LMFDB - Embedded newform 169.8.b.e.168.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,8,Mod(168,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.168"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$N$	$=$	$169 = 13^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	169.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$52.7930693068$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 1591 x^{14} + 998837 x^{12} + 319862003 x^{10} + 57017400035 x^{8} + 5819167911653 x^{6} + \cdots + 11\!\cdots\!00$ x^16 + 1591x^14 + 998837x^12 + 319862003x^10 + 57017400035x^8 + 5819167911653x^6 + 333227942883751x^4 + 9796730453852209*x^2 + 113340278793830400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{15}\cdot 13^{8}$
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.12
Root			$11.1141i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.8.b.e.168.5

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+10.1141i q^{2} -70.4106 q^{3} +25.7042 q^{4} +163.930i q^{5} -712.143i q^{6} +246.938i q^{7} +1554.59i q^{8} +2770.65 q^{9} -1658.01 q^{10} +2031.13i q^{11} -1809.85 q^{12} -2497.57 q^{14} -11542.4i q^{15} -12433.2 q^{16} +34969.8 q^{17} +28022.8i q^{18} -21478.2i q^{19} +4213.68i q^{20} -17387.1i q^{21} -20543.1 q^{22} +70293.3 q^{23} -109459. i q^{24} +51252.0 q^{25} -41095.4 q^{27} +6347.33i q^{28} +206196. q^{29} +116742. q^{30} +304382. i q^{31} +73236.2i q^{32} -143013. i q^{33} +353690. i q^{34} -40480.5 q^{35} +71217.3 q^{36} -122656. i q^{37} +217233. q^{38} -254843. q^{40} -235517. i q^{41} +175855. q^{42} -91350.2 q^{43} +52208.5i q^{44} +454193. i q^{45} +710956. i q^{46} -967683. i q^{47} +875427. q^{48} +762565. q^{49} +518370. i q^{50} -2.46225e6 q^{51} +28768.9 q^{53} -415645. i q^{54} -332963. q^{55} -383886. q^{56} +1.51229e6i q^{57} +2.08550e6i q^{58} +76497.0i q^{59} -296688. i q^{60} -593112. q^{61} -3.07857e6 q^{62} +684180. i q^{63} -2.33217e6 q^{64} +1.44645e6 q^{66} +1.83002e6i q^{67} +898870. q^{68} -4.94939e6 q^{69} -409426. i q^{70} +2.83878e6i q^{71} +4.30722e6i q^{72} +1.74309e6i q^{73} +1.24056e6 q^{74} -3.60868e6 q^{75} -552079. i q^{76} -501563. q^{77} +3.23539e6 q^{79} -2.03817e6i q^{80} -3.16587e6 q^{81} +2.38206e6 q^{82} +4.95198e6i q^{83} -446920. i q^{84} +5.73260e6i q^{85} -923929. i q^{86} -1.45184e7 q^{87} -3.15756e6 q^{88} -5.08510e6i q^{89} -4.59377e6 q^{90} +1.80683e6 q^{92} -2.14317e7i q^{93} +9.78728e6 q^{94} +3.52092e6 q^{95} -5.15660e6i q^{96} -1.50757e7i q^{97} +7.71269e6i q^{98} +5.62756e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 56 q^{3} - 1154 q^{4} + 11976 q^{9} - 13626 q^{10} + 15816 q^{12} + 14484 q^{14} + 122754 q^{16} - 45648 q^{17} + 147452 q^{22} - 144936 q^{23} - 58488 q^{25} + 358616 q^{27} + 443544 q^{29} - 516232 q^{30}+ \cdots - 32534160 q^{95}+O(q^{100})$ 16 * q + 56 * q^3 - 1154 * q^4 + 11976 * q^9 - 13626 * q^10 + 15816 * q^12 + 14484 * q^14 + 122754 * q^16 - 45648 * q^17 + 147452 * q^22 - 144936 * q^23 - 58488 * q^25 + 358616 * q^27 + 443544 * q^29 - 516232 * q^30 - 63120 * q^35 - 2325986 * q^36 - 2364492 * q^38 + 766030 * q^40 - 3121908 * q^42 + 782984 * q^43 - 9171424 * q^48 - 2217560 * q^49 - 8040712 * q^51 + 3329328 * q^53 + 836432 * q^55 - 7218312 * q^56 - 3527096 * q^61 - 25996776 * q^62 - 32227074 * q^64 + 16245108 * q^66 + 20236182 * q^68 - 9236984 * q^69 - 4412442 * q^74 - 43669976 * q^75 - 35263992 * q^77 + 12644416 * q^79 + 18632544 * q^81 - 18695502 * q^82 - 10399976 * q^87 - 73352808 * q^88 - 132263562 * q^90 + 64127424 * q^92 + 12052256 * q^94 - 32534160 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$ .

$n$	$2$
$\chi(n)$	$-1$

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

)

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

$2$	10.1141i	0.893972i	0.894541	+	0.446986i	$0.147503\pi$
$2$	10.1141i	0.893972i	−0.894541	+	0.446986i	$0.852497\pi$
$3$	−70.4106	−1.50561	−0.752807	−	0.658241i	$-0.771301\pi$
$3$	−70.4106	−1.50561	−0.752807	+	0.658241i	$0.771301\pi$
$4$	25.7042	0.200814
$5$	163.930i	0.586494i	0.956037	+	0.293247i	$0.0947358\pi$
$5$	163.930i	0.586494i	−0.956037	+	0.293247i	$0.905264\pi$
$6$	− 712.143i	− 1.34598i
$7$	246.938i	0.272110i	0.990701	+	0.136055i	$0.0434424\pi$
$7$	246.938i	0.272110i	−0.990701	+	0.136055i	$0.956558\pi$
$8$	1554.59i	1.07349i
$9$	2770.65	1.26687
$10$	−1658.01	−0.524309
$11$	2031.13i	0.460112i	0.973177	+	0.230056i	$0.0738909\pi$
$11$	2031.13i	0.460112i	−0.973177	+	0.230056i	$0.926109\pi$
$12$	−1809.85	−0.302348
$13$	0	0
$13$	0	0
$14$	−2497.57	−0.243259
$15$	− 11542.4i	− 0.883033i
$16$	−12433.2	−0.758860
$17$	34969.8	1.72632	0.863162	−	0.504927i	$-0.168481\pi$
$17$	34969.8	1.72632	0.863162	+	0.504927i	$0.168481\pi$
$18$	28022.8i	1.13255i
$19$	− 21478.2i	− 0.718389i	−0.933263	−	0.359195i	$-0.883051\pi$
$19$	− 21478.2i	− 0.718389i	0.933263	−	0.359195i	$-0.116949\pi$
$20$	4213.68i	0.117776i
$21$	− 17387.1i	− 0.409693i
$22$	−20543.1	−0.411327
$23$	70293.3	1.20467	0.602333	−	0.798245i	$-0.294238\pi$
$23$	70293.3	1.20467	0.602333	+	0.798245i	$0.294238\pi$
$24$	− 109459.i	− 1.61627i
$25$	51252.0	0.656025
$26$	0	0
$27$	−41095.4	−0.401810
$28$	6347.33i	0.0546434i
$29$	206196.	1.56996	0.784979	−	0.619523i	$-0.212674\pi$
$29$	206196.	1.56996	0.784979	+	0.619523i	$0.212674\pi$
$30$	116742.	0.789407
$31$	304382.i	1.83507i	0.397651	+	0.917537i	$0.369825\pi$
$31$	304382.i	1.83507i	−0.397651	+	0.917537i	$0.630175\pi$
$32$	73236.2i	0.395094i
$33$	− 143013.i	− 0.692751i
$34$	353690.i	1.54329i
$35$	−40480.5	−0.159591
$36$	71217.3	0.254406
$37$	− 122656.i	− 0.398090i	−0.979990	−	0.199045i	$-0.936216\pi$
$37$	− 122656.i	− 0.398090i	0.979990	−	0.199045i	$-0.0637840\pi$
$38$	217233.	0.642220
$39$	0	0
$40$	−254843.	−0.629598
$41$	− 235517.i	− 0.533678i	−0.963741	−	0.266839i	$-0.914021\pi$
$41$	− 235517.i	− 0.533678i	0.963741	−	0.266839i	$-0.0859792\pi$
$42$	175855.	0.366254
$43$	−91350.2	−0.175214	−0.0876072	−	0.996155i	$-0.527922\pi$
$43$	−91350.2	−0.175214	−0.0876072	+	0.996155i	$0.527922\pi$
