Space of modular forms of level 1050, weight 4, and trivial character

• Label: 1050.4.a
• Level: $1050$
• Weight: $4$
• Character orbit: 1050.a
• Rep. character: $\chi_{1050}(1,\cdot)$
• Character field: $\Q$
• Dimension: $58$
• Newform subspaces: $39$
• Sturm bound: $960$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 39 \)
Sturm bound:			\(960\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1050))\).

	Total	New	Old
Modular forms	744	58	686
Cusp forms	696	58	638
Eisenstein series	48	0	48

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

\(2\)	\(3\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(51\)	\(4\)	\(47\)		\(48\)	\(4\)	\(44\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(45\)	\(3\)	\(42\)		\(42\)	\(3\)	\(39\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(44\)	\(4\)	\(40\)		\(41\)	\(4\)	\(37\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(47\)	\(4\)	\(43\)		\(44\)	\(4\)	\(40\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(45\)	\(3\)	\(42\)		\(42\)	\(3\)	\(39\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(45\)	\(4\)	\(41\)		\(42\)	\(4\)	\(38\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(48\)	\(4\)	\(44\)		\(45\)	\(4\)	\(41\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(47\)	\(4\)	\(43\)		\(44\)	\(4\)	\(40\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(45\)	\(2\)	\(43\)		\(42\)	\(2\)	\(40\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(48\)	\(4\)	\(44\)		\(45\)	\(4\)	\(41\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(47\)	\(5\)	\(42\)		\(44\)	\(5\)	\(39\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(45\)	\(3\)	\(42\)		\(42\)	\(3\)	\(39\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(45\)	\(4\)	\(41\)		\(42\)	\(4\)	\(38\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(48\)	\(2\)	\(46\)		\(45\)	\(2\)	\(43\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(47\)	\(3\)	\(44\)		\(44\)	\(3\)	\(41\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(47\)	\(5\)	\(42\)		\(44\)	\(5\)	\(39\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(378\)	\(34\)	\(344\)		\(354\)	\(34\)	\(320\)		\(24\)	\(0\)	\(24\)
Minus space				\(-\)		\(366\)	\(24\)	\(342\)		\(342\)	\(24\)	\(318\)		\(24\)	\(0\)	\(24\)

