LMFDB - Embedded newform 6003.2.a.u.1.21

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6003,2,Mod(1,6003)] 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("6003.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6003, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.a (trivial)

Newform invariants

comment:select newform

sage:traces = [22,3,0,17,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	6003.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.33095 q^{2} +3.43331 q^{4} -2.03702 q^{5} +4.28183 q^{7} +3.34096 q^{8} -4.74817 q^{10} -3.57655 q^{11} -5.73812 q^{13} +9.98070 q^{14} +0.920976 q^{16} +4.13707 q^{17} -8.31788 q^{19} -6.99370 q^{20} -8.33675 q^{22} -1.00000 q^{23} -0.850567 q^{25} -13.3752 q^{26} +14.7008 q^{28} -1.00000 q^{29} +4.54582 q^{31} -4.53517 q^{32} +9.64327 q^{34} -8.72215 q^{35} -9.15659 q^{37} -19.3885 q^{38} -6.80558 q^{40} -3.77238 q^{41} -4.11401 q^{43} -12.2794 q^{44} -2.33095 q^{46} +6.22762 q^{47} +11.3340 q^{49} -1.98263 q^{50} -19.7007 q^{52} +1.32969 q^{53} +7.28549 q^{55} +14.3054 q^{56} -2.33095 q^{58} -12.0838 q^{59} +4.42239 q^{61} +10.5961 q^{62} -12.4132 q^{64} +11.6886 q^{65} -5.73198 q^{67} +14.2038 q^{68} -20.3309 q^{70} +9.12802 q^{71} -11.8581 q^{73} -21.3435 q^{74} -28.5578 q^{76} -15.3142 q^{77} +2.03687 q^{79} -1.87604 q^{80} -8.79321 q^{82} +2.51017 q^{83} -8.42727 q^{85} -9.58952 q^{86} -11.9491 q^{88} +10.0168 q^{89} -24.5696 q^{91} -3.43331 q^{92} +14.5163 q^{94} +16.9436 q^{95} +0.738955 q^{97} +26.4190 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34}+ \cdots + 28 q^{98}+O(q^{100}) $ 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * 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 $

$2$	2.33095	1.64823	0.824114	−	0.566425i	$-0.191674\pi$
$2$	2.33095	1.64823	0.824114	+	0.566425i	$0.191674\pi$
$3$	0	0
$3$	0	0
$4$	3.43331	1.71665
$5$	−2.03702	−0.910981	−0.455491	−	0.890241i	$-0.650536\pi$
$5$	−2.03702	−0.910981	−0.455491	+	0.890241i	$0.650536\pi$
$6$	0	0
$7$	4.28183	1.61838	0.809189	−	0.587548i	$-0.199907\pi$
$7$	4.28183	1.61838	0.809189	+	0.587548i	$0.199907\pi$
$8$	3.34096	1.18121
$9$	0	0
$10$	−4.74817	−1.50150
$11$	−3.57655	−1.07837	−0.539186	−	0.842187i	$-0.681268\pi$
$11$	−3.57655	−1.07837	−0.539186	+	0.842187i	$0.681268\pi$
$12$	0	0
$13$	−5.73812	−1.59147	−0.795734	−	0.605646i	$-0.792915\pi$
$13$	−5.73812	−1.59147	−0.795734	+	0.605646i	$0.792915\pi$
$14$	9.98070	2.66746
$15$	0	0
$16$	0.920976	0.230244
$17$	4.13707	1.00339	0.501693	−	0.865046i	$-0.332711\pi$
$17$	4.13707	1.00339	0.501693	+	0.865046i	$0.332711\pi$
$18$	0	0
$19$	−8.31788	−1.90825	−0.954126	−	0.299406i	$-0.903212\pi$
$19$	−8.31788	−1.90825	−0.954126	+	0.299406i	$0.903212\pi$
$20$	−6.99370	−1.56384
$21$	0	0
$22$	−8.33675	−1.77740
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−0.850567	−0.170113
$26$	−13.3752	−2.62310
$27$	0	0
$28$	14.7008	2.77819
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.54582	0.816454	0.408227	−	0.912880i	$-0.366147\pi$
$31$	4.54582	0.816454	0.408227	+	0.912880i	$0.366147\pi$
$32$	−4.53517	−0.801712
$33$	0	0
$34$	9.64327	1.65381
$35$	−8.72215	−1.47431
$36$	0	0
$37$	−9.15659	−1.50533	−0.752667	−	0.658401i	$-0.771233\pi$
$37$	−9.15659	−1.50533	−0.752667	+	0.658401i	$0.771233\pi$
$38$	−19.3885	−3.14523
$39$	0	0
$40$	−6.80558	−1.07606
$41$	−3.77238	−0.589146	−0.294573	−	0.955629i	$-0.595177\pi$
$41$	−3.77238	−0.589146	−0.294573	+	0.955629i	$0.595177\pi$
$42$	0	0
$43$	−4.11401	−0.627380	−0.313690	−	0.949525i	$-0.601565\pi$
$43$	−4.11401	−0.627380	−0.313690	+	0.949525i	$0.601565\pi$
$44$	−12.2794	−1.85119
$45$	0	0
$46$	−2.33095	−0.343679
$47$	6.22762	0.908392	0.454196	−	0.890902i	$-0.349927\pi$
$47$	6.22762	0.908392	0.454196	+	0.890902i	$0.349927\pi$
$48$	0	0
$49$	11.3340	1.61915
$50$	−1.98263	−0.280386
$51$	0	0
$52$	−19.7007	−2.73200
$53$	1.32969	0.182647	0.0913237	−	0.995821i	$-0.470890\pi$
$53$	1.32969	0.182647	0.0913237	+	0.995821i	$0.470890\pi$
$54$	0	0
$55$	7.28549	0.982376
$56$	14.3054	1.91164
$57$	0	0
$58$	−2.33095	−0.306068
$59$	−12.0838	−1.57318	−0.786591	−	0.617475i	$-0.788156\pi$
$59$	−12.0838	−1.57318	−0.786591	+	0.617475i	$0.788156\pi$
$60$	0	0
$61$	4.42239	0.566229	0.283115	−	0.959086i	$-0.408632\pi$
$61$	4.42239	0.566229	0.283115	+	0.959086i	$0.408632\pi$
$62$	10.5961	1.34570
$63$	0	0
$64$	−12.4132	−1.55165
$65$	11.6886	1.44980
$66$	0	0
$67$	−5.73198	−0.700273	−0.350136	−	0.936699i	$-0.613865\pi$
$67$	−5.73198	−0.700273	−0.350136	+	0.936699i	$0.613865\pi$
$68$	14.2038	1.72246
$69$	0	0
$70$	−20.3309	−2.43000
$71$	9.12802	1.08330	0.541648	−	0.840605i	$-0.317801\pi$
$71$	9.12802	1.08330	0.541648	+	0.840605i	$0.317801\pi$
$72$	0	0
$73$	−11.8581	−1.38788	−0.693940	−	0.720033i	$-0.744127\pi$
$73$	−11.8581	−1.38788	−0.693940	+	0.720033i	$0.744127\pi$
$74$	−21.3435	−2.48113
$75$	0	0
$76$	−28.5578	−3.27581
$77$	−15.3142	−1.74521
$78$	0	0
