LMFDB - Embedded newform 5445.2.a.bi.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5445,2,Mod(1,5445)] 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("5445.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-1,0,-1,4,0,3,-3,0,-1,0,0,1,-2] 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:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.35567$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35567 q^{2} -0.162147 q^{4} +1.00000 q^{5} +3.64941 q^{7} -2.93117 q^{8} +1.35567 q^{10} +2.83095 q^{13} +4.94742 q^{14} -3.64941 q^{16} +3.69195 q^{17} +0.0951243 q^{19} -0.162147 q^{20} -1.16215 q^{23} +1.00000 q^{25} +3.83785 q^{26} -0.591742 q^{28} -6.75389 q^{29} +6.77837 q^{31} +0.914918 q^{32} +5.00509 q^{34} +3.64941 q^{35} +9.83980 q^{37} +0.128958 q^{38} -2.93117 q^{40} -8.31822 q^{41} +2.96862 q^{43} -1.57549 q^{46} +2.22491 q^{47} +6.31822 q^{49} +1.35567 q^{50} -0.459031 q^{52} -2.99393 q^{53} -10.6970 q^{56} -9.15607 q^{58} +8.50860 q^{59} +8.48037 q^{61} +9.18926 q^{62} +8.53916 q^{64} +2.83095 q^{65} -13.4153 q^{67} -0.598640 q^{68} +4.94742 q^{70} -8.30309 q^{71} +1.32003 q^{73} +13.3396 q^{74} -0.0154241 q^{76} +13.8661 q^{79} -3.64941 q^{80} -11.2768 q^{82} -10.6445 q^{83} +3.69195 q^{85} +4.02448 q^{86} +12.1612 q^{89} +10.3313 q^{91} +0.188439 q^{92} +3.01625 q^{94} +0.0951243 q^{95} -4.33133 q^{97} +8.56545 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{4} + 4 q^{5} + 3 q^{7} - 3 q^{8} - q^{10} + q^{13} - 2 q^{14} - 3 q^{16} + q^{17} + 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} + 13 q^{28} - 12 q^{29} - 5 q^{31} + 8 q^{32}+ \cdots + 8 q^{98}+O(q^{100}) $ 4 * q - q^2 - q^4 + 4 * q^5 + 3 * q^7 - 3 * q^8 - q^10 + q^13 - 2 * q^14 - 3 * q^16 + q^17 + 20 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 15 * q^26 + 13 * q^28 - 12 * q^29 - 5 * q^31 + 8 * q^32 + 2 * q^34 + 3 * q^35 + 7 * q^37 - 20 * q^38 - 3 * q^40 - 11 * q^41 + 19 * q^43 - 4 * q^46 - 5 * q^47 + 3 * q^49 - q^50 - 11 * q^52 + 11 * q^53 - 11 * q^56 - 14 * q^58 - 9 * q^59 + 12 * q^61 + 35 * q^62 - 3 * q^64 + q^65 - 19 * q^67 + 3 * q^68 - 2 * q^70 - 5 * q^71 + 11 * q^73 + 34 * q^79 - 3 * q^80 - 6 * q^82 + 11 * q^83 + q^85 - q^86 + 8 * q^89 - 8 * q^91 + 12 * q^92 - q^94 + 20 * q^95 + 32 * q^97 + 8 * q^98

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 $

$2$	1.35567	0.958606	0.479303	−	0.877649i	$-0.340889\pi$
$2$	1.35567	0.958606	0.479303	+	0.877649i	$0.340889\pi$
$3$	0	0
$3$	0	0
$4$	−0.162147	−0.0810736
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.64941	1.37935	0.689674	−	0.724120i	$-0.257754\pi$
$7$	3.64941	1.37935	0.689674	+	0.724120i	$0.257754\pi$
$8$	−2.93117	−1.03632
$9$	0	0
$10$	1.35567	0.428702
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.83095	0.785166	0.392583	−	0.919717i	$-0.371582\pi$
$13$	2.83095	0.785166	0.392583	+	0.919717i	$0.371582\pi$
$14$	4.94742	1.32225
$15$	0	0
$16$	−3.64941	−0.912353
$17$	3.69195	0.895431	0.447715	−	0.894176i	$-0.352238\pi$
$17$	3.69195	0.895431	0.447715	+	0.894176i	$0.352238\pi$
$18$	0	0
$19$	0.0951243	0.0218230	0.0109115	−	0.999940i	$-0.496527\pi$
$19$	0.0951243	0.0218230	0.0109115	+	0.999940i	$0.496527\pi$
$20$	−0.162147	−0.0362572
$21$	0	0
$22$	0	0
$23$	−1.16215	−0.242324	−0.121162	−	0.992633i	$-0.538662\pi$
$23$	−1.16215	−0.242324	−0.121162	+	0.992633i	$0.538662\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.83785	0.752665
$27$	0	0
$28$	−0.591742	−0.111829
$29$	−6.75389	−1.25417	−0.627083	−	0.778953i	$-0.715751\pi$
$29$	−6.75389	−1.25417	−0.627083	+	0.778953i	$0.715751\pi$
$30$	0	0
$31$	6.77837	1.21743	0.608716	−	0.793388i	$-0.291685\pi$
$31$	6.77837	1.21743	0.608716	+	0.793388i	$0.291685\pi$
$32$	0.914918	0.161736
$33$	0	0
$34$	5.00509	0.858366
$35$	3.64941	0.616864
$36$	0	0
$37$	9.83980	1.61765	0.808826	−	0.588048i	$-0.200103\pi$
$37$	9.83980	1.61765	0.808826	+	0.588048i	$0.200103\pi$
$38$	0.128958	0.0209197
$39$	0	0
$40$	−2.93117	−0.463458
$41$	−8.31822	−1.29909	−0.649544	−	0.760324i	$-0.725040\pi$
$41$	−8.31822	−1.29909	−0.649544	+	0.760324i	$0.725040\pi$
$42$	0	0
$43$	2.96862	0.452710	0.226355	−	0.974045i	$-0.427319\pi$
$43$	2.96862	0.452710	0.226355	+	0.974045i	$0.427319\pi$
$44$	0	0
$45$	0	0
$46$	−1.57549	−0.232294
$47$	2.22491	0.324536	0.162268	−	0.986747i	$-0.448119\pi$
$47$	2.22491	0.324536	0.162268	+	0.986747i	$0.448119\pi$
$48$	0	0
$49$	6.31822	0.902603
$50$	1.35567	0.191721
$51$	0	0
$52$	−0.459031	−0.0636562
$53$	−2.99393	−0.411248	−0.205624	−	0.978631i	$-0.565922\pi$
$53$	−2.99393	−0.411248	−0.205624	+	0.978631i	$0.565922\pi$
$54$	0	0
$55$	0	0
$56$	−10.6970	−1.42945
$57$	0	0
$58$	−9.15607	−1.20225
$59$	8.50860	1.10773	0.553863	−	0.832608i	$-0.313153\pi$
$59$	8.50860	1.10773	0.553863	+	0.832608i	$0.313153\pi$
$60$	0	0
$61$	8.48037	1.08580	0.542900	−	0.839797i	$-0.317326\pi$
$61$	8.48037	1.08580	0.542900	+	0.839797i	$0.317326\pi$
$62$	9.18926	1.16704
$63$	0	0
$64$	8.53916	1.06739
$65$	2.83095	0.351137
$66$	0	0
$67$	−13.4153	−1.63894	−0.819469	−	0.573123i	$-0.805732\pi$
$67$	−13.4153	−1.63894	−0.819469	+	0.573123i	$0.805732\pi$
$68$	−0.598640	−0.0725958
