LMFDB - Embedded newform 1620.3.b.a.809.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(809,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1620.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.6
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.a.809.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.23653 + 3.81115i) q^{5} -12.6606i q^{7} +O(q^{10})$	$q+(-3.23653 + 3.81115i) q^{5} -12.6606i q^{7} +18.7009i q^{11} -6.79817i q^{13} -11.5455 q^{17} +24.2335 q^{19} -29.2202 q^{23} +(-4.04977 - 24.6698i) q^{25} -14.9461i q^{29} +13.8362 q^{31} +(48.2515 + 40.9764i) q^{35} +38.2737i q^{37} +30.9192i q^{41} -13.0796i q^{43} -5.68939 q^{47} -111.291 q^{49} +57.7486 q^{53} +(-71.2720 - 60.5260i) q^{55} -30.3390i q^{59} +77.0497 q^{61} +(25.9089 + 22.0025i) q^{65} +128.396i q^{67} +35.8765i q^{71} -40.6558i q^{73} +236.765 q^{77} +140.080 q^{79} +118.832 q^{83} +(37.3672 - 44.0015i) q^{85} -75.0520i q^{89} -86.0690 q^{91} +(-78.4325 + 92.3577i) q^{95} +84.5227i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) $ 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	−3.23653	+	3.81115i	−0.647306	+	0.762231i
$5$	−3.23653	+	3.81115i	−0.647306	+	0.762231i
$6$		0			0
$7$		−	12.6606i		−	1.80866i	−0.426835	−	0.904330i	$-0.640371\pi$
$7$		−	12.6606i		−	1.80866i	0.426835	−	0.904330i	$-0.359629\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			18.7009i			1.70008i	0.526716	+	0.850041i	$0.323423\pi$
$11$			18.7009i			1.70008i	−0.526716	+	0.850041i	$0.676577\pi$
$12$		0			0
$13$		−	6.79817i		−	0.522936i	−0.965212	−	0.261468i	$-0.915793\pi$
$13$		−	6.79817i		−	0.522936i	0.965212	−	0.261468i	$-0.0842066\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−11.5455			−0.679145			−0.339573	−	0.940580i	$-0.610282\pi$
$17$	−11.5455			−0.679145			−0.339573	+	0.940580i	$0.610282\pi$
$18$		0			0
$19$	24.2335			1.27545			0.637725	−	0.770265i	$-0.279876\pi$
$19$	24.2335			1.27545			0.637725	+	0.770265i	$0.279876\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−29.2202			−1.27044			−0.635221	−	0.772330i	$-0.719091\pi$
$23$	−29.2202			−1.27044			−0.635221	+	0.772330i	$0.719091\pi$
$24$		0			0
$25$	−4.04977	−	24.6698i	−0.161991	−	0.986792i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	14.9461i		−	0.515381i	−0.966227	−	0.257691i	$-0.917038\pi$
$29$		−	14.9461i		−	0.515381i	0.966227	−	0.257691i	$-0.0829616\pi$
$30$		0			0
$31$	13.8362			0.446329			0.223164	−	0.974781i	$-0.428361\pi$
$31$	13.8362			0.446329			0.223164	+	0.974781i	$0.428361\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	48.2515	+	40.9764i	1.37862	+	1.17076i
$36$		0			0
$37$			38.2737i			1.03443i	0.855857	+	0.517213i	$0.173030\pi$
$37$			38.2737i			1.03443i	−0.855857	+	0.517213i	$0.826970\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			30.9192i			0.754128i	0.926187	+	0.377064i	$0.123066\pi$
$41$			30.9192i			0.754128i	−0.926187	+	0.377064i	$0.876934\pi$
$42$		0			0
$43$		−	13.0796i		−	0.304176i	−0.988367	−	0.152088i	$-0.951400\pi$
$43$		−	13.0796i		−	0.304176i	0.988367	−	0.152088i	$-0.0485997\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.68939			−0.121051			−0.0605255	−	0.998167i	$-0.519278\pi$
$47$	−5.68939			−0.121051			−0.0605255	+	0.998167i	$0.519278\pi$
$48$		0			0
$49$	−111.291			−2.27125
$50$		0			0
$51$		0			0
$52$		0			0
$53$	57.7486			1.08960			0.544798	−	0.838567i	$-0.316606\pi$
$53$	57.7486			1.08960			0.544798	+	0.838567i	$0.316606\pi$
$54$		0			0
$55$	−71.2720	−	60.5260i	−1.29585	−	1.10047i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	30.3390i		−	0.514220i	−0.966382	−	0.257110i	$-0.917230\pi$
$59$		−	30.3390i		−	0.514220i	0.966382	−	0.257110i	$-0.0827703\pi$
$60$		0			0
$61$	77.0497			1.26311			0.631555	−	0.775331i	$-0.282417\pi$
$61$	77.0497			1.26311			0.631555	+	0.775331i	$0.282417\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	25.9089	+	22.0025i	0.398598	+	0.338499i
$66$		0			0
$67$			128.396i			1.91636i	0.286164	+	0.958181i	$0.407620\pi$
$67$			128.396i			1.91636i	−0.286164	+	0.958181i	$0.592380\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			35.8765i			0.505303i	0.967557	+	0.252652i	$0.0813026\pi$
$71$			35.8765i			0.505303i	−0.967557	+	0.252652i	$0.918697\pi$
$72$		0			0
$73$		−	40.6558i		−	0.556929i	−0.960447	−	0.278464i	$-0.910175\pi$
$73$		−	40.6558i		−	0.556929i	0.960447	−	0.278464i	$-0.0898254\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	236.765			3.07487
$78$		0			0
$79$	140.080			1.77317			0.886585	−	0.462565i	$-0.153071\pi$
$79$	140.080			1.77317			0.886585	+	0.462565i	$0.153071\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	118.832			1.43171			0.715853	−	0.698251i	$-0.246038\pi$
$83$	118.832			1.43171			0.715853	+	0.698251i	$0.246038\pi$
$84$		0			0
$85$	37.3672	−	44.0015i	0.439614	−	0.517665i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	75.0520i		−	0.843281i	−0.906763	−	0.421641i	$-0.861454\pi$
$89$		−	75.0520i		−	0.843281i	0.906763	−	0.421641i	$-0.138546\pi$
$90$		0			0
$91$	−86.0690			−0.945813