$79$	2.03687	0.229166	0.114583	−	0.993414i	$-0.463447\pi$
$79$	2.03687	0.229166	0.114583	+	0.993414i	$0.463447\pi$
$80$	−1.87604	−0.209748
$81$	0	0
$82$	−8.79321	−0.971047
$83$	2.51017	0.275527	0.137764	−	0.990465i	$-0.456009\pi$
$83$	2.51017	0.275527	0.137764	+	0.990465i	$0.456009\pi$
$84$	0	0
$85$	−8.42727	−0.914065
$86$	−9.58952	−1.03406
$87$	0	0
$88$	−11.9491	−1.27378
$89$	10.0168	1.06178	0.530889	−	0.847441i	$-0.321858\pi$
$89$	10.0168	1.06178	0.530889	+	0.847441i	$0.321858\pi$
$90$	0	0
$91$	−24.5696	−2.57560
$92$	−3.43331	−0.357947
$93$	0	0
$94$	14.5163	1.49724
$95$	16.9436	1.73838
$96$	0	0
$97$	0.738955	0.0750295	0.0375147	−	0.999296i	$-0.488056\pi$
$97$	0.738955	0.0750295	0.0375147	+	0.999296i	$0.488056\pi$
$98$	26.4190	2.66873
$99$	0	0
$100$	−2.92026	−0.292026
$101$	14.1605	1.40902	0.704509	−	0.709695i	$-0.251167\pi$
$101$	14.1605	1.40902	0.704509	+	0.709695i	$0.251167\pi$
$102$	0	0
$103$	−15.3578	−1.51325	−0.756626	−	0.653848i	$-0.773154\pi$
$103$	−15.3578	−1.51325	−0.756626	+	0.653848i	$0.773154\pi$
$104$	−19.1708	−1.87985
$105$	0	0
$106$	3.09944	0.301044
$107$	13.1005	1.26647	0.633237	−	0.773958i	$-0.281726\pi$
$107$	13.1005	1.26647	0.633237	+	0.773958i	$0.281726\pi$
$108$	0	0
$109$	17.4120	1.66777	0.833883	−	0.551941i	$-0.186113\pi$
$109$	17.4120	1.66777	0.833883	+	0.551941i	$0.186113\pi$
$110$	16.9821	1.61918
$111$	0	0
$112$	3.94346	0.372622
$113$	−13.9526	−1.31255	−0.656276	−	0.754521i	$-0.727869\pi$
$113$	−13.9526	−1.31255	−0.656276	+	0.754521i	$0.727869\pi$
$114$	0	0
$115$	2.03702	0.189953
$116$	−3.43331	−0.318774
$117$	0	0
$118$	−28.1668	−2.59296
$119$	17.7142	1.62386
$120$	0	0
$121$	1.79173	0.162884
$122$	10.3084	0.933275
$123$	0	0
$124$	15.6072	1.40157
$125$	11.9177	1.06595
$126$	0	0
$127$	9.48833	0.841953	0.420976	−	0.907072i	$-0.361687\pi$
$127$	9.48833	0.841953	0.420976	+	0.907072i	$0.361687\pi$
$128$	−19.8641	−1.75576
$129$	0	0
$130$	27.2456	2.38959
$131$	8.07820	0.705795	0.352898	−	0.935662i	$-0.385196\pi$
$131$	8.07820	0.705795	0.352898	+	0.935662i	$0.385196\pi$
$132$	0	0
$133$	−35.6157	−3.08827
$134$	−13.3609	−1.15421
$135$	0	0
$136$	13.8218	1.18521
$137$	−14.5574	−1.24372	−0.621861	−	0.783128i	$-0.713623\pi$
$137$	−14.5574	−1.24372	−0.621861	+	0.783128i	$0.713623\pi$
$138$	0	0
$139$	−16.1942	−1.37358	−0.686788	−	0.726858i	$-0.740980\pi$
$139$	−16.1942	−1.37358	−0.686788	+	0.726858i	$0.740980\pi$
$140$	−29.9458	−2.53088
$141$	0	0
$142$	21.2769	1.78552
$143$	20.5227	1.71619
$144$	0	0
$145$	2.03702	0.169165
$146$	−27.6405	−2.28754
$147$	0	0
$148$	−31.4374	−2.58414
$149$	−8.75292	−0.717067	−0.358533	−	0.933517i	$-0.616723\pi$
$149$	−8.75292	−0.717067	−0.358533	+	0.933517i	$0.616723\pi$
$150$	0	0
$151$	20.0492	1.63158	0.815791	−	0.578347i	$-0.196302\pi$
$151$	20.0492	1.63158	0.815791	+	0.578347i	$0.196302\pi$
$152$	−27.7897	−2.25404
$153$	0	0
$154$	−35.6965	−2.87651
$155$	−9.25991	−0.743774
$156$	0	0
$157$	−9.08819	−0.725316	−0.362658	−	0.931922i	$-0.618131\pi$
$157$	−9.08819	−0.725316	−0.362658	+	0.931922i	$0.618131\pi$
$158$	4.74783	0.377717
$159$	0	0
$160$	9.23821	0.730345
$161$	−4.28183	−0.337455
$162$	0	0
$163$	−11.1959	−0.876933	−0.438467	−	0.898747i	$-0.644478\pi$
$163$	−11.1959	−0.876933	−0.438467	+	0.898747i	$0.644478\pi$
$164$	−12.9517	−1.01136
$165$	0	0
$166$	5.85107	0.454131
$167$	−11.7353	−0.908102	−0.454051	−	0.890976i	$-0.650022\pi$
$167$	−11.7353	−0.908102	−0.454051	+	0.890976i	$0.650022\pi$
$168$	0	0
$169$	19.9260	1.53277
$170$	−19.6435	−1.50659
$171$	0	0
$172$	−14.1246	−1.07699
$173$	−20.2443	−1.53915	−0.769573	−	0.638558i	$-0.779531\pi$
$173$	−20.2443	−1.53915	−0.769573	+	0.638558i	$0.779531\pi$
$174$	0	0
$175$	−3.64198	−0.275308
$176$	−3.29392	−0.248288
$177$	0	0
$178$	23.3486	1.75005
$179$	8.16709	0.610437	0.305219	−	0.952282i	$-0.401270\pi$
$179$	8.16709	0.610437	0.305219	+	0.952282i	$0.401270\pi$
$180$	0	0
$181$	25.1168	1.86692	0.933459	−	0.358684i	$-0.116774\pi$
$181$	25.1168	1.86692	0.933459	+	0.358684i	$0.116774\pi$
$182$	−57.2705	−4.24517
$183$	0	0
$184$	−3.34096	−0.246299
$185$	18.6521	1.37133
$186$	0	0
$187$	−14.7964	−1.08202
$188$	21.3813	1.55939
$189$	0	0
$190$	39.4947	2.86525
$191$	24.6470	1.78340	0.891699	−	0.452629i	$-0.149514\pi$
$191$	24.6470	1.78340	0.891699	+	0.452629i	$0.149514\pi$
$192$	0	0
$193$	1.35264	0.0973648	0.0486824	−	0.998814i	$-0.484498\pi$
$193$	1.35264	0.0973648	0.0486824	+	0.998814i	$0.484498\pi$
$194$	1.72246	0.123666
$195$	0	0
$196$	38.9132	2.77952
$197$	6.03899	0.430260	0.215130	−	0.976585i	$-0.430982\pi$
$197$	6.03899	0.430260	0.215130	+	0.976585i	$0.430982\pi$
$198$	0	0
$199$	−21.5865	−1.53023	−0.765114	−	0.643895i	$-0.777317\pi$
$199$	−21.5865	−1.53023	−0.765114	+	0.643895i	$0.777317\pi$
$200$	−2.84171	−0.200939
$201$	0	0
$202$	33.0072	2.32238
$203$	−4.28183	−0.300525
$204$	0	0
$205$	7.68439	0.536701
$206$	−35.7983	−2.49418
$207$	0	0
$208$	−5.28467	−0.366426
$209$	29.7493	2.05780
$210$	0	0
