Space of modular forms of level 2070, weight 4, and trivial character

• Label: 2070.4.a
• Level: $2070$
• Weight: $4$
• Character orbit: 2070.a
• Rep. character: $\chi_{2070}(1,\cdot)$
• Character field: $\Q$
• Dimension: $110$
• Newform subspaces: $42$
• Sturm bound: $1728$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 42 \)
Sturm bound:			\(1728\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2070))\).

	Total	New	Old
Modular forms	1312	110	1202
Cusp forms	1280	110	1170
Eisenstein series	32	0	32

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

\(2\)	\(3\)	\(5\)	\(23\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(88\)	\(5\)	\(83\)		\(86\)	\(5\)	\(81\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(78\)	\(6\)	\(72\)		\(76\)	\(6\)	\(70\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(78\)	\(6\)	\(72\)		\(76\)	\(6\)	\(70\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(84\)	\(5\)	\(79\)		\(82\)	\(5\)	\(77\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(79\)	\(7\)	\(72\)		\(77\)	\(7\)	\(70\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(84\)	\(9\)	\(75\)		\(82\)	\(9\)	\(73\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(83\)	\(9\)	\(74\)		\(81\)	\(9\)	\(72\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(82\)	\(7\)	\(75\)		\(80\)	\(7\)	\(73\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(80\)	\(5\)	\(75\)		\(78\)	\(5\)	\(73\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(84\)	\(6\)	\(78\)		\(82\)	\(6\)	\(76\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(82\)	\(6\)	\(76\)		\(80\)	\(6\)	\(74\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(82\)	\(5\)	\(77\)		\(80\)	\(5\)	\(75\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(82\)	\(10\)	\(72\)		\(80\)	\(10\)	\(70\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(83\)	\(7\)	\(76\)		\(81\)	\(7\)	\(74\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(84\)	\(7\)	\(77\)		\(82\)	\(7\)	\(75\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(79\)	\(10\)	\(69\)		\(77\)	\(10\)	\(67\)		\(2\)	\(0\)	\(2\)
Plus space				\(+\)		\(666\)	\(60\)	\(606\)		\(650\)	\(60\)	\(590\)		\(16\)	\(0\)	\(16\)
Minus space				\(-\)		\(646\)	\(50\)	\(596\)		\(630\)	\(50\)	\(580\)		\(16\)	\(0\)	\(16\)

