LMFDB - Embedded newform 5200.2.a.cc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5200,2,Mod(1,5200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("5200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-2,0,0,0,2,0,-1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.5222090511$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17009 q^{3} -0.630898 q^{7} -1.63090 q^{9} +0.539189 q^{11} -1.00000 q^{13} +3.07838 q^{17} -0.879362 q^{19} -0.738205 q^{21} -2.58864 q^{23} -5.41855 q^{27} -2.29072 q^{29} +7.95774 q^{31} +0.630898 q^{33} +4.78765 q^{37} -1.17009 q^{39} +3.41855 q^{41} +3.17009 q^{43} +8.94441 q^{47} -6.60197 q^{49} +3.60197 q^{51} -0.496928 q^{53} -1.02893 q^{57} +8.29791 q^{59} +4.04945 q^{61} +1.02893 q^{63} -3.36910 q^{67} -3.02893 q^{69} +9.06278 q^{71} +12.2062 q^{73} -0.340173 q^{77} +5.65983 q^{79} -1.44748 q^{81} +1.02893 q^{83} -2.68035 q^{87} -16.8371 q^{89} +0.630898 q^{91} +9.31124 q^{93} -5.23513 q^{97} -0.879362 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 2 q^{7} - q^{9} - 3 q^{13} + 6 q^{17} + 10 q^{19} - 10 q^{21} + 12 q^{23} - 2 q^{27} - 14 q^{29} + 8 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} - 4 q^{41} + 4 q^{43} + 10 q^{47} - q^{49}+ \cdots + 10 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 2 * q^7 - q^9 - 3 * q^13 + 6 * q^17 + 10 * q^19 - 10 * q^21 + 12 * q^23 - 2 * q^27 - 14 * q^29 + 8 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 - 4 * q^41 + 4 * q^43 + 10 * q^47 - q^49 - 8 * q^51 + 16 * q^53 - 18 * q^57 - 2 * q^59 - 6 * q^61 + 18 * q^63 - 14 * q^67 - 24 * q^69 + 10 * q^71 + 12 * q^73 + 10 * q^77 + 28 * q^79 - 5 * q^81 + 18 * q^83 + 14 * q^87 - 22 * q^89 - 2 * q^91 + 2 * q^93 - 6 * q^97 + 10 * q^99

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
$3$	1.17009	0.675550	0.337775	−	0.941227i	$-0.390326\pi$
$3$	1.17009	0.675550	0.337775	+	0.941227i	$0.390326\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.630898	−0.238457	−0.119228	−	0.992867i	$-0.538042\pi$
$7$	−0.630898	−0.238457	−0.119228	+	0.992867i	$0.538042\pi$
$8$	0	0
$9$	−1.63090	−0.543633
$10$	0	0
$11$	0.539189	0.162572	0.0812858	−	0.996691i	$-0.474097\pi$
$11$	0.539189	0.162572	0.0812858	+	0.996691i	$0.474097\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.07838	0.746616	0.373308	−	0.927707i	$-0.378223\pi$
$17$	3.07838	0.746616	0.373308	+	0.927707i	$0.378223\pi$
$18$	0	0
$19$	−0.879362	−0.201739	−0.100870	−	0.994900i	$-0.532163\pi$
$19$	−0.879362	−0.201739	−0.100870	+	0.994900i	$0.532163\pi$
$20$	0	0
$21$	−0.738205	−0.161089
$22$	0	0
$23$	−2.58864	−0.539768	−0.269884	−	0.962893i	$-0.586985\pi$
$23$	−2.58864	−0.539768	−0.269884	+	0.962893i	$0.586985\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.41855	−1.04280
$28$	0	0
$29$	−2.29072	−0.425377	−0.212688	−	0.977120i	$-0.568222\pi$
$29$	−2.29072	−0.425377	−0.212688	+	0.977120i	$0.568222\pi$
$30$	0	0
$31$	7.95774	1.42925	0.714626	−	0.699507i	$-0.246597\pi$
$31$	7.95774	1.42925	0.714626	+	0.699507i	$0.246597\pi$
$32$	0	0
$33$	0.630898	0.109825
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.78765	0.787085	0.393543	−	0.919306i	$-0.371249\pi$
$37$	4.78765	0.787085	0.393543	+	0.919306i	$0.371249\pi$
$38$	0	0
$39$	−1.17009	−0.187364
$40$	0	0
$41$	3.41855	0.533888	0.266944	−	0.963712i	$-0.413986\pi$
$41$	3.41855	0.533888	0.266944	+	0.963712i	$0.413986\pi$
$42$	0	0
$43$	3.17009	0.483434	0.241717	−	0.970347i	$-0.422289\pi$
$43$	3.17009	0.483434	0.241717	+	0.970347i	$0.422289\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.94441	1.30468	0.652338	−	0.757928i	$-0.273788\pi$
$47$	8.94441	1.30468	0.652338	+	0.757928i	$0.273788\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	3.60197	0.504376
$52$	0	0
$53$	−0.496928	−0.0682584	−0.0341292	−	0.999417i	$-0.510866\pi$
$53$	−0.496928	−0.0682584	−0.0341292	+	0.999417i	$0.510866\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.02893	−0.136285
$58$	0	0
$59$	8.29791	1.08030	0.540148	−	0.841570i	$-0.318368\pi$
$59$	8.29791	1.08030	0.540148	+	0.841570i	$0.318368\pi$
$60$	0	0
$61$	4.04945	0.518479	0.259239	−	0.965813i	$-0.416528\pi$
$61$	4.04945	0.518479	0.259239	+	0.965813i	$0.416528\pi$
$62$	0	0
$63$	1.02893	0.129633
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.36910	−0.411601	−0.205801	−	0.978594i	$-0.565980\pi$
$67$	−3.36910	−0.411601	−0.205801	+	0.978594i	$0.565980\pi$
$68$	0	0
$69$	−3.02893	−0.364640
$70$	0	0
$71$	9.06278	1.07555	0.537777	−	0.843087i	$-0.319264\pi$
$71$	9.06278	1.07555	0.537777	+	0.843087i	$0.319264\pi$
$72$	0	0
$73$	12.2062	1.42863	0.714314	−	0.699825i	$-0.246739\pi$
