Space of modular forms of level 5850, weight 2, and trivial character

• Label: 5850.2.a
• Level: $5850$
• Weight: $2$
• Character orbit: 5850.a
• Rep. character: $\chi_{5850}(1,\cdot)$
• Character field: $\Q$
• Dimension: $95$
• Newform subspaces: $72$
• Sturm bound: $2520$
• Trace bound: $17$

Defining parameters

Level:	$N$	$=$	$5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5850.a (trivial)
Character field:			$\Q$
Newform subspaces:			$72$
Sturm bound:			$2520$
Trace bound:			$17$
Distinguishing $T_p$:			$7$, $11$, $17$, $23$, $31$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(5850))$.

	Total	New	Old
Modular forms	1308	95	1213
Cusp forms	1213	95	1118
Eisenstein series	95	0	95

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

$2$	$3$	$5$	$13$	Fricke	Dim
$+$	$+$	$+$	$+$	$+$	$6$
$+$	$+$	$+$	$-$	$-$	$3$
$+$	$+$	$-$	$+$	$-$	$4$
$+$	$+$	$-$	$-$	$+$	$6$
$+$	$-$	$+$	$+$	$-$	$6$
$+$	$-$	$+$	$-$	$+$	$6$
$+$	$-$	$-$	$+$	$+$	$7$
$+$	$-$	$-$	$-$	$-$	$9$
$-$	$+$	$+$	$+$	$-$	$6$
$-$	$+$	$+$	$-$	$+$	$3$
$-$	$+$	$-$	$+$	$+$	$4$
$-$	$+$	$-$	$-$	$-$	$6$
$-$	$-$	$+$	$+$	$+$	$6$
$-$	$-$	$+$	$-$	$-$	$9$
$-$	$-$	$-$	$+$	$-$	$9$
$-$	$-$	$-$	$-$	$+$	$5$
Plus space				$+$	$43$
Minus space				$-$	$52$

Trace form

95 * q + q^2 + 95 * q^4 + q^8 + 4 * q^11 - q^13 - 6 * q^14 + 95 * q^16 - 12 * q^17 - 8 * q^19 - 20 * q^22 + 12 * q^23 - 3 * q^26 + 2 * q^29 + 24 * q^31 + q^32 + 6 * q^34 + 14 * q^37 + 4 * q^38 + 2 * q^41 + 14 * q^43 + 4 * q^44 + 4 * q^46 - 8 * q^47 + 105 * q^49 - q^52 - 2 * q^53 - 6 * q^56 + 22 * q^58 + 36 * q^59 + 2 * q^61 - 8 * q^62 + 95 * q^64 - 16 * q^67 - 12 * q^68 + 80 * q^71 - 10 * q^73 - 12 * q^74 - 8 * q^76 - 24 * q^77 - 4 * q^79 + 14 * q^82 + 24 * q^83 + 24 * q^86 - 20 * q^88 + 34 * q^89 - 6 * q^91 + 12 * q^92 + 26 * q^94 + 2 * q^97 - 23 * q^98