Trace form

58 * q - 4 * q^2 + 232 * q^4 - 16 * q^8 + 522 * q^9 - 80 * q^11 + 12 * q^13 + 928 * q^16 - 132 * q^17 - 36 * q^18 - 128 * q^19 + 42 * q^21 - 16 * q^22 + 248 * q^23 - 888 * q^26 - 340 * q^29 + 416 * q^31 - 64 * q^32 - 576 * q^33 + 72 * q^34 + 2088 * q^36 - 148 * q^37 - 64 * q^38 + 24 * q^39 + 692 * q^41 - 84 * q^42 - 1480 * q^43 - 320 * q^44 - 192 * q^46 + 1280 * q^47 + 2842 * q^49 + 384 * q^51 + 48 * q^52 + 1476 * q^53 + 192 * q^57 - 184 * q^58 + 2224 * q^59 + 4100 * q^61 - 96 * q^62 + 3712 * q^64 - 864 * q^66 + 1592 * q^67 - 528 * q^68 - 1728 * q^69 + 2120 * q^71 - 144 * q^72 + 2764 * q^73 + 3992 * q^74 - 512 * q^76 + 448 * q^77 - 48 * q^78 - 1248 * q^79 + 4698 * q^81 - 1704 * q^82 + 3376 * q^83 + 168 * q^84 + 3728 * q^86 + 192 * q^87 - 64 * q^88 + 8100 * q^89 + 784 * q^91 + 992 * q^92 + 456 * q^93 - 928 * q^94 - 116 * q^97 - 196 * q^98 - 720 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1050))\) 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	7
1050.4.a.a	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.a	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(-7\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.b	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.b	$1$	$0$	\(-2\)	\(-3\)	\(0\)	\(7\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.c	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.c	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(7\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.d	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				42.4.a.b	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(7\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.e	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.e	$1$	$0$	\(-2\)	\(-3\)	\(0\)	\(7\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.f	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.j	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(7\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.g	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				42.4.a.a	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-7\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.h	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.h	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-7\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.i	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.h	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(-7\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.j	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.j	$1$	$1$	\(-2\)	\(3\)	\(0\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.k	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.g	$1$	$0$	\(-2\)	\(3\)	\(0\)	\(7\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.l	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.i	$1$	$0$	\(-2\)	\(3\)	\(0\)	\(7\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.m	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.j	$1$	$1$	\(2\)	\(-3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.n	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.e	$1$	$1$	\(2\)	\(-3\)	\(0\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.o	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.h	$1$	$1$	\(2\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.p	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.f	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.q	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.d	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.r	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.b	$1$	$0$	\(2\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.s	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.c	$1$	$1$	\(2\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.t	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.a	$1$	$0$	\(2\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.u	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.c	$1$	$0$	\(2\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.v	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.e	$1$	$0$	\(2\)	\(3\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.w	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓				210.4.a.b	$1$	$1$	\(2\)	\(3\)	\(0\)	\(7\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.x	$1050$	$4$	1050.a	1.a	$1$	$1$	$1$	$61.952$	\(\Q\)	None	✓	✓			1050.4.a.a	$1$	$1$	\(2\)	\(3\)	\(0\)	\(7\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.y	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{8761}) \)	None	✓	✓			1050.4.a.y	$1$	$0$	\(-4\)	\(-6\)	\(0\)	\(-14\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.z	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{106}) \)	None	✓				210.4.a.k	$1$	$0$	\(-4\)	\(-6\)	\(0\)	\(-14\)		$+$	$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.ba	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{21}) \)	None	✓		✓		210.4.g.b	$1$	$0$	\(-4\)	\(-6\)	\(0\)	\(14\)		$+$	$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.bb	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{1129}) \)	None	✓	✓			1050.4.a.bb	$1$	$0$	\(-4\)	\(6\)	\(0\)	\(-14\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.bc	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{6}) \)	None	✓				210.4.g.a	$1$	$0$	\(-4\)	\(6\)	\(0\)	\(-14\)		$+$	$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.bd	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{6001}) \)	None	✓	✓	✓		1050.4.a.bd	$1$	$0$	\(-4\)	\(6\)	\(0\)	\(14\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.be	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{6001}) \)	None	✓	✓			1050.4.a.bd	$1$	$0$	\(4\)	\(-6\)	\(0\)	\(-14\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.bf	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{1129}) \)	None	✓	✓	✓		1050.4.a.bb	$1$	$0$	\(4\)	\(-6\)	\(0\)	\(14\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.bg	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{6}) \)	None	✓		✓		210.4.g.a	$1$	$1$	\(4\)	\(-6\)	\(0\)	\(14\)		$-$	$+$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.bh	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{21}) \)	None	✓		✓		210.4.g.b	$1$	$1$	\(4\)	\(6\)	\(0\)	\(-14\)		$-$	$-$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.bi	$1050$	$4$	1050.a	1.a	$1$	$2$	$2$	$61.952$	\(\Q(\sqrt{8761}) \)	None	✓	✓			1050.4.a.y	$1$	$0$	\(4\)	\(6\)	\(0\)	\(14\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)
1050.4.a.bj	$1050$	$4$	1050.a	1.a	$1$	$3$	$3$	$61.952$	3.3.104916.1	None	✓		✓		210.4.g.c	$1$	$1$	\(-6\)	\(-9\)	\(0\)	\(-21\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots\)
1050.4.a.bk	$1050$	$4$	1050.a	1.a	$1$	$3$	$3$	$61.952$	3.3.62004.1	None	✓		✓		210.4.g.d	$1$	$1$	\(-6\)	\(9\)	\(0\)	\(21\)		$+$	$-$	$-$	$-$	$-$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots\)
1050.4.a.bl	$1050$	$4$	1050.a	1.a	$1$	$3$	$3$	$61.952$	3.3.62004.1	None	✓		✓		210.4.g.d	$1$	$0$	\(6\)	\(-9\)	\(0\)	\(-21\)		$-$	$+$	$-$	$+$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}-7q^{7}+\cdots\)
1050.4.a.bm	$1050$	$4$	1050.a	1.a	$1$	$3$	$3$	$61.952$	3.3.104916.1	None	✓		✓		210.4.g.c	$1$	$0$	\(6\)	\(9\)	\(0\)	\(21\)		$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1050))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1050)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)