LMFDB - Embedded newform 4232.2.a.x.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4232,2,Mod(1,4232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("4232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.7926901354$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 157x^{8} - 616x^{6} + 1264x^{4} - 1272x^{2} + 484 $ x^12 - 20x^10 + 157x^8 - 616x^6 + 1264x^4 - 1272*x^2 + 484
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.12938$ of defining polynomial
Character	$\chi$	$=$	4232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.534266 q^{3} +2.85360 q^{5} -1.52458 q^{7} -2.71456 q^{9} +3.72860 q^{11} +1.48994 q^{13} -1.52458 q^{15} +6.09414 q^{17} +8.39902 q^{19} +0.814534 q^{21} +3.14306 q^{25} +3.05310 q^{27} -1.98697 q^{29} +3.69466 q^{31} -1.99206 q^{33} -4.35056 q^{35} -9.05221 q^{37} -0.796024 q^{39} -2.86381 q^{41} +2.13351 q^{43} -7.74628 q^{45} -9.05831 q^{47} -4.67564 q^{49} -3.25589 q^{51} +1.56105 q^{53} +10.6399 q^{55} -4.48732 q^{57} +8.98316 q^{59} -14.4631 q^{61} +4.13858 q^{63} +4.25169 q^{65} +4.29489 q^{67} -0.584685 q^{71} +3.96707 q^{73} -1.67923 q^{75} -5.68456 q^{77} +6.05191 q^{79} +6.51251 q^{81} -11.9759 q^{83} +17.3902 q^{85} +1.06157 q^{87} +18.2247 q^{89} -2.27154 q^{91} -1.97394 q^{93} +23.9675 q^{95} +10.5468 q^{97} -10.1215 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} + 8 q^{27} + 8 q^{31} + 56 q^{35} + 64 q^{39} - 40 q^{41} + 32 q^{47} + 28 q^{49} + 64 q^{55} + 60 q^{59} - 32 q^{71} + 28 q^{73} + 16 q^{75} + 24 q^{77}+ \cdots + 16 q^{95}+O(q^{100}) $ 12 * q + 8 * q^3 + 8 * q^9 + 16 * q^13 + 4 * q^25 + 8 * q^27 + 8 * q^31 + 56 * q^35 + 64 * q^39 - 40 * q^41 + 32 * q^47 + 28 * q^49 + 64 * q^55 + 60 * q^59 - 32 * q^71 + 28 * q^73 + 16 * q^75 + 24 * q^77 + 4 * q^81 + 64 * q^85 - 16 * q^87 + 24 * q^93 + 16 * q^95

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
$3$	−0.534266	−0.308459	−0.154229	−	0.988035i	$-0.549289\pi$
$3$	−0.534266	−0.308459	−0.154229	+	0.988035i	$0.549289\pi$
$4$	0	0
$5$	2.85360	1.27617	0.638085	−	0.769966i	$-0.279727\pi$
$5$	2.85360	1.27617	0.638085	+	0.769966i	$0.279727\pi$
$6$	0	0
$7$	−1.52458	−0.576239	−0.288119	−	0.957594i	$-0.593030\pi$
$7$	−1.52458	−0.576239	−0.288119	+	0.957594i	$0.593030\pi$
$8$	0	0
$9$	−2.71456	−0.904853
$10$	0	0
$11$	3.72860	1.12421	0.562107	−	0.827065i	$-0.309991\pi$
$11$	3.72860	1.12421	0.562107	+	0.827065i	$0.309991\pi$
$12$	0	0
$13$	1.48994	0.413235	0.206617	−	0.978422i	$-0.433754\pi$
$13$	1.48994	0.413235	0.206617	+	0.978422i	$0.433754\pi$
$14$	0	0
$15$	−1.52458	−0.393646
$16$	0	0
$17$	6.09414	1.47804	0.739022	−	0.673681i	$-0.235288\pi$
$17$	6.09414	1.47804	0.739022	+	0.673681i	$0.235288\pi$
$18$	0	0
$19$	8.39902	1.92687	0.963434	−	0.267946i	$-0.0863447\pi$
$19$	8.39902	1.92687	0.963434	+	0.267946i	$0.0863447\pi$
$20$	0	0
$21$	0.814534	0.177746
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	3.14306	0.628611
$26$	0	0
$27$	3.05310	0.587569
$28$	0	0
$29$	−1.98697	−0.368971	−0.184485	−	0.982835i	$-0.559062\pi$
$29$	−1.98697	−0.368971	−0.184485	+	0.982835i	$0.559062\pi$
$30$	0	0
$31$	3.69466	0.663581	0.331791	−	0.943353i	$-0.392347\pi$
$31$	3.69466	0.663581	0.331791	+	0.943353i	$0.392347\pi$
$32$	0	0
$33$	−1.99206	−0.346774
$34$	0	0
$35$	−4.35056	−0.735379
$36$	0	0
$37$	−9.05221	−1.48817	−0.744087	−	0.668083i	$-0.767115\pi$
$37$	−9.05221	−1.48817	−0.744087	+	0.668083i	$0.767115\pi$
$38$	0	0
$39$	−0.796024	−0.127466
$40$	0	0
$41$	−2.86381	−0.447251	−0.223626	−	0.974675i	$-0.571789\pi$
$41$	−2.86381	−0.447251	−0.223626	+	0.974675i	$0.571789\pi$
$42$	0	0
$43$	2.13351	0.325358	0.162679	−	0.986679i	$-0.447987\pi$
$43$	2.13351	0.325358	0.162679	+	0.986679i	$0.447987\pi$
$44$	0	0
$45$	−7.74628	−1.15475
$46$	0	0
$47$	−9.05831	−1.32129	−0.660645	−	0.750699i	$-0.729717\pi$
$47$	−9.05831	−1.32129	−0.660645	+	0.750699i	$0.729717\pi$
$48$	0	0
$49$	−4.67564	−0.667949
$50$	0	0
$51$	−3.25589	−0.455916
$52$	0	0
$53$	1.56105	0.214426	0.107213	−	0.994236i	$-0.465807\pi$
$53$	1.56105	0.214426	0.107213	+	0.994236i	$0.465807\pi$
$54$	0	0
$55$	10.6399	1.43469
$56$	0	0
$57$	−4.48732	−0.594359
$58$	0	0
$59$	8.98316	1.16951	0.584754	−	0.811211i	$-0.301191\pi$
$59$	8.98316	1.16951	0.584754	+	0.811211i	$0.301191\pi$
$60$	0	0
$61$	−14.4631	−1.85182	−0.925908	−	0.377749i	$-0.876698\pi$
$61$	−14.4631	−1.85182	−0.925908	+	0.377749i	$0.876698\pi$
$62$	0	0
$63$	4.13858	0.521411
$64$	0	0
$65$	4.25169	0.527358
$66$	0	0
$67$	4.29489	0.524704	0.262352	−	0.964972i	$-0.415502\pi$
$67$	4.29489	0.524704	0.262352	+	0.964972i	$0.415502\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.584685	−0.0693893	−0.0346947	−	0.999398i	$-0.511046\pi$
