LMFDB - Embedded newform 230.2.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [230,2,Mod(1,230)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("230.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$230 = 2 \cdot 5 \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	230.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$1.83655924649$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 9x + 12$ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.43163$ of defining polynomial
Character	$\chi$	$=$	230.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.43163 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.43163 q^{6} +3.08719 q^{7} +1.00000 q^{8} -0.950444 q^{9} -1.00000 q^{10} -6.46926 q^{11} +1.43163 q^{12} +3.95044 q^{13} +3.08719 q^{14} -1.43163 q^{15} +1.00000 q^{16} -3.43163 q^{17} -0.950444 q^{18} +3.08719 q^{19} -1.00000 q^{20} +4.41970 q^{21} -6.46926 q^{22} -1.00000 q^{23} +1.43163 q^{24} +1.00000 q^{25} +3.95044 q^{26} -5.65556 q^{27} +3.08719 q^{28} +0.863254 q^{29} -1.43163 q^{30} -5.95044 q^{31} +1.00000 q^{32} -9.26157 q^{33} -3.43163 q^{34} -3.08719 q^{35} -0.950444 q^{36} -7.03763 q^{37} +3.08719 q^{38} +5.65556 q^{39} -1.00000 q^{40} +5.60601 q^{41} +4.41970 q^{42} +8.00000 q^{43} -6.46926 q^{44} +0.950444 q^{45} -1.00000 q^{46} +3.90089 q^{47} +1.43163 q^{48} +2.53074 q^{49} +1.00000 q^{50} -4.91281 q^{51} +3.95044 q^{52} -6.00000 q^{53} -5.65556 q^{54} +6.46926 q^{55} +3.08719 q^{56} +4.41970 q^{57} +0.863254 q^{58} +6.86325 q^{59} -1.43163 q^{60} -13.5069 q^{61} -5.95044 q^{62} -2.93420 q^{63} +1.00000 q^{64} -3.95044 q^{65} -9.26157 q^{66} -10.0753 q^{67} -3.43163 q^{68} -1.43163 q^{69} -3.08719 q^{70} +2.56837 q^{71} -0.950444 q^{72} +5.90089 q^{73} -7.03763 q^{74} +1.43163 q^{75} +3.08719 q^{76} -19.9718 q^{77} +5.65556 q^{78} +15.8018 q^{79} -1.00000 q^{80} -5.24533 q^{81} +5.60601 q^{82} +9.03763 q^{83} +4.41970 q^{84} +3.43163 q^{85} +8.00000 q^{86} +1.23586 q^{87} -6.46926 q^{88} +16.7641 q^{89} +0.950444 q^{90} +12.1958 q^{91} -1.00000 q^{92} -8.51882 q^{93} +3.90089 q^{94} -3.08719 q^{95} +1.43163 q^{96} -14.2949 q^{97} +2.53074 q^{98} +6.14867 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{2} + q^{3} + 3 q^{4} - 3 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9} - 3 q^{10} + 3 q^{11} + q^{12} - q^{13} + 3 q^{14} - q^{15} + 3 q^{16} - 7 q^{17} + 10 q^{18} + 3 q^{19} - 3 q^{20}+ \cdots + 57 q^{99}+O(q^{100})$ 3 * q + 3 * q^2 + q^3 + 3 * q^4 - 3 * q^5 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9 - 3 * q^10 + 3 * q^11 + q^12 - q^13 + 3 * q^14 - q^15 + 3 * q^16 - 7 * q^17 + 10 * q^18 + 3 * q^19 - 3 * q^20 - 22 * q^21 + 3 * q^22 - 3 * q^23 + q^24 + 3 * q^25 - q^26 - 14 * q^27 + 3 * q^28 - 4 * q^29 - q^30 - 5 * q^31 + 3 * q^32 - 9 * q^33 - 7 * q^34 - 3 * q^35 + 10 * q^36 - 2 * q^37 + 3 * q^38 + 14 * q^39 - 3 * q^40 + q^41 - 22 * q^42 + 24 * q^43 + 3 * q^44 - 10 * q^45 - 3 * q^46 - 14 * q^47 + q^48 + 30 * q^49 + 3 * q^50 - 21 * q^51 - q^52 - 18 * q^53 - 14 * q^54 - 3 * q^55 + 3 * q^56 - 22 * q^57 - 4 * q^58 + 14 * q^59 - q^60 + q^61 - 5 * q^62 + 8 * q^63 + 3 * q^64 + q^65 - 9 * q^66 + 8 * q^67 - 7 * q^68 - q^69 - 3 * q^70 + 11 * q^71 + 10 * q^72 - 8 * q^73 - 2 * q^74 + q^75 + 3 * q^76 - 24 * q^77 + 14 * q^78 - 4 * q^79 - 3 * q^80 + 7 * q^81 + q^82 + 8 * q^83 - 22 * q^84 + 7 * q^85 + 24 * q^86 + 36 * q^87 + 3 * q^88 + 18 * q^89 - 10 * q^90 + q^91 - 3 * q^92 - 16 * q^93 - 14 * q^94 - 3 * q^95 + q^96 - 33 * q^97 + 30 * q^98 + 57 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.43163	0.826550	0.413275	−	0.910606i	$-0.364385\pi$
$3$	1.43163	0.826550	0.413275	+	0.910606i	$0.364385\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.43163	0.584459
$7$	3.08719	1.16685	0.583424	−	0.812168i	$-0.301713\pi$
$7$	3.08719	1.16685	0.583424	+	0.812168i	$0.301713\pi$
$8$	1.00000	0.353553
$9$	−0.950444	−0.316815
$10$	−1.00000	−0.316228
$11$	−6.46926	−1.95056	−0.975278	−	0.220983i	$-0.929074\pi$
$11$	−6.46926	−1.95056	−0.975278	+	0.220983i	$0.929074\pi$
$12$	1.43163	0.413275
$13$	3.95044	1.09566	0.547828	−	0.836591i	$-0.315455\pi$
$13$	3.95044	1.09566	0.547828	+	0.836591i	$0.315455\pi$
$14$	3.08719	0.825086
$15$	−1.43163	−0.369645
$16$	1.00000	0.250000
$17$	−3.43163	−0.832292	−0.416146	−	0.909298i	$-0.636619\pi$
$17$	−3.43163	−0.832292	−0.416146	+	0.909298i	$0.636619\pi$
$18$	−0.950444	−0.224022
$19$	3.08719	0.708250	0.354125	−	0.935198i	$-0.384779\pi$
$19$	3.08719	0.708250	0.354125	+	0.935198i	$0.384779\pi$
$20$	−1.00000	−0.223607
$21$	4.41970	0.964459
$22$	−6.46926	−1.37925
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.43163	0.292230
$25$	1.00000	0.200000
$26$	3.95044	0.774746
$27$	−5.65556	−1.08841
$28$	3.08719	0.583424
$29$	0.863254	0.160302	0.0801511	−	0.996783i	$-0.474460\pi$
$29$	0.863254	0.160302	0.0801511	+	0.996783i	$0.474460\pi$
$30$	−1.43163	−0.261378
$31$	−5.95044	−1.06873	−0.534366	−	0.845253i	$-0.679449\pi$
$31$	−5.95044	−1.06873	−0.534366	+	0.845253i	$0.679449\pi$
$32$	1.00000	0.176777
$33$	−9.26157	−1.61223
$34$	−3.43163	−0.588519
$35$	−3.08719	−0.521830
$36$	−0.950444	−0.158407
$37$	−7.03763	−1.15698	−0.578490	−	0.815690i	$-0.696358\pi$
$37$	−7.03763	−1.15698	−0.578490	+	0.815690i	$0.696358\pi$
$38$	3.08719	0.500808
$39$	5.65556	0.905615
$40$	−1.00000	−0.158114
$41$	5.60601	0.875511	0.437756	−	0.899094i	$-0.355774\pi$
$41$	5.60601	0.875511	0.437756	+	0.899094i	$0.355774\pi$
$42$	4.41970	0.681975
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−6.46926	−0.975278