$69$	0	0
$70$	4.94742	0.591329
$71$	−8.30309	−0.985396	−0.492698	−	0.870200i	$-0.663989\pi$
$71$	−8.30309	−0.985396	−0.492698	+	0.870200i	$0.663989\pi$
$72$	0	0
$73$	1.32003	0.154498	0.0772490	−	0.997012i	$-0.475386\pi$
$73$	1.32003	0.154498	0.0772490	+	0.997012i	$0.475386\pi$
$74$	13.3396	1.55069
$75$	0	0
$76$	−0.0154241	−0.00176927
$77$	0	0
$78$	0	0
$79$	13.8661	1.56006	0.780028	−	0.625744i	$-0.215205\pi$
$79$	13.8661	1.56006	0.780028	+	0.625744i	$0.215205\pi$
$80$	−3.64941	−0.408017
$81$	0	0
$82$	−11.2768	−1.24531
$83$	−10.6445	−1.16838	−0.584191	−	0.811616i	$-0.698588\pi$
$83$	−10.6445	−1.16838	−0.584191	+	0.811616i	$0.698588\pi$
$84$	0	0
$85$	3.69195	0.400449
$86$	4.02448	0.433971
$87$	0	0
$88$	0	0
$89$	12.1612	1.28908	0.644540	−	0.764570i	$-0.277049\pi$
$89$	12.1612	1.28908	0.644540	+	0.764570i	$0.277049\pi$
$90$	0	0
$91$	10.3313	1.08302
$92$	0.188439	0.0196461
$93$	0	0
$94$	3.01625	0.311102
$95$	0.0951243	0.00975955
$96$	0	0
$97$	−4.33133	−0.439780	−0.219890	−	0.975525i	$-0.570570\pi$
$97$	−4.33133	−0.439780	−0.219890	+	0.975525i	$0.570570\pi$
$98$	8.56545	0.865241
$99$	0	0
$100$	−0.162147	−0.0162147
$101$	9.90570	0.985654	0.492827	−	0.870127i	$-0.335964\pi$
$101$	9.90570	0.985654	0.492827	+	0.870127i	$0.335964\pi$
$102$	0	0
$103$	4.06590	0.400625	0.200313	−	0.979732i	$-0.435804\pi$
$103$	4.06590	0.400625	0.200313	+	0.979732i	$0.435804\pi$
$104$	−8.29800	−0.813686
$105$	0	0
$106$	−4.05879	−0.394225
$107$	1.93858	0.187409	0.0937046	−	0.995600i	$-0.470129\pi$
$107$	1.93858	0.187409	0.0937046	+	0.995600i	$0.470129\pi$
$108$	0	0
$109$	−6.12664	−0.586825	−0.293413	−	0.955986i	$-0.594791\pi$
$109$	−6.12664	−0.586825	−0.293413	+	0.955986i	$0.594791\pi$
$110$	0	0
$111$	0	0
$112$	−13.3182	−1.25845
$113$	−5.78527	−0.544232	−0.272116	−	0.962264i	$-0.587724\pi$
$113$	−5.78527	−0.544232	−0.272116	+	0.962264i	$0.587724\pi$
$114$	0	0
$115$	−1.16215	−0.108371
$116$	1.09512	0.101680
$117$	0	0
$118$	11.5349	1.06187
$119$	13.4735	1.23511
$120$	0	0
$121$	0	0
$122$	11.4966	1.04085
$123$	0	0
$124$	−1.09909	−0.0987016
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.43783	−0.216322	−0.108161	−	0.994133i	$-0.534496\pi$
$127$	−2.43783	−0.216322	−0.108161	+	0.994133i	$0.534496\pi$
$128$	9.74648	0.861475
$129$	0	0
$130$	3.83785	0.336602
$131$	7.04156	0.615224	0.307612	−	0.951512i	$-0.400470\pi$
$131$	7.04156	0.615224	0.307612	+	0.951512i	$0.400470\pi$
$132$	0	0
$133$	0.347148	0.0301016
$134$	−18.1868	−1.57110
$135$	0	0
$136$	−10.8217	−0.927956
$137$	9.57286	0.817864	0.408932	−	0.912565i	$-0.365901\pi$
$137$	9.57286	0.817864	0.408932	+	0.912565i	$0.365901\pi$
$138$	0	0
$139$	−0.515502	−0.0437243	−0.0218621	−	0.999761i	$-0.506959\pi$
$139$	−0.515502	−0.0437243	−0.0218621	+	0.999761i	$0.506959\pi$
$140$	−0.591742	−0.0500113
$141$	0	0
$142$	−11.2563	−0.944607
$143$	0	0
$144$	0	0
$145$	−6.75389	−0.560880
$146$	1.78953	0.148103
$147$	0	0
$148$	−1.59550	−0.131149
$149$	−8.15983	−0.668479	−0.334240	−	0.942488i	$-0.608479\pi$
$149$	−8.15983	−0.668479	−0.334240	+	0.942488i	$0.608479\pi$
$150$	0	0
$151$	−1.94023	−0.157893	−0.0789466	−	0.996879i	$-0.525156\pi$
$151$	−1.94023	−0.157893	−0.0789466	+	0.996879i	$0.525156\pi$
$152$	−0.278825	−0.0226157
$153$	0	0
$154$	0	0
$155$	6.77837	0.544452
$156$	0	0
$157$	21.2745	1.69789	0.848944	−	0.528483i	$-0.177239\pi$
$157$	21.2745	1.69789	0.848944	+	0.528483i	$0.177239\pi$
$158$	18.7979	1.49548
$159$	0	0
$160$	0.914918	0.0723306
$161$	−4.24116	−0.334250
$162$	0	0
$163$	−15.9810	−1.25173	−0.625863	−	0.779933i	$-0.715253\pi$
$163$	−15.9810	−1.25173	−0.625863	+	0.779933i	$0.715253\pi$
$164$	1.34878	0.105322
$165$	0	0
$166$	−14.4304	−1.12002
$167$	17.7090	1.37037	0.685183	−	0.728371i	$-0.259722\pi$
$167$	17.7090	1.37037	0.685183	+	0.728371i	$0.259722\pi$
$168$	0	0
$169$	−4.98569	−0.383515
$170$	5.00509	0.383873
$171$	0	0
$172$	−0.481353	−0.0367029
$173$	15.8855	1.20775	0.603875	−	0.797079i	$-0.293622\pi$
$173$	15.8855	1.20775	0.603875	+	0.797079i	$0.293622\pi$
$174$	0	0
$175$	3.64941	0.275870
$176$	0	0
$177$	0	0
$178$	16.4866	1.23572
$179$	16.8810	1.26175	0.630874	−	0.775885i	$-0.282696\pi$
$179$	16.8810	1.26175	0.630874	+	0.775885i	$0.282696\pi$
$180$	0	0
$181$	−24.0874	−1.79040	−0.895200	−	0.445664i	$-0.852968\pi$
$181$	−24.0874	−1.79040	−0.895200	+	0.445664i	$0.852968\pi$
$182$	14.0059	1.03819
$183$	0	0
$184$	3.40645	0.251127
$185$	9.83980	0.723436
$186$	0	0
$187$	0	0
$188$	−0.360762	−0.0263113
$189$	0	0
$190$	0.128958	0.00935557
$191$	−5.38279	−0.389485	−0.194743	−	0.980854i	$-0.562387\pi$
$191$	−5.38279	−0.389485	−0.194743	+	0.980854i	$0.562387\pi$
$192$	0	0
$193$	−18.2840	−1.31611	−0.658057	−	0.752968i	$-0.728622\pi$
$193$	−18.2840	−1.31611	−0.658057	+	0.752968i	$0.728622\pi$
$194$	−5.87187	−0.421576
$195$	0	0
$196$	−1.02448	−0.0731773
$197$	2.64566	0.188496	0.0942478	−	0.995549i	$-0.469955\pi$
$197$	2.64566	0.188496	0.0942478	+	0.995549i	$0.469955\pi$
$198$	0	0
$199$	6.52800	0.462757	0.231379	−	0.972864i	$-0.425676\pi$
$199$	6.52800	0.462757	0.231379	+	0.972864i	$0.425676\pi$
$200$	−2.93117	−0.207265
$201$	0	0
$202$	13.4289	0.944854
