:			\(1620\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(23\), \(41\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4050))\).

	Total	New	Old
Modular forms	882	76	806
Cusp forms	739	76	663
Eisenstein series	143	0	143

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
\(+\)	\(+\)	\(+\)	\(+\)		\(105\)	\(8\)	\(97\)		\(88\)	\(8\)	\(80\)		\(17\)	\(0\)	\(17\)
\(+\)	\(+\)	\(-\)	\(-\)		\(114\)	\(10\)	\(104\)		\(96\)	\(10\)	\(86\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(+\)	\(-\)		\(114\)	\(10\)	\(104\)		\(96\)	\(10\)	\(86\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(-\)	\(+\)		\(108\)	\(10\)	\(98\)		\(90\)	\(10\)	\(80\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(+\)	\(-\)		\(111\)	\(11\)	\(100\)		\(93\)	\(11\)	\(82\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)	\(+\)		\(108\)	\(8\)	\(100\)		\(90\)	\(8\)	\(82\)		\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(+\)	\(+\)		\(111\)	\(7\)	\(104\)		\(93\)	\(7\)	\(86\)		\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(-\)	\(-\)		\(111\)	\(12\)	\(99\)		\(93\)	\(12\)	\(81\)		\(18\)	\(0\)	\(18\)
Plus space			\(+\)		\(432\)	\(33\)	\(399\)		\(361\)	\(33\)	\(328\)		\(71\)	\(0\)	\(71\)
Minus space			\(-\)		\(450\)	\(43\)	\(407\)		\(378\)	\(43\)	\(335\)		\(72\)	\(0\)	\(72\)