$44$	52208.5i	0.0923967i
$45$	454193.i	0.743014i
$46$	710956.i	1.07694i
$47$	− 967683.i	− 1.35954i	−0.733427	−	0.679768i	$-0.762081\pi$
$47$	− 967683.i	− 1.35954i	0.733427	−	0.679768i	$-0.237919\pi$
$48$	875427.	1.14255
$49$	762565.	0.925956
$50$	518370.i	0.586468i
$51$	−2.46225e6	−2.59918
$52$	0	0
$53$	28768.9	0.0265435	0.0132717	−	0.999912i	$-0.495775\pi$
$53$	28768.9	0.0265435	0.0132717	+	0.999912i	$0.495775\pi$
$54$	− 415645.i	− 0.359207i
$55$	−332963.	−0.269853
$56$	−383886.	−0.292109
$57$	1.51229e6i	1.08162i
$58$	2.08550e6i	1.40350i
$59$	76497.0i	0.0484912i	0.999706	+	0.0242456i	$0.00771836\pi$
$59$	76497.0i	0.0484912i	−0.999706	+	0.0242456i	$0.992282\pi$
$60$	− 296688.i	− 0.177325i
$61$	−593112.	−0.334567	−0.167283	−	0.985909i	$-0.553499\pi$
$61$	−593112.	−0.334567	−0.167283	+	0.985909i	$0.553499\pi$
$62$	−3.07857e6	−1.64050
$63$	684180.i	0.344729i
$64$	−2.33217e6	−1.11206
$65$	0	0
$66$	1.44645e6	0.619300
$67$	1.83002e6i	0.743353i	0.928362	+	0.371676i	$0.121217\pi$
$67$	1.83002e6i	0.743353i	−0.928362	+	0.371676i	$0.878783\pi$
$68$	898870.	0.346670
$69$	−4.94939e6	−1.81376
$70$	− 409426.i	− 0.142670i
$71$	2.83878e6i	0.941298i	0.882321	+	0.470649i	$0.155980\pi$
$71$	2.83878e6i	0.941298i	−0.882321	+	0.470649i	$0.844020\pi$
$72$	4.30722e6i	1.35998i
$73$	1.74309e6i	0.524434i	0.965009	+	0.262217i	$0.0844536\pi$
$73$	1.74309e6i	0.524434i	−0.965009	+	0.262217i	$0.915546\pi$
$74$	1.24056e6	0.355881
$75$	−3.60868e6	−0.987721
$76$	− 552079.i	− 0.144262i
$77$	−501563.	−0.125201
$78$	0	0
$79$	3.23539e6	0.738298	0.369149	−	0.929370i	$-0.379649\pi$
$79$	3.23539e6	0.738298	0.369149	+	0.929370i	$0.379649\pi$
$80$	− 2.03817e6i	− 0.445067i
$81$	−3.16587e6	−0.661904
$82$	2.38206e6	0.477094
$83$	4.95198e6i	0.950617i	0.879819	+	0.475309i	$0.157664\pi$
$83$	4.95198e6i	0.950617i	−0.879819	+	0.475309i	$0.842336\pi$
$84$	− 446920.i	− 0.0822720i
$85$	5.73260e6i	1.01248i
$86$	− 923929.i	− 0.156637i
$87$	−1.45184e7	−2.36375
$88$	−3.15756e6	−0.493927
$89$	− 5.08510e6i	− 0.764600i	−0.924038	−	0.382300i	$-0.875132\pi$
$89$	− 5.08510e6i	− 0.764600i	0.924038	−	0.382300i	$-0.124868\pi$
$90$	−4.59377e6	−0.664234
$91$	0	0
$92$	1.80683e6	0.241913
$93$	− 2.14317e7i	− 2.76291i
$94$	9.78728e6	1.21539
$95$	3.52092e6	0.421331
$96$	− 5.15660e6i	− 0.594860i
$97$	− 1.50757e7i	− 1.67717i	−0.544774	−	0.838583i	$-0.683385\pi$
$97$	− 1.50757e7i	− 1.67717i	0.544774	−	0.838583i	$-0.316615\pi$
$98$	7.71269e6i	0.827779i
$99$	5.62756e6i	0.582904i
$100$	1.31739e6	0.131739
$101$	−1.16655e7	−1.12662	−0.563312	−	0.826244i	$-0.690473\pi$
$101$	−1.16655e7	−1.12662	−0.563312	+	0.826244i	$0.690473\pi$
$102$	− 2.49035e7i	− 2.32359i
$103$	−6.15835e6	−0.555308	−0.277654	−	0.960681i	$-0.589557\pi$
$103$	−6.15835e6	−0.555308	−0.277654	+	0.960681i	$0.589557\pi$
$104$	0	0
$105$	2.85026e6	0.240282
$106$	290973.i	0.0237291i
$107$	−1.61162e7	−1.27180	−0.635902	−	0.771770i	$-0.719372\pi$
$107$	−1.61162e7	−1.27180	−0.635902	+	0.771770i	$0.719372\pi$
$108$	−1.05632e6	−0.0806889
$109$	2.48704e7i	1.83946i	0.392553	+	0.919729i	$0.371592\pi$
$109$	2.48704e7i	1.83946i	−0.392553	+	0.919729i	$0.628408\pi$
$110$	− 3.36763e6i	− 0.241241i
$111$	8.63625e6i	0.599370i
$112$	− 3.07022e6i	− 0.206494i
$113$	−1.45555e6	−0.0948972	−0.0474486	−	0.998874i	$-0.515109\pi$
$113$	−1.45555e6	−0.0948972	−0.0474486	+	0.998874i	$0.515109\pi$
$114$	−1.52955e7	−0.966936
$115$	1.15232e7i	0.706529i
$116$	5.30010e6	0.315269
$117$	0	0
$118$	−773702.	−0.0433497
$119$	8.63538e6i	0.469750i
$120$	1.79437e7	0.947931
$121$	1.53617e7	0.788297
$122$	− 5.99882e6i	− 0.299093i
$123$	1.65829e7i	0.803514i
$124$	7.82389e6i	0.368508i
$125$	2.12088e7i	0.971248i
$126$	−6.91989e6	−0.308178
$127$	1.39029e7	0.602271	0.301136	−	0.953581i	$-0.402634\pi$
$127$	1.39029e7	0.602271	0.301136	+	0.953581i	$0.402634\pi$
$128$	− 1.42136e7i	− 0.599059i
$129$	6.43202e6	0.263805
$130$	0	0
$131$	2.68014e7	1.04162	0.520809	−	0.853673i	$-0.325631\pi$
$131$	2.68014e7	1.04162	0.520809	+	0.853673i	$0.325631\pi$
$132$	− 3.67603e6i	− 0.139114i
$133$	5.30378e6	0.195481
$134$	−1.85091e7	−0.664537
$135$	− 6.73678e6i	− 0.235659i
$136$	5.43636e7i	1.85320i
$137$	− 7.18419e6i	− 0.238702i	−0.992852	−	0.119351i	$-0.961919\pi$
$137$	− 7.18419e6i	− 0.238702i	0.992852	−	0.119351i	$-0.0380813\pi$
$138$	− 5.00589e7i	− 1.62145i
$139$	1.83360e7	0.579098	0.289549	−	0.957163i	$-0.406495\pi$
$139$	1.83360e7	0.579098	0.289549	+	0.957163i	$0.406495\pi$
$140$	−1.04052e6	−0.0320480
$141$	6.81351e7i	2.04694i
$142$	−2.87118e7	−0.841494
$143$	0	0
$144$	−3.44480e7	−0.961380
$145$	3.38018e7i	0.920770i
$146$	−1.76299e7	−0.468829
$147$	−5.36926e7	−1.39413
$148$	− 3.15276e6i	− 0.0799419i
$149$	− 1.02171e7i	− 0.253033i	−0.991965	−	0.126516i	$-0.959620\pi$
$149$	− 1.02171e7i	− 0.253033i	0.991965	−	0.126516i	$-0.0403796\pi$
$150$	− 3.64987e7i	− 0.882995i
$151$	1.08524e7i	0.256512i	0.991741	+	0.128256i	$0.0409379\pi$
$151$	1.08524e7i	0.256512i	−0.991741	+	0.128256i	$0.959062\pi$
$152$	3.33897e7	0.771187
$153$	9.68893e7	2.18704
$154$	− 5.07288e6i	− 0.111926i
$155$	−4.98974e7	−1.07626
$156$	0	0
$157$	−3.58959e7	−0.740280	−0.370140	−	0.928976i	$-0.620690\pi$
$157$	−3.58959e7	−0.740280	−0.370140	+	0.928976i	$0.620690\pi$
$158$	3.27232e7i	0.660018i
$159$	−2.02564e6	−0.0399642
$160$	−1.20056e7	−0.231720
$161$	1.73581e7i	0.327802i
$162$	− 3.20200e7i	− 0.591724i
$163$	− 5.18364e7i	− 0.937515i	−0.883327	−	0.468758i	$-0.844702\pi$
$163$	− 5.18364e7i	− 0.937515i	0.883327	−	0.468758i	$-0.155298\pi$
$164$	− 6.05378e6i	− 0.107170i
$165$	2.34441e7	0.406294
$166$	−5.00850e7	−0.849825
$167$	6.97475e7i	1.15883i	0.815031	+	0.579417i	$0.196720\pi$
$167$	6.97475e7i	1.15883i	−0.815031	+	0.579417i	$0.803280\pi$
$168$	2.70297e7	0.439803
$169$	0	0
$170$	−5.79804e7	−0.905127
$171$	− 5.95086e7i	− 0.910109i
$172$	−2.34808e6	−0.0351855
$173$	4.34880e7	0.638570	0.319285	−	0.947659i	$-0.396557\pi$
$173$	4.34880e7	0.638570	0.319285	+	0.947659i	$0.396557\pi$
$174$	− 1.46841e8i	− 2.11313i
$175$	1.26561e7i	0.178511i
