LMFDB - Embedded newform 4096.2.a.s.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4096,2,Mod(1,4096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4096, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4096 = 2^{12} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4096.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:	$32.7067246679$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{48})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 1 $ x^8 - 8x^6 + 20x^4 - 16*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.21752$ of defining polynomial
Character	$\chi$	$=$	4096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.482362 q^{3} +1.47858 q^{5} -0.191104 q^{7} -2.76733 q^{9} +O(q^{10})$	$q+0.482362 q^{3} +1.47858 q^{5} -0.191104 q^{7} -2.76733 q^{9} -3.38134 q^{11} +2.48720 q^{13} +0.713208 q^{15} +3.11099 q^{17} +6.49233 q^{19} -0.0921811 q^{21} +7.33030 q^{23} -2.81382 q^{25} -2.78194 q^{27} -4.69017 q^{29} +7.44503 q^{31} -1.63103 q^{33} -0.282561 q^{35} -9.13987 q^{37} +1.19973 q^{39} +6.04989 q^{41} +4.68921 q^{43} -4.09170 q^{45} -12.0952 q^{47} -6.96348 q^{49} +1.50062 q^{51} +3.70949 q^{53} -4.99957 q^{55} +3.13165 q^{57} +3.04232 q^{59} +9.44832 q^{61} +0.528846 q^{63} +3.67752 q^{65} +7.73961 q^{67} +3.53586 q^{69} +4.04494 q^{71} -3.53125 q^{73} -1.35728 q^{75} +0.646187 q^{77} -8.39967 q^{79} +6.96008 q^{81} +14.1633 q^{83} +4.59983 q^{85} -2.26236 q^{87} +7.02458 q^{89} -0.475314 q^{91} +3.59120 q^{93} +9.59940 q^{95} +2.87492 q^{97} +9.35728 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3}+O(q^{10}) $ 8 * q + 8 * q^3	$ 8 q + 8 q^{3} + 8 q^{11} + 8 q^{17} - 8 q^{25} + 8 q^{27} + 40 q^{33} - 8 q^{35} + 32 q^{41} + 56 q^{43} + 8 q^{49} + 48 q^{51} + 8 q^{57} + 32 q^{59} - 16 q^{65} + 24 q^{67} - 8 q^{73} + 16 q^{81} + 48 q^{83} - 8 q^{89} + 8 q^{91} + 8 q^{97} + 64 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^11 + 8 * q^17 - 8 * q^25 + 8 * q^27 + 40 * q^33 - 8 * q^35 + 32 * q^41 + 56 * q^43 + 8 * q^49 + 48 * q^51 + 8 * q^57 + 32 * q^59 - 16 * q^65 + 24 * q^67 - 8 * q^73 + 16 * q^81 + 48 * q^83 - 8 * q^89 + 8 * q^91 + 8 * q^97 + 64 * 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
$3$	0.482362	0.278492	0.139246	−	0.990258i	$-0.455532\pi$
$3$	0.482362	0.278492	0.139246	+	0.990258i	$0.455532\pi$
$4$	0	0
$5$	1.47858	0.661239	0.330619	−	0.943764i	$-0.392742\pi$
$5$	1.47858	0.661239	0.330619	+	0.943764i	$0.392742\pi$
$6$	0	0
$7$	−0.191104	−0.0722304	−0.0361152	−	0.999348i	$-0.511498\pi$
$7$	−0.191104	−0.0722304	−0.0361152	+	0.999348i	$0.511498\pi$
$8$	0	0
$9$	−2.76733	−0.922442
$10$	0	0
$11$	−3.38134	−1.01951	−0.509756	−	0.860319i	$-0.670264\pi$
$11$	−3.38134	−1.01951	−0.509756	+	0.860319i	$0.670264\pi$
$12$	0	0
$13$	2.48720	0.689826	0.344913	−	0.938635i	$-0.387908\pi$
$13$	2.48720	0.689826	0.344913	+	0.938635i	$0.387908\pi$
$14$	0	0
$15$	0.713208	0.184150
$16$	0	0
$17$	3.11099	0.754525	0.377263	−	0.926106i	$-0.376865\pi$
$17$	3.11099	0.754525	0.377263	+	0.926106i	$0.376865\pi$
$18$	0	0
$19$	6.49233	1.48944	0.744721	−	0.667376i	$-0.232582\pi$
$19$	6.49233	1.48944	0.744721	+	0.667376i	$0.232582\pi$
$20$	0	0
$21$	−0.0921811	−0.0201156
$22$	0	0
$23$	7.33030	1.52847	0.764236	−	0.644936i	$-0.223116\pi$
$23$	7.33030	1.52847	0.764236	+	0.644936i	$0.223116\pi$
$24$	0	0
$25$	−2.81382	−0.562763
$26$	0	0
$27$	−2.78194	−0.535384
$28$	0	0
$29$	−4.69017	−0.870943	−0.435471	−	0.900203i	$-0.643418\pi$
$29$	−4.69017	−0.870943	−0.435471	+	0.900203i	$0.643418\pi$
$30$	0	0
$31$	7.44503	1.33717	0.668584	−	0.743637i	$-0.266901\pi$
$31$	7.44503	1.33717	0.668584	+	0.743637i	$0.266901\pi$
$32$	0	0
$33$	−1.63103	−0.283926
$34$	0	0
$35$	−0.282561	−0.0477615
$36$	0	0
$37$	−9.13987	−1.50259	−0.751293	−	0.659969i	$-0.770569\pi$
$37$	−9.13987	−1.50259	−0.751293	+	0.659969i	$0.770569\pi$
$38$	0	0
$39$	1.19973	0.192111
$40$	0	0
$41$	6.04989	0.944834	0.472417	−	0.881375i	$-0.343382\pi$
$41$	6.04989	0.944834	0.472417	+	0.881375i	$0.343382\pi$
$42$	0	0
$43$	4.68921	0.715098	0.357549	−	0.933894i	$-0.383613\pi$
$43$	4.68921	0.715098	0.357549	+	0.933894i	$0.383613\pi$
$44$	0	0
$45$	−4.09170	−0.609955
$46$	0	0
$47$	−12.0952	−1.76426	−0.882132	−	0.471002i	$-0.843892\pi$
$47$	−12.0952	−1.76426	−0.882132	+	0.471002i	$0.843892\pi$
$48$	0	0
$49$	−6.96348	−0.994783
$50$	0	0
$51$	1.50062	0.210129
$52$	0	0
$53$	3.70949	0.509538	0.254769	−	0.967002i	$-0.418001\pi$
$53$	3.70949	0.509538	0.254769	+	0.967002i	$0.418001\pi$
$54$	0	0
$55$	−4.99957	−0.674142
$56$	0	0
$57$	3.13165	0.414798
$58$	0	0
$59$	3.04232	0.396077	0.198038	−	0.980194i	$-0.436543\pi$
$59$	3.04232	0.396077	0.198038	+	0.980194i	$0.436543\pi$
$60$	0	0
$61$	9.44832	1.20973	0.604867	−	0.796327i	$-0.293226\pi$
$61$	9.44832	1.20973	0.604867	+	0.796327i	$0.293226\pi$
$62$	0	0
$63$	0.528846	0.0666284
$64$	0	0
$65$	3.67752	0.456140
