LMFDB - Embedded newform 513.2.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(1,513)] 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("513.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	513.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.34730 q^{2} -0.184793 q^{4} +1.34730 q^{5} +1.22668 q^{7} -2.94356 q^{8} +1.81521 q^{10} +4.59627 q^{11} +3.59627 q^{13} +1.65270 q^{14} -3.59627 q^{16} +0.305407 q^{17} +1.00000 q^{19} -0.248970 q^{20} +6.19253 q^{22} +4.65270 q^{23} -3.18479 q^{25} +4.84524 q^{26} -0.226682 q^{28} +1.65270 q^{29} -7.04963 q^{31} +1.04189 q^{32} +0.411474 q^{34} +1.65270 q^{35} +0.184793 q^{37} +1.34730 q^{38} -3.96585 q^{40} +3.43107 q^{41} -11.2344 q^{43} -0.849356 q^{44} +6.26857 q^{46} -5.58172 q^{47} -5.49525 q^{49} -4.29086 q^{50} -0.664563 q^{52} -2.94356 q^{53} +6.19253 q^{55} -3.61081 q^{56} +2.22668 q^{58} +11.9436 q^{59} -2.40373 q^{61} -9.49794 q^{62} +8.59627 q^{64} +4.84524 q^{65} +7.63816 q^{67} -0.0564370 q^{68} +2.22668 q^{70} -1.29086 q^{71} -12.2763 q^{73} +0.248970 q^{74} -0.184793 q^{76} +5.63816 q^{77} -12.6382 q^{79} -4.84524 q^{80} +4.62267 q^{82} -8.27631 q^{83} +0.411474 q^{85} -15.1361 q^{86} -13.5294 q^{88} -7.90167 q^{89} +4.41147 q^{91} -0.859785 q^{92} -7.52023 q^{94} +1.34730 q^{95} +7.00774 q^{97} -7.40373 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} + 9 q^{10} - 3 q^{13} + 6 q^{14} + 3 q^{16} + 3 q^{17} + 3 q^{19} + 12 q^{20} - 9 q^{22} + 15 q^{23} - 6 q^{25} - 12 q^{26} + 6 q^{28} + 6 q^{29} + 6 q^{31}+ \cdots - 36 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 + 9 * q^10 - 3 * q^13 + 6 * q^14 + 3 * q^16 + 3 * q^17 + 3 * q^19 + 12 * q^20 - 9 * q^22 + 15 * q^23 - 6 * q^25 - 12 * q^26 + 6 * q^28 + 6 * q^29 + 6 * q^31 - 9 * q^34 + 6 * q^35 - 3 * q^37 + 3 * q^38 + 9 * q^40 + 3 * q^41 - 3 * q^43 - 27 * q^44 + 9 * q^46 + 15 * q^47 + 3 * q^50 - 30 * q^52 + 6 * q^53 - 9 * q^55 - 15 * q^56 + 21 * q^59 - 21 * q^61 - 3 * q^62 + 12 * q^64 - 12 * q^65 + 6 * q^67 - 15 * q^68 + 12 * q^71 - 3 * q^73 - 12 * q^74 + 3 * q^76 - 21 * q^79 + 12 * q^80 + 45 * q^82 + 9 * q^83 - 9 * q^85 - 3 * q^86 - 45 * q^88 - 12 * q^89 + 3 * q^91 + 6 * q^92 + 9 * q^94 + 3 * q^95 - 3 * q^97 - 36 * q^98

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 $

$2$	1.34730	0.952682	0.476341	−	0.879261i	$-0.341963\pi$
$2$	1.34730	0.952682	0.476341	+	0.879261i	$0.341963\pi$
$3$	0	0
$3$	0	0
$4$	−0.184793	−0.0923963
$5$	1.34730	0.602529	0.301265	−	0.953541i	$-0.402591\pi$
$5$	1.34730	0.602529	0.301265	+	0.953541i	$0.402591\pi$
$6$	0	0
$7$	1.22668	0.463642	0.231821	−	0.972758i	$-0.425532\pi$
$7$	1.22668	0.463642	0.231821	+	0.972758i	$0.425532\pi$
$8$	−2.94356	−1.04071
$9$	0	0
$10$	1.81521	0.574019
$11$	4.59627	1.38583	0.692913	−	0.721021i	$-0.256327\pi$
$11$	4.59627	1.38583	0.692913	+	0.721021i	$0.256327\pi$
$12$	0	0
$13$	3.59627	0.997425	0.498712	−	0.866767i	$-0.333806\pi$
$13$	3.59627	0.997425	0.498712	+	0.866767i	$0.333806\pi$
$14$	1.65270	0.441704
$15$	0	0
$16$	−3.59627	−0.899067
$17$	0.305407	0.0740721	0.0370361	−	0.999314i	$-0.488208\pi$
$17$	0.305407	0.0740721	0.0370361	+	0.999314i	$0.488208\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.248970	−0.0556715
$21$	0	0
$22$	6.19253	1.32025
$23$	4.65270	0.970156	0.485078	−	0.874471i	$-0.338791\pi$
$23$	4.65270	0.970156	0.485078	+	0.874471i	$0.338791\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	4.84524	0.950229
$27$	0	0
$28$	−0.226682	−0.0428388
$29$	1.65270	0.306899	0.153450	−	0.988156i	$-0.450962\pi$
$29$	1.65270	0.306899	0.153450	+	0.988156i	$0.450962\pi$
$30$	0	0
$31$	−7.04963	−1.26615	−0.633075	−	0.774090i	$-0.718208\pi$
$31$	−7.04963	−1.26615	−0.633075	+	0.774090i	$0.718208\pi$
$32$	1.04189	0.184182
$33$	0	0
$34$	0.411474	0.0705672
$35$	1.65270	0.279358
$36$	0	0
$37$	0.184793	0.0303797	0.0151899	−	0.999885i	$-0.495165\pi$
$37$	0.184793	0.0303797	0.0151899	+	0.999885i	$0.495165\pi$
$38$	1.34730	0.218560
$39$	0	0
$40$	−3.96585	−0.627056
$41$	3.43107	0.535844	0.267922	−	0.963441i	$-0.413663\pi$
$41$	3.43107	0.535844	0.267922	+	0.963441i	$0.413663\pi$
$42$	0	0
$43$	−11.2344	−1.71323	−0.856617	−	0.515953i	$-0.827438\pi$
$43$	−11.2344	−1.71323	−0.856617	+	0.515953i	$0.827438\pi$
$44$	−0.849356	−0.128045
$45$	0	0
$46$	6.26857	0.924250
$47$	−5.58172	−0.814177	−0.407089	−	0.913389i	$-0.633456\pi$
$47$	−5.58172	−0.814177	−0.407089	+	0.913389i	$0.633456\pi$
$48$	0	0
$49$	−5.49525	−0.785036
$50$	−4.29086	−0.606819
$51$	0	0
$52$	−0.664563	−0.0921583
$53$	−2.94356	−0.404329	−0.202165	−	0.979352i	$-0.564798\pi$
$53$	−2.94356	−0.404329	−0.202165	+	0.979352i	$0.564798\pi$
$54$	0	0
$55$	6.19253	0.835001
$56$	−3.61081	−0.482515
$57$	0	0
$58$	2.22668	0.292378
$59$	11.9436	1.55492	0.777460	−	0.628933i	$-0.216508\pi$
$59$	11.9436	1.55492	0.777460	+	0.628933i	$0.216508\pi$
$60$	0	0
$61$	−2.40373	−0.307767	−0.153883	−	0.988089i	$-0.549178\pi$
$61$	−2.40373	−0.307767	−0.153883	+	0.988089i	$0.549178\pi$
$62$	−9.49794	−1.20624
$63$	0	0
$64$	8.59627	1.07453
$65$	4.84524	0.600978
$66$	0	0