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−78.4325	+	92.3577i	−0.825605	+	0.972186i
$96$		0			0
$97$			84.5227i			0.871368i	0.900100	+	0.435684i	$0.143493\pi$
$97$			84.5227i			0.871368i	−0.900100	+	0.435684i	$0.856507\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			50.1020i			0.496059i	0.968752	+	0.248029i	$0.0797830\pi$
$101$			50.1020i			0.496059i	−0.968752	+	0.248029i	$0.920217\pi$
$102$		0			0
$103$			145.033i			1.40808i	0.710159	+	0.704042i	$0.248623\pi$
$103$			145.033i			1.40808i	−0.710159	+	0.704042i	$0.751377\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−209.074			−1.95396			−0.976980	−	0.213330i	$-0.931569\pi$
$107$	−209.074			−1.95396			−0.976980	+	0.213330i	$0.931569\pi$
$108$		0			0
$109$	113.447			1.04079			0.520397	−	0.853924i	$-0.325784\pi$
$109$	113.447			1.04079			0.520397	+	0.853924i	$0.325784\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	90.8349			0.803849			0.401924	−	0.915673i	$-0.368341\pi$
$113$	90.8349			0.803849			0.401924	+	0.915673i	$0.368341\pi$
$114$		0			0
$115$	94.5719	−	111.363i	0.822364	−	0.968370i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			146.173i			1.22834i
$120$		0			0
$121$	−228.724			−1.89028
$122$		0			0
$123$		0			0
$124$		0			0
$125$	107.128	+	64.4102i	0.857021	+	0.515282i
$126$		0			0
$127$			150.585i			1.18571i	0.805311	+	0.592853i	$0.201999\pi$
$127$			150.585i			1.18571i	−0.805311	+	0.592853i	$0.798001\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			74.4968i			0.568678i	0.958724	+	0.284339i	$0.0917741\pi$
$131$			74.4968i			0.568678i	−0.958724	+	0.284339i	$0.908226\pi$
$132$		0			0
$133$		−	306.811i		−	2.30685i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	95.8194			0.699411			0.349706	−	0.936860i	$-0.386282\pi$
$137$	95.8194			0.699411			0.349706	+	0.936860i	$0.386282\pi$
$138$		0			0
$139$	−50.3574			−0.362284			−0.181142	−	0.983457i	$-0.557979\pi$
$139$	−50.3574			−0.362284			−0.181142	+	0.983457i	$0.557979\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	127.132			0.889035
$144$		0			0
$145$	56.9617	+	48.3733i	0.392839	+	0.333609i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			211.533i			1.41968i	0.704361	+	0.709842i	$0.251234\pi$
$149$			211.533i			1.41968i	−0.704361	+	0.709842i	$0.748766\pi$
$150$		0			0
$151$	−4.72563			−0.0312956			−0.0156478	−	0.999878i	$-0.504981\pi$
$151$	−4.72563			−0.0312956			−0.0156478	+	0.999878i	$0.504981\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−44.7812	+	52.7318i	−0.288911	+	0.340205i
$156$		0			0
$157$			228.184i			1.45340i	0.686954	+	0.726701i	$0.258947\pi$
$157$			228.184i			1.45340i	−0.686954	+	0.726701i	$0.741053\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			369.945i			2.29780i
$162$		0			0
$163$			160.004i			0.981618i	0.871267	+	0.490809i	$0.163299\pi$
$163$			160.004i			0.981618i	−0.871267	+	0.490809i	$0.836701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	170.528			1.02113			0.510563	−	0.859840i	$-0.329437\pi$
$167$	170.528			1.02113			0.510563	+	0.859840i	$0.329437\pi$
$168$		0			0
$169$	122.785			0.726538
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−161.303			−0.932389			−0.466195	−	0.884682i	$-0.654375\pi$
$173$	−161.303			−0.932389			−0.466195	+	0.884682i	$0.654375\pi$
$174$		0			0
$175$	−312.335	+	51.2726i	−1.78477	+	0.292986i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			92.7337i			0.518065i	0.965869	+	0.259033i	$0.0834037\pi$
$179$			92.7337i			0.518065i	−0.965869	+	0.259033i	$0.916596\pi$
$180$		0			0
$181$	70.5166			0.389595			0.194797	−	0.980844i	$-0.437595\pi$
$181$	70.5166			0.389595			0.194797	+	0.980844i	$0.437595\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−145.867	−	123.874i	−0.788471	−	0.669589i
$186$		0			0
$187$		−	215.911i		−	1.15460i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	166.924i		−	0.873945i	−0.899475	−	0.436973i	$-0.856051\pi$
$191$		−	166.924i		−	0.873945i	0.899475	−	0.436973i	$-0.143949\pi$
$192$		0			0
$193$		−	85.3304i		−	0.442126i	−0.975260	−	0.221063i	$-0.929047\pi$
$193$		−	85.3304i		−	0.442126i	0.975260	−	0.221063i	$-0.0709527\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	246.230			1.24990			0.624950	−	0.780665i	$-0.285119\pi$
$197$	246.230			1.24990			0.624950	+	0.780665i	$0.285119\pi$
$198$		0			0
$199$	−143.191			−0.719553			−0.359777	−	0.933038i	$-0.617147\pi$
$199$	−143.191			−0.719553			−0.359777	+	0.933038i	$0.617147\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−189.226			−0.932149
$204$		0			0
$205$	−117.838	−	100.071i	−0.574819	−	0.488151i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			453.189i			2.16837i
$210$		0			0
$211$	−103.601			−0.491002			−0.245501	−	0.969396i	$-0.578952\pi$
$211$	−103.601			−0.491002			−0.245501	+	0.969396i	$0.578952\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	49.8482	+	42.3324i	0.231852	+	0.196895i
$216$		0			0
$217$		−	175.175i		−	0.807257i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			78.4880i			0.355150i
