Space of modular forms of level 1600, weight 4, and trivial character

• Label: 1600.4.a
• Level: $1600$
• Weight: $4$
• Character orbit: 1600.a
• Rep. character: $\chi_{1600}(1,\cdot)$
• Character field: $\Q$
• Dimension: $111$
• Newform subspaces: $76$
• Sturm bound: $960$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 76 \)
Sturm bound:			\(960\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1600))\).

	Total	New	Old
Modular forms	756	117	639
Cusp forms	684	111	573
Eisenstein series	72	6	66

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

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(27\)
\(+\)	\(-\)	\(-\)	\(28\)
\(-\)	\(+\)	\(-\)	\(26\)
\(-\)	\(-\)	\(+\)	\(30\)
Plus space		\(+\)	\(57\)
Minus space		\(-\)	\(54\)

Trace form

111 * q + 947 * q^9 + 70 * q^13 + 54 * q^17 - 168 * q^21 - 202 * q^29 - 176 * q^33 - 514 * q^37 - 282 * q^41 + 4559 * q^49 - 34 * q^53 + 400 * q^57 - 162 * q^61 + 1432 * q^69 - 146 * q^73 - 1488 * q^77 + 9247 * q^81 + 894 * q^89 - 1568 * q^93 + 582 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\) 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
1600.4.a.a	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				40.4.a.c	$1$	$1$	\(0\)	\(-10\)	\(0\)	\(-18\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}-18q^{7}+73q^{9}-2^{4}q^{11}+\cdots\)
1600.4.a.b	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.b	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+26q^{7}+54q^{9}-59q^{11}+\cdots\)
1600.4.a.c	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.b	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+26q^{7}+54q^{9}+59q^{11}+\cdots\)
1600.4.a.d	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				10.4.a.a	$1$	$0$	\(0\)	\(-8\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+4q^{7}+37q^{9}-12q^{11}-58q^{13}+\cdots\)
1600.4.a.e	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				32.4.a.a	$1$	$0$	\(0\)	\(-8\)	\(0\)	\(16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+2^{4}q^{7}+37q^{9}-40q^{11}+\cdots\)
1600.4.a.f	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				50.4.a.a	$1$	$1$	\(0\)	\(-7\)	\(0\)	\(-34\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-34q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
1600.4.a.g	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				50.4.a.a	$1$	$1$	\(0\)	\(-7\)	\(0\)	\(-34\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-34q^{7}+22q^{9}+3^{3}q^{11}+\cdots\)
1600.4.a.h	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				25.4.a.a	$1$	$0$	\(0\)	\(-7\)	\(0\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+6q^{7}+22q^{9}-43q^{11}-28q^{13}+\cdots\)
1600.4.a.i	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				25.4.a.a	$1$	$0$	\(0\)	\(-7\)	\(0\)	\(6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+6q^{7}+22q^{9}+43q^{11}+28q^{13}+\cdots\)
1600.4.a.j	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				40.4.a.a	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(34\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}+34q^{7}+9q^{9}-2^{4}q^{11}+58q^{13}+\cdots\)
1600.4.a.k	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				800.4.a.b	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-10q^{7}-2q^{9}-15q^{11}-8q^{13}+\cdots\)
1600.4.a.l	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				800.4.a.b	$1$	$0$	\(0\)	\(-5\)	\(0\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-10q^{7}-2q^{9}+15q^{11}+8q^{13}+\cdots\)
1600.4.a.m	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.c	$1$	$2$	\(0\)	\(-5\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+2q^{7}-2q^{9}-39q^{11}-84q^{13}+\cdots\)
1600.4.a.n	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.c	$1$	$0$	\(0\)	\(-5\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+2q^{7}-2q^{9}+39q^{11}+84q^{13}+\cdots\)
1600.4.a.o	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				8.4.a.a	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-24q^{7}-11q^{9}+44q^{11}+\cdots\)
1600.4.a.p	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				20.4.a.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}-2^{4}q^{7}-11q^{9}-60q^{11}+\cdots\)
1600.4.a.q	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				40.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+2^{4}q^{7}-11q^{9}+6^{2}q^{11}+\cdots\)
1600.4.a.r	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				160.4.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-6q^{7}-23q^{9}-60q^{11}+50q^{13}+\cdots\)
1600.4.a.s	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				5.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+6q^{7}-23q^{9}+2^{5}q^{11}-38q^{13}+\cdots\)
1600.4.a.t	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				10.4.b.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+26q^{7}-23q^{9}-28q^{11}+\cdots\)
1600.4.a.u	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				10.4.b.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+26q^{7}-23q^{9}+28q^{11}+\cdots\)
1600.4.a.v	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-6q^{7}-26q^{9}-19q^{11}+12q^{13}+\cdots\)
1600.4.a.w	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-6q^{7}-26q^{9}+19q^{11}-12q^{13}+\cdots\)