$176$	− 2.52534e7i	− 0.349160i
$177$	− 5.38620e6i	− 0.0730090i
$178$	5.14314e7	0.683531
$179$	3.08734e7	0.402345	0.201173	−	0.979556i	$-0.435525\pi$
$179$	3.08734e7	0.402345	0.201173	+	0.979556i	$0.435525\pi$
$180$	1.16747e7i	0.149207i
$181$	1.39507e8	1.74872	0.874360	−	0.485278i	$-0.161282\pi$
$181$	1.39507e8	1.74872	0.874360	+	0.485278i	$0.161282\pi$
$182$	0	0
$183$	4.17614e7	0.503728
$184$	1.09277e8i	1.29320i
$185$	2.01069e7	0.233477
$186$	2.16764e8	2.46997
$187$	7.10282e7i	0.794302i
$188$	− 2.48735e7i	− 0.273013i
$189$	− 1.01480e7i	− 0.109336i
$190$	3.56111e7i	0.376658i
$191$	−4.12386e7	−0.428240	−0.214120	−	0.976807i	$-0.568688\pi$
$191$	−4.12386e7	−0.428240	−0.214120	+	0.976807i	$0.568688\pi$
$192$	1.64209e8	1.67434
$193$	− 1.38800e8i	− 1.38976i	−0.719127	−	0.694879i	$-0.755458\pi$
$193$	− 1.38800e8i	− 1.38976i	0.719127	−	0.694879i	$-0.244542\pi$
$194$	1.52478e8	1.49934
$195$	0	0
$196$	1.96011e7	0.185945
$197$	− 5.13435e6i	− 0.0478468i	−0.999714	−	0.0239234i	$-0.992384\pi$
$197$	− 5.13435e6i	− 0.0478468i	0.999714	−	0.0239234i	$-0.00761579\pi$
$198$	−5.69179e7	−0.521100
$199$	−9.62755e7	−0.866024	−0.433012	−	0.901388i	$-0.642549\pi$
$199$	−9.62755e7	−0.866024	−0.433012	+	0.901388i	$0.642549\pi$
$200$	7.96756e7i	0.704239i
$201$	− 1.28853e8i	− 1.11920i
$202$	− 1.17987e8i	− 1.00717i
$203$	5.09177e7i	0.427201i
$204$	−6.32900e7	−0.521951
$205$	3.86084e7	0.312999
$206$	− 6.22864e7i	− 0.496430i
$207$	1.94758e8	1.52616
$208$	0	0
$209$	4.36250e7	0.330539
$210$	2.88279e7i	0.214806i
$211$	1.07796e8	0.789976	0.394988	−	0.918686i	$-0.370749\pi$
$211$	1.07796e8	0.789976	0.394988	+	0.918686i	$0.370749\pi$
$212$	739480.	0.00533029
$213$	− 1.99880e8i	− 1.41723i
$214$	− 1.63002e8i	− 1.13696i
$215$	− 1.49750e7i	− 0.102762i
$216$	− 6.38864e7i	− 0.431340i
$217$	−7.51635e7	−0.499342
$218$	−2.51543e8	−1.64442
$219$	− 1.22732e8i	− 0.789595i
$220$	−8.55853e6	−0.0541901
$221$	0	0
$222$	−8.73483e7	−0.535820
$223$	− 7.54899e6i	− 0.0455850i	−0.999740	−	0.0227925i	$-0.992744\pi$
$223$	− 7.54899e6i	− 0.0455850i	0.999740	−	0.0227925i	$-0.00725570\pi$
$224$	−1.80848e7	−0.107509
$225$	1.42001e8	0.831101
$226$	− 1.47217e7i	− 0.0848355i
$227$	− 1.61179e8i	− 0.914572i	−0.889320	−	0.457286i	$-0.848822\pi$
$227$	− 1.61179e8i	− 0.914572i	0.889320	−	0.457286i	$-0.151178\pi$
$228$	3.88722e7i	0.217204i
$229$	4.77059e7i	0.262511i	0.991349	+	0.131256i	$0.0419009\pi$
$229$	4.77059e7i	0.262511i	−0.991349	+	0.131256i	$0.958099\pi$
$230$	−1.16547e8	−0.631617
$231$	3.53153e7	0.188504
$232$	3.20550e8i	1.68534i
$233$	−1.71967e8	−0.890632	−0.445316	−	0.895373i	$-0.646909\pi$
$233$	−1.71967e8	−0.890632	−0.445316	+	0.895373i	$0.646909\pi$
$234$	0	0
$235$	1.58632e8	0.797359
$236$	1.96629e6i	0.00973769i
$237$	−2.27806e8	−1.11159
$238$	−8.73394e7	−0.419944
$239$	− 2.50000e8i	− 1.18453i	−0.805743	−	0.592266i	$-0.798234\pi$
$239$	− 2.50000e8i	− 1.18453i	0.805743	−	0.592266i	$-0.201766\pi$
$240$	1.43509e8i	0.670099i
$241$	− 7.12253e7i	− 0.327774i	−0.986479	−	0.163887i	$-0.947597\pi$
$241$	− 7.12253e7i	− 0.327774i	0.986479	−	0.163887i	$-0.0524032\pi$
$242$	1.55370e8i	0.704716i
$243$	3.12786e8	1.39838
$244$	−1.52455e7	−0.0671856
$245$	1.25007e8i	0.543067i
$246$	−1.67722e8	−0.718319
$247$	0	0
$248$	−4.73188e8	−1.96994
$249$	− 3.48672e8i	− 1.43126i
$250$	−2.14508e8	−0.868269
$251$	−3.80815e8	−1.52004	−0.760022	−	0.649898i	$-0.774812\pi$
$251$	−3.80815e8	−1.52004	−0.760022	+	0.649898i	$0.774812\pi$
$252$	1.75863e7i	0.0692264i
$253$	1.42775e8i	0.554281i
$254$	1.40616e8i	0.538414i
$255$	− 4.03636e8i	− 1.52440i
$256$	−1.54759e8	−0.576521
$257$	−4.85828e7	−0.178532	−0.0892662	−	0.996008i	$-0.528452\pi$
$257$	−4.85828e7	−0.178532	−0.0892662	+	0.996008i	$0.528452\pi$
$258$	6.50544e7i	0.235835i
$259$	3.02883e7	0.108324
$260$	0	0
$261$	5.71299e8	1.98894
$262$	2.71073e8i	0.931177i
$263$	2.49404e8	0.845393	0.422697	−	0.906271i	$-0.361084\pi$
$263$	2.49404e8	0.845393	0.422697	+	0.906271i	$0.361084\pi$
$264$	2.22326e8	0.743664
$265$	4.71609e6i	0.0155676i
$266$	5.36432e7i	0.174755i
$267$	3.58045e8i	1.15119i
$268$	4.70392e7i	0.149275i
$269$	4.73557e8	1.48333	0.741667	−	0.670768i	$-0.234036\pi$
$269$	4.73557e8	1.48333	0.741667	+	0.670768i	$0.234036\pi$
$270$	6.81367e7	0.210673
$271$	− 1.44445e8i	− 0.440871i	−0.975402	−	0.220435i	$-0.929252\pi$
$271$	− 1.44445e8i	− 0.440871i	0.975402	−	0.220435i	$-0.0707478\pi$
$272$	−4.34786e8	−1.31004
$273$	0	0
$274$	7.26619e7	0.213393
$275$	1.04099e8i	0.301845i
$276$	−1.27220e8	−0.364228
$277$	−4.52170e8	−1.27827	−0.639134	−	0.769095i	$-0.720707\pi$
$277$	−4.52170e8	−1.27827	−0.639134	+	0.769095i	$0.720707\pi$
$278$	1.85453e8i	0.517697i
$279$	8.43338e8i	2.32481i
$280$	− 6.29304e7i	− 0.171320i
$281$	1.80819e8i	0.486151i	0.970007	+	0.243075i	$0.0781563\pi$
$281$	1.80819e8i	0.486151i	−0.970007	+	0.243075i	$0.921844\pi$
$282$	−6.89128e8	−1.82990
$283$	−5.59469e8	−1.46732	−0.733658	−	0.679519i	$-0.762188\pi$
$283$	−5.59469e8	−1.46732	−0.733658	+	0.679519i	$0.762188\pi$
$284$	7.29684e7i	0.189026i
$285$	−2.47910e8	−0.634362
$286$	0	0
$287$	5.81582e7	0.145219
$288$	2.02912e8i	0.500535i
$289$	8.12550e8	1.98019
$290$	−3.41876e8	−0.823143
$291$	1.06149e9i	2.52517i
$292$	4.48048e7i	0.105314i
$293$	− 3.79042e8i	− 0.880340i	−0.897914	−	0.440170i	$-0.854918\pi$
$293$	− 3.79042e8i	− 0.880340i	0.897914	−	0.440170i	$-0.145082\pi$
$294$	− 5.43055e8i	− 1.24632i
$295$	−1.25402e7	−0.0284398
$296$	1.90679e8	0.427347
$297$	− 8.34702e7i	− 0.184877i
$298$	1.03337e8	0.226204
$299$	0	0
$300$	−9.27581e7	−0.198348
$301$	− 2.25578e7i	− 0.0476776i
$302$	−1.09763e8	−0.229315
$303$	8.21375e8	1.69626
$304$	2.67042e8i	0.545157i
$305$	− 9.72289e7i	− 0.196221i
$306$	9.79952e8i	1.95515i
$307$	1.00446e9i	1.98130i	0.136433	+	0.990649i	$0.456436\pi$
$307$	1.00446e9i	1.98130i	−0.136433	+	0.990649i	$0.543564\pi$
$308$	−1.28923e7	−0.0251421
$309$	4.33613e8	0.836080
$310$	− 5.04669e8i	− 0.962146i
$311$	−4.57084e8	−0.861658	−0.430829	−	0.902434i	$-0.641779\pi$
$311$	−4.57084e8	−0.861658	−0.430829	+	0.902434i	$0.641779\pi$