$222$		0			0
$223$			27.3801i			0.122781i	0.998114	+	0.0613904i	$0.0195535\pi$
$223$			27.3801i			0.122781i	−0.998114	+	0.0613904i	$0.980447\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	60.7995			0.267839			0.133919	−	0.990992i	$-0.457244\pi$
$227$	60.7995			0.267839			0.133919	+	0.990992i	$0.457244\pi$
$228$		0			0
$229$	421.560			1.84087			0.920436	−	0.390894i	$-0.127834\pi$
$229$	421.560			1.84087			0.920436	+	0.390894i	$0.127834\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	34.0839			0.146283			0.0731415	−	0.997322i	$-0.476698\pi$
$233$	34.0839			0.146283			0.0731415	+	0.997322i	$0.476698\pi$
$234$		0			0
$235$	18.4139	−	21.6831i	0.0783569	−	0.0922687i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			109.397i			0.457729i	0.973458	+	0.228864i	$0.0735012\pi$
$239$			109.397i			0.457729i	−0.973458	+	0.228864i	$0.926499\pi$
$240$		0			0
$241$	285.068			1.18286			0.591428	−	0.806358i	$-0.298565\pi$
$241$	285.068			1.18286			0.591428	+	0.806358i	$0.298565\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	360.197	−	424.148i	1.47019	−	1.73121i
$246$		0			0
$247$		−	164.744i		−	0.666978i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	440.040i		−	1.75315i	−0.481268	−	0.876574i	$-0.659823\pi$
$251$		−	440.040i		−	1.75315i	0.481268	−	0.876574i	$-0.340177\pi$
$252$		0			0
$253$		−	546.444i		−	2.15986i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−424.196			−1.65057			−0.825285	−	0.564717i	$-0.808985\pi$
$257$	−424.196			−1.65057			−0.825285	+	0.564717i	$0.808985\pi$
$258$		0			0
$259$	484.569			1.87092
$260$		0			0
$261$		0			0
$262$		0			0
$263$	263.386			1.00147			0.500734	−	0.865601i	$-0.333063\pi$
$263$	263.386			1.00147			0.500734	+	0.865601i	$0.333063\pi$
$264$		0			0
$265$	−186.905	+	220.089i	−0.705302	+	0.830523i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	169.593i		−	0.630456i	−0.949016	−	0.315228i	$-0.897919\pi$
$269$		−	169.593i		−	0.630456i	0.949016	−	0.315228i	$-0.102081\pi$
$270$		0			0
$271$	−72.1924			−0.266393			−0.133196	−	0.991090i	$-0.542524\pi$
$271$	−72.1924			−0.266393			−0.133196	+	0.991090i	$0.542524\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	461.348	−	75.7344i	1.67763	−	0.275398i
$276$		0			0
$277$		−	24.5117i		−	0.0884899i	−0.999021	−	0.0442449i	$-0.985912\pi$
$277$		−	24.5117i		−	0.0884899i	0.999021	−	0.0442449i	$-0.0140882\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			288.862i			1.02798i	0.857797	+	0.513989i	$0.171833\pi$
$281$			288.862i			1.02798i	−0.857797	+	0.513989i	$0.828167\pi$
$282$		0			0
$283$		−	81.5992i		−	0.288336i	−0.989553	−	0.144168i	$-0.953949\pi$
$283$		−	81.5992i		−	0.288336i	0.989553	−	0.144168i	$-0.0460506\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	391.457			1.36396
$288$		0			0
$289$	−155.702			−0.538762
$290$		0			0
$291$		0			0
$292$		0			0
$293$	547.227			1.86767			0.933835	−	0.357704i	$-0.116440\pi$
$293$	547.227			1.86767			0.933835	+	0.357704i	$0.116440\pi$
$294$		0			0
$295$	115.627	+	98.1930i	0.391954	+	0.332858i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			198.644i			0.664360i
$300$		0			0
$301$	−165.595			−0.550151
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−249.373	+	293.648i	−0.817618	+	0.962780i
$306$		0			0
$307$		−	145.422i		−	0.473686i	−0.971548	−	0.236843i	$-0.923887\pi$
$307$		−	145.422i		−	0.473686i	0.971548	−	0.236843i	$-0.0761127\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	499.738i		−	1.60688i	−0.595388	−	0.803438i	$-0.703002\pi$
$311$		−	499.738i		−	1.60688i	0.595388	−	0.803438i	$-0.296998\pi$
$312$		0			0
$313$			478.102i			1.52748i	0.645523	+	0.763741i	$0.276639\pi$
$313$			478.102i			1.52748i	−0.645523	+	0.763741i	$0.723361\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	106.348			0.335484			0.167742	−	0.985831i	$-0.446352\pi$
$317$	106.348			0.335484			0.167742	+	0.985831i	$0.446352\pi$
$318$		0			0
$319$	279.505			0.876191
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−279.787			−0.866215
$324$		0			0
$325$	−167.710	+	27.5310i	−0.516029	+	0.0847109i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			72.0312i			0.218940i
$330$		0			0
$331$	−263.287			−0.795429			−0.397714	−	0.917509i	$-0.630197\pi$
$331$	−263.287			−0.795429			−0.397714	+	0.917509i	$0.630197\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−489.338	−	415.558i	−1.46071	−	1.24047i
$336$		0			0
$337$		−	162.235i		−	0.481409i	−0.970599	−	0.240704i	$-0.922622\pi$
$337$		−	162.235i		−	0.481409i	0.970599	−	0.240704i	$-0.0773784\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			258.749i			0.758796i
$342$		0			0
$343$			788.644i			2.29925i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−339.297			−0.977801			−0.488901	−	0.872340i	$-0.662602\pi$
$347$	−339.297			−0.977801			−0.488901	+	0.872340i	$0.662602\pi$
$348$		0			0
$349$	−553.637			−1.58635			−0.793177	−	0.608992i	$-0.791574\pi$
$349$	−553.637			−1.58635			−0.793177	+	0.608992i	$0.791574\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−344.942			−0.977173			−0.488587	−	0.872515i	$-0.662487\pi$
