Space of modular forms of level 5175, weight 2, and trivial character

• Label: 5175.2.a
• Level: $5175$
• Weight: $2$
• Character orbit: 5175.a
• Rep. character: $\chi_{5175}(1,\cdot)$
• Character field: $\Q$
• Dimension: $173$
• Newform subspaces: $62$
• Sturm bound: $1440$
• Trace bound: $11$

Defining parameters

Level:	$N$	$=$	$5175 = 3^{2} \cdot 5^{2} \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5175.a (trivial)
Character field:			$\Q$
Newform subspaces:			$62$
Sturm bound:			$1440$
Trace bound:			$11$
Distinguishing $T_p$:			$2$, $7$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(5175))$.

	Total	New	Old
Modular forms	744	173	571
Cusp forms	697	173	524
Eisenstein series	47	0	47

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

$3$	$5$	$23$	Fricke	Dim
$+$	$+$	$+$	$+$	$17$
$+$	$+$	$-$	$-$	$17$
$+$	$-$	$+$	$-$	$17$
$+$	$-$	$-$	$+$	$17$
$-$	$+$	$+$	$-$	$29$
$-$	$+$	$-$	$+$	$20$
$-$	$-$	$+$	$+$	$25$
$-$	$-$	$-$	$-$	$31$
Plus space			$+$	$79$
Minus space			$-$	$94$

Trace form

173 * q - 2 * q^2 + 170 * q^4 + 2 * q^7 - 15 * q^8 + 2 * q^11 - 8 * q^13 - 20 * q^14 + 164 * q^16 - 4 * q^17 - 4 * q^19 + 10 * q^22 - 3 * q^23 + 25 * q^26 - 10 * q^28 + 24 * q^29 - 12 * q^31 - 12 * q^32 + 10 * q^34 - 24 * q^37 + 18 * q^38 - 4 * q^41 + 20 * q^43 - 8 * q^44 - 2 * q^46 - 16 * q^47 + 161 * q^49 - q^52 + 10 * q^53 - 2 * q^56 + 17 * q^58 - 36 * q^59 + 22 * q^61 + 37 * q^62 + 191 * q^64 + 2 * q^67 - 28 * q^68 - 16 * q^71 + 8 * q^73 + 40 * q^74 - 6 * q^76 + 36 * q^77 - 68 * q^79 - 17 * q^82 - 22 * q^83 + 12 * q^86 + 10 * q^88 - 2 * q^89 - 90 * q^91 - 6 * q^92 - 25 * q^94 + 12 * q^97 - 46 * q^98

