LMFDB - Embedded newform 1338.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1338,2,Mod(1,1338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1338, 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("1338.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1338 = 2 \cdot 3 \cdot 223$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1338.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,3] 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:	$10.6839837904$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 2x + 1$ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	1338.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.80194 q^{5} +1.00000 q^{6} +0.0489173 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.80194 q^{10} -4.82908 q^{11} +1.00000 q^{12} +2.29590 q^{13} +0.0489173 q^{14} -3.80194 q^{15} +1.00000 q^{16} -7.51573 q^{17} +1.00000 q^{18} -1.46681 q^{19} -3.80194 q^{20} +0.0489173 q^{21} -4.82908 q^{22} -0.951083 q^{23} +1.00000 q^{24} +9.45473 q^{25} +2.29590 q^{26} +1.00000 q^{27} +0.0489173 q^{28} -8.54288 q^{29} -3.80194 q^{30} -3.24698 q^{31} +1.00000 q^{32} -4.82908 q^{33} -7.51573 q^{34} -0.185981 q^{35} +1.00000 q^{36} +1.26875 q^{37} -1.46681 q^{38} +2.29590 q^{39} -3.80194 q^{40} -2.91723 q^{41} +0.0489173 q^{42} -3.35690 q^{43} -4.82908 q^{44} -3.80194 q^{45} -0.951083 q^{46} +9.67994 q^{47} +1.00000 q^{48} -6.99761 q^{49} +9.45473 q^{50} -7.51573 q^{51} +2.29590 q^{52} +8.94869 q^{53} +1.00000 q^{54} +18.3599 q^{55} +0.0489173 q^{56} -1.46681 q^{57} -8.54288 q^{58} +1.86831 q^{59} -3.80194 q^{60} -2.62133 q^{61} -3.24698 q^{62} +0.0489173 q^{63} +1.00000 q^{64} -8.72886 q^{65} -4.82908 q^{66} +6.47219 q^{67} -7.51573 q^{68} -0.951083 q^{69} -0.185981 q^{70} -10.9215 q^{71} +1.00000 q^{72} -2.18598 q^{73} +1.26875 q^{74} +9.45473 q^{75} -1.46681 q^{76} -0.236226 q^{77} +2.29590 q^{78} -17.2446 q^{79} -3.80194 q^{80} +1.00000 q^{81} -2.91723 q^{82} -4.85086 q^{83} +0.0489173 q^{84} +28.5743 q^{85} -3.35690 q^{86} -8.54288 q^{87} -4.82908 q^{88} -8.66487 q^{89} -3.80194 q^{90} +0.112309 q^{91} -0.951083 q^{92} -3.24698 q^{93} +9.67994 q^{94} +5.57673 q^{95} +1.00000 q^{96} +9.12498 q^{97} -6.99761 q^{98} -4.82908 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 7 q^{5} + 3 q^{6} - 9 q^{7} + 3 q^{8} + 3 q^{9} - 7 q^{10} - 4 q^{11} + 3 q^{12} - 7 q^{13} - 9 q^{14} - 7 q^{15} + 3 q^{16} - 10 q^{17} + 3 q^{18} - q^{19} - 7 q^{20}+ \cdots - 4 q^{99}+O(q^{100})$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 7 * q^5 + 3 * q^6 - 9 * q^7 + 3 * q^8 + 3 * q^9 - 7 * q^10 - 4 * q^11 + 3 * q^12 - 7 * q^13 - 9 * q^14 - 7 * q^15 + 3 * q^16 - 10 * q^17 + 3 * q^18 - q^19 - 7 * q^20 - 9 * q^21 - 4 * q^22 - 12 * q^23 + 3 * q^24 + 6 * q^25 - 7 * q^26 + 3 * q^27 - 9 * q^28 - 7 * q^29 - 7 * q^30 - 5 * q^31 + 3 * q^32 - 4 * q^33 - 10 * q^34 + 14 * q^35 + 3 * q^36 - 4 * q^37 - q^38 - 7 * q^39 - 7 * q^40 - 2 * q^41 - 9 * q^42 - 6 * q^43 - 4 * q^44 - 7 * q^45 - 12 * q^46 + 5 * q^47 + 3 * q^48 + 20 * q^49 + 6 * q^50 - 10 * q^51 - 7 * q^52 - 5 * q^53 + 3 * q^54 + 7 * q^55 - 9 * q^56 - q^57 - 7 * q^58 + 8 * q^59 - 7 * q^60 - 15 * q^61 - 5 * q^62 - 9 * q^63 + 3 * q^64 + 7 * q^65 - 4 * q^66 + 13 * q^67 - 10 * q^68 - 12 * q^69 + 14 * q^70 - 7 * q^71 + 3 * q^72 + 8 * q^73 - 4 * q^74 + 6 * q^75 - q^76 - 2 * q^77 - 7 * q^78 - 6 * q^79 - 7 * q^80 + 3 * q^81 - 2 * q^82 - q^83 - 9 * q^84 + 42 * q^85 - 6 * q^86 - 7 * q^87 - 4 * q^88 - 27 * q^89 - 7 * q^90 + 42 * q^91 - 12 * q^92 - 5 * q^93 + 5 * q^94 + 14 * q^95 + 3 * q^96 + 3 * q^97 + 20 * q^98 - 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.80194	−1.70028	−0.850139	−	0.526558i	$-0.823482\pi$
$5$	−3.80194	−1.70028	−0.850139	+	0.526558i	$0.823482\pi$
$6$	1.00000	0.408248
$7$	0.0489173	0.0184890	0.00924451	−	0.999957i	$-0.497057\pi$
$7$	0.0489173	0.0184890	0.00924451	+	0.999957i	$0.497057\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.80194	−1.20228
$11$	−4.82908	−1.45602	−0.728012	−	0.685564i	$-0.759555\pi$
$11$	−4.82908	−1.45602	−0.728012	+	0.685564i	$0.759555\pi$
$12$	1.00000	0.288675
$13$	2.29590	0.636767	0.318384	−	0.947962i	$-0.396860\pi$
$13$	2.29590	0.636767	0.318384	+	0.947962i	$0.396860\pi$
$14$	0.0489173	0.0130737
$15$	−3.80194	−0.981656
$16$	1.00000	0.250000
$17$	−7.51573	−1.82283	−0.911416	−	0.411486i	$-0.865010\pi$
$17$	−7.51573	−1.82283	−0.911416	+	0.411486i	$0.865010\pi$
$18$	1.00000	0.235702
$19$	−1.46681	−0.336510	−0.168255	−	0.985744i	$-0.553813\pi$
$19$	−1.46681	−0.336510	−0.168255	+	0.985744i	$0.553813\pi$
$20$	−3.80194	−0.850139
$21$	0.0489173	0.0106746
$22$	−4.82908	−1.02956
$23$	−0.951083	−0.198314	−0.0991572	−	0.995072i	$-0.531615\pi$
$23$	−0.951083	−0.198314	−0.0991572	+	0.995072i	$0.531615\pi$
$24$	1.00000	0.204124
$25$	9.45473	1.89095
$26$	2.29590	0.450262
$27$	1.00000	0.192450
$28$	0.0489173	0.00924451
$29$	−8.54288	−1.58637	−0.793186	−	0.608979i	$-0.791579\pi$
$29$	−8.54288	−1.58637	−0.793186	+	0.608979i	$0.791579\pi$
$30$	−3.80194	−0.694136
$31$	−3.24698	−0.583175	−0.291587	−	0.956544i	$-0.594183\pi$
$31$	−3.24698	−0.583175	−0.291587	+	0.956544i	$0.594183\pi$
$32$	1.00000	0.176777
$33$	−4.82908	−0.840636
$34$	−7.51573	−1.28894
$35$	−0.185981	−0.0314365
$36$	1.00000	0.166667
$37$	1.26875	0.208581	0.104291	−	0.994547i	$-0.466743\pi$
$37$	1.26875	0.208581	0.104291	+	0.994547i	$0.466743\pi$
$38$	−1.46681	−0.237948
$39$	2.29590	0.367638
$40$	−3.80194	−0.601139
$41$	−2.91723	−0.455595	−0.227797	−	0.973709i	$-0.573152\pi$
$41$	−2.91723	−0.455595	−0.227797	+	0.973709i	$0.573152\pi$
$42$	0.0489173	0.00754811
