LMFDB - Embedded newform 8464.2.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,-2,0,-2,0,-2,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -2.00000 q^{5} -2.00000 q^{7} -2.00000 q^{9} -4.00000 q^{11} +4.00000 q^{13} -2.00000 q^{15} -7.00000 q^{17} -3.00000 q^{19} -2.00000 q^{21} -1.00000 q^{25} -5.00000 q^{27} -4.00000 q^{29} -6.00000 q^{31} -4.00000 q^{33} +4.00000 q^{35} +2.00000 q^{37} +4.00000 q^{39} +6.00000 q^{41} -5.00000 q^{43} +4.00000 q^{45} +10.0000 q^{47} -3.00000 q^{49} -7.00000 q^{51} +8.00000 q^{55} -3.00000 q^{57} +5.00000 q^{59} -4.00000 q^{61} +4.00000 q^{63} -8.00000 q^{65} -5.00000 q^{67} -14.0000 q^{71} +15.0000 q^{73} -1.00000 q^{75} +8.00000 q^{77} +12.0000 q^{79} +1.00000 q^{81} +15.0000 q^{83} +14.0000 q^{85} -4.00000 q^{87} -10.0000 q^{89} -8.00000 q^{91} -6.00000 q^{93} +6.00000 q^{95} +10.0000 q^{97} +8.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−7.00000	−0.980196
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	15.0000	1.75562	0.877809	−	0.479012i	$-0.159005\pi$
$73$	15.0000	1.75562	0.877809	+	0.479012i	$0.159005\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	14.0000	1.51851
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	8.00000	0.804030
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.00000	−0.739600
$118$	0	0
$119$	14.0000	1.28338
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	0	0
$129$	−5.00000	−0.440225
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	5.00000	0.427179	0.213589	−	0.976924i	$-0.431485\pi$
$137$	5.00000	0.427179	0.213589	+	0.976924i	$0.431485\pi$
$138$	0	0
$139$	−17.0000	−1.44192	−0.720961	−	0.692976i	$-0.756299\pi$
$139$	−17.0000	−1.44192	−0.720961	+	0.692976i	$0.756299\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	0	0
$143$	−16.0000	−1.33799
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	0	0
$153$	14.0000	1.13183
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	5.00000	0.375823
$178$	0	0
$179$	−17.0000	−1.27064	−0.635320	−	0.772249i	$-0.719132\pi$
$179$	−17.0000	−1.27064	−0.635320	+	0.772249i	$0.719132\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	28.0000	2.04756
$188$	0	0
$189$	10.0000	0.727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	25.0000	1.79954	0.899770	−	0.436365i	$-0.143734\pi$
$193$	25.0000	1.79954	0.899770	+	0.436365i	$0.143734\pi$
$194$	0	0
$195$	−8.00000	−0.572892
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	0	0
$201$	−5.00000	−0.352673
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	0	0
$213$	−14.0000	−0.959264
$214$	0	0
$215$	10.0000	0.681994
$216$	0	0
$217$	12.0000	0.814613
$218$	0	0
$219$	15.0000	1.01361
$220$	0	0
$221$	−28.0000	−1.88348
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	−17.0000	−1.12833	−0.564165	−	0.825662i	$-0.690802\pi$
$227$	−17.0000	−1.12833	−0.564165	+	0.825662i	$0.690802\pi$
$228$	0	0
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	0	0
$235$	−20.0000	−1.30466
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	15.0000	0.950586
$250$	0	0
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	14.0000	0.876714
$256$	0	0
$257$	15.0000	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$257$	15.0000	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−10.0000	−0.611990
$268$	0	0
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	12.0000	0.718421
$280$	0	0
$281$	−9.00000	−0.536895	−0.268447	−	0.963294i	$-0.586511\pi$
$281$	−9.00000	−0.536895	−0.268447	+	0.963294i	$0.586511\pi$
$282$	0	0
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	32.0000	1.88235
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	0	0
$297$	20.0000	1.16052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	10.0000	0.576390
$302$	0	0
$303$	4.00000	0.229794
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	5.00000	0.282617	0.141308	−	0.989966i	$-0.454869\pi$
$313$	5.00000	0.282617	0.141308	+	0.989966i	$0.454869\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	21.0000	1.16847
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−18.0000	−0.995402
$328$	0	0
$329$	−20.0000	−1.10264
$330$	0	0
$331$	29.0000	1.59398	0.796992	−	0.603990i	$-0.206423\pi$