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(5850))$ 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	13
5850.2.a.a	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.e	$1$	$1$	$-1$	$0$	$0$	$-4$		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}-4q^{11}+q^{13}+\cdots$
5850.2.a.b	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				650.2.a.e	$1$	$1$	$-1$	$0$	$0$	$-4$		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}-q^{11}-q^{13}+\cdots$
5850.2.a.c	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.a.c	$1$	$1$	$-1$	$0$	$0$	$-4$		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}+q^{13}+4q^{14}+\cdots$
5850.2.a.d	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				78.2.a.a	$1$	$0$	$-1$	$0$	$0$	$-4$		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}+4q^{11}-q^{13}+\cdots$
5850.2.a.e	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.e.d	$1$	$0$	$-1$	$0$	$0$	$-4$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}+6q^{11}+q^{13}+\cdots$
5850.2.a.f	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓	✓			5850.2.a.f	$1$	$0$	$-1$	$0$	$0$	$-2$		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}-2q^{11}+q^{13}+\cdots$
5850.2.a.g	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.a.d	$1$	$0$	$-1$	$0$	$0$	$-2$		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}-q^{13}+2q^{14}+\cdots$
5850.2.a.h	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1170.2.a.f	$1$	$0$	$-1$	$0$	$0$	$-2$		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}+4q^{11}+q^{13}+\cdots$
5850.2.a.i	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓	✓			5850.2.a.i	$1$	$1$	$-1$	$0$	$0$	$-2$		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}+6q^{11}-q^{13}+\cdots$
5850.2.a.j	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.m	$1$	$1$	$-1$	$0$	$0$	$-1$		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{7}-q^{8}+3q^{11}+q^{13}+\cdots$
5850.2.a.k	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.l	$1$	$1$	$-1$	$0$	$0$	$-1$		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{7}-q^{8}+5q^{11}-q^{13}+\cdots$
5850.2.a.l	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.j	$1$	$0$	$-1$	$0$	$0$	$0$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}-4q^{11}+q^{13}+q^{16}+\cdots$
5850.2.a.m	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.a.a	$1$	$1$	$-1$	$0$	$0$	$0$		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}+q^{13}+q^{16}-6q^{17}+\cdots$
5850.2.a.n	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				650.2.a.a	$1$	$0$	$-1$	$0$	$0$	$0$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}+3q^{11}+q^{13}+q^{16}+\cdots$
5850.2.a.o	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.e.a	$1$	$0$	$-1$	$0$	$0$	$0$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}+6q^{11}+q^{13}+q^{16}+\cdots$
5850.2.a.p	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				26.2.a.a	$1$	$0$	$-1$	$0$	$0$	$1$		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{7}-q^{8}-6q^{11}-q^{13}+\cdots$
5850.2.a.q	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				650.2.a.b	$1$	$0$	$-1$	$0$	$0$	$1$		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{11}-q^{13}+\cdots$
5850.2.a.r	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓	✓			5850.2.a.i	$1$	$1$	$-1$	$0$	$0$	$2$		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+2q^{7}-q^{8}-6q^{11}+q^{13}+\cdots$
5850.2.a.s	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.a.b	$1$	$1$	$-1$	$0$	$0$	$2$		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+2q^{7}-q^{8}-4q^{11}+q^{13}+\cdots$
5850.2.a.t	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.e.c	$1$	$1$	$-1$	$0$	$0$	$2$		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+2q^{7}-q^{8}-2q^{11}-q^{13}+\cdots$
5850.2.a.u	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓	✓			5850.2.a.f	$1$	$0$	$-1$	$0$	$0$	$2$		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+2q^{7}-q^{8}+2q^{11}-q^{13}+\cdots$
5850.2.a.v	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				234.2.a.a	$1$	$0$	$-1$	$0$	$0$	$2$		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+2q^{7}-q^{8}+4q^{11}+q^{13}+\cdots$
5850.2.a.w	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.a	$1$	$1$	$-1$	$0$	$0$	$3$		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+3q^{7}-q^{8}+3q^{11}+q^{13}+\cdots$
5850.2.a.x	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.e.b	$1$	$0$	$-1$	$0$	$0$	$4$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+4q^{7}-q^{8}-2q^{11}+q^{13}+\cdots$
5850.2.a.y	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.f	$1$	$0$	$-1$	$0$	$0$	$4$		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+4q^{7}-q^{8}-q^{13}-4q^{14}+\cdots$
5850.2.a.z	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1170.2.a.c	$1$	$1$	$-1$	$0$	$0$	$4$		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+4q^{7}-q^{8}-q^{13}-4q^{14}+\cdots$
5850.2.a.ba	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				130.2.a.a	$1$	$0$	$-1$	$0$	$0$	$4$		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+4q^{7}-q^{8}+6q^{11}-q^{13}+\cdots$
