Space of modular forms of level 1575, weight 4, and trivial character

• Label: 1575.4.a
• Level: $1575$
• Weight: $4$
• Character orbit: 1575.a
• Rep. character: $\chi_{1575}(1,\cdot)$
• Character field: $\Q$
• Dimension: $143$
• Newform subspaces: $47$
• Sturm bound: $960$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 47 \)
Sturm bound:			\(960\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1575))\).

	Total	New	Old
Modular forms	744	143	601
Cusp forms	696	143	553
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.

\(3\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(96\)	\(16\)	\(80\)		\(90\)	\(16\)	\(74\)		\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)		\(90\)	\(10\)	\(80\)		\(84\)	\(10\)	\(74\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(90\)	\(14\)	\(76\)		\(84\)	\(14\)	\(70\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(96\)	\(18\)	\(78\)		\(90\)	\(18\)	\(72\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(90\)	\(19\)	\(71\)		\(84\)	\(19\)	\(65\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(96\)	\(22\)	\(74\)		\(90\)	\(22\)	\(68\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(96\)	\(23\)	\(73\)		\(90\)	\(23\)	\(67\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(90\)	\(21\)	\(69\)		\(84\)	\(21\)	\(63\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(384\)	\(79\)	\(305\)		\(360\)	\(79\)	\(281\)		\(24\)	\(0\)	\(24\)
Minus space			\(-\)		\(360\)	\(64\)	\(296\)		\(336\)	\(64\)	\(272\)		\(24\)	\(0\)	\(24\)

