LMFDB - Embedded newform 8208.2.a.cf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] 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("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,2,0,-2,0,0,0,3] 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:	$65.5412099791$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 18x^{4} + 17x^{3} + 72x^{2} + 29x - 3 $ x^6 - 2x^5 - 18x^4 + 17x^3 + 72x^2 + 29*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.0851277$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0851277 q^{5} -5.11790 q^{7} -4.72733 q^{11} -2.04113 q^{13} +1.60943 q^{17} -1.00000 q^{19} -7.24416 q^{23} -4.99275 q^{25} -8.68334 q^{29} -6.55708 q^{31} -0.435675 q^{35} +7.55358 q^{37} +0.564325 q^{41} -11.6715 q^{43} +2.35055 q^{47} +19.1929 q^{49} -10.1918 q^{53} -0.402427 q^{55} +6.17487 q^{59} +6.17137 q^{61} -0.173757 q^{65} +11.2453 q^{67} +7.76408 q^{71} -6.55819 q^{73} +24.1940 q^{77} -0.406402 q^{79} +2.40243 q^{83} +0.137007 q^{85} -3.16651 q^{89} +10.4463 q^{91} -0.0851277 q^{95} -11.6310 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 2 q^{7} + 3 q^{11} + q^{13} + 7 q^{17} - 6 q^{19} - 3 q^{23} + 10 q^{25} - 6 q^{29} - 5 q^{31} + 3 q^{35} + 11 q^{37} + 9 q^{41} - 7 q^{43} + 7 q^{47} + 20 q^{49} + 11 q^{53} - 22 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) $ 6 * q + 2 * q^5 - 2 * q^7 + 3 * q^11 + q^13 + 7 * q^17 - 6 * q^19 - 3 * q^23 + 10 * q^25 - 6 * q^29 - 5 * q^31 + 3 * q^35 + 11 * q^37 + 9 * q^41 - 7 * q^43 + 7 * q^47 + 20 * q^49 + 11 * q^53 - 22 * q^55 + 17 * q^59 + 17 * q^61 - 4 * q^65 + 4 * q^67 - 10 * q^71 + 18 * q^73 + 27 * q^77 - 27 * q^79 + 34 * q^83 + 29 * q^85 + 18 * q^89 - 17 * q^91 - 2 * q^95 + 8 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	0.0851277	0.0380703	0.0190351	−	0.999819i	$-0.493941\pi$
$5$	0.0851277	0.0380703	0.0190351	+	0.999819i	$0.493941\pi$
$6$	0	0
$7$	−5.11790	−1.93439	−0.967193	−	0.254045i	$-0.918239\pi$
$7$	−5.11790	−1.93439	−0.967193	+	0.254045i	$0.918239\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.72733	−1.42534	−0.712672	−	0.701497i	$-0.752515\pi$
$11$	−4.72733	−1.42534	−0.712672	+	0.701497i	$0.752515\pi$
$12$	0	0
$13$	−2.04113	−0.566108	−0.283054	−	0.959104i	$-0.591348\pi$
$13$	−2.04113	−0.566108	−0.283054	+	0.959104i	$0.591348\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.60943	0.390345	0.195172	−	0.980769i	$-0.437473\pi$
$17$	1.60943	0.390345	0.195172	+	0.980769i	$0.437473\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.24416	−1.51051	−0.755256	−	0.655430i	$-0.772487\pi$
$23$	−7.24416	−1.51051	−0.755256	+	0.655430i	$0.772487\pi$
$24$	0	0
$25$	−4.99275	−0.998551
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.68334	−1.61246	−0.806228	−	0.591605i	$-0.798494\pi$
$29$	−8.68334	−1.61246	−0.806228	+	0.591605i	$0.798494\pi$
$30$	0	0
$31$	−6.55708	−1.17769	−0.588843	−	0.808247i	$-0.700416\pi$
$31$	−6.55708	−1.17769	−0.588843	+	0.808247i	$0.700416\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.435675	−0.0736425
$36$	0	0
$37$	7.55358	1.24180	0.620900	−	0.783890i	$-0.286767\pi$
$37$	7.55358	1.24180	0.620900	+	0.783890i	$0.286767\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.564325	0.0881327	0.0440664	−	0.999029i	$-0.485969\pi$
$41$	0.564325	0.0881327	0.0440664	+	0.999029i	$0.485969\pi$
$42$	0	0
$43$	−11.6715	−1.77988	−0.889942	−	0.456074i	$-0.849255\pi$
$43$	−11.6715	−1.77988	−0.889942	+	0.456074i	$0.849255\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.35055	0.342863	0.171431	−	0.985196i	$-0.445161\pi$
$47$	2.35055	0.342863	0.171431	+	0.985196i	$0.445161\pi$
$48$	0	0
$49$	19.1929	2.74185
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.1918	−1.39995	−0.699976	−	0.714166i	$-0.746806\pi$
$53$	−10.1918	−1.39995	−0.699976	+	0.714166i	$0.746806\pi$
$54$	0	0
$55$	−0.402427	−0.0542632
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.17487	0.803899	0.401950	−	0.915662i	$-0.368333\pi$
$59$	6.17487	0.803899	0.401950	+	0.915662i	$0.368333\pi$
$60$	0	0
$61$	6.17137	0.790162	0.395081	−	0.918646i	$-0.370716\pi$
$61$	6.17137	0.790162	0.395081	+	0.918646i	$0.370716\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.173757	−0.0215519
$66$	0	0
$67$	11.2453	1.37383	0.686914	−	0.726738i	$-0.258965\pi$
$67$	11.2453	1.37383	0.686914	+	0.726738i	$0.258965\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.76408	0.921427	0.460714	−	0.887549i	$-0.347594\pi$
$71$	7.76408	0.921427	0.460714	+	0.887549i	$0.347594\pi$
$72$	0	0