$43$	−3.35690	−0.511922	−0.255961	−	0.966687i	$-0.582392\pi$
$43$	−3.35690	−0.511922	−0.255961	+	0.966687i	$0.582392\pi$
$44$	−4.82908	−0.728012
$45$	−3.80194	−0.566759
$46$	−0.951083	−0.140229
$47$	9.67994	1.41196	0.705982	−	0.708230i	$-0.250506\pi$
$47$	9.67994	1.41196	0.705982	+	0.708230i	$0.250506\pi$
$48$	1.00000	0.144338
$49$	−6.99761	−0.999658
$50$	9.45473	1.33710
$51$	−7.51573	−1.05241
$52$	2.29590	0.318384
$53$	8.94869	1.22920	0.614599	−	0.788840i	$-0.289318\pi$
$53$	8.94869	1.22920	0.614599	+	0.788840i	$0.289318\pi$
$54$	1.00000	0.136083
$55$	18.3599	2.47565
$56$	0.0489173	0.00653685
$57$	−1.46681	−0.194284
$58$	−8.54288	−1.12173
$59$	1.86831	0.243234	0.121617	−	0.992577i	$-0.461192\pi$
$59$	1.86831	0.243234	0.121617	+	0.992577i	$0.461192\pi$
$60$	−3.80194	−0.490828
$61$	−2.62133	−0.335627	−0.167814	−	0.985819i	$-0.553671\pi$
$61$	−2.62133	−0.335627	−0.167814	+	0.985819i	$0.553671\pi$
$62$	−3.24698	−0.412367
$63$	0.0489173	0.00616301
$64$	1.00000	0.125000
$65$	−8.72886	−1.08268
$66$	−4.82908	−0.594419
$67$	6.47219	0.790704	0.395352	−	0.918530i	$-0.370623\pi$
$67$	6.47219	0.790704	0.395352	+	0.918530i	$0.370623\pi$
$68$	−7.51573	−0.911416
$69$	−0.951083	−0.114497
$70$	−0.185981	−0.0222289
$71$	−10.9215	−1.29615	−0.648074	−	0.761577i	$-0.724425\pi$
$71$	−10.9215	−1.29615	−0.648074	+	0.761577i	$0.724425\pi$
$72$	1.00000	0.117851
$73$	−2.18598	−0.255850	−0.127925	−	0.991784i	$-0.540832\pi$
$73$	−2.18598	−0.255850	−0.127925	+	0.991784i	$0.540832\pi$
$74$	1.26875	0.147489
$75$	9.45473	1.09174
$76$	−1.46681	−0.168255
$77$	−0.236226	−0.0269204
$78$	2.29590	0.259959
$79$	−17.2446	−1.94017	−0.970084	−	0.242770i	$-0.921944\pi$
$79$	−17.2446	−1.94017	−0.970084	+	0.242770i	$0.921944\pi$
$80$	−3.80194	−0.425070
$81$	1.00000	0.111111
$82$	−2.91723	−0.322154
$83$	−4.85086	−0.532451	−0.266225	−	0.963911i	$-0.585777\pi$
$83$	−4.85086	−0.532451	−0.266225	+	0.963911i	$0.585777\pi$
$84$	0.0489173	0.00533732
$85$	28.5743	3.09932
$86$	−3.35690	−0.361983
$87$	−8.54288	−0.915893
$88$	−4.82908	−0.514782
$89$	−8.66487	−0.918475	−0.459237	−	0.888314i	$-0.651877\pi$
$89$	−8.66487	−0.918475	−0.459237	+	0.888314i	$0.651877\pi$
$90$	−3.80194	−0.400759
$91$	0.112309	0.0117732
$92$	−0.951083	−0.0991572
$93$	−3.24698	−0.336696
$94$	9.67994	0.998410
$95$	5.57673	0.572160
$96$	1.00000	0.102062
$97$	9.12498	0.926502	0.463251	−	0.886227i	$-0.346683\pi$
$97$	9.12498	0.926502	0.463251	+	0.886227i	$0.346683\pi$
$98$	−6.99761	−0.706865
$99$	−4.82908	−0.485341
$100$	9.45473	0.945473
$101$	15.1347	1.50596	0.752978	−	0.658046i	$-0.228617\pi$
$101$	15.1347	1.50596	0.752978	+	0.658046i	$0.228617\pi$
$102$	−7.51573	−0.744168
$103$	−16.9638	−1.67149	−0.835744	−	0.549119i	$-0.814963\pi$
$103$	−16.9638	−1.67149	−0.835744	+	0.549119i	$0.814963\pi$
$104$	2.29590	0.225131
$105$	−0.185981	−0.0181499
$106$	8.94869	0.869174
$107$	12.7724	1.23475	0.617377	−	0.786667i	$-0.288195\pi$
$107$	12.7724	1.23475	0.617377	+	0.786667i	$0.288195\pi$
$108$	1.00000	0.0962250
$109$	6.80731	0.652022	0.326011	−	0.945366i	$-0.394295\pi$
$109$	6.80731	0.652022	0.326011	+	0.945366i	$0.394295\pi$
$110$	18.3599	1.75055
$111$	1.26875	0.120424
$112$	0.0489173	0.00462225
$113$	9.97046	0.937942	0.468971	−	0.883214i	$-0.344625\pi$
$113$	9.97046	0.937942	0.468971	+	0.883214i	$0.344625\pi$
$114$	−1.46681	−0.137380
$115$	3.61596	0.337190
$116$	−8.54288	−0.793186
$117$	2.29590	0.212256
$118$	1.86831	0.171992
$119$	−0.367649	−0.0337024
$120$	−3.80194	−0.347068
$121$	12.3201	1.12001
$122$	−2.62133	−0.237324
$123$	−2.91723	−0.263038
$124$	−3.24698	−0.291587
$125$	−16.9366	−1.51486
$126$	0.0489173	0.00435790
$127$	−6.74094	−0.598162	−0.299081	−	0.954228i	$-0.596680\pi$
$127$	−6.74094	−0.598162	−0.299081	+	0.954228i	$0.596680\pi$
$128$	1.00000	0.0883883
$129$	−3.35690	−0.295558
$130$	−8.72886	−0.765571
$131$	15.9879	1.39687	0.698435	−	0.715673i	$-0.253880\pi$
$131$	15.9879	1.39687	0.698435	+	0.715673i	$0.253880\pi$
$132$	−4.82908	−0.420318
$133$	−0.0717525	−0.00622173
$134$	6.47219	0.559112
$135$	−3.80194	−0.327219
$136$	−7.51573	−0.644468
$137$	12.8823	1.10061	0.550305	−	0.834964i	$-0.314511\pi$
$137$	12.8823	1.10061	0.550305	+	0.834964i	$0.314511\pi$
$138$	−0.951083	−0.0809615
$139$	12.1860	1.03360	0.516801	−	0.856106i	$-0.327123\pi$
$139$	12.1860	1.03360	0.516801	+	0.856106i	$0.327123\pi$
$140$	−0.185981	−0.0157182
$141$	9.67994	0.815198
$142$	−10.9215	−0.916516
$143$	−11.0871	−0.927148
$144$	1.00000	0.0833333
$145$	32.4795	2.69727
$146$	−2.18598	−0.180913
$147$	−6.99761	−0.577153
$148$	1.26875	0.104291
$149$	−1.36898	−0.112151	−0.0560755	−	0.998427i	$-0.517859\pi$
$149$	−1.36898	−0.112151	−0.0560755	+	0.998427i	$0.517859\pi$
$150$	9.45473	0.771976
$151$	13.0707	1.06368	0.531839	−	0.846846i	$-0.321501\pi$
$151$	13.0707	1.06368	0.531839	+	0.846846i	$0.321501\pi$
$152$	−1.46681	−0.118974
$153$	−7.51573	−0.607611
$154$	−0.236226	−0.0190356
$155$	12.3448	0.991559
$156$	2.29590	0.183819
$157$	−12.1347	−0.968452	−0.484226	−	0.874943i	$-0.660899\pi$
$157$	−12.1347	−0.968452	−0.484226	+	0.874943i	$0.660899\pi$
$158$	−17.2446	−1.37191
$159$	8.94869	0.709677
$160$	−3.80194	−0.300570
$161$	−0.0465244	−0.00366664
$162$	1.00000	0.0785674
$163$	−9.86725	−0.772863	−0.386431	−	0.922318i	$-0.626292\pi$
$163$	−9.86725	−0.772863	−0.386431	+	0.922318i	$0.626292\pi$
$164$	−2.91723	−0.227797
$165$	18.3599	1.42931
$166$	−4.85086	−0.376499
$167$	−11.4306	−0.884524	−0.442262	−	0.896886i	$-0.645824\pi$
$167$	−11.4306	−0.884524	−0.442262	+	0.896886i	$0.645824\pi$
$168$	0.0489173	0.00377405
$169$	−7.72886	−0.594527
