LMFDB - Embedded newform 9801.2.a.ca.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9801,2,Mod(1,9801)] 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("9801.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9801, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,16,4,0,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$78.2613790211$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.36514623744.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 13x^{6} + 49x^{4} - 45x^{2} + 9 $ x^8 - 13x^6 + 49x^4 - 45*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 1089)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.970194$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.81578 q^{2} +1.29706 q^{4} -3.44209 q^{5} +1.15177 q^{7} +1.27638 q^{8} +6.25008 q^{10} -2.20570 q^{13} -2.09137 q^{14} -4.91175 q^{16} +7.48558 q^{17} +1.52371 q^{19} -4.46460 q^{20} +7.38336 q^{23} +6.84796 q^{25} +4.00507 q^{26} +1.49392 q^{28} +6.49741 q^{29} -4.05872 q^{31} +6.36591 q^{32} -13.5922 q^{34} -3.96451 q^{35} -4.43702 q^{37} -2.76673 q^{38} -4.39342 q^{40} +1.27638 q^{41} +7.23825 q^{43} -13.4066 q^{46} +2.02758 q^{47} -5.67342 q^{49} -12.4344 q^{50} -2.86093 q^{52} +12.5059 q^{53} +1.47010 q^{56} -11.7979 q^{58} +13.1812 q^{59} +8.93434 q^{61} +7.36975 q^{62} -1.73559 q^{64} +7.59221 q^{65} +5.08630 q^{67} +9.70926 q^{68} +7.19868 q^{70} -4.69787 q^{71} -0.163600 q^{73} +8.05666 q^{74} +1.97635 q^{76} +3.28252 q^{79} +16.9067 q^{80} -2.31763 q^{82} -12.1015 q^{83} -25.7660 q^{85} -13.1431 q^{86} +8.15366 q^{89} -2.54047 q^{91} +9.57668 q^{92} -3.68164 q^{94} -5.24475 q^{95} -3.38693 q^{97} +10.3017 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} + 4 q^{5} + 4 q^{14} + 24 q^{16} - 6 q^{20} + 46 q^{23} + 12 q^{25} + 30 q^{26} - 14 q^{31} - 38 q^{34} - 6 q^{37} + 4 q^{38} + 16 q^{47} + 42 q^{49} + 48 q^{53} + 46 q^{56} - 50 q^{58} + 48 q^{59}+ \cdots + 4 q^{97}+O(q^{100}) $ 8 * q + 16 * q^4 + 4 * q^5 + 4 * q^14 + 24 * q^16 - 6 * q^20 + 46 * q^23 + 12 * q^25 + 30 * q^26 - 14 * q^31 - 38 * q^34 - 6 * q^37 + 4 * q^38 + 16 * q^47 + 42 * q^49 + 48 * q^53 + 46 * q^56 - 50 * q^58 + 48 * q^59 - 6 * q^64 + 22 * q^67 + 4 * q^70 - 34 * q^71 + 74 * q^80 + 56 * q^82 + 14 * q^86 + 8 * q^89 - 48 * q^91 + 84 * q^92 + 4 * 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 $

$2$	−1.81578	−1.28395	−0.641976	−	0.766725i	$-0.721885\pi$
$2$	−1.81578	−1.28395	−0.641976	+	0.766725i	$0.721885\pi$
$3$	0	0
$3$	0	0
$4$	1.29706	0.648531
$5$	−3.44209	−1.53935	−0.769674	−	0.638437i	$-0.779581\pi$
$5$	−3.44209	−1.53935	−0.769674	+	0.638437i	$0.779581\pi$
$6$	0	0
$7$	1.15177	0.435330	0.217665	−	0.976024i	$-0.430156\pi$
$7$	1.15177	0.435330	0.217665	+	0.976024i	$0.430156\pi$
$8$	1.27638	0.451269
$9$	0	0
$10$	6.25008	1.97645
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.20570	−0.611751	−0.305875	−	0.952072i	$-0.598949\pi$
$13$	−2.20570	−0.611751	−0.305875	+	0.952072i	$0.598949\pi$
$14$	−2.09137	−0.558942
$15$	0	0
$16$	−4.91175	−1.22794
$17$	7.48558	1.81552	0.907760	−	0.419489i	$-0.137791\pi$
$17$	7.48558	1.81552	0.907760	+	0.419489i	$0.137791\pi$
$18$	0	0
$19$	1.52371	0.349564	0.174782	−	0.984607i	$-0.444078\pi$
$19$	1.52371	0.349564	0.174782	+	0.984607i	$0.444078\pi$
$20$	−4.46460	−0.998315
$21$	0	0
$22$	0	0
$23$	7.38336	1.53954	0.769769	−	0.638323i	$-0.220371\pi$
$23$	7.38336	1.53954	0.769769	+	0.638323i	$0.220371\pi$
$24$	0	0
$25$	6.84796	1.36959
$26$	4.00507	0.785458
$27$	0	0
$28$	1.49392	0.282325
$29$	6.49741	1.20654	0.603269	−	0.797538i	$-0.293864\pi$
$29$	6.49741	1.20654	0.603269	+	0.797538i	$0.293864\pi$
$30$	0	0
$31$	−4.05872	−0.728968	−0.364484	−	0.931210i	$-0.618755\pi$
$31$	−4.05872	−0.728968	−0.364484	+	0.931210i	$0.618755\pi$
$32$	6.36591	1.12534
$33$	0	0
$34$	−13.5922	−2.33104
$35$	−3.96451	−0.670124
$36$	0	0
$37$	−4.43702	−0.729441	−0.364721	−	0.931117i	$-0.618836\pi$
$37$	−4.43702	−0.729441	−0.364721	+	0.931117i	$0.618836\pi$
$38$	−2.76673	−0.448823
$39$	0	0
$40$	−4.39342	−0.694661
$41$	1.27638	0.199337	0.0996687	−	0.995021i	$-0.468222\pi$
$41$	1.27638	0.199337	0.0996687	+	0.995021i	$0.468222\pi$
$42$	0	0
$43$	7.23825	1.10382	0.551912	−	0.833903i	$-0.313899\pi$
$43$	7.23825	1.10382	0.551912	+	0.833903i	$0.313899\pi$
$44$	0	0
$45$	0	0
$46$	−13.4066	−1.97669
$47$	2.02758	0.295753	0.147876	−	0.989006i	$-0.452756\pi$
$47$	2.02758	0.295753	0.147876	+	0.989006i	$0.452756\pi$
$48$	0	0
$49$	−5.67342	−0.810488
$50$	−12.4344	−1.75849
$51$	0	0
$52$	−2.86093	−0.396739
$53$	12.5059	1.71781	0.858907	−	0.512131i	$-0.171144\pi$
$53$	12.5059	1.71781	0.858907	+	0.512131i	$0.171144\pi$
$54$	0	0
$55$	0	0
$56$	1.47010	0.196451
$57$	0	0
$58$	−11.7979	−1.54914
$59$	13.1812	1.71605	0.858025	−	0.513607i	$-0.171691\pi$
$59$	13.1812	1.71605	0.858025	+	0.513607i	$0.171691\pi$
$60$	0	0
$61$	8.93434	1.14393	0.571963	−	0.820280i	$-0.306182\pi$