$203$	−24.6477	−1.72993
$204$	0	0
$205$	−8.31822	−0.580970
$206$	5.51204	0.384042
$207$	0	0
$208$	−10.3313	−0.716349
$209$	0	0
$210$	0	0
$211$	27.4478	1.88958	0.944792	−	0.327671i	$-0.106264\pi$
$211$	27.4478	1.88958	0.944792	+	0.327671i	$0.106264\pi$
$212$	0.485457	0.0333413
$213$	0	0
$214$	2.62808	0.179652
$215$	2.96862	0.202458
$216$	0	0
$217$	24.7371	1.67926
$218$	−8.30573	−0.562535
$219$	0	0
$220$	0	0
$221$	10.4518	0.703061
$222$	0	0
$223$	−5.08194	−0.340312	−0.170156	−	0.985417i	$-0.554427\pi$
$223$	−5.08194	−0.340312	−0.170156	+	0.985417i	$0.554427\pi$
$224$	3.33892	0.223091
$225$	0	0
$226$	−7.84294	−0.521705
$227$	3.73980	0.248219	0.124110	−	0.992269i	$-0.460393\pi$
$227$	3.73980	0.248219	0.124110	+	0.992269i	$0.460393\pi$
$228$	0	0
$229$	26.9241	1.77920	0.889598	−	0.456745i	$-0.150985\pi$
$229$	26.9241	1.77920	0.889598	+	0.456745i	$0.150985\pi$
$230$	−1.57549	−0.103885
$231$	0	0
$232$	19.7968	1.29972
$233$	18.3167	1.19997	0.599984	−	0.800012i	$-0.295174\pi$
$233$	18.3167	1.19997	0.599984	+	0.800012i	$0.295174\pi$
$234$	0	0
$235$	2.22491	0.145137
$236$	−1.37965	−0.0898073
$237$	0	0
$238$	18.2656	1.18399
$239$	−10.9559	−0.708676	−0.354338	−	0.935117i	$-0.615294\pi$
$239$	−10.9559	−0.708676	−0.354338	+	0.935117i	$0.615294\pi$
$240$	0	0
$241$	9.99444	0.643798	0.321899	−	0.946774i	$-0.395679\pi$
$241$	9.99444	0.643798	0.321899	+	0.946774i	$0.395679\pi$
$242$	0	0
$243$	0	0
$244$	−1.37507	−0.0880297
$245$	6.31822	0.403656
$246$	0	0
$247$	0.269293	0.0171347
$248$	−19.8685	−1.26165
$249$	0	0
$250$	1.35567	0.0857404
$251$	−9.65743	−0.609572	−0.304786	−	0.952421i	$-0.598585\pi$
$251$	−9.65743	−0.609572	−0.304786	+	0.952421i	$0.598585\pi$
$252$	0	0
$253$	0	0
$254$	−3.30490	−0.207368
$255$	0	0
$256$	−3.86526	−0.241579
$257$	−10.4276	−0.650454	−0.325227	−	0.945636i	$-0.605441\pi$
$257$	−10.4276	−0.650454	−0.325227	+	0.945636i	$0.605441\pi$
$258$	0	0
$259$	35.9095	2.23131
$260$	−0.459031	−0.0284679
$261$	0	0
$262$	9.54606	0.589757
$263$	−10.9619	−0.675937	−0.337968	−	0.941157i	$-0.609740\pi$
$263$	−10.9619	−0.675937	−0.337968	+	0.941157i	$0.609740\pi$
$264$	0	0
$265$	−2.99393	−0.183915
$266$	0.470620	0.0288555
$267$	0	0
$268$	2.17525	0.132875
$269$	0.0893449	0.00544746	0.00272373	−	0.999996i	$-0.499133\pi$
$269$	0.0893449	0.00544746	0.00272373	+	0.999996i	$0.499133\pi$
$270$	0	0
$271$	13.3996	0.813965	0.406982	−	0.913436i	$-0.366581\pi$
$271$	13.3996	0.813965	0.406982	+	0.913436i	$0.366581\pi$
$272$	−13.4735	−0.816949
$273$	0	0
$274$	12.9777	0.784010
$275$	0	0
$276$	0	0
$277$	3.90669	0.234730	0.117365	−	0.993089i	$-0.462555\pi$
$277$	3.90669	0.234730	0.117365	+	0.993089i	$0.462555\pi$
$278$	−0.698853	−0.0419144
$279$	0	0
$280$	−10.6970	−0.639271
$281$	−1.53743	−0.0917155	−0.0458577	−	0.998948i	$-0.514602\pi$
$281$	−1.53743	−0.0917155	−0.0458577	+	0.998948i	$0.514602\pi$
$282$	0	0
$283$	5.41170	0.321692	0.160846	−	0.986980i	$-0.448578\pi$
$283$	5.41170	0.321692	0.160846	+	0.986980i	$0.448578\pi$
$284$	1.34632	0.0798896
$285$	0	0
$286$	0	0
$287$	−30.3566	−1.79190
$288$	0	0
$289$	−3.36947	−0.198204
$290$	−9.15607	−0.537663
$291$	0	0
$292$	−0.214039	−0.0125257
$293$	11.4165	0.666958	0.333479	−	0.942757i	$-0.391777\pi$
$293$	11.4165	0.666958	0.333479	+	0.942757i	$0.391777\pi$
$294$	0	0
$295$	8.50860	0.495390
$296$	−28.8421	−1.67641
$297$	0	0
$298$	−11.0621	−0.640808
$299$	−3.28999	−0.190265
$300$	0	0
$301$	10.8337	0.624445
$302$	−2.63031	−0.151358
$303$	0	0
$304$	−0.347148	−0.0199103
$305$	8.48037	0.485585
$306$	0	0
$307$	4.25008	0.242565	0.121282	−	0.992618i	$-0.461299\pi$
$307$	4.25008	0.242565	0.121282	+	0.992618i	$0.461299\pi$
$308$	0	0
$309$	0	0
$310$	9.18926	0.521915
$311$	16.6195	0.942404	0.471202	−	0.882025i	$-0.343820\pi$
$311$	16.6195	0.942404	0.471202	+	0.882025i	$0.343820\pi$
$312$	0	0
$313$	−26.5770	−1.50222	−0.751109	−	0.660178i	$-0.770481\pi$
$313$	−26.5770	−1.50222	−0.751109	+	0.660178i	$0.770481\pi$
$314$	28.8413	1.62761
$315$	0	0
$316$	−2.24835	−0.126479
$317$	−5.79694	−0.325589	−0.162794	−	0.986660i	$-0.552051\pi$
$317$	−5.79694	−0.325589	−0.162794	+	0.986660i	$0.552051\pi$
$318$	0	0
$319$	0	0
$320$	8.53916	0.477353
$321$	0	0
$322$	−5.74963	−0.320414
$323$	0.351195	0.0195410
$324$	0	0
$325$	2.83095	0.157033
$326$	−21.6650	−1.19991
$327$	0	0
$328$	24.3821	1.34628
$329$	8.11961	0.447648
$330$	0	0
$331$	−12.9230	−0.710311	−0.355155	−	0.934807i	$-0.615572\pi$
$331$	−12.9230	−0.710311	−0.355155	+	0.934807i	$0.615572\pi$
$332$	1.72597	0.0947249
$333$	0	0
$334$	24.0077	1.31364
$335$	−13.4153	−0.732956
$336$	0	0
$337$	−13.3854	−0.729148	−0.364574	−	0.931174i	$-0.618785\pi$
$337$	−13.3854	−0.729148	−0.364574	+	0.931174i	$0.618785\pi$
$338$	−6.75898	−0.367640
$339$	0	0
$340$	−0.598640	−0.0324658
$341$	0	0
$342$	0	0
$343$	−2.48809	−0.134344
$344$	−8.70152	−0.469155
$345$	0	0
$346$	21.5355	1.15776
$347$	−8.44899	−0.453565	−0.226783	−	0.973945i	$-0.572821\pi$
$347$	−8.44899	−0.453565	−0.226783	+	0.973945i	$0.572821\pi$
$348$	0	0
$349$	10.3988	0.556636	0.278318	−	0.960489i	$-0.410223\pi$
$349$	10.3988	0.556636	0.278318	+	0.960489i	$0.410223\pi$
