LMFDB - Embedded newform 1629.2.a.b.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1629,2,Mod(1,1629)] 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("1629.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1629, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1629 = 3^{2} \cdot 181 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1629.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.0076304893$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.170701.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 4x + 1 $ x^5 - x^4 - 6x^3 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 543)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.89398$ of defining polynomial
Character	$\chi$	$=$	1629.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.34554 q^{2} -0.189509 q^{4} -4.08349 q^{5} +0.669650 q^{7} -2.94608 q^{8} -5.49452 q^{10} -1.08874 q^{11} +1.39765 q^{13} +0.901044 q^{14} -3.58507 q^{16} +2.39556 q^{17} +1.54030 q^{19} +0.773856 q^{20} -1.46495 q^{22} +5.64530 q^{23} +11.6749 q^{25} +1.88060 q^{26} -0.126904 q^{28} +5.54843 q^{29} -3.57800 q^{31} +1.06829 q^{32} +3.22333 q^{34} -2.73451 q^{35} +2.52980 q^{37} +2.07255 q^{38} +12.0303 q^{40} -6.18036 q^{41} +12.3660 q^{43} +0.206325 q^{44} +7.59601 q^{46} +9.28737 q^{47} -6.55157 q^{49} +15.7091 q^{50} -0.264866 q^{52} -3.33560 q^{53} +4.44585 q^{55} -1.97284 q^{56} +7.46567 q^{58} -5.41988 q^{59} +4.93270 q^{61} -4.81437 q^{62} +8.60757 q^{64} -5.70728 q^{65} -3.07018 q^{67} -0.453979 q^{68} -3.67940 q^{70} -4.02992 q^{71} +4.74454 q^{73} +3.40396 q^{74} -0.291901 q^{76} -0.729073 q^{77} -13.7984 q^{79} +14.6396 q^{80} -8.31595 q^{82} +6.47380 q^{83} -9.78224 q^{85} +16.6390 q^{86} +3.20751 q^{88} -3.68537 q^{89} +0.935935 q^{91} -1.06983 q^{92} +12.4966 q^{94} -6.28981 q^{95} +2.70412 q^{97} -8.81543 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 3 q^{4} - 3 q^{5} + 9 q^{7} - 9 q^{8} - 14 q^{10} + 4 q^{11} + 9 q^{13} - 3 q^{14} + 7 q^{16} - 9 q^{17} + 6 q^{19} + 25 q^{20} - 17 q^{22} + 5 q^{23} + 12 q^{25} - 6 q^{26} + 27 q^{28} + 20 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) $ 5 * q + q^2 + 3 * q^4 - 3 * q^5 + 9 * q^7 - 9 * q^8 - 14 * q^10 + 4 * q^11 + 9 * q^13 - 3 * q^14 + 7 * q^16 - 9 * q^17 + 6 * q^19 + 25 * q^20 - 17 * q^22 + 5 * q^23 + 12 * q^25 - 6 * q^26 + 27 * q^28 + 20 * q^29 + 13 * q^31 - 12 * q^32 + 22 * q^34 + 8 * q^35 - 5 * q^37 + 26 * q^38 - 13 * q^40 + 2 * q^41 + 4 * q^43 + 9 * q^44 - 20 * q^46 + 2 * q^47 + 50 * q^49 - 20 * q^50 + 14 * q^52 - 19 * q^53 + 11 * q^55 - 8 * q^56 - 3 * q^58 + 25 * q^59 + 17 * q^61 + 11 * q^62 - 11 * q^64 + 8 * q^65 - 9 * q^67 + 15 * q^68 - 30 * q^70 - 9 * q^71 + 7 * q^73 + 15 * q^74 - 15 * q^76 + 38 * q^77 + 2 * q^79 + 31 * q^80 + q^82 - 14 * q^83 - 3 * q^85 + 4 * q^86 - 15 * q^88 + 12 * q^89 - 5 * q^91 + 39 * q^92 + 12 * q^94 - 22 * q^95 - 8 * q^97 + 3 * 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 $

$2$	1.34554	0.951444	0.475722	−	0.879596i	$-0.342187\pi$
$2$	1.34554	0.951444	0.475722	+	0.879596i	$0.342187\pi$
$3$	0	0
$3$	0	0
$4$	−0.189509	−0.0947543
$5$	−4.08349	−1.82619	−0.913096	−	0.407745i	$-0.866315\pi$
$5$	−4.08349	−1.82619	−0.913096	+	0.407745i	$0.866315\pi$
$6$	0	0
$7$	0.669650	0.253104	0.126552	−	0.991960i	$-0.459609\pi$
$7$	0.669650	0.253104	0.126552	+	0.991960i	$0.459609\pi$
$8$	−2.94608	−1.04160
$9$	0	0
$10$	−5.49452	−1.73752
$11$	−1.08874	−0.328267	−0.164133	−	0.986438i	$-0.552483\pi$
$11$	−1.08874	−0.328267	−0.164133	+	0.986438i	$0.552483\pi$
$12$	0	0
$13$	1.39765	0.387638	0.193819	−	0.981037i	$-0.437913\pi$
$13$	1.39765	0.387638	0.193819	+	0.981037i	$0.437913\pi$
$14$	0.901044	0.240814
$15$	0	0
$16$	−3.58507	−0.896267
$17$	2.39556	0.581009	0.290504	−	0.956874i	$-0.406177\pi$
$17$	2.39556	0.581009	0.290504	+	0.956874i	$0.406177\pi$
$18$	0	0
$19$	1.54030	0.353370	0.176685	−	0.984267i	$-0.443463\pi$
$19$	1.54030	0.353370	0.176685	+	0.984267i	$0.443463\pi$
$20$	0.773856	0.173040
$21$	0	0
$22$	−1.46495	−0.312328
$23$	5.64530	1.17713	0.588563	−	0.808451i	$-0.299694\pi$
$23$	5.64530	1.17713	0.588563	+	0.808451i	$0.299694\pi$
$24$	0	0
$25$	11.6749	2.33497
$26$	1.88060	0.368816
$27$	0	0
$28$	−0.126904	−0.0239827
$29$	5.54843	1.03032	0.515159	−	0.857095i	$-0.327733\pi$
$29$	5.54843	1.03032	0.515159	+	0.857095i	$0.327733\pi$
$30$	0	0
$31$	−3.57800	−0.642629	−0.321314	−	0.946973i	$-0.604125\pi$
$31$	−3.57800	−0.642629	−0.321314	+	0.946973i	$0.604125\pi$
$32$	1.06829	0.188849
$33$	0	0
$34$	3.22333	0.552797
$35$	−2.73451	−0.462216
$36$	0	0
$37$	2.52980	0.415897	0.207949	−	0.978140i	$-0.433321\pi$
$37$	2.52980	0.415897	0.207949	+	0.978140i	$0.433321\pi$
$38$	2.07255	0.336212
$39$	0	0
$40$	12.0303	1.90216
$41$	−6.18036	−0.965209	−0.482605	−	0.875838i	$-0.660309\pi$
$41$	−6.18036	−0.965209	−0.482605	+	0.875838i	$0.660309\pi$
$42$	0	0
$43$	12.3660	1.88579	0.942896	−	0.333088i	$-0.108091\pi$
$43$	12.3660	1.88579	0.942896	+	0.333088i	$0.108091\pi$
$44$	0.206325	0.0311047
$45$	0	0
$46$	7.59601	1.11997
$47$	9.28737	1.35470	0.677351	−	0.735660i	$-0.263128\pi$
$47$	9.28737	1.35470	0.677351	+	0.735660i	$0.263128\pi$