$71$	−0.584685	−0.0693893	−0.0346947	+	0.999398i	$0.511046\pi$
$72$	0	0
$73$	3.96707	0.464311	0.232155	−	0.972679i	$-0.425422\pi$
$73$	3.96707	0.464311	0.232155	+	0.972679i	$0.425422\pi$
$74$	0	0
$75$	−1.67923	−0.193901
$76$	0	0
$77$	−5.68456	−0.647816
$78$	0	0
$79$	6.05191	0.680894	0.340447	−	0.940264i	$-0.389422\pi$
$79$	6.05191	0.680894	0.340447	+	0.940264i	$0.389422\pi$
$80$	0	0
$81$	6.51251	0.723612
$82$	0	0
$83$	−11.9759	−1.31452	−0.657261	−	0.753663i	$-0.728285\pi$
$83$	−11.9759	−1.31452	−0.657261	+	0.753663i	$0.728285\pi$
$84$	0	0
$85$	17.3902	1.88624
$86$	0	0
$87$	1.06157	0.113812
$88$	0	0
$89$	18.2247	1.93182	0.965908	−	0.258886i	$-0.0833554\pi$
$89$	18.2247	1.93182	0.965908	+	0.258886i	$0.0833554\pi$
$90$	0	0
$91$	−2.27154	−0.238122
$92$	0	0
$93$	−1.97394	−0.204688
$94$	0	0
$95$	23.9675	2.45901
$96$	0	0
$97$	10.5468	1.07086	0.535432	−	0.844579i	$-0.320149\pi$
$97$	10.5468	1.07086	0.535432	+	0.844579i	$0.320149\pi$
$98$	0	0
$99$	−10.1215	−1.01725
$100$	0	0
$101$	0.672736	0.0669397	0.0334699	−	0.999440i	$-0.489344\pi$
$101$	0.672736	0.0669397	0.0334699	+	0.999440i	$0.489344\pi$
$102$	0	0
$103$	6.84834	0.674787	0.337394	−	0.941364i	$-0.390455\pi$
$103$	6.84834	0.674787	0.337394	+	0.941364i	$0.390455\pi$
$104$	0	0
$105$	2.32436	0.226834
$106$	0	0
$107$	14.7117	1.42224	0.711119	−	0.703072i	$-0.248189\pi$
$107$	14.7117	1.42224	0.711119	+	0.703072i	$0.248189\pi$
$108$	0	0
$109$	−15.0822	−1.44461	−0.722305	−	0.691575i	$-0.756917\pi$
$109$	−15.0822	−1.44461	−0.722305	+	0.691575i	$0.756917\pi$
$110$	0	0
$111$	4.83629	0.459040
$112$	0	0
$113$	4.58025	0.430874	0.215437	−	0.976518i	$-0.430882\pi$
$113$	4.58025	0.430874	0.215437	+	0.976518i	$0.430882\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.04453	−0.373917
$118$	0	0
$119$	−9.29102	−0.851707
$120$	0	0
$121$	2.90243	0.263857
$122$	0	0
$123$	1.53004	0.137959
$124$	0	0
$125$	−5.29898	−0.473956
$126$	0	0
$127$	17.0255	1.51077	0.755383	−	0.655283i	$-0.227451\pi$
$127$	17.0255	1.51077	0.755383	+	0.655283i	$0.227451\pi$
$128$	0	0
$129$	−1.13986	−0.100359
$130$	0	0
$131$	17.7284	1.54893	0.774467	−	0.632614i	$-0.218018\pi$
$131$	17.7284	1.54893	0.774467	+	0.632614i	$0.218018\pi$
$132$	0	0
$133$	−12.8050	−1.11034
$134$	0	0
$135$	8.71233	0.749838
$136$	0	0
$137$	11.8729	1.01437	0.507187	−	0.861836i	$-0.330685\pi$
$137$	11.8729	1.01437	0.507187	+	0.861836i	$0.330685\pi$
$138$	0	0
$139$	3.16290	0.268274	0.134137	−	0.990963i	$-0.457174\pi$
$139$	3.16290	0.268274	0.134137	+	0.990963i	$0.457174\pi$
$140$	0	0
$141$	4.83955	0.407563
$142$	0	0
$143$	5.55538	0.464564
$144$	0	0
$145$	−5.67002	−0.470869
$146$	0	0
$147$	2.49804	0.206035
$148$	0	0
$149$	−6.22118	−0.509659	−0.254829	−	0.966986i	$-0.582019\pi$
$149$	−6.22118	−0.509659	−0.254829	+	0.966986i	$0.582019\pi$
$150$	0	0
$151$	−16.6063	−1.35140	−0.675699	−	0.737178i	$-0.736158\pi$
$151$	−16.6063	−1.35140	−0.675699	+	0.737178i	$0.736158\pi$
$152$	0	0
$153$	−16.5429	−1.33741
$154$	0	0
$155$	10.5431	0.846843
$156$	0	0
$157$	0.449800	0.0358979	0.0179490	−	0.999839i	$-0.494286\pi$
$157$	0.449800	0.0358979	0.0179490	+	0.999839i	$0.494286\pi$
$158$	0	0
$159$	−0.834015	−0.0661417
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.1529	0.951889	0.475944	−	0.879475i	$-0.342106\pi$
$163$	12.1529	0.951889	0.475944	+	0.879475i	$0.342106\pi$
$164$	0	0
$165$	−5.68456	−0.442542
$166$	0	0
$167$	19.6914	1.52377	0.761883	−	0.647714i	$-0.224275\pi$
$167$	19.6914	1.52377	0.761883	+	0.647714i	$0.224275\pi$
$168$	0	0
$169$	−10.7801	−0.829237
$170$	0	0
$171$	−22.7996	−1.74353
$172$	0	0
$173$	10.9868	0.835314	0.417657	−	0.908605i	$-0.362851\pi$
$173$	10.9868	0.835314	0.417657	+	0.908605i	$0.362851\pi$
$174$	0	0
$175$	−4.79185	−0.362230
$176$	0	0
$177$	−4.79940	−0.360745
$178$	0	0
$179$	−21.4750	−1.60512	−0.802560	−	0.596571i	$-0.796529\pi$
$179$	−21.4750	−1.60512	−0.802560	+	0.596571i	$0.796529\pi$
$180$	0	0
$181$	7.94598	0.590620	0.295310	−	0.955401i	$-0.404577\pi$
$181$	7.94598	0.590620	0.295310	+	0.955401i	$0.404577\pi$
$182$	0	0
$183$	7.72717	0.571209
$184$	0	0
$185$	−25.8314	−1.89916
$186$	0	0
$187$	22.7226	1.66164
$188$	0	0
$189$	−4.65470	−0.338580
$190$	0	0
$191$	−13.0951	−0.947525	−0.473763	−	0.880653i	$-0.657104\pi$
$191$	−13.0951	−0.947525	−0.473763	+	0.880653i	$0.657104\pi$
$192$	0	0
$193$	−3.35883	−0.241774	−0.120887	−	0.992666i	$-0.538574\pi$
$193$	−3.35883	−0.241774	−0.120887	+	0.992666i	$0.538574\pi$
$194$	0	0
$195$	−2.27154	−0.162668
$196$	0	0
$197$	20.7798	1.48050	0.740250	−	0.672332i	$-0.234707\pi$
