Space of modular forms of level 1680, weight 4, and trivial character

• Label: 1680.4.a
• Level: $1680$
• Weight: $4$
• Character orbit: 1680.a
• Rep. character: $\chi_{1680}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $46$
• Sturm bound: $1536$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 46 \)
Sturm bound:			\(1536\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1680))\).

	Total	New	Old
Modular forms	1176	72	1104
Cusp forms	1128	72	1056
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
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(78\)	\(5\)	\(73\)		\(75\)	\(5\)	\(70\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(71\)	\(4\)	\(67\)		\(68\)	\(4\)	\(64\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(71\)	\(4\)	\(67\)		\(68\)	\(4\)	\(64\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(74\)	\(5\)	\(69\)		\(71\)	\(5\)	\(66\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(74\)	\(4\)	\(70\)		\(71\)	\(4\)	\(67\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(75\)	\(5\)	\(70\)		\(72\)	\(5\)	\(67\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(71\)	\(5\)	\(66\)		\(68\)	\(5\)	\(63\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(74\)	\(4\)	\(70\)		\(71\)	\(4\)	\(67\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(69\)	\(4\)	\(65\)		\(66\)	\(4\)	\(62\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(76\)	\(6\)	\(70\)		\(73\)	\(6\)	\(67\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(76\)	\(4\)	\(72\)		\(73\)	\(4\)	\(69\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(73\)	\(4\)	\(69\)		\(70\)	\(4\)	\(66\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(73\)	\(4\)	\(69\)		\(70\)	\(4\)	\(66\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(72\)	\(4\)	\(68\)		\(69\)	\(4\)	\(65\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(76\)	\(4\)	\(72\)		\(73\)	\(4\)	\(69\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(73\)	\(6\)	\(67\)		\(70\)	\(6\)	\(64\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(596\)	\(40\)	\(556\)		\(572\)	\(40\)	\(532\)		\(24\)	\(0\)	\(24\)
Minus space				\(-\)		\(580\)	\(32\)	\(548\)		\(556\)	\(32\)	\(524\)		\(24\)	\(0\)	\(24\)

Trace form

72 * q + 28 * q^7 + 648 * q^9 - 40 * q^11 + 60 * q^15 + 24 * q^19 - 328 * q^23 + 1800 * q^25 - 400 * q^29 - 264 * q^31 - 16 * q^37 - 1840 * q^43 - 1488 * q^47 + 3528 * q^49 - 744 * q^51 + 752 * q^53 + 240 * q^59 - 912 * q^61 + 252 * q^63 - 1200 * q^67 - 528 * q^69 - 4120 * q^71 + 1904 * q^77 + 1824 * q^79 + 5832 * q^81 - 912 * q^83 + 240 * q^85 + 2720 * q^89 + 1952 * q^97 - 360 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1680))\) 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
1680.4.a.a	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.e	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-7q^{7}+9q^{9}+26q^{13}+\cdots\)
1680.4.a.b	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				420.4.a.c	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(-7\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
1680.4.a.c	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.j	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}-56q^{11}+\cdots\)
1680.4.a.d	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}-28q^{11}+\cdots\)
1680.4.a.e	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				420.4.a.f	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(-7\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-7q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
1680.4.a.f	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				840.4.a.f	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(-7\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-7q^{7}+9q^{9}+2^{4}q^{11}+\cdots\)
1680.4.a.g	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				420.4.a.e	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(-7\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-7q^{7}+9q^{9}+2^{4}q^{11}+\cdots\)
