LMFDB - Space of modular forms of level 8550, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 76 $
Sturm bound:			$3600$
Trace bound:			$23$
Distinguishing $T_p$:			$7$, $11$, $13$, $17$, $23$, $53$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(8550))$.

	Total	New	Old
Modular forms	1848	143	1705
Cusp forms	1753	143	1610
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.

$2$	$3$	$5$	$19$	Fricke	Total			Cusp			Eisenstein
$2$	$3$	$5$	$19$	Fricke	All	New	Old	All	New	Old	All	New	Old
$+$	$+$	$+$	$+$	$+$	$105$	$7$	$98$	$100$	$7$	$93$	$5$	$0$	$5$
$+$	$+$	$+$	$-$	$-$	$123$	$6$	$117$	$117$	$6$	$111$	$6$	$0$	$6$
$+$	$+$	$-$	$+$	$-$	$123$	$7$	$116$	$117$	$7$	$110$	$6$	$0$	$6$
$+$	$+$	$-$	$-$	$+$	$111$	$9$	$102$	$105$	$9$	$96$	$6$	$0$	$6$
$+$	$-$	$+$	$+$	$-$	$120$	$12$	$108$	$114$	$12$	$102$	$6$	$0$	$6$
$+$	$-$	$+$	$-$	$+$	$111$	$8$	$103$	$105$	$8$	$97$	$6$	$0$	$6$
$+$	$-$	$-$	$+$	$+$	$114$	$10$	$104$	$108$	$10$	$98$	$6$	$0$	$6$
$+$	$-$	$-$	$-$	$-$	$117$	$12$	$105$	$111$	$12$	$99$	$6$	$0$	$6$
$-$	$+$	$+$	$+$	$-$	$117$	$7$	$110$	$111$	$7$	$104$	$6$	$0$	$6$
$-$	$+$	$+$	$-$	$+$	$117$	$6$	$111$	$111$	$6$	$105$	$6$	$0$	$6$
$-$	$+$	$-$	$+$	$+$	$111$	$7$	$104$	$105$	$7$	$98$	$6$	$0$	$6$
$-$	$+$	$-$	$-$	$-$	$117$	$9$	$108$	$111$	$9$	$102$	$6$	$0$	$6$
$-$	$-$	$+$	$+$	$+$	$120$	$8$	$112$	$114$	$8$	$106$	$6$	$0$	$6$
$-$	$-$	$+$	$-$	$-$	$111$	$13$	$98$	$105$	$13$	$92$	$6$	$0$	$6$
$-$	$-$	$-$	$+$	$-$	$114$	$13$	$101$	$108$	$13$	$95$	$6$	$0$	$6$
$-$	$-$	$-$	$-$	$+$	$117$	$9$	$108$	$111$	$9$	$102$	$6$	$0$	$6$
Plus space				$+$	$906$	$64$	$842$	$859$	$64$	$795$	$47$	$0$	$47$
Minus space				$-$	$942$	$79$	$863$	$894$	$79$	$815$	$48$	$0$	$48$

Trace form

$ 143 q + q^{2} + 143 q^{4} - 2 q^{7} + q^{8} - 12 q^{11} - 2 q^{13} - 4 q^{14} + 143 q^{16} + 4 q^{17} + q^{19} - 4 q^{22} + 6 q^{23} - 12 q^{26} - 2 q^{28} - 18 q^{29} + 12 q^{31} + q^{32} - 22 q^{34}+ \cdots + 17 q^{98}+O(q^{100}) $

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(8550))$ 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
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	2	3	5	19	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
8550.2.a.a	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				114.2.a.a	$1$	$1$	$-1$	$0$	$0$	$-4$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}-4q^{11}+4q^{14}+\cdots$
8550.2.a.b	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.d	$1$	$0$	$-1$	$0$	$0$	$-4$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}-2q^{13}+4q^{14}+\cdots$
8550.2.a.c	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				2850.2.a.f	$1$	$0$	$-1$	$0$	$0$	$-4$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}+q^{11}+4q^{14}+\cdots$
8550.2.a.d	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				2850.2.a.o	$1$	$1$	$-1$	$0$	$0$	$-2$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}-3q^{11}-2q^{13}+\cdots$
8550.2.a.e	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.f	$1$	$1$	$-1$	$0$	$0$	$-2$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}-2q^{13}+2q^{14}+\cdots$
8550.2.a.f	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				2850.2.a.d	$1$	$0$	$-1$	$0$	$0$	$-2$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}+3q^{11}-6q^{13}+\cdots$
8550.2.a.g	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.b	$1$	$0$	$-1$	$0$	$0$	$-2$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}+6q^{11}+2q^{14}+\cdots$
8550.2.a.h	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				2850.2.a.l	$1$	$1$	$-1$	$0$	$0$	$0$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}-q^{11}-4q^{13}+q^{16}+\cdots$
8550.2.a.i	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				342.2.a.a	$1$	$1$	$-1$	$0$	$0$	$0$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}+2q^{11}+4q^{13}+\cdots$
8550.2.a.j	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.d.b	$1$	$1$	$-1$	$0$	$0$	$0$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}+4q^{11}-4q^{13}+\cdots$
