LMFDB - Space of modular forms of level 8450, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 79 $
Sturm bound:			$2730$
Trace bound:			$17$
Distinguishing $T_p$:			$3$, $7$, $11$, $17$, $31$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(8450))$.

	Total	New	Old
Modular forms	1448	245	1203
Cusp forms	1281	245	1036
Eisenstein series	167	0	167

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

$2$	$5$	$13$	Fricke	Total			Cusp			Eisenstein
$2$	$5$	$13$	Fricke	All	New	Old	All	New	Old	All	New	Old
$+$	$+$	$+$	$+$	$173$	$24$	$149$	$153$	$24$	$129$	$20$	$0$	$20$
$+$	$+$	$-$	$-$	$188$	$33$	$155$	$167$	$33$	$134$	$21$	$0$	$21$
$+$	$-$	$+$	$-$	$187$	$34$	$153$	$166$	$34$	$132$	$21$	$0$	$21$
$+$	$-$	$-$	$+$	$175$	$31$	$144$	$154$	$31$	$123$	$21$	$0$	$21$
$-$	$+$	$+$	$-$	$184$	$35$	$149$	$163$	$35$	$128$	$21$	$0$	$21$
$-$	$+$	$-$	$+$	$178$	$24$	$154$	$157$	$24$	$133$	$21$	$0$	$21$
$-$	$-$	$+$	$+$	$177$	$27$	$150$	$156$	$27$	$129$	$21$	$0$	$21$
$-$	$-$	$-$	$-$	$186$	$37$	$149$	$165$	$37$	$128$	$21$	$0$	$21$
Plus space			$+$	$703$	$106$	$597$	$620$	$106$	$514$	$83$	$0$	$83$
Minus space			$-$	$745$	$139$	$606$	$661$	$139$	$522$	$84$	$0$	$84$

Trace form

$ 245 q + q^{2} - 2 q^{3} + 245 q^{4} - 2 q^{6} - 8 q^{7} + q^{8} + 245 q^{9} + 2 q^{11} - 2 q^{12} + 245 q^{16} - 10 q^{17} + 5 q^{18} - 2 q^{19} - 16 q^{21} - 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{27}+ \cdots + 40 q^{99}+O(q^{100}) $