$312$	0	0
$313$	−6.79628e8	−1.25275	−0.626377	−	0.779520i	$-0.715463\pi$
$313$	−6.79628e8	−1.25275	−0.626377	+	0.779520i	$0.715463\pi$
$314$	− 3.63056e8i	− 0.661790i
$315$	−1.12158e8	−0.202182
$316$	8.31630e7	0.148260
$317$	− 1.42499e8i	− 0.251249i	−0.992078	−	0.125625i	$-0.959907\pi$
$317$	− 1.42499e8i	− 0.251249i	0.992078	−	0.125625i	$-0.0400935\pi$
$318$	− 2.04876e7i	− 0.0357269i
$319$	4.18811e8i	0.722356i
$320$	− 3.82312e8i	− 0.652218i
$321$	1.13475e9	1.91485
$322$	−1.75562e8	−0.293046
$323$	− 7.51088e8i	− 1.24017i
$324$	−8.13759e7	−0.132919
$325$	0	0
$326$	5.24281e8	0.838113
$327$	− 1.75114e9i	− 2.76952i
$328$	3.66132e8	0.572900
$329$	2.38958e8	0.369943
$330$	2.37117e8i	0.363215i
$331$	− 1.19105e8i	− 0.180523i	−0.995918	−	0.0902614i	$-0.971230\pi$
$331$	− 1.19105e8i	− 0.180523i	0.995918	−	0.0902614i	$-0.0287703\pi$
$332$	1.27287e8i	0.190897i
$333$	− 3.39836e8i	− 0.504330i
$334$	−7.05436e8	−1.03596
$335$	−2.99996e8	−0.435972
$336$	2.16176e8i	0.310900i
$337$	3.15254e8	0.448700	0.224350	−	0.974509i	$-0.427974\pi$
$337$	3.15254e8	0.448700	0.224350	+	0.974509i	$0.427974\pi$
$338$	0	0
$339$	1.02486e8	0.142879
$340$	1.47352e8i	0.203320i
$341$	−6.18240e8	−0.844339
$342$	6.01879e8	0.813612
$343$	3.91670e8i	0.524072i
$344$	− 1.42012e8i	− 0.188092i
$345$	− 8.11354e8i	− 1.06376i
$346$	4.39844e8i	0.570864i
$347$	−5.40640e8	−0.694632	−0.347316	−	0.937748i	$-0.612907\pi$
$347$	−5.40640e8	−0.694632	−0.347316	+	0.937748i	$0.612907\pi$
$348$	−3.73183e8	−0.474674
$349$	− 2.77488e8i	− 0.349426i	−0.984619	−	0.174713i	$-0.944100\pi$
$349$	− 2.77488e8i	− 0.349426i	0.984619	−	0.174713i	$-0.0558998\pi$
$350$	−1.28005e8	−0.159584
$351$	0	0
$352$	−1.48752e8	−0.181787
$353$	− 6.59769e6i	− 0.00798327i	−0.999992	−	0.00399163i	$-0.998729\pi$
$353$	− 6.59769e6i	− 0.00798327i	0.999992	−	0.00399163i	$-0.00127058\pi$
$354$	5.44768e7	0.0652680
$355$	−4.65361e8	−0.552065
$356$	− 1.30708e8i	− 0.153542i
$357$	− 6.08022e8i	− 0.707263i
$358$	3.12258e8i	0.359686i
$359$	6.67727e8i	0.761672i	0.924643	+	0.380836i	$0.124364\pi$
$359$	6.67727e8i	0.761672i	−0.924643	+	0.380836i	$0.875636\pi$
$360$	−7.06082e8	−0.797621
$361$	4.32559e8	0.483917
$362$	1.41099e9i	1.56331i
$363$	−1.08163e9	−1.18687
$364$	0	0
$365$	−2.85745e8	−0.307577
$366$	4.22381e8i	0.450319i
$367$	8.10048e8	0.855421	0.427710	−	0.903916i	$-0.359320\pi$
$367$	8.10048e8	0.855421	0.427710	+	0.903916i	$0.359320\pi$
$368$	−8.73968e8	−0.914173
$369$	− 6.52537e8i	− 0.676103i
$370$	2.03364e8i	0.208722i
$371$	7.10413e6i	0.00722275i
$372$	− 5.50885e8i	− 0.554831i
$373$	−6.52847e8	−0.651374	−0.325687	−	0.945478i	$-0.605596\pi$
$373$	−6.52847e8	−0.651374	−0.325687	+	0.945478i	$0.605596\pi$
$374$	−7.18390e8	−0.710084
$375$	− 1.49332e9i	− 1.46233i
$376$	1.50435e9	1.45945
$377$	0	0
$378$	1.02639e8	0.0977438
$379$	1.48973e9i	1.40563i	0.711372	+	0.702816i	$0.248074\pi$
$379$	1.48973e9i	1.40563i	−0.711372	+	0.702816i	$0.751926\pi$
$380$	9.05022e7	0.0846090
$381$	−9.78911e8	−0.906788
$382$	− 4.17093e8i	− 0.382834i
$383$	4.40576e7i	0.0400705i	0.999799	+	0.0200353i	$0.00637785\pi$
$383$	4.40576e7i	0.0400705i	−0.999799	+	0.0200353i	$0.993622\pi$
$384$	1.00079e9i	0.901952i
$385$	− 8.22212e7i	− 0.0734296i
$386$	1.40384e9	1.24241
$387$	−2.53100e8	−0.221975
$388$	− 3.87508e8i	− 0.336798i
$389$	742221.	0.000639307 0	0.000319654	−	1.00000i	$-0.499898\pi$
$389$	742221.	0.000639307 0	0.000319654		1.00000i	$0.499898\pi$
$390$	0	0
$391$	2.45814e9	2.07964
$392$	1.18547e9i	0.994008i
$393$	−1.88710e9	−1.56827
$394$	5.19295e7	0.0427737
$395$	5.30377e8i	0.433007i
$396$	1.44652e8i	0.117055i
$397$	− 1.29390e9i	− 1.03785i	−0.854819	−	0.518926i	$-0.826332\pi$
$397$	− 1.29390e9i	− 1.03785i	0.854819	−	0.518926i	$-0.173668\pi$
$398$	− 9.73744e8i	− 0.774201i
$399$	−3.73442e8	−0.294319
$400$	−6.37224e8	−0.497831
$401$	2.27693e9i	1.76338i	0.471834	+	0.881688i	$0.343592\pi$
$401$	2.27693e9i	1.76338i	−0.471834	+	0.881688i	$0.656408\pi$
$402$	1.30324e9	1.00054
$403$	0	0
$404$	−2.99852e8	−0.226241
$405$	− 5.18980e8i	− 0.388202i
$406$	−5.14989e8	−0.381906
$407$	2.49129e8	0.183166
$408$	− 3.82777e9i	− 2.79020i
$409$	− 2.43493e8i	− 0.175976i	−0.996122	−	0.0879882i	$-0.971956\pi$
$409$	− 2.43493e8i	− 0.175976i	0.996122	−	0.0879882i	$-0.0280438\pi$
$410$	3.90490e8i	0.279812i
$411$	5.05843e8i	0.359393i
$412$	−1.58295e8	−0.111513
$413$	−1.88900e7	−0.0131949
$414$	1.96981e9i	1.36434i
$415$	−8.11778e8	−0.557531
$416$	0	0
$417$	−1.29105e9	−0.871898
$418$	4.41229e8i	0.295493i
$419$	−8.61060e8	−0.571853	−0.285926	−	0.958252i	$-0.592301\pi$
$419$	−8.61060e8	−0.571853	−0.285926	+	0.958252i	$0.592301\pi$
$420$	7.32635e7	0.0482520
$421$	− 3.23521e8i	− 0.211308i	−0.994403	−	0.105654i	$-0.966306\pi$
$421$	− 3.23521e8i	− 0.211308i	0.994403	−	0.105654i	$-0.0336936\pi$
$422$	1.09026e9i	0.706217i
$423$	− 2.68111e9i	− 1.72236i
$424$	4.47237e7i	0.0284943i
$425$	1.79227e9	1.13251
$426$	2.02161e9	1.26697
$427$	− 1.46462e8i	− 0.0910390i
$428$	−4.14254e8	−0.255396
$429$	0	0
$430$	1.51460e8	0.0918665
$431$	− 2.22285e9i	− 1.33733i	−0.743562	−	0.668667i	$-0.766865\pi$
$431$	− 2.22285e9i	− 1.33733i	0.743562	−	0.668667i	$-0.233135\pi$
$432$	5.10946e8	0.304917
$433$	2.05441e9	1.21613	0.608066	−	0.793887i	$-0.291946\pi$
$433$	2.05441e9	1.21613	0.608066	+	0.793887i	$0.291946\pi$
$434$	− 7.60215e8i	− 0.446398i
$435$	− 2.38000e9i	− 1.38632i
$436$	6.39273e8i	0.369389i
$437$	− 1.50977e9i	− 0.865419i
$438$	1.24133e9	0.705876
$439$	2.35136e8	0.132646	0.0663229	−	0.997798i	$-0.478873\pi$
$439$	2.35136e8	0.132646	0.0663229	+	0.997798i	$0.478873\pi$
$440$	− 5.17619e8i	− 0.289685i
$441$	2.11280e9	1.17307
$442$	0	0
$443$	−1.35426e9	−0.740100	−0.370050	−	0.929012i	$-0.620659\pi$
$443$	−1.35426e9	−0.740100	−0.370050	+	0.929012i	$0.620659\pi$
$444$	2.21988e8i	0.120362i
$445$	8.33600e8	0.448433
$446$	7.63516e7	0.0407517
$447$	7.19394e8i	0.380970i
$448$	− 5.75900e8i	− 0.302604i
$449$	− 1.91377e9i	− 0.997764i	−0.866670	−	0.498882i	$-0.833744\pi$
