Space of modular forms of level 2178, weight 4, and trivial character

• Label: 2178.4.a
• Level: $2178$
• Weight: $4$
• Character orbit: 2178.a
• Rep. character: $\chi_{2178}(1,\cdot)$
• Character field: $\Q$
• Dimension: $136$
• Newform subspaces: $59$
• Sturm bound: $1584$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 59 \)
Sturm bound:			\(1584\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2178))\).

	Total	New	Old
Modular forms	1236	136	1100
Cusp forms	1140	136	1004
Eisenstein series	96	0	96

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

\(2\)	\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(159\)	\(12\)	\(147\)		\(147\)	\(12\)	\(135\)		\(12\)	\(0\)	\(12\)
\(+\)	\(+\)	\(-\)	\(-\)		\(151\)	\(15\)	\(136\)		\(139\)	\(15\)	\(124\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(153\)	\(20\)	\(133\)		\(141\)	\(20\)	\(121\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(156\)	\(21\)	\(135\)		\(144\)	\(21\)	\(123\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(153\)	\(12\)	\(141\)		\(141\)	\(12\)	\(129\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(155\)	\(15\)	\(140\)		\(143\)	\(15\)	\(128\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(153\)	\(22\)	\(131\)		\(141\)	\(22\)	\(119\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(156\)	\(19\)	\(137\)		\(144\)	\(19\)	\(125\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(623\)	\(70\)	\(553\)		\(575\)	\(70\)	\(505\)		\(48\)	\(0\)	\(48\)
Minus space			\(-\)		\(613\)	\(66\)	\(547\)		\(565\)	\(66\)	\(499\)		\(48\)	\(0\)	\(48\)

Trace form

136 * q + 544 * q^4 + 26 * q^5 - 36 * q^7 + 32 * q^10 - 16 * q^13 + 2176 * q^16 - 60 * q^17 + 128 * q^19 + 104 * q^20 - 104 * q^23 + 3386 * q^25 + 44 * q^26 - 144 * q^28 - 168 * q^29 - 416 * q^31 - 312 * q^34 - 876 * q^35 + 398 * q^37 - 124 * q^38 + 128 * q^40 + 120 * q^41 + 532 * q^43 + 32 * q^46 + 1004 * q^47 + 6172 * q^49 - 480 * q^50 - 64 * q^52 + 1602 * q^53 - 164 * q^58 - 78 * q^59 - 1376 * q^61 - 528 * q^62 + 8704 * q^64 + 2472 * q^65 - 2250 * q^67 - 240 * q^68 + 1536 * q^70 - 1164 * q^71 + 2592 * q^73 - 1008 * q^74 + 512 * q^76 + 3372 * q^79 + 416 * q^80 - 488 * q^82 + 372 * q^83 + 3876 * q^85 - 284 * q^86 + 1156 * q^89 + 4760 * q^91 - 416 * q^92 + 2096 * q^94 - 5040 * q^95 + 5172 * q^97 + 2784 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\) 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	11
2178.4.a.a	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				22.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(-14\)	\(8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-14q^{5}+8q^{7}-8q^{8}+\cdots\)
2178.4.a.b	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				198.4.a.c	$1$	$1$	\(-2\)	\(0\)	\(-8\)	\(22\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-8q^{5}+22q^{7}-8q^{8}+\cdots\)
2178.4.a.c	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓	✓			2178.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(-6\)	\(-18\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-6q^{5}-18q^{7}-8q^{8}+\cdots\)
2178.4.a.d	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				726.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(-6\)	\(6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-6q^{5}+6q^{7}-8q^{8}+\cdots\)
2178.4.a.e	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				6.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(-6\)	\(16\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-6q^{5}+2^{4}q^{7}-8q^{8}+\cdots\)
2178.4.a.f	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				242.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(-3\)	\(8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-3q^{5}+8q^{7}-8q^{8}+\cdots\)
2178.4.a.g	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				66.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-14\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-14q^{7}-8q^{8}-80q^{13}+\cdots\)
2178.4.a.h	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				726.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-11\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-11q^{7}-8q^{8}+34q^{13}+\cdots\)
