LMFDB - Space of modular forms of level 8624, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 85 $
Sturm bound:			$2688$
Trace bound:			$37$
Distinguishing $T_p$:			$3$, $5$, $13$, $17$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(8624))$.

	Total	New	Old
Modular forms	1392	205	1187
Cusp forms	1297	205	1092
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.

$2$	$7$	$11$	Fricke	Total			Cusp			Eisenstein
$2$	$7$	$11$	Fricke	All	New	Old	All	New	Old	All	New	Old
$+$	$+$	$+$	$+$	$168$	$20$	$148$	$157$	$20$	$137$	$11$	$0$	$11$
$+$	$+$	$-$	$-$	$180$	$32$	$148$	$168$	$32$	$136$	$12$	$0$	$12$
$+$	$-$	$+$	$-$	$180$	$30$	$150$	$168$	$30$	$138$	$12$	$0$	$12$
$+$	$-$	$-$	$+$	$168$	$21$	$147$	$156$	$21$	$135$	$12$	$0$	$12$
$-$	$+$	$+$	$-$	$180$	$24$	$156$	$168$	$24$	$144$	$12$	$0$	$12$
$-$	$+$	$-$	$+$	$168$	$24$	$144$	$156$	$24$	$132$	$12$	$0$	$12$
$-$	$-$	$+$	$+$	$168$	$27$	$141$	$156$	$27$	$129$	$12$	$0$	$12$
$-$	$-$	$-$	$-$	$180$	$27$	$153$	$168$	$27$	$141$	$12$	$0$	$12$
Plus space			$+$	$672$	$92$	$580$	$625$	$92$	$533$	$47$	$0$	$47$
Minus space			$-$	$720$	$113$	$607$	$672$	$113$	$559$	$48$	$0$	$48$

Trace form

$ 205 q - 2 q^{3} - 2 q^{5} + 205 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} + 2 q^{17} - 12 q^{19} - 6 q^{23} + 207 q^{25} - 14 q^{27} + 6 q^{29} - 18 q^{31} - 10 q^{37} - 16 q^{39} + 2 q^{41} - 16 q^{43} - 2 q^{45}+ \cdots + 9 q^{99}+O(q^{100}) $