$61$	8.93434	1.14393	0.571963	+	0.820280i	$0.306182\pi$
$62$	7.36975	0.935960
$63$	0	0
$64$	−1.73559	−0.216948
$65$	7.59221	0.941698
$66$	0	0
$67$	5.08630	0.621391	0.310695	−	0.950510i	$-0.399438\pi$
$67$	5.08630	0.621391	0.310695	+	0.950510i	$0.399438\pi$
$68$	9.70926	1.17742
$69$	0	0
$70$	7.19868	0.860407
$71$	−4.69787	−0.557535	−0.278767	−	0.960359i	$-0.589926\pi$
$71$	−4.69787	−0.557535	−0.278767	+	0.960359i	$0.589926\pi$
$72$	0	0
$73$	−0.163600	−0.0191479	−0.00957396	−	0.999954i	$-0.503048\pi$
$73$	−0.163600	−0.0191479	−0.00957396	+	0.999954i	$0.503048\pi$
$74$	8.05666	0.936567
$75$	0	0
$76$	1.97635	0.226703
$77$	0	0
$78$	0	0
$79$	3.28252	0.369313	0.184656	−	0.982803i	$-0.440883\pi$
$79$	3.28252	0.369313	0.184656	+	0.982803i	$0.440883\pi$
$80$	16.9067	1.89023
$81$	0	0
$82$	−2.31763	−0.255940
$83$	−12.1015	−1.32831	−0.664154	−	0.747596i	$-0.731208\pi$
$83$	−12.1015	−1.32831	−0.664154	+	0.747596i	$0.731208\pi$
$84$	0	0
$85$	−25.7660	−2.79472
$86$	−13.1431	−1.41726
$87$	0	0
$88$	0	0
$89$	8.15366	0.864286	0.432143	−	0.901805i	$-0.357758\pi$
$89$	8.15366	0.864286	0.432143	+	0.901805i	$0.357758\pi$
$90$	0	0
$91$	−2.54047	−0.266313
$92$	9.57668	0.998438
$93$	0	0
$94$	−3.68164	−0.379732
$95$	−5.24475	−0.538100
$96$	0	0
$97$	−3.38693	−0.343890	−0.171945	−	0.985107i	$-0.555005\pi$
$97$	−3.38693	−0.343890	−0.171945	+	0.985107i	$0.555005\pi$
$98$	10.3017	1.04063
$99$	0	0
$100$	8.88223	0.888223
$101$	−1.27638	−0.127005	−0.0635024	−	0.997982i	$-0.520227\pi$
$101$	−1.27638	−0.127005	−0.0635024	+	0.997982i	$0.520227\pi$
$102$	0	0
$103$	2.08123	0.205070	0.102535	−	0.994729i	$-0.467305\pi$
$103$	2.08123	0.205070	0.102535	+	0.994729i	$0.467305\pi$
$104$	−2.81532	−0.276064
$105$	0	0
$106$	−22.7079	−2.20559
$107$	3.67244	0.355028	0.177514	−	0.984118i	$-0.443194\pi$
$107$	3.67244	0.355028	0.177514	+	0.984118i	$0.443194\pi$
$108$	0	0
$109$	−10.7321	−1.02795	−0.513976	−	0.857804i	$-0.671828\pi$
$109$	−10.7321	−1.02795	−0.513976	+	0.857804i	$0.671828\pi$
$110$	0	0
$111$	0	0
$112$	−5.65723	−0.534558
$113$	−1.56654	−0.147368	−0.0736840	−	0.997282i	$-0.523476\pi$
$113$	−1.56654	−0.147368	−0.0736840	+	0.997282i	$0.523476\pi$
$114$	0	0
$115$	−25.4142	−2.36988
$116$	8.42754	0.782477
$117$	0	0
$118$	−23.9342	−2.20333
$119$	8.62170	0.790350
$120$	0	0
$121$	0	0
$122$	−16.2228	−1.46874
$123$	0	0
$124$	−5.26441	−0.472758
$125$	−6.36085	−0.568932
$126$	0	0
$127$	21.0255	1.86572	0.932858	−	0.360245i	$-0.117307\pi$
$127$	21.0255	1.86572	0.932858	+	0.360245i	$0.117307\pi$
$128$	−9.58037	−0.846793
$129$	0	0
$130$	−13.7858	−1.20909
$131$	−17.6146	−1.53899	−0.769495	−	0.638652i	$-0.779492\pi$
$131$	−17.6146	−1.53899	−0.769495	+	0.638652i	$0.779492\pi$
$132$	0	0
$133$	1.75497	0.152175
$134$	−9.23561	−0.797835
$135$	0	0
$136$	9.55446	0.819289
$137$	7.26247	0.620475	0.310237	−	0.950659i	$-0.399591\pi$
$137$	7.26247	0.620475	0.310237	+	0.950659i	$0.399591\pi$
$138$	0	0
$139$	3.13993	0.266326	0.133163	−	0.991094i	$-0.457487\pi$
$139$	3.13993	0.266326	0.133163	+	0.991094i	$0.457487\pi$
$140$	−5.14221	−0.434596
$141$	0	0
$142$	8.53031	0.715847
$143$	0	0
$144$	0	0
$145$	−22.3646	−1.85728
$146$	0.297062	0.0245850
$147$	0	0
$148$	−5.75509	−0.473065
$149$	−21.9778	−1.80049	−0.900244	−	0.435386i	$-0.856612\pi$
$149$	−21.9778	−1.80049	−0.900244	+	0.435386i	$0.856612\pi$
$150$	0	0
$151$	2.13609	0.173832	0.0869162	−	0.996216i	$-0.472299\pi$
$151$	2.13609	0.173832	0.0869162	+	0.996216i	$0.472299\pi$
$152$	1.94484	0.157747
$153$	0	0
$154$	0	0
$155$	13.9705	1.12214
$156$	0	0
$157$	17.9635	1.43364	0.716820	−	0.697258i	$-0.245597\pi$
$157$	17.9635	1.43364	0.716820	+	0.697258i	$0.245597\pi$
$158$	−5.96034	−0.474179
$159$	0	0
$160$	−21.9120	−1.73230
$161$	8.50397	0.670207
$162$	0	0
$163$	−14.4887	−1.13485	−0.567423	−	0.823427i	$-0.692059\pi$
$163$	−14.4887	−1.13485	−0.567423	+	0.823427i	$0.692059\pi$
$164$	1.65555	0.129276
$165$	0	0
$166$	21.9736	1.70548
$167$	1.59719	0.123594	0.0617969	−	0.998089i	$-0.480317\pi$
$167$	1.59719	0.123594	0.0617969	+	0.998089i	$0.480317\pi$
$168$	0	0
$169$	−8.13489	−0.625761
$170$	46.7855	3.58828
$171$	0	0
$172$	9.38846	0.715863
$173$	−3.51187	−0.267003	−0.133501	−	0.991049i	$-0.542622\pi$
$173$	−3.51187	−0.267003	−0.133501	+	0.991049i	$0.542622\pi$
$174$	0	0
$175$	7.88731	0.596225
$176$	0	0
$177$	0	0
$178$	−14.8053	−1.10970
$179$	−0.555970	−0.0415551	−0.0207776	−	0.999784i	$-0.506614\pi$
$179$	−0.555970	−0.0415551	−0.0207776	+	0.999784i	$0.506614\pi$
$180$	0	0
$181$	3.97048	0.295123	0.147562	−	0.989053i	$-0.452858\pi$
$181$	3.97048	0.295123	0.147562	+	0.989053i	$0.452858\pi$
$182$	4.61294	0.341934
$183$	0	0
$184$	9.42400	0.694746
$185$	15.2726	1.12286
$186$	0	0
$187$	0	0
$188$	2.62990	0.191805
$189$	0	0
$190$	9.52332	0.690894
$191$	−10.2276	−0.740042	−0.370021	−	0.929023i	$-0.620649\pi$
$191$	−10.2276	−0.740042	−0.370021	+	0.929023i	$0.620649\pi$
$192$	0	0
$193$	2.87357	0.206844	0.103422	−	0.994638i	$-0.467021\pi$
