LMFDB - Embedded newform 8450.2.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-3,1,0,-3,0,1,6,0,3,-3,0,0,0,1,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 650)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +3.00000 q^{11} -3.00000 q^{12} +1.00000 q^{16} +7.00000 q^{17} +6.00000 q^{18} -1.00000 q^{19} +3.00000 q^{22} +4.00000 q^{23} -3.00000 q^{24} -9.00000 q^{27} +4.00000 q^{29} +10.0000 q^{31} +1.00000 q^{32} -9.00000 q^{33} +7.00000 q^{34} +6.00000 q^{36} +12.0000 q^{37} -1.00000 q^{38} +5.00000 q^{41} -12.0000 q^{43} +3.00000 q^{44} +4.00000 q^{46} -4.00000 q^{47} -3.00000 q^{48} -7.00000 q^{49} -21.0000 q^{51} -6.00000 q^{53} -9.00000 q^{54} +3.00000 q^{57} +4.00000 q^{58} +4.00000 q^{59} +4.00000 q^{61} +10.0000 q^{62} +1.00000 q^{64} -9.00000 q^{66} -5.00000 q^{67} +7.00000 q^{68} -12.0000 q^{69} +6.00000 q^{72} -11.0000 q^{73} +12.0000 q^{74} -1.00000 q^{76} +4.00000 q^{79} +9.00000 q^{81} +5.00000 q^{82} +15.0000 q^{83} -12.0000 q^{86} -12.0000 q^{87} +3.00000 q^{88} +11.0000 q^{89} +4.00000 q^{92} -30.0000 q^{93} -4.00000 q^{94} -3.00000 q^{96} -2.00000 q^{97} -7.00000 q^{98} +18.0000 q^{99} +O(q^{100})$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.00000	−1.22474
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−3.00000	−0.866025
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	6.00000	1.41421
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	3.00000	0.639602
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000	0.176777
$33$	−9.00000	−1.56670
$34$	7.00000	1.20049
$35$	0	0
$36$	6.00000	1.00000
$37$	12.0000	1.97279	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	12.0000	1.97279	0.986394	+	0.164399i	$0.0525685\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	−3.00000	−0.433013
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−21.0000	−2.94059
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−9.00000	−1.22474
$55$	0	0
$56$	0	0
$57$	3.00000	0.397360
$58$	4.00000	0.525226
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	10.0000	1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−9.00000	−1.10782
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	7.00000	0.848875
$69$	−12.0000	−1.44463
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	6.00000	0.707107
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	−1.00000	−0.114708
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	5.00000	0.552158
$83$	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
$85$	0	0
$86$	−12.0000	−1.29399
$87$	−12.0000	−1.28654
$88$	3.00000	0.319801
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	−30.0000	−3.11086
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−3.00000	−0.306186
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−7.00000	−0.707107
$99$	18.0000	1.80907
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	−21.0000	−2.07931
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	9.00000	0.870063	0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000	0.870063	0.435031	+	0.900415i	$0.356737\pi$
$108$	−9.00000	−0.866025
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	−36.0000	−3.41697
$112$	0	0
$113$	−19.0000	−1.78737	−0.893685	−	0.448695i	$-0.851889\pi$
$113$	−19.0000	−1.78737	−0.893685	+	0.448695i	$0.851889\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	4.00000	0.371391
$117$	0	0
$118$	4.00000	0.368230
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	4.00000	0.362143
$123$	−15.0000	−1.35250
$124$	10.0000	0.898027
$125$	0	0
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	1.00000	0.0883883