Trace form

110 * q + 4 * q^2 + 440 * q^4 - 104 * q^7 + 16 * q^8 - 84 * q^11 - 60 * q^13 - 80 * q^14 + 1760 * q^16 - 84 * q^17 - 92 * q^19 + 40 * q^22 + 2750 * q^25 - 416 * q^28 - 248 * q^29 + 24 * q^31 + 64 * q^32 + 360 * q^34 + 100 * q^35 + 1144 * q^37 + 72 * q^38 - 268 * q^41 + 428 * q^43 - 336 * q^44 - 912 * q^47 + 6206 * q^49 + 100 * q^50 - 240 * q^52 - 872 * q^53 - 320 * q^56 - 232 * q^58 - 260 * q^59 - 3408 * q^61 + 1696 * q^62 + 7040 * q^64 - 1280 * q^65 - 1220 * q^67 - 336 * q^68 - 280 * q^70 + 792 * q^71 - 4300 * q^73 - 1664 * q^74 - 368 * q^76 + 752 * q^77 - 3088 * q^79 + 1752 * q^82 - 428 * q^83 - 1220 * q^85 - 584 * q^86 + 160 * q^88 + 1596 * q^89 + 1816 * q^91 + 3696 * q^94 + 40 * q^95 + 5612 * q^97 - 220 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2070))\) 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	23
2070.4.a.a	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				230.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(-5\)	\(-32\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}-2^{5}q^{7}-8q^{8}+\cdots\)
2070.4.a.b	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				690.4.a.h	$1$	$0$	\(-2\)	\(0\)	\(-5\)	\(-7\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}-7q^{7}-8q^{8}+\cdots\)
2070.4.a.c	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓	✓			2070.4.a.c	$1$	$1$	\(-2\)	\(0\)	\(-5\)	\(2\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+2q^{7}-8q^{8}+\cdots\)
2070.4.a.d	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				690.4.a.f	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(-20\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-20q^{7}-8q^{8}+\cdots\)
2070.4.a.e	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				230.4.a.e	$1$	$1$	\(-2\)	\(0\)	\(5\)	\(-18\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-18q^{7}-8q^{8}+\cdots\)
2070.4.a.f	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				690.4.a.g	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(-5\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-5q^{7}-8q^{8}+\cdots\)
2070.4.a.g	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				690.4.a.a	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(-19\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-19q^{7}+8q^{8}+\cdots\)
2070.4.a.h	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				690.4.a.b	$1$	$0$	\(2\)	\(0\)	\(-5\)	\(-18\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-18q^{7}+8q^{8}+\cdots\)
2070.4.a.i	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				690.4.a.c	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(16\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+2^{4}q^{7}+8q^{8}+\cdots\)
2070.4.a.j	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				230.4.a.c	$1$	$0$	\(2\)	\(0\)	\(-5\)	\(20\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+20q^{7}+8q^{8}+\cdots\)
2070.4.a.k	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				690.4.a.d	$1$	$1$	\(2\)	\(0\)	\(5\)	\(-16\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}-2^{4}q^{7}+8q^{8}+\cdots\)
2070.4.a.l	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				690.4.a.e	$1$	$1$	\(2\)	\(0\)	\(5\)	\(-5\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}-5q^{7}+8q^{8}+\cdots\)
2070.4.a.m	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓	✓			2070.4.a.c	$1$	$0$	\(2\)	\(0\)	\(5\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+2q^{7}+8q^{8}+\cdots\)
2070.4.a.n	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				230.4.a.b	$1$	$1$	\(2\)	\(0\)	\(5\)	\(3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+3q^{7}+8q^{8}+\cdots\)
2070.4.a.o	$2070$	$4$	2070.a	1.a	$1$	$1$	$1$	$122.134$	\(\Q\)	None	✓				230.4.a.a	$1$	$1$	\(2\)	\(0\)	\(5\)	\(12\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+12q^{7}+8q^{8}+\cdots\)
2070.4.a.p	$2070$	$4$	2070.a	1.a	$1$	$2$	$2$	$122.134$	\(\Q(\sqrt{41}) \)	None	✓	✓			2070.4.a.p	$1$	$0$	\(-4\)	\(0\)	\(-10\)	\(-26\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(-13-\beta )q^{7}+\cdots\)
2070.4.a.q	$2070$	$4$	2070.a	1.a	$1$	$2$	$2$	$122.134$	\(\Q(\sqrt{14}) \)	None	✓				690.4.a.j	$1$	$1$	\(-4\)	\(0\)	\(-10\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(-1+4\beta )q^{7}+\cdots\)
2070.4.a.r	$2070$	$4$	2070.a	1.a	$1$	$2$	$2$	$122.134$	\(\Q(\sqrt{6}) \)	None	✓				690.4.a.i	$1$	$0$	\(4\)	\(0\)	\(-10\)	\(-26\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(-13+2\beta )q^{7}+\cdots\)
2070.4.a.s	$2070$	$4$	2070.a	1.a	$1$	$2$	$2$	$122.134$	\(\Q(\sqrt{73}) \)	None	✓				230.4.a.f	$1$	$1$	\(4\)	\(0\)	\(-10\)	\(-17\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(-8-\beta )q^{7}+\cdots\)
2070.4.a.t	$2070$	$4$	2070.a	1.a	$1$	$2$	$2$	$122.134$	\(\Q(\sqrt{41}) \)	None	✓	✓			2070.4.a.p	$1$	$1$	\(4\)	\(0\)	\(10\)	\(-26\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(-13-\beta )q^{7}+\cdots\)
2070.4.a.u	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.19544.1	None	✓	✓	✓		2070.4.a.u	$1$	$0$	\(-6\)	\(0\)	\(-15\)	\(-16\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(-6+3\beta _{1}+\cdots)q^{7}+\cdots\)