$211$	−16.2713	−1.12016	−0.560081	−	0.828438i	$-0.689230\pi$
$211$	−16.2713	−1.12016	−0.560081	+	0.828438i	$0.689230\pi$
$212$	4.56524	0.313542
$213$	0	0
$214$	30.5366	2.08744
$215$	8.38029	0.571531
$216$	0	0
$217$	19.4644	1.32133
$218$	40.5864	2.74886
$219$	0	0
$220$	25.0133	1.68640
$221$	−23.7390	−1.59686
$222$	0	0
$223$	−10.7544	−0.720165	−0.360083	−	0.932920i	$-0.617252\pi$
$223$	−10.7544	−0.720165	−0.360083	+	0.932920i	$0.617252\pi$
$224$	−19.4188	−1.29747
$225$	0	0
$226$	−32.5228	−2.16338
$227$	−8.21789	−0.545441	−0.272720	−	0.962093i	$-0.587923\pi$
$227$	−8.21789	−0.545441	−0.272720	+	0.962093i	$0.587923\pi$
$228$	0	0
$229$	−28.0859	−1.85597	−0.927984	−	0.372620i	$-0.878460\pi$
$229$	−28.0859	−1.85597	−0.927984	+	0.372620i	$0.878460\pi$
$230$	4.74817	0.313085
$231$	0	0
$232$	−3.34096	−0.219345
$233$	−3.17124	−0.207755	−0.103878	−	0.994590i	$-0.533125\pi$
$233$	−3.17124	−0.207755	−0.103878	+	0.994590i	$0.533125\pi$
$234$	0	0
$235$	−12.6858	−0.827528
$236$	−41.4875	−2.70061
$237$	0	0
$238$	41.2908	2.67649
$239$	17.8380	1.15384	0.576921	−	0.816800i	$-0.304254\pi$
$239$	17.8380	1.15384	0.576921	+	0.816800i	$0.304254\pi$
$240$	0	0
$241$	−23.9855	−1.54504	−0.772522	−	0.634988i	$-0.781005\pi$
$241$	−23.9855	−1.54504	−0.772522	+	0.634988i	$0.781005\pi$
$242$	4.17642	0.268470
$243$	0	0
$244$	15.1834	0.972019
$245$	−23.0876	−1.47501
$246$	0	0
$247$	47.7290	3.03692
$248$	15.1874	0.964401
$249$	0	0
$250$	27.7795	1.75693
$251$	10.0692	0.635564	0.317782	−	0.948164i	$-0.397062\pi$
$251$	10.0692	0.635564	0.317782	+	0.948164i	$0.397062\pi$
$252$	0	0
$253$	3.57655	0.224856
$254$	22.1168	1.38773
$255$	0	0
$256$	−21.4758	−1.34224
$257$	7.58949	0.473419	0.236710	−	0.971580i	$-0.423931\pi$
$257$	7.58949	0.473419	0.236710	+	0.971580i	$0.423931\pi$
$258$	0	0
$259$	−39.2069	−2.43620
$260$	40.1307	2.48880
$261$	0	0
$262$	18.8298	1.16331
$263$	−12.8724	−0.793746	−0.396873	−	0.917874i	$-0.629905\pi$
$263$	−12.8724	−0.793746	−0.396873	+	0.917874i	$0.629905\pi$
$264$	0	0
$265$	−2.70861	−0.166388
$266$	−83.0183	−5.09018
$267$	0	0
$268$	−19.6796	−1.20213
$269$	16.0869	0.980835	0.490417	−	0.871488i	$-0.336844\pi$
$269$	16.0869	0.980835	0.490417	+	0.871488i	$0.336844\pi$
$270$	0	0
$271$	−5.79337	−0.351922	−0.175961	−	0.984397i	$-0.556303\pi$
$271$	−5.79337	−0.351922	−0.175961	+	0.984397i	$0.556303\pi$
$272$	3.81014	0.231023
$273$	0	0
$274$	−33.9325	−2.04993
$275$	3.04210	0.183445
$276$	0	0
$277$	4.05254	0.243494	0.121747	−	0.992561i	$-0.461150\pi$
$277$	4.05254	0.243494	0.121747	+	0.992561i	$0.461150\pi$
$278$	−37.7478	−2.26396
$279$	0	0
$280$	−29.1403	−1.74147
$281$	−30.6926	−1.83096	−0.915482	−	0.402359i	$-0.868190\pi$
$281$	−30.6926	−1.83096	−0.915482	+	0.402359i	$0.868190\pi$
$282$	0	0
$283$	25.6797	1.52650	0.763250	−	0.646103i	$-0.223602\pi$
$283$	25.6797	1.52650	0.763250	+	0.646103i	$0.223602\pi$
$284$	31.3393	1.85964
$285$	0	0
$286$	47.8372	2.82868
$287$	−16.1527	−0.953462
$288$	0	0
$289$	0.115308	0.00678284
$290$	4.74817	0.278822
$291$	0	0
$292$	−40.7123	−2.38251
$293$	16.9783	0.991886	0.495943	−	0.868355i	$-0.334823\pi$
$293$	16.9783	0.991886	0.495943	+	0.868355i	$0.334823\pi$
$294$	0	0
$295$	24.6150	1.43314
$296$	−30.5918	−1.77811
$297$	0	0
$298$	−20.4026	−1.18189
$299$	5.73812	0.331844
$300$	0	0
$301$	−17.6155	−1.01534
$302$	46.7336	2.68922
$303$	0	0
$304$	−7.66056	−0.439363
$305$	−9.00848	−0.515824
$306$	0	0
$307$	5.04995	0.288216	0.144108	−	0.989562i	$-0.453969\pi$
$307$	5.04995	0.288216	0.144108	+	0.989562i	$0.453969\pi$
$308$	−52.5783	−2.99592
$309$	0	0
$310$	−21.5843	−1.22591
$311$	−11.5885	−0.657124	−0.328562	−	0.944482i	$-0.606564\pi$
$311$	−11.5885	−0.657124	−0.328562	+	0.944482i	$0.606564\pi$
$312$	0	0
$313$	−17.4835	−0.988226	−0.494113	−	0.869398i	$-0.664507\pi$
$313$	−17.4835	−0.988226	−0.494113	+	0.869398i	$0.664507\pi$
$314$	−21.1841	−1.19549
$315$	0	0
$316$	6.99320	0.393398
$317$	17.8893	1.00476	0.502382	−	0.864646i	$-0.332457\pi$
$317$	17.8893	1.00476	0.502382	+	0.864646i	$0.332457\pi$
$318$	0	0
$319$	3.57655	0.200248
$320$	25.2858	1.41352
$321$	0	0
$322$	−9.98070	−0.556203
$323$	−34.4116	−1.91471
$324$	0	0
$325$	4.88066	0.270730
$326$	−26.0971	−1.44539
$327$	0	0
$328$	−12.6034	−0.695904
$329$	26.6656	1.47012
$330$	0	0
$331$	−4.66032	−0.256154	−0.128077	−	0.991764i	$-0.540880\pi$
$331$	−4.66032	−0.256154	−0.128077	+	0.991764i	$0.540880\pi$
$332$	8.61819	0.472985
$333$	0	0
$334$	−27.3543	−1.49676
$335$	11.6761	0.637935
$336$	0	0
$337$	−7.31254	−0.398339	−0.199170	−	0.979965i	$-0.563824\pi$
$337$	−7.31254	−0.398339	−0.199170	+	0.979965i	$0.563824\pi$
$338$	46.4464	2.52635
$339$	0	0
$340$	−28.9334	−1.56913
$341$	−16.2584	−0.880440
$342$	0	0
$343$	18.5576	1.00202
$344$	−13.7447	−0.741065
$345$	0	0
$346$	−47.1884	−2.53686
$347$	−27.3907	−1.47041	−0.735204	−	0.677846i	$-0.762914\pi$
$347$	−27.3907	−1.47041	−0.735204	+	0.677846i	$0.762914\pi$
$348$	0	0