$48$	0	0
$49$	−6.55157	−0.935938
$50$	15.7091	2.22160
$51$	0	0
$52$	−0.264866	−0.0367304
$53$	−3.33560	−0.458180	−0.229090	−	0.973405i	$-0.573575\pi$
$53$	−3.33560	−0.458180	−0.229090	+	0.973405i	$0.573575\pi$
$54$	0	0
$55$	4.44585	0.599478
$56$	−1.97284	−0.263632
$57$	0	0
$58$	7.46567	0.980290
$59$	−5.41988	−0.705608	−0.352804	−	0.935697i	$-0.614772\pi$
$59$	−5.41988	−0.705608	−0.352804	+	0.935697i	$0.614772\pi$
$60$	0	0
$61$	4.93270	0.631568	0.315784	−	0.948831i	$-0.397733\pi$
$61$	4.93270	0.631568	0.315784	+	0.948831i	$0.397733\pi$
$62$	−4.81437	−0.611425
$63$	0	0
$64$	8.60757	1.07595
$65$	−5.70728	−0.707901
$66$	0	0
$67$	−3.07018	−0.375082	−0.187541	−	0.982257i	$-0.560052\pi$
$67$	−3.07018	−0.375082	−0.187541	+	0.982257i	$0.560052\pi$
$68$	−0.453979	−0.0550531
$69$	0	0
$70$	−3.67940	−0.439773
$71$	−4.02992	−0.478264	−0.239132	−	0.970987i	$-0.576863\pi$
$71$	−4.02992	−0.478264	−0.239132	+	0.970987i	$0.576863\pi$
$72$	0	0
$73$	4.74454	0.555307	0.277653	−	0.960681i	$-0.410443\pi$
$73$	4.74454	0.555307	0.277653	+	0.960681i	$0.410443\pi$
$74$	3.40396	0.395703
$75$	0	0
$76$	−0.291901	−0.0334833
$77$	−0.729073	−0.0830856
$78$	0	0
$79$	−13.7984	−1.55244	−0.776222	−	0.630459i	$-0.782867\pi$
$79$	−13.7984	−1.55244	−0.776222	+	0.630459i	$0.782867\pi$
$80$	14.6396	1.63676
$81$	0	0
$82$	−8.31595	−0.918343
$83$	6.47380	0.710592	0.355296	−	0.934754i	$-0.384380\pi$
$83$	6.47380	0.710592	0.355296	+	0.934754i	$0.384380\pi$
$84$	0	0
$85$	−9.78224	−1.06103
$86$	16.6390	1.79423
$87$	0	0
$88$	3.20751	0.341922
$89$	−3.68537	−0.390649	−0.195324	−	0.980739i	$-0.562576\pi$
$89$	−3.68537	−0.390649	−0.195324	+	0.980739i	$0.562576\pi$
$90$	0	0
$91$	0.935935	0.0981126
$92$	−1.06983	−0.111538
$93$	0	0
$94$	12.4966	1.28892
$95$	−6.28981	−0.645321
$96$	0	0
$97$	2.70412	0.274561	0.137281	−	0.990532i	$-0.456164\pi$
$97$	2.70412	0.274561	0.137281	+	0.990532i	$0.456164\pi$
$98$	−8.81543	−0.890493
$99$	0	0
$100$	−2.21249	−0.221249
$101$	4.70285	0.467951	0.233975	−	0.972243i	$-0.424826\pi$
$101$	4.70285	0.467951	0.233975	+	0.972243i	$0.424826\pi$
$102$	0	0
$103$	3.56750	0.351517	0.175758	−	0.984433i	$-0.443762\pi$
$103$	3.56750	0.351517	0.175758	+	0.984433i	$0.443762\pi$
$104$	−4.11759	−0.403763
$105$	0	0
$106$	−4.48820	−0.435933
$107$	14.5489	1.40649	0.703246	−	0.710946i	$-0.251733\pi$
$107$	14.5489	1.40649	0.703246	+	0.710946i	$0.251733\pi$
$108$	0	0
$109$	19.3555	1.85392	0.926958	−	0.375164i	$-0.122414\pi$
$109$	19.3555	1.85392	0.926958	+	0.375164i	$0.122414\pi$
$110$	5.98209	0.570370
$111$	0	0
$112$	−2.40074	−0.226849
$113$	18.7806	1.76673	0.883365	−	0.468686i	$-0.155272\pi$
$113$	18.7806	1.76673	0.883365	+	0.468686i	$0.155272\pi$
$114$	0	0
$115$	−23.0525	−2.14966
$116$	−1.05148	−0.0976271
$117$	0	0
$118$	−7.29269	−0.671347
$119$	1.60419	0.147056
$120$	0	0
$121$	−9.81465	−0.892241
$122$	6.63717	0.600901
$123$	0	0
$124$	0.678063	0.0608918
$125$	−27.2568	−2.43792
$126$	0	0
$127$	−11.4442	−1.01551	−0.507755	−	0.861501i	$-0.669525\pi$
$127$	−11.4442	−1.01551	−0.507755	+	0.861501i	$0.669525\pi$
$128$	9.44529	0.834854
$129$	0	0
$130$	−7.67940	−0.673528
$131$	12.8950	1.12664	0.563320	−	0.826239i	$-0.309524\pi$
$131$	12.8950	1.12664	0.563320	+	0.826239i	$0.309524\pi$
$132$	0	0
$133$	1.03146	0.0894393
$134$	−4.13106	−0.356869
$135$	0	0
$136$	−7.05752	−0.605177
$137$	4.00363	0.342053	0.171027	−	0.985266i	$-0.445292\pi$
$137$	4.00363	0.342053	0.171027	+	0.985266i	$0.445292\pi$
$138$	0	0
$139$	−6.78922	−0.575854	−0.287927	−	0.957652i	$-0.592966\pi$
$139$	−6.78922	−0.575854	−0.287927	+	0.957652i	$0.592966\pi$
$140$	0.518213	0.0437970
$141$	0	0
$142$	−5.42244	−0.455041
$143$	−1.52167	−0.127249
$144$	0	0
$145$	−22.6570	−1.88156
$146$	6.38399	0.528343
$147$	0	0
$148$	−0.479420	−0.0394081
$149$	−5.98868	−0.490612	−0.245306	−	0.969446i	$-0.578888\pi$
$149$	−5.98868	−0.490612	−0.245306	+	0.969446i	$0.578888\pi$
$150$	0	0
$151$	−2.38661	−0.194220	−0.0971098	−	0.995274i	$-0.530960\pi$
$151$	−2.38661	−0.194220	−0.0971098	+	0.995274i	$0.530960\pi$
$152$	−4.53786	−0.368069
$153$	0	0
$154$	−0.981001	−0.0790513
$155$	14.6107	1.17356
$156$	0	0
$157$	4.38478	0.349943	0.174972	−	0.984573i	$-0.444017\pi$
$157$	4.38478	0.349943	0.174972	+	0.984573i	$0.444017\pi$
$158$	−18.5664	−1.47706
$159$	0	0
$160$	−4.36236	−0.344875
$161$	3.78038	0.297935
$162$	0	0
$163$	13.0225	1.02000	0.510001	−	0.860174i	$-0.329645\pi$
$163$	13.0225	1.02000	0.510001	+	0.860174i	$0.329645\pi$
$164$	1.17123	0.0914578
$165$	0	0
$166$	8.71079	0.676088
$167$	−1.06884	−0.0827092	−0.0413546	−	0.999145i	$-0.513167\pi$
$167$	−1.06884	−0.0827092	−0.0413546	+	0.999145i	$0.513167\pi$
$168$	0	0
$169$	−11.0466	−0.849737
$170$	−13.1624	−1.00951
$171$	0	0
$172$	−2.34346	−0.178687
$173$	8.05626	0.612506	0.306253	−	0.951950i	$-0.400925\pi$
$173$	8.05626	0.612506	0.306253	+	0.951950i	$0.400925\pi$
$174$	0	0
$175$	7.81808	0.590991
$176$	3.90320	0.294215
$177$	0	0
$178$	−4.95884	−0.371680
$179$	6.92281	0.517435	0.258718	−	0.965953i	$-0.416700\pi$
$179$	6.92281	0.517435	0.258718	+	0.965953i	$0.416700\pi$
$180$	0	0
$181$	−1.00000	−0.0743294
$181$	−1.00000	−0.0743294
$182$	1.25934	0.0933487