$73$	12.2062	1.42863	0.714314	+	0.699825i	$0.246739\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.340173	−0.0387663
$78$	0	0
$79$	5.65983	0.636780	0.318390	−	0.947960i	$-0.396858\pi$
$79$	5.65983	0.636780	0.318390	+	0.947960i	$0.396858\pi$
$80$	0	0
$81$	−1.44748	−0.160831
$82$	0	0
$83$	1.02893	0.112940	0.0564698	−	0.998404i	$-0.482016\pi$
$83$	1.02893	0.112940	0.0564698	+	0.998404i	$0.482016\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.68035	−0.287363
$88$	0	0
$89$	−16.8371	−1.78473	−0.892365	−	0.451315i	$-0.850955\pi$
$89$	−16.8371	−1.78473	−0.892365	+	0.451315i	$0.850955\pi$
$90$	0	0
$91$	0.630898	0.0661360
$92$	0	0
$93$	9.31124	0.965531
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.23513	−0.531547	−0.265774	−	0.964035i	$-0.585627\pi$
$97$	−5.23513	−0.531547	−0.265774	+	0.964035i	$0.585627\pi$
$98$	0	0
$99$	−0.879362	−0.0883792
$100$	0	0
$101$	5.23513	0.520915	0.260458	−	0.965485i	$-0.416127\pi$
$101$	5.23513	0.520915	0.260458	+	0.965485i	$0.416127\pi$
$102$	0	0
$103$	3.69368	0.363949	0.181974	−	0.983303i	$-0.441751\pi$
$103$	3.69368	0.363949	0.181974	+	0.983303i	$0.441751\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.8504	1.53232	0.766160	−	0.642650i	$-0.222165\pi$
$107$	15.8504	1.53232	0.766160	+	0.642650i	$0.222165\pi$
$108$	0	0
$109$	−3.81658	−0.365562	−0.182781	−	0.983154i	$-0.558510\pi$
$109$	−3.81658	−0.365562	−0.182781	+	0.983154i	$0.558510\pi$
$110$	0	0
$111$	5.60197	0.531715
$112$	0	0
$113$	11.4452	1.07668	0.538338	−	0.842729i	$-0.319053\pi$
$113$	11.4452	1.07668	0.538338	+	0.842729i	$0.319053\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.63090	0.150777
$118$	0	0
$119$	−1.94214	−0.178036
$120$	0	0
$121$	−10.7093	−0.973570
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	5.35350	0.475047	0.237523	−	0.971382i	$-0.423664\pi$
$127$	5.35350	0.475047	0.237523	+	0.971382i	$0.423664\pi$
$128$	0	0
$129$	3.70928	0.326583
$130$	0	0
$131$	−18.9360	−1.65445	−0.827223	−	0.561874i	$-0.810081\pi$
$131$	−18.9360	−1.65445	−0.827223	+	0.561874i	$0.810081\pi$
$132$	0	0
$133$	0.554787	0.0481062
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.09890	−0.521064	−0.260532	−	0.965465i	$-0.583898\pi$
$137$	−6.09890	−0.521064	−0.260532	+	0.965465i	$0.583898\pi$
$138$	0	0
$139$	21.5441	1.82735	0.913674	−	0.406448i	$-0.133233\pi$
$139$	21.5441	1.82735	0.913674	+	0.406448i	$0.133233\pi$
$140$	0	0
$141$	10.4657	0.881374
$142$	0	0
$143$	−0.539189	−0.0450892
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.72487	−0.637137
$148$	0	0
$149$	8.28231	0.678514	0.339257	−	0.940694i	$-0.389824\pi$
$149$	8.28231	0.678514	0.339257	+	0.940694i	$0.389824\pi$
$150$	0	0
$151$	7.61757	0.619909	0.309954	−	0.950751i	$-0.399686\pi$
$151$	7.61757	0.619909	0.309954	+	0.950751i	$0.399686\pi$
$152$	0	0
$153$	−5.02052	−0.405885
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.47641	0.277448	0.138724	−	0.990331i	$-0.455700\pi$
$157$	3.47641	0.277448	0.138724	+	0.990331i	$0.455700\pi$
$158$	0	0
$159$	−0.581449	−0.0461119
$160$	0	0
$161$	1.63317	0.128711
$162$	0	0
$163$	9.12783	0.714947	0.357473	−	0.933923i	$-0.383638\pi$
$163$	9.12783	0.714947	0.357473	+	0.933923i	$0.383638\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.8143	0.991601	0.495801	−	0.868436i	$-0.334875\pi$
$167$	12.8143	0.991601	0.495801	+	0.868436i	$0.334875\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.43415	0.109672
$172$	0	0
$173$	4.18342	0.318059	0.159030	−	0.987274i	$-0.449163\pi$
$173$	4.18342	0.318059	0.159030	+	0.987274i	$0.449163\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.70928	0.729794
$178$	0	0
$179$	13.7587	1.02838	0.514188	−	0.857678i	$-0.328093\pi$
$179$	13.7587	1.02838	0.514188	+	0.857678i	$0.328093\pi$
$180$	0	0
$181$	−21.8082	−1.62099	−0.810494	−	0.585746i	$-0.800801\pi$
$181$	−21.8082	−1.62099	−0.810494	+	0.585746i	$0.800801\pi$
$182$	0	0
$183$	4.73820	0.350258
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.65983	0.121379
$188$	0	0
$189$	3.41855	0.248663
$190$	0	0
$191$	2.83710	0.205285	0.102643	−	0.994718i	$-0.467270\pi$
$191$	2.83710	0.205285	0.102643	+	0.994718i	$0.467270\pi$
$192$	0	0
$193$	−21.6163	−1.55598	−0.777989	−	0.628278i	$-0.783760\pi$
$193$	−21.6163	−1.55598	−0.777989	+	0.628278i	$0.783760\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	14.4391	1.02356	0.511779	−	0.859117i	$-0.328987\pi$
$199$	14.4391	1.02356	0.511779	+	0.859117i	$0.328987\pi$
$200$	0	0
$201$	−3.94214	−0.278057
$202$	0	0
$203$	1.44521	0.101434
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.22180	0.293436
$208$	0	0
$209$	−0.474142	−0.0327971
$210$	0	0