$350$	4.94742	0.264451
$351$	0	0
$352$	0	0
$353$	−19.1073	−1.01698	−0.508489	−	0.861069i	$-0.669796\pi$
$353$	−19.1073	−1.01698	−0.508489	+	0.861069i	$0.669796\pi$
$354$	0	0
$355$	−8.30309	−0.440682
$356$	−1.97190	−0.104510
$357$	0	0
$358$	22.8852	1.20952
$359$	−4.41417	−0.232971	−0.116486	−	0.993192i	$-0.537163\pi$
$359$	−4.41417	−0.232971	−0.116486	+	0.993192i	$0.537163\pi$
$360$	0	0
$361$	−18.9910	−0.999524
$362$	−32.6546	−1.71629
$363$	0	0
$364$	−1.67520	−0.0878041
$365$	1.32003	0.0690936
$366$	0	0
$367$	−29.3617	−1.53267	−0.766335	−	0.642442i	$-0.777922\pi$
$367$	−29.3617	−1.53267	−0.766335	+	0.642442i	$0.777922\pi$
$368$	4.24116	0.221086
$369$	0	0
$370$	13.3396	0.693491
$371$	−10.9261	−0.567254
$372$	0	0
$373$	4.96478	0.257067	0.128533	−	0.991705i	$-0.458973\pi$
$373$	4.96478	0.257067	0.128533	+	0.991705i	$0.458973\pi$
$374$	0	0
$375$	0	0
$376$	−6.52157	−0.336325
$377$	−19.1200	−0.984728
$378$	0	0
$379$	7.92315	0.406985	0.203492	−	0.979077i	$-0.434771\pi$
$379$	7.92315	0.406985	0.203492	+	0.979077i	$0.434771\pi$
$380$	−0.0154241	−0.000791242 0
$381$	0	0
$382$	−7.29731	−0.373363
$383$	24.5155	1.25268	0.626342	−	0.779549i	$-0.284551\pi$
$383$	24.5155	1.25268	0.626342	+	0.779549i	$0.284551\pi$
$384$	0	0
$385$	0	0
$386$	−24.7872	−1.26164
$387$	0	0
$388$	0.702312	0.0356545
$389$	−5.46094	−0.276881	−0.138440	−	0.990371i	$-0.544209\pi$
$389$	−5.46094	−0.276881	−0.138440	+	0.990371i	$0.544209\pi$
$390$	0	0
$391$	−4.29059	−0.216985
$392$	−18.5198	−0.935389
$393$	0	0
$394$	3.58665	0.180693
$395$	13.8661	0.697679
$396$	0	0
$397$	−6.43455	−0.322941	−0.161470	−	0.986878i	$-0.551624\pi$
$397$	−6.43455	−0.322941	−0.161470	+	0.986878i	$0.551624\pi$
$398$	8.84984	0.443602
$399$	0	0
$400$	−3.64941	−0.182471
$401$	14.7026	0.734213	0.367107	−	0.930179i	$-0.380348\pi$
$401$	14.7026	0.734213	0.367107	+	0.930179i	$0.380348\pi$
$402$	0	0
$403$	19.1893	0.955885
$404$	−1.60618	−0.0799105
$405$	0	0
$406$	−33.4143	−1.65832
$407$	0	0
$408$	0	0
$409$	−4.39576	−0.217356	−0.108678	−	0.994077i	$-0.534662\pi$
$409$	−4.39576	−0.217356	−0.108678	+	0.994077i	$0.534662\pi$
$410$	−11.2768	−0.556921
$411$	0	0
$412$	−0.659275	−0.0324802
$413$	31.0514	1.52794
$414$	0	0
$415$	−10.6445	−0.522516
$416$	2.59009	0.126990
$417$	0	0
$418$	0	0
$419$	−17.8526	−0.872159	−0.436079	−	0.899908i	$-0.643633\pi$
$419$	−17.8526	−0.872159	−0.436079	+	0.899908i	$0.643633\pi$
$420$	0	0
$421$	−4.82854	−0.235328	−0.117664	−	0.993053i	$-0.537541\pi$
$421$	−4.82854	−0.235328	−0.117664	+	0.993053i	$0.537541\pi$
$422$	37.2103	1.81137
$423$	0	0
$424$	8.77570	0.426186
$425$	3.69195	0.179086
$426$	0	0
$427$	30.9484	1.49770
$428$	−0.314335	−0.0151939
$429$	0	0
$430$	4.02448	0.194078
$431$	−24.8739	−1.19814	−0.599068	−	0.800698i	$-0.704462\pi$
$431$	−24.8739	−1.19814	−0.599068	+	0.800698i	$0.704462\pi$
$432$	0	0
$433$	−21.2502	−1.02122	−0.510611	−	0.859812i	$-0.670581\pi$
$433$	−21.2502	−1.02122	−0.510611	+	0.859812i	$0.670581\pi$
$434$	33.5354	1.60975
$435$	0	0
$436$	0.993417	0.0475761
$437$	−0.110548	−0.00528825
$438$	0	0
$439$	15.9119	0.759434	0.379717	−	0.925103i	$-0.376021\pi$
$439$	15.9119	0.759434	0.379717	+	0.925103i	$0.376021\pi$
$440$	0	0
$441$	0	0
$442$	14.1692	0.673959
$443$	−26.2876	−1.24896	−0.624481	−	0.781040i	$-0.714690\pi$
$443$	−26.2876	−1.24896	−0.624481	+	0.781040i	$0.714690\pi$
$444$	0	0
$445$	12.1612	0.576494
$446$	−6.88945	−0.326225
$447$	0	0
$448$	31.1629	1.47231
$449$	8.18961	0.386492	0.193246	−	0.981150i	$-0.438098\pi$
$449$	8.18961	0.386492	0.193246	+	0.981150i	$0.438098\pi$
$450$	0	0
$451$	0	0
$452$	0.938065	0.0441229
$453$	0	0
$454$	5.06995	0.237945
$455$	10.3313	0.484340
$456$	0	0
$457$	−11.9164	−0.557425	−0.278713	−	0.960375i	$-0.589908\pi$
$457$	−11.9164	−0.557425	−0.278713	+	0.960375i	$0.589908\pi$
$458$	36.5003	1.70555
$459$	0	0
$460$	0.188439	0.00878601
$461$	6.96172	0.324240	0.162120	−	0.986771i	$-0.448167\pi$
$461$	6.96172	0.324240	0.162120	+	0.986771i	$0.448167\pi$
$462$	0	0
$463$	12.4762	0.579817	0.289909	−	0.957054i	$-0.406375\pi$
$463$	12.4762	0.579817	0.289909	+	0.957054i	$0.406375\pi$
$464$	24.6477	1.14424
$465$	0	0
$466$	24.8315	1.15030
$467$	6.14617	0.284411	0.142205	−	0.989837i	$-0.454581\pi$
$467$	6.14617	0.284411	0.142205	+	0.989837i	$0.454581\pi$
$468$	0	0
$469$	−48.9579	−2.26067
$470$	3.01625	0.139129
$471$	0	0
$472$	−24.9401	−1.14796
$473$	0	0
$474$	0	0
$475$	0.0951243	0.00436460
$476$	−2.18469	−0.100135
$477$	0	0
$478$	−14.8526	−0.679341
$479$	22.1942	1.01408	0.507039	−	0.861923i	$-0.330740\pi$
$479$	22.1942	1.01408	0.507039	+	0.861923i	$0.330740\pi$
$480$	0	0
$481$	27.8560	1.27013
$482$	13.5492	0.617149
$483$	0	0
$484$	0	0
$485$	−4.33133	−0.196675
$486$	0	0
$487$	−34.2306	−1.55114	−0.775569	−	0.631263i	$-0.782537\pi$
$487$	−34.2306	−1.55114	−0.775569	+	0.631263i	$0.782537\pi$
$488$	−24.8574	−1.12524
$489$	0	0
$490$	8.56545	0.386948
$491$	−16.9957	−0.767007	−0.383503	−	0.923539i	$-0.625283\pi$
$491$	−16.9957	−0.767007	−0.383503	+	0.923539i	$0.625283\pi$
$492$	0	0
$493$	−24.9351	−1.12302
$494$	0.365073	0.0164254
$495$	0	0
$496$	−24.7371	−1.11073
$497$	−30.3014	−1.35920