Trace form

143 * q - q^2 + 569 * q^4 - 7 * q^7 + 3 * q^8 - 108 * q^11 - 128 * q^13 + 7 * q^14 + 2469 * q^16 - 86 * q^17 - 110 * q^19 + 256 * q^22 + 412 * q^23 - 1036 * q^26 - 35 * q^28 - 410 * q^29 + 324 * q^31 - 597 * q^32 + 362 * q^34 + 86 * q^37 + 722 * q^38 - 274 * q^41 + 240 * q^43 - 870 * q^44 + 1406 * q^46 + 604 * q^47 + 7007 * q^49 - 836 * q^52 + 394 * q^53 - 147 * q^56 - 658 * q^58 + 562 * q^59 + 296 * q^61 - 3516 * q^62 + 9031 * q^64 + 156 * q^67 - 490 * q^68 + 2248 * q^71 - 1026 * q^73 + 6316 * q^74 - 738 * q^76 + 896 * q^77 + 4260 * q^79 + 6650 * q^82 + 422 * q^83 + 4222 * q^86 + 4204 * q^88 + 1530 * q^89 + 196 * q^91 + 1368 * q^92 + 10024 * q^94 - 1270 * q^97 - 49 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	7
1575.4.a.a	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				525.4.a.c	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6q^{11}+\cdots\)
1575.4.a.b	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				21.4.a.a	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+6^{2}q^{11}+\cdots\)
1575.4.a.c	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				315.4.a.b	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}-7q^{7}+21q^{8}+60q^{11}+\cdots\)
1575.4.a.d	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				525.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-4q^{4}+7q^{7}+24q^{8}+21q^{11}+\cdots\)
1575.4.a.e	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				7.4.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+7q^{7}+15q^{8}+8q^{11}+\cdots\)
1575.4.a.f	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				105.4.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}-7q^{7}-42q^{11}-20q^{13}+\cdots\)
1575.4.a.g	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				35.4.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}-7q^{7}-15q^{8}-12q^{11}+\cdots\)
1575.4.a.h	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				525.4.a.d	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{4}-7q^{7}-24q^{8}+21q^{11}+\cdots\)
1575.4.a.i	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				315.4.a.b	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-7q^{7}-21q^{8}-60q^{11}+\cdots\)
1575.4.a.j	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				525.4.a.c	$1$	$1$	\(3\)	\(0\)	\(0\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+7q^{7}-21q^{8}+6q^{11}+\cdots\)
1575.4.a.k	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				21.4.a.b	$1$	$0$	\(4\)	\(0\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+8q^{4}+7q^{7}-62q^{11}+62q^{13}+\cdots\)
1575.4.a.l	$1575$	$4$	1575.a	1.a	$1$	$1$	$1$	$92.928$	\(\Q\)	None	✓				105.4.a.b	$1$	$1$	\(5\)	\(0\)	\(0\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+17q^{4}-7q^{7}+45q^{8}-12q^{11}+\cdots\)
1575.4.a.m	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	✓				105.4.a.c	$1$	$0$	\(-7\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+(5+7\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.n	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{5}) \)	None	✓				105.4.a.d	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}+(1+4\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.o	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	✓				525.4.a.j	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(-3+3\beta )q^{4}-7q^{7}+\cdots\)
1575.4.a.p	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{57}) \)	None	✓				21.4.a.c	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(7+3\beta )q^{4}-7q^{7}+\cdots\)
1575.4.a.q	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{2}) \)	None	✓				105.4.a.e	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.r	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	✓				315.4.a.h	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(-4+\cdots)q^{8}+\cdots\)
1575.4.a.s	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{41}) \)	None	✓				175.4.a.d	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{4}+7q^{7}+(-10+5\beta )q^{8}+\cdots\)
1575.4.a.t	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{19}) \)	None	✓				63.4.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+11q^{4}+7q^{7}+3\beta q^{8}+10\beta q^{11}+\cdots\)
1575.4.a.u	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	✓				315.4.a.h	$1$	$1$	\(1\)	\(0\)	\(0\)	\(14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-4+\beta )q^{4}+7q^{7}+(4-11\beta )q^{8}+\cdots\)
1575.4.a.v	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{41}) \)	None	✓				175.4.a.d	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2+\beta )q^{4}-7q^{7}+(10-5\beta )q^{8}+\cdots\)
1575.4.a.w	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{65}) \)	None	✓				105.4.a.f	$1$	$0$	\(1\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(8+\beta )q^{4}+7q^{7}+(2^{4}+\beta )q^{8}+\cdots\)
1575.4.a.x	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{17}) \)	None	✓				525.4.a.j	$1$	$1$	\(3\)	\(0\)	\(0\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+7q^{7}+\cdots\)
1575.4.a.y	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{41}) \)	None	✓				105.4.a.g	$1$	$1$	\(3\)	\(0\)	\(0\)	\(-14\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(3+3\beta )q^{4}-7q^{7}+(5^{2}+\cdots)q^{8}+\cdots\)