$45$	0.950444	0.141684
$46$	−1.00000	−0.147442
$47$	3.90089	0.569003	0.284501	−	0.958676i	$-0.408172\pi$
$47$	3.90089	0.569003	0.284501	+	0.958676i	$0.408172\pi$
$48$	1.43163	0.206638
$49$	2.53074	0.361534
$50$	1.00000	0.141421
$51$	−4.91281	−0.687931
$52$	3.95044	0.547828
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−5.65556	−0.769625
$55$	6.46926	0.872315
$56$	3.08719	0.412543
$57$	4.41970	0.585404
$58$	0.863254	0.113351
$59$	6.86325	0.893520	0.446760	−	0.894654i	$-0.352578\pi$
$59$	6.86325	0.893520	0.446760	+	0.894654i	$0.352578\pi$
$60$	−1.43163	−0.184822
$61$	−13.5069	−1.72938	−0.864690	−	0.502305i	$-0.832485\pi$
$61$	−13.5069	−1.72938	−0.864690	+	0.502305i	$0.832485\pi$
$62$	−5.95044	−0.755707
$63$	−2.93420	−0.369674
$64$	1.00000	0.125000
$65$	−3.95044	−0.489992
$66$	−9.26157	−1.14002
$67$	−10.0753	−1.23089	−0.615445	−	0.788180i	$-0.711024\pi$
$67$	−10.0753	−1.23089	−0.615445	+	0.788180i	$0.711024\pi$
$68$	−3.43163	−0.416146
$69$	−1.43163	−0.172348
$70$	−3.08719	−0.368990
$71$	2.56837	0.304810	0.152405	−	0.988318i	$-0.451298\pi$
$71$	2.56837	0.304810	0.152405	+	0.988318i	$0.451298\pi$
$72$	−0.950444	−0.112011
$73$	5.90089	0.690647	0.345323	−	0.938484i	$-0.387769\pi$
$73$	5.90089	0.690647	0.345323	+	0.938484i	$0.387769\pi$
$74$	−7.03763	−0.818108
$75$	1.43163	0.165310
$76$	3.08719	0.354125
$77$	−19.9718	−2.27600
$78$	5.65556	0.640366
$79$	15.8018	1.77784	0.888919	−	0.458064i	$-0.151457\pi$
$79$	15.8018	1.77784	0.888919	+	0.458064i	$0.151457\pi$
$80$	−1.00000	−0.111803
$81$	−5.24533	−0.582814
$82$	5.60601	0.619080
$83$	9.03763	0.992009	0.496005	−	0.868320i	$-0.334800\pi$
$83$	9.03763	0.992009	0.496005	+	0.868320i	$0.334800\pi$
$84$	4.41970	0.482229
$85$	3.43163	0.372212
$86$	8.00000	0.862662
$87$	1.23586	0.132498
$88$	−6.46926	−0.689625
$89$	16.7641	1.77700	0.888498	−	0.458881i	$-0.151750\pi$
$89$	16.7641	1.77700	0.888498	+	0.458881i	$0.151750\pi$
$90$	0.950444	0.100186
$91$	12.1958	1.27846
$92$	−1.00000	−0.104257
$93$	−8.51882	−0.883360
$94$	3.90089	0.402346
$95$	−3.08719	−0.316739
$96$	1.43163	0.146115
$97$	−14.2949	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$97$	−14.2949	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$98$	2.53074	0.255643
$99$	6.14867	0.617964
$100$	1.00000	0.100000
$101$	0.863254	0.0858970	0.0429485	−	0.999077i	$-0.486325\pi$
$101$	0.863254	0.0858970	0.0429485	+	0.999077i	$0.486325\pi$
$102$	−4.91281	−0.486441
$103$	1.53074	0.150828	0.0754141	−	0.997152i	$-0.475972\pi$
$103$	1.53074	0.150828	0.0754141	+	0.997152i	$0.475972\pi$
$104$	3.95044	0.387373
$105$	−4.41970	−0.431319
$106$	−6.00000	−0.582772
$107$	17.6274	1.70410	0.852052	−	0.523457i	$-0.175358\pi$
$107$	17.6274	1.70410	0.852052	+	0.523457i	$0.175358\pi$
$108$	−5.65556	−0.544207
$109$	−2.91281	−0.278997	−0.139498	−	0.990222i	$-0.544549\pi$
$109$	−2.91281	−0.278997	−0.139498	+	0.990222i	$0.544549\pi$
$110$	6.46926	0.616820
$111$	−10.0753	−0.956302
$112$	3.08719	0.291712
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.41970	0.413943
$115$	1.00000	0.0932505
$116$	0.863254	0.0801511
$117$	−3.75467	−0.347120
$118$	6.86325	0.631814
$119$	−10.5941	−0.971158
$120$	−1.43163	−0.130689
$121$	30.8513	2.80467
$122$	−13.5069	−1.22286
$123$	8.02571	0.723654
$124$	−5.95044	−0.534366
$125$	−1.00000	−0.0894427
$126$	−2.93420	−0.261399
$127$	20.9385	1.85799	0.928997	−	0.370088i	$-0.120673\pi$
$127$	20.9385	1.85799	0.928997	+	0.370088i	$0.120673\pi$
$128$	1.00000	0.0883883
$129$	11.4530	1.00838
$130$	−3.95044	−0.346477
$131$	10.7641	0.940467	0.470234	−	0.882542i	$-0.344170\pi$
$131$	10.7641	0.940467	0.470234	+	0.882542i	$0.344170\pi$
$132$	−9.26157	−0.806116
$133$	9.53074	0.826420
$134$	−10.0753	−0.870370
$135$	5.65556	0.486753
$136$	−3.43163	−0.294260
$137$	3.26157	0.278655	0.139327	−	0.990246i	$-0.455506\pi$
$137$	3.26157	0.278655	0.139327	+	0.990246i	$0.455506\pi$
$138$	−1.43163	−0.121868
$139$	−16.9385	−1.43671	−0.718353	−	0.695678i	$-0.755104\pi$
$139$	−16.9385	−1.43671	−0.718353	+	0.695678i	$0.755104\pi$
$140$	−3.08719	−0.260915
$141$	5.58462	0.470310
$142$	2.56837	0.215533
$143$	−25.5565	−2.13714
$144$	−0.950444	−0.0792037
$145$	−0.863254	−0.0716894
$146$	5.90089	0.488361
$147$	3.62308	0.298826
$148$	−7.03763	−0.578490
$149$	3.26157	0.267198	0.133599	−	0.991035i	$-0.457347\pi$
$149$	3.26157	0.267198	0.133599	+	0.991035i	$0.457347\pi$
$150$	1.43163	0.116892
$151$	0.294881	0.0239971	0.0119986	−	0.999928i	$-0.496181\pi$
$151$	0.294881	0.0239971	0.0119986	+	0.999928i	$0.496181\pi$
$152$	3.08719	0.250404
$153$	3.26157	0.263682
$154$	−19.9718	−1.60938
$155$	5.95044	0.477951
$156$	5.65556	0.452807
$157$	7.13675	0.569574	0.284787	−	0.958591i	$-0.408077\pi$
$157$	7.13675	0.569574	0.284787	+	0.958591i	$0.408077\pi$
$158$	15.8018	1.25712
$159$	−8.58976	−0.681212
$160$	−1.00000	−0.0790569
$161$	−3.08719	−0.243305
$162$	−5.24533	−0.412112
$163$	−8.12482	−0.636385	−0.318193	−	0.948026i	$-0.603076\pi$
$163$	−8.12482	−0.636385	−0.318193	+	0.948026i	$0.603076\pi$
$164$	5.60601	0.437756
$165$	9.26157	0.721012
$166$	9.03763	0.701456
$167$	−7.80178	−0.603719	−0.301860	−	0.953352i	$-0.597607\pi$
$167$	−7.80178	−0.603719	−0.301860	+	0.953352i	$0.597607\pi$
$168$	4.41970	0.340988
$169$	2.60601	0.200462
$170$	3.43163	0.263194
$171$	−2.93420	−0.224384
$172$	8.00000	0.609994
$173$	−19.3325	−1.46982	−0.734912	−	0.678163i	$-0.762777\pi$
$173$	−19.3325	−1.46982	−0.734912	+	0.678163i	$0.762777\pi$
$174$	1.23586	0.0936902
$175$	3.08719	0.233370
$176$	−6.46926	−0.487639
$177$	9.82562	0.738539