$193$	2.87357	0.206844	0.103422	+	0.994638i	$0.467021\pi$
$194$	6.14992	0.441538
$195$	0	0
$196$	−7.35877	−0.525626
$197$	6.57236	0.468261	0.234131	−	0.972205i	$-0.424776\pi$
$197$	6.57236	0.468261	0.234131	+	0.972205i	$0.424776\pi$
$198$	0	0
$199$	−1.02758	−0.0728432	−0.0364216	−	0.999337i	$-0.511596\pi$
$199$	−1.02758	−0.0728432	−0.0364216	+	0.999337i	$0.511596\pi$
$200$	8.74062	0.618055
$201$	0	0
$202$	2.31763	0.163068
$203$	7.48355	0.525242
$204$	0	0
$205$	−4.39342	−0.306850
$206$	−3.77907	−0.263300
$207$	0	0
$208$	10.8339	0.751193
$209$	0	0
$210$	0	0
$211$	−14.4279	−0.993258	−0.496629	−	0.867963i	$-0.665429\pi$
$211$	−14.4279	−0.993258	−0.496629	+	0.867963i	$0.665429\pi$
$212$	16.2209	1.11406
$213$	0	0
$214$	−6.66835	−0.455839
$215$	−24.9147	−1.69917
$216$	0	0
$217$	−4.67473	−0.317342
$218$	19.4872	1.31984
$219$	0	0
$220$	0	0
$221$	−16.5109	−1.11065
$222$	0	0
$223$	18.1568	1.21587	0.607934	−	0.793988i	$-0.291999\pi$
$223$	18.1568	1.21587	0.607934	+	0.793988i	$0.291999\pi$
$224$	7.33209	0.489896
$225$	0	0
$226$	2.84450	0.189213
$227$	−5.49208	−0.364522	−0.182261	−	0.983250i	$-0.558342\pi$
$227$	−5.49208	−0.364522	−0.182261	+	0.983250i	$0.558342\pi$
$228$	0	0
$229$	4.89119	0.323219	0.161609	−	0.986855i	$-0.448332\pi$
$229$	4.89119	0.323219	0.161609	+	0.986855i	$0.448332\pi$
$230$	46.1466	3.04282
$231$	0	0
$232$	8.29318	0.544474
$233$	−22.8673	−1.49808	−0.749042	−	0.662523i	$-0.769486\pi$
$233$	−22.8673	−1.49808	−0.749042	+	0.662523i	$0.769486\pi$
$234$	0	0
$235$	−6.97911	−0.455267
$236$	17.0969	1.11291
$237$	0	0
$238$	−15.6551	−1.01477
$239$	0.247330	0.0159985	0.00799924	−	0.999968i	$-0.497454\pi$
$239$	0.247330	0.0159985	0.00799924	+	0.999968i	$0.497454\pi$
$240$	0	0
$241$	−1.79969	−0.115928	−0.0579640	−	0.998319i	$-0.518461\pi$
$241$	−1.79969	−0.115928	−0.0579640	+	0.998319i	$0.518461\pi$
$242$	0	0
$243$	0	0
$244$	11.5884	0.741871
$245$	19.5284	1.24762
$246$	0	0
$247$	−3.36085	−0.213846
$248$	−5.18048	−0.328961
$249$	0	0
$250$	11.5499	0.730481
$251$	−1.28843	−0.0813251	−0.0406625	−	0.999173i	$-0.512947\pi$
$251$	−1.28843	−0.0813251	−0.0406625	+	0.999173i	$0.512947\pi$
$252$	0	0
$253$	0	0
$254$	−38.1778	−2.39549
$255$	0	0
$256$	20.8670	1.30419
$257$	11.6321	0.725593	0.362796	−	0.931868i	$-0.381822\pi$
$257$	11.6321	0.725593	0.362796	+	0.931868i	$0.381822\pi$
$258$	0	0
$259$	−5.11045	−0.317548
$260$	9.84756	0.610720
$261$	0	0
$262$	31.9842	1.97599
$263$	9.69086	0.597564	0.298782	−	0.954321i	$-0.403420\pi$
$263$	9.69086	0.597564	0.298782	+	0.954321i	$0.403420\pi$
$264$	0	0
$265$	−43.0463	−2.64431
$266$	−3.18665	−0.195386
$267$	0	0
$268$	6.59725	0.402991
$269$	1.73753	0.105939	0.0529695	−	0.998596i	$-0.483131\pi$
$269$	1.73753	0.105939	0.0529695	+	0.998596i	$0.483131\pi$
$270$	0	0
$271$	−18.9387	−1.15044	−0.575222	−	0.817997i	$-0.695084\pi$
$271$	−18.9387	−1.15044	−0.575222	+	0.817997i	$0.695084\pi$
$272$	−36.7673	−2.22935
$273$	0	0
$274$	−13.1871	−0.796659
$275$	0	0
$276$	0	0
$277$	4.79400	0.288044	0.144022	−	0.989575i	$-0.453996\pi$
$277$	4.79400	0.288044	0.144022	+	0.989575i	$0.453996\pi$
$278$	−5.70143	−0.341949
$279$	0	0
$280$	−5.06023	−0.302406
$281$	−17.2961	−1.03180	−0.515900	−	0.856649i	$-0.672543\pi$
$281$	−17.2961	−1.03180	−0.515900	+	0.856649i	$0.672543\pi$
$282$	0	0
$283$	17.4918	1.03978	0.519891	−	0.854233i	$-0.325973\pi$
$283$	17.4918	1.03978	0.519891	+	0.854233i	$0.325973\pi$
$284$	−6.09343	−0.361578
$285$	0	0
$286$	0	0
$287$	1.47010	0.0867775
$288$	0	0
$289$	39.0339	2.29611
$290$	40.6093	2.38466
$291$	0	0
$292$	−0.212199	−0.0124180
$293$	0.448776	0.0262178	0.0131089	−	0.999914i	$-0.495827\pi$
$293$	0.448776	0.0262178	0.0131089	+	0.999914i	$0.495827\pi$
$294$	0	0
$295$	−45.3710	−2.64160
$296$	−5.66333	−0.329175
$297$	0	0
$298$	39.9068	2.31174
$299$	−16.2855	−0.941814
$300$	0	0
$301$	8.33683	0.480527
$302$	−3.87867	−0.223192
$303$	0	0
$304$	−7.48410	−0.429243
$305$	−30.7528	−1.76090
$306$	0	0
$307$	−12.4154	−0.708583	−0.354291	−	0.935135i	$-0.615278\pi$
$307$	−12.4154	−0.708583	−0.354291	+	0.935135i	$0.615278\pi$
$308$	0	0
$309$	0	0
$310$	−25.3673	−1.44077
$311$	23.3002	1.32123	0.660616	−	0.750724i	$-0.270295\pi$
$311$	23.3002	1.32123	0.660616	+	0.750724i	$0.270295\pi$
$312$	0	0
$313$	18.4938	1.04533	0.522666	−	0.852538i	$-0.324938\pi$
$313$	18.4938	1.04533	0.522666	+	0.852538i	$0.324938\pi$
$314$	−32.6177	−1.84072
$315$	0	0
$316$	4.25763	0.239511
$317$	−21.9966	−1.23545	−0.617725	−	0.786394i	$-0.711945\pi$
$317$	−21.9966	−1.23545	−0.617725	+	0.786394i	$0.711945\pi$
$318$	0	0
$319$	0	0
$320$	5.97404	0.333959
$321$	0	0
$322$	−15.4414	−0.860513
$323$	11.4059	0.634640
$324$	0	0
$325$	−15.1046	−0.837850
$326$	26.3084	1.45709
$327$	0	0
$328$	1.62915	0.0899549
$329$	2.33532	0.128750
$330$	0	0
$331$	−19.9273	−1.09530	−0.547651	−	0.836707i	$-0.684478\pi$
$331$	−19.9273	−1.09530	−0.547651	+	0.836707i	$0.684478\pi$
$332$	−15.6963	−0.861448