$183$	0	0
$184$	−16.6315	−1.22609
$185$	−10.3304	−0.759508
$186$	0	0
$187$	−2.60814	−0.190726
$188$	−1.76004	−0.128364
$189$	0	0
$190$	−8.46323	−0.613987
$191$	26.8490	1.94273	0.971363	−	0.237601i	$-0.0763610\pi$
$191$	26.8490	1.94273	0.971363	+	0.237601i	$0.0763610\pi$
$192$	0	0
$193$	19.6947	1.41765	0.708827	−	0.705382i	$-0.249224\pi$
$193$	19.6947	1.41765	0.708827	+	0.705382i	$0.249224\pi$
$194$	3.63851	0.261230
$195$	0	0
$196$	1.24158	0.0886842
$197$	−4.83087	−0.344185	−0.172092	−	0.985081i	$-0.555053\pi$
$197$	−4.83087	−0.344185	−0.172092	+	0.985081i	$0.555053\pi$
$198$	0	0
$199$	−12.1320	−0.860013	−0.430006	−	0.902826i	$-0.641489\pi$
$199$	−12.1320	−0.860013	−0.430006	+	0.902826i	$0.641489\pi$
$200$	−34.3951	−2.43210
$201$	0	0
$202$	6.32789	0.445229
$203$	3.71551	0.260778
$204$	0	0
$205$	25.2374	1.76266
$206$	4.80024	0.334448
$207$	0	0
$208$	−5.01067	−0.347427
$209$	−1.67699	−0.116000
$210$	0	0
$211$	−11.1701	−0.768984	−0.384492	−	0.923128i	$-0.625623\pi$
$211$	−11.1701	−0.768984	−0.384492	+	0.923128i	$0.625623\pi$
$212$	0.632125	0.0434145
$213$	0	0
$214$	19.5762	1.33820
$215$	−50.4963	−3.44382
$216$	0	0
$217$	−2.39601	−0.162652
$218$	26.0436	1.76390
$219$	0	0
$220$	−0.842527	−0.0568032
$221$	3.34815	0.225221
$222$	0	0
$223$	−12.7690	−0.855078	−0.427539	−	0.903997i	$-0.640619\pi$
$223$	−12.7690	−0.855078	−0.427539	+	0.903997i	$0.640619\pi$
$224$	0.715382	0.0477985
$225$	0	0
$226$	25.2701	1.68094
$227$	−25.8466	−1.71550	−0.857750	−	0.514068i	$-0.828138\pi$
$227$	−25.8466	−1.71550	−0.857750	+	0.514068i	$0.828138\pi$
$228$	0	0
$229$	13.3599	0.882848	0.441424	−	0.897299i	$-0.354473\pi$
$229$	13.3599	0.882848	0.441424	+	0.897299i	$0.354473\pi$
$230$	−31.0182	−2.04528
$231$	0	0
$232$	−16.3461	−1.07318
$233$	−20.4320	−1.33854	−0.669272	−	0.743017i	$-0.733394\pi$
$233$	−20.4320	−1.33854	−0.669272	+	0.743017i	$0.733394\pi$
$234$	0	0
$235$	−37.9249	−2.47395
$236$	1.02711	0.0668594
$237$	0	0
$238$	2.15851	0.139915
$239$	−1.89949	−0.122868	−0.0614340	−	0.998111i	$-0.519567\pi$
$239$	−1.89949	−0.122868	−0.0614340	+	0.998111i	$0.519567\pi$
$240$	0	0
$241$	4.83473	0.311432	0.155716	−	0.987802i	$-0.450231\pi$
$241$	4.83473	0.311432	0.155716	+	0.987802i	$0.450231\pi$
$242$	−13.2061	−0.848917
$243$	0	0
$244$	−0.934790	−0.0598438
$245$	26.7533	1.70920
$246$	0	0
$247$	2.15280	0.136980
$248$	10.5411	0.669360
$249$	0	0
$250$	−36.6752	−2.31954
$251$	−12.2118	−0.770804	−0.385402	−	0.922749i	$-0.625937\pi$
$251$	−12.2118	−0.770804	−0.385402	+	0.922749i	$0.625937\pi$
$252$	0	0
$253$	−6.14626	−0.386412
$254$	−15.3987	−0.966202
$255$	0	0
$256$	−4.50608	−0.281630
$257$	29.5345	1.84231	0.921155	−	0.389197i	$-0.127247\pi$
$257$	29.5345	1.84231	0.921155	+	0.389197i	$0.127247\pi$
$258$	0	0
$259$	1.69408	0.105265
$260$	1.08158	0.0670767
$261$	0	0
$262$	17.3508	1.07193
$263$	−29.3589	−1.81035	−0.905173	−	0.425043i	$-0.860259\pi$
$263$	−29.3589	−1.81035	−0.905173	+	0.425043i	$0.860259\pi$
$264$	0	0
$265$	13.6209	0.836724
$266$	1.38788	0.0850965
$267$	0	0
$268$	0.581825	0.0355406
$269$	5.39718	0.329072	0.164536	−	0.986371i	$-0.447387\pi$
$269$	5.39718	0.329072	0.164536	+	0.986371i	$0.447387\pi$
$270$	0	0
$271$	−7.08604	−0.430446	−0.215223	−	0.976565i	$-0.569048\pi$
$271$	−7.08604	−0.430446	−0.215223	+	0.976565i	$0.569048\pi$
$272$	−8.58825	−0.520739
$273$	0	0
$274$	5.38706	0.325444
$275$	−12.7109	−0.766495
$276$	0	0
$277$	20.9729	1.26014	0.630071	−	0.776538i	$-0.283026\pi$
$277$	20.9729	1.26014	0.630071	+	0.776538i	$0.283026\pi$
$278$	−9.13520	−0.547893
$279$	0	0
$280$	8.05608	0.481443
$281$	14.1393	0.843478	0.421739	−	0.906717i	$-0.361420\pi$
$281$	14.1393	0.843478	0.421739	+	0.906717i	$0.361420\pi$
$282$	0	0
$283$	30.0104	1.78393	0.891966	−	0.452103i	$-0.149326\pi$
$283$	30.0104	1.78393	0.891966	+	0.452103i	$0.149326\pi$
$284$	0.763705	0.0453176
$285$	0	0
$286$	−2.04748	−0.121070
$287$	−4.13867	−0.244298
$288$	0	0
$289$	−11.2613	−0.662429
$290$	−30.4860	−1.79020
$291$	0	0
$292$	−0.899132	−0.0526177
$293$	−18.4790	−1.07956	−0.539779	−	0.841807i	$-0.681492\pi$
$293$	−18.4790	−1.07956	−0.539779	+	0.841807i	$0.681492\pi$
$294$	0	0
$295$	22.1320	1.28858
$296$	−7.45301	−0.433197
$297$	0	0
$298$	−8.05804	−0.466790
$299$	7.89015	0.456299
$300$	0	0
$301$	8.28086	0.477301
$302$	−3.21129	−0.184789
$303$	0	0
$304$	−5.52210	−0.316714
$305$	−20.1426	−1.15336
$306$	0	0
$307$	17.6509	1.00739	0.503696	−	0.863881i	$-0.331973\pi$
$307$	17.6509	1.00739	0.503696	+	0.863881i	$0.331973\pi$
$308$	0.138166	0.00787272
$309$	0	0
$310$	19.6594	1.11658
$311$	24.3424	1.38033	0.690164	−	0.723653i	$-0.257538\pi$
$311$	24.3424	1.38033	0.690164	+	0.723653i	$0.257538\pi$
$312$	0	0
$313$	−22.4588	−1.26945	−0.634725	−	0.772738i	$-0.718886\pi$
$313$	−22.4588	−1.26945	−0.634725	+	0.772738i	$0.718886\pi$
$314$	5.89991	0.332951
$315$	0	0
$316$	2.61492	0.147101
$317$	13.7599	0.772835	0.386417	−	0.922324i	$-0.373712\pi$
$317$	13.7599	0.772835	0.386417	+	0.922324i	$0.373712\pi$
$318$	0	0
$319$	−6.04079	−0.338219
$320$	−35.1489	−1.96488
$321$	0	0
$322$	5.08667	0.283469
$323$	3.68989	0.205311
$324$	0	0
$325$	16.3174	0.905125
$326$	17.5224	0.970475