$449$	− 1.91377e9i	− 0.997764i	0.866670	−	0.498882i	$-0.166256\pi$
$450$	1.43622e9i	0.742981i
$451$	4.78366e8	0.245552
$452$	−3.74138e7	−0.0190567
$453$	− 7.64126e8i	− 0.386208i
$454$	1.63019e9	0.817602
$455$	0	0
$456$	−2.35099e9	−1.16111
$457$	3.29110e9i	1.61300i	0.591234	+	0.806500i	$0.298641\pi$
$457$	3.29110e9i	1.61300i	−0.591234	+	0.806500i	$0.701359\pi$
$458$	−4.82504e8	−0.234678
$459$	−1.43710e9	−0.693654
$460$	2.96194e8i	0.141881i
$461$	− 1.74075e9i	− 0.827530i	−0.910384	−	0.413765i	$-0.864214\pi$
$461$	− 1.74075e9i	− 0.827530i	0.910384	−	0.413765i	$-0.135786\pi$
$462$	3.57184e8i	0.168518i
$463$	5.88177e7i	0.0275407i	0.999905	+	0.0137703i	$0.00438337\pi$
$463$	5.88177e7i	0.0275407i	−0.999905	+	0.0137703i	$0.995617\pi$
$464$	−2.56367e9	−1.19138
$465$	3.51331e9	1.62043
$466$	− 1.73929e9i	− 0.796200i
$467$	−1.47573e9	−0.670499	−0.335249	−	0.942129i	$-0.608821\pi$
$467$	−1.47573e9	−0.670499	−0.335249	+	0.942129i	$0.608821\pi$
$468$	0	0
$469$	−4.51902e8	−0.202274
$470$	1.60443e9i	0.712817i
$471$	2.52745e9	1.11458
$472$	−1.18921e8	−0.0520550
$473$	− 1.85544e8i	− 0.0806182i
$474$	− 2.30406e9i	− 0.993732i
$475$	− 1.10080e9i	− 0.471281i
$476$	2.21965e8i	0.0943323i
$477$	7.97087e7	0.0336272
$478$	2.52853e9	1.05894
$479$	1.15115e9i	0.478581i	0.970948	+	0.239291i	$0.0769149\pi$
$479$	1.15115e9i	0.478581i	−0.970948	+	0.239291i	$0.923085\pi$
$480$	8.45322e8	0.348881
$481$	0	0
$482$	7.20382e8	0.293021
$483$	− 1.22219e9i	− 0.493543i
$484$	3.94859e8	0.158301
$485$	2.47136e9	0.983647
$486$	3.16356e9i	1.25011i
$487$	4.88210e9i	1.91538i	0.287796	+	0.957692i	$0.407077\pi$
$487$	4.88210e9i	1.91538i	−0.287796	+	0.957692i	$0.592923\pi$
$488$	− 9.22044e8i	− 0.359155i
$489$	3.64983e9i	1.41154i
$490$	−1.26434e9	−0.485487
$491$	1.20367e9	0.458903	0.229452	−	0.973320i	$-0.426307\pi$
$491$	1.20367e9	0.458903	0.229452	+	0.973320i	$0.426307\pi$
$492$	4.26250e8i	0.161357i
$493$	7.21065e9	2.71025
$494$	0	0
$495$	−9.22525e8	−0.341869
$496$	− 3.78443e9i	− 1.39256i
$497$	−7.01002e8	−0.256137
$498$	3.52652e9	1.27951
$499$	3.64043e9i	1.31160i	0.754935	+	0.655799i	$0.227668\pi$
$499$	3.64043e9i	1.31160i	−0.754935	+	0.655799i	$0.772332\pi$
$500$	5.45153e8i	0.195040i
$501$	− 4.91096e9i	− 1.74476i
$502$	− 3.85162e9i	− 1.35888i
$503$	3.30636e9	1.15841	0.579206	−	0.815181i	$-0.303363\pi$
$503$	3.30636e9	1.15841	0.579206	+	0.815181i	$0.303363\pi$
$504$	−1.06362e9	−0.370065
$505$	− 1.91233e9i	− 0.660758i
$506$	−1.44404e9	−0.495512
$507$	0	0
$508$	3.57362e8	0.120944
$509$	1.10135e9i	0.370179i	0.982722	+	0.185089i	$0.0592575\pi$
$509$	1.10135e9i	0.370179i	−0.982722	+	0.185089i	$0.940743\pi$
$510$	4.08243e9	1.36277
$511$	−4.30436e8	−0.142704
$512$	− 3.38459e9i	− 1.11445i
$513$	8.82655e8i	0.288656i
$514$	− 4.91374e8i	− 0.159603i
$515$	− 1.00954e9i	− 0.325685i
$516$	1.65330e8	0.0529757
$517$	1.96549e9	0.625538
$518$	3.06340e8i	0.0968389i
$519$	−3.06202e9	−0.961440
$520$	0	0
$521$	4.27358e9	1.32391	0.661957	−	0.749541i	$-0.269726\pi$
$521$	4.27358e9	1.32391	0.661957	+	0.749541i	$0.269726\pi$
$522$	5.77819e9i	1.77806i
$523$	−3.01897e9	−0.922790	−0.461395	−	0.887195i	$-0.652651\pi$
$523$	−3.01897e9	−0.922790	−0.461395	+	0.887195i	$0.652651\pi$
$524$	6.88908e8	0.209171
$525$	− 8.91120e8i	− 0.268769i
$526$	2.52251e9i	0.755758i
$527$	1.06442e10i	3.16793i
$528$	1.77810e9i	0.525701i
$529$	1.53632e9	0.451220
$530$	−4.76992e7	−0.0139170
$531$	2.11947e8i	0.0614322i
$532$	1.36329e8	0.0392553
$533$	0	0
$534$	−3.62132e9	−1.02913
$535$	− 2.64193e9i	− 0.745905i
$536$	−2.84493e9	−0.797985
$537$	−2.17382e9	−0.605777
$538$	4.78962e9i	1.32606i
$539$	1.54887e9i	0.426043i
$540$	− 1.73163e8i	− 0.0473235i
$541$	2.08205e9i	0.565329i	0.959219	+	0.282664i	$0.0912183\pi$
$541$	2.08205e9i	0.565329i	−0.959219	+	0.282664i	$0.908782\pi$
$542$	1.46094e9	0.394126
$543$	−9.82275e9	−2.63290
$544$	2.56106e9i	0.682061i
$545$	−4.07700e9	−1.07883
$546$	0	0
$547$	−1.26947e9	−0.331641	−0.165820	−	0.986156i	$-0.553027\pi$
$547$	−1.26947e9	−0.331641	−0.165820	+	0.986156i	$0.553027\pi$
$548$	− 1.84663e8i	− 0.0479346i
$549$	−1.64331e9	−0.423854
$550$	−1.05288e9	−0.269841
$551$	− 4.42872e9i	− 1.12784i
$552$	− 7.69426e9i	− 1.94706i
$553$	7.98940e8i	0.200898i
$554$	− 4.57331e9i	− 1.14274i
$555$	−1.41574e9	−0.351527
$556$	4.71311e8	0.116291
$557$	2.92572e9i	0.717365i	0.933460	+	0.358682i	$0.116774\pi$
$557$	2.92572e9i	0.717365i	−0.933460	+	0.358682i	$0.883226\pi$
$558$	−8.52964e9	−2.07831
$559$	0	0
$560$	5.03301e8	0.121107
$561$	− 5.00114e9i	− 1.19591i
$562$	−1.82882e9	−0.434605
$563$	−8.16638e9	−1.92864	−0.964318	−	0.264746i	$-0.914712\pi$
$563$	−8.16638e9	−1.92864	−0.964318	+	0.264746i	$0.914712\pi$
$564$	1.75136e9i	0.411053i
$565$	− 2.38609e8i	− 0.0556566i
$566$	− 5.65855e9i	− 1.31174i
$567$	− 7.81772e8i	− 0.180111i
$568$	−4.41312e9	−1.01048
$569$	2.02054e9	0.459805	0.229903	−	0.973214i	$-0.426159\pi$
$569$	2.02054e9	0.459805	0.229903	+	0.973214i	$0.426159\pi$
$570$	− 2.50740e9i	− 0.567102i
$571$	−4.65062e9	−1.04540	−0.522702	−	0.852515i	$-0.675076\pi$
$571$	−4.65062e9	−1.04540	−0.522702	+	0.852515i	$0.675076\pi$
$572$	0	0
$573$	2.90363e9	0.644764
$574$	5.88220e8i	0.129822i
$575$	3.60267e9	0.790291
$576$	−6.46163e9	−1.40884
$577$	− 5.94249e9i	− 1.28781i	−0.765104	−	0.643907i	$-0.777312\pi$
$577$	− 5.94249e9i	− 1.28781i	0.765104	−	0.643907i	$-0.222688\pi$
$578$	8.21825e9i	1.77024i
$579$	9.77300e9i	2.09244i
$580$	8.68846e8i	0.184903i
$581$	−1.22283e9	−0.258673
$582$	−1.07360e10	−2.25743
$583$	5.84334e7i	0.0122130i
$584$	−2.70979e9	−0.562977
$585$	0	0
$586$	3.83368e9	0.787000
$587$	7.96514e8i	0.162540i	0.996692	+	0.0812700i	$0.0258976\pi$
$587$	7.96514e8i	0.162540i	−0.996692	+	0.0812700i	$0.974102\pi$
$588$	−1.38012e9	−0.279961
$589$	6.53758e9	1.31830
$590$	− 1.26833e8i	− 0.0254244i
$591$	3.61512e8i	0.0720389i
$592$	1.52500e9i	0.302095i
$593$	− 6.11768e9i	− 1.20475i	−0.798215	−	0.602373i	$-0.794222\pi$
$593$	− 6.11768e9i	− 1.20475i	0.798215	−	0.602373i	$-0.205778\pi$
$594$	8.44229e8	0.165275
$595$	−1.41560e9	−0.275506