$333$	0	0
$334$	−2.90014	−0.158689
$335$	−17.5075	−0.956537
$336$	0	0
$337$	−5.91713	−0.322327	−0.161163	−	0.986928i	$-0.551525\pi$
$337$	−5.91713	−0.322327	−0.161163	+	0.986928i	$0.551525\pi$
$338$	14.7712	0.803446
$339$	0	0
$340$	−33.4201	−1.81246
$341$	0	0
$342$	0	0
$343$	−14.5969	−0.788159
$344$	9.23878	0.498121
$345$	0	0
$346$	6.37679	0.342818
$347$	24.9988	1.34201	0.671004	−	0.741454i	$-0.265863\pi$
$347$	24.9988	1.34201	0.671004	+	0.741454i	$0.265863\pi$
$348$	0	0
$349$	−25.7583	−1.37881	−0.689405	−	0.724376i	$-0.742128\pi$
$349$	−25.7583	−1.37881	−0.689405	+	0.724376i	$0.742128\pi$
$350$	−14.3216	−0.765523
$351$	0	0
$352$	0	0
$353$	2.55417	0.135945	0.0679723	−	0.997687i	$-0.478347\pi$
$353$	2.55417	0.135945	0.0679723	+	0.997687i	$0.478347\pi$
$354$	0	0
$355$	16.1705	0.858240
$356$	10.5758	0.560516
$357$	0	0
$358$	1.00952	0.0533548
$359$	6.28175	0.331538	0.165769	−	0.986165i	$-0.446989\pi$
$359$	6.28175	0.331538	0.165769	+	0.986165i	$0.446989\pi$
$360$	0	0
$361$	−16.6783	−0.877805
$362$	−7.20952	−0.378924
$363$	0	0
$364$	−3.29514	−0.172713
$365$	0.563125	0.0294753
$366$	0	0
$367$	−17.5732	−0.917315	−0.458658	−	0.888613i	$-0.651670\pi$
$367$	−17.5732	−0.917315	−0.458658	+	0.888613i	$0.651670\pi$
$368$	−36.2653	−1.89046
$369$	0	0
$370$	−27.7317	−1.44170
$371$	14.4040	0.747816
$372$	0	0
$373$	−6.64920	−0.344282	−0.172141	−	0.985072i	$-0.555069\pi$
$373$	−6.64920	−0.344282	−0.172141	+	0.985072i	$0.555069\pi$
$374$	0	0
$375$	0	0
$376$	2.58797	0.133464
$377$	−14.3313	−0.738101
$378$	0	0
$379$	−2.34371	−0.120388	−0.0601940	−	0.998187i	$-0.519172\pi$
$379$	−2.34371	−0.120388	−0.0601940	+	0.998187i	$0.519172\pi$
$380$	−6.80277	−0.348975
$381$	0	0
$382$	18.5711	0.950178
$383$	18.1762	0.928759	0.464379	−	0.885636i	$-0.346277\pi$
$383$	18.1762	0.928759	0.464379	+	0.885636i	$0.346277\pi$
$384$	0	0
$385$	0	0
$386$	−5.21777	−0.265577
$387$	0	0
$388$	−4.39305	−0.223023
$389$	−13.0047	−0.659367	−0.329683	−	0.944092i	$-0.606942\pi$
$389$	−13.0047	−0.659367	−0.329683	+	0.944092i	$0.606942\pi$
$390$	0	0
$391$	55.2688	2.79506
$392$	−7.24145	−0.365748
$393$	0	0
$394$	−11.9340	−0.601225
$395$	−11.2987	−0.568501
$396$	0	0
$397$	−34.8041	−1.74677	−0.873384	−	0.487032i	$-0.838080\pi$
$397$	−34.8041	−1.74677	−0.873384	+	0.487032i	$0.838080\pi$
$398$	1.86586	0.0935271
$399$	0	0
$400$	−33.6355	−1.68178
$401$	4.59769	0.229597	0.114799	−	0.993389i	$-0.463378\pi$
$401$	4.59769	0.229597	0.114799	+	0.993389i	$0.463378\pi$
$402$	0	0
$403$	8.95232	0.445947
$404$	−1.65555	−0.0823665
$405$	0	0
$406$	−13.5885	−0.674385
$407$	0	0
$408$	0	0
$409$	32.7492	1.61935	0.809673	−	0.586882i	$-0.199645\pi$
$409$	32.7492	1.61935	0.809673	+	0.586882i	$0.199645\pi$
$410$	7.97749	0.393980
$411$	0	0
$412$	2.69949	0.132994
$413$	15.1818	0.747048
$414$	0	0
$415$	41.6543	2.04473
$416$	−14.0413	−0.688430
$417$	0	0
$418$	0	0
$419$	−25.8563	−1.26316	−0.631581	−	0.775310i	$-0.717594\pi$
$419$	−25.8563	−1.26316	−0.631581	+	0.775310i	$0.717594\pi$
$420$	0	0
$421$	28.9324	1.41008	0.705040	−	0.709167i	$-0.250929\pi$
$421$	28.9324	1.41008	0.705040	+	0.709167i	$0.250929\pi$
$422$	26.1979	1.27529
$423$	0	0
$424$	15.9623	0.775197
$425$	51.2610	2.48652
$426$	0	0
$427$	10.2903	0.497985
$428$	4.76338	0.230247
$429$	0	0
$430$	45.2396	2.18165
$431$	21.9002	1.05490	0.527448	−	0.849588i	$-0.323149\pi$
$431$	21.9002	1.05490	0.527448	+	0.849588i	$0.323149\pi$
$432$	0	0
$433$	22.7804	1.09476	0.547379	−	0.836885i	$-0.315626\pi$
$433$	22.7804	1.09476	0.547379	+	0.836885i	$0.315626\pi$
$434$	8.48829	0.407451
$435$	0	0
$436$	−13.9203	−0.666659
$437$	11.2501	0.538167
$438$	0	0
$439$	−12.6982	−0.606054	−0.303027	−	0.952982i	$-0.597997\pi$
$439$	−12.6982	−0.606054	−0.303027	+	0.952982i	$0.597997\pi$
$440$	0	0
$441$	0	0
$442$	29.9803	1.42602
$443$	−3.46280	−0.164522	−0.0822612	−	0.996611i	$-0.526214\pi$
$443$	−3.46280	−0.164522	−0.0822612	+	0.996611i	$0.526214\pi$
$444$	0	0
$445$	−28.0656	−1.33044
$446$	−32.9687	−1.56112
$447$	0	0
$448$	−1.99900	−0.0944441
$449$	−32.0547	−1.51275	−0.756376	−	0.654137i	$-0.773032\pi$
$449$	−32.0547	−1.51275	−0.756376	+	0.654137i	$0.773032\pi$
$450$	0	0
$451$	0	0
$452$	−2.03190	−0.0955727
$453$	0	0
$454$	9.97242	0.468029
$455$	8.74451	0.409949
$456$	0	0
$457$	−13.7765	−0.644439	−0.322220	−	0.946665i	$-0.604429\pi$
$457$	−13.7765	−0.644439	−0.322220	+	0.946665i	$0.604429\pi$
$458$	−8.88132	−0.414997
$459$	0	0
$460$	−32.9638	−1.53694
$461$	−3.59215	−0.167303	−0.0836515	−	0.996495i	$-0.526658\pi$
$461$	−3.59215	−0.167303	−0.0836515	+	0.996495i	$0.526658\pi$
$462$	0	0
$463$	24.8892	1.15670	0.578351	−	0.815788i	$-0.303697\pi$
$463$	24.8892	1.15670	0.578351	+	0.815788i	$0.303697\pi$
$464$	−31.9137	−1.48155
$465$	0	0
$466$	41.5219	1.92347
$467$	23.8940	1.10568	0.552841	−	0.833287i	$-0.313544\pi$
$467$	23.8940	1.10568	0.552841	+	0.833287i	$0.313544\pi$
$468$	0	0
$469$	5.85827	0.270510