$129$	36.0000	3.16962
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	−9.00000	−0.783349
$133$	0	0
$134$	−5.00000	−0.431934
$135$	0	0
$136$	7.00000	0.600245
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	−12.0000	−1.02151
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	6.00000	0.500000
$145$	0	0
$146$	−11.0000	−0.910366
$147$	21.0000	1.73205
$148$	12.0000	0.986394
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−1.00000	−0.0811107
$153$	42.0000	3.39550
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.0000	−0.957704	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	−12.0000	−0.957704	−0.478852	+	0.877896i	$0.658947\pi$
$158$	4.00000	0.318223
$159$	18.0000	1.42749
$160$	0	0
$161$	0	0
$162$	9.00000	0.707107
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	15.0000	1.16423
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−6.00000	−0.458831
$172$	−12.0000	−0.914991
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−12.0000	−0.909718
$175$	0	0
$176$	3.00000	0.226134
$177$	−12.0000	−0.901975
$178$	11.0000	0.824485
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	−12.0000	−0.887066
$184$	4.00000	0.294884
$185$	0	0
$186$	−30.0000	−2.19971
$187$	21.0000	1.53567
$188$	−4.00000	−0.291730
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−3.00000	−0.216506
$193$	15.0000	1.07972	0.539862	−	0.841754i	$-0.318476\pi$
$193$	15.0000	1.07972	0.539862	+	0.841754i	$0.318476\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	16.0000	1.13995	0.569976	−	0.821661i	$-0.306952\pi$
$197$	16.0000	1.13995	0.569976	+	0.821661i	$0.306952\pi$
$198$	18.0000	1.27920
$199$	−6.00000	−0.425329	−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000	−0.425329	−0.212664	+	0.977125i	$0.568214\pi$
$200$	0	0
$201$	15.0000	1.05802
$202$	12.0000	0.844317
$203$	0	0
$204$	−21.0000	−1.47029
$205$	0	0
$206$	−14.0000	−0.975426
$207$	24.0000	1.66812
$208$	0	0
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	9.00000	0.615227
$215$	0	0
$216$	−9.00000	−0.612372
$217$	0	0
$218$	8.00000	0.541828
$219$	33.0000	2.22993
$220$	0	0
$221$	0	0
$222$	−36.0000	−2.41616
$223$	−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
$225$	0	0
$226$	−19.0000	−1.26386
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	3.00000	0.198680
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	0	0
$232$	4.00000	0.262613
$233$	−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
$236$	4.00000	0.260378
$237$	−12.0000	−0.779484
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	4.00000	0.256074
$245$	0	0
$246$	−15.0000	−0.956365
$247$	0	0
$248$	10.0000	0.635001
$249$	−45.0000	−2.85176
$250$	0	0
$251$	−19.0000	−1.19927	−0.599635	−	0.800274i	$-0.704687\pi$
$251$	−19.0000	−1.19927	−0.599635	+	0.800274i	$0.704687\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	−10.0000	−0.627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	36.0000	2.24126
$259$	0	0
$260$	0	0
$261$	24.0000	1.48556
$262$	4.00000	0.247121
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−9.00000	−0.553912
$265$	0	0
$266$	0	0
$267$	−33.0000	−2.01957
$268$	−5.00000	−0.305424
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	−9.00000	−0.543710
$275$	0	0
$276$	−12.0000	−0.722315
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−5.00000	−0.299880
$279$	60.0000	3.59211
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	12.0000	0.714590
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	6.00000	0.353553
$289$	32.0000	1.88235
$290$	0	0
$291$	6.00000	0.351726