$73$	−6.55819	−0.767578	−0.383789	−	0.923421i	$-0.625381\pi$
$73$	−6.55819	−0.767578	−0.383789	+	0.923421i	$0.625381\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	24.1940	2.75717
$78$	0	0
$79$	−0.406402	−0.0457239	−0.0228619	−	0.999739i	$-0.507278\pi$
$79$	−0.406402	−0.0457239	−0.0228619	+	0.999739i	$0.507278\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.40243	0.263701	0.131850	−	0.991270i	$-0.457908\pi$
$83$	2.40243	0.263701	0.131850	+	0.991270i	$0.457908\pi$
$84$	0	0
$85$	0.137007	0.0148605
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.16651	−0.335649	−0.167825	−	0.985817i	$-0.553674\pi$
$89$	−3.16651	−0.335649	−0.167825	+	0.985817i	$0.553674\pi$
$90$	0	0
$91$	10.4463	1.09507
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.0851277	−0.00873392
$96$	0	0
$97$	−11.6310	−1.18095	−0.590474	−	0.807057i	$-0.701059\pi$
$97$	−11.6310	−1.18095	−0.590474	+	0.807057i	$0.701059\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.4839	−1.54071	−0.770355	−	0.637615i	$-0.779921\pi$
$101$	−15.4839	−1.54071	−0.770355	+	0.637615i	$0.779921\pi$
$102$	0	0
$103$	−12.4731	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$103$	−12.4731	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.11551	0.881230	0.440615	−	0.897696i	$-0.354760\pi$
$107$	9.11551	0.881230	0.440615	+	0.897696i	$0.354760\pi$
$108$	0	0
$109$	−3.98528	−0.381720	−0.190860	−	0.981617i	$-0.561128\pi$
$109$	−3.98528	−0.381720	−0.190860	+	0.981617i	$0.561128\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.5487	0.992340	0.496170	−	0.868225i	$-0.334739\pi$
$113$	10.5487	0.992340	0.496170	+	0.868225i	$0.334739\pi$
$114$	0	0
$115$	−0.616679	−0.0575056
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.23691	−0.755077
$120$	0	0
$121$	11.3477	1.03161
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−0.850660	−0.0760853
$126$	0	0
$127$	0.848506	0.0752928	0.0376464	−	0.999291i	$-0.488014\pi$
$127$	0.848506	0.0752928	0.0376464	+	0.999291i	$0.488014\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.15815	−0.450670	−0.225335	−	0.974281i	$-0.572348\pi$
$131$	−5.15815	−0.450670	−0.225335	+	0.974281i	$0.572348\pi$
$132$	0	0
$133$	5.11790	0.443778
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.5358	0.985572	0.492786	−	0.870151i	$-0.335979\pi$
$137$	11.5358	0.985572	0.492786	+	0.870151i	$0.335979\pi$
$138$	0	0
$139$	−0.834601	−0.0707899	−0.0353949	−	0.999373i	$-0.511269\pi$
$139$	−0.834601	−0.0707899	−0.0353949	+	0.999373i	$0.511269\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.64911	0.806899
$144$	0	0
$145$	−0.739192	−0.0613866
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.3559	−1.58570	−0.792850	−	0.609417i	$-0.791403\pi$
$149$	−19.3559	−1.58570	−0.792850	+	0.609417i	$0.791403\pi$
$150$	0	0
$151$	−3.52319	−0.286713	−0.143357	−	0.989671i	$-0.545790\pi$
$151$	−3.52319	−0.286713	−0.143357	+	0.989671i	$0.545790\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.558189	−0.0448348
$156$	0	0
$157$	9.76297	0.779170	0.389585	−	0.920991i	$-0.372618\pi$
$157$	9.76297	0.779170	0.389585	+	0.920991i	$0.372618\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	37.0749	2.92191
$162$	0	0
$163$	−13.2809	−1.04024	−0.520121	−	0.854093i	$-0.674113\pi$
$163$	−13.2809	−1.04024	−0.520121	+	0.854093i	$0.674113\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.0082	−1.16137	−0.580686	−	0.814127i	$-0.697216\pi$
$167$	−15.0082	−1.16137	−0.580686	+	0.814127i	$0.697216\pi$
$168$	0	0
$169$	−8.83378	−0.679522
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.08832	0.462887	0.231443	−	0.972848i	$-0.425655\pi$
$173$	6.08832	0.462887	0.231443	+	0.972848i	$0.425655\pi$
$174$	0	0
$175$	25.5524	1.93158
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.21657	−0.165674	−0.0828370	−	0.996563i	$-0.526398\pi$
$179$	−2.21657	−0.165674	−0.0828370	+	0.996563i	$0.526398\pi$
$180$	0	0
$181$	20.6979	1.53847	0.769233	−	0.638968i	$-0.220638\pi$
$181$	20.6979	1.53847	0.769233	+	0.638968i	$0.220638\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.643018	0.0472757
$186$	0	0
$187$	−7.60832	−0.556376
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.8800	1.94497	0.972484	−	0.232969i	$-0.0748439\pi$
$191$	26.8800	1.94497	0.972484	+	0.232969i	$0.0748439\pi$
$192$	0	0
$193$	−16.6548	−1.19884	−0.599421	−	0.800434i	$-0.704602\pi$
$193$	−16.6548	−1.19884	−0.599421	+	0.800434i	$0.704602\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.25538	0.303183	0.151592	−	0.988443i	$-0.451560\pi$
$197$	4.25538	0.303183	0.151592	+	0.988443i	$0.451560\pi$
