LMFDB - Embedded newform 5103.2.a.i.1.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5103,2,Mod(1,5103)] 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("5103.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5103, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5103 = 3^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5103.a (trivial)

Newform invariants

comment:select newform

sage:traces = [27,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.7476601515$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	5103.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.26564 q^{2} -0.398155 q^{4} +0.909945 q^{5} +1.00000 q^{7} -3.03520 q^{8} +1.15166 q^{10} -5.96120 q^{11} -3.32145 q^{13} +1.26564 q^{14} -3.04516 q^{16} -1.03692 q^{17} +2.05933 q^{19} -0.362299 q^{20} -7.54474 q^{22} +3.19257 q^{23} -4.17200 q^{25} -4.20376 q^{26} -0.398155 q^{28} +7.30992 q^{29} +6.98197 q^{31} +2.21632 q^{32} -1.31237 q^{34} +0.909945 q^{35} -3.87452 q^{37} +2.60637 q^{38} -2.76187 q^{40} +5.70449 q^{41} +4.43844 q^{43} +2.37348 q^{44} +4.04065 q^{46} +6.27814 q^{47} +1.00000 q^{49} -5.28025 q^{50} +1.32245 q^{52} +1.64075 q^{53} -5.42437 q^{55} -3.03520 q^{56} +9.25172 q^{58} +9.59238 q^{59} +12.6249 q^{61} +8.83666 q^{62} +8.89539 q^{64} -3.02233 q^{65} -10.0827 q^{67} +0.412856 q^{68} +1.15166 q^{70} +4.22229 q^{71} -4.58759 q^{73} -4.90375 q^{74} -0.819933 q^{76} -5.96120 q^{77} +14.3775 q^{79} -2.77093 q^{80} +7.21984 q^{82} +17.2062 q^{83} -0.943542 q^{85} +5.61747 q^{86} +18.0934 q^{88} +0.759027 q^{89} -3.32145 q^{91} -1.27114 q^{92} +7.94587 q^{94} +1.87388 q^{95} +10.3864 q^{97} +1.26564 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 9 q^{2} + 27 q^{4} + 12 q^{5} + 27 q^{7} + 27 q^{8} + 24 q^{11} + 9 q^{14} + 27 q^{16} + 30 q^{17} + 30 q^{20} + 39 q^{23} + 27 q^{25} + 9 q^{26} + 27 q^{28} + 39 q^{29} + 63 q^{32} + 12 q^{35} + 9 q^{38}+ \cdots + 9 q^{98}+O(q^{100}) $ 27 * q + 9 * q^2 + 27 * q^4 + 12 * q^5 + 27 * q^7 + 27 * q^8 + 24 * q^11 + 9 * q^14 + 27 * q^16 + 30 * q^17 + 30 * q^20 + 39 * q^23 + 27 * q^25 + 9 * q^26 + 27 * q^28 + 39 * q^29 + 63 * q^32 + 12 * q^35 + 9 * q^38 + 42 * q^41 + 42 * q^44 + 27 * q^47 + 27 * q^49 + 36 * q^50 + 66 * q^53 + 27 * q^56 + 18 * q^59 + 36 * q^62 + 27 * q^64 + 69 * q^65 + 21 * q^68 + 72 * q^71 + 54 * q^74 + 24 * q^77 + 21 * q^80 + 39 * q^83 + 27 * q^86 + 42 * q^89 + 75 * q^92 + 78 * q^95 + 9 * 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 $

$2$	1.26564	0.894943	0.447471	−	0.894298i	$-0.352325\pi$
$2$	1.26564	0.894943	0.447471	+	0.894298i	$0.352325\pi$
$3$	0	0
$3$	0	0
$4$	−0.398155	−0.199077
$5$	0.909945	0.406940	0.203470	−	0.979081i	$-0.434778\pi$
$5$	0.909945	0.406940	0.203470	+	0.979081i	$0.434778\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.03520	−1.07311
$9$	0	0
$10$	1.15166	0.364188
$11$	−5.96120	−1.79737	−0.898685	−	0.438595i	$-0.855476\pi$
$11$	−5.96120	−1.79737	−0.898685	+	0.438595i	$0.855476\pi$
$12$	0	0
$13$	−3.32145	−0.921203	−0.460602	−	0.887607i	$-0.652366\pi$
$13$	−3.32145	−0.921203	−0.460602	+	0.887607i	$0.652366\pi$
$14$	1.26564	0.338257
$15$	0	0
$16$	−3.04516	−0.761291
$17$	−1.03692	−0.251491	−0.125745	−	0.992063i	$-0.540132\pi$
$17$	−1.03692	−0.251491	−0.125745	+	0.992063i	$0.540132\pi$
$18$	0	0
$19$	2.05933	0.472443	0.236222	−	0.971699i	$-0.424091\pi$
$19$	2.05933	0.472443	0.236222	+	0.971699i	$0.424091\pi$
$20$	−0.362299	−0.0810126
$21$	0	0
$22$	−7.54474	−1.60854
$23$	3.19257	0.665697	0.332849	−	0.942980i	$-0.391990\pi$
$23$	3.19257	0.665697	0.332849	+	0.942980i	$0.391990\pi$
$24$	0	0
$25$	−4.17200	−0.834400
$26$	−4.20376	−0.824424
$27$	0	0
$28$	−0.398155	−0.0752442
$29$	7.30992	1.35742	0.678709	−	0.734408i	$-0.262540\pi$
$29$	7.30992	1.35742	0.678709	+	0.734408i	$0.262540\pi$
$30$	0	0
$31$	6.98197	1.25400	0.626999	−	0.779020i	$-0.284283\pi$
$31$	6.98197	1.25400	0.626999	+	0.779020i	$0.284283\pi$
$32$	2.21632	0.391794
$33$	0	0
$34$	−1.31237	−0.225070
$35$	0.909945	0.153809
$36$	0	0
$37$	−3.87452	−0.636967	−0.318483	−	0.947928i	$-0.603174\pi$
$37$	−3.87452	−0.636967	−0.318483	+	0.947928i	$0.603174\pi$
$38$	2.60637	0.422810
$39$	0	0
$40$	−2.76187	−0.436689
$41$	5.70449	0.890892	0.445446	−	0.895309i	$-0.353045\pi$
$41$	5.70449	0.890892	0.445446	+	0.895309i	$0.353045\pi$
$42$	0	0
$43$	4.43844	0.676856	0.338428	−	0.940992i	$-0.390105\pi$
$43$	4.43844	0.676856	0.338428	+	0.940992i	$0.390105\pi$
$44$	2.37348	0.357816
$45$	0	0
$46$	4.04065	0.595761
$47$	6.27814	0.915761	0.457881	−	0.889014i	$-0.348609\pi$
$47$	6.27814	0.915761	0.457881	+	0.889014i	$0.348609\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.28025	−0.746740
$51$	0	0
$52$	1.32245	0.183391
$53$	1.64075	0.225374	0.112687	−	0.993631i	$-0.464054\pi$
$53$	1.64075	0.225374	0.112687	+	0.993631i	$0.464054\pi$
$54$	0	0
$55$	−5.42437	−0.731421
$56$	−3.03520	−0.405596
$57$	0	0
$58$	9.25172	1.21481
$59$	9.59238	1.24882	0.624411	−	0.781096i	$-0.285339\pi$
$59$	9.59238	1.24882	0.624411	+	0.781096i	$0.285339\pi$
$60$	0	0
$61$	12.6249	1.61645	0.808223	−	0.588876i	$-0.200429\pi$
