LMFDB - Embedded newform 169.8.b.d.168.2

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)

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);

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")

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 = [14] 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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 $ x^14 + 1279x^12 + 629380x^10 + 148562016x^8 + 16872573312x^6 + 790180980480x^4 + 10484997239808x^2 + 4669637050368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{17}\cdot 3^{3}\cdot 13^{6} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.2
Root			$-16.7657i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.8.b.d.168.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-16.7657i q^{2} -81.0229 q^{3} -153.090 q^{4} +223.318i q^{5} +1358.41i q^{6} +666.907i q^{7} +420.649i q^{8} +4377.70 q^{9} +3744.09 q^{10} +7590.80i q^{11} +12403.8 q^{12} +11181.2 q^{14} -18093.9i q^{15} -12543.0 q^{16} +14102.9 q^{17} -73395.4i q^{18} +8343.98i q^{19} -34187.7i q^{20} -54034.7i q^{21} +127265. q^{22} -58145.3 q^{23} -34082.2i q^{24} +28254.0 q^{25} -177497. q^{27} -102097. i q^{28} -33045.0 q^{29} -303357. q^{30} +91450.2i q^{31} +264136. i q^{32} -615028. i q^{33} -236445. i q^{34} -148932. q^{35} -670182. q^{36} -355545. i q^{37} +139893. q^{38} -93938.5 q^{40} +582075. i q^{41} -905932. q^{42} +104498. q^{43} -1.16207e6i q^{44} +977620. i q^{45} +974848. i q^{46} +398024. i q^{47} +1.01627e6 q^{48} +378778. q^{49} -473700. i q^{50} -1.14266e6 q^{51} +1.49881e6 q^{53} +2.97587e6i q^{54} -1.69516e6 q^{55} -280534. q^{56} -676053. i q^{57} +554024. i q^{58} -330628. i q^{59} +2.76999e6i q^{60} -1.44273e6 q^{61} +1.53323e6 q^{62} +2.91952e6i q^{63} +2.82292e6 q^{64} -1.03114e7 q^{66} +2.56680e6i q^{67} -2.15901e6 q^{68} +4.71110e6 q^{69} +2.49696e6i q^{70} -1.29773e6i q^{71} +1.84148e6i q^{72} -2.66195e6i q^{73} -5.96097e6 q^{74} -2.28922e6 q^{75} -1.27738e6i q^{76} -5.06236e6 q^{77} +2.13906e6 q^{79} -2.80108e6i q^{80} +4.80728e6 q^{81} +9.75892e6 q^{82} +1.63158e6i q^{83} +8.27217e6i q^{84} +3.14943e6i q^{85} -1.75198e6i q^{86} +2.67740e6 q^{87} -3.19306e6 q^{88} -401206. i q^{89} +1.63905e7 q^{90} +8.90145e6 q^{92} -7.40955e6i q^{93} +6.67316e6 q^{94} -1.86336e6 q^{95} -2.14010e7i q^{96} +1.10423e7i q^{97} -6.35048e6i q^{98} +3.32303e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 52 q^{3} - 766 q^{4} + 6982 q^{9} + 1018 q^{10} + 38380 q^{12} - 47916 q^{14} + 1266 q^{16} + 76806 q^{17} + 251764 q^{22} + 137100 q^{23} + 39380 q^{25} - 432400 q^{27} - 443166 q^{29} + 315780 q^{30}+ \cdots + 86840772 q^{95}+O(q^{100}) $ 14 * q - 52 * q^3 - 766 * q^4 + 6982 * q^9 + 1018 * q^10 + 38380 * q^12 - 47916 * q^14 + 1266 * q^16 + 76806 * q^17 + 251764 * q^22 + 137100 * q^23 + 39380 * q^25 - 432400 * q^27 - 443166 * q^29 + 315780 * q^30 - 1319232 * q^35 - 1271350 * q^36 + 953640 * q^38 - 77182 * q^40 - 412812 * q^42 + 1765384 * q^43 - 2267260 * q^48 - 5307958 * q^49 - 38964 * q^51 - 1791210 * q^53 - 4907072 * q^55 + 14915904 * q^56 - 7717050 * q^61 - 3808284 * q^62 + 14570922 * q^64 - 31956396 * q^66 - 19139238 * q^68 + 27487680 * q^69 - 17280210 * q^74 - 20632304 * q^75 + 19599336 * q^77 - 30688440 * q^79 - 27772442 * q^81 + 74719438 * q^82 - 25539372 * q^87 - 54091072 * q^88 + 41193546 * q^90 - 47685588 * q^92 - 38277620 * q^94 + 86840772 * 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 $

$2$	− 16.7657i	− 1.48190i	−0.671563	−	0.740948i	$-0.734377\pi$
$2$	− 16.7657i	− 1.48190i	0.671563	−	0.740948i	$-0.265623\pi$
$3$	−81.0229	−1.73254	−0.866270	−	0.499576i	$-0.833489\pi$
$3$	−81.0229	−1.73254	−0.866270	+	0.499576i	$0.833489\pi$
$4$	−153.090	−1.19601
$5$	223.318i	0.798967i	0.916740	+	0.399483i	$0.130811\pi$
$5$	223.318i	0.798967i	−0.916740	+	0.399483i	$0.869189\pi$
$6$	1358.41i	2.56744i
$7$	666.907i	0.734890i	0.930045	+	0.367445i	$0.119767\pi$
$7$	666.907i	0.734890i	−0.930045	+	0.367445i	$0.880233\pi$
$8$	420.649i	0.290472i
$9$	4377.70	2.00169
$10$	3744.09	1.18399
$11$	7590.80i	1.71954i	0.510679	+	0.859771i	$0.329394\pi$
$11$	7590.80i	1.71954i	−0.510679	+	0.859771i	$0.670606\pi$
$12$	12403.8	2.07214
$13$	0	0
$13$	0	0
$14$	11181.2	1.08903
$15$	− 18093.9i	− 1.38424i
$16$	−12543.0	−0.765564
$17$	14102.9	0.696204	0.348102	−	0.937457i	$-0.386826\pi$
$17$	14102.9	0.696204	0.348102	+	0.937457i	$0.386826\pi$
$18$	− 73395.4i	− 2.96630i
$19$	8343.98i	0.279085i	0.990216	+	0.139542i	$0.0445631\pi$
$19$	8343.98i	0.279085i	−0.990216	+	0.139542i	$0.955437\pi$