$353$	−344.942			−0.977173			−0.488587	+	0.872515i	$0.662487\pi$
$354$		0			0
$355$	−136.731	−	116.115i	−0.385158	−	0.327086i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	679.001i		−	1.89137i	−0.325088	−	0.945684i	$-0.605394\pi$
$359$		−	679.001i		−	1.89137i	0.325088	−	0.945684i	$-0.394606\pi$
$360$		0			0
$361$	226.264			0.626770
$362$		0			0
$363$		0			0
$364$		0			0
$365$	154.945	+	131.584i	0.424508	+	0.360503i
$366$		0			0
$367$		−	468.698i		−	1.27710i	−0.769578	−	0.638552i	$-0.779534\pi$
$367$		−	468.698i		−	1.27710i	0.769578	−	0.638552i	$-0.220466\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	731.133i		−	1.97071i
$372$		0			0
$373$			467.030i			1.25209i	0.779786	+	0.626046i	$0.215328\pi$
$373$			467.030i			1.25209i	−0.779786	+	0.626046i	$0.784672\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−101.606			−0.269512
$378$		0			0
$379$	485.497			1.28100			0.640498	−	0.767960i	$-0.278728\pi$
$379$	485.497			1.28100			0.640498	+	0.767960i	$0.278728\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−259.490			−0.677519			−0.338760	−	0.940873i	$-0.610007\pi$
$383$	−259.490			−0.677519			−0.338760	+	0.940873i	$0.610007\pi$
$384$		0			0
$385$	−766.297	+	902.348i	−1.99038	+	2.34376i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			232.440i			0.597533i	0.954326	+	0.298766i	$0.0965752\pi$
$389$			232.440i			0.597533i	−0.954326	+	0.298766i	$0.903425\pi$
$390$		0			0
$391$	337.361			0.862815
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−453.374	+	533.868i	−1.14778	+	1.35156i
$396$		0			0
$397$			58.1107i			0.146374i	0.997318	+	0.0731872i	$0.0233171\pi$
$397$			58.1107i			0.146374i	−0.997318	+	0.0731872i	$0.976683\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			619.430i			1.54471i	0.635189	+	0.772356i	$0.280922\pi$
$401$			619.430i			1.54471i	−0.635189	+	0.772356i	$0.719078\pi$
$402$		0			0
$403$		−	94.0608i		−	0.233401i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−715.754			−1.75861
$408$		0			0
$409$	229.647			0.561485			0.280743	−	0.959783i	$-0.409419\pi$
$409$	229.647			0.561485			0.280743	+	0.959783i	$0.409419\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−384.110			−0.930049
$414$		0			0
$415$	−384.602	+	452.885i	−0.926751	+	1.09129i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	378.523i		−	0.903397i	−0.892171	−	0.451699i	$-0.850818\pi$
$419$		−	378.523i		−	0.903397i	0.892171	−	0.451699i	$-0.149182\pi$
$420$		0			0
$421$	−735.935			−1.74806			−0.874032	−	0.485869i	$-0.838503\pi$
$421$	−735.935			−1.74806			−0.874032	+	0.485869i	$0.838503\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	46.7565	+	284.824i	0.110015	+	0.670175i
$426$		0			0
$427$		−	975.496i		−	2.28453i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			352.044i			0.816807i	0.912801	+	0.408404i	$0.133914\pi$
$431$			352.044i			0.816807i	−0.912801	+	0.408404i	$0.866086\pi$
$432$		0			0
$433$			186.839i			0.431498i	0.976449	+	0.215749i	$0.0692193\pi$
$433$			186.839i			0.431498i	−0.976449	+	0.215749i	$0.930781\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−708.108			−1.62038
$438$		0			0
$439$	402.728			0.917375			0.458687	−	0.888598i	$-0.348320\pi$
$439$	402.728			0.917375			0.458687	+	0.888598i	$0.348320\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	134.835			0.304369			0.152184	−	0.988352i	$-0.451369\pi$
$443$	134.835			0.304369			0.152184	+	0.988352i	$0.451369\pi$
$444$		0			0
$445$	286.035	+	242.908i	0.642775	+	0.545861i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	763.209i		−	1.69980i	−0.526947	−	0.849898i	$-0.676663\pi$
$449$		−	763.209i		−	1.69980i	0.526947	−	0.849898i	$-0.323337\pi$
$450$		0			0
$451$	−578.218			−1.28208
$452$		0			0
$453$		0			0
$454$		0			0
$455$	278.565	−	328.022i	0.612230	−	0.720928i
$456$		0			0
$457$			463.889i			1.01508i	0.861630	+	0.507538i	$0.169444\pi$
$457$			463.889i			1.01508i	−0.861630	+	0.507538i	$0.830556\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	158.704i		−	0.344260i	−0.985074	−	0.172130i	$-0.944935\pi$
$461$		−	158.704i		−	0.344260i	0.985074	−	0.172130i	$-0.0550649\pi$
$462$		0			0
$463$			141.375i			0.305346i	0.988277	+	0.152673i	$0.0487882\pi$
$463$			141.375i			0.305346i	−0.988277	+	0.152673i	$0.951212\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−139.024			−0.297697			−0.148848	−	0.988860i	$-0.547557\pi$
$467$	−139.024			−0.297697			−0.148848	+	0.988860i	$0.547557\pi$
$468$		0			0
$469$	1625.57			3.46604
$470$		0			0
$471$		0			0
$472$		0			0
$473$	244.600			0.517124
$474$		0			0
$475$	−98.1403	−	597.837i	−0.206611	−	1.25860i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			382.476i			0.798488i	0.916845	+	0.399244i	$0.130727\pi$
$479$			382.476i			0.798488i	−0.916845	+	0.399244i	$0.869273\pi$
$480$		0			0
$481$	260.191			0.540938
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−322.129	−	273.560i	−0.664183	−	0.564041i
$486$		0			0