1600.4.a.x	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				100.4.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+26q^{7}-26q^{9}-45q^{11}-44q^{13}+\cdots\)
1600.4.a.y	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				100.4.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+26q^{7}-26q^{9}+45q^{11}+44q^{13}+\cdots\)
1600.4.a.z	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.4.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3^{3}q^{9}-92q^{13}-104q^{17}-130q^{29}+\cdots\)
1600.4.a.ba	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.4.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3^{3}q^{9}-18q^{13}+94q^{17}+130q^{29}+\cdots\)
1600.4.a.bb	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.4.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3^{3}q^{9}+92q^{13}+104q^{17}-130q^{29}+\cdots\)
1600.4.a.bc	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				100.4.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-26\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-26q^{7}-26q^{9}-45q^{11}+44q^{13}+\cdots\)
1600.4.a.bd	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				100.4.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-26q^{7}-26q^{9}+45q^{11}-44q^{13}+\cdots\)
1600.4.a.be	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+6q^{7}-26q^{9}-19q^{11}-12q^{13}+\cdots\)
1600.4.a.bf	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+6q^{7}-26q^{9}+19q^{11}+12q^{13}+\cdots\)
1600.4.a.bg	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				10.4.b.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-26q^{7}-23q^{9}-28q^{11}+\cdots\)
1600.4.a.bh	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				10.4.b.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-26q^{7}-23q^{9}+28q^{11}+\cdots\)
1600.4.a.bi	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				5.4.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-6q^{7}-23q^{9}-2^{5}q^{11}-38q^{13}+\cdots\)
1600.4.a.bj	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				160.4.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+6q^{7}-23q^{9}+60q^{11}+50q^{13}+\cdots\)
1600.4.a.bk	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				40.4.a.b	$1$	$0$	\(0\)	\(4\)	\(0\)	\(-16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-2^{4}q^{7}-11q^{9}-6^{2}q^{11}+\cdots\)
1600.4.a.bl	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				20.4.a.a	$1$	$0$	\(0\)	\(4\)	\(0\)	\(16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+2^{4}q^{7}-11q^{9}+60q^{11}+\cdots\)
1600.4.a.bm	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				8.4.a.a	$1$	$1$	\(0\)	\(4\)	\(0\)	\(24\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+24q^{7}-11q^{9}-44q^{11}+\cdots\)
1600.4.a.bn	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.c	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-2q^{7}-2q^{9}-39q^{11}+84q^{13}+\cdots\)
1600.4.a.bo	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.c	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-2q^{7}-2q^{9}+39q^{11}-84q^{13}+\cdots\)
1600.4.a.bp	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				800.4.a.b	$1$	$0$	\(0\)	\(5\)	\(0\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+10q^{7}-2q^{9}-15q^{11}+8q^{13}+\cdots\)
1600.4.a.bq	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				800.4.a.b	$1$	$1$	\(0\)	\(5\)	\(0\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+10q^{7}-2q^{9}+15q^{11}-8q^{13}+\cdots\)
1600.4.a.br	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				40.4.a.a	$1$	$1$	\(0\)	\(6\)	\(0\)	\(-34\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}-34q^{7}+9q^{9}+2^{4}q^{11}+58q^{13}+\cdots\)
1600.4.a.bs	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				25.4.a.a	$1$	$1$	\(0\)	\(7\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-6q^{7}+22q^{9}-43q^{11}+28q^{13}+\cdots\)
1600.4.a.bt	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				25.4.a.a	$1$	$1$	\(0\)	\(7\)	\(0\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-6q^{7}+22q^{9}+43q^{11}-28q^{13}+\cdots\)
1600.4.a.bu	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				50.4.a.a	$1$	$0$	\(0\)	\(7\)	\(0\)	\(34\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+34q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
1600.4.a.bv	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				50.4.a.a	$1$	$0$	\(0\)	\(7\)	\(0\)	\(34\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+34q^{7}+22q^{9}+3^{3}q^{11}+\cdots\)
1600.4.a.bw	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				32.4.a.a	$1$	$0$	\(0\)	\(8\)	\(0\)	\(-16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-2^{4}q^{7}+37q^{9}+40q^{11}+\cdots\)
1600.4.a.bx	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				10.4.a.a	$1$	$1$	\(0\)	\(8\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-4q^{7}+37q^{9}+12q^{11}-58q^{13}+\cdots\)
1600.4.a.by	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.b	$1$	$0$	\(0\)	\(9\)	\(0\)	\(-26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-26q^{7}+54q^{9}-59q^{11}+\cdots\)
1600.4.a.bz	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				200.4.a.b	$1$	$0$	\(0\)	\(9\)	\(0\)	\(-26\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-26q^{7}+54q^{9}+59q^{11}+\cdots\)