$20$	− 34187.7i	− 0.955576i
$21$	− 54034.7i	− 1.27323i
$22$	127265.	2.54818
$23$	−58145.3	−0.996476	−0.498238	−	0.867040i	$-0.666019\pi$
$23$	−58145.3	−0.996476	−0.498238	+	0.867040i	$0.666019\pi$
$24$	− 34082.2i	− 0.503255i
$25$	28254.0	0.361652
$26$	0	0
$27$	−177497.	−1.73547
$28$	− 102097.i	− 0.878939i
$29$	−33045.0	−0.251601	−0.125801	−	0.992056i	$-0.540150\pi$
$29$	−33045.0	−0.251601	−0.125801	+	0.992056i	$0.540150\pi$
$30$	−303357.	−2.05130
$31$	91450.2i	0.551339i	0.961252	+	0.275669i	$0.0888995\pi$
$31$	91450.2i	0.551339i	−0.961252	+	0.275669i	$0.911100\pi$
$32$	264136.i	1.42496i
$33$	− 615028.i	− 2.97918i
$34$	− 236445.i	− 1.03170i
$35$	−148932.	−0.587153
$36$	−670182.	−2.39405
$37$	− 355545.i	− 1.15395i	−0.816760	−	0.576977i	$-0.804232\pi$
$37$	− 355545.i	− 1.15395i	0.816760	−	0.576977i	$-0.195768\pi$
$38$	139893.	0.413574
$39$	0	0
$40$	−93938.5	−0.232078
$41$	582075.i	1.31897i	0.751717	+	0.659486i	$0.229226\pi$
$41$	582075.i	1.31897i	−0.751717	+	0.659486i	$0.770774\pi$
$42$	−905932.	−1.88679
$43$	104498.	0.200432	0.100216	−	0.994966i	$-0.468047\pi$
$43$	104498.	0.200432	0.100216	+	0.994966i	$0.468047\pi$
$44$	− 1.16207e6i	− 2.05660i
$45$	977620.i	1.59929i
$46$	974848.i	1.47667i
$47$	398024.i	0.559199i	0.960117	+	0.279600i	$0.0902017\pi$
$47$	398024.i	0.559199i	−0.960117	+	0.279600i	$0.909798\pi$
$48$	1.01627e6	1.32637
$49$	378778.	0.459937
$50$	− 473700.i	− 0.535930i
$51$	−1.14266e6	−1.20620
$52$	0	0
$53$	1.49881e6	1.38287	0.691433	−	0.722441i	$-0.256980\pi$
$53$	1.49881e6	1.38287	0.691433	+	0.722441i	$0.256980\pi$
$54$	2.97587e6i	2.57179i
$55$	−1.69516e6	−1.37386
$56$	−280534.	−0.213465
$57$	− 676053.i	− 0.483525i
$58$	554024.i	0.372847i
$59$	− 330628.i	− 0.209584i	−0.994494	−	0.104792i	$-0.966582\pi$
$59$	− 330628.i	− 0.209584i	0.994494	−	0.104792i	$-0.0334176\pi$
$60$	2.76999e6i	1.65557i
$61$	−1.44273e6	−0.813825	−0.406912	−	0.913467i	$-0.633395\pi$
$61$	−1.44273e6	−0.813825	−0.406912	+	0.913467i	$0.633395\pi$
$62$	1.53323e6	0.817027
$63$	2.91952e6i	1.47102i
$64$	2.82292e6	1.34608
$65$	0	0
$66$	−1.03114e7	−4.41483
$67$	2.56680e6i	1.04263i	0.853364	+	0.521316i	$0.174559\pi$
$67$	2.56680e6i	1.04263i	−0.853364	+	0.521316i	$0.825441\pi$
$68$	−2.15901e6	−0.832670
$69$	4.71110e6	1.72643
$70$	2.49696e6i	0.870099i
$71$	− 1.29773e6i	− 0.430307i	−0.976580	−	0.215154i	$-0.930975\pi$
$71$	− 1.29773e6i	− 0.430307i	0.976580	−	0.215154i	$-0.0690252\pi$
$72$	1.84148e6i	0.581437i
$73$	− 2.66195e6i	− 0.800885i	−0.916322	−	0.400442i	$-0.868856\pi$
$73$	− 2.66195e6i	− 0.800885i	0.916322	−	0.400442i	$-0.131144\pi$
$74$	−5.96097e6	−1.71004
$75$	−2.28922e6	−0.626576
$76$	− 1.27738e6i	− 0.333789i
$77$	−5.06236e6	−1.26367
$78$	0	0
$79$	2.13906e6	0.488121	0.244060	−	0.969760i	$-0.421520\pi$
$79$	2.13906e6	0.488121	0.244060	+	0.969760i	$0.421520\pi$
$80$	− 2.80108e6i	− 0.611661i
$81$	4.80728e6	1.00508
$82$	9.75892e6	1.95458
$83$	1.63158e6i	0.313211i	0.987661	+	0.156605i	$0.0500550\pi$
$83$	1.63158e6i	0.313211i	−0.987661	+	0.156605i	$0.949945\pi$
$84$	8.27217e6i	1.52280i
$85$	3.14943e6i	0.556244i
$86$	− 1.75198e6i	− 0.297020i
$87$	2.67740e6	0.435909
$88$	−3.19306e6	−0.499480
$89$	− 401206.i	− 0.0603258i	−0.999545	−	0.0301629i	$-0.990397\pi$
$89$	− 401206.i	− 0.0603258i	0.999545	−	0.0301629i	$-0.00960260\pi$
$90$	1.63905e7	2.36998
$91$	0	0
$92$	8.90145e6	1.19180
$93$	− 7.40955e6i	− 0.955216i
$94$	6.67316e6	0.828675
$95$	−1.86336e6	−0.222979
$96$	− 2.14010e7i	− 2.46880i
$97$	1.10423e7i	1.22846i	0.789128	+	0.614228i	$0.210533\pi$
$97$	1.10423e7i	1.22846i	−0.789128	+	0.614228i	$0.789467\pi$
$98$	− 6.35048e6i	− 0.681578i
$99$	3.32303e7i	3.44200i
$100$	−4.32541e6	−0.432541
$101$	−7.68319e6	−0.742022	−0.371011	−	0.928629i	$-0.620989\pi$
$101$	−7.68319e6	−0.742022	−0.371011	+	0.928629i	$0.620989\pi$
$102$	1.91575e7i	1.78746i
$103$	485826.	0.0438077	0.0219039	−	0.999760i	$-0.493027\pi$
$103$	485826.	0.0438077	0.0219039	+	0.999760i	$0.493027\pi$
$104$	0	0
$105$	1.20669e7	1.01727
$106$	− 2.51286e7i	− 2.04926i
$107$	−69974.3	−0.00552198	−0.00276099	−	0.999996i	$-0.500879\pi$
$107$	−69974.3	−0.00552198	−0.00276099	+	0.999996i	$0.500879\pi$
$108$	2.71730e7	2.07565
$109$	4.06366e6i	0.300556i	0.988644	+	0.150278i	$0.0480168\pi$
$109$	4.06366e6i	0.300556i	−0.988644	+	0.150278i	$0.951983\pi$
$110$	2.84206e7i	2.03591i
$111$	2.88073e7i	1.99927i
$112$	− 8.36502e6i	− 0.562606i
$113$	−2.09954e7	−1.36883	−0.684417	−	0.729091i	$-0.739943\pi$
$113$	−2.09954e7	−1.36883	−0.684417	+	0.729091i	$0.739943\pi$
