LMFDB - Embedded newform 9680.2.a.bl.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9680,2,Mod(1,9680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-2,0,3,0,3,0,0,0,-2,0,1,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} -1.00000 q^{5} +3.56155 q^{7} -0.561553 q^{9} +3.12311 q^{13} -1.56155 q^{15} -5.56155 q^{17} +2.43845 q^{19} +5.56155 q^{21} -7.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} +0.438447 q^{29} -8.68466 q^{31} -3.56155 q^{35} +9.80776 q^{37} +4.87689 q^{39} +10.0000 q^{41} +5.12311 q^{43} +0.561553 q^{45} +7.12311 q^{47} +5.68466 q^{49} -8.68466 q^{51} -4.43845 q^{53} +3.80776 q^{57} +13.3693 q^{59} +3.56155 q^{61} -2.00000 q^{63} -3.12311 q^{65} -11.1231 q^{69} +2.43845 q^{71} -4.87689 q^{73} +1.56155 q^{75} +0.876894 q^{79} -7.00000 q^{81} +10.0000 q^{83} +5.56155 q^{85} +0.684658 q^{87} +9.80776 q^{89} +11.1231 q^{91} -13.5616 q^{93} -2.43845 q^{95} +17.1231 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{13} + q^{15} - 7 q^{17} + 9 q^{19} + 7 q^{21} - 6 q^{23} + 2 q^{25} - 7 q^{27} + 5 q^{29} - 5 q^{31} - 3 q^{35} - q^{37} + 18 q^{39} + 20 q^{41} + 2 q^{43}+ \cdots + 26 q^{97}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 - 2 * q^13 + q^15 - 7 * q^17 + 9 * q^19 + 7 * q^21 - 6 * q^23 + 2 * q^25 - 7 * q^27 + 5 * q^29 - 5 * q^31 - 3 * q^35 - q^37 + 18 * q^39 + 20 * q^41 + 2 * q^43 - 3 * q^45 + 6 * q^47 - q^49 - 5 * q^51 - 13 * q^53 - 13 * q^57 + 2 * q^59 + 3 * q^61 - 4 * q^63 + 2 * q^65 - 14 * q^69 + 9 * q^71 - 18 * q^73 - q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 7 * q^85 - 11 * q^87 - q^89 + 14 * q^91 - 23 * q^93 - 9 * q^95 + 26 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.56155	1.34614	0.673070	−	0.739579i	$-0.264975\pi$
$7$	3.56155	1.34614	0.673070	+	0.739579i	$0.264975\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.12311	0.866194	0.433097	−	0.901347i	$-0.357421\pi$
$13$	3.12311	0.866194	0.433097	+	0.901347i	$0.357421\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	−5.56155	−1.34887	−0.674437	−	0.738332i	$-0.735614\pi$
$17$	−5.56155	−1.34887	−0.674437	+	0.738332i	$0.735614\pi$
$18$	0	0
$19$	2.43845	0.559418	0.279709	−	0.960085i	$-0.409762\pi$
$19$	2.43845	0.559418	0.279709	+	0.960085i	$0.409762\pi$
$20$	0	0
$21$	5.56155	1.21363
$22$	0	0
$23$	−7.12311	−1.48527	−0.742635	−	0.669696i	$-0.766424\pi$
$23$	−7.12311	−1.48527	−0.742635	+	0.669696i	$0.766424\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	0.438447	0.0814176	0.0407088	−	0.999171i	$-0.487038\pi$
$29$	0.438447	0.0814176	0.0407088	+	0.999171i	$0.487038\pi$
$30$	0	0
$31$	−8.68466	−1.55981	−0.779905	−	0.625897i	$-0.784733\pi$
$31$	−8.68466	−1.55981	−0.779905	+	0.625897i	$0.784733\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.56155	−0.602012
$36$	0	0
$37$	9.80776	1.61239	0.806193	−	0.591652i	$-0.201524\pi$
$37$	9.80776	1.61239	0.806193	+	0.591652i	$0.201524\pi$
$38$	0	0
$39$	4.87689	0.780928
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	5.12311	0.781266	0.390633	−	0.920546i	$-0.372256\pi$
$43$	5.12311	0.781266	0.390633	+	0.920546i	$0.372256\pi$
$44$	0	0
$45$	0.561553	0.0837114
$46$	0	0
$47$	7.12311	1.03901	0.519506	−	0.854467i	$-0.326116\pi$
$47$	7.12311	1.03901	0.519506	+	0.854467i	$0.326116\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	0	0
$51$	−8.68466	−1.21610
$52$	0	0
$53$	−4.43845	−0.609668	−0.304834	−	0.952406i	$-0.598601\pi$
$53$	−4.43845	−0.609668	−0.304834	+	0.952406i	$0.598601\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.80776	0.504351
$58$	0	0
$59$	13.3693	1.74054	0.870268	−	0.492578i	$-0.163945\pi$
$59$	13.3693	1.74054	0.870268	+	0.492578i	$0.163945\pi$
$60$	0	0
$61$	3.56155	0.456010	0.228005	−	0.973660i	$-0.426780\pi$
$61$	3.56155	0.456010	0.228005	+	0.973660i	$0.426780\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−3.12311	−0.387374
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−11.1231	−1.33906
$70$	0	0
$71$	2.43845	0.289390	0.144695	−	0.989476i	$-0.453780\pi$
$71$	2.43845	0.289390	0.144695	+	0.989476i	$0.453780\pi$
$72$	0	0
$73$	−4.87689	−0.570797	−0.285399	−	0.958409i	$-0.592126\pi$
$73$	−4.87689	−0.570797	−0.285399	+	0.958409i	$0.592126\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.876894	0.0986583	0.0493292	−	0.998783i	$-0.484292\pi$
$79$	0.876894	0.0986583	0.0493292	+	0.998783i	$0.484292\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	5.56155	0.603235