$211$	−4.28231	−0.294807	−0.147403	−	0.989076i	$-0.547092\pi$
$211$	−4.28231	−0.294807	−0.147403	+	0.989076i	$0.547092\pi$
$212$	0	0
$213$	10.6042	0.726590
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.02052	−0.340815
$218$	0	0
$219$	14.2823	0.965109
$220$	0	0
$221$	−3.07838	−0.207074
$222$	0	0
$223$	22.8865	1.53260	0.766298	−	0.642485i	$-0.222096\pi$
$223$	22.8865	1.53260	0.766298	+	0.642485i	$0.222096\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.89269	0.391112	0.195556	−	0.980693i	$-0.437349\pi$
$227$	5.89269	0.391112	0.195556	+	0.980693i	$0.437349\pi$
$228$	0	0
$229$	17.1194	1.13128	0.565641	−	0.824651i	$-0.308629\pi$
$229$	17.1194	1.13128	0.565641	+	0.824651i	$0.308629\pi$
$230$	0	0
$231$	−0.398032	−0.0261886
$232$	0	0
$233$	13.8888	0.909887	0.454943	−	0.890520i	$-0.349660\pi$
$233$	13.8888	0.909887	0.454943	+	0.890520i	$0.349660\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.62249	0.430177
$238$	0	0
$239$	−20.9516	−1.35525	−0.677623	−	0.735409i	$-0.736990\pi$
$239$	−20.9516	−1.35525	−0.677623	+	0.735409i	$0.736990\pi$
$240$	0	0
$241$	−8.68035	−0.559150	−0.279575	−	0.960124i	$-0.590194\pi$
$241$	−8.68035	−0.559150	−0.279575	+	0.960124i	$0.590194\pi$
$242$	0	0
$243$	14.5620	0.934151
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.879362	0.0559525
$248$	0	0
$249$	1.20394	0.0762964
$250$	0	0
$251$	−8.58145	−0.541656	−0.270828	−	0.962628i	$-0.587298\pi$
$251$	−8.58145	−0.541656	−0.270828	+	0.962628i	$0.587298\pi$
$252$	0	0
$253$	−1.39576	−0.0877510
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.2557	0.889243	0.444622	−	0.895719i	$-0.353338\pi$
$257$	14.2557	0.889243	0.444622	+	0.895719i	$0.353338\pi$
$258$	0	0
$259$	−3.02052	−0.187686
$260$	0	0
$261$	3.73594	0.231249
$262$	0	0
$263$	10.8299	0.667801	0.333901	−	0.942608i	$-0.391635\pi$
$263$	10.8299	0.667801	0.333901	+	0.942608i	$0.391635\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−19.7009	−1.20567
$268$	0	0
$269$	−21.4329	−1.30679	−0.653394	−	0.757018i	$-0.726656\pi$
$269$	−21.4329	−1.30679	−0.653394	+	0.757018i	$0.726656\pi$
$270$	0	0
$271$	−21.4752	−1.30452	−0.652262	−	0.757993i	$-0.726180\pi$
$271$	−21.4752	−1.30452	−0.652262	+	0.757993i	$0.726180\pi$
$272$	0	0
$273$	0.738205	0.0446782
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.28846	0.197584	0.0987921	−	0.995108i	$-0.468502\pi$
$277$	3.28846	0.197584	0.0987921	+	0.995108i	$0.468502\pi$
$278$	0	0
$279$	−12.9783	−0.776988
$280$	0	0
$281$	−28.1978	−1.68214	−0.841070	−	0.540927i	$-0.818074\pi$
$281$	−28.1978	−1.68214	−0.841070	+	0.540927i	$0.818074\pi$
$282$	0	0
$283$	12.1639	0.723071	0.361536	−	0.932358i	$-0.382253\pi$
$283$	12.1639	0.723071	0.361536	+	0.932358i	$0.382253\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.15676	−0.127309
$288$	0	0
$289$	−7.52359	−0.442564
$290$	0	0
$291$	−6.12556	−0.359087
$292$	0	0
$293$	9.36910	0.547349	0.273674	−	0.961822i	$-0.411761\pi$
$293$	9.36910	0.547349	0.273674	+	0.961822i	$0.411761\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.92162	−0.169530
$298$	0	0
$299$	2.58864	0.149705
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	6.12556	0.351904
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−31.0700	−1.77326	−0.886628	−	0.462483i	$-0.846959\pi$
$307$	−31.0700	−1.77326	−0.886628	+	0.462483i	$0.846959\pi$
$308$	0	0
$309$	4.32192	0.245866
$310$	0	0
$311$	3.68649	0.209042	0.104521	−	0.994523i	$-0.466669\pi$
$311$	3.68649	0.209042	0.104521	+	0.994523i	$0.466669\pi$
$312$	0	0
$313$	−2.49693	−0.141135	−0.0705674	−	0.997507i	$-0.522481\pi$
$313$	−2.49693	−0.141135	−0.0705674	+	0.997507i	$0.522481\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.5090	−1.54506	−0.772531	−	0.634977i	$-0.781009\pi$
$317$	−27.5090	−1.54506	−0.772531	+	0.634977i	$0.781009\pi$
$318$	0	0
$319$	−1.23513	−0.0691542
$320$	0	0
$321$	18.5464	1.03516
$322$	0	0
$323$	−2.70701	−0.150622
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−4.46573	−0.246956
$328$	0	0
$329$	−5.64301	−0.311109
$330$	0	0
$331$	9.30406	0.511397	0.255699	−	0.966757i	$-0.417695\pi$
$331$	9.30406	0.511397	0.255699	+	0.966757i	$0.417695\pi$
$332$	0	0
$333$	−7.80817	−0.427885
$334$	0	0
$335$	0	0
$336$	0	0
$337$	21.0205	1.14506	0.572530	−	0.819884i	$-0.305962\pi$
$337$	21.0205	1.14506	0.572530	+	0.819884i	$0.305962\pi$
$338$	0	0
$339$	13.3919	0.727348
$340$	0	0
$341$	4.29072	0.232356
$342$	0	0
$343$	8.58145	0.463355
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.9083	−0.639271	−0.319635	−	0.947541i	$-0.603560\pi$
$347$	−11.9083	−0.639271	−0.319635	+	0.947541i	$0.603560\pi$
$348$	0	0