$198$	0	0
$199$	1.48428	0.105218	0.0526090	−	0.998615i	$-0.483246\pi$
$199$	1.48428	0.105218	0.0526090	+	0.998615i	$0.483246\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	44.4405	3.11911
$204$	0	0
$205$	0.0480397	0.00335524
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.72733	0.326996
$210$	0	0
$211$	−18.4395	−1.26943	−0.634713	−	0.772748i	$-0.718882\pi$
$211$	−18.4395	−1.26943	−0.634713	+	0.772748i	$0.718882\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.993566	−0.0677606
$216$	0	0
$217$	33.5585	2.27810
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.28506	−0.220977
$222$	0	0
$223$	−2.43147	−0.162823	−0.0814116	−	0.996681i	$-0.525943\pi$
$223$	−2.43147	−0.162823	−0.0814116	+	0.996681i	$0.525943\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.4201	1.55444	0.777222	−	0.629226i	$-0.216628\pi$
$227$	23.4201	1.55444	0.777222	+	0.629226i	$0.216628\pi$
$228$	0	0
$229$	−26.2358	−1.73371	−0.866854	−	0.498562i	$-0.833861\pi$
$229$	−26.2358	−1.73371	−0.866854	+	0.498562i	$0.833861\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.0480	1.11685	0.558427	−	0.829554i	$-0.311405\pi$
$233$	17.0480	1.11685	0.558427	+	0.829554i	$0.311405\pi$
$234$	0	0
$235$	0.200097	0.0130529
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.2128	−1.63088	−0.815440	−	0.578842i	$-0.803505\pi$
$239$	−25.2128	−1.63088	−0.815440	+	0.578842i	$0.803505\pi$
$240$	0	0
$241$	4.10702	0.264557	0.132278	−	0.991213i	$-0.457771\pi$
$241$	4.10702	0.264557	0.132278	+	0.991213i	$0.457771\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.63385	0.104383
$246$	0	0
$247$	2.04113	0.129874
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.8618	−1.25367	−0.626834	−	0.779153i	$-0.715650\pi$
$251$	−19.8618	−1.25367	−0.626834	+	0.779153i	$0.715650\pi$
$252$	0	0
$253$	34.2456	2.15300
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.1038	1.69069	0.845345	−	0.534221i	$-0.179395\pi$
$257$	27.1038	1.69069	0.845345	+	0.534221i	$0.179395\pi$
$258$	0	0
$259$	−38.6585	−2.40212
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.3577	1.07032	0.535160	−	0.844751i	$-0.320251\pi$
$263$	17.3577	1.07032	0.535160	+	0.844751i	$0.320251\pi$
$264$	0	0
$265$	−0.867605	−0.0532965
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.6594	1.93031	0.965153	−	0.261686i	$-0.0842784\pi$
$269$	31.6594	1.93031	0.965153	+	0.261686i	$0.0842784\pi$
$270$	0	0
$271$	−13.1823	−0.800770	−0.400385	−	0.916347i	$-0.631124\pi$
$271$	−13.1823	−0.800770	−0.400385	+	0.916347i	$0.631124\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	23.6024	1.42328
$276$	0	0
$277$	7.52891	0.452368	0.226184	−	0.974085i	$-0.427375\pi$
$277$	7.52891	0.452368	0.226184	+	0.974085i	$0.427375\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.06410	0.540719	0.270359	−	0.962759i	$-0.412857\pi$
$281$	9.06410	0.540719	0.270359	+	0.962759i	$0.412857\pi$
$282$	0	0
$283$	−7.80148	−0.463750	−0.231875	−	0.972746i	$-0.574486\pi$
$283$	−7.80148	−0.463750	−0.231875	+	0.972746i	$0.574486\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.88816	−0.170483
$288$	0	0
$289$	−14.4097	−0.847631
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−13.7093	−0.800908	−0.400454	−	0.916317i	$-0.631148\pi$
$293$	−13.7093	−0.800908	−0.400454	+	0.916317i	$0.631148\pi$
$294$	0	0
$295$	0.525652	0.0306046
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.7863	0.855113
$300$	0	0
$301$	59.7335	3.44298
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.525354	0.0300817
$306$	0	0
$307$	28.6862	1.63721	0.818604	−	0.574359i	$-0.194749\pi$
$307$	28.6862	1.63721	0.818604	+	0.574359i	$0.194749\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	10.1176	0.573716	0.286858	−	0.957973i	$-0.407389\pi$
$311$	10.1176	0.573716	0.286858	+	0.957973i	$0.407389\pi$
$312$	0	0
$313$	4.88829	0.276302	0.138151	−	0.990411i	$-0.455884\pi$
$313$	4.88829	0.276302	0.138151	+	0.990411i	$0.455884\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.33232	0.299493	0.149747	−	0.988724i	$-0.452154\pi$
$317$	5.33232	0.299493	0.149747	+	0.988724i	$0.452154\pi$
$318$	0	0
$319$	41.0490	2.29830
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.60943	−0.0895512
$324$	0	0
$325$	10.1909	0.565288
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.0299	−0.663228
$330$	0	0
$331$	−14.5861	−0.801726	−0.400863	−	0.916138i	$-0.631290\pi$
$331$	−14.5861	−0.801726	−0.400863	+	0.916138i	$0.631290\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.957284	0.0523020
$336$	0	0