$86$	0	0
$87$	0.684658	0.0734031
$88$	0	0
$89$	9.80776	1.03962	0.519810	−	0.854282i	$-0.326003\pi$
$89$	9.80776	1.03962	0.519810	+	0.854282i	$0.326003\pi$
$90$	0	0
$91$	11.1231	1.16602
$92$	0	0
$93$	−13.5616	−1.40627
$94$	0	0
$95$	−2.43845	−0.250179
$96$	0	0
$97$	17.1231	1.73859	0.869294	−	0.494295i	$-0.164574\pi$
$97$	17.1231	1.73859	0.869294	+	0.494295i	$0.164574\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	14.2462	1.40372	0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462	1.40372	0.701860	+	0.712314i	$0.252353\pi$
$104$	0	0
$105$	−5.56155	−0.542752
$106$	0	0
$107$	5.12311	0.495269	0.247635	−	0.968853i	$-0.420347\pi$
$107$	5.12311	0.495269	0.247635	+	0.968853i	$0.420347\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	15.3153	1.45367
$112$	0	0
$113$	1.12311	0.105653	0.0528264	−	0.998604i	$-0.483177\pi$
$113$	1.12311	0.105653	0.0528264	+	0.998604i	$0.483177\pi$
$114$	0	0
$115$	7.12311	0.664233
$116$	0	0
$117$	−1.75379	−0.162138
$118$	0	0
$119$	−19.8078	−1.81577
$120$	0	0
$121$	0	0
$122$	0	0
$123$	15.6155	1.40800
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.36932	−0.653921	−0.326961	−	0.945038i	$-0.606024\pi$
$127$	−7.36932	−0.653921	−0.326961	+	0.945038i	$0.606024\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	0.684658	0.0598189	0.0299094	−	0.999553i	$-0.490478\pi$
$131$	0.684658	0.0598189	0.0299094	+	0.999553i	$0.490478\pi$
$132$	0	0
$133$	8.68466	0.753055
$134$	0	0
$135$	5.56155	0.478662
$136$	0	0
$137$	−19.3693	−1.65483	−0.827416	−	0.561589i	$-0.810190\pi$
$137$	−19.3693	−1.65483	−0.827416	+	0.561589i	$0.810190\pi$
$138$	0	0
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	0	0
$141$	11.1231	0.936734
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.438447	−0.0364111
$146$	0	0
$147$	8.87689	0.732154
$148$	0	0
$149$	1.31534	0.107757	0.0538785	−	0.998547i	$-0.482842\pi$
$149$	1.31534	0.107757	0.0538785	+	0.998547i	$0.482842\pi$
$150$	0	0
$151$	−0.876894	−0.0713607	−0.0356803	−	0.999363i	$-0.511360\pi$
$151$	−0.876894	−0.0713607	−0.0356803	+	0.999363i	$0.511360\pi$
$152$	0	0
$153$	3.12311	0.252488
$154$	0	0
$155$	8.68466	0.697569
$156$	0	0
$157$	−12.9309	−1.03200	−0.515998	−	0.856590i	$-0.672579\pi$
$157$	−12.9309	−1.03200	−0.515998	+	0.856590i	$0.672579\pi$
$158$	0	0
$159$	−6.93087	−0.549654
$160$	0	0
$161$	−25.3693	−1.99938
$162$	0	0
$163$	−0.192236	−0.0150571	−0.00752854	−	0.999972i	$-0.502396\pi$
$163$	−0.192236	−0.0150571	−0.00752854	+	0.999972i	$0.502396\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.31534	0.101784	0.0508921	−	0.998704i	$-0.483794\pi$
$167$	1.31534	0.101784	0.0508921	+	0.998704i	$0.483794\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	−1.36932	−0.104714
$172$	0	0
$173$	21.3693	1.62468	0.812340	−	0.583185i	$-0.198194\pi$
$173$	21.3693	1.62468	0.812340	+	0.583185i	$0.198194\pi$
$174$	0	0
$175$	3.56155	0.269228
$176$	0	0
$177$	20.8769	1.56920
$178$	0	0
$179$	−8.49242	−0.634753	−0.317377	−	0.948300i	$-0.602802\pi$
$179$	−8.49242	−0.634753	−0.317377	+	0.948300i	$0.602802\pi$
$180$	0	0
$181$	14.4924	1.07721	0.538607	−	0.842557i	$-0.318951\pi$
$181$	14.4924	1.07721	0.538607	+	0.842557i	$0.318951\pi$
$182$	0	0
$183$	5.56155	0.411122
$184$	0	0
$185$	−9.80776	−0.721081
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−19.8078	−1.44080
$190$	0	0
$191$	−1.75379	−0.126900	−0.0634499	−	0.997985i	$-0.520210\pi$
$191$	−1.75379	−0.126900	−0.0634499	+	0.997985i	$0.520210\pi$
$192$	0	0
$193$	−19.8078	−1.42579	−0.712897	−	0.701269i	$-0.752617\pi$
$193$	−19.8078	−1.42579	−0.712897	+	0.701269i	$0.752617\pi$
$194$	0	0
$195$	−4.87689	−0.349242
$196$	0	0
$197$	−18.2462	−1.29999	−0.649994	−	0.759939i	$-0.725229\pi$
$197$	−18.2462	−1.29999	−0.649994	+	0.759939i	$0.725229\pi$
$198$	0	0
$199$	6.93087	0.491316	0.245658	−	0.969357i	$-0.420996\pi$
$199$	6.93087	0.491316	0.245658	+	0.969357i	$0.420996\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.56155	0.109600
$204$	0	0
$205$	−10.0000	−0.698430
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0.684658	0.0471338	0.0235669	−	0.999722i	$-0.492498\pi$
$211$	0.684658	0.0471338	0.0235669	+	0.999722i	$0.492498\pi$
$212$	0	0
$213$	3.80776	0.260904
$214$	0	0
$215$	−5.12311	−0.349393
$216$	0	0
$217$	−30.9309	−2.09972
$218$	0	0
$219$	−7.61553	−0.514610
$220$	0	0
$221$	−17.3693	−1.16839
$222$	0	0