$498$	0	0
$499$	5.22946	0.234103	0.117051	−	0.993126i	$-0.462656\pi$
$499$	5.22946	0.234103	0.117051	+	0.993126i	$0.462656\pi$
$500$	−0.162147	−0.00725144
$501$	0	0
$502$	−13.0923	−0.584339
$503$	41.9448	1.87023	0.935113	−	0.354350i	$-0.115298\pi$
$503$	41.9448	1.87023	0.935113	+	0.354350i	$0.115298\pi$
$504$	0	0
$505$	9.90570	0.440798
$506$	0	0
$507$	0	0
$508$	0.395287	0.0175380
$509$	−20.3678	−0.902787	−0.451393	−	0.892325i	$-0.649073\pi$
$509$	−20.3678	−0.902787	−0.451393	+	0.892325i	$0.649073\pi$
$510$	0	0
$511$	4.81734	0.213107
$512$	−24.7330	−1.09305
$513$	0	0
$514$	−14.1364	−0.623529
$515$	4.06590	0.179165
$516$	0	0
$517$	0	0
$518$	48.6816	2.13895
$519$	0	0
$520$	−8.29800	−0.363891
$521$	−14.4779	−0.634287	−0.317143	−	0.948378i	$-0.602724\pi$
$521$	−14.4779	−0.634287	−0.317143	+	0.948378i	$0.602724\pi$
$522$	0	0
$523$	−11.1601	−0.487998	−0.243999	−	0.969775i	$-0.578459\pi$
$523$	−11.1601	−0.487998	−0.243999	+	0.969775i	$0.578459\pi$
$524$	−1.14177	−0.0498784
$525$	0	0
$526$	−14.8607	−0.647958
$527$	25.0254	1.09013
$528$	0	0
$529$	−21.6494	−0.941279
$530$	−4.05879	−0.176303
$531$	0	0
$532$	−0.0562891	−0.00244044
$533$	−23.5485	−1.02000
$534$	0	0
$535$	1.93858	0.0838119
$536$	39.3225	1.69847
$537$	0	0
$538$	0.121123	0.00522197
$539$	0	0
$540$	0	0
$541$	−10.6808	−0.459203	−0.229602	−	0.973285i	$-0.573742\pi$
$541$	−10.6808	−0.459203	−0.229602	+	0.973285i	$0.573742\pi$
$542$	18.1654	0.780272
$543$	0	0
$544$	3.37784	0.144824
$545$	−6.12664	−0.262436
$546$	0	0
$547$	−1.74760	−0.0747220	−0.0373610	−	0.999302i	$-0.511895\pi$
$547$	−1.74760	−0.0747220	−0.0373610	+	0.999302i	$0.511895\pi$
$548$	−1.55221	−0.0663072
$549$	0	0
$550$	0	0
$551$	−0.642459	−0.0273697
$552$	0	0
$553$	50.6031	2.15186
$554$	5.29619	0.225014
$555$	0	0
$556$	0.0835872	0.00354489
$557$	−19.4844	−0.825579	−0.412790	−	0.910826i	$-0.635446\pi$
$557$	−19.4844	−0.825579	−0.412790	+	0.910826i	$0.635446\pi$
$558$	0	0
$559$	8.40403	0.355453
$560$	−13.3182	−0.562798
$561$	0	0
$562$	−2.08426	−0.0879191
$563$	−14.6892	−0.619074	−0.309537	−	0.950887i	$-0.600174\pi$
$563$	−14.6892	−0.619074	−0.309537	+	0.950887i	$0.600174\pi$
$564$	0	0
$565$	−5.78527	−0.243388
$566$	7.33650	0.308376
$567$	0	0
$568$	24.3377	1.02119
$569$	19.9335	0.835658	0.417829	−	0.908526i	$-0.362791\pi$
$569$	19.9335	0.835658	0.417829	+	0.908526i	$0.362791\pi$
$570$	0	0
$571$	−5.24422	−0.219464	−0.109732	−	0.993961i	$-0.534999\pi$
$571$	−5.24422	−0.219464	−0.109732	+	0.993961i	$0.534999\pi$
$572$	0	0
$573$	0	0
$574$	−41.1537	−1.71772
$575$	−1.16215	−0.0484649
$576$	0	0
$577$	37.6004	1.56533	0.782663	−	0.622446i	$-0.213861\pi$
$577$	37.6004	1.56533	0.782663	+	0.622446i	$0.213861\pi$
$578$	−4.56790	−0.190000
$579$	0	0
$580$	1.09512	0.0454726
$581$	−38.8460	−1.61161
$582$	0	0
$583$	0	0
$584$	−3.86923	−0.160110
$585$	0	0
$586$	15.4770	0.639351
$587$	−25.5711	−1.05543	−0.527716	−	0.849421i	$-0.676952\pi$
$587$	−25.5711	−1.05543	−0.527716	+	0.849421i	$0.676952\pi$
$588$	0	0
$589$	0.644788	0.0265680
$590$	11.5349	0.474884
$591$	0	0
$592$	−35.9095	−1.47587
$593$	40.2260	1.65188	0.825942	−	0.563754i	$-0.190644\pi$
$593$	40.2260	1.65188	0.825942	+	0.563754i	$0.190644\pi$
$594$	0	0
$595$	13.4735	0.552358
$596$	1.32309	0.0541960
$597$	0	0
$598$	−4.46015	−0.182389
$599$	−4.92997	−0.201433	−0.100716	−	0.994915i	$-0.532114\pi$
$599$	−4.92997	−0.201433	−0.100716	+	0.994915i	$0.532114\pi$
$600$	0	0
$601$	−46.0896	−1.88003	−0.940017	−	0.341127i	$-0.889191\pi$
$601$	−46.0896	−1.88003	−0.940017	+	0.341127i	$0.889191\pi$
$602$	14.6870	0.598597
$603$	0	0
$604$	0.314602	0.0128010
$605$	0	0
$606$	0	0
$607$	45.1365	1.83203	0.916016	−	0.401141i	$-0.131386\pi$
$607$	45.1365	1.83203	0.916016	+	0.401141i	$0.131386\pi$
$608$	0.0870310	0.00352957
$609$	0	0
$610$	11.4966	0.465484
$611$	6.29861	0.254815
$612$	0	0
$613$	−4.73418	−0.191212	−0.0956059	−	0.995419i	$-0.530479\pi$
$613$	−4.73418	−0.191212	−0.0956059	+	0.995419i	$0.530479\pi$
$614$	5.76172	0.232524
$615$	0	0
$616$	0	0
$617$	−17.8468	−0.718486	−0.359243	−	0.933244i	$-0.616965\pi$
$617$	−17.8468	−0.718486	−0.359243	+	0.933244i	$0.616965\pi$
$618$	0	0
$619$	0.356952	0.0143471	0.00717356	−	0.999974i	$-0.497717\pi$
$619$	0.356952	0.0143471	0.00717356	+	0.999974i	$0.497717\pi$
$620$	−1.09909	−0.0441407
$621$	0	0
$622$	22.5306	0.903394
$623$	44.3811	1.77809
$624$	0	0
$625$	1.00000	0.0400000
$626$	−36.0297	−1.44004
$627$	0	0
$628$	−3.44960	−0.137654
$629$	36.3281	1.44850
$630$	0	0
$631$	−31.9922	−1.27359	−0.636795	−	0.771033i	$-0.719740\pi$
$631$	−31.9922	−1.27359	−0.636795	+	0.771033i	$0.719740\pi$
$632$	−40.6438	−1.61672
$633$	0	0
$634$	−7.85876	−0.312111
$635$	−2.43783	−0.0967422
$636$	0	0
$637$	17.8866	0.708693
$638$	0	0
$639$	0	0
$640$	9.74648	0.385264
$641$	−1.01285	−0.0400050	−0.0200025	−	0.999800i	$-0.506367\pi$
$641$	−1.01285	−0.0400050	−0.0200025	+	0.999800i	$0.506367\pi$
$642$	0	0
$643$	−14.9724	−0.590455	−0.295228	−	0.955427i	$-0.595395\pi$
$643$	−14.9724	−0.590455	−0.295228	+	0.955427i	$0.595395\pi$
$644$	0.687692	0.0270988
$645$	0	0
$646$	0.476106	0.0187321
$647$	−17.8873	−0.703224	−0.351612	−	0.936146i	$-0.614366\pi$