$337$	16.2295	0.884079	0.442040	−	0.896996i	$-0.354255\pi$
$337$	16.2295	0.884079	0.442040	+	0.896996i	$0.354255\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	30.9975	1.67861
$342$	0	0
$343$	−62.4022	−3.36940
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.80135	0.0967017	0.0483508	−	0.998830i	$-0.484603\pi$
$347$	1.80135	0.0967017	0.0483508	+	0.998830i	$0.484603\pi$
$348$	0	0
$349$	−26.7712	−1.43303	−0.716515	−	0.697572i	$-0.754264\pi$
$349$	−26.7712	−1.43303	−0.716515	+	0.697572i	$0.754264\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.1968	−0.755621	−0.377810	−	0.925883i	$-0.623323\pi$
$353$	−14.1968	−0.755621	−0.377810	+	0.925883i	$0.623323\pi$
$354$	0	0
$355$	0.660938	0.0350790
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.48755	−0.131288	−0.0656441	−	0.997843i	$-0.520910\pi$
$359$	−2.48755	−0.131288	−0.0656441	+	0.997843i	$0.520910\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−0.558283	−0.0292219
$366$	0	0
$367$	−35.8358	−1.87061	−0.935306	−	0.353841i	$-0.884876\pi$
$367$	−35.8358	−1.87061	−0.935306	+	0.353841i	$0.884876\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	52.1607	2.70805
$372$	0	0
$373$	7.82886	0.405363	0.202682	−	0.979245i	$-0.435034\pi$
$373$	7.82886	0.405363	0.202682	+	0.979245i	$0.435034\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.7238	0.912824
$378$	0	0
$379$	17.9528	0.922173	0.461087	−	0.887355i	$-0.347460\pi$
$379$	17.9528	0.922173	0.461087	+	0.887355i	$0.347460\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.7299	0.752660	0.376330	−	0.926486i	$-0.377186\pi$
$383$	14.7299	0.752660	0.376330	+	0.926486i	$0.377186\pi$
$384$	0	0
$385$	2.05958	0.104966
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−19.7863	−1.00320	−0.501602	−	0.865098i	$-0.667256\pi$
$389$	−19.7863	−1.00320	−0.501602	+	0.865098i	$0.667256\pi$
$390$	0	0
$391$	−11.6590	−0.589620
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.0345961	−0.00174072
$396$	0	0
$397$	−8.31807	−0.417472	−0.208736	−	0.977972i	$-0.566935\pi$
$397$	−8.31807	−0.417472	−0.208736	+	0.977972i	$0.566935\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.9073	−1.19388	−0.596938	−	0.802287i	$-0.703616\pi$
$401$	−23.9073	−1.19388	−0.596938	+	0.802287i	$0.703616\pi$
$402$	0	0
$403$	13.3839	0.666697
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−35.7083	−1.76999
$408$	0	0
$409$	8.67148	0.428777	0.214389	−	0.976748i	$-0.431224\pi$
$409$	8.67148	0.428777	0.214389	+	0.976748i	$0.431224\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−31.6024	−1.55505
$414$	0	0
$415$	0.204513	0.0100391
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.2585	1.03855	0.519274	−	0.854608i	$-0.326202\pi$
$419$	21.2585	1.03855	0.519274	+	0.854608i	$0.326202\pi$
$420$	0	0
$421$	−37.8218	−1.84332	−0.921661	−	0.387996i	$-0.873168\pi$
$421$	−37.8218	−1.84332	−0.921661	+	0.387996i	$0.873168\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−8.03550	−0.389779
$426$	0	0
$427$	−31.5844	−1.52848
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.0966	−1.35337	−0.676684	−	0.736274i	$-0.736584\pi$
$431$	−28.0966	−1.35337	−0.676684	+	0.736274i	$0.736584\pi$
$432$	0	0
$433$	−37.2548	−1.79035	−0.895176	−	0.445714i	$-0.852950\pi$
$433$	−37.2548	−1.79035	−0.895176	+	0.445714i	$0.852950\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.24416	0.346535
$438$	0	0
$439$	−12.2381	−0.584092	−0.292046	−	0.956404i	$-0.594336\pi$
$439$	−12.2381	−0.584092	−0.292046	+	0.956404i	$0.594336\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.2116	0.770237	0.385119	−	0.922867i	$-0.374161\pi$
$443$	16.2116	0.770237	0.385119	+	0.922867i	$0.374161\pi$
$444$	0	0
$445$	−0.269558	−0.0127783
$446$	0	0
$447$	0	0
$448$	0	0
$449$	21.6456	1.02152	0.510760	−	0.859724i	$-0.329364\pi$
$449$	21.6456	1.02152	0.510760	+	0.859724i	$0.329364\pi$
$450$	0	0
$451$	−2.66775	−0.125620
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.889270	0.0416896
$456$	0	0
$457$	−8.59278	−0.401954	−0.200977	−	0.979596i	$-0.564412\pi$
$457$	−8.59278	−0.401954	−0.200977	+	0.979596i	$0.564412\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.34430	0.388633	0.194317	−	0.980939i	$-0.437751\pi$
$461$	8.34430	0.388633	0.194317	+	0.980939i	$0.437751\pi$
$462$	0	0
$463$	33.1955	1.54272	0.771362	−	0.636396i	$-0.219576\pi$
$463$	33.1955	1.54272	0.771362	+	0.636396i	$0.219576\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.3072	0.985979	0.492990	−	0.870035i	$-0.335904\pi$
$467$	21.3072	0.985979	0.492990	+	0.870035i	$0.335904\pi$
$468$	0	0