$114$	−1.13345e7	−0.716534
$115$	− 1.29849e7i	− 0.796152i
$116$	5.05885e6	0.300919
$117$	0	0
$118$	−5.54322e6	−0.310581
$119$	9.40531e6i	0.511633i
$120$	7.61117e6	0.402084
$121$	−3.81330e7	−1.95683
$122$	2.41884e7i	1.20600i
$123$	− 4.71614e7i	− 2.28517i
$124$	− 1.40001e7i	− 0.659409i
$125$	2.37564e7i	1.08791i
$126$	4.89479e7	2.17990
$127$	−9.33567e6	−0.404420	−0.202210	−	0.979342i	$-0.564812\pi$
$127$	−9.33567e6	−0.404420	−0.202210	+	0.979342i	$0.564812\pi$
$128$	− 1.35190e7i	− 0.569784i
$129$	−8.46671e6	−0.347257
$130$	0	0
$131$	−4.07546e7	−1.58390	−0.791948	−	0.610588i	$-0.790933\pi$
$131$	−4.07546e7	−1.58390	−0.791948	+	0.610588i	$0.790933\pi$
$132$	9.41545e7i	3.56314i
$133$	−5.56466e6	−0.205096
$134$	4.30343e7	1.54507
$135$	− 3.96383e7i	− 1.38659i
$136$	5.93236e6i	0.202228i
$137$	− 4.87238e7i	− 1.61890i	−0.587190	−	0.809449i	$-0.699766\pi$
$137$	− 4.87238e7i	− 1.61890i	0.587190	−	0.809449i	$-0.300234\pi$
$138$	− 7.89850e7i	− 2.55840i
$139$	−5.32895e7	−1.68302	−0.841511	−	0.540240i	$-0.818334\pi$
$139$	−5.32895e7	−1.68302	−0.841511	+	0.540240i	$0.818334\pi$
$140$	2.28000e7	0.702243
$141$	− 3.22490e7i	− 0.968835i
$142$	−2.17573e7	−0.637670
$143$	0	0
$144$	−5.49096e7	−1.53243
$145$	− 7.37955e6i	− 0.201021i
$146$	−4.46296e7	−1.18683
$147$	−3.06896e7	−0.796858
$148$	5.44303e7i	1.38015i
$149$	3.62391e7i	0.897480i	0.893662	+	0.448740i	$0.148127\pi$
$149$	3.62391e7i	0.897480i	−0.893662	+	0.448740i	$0.851873\pi$
$150$	3.83805e7i	0.928520i
$151$	2.55676e7i	0.604326i	0.953256	+	0.302163i	$0.0977087\pi$
$151$	2.55676e7i	0.604326i	−0.953256	+	0.302163i	$0.902291\pi$
$152$	−3.50989e6	−0.0810664
$153$	6.17382e7	1.39359
$154$	8.48741e7i	1.87263i
$155$	−2.04225e7	−0.440502
$156$	0	0
$157$	4.23295e7	0.872960	0.436480	−	0.899714i	$-0.356225\pi$
$157$	4.23295e7	0.872960	0.436480	+	0.899714i	$0.356225\pi$
$158$	− 3.58629e7i	− 0.723344i
$159$	−1.21438e8	−2.39587
$160$	−5.89863e7	−1.13849
$161$	− 3.87775e7i	− 0.732301i
$162$	− 8.05976e7i	− 1.48943i
$163$	3.44898e7i	0.623784i	0.950118	+	0.311892i	$0.100963\pi$
$163$	3.44898e7i	0.623784i	−0.950118	+	0.311892i	$0.899037\pi$
$164$	− 8.91098e7i	− 1.57751i
$165$	1.37347e8	2.38026
$166$	2.73547e7	0.464145
$167$	8.00957e6i	0.133077i	0.997784	+	0.0665383i	$0.0211955\pi$
$167$	8.00957e6i	0.133077i	−0.997784	+	0.0665383i	$0.978805\pi$
$168$	2.27297e7	0.369837
$169$	0	0
$170$	5.28024e7	0.824295
$171$	3.65275e7i	0.558642i
$172$	−1.59975e7	−0.239720
$173$	−3.73141e7	−0.547912	−0.273956	−	0.961742i	$-0.588332\pi$
$173$	−3.73141e7	−0.547912	−0.273956	+	0.961742i	$0.588332\pi$
$174$	− 4.48886e7i	− 0.645972i
$175$	1.88428e7i	0.265774i
$176$	− 9.52114e7i	− 1.31642i
$177$	2.67884e7i	0.363112i
$178$	−6.72652e6	−0.0893965
$179$	6.85207e7	0.892968	0.446484	−	0.894792i	$-0.352676\pi$
$179$	6.85207e7	0.892968	0.446484	+	0.894792i	$0.352676\pi$
$180$	− 1.49664e8i	− 1.91277i
$181$	−1.20860e8	−1.51498	−0.757489	−	0.652848i	$-0.773574\pi$
$181$	−1.20860e8	−1.51498	−0.757489	+	0.652848i	$0.773574\pi$
$182$	0	0
$183$	1.16894e8	1.40998
$184$	− 2.44588e7i	− 0.289449i
$185$	7.93996e7	0.921972
$186$	−1.24227e8	−1.41553
$187$	1.07052e8i	1.19715i
$188$	− 6.09334e7i	− 0.668810i
$189$	− 1.18374e8i	− 1.27538i
$190$	3.12406e7i	0.330432i
$191$	−8.40445e7	−0.872755	−0.436377	−	0.899764i	$-0.643739\pi$
$191$	−8.40445e7	−0.872755	−0.436377	+	0.899764i	$0.643739\pi$
$192$	−2.28721e8	−2.33213
$193$	− 4.45097e7i	− 0.445661i	−0.974857	−	0.222830i	$-0.928470\pi$
$193$	− 4.45097e7i	− 0.445661i	0.974857	−	0.222830i	$-0.0715296\pi$
$194$	1.85133e8	1.82044
$195$	0	0
$196$	−5.79870e7	−0.550091
$197$	− 1.26502e8i	− 1.17887i	−0.807815	−	0.589437i	$-0.799350\pi$
$197$	− 1.26502e8i	− 1.17887i	0.807815	−	0.589437i	$-0.200650\pi$
$198$	5.57130e8	5.10068
$199$	9.62184e7	0.865511	0.432755	−	0.901511i	$-0.357541\pi$
$199$	9.62184e7	0.865511	0.432755	+	0.901511i	$0.357541\pi$
$200$	1.18850e7i	0.105050i
$201$	− 2.07970e8i	− 1.80640i
$202$	1.28814e8i	1.09960i
$203$	− 2.20380e7i	− 0.184899i
$204$	1.74929e8	1.44263
$205$	−1.29988e8	−1.05382
$206$	− 8.14523e6i	− 0.0649185i
$207$	−2.54543e8	−1.99464
$208$	0	0
$209$	−6.33375e7	−0.479898
$210$	− 2.02311e8i	− 1.50748i
$211$	−5.03533e7	−0.369011	−0.184505	−	0.982832i	$-0.559068\pi$
$211$	−5.03533e7	−0.369011	−0.184505	+	0.982832i	$0.559068\pi$
$212$	−2.29452e8	−1.65393
$213$	1.05145e8i	0.745524i
$214$	1.17317e6i	0.00818300i
$215$	2.33362e7i	0.160139i
$216$	− 7.46640e7i	− 0.504107i
$217$	−6.09888e7	−0.405173
$218$	6.81303e7	0.445392
$219$	2.15679e8i	1.38756i
$220$	2.59512e8	1.64315
