LMFDB - Embedded newform 8550.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.2720937282$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 570)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} +2.00000 q^{11} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{22} -8.00000 q^{23} +2.00000 q^{28} -1.00000 q^{32} +2.00000 q^{34} -4.00000 q^{37} -1.00000 q^{38} +8.00000 q^{41} +6.00000 q^{43} +2.00000 q^{44} +8.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -10.0000 q^{53} -2.00000 q^{56} +8.00000 q^{59} +2.00000 q^{61} +1.00000 q^{64} -2.00000 q^{68} -8.00000 q^{71} +2.00000 q^{73} +4.00000 q^{74} +1.00000 q^{76} +4.00000 q^{77} -8.00000 q^{79} -8.00000 q^{82} -16.0000 q^{83} -6.00000 q^{86} -2.00000 q^{88} -16.0000 q^{89} -8.00000 q^{92} +8.00000 q^{94} -8.00000 q^{97} +3.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	1.00000	0.114708
$77$	4.00000	0.455842
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	−8.00000	−0.883452
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	0	0
$86$	−6.00000	−0.646997
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	0	0
$92$	−8.00000	−0.834058
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	0	0
$105$	0	0
$106$	10.0000	0.971286
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	2.00000	0.188982
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	0	0
$136$	2.00000	0.171499
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	8.00000	0.624695
$165$	0	0
$166$	16.0000	1.24184
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	6.00000	0.457496
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	16.0000	1.19925
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	8.00000	0.589768
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−8.00000	−0.583460
$189$	0	0
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	2.00000	0.140720
$203$	0	0
$204$	0	0
$205$	0	0
$206$	12.0000	0.836080
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−10.0000	−0.686803
$213$	0	0
$214$	−20.0000	−1.36717
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−2.00000	−0.135457
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	2.00000	0.133038
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	8.00000	0.520756
$237$	0	0
$238$	4.00000	0.259281
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	−6.00000	−0.370681
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	0	0
$269$	−8.00000	−0.487769	−0.243884	−	0.969804i	$-0.578422\pi$
$269$	−8.00000	−0.487769	−0.243884	+	0.969804i	$0.578422\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	−8.00000	−0.479808
$279$	0	0
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	0	0
$287$	16.0000	0.944450
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	2.00000	0.117041
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	2.00000	0.115857
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	8.00000	0.460348
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	24.0000	1.36975	0.684876	−	0.728659i	$-0.259856\pi$
$307$	24.0000	1.36975	0.684876	+	0.728659i	$0.259856\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	−18.0000	−1.01742	−0.508710	−	0.860938i	$-0.669877\pi$
$313$	−18.0000	−1.01742	−0.508710	+	0.860938i	$0.669877\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−8.00000	−0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	16.0000	0.891645
$323$	−2.00000	−0.111283
$324$	0	0
$325$	0	0
$326$	−10.0000	−0.553849
$327$	0	0
$328$	−8.00000	−0.441726
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−16.0000	−0.878114
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	13.0000	0.707107
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−6.00000	−0.323498
$345$	0	0
$346$	6.00000	0.322562
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	−16.0000	−0.847998
$357$	0	0
$358$	−8.00000	−0.422813
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	6.00000	0.315353
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−26.0000	−1.35719	−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000	−1.35719	−0.678594	+	0.734513i	$0.737411\pi$
$368$	−8.00000	−0.417029
$369$	0	0
$370$	0	0
$371$	−20.0000	−1.03835
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	10.0000	0.511645
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−8.00000	−0.406138
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	3.00000	0.151523
$393$	0	0
$394$	−10.0000	−0.503793
$395$	0	0
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	−12.0000	−0.591198
$413$	16.0000	0.787309
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	0	0