$67$	7.63816	0.933149	0.466575	−	0.884482i	$-0.345488\pi$
$67$	7.63816	0.933149	0.466575	+	0.884482i	$0.345488\pi$
$68$	−0.0564370	−0.00684399
$69$	0	0
$70$	2.22668	0.266139
$71$	−1.29086	−0.153197	−0.0765984	−	0.997062i	$-0.524406\pi$
$71$	−1.29086	−0.153197	−0.0765984	+	0.997062i	$0.524406\pi$
$72$	0	0
$73$	−12.2763	−1.43683	−0.718417	−	0.695613i	$-0.755133\pi$
$73$	−12.2763	−1.43683	−0.718417	+	0.695613i	$0.755133\pi$
$74$	0.248970	0.0289422
$75$	0	0
$76$	−0.184793	−0.0211972
$77$	5.63816	0.642527
$78$	0	0
$79$	−12.6382	−1.42190	−0.710952	−	0.703241i	$-0.751736\pi$
$79$	−12.6382	−1.42190	−0.710952	+	0.703241i	$0.751736\pi$
$80$	−4.84524	−0.541714
$81$	0	0
$82$	4.62267	0.510489
$83$	−8.27631	−0.908443	−0.454222	−	0.890889i	$-0.650083\pi$
$83$	−8.27631	−0.908443	−0.454222	+	0.890889i	$0.650083\pi$
$84$	0	0
$85$	0.411474	0.0446306
$86$	−15.1361	−1.63217
$87$	0	0
$88$	−13.5294	−1.44224
$89$	−7.90167	−0.837576	−0.418788	−	0.908084i	$-0.637545\pi$
$89$	−7.90167	−0.837576	−0.418788	+	0.908084i	$0.637545\pi$
$90$	0	0
$91$	4.41147	0.462448
$92$	−0.859785	−0.0896388
$93$	0	0
$94$	−7.52023	−0.775652
$95$	1.34730	0.138230
$96$	0	0
$97$	7.00774	0.711528	0.355764	−	0.934576i	$-0.384221\pi$
$97$	7.00774	0.711528	0.355764	+	0.934576i	$0.384221\pi$
$98$	−7.40373	−0.747890
$99$	0	0
$100$	0.588526	0.0588526
$101$	−7.70914	−0.767088	−0.383544	−	0.923523i	$-0.625297\pi$
$101$	−7.70914	−0.767088	−0.383544	+	0.923523i	$0.625297\pi$
$102$	0	0
$103$	4.58853	0.452121	0.226060	−	0.974113i	$-0.427415\pi$
$103$	4.58853	0.452121	0.226060	+	0.974113i	$0.427415\pi$
$104$	−10.5858	−1.03803
$105$	0	0
$106$	−3.96585	−0.385198
$107$	19.4270	1.87807	0.939037	−	0.343815i	$-0.111719\pi$
$107$	19.4270	1.87807	0.939037	+	0.343815i	$0.111719\pi$
$108$	0	0
$109$	−10.9486	−1.04869	−0.524344	−	0.851507i	$-0.675689\pi$
$109$	−10.9486	−1.04869	−0.524344	+	0.851507i	$0.675689\pi$
$110$	8.34318	0.795491
$111$	0	0
$112$	−4.41147	−0.416845
$113$	−3.55438	−0.334368	−0.167184	−	0.985926i	$-0.553467\pi$
$113$	−3.55438	−0.334368	−0.167184	+	0.985926i	$0.553467\pi$
$114$	0	0
$115$	6.26857	0.584547
$116$	−0.305407	−0.0283564
$117$	0	0
$118$	16.0915	1.48134
$119$	0.374638	0.0343430
$120$	0	0
$121$	10.1257	0.920515
$122$	−3.23854	−0.293204
$123$	0	0
$124$	1.30272	0.116988
$125$	−11.0273	−0.986315
$126$	0	0
$127$	5.53714	0.491342	0.245671	−	0.969353i	$-0.420992\pi$
$127$	5.53714	0.491342	0.245671	+	0.969353i	$0.420992\pi$
$128$	9.49794	0.839507
$129$	0	0
$130$	6.52797	0.572541
$131$	−17.5253	−1.53119	−0.765595	−	0.643322i	$-0.777556\pi$
$131$	−17.5253	−1.53119	−0.765595	+	0.643322i	$0.777556\pi$
$132$	0	0
$133$	1.22668	0.106367
$134$	10.2909	0.888995
$135$	0	0
$136$	−0.898986	−0.0770874
$137$	−6.98545	−0.596807	−0.298404	−	0.954440i	$-0.596454\pi$
$137$	−6.98545	−0.596807	−0.298404	+	0.954440i	$0.596454\pi$
$138$	0	0
$139$	18.3601	1.55728	0.778641	−	0.627469i	$-0.215909\pi$
$139$	18.3601	1.55728	0.778641	+	0.627469i	$0.215909\pi$
$140$	−0.305407	−0.0258116
$141$	0	0
$142$	−1.73917	−0.145948
$143$	16.5294	1.38226
$144$	0	0
$145$	2.22668	0.184916
$146$	−16.5398	−1.36885
$147$	0	0
$148$	−0.0341483	−0.00280697
$149$	−6.37464	−0.522231	−0.261115	−	0.965308i	$-0.584090\pi$
$149$	−6.37464	−0.522231	−0.261115	+	0.965308i	$0.584090\pi$
$150$	0	0
$151$	17.7469	1.44422	0.722112	−	0.691777i	$-0.243172\pi$
$151$	17.7469	1.44422	0.722112	+	0.691777i	$0.243172\pi$
$152$	−2.94356	−0.238754
$153$	0	0
$154$	7.59627	0.612125
$155$	−9.49794	−0.762893
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	−17.0273	−1.35462
$159$	0	0
$160$	1.40373	0.110975
$161$	5.70739	0.449805
$162$	0	0
$163$	−10.2189	−0.800409	−0.400205	−	0.916426i	$-0.631061\pi$
$163$	−10.2189	−0.800409	−0.400205	+	0.916426i	$0.631061\pi$
$164$	−0.634037	−0.0495100
$165$	0	0
$166$	−11.1506	−0.865458
$167$	17.3327	1.34125	0.670624	−	0.741797i	$-0.266026\pi$
$167$	17.3327	1.34125	0.670624	+	0.741797i	$0.266026\pi$
$168$	0	0
$169$	−0.0668661	−0.00514355
$170$	0.554378	0.0425188
$171$	0	0
$172$	2.07604	0.158296
$173$	6.66725	0.506902	0.253451	−	0.967348i	$-0.418434\pi$
$173$	6.66725	0.506902	0.253451	+	0.967348i	$0.418434\pi$
$174$	0	0
$175$	−3.90673	−0.295321
$176$	−16.5294	−1.24595
$177$	0	0
$178$	−10.6459	−0.797944
$179$	4.59627	0.343541	0.171771	−	0.985137i	$-0.445051\pi$
$179$	4.59627	0.343541	0.171771	+	0.985137i	$0.445051\pi$
$180$	0	0
$181$	6.94087	0.515911	0.257956	−	0.966157i	$-0.416951\pi$
$181$	6.94087	0.515911	0.257956	+	0.966157i	$0.416951\pi$
$182$	5.94356	0.440566
$183$	0	0
$184$	−13.6955	−1.00965
$185$	0.248970	0.0183047
$186$	0	0
$187$	1.40373	0.102651
$188$	1.03146	0.0752269
$189$	0	0
$190$	1.81521	0.131689
$191$	−0.680045	−0.0492063	−0.0246032	−	0.999697i	$-0.507832\pi$
$191$	−0.680045	−0.0492063	−0.0246032	+	0.999697i	$0.507832\pi$
$192$	0	0
$193$	−24.9564	−1.79640	−0.898199	−	0.439589i	$-0.855124\pi$
$193$	−24.9564	−1.79640	−0.898199	+	0.439589i	$0.855124\pi$
$194$	9.44150	0.677860
$195$	0	0
$196$	1.01548	0.0725344
$197$	22.4270	1.59785	0.798927	−	0.601428i	$-0.205401\pi$