$197$	20.7798	1.48050	0.740250	+	0.672332i	$0.234707\pi$
$198$	0	0
$199$	−21.4565	−1.52101	−0.760506	−	0.649330i	$-0.775049\pi$
$199$	−21.4565	−1.52101	−0.760506	+	0.649330i	$0.775049\pi$
$200$	0	0
$201$	−2.29461	−0.161850
$202$	0	0
$203$	3.02930	0.212615
$204$	0	0
$205$	−8.17217	−0.570769
$206$	0	0
$207$	0	0
$208$	0	0
$209$	31.3166	2.16621
$210$	0	0
$211$	20.1736	1.38881	0.694405	−	0.719584i	$-0.255667\pi$
$211$	20.1736	1.38881	0.694405	+	0.719584i	$0.255667\pi$
$212$	0	0
$213$	0.312377	0.0214038
$214$	0	0
$215$	6.08820	0.415212
$216$	0	0
$217$	−5.63283	−0.382381
$218$	0	0
$219$	−2.11947	−0.143221
$220$	0	0
$221$	9.07989	0.610779
$222$	0	0
$223$	5.96866	0.399691	0.199845	−	0.979827i	$-0.435956\pi$
$223$	5.96866	0.399691	0.199845	+	0.979827i	$0.435956\pi$
$224$	0	0
$225$	−8.53201	−0.568801
$226$	0	0
$227$	3.89888	0.258778	0.129389	−	0.991594i	$-0.458698\pi$
$227$	3.89888	0.258778	0.129389	+	0.991594i	$0.458698\pi$
$228$	0	0
$229$	7.55734	0.499403	0.249702	−	0.968323i	$-0.419667\pi$
$229$	7.55734	0.499403	0.249702	+	0.968323i	$0.419667\pi$
$230$	0	0
$231$	3.03707	0.199824
$232$	0	0
$233$	15.2400	0.998405	0.499203	−	0.866485i	$-0.333626\pi$
$233$	15.2400	0.998405	0.499203	+	0.866485i	$0.333626\pi$
$234$	0	0
$235$	−25.8488	−1.68619
$236$	0	0
$237$	−3.23333	−0.210028
$238$	0	0
$239$	22.7686	1.47278	0.736388	−	0.676560i	$-0.236530\pi$
$239$	22.7686	1.47278	0.736388	+	0.676560i	$0.236530\pi$
$240$	0	0
$241$	−8.44396	−0.543923	−0.271962	−	0.962308i	$-0.587672\pi$
$241$	−8.44396	−0.543923	−0.271962	+	0.962308i	$0.587672\pi$
$242$	0	0
$243$	−12.6387	−0.810773
$244$	0	0
$245$	−13.3424	−0.852417
$246$	0	0
$247$	12.5140	0.796249
$248$	0	0
$249$	6.39831	0.405476
$250$	0	0
$251$	−6.88513	−0.434585	−0.217293	−	0.976107i	$-0.569723\pi$
$251$	−6.88513	−0.434585	−0.217293	+	0.976107i	$0.569723\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9.29102	−0.581827
$256$	0	0
$257$	−27.5891	−1.72096	−0.860480	−	0.509485i	$-0.829836\pi$
$257$	−27.5891	−1.72096	−0.860480	+	0.509485i	$0.829836\pi$
$258$	0	0
$259$	13.8009	0.857544
$260$	0	0
$261$	5.39374	0.333864
$262$	0	0
$263$	10.4539	0.644613	0.322307	−	0.946635i	$-0.395542\pi$
$263$	10.4539	0.644613	0.322307	+	0.946635i	$0.395542\pi$
$264$	0	0
$265$	4.45461	0.273644
$266$	0	0
$267$	−9.73685	−0.595886
$268$	0	0
$269$	−6.38665	−0.389401	−0.194701	−	0.980863i	$-0.562373\pi$
$269$	−6.38665	−0.389401	−0.194701	+	0.980863i	$0.562373\pi$
$270$	0	0
$271$	22.7152	1.37985	0.689926	−	0.723880i	$-0.257643\pi$
$271$	22.7152	1.37985	0.689926	+	0.723880i	$0.257643\pi$
$272$	0	0
$273$	1.21361	0.0734508
$274$	0	0
$275$	11.7192	0.706693
$276$	0	0
$277$	−26.2170	−1.57523	−0.787613	−	0.616170i	$-0.788683\pi$
$277$	−26.2170	−1.57523	−0.787613	+	0.616170i	$0.788683\pi$
$278$	0	0
$279$	−10.0294	−0.600444
$280$	0	0
$281$	5.70769	0.340492	0.170246	−	0.985402i	$-0.445544\pi$
$281$	5.70769	0.340492	0.170246	+	0.985402i	$0.445544\pi$
$282$	0	0
$283$	−26.0990	−1.55142	−0.775712	−	0.631088i	$-0.782609\pi$
$283$	−26.0990	−1.55142	−0.775712	+	0.631088i	$0.782609\pi$
$284$	0	0
$285$	−12.8050	−0.758504
$286$	0	0
$287$	4.36612	0.257724
$288$	0	0
$289$	20.1385	1.18462
$290$	0	0
$291$	−5.63479	−0.330317
$292$	0	0
$293$	5.22212	0.305079	0.152540	−	0.988297i	$-0.451255\pi$
$293$	5.22212	0.305079	0.152540	+	0.988297i	$0.451255\pi$
$294$	0	0
$295$	25.6344	1.49249
$296$	0	0
$297$	11.3838	0.660553
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.25272	−0.187484
$302$	0	0
$303$	−0.359420	−0.0206481
$304$	0	0
$305$	−41.2721	−2.36323
$306$	0	0
$307$	−31.9206	−1.82181	−0.910904	−	0.412619i	$-0.864614\pi$
$307$	−31.9206	−1.82181	−0.910904	+	0.412619i	$0.864614\pi$
$308$	0	0
$309$	−3.65884	−0.208144
$310$	0	0
$311$	14.8254	0.840672	0.420336	−	0.907369i	$-0.361912\pi$
$311$	14.8254	0.840672	0.420336	+	0.907369i	$0.361912\pi$
$312$	0	0
$313$	−2.21209	−0.125035	−0.0625175	−	0.998044i	$-0.519913\pi$
$313$	−2.21209	−0.125035	−0.0625175	+	0.998044i	$0.519913\pi$
$314$	0	0
$315$	11.8099	0.665410
$316$	0	0
$317$	−10.1357	−0.569279	−0.284639	−	0.958635i	$-0.591874\pi$
$317$	−10.1357	−0.569279	−0.284639	+	0.958635i	$0.591874\pi$
$318$	0	0
$319$	−7.40860	−0.414802
$320$	0	0
$321$	−7.85998	−0.438702
$322$	0	0
$323$	51.1848	2.84800
$324$	0	0
$325$	4.68296	0.259764
$326$	0	0
$327$	8.05789	0.445603
$328$	0	0
$329$	13.8102	0.761378
$330$	0	0
$331$	−6.41117	−0.352390	−0.176195	−	0.984355i	$-0.556379\pi$
$331$	−6.41117	−0.352390	−0.176195	+	0.984355i	$0.556379\pi$
$332$	0	0
$333$	24.5728	1.34658
$334$	0	0
$335$	12.2559	0.669612
$336$	0	0
$337$	−1.73511	−0.0945172	−0.0472586	−	0.998883i	$-0.515048\pi$