$178$	16.7641	1.25653
$179$	2.17438	0.162521	0.0812604	−	0.996693i	$-0.474105\pi$
$179$	2.17438	0.162521	0.0812604	+	0.996693i	$0.474105\pi$
$180$	0.950444	0.0708419
$181$	−7.26157	−0.539748	−0.269874	−	0.962896i	$-0.586982\pi$
$181$	−7.26157	−0.539748	−0.269874	+	0.962896i	$0.586982\pi$
$182$	12.1958	0.904011
$183$	−19.3368	−1.42942
$184$	−1.00000	−0.0737210
$185$	7.03763	0.517417
$186$	−8.51882	−0.624630
$187$	22.2001	1.62343
$188$	3.90089	0.284501
$189$	−17.4598	−1.27001
$190$	−3.08719	−0.223968
$191$	8.58976	0.621533	0.310767	−	0.950486i	$-0.399414\pi$
$191$	8.58976	0.621533	0.310767	+	0.950486i	$0.399414\pi$
$192$	1.43163	0.103319
$193$	2.44787	0.176202	0.0881008	−	0.996112i	$-0.471920\pi$
$193$	2.44787	0.176202	0.0881008	+	0.996112i	$0.471920\pi$
$194$	−14.2949	−1.02631
$195$	−5.65556	−0.405003
$196$	2.53074	0.180767
$197$	−10.7428	−0.765389	−0.382695	−	0.923875i	$-0.625004\pi$
$197$	−10.7428	−0.765389	−0.382695	+	0.923875i	$0.625004\pi$
$198$	6.14867	0.436967
$199$	−11.3111	−0.801824	−0.400912	−	0.916116i	$-0.631307\pi$
$199$	−11.3111	−0.801824	−0.400912	+	0.916116i	$0.631307\pi$
$200$	1.00000	0.0707107
$201$	−14.4240	−1.01739
$202$	0.863254	0.0607384
$203$	2.66503	0.187048
$204$	−4.91281	−0.343966
$205$	−5.60601	−0.391540
$206$	1.53074	0.106652
$207$	0.950444	0.0660604
$208$	3.95044	0.273914
$209$	−19.9718	−1.38148
$210$	−4.41970	−0.304989
$211$	23.1129	1.59116	0.795579	−	0.605850i	$-0.207167\pi$
$211$	23.1129	1.59116	0.795579	+	0.605850i	$0.207167\pi$
$212$	−6.00000	−0.412082
$213$	3.67695	0.251941
$214$	17.6274	1.20498
$215$	−8.00000	−0.545595
$216$	−5.65556	−0.384812
$217$	−18.3701	−1.24705
$218$	−2.91281	−0.197280
$219$	8.44787	0.570854
$220$	6.46926	0.436157
$221$	−13.5565	−0.911906
$222$	−10.0753	−0.676208
$223$	−5.72651	−0.383475	−0.191738	−	0.981446i	$-0.561412\pi$
$223$	−5.72651	−0.383475	−0.191738	+	0.981446i	$0.561412\pi$
$224$	3.08719	0.206272
$225$	−0.950444	−0.0633629
$226$	−6.00000	−0.399114
$227$	−17.6274	−1.16997	−0.584986	−	0.811044i	$-0.698900\pi$
$227$	−17.6274	−1.16997	−0.584986	+	0.811044i	$0.698900\pi$
$228$	4.41970	0.292702
$229$	−16.0753	−1.06228	−0.531142	−	0.847283i	$-0.678237\pi$
$229$	−16.0753	−1.06228	−0.531142	+	0.847283i	$0.678237\pi$
$230$	1.00000	0.0659380
$231$	−28.5922	−1.88123
$232$	0.863254	0.0566754
$233$	−6.44787	−0.422414	−0.211207	−	0.977441i	$-0.567739\pi$
$233$	−6.44787	−0.422414	−0.211207	+	0.977441i	$0.567739\pi$
$234$	−3.75467	−0.245451
$235$	−3.90089	−0.254466
$236$	6.86325	0.446760
$237$	22.6222	1.46947
$238$	−10.5941	−0.686712
$239$	4.34876	0.281298	0.140649	−	0.990060i	$-0.455081\pi$
$239$	4.34876	0.281298	0.140649	+	0.990060i	$0.455081\pi$
$240$	−1.43163	−0.0924111
$241$	0.764142	0.0492227	0.0246114	−	0.999697i	$-0.492165\pi$
$241$	0.764142	0.0492227	0.0246114	+	0.999697i	$0.492165\pi$
$242$	30.8513	1.98320
$243$	9.45734	0.606688
$244$	−13.5069	−0.864690
$245$	−2.53074	−0.161683
$246$	8.02571	0.511701
$247$	12.1958	0.775998
$248$	−5.95044	−0.377854
$249$	12.9385	0.819945
$250$	−1.00000	−0.0632456
$251$	−22.5402	−1.42273	−0.711363	−	0.702825i	$-0.751922\pi$
$251$	−22.5402	−1.42273	−0.711363	+	0.702825i	$0.751922\pi$
$252$	−2.93420	−0.184837
$253$	6.46926	0.406719
$254$	20.9385	1.31380
$255$	4.91281	0.307652
$256$	1.00000	0.0625000
$257$	−9.90089	−0.617600	−0.308800	−	0.951127i	$-0.599927\pi$
$257$	−9.90089	−0.617600	−0.308800	+	0.951127i	$0.599927\pi$
$258$	11.4530	0.713034
$259$	−21.7265	−1.35002
$260$	−3.95044	−0.244996
$261$	−0.820475	−0.0507861
$262$	10.7641	0.665011
$263$	−7.25725	−0.447501	−0.223751	−	0.974646i	$-0.571830\pi$
$263$	−7.25725	−0.447501	−0.223751	+	0.974646i	$0.571830\pi$
$264$	−9.26157	−0.570010
$265$	6.00000	0.368577
$266$	9.53074	0.584367
$267$	24.0000	1.46878
$268$	−10.0753	−0.615445
$269$	−6.93852	−0.423049	−0.211525	−	0.977373i	$-0.567843\pi$
$269$	−6.93852	−0.423049	−0.211525	+	0.977373i	$0.567843\pi$
$270$	5.65556	0.344187
$271$	−10.2992	−0.625632	−0.312816	−	0.949814i	$-0.601272\pi$
$271$	−10.2992	−0.625632	−0.312816	+	0.949814i	$0.601272\pi$
$272$	−3.43163	−0.208073
$273$	17.4598	1.05671
$274$	3.26157	0.197039
$275$	−6.46926	−0.390111
$276$	−1.43163	−0.0861738
$277$	−17.8018	−1.06961	−0.534803	−	0.844977i	$-0.679614\pi$
$277$	−17.8018	−1.06961	−0.534803	+	0.844977i	$0.679614\pi$
$278$	−16.9385	−1.01590
$279$	5.65556	0.338590
$280$	−3.08719	−0.184495
$281$	−18.9385	−1.12978	−0.564889	−	0.825167i	$-0.691081\pi$
$281$	−18.9385	−1.12978	−0.564889	+	0.825167i	$0.691081\pi$
$282$	5.58462	0.332559
$283$	12.6889	0.754275	0.377138	−	0.926157i	$-0.376908\pi$
$283$	12.6889	0.754275	0.377138	+	0.926157i	$0.376908\pi$
$284$	2.56837	0.152405
$285$	−4.41970	−0.261801
$286$	−25.5565	−1.51118
$287$	17.3068	1.02159
$288$	−0.950444	−0.0560054
$289$	−5.22394	−0.307290
$290$	−0.863254	−0.0506920
$291$	−20.4649	−1.19968
$292$	5.90089	0.345323
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	3.62308	0.211302
$295$	−6.86325	−0.399594
$296$	−7.03763	−0.409054
$297$	36.5873	2.12301
$298$	3.26157	0.188938
$299$	−3.95044	−0.228460
$300$	1.43163	0.0826550
$301$	24.6975	1.42354
$302$	0.294881	0.0169685
$303$	1.23586	0.0709982
$304$	3.08719	0.177062
$305$	13.5069	0.773402
$306$	3.26157	0.186451
$307$	−22.9599	−1.31039	−0.655196	−	0.755459i	$-0.727414\pi$
$307$	−22.9599	−1.31039	−0.655196	+	0.755459i	$0.727414\pi$
$308$	−19.9718	−1.13800
$309$	2.19145	0.124667
$310$	5.95044	0.337962
$311$	18.0753	1.02495	0.512477	−	0.858701i	$-0.328728\pi$
$311$	18.0753	1.02495	0.512477	+	0.858701i	$0.328728\pi$
