LMFDB - Embedded newform 208.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,2,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, 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("208.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	208.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:	$1.66088836204$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -5.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} -3.00000 q^{17} +2.00000 q^{19} +5.00000 q^{21} -4.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} -2.00000 q^{33} +5.00000 q^{35} +11.0000 q^{37} +1.00000 q^{39} +8.00000 q^{41} +1.00000 q^{43} +2.00000 q^{45} -9.00000 q^{47} +18.0000 q^{49} +3.00000 q^{51} -12.0000 q^{53} -2.00000 q^{55} -2.00000 q^{57} -6.00000 q^{59} +10.0000 q^{63} +1.00000 q^{65} -6.00000 q^{67} +4.00000 q^{69} -7.00000 q^{71} -2.00000 q^{73} +4.00000 q^{75} -10.0000 q^{77} -12.0000 q^{79} +1.00000 q^{81} +16.0000 q^{83} +3.00000 q^{85} +6.00000 q^{87} -10.0000 q^{89} +5.00000 q^{91} -4.00000 q^{93} -2.00000 q^{95} -10.0000 q^{97} -4.00000 q^{99} +O(q^{100})$

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	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−5.00000	−1.88982	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−5.00000	−1.88982	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	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
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	5.00000	0.845154
$36$	0	0
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	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
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	18.0000	2.57143
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	10.0000	1.25988
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	−10.0000	−1.13961
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	6.00000	0.643268
$88$	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
$91$	5.00000	0.524142
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	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
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−5.00000	−0.487950
$106$	0	0
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	15.0000	1.37505
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−8.00000	−0.721336
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	−10.0000	−0.867110
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	20.0000	1.57622
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	20.0000	1.51186
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	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
$183$	0	0
$184$	0	0
$185$	−11.0000	−0.808736
$186$	0	0
$187$	−6.00000	−0.438763
$188$	0	0
$189$	−25.0000	−1.81848
$190$	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
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	17.0000	1.21120	0.605600	−	0.795769i	$-0.292933\pi$
$197$	17.0000	1.21120	0.605600	+	0.795769i	$0.292933\pi$
$198$	0	0
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	30.0000	2.10559
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	0	0
$213$	7.00000	0.479632
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	16.0000	1.06196	0.530979	−	0.847385i	$-0.321824\pi$
$227$	16.0000	1.06196	0.530979	+	0.847385i	$0.321824\pi$
$228$	0	0
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	10.0000	0.657952
$232$	0	0
$233$	13.0000	0.851658	0.425829	−	0.904804i	$-0.359982\pi$
$233$	13.0000	0.851658	0.425829	+	0.904804i	$0.359982\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−18.0000	−1.14998
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−8.00000	−0.504956	−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000	−0.504956	−0.252478	+	0.967603i	$0.581245\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	−31.0000	−1.93373	−0.966863	−	0.255294i	$-0.917828\pi$
$257$	−31.0000	−1.93373	−0.966863	+	0.255294i	$0.917828\pi$
$258$	0	0
$259$	−55.0000	−3.41753
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	0	0
$273$	−5.00000	−0.302614
$274$	0	0
$275$	−8.00000	−0.482418
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−40.0000	−2.36113
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	15.0000	0.876309	0.438155	−	0.898900i	$-0.355632\pi$
$293$	15.0000	0.876309	0.438155	+	0.898900i	$0.355632\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	10.0000	0.580259
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	22.0000	1.24751	0.623753	−	0.781622i	$-0.285607\pi$
$311$	22.0000	1.24751	0.623753	+	0.781622i	$0.285607\pi$
$312$	0	0
$313$	−9.00000	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$314$	0	0