$337$	−1.73511	−0.0945172	−0.0472586	+	0.998883i	$0.515048\pi$
$338$	0	0
$339$	−2.44708	−0.132907
$340$	0	0
$341$	13.7759	0.746007
$342$	0	0
$343$	17.8005	0.961137
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.70398	0.252523	0.126262	−	0.991997i	$-0.459702\pi$
$347$	4.70398	0.252523	0.126262	+	0.991997i	$0.459702\pi$
$348$	0	0
$349$	14.2876	0.764795	0.382398	−	0.923998i	$-0.375098\pi$
$349$	14.2876	0.764795	0.382398	+	0.923998i	$0.375098\pi$
$350$	0	0
$351$	4.54893	0.242804
$352$	0	0
$353$	−9.50758	−0.506037	−0.253019	−	0.967461i	$-0.581423\pi$
$353$	−9.50758	−0.506037	−0.253019	+	0.967461i	$0.581423\pi$
$354$	0	0
$355$	−1.66846	−0.0885526
$356$	0	0
$357$	4.96388	0.262716
$358$	0	0
$359$	−35.1667	−1.85603	−0.928015	−	0.372543i	$-0.878486\pi$
$359$	−35.1667	−1.85603	−0.928015	+	0.372543i	$0.878486\pi$
$360$	0	0
$361$	51.5436	2.71282
$362$	0	0
$363$	−1.55067	−0.0813890
$364$	0	0
$365$	11.3205	0.592540
$366$	0	0
$367$	−36.6112	−1.91109	−0.955545	−	0.294846i	$-0.904732\pi$
$367$	−36.6112	−1.91109	−0.955545	+	0.294846i	$0.904732\pi$
$368$	0	0
$369$	7.77397	0.404697
$370$	0	0
$371$	−2.37995	−0.123561
$372$	0	0
$373$	23.0471	1.19333	0.596667	−	0.802489i	$-0.296491\pi$
$373$	23.0471	1.19333	0.596667	+	0.802489i	$0.296491\pi$
$374$	0	0
$375$	2.83107	0.146196
$376$	0	0
$377$	−2.96046	−0.152471
$378$	0	0
$379$	−18.5575	−0.953234	−0.476617	−	0.879111i	$-0.658137\pi$
$379$	−18.5575	−0.953234	−0.476617	+	0.879111i	$0.658137\pi$
$380$	0	0
$381$	−9.09614	−0.466009
$382$	0	0
$383$	19.7753	1.01047	0.505236	−	0.862981i	$-0.331405\pi$
$383$	19.7753	1.01047	0.505236	+	0.862981i	$0.331405\pi$
$384$	0	0
$385$	−16.2215	−0.826723
$386$	0	0
$387$	−5.79155	−0.294401
$388$	0	0
$389$	5.06128	0.256617	0.128309	−	0.991734i	$-0.459045\pi$
$389$	5.06128	0.256617	0.128309	+	0.991734i	$0.459045\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.47167	−0.477782
$394$	0	0
$395$	17.2698	0.868936
$396$	0	0
$397$	−25.6294	−1.28630	−0.643150	−	0.765740i	$-0.722373\pi$
$397$	−25.6294	−1.28630	−0.643150	+	0.765740i	$0.722373\pi$
$398$	0	0
$399$	6.84129	0.342493
$400$	0	0
$401$	18.5145	0.924570	0.462285	−	0.886731i	$-0.347030\pi$
$401$	18.5145	0.924570	0.462285	+	0.886731i	$0.347030\pi$
$402$	0	0
$403$	5.50482	0.274215
$404$	0	0
$405$	18.5841	0.923453
$406$	0	0
$407$	−33.7520	−1.67303
$408$	0	0
$409$	6.78632	0.335562	0.167781	−	0.985824i	$-0.446340\pi$
$409$	6.78632	0.335562	0.167781	+	0.985824i	$0.446340\pi$
$410$	0	0
$411$	−6.34331	−0.312893
$412$	0	0
$413$	−13.6956	−0.673916
$414$	0	0
$415$	−34.1744	−1.67756
$416$	0	0
$417$	−1.68983	−0.0827514
$418$	0	0
$419$	−9.80857	−0.479180	−0.239590	−	0.970874i	$-0.577013\pi$
$419$	−9.80857	−0.479180	−0.239590	+	0.970874i	$0.577013\pi$
$420$	0	0
$421$	22.5737	1.10017	0.550086	−	0.835108i	$-0.314595\pi$
$421$	22.5737	1.10017	0.550086	+	0.835108i	$0.314595\pi$
$422$	0	0
$423$	24.5893	1.19557
$424$	0	0
$425$	19.1542	0.929116
$426$	0	0
$427$	22.0503	1.06709
$428$	0	0
$429$	−2.96805	−0.143299
$430$	0	0
$431$	32.8771	1.58363	0.791817	−	0.610759i	$-0.209136\pi$
$431$	32.8771	1.58363	0.791817	+	0.610759i	$0.209136\pi$
$432$	0	0
$433$	−2.35694	−0.113267	−0.0566337	−	0.998395i	$-0.518037\pi$
$433$	−2.35694	−0.113267	−0.0566337	+	0.998395i	$0.518037\pi$
$434$	0	0
$435$	3.02930	0.145244
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−18.6307	−0.889195	−0.444597	−	0.895731i	$-0.646653\pi$
$439$	−18.6307	−0.889195	−0.444597	+	0.895731i	$0.646653\pi$
$440$	0	0
$441$	12.6923	0.604396
$442$	0	0
$443$	−3.40046	−0.161561	−0.0807803	−	0.996732i	$-0.525741\pi$
$443$	−3.40046	−0.161561	−0.0807803	+	0.996732i	$0.525741\pi$
$444$	0	0
$445$	52.0061	2.46533
$446$	0	0
$447$	3.32376	0.157209
$448$	0	0
$449$	−31.8441	−1.50282	−0.751409	−	0.659837i	$-0.770625\pi$
$449$	−31.8441	−1.50282	−0.751409	+	0.659837i	$0.770625\pi$
$450$	0	0
$451$	−10.6780	−0.502806
$452$	0	0
$453$	8.87216	0.416851
$454$	0	0
$455$	−6.48207	−0.303884
$456$	0	0
$457$	−23.5089	−1.09970	−0.549849	−	0.835264i	$-0.685315\pi$
$457$	−23.5089	−1.09970	−0.549849	+	0.835264i	$0.685315\pi$
$458$	0	0
$459$	18.6060	0.868453
$460$	0	0
$461$	19.0051	0.885154	0.442577	−	0.896730i	$-0.354064\pi$
$461$	19.0051	0.885154	0.442577	+	0.896730i	$0.354064\pi$
$462$	0	0
$463$	−13.8723	−0.644702	−0.322351	−	0.946620i	$-0.604473\pi$
$463$	−13.8723	−0.644702	−0.322351	+	0.946620i	$0.604473\pi$
$464$	0	0
$465$	−5.63283	−0.261216
$466$	0	0
$467$	−3.35064	−0.155049	−0.0775246	−	0.996990i	$-0.524702\pi$
$467$	−3.35064	−0.155049	−0.0775246	+	0.996990i	$0.524702\pi$
$468$	0	0
$469$	−6.54792	−0.302355
$470$	0	0
$471$	−0.240313	−0.0110730