$197$	22.4270	1.59785	0.798927	+	0.601428i	$0.205401\pi$
$198$	0	0
$199$	−16.3182	−1.15677	−0.578383	−	0.815765i	$-0.696316\pi$
$199$	−16.3182	−1.15677	−0.578383	+	0.815765i	$0.696316\pi$
$200$	9.37464	0.662887
$201$	0	0
$202$	−10.3865	−0.730791
$203$	2.02734	0.142291
$204$	0	0
$205$	4.62267	0.322862
$206$	6.18210	0.430728
$207$	0	0
$208$	−12.9331	−0.896751
$209$	4.59627	0.317930
$210$	0	0
$211$	15.7219	1.08234	0.541171	−	0.840912i	$-0.317981\pi$
$211$	15.7219	1.08234	0.541171	+	0.840912i	$0.317981\pi$
$212$	0.543948	0.0373585
$213$	0	0
$214$	26.1739	1.78921
$215$	−15.1361	−1.03227
$216$	0	0
$217$	−8.64765	−0.587041
$218$	−14.7510	−0.999066
$219$	0	0
$220$	−1.14433	−0.0771510
$221$	1.09833	0.0738814
$222$	0	0
$223$	−25.9982	−1.74097	−0.870486	−	0.492194i	$-0.836195\pi$
$223$	−25.9982	−1.74097	−0.870486	+	0.492194i	$0.836195\pi$
$224$	1.27807	0.0853944
$225$	0	0
$226$	−4.78880	−0.318546
$227$	15.5544	1.03238	0.516190	−	0.856474i	$-0.327350\pi$
$227$	15.5544	1.03238	0.516190	+	0.856474i	$0.327350\pi$
$228$	0	0
$229$	16.0574	1.06110	0.530550	−	0.847653i	$-0.321985\pi$
$229$	16.0574	1.06110	0.530550	+	0.847653i	$0.321985\pi$
$230$	8.44562	0.556888
$231$	0	0
$232$	−4.86484	−0.319392
$233$	−19.2909	−1.26379	−0.631893	−	0.775056i	$-0.717722\pi$
$233$	−19.2909	−1.26379	−0.631893	+	0.775056i	$0.717722\pi$
$234$	0	0
$235$	−7.52023	−0.490566
$236$	−2.20708	−0.143669
$237$	0	0
$238$	0.504748	0.0327179
$239$	10.4706	0.677287	0.338643	−	0.940915i	$-0.390032\pi$
$239$	10.4706	0.677287	0.338643	+	0.940915i	$0.390032\pi$
$240$	0	0
$241$	12.9581	0.834705	0.417353	−	0.908745i	$-0.362958\pi$
$241$	12.9581	0.834705	0.417353	+	0.908745i	$0.362958\pi$
$242$	13.6423	0.876959
$243$	0	0
$244$	0.444192	0.0284365
$245$	−7.40373	−0.473007
$246$	0	0
$247$	3.59627	0.228825
$248$	20.7510	1.31769
$249$	0	0
$250$	−14.8571	−0.939645
$251$	23.6509	1.49283	0.746417	−	0.665478i	$-0.231772\pi$
$251$	23.6509	1.49283	0.746417	+	0.665478i	$0.231772\pi$
$252$	0	0
$253$	21.3851	1.34447
$254$	7.46017	0.468093
$255$	0	0
$256$	−4.39599	−0.274750
$257$	7.16519	0.446952	0.223476	−	0.974709i	$-0.428260\pi$
$257$	7.16519	0.446952	0.223476	+	0.974709i	$0.428260\pi$
$258$	0	0
$259$	0.226682	0.0140853
$260$	−0.895364	−0.0555281
$261$	0	0
$262$	−23.6117	−1.45874
$263$	19.7324	1.21675	0.608375	−	0.793650i	$-0.291822\pi$
$263$	19.7324	1.21675	0.608375	+	0.793650i	$0.291822\pi$
$264$	0	0
$265$	−3.96585	−0.243620
$266$	1.65270	0.101334
$267$	0	0
$268$	−1.41147	−0.0862195
$269$	−31.9273	−1.94664	−0.973320	−	0.229453i	$-0.926306\pi$
$269$	−31.9273	−1.94664	−0.973320	+	0.229453i	$0.926306\pi$
$270$	0	0
$271$	8.92221	0.541986	0.270993	−	0.962581i	$-0.412648\pi$
$271$	8.92221	0.541986	0.270993	+	0.962581i	$0.412648\pi$
$272$	−1.09833	−0.0665958
$273$	0	0
$274$	−9.41147	−0.568568
$275$	−14.6382	−0.882714
$276$	0	0
$277$	−9.83069	−0.590669	−0.295334	−	0.955394i	$-0.595431\pi$
$277$	−9.83069	−0.590669	−0.295334	+	0.955394i	$0.595431\pi$
$278$	24.7365	1.48360
$279$	0	0
$280$	−4.86484	−0.290730
$281$	−0.305407	−0.0182191	−0.00910954	−	0.999959i	$-0.502900\pi$
$281$	−0.305407	−0.0182191	−0.00910954	+	0.999959i	$0.502900\pi$
$282$	0	0
$283$	10.3696	0.616408	0.308204	−	0.951320i	$-0.400272\pi$
$283$	10.3696	0.616408	0.308204	+	0.951320i	$0.400272\pi$
$284$	0.238541	0.0141548
$285$	0	0
$286$	22.2700	1.31685
$287$	4.20884	0.248440
$288$	0	0
$289$	−16.9067	−0.994513
$290$	3.00000	0.176166
$291$	0	0
$292$	2.26857	0.132758
$293$	−9.74691	−0.569421	−0.284710	−	0.958614i	$-0.591897\pi$
$293$	−9.74691	−0.569421	−0.284710	+	0.958614i	$0.591897\pi$
$294$	0	0
$295$	16.0915	0.936884
$296$	−0.543948	−0.0316164
$297$	0	0
$298$	−8.58853	−0.497520
$299$	16.7324	0.967658
$300$	0	0
$301$	−13.7811	−0.794327
$302$	23.9103	1.37589
$303$	0	0
$304$	−3.59627	−0.206260
$305$	−3.23854	−0.185438
$306$	0	0
$307$	−18.3756	−1.04875	−0.524375	−	0.851488i	$-0.675701\pi$
$307$	−18.3756	−1.04875	−0.524375	+	0.851488i	$0.675701\pi$
$308$	−1.04189	−0.0593671
$309$	0	0
$310$	−12.7965	−0.726795
$311$	10.9581	0.621377	0.310689	−	0.950512i	$-0.399440\pi$
$311$	10.9581	0.621377	0.310689	+	0.950512i	$0.399440\pi$
$312$	0	0
$313$	25.7547	1.45574	0.727869	−	0.685716i	$-0.240511\pi$
$313$	25.7547	1.45574	0.727869	+	0.685716i	$0.240511\pi$
$314$	6.73648	0.380162
$315$	0	0
$316$	2.33544	0.131379
$317$	−22.8580	−1.28383	−0.641917	−	0.766774i	$-0.721861\pi$
$317$	−22.8580	−1.28383	−0.641917	+	0.766774i	$0.721861\pi$
$318$	0	0
$319$	7.59627	0.425309
$320$	11.5817	0.647438
$321$	0	0
$322$	7.68954	0.428521
$323$	0.305407	0.0169933
$324$	0	0
$325$	−11.4534	−0.635318
$326$	−13.7679	−0.762536
$327$	0	0
$328$	−10.0996	−0.557656
$329$	−6.84699	−0.377487
$330$	0	0
$331$	20.8898	1.14821	0.574104	−	0.818782i	$-0.305350\pi$
$331$	20.8898	1.14821	0.574104	+	0.818782i	$0.305350\pi$
$332$	1.52940	0.0839368
$333$	0	0
$334$	23.3523	1.27778
$335$	10.2909	0.562250
$336$	0	0
$337$	3.16756	0.172548	0.0862739	−	0.996271i	$-0.472504\pi$
$337$	3.16756	0.172548	0.0862739	+	0.996271i	$0.472504\pi$
$338$	−0.0900885	−0.00490017
$339$	0	0