$647$	−17.8873	−0.703224	−0.351612	+	0.936146i	$0.614366\pi$
$648$	0	0
$649$	0	0
$650$	3.83785	0.150533
$651$	0	0
$652$	2.59127	0.101482
$653$	−45.7642	−1.79089	−0.895446	−	0.445169i	$-0.853144\pi$
$653$	−45.7642	−1.79089	−0.895446	+	0.445169i	$0.853144\pi$
$654$	0	0
$655$	7.04156	0.275136
$656$	30.3566	1.18523
$657$	0	0
$658$	11.0075	0.429119
$659$	−9.54036	−0.371640	−0.185820	−	0.982584i	$-0.559494\pi$
$659$	−9.54036	−0.371640	−0.185820	+	0.982584i	$0.559494\pi$
$660$	0	0
$661$	15.7769	0.613651	0.306825	−	0.951766i	$-0.400733\pi$
$661$	15.7769	0.613651	0.306825	+	0.951766i	$0.400733\pi$
$662$	−17.5193	−0.680908
$663$	0	0
$664$	31.2007	1.21082
$665$	0.347148	0.0134618
$666$	0	0
$667$	7.84901	0.303915
$668$	−2.87147	−0.111100
$669$	0	0
$670$	−18.1868	−0.702616
$671$	0	0
$672$	0	0
$673$	−47.3031	−1.82340	−0.911700	−	0.410856i	$-0.865230\pi$
$673$	−47.3031	−1.82340	−0.911700	+	0.410856i	$0.865230\pi$
$674$	−18.1462	−0.698966
$675$	0	0
$676$	0.808416	0.0310929
$677$	27.5431	1.05857	0.529284	−	0.848445i	$-0.322461\pi$
$677$	27.5431	1.05857	0.529284	+	0.848445i	$0.322461\pi$
$678$	0	0
$679$	−15.8068	−0.606609
$680$	−10.8217	−0.414995
$681$	0	0
$682$	0	0
$683$	27.1617	1.03931	0.519656	−	0.854375i	$-0.326060\pi$
$683$	27.1617	1.03931	0.519656	+	0.854375i	$0.326060\pi$
$684$	0	0
$685$	9.57286	0.365760
$686$	−3.37304	−0.128783
$687$	0	0
$688$	−10.8337	−0.413032
$689$	−8.47567	−0.322897
$690$	0	0
$691$	−7.52680	−0.286333	−0.143166	−	0.989699i	$-0.545728\pi$
$691$	−7.52680	−0.286333	−0.143166	+	0.989699i	$0.545728\pi$
$692$	−2.57579	−0.0979167
$693$	0	0
$694$	−11.4541	−0.434791
$695$	−0.515502	−0.0195541
$696$	0	0
$697$	−30.7105	−1.16324
$698$	14.0974	0.533595
$699$	0	0
$700$	−0.591742	−0.0223658
$701$	−31.8207	−1.20185	−0.600926	−	0.799305i	$-0.705201\pi$
$701$	−31.8207	−1.20185	−0.600926	+	0.799305i	$0.705201\pi$
$702$	0	0
$703$	0.936004	0.0353021
$704$	0	0
$705$	0	0
$706$	−25.9032	−0.974882
$707$	36.1500	1.35956
$708$	0	0
$709$	−14.4381	−0.542235	−0.271118	−	0.962546i	$-0.587393\pi$
$709$	−14.4381	−0.542235	−0.271118	+	0.962546i	$0.587393\pi$
$710$	−11.2563	−0.422441
$711$	0	0
$712$	−35.6464	−1.33591
$713$	−7.87747	−0.295013
$714$	0	0
$715$	0	0
$716$	−2.73721	−0.102294
$717$	0	0
$718$	−5.98418	−0.223328
$719$	−5.41004	−0.201761	−0.100880	−	0.994899i	$-0.532166\pi$
$719$	−5.41004	−0.201761	−0.100880	+	0.994899i	$0.532166\pi$
$720$	0	0
$721$	14.8382	0.552602
$722$	−25.7455	−0.958150
$723$	0	0
$724$	3.90570	0.145154
$725$	−6.75389	−0.250833
$726$	0	0
$727$	−16.7753	−0.622161	−0.311080	−	0.950384i	$-0.600691\pi$
$727$	−16.7753	−0.622161	−0.311080	+	0.950384i	$0.600691\pi$
$728$	−30.2828	−1.12236
$729$	0	0
$730$	1.78953	0.0662336
$731$	10.9600	0.405371
$732$	0	0
$733$	14.0851	0.520243	0.260122	−	0.965576i	$-0.416237\pi$
$733$	14.0851	0.520243	0.260122	+	0.965576i	$0.416237\pi$
$734$	−39.8049	−1.46923
$735$	0	0
$736$	−1.06327	−0.0391926
$737$	0	0
$738$	0	0
$739$	36.3457	1.33700	0.668499	−	0.743713i	$-0.266937\pi$
$739$	36.3457	1.33700	0.668499	+	0.743713i	$0.266937\pi$
$740$	−1.59550	−0.0586516
$741$	0	0
$742$	−14.8122	−0.543773
$743$	−1.95716	−0.0718012	−0.0359006	−	0.999355i	$-0.511430\pi$
$743$	−1.95716	−0.0718012	−0.0359006	+	0.999355i	$0.511430\pi$
$744$	0	0
$745$	−8.15983	−0.298953
$746$	6.73063	0.246426
$747$	0	0
$748$	0	0
$749$	7.07466	0.258503
$750$	0	0
$751$	18.7106	0.682759	0.341379	−	0.939926i	$-0.389106\pi$
$751$	18.7106	0.682759	0.341379	+	0.939926i	$0.389106\pi$
$752$	−8.11961	−0.296092
$753$	0	0
$754$	−25.9204	−0.943967
$755$	−1.94023	−0.0706120
$756$	0	0
$757$	−14.5470	−0.528721	−0.264361	−	0.964424i	$-0.585161\pi$
$757$	−14.5470	−0.528721	−0.264361	+	0.964424i	$0.585161\pi$
$758$	10.7412	0.390138
$759$	0	0
$760$	−0.278825	−0.0101141
$761$	13.1406	0.476345	0.238173	−	0.971223i	$-0.423452\pi$
$761$	13.1406	0.476345	0.238173	+	0.971223i	$0.423452\pi$
$762$	0	0
$763$	−22.3586	−0.809437
$764$	0.872805	0.0315770
$765$	0	0
$766$	33.2350	1.20083
$767$	24.0875	0.869748
$768$	0	0
$769$	−38.9767	−1.40554	−0.702768	−	0.711419i	$-0.748053\pi$
$769$	−38.9767	−1.40554	−0.702768	+	0.711419i	$0.748053\pi$
$770$	0	0
$771$	0	0
$772$	2.96471	0.106702
$773$	38.7539	1.39388	0.696940	−	0.717129i	$-0.254544\pi$
$773$	38.7539	1.39388	0.696940	+	0.717129i	$0.254544\pi$
$774$	0	0
$775$	6.77837	0.243486
$776$	12.6958	0.455754
$777$	0	0
$778$	−7.40326	−0.265420
$779$	−0.791265	−0.0283500
$780$	0	0
$781$	0	0
$782$	−5.81665	−0.208003
$783$	0	0
$784$	−23.0578	−0.823493
$785$	21.2745	0.759319
$786$	0	0
$787$	21.3842	0.762265	0.381132	−	0.924521i	$-0.375534\pi$
$787$	21.3842	0.762265	0.381132	+	0.924521i	$0.375534\pi$
$788$	−0.428986	−0.0152820
$789$	0	0
$790$	18.7979	0.668799
$791$	−21.1128	−0.750686
$792$	0	0
$793$	24.0075	0.852533
$794$	−8.72315	−0.309573
$795$	0	0
$796$	−1.05850	−0.0375174
$797$	2.22456	0.0787978	0.0393989	−	0.999224i	$-0.487456\pi$
$797$	2.22456	0.0787978	0.0393989	+	0.999224i	$0.487456\pi$
$798$	0	0
$799$	8.21426	0.290599
$800$	0.914918	0.0323472
$801$	0	0
$802$	19.9319	0.703821
$803$	0	0
$804$	0	0
$805$	−4.24116	−0.149481