$315$	−10.0000	−0.563436
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	20.0000	1.11629
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	5.00000	0.276501
$328$	0	0
$329$	45.0000	2.48093
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	−22.0000	−1.20559
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	15.0000	0.817102	0.408551	−	0.912735i	$-0.366034\pi$
$337$	15.0000	0.817102	0.408551	+	0.912735i	$0.366034\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	−55.0000	−2.96972
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	0	0
$349$	−9.00000	−0.481759	−0.240879	−	0.970555i	$-0.577436\pi$
$349$	−9.00000	−0.481759	−0.240879	+	0.970555i	$0.577436\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−28.0000	−1.49029	−0.745145	−	0.666903i	$-0.767620\pi$
$353$	−28.0000	−1.49029	−0.745145	+	0.666903i	$0.767620\pi$
$354$	0	0
$355$	7.00000	0.371521
$356$	0	0
$357$	−15.0000	−0.793884
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	−16.0000	−0.832927
$370$	0	0
$371$	60.0000	3.11504
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	−15.0000	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$383$	−15.0000	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$384$	0	0
$385$	10.0000	0.509647
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	0	0
$399$	10.0000	0.500626
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	22.0000	1.09050
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	30.0000	1.47620
$414$	0	0
$415$	−16.0000	−0.785409
$416$	0	0
$417$	3.00000	0.146911
$418$	0	0
$419$	39.0000	1.90527	0.952637	−	0.304109i	$-0.0983586\pi$
$419$	39.0000	1.90527	0.952637	+	0.304109i	$0.0983586\pi$
$420$	0	0
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	0	0
$423$	18.0000	0.875190
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	11.0000	0.529851	0.264926	−	0.964269i	$-0.414653\pi$
$431$	11.0000	0.529851	0.264926	+	0.964269i	$0.414653\pi$
$432$	0	0
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	27.0000	1.28281	0.641404	−	0.767203i	$-0.278352\pi$
$443$	27.0000	1.28281	0.641404	+	0.767203i	$0.278352\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	−5.00000	−0.234920
$454$	0	0
$455$	−5.00000	−0.234404
$456$	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
$459$	−15.0000	−0.700140
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	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
$469$	30.0000	1.38527
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	24.0000	1.09888
$478$	0	0
$479$	−9.00000	−0.411220	−0.205610	−	0.978634i	$-0.565918\pi$
$479$	−9.00000	−0.411220	−0.205610	+	0.978634i	$0.565918\pi$
$480$	0	0
$481$	−11.0000	−0.501557
$482$	0	0
$483$	−20.0000	−0.910032
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	35.0000	1.56996
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	−26.0000	−1.15928	−0.579641	−	0.814872i	$-0.696807\pi$
$503$	−26.0000	−1.15928	−0.579641	+	0.814872i	$0.696807\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	10.0000	0.441511
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−18.0000	−0.791639
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	−20.0000	−0.872872
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	20.0000	0.864675
$536$	0	0
$537$	3.00000	0.129460
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	9.00000	0.386940	0.193470	−	0.981106i	$-0.438026\pi$
$541$	9.00000	0.386940	0.193470	+	0.981106i	$0.438026\pi$
$542$	0	0
$543$	−16.0000	−0.686626
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	39.0000	1.66752	0.833760	−	0.552127i	$-0.186184\pi$
$547$	39.0000	1.66752	0.833760	+	0.552127i	$0.186184\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	60.0000	2.55146
$554$	0	0
$555$	11.0000	0.466924
$556$	0	0
$557$	−23.0000	−0.974541	−0.487271	−	0.873251i	$-0.662007\pi$
$557$	−23.0000	−0.974541	−0.487271	+	0.873251i	$0.662007\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	6.00000	0.253320
$562$	0	0
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	0	0
$567$	−5.00000	−0.209980
$568$	0	0
$569$	−17.0000	−0.712677	−0.356339	−	0.934357i	$-0.615975\pi$
$569$	−17.0000	−0.712677	−0.356339	+	0.934357i	$0.615975\pi$
$570$	0	0
$571$	−37.0000	−1.54840	−0.774201	−	0.632940i	$-0.781848\pi$
$571$	−37.0000	−1.54840	−0.774201	+	0.632940i	$0.781848\pi$
$572$	0	0
$573$	10.0000	0.417756
$574$	0	0
$575$	16.0000	0.667246
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	−80.0000	−3.31896
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−17.0000	−0.699287
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	−15.0000	−0.614940