2070.4.a.v	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				690.4.a.r	$1$	$1$	\(-6\)	\(0\)	\(15\)	\(-2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+(-1+\beta _{1}-\beta _{2})q^{7}+\cdots\)
2070.4.a.w	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.162793.1	None	✓				690.4.a.p	$1$	$0$	\(-6\)	\(0\)	\(15\)	\(3\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+(1-\beta _{1}+\beta _{2})q^{7}+\cdots\)
2070.4.a.x	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.396732.1	None	✓				690.4.a.q	$1$	$1$	\(-6\)	\(0\)	\(15\)	\(12\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+(4-\beta _{1}+\beta _{2})q^{7}+\cdots\)
2070.4.a.y	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.460593.1	None	✓		✓		690.4.a.o	$1$	$1$	\(6\)	\(0\)	\(-15\)	\(-3\)		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
2070.4.a.z	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.931848.1	None	✓				690.4.a.m	$1$	$0$	\(6\)	\(0\)	\(-15\)	\(6\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(1+2\beta _{1}-\beta _{2})q^{7}+\cdots\)
2070.4.a.ba	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.318165.1	None	✓				230.4.a.g	$1$	$0$	\(6\)	\(0\)	\(-15\)	\(7\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(1+3\beta _{1}-\beta _{2})q^{7}+\cdots\)
2070.4.a.bb	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.471057.3	None	✓		✓		690.4.a.k	$1$	$1$	\(6\)	\(0\)	\(15\)	\(-35\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(-12-\beta _{1}+\cdots)q^{7}+\cdots\)
2070.4.a.bc	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.19544.1	None	✓	✓	✓		2070.4.a.u	$1$	$1$	\(6\)	\(0\)	\(15\)	\(-16\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(-6+3\beta _{1}+\cdots)q^{7}+\cdots\)
2070.4.a.bd	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.207308.1	None	✓				690.4.a.n	$1$	$0$	\(6\)	\(0\)	\(15\)	\(2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(1-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
2070.4.a.be	$2070$	$4$	2070.a	1.a	$1$	$3$	$3$	$122.134$	3.3.617756.1	None	✓				690.4.a.l	$1$	$0$	\(6\)	\(0\)	\(15\)	\(30\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(10+\beta _{2})q^{7}+\cdots\)
2070.4.a.bf	$2070$	$4$	2070.a	1.a	$1$	$4$	$4$	$122.134$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				690.4.a.s	$1$	$0$	\(-8\)	\(0\)	\(-20\)	\(7\)		$+$	$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(2-\beta _{1})q^{7}+\cdots\)
2070.4.a.bg	$2070$	$4$	2070.a	1.a	$1$	$4$	$4$	$122.134$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				230.4.a.j	$1$	$0$	\(-8\)	\(0\)	\(-20\)	\(8\)		$+$	$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(1+\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
2070.4.a.bh	$2070$	$4$	2070.a	1.a	$1$	$4$	$4$	$122.134$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		690.4.a.t	$1$	$1$	\(-8\)	\(0\)	\(-20\)	\(26\)		$+$	$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(6-\beta _{1})q^{7}+\cdots\)
2070.4.a.bi	$2070$	$4$	2070.a	1.a	$1$	$4$	$4$	$122.134$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		230.4.a.i	$1$	$0$	\(-8\)	\(0\)	\(20\)	\(26\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+(6-\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
2070.4.a.bj	$2070$	$4$	2070.a	1.a	$1$	$4$	$4$	$122.134$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		230.4.a.h	$1$	$0$	\(8\)	\(0\)	\(20\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(2\beta _{1}+\beta _{2})q^{7}+\cdots\)
2070.4.a.bk	$2070$	$4$	2070.a	1.a	$1$	$5$	$5$	$122.134$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		2070.4.a.bk	$1$	$1$	\(-10\)	\(0\)	\(-25\)	\(12\)		$+$	$+$	$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(2-\beta _{2})q^{7}+\cdots\)
2070.4.a.bl	$2070$	$4$	2070.a	1.a	$1$	$5$	$5$	$122.134$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	2070.4.a.bl	$1$	$0$	\(-10\)	\(0\)	\(25\)	\(0\)		$+$	$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+(-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
2070.4.a.bm	$2070$	$4$	2070.a	1.a	$1$	$5$	$5$	$122.134$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	2070.4.a.bl	$1$	$1$	\(10\)	\(0\)	\(-25\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
2070.4.a.bn	$2070$	$4$	2070.a	1.a	$1$	$5$	$5$	$122.134$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		2070.4.a.bk	$1$	$0$	\(10\)	\(0\)	\(25\)	\(12\)		$-$	$+$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(2-\beta _{2})q^{7}+\cdots\)
2070.4.a.bo	$2070$	$4$	2070.a	1.a	$1$	$6$	$6$	$122.134$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	2070.4.a.bo	$1$	$1$	\(-12\)	\(0\)	\(30\)	\(0\)		$+$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-\beta _{1}q^{7}-8q^{8}+\cdots\)
2070.4.a.bp	$2070$	$4$	2070.a	1.a	$1$	$6$	$6$	$122.134$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	2070.4.a.bo	$1$	$0$	\(12\)	\(0\)	\(-30\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-\beta _{1}q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2070))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2070)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(414))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(1035))\)\(^{\oplus 2}\)