$427$	4.00000	0.193574
$428$	20.0000	0.966736
$429$	0	0
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	12.0000	0.576683	0.288342	−	0.957528i	$-0.406896\pi$
$433$	12.0000	0.576683	0.288342	+	0.957528i	$0.406896\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	−8.00000	−0.382692
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	0	0
$446$	12.0000	0.568216
$447$	0	0
$448$	2.00000	0.0944911
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−18.0000	−0.836531	−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000	−0.836531	−0.418265	+	0.908325i	$0.637362\pi$
$464$	0	0
$465$	0	0
$466$	6.00000	0.277945
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−8.00000	−0.368230
$473$	12.0000	0.551761
$474$	0	0
$475$	0	0
$476$	−4.00000	−0.183340
$477$	0	0
$478$	6.00000	0.274434
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	0	0
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	0	0
$502$	6.00000	0.267793
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	0	0
$505$	0	0
$506$	16.0000	0.711287
$507$	0	0
$508$	−16.0000	−0.709885
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−30.0000	−1.32324
$515$	0	0
$516$	0	0
$517$	−16.0000	−0.703679
$518$	8.00000	0.351500
$519$	0	0
$520$	0	0
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	12.0000	0.523225
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	0	0
$532$	2.00000	0.0867110
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	8.00000	0.344904
$539$	−6.00000	−0.258438
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	4.00000	0.171815
$543$	0	0
$544$	2.00000	0.0857493
$545$	0	0
$546$	0	0
$547$	40.0000	1.71028	0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000	1.71028	0.855138	+	0.518400i	$0.173472\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−14.0000	−0.594803
$555$	0	0
$556$	8.00000	0.339276
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−20.0000	−0.843649
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	8.00000	0.335673
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	−16.0000	−0.667827
$575$	0	0
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	0	0
$581$	−32.0000	−1.32758
$582$	0	0
$583$	−20.0000	−0.828315
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	22.0000	0.908812
$587$	−44.0000	−1.81607	−0.908037	−	0.418890i	$-0.862419\pi$
$587$	−44.0000	−1.81607	−0.908037	+	0.418890i	$0.862419\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−4.00000	−0.164399
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	0	0
$596$	−2.00000	−0.0819232
$597$	0	0
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	−44.0000	−1.78590	−0.892952	−	0.450151i	$-0.851370\pi$
$607$	−44.0000	−1.78590	−0.892952	+	0.450151i	$0.851370\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	−24.0000	−0.968561
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	0	0
$622$	−14.0000	−0.561349
$623$	−32.0000	−1.28205
$624$	0	0
$625$	0	0
$626$	18.0000	0.719425
$627$	0	0
$628$	6.00000	0.239426
$629$	8.00000	0.318981
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−16.0000	−0.631962	−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000	−0.631962	−0.315981	+	0.948766i	$0.602334\pi$
$642$	0	0
$643$	−46.0000	−1.81406	−0.907031	−	0.421063i	$-0.861657\pi$
$643$	−46.0000	−1.81406	−0.907031	+	0.421063i	$0.861657\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	2.00000	0.0786889
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	10.0000	0.391630
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	0	0
$656$	8.00000	0.312348
$657$	0	0
$658$	16.0000	0.623745
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	16.0000	0.620920
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	−16.0000	−0.616297
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	6.00000	0.228748
$689$	0	0
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	8.00000	0.303676
$695$	0	0
$696$	0	0
$697$	−16.0000	−0.606043
$698$	26.0000	0.984115
$699$	0	0
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	2.00000	0.0753778
$705$	0	0
$706$	−18.0000	−0.677439
$707$	−4.00000	−0.150435
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	0	0
$712$	16.0000	0.599625
$713$	0	0
$714$	0	0
$715$	0	0
$716$	8.00000	0.298974
$717$	0	0
$718$	−30.0000	−1.11959
$719$	−22.0000	−0.820462	−0.410231	−	0.911982i	$-0.634552\pi$
$719$	−22.0000	−0.820462	−0.410231	+	0.911982i	$0.634552\pi$
$720$	0	0
$721$	−24.0000	−0.893807
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−6.00000	−0.222988
$725$	0	0
$726$	0	0
$727$	22.0000	0.815935	0.407967	−	0.912996i	$-0.366238\pi$