$596$	0	0
$597$	−18.0000	−0.736691
$598$	0	0
$599$	2.00000	0.0817178	0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000	0.0817178	0.0408589	+	0.999165i	$0.486991\pi$
$600$	0	0
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	0	0
$609$	−30.0000	−1.21566
$610$	0	0
$611$	9.00000	0.364101
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	32.0000	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$618$	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
$621$	−20.0000	−0.802572
$622$	0	0
$623$	50.0000	2.00321
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	−33.0000	−1.31580
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	0	0
$633$	−5.00000	−0.198732
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−18.0000	−0.713186
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	0	0
$645$	1.00000	0.0393750
$646$	0	0
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	20.0000	0.783862
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	3.00000	0.117220
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	10.0000	0.387783
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	−17.0000	−0.657258
$670$	0	0
$671$	0	0
$672$	0	0
$673$	21.0000	0.809491	0.404745	−	0.914429i	$-0.367360\pi$
$673$	21.0000	0.809491	0.404745	+	0.914429i	$0.367360\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	0	0
$679$	50.0000	1.91882
$680$	0	0
$681$	−16.0000	−0.613121
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	15.0000	0.572286
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	0	0
$693$	20.0000	0.759737
$694$	0	0
$695$	3.00000	0.113796
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	−13.0000	−0.491705
$700$	0	0
$701$	−4.00000	−0.151078	−0.0755390	−	0.997143i	$-0.524068\pi$
$701$	−4.00000	−0.151078	−0.0755390	+	0.997143i	$0.524068\pi$
$702$	0	0
$703$	22.0000	0.829746
$704$	0	0
$705$	−9.00000	−0.338960
$706$	0	0
$707$	−20.0000	−0.752177
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	2.00000	0.0747958
$716$	0	0
$717$	−3.00000	−0.112037
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	24.0000	0.891338
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	37.0000	1.36663	0.683313	−	0.730125i	$-0.260538\pi$
$733$	37.0000	1.36663	0.683313	+	0.730125i	$0.260538\pi$
$734$	0	0
$735$	18.0000	0.663940
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−29.0000	−1.06391	−0.531953	−	0.846774i	$-0.678542\pi$
$743$	−29.0000	−1.06391	−0.531953	+	0.846774i	$0.678542\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	−32.0000	−1.17082
$748$	0	0
$749$	100.000	3.65392
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	8.00000	0.291536
$754$	0	0
$755$	−5.00000	−0.181969
$756$	0	0
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	25.0000	0.905061
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	31.0000	1.11644
$772$	0	0
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	55.0000	1.97311
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	−14.0000	−0.500959
$782$	0	0
$783$	−30.0000	−1.07211
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	0	0
$789$	20.0000	0.712019
$790$	0	0
$791$	−10.0000	−0.355559
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	0	0
$799$	27.0000	0.955191
$800$	0	0
$801$	20.0000	0.706665
$802$	0	0
$803$	−4.00000	−0.141157
$804$	0	0
$805$	−20.0000	−0.704907
$806$	0	0
$807$	−16.0000	−0.563227
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	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
$813$	−7.00000	−0.245501
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	−10.0000	−0.349428
$820$	0	0
$821$	−41.0000	−1.43091	−0.715455	−	0.698659i	$-0.753781\pi$
$821$	−41.0000	−1.43091	−0.715455	+	0.698659i	$0.753781\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	0	0
$825$	8.00000	0.278524
$826$	0	0
$827$	10.0000	0.347734	0.173867	−	0.984769i	$-0.444374\pi$
$827$	10.0000	0.347734	0.173867	+	0.984769i	$0.444374\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	12.0000	0.416275
$832$	0	0
$833$	−54.0000	−1.87099
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	10.0000	0.344418
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	35.0000	1.20261
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−44.0000	−1.50830
$852$	0	0
$853$	−47.0000	−1.60925	−0.804625	−	0.593784i	$-0.797633\pi$
$853$	−47.0000	−1.60925	−0.804625	+	0.593784i	$0.797633\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	40.0000	1.36320
$862$	0	0
$863$	7.00000	0.238283	0.119141	−	0.992877i	$-0.461986\pi$
$863$	7.00000	0.238283	0.119141	+	0.992877i	$0.461986\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	6.00000	0.203302