$61$	12.6249	1.61645	0.808223	+	0.588876i	$0.200429\pi$
$62$	8.83666	1.12226
$63$	0	0
$64$	8.89539	1.11192
$65$	−3.02233	−0.374874
$66$	0	0
$67$	−10.0827	−1.23180	−0.615899	−	0.787825i	$-0.711207\pi$
$67$	−10.0827	−1.23180	−0.615899	+	0.787825i	$0.711207\pi$
$68$	0.412856	0.0500661
$69$	0	0
$70$	1.15166	0.137650
$71$	4.22229	0.501094	0.250547	−	0.968104i	$-0.419390\pi$
$71$	4.22229	0.501094	0.250547	+	0.968104i	$0.419390\pi$
$72$	0	0
$73$	−4.58759	−0.536937	−0.268469	−	0.963288i	$-0.586518\pi$
$73$	−4.58759	−0.536937	−0.268469	+	0.963288i	$0.586518\pi$
$74$	−4.90375	−0.570049
$75$	0	0
$76$	−0.819933	−0.0940528
$77$	−5.96120	−0.679342
$78$	0	0
$79$	14.3775	1.61759	0.808796	−	0.588090i	$-0.200120\pi$
$79$	14.3775	1.61759	0.808796	+	0.588090i	$0.200120\pi$
$80$	−2.77093	−0.309800
$81$	0	0
$82$	7.21984	0.797298
$83$	17.2062	1.88863	0.944313	−	0.329048i	$-0.106728\pi$
$83$	17.2062	1.88863	0.944313	+	0.329048i	$0.106728\pi$
$84$	0	0
$85$	−0.943542	−0.102342
$86$	5.61747	0.605747
$87$	0	0
$88$	18.0934	1.92877
$89$	0.759027	0.0804568	0.0402284	−	0.999191i	$-0.487191\pi$
$89$	0.759027	0.0804568	0.0402284	+	0.999191i	$0.487191\pi$
$90$	0	0
$91$	−3.32145	−0.348182
$92$	−1.27114	−0.132525
$93$	0	0
$94$	7.94587	0.819554
$95$	1.87388	0.192256
$96$	0	0
$97$	10.3864	1.05458	0.527288	−	0.849686i	$-0.323209\pi$
$97$	10.3864	1.05458	0.527288	+	0.849686i	$0.323209\pi$
$98$	1.26564	0.127849
$99$	0	0
$100$	1.66110	0.166110
$101$	−14.4260	−1.43544	−0.717720	−	0.696332i	$-0.754814\pi$
$101$	−14.4260	−1.43544	−0.717720	+	0.696332i	$0.754814\pi$
$102$	0	0
$103$	−2.75014	−0.270979	−0.135489	−	0.990779i	$-0.543261\pi$
$103$	−2.75014	−0.270979	−0.135489	+	0.990779i	$0.543261\pi$
$104$	10.0813	0.988548
$105$	0	0
$106$	2.07660	0.201697
$107$	−4.35433	−0.420949	−0.210475	−	0.977599i	$-0.567501\pi$
$107$	−4.35433	−0.420949	−0.210475	+	0.977599i	$0.567501\pi$
$108$	0	0
$109$	−16.1963	−1.55132	−0.775660	−	0.631151i	$-0.782583\pi$
$109$	−16.1963	−1.55132	−0.775660	+	0.631151i	$0.782583\pi$
$110$	−6.86530	−0.654580
$111$	0	0
$112$	−3.04516	−0.287741
$113$	5.74343	0.540297	0.270148	−	0.962819i	$-0.412927\pi$
$113$	5.74343	0.540297	0.270148	+	0.962819i	$0.412927\pi$
$114$	0	0
$115$	2.90507	0.270899
$116$	−2.91048	−0.270231
$117$	0	0
$118$	12.1405	1.11762
$119$	−1.03692	−0.0950545
$120$	0	0
$121$	24.5359	2.23054
$122$	15.9785	1.44663
$123$	0	0
$124$	−2.77991	−0.249643
$125$	−8.34602	−0.746490
$126$	0	0
$127$	2.53565	0.225003	0.112501	−	0.993652i	$-0.464114\pi$
$127$	2.53565	0.225003	0.112501	+	0.993652i	$0.464114\pi$
$128$	6.82572	0.603314
$129$	0	0
$130$	−3.82519	−0.335491
$131$	2.29054	0.200126	0.100063	−	0.994981i	$-0.468096\pi$
$131$	2.29054	0.200126	0.100063	+	0.994981i	$0.468096\pi$
$132$	0	0
$133$	2.05933	0.178567
$134$	−12.7611	−1.10239
$135$	0	0
$136$	3.14727	0.269876
$137$	−11.0188	−0.941397	−0.470699	−	0.882294i	$-0.655998\pi$
$137$	−11.0188	−0.941397	−0.470699	+	0.882294i	$0.655998\pi$
$138$	0	0
$139$	−21.3661	−1.81225	−0.906126	−	0.423007i	$-0.860975\pi$
$139$	−21.3661	−1.81225	−0.906126	+	0.423007i	$0.860975\pi$
$140$	−0.362299	−0.0306199
$141$	0	0
$142$	5.34390	0.448450
$143$	19.7998	1.65574
$144$	0	0
$145$	6.65162	0.552387
$146$	−5.80624	−0.480528
$147$	0	0
$148$	1.54266	0.126806
$149$	−7.93122	−0.649751	−0.324875	−	0.945757i	$-0.605322\pi$
$149$	−7.93122	−0.649751	−0.324875	+	0.945757i	$0.605322\pi$
$150$	0	0
$151$	−4.64445	−0.377960	−0.188980	−	0.981981i	$-0.560518\pi$
$151$	−4.64445	−0.377960	−0.188980	+	0.981981i	$0.560518\pi$
$152$	−6.25049	−0.506982
$153$	0	0
$154$	−7.54474	−0.607972
$155$	6.35321	0.510302
$156$	0	0
$157$	21.8510	1.74390	0.871950	−	0.489596i	$-0.162856\pi$
$157$	21.8510	1.74390	0.871950	+	0.489596i	$0.162856\pi$
$158$	18.1967	1.44765
$159$	0	0
$160$	2.01673	0.159437
$161$	3.19257	0.251610
$162$	0	0
$163$	−13.0830	−1.02474	−0.512369	−	0.858765i	$-0.671232\pi$
$163$	−13.0830	−1.02474	−0.512369	+	0.858765i	$0.671232\pi$
$164$	−2.27127	−0.177357
$165$	0	0
$166$	21.7769	1.69021
$167$	3.57831	0.276898	0.138449	−	0.990370i	$-0.455788\pi$
$167$	3.57831	0.276898	0.138449	+	0.990370i	$0.455788\pi$
$168$	0	0
$169$	−1.96800	−0.151385
$170$	−1.19419	−0.0915898
$171$	0	0
$172$	−1.76719	−0.134747
$173$	4.28671	0.325913	0.162956	−	0.986633i	$-0.447897\pi$
$173$	4.28671	0.325913	0.162956	+	0.986633i	$0.447897\pi$
$174$	0	0
$175$	−4.17200	−0.315374
$176$	18.1528	1.36832
$177$	0	0
$178$	0.960656	0.0720042
$179$	−17.5261	−1.30996	−0.654981	−	0.755645i	$-0.727323\pi$
$179$	−17.5261	−1.30996	−0.654981	+	0.755645i	$0.727323\pi$
$180$	0	0
$181$	1.37424	0.102146	0.0510731	−	0.998695i	$-0.483736\pi$
$181$	1.37424	0.102146	0.0510731	+	0.998695i	$0.483736\pi$
$182$	−4.20376	−0.311603
$183$	0	0
$184$	−9.69010	−0.714364
$185$	−3.52560	−0.259207
$186$	0	0
$187$	6.18130	0.452021
$188$	−2.49967	−0.182307
$189$	0	0
$190$	2.37166	0.172058
$191$	6.83185	0.494335	0.247168	−	0.968973i	$-0.420500\pi$
$191$	6.83185	0.494335	0.247168	+	0.968973i	$0.420500\pi$
$192$	0	0
$193$	−2.63342	−0.189558	−0.0947788	−	0.995498i	$-0.530214\pi$