1600.4.a.ca	$1600$	$4$	1600.a	1.a	$1$	$1$	$1$	$94.403$	\(\Q\)	None	✓				40.4.a.c	$1$	$0$	\(0\)	\(10\)	\(0\)	\(18\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}+18q^{7}+73q^{9}+2^{4}q^{11}+\cdots\)
1600.4.a.cb	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.4.c.c	$2$	$1$	\(0\)	\(-14\)	\(0\)	\(18\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(-7-\beta )q^{3}+(9+11\beta )q^{7}+(3^{3}+14\beta )q^{9}+\cdots\)
1600.4.a.cc	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{29}) \)	None	✓				160.4.c.b	$2$	$1$	\(0\)	\(-8\)	\(0\)	\(-8\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-4q^{3}-4q^{7}-11q^{9}-2\beta q^{11}+\beta q^{13}+\cdots\)
1600.4.a.cd	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{6}) \)	None	✓				160.4.a.c	$1$	$0$	\(0\)	\(-8\)	\(0\)	\(8\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{3}+(4+5\beta )q^{7}+(13-8\beta )q^{9}+\cdots\)
1600.4.a.ce	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{6}) \)	None	✓				40.4.c.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(2+3\beta )q^{7}+(1-4\beta )q^{9}+\cdots\)
1600.4.a.cf	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{6}) \)	None	✓				40.4.c.a	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(2+3\beta )q^{7}+(1-4\beta )q^{9}+\cdots\)
1600.4.a.cg	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{5}) \)	None	✓				160.4.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-7\beta q^{7}-7q^{9}-2\beta q^{11}-62q^{13}+\cdots\)
1600.4.a.ch	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{10}) \)	None	✓				160.4.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-3\beta q^{7}+13q^{9}+2\beta q^{11}+\cdots\)
1600.4.a.ci	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{13}) \)	None	✓				160.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+\beta q^{7}+5^{2}q^{9}+6\beta q^{11}+34q^{13}+\cdots\)
1600.4.a.cj	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{19}) \)	None	✓				20.4.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+7^{2}q^{9}-20q^{11}-6\beta q^{13}+\cdots\)
1600.4.a.ck	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{19}) \)	None	✓				20.4.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+7^{2}q^{9}+20q^{11}+6\beta q^{13}+\cdots\)
1600.4.a.cl	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{6}) \)	None	✓				40.4.c.a	$1$	$1$	\(0\)	\(4\)	\(0\)	\(-4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(-2+3\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
1600.4.a.cm	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{6}) \)	None	✓				40.4.c.a	$1$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(-2+3\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)
1600.4.a.cn	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{6}) \)	None	✓				160.4.a.c	$1$	$0$	\(0\)	\(8\)	\(0\)	\(-8\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{3}+(-4+5\beta )q^{7}+(13+8\beta )q^{9}+\cdots\)
1600.4.a.co	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{29}) \)	None	✓				160.4.c.b	$2$	$1$	\(0\)	\(8\)	\(0\)	\(8\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+4q^{3}+4q^{7}-11q^{9}+2\beta q^{11}+\beta q^{13}+\cdots\)
1600.4.a.cp	$1600$	$4$	1600.a	1.a	$1$	$2$	$2$	$94.403$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.4.c.c	$2$	$1$	\(0\)	\(14\)	\(0\)	\(-18\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(7-\beta )q^{3}+(-9+11\beta )q^{7}+(3^{3}-14\beta )q^{9}+\cdots\)
1600.4.a.cq	$1600$	$4$	1600.a	1.a	$1$	$3$	$3$	$94.403$	3.3.5685.1	None	✓				800.4.a.u	$1$	$0$	\(0\)	\(-5\)	\(0\)	\(-2\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1600.4.a.cr	$1600$	$4$	1600.a	1.a	$1$	$3$	$3$	$94.403$	3.3.5685.1	None	✓				800.4.a.u	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(-2\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
1600.4.a.cs	$1600$	$4$	1600.a	1.a	$1$	$3$	$3$	$94.403$	3.3.5685.1	None	✓				800.4.a.u	$1$	$1$	\(0\)	\(5\)	\(0\)	\(2\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+(7+\cdots)q^{9}+\cdots\)
1600.4.a.ct	$1600$	$4$	1600.a	1.a	$1$	$3$	$3$	$94.403$	3.3.5685.1	None	✓				800.4.a.u	$1$	$0$	\(0\)	\(5\)	\(0\)	\(2\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+(7+\cdots)q^{9}+\cdots\)
1600.4.a.cu	$1600$	$4$	1600.a	1.a	$1$	$4$	$4$	$94.403$	4.4.37485.2	None	✓				160.4.c.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(19-3\beta _{2}+\cdots)q^{9}+\cdots\)
1600.4.a.cv	$1600$	$4$	1600.a	1.a	$1$	$4$	$4$	$94.403$	4.4.37485.2	None	✓				160.4.c.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(19-3\beta _{2}+\cdots)q^{9}+\cdots\)
1600.4.a.cw	$1600$	$4$	1600.a	1.a	$1$	$4$	$4$	$94.403$	4.4.2106005.1	None	✓		✓		800.4.a.ba	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{7}+(24+\beta _{2}+\cdots)q^{9}+\cdots\)
1600.4.a.cx	$1600$	$4$	1600.a	1.a	$1$	$4$	$4$	$94.403$	4.4.2106005.1	None	✓				800.4.a.ba	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{7}+(24+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1600))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1600)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(320))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(800))\)\(^{\oplus 2}\)