$469$	−57.5522	−2.65751
$470$	0	0
$471$	0	0
$472$	0	0
$473$	55.1750	2.53695
$474$	0	0
$475$	4.99275	0.229083
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.4722	1.62077	0.810384	−	0.585899i	$-0.199258\pi$
$479$	35.4722	1.62077	0.810384	+	0.585899i	$0.199258\pi$
$480$	0	0
$481$	−15.4178	−0.702993
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.990119	−0.0449590
$486$	0	0
$487$	−4.10446	−0.185991	−0.0929954	−	0.995667i	$-0.529644\pi$
$487$	−4.10446	−0.185991	−0.0929954	+	0.995667i	$0.529644\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.5609	−1.19868	−0.599339	−	0.800495i	$-0.704570\pi$
$491$	−26.5609	−1.19868	−0.599339	+	0.800495i	$0.704570\pi$
$492$	0	0
$493$	−13.9752	−0.629413
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−39.7358	−1.78239
$498$	0	0
$499$	11.9735	0.536008	0.268004	−	0.963418i	$-0.413636\pi$
$499$	11.9735	0.536008	0.268004	+	0.963418i	$0.413636\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.0890	−1.29701	−0.648507	−	0.761209i	$-0.724606\pi$
$503$	−29.0890	−1.29701	−0.648507	+	0.761209i	$0.724606\pi$
$504$	0	0
$505$	−1.31811	−0.0586552
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.3780	1.21351	0.606753	−	0.794890i	$-0.292472\pi$
$509$	27.3780	1.21351	0.606753	+	0.794890i	$0.292472\pi$
$510$	0	0
$511$	33.5642	1.48479
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.06180	−0.0467886
$516$	0	0
$517$	−11.1118	−0.488697
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.69574	−0.0742917	−0.0371459	−	0.999310i	$-0.511827\pi$
$521$	−1.69574	−0.0742917	−0.0371459	+	0.999310i	$0.511827\pi$
$522$	0	0
$523$	12.4497	0.544387	0.272193	−	0.962243i	$-0.412251\pi$
$523$	12.4497	0.544387	0.272193	+	0.962243i	$0.412251\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.5532	−0.459703
$528$	0	0
$529$	29.4779	1.28165
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.15186	−0.0498927
$534$	0	0
$535$	0.775982	0.0335486
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−90.7313	−3.90807
$540$	0	0
$541$	−7.66471	−0.329532	−0.164766	−	0.986333i	$-0.552687\pi$
$541$	−7.66471	−0.329532	−0.164766	+	0.986333i	$0.552687\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.339257	−0.0145322
$546$	0	0
$547$	23.8026	1.01773	0.508864	−	0.860847i	$-0.330066\pi$
$547$	23.8026	1.01773	0.508864	+	0.860847i	$0.330066\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.68334	0.369923
$552$	0	0
$553$	2.07993	0.0884475
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.06246	0.383989	0.191994	−	0.981396i	$-0.438504\pi$
$557$	9.06246	0.383989	0.191994	+	0.981396i	$0.438504\pi$
$558$	0	0
$559$	23.8230	1.00761
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.8789	−1.17495	−0.587477	−	0.809241i	$-0.699879\pi$
$563$	−27.8789	−1.17495	−0.587477	+	0.809241i	$0.699879\pi$
$564$	0	0
$565$	0.897988	0.0377786
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.2573	0.471929	0.235964	−	0.971762i	$-0.424175\pi$
$569$	11.2573	0.471929	0.235964	+	0.971762i	$0.424175\pi$
$570$	0	0
$571$	14.4809	0.606007	0.303004	−	0.952989i	$-0.402011\pi$
$571$	14.4809	0.606007	0.303004	+	0.952989i	$0.402011\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	36.1683	1.50832
$576$	0	0
$577$	−25.0761	−1.04393	−0.521966	−	0.852967i	$-0.674801\pi$
$577$	−25.0761	−1.04393	−0.521966	+	0.852967i	$0.674801\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.2954	−0.510099
$582$	0	0
$583$	48.1801	1.99541
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.2078	−0.751515	−0.375757	−	0.926718i	$-0.622617\pi$
$587$	−18.2078	−0.751515	−0.375757	+	0.926718i	$0.622617\pi$
$588$	0	0
$589$	6.55708	0.270180
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.88553	0.405950	0.202975	−	0.979184i	$-0.434939\pi$
$593$	9.88553	0.405950	0.202975	+	0.979184i	$0.434939\pi$
$594$	0	0
$595$	−0.701189	−0.0287460
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.6772	0.803989	0.401994	−	0.915642i	$-0.368317\pi$
$599$	19.6772	0.803989	0.401994	+	0.915642i	$0.368317\pi$
$600$	0	0
$601$	5.62072	0.229274	0.114637	−	0.993407i	$-0.463430\pi$
$601$	5.62072	0.229274	0.114637	+	0.993407i	$0.463430\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.966002	0.0392736
$606$	0	0
$607$	−12.8989	−0.523549	−0.261774	−	0.965129i	$-0.584308\pi$
$607$	−12.8989	−0.523549	−0.261774	+	0.965129i	$0.584308\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.79778	−0.194097
$612$	0	0
$613$	15.9193	0.642973	0.321487	−	0.946914i	$-0.395817\pi$