$327$	0	0
$328$	18.2078	1.00536
$329$	6.21929	0.342880
$330$	0	0
$331$	27.9101	1.53408	0.767040	−	0.641600i	$-0.221729\pi$
$331$	27.9101	1.53408	0.767040	+	0.641600i	$0.221729\pi$
$332$	−1.22684	−0.0673316
$333$	0	0
$334$	−1.43817	−0.0786932
$335$	12.5370	0.684971
$336$	0	0
$337$	28.6615	1.56129	0.780645	−	0.624974i	$-0.214890\pi$
$337$	28.6615	1.56129	0.780645	+	0.624974i	$0.214890\pi$
$338$	−14.8637	−0.808477
$339$	0	0
$340$	1.85382	0.100537
$341$	3.89551	0.210954
$342$	0	0
$343$	−9.07481	−0.489993
$344$	−36.4311	−1.96424
$345$	0	0
$346$	10.8401	0.582765
$347$	10.2561	0.550575	0.275287	−	0.961362i	$-0.411227\pi$
$347$	10.2561	0.550575	0.275287	+	0.961362i	$0.411227\pi$
$348$	0	0
$349$	−6.11407	−0.327279	−0.163639	−	0.986520i	$-0.552323\pi$
$349$	−6.11407	−0.327279	−0.163639	+	0.986520i	$0.552323\pi$
$350$	10.5196	0.562295
$351$	0	0
$352$	−1.16309	−0.0619930
$353$	22.2256	1.18295	0.591474	−	0.806324i	$-0.298546\pi$
$353$	22.2256	1.18295	0.591474	+	0.806324i	$0.298546\pi$
$354$	0	0
$355$	16.4561	0.873401
$356$	0.698410	0.0370157
$357$	0	0
$358$	9.31495	0.492311
$359$	23.0110	1.21447	0.607237	−	0.794521i	$-0.292278\pi$
$359$	23.0110	1.21447	0.607237	+	0.794521i	$0.292278\pi$
$360$	0	0
$361$	−16.6275	−0.875130
$362$	−1.34554	−0.0707203
$363$	0	0
$364$	−0.177368	−0.00929660
$365$	−19.3743	−1.01410
$366$	0	0
$367$	−35.7818	−1.86779	−0.933897	−	0.357543i	$-0.883615\pi$
$367$	−35.7818	−1.86779	−0.933897	+	0.357543i	$0.883615\pi$
$368$	−20.2388	−1.05502
$369$	0	0
$370$	−13.9000	−0.722629
$371$	−2.23368	−0.115967
$372$	0	0
$373$	16.7186	0.865655	0.432827	−	0.901477i	$-0.357516\pi$
$373$	16.7186	0.865655	0.432827	+	0.901477i	$0.357516\pi$
$374$	−3.50937	−0.181465
$375$	0	0
$376$	−27.3614	−1.41105
$377$	7.75476	0.399390
$378$	0	0
$379$	10.2247	0.525206	0.262603	−	0.964904i	$-0.415419\pi$
$379$	10.2247	0.525206	0.262603	+	0.964904i	$0.415419\pi$
$380$	1.19197	0.0611470
$381$	0	0
$382$	36.1265	1.84839
$383$	−25.6793	−1.31215	−0.656075	−	0.754696i	$-0.727784\pi$
$383$	−25.6793	−1.31215	−0.656075	+	0.754696i	$0.727784\pi$
$384$	0	0
$385$	2.97716	0.151730
$386$	26.5001	1.34882
$387$	0	0
$388$	−0.512454	−0.0260159
$389$	−12.8963	−0.653868	−0.326934	−	0.945047i	$-0.606016\pi$
$389$	−12.8963	−0.653868	−0.326934	+	0.945047i	$0.606016\pi$
$390$	0	0
$391$	13.5237	0.683921
$392$	19.3015	0.974871
$393$	0	0
$394$	−6.50015	−0.327473
$395$	56.3457	2.83506
$396$	0	0
$397$	−15.2937	−0.767569	−0.383784	−	0.923423i	$-0.625379\pi$
$397$	−15.2937	−0.767569	−0.383784	+	0.923423i	$0.625379\pi$
$398$	−16.3241	−0.818254
$399$	0	0
$400$	−41.8552	−2.09276
$401$	35.5621	1.77589	0.887943	−	0.459954i	$-0.152134\pi$
$401$	35.5621	1.77589	0.887943	+	0.459954i	$0.152134\pi$
$402$	0	0
$403$	−5.00079	−0.249107
$404$	−0.891231	−0.0443404
$405$	0	0
$406$	4.99938	0.248115
$407$	−2.75429	−0.136525
$408$	0	0
$409$	−26.1340	−1.29224	−0.646121	−	0.763235i	$-0.723610\pi$
$409$	−26.1340	−1.29224	−0.646121	+	0.763235i	$0.723610\pi$
$410$	33.9581	1.67707
$411$	0	0
$412$	−0.676073	−0.0333077
$413$	−3.62942	−0.178592
$414$	0	0
$415$	−26.4357	−1.29768
$416$	1.49310	0.0732052
$417$	0	0
$418$	−2.25646	−0.110367
$419$	6.43693	0.314464	0.157232	−	0.987562i	$-0.449743\pi$
$419$	6.43693	0.314464	0.157232	+	0.987562i	$0.449743\pi$
$420$	0	0
$421$	−37.1347	−1.80984	−0.904918	−	0.425586i	$-0.860068\pi$
$421$	−37.1347	−1.80984	−0.904918	+	0.425586i	$0.860068\pi$
$422$	−15.0299	−0.731645
$423$	0	0
$424$	9.82695	0.477239
$425$	27.9679	1.35664
$426$	0	0
$427$	3.30318	0.159852
$428$	−2.75714	−0.133271
$429$	0	0
$430$	−67.9450	−3.27660
$431$	12.9880	0.625608	0.312804	−	0.949818i	$-0.398732\pi$
$431$	12.9880	0.625608	0.312804	+	0.949818i	$0.398732\pi$
$432$	0	0
$433$	32.9228	1.58217	0.791085	−	0.611707i	$-0.209517\pi$
$433$	32.9228	1.58217	0.791085	+	0.611707i	$0.209517\pi$
$434$	−3.22394	−0.154754
$435$	0	0
$436$	−3.66803	−0.175667
$437$	8.69548	0.415961
$438$	0	0
$439$	−27.3876	−1.30714	−0.653568	−	0.756867i	$-0.726729\pi$
$439$	−27.3876	−1.30714	−0.653568	+	0.756867i	$0.726729\pi$
$440$	−13.0978	−0.624415
$441$	0	0
$442$	4.50509	0.214285
$443$	17.5116	0.832001	0.416001	−	0.909364i	$-0.363431\pi$
$443$	17.5116	0.832001	0.416001	+	0.909364i	$0.363431\pi$
$444$	0	0
$445$	15.0492	0.713399
$446$	−17.1813	−0.813559
$447$	0	0
$448$	5.76406	0.272326
$449$	−41.2447	−1.94646	−0.973229	−	0.229837i	$-0.926181\pi$
$449$	−41.2447	−1.94646	−0.973229	+	0.229837i	$0.926181\pi$
$450$	0	0
$451$	6.72879	0.316846
$452$	−3.55909	−0.167405
$453$	0	0
$454$	−34.7778	−1.63220
$455$	−3.82188	−0.179172
$456$	0	0
$457$	5.02878	0.235236	0.117618	−	0.993059i	$-0.462474\pi$
$457$	5.02878	0.235236	0.117618	+	0.993059i	$0.462474\pi$
$458$	17.9764	0.839981
$459$	0	0
$460$	4.36865	0.203690
$461$	34.5652	1.60986	0.804932	−	0.593367i	$-0.202202\pi$
$461$	34.5652	1.60986	0.804932	+	0.593367i	$0.202202\pi$
$462$	0	0
$463$	10.8253	0.503096	0.251548	−	0.967845i	$-0.419060\pi$
$463$	10.8253	0.503096	0.251548	+	0.967845i	$0.419060\pi$
$464$	−19.8915	−0.923441
$465$	0	0
$466$	−27.4922	−1.27355
$467$	13.8713	0.641889	0.320945	−	0.947098i	$-0.396000\pi$