$487$		−	522.699i		−	1.07330i	−0.843804	−	0.536652i	$-0.819689\pi$
$487$		−	522.699i		−	1.07330i	0.843804	−	0.536652i	$-0.180311\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	280.674i		−	0.571637i	−0.958284	−	0.285819i	$-0.907734\pi$
$491$		−	280.674i		−	0.571637i	0.958284	−	0.285819i	$-0.0922655\pi$
$492$		0			0
$493$			172.559i			0.350019i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	454.219			0.913921
$498$		0			0
$499$	280.725			0.562576			0.281288	−	0.959623i	$-0.409238\pi$
$499$	280.725			0.562576			0.281288	+	0.959623i	$0.409238\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−336.619			−0.669222			−0.334611	−	0.942356i	$-0.608605\pi$
$503$	−336.619			−0.669222			−0.334611	+	0.942356i	$0.608605\pi$
$504$		0			0
$505$	−190.946	−	162.156i	−0.378111	−	0.321102i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	362.181i		−	0.711555i	−0.934571	−	0.355777i	$-0.884216\pi$
$509$		−	362.181i		−	0.711555i	0.934571	−	0.355777i	$-0.115784\pi$
$510$		0			0
$511$	−514.727			−1.00729
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−552.741	−	469.402i	−1.07328	−	0.911460i
$516$		0			0
$517$		−	106.397i		−	0.205797i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			338.445i			0.649607i	0.945782	+	0.324803i	$0.105298\pi$
$521$			338.445i			0.649607i	−0.945782	+	0.324803i	$0.894702\pi$
$522$		0			0
$523$		−	460.915i		−	0.881291i	−0.897681	−	0.440646i	$-0.854750\pi$
$523$		−	460.915i		−	0.881291i	0.897681	−	0.440646i	$-0.145250\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−159.745			−0.303122
$528$		0			0
$529$	324.819			0.614024
$530$		0			0
$531$		0			0
$532$		0			0
$533$	210.194			0.394361
$534$		0			0
$535$	676.673	−	796.812i	1.26481	−	1.48937i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	2081.25i		−	3.86131i
$540$		0			0
$541$	−434.855			−0.803799			−0.401899	−	0.915684i	$-0.631650\pi$
$541$	−434.855			−0.803799			−0.401899	+	0.915684i	$0.631650\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−367.173	+	432.362i	−0.673712	+	0.793326i
$546$		0			0
$547$			426.428i			0.779576i	0.920905	+	0.389788i	$0.127452\pi$
$547$			426.428i			0.779576i	−0.920905	+	0.389788i	$0.872548\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	362.196i		−	0.657343i
$552$		0			0
$553$		−	1773.50i		−	3.20706i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	841.776			1.51127			0.755634	−	0.654994i	$-0.227329\pi$
$557$	841.776			1.51127			0.755634	+	0.654994i	$0.227329\pi$
$558$		0			0
$559$	−88.9171			−0.159065
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−977.610			−1.73643			−0.868215	−	0.496189i	$-0.834732\pi$
$563$	−977.610			−1.73643			−0.868215	+	0.496189i	$0.834732\pi$
$564$		0			0
$565$	−293.990	+	346.186i	−0.520336	+	0.612718i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	16.3455i		−	0.0287267i	−0.999897	−	0.0143634i	$-0.995428\pi$
$569$		−	16.3455i		−	0.0287267i	0.999897	−	0.0143634i	$-0.00457216\pi$
$570$		0			0
$571$	297.635			0.521252			0.260626	−	0.965440i	$-0.416071\pi$
$571$	297.635			0.521252			0.260626	+	0.965440i	$0.416071\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	118.335	+	720.856i	0.205800	+	1.25366i
$576$		0			0
$577$		−	348.674i		−	0.604287i	−0.953262	−	0.302144i	$-0.902298\pi$
$577$		−	348.674i		−	0.604287i	0.953262	−	0.302144i	$-0.0977022\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	1504.48i		−	2.58947i
$582$		0			0
$583$			1079.95i			1.85240i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−135.366			−0.230606			−0.115303	−	0.993330i	$-0.536784\pi$
$587$	−135.366			−0.230606			−0.115303	+	0.993330i	$0.536784\pi$
$588$		0			0
$589$	335.300			0.569270
$590$		0			0
$591$		0			0
$592$		0			0
$593$	871.003			1.46881			0.734404	−	0.678713i	$-0.237462\pi$
$593$	871.003			1.46881			0.734404	+	0.678713i	$0.237462\pi$
$594$		0			0
$595$	−557.087	−	473.092i	−0.936280	−	0.795113i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	201.931i		−	0.337114i	−0.985692	−	0.168557i	$-0.946089\pi$
$599$		−	201.931i		−	0.337114i	0.985692	−	0.168557i	$-0.0539107\pi$
$600$		0			0
$601$	−589.720			−0.981232			−0.490616	−	0.871376i	$-0.663228\pi$
$601$	−589.720			−0.981232			−0.490616	+	0.871376i	$0.663228\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	740.272	−	871.702i	1.22359	−	1.44083i
$606$		0			0
$607$			532.581i			0.877399i	0.898634	+	0.438699i	$0.144561\pi$
$607$			532.581i			0.877399i	−0.898634	+	0.438699i	$0.855439\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			38.6775i			0.0633019i
$612$		0			0
$613$		−	753.304i		−	1.22888i	−0.788963	−	0.614440i	$-0.789382\pi$
$613$		−	753.304i		−	1.22888i	0.788963	−	0.614440i	$-0.210618\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	611.833			0.991625			0.495813	−	0.868430i	$-0.334870\pi$
$617$	611.833			0.991625			0.495813	+	0.868430i	$0.334870\pi$
$618$		0			0
$619$	152.987			0.247152			0.123576	−	0.992335i	$-0.460564\pi$
$619$	152.987			0.247152			0.123576	+	0.992335i	$0.460564\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−950.205			−1.52521