1575.4.a.z	$1575$	$4$	1575.a	1.a	$1$	$2$	$2$	$92.928$	\(\Q(\sqrt{2}) \)	None	✓				35.4.a.b	$1$	$0$	\(8\)	\(0\)	\(0\)	\(14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{2}+(10+8\beta )q^{4}+7q^{7}+(24+\cdots)q^{8}+\cdots\)
1575.4.a.ba	$1575$	$4$	1575.a	1.a	$1$	$3$	$3$	$92.928$	3.3.14360.1	None	✓				35.4.a.c	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-21\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bb	$1575$	$4$	1575.a	1.a	$1$	$3$	$3$	$92.928$	3.3.22952.1	None	✓				315.4.a.n	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-21\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bc	$1575$	$4$	1575.a	1.a	$1$	$3$	$3$	$92.928$	3.3.2292.1	None	✓				105.4.d.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-21\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots\)
1575.4.a.bd	$1575$	$4$	1575.a	1.a	$1$	$3$	$3$	$92.928$	3.3.2292.1	None	✓				105.4.d.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(21\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots\)
1575.4.a.be	$1575$	$4$	1575.a	1.a	$1$	$3$	$3$	$92.928$	3.3.22952.1	None	✓				315.4.a.n	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-21\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(5-2\beta _{1}+\beta _{2})q^{4}-7q^{7}+\cdots\)
1575.4.a.bf	$1575$	$4$	1575.a	1.a	$1$	$4$	$4$	$92.928$	\(\Q(\sqrt{39 +2 \sqrt{185}})\)	None	✓				525.4.a.s	$1$	$0$	\(-6\)	\(0\)	\(0\)	\(-28\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
1575.4.a.bg	$1575$	$4$	1575.a	1.a	$1$	$4$	$4$	$92.928$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				175.4.a.g	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(28\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1575.4.a.bh	$1575$	$4$	1575.a	1.a	$1$	$4$	$4$	$92.928$	\(\Q(\sqrt{42 +2 \sqrt{329}})\)	None	✓	✓			1575.4.a.bh	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-28\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}-7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bi	$1575$	$4$	1575.a	1.a	$1$	$4$	$4$	$92.928$	\(\Q(\sqrt{42 +2 \sqrt{329}})\)	None	✓	✓	✓		1575.4.a.bh	$2$	$1$	\(0\)	\(0\)	\(0\)	\(28\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{3})q^{4}+7q^{7}+(3\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bj	$1575$	$4$	1575.a	1.a	$1$	$4$	$4$	$92.928$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				525.4.a.t	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-28\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}-7q^{7}+(2+4\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bk	$1575$	$4$	1575.a	1.a	$1$	$4$	$4$	$92.928$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				525.4.a.t	$1$	$0$	\(0\)	\(0\)	\(0\)	\(28\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+7q^{7}+(-2+\cdots)q^{8}+\cdots\)
1575.4.a.bl	$1575$	$4$	1575.a	1.a	$1$	$4$	$4$	$92.928$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		175.4.a.g	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-28\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(9+\beta _{2}-\beta _{3})q^{4}-7q^{7}+\cdots\)
1575.4.a.bm	$1575$	$4$	1575.a	1.a	$1$	$4$	$4$	$92.928$	\(\Q(\sqrt{39 +2 \sqrt{185}})\)	None	✓				525.4.a.s	$1$	$0$	\(6\)	\(0\)	\(0\)	\(28\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(3+3\beta _{1}+\beta _{2})q^{4}+7q^{7}+\cdots\)
1575.4.a.bn	$1575$	$4$	1575.a	1.a	$1$	$5$	$5$	$92.928$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				35.4.b.a	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(35\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
1575.4.a.bo	$1575$	$4$	1575.a	1.a	$1$	$5$	$5$	$92.928$	5.5.78066700.1	None	✓				105.4.d.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(35\)		$-$	$-$	$-$	$-$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}+7q^{7}+\cdots\)
1575.4.a.bp	$1575$	$4$	1575.a	1.a	$1$	$5$	$5$	$92.928$	5.5.78066700.1	None	✓				105.4.d.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-35\)		$-$	$-$	$+$	$+$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(5-\beta _{1}-\beta _{3})q^{4}-7q^{7}+\cdots\)
1575.4.a.bq	$1575$	$4$	1575.a	1.a	$1$	$5$	$5$	$92.928$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				35.4.b.a	$1$	$0$	\(4\)	\(0\)	\(0\)	\(-35\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1575.4.a.br	$1575$	$4$	1575.a	1.a	$1$	$8$	$8$	$92.928$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		1575.4.a.br	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-56\)		$+$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bs	$1575$	$4$	1575.a	1.a	$1$	$8$	$8$	$92.928$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			1575.4.a.br	$2$	$0$	\(0\)	\(0\)	\(0\)	\(56\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bt	$1575$	$4$	1575.a	1.a	$1$	$10$	$10$	$92.928$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		315.4.d.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-70\)		$+$	$-$	$+$	$-$	$2^{12}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}-7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)
1575.4.a.bu	$1575$	$4$	1575.a	1.a	$1$	$10$	$10$	$92.928$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		315.4.d.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(70\)		$+$	$-$	$-$	$+$	$2^{12}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+7q^{7}+(6\beta _{1}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1575))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1575)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)