2178.4.a.i	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				726.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(-16\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-2^{4}q^{7}-8q^{8}+\cdots\)
2178.4.a.j	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓	✓			2178.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(6\)	\(18\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+6q^{5}+18q^{7}-8q^{8}+\cdots\)
2178.4.a.k	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				242.4.a.c	$1$	$1$	\(-2\)	\(0\)	\(15\)	\(-36\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+15q^{5}-6^{2}q^{7}-8q^{8}+\cdots\)
2178.4.a.l	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				22.4.a.a	$1$	$0$	\(-2\)	\(0\)	\(19\)	\(-14\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+19q^{5}-14q^{7}-8q^{8}+\cdots\)
2178.4.a.m	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				66.4.a.b	$1$	$1$	\(2\)	\(0\)	\(-10\)	\(-16\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-10q^{5}-2^{4}q^{7}+8q^{8}+\cdots\)
2178.4.a.n	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				726.4.a.d	$1$	$0$	\(2\)	\(0\)	\(-6\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-6q^{5}-6q^{7}+8q^{8}+\cdots\)
2178.4.a.o	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓	✓			2178.4.a.c	$1$	$1$	\(2\)	\(0\)	\(-6\)	\(18\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-6q^{5}+18q^{7}+8q^{8}+\cdots\)
2178.4.a.p	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				242.4.a.b	$1$	$1$	\(2\)	\(0\)	\(-3\)	\(-8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-3q^{5}-8q^{7}+8q^{8}+\cdots\)
2178.4.a.q	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				726.4.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(11\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+11q^{7}+8q^{8}-34q^{13}+\cdots\)
2178.4.a.r	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				22.4.a.c	$1$	$1$	\(2\)	\(0\)	\(3\)	\(10\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+3q^{5}+10q^{7}+8q^{8}+\cdots\)
2178.4.a.s	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				726.4.a.c	$1$	$1$	\(2\)	\(0\)	\(5\)	\(16\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+2^{4}q^{7}+8q^{8}+\cdots\)
2178.4.a.t	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓	✓			2178.4.a.c	$1$	$1$	\(2\)	\(0\)	\(6\)	\(-18\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+6q^{5}-18q^{7}+8q^{8}+\cdots\)
2178.4.a.u	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				198.4.a.c	$1$	$0$	\(2\)	\(0\)	\(8\)	\(22\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+8q^{5}+22q^{7}+8q^{8}+\cdots\)
2178.4.a.v	$2178$	$4$	2178.a	1.a	$1$	$1$	$1$	$128.506$	\(\Q\)	None	✓				242.4.a.c	$1$	$0$	\(2\)	\(0\)	\(15\)	\(36\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+15q^{5}+6^{2}q^{7}+8q^{8}+\cdots\)
2178.4.a.w	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{70}) \)	None	✓				198.4.a.i	$1$	$1$	\(-4\)	\(0\)	\(-8\)	\(-36\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-4+\beta )q^{5}+(-18+\cdots)q^{7}+\cdots\)
2178.4.a.x	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{273}) \)	None	✓				726.4.a.m	$1$	$1$	\(-4\)	\(0\)	\(-6\)	\(-6\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-3-\beta )q^{5}+(-3+\cdots)q^{7}+\cdots\)
2178.4.a.y	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{37}) \)	None	✓				242.4.a.j	$1$	$0$	\(-4\)	\(0\)	\(-4\)	\(42\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-2-3\beta )q^{5}+(21+\cdots)q^{7}+\cdots\)
2178.4.a.z	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{5}) \)	None	✓				22.4.c.a	$1$	$1$	\(-4\)	\(0\)	\(1\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+\beta q^{5}+(-3+11\beta )q^{7}+\cdots\)
2178.4.a.ba	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{5}) \)	None	✓				66.4.e.a	$1$	$1$	\(-4\)	\(0\)	\(5\)	\(-28\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-1+7\beta )q^{5}+(-23+\cdots)q^{7}+\cdots\)