$349$	4.07223	0.217982	0.108991	−	0.994043i	$-0.465238\pi$
$349$	4.07223	0.217982	0.108991	+	0.994043i	$0.465238\pi$
$350$	0	0
$351$	5.41855	0.289221
$352$	0	0
$353$	26.5152	1.41126	0.705630	−	0.708580i	$-0.250664\pi$
$353$	26.5152	1.41126	0.705630	+	0.708580i	$0.250664\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.27247	−0.120272
$358$	0	0
$359$	−28.4501	−1.50154	−0.750770	−	0.660563i	$-0.770317\pi$
$359$	−28.4501	−1.50154	−0.750770	+	0.660563i	$0.770317\pi$
$360$	0	0
$361$	−18.2267	−0.959301
$362$	0	0
$363$	−12.5308	−0.657695
$364$	0	0
$365$	0	0
$366$	0	0
$367$	36.3740	1.89871	0.949354	−	0.314208i	$-0.101739\pi$
$367$	36.3740	1.89871	0.949354	+	0.314208i	$0.101739\pi$
$368$	0	0
$369$	−5.57531	−0.290239
$370$	0	0
$371$	0.313511	0.0162767
$372$	0	0
$373$	23.3607	1.20957	0.604785	−	0.796388i	$-0.293259\pi$
$373$	23.3607	1.20957	0.604785	+	0.796388i	$0.293259\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.29072	0.117978
$378$	0	0
$379$	−30.0300	−1.54254	−0.771268	−	0.636510i	$-0.780377\pi$
$379$	−30.0300	−1.54254	−0.771268	+	0.636510i	$0.780377\pi$
$380$	0	0
$381$	6.26406	0.320918
$382$	0	0
$383$	−2.02279	−0.103360	−0.0516798	−	0.998664i	$-0.516458\pi$
$383$	−2.02279	−0.103360	−0.0516798	+	0.998664i	$0.516458\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.17009	−0.262810
$388$	0	0
$389$	35.6742	1.80875	0.904377	−	0.426735i	$-0.140336\pi$
$389$	35.6742	1.80875	0.904377	+	0.426735i	$0.140336\pi$
$390$	0	0
$391$	−7.96880	−0.403000
$392$	0	0
$393$	−22.1568	−1.11766
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9.05559	−0.454487	−0.227244	−	0.973838i	$-0.572971\pi$
$397$	−9.05559	−0.454487	−0.227244	+	0.973838i	$0.572971\pi$
$398$	0	0
$399$	0.649149	0.0324981
$400$	0	0
$401$	27.9421	1.39536	0.697682	−	0.716408i	$-0.254215\pi$
$401$	27.9421	1.39536	0.697682	+	0.716408i	$0.254215\pi$
$402$	0	0
$403$	−7.95774	−0.396403
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.58145	0.127958
$408$	0	0
$409$	32.0989	1.58719	0.793594	−	0.608447i	$-0.208207\pi$
$409$	32.0989	1.58719	0.793594	+	0.608447i	$0.208207\pi$
$410$	0	0
$411$	−7.13624	−0.352005
$412$	0	0
$413$	−5.23513	−0.257604
$414$	0	0
$415$	0	0
$416$	0	0
$417$	25.2085	1.23446
$418$	0	0
$419$	−18.3090	−0.894452	−0.447226	−	0.894421i	$-0.647588\pi$
$419$	−18.3090	−0.894452	−0.447226	+	0.894421i	$0.647588\pi$
$420$	0	0
$421$	−27.0205	−1.31690	−0.658450	−	0.752625i	$-0.728787\pi$
$421$	−27.0205	−1.31690	−0.658450	+	0.752625i	$0.728787\pi$
$422$	0	0
$423$	−14.5874	−0.709264
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.55479	−0.123635
$428$	0	0
$429$	−0.630898	−0.0304600
$430$	0	0
$431$	−14.7636	−0.711140	−0.355570	−	0.934650i	$-0.615713\pi$
$431$	−14.7636	−0.711140	−0.355570	+	0.934650i	$0.615713\pi$
$432$	0	0
$433$	26.7214	1.28415	0.642074	−	0.766643i	$-0.278074\pi$
$433$	26.7214	1.28415	0.642074	+	0.766643i	$0.278074\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.27635	0.108893
$438$	0	0
$439$	23.6475	1.12864	0.564318	−	0.825558i	$-0.309139\pi$
$439$	23.6475	1.12864	0.564318	+	0.825558i	$0.309139\pi$
$440$	0	0
$441$	10.7671	0.512721
$442$	0	0
$443$	−25.4524	−1.20928	−0.604640	−	0.796499i	$-0.706683\pi$
$443$	−25.4524	−1.20928	−0.604640	+	0.796499i	$0.706683\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.69102	0.458370
$448$	0	0
$449$	−28.2967	−1.33540	−0.667702	−	0.744429i	$-0.732722\pi$
$449$	−28.2967	−1.33540	−0.667702	+	0.744429i	$0.732722\pi$
$450$	0	0
$451$	1.84324	0.0867950
$452$	0	0
$453$	8.91321	0.418779
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.9155	−1.02516	−0.512581	−	0.858639i	$-0.671311\pi$
$457$	−21.9155	−1.02516	−0.512581	+	0.858639i	$0.671311\pi$
$458$	0	0
$459$	−16.6803	−0.778572
$460$	0	0
$461$	20.3402	0.947336	0.473668	−	0.880703i	$-0.342930\pi$
$461$	20.3402	0.947336	0.473668	+	0.880703i	$0.342930\pi$
$462$	0	0
$463$	−11.7503	−0.546083	−0.273042	−	0.962002i	$-0.588030\pi$
$463$	−11.7503	−0.546083	−0.273042	+	0.962002i	$0.588030\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.1578	0.886517	0.443259	−	0.896394i	$-0.353822\pi$
$467$	19.1578	0.886517	0.443259	+	0.896394i	$0.353822\pi$
$468$	0	0
$469$	2.12556	0.0981492
$470$	0	0
$471$	4.06770	0.187430
$472$	0	0
$473$	1.70928	0.0785926
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.810439	0.0371075
$478$	0	0
$479$	−1.72875	−0.0789886	−0.0394943	−	0.999220i	$-0.512575\pi$
$479$	−1.72875	−0.0789886	−0.0394943	+	0.999220i	$0.512575\pi$
$480$	0	0
$481$	−4.78765	−0.218298
$482$	0	0
$483$	1.91094	0.0869510
$484$	0	0
$485$	0	0
$486$	0	0
$487$	42.4885	1.92534	0.962669	−	0.270680i	$-0.0872487\pi$