$66$	0	0
$67$	7.73961	0.945544	0.472772	−	0.881185i	$-0.343253\pi$
$67$	7.73961	0.945544	0.472772	+	0.881185i	$0.343253\pi$
$68$	0	0
$69$	3.53586	0.425667
$70$	0	0
$71$	4.04494	0.480046	0.240023	−	0.970767i	$-0.422845\pi$
$71$	4.04494	0.480046	0.240023	+	0.970767i	$0.422845\pi$
$72$	0	0
$73$	−3.53125	−0.413302	−0.206651	−	0.978415i	$-0.566256\pi$
$73$	−3.53125	−0.413302	−0.206651	+	0.978415i	$0.566256\pi$
$74$	0	0
$75$	−1.35728	−0.156725
$76$	0	0
$77$	0.646187	0.0736398
$78$	0	0
$79$	−8.39967	−0.945036	−0.472518	−	0.881321i	$-0.656655\pi$
$79$	−8.39967	−0.945036	−0.472518	+	0.881321i	$0.656655\pi$
$80$	0	0
$81$	6.96008	0.773342
$82$	0	0
$83$	14.1633	1.55462	0.777311	−	0.629117i	$-0.216583\pi$
$83$	14.1633	1.55462	0.777311	+	0.629117i	$0.216583\pi$
$84$	0	0
$85$	4.59983	0.498922
$86$	0	0
$87$	−2.26236	−0.242550
$88$	0	0
$89$	7.02458	0.744604	0.372302	−	0.928112i	$-0.378569\pi$
$89$	7.02458	0.744604	0.372302	+	0.928112i	$0.378569\pi$
$90$	0	0
$91$	−0.475314	−0.0498264
$92$	0	0
$93$	3.59120	0.372390
$94$	0	0
$95$	9.59940	0.984877
$96$	0	0
$97$	2.87492	0.291903	0.145952	−	0.989292i	$-0.453376\pi$
$97$	2.87492	0.291903	0.145952	+	0.989292i	$0.453376\pi$
$98$	0	0
$99$	9.35728	0.940442
$100$	0	0
$101$	7.84382	0.780489	0.390245	−	0.920711i	$-0.372390\pi$
$101$	7.84382	0.780489	0.390245	+	0.920711i	$0.372390\pi$
$102$	0	0
$103$	5.82751	0.574202	0.287101	−	0.957900i	$-0.407308\pi$
$103$	5.82751	0.574202	0.287101	+	0.957900i	$0.407308\pi$
$104$	0	0
$105$	−0.136297	−0.0133012
$106$	0	0
$107$	6.01702	0.581687	0.290843	−	0.956771i	$-0.406064\pi$
$107$	6.01702	0.581687	0.290843	+	0.956771i	$0.406064\pi$
$108$	0	0
$109$	−4.54004	−0.434857	−0.217429	−	0.976076i	$-0.569767\pi$
$109$	−4.54004	−0.434857	−0.217429	+	0.976076i	$0.569767\pi$
$110$	0	0
$111$	−4.40873	−0.418458
$112$	0	0
$113$	9.86370	0.927899	0.463950	−	0.885862i	$-0.346432\pi$
$113$	9.86370	0.927899	0.463950	+	0.885862i	$0.346432\pi$
$114$	0	0
$115$	10.8384	1.01069
$116$	0	0
$117$	−6.88291	−0.636325
$118$	0	0
$119$	−0.594521	−0.0544997
$120$	0	0
$121$	0.433470	0.0394063
$122$	0	0
$123$	2.91824	0.263128
$124$	0	0
$125$	−11.5533	−1.03336
$126$	0	0
$127$	15.4530	1.37123	0.685614	−	0.727965i	$-0.259534\pi$
$127$	15.4530	1.37123	0.685614	+	0.727965i	$0.259534\pi$
$128$	0	0
$129$	2.26190	0.199149
$130$	0	0
$131$	15.7809	1.37879	0.689394	−	0.724387i	$-0.257877\pi$
$131$	15.7809	1.37879	0.689394	+	0.724387i	$0.257877\pi$
$132$	0	0
$133$	−1.24071	−0.107583
$134$	0	0
$135$	−4.11331	−0.354017
$136$	0	0
$137$	8.24745	0.704627	0.352314	−	0.935882i	$-0.385395\pi$
$137$	8.24745	0.704627	0.352314	+	0.935882i	$0.385395\pi$
$138$	0	0
$139$	7.91724	0.671532	0.335766	−	0.941946i	$-0.391005\pi$
$139$	7.91724	0.671532	0.335766	+	0.941946i	$0.391005\pi$
$140$	0	0
$141$	−5.83426	−0.491333
$142$	0	0
$143$	−8.41009	−0.703287
$144$	0	0
$145$	−6.93477	−0.575901
$146$	0	0
$147$	−3.35892	−0.277039
$148$	0	0
$149$	−7.10680	−0.582212	−0.291106	−	0.956691i	$-0.594023\pi$
$149$	−7.10680	−0.582212	−0.291106	+	0.956691i	$0.594023\pi$
$150$	0	0
$151$	−15.2358	−1.23987	−0.619936	−	0.784652i	$-0.712842\pi$
$151$	−15.2358	−1.23987	−0.619936	+	0.784652i	$0.712842\pi$
$152$	0	0
$153$	−8.60912	−0.696006
$154$	0	0
$155$	11.0080	0.884187
$156$	0	0
$157$	0.576865	0.0460389	0.0230194	−	0.999735i	$-0.492672\pi$
$157$	0.576865	0.0460389	0.0230194	+	0.999735i	$0.492672\pi$
$158$	0	0
$159$	1.78932	0.141902
$160$	0	0
$161$	−1.40085	−0.110402
$162$	0	0
$163$	24.2025	1.89568	0.947842	−	0.318741i	$-0.103260\pi$
$163$	24.2025	1.89568	0.947842	+	0.318741i	$0.103260\pi$
$164$	0	0
$165$	−2.41160	−0.187743
$166$	0	0
$167$	−15.8591	−1.22721	−0.613606	−	0.789612i	$-0.710282\pi$
$167$	−15.8591	−1.22721	−0.613606	+	0.789612i	$0.710282\pi$
$168$	0	0
$169$	−6.81382	−0.524140
$170$	0	0
$171$	−17.9664	−1.37392
$172$	0	0
$173$	21.8615	1.66210	0.831049	−	0.556200i	$-0.187741\pi$
$173$	21.8615	1.66210	0.831049	+	0.556200i	$0.187741\pi$
$174$	0	0
$175$	0.537730	0.0406486
$176$	0	0
$177$	1.46750	0.110304
$178$	0	0
$179$	15.3208	1.14513	0.572564	−	0.819860i	$-0.305949\pi$
$179$	15.3208	1.14513	0.572564	+	0.819860i	$0.305949\pi$
$180$	0	0
$181$	10.5505	0.784211	0.392105	−	0.919920i	$-0.371747\pi$
$181$	10.5505	0.784211	0.392105	+	0.919920i	$0.371747\pi$
$182$	0	0
$183$	4.55751	0.336901
$184$	0	0
$185$	−13.5140	−0.993568
$186$	0	0
$187$	−10.5193	−0.769248
$188$	0	0
$189$	0.531639	0.0386710
$190$	0	0
$191$	22.9763	1.66250	0.831252	−	0.555896i	$-0.187625\pi$
$191$	22.9763	1.66250	0.831252	+	0.555896i	$0.187625\pi$
$192$	0	0
$193$	−18.2368	−1.31271	−0.656355	−	0.754452i	$-0.727903\pi$
$193$	−18.2368	−1.31271	−0.656355	+	0.754452i	$0.727903\pi$
$194$	0	0
$195$	1.77389	0.127031
$196$	0	0
$197$	−22.3817	−1.59463	−0.797314	−	0.603564i	$-0.793747\pi$
$197$	−22.3817	−1.59463	−0.797314	+	0.603564i	$0.793747\pi$
$198$	0	0