205 * q - 2 * q^3 - 2 * q^5 + 205 * q^9 + 3 * q^11 - 2 * q^13 + 2 * q^15 + 2 * q^17 - 12 * q^19 - 6 * q^23 + 207 * q^25 - 14 * q^27 + 6 * q^29 - 18 * q^31 - 10 * q^37 - 16 * q^39 + 2 * q^41 - 16 * q^43 - 2 * q^45 + 28 * q^47 + 12 * q^51 - 2 * q^53 + 4 * q^55 + 8 * q^57 - 22 * q^59 + 6 * q^61 + 28 * q^65 + 14 * q^67 + 24 * q^69 + 6 * q^71 + 2 * q^73 - 72 * q^75 - 36 * q^79 + 213 * q^81 + 32 * q^83 + 4 * q^85 - 40 * q^87 - 2 * q^89 + 24 * q^93 - 48 * q^95 - 10 * q^97 + 9 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(8624))$ 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	7	11	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
8624.2.a.a	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				77.2.a.a	$1$	$0$	$0$	$-3$	$1$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}+q^{5}+6q^{9}+q^{11}+4q^{13}+\cdots$
8624.2.a.b	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.e.a	$1$	$1$	$0$	$-3$	$2$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}+2q^{5}+6q^{9}+q^{11}-7q^{13}+\cdots$
8624.2.a.c	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				88.2.a.a	$1$	$1$	$0$	$-3$	$3$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}+3q^{5}+6q^{9}+q^{11}-9q^{15}+\cdots$
8624.2.a.d	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.e.d	$1$	$0$	$0$	$-3$	$4$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}+4q^{5}+6q^{9}+q^{11}+q^{13}+\cdots$
8624.2.a.e	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				616.2.a.a	$1$	$1$	$0$	$-2$	$-2$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}-2q^{5}+q^{9}+q^{11}-4q^{13}+\cdots$
8624.2.a.f	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				4312.2.a.c	$1$	$1$	$0$	$-2$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}+q^{9}+q^{11}-2q^{13}+2q^{17}+\cdots$
8624.2.a.g	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				1078.2.a.h	$1$	$1$	$0$	$-2$	$2$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}+2q^{5}+q^{9}-q^{11}-2q^{13}+\cdots$
8624.2.a.h	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				4312.2.a.a	$1$	$0$	$0$	$-2$	$4$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}+4q^{5}+q^{9}-q^{11}+2q^{13}+\cdots$
8624.2.a.i	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				616.2.q.a	$1$	$1$	$0$	$-1$	$-2$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}-2q^{5}-2q^{9}+q^{11}-3q^{13}+\cdots$
8624.2.a.j	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				11.2.a.a	$1$	$1$	$0$	$-1$	$-1$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}-q^{5}-2q^{9}-q^{11}-4q^{13}+\cdots$
8624.2.a.k	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.e.c	$1$	$1$	$0$	$-1$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}-2q^{9}+q^{11}-q^{13}-6q^{17}+\cdots$
8624.2.a.l	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.e.b	$1$	$1$	$0$	$-1$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}-2q^{9}+q^{11}+5q^{13}+6q^{17}+\cdots$
8624.2.a.m	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				616.2.a.b	$1$	$0$	$0$	$-1$	$1$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+q^{5}-2q^{9}-q^{11}-q^{15}+2q^{17}+\cdots$
8624.2.a.n	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				308.2.a.a	$1$	$1$	$0$	$-1$	$1$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+q^{5}-2q^{9}-q^{11}+4q^{13}+\cdots$
8624.2.a.o	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.a.c	$1$	$0$	$0$	$0$	$-2$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{5}-3q^{9}+q^{11}-2q^{13}-2q^{17}+\cdots$
8624.2.a.p	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				616.2.a.d	$1$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-3q^{9}+q^{11}+6q^{13}-2q^{19}-4q^{23}+\cdots$
8624.2.a.q	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				616.2.a.c	$1$	$1$	$0$	$0$	$2$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{5}-3q^{9}+q^{11}-2q^{13}+2q^{17}+\cdots$
8624.2.a.r	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.a.a	$1$	$0$	$0$	$0$	$4$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+4q^{5}-3q^{9}+q^{11}-2q^{13}+4q^{17}+\cdots$
8624.2.a.s	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				77.2.a.b	$1$	$0$	$0$	$1$	$-3$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-3q^{5}-2q^{9}+q^{11}+4q^{13}+\cdots$
8624.2.a.t	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.e.b	$1$	$0$	$0$	$1$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-2q^{9}+q^{11}-5q^{13}-6q^{17}+\cdots$
8624.2.a.u	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.e.c	$1$	$0$	$0$	$1$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-2q^{9}+q^{11}+q^{13}+6q^{17}+\cdots$
8624.2.a.v	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				616.2.q.a	$1$	$0$	$0$	$1$	$2$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}+2q^{5}-2q^{9}+q^{11}+3q^{13}+\cdots$
8624.2.a.w	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				44.2.a.a	$1$	$0$	$0$	$1$	$3$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}+3q^{5}-2q^{9}+q^{11}+4q^{13}+\cdots$
8624.2.a.x	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				4312.2.a.a	$1$	$0$	$0$	$2$	$-4$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-4q^{5}+q^{9}-q^{11}-2q^{13}+\cdots$
8624.2.a.y	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				1078.2.a.h	$1$	$1$	$0$	$2$	$-2$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-2q^{5}+q^{9}-q^{11}+2q^{13}+\cdots$
8624.2.a.z	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.a.b	$1$	$1$	$0$	$2$	$-2$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-2q^{5}+q^{9}-q^{11}+4q^{13}+\cdots$
8624.2.a.ba	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				616.2.a.e	$1$	$1$	$0$	$2$	$-2$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-2q^{5}+q^{9}+q^{11}-4q^{15}+\cdots$
8624.2.a.bb	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				4312.2.a.c	$1$	$1$	$0$	$2$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}+q^{9}+q^{11}+2q^{13}-2q^{17}+\cdots$