$223$	23.1231	1.54844	0.774219	−	0.632918i	$-0.218143\pi$
$223$	23.1231	1.54844	0.774219	+	0.632918i	$0.218143\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	−6.49242	−0.429031	−0.214516	−	0.976721i	$-0.568817\pi$
$229$	−6.49242	−0.429031	−0.214516	+	0.976721i	$0.568817\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.3153	−0.741293	−0.370646	−	0.928774i	$-0.620864\pi$
$233$	−11.3153	−0.741293	−0.370646	+	0.928774i	$0.620864\pi$
$234$	0	0
$235$	−7.12311	−0.464660
$236$	0	0
$237$	1.36932	0.0889467
$238$	0	0
$239$	0.876894	0.0567216	0.0283608	−	0.999598i	$-0.490971\pi$
$239$	0.876894	0.0567216	0.0283608	+	0.999598i	$0.490971\pi$
$240$	0	0
$241$	28.7386	1.85122	0.925609	−	0.378481i	$-0.123553\pi$
$241$	28.7386	1.85122	0.925609	+	0.378481i	$0.123553\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	−5.68466	−0.363180
$246$	0	0
$247$	7.61553	0.484564
$248$	0	0
$249$	15.6155	0.989594
$250$	0	0
$251$	−23.1231	−1.45952	−0.729759	−	0.683705i	$-0.760368\pi$
$251$	−23.1231	−1.45952	−0.729759	+	0.683705i	$0.760368\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	8.68466	0.543854
$256$	0	0
$257$	−7.75379	−0.483668	−0.241834	−	0.970318i	$-0.577749\pi$
$257$	−7.75379	−0.483668	−0.241834	+	0.970318i	$0.577749\pi$
$258$	0	0
$259$	34.9309	2.17050
$260$	0	0
$261$	−0.246211	−0.0152401
$262$	0	0
$263$	24.4384	1.50694	0.753470	−	0.657483i	$-0.228379\pi$
$263$	24.4384	1.50694	0.753470	+	0.657483i	$0.228379\pi$
$264$	0	0
$265$	4.43845	0.272652
$266$	0	0
$267$	15.3153	0.937284
$268$	0	0
$269$	−22.8769	−1.39483	−0.697414	−	0.716668i	$-0.745666\pi$
$269$	−22.8769	−1.39483	−0.697414	+	0.716668i	$0.745666\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	17.3693	1.05124
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.7386	−1.12590	−0.562948	−	0.826493i	$-0.690333\pi$
$277$	−18.7386	−1.12590	−0.562948	+	0.826493i	$0.690333\pi$
$278$	0	0
$279$	4.87689	0.291972
$280$	0	0
$281$	−18.4924	−1.10317	−0.551583	−	0.834120i	$-0.685976\pi$
$281$	−18.4924	−1.10317	−0.551583	+	0.834120i	$0.685976\pi$
$282$	0	0
$283$	−18.4924	−1.09926	−0.549630	−	0.835408i	$-0.685231\pi$
$283$	−18.4924	−1.09926	−0.549630	+	0.835408i	$0.685231\pi$
$284$	0	0
$285$	−3.80776	−0.225552
$286$	0	0
$287$	35.6155	2.10232
$288$	0	0
$289$	13.9309	0.819463
$290$	0	0
$291$	26.7386	1.56745
$292$	0	0
$293$	11.1231	0.649819	0.324909	−	0.945745i	$-0.394666\pi$
$293$	11.1231	0.649819	0.324909	+	0.945745i	$0.394666\pi$
$294$	0	0
$295$	−13.3693	−0.778392
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−22.2462	−1.28653
$300$	0	0
$301$	18.2462	1.05169
$302$	0	0
$303$	25.3693	1.45743
$304$	0	0
$305$	−3.56155	−0.203934
$306$	0	0
$307$	20.7386	1.18362	0.591808	−	0.806079i	$-0.298414\pi$
$307$	20.7386	1.18362	0.591808	+	0.806079i	$0.298414\pi$
$308$	0	0
$309$	22.2462	1.26554
$310$	0	0
$311$	−21.1771	−1.20084	−0.600421	−	0.799684i	$-0.705000\pi$
$311$	−21.1771	−1.20084	−0.600421	+	0.799684i	$0.705000\pi$
$312$	0	0
$313$	16.2462	0.918290	0.459145	−	0.888361i	$-0.348156\pi$
$313$	16.2462	0.918290	0.459145	+	0.888361i	$0.348156\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	−3.56155	−0.200037	−0.100018	−	0.994986i	$-0.531890\pi$
$317$	−3.56155	−0.200037	−0.100018	+	0.994986i	$0.531890\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	−13.5616	−0.754585
$324$	0	0
$325$	3.12311	0.173239
$326$	0	0
$327$	3.12311	0.172708
$328$	0	0
$329$	25.3693	1.39866
$330$	0	0
$331$	2.24621	0.123463	0.0617315	−	0.998093i	$-0.480338\pi$
$331$	2.24621	0.123463	0.0617315	+	0.998093i	$0.480338\pi$
$332$	0	0
$333$	−5.50758	−0.301813
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−6.93087	−0.377549	−0.188774	−	0.982021i	$-0.560451\pi$
$337$	−6.93087	−0.377549	−0.188774	+	0.982021i	$0.560451\pi$
$338$	0	0
$339$	1.75379	0.0952527
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−4.68466	−0.252948
$344$	0	0
$345$	11.1231	0.598848
$346$	0	0
$347$	24.2462	1.30160	0.650802	−	0.759247i	$-0.274433\pi$
$347$	24.2462	1.30160	0.650802	+	0.759247i	$0.274433\pi$
$348$	0	0
$349$	24.7386	1.32423	0.662114	−	0.749403i	$-0.269659\pi$
$349$	24.7386	1.32423	0.662114	+	0.749403i	$0.269659\pi$
$350$	0	0
$351$	−17.3693	−0.927106
$352$	0	0
$353$	3.75379	0.199794	0.0998970	−	0.994998i	$-0.468149\pi$
$353$	3.75379	0.199794	0.0998970	+	0.994998i	$0.468149\pi$
$354$	0	0
$355$	−2.43845	−0.129419
$356$	0	0
$357$	−30.9309	−1.63704
$358$	0	0