1680.4.a.h	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.f	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(7\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}-28q^{11}+\cdots\)
1680.4.a.i	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				840.4.a.e	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(7\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}-22q^{11}+\cdots\)
1680.4.a.j	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				840.4.a.d	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(7\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}-20q^{11}+\cdots\)
1680.4.a.k	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				420.4.a.d	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(7\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}+44q^{11}+\cdots\)
1680.4.a.l	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				840.4.a.c	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(7\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}+44q^{11}+\cdots\)
1680.4.a.m	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.h	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-7q^{7}+9q^{9}-24q^{11}+\cdots\)
1680.4.a.n	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(-7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-7q^{7}+9q^{9}-12q^{11}+\cdots\)
1680.4.a.o	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				420.4.a.a	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(7\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+7q^{7}+9q^{9}-2^{5}q^{11}+\cdots\)
1680.4.a.p	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				840.4.a.b	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(7\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+7q^{7}+9q^{9}-54q^{13}+\cdots\)
1680.4.a.q	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.g	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(7\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+7q^{7}+9q^{9}+4q^{11}+\cdots\)
1680.4.a.r	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				840.4.a.a	$1$	$0$	\(0\)	\(3\)	\(-5\)	\(7\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+7q^{7}+9q^{9}+58q^{11}+\cdots\)
1680.4.a.s	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				105.4.a.a	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}-42q^{11}+\cdots\)
1680.4.a.t	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.c	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}-12q^{11}+\cdots\)
1680.4.a.u	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				105.4.a.b	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}-12q^{11}+\cdots\)
1680.4.a.v	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				420.4.a.b	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
1680.4.a.w	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.i	$1$	$0$	\(0\)	\(3\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}+9q^{9}-2^{4}q^{11}+\cdots\)
1680.4.a.x	$1680$	$4$	1680.a	1.a	$1$	$1$	$1$	$99.123$	\(\Q\)	None	✓				210.4.a.b	$1$	$0$	\(0\)	\(3\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}+9q^{9}+44q^{11}+\cdots\)
1680.4.a.y	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{41}) \)	None	✓		✓		105.4.a.g	$1$	$1$	\(0\)	\(-6\)	\(-10\)	\(-14\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-7q^{7}+9q^{9}+(-31+\cdots)q^{11}+\cdots\)
1680.4.a.z	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{193}) \)	None	✓				840.4.a.n	$1$	$0$	\(0\)	\(-6\)	\(-10\)	\(-14\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-7q^{7}+9q^{9}+(17-3\beta )q^{11}+\cdots\)
1680.4.a.ba	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{21}) \)	None	✓				840.4.a.m	$1$	$1$	\(0\)	\(-6\)	\(-10\)	\(14\)		$+$	$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}+(-14+\cdots)q^{11}+\cdots\)
1680.4.a.bb	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{309}) \)	None	✓				840.4.a.l	$1$	$1$	\(0\)	\(-6\)	\(-10\)	\(14\)		$+$	$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}+(2+\beta )q^{11}+\cdots\)
1680.4.a.bc	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{421}) \)	None	✓				420.4.a.i	$1$	$0$	\(0\)	\(-6\)	\(-10\)	\(14\)		$-$	$+$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}+(6+\beta )q^{11}+\cdots\)