5850.2.a.bb	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				650.2.a.f	$1$	$0$	$-1$	$0$	$0$	$5$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+5q^{7}-q^{8}+3q^{11}+q^{13}+\cdots$
5850.2.a.bc	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				650.2.a.f	$1$	$1$	$1$	$0$	$0$	$-5$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-5q^{7}+q^{8}+3q^{11}-q^{13}+\cdots$
5850.2.a.bd	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.e.b	$1$	$0$	$1$	$0$	$0$	$-4$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-4q^{7}+q^{8}-2q^{11}-q^{13}+\cdots$
5850.2.a.be	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.f	$1$	$1$	$1$	$0$	$0$	$-4$		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-4q^{7}+q^{8}+q^{13}-4q^{14}+\cdots$
5850.2.a.bf	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.a	$1$	$0$	$1$	$0$	$0$	$-3$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-3q^{7}+q^{8}+3q^{11}-q^{13}+\cdots$
5850.2.a.bg	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓	✓			5850.2.a.i	$1$	$0$	$1$	$0$	$0$	$-2$		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}-6q^{11}-q^{13}+\cdots$
5850.2.a.bh	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1170.2.a.f	$1$	$1$	$1$	$0$	$0$	$-2$		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}-4q^{11}+q^{13}+\cdots$
5850.2.a.bi	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.a.e	$1$	$0$	$1$	$0$	$0$	$-2$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}-4q^{11}+q^{13}+\cdots$
5850.2.a.bj	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.e.c	$1$	$1$	$1$	$0$	$0$	$-2$		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}-2q^{11}+q^{13}+\cdots$
5850.2.a.bk	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.a.g	$1$	$1$	$1$	$0$	$0$	$-2$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}-q^{13}-2q^{14}+\cdots$
5850.2.a.bl	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓	✓			5850.2.a.f	$1$	$1$	$1$	$0$	$0$	$-2$		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}+2q^{11}+q^{13}+\cdots$
5850.2.a.bm	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				650.2.a.b	$1$	$1$	$1$	$0$	$0$	$-1$		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{7}+q^{8}-3q^{11}+q^{13}+\cdots$
5850.2.a.bn	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				26.2.a.b	$1$	$0$	$1$	$0$	$0$	$-1$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{7}+q^{8}+2q^{11}+q^{13}+\cdots$
5850.2.a.bo	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.a.f	$1$	$1$	$1$	$0$	$0$	$0$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}-4q^{11}-q^{13}+q^{16}+\cdots$
5850.2.a.bp	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.j	$1$	$1$	$1$	$0$	$0$	$0$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}-4q^{11}-q^{13}+q^{16}+\cdots$
5850.2.a.bq	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				130.2.a.b	$1$	$1$	$1$	$0$	$0$	$0$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}-q^{13}+q^{16}+2q^{17}+\cdots$
5850.2.a.br	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				650.2.a.a	$1$	$1$	$1$	$0$	$0$	$0$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}+3q^{11}-q^{13}+q^{16}+\cdots$
5850.2.a.bs	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.e.a	$1$	$0$	$1$	$0$	$0$	$0$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}+6q^{11}-q^{13}+q^{16}+\cdots$
5850.2.a.bt	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.m	$1$	$0$	$1$	$0$	$0$	$1$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{7}+q^{8}+3q^{11}-q^{13}+\cdots$
5850.2.a.bu	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.l	$1$	$0$	$1$	$0$	$0$	$1$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{7}+q^{8}+5q^{11}+q^{13}+\cdots$
5850.2.a.bv	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				234.2.a.a	$1$	$1$	$1$	$0$	$0$	$2$		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}-4q^{11}+q^{13}+\cdots$
5850.2.a.bw	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓	✓			5850.2.a.f	$1$	$1$	$1$	$0$	$0$	$2$		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}-q^{13}+\cdots$
5850.2.a.bx	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓	✓			5850.2.a.i	$1$	$0$	$1$	$0$	$0$	$2$		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}+6q^{11}+q^{13}+\cdots$
5850.2.a.by	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1950.2.a.e	$1$	$0$	$1$	$0$	$0$	$4$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}-4q^{11}-q^{13}+\cdots$
5850.2.a.bz	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				650.2.a.e	$1$	$0$	$1$	$0$	$0$	$4$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}-q^{11}+q^{13}+\cdots$
5850.2.a.ca	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				1170.2.a.c	$1$	$0$	$1$	$0$	$0$	$4$		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}-q^{13}+4q^{14}+\cdots$
5850.2.a.cb	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				130.2.a.c	$1$	$0$	$1$	$0$	$0$	$4$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}+2q^{11}+q^{13}+\cdots$
5850.2.a.cc	$5850$	$2$	5850.a	1.a	$1$	$1$	$1$	$46.712$	$\Q$	None	✓				390.2.e.d	$1$	$0$	$1$	$0$	$0$	$4$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}+6q^{11}-q^{13}+\cdots$