$467$	13.8713	0.641889	0.320945	+	0.947098i	$0.396000\pi$
$468$	0	0
$469$	−2.05594	−0.0949347
$470$	−51.0296	−2.35382
$471$	0	0
$472$	15.9674	0.734960
$473$	−13.4633	−0.619043
$474$	0	0
$475$	17.9829	0.825110
$476$	−0.304007	−0.0139341
$477$	0	0
$478$	−2.55585	−0.116902
$479$	27.9865	1.27873	0.639367	−	0.768901i	$-0.279196\pi$
$479$	27.9865	1.27873	0.639367	+	0.768901i	$0.279196\pi$
$480$	0	0
$481$	3.53578	0.161217
$482$	6.50535	0.296311
$483$	0	0
$484$	1.85996	0.0845437
$485$	−11.0422	−0.501402
$486$	0	0
$487$	−27.5749	−1.24954	−0.624769	−	0.780810i	$-0.714807\pi$
$487$	−27.5749	−1.24954	−0.624769	+	0.780810i	$0.714807\pi$
$488$	−14.5321	−0.657839
$489$	0	0
$490$	35.9977	1.62621
$491$	7.54344	0.340431	0.170215	−	0.985407i	$-0.445554\pi$
$491$	7.54344	0.340431	0.170215	+	0.985407i	$0.445554\pi$
$492$	0	0
$493$	13.2916	0.598624
$494$	2.89669	0.130328
$495$	0	0
$496$	12.8274	0.575967
$497$	−2.69864	−0.121050
$498$	0	0
$499$	8.52283	0.381534	0.190767	−	0.981635i	$-0.438902\pi$
$499$	8.52283	0.381534	0.190767	+	0.981635i	$0.438902\pi$
$500$	5.16539	0.231003
$501$	0	0
$502$	−16.4316	−0.733377
$503$	−1.82245	−0.0812592	−0.0406296	−	0.999174i	$-0.512936\pi$
$503$	−1.82245	−0.0812592	−0.0406296	+	0.999174i	$0.512936\pi$
$504$	0	0
$505$	−19.2040	−0.854568
$506$	−8.27007	−0.367649
$507$	0	0
$508$	2.16878	0.0962241
$509$	−38.3732	−1.70086	−0.850432	−	0.526085i	$-0.823659\pi$
$509$	−38.3732	−1.70086	−0.850432	+	0.526085i	$0.823659\pi$
$510$	0	0
$511$	3.17718	0.140550
$512$	−24.9537	−1.10281
$513$	0	0
$514$	39.7399	1.75285
$515$	−14.5679	−0.641936
$516$	0	0
$517$	−10.1115	−0.444704
$518$	2.27946	0.100154
$519$	0	0
$520$	16.8141	0.737348
$521$	−20.2464	−0.887012	−0.443506	−	0.896271i	$-0.646266\pi$
$521$	−20.2464	−0.887012	−0.443506	+	0.896271i	$0.646266\pi$
$522$	0	0
$523$	−15.7698	−0.689567	−0.344784	−	0.938682i	$-0.612048\pi$
$523$	−15.7698	−0.689567	−0.344784	+	0.938682i	$0.612048\pi$
$524$	−2.44371	−0.106754
$525$	0	0
$526$	−39.5037	−1.72244
$527$	−8.57133	−0.373373
$528$	0	0
$529$	8.86944	0.385628
$530$	18.3275	0.796096
$531$	0	0
$532$	−0.195471	−0.00847476
$533$	−8.63796	−0.374152
$534$	0	0
$535$	−59.4101	−2.56852
$536$	9.04500	0.390684
$537$	0	0
$538$	7.26215	0.313094
$539$	7.13294	0.307238
$540$	0	0
$541$	14.1376	0.607822	0.303911	−	0.952700i	$-0.401707\pi$
$541$	14.1376	0.607822	0.303911	+	0.952700i	$0.401707\pi$
$542$	−9.53459	−0.409546
$543$	0	0
$544$	2.55916	0.109723
$545$	−79.0378	−3.38561
$546$	0	0
$547$	35.9270	1.53613	0.768064	−	0.640373i	$-0.221220\pi$
$547$	35.9270	1.53613	0.768064	+	0.640373i	$0.221220\pi$
$548$	−0.758722	−0.0324110
$549$	0	0
$550$	−17.1031	−0.729277
$551$	8.54628	0.364084
$552$	0	0
$553$	−9.24012	−0.392930
$554$	28.2200	1.19895
$555$	0	0
$556$	1.28662	0.0545646
$557$	−32.6234	−1.38230	−0.691149	−	0.722713i	$-0.742895\pi$
$557$	−32.6234	−1.38230	−0.691149	+	0.722713i	$0.742895\pi$
$558$	0	0
$559$	17.2833	0.731004
$560$	9.80340	0.414269
$561$	0	0
$562$	19.0250	0.802522
$563$	20.8936	0.880559	0.440279	−	0.897861i	$-0.354879\pi$
$563$	20.8936	0.880559	0.440279	+	0.897861i	$0.354879\pi$
$564$	0	0
$565$	−76.6903	−3.22639
$566$	40.3803	1.69731
$567$	0	0
$568$	11.8725	0.498158
$569$	−25.2856	−1.06003	−0.530015	−	0.847989i	$-0.677814\pi$
$569$	−25.2856	−1.06003	−0.530015	+	0.847989i	$0.677814\pi$
$570$	0	0
$571$	25.0196	1.04704	0.523518	−	0.852014i	$-0.324619\pi$
$571$	25.0196	1.04704	0.523518	+	0.852014i	$0.324619\pi$
$572$	0.288370	0.0120574
$573$	0	0
$574$	−5.56877	−0.232436
$575$	65.9082	2.74856
$576$	0	0
$577$	30.9305	1.28765	0.643827	−	0.765171i	$-0.277346\pi$
$577$	30.9305	1.28765	0.643827	+	0.765171i	$0.277346\pi$
$578$	−15.1526	−0.630264
$579$	0	0
$580$	4.29369	0.178286
$581$	4.33518	0.179853
$582$	0	0
$583$	3.63160	0.150405
$584$	−13.9778	−0.578406
$585$	0	0
$586$	−24.8644	−1.02714
$587$	−23.5644	−0.972608	−0.486304	−	0.873790i	$-0.661655\pi$
$587$	−23.5644	−0.972608	−0.486304	+	0.873790i	$0.661655\pi$
$588$	0	0
$589$	−5.51121	−0.227086
$590$	29.7796	1.22601
$591$	0	0
$592$	−9.06952	−0.372755
$593$	−28.5071	−1.17065	−0.585323	−	0.810800i	$-0.699032\pi$
$593$	−28.5071	−1.17065	−0.585323	+	0.810800i	$0.699032\pi$
$594$	0	0
$595$	−6.55068	−0.268552
$596$	1.13491	0.0464876
$597$	0	0
$598$	10.6165	0.434143
$599$	−7.64261	−0.312269	−0.156134	−	0.987736i	$-0.549903\pi$
$599$	−7.64261	−0.312269	−0.156134	+	0.987736i	$0.549903\pi$
$600$	0	0
$601$	4.90792	0.200198	0.100099	−	0.994977i	$-0.468084\pi$
$601$	4.90792	0.200198	0.100099	+	0.994977i	$0.468084\pi$
$602$	11.1423	0.454125
$603$	0	0
$604$	0.452284	0.0184032
$605$	40.0780	1.62940
$606$	0	0
$607$	0.275203	0.0111701	0.00558507	−	0.999984i	$-0.498222\pi$
$607$	0.275203	0.0111701	0.00558507	+	0.999984i	$0.498222\pi$
$608$	1.64550	0.0667337
$609$	0	0
$610$	−27.1028	−1.09736
$611$	12.9805	0.525134
$612$	0	0
$613$	26.0551	1.05235	0.526177	−	0.850375i	$-0.323625\pi$
$613$	26.0551	1.05235	0.526177	+	0.850375i	$0.323625\pi$
$614$	23.7501	0.958476
$615$	0	0
$616$	2.14791	0.0865418
$617$	−47.5994	−1.91628	−0.958139	−	0.286304i	$-0.907573\pi$
$617$	−47.5994	−1.91628	−0.958139	+	0.286304i	$0.907573\pi$
$618$	0	0
$619$	−2.33389	−0.0938069	−0.0469035	−	0.998899i	$-0.514935\pi$