$596$	− 2.62623e8i	− 0.0508125i
$597$	6.77881e9	1.30390
$598$	0	0
$599$	4.08018e9	0.775685	0.387843	−	0.921726i	$-0.373220\pi$
$599$	4.08018e9	0.775685	0.387843	+	0.921726i	$0.373220\pi$
$600$	− 5.61000e9i	− 1.06031i
$601$	1.01331e10	1.90407	0.952035	−	0.305990i	$-0.0989875\pi$
$601$	1.01331e10	1.90407	0.952035	+	0.305990i	$0.0989875\pi$
$602$	2.28153e8	0.0426225
$603$	5.07036e9i	0.941734i
$604$	2.78953e8i	0.0515111i
$605$	2.51824e9i	0.462331i
$606$	8.30751e9i	1.51641i
$607$	−4.52529e9	−0.821269	−0.410635	−	0.911800i	$-0.634693\pi$
$607$	−4.52529e9	−0.821269	−0.410635	+	0.911800i	$0.634693\pi$
$608$	1.57298e9	0.283832
$609$	− 3.58515e9i	− 0.643200i
$610$	9.83387e8	0.175416
$611$	0	0
$612$	2.49046e9	0.439187
$613$	− 8.24422e7i	− 0.0144557i	−0.999974	−	0.00722783i	$-0.997699\pi$
$613$	− 8.24422e7i	− 0.0144557i	0.999974	−	0.00722783i	$-0.00230071\pi$
$614$	−1.01593e10	−1.77123
$615$	−2.71844e9	−0.471256
$616$	− 7.79722e8i	− 0.134403i
$617$	1.06580e9i	0.182674i	0.995820	+	0.0913372i	$0.0291141\pi$
$617$	1.06580e9i	0.182674i	−0.995820	+	0.0913372i	$0.970886\pi$
$618$	4.38562e9i	0.747432i
$619$	− 7.35459e8i	− 0.124635i	−0.998056	−	0.0623177i	$-0.980151\pi$
$619$	− 7.35459e8i	− 0.124635i	0.998056	−	0.0623177i	$-0.0198492\pi$
$620$	−1.28257e9	−0.216128
$621$	−2.88873e9	−0.484046
$622$	− 4.62301e9i	− 0.770298i
$623$	1.25570e9	0.208055
$624$	0	0
$625$	5.27307e8	0.0863940
$626$	− 6.87385e9i	− 1.11993i
$627$	−3.07166e9	−0.497665
$628$	−9.22674e8	−0.148658
$629$	− 4.28924e9i	− 0.687232i
$630$	− 1.13438e9i	− 0.180745i
$631$	4.15983e8i	0.0659132i	0.999457	+	0.0329566i	$0.0104923\pi$
$631$	4.15983e8i	0.0659132i	−0.999457	+	0.0329566i	$0.989508\pi$
$632$	5.02969e9i	0.792559i
$633$	−7.58998e9	−1.18940
$634$	1.44126e9	0.224610
$635$	2.27910e9i	0.353228i
$636$	−5.20673e7	−0.00802537
$637$	0	0
$638$	−4.23592e9	−0.645766
$639$	7.86527e9i	1.19251i
$640$	2.33004e9	0.351345
$641$	−5.62314e9	−0.843288	−0.421644	−	0.906762i	$-0.638547\pi$
$641$	−5.62314e9	−0.843288	−0.421644	+	0.906762i	$0.638547\pi$
$642$	1.14771e10i	1.71182i
$643$	− 8.95418e9i	− 1.32827i	−0.747611	−	0.664137i	$-0.768799\pi$
$643$	− 8.95418e9i	− 1.32827i	0.747611	−	0.664137i	$-0.231201\pi$
$644$	4.46175e8i	0.0658271i
$645$	1.05440e9i	0.154720i
$646$	7.59661e9	1.10868
$647$	−8.72793e9	−1.26691	−0.633456	−	0.773779i	$-0.718364\pi$
$647$	−8.72793e9	−1.26691	−0.633456	+	0.773779i	$0.718364\pi$
$648$	− 4.92161e9i	− 0.710550i
$649$	−1.55375e8	−0.0223113
$650$	0	0
$651$	5.29231e9	0.751816
$652$	− 1.33241e9i	− 0.188266i
$653$	4.03484e9	0.567062	0.283531	−	0.958963i	$-0.408494\pi$
$653$	4.03484e9	0.567062	0.283531	+	0.958963i	$0.408494\pi$
$654$	1.77113e10	2.47587
$655$	4.39355e9i	0.610902i
$656$	2.92823e9i	0.404987i
$657$	4.82951e9i	0.664392i
$658$	2.41685e9i	0.330719i
$659$	−4.47266e9	−0.608789	−0.304394	−	0.952546i	$-0.598454\pi$
$659$	−4.47266e9	−0.608789	−0.304394	+	0.952546i	$0.598454\pi$
$660$	6.02612e8	0.0815894
$661$	− 9.07867e9i	− 1.22269i	−0.791363	−	0.611346i	$-0.790628\pi$
$661$	− 9.07867e9i	− 1.22269i	0.791363	−	0.611346i	$-0.209372\pi$
$662$	1.20465e9	0.161382
$663$	0	0
$664$	−7.69828e9	−1.02048
$665$	8.69448e8i	0.114648i
$666$	3.43715e9	0.450857
$667$	1.44942e10	1.89127
$668$	1.79280e9i	0.232710i
$669$	5.31529e8i	0.0686334i
$670$	− 3.03420e9i	− 0.389747i
$671$	− 1.20469e9i	− 0.153938i
$672$	1.27336e9	0.161867
$673$	8.27137e9	1.04598	0.522991	−	0.852338i	$-0.324816\pi$
$673$	8.27137e9	1.04598	0.522991	+	0.852338i	$0.324816\pi$
$674$	3.18852e9i	0.401125i
$675$	−2.10622e9	−0.263597
$676$	0	0
$677$	−7.18333e9	−0.889744	−0.444872	−	0.895594i	$-0.646751\pi$
$677$	−7.18333e9	−0.889744	−0.444872	+	0.895594i	$0.646751\pi$
$678$	1.03656e9i	0.127730i
$679$	3.72276e9	0.456374
$680$	−8.91182e9	−1.08689
$681$	1.13487e10i	1.37699i
$682$	− 6.25296e9i	− 0.754815i
$683$	− 4.19859e9i	− 0.504233i	−0.967697	−	0.252117i	$-0.918873\pi$
$683$	− 4.19859e9i	− 0.504233i	0.967697	−	0.252117i	$-0.0811267\pi$
$684$	− 1.52962e9i	− 0.182762i
$685$	1.17770e9	0.139997
$686$	−3.96141e9	−0.468506
$687$	− 3.35900e9i	− 0.395241i
$688$	1.13577e9	0.132963
$689$	0	0
$690$	8.20615e9	0.950972
$691$	− 1.39043e10i	− 1.60316i	−0.597889	−	0.801579i	$-0.703994\pi$
$691$	− 1.39043e10i	− 1.60316i	0.597889	−	0.801579i	$-0.296006\pi$
$692$	1.11782e9	0.128234
$693$	−1.38966e9	−0.158614
$694$	− 5.46811e9i	− 0.620982i
$695$	3.00581e9i	0.339637i
$696$	− 2.25701e10i	− 2.53747i
$697$	− 8.23600e9i	− 0.921302i
$698$	2.80656e9	0.312377
$699$	1.21083e10	1.34095
$700$	3.25313e8i	0.0358475i
$701$	−9.20624e9	−1.00941	−0.504707	−	0.863291i	$-0.668399\pi$
$701$	−9.20624e9	−1.00941	−0.504707	+	0.863291i	$0.668399\pi$
$702$	0	0
$703$	−2.63442e9	−0.285984
$704$	− 4.73693e9i	− 0.511673i
$705$	−1.11694e10	−1.20052
$706$	6.67300e7	0.00713682
$707$	− 2.88066e9i	− 0.306566i
$708$	− 1.38448e8i	− 0.0146612i
$709$	1.57099e10i	1.65543i	0.561148	+	0.827716i	$0.310360\pi$
$709$	1.57099e10i	1.65543i	−0.561148	+	0.827716i	$0.689640\pi$
$710$	− 4.70672e9i	− 0.493531i
$711$	8.96415e9	0.935331
$712$	7.90522e9	0.820794
$713$	2.13960e10i	2.21065i
$714$	6.14962e9	0.632273
$715$	0	0
$716$	7.93575e8	0.0807965
$717$	1.76026e10i	1.78345i
$718$	−6.75348e9	−0.680914
$719$	−5.85775e9	−0.587733	−0.293866	−	0.955846i	$-0.594942\pi$
$719$	−5.85775e9	−0.587733	−0.293866	+	0.955846i	$0.594942\pi$
$720$	− 5.64706e9i	− 0.563844i
$721$	− 1.52073e9i	− 0.151105i
$722$	4.37497e9i	0.432608i
$723$	5.01501e9i	0.493501i
$724$	3.58590e9	0.351167
$725$	1.05680e10	1.02993
$726$	− 1.09397e10i	− 1.06103i
$727$	−7.95539e8	−0.0767876	−0.0383938	−	0.999263i	$-0.512224\pi$
$727$	−7.95539e8	−0.0767876	−0.0383938	+	0.999263i	$0.512224\pi$
$728$	0	0
$729$	−1.50997e10	−1.44352
$730$	− 2.89007e9i	− 0.274966i
$731$	−3.19450e9	−0.302477
$732$	1.07344e9	0.101156
$733$	− 5.89067e9i	− 0.552460i	−0.961092	−	0.276230i	$-0.910915\pi$
$733$	− 5.89067e9i	− 0.552460i	0.961092	−	0.276230i	$-0.0890851\pi$
$734$	8.19294e9i	0.764722i
$735$	− 8.80183e9i	− 0.817650i
$736$	5.14801e9i	0.475957i
$737$	−3.71701e9	−0.342025