$624$		0			0
$625$	−592.199	+	199.814i	−0.947518	+	0.319703i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	441.888i		−	0.702525i
$630$		0			0
$631$	−888.882			−1.40869			−0.704344	−	0.709859i	$-0.748759\pi$
$631$	−888.882			−1.40869			−0.704344	+	0.709859i	$0.748759\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−573.901	−	487.372i	−0.903782	−	0.767514i
$636$		0			0
$637$			756.576i			1.18772i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	983.352i		−	1.53409i	−0.641593	−	0.767045i	$-0.721726\pi$
$641$		−	983.352i		−	1.53409i	0.641593	−	0.767045i	$-0.278274\pi$
$642$		0			0
$643$			600.569i			0.934011i	0.884254	+	0.467005i	$0.154667\pi$
$643$			600.569i			0.934011i	−0.884254	+	0.467005i	$0.845333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−315.678			−0.487911			−0.243955	−	0.969786i	$-0.578445\pi$
$647$	−315.678			−0.487911			−0.243955	+	0.969786i	$0.578445\pi$
$648$		0			0
$649$	567.367			0.874217
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−83.1761			−0.127375			−0.0636877	−	0.997970i	$-0.520286\pi$
$653$	−83.1761			−0.127375			−0.0636877	+	0.997970i	$0.520286\pi$
$654$		0			0
$655$	−283.919	−	241.111i	−0.433464	−	0.368109i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			104.212i			0.158136i	0.996869	+	0.0790681i	$0.0251944\pi$
$659$			104.212i			0.158136i	−0.996869	+	0.0790681i	$0.974806\pi$
$660$		0			0
$661$	−26.5966			−0.0402369			−0.0201185	−	0.999798i	$-0.506404\pi$
$661$	−26.5966			−0.0402369			−0.0201185	+	0.999798i	$0.506404\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1169.31	+	993.004i	1.75835	+	1.49324i
$666$		0			0
$667$			436.727i			0.654762i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1440.90i			2.14739i
$672$		0			0
$673$		−	138.616i		−	0.205967i	−0.994683	−	0.102984i	$-0.967161\pi$
$673$		−	138.616i		−	0.205967i	0.994683	−	0.102984i	$-0.0328389\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−219.726			−0.324559			−0.162280	−	0.986745i	$-0.551885\pi$
$677$	−219.726			−0.324559			−0.162280	+	0.986745i	$0.551885\pi$
$678$		0			0
$679$	1070.11			1.57601
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−897.042			−1.31338			−0.656692	−	0.754159i	$-0.728045\pi$
$683$	−897.042			−1.31338			−0.656692	+	0.754159i	$0.728045\pi$
$684$		0			0
$685$	−310.122	+	365.182i	−0.452733	+	0.533113i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	392.585i		−	0.569789i
$690$		0			0
$691$	530.181			0.767267			0.383633	−	0.923486i	$-0.374673\pi$
$691$	530.181			0.767267			0.383633	+	0.923486i	$0.374673\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	162.983	−	191.920i	0.234508	−	0.276144i
$696$		0			0
$697$		−	356.977i		−	0.512162i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			241.071i			0.343895i	0.985106	+	0.171948i	$0.0550060\pi$
$701$			241.071i			0.343895i	−0.985106	+	0.171948i	$0.944994\pi$
$702$		0			0
$703$			927.508i			1.31936i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	634.321			0.897202
$708$		0			0
$709$	−186.005			−0.262349			−0.131174	−	0.991359i	$-0.541875\pi$
$709$	−186.005			−0.262349			−0.131174	+	0.991359i	$0.541875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−404.296			−0.567035
$714$		0			0
$715$	−411.466	+	484.519i	−0.575477	+	0.677649i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			193.919i			0.269707i	0.990866	+	0.134854i	$0.0430564\pi$
$719$			193.919i			0.269707i	−0.990866	+	0.134854i	$0.956944\pi$
$720$		0			0
$721$	1836.20			2.54674
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−368.716	+	60.5281i	−0.508574	+	0.0834871i
$726$		0			0
$727$			236.794i			0.325714i	0.986650	+	0.162857i	$0.0520709\pi$
$727$			236.794i			0.325714i	−0.986650	+	0.162857i	$0.947929\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			151.010i			0.206580i
$732$		0			0
$733$		−	580.637i		−	0.792137i	−0.918221	−	0.396069i	$-0.870374\pi$
$733$		−	580.637i		−	0.792137i	0.918221	−	0.396069i	$-0.129626\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2401.13			−3.25797
$738$		0			0
$739$	−419.984			−0.568314			−0.284157	−	0.958778i	$-0.591714\pi$
$739$	−419.984			−0.568314			−0.284157	+	0.958778i	$0.591714\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−500.019			−0.672974			−0.336487	−	0.941688i	$-0.609239\pi$
$743$	−500.019			−0.672974			−0.336487	+	0.941688i	$0.609239\pi$
$744$		0			0
$745$	−806.184	−	684.632i	−1.08213	−	0.918969i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			2647.00i			3.53405i
$750$		0			0
$751$	−699.344			−0.931217			−0.465609	−	0.884991i	$-0.654165\pi$
$751$	−699.344			−0.931217			−0.465609	+	0.884991i	$0.654165\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.2946	−	18.0101i	0.0202578	−	0.0238544i
$756$		0			0
$757$			741.479i			0.979496i	0.871864	+	0.489748i	$0.162911\pi$
$757$			741.479i			0.979496i	−0.871864	+	0.489748i	$0.837089\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			106.232i			0.139595i	0.997561	+	0.0697977i	$0.0222354\pi$
$761$			106.232i			0.139595i	−0.997561	+	0.0697977i	$0.977765\pi$