$872$	0	0
$873$	20.0000	0.676897
$874$	0	0
$875$	−45.0000	−1.52128
$876$	0	0
$877$	−7.00000	−0.236373	−0.118187	−	0.992991i	$-0.537708\pi$
$877$	−7.00000	−0.236373	−0.118187	+	0.992991i	$0.537708\pi$
$878$	0	0
$879$	−15.0000	−0.505937
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	51.0000	1.71629	0.858143	−	0.513410i	$-0.171618\pi$
$883$	51.0000	1.71629	0.858143	+	0.513410i	$0.171618\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	−40.0000	−1.34156
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−18.0000	−0.602347
$894$	0	0
$895$	3.00000	0.100279
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	5.00000	0.166390
$904$	0	0
$905$	−16.0000	−0.531858
$906$	0	0
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−26.0000	−0.861418	−0.430709	−	0.902491i	$-0.641737\pi$
$911$	−26.0000	−0.861418	−0.430709	+	0.902491i	$0.641737\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.0000	0.495344
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	7.00000	0.230408
$924$	0	0
$925$	−44.0000	−1.44671
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	−22.0000	−0.720248
$934$	0	0
$935$	6.00000	0.196221
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	9.00000	0.293704
$940$	0	0
$941$	−7.00000	−0.228193	−0.114097	−	0.993470i	$-0.536397\pi$
$941$	−7.00000	−0.228193	−0.114097	+	0.993470i	$0.536397\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	0	0
$945$	25.0000	0.813250
$946$	0	0
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	0	0
$955$	10.0000	0.323592
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	60.0000	1.93750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	40.0000	1.28898
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	13.0000	0.418052	0.209026	−	0.977910i	$-0.432971\pi$
$967$	13.0000	0.418052	0.209026	+	0.977910i	$0.432971\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	−45.0000	−1.44412	−0.722059	−	0.691831i	$-0.756804\pi$
$971$	−45.0000	−1.44412	−0.722059	+	0.691831i	$0.756804\pi$
$972$	0	0
$973$	15.0000	0.480878
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−5.00000	−0.159475	−0.0797376	−	0.996816i	$-0.525408\pi$
$983$	−5.00000	−0.159475	−0.0797376	+	0.996816i	$0.525408\pi$
$984$	0	0
$985$	−17.0000	−0.541665
$986$	0	0
$987$	−45.0000	−1.43237
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	−18.0000	−0.570638
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	0	0
$999$	55.0000	1.74012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.2.a.b.1.1		1
3.2	odd	2		1872.2.a.l.1.1		1
4.3	odd	2		104.2.a.a.1.1	✓	1
5.4	even	2		5200.2.a.bb.1.1		1
8.3	odd	2		832.2.a.c.1.1		1
8.5	even	2		832.2.a.h.1.1		1
12.11	even	2		936.2.a.f.1.1		1
13.5	odd	4		2704.2.f.e.337.2		2
13.8	odd	4		2704.2.f.e.337.1		2
13.12	even	2		2704.2.a.d.1.1		1
16.3	odd	4		3328.2.b.a.1665.2		2
16.5	even	4		3328.2.b.t.1665.2		2
16.11	odd	4		3328.2.b.a.1665.1		2
16.13	even	4		3328.2.b.t.1665.1		2
20.3	even	4		2600.2.d.f.1249.2		2
20.7	even	4		2600.2.d.f.1249.1		2
20.19	odd	2		2600.2.a.e.1.1		1
24.5	odd	2		7488.2.a.u.1.1		1
24.11	even	2		7488.2.a.x.1.1		1
28.27	even	2		5096.2.a.c.1.1		1
52.3	odd	6		1352.2.i.b.529.1		2
52.7	even	12		1352.2.o.a.361.1		4
52.11	even	12		1352.2.o.a.1161.1		4
52.15	even	12		1352.2.o.a.1161.2		4
52.19	even	12		1352.2.o.a.361.2		4
52.23	odd	6		1352.2.i.c.529.1		2
52.31	even	4		1352.2.f.b.337.2		2
52.35	odd	6		1352.2.i.b.1329.1		2
52.43	odd	6		1352.2.i.c.1329.1		2
52.47	even	4		1352.2.f.b.337.1		2
52.51	odd	2		1352.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.a.a.1.1	✓	1	4.3	odd	2
208.2.a.b.1.1		1	1.1	even	1	trivial
832.2.a.c.1.1		1	8.3	odd	2
832.2.a.h.1.1		1	8.5	even	2
936.2.a.f.1.1		1	12.11	even	2
1352.2.a.b.1.1		1	52.51	odd	2
1352.2.f.b.337.1		2	52.47	even	4
1352.2.f.b.337.2		2	52.31	even	4
1352.2.i.b.529.1		2	52.3	odd	6
1352.2.i.b.1329.1		2	52.35	odd	6
1352.2.i.c.529.1		2	52.23	odd	6
1352.2.i.c.1329.1		2	52.43	odd	6
1352.2.o.a.361.1		4	52.7	even	12
1352.2.o.a.361.2		4	52.19	even	12
1352.2.o.a.1161.1		4	52.11	even	12
1352.2.o.a.1161.2		4	52.15	even	12
1872.2.a.l.1.1		1	3.2	odd	2
2600.2.a.e.1.1		1	20.19	odd	2
2600.2.d.f.1249.1		2	20.7	even	4
2600.2.d.f.1249.2		2	20.3	even	4
2704.2.a.d.1.1		1	13.12	even	2
2704.2.f.e.337.1		2	13.8	odd	4
2704.2.f.e.337.2		2	13.5	odd	4
3328.2.b.a.1665.1		2	16.11	odd	4
3328.2.b.a.1665.2		2	16.3	odd	4
3328.2.b.t.1665.1		2	16.13	even	4
3328.2.b.t.1665.2		2	16.5	even	4
5096.2.a.c.1.1		1	28.27	even	2
5200.2.a.bb.1.1		1	5.4	even	2
7488.2.a.u.1.1		1	24.5	odd	2
7488.2.a.x.1.1		1	24.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

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