$292$	−11.0000	−0.643726
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	21.0000	1.22474
$295$	0	0
$296$	12.0000	0.697486
$297$	−27.0000	−1.56670
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	2.00000	0.115087
$303$	−36.0000	−2.06815
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	42.0000	2.40098
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	0	0
$309$	42.0000	2.38930
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−12.0000	−0.677199
$315$	0	0
$316$	4.00000	0.225018
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	18.0000	1.00939
$319$	12.0000	0.671871
$320$	0	0
$321$	−27.0000	−1.50699
$322$	0	0
$323$	−7.00000	−0.389490
$324$	9.00000	0.500000
$325$	0	0
$326$	−1.00000	−0.0553849
$327$	−24.0000	−1.32720
$328$	5.00000	0.276079
$329$	0	0
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	15.0000	0.823232
$333$	72.0000	3.94558
$334$	−14.0000	−0.766046
$335$	0	0
$336$	0	0
$337$	−27.0000	−1.47078	−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000	−1.47078	−0.735392	+	0.677642i	$0.763002\pi$
$338$	0	0
$339$	57.0000	3.09582
$340$	0	0
$341$	30.0000	1.62459
$342$	−6.00000	−0.324443
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	18.0000	0.967686
$347$	25.0000	1.34207	0.671035	−	0.741426i	$-0.265850\pi$
$347$	25.0000	1.34207	0.671035	+	0.741426i	$0.265850\pi$
$348$	−12.0000	−0.643268
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	3.00000	0.159901
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	11.0000	0.582999
$357$	0	0
$358$	−5.00000	−0.264258
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	16.0000	0.840941
$363$	6.00000	0.314918
$364$	0	0
$365$	0	0
$366$	−12.0000	−0.627250
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	4.00000	0.208514
$369$	30.0000	1.56174
$370$	0	0
$371$	0	0
$372$	−30.0000	−1.55543
$373$	−24.0000	−1.24267	−0.621336	−	0.783544i	$-0.713410\pi$
$373$	−24.0000	−1.24267	−0.621336	+	0.783544i	$0.713410\pi$
$374$	21.0000	1.08588
$375$	0	0
$376$	−4.00000	−0.206284
$377$	0	0
$378$	0	0
$379$	21.0000	1.07870	0.539349	−	0.842082i	$-0.318670\pi$
$379$	21.0000	1.07870	0.539349	+	0.842082i	$0.318670\pi$
$380$	0	0
$381$	30.0000	1.53695
$382$	−18.0000	−0.920960
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	15.0000	0.763480
$387$	−72.0000	−3.65997
$388$	−2.00000	−0.101535
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	28.0000	1.41602
$392$	−7.00000	−0.353553
$393$	−12.0000	−0.605320
$394$	16.0000	0.806068
$395$	0	0
$396$	18.0000	0.904534
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−6.00000	−0.300753
$399$	0	0
$400$	0	0
$401$	33.0000	1.64794	0.823971	−	0.566632i	$-0.191754\pi$
$401$	33.0000	1.64794	0.823971	+	0.566632i	$0.191754\pi$
$402$	15.0000	0.748132
$403$	0	0
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	36.0000	1.78445
$408$	−21.0000	−1.03965
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	27.0000	1.33181
$412$	−14.0000	−0.689730
$413$	0	0
$414$	24.0000	1.17954
$415$	0	0
$416$	0	0
$417$	15.0000	0.734553
$418$	−3.00000	−0.146735
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	−11.0000	−0.535472
$423$	−24.0000	−1.16692
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	9.00000	0.435031
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	−9.00000	−0.433013
$433$	23.0000	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$433$	23.0000	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$434$	0	0
$435$	0	0
$436$	8.00000	0.383131
$437$	−4.00000	−0.191346
$438$	33.0000	1.57680
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	0	0