$170$	28.5743	2.19155
$171$	−1.46681	−0.112170
$172$	−3.35690	−0.255961
$173$	−7.91185	−0.601527	−0.300764	−	0.953699i	$-0.597242\pi$
$173$	−7.91185	−0.601527	−0.300764	+	0.953699i	$0.597242\pi$
$174$	−8.54288	−0.647634
$175$	0.462500	0.0349617
$176$	−4.82908	−0.364006
$177$	1.86831	0.140431
$178$	−8.66487	−0.649460
$179$	11.9162	0.890656	0.445328	−	0.895367i	$-0.353087\pi$
$179$	11.9162	0.890656	0.445328	+	0.895367i	$0.353087\pi$
$180$	−3.80194	−0.283380
$181$	−17.0097	−1.26432	−0.632160	−	0.774838i	$-0.717832\pi$
$181$	−17.0097	−1.26432	−0.632160	+	0.774838i	$0.717832\pi$
$182$	0.112309	0.00832491
$183$	−2.62133	−0.193775
$184$	−0.951083	−0.0701147
$185$	−4.82371	−0.354646
$186$	−3.24698	−0.238080
$187$	36.2941	2.65409
$188$	9.67994	0.705982
$189$	0.0489173	0.00355821
$190$	5.57673	0.404578
$191$	8.43057	0.610014	0.305007	−	0.952350i	$-0.401341\pi$
$191$	8.43057	0.610014	0.305007	+	0.952350i	$0.401341\pi$
$192$	1.00000	0.0721688
$193$	21.8920	1.57582	0.787910	−	0.615790i	$-0.211163\pi$
$193$	21.8920	1.57582	0.787910	+	0.615790i	$0.211163\pi$
$194$	9.12498	0.655136
$195$	−8.72886	−0.625086
$196$	−6.99761	−0.499829
$197$	−1.07606	−0.0766664	−0.0383332	−	0.999265i	$-0.512205\pi$
$197$	−1.07606	−0.0766664	−0.0383332	+	0.999265i	$0.512205\pi$
$198$	−4.82908	−0.343188
$199$	−19.1588	−1.35813	−0.679067	−	0.734076i	$-0.737615\pi$
$199$	−19.1588	−1.35813	−0.679067	+	0.734076i	$0.737615\pi$
$200$	9.45473	0.668550
$201$	6.47219	0.456513
$202$	15.1347	1.06487
$203$	−0.417895	−0.0293305
$204$	−7.51573	−0.526206
$205$	11.0911	0.774638
$206$	−16.9638	−1.18192
$207$	−0.951083	−0.0661048
$208$	2.29590	0.159192
$209$	7.08336	0.489966
$210$	−0.185981	−0.0128339
$211$	17.1414	1.18006	0.590030	−	0.807381i	$-0.299116\pi$
$211$	17.1414	1.18006	0.590030	+	0.807381i	$0.299116\pi$
$212$	8.94869	0.614599
$213$	−10.9215	−0.748332
$214$	12.7724	0.873103
$215$	12.7627	0.870410
$216$	1.00000	0.0680414
$217$	−0.158834	−0.0107823
$218$	6.80731	0.461050
$219$	−2.18598	−0.147715
$220$	18.3599	1.23782
$221$	−17.2553	−1.16072
$222$	1.26875	0.0851529
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	0.0489173	0.00326843
$225$	9.45473	0.630315
$226$	9.97046	0.663225
$227$	13.8605	0.919957	0.459978	−	0.887930i	$-0.347857\pi$
$227$	13.8605	0.919957	0.459978	+	0.887930i	$0.347857\pi$
$228$	−1.46681	−0.0971420
$229$	−1.16852	−0.0772181	−0.0386091	−	0.999254i	$-0.512293\pi$
$229$	−1.16852	−0.0772181	−0.0386091	+	0.999254i	$0.512293\pi$
$230$	3.61596	0.238429
$231$	−0.236226	−0.0155425
$232$	−8.54288	−0.560867
$233$	21.3381	1.39791	0.698953	−	0.715168i	$-0.253650\pi$
$233$	21.3381	1.39791	0.698953	+	0.715168i	$0.253650\pi$
$234$	2.29590	0.150087
$235$	−36.8025	−2.40073
$236$	1.86831	0.121617
$237$	−17.2446	−1.12016
$238$	−0.367649	−0.0238312
$239$	−2.81940	−0.182372	−0.0911858	−	0.995834i	$-0.529066\pi$
$239$	−2.81940	−0.182372	−0.0911858	+	0.995834i	$0.529066\pi$
$240$	−3.80194	−0.245414
$241$	−28.5555	−1.83942	−0.919712	−	0.392593i	$-0.871578\pi$
$241$	−28.5555	−1.83942	−0.919712	+	0.392593i	$0.871578\pi$
$242$	12.3201	0.791963
$243$	1.00000	0.0641500
$244$	−2.62133	−0.167814
$245$	26.6045	1.69970
$246$	−2.91723	−0.185996
$247$	−3.36765	−0.214278
$248$	−3.24698	−0.206183
$249$	−4.85086	−0.307410
$250$	−16.9366	−1.07117
$251$	−29.3260	−1.85104	−0.925521	−	0.378696i	$-0.876373\pi$
$251$	−29.3260	−1.85104	−0.925521	+	0.378696i	$0.876373\pi$
$252$	0.0489173	0.00308150
$253$	4.59286	0.288751
$254$	−6.74094	−0.422964
$255$	28.5743	1.78939
$256$	1.00000	0.0625000
$257$	−25.6256	−1.59848	−0.799242	−	0.601009i	$-0.794765\pi$
$257$	−25.6256	−1.59848	−0.799242	+	0.601009i	$0.794765\pi$
$258$	−3.35690	−0.208991
$259$	0.0620639	0.00385646
$260$	−8.72886	−0.541341
$261$	−8.54288	−0.528791
$262$	15.9879	0.987737
$263$	−21.5066	−1.32616	−0.663078	−	0.748550i	$-0.730750\pi$
$263$	−21.5066	−1.32616	−0.663078	+	0.748550i	$0.730750\pi$
$264$	−4.82908	−0.297210
$265$	−34.0224	−2.08998
$266$	−0.0717525	−0.00439943
$267$	−8.66487	−0.530282
$268$	6.47219	0.395352
$269$	−2.54527	−0.155188	−0.0775939	−	0.996985i	$-0.524724\pi$
$269$	−2.54527	−0.155188	−0.0775939	+	0.996985i	$0.524724\pi$
$270$	−3.80194	−0.231379
$271$	−16.6896	−1.01382	−0.506911	−	0.861998i	$-0.669213\pi$
$271$	−16.6896	−1.01382	−0.506911	+	0.861998i	$0.669213\pi$
$272$	−7.51573	−0.455708
$273$	0.112309	0.00679726
$274$	12.8823	0.778249
$275$	−45.6577	−2.75326
$276$	−0.951083	−0.0572484
$277$	−21.1957	−1.27352	−0.636762	−	0.771060i	$-0.719727\pi$
$277$	−21.1957	−1.27352	−0.636762	+	0.771060i	$0.719727\pi$
$278$	12.1860	0.730867
$279$	−3.24698	−0.194392
$280$	−0.185981	−0.0111145
$281$	−3.09246	−0.184481	−0.0922403	−	0.995737i	$-0.529403\pi$
$281$	−3.09246	−0.184481	−0.0922403	+	0.995737i	$0.529403\pi$
$282$	9.67994	0.576432
$283$	0.987918	0.0587257	0.0293628	−	0.999569i	$-0.490652\pi$
$283$	0.987918	0.0587257	0.0293628	+	0.999569i	$0.490652\pi$
$284$	−10.9215	−0.648074
$285$	5.57673	0.330337
$286$	−11.0871	−0.655593
$287$	−0.142703	−0.00842350
$288$	1.00000	0.0589256
$289$	39.4862	2.32272
$290$	32.4795	1.90726
$291$	9.12498	0.534916
$292$	−2.18598	−0.127925
$293$	12.7657	0.745780	0.372890	−	0.927876i	$-0.378367\pi$
$293$	12.7657	0.745780	0.372890	+	0.927876i	$0.378367\pi$
$294$	−6.99761	−0.408109
$295$	−7.10321	−0.413565
$296$	1.26875	0.0737446
$297$	−4.82908	−0.280212
$298$	−1.36898	−0.0793027
$299$	−2.18359	−0.126280
$300$	9.45473	0.545869
$301$	−0.164210	−0.00946493
$302$	13.0707	0.752134
$303$	15.1347	0.869464
$304$	−1.46681	−0.0841274
$305$	9.96615	0.570660
$306$	−7.51573	−0.429646