$331$	29.0000	1.59398	0.796992	+	0.603990i	$0.206423\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	10.0000	0.546358
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	15.0000	0.814688
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.00000	0.375780	0.187890	−	0.982190i	$-0.439835\pi$
$347$	7.00000	0.375780	0.187890	+	0.982190i	$0.439835\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	−20.0000	−1.06752
$352$	0	0
$353$	15.0000	0.798369	0.399185	−	0.916871i	$-0.369293\pi$
$353$	15.0000	0.798369	0.399185	+	0.916871i	$0.369293\pi$
$354$	0	0
$355$	28.0000	1.48609
$356$	0	0
$357$	14.0000	0.740959
$358$	0	0
$359$	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
$361$	−10.0000	−0.526316
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−30.0000	−1.57027
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	−16.0000	−0.824042
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	10.0000	0.512316
$382$	0	0
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.00000	−0.453990
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	9.00000	0.449439	0.224719	−	0.974424i	$-0.427853\pi$
$401$	9.00000	0.449439	0.224719	+	0.974424i	$0.427853\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	5.00000	0.246632
$412$	0	0
$413$	−10.0000	−0.492068
$414$	0	0
$415$	−30.0000	−1.47264
$416$	0	0
$417$	−17.0000	−0.832494
$418$	0	0
$419$	13.0000	0.635092	0.317546	−	0.948243i	$-0.397141\pi$
$419$	13.0000	0.635092	0.317546	+	0.948243i	$0.397141\pi$
$420$	0	0
$421$	16.0000	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$422$	0	0
$423$	−20.0000	−0.972433
$424$	0	0
$425$	7.00000	0.339550
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	−16.0000	−0.772487
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−34.0000	−1.62273	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	−34.0000	−1.62273	−0.811366	+	0.584539i	$0.801275\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	31.0000	1.47285	0.736427	−	0.676517i	$-0.236511\pi$
$443$	31.0000	1.47285	0.736427	+	0.676517i	$0.236511\pi$
$444$	0	0
$445$	20.0000	0.948091
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	25.0000	1.17982	0.589911	−	0.807468i	$-0.299163\pi$
$449$	25.0000	1.17982	0.589911	+	0.807468i	$0.299163\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	−14.0000	−0.657777
$454$	0	0
$455$	16.0000	0.750092
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	35.0000	1.63366
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	0	0
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	10.0000	0.461757
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	20.0000	0.919601
$474$	0	0
$475$	3.00000	0.137649
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.0000	−0.908153
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	28.0000	1.26106
$494$	0	0
$495$	−16.0000	−0.719147
$496$	0	0
$497$	28.0000	1.25597
$498$	0	0
$499$	43.0000	1.92494	0.962472	−	0.271380i	$-0.0874801\pi$
$499$	43.0000	1.92494	0.962472	+	0.271380i	$0.0874801\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−30.0000	−1.32712
$512$	0	0
$513$	15.0000	0.662266
$514$	0	0
$515$	20.0000	0.881305
$516$	0	0
$517$	−40.0000	−1.75920
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	42.0000	1.82955
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−10.0000	−0.433963
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	−24.0000	−1.03761
$536$	0	0
$537$	−17.0000	−0.733604
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	−36.0000	−1.54776	−0.773880	−	0.633332i	$-0.781687\pi$
$541$	−36.0000	−1.54776	−0.773880	+	0.633332i	$0.781687\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	36.0000	1.54207
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	28.0000	1.18216
$562$	0	0
$563$	−15.0000	−0.632175	−0.316087	−	0.948730i	$-0.602369\pi$
$563$	−15.0000	−0.632175	−0.316087	+	0.948730i	$0.602369\pi$
$564$	0	0
$565$	−30.0000	−1.26211
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	13.0000	0.544988	0.272494	−	0.962157i	$-0.412151\pi$
$569$	13.0000	0.544988	0.272494	+	0.962157i	$0.412151\pi$
$570$	0	0
$571$	11.0000	0.460336	0.230168	−	0.973151i	$-0.426072\pi$