$340$	−0.0760373	−0.00412370
$341$	−32.4020	−1.75467
$342$	0	0
$343$	−15.3277	−0.827618
$344$	33.0692	1.78297
$345$	0	0
$346$	8.98276	0.482916
$347$	−23.7615	−1.27558	−0.637791	−	0.770210i	$-0.720152\pi$
$347$	−23.7615	−1.27558	−0.637791	+	0.770210i	$0.720152\pi$
$348$	0	0
$349$	25.7374	1.37769	0.688846	−	0.724908i	$-0.258118\pi$
$349$	25.7374	1.37769	0.688846	+	0.724908i	$0.258118\pi$
$350$	−5.26352	−0.281347
$351$	0	0
$352$	4.78880	0.255244
$353$	−3.06687	−0.163233	−0.0816164	−	0.996664i	$-0.526008\pi$
$353$	−3.06687	−0.163233	−0.0816164	+	0.996664i	$0.526008\pi$
$354$	0	0
$355$	−1.73917	−0.0923056
$356$	1.46017	0.0773889
$357$	0	0
$358$	6.19253	0.327286
$359$	20.8999	1.10306	0.551528	−	0.834157i	$-0.314045\pi$
$359$	20.8999	1.10306	0.551528	+	0.834157i	$0.314045\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	9.35142	0.491499
$363$	0	0
$364$	−0.815207	−0.0427285
$365$	−16.5398	−0.865734
$366$	0	0
$367$	−0.512489	−0.0267517	−0.0133758	−	0.999911i	$-0.504258\pi$
$367$	−0.512489	−0.0267517	−0.0133758	+	0.999911i	$0.504258\pi$
$368$	−16.7324	−0.872235
$369$	0	0
$370$	0.335437	0.0174385
$371$	−3.61081	−0.187464
$372$	0	0
$373$	14.7469	0.763566	0.381783	−	0.924252i	$-0.375310\pi$
$373$	14.7469	0.763566	0.381783	+	0.924252i	$0.375310\pi$
$374$	1.89124	0.0977939
$375$	0	0
$376$	16.4301	0.847320
$377$	5.94356	0.306109
$378$	0	0
$379$	−29.7128	−1.52624	−0.763121	−	0.646256i	$-0.776334\pi$
$379$	−29.7128	−1.52624	−0.763121	+	0.646256i	$0.776334\pi$
$380$	−0.248970	−0.0127719
$381$	0	0
$382$	−0.916222	−0.0468780
$383$	37.1343	1.89748	0.948738	−	0.316063i	$-0.102361\pi$
$383$	37.1343	1.89748	0.948738	+	0.316063i	$0.102361\pi$
$384$	0	0
$385$	7.59627	0.387142
$386$	−33.6236	−1.71140
$387$	0	0
$388$	−1.29498	−0.0657426
$389$	−13.7760	−0.698471	−0.349236	−	0.937035i	$-0.613559\pi$
$389$	−13.7760	−0.698471	−0.349236	+	0.937035i	$0.613559\pi$
$390$	0	0
$391$	1.42097	0.0718615
$392$	16.1756	0.816992
$393$	0	0
$394$	30.2158	1.52225
$395$	−17.0273	−0.856739
$396$	0	0
$397$	18.1676	0.911804	0.455902	−	0.890030i	$-0.349317\pi$
$397$	18.1676	0.911804	0.455902	+	0.890030i	$0.349317\pi$
$398$	−21.9855	−1.10203
$399$	0	0
$400$	11.4534	0.572668
$401$	26.2867	1.31270	0.656349	−	0.754458i	$-0.272100\pi$
$401$	26.2867	1.31270	0.656349	+	0.754458i	$0.272100\pi$
$402$	0	0
$403$	−25.3523	−1.26289
$404$	1.42459	0.0708761
$405$	0	0
$406$	2.73143	0.135559
$407$	0.849356	0.0421010
$408$	0	0
$409$	−4.50475	−0.222745	−0.111373	−	0.993779i	$-0.535525\pi$
$409$	−4.50475	−0.222745	−0.111373	+	0.993779i	$0.535525\pi$
$410$	6.22811	0.307585
$411$	0	0
$412$	−0.847925	−0.0417743
$413$	14.6509	0.720926
$414$	0	0
$415$	−11.1506	−0.547364
$416$	3.74691	0.183707
$417$	0	0
$418$	6.19253	0.302887
$419$	−0.554378	−0.0270831	−0.0135416	−	0.999908i	$-0.504311\pi$
$419$	−0.554378	−0.0270831	−0.0135416	+	0.999908i	$0.504311\pi$
$420$	0	0
$421$	−3.22668	−0.157259	−0.0786294	−	0.996904i	$-0.525054\pi$
$421$	−3.22668	−0.157259	−0.0786294	+	0.996904i	$0.525054\pi$
$422$	21.1821	1.03113
$423$	0	0
$424$	8.66456	0.420788
$425$	−0.972659	−0.0471809
$426$	0	0
$427$	−2.94862	−0.142694
$428$	−3.58996	−0.173527
$429$	0	0
$430$	−20.3928	−0.983429
$431$	−40.3705	−1.94458	−0.972290	−	0.233779i	$-0.924891\pi$
$431$	−40.3705	−1.94458	−0.972290	+	0.233779i	$0.924891\pi$
$432$	0	0
$433$	−20.9581	−1.00718	−0.503591	−	0.863942i	$-0.667988\pi$
$433$	−20.9581	−1.00718	−0.503591	+	0.863942i	$0.667988\pi$
$434$	−11.6509	−0.559263
$435$	0	0
$436$	2.02322	0.0968948
$437$	4.65270	0.222569
$438$	0	0
$439$	−24.2763	−1.15865	−0.579323	−	0.815098i	$-0.696683\pi$
$439$	−24.2763	−1.15865	−0.579323	+	0.815098i	$0.696683\pi$
$440$	−18.2281	−0.868991
$441$	0	0
$442$	1.47977	0.0703855
$443$	−14.9436	−0.709990	−0.354995	−	0.934868i	$-0.615517\pi$
$443$	−14.9436	−0.709990	−0.354995	+	0.934868i	$0.615517\pi$
$444$	0	0
$445$	−10.6459	−0.504664
$446$	−35.0273	−1.65859
$447$	0	0
$448$	10.5449	0.498199
$449$	−37.0651	−1.74921	−0.874605	−	0.484835i	$-0.838879\pi$
$449$	−37.0651	−1.74921	−0.874605	+	0.484835i	$0.838879\pi$
$450$	0	0
$451$	15.7701	0.742587
$452$	0.656822	0.0308943
$453$	0	0
$454$	20.9564	0.983531
$455$	5.94356	0.278639
$456$	0	0
$457$	28.4861	1.33252	0.666261	−	0.745718i	$-0.267894\pi$
$457$	28.4861	1.33252	0.666261	+	0.745718i	$0.267894\pi$
$458$	21.6340	1.01089
$459$	0	0
$460$	−1.15839	−0.0540100
$461$	−40.4834	−1.88550	−0.942750	−	0.333501i	$-0.891770\pi$
$461$	−40.4834	−1.88550	−0.942750	+	0.333501i	$0.891770\pi$
$462$	0	0
$463$	−0.200274	−0.00930753	−0.00465376	−	0.999989i	$-0.501481\pi$
$463$	−0.200274	−0.00930753	−0.00465376	+	0.999989i	$0.501481\pi$
$464$	−5.94356	−0.275923
$465$	0	0
$466$	−25.9905	−1.20399
$467$	−17.1070	−0.791618	−0.395809	−	0.918333i	$-0.629536\pi$
$467$	−17.1070	−0.791618	−0.395809	+	0.918333i	$0.629536\pi$
$468$	0	0
$469$	9.36959	0.432647
$470$	−10.1320	−0.467353
$471$	0	0
$472$	−35.1566	−1.61822
$473$	−51.6364	−2.37424
$474$	0	0
$475$	−3.18479	−0.146128
$476$	−0.0692302	−0.00317316
$477$	0	0
$478$	14.1070	0.645239
$479$	21.8033	0.996220	0.498110	−	0.867114i	$-0.334027\pi$