$193$	−2.63342	−0.189558	−0.0947788	+	0.995498i	$0.530214\pi$
$194$	13.1454	0.943786
$195$	0	0
$196$	−0.398155	−0.0284396
$197$	0.439831	0.0313366	0.0156683	−	0.999877i	$-0.495012\pi$
$197$	0.439831	0.0313366	0.0156683	+	0.999877i	$0.495012\pi$
$198$	0	0
$199$	−21.4394	−1.51980	−0.759899	−	0.650041i	$-0.774752\pi$
$199$	−21.4394	−1.51980	−0.759899	+	0.650041i	$0.774752\pi$
$200$	12.6629	0.895399
$201$	0	0
$202$	−18.2581	−1.28464
$203$	7.30992	0.513055
$204$	0	0
$205$	5.19078	0.362540
$206$	−3.48068	−0.242511
$207$	0	0
$208$	10.1143	0.701303
$209$	−12.2761	−0.849155
$210$	0	0
$211$	10.5126	0.723717	0.361858	−	0.932233i	$-0.382142\pi$
$211$	10.5126	0.723717	0.361858	+	0.932233i	$0.382142\pi$
$212$	−0.653272	−0.0448669
$213$	0	0
$214$	−5.51102	−0.376725
$215$	4.03874	0.275440
$216$	0	0
$217$	6.98197	0.473967
$218$	−20.4986	−1.38834
$219$	0	0
$220$	2.15974	0.145610
$221$	3.44408	0.231674
$222$	0	0
$223$	−12.9412	−0.866607	−0.433303	−	0.901248i	$-0.642652\pi$
$223$	−12.9412	−0.866607	−0.433303	+	0.901248i	$0.642652\pi$
$224$	2.21632	0.148084
$225$	0	0
$226$	7.26912	0.483535
$227$	6.09253	0.404375	0.202188	−	0.979347i	$-0.435195\pi$
$227$	6.09253	0.404375	0.202188	+	0.979347i	$0.435195\pi$
$228$	0	0
$229$	20.0048	1.32196	0.660979	−	0.750405i	$-0.270141\pi$
$229$	20.0048	1.32196	0.660979	+	0.750405i	$0.270141\pi$
$230$	3.67677	0.242439
$231$	0	0
$232$	−22.1871	−1.45665
$233$	27.1591	1.77925	0.889625	−	0.456691i	$-0.150966\pi$
$233$	27.1591	1.77925	0.889625	+	0.456691i	$0.150966\pi$
$234$	0	0
$235$	5.71277	0.372660
$236$	−3.81925	−0.248612
$237$	0	0
$238$	−1.31237	−0.0850683
$239$	11.3592	0.734765	0.367382	−	0.930070i	$-0.380254\pi$
$239$	11.3592	0.734765	0.367382	+	0.930070i	$0.380254\pi$
$240$	0	0
$241$	−6.61577	−0.426159	−0.213079	−	0.977035i	$-0.568349\pi$
$241$	−6.61577	−0.426159	−0.213079	+	0.977035i	$0.568349\pi$
$242$	31.0536	1.99620
$243$	0	0
$244$	−5.02665	−0.321798
$245$	0.909945	0.0581343
$246$	0	0
$247$	−6.83996	−0.435216
$248$	−21.1917	−1.34567
$249$	0	0
$250$	−10.5631	−0.668066
$251$	−16.2724	−1.02710	−0.513552	−	0.858058i	$-0.671671\pi$
$251$	−16.2724	−1.02710	−0.513552	+	0.858058i	$0.671671\pi$
$252$	0	0
$253$	−19.0316	−1.19650
$254$	3.20922	0.201364
$255$	0	0
$256$	−9.15188	−0.571992
$257$	13.3437	0.832359	0.416179	−	0.909283i	$-0.363369\pi$
$257$	13.3437	0.832359	0.416179	+	0.909283i	$0.363369\pi$
$258$	0	0
$259$	−3.87452	−0.240751
$260$	1.20336	0.0746290
$261$	0	0
$262$	2.89900	0.179101
$263$	22.5865	1.39274	0.696371	−	0.717682i	$-0.254797\pi$
$263$	22.5865	1.39274	0.696371	+	0.717682i	$0.254797\pi$
$264$	0	0
$265$	1.49299	0.0917137
$266$	2.60637	0.159807
$267$	0	0
$268$	4.01448	0.245223
$269$	1.28401	0.0782877	0.0391439	−	0.999234i	$-0.487537\pi$
$269$	1.28401	0.0782877	0.0391439	+	0.999234i	$0.487537\pi$
$270$	0	0
$271$	13.7466	0.835049	0.417525	−	0.908666i	$-0.362898\pi$
$271$	13.7466	0.835049	0.417525	+	0.908666i	$0.362898\pi$
$272$	3.15760	0.191457
$273$	0	0
$274$	−13.9458	−0.842497
$275$	24.8701	1.49973
$276$	0	0
$277$	−9.32468	−0.560265	−0.280133	−	0.959961i	$-0.590378\pi$
$277$	−9.32468	−0.560265	−0.280133	+	0.959961i	$0.590378\pi$
$278$	−27.0419	−1.62186
$279$	0	0
$280$	−2.76187	−0.165053
$281$	−9.88864	−0.589907	−0.294953	−	0.955512i	$-0.595304\pi$
$281$	−9.88864	−0.589907	−0.294953	+	0.955512i	$0.595304\pi$
$282$	0	0
$283$	−16.7477	−0.995550	−0.497775	−	0.867306i	$-0.665849\pi$
$283$	−16.7477	−0.995550	−0.497775	+	0.867306i	$0.665849\pi$
$284$	−1.68113	−0.0997565
$285$	0	0
$286$	25.0594	1.48180
$287$	5.70449	0.336726
$288$	0	0
$289$	−15.9248	−0.936753
$290$	8.41856	0.494355
$291$	0	0
$292$	1.82657	0.106892
$293$	15.5244	0.906945	0.453473	−	0.891270i	$-0.350185\pi$
$293$	15.5244	0.906945	0.453473	+	0.891270i	$0.350185\pi$
$294$	0	0
$295$	8.72854	0.508195
$296$	11.7599	0.683533
$297$	0	0
$298$	−10.0381	−0.581490
$299$	−10.6040	−0.613243
$300$	0	0
$301$	4.43844	0.255827
$302$	−5.87820	−0.338253
$303$	0	0
$304$	−6.27100	−0.359667
$305$	11.4879	0.657797
$306$	0	0
$307$	3.48093	0.198667	0.0993335	−	0.995054i	$-0.468329\pi$
$307$	3.48093	0.198667	0.0993335	+	0.995054i	$0.468329\pi$
$308$	2.37348	0.135242
$309$	0	0
$310$	8.04088	0.456691
$311$	13.8524	0.785496	0.392748	−	0.919646i	$-0.371525\pi$
$311$	13.8524	0.785496	0.392748	+	0.919646i	$0.371525\pi$
$312$	0	0
$313$	10.2958	0.581951	0.290975	−	0.956731i	$-0.406020\pi$
$313$	10.2958	0.581951	0.290975	+	0.956731i	$0.406020\pi$
$314$	27.6555	1.56069
$315$	0	0
$316$	−5.72446	−0.322026
$317$	28.4642	1.59871	0.799354	−	0.600861i	$-0.205175\pi$
$317$	28.4642	1.59871	0.799354	+	0.600861i	$0.205175\pi$
$318$	0	0
$319$	−43.5759	−2.43978
$320$	8.09432	0.452486
$321$	0	0
$322$	4.04065	0.225177
$323$	−2.13537	−0.118815
$324$	0	0
$325$	13.8571	0.768652
$326$	−16.5583	−0.917082
$327$	0	0
$328$	−17.3143	−0.956022
$329$	6.27814	0.346125
$330$	0	0
$331$	−26.9239	−1.47987	−0.739937	−	0.672677i	$-0.765145\pi$
$331$	−26.9239	−1.47987	−0.739937	+	0.672677i	$0.765145\pi$
$332$	−6.85073	−0.375983
$333$	0	0
$334$	4.52886	0.247808
$335$	−9.17470	−0.501267
$336$	0	0