245 * q + q^2 - 2 * q^3 + 245 * q^4 - 2 * q^6 - 8 * q^7 + q^8 + 245 * q^9 + 2 * q^11 - 2 * q^12 + 245 * q^16 - 10 * q^17 + 5 * q^18 - 2 * q^19 - 16 * q^21 - 6 * q^22 + 8 * q^23 - 2 * q^24 - 8 * q^27 - 8 * q^28 - 8 * q^29 - 20 * q^31 + q^32 + 20 * q^33 + 4 * q^34 + 245 * q^36 + 2 * q^37 - 2 * q^38 + 8 * q^41 - 12 * q^42 - 10 * q^43 + 2 * q^44 - 8 * q^46 - 2 * q^48 + 225 * q^49 + 70 * q^51 + 28 * q^53 - 2 * q^54 - 8 * q^57 + 2 * q^58 + 36 * q^59 - 16 * q^61 + 245 * q^64 + 14 * q^66 - 10 * q^68 + 72 * q^69 + 24 * q^71 + 5 * q^72 + 2 * q^73 + 8 * q^74 - 2 * q^76 + 36 * q^77 + 317 * q^81 + 22 * q^82 + 24 * q^83 - 16 * q^84 - 8 * q^86 + 28 * q^87 - 6 * q^88 + 28 * q^89 + 8 * q^92 - 32 * q^93 + 16 * q^94 - 2 * q^96 + 6 * q^97 - 7 * q^98 + 40 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(8450))$ 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	5	13	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
8450.2.a.a	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				650.2.a.f	$1$	$0$	$-1$	$-2$	$0$	$-5$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}-5q^{7}-q^{8}+\cdots$
8450.2.a.b	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.d.a	$1$	$0$	$-1$	$-2$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots$
8450.2.a.c	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				26.2.a.a	$1$	$1$	$-1$	$-1$	$0$	$-1$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots$
8450.2.a.d	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				50.2.a.a	$1$	$1$	$-1$	$-1$	$0$	$2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots$
8450.2.a.e	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				650.2.a.e	$1$	$0$	$-1$	$-1$	$0$	$4$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots$
8450.2.a.f	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				26.2.c.a	$1$	$1$	$-1$	$0$	$0$	$-4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-4q^{7}-q^{8}-3q^{9}+4q^{11}+\cdots$
8450.2.a.g	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.e.a	$1$	$1$	$-1$	$0$	$0$	$3$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+3q^{7}-q^{8}-3q^{9}-3q^{11}+\cdots$
8450.2.a.h	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				26.2.b.a	$1$	$0$	$-1$	$1$	$0$	$3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}-q^{6}+3q^{7}-q^{8}+\cdots$
8450.2.a.i	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.a.a	$1$	$1$	$-1$	$2$	$0$	$-4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-4q^{7}-q^{8}+\cdots$
8450.2.a.j	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				650.2.a.b	$1$	$1$	$-1$	$2$	$0$	$-1$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots$
8450.2.a.k	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.e.b	$1$	$1$	$-1$	$2$	$0$	$-1$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots$
8450.2.a.l	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				650.2.a.a	$1$	$0$	$-1$	$3$	$0$	$0$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots$
8450.2.a.m	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				650.2.a.a	$1$	$0$	$1$	$-3$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots$
8450.2.a.n	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.a.c	$1$	$0$	$1$	$-2$	$0$	$-4$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}-4q^{7}+q^{8}+\cdots$
8450.2.a.o	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.d.a	$1$	$1$	$1$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{8}+q^{9}+\cdots$
8450.2.a.p	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				650.2.a.b	$1$	$1$	$1$	$-2$	$0$	$1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{7}+q^{8}+\cdots$
8450.2.a.q	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.e.a	$1$	$0$	$1$	$0$	$0$	$-3$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-3q^{7}+q^{8}-3q^{9}+3q^{11}+\cdots$
8450.2.a.r	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.a.b	$1$	$0$	$1$	$0$	$0$	$0$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+q^{8}-3q^{9}+q^{16}-2q^{17}+\cdots$
8450.2.a.s	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				26.2.c.a	$1$	$0$	$1$	$0$	$0$	$4$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+4q^{7}+q^{8}-3q^{9}-4q^{11}+\cdots$
8450.2.a.t	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				650.2.a.e	$1$	$0$	$1$	$1$	$0$	$-4$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots$
8450.2.a.u	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				26.2.b.a	$1$	$1$	$1$	$1$	$0$	$-3$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-3q^{7}+q^{8}+\cdots$
8450.2.a.v	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				50.2.a.a	$1$	$1$	$1$	$1$	$0$	$-2$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots$
8450.2.a.w	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				130.2.e.b	$1$	$0$	$1$	$2$	$0$	$1$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots$
8450.2.a.x	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				650.2.a.f	$1$	$0$	$1$	$2$	$0$	$5$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}+2q^{6}+5q^{7}+q^{8}+\cdots$
8450.2.a.y	$8450$	$2$	8450.a	1.a	$1$	$1$	$1$	$67.474$	$\Q$	None	✓				26.2.a.b	$1$	$0$	$1$	$3$	$0$	$1$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+3q^{3}+q^{4}+3q^{6}+q^{7}+q^{8}+\cdots$