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(5175))$ 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	23
5175.2.a.a	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				115.2.b.a	$1$	$0$	$-2$	$0$	$0$	$1$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+2q^{4}+q^{7}+2q^{13}-2q^{14}+\cdots$
5175.2.a.b	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.a.a	$1$	$0$	$-2$	$0$	$0$	$5$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+2q^{4}+5q^{7}+2q^{11}+6q^{13}+\cdots$
5175.2.a.c	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.a.b	$1$	$0$	$-1$	$0$	$0$	$-4$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}-4q^{7}+3q^{8}+4q^{11}+\cdots$
5175.2.a.d	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.f	$1$	$0$	$-1$	$0$	$0$	$-3$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}-3q^{7}+3q^{8}-5q^{11}+\cdots$
5175.2.a.e	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				575.2.a.c	$1$	$1$	$-1$	$0$	$0$	$-1$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}-q^{7}+3q^{8}+q^{11}-q^{13}+\cdots$
5175.2.a.f	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.b.a	$1$	$1$	$-1$	$0$	$0$	$0$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}+3q^{8}+4q^{13}-q^{16}+\cdots$
5175.2.a.g	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.b.b	$1$	$1$	$-1$	$0$	$0$	$2$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}+2q^{7}+3q^{8}+4q^{13}+\cdots$
5175.2.a.h	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.c	$1$	$1$	$-1$	$0$	$0$	$3$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}+3q^{7}+3q^{8}+q^{11}-q^{13}+\cdots$
5175.2.a.i	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.g	$1$	$0$	$-1$	$0$	$0$	$5$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}+5q^{7}+3q^{8}-5q^{11}+\cdots$
5175.2.a.j	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.i	$1$	$1$	$0$	$0$	$0$	$-3$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}-3q^{7}-2q^{11}+3q^{13}+4q^{16}+\cdots$
5175.2.a.k	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.a.c	$1$	$1$	$0$	$0$	$0$	$-1$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}-q^{7}-4q^{11}+4q^{16}+5q^{17}+\cdots$
5175.2.a.l	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.j	$1$	$0$	$0$	$0$	$0$	$-1$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}-q^{7}+6q^{11}+5q^{13}+4q^{16}+\cdots$
5175.2.a.m	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.j	$1$	$0$	$0$	$0$	$0$	$1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}+q^{7}+6q^{11}-5q^{13}+4q^{16}+\cdots$
5175.2.a.n	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.i	$1$	$1$	$0$	$0$	$0$	$3$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}+3q^{7}-2q^{11}-3q^{13}+4q^{16}+\cdots$
5175.2.a.o	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.a.d	$1$	$1$	$0$	$0$	$0$	$3$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}+3q^{7}+4q^{11}+4q^{16}-3q^{17}+\cdots$
5175.2.a.p	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.g	$1$	$0$	$1$	$0$	$0$	$-5$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-5q^{7}-3q^{8}-5q^{11}+\cdots$
5175.2.a.q	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.a.e	$1$	$0$	$1$	$0$	$0$	$-4$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-4q^{7}-3q^{8}-4q^{11}+\cdots$
5175.2.a.r	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.c	$1$	$1$	$1$	$0$	$0$	$-3$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-3q^{7}-3q^{8}+q^{11}+q^{13}+\cdots$
5175.2.a.s	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.b.b	$1$	$0$	$1$	$0$	$0$	$-2$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-2q^{7}-3q^{8}-4q^{13}+\cdots$
5175.2.a.t	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.b.a	$1$	$0$	$1$	$0$	$0$	$0$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-3q^{8}-4q^{13}-q^{16}+\cdots$
5175.2.a.u	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				575.2.a.c	$1$	$1$	$1$	$0$	$0$	$1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}+q^{7}-3q^{8}+q^{11}+q^{13}+\cdots$
5175.2.a.v	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				69.2.a.a	$1$	$0$	$1$	$0$	$0$	$2$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}+2q^{7}-3q^{8}-4q^{11}+\cdots$
5175.2.a.w	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				1725.2.a.f	$1$	$0$	$1$	$0$	$0$	$3$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}+3q^{7}-3q^{8}-5q^{11}+\cdots$
5175.2.a.x	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				345.2.a.f	$1$	$0$	$2$	$0$	$0$	$-3$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+2q^{4}-3q^{7}-2q^{11}+2q^{13}+\cdots$