$312$	5.65556	0.320183
$313$	15.1625	0.857033	0.428516	−	0.903534i	$-0.359036\pi$
$313$	15.1625	0.857033	0.428516	+	0.903534i	$0.359036\pi$
$314$	7.13675	0.402750
$315$	2.93420	0.165323
$316$	15.8018	0.888919
$317$	14.0257	0.787762	0.393881	−	0.919161i	$-0.371132\pi$
$317$	14.0257	0.787762	0.393881	+	0.919161i	$0.371132\pi$
$318$	−8.58976	−0.481690
$319$	−5.58462	−0.312678
$320$	−1.00000	−0.0559017
$321$	25.2359	1.40853
$322$	−3.08719	−0.172042
$323$	−10.5941	−0.589471
$324$	−5.24533	−0.291407
$325$	3.95044	0.219131
$326$	−8.12482	−0.449992
$327$	−4.17006	−0.230605
$328$	5.60601	0.309540
$329$	12.0428	0.663940
$330$	9.26157	0.509833
$331$	−24.7403	−1.35985	−0.679925	−	0.733282i	$-0.737988\pi$
$331$	−24.7403	−1.35985	−0.679925	+	0.733282i	$0.737988\pi$
$332$	9.03763	0.496005
$333$	6.68888	0.366548
$334$	−7.80178	−0.426894
$335$	10.0753	0.550471
$336$	4.41970	0.241115
$337$	14.7146	0.801555	0.400777	−	0.916176i	$-0.368740\pi$
$337$	14.7146	0.801555	0.400777	+	0.916176i	$0.368740\pi$
$338$	2.60601	0.141748
$339$	−8.58976	−0.466532
$340$	3.43163	0.186106
$341$	38.4950	2.08462
$342$	−2.93420	−0.158663
$343$	−13.7975	−0.744993
$344$	8.00000	0.431331
$345$	1.43163	0.0770762
$346$	−19.3325	−1.03932
$347$	9.43163	0.506316	0.253158	−	0.967425i	$-0.418531\pi$
$347$	9.43163	0.506316	0.253158	+	0.967425i	$0.418531\pi$
$348$	1.23586	0.0662490
$349$	18.3488	0.982187	0.491093	−	0.871107i	$-0.336597\pi$
$349$	18.3488	0.982187	0.491093	+	0.871107i	$0.336597\pi$
$350$	3.08719	0.165017
$351$	−22.3420	−1.19253
$352$	−6.46926	−0.344813
$353$	−33.1129	−1.76242	−0.881211	−	0.472723i	$-0.843271\pi$
$353$	−33.1129	−1.76242	−0.881211	+	0.472723i	$0.843271\pi$
$354$	9.82562	0.522226
$355$	−2.56837	−0.136315
$356$	16.7641	0.888498
$357$	−15.1668	−0.802711
$358$	2.17438	0.114920
$359$	33.0376	1.74366	0.871830	−	0.489809i	$-0.162933\pi$
$359$	33.0376	1.74366	0.871830	+	0.489809i	$0.162933\pi$
$360$	0.950444	0.0500928
$361$	−9.46926	−0.498382
$362$	−7.26157	−0.381660
$363$	44.1676	2.31820
$364$	12.1958	0.639232
$365$	−5.90089	−0.308867
$366$	−19.3368	−1.01075
$367$	−2.27349	−0.118675	−0.0593376	−	0.998238i	$-0.518899\pi$
$367$	−2.27349	−0.118675	−0.0593376	+	0.998238i	$0.518899\pi$
$368$	−1.00000	−0.0521286
$369$	−5.32819	−0.277375
$370$	7.03763	0.365869
$371$	−18.5231	−0.961673
$372$	−8.51882	−0.441680
$373$	23.9762	1.24144	0.620719	−	0.784033i	$-0.286841\pi$
$373$	23.9762	1.24144	0.620719	+	0.784033i	$0.286841\pi$
$374$	22.2001	1.14794
$375$	−1.43163	−0.0739289
$376$	3.90089	0.201173
$377$	3.41024	0.175636
$378$	−17.4598	−0.898035
$379$	−13.8795	−0.712942	−0.356471	−	0.934306i	$-0.616020\pi$
$379$	−13.8795	−0.712942	−0.356471	+	0.934306i	$0.616020\pi$
$380$	−3.08719	−0.158369
$381$	29.9762	1.53572
$382$	8.58976	0.439490
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.43163	0.0730574
$385$	19.9718	1.01786
$386$	2.44787	0.124593
$387$	−7.60355	−0.386510
$388$	−14.2949	−0.725713
$389$	−26.6436	−1.35089	−0.675443	−	0.737412i	$-0.736048\pi$
$389$	−26.6436	−1.35089	−0.675443	+	0.737412i	$0.736048\pi$
$390$	−5.65556	−0.286381
$391$	3.43163	0.173545
$392$	2.53074	0.127822
$393$	15.4102	0.777344
$394$	−10.7428	−0.541212
$395$	−15.8018	−0.795074
$396$	6.14867	0.308982
$397$	−8.66749	−0.435009	−0.217504	−	0.976059i	$-0.569792\pi$
$397$	−8.66749	−0.435009	−0.217504	+	0.976059i	$0.569792\pi$
$398$	−11.3111	−0.566975
$399$	13.6445	0.683078
$400$	1.00000	0.0500000
$401$	39.9762	1.99631	0.998157	−	0.0606854i	$-0.0193286\pi$
$401$	39.9762	1.99631	0.998157	+	0.0606854i	$0.0193286\pi$
$402$	−14.4240	−0.719405
$403$	−23.5069	−1.17096
$404$	0.863254	0.0429485
$405$	5.24533	0.260642
$406$	2.66503	0.132263
$407$	45.5283	2.25675
$408$	−4.91281	−0.243220
$409$	30.1248	1.48958	0.744788	−	0.667301i	$-0.232550\pi$
$409$	30.1248	1.48958	0.744788	+	0.667301i	$0.232550\pi$
$410$	−5.60601	−0.276861
$411$	4.66935	0.230322
$412$	1.53074	0.0754141
$413$	21.1882	1.04260
$414$	0.950444	0.0467118
$415$	−9.03763	−0.443640
$416$	3.95044	0.193686
$417$	−24.2496	−1.18751
$418$	−19.9718	−0.976854
$419$	25.8770	1.26418	0.632088	−	0.774897i	$-0.282198\pi$
$419$	25.8770	1.26418	0.632088	+	0.774897i	$0.282198\pi$
$420$	−4.41970	−0.215659
$421$	−10.7146	−0.522197	−0.261098	−	0.965312i	$-0.584085\pi$
$421$	−10.7146	−0.522197	−0.261098	+	0.965312i	$0.584085\pi$
$422$	23.1129	1.12512
$423$	−3.70757	−0.180268
$424$	−6.00000	−0.291386
$425$	−3.43163	−0.166458
$426$	3.67695	0.178149
$427$	−41.6983	−2.01792
$428$	17.6274	0.852052
$429$	−36.5873	−1.76645
$430$	−8.00000	−0.385794
$431$	32.2496	1.55341	0.776705	−	0.629864i	$-0.216889\pi$
$431$	32.2496	1.55341	0.776705	+	0.629864i	$0.216889\pi$
$432$	−5.65556	−0.272103
$433$	20.6393	0.991862	0.495931	−	0.868362i	$-0.334827\pi$
$433$	20.6393	0.991862	0.495931	+	0.868362i	$0.334827\pi$
$434$	−18.3701	−0.881795
$435$	−1.23586	−0.0592549
$436$	−2.91281	−0.139498
$437$	−3.08719	−0.147680
$438$	8.44787	0.403655
$439$	−16.2239	−0.774326	−0.387163	−	0.922011i	$-0.626545\pi$
$439$	−16.2239	−0.774326	−0.387163	+	0.922011i	$0.626545\pi$
$440$	6.46926	0.308410
$441$	−2.40533	−0.114539
$442$	−13.5565	−0.644815
$443$	−15.6770	−0.744834	−0.372417	−	0.928065i	$-0.621471\pi$
$443$	−15.6770	−0.744834	−0.372417	+	0.928065i	$0.621471\pi$
$444$	−10.0753	−0.478151
$445$	−16.7641	−0.794697
$446$	−5.72651	−0.271158
$447$	4.66935	0.220853
$448$	3.08719	0.145856
$449$	22.5727	1.06527	0.532636	−	0.846345i	$-0.321202\pi$
$449$	22.5727	1.06527	0.532636	+	0.846345i	$0.321202\pi$
$450$	−0.950444	−0.0448044