$806$	26.0144	0.916318
$807$	0	0
$808$	−29.0353	−1.02146
$809$	21.1682	0.744234	0.372117	−	0.928186i	$-0.378632\pi$
$809$	21.1682	0.744234	0.372117	+	0.928186i	$0.378632\pi$
$810$	0	0
$811$	36.7172	1.28932	0.644658	−	0.764471i	$-0.277000\pi$
$811$	36.7172	1.28932	0.644658	+	0.764471i	$0.277000\pi$
$812$	3.99656	0.140252
$813$	0	0
$814$	0	0
$815$	−15.9810	−0.559788
$816$	0	0
$817$	0.282388	0.00987951
$818$	−5.95922	−0.208359
$819$	0	0
$820$	1.34878	0.0471013
$821$	−39.6693	−1.38447	−0.692235	−	0.721673i	$-0.743374\pi$
$821$	−39.6693	−1.38447	−0.692235	+	0.721673i	$0.743374\pi$
$822$	0	0
$823$	−45.9283	−1.60096	−0.800480	−	0.599359i	$-0.795422\pi$
$823$	−45.9283	−1.60096	−0.800480	+	0.599359i	$0.795422\pi$
$824$	−11.9178	−0.415178
$825$	0	0
$826$	42.0956	1.46469
$827$	−39.6949	−1.38033	−0.690164	−	0.723653i	$-0.742462\pi$
$827$	−39.6949	−1.38033	−0.690164	+	0.723653i	$0.742462\pi$
$828$	0	0
$829$	7.65584	0.265898	0.132949	−	0.991123i	$-0.457555\pi$
$829$	7.65584	0.265898	0.132949	+	0.991123i	$0.457555\pi$
$830$	−14.4304	−0.500887
$831$	0	0
$832$	24.1740	0.838082
$833$	23.3266	0.808218
$834$	0	0
$835$	17.7090	0.612846
$836$	0	0
$837$	0	0
$838$	−24.2024	−0.836057
$839$	−27.5886	−0.952465	−0.476233	−	0.879319i	$-0.657998\pi$
$839$	−27.5886	−0.952465	−0.476233	+	0.879319i	$0.657998\pi$
$840$	0	0
$841$	16.6150	0.572932
$842$	−6.54592	−0.225587
$843$	0	0
$844$	−4.45058	−0.153195
$845$	−4.98569	−0.171513
$846$	0	0
$847$	0	0
$848$	10.9261	0.375203
$849$	0	0
$850$	5.00509	0.171673
$851$	−11.4353	−0.391997
$852$	0	0
$853$	−42.1496	−1.44318	−0.721588	−	0.692323i	$-0.756587\pi$
$853$	−42.1496	−1.44318	−0.721588	+	0.692323i	$0.756587\pi$
$854$	41.9559	1.43570
$855$	0	0
$856$	−5.68229	−0.194217
$857$	−45.0850	−1.54008	−0.770038	−	0.637998i	$-0.779763\pi$
$857$	−45.0850	−1.54008	−0.770038	+	0.637998i	$0.779763\pi$
$858$	0	0
$859$	−11.8257	−0.403488	−0.201744	−	0.979438i	$-0.564661\pi$
$859$	−11.8257	−0.403488	−0.201744	+	0.979438i	$0.564661\pi$
$860$	−0.481353	−0.0164140
$861$	0	0
$862$	−33.7210	−1.14854
$863$	−27.8713	−0.948750	−0.474375	−	0.880323i	$-0.657326\pi$
$863$	−27.8713	−0.948750	−0.474375	+	0.880323i	$0.657326\pi$
$864$	0	0
$865$	15.8855	0.540123
$866$	−28.8084	−0.978950
$867$	0	0
$868$	−4.01105	−0.136144
$869$	0	0
$870$	0	0
$871$	−37.9781	−1.28684
$872$	17.9582	0.608141
$873$	0	0
$874$	−0.149868	−0.00506935
$875$	3.64941	0.123373
$876$	0	0
$877$	11.4471	0.386543	0.193271	−	0.981145i	$-0.438090\pi$
$877$	11.4471	0.386543	0.193271	+	0.981145i	$0.438090\pi$
$878$	21.5714	0.727998
$879$	0	0
$880$	0	0
$881$	−47.0037	−1.58360	−0.791798	−	0.610783i	$-0.790855\pi$
$881$	−47.0037	−1.58360	−0.791798	+	0.610783i	$0.790855\pi$
$882$	0	0
$883$	−46.9146	−1.57880	−0.789401	−	0.613877i	$-0.789609\pi$
$883$	−46.9146	−1.57880	−0.789401	+	0.613877i	$0.789609\pi$
$884$	−1.69472	−0.0569997
$885$	0	0
$886$	−35.6374	−1.19726
$887$	−27.8427	−0.934866	−0.467433	−	0.884029i	$-0.654821\pi$
$887$	−27.8427	−0.934866	−0.467433	+	0.884029i	$0.654821\pi$
$888$	0	0
$889$	−8.89664	−0.298384
$890$	16.4866	0.552631
$891$	0	0
$892$	0.824022	0.0275903
$893$	0.211643	0.00708236
$894$	0	0
$895$	16.8810	0.564271
$896$	35.5689	1.18828
$897$	0	0
$898$	11.1024	0.370494
$899$	−45.7804	−1.52686
$900$	0	0
$901$	−11.0534	−0.368244
$902$	0	0
$903$	0	0
$904$	16.9576	0.564001
$905$	−24.0874	−0.800691
$906$	0	0
$907$	−28.6233	−0.950421	−0.475210	−	0.879872i	$-0.657628\pi$
$907$	−28.6233	−0.950421	−0.475210	+	0.879872i	$0.657628\pi$
$908$	−0.606398	−0.0201240
$909$	0	0
$910$	14.0059	0.464292
$911$	−5.12823	−0.169906	−0.0849529	−	0.996385i	$-0.527074\pi$
$911$	−5.12823	−0.169906	−0.0849529	+	0.996385i	$0.527074\pi$
$912$	0	0
$913$	0	0
$914$	−16.1547	−0.534351
$915$	0	0
$916$	−4.36567	−0.144246
$917$	25.6976	0.848608
$918$	0	0
$919$	−35.2810	−1.16381	−0.581906	−	0.813256i	$-0.697693\pi$
$919$	−35.2810	−1.16381	−0.581906	+	0.813256i	$0.697693\pi$
$920$	3.40645	0.112307
$921$	0	0
$922$	9.43783	0.310818
$923$	−23.5057	−0.773699
$924$	0	0
$925$	9.83980	0.323531
$926$	16.9136	0.555817
$927$	0	0
$928$	−6.17926	−0.202844
$929$	59.1427	1.94041	0.970204	−	0.242289i	$-0.0778981\pi$
$929$	59.1427	1.94041	0.970204	+	0.242289i	$0.0778981\pi$
$930$	0	0
$931$	0.601017	0.0196975
$932$	−2.97000	−0.0972857
$933$	0	0
$934$	8.33220	0.272638
$935$	0	0
$936$	0	0
$937$	−14.4425	−0.471817	−0.235909	−	0.971775i	$-0.575807\pi$
$937$	−14.4425	−0.471817	−0.235909	+	0.971775i	$0.575807\pi$
$938$	−66.3710	−2.16709
$939$	0	0
$940$	−0.360762	−0.0117668
$941$	−18.6591	−0.608269	−0.304135	−	0.952629i	$-0.598367\pi$
$941$	−18.6591	−0.608269	−0.304135	+	0.952629i	$0.598367\pi$
$942$	0	0
$943$	9.66700	0.314801
$944$	−31.0514	−1.01064
$945$	0	0
$946$	0	0
$947$	0.991391	0.0322159	0.0161079	−	0.999870i	$-0.494872\pi$
$947$	0.991391	0.0322159	0.0161079	+	0.999870i	$0.494872\pi$
$948$	0	0
$949$	3.73695	0.121306
$950$	0.128958	0.00418394
$951$	0	0
$952$	−39.4930	−1.27998
$953$	−8.26404	−0.267699	−0.133849	−	0.991002i	$-0.542734\pi$
$953$	−8.26404	−0.267699	−0.133849	+	0.991002i	$0.542734\pi$
$954$	0	0
$955$	−5.38279	−0.174183
$956$	1.77646	0.0574549
$957$	0	0