$470$	12.6725	0.584540
$471$	0	0
$472$	16.8243	0.774401
$473$	0	0
$474$	0	0
$475$	10.4343	0.478760
$476$	11.1829	0.512567
$477$	0	0
$478$	−0.449098	−0.0205413
$479$	18.5977	0.849751	0.424876	−	0.905252i	$-0.360318\pi$
$479$	18.5977	0.849751	0.424876	+	0.905252i	$0.360318\pi$
$480$	0	0
$481$	9.78673	0.446237
$482$	3.26784	0.148846
$483$	0	0
$484$	0	0
$485$	11.6581	0.529367
$486$	0	0
$487$	−19.3139	−0.875196	−0.437598	−	0.899171i	$-0.644171\pi$
$487$	−19.3139	−0.875196	−0.437598	+	0.899171i	$0.644171\pi$
$488$	11.4036	0.516218
$489$	0	0
$490$	−35.4593	−1.60189
$491$	42.0595	1.89812	0.949058	−	0.315100i	$-0.102038\pi$
$491$	42.0595	1.89812	0.949058	+	0.315100i	$0.102038\pi$
$492$	0	0
$493$	48.6369	2.19049
$494$	6.10257	0.274568
$495$	0	0
$496$	19.9354	0.895128
$497$	−5.41089	−0.242711
$498$	0	0
$499$	−1.76823	−0.0791570	−0.0395785	−	0.999216i	$-0.512602\pi$
$499$	−1.76823	−0.0791570	−0.0395785	+	0.999216i	$0.512602\pi$
$500$	−8.25042	−0.368970
$501$	0	0
$502$	2.33951	0.104417
$503$	18.6392	0.831080	0.415540	−	0.909575i	$-0.363593\pi$
$503$	18.6392	0.831080	0.415540	+	0.909575i	$0.363593\pi$
$504$	0	0
$505$	4.39342	0.195505
$506$	0	0
$507$	0	0
$508$	27.2714	1.20997
$509$	9.16096	0.406053	0.203026	−	0.979173i	$-0.434922\pi$
$509$	9.16096	0.406053	0.203026	+	0.979173i	$0.434922\pi$
$510$	0	0
$511$	−0.188430	−0.00833566
$512$	−18.7292	−0.827722
$513$	0	0
$514$	−21.1214	−0.931626
$515$	−7.16379	−0.315674
$516$	0	0
$517$	0	0
$518$	9.27945	0.407716
$519$	0	0
$520$	9.69056	0.424959
$521$	2.64766	0.115996	0.0579981	−	0.998317i	$-0.481528\pi$
$521$	2.64766	0.115996	0.0579981	+	0.998317i	$0.481528\pi$
$522$	0	0
$523$	−27.7951	−1.21539	−0.607697	−	0.794169i	$-0.707906\pi$
$523$	−27.7951	−1.21539	−0.607697	+	0.794169i	$0.707906\pi$
$524$	−22.8472	−0.998083
$525$	0	0
$526$	−17.5965	−0.767243
$527$	−30.3819	−1.32346
$528$	0	0
$529$	31.5141	1.37018
$530$	78.1627	3.39517
$531$	0	0
$532$	2.27631	0.0986905
$533$	−2.81532	−0.121945
$534$	0	0
$535$	−12.6409	−0.546512
$536$	6.49207	0.280415
$537$	0	0
$538$	−3.15497	−0.136021
$539$	0	0
$540$	0	0
$541$	−8.70614	−0.374306	−0.187153	−	0.982331i	$-0.559926\pi$
$541$	−8.70614	−0.374306	−0.187153	+	0.982331i	$0.559926\pi$
$542$	34.3885	1.47711
$543$	0	0
$544$	47.6525	2.04309
$545$	36.9410	1.58238
$546$	0	0
$547$	−6.59025	−0.281779	−0.140889	−	0.990025i	$-0.544996\pi$
$547$	−6.59025	−0.281779	−0.140889	+	0.990025i	$0.544996\pi$
$548$	9.41987	0.402397
$549$	0	0
$550$	0	0
$551$	9.90018	0.421762
$552$	0	0
$553$	3.78072	0.160773
$554$	−8.70486	−0.369834
$555$	0	0
$556$	4.07269	0.172720
$557$	43.7019	1.85171	0.925855	−	0.377878i	$-0.123346\pi$
$557$	43.7019	1.85171	0.925855	+	0.377878i	$0.123346\pi$
$558$	0	0
$559$	−15.9654	−0.675265
$560$	19.4727	0.822871
$561$	0	0
$562$	31.4060	1.32478
$563$	−44.9334	−1.89372	−0.946858	−	0.321653i	$-0.895762\pi$
$563$	−44.9334	−1.89372	−0.946858	+	0.321653i	$0.895762\pi$
$564$	0	0
$565$	5.39218	0.226851
$566$	−31.7613	−1.33503
$567$	0	0
$568$	−5.99628	−0.251598
$569$	30.2344	1.26749	0.633746	−	0.773541i	$-0.281516\pi$
$569$	30.2344	1.26749	0.633746	+	0.773541i	$0.281516\pi$
$570$	0	0
$571$	−42.1562	−1.76418	−0.882090	−	0.471080i	$-0.843864\pi$
$571$	−42.1562	−1.76418	−0.882090	+	0.471080i	$0.843864\pi$
$572$	0	0
$573$	0	0
$574$	−2.66939	−0.111418
$575$	50.5610	2.10854
$576$	0	0
$577$	22.5059	0.936932	0.468466	−	0.883482i	$-0.344807\pi$
$577$	22.5059	0.936932	0.468466	+	0.883482i	$0.344807\pi$
$578$	−70.8771	−2.94810
$579$	0	0
$580$	−29.0083	−1.20451
$581$	−13.9382	−0.578252
$582$	0	0
$583$	0	0
$584$	−0.208816	−0.00864087
$585$	0	0
$586$	−0.814878	−0.0336623
$587$	9.35535	0.386136	0.193068	−	0.981185i	$-0.438156\pi$
$587$	9.35535	0.386136	0.193068	+	0.981185i	$0.438156\pi$
$588$	0	0
$589$	−6.18433	−0.254821
$590$	82.3837	3.39169
$591$	0	0
$592$	21.7935	0.895709
$593$	16.8332	0.691255	0.345628	−	0.938372i	$-0.387666\pi$
$593$	16.8332	0.691255	0.345628	+	0.938372i	$0.387666\pi$
$594$	0	0
$595$	−29.6767	−1.21662
$596$	−28.5065	−1.16767
$597$	0	0
$598$	29.5709	1.20924
$599$	26.3465	1.07649	0.538245	−	0.842788i	$-0.319087\pi$
$599$	26.3465	1.07649	0.538245	+	0.842788i	$0.319087\pi$
$600$	0	0
$601$	−2.95983	−0.120734	−0.0603671	−	0.998176i	$-0.519227\pi$
$601$	−2.95983	−0.120734	−0.0603671	+	0.998176i	$0.519227\pi$
$602$	−15.1379	−0.616973
$603$	0	0
$604$	2.77064	0.112736
$605$	0	0
$606$	0	0
$607$	−3.91894	−0.159065	−0.0795325	−	0.996832i	$-0.525343\pi$
$607$	−3.91894	−0.159065	−0.0795325	+	0.996832i	$0.525343\pi$
$608$	9.69981	0.393379
$609$	0	0
$610$	55.8403	2.26091
$611$	−4.47223	−0.180927
$612$	0	0
$613$	12.7621	0.515456	0.257728	−	0.966217i	$-0.417026\pi$
$613$	12.7621	0.515456	0.257728	+	0.966217i	$0.417026\pi$
$614$	22.5436	0.909785
$615$	0	0
$616$	0	0
$617$	29.4259	1.18464	0.592322	−	0.805701i	$-0.298211\pi$
$617$	29.4259	1.18464	0.592322	+	0.805701i	$0.298211\pi$
$618$	0	0
$619$	−28.1001	−1.12944	−0.564719	−	0.825283i	$-0.691016\pi$