$199$	−6.53793	−0.463462	−0.231731	−	0.972780i	$-0.574439\pi$
$199$	−6.53793	−0.463462	−0.231731	+	0.972780i	$0.574439\pi$
$200$	0	0
$201$	3.73330	0.263326
$202$	0	0
$203$	0.896309	0.0629085
$204$	0	0
$205$	8.94521	0.624761
$206$	0	0
$207$	−20.2853	−1.40993
$208$	0	0
$209$	−21.9528	−1.51851
$210$	0	0
$211$	−8.39935	−0.578235	−0.289118	−	0.957294i	$-0.593362\pi$
$211$	−8.39935	−0.578235	−0.289118	+	0.957294i	$0.593362\pi$
$212$	0	0
$213$	1.95112	0.133689
$214$	0	0
$215$	6.93335	0.472851
$216$	0	0
$217$	−1.42277	−0.0965841
$218$	0	0
$219$	−1.70334	−0.115101
$220$	0	0
$221$	7.73766	0.520492
$222$	0	0
$223$	6.64899	0.445249	0.222625	−	0.974904i	$-0.428538\pi$
$223$	6.64899	0.445249	0.222625	+	0.974904i	$0.428538\pi$
$224$	0	0
$225$	7.78675	0.519116
$226$	0	0
$227$	6.30474	0.418460	0.209230	−	0.977866i	$-0.432904\pi$
$227$	6.30474	0.418460	0.209230	+	0.977866i	$0.432904\pi$
$228$	0	0
$229$	16.2886	1.07638	0.538190	−	0.842823i	$-0.319108\pi$
$229$	16.2886	1.07638	0.538190	+	0.842823i	$0.319108\pi$
$230$	0	0
$231$	0.311696	0.0205081
$232$	0	0
$233$	−25.2148	−1.65188	−0.825940	−	0.563759i	$-0.809355\pi$
$233$	−25.2148	−1.65188	−0.825940	+	0.563759i	$0.809355\pi$
$234$	0	0
$235$	−17.8836	−1.16660
$236$	0	0
$237$	−4.05168	−0.263185
$238$	0	0
$239$	−14.7833	−0.956254	−0.478127	−	0.878291i	$-0.658684\pi$
$239$	−14.7833	−0.956254	−0.478127	+	0.878291i	$0.658684\pi$
$240$	0	0
$241$	2.03919	0.131356	0.0656779	−	0.997841i	$-0.479079\pi$
$241$	2.03919	0.131356	0.0656779	+	0.997841i	$0.479079\pi$
$242$	0	0
$243$	11.7031	0.750754
$244$	0	0
$245$	−10.2960	−0.657789
$246$	0	0
$247$	16.1477	1.02746
$248$	0	0
$249$	6.83183	0.432949
$250$	0	0
$251$	−8.69188	−0.548627	−0.274313	−	0.961640i	$-0.588451\pi$
$251$	−8.69188	−0.548627	−0.274313	+	0.961640i	$0.588451\pi$
$252$	0	0
$253$	−24.7862	−1.55830
$254$	0	0
$255$	2.21878	0.138946
$256$	0	0
$257$	2.91308	0.181713	0.0908563	−	0.995864i	$-0.471040\pi$
$257$	2.91308	0.181713	0.0908563	+	0.995864i	$0.471040\pi$
$258$	0	0
$259$	1.74666	0.108532
$260$	0	0
$261$	12.9792	0.803395
$262$	0	0
$263$	−0.168737	−0.0104048	−0.00520238	−	0.999986i	$-0.501656\pi$
$263$	−0.168737	−0.0104048	−0.00520238	+	0.999986i	$0.501656\pi$
$264$	0	0
$265$	5.48477	0.336926
$266$	0	0
$267$	3.38839	0.207366
$268$	0	0
$269$	−31.3574	−1.91190	−0.955949	−	0.293534i	$-0.905169\pi$
$269$	−31.3574	−1.91190	−0.955949	+	0.293534i	$0.905169\pi$
$270$	0	0
$271$	−18.1938	−1.10520	−0.552599	−	0.833447i	$-0.686364\pi$
$271$	−18.1938	−1.10520	−0.552599	+	0.833447i	$0.686364\pi$
$272$	0	0
$273$	−0.229273	−0.0138762
$274$	0	0
$275$	9.51447	0.573744
$276$	0	0
$277$	10.1472	0.609683	0.304842	−	0.952403i	$-0.401396\pi$
$277$	10.1472	0.609683	0.304842	+	0.952403i	$0.401396\pi$
$278$	0	0
$279$	−20.6028	−1.23346
$280$	0	0
$281$	17.8072	1.06229	0.531144	−	0.847281i	$-0.321762\pi$
$281$	17.8072	1.06229	0.531144	+	0.847281i	$0.321762\pi$
$282$	0	0
$283$	−16.2490	−0.965900	−0.482950	−	0.875648i	$-0.660435\pi$
$283$	−16.2490	−0.965900	−0.482950	+	0.875648i	$0.660435\pi$
$284$	0	0
$285$	4.63038	0.274280
$286$	0	0
$287$	−1.15616	−0.0682457
$288$	0	0
$289$	−7.32175	−0.430691
$290$	0	0
$291$	1.38675	0.0812927
$292$	0	0
$293$	14.6588	0.856373	0.428187	−	0.903690i	$-0.359153\pi$
$293$	14.6588	0.856373	0.428187	+	0.903690i	$0.359153\pi$
$294$	0	0
$295$	4.49831	0.261901
$296$	0	0
$297$	9.40669	0.545831
$298$	0	0
$299$	18.2319	1.05438
$300$	0	0
$301$	−0.896125	−0.0516518
$302$	0	0
$303$	3.78356	0.217360
$304$	0	0
$305$	13.9700	0.799923
$306$	0	0
$307$	12.0834	0.689638	0.344819	−	0.938669i	$-0.387940\pi$
$307$	12.0834	0.689638	0.344819	+	0.938669i	$0.387940\pi$
$308$	0	0
$309$	2.81097	0.159911
$310$	0	0
$311$	7.11798	0.403624	0.201812	−	0.979424i	$-0.435317\pi$
$311$	7.11798	0.403624	0.201812	+	0.979424i	$0.435317\pi$
$312$	0	0
$313$	3.65509	0.206598	0.103299	−	0.994650i	$-0.467060\pi$
$313$	3.65509	0.206598	0.103299	+	0.994650i	$0.467060\pi$
$314$	0	0
$315$	0.781939	0.0440573
$316$	0	0
$317$	21.4635	1.20551	0.602754	−	0.797927i	$-0.294070\pi$
$317$	21.4635	1.20551	0.602754	+	0.797927i	$0.294070\pi$
$318$	0	0
$319$	15.8591	0.887937
$320$	0	0
$321$	2.90238	0.161995
$322$	0	0
$323$	20.1976	1.12382
$324$	0	0
$325$	−6.99853	−0.388209
$326$	0	0
$327$	−2.18994	−0.121104
$328$	0	0
$329$	2.31143	0.127433
$330$	0	0
$331$	12.6454	0.695054	0.347527	−	0.937670i	$-0.387022\pi$
$331$	12.6454	0.695054	0.347527	+	0.937670i	$0.387022\pi$
$332$	0	0
$333$	25.2930	1.38605
$334$	0	0
$335$	11.4436	0.625231
$336$	0	0
$337$	17.3525	0.945254	0.472627	−	0.881263i	$-0.343306\pi$
$337$	17.3525	0.945254	0.472627	+	0.881263i	$0.343306\pi$
$338$	0	0
$339$	4.75787	0.258412
$340$	0	0
$341$	−25.1742	−1.36326
$342$	0	0
$343$	2.66847	0.144084
$344$	0	0
$345$	5.22803	0.281468
$346$	0	0
$347$	17.4430	0.936387	0.468193	−	0.883626i	$-0.344905\pi$
$347$	17.4430	0.936387	0.468193	+	0.883626i	$0.344905\pi$