$762$		0			0
$763$		−	1436.30i		−	1.88244i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−206.250			−0.268904
$768$		0			0
$769$	801.192			1.04186			0.520931	−	0.853599i	$-0.325585\pi$
$769$	801.192			1.04186			0.520931	+	0.853599i	$0.325585\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	763.281			0.987427			0.493713	−	0.869625i	$-0.335639\pi$
$773$	763.281			0.987427			0.493713	+	0.869625i	$0.335639\pi$
$774$		0			0
$775$	−56.0334	−	341.336i	−0.0723012	−	0.440434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			749.282i			0.961851i
$780$		0			0
$781$	−670.924			−0.859057
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−869.644	−	738.524i	−1.10783	−	0.940795i
$786$		0			0
$787$			1035.23i			1.31541i	0.753275	+	0.657706i	$0.228473\pi$
$787$			1035.23i			1.31541i	−0.753275	+	0.657706i	$0.771527\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1150.03i		−	1.45389i
$792$		0			0
$793$		−	523.797i		−	0.660525i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1021.03			1.28110			0.640549	−	0.767917i	$-0.278707\pi$
$797$	1021.03			1.28110			0.640549	+	0.767917i	$0.278707\pi$
$798$		0			0
$799$	65.6867			0.0822112
$800$		0			0
$801$		0			0
$802$		0			0
$803$	760.300			0.946825
$804$		0			0
$805$	−1409.92	−	1197.34i	−1.75145	−	1.48738i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	933.886i		−	1.15437i	−0.816613	−	0.577186i	$-0.804151\pi$
$809$		−	933.886i		−	1.15437i	0.816613	−	0.577186i	$-0.195849\pi$
$810$		0			0
$811$	194.761			0.240149			0.120075	−	0.992765i	$-0.461687\pi$
$811$	194.761			0.240149			0.120075	+	0.992765i	$0.461687\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−609.799	−	517.857i	−0.748219	−	0.635407i
$816$		0			0
$817$		−	316.964i		−	0.387961i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			535.638i			0.652421i	0.945297	+	0.326210i	$0.105772\pi$
$821$			535.638i			0.652421i	−0.945297	+	0.326210i	$0.894228\pi$
$822$		0			0
$823$		−	449.845i		−	0.546592i	−0.961930	−	0.273296i	$-0.911886\pi$
$823$		−	449.845i		−	0.546592i	0.961930	−	0.273296i	$-0.0881138\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1623.34			−1.96292			−0.981462	−	0.191656i	$-0.938614\pi$
$827$	−1623.34			−1.96292			−0.981462	+	0.191656i	$0.938614\pi$
$828$		0			0
$829$	746.355			0.900307			0.450154	−	0.892951i	$-0.351369\pi$
$829$	746.355			0.900307			0.450154	+	0.892951i	$0.351369\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1284.91			1.54251
$834$		0			0
$835$	−551.919	+	649.908i	−0.660981	+	0.778334i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			482.656i			0.575275i	0.957739	+	0.287638i	$0.0928698\pi$
$839$			482.656i			0.575275i	−0.957739	+	0.287638i	$0.907130\pi$
$840$		0			0
$841$	617.615			0.734382
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−397.397	+	467.952i	−0.470292	+	0.553789i
$846$		0			0
$847$			2895.79i			3.41887i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1118.37i		−	1.31418i
$852$		0			0
$853$			349.830i			0.410118i	0.978750	+	0.205059i	$0.0657386\pi$
$853$			349.830i			0.410118i	−0.978750	+	0.205059i	$0.934261\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	712.534			0.831429			0.415714	−	0.909495i	$-0.363532\pi$
$857$	712.534			0.831429			0.415714	+	0.909495i	$0.363532\pi$
$858$		0			0
$859$	1086.57			1.26492			0.632460	−	0.774593i	$-0.282045\pi$
$859$	1086.57			1.26492			0.632460	+	0.774593i	$0.282045\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	126.997			0.147158			0.0735789	−	0.997289i	$-0.476558\pi$
$863$	126.997			0.147158			0.0735789	+	0.997289i	$0.476558\pi$
$864$		0			0
$865$	522.063	−	614.752i	0.603541	−	0.710696i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			2619.63i			3.01454i
$870$		0			0
$871$	872.859			1.00213
$872$		0			0
$873$		0			0
$874$		0			0
$875$	815.473	−	1356.30i	0.931969	−	1.55006i
$876$		0			0
$877$			1288.77i			1.46952i	0.678326	+	0.734761i	$0.262706\pi$
$877$			1288.77i			1.46952i	−0.678326	+	0.734761i	$0.737294\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1285.57i			1.45922i	0.683863	+	0.729611i	$0.260299\pi$
$881$			1285.57i			1.45922i	−0.683863	+	0.729611i	$0.739701\pi$
$882$		0			0
$883$		−	429.815i		−	0.486767i	−0.969930	−	0.243384i	$-0.921743\pi$
$883$		−	429.815i		−	0.486767i	0.969930	−	0.243384i	$-0.0782574\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	434.522			0.489878			0.244939	−	0.969538i	$-0.421232\pi$
$887$	434.522			0.489878			0.244939	+	0.969538i	$0.421232\pi$
$888$		0			0
$889$	1906.49			2.14454
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−137.874			−0.154394
$894$		0			0
$895$	−353.422	−	300.135i	−0.394885	−	0.335347i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	206.797i		−	0.230030i
$900$		0			0
$901$	−666.734			−0.739994
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−228.229	+	268.750i	−0.252187	+	0.296961i
$906$		0			0
$907$		−	256.632i		−	0.282946i	−0.989942	−	0.141473i	$-0.954816\pi$