$451$	−36.2667	−1.70773
$452$	−6.00000	−0.282216
$453$	0.422160	0.0198348
$454$	−17.6274	−0.827295
$455$	−12.1958	−0.571746
$456$	4.41970	0.206972
$457$	−1.45302	−0.0679693	−0.0339846	−	0.999422i	$-0.510820\pi$
$457$	−1.45302	−0.0679693	−0.0339846	+	0.999422i	$0.510820\pi$
$458$	−16.0753	−0.751148
$459$	19.4078	0.905878
$460$	1.00000	0.0466252
$461$	−29.7027	−1.38339	−0.691695	−	0.722189i	$-0.743136\pi$
$461$	−29.7027	−1.38339	−0.691695	+	0.722189i	$0.743136\pi$
$462$	−28.5922	−1.33023
$463$	22.9624	1.06715	0.533576	−	0.845752i	$-0.320848\pi$
$463$	22.9624	1.06715	0.533576	+	0.845752i	$0.320848\pi$
$464$	0.863254	0.0400756
$465$	8.51882	0.395051
$466$	−6.44787	−0.298692
$467$	9.48550	0.438937	0.219468	−	0.975620i	$-0.429568\pi$
$467$	9.48550	0.438937	0.219468	+	0.975620i	$0.429568\pi$
$468$	−3.75467	−0.173560
$469$	−31.1043	−1.43626
$470$	−3.90089	−0.179935
$471$	10.2172	0.470782
$472$	6.86325	0.315907
$473$	−51.7541	−2.37966
$474$	22.6222	1.03907
$475$	3.08719	0.141650
$476$	−10.5941	−0.485579
$477$	5.70266	0.261107
$478$	4.34876	0.198908
$479$	−30.5659	−1.39659	−0.698296	−	0.715809i	$-0.746058\pi$
$479$	−30.5659	−1.39659	−0.698296	+	0.715809i	$0.746058\pi$
$480$	−1.43163	−0.0653445
$481$	−27.8018	−1.26765
$482$	0.764142	0.0348057
$483$	−4.41970	−0.201104
$484$	30.8513	1.40233
$485$	14.2949	0.649097
$486$	9.45734	0.428994
$487$	−21.1367	−0.957797	−0.478899	−	0.877870i	$-0.658964\pi$
$487$	−21.1367	−0.957797	−0.478899	+	0.877870i	$0.658964\pi$
$488$	−13.5069	−0.611428
$489$	−11.6317	−0.526004
$490$	−2.53074	−0.114327
$491$	−28.0514	−1.26594	−0.632971	−	0.774175i	$-0.718165\pi$
$491$	−28.0514	−1.26594	−0.632971	+	0.774175i	$0.718165\pi$
$492$	8.02571	0.361827
$493$	−2.96237	−0.133418
$494$	12.1958	0.548714
$495$	−6.14867	−0.276362
$496$	−5.95044	−0.267183
$497$	7.92905	0.355667
$498$	12.9385	0.579789
$499$	3.80178	0.170191	0.0850954	−	0.996373i	$-0.472880\pi$
$499$	3.80178	0.170191	0.0850954	+	0.996373i	$0.472880\pi$
$500$	−1.00000	−0.0447214
$501$	−11.1692	−0.499005
$502$	−22.5402	−1.00602
$503$	19.0872	0.851056	0.425528	−	0.904945i	$-0.360088\pi$
$503$	19.0872	0.851056	0.425528	+	0.904945i	$0.360088\pi$
$504$	−2.93420	−0.130700
$505$	−0.863254	−0.0384143
$506$	6.46926	0.287594
$507$	3.73083	0.165692
$508$	20.9385	0.928997
$509$	−9.41024	−0.417101	−0.208551	−	0.978012i	$-0.566875\pi$
$509$	−9.41024	−0.417101	−0.208551	+	0.978012i	$0.566875\pi$
$510$	4.91281	0.217543
$511$	18.2172	0.805880
$512$	1.00000	0.0441942
$513$	−17.4598	−0.770869
$514$	−9.90089	−0.436709
$515$	−1.53074	−0.0674524
$516$	11.4530	0.504191
$517$	−25.2359	−1.10987
$518$	−21.7265	−0.954608
$519$	−27.6770	−1.21488
$520$	−3.95044	−0.173238
$521$	25.8018	1.13040	0.565198	−	0.824955i	$-0.308800\pi$
$521$	25.8018	1.13040	0.565198	+	0.824955i	$0.308800\pi$
$522$	−0.820475	−0.0359112
$523$	−19.1129	−0.835749	−0.417874	−	0.908505i	$-0.637225\pi$
$523$	−19.1129	−0.835749	−0.417874	+	0.908505i	$0.637225\pi$
$524$	10.7641	0.470234
$525$	4.41970	0.192892
$526$	−7.25725	−0.316431
$527$	20.4197	0.889496
$528$	−9.26157	−0.403058
$529$	1.00000	0.0434783
$530$	6.00000	0.260623
$531$	−6.52314	−0.283080
$532$	9.53074	0.413210
$533$	22.1462	0.959259
$534$	24.0000	1.03858
$535$	−17.6274	−0.762099
$536$	−10.0753	−0.435185
$537$	3.11290	0.134332
$538$	−6.93852	−0.299141
$539$	−16.3720	−0.705193
$540$	5.65556	0.243377
$541$	30.8394	1.32589	0.662945	−	0.748668i	$-0.269306\pi$
$541$	30.8394	1.32589	0.662945	+	0.748668i	$0.269306\pi$
$542$	−10.2992	−0.442389
$543$	−10.3959	−0.446129
$544$	−3.43163	−0.147130
$545$	2.91281	0.124771
$546$	17.4598	0.747210
$547$	13.8513	0.592240	0.296120	−	0.955151i	$-0.404307\pi$
$547$	13.8513	0.592240	0.296120	+	0.955151i	$0.404307\pi$
$548$	3.26157	0.139327
$549$	12.8375	0.547893
$550$	−6.46926	−0.275850
$551$	2.66503	0.113534
$552$	−1.43163	−0.0609341
$553$	48.7831	2.07447
$554$	−17.8018	−0.756325
$555$	10.0753	0.427671
$556$	−16.9385	−0.718353
$557$	−28.7641	−1.21878	−0.609388	−	0.792872i	$-0.708585\pi$
$557$	−28.7641	−1.21878	−0.609388	+	0.792872i	$0.708585\pi$
$558$	5.65556	0.239419
$559$	31.6036	1.33669
$560$	−3.08719	−0.130458
$561$	31.7823	1.34185
$562$	−18.9385	−0.798873
$563$	36.1505	1.52356	0.761782	−	0.647834i	$-0.224325\pi$
$563$	36.1505	1.52356	0.761782	+	0.647834i	$0.224325\pi$
$564$	5.58462	0.235155
$565$	6.00000	0.252422
$566$	12.6889	0.533353
$567$	−16.1933	−0.680055
$568$	2.56837	0.107767
$569$	−10.3488	−0.433843	−0.216921	−	0.976189i	$-0.569601\pi$
$569$	−10.3488	−0.433843	−0.216921	+	0.976189i	$0.569601\pi$
$570$	−4.41970	−0.185121
$571$	19.6060	0.820486	0.410243	−	0.911976i	$-0.365444\pi$
$571$	19.6060	0.820486	0.410243	+	0.911976i	$0.365444\pi$
$572$	−25.5565	−1.06857
$573$	12.2973	0.513729
$574$	17.3068	0.722372
$575$	−1.00000	−0.0417029
$576$	−0.950444	−0.0396018
$577$	−34.1505	−1.42171	−0.710853	−	0.703341i	$-0.751691\pi$
$577$	−34.1505	−1.42171	−0.710853	+	0.703341i	$0.751691\pi$
$578$	−5.22394	−0.217287
$579$	3.50444	0.145639
$580$	−0.863254	−0.0358447
$581$	27.9009	1.15752
$582$	−20.4649	−0.848299
$583$	38.8156	1.60758
$584$	5.90089	0.244180
$585$	3.75467	0.155237
$586$	−6.00000	−0.247858
$587$	−5.19062	−0.214240	−0.107120	−	0.994246i	$-0.534163\pi$
$587$	−5.19062	−0.214240	−0.107120	+	0.994246i	$0.534163\pi$
$588$	3.62308	0.149413
$589$	−18.3701	−0.756929
$590$	−6.86325	−0.282556
$591$	−15.3796	−0.632633
$592$	−7.03763	−0.289245
$593$	−2.09911	−0.0862002	−0.0431001	−	0.999071i	$-0.513723\pi$
$593$	−2.09911	−0.0862002	−0.0431001	+	0.999071i	$0.513723\pi$