$221$	0	0
$222$	4.82975e8	2.96271
$223$	− 2.45161e8i	− 1.48042i	−0.672378	−	0.740208i	$-0.734727\pi$
$223$	− 2.45161e8i	− 1.48042i	0.672378	−	0.740208i	$-0.265273\pi$
$224$	−1.76154e8	−1.04719
$225$	1.23688e8	0.723916
$226$	3.52004e8i	2.02847i
$227$	1.90894e6i	0.0108318i	0.999985	+	0.00541591i	$0.00172395\pi$
$227$	1.90894e6i	0.0108318i	−0.999985	+	0.00541591i	$0.998276\pi$
$228$	1.03497e8i	0.578303i
$229$	− 2.07407e8i	− 1.14130i	−0.821194	−	0.570650i	$-0.806691\pi$
$229$	− 2.07407e8i	− 1.14130i	0.821194	−	0.570650i	$-0.193309\pi$
$230$	−2.17701e8	−1.17981
$231$	4.10167e8	2.18937
$232$	− 1.39004e7i	− 0.0730832i
$233$	3.42955e8	1.77620	0.888099	−	0.459652i	$-0.152026\pi$
$233$	3.42955e8	1.77620	0.888099	+	0.459652i	$0.152026\pi$
$234$	0	0
$235$	−8.88859e7	−0.446782
$236$	5.06158e7i	0.250665i
$237$	−1.73312e8	−0.845689
$238$	1.57687e8	0.758187
$239$	− 3.94963e8i	− 1.87139i	−0.352815	−	0.935693i	$-0.614775\pi$
$239$	− 3.94963e8i	− 1.87139i	0.352815	−	0.935693i	$-0.385225\pi$
$240$	2.26952e8i	1.05973i
$241$	− 7.77374e7i	− 0.357742i	−0.983873	−	0.178871i	$-0.942755\pi$
$241$	− 7.77374e7i	− 0.357742i	0.983873	−	0.178871i	$-0.0572445\pi$
$242$	6.39328e8i	2.89981i
$243$	−1.31373e6	−0.00587334
$244$	2.20867e8	0.973346
$245$	8.45879e7i	0.367474i
$246$	−7.90695e8	−3.38638
$247$	0	0
$248$	−3.84684e7	−0.160149
$249$	− 1.32196e8i	− 0.542650i
$250$	3.98293e8	1.61218
$251$	3.50877e8	1.40054	0.700272	−	0.713877i	$-0.253062\pi$
$251$	3.50877e8	1.40054	0.700272	+	0.713877i	$0.253062\pi$
$252$	− 4.46949e8i	− 1.75937i
$253$	− 4.41369e8i	− 1.71348i
$254$	1.56519e8i	0.599308i
$255$	− 2.55176e8i	− 0.963715i
$256$	1.34678e8	0.501715
$257$	3.59200e8	1.31999	0.659995	−	0.751270i	$-0.270558\pi$
$257$	3.59200e8	1.31999	0.659995	+	0.751270i	$0.270558\pi$
$258$	1.41951e8i	0.514598i
$259$	2.37116e8	0.848030
$260$	0	0
$261$	−1.44661e8	−0.503629
$262$	6.83280e8i	2.34717i
$263$	−1.99512e8	−0.676277	−0.338139	−	0.941096i	$-0.609797\pi$
$263$	−1.99512e8	−0.676277	−0.338139	+	0.941096i	$0.609797\pi$
$264$	2.58711e8	0.865368
$265$	3.34711e8i	1.10486i
$266$	9.32957e7i	0.303932i
$267$	3.25069e7i	0.104517i
$268$	− 3.92951e8i	− 1.24700i
$269$	−6.44857e7	−0.201990	−0.100995	−	0.994887i	$-0.532203\pi$
$269$	−6.44857e7	−0.201990	−0.100995	+	0.994887i	$0.532203\pi$
$270$	−6.64565e8	−2.05478
$271$	− 4.79494e7i	− 0.146349i	−0.997319	−	0.0731747i	$-0.976687\pi$
$271$	− 4.79494e7i	− 0.146349i	0.997319	−	0.0731747i	$-0.0233131\pi$
$272$	−1.76892e8	−0.532989
$273$	0	0
$274$	−8.16891e8	−2.39904
$275$	2.14471e8i	0.621876i
$276$	−7.21221e8	−2.06484
$277$	−5.93018e8	−1.67644	−0.838221	−	0.545331i	$-0.816404\pi$
$277$	−5.93018e8	−1.67644	−0.838221	+	0.545331i	$0.816404\pi$
$278$	8.93438e8i	2.49406i
$279$	4.00342e8i	1.10361i
$280$	− 6.26483e7i	− 0.170552i
$281$	1.74758e8i	0.469856i	0.972013	+	0.234928i	$0.0754854\pi$
$281$	1.74758e8i	0.469856i	−0.972013	+	0.234928i	$0.924515\pi$
$282$	−5.40679e8	−1.43571
$283$	−6.61117e8	−1.73391	−0.866954	−	0.498389i	$-0.833925\pi$
$283$	−6.61117e8	−1.73391	−0.866954	+	0.498389i	$0.833925\pi$
$284$	1.98669e8i	0.514653i
$285$	1.50975e8	0.386321
$286$	0	0
$287$	−3.88190e8	−0.969299
$288$	1.15631e9i	2.85233i
$289$	−2.11448e8	−0.515300
$290$	−1.23724e8	−0.297892
$291$	− 8.94681e8i	− 2.12835i
$292$	4.07518e8i	0.957869i
$293$	5.36743e8i	1.24661i	0.781980	+	0.623304i	$0.214210\pi$
$293$	5.36743e8i	1.24661i	−0.781980	+	0.623304i	$0.785790\pi$
$294$	5.14534e8i	1.18086i
$295$	7.38352e7	0.167451
$296$	1.49560e8	0.335192
$297$	− 1.34734e9i	− 2.98422i
$298$	6.07574e8	1.32997
$299$	0	0
$300$	3.50457e8	0.749394
$301$	6.96903e7i	0.147296i
$302$	4.28660e8	0.895548
$303$	6.22514e8	1.28558
$304$	− 1.04659e8i	− 0.213657i
$305$	− 3.22188e8i	− 0.650219i
$306$	− 1.03509e9i	− 2.06515i
$307$	3.58135e8i	0.706419i	0.935544	+	0.353209i	$0.114910\pi$
$307$	3.58135e8i	0.706419i	−0.935544	+	0.353209i	$0.885090\pi$
$308$	7.74995e8	1.51137
$309$	−3.93630e7	−0.0758986
$310$	3.42398e8i	0.652777i
$311$	4.95093e8	0.933309	0.466655	−	0.884440i	$-0.345459\pi$
$311$	4.95093e8	0.933309	0.466655	+	0.884440i	$0.345459\pi$
$312$	0	0
$313$	−3.14797e8	−0.580263	−0.290131	−	0.956987i	$-0.593699\pi$
$313$	−3.14797e8	−0.580263	−0.290131	+	0.956987i	$0.593699\pi$
$314$	− 7.09685e8i	− 1.29364i
$315$	−6.51982e8	−1.17530
$316$	−3.27468e8	−0.583800
$317$	− 6.24177e8i	− 1.10053i	−0.834991	−	0.550263i	$-0.814527\pi$
$317$	− 6.24177e8i	− 1.10053i	0.834991	−	0.550263i	$-0.185473\pi$
$318$	2.03599e9i	3.55043i
$319$	− 2.50838e8i	− 0.432639i
$320$	6.30410e8i	1.07547i