$619$	−2.33389	−0.0938069	−0.0469035	+	0.998899i	$0.514935\pi$
$620$	−2.76886	−0.111200
$621$	0	0
$622$	32.7537	1.31331
$623$	−2.46791	−0.0988747
$624$	0	0
$625$	52.9283	2.11713
$626$	−30.2194	−1.20781
$627$	0	0
$628$	−0.830953	−0.0331586
$629$	6.06030	0.241640
$630$	0	0
$631$	18.0965	0.720409	0.360204	−	0.932873i	$-0.382707\pi$
$631$	18.0965	0.720409	0.360204	+	0.932873i	$0.382707\pi$
$632$	40.6513	1.61702
$633$	0	0
$634$	18.5146	0.735309
$635$	46.7324	1.85452
$636$	0	0
$637$	−9.15679	−0.362805
$638$	−8.12816	−0.321797
$639$	0	0
$640$	−38.5697	−1.52460
$641$	21.5079	0.849509	0.424755	−	0.905308i	$-0.360360\pi$
$641$	21.5079	0.849509	0.424755	+	0.905308i	$0.360360\pi$
$642$	0	0
$643$	−18.7956	−0.741227	−0.370613	−	0.928787i	$-0.620852\pi$
$643$	−18.7956	−0.741227	−0.370613	+	0.928787i	$0.620852\pi$
$644$	−0.716414	−0.0282307
$645$	0	0
$646$	4.96492	0.195342
$647$	−1.56661	−0.0615899	−0.0307950	−	0.999526i	$-0.509804\pi$
$647$	−1.56661	−0.0615899	−0.0307950	+	0.999526i	$0.509804\pi$
$648$	0	0
$649$	5.90083	0.231628
$650$	21.9557	0.861175
$651$	0	0
$652$	−2.46788	−0.0966496
$653$	8.95996	0.350630	0.175315	−	0.984512i	$-0.443906\pi$
$653$	8.95996	0.350630	0.175315	+	0.984512i	$0.443906\pi$
$654$	0	0
$655$	−52.6565	−2.05746
$656$	22.1570	0.865086
$657$	0	0
$658$	8.36833	0.326231
$659$	−49.0075	−1.90906	−0.954531	−	0.298112i	$-0.903643\pi$
$659$	−49.0075	−1.90906	−0.954531	+	0.298112i	$0.903643\pi$
$660$	0	0
$661$	−7.84298	−0.305056	−0.152528	−	0.988299i	$-0.548742\pi$
$661$	−7.84298	−0.305056	−0.152528	+	0.988299i	$0.548742\pi$
$662$	37.5543	1.45959
$663$	0	0
$664$	−19.0723	−0.740150
$665$	−4.21197	−0.163333
$666$	0	0
$667$	31.3226	1.21282
$668$	0.202554	0.00783706
$669$	0	0
$670$	16.8691	0.651712
$671$	−5.37042	−0.207323
$672$	0	0
$673$	18.3858	0.708722	0.354361	−	0.935109i	$-0.384698\pi$
$673$	18.3858	0.708722	0.354361	+	0.935109i	$0.384698\pi$
$674$	38.5653	1.48548
$675$	0	0
$676$	2.09342	0.0805163
$677$	33.0120	1.26876	0.634378	−	0.773023i	$-0.281256\pi$
$677$	33.0120	1.26876	0.634378	+	0.773023i	$0.281256\pi$
$678$	0	0
$679$	1.81081	0.0694926
$680$	28.8193	1.10517
$681$	0	0
$682$	5.24158	0.200711
$683$	21.0365	0.804940	0.402470	−	0.915433i	$-0.368152\pi$
$683$	21.0365	0.804940	0.402470	+	0.915433i	$0.368152\pi$
$684$	0	0
$685$	−16.3488	−0.624654
$686$	−12.2106	−0.466201
$687$	0	0
$688$	−44.3328	−1.69017
$689$	−4.66200	−0.177608
$690$	0	0
$691$	35.0190	1.33219	0.666093	−	0.745869i	$-0.267965\pi$
$691$	35.0190	1.33219	0.666093	+	0.745869i	$0.267965\pi$
$692$	−1.52673	−0.0580376
$693$	0	0
$694$	13.8000	0.523841
$695$	27.7237	1.05162
$696$	0	0
$697$	−14.8054	−0.560795
$698$	−8.22676	−0.311387
$699$	0	0
$700$	−1.48159	−0.0559990
$701$	−46.3525	−1.75071	−0.875354	−	0.483482i	$-0.839372\pi$
$701$	−46.3525	−1.75071	−0.875354	+	0.483482i	$0.839372\pi$
$702$	0	0
$703$	3.89667	0.146966
$704$	−9.37140	−0.353198
$705$	0	0
$706$	29.9055	1.12551
$707$	3.14926	0.118440
$708$	0	0
$709$	1.37355	0.0515849	0.0257925	−	0.999667i	$-0.491789\pi$
$709$	1.37355	0.0515849	0.0257925	+	0.999667i	$0.491789\pi$
$710$	22.1425	0.830993
$711$	0	0
$712$	10.8574	0.406899
$713$	−20.1989	−0.756455
$714$	0	0
$715$	6.21373	0.232380
$716$	−1.31193	−0.0490292
$717$	0	0
$718$	30.9623	1.15550
$719$	−1.20707	−0.0450163	−0.0225081	−	0.999747i	$-0.507165\pi$
$719$	−1.20707	−0.0450163	−0.0225081	+	0.999747i	$0.507165\pi$
$720$	0	0
$721$	2.38898	0.0889702
$722$	−22.3730	−0.832637
$723$	0	0
$724$	0.189509	0.00704303
$725$	64.7773	2.40577
$726$	0	0
$727$	4.63685	0.171971	0.0859856	−	0.996296i	$-0.472596\pi$
$727$	4.63685	0.171971	0.0859856	+	0.996296i	$0.472596\pi$
$728$	−2.75734	−0.102194
$729$	0	0
$730$	−26.0690	−0.964856
$731$	29.6234	1.09566
$732$	0	0
$733$	−34.8149	−1.28592	−0.642960	−	0.765900i	$-0.722294\pi$
$733$	−34.8149	−1.28592	−0.642960	+	0.765900i	$0.722294\pi$
$734$	−48.1460	−1.77710
$735$	0	0
$736$	6.03084	0.222300
$737$	3.34262	0.123127
$738$	0	0
$739$	−9.09889	−0.334708	−0.167354	−	0.985897i	$-0.553522\pi$
$739$	−9.09889	−0.334708	−0.167354	+	0.985897i	$0.553522\pi$
$740$	1.95770	0.0719666
$741$	0	0
$742$	−3.00552	−0.110336
$743$	−32.6879	−1.19920	−0.599601	−	0.800299i	$-0.704674\pi$
$743$	−32.6879	−1.19920	−0.599601	+	0.800299i	$0.704674\pi$
$744$	0	0
$745$	24.4547	0.895951
$746$	22.4956	0.823622
$747$	0	0
$748$	0.494265	0.0180721
$749$	9.74265	0.355989
$750$	0	0
$751$	36.7776	1.34203	0.671017	−	0.741442i	$-0.265858\pi$
$751$	36.7776	1.34203	0.671017	+	0.741442i	$0.265858\pi$
$752$	−33.2959	−1.21418
$753$	0	0
$754$	10.4344	0.379998
$755$	9.74570	0.354682
$756$	0	0
$757$	−21.1101	−0.767260	−0.383630	−	0.923487i	$-0.625326\pi$
$757$	−21.1101	−0.767260	−0.383630	+	0.923487i	$0.625326\pi$
$758$	13.7577	0.499704
$759$	0	0
$760$	18.5303	0.672165
$761$	−15.4759	−0.561002	−0.280501	−	0.959854i	$-0.590501\pi$
$761$	−15.4759	−0.561002	−0.280501	+	0.959854i	$0.590501\pi$
$762$	0	0
$763$	12.9614	0.469233
$764$	−5.08812	−0.184082
$765$	0	0
$766$	−34.5526	−1.24844
$767$	−7.57509	−0.273521
$768$	0	0
$769$	−10.1275	−0.365205	−0.182603	−	0.983187i	$-0.558452\pi$