$619$	−28.1001	−1.12944	−0.564719	+	0.825283i	$0.691016\pi$
$620$	18.1206	0.727740
$621$	0	0
$622$	−42.3080	−1.69640
$623$	9.39117	0.376249
$624$	0	0
$625$	−12.3452	−0.493808
$626$	−33.5807	−1.34215
$627$	0	0
$628$	23.2997	0.929760
$629$	−33.2137	−1.32432
$630$	0	0
$631$	−28.4345	−1.13196	−0.565979	−	0.824420i	$-0.691502\pi$
$631$	−28.4345	−1.13196	−0.565979	+	0.824420i	$0.691502\pi$
$632$	4.18975	0.166659
$633$	0	0
$634$	39.9409	1.58626
$635$	−72.3718	−2.87199
$636$	0	0
$637$	12.5138	0.495817
$638$	0	0
$639$	0	0
$640$	32.9765	1.30351
$641$	−16.6256	−0.656670	−0.328335	−	0.944561i	$-0.606488\pi$
$641$	−16.6256	−0.656670	−0.328335	+	0.944561i	$0.606488\pi$
$642$	0	0
$643$	15.9720	0.629874	0.314937	−	0.949113i	$-0.398017\pi$
$643$	15.9720	0.629874	0.314937	+	0.949113i	$0.398017\pi$
$644$	11.0302	0.434650
$645$	0	0
$646$	−20.7106	−0.814847
$647$	15.0590	0.592031	0.296016	−	0.955183i	$-0.404342\pi$
$647$	15.0590	0.592031	0.296016	+	0.955183i	$0.404342\pi$
$648$	0	0
$649$	0	0
$650$	27.4266	1.07576
$651$	0	0
$652$	−18.7928	−0.735982
$653$	31.3586	1.22716	0.613578	−	0.789634i	$-0.289730\pi$
$653$	31.3586	1.22716	0.613578	+	0.789634i	$0.289730\pi$
$654$	0	0
$655$	60.6308	2.36904
$656$	−6.26928	−0.244774
$657$	0	0
$658$	−4.24042	−0.165309
$659$	18.1039	0.705227	0.352613	−	0.935769i	$-0.385293\pi$
$659$	18.1039	0.705227	0.352613	+	0.935769i	$0.385293\pi$
$660$	0	0
$661$	−31.9775	−1.24378	−0.621890	−	0.783105i	$-0.713635\pi$
$661$	−31.9775	−1.24378	−0.621890	+	0.783105i	$0.713635\pi$
$662$	36.1835	1.40631
$663$	0	0
$664$	−15.4461	−0.599424
$665$	−6.04077	−0.234251
$666$	0	0
$667$	47.9727	1.85751
$668$	2.07165	0.0801544
$669$	0	0
$670$	31.7898	1.22815
$671$	0	0
$672$	0	0
$673$	−33.6762	−1.29812	−0.649062	−	0.760736i	$-0.724838\pi$
$673$	−33.6762	−1.29812	−0.649062	+	0.760736i	$0.724838\pi$
$674$	10.7442	0.413852
$675$	0	0
$676$	−10.5515	−0.405825
$677$	38.1675	1.46690	0.733448	−	0.679745i	$-0.237910\pi$
$677$	38.1675	1.46690	0.733448	+	0.679745i	$0.237910\pi$
$678$	0	0
$679$	−3.90098	−0.149706
$680$	−32.8873	−1.26117
$681$	0	0
$682$	0	0
$683$	−32.1292	−1.22939	−0.614695	−	0.788765i	$-0.710721\pi$
$683$	−32.1292	−1.22939	−0.614695	+	0.788765i	$0.710721\pi$
$684$	0	0
$685$	−24.9981	−0.955127
$686$	26.5048	1.01196
$687$	0	0
$688$	−35.5525	−1.35543
$689$	−27.5842	−1.05087
$690$	0	0
$691$	8.62869	0.328251	0.164125	−	0.986439i	$-0.447520\pi$
$691$	8.62869	0.328251	0.164125	+	0.986439i	$0.447520\pi$
$692$	−4.55511	−0.173159
$693$	0	0
$694$	−45.3924	−1.72307
$695$	−10.8079	−0.409968
$696$	0	0
$697$	9.55446	0.361901
$698$	46.7714	1.77033
$699$	0	0
$700$	10.2303	0.386670
$701$	12.7081	0.479977	0.239989	−	0.970776i	$-0.422856\pi$
$701$	12.7081	0.479977	0.239989	+	0.970776i	$0.422856\pi$
$702$	0	0
$703$	−6.76074	−0.254986
$704$	0	0
$705$	0	0
$706$	−4.63781	−0.174546
$707$	−1.47010	−0.0552890
$708$	0	0
$709$	21.0465	0.790416	0.395208	−	0.918592i	$-0.370672\pi$
$709$	21.0465	0.790416	0.395208	+	0.918592i	$0.370672\pi$
$710$	−29.3621	−1.10194
$711$	0	0
$712$	10.4072	0.390026
$713$	−29.9670	−1.12227
$714$	0	0
$715$	0	0
$716$	−0.721127	−0.0269498
$717$	0	0
$718$	−11.4063	−0.425679
$719$	40.6509	1.51602	0.758011	−	0.652242i	$-0.226171\pi$
$719$	40.6509	1.51602	0.758011	+	0.652242i	$0.226171\pi$
$720$	0	0
$721$	2.39711	0.0892732
$722$	30.2841	1.12706
$723$	0	0
$724$	5.14995	0.191397
$725$	44.4940	1.65247
$726$	0	0
$727$	−38.1637	−1.41541	−0.707706	−	0.706507i	$-0.750270\pi$
$727$	−38.1637	−1.41541	−0.707706	+	0.706507i	$0.750270\pi$
$728$	−3.24261	−0.120179
$729$	0	0
$730$	−1.02251	−0.0378449
$731$	54.1825	2.00401
$732$	0	0
$733$	21.0684	0.778179	0.389089	−	0.921200i	$-0.372790\pi$
$733$	21.0684	0.778179	0.389089	+	0.921200i	$0.372790\pi$
$734$	31.9091	1.17779
$735$	0	0
$736$	47.0018	1.73251
$737$	0	0
$738$	0	0
$739$	−17.3113	−0.636807	−0.318403	−	0.947955i	$-0.603147\pi$
$739$	−17.3113	−0.636807	−0.318403	+	0.947955i	$0.603147\pi$
$740$	19.8095	0.728212
$741$	0	0
$742$	−26.1544	−0.960159
$743$	−5.29450	−0.194236	−0.0971181	−	0.995273i	$-0.530962\pi$
$743$	−5.29450	−0.194236	−0.0971181	+	0.995273i	$0.530962\pi$
$744$	0	0
$745$	75.6494	2.77158
$746$	12.0735	0.442042
$747$	0	0
$748$	0	0
$749$	4.22982	0.154554
$750$	0	0
$751$	−15.6266	−0.570222	−0.285111	−	0.958494i	$-0.592031\pi$
$751$	−15.6266	−0.570222	−0.285111	+	0.958494i	$0.592031\pi$
$752$	−9.95898	−0.363166
$753$	0	0
$754$	26.0226	0.947686
$755$	−7.35260	−0.267589
$756$	0	0
$757$	21.9655	0.798351	0.399175	−	0.916875i	$-0.369296\pi$
$757$	21.9655	0.798351	0.399175	+	0.916875i	$0.369296\pi$
$758$	4.25566	0.154572
$759$	0	0
$760$	−6.69431	−0.242828
$761$	27.2883	0.989199	0.494600	−	0.869121i	$-0.335315\pi$
$761$	27.2883	0.989199	0.494600	+	0.869121i	$0.335315\pi$
$762$	0	0
$763$	−12.3610	−0.447499
$764$	−13.2658	−0.479940
$765$	0	0
$766$	−33.0039	−1.19248
$767$	−29.0738	−1.04980
$768$	0	0