$349$	20.2650	1.08476	0.542379	−	0.840134i	$-0.317524\pi$
$349$	20.2650	1.08476	0.542379	+	0.840134i	$0.317524\pi$
$350$	−8.48926	−0.453770
$351$	0	0
$352$	16.2203	0.864543
$353$	9.51815	0.506600	0.253300	−	0.967388i	$-0.418484\pi$
$353$	9.51815	0.506600	0.253300	+	0.967388i	$0.418484\pi$
$354$	0	0
$355$	−18.5939	−0.986863
$356$	34.3907	1.82271
$357$	0	0
$358$	19.0370	1.00614
$359$	8.98273	0.474090	0.237045	−	0.971499i	$-0.423821\pi$
$359$	8.98273	0.474090	0.237045	+	0.971499i	$0.423821\pi$
$360$	0	0
$361$	50.1871	2.64142
$362$	58.5459	3.07711
$363$	0	0
$364$	−84.3550	−4.42141
$365$	24.1550	1.26433
$366$	0	0
$367$	−25.2777	−1.31949	−0.659744	−	0.751491i	$-0.729335\pi$
$367$	−25.2777	−1.31949	−0.659744	+	0.751491i	$0.729335\pi$
$368$	−0.920976	−0.0480092
$369$	0	0
$370$	43.4771	2.26027
$371$	5.69352	0.295593
$372$	0	0
$373$	2.48700	0.128772	0.0643859	−	0.997925i	$-0.479491\pi$
$373$	2.48700	0.128772	0.0643859	+	0.997925i	$0.479491\pi$
$374$	−34.4897	−1.78342
$375$	0	0
$376$	20.8062	1.07300
$377$	5.73812	0.295528
$378$	0	0
$379$	32.0311	1.64533	0.822664	−	0.568528i	$-0.192487\pi$
$379$	32.0311	1.64533	0.822664	+	0.568528i	$0.192487\pi$
$380$	58.1727	2.98420
$381$	0	0
$382$	57.4509	2.93944
$383$	−0.611138	−0.0312277	−0.0156139	−	0.999878i	$-0.504970\pi$
$383$	−0.611138	−0.0312277	−0.0156139	+	0.999878i	$0.504970\pi$
$384$	0	0
$385$	31.1952	1.58986
$386$	3.15292	0.160479
$387$	0	0
$388$	2.53706	0.128800
$389$	20.0636	1.01727	0.508633	−	0.860984i	$-0.330151\pi$
$389$	20.0636	1.01727	0.508633	+	0.860984i	$0.330151\pi$
$390$	0	0
$391$	−4.13707	−0.209220
$392$	37.8665	1.91255
$393$	0	0
$394$	14.0765	0.709166
$395$	−4.14914	−0.208766
$396$	0	0
$397$	0.630252	0.0316314	0.0158157	−	0.999875i	$-0.494965\pi$
$397$	0.630252	0.0316314	0.0158157	+	0.999875i	$0.494965\pi$
$398$	−50.3170	−2.52216
$399$	0	0
$400$	−0.783352	−0.0391676
$401$	−23.2037	−1.15874	−0.579368	−	0.815066i	$-0.696701\pi$
$401$	−23.2037	−1.15874	−0.579368	+	0.815066i	$0.696701\pi$
$402$	0	0
$403$	−26.0845	−1.29936
$404$	48.6172	2.41879
$405$	0	0
$406$	−9.98070	−0.495334
$407$	32.7490	1.62331
$408$	0	0
$409$	13.6159	0.673265	0.336632	−	0.941636i	$-0.390712\pi$
$409$	13.6159	0.673265	0.336632	+	0.941636i	$0.390712\pi$
$410$	17.9119	0.884606
$411$	0	0
$412$	−52.7281	−2.59773
$413$	−51.7409	−2.54600
$414$	0	0
$415$	−5.11326	−0.251000
$416$	26.0233	1.27590
$417$	0	0
$418$	69.3440	3.39173
$419$	−29.5393	−1.44309	−0.721543	−	0.692369i	$-0.756567\pi$
$419$	−29.5393	−1.44309	−0.721543	+	0.692369i	$0.756567\pi$
$420$	0	0
$421$	10.1117	0.492816	0.246408	−	0.969166i	$-0.420750\pi$
$421$	10.1117	0.492816	0.246408	+	0.969166i	$0.420750\pi$
$422$	−37.9275	−1.84628
$423$	0	0
$424$	4.44245	0.215744
$425$	−3.51885	−0.170689
$426$	0	0
$427$	18.9359	0.916374
$428$	44.9780	2.17410
$429$	0	0
$430$	19.5340	0.942013
$431$	−0.288442	−0.0138937	−0.00694687	−	0.999976i	$-0.502211\pi$
$431$	−0.288442	−0.0138937	−0.00694687	+	0.999976i	$0.502211\pi$
$432$	0	0
$433$	−12.5475	−0.602993	−0.301496	−	0.953467i	$-0.597486\pi$
$433$	−12.5475	−0.602993	−0.301496	+	0.953467i	$0.597486\pi$
$434$	45.3705	2.17785
$435$	0	0
$436$	59.7807	2.86298
$437$	8.31788	0.397898
$438$	0	0
$439$	5.33735	0.254738	0.127369	−	0.991855i	$-0.459347\pi$
$439$	5.33735	0.254738	0.127369	+	0.991855i	$0.459347\pi$
$440$	24.3405	1.16039
$441$	0	0
$442$	−55.3342	−2.63198
$443$	22.8067	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$443$	22.8067	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$444$	0	0
$445$	−20.4044	−0.967260
$446$	−25.0678	−1.18700
$447$	0	0
$448$	−53.1511	−2.51115
$449$	23.8512	1.12561	0.562805	−	0.826590i	$-0.309722\pi$
$449$	23.8512	1.12561	0.562805	+	0.826590i	$0.309722\pi$
$450$	0	0
$451$	13.4921	0.635319
$452$	−47.9036	−2.25320
$453$	0	0
$454$	−19.1555	−0.899010
$455$	50.0487	2.34632
$456$	0	0
$457$	22.1886	1.03794	0.518969	−	0.854793i	$-0.326316\pi$
$457$	22.1886	1.03794	0.518969	+	0.854793i	$0.326316\pi$
$458$	−65.4667	−3.05906
$459$	0	0
$460$	6.99370	0.326083
$461$	−18.8126	−0.876192	−0.438096	−	0.898928i	$-0.644347\pi$
$461$	−18.8126	−0.876192	−0.438096	+	0.898928i	$0.644347\pi$
$462$	0	0
$463$	−5.42370	−0.252061	−0.126030	−	0.992026i	$-0.540224\pi$
$463$	−5.42370	−0.252061	−0.126030	+	0.992026i	$0.540224\pi$
$464$	−0.920976	−0.0427552
$465$	0	0
$466$	−7.39200	−0.342428
$467$	−16.8183	−0.778257	−0.389129	−	0.921183i	$-0.627224\pi$
$467$	−16.8183	−0.778257	−0.389129	+	0.921183i	$0.627224\pi$
$468$	0	0
$469$	−24.5434	−1.13331
$470$	−29.5698	−1.36395
$471$	0	0
$472$	−40.3716	−1.85825
$473$	14.7140	0.676548
$474$	0	0
$475$	7.07491	0.324619
$476$	60.8183	2.78760
$477$	0	0
$478$	41.5793	1.90179
$479$	−32.8574	−1.50129	−0.750647	−	0.660704i	$-0.770258\pi$
$479$	−32.8574	−1.50129	−0.750647	+	0.660704i	$0.770258\pi$
$480$	0	0
$481$	52.5416	2.39569
$482$	−55.9090	−2.54658
$483$	0	0
$484$	6.15155	0.279616
$485$	−1.50526	−0.0683504
$486$	0	0