$441$	−42.0000	−2.00000
$442$	0	0
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	−36.0000	−1.70848
$445$	0	0
$446$	−14.0000	−0.662919
$447$	0	0
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	15.0000	0.706322
$452$	−19.0000	−0.893685
$453$	−6.00000	−0.281905
$454$	−4.00000	−0.187729
$455$	0	0
$456$	3.00000	0.140488
$457$	31.0000	1.45012	0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000	1.45012	0.725059	+	0.688686i	$0.241812\pi$
$458$	16.0000	0.747631
$459$	−63.0000	−2.94059
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	36.0000	1.65879
$472$	4.00000	0.184115
$473$	−36.0000	−1.65528
$474$	−12.0000	−0.551178
$475$	0	0
$476$	0	0
$477$	−36.0000	−1.64833
$478$	−20.0000	−0.914779
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	0	0
$482$	−23.0000	−1.04762
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	4.00000	0.181071
$489$	3.00000	0.135665
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	−15.0000	−0.676252
$493$	28.0000	1.26106
$494$	0	0
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	−45.0000	−2.01650
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	42.0000	1.87642
$502$	−19.0000	−0.848012
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	0	0
$506$	12.0000	0.533465
$507$	0	0
$508$	−10.0000	−0.443678
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	9.00000	0.397360
$514$	10.0000	0.441081
$515$	0	0
$516$	36.0000	1.58481
$517$	−12.0000	−0.527759
$518$	0	0
$519$	−54.0000	−2.37034
$520$	0	0
$521$	−7.00000	−0.306676	−0.153338	−	0.988174i	$-0.549002\pi$
$521$	−7.00000	−0.306676	−0.153338	+	0.988174i	$0.549002\pi$
$522$	24.0000	1.05045
$523$	5.00000	0.218635	0.109317	−	0.994007i	$-0.465134\pi$
$523$	5.00000	0.218635	0.109317	+	0.994007i	$0.465134\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−8.00000	−0.348817
$527$	70.0000	3.04925
$528$	−9.00000	−0.391675
$529$	−7.00000	−0.304348
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	0	0
$534$	−33.0000	−1.42805
$535$	0	0
$536$	−5.00000	−0.215967
$537$	15.0000	0.647298
$538$	24.0000	1.03471
$539$	−21.0000	−0.904534
$540$	0	0
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	4.00000	0.171815
$543$	−48.0000	−2.05988
$544$	7.00000	0.300123
$545$	0	0
$546$	0	0
$547$	13.0000	0.555840	0.277920	−	0.960604i	$-0.410355\pi$
$547$	13.0000	0.555840	0.277920	+	0.960604i	$0.410355\pi$
$548$	−9.00000	−0.384461
$549$	24.0000	1.02430
$550$	0	0
$551$	−4.00000	−0.170406
$552$	−12.0000	−0.510754
$553$	0	0
$554$	2.00000	0.0849719
$555$	0	0
$556$	−5.00000	−0.212047
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	60.0000	2.54000
$559$	0	0
$560$	0	0
$561$	−63.0000	−2.65986
$562$	−2.00000	−0.0843649
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	11.0000	0.462364
$567$	0	0
$568$	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
$571$	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
$573$	54.0000	2.25588
$574$	0	0
$575$	0	0
$576$	6.00000	0.250000
$577$	−19.0000	−0.790980	−0.395490	−	0.918470i	$-0.629425\pi$
$577$	−19.0000	−0.790980	−0.395490	+	0.918470i	$0.629425\pi$
$578$	32.0000	1.33102
$579$	−45.0000	−1.87014
$580$	0	0
$581$	0	0
$582$	6.00000	0.248708
$583$	−18.0000	−0.745484
$584$	−11.0000	−0.455183
$585$	0	0
$586$	30.0000	1.23929
$587$	−5.00000	−0.206372	−0.103186	−	0.994662i	$-0.532904\pi$
$587$	−5.00000	−0.206372	−0.103186	+	0.994662i	$0.532904\pi$
$588$	21.0000	0.866025
$589$	−10.0000	−0.412043
$590$	0	0
$591$	−48.0000	−1.97446
$592$	12.0000	0.493197
$593$	29.0000	1.19089	0.595444	−	0.803397i	$-0.296976\pi$
$593$	29.0000	1.19089	0.595444	+	0.803397i	$0.296976\pi$