8550.2.a.k	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.d.a	$1$	$1$	$-1$	$0$	$0$	$0$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-q^{8}+4q^{11}+2q^{13}+\cdots$
8550.2.a.l	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				190.2.a.a	$1$	$0$	$-1$	$0$	$0$	$1$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{7}-q^{8}+3q^{13}-q^{14}+\cdots$
8550.2.a.m	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				38.2.a.a	$1$	$1$	$-1$	$0$	$0$	$1$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{7}-q^{8}+6q^{11}-5q^{13}+\cdots$
8550.2.a.n	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.a	$1$	$1$	$-1$	$0$	$0$	$2$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+2q^{7}-q^{8}-4q^{11}+6q^{13}+\cdots$
8550.2.a.o	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.c	$1$	$1$	$-1$	$0$	$0$	$2$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+2q^{7}-q^{8}+2q^{11}-2q^{14}+\cdots$
8550.2.a.p	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				1710.2.a.h	$1$	$0$	$-1$	$0$	$0$	$2$	$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+2q^{7}-q^{8}+2q^{11}+4q^{13}+\cdots$
8550.2.a.q	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				1710.2.a.a	$1$	$1$	$-1$	$0$	$0$	$4$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+4q^{7}-q^{8}-6q^{11}-4q^{14}+\cdots$
8550.2.a.r	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.e	$1$	$0$	$-1$	$0$	$0$	$4$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+4q^{7}-q^{8}+4q^{11}+6q^{13}+\cdots$
8550.2.a.s	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.i	$1$	$0$	$1$	$0$	$0$	$-4$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-4q^{7}+q^{8}+6q^{13}-4q^{14}+\cdots$
8550.2.a.t	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.m	$1$	$1$	$1$	$0$	$0$	$-4$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-4q^{7}+q^{8}+4q^{11}+2q^{13}+\cdots$
8550.2.a.u	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				38.2.a.b	$1$	$1$	$1$	$0$	$0$	$-3$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-3q^{7}+q^{8}-2q^{11}+q^{13}+\cdots$
8550.2.a.v	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.k	$1$	$0$	$1$	$0$	$0$	$-2$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}-6q^{11}+4q^{13}+\cdots$
8550.2.a.w	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.j	$1$	$0$	$1$	$0$	$0$	$-2$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}+4q^{11}-6q^{13}+\cdots$
8550.2.a.x	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.g	$1$	$1$	$1$	$0$	$0$	$0$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}-4q^{11}-2q^{13}+\cdots$
8550.2.a.y	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				342.2.a.a	$1$	$0$	$1$	$0$	$0$	$0$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}-2q^{11}+4q^{13}+\cdots$
8550.2.a.z	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				2850.2.a.l	$1$	$1$	$1$	$0$	$0$	$0$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}-q^{11}+4q^{13}+q^{16}+\cdots$
8550.2.a.ba	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				114.2.a.b	$1$	$1$	$1$	$0$	$0$	$0$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}+4q^{11}-2q^{13}+\cdots$
8550.2.a.bb	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.d.a	$1$	$0$	$1$	$0$	$0$	$0$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}+4q^{11}-2q^{13}+\cdots$
8550.2.a.bc	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.d.b	$1$	$0$	$1$	$0$	$0$	$0$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}+4q^{11}+4q^{13}+\cdots$
8550.2.a.bd	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				190.2.a.c	$1$	$0$	$1$	$0$	$0$	$1$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{7}+q^{8}+q^{13}+q^{14}+\cdots$
8550.2.a.be	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				2850.2.a.o	$1$	$1$	$1$	$0$	$0$	$2$	$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}-3q^{11}+2q^{13}+\cdots$
8550.2.a.bf	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.l	$1$	$1$	$1$	$0$	$0$	$2$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}-4q^{13}+\cdots$