$337$	4.65503	0.253576	0.126788	−	0.991930i	$-0.459533\pi$
$337$	4.65503	0.253576	0.126788	+	0.991930i	$0.459533\pi$
$338$	−2.49078	−0.135480
$339$	0	0
$340$	0.375676	0.0203739
$341$	−41.6209	−2.25390
$342$	0	0
$343$	1.00000	0.0539949
$344$	−13.4716	−0.726338
$345$	0	0
$346$	5.42543	0.291673
$347$	7.88485	0.423281	0.211641	−	0.977348i	$-0.432119\pi$
$347$	7.88485	0.423281	0.211641	+	0.977348i	$0.432119\pi$
$348$	0	0
$349$	11.2715	0.603348	0.301674	−	0.953411i	$-0.402455\pi$
$349$	11.2715	0.603348	0.301674	+	0.953411i	$0.402455\pi$
$350$	−5.28025	−0.282241
$351$	0	0
$352$	−13.2119	−0.704199
$353$	18.4204	0.980417	0.490209	−	0.871605i	$-0.336921\pi$
$353$	18.4204	0.980417	0.490209	+	0.871605i	$0.336921\pi$
$354$	0	0
$355$	3.84205	0.203915
$356$	−0.302211	−0.0160171
$357$	0	0
$358$	−22.1817	−1.17234
$359$	−19.1619	−1.01132	−0.505662	−	0.862732i	$-0.668752\pi$
$359$	−19.1619	−1.01132	−0.505662	+	0.862732i	$0.668752\pi$
$360$	0	0
$361$	−14.7592	−0.776797
$362$	1.73929	0.0914149
$363$	0	0
$364$	1.32245	0.0693152
$365$	−4.17446	−0.218501
$366$	0	0
$367$	6.26421	0.326989	0.163495	−	0.986544i	$-0.447723\pi$
$367$	6.26421	0.326989	0.163495	+	0.986544i	$0.447723\pi$
$368$	−9.72190	−0.506789
$369$	0	0
$370$	−4.46214	−0.231976
$371$	1.64075	0.0851834
$372$	0	0
$373$	19.2214	0.995248	0.497624	−	0.867393i	$-0.334206\pi$
$373$	19.2214	0.995248	0.497624	+	0.867393i	$0.334206\pi$
$374$	7.82330	0.404533
$375$	0	0
$376$	−19.0554	−0.982709
$377$	−24.2795	−1.25046
$378$	0	0
$379$	30.6727	1.57555	0.787775	−	0.615963i	$-0.211233\pi$
$379$	30.6727	1.57555	0.787775	+	0.615963i	$0.211233\pi$
$380$	−0.746094	−0.0382738
$381$	0	0
$382$	8.64666	0.442402
$383$	−3.91379	−0.199986	−0.0999928	−	0.994988i	$-0.531882\pi$
$383$	−3.91379	−0.199986	−0.0999928	+	0.994988i	$0.531882\pi$
$384$	0	0
$385$	−5.42437	−0.276451
$386$	−3.33296	−0.169643
$387$	0	0
$388$	−4.13539	−0.209942
$389$	26.9603	1.36694	0.683471	−	0.729978i	$-0.260470\pi$
$389$	26.9603	1.36694	0.683471	+	0.729978i	$0.260470\pi$
$390$	0	0
$391$	−3.31045	−0.167417
$392$	−3.03520	−0.153301
$393$	0	0
$394$	0.556668	0.0280445
$395$	13.0827	0.658262
$396$	0	0
$397$	3.98546	0.200025	0.100012	−	0.994986i	$-0.468112\pi$
$397$	3.98546	0.200025	0.100012	+	0.994986i	$0.468112\pi$
$398$	−27.1346	−1.36013
$399$	0	0
$400$	12.7044	0.635221
$401$	30.5526	1.52572	0.762861	−	0.646562i	$-0.223794\pi$
$401$	30.5526	1.52572	0.762861	+	0.646562i	$0.223794\pi$
$402$	0	0
$403$	−23.1902	−1.15519
$404$	5.74378	0.285764
$405$	0	0
$406$	9.25172	0.459155
$407$	23.0968	1.14487
$408$	0	0
$409$	−32.1305	−1.58875	−0.794376	−	0.607426i	$-0.792202\pi$
$409$	−32.1305	−1.58875	−0.794376	+	0.607426i	$0.792202\pi$
$410$	6.56966	0.324452
$411$	0	0
$412$	1.09498	0.0539458
$413$	9.59238	0.472010
$414$	0	0
$415$	15.6567	0.768557
$416$	−7.36139	−0.360922
$417$	0	0
$418$	−15.5371	−0.759945
$419$	−5.06445	−0.247415	−0.123707	−	0.992319i	$-0.539478\pi$
$419$	−5.06445	−0.247415	−0.123707	+	0.992319i	$0.539478\pi$
$420$	0	0
$421$	10.7340	0.523143	0.261572	−	0.965184i	$-0.415759\pi$
$421$	10.7340	0.523143	0.261572	+	0.965184i	$0.415759\pi$
$422$	13.3052	0.647685
$423$	0	0
$424$	−4.98000	−0.241850
$425$	4.32604	0.209844
$426$	0	0
$427$	12.6249	0.610959
$428$	1.73370	0.0838015
$429$	0	0
$430$	5.11159	0.246503
$431$	34.2002	1.64736	0.823682	−	0.567052i	$-0.191916\pi$
$431$	34.2002	1.64736	0.823682	+	0.567052i	$0.191916\pi$
$432$	0	0
$433$	36.4825	1.75324	0.876619	−	0.481185i	$-0.159793\pi$
$433$	36.4825	1.75324	0.876619	+	0.481185i	$0.159793\pi$
$434$	8.83666	0.424173
$435$	0	0
$436$	6.44862	0.308833
$437$	6.57457	0.314504
$438$	0	0
$439$	39.7337	1.89639	0.948194	−	0.317693i	$-0.102908\pi$
$439$	39.7337	1.89639	0.948194	+	0.317693i	$0.102908\pi$
$440$	16.4640	0.784892
$441$	0	0
$442$	4.35897	0.207335
$443$	−4.67541	−0.222136	−0.111068	−	0.993813i	$-0.535427\pi$
$443$	−4.67541	−0.222136	−0.111068	+	0.993813i	$0.535427\pi$
$444$	0	0
$445$	0.690673	0.0327411
$446$	−16.3789	−0.775563
$447$	0	0
$448$	8.89539	0.420268
$449$	23.5988	1.11370	0.556848	−	0.830615i	$-0.312011\pi$
$449$	23.5988	1.11370	0.556848	+	0.830615i	$0.312011\pi$
$450$	0	0
$451$	−34.0056	−1.60126
$452$	−2.28678	−0.107561
$453$	0	0
$454$	7.71095	0.361893
$455$	−3.02233	−0.141689
$456$	0	0
$457$	−18.4002	−0.860724	−0.430362	−	0.902656i	$-0.641614\pi$
$457$	−18.4002	−0.860724	−0.430362	+	0.902656i	$0.641614\pi$
$458$	25.3189	1.18308
$459$	0	0
$460$	−1.15667	−0.0539299
$461$	0.561870	0.0261689	0.0130844	−	0.999914i	$-0.495835\pi$
$461$	0.561870	0.0261689	0.0130844	+	0.999914i	$0.495835\pi$
$462$	0	0
$463$	24.5820	1.14242	0.571210	−	0.820804i	$-0.306474\pi$
$463$	24.5820	1.14242	0.571210	+	0.820804i	$0.306474\pi$
$464$	−22.2599	−1.03339
$465$	0	0
$466$	34.3736	1.59233
$467$	−14.7841	−0.684125	−0.342062	−	0.939677i	$-0.611125\pi$
$467$	−14.7841	−0.684125	−0.342062	+	0.939677i	$0.611125\pi$
$468$	0	0
$469$	−10.0827	−0.465576
$470$	7.23031	0.333509
$471$	0	0
$472$	−29.1148	−1.34012
$473$	−26.4584	−1.21656
$474$	0	0
$475$	−8.59153	−0.394207
$476$	0.412856	0.0189232
$477$	0	0