5175.2.a.y	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				115.2.a.a	$1$	$1$	$2$	$0$	$0$	$-1$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+2q^{4}-q^{7}-2q^{11}+2q^{13}+\cdots$
5175.2.a.z	$5175$	$2$	5175.a	1.a	$1$	$1$	$1$	$41.323$	$\Q$	None	✓				115.2.b.a	$1$	$1$	$2$	$0$	$0$	$-1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+2q^{4}-q^{7}-2q^{13}-2q^{14}+\cdots$
5175.2.a.ba	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{5})$	None	✓				1725.2.a.v	$1$	$0$	$-3$	$0$	$0$	$1$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta )q^{2}+3\beta q^{4}+\beta q^{7}+(-1+\cdots)q^{8}+\cdots$
5175.2.a.bb	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{5})$	None	✓				115.2.a.b	$1$	$0$	$-3$	$0$	$0$	$2$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta )q^{2}+3\beta q^{4}+(2-2\beta )q^{7}+\cdots$
5175.2.a.bc	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{2})$	None	✓				207.2.a.b	$1$	$1$	$-2$	$0$	$0$	$4$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+(2-\beta )q^{7}+\cdots$
5175.2.a.bd	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{3})$	None	✓				345.2.a.g	$1$	$1$	$-2$	$0$	$0$	$6$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{2}+(2-2\beta )q^{4}+3q^{7}+\cdots$
5175.2.a.be	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{5})$	None	✓				23.2.a.a	$1$	$1$	$-1$	$0$	$0$	$-2$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{2}+(-1+\beta )q^{4}+(-2+2\beta )q^{7}+\cdots$
5175.2.a.bf	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{13})$	None	✓				1725.2.a.w	$1$	$1$	$-1$	$0$	$0$	$-1$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{2}+(1+\beta )q^{4}+(-1+\beta )q^{7}-3q^{8}+\cdots$
5175.2.a.bg	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{21})$	None	✓				1725.2.a.x	$1$	$0$	$-1$	$0$	$0$	$1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{2}+(3+\beta )q^{4}+(1-\beta )q^{7}+(-5+\cdots)q^{8}+\cdots$
5175.2.a.bh	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{2})$	None	✓				1035.2.a.h	$1$	$1$	$0$	$0$	$0$	$2$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(1-\beta )q^{7}-2\beta q^{8}+(-2+\beta )q^{11}+\cdots$
5175.2.a.bi	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{2})$	None	✓				1035.2.a.h	$1$	$0$	$0$	$0$	$0$	$2$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(1+\beta )q^{7}-2\beta q^{8}+(2+\beta )q^{11}+\cdots$
5175.2.a.bj	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{2})$	None	✓				345.2.a.h	$1$	$0$	$0$	$0$	$0$	$2$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(1+2\beta )q^{7}-2\beta q^{8}+(4+\beta )q^{11}+\cdots$
5175.2.a.bk	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{5})$	None	✓				69.2.a.b	$1$	$1$	$0$	$0$	$0$	$-2$		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-\beta q^{2}+3q^{4}+(-1-\beta )q^{7}-\beta q^{8}+\cdots$
5175.2.a.bl	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{6})$	None	✓				345.2.a.i	$1$	$0$	$0$	$0$	$0$	$2$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+4q^{4}+q^{7}+2\beta q^{8}+\beta q^{11}+\cdots$
5175.2.a.bm	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{13})$	None	✓				1725.2.a.w	$1$	$1$	$1$	$0$	$0$	$1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(1+\beta )q^{4}+(1-\beta )q^{7}+3q^{8}+\cdots$
5175.2.a.bn	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{21})$	None	✓				1725.2.a.x	$1$	$0$	$1$	$0$	$0$	$-1$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(3+\beta )q^{4}+(-1+\beta )q^{7}+(5+\cdots)q^{8}+\cdots$
5175.2.a.bo	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{2})$	None	✓				207.2.a.b	$1$	$0$	$2$	$0$	$0$	$4$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(2+\beta )q^{7}+\cdots$
5175.2.a.bp	$5175$	$2$	5175.a	1.a	$1$	$2$	$2$	$41.323$	$\Q(\sqrt{5})$	None	✓				1725.2.a.v	$1$	$0$	$3$	$0$	$0$	$-1$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+3\beta q^{4}-\beta q^{7}+(1+4\beta )q^{8}+\cdots$
5175.2.a.bq	$5175$	$2$	5175.a	1.a	$1$	$3$	$3$	$41.323$	3.3.229.1	None	✓				1725.2.a.bf	$1$	$1$	$-2$	$0$	$0$	$0$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{2}+(2+\beta _{1})q^{4}+\beta _{1}q^{7}+\cdots$
5175.2.a.br	$5175$	$2$	5175.a	1.a	$1$	$3$	$3$	$41.323$	3.3.316.1	None	✓				345.2.a.j	$1$	$1$	$-1$	$0$	$0$	$-6$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-2-\beta _{2})q^{7}+\cdots$
5175.2.a.bs	$5175$	$2$	5175.a	1.a	$1$	$3$	$3$	$41.323$	3.3.148.1	None	✓				345.2.b.c	$1$	$0$	$0$	$0$	$0$	$-1$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots$