8450.2.a.z	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{3}) $	None	✓				130.2.e.d	$1$	$1$	$-2$	$-2$	$0$	$-2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots$
8450.2.a.ba	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{13}) $	None	✓				650.2.e.e	$1$	$0$	$-2$	$-1$	$0$	$-5$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+(-3+\beta )q^{7}+\cdots$
8450.2.a.bb	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{21}) $	None	✓				650.2.e.d	$1$	$0$	$-2$	$-1$	$0$	$1$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta q^{3}+q^{4}+\beta q^{6}+(1-\beta )q^{7}+\cdots$
8450.2.a.bc	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{10}) $	None	✓				130.2.e.c	$1$	$1$	$-2$	$0$	$0$	$2$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots$
8450.2.a.bd	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{13}) $	None	✓				650.2.e.e	$1$	$1$	$-2$	$1$	$0$	$-5$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(-3+\beta )q^{7}+\cdots$
8450.2.a.be	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{21}) $	None	✓				650.2.e.d	$1$	$1$	$-2$	$1$	$0$	$1$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(1-\beta )q^{7}+\cdots$
8450.2.a.bf	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{3}) $	None	✓				130.2.l.a	$1$	$0$	$-2$	$2$	$0$	$6$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots$
8450.2.a.bg	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{3}) $	None	✓				130.2.e.d	$1$	$0$	$2$	$-2$	$0$	$2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots$
8450.2.a.bh	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{21}) $	None	✓				650.2.e.d	$1$	$1$	$2$	$-1$	$0$	$-1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+(-1+\beta )q^{7}+\cdots$
8450.2.a.bi	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{13}) $	None	✓				650.2.e.e	$1$	$1$	$2$	$-1$	$0$	$5$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+(3-\beta )q^{7}+\cdots$
8450.2.a.bj	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{10}) $	None	✓				130.2.e.c	$1$	$0$	$2$	$0$	$0$	$-2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}-q^{7}+q^{8}+\cdots$
8450.2.a.bk	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{21}) $	None	✓				650.2.e.d	$1$	$0$	$2$	$1$	$0$	$-1$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(-1+\beta )q^{7}+\cdots$
8450.2.a.bl	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{13}) $	None	✓				650.2.e.e	$1$	$0$	$2$	$1$	$0$	$5$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(3-\beta )q^{7}+\cdots$
8450.2.a.bm	$8450$	$2$	8450.a	1.a	$1$	$2$	$2$	$67.474$	$\Q(\sqrt{3}) $	None	✓				130.2.l.a	$1$	$1$	$2$	$2$	$0$	$-6$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots$
8450.2.a.bn	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓				338.2.a.g	$1$	$1$	$-3$	$-3$	$0$	$4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(1+\cdots)q^{6}+\cdots$
8450.2.a.bo	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓				1690.2.a.q	$1$	$1$	$-3$	$-1$	$0$	$-5$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.bp	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	✓			8450.2.a.bp	$1$	$1$	$-3$	$-1$	$0$	$4$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bq	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.2808.1	None	✓				650.2.d.c	$1$	$1$	$-3$	$0$	$0$	$-3$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.br	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.2808.1	None	✓				650.2.d.c	$1$	$0$	$-3$	$0$	$0$	$-3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.bs	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.940.1	None	✓				130.2.b.a	$1$	$0$	$-3$	$0$	$0$	$2$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bt	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.148.1	None	✓				130.2.n.a	$1$	$0$	$-3$	$0$	$0$	$5$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bu	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.148.1	None	✓				130.2.n.a	$1$	$0$	$-3$	$0$	$0$	$5$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bv	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓				1690.2.a.p	$1$	$0$	$-3$	$1$	$0$	$-2$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-2\beta _{1}q^{7}+\cdots$
8450.2.a.bw	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	✓			8450.2.a.bp	$1$	$1$	$-3$	$1$	$0$	$4$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(2-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.bx	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓				338.2.a.g	$1$	$1$	$3$	$-3$	$0$	$-4$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-1+\cdots)q^{6}+\cdots$
8450.2.a.by	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	✓			8450.2.a.bp	$1$	$1$	$3$	$-1$	$0$	$-4$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-2+\cdots)q^{7}+\cdots$
8450.2.a.bz	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓				1690.2.a.q	$1$	$1$	$3$	$-1$	$0$	$5$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.ca	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.148.1	None	✓				130.2.n.a	$1$	$1$	$3$	$0$	$0$	$-5$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+(-2+\cdots)q^{7}+\cdots$