$472$	0	0
$473$	7.95501	0.365772
$474$	0	0
$475$	26.3986	1.21125
$476$	0	0
$477$	−4.23755	−0.194024
$478$	0	0
$479$	−10.0565	−0.459493	−0.229746	−	0.973251i	$-0.573790\pi$
$479$	−10.0565	−0.459493	−0.229746	+	0.973251i	$0.573790\pi$
$480$	0	0
$481$	−13.4872	−0.614965
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.0963	1.36660
$486$	0	0
$487$	26.0598	1.18088	0.590440	−	0.807081i	$-0.298954\pi$
$487$	26.0598	1.18088	0.590440	+	0.807081i	$0.298954\pi$
$488$	0	0
$489$	−6.49289	−0.293619
$490$	0	0
$491$	0.547879	0.0247254	0.0123627	−	0.999924i	$-0.496065\pi$
$491$	0.547879	0.0247254	0.0123627	+	0.999924i	$0.496065\pi$
$492$	0	0
$493$	−12.1088	−0.545355
$494$	0	0
$495$	−28.8827	−1.29818
$496$	0	0
$497$	0.891402	0.0399848
$498$	0	0
$499$	−8.09369	−0.362323	−0.181162	−	0.983453i	$-0.557986\pi$
$499$	−8.09369	−0.362323	−0.181162	+	0.983453i	$0.557986\pi$
$500$	0	0
$501$	−10.5205	−0.470019
$502$	0	0
$503$	9.46578	0.422058	0.211029	−	0.977480i	$-0.432318\pi$
$503$	9.46578	0.422058	0.211029	+	0.977480i	$0.432318\pi$
$504$	0	0
$505$	1.91972	0.0854265
$506$	0	0
$507$	5.75944	0.255786
$508$	0	0
$509$	−23.8605	−1.05760	−0.528800	−	0.848747i	$-0.677358\pi$
$509$	−23.8605	−1.05760	−0.528800	+	0.848747i	$0.677358\pi$
$510$	0	0
$511$	−6.04814	−0.267554
$512$	0	0
$513$	25.6430	1.13217
$514$	0	0
$515$	19.5425	0.861143
$516$	0	0
$517$	−33.7748	−1.48541
$518$	0	0
$519$	−5.86990	−0.257660
$520$	0	0
$521$	−19.2303	−0.842495	−0.421248	−	0.906946i	$-0.638408\pi$
$521$	−19.2303	−0.842495	−0.421248	+	0.906946i	$0.638408\pi$
$522$	0	0
$523$	7.55486	0.330351	0.165176	−	0.986264i	$-0.447181\pi$
$523$	7.55486	0.330351	0.165176	+	0.986264i	$0.447181\pi$
$524$	0	0
$525$	2.56013	0.111733
$526$	0	0
$527$	22.5158	0.980803
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−24.3853	−1.05823
$532$	0	0
$533$	−4.26690	−0.184820
$534$	0	0
$535$	41.9815	1.81502
$536$	0	0
$537$	11.4734	0.495113
$538$	0	0
$539$	−17.4336	−0.750917
$540$	0	0
$541$	−11.7376	−0.504640	−0.252320	−	0.967644i	$-0.581194\pi$
$541$	−11.7376	−0.504640	−0.252320	+	0.967644i	$0.581194\pi$
$542$	0	0
$543$	−4.24527	−0.182182
$544$	0	0
$545$	−43.0385	−1.84357
$546$	0	0
$547$	21.7408	0.929571	0.464786	−	0.885423i	$-0.346131\pi$
$547$	21.7408	0.929571	0.464786	+	0.885423i	$0.346131\pi$
$548$	0	0
$549$	39.2611	1.67562
$550$	0	0
$551$	−16.6886	−0.710958
$552$	0	0
$553$	−9.22666	−0.392357
$554$	0	0
$555$	13.8009	0.585814
$556$	0	0
$557$	12.1428	0.514508	0.257254	−	0.966344i	$-0.417182\pi$
$557$	12.1428	0.514508	0.257254	+	0.966344i	$0.417182\pi$
$558$	0	0
$559$	3.17880	0.134449
$560$	0	0
$561$	−12.1399	−0.512547
$562$	0	0
$563$	−4.09095	−0.172413	−0.0862066	−	0.996277i	$-0.527475\pi$
$563$	−4.09095	−0.172413	−0.0862066	+	0.996277i	$0.527475\pi$
$564$	0	0
$565$	13.0702	0.549869
$566$	0	0
$567$	−9.92887	−0.416974
$568$	0	0
$569$	−8.26623	−0.346538	−0.173269	−	0.984875i	$-0.555433\pi$
$569$	−8.26623	−0.346538	−0.173269	+	0.984875i	$0.555433\pi$
$570$	0	0
$571$	3.12975	0.130976	0.0654880	−	0.997853i	$-0.479140\pi$
$571$	3.12975	0.130976	0.0654880	+	0.997853i	$0.479140\pi$
$572$	0	0
$573$	6.99625	0.292272
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−6.14992	−0.256024	−0.128012	−	0.991773i	$-0.540860\pi$
$577$	−6.14992	−0.256024	−0.128012	+	0.991773i	$0.540860\pi$
$578$	0	0
$579$	1.79451	0.0745772
$580$	0	0
$581$	18.2582	0.757479
$582$	0	0
$583$	5.82051	0.241061
$584$	0	0
$585$	−11.5415	−0.477181
$586$	0	0
$587$	21.7356	0.897123	0.448562	−	0.893752i	$-0.351936\pi$
$587$	21.7356	0.897123	0.448562	+	0.893752i	$0.351936\pi$
$588$	0	0
$589$	31.0316	1.27863
$590$	0	0
$591$	−11.1020	−0.456673
$592$	0	0
$593$	2.93778	0.120640	0.0603202	−	0.998179i	$-0.480788\pi$
$593$	2.93778	0.120640	0.0603202	+	0.998179i	$0.480788\pi$
$594$	0	0
$595$	−26.5129	−1.08692
$596$	0	0
$597$	11.4635	0.469170
$598$	0	0
$599$	−7.58964	−0.310104	−0.155052	−	0.987906i	$-0.549555\pi$
$599$	−7.58964	−0.310104	−0.155052	+	0.987906i	$0.549555\pi$
$600$	0	0
$601$	−22.4842	−0.917151	−0.458576	−	0.888655i	$-0.651640\pi$
$601$	−22.4842	−0.917151	−0.458576	+	0.888655i	$0.651640\pi$
$602$	0	0
$603$	−11.6587	−0.474780
$604$	0	0
$605$	8.28238	0.336727
$606$	0	0
$607$	−23.9936	−0.973872	−0.486936	−	0.873438i	$-0.661886\pi$
$607$	−23.9936	−0.973872	−0.486936	+	0.873438i	$0.661886\pi$
$608$	0	0
$609$	−1.61845	−0.0655830
$610$	0	0
$611$	−13.4963	−0.546003
$612$	0	0
$613$	40.8425	1.64961	0.824806	−	0.565416i	$-0.191284\pi$
$613$	40.8425	1.64961	0.824806	+	0.565416i	$0.191284\pi$
$614$	0	0
$615$	4.36612	0.176059
$616$	0	0
$617$	−10.3708	−0.417514	−0.208757	−	0.977968i	$-0.566942\pi$
