LMFDB - Embedded newform 6728.2.a.be.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6728,2,Mod(1,6728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6728 = 2^{3} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6728.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$53.7233504799$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	6728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21654 q^{3} -2.96314 q^{5} +0.724653 q^{7} -1.52003 q^{9} +O(q^{10})$	$q-1.21654 q^{3} -2.96314 q^{5} +0.724653 q^{7} -1.52003 q^{9} -2.82000 q^{11} +3.52031 q^{13} +3.60477 q^{15} +4.89842 q^{17} -3.90786 q^{19} -0.881568 q^{21} -2.05228 q^{23} +3.78019 q^{25} +5.49880 q^{27} +1.44812 q^{31} +3.43064 q^{33} -2.14725 q^{35} -4.42604 q^{37} -4.28259 q^{39} +6.63260 q^{41} -12.2437 q^{43} +4.50407 q^{45} +3.82308 q^{47} -6.47488 q^{49} -5.95912 q^{51} +3.54901 q^{53} +8.35604 q^{55} +4.75406 q^{57} +11.0729 q^{59} +2.42780 q^{61} -1.10150 q^{63} -10.4312 q^{65} +10.9099 q^{67} +2.49668 q^{69} +5.73474 q^{71} +16.0173 q^{73} -4.59875 q^{75} -2.04352 q^{77} -6.52280 q^{79} -2.12940 q^{81} +10.1433 q^{83} -14.5147 q^{85} -6.50429 q^{89} +2.55100 q^{91} -1.76169 q^{93} +11.5795 q^{95} +2.96320 q^{97} +4.28648 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9}+O(q^{10}) $ 24 * q - 4 * q^3 + 8 * q^5 - 2 * q^7 + 32 * q^9	$ 24 q - 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9} - 6 q^{11} + 2 q^{13} - 32 q^{15} - 20 q^{17} - 44 q^{19} - 4 q^{21} + 2 q^{23} + 32 q^{25} - 34 q^{27} - 10 q^{31} - 12 q^{33} - 32 q^{35} - 40 q^{37} - 18 q^{39} - 8 q^{41} - 30 q^{43} + 16 q^{45} - 34 q^{47} + 26 q^{49} - 28 q^{51} + 18 q^{53} - 50 q^{55} + 6 q^{57} - 64 q^{59} - 6 q^{61} - 28 q^{63} + 14 q^{65} - 40 q^{67} - 38 q^{69} - 52 q^{73} - 66 q^{75} - 90 q^{77} - 4 q^{79} + 40 q^{81} - 44 q^{83} - 40 q^{85} + 40 q^{89} - 28 q^{91} - 10 q^{93} - 84 q^{97} - 60 q^{99}+O(q^{100}) $ 24 * q - 4 * q^3 + 8 * q^5 - 2 * q^7 + 32 * q^9 - 6 * q^11 + 2 * q^13 - 32 * q^15 - 20 * q^17 - 44 * q^19 - 4 * q^21 + 2 * q^23 + 32 * q^25 - 34 * q^27 - 10 * q^31 - 12 * q^33 - 32 * q^35 - 40 * q^37 - 18 * q^39 - 8 * q^41 - 30 * q^43 + 16 * q^45 - 34 * q^47 + 26 * q^49 - 28 * q^51 + 18 * q^53 - 50 * q^55 + 6 * q^57 - 64 * q^59 - 6 * q^61 - 28 * q^63 + 14 * q^65 - 40 * q^67 - 38 * q^69 - 52 * q^73 - 66 * q^75 - 90 * q^77 - 4 * q^79 + 40 * q^81 - 44 * q^83 - 40 * q^85 + 40 * q^89 - 28 * q^91 - 10 * q^93 - 84 * q^97 - 60 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.21654	−0.702369	−0.351185	−	0.936306i	$-0.614221\pi$
$3$	−1.21654	−0.702369	−0.351185	+	0.936306i	$0.614221\pi$
$4$	0	0
$5$	−2.96314	−1.32516	−0.662578	−	0.748993i	$-0.730538\pi$
$5$	−2.96314	−1.32516	−0.662578	+	0.748993i	$0.730538\pi$
$6$	0	0
$7$	0.724653	0.273893	0.136946	−	0.990578i	$-0.456271\pi$
$7$	0.724653	0.273893	0.136946	+	0.990578i	$0.456271\pi$
$8$	0	0
$9$	−1.52003	−0.506677
$10$	0	0
$11$	−2.82000	−0.850261	−0.425130	−	0.905132i	$-0.639772\pi$
$11$	−2.82000	−0.850261	−0.425130	+	0.905132i	$0.639772\pi$
$12$	0	0
$13$	3.52031	0.976358	0.488179	−	0.872744i	$-0.337661\pi$
$13$	3.52031	0.976358	0.488179	+	0.872744i	$0.337661\pi$
$14$	0	0
$15$	3.60477	0.930749
$16$	0	0
$17$	4.89842	1.18804	0.594020	−	0.804450i	$-0.297540\pi$
$17$	4.89842	1.18804	0.594020	+	0.804450i	$0.297540\pi$
$18$	0	0
$19$	−3.90786	−0.896524	−0.448262	−	0.893902i	$-0.647957\pi$
$19$	−3.90786	−0.896524	−0.448262	+	0.893902i	$0.647957\pi$
$20$	0	0
$21$	−0.881568	−0.192374
$22$	0	0
$23$	−2.05228	−0.427930	−0.213965	−	0.976841i	$-0.568638\pi$
$23$	−2.05228	−0.427930	−0.213965	+	0.976841i	$0.568638\pi$
$24$	0	0
$25$	3.78019	0.756038
$26$	0	0
$27$	5.49880	1.05824
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	1.44812	0.260090	0.130045	−	0.991508i	$-0.458488\pi$
$31$	1.44812	0.260090	0.130045	+	0.991508i	$0.458488\pi$
$32$	0	0
$33$	3.43064	0.597197
$34$	0	0
$35$	−2.14725	−0.362951
$36$	0	0
$37$	−4.42604	−0.727636	−0.363818	−	0.931470i	$-0.618527\pi$
$37$	−4.42604	−0.727636	−0.363818	+	0.931470i	$0.618527\pi$
$38$	0	0
$39$	−4.28259	−0.685764
$40$	0	0
$41$	6.63260	1.03584	0.517919	−	0.855429i	$-0.326707\pi$
$41$	6.63260	1.03584	0.517919	+	0.855429i	$0.326707\pi$
$42$	0	0
$43$	−12.2437	−1.86715	−0.933573	−	0.358387i	$-0.883327\pi$
$43$	−12.2437	−1.86715	−0.933573	+	0.358387i	$0.883327\pi$
$44$	0	0
$45$	4.50407	0.671427
$46$	0	0
$47$	3.82308	0.557654	0.278827	−	0.960341i	$-0.410054\pi$
$47$	3.82308	0.557654	0.278827	+	0.960341i	$0.410054\pi$
$48$	0	0
$49$	−6.47488	−0.924983
$50$	0	0
$51$	−5.95912	−0.834443
$52$	0	0
$53$	3.54901	0.487494	0.243747	−	0.969839i	$-0.421623\pi$
$53$	3.54901	0.487494	0.243747	+	0.969839i	$0.421623\pi$
$54$	0	0
$55$	8.35604	1.12673
$56$	0	0
$57$	4.75406	0.629691
$58$	0	0
$59$	11.0729	1.44157	0.720784	−	0.693160i	$-0.243782\pi$