8624.2.a.bc	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				77.2.a.c	$1$	$1$	$0$	$2$	$2$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}+2q^{5}+q^{9}-q^{11}-4q^{13}+\cdots$
8624.2.a.bd	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.e.d	$1$	$1$	$0$	$3$	$-4$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+3q^{3}-4q^{5}+6q^{9}+q^{11}-q^{13}+\cdots$
8624.2.a.be	$8624$	$2$	8624.a	1.a	$1$	$1$	$1$	$68.863$	$\Q$	None	✓				154.2.e.a	$1$	$0$	$0$	$3$	$-2$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+3q^{3}-2q^{5}+6q^{9}+q^{11}+7q^{13}+\cdots$
8624.2.a.bf	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{5}) $	None	✓				154.2.a.d	$1$	$1$	$0$	$-2$	$-2$	$0$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q+(-1-\beta )q^{3}+(-1-\beta )q^{5}+(3+2\beta )q^{9}+\cdots$
8624.2.a.bg	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				616.2.q.b	$1$	$1$	$0$	$-2$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{3}+\beta q^{5}-2\beta q^{9}-q^{11}+\cdots$
8624.2.a.bh	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				154.2.e.e	$1$	$1$	$0$	$-2$	$4$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{3}+(2+\beta )q^{5}-2\beta q^{9}+\cdots$
8624.2.a.bi	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{17}) $	None	✓				616.2.a.f	$1$	$0$	$0$	$-1$	$3$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{3}+(2-\beta )q^{5}+(1+\beta )q^{9}-q^{11}+\cdots$
8624.2.a.bj	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{6}) $	None	✓				308.2.a.b	$1$	$0$	$0$	$0$	$-4$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-2q^{5}+3q^{9}+q^{11}+(-2+\cdots)q^{13}+\cdots$
8624.2.a.bk	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{7}) $	None	✓				154.2.e.f	$1$	$1$	$0$	$0$	$-2$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+(-1+\beta )q^{5}+4q^{9}-q^{11}+\cdots$
8624.2.a.bl	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				1078.2.a.p	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2\beta q^{5}-3q^{9}-q^{11}+3\beta q^{13}-2\beta q^{17}+\cdots$
8624.2.a.bm	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				2156.2.a.d	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{5}-3q^{9}-q^{11}-3\beta q^{13}-\beta q^{17}+\cdots$
8624.2.a.bn	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				4312.2.a.p	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2\beta q^{5}-3q^{9}-q^{11}-\beta q^{13}+2\beta q^{17}+\cdots$
8624.2.a.bo	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				1078.2.a.o	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q+\beta q^{5}-3q^{9}-q^{11}-2\beta q^{13}-\beta q^{17}+\cdots$
8624.2.a.bp	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				4312.2.a.o	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+\beta q^{5}-3q^{9}+q^{11}+\beta q^{17}-\beta q^{19}+\cdots$
8624.2.a.bq	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				4312.2.a.s	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-\beta q^{5}-q^{9}-q^{11}-2q^{15}+\cdots$
8624.2.a.br	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				4312.2.a.r	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+\beta q^{5}-q^{9}-q^{11}-2\beta q^{13}+\cdots$
8624.2.a.bs	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				1078.2.a.u	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-3\beta q^{5}-q^{9}+q^{11}-6q^{15}+\cdots$
8624.2.a.bt	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				4312.2.a.q	$2$	$0$	$0$	$0$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-\beta q^{5}-q^{9}+q^{11}-2q^{15}+\cdots$
8624.2.a.bu	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				2156.2.a.e	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-q^{9}+q^{11}+\beta q^{13}+5\beta q^{17}+\cdots$
8624.2.a.bv	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				539.2.a.e	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+\beta q^{5}-q^{9}+q^{11}+2q^{15}+\cdots$
8624.2.a.bw	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				1078.2.a.q	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+\beta q^{5}-q^{9}+q^{11}+2\beta q^{13}+\cdots$
8624.2.a.bx	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				2156.2.a.f	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2\beta q^{3}-\beta q^{5}+5q^{9}+q^{11}+\beta q^{13}+\cdots$
8624.2.a.by	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				1078.2.a.r	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta q^{3}+5q^{9}+q^{11}+\beta q^{13}-\beta q^{17}+\cdots$
8624.2.a.bz	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				1078.2.a.v	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2\beta q^{3}+5q^{9}+q^{11}-3\beta q^{13}+2\beta q^{17}+\cdots$
8624.2.a.ca	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{7}) $	None	✓				154.2.e.f	$1$	$0$	$0$	$0$	$2$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+(1+\beta )q^{5}+4q^{9}-q^{11}+5q^{13}+\cdots$
8624.2.a.cb	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{17}) $	None	✓				88.2.a.b	$1$	$1$	$0$	$1$	$-3$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{9}+q^{11}+\cdots$
8624.2.a.cc	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				154.2.e.e	$1$	$0$	$0$	$2$	$-4$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(-2+\beta )q^{5}+2\beta q^{9}+\cdots$
8624.2.a.cd	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{2}) $	None	✓				616.2.q.b	$1$	$0$	$0$	$2$	$0$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+\beta q^{5}+2\beta q^{9}-q^{11}+\cdots$
8624.2.a.ce	$8624$	$2$	8624.a	1.a	$1$	$2$	$2$	$68.863$	$\Q(\sqrt{5}) $	None	✓				77.2.a.d	$1$	$0$	$0$	$2$	$4$	$0$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+2q^{5}+(3+2\beta )q^{9}+q^{11}+\cdots$
8624.2.a.cf	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	$\Q(\zeta_{14})^+$	None	✓				616.2.q.c	$1$	$0$	$0$	$-5$	$2$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-2-\beta _{2})q^{3}+(-1+3\beta _{1}-2\beta _{2})q^{5}+\cdots$
8624.2.a.cg	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	3.3.257.1	None	✓				308.2.i.b	$1$	$1$	$0$	$-3$	$2$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(1+\beta _{1}+\cdots)q^{9}+\cdots$