$769$	−10.1275	−0.365205	−0.182603	+	0.983187i	$0.558452\pi$
$770$	4.00591	0.144363
$771$	0	0
$772$	−3.73231	−0.134329
$773$	−9.39181	−0.337800	−0.168900	−	0.985633i	$-0.554021\pi$
$773$	−9.39181	−0.337800	−0.168900	+	0.985633i	$0.554021\pi$
$774$	0	0
$775$	−41.7727	−1.50052
$776$	−7.96655	−0.285983
$777$	0	0
$778$	−17.3526	−0.622119
$779$	−9.51963	−0.341076
$780$	0	0
$781$	4.38753	0.156998
$782$	18.1967	0.650713
$783$	0	0
$784$	23.4878	0.838851
$785$	−17.9052	−0.639063
$786$	0	0
$787$	−19.6658	−0.701009	−0.350505	−	0.936561i	$-0.613990\pi$
$787$	−19.6658	−0.701009	−0.350505	+	0.936561i	$0.613990\pi$
$788$	0.915491	0.0326130
$789$	0	0
$790$	75.8157	2.69740
$791$	12.5764	0.447166
$792$	0	0
$793$	6.89418	0.244820
$794$	−20.5784	−0.730299
$795$	0	0
$796$	2.29911	0.0814899
$797$	−40.7846	−1.44467	−0.722333	−	0.691545i	$-0.756930\pi$
$797$	−40.7846	−1.44467	−0.722333	+	0.691545i	$0.756930\pi$
$798$	0	0
$799$	22.2485	0.787094
$800$	12.4722	0.440958
$801$	0	0
$802$	47.8504	1.68966
$803$	−5.16556	−0.182289
$804$	0	0
$805$	−15.4371	−0.544087
$806$	−6.72879	−0.237012
$807$	0	0
$808$	−13.8550	−0.487416
$809$	52.6652	1.85161	0.925805	−	0.378002i	$-0.123389\pi$
$809$	52.6652	1.85161	0.925805	+	0.378002i	$0.123389\pi$
$810$	0	0
$811$	−43.3746	−1.52309	−0.761544	−	0.648113i	$-0.775559\pi$
$811$	−43.3746	−1.52309	−0.761544	+	0.648113i	$0.775559\pi$
$812$	−0.704121	−0.0247098
$813$	0	0
$814$	−3.70603	−0.129896
$815$	−53.1773	−1.86272
$816$	0	0
$817$	19.0473	0.666382
$818$	−35.1644	−1.22950
$819$	0	0
$820$	−4.78271	−0.167019
$821$	0.995948	0.0347588	0.0173794	−	0.999849i	$-0.494468\pi$
$821$	0.995948	0.0347588	0.0173794	+	0.999849i	$0.494468\pi$
$822$	0	0
$823$	−19.6725	−0.685740	−0.342870	−	0.939383i	$-0.611399\pi$
$823$	−19.6725	−0.685740	−0.342870	+	0.939383i	$0.611399\pi$
$824$	−10.5102	−0.366139
$825$	0	0
$826$	−4.88355	−0.169920
$827$	−1.01496	−0.0352938	−0.0176469	−	0.999844i	$-0.505617\pi$
$827$	−1.01496	−0.0352938	−0.0176469	+	0.999844i	$0.505617\pi$
$828$	0	0
$829$	1.60769	0.0558373	0.0279186	−	0.999610i	$-0.491112\pi$
$829$	1.60769	0.0558373	0.0279186	+	0.999610i	$0.491112\pi$
$830$	−35.5704	−1.23467
$831$	0	0
$832$	12.0304	0.417078
$833$	−15.6947	−0.543788
$834$	0	0
$835$	4.36459	0.151043
$836$	0.317804	0.0109915
$837$	0	0
$838$	8.66118	0.299195
$839$	2.56818	0.0886634	0.0443317	−	0.999017i	$-0.485884\pi$
$839$	2.56818	0.0886634	0.0443317	+	0.999017i	$0.485884\pi$
$840$	0	0
$841$	1.78512	0.0615559
$842$	−49.9664	−1.72196
$843$	0	0
$844$	2.11684	0.0728646
$845$	45.1086	1.55178
$846$	0	0
$847$	−6.57238	−0.225830
$848$	11.9584	0.410652
$849$	0	0
$850$	37.6320	1.29077
$851$	14.2815	0.489564
$852$	0	0
$853$	14.7385	0.504638	0.252319	−	0.967644i	$-0.418807\pi$
$853$	14.7385	0.504638	0.252319	+	0.967644i	$0.418807\pi$
$854$	4.44458	0.152090
$855$	0	0
$856$	−42.8622	−1.46500
$857$	1.19358	0.0407719	0.0203859	−	0.999792i	$-0.493511\pi$
$857$	1.19358	0.0407719	0.0203859	+	0.999792i	$0.493511\pi$
$858$	0	0
$859$	30.3736	1.03633	0.518167	−	0.855279i	$-0.326614\pi$
$859$	30.3736	1.03633	0.518167	+	0.855279i	$0.326614\pi$
$860$	9.56948	0.326317
$861$	0	0
$862$	17.4759	0.595231
$863$	15.6622	0.533148	0.266574	−	0.963814i	$-0.414108\pi$
$863$	15.6622	0.533148	0.266574	+	0.963814i	$0.414108\pi$
$864$	0	0
$865$	−32.8976	−1.11855
$866$	44.2991	1.50535
$867$	0	0
$868$	0.454065	0.0154120
$869$	15.0229	0.509616
$870$	0	0
$871$	−4.29103	−0.145396
$872$	−57.0228	−1.93104
$873$	0	0
$874$	11.7002	0.395764
$875$	−18.2525	−0.617047
$876$	0	0
$877$	21.7304	0.733784	0.366892	−	0.930263i	$-0.380422\pi$
$877$	21.7304	0.733784	0.366892	+	0.930263i	$0.380422\pi$
$878$	−36.8512	−1.24367
$879$	0	0
$880$	−15.9387	−0.537293
$881$	19.2140	0.647335	0.323668	−	0.946171i	$-0.395084\pi$
$881$	19.2140	0.647335	0.323668	+	0.946171i	$0.395084\pi$
$882$	0	0
$883$	38.8852	1.30859	0.654295	−	0.756239i	$-0.272965\pi$
$883$	38.8852	1.30859	0.654295	+	0.756239i	$0.272965\pi$
$884$	−0.634504	−0.0213407
$885$	0	0
$886$	23.5626	0.791602
$887$	26.2541	0.881526	0.440763	−	0.897624i	$-0.354708\pi$
$887$	26.2541	0.881526	0.440763	+	0.897624i	$0.354708\pi$
$888$	0	0
$889$	−7.66362	−0.257030
$890$	20.2493	0.678760
$891$	0	0
$892$	2.41984	0.0810224
$893$	14.3054	0.478711
$894$	0	0
$895$	−28.2692	−0.944935
$896$	6.32504	0.211305
$897$	0	0
$898$	−55.4966	−1.85195
$899$	−19.8523	−0.662112
$900$	0	0
$901$	−7.99063	−0.266207
$902$	9.05389	0.301462
$903$	0	0
$904$	−55.3292	−1.84022
$905$	4.08349	0.135740
$906$	0	0
$907$	26.8223	0.890621	0.445310	−	0.895376i	$-0.353093\pi$
$907$	26.8223	0.890621	0.445310	+	0.895376i	$0.353093\pi$
$908$	4.89816	0.162551
$909$	0	0
$910$	−5.14251	−0.170473
$911$	4.39537	0.145625	0.0728125	−	0.997346i	$-0.476803\pi$
$911$	4.39537	0.145625	0.0728125	+	0.997346i	$0.476803\pi$
$912$	0	0
$913$	−7.04827	−0.233264
$914$	6.76645	0.223814
$915$	0	0
$916$	−2.53182	−0.0836537
$917$	8.63512	0.285157
$918$	0	0
$919$	27.0703	0.892967	0.446484	−	0.894792i	$-0.352676\pi$
$919$	27.0703	0.892967	0.446484	+	0.894792i	$0.352676\pi$
$920$	67.9146	2.23908
$921$	0	0
$922$	46.5091	1.53169
$923$	−5.63242	−0.185393
$924$	0	0
$925$	29.5351	0.971109
$926$	14.5660	0.478668