$738$	6.59985e9	0.604418
$739$	8.06680e9i	0.735268i	0.929970	+	0.367634i	$0.119832\pi$
$739$	8.06680e9i	0.735268i	−0.929970	+	0.367634i	$0.880168\pi$
$740$	5.16832e8	0.0468854
$741$	0	0
$742$	−7.18522e7	−0.00645693
$743$	1.84697e10i	1.65196i	0.563699	+	0.825980i	$0.309378\pi$
$743$	1.84697e10i	1.65196i	−0.563699	+	0.825980i	$0.690622\pi$
$744$	3.33175e10	2.96597
$745$	1.67489e9	0.148402
$746$	− 6.60299e9i	− 0.582310i
$747$	1.37202e10i	1.20431i
$748$	1.82572e9i	0.159507i
$749$	− 3.97971e9i	− 0.346071i
$750$	1.51037e10	1.30728
$751$	−4.11526e9	−0.354533	−0.177267	−	0.984163i	$-0.556726\pi$
$751$	−4.11526e9	−0.354533	−0.177267	+	0.984163i	$0.556726\pi$
$752$	1.20314e10i	1.03170i
$753$	2.68134e10	2.28860
$754$	0	0
$755$	−1.77904e9	−0.150443
$756$	− 2.60846e8i	− 0.0219563i
$757$	−1.00115e10	−0.838812	−0.419406	−	0.907799i	$-0.637762\pi$
$757$	−1.00115e10	−0.838812	−0.419406	+	0.907799i	$0.637762\pi$
$758$	−1.50674e10	−1.25660
$759$	− 1.00529e10i	− 0.834533i
$760$	5.47357e9i	0.452296i
$761$	− 1.88428e10i	− 1.54988i	−0.632032	−	0.774942i	$-0.717779\pi$
$761$	− 1.88428e10i	− 1.54988i	0.632032	−	0.774942i	$-0.282221\pi$
$762$	− 9.90085e9i	− 0.810644i
$763$	−6.14145e9	−0.500535
$764$	−1.06000e9	−0.0859964
$765$	1.58831e10i	1.28268i
$766$	−4.45605e8	−0.0358219
$767$	0	0
$768$	1.08967e10	0.868018
$769$	7.85177e9i	0.622623i	0.950308	+	0.311312i	$0.100768\pi$
$769$	7.85177e9i	0.622623i	−0.950308	+	0.311312i	$0.899232\pi$
$770$	8.31597e8	0.0656440
$771$	3.42075e9	0.268801
$772$	− 3.56774e9i	− 0.279083i
$773$	1.94052e10i	1.51109i	0.655098	+	0.755544i	$0.272627\pi$
$773$	1.94052e10i	1.51109i	−0.655098	+	0.755544i	$0.727373\pi$
$774$	− 2.55989e9i	− 0.198439i
$775$	1.56002e10i	1.20385i
$776$	2.34364e10	1.80043
$777$	−2.13262e9	−0.163095
$778$	7.50693e6i	0 0.000571523i
$779$	−5.05849e9	−0.383389
$780$	0	0
$781$	−5.76592e9	−0.433102
$782$	2.48620e10i	1.85914i
$783$	−8.47373e9	−0.630824
$784$	−9.48109e9	−0.702671
$785$	− 5.88442e9i	− 0.434170i
$786$	− 1.90864e10i	− 1.40199i
$787$	− 2.52499e10i	− 1.84649i	−0.384210	−	0.923246i	$-0.625526\pi$
$787$	− 2.52499e10i	− 1.84649i	0.384210	−	0.923246i	$-0.374474\pi$
$788$	− 1.31974e8i	− 0.00960830i
$789$	−1.75607e10	−1.27284
$790$	−5.36431e9	−0.387096
$791$	− 3.59431e8i	− 0.0258225i
$792$	−8.74852e9	−0.625744
$793$	0	0
$794$	1.30867e10	0.927810
$795$	− 3.32062e8i	− 0.0234388i
$796$	−2.47468e9	−0.173910
$797$	−2.34970e9	−0.164402	−0.0822011	−	0.996616i	$-0.526195\pi$
$797$	−2.34970e9	−0.164402	−0.0822011	+	0.996616i	$0.526195\pi$
$798$	− 3.77705e9i	− 0.263113i
$799$	− 3.38397e10i	− 2.34700i
$800$	3.75350e9i	0.259192i
$801$	− 1.40890e10i	− 0.968652i
$802$	−2.30292e10	−1.57641
$803$	−3.54045e9	−0.241298
$804$	− 3.31206e9i	− 0.224751i
$805$	−2.84551e9	−0.192254
$806$	0	0
$807$	−3.33434e10	−2.23333
$808$	− 1.81350e10i	− 1.20942i
$809$	−1.05873e10	−0.703019	−0.351510	−	0.936184i	$-0.614332\pi$
$809$	−1.05873e10	−0.703019	−0.351510	+	0.936184i	$0.614332\pi$
$810$	5.24904e9	0.347042
$811$	8.44691e9i	0.556064i	0.960572	+	0.278032i	$0.0896821\pi$
$811$	8.44691e9i	0.556064i	−0.960572	+	0.278032i	$0.910318\pi$
$812$	1.30880e9i	0.0857879i
$813$	1.01705e10i	0.663782i
$814$	2.51973e9i	0.163745i
$815$	8.49754e9	0.549847
$816$	3.06135e10	1.97241
$817$	1.96204e9i	0.125872i
$818$	2.46272e9	0.157318
$819$	0	0
$820$	9.92396e8	0.0628545
$821$	9.23728e8i	0.0582563i	0.999576	+	0.0291281i	$0.00927309\pi$
$821$	9.23728e8i	0.0582563i	−0.999576	+	0.0291281i	$0.990727\pi$
$822$	−5.11617e9	−0.321287
$823$	5.95548e9	0.372407	0.186203	−	0.982511i	$-0.440382\pi$
$823$	5.95548e9	0.372407	0.186203	+	0.982511i	$0.440382\pi$
$824$	− 9.57368e9i	− 0.596120i
$825$	− 7.32970e9i	− 0.454462i
$826$	− 1.91056e8i	− 0.0117959i
$827$	− 7.33081e9i	− 0.450695i	−0.974278	−	0.225348i	$-0.927648\pi$
$827$	− 7.33081e9i	− 0.450695i	0.974278	−	0.225348i	$-0.0723518\pi$
$828$	5.00610e9	0.306474
$829$	−2.02769e10	−1.23612	−0.618059	−	0.786131i	$-0.712081\pi$
$829$	−2.02769e10	−1.23612	−0.618059	+	0.786131i	$0.712081\pi$
$830$	− 8.21044e9i	− 0.498417i
$831$	3.18375e10	1.92458
$832$	0	0
$833$	2.66668e10	1.59850
$834$	− 1.30578e10i	− 0.779453i
$835$	−1.14337e10	−0.679649
$836$	1.12134e9	0.0663768
$837$	− 1.25087e10i	− 0.737350i
$838$	− 8.70888e9i	− 0.511220i
$839$	1.19678e10i	0.699595i	0.936825	+	0.349798i	$0.113750\pi$
$839$	1.19678e10i	0.699595i	−0.936825	+	0.349798i	$0.886250\pi$
$840$	4.43097e9i	0.257942i
$841$	2.52670e10	1.46477
$842$	3.27214e9	0.188903
$843$	− 1.27315e10i	− 0.731956i
$844$	2.77081e9	0.158638
$845$	0	0
$846$	2.71172e10	1.53974
$847$	3.79338e9i	0.214504i
$848$	−3.57688e8	−0.0201428
$849$	3.93926e10	2.20921
$850$	1.81273e10i	1.01243i
$851$	− 8.62186e9i	− 0.479565i
$852$	− 5.13775e9i	− 0.284600i
$853$	− 1.44625e10i	− 0.797853i	−0.916983	−	0.398927i	$-0.869383\pi$
$853$	− 1.44625e10i	− 0.797853i	0.916983	−	0.398927i	$-0.130617\pi$
$854$	1.48134e9	0.0813863
$855$	9.75525e9	0.533773
$856$	− 2.50540e10i	− 1.36527i
$857$	−1.74997e10	−0.949723	−0.474861	−	0.880061i	$-0.657502\pi$
$857$	−1.74997e10	−0.949723	−0.474861	+	0.880061i	$0.657502\pi$
$858$	0	0
$859$	−1.21484e10	−0.653945	−0.326973	−	0.945034i	$-0.606028\pi$
$859$	−1.21484e10	−0.653945	−0.326973	+	0.945034i	$0.606028\pi$
$860$	− 3.84921e8i	− 0.0206361i
$861$	−4.09495e9	−0.218644
$862$	2.24822e10	1.19554
$863$	− 5.06062e9i	− 0.268019i	−0.990980	−	0.134010i	$-0.957215\pi$
$863$	− 5.06062e9i	− 0.268019i	0.990980	−	0.134010i	$-0.0427853\pi$
$864$	− 3.00967e9i	− 0.158753i
$865$	7.12899e9i	0.374517i
$866$	2.07786e10i	1.08719i
$867$	−5.72122e10	−2.98141
$868$	−1.93202e9	−0.100275
$869$	6.57149e9i	0.339700i
$870$	2.40717e10	1.23934
$871$	0	0
$872$	−3.86632e10	−1.97465
$873$	− 4.17695e10i	− 2.12476i
$874$	1.52701e10	0.773661
$875$	−5.23725e9	−0.264286
$876$	− 3.15473e9i	− 0.158562i
$877$	− 2.28400e10i	− 1.14340i	−0.820463	−	0.571700i	$-0.806284\pi$
$877$	− 2.28400e10i	− 1.14340i	0.820463	−	0.571700i	$-0.193716\pi$
$878$	2.37820e9i	0.118582i
$879$	2.66886e10i	1.32545i
$880$	4.13978e9	0.204780