8624.2.a.ch	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	$\Q(\zeta_{18})^+$	None	✓				77.2.e.a	$1$	$0$	$0$	$-3$	$6$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{2})q^{5}+\cdots$
8624.2.a.ci	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	3.3.321.1	None	✓				308.2.i.a	$1$	$1$	$0$	$-1$	$-2$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(1-\beta _{1}+\cdots)q^{9}+\cdots$
8624.2.a.cj	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	3.3.1016.1	None	✓				308.2.a.c	$1$	$1$	$0$	$-1$	$1$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots$
8624.2.a.ck	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	3.3.257.1	None	✓				77.2.e.b	$1$	$0$	$0$	$-1$	$2$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\beta _{2}q^{9}+\cdots$
8624.2.a.cl	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	3.3.257.1	None	✓				77.2.e.b	$1$	$1$	$0$	$1$	$-2$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\beta _{2}q^{9}+\cdots$
8624.2.a.cm	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	3.3.229.1	None	✓				616.2.a.g	$1$	$0$	$0$	$1$	$-1$	$0$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{3}+\beta _{2}q^{5}+(1-\beta _{1})q^{9}-q^{11}+\cdots$
8624.2.a.cn	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	3.3.321.1	None	✓				308.2.i.a	$1$	$0$	$0$	$1$	$2$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(1-\beta _{1}-2\beta _{2})q^{9}+\cdots$
8624.2.a.co	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	$\Q(\zeta_{18})^+$	None	✓				77.2.e.a	$1$	$1$	$0$	$3$	$-6$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta _{1})q^{3}+(-2-\beta _{1}+\beta _{2})q^{5}+\cdots$
8624.2.a.cp	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	3.3.257.1	None	✓				308.2.i.b	$1$	$0$	$0$	$3$	$-2$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(1+\beta _{1}+\cdots)q^{9}+\cdots$
8624.2.a.cq	$8624$	$2$	8624.a	1.a	$1$	$3$	$3$	$68.863$	$\Q(\zeta_{14})^+$	None	✓				616.2.q.c	$1$	$1$	$0$	$5$	$-2$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(2-\beta _{1})q^{3}+(-2+\beta _{1}-3\beta _{2})q^{5}+\cdots$
8624.2.a.cr	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	4.4.89289.1	None	✓				616.2.q.d	$1$	$1$	$0$	$-2$	$4$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(3+\beta _{1}+\cdots)q^{9}+\cdots$
8624.2.a.cs	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	$\Q(\zeta_{24})^+$	None	✓		✓		2156.2.a.l	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(-2\beta _{1}-\beta _{3})q^{5}+(-1+\beta _{2}+\cdots)q^{9}+\cdots$
8624.2.a.ct	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	4.4.9248.1	None	✓				4312.2.a.ba	$2$	$0$	$0$	$0$	$0$	$0$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{9}-q^{11}+(\beta _{1}+\cdots)q^{13}+\cdots$
8624.2.a.cu	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	4.4.9248.1	None	✓		✓		539.2.a.k	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{5}-\beta _{3}q^{9}+q^{11}+\cdots$
8624.2.a.cv	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	$\Q(\sqrt{7}, \sqrt{15})$	None	✓				4312.2.a.bb	$2$	$0$	$0$	$0$	$0$	$0$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{3}+\beta _{2}q^{5}+(2-\beta _{3})q^{9}-q^{11}+\cdots$
8624.2.a.cw	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	$\Q(\zeta_{24})^+$	None	✓				4312.2.a.bc	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{3}-\beta _{1}q^{5}+3q^{9}-q^{11}+(\beta _{1}+\cdots)q^{13}+\cdots$
8624.2.a.cx	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	4.4.301088.1	None	✓		✓		2156.2.a.m	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}-\beta _{2}q^{5}+(4+\beta _{3})q^{9}-q^{11}+\cdots$
8624.2.a.cy	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	4.4.11348.1	None	✓				616.2.a.h	$1$	$0$	$0$	$1$	$-5$	$0$	$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q+\beta _{2}q^{3}+(-1+\beta _{1})q^{5}+(2+\beta _{2}-\beta _{3})q^{9}+\cdots$
8624.2.a.cz	$8624$	$2$	8624.a	1.a	$1$	$4$	$4$	$68.863$	4.4.89289.1	None	✓				616.2.q.d	$1$	$0$	$0$	$2$	$-4$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(3+\beta _{1}+\cdots)q^{9}+\cdots$
8624.2.a.da	$8624$	$2$	8624.a	1.a	$1$	$5$	$5$	$68.863$	5.5.352076.1	None	✓		✓		616.2.q.e	$1$	$1$	$0$	$-3$	$4$	$0$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{3}+(1-\beta _{1}+\beta _{4})q^{5}+\cdots$
8624.2.a.db	$8624$	$2$	8624.a	1.a	$1$	$5$	$5$	$68.863$	5.5.559701.1	None	✓		✓		616.2.q.f	$1$	$1$	$0$	$-1$	$-4$	$0$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\beta _{2}q^{9}+\cdots$
8624.2.a.dc	$8624$	$2$	8624.a	1.a	$1$	$5$	$5$	$68.863$	5.5.559701.1	None	✓				616.2.q.f	$1$	$0$	$0$	$1$	$4$	$0$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\beta _{2}q^{9}+\cdots$
8624.2.a.dd	$8624$	$2$	8624.a	1.a	$1$	$5$	$5$	$68.863$	5.5.352076.1	None	✓		✓		616.2.q.e	$1$	$0$	$0$	$3$	$-4$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta _{2})q^{3}+(-1+\beta _{1}-\beta _{4})q^{5}+\cdots$
8624.2.a.de	$8624$	$2$	8624.a	1.a	$1$	$8$	$8$	$68.863$	8.8.$\cdots$.1	None	✓				4312.2.a.bi	$2$	$0$	$0$	$0$	$0$	$0$	$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q-\beta _{3}q^{3}+(-\beta _{3}-\beta _{4}+\beta _{5})q^{5}+(1+\cdots)q^{9}+\cdots$
8624.2.a.df	$8624$	$2$	8624.a	1.a	$1$	$10$	$10$	$68.863$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓		✓		539.2.a.l	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$+$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+\beta _{6}q^{5}+(2+\beta _{2})q^{9}-q^{11}+\cdots$
8624.2.a.dg	$8624$	$2$	8624.a	1.a	$1$	$12$	$12$	$68.863$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓		✓		4312.2.a.bj	$2$	$0$	$0$	$0$	$0$	$0$	$+$	$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+\beta _{9}q^{5}+(3+\beta _{2})q^{9}+q^{11}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(8624))$ 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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 85 \)
Sturm bound:			\(2688\)
Trace bound:			\(37\)
Distinguishing \(T_p\):			\(3\), \(5\), \(13\), \(17\)