$348$	0	0
$349$	−0.518918	−0.0277770	−0.0138885	−	0.999904i	$-0.504421\pi$
$349$	−0.518918	−0.0277770	−0.0138885	+	0.999904i	$0.504421\pi$
$350$	0	0
$351$	−6.91925	−0.369322
$352$	0	0
$353$	2.30663	0.122769	0.0613846	−	0.998114i	$-0.480448\pi$
$353$	2.30663	0.122769	0.0613846	+	0.998114i	$0.480448\pi$
$354$	0	0
$355$	5.98074	0.317425
$356$	0	0
$357$	−0.286774	−0.0151777
$358$	0	0
$359$	−8.52203	−0.449776	−0.224888	−	0.974385i	$-0.572202\pi$
$359$	−8.52203	−0.449776	−0.224888	+	0.974385i	$0.572202\pi$
$360$	0	0
$361$	23.1503	1.21844
$362$	0	0
$363$	0.209089	0.0109743
$364$	0	0
$365$	−5.22123	−0.273291
$366$	0	0
$367$	5.67199	0.296076	0.148038	−	0.988982i	$-0.452704\pi$
$367$	5.67199	0.296076	0.148038	+	0.988982i	$0.452704\pi$
$368$	0	0
$369$	−16.7420	−0.871555
$370$	0	0
$371$	−0.708898	−0.0368041
$372$	0	0
$373$	−34.8096	−1.80237	−0.901186	−	0.433433i	$-0.857302\pi$
$373$	−34.8096	−1.80237	−0.901186	+	0.433433i	$0.857302\pi$
$374$	0	0
$375$	−5.57288	−0.287782
$376$	0	0
$377$	−11.6654	−0.600799
$378$	0	0
$379$	3.14210	0.161399	0.0806994	−	0.996738i	$-0.474285\pi$
$379$	3.14210	0.161399	0.0806994	+	0.996738i	$0.474285\pi$
$380$	0	0
$381$	7.45392	0.381876
$382$	0	0
$383$	−26.6159	−1.36001	−0.680004	−	0.733208i	$-0.738022\pi$
$383$	−26.6159	−1.36001	−0.680004	+	0.733208i	$0.738022\pi$
$384$	0	0
$385$	0.955435	0.0486935
$386$	0	0
$387$	−12.9766	−0.659637
$388$	0	0
$389$	12.2725	0.622241	0.311120	−	0.950371i	$-0.399296\pi$
$389$	12.2725	0.622241	0.311120	+	0.950371i	$0.399296\pi$
$390$	0	0
$391$	22.8045	1.15327
$392$	0	0
$393$	7.61213	0.383981
$394$	0	0
$395$	−12.4195	−0.624895
$396$	0	0
$397$	−14.3026	−0.717827	−0.358913	−	0.933371i	$-0.616853\pi$
$397$	−14.3026	−0.717827	−0.358913	+	0.933371i	$0.616853\pi$
$398$	0	0
$399$	−0.598470	−0.0299610
$400$	0	0
$401$	−12.8160	−0.639999	−0.320000	−	0.947418i	$-0.603683\pi$
$401$	−12.8160	−0.639999	−0.320000	+	0.947418i	$0.603683\pi$
$402$	0	0
$403$	18.5173	0.922413
$404$	0	0
$405$	10.2910	0.511364
$406$	0	0
$407$	30.9050	1.53191
$408$	0	0
$409$	−14.9207	−0.737780	−0.368890	−	0.929473i	$-0.620262\pi$
$409$	−14.9207	−0.737780	−0.368890	+	0.929473i	$0.620262\pi$
$410$	0	0
$411$	3.97826	0.196233
$412$	0	0
$413$	−0.581399	−0.0286088
$414$	0	0
$415$	20.9415	1.02798
$416$	0	0
$417$	3.81897	0.187016
$418$	0	0
$419$	−23.3872	−1.14254	−0.571269	−	0.820763i	$-0.693549\pi$
$419$	−23.3872	−1.14254	−0.571269	+	0.820763i	$0.693549\pi$
$420$	0	0
$421$	−37.1764	−1.81187	−0.905935	−	0.423418i	$-0.860830\pi$
$421$	−37.1764	−1.81187	−0.905935	+	0.423418i	$0.860830\pi$
$422$	0	0
$423$	33.4713	1.62743
$424$	0	0
$425$	−8.75375	−0.424619
$426$	0	0
$427$	−1.80561	−0.0873795
$428$	0	0
$429$	−4.05671	−0.195860
$430$	0	0
$431$	−27.4006	−1.31984	−0.659921	−	0.751335i	$-0.729410\pi$
$431$	−27.4006	−1.31984	−0.659921	+	0.751335i	$0.729410\pi$
$432$	0	0
$433$	−17.6255	−0.847027	−0.423514	−	0.905890i	$-0.639203\pi$
$433$	−17.6255	−0.847027	−0.423514	+	0.905890i	$0.639203\pi$
$434$	0	0
$435$	−3.34507	−0.160384
$436$	0	0
$437$	47.5907	2.27657
$438$	0	0
$439$	−18.3089	−0.873834	−0.436917	−	0.899502i	$-0.643930\pi$
$439$	−18.3089	−0.873834	−0.436917	+	0.899502i	$0.643930\pi$
$440$	0	0
$441$	19.2702	0.917630
$442$	0	0
$443$	33.4020	1.58697	0.793487	−	0.608587i	$-0.208263\pi$
$443$	33.4020	1.58697	0.793487	+	0.608587i	$0.208263\pi$
$444$	0	0
$445$	10.3864	0.492361
$446$	0	0
$447$	−3.42805	−0.162141
$448$	0	0
$449$	0.0878169	0.00414433	0.00207217	−	0.999998i	$-0.499340\pi$
$449$	0.0878169	0.00414433	0.00207217	+	0.999998i	$0.499340\pi$
$450$	0	0
$451$	−20.4567	−0.963270
$452$	0	0
$453$	−7.34917	−0.345294
$454$	0	0
$455$	−0.702787	−0.0329472
$456$	0	0
$457$	−16.0275	−0.749733	−0.374867	−	0.927079i	$-0.622311\pi$
$457$	−16.0275	−0.749733	−0.374867	+	0.927079i	$0.622311\pi$
$458$	0	0
$459$	−8.65458	−0.403961
$460$	0	0
$461$	15.7965	0.735716	0.367858	−	0.929882i	$-0.380091\pi$
$461$	15.7965	0.735716	0.367858	+	0.929882i	$0.380091\pi$
$462$	0	0
$463$	21.2329	0.986779	0.493389	−	0.869809i	$-0.335758\pi$
$463$	21.2329	0.986779	0.493389	+	0.869809i	$0.335758\pi$
$464$	0	0
$465$	5.30986	0.246239
$466$	0	0
$467$	9.98565	0.462081	0.231041	−	0.972944i	$-0.425787\pi$
$467$	9.98565	0.462081	0.231041	+	0.972944i	$0.425787\pi$
$468$	0	0
$469$	−1.47907	−0.0682970
$470$	0	0
$471$	0.278258	0.0128214
$472$	0	0
$473$	−15.8558	−0.729052
$474$	0	0
$475$	−18.2682	−0.838203
$476$	0	0
$477$	−10.2654	−0.470020
$478$	0	0
$479$	4.02741	0.184017	0.0920085	−	0.995758i	$-0.470671\pi$
$479$	4.02741	0.184017	0.0920085	+	0.995758i	$0.470671\pi$
$480$	0	0
$481$	−22.7327	−1.03652
$482$	0	0
$483$	−0.675715	−0.0307461
$484$	0	0
$485$	4.25078	0.193018
$486$	0	0
$487$	9.46713	0.428997	0.214498	−	0.976724i	$-0.431188\pi$
$487$	9.46713	0.428997	0.214498	+	0.976724i	$0.431188\pi$
$488$	0	0
$489$	11.6744	0.527932
$490$	0	0