$359$	−8.49242	−0.448213	−0.224106	−	0.974565i	$-0.571946\pi$
$359$	−8.49242	−0.448213	−0.224106	+	0.974565i	$0.571946\pi$
$360$	0	0
$361$	−13.0540	−0.687051
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.87689	0.255268
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	−5.61553	−0.292333
$370$	0	0
$371$	−15.8078	−0.820698
$372$	0	0
$373$	29.3693	1.52069	0.760343	−	0.649522i	$-0.225031\pi$
$373$	29.3693	1.52069	0.760343	+	0.649522i	$0.225031\pi$
$374$	0	0
$375$	−1.56155	−0.0806382
$376$	0	0
$377$	1.36932	0.0705234
$378$	0	0
$379$	−27.6155	−1.41851	−0.709257	−	0.704950i	$-0.750970\pi$
$379$	−27.6155	−1.41851	−0.709257	+	0.704950i	$0.750970\pi$
$380$	0	0
$381$	−11.5076	−0.589551
$382$	0	0
$383$	29.8617	1.52586	0.762932	−	0.646479i	$-0.223759\pi$
$383$	29.8617	1.52586	0.762932	+	0.646479i	$0.223759\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.87689	−0.146241
$388$	0	0
$389$	−32.2462	−1.63495	−0.817474	−	0.575966i	$-0.804626\pi$
$389$	−32.2462	−1.63495	−0.817474	+	0.575966i	$0.804626\pi$
$390$	0	0
$391$	39.6155	2.00344
$392$	0	0
$393$	1.06913	0.0539305
$394$	0	0
$395$	−0.876894	−0.0441213
$396$	0	0
$397$	5.50758	0.276417	0.138209	−	0.990403i	$-0.455866\pi$
$397$	5.50758	0.276417	0.138209	+	0.990403i	$0.455866\pi$
$398$	0	0
$399$	13.5616	0.678927
$400$	0	0
$401$	−21.8078	−1.08903	−0.544514	−	0.838752i	$-0.683286\pi$
$401$	−21.8078	−1.08903	−0.544514	+	0.838752i	$0.683286\pi$
$402$	0	0
$403$	−27.1231	−1.35110
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	0	0
$408$	0	0
$409$	17.1231	0.846683	0.423342	−	0.905970i	$-0.360857\pi$
$409$	17.1231	0.846683	0.423342	+	0.905970i	$0.360857\pi$
$410$	0	0
$411$	−30.2462	−1.49194
$412$	0	0
$413$	47.6155	2.34301
$414$	0	0
$415$	−10.0000	−0.490881
$416$	0	0
$417$	25.7538	1.26117
$418$	0	0
$419$	5.75379	0.281091	0.140545	−	0.990074i	$-0.455114\pi$
$419$	5.75379	0.281091	0.140545	+	0.990074i	$0.455114\pi$
$420$	0	0
$421$	21.1231	1.02948	0.514739	−	0.857347i	$-0.327889\pi$
$421$	21.1231	1.02948	0.514739	+	0.857347i	$0.327889\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	−5.56155	−0.269775
$426$	0	0
$427$	12.6847	0.613854
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.4924	0.987085	0.493543	−	0.869722i	$-0.335702\pi$
$431$	20.4924	0.987085	0.493543	+	0.869722i	$0.335702\pi$
$432$	0	0
$433$	−23.8617	−1.14672	−0.573361	−	0.819303i	$-0.694361\pi$
$433$	−23.8617	−1.14672	−0.573361	+	0.819303i	$0.694361\pi$
$434$	0	0
$435$	−0.684658	−0.0328269
$436$	0	0
$437$	−17.3693	−0.830887
$438$	0	0
$439$	−24.9848	−1.19246	−0.596231	−	0.802813i	$-0.703336\pi$
$439$	−24.9848	−1.19246	−0.596231	+	0.802813i	$0.703336\pi$
$440$	0	0
$441$	−3.19224	−0.152011
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	−9.80776	−0.464933
$446$	0	0
$447$	2.05398	0.0971497
$448$	0	0
$449$	−28.7386	−1.35626	−0.678130	−	0.734942i	$-0.737209\pi$
$449$	−28.7386	−1.35626	−0.678130	+	0.734942i	$0.737209\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−1.36932	−0.0643361
$454$	0	0
$455$	−11.1231	−0.521459
$456$	0	0
$457$	38.0540	1.78009	0.890045	−	0.455873i	$-0.150673\pi$
$457$	38.0540	1.78009	0.890045	+	0.455873i	$0.150673\pi$
$458$	0	0
$459$	30.9309	1.44373
$460$	0	0
$461$	−28.9309	−1.34744	−0.673722	−	0.738984i	$-0.735306\pi$
$461$	−28.9309	−1.34744	−0.673722	+	0.738984i	$0.735306\pi$
$462$	0	0
$463$	0.876894	0.0407527	0.0203764	−	0.999792i	$-0.493514\pi$
$463$	0.876894	0.0407527	0.0203764	+	0.999792i	$0.493514\pi$
$464$	0	0
$465$	13.5616	0.628902
$466$	0	0
$467$	−8.68466	−0.401878	−0.200939	−	0.979604i	$-0.564399\pi$
$467$	−8.68466	−0.401878	−0.200939	+	0.979604i	$0.564399\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−20.1922	−0.930409
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.43845	0.111884
$476$	0	0
$477$	2.49242	0.114120
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	30.6307	1.39664
$482$	0	0
$483$	−39.6155	−1.80257
$484$	0	0
$485$	−17.1231	−0.777520
$486$	0	0
$487$	34.2462	1.55184	0.775922	−	0.630829i	$-0.217285\pi$
$487$	34.2462	1.55184	0.775922	+	0.630829i	$0.217285\pi$
$488$	0	0
$489$	−0.300187	−0.0135749
$490$	0	0
$491$	−15.8078	−0.713394	−0.356697	−	0.934220i	$-0.616097\pi$
$491$	−15.8078	−0.713394	−0.356697	+	0.934220i	$0.616097\pi$
$492$	0	0
$493$	−2.43845	−0.109822
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.68466	0.389560
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	2.05398	0.0917648