\(2\)	\(7\)	\(11\)	Fricke	Total			Cusp			Eisenstein
\(2\)	\(7\)	\(11\)	Fricke	All	New	Old	All	New	Old	All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(168\)	\(20\)	\(148\)	\(157\)	\(20\)	\(137\)	\(11\)	\(0\)	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)	\(180\)	\(32\)	\(148\)	\(168\)	\(32\)	\(136\)	\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)	\(180\)	\(30\)	\(150\)	\(168\)	\(30\)	\(138\)	\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)	\(168\)	\(21\)	\(147\)	\(156\)	\(21\)	\(135\)	\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)	\(180\)	\(24\)	\(156\)	\(168\)	\(24\)	\(144\)	\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)	\(168\)	\(24\)	\(144\)	\(156\)	\(24\)	\(132\)	\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)	\(168\)	\(27\)	\(141\)	\(156\)	\(27\)	\(129\)	\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)	\(180\)	\(27\)	\(153\)	\(168\)	\(27\)	\(141\)	\(12\)	\(0\)	\(12\)
Plus space			\(+\)	\(672\)	\(92\)	\(580\)	\(625\)	\(92\)	\(533\)	\(47\)	\(0\)	\(47\)
Minus space			\(-\)	\(720\)	\(113\)	\(607\)	\(672\)	\(113\)	\(559\)	\(48\)	\(0\)	\(48\)

Introduction

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Groups

Database

Properties

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8624))\) 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	8624.2.a

Level	$8624$
Weight	$2$
Character orbit	8624.a
Rep. character	$\chi_{8624}(1,\cdot)$
Character field	$\Q$
Dimension	$205$
Newform subspaces	$85$
Sturm bound	$2688$
Trace bound	$37$