$617$	−10.3708	−0.417514	−0.208757	+	0.977968i	$0.566942\pi$
$618$	0	0
$619$	−30.2348	−1.21524	−0.607619	−	0.794229i	$-0.707875\pi$
$619$	−30.2348	−1.21524	−0.607619	+	0.794229i	$0.707875\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−27.7851	−1.11319
$624$	0	0
$625$	−30.8365	−1.23346
$626$	0	0
$627$	−16.7314	−0.668187
$628$	0	0
$629$	−55.1654	−2.19959
$630$	0	0
$631$	33.2185	1.32241	0.661204	−	0.750206i	$-0.270046\pi$
$631$	33.2185	1.32241	0.661204	+	0.750206i	$0.270046\pi$
$632$	0	0
$633$	−10.7781	−0.428391
$634$	0	0
$635$	48.5840	1.92800
$636$	0	0
$637$	−6.96642	−0.276020
$638$	0	0
$639$	1.58716	0.0627872
$640$	0	0
$641$	−15.9392	−0.629562	−0.314781	−	0.949164i	$-0.601931\pi$
$641$	−15.9392	−0.629562	−0.314781	+	0.949164i	$0.601931\pi$
$642$	0	0
$643$	30.0900	1.18663	0.593316	−	0.804969i	$-0.297818\pi$
$643$	30.0900	1.18663	0.593316	+	0.804969i	$0.297818\pi$
$644$	0	0
$645$	−3.25272	−0.128076
$646$	0	0
$647$	−36.8502	−1.44873	−0.724366	−	0.689416i	$-0.757867\pi$
$647$	−36.8502	−1.44873	−0.724366	+	0.689416i	$0.757867\pi$
$648$	0	0
$649$	33.4946	1.31478
$650$	0	0
$651$	3.00943	0.117949
$652$	0	0
$653$	−15.1567	−0.593128	−0.296564	−	0.955013i	$-0.595841\pi$
$653$	−15.1567	−0.593128	−0.296564	+	0.955013i	$0.595841\pi$
$654$	0	0
$655$	50.5897	1.97670
$656$	0	0
$657$	−10.7689	−0.420133
$658$	0	0
$659$	4.70664	0.183345	0.0916724	−	0.995789i	$-0.470779\pi$
$659$	4.70664	0.183345	0.0916724	+	0.995789i	$0.470779\pi$
$660$	0	0
$661$	−17.4347	−0.678131	−0.339066	−	0.940763i	$-0.610111\pi$
$661$	−17.4347	−0.678131	−0.339066	+	0.940763i	$0.610111\pi$
$662$	0	0
$663$	−4.85108	−0.188400
$664$	0	0
$665$	−36.5405	−1.41698
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−3.18885	−0.123288
$670$	0	0
$671$	−53.9272	−2.08184
$672$	0	0
$673$	39.5855	1.52591	0.762954	−	0.646453i	$-0.223748\pi$
$673$	39.5855	1.52591	0.762954	+	0.646453i	$0.223748\pi$
$674$	0	0
$675$	9.59605	0.369352
$676$	0	0
$677$	9.60757	0.369249	0.184624	−	0.982809i	$-0.440893\pi$
$677$	9.60757	0.369249	0.184624	+	0.982809i	$0.440893\pi$
$678$	0	0
$679$	−16.0795	−0.617073
$680$	0	0
$681$	−2.08304	−0.0798223
$682$	0	0
$683$	7.55260	0.288992	0.144496	−	0.989505i	$-0.453844\pi$
$683$	7.55260	0.288992	0.144496	+	0.989505i	$0.453844\pi$
$684$	0	0
$685$	33.8807	1.29451
$686$	0	0
$687$	−4.03763	−0.154045
$688$	0	0
$689$	2.32586	0.0886084
$690$	0	0
$691$	23.5083	0.894298	0.447149	−	0.894459i	$-0.352439\pi$
$691$	23.5083	0.894298	0.447149	+	0.894459i	$0.352439\pi$
$692$	0	0
$693$	15.4311	0.586178
$694$	0	0
$695$	9.02567	0.342363
$696$	0	0
$697$	−17.4524	−0.661058
$698$	0	0
$699$	−8.14222	−0.307967
$700$	0	0
$701$	−16.3848	−0.618845	−0.309423	−	0.950925i	$-0.600136\pi$
$701$	−16.3848	−0.618845	−0.309423	+	0.950925i	$0.600136\pi$
$702$	0	0
$703$	−76.0297	−2.86752
$704$	0	0
$705$	13.8102	0.520120
$706$	0	0
$707$	−1.02564	−0.0385733
$708$	0	0
$709$	−22.5110	−0.845420	−0.422710	−	0.906265i	$-0.638921\pi$
$709$	−22.5110	−0.845420	−0.422710	+	0.906265i	$0.638921\pi$
$710$	0	0
$711$	−16.4283	−0.616109
$712$	0	0
$713$	0	0
$714$	0	0
$715$	15.8529	0.592863
$716$	0	0
$717$	−12.1645	−0.454291
$718$	0	0
$719$	−12.2464	−0.456712	−0.228356	−	0.973578i	$-0.573335\pi$
$719$	−12.2464	−0.456712	−0.228356	+	0.973578i	$0.573335\pi$
$720$	0	0
$721$	−10.4409	−0.388838
$722$	0	0
$723$	4.51132	0.167778
$724$	0	0
$725$	−6.24515	−0.231939
$726$	0	0
$727$	23.9253	0.887340	0.443670	−	0.896190i	$-0.353676\pi$
$727$	23.9253	0.887340	0.443670	+	0.896190i	$0.353676\pi$
$728$	0	0
$729$	−12.7851	−0.473522
$730$	0	0
$731$	13.0019	0.480893
$732$	0	0
$733$	−35.4522	−1.30946	−0.654729	−	0.755864i	$-0.727217\pi$
$733$	−35.4522	−1.30946	−0.654729	+	0.755864i	$0.727217\pi$
$734$	0	0
$735$	7.12841	0.262935
$736$	0	0
$737$	16.0139	0.589880
$738$	0	0
$739$	7.23658	0.266202	0.133101	−	0.991102i	$-0.457507\pi$
$739$	7.23658	0.266202	0.133101	+	0.991102i	$0.457507\pi$
$740$	0	0
$741$	−6.68582	−0.245610
$742$	0	0
$743$	−22.3672	−0.820573	−0.410286	−	0.911957i	$-0.634571\pi$
$743$	−22.3672	−0.820573	−0.410286	+	0.911957i	$0.634571\pi$
$744$	0	0
$745$	−17.7528	−0.650411
$746$	0	0
$747$	32.5092	1.18945
$748$	0	0
$749$	−22.4293	−0.819548
$750$	0	0
$751$	37.1390	1.35522	0.677611	−	0.735420i	$-0.263015\pi$
$751$	37.1390	1.35522	0.677611	+	0.735420i	$0.263015\pi$
$752$	0	0
$753$	3.67849	0.134052
$754$	0	0
$755$	−47.3877	−1.72461
$756$	0	0
$757$	11.4260	0.415286	0.207643	−	0.978205i	$-0.433421\pi$
$757$	11.4260	0.415286	0.207643	+	0.978205i	$0.433421\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.5484	−0.418630	−0.209315	−	0.977848i	$-0.567123\pi$