$502$	0	0
$503$	29.1231	1.29854	0.649268	−	0.760560i	$-0.275076\pi$
$503$	29.1231	1.29854	0.649268	+	0.760560i	$0.275076\pi$
$504$	0	0
$505$	−16.2462	−0.722947
$506$	0	0
$507$	−5.06913	−0.225128
$508$	0	0
$509$	−35.3693	−1.56772	−0.783859	−	0.620939i	$-0.786751\pi$
$509$	−35.3693	−1.56772	−0.783859	+	0.620939i	$0.786751\pi$
$510$	0	0
$511$	−17.3693	−0.768373
$512$	0	0
$513$	−13.5616	−0.598757
$514$	0	0
$515$	−14.2462	−0.627763
$516$	0	0
$517$	0	0
$518$	0	0
$519$	33.3693	1.46475
$520$	0	0
$521$	9.50758	0.416535	0.208267	−	0.978072i	$-0.433218\pi$
$521$	9.50758	0.416535	0.208267	+	0.978072i	$0.433218\pi$
$522$	0	0
$523$	23.3693	1.02187	0.510934	−	0.859620i	$-0.329299\pi$
$523$	23.3693	1.02187	0.510934	+	0.859620i	$0.329299\pi$
$524$	0	0
$525$	5.56155	0.242726
$526$	0	0
$527$	48.3002	2.10399
$528$	0	0
$529$	27.7386	1.20603
$530$	0	0
$531$	−7.50758	−0.325801
$532$	0	0
$533$	31.2311	1.35277
$534$	0	0
$535$	−5.12311	−0.221491
$536$	0	0
$537$	−13.2614	−0.572270
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.5616	−0.497070	−0.248535	−	0.968623i	$-0.579949\pi$
$541$	−11.5616	−0.497070	−0.248535	+	0.968623i	$0.579949\pi$
$542$	0	0
$543$	22.6307	0.971176
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−13.1231	−0.561103	−0.280552	−	0.959839i	$-0.590517\pi$
$547$	−13.1231	−0.561103	−0.280552	+	0.959839i	$0.590517\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	1.06913	0.0455465
$552$	0	0
$553$	3.12311	0.132808
$554$	0	0
$555$	−15.3153	−0.650100
$556$	0	0
$557$	−7.12311	−0.301816	−0.150908	−	0.988548i	$-0.548220\pi$
$557$	−7.12311	−0.301816	−0.150908	+	0.988548i	$0.548220\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.24621	−0.347536	−0.173768	−	0.984787i	$-0.555594\pi$
$563$	−8.24621	−0.347536	−0.173768	+	0.984787i	$0.555594\pi$
$564$	0	0
$565$	−1.12311	−0.0472494
$566$	0	0
$567$	−24.9309	−1.04700
$568$	0	0
$569$	2.87689	0.120606	0.0603028	−	0.998180i	$-0.480793\pi$
$569$	2.87689	0.120606	0.0603028	+	0.998180i	$0.480793\pi$
$570$	0	0
$571$	31.8078	1.33111	0.665557	−	0.746347i	$-0.268194\pi$
$571$	31.8078	1.33111	0.665557	+	0.746347i	$0.268194\pi$
$572$	0	0
$573$	−2.73863	−0.114408
$574$	0	0
$575$	−7.12311	−0.297054
$576$	0	0
$577$	−30.4924	−1.26942	−0.634708	−	0.772752i	$-0.718880\pi$
$577$	−30.4924	−1.26942	−0.634708	+	0.772752i	$0.718880\pi$
$578$	0	0
$579$	−30.9309	−1.28544
$580$	0	0
$581$	35.6155	1.47758
$582$	0	0
$583$	0	0
$584$	0	0
$585$	1.75379	0.0725102
$586$	0	0
$587$	−21.5616	−0.889941	−0.444970	−	0.895545i	$-0.646786\pi$
$587$	−21.5616	−0.889941	−0.444970	+	0.895545i	$0.646786\pi$
$588$	0	0
$589$	−21.1771	−0.872586
$590$	0	0
$591$	−28.4924	−1.17202
$592$	0	0
$593$	7.61553	0.312732	0.156366	−	0.987699i	$-0.450022\pi$
$593$	7.61553	0.312732	0.156366	+	0.987699i	$0.450022\pi$
$594$	0	0
$595$	19.8078	0.812039
$596$	0	0
$597$	10.8229	0.442953
$598$	0	0
$599$	15.3153	0.625768	0.312884	−	0.949791i	$-0.398705\pi$
$599$	15.3153	0.625768	0.312884	+	0.949791i	$0.398705\pi$
$600$	0	0
$601$	−37.2311	−1.51869	−0.759343	−	0.650690i	$-0.774480\pi$
$601$	−37.2311	−1.51869	−0.759343	+	0.650690i	$0.774480\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	34.6847	1.40781	0.703903	−	0.710296i	$-0.251439\pi$
$607$	34.6847	1.40781	0.703903	+	0.710296i	$0.251439\pi$
$608$	0	0
$609$	2.43845	0.0988109
$610$	0	0
$611$	22.2462	0.899985
$612$	0	0
$613$	−27.6155	−1.11538	−0.557690	−	0.830049i	$-0.688312\pi$
$613$	−27.6155	−1.11538	−0.557690	+	0.830049i	$0.688312\pi$
$614$	0	0
$615$	−15.6155	−0.629679
$616$	0	0
$617$	2.87689	0.115819	0.0579097	−	0.998322i	$-0.481556\pi$
$617$	2.87689	0.115819	0.0579097	+	0.998322i	$0.481556\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	39.6155	1.58972
$622$	0	0
$623$	34.9309	1.39948
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−54.5464	−2.17491
$630$	0	0
$631$	2.43845	0.0970730	0.0485365	−	0.998821i	$-0.484544\pi$
$631$	2.43845	0.0970730	0.0485365	+	0.998821i	$0.484544\pi$
$632$	0	0
$633$	1.06913	0.0424941
$634$	0	0
$635$	7.36932	0.292442
$636$	0	0
$637$	17.7538	0.703431
$638$	0	0
$639$	−1.36932	−0.0541693
$640$	0	0
$641$	3.94602	0.155859	0.0779293	−	0.996959i	$-0.475169\pi$
$641$	3.94602	0.155859	0.0779293	+	0.996959i	$0.475169\pi$
$642$	0	0
$643$	19.3153	0.761723	0.380861	−	0.924632i	$-0.375628\pi$
$643$	19.3153	0.761723	0.380861	+	0.924632i	$0.375628\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	9.36932	0.368346	0.184173	−	0.982894i	$-0.441039\pi$