$613$	15.9193	0.642973	0.321487	+	0.946914i	$0.395817\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.3987	−1.42510	−0.712549	−	0.701622i	$-0.752459\pi$
$617$	−35.3987	−1.42510	−0.712549	+	0.701622i	$0.752459\pi$
$618$	0	0
$619$	21.3608	0.858565	0.429282	−	0.903170i	$-0.358767\pi$
$619$	21.3608	0.858565	0.429282	+	0.903170i	$0.358767\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	16.2059	0.649275
$624$	0	0
$625$	24.8914	0.995654
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.1570	0.484730
$630$	0	0
$631$	−34.2860	−1.36490	−0.682452	−	0.730931i	$-0.739086\pi$
$631$	−34.2860	−1.36490	−0.682452	+	0.730931i	$0.739086\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.0722314	0.00286641
$636$	0	0
$637$	−39.1753	−1.55218
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.10668	0.280697	0.140349	−	0.990102i	$-0.455178\pi$
$641$	7.10668	0.280697	0.140349	+	0.990102i	$0.455178\pi$
$642$	0	0
$643$	−45.8917	−1.80979	−0.904897	−	0.425632i	$-0.860052\pi$
$643$	−45.8917	−1.80979	−0.904897	+	0.425632i	$0.860052\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.8067	−0.739366	−0.369683	−	0.929158i	$-0.620534\pi$
$647$	−18.8067	−0.739366	−0.369683	+	0.929158i	$0.620534\pi$
$648$	0	0
$649$	−29.1907	−1.14583
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.1271	−0.865901	−0.432951	−	0.901418i	$-0.642528\pi$
$653$	−22.1271	−0.865901	−0.432951	+	0.901418i	$0.642528\pi$
$654$	0	0
$655$	−0.439102	−0.0171571
$656$	0	0
$657$	0	0
$658$	0	0
$659$	35.8466	1.39638	0.698192	−	0.715910i	$-0.253988\pi$
$659$	35.8466	1.39638	0.698192	+	0.715910i	$0.253988\pi$
$660$	0	0
$661$	−7.50730	−0.292000	−0.146000	−	0.989285i	$-0.546640\pi$
$661$	−7.50730	−0.292000	−0.146000	+	0.989285i	$0.546640\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.435675	0.0168948
$666$	0	0
$667$	62.9035	2.43563
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−29.1741	−1.12625
$672$	0	0
$673$	39.9266	1.53906	0.769529	−	0.638612i	$-0.220491\pi$
$673$	39.9266	1.53906	0.769529	+	0.638612i	$0.220491\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.7975	−0.607148	−0.303574	−	0.952808i	$-0.598180\pi$
$677$	−15.7975	−0.607148	−0.303574	+	0.952808i	$0.598180\pi$
$678$	0	0
$679$	59.5262	2.28441
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.9134	0.647174	0.323587	−	0.946198i	$-0.395111\pi$
$683$	16.9134	0.647174	0.323587	+	0.946198i	$0.395111\pi$
$684$	0	0
$685$	0.982017	0.0375210
$686$	0	0
$687$	0	0
$688$	0	0
$689$	20.8028	0.792524
$690$	0	0
$691$	−30.4001	−1.15647	−0.578236	−	0.815869i	$-0.696259\pi$
$691$	−30.4001	−1.15647	−0.578236	+	0.815869i	$0.696259\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.0710476	−0.00269499
$696$	0	0
$697$	0.908242	0.0344021
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.1930	−0.724911	−0.362456	−	0.932001i	$-0.618062\pi$
$701$	−19.1930	−0.724911	−0.362456	+	0.932001i	$0.618062\pi$
$702$	0	0
$703$	−7.55358	−0.284889
$704$	0	0
$705$	0	0
$706$	0	0
$707$	79.2453	2.98033
$708$	0	0
$709$	39.7184	1.49166	0.745829	−	0.666138i	$-0.232054\pi$
$709$	39.7184	1.49166	0.745829	+	0.666138i	$0.232054\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	47.5005	1.77891
$714$	0	0
$715$	0.821406	0.0307189
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.3182	0.422098	0.211049	−	0.977476i	$-0.432312\pi$
$719$	11.3182	0.422098	0.211049	+	0.977476i	$0.432312\pi$
$720$	0	0
$721$	63.8359	2.37737
$722$	0	0
$723$	0	0
$724$	0	0
$725$	43.3538	1.61012
$726$	0	0
$727$	−27.8190	−1.03175	−0.515876	−	0.856663i	$-0.672533\pi$
$727$	−27.8190	−1.03175	−0.515876	+	0.856663i	$0.672533\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−18.7844	−0.694768
$732$	0	0
$733$	−24.5066	−0.905172	−0.452586	−	0.891721i	$-0.649498\pi$
$733$	−24.5066	−0.905172	−0.452586	+	0.891721i	$0.649498\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−53.1601	−1.95818
$738$	0	0
$739$	0.634560	0.0233427	0.0116713	−	0.999932i	$-0.496285\pi$
$739$	0.634560	0.0233427	0.0116713	+	0.999932i	$0.496285\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.22678	0.118379	0.0591895	−	0.998247i	$-0.481148\pi$
$743$	3.22678	0.118379	0.0591895	+	0.998247i	$0.481148\pi$
$744$	0	0
$745$	−1.64773	−0.0603680
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−46.6523	−1.70464
$750$	0	0
$751$	38.5227	1.40571	0.702857	−	0.711331i	$-0.251907\pi$
$751$	38.5227	1.40571	0.702857	+	0.711331i	$0.251907\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.299921	−0.0109153
$756$	0	0
$757$	−21.5627	−0.783709	−0.391855	−	0.920027i	$-0.628166\pi$