8550.2.a.bg	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				1710.2.a.h	$1$	$1$	$1$	$0$	$0$	$2$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}-2q^{11}+4q^{13}+\cdots$
8550.2.a.bh	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				570.2.a.h	$1$	$0$	$1$	$0$	$0$	$2$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}-6q^{13}+2q^{14}+\cdots$
8550.2.a.bi	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				2850.2.a.d	$1$	$0$	$1$	$0$	$0$	$2$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}+3q^{11}+6q^{13}+\cdots$
8550.2.a.bj	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				114.2.a.c	$1$	$0$	$1$	$0$	$0$	$4$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}+4q^{13}+4q^{14}+\cdots$
8550.2.a.bk	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				2850.2.a.f	$1$	$0$	$1$	$0$	$0$	$4$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}+q^{11}+4q^{14}+\cdots$
8550.2.a.bl	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				1710.2.a.a	$1$	$0$	$1$	$0$	$0$	$4$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}+6q^{11}+4q^{14}+\cdots$
8550.2.a.bm	$8550$	$2$	8550.a	1.a	$1$	$1$	$1$	$68.272$	$\Q$	None	✓				190.2.a.b	$1$	$0$	$1$	$0$	$0$	$5$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+5q^{7}+q^{8}+4q^{11}+q^{13}+\cdots$
8550.2.a.bn	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{2}) $	None	✓				190.2.b.a	$1$	$1$	$-2$	$0$	$0$	$-6$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-3+\beta )q^{7}-q^{8}-\beta q^{11}+\cdots$
8550.2.a.bo	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{3}) $	None	✓				1710.2.a.u	$1$	$0$	$-2$	$0$	$0$	$-4$	$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{7}-q^{8}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots$
8550.2.a.bp	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{17}) $	None	✓				1710.2.a.v	$1$	$1$	$-2$	$0$	$0$	$-2$	$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-1-\beta )q^{7}-q^{8}+2q^{11}+\cdots$
8550.2.a.bq	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{10}) $	None	✓				2850.2.a.bf	$1$	$0$	$-2$	$0$	$0$	$0$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+\beta q^{7}-q^{8}+(-1-\beta )q^{11}+\cdots$
8550.2.a.br	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{17}) $	None	✓				190.2.a.d	$1$	$0$	$-2$	$0$	$0$	$1$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+\beta q^{7}-q^{8}-4q^{11}+(2+\cdots)q^{13}+\cdots$
8550.2.a.bs	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{3}) $	None	✓				2850.2.a.bd	$1$	$1$	$-2$	$0$	$0$	$2$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1+\beta )q^{7}-q^{8}+(-2+\cdots)q^{11}+\cdots$
8550.2.a.bt	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{3}) $	None	✓				2850.2.a.be	$1$	$0$	$-2$	$0$	$0$	$2$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1+\beta )q^{7}-q^{8}-3\beta q^{11}+\cdots$
8550.2.a.bu	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{6}) $	None	✓		✓		2850.2.a.bc	$1$	$1$	$-2$	$0$	$0$	$4$	$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(2+\beta )q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots$
8550.2.a.bv	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{6}) $	None	✓				2850.2.a.bc	$1$	$1$	$2$	$0$	$0$	$-4$	$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-2+\beta )q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots$
8550.2.a.bw	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{3}) $	None	✓				1710.2.a.u	$1$	$1$	$2$	$0$	$0$	$-4$	$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-2q^{7}+q^{8}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots$
8550.2.a.bx	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{17}) $	None	✓				1710.2.a.v	$1$	$0$	$2$	$0$	$0$	$-2$	$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1-\beta )q^{7}+q^{8}-2q^{11}+\cdots$
8550.2.a.by	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{3}) $	None	✓		✓		2850.2.a.bd	$1$	$1$	$2$	$0$	$0$	$-2$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1+\beta )q^{7}+q^{8}+(-2+\cdots)q^{11}+\cdots$