$321$	5.66952e6	0.00956706
$322$	−6.50133e8	−1.08519
$323$	1.17674e8i	0.194300i
$324$	−7.35946e8	−1.20209
$325$	0	0
$326$	5.78247e8	0.924382
$327$	− 3.29250e8i	− 0.520725i
$328$	−2.44849e8	−0.383125
$329$	−2.65445e8	−0.410950
$330$	− 2.30272e9i	− 3.52730i
$331$	5.03800e8i	0.763590i	0.924247	+	0.381795i	$0.124694\pi$
$331$	5.03800e8i	0.763590i	−0.924247	+	0.381795i	$0.875306\pi$
$332$	− 2.49779e8i	− 0.374604i
$333$	− 1.55647e9i	− 2.30986i
$334$	1.34286e8	0.197206
$335$	−5.73213e8	−0.833028
$336$	6.77758e8i	0.974737i
$337$	−9.64189e8	−1.37233	−0.686164	−	0.727447i	$-0.740707\pi$
$337$	−9.64189e8	−1.37233	−0.686164	+	0.727447i	$0.740707\pi$
$338$	0	0
$339$	1.70111e9	2.37156
$340$	− 4.82145e8i	− 0.665276i
$341$	−6.94180e8	−0.948051
$342$	6.12410e8	0.827849
$343$	8.01836e8i	1.07289i
$344$	4.39569e7i	0.0582200i
$345$	1.05207e9i	1.37936i
$346$	6.25598e8i	0.811949i
$347$	4.60468e8	0.591624	0.295812	−	0.955246i	$-0.404410\pi$
$347$	4.60468e8	0.591624	0.295812	+	0.955246i	$0.404410\pi$
$348$	−4.09883e8	−0.521354
$349$	− 2.44565e8i	− 0.307967i	−0.988073	−	0.153984i	$-0.950790\pi$
$349$	− 2.44565e8i	− 0.307967i	0.988073	−	0.153984i	$-0.0492103\pi$
$350$	3.15914e8	0.393850
$351$	0	0
$352$	−2.00500e9	−2.45028
$353$	5.42061e8i	0.655899i	0.944695	+	0.327949i	$0.106358\pi$
$353$	5.42061e8i	0.655899i	−0.944695	+	0.327949i	$0.893642\pi$
$354$	4.49128e8	0.538094
$355$	2.89806e8	0.343801
$356$	6.14206e7i	0.0721505i
$357$	− 7.62045e8i	− 0.886425i
$358$	− 1.14880e9i	− 1.32329i
$359$	− 5.16223e8i	− 0.588853i	−0.955674	−	0.294427i	$-0.904871\pi$
$359$	− 5.16223e8i	− 0.588853i	0.955674	−	0.294427i	$-0.0951286\pi$
$360$	−4.11235e8	−0.464549
$361$	8.24250e8	0.922112
$362$	2.02630e9i	2.24504i
$363$	3.08965e9	3.39028
$364$	0	0
$365$	5.94462e8	0.639880
$366$	− 1.95982e9i	− 2.08945i
$367$	1.03213e9	1.08994	0.544969	−	0.838456i	$-0.316541\pi$
$367$	1.03213e9	1.08994	0.544969	+	0.838456i	$0.316541\pi$
$368$	7.29317e8	0.762867
$369$	2.54815e9i	2.64018i
$370$	− 1.33119e9i	− 1.36627i
$371$	9.99565e8i	1.01625i
$372$	1.13433e9i	1.14245i
$373$	−1.80008e9	−1.79602	−0.898012	−	0.439971i	$-0.854989\pi$
$373$	−1.80008e9	−1.79602	−0.898012	+	0.439971i	$0.854989\pi$
$374$	1.79481e9	1.77405
$375$	− 1.92481e9i	− 1.88486i
$376$	−1.67428e8	−0.162432
$377$	0	0
$378$	−1.98463e9	−1.88998
$379$	7.94428e8i	0.749579i	0.927110	+	0.374790i	$0.122285\pi$
$379$	7.94428e8i	0.749579i	−0.927110	+	0.374790i	$0.877715\pi$
$380$	2.85262e8	0.266686
$381$	7.56403e8	0.700674
$382$	1.40907e9i	1.29333i
$383$	1.89190e9i	1.72069i	0.509710	+	0.860346i	$0.329753\pi$
$383$	1.89190e9i	1.72069i	−0.509710	+	0.860346i	$0.670247\pi$
$384$	1.09535e9i	0.987174i
$385$	− 1.13052e9i	− 1.00963i
$386$	−7.46239e8	−0.660423
$387$	4.57460e8	0.401204
$388$	− 1.69047e9i	− 1.46925i
$389$	−9.79194e8	−0.843422	−0.421711	−	0.906730i	$-0.638570\pi$
$389$	−9.79194e8	−0.843422	−0.421711	+	0.906730i	$0.638570\pi$
$390$	0	0
$391$	−8.20016e8	−0.693751
$392$	1.59332e8i	0.133599i
$393$	3.30205e9	2.74416
$394$	−2.12091e9	−1.74697
$395$	4.77690e8i	0.389993i
$396$	− 5.08721e9i	− 4.11668i
$397$	− 1.17845e9i	− 0.945244i	−0.881265	−	0.472622i	$-0.843308\pi$
$397$	− 1.17845e9i	− 0.945244i	0.881265	−	0.472622i	$-0.156692\pi$
$398$	− 1.61317e9i	− 1.28260i
$399$	4.50865e8	0.355338
$400$	−3.54391e8	−0.276868
$401$	9.42929e8i	0.730254i	0.930958	+	0.365127i	$0.118974\pi$
$401$	9.42929e8i	0.730254i	−0.930958	+	0.365127i	$0.881026\pi$
$402$	−3.48676e9	−2.67690
$403$	0	0
$404$	1.17622e9	0.887468
$405$	1.07355e9i	0.803029i
$406$	−3.69483e8	−0.274001
$407$	2.69887e9	1.98427
$408$	− 4.80657e8i	− 0.350368i
$409$	− 1.96849e9i	− 1.42266i	−0.702858	−	0.711331i	$-0.748093\pi$
$409$	− 1.96849e9i	− 1.42266i	0.702858	−	0.711331i	$-0.251907\pi$
$410$	2.17934e9i	1.56164i
$411$	3.94774e9i	2.80480i
$412$	−7.43751e7	−0.0523947
$413$	2.20498e8	0.154021
$414$	4.26760e9i	2.95585i
$415$	−3.64362e8	−0.250245
$416$	0	0
$417$	4.31767e9	2.91590
$418$	1.06190e9i	0.711159i
$419$	−1.95673e9	−1.29952	−0.649758	−	0.760141i	$-0.725130\pi$
$419$	−1.95673e9	−1.29952	−0.649758	+	0.760141i	$0.725130\pi$
$420$	−1.84732e9	−1.21666
$421$	− 2.92602e8i	− 0.191113i	−0.995424	−	0.0955563i	$-0.969537\pi$
$421$	− 2.92602e8i	− 0.191113i	0.995424	−	0.0955563i	$-0.0304630\pi$
$422$	8.44209e8i	0.546835i
$423$	1.74243e9i	1.11935i
$424$	6.30471e8i	0.401684i
$425$	3.98463e8	0.251783
$426$	1.76284e9	1.10479
$427$	− 9.62168e8i	− 0.598072i
$428$	1.07123e7	0.00660437
$429$	0	0
$430$	3.91249e8	0.237309
$431$	− 9.77619e7i	− 0.0588165i	−0.999567	−	0.0294083i	$-0.990638\pi$