$647$	9.36932	0.368346	0.184173	+	0.982894i	$0.441039\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−48.3002	−1.89303
$652$	0	0
$653$	10.1922	0.398853	0.199427	−	0.979913i	$-0.436092\pi$
$653$	10.1922	0.398853	0.199427	+	0.979913i	$0.436092\pi$
$654$	0	0
$655$	−0.684658	−0.0267518
$656$	0	0
$657$	2.73863	0.106844
$658$	0	0
$659$	5.06913	0.197465	0.0987326	−	0.995114i	$-0.468521\pi$
$659$	5.06913	0.197465	0.0987326	+	0.995114i	$0.468521\pi$
$660$	0	0
$661$	−14.4924	−0.563690	−0.281845	−	0.959460i	$-0.590946\pi$
$661$	−14.4924	−0.563690	−0.281845	+	0.959460i	$0.590946\pi$
$662$	0	0
$663$	−27.1231	−1.05337
$664$	0	0
$665$	−8.68466	−0.336777
$666$	0	0
$667$	−3.12311	−0.120927
$668$	0	0
$669$	36.1080	1.39601
$670$	0	0
$671$	0	0
$672$	0	0
$673$	21.1771	0.816316	0.408158	−	0.912911i	$-0.366171\pi$
$673$	21.1771	0.816316	0.408158	+	0.912911i	$0.366171\pi$
$674$	0	0
$675$	−5.56155	−0.214064
$676$	0	0
$677$	6.24621	0.240061	0.120031	−	0.992770i	$-0.461701\pi$
$677$	6.24621	0.240061	0.120031	+	0.992770i	$0.461701\pi$
$678$	0	0
$679$	60.9848	2.34038
$680$	0	0
$681$	−28.1080	−1.07710
$682$	0	0
$683$	−3.80776	−0.145700	−0.0728500	−	0.997343i	$-0.523209\pi$
$683$	−3.80776	−0.145700	−0.0728500	+	0.997343i	$0.523209\pi$
$684$	0	0
$685$	19.3693	0.740064
$686$	0	0
$687$	−10.1383	−0.386799
$688$	0	0
$689$	−13.8617	−0.528090
$690$	0	0
$691$	35.6155	1.35488	0.677439	−	0.735579i	$-0.263090\pi$
$691$	35.6155	1.35488	0.677439	+	0.735579i	$0.263090\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.4924	−0.625593
$696$	0	0
$697$	−55.6155	−2.10659
$698$	0	0
$699$	−17.6695	−0.668322
$700$	0	0
$701$	36.5464	1.38034	0.690169	−	0.723648i	$-0.257536\pi$
$701$	36.5464	1.38034	0.690169	+	0.723648i	$0.257536\pi$
$702$	0	0
$703$	23.9157	0.901998
$704$	0	0
$705$	−11.1231	−0.418920
$706$	0	0
$707$	57.8617	2.17611
$708$	0	0
$709$	48.2462	1.81192	0.905962	−	0.423358i	$-0.139149\pi$
$709$	48.2462	1.81192	0.905962	+	0.423358i	$0.139149\pi$
$710$	0	0
$711$	−0.492423	−0.0184673
$712$	0	0
$713$	61.8617	2.31674
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.36932	0.0511381
$718$	0	0
$719$	42.0540	1.56835	0.784174	−	0.620541i	$-0.213087\pi$
$719$	42.0540	1.56835	0.784174	+	0.620541i	$0.213087\pi$
$720$	0	0
$721$	50.7386	1.88961
$722$	0	0
$723$	44.8769	1.66899
$724$	0	0
$725$	0.438447	0.0162835
$726$	0	0
$727$	−23.6155	−0.875851	−0.437926	−	0.899011i	$-0.644287\pi$
$727$	−23.6155	−0.875851	−0.437926	+	0.899011i	$0.644287\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−28.4924	−1.05383
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	−8.87689	−0.327429
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−46.7386	−1.71931	−0.859654	−	0.510876i	$-0.829321\pi$
$739$	−46.7386	−1.71931	−0.859654	+	0.510876i	$0.829321\pi$
$740$	0	0
$741$	11.8920	0.436865
$742$	0	0
$743$	−28.4384	−1.04331	−0.521653	−	0.853158i	$-0.674684\pi$
$743$	−28.4384	−1.04331	−0.521653	+	0.853158i	$0.674684\pi$
$744$	0	0
$745$	−1.31534	−0.0481904
$746$	0	0
$747$	−5.61553	−0.205461
$748$	0	0
$749$	18.2462	0.666702
$750$	0	0
$751$	−18.4384	−0.672828	−0.336414	−	0.941714i	$-0.609214\pi$
$751$	−18.4384	−0.672828	−0.336414	+	0.941714i	$0.609214\pi$
$752$	0	0
$753$	−36.1080	−1.31585
$754$	0	0
$755$	0.876894	0.0319135
$756$	0	0
$757$	−26.4924	−0.962883	−0.481442	−	0.876478i	$-0.659887\pi$
$757$	−26.4924	−0.962883	−0.481442	+	0.876478i	$0.659887\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	7.12311	0.257874
$764$	0	0
$765$	−3.12311	−0.112916
$766$	0	0
$767$	41.7538	1.50764
$768$	0	0
$769$	−30.9848	−1.11734	−0.558671	−	0.829389i	$-0.688689\pi$
$769$	−30.9848	−1.11734	−0.558671	+	0.829389i	$0.688689\pi$
$770$	0	0
$771$	−12.1080	−0.436057
$772$	0	0
$773$	0.822919	0.0295983	0.0147992	−	0.999890i	$-0.495289\pi$
$773$	0.822919	0.0295983	0.0147992	+	0.999890i	$0.495289\pi$
$774$	0	0
$775$	−8.68466	−0.311962
$776$	0	0
$777$	54.5464	1.95684
$778$	0	0
$779$	24.3845	0.873664
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.43845	−0.0871430
$784$	0	0
$785$	12.9309	0.461523
$786$	0	0
$787$	−26.8769	−0.958058	−0.479029	−	0.877799i	$-0.659011\pi$
$787$	−26.8769	−0.958058	−0.479029	+	0.877799i	$0.659011\pi$
$788$	0	0
$789$	38.1619	1.35860
$790$	0	0
$791$	4.00000	0.142224
$792$	0	0
$793$	11.1231	0.394993
$794$	0	0
$795$	6.93087	0.245813
$796$	0	0