$491$	34.2345	1.54498	0.772491	−	0.635026i	$-0.219011\pi$
$491$	34.2345	1.54498	0.772491	+	0.635026i	$0.219011\pi$
$492$	0	0
$493$	−14.5911	−0.657149
$494$	0	0
$495$	13.8354	0.621857
$496$	0	0
$497$	−0.773002	−0.0346739
$498$	0	0
$499$	−15.8718	−0.710519	−0.355259	−	0.934768i	$-0.615607\pi$
$499$	−15.8718	−0.710519	−0.355259	+	0.934768i	$0.615607\pi$
$500$	0	0
$501$	−7.64981	−0.341768
$502$	0	0
$503$	−23.7492	−1.05892	−0.529462	−	0.848334i	$-0.677606\pi$
$503$	−23.7492	−1.05892	−0.529462	+	0.848334i	$0.677606\pi$
$504$	0	0
$505$	11.5977	0.516090
$506$	0	0
$507$	−3.28672	−0.145969
$508$	0	0
$509$	−9.63458	−0.427045	−0.213523	−	0.976938i	$-0.568494\pi$
$509$	−9.63458	−0.427045	−0.213523	+	0.976938i	$0.568494\pi$
$510$	0	0
$511$	0.674835	0.0298530
$512$	0	0
$513$	−18.0613	−0.797424
$514$	0	0
$515$	8.61642	0.379685
$516$	0	0
$517$	40.8979	1.79869
$518$	0	0
$519$	10.5451	0.462880
$520$	0	0
$521$	−35.9535	−1.57515	−0.787575	−	0.616219i	$-0.788664\pi$
$521$	−35.9535	−1.57515	−0.787575	+	0.616219i	$0.788664\pi$
$522$	0	0
$523$	−9.40377	−0.411198	−0.205599	−	0.978636i	$-0.565914\pi$
$523$	−9.40377	−0.411198	−0.205599	+	0.978636i	$0.565914\pi$
$524$	0	0
$525$	0.259381	0.0113203
$526$	0	0
$527$	23.1614	1.00893
$528$	0	0
$529$	30.7332	1.33623
$530$	0	0
$531$	−8.41911	−0.365358
$532$	0	0
$533$	15.0473	0.651771
$534$	0	0
$535$	8.89661	0.384634
$536$	0	0
$537$	7.39015	0.318908
$538$	0	0
$539$	23.5459	1.01419
$540$	0	0
$541$	−32.0540	−1.37811	−0.689054	−	0.724710i	$-0.741974\pi$
$541$	−32.0540	−1.37811	−0.689054	+	0.724710i	$0.741974\pi$
$542$	0	0
$543$	5.08915	0.218396
$544$	0	0
$545$	−6.71279	−0.287545
$546$	0	0
$547$	−12.3529	−0.528170	−0.264085	−	0.964499i	$-0.585070\pi$
$547$	−12.3529	−0.528170	−0.264085	+	0.964499i	$0.585070\pi$
$548$	0	0
$549$	−26.1466	−1.11591
$550$	0	0
$551$	−30.4501	−1.29722
$552$	0	0
$553$	1.60521	0.0682603
$554$	0	0
$555$	−6.51863	−0.276701
$556$	0	0
$557$	15.2297	0.645301	0.322651	−	0.946518i	$-0.395426\pi$
$557$	15.2297	0.645301	0.322651	+	0.946518i	$0.395426\pi$
$558$	0	0
$559$	11.6630	0.493293
$560$	0	0
$561$	−5.07412	−0.214229
$562$	0	0
$563$	−47.1903	−1.98883	−0.994417	−	0.105522i	$-0.966349\pi$
$563$	−47.1903	−1.98883	−0.994417	+	0.105522i	$0.966349\pi$
$564$	0	0
$565$	14.5842	0.613563
$566$	0	0
$567$	−1.33010	−0.0558588
$568$	0	0
$569$	−23.6128	−0.989900	−0.494950	−	0.868921i	$-0.664814\pi$
$569$	−23.6128	−0.989900	−0.494950	+	0.868921i	$0.664814\pi$
$570$	0	0
$571$	−27.8322	−1.16474	−0.582371	−	0.812923i	$-0.697875\pi$
$571$	−27.8322	−1.16474	−0.582371	+	0.812923i	$0.697875\pi$
$572$	0	0
$573$	11.0829	0.462994
$574$	0	0
$575$	−20.6261	−0.860168
$576$	0	0
$577$	42.1981	1.75673	0.878365	−	0.477991i	$-0.158635\pi$
$577$	42.1981	1.75673	0.878365	+	0.477991i	$0.158635\pi$
$578$	0	0
$579$	−8.79671	−0.365579
$580$	0	0
$581$	−2.70665	−0.112291
$582$	0	0
$583$	−12.5431	−0.519481
$584$	0	0
$585$	−10.1769	−0.420763
$586$	0	0
$587$	−18.7845	−0.775320	−0.387660	−	0.921803i	$-0.626716\pi$
$587$	−18.7845	−0.775320	−0.387660	+	0.921803i	$0.626716\pi$
$588$	0	0
$589$	48.3356	1.99163
$590$	0	0
$591$	−10.7961	−0.444091
$592$	0	0
$593$	−0.516291	−0.0212015	−0.0106008	−	0.999944i	$-0.503374\pi$
$593$	−0.516291	−0.0212015	−0.0106008	+	0.999944i	$0.503374\pi$
$594$	0	0
$595$	−0.879044	−0.0360373
$596$	0	0
$597$	−3.15365	−0.129070
$598$	0	0
$599$	19.3135	0.789130	0.394565	−	0.918868i	$-0.370895\pi$
$599$	19.3135	0.789130	0.394565	+	0.918868i	$0.370895\pi$
$600$	0	0
$601$	−4.58992	−0.187227	−0.0936133	−	0.995609i	$-0.529842\pi$
$601$	−4.58992	−0.187227	−0.0936133	+	0.995609i	$0.529842\pi$
$602$	0	0
$603$	−21.4180	−0.872210
$604$	0	0
$605$	0.640917	0.0260570
$606$	0	0
$607$	−13.8854	−0.563591	−0.281795	−	0.959475i	$-0.590930\pi$
$607$	−13.8854	−0.563591	−0.281795	+	0.959475i	$0.590930\pi$
$608$	0	0
$609$	0.432345	0.0175195
$610$	0	0
$611$	−30.0832	−1.21704
$612$	0	0
$613$	22.7226	0.917756	0.458878	−	0.888499i	$-0.348251\pi$
$613$	22.7226	0.917756	0.458878	+	0.888499i	$0.348251\pi$
$614$	0	0
$615$	4.31483	0.173991
$616$	0	0
$617$	15.7752	0.635084	0.317542	−	0.948244i	$-0.397143\pi$
$617$	15.7752	0.635084	0.317542	+	0.948244i	$0.397143\pi$
$618$	0	0
$619$	−0.432706	−0.0173919	−0.00869595	−	0.999962i	$-0.502768\pi$
$619$	−0.432706	−0.0173919	−0.00869595	+	0.999962i	$0.502768\pi$
$620$	0	0
$621$	−20.3924	−0.818320
$622$	0	0
$623$	−1.34242	−0.0537830
$624$	0	0
$625$	−3.01337	−0.120535
$626$	0	0
$627$	−10.5892	−0.422891
$628$	0	0
$629$	−28.4340	−1.13374
$630$	0	0
$631$	12.0025	0.477814	0.238907	−	0.971042i	$-0.423211\pi$
$631$	12.0025	0.477814	0.238907	+	0.971042i	$0.423211\pi$
$632$	0	0
$633$	−4.05153	−0.161034
$634$	0	0
$635$	22.8484	0.906710
$636$	0	0
$637$	−17.3196	−0.686227
$638$	0	0
$639$	−11.1937	−0.442814
$640$	0	0
$641$	22.4227	0.885644	0.442822	−	0.896610i	$-0.353977\pi$
$641$	22.4227	0.885644	0.442822	+	0.896610i	$0.353977\pi$