$59$	11.0729	1.44157	0.720784	+	0.693160i	$0.243782\pi$
$60$	0	0
$61$	2.42780	0.310848	0.155424	−	0.987848i	$-0.450326\pi$
$61$	2.42780	0.310848	0.155424	+	0.987848i	$0.450326\pi$
$62$	0	0
$63$	−1.10150	−0.138775
$64$	0	0
$65$	−10.4312	−1.29383
$66$	0	0
$67$	10.9099	1.33285	0.666427	−	0.745570i	$-0.267823\pi$
$67$	10.9099	1.33285	0.666427	+	0.745570i	$0.267823\pi$
$68$	0	0
$69$	2.49668	0.300565
$70$	0	0
$71$	5.73474	0.680588	0.340294	−	0.940319i	$-0.389473\pi$
$71$	5.73474	0.680588	0.340294	+	0.940319i	$0.389473\pi$
$72$	0	0
$73$	16.0173	1.87469	0.937344	−	0.348406i	$-0.113277\pi$
$73$	16.0173	1.87469	0.937344	+	0.348406i	$0.113277\pi$
$74$	0	0
$75$	−4.59875	−0.531018
$76$	0	0
$77$	−2.04352	−0.232880
$78$	0	0
$79$	−6.52280	−0.733872	−0.366936	−	0.930246i	$-0.619593\pi$
$79$	−6.52280	−0.733872	−0.366936	+	0.930246i	$0.619593\pi$
$80$	0	0
$81$	−2.12940	−0.236600
$82$	0	0
$83$	10.1433	1.11337	0.556687	−	0.830723i	$-0.312072\pi$
$83$	10.1433	1.11337	0.556687	+	0.830723i	$0.312072\pi$
$84$	0	0
$85$	−14.5147	−1.57434
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.50429	−0.689453	−0.344726	−	0.938703i	$-0.612028\pi$
$89$	−6.50429	−0.689453	−0.344726	+	0.938703i	$0.612028\pi$
$90$	0	0
$91$	2.55100	0.267418
$92$	0	0
$93$	−1.76169	−0.182679
$94$	0	0
$95$	11.5795	1.18803
$96$	0	0
$97$	2.96320	0.300868	0.150434	−	0.988620i	$-0.451933\pi$
$97$	2.96320	0.300868	0.150434	+	0.988620i	$0.451933\pi$
$98$	0	0
$99$	4.28648	0.430808
$100$	0	0
$101$	−4.42041	−0.439847	−0.219924	−	0.975517i	$-0.570581\pi$
$101$	−4.42041	−0.439847	−0.219924	+	0.975517i	$0.570581\pi$
$102$	0	0
$103$	15.1074	1.48858	0.744288	−	0.667859i	$-0.232789\pi$
$103$	15.1074	1.48858	0.744288	+	0.667859i	$0.232789\pi$
$104$	0	0
$105$	2.61221	0.254926
$106$	0	0
$107$	−15.0111	−1.45117	−0.725587	−	0.688131i	$-0.758432\pi$
$107$	−15.0111	−1.45117	−0.725587	+	0.688131i	$0.758432\pi$
$108$	0	0
$109$	−0.172046	−0.0164791	−0.00823953	−	0.999966i	$-0.502623\pi$
$109$	−0.172046	−0.0164791	−0.00823953	+	0.999966i	$0.502623\pi$
$110$	0	0
$111$	5.38445	0.511069
$112$	0	0
$113$	11.0697	1.04135	0.520675	−	0.853755i	$-0.325680\pi$
$113$	11.0697	1.04135	0.520675	+	0.853755i	$0.325680\pi$
$114$	0	0
$115$	6.08118	0.567073
$116$	0	0
$117$	−5.35098	−0.494699
$118$	0	0
$119$	3.54965	0.325396
$120$	0	0
$121$	−3.04763	−0.277057
$122$	0	0
$123$	−8.06882	−0.727541
$124$	0	0
$125$	3.61446	0.323288
$126$	0	0
$127$	−13.3365	−1.18342	−0.591712	−	0.806149i	$-0.701548\pi$
$127$	−13.3365	−1.18342	−0.591712	+	0.806149i	$0.701548\pi$
$128$	0	0
$129$	14.8949	1.31143
$130$	0	0
$131$	−17.5929	−1.53710	−0.768550	−	0.639790i	$-0.779021\pi$
$131$	−17.5929	−1.53710	−0.768550	+	0.639790i	$0.779021\pi$
$132$	0	0
$133$	−2.83184	−0.245552
$134$	0	0
$135$	−16.2937	−1.40234
$136$	0	0
$137$	6.27142	0.535804	0.267902	−	0.963446i	$-0.413670\pi$
$137$	6.27142	0.535804	0.267902	+	0.963446i	$0.413670\pi$
$138$	0	0
$139$	9.18488	0.779052	0.389526	−	0.921016i	$-0.372639\pi$
$139$	9.18488	0.779052	0.389526	+	0.921016i	$0.372639\pi$
$140$	0	0
$141$	−4.65093	−0.391679
$142$	0	0
$143$	−9.92725	−0.830159
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.87694	0.649679
$148$	0	0
$149$	−23.7176	−1.94302	−0.971512	−	0.236992i	$-0.923838\pi$
$149$	−23.7176	−1.94302	−0.971512	+	0.236992i	$0.923838\pi$
$150$	0	0
$151$	8.11225	0.660166	0.330083	−	0.943952i	$-0.392923\pi$
$151$	8.11225	0.660166	0.330083	+	0.943952i	$0.392923\pi$
$152$	0	0
$153$	−7.44575	−0.601953
$154$	0	0
$155$	−4.29098	−0.344659
$156$	0	0
$157$	−1.65786	−0.132312	−0.0661559	−	0.997809i	$-0.521073\pi$
$157$	−1.65786	−0.132312	−0.0661559	+	0.997809i	$0.521073\pi$
$158$	0	0
$159$	−4.31751	−0.342401
$160$	0	0
$161$	−1.48719	−0.117207
$162$	0	0
$163$	−1.20481	−0.0943678	−0.0471839	−	0.998886i	$-0.515025\pi$
$163$	−1.20481	−0.0943678	−0.0471839	+	0.998886i	$0.515025\pi$
$164$	0	0
$165$	−10.1654	−0.791379
$166$	0	0
$167$	−17.5096	−1.35493	−0.677465	−	0.735555i	$-0.736922\pi$
$167$	−17.5096	−1.35493	−0.677465	+	0.735555i	$0.736922\pi$
$168$	0	0
$169$	−0.607427	−0.0467252
$170$	0	0
$171$	5.94007	0.454248
$172$	0	0
$173$	23.7453	1.80532	0.902661	−	0.430351i	$-0.141610\pi$
$173$	23.7453	1.80532	0.902661	+	0.430351i	$0.141610\pi$
$174$	0	0
$175$	2.73933	0.207074
$176$	0	0
$177$	−13.4706	−1.01251
$178$	0	0
$179$	−23.3002	−1.74154	−0.870769	−	0.491693i	$-0.836378\pi$
$179$	−23.3002	−1.74154	−0.870769	+	0.491693i	$0.836378\pi$
$180$	0	0
$181$	−4.14992	−0.308461	−0.154230	−	0.988035i	$-0.549290\pi$
$181$	−4.14992	−0.308461	−0.154230	+	0.988035i	$0.549290\pi$
$182$	0	0
$183$	−2.95352	−0.218330
$184$	0	0
$185$	13.1150	0.964231
$186$	0	0
$187$	−13.8135	−1.01014
$188$	0	0
$189$	3.98472	0.289846
$190$	0	0
$191$	−15.2057	−1.10025	−0.550123	−	0.835084i	$-0.685419\pi$
$191$	−15.2057	−1.10025	−0.550123	+	0.835084i	$0.685419\pi$