$594$	36.5873	1.50120
$595$	10.5941	0.434315
$596$	3.26157	0.133599
$597$	−16.1933	−0.662748
$598$	−3.95044	−0.161546
$599$	11.1581	0.455909	0.227955	−	0.973672i	$-0.426796\pi$
$599$	11.1581	0.455909	0.227955	+	0.973672i	$0.426796\pi$
$600$	1.43163	0.0584459
$601$	9.57784	0.390688	0.195344	−	0.980735i	$-0.437418\pi$
$601$	9.57784	0.390688	0.195344	+	0.980735i	$0.437418\pi$
$602$	24.6975	1.00660
$603$	9.57597	0.389964
$604$	0.294881	0.0119986
$605$	−30.8513	−1.25428
$606$	1.23586	0.0502033
$607$	−24.6975	−1.00244	−0.501221	−	0.865320i	$-0.667115\pi$
$607$	−24.6975	−1.00244	−0.501221	+	0.865320i	$0.667115\pi$
$608$	3.08719	0.125202
$609$	3.81533	0.154605
$610$	13.5069	0.546878
$611$	15.4102	0.623431
$612$	3.26157	0.131841
$613$	−26.3488	−1.06422	−0.532108	−	0.846676i	$-0.678600\pi$
$613$	−26.3488	−1.06422	−0.532108	+	0.846676i	$0.678600\pi$
$614$	−22.9599	−0.926587
$615$	−8.02571	−0.323628
$616$	−19.9718	−0.804688
$617$	5.94612	0.239382	0.119691	−	0.992811i	$-0.461810\pi$
$617$	5.94612	0.239382	0.119691	+	0.992811i	$0.461810\pi$
$618$	2.19145	0.0881530
$619$	39.6856	1.59510	0.797549	−	0.603254i	$-0.206129\pi$
$619$	39.6856	1.59510	0.797549	+	0.603254i	$0.206129\pi$
$620$	5.95044	0.238976
$621$	5.65556	0.226950
$622$	18.0753	0.724752
$623$	51.7541	2.07348
$624$	5.65556	0.226404
$625$	1.00000	0.0400000
$626$	15.1625	0.606014
$627$	−28.5922	−1.14186
$628$	7.13675	0.284787
$629$	24.1505	0.962945
$630$	2.93420	0.116901
$631$	−23.0138	−0.916164	−0.458082	−	0.888910i	$-0.651463\pi$
$631$	−23.0138	−0.916164	−0.458082	+	0.888910i	$0.651463\pi$
$632$	15.8018	0.628561
$633$	33.0891	1.31517
$634$	14.0257	0.557032
$635$	−20.9385	−0.830920
$636$	−8.58976	−0.340606
$637$	9.99754	0.396117
$638$	−5.58462	−0.221097
$639$	−2.44109	−0.0965682
$640$	−1.00000	−0.0395285
$641$	25.4617	1.00568	0.502838	−	0.864381i	$-0.332289\pi$
$641$	25.4617	1.00568	0.502838	+	0.864381i	$0.332289\pi$
$642$	25.2359	0.995980
$643$	−16.4479	−0.648641	−0.324320	−	0.945947i	$-0.605136\pi$
$643$	−16.4479	−0.648641	−0.324320	+	0.945947i	$0.605136\pi$
$644$	−3.08719	−0.121652
$645$	−11.4530	−0.450962
$646$	−10.5941	−0.416819
$647$	−34.9147	−1.37264	−0.686319	−	0.727301i	$-0.740774\pi$
$647$	−34.9147	−1.37264	−0.686319	+	0.727301i	$0.740774\pi$
$648$	−5.24533	−0.206056
$649$	−44.4002	−1.74286
$650$	3.95044	0.154949
$651$	−26.2992	−1.03075
$652$	−8.12482	−0.318193
$653$	−34.2754	−1.34130	−0.670649	−	0.741775i	$-0.733984\pi$
$653$	−34.2754	−1.34130	−0.670649	+	0.741775i	$0.733984\pi$
$654$	−4.17006	−0.163062
$655$	−10.7641	−0.420590
$656$	5.60601	0.218878
$657$	−5.60846	−0.218807
$658$	12.0428	0.469476
$659$	35.2548	1.37333	0.686666	−	0.726973i	$-0.259074\pi$
$659$	35.2548	1.37333	0.686666	+	0.726973i	$0.259074\pi$
$660$	9.26157	0.360506
$661$	28.5684	1.11118	0.555590	−	0.831456i	$-0.312492\pi$
$661$	28.5684	1.11118	0.555590	+	0.831456i	$0.312492\pi$
$662$	−24.7403	−0.961559
$663$	−19.4078	−0.753736
$664$	9.03763	0.350728
$665$	−9.53074	−0.369586
$666$	6.68888	0.259189
$667$	−0.863254	−0.0334253
$668$	−7.80178	−0.301860
$669$	−8.19822	−0.316962
$670$	10.0753	0.389242
$671$	87.3796	3.37325
$672$	4.41970	0.170494
$673$	−23.8770	−0.920392	−0.460196	−	0.887817i	$-0.652221\pi$
$673$	−23.8770	−0.920392	−0.460196	+	0.887817i	$0.652221\pi$
$674$	14.7146	0.566785
$675$	−5.65556	−0.217683
$676$	2.60601	0.100231
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−8.58976	−0.329888
$679$	−44.1310	−1.69359
$680$	3.43163	0.131597
$681$	−25.2359	−0.967040
$682$	38.4950	1.47405
$683$	34.6480	1.32577	0.662884	−	0.748722i	$-0.269332\pi$
$683$	34.6480	1.32577	0.662884	+	0.748722i	$0.269332\pi$
$684$	−2.93420	−0.112192
$685$	−3.26157	−0.124618
$686$	−13.7975	−0.526789
$687$	−23.0138	−0.878031
$688$	8.00000	0.304997
$689$	−23.7027	−0.903000
$690$	1.43163	0.0545011
$691$	−18.1744	−0.691386	−0.345693	−	0.938348i	$-0.612356\pi$
$691$	−18.1744	−0.691386	−0.345693	+	0.938348i	$0.612356\pi$
$692$	−19.3325	−0.734912
$693$	18.9821	0.721071
$694$	9.43163	0.358020
$695$	16.9385	0.642515
$696$	1.23586	0.0468451
$697$	−19.2377	−0.728681
$698$	18.3488	0.694511
$699$	−9.23095	−0.349146
$700$	3.08719	0.116685
$701$	−40.6907	−1.53687	−0.768434	−	0.639929i	$-0.778964\pi$
$701$	−40.6907	−1.53687	−0.768434	+	0.639929i	$0.778964\pi$
$702$	−22.3420	−0.843244
$703$	−21.7265	−0.819431
$704$	−6.46926	−0.243819
$705$	−5.58462	−0.210329
$706$	−33.1129	−1.24622
$707$	2.66503	0.100229
$708$	9.82562	0.369269
$709$	−26.4454	−0.993178	−0.496589	−	0.867986i	$-0.665414\pi$
$709$	−26.4454	−0.993178	−0.496589	+	0.867986i	$0.665414\pi$
$710$	−2.56837	−0.0963893
$711$	−15.0187	−0.563245
$712$	16.7641	0.628263
$713$	5.95044	0.222846
$714$	−15.1668	−0.567602
$715$	25.5565	0.955757
$716$	2.17438	0.0812604
$717$	6.22580	0.232507
$718$	33.0376	1.23295
$719$	−2.24778	−0.0838281	−0.0419140	−	0.999121i	$-0.513346\pi$
$719$	−2.24778	−0.0838281	−0.0419140	+	0.999121i	$0.513346\pi$
$720$	0.950444	0.0354209
$721$	4.72568	0.175994
$722$	−9.46926	−0.352409
$723$	1.09397	0.0406850
$724$	−7.26157	−0.269874
$725$	0.863254	0.0320605
$726$	44.1676	1.63921
$727$	21.4830	0.796762	0.398381	−	0.917220i	$-0.369572\pi$
$727$	21.4830	0.796762	0.398381	+	0.917220i	$0.369572\pi$
$728$	12.1958	0.452005
$729$	29.2754	1.08427
$730$	−5.90089	−0.218402
$731$	−27.4530	−1.01539
$732$	−19.3368	−0.714710
$733$	11.3778	0.420247	0.210123	−	0.977675i	$-0.432613\pi$
$733$	11.3778	0.420247	0.210123	+	0.977675i	$0.432613\pi$
$734$	−2.27349	−0.0839161
$735$	−3.62308	−0.133639
$736$	−1.00000	−0.0368605