$594$	−27.0000	−1.10782
$595$	0	0
$596$	0	0
$597$	18.0000	0.736691
$598$	0	0
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	0	0
$603$	−30.0000	−1.22169
$604$	2.00000	0.0813788
$605$	0	0
$606$	−36.0000	−1.46240
$607$	42.0000	1.70473	0.852364	−	0.522949i	$-0.175168\pi$
$607$	42.0000	1.70473	0.852364	+	0.522949i	$0.175168\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	0	0
$612$	42.0000	1.69775
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	25.0000	1.00892
$615$	0	0
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	42.0000	1.68949
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	−36.0000	−1.44463
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	9.00000	0.359425
$628$	−12.0000	−0.478852
$629$	84.0000	3.34930
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	4.00000	0.159111
$633$	33.0000	1.31163
$634$	10.0000	0.397151
$635$	0	0
$636$	18.0000	0.713746
$637$	0	0
$638$	12.0000	0.475085
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−27.0000	−1.06561
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	0	0
$646$	−7.00000	−0.275411
$647$	46.0000	1.80845	0.904223	−	0.427060i	$-0.140451\pi$
$647$	46.0000	1.80845	0.904223	+	0.427060i	$0.140451\pi$
$648$	9.00000	0.353553
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	−24.0000	−0.938474
$655$	0	0
$656$	5.00000	0.195217
$657$	−66.0000	−2.57491
$658$	0	0
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−25.0000	−0.971653
$663$	0	0
$664$	15.0000	0.582113
$665$	0	0
$666$	72.0000	2.78994
$667$	16.0000	0.619522
$668$	−14.0000	−0.541676
$669$	42.0000	1.62381
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−27.0000	−1.04000
$675$	0	0
$676$	0	0
$677$	44.0000	1.69106	0.845529	−	0.533930i	$-0.179285\pi$
$677$	44.0000	1.69106	0.845529	+	0.533930i	$0.179285\pi$
$678$	57.0000	2.18907
$679$	0	0
$680$	0	0
$681$	12.0000	0.459841
$682$	30.0000	1.14876
$683$	−35.0000	−1.33924	−0.669619	−	0.742705i	$-0.733543\pi$
$683$	−35.0000	−1.33924	−0.669619	+	0.742705i	$0.733543\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	0	0
$687$	−48.0000	−1.83131
$688$	−12.0000	−0.457496
$689$	0	0
$690$	0	0
$691$	31.0000	1.17930	0.589648	−	0.807661i	$-0.299267\pi$
$691$	31.0000	1.17930	0.589648	+	0.807661i	$0.299267\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	25.0000	0.948987
$695$	0	0
$696$	−12.0000	−0.454859
$697$	35.0000	1.32572
$698$	10.0000	0.378506
$699$	18.0000	0.680823
$700$	0	0
$701$	32.0000	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	3.00000	0.113067
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	−12.0000	−0.450988
$709$	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
$712$	11.0000	0.412242
$713$	40.0000	1.49801
$714$	0	0
$715$	0	0
$716$	−5.00000	−0.186859
$717$	60.0000	2.24074
$718$	−4.00000	−0.149279
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	0	0
$721$	0	0
$722$	−18.0000	−0.669891
$723$	69.0000	2.56614
$724$	16.0000	0.594635
$725$	0	0
$726$	6.00000	0.222681
$727$	10.0000	0.370879	0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000	0.370879	0.185440	+	0.982656i	$0.440629\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−84.0000	−3.10685
$732$	−12.0000	−0.443533
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	0	0
$736$	4.00000	0.147442
$737$	−15.0000	−0.552532
$738$	30.0000	1.10432
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	−30.0000	−1.09985
$745$	0	0
$746$	−24.0000	−0.878702
$747$	90.0000	3.29293
$748$	21.0000	0.767836
$749$	0	0
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	−4.00000	−0.145865
$753$	57.0000	2.07720