$769$	12.2926	0.443284	0.221642	−	0.975128i	$-0.428858\pi$
$769$	12.2926	0.443284	0.221642	+	0.975128i	$0.428858\pi$
$770$	0	0
$771$	0	0
$772$	3.72719	0.134145
$773$	11.9400	0.429450	0.214725	−	0.976675i	$-0.431114\pi$
$773$	11.9400	0.429450	0.214725	+	0.976675i	$0.431114\pi$
$774$	0	0
$775$	−27.7940	−0.998389
$776$	−4.32301	−0.155187
$777$	0	0
$778$	23.6138	0.846595
$779$	1.94484	0.0696811
$780$	0	0
$781$	0	0
$782$	−100.356	−3.58872
$783$	0	0
$784$	27.8664	0.995229
$785$	−61.8318	−2.20687
$786$	0	0
$787$	7.06242	0.251748	0.125874	−	0.992046i	$-0.459826\pi$
$787$	7.06242	0.251748	0.125874	+	0.992046i	$0.459826\pi$
$788$	8.52476	0.303682
$789$	0	0
$790$	20.5160	0.729927
$791$	−1.80430	−0.0641537
$792$	0	0
$793$	−19.7065	−0.699797
$794$	63.1967	2.24277
$795$	0	0
$796$	−1.33283	−0.0472411
$797$	9.87598	0.349825	0.174913	−	0.984584i	$-0.444036\pi$
$797$	9.87598	0.349825	0.174913	+	0.984584i	$0.444036\pi$
$798$	0	0
$799$	15.1776	0.536946
$800$	43.5935	1.54126
$801$	0	0
$802$	−8.34839	−0.294792
$803$	0	0
$804$	0	0
$805$	−29.2714	−1.03168
$806$	−16.2555	−0.572574
$807$	0	0
$808$	−1.62915	−0.0573134
$809$	−0.274918	−0.00966559	−0.00483279	−	0.999988i	$-0.501538\pi$
$809$	−0.274918	−0.00966559	−0.00483279	+	0.999988i	$0.501538\pi$
$810$	0	0
$811$	19.5890	0.687863	0.343932	−	0.938995i	$-0.388241\pi$
$811$	19.5890	0.687863	0.343932	+	0.938995i	$0.388241\pi$
$812$	9.70662	0.340636
$813$	0	0
$814$	0	0
$815$	49.8715	1.74692
$816$	0	0
$817$	11.0290	0.385856
$818$	−59.4654	−2.07916
$819$	0	0
$820$	−5.69854	−0.199001
$821$	−36.3320	−1.26799	−0.633997	−	0.773335i	$-0.718587\pi$
$821$	−36.3320	−1.26799	−0.633997	+	0.773335i	$0.718587\pi$
$822$	0	0
$823$	27.6334	0.963241	0.481620	−	0.876380i	$-0.340048\pi$
$823$	27.6334	0.963241	0.481620	+	0.876380i	$0.340048\pi$
$824$	2.65645	0.0925419
$825$	0	0
$826$	−27.5669	−0.959173
$827$	31.9450	1.11084	0.555418	−	0.831571i	$-0.312558\pi$
$827$	31.9450	1.11084	0.555418	+	0.831571i	$0.312558\pi$
$828$	0	0
$829$	−24.0996	−0.837012	−0.418506	−	0.908214i	$-0.637446\pi$
$829$	−24.0996	−0.837012	−0.418506	+	0.908214i	$0.637446\pi$
$830$	−75.6351	−2.62533
$831$	0	0
$832$	3.82818	0.132718
$833$	−42.4688	−1.47146
$834$	0	0
$835$	−5.49765	−0.190254
$836$	0	0
$837$	0	0
$838$	46.9493	1.62184
$839$	−46.6658	−1.61108	−0.805540	−	0.592541i	$-0.798125\pi$
$839$	−46.6658	−1.61108	−0.805540	+	0.592541i	$0.798125\pi$
$840$	0	0
$841$	13.2163	0.455735
$842$	−52.5350	−1.81047
$843$	0	0
$844$	−18.7139	−0.644158
$845$	28.0010	0.963264
$846$	0	0
$847$	0	0
$848$	−61.4258	−2.10937
$849$	0	0
$850$	−93.0788	−3.19258
$851$	−32.7601	−1.12300
$852$	0	0
$853$	19.9192	0.682019	0.341009	−	0.940060i	$-0.389231\pi$
$853$	19.9192	0.682019	0.341009	+	0.940060i	$0.389231\pi$
$854$	−18.6850	−0.639388
$855$	0	0
$856$	4.68744	0.160213
$857$	−39.1187	−1.33627	−0.668134	−	0.744041i	$-0.732907\pi$
$857$	−39.1187	−1.33627	−0.668134	+	0.744041i	$0.732907\pi$
$858$	0	0
$859$	14.8984	0.508325	0.254163	−	0.967161i	$-0.418200\pi$
$859$	14.8984	0.508325	0.254163	+	0.967161i	$0.418200\pi$
$860$	−32.3159	−1.10196
$861$	0	0
$862$	−39.7660	−1.35443
$863$	7.92445	0.269752	0.134876	−	0.990863i	$-0.456936\pi$
$863$	7.92445	0.269752	0.134876	+	0.990863i	$0.456936\pi$
$864$	0	0
$865$	12.0882	0.411010
$866$	−41.3643	−1.40562
$867$	0	0
$868$	−6.06342	−0.205806
$869$	0	0
$870$	0	0
$871$	−11.2189	−0.380136
$872$	−13.6983	−0.463884
$873$	0	0
$874$	−20.4278	−0.690980
$875$	−7.32627	−0.247673
$876$	0	0
$877$	−36.4294	−1.23013	−0.615067	−	0.788475i	$-0.710871\pi$
$877$	−36.4294	−1.23013	−0.615067	+	0.788475i	$0.710871\pi$
$878$	23.0572	0.778143
$879$	0	0
$880$	0	0
$881$	−25.7097	−0.866183	−0.433092	−	0.901350i	$-0.642577\pi$
$881$	−25.7097	−0.866183	−0.433092	+	0.901350i	$0.642577\pi$
$882$	0	0
$883$	−7.91844	−0.266477	−0.133238	−	0.991084i	$-0.542538\pi$
$883$	−7.91844	−0.266477	−0.133238	+	0.991084i	$0.542538\pi$
$884$	−21.4157	−0.720288
$885$	0	0
$886$	6.28768	0.211239
$887$	12.0079	0.403186	0.201593	−	0.979469i	$-0.435388\pi$
$887$	12.0079	0.403186	0.201593	+	0.979469i	$0.435388\pi$
$888$	0	0
$889$	24.2167	0.812202
$890$	50.9610	1.70822
$891$	0	0
$892$	23.5505	0.788528
$893$	3.08945	0.103384
$894$	0	0
$895$	1.91370	0.0639678
$896$	−11.0344	−0.368634
$897$	0	0
$898$	58.2042	1.94230
$899$	−26.3712	−0.879528
$900$	0	0
$901$	93.6138	3.11873
$902$	0	0
$903$	0	0
$904$	−1.99951	−0.0665026
$905$	−13.6667	−0.454297
$906$	0	0
$907$	−40.5608	−1.34680	−0.673400	−	0.739278i	$-0.735167\pi$
$907$	−40.5608	−1.34680	−0.673400	+	0.739278i	$0.735167\pi$
$908$	−7.12357	−0.236404
$909$	0	0
$910$	−15.8781	−0.526355
$911$	35.5287	1.17712	0.588559	−	0.808454i	$-0.299695\pi$
$911$	35.5287	1.17712	0.588559	+	0.808454i	$0.299695\pi$
$912$	0	0
$913$	0	0
$914$	25.0152	0.827429
$915$	0	0
$916$	6.34417	0.209617
$917$	−20.2880	−0.669969
$918$	0	0
$919$	51.4484	1.69713	0.848563	−	0.529094i	$-0.177468\pi$
$919$	51.4484	1.69713	0.848563	+	0.529094i	$0.177468\pi$