$487$	42.4885	1.92534	0.962669	+	0.270680i	$0.0872487\pi$
$488$	0	0
$489$	10.6803	0.482982
$490$	0	0
$491$	0.398032	0.0179629	0.00898146	−	0.999960i	$-0.497141\pi$
$491$	0.398032	0.0179629	0.00898146	+	0.999960i	$0.497141\pi$
$492$	0	0
$493$	−7.05172	−0.317593
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.71769	−0.256473
$498$	0	0
$499$	−8.56585	−0.383460	−0.191730	−	0.981448i	$-0.561410\pi$
$499$	−8.56585	−0.383460	−0.191730	+	0.981448i	$0.561410\pi$
$500$	0	0
$501$	14.9939	0.669876
$502$	0	0
$503$	35.7347	1.59333	0.796666	−	0.604420i	$-0.206595\pi$
$503$	35.7347	1.59333	0.796666	+	0.604420i	$0.206595\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.17009	0.0519654
$508$	0	0
$509$	−6.13009	−0.271712	−0.135856	−	0.990729i	$-0.543378\pi$
$509$	−6.13009	−0.271712	−0.135856	+	0.990729i	$0.543378\pi$
$510$	0	0
$511$	−7.70086	−0.340666
$512$	0	0
$513$	4.76487	0.210374
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.82273	0.212103
$518$	0	0
$519$	4.89496	0.214865
$520$	0	0
$521$	24.7442	1.08406	0.542031	−	0.840359i	$-0.317656\pi$
$521$	24.7442	1.08406	0.542031	+	0.840359i	$0.317656\pi$
$522$	0	0
$523$	−23.7081	−1.03668	−0.518340	−	0.855174i	$-0.673450\pi$
$523$	−23.7081	−1.03668	−0.518340	+	0.855174i	$0.673450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.4969	1.06710
$528$	0	0
$529$	−16.2990	−0.708650
$530$	0	0
$531$	−13.5330	−0.587284
$532$	0	0
$533$	−3.41855	−0.148074
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0989	0.694719
$538$	0	0
$539$	−3.55971	−0.153327
$540$	0	0
$541$	17.6475	0.758727	0.379364	−	0.925248i	$-0.376143\pi$
$541$	17.6475	0.758727	0.379364	+	0.925248i	$0.376143\pi$
$542$	0	0
$543$	−25.5174	−1.09506
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24.8299	−1.06165	−0.530825	−	0.847481i	$-0.678118\pi$
$547$	−24.8299	−1.06165	−0.530825	+	0.847481i	$0.678118\pi$
$548$	0	0
$549$	−6.60424	−0.281862
$550$	0	0
$551$	2.01438	0.0858153
$552$	0	0
$553$	−3.57077	−0.151845
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.0950	0.766711	0.383355	−	0.923601i	$-0.374768\pi$
$557$	18.0950	0.766711	0.383355	+	0.923601i	$0.374768\pi$
$558$	0	0
$559$	−3.17009	−0.134080
$560$	0	0
$561$	1.94214	0.0819973
$562$	0	0
$563$	−6.67316	−0.281240	−0.140620	−	0.990064i	$-0.544910\pi$
$563$	−6.67316	−0.281240	−0.140620	+	0.990064i	$0.544910\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.913212	0.0383513
$568$	0	0
$569$	10.7877	0.452242	0.226121	−	0.974099i	$-0.427396\pi$
$569$	10.7877	0.452242	0.226121	+	0.974099i	$0.427396\pi$
$570$	0	0
$571$	22.2245	0.930065	0.465032	−	0.885294i	$-0.346043\pi$
$571$	22.2245	0.930065	0.465032	+	0.885294i	$0.346043\pi$
$572$	0	0
$573$	3.31965	0.138681
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.6309	−1.44170	−0.720852	−	0.693089i	$-0.756249\pi$
$577$	−34.6309	−1.44170	−0.720852	+	0.693089i	$0.756249\pi$
$578$	0	0
$579$	−25.2930	−1.05114
$580$	0	0
$581$	−0.649149	−0.0269312
$582$	0	0
$583$	−0.267938	−0.0110969
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.8660	−0.985057	−0.492528	−	0.870296i	$-0.663927\pi$
$587$	−23.8660	−0.985057	−0.492528	+	0.870296i	$0.663927\pi$
$588$	0	0
$589$	−6.99773	−0.288337
$590$	0	0
$591$	2.34017	0.0962619
$592$	0	0
$593$	2.39803	0.0984754	0.0492377	−	0.998787i	$-0.484321\pi$
$593$	2.39803	0.0984754	0.0492377	+	0.998787i	$0.484321\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.8950	0.691465
$598$	0	0
$599$	−40.8248	−1.66806	−0.834028	−	0.551722i	$-0.813971\pi$
$599$	−40.8248	−1.66806	−0.834028	+	0.551722i	$0.813971\pi$
$600$	0	0
$601$	−20.4703	−0.835000	−0.417500	−	0.908677i	$-0.637094\pi$
$601$	−20.4703	−0.835000	−0.417500	+	0.908677i	$0.637094\pi$
$602$	0	0
$603$	5.49466	0.223760
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−37.1578	−1.50819	−0.754094	−	0.656766i	$-0.771924\pi$
$607$	−37.1578	−1.50819	−0.754094	+	0.656766i	$0.771924\pi$
$608$	0	0
$609$	1.69102	0.0685237
$610$	0	0
$611$	−8.94441	−0.361852
$612$	0	0
$613$	20.1711	0.814704	0.407352	−	0.913271i	$-0.366452\pi$
$613$	20.1711	0.814704	0.407352	+	0.913271i	$0.366452\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.4657	−0.582368	−0.291184	−	0.956667i	$-0.594049\pi$
$617$	−14.4657	−0.582368	−0.291184	+	0.956667i	$0.594049\pi$
$618$	0	0
$619$	−19.5330	−0.785099	−0.392550	−	0.919731i	$-0.628407\pi$
$619$	−19.5330	−0.785099	−0.392550	+	0.919731i	$0.628407\pi$
$620$	0	0
$621$	14.0267	0.562871
$622$	0	0
$623$	10.6225	0.425581
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.554787	−0.0221561
$628$	0	0
$629$	14.7382	0.587651
$630$	0	0
$631$	22.9783	0.914750	0.457375	−	0.889274i	$-0.348790\pi$