$478$	14.3766	0.657572
$479$	9.59264	0.438299	0.219150	−	0.975691i	$-0.429672\pi$
$479$	9.59264	0.438299	0.219150	+	0.975691i	$0.429672\pi$
$480$	0	0
$481$	12.8690	0.586776
$482$	−8.37318	−0.381388
$483$	0	0
$484$	−9.76910	−0.444050
$485$	9.45103	0.429149
$486$	0	0
$487$	−10.5107	−0.476284	−0.238142	−	0.971230i	$-0.576538\pi$
$487$	−10.5107	−0.476284	−0.238142	+	0.971230i	$0.576538\pi$
$488$	−38.3190	−1.73462
$489$	0	0
$490$	1.15166	0.0520268
$491$	32.3760	1.46111	0.730553	−	0.682855i	$-0.239262\pi$
$491$	32.3760	1.46111	0.730553	+	0.682855i	$0.239262\pi$
$492$	0	0
$493$	−7.57981	−0.341378
$494$	−8.65693	−0.389494
$495$	0	0
$496$	−21.2612	−0.954657
$497$	4.22229	0.189396
$498$	0	0
$499$	11.1028	0.497031	0.248515	−	0.968628i	$-0.420057\pi$
$499$	11.1028	0.497031	0.248515	+	0.968628i	$0.420057\pi$
$500$	3.32301	0.148609
$501$	0	0
$502$	−20.5950	−0.919200
$503$	−16.4857	−0.735063	−0.367531	−	0.930011i	$-0.619797\pi$
$503$	−16.4857	−0.735063	−0.367531	+	0.930011i	$0.619797\pi$
$504$	0	0
$505$	−13.1269	−0.584138
$506$	−24.0871	−1.07080
$507$	0	0
$508$	−1.00958	−0.0447929
$509$	−0.574745	−0.0254751	−0.0127376	−	0.999919i	$-0.504055\pi$
$509$	−0.574745	−0.0254751	−0.0127376	+	0.999919i	$0.504055\pi$
$510$	0	0
$511$	−4.58759	−0.202943
$512$	−25.2344	−1.11521
$513$	0	0
$514$	16.8884	0.744913
$515$	−2.50247	−0.110272
$516$	0	0
$517$	−37.4253	−1.64596
$518$	−4.90375	−0.215458
$519$	0	0
$520$	9.17339	0.402280
$521$	1.43449	0.0628462	0.0314231	−	0.999506i	$-0.489996\pi$
$521$	1.43449	0.0628462	0.0314231	+	0.999506i	$0.489996\pi$
$522$	0	0
$523$	−17.8163	−0.779054	−0.389527	−	0.921015i	$-0.627361\pi$
$523$	−17.8163	−0.779054	−0.389527	+	0.921015i	$0.627361\pi$
$524$	−0.911991	−0.0398405
$525$	0	0
$526$	28.5864	1.24642
$527$	−7.23976	−0.315369
$528$	0	0
$529$	−12.8075	−0.556847
$530$	1.88959	0.0820785
$531$	0	0
$532$	−0.819933	−0.0355486
$533$	−18.9472	−0.820693
$534$	0	0
$535$	−3.96220	−0.171301
$536$	30.6030	1.32185
$537$	0	0
$538$	1.62510	0.0700630
$539$	−5.96120	−0.256767
$540$	0	0
$541$	2.90252	0.124789	0.0623946	−	0.998052i	$-0.480126\pi$
$541$	2.90252	0.124789	0.0623946	+	0.998052i	$0.480126\pi$
$542$	17.3983	0.747321
$543$	0	0
$544$	−2.29815	−0.0985325
$545$	−14.7377	−0.631294
$546$	0	0
$547$	−29.0335	−1.24138	−0.620691	−	0.784055i	$-0.713148\pi$
$547$	−29.0335	−1.24138	−0.620691	+	0.784055i	$0.713148\pi$
$548$	4.38718	0.187411
$549$	0	0
$550$	31.4766	1.34217
$551$	15.0535	0.641303
$552$	0	0
$553$	14.3775	0.611392
$554$	−11.8017	−0.501405
$555$	0	0
$556$	8.50704	0.360779
$557$	33.3661	1.41376	0.706882	−	0.707331i	$-0.250101\pi$
$557$	33.3661	1.41376	0.706882	+	0.707331i	$0.250101\pi$
$558$	0	0
$559$	−14.7420	−0.623522
$560$	−2.77093	−0.117093
$561$	0	0
$562$	−12.5155	−0.527933
$563$	−25.1651	−1.06058	−0.530290	−	0.847816i	$-0.677917\pi$
$563$	−25.1651	−1.06058	−0.530290	+	0.847816i	$0.677917\pi$
$564$	0	0
$565$	5.22621	0.219868
$566$	−21.1966	−0.890960
$567$	0	0
$568$	−12.8155	−0.537726
$569$	−30.8874	−1.29487	−0.647435	−	0.762121i	$-0.724158\pi$
$569$	−30.8874	−1.29487	−0.647435	+	0.762121i	$0.724158\pi$
$570$	0	0
$571$	36.9567	1.54659	0.773295	−	0.634046i	$-0.218607\pi$
$571$	36.9567	1.54659	0.773295	+	0.634046i	$0.218607\pi$
$572$	−7.88339	−0.329621
$573$	0	0
$574$	7.21984	0.301350
$575$	−13.3194	−0.555458
$576$	0	0
$577$	37.9663	1.58056	0.790278	−	0.612748i	$-0.209936\pi$
$577$	37.9663	1.58056	0.790278	+	0.612748i	$0.209936\pi$
$578$	−20.1551	−0.838340
$579$	0	0
$580$	−2.64838	−0.109968
$581$	17.2062	0.713834
$582$	0	0
$583$	−9.78083	−0.405081
$584$	13.9243	0.576190
$585$	0	0
$586$	19.6483	0.811664
$587$	37.1033	1.53142	0.765709	−	0.643187i	$-0.222388\pi$
$587$	37.1033	1.53142	0.765709	+	0.643187i	$0.222388\pi$
$588$	0	0
$589$	14.3782	0.592443
$590$	11.0472	0.454806
$591$	0	0
$592$	11.7985	0.484917
$593$	−26.3991	−1.08408	−0.542041	−	0.840352i	$-0.682348\pi$
$593$	−26.3991	−1.08408	−0.542041	+	0.840352i	$0.682348\pi$
$594$	0	0
$595$	−0.943542	−0.0386815
$596$	3.15785	0.129351
$597$	0	0
$598$	−13.4208	−0.548817
$599$	2.49643	0.102001	0.0510006	−	0.998699i	$-0.483759\pi$
$599$	2.49643	0.102001	0.0510006	+	0.998699i	$0.483759\pi$
$600$	0	0
$601$	−39.6056	−1.61554	−0.807772	−	0.589494i	$-0.799327\pi$
$601$	−39.6056	−1.61554	−0.807772	+	0.589494i	$0.799327\pi$
$602$	5.61747	0.228951
$603$	0	0
$604$	1.84921	0.0752433
$605$	22.3263	0.907695
$606$	0	0
$607$	−7.24219	−0.293951	−0.146976	−	0.989140i	$-0.546954\pi$
$607$	−7.24219	−0.293951	−0.146976	+	0.989140i	$0.546954\pi$
$608$	4.56414	0.185100
$609$	0	0
$610$	14.5396	0.588690
$611$	−20.8525	−0.843602
$612$	0	0
$613$	17.9455	0.724811	0.362405	−	0.932021i	$-0.381956\pi$
$613$	17.9455	0.724811	0.362405	+	0.932021i	$0.381956\pi$
$614$	4.40560	0.177796
$615$	0	0
$616$	18.0934	0.729006
$617$	1.43309	0.0576942	0.0288471	−	0.999584i	$-0.490816\pi$
$617$	1.43309	0.0576942	0.0288471	+	0.999584i	$0.490816\pi$
$618$	0	0
$619$	−41.4789	−1.66718	−0.833589	−	0.552385i	$-0.813718\pi$
$619$	−41.4789	−1.66718	−0.833589	+	0.552385i	$0.813718\pi$
$620$	−2.52956	−0.101590
$621$	0	0
$622$	17.5321	0.702974