$307$	18.7211	1.06847	0.534234	−	0.845336i	$-0.320600\pi$
$307$	18.7211	1.06847	0.534234	+	0.845336i	$0.320600\pi$
$308$	−0.236226	−0.0134602
$309$	−16.9638	−0.965034
$310$	12.3448	0.701138
$311$	7.89977	0.447955	0.223977	−	0.974594i	$-0.428096\pi$
$311$	7.89977	0.447955	0.223977	+	0.974594i	$0.428096\pi$
$312$	2.29590	0.129980
$313$	−32.3448	−1.82824	−0.914118	−	0.405447i	$-0.867116\pi$
$313$	−32.3448	−1.82824	−0.914118	+	0.405447i	$0.867116\pi$
$314$	−12.1347	−0.684799
$315$	−0.185981	−0.0104788
$316$	−17.2446	−0.970084
$317$	−27.3967	−1.53875	−0.769376	−	0.638796i	$-0.779433\pi$
$317$	−27.3967	−1.53875	−0.769376	+	0.638796i	$0.779433\pi$
$318$	8.94869	0.501818
$319$	41.2543	2.30980
$320$	−3.80194	−0.212535
$321$	12.7724	0.712886
$322$	−0.0465244	−0.00259271
$323$	11.0242	0.613401
$324$	1.00000	0.0555556
$325$	21.7071	1.20409
$326$	−9.86725	−0.546496
$327$	6.80731	0.376445
$328$	−2.91723	−0.161077
$329$	0.473517	0.0261058
$330$	18.3599	1.01068
$331$	27.7071	1.52292	0.761460	−	0.648212i	$-0.224483\pi$
$331$	27.7071	1.52292	0.761460	+	0.648212i	$0.224483\pi$
$332$	−4.85086	−0.266225
$333$	1.26875	0.0695271
$334$	−11.4306	−0.625453
$335$	−24.6069	−1.34442
$336$	0.0489173	0.00266866
$337$	25.2403	1.37493	0.687463	−	0.726220i	$-0.258724\pi$
$337$	25.2403	1.37493	0.687463	+	0.726220i	$0.258724\pi$
$338$	−7.72886	−0.420394
$339$	9.97046	0.541521
$340$	28.5743	1.54966
$341$	15.6799	0.849116
$342$	−1.46681	−0.0793161
$343$	−0.684726	−0.0369717
$344$	−3.35690	−0.180992
$345$	3.61596	0.194677
$346$	−7.91185	−0.425344
$347$	4.34242	0.233113	0.116557	−	0.993184i	$-0.462814\pi$
$347$	4.34242	0.233113	0.116557	+	0.993184i	$0.462814\pi$
$348$	−8.54288	−0.457946
$349$	−17.7942	−0.952500	−0.476250	−	0.879310i	$-0.658004\pi$
$349$	−17.7942	−0.952500	−0.476250	+	0.879310i	$0.658004\pi$
$350$	0.462500	0.0247217
$351$	2.29590	0.122546
$352$	−4.82908	−0.257391
$353$	3.17821	0.169159	0.0845796	−	0.996417i	$-0.473045\pi$
$353$	3.17821	0.169159	0.0845796	+	0.996417i	$0.473045\pi$
$354$	1.86831	0.0992997
$355$	41.5230	2.20381
$356$	−8.66487	−0.459237
$357$	−0.367649	−0.0194581
$358$	11.9162	0.629789
$359$	6.82238	0.360071	0.180036	−	0.983660i	$-0.442379\pi$
$359$	6.82238	0.360071	0.180036	+	0.983660i	$0.442379\pi$
$360$	−3.80194	−0.200380
$361$	−16.8485	−0.886761
$362$	−17.0097	−0.894009
$363$	12.3201	0.646635
$364$	0.112309	0.00588660
$365$	8.31096	0.435016
$366$	−2.62133	−0.137019
$367$	7.61655	0.397581	0.198790	−	0.980042i	$-0.436299\pi$
$367$	7.61655	0.397581	0.198790	+	0.980042i	$0.436299\pi$
$368$	−0.951083	−0.0495786
$369$	−2.91723	−0.151865
$370$	−4.82371	−0.250773
$371$	0.437746	0.0227266
$372$	−3.24698	−0.168348
$373$	9.86161	0.510615	0.255307	−	0.966860i	$-0.417823\pi$
$373$	9.86161	0.510615	0.255307	+	0.966860i	$0.417823\pi$
$374$	36.2941	1.87672
$375$	−16.9366	−0.874603
$376$	9.67994	0.499205
$377$	−19.6136	−1.01015
$378$	0.0489173	0.00251604
$379$	−9.93469	−0.510311	−0.255155	−	0.966900i	$-0.582127\pi$
$379$	−9.93469	−0.510311	−0.255155	+	0.966900i	$0.582127\pi$
$380$	5.57673	0.286080
$381$	−6.74094	−0.345349
$382$	8.43057	0.431345
$383$	12.9245	0.660412	0.330206	−	0.943909i	$-0.392882\pi$
$383$	12.9245	0.660412	0.330206	+	0.943909i	$0.392882\pi$
$384$	1.00000	0.0510310
$385$	0.898116	0.0457723
$386$	21.8920	1.11427
$387$	−3.35690	−0.170641
$388$	9.12498	0.463251
$389$	−28.8726	−1.46390	−0.731950	−	0.681358i	$-0.761390\pi$
$389$	−28.8726	−1.46390	−0.731950	+	0.681358i	$0.761390\pi$
$390$	−8.72886	−0.442003
$391$	7.14808	0.361494
$392$	−6.99761	−0.353433
$393$	15.9879	0.806484
$394$	−1.07606	−0.0542113
$395$	65.5628	3.29882
$396$	−4.82908	−0.242671
$397$	0.423272	0.0212434	0.0106217	−	0.999944i	$-0.496619\pi$
$397$	0.423272	0.0212434	0.0106217	+	0.999944i	$0.496619\pi$
$398$	−19.1588	−0.960346
$399$	−0.0717525	−0.00359212
$400$	9.45473	0.472737
$401$	−27.8998	−1.39325	−0.696624	−	0.717436i	$-0.745315\pi$
$401$	−27.8998	−1.39325	−0.696624	+	0.717436i	$0.745315\pi$
$402$	6.47219	0.322803
$403$	−7.45473	−0.371347
$404$	15.1347	0.752978
$405$	−3.80194	−0.188920
$406$	−0.417895	−0.0207398
$407$	−6.12690	−0.303699
$408$	−7.51573	−0.372084
$409$	7.20882	0.356453	0.178227	−	0.983989i	$-0.442964\pi$
$409$	7.20882	0.356453	0.178227	+	0.983989i	$0.442964\pi$
$410$	11.0911	0.547752
$411$	12.8823	0.635438
$412$	−16.9638	−0.835744
$413$	0.0913929	0.00449715
$414$	−0.951083	−0.0467432
$415$	18.4426	0.905314
$416$	2.29590	0.112566
$417$	12.1860	0.596750
$418$	7.08336	0.346458
$419$	−3.50365	−0.171164	−0.0855822	−	0.996331i	$-0.527275\pi$
$419$	−3.50365	−0.171164	−0.0855822	+	0.996331i	$0.527275\pi$
$420$	−0.185981	−0.00907493
$421$	25.0965	1.22313	0.611564	−	0.791195i	$-0.290541\pi$
$421$	25.0965	1.22313	0.611564	+	0.791195i	$0.290541\pi$
$422$	17.1414	0.834429
$423$	9.67994	0.470655
$424$	8.94869	0.434587
$425$	−71.0592	−3.44688
$426$	−10.9215	−0.529150
$427$	−0.128229	−0.00620542
$428$	12.7724	0.617377
$429$	−11.0871	−0.535289
$430$	12.7627	0.615472
$431$	−3.33944	−0.160855	−0.0804275	−	0.996760i	$-0.525629\pi$
$431$	−3.33944	−0.160855	−0.0804275	+	0.996760i	$0.525629\pi$
$432$	1.00000	0.0481125
$433$	17.3980	0.836097	0.418048	−	0.908425i	$-0.362714\pi$
$433$	17.3980	0.836097	0.418048	+	0.908425i	$0.362714\pi$
$434$	−0.158834	−0.00762426
$435$	32.4795	1.55727
$436$	6.80731	0.326011
$437$	1.39506	0.0667348
$438$	−2.18598	−0.104450
$439$	−14.7869	−0.705739	−0.352869	−	0.935673i	$-0.614794\pi$
$439$	−14.7869	−0.705739	−0.352869	+	0.935673i	$0.614794\pi$
$440$	18.3599	0.875273
$441$	−6.99761	−0.333219
$442$	−17.2553	−0.820753
$443$	−6.95407	−0.330398	−0.165199	−	0.986260i	$-0.552827\pi$