$571$	11.0000	0.460336	0.230168	+	0.973151i	$0.426072\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	27.0000	1.12402	0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000	1.12402	0.562012	+	0.827129i	$0.310027\pi$
$578$	0	0
$579$	25.0000	1.03896
$580$	0	0
$581$	−30.0000	−1.24461
$582$	0	0
$583$	0	0
$584$	0	0
$585$	16.0000	0.661519
$586$	0	0
$587$	35.0000	1.44460	0.722302	−	0.691577i	$-0.243084\pi$
$587$	35.0000	1.44460	0.722302	+	0.691577i	$0.243084\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	−25.0000	−1.02663	−0.513313	−	0.858201i	$-0.671582\pi$
$593$	−25.0000	−1.02663	−0.513313	+	0.858201i	$0.671582\pi$
$594$	0	0
$595$	−28.0000	−1.14789
$596$	0	0
$597$	−22.0000	−0.900400
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	10.0000	0.407231
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	40.0000	1.61823
$612$	0	0
$613$	4.00000	0.161558	0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000	0.161558	0.0807792	+	0.996732i	$0.474259\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	3.00000	0.120775	0.0603877	−	0.998175i	$-0.480766\pi$
$617$	3.00000	0.120775	0.0603877	+	0.998175i	$0.480766\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	20.0000	0.801283
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	−14.0000	−0.558217
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	−11.0000	−0.437211
$634$	0	0
$635$	−20.0000	−0.793676
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	28.0000	1.10766
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	10.0000	0.393750
$646$	0	0
$647$	20.0000	0.786281	0.393141	−	0.919478i	$-0.371389\pi$
$647$	20.0000	0.786281	0.393141	+	0.919478i	$0.371389\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	18.0000	0.703318
$656$	0	0
$657$	−30.0000	−1.17041
$658$	0	0
$659$	7.00000	0.272681	0.136341	−	0.990662i	$-0.456466\pi$
$659$	7.00000	0.272681	0.136341	+	0.990662i	$0.456466\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	0	0
$663$	−28.0000	−1.08743
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	16.0000	0.617673
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	10.0000	0.384331	0.192166	−	0.981363i	$-0.438449\pi$
$677$	10.0000	0.384331	0.192166	+	0.981363i	$0.438449\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−17.0000	−0.651441
$682$	0	0
$683$	11.0000	0.420903	0.210452	−	0.977604i	$-0.432507\pi$
$683$	11.0000	0.420903	0.210452	+	0.977604i	$0.432507\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	28.0000	1.06827
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	0	0
$693$	−16.0000	−0.607790
$694$	0	0
$695$	34.0000	1.28969
$696$	0	0
$697$	−42.0000	−1.59086
$698$	0	0
$699$	15.0000	0.567352
$700$	0	0
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	−20.0000	−0.753244
$706$	0	0
$707$	−8.00000	−0.300871
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	0	0
$713$	0	0
$714$	0	0
$715$	32.0000	1.19673
$716$	0	0
$717$	8.00000	0.298765
$718$	0	0
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	0	0
$723$	−19.0000	−0.706618
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	35.0000	1.29452
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	20.0000	0.736709
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	0	0
$747$	−30.0000	−1.09764
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	21.0000	0.765283
$754$	0	0
$755$	28.0000	1.01902
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.0000	−1.12375	−0.561875	−	0.827222i	$-0.689920\pi$
$761$	−31.0000	−1.12375	−0.561875	+	0.827222i	$0.689920\pi$
$762$	0	0
$763$	36.0000	1.30329
$764$	0	0
$765$	−28.0000	−1.01234
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	15.0000	0.540212
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	56.0000	2.00384
$782$	0	0
$783$	20.0000	0.714742
$784$	0	0
$785$	24.0000	0.856597
$786$	0	0
$787$	33.0000	1.17632	0.588161	−	0.808744i	$-0.299852\pi$
$787$	33.0000	1.17632	0.588161	+	0.808744i	$0.299852\pi$
$788$	0	0
$789$	10.0000	0.356009
$790$	0	0
$791$	−30.0000	−1.06668
$792$	0	0
$793$	−16.0000	−0.568177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	−70.0000	−2.47642
$800$	0	0
$801$	20.0000	0.706665
$802$	0	0
$803$	−60.0000	−2.11735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	−23.0000	−0.808637	−0.404318	−	0.914618i	$-0.632491\pi$