$192$	0	0
$193$	−18.6492	−1.34240	−0.671198	−	0.741278i	$-0.734220\pi$
$193$	−18.6492	−1.34240	−0.671198	+	0.741278i	$0.734220\pi$
$194$	0	0
$195$	12.6899	0.908744
$196$	0	0
$197$	6.95774	0.495719	0.247859	−	0.968796i	$-0.420273\pi$
$197$	6.95774	0.495719	0.247859	+	0.968796i	$0.420273\pi$
$198$	0	0
$199$	−2.79995	−0.198483	−0.0992416	−	0.995063i	$-0.531642\pi$
$199$	−2.79995	−0.198483	−0.0992416	+	0.995063i	$0.531642\pi$
$200$	0	0
$201$	−13.2723	−0.936155
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−19.6533	−1.37265
$206$	0	0
$207$	3.11953	0.216822
$208$	0	0
$209$	11.0201	0.762279
$210$	0	0
$211$	−25.5415	−1.75835	−0.879174	−	0.476501i	$-0.841905\pi$
$211$	−25.5415	−1.75835	−0.879174	+	0.476501i	$0.841905\pi$
$212$	0	0
$213$	−6.97654	−0.478024
$214$	0	0
$215$	36.2798	2.47426
$216$	0	0
$217$	1.04938	0.0712368
$218$	0	0
$219$	−19.4857	−1.31672
$220$	0	0
$221$	17.2439	1.15995
$222$	0	0
$223$	7.65748	0.512783	0.256392	−	0.966573i	$-0.417466\pi$
$223$	7.65748	0.512783	0.256392	+	0.966573i	$0.417466\pi$
$224$	0	0
$225$	−5.74601	−0.383067
$226$	0	0
$227$	3.33670	0.221465	0.110732	−	0.993850i	$-0.464680\pi$
$227$	3.33670	0.221465	0.110732	+	0.993850i	$0.464680\pi$
$228$	0	0
$229$	−26.4170	−1.74568	−0.872842	−	0.488002i	$-0.837726\pi$
$229$	−26.4170	−1.74568	−0.872842	+	0.488002i	$0.837726\pi$
$230$	0	0
$231$	2.48602	0.163568
$232$	0	0
$233$	11.1059	0.727574	0.363787	−	0.931482i	$-0.381484\pi$
$233$	11.1059	0.727574	0.363787	+	0.931482i	$0.381484\pi$
$234$	0	0
$235$	−11.3283	−0.738978
$236$	0	0
$237$	7.93524	0.515449
$238$	0	0
$239$	23.3185	1.50835	0.754174	−	0.656674i	$-0.228037\pi$
$239$	23.3185	1.50835	0.754174	+	0.656674i	$0.228037\pi$
$240$	0	0
$241$	1.83411	0.118145	0.0590727	−	0.998254i	$-0.481186\pi$
$241$	1.83411	0.118145	0.0590727	+	0.998254i	$0.481186\pi$
$242$	0	0
$243$	−13.9059	−0.892063
$244$	0	0
$245$	19.1860	1.22575
$246$	0	0
$247$	−13.7569	−0.875328
$248$	0	0
$249$	−12.3397	−0.781999
$250$	0	0
$251$	9.57961	0.604660	0.302330	−	0.953203i	$-0.402236\pi$
$251$	9.57961	0.604660	0.302330	+	0.953203i	$0.402236\pi$
$252$	0	0
$253$	5.78741	0.363852
$254$	0	0
$255$	17.6577	1.10577
$256$	0	0
$257$	3.62524	0.226136	0.113068	−	0.993587i	$-0.463932\pi$
$257$	3.62524	0.226136	0.113068	+	0.993587i	$0.463932\pi$
$258$	0	0
$259$	−3.20734	−0.199294
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0765	0.744669	0.372334	−	0.928099i	$-0.378557\pi$
$263$	12.0765	0.744669	0.372334	+	0.928099i	$0.378557\pi$
$264$	0	0
$265$	−10.5162	−0.646006
$266$	0	0
$267$	7.91272	0.484250
$268$	0	0
$269$	−19.7000	−1.20113	−0.600565	−	0.799576i	$-0.705058\pi$
$269$	−19.7000	−1.20113	−0.600565	+	0.799576i	$0.705058\pi$
$270$	0	0
$271$	−31.7045	−1.92591	−0.962956	−	0.269660i	$-0.913089\pi$
$271$	−31.7045	−1.92591	−0.962956	+	0.269660i	$0.913089\pi$
$272$	0	0
$273$	−3.10339	−0.187826
$274$	0	0
$275$	−10.6601	−0.642829
$276$	0	0
$277$	−19.6610	−1.18132	−0.590659	−	0.806922i	$-0.701132\pi$
$277$	−19.6610	−1.18132	−0.590659	+	0.806922i	$0.701132\pi$
$278$	0	0
$279$	−2.20119	−0.131782
$280$	0	0
$281$	7.09124	0.423028	0.211514	−	0.977375i	$-0.432161\pi$
$281$	7.09124	0.423028	0.211514	+	0.977375i	$0.432161\pi$
$282$	0	0
$283$	−3.61343	−0.214796	−0.107398	−	0.994216i	$-0.534252\pi$
$283$	−3.61343	−0.214796	−0.107398	+	0.994216i	$0.534252\pi$
$284$	0	0
$285$	−14.0869	−0.834438
$286$	0	0
$287$	4.80633	0.283709
$288$	0	0
$289$	6.99449	0.411441
$290$	0	0
$291$	−3.60485	−0.211320
$292$	0	0
$293$	21.6077	1.26234	0.631169	−	0.775646i	$-0.282576\pi$
$293$	21.6077	1.26234	0.631169	+	0.775646i	$0.282576\pi$
$294$	0	0
$295$	−32.8105	−1.91030
$296$	0	0
$297$	−15.5066	−0.899783
$298$	0	0
$299$	−7.22465	−0.417812
$300$	0	0
$301$	−8.87243	−0.511398
$302$	0	0
$303$	5.37760	0.308935
$304$	0	0
$305$	−7.19391	−0.411922
$306$	0	0
$307$	−22.3962	−1.27822	−0.639109	−	0.769116i	$-0.720697\pi$
$307$	−22.3962	−1.27822	−0.639109	+	0.769116i	$0.720697\pi$
$308$	0	0
$309$	−18.3787	−1.04553
$310$	0	0
$311$	−23.1481	−1.31261	−0.656305	−	0.754495i	$-0.727882\pi$
$311$	−23.1481	−1.31261	−0.656305	+	0.754495i	$0.727882\pi$
$312$	0	0
$313$	16.2377	0.917811	0.458906	−	0.888485i	$-0.348242\pi$
$313$	16.2377	0.917811	0.458906	+	0.888485i	$0.348242\pi$
$314$	0	0
$315$	3.26388	0.183899
$316$	0	0
$317$	−25.5464	−1.43483	−0.717415	−	0.696646i	$-0.754675\pi$
$317$	−25.5464	−1.43483	−0.717415	+	0.696646i	$0.754675\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	18.2615	1.01926
$322$	0	0
$323$	−19.1423	−1.06511
$324$	0	0
$325$	13.3074	0.738164
$326$	0	0
$327$	0.209301	0.0115744
$328$	0	0
$329$	2.77041	0.152737
$330$	0	0
$331$	−10.1111	−0.555755	−0.277877	−	0.960617i	$-0.589631\pi$
$331$	−10.1111	−0.555755	−0.277877	+	0.960617i	$0.589631\pi$
$332$	0	0
$333$	6.72772	0.368677
$334$	0	0