$443$	−6.95407	−0.330398	−0.165199	+	0.986260i	$0.552827\pi$
$444$	1.26875	0.0602122
$445$	32.9433	1.56166
$446$	−1.00000	−0.0473514
$447$	−1.36898	−0.0647504
$448$	0.0489173	0.00231113
$449$	−39.0930	−1.84492	−0.922458	−	0.386099i	$-0.873822\pi$
$449$	−39.0930	−1.84492	−0.922458	+	0.386099i	$0.873822\pi$
$450$	9.45473	0.445700
$451$	14.0876	0.663357
$452$	9.97046	0.468971
$453$	13.0707	0.614114
$454$	13.8605	0.650508
$455$	−0.426992	−0.0200177
$456$	−1.46681	−0.0686898
$457$	−15.6896	−0.733930	−0.366965	−	0.930235i	$-0.619603\pi$
$457$	−15.6896	−0.733930	−0.366965	+	0.930235i	$0.619603\pi$
$458$	−1.16852	−0.0546015
$459$	−7.51573	−0.350804
$460$	3.61596	0.168595
$461$	4.28382	0.199517	0.0997586	−	0.995012i	$-0.468193\pi$
$461$	4.28382	0.199517	0.0997586	+	0.995012i	$0.468193\pi$
$462$	−0.236226	−0.0109902
$463$	1.21014	0.0562402	0.0281201	−	0.999605i	$-0.491048\pi$
$463$	1.21014	0.0562402	0.0281201	+	0.999605i	$0.491048\pi$
$464$	−8.54288	−0.396593
$465$	12.3448	0.572477
$466$	21.3381	0.988469
$467$	−24.7006	−1.14301	−0.571505	−	0.820599i	$-0.693640\pi$
$467$	−24.7006	−1.14301	−0.571505	+	0.820599i	$0.693640\pi$
$468$	2.29590	0.106128
$469$	0.316602	0.0146193
$470$	−36.8025	−1.69757
$471$	−12.1347	−0.559136
$472$	1.86831	0.0859961
$473$	16.2107	0.745370
$474$	−17.2446	−0.792070
$475$	−13.8683	−0.636322
$476$	−0.367649	−0.0168512
$477$	8.94869	0.409732
$478$	−2.81940	−0.128956
$479$	−13.9366	−0.636780	−0.318390	−	0.947960i	$-0.603142\pi$
$479$	−13.9366	−0.636780	−0.318390	+	0.947960i	$0.603142\pi$
$480$	−3.80194	−0.173534
$481$	2.91292	0.132818
$482$	−28.5555	−1.30067
$483$	−0.0465244	−0.00211694
$484$	12.3201	0.560003
$485$	−34.6926	−1.57531
$486$	1.00000	0.0453609
$487$	−11.1304	−0.504365	−0.252182	−	0.967680i	$-0.581148\pi$
$487$	−11.1304	−0.504365	−0.252182	+	0.967680i	$0.581148\pi$
$488$	−2.62133	−0.118662
$489$	−9.86725	−0.446212
$490$	26.6045	1.20187
$491$	1.41550	0.0638807	0.0319404	−	0.999490i	$-0.489831\pi$
$491$	1.41550	0.0638807	0.0319404	+	0.999490i	$0.489831\pi$
$492$	−2.91723	−0.131519
$493$	64.2059	2.89169
$494$	−3.36765	−0.151518
$495$	18.3599	0.825215
$496$	−3.24698	−0.145794
$497$	−0.534253	−0.0239645
$498$	−4.85086	−0.217372
$499$	37.2771	1.66875	0.834376	−	0.551195i	$-0.185828\pi$
$499$	37.2771	1.66875	0.834376	+	0.551195i	$0.185828\pi$
$500$	−16.9366	−0.757428
$501$	−11.4306	−0.510680
$502$	−29.3260	−1.30888
$503$	−1.42865	−0.0637003	−0.0318501	−	0.999493i	$-0.510140\pi$
$503$	−1.42865	−0.0637003	−0.0318501	+	0.999493i	$0.510140\pi$
$504$	0.0489173	0.00217895
$505$	−57.5411	−2.56054
$506$	4.59286	0.204177
$507$	−7.72886	−0.343251
$508$	−6.74094	−0.299081
$509$	−31.7802	−1.40863	−0.704316	−	0.709887i	$-0.748746\pi$
$509$	−31.7802	−1.40863	−0.704316	+	0.709887i	$0.748746\pi$
$510$	28.5743	1.26529
$511$	−0.106932	−0.00473041
$512$	1.00000	0.0441942
$513$	−1.46681	−0.0647613
$514$	−25.6256	−1.13030
$515$	64.4951	2.84200
$516$	−3.35690	−0.147779
$517$	−46.7453	−2.05585
$518$	0.0620639	0.00272693
$519$	−7.91185	−0.347292
$520$	−8.72886	−0.382786
$521$	−42.3099	−1.85363	−0.926815	−	0.375518i	$-0.877465\pi$
$521$	−42.3099	−1.85363	−0.926815	+	0.375518i	$0.877465\pi$
$522$	−8.54288	−0.373912
$523$	2.18731	0.0956443	0.0478222	−	0.998856i	$-0.484772\pi$
$523$	2.18731	0.0956443	0.0478222	+	0.998856i	$0.484772\pi$
$524$	15.9879	0.698435
$525$	0.462500	0.0201852
$526$	−21.5066	−0.937734
$527$	24.4034	1.06303
$528$	−4.82908	−0.210159
$529$	−22.0954	−0.960671
$530$	−34.0224	−1.47784
$531$	1.86831	0.0810779
$532$	−0.0717525	−0.00311087
$533$	−6.69766	−0.290108
$534$	−8.66487	−0.374966
$535$	−48.5599	−2.09943
$536$	6.47219	0.279556
$537$	11.9162	0.514221
$538$	−2.54527	−0.109734
$539$	33.7920	1.45553
$540$	−3.80194	−0.163609
$541$	1.76941	0.0760730	0.0380365	−	0.999276i	$-0.487890\pi$
$541$	1.76941	0.0760730	0.0380365	+	0.999276i	$0.487890\pi$
$542$	−16.6896	−0.716881
$543$	−17.0097	−0.729956
$544$	−7.51573	−0.322234
$545$	−25.8810	−1.10862
$546$	0.112309	0.00480639
$547$	−34.7657	−1.48647	−0.743237	−	0.669028i	$-0.766710\pi$
$547$	−34.7657	−1.48647	−0.743237	+	0.669028i	$0.766710\pi$
$548$	12.8823	0.550305
$549$	−2.62133	−0.111876
$550$	−45.6577	−1.94685
$551$	12.5308	0.533830
$552$	−0.951083	−0.0404808
$553$	−0.843559	−0.0358718
$554$	−21.1957	−0.900518
$555$	−4.82371	−0.204755
$556$	12.1860	0.516801
$557$	18.4179	0.780391	0.390196	−	0.920732i	$-0.372407\pi$
$557$	18.4179	0.780391	0.390196	+	0.920732i	$0.372407\pi$
$558$	−3.24698	−0.137456
$559$	−7.70709	−0.325975
$560$	−0.185981	−0.00785912
$561$	36.2941	1.53234
$562$	−3.09246	−0.130447
$563$	42.1836	1.77783	0.888913	−	0.458076i	$-0.151461\pi$
$563$	42.1836	1.77783	0.888913	+	0.458076i	$0.151461\pi$
$564$	9.67994	0.407599
$565$	−37.9071	−1.59476
$566$	0.987918	0.0415253
$567$	0.0489173	0.00205434
$568$	−10.9215	−0.458258
$569$	−33.2121	−1.39232	−0.696161	−	0.717886i	$-0.745110\pi$
$569$	−33.2121	−1.39232	−0.696161	+	0.717886i	$0.745110\pi$
$570$	5.57673	0.233583
$571$	28.1540	1.17821	0.589105	−	0.808056i	$-0.299480\pi$
$571$	28.1540	1.17821	0.589105	+	0.808056i	$0.299480\pi$
$572$	−11.0871	−0.463574
$573$	8.43057	0.352192
$574$	−0.142703	−0.00595632
$575$	−8.99223	−0.375002
$576$	1.00000	0.0416667
$577$	18.3448	0.763705	0.381852	−	0.924223i	$-0.375286\pi$
$577$	18.3448	0.763705	0.381852	+	0.924223i	$0.375286\pi$
$578$	39.4862	1.64241
$579$	21.8920	0.909801
$580$	32.4795	1.34864
$581$	−0.237291	−0.00984449
$582$	9.12498	0.378243
$583$	−43.2140	−1.78974
$584$	−2.18598	−0.0904565
$585$	−8.72886	−0.360894
$586$	12.7657	0.527346