8550.2.a.bz	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{3}) $	None	✓				2850.2.a.be	$1$	$0$	$2$	$0$	$0$	$-2$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1+\beta )q^{7}+q^{8}+3\beta q^{11}+\cdots$
8550.2.a.ca	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{10}) $	None	✓				2850.2.a.bf	$1$	$0$	$2$	$0$	$0$	$0$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+\beta q^{7}+q^{8}+(-1+\beta )q^{11}+\cdots$
8550.2.a.cb	$8550$	$2$	8550.a	1.a	$1$	$2$	$2$	$68.272$	$\Q(\sqrt{2}) $	None	✓				190.2.b.a	$1$	$0$	$2$	$0$	$0$	$6$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(3+\beta )q^{7}+q^{8}+\beta q^{11}+\cdots$
8550.2.a.cc	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.788.1	None	✓	✓			8550.2.a.cc	$1$	$0$	$-3$	$0$	$0$	$-2$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-1+\beta _{1})q^{7}-q^{8}+(1+\cdots)q^{11}+\cdots$
8550.2.a.cd	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.564.1	None	✓	✓	✓		8550.2.a.cd	$1$	$0$	$-3$	$0$	$0$	$-2$	$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-\beta _{1}+\beta _{2})q^{7}-q^{8}+\cdots$
8550.2.a.ce	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.148.1	None	✓				570.2.d.c	$1$	$0$	$-3$	$0$	$0$	$0$	$+$	$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+\beta _{2}q^{7}-q^{8}+(-3-\beta _{1}+\cdots)q^{11}+\cdots$
8550.2.a.cf	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.148.1	None	✓		✓		570.2.d.d	$1$	$1$	$-3$	$0$	$0$	$0$	$+$	$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+\beta _{2}q^{7}-q^{8}+(1-\beta _{1}+\cdots)q^{11}+\cdots$
8550.2.a.cg	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.564.1	None	✓	✓			8550.2.a.cd	$1$	$1$	$-3$	$0$	$0$	$2$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(\beta _{1}-\beta _{2})q^{7}-q^{8}+(-2+\cdots)q^{11}+\cdots$
8550.2.a.ch	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.788.1	None	✓	✓	✓		8550.2.a.cc	$1$	$1$	$-3$	$0$	$0$	$2$	$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1-\beta _{1})q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots$
8550.2.a.ci	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.993.1	None	✓				950.2.a.j	$1$	$0$	$-3$	$0$	$0$	$2$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1-\beta _{2})q^{7}-q^{8}-2\beta _{1}q^{11}+\cdots$
8550.2.a.cj	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.257.1	None	✓		✓		950.2.a.k	$1$	$0$	$-3$	$0$	$0$	$2$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1-\beta _{1}+2\beta _{2})q^{7}-q^{8}+\cdots$
8550.2.a.ck	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.568.1	None	✓				190.2.b.b	$1$	$0$	$-3$	$0$	$0$	$4$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1+\beta _{1})q^{7}-q^{8}+(\beta _{1}+\cdots)q^{11}+\cdots$
8550.2.a.cl	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.568.1	None	✓				190.2.b.b	$1$	$1$	$3$	$0$	$0$	$-4$	$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1-\beta _{1})q^{7}+q^{8}+(\beta _{1}+\cdots)q^{11}+\cdots$
8550.2.a.cm	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.788.1	None	✓	✓			8550.2.a.cc	$1$	$1$	$3$	$0$	$0$	$-2$	$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1+\beta _{1})q^{7}+q^{8}+(-1+\cdots)q^{11}+\cdots$
8550.2.a.cn	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.564.1	None	✓	✓	✓		8550.2.a.cd	$1$	$1$	$3$	$0$	$0$	$-2$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-\beta _{1}+\beta _{2})q^{7}+q^{8}+\cdots$
8550.2.a.co	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.257.1	None	✓				950.2.a.k	$1$	$0$	$3$	$0$	$0$	$-2$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1+\beta _{1}-2\beta _{2})q^{7}+\cdots$
8550.2.a.cp	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.993.1	None	✓		✓		950.2.a.j	$1$	$0$	$3$	$0$	$0$	$-2$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(-1+\beta _{2})q^{7}+q^{8}-2\beta _{1}q^{11}+\cdots$
8550.2.a.cq	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.148.1	None	✓				570.2.d.c	$1$	$1$	$3$	$0$	$0$	$0$	$-$	$-$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-\beta _{2}q^{7}+q^{8}+(-3-\beta _{1}+\cdots)q^{11}+\cdots$