$487$	2.35263	0.106608	0.0533039	−	0.998578i	$-0.483025\pi$
$487$	2.35263	0.106608	0.0533039	+	0.998578i	$0.483025\pi$
$488$	14.7750	0.668834
$489$	0	0
$490$	−53.8160	−2.43116
$491$	−14.0933	−0.636023	−0.318011	−	0.948087i	$-0.603015\pi$
$491$	−14.0933	−0.636023	−0.318011	+	0.948087i	$0.603015\pi$
$492$	0	0
$493$	−4.13707	−0.186324
$494$	111.254	5.00554
$495$	0	0
$496$	4.18659	0.187984
$497$	39.0846	1.75318
$498$	0	0
$499$	−7.21406	−0.322946	−0.161473	−	0.986877i	$-0.551624\pi$
$499$	−7.21406	−0.322946	−0.161473	+	0.986877i	$0.551624\pi$
$500$	40.9171	1.82987
$501$	0	0
$502$	23.4708	1.04755
$503$	−35.3702	−1.57708	−0.788541	−	0.614983i	$-0.789163\pi$
$503$	−35.3702	−1.57708	−0.788541	+	0.614983i	$0.789163\pi$
$504$	0	0
$505$	−28.8451	−1.28359
$506$	8.33675	0.370614
$507$	0	0
$508$	32.5763	1.44534
$509$	7.23766	0.320803	0.160402	−	0.987052i	$-0.448721\pi$
$509$	7.23766	0.320803	0.160402	+	0.987052i	$0.448721\pi$
$510$	0	0
$511$	−50.7741	−2.24612
$512$	−10.3307	−0.456555
$513$	0	0
$514$	17.6907	0.780303
$515$	31.2841	1.37854
$516$	0	0
$517$	−22.2734	−0.979584
$518$	−91.3892	−4.01541
$519$	0	0
$520$	39.0512	1.71251
$521$	−41.0518	−1.79851	−0.899256	−	0.437422i	$-0.855892\pi$
$521$	−41.0518	−1.79851	−0.899256	+	0.437422i	$0.855892\pi$
$522$	0	0
$523$	−29.1248	−1.27354	−0.636770	−	0.771054i	$-0.719730\pi$
$523$	−29.1248	−1.27354	−0.636770	+	0.771054i	$0.719730\pi$
$524$	27.7349	1.21161
$525$	0	0
$526$	−30.0048	−1.30827
$527$	18.8064	0.819218
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.31361	−0.274246
$531$	0	0
$532$	−122.280	−5.30149
$533$	21.6464	0.937608
$534$	0	0
$535$	−26.6859	−1.15373
$536$	−19.1503	−0.827167
$537$	0	0
$538$	37.4976	1.61664
$539$	−40.5368	−1.74604
$540$	0	0
$541$	26.1303	1.12343	0.561715	−	0.827331i	$-0.310142\pi$
$541$	26.1303	1.12343	0.561715	+	0.827331i	$0.310142\pi$
$542$	−13.5040	−0.580048
$543$	0	0
$544$	−18.7623	−0.804427
$545$	−35.4685	−1.51930
$546$	0	0
$547$	−25.3917	−1.08567	−0.542835	−	0.839839i	$-0.682649\pi$
$547$	−25.3917	−1.08567	−0.542835	+	0.839839i	$0.682649\pi$
$548$	−49.9799	−2.13504
$549$	0	0
$550$	7.09097	0.302360
$551$	8.31788	0.354353
$552$	0	0
$553$	8.72153	0.370877
$554$	9.44626	0.401333
$555$	0	0
$556$	−55.5997	−2.35795
$557$	29.2515	1.23943	0.619714	−	0.784828i	$-0.287249\pi$
$557$	29.2515	1.23943	0.619714	+	0.784828i	$0.287249\pi$
$558$	0	0
$559$	23.6067	0.998455
$560$	−8.03289	−0.339451
$561$	0	0
$562$	−71.5427	−3.01784
$563$	28.7525	1.21177	0.605886	−	0.795551i	$-0.292819\pi$
$563$	28.7525	1.21177	0.605886	+	0.795551i	$0.292819\pi$
$564$	0	0
$565$	28.4217	1.19571
$566$	59.8580	2.51602
$567$	0	0
$568$	30.4963	1.27960
$569$	13.9221	0.583647	0.291823	−	0.956472i	$-0.405738\pi$
$569$	13.9221	0.583647	0.291823	+	0.956472i	$0.405738\pi$
$570$	0	0
$571$	7.66013	0.320566	0.160283	−	0.987071i	$-0.448759\pi$
$571$	7.66013	0.320566	0.160283	+	0.987071i	$0.448759\pi$
$572$	70.4606	2.94611
$573$	0	0
$574$	−37.6510	−1.57152
$575$	0.850567	0.0354711
$576$	0	0
$577$	−28.4816	−1.18570	−0.592851	−	0.805312i	$-0.701998\pi$
$577$	−28.4816	−1.18570	−0.592851	+	0.805312i	$0.701998\pi$
$578$	0.268777	0.0111797
$579$	0	0
$580$	6.99370	0.290397
$581$	10.7481	0.445907
$582$	0	0
$583$	−4.75572	−0.196962
$584$	−39.6172	−1.63937
$585$	0	0
$586$	39.5756	1.63485
$587$	28.0018	1.15576	0.577879	−	0.816122i	$-0.303880\pi$
$587$	28.0018	1.15576	0.577879	+	0.816122i	$0.303880\pi$
$588$	0	0
$589$	−37.8116	−1.55800
$590$	57.3761	2.36214
$591$	0	0
$592$	−8.43300	−0.346594
$593$	37.7603	1.55063	0.775315	−	0.631575i	$-0.217591\pi$
$593$	37.7603	1.55063	0.775315	+	0.631575i	$0.217591\pi$
$594$	0	0
$595$	−36.0841	−1.47930
$596$	−30.0514	−1.23095
$597$	0	0
$598$	13.3752	0.546954
$599$	−15.7412	−0.643168	−0.321584	−	0.946881i	$-0.604215\pi$
$599$	−15.7412	−0.643168	−0.321584	+	0.946881i	$0.604215\pi$
$600$	0	0
$601$	−17.6070	−0.718204	−0.359102	−	0.933298i	$-0.616917\pi$
$601$	−17.6070	−0.718204	−0.359102	+	0.933298i	$0.616917\pi$
$602$	−41.0607	−1.67351
$603$	0	0
$604$	68.8351	2.80086
$605$	−3.64978	−0.148385
$606$	0	0
$607$	33.3243	1.35259	0.676297	−	0.736629i	$-0.263584\pi$
$607$	33.3243	1.35259	0.676297	+	0.736629i	$0.263584\pi$
$608$	37.7230	1.52987
$609$	0	0
$610$	−20.9983	−0.850196
$611$	−35.7348	−1.44568
$612$	0	0
$613$	−19.6103	−0.792053	−0.396027	−	0.918239i	$-0.629611\pi$
$613$	−19.6103	−0.792053	−0.396027	+	0.918239i	$0.629611\pi$
$614$	11.7712	0.475046
$615$	0	0
$616$	−51.1640	−2.06146
$617$	1.16373	0.0468500	0.0234250	−	0.999726i	$-0.492543\pi$
$617$	1.16373	0.0468500	0.0234250	+	0.999726i	$0.492543\pi$
$618$	0	0
$619$	−21.1267	−0.849154	−0.424577	−	0.905392i	$-0.639577\pi$
$619$	−21.1267	−0.849154	−0.424577	+	0.905392i	$0.639577\pi$
$620$	−31.7921	−1.27680
$621$	0	0
$622$	−27.0122	−1.08309
$623$	42.8902	1.71836
$624$	0	0
$625$	−20.0237	−0.800948
$626$	−40.7531	−1.62882
$627$	0	0
$628$	−31.2025	−1.24512
$629$	−37.8814	−1.51043
$630$	0	0