$631$	22.9783	0.914750	0.457375	+	0.889274i	$0.348790\pi$
$632$	0	0
$633$	−5.01068	−0.199157
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.60197	0.261580
$638$	0	0
$639$	−14.7805	−0.584706
$640$	0	0
$641$	5.00614	0.197731	0.0988654	−	0.995101i	$-0.468479\pi$
$641$	5.00614	0.197731	0.0988654	+	0.995101i	$0.468479\pi$
$642$	0	0
$643$	3.95055	0.155795	0.0778973	−	0.996961i	$-0.475179\pi$
$643$	3.95055	0.155795	0.0778973	+	0.996961i	$0.475179\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.8710	1.29229	0.646145	−	0.763214i	$-0.276380\pi$
$647$	32.8710	1.29229	0.646145	+	0.763214i	$0.276380\pi$
$648$	0	0
$649$	4.47414	0.175625
$650$	0	0
$651$	−5.87444	−0.230238
$652$	0	0
$653$	−24.4703	−0.957596	−0.478798	−	0.877925i	$-0.658927\pi$
$653$	−24.4703	−0.957596	−0.478798	+	0.877925i	$0.658927\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−19.9071	−0.776649
$658$	0	0
$659$	−21.0928	−0.821657	−0.410829	−	0.911713i	$-0.634761\pi$
$659$	−21.0928	−0.821657	−0.410829	+	0.911713i	$0.634761\pi$
$660$	0	0
$661$	−28.5958	−1.11225	−0.556124	−	0.831099i	$-0.687712\pi$
$661$	−28.5958	−1.11225	−0.556124	+	0.831099i	$0.687712\pi$
$662$	0	0
$663$	−3.60197	−0.139889
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.92986	0.229605
$668$	0	0
$669$	26.7792	1.03535
$670$	0	0
$671$	2.18342	0.0842899
$672$	0	0
$673$	1.02052	0.0393381	0.0196691	−	0.999807i	$-0.493739\pi$
$673$	1.02052	0.0393381	0.0196691	+	0.999807i	$0.493739\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.3295	1.28096	0.640478	−	0.767976i	$-0.278736\pi$
$677$	33.3295	1.28096	0.640478	+	0.767976i	$0.278736\pi$
$678$	0	0
$679$	3.30283	0.126751
$680$	0	0
$681$	6.89496	0.264215
$682$	0	0
$683$	−1.52586	−0.0583853	−0.0291927	−	0.999574i	$-0.509294\pi$
$683$	−1.52586	−0.0583853	−0.0291927	+	0.999574i	$0.509294\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	20.0312	0.764238
$688$	0	0
$689$	0.496928	0.0189315
$690$	0	0
$691$	21.2339	0.807776	0.403888	−	0.914808i	$-0.367659\pi$
$691$	21.2339	0.807776	0.403888	+	0.914808i	$0.367659\pi$
$692$	0	0
$693$	0.554787	0.0210746
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.5236	0.398609
$698$	0	0
$699$	16.2511	0.614674
$700$	0	0
$701$	0.974946	0.0368232	0.0184116	−	0.999830i	$-0.494139\pi$
$701$	0.974946	0.0368232	0.0184116	+	0.999830i	$0.494139\pi$
$702$	0	0
$703$	−4.21008	−0.158786
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.30283	−0.124216
$708$	0	0
$709$	4.14238	0.155570	0.0777852	−	0.996970i	$-0.475215\pi$
$709$	4.14238	0.155570	0.0777852	+	0.996970i	$0.475215\pi$
$710$	0	0
$711$	−9.23060	−0.346174
$712$	0	0
$713$	−20.5997	−0.771465
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.5152	−0.915536
$718$	0	0
$719$	9.26180	0.345407	0.172703	−	0.984974i	$-0.444750\pi$
$719$	9.26180	0.345407	0.172703	+	0.984974i	$0.444750\pi$
$720$	0	0
$721$	−2.33033	−0.0867861
$722$	0	0
$723$	−10.1568	−0.377734
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−38.9165	−1.44333	−0.721667	−	0.692240i	$-0.756624\pi$
$727$	−38.9165	−1.44333	−0.721667	+	0.692240i	$0.756624\pi$
$728$	0	0
$729$	21.3812	0.791897
$730$	0	0
$731$	9.75872	0.360939
$732$	0	0
$733$	−51.1917	−1.89081	−0.945403	−	0.325903i	$-0.894332\pi$
$733$	−51.1917	−1.89081	−0.945403	+	0.325903i	$0.894332\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.81658	−0.0669147
$738$	0	0
$739$	10.8260	0.398242	0.199121	−	0.979975i	$-0.436191\pi$
$739$	10.8260	0.398242	0.199121	+	0.979975i	$0.436191\pi$
$740$	0	0
$741$	1.02893	0.0377987
$742$	0	0
$743$	43.3667	1.59097	0.795484	−	0.605974i	$-0.207217\pi$
$743$	43.3667	1.59097	0.795484	+	0.605974i	$0.207217\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.67808	−0.0613977
$748$	0	0
$749$	−10.0000	−0.365392
$750$	0	0
$751$	−23.9299	−0.873213	−0.436606	−	0.899653i	$-0.643820\pi$
$751$	−23.9299	−0.873213	−0.436606	+	0.899653i	$0.643820\pi$
$752$	0	0
$753$	−10.0410	−0.365916
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.9733	−0.944017	−0.472009	−	0.881594i	$-0.656471\pi$
$757$	−25.9733	−0.944017	−0.472009	+	0.881594i	$0.656471\pi$
$758$	0	0
$759$	−1.63317	−0.0592801
$760$	0	0
$761$	17.7854	0.644720	0.322360	−	0.946617i	$-0.395524\pi$
$761$	17.7854	0.644720	0.322360	+	0.946617i	$0.395524\pi$
$762$	0	0
$763$	2.40787	0.0871708
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.29791	−0.299620
$768$	0	0
$769$	−13.9011	−0.501287	−0.250643	−	0.968080i	$-0.580642\pi$
$769$	−13.9011	−0.501287	−0.250643	+	0.968080i	$0.580642\pi$
$770$	0	0
$771$	16.6803	0.600728
$772$	0	0
$773$	−10.7298	−0.385924	−0.192962	−	0.981206i	$-0.561809\pi$