$927$	0	0
$928$	5.92735	0.194575
$929$	27.5526	0.903972	0.451986	−	0.892025i	$-0.350716\pi$
$929$	27.5526	0.903972	0.451986	+	0.892025i	$0.350716\pi$
$930$	0	0
$931$	−10.0914	−0.330733
$932$	3.87204	0.126833
$933$	0	0
$934$	18.6645	0.610722
$935$	10.6503	0.348302
$936$	0	0
$937$	−46.7565	−1.52747	−0.763734	−	0.645531i	$-0.776636\pi$
$937$	−46.7565	−1.52747	−0.763734	+	0.645531i	$0.776636\pi$
$938$	−2.76636	−0.0903250
$939$	0	0
$940$	7.18709	0.234417
$941$	−55.0405	−1.79427	−0.897135	−	0.441757i	$-0.854355\pi$
$941$	−55.0405	−1.79427	−0.897135	+	0.441757i	$0.854355\pi$
$942$	0	0
$943$	−34.8900	−1.13617
$944$	19.4306	0.632414
$945$	0	0
$946$	−18.1155	−0.588985
$947$	35.1897	1.14351	0.571755	−	0.820424i	$-0.306263\pi$
$947$	35.1897	1.14351	0.571755	+	0.820424i	$0.306263\pi$
$948$	0	0
$949$	6.63120	0.215258
$950$	24.1967	0.785046
$951$	0	0
$952$	−4.72607	−0.153173
$953$	6.39216	0.207062	0.103531	−	0.994626i	$-0.466986\pi$
$953$	6.39216	0.207062	0.103531	+	0.994626i	$0.466986\pi$
$954$	0	0
$955$	−109.638	−3.54779
$956$	0.359970	0.0116423
$957$	0	0
$958$	37.6571	1.21664
$959$	2.68103	0.0865749
$960$	0	0
$961$	−18.1979	−0.587029
$962$	4.75754	0.153389
$963$	0	0
$964$	−0.916224	−0.0295096
$965$	−80.4230	−2.58891
$966$	0	0
$967$	36.7900	1.18309	0.591543	−	0.806273i	$-0.298519\pi$
$967$	36.7900	1.18309	0.591543	+	0.806273i	$0.298519\pi$
$968$	28.9148	0.929356
$969$	0	0
$970$	−14.8578	−0.477056
$971$	−9.76929	−0.313511	−0.156756	−	0.987637i	$-0.550104\pi$
$971$	−9.76929	−0.313511	−0.156756	+	0.987637i	$0.550104\pi$
$972$	0	0
$973$	−4.54640	−0.145751
$974$	−37.1033	−1.18887
$975$	0	0
$976$	−17.6841	−0.566054
$977$	−1.41735	−0.0453452	−0.0226726	−	0.999743i	$-0.507218\pi$
$977$	−1.41735	−0.0453452	−0.0226726	+	0.999743i	$0.507218\pi$
$978$	0	0
$979$	4.01241	0.128237
$980$	−5.06997	−0.161954
$981$	0	0
$982$	10.1500	0.323901
$983$	−26.9669	−0.860110	−0.430055	−	0.902803i	$-0.641506\pi$
$983$	−26.9669	−0.860110	−0.430055	+	0.902803i	$0.641506\pi$
$984$	0	0
$985$	19.7268	0.628548
$986$	17.8845	0.569557
$987$	0	0
$988$	−0.407975	−0.0129794
$989$	69.8096	2.21982
$990$	0	0
$991$	40.6222	1.29041	0.645203	−	0.764011i	$-0.276773\pi$
$991$	40.6222	1.29041	0.645203	+	0.764011i	$0.276773\pi$
$992$	−3.82236	−0.121360
$993$	0	0
$994$	−3.63114	−0.115173
$995$	49.5408	1.57055
$996$	0	0
$997$	12.2955	0.389401	0.194701	−	0.980863i	$-0.437626\pi$
$997$	12.2955	0.389401	0.194701	+	0.980863i	$0.437626\pi$
$998$	11.4679	0.363008
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1629.2.a.b.1.4		5
3.2	odd	2		543.2.a.b.1.2	✓	5
12.11	even	2		8688.2.a.z.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
543.2.a.b.1.2	✓	5	3.2	odd	2
1629.2.a.b.1.4		5	1.1	even	1	trivial
8688.2.a.z.1.5		5	12.11	even	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:	\( N \)	\(=\)	\( 1629 = 3^{2} \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1629.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34554 q^{2} -0.189509 q^{4} -4.08349 q^{5} +0.669650 q^{7} -2.94608 q^{8} -5.49452 q^{10} -1.08874 q^{11} +1.39765 q^{13} +0.901044 q^{14} -3.58507 q^{16} +2.39556 q^{17} +1.54030 q^{19} +0.773856 q^{20} -1.46495 q^{22} +5.64530 q^{23} +11.6749 q^{25} +1.88060 q^{26} -0.126904 q^{28} +5.54843 q^{29} -3.57800 q^{31} +1.06829 q^{32} +3.22333 q^{34} -2.73451 q^{35} +2.52980 q^{37} +2.07255 q^{38} +12.0303 q^{40} -6.18036 q^{41} +12.3660 q^{43} +0.206325 q^{44} +7.59601 q^{46} +9.28737 q^{47} -6.55157 q^{49} +15.7091 q^{50} -0.264866 q^{52} -3.33560 q^{53} +4.44585 q^{55} -1.97284 q^{56} +7.46567 q^{58} -5.41988 q^{59} +4.93270 q^{61} -4.81437 q^{62} +8.60757 q^{64} -5.70728 q^{65} -3.07018 q^{67} -0.453979 q^{68} -3.67940 q^{70} -4.02992 q^{71} +4.74454 q^{73} +3.40396 q^{74} -0.291901 q^{76} -0.729073 q^{77} -13.7984 q^{79} +14.6396 q^{80} -8.31595 q^{82} +6.47380 q^{83} -9.78224 q^{85} +16.6390 q^{86} +3.20751 q^{88} -3.68537 q^{89} +0.935935 q^{91} -1.06983 q^{92} +12.4966 q^{94} -6.28981 q^{95} +2.70412 q^{97} -8.81543 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 3 q^{4} - 3 q^{5} + 9 q^{7} - 9 q^{8} - 14 q^{10} + 4 q^{11} + 9 q^{13} - 3 q^{14} + 7 q^{16} - 9 q^{17} + 6 q^{19} + 25 q^{20} - 17 q^{22} + 5 q^{23} + 12 q^{25} - 6 q^{26} + 27 q^{28} + 20 q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) 5 * q + q^2 + 3 * q^4 - 3 * q^5 + 9 * q^7 - 9 * q^8 - 14 * q^10 + 4 * q^11 + 9 * q^13 - 3 * q^14 + 7 * q^16 - 9 * q^17 + 6 * q^19 + 25 * q^20 - 17 * q^22 + 5 * q^23 + 12 * q^25 - 6 * q^26 + 27 * q^28 + 20 * q^29 + 13 * q^31 - 12 * q^32 + 22 * q^34 + 8 * q^35 - 5 * q^37 + 26 * q^38 - 13 * q^40 + 2 * q^41 + 4 * q^43 + 9 * q^44 - 20 * q^46 + 2 * q^47 + 50 * q^49 - 20 * q^50 + 14 * q^52 - 19 * q^53 + 11 * q^55 - 8 * q^56 - 3 * q^58 + 25 * q^59 + 17 * q^61 + 11 * q^62 - 11 * q^64 + 8 * q^65 - 9 * q^67 + 15 * q^68 - 30 * q^70 - 9 * q^71 + 7 * q^73 + 15 * q^74 - 15 * q^76 + 38 * q^77 + 2 * q^79 + 31 * q^80 + q^82 - 14 * q^83 - 3 * q^85 + 4 * q^86 - 15 * q^88 + 12 * q^89 - 5 * q^91 + 39 * q^92 + 12 * q^94 - 22 * q^95 - 8 * q^97 + 3 * q^98

Introduction

L-functions

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