$587$	−36.2252	−1.49517	−0.747587	−	0.664164i	$-0.768788\pi$
$587$	−36.2252	−1.49517	−0.747587	+	0.664164i	$0.768788\pi$
$588$	−6.99761	−0.288576
$589$	4.76271	0.196244
$590$	−7.10321	−0.292435
$591$	−1.07606	−0.0442634
$592$	1.26875	0.0521453
$593$	12.5399	0.514952	0.257476	−	0.966285i	$-0.417109\pi$
$593$	12.5399	0.514952	0.257476	+	0.966285i	$0.417109\pi$
$594$	−4.82908	−0.198140
$595$	1.39778	0.0573034
$596$	−1.36898	−0.0560755
$597$	−19.1588	−0.784119
$598$	−2.18359	−0.0892935
$599$	25.6612	1.04849	0.524243	−	0.851569i	$-0.324348\pi$
$599$	25.6612	1.04849	0.524243	+	0.851569i	$0.324348\pi$
$600$	9.45473	0.385988
$601$	−0.963164	−0.0392883	−0.0196441	−	0.999807i	$-0.506253\pi$
$601$	−0.963164	−0.0392883	−0.0196441	+	0.999807i	$0.506253\pi$
$602$	−0.164210	−0.00669272
$603$	6.47219	0.263568
$604$	13.0707	0.531839
$605$	−46.8401	−1.90432
$606$	15.1347	0.614804
$607$	−4.67217	−0.189638	−0.0948188	−	0.995495i	$-0.530227\pi$
$607$	−4.67217	−0.189638	−0.0948188	+	0.995495i	$0.530227\pi$
$608$	−1.46681	−0.0594871
$609$	−0.417895	−0.0169340
$610$	9.96615	0.403518
$611$	22.2241	0.899093
$612$	−7.51573	−0.303805
$613$	−14.2392	−0.575116	−0.287558	−	0.957763i	$-0.592843\pi$
$613$	−14.2392	−0.575116	−0.287558	+	0.957763i	$0.592843\pi$
$614$	18.7211	0.755522
$615$	11.0911	0.447238
$616$	−0.236226	−0.00951782
$617$	−6.56571	−0.264326	−0.132163	−	0.991228i	$-0.542192\pi$
$617$	−6.56571	−0.264326	−0.132163	+	0.991228i	$0.542192\pi$
$618$	−16.9638	−0.682382
$619$	15.6552	0.629235	0.314618	−	0.949219i	$-0.398124\pi$
$619$	15.6552	0.629235	0.314618	+	0.949219i	$0.398124\pi$
$620$	12.3448	0.495780
$621$	−0.951083	−0.0381656
$622$	7.89977	0.316752
$623$	−0.423863	−0.0169817
$624$	2.29590	0.0919094
$625$	17.1183	0.684731
$626$	−32.3448	−1.29276
$627$	7.08336	0.282882
$628$	−12.1347	−0.484226
$629$	−9.53558	−0.380209
$630$	−0.185981	−0.00740965
$631$	31.5013	1.25405	0.627023	−	0.779001i	$-0.284273\pi$
$631$	31.5013	1.25405	0.627023	+	0.779001i	$0.284273\pi$
$632$	−17.2446	−0.685953
$633$	17.1414	0.681308
$634$	−27.3967	−1.08806
$635$	25.6286	1.01704
$636$	8.94869	0.354839
$637$	−16.0658	−0.636550
$638$	41.2543	1.63327
$639$	−10.9215	−0.432050
$640$	−3.80194	−0.150285
$641$	35.6316	1.40736	0.703682	−	0.710515i	$-0.251538\pi$
$641$	35.6316	1.40736	0.703682	+	0.710515i	$0.251538\pi$
$642$	12.7724	0.504086
$643$	11.1142	0.438302	0.219151	−	0.975691i	$-0.429671\pi$
$643$	11.1142	0.438302	0.219151	+	0.975691i	$0.429671\pi$
$644$	−0.0465244	−0.00183332
$645$	12.7627	0.502531
$646$	11.0242	0.433740
$647$	24.9729	0.981784	0.490892	−	0.871220i	$-0.336671\pi$
$647$	24.9729	0.981784	0.490892	+	0.871220i	$0.336671\pi$
$648$	1.00000	0.0392837
$649$	−9.02224	−0.354154
$650$	21.7071	0.851422
$651$	−0.158834	−0.00622518
$652$	−9.86725	−0.386431
$653$	−33.9245	−1.32757	−0.663785	−	0.747924i	$-0.731051\pi$
$653$	−33.9245	−1.32757	−0.663785	+	0.747924i	$0.731051\pi$
$654$	6.80731	0.266187
$655$	−60.7851	−2.37507
$656$	−2.91723	−0.113899
$657$	−2.18598	−0.0852832
$658$	0.473517	0.0184596
$659$	−7.03923	−0.274209	−0.137105	−	0.990557i	$-0.543780\pi$
$659$	−7.03923	−0.274209	−0.137105	+	0.990557i	$0.543780\pi$
$660$	18.3599	0.714657
$661$	−3.83685	−0.149236	−0.0746182	−	0.997212i	$-0.523774\pi$
$661$	−3.83685	−0.149236	−0.0746182	+	0.997212i	$0.523774\pi$
$662$	27.7071	1.07687
$663$	−17.2553	−0.670142
$664$	−4.85086	−0.188250
$665$	0.272799	0.0105787
$666$	1.26875	0.0491631
$667$	8.12498	0.314601
$668$	−11.4306	−0.442262
$669$	−1.00000	−0.0386622
$670$	−24.6069	−0.950646
$671$	12.6586	0.488682
$672$	0.0489173	0.00188703
$673$	−46.9004	−1.80788	−0.903938	−	0.427663i	$-0.859337\pi$
$673$	−46.9004	−1.80788	−0.903938	+	0.427663i	$0.859337\pi$
$674$	25.2403	0.972219
$675$	9.45473	0.363913
$676$	−7.72886	−0.297264
$677$	−7.27519	−0.279608	−0.139804	−	0.990179i	$-0.544647\pi$
$677$	−7.27519	−0.279608	−0.139804	+	0.990179i	$0.544647\pi$
$678$	9.97046	0.382913
$679$	0.446370	0.0171301
$680$	28.5743	1.09578
$681$	13.8605	0.531137
$682$	15.6799	0.600416
$683$	8.80061	0.336746	0.168373	−	0.985723i	$-0.446149\pi$
$683$	8.80061	0.336746	0.168373	+	0.985723i	$0.446149\pi$
$684$	−1.46681	−0.0560850
$685$	−48.9778	−1.87134
$686$	−0.684726	−0.0261429
$687$	−1.16852	−0.0445819
$688$	−3.35690	−0.127980
$689$	20.5453	0.782712
$690$	3.61596	0.137657
$691$	−18.9084	−0.719309	−0.359655	−	0.933085i	$-0.617105\pi$
$691$	−18.9084	−0.719309	−0.359655	+	0.933085i	$0.617105\pi$
$692$	−7.91185	−0.300764
$693$	−0.236226	−0.00897348
$694$	4.34242	0.164836
$695$	−46.3303	−1.75741
$696$	−8.54288	−0.323817
$697$	21.9251	0.830473
$698$	−17.7942	−0.673519
$699$	21.3381	0.807081
$700$	0.462500	0.0174809
$701$	3.34721	0.126422	0.0632111	−	0.998000i	$-0.479866\pi$
$701$	3.34721	0.126422	0.0632111	+	0.998000i	$0.479866\pi$
$702$	2.29590	0.0866530
$703$	−1.86102	−0.0701896
$704$	−4.82908	−0.182003
$705$	−36.8025	−1.38606
$706$	3.17821	0.119614
$707$	0.740348	0.0278436
$708$	1.86831	0.0702155
$709$	18.5907	0.698189	0.349095	−	0.937087i	$-0.386489\pi$
$709$	18.5907	0.698189	0.349095	+	0.937087i	$0.386489\pi$
$710$	41.5230	1.55833
$711$	−17.2446	−0.646723
$712$	−8.66487	−0.324730
$713$	3.08815	0.115652
$714$	−0.367649	−0.0137589
$715$	42.1524	1.57641
$716$	11.9162	0.445328
$717$	−2.81940	−0.105292
$718$	6.82238	0.254609
$719$	−23.3787	−0.871877	−0.435939	−	0.899976i	$-0.643584\pi$
$719$	−23.3787	−0.871877	−0.435939	+	0.899976i	$0.643584\pi$
$720$	−3.80194	−0.141690
$721$	−0.829822	−0.0309042
$722$	−16.8485	−0.627035
$723$	−28.5555	−1.06199
$724$	−17.0097	−0.632160
$725$	−80.7706	−2.99974
$726$	12.3201	0.457240