$773$	−10.7298	−0.385924	−0.192962	+	0.981206i	$0.561809\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.53427	−0.126791
$778$	0	0
$779$	−3.00614	−0.107706
$780$	0	0
$781$	4.88655	0.174854
$782$	0	0
$783$	12.4124	0.443583
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9.76260	−0.347999	−0.174000	−	0.984746i	$-0.555669\pi$
$787$	−9.76260	−0.347999	−0.174000	+	0.984746i	$0.555669\pi$
$788$	0	0
$789$	12.6719	0.451133
$790$	0	0
$791$	−7.22076	−0.256741
$792$	0	0
$793$	−4.04945	−0.143800
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.7792	1.51532	0.757659	−	0.652650i	$-0.226343\pi$
$797$	42.7792	1.51532	0.757659	+	0.652650i	$0.226343\pi$
$798$	0	0
$799$	27.5343	0.974092
$800$	0	0
$801$	27.4596	0.970237
$802$	0	0
$803$	6.58145	0.232254
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−25.0784	−0.882801
$808$	0	0
$809$	23.1155	0.812699	0.406350	−	0.913718i	$-0.366802\pi$
$809$	23.1155	0.812699	0.406350	+	0.913718i	$0.366802\pi$
$810$	0	0
$811$	−7.59090	−0.266553	−0.133276	−	0.991079i	$-0.542550\pi$
$811$	−7.59090	−0.266553	−0.133276	+	0.991079i	$0.542550\pi$
$812$	0	0
$813$	−25.1278	−0.881271
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.78765	−0.0975276
$818$	0	0
$819$	−1.02893	−0.0359537
$820$	0	0
$821$	14.0722	0.491124	0.245562	−	0.969381i	$-0.421027\pi$
$821$	14.0722	0.491124	0.245562	+	0.969381i	$0.421027\pi$
$822$	0	0
$823$	8.10608	0.282560	0.141280	−	0.989970i	$-0.454878\pi$
$823$	8.10608	0.282560	0.141280	+	0.989970i	$0.454878\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.1750	−1.74476	−0.872378	−	0.488832i	$-0.837423\pi$
$827$	−50.1750	−1.74476	−0.872378	+	0.488832i	$0.837423\pi$
$828$	0	0
$829$	21.2123	0.736735	0.368368	−	0.929680i	$-0.379917\pi$
$829$	21.2123	0.736735	0.368368	+	0.929680i	$0.379917\pi$
$830$	0	0
$831$	3.84778	0.133478
$832$	0	0
$833$	−20.3234	−0.704162
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−43.1194	−1.49043
$838$	0	0
$839$	46.4645	1.60413	0.802066	−	0.597235i	$-0.203734\pi$
$839$	46.4645	1.60413	0.802066	+	0.597235i	$0.203734\pi$
$840$	0	0
$841$	−23.7526	−0.819055
$842$	0	0
$843$	−32.9939	−1.13637
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.75646	0.232155
$848$	0	0
$849$	14.2329	0.488471
$850$	0	0
$851$	−12.3935	−0.424844
$852$	0	0
$853$	−3.99159	−0.136669	−0.0683347	−	0.997662i	$-0.521769\pi$
$853$	−3.99159	−0.136669	−0.0683347	+	0.997662i	$0.521769\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.8843	1.08915	0.544573	−	0.838713i	$-0.316692\pi$
$857$	31.8843	1.08915	0.544573	+	0.838713i	$0.316692\pi$
$858$	0	0
$859$	−26.8059	−0.914606	−0.457303	−	0.889311i	$-0.651184\pi$
$859$	−26.8059	−0.914606	−0.457303	+	0.889311i	$0.651184\pi$
$860$	0	0
$861$	−2.52359	−0.0860037
$862$	0	0
$863$	41.5259	1.41356	0.706778	−	0.707435i	$-0.250148\pi$
$863$	41.5259	1.41356	0.706778	+	0.707435i	$0.250148\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−8.80325	−0.298974
$868$	0	0
$869$	3.05172	0.103522
$870$	0	0
$871$	3.36910	0.114158
$872$	0	0
$873$	8.53797	0.288966
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−9.98562	−0.337191	−0.168595	−	0.985685i	$-0.553923\pi$
$877$	−9.98562	−0.337191	−0.168595	+	0.985685i	$0.553923\pi$
$878$	0	0
$879$	10.9627	0.369761
$880$	0	0
$881$	8.41628	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$881$	8.41628	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$882$	0	0
$883$	−30.4585	−1.02501	−0.512506	−	0.858684i	$-0.671283\pi$
$883$	−30.4585	−1.02501	−0.512506	+	0.858684i	$0.671283\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	54.7454	1.83817	0.919085	−	0.394059i	$-0.128929\pi$
$887$	54.7454	1.83817	0.919085	+	0.394059i	$0.128929\pi$
$888$	0	0
$889$	−3.37751	−0.113278
$890$	0	0
$891$	−0.780465	−0.0261466
$892$	0	0
$893$	−7.86537	−0.263205
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.02893	0.101133
$898$	0	0
$899$	−18.2290	−0.607971
$900$	0	0
$901$	−1.52973	−0.0509628
$902$	0	0
$903$	−2.34017	−0.0778761
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30.4007	−1.00944	−0.504719	−	0.863284i	$-0.668404\pi$
$907$	−30.4007	−1.00944	−0.504719	+	0.863284i	$0.668404\pi$
$908$	0	0
$909$	−8.53797	−0.283186
$910$	0	0
$911$	10.7526	0.356249	0.178124	−	0.984008i	$-0.442997\pi$
$911$	10.7526	0.356249	0.178124	+	0.984008i	$0.442997\pi$
$912$	0	0
$913$	0.554787	0.0183608
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.9467	0.394514
$918$	0	0
$919$	37.3607	1.23242	0.616208	−	0.787584i	$-0.288668\pi$
$919$	37.3607	1.23242	0.616208	+	0.787584i	$0.288668\pi$
$920$	0	0
$921$	−36.3545	−1.19792
$922$	0	0
$923$	−9.06278	−0.298305
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6.02401	−0.197854