$761$	−11.5484	−0.418630	−0.209315	+	0.977848i	$0.567123\pi$
$762$	0	0
$763$	22.9940	0.832440
$764$	0	0
$765$	−47.2069	−1.70677
$766$	0	0
$767$	13.3844	0.483281
$768$	0	0
$769$	2.69911	0.0973323	0.0486662	−	0.998815i	$-0.484503\pi$
$769$	2.69911	0.0973323	0.0486662	+	0.998815i	$0.484503\pi$
$770$	0	0
$771$	14.7399	0.530845
$772$	0	0
$773$	−33.4401	−1.20276	−0.601378	−	0.798965i	$-0.705381\pi$
$773$	−33.4401	−1.20276	−0.601378	+	0.798965i	$0.705381\pi$
$774$	0	0
$775$	11.6125	0.417135
$776$	0	0
$777$	−7.37333	−0.264517
$778$	0	0
$779$	−24.0532	−0.861795
$780$	0	0
$781$	−2.18005	−0.0780085
$782$	0	0
$783$	−6.06640	−0.216796
$784$	0	0
$785$	1.28355	0.0458119
$786$	0	0
$787$	−38.4522	−1.37067	−0.685337	−	0.728226i	$-0.740345\pi$
$787$	−38.4522	−1.37067	−0.685337	+	0.728226i	$0.740345\pi$
$788$	0	0
$789$	−5.58515	−0.198837
$790$	0	0
$791$	−6.98298	−0.248286
$792$	0	0
$793$	−21.5492	−0.765234
$794$	0	0
$795$	−2.37995	−0.0844080
$796$	0	0
$797$	−32.1667	−1.13940	−0.569702	−	0.821851i	$-0.692941\pi$
$797$	−32.1667	−1.13940	−0.569702	+	0.821851i	$0.692941\pi$
$798$	0	0
$799$	−55.2025	−1.95293
$800$	0	0
$801$	−49.4721	−1.74801
$802$	0	0
$803$	14.7916	0.521985
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.41217	0.120114
$808$	0	0
$809$	17.2568	0.606717	0.303358	−	0.952876i	$-0.401892\pi$
$809$	17.2568	0.606717	0.303358	+	0.952876i	$0.401892\pi$
$810$	0	0
$811$	4.43848	0.155856	0.0779280	−	0.996959i	$-0.475170\pi$
$811$	4.43848	0.155856	0.0779280	+	0.996959i	$0.475170\pi$
$812$	0	0
$813$	−12.1360	−0.425627
$814$	0	0
$815$	34.6796	1.21477
$816$	0	0
$817$	17.9194	0.626921
$818$	0	0
$819$	6.16622	0.215465
$820$	0	0
$821$	−41.4228	−1.44567	−0.722833	−	0.691023i	$-0.757161\pi$
$821$	−41.4228	−1.44567	−0.722833	+	0.691023i	$0.757161\pi$
$822$	0	0
$823$	−4.93096	−0.171882	−0.0859412	−	0.996300i	$-0.527390\pi$
$823$	−4.93096	−0.171882	−0.0859412	+	0.996300i	$0.527390\pi$
$824$	0	0
$825$	−6.26117	−0.217986
$826$	0	0
$827$	8.67963	0.301820	0.150910	−	0.988547i	$-0.451780\pi$
$827$	8.67963	0.301820	0.150910	+	0.988547i	$0.451780\pi$
$828$	0	0
$829$	−12.3701	−0.429630	−0.214815	−	0.976655i	$-0.568915\pi$
$829$	−12.3701	−0.429630	−0.214815	+	0.976655i	$0.568915\pi$
$830$	0	0
$831$	14.0069	0.485892
$832$	0	0
$833$	−28.4940	−0.987258
$834$	0	0
$835$	56.1915	1.94459
$836$	0	0
$837$	11.2802	0.389900
$838$	0	0
$839$	−30.9958	−1.07009	−0.535047	−	0.844822i	$-0.679706\pi$
$839$	−30.9958	−1.07009	−0.535047	+	0.844822i	$0.679706\pi$
$840$	0	0
$841$	−25.0520	−0.863861
$842$	0	0
$843$	−3.04943	−0.105028
$844$	0	0
$845$	−30.7621	−1.05825
$846$	0	0
$847$	−4.42500	−0.152045
$848$	0	0
$849$	13.9438	0.478550
$850$	0	0
$851$	0	0
$852$	0	0
$853$	31.3510	1.07344	0.536719	−	0.843761i	$-0.319664\pi$
$853$	31.3510	1.07344	0.536719	+	0.843761i	$0.319664\pi$
$854$	0	0
$855$	−65.0612	−2.22504
$856$	0	0
$857$	35.0754	1.19815	0.599076	−	0.800692i	$-0.295535\pi$
$857$	35.0754	1.19815	0.599076	+	0.800692i	$0.295535\pi$
$858$	0	0
$859$	3.68637	0.125777	0.0628887	−	0.998021i	$-0.479969\pi$
$859$	3.68637	0.125777	0.0628887	+	0.998021i	$0.479969\pi$
$860$	0	0
$861$	−2.33267	−0.0794971
$862$	0	0
$863$	−22.4200	−0.763185	−0.381593	−	0.924331i	$-0.624624\pi$
$863$	−22.4200	−0.763185	−0.381593	+	0.924331i	$0.624624\pi$
$864$	0	0
$865$	31.3521	1.06600
$866$	0	0
$867$	−10.7593	−0.365406
$868$	0	0
$869$	22.5651	0.765470
$870$	0	0
$871$	6.39912	0.216826
$872$	0	0
$873$	−28.6299	−0.968974
$874$	0	0
$875$	8.07875	0.273112
$876$	0	0
$877$	−55.2027	−1.86406	−0.932031	−	0.362379i	$-0.881965\pi$
$877$	−55.2027	−1.86406	−0.932031	+	0.362379i	$0.881965\pi$
$878$	0	0
$879$	−2.79000	−0.0941044
$880$	0	0
$881$	4.98160	0.167834	0.0839172	−	0.996473i	$-0.473257\pi$
$881$	4.98160	0.167834	0.0839172	+	0.996473i	$0.473257\pi$
$882$	0	0
$883$	−1.94580	−0.0654813	−0.0327406	−	0.999464i	$-0.510424\pi$
$883$	−1.94580	−0.0654813	−0.0327406	+	0.999464i	$0.510424\pi$
$884$	0	0
$885$	−13.6956	−0.460372
$886$	0	0
$887$	−38.8970	−1.30603	−0.653017	−	0.757344i	$-0.726497\pi$
$887$	−38.8970	−1.30603	−0.653017	+	0.757344i	$0.726497\pi$
$888$	0	0
$889$	−25.9568	−0.870563
$890$	0	0
$891$	24.2825	0.813495
$892$	0	0
$893$	−76.0809	−2.54595
$894$	0	0
$895$	−61.2813	−2.04841
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.34118	−0.244842
$900$	0	0
$901$	9.51323	0.316932
$902$	0	0
$903$	1.73782	0.0578310
$904$	0	0
$905$	22.6747	0.753732
$906$	0	0
$907$	41.0340	1.36251	0.681256	−	0.732045i	$-0.261434\pi$
$907$	41.0340	1.36251	0.681256	+	0.732045i	$0.261434\pi$
$908$	0	0
$909$	−1.82618	−0.0605706
$910$	0	0
$911$	−16.7967	−0.556499	−0.278249	−	0.960509i	$-0.589754\pi$