$623$	0.759027	0.0304098
$624$	0	0
$625$	13.2656	0.530623
$626$	13.0307	0.520813
$627$	0	0
$628$	−8.70008	−0.347171
$629$	4.01757	0.160191
$630$	0	0
$631$	22.4692	0.894486	0.447243	−	0.894413i	$-0.352406\pi$
$631$	22.4692	0.894486	0.447243	+	0.894413i	$0.352406\pi$
$632$	−43.6385	−1.73585
$633$	0	0
$634$	36.0254	1.43075
$635$	2.30730	0.0915625
$636$	0	0
$637$	−3.32145	−0.131600
$638$	−55.1514	−2.18346
$639$	0	0
$640$	6.21103	0.245513
$641$	22.4608	0.887150	0.443575	−	0.896237i	$-0.353710\pi$
$641$	22.4608	0.887150	0.443575	+	0.896237i	$0.353710\pi$
$642$	0	0
$643$	−30.5513	−1.20483	−0.602413	−	0.798184i	$-0.705794\pi$
$643$	−30.5513	−1.20483	−0.602413	+	0.798184i	$0.705794\pi$
$644$	−1.27114	−0.0500899
$645$	0	0
$646$	−2.70261	−0.106333
$647$	−36.1118	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$647$	−36.1118	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$648$	0	0
$649$	−57.1821	−2.24459
$650$	17.5381	0.687899
$651$	0	0
$652$	5.20905	0.204002
$653$	−13.8060	−0.540271	−0.270135	−	0.962822i	$-0.587068\pi$
$653$	−13.8060	−0.540271	−0.270135	+	0.962822i	$0.587068\pi$
$654$	0	0
$655$	2.08427	0.0814391
$656$	−17.3711	−0.678228
$657$	0	0
$658$	7.94587	0.309762
$659$	0.179145	0.00697850	0.00348925	−	0.999994i	$-0.498889\pi$
$659$	0.179145	0.00697850	0.00348925	+	0.999994i	$0.498889\pi$
$660$	0	0
$661$	−2.61234	−0.101608	−0.0508042	−	0.998709i	$-0.516178\pi$
$661$	−2.61234	−0.101608	−0.0508042	+	0.998709i	$0.516178\pi$
$662$	−34.0760	−1.32440
$663$	0	0
$664$	−52.2243	−2.02670
$665$	1.87388	0.0726659
$666$	0	0
$667$	23.3374	0.903629
$668$	−1.42472	−0.0551242
$669$	0	0
$670$	−11.6119	−0.448606
$671$	−75.2593	−2.90535
$672$	0	0
$673$	−24.3464	−0.938485	−0.469243	−	0.883069i	$-0.655473\pi$
$673$	−24.3464	−0.938485	−0.469243	+	0.883069i	$0.655473\pi$
$674$	5.89160	0.226936
$675$	0	0
$676$	0.783568	0.0301372
$677$	30.0744	1.15585	0.577927	−	0.816089i	$-0.303862\pi$
$677$	30.0744	1.15585	0.577927	+	0.816089i	$0.303862\pi$
$678$	0	0
$679$	10.3864	0.398592
$680$	2.86384	0.109823
$681$	0	0
$682$	−52.6771	−2.01711
$683$	1.97381	0.0755257	0.0377628	−	0.999287i	$-0.487977\pi$
$683$	1.97381	0.0755257	0.0377628	+	0.999287i	$0.487977\pi$
$684$	0	0
$685$	−10.0265	−0.383092
$686$	1.26564	0.0483224
$687$	0	0
$688$	−13.5158	−0.515284
$689$	−5.44966	−0.207615
$690$	0	0
$691$	−26.5560	−1.01024	−0.505119	−	0.863050i	$-0.668551\pi$
$691$	−26.5560	−1.01024	−0.505119	+	0.863050i	$0.668551\pi$
$692$	−1.70677	−0.0648818
$693$	0	0
$694$	9.97939	0.378812
$695$	−19.4420	−0.737478
$696$	0	0
$697$	−5.91512	−0.224051
$698$	14.2656	0.539962
$699$	0	0
$700$	1.66110	0.0627838
$701$	6.58398	0.248673	0.124337	−	0.992240i	$-0.460320\pi$
$701$	6.58398	0.248673	0.124337	+	0.992240i	$0.460320\pi$
$702$	0	0
$703$	−7.97892	−0.300931
$704$	−53.0272	−1.99854
$705$	0	0
$706$	23.3136	0.877417
$707$	−14.4260	−0.542545
$708$	0	0
$709$	11.4922	0.431598	0.215799	−	0.976438i	$-0.430764\pi$
$709$	11.4922	0.431598	0.215799	+	0.976438i	$0.430764\pi$
$710$	4.86266	0.182492
$711$	0	0
$712$	−2.30380	−0.0863386
$713$	22.2904	0.834784
$714$	0	0
$715$	18.0167	0.673788
$716$	6.97810	0.260784
$717$	0	0
$718$	−24.2520	−0.905077
$719$	−45.4467	−1.69488	−0.847438	−	0.530895i	$-0.821856\pi$
$719$	−45.4467	−1.69488	−0.847438	+	0.530895i	$0.821856\pi$
$720$	0	0
$721$	−2.75014	−0.102420
$722$	−18.6798	−0.695189
$723$	0	0
$724$	−0.547159	−0.0203350
$725$	−30.4970	−1.13263
$726$	0	0
$727$	25.7698	0.955749	0.477875	−	0.878428i	$-0.341407\pi$
$727$	25.7698	0.955749	0.477875	+	0.878428i	$0.341407\pi$
$728$	10.0813	0.373636
$729$	0	0
$730$	−5.28336	−0.195546
$731$	−4.60232	−0.170223
$732$	0	0
$733$	−13.0815	−0.483178	−0.241589	−	0.970379i	$-0.577669\pi$
$733$	−13.0815	−0.483178	−0.241589	+	0.970379i	$0.577669\pi$
$734$	7.92824	0.292637
$735$	0	0
$736$	7.07577	0.260816
$737$	60.1050	2.21400
$738$	0	0
$739$	−20.8767	−0.767962	−0.383981	−	0.923341i	$-0.625447\pi$
$739$	−20.8767	−0.767962	−0.383981	+	0.923341i	$0.625447\pi$
$740$	1.40373	0.0516023
$741$	0	0
$742$	2.07660	0.0762343
$743$	17.3527	0.636610	0.318305	−	0.947988i	$-0.396886\pi$
$743$	17.3527	0.636610	0.318305	+	0.947988i	$0.396886\pi$
$744$	0	0
$745$	−7.21698	−0.264410
$746$	24.3274	0.890690
$747$	0	0
$748$	−2.46112	−0.0899873
$749$	−4.35433	−0.159104
$750$	0	0
$751$	6.38115	0.232852	0.116426	−	0.993199i	$-0.462856\pi$
$751$	6.38115	0.232852	0.116426	+	0.993199i	$0.462856\pi$
$752$	−19.1180	−0.697161
$753$	0	0
$754$	−30.7291	−1.11909
$755$	−4.22620	−0.153807
$756$	0	0
$757$	−8.80999	−0.320205	−0.160102	−	0.987100i	$-0.551182\pi$
$757$	−8.80999	−0.320205	−0.160102	+	0.987100i	$0.551182\pi$
$758$	38.8206	1.41003
$759$	0	0
$760$	−5.68760	−0.206311
$761$	6.84845	0.248256	0.124128	−	0.992266i	$-0.460387\pi$
$761$	6.84845	0.248256	0.124128	+	0.992266i	$0.460387\pi$
$762$	0	0
$763$	−16.1963	−0.586344
$764$	−2.72013	−0.0984110
$765$	0	0
$766$	−4.95346	−0.178976
$767$	−31.8606	−1.15042
$768$	0	0
$769$	−19.8515	−0.715863	−0.357931	−	0.933748i	$-0.616518\pi$
$769$	−19.8515	−0.715863	−0.357931	+	0.933748i	$0.616518\pi$