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	21.0000	0.762754
$759$	−36.0000	−1.30672
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	30.0000	1.08679
$763$	0	0
$764$	−18.0000	−0.651217
$765$	0	0
$766$	26.0000	0.939418
$767$	0	0
$768$	−3.00000	−0.108253
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	15.0000	0.539862
$773$	−38.0000	−1.36677	−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000	−1.36677	−0.683383	+	0.730061i	$0.739492\pi$
$774$	−72.0000	−2.58799
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−5.00000	−0.179144
$780$	0	0
$781$	0	0
$782$	28.0000	1.00128
$783$	−36.0000	−1.28654
$784$	−7.00000	−0.250000
$785$	0	0
$786$	−12.0000	−0.428026
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	16.0000	0.569976
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	18.0000	0.639602
$793$	0	0
$794$	18.0000	0.638796
$795$	0	0
$796$	−6.00000	−0.212664
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	−28.0000	−0.990569
$800$	0	0
$801$	66.0000	2.33200
$802$	33.0000	1.16527
$803$	−33.0000	−1.16454
$804$	15.0000	0.529009
$805$	0	0
$806$	0	0
$807$	−72.0000	−2.53452
$808$	12.0000	0.422159
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	−12.0000	−0.420858
$814$	36.0000	1.26180
$815$	0	0
$816$	−21.0000	−0.735147
$817$	12.0000	0.419827
$818$	−23.0000	−0.804176
$819$	0	0
$820$	0	0
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	27.0000	0.941733
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	0	0
$827$	−21.0000	−0.730242	−0.365121	−	0.930960i	$-0.618972\pi$
$827$	−21.0000	−0.730242	−0.365121	+	0.930960i	$0.618972\pi$
$828$	24.0000	0.834058
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	−49.0000	−1.69775
$834$	15.0000	0.519408
$835$	0	0
$836$	−3.00000	−0.103757
$837$	−90.0000	−3.11086
$838$	−9.00000	−0.310900
$839$	−10.0000	−0.345238	−0.172619	−	0.984989i	$-0.555223\pi$
$839$	−10.0000	−0.345238	−0.172619	+	0.984989i	$0.555223\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−16.0000	−0.551396
$843$	6.00000	0.206651
$844$	−11.0000	−0.378636
$845$	0	0
$846$	−24.0000	−0.825137
$847$	0	0
$848$	−6.00000	−0.206041
$849$	−33.0000	−1.13256
$850$	0	0
$851$	48.0000	1.64542
$852$	0	0
$853$	−20.0000	−0.684787	−0.342393	−	0.939557i	$-0.611238\pi$
$853$	−20.0000	−0.684787	−0.342393	+	0.939557i	$0.611238\pi$
$854$	0	0
$855$	0	0
$856$	9.00000	0.307614
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	−13.0000	−0.443554	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	−13.0000	−0.443554	−0.221777	+	0.975097i	$0.571186\pi$
$860$	0	0
$861$	0	0
$862$	24.0000	0.817443
$863$	−50.0000	−1.70202	−0.851010	−	0.525150i	$-0.824009\pi$
$863$	−50.0000	−1.70202	−0.851010	+	0.525150i	$0.824009\pi$
$864$	−9.00000	−0.306186
$865$	0	0
$866$	23.0000	0.781572
$867$	−96.0000	−3.26033
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	0	0
$872$	8.00000	0.270914
$873$	−12.0000	−0.406138
$874$	−4.00000	−0.135302
$875$	0	0
$876$	33.0000	1.11497
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	30.0000	1.01245
$879$	−90.0000	−3.03562
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	−42.0000	−1.41421
$883$	−55.0000	−1.85090	−0.925449	−	0.378873i	$-0.876312\pi$
$883$	−55.0000	−1.85090	−0.925449	+	0.378873i	$0.876312\pi$
$884$	0	0
$885$	0	0
$886$	39.0000	1.31023
$887$	−44.0000	−1.47738	−0.738688	−	0.674048i	$-0.764554\pi$
$887$	−44.0000	−1.47738	−0.738688	+	0.674048i	$0.764554\pi$
$888$	−36.0000	−1.20808
$889$	0	0
$890$	0	0
$891$	27.0000	0.904534
$892$	−14.0000	−0.468755
$893$	4.00000	0.133855