$642$	0	0
$643$	−10.6035	−0.418162	−0.209081	−	0.977898i	$-0.567047\pi$
$643$	−10.6035	−0.418162	−0.209081	+	0.977898i	$0.567047\pi$
$644$	0	0
$645$	3.34438	0.131685
$646$	0	0
$647$	−37.3378	−1.46790	−0.733949	−	0.679204i	$-0.762325\pi$
$647$	−37.3378	−1.46790	−0.733949	+	0.679204i	$0.762325\pi$
$648$	0	0
$649$	−10.2871	−0.403805
$650$	0	0
$651$	−0.686292	−0.0268979
$652$	0	0
$653$	−28.2960	−1.10731	−0.553654	−	0.832747i	$-0.686767\pi$
$653$	−28.2960	−1.10731	−0.553654	+	0.832747i	$0.686767\pi$
$654$	0	0
$655$	23.3333	0.911708
$656$	0	0
$657$	9.77214	0.381247
$658$	0	0
$659$	−16.3911	−0.638506	−0.319253	−	0.947669i	$-0.603432\pi$
$659$	−16.3911	−0.638506	−0.319253	+	0.947669i	$0.603432\pi$
$660$	0	0
$661$	−17.6986	−0.688394	−0.344197	−	0.938897i	$-0.611849\pi$
$661$	−17.6986	−0.688394	−0.344197	+	0.938897i	$0.611849\pi$
$662$	0	0
$663$	3.73235	0.144953
$664$	0	0
$665$	−1.83448	−0.0711381
$666$	0	0
$667$	−34.3803	−1.33121
$668$	0	0
$669$	3.20722	0.123998
$670$	0	0
$671$	−31.9480	−1.23334
$672$	0	0
$673$	49.6916	1.91547	0.957735	−	0.287654i	$-0.0928752\pi$
$673$	49.6916	1.91547	0.957735	+	0.287654i	$0.0928752\pi$
$674$	0	0
$675$	7.82786	0.301295
$676$	0	0
$677$	−24.6711	−0.948187	−0.474094	−	0.880474i	$-0.657224\pi$
$677$	−24.6711	−0.948187	−0.474094	+	0.880474i	$0.657224\pi$
$678$	0	0
$679$	−0.549407	−0.0210843
$680$	0	0
$681$	3.04116	0.116538
$682$	0	0
$683$	28.1330	1.07648	0.538239	−	0.842792i	$-0.319090\pi$
$683$	28.1330	1.07648	0.538239	+	0.842792i	$0.319090\pi$
$684$	0	0
$685$	12.1945	0.465927
$686$	0	0
$687$	7.85700	0.299763
$688$	0	0
$689$	9.22627	0.351493
$690$	0	0
$691$	1.65710	0.0630392	0.0315196	−	0.999503i	$-0.489965\pi$
$691$	1.65710	0.0630392	0.0315196	+	0.999503i	$0.489965\pi$
$692$	0	0
$693$	−1.78821	−0.0679285
$694$	0	0
$695$	11.7062	0.444043
$696$	0	0
$697$	18.8211	0.712901
$698$	0	0
$699$	−12.1627	−0.460035
$700$	0	0
$701$	24.1405	0.911776	0.455888	−	0.890037i	$-0.349322\pi$
$701$	24.1405	0.911776	0.455888	+	0.890037i	$0.349322\pi$
$702$	0	0
$703$	−59.3391	−2.23802
$704$	0	0
$705$	−8.62639	−0.324888
$706$	0	0
$707$	−1.49898	−0.0563750
$708$	0	0
$709$	−12.2580	−0.460360	−0.230180	−	0.973148i	$-0.573932\pi$
$709$	−12.2580	−0.460360	−0.230180	+	0.973148i	$0.573932\pi$
$710$	0	0
$711$	23.2446	0.871741
$712$	0	0
$713$	54.5743	2.04382
$714$	0	0
$715$	−12.4349	−0.465041
$716$	0	0
$717$	−7.13091	−0.266309
$718$	0	0
$719$	18.8205	0.701888	0.350944	−	0.936397i	$-0.385861\pi$
$719$	18.8205	0.701888	0.350944	+	0.936397i	$0.385861\pi$
$720$	0	0
$721$	−1.11366	−0.0414748
$722$	0	0
$723$	0.983628	0.0365815
$724$	0	0
$725$	13.1973	0.490135
$726$	0	0
$727$	−23.7403	−0.880479	−0.440240	−	0.897880i	$-0.645107\pi$
$727$	−23.7403	−0.880479	−0.440240	+	0.897880i	$0.645107\pi$
$728$	0	0
$729$	−15.2351	−0.564263
$730$	0	0
$731$	14.5881	0.539560
$732$	0	0
$733$	33.4242	1.23455	0.617275	−	0.786747i	$-0.288236\pi$
$733$	33.4242	1.23455	0.617275	+	0.786747i	$0.288236\pi$
$734$	0	0
$735$	−4.96641	−0.183189
$736$	0	0
$737$	−26.1703	−0.963995
$738$	0	0
$739$	−13.2017	−0.485633	−0.242817	−	0.970072i	$-0.578071\pi$
$739$	−13.2017	−0.485633	−0.242817	+	0.970072i	$0.578071\pi$
$740$	0	0
$741$	7.78906	0.286138
$742$	0	0
$743$	−20.1022	−0.737479	−0.368739	−	0.929533i	$-0.620211\pi$
$743$	−20.1022	−0.737479	−0.368739	+	0.929533i	$0.620211\pi$
$744$	0	0
$745$	−10.5079	−0.384981
$746$	0	0
$747$	−39.1944	−1.43405
$748$	0	0
$749$	−1.14987	−0.0420155
$750$	0	0
$751$	17.6604	0.644438	0.322219	−	0.946665i	$-0.395571\pi$
$751$	17.6604	0.644438	0.322219	+	0.946665i	$0.395571\pi$
$752$	0	0
$753$	−4.19263	−0.152788
$754$	0	0
$755$	−22.5273	−0.819852
$756$	0	0
$757$	−5.91874	−0.215120	−0.107560	−	0.994199i	$-0.534304\pi$
$757$	−5.91874	−0.215120	−0.107560	+	0.994199i	$0.534304\pi$
$758$	0	0
$759$	−11.9559	−0.433973
$760$	0	0
$761$	−12.5059	−0.453340	−0.226670	−	0.973972i	$-0.572784\pi$
$761$	−12.5059	−0.453340	−0.226670	+	0.973972i	$0.572784\pi$
$762$	0	0
$763$	0.867619	0.0314099
$764$	0	0
$765$	−12.7292	−0.460226
$766$	0	0
$767$	7.56688	0.273224
$768$	0	0
$769$	−32.3761	−1.16751	−0.583755	−	0.811930i	$-0.698417\pi$
$769$	−32.3761	−1.16751	−0.583755	+	0.811930i	$0.698417\pi$
$770$	0	0
$771$	1.40516	0.0506055
$772$	0	0
$773$	−23.5039	−0.845376	−0.422688	−	0.906275i	$-0.638913\pi$
$773$	−23.5039	−0.845376	−0.422688	+	0.906275i	$0.638913\pi$
$774$	0	0
$775$	−20.9490	−0.752509
$776$	0	0
$777$	0.842524	0.0302254
$778$	0	0
$779$	39.2779	1.40728
$780$	0	0
$781$	−13.6773	−0.489413
$782$	0	0
$783$	13.0478	0.466289
$784$	0	0
$785$	0.852939	0.0304427
$786$	0	0
$787$	−54.9167	−1.95757	−0.978784	−	0.204895i	$-0.934315\pi$
$787$	−54.9167	−1.95757	−0.978784	+	0.204895i	$0.934315\pi$
$788$	0	0
$789$	−0.0813922	−0.00289764
$790$	0	0
$791$	−1.88499	−0.0670225
$792$	0	0
$793$	23.4999	0.834506
$794$	0	0
$795$	2.64564	0.0938313