$631$	22.5717	0.898565	0.449282	−	0.893390i	$-0.351680\pi$
$631$	22.5717	0.898565	0.449282	+	0.893390i	$0.351680\pi$
$632$	6.80510	0.270692
$633$	0	0
$634$	41.6990	1.65608
$635$	−19.3279	−0.767003
$636$	0	0
$637$	−65.0361	−2.57682
$638$	8.33675	0.330055
$639$	0	0
$640$	40.4635	1.59946
$641$	−1.63884	−0.0647303	−0.0323651	−	0.999476i	$-0.510304\pi$
$641$	−1.63884	−0.0647303	−0.0323651	+	0.999476i	$0.510304\pi$
$642$	0	0
$643$	−17.0563	−0.672634	−0.336317	−	0.941749i	$-0.609181\pi$
$643$	−17.0563	−0.672634	−0.336317	+	0.941749i	$0.609181\pi$
$644$	−14.7008	−0.579293
$645$	0	0
$646$	−80.2115	−3.15588
$647$	−42.8199	−1.68342	−0.841712	−	0.539926i	$-0.818452\pi$
$647$	−42.8199	−1.68342	−0.841712	+	0.539926i	$0.818452\pi$
$648$	0	0
$649$	43.2185	1.69647
$650$	11.3765	0.446225
$651$	0	0
$652$	−38.4391	−1.50539
$653$	13.5203	0.529091	0.264545	−	0.964373i	$-0.414778\pi$
$653$	13.5203	0.529091	0.264545	+	0.964373i	$0.414778\pi$
$654$	0	0
$655$	−16.4554	−0.642966
$656$	−3.47427	−0.135647
$657$	0	0
$658$	62.1561	2.42310
$659$	3.50630	0.136586	0.0682929	−	0.997665i	$-0.478245\pi$
$659$	3.50630	0.136586	0.0682929	+	0.997665i	$0.478245\pi$
$660$	0	0
$661$	−24.3043	−0.945328	−0.472664	−	0.881243i	$-0.656708\pi$
$661$	−24.3043	−0.945328	−0.472664	+	0.881243i	$0.656708\pi$
$662$	−10.8629	−0.422200
$663$	0	0
$664$	8.38638	0.325455
$665$	72.5498	2.81336
$666$	0	0
$667$	1.00000	0.0387202
$668$	−40.2907	−1.55890
$669$	0	0
$670$	27.2164	1.05146
$671$	−15.8169	−0.610606
$672$	0	0
$673$	−8.49381	−0.327412	−0.163706	−	0.986509i	$-0.552345\pi$
$673$	−8.49381	−0.327412	−0.163706	+	0.986509i	$0.552345\pi$
$674$	−17.0451	−0.656553
$675$	0	0
$676$	68.4120	2.63123
$677$	6.22294	0.239167	0.119584	−	0.992824i	$-0.461844\pi$
$677$	6.22294	0.239167	0.119584	+	0.992824i	$0.461844\pi$
$678$	0	0
$679$	3.16408	0.121426
$680$	−28.1551	−1.07970
$681$	0	0
$682$	−37.8974	−1.45117
$683$	24.5440	0.939152	0.469576	−	0.882892i	$-0.344407\pi$
$683$	24.5440	0.939152	0.469576	+	0.882892i	$0.344407\pi$
$684$	0	0
$685$	29.6536	1.13301
$686$	43.2568	1.65155
$687$	0	0
$688$	−3.78890	−0.144450
$689$	−7.62994	−0.290678
$690$	0	0
$691$	−13.7582	−0.523386	−0.261693	−	0.965151i	$-0.584281\pi$
$691$	−13.7582	−0.523386	−0.261693	+	0.965151i	$0.584281\pi$
$692$	−69.5049	−2.64218
$693$	0	0
$694$	−63.8462	−2.42357
$695$	32.9879	1.25130
$696$	0	0
$697$	−15.6066	−0.591141
$698$	47.2365	1.78793
$699$	0	0
$700$	−12.5040	−0.472608
$701$	−38.9004	−1.46925	−0.734625	−	0.678474i	$-0.762642\pi$
$701$	−38.9004	−1.46925	−0.734625	+	0.678474i	$0.762642\pi$
$702$	0	0
$703$	76.1634	2.87256
$704$	44.3964	1.67325
$705$	0	0
$706$	22.1863	0.834991
$707$	60.6326	2.28032
$708$	0	0
$709$	49.1600	1.84624	0.923121	−	0.384510i	$-0.125629\pi$
$709$	49.1600	1.84624	0.923121	+	0.384510i	$0.125629\pi$
$710$	−43.3414	−1.62657
$711$	0	0
$712$	33.4657	1.25418
$713$	−4.54582	−0.170242
$714$	0	0
$715$	−41.8050	−1.56342
$716$	28.0401	1.04791
$717$	0	0
$718$	20.9382	0.781408
$719$	26.3301	0.981947	0.490973	−	0.871175i	$-0.336641\pi$
$719$	26.3301	0.981947	0.490973	+	0.871175i	$0.336641\pi$
$720$	0	0
$721$	−65.7596	−2.44901
$722$	116.983	4.35367
$723$	0	0
$724$	86.2337	3.20485
$725$	0.850567	0.0315893
$726$	0	0
$727$	30.5986	1.13484	0.567420	−	0.823428i	$-0.307942\pi$
$727$	30.5986	1.13484	0.567420	+	0.823428i	$0.307942\pi$
$728$	−82.0861	−3.04231
$729$	0	0
$730$	56.3041	2.08391
$731$	−17.0199	−0.629504
$732$	0	0
$733$	−47.0714	−1.73862	−0.869311	−	0.494266i	$-0.835437\pi$
$733$	−47.0714	−1.73862	−0.869311	+	0.494266i	$0.835437\pi$
$734$	−58.9210	−2.17482
$735$	0	0
$736$	4.53517	0.167169
$737$	20.5007	0.755154
$738$	0	0
$739$	−35.4173	−1.30285	−0.651424	−	0.758714i	$-0.725828\pi$
$739$	−35.4173	−1.30285	−0.651424	+	0.758714i	$0.725828\pi$
$740$	64.0384	2.35410
$741$	0	0
$742$	13.2713	0.487204
$743$	11.1959	0.410738	0.205369	−	0.978685i	$-0.434161\pi$
$743$	11.1959	0.410738	0.205369	+	0.978685i	$0.434161\pi$
$744$	0	0
$745$	17.8298	0.653234
$746$	5.79706	0.212245
$747$	0	0
$748$	−50.8007	−1.85746
$749$	56.0941	2.04963
$750$	0	0
$751$	−19.4942	−0.711352	−0.355676	−	0.934609i	$-0.615749\pi$
$751$	−19.4942	−0.711352	−0.355676	+	0.934609i	$0.615749\pi$
$752$	5.73549	0.209152
$753$	0	0
$754$	13.3752	0.487097
$755$	−40.8406	−1.48634
$756$	0	0
$757$	34.4467	1.25199	0.625993	−	0.779829i	$-0.284694\pi$
$757$	34.4467	1.25199	0.625993	+	0.779829i	$0.284694\pi$
$758$	74.6628	2.71187
$759$	0	0
$760$	56.6080	2.05339
$761$	26.7158	0.968447	0.484223	−	0.874944i	$-0.339102\pi$
$761$	26.7158	0.968447	0.484223	+	0.874944i	$0.339102\pi$
$762$	0	0
$763$	74.5551	2.69908
$764$	84.6208	3.06147
$765$	0	0
$766$	−1.42453	−0.0514704
$767$	69.3385	2.50367
$768$	0	0
$769$	23.6647	0.853372	0.426686	−	0.904400i	$-0.359681\pi$
$769$	23.6647	0.853372	0.426686	+	0.904400i	$0.359681\pi$
$770$	72.7144	2.62044
$771$	0	0
$772$	4.64401	0.167142
$773$	49.9240	1.79564	0.897822	−	0.440359i	$-0.145149\pi$
