Space of modular forms of level 2475, weight 4, and trivial character

• Label: 2475.4.a
• Level: $2475$
• Weight: $4$
• Character orbit: 2475.a
• Rep. character: $\chi_{2475}(1,\cdot)$
• Character field: $\Q$
• Dimension: $238$
• Newform subspaces: $53$
• Sturm bound: $1440$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 53 \)
Sturm bound:			\(1440\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(2475))\).

	Total	New	Old
Modular forms	1104	238	866
Cusp forms	1056	238	818
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.

\(3\)	\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(22\)
\(+\)	\(+\)	\(-\)	$-$	\(22\)
\(+\)	\(-\)	\(+\)	$-$	\(26\)
\(+\)	\(-\)	\(-\)	$+$	\(26\)
\(-\)	\(+\)	\(+\)	$-$	\(33\)
\(-\)	\(+\)	\(-\)	$+$	\(35\)
\(-\)	\(-\)	\(+\)	$+$	\(39\)
\(-\)	\(-\)	\(-\)	$-$	\(35\)
Plus space			\(+\)	\(122\)
Minus space			\(-\)	\(116\)

Trace form

\( 238 q - 2 q^{2} + 944 q^{4} - 44 q^{7} + 12 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	11
2475.4.a.a	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}+3q^{7}-45q^{8}+11q^{11}+\cdots\)
2475.4.a.b	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$0$	\(-5\)	\(0\)	\(0\)	\(32\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}+2^{5}q^{7}-45q^{8}+11q^{11}+\cdots\)
2475.4.a.c	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(-21\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{4}-21q^{7}-11q^{11}+68q^{13}+\cdots\)
2475.4.a.d	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+7q^{7}+21q^{8}-11q^{11}+\cdots\)
2475.4.a.e	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(26\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-7q^{4}+26q^{7}+15q^{8}-11q^{11}+\cdots\)
2475.4.a.f	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}-2q^{7}+11q^{11}+22q^{13}+\cdots\)
2475.4.a.g	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-36\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}-6^{2}q^{7}-15q^{8}-11q^{11}+\cdots\)
2475.4.a.h	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(9\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}+9q^{7}-15q^{8}-11q^{11}+\cdots\)
2475.4.a.i	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(0\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-7q^{7}-21q^{8}-11q^{11}+\cdots\)
2475.4.a.j	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(0\)	\(0\)	\(21\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+8q^{4}+21q^{7}-11q^{11}-68q^{13}+\cdots\)
2475.4.a.k	$2475$	$4$	2475.a	1.a	$1$	$1$	$1$	$146.030$	\(\Q\)	None	✓	$1$	$0$	\(5\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+17q^{4}-3q^{7}+45q^{8}+11q^{11}+\cdots\)
2475.4.a.l	$2475$	$4$	2475.a	1.a	$1$	$2$	$2$	$146.030$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-7\)	\(0\)	\(0\)	\(25\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}+(5+7\beta )q^{4}+(17-9\beta )q^{7}+\cdots\)
2475.4.a.m	$2475$	$4$	2475.a	1.a	$1$	$2$	$2$	$146.030$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(16\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(5-2\beta )q^{4}+(8+5\beta )q^{7}+\cdots\)
2475.4.a.n	$2475$	$4$	2475.a	1.a	$1$	$2$	$2$	$146.030$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-4+\beta )q^{4}+(4-4\beta )q^{7}+\cdots\)
2475.4.a.o	$2475$	$4$	2475.a	1.a	$1$	$2$	$2$	$146.030$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+\beta q^{4}+(-2+2\beta )q^{7}+(8-7\beta )q^{8}+\cdots\)
2475.4.a.p	$2475$	$4$	2475.a	1.a	$1$	$2$	$2$	$146.030$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-24\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2^{4}+\beta )q^{4}+(-14+4\beta )q^{7}+\cdots\)
2475.4.a.q	$2475$	$4$	2475.a	1.a	$1$	$2$	$2$	$146.030$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-20\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-4+2\beta )q^{4}+(-10+\cdots)q^{7}+\cdots\)
2475.4.a.r	$2475$	$4$	2475.a	1.a	$1$	$2$	$2$	$146.030$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(16\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(5+2\beta )q^{4}+(8-5\beta )q^{7}+\cdots\)
2475.4.a.s	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.23612.1	None	✓	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(7+\beta _{1}+\beta _{2})q^{4}+\cdots\)
2475.4.a.t	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.47528.1	None	✓	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-10\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(10+\beta _{2})q^{4}+(-4+\cdots)q^{7}+\cdots\)
2475.4.a.u	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.1772.1	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+2\beta _{2})q^{4}+(-2+4\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.v	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.3368.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(16\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(3+2\beta _{2})q^{4}+(7-4\beta _{1}+\beta _{2})q^{7}+\cdots\)
2475.4.a.w	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.788.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(16\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(-2-\beta _{1})q^{4}+(3-5\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.x	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.1772.1	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+2\beta _{2})q^{4}+(-2+4\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.y	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.3368.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-16\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+2\beta _{2})q^{4}+(-7+4\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.z	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.1957.1	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-6\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(6+\beta _{1}+2\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