$479$	21.8033	0.996220	0.498110	+	0.867114i	$0.334027\pi$
$480$	0	0
$481$	0.664563	0.0303015
$482$	17.4584	0.795209
$483$	0	0
$484$	−1.87115	−0.0850522
$485$	9.44150	0.428717
$486$	0	0
$487$	−12.7050	−0.575719	−0.287860	−	0.957673i	$-0.592944\pi$
$487$	−12.7050	−0.575719	−0.287860	+	0.957673i	$0.592944\pi$
$488$	7.07554	0.320295
$489$	0	0
$490$	−9.97502	−0.450626
$491$	−35.1634	−1.58690	−0.793452	−	0.608633i	$-0.791718\pi$
$491$	−35.1634	−1.58690	−0.793452	+	0.608633i	$0.791718\pi$
$492$	0	0
$493$	0.504748	0.0227327
$494$	4.84524	0.217998
$495$	0	0
$496$	25.3523	1.13835
$497$	−1.58347	−0.0710285
$498$	0	0
$499$	1.87433	0.0839067	0.0419533	−	0.999120i	$-0.486642\pi$
$499$	1.87433	0.0839067	0.0419533	+	0.999120i	$0.486642\pi$
$500$	2.03777	0.0911319
$501$	0	0
$502$	31.8648	1.42220
$503$	26.2891	1.17217	0.586087	−	0.810248i	$-0.300668\pi$
$503$	26.2891	1.17217	0.586087	+	0.810248i	$0.300668\pi$
$504$	0	0
$505$	−10.3865	−0.462193
$506$	28.8120	1.28085
$507$	0	0
$508$	−1.02322	−0.0453982
$509$	27.0360	1.19835	0.599175	−	0.800618i	$-0.295495\pi$
$509$	27.0360	1.19835	0.599175	+	0.800618i	$0.295495\pi$
$510$	0	0
$511$	−15.0591	−0.666176
$512$	−24.9186	−1.10126
$513$	0	0
$514$	9.65364	0.425804
$515$	6.18210	0.272416
$516$	0	0
$517$	−25.6551	−1.12831
$518$	0.305407	0.0134188
$519$	0	0
$520$	−14.2623	−0.625441
$521$	16.1676	0.708314	0.354157	−	0.935186i	$-0.384768\pi$
$521$	16.1676	0.708314	0.354157	+	0.935186i	$0.384768\pi$
$522$	0	0
$523$	19.5125	0.853222	0.426611	−	0.904435i	$-0.359707\pi$
$523$	19.5125	0.853222	0.426611	+	0.904435i	$0.359707\pi$
$524$	3.23854	0.141476
$525$	0	0
$526$	26.5853	1.15918
$527$	−2.15301	−0.0937865
$528$	0	0
$529$	−1.35235	−0.0587978
$530$	−5.34318	−0.232093
$531$	0	0
$532$	−0.226682	−0.00982789
$533$	12.3391	0.534464
$534$	0	0
$535$	26.1739	1.13159
$536$	−22.4834	−0.971135
$537$	0	0
$538$	−43.0155	−1.85453
$539$	−25.2576	−1.08792
$540$	0	0
$541$	−34.2594	−1.47293	−0.736463	−	0.676477i	$-0.763506\pi$
$541$	−34.2594	−1.47293	−0.736463	+	0.676477i	$0.763506\pi$
$542$	12.0209	0.516340
$543$	0	0
$544$	0.318201	0.0136427
$545$	−14.7510	−0.631865
$546$	0	0
$547$	38.2746	1.63650	0.818251	−	0.574861i	$-0.194944\pi$
$547$	38.2746	1.63650	0.818251	+	0.574861i	$0.194944\pi$
$548$	1.29086	0.0551428
$549$	0	0
$550$	−19.7219	−0.840946
$551$	1.65270	0.0704075
$552$	0	0
$553$	−15.5030	−0.659254
$554$	−13.2449	−0.562720
$555$	0	0
$556$	−3.39281	−0.143887
$557$	−11.9872	−0.507914	−0.253957	−	0.967215i	$-0.581732\pi$
$557$	−11.9872	−0.507914	−0.253957	+	0.967215i	$0.581732\pi$
$558$	0	0
$559$	−40.4020	−1.70882
$560$	−5.94356	−0.251161
$561$	0	0
$562$	−0.411474	−0.0173570
$563$	13.4270	0.565879	0.282939	−	0.959138i	$-0.408690\pi$
$563$	13.4270	0.565879	0.282939	+	0.959138i	$0.408690\pi$
$564$	0	0
$565$	−4.78880	−0.201466
$566$	13.9709	0.587241
$567$	0	0
$568$	3.79973	0.159433
$569$	2.57129	0.107794	0.0538970	−	0.998546i	$-0.482836\pi$
$569$	2.57129	0.107794	0.0538970	+	0.998546i	$0.482836\pi$
$570$	0	0
$571$	−39.3259	−1.64574	−0.822870	−	0.568230i	$-0.807628\pi$
$571$	−39.3259	−1.64574	−0.822870	+	0.568230i	$0.807628\pi$
$572$	−3.05451	−0.127715
$573$	0	0
$574$	5.67055	0.236684
$575$	−14.8179	−0.617949
$576$	0	0
$577$	34.5357	1.43774	0.718870	−	0.695144i	$-0.244660\pi$
$577$	34.5357	1.43774	0.718870	+	0.695144i	$0.244660\pi$
$578$	−22.7784	−0.947455
$579$	0	0
$580$	−0.411474	−0.0170855
$581$	−10.1524	−0.421192
$582$	0	0
$583$	−13.5294	−0.560331
$584$	36.1361	1.49532
$585$	0	0
$586$	−13.1320	−0.542477
$587$	−25.8016	−1.06495	−0.532473	−	0.846447i	$-0.678737\pi$
$587$	−25.8016	−1.06495	−0.532473	+	0.846447i	$0.678737\pi$
$588$	0	0
$589$	−7.04963	−0.290475
$590$	21.6800	0.892553
$591$	0	0
$592$	−0.664563	−0.0273134
$593$	42.0797	1.72800	0.864002	−	0.503488i	$-0.167950\pi$
$593$	42.0797	1.72800	0.864002	+	0.503488i	$0.167950\pi$
$594$	0	0
$595$	0.504748	0.0206926
$596$	1.17799	0.0482522
$597$	0	0
$598$	22.5435	0.921870
$599$	−8.64052	−0.353042	−0.176521	−	0.984297i	$-0.556484\pi$
$599$	−8.64052	−0.353042	−0.176521	+	0.984297i	$0.556484\pi$
$600$	0	0
$601$	12.1179	0.494301	0.247150	−	0.968977i	$-0.420506\pi$
$601$	12.1179	0.494301	0.247150	+	0.968977i	$0.420506\pi$
$602$	−18.5672	−0.756741
$603$	0	0
$604$	−3.27950	−0.133441
$605$	13.6423	0.554637
$606$	0	0
$607$	26.9486	1.09381	0.546905	−	0.837194i	$-0.315806\pi$
$607$	26.9486	1.09381	0.546905	+	0.837194i	$0.315806\pi$
$608$	1.04189	0.0422542
$609$	0	0
$610$	−4.36327	−0.176664
$611$	−20.0733	−0.812081
$612$	0	0
$613$	−40.7357	−1.64530	−0.822649	−	0.568550i	$-0.807505\pi$
$613$	−40.7357	−1.64530	−0.822649	+	0.568550i	$0.807505\pi$
$614$	−24.7573	−0.999125
$615$	0	0
$616$	−16.5963	−0.668683
$617$	−48.3438	−1.94625	−0.973124	−	0.230283i	$-0.926035\pi$
$617$	−48.3438	−1.94625	−0.973124	+	0.230283i	$0.926035\pi$
$618$	0	0
$619$	−27.9240	−1.12236	−0.561179	−	0.827694i	$-0.689652\pi$
$619$	−27.9240	−1.12236	−0.561179	+	0.827694i	$0.689652\pi$
$620$	1.75515	0.0704885
$621$	0	0
$622$	14.7638	0.591975
$623$	−9.69284	−0.388335
$624$	0	0