5175.2.a.bt	$5175$	$2$	5175.a	1.a	$1$	$3$	$3$	$41.323$	3.3.148.1	None	✓				345.2.b.c	$1$	$1$	$0$	$0$	$0$	$1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+(1-2\beta _{1}+\cdots)q^{7}+\cdots$
5175.2.a.bu	$5175$	$2$	5175.a	1.a	$1$	$3$	$3$	$41.323$	3.3.229.1	None	✓				1725.2.a.bf	$1$	$1$	$2$	$0$	$0$	$0$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta _{2})q^{2}+(2+\beta _{1})q^{4}-\beta _{1}q^{7}+\cdots$
5175.2.a.bv	$5175$	$2$	5175.a	1.a	$1$	$4$	$4$	$41.323$	4.4.5744.1	None	✓				115.2.b.b	$1$	$1$	$0$	$0$	$0$	$-6$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{4}+(-1+\cdots)q^{7}+\cdots$
5175.2.a.bw	$5175$	$2$	5175.a	1.a	$1$	$4$	$4$	$41.323$	4.4.5744.1	None	✓				115.2.b.b	$1$	$0$	$0$	$0$	$0$	$6$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{4}+(1+\beta _{2}+\cdots)q^{7}+\cdots$
5175.2.a.bx	$5175$	$2$	5175.a	1.a	$1$	$4$	$4$	$41.323$	4.4.15317.1	None	✓				115.2.a.c	$1$	$0$	$2$	$0$	$0$	$3$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
5175.2.a.by	$5175$	$2$	5175.a	1.a	$1$	$6$	$6$	$41.323$	6.6.98838128.1	None	✓				1035.2.a.p	$1$	$0$	$0$	$0$	$0$	$-6$		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}+(2+\beta _{4})q^{4}+(-1-\beta _{3})q^{7}+\cdots$
5175.2.a.bz	$5175$	$2$	5175.a	1.a	$1$	$6$	$6$	$41.323$	6.6.98838128.1	None	✓				1035.2.a.p	$1$	$1$	$0$	$0$	$0$	$-6$		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+(2+\beta _{4})q^{4}+(-1-\beta _{3})q^{7}+\cdots$
5175.2.a.ca	$5175$	$2$	5175.a	1.a	$1$	$7$	$7$	$41.323$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓		✓		345.2.b.d	$1$	$1$	$-2$	$0$	$0$	$-5$		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+(2-\beta _{3})q^{4}+(-1-\beta _{5})q^{7}+\cdots$
5175.2.a.cb	$5175$	$2$	5175.a	1.a	$1$	$7$	$7$	$41.323$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓		✓		575.2.a.k	$1$	$0$	$-1$	$0$	$0$	$-3$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}+\beta _{4}+\beta _{6})q^{7}+\cdots$
5175.2.a.cc	$5175$	$2$	5175.a	1.a	$1$	$7$	$7$	$41.323$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓			5175.2.a.cc	$1$	$1$	$0$	$0$	$0$	$-1$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{8}+\cdots$
5175.2.a.cd	$5175$	$2$	5175.a	1.a	$1$	$7$	$7$	$41.323$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓			5175.2.a.cc	$1$	$0$	$0$	$0$	$0$	$-1$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{7}+(-\beta _{2}+\cdots)q^{8}+\cdots$
5175.2.a.ce	$5175$	$2$	5175.a	1.a	$1$	$7$	$7$	$41.323$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓	✓		5175.2.a.cc	$1$	$1$	$0$	$0$	$0$	$1$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{7}+(-\beta _{2}+\cdots)q^{8}+\cdots$
5175.2.a.cf	$5175$	$2$	5175.a	1.a	$1$	$7$	$7$	$41.323$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	✓	✓		5175.2.a.cc	$1$	$0$	$0$	$0$	$0$	$1$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{8}+\cdots$
5175.2.a.cg	$5175$	$2$	5175.a	1.a	$1$	$7$	$7$	$41.323$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓				575.2.a.k	$1$	$0$	$1$	$0$	$0$	$3$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots$
5175.2.a.ch	$5175$	$2$	5175.a	1.a	$1$	$7$	$7$	$41.323$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓				345.2.b.d	$1$	$0$	$2$	$0$	$0$	$5$		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{2}+(2-\beta _{3})q^{4}+(1+\beta _{5})q^{7}+\cdots$
5175.2.a.ci	$5175$	$2$	5175.a	1.a	$1$	$10$	$10$	$41.323$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓		✓		1035.2.b.g	$2$	$1$	$-4$	$0$	$0$	$0$		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+\beta _{6}q^{2}+(-\beta _{6}-\beta _{7})q^{4}-\beta _{2}q^{7}+\cdots$
5175.2.a.cj	$5175$	$2$	5175.a	1.a	$1$	$10$	$10$	$41.323$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓		✓		1035.2.b.g	$2$	$0$	$4$	$0$	$0$	$0$		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q-\beta _{6}q^{2}+(-\beta _{6}-\beta _{7})q^{4}+\beta _{2}q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(5175))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(5175)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(23))$$^{\oplus 9}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(45))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(69))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(75))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(115))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(207))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(225))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(345))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(575))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1035))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1725))$$^{\oplus 2}$