$773$	49.9240	1.79564	0.897822	+	0.440359i	$0.145149\pi$
$774$	0	0
$775$	−3.86653	−0.138890
$776$	2.46882	0.0886253
$777$	0	0
$778$	46.7672	1.67668
$779$	31.3782	1.12424
$780$	0	0
$781$	−32.6468	−1.16820
$782$	−9.64327	−0.344843
$783$	0	0
$784$	10.4384	0.372799
$785$	18.5128	0.660749
$786$	0	0
$787$	−7.48300	−0.266740	−0.133370	−	0.991066i	$-0.542580\pi$
$787$	−7.48300	−0.266740	−0.133370	+	0.991066i	$0.542580\pi$
$788$	20.7337	0.738607
$789$	0	0
$790$	−9.67141	−0.344093
$791$	−59.7427	−2.12421
$792$	0	0
$793$	−25.3762	−0.901136
$794$	1.46908	0.0521358
$795$	0	0
$796$	−74.1131	−2.62687
$797$	−36.0709	−1.27770	−0.638849	−	0.769332i	$-0.720589\pi$
$797$	−36.0709	−1.27770	−0.638849	+	0.769332i	$0.720589\pi$
$798$	0	0
$799$	25.7641	0.911468
$800$	3.85747	0.136382
$801$	0	0
$802$	−54.0865	−1.90986
$803$	42.4110	1.49665
$804$	0	0
$805$	8.72215	0.307415
$806$	−60.8015	−2.14164
$807$	0	0
$808$	47.3095	1.66434
$809$	15.9545	0.560929	0.280465	−	0.959864i	$-0.409511\pi$
$809$	15.9545	0.560929	0.280465	+	0.959864i	$0.409511\pi$
$810$	0	0
$811$	36.2089	1.27147	0.635733	−	0.771909i	$-0.280698\pi$
$811$	36.2089	1.27147	0.635733	+	0.771909i	$0.280698\pi$
$812$	−14.7008	−0.515898
$813$	0	0
$814$	76.3362	2.67558
$815$	22.8063	0.798870
$816$	0	0
$817$	34.2198	1.19720
$818$	31.7380	1.10969
$819$	0	0
$820$	26.3829	0.921330
$821$	−5.97804	−0.208635	−0.104317	−	0.994544i	$-0.533266\pi$
$821$	−5.97804	−0.208635	−0.104317	+	0.994544i	$0.533266\pi$
$822$	0	0
$823$	21.2654	0.741264	0.370632	−	0.928780i	$-0.379141\pi$
$823$	21.2654	0.741264	0.370632	+	0.928780i	$0.379141\pi$
$824$	−51.3098	−1.78746
$825$	0	0
$826$	−120.605	−4.19639
$827$	−35.3932	−1.23074	−0.615371	−	0.788237i	$-0.710994\pi$
$827$	−35.3932	−1.23074	−0.615371	+	0.788237i	$0.710994\pi$
$828$	0	0
$829$	10.3204	0.358444	0.179222	−	0.983809i	$-0.442642\pi$
$829$	10.3204	0.358444	0.179222	+	0.983809i	$0.442642\pi$
$830$	−11.9187	−0.413705
$831$	0	0
$832$	71.2283	2.46940
$833$	46.8897	1.62463
$834$	0	0
$835$	23.9049	0.827264
$836$	102.139	3.53253
$837$	0	0
$838$	−68.8544	−2.37853
$839$	23.4456	0.809433	0.404717	−	0.914442i	$-0.367370\pi$
$839$	23.4456	0.809433	0.404717	+	0.914442i	$0.367370\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	23.5699	0.812272
$843$	0	0
$844$	−55.8643	−1.92293
$845$	−40.5896	−1.39632
$846$	0	0
$847$	7.67187	0.263608
$848$	1.22462	0.0420535
$849$	0	0
$850$	−8.20225	−0.281335
$851$	9.15659	0.313884
$852$	0	0
$853$	23.9444	0.819842	0.409921	−	0.912121i	$-0.365556\pi$
$853$	23.9444	0.819842	0.409921	+	0.912121i	$0.365556\pi$
$854$	44.1386	1.51039
$855$	0	0
$856$	43.7682	1.49597
$857$	−13.0836	−0.446927	−0.223464	−	0.974712i	$-0.571736\pi$
$857$	−13.0836	−0.446927	−0.223464	+	0.974712i	$0.571736\pi$
$858$	0	0
$859$	41.8025	1.42628	0.713141	−	0.701020i	$-0.247272\pi$
$859$	41.8025	1.42628	0.713141	+	0.701020i	$0.247272\pi$
$860$	28.7721	0.981121
$861$	0	0
$862$	−0.672342	−0.0229000
$863$	10.4612	0.356105	0.178052	−	0.984021i	$-0.443020\pi$
$863$	10.4612	0.356105	0.178052	+	0.984021i	$0.443020\pi$
$864$	0	0
$865$	41.2380	1.40213
$866$	−29.2474	−0.993869
$867$	0	0
$868$	66.8273	2.26827
$869$	−7.28498	−0.247126
$870$	0	0
$871$	32.8908	1.11446
$872$	58.1727	1.96998
$873$	0	0
$874$	19.3885	0.655826
$875$	51.0295	1.72511
$876$	0	0
$877$	28.4848	0.961862	0.480931	−	0.876758i	$-0.340299\pi$
$877$	28.4848	0.961862	0.480931	+	0.876758i	$0.340299\pi$
$878$	12.4411	0.419866
$879$	0	0
$880$	6.70976	0.226186
$881$	−38.9665	−1.31281	−0.656407	−	0.754407i	$-0.727925\pi$
$881$	−38.9665	−1.31281	−0.656407	+	0.754407i	$0.727925\pi$
$882$	0	0
$883$	10.6192	0.357365	0.178683	−	0.983907i	$-0.442816\pi$
$883$	10.6192	0.357365	0.178683	+	0.983907i	$0.442816\pi$
$884$	−81.5031	−2.74125
$885$	0	0
$886$	53.1612	1.78599
$887$	−0.468182	−0.0157200	−0.00786000	−	0.999969i	$-0.502502\pi$
$887$	−0.468182	−0.0157200	−0.00786000	+	0.999969i	$0.502502\pi$
$888$	0	0
$889$	40.6274	1.36260
$890$	−47.5615	−1.59426
$891$	0	0
$892$	−36.9230	−1.23627
$893$	−51.8006	−1.73344
$894$	0	0
$895$	−16.6365	−0.556097
$896$	−85.0547	−2.84148
$897$	0	0
$898$	55.5959	1.85526
$899$	−4.54582	−0.151612
$900$	0	0
$901$	5.50103	0.183266
$902$	31.4494	1.04715
$903$	0	0
$904$	−46.6151	−1.55040
$905$	−51.1633	−1.70073
$906$	0	0
$907$	11.6922	0.388233	0.194117	−	0.980978i	$-0.437816\pi$
$907$	11.6922	0.388233	0.194117	+	0.980978i	$0.437816\pi$
$908$	−28.2145	−0.936332
$909$	0	0
$910$	116.661	3.86727
$911$	−29.7390	−0.985298	−0.492649	−	0.870228i	$-0.663971\pi$
$911$	−29.7390	−0.985298	−0.492649	+	0.870228i	$0.663971\pi$
$912$	0	0
$913$	−8.97776	−0.297121
$914$	51.7203	1.71076
$915$	0	0
$916$	−96.4275	−3.18605
$917$	34.5894	1.14224
$918$	0	0
$919$	−16.8566	−0.556048	−0.278024	−	0.960574i	$-0.589679\pi$
$919$	−16.8566	−0.556048	−0.278024	+	0.960574i	$0.589679\pi$
$920$	6.80558	0.224373
$921$	0	0
$922$	−43.8512	−1.44416
$923$	−52.3776	−1.72403