$625$	1.06687	0.0426746
$626$	34.6991	1.38686
$627$	0	0
$628$	−0.923963	−0.0368701
$629$	0.0564370	0.00225029
$630$	0	0
$631$	27.1088	1.07918	0.539591	−	0.841927i	$-0.318579\pi$
$631$	27.1088	1.07918	0.539591	+	0.841927i	$0.318579\pi$
$632$	37.2012	1.47978
$633$	0	0
$634$	−30.7965	−1.22309
$635$	7.46017	0.296048
$636$	0	0
$637$	−19.7624	−0.783014
$638$	10.2344	0.405185
$639$	0	0
$640$	12.7965	0.505828
$641$	−26.7178	−1.05529	−0.527645	−	0.849465i	$-0.676925\pi$
$641$	−26.7178	−1.05529	−0.527645	+	0.849465i	$0.676925\pi$
$642$	0	0
$643$	31.8307	1.25528	0.627640	−	0.778504i	$-0.284021\pi$
$643$	31.8307	1.25528	0.627640	+	0.778504i	$0.284021\pi$
$644$	−1.05468	−0.0415603
$645$	0	0
$646$	0.411474	0.0161892
$647$	16.8912	0.664063	0.332032	−	0.943268i	$-0.392266\pi$
$647$	16.8912	0.664063	0.332032	+	0.943268i	$0.392266\pi$
$648$	0	0
$649$	54.8958	2.15485
$650$	−15.4311	−0.605257
$651$	0	0
$652$	1.88838	0.0739548
$653$	18.7469	0.733623	0.366812	−	0.930295i	$-0.380449\pi$
$653$	18.7469	0.733623	0.366812	+	0.930295i	$0.380449\pi$
$654$	0	0
$655$	−23.6117	−0.922587
$656$	−12.3391	−0.481759
$657$	0	0
$658$	−9.22493	−0.359625
$659$	20.1198	0.783756	0.391878	−	0.920017i	$-0.371826\pi$
$659$	20.1198	0.783756	0.391878	+	0.920017i	$0.371826\pi$
$660$	0	0
$661$	−29.9317	−1.16421	−0.582104	−	0.813114i	$-0.697770\pi$
$661$	−29.9317	−1.16421	−0.582104	+	0.813114i	$0.697770\pi$
$662$	28.1448	1.09388
$663$	0	0
$664$	24.3618	0.945423
$665$	1.65270	0.0640891
$666$	0	0
$667$	7.68954	0.297740
$668$	−3.20296	−0.123926
$669$	0	0
$670$	13.8648	0.535645
$671$	−11.0482	−0.426511
$672$	0	0
$673$	8.99226	0.346626	0.173313	−	0.984867i	$-0.444553\pi$
$673$	8.99226	0.346626	0.173313	+	0.984867i	$0.444553\pi$
$674$	4.26764	0.164383
$675$	0	0
$676$	0.0123564	0.000475245 0
$677$	15.6212	0.600373	0.300187	−	0.953881i	$-0.402951\pi$
$677$	15.6212	0.600373	0.300187	+	0.953881i	$0.402951\pi$
$678$	0	0
$679$	8.59627	0.329894
$680$	−1.21120	−0.0464474
$681$	0	0
$682$	−43.6551	−1.67164
$683$	−1.10876	−0.0424253	−0.0212127	−	0.999775i	$-0.506753\pi$
$683$	−1.10876	−0.0424253	−0.0212127	+	0.999775i	$0.506753\pi$
$684$	0	0
$685$	−9.41147	−0.359594
$686$	−20.6509	−0.788457
$687$	0	0
$688$	40.4020	1.54031
$689$	−10.5858	−0.403288
$690$	0	0
$691$	−10.9162	−0.415273	−0.207636	−	0.978206i	$-0.566577\pi$
$691$	−10.9162	−0.415273	−0.207636	+	0.978206i	$0.566577\pi$
$692$	−1.23206	−0.0468358
$693$	0	0
$694$	−32.0137	−1.21522
$695$	24.7365	0.938308
$696$	0	0
$697$	1.04788	0.0396911
$698$	34.6759	1.31250
$699$	0	0
$700$	0.721934	0.0272865
$701$	25.9168	0.978865	0.489433	−	0.872041i	$-0.337204\pi$
$701$	25.9168	0.978865	0.489433	+	0.872041i	$0.337204\pi$
$702$	0	0
$703$	0.184793	0.00696958
$704$	39.5107	1.48912
$705$	0	0
$706$	−4.13198	−0.155509
$707$	−9.45666	−0.355654
$708$	0	0
$709$	−41.6195	−1.56305	−0.781526	−	0.623872i	$-0.785559\pi$
$709$	−41.6195	−1.56305	−0.781526	+	0.623872i	$0.785559\pi$
$710$	−2.34318	−0.0879379
$711$	0	0
$712$	23.2591	0.871671
$713$	−32.7998	−1.22836
$714$	0	0
$715$	22.2700	0.832851
$716$	−0.849356	−0.0317419
$717$	0	0
$718$	28.1584	1.05086
$719$	29.4561	1.09853	0.549263	−	0.835650i	$-0.314909\pi$
$719$	29.4561	1.09853	0.549263	+	0.835650i	$0.314909\pi$
$720$	0	0
$721$	5.62866	0.209622
$722$	1.34730	0.0501412
$723$	0	0
$724$	−1.28262	−0.0476683
$725$	−5.26352	−0.195482
$726$	0	0
$727$	3.78880	0.140519	0.0702594	−	0.997529i	$-0.477617\pi$
$727$	3.78880	0.140519	0.0702594	+	0.997529i	$0.477617\pi$
$728$	−12.9855	−0.481273
$729$	0	0
$730$	−22.2841	−0.824770
$731$	−3.43107	−0.126903
$732$	0	0
$733$	15.1239	0.558614	0.279307	−	0.960202i	$-0.409895\pi$
$733$	15.1239	0.558614	0.279307	+	0.960202i	$0.409895\pi$
$734$	−0.690474	−0.0254859
$735$	0	0
$736$	4.84760	0.178685
$737$	35.1070	1.29318
$738$	0	0
$739$	48.4938	1.78387	0.891937	−	0.452160i	$-0.149346\pi$
$739$	48.4938	1.78387	0.891937	+	0.452160i	$0.149346\pi$
$740$	−0.0460079	−0.00169128
$741$	0	0
$742$	−4.86484	−0.178594
$743$	16.1780	0.593513	0.296756	−	0.954953i	$-0.404095\pi$
$743$	16.1780	0.593513	0.296756	+	0.954953i	$0.404095\pi$
$744$	0	0
$745$	−8.58853	−0.314659
$746$	19.8685	0.727436
$747$	0	0
$748$	−0.259399	−0.00948458
$749$	23.8307	0.870754
$750$	0	0
$751$	8.55438	0.312154	0.156077	−	0.987745i	$-0.450115\pi$
$751$	8.55438	0.312154	0.156077	+	0.987745i	$0.450115\pi$
$752$	20.0733	0.732000
$753$	0	0
$754$	8.00774	0.291625
$755$	23.9103	0.870187
$756$	0	0
$757$	45.1471	1.64090	0.820450	−	0.571718i	$-0.193723\pi$
$757$	45.1471	1.64090	0.820450	+	0.571718i	$0.193723\pi$
$758$	−40.0319	−1.45402
$759$	0	0
$760$	−3.96585	−0.143857
$761$	−12.5544	−0.455096	−0.227548	−	0.973767i	$-0.573071\pi$
$761$	−12.5544	−0.455096	−0.227548	+	0.973767i	$0.573071\pi$
$762$	0	0
$763$	−13.4305	−0.486215
$764$	0.125667	0.00454648
$765$	0	0
$766$	50.0310	1.80769
$767$	42.9522	1.55092
$768$	0	0
$769$	−15.2094	−0.548467	−0.274233	−	0.961663i	$-0.588424\pi$
$769$	−15.2094	−0.548467	−0.274233	+	0.961663i	$0.588424\pi$
$770$	10.2344	0.368823
$771$	0	0
$772$	4.61175	0.165980