$911$	−16.7967	−0.556499	−0.278249	+	0.960509i	$0.589754\pi$
$912$	0	0
$913$	−44.6532	−1.47780
$914$	0	0
$915$	22.0503	0.728960
$916$	0	0
$917$	−27.0284	−0.892556
$918$	0	0
$919$	−23.2745	−0.767753	−0.383877	−	0.923384i	$-0.625411\pi$
$919$	−23.2745	−0.767753	−0.383877	+	0.923384i	$0.625411\pi$
$920$	0	0
$921$	17.0541	0.561953
$922$	0	0
$923$	−0.871145	−0.0286741
$924$	0	0
$925$	−28.4516	−0.935483
$926$	0	0
$927$	−18.5902	−0.610583
$928$	0	0
$929$	−21.7654	−0.714098	−0.357049	−	0.934086i	$-0.616217\pi$
$929$	−21.7654	−0.714098	−0.357049	+	0.934086i	$0.616217\pi$
$930$	0	0
$931$	−39.2708	−1.28705
$932$	0	0
$933$	−7.92072	−0.259313
$934$	0	0
$935$	64.8412	2.12053
$936$	0	0
$937$	−46.1603	−1.50799	−0.753995	−	0.656880i	$-0.771876\pi$
$937$	−46.1603	−1.50799	−0.753995	+	0.656880i	$0.771876\pi$
$938$	0	0
$939$	1.18185	0.0385681
$940$	0	0
$941$	8.17739	0.266575	0.133288	−	0.991077i	$-0.457447\pi$
$941$	8.17739	0.266575	0.133288	+	0.991077i	$0.457447\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−13.2827	−0.432086
$946$	0	0
$947$	8.29747	0.269632	0.134816	−	0.990871i	$-0.456956\pi$
$947$	8.29747	0.269632	0.134816	+	0.990871i	$0.456956\pi$
$948$	0	0
$949$	5.91070	0.191869
$950$	0	0
$951$	5.41517	0.175599
$952$	0	0
$953$	−53.0822	−1.71950	−0.859751	−	0.510713i	$-0.829381\pi$
$953$	−53.0822	−1.71950	−0.859751	+	0.510713i	$0.829381\pi$
$954$	0	0
$955$	−37.3681	−1.20920
$956$	0	0
$957$	3.95816	0.127949
$958$	0	0
$959$	−18.1013	−0.584522
$960$	0	0
$961$	−17.3495	−0.559660
$962$	0	0
$963$	−39.9359	−1.28692
$964$	0	0
$965$	−9.58476	−0.308544
$966$	0	0
$967$	−3.85444	−0.123950	−0.0619752	−	0.998078i	$-0.519740\pi$
$967$	−3.85444	−0.123950	−0.0619752	+	0.998078i	$0.519740\pi$
$968$	0	0
$969$	−27.3463	−0.878490
$970$	0	0
$971$	−2.27001	−0.0728480	−0.0364240	−	0.999336i	$-0.511597\pi$
$971$	−2.27001	−0.0728480	−0.0364240	+	0.999336i	$0.511597\pi$
$972$	0	0
$973$	−4.82211	−0.154590
$974$	0	0
$975$	−2.50195	−0.0801265
$976$	0	0
$977$	35.5882	1.13857	0.569283	−	0.822142i	$-0.307221\pi$
$977$	35.5882	1.13857	0.569283	+	0.822142i	$0.307221\pi$
$978$	0	0
$979$	67.9526	2.17177
$980$	0	0
$981$	40.9414	1.30716
$982$	0	0
$983$	−31.1891	−0.994776	−0.497388	−	0.867528i	$-0.665708\pi$
$983$	−31.1891	−0.994776	−0.497388	+	0.867528i	$0.665708\pi$
$984$	0	0
$985$	59.2973	1.88937
$986$	0	0
$987$	−7.37830	−0.234854
$988$	0	0
$989$	0	0
$990$	0	0
$991$	51.2528	1.62810	0.814050	−	0.580795i	$-0.197258\pi$
$991$	51.2528	1.62810	0.814050	+	0.580795i	$0.197258\pi$
$992$	0	0
$993$	3.42527	0.108698
$994$	0	0
$995$	−61.2284	−1.94107
$996$	0	0
$997$	16.4889	0.522208	0.261104	−	0.965311i	$-0.415913\pi$
$997$	16.4889	0.522208	0.261104	+	0.965311i	$0.415913\pi$
$998$	0	0
$999$	−27.6373	−0.874405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4232.2.a.x.1.4	yes	12
4.3	odd	2		8464.2.a.cf.1.10		12
23.22	odd	2	inner	4232.2.a.x.1.3	✓	12
92.91	even	2		8464.2.a.cf.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.x.1.3	✓	12	23.22	odd	2	inner
4232.2.a.x.1.4	yes	12	1.1	even	1	trivial
8464.2.a.cf.1.9		12	92.91	even	2
8464.2.a.cf.1.10		12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q-0.534266 q^{3} +2.85360 q^{5} -1.52458 q^{7} -2.71456 q^{9} +3.72860 q^{11} +1.48994 q^{13} -1.52458 q^{15} +6.09414 q^{17} +8.39902 q^{19} +0.814534 q^{21} +3.14306 q^{25} +3.05310 q^{27} -1.98697 q^{29} +3.69466 q^{31} -1.99206 q^{33} -4.35056 q^{35} -9.05221 q^{37} -0.796024 q^{39} -2.86381 q^{41} +2.13351 q^{43} -7.74628 q^{45} -9.05831 q^{47} -4.67564 q^{49} -3.25589 q^{51} +1.56105 q^{53} +10.6399 q^{55} -4.48732 q^{57} +8.98316 q^{59} -14.4631 q^{61} +4.13858 q^{63} +4.25169 q^{65} +4.29489 q^{67} -0.584685 q^{71} +3.96707 q^{73} -1.67923 q^{75} -5.68456 q^{77} +6.05191 q^{79} +6.51251 q^{81} -11.9759 q^{83} +17.3902 q^{85} +1.06157 q^{87} +18.2247 q^{89} -2.27154 q^{91} -1.97394 q^{93} +23.9675 q^{95} +10.5468 q^{97} -10.1215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{3} + 8 q^{9} + 16 q^{13} + 4 q^{25} + 8 q^{27} + 8 q^{31} + 56 q^{35} + 64 q^{39} - 40 q^{41} + 32 q^{47} + 28 q^{49} + 64 q^{55} + 60 q^{59} - 32 q^{71} + 28 q^{73} + 16 q^{75} + 24 q^{77}+ \cdots + 16 q^{95}+O(q^{100}) \) 12 * q + 8 * q^3 + 8 * q^9 + 16 * q^13 + 4 * q^25 + 8 * q^27 + 8 * q^31 + 56 * q^35 + 64 * q^39 - 40 * q^41 + 32 * q^47 + 28 * q^49 + 64 * q^55 + 60 * q^59 - 32 * q^71 + 28 * q^73 + 16 * q^75 + 24 * q^77 + 4 * q^81 + 64 * q^85 - 16 * q^87 + 24 * q^93 + 16 * q^95

Introduction

L-functions

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Database

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