2178.4.a.bb	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{1081}) \)	None	✓				726.4.a.k	$1$	$0$	\(-4\)	\(0\)	\(5\)	\(27\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(3-\beta )q^{5}+(14-\beta )q^{7}+\cdots\)
2178.4.a.bc	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{3}) \)	None	✓				726.4.a.j	$1$	$1$	\(-4\)	\(0\)	\(6\)	\(-12\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(3+4\beta )q^{5}-6q^{7}+\cdots\)
2178.4.a.bd	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{3}) \)	None	✓				242.4.a.i	$1$	$1$	\(-4\)	\(0\)	\(12\)	\(-6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(6+7\beta )q^{5}+(-3+11\beta )q^{7}+\cdots\)
2178.4.a.be	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{5}) \)	None	✓				66.4.e.b	$1$	$0$	\(-4\)	\(0\)	\(25\)	\(-8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(13-\beta )q^{5}+(-1-6\beta )q^{7}+\cdots\)
2178.4.a.bf	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{97}) \)	None	✓				66.4.a.c	$1$	$1$	\(4\)	\(0\)	\(-10\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-5-\beta )q^{5}+(1+3\beta )q^{7}+\cdots\)
2178.4.a.bg	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{273}) \)	None	✓				726.4.a.m	$1$	$0$	\(4\)	\(0\)	\(-6\)	\(6\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-3-\beta )q^{5}+(3+\beta )q^{7}+\cdots\)
2178.4.a.bh	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{37}) \)	None	✓				242.4.a.j	$1$	$1$	\(4\)	\(0\)	\(-4\)	\(-42\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-2-3\beta )q^{5}+(-21+\cdots)q^{7}+\cdots\)
2178.4.a.bi	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{5}) \)	None	✓				22.4.c.a	$1$	$1$	\(4\)	\(0\)	\(1\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+\beta q^{5}+(3-11\beta )q^{7}+\cdots\)
2178.4.a.bj	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{1081}) \)	None	✓				726.4.a.k	$1$	$1$	\(4\)	\(0\)	\(5\)	\(-27\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(3-\beta )q^{5}+(-14+\beta )q^{7}+\cdots\)
2178.4.a.bk	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{5}) \)	None	✓				66.4.e.a	$1$	$1$	\(4\)	\(0\)	\(5\)	\(28\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-1+7\beta )q^{5}+(23+\cdots)q^{7}+\cdots\)
2178.4.a.bl	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{3}) \)	None	✓				726.4.a.j	$1$	$0$	\(4\)	\(0\)	\(6\)	\(12\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(3+4\beta )q^{5}+6q^{7}+\cdots\)
2178.4.a.bm	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{70}) \)	None	✓				198.4.a.i	$1$	$0$	\(4\)	\(0\)	\(8\)	\(-36\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(4+\beta )q^{5}+(-18-\beta )q^{7}+\cdots\)
2178.4.a.bn	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{3}) \)	None	✓				242.4.a.i	$1$	$0$	\(4\)	\(0\)	\(12\)	\(6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(6+7\beta )q^{5}+(3-11\beta )q^{7}+\cdots\)
2178.4.a.bo	$2178$	$4$	2178.a	1.a	$1$	$2$	$2$	$128.506$	\(\Q(\sqrt{5}) \)	None	✓				66.4.e.b	$1$	$0$	\(4\)	\(0\)	\(25\)	\(8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(13-\beta )q^{5}+(1+6\beta )q^{7}+\cdots\)
2178.4.a.bp	$2178$	$4$	2178.a	1.a	$1$	$3$	$3$	$128.506$	3.3.2117020.1	None	✓	✓			2178.4.a.bp	$1$	$1$	\(-6\)	\(0\)	\(-8\)	\(11\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-3+\beta _{1})q^{5}+(4-\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.bq	$2178$	$4$	2178.a	1.a	$1$	$3$	$3$	$128.506$	3.3.2117020.1	None	✓	✓			2178.4.a.bp	$1$	$1$	\(-6\)	\(0\)	\(8\)	\(-11\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(3-\beta _{1})q^{5}+(-4+\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.br	$2178$	$4$	2178.a	1.a	$1$	$3$	$3$	$128.506$	3.3.2117020.1	None	✓	✓			2178.4.a.bp	$1$	$0$	\(6\)	\(0\)	\(-8\)	\(-11\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-3+\beta _{1})q^{5}+(-4+\cdots)q^{7}+\cdots\)