$773$	38.2303	1.37505	0.687524	−	0.726161i	$-0.258698\pi$
$773$	38.2303	1.37505	0.687524	+	0.726161i	$0.258698\pi$
$774$	0	0
$775$	22.4516	0.806486
$776$	−20.6277	−0.740492
$777$	0	0
$778$	−18.5604	−0.665421
$779$	3.43107	0.122931
$780$	0	0
$781$	−5.93313	−0.212304
$782$	1.91447	0.0684612
$783$	0	0
$784$	19.7624	0.705800
$785$	6.73648	0.240435
$786$	0	0
$787$	16.8631	0.601104	0.300552	−	0.953765i	$-0.402829\pi$
$787$	16.8631	0.601104	0.300552	+	0.953765i	$0.402829\pi$
$788$	−4.14433	−0.147636
$789$	0	0
$790$	−22.9409	−0.816200
$791$	−4.36009	−0.155027
$792$	0	0
$793$	−8.64447	−0.306974
$794$	24.4771	0.868659
$795$	0	0
$796$	3.01548	0.106881
$797$	32.4688	1.15011	0.575053	−	0.818116i	$-0.304982\pi$
$797$	32.4688	1.15011	0.575053	+	0.818116i	$0.304982\pi$
$798$	0	0
$799$	−1.70470	−0.0603079
$800$	−3.31820	−0.117316
$801$	0	0
$802$	35.4160	1.25058
$803$	−56.4252	−1.99120
$804$	0	0
$805$	7.68954	0.271021
$806$	−34.1571	−1.20313
$807$	0	0
$808$	22.6923	0.798314
$809$	−7.93489	−0.278976	−0.139488	−	0.990224i	$-0.544546\pi$
$809$	−7.93489	−0.278976	−0.139488	+	0.990224i	$0.544546\pi$
$810$	0	0
$811$	−40.0788	−1.40736	−0.703679	−	0.710518i	$-0.748461\pi$
$811$	−40.0788	−1.40736	−0.703679	+	0.710518i	$0.748461\pi$
$812$	−0.374638	−0.0131472
$813$	0	0
$814$	1.14433	0.0401089
$815$	−13.7679	−0.482270
$816$	0	0
$817$	−11.2344	−0.393043
$818$	−6.06923	−0.212206
$819$	0	0
$820$	−0.854236	−0.0298312
$821$	−3.98545	−0.139093	−0.0695466	−	0.997579i	$-0.522155\pi$
$821$	−3.98545	−0.139093	−0.0695466	+	0.997579i	$0.522155\pi$
$822$	0	0
$823$	31.0077	1.08086	0.540431	−	0.841388i	$-0.318261\pi$
$823$	31.0077	1.08086	0.540431	+	0.841388i	$0.318261\pi$
$824$	−13.5066	−0.470525
$825$	0	0
$826$	19.7392	0.686814
$827$	52.2036	1.81530	0.907648	−	0.419732i	$-0.137876\pi$
$827$	52.2036	1.81530	0.907648	+	0.419732i	$0.137876\pi$
$828$	0	0
$829$	35.2686	1.22493	0.612464	−	0.790498i	$-0.290178\pi$
$829$	35.2686	1.22493	0.612464	+	0.790498i	$0.290178\pi$
$830$	−15.0232	−0.521464
$831$	0	0
$832$	30.9145	1.07177
$833$	−1.67829	−0.0581493
$834$	0	0
$835$	23.3523	0.808141
$836$	−0.849356	−0.0293756
$837$	0	0
$838$	−0.746911	−0.0258016
$839$	37.9837	1.31134	0.655671	−	0.755046i	$-0.272386\pi$
$839$	37.9837	1.31134	0.655671	+	0.755046i	$0.272386\pi$
$840$	0	0
$841$	−26.2686	−0.905813
$842$	−4.34730	−0.149818
$843$	0	0
$844$	−2.90530	−0.100004
$845$	−0.0900885	−0.00309914
$846$	0	0
$847$	12.4210	0.426790
$848$	10.5858	0.363519
$849$	0	0
$850$	−1.31046	−0.0449484
$851$	0.859785	0.0294730
$852$	0	0
$853$	−36.5125	−1.25016	−0.625082	−	0.780559i	$-0.714934\pi$
$853$	−36.5125	−1.25016	−0.625082	+	0.780559i	$0.714934\pi$
$854$	−3.97266	−0.135942
$855$	0	0
$856$	−57.1845	−1.95452
$857$	51.8394	1.77080	0.885399	−	0.464831i	$-0.153885\pi$
$857$	51.8394	1.77080	0.885399	+	0.464831i	$0.153885\pi$
$858$	0	0
$859$	5.80571	0.198088	0.0990442	−	0.995083i	$-0.468421\pi$
$859$	5.80571	0.198088	0.0990442	+	0.995083i	$0.468421\pi$
$860$	2.79704	0.0953782
$861$	0	0
$862$	−54.3911	−1.85257
$863$	38.0250	1.29439	0.647193	−	0.762326i	$-0.275943\pi$
$863$	38.0250	1.29439	0.647193	+	0.762326i	$0.275943\pi$
$864$	0	0
$865$	8.98276	0.305423
$866$	−28.2368	−0.959525
$867$	0	0
$868$	1.59802	0.0542404
$869$	−58.0883	−1.97051
$870$	0	0
$871$	27.4688	0.930746
$872$	32.2279	1.09138
$873$	0	0
$874$	6.26857	0.212038
$875$	−13.5270	−0.457297
$876$	0	0
$877$	3.16756	0.106961	0.0534804	−	0.998569i	$-0.482969\pi$
$877$	3.16756	0.106961	0.0534804	+	0.998569i	$0.482969\pi$
$878$	−32.7074	−1.10382
$879$	0	0
$880$	−22.2700	−0.750722
$881$	−6.38507	−0.215118	−0.107559	−	0.994199i	$-0.534304\pi$
$881$	−6.38507	−0.215118	−0.107559	+	0.994199i	$0.534304\pi$
$882$	0	0
$883$	0.426956	0.0143682	0.00718410	−	0.999974i	$-0.497713\pi$
$883$	0.426956	0.0143682	0.00718410	+	0.999974i	$0.497713\pi$
$884$	−0.202962	−0.00682637
$885$	0	0
$886$	−20.1334	−0.676395
$887$	−5.64858	−0.189661	−0.0948305	−	0.995493i	$-0.530231\pi$
$887$	−5.64858	−0.189661	−0.0948305	+	0.995493i	$0.530231\pi$
$888$	0	0
$889$	6.79231	0.227807
$890$	−14.3432	−0.480784
$891$	0	0
$892$	4.80428	0.160859
$893$	−5.58172	−0.186785
$894$	0	0
$895$	6.19253	0.206994
$896$	11.6509	0.389231
$897$	0	0
$898$	−49.9377	−1.66644
$899$	−11.6509	−0.388581
$900$	0	0
$901$	−0.898986	−0.0299496
$902$	21.2470	0.707449
$903$	0	0
$904$	10.4625	0.347979
$905$	9.35142	0.310852
$906$	0	0
$907$	−6.83986	−0.227114	−0.113557	−	0.993531i	$-0.536224\pi$
$907$	−6.83986	−0.227114	−0.113557	+	0.993531i	$0.536224\pi$
$908$	−2.87433	−0.0953881
$909$	0	0
$910$	8.00774	0.265454
$911$	27.8057	0.921244	0.460622	−	0.887596i	$-0.347626\pi$
$911$	27.8057	0.921244	0.460622	+	0.887596i	$0.347626\pi$
$912$	0	0
$913$	−38.0401	−1.25894
$914$	38.3792	1.26947
$915$	0	0
$916$	−2.96728	−0.0980418
$917$	−21.4979	−0.709925
$918$	0	0
$919$	0.184793	0.00609574	0.00304787	−	0.999995i	$-0.499030\pi$
$919$	0.184793	0.00609574	0.00304787	+	0.999995i	$0.499030\pi$
$920$	−18.4519	−0.608342
$921$	0	0
$922$	−54.5431	−1.79628
$923$	−4.64227	−0.152802
$924$	0	0
$925$	−0.588526	−0.0193506