$907$		−	256.632i		−	0.282946i	0.989942	−	0.141473i	$-0.0451838\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			314.828i			0.345585i	0.984958	+	0.172792i	$0.0552790\pi$
$911$			314.828i			0.345585i	−0.984958	+	0.172792i	$0.944721\pi$
$912$		0			0
$913$			2222.26i			2.43402i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	943.176			1.02854
$918$		0			0
$919$	−440.096			−0.478886			−0.239443	−	0.970910i	$-0.576965\pi$
$919$	−440.096			−0.478886			−0.239443	+	0.970910i	$0.576965\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	243.895			0.264241
$924$		0			0
$925$	944.206	−	155.000i	1.02076	−	0.167567i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			707.210i			0.761260i	0.924728	+	0.380630i	$0.124293\pi$
$929$			707.210i			0.761260i	−0.924728	+	0.380630i	$0.875707\pi$
$930$		0			0
$931$	−2696.98			−2.89686
$932$		0			0
$933$		0			0
$934$		0			0
$935$	822.869	+	698.801i	0.880074	+	0.747381i
$936$		0			0
$937$		−	1751.27i		−	1.86902i	−0.355941	−	0.934508i	$-0.615840\pi$
$937$		−	1751.27i		−	1.86902i	0.355941	−	0.934508i	$-0.384160\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			3.50123i			0.00372076i	0.999998	+	0.00186038i	$0.000592178\pi$
$941$			3.50123i			0.00372076i	−0.999998	+	0.00186038i	$0.999408\pi$
$942$		0			0
$943$		−	903.465i		−	0.958076i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−902.900			−0.953432			−0.476716	−	0.879057i	$-0.658173\pi$
$947$	−902.900			−0.953432			−0.476716	+	0.879057i	$0.658173\pi$
$948$		0			0
$949$	−276.385			−0.291238
$950$		0			0
$951$		0			0
$952$		0			0
$953$	364.685			0.382670			0.191335	−	0.981525i	$-0.438718\pi$
$953$	364.685			0.382670			0.191335	+	0.981525i	$0.438718\pi$
$954$		0			0
$955$	636.171	+	540.253i	0.666148	+	0.565710i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1213.13i		−	1.26500i
$960$		0			0
$961$	−769.560			−0.800791
$962$		0			0
$963$		0			0
$964$		0			0
$965$	325.207	+	276.174i	0.337002	+	0.286191i
$966$		0			0
$967$		−	303.856i		−	0.314225i	−0.987581	−	0.157113i	$-0.949781\pi$
$967$		−	303.856i		−	0.314225i	0.987581	−	0.157113i	$-0.0502186\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			987.430i			1.01692i	0.861086	+	0.508460i	$0.169785\pi$
$971$			987.430i			1.01692i	−0.861086	+	0.508460i	$0.830215\pi$
$972$		0			0
$973$			637.556i			0.655248i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1564.64			1.60147			0.800735	−	0.599019i	$-0.204443\pi$
$977$	1564.64			1.60147			0.800735	+	0.599019i	$0.204443\pi$
$978$		0			0
$979$	1403.54			1.43365
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1516.76			1.54299			0.771496	−	0.636234i	$-0.219509\pi$
$983$	1516.76			1.54299			0.771496	+	0.636234i	$0.219509\pi$
$984$		0			0
$985$	−796.931	+	938.421i	−0.809067	+	0.952712i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			382.187i			0.386438i
$990$		0			0
$991$	−1864.50			−1.88144			−0.940718	−	0.339189i	$-0.889847\pi$
$991$	−1864.50			−1.88144			−0.940718	+	0.339189i	$0.889847\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	463.442	−	545.723i	0.465771	−	0.548465i
$996$		0			0
$997$			1438.27i			1.44259i	0.692626	+	0.721296i	$0.256453\pi$
$997$			1438.27i			1.44259i	−0.692626	+	0.721296i	$0.743547\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.a.809.6	yes	24
3.2	odd	2	inner	1620.3.b.a.809.19	yes	24
5.4	even	2	inner	1620.3.b.a.809.20	yes	24
9.2	odd	6		1620.3.t.f.1349.14		48
9.4	even	3		1620.3.t.f.269.22		48
9.5	odd	6		1620.3.t.f.269.3		48
9.7	even	3		1620.3.t.f.1349.11		48
15.14	odd	2	inner	1620.3.b.a.809.5	✓	24
45.4	even	6		1620.3.t.f.269.14		48
45.14	odd	6		1620.3.t.f.269.11		48
45.29	odd	6		1620.3.t.f.1349.22		48
45.34	even	6		1620.3.t.f.1349.3		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.5	✓	24	15.14	odd	2	inner
1620.3.b.a.809.6	yes	24	1.1	even	1	trivial
1620.3.b.a.809.19	yes	24	3.2	odd	2	inner
1620.3.b.a.809.20	yes	24	5.4	even	2	inner
1620.3.t.f.269.3		48	9.5	odd	6
1620.3.t.f.269.11		48	45.14	odd	6
1620.3.t.f.269.14		48	45.4	even	6
1620.3.t.f.269.22		48	9.4	even	3
1620.3.t.f.1349.3		48	45.34	even	6
1620.3.t.f.1349.11		48	9.7	even	3
1620.3.t.f.1349.14		48	9.2	odd	6
1620.3.t.f.1349.22		48	45.29	odd	6

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.23653 + 3.81115i) q^{5} -12.6606i q^{7} +O(q^{10})\)	\(q+(-3.23653 + 3.81115i) q^{5} -12.6606i q^{7} +18.7009i q^{11} -6.79817i q^{13} -11.5455 q^{17} +24.2335 q^{19} -29.2202 q^{23} +(-4.04977 - 24.6698i) q^{25} -14.9461i q^{29} +13.8362 q^{31} +(48.2515 + 40.9764i) q^{35} +38.2737i q^{37} +30.9192i q^{41} -13.0796i q^{43} -5.68939 q^{47} -111.291 q^{49} +57.7486 q^{53} +(-71.2720 - 60.5260i) q^{55} -30.3390i q^{59} +77.0497 q^{61} +(25.9089 + 22.0025i) q^{65} +128.396i q^{67} +35.8765i q^{71} -40.6558i q^{73} +236.765 q^{77} +140.080 q^{79} +118.832 q^{83} +(37.3672 - 44.0015i) q^{85} -75.0520i q^{89} -86.0690 q^{91} +(-78.4325 + 92.3577i) q^{95} +84.5227i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) \) 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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