Trace form

76 * q + 76 * q^4 - 4 * q^7 - 10 * q^13 + 76 * q^16 - 10 * q^19 - 6 * q^22 - 4 * q^28 + 8 * q^31 - 12 * q^34 + 2 * q^37 - 34 * q^43 + 12 * q^46 + 60 * q^49 - 10 * q^52 + 6 * q^58 + 26 * q^61 + 76 * q^64 + 26 * q^67 - 16 * q^73 - 10 * q^76 + 32 * q^79 - 18 * q^82 - 6 * q^88 + 16 * q^91 - 12 * q^94 + 14 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4050))\) 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
4050.2.a.a	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				810.2.a.d	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-5q^{7}-q^{8}+q^{13}+5q^{14}+\cdots\)
4050.2.a.b	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				450.2.e.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{7}-q^{8}+3q^{11}-4q^{13}+\cdots\)
4050.2.a.c	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				18.2.c.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{11}-2q^{13}+\cdots\)
4050.2.a.d	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				450.2.e.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-2q^{7}-q^{8}+4q^{13}+2q^{14}+\cdots\)
4050.2.a.e	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				810.2.c.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-4q^{11}-3q^{13}+\cdots\)
4050.2.a.f	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓	✓			4050.2.a.f	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-4q^{13}+q^{14}+\cdots\)
4050.2.a.g	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				90.2.i.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}+2q^{11}-6q^{13}+\cdots\)
4050.2.a.h	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓	✓			4050.2.a.h	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}+6q^{11}+2q^{13}+\cdots\)
4050.2.a.i	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓	✓			4050.2.a.h	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-6q^{11}-2q^{13}+\cdots\)
4050.2.a.j	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				90.2.i.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-2q^{11}+6q^{13}+\cdots\)
4050.2.a.k	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				810.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-5q^{13}-q^{14}+\cdots\)
4050.2.a.l	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓	✓			4050.2.a.f	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+4q^{13}-q^{14}+\cdots\)
4050.2.a.m	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				810.2.c.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+4q^{11}+3q^{13}+\cdots\)
4050.2.a.n	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				90.2.e.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+6q^{11}-2q^{13}+\cdots\)
4050.2.a.o	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				450.2.e.c	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}-4q^{13}-2q^{14}+\cdots\)
4050.2.a.p	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				450.2.e.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}-3q^{11}+4q^{13}+\cdots\)
4050.2.a.q	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				90.2.e.b	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}-3q^{11}+4q^{13}+\cdots\)
4050.2.a.r	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				162.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+4q^{7}-q^{8}+q^{13}-4q^{14}+\cdots\)
4050.2.a.s	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				810.2.a.d	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-5q^{7}+q^{8}+q^{13}-5q^{14}+\cdots\)
4050.2.a.t	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				450.2.e.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{7}+q^{8}-3q^{11}-4q^{13}+\cdots\)
4050.2.a.u	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				450.2.e.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}+4q^{13}-2q^{14}+\cdots\)
4050.2.a.v	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				18.2.c.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}+3q^{11}-2q^{13}+\cdots\)
4050.2.a.w	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓	✓			4050.2.a.h	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}-6q^{11}+2q^{13}+\cdots\)
4050.2.a.x	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				90.2.i.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}-2q^{11}-6q^{13}+\cdots\)
4050.2.a.y	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓	✓			4050.2.a.f	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}-4q^{13}-q^{14}+\cdots\)
4050.2.a.z	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				810.2.c.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}+4q^{11}-3q^{13}+\cdots\)
4050.2.a.ba	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				90.2.e.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-6q^{11}-2q^{13}+\cdots\)
4050.2.a.bb	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				810.2.c.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-4q^{11}+3q^{13}+\cdots\)
4050.2.a.bc	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				810.2.a.c	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-5q^{13}+q^{14}+\cdots\)
4050.2.a.bd	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓	✓			4050.2.a.f	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+4q^{13}+q^{14}+\cdots\)
4050.2.a.be	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				90.2.i.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+2q^{11}+6q^{13}+\cdots\)
4050.2.a.bf	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓	✓			4050.2.a.h	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+6q^{11}-2q^{13}+\cdots\)
4050.2.a.bg	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				450.2.e.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}-4q^{13}+2q^{14}+\cdots\)
4050.2.a.bh	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				162.2.a.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+q^{13}+4q^{14}+\cdots\)
4050.2.a.bi	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				90.2.e.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+3q^{11}+4q^{13}+\cdots\)
4050.2.a.bj	$4050$	$2$	4050.a	1.a	$1$	$1$	$1$	$32.339$	\(\Q\)	None	✓				450.2.e.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{7}+q^{8}+3q^{11}+4q^{13}+\cdots\)
4050.2.a.bk	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{3}) \)	None	✓				810.2.a.j	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-2+\beta )q^{7}-q^{8}+2\beta q^{11}+\cdots\)
4050.2.a.bl	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{6}) \)	None	✓				450.2.e.l	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-2+\beta )q^{7}-q^{8}+2\beta q^{11}+\cdots\)
4050.2.a.bm	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{6}) \)	None	✓				90.2.i.b	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-2+\beta )q^{7}-q^{8}+(1+\cdots)q^{11}+\cdots\)
4050.2.a.bn	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{3}) \)	None	✓	✓			4050.2.a.bn	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+2\beta )q^{7}-q^{8}+(-3+\cdots)q^{11}+\cdots\)
4050.2.a.bo	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{33}) \)	None	✓				90.2.e.c	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta q^{7}-q^{8}+(1+\beta )q^{11}+\cdots\)
4050.2.a.bp	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{3}) \)	None	✓	✓			4050.2.a.bn	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+2\beta )q^{7}-q^{8}+(3+\beta )q^{11}+\cdots\)
4050.2.a.bq	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{6}) \)	None	✓				90.2.i.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(2+\beta )q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots\)
4050.2.a.br	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{6}) \)	None	✓				450.2.e.l	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(2+\beta )q^{7}-q^{8}+2\beta q^{11}+\cdots\)
4050.2.a.bs	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{6}) \)	None	✓		✓		90.2.i.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots\)
4050.2.a.bt	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{3}) \)	None	✓				810.2.a.j	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}-2\beta q^{11}+\cdots\)
4050.2.a.bu	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{6}) \)	None	✓				450.2.e.l	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}-2\beta q^{11}+\cdots\)
4050.2.a.bv	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{3}) \)	None	✓	✓			4050.2.a.bn	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+2\beta )q^{7}+q^{8}+(3+\cdots)q^{11}+\cdots\)
4050.2.a.bw	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{33}) \)	None	✓				90.2.e.c	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta q^{7}+q^{8}+(-1-\beta )q^{11}+\cdots\)
4050.2.a.bx	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{3}) \)	None	✓	✓			4050.2.a.bn	$1$	$1$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+2\beta )q^{7}+q^{8}+(-3+\cdots)q^{11}+\cdots\)
4050.2.a.by	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{6}) \)	None	✓				450.2.e.l	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(2+\beta )q^{7}+q^{8}-2\beta q^{11}+\cdots\)
4050.2.a.bz	$4050$	$2$	4050.a	1.a	$1$	$2$	$2$	$32.339$	\(\Q(\sqrt{6}) \)	None	✓				90.2.i.b	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(2+\beta )q^{7}+q^{8}+(1-\beta )q^{11}+\cdots\)
4050.2.a.ca	$4050$	$2$	4050.a	1.a	$1$	$4$	$4$	$32.339$	\(\Q(\sqrt{3}, \sqrt{19})\)	None	✓		✓		810.2.c.g	$2$	$1$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}-2\beta _{1}q^{11}+\cdots\)
4050.2.a.cb	$4050$	$2$	4050.a	1.a	$1$	$4$	$4$	$32.339$	\(\Q(\sqrt{3}, \sqrt{19})\)	None	✓		✓		810.2.c.g	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+2\beta _{1}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4050))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4050)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(810))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2025))\)\(^{\oplus 2}\)