$926$	−0.269829	−0.00886712
$927$	0	0
$928$	1.72193	0.0565252
$929$	50.0506	1.64211	0.821053	−	0.570852i	$-0.193387\pi$
$929$	50.0506	1.64211	0.821053	+	0.570852i	$0.193387\pi$
$930$	0	0
$931$	−5.49525	−0.180100
$932$	3.56481	0.116769
$933$	0	0
$934$	−23.0482	−0.754160
$935$	1.89124	0.0618503
$936$	0	0
$937$	−9.95636	−0.325260	−0.162630	−	0.986687i	$-0.551998\pi$
$937$	−9.95636	−0.325260	−0.162630	+	0.986687i	$0.551998\pi$
$938$	12.6236	0.412175
$939$	0	0
$940$	1.38968	0.0453264
$941$	3.61081	0.117709	0.0588546	−	0.998267i	$-0.481255\pi$
$941$	3.61081	0.117709	0.0588546	+	0.998267i	$0.481255\pi$
$942$	0	0
$943$	15.9638	0.519852
$944$	−42.9522	−1.39798
$945$	0	0
$946$	−69.5695	−2.26190
$947$	25.5550	0.830425	0.415213	−	0.909724i	$-0.363707\pi$
$947$	25.5550	0.830425	0.415213	+	0.909724i	$0.363707\pi$
$948$	0	0
$949$	−44.1489	−1.43313
$950$	−4.29086	−0.139214
$951$	0	0
$952$	−1.10277	−0.0357410
$953$	−40.9505	−1.32652	−0.663258	−	0.748391i	$-0.730827\pi$
$953$	−40.9505	−1.32652	−0.663258	+	0.748391i	$0.730827\pi$
$954$	0	0
$955$	−0.916222	−0.0296482
$956$	−1.93489	−0.0625788
$957$	0	0
$958$	29.3756	0.949082
$959$	−8.56893	−0.276705
$960$	0	0
$961$	18.6973	0.603138
$962$	0.895364	0.0288677
$963$	0	0
$964$	−2.39456	−0.0771237
$965$	−33.6236	−1.08238
$966$	0	0
$967$	−55.0028	−1.76877	−0.884385	−	0.466757i	$-0.845422\pi$
$967$	−55.0028	−1.76877	−0.884385	+	0.466757i	$0.845422\pi$
$968$	−29.8055	−0.957986
$969$	0	0
$970$	12.7205	0.408431
$971$	−11.6714	−0.374552	−0.187276	−	0.982307i	$-0.559966\pi$
$971$	−11.6714	−0.374552	−0.187276	+	0.982307i	$0.559966\pi$
$972$	0	0
$973$	22.5220	0.722022
$974$	−17.1174	−0.548478
$975$	0	0
$976$	8.64447	0.276703
$977$	27.5107	0.880146	0.440073	−	0.897962i	$-0.354952\pi$
$977$	27.5107	0.880146	0.440073	+	0.897962i	$0.354952\pi$
$978$	0	0
$979$	−36.3182	−1.16073
$980$	1.36815	0.0437041
$981$	0	0
$982$	−47.3756	−1.51182
$983$	−4.60906	−0.147006	−0.0735031	−	0.997295i	$-0.523418\pi$
$983$	−4.60906	−0.147006	−0.0735031	+	0.997295i	$0.523418\pi$
$984$	0	0
$985$	30.2158	0.962754
$986$	0.680045	0.0216570
$987$	0	0
$988$	−0.664563	−0.0211426
$989$	−52.2704	−1.66210
$990$	0	0
$991$	49.9564	1.58692	0.793458	−	0.608625i	$-0.208279\pi$
$991$	49.9564	1.58692	0.793458	+	0.608625i	$0.208279\pi$
$992$	−7.34493	−0.233202
$993$	0	0
$994$	−2.13341	−0.0676676
$995$	−21.9855	−0.696986
$996$	0	0
$997$	49.5357	1.56881	0.784406	−	0.620248i	$-0.212968\pi$
$997$	49.5357	1.56881	0.784406	+	0.620248i	$0.212968\pi$
$998$	2.52528	0.0799364
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.g.1.2	yes	3
3.2	odd	2		513.2.a.d.1.2	✓	3
4.3	odd	2		8208.2.a.bn.1.2		3
12.11	even	2		8208.2.a.bh.1.2		3
19.18	odd	2		9747.2.a.w.1.2		3
57.56	even	2		9747.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.d.1.2	✓	3	3.2	odd	2
513.2.a.g.1.2	yes	3	1.1	even	1	trivial
8208.2.a.bh.1.2		3	12.11	even	2
8208.2.a.bn.1.2		3	4.3	odd	2
9747.2.a.w.1.2		3	19.18	odd	2
9747.2.a.bc.1.2		3	57.56	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34730 q^{2} -0.184793 q^{4} +1.34730 q^{5} +1.22668 q^{7} -2.94356 q^{8} +1.81521 q^{10} +4.59627 q^{11} +3.59627 q^{13} +1.65270 q^{14} -3.59627 q^{16} +0.305407 q^{17} +1.00000 q^{19} -0.248970 q^{20} +6.19253 q^{22} +4.65270 q^{23} -3.18479 q^{25} +4.84524 q^{26} -0.226682 q^{28} +1.65270 q^{29} -7.04963 q^{31} +1.04189 q^{32} +0.411474 q^{34} +1.65270 q^{35} +0.184793 q^{37} +1.34730 q^{38} -3.96585 q^{40} +3.43107 q^{41} -11.2344 q^{43} -0.849356 q^{44} +6.26857 q^{46} -5.58172 q^{47} -5.49525 q^{49} -4.29086 q^{50} -0.664563 q^{52} -2.94356 q^{53} +6.19253 q^{55} -3.61081 q^{56} +2.22668 q^{58} +11.9436 q^{59} -2.40373 q^{61} -9.49794 q^{62} +8.59627 q^{64} +4.84524 q^{65} +7.63816 q^{67} -0.0564370 q^{68} +2.22668 q^{70} -1.29086 q^{71} -12.2763 q^{73} +0.248970 q^{74} -0.184793 q^{76} +5.63816 q^{77} -12.6382 q^{79} -4.84524 q^{80} +4.62267 q^{82} -8.27631 q^{83} +0.411474 q^{85} -15.1361 q^{86} -13.5294 q^{88} -7.90167 q^{89} +4.41147 q^{91} -0.859785 q^{92} -7.52023 q^{94} +1.34730 q^{95} +7.00774 q^{97} -7.40373 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} + 9 q^{10} - 3 q^{13} + 6 q^{14} + 3 q^{16} + 3 q^{17} + 3 q^{19} + 12 q^{20} - 9 q^{22} + 15 q^{23} - 6 q^{25} - 12 q^{26} + 6 q^{28} + 6 q^{29} + 6 q^{31}+ \cdots - 36 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 + 9 * q^10 - 3 * q^13 + 6 * q^14 + 3 * q^16 + 3 * q^17 + 3 * q^19 + 12 * q^20 - 9 * q^22 + 15 * q^23 - 6 * q^25 - 12 * q^26 + 6 * q^28 + 6 * q^29 + 6 * q^31 - 9 * q^34 + 6 * q^35 - 3 * q^37 + 3 * q^38 + 9 * q^40 + 3 * q^41 - 3 * q^43 - 27 * q^44 + 9 * q^46 + 15 * q^47 + 3 * q^50 - 30 * q^52 + 6 * q^53 - 9 * q^55 - 15 * q^56 + 21 * q^59 - 21 * q^61 - 3 * q^62 + 12 * q^64 - 12 * q^65 + 6 * q^67 - 15 * q^68 + 12 * q^71 - 3 * q^73 - 12 * q^74 + 3 * q^76 - 21 * q^79 + 12 * q^80 + 45 * q^82 + 9 * q^83 - 9 * q^85 - 3 * q^86 - 45 * q^88 - 12 * q^89 + 3 * q^91 + 6 * q^92 + 9 * q^94 + 3 * q^95 - 3 * q^97 - 36 * q^98

Introduction

L-functions

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