$920$	−32.4382	−1.06946
$921$	0	0
$922$	6.52255	0.214809
$923$	10.3621	0.341072
$924$	0	0
$925$	−30.3845	−0.999038
$926$	−45.1934	−1.48515
$927$	0	0
$928$	41.3619	1.35777
$929$	48.2748	1.58385	0.791923	−	0.610621i	$-0.209080\pi$
$929$	48.2748	1.58385	0.791923	+	0.610621i	$0.209080\pi$
$930$	0	0
$931$	−8.64465	−0.283317
$932$	−29.6602	−0.971553
$933$	0	0
$934$	−43.3863	−1.41964
$935$	0	0
$936$	0	0
$937$	44.8251	1.46437	0.732185	−	0.681105i	$-0.238500\pi$
$937$	44.8251	1.46437	0.732185	+	0.681105i	$0.238500\pi$
$938$	−10.6373	−0.347322
$939$	0	0
$940$	−9.05233	−0.295255
$941$	−0.273684	−0.00892183	−0.00446092	−	0.999990i	$-0.501420\pi$
$941$	−0.273684	−0.00892183	−0.00446092	+	0.999990i	$0.501420\pi$
$942$	0	0
$943$	9.42400	0.306888
$944$	−64.7430	−2.10720
$945$	0	0
$946$	0	0
$947$	−22.3160	−0.725172	−0.362586	−	0.931950i	$-0.618106\pi$
$947$	−22.3160	−0.725172	−0.362586	+	0.931950i	$0.618106\pi$
$948$	0	0
$949$	0.360852	0.0117138
$950$	−18.9465	−0.614704
$951$	0	0
$952$	11.0046	0.356661
$953$	−38.9268	−1.26096	−0.630481	−	0.776205i	$-0.717142\pi$
$953$	−38.9268	−1.26096	−0.630481	+	0.776205i	$0.717142\pi$
$954$	0	0
$955$	35.2042	1.13918
$956$	0.320803	0.0103755
$957$	0	0
$958$	−33.7694	−1.09104
$959$	8.36473	0.270111
$960$	0	0
$961$	−14.5268	−0.468605
$962$	−17.7706	−0.572946
$963$	0	0
$964$	−2.33430	−0.0751829
$965$	−9.89107	−0.318405
$966$	0	0
$967$	−26.7607	−0.860565	−0.430283	−	0.902694i	$-0.641586\pi$
$967$	−26.7607	−0.860565	−0.430283	+	0.902694i	$0.641586\pi$
$968$	0	0
$969$	0	0
$970$	−21.1686	−0.679681
$971$	−1.83070	−0.0587501	−0.0293750	−	0.999568i	$-0.509352\pi$
$971$	−1.83070	−0.0587501	−0.0293750	+	0.999568i	$0.509352\pi$
$972$	0	0
$973$	3.61650	0.115940
$974$	35.0698	1.12371
$975$	0	0
$976$	−43.8833	−1.40467
$977$	39.3710	1.25959	0.629794	−	0.776762i	$-0.283139\pi$
$977$	39.3710	1.25959	0.629794	+	0.776762i	$0.283139\pi$
$978$	0	0
$979$	0	0
$980$	25.3295	0.809122
$981$	0	0
$982$	−76.3708	−2.43709
$983$	35.9974	1.14814	0.574070	−	0.818806i	$-0.305364\pi$
$983$	35.9974	1.14814	0.574070	+	0.818806i	$0.305364\pi$
$984$	0	0
$985$	−22.6226	−0.720817
$986$	−88.3139	−2.81249
$987$	0	0
$988$	−4.35923	−0.138686
$989$	53.4427	1.69938
$990$	0	0
$991$	23.0540	0.732334	0.366167	−	0.930549i	$-0.380670\pi$
$991$	23.0540	0.732334	0.366167	+	0.930549i	$0.380670\pi$
$992$	−25.8375	−0.820340
$993$	0	0
$994$	9.82499	0.311630
$995$	3.53702	0.112131
$996$	0	0
$997$	37.6561	1.19258	0.596291	−	0.802769i	$-0.296641\pi$
$997$	37.6561	1.19258	0.596291	+	0.802769i	$0.296641\pi$
$998$	3.21073	0.101634
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.ca.1.3		8
3.2	odd	2		9801.2.a.bz.1.6		8
9.2	odd	6		1089.2.e.k.364.3	✓	16
9.5	odd	6		1089.2.e.k.727.3	yes	16
11.10	odd	2	inner	9801.2.a.ca.1.6		8
33.32	even	2		9801.2.a.bz.1.3		8
99.32	even	6		1089.2.e.k.727.6	yes	16
99.65	even	6		1089.2.e.k.364.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.2.e.k.364.3	✓	16	9.2	odd	6
1089.2.e.k.364.6	yes	16	99.65	even	6
1089.2.e.k.727.3	yes	16	9.5	odd	6
1089.2.e.k.727.6	yes	16	99.32	even	6
9801.2.a.bz.1.3		8	33.32	even	2
9801.2.a.bz.1.6		8	3.2	odd	2
9801.2.a.ca.1.3		8	1.1	even	1	trivial
9801.2.a.ca.1.6		8	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q-1.81578 q^{2} +1.29706 q^{4} -3.44209 q^{5} +1.15177 q^{7} +1.27638 q^{8} +6.25008 q^{10} -2.20570 q^{13} -2.09137 q^{14} -4.91175 q^{16} +7.48558 q^{17} +1.52371 q^{19} -4.46460 q^{20} +7.38336 q^{23} +6.84796 q^{25} +4.00507 q^{26} +1.49392 q^{28} +6.49741 q^{29} -4.05872 q^{31} +6.36591 q^{32} -13.5922 q^{34} -3.96451 q^{35} -4.43702 q^{37} -2.76673 q^{38} -4.39342 q^{40} +1.27638 q^{41} +7.23825 q^{43} -13.4066 q^{46} +2.02758 q^{47} -5.67342 q^{49} -12.4344 q^{50} -2.86093 q^{52} +12.5059 q^{53} +1.47010 q^{56} -11.7979 q^{58} +13.1812 q^{59} +8.93434 q^{61} +7.36975 q^{62} -1.73559 q^{64} +7.59221 q^{65} +5.08630 q^{67} +9.70926 q^{68} +7.19868 q^{70} -4.69787 q^{71} -0.163600 q^{73} +8.05666 q^{74} +1.97635 q^{76} +3.28252 q^{79} +16.9067 q^{80} -2.31763 q^{82} -12.1015 q^{83} -25.7660 q^{85} -13.1431 q^{86} +8.15366 q^{89} -2.54047 q^{91} +9.57668 q^{92} -3.68164 q^{94} -5.24475 q^{95} -3.38693 q^{97} +10.3017 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} + 4 q^{5} + 4 q^{14} + 24 q^{16} - 6 q^{20} + 46 q^{23} + 12 q^{25} + 30 q^{26} - 14 q^{31} - 38 q^{34} - 6 q^{37} + 4 q^{38} + 16 q^{47} + 42 q^{49} + 48 q^{53} + 46 q^{56} - 50 q^{58} + 48 q^{59}+ \cdots + 4 q^{97}+O(q^{100}) \) 8 * q + 16 * q^4 + 4 * q^5 + 4 * q^14 + 24 * q^16 - 6 * q^20 + 46 * q^23 + 12 * q^25 + 30 * q^26 - 14 * q^31 - 38 * q^34 - 6 * q^37 + 4 * q^38 + 16 * q^47 + 42 * q^49 + 48 * q^53 + 46 * q^56 - 50 * q^58 + 48 * q^59 - 6 * q^64 + 22 * q^67 + 4 * q^70 - 34 * q^71 + 74 * q^80 + 56 * q^82 + 14 * q^86 + 8 * q^89 - 48 * q^91 + 84 * q^92 + 4 * q^97

Introduction

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