$737$	65.1795	2.40092
$738$	−5.32819	−0.196134
$739$	21.8770	0.804760	0.402380	−	0.915473i	$-0.368183\pi$
$739$	21.8770	0.804760	0.402380	+	0.915473i	$0.368183\pi$
$740$	7.03763	0.258709
$741$	17.4598	0.641402
$742$	−18.5231	−0.680006
$743$	5.36068	0.196664	0.0983322	−	0.995154i	$-0.468649\pi$
$743$	5.36068	0.196664	0.0983322	+	0.995154i	$0.468649\pi$
$744$	−8.51882	−0.312315
$745$	−3.26157	−0.119495
$746$	23.9762	0.877829
$747$	−8.58976	−0.314283
$748$	22.2001	0.811716
$749$	54.4191	1.98843
$750$	−1.43163	−0.0522756
$751$	24.3915	0.890060	0.445030	−	0.895516i	$-0.353193\pi$
$751$	24.3915	0.890060	0.445030	+	0.895516i	$0.353193\pi$
$752$	3.90089	0.142251
$753$	−32.2692	−1.17595
$754$	3.41024	0.124194
$755$	−0.294881	−0.0107318
$756$	−17.4598	−0.635007
$757$	41.9437	1.52447	0.762234	−	0.647301i	$-0.224102\pi$
$757$	41.9437	1.52447	0.762234	+	0.647301i	$0.224102\pi$
$758$	−13.8795	−0.504126
$759$	9.26157	0.336174
$760$	−3.08719	−0.111984
$761$	−18.3745	−0.666074	−0.333037	−	0.942914i	$-0.608073\pi$
$761$	−18.3745	−0.666074	−0.333037	+	0.942914i	$0.608073\pi$
$762$	29.9762	1.08592
$763$	−8.99240	−0.325547
$764$	8.58976	0.310767
$765$	−3.26157	−0.117922
$766$	0	0
$767$	27.1129	0.978990
$768$	1.43163	0.0516594
$769$	42.4993	1.53256	0.766282	−	0.642505i	$-0.222105\pi$
$769$	42.4993	1.53256	0.766282	+	0.642505i	$0.222105\pi$
$770$	19.9718	0.719735
$771$	−14.1744	−0.510478
$772$	2.44787	0.0881008
$773$	−15.0376	−0.540866	−0.270433	−	0.962739i	$-0.587167\pi$
$773$	−15.0376	−0.540866	−0.270433	+	0.962739i	$0.587167\pi$
$774$	−7.60355	−0.273304
$775$	−5.95044	−0.213746
$776$	−14.2949	−0.513156
$777$	−31.1043	−1.11586
$778$	−26.6436	−0.955221
$779$	17.3068	0.620081
$780$	−5.65556	−0.202502
$781$	−16.6155	−0.594548
$782$	3.43163	0.122715
$783$	−4.88219	−0.174475
$784$	2.53074	0.0903836
$785$	−7.13675	−0.254721
$786$	15.4102	0.549665
$787$	−8.39154	−0.299126	−0.149563	−	0.988752i	$-0.547787\pi$
$787$	−8.39154	−0.299126	−0.149563	+	0.988752i	$0.547787\pi$
$788$	−10.7428	−0.382695
$789$	−10.3897	−0.369882
$790$	−15.8018	−0.562202
$791$	−18.5231	−0.658607
$792$	6.14867	0.218483
$793$	−53.3582	−1.89481
$794$	−8.66749	−0.307598
$795$	8.58976	0.304647
$796$	−11.3111	−0.400912
$797$	−22.0514	−0.781101	−0.390551	−	0.920581i	$-0.627715\pi$
$797$	−22.0514	−0.781101	−0.390551	+	0.920581i	$0.627715\pi$
$798$	13.6445	0.483009
$799$	−13.3864	−0.473576
$800$	1.00000	0.0353553
$801$	−15.9334	−0.562978
$802$	39.9762	1.41161
$803$	−38.1744	−1.34714
$804$	−14.4240	−0.508696
$805$	3.08719	0.108809
$806$	−23.5069	−0.827995
$807$	−9.93337	−0.349671
$808$	0.863254	0.0303692
$809$	2.04524	0.0719066	0.0359533	−	0.999353i	$-0.488553\pi$
$809$	2.04524	0.0719066	0.0359533	+	0.999353i	$0.488553\pi$
$810$	5.24533	0.184302
$811$	4.24965	0.149225	0.0746126	−	0.997213i	$-0.476228\pi$
$811$	4.24965	0.149225	0.0746126	+	0.997213i	$0.476228\pi$
$812$	2.66503	0.0935242
$813$	−14.7446	−0.517116
$814$	45.5283	1.59577
$815$	8.12482	0.284600
$816$	−4.91281	−0.171983
$817$	24.6975	0.864057
$818$	30.1248	1.05329
$819$	−11.5914	−0.405036
$820$	−5.60601	−0.195770
$821$	−29.2548	−1.02100	−0.510500	−	0.859878i	$-0.670540\pi$
$821$	−29.2548	−1.02100	−0.510500	+	0.859878i	$0.670540\pi$
$822$	4.66935	0.162862
$823$	13.5846	0.473530	0.236765	−	0.971567i	$-0.423913\pi$
$823$	13.5846	0.473530	0.236765	+	0.971567i	$0.423913\pi$
$824$	1.53074	0.0533258
$825$	−9.26157	−0.322446
$826$	21.1882	0.737231
$827$	24.7880	0.861963	0.430981	−	0.902361i	$-0.358167\pi$
$827$	24.7880	0.861963	0.430981	+	0.902361i	$0.358167\pi$
$828$	0.950444	0.0330302
$829$	26.1505	0.908246	0.454123	−	0.890939i	$-0.349953\pi$
$829$	26.1505	0.908246	0.454123	+	0.890939i	$0.349953\pi$
$830$	−9.03763	−0.313701
$831$	−25.4855	−0.884082
$832$	3.95044	0.136957
$833$	−8.68455	−0.300902
$834$	−24.2496	−0.839697
$835$	7.80178	0.269992
$836$	−19.9718	−0.690740
$837$	33.6531	1.16322
$838$	25.8770	0.893908
$839$	6.37260	0.220007	0.110003	−	0.993931i	$-0.464914\pi$
$839$	6.37260	0.220007	0.110003	+	0.993931i	$0.464914\pi$
$840$	−4.41970	−0.152494
$841$	−28.2548	−0.974303
$842$	−10.7146	−0.369249
$843$	−27.1129	−0.933818
$844$	23.1129	0.795579
$845$	−2.60601	−0.0896493
$846$	−3.70757	−0.127469
$847$	95.2439	3.27262
$848$	−6.00000	−0.206041
$849$	18.1657	0.623447
$850$	−3.43163	−0.117704
$851$	7.03763	0.241247
$852$	3.67695	0.125970
$853$	−33.6293	−1.15144	−0.575722	−	0.817646i	$-0.695279\pi$
$853$	−33.6293	−1.15144	−0.575722	+	0.817646i	$0.695279\pi$
$854$	−41.6983	−1.42689
$855$	2.93420	0.100348
$856$	17.6274	0.602492
$857$	11.1795	0.381885	0.190943	−	0.981601i	$-0.438846\pi$
$857$	11.1795	0.381885	0.190943	+	0.981601i	$0.438846\pi$
$858$	−36.5873	−1.24907
$859$	47.9009	1.63436	0.817179	−	0.576385i	$-0.195537\pi$
$859$	47.9009	1.63436	0.817179	+	0.576385i	$0.195537\pi$
$860$	−8.00000	−0.272798
$861$	24.7769	0.844394
$862$	32.2496	1.09843
$863$	−30.1180	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$863$	−30.1180	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$864$	−5.65556	−0.192406
$865$	19.3325	0.657325
$866$	20.6393	0.701353
$867$	−7.47873	−0.253991
$868$	−18.3701	−0.623523
$869$	−102.226	−3.46777
$870$	−1.23586	−0.0418995
$871$	−39.8018	−1.34863
$872$	−2.91281	−0.0986402
$873$	13.5865	0.459833
$874$	−3.08719	−0.104426
$875$	−3.08719	−0.104366
$876$	8.44787	0.285427
$877$	−22.3940	−0.756191	−0.378096	−	0.925767i	$-0.623421\pi$
$877$	−22.3940	−0.756191	−0.378096	+	0.925767i	$0.623421\pi$
$878$	−16.2239	−0.547531
$879$	−8.58976	−0.289726