5850.2.a.cd	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{7})$	None	✓	✓			5850.2.a.cd	$1$	$1$	$-2$	$0$	$0$	$-4$		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-2+\beta )q^{7}-q^{8}+3q^{11}+\cdots$
5850.2.a.ce	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{13})$	None	✓				1170.2.a.p	$1$	$1$	$-2$	$0$	$0$	$-2$		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-1-\beta )q^{7}-q^{8}-q^{13}+\cdots$
5850.2.a.cf	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{5})$	None	✓				390.2.e.e	$1$	$1$	$-2$	$0$	$0$	$-2$		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-1-\beta )q^{7}-q^{8}-2\beta q^{11}+\cdots$
5850.2.a.cg	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{41})$	None	✓				1950.2.a.bc	$1$	$1$	$-2$	$0$	$0$	$3$		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1+\beta )q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots$
5850.2.a.ch	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{7})$	None	✓	✓			5850.2.a.cd	$1$	$1$	$-2$	$0$	$0$	$4$		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(2+\beta )q^{7}-q^{8}-3q^{11}+\cdots$
5850.2.a.ci	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{7})$	None	✓	✓			5850.2.a.cd	$1$	$0$	$2$	$0$	$0$	$-4$		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}-3q^{11}+\cdots$
5850.2.a.cj	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{41})$	None	✓				1950.2.a.bc	$1$	$0$	$2$	$0$	$0$	$-3$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1-\beta )q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots$
5850.2.a.ck	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{13})$	None	✓				1170.2.a.p	$1$	$0$	$2$	$0$	$0$	$-2$		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1-\beta )q^{7}+q^{8}-q^{13}+\cdots$
5850.2.a.cl	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{2})$	None	✓				390.2.a.h	$1$	$0$	$2$	$0$	$0$	$0$		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+\beta q^{7}+q^{8}-2\beta q^{11}+\cdots$
5850.2.a.cm	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{5})$	None	✓		✓		390.2.e.e	$1$	$1$	$2$	$0$	$0$	$2$		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(1+\beta )q^{7}+q^{8}-2\beta q^{11}+\cdots$
5850.2.a.cn	$5850$	$2$	5850.a	1.a	$1$	$2$	$2$	$46.712$	$\Q(\sqrt{7})$	None	✓	✓			5850.2.a.cd	$1$	$0$	$2$	$0$	$0$	$4$		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(2+\beta )q^{7}+q^{8}+3q^{11}+\cdots$
5850.2.a.co	$5850$	$2$	5850.a	1.a	$1$	$3$	$3$	$46.712$	3.3.148.1	None	✓		✓		1170.2.e.g	$1$	$0$	$-3$	$0$	$0$	$-4$		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-1-\beta _{1})q^{7}-q^{8}+2q^{11}+\cdots$
5850.2.a.cp	$5850$	$2$	5850.a	1.a	$1$	$3$	$3$	$46.712$	3.3.940.1	None	✓		✓		130.2.b.a	$1$	$0$	$-3$	$0$	$0$	$-2$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-1-\beta _{1}-\beta _{2})q^{7}-q^{8}+\cdots$
5850.2.a.cq	$5850$	$2$	5850.a	1.a	$1$	$3$	$3$	$46.712$	3.3.148.1	None	✓		✓		1170.2.e.g	$1$	$1$	$-3$	$0$	$0$	$4$		$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1+\beta _{1})q^{7}-q^{8}-2q^{11}+\cdots$
5850.2.a.cr	$5850$	$2$	5850.a	1.a	$1$	$3$	$3$	$46.712$	3.3.148.1	None	✓		✓		1170.2.e.g	$1$	$1$	$3$	$0$	$0$	$-4$		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1-\beta _{1})q^{7}+q^{8}-2q^{11}+\cdots$
5850.2.a.cs	$5850$	$2$	5850.a	1.a	$1$	$3$	$3$	$46.712$	3.3.940.1	None	✓		✓		130.2.b.a	$1$	$0$	$3$	$0$	$0$	$2$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(1+\beta _{1}+\beta _{2})q^{7}+q^{8}+\cdots$
5850.2.a.ct	$5850$	$2$	5850.a	1.a	$1$	$3$	$3$	$46.712$	3.3.148.1	None	✓		✓		1170.2.e.g	$1$	$0$	$3$	$0$	$0$	$4$		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(1+\beta _{1})q^{7}+q^{8}+2q^{11}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(5850))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(5850)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 16}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(26))$$^{\oplus 9}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(30))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(39))$$^{\oplus 12}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(45))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(50))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(65))$$^{\oplus 12}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(75))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(78))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(90))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(117))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(130))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(150))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(195))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(225))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(234))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(325))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(390))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(450))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(585))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(650))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(975))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1170))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1950))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(2925))$$^{\oplus 2}$