$928$	0	0
$929$	33.6742	1.10481	0.552407	−	0.833574i	$-0.313709\pi$
$929$	33.6742	1.10481	0.552407	+	0.833574i	$0.313709\pi$
$930$	0	0
$931$	5.80552	0.190268
$932$	0	0
$933$	4.31351	0.141218
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25.0394	−0.818003	−0.409001	−	0.912534i	$-0.634123\pi$
$937$	−25.0394	−0.818003	−0.409001	+	0.912534i	$0.634123\pi$
$938$	0	0
$939$	−2.92162	−0.0953435
$940$	0	0
$941$	7.07838	0.230749	0.115374	−	0.993322i	$-0.463193\pi$
$941$	7.07838	0.230749	0.115374	+	0.993322i	$0.463193\pi$
$942$	0	0
$943$	−8.84939	−0.288176
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.1795	0.785730	0.392865	−	0.919596i	$-0.371484\pi$
$947$	24.1795	0.785730	0.392865	+	0.919596i	$0.371484\pi$
$948$	0	0
$949$	−12.2062	−0.396230
$950$	0	0
$951$	−32.1880	−1.04377
$952$	0	0
$953$	−21.6742	−0.702096	−0.351048	−	0.936357i	$-0.614175\pi$
$953$	−21.6742	−0.702096	−0.351048	+	0.936357i	$0.614175\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.44521	−0.0467171
$958$	0	0
$959$	3.84778	0.124251
$960$	0	0
$961$	32.3256	1.04276
$962$	0	0
$963$	−25.8504	−0.833019
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.14220	−0.0367307	−0.0183654	−	0.999831i	$-0.505846\pi$
$967$	−1.14220	−0.0367307	−0.0183654	+	0.999831i	$0.505846\pi$
$968$	0	0
$969$	−3.16743	−0.101753
$970$	0	0
$971$	41.9565	1.34645	0.673224	−	0.739438i	$-0.264909\pi$
$971$	41.9565	1.34645	0.673224	+	0.739438i	$0.264909\pi$
$972$	0	0
$973$	−13.5921	−0.435744
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.0023	0.671922	0.335961	−	0.941876i	$-0.390939\pi$
$977$	21.0023	0.671922	0.335961	+	0.941876i	$0.390939\pi$
$978$	0	0
$979$	−9.07838	−0.290146
$980$	0	0
$981$	6.22446	0.198732
$982$	0	0
$983$	−44.1171	−1.40712	−0.703559	−	0.710637i	$-0.748407\pi$
$983$	−44.1171	−1.40712	−0.703559	+	0.710637i	$0.748407\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.60281	−0.210170
$988$	0	0
$989$	−8.20620	−0.260942
$990$	0	0
$991$	−48.9549	−1.55510	−0.777552	−	0.628819i	$-0.783539\pi$
$991$	−48.9549	−1.55510	−0.777552	+	0.628819i	$0.783539\pi$
$992$	0	0
$993$	10.8865	0.345474
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−52.9914	−1.67825	−0.839127	−	0.543935i	$-0.816934\pi$
$997$	−52.9914	−1.67825	−0.839127	+	0.543935i	$0.816934\pi$
$998$	0	0
$999$	−25.9421	−0.820773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.cc.1.3		3
4.3	odd	2		2600.2.a.y.1.1		3
5.2	odd	4		1040.2.d.e.209.2		6
5.3	odd	4		1040.2.d.e.209.5		6
5.4	even	2		5200.2.a.ch.1.1		3
20.3	even	4		520.2.d.b.209.2	✓	6
20.7	even	4		520.2.d.b.209.5	yes	6
20.19	odd	2		2600.2.a.x.1.3		3
60.23	odd	4		4680.2.l.d.2809.4		6
60.47	odd	4		4680.2.l.d.2809.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.d.b.209.2	✓	6	20.3	even	4
520.2.d.b.209.5	yes	6	20.7	even	4
1040.2.d.e.209.2		6	5.2	odd	4
1040.2.d.e.209.5		6	5.3	odd	4
2600.2.a.x.1.3		3	20.19	odd	2
2600.2.a.y.1.1		3	4.3	odd	2
4680.2.l.d.2809.3		6	60.47	odd	4
4680.2.l.d.2809.4		6	60.23	odd	4
5200.2.a.cc.1.3		3	1.1	even	1	trivial
5200.2.a.ch.1.1		3	5.4	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:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17009 q^{3} -0.630898 q^{7} -1.63090 q^{9} +0.539189 q^{11} -1.00000 q^{13} +3.07838 q^{17} -0.879362 q^{19} -0.738205 q^{21} -2.58864 q^{23} -5.41855 q^{27} -2.29072 q^{29} +7.95774 q^{31} +0.630898 q^{33} +4.78765 q^{37} -1.17009 q^{39} +3.41855 q^{41} +3.17009 q^{43} +8.94441 q^{47} -6.60197 q^{49} +3.60197 q^{51} -0.496928 q^{53} -1.02893 q^{57} +8.29791 q^{59} +4.04945 q^{61} +1.02893 q^{63} -3.36910 q^{67} -3.02893 q^{69} +9.06278 q^{71} +12.2062 q^{73} -0.340173 q^{77} +5.65983 q^{79} -1.44748 q^{81} +1.02893 q^{83} -2.68035 q^{87} -16.8371 q^{89} +0.630898 q^{91} +9.31124 q^{93} -5.23513 q^{97} -0.879362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 2 q^{7} - q^{9} - 3 q^{13} + 6 q^{17} + 10 q^{19} - 10 q^{21} + 12 q^{23} - 2 q^{27} - 14 q^{29} + 8 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} - 4 q^{41} + 4 q^{43} + 10 q^{47} - q^{49}+ \cdots + 10 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 2 * q^7 - q^9 - 3 * q^13 + 6 * q^17 + 10 * q^19 - 10 * q^21 + 12 * q^23 - 2 * q^27 - 14 * q^29 + 8 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 - 4 * q^41 + 4 * q^43 + 10 * q^47 - q^49 - 8 * q^51 + 16 * q^53 - 18 * q^57 - 2 * q^59 - 6 * q^61 + 18 * q^63 - 14 * q^67 - 24 * q^69 + 10 * q^71 + 12 * q^73 + 10 * q^77 + 28 * q^79 - 5 * q^81 + 18 * q^83 + 14 * q^87 - 22 * q^89 - 2 * q^91 + 2 * q^93 - 6 * q^97 + 10 * q^99

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