$757$	−21.5627	−0.783709	−0.391855	+	0.920027i	$0.628166\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−50.8712	−1.84408	−0.922040	−	0.387096i	$-0.873478\pi$
$761$	−50.8712	−1.84408	−0.922040	+	0.387096i	$0.873478\pi$
$762$	0	0
$763$	20.3963	0.738394
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.6037	−0.455094
$768$	0	0
$769$	−22.2344	−0.801795	−0.400897	−	0.916123i	$-0.631302\pi$
$769$	−22.2344	−0.801795	−0.400897	+	0.916123i	$0.631302\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.5774	1.17173	0.585864	−	0.810409i	$-0.300755\pi$
$773$	32.5774	1.17173	0.585864	+	0.810409i	$0.300755\pi$
$774$	0	0
$775$	32.7379	1.17598
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.564325	−0.0202190
$780$	0	0
$781$	−36.7034	−1.31335
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.831099	0.0296632
$786$	0	0
$787$	23.1605	0.825584	0.412792	−	0.910825i	$-0.364554\pi$
$787$	23.1605	0.825584	0.412792	+	0.910825i	$0.364554\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−53.9873	−1.91957
$792$	0	0
$793$	−12.5966	−0.447317
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.42143	0.227459	0.113729	−	0.993512i	$-0.463720\pi$
$797$	6.42143	0.227459	0.113729	+	0.993512i	$0.463720\pi$
$798$	0	0
$799$	3.78305	0.133835
$800$	0	0
$801$	0	0
$802$	0	0
$803$	31.0027	1.09406
$804$	0	0
$805$	3.15610	0.111238
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−47.6346	−1.67474	−0.837371	−	0.546635i	$-0.815909\pi$
$809$	−47.6346	−1.67474	−0.837371	+	0.546635i	$0.815909\pi$
$810$	0	0
$811$	15.0006	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$811$	15.0006	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.13057	−0.0396022
$816$	0	0
$817$	11.6715	0.408333
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.4482	0.923048	0.461524	−	0.887128i	$-0.347303\pi$
$821$	26.4482	0.923048	0.461524	+	0.887128i	$0.347303\pi$
$822$	0	0
$823$	24.4311	0.851613	0.425807	−	0.904814i	$-0.359990\pi$
$823$	24.4311	0.851613	0.425807	+	0.904814i	$0.359990\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21.8517	0.759857	0.379928	−	0.925016i	$-0.375949\pi$
$827$	21.8517	0.759857	0.379928	+	0.925016i	$0.375949\pi$
$828$	0	0
$829$	−26.6102	−0.924210	−0.462105	−	0.886825i	$-0.652906\pi$
$829$	−26.6102	−0.924210	−0.462105	+	0.886825i	$0.652906\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	30.8897	1.07026
$834$	0	0
$835$	−1.27762	−0.0442138
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.1380	0.350001	0.175001	−	0.984568i	$-0.444007\pi$
$839$	10.1380	0.350001	0.175001	+	0.984568i	$0.444007\pi$
$840$	0	0
$841$	46.4003	1.60001
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.751999	−0.0258696
$846$	0	0
$847$	−58.0763	−1.99553
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−54.7193	−1.87575
$852$	0	0
$853$	13.5634	0.464403	0.232202	−	0.972668i	$-0.425407\pi$
$853$	13.5634	0.464403	0.232202	+	0.972668i	$0.425407\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.0025	−0.375839	−0.187919	−	0.982184i	$-0.560174\pi$
$857$	−11.0025	−0.375839	−0.187919	+	0.982184i	$0.560174\pi$
$858$	0	0
$859$	27.7278	0.946059	0.473030	−	0.881046i	$-0.343160\pi$
$859$	27.7278	0.946059	0.473030	+	0.881046i	$0.343160\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.8775	1.01704	0.508521	−	0.861050i	$-0.330192\pi$
$863$	29.8775	1.01704	0.508521	+	0.861050i	$0.330192\pi$
$864$	0	0
$865$	0.518285	0.0176222
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.92120	0.0651723
$870$	0	0
$871$	−22.9531	−0.777735
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.35359	0.147178
$876$	0	0
$877$	−0.325882	−0.0110042	−0.00550212	−	0.999985i	$-0.501751\pi$
$877$	−0.325882	−0.0110042	−0.00550212	+	0.999985i	$0.501751\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−15.5697	−0.524558	−0.262279	−	0.964992i	$-0.584474\pi$
$881$	−15.5697	−0.524558	−0.262279	+	0.964992i	$0.584474\pi$
$882$	0	0
$883$	−23.6825	−0.796979	−0.398489	−	0.917173i	$-0.630465\pi$
$883$	−23.6825	−0.796979	−0.398489	+	0.917173i	$0.630465\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.8677	−1.57366	−0.786831	−	0.617169i	$-0.788280\pi$
$887$	−46.8677	−1.57366	−0.786831	+	0.617169i	$0.788280\pi$
$888$	0	0
$889$	−4.34257	−0.145645
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.35055	−0.0786581
$894$	0	0
$895$	−0.188691	−0.00630725
$896$	0	0
$897$	0	0
$898$	0	0
$899$	56.9373	1.89897
$900$	0	0
$901$	−16.4030	−0.546464
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.76197	0.0585698
$906$	0	0
$907$	−4.30477	−0.142937	−0.0714687	−	0.997443i	$-0.522769\pi$