8550.2.a.cr	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.148.1	None	✓				570.2.d.d	$1$	$0$	$3$	$0$	$0$	$0$	$-$	$-$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-\beta _{2}q^{7}+q^{8}+(1-\beta _{1}+\cdots)q^{11}+\cdots$
8550.2.a.cs	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.564.1	None	✓	✓			8550.2.a.cd	$1$	$0$	$3$	$0$	$0$	$2$	$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(\beta _{1}-\beta _{2})q^{7}+q^{8}+(2+\cdots)q^{11}+\cdots$
8550.2.a.ct	$8550$	$2$	8550.a	1.a	$1$	$3$	$3$	$68.272$	3.3.788.1	None	✓	✓	✓		8550.2.a.cc	$1$	$0$	$3$	$0$	$0$	$2$	$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(1-\beta _{1})q^{7}+q^{8}+(1+\beta _{1}+\cdots)q^{11}+\cdots$
8550.2.a.cu	$8550$	$2$	8550.a	1.a	$1$	$4$	$4$	$68.272$	$\Q(\zeta_{24})^+$	None	✓		✓		1710.2.d.g	$2$	$0$	$-4$	$0$	$0$	$0$	$+$	$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}-\beta _{3}q^{11}+\cdots$
8550.2.a.cv	$8550$	$2$	8550.a	1.a	$1$	$4$	$4$	$68.272$	$\Q(\zeta_{24})^+$	None	✓		✓		1710.2.d.g	$2$	$1$	$4$	$0$	$0$	$0$	$-$	$+$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+\beta _{3}q^{11}+\cdots$
8550.2.a.cw	$8550$	$2$	8550.a	1.a	$1$	$6$	$6$	$68.272$	6.6.3356224.1	None	✓		✓		1710.2.d.h	$2$	$1$	$-6$	$0$	$0$	$0$	$+$	$+$	$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}+(-\beta _{1}+\beta _{4}+\cdots)q^{11}+\cdots$
8550.2.a.cx	$8550$	$2$	8550.a	1.a	$1$	$6$	$6$	$68.272$	6.6.3356224.1	None	✓		✓		1710.2.d.h	$2$	$0$	$6$	$0$	$0$	$0$	$-$	$+$	$-$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+(\beta _{1}-\beta _{4}+\cdots)q^{11}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(8550))$ into lower level spaces

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 76 \)
Sturm bound:			\(3600\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(23\), \(53\)

\(2\)	\(3\)	\(5\)	\(19\)	Fricke	Total			Cusp			Eisenstein
\(2\)	\(3\)	\(5\)	\(19\)	Fricke	All	New	Old	All	New	Old	All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(105\)	\(7\)	\(98\)	\(100\)	\(7\)	\(93\)	\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(123\)	\(6\)	\(117\)	\(117\)	\(6\)	\(111\)	\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(123\)	\(7\)	\(116\)	\(117\)	\(7\)	\(110\)	\(6\)	\(0\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(111\)	\(9\)	\(102\)	\(105\)	\(9\)	\(96\)	\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(120\)	\(12\)	\(108\)	\(114\)	\(12\)	\(102\)	\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(111\)	\(8\)	\(103\)	\(105\)	\(8\)	\(97\)	\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(114\)	\(10\)	\(104\)	\(108\)	\(10\)	\(98\)	\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(117\)	\(12\)	\(105\)	\(111\)	\(12\)	\(99\)	\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(117\)	\(7\)	\(110\)	\(111\)	\(7\)	\(104\)	\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(117\)	\(6\)	\(111\)	\(111\)	\(6\)	\(105\)	\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(111\)	\(7\)	\(104\)	\(105\)	\(7\)	\(98\)	\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(117\)	\(9\)	\(108\)	\(111\)	\(9\)	\(102\)	\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(120\)	\(8\)	\(112\)	\(114\)	\(8\)	\(106\)	\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(111\)	\(13\)	\(98\)	\(105\)	\(13\)	\(92\)	\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(114\)	\(13\)	\(101\)	\(108\)	\(13\)	\(95\)	\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(117\)	\(9\)	\(108\)	\(111\)	\(9\)	\(102\)	\(6\)	\(0\)	\(6\)
Plus space				\(+\)	\(906\)	\(64\)	\(842\)	\(859\)	\(64\)	\(795\)	\(47\)	\(0\)	\(47\)
Minus space				\(-\)	\(942\)	\(79\)	\(863\)	\(894\)	\(79\)	\(815\)	\(48\)	\(0\)	\(48\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8550))\) into lower level spaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8550.2.a

Level	$8550$
Weight	$2$
Character orbit	8550.a
Rep. character	$\chi_{8550}(1,\cdot)$
Character field	$\Q$
Dimension	$143$
Newform subspaces	$76$
Sturm bound	$3600$
Trace bound	$23$