$770$	−6.86530	−0.247408
$771$	0	0
$772$	1.04851	0.0377366
$773$	14.5090	0.521851	0.260926	−	0.965359i	$-0.415972\pi$
$773$	14.5090	0.521851	0.260926	+	0.965359i	$0.415972\pi$
$774$	0	0
$775$	−29.1288	−1.04634
$776$	−31.5247	−1.13167
$777$	0	0
$778$	34.1220	1.22333
$779$	11.7474	0.420896
$780$	0	0
$781$	−25.1699	−0.900651
$782$	−4.18984	−0.149828
$783$	0	0
$784$	−3.04516	−0.108756
$785$	19.8832	0.709662
$786$	0	0
$787$	−18.5883	−0.662601	−0.331300	−	0.943525i	$-0.607487\pi$
$787$	−18.5883	−0.662601	−0.331300	+	0.943525i	$0.607487\pi$
$788$	−0.175121	−0.00623842
$789$	0	0
$790$	16.5580	0.589107
$791$	5.74343	0.204213
$792$	0	0
$793$	−41.9328	−1.48908
$794$	5.04416	0.179011
$795$	0	0
$796$	8.53620	0.302558
$797$	51.6930	1.83106	0.915531	−	0.402248i	$-0.131771\pi$
$797$	51.6930	1.83106	0.915531	+	0.402248i	$0.131771\pi$
$798$	0	0
$799$	−6.50995	−0.230305
$800$	−9.24650	−0.326913
$801$	0	0
$802$	38.6686	1.36543
$803$	27.3476	0.965075
$804$	0	0
$805$	2.90507	0.102390
$806$	−29.3505	−1.03383
$807$	0	0
$808$	43.7858	1.54038
$809$	28.1422	0.989426	0.494713	−	0.869057i	$-0.335273\pi$
$809$	28.1422	0.989426	0.494713	+	0.869057i	$0.335273\pi$
$810$	0	0
$811$	−35.1405	−1.23395	−0.616975	−	0.786983i	$-0.711642\pi$
$811$	−35.1405	−1.23395	−0.616975	+	0.786983i	$0.711642\pi$
$812$	−2.91048	−0.102138
$813$	0	0
$814$	29.2322	1.02459
$815$	−11.9048	−0.417007
$816$	0	0
$817$	9.14022	0.319776
$818$	−40.6657	−1.42184
$819$	0	0
$820$	−2.06673	−0.0721735
$821$	33.0788	1.15446	0.577229	−	0.816583i	$-0.304134\pi$
$821$	33.0788	1.15446	0.577229	+	0.816583i	$0.304134\pi$
$822$	0	0
$823$	−34.9144	−1.21704	−0.608520	−	0.793538i	$-0.708237\pi$
$823$	−34.9144	−1.21704	−0.608520	+	0.793538i	$0.708237\pi$
$824$	8.34722	0.290789
$825$	0	0
$826$	12.1405	0.422422
$827$	−28.8272	−1.00242	−0.501211	−	0.865325i	$-0.667112\pi$
$827$	−28.8272	−1.00242	−0.501211	+	0.865325i	$0.667112\pi$
$828$	0	0
$829$	19.9217	0.691909	0.345954	−	0.938251i	$-0.387555\pi$
$829$	19.9217	0.691909	0.345954	+	0.938251i	$0.387555\pi$
$830$	19.8158	0.687815
$831$	0	0
$832$	−29.5456	−1.02431
$833$	−1.03692	−0.0359272
$834$	0	0
$835$	3.25607	0.112681
$836$	4.88779	0.169048
$837$	0	0
$838$	−6.40977	−0.221422
$839$	−31.1919	−1.07686	−0.538432	−	0.842669i	$-0.680983\pi$
$839$	−31.1919	−1.07686	−0.538432	+	0.842669i	$0.680983\pi$
$840$	0	0
$841$	24.4349	0.842582
$842$	13.5854	0.468183
$843$	0	0
$844$	−4.18564	−0.144076
$845$	−1.79077	−0.0616044
$846$	0	0
$847$	24.5359	0.843064
$848$	−4.99635	−0.171575
$849$	0	0
$850$	5.47521	0.187798
$851$	−12.3697	−0.424027
$852$	0	0
$853$	28.1808	0.964892	0.482446	−	0.875926i	$-0.339749\pi$
$853$	28.1808	0.964892	0.482446	+	0.875926i	$0.339749\pi$
$854$	15.9785	0.546774
$855$	0	0
$856$	13.2163	0.451723
$857$	−12.4323	−0.424678	−0.212339	−	0.977196i	$-0.568108\pi$
$857$	−12.4323	−0.424678	−0.212339	+	0.977196i	$0.568108\pi$
$858$	0	0
$859$	−10.9067	−0.372131	−0.186065	−	0.982537i	$-0.559574\pi$
$859$	−10.9067	−0.372131	−0.186065	+	0.982537i	$0.559574\pi$
$860$	−1.60804	−0.0548338
$861$	0	0
$862$	43.2851	1.47430
$863$	57.0196	1.94097	0.970485	−	0.241162i	$-0.0775287\pi$
$863$	57.0196	1.94097	0.970485	+	0.241162i	$0.0775287\pi$
$864$	0	0
$865$	3.90067	0.132627
$866$	46.1738	1.56905
$867$	0	0
$868$	−2.77991	−0.0943561
$869$	−85.7070	−2.90741
$870$	0	0
$871$	33.4891	1.13474
$872$	49.1589	1.66473
$873$	0	0
$874$	8.32104	0.281463
$875$	−8.34602	−0.282147
$876$	0	0
$877$	0.748959	0.0252906	0.0126453	−	0.999920i	$-0.495975\pi$
$877$	0.748959	0.0252906	0.0126453	+	0.999920i	$0.495975\pi$
$878$	50.2886	1.69716
$879$	0	0
$880$	16.5181	0.556824
$881$	11.6792	0.393481	0.196741	−	0.980456i	$-0.436964\pi$
$881$	11.6792	0.393481	0.196741	+	0.980456i	$0.436964\pi$
$882$	0	0
$883$	−18.9696	−0.638378	−0.319189	−	0.947691i	$-0.603410\pi$
$883$	−18.9696	−0.638378	−0.319189	+	0.947691i	$0.603410\pi$
$884$	−1.37128	−0.0461211
$885$	0	0
$886$	−5.91739	−0.198799
$887$	−7.29628	−0.244985	−0.122493	−	0.992469i	$-0.539089\pi$
$887$	−7.29628	−0.244985	−0.122493	+	0.992469i	$0.539089\pi$
$888$	0	0
$889$	2.53565	0.0850430
$890$	0.874144	0.0293014
$891$	0	0
$892$	5.15260	0.172522
$893$	12.9288	0.432645
$894$	0	0
$895$	−15.9478	−0.533076
$896$	6.82572	0.228031
$897$	0	0
$898$	29.8676	0.996694
$899$	51.0376	1.70220
$900$	0	0
$901$	−1.70133	−0.0566795
$902$	−43.0389	−1.43304
$903$	0	0
$904$	−17.4325	−0.579796
$905$	1.25048	0.0415673
$906$	0	0
$907$	−46.4417	−1.54207	−0.771035	−	0.636793i	$-0.780261\pi$
$907$	−46.4417	−1.54207	−0.771035	+	0.636793i	$0.780261\pi$
$908$	−2.42577	−0.0805020
$909$	0	0
$910$	−3.82519	−0.126804
$911$	39.1211	1.29614	0.648070	−	0.761581i	$-0.275576\pi$
$911$	39.1211	1.29614	0.648070	+	0.761581i	$0.275576\pi$
$912$	0	0
$913$	−102.570	−3.39456
$914$	−23.2880	−0.770299
$915$	0	0
$916$	−7.96503	−0.263172
$917$	2.29054	0.0756404
$918$	0	0
$919$	1.47738	0.0487343	0.0243671	−	0.999703i	$-0.492243\pi$
$919$	1.47738	0.0487343	0.0243671	+	0.999703i	$0.492243\pi$
$920$	−8.81746	−0.290703
$921$	0	0
$922$	0.711125	0.0234197
$923$	−14.0241	−0.461609
$924$	0	0