$335$	−32.3275	−1.76624
$336$	0	0
$337$	20.1468	1.09747	0.548734	−	0.835997i	$-0.315110\pi$
$337$	20.1468	1.09747	0.548734	+	0.835997i	$0.315110\pi$
$338$	0	0
$339$	−13.4667	−0.731413
$340$	0	0
$341$	−4.08369	−0.221144
$342$	0	0
$343$	−9.76461	−0.527239
$344$	0	0
$345$	−7.39800	−0.398295
$346$	0	0
$347$	17.8444	0.957938	0.478969	−	0.877832i	$-0.341011\pi$
$347$	17.8444	0.957938	0.478969	+	0.877832i	$0.341011\pi$
$348$	0	0
$349$	−7.19398	−0.385085	−0.192543	−	0.981289i	$-0.561673\pi$
$349$	−7.19398	−0.385085	−0.192543	+	0.981289i	$0.561673\pi$
$350$	0	0
$351$	19.3575	1.03322
$352$	0	0
$353$	19.3841	1.03171	0.515855	−	0.856676i	$-0.327474\pi$
$353$	19.3841	1.03171	0.515855	+	0.856676i	$0.327474\pi$
$354$	0	0
$355$	−16.9928	−0.901886
$356$	0	0
$357$	−4.31829	−0.228548
$358$	0	0
$359$	3.33351	0.175936	0.0879680	−	0.996123i	$-0.471963\pi$
$359$	3.33351	0.175936	0.0879680	+	0.996123i	$0.471963\pi$
$360$	0	0
$361$	−3.72866	−0.196245
$362$	0	0
$363$	3.70756	0.194596
$364$	0	0
$365$	−47.4616	−2.48425
$366$	0	0
$367$	−33.8159	−1.76517	−0.882587	−	0.470149i	$-0.844200\pi$
$367$	−33.8159	−1.76517	−0.882587	+	0.470149i	$0.844200\pi$
$368$	0	0
$369$	−10.0818	−0.524836
$370$	0	0
$371$	2.57180	0.133521
$372$	0	0
$373$	37.4600	1.93961	0.969803	−	0.243891i	$-0.0784238\pi$
$373$	37.4600	1.93961	0.969803	+	0.243891i	$0.0784238\pi$
$374$	0	0
$375$	−4.39714	−0.227067
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−12.4635	−0.640209	−0.320104	−	0.947382i	$-0.603718\pi$
$379$	−12.4635	−0.640209	−0.320104	+	0.947382i	$0.603718\pi$
$380$	0	0
$381$	16.2244	0.831201
$382$	0	0
$383$	9.15521	0.467809	0.233905	−	0.972260i	$-0.424850\pi$
$383$	9.15521	0.467809	0.233905	+	0.972260i	$0.424850\pi$
$384$	0	0
$385$	6.05522	0.308603
$386$	0	0
$387$	18.6108	0.946041
$388$	0	0
$389$	−14.6428	−0.742417	−0.371209	−	0.928549i	$-0.621057\pi$
$389$	−14.6428	−0.742417	−0.371209	+	0.928549i	$0.621057\pi$
$390$	0	0
$391$	−10.0529	−0.508398
$392$	0	0
$393$	21.4025	1.07961
$394$	0	0
$395$	19.3280	0.972495
$396$	0	0
$397$	2.03791	0.102280	0.0511400	−	0.998691i	$-0.483715\pi$
$397$	2.03791	0.102280	0.0511400	+	0.998691i	$0.483715\pi$
$398$	0	0
$399$	3.44504	0.172468
$400$	0	0
$401$	−18.7558	−0.936618	−0.468309	−	0.883565i	$-0.655137\pi$
$401$	−18.7558	−0.936618	−0.468309	+	0.883565i	$0.655137\pi$
$402$	0	0
$403$	5.09782	0.253941
$404$	0	0
$405$	6.30972	0.313533
$406$	0	0
$407$	12.4814	0.618680
$408$	0	0
$409$	11.9159	0.589203	0.294601	−	0.955620i	$-0.404813\pi$
$409$	11.9159	0.589203	0.294601	+	0.955620i	$0.404813\pi$
$410$	0	0
$411$	−7.62943	−0.376332
$412$	0	0
$413$	8.02400	0.394835
$414$	0	0
$415$	−30.0560	−1.47539
$416$	0	0
$417$	−11.1738	−0.547182
$418$	0	0
$419$	−13.1897	−0.644357	−0.322179	−	0.946679i	$-0.604415\pi$
$419$	−13.1897	−0.644357	−0.322179	+	0.946679i	$0.604415\pi$
$420$	0	0
$421$	15.5337	0.757067	0.378534	−	0.925587i	$-0.376428\pi$
$421$	15.5337	0.757067	0.378534	+	0.925587i	$0.376428\pi$
$422$	0	0
$423$	−5.81121	−0.282551
$424$	0	0
$425$	18.5170	0.898204
$426$	0	0
$427$	1.75931	0.0851391
$428$	0	0
$429$	12.0769	0.583078
$430$	0	0
$431$	−37.3480	−1.79899	−0.899496	−	0.436930i	$-0.856066\pi$
$431$	−37.3480	−1.79899	−0.899496	+	0.436930i	$0.856066\pi$
$432$	0	0
$433$	−23.2548	−1.11755	−0.558777	−	0.829318i	$-0.688729\pi$
$433$	−23.2548	−1.11755	−0.558777	+	0.829318i	$0.688729\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.02001	0.383649
$438$	0	0
$439$	22.8331	1.08976	0.544882	−	0.838513i	$-0.316575\pi$
$439$	22.8331	1.08976	0.544882	+	0.838513i	$0.316575\pi$
$440$	0	0
$441$	9.84203	0.468668
$442$	0	0
$443$	0.565378	0.0268619	0.0134310	−	0.999910i	$-0.495725\pi$
$443$	0.565378	0.0268619	0.0134310	+	0.999910i	$0.495725\pi$
$444$	0	0
$445$	19.2731	0.913633
$446$	0	0
$447$	28.8534	1.36472
$448$	0	0
$449$	29.9326	1.41261	0.706304	−	0.707909i	$-0.250361\pi$
$449$	29.9326	1.41261	0.706304	+	0.707909i	$0.250361\pi$
$450$	0	0
$451$	−18.7039	−0.880733
$452$	0	0
$453$	−9.86887	−0.463680
$454$	0	0
$455$	−7.55897	−0.354370
$456$	0	0
$457$	−26.3292	−1.23163	−0.615815	−	0.787891i	$-0.711173\pi$
$457$	−26.3292	−1.23163	−0.615815	+	0.787891i	$0.711173\pi$
$458$	0	0
$459$	26.9354	1.25724
$460$	0	0
$461$	4.16607	0.194033	0.0970166	−	0.995283i	$-0.469070\pi$
$461$	4.16607	0.194033	0.0970166	+	0.995283i	$0.469070\pi$
$462$	0	0
$463$	−3.39217	−0.157648	−0.0788239	−	0.996889i	$-0.525116\pi$
$463$	−3.39217	−0.157648	−0.0788239	+	0.996889i	$0.525116\pi$
$464$	0	0
$465$	5.22014	0.242078
$466$	0	0
$467$	−30.9668	−1.43297	−0.716486	−	0.697602i	$-0.754251\pi$
$467$	−30.9668	−1.43297	−0.716486	+	0.697602i	$0.754251\pi$
$468$	0	0
$469$	7.90587	0.365059
$470$	0	0
$471$	2.01685	0.0929317
$472$	0	0
$473$	34.5272	1.58756
$474$	0	0
$475$	−14.7724	−0.677806
$476$	0	0
$477$	−5.39461	−0.247002
$478$	0	0