8450.2.a.cb	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.148.1	None	✓				130.2.n.a	$1$	$1$	$3$	$0$	$0$	$-5$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+(-2+\cdots)q^{7}+\cdots$
8450.2.a.cc	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.940.1	None	✓				130.2.b.a	$1$	$1$	$3$	$0$	$0$	$-2$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.cd	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.2808.1	None	✓				650.2.d.c	$1$	$1$	$3$	$0$	$0$	$3$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.ce	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	3.3.2808.1	None	✓				650.2.d.c	$1$	$0$	$3$	$0$	$0$	$3$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.cf	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓	✓			8450.2.a.bp	$1$	$1$	$3$	$1$	$0$	$-4$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-2+\cdots)q^{7}+\cdots$
8450.2.a.cg	$8450$	$2$	8450.a	1.a	$1$	$3$	$3$	$67.474$	$\Q(\zeta_{14})^+$	None	✓				1690.2.a.p	$1$	$0$	$3$	$1$	$0$	$2$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+2\beta _{1}q^{7}+\cdots$
8450.2.a.ch	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓				650.2.m.b	$1$	$0$	$-4$	$-2$	$0$	$0$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(\beta _{1}-\beta _{3})q^{7}+\cdots$
8450.2.a.ci	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓				130.2.l.b	$1$	$0$	$-4$	$-2$	$0$	$0$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+q^{4}+(1+\cdots)q^{6}+\cdots$
8450.2.a.cj	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	$\Q(\sqrt{3}, \sqrt{11})$	None	✓				130.2.c.a	$2$	$1$	$-4$	$0$	$0$	$-2$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+\beta _{2}q^{7}+\cdots$
8450.2.a.ck	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓				650.2.m.b	$1$	$1$	$-4$	$2$	$0$	$0$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{1}-\beta _{3})q^{7}+\cdots$
8450.2.a.cl	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓				650.2.m.b	$1$	$1$	$4$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.cm	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓				130.2.l.b	$1$	$1$	$4$	$-2$	$0$	$0$	$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+(-1-\beta _{2}+\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{6}+\cdots$
8450.2.a.cn	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	$\Q(\sqrt{3}, \sqrt{11})$	None	✓				130.2.c.a	$2$	$0$	$4$	$0$	$0$	$2$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}-\beta _{2}q^{7}+\cdots$
8450.2.a.co	$8450$	$2$	8450.a	1.a	$1$	$4$	$4$	$67.474$	4.4.4752.1	None	✓				650.2.m.b	$1$	$0$	$4$	$2$	$0$	$0$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$
8450.2.a.cp	$8450$	$2$	8450.a	1.a	$1$	$6$	$6$	$67.474$	6.6.20439713.1	None	✓				1690.2.a.v	$1$	$0$	$-6$	$2$	$0$	$-3$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots$
8450.2.a.cq	$8450$	$2$	8450.a	1.a	$1$	$6$	$6$	$67.474$	6.6.20439713.1	None	✓				1690.2.a.v	$1$	$0$	$6$	$2$	$0$	$3$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-\beta _{2}+\cdots)q^{7}+\cdots$
8450.2.a.cr	$8450$	$2$	8450.a	1.a	$1$	$8$	$8$	$67.474$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				130.2.m.a	$2$	$1$	$-8$	$0$	$0$	$-10$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots$
8450.2.a.cs	$8450$	$2$	8450.a	1.a	$1$	$8$	$8$	$67.474$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓				130.2.m.a	$2$	$0$	$8$	$0$	$0$	$10$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{7})q^{7}+\cdots$
8450.2.a.ct	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓		✓		1690.2.b.f	$1$	$1$	$-9$	$-7$	$0$	$-1$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1-\beta _{4})q^{3}+q^{4}+(1+\beta _{4}+\cdots)q^{6}+\cdots$
8450.2.a.cu	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	✓	✓		8450.2.a.cu	$1$	$0$	$-9$	$0$	$0$	$4$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-\beta _{8}q^{7}+\cdots$
8450.2.a.cv	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	✓			8450.2.a.cu	$1$	$0$	$-9$	$0$	$0$	$4$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}-\beta _{8}q^{7}+\cdots$
8450.2.a.cw	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓				1690.2.b.f	$1$	$0$	$-9$	$7$	$0$	$-1$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta _{4})q^{3}+q^{4}+(-1-\beta _{4}+\cdots)q^{6}+\cdots$
8450.2.a.cx	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓		✓		1690.2.b.f	$1$	$1$	$9$	$-7$	$0$	$1$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1-\beta _{4})q^{3}+q^{4}+(-1-\beta _{4}+\cdots)q^{6}+\cdots$
8450.2.a.cy	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	✓			8450.2.a.cu	$1$	$0$	$9$	$0$	$0$	$-4$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+\beta _{8}q^{7}+\cdots$
8450.2.a.cz	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	✓	✓		8450.2.a.cu	$1$	$0$	$9$	$0$	$0$	$-4$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+\beta _{8}q^{7}+\cdots$
8450.2.a.da	$8450$	$2$	8450.a	1.a	$1$	$9$	$9$	$67.474$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓				1690.2.b.f	$1$	$0$	$9$	$7$	$0$	$1$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta _{4})q^{3}+q^{4}+(1+\beta _{4})q^{6}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(8450))$ into lower level spaces