$797$	12.7386	0.451226	0.225613	−	0.974217i	$-0.427562\pi$
$797$	12.7386	0.451226	0.225613	+	0.974217i	$0.427562\pi$
$798$	0	0
$799$	−39.6155	−1.40150
$800$	0	0
$801$	−5.50758	−0.194601
$802$	0	0
$803$	0	0
$804$	0	0
$805$	25.3693	0.894151
$806$	0	0
$807$	−35.7235	−1.25753
$808$	0	0
$809$	52.7386	1.85419	0.927096	−	0.374824i	$-0.122297\pi$
$809$	52.7386	1.85419	0.927096	+	0.374824i	$0.122297\pi$
$810$	0	0
$811$	−46.4384	−1.63067	−0.815337	−	0.578986i	$-0.803448\pi$
$811$	−46.4384	−1.63067	−0.815337	+	0.578986i	$0.803448\pi$
$812$	0	0
$813$	18.7386	0.657193
$814$	0	0
$815$	0.192236	0.00673373
$816$	0	0
$817$	12.4924	0.437055
$818$	0	0
$819$	−6.24621	−0.218260
$820$	0	0
$821$	−19.7538	−0.689412	−0.344706	−	0.938711i	$-0.612021\pi$
$821$	−19.7538	−0.689412	−0.344706	+	0.938711i	$0.612021\pi$
$822$	0	0
$823$	−18.7386	−0.653188	−0.326594	−	0.945165i	$-0.605901\pi$
$823$	−18.7386	−0.653188	−0.326594	+	0.945165i	$0.605901\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.3845	0.361103	0.180552	−	0.983565i	$-0.442212\pi$
$827$	10.3845	0.361103	0.180552	+	0.983565i	$0.442212\pi$
$828$	0	0
$829$	47.8617	1.66231	0.831153	−	0.556043i	$-0.187681\pi$
$829$	47.8617	1.66231	0.831153	+	0.556043i	$0.187681\pi$
$830$	0	0
$831$	−29.2614	−1.01507
$832$	0	0
$833$	−31.6155	−1.09541
$834$	0	0
$835$	−1.31534	−0.0455193
$836$	0	0
$837$	48.3002	1.66950
$838$	0	0
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−28.8078	−0.993371
$842$	0	0
$843$	−28.8769	−0.994573
$844$	0	0
$845$	3.24621	0.111673
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−28.8769	−0.991052
$850$	0	0
$851$	−69.8617	−2.39483
$852$	0	0
$853$	40.1080	1.37327	0.686635	−	0.727002i	$-0.259087\pi$
$853$	40.1080	1.37327	0.686635	+	0.727002i	$0.259087\pi$
$854$	0	0
$855$	1.36932	0.0468296
$856$	0	0
$857$	−50.0540	−1.70981	−0.854906	−	0.518784i	$-0.826385\pi$
$857$	−50.0540	−1.70981	−0.854906	+	0.518784i	$0.826385\pi$
$858$	0	0
$859$	−21.3693	−0.729112	−0.364556	−	0.931182i	$-0.618779\pi$
$859$	−21.3693	−0.729112	−0.364556	+	0.931182i	$0.618779\pi$
$860$	0	0
$861$	55.6155	1.89537
$862$	0	0
$863$	−9.36932	−0.318935	−0.159468	−	0.987203i	$-0.550978\pi$
$863$	−9.36932	−0.318935	−0.159468	+	0.987203i	$0.550978\pi$
$864$	0	0
$865$	−21.3693	−0.726579
$866$	0	0
$867$	21.7538	0.738797
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−9.61553	−0.325436
$874$	0	0
$875$	−3.56155	−0.120402
$876$	0	0
$877$	−39.1231	−1.32109	−0.660547	−	0.750785i	$-0.729675\pi$
$877$	−39.1231	−1.32109	−0.660547	+	0.750785i	$0.729675\pi$
$878$	0	0
$879$	17.3693	0.585853
$880$	0	0
$881$	−34.4924	−1.16208	−0.581040	−	0.813875i	$-0.697354\pi$
$881$	−34.4924	−1.16208	−0.581040	+	0.813875i	$0.697354\pi$
$882$	0	0
$883$	17.9460	0.603932	0.301966	−	0.953319i	$-0.402357\pi$
$883$	17.9460	0.603932	0.301966	+	0.953319i	$0.402357\pi$
$884$	0	0
$885$	−20.8769	−0.701769
$886$	0	0
$887$	−55.3693	−1.85912	−0.929560	−	0.368671i	$-0.879813\pi$
$887$	−55.3693	−1.85912	−0.929560	+	0.368671i	$0.879813\pi$
$888$	0	0
$889$	−26.2462	−0.880270
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17.3693	0.581242
$894$	0	0
$895$	8.49242	0.283870
$896$	0	0
$897$	−34.7386	−1.15989
$898$	0	0
$899$	−3.80776	−0.126996
$900$	0	0
$901$	24.6847	0.822365
$902$	0	0
$903$	28.4924	0.948168
$904$	0	0
$905$	−14.4924	−0.481744
$906$	0	0
$907$	−45.1771	−1.50008	−0.750040	−	0.661392i	$-0.769966\pi$
$907$	−45.1771	−1.50008	−0.750040	+	0.661392i	$0.769966\pi$
$908$	0	0
$909$	−9.12311	−0.302594
$910$	0	0
$911$	41.6695	1.38057	0.690286	−	0.723537i	$-0.257485\pi$
$911$	41.6695	1.38057	0.690286	+	0.723537i	$0.257485\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−5.56155	−0.183859
$916$	0	0
$917$	2.43845	0.0805246
$918$	0	0
$919$	−9.75379	−0.321748	−0.160874	−	0.986975i	$-0.551431\pi$
$919$	−9.75379	−0.321748	−0.160874	+	0.986975i	$0.551431\pi$
$920$	0	0
$921$	32.3845	1.06710
$922$	0	0
$923$	7.61553	0.250668
$924$	0	0
$925$	9.80776	0.322477
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	0.930870	0.0305408	0.0152704	−	0.999883i	$-0.495139\pi$
$929$	0.930870	0.0305408	0.0152704	+	0.999883i	$0.495139\pi$
$930$	0	0
$931$	13.8617	0.454300
$932$	0	0
$933$	−33.0691	−1.08263
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.3693	0.959454	0.479727	−	0.877418i	$-0.340736\pi$
$937$	29.3693	0.959454	0.479727	+	0.877418i	$0.340736\pi$
$938$	0	0
$939$	25.3693	0.827896
$940$	0	0