$479$	−9.60433	−0.438833	−0.219417	−	0.975631i	$-0.570415\pi$
$479$	−9.60433	−0.438833	−0.219417	+	0.975631i	$0.570415\pi$
$480$	0	0
$481$	−15.5810	−0.710433
$482$	0	0
$483$	1.80922	0.0823225
$484$	0	0
$485$	−8.78038	−0.398697
$486$	0	0
$487$	−26.2134	−1.18784	−0.593921	−	0.804524i	$-0.702421\pi$
$487$	−26.2134	−1.18784	−0.593921	+	0.804524i	$0.702421\pi$
$488$	0	0
$489$	1.46569	0.0662810
$490$	0	0
$491$	15.0309	0.678335	0.339168	−	0.940726i	$-0.389855\pi$
$491$	15.0309	0.678335	0.339168	+	0.940726i	$0.389855\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−12.7014	−0.570888
$496$	0	0
$497$	4.15569	0.186408
$498$	0	0
$499$	8.57890	0.384044	0.192022	−	0.981391i	$-0.438495\pi$
$499$	8.57890	0.384044	0.192022	+	0.981391i	$0.438495\pi$
$500$	0	0
$501$	21.3011	0.951661
$502$	0	0
$503$	−20.8453	−0.929448	−0.464724	−	0.885456i	$-0.653846\pi$
$503$	−20.8453	−0.929448	−0.464724	+	0.885456i	$0.653846\pi$
$504$	0	0
$505$	13.0983	0.582866
$506$	0	0
$507$	0.738959	0.0328183
$508$	0	0
$509$	37.4895	1.66169	0.830847	−	0.556501i	$-0.187857\pi$
$509$	37.4895	1.66169	0.830847	+	0.556501i	$0.187857\pi$
$510$	0	0
$511$	11.6070	0.513464
$512$	0	0
$513$	−21.4885	−0.948741
$514$	0	0
$515$	−44.7653	−1.97259
$516$	0	0
$517$	−10.7811	−0.474151
$518$	0	0
$519$	−28.8871	−1.26800
$520$	0	0
$521$	38.5983	1.69102	0.845512	−	0.533957i	$-0.179296\pi$
$521$	38.5983	1.69102	0.845512	+	0.533957i	$0.179296\pi$
$522$	0	0
$523$	−6.47374	−0.283077	−0.141538	−	0.989933i	$-0.545205\pi$
$523$	−6.47374	−0.283077	−0.141538	+	0.989933i	$0.545205\pi$
$524$	0	0
$525$	−3.33250	−0.145442
$526$	0	0
$527$	7.09349	0.308997
$528$	0	0
$529$	−18.7882	−0.816876
$530$	0	0
$531$	−16.8312	−0.730410
$532$	0	0
$533$	23.3488	1.01135
$534$	0	0
$535$	44.4798	1.92303
$536$	0	0
$537$	28.3456	1.22320
$538$	0	0
$539$	18.2591	0.786476
$540$	0	0
$541$	29.5094	1.26871	0.634355	−	0.773042i	$-0.281266\pi$
$541$	29.5094	1.26871	0.634355	+	0.773042i	$0.281266\pi$
$542$	0	0
$543$	5.04854	0.216653
$544$	0	0
$545$	0.509797	0.0218373
$546$	0	0
$547$	−31.8677	−1.36256	−0.681282	−	0.732021i	$-0.738577\pi$
$547$	−31.8677	−1.36256	−0.681282	+	0.732021i	$0.738577\pi$
$548$	0	0
$549$	−3.69034	−0.157500
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.72676	−0.201002
$554$	0	0
$555$	−15.9549	−0.677246
$556$	0	0
$557$	14.9416	0.633094	0.316547	−	0.948577i	$-0.397477\pi$
$557$	14.9416	0.633094	0.316547	+	0.948577i	$0.397477\pi$
$558$	0	0
$559$	−43.1016	−1.82300
$560$	0	0
$561$	16.8047	0.709494
$562$	0	0
$563$	−4.93691	−0.208066	−0.104033	−	0.994574i	$-0.533175\pi$
$563$	−4.93691	−0.208066	−0.104033	+	0.994574i	$0.533175\pi$
$564$	0	0
$565$	−32.8011	−1.37995
$566$	0	0
$567$	−1.54308	−0.0648032
$568$	0	0
$569$	−2.02698	−0.0849752	−0.0424876	−	0.999097i	$-0.513528\pi$
$569$	−2.02698	−0.0849752	−0.0424876	+	0.999097i	$0.513528\pi$
$570$	0	0
$571$	−44.9411	−1.88073	−0.940364	−	0.340170i	$-0.889515\pi$
$571$	−44.9411	−1.88073	−0.940364	+	0.340170i	$0.889515\pi$
$572$	0	0
$573$	18.4983	0.772779
$574$	0	0
$575$	−7.75800	−0.323531
$576$	0	0
$577$	−2.03773	−0.0848319	−0.0424159	−	0.999100i	$-0.513505\pi$
$577$	−2.03773	−0.0848319	−0.0424159	+	0.999100i	$0.513505\pi$
$578$	0	0
$579$	22.6874	0.942857
$580$	0	0
$581$	7.35038	0.304945
$582$	0	0
$583$	−10.0082	−0.414497
$584$	0	0
$585$	15.8557	0.655553
$586$	0	0
$587$	−6.39933	−0.264128	−0.132064	−	0.991241i	$-0.542161\pi$
$587$	−6.39933	−0.264128	−0.132064	+	0.991241i	$0.542161\pi$
$588$	0	0
$589$	−5.65904	−0.233177
$590$	0	0
$591$	−8.46437	−0.348177
$592$	0	0
$593$	−21.6680	−0.889798	−0.444899	−	0.895581i	$-0.646760\pi$
$593$	−21.6680	−0.889798	−0.444899	+	0.895581i	$0.646760\pi$
$594$	0	0
$595$	−10.5181	−0.431200
$596$	0	0
$597$	3.40625	0.139408
$598$	0	0
$599$	6.72575	0.274807	0.137403	−	0.990515i	$-0.456124\pi$
$599$	6.72575	0.274807	0.137403	+	0.990515i	$0.456124\pi$
$600$	0	0
$601$	−15.7711	−0.643317	−0.321658	−	0.946856i	$-0.604240\pi$
$601$	−15.7711	−0.643317	−0.321658	+	0.946856i	$0.604240\pi$
$602$	0	0
$603$	−16.5834	−0.675327
$604$	0	0
$605$	9.03054	0.367144
$606$	0	0
$607$	48.5384	1.97011	0.985056	−	0.172237i	$-0.0550995\pi$
$607$	48.5384	1.97011	0.985056	+	0.172237i	$0.0550995\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.4584	0.544470
$612$	0	0
$613$	21.4459	0.866190	0.433095	−	0.901348i	$-0.357421\pi$
$613$	21.4459	0.866190	0.433095	+	0.901348i	$0.357421\pi$
$614$	0	0
$615$	23.9090	0.964106
$616$	0	0
$617$	9.75777	0.392833	0.196417	−	0.980521i	$-0.437070\pi$
$617$	9.75777	0.392833	0.196417	+	0.980521i	$0.437070\pi$
$618$	0	0
$619$	16.9111	0.679713	0.339856	−	0.940477i	$-0.389622\pi$
$619$	16.9111	0.679713	0.339856	+	0.940477i	$0.389622\pi$
$620$	0	0
$621$	−11.2851	−0.452854
$622$	0	0
$623$	−4.71335	−0.188836
$624$	0	0
$625$	−29.6111	−1.18444
$626$	0	0
$627$	−13.4064	−0.535401