$431$	− 9.77619e7i	− 0.0588165i	0.999567	−	0.0294083i	$-0.00936229\pi$
$432$	2.22635e9	1.32862
$433$	6.53277e8	0.386714	0.193357	−	0.981128i	$-0.438062\pi$
$433$	6.53277e8	0.386714	0.193357	+	0.981128i	$0.438062\pi$
$434$	1.02252e9i	0.600425i
$435$	5.97912e8i	0.348277i
$436$	− 6.22105e8i	− 0.359469i
$437$	− 4.85163e8i	− 0.278101i
$438$	3.61601e9	2.05623
$439$	−1.57565e9	−0.888862	−0.444431	−	0.895813i	$-0.646594\pi$
$439$	−1.57565e9	−0.888862	−0.444431	+	0.895813i	$0.646594\pi$
$440$	− 7.13068e8i	− 0.399068i
$441$	1.65818e9	0.920652
$442$	0	0
$443$	2.36999e9	1.29519	0.647594	−	0.761985i	$-0.275775\pi$
$443$	2.36999e9	1.29519	0.647594	+	0.761985i	$0.275775\pi$
$444$	− 4.41010e9i	− 2.39116i
$445$	8.95966e7	0.0481983
$446$	−4.11030e9	−2.19382
$447$	− 2.93619e9i	− 1.55492i
$448$	1.88263e9i	0.989217i
$449$	− 2.05507e8i	− 0.107143i	−0.998564	−	0.0535716i	$-0.982939\pi$
$449$	− 2.05507e8i	− 0.107143i	0.998564	−	0.0535716i	$-0.0170605\pi$
$450$	− 2.07372e9i	− 1.07277i
$451$	−4.41841e9	−2.26803
$452$	3.21419e9	1.63714
$453$	− 2.07156e9i	− 1.04702i
$454$	3.20048e7	0.0160516
$455$	0	0
$456$	2.84381e8	0.140451
$457$	− 2.69717e9i	− 1.32191i	−0.750426	−	0.660954i	$-0.770152\pi$
$457$	− 2.69717e9i	− 1.32191i	0.750426	−	0.660954i	$-0.229848\pi$
$458$	−3.47733e9	−1.69129
$459$	−2.50322e9	−1.20824
$460$	1.98785e9i	0.952209i
$461$	− 1.92610e9i	− 0.915642i	−0.889044	−	0.457821i	$-0.848630\pi$
$461$	− 1.92610e9i	− 0.915642i	0.889044	−	0.457821i	$-0.151370\pi$
$462$	− 6.87675e9i	− 3.24441i
$463$	− 2.34776e9i	− 1.09931i	−0.835391	−	0.549655i	$-0.814759\pi$
$463$	− 2.34776e9i	− 1.09931i	0.835391	−	0.549655i	$-0.185241\pi$
$464$	4.14484e8	0.192617
$465$	1.65469e9	0.763186
$466$	− 5.74989e9i	− 2.63214i
$467$	−1.71342e9	−0.778493	−0.389246	−	0.921134i	$-0.627264\pi$
$467$	−1.71342e9	−0.778493	−0.389246	+	0.921134i	$0.627264\pi$
$468$	0	0
$469$	−1.71182e9	−0.766219
$470$	1.49024e9i	0.662084i
$471$	−3.42966e9	−1.51244
$472$	1.39078e8	0.0608783
$473$	7.93221e8i	0.344652i
$474$	2.90571e9i	1.25322i
$475$	2.35751e8i	0.100931i
$476$	− 1.43986e9i	− 0.611921i
$477$	6.56133e9	2.76807
$478$	−6.62184e9	−2.77320
$479$	1.13940e8i	0.0473697i	0.999719	+	0.0236848i	$0.00753982\pi$
$479$	1.13940e8i	0.0473697i	−0.999719	+	0.0236848i	$0.992460\pi$
$480$	4.77924e9	1.97249
$481$	0	0
$482$	−1.30332e9	−0.530137
$483$	3.14186e9i	1.26874i
$484$	5.83778e9	2.34039
$485$	−2.46595e9	−0.981496
$486$	2.20257e7i	0.00870367i
$487$	− 7.96258e8i	− 0.312394i	−0.987726	−	0.156197i	$-0.950077\pi$
$487$	− 7.96258e8i	− 0.312394i	0.987726	−	0.156197i	$-0.0499234\pi$
$488$	− 6.06883e8i	− 0.236394i
$489$	− 2.79446e9i	− 1.08073i
$490$	1.41818e9	0.544558
$491$	−3.65621e9	−1.39395	−0.696973	−	0.717097i	$-0.745470\pi$
$491$	−3.65621e9	−1.39395	−0.696973	+	0.717097i	$0.745470\pi$
$492$	7.21993e9i	2.73310i
$493$	−4.66030e8	−0.175166
$494$	0	0
$495$	−7.42092e9	−2.75004
$496$	− 1.14706e9i	− 0.422085i
$497$	8.65463e8	0.316228
$498$	−2.21636e9	−0.804150
$499$	− 7.46131e8i	− 0.268821i	−0.990926	−	0.134411i	$-0.957086\pi$
$499$	− 7.46131e8i	− 0.268821i	0.990926	−	0.134411i	$-0.0429141\pi$
$500$	− 3.63686e9i	− 1.30116i
$501$	− 6.48959e8i	− 0.230560i
$502$	− 5.88270e9i	− 2.07546i
$503$	−4.68159e8	−0.164023	−0.0820117	−	0.996631i	$-0.526134\pi$
$503$	−4.68159e8	−0.164023	−0.0820117	+	0.996631i	$0.526134\pi$
$504$	−1.22809e9	−0.427292
$505$	− 1.71579e9i	− 0.592851i
$506$	−7.39987e9	−2.53920
$507$	0	0
$508$	1.42920e9	0.483692
$509$	− 5.93386e8i	− 0.199446i	−0.995015	−	0.0997229i	$-0.968204\pi$
$509$	− 5.93386e8i	− 0.199446i	0.995015	−	0.0997229i	$-0.0317956\pi$
$510$	−4.27821e9	−1.42812
$511$	1.77527e9	0.588562
$512$	− 3.98841e9i	− 1.31327i
$513$	− 1.48103e9i	− 0.484344i
$514$	− 6.02225e9i	− 1.95609i
$515$	1.08494e8i	0.0350009i
$516$	1.29617e9	0.415324
$517$	−3.02132e9	−0.961567
$518$	− 3.97542e9i	− 1.25669i
$519$	3.02329e9	0.949280
$520$	0	0
$521$	2.35950e9	0.730950	0.365475	−	0.930821i	$-0.380907\pi$
$521$	2.35950e9	0.730950	0.365475	+	0.930821i	$0.380907\pi$
$522$	2.42535e9i	0.746325i
$523$	5.49264e8	0.167890	0.0839451	−	0.996470i	$-0.473248\pi$
$523$	5.49264e8	0.167890	0.0839451	+	0.996470i	$0.473248\pi$
$524$	6.23911e9	1.89436
$525$	− 1.52670e9i	− 0.460465i
$526$	3.34497e9i	1.00217i
$527$	1.28971e9i	0.383844i
$528$	7.71430e9i	2.28075i
$529$	−2.39521e7	−0.00703475
$530$	5.61167e9	1.63729
$531$	− 1.44739e9i	− 0.419522i
$532$	8.51893e8	0.245298
$533$	0	0
$534$	5.45002e8	0.154883
$535$	− 1.56265e7i	− 0.00441188i
$536$	−1.07972e9	−0.302856
$537$	−5.55174e9	−1.54710
$538$	1.08115e9i	0.299328i