$925$	16.1645	0.531485
$926$	31.1119	1.02240
$927$	0	0
$928$	16.2011	0.531828
$929$	7.07631	0.232166	0.116083	−	0.993240i	$-0.462966\pi$
$929$	7.07631	0.232166	0.116083	+	0.993240i	$0.462966\pi$
$930$	0	0
$931$	2.05933	0.0674919
$932$	−10.8135	−0.354209
$933$	0	0
$934$	−18.7113	−0.612253
$935$	5.62465	0.183946
$936$	0	0
$937$	−21.3475	−0.697393	−0.348696	−	0.937236i	$-0.613376\pi$
$937$	−21.3475	−0.697393	−0.348696	+	0.937236i	$0.613376\pi$
$938$	−12.7611	−0.416664
$939$	0	0
$940$	−2.27457	−0.0741882
$941$	−13.6692	−0.445604	−0.222802	−	0.974864i	$-0.571520\pi$
$941$	−13.6692	−0.445604	−0.222802	+	0.974864i	$0.571520\pi$
$942$	0	0
$943$	18.2120	0.593065
$944$	−29.2104	−0.950716
$945$	0	0
$946$	−33.4868	−1.08875
$947$	−16.4608	−0.534903	−0.267452	−	0.963571i	$-0.586182\pi$
$947$	−16.4608	−0.534903	−0.267452	+	0.963571i	$0.586182\pi$
$948$	0	0
$949$	15.2374	0.494628
$950$	−10.8738	−0.352792
$951$	0	0
$952$	3.14727	0.102004
$953$	−6.19246	−0.200593	−0.100297	−	0.994958i	$-0.531979\pi$
$953$	−6.19246	−0.200593	−0.100297	+	0.994958i	$0.531979\pi$
$954$	0	0
$955$	6.21661	0.201165
$956$	−4.52272	−0.146275
$957$	0	0
$958$	12.1408	0.392253
$959$	−11.0188	−0.355815
$960$	0	0
$961$	17.7479	0.572512
$962$	16.2875	0.525131
$963$	0	0
$964$	2.63410	0.0848387
$965$	−2.39627	−0.0771385
$966$	0	0
$967$	41.0765	1.32093	0.660465	−	0.750856i	$-0.270359\pi$
$967$	41.0765	1.32093	0.660465	+	0.750856i	$0.270359\pi$
$968$	−74.4714	−2.39360
$969$	0	0
$970$	11.9616	0.384064
$971$	−53.7396	−1.72459	−0.862293	−	0.506409i	$-0.830973\pi$
$971$	−53.7396	−1.72459	−0.862293	+	0.506409i	$0.830973\pi$
$972$	0	0
$973$	−21.3661	−0.684967
$974$	−13.3027	−0.426247
$975$	0	0
$976$	−38.4447	−1.23059
$977$	−8.05649	−0.257750	−0.128875	−	0.991661i	$-0.541137\pi$
$977$	−8.05649	−0.257750	−0.128875	+	0.991661i	$0.541137\pi$
$978$	0	0
$979$	−4.52472	−0.144611
$980$	−0.362299	−0.0115732
$981$	0	0
$982$	40.9763	1.30761
$983$	−49.0492	−1.56443	−0.782213	−	0.623011i	$-0.785909\pi$
$983$	−49.0492	−1.56443	−0.782213	+	0.623011i	$0.785909\pi$
$984$	0	0
$985$	0.400222	0.0127521
$986$	−9.59332	−0.305513
$987$	0	0
$988$	2.72336	0.0866417
$989$	14.1700	0.450581
$990$	0	0
$991$	10.6031	0.336819	0.168409	−	0.985717i	$-0.446137\pi$
$991$	10.6031	0.336819	0.168409	+	0.985717i	$0.446137\pi$
$992$	15.4743	0.491309
$993$	0	0
$994$	5.34390	0.169498
$995$	−19.5087	−0.618466
$996$	0	0
$997$	57.6296	1.82515	0.912574	−	0.408912i	$-0.134092\pi$
$997$	57.6296	1.82515	0.912574	+	0.408912i	$0.134092\pi$
$998$	14.0522	0.444814
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5103.2.a.i.1.18		27
3.2	odd	2		5103.2.a.f.1.10		27
27.4	even	9		189.2.v.a.43.7	yes	54
27.7	even	9		189.2.v.a.22.7	✓	54
27.20	odd	18		567.2.v.b.442.3		54
27.23	odd	18		567.2.v.b.127.3		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.a.22.7	✓	54	27.7	even	9
189.2.v.a.43.7	yes	54	27.4	even	9
567.2.v.b.127.3		54	27.23	odd	18
567.2.v.b.442.3		54	27.20	odd	18
5103.2.a.f.1.10		27	3.2	odd	2
5103.2.a.i.1.18		27	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 5103 = 3^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5103.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26564 q^{2} -0.398155 q^{4} +0.909945 q^{5} +1.00000 q^{7} -3.03520 q^{8} +1.15166 q^{10} -5.96120 q^{11} -3.32145 q^{13} +1.26564 q^{14} -3.04516 q^{16} -1.03692 q^{17} +2.05933 q^{19} -0.362299 q^{20} -7.54474 q^{22} +3.19257 q^{23} -4.17200 q^{25} -4.20376 q^{26} -0.398155 q^{28} +7.30992 q^{29} +6.98197 q^{31} +2.21632 q^{32} -1.31237 q^{34} +0.909945 q^{35} -3.87452 q^{37} +2.60637 q^{38} -2.76187 q^{40} +5.70449 q^{41} +4.43844 q^{43} +2.37348 q^{44} +4.04065 q^{46} +6.27814 q^{47} +1.00000 q^{49} -5.28025 q^{50} +1.32245 q^{52} +1.64075 q^{53} -5.42437 q^{55} -3.03520 q^{56} +9.25172 q^{58} +9.59238 q^{59} +12.6249 q^{61} +8.83666 q^{62} +8.89539 q^{64} -3.02233 q^{65} -10.0827 q^{67} +0.412856 q^{68} +1.15166 q^{70} +4.22229 q^{71} -4.58759 q^{73} -4.90375 q^{74} -0.819933 q^{76} -5.96120 q^{77} +14.3775 q^{79} -2.77093 q^{80} +7.21984 q^{82} +17.2062 q^{83} -0.943542 q^{85} +5.61747 q^{86} +18.0934 q^{88} +0.759027 q^{89} -3.32145 q^{91} -1.27114 q^{92} +7.94587 q^{94} +1.87388 q^{95} +10.3864 q^{97} +1.26564 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 9 q^{2} + 27 q^{4} + 12 q^{5} + 27 q^{7} + 27 q^{8} + 24 q^{11} + 9 q^{14} + 27 q^{16} + 30 q^{17} + 30 q^{20} + 39 q^{23} + 27 q^{25} + 9 q^{26} + 27 q^{28} + 39 q^{29} + 63 q^{32} + 12 q^{35} + 9 q^{38}+ \cdots + 9 q^{98}+O(q^{100}) \) 27 * q + 9 * q^2 + 27 * q^4 + 12 * q^5 + 27 * q^7 + 27 * q^8 + 24 * q^11 + 9 * q^14 + 27 * q^16 + 30 * q^17 + 30 * q^20 + 39 * q^23 + 27 * q^25 + 9 * q^26 + 27 * q^28 + 39 * q^29 + 63 * q^32 + 12 * q^35 + 9 * q^38 + 42 * q^41 + 42 * q^44 + 27 * q^47 + 27 * q^49 + 36 * q^50 + 66 * q^53 + 27 * q^56 + 18 * q^59 + 36 * q^62 + 27 * q^64 + 69 * q^65 + 21 * q^68 + 72 * q^71 + 54 * q^74 + 24 * q^77 + 21 * q^80 + 39 * q^83 + 27 * q^86 + 42 * q^89 + 75 * q^92 + 78 * q^95 + 9 * q^98

Introduction

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