$881$	−2.34736e10	−1.15655	−0.578273	−	0.815843i	$-0.696273\pi$
$881$	−2.34736e10	−1.15655	−0.578273	+	0.815843i	$0.696273\pi$
$882$	2.13692e10i	1.04869i
$883$	1.70477e10	0.833302	0.416651	−	0.909067i	$-0.363204\pi$
$883$	1.70477e10	0.833302	0.416651	+	0.909067i	$0.363204\pi$
$884$	0	0
$885$	8.82960e8	0.0428193
$886$	− 1.36972e10i	− 0.661629i
$887$	1.45686e10	0.700949	0.350474	−	0.936572i	$-0.386020\pi$
$887$	1.45686e10	0.700949	0.350474	+	0.936572i	$0.386020\pi$
$888$	−1.34258e10	−0.643420
$889$	3.43315e9i	0.163884i
$890$	8.43115e9i	0.400887i
$891$	− 6.43028e9i	− 0.304550i
$892$	− 1.94040e8i	− 0.00915409i
$893$	−2.07841e10	−0.976676
$894$	−7.27605e9	−0.340576
$895$	5.06108e9i	0.235973i
$896$	3.50988e9	0.163010
$897$	0	0
$898$	1.93562e10	0.891974
$899$	6.27625e10i	2.88099i
$900$	3.65003e9	0.166897
$901$	1.00604e9	0.0458226
$902$	4.83826e9i	0.219516i
$903$	1.58831e9i	0.0717841i
$904$	− 2.26278e9i	− 0.101872i
$905$	2.28693e10i	1.02561i
$906$	7.72848e9	0.345259
$907$	−1.93215e10	−0.859836	−0.429918	−	0.902868i	$-0.641457\pi$
$907$	−1.93215e10	−0.859836	−0.429918	+	0.902868i	$0.641457\pi$
$908$	− 4.14297e9i	− 0.183659i
$909$	−3.23211e10	−1.42729
$910$	0	0
$911$	1.95446e10	0.856471	0.428235	−	0.903667i	$-0.359135\pi$
$911$	1.95446e10	0.856471	0.428235	+	0.903667i	$0.359135\pi$
$912$	− 1.88026e10i	− 0.820796i
$913$	−1.00581e10	−0.437390
$914$	−3.32866e10	−1.44198
$915$	6.84595e9i	0.295433i
$916$	1.22624e9i	0.0527159i
$917$	6.61829e9i	0.283435i
$918$	− 1.45350e10i	− 0.620107i
$919$	1.96336e10	0.834439	0.417220	−	0.908806i	$-0.363005\pi$
$919$	1.96336e10	0.834439	0.417220	+	0.908806i	$0.363005\pi$
$920$	−1.79138e10	−0.758455
$921$	− 7.07249e10i	− 2.98307i
$922$	1.76062e10	0.739789
$923$	0	0
$924$	9.07751e8	0.0378543
$925$	− 6.28634e9i	− 0.261157i
$926$	−5.94890e8	−0.0246206
$927$	−1.70627e10	−0.703506
$928$	1.51010e10i	0.620281i
$929$	2.66303e10i	1.08973i	0.838522	+	0.544867i	$0.183420\pi$
$929$	2.66303e10i	1.08973i	−0.838522	+	0.544867i	$0.816580\pi$
$930$	3.55341e10i	1.44862i
$931$	− 1.63785e10i	− 0.665197i
$932$	−4.42026e9	−0.178851
$933$	3.21836e10	1.29732
$934$	− 1.49257e10i	− 0.599407i
$935$	−1.16437e10	−0.465853
$936$	0	0
$937$	1.47566e9	0.0585999	0.0292999	−	0.999571i	$-0.490672\pi$
$937$	1.47566e9	0.0585999	0.0292999	+	0.999571i	$0.490672\pi$
$938$	− 4.57060e9i	− 0.180827i
$939$	4.78530e10	1.88617
$940$	4.07751e9	0.160121
$941$	2.25755e10i	0.883231i	0.897204	+	0.441616i	$0.145595\pi$
$941$	2.25755e10i	0.883231i	−0.897204	+	0.441616i	$0.854405\pi$
$942$	2.55630e10i	0.996400i
$943$	− 1.65553e10i	− 0.642904i
$944$	− 9.51100e8i	− 0.0367980i
$945$	1.66357e9	0.0641252
$946$	1.87662e9	0.0720704
$947$	1.98795e10i	0.760642i	0.924855	+	0.380321i	$0.124186\pi$
$947$	1.98795e10i	0.760642i	−0.924855	+	0.380321i	$0.875814\pi$
$948$	−5.85556e9	−0.223223
$949$	0	0
$950$	1.11336e10	0.421313
$951$	1.00335e10i	0.378285i
$952$	−1.34244e10	−0.504274
$953$	−2.46036e10	−0.920817	−0.460408	−	0.887707i	$-0.652297\pi$
$953$	−2.46036e10	−0.920817	−0.460408	+	0.887707i	$0.652297\pi$
$954$	8.06185e8i	0.0300618i
$955$	− 6.76024e9i	− 0.251160i
$956$	− 6.42603e9i	− 0.237870i
$957$	− 2.94888e10i	− 1.08759i
$958$	−1.16428e10	−0.427838
$959$	1.77405e9	0.0649532
$960$	2.69188e10i	0.981989i
$961$	−6.51360e10	−2.36749
$962$	0	0
$963$	−4.46525e10	−1.61122
$964$	− 1.83079e9i	− 0.0658215i
$965$	2.27535e10	0.815084
$966$	1.23614e10	0.441214
$967$	− 1.21598e10i	− 0.432449i	−0.976344	−	0.216224i	$-0.930626\pi$
$967$	− 1.21598e10i	− 0.432449i	0.976344	−	0.216224i	$-0.0693742\pi$
$968$	2.38811e10i	0.846232i
$969$	5.28846e10i	1.86722i
$970$	2.49957e10i	0.879353i
$971$	3.44913e10	1.20904	0.604522	−	0.796588i	$-0.293364\pi$
$971$	3.44913e10	1.20904	0.604522	+	0.796588i	$0.293364\pi$
$972$	8.03991e9	0.280814
$973$	4.52785e9i	0.157578i
$974$	−4.93783e10	−1.71230
$975$	0	0
$976$	7.37426e9	0.253889
$977$	− 3.29649e10i	− 1.13089i	−0.824786	−	0.565445i	$-0.808704\pi$
$977$	− 3.29649e10i	− 1.13089i	0.824786	−	0.565445i	$-0.191296\pi$
$978$	−3.69149e10	−1.26187
$979$	1.03285e10	0.351801
$980$	3.21321e9i	0.109055i
$981$	6.89073e10i	2.33036i
$982$	1.21741e10i	0.410247i
$983$	− 4.71393e10i	− 1.58287i	−0.611253	−	0.791435i	$-0.709334\pi$
$983$	− 4.71393e10i	− 1.58287i	0.611253	−	0.791435i	$-0.290666\pi$
$984$	−2.57796e10	−0.862567
$985$	8.41673e8	0.0280619
$986$	7.29295e10i	2.42289i
$987$	−1.68252e10	−0.556992
$988$	0	0
$989$	−6.42131e9	−0.211075
$990$	− 9.33055e9i	− 0.305622i
$991$	−2.37320e10	−0.774600	−0.387300	−	0.921954i	$-0.626592\pi$
$991$	−2.37320e10	−0.774600	−0.387300	+	0.921954i	$0.626592\pi$
$992$	−2.22918e10	−0.725027
$993$	8.38626e9i	0.271798i
$994$	− 7.09003e9i	− 0.228979i
$995$	− 1.57824e10i	− 0.507918i
$996$	− 8.96232e9i	− 0.287417i
$997$	1.20079e10	0.383737	0.191869	−	0.981421i	$-0.438545\pi$
$997$	1.20079e10	0.383737	0.191869	+	0.981421i	$0.438545\pi$
$998$	−3.68198e10	−1.17253
$999$	5.04058e9i	0.159956i

Display

a_p

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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	169.8.b.e.168.12		16
13.5	odd	4		169.8.a.e.1.7		8
13.7	odd	12		13.8.c.a.3.7	✓	16
13.8	odd	4		169.8.a.f.1.2		8
13.11	odd	12		13.8.c.a.9.7	yes	16
13.12	even	2	inner	169.8.b.e.168.5		16
39.11	even	12		117.8.g.d.100.2		16
39.20	even	12		117.8.g.d.55.2		16
52.7	even	12		208.8.i.d.81.1		16
52.11	even	12		208.8.i.d.113.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.c.a.3.7	✓	16	13.7	odd	12
13.8.c.a.9.7	yes	16	13.11	odd	12
117.8.g.d.55.2		16	39.20	even	12
117.8.g.d.100.2		16	39.11	even	12
169.8.a.e.1.7		8	13.5	odd	4
169.8.a.f.1.2		8	13.8	odd	4
169.8.b.e.168.5		16	13.12	even	2	inner
169.8.b.e.168.12		16	1.1	even	1	trivial
208.8.i.d.81.1		16	52.7	even	12
208.8.i.d.113.1		16	52.11	even	12

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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	169.8.b.e.168.12

Level	$169$
Weight	$8$
Character	169.168
Analytic conductor	$52.793$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$