$539$	2.87522e9i	0.790881i
$540$	6.06822e9i	1.65838i
$541$	− 6.19192e7i	− 0.0168126i	−0.999965	−	0.00840630i	$-0.997324\pi$
$541$	− 6.19192e7i	− 0.0168126i	0.999965	−	0.00840630i	$-0.00267584\pi$
$542$	−8.03907e8	−0.216874
$543$	9.79240e9	2.62476
$544$	3.72507e9i	0.992062i
$545$	−9.07489e8	−0.240134
$546$	0	0
$547$	−4.11747e9	−1.07566	−0.537830	−	0.843053i	$-0.680756\pi$
$547$	−4.11747e9	−1.07566	−0.537830	+	0.843053i	$0.680756\pi$
$548$	7.45912e9i	1.93622i
$549$	−6.31585e9	−1.62903
$550$	3.59576e9	0.921555
$551$	− 2.75727e8i	− 0.0702181i
$552$	1.98172e9i	0.501482i
$553$	1.42655e9i	0.358715i
$554$	9.94238e9i	2.48431i
$555$	−6.43319e9	−1.59735
$556$	8.15808e9	2.01292
$557$	4.03807e9i	0.990105i	0.868863	+	0.495052i	$0.164851\pi$
$557$	4.03807e9i	0.990105i	−0.868863	+	0.495052i	$0.835149\pi$
$558$	6.71202e9	1.63544
$559$	0	0
$560$	1.86806e9	0.449503
$561$	− 8.67366e9i	− 2.07411i
$562$	2.92994e9	0.696277
$563$	6.53949e8	0.154442	0.0772209	−	0.997014i	$-0.475395\pi$
$563$	6.53949e8	0.154442	0.0772209	+	0.997014i	$0.475395\pi$
$564$	4.93700e9i	1.15874i
$565$	− 4.68866e9i	− 1.09365i
$566$	1.10841e10i	2.56947i
$567$	3.20601e9i	0.738626i
$568$	5.45887e8	0.124992
$569$	−3.78098e9	−0.860423	−0.430211	−	0.902728i	$-0.641561\pi$
$569$	−3.78098e9	−0.860423	−0.430211	+	0.902728i	$0.641561\pi$
$570$	− 2.53121e9i	− 0.572487i
$571$	−1.41244e9	−0.317499	−0.158750	−	0.987319i	$-0.550746\pi$
$571$	−1.41244e9	−0.317499	−0.158750	+	0.987319i	$0.550746\pi$
$572$	0	0
$573$	6.80952e9	1.51208
$574$	6.50829e9i	1.43640i
$575$	−1.64284e9	−0.360377
$576$	1.23579e10	2.69443
$577$	− 2.80901e9i	− 0.608749i	−0.952553	−	0.304374i	$-0.901553\pi$
$577$	− 2.80901e9i	− 0.608749i	0.952553	−	0.304374i	$-0.0984474\pi$
$578$	3.54507e9i	0.763621i
$579$	3.60631e9i	0.772125i
$580$	1.12973e9i	0.240424i
$581$	−1.08812e9	−0.230175
$582$	−1.50000e10	−3.15399
$583$	1.13771e10i	2.37790i
$584$	1.11975e9	0.232635
$585$	0	0
$586$	8.99889e9	1.84734
$587$	5.36072e9i	1.09393i	0.837156	+	0.546965i	$0.184217\pi$
$587$	5.36072e9i	1.09393i	−0.837156	+	0.546965i	$0.815783\pi$
$588$	4.69827e9	0.953054
$589$	−7.63059e8	−0.153870
$590$	− 1.23790e9i	− 0.248144i
$591$	1.02496e10i	2.04245i
$592$	4.45960e9i	0.883426i
$593$	2.19757e9i	0.432765i	0.976309	+	0.216382i	$0.0694258\pi$
$593$	2.19757e9i	0.432765i	−0.976309	+	0.216382i	$0.930574\pi$
$594$	−2.25892e10	−4.42230
$595$	−2.10038e9	−0.408778
$596$	− 5.54783e9i	− 1.07340i
$597$	−7.79589e9	−1.49953
$598$	0	0
$599$	−9.75825e9	−1.85515	−0.927573	−	0.373642i	$-0.878109\pi$
$599$	−9.75825e9	−1.85515	−0.927573	+	0.373642i	$0.878109\pi$
$600$	− 9.62960e8i	− 0.182003i
$601$	3.68732e9	0.692868	0.346434	−	0.938074i	$-0.387393\pi$
$601$	3.68732e9	0.692868	0.346434	+	0.938074i	$0.387393\pi$
$602$	1.16841e9	0.218277
$603$	1.12367e10i	2.08703i
$604$	− 3.91414e9i	− 0.722782i
$605$	− 8.51579e9i	− 1.56344i
$606$	− 1.04369e10i	− 1.90510i
$607$	5.90498e9	1.07166	0.535831	−	0.844325i	$-0.319998\pi$
$607$	5.90498e9	1.07166	0.535831	+	0.844325i	$0.319998\pi$
$608$	−2.20394e9	−0.397684
$609$	1.78558e9i	0.320345i
$610$	−5.40171e9	−0.963557
$611$	0	0
$612$	−9.45149e9	−1.66675
$613$	− 8.40999e9i	− 1.47463i	−0.675548	−	0.737316i	$-0.736093\pi$
$613$	− 8.40999e9i	− 1.47463i	0.675548	−	0.737316i	$-0.263907\pi$
$614$	6.00439e9	1.04684
$615$	1.05320e10	1.82578
$616$	− 2.12948e9i	− 0.367063i
$617$	− 8.51257e9i	− 1.45902i	−0.683967	−	0.729512i	$-0.739747\pi$
$617$	− 8.51257e9i	− 1.45902i	0.683967	−	0.729512i	$-0.260253\pi$
$618$	6.59950e8i	0.112474i
$619$	8.70392e9i	1.47502i	0.675337	+	0.737509i	$0.263998\pi$
$619$	8.70392e9i	1.47502i	−0.675337	+	0.737509i	$0.736002\pi$
$620$	3.12647e9	0.526846
$621$	1.03206e10	1.72936
$622$	− 8.30060e9i	− 1.38307i
$623$	2.67568e8	0.0443328
$624$	0	0
$625$	−3.09788e9	−0.507556
$626$	5.27780e9i	0.859889i
$627$	5.13178e9	0.831442
$628$	−6.48022e9	−1.04407
$629$	− 5.01421e9i	− 0.803388i
$630$	1.09310e10i	1.74167i
$631$	− 9.88656e9i	− 1.56654i	−0.621679	−	0.783272i	$-0.713549\pi$
$631$	− 9.88656e9i	− 1.56654i	0.621679	−	0.783272i	$-0.286451\pi$
$632$	8.99792e8i	0.141786i
$633$	4.07976e9	0.639326
$634$	−1.04648e10	−1.63086
$635$	− 2.08482e9i	− 0.323118i
$636$	1.85909e10	2.86549
$637$	0	0
$638$	−4.20548e9	−0.641126
$639$	− 5.68106e9i	− 0.861343i
$640$	3.01904e9	0.455239
$641$	6.76169e9	1.01403	0.507017	−	0.861936i	$-0.330748\pi$
$641$	6.76169e9	1.01403	0.507017	+	0.861936i	$0.330748\pi$
$642$	− 9.50536e7i	− 0.0141774i
$643$	− 2.81532e9i	− 0.417627i	−0.977955	−	0.208814i	$-0.933040\pi$