$727$	−18.4993	−0.686102	−0.343051	−	0.939317i	$-0.611460\pi$
$727$	−18.4993	−0.686102	−0.343051	+	0.939317i	$0.611460\pi$
$728$	0.112309	0.00416245
$729$	1.00000	0.0370370
$730$	8.31096	0.307603
$731$	25.2295	0.933148
$732$	−2.62133	−0.0968873
$733$	−26.1575	−0.966150	−0.483075	−	0.875579i	$-0.660480\pi$
$733$	−26.1575	−0.966150	−0.483075	+	0.875579i	$0.660480\pi$
$734$	7.61655	0.281132
$735$	26.6045	0.981321
$736$	−0.951083	−0.0350574
$737$	−31.2547	−1.15128
$738$	−2.91723	−0.107385
$739$	−45.8316	−1.68594	−0.842971	−	0.537959i	$-0.819196\pi$
$739$	−45.8316	−1.68594	−0.842971	+	0.537959i	$0.819196\pi$
$740$	−4.82371	−0.177323
$741$	−3.36765	−0.123714
$742$	0.437746	0.0160702
$743$	26.4058	0.968735	0.484368	−	0.874865i	$-0.339050\pi$
$743$	26.4058	0.968735	0.484368	+	0.874865i	$0.339050\pi$
$744$	−3.24698	−0.119040
$745$	5.20477	0.190688
$746$	9.86161	0.361059
$747$	−4.85086	−0.177484
$748$	36.2941	1.32704
$749$	0.624792	0.0228294
$750$	−16.9366	−0.618437
$751$	3.32842	0.121456	0.0607279	−	0.998154i	$-0.480658\pi$
$751$	3.32842	0.121456	0.0607279	+	0.998154i	$0.480658\pi$
$752$	9.67994	0.352991
$753$	−29.3260	−1.06870
$754$	−19.6136	−0.714284
$755$	−49.6939	−1.80855
$756$	0.0489173	0.00177911
$757$	−29.7875	−1.08264	−0.541322	−	0.840815i	$-0.682076\pi$
$757$	−29.7875	−1.08264	−0.541322	+	0.840815i	$0.682076\pi$
$758$	−9.93469	−0.360844
$759$	4.59286	0.166710
$760$	5.57673	0.202289
$761$	31.2194	1.13170	0.565850	−	0.824508i	$-0.308548\pi$
$761$	31.2194	1.13170	0.565850	+	0.824508i	$0.308548\pi$
$762$	−6.74094	−0.244198
$763$	0.332996	0.0120553
$764$	8.43057	0.305007
$765$	28.5743	1.03311
$766$	12.9245	0.466982
$767$	4.28946	0.154883
$768$	1.00000	0.0360844
$769$	−14.7178	−0.530739	−0.265369	−	0.964147i	$-0.585494\pi$
$769$	−14.7178	−0.530739	−0.265369	+	0.964147i	$0.585494\pi$
$770$	0.898116	0.0323659
$771$	−25.6256	−0.922885
$772$	21.8920	0.787910
$773$	18.5670	0.667810	0.333905	−	0.942607i	$-0.391634\pi$
$773$	18.5670	0.667810	0.333905	+	0.942607i	$0.391634\pi$
$774$	−3.35690	−0.120661
$775$	−30.6993	−1.10275
$776$	9.12498	0.327568
$777$	0.0620639	0.00222653
$778$	−28.8726	−1.03513
$779$	4.27903	0.153312
$780$	−8.72886	−0.312543
$781$	52.7411	1.88722
$782$	7.14808	0.255615
$783$	−8.54288	−0.305298
$784$	−6.99761	−0.249915
$785$	46.1353	1.64664
$786$	15.9879	0.570270
$787$	36.9670	1.31773	0.658866	−	0.752261i	$-0.271037\pi$
$787$	36.9670	1.31773	0.658866	+	0.752261i	$0.271037\pi$
$788$	−1.07606	−0.0383332
$789$	−21.5066	−0.765656
$790$	65.5628	2.33262
$791$	0.487728	0.0173416
$792$	−4.82908	−0.171594
$793$	−6.01831	−0.213717
$794$	0.423272	0.0150213
$795$	−34.0224	−1.20665
$796$	−19.1588	−0.679067
$797$	−35.2717	−1.24939	−0.624694	−	0.780869i	$-0.714776\pi$
$797$	−35.2717	−1.24939	−0.624694	+	0.780869i	$0.714776\pi$
$798$	−0.0717525	−0.00254001
$799$	−72.7518	−2.57377
$800$	9.45473	0.334275
$801$	−8.66487	−0.306158
$802$	−27.8998	−0.985175
$803$	10.5563	0.372523
$804$	6.47219	0.228257
$805$	0.176883	0.00623431
$806$	−7.45473	−0.262582
$807$	−2.54527	−0.0895977
$808$	15.1347	0.532436
$809$	36.1430	1.27072	0.635361	−	0.772216i	$-0.280851\pi$
$809$	36.1430	1.27072	0.635361	+	0.772216i	$0.280851\pi$
$810$	−3.80194	−0.133586
$811$	−22.4239	−0.787408	−0.393704	−	0.919237i	$-0.628807\pi$
$811$	−22.4239	−0.787408	−0.393704	+	0.919237i	$0.628807\pi$
$812$	−0.417895	−0.0146652
$813$	−16.6896	−0.585331
$814$	−6.12690	−0.214748
$815$	37.5147	1.31408
$816$	−7.51573	−0.263103
$817$	4.92394	0.172267
$818$	7.20882	0.252050
$819$	0.112309	0.00392440
$820$	11.0911	0.387319
$821$	−53.6577	−1.87267	−0.936333	−	0.351113i	$-0.885803\pi$
$821$	−53.6577	−1.87267	−0.936333	+	0.351113i	$0.885803\pi$
$822$	12.8823	0.449322
$823$	−42.9144	−1.49590	−0.747950	−	0.663755i	$-0.768962\pi$
$823$	−42.9144	−1.49590	−0.747950	+	0.663755i	$0.768962\pi$
$824$	−16.9638	−0.590960
$825$	−45.6577	−1.58960
$826$	0.0913929	0.00317997
$827$	−17.7657	−0.617774	−0.308887	−	0.951099i	$-0.599956\pi$
$827$	−17.7657	−0.617774	−0.308887	+	0.951099i	$0.599956\pi$
$828$	−0.951083	−0.0330524
$829$	−7.95300	−0.276219	−0.138110	−	0.990417i	$-0.544103\pi$
$829$	−7.95300	−0.276219	−0.138110	+	0.990417i	$0.544103\pi$
$830$	18.4426	0.640154
$831$	−21.1957	−0.735270
$832$	2.29590	0.0795959
$833$	52.5921	1.82221
$834$	12.1860	0.421966
$835$	43.4583	1.50394
$836$	7.08336	0.244983
$837$	−3.24698	−0.112232
$838$	−3.50365	−0.121032
$839$	19.5894	0.676301	0.338151	−	0.941092i	$-0.390199\pi$
$839$	19.5894	0.676301	0.338151	+	0.941092i	$0.390199\pi$
$840$	−0.185981	−0.00641694
$841$	43.9807	1.51658
$842$	25.0965	0.864883
$843$	−3.09246	−0.106510
$844$	17.1414	0.590030
$845$	29.3846	1.01086
$846$	9.67994	0.332803
$847$	0.602665	0.0207078
$848$	8.94869	0.307299
$849$	0.987918	0.0339053
$850$	−71.0592	−2.43731
$851$	−1.20669	−0.0413647
$852$	−10.9215	−0.374166
$853$	−16.5810	−0.567724	−0.283862	−	0.958865i	$-0.591616\pi$
$853$	−16.5810	−0.567724	−0.283862	+	0.958865i	$0.591616\pi$
$854$	−0.128229	−0.00438790
$855$	5.57673	0.190720
$856$	12.7724	0.436552
$857$	11.4558	0.391323	0.195661	−	0.980672i	$-0.437315\pi$
$857$	11.4558	0.391323	0.195661	+	0.980672i	$0.437315\pi$
$858$	−11.0871	−0.378507
$859$	−26.0549	−0.888981	−0.444491	−	0.895784i	$-0.646615\pi$
$859$	−26.0549	−0.888981	−0.444491	+	0.895784i	$0.646615\pi$
$860$	12.7627	0.435205
$861$	−0.142703	−0.00486331
$862$	−3.33944	−0.113742
$863$	35.0586	1.19341	0.596704	−	0.802461i	$-0.296476\pi$
$863$	35.0586	1.19341	0.596704	+	0.802461i	$0.296476\pi$
$864$	1.00000	0.0340207
$865$	30.0804	1.02276
$866$	17.3980	0.591210
$867$	39.4862	1.34102