$727$	22.0000	0.815935	0.407967	+	0.912996i	$0.366238\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	8.00000	0.294884
$737$	0	0
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	20.0000	0.734223
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	32.0000	1.17160
$747$	0	0
$748$	−4.00000	−0.146254
$749$	40.0000	1.46157
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	−10.0000	−0.361787
$765$	0	0
$766$	−8.00000	−0.289052
$767$	0	0
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	0	0
$772$	4.00000	0.143963
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	0	0
$775$	0	0
$776$	8.00000	0.287183
$777$	0	0
$778$	14.0000	0.501924
$779$	8.00000	0.286630
$780$	0	0
$781$	−16.0000	−0.572525
$782$	−16.0000	−0.572159
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	−52.0000	−1.85360	−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000	−1.85360	−0.926800	+	0.375555i	$0.877452\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	0	0
$794$	−6.00000	−0.212932
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	−12.0000	−0.423735
$803$	4.00000	0.141157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	2.00000	0.0703598
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	8.00000	0.280400
$815$	0	0
$816$	0	0
$817$	6.00000	0.209913
$818$	18.0000	0.629355
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	−16.0000	−0.556711
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	2.00000	0.0691714
$837$	0	0
$838$	30.0000	1.03633
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	26.0000	0.896019
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	−14.0000	−0.481046
$848$	−10.0000	−0.343401
$849$	0	0
$850$	0	0
$851$	32.0000	1.09695
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−20.0000	−0.683586
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	20.0000	0.681203
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	0	0
$866$	−12.0000	−0.407777
$867$	0	0
$868$	0	0
$869$	−16.0000	−0.542763
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	0	0
$874$	8.00000	0.270604
$875$	0	0
$876$	0	0
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	−16.0000	−0.539974
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	18.0000	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$883$	18.0000	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	0	0
$891$	0	0
$892$	−12.0000	−0.401790
$893$	−8.00000	−0.267710
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	24.0000	0.800890
$899$	0	0
$900$	0	0
$901$	20.0000	0.666297
$902$	−16.0000	−0.532742
$903$	0	0
$904$	2.00000	0.0665190
$905$	0	0
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	−22.0000	−0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	12.0000	0.396275
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	0	0
$922$	−6.00000	−0.197599
$923$	0	0
$924$	0	0
$925$	0	0
$926$	18.0000	0.591517
$927$	0	0
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−20.0000	−0.654420
$935$	0	0
$936$	0	0
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−16.0000	−0.521585	−0.260793	−	0.965395i	$-0.583984\pi$
$941$	−16.0000	−0.521585	−0.260793	+	0.965395i	$0.583984\pi$
$942$	0	0
$943$	−64.0000	−2.08413
$944$	8.00000	0.260378
$945$	0	0
$946$	−12.0000	−0.390154
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	4.00000	0.129641
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	−30.0000	−0.969256
$959$	36.0000	1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	6.00000	0.192947	0.0964735	−	0.995336i	$-0.469244\pi$
$967$	6.00000	0.192947	0.0964735	+	0.995336i	$0.469244\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	20.0000	0.640841
$975$	0	0
$976$	2.00000	0.0640184
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	0	0
$982$	−30.0000	−0.957338
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	0	0
$994$	16.0000	0.507489
$995$	0	0
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	16.0000	0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.o.1.1		1
3.2	odd	2		2850.2.a.ba.1.1		1
5.4	even	2		1710.2.a.n.1.1		1
15.2	even	4		2850.2.d.n.799.2		2
15.8	even	4		2850.2.d.n.799.1		2
15.14	odd	2		570.2.a.c.1.1	✓	1
60.59	even	2		4560.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.a.c.1.1	✓	1	15.14	odd	2
1710.2.a.n.1.1		1	5.4	even	2
2850.2.a.ba.1.1		1	3.2	odd	2
2850.2.d.n.799.1		2	15.8	even	4
2850.2.d.n.799.2		2	15.2	even	4
4560.2.a.bd.1.1		1	60.59	even	2
8550.2.a.o.1.1		1	1.1	even	1	trivial

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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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