$958$	30.0881	0.972101
$959$	34.9353	1.12812
$960$	0	0
$961$	14.9463	0.482139
$962$	37.7637	1.21755
$963$	0	0
$964$	−1.62057	−0.0521950
$965$	−18.2840	−0.588584
$966$	0	0
$967$	−7.36029	−0.236691	−0.118345	−	0.992972i	$-0.537759\pi$
$967$	−7.36029	−0.236691	−0.118345	+	0.992972i	$0.537759\pi$
$968$	0	0
$969$	0	0
$970$	−5.87187	−0.188534
$971$	4.97733	0.159730	0.0798650	−	0.996806i	$-0.474551\pi$
$971$	4.97733	0.159730	0.0798650	+	0.996806i	$0.474551\pi$
$972$	0	0
$973$	−1.88128	−0.0603111
$974$	−46.4056	−1.48693
$975$	0	0
$976$	−30.9484	−0.990633
$977$	−10.3368	−0.330704	−0.165352	−	0.986235i	$-0.552876\pi$
$977$	−10.3368	−0.330704	−0.165352	+	0.986235i	$0.552876\pi$
$978$	0	0
$979$	0	0
$980$	−1.02448	−0.0327259
$981$	0	0
$982$	−23.0407	−0.735258
$983$	29.0614	0.926913	0.463457	−	0.886120i	$-0.346609\pi$
$983$	29.0614	0.926913	0.463457	+	0.886120i	$0.346609\pi$
$984$	0	0
$985$	2.64566	0.0842978
$986$	−33.8038	−1.07653
$987$	0	0
$988$	−0.0436651	−0.00138917
$989$	−3.44997	−0.109703
$990$	0	0
$991$	7.70381	0.244719	0.122360	−	0.992486i	$-0.460954\pi$
$991$	7.70381	0.244719	0.122360	+	0.992486i	$0.460954\pi$
$992$	6.20166	0.196903
$993$	0	0
$994$	−41.0788	−1.30294
$995$	6.52800	0.206951
$996$	0	0
$997$	2.86418	0.0907095	0.0453547	−	0.998971i	$-0.485558\pi$
$997$	2.86418	0.0907095	0.0453547	+	0.998971i	$0.485558\pi$
$998$	7.08945	0.224413
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bi.1.4		4
3.2	odd	2		605.2.a.k.1.1		4
11.2	odd	10		495.2.n.e.136.2		8
11.6	odd	10		495.2.n.e.91.2		8
11.10	odd	2		5445.2.a.bp.1.1		4
12.11	even	2		9680.2.a.cm.1.2		4
15.14	odd	2		3025.2.a.w.1.4		4
33.2	even	10		55.2.g.b.26.1	✓	8
33.5	odd	10		605.2.g.k.366.2		8
33.8	even	10		605.2.g.m.251.2		8
33.14	odd	10		605.2.g.e.251.1		8
33.17	even	10		55.2.g.b.36.1	yes	8
33.20	odd	10		605.2.g.k.81.2		8
33.26	odd	10		605.2.g.e.511.1		8
33.29	even	10		605.2.g.m.511.2		8
33.32	even	2		605.2.a.j.1.4		4
132.35	odd	10		880.2.bo.h.81.1		8
132.83	odd	10		880.2.bo.h.641.1		8
132.131	odd	2		9680.2.a.cn.1.2		4
165.2	odd	20		275.2.z.a.224.4		16
165.17	odd	20		275.2.z.a.124.1		16
165.68	odd	20		275.2.z.a.224.1		16
165.83	odd	20		275.2.z.a.124.4		16
165.134	even	10		275.2.h.a.26.2		8
165.149	even	10		275.2.h.a.201.2		8
165.164	even	2		3025.2.a.bd.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.26.1	✓	8	33.2	even	10
55.2.g.b.36.1	yes	8	33.17	even	10
275.2.h.a.26.2		8	165.134	even	10
275.2.h.a.201.2		8	165.149	even	10
275.2.z.a.124.1		16	165.17	odd	20
275.2.z.a.124.4		16	165.83	odd	20
275.2.z.a.224.1		16	165.68	odd	20
275.2.z.a.224.4		16	165.2	odd	20
495.2.n.e.91.2		8	11.6	odd	10
495.2.n.e.136.2		8	11.2	odd	10
605.2.a.j.1.4		4	33.32	even	2
605.2.a.k.1.1		4	3.2	odd	2
605.2.g.e.251.1		8	33.14	odd	10
605.2.g.e.511.1		8	33.26	odd	10
605.2.g.k.81.2		8	33.20	odd	10
605.2.g.k.366.2		8	33.5	odd	10
605.2.g.m.251.2		8	33.8	even	10
605.2.g.m.511.2		8	33.29	even	10
880.2.bo.h.81.1		8	132.35	odd	10
880.2.bo.h.641.1		8	132.83	odd	10
3025.2.a.w.1.4		4	15.14	odd	2
3025.2.a.bd.1.1		4	165.164	even	2
5445.2.a.bi.1.4		4	1.1	even	1	trivial
5445.2.a.bp.1.1		4	11.10	odd	2
9680.2.a.cm.1.2		4	12.11	even	2
9680.2.a.cn.1.2		4	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35567 q^{2} -0.162147 q^{4} +1.00000 q^{5} +3.64941 q^{7} -2.93117 q^{8} +1.35567 q^{10} +2.83095 q^{13} +4.94742 q^{14} -3.64941 q^{16} +3.69195 q^{17} +0.0951243 q^{19} -0.162147 q^{20} -1.16215 q^{23} +1.00000 q^{25} +3.83785 q^{26} -0.591742 q^{28} -6.75389 q^{29} +6.77837 q^{31} +0.914918 q^{32} +5.00509 q^{34} +3.64941 q^{35} +9.83980 q^{37} +0.128958 q^{38} -2.93117 q^{40} -8.31822 q^{41} +2.96862 q^{43} -1.57549 q^{46} +2.22491 q^{47} +6.31822 q^{49} +1.35567 q^{50} -0.459031 q^{52} -2.99393 q^{53} -10.6970 q^{56} -9.15607 q^{58} +8.50860 q^{59} +8.48037 q^{61} +9.18926 q^{62} +8.53916 q^{64} +2.83095 q^{65} -13.4153 q^{67} -0.598640 q^{68} +4.94742 q^{70} -8.30309 q^{71} +1.32003 q^{73} +13.3396 q^{74} -0.0154241 q^{76} +13.8661 q^{79} -3.64941 q^{80} -11.2768 q^{82} -10.6445 q^{83} +3.69195 q^{85} +4.02448 q^{86} +12.1612 q^{89} +10.3313 q^{91} +0.188439 q^{92} +3.01625 q^{94} +0.0951243 q^{95} -4.33133 q^{97} +8.56545 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{4} + 4 q^{5} + 3 q^{7} - 3 q^{8} - q^{10} + q^{13} - 2 q^{14} - 3 q^{16} + q^{17} + 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} + 13 q^{28} - 12 q^{29} - 5 q^{31} + 8 q^{32}+ \cdots + 8 q^{98}+O(q^{100}) \) 4 * q - q^2 - q^4 + 4 * q^5 + 3 * q^7 - 3 * q^8 - q^10 + q^13 - 2 * q^14 - 3 * q^16 + q^17 + 20 * q^19 - q^20 - 5 * q^23 + 4 * q^25 + 15 * q^26 + 13 * q^28 - 12 * q^29 - 5 * q^31 + 8 * q^32 + 2 * q^34 + 3 * q^35 + 7 * q^37 - 20 * q^38 - 3 * q^40 - 11 * q^41 + 19 * q^43 - 4 * q^46 - 5 * q^47 + 3 * q^49 - q^50 - 11 * q^52 + 11 * q^53 - 11 * q^56 - 14 * q^58 - 9 * q^59 + 12 * q^61 + 35 * q^62 - 3 * q^64 + q^65 - 19 * q^67 + 3 * q^68 - 2 * q^70 - 5 * q^71 + 11 * q^73 + 34 * q^79 - 3 * q^80 - 6 * q^82 + 11 * q^83 + q^85 - q^86 + 8 * q^89 - 8 * q^91 + 12 * q^92 - q^94 + 20 * q^95 + 32 * q^97 + 8 * q^98

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