$924$	0	0
$925$	7.78830	0.256078
$926$	−12.6423	−0.415453
$927$	0	0
$928$	4.53517	0.148874
$929$	24.0401	0.788730	0.394365	−	0.918954i	$-0.370965\pi$
$929$	24.0401	0.788730	0.394365	+	0.918954i	$0.370965\pi$
$930$	0	0
$931$	−94.2752	−3.08974
$932$	−10.8878	−0.356643
$933$	0	0
$934$	−39.2025	−1.28275
$935$	30.1406	0.985702
$936$	0	0
$937$	5.39247	0.176164	0.0880821	−	0.996113i	$-0.471926\pi$
$937$	5.39247	0.176164	0.0880821	+	0.996113i	$0.471926\pi$
$938$	−57.2092	−1.86795
$939$	0	0
$940$	−43.5541	−1.42058
$941$	−8.82008	−0.287527	−0.143763	−	0.989612i	$-0.545920\pi$
$941$	−8.82008	−0.287527	−0.143763	+	0.989612i	$0.545920\pi$
$942$	0	0
$943$	3.77238	0.122846
$944$	−11.1289	−0.362215
$945$	0	0
$946$	34.2974	1.11511
$947$	9.44632	0.306964	0.153482	−	0.988151i	$-0.450951\pi$
$947$	9.44632	0.306964	0.153482	+	0.988151i	$0.450951\pi$
$948$	0	0
$949$	68.0429	2.20877
$950$	16.4912	0.535046
$951$	0	0
$952$	59.1824	1.91811
$953$	−52.6988	−1.70708	−0.853541	−	0.521025i	$-0.825550\pi$
$953$	−52.6988	−1.70708	−0.853541	+	0.521025i	$0.825550\pi$
$954$	0	0
$955$	−50.2064	−1.62464
$956$	61.2432	1.98075
$957$	0	0
$958$	−76.5888	−2.47447
$959$	−62.3322	−2.01281
$960$	0	0
$961$	−10.3355	−0.333403
$962$	122.472	3.94864
$963$	0	0
$964$	−82.3497	−2.65230
$965$	−2.75534	−0.0886975
$966$	0	0
$967$	−37.7635	−1.21439	−0.607197	−	0.794552i	$-0.707706\pi$
$967$	−37.7635	−1.21439	−0.607197	+	0.794552i	$0.707706\pi$
$968$	5.98608	0.192400
$969$	0	0
$970$	−3.50868	−0.112657
$971$	10.8940	0.349605	0.174803	−	0.984604i	$-0.444071\pi$
$971$	10.8940	0.349605	0.174803	+	0.984604i	$0.444071\pi$
$972$	0	0
$973$	−69.3408	−2.22297
$974$	5.48385	0.175714
$975$	0	0
$976$	4.07292	0.130371
$977$	57.4977	1.83951	0.919756	−	0.392489i	$-0.128386\pi$
$977$	57.4977	1.83951	0.919756	+	0.392489i	$0.128386\pi$
$978$	0	0
$979$	−35.8256	−1.14499
$980$	−79.2669	−2.53209
$981$	0	0
$982$	−32.8508	−1.04831
$983$	−48.8762	−1.55891	−0.779454	−	0.626459i	$-0.784504\pi$
$983$	−48.8762	−1.55891	−0.779454	+	0.626459i	$0.784504\pi$
$984$	0	0
$985$	−12.3015	−0.391959
$986$	−9.64327	−0.307104
$987$	0	0
$988$	163.868	5.21334
$989$	4.11401	0.130818
$990$	0	0
$991$	39.1297	1.24300	0.621498	−	0.783416i	$-0.286524\pi$
$991$	39.1297	1.24300	0.621498	+	0.783416i	$0.286524\pi$
$992$	−20.6161	−0.654561
$993$	0	0
$994$	91.1040	2.88964
$995$	43.9721	1.39401
$996$	0	0
$997$	2.04245	0.0646851	0.0323425	−	0.999477i	$-0.489703\pi$
$997$	2.04245	0.0646851	0.0323425	+	0.999477i	$0.489703\pi$
$998$	−16.8156	−0.532288
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.u.1.21	yes	22
3.2	odd	2		6003.2.a.t.1.2	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.2	✓	22	3.2	odd	2
6003.2.a.u.1.21	yes	22	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+2.33095 q^{2} +3.43331 q^{4} -2.03702 q^{5} +4.28183 q^{7} +3.34096 q^{8} -4.74817 q^{10} -3.57655 q^{11} -5.73812 q^{13} +9.98070 q^{14} +0.920976 q^{16} +4.13707 q^{17} -8.31788 q^{19} -6.99370 q^{20} -8.33675 q^{22} -1.00000 q^{23} -0.850567 q^{25} -13.3752 q^{26} +14.7008 q^{28} -1.00000 q^{29} +4.54582 q^{31} -4.53517 q^{32} +9.64327 q^{34} -8.72215 q^{35} -9.15659 q^{37} -19.3885 q^{38} -6.80558 q^{40} -3.77238 q^{41} -4.11401 q^{43} -12.2794 q^{44} -2.33095 q^{46} +6.22762 q^{47} +11.3340 q^{49} -1.98263 q^{50} -19.7007 q^{52} +1.32969 q^{53} +7.28549 q^{55} +14.3054 q^{56} -2.33095 q^{58} -12.0838 q^{59} +4.42239 q^{61} +10.5961 q^{62} -12.4132 q^{64} +11.6886 q^{65} -5.73198 q^{67} +14.2038 q^{68} -20.3309 q^{70} +9.12802 q^{71} -11.8581 q^{73} -21.3435 q^{74} -28.5578 q^{76} -15.3142 q^{77} +2.03687 q^{79} -1.87604 q^{80} -8.79321 q^{82} +2.51017 q^{83} -8.42727 q^{85} -9.58952 q^{86} -11.9491 q^{88} +10.0168 q^{89} -24.5696 q^{91} -3.43331 q^{92} +14.5163 q^{94} +16.9436 q^{95} +0.738955 q^{97} +26.4190 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 3 q^{2} + 17 q^{4} - 6 q^{7} + 6 q^{8} - 12 q^{10} - 28 q^{13} + q^{14} + 3 q^{16} + 10 q^{17} - 8 q^{19} - 11 q^{22} - 22 q^{23} - 11 q^{26} - 21 q^{28} - 22 q^{29} - 18 q^{31} - 5 q^{32} - 33 q^{34}+ \cdots + 28 q^{98}+O(q^{100}) \) 22 * q + 3 * q^2 + 17 * q^4 - 6 * q^7 + 6 * q^8 - 12 * q^10 - 28 * q^13 + q^14 + 3 * q^16 + 10 * q^17 - 8 * q^19 - 11 * q^22 - 22 * q^23 - 11 * q^26 - 21 * q^28 - 22 * q^29 - 18 * q^31 - 5 * q^32 - 33 * q^34 - 2 * q^35 - 28 * q^37 - 14 * q^38 - 30 * q^40 + 10 * q^41 - 14 * q^43 - 37 * q^44 - 3 * q^46 + 18 * q^47 + 2 * q^49 - 7 * q^50 - 57 * q^52 - 20 * q^53 - 42 * q^55 + 2 * q^56 - 3 * q^58 + 20 * q^59 - 38 * q^61 - 4 * q^62 - 24 * q^64 - 12 * q^65 - 50 * q^67 - 11 * q^68 - 48 * q^70 - 12 * q^71 - 46 * q^73 + 6 * q^74 - 16 * q^76 + 14 * q^77 - 20 * q^79 + 58 * q^80 - 42 * q^82 - 22 * q^83 - 66 * q^85 - 22 * q^86 - 68 * q^88 + 14 * q^89 - 16 * q^91 - 17 * q^92 - 27 * q^94 + 20 * q^95 - 48 * q^97 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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