$809$	−23.0000	−0.808637	−0.404318	+	0.914618i	$0.632491\pi$
$810$	0	0
$811$	−1.00000	−0.0351147	−0.0175574	−	0.999846i	$-0.505589\pi$
$811$	−1.00000	−0.0351147	−0.0175574	+	0.999846i	$0.505589\pi$
$812$	0	0
$813$	−10.0000	−0.350715
$814$	0	0
$815$	32.0000	1.12091
$816$	0	0
$817$	15.0000	0.524784
$818$	0	0
$819$	16.0000	0.559085
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	0	0
$833$	21.0000	0.727607
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	30.0000	1.03695
$838$	0	0
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−9.00000	−0.309976
$844$	0	0
$845$	−6.00000	−0.206406
$846$	0	0
$847$	−10.0000	−0.343604
$848$	0	0
$849$	1.00000	0.0343199
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	−35.0000	−1.19558	−0.597789	−	0.801654i	$-0.703954\pi$
$857$	−35.0000	−1.19558	−0.597789	+	0.801654i	$0.703954\pi$
$858$	0	0
$859$	19.0000	0.648272	0.324136	−	0.946011i	$-0.394927\pi$
$859$	19.0000	0.648272	0.324136	+	0.946011i	$0.394927\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	0	0
$867$	32.0000	1.08678
$868$	0	0
$869$	−48.0000	−1.62829
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	−20.0000	−0.676897
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	40.0000	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$877$	40.0000	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$878$	0	0
$879$	10.0000	0.337292
$880$	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
$883$	15.0000	0.504790	0.252395	−	0.967624i	$-0.418782\pi$
$883$	15.0000	0.504790	0.252395	+	0.967624i	$0.418782\pi$
$884$	0	0
$885$	−10.0000	−0.336146
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	−30.0000	−1.00391
$894$	0	0
$895$	34.0000	1.13649
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	10.0000	0.332779
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	−23.0000	−0.763702	−0.381851	−	0.924224i	$-0.624713\pi$
$907$	−23.0000	−0.763702	−0.381851	+	0.924224i	$0.624713\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	0	0
$915$	8.00000	0.264472
$916$	0	0
$917$	18.0000	0.594412
$918$	0	0
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	0	0
$923$	−56.0000	−1.84326
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	20.0000	0.656886
$928$	0	0
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−56.0000	−1.83140
$936$	0	0
$937$	55.0000	1.79677	0.898386	−	0.439207i	$-0.144741\pi$
$937$	55.0000	1.79677	0.898386	+	0.439207i	$0.144741\pi$
$938$	0	0
$939$	5.00000	0.163169
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−20.0000	−0.650600
$946$	0	0
$947$	−5.00000	−0.162478	−0.0812391	−	0.996695i	$-0.525888\pi$
$947$	−5.00000	−0.162478	−0.0812391	+	0.996695i	$0.525888\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	50.0000	1.61966	0.809829	−	0.586665i	$-0.199560\pi$
$953$	50.0000	1.61966	0.809829	+	0.586665i	$0.199560\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.0000	0.517207
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−24.0000	−0.773389
$964$	0	0
$965$	−50.0000	−1.60956
$966$	0	0
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	0	0
$969$	21.0000	0.674617
$970$	0	0
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	34.0000	1.08999
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	36.0000	1.14939
$982$	0	0
$983$	−10.0000	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$983$	−10.0000	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	−20.0000	−0.636607
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−30.0000	−0.952981	−0.476491	−	0.879180i	$-0.658091\pi$
$991$	−30.0000	−0.952981	−0.476491	+	0.879180i	$0.658091\pi$
$992$	0	0
$993$	29.0000	0.920287
$994$	0	0
$995$	44.0000	1.39489
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.k.1.1		1
4.3	odd	2		4232.2.a.b.1.1	✓	1
23.22	odd	2		8464.2.a.o.1.1		1
92.91	even	2		4232.2.a.c.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.b.1.1	✓	1	4.3	odd	2
4232.2.a.c.1.1	yes	1	92.91	even	2
8464.2.a.k.1.1		1	1.1	even	1	trivial
8464.2.a.o.1.1		1	23.22	odd	2

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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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