2475.4.a.ba	$2475$	$4$	2475.a	1.a	$1$	$3$	$3$	$146.030$	3.3.568.1	None	✓	$1$	$0$	\(5\)	\(0\)	\(0\)	\(15\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}-\beta _{2})q^{2}+(7+2\beta _{1})q^{4}+(8+\cdots)q^{7}+\cdots\)
2475.4.a.bb	$2475$	$4$	2475.a	1.a	$1$	$4$	$4$	$146.030$	4.4.91035289.2	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(11\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+(-4-\beta _{3})q^{4}+(3-\beta _{2}+\beta _{3})q^{7}+\cdots\)
2475.4.a.bc	$2475$	$4$	2475.a	1.a	$1$	$4$	$4$	$146.030$	4.4.1539480.1	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-9\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{1}+\beta _{3})q^{4}+(-2+2\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.bd	$2475$	$4$	2475.a	1.a	$1$	$4$	$4$	$146.030$	4.4.91035289.2	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-11\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+(-4-\beta _{3})q^{4}+(-3-\beta _{2}+\cdots)q^{7}+\cdots\)
2475.4.a.be	$2475$	$4$	2475.a	1.a	$1$	$4$	$4$	$146.030$	4.4.1540841.1	None	✓	$1$	$0$	\(4\)	\(0\)	\(0\)	\(-34\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(7-\beta _{1}+\beta _{3})q^{4}+(-8+\cdots)q^{7}+\cdots\)
2475.4.a.bf	$2475$	$4$	2475.a	1.a	$1$	$5$	$5$	$146.030$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-38\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(9-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2475.4.a.bg	$2475$	$4$	2475.a	1.a	$1$	$5$	$5$	$146.030$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(24\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(5+\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.bh	$2475$	$4$	2475.a	1.a	$1$	$5$	$5$	$146.030$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(40\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(8+\beta _{2}+\beta _{3})q^{4}+(9-2\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.bi	$2475$	$4$	2475.a	1.a	$1$	$5$	$5$	$146.030$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(18\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(4+\beta _{2}-\beta _{4})q^{7}+\cdots\)
2475.4.a.bj	$2475$	$4$	2475.a	1.a	$1$	$5$	$5$	$146.030$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-18\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-4-\beta _{2}+\beta _{4})q^{7}+\cdots\)
2475.4.a.bk	$2475$	$4$	2475.a	1.a	$1$	$5$	$5$	$146.030$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-24\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(-5-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.bl	$2475$	$4$	2475.a	1.a	$1$	$5$	$5$	$146.030$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-40\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(8+\beta _{2}+\beta _{3})q^{4}+(-9+2\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.bm	$2475$	$4$	2475.a	1.a	$1$	$5$	$5$	$146.030$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(0\)	\(38\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(9-\beta _{1}+\beta _{2})q^{4}+(7+\cdots)q^{7}+\cdots\)
2475.4.a.bn	$2475$	$4$	2475.a	1.a	$1$	$6$	$6$	$146.030$	6.6.2301792529.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
2475.4.a.bo	$2475$	$4$	2475.a	1.a	$1$	$7$	$7$	$146.030$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(-5\)	\(0\)	\(0\)	\(34\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
2475.4.a.bp	$2475$	$4$	2475.a	1.a	$1$	$7$	$7$	$146.030$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(0\)	\(0\)	\(-30\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(5-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2475.4.a.bq	$2475$	$4$	2475.a	1.a	$1$	$7$	$7$	$146.030$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(50\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(7+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
2475.4.a.br	$2475$	$4$	2475.a	1.a	$1$	$7$	$7$	$146.030$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(3\)	\(0\)	\(0\)	\(-50\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(-7-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
2475.4.a.bs	$2475$	$4$	2475.a	1.a	$1$	$7$	$7$	$146.030$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(0\)	\(0\)	\(-34\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-2\beta _{1}+\beta _{2})q^{4}+(-5+\cdots)q^{7}+\cdots\)
2475.4.a.bt	$2475$	$4$	2475.a	1.a	$1$	$7$	$7$	$146.030$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(0\)	\(0\)	\(-30\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(5-\beta _{1}+\beta _{2})q^{4}+(-4+\cdots)q^{7}+\cdots\)
2475.4.a.bu	$2475$	$4$	2475.a	1.a	$1$	$10$	$10$	$146.030$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(3+\beta _{1}+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.bv	$2475$	$4$	2475.a	1.a	$1$	$10$	$10$	$146.030$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(3+\beta _{1}+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.bw	$2475$	$4$	2475.a	1.a	$1$	$10$	$10$	$146.030$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+(\beta _{6}+\beta _{7})q^{7}+\cdots\)
2475.4.a.bx	$2475$	$4$	2475.a	1.a	$1$	$10$	$10$	$146.030$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{1}+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.by	$2475$	$4$	2475.a	1.a	$1$	$10$	$10$	$146.030$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{1}+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
2475.4.a.bz	$2475$	$4$	2475.a	1.a	$1$	$16$	$16$	$146.030$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2^{14}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(-\beta _{1}-\beta _{12}+\cdots)q^{7}+\cdots\)
2475.4.a.ca	$2475$	$4$	2475.a	1.a	$1$	$16$	$16$	$146.030$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2^{14}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5+\beta _{2})q^{4}+(\beta _{1}+\beta _{12})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(2475))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(2475)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(495))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(825))\)\(^{\oplus 2}\)