$628$	0	0
$629$	−21.6806	−0.864461
$630$	0	0
$631$	−9.58087	−0.381408	−0.190704	−	0.981648i	$-0.561077\pi$
$631$	−9.58087	−0.381408	−0.190704	+	0.981648i	$0.561077\pi$
$632$	0	0
$633$	31.0722	1.23501
$634$	0	0
$635$	39.5180	1.56822
$636$	0	0
$637$	−22.7936	−0.903114
$638$	0	0
$639$	−8.71699	−0.344839
$640$	0	0
$641$	10.0494	0.396926	0.198463	−	0.980108i	$-0.436405\pi$
$641$	10.0494	0.396926	0.198463	+	0.980108i	$0.436405\pi$
$642$	0	0
$643$	−47.7664	−1.88372	−0.941861	−	0.336002i	$-0.890925\pi$
$643$	−47.7664	−1.88372	−0.941861	+	0.336002i	$0.890925\pi$
$644$	0	0
$645$	−44.1358	−1.73784
$646$	0	0
$647$	−15.6271	−0.614366	−0.307183	−	0.951650i	$-0.599386\pi$
$647$	−15.6271	−0.614366	−0.307183	+	0.951650i	$0.599386\pi$
$648$	0	0
$649$	−31.2255	−1.22571
$650$	0	0
$651$	−1.27662	−0.0500345
$652$	0	0
$653$	−3.89573	−0.152452	−0.0762258	−	0.997091i	$-0.524287\pi$
$653$	−3.89573	−0.152452	−0.0762258	+	0.997091i	$0.524287\pi$
$654$	0	0
$655$	52.1303	2.03690
$656$	0	0
$657$	−24.3469	−0.949862
$658$	0	0
$659$	16.0552	0.625420	0.312710	−	0.949849i	$-0.398763\pi$
$659$	16.0552	0.625420	0.312710	+	0.949849i	$0.398763\pi$
$660$	0	0
$661$	−45.8698	−1.78413	−0.892064	−	0.451909i	$-0.850743\pi$
$661$	−45.8698	−1.78413	−0.892064	+	0.451909i	$0.850743\pi$
$662$	0	0
$663$	−20.9779	−0.814715
$664$	0	0
$665$	8.39113	0.325394
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−9.31563	−0.360163
$670$	0	0
$671$	−6.84639	−0.264302
$672$	0	0
$673$	−31.8163	−1.22643	−0.613214	−	0.789917i	$-0.710124\pi$
$673$	−31.8163	−1.22643	−0.613214	+	0.789917i	$0.710124\pi$
$674$	0	0
$675$	20.7865	0.800073
$676$	0	0
$677$	−16.0243	−0.615865	−0.307933	−	0.951408i	$-0.599637\pi$
$677$	−16.0243	−0.615865	−0.307933	+	0.951408i	$0.599637\pi$
$678$	0	0
$679$	2.14729	0.0824055
$680$	0	0
$681$	−4.05923	−0.155550
$682$	0	0
$683$	12.6410	0.483695	0.241848	−	0.970314i	$-0.422247\pi$
$683$	12.6410	0.483695	0.241848	+	0.970314i	$0.422247\pi$
$684$	0	0
$685$	−18.5831	−0.710024
$686$	0	0
$687$	32.1373	1.22612
$688$	0	0
$689$	12.4936	0.475969
$690$	0	0
$691$	2.48717	0.0946163	0.0473081	−	0.998880i	$-0.484936\pi$
$691$	2.48717	0.0946163	0.0473081	+	0.998880i	$0.484936\pi$
$692$	0	0
$693$	3.10621	0.117995
$694$	0	0
$695$	−27.2161	−1.03236
$696$	0	0
$697$	32.4893	1.23062
$698$	0	0
$699$	−13.5108	−0.511025
$700$	0	0
$701$	−37.1716	−1.40395	−0.701976	−	0.712200i	$-0.747699\pi$
$701$	−37.1716	−1.40395	−0.701976	+	0.712200i	$0.747699\pi$
$702$	0	0
$703$	17.2963	0.652343
$704$	0	0
$705$	13.7813	0.519035
$706$	0	0
$707$	−3.20326	−0.120471
$708$	0	0
$709$	−2.19730	−0.0825215	−0.0412607	−	0.999148i	$-0.513137\pi$
$709$	−2.19730	−0.0825215	−0.0412607	+	0.999148i	$0.513137\pi$
$710$	0	0
$711$	9.91487	0.371837
$712$	0	0
$713$	−2.97194	−0.111300
$714$	0	0
$715$	29.4158	1.10009
$716$	0	0
$717$	−28.3679	−1.05942
$718$	0	0
$719$	−20.5167	−0.765143	−0.382572	−	0.923926i	$-0.624961\pi$
$719$	−20.5167	−0.765143	−0.382572	+	0.923926i	$0.624961\pi$
$720$	0	0
$721$	10.9476	0.407710
$722$	0	0
$723$	−2.23127	−0.0829817
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.3342	1.64426	0.822132	−	0.569297i	$-0.192784\pi$
$727$	44.3342	1.64426	0.822132	+	0.569297i	$0.192784\pi$
$728$	0	0
$729$	23.3053	0.863158
$730$	0	0
$731$	−59.9747	−2.21825
$732$	0	0
$733$	−1.73739	−0.0641719	−0.0320860	−	0.999485i	$-0.510215\pi$
$733$	−1.73739	−0.0641719	−0.0320860	+	0.999485i	$0.510215\pi$
$734$	0	0
$735$	−23.3405	−0.860926
$736$	0	0
$737$	−30.7658	−1.13327
$738$	0	0
$739$	−10.9491	−0.402770	−0.201385	−	0.979512i	$-0.564544\pi$
$739$	−10.9491	−0.402770	−0.201385	+	0.979512i	$0.564544\pi$
$740$	0	0
$741$	16.7358	0.614803
$742$	0	0
$743$	23.6581	0.867931	0.433966	−	0.900929i	$-0.357114\pi$
$743$	23.6581	0.867931	0.433966	+	0.900929i	$0.357114\pi$
$744$	0	0
$745$	70.2786	2.57481
$746$	0	0
$747$	−15.4182	−0.564121
$748$	0	0
$749$	−10.8778	−0.397466
$750$	0	0
$751$	−3.49190	−0.127421	−0.0637106	−	0.997968i	$-0.520293\pi$
$751$	−3.49190	−0.127421	−0.0637106	+	0.997968i	$0.520293\pi$
$752$	0	0
$753$	−11.6540	−0.424694
$754$	0	0
$755$	−24.0377	−0.874822
$756$	0	0
$757$	−4.40722	−0.160183	−0.0800915	−	0.996788i	$-0.525521\pi$
$757$	−4.40722	−0.160183	−0.0800915	+	0.996788i	$0.525521\pi$
$758$	0	0
$759$	−7.04062	−0.255558
$760$	0	0
$761$	−6.31984	−0.229094	−0.114547	−	0.993418i	$-0.536542\pi$
$761$	−6.31984	−0.229094	−0.114547	+	0.993418i	$0.536542\pi$
$762$	0	0
$763$	−0.124674	−0.00451350
$764$	0	0
$765$	22.0628	0.797682
$766$	0	0
$767$	38.9800	1.40749
$768$	0	0
$769$	−9.73258	−0.350966	−0.175483	−	0.984482i	$-0.556149\pi$
$769$	−9.73258	−0.350966	−0.175483	+	0.984482i	$0.556149\pi$
$770$	0	0
$771$	−4.41025	−0.158831
$772$	0	0
$773$	10.2603	0.369038	0.184519	−	0.982829i	$-0.440927\pi$
$773$	10.2603	0.369038	0.184519	+	0.982829i	$0.440927\pi$