Overview	Random
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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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 79 \)
Sturm bound:			\(2730\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(17\), \(31\)

\(2\)	\(5\)	\(13\)	Fricke	Total			Cusp			Eisenstein
\(2\)	\(5\)	\(13\)	Fricke	All	New	Old	All	New	Old	All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(173\)	\(24\)	\(149\)	\(153\)	\(24\)	\(129\)	\(20\)	\(0\)	\(20\)
\(+\)	\(+\)	\(-\)	\(-\)	\(188\)	\(33\)	\(155\)	\(167\)	\(33\)	\(134\)	\(21\)	\(0\)	\(21\)
\(+\)	\(-\)	\(+\)	\(-\)	\(187\)	\(34\)	\(153\)	\(166\)	\(34\)	\(132\)	\(21\)	\(0\)	\(21\)
\(+\)	\(-\)	\(-\)	\(+\)	\(175\)	\(31\)	\(144\)	\(154\)	\(31\)	\(123\)	\(21\)	\(0\)	\(21\)
\(-\)	\(+\)	\(+\)	\(-\)	\(184\)	\(35\)	\(149\)	\(163\)	\(35\)	\(128\)	\(21\)	\(0\)	\(21\)
\(-\)	\(+\)	\(-\)	\(+\)	\(178\)	\(24\)	\(154\)	\(157\)	\(24\)	\(133\)	\(21\)	\(0\)	\(21\)
\(-\)	\(-\)	\(+\)	\(+\)	\(177\)	\(27\)	\(150\)	\(156\)	\(27\)	\(129\)	\(21\)	\(0\)	\(21\)
\(-\)	\(-\)	\(-\)	\(-\)	\(186\)	\(37\)	\(149\)	\(165\)	\(37\)	\(128\)	\(21\)	\(0\)	\(21\)
Plus space			\(+\)	\(703\)	\(106\)	\(597\)	\(620\)	\(106\)	\(514\)	\(83\)	\(0\)	\(83\)
Minus space			\(-\)	\(745\)	\(139\)	\(606\)	\(661\)	\(139\)	\(522\)	\(84\)	\(0\)	\(84\)

Introduction

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8450))\) 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	8450.2.a

Level	$8450$
Weight	$2$
Character orbit	8450.a
Rep. character	$\chi_{8450}(1,\cdot)$
Character field	$\Q$
Dimension	$245$
Newform subspaces	$79$
Sturm bound	$2730$
Trace bound	$17$