$796$	0	0
$797$	−17.6884	−0.626556	−0.313278	−	0.949662i	$-0.601427\pi$
$797$	−17.6884	−0.626556	−0.313278	+	0.949662i	$0.601427\pi$
$798$	0	0
$799$	−37.6280	−1.33118
$800$	0	0
$801$	−19.4393	−0.686854
$802$	0	0
$803$	11.9404	0.421367
$804$	0	0
$805$	−2.07126	−0.0730022
$806$	0	0
$807$	−15.1256	−0.532448
$808$	0	0
$809$	−6.21239	−0.218416	−0.109208	−	0.994019i	$-0.534831\pi$
$809$	−6.21239	−0.218416	−0.109208	+	0.994019i	$0.534831\pi$
$810$	0	0
$811$	−29.3516	−1.03067	−0.515337	−	0.856987i	$-0.672333\pi$
$811$	−29.3516	−1.03067	−0.515337	+	0.856987i	$0.672333\pi$
$812$	0	0
$813$	−8.77602	−0.307788
$814$	0	0
$815$	35.7852	1.25350
$816$	0	0
$817$	30.4439	1.06510
$818$	0	0
$819$	1.31535	0.0459620
$820$	0	0
$821$	49.2023	1.71717	0.858587	−	0.512669i	$-0.171343\pi$
$821$	49.2023	1.71717	0.858587	+	0.512669i	$0.171343\pi$
$822$	0	0
$823$	−20.3730	−0.710158	−0.355079	−	0.934836i	$-0.615546\pi$
$823$	−20.3730	−0.710158	−0.355079	+	0.934836i	$0.615546\pi$
$824$	0	0
$825$	4.58942	0.159783
$826$	0	0
$827$	−13.7934	−0.479643	−0.239822	−	0.970817i	$-0.577089\pi$
$827$	−13.7934	−0.479643	−0.239822	+	0.970817i	$0.577089\pi$
$828$	0	0
$829$	52.0283	1.80702	0.903509	−	0.428568i	$-0.140982\pi$
$829$	52.0283	1.80702	0.903509	+	0.428568i	$0.140982\pi$
$830$	0	0
$831$	4.89460	0.169792
$832$	0	0
$833$	−21.6633	−0.750589
$834$	0	0
$835$	−23.4488	−0.811480
$836$	0	0
$837$	−20.7116	−0.715899
$838$	0	0
$839$	7.76092	0.267937	0.133968	−	0.990986i	$-0.457228\pi$
$839$	7.76092	0.267937	0.133968	+	0.990986i	$0.457228\pi$
$840$	0	0
$841$	−7.00229	−0.241458
$842$	0	0
$843$	8.58951	0.295839
$844$	0	0
$845$	−10.0747	−0.346582
$846$	0	0
$847$	−0.0828376	−0.00284633
$848$	0	0
$849$	−7.83788	−0.268995
$850$	0	0
$851$	−66.9980	−2.29666
$852$	0	0
$853$	21.4244	0.733559	0.366780	−	0.930308i	$-0.380460\pi$
$853$	21.4244	0.733559	0.366780	+	0.930308i	$0.380460\pi$
$854$	0	0
$855$	−26.5647	−0.908493
$856$	0	0
$857$	36.0746	1.23229	0.616143	−	0.787634i	$-0.288694\pi$
$857$	36.0746	1.23229	0.616143	+	0.787634i	$0.288694\pi$
$858$	0	0
$859$	22.5828	0.770516	0.385258	−	0.922809i	$-0.374112\pi$
$859$	22.5828	0.770516	0.385258	+	0.922809i	$0.374112\pi$
$860$	0	0
$861$	−0.557685	−0.0190059
$862$	0	0
$863$	0.587161	0.0199872	0.00999360	−	0.999950i	$-0.496819\pi$
$863$	0.587161	0.0199872	0.00999360	+	0.999950i	$0.496819\pi$
$864$	0	0
$865$	32.3238	1.09904
$866$	0	0
$867$	−3.53173	−0.119944
$868$	0	0
$869$	28.4021	0.963476
$870$	0	0
$871$	19.2500	0.652261
$872$	0	0
$873$	−7.95583	−0.269264
$874$	0	0
$875$	2.20788	0.0746400
$876$	0	0
$877$	−2.09546	−0.0707588	−0.0353794	−	0.999374i	$-0.511264\pi$
$877$	−2.09546	−0.0707588	−0.0353794	+	0.999374i	$0.511264\pi$
$878$	0	0
$879$	7.07082	0.238493
$880$	0	0
$881$	−6.47745	−0.218231	−0.109115	−	0.994029i	$-0.534802\pi$
$881$	−6.47745	−0.218231	−0.109115	+	0.994029i	$0.534802\pi$
$882$	0	0
$883$	−15.2716	−0.513930	−0.256965	−	0.966421i	$-0.582723\pi$
$883$	−15.2716	−0.513930	−0.256965	+	0.966421i	$0.582723\pi$
$884$	0	0
$885$	2.16981	0.0729374
$886$	0	0
$887$	38.1041	1.27941	0.639705	−	0.768621i	$-0.279057\pi$
$887$	38.1041	1.27941	0.639705	+	0.768621i	$0.279057\pi$
$888$	0	0
$889$	−2.95312	−0.0990443
$890$	0	0
$891$	−23.5344	−0.788432
$892$	0	0
$893$	−78.5259	−2.62777
$894$	0	0
$895$	22.6529	0.757203
$896$	0	0
$897$	8.79440	0.293636
$898$	0	0
$899$	−34.9185	−1.16460
$900$	0	0
$901$	11.5402	0.384460
$902$	0	0
$903$	−0.432257	−0.0143846
$904$	0	0
$905$	15.5997	0.518551
$906$	0	0
$907$	−10.0603	−0.334048	−0.167024	−	0.985953i	$-0.553416\pi$
$907$	−10.0603	−0.334048	−0.167024	+	0.985953i	$0.553416\pi$
$908$	0	0
$909$	−21.7064	−0.719956
$910$	0	0
$911$	17.1254	0.567389	0.283694	−	0.958915i	$-0.408440\pi$
$911$	17.1254	0.567389	0.283694	+	0.958915i	$0.408440\pi$
$912$	0	0
$913$	−47.8909	−1.58496
$914$	0	0
$915$	6.73862	0.222772
$916$	0	0
$917$	−3.01580	−0.0995903
$918$	0	0
$919$	−41.1789	−1.35837	−0.679183	−	0.733969i	$-0.737666\pi$
$919$	−41.1789	−1.35837	−0.679183	+	0.733969i	$0.737666\pi$
$920$	0	0
$921$	5.82859	0.192059
$922$	0	0
$923$	10.0606	0.331148
$924$	0	0
$925$	25.7179	0.845600
$926$	0	0
$927$	−16.1266	−0.529668
$928$	0	0
$929$	−2.16235	−0.0709445	−0.0354722	−	0.999371i	$-0.511294\pi$
$929$	−2.16235	−0.0709445	−0.0354722	+	0.999371i	$0.511294\pi$
$930$	0	0
$931$	−45.2092	−1.48167
$932$	0	0
$933$	3.43344	0.112406
$934$	0	0
$935$	−15.5536	−0.508657
$936$	0	0
$937$	−37.8729	−1.23726	−0.618628	−	0.785684i	$-0.712311\pi$
$937$	−37.8729	−1.23726	−0.618628	+	0.785684i	$0.712311\pi$
$938$	0	0
$939$	1.76308	0.0575359
$940$	0	0
$941$	42.6094	1.38903	0.694514	−	0.719479i	$-0.255620\pi$
$941$	42.6094	1.38903	0.694514	+	0.719479i	$0.255620\pi$
$942$	0	0
$943$	44.3475	1.44415
$944$	0	0
$945$	0.786068	0.0255708
$946$	0	0
$947$	−1.21816	−0.0395850	−0.0197925	−	0.999804i	$-0.506301\pi$