$774$	0	0
$775$	5.47416	0.196638
$776$	0	0
$777$	3.90185	0.139978
$778$	0	0
$779$	−25.9193	−0.928654
$780$	0	0
$781$	−16.1719	−0.578678
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.91248	0.175334
$786$	0	0
$787$	23.4279	0.835114	0.417557	−	0.908651i	$-0.362886\pi$
$787$	23.4279	0.835114	0.417557	+	0.908651i	$0.362886\pi$
$788$	0	0
$789$	−14.6915	−0.523033
$790$	0	0
$791$	8.02169	0.285219
$792$	0	0
$793$	8.54661	0.303499
$794$	0	0
$795$	12.7934	0.453734
$796$	0	0
$797$	10.7261	0.379937	0.189969	−	0.981790i	$-0.439161\pi$
$797$	10.7261	0.379937	0.189969	+	0.981790i	$0.439161\pi$
$798$	0	0
$799$	18.7270	0.662515
$800$	0	0
$801$	9.88672	0.349330
$802$	0	0
$803$	−45.1688	−1.59397
$804$	0	0
$805$	4.40675	0.155317
$806$	0	0
$807$	23.9658	0.843636
$808$	0	0
$809$	18.4121	0.647336	0.323668	−	0.946171i	$-0.395084\pi$
$809$	18.4121	0.647336	0.323668	+	0.946171i	$0.395084\pi$
$810$	0	0
$811$	−25.8805	−0.908789	−0.454394	−	0.890801i	$-0.650144\pi$
$811$	−25.8805	−0.908789	−0.454394	+	0.890801i	$0.650144\pi$
$812$	0	0
$813$	38.5698	1.35270
$814$	0	0
$815$	3.57001	0.125052
$816$	0	0
$817$	47.8466	1.67394
$818$	0	0
$819$	−3.87760	−0.135494
$820$	0	0
$821$	44.8517	1.56533	0.782667	−	0.622441i	$-0.213859\pi$
$821$	44.8517	1.56533	0.782667	+	0.622441i	$0.213859\pi$
$822$	0	0
$823$	−28.3904	−0.989625	−0.494813	−	0.869000i	$-0.664763\pi$
$823$	−28.3904	−0.989625	−0.494813	+	0.869000i	$0.664763\pi$
$824$	0	0
$825$	12.9685	0.451504
$826$	0	0
$827$	12.7427	0.443108	0.221554	−	0.975148i	$-0.428887\pi$
$827$	12.7427	0.443108	0.221554	+	0.975148i	$0.428887\pi$
$828$	0	0
$829$	−3.79481	−0.131799	−0.0658996	−	0.997826i	$-0.520992\pi$
$829$	−3.79481	−0.131799	−0.0658996	+	0.997826i	$0.520992\pi$
$830$	0	0
$831$	23.9184	0.829721
$832$	0	0
$833$	−31.7167	−1.09892
$834$	0	0
$835$	51.8832	1.79549
$836$	0	0
$837$	7.96291	0.275238
$838$	0	0
$839$	3.10964	0.107357	0.0536783	−	0.998558i	$-0.482905\pi$
$839$	3.10964	0.107357	0.0536783	+	0.998558i	$0.482905\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	−8.62677	−0.297122
$844$	0	0
$845$	1.79989	0.0619181
$846$	0	0
$847$	−2.20847	−0.0758839
$848$	0	0
$849$	4.39588	0.150866
$850$	0	0
$851$	9.08346	0.311377
$852$	0	0
$853$	0.125350	0.00429189	0.00214594	−	0.999998i	$-0.499317\pi$
$853$	0.125350	0.00429189	0.00214594	+	0.999998i	$0.499317\pi$
$854$	0	0
$855$	−17.6012	−0.601950
$856$	0	0
$857$	16.8270	0.574798	0.287399	−	0.957811i	$-0.407209\pi$
$857$	16.8270	0.574798	0.287399	+	0.957811i	$0.407209\pi$
$858$	0	0
$859$	−19.2539	−0.656934	−0.328467	−	0.944515i	$-0.606532\pi$
$859$	−19.2539	−0.656934	−0.328467	+	0.944515i	$0.606532\pi$
$860$	0	0
$861$	−5.84709	−0.199268
$862$	0	0
$863$	22.4777	0.765148	0.382574	−	0.923925i	$-0.375038\pi$
$863$	22.4777	0.765148	0.382574	+	0.923925i	$0.375038\pi$
$864$	0	0
$865$	−70.3607	−2.39233
$866$	0	0
$867$	−8.50907	−0.288983
$868$	0	0
$869$	18.3943	0.623983
$870$	0	0
$871$	38.4061	1.30134
$872$	0	0
$873$	−4.50416	−0.152443
$874$	0	0
$875$	2.61923	0.0885462
$876$	0	0
$877$	17.5811	0.593673	0.296836	−	0.954928i	$-0.404068\pi$
$877$	17.5811	0.593673	0.296836	+	0.954928i	$0.404068\pi$
$878$	0	0
$879$	−26.2867	−0.886627
$880$	0	0
$881$	46.1375	1.55441	0.777206	−	0.629247i	$-0.216636\pi$
$881$	46.1375	1.55441	0.777206	+	0.629247i	$0.216636\pi$
$882$	0	0
$883$	−33.4919	−1.12709	−0.563546	−	0.826085i	$-0.690563\pi$
$883$	−33.4919	−1.12709	−0.563546	+	0.826085i	$0.690563\pi$
$884$	0	0
$885$	39.9153	1.34174
$886$	0	0
$887$	12.7174	0.427007	0.213504	−	0.976942i	$-0.431512\pi$
$887$	12.7174	0.427007	0.213504	+	0.976942i	$0.431512\pi$
$888$	0	0
$889$	−9.66434	−0.324132
$890$	0	0
$891$	6.00491	0.201172
$892$	0	0
$893$	−14.9400	−0.499950
$894$	0	0
$895$	69.0417	2.30781
$896$	0	0
$897$	8.78907	0.293459
$898$	0	0
$899$	0	0
$900$	0	0
$901$	17.3845	0.579163
$902$	0	0
$903$	10.7937	0.359190
$904$	0	0
$905$	12.2968	0.408759
$906$	0	0
$907$	−34.1270	−1.13317	−0.566585	−	0.824004i	$-0.691736\pi$
$907$	−34.1270	−1.13317	−0.566585	+	0.824004i	$0.691736\pi$
$908$	0	0
$909$	6.71916	0.222861
$910$	0	0
$911$	−22.9200	−0.759372	−0.379686	−	0.925115i	$-0.623968\pi$
$911$	−22.9200	−0.759372	−0.379686	+	0.925115i	$0.623968\pi$
$912$	0	0
$913$	−28.6041	−0.946657
$914$	0	0
$915$	8.75168	0.289322
$916$	0	0
$917$	−12.7488	−0.421001
$918$	0	0
$919$	43.3126	1.42875	0.714376	−	0.699763i	$-0.246711\pi$
$919$	43.3126	1.42875	0.714376	+	0.699763i	$0.246711\pi$
$920$	0	0
$921$	27.2458	0.897781
$922$	0	0
$923$	20.1881	0.664498
$924$	0	0
$925$	−16.7313	−0.550120
$926$	0	0
$927$	−22.9637	−0.754228
$928$	0	0
$929$	−42.9957	−1.41064	−0.705321	−	0.708888i	$-0.749197\pi$
$929$	−42.9957	−1.41064	−0.705321	+	0.708888i	$0.749197\pi$
$930$	0	0
$931$	25.3029	0.829269
$932$	0	0
$933$	28.1606	0.921938
$934$	0	0
$935$	40.9314	1.33860
$936$	0	0