2178.4.a.bs	$2178$	$4$	2178.a	1.a	$1$	$3$	$3$	$128.506$	3.3.2117020.1	None	✓	✓			2178.4.a.bp	$1$	$0$	\(6\)	\(0\)	\(8\)	\(11\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(3-\beta _{1})q^{5}+(4-\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.bt	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{201 +16 \sqrt{5}})\)	None	✓				22.4.c.b	$1$	$0$	\(-8\)	\(0\)	\(-25\)	\(3\)		$+$	$-$	$-$	$+$	$11$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-6+\beta _{2})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.bu	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{713 +48 \sqrt{5}})\)	None	✓				66.4.e.c	$1$	$1$	\(-8\)	\(0\)	\(-20\)	\(21\)		$+$	$-$	$+$	$-$	$11$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-5+\beta _{2})q^{5}+(5-\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.bv	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{3}, \sqrt{67})\)	None	✓	✓			2178.4.a.bv	$2$	$0$	\(-8\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-2\beta _{1}-\beta _{2})q^{5}+(3\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.bw	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{422 +66 \sqrt{5}})\)	None	✓				66.4.e.d	$1$	$0$	\(-8\)	\(0\)	\(6\)	\(-1\)		$+$	$-$	$-$	$+$	$5\cdot 11$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(2+\beta _{1}-\beta _{3})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.bx	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{243 +88 \sqrt{3}})\)	None	✓				726.4.a.v	$1$	$1$	\(-8\)	\(0\)	\(6\)	\(12\)		$+$	$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(2-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
2178.4.a.by	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{201 +16 \sqrt{5}})\)	None	✓				22.4.c.b	$1$	$0$	\(8\)	\(0\)	\(-25\)	\(-3\)		$-$	$-$	$+$	$+$	$11$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-6+\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
2178.4.a.bz	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{713 +48 \sqrt{5}})\)	None	✓		✓		66.4.e.c	$1$	$1$	\(8\)	\(0\)	\(-20\)	\(-21\)		$-$	$-$	$-$	$-$	$11$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-5+\beta _{2})q^{5}+(-5+\cdots)q^{7}+\cdots\)
2178.4.a.ca	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{3}, \sqrt{67})\)	None	✓	✓			2178.4.a.bv	$2$	$1$	\(8\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-2\beta _{1}-\beta _{2})q^{5}+(-3\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.cb	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{243 +88 \sqrt{3}})\)	None	✓				726.4.a.v	$1$	$0$	\(8\)	\(0\)	\(6\)	\(-12\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(2-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
2178.4.a.cc	$2178$	$4$	2178.a	1.a	$1$	$4$	$4$	$128.506$	\(\Q(\sqrt{422 +66 \sqrt{5}})\)	None	✓				66.4.e.d	$1$	$0$	\(8\)	\(0\)	\(6\)	\(1\)		$-$	$-$	$+$	$+$	$5\cdot 11$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(2+\beta _{1}-\beta _{3})q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
2178.4.a.cd	$2178$	$4$	2178.a	1.a	$1$	$6$	$6$	$128.506$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		198.4.f.g	$1$	$1$	\(-12\)	\(0\)	\(-17\)	\(7\)		$+$	$+$	$-$	$-$	$11$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-3+\beta _{3})q^{5}+(1-\beta _{3}+\cdots)q^{7}+\cdots\)
2178.4.a.ce	$2178$	$4$	2178.a	1.a	$1$	$6$	$6$	$128.506$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		198.4.f.g	$1$	$0$	\(-12\)	\(0\)	\(17\)	\(-7\)		$+$	$+$	$+$	$+$	$11$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(3-\beta _{3})q^{5}+(-1+\beta _{3}+\cdots)q^{7}+\cdots\)
2178.4.a.cf	$2178$	$4$	2178.a	1.a	$1$	$6$	$6$	$128.506$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		198.4.f.g	$1$	$1$	\(12\)	\(0\)	\(-17\)	\(-7\)		$-$	$+$	$+$	$-$	$11$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(-3+\beta _{3})q^{5}+(-1+\cdots)q^{7}+\cdots\)
2178.4.a.cg	$2178$	$4$	2178.a	1.a	$1$	$6$	$6$	$128.506$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		198.4.f.g	$1$	$0$	\(12\)	\(0\)	\(17\)	\(7\)		$-$	$+$	$-$	$+$	$11$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(3-\beta _{3})q^{5}+(1-\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2178))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2178)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(726))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\)\(^{\oplus 2}\)