$947$	−1.21816	−0.0395850	−0.0197925	+	0.999804i	$0.506301\pi$
$948$	0	0
$949$	−8.78295	−0.285107
$950$	0	0
$951$	10.3532	0.335724
$952$	0	0
$953$	44.4387	1.43951	0.719756	−	0.694227i	$-0.244254\pi$
$953$	44.4387	1.43951	0.719756	+	0.694227i	$0.244254\pi$
$954$	0	0
$955$	33.9721	1.09931
$956$	0	0
$957$	7.64981	0.247283
$958$	0	0
$959$	−1.57612	−0.0508955
$960$	0	0
$961$	24.4285	0.788017
$962$	0	0
$963$	−16.6511	−0.536573
$964$	0	0
$965$	−26.9644	−0.868015
$966$	0	0
$967$	25.1250	0.807967	0.403983	−	0.914766i	$-0.367625\pi$
$967$	25.1250	0.807967	0.403983	+	0.914766i	$0.367625\pi$
$968$	0	0
$969$	9.74253	0.312975
$970$	0	0
$971$	58.1178	1.86509	0.932545	−	0.361054i	$-0.117583\pi$
$971$	58.1178	1.86509	0.932545	+	0.361054i	$0.117583\pi$
$972$	0	0
$973$	−1.51301	−0.0485050
$974$	0	0
$975$	−3.37583	−0.108113
$976$	0	0
$977$	−16.3541	−0.523215	−0.261608	−	0.965174i	$-0.584253\pi$
$977$	−16.3541	−0.523215	−0.261608	+	0.965174i	$0.584253\pi$
$978$	0	0
$979$	−23.7525	−0.759133
$980$	0	0
$981$	12.5638	0.401131
$982$	0	0
$983$	7.40336	0.236131	0.118065	−	0.993006i	$-0.462331\pi$
$983$	7.40336	0.236131	0.118065	+	0.993006i	$0.462331\pi$
$984$	0	0
$985$	−33.0930	−1.05443
$986$	0	0
$987$	1.11495	0.0354892
$988$	0	0
$989$	34.3733	1.09301
$990$	0	0
$991$	−41.6039	−1.32159	−0.660796	−	0.750566i	$-0.729781\pi$
$991$	−41.6039	−1.32159	−0.660796	+	0.750566i	$0.729781\pi$
$992$	0	0
$993$	6.09966	0.193567
$994$	0	0
$995$	−9.66682	−0.306459
$996$	0	0
$997$	31.7968	1.00702	0.503508	−	0.863991i	$-0.332043\pi$
$997$	31.7968	1.00702	0.503508	+	0.863991i	$0.332043\pi$
$998$	0	0
$999$	25.4266	0.804461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4096.2.a.s.1.4		8
4.3	odd	2		4096.2.a.i.1.6		8
8.3	odd	2	inner	4096.2.a.s.1.3		8
8.5	even	2		4096.2.a.i.1.5		8
64.3	odd	16		1024.2.g.d.641.3	yes	16
64.5	even	16		1024.2.g.f.897.3	yes	16
64.11	odd	16		1024.2.g.g.385.2	yes	16
64.13	even	16		1024.2.g.f.129.3	yes	16
64.19	odd	16		1024.2.g.a.129.3	yes	16
64.21	even	16		1024.2.g.d.385.2	yes	16
64.27	odd	16		1024.2.g.a.897.3	yes	16
64.29	even	16		1024.2.g.g.641.3	yes	16
64.35	odd	16		1024.2.g.g.641.2	yes	16
64.37	even	16		1024.2.g.a.897.2	yes	16
64.43	odd	16		1024.2.g.d.385.3	yes	16
64.45	even	16		1024.2.g.a.129.2	✓	16
64.51	odd	16		1024.2.g.f.129.2	yes	16
64.53	even	16		1024.2.g.g.385.3	yes	16
64.59	odd	16		1024.2.g.f.897.2	yes	16
64.61	even	16		1024.2.g.d.641.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1024.2.g.a.129.2	✓	16	64.45	even	16
1024.2.g.a.129.3	yes	16	64.19	odd	16
1024.2.g.a.897.2	yes	16	64.37	even	16
1024.2.g.a.897.3	yes	16	64.27	odd	16
1024.2.g.d.385.2	yes	16	64.21	even	16
1024.2.g.d.385.3	yes	16	64.43	odd	16
1024.2.g.d.641.2	yes	16	64.61	even	16
1024.2.g.d.641.3	yes	16	64.3	odd	16
1024.2.g.f.129.2	yes	16	64.51	odd	16
1024.2.g.f.129.3	yes	16	64.13	even	16
1024.2.g.f.897.2	yes	16	64.59	odd	16
1024.2.g.f.897.3	yes	16	64.5	even	16
1024.2.g.g.385.2	yes	16	64.11	odd	16
1024.2.g.g.385.3	yes	16	64.53	even	16
1024.2.g.g.641.2	yes	16	64.35	odd	16
1024.2.g.g.641.3	yes	16	64.29	even	16
4096.2.a.i.1.5		8	8.5	even	2
4096.2.a.i.1.6		8	4.3	odd	2
4096.2.a.s.1.3		8	8.3	odd	2	inner
4096.2.a.s.1.4		8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4096 = 2^{12} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4096.a (trivial)

\(f(q)\)	\(=\)	\(q+0.482362 q^{3} +1.47858 q^{5} -0.191104 q^{7} -2.76733 q^{9} +O(q^{10})\)	\(q+0.482362 q^{3} +1.47858 q^{5} -0.191104 q^{7} -2.76733 q^{9} -3.38134 q^{11} +2.48720 q^{13} +0.713208 q^{15} +3.11099 q^{17} +6.49233 q^{19} -0.0921811 q^{21} +7.33030 q^{23} -2.81382 q^{25} -2.78194 q^{27} -4.69017 q^{29} +7.44503 q^{31} -1.63103 q^{33} -0.282561 q^{35} -9.13987 q^{37} +1.19973 q^{39} +6.04989 q^{41} +4.68921 q^{43} -4.09170 q^{45} -12.0952 q^{47} -6.96348 q^{49} +1.50062 q^{51} +3.70949 q^{53} -4.99957 q^{55} +3.13165 q^{57} +3.04232 q^{59} +9.44832 q^{61} +0.528846 q^{63} +3.67752 q^{65} +7.73961 q^{67} +3.53586 q^{69} +4.04494 q^{71} -3.53125 q^{73} -1.35728 q^{75} +0.646187 q^{77} -8.39967 q^{79} +6.96008 q^{81} +14.1633 q^{83} +4.59983 q^{85} -2.26236 q^{87} +7.02458 q^{89} -0.475314 q^{91} +3.59120 q^{93} +9.59940 q^{95} +2.87492 q^{97} +9.35728 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3}+O(q^{10}) \) 8 * q + 8 * q^3	\( 8 q + 8 q^{3} + 8 q^{11} + 8 q^{17} - 8 q^{25} + 8 q^{27} + 40 q^{33} - 8 q^{35} + 32 q^{41} + 56 q^{43} + 8 q^{49} + 48 q^{51} + 8 q^{57} + 32 q^{59} - 16 q^{65} + 24 q^{67} - 8 q^{73} + 16 q^{81} + 48 q^{83} - 8 q^{89} + 8 q^{91} + 8 q^{97} + 64 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^11 + 8 * q^17 - 8 * q^25 + 8 * q^27 + 40 * q^33 - 8 * q^35 + 32 * q^41 + 56 * q^43 + 8 * q^49 + 48 * q^51 + 8 * q^57 + 32 * q^59 - 16 * q^65 + 24 * q^67 - 8 * q^73 + 16 * q^81 + 48 * q^83 - 8 * q^89 + 8 * q^91 + 8 * q^97 + 64 * q^99

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