1680.4.a.bd	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{5}) \)	None	✓				105.4.a.d	$1$	$0$	\(0\)	\(-6\)	\(-10\)	\(14\)		$-$	$+$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+7q^{7}+9q^{9}+(46-\beta )q^{11}+\cdots\)
1680.4.a.be	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{106}) \)	None	✓		✓		210.4.a.k	$1$	$0$	\(0\)	\(-6\)	\(10\)	\(-14\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-7q^{7}+9q^{9}+(-2^{4}+\cdots)q^{11}+\cdots\)
1680.4.a.bf	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{2881}) \)	None	✓		✓		840.4.a.o	$1$	$0$	\(0\)	\(-6\)	\(10\)	\(14\)		$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}+(11-\beta )q^{11}+\cdots\)
1680.4.a.bg	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{65}) \)	None	✓		✓		105.4.a.f	$1$	$1$	\(0\)	\(-6\)	\(10\)	\(14\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+7q^{7}+9q^{9}+(11-\beta )q^{11}+\cdots\)
1680.4.a.bh	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{61}) \)	None	✓				840.4.a.h	$1$	$1$	\(0\)	\(6\)	\(-10\)	\(-14\)		$+$	$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-7q^{7}+9q^{9}+(-26+\cdots)q^{11}+\cdots\)
1680.4.a.bi	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{109}) \)	None	✓		✓		420.4.a.g	$1$	$0$	\(0\)	\(6\)	\(-10\)	\(-14\)		$-$	$-$	$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-7q^{7}+9q^{9}+(-6+\cdots)q^{11}+\cdots\)
1680.4.a.bj	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{29}) \)	None	✓				840.4.a.g	$1$	$1$	\(0\)	\(6\)	\(-10\)	\(-14\)		$+$	$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}-7q^{7}+9q^{9}+(14-\beta )q^{11}+\cdots\)
1680.4.a.bk	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{17}) \)	None	✓		✓		105.4.a.c	$1$	$1$	\(0\)	\(6\)	\(-10\)	\(14\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+7q^{7}+9q^{9}+(13+5\beta )q^{11}+\cdots\)
1680.4.a.bl	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{401}) \)	None	✓				840.4.a.k	$1$	$0$	\(0\)	\(6\)	\(10\)	\(-14\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}+(13-\beta )q^{11}+\cdots\)
1680.4.a.bm	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{22}) \)	None	✓				840.4.a.j	$1$	$1$	\(0\)	\(6\)	\(10\)	\(14\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}+9q^{9}+(-24+\cdots)q^{11}+\cdots\)
1680.4.a.bn	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{6}) \)	None	✓				840.4.a.i	$1$	$1$	\(0\)	\(6\)	\(10\)	\(14\)		$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}+9q^{9}+(-8+\cdots)q^{11}+\cdots\)
1680.4.a.bo	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{2}) \)	None	✓				105.4.a.e	$1$	$0$	\(0\)	\(6\)	\(10\)	\(14\)		$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}+9q^{9}+(8+10\beta )q^{11}+\cdots\)
1680.4.a.bp	$1680$	$4$	1680.a	1.a	$1$	$2$	$2$	$99.123$	\(\Q(\sqrt{130}) \)	None	✓				420.4.a.h	$1$	$0$	\(0\)	\(6\)	\(10\)	\(14\)		$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}+9q^{9}+8q^{11}+\cdots\)
1680.4.a.bq	$1680$	$4$	1680.a	1.a	$1$	$3$	$3$	$99.123$	3.3.851417.1	None	✓		✓		840.4.a.r	$1$	$0$	\(0\)	\(-9\)	\(-15\)	\(-21\)		$+$	$+$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-7q^{7}+9q^{9}+(-5+\cdots)q^{11}+\cdots\)
1680.4.a.br	$1680$	$4$	1680.a	1.a	$1$	$3$	$3$	$99.123$	3.3.79853.1	None	✓		✓		840.4.a.s	$1$	$1$	\(0\)	\(-9\)	\(15\)	\(-21\)		$+$	$+$	$-$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-7q^{7}+9q^{9}+(-12+\cdots)q^{11}+\cdots\)
1680.4.a.bs	$1680$	$4$	1680.a	1.a	$1$	$3$	$3$	$99.123$	3.3.821313.1	None	✓		✓		840.4.a.p	$1$	$0$	\(0\)	\(9\)	\(-15\)	\(21\)		$+$	$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+7q^{7}+9q^{9}+(2+\beta _{2})q^{11}+\cdots\)
1680.4.a.bt	$1680$	$4$	1680.a	1.a	$1$	$3$	$3$	$99.123$	3.3.661769.1	None	✓		✓		840.4.a.q	$1$	$0$	\(0\)	\(9\)	\(15\)	\(-21\)		$+$	$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-7q^{7}+9q^{9}+(-1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1680))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1680)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(840))\)\(^{\oplus 2}\)