$868$	−0.158834	−0.00539116
$869$	83.2756	2.82493
$870$	32.4795	1.10116
$871$	14.8595	0.503494
$872$	6.80731	0.230525
$873$	9.12498	0.308834
$874$	1.39506	0.0471886
$875$	−0.828494	−0.0280082
$876$	−2.18598	−0.0738574
$877$	11.3860	0.384477	0.192238	−	0.981348i	$-0.438425\pi$
$877$	11.3860	0.384477	0.192238	+	0.981348i	$0.438425\pi$
$878$	−14.7869	−0.499033
$879$	12.7657	0.430576
$880$	18.3599	0.618911
$881$	36.0858	1.21576	0.607880	−	0.794029i	$-0.292020\pi$
$881$	36.0858	1.21576	0.607880	+	0.794029i	$0.292020\pi$
$882$	−6.99761	−0.235622
$883$	17.4432	0.587012	0.293506	−	0.955957i	$-0.405178\pi$
$883$	17.4432	0.587012	0.293506	+	0.955957i	$0.405178\pi$
$884$	−17.2553	−0.580360
$885$	−7.10321	−0.238772
$886$	−6.95407	−0.233626
$887$	−0.745251	−0.0250231	−0.0125115	−	0.999922i	$-0.503983\pi$
$887$	−0.745251	−0.0250231	−0.0125115	+	0.999922i	$0.503983\pi$
$888$	1.26875	0.0425765
$889$	−0.329749	−0.0110594
$890$	32.9433	1.10426
$891$	−4.82908	−0.161780
$892$	−1.00000	−0.0334825
$893$	−14.1987	−0.475140
$894$	−1.36898	−0.0457855
$895$	−45.3045	−1.51436
$896$	0.0489173	0.00163421
$897$	−2.18359	−0.0729079
$898$	−39.0930	−1.30455
$899$	27.7385	0.925132
$900$	9.45473	0.315158
$901$	−67.2559	−2.24062
$902$	14.0876	0.469064
$903$	−0.164210	−0.00546458
$904$	9.97046	0.331613
$905$	64.6698	2.14970
$906$	13.0707	0.434245
$907$	−8.46980	−0.281235	−0.140617	−	0.990064i	$-0.544909\pi$
$907$	−8.46980	−0.281235	−0.140617	+	0.990064i	$0.544909\pi$
$908$	13.8605	0.459978
$909$	15.1347	0.501985
$910$	−0.426992	−0.0141547
$911$	−34.1075	−1.13003	−0.565016	−	0.825080i	$-0.691130\pi$
$911$	−34.1075	−1.13003	−0.565016	+	0.825080i	$0.691130\pi$
$912$	−1.46681	−0.0485710
$913$	23.4252	0.775261
$914$	−15.6896	−0.518967
$915$	9.96615	0.329471
$916$	−1.16852	−0.0386091
$917$	0.782086	0.0258268
$918$	−7.51573	−0.248056
$919$	−6.24804	−0.206104	−0.103052	−	0.994676i	$-0.532861\pi$
$919$	−6.24804	−0.206104	−0.103052	+	0.994676i	$0.532861\pi$
$920$	3.61596	0.119215
$921$	18.7211	0.616881
$922$	4.28382	0.141080
$923$	−25.0747	−0.825345
$924$	−0.236226	−0.00777126
$925$	11.9957	0.394416
$926$	1.21014	0.0397678
$927$	−16.9638	−0.557163
$928$	−8.54288	−0.280434
$929$	−32.9071	−1.07965	−0.539823	−	0.841779i	$-0.681509\pi$
$929$	−32.9071	−1.07965	−0.539823	+	0.841779i	$0.681509\pi$
$930$	12.3448	0.404802
$931$	10.2642	0.336395
$932$	21.3381	0.698953
$933$	7.89977	0.258627
$934$	−24.7006	−0.808230
$935$	−137.988	−4.51269
$936$	2.29590	0.0750437
$937$	−56.2804	−1.83860	−0.919300	−	0.393559i	$-0.871244\pi$
$937$	−56.2804	−1.83860	−0.919300	+	0.393559i	$0.871244\pi$
$938$	0.316602	0.0103374
$939$	−32.3448	−1.05553
$940$	−36.8025	−1.20037
$941$	14.6890	0.478849	0.239424	−	0.970915i	$-0.423041\pi$
$941$	14.6890	0.478849	0.239424	+	0.970915i	$0.423041\pi$
$942$	−12.1347	−0.395369
$943$	2.77453	0.0903511
$944$	1.86831	0.0608084
$945$	−0.185981	−0.00604995
$946$	16.2107	0.527056
$947$	19.2010	0.623950	0.311975	−	0.950090i	$-0.399009\pi$
$947$	19.2010	0.623950	0.311975	+	0.950090i	$0.399009\pi$
$948$	−17.2446	−0.560078
$949$	−5.01879	−0.162917
$950$	−13.8683	−0.449948
$951$	−27.3967	−0.888399
$952$	−0.367649	−0.0119156
$953$	−10.3709	−0.335946	−0.167973	−	0.985792i	$-0.553722\pi$
$953$	−10.3709	−0.335946	−0.167973	+	0.985792i	$0.553722\pi$
$954$	8.94869	0.289725
$955$	−32.0525	−1.03719
$956$	−2.81940	−0.0911858
$957$	41.2543	1.33356
$958$	−13.9366	−0.450271
$959$	0.630169	0.0203492
$960$	−3.80194	−0.122707
$961$	−20.4571	−0.659907
$962$	2.91292	0.0939163
$963$	12.7724	0.411585
$964$	−28.5555	−0.919712
$965$	−83.2320	−2.67933
$966$	−0.0465244	−0.00149690
$967$	−1.36227	−0.0438077	−0.0219039	−	0.999760i	$-0.506973\pi$
$967$	−1.36227	−0.0438077	−0.0219039	+	0.999760i	$0.506973\pi$
$968$	12.3201	0.395982
$969$	11.0242	0.354147
$970$	−34.6926	−1.11391
$971$	−4.70304	−0.150928	−0.0754638	−	0.997149i	$-0.524044\pi$
$971$	−4.70304	−0.150928	−0.0754638	+	0.997149i	$0.524044\pi$
$972$	1.00000	0.0320750
$973$	0.596106	0.0191103
$974$	−11.1304	−0.356640
$975$	21.7071	0.695183
$976$	−2.62133	−0.0839069
$977$	42.6746	1.36528	0.682640	−	0.730755i	$-0.260832\pi$
$977$	42.6746	1.36528	0.682640	+	0.730755i	$0.260832\pi$
$978$	−9.86725	−0.315520
$979$	41.8434	1.33732
$980$	26.6045	0.849849
$981$	6.80731	0.217341
$982$	1.41550	0.0451705
$983$	−39.2379	−1.25149	−0.625747	−	0.780026i	$-0.715206\pi$
$983$	−39.2379	−1.25149	−0.625747	+	0.780026i	$0.715206\pi$
$984$	−2.91723	−0.0929979
$985$	4.09113	0.130354
$986$	64.2059	2.04473
$987$	0.473517	0.0150722
$988$	−3.36765	−0.107139
$989$	3.19269	0.101521
$990$	18.3599	0.583515
$991$	41.8544	1.32955	0.664775	−	0.747044i	$-0.268527\pi$
$991$	41.8544	1.32955	0.664775	+	0.747044i	$0.268527\pi$
$992$	−3.24698	−0.103092
$993$	27.7071	0.879258
$994$	−0.534253	−0.0169455
$995$	72.8407	2.30921
$996$	−4.85086	−0.153705
$997$	10.7283	0.339768	0.169884	−	0.985464i	$-0.445661\pi$
$997$	10.7283	0.339768	0.169884	+	0.985464i	$0.445661\pi$
$998$	37.2771	1.17999
$999$	1.26875	0.0401415

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1338.2.a.f.1.1	✓	3
3.2	odd	2		4014.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.f.1.1	✓	3	1.1	even	1	trivial
4014.2.a.n.1.3		3	3.2	odd	2

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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

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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	1338.2.a.f.1.1

Level	$1338$
Weight	$2$
Character	1338.1
Self dual	yes
Analytic conductor	$10.684$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$