$941$	28.4384	0.927067	0.463533	−	0.886079i	$-0.346581\pi$
$941$	28.4384	0.927067	0.463533	+	0.886079i	$0.346581\pi$
$942$	0	0
$943$	−71.2311	−2.31960
$944$	0	0
$945$	19.8078	0.644347
$946$	0	0
$947$	−50.0540	−1.62654	−0.813268	−	0.581890i	$-0.802314\pi$
$947$	−50.0540	−1.62654	−0.813268	+	0.581890i	$0.802314\pi$
$948$	0	0
$949$	−15.2311	−0.494421
$950$	0	0
$951$	−5.56155	−0.180346
$952$	0	0
$953$	−37.6695	−1.22023	−0.610117	−	0.792311i	$-0.708878\pi$
$953$	−37.6695	−1.22023	−0.610117	+	0.792311i	$0.708878\pi$
$954$	0	0
$955$	1.75379	0.0567513
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−68.9848	−2.22764
$960$	0	0
$961$	44.4233	1.43301
$962$	0	0
$963$	−2.87689	−0.0927066
$964$	0	0
$965$	19.8078	0.637634
$966$	0	0
$967$	−8.05398	−0.258998	−0.129499	−	0.991580i	$-0.541337\pi$
$967$	−8.05398	−0.258998	−0.129499	+	0.991580i	$0.541337\pi$
$968$	0	0
$969$	−21.1771	−0.680306
$970$	0	0
$971$	−2.24621	−0.0720843	−0.0360422	−	0.999350i	$-0.511475\pi$
$971$	−2.24621	−0.0720843	−0.0360422	+	0.999350i	$0.511475\pi$
$972$	0	0
$973$	58.7386	1.88307
$974$	0	0
$975$	4.87689	0.156186
$976$	0	0
$977$	−51.3693	−1.64345	−0.821725	−	0.569884i	$-0.806988\pi$
$977$	−51.3693	−1.64345	−0.821725	+	0.569884i	$0.806988\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−1.12311	−0.0358580
$982$	0	0
$983$	−20.8769	−0.665870	−0.332935	−	0.942950i	$-0.608039\pi$
$983$	−20.8769	−0.665870	−0.332935	+	0.942950i	$0.608039\pi$
$984$	0	0
$985$	18.2462	0.581373
$986$	0	0
$987$	39.6155	1.26098
$988$	0	0
$989$	−36.4924	−1.16039
$990$	0	0
$991$	52.4924	1.66748	0.833738	−	0.552160i	$-0.186196\pi$
$991$	52.4924	1.66748	0.833738	+	0.552160i	$0.186196\pi$
$992$	0	0
$993$	3.50758	0.111310
$994$	0	0
$995$	−6.93087	−0.219723
$996$	0	0
$997$	−12.9848	−0.411234	−0.205617	−	0.978633i	$-0.565920\pi$
$997$	−12.9848	−0.411234	−0.205617	+	0.978633i	$0.565920\pi$
$998$	0	0
$999$	−54.5464	−1.72577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.bl.1.2		2
4.3	odd	2		4840.2.a.n.1.1		2
11.10	odd	2		880.2.a.l.1.2		2
33.32	even	2		7920.2.a.ca.1.1		2
44.43	even	2		440.2.a.f.1.1	✓	2
55.32	even	4		4400.2.b.u.4049.2		4
55.43	even	4		4400.2.b.u.4049.3		4
55.54	odd	2		4400.2.a.br.1.1		2
88.21	odd	2		3520.2.a.bs.1.1		2
88.43	even	2		3520.2.a.bl.1.2		2
132.131	odd	2		3960.2.a.be.1.2		2
220.43	odd	4		2200.2.b.h.1849.2		4
220.87	odd	4		2200.2.b.h.1849.3		4
220.219	even	2		2200.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.f.1.1	✓	2	44.43	even	2
880.2.a.l.1.2		2	11.10	odd	2
2200.2.a.m.1.2		2	220.219	even	2
2200.2.b.h.1849.2		4	220.43	odd	4
2200.2.b.h.1849.3		4	220.87	odd	4
3520.2.a.bl.1.2		2	88.43	even	2
3520.2.a.bs.1.1		2	88.21	odd	2
3960.2.a.be.1.2		2	132.131	odd	2
4400.2.a.br.1.1		2	55.54	odd	2
4400.2.b.u.4049.2		4	55.32	even	4
4400.2.b.u.4049.3		4	55.43	even	4
4840.2.a.n.1.1		2	4.3	odd	2
7920.2.a.ca.1.1		2	33.32	even	2
9680.2.a.bl.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -1.00000 q^{5} +3.56155 q^{7} -0.561553 q^{9} +3.12311 q^{13} -1.56155 q^{15} -5.56155 q^{17} +2.43845 q^{19} +5.56155 q^{21} -7.12311 q^{23} +1.00000 q^{25} -5.56155 q^{27} +0.438447 q^{29} -8.68466 q^{31} -3.56155 q^{35} +9.80776 q^{37} +4.87689 q^{39} +10.0000 q^{41} +5.12311 q^{43} +0.561553 q^{45} +7.12311 q^{47} +5.68466 q^{49} -8.68466 q^{51} -4.43845 q^{53} +3.80776 q^{57} +13.3693 q^{59} +3.56155 q^{61} -2.00000 q^{63} -3.12311 q^{65} -11.1231 q^{69} +2.43845 q^{71} -4.87689 q^{73} +1.56155 q^{75} +0.876894 q^{79} -7.00000 q^{81} +10.0000 q^{83} +5.56155 q^{85} +0.684658 q^{87} +9.80776 q^{89} +11.1231 q^{91} -13.5616 q^{93} -2.43845 q^{95} +17.1231 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{13} + q^{15} - 7 q^{17} + 9 q^{19} + 7 q^{21} - 6 q^{23} + 2 q^{25} - 7 q^{27} + 5 q^{29} - 5 q^{31} - 3 q^{35} - q^{37} + 18 q^{39} + 20 q^{41} + 2 q^{43}+ \cdots + 26 q^{97}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 - 2 * q^13 + q^15 - 7 * q^17 + 9 * q^19 + 7 * q^21 - 6 * q^23 + 2 * q^25 - 7 * q^27 + 5 * q^29 - 5 * q^31 - 3 * q^35 - q^37 + 18 * q^39 + 20 * q^41 + 2 * q^43 - 3 * q^45 + 6 * q^47 - q^49 - 5 * q^51 - 13 * q^53 - 13 * q^57 + 2 * q^59 + 3 * q^61 - 4 * q^63 + 2 * q^65 - 14 * q^69 + 9 * q^71 - 18 * q^73 - q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 7 * q^85 - 11 * q^87 - q^89 + 14 * q^91 - 23 * q^93 - 9 * q^95 + 26 * q^97

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