$937$	18.3127	0.598251	0.299126	−	0.954214i	$-0.403305\pi$
$937$	18.3127	0.598251	0.299126	+	0.954214i	$0.403305\pi$
$938$	0	0
$939$	−19.7538	−0.644642
$940$	0	0
$941$	−4.62620	−0.150810	−0.0754048	−	0.997153i	$-0.524025\pi$
$941$	−4.62620	−0.150810	−0.0754048	+	0.997153i	$0.524025\pi$
$942$	0	0
$943$	−13.6119	−0.443266
$944$	0	0
$945$	−11.8073	−0.384091
$946$	0	0
$947$	−24.3131	−0.790071	−0.395035	−	0.918666i	$-0.629268\pi$
$947$	−24.3131	−0.790071	−0.395035	+	0.918666i	$0.629268\pi$
$948$	0	0
$949$	56.3860	1.83037
$950$	0	0
$951$	31.0782	1.00778
$952$	0	0
$953$	−1.16569	−0.0377605	−0.0188803	−	0.999822i	$-0.506010\pi$
$953$	−1.16569	−0.0377605	−0.0188803	+	0.999822i	$0.506010\pi$
$954$	0	0
$955$	45.0566	1.45800
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.54460	0.146753
$960$	0	0
$961$	−28.9030	−0.932353
$962$	0	0
$963$	22.8173	0.735277
$964$	0	0
$965$	55.2600	1.77888
$966$	0	0
$967$	−25.3184	−0.814184	−0.407092	−	0.913387i	$-0.633457\pi$
$967$	−25.3184	−0.814184	−0.407092	+	0.913387i	$0.633457\pi$
$968$	0	0
$969$	23.2874	0.748098
$970$	0	0
$971$	24.5334	0.787314	0.393657	−	0.919257i	$-0.371210\pi$
$971$	24.5334	0.787314	0.393657	+	0.919257i	$0.371210\pi$
$972$	0	0
$973$	6.65585	0.213377
$974$	0	0
$975$	−16.1890	−0.518464
$976$	0	0
$977$	14.7494	0.471875	0.235938	−	0.971768i	$-0.424184\pi$
$977$	14.7494	0.471875	0.235938	+	0.971768i	$0.424184\pi$
$978$	0	0
$979$	18.3421	0.586215
$980$	0	0
$981$	0.261516	0.00834956
$982$	0	0
$983$	25.8106	0.823230	0.411615	−	0.911358i	$-0.364965\pi$
$983$	25.8106	0.823230	0.411615	+	0.911358i	$0.364965\pi$
$984$	0	0
$985$	−20.6168	−0.656904
$986$	0	0
$987$	−3.37031	−0.107278
$988$	0	0
$989$	25.1275	0.799007
$990$	0	0
$991$	25.5590	0.811909	0.405955	−	0.913893i	$-0.366939\pi$
$991$	25.5590	0.811909	0.405955	+	0.913893i	$0.366939\pi$
$992$	0	0
$993$	12.3005	0.390345
$994$	0	0
$995$	8.29664	0.263021
$996$	0	0
$997$	40.0196	1.26743	0.633717	−	0.773565i	$-0.281528\pi$
$997$	40.0196	1.26743	0.633717	+	0.773565i	$0.281528\pi$
$998$	0	0
$999$	−24.3379	−0.770016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6728.2.a.be.1.9		24
29.18	odd	28		232.2.q.a.121.3	✓	48
29.21	odd	28		232.2.q.a.209.3	yes	48
29.28	even	2		6728.2.a.bf.1.16		24
116.47	even	28		464.2.y.e.353.6		48
116.79	even	28		464.2.y.e.209.6		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.2.q.a.121.3	✓	48	29.18	odd	28
232.2.q.a.209.3	yes	48	29.21	odd	28
464.2.y.e.209.6		48	116.79	even	28
464.2.y.e.353.6		48	116.47	even	28
6728.2.a.be.1.9		24	1.1	even	1	trivial
6728.2.a.bf.1.16		24	29.28	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6728 = 2^{3} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21654 q^{3} -2.96314 q^{5} +0.724653 q^{7} -1.52003 q^{9} +O(q^{10})\)	\(q-1.21654 q^{3} -2.96314 q^{5} +0.724653 q^{7} -1.52003 q^{9} -2.82000 q^{11} +3.52031 q^{13} +3.60477 q^{15} +4.89842 q^{17} -3.90786 q^{19} -0.881568 q^{21} -2.05228 q^{23} +3.78019 q^{25} +5.49880 q^{27} +1.44812 q^{31} +3.43064 q^{33} -2.14725 q^{35} -4.42604 q^{37} -4.28259 q^{39} +6.63260 q^{41} -12.2437 q^{43} +4.50407 q^{45} +3.82308 q^{47} -6.47488 q^{49} -5.95912 q^{51} +3.54901 q^{53} +8.35604 q^{55} +4.75406 q^{57} +11.0729 q^{59} +2.42780 q^{61} -1.10150 q^{63} -10.4312 q^{65} +10.9099 q^{67} +2.49668 q^{69} +5.73474 q^{71} +16.0173 q^{73} -4.59875 q^{75} -2.04352 q^{77} -6.52280 q^{79} -2.12940 q^{81} +10.1433 q^{83} -14.5147 q^{85} -6.50429 q^{89} +2.55100 q^{91} -1.76169 q^{93} +11.5795 q^{95} +2.96320 q^{97} +4.28648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9}+O(q^{10}) \) 24 * q - 4 * q^3 + 8 * q^5 - 2 * q^7 + 32 * q^9	\( 24 q - 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9} - 6 q^{11} + 2 q^{13} - 32 q^{15} - 20 q^{17} - 44 q^{19} - 4 q^{21} + 2 q^{23} + 32 q^{25} - 34 q^{27} - 10 q^{31} - 12 q^{33} - 32 q^{35} - 40 q^{37} - 18 q^{39} - 8 q^{41} - 30 q^{43} + 16 q^{45} - 34 q^{47} + 26 q^{49} - 28 q^{51} + 18 q^{53} - 50 q^{55} + 6 q^{57} - 64 q^{59} - 6 q^{61} - 28 q^{63} + 14 q^{65} - 40 q^{67} - 38 q^{69} - 52 q^{73} - 66 q^{75} - 90 q^{77} - 4 q^{79} + 40 q^{81} - 44 q^{83} - 40 q^{85} + 40 q^{89} - 28 q^{91} - 10 q^{93} - 84 q^{97} - 60 q^{99}+O(q^{100}) \) 24 * q - 4 * q^3 + 8 * q^5 - 2 * q^7 + 32 * q^9 - 6 * q^11 + 2 * q^13 - 32 * q^15 - 20 * q^17 - 44 * q^19 - 4 * q^21 + 2 * q^23 + 32 * q^25 - 34 * q^27 - 10 * q^31 - 12 * q^33 - 32 * q^35 - 40 * q^37 - 18 * q^39 - 8 * q^41 - 30 * q^43 + 16 * q^45 - 34 * q^47 + 26 * q^49 - 28 * q^51 + 18 * q^53 - 50 * q^55 + 6 * q^57 - 64 * q^59 - 6 * q^61 - 28 * q^63 + 14 * q^65 - 40 * q^67 - 38 * q^69 - 52 * q^73 - 66 * q^75 - 90 * q^77 - 4 * q^79 + 40 * q^81 - 44 * q^83 - 40 * q^85 + 40 * q^89 - 28 * q^91 - 10 * q^93 - 84 * q^97 - 60 * q^99

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