$880$	6.46926	0.218079
$881$	21.1129	0.711312	0.355656	−	0.934617i	$-0.384258\pi$
$881$	21.1129	0.711312	0.355656	+	0.934617i	$0.384258\pi$
$882$	−2.40533	−0.0809915
$883$	49.1061	1.65255	0.826276	−	0.563265i	$-0.190455\pi$
$883$	49.1061	1.65255	0.826276	+	0.563265i	$0.190455\pi$
$884$	−13.5565	−0.455953
$885$	−9.82562	−0.330285
$886$	−15.6770	−0.526678
$887$	−43.0566	−1.44570	−0.722849	−	0.691006i	$-0.757168\pi$
$887$	−43.0566	−1.44570	−0.722849	+	0.691006i	$0.757168\pi$
$888$	−10.0753	−0.338104
$889$	64.6412	2.16800
$890$	−16.7641	−0.561935
$891$	33.9334	1.13681
$892$	−5.72651	−0.191738
$893$	12.0428	0.402996
$894$	4.66935	0.156166
$895$	−2.17438	−0.0726815
$896$	3.08719	0.103136
$897$	−5.65556	−0.188834
$898$	22.5727	0.753261
$899$	−5.13675	−0.171320
$900$	−0.950444	−0.0316815
$901$	20.5898	0.685944
$902$	−36.2667	−1.20755
$903$	35.3576	1.17663
$904$	−6.00000	−0.199557
$905$	7.26157	0.241383
$906$	0.422160	0.0140253
$907$	−37.9762	−1.26098	−0.630489	−	0.776198i	$-0.717146\pi$
$907$	−37.9762	−1.26098	−0.630489	+	0.776198i	$0.717146\pi$
$908$	−17.6274	−0.584986
$909$	−0.820475	−0.0272134
$910$	−12.1958	−0.404286
$911$	−15.7504	−0.521833	−0.260916	−	0.965361i	$-0.584025\pi$
$911$	−15.7504	−0.521833	−0.260916	+	0.965361i	$0.584025\pi$
$912$	4.41970	0.146351
$913$	−58.4668	−1.93497
$914$	−1.45302	−0.0480615
$915$	19.3368	0.639256
$916$	−16.0753	−0.531142
$917$	33.2309	1.09738
$918$	19.4078	0.640552
$919$	30.4240	1.00360	0.501798	−	0.864985i	$-0.332672\pi$
$919$	30.4240	1.00360	0.501798	+	0.864985i	$0.332672\pi$
$920$	1.00000	0.0329690
$921$	−32.8700	−1.08310
$922$	−29.7027	−0.978205
$923$	10.1462	0.333967
$924$	−28.5922	−0.940615
$925$	−7.03763	−0.231396
$926$	22.9624	0.754590
$927$	−1.45488	−0.0477846
$928$	0.863254	0.0283377
$929$	−37.8018	−1.24024	−0.620118	−	0.784509i	$-0.712915\pi$
$929$	−37.8018	−1.24024	−0.620118	+	0.784509i	$0.712915\pi$
$930$	8.51882	0.279343
$931$	7.81287	0.256057
$932$	−6.44787	−0.211207
$933$	25.8770	0.847176
$934$	9.48550	0.310375
$935$	−22.2001	−0.726021
$936$	−3.75467	−0.122725
$937$	−32.6907	−1.06796	−0.533980	−	0.845497i	$-0.679304\pi$
$937$	−32.6907	−1.06796	−0.533980	+	0.845497i	$0.679304\pi$
$938$	−31.1043	−1.01559
$939$	21.7070	0.708381
$940$	−3.90089	−0.127233
$941$	−46.4882	−1.51547	−0.757736	−	0.652561i	$-0.773694\pi$
$941$	−46.4882	−1.51547	−0.757736	+	0.652561i	$0.773694\pi$
$942$	10.2172	0.332893
$943$	−5.60601	−0.182557
$944$	6.86325	0.223380
$945$	17.4598	0.567967
$946$	−51.7541	−1.68267
$947$	−37.2052	−1.20901	−0.604504	−	0.796602i	$-0.706629\pi$
$947$	−37.2052	−1.20901	−0.604504	+	0.796602i	$0.706629\pi$
$948$	22.6222	0.734737
$949$	23.3111	0.756711
$950$	3.08719	0.100162
$951$	20.0796	0.651125
$952$	−10.5941	−0.343356
$953$	49.1104	1.59084	0.795422	−	0.606056i	$-0.207249\pi$
$953$	49.1104	1.59084	0.795422	+	0.606056i	$0.207249\pi$
$954$	5.70266	0.184631
$955$	−8.58976	−0.277958
$956$	4.34876	0.140649
$957$	−7.99509	−0.258445
$958$	−30.5659	−0.987540
$959$	10.0691	0.325148
$960$	−1.43163	−0.0462056
$961$	4.40778	0.142187
$962$	−27.8018	−0.896365
$963$	−16.7538	−0.539885
$964$	0.764142	0.0246114
$965$	−2.44787	−0.0787997
$966$	−4.41970	−0.142202
$967$	1.96751	0.0632709	0.0316355	−	0.999499i	$-0.489928\pi$
$967$	1.96751	0.0632709	0.0316355	+	0.999499i	$0.489928\pi$
$968$	30.8513	0.991599
$969$	−15.1668	−0.487227
$970$	14.2949	0.458981
$971$	26.1205	0.838247	0.419123	−	0.907929i	$-0.362337\pi$
$971$	26.1205	0.838247	0.419123	+	0.907929i	$0.362337\pi$
$972$	9.45734	0.303344
$973$	−52.2924	−1.67642
$974$	−21.1367	−0.677265
$975$	5.65556	0.181123
$976$	−13.5069	−0.432345
$977$	−0.639319	−0.0204536	−0.0102268	−	0.999948i	$-0.503255\pi$
$977$	−0.639319	−0.0204536	−0.0102268	+	0.999948i	$0.503255\pi$
$978$	−11.6317	−0.371941
$979$	−108.452	−3.46613
$980$	−2.53074	−0.0808415
$981$	2.76846	0.0883902
$982$	−28.0514	−0.895157
$983$	−6.46926	−0.206337	−0.103169	−	0.994664i	$-0.532898\pi$
$983$	−6.46926	−0.206337	−0.103169	+	0.994664i	$0.532898\pi$
$984$	8.02571	0.255850
$985$	10.7428	0.342293
$986$	−2.96237	−0.0943410
$987$	17.2408	0.548780
$988$	12.1958	0.387999
$989$	−8.00000	−0.254385
$990$	−6.14867	−0.195418
$991$	37.2334	1.18276	0.591379	−	0.806394i	$-0.298584\pi$
$991$	37.2334	1.18276	0.591379	+	0.806394i	$0.298584\pi$
$992$	−5.95044	−0.188927
$993$	−35.4189	−1.12398
$994$	7.92905	0.251494
$995$	11.3111	0.358587
$996$	12.9385	0.409973
$997$	9.65124	0.305658	0.152829	−	0.988253i	$-0.451162\pi$
$997$	9.65124	0.305658	0.152829	+	0.988253i	$0.451162\pi$
$998$	3.80178	0.120343
$999$	39.8018	1.25927

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.2.a.d.1.2	✓	3
3.2	odd	2		2070.2.a.z.1.2		3
4.3	odd	2		1840.2.a.r.1.2		3
5.2	odd	4		1150.2.b.j.599.5		6
5.3	odd	4		1150.2.b.j.599.2		6
5.4	even	2		1150.2.a.q.1.2		3
8.3	odd	2		7360.2.a.ce.1.2		3
8.5	even	2		7360.2.a.bz.1.2		3
20.19	odd	2		9200.2.a.cf.1.2		3
23.22	odd	2		5290.2.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.d.1.2	✓	3	1.1	even	1	trivial
1150.2.a.q.1.2		3	5.4	even	2
1150.2.b.j.599.2		6	5.3	odd	4
1150.2.b.j.599.5		6	5.2	odd	4
1840.2.a.r.1.2		3	4.3	odd	2
2070.2.a.z.1.2		3	3.2	odd	2
5290.2.a.r.1.2		3	23.22	odd	2
7360.2.a.bz.1.2		3	8.5	even	2
7360.2.a.ce.1.2		3	8.3	odd	2
9200.2.a.cf.1.2		3	20.19	odd	2

Overview	Random
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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	230.2.a.d.1.2

Level	$230$
Weight	$2$
Character	230.1
Self dual	yes
Analytic conductor	$1.837$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$