$907$	−4.30477	−0.142937	−0.0714687	+	0.997443i	$0.522769\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.0623	−0.830350	−0.415175	−	0.909741i	$-0.636280\pi$
$911$	−25.0623	−0.830350	−0.415175	+	0.909741i	$0.636280\pi$
$912$	0	0
$913$	−11.3571	−0.375864
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.3989	0.871769
$918$	0	0
$919$	−35.1544	−1.15964	−0.579818	−	0.814746i	$-0.696876\pi$
$919$	−35.1544	−1.15964	−0.579818	+	0.814746i	$0.696876\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−15.8475	−0.521627
$924$	0	0
$925$	−37.7131	−1.24000
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.8505	−0.946555	−0.473277	−	0.880913i	$-0.656929\pi$
$929$	−28.8505	−0.946555	−0.473277	+	0.880913i	$0.656929\pi$
$930$	0	0
$931$	−19.1929	−0.629022
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.647679	−0.0211814
$936$	0	0
$937$	−14.0488	−0.458953	−0.229477	−	0.973314i	$-0.573701\pi$
$937$	−14.0488	−0.458953	−0.229477	+	0.973314i	$0.573701\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	35.6039	1.16065	0.580327	−	0.814384i	$-0.302925\pi$
$941$	35.6039	1.16065	0.580327	+	0.814384i	$0.302925\pi$
$942$	0	0
$943$	−4.08806	−0.133126
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.7225	1.35580	0.677899	−	0.735155i	$-0.262891\pi$
$947$	41.7225	1.35580	0.677899	+	0.735155i	$0.262891\pi$
$948$	0	0
$949$	13.3861	0.434532
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.9744	−0.485067	−0.242534	−	0.970143i	$-0.577978\pi$
$953$	−14.9744	−0.485067	−0.242534	+	0.970143i	$0.577978\pi$
$954$	0	0
$955$	2.28823	0.0740454
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−59.0392	−1.90647
$960$	0	0
$961$	11.9953	0.386944
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.41779	−0.0456402
$966$	0	0
$967$	11.3537	0.365111	0.182555	−	0.983196i	$-0.441563\pi$
$967$	11.3537	0.365111	0.182555	+	0.983196i	$0.441563\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−26.1383	−0.838819	−0.419410	−	0.907797i	$-0.637763\pi$
$971$	−26.1383	−0.838819	−0.419410	+	0.907797i	$0.637763\pi$
$972$	0	0
$973$	4.27140	0.136935
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.8859	0.508236	0.254118	−	0.967173i	$-0.418215\pi$
$977$	15.8859	0.508236	0.254118	+	0.967173i	$0.418215\pi$
$978$	0	0
$979$	14.9691	0.478416
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.9904	−0.701383	−0.350692	−	0.936491i	$-0.614053\pi$
$983$	−21.9904	−0.701383	−0.350692	+	0.936491i	$0.614053\pi$
$984$	0	0
$985$	0.362251	0.0115423
$986$	0	0
$987$	0	0
$988$	0	0
$989$	84.5501	2.68854
$990$	0	0
$991$	−0.568262	−0.0180514	−0.00902572	−	0.999959i	$-0.502873\pi$
$991$	−0.568262	−0.0180514	−0.00902572	+	0.999959i	$0.502873\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.126354	0.00400568
$996$	0	0
$997$	−10.3329	−0.327245	−0.163623	−	0.986523i	$-0.552318\pi$
$997$	−10.3329	−0.327245	−0.163623	+	0.986523i	$0.552318\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.cf.1.4		6
3.2	odd	2		8208.2.a.ce.1.3		6
4.3	odd	2		4104.2.a.t.1.4	yes	6
12.11	even	2		4104.2.a.s.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.s.1.3	✓	6	12.11	even	2
4104.2.a.t.1.4	yes	6	4.3	odd	2
8208.2.a.ce.1.3		6	3.2	odd	2
8208.2.a.cf.1.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0851277 q^{5} -5.11790 q^{7} -4.72733 q^{11} -2.04113 q^{13} +1.60943 q^{17} -1.00000 q^{19} -7.24416 q^{23} -4.99275 q^{25} -8.68334 q^{29} -6.55708 q^{31} -0.435675 q^{35} +7.55358 q^{37} +0.564325 q^{41} -11.6715 q^{43} +2.35055 q^{47} +19.1929 q^{49} -10.1918 q^{53} -0.402427 q^{55} +6.17487 q^{59} +6.17137 q^{61} -0.173757 q^{65} +11.2453 q^{67} +7.76408 q^{71} -6.55819 q^{73} +24.1940 q^{77} -0.406402 q^{79} +2.40243 q^{83} +0.137007 q^{85} -3.16651 q^{89} +10.4463 q^{91} -0.0851277 q^{95} -11.6310 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 2 q^{7} + 3 q^{11} + q^{13} + 7 q^{17} - 6 q^{19} - 3 q^{23} + 10 q^{25} - 6 q^{29} - 5 q^{31} + 3 q^{35} + 11 q^{37} + 9 q^{41} - 7 q^{43} + 7 q^{47} + 20 q^{49} + 11 q^{53} - 22 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) \) 6 * q + 2 * q^5 - 2 * q^7 + 3 * q^11 + q^13 + 7 * q^17 - 6 * q^19 - 3 * q^23 + 10 * q^25 - 6 * q^29 - 5 * q^31 + 3 * q^35 + 11 * q^37 + 9 * q^41 - 7 * q^43 + 7 * q^47 + 20 * q^49 + 11 * q^53 - 22 * q^55 + 17 * q^59 + 17 * q^61 - 4 * q^65 + 4 * q^67 - 10 * q^71 + 18 * q^73 + 27 * q^77 - 27 * q^79 + 34 * q^83 + 29 * q^85 + 18 * q^89 - 17 * q^91 - 2 * q^95 + 8 * q^97

Introduction

L-functions

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