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	9.00000	0.300334
$899$	40.0000	1.33407
$900$	0	0
$901$	−42.0000	−1.39922
$902$	15.0000	0.499445
$903$	0	0
$904$	−19.0000	−0.631931
$905$	0	0
$906$	−6.00000	−0.199337
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	−4.00000	−0.132745
$909$	72.0000	2.38809
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	3.00000	0.0993399
$913$	45.0000	1.48928
$914$	31.0000	1.02539
$915$	0	0
$916$	16.0000	0.528655
$917$	0	0
$918$	−63.0000	−2.07931
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	−75.0000	−2.47133
$922$	10.0000	0.329332
$923$	0	0
$924$	0	0
$925$	0	0
$926$	14.0000	0.460069
$927$	−84.0000	−2.75892
$928$	4.00000	0.131306
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	−6.00000	−0.196537
$933$	−36.0000	−1.17859
$934$	−20.0000	−0.654420
$935$	0	0
$936$	0	0
$937$	13.0000	0.424691	0.212346	−	0.977195i	$-0.431890\pi$
$937$	13.0000	0.424691	0.212346	+	0.977195i	$0.431890\pi$
$938$	0	0
$939$	18.0000	0.587408
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	36.0000	1.17294
$943$	20.0000	0.651290
$944$	4.00000	0.130189
$945$	0	0
$946$	−36.0000	−1.17046
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	−12.0000	−0.389742
$949$	0	0
$950$	0	0
$951$	−30.0000	−0.972817
$952$	0	0
$953$	7.00000	0.226752	0.113376	−	0.993552i	$-0.463833\pi$
$953$	7.00000	0.226752	0.113376	+	0.993552i	$0.463833\pi$
$954$	−36.0000	−1.16554
$955$	0	0
$956$	−20.0000	−0.646846
$957$	−36.0000	−1.16371
$958$	10.0000	0.323085
$959$	0	0
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	54.0000	1.74013
$964$	−23.0000	−0.740780
$965$	0	0
$966$	0	0
$967$	−36.0000	−1.15768	−0.578841	−	0.815440i	$-0.696495\pi$
$967$	−36.0000	−1.15768	−0.578841	+	0.815440i	$0.696495\pi$
$968$	−2.00000	−0.0642824
$969$	21.0000	0.674617
$970$	0	0
$971$	−13.0000	−0.417190	−0.208595	−	0.978002i	$-0.566889\pi$
$971$	−13.0000	−0.417190	−0.208595	+	0.978002i	$0.566889\pi$
$972$	0	0
$973$	0	0
$974$	−32.0000	−1.02535
$975$	0	0
$976$	4.00000	0.128037
$977$	−9.00000	−0.287936	−0.143968	−	0.989582i	$-0.545986\pi$
$977$	−9.00000	−0.287936	−0.143968	+	0.989582i	$0.545986\pi$
$978$	3.00000	0.0959294
$979$	33.0000	1.05468
$980$	0	0
$981$	48.0000	1.53252
$982$	−4.00000	−0.127645
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−15.0000	−0.478183
$985$	0	0
$986$	28.0000	0.891702
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	10.0000	0.317500
$993$	75.0000	2.38005
$994$	0	0
$995$	0	0
$996$	−45.0000	−1.42588
$997$	56.0000	1.77354	0.886769	−	0.462213i	$-0.152944\pi$
$997$	56.0000	1.77354	0.886769	+	0.462213i	$0.152944\pi$
$998$	−20.0000	−0.633089
$999$	−108.000	−3.41697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.m.1.1		1
5.4	even	2		8450.2.a.l.1.1		1
13.12	even	2		650.2.a.a.1.1	✓	1
39.38	odd	2		5850.2.a.br.1.1		1
52.51	odd	2		5200.2.a.bj.1.1		1
65.12	odd	4		650.2.b.i.599.1		2
65.38	odd	4		650.2.b.i.599.2		2
65.64	even	2		650.2.a.m.1.1	yes	1
195.38	even	4		5850.2.e.y.5149.1		2
195.77	even	4		5850.2.e.y.5149.2		2
195.194	odd	2		5850.2.a.n.1.1		1
260.259	odd	2		5200.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
650.2.a.a.1.1	✓	1	13.12	even	2
650.2.a.m.1.1	yes	1	65.64	even	2
650.2.b.i.599.1		2	65.12	odd	4
650.2.b.i.599.2		2	65.38	odd	4
5200.2.a.b.1.1		1	260.259	odd	2
5200.2.a.bj.1.1		1	52.51	odd	2
5850.2.a.n.1.1		1	195.194	odd	2
5850.2.a.br.1.1		1	39.38	odd	2
5850.2.e.y.5149.1		2	195.38	even	4
5850.2.e.y.5149.2		2	195.77	even	4
8450.2.a.l.1.1		1	5.4	even	2
8450.2.a.m.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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