$643$	− 2.81532e9i	− 0.417627i	0.977955	−	0.208814i	$-0.0669602\pi$
$644$	5.93644e9i	0.875842i
$645$	− 1.89077e9i	− 0.277447i
$646$	1.97289e9	0.287932
$647$	1.19349e10	1.73243	0.866213	−	0.499674i	$-0.166547\pi$
$647$	1.19349e10	1.73243	0.866213	+	0.499674i	$0.166547\pi$
$648$	2.02218e9i	0.291949i
$649$	2.50973e9	0.360388
$650$	0	0
$651$	4.94149e9	0.701979
$652$	− 5.28004e9i	− 0.746054i
$653$	1.28240e9	0.180231	0.0901153	−	0.995931i	$-0.471276\pi$
$653$	1.28240e9	0.180231	0.0901153	+	0.995931i	$0.471276\pi$
$654$	−5.52011e9	−0.771659
$655$	− 9.10123e9i	− 1.26548i
$656$	− 7.30097e9i	− 1.00976i
$657$	− 1.16532e10i	− 1.60313i
$658$	4.45038e9i	0.608985i
$659$	−4.85245e9	−0.660484	−0.330242	−	0.943896i	$-0.607130\pi$
$659$	−4.85245e9	−0.660484	−0.330242	+	0.943896i	$0.607130\pi$
$660$	−2.10264e10	−2.84683
$661$	− 4.76107e9i	− 0.641209i	−0.947213	−	0.320605i	$-0.896114\pi$
$661$	− 4.76107e9i	− 0.641209i	0.947213	−	0.320605i	$-0.103886\pi$
$662$	8.44658e9	1.13156
$663$	0	0
$664$	−6.86325e8	−0.0909790
$665$	− 1.24269e9i	− 0.163865i
$666$	−2.60954e10	−3.42298
$667$	1.92141e9	0.250715
$668$	− 1.22618e9i	− 0.159162i
$669$	1.98636e10i	2.56488i
$670$	9.61034e9i	1.23446i
$671$	− 1.09515e10i	− 1.39941i
$672$	1.42725e10	1.81429
$673$	1.03481e10	1.30861	0.654303	−	0.756233i	$-0.272962\pi$
$673$	1.03481e10	1.30861	0.654303	+	0.756233i	$0.272962\pi$
$674$	1.61653e10i	2.03365i
$675$	−5.01501e9	−0.627637
$676$	0	0
$677$	−1.09230e9	−0.135295	−0.0676477	−	0.997709i	$-0.521549\pi$
$677$	−1.09230e9	−0.135295	−0.0676477	+	0.997709i	$0.521549\pi$
$678$	− 2.85204e10i	− 3.51440i
$679$	−7.36421e9	−0.902780
$680$	−1.32480e9	−0.161573
$681$	− 1.54668e8i	− 0.0187666i
$682$	1.16384e10i	1.40491i
$683$	− 7.72912e9i	− 0.928234i	−0.885774	−	0.464117i	$-0.846372\pi$
$683$	− 7.72912e9i	− 0.928234i	0.885774	−	0.464117i	$-0.153628\pi$
$684$	− 5.59199e9i	− 0.668144i
$685$	1.08809e10	1.29345
$686$	1.34434e10	1.58992
$687$	1.68047e10i	1.97735i
$688$	−1.31072e9	−0.153444
$689$	0	0
$690$	1.76388e10	2.04407
$691$	8.59771e9i	0.991310i	0.868519	+	0.495655i	$0.165072\pi$
$691$	8.59771e9i	0.991310i	−0.868519	+	0.495655i	$0.834928\pi$
$692$	5.71240e9	0.655311
$693$	−2.21615e10	−2.52949
$694$	− 7.72008e9i	− 0.876725i
$695$	− 1.19005e10i	− 1.34468i
$696$	1.12625e9i	0.126620i
$697$	8.20893e9i	0.918273i
$698$	−4.10030e9	−0.456375
$699$	−2.77872e10	−3.07733
$700$	− 2.88465e9i	− 0.317870i
$701$	4.34708e9	0.476633	0.238317	−	0.971188i	$-0.423404\pi$
$701$	4.34708e9	0.476633	0.238317	+	0.971188i	$0.423404\pi$
$702$	0	0
$703$	2.96666e9	0.322051
$704$	2.14282e10i	2.31463i
$705$	7.20179e9	0.774067
$706$	9.08805e9	0.971973
$707$	− 5.12397e9i	− 0.545304i
$708$	− 4.10104e9i	− 0.434287i
$709$	1.22439e10i	1.29020i	0.764098	+	0.645100i	$0.223184\pi$
$709$	1.22439e10i	1.29020i	−0.764098	+	0.645100i	$0.776816\pi$
$710$	− 4.85880e9i	− 0.509477i
$711$	9.36416e9	0.977069
$712$	1.68767e8	0.0175230
$713$	− 5.31740e9i	− 0.549396i
$714$	−1.27762e10	−1.31359
$715$	0	0
$716$	−1.04898e10	−1.06800
$717$	3.20010e10i	3.24225i
$718$	−8.65486e9	−0.872619
$719$	2.29472e9	0.230239	0.115119	−	0.993352i	$-0.463275\pi$
$719$	2.29472e9	0.230239	0.115119	+	0.993352i	$0.463275\pi$
$720$	− 1.22623e10i	− 1.22436i
$721$	3.24001e8i	0.0321939i
$722$	− 1.38191e10i	− 1.36647i
$723$	6.29851e9i	0.619803i
$724$	1.85024e10	1.81194
$725$	−9.33655e8	−0.0909921
$726$	− 5.18002e10i	− 5.02404i
$727$	−5.74619e9	−0.554638	−0.277319	−	0.960778i	$-0.589446\pi$
$727$	−5.74619e9	−0.554638	−0.277319	+	0.960778i	$0.589446\pi$
$728$	0	0
$729$	−1.04071e10	−0.994908
$730$	− 9.96659e9i	− 0.948236i
$731$	1.47372e9	0.139542
$732$	−1.78953e10	−1.68636
$733$	− 1.26955e9i	− 0.119065i	−0.998226	−	0.0595326i	$-0.981039\pi$
$733$	− 1.26955e9i	− 0.119065i	0.998226	−	0.0595326i	$-0.0189610\pi$
$734$	− 1.73044e10i	− 1.61518i
$735$	− 6.85355e9i	− 0.636664i
$736$	− 1.53582e10i	− 1.41994i
$737$	−1.94841e10	−1.79285
$738$	4.27216e10	3.91247
$739$	2.57407e9i	0.234620i	0.993095	+	0.117310i	$0.0374270\pi$
$739$	2.57407e9i	0.234620i	−0.993095	+	0.117310i	$0.962573\pi$
$740$	−1.21553e10	−1.10269
$741$	0	0
$742$	1.67584e10	1.50598
$743$	− 4.84335e9i	− 0.433196i	−0.976261	−	0.216598i	$-0.930504\pi$
$743$	− 4.84335e9i	− 0.433196i	0.976261	−	0.216598i	$-0.0694961\pi$
$744$	3.11682e9	0.277464
$745$	−8.09283e9	−0.717057
$746$	3.01797e10i	2.66152i
$747$	7.14260e9i	0.626952i
$748$	− 1.63886e10i	− 1.43181i
$749$	− 4.66664e7i	− 0.00405805i
$750$	−3.22708e10	−2.79316
$751$	−2.07398e10	−1.78676	−0.893378	−	0.449307i	$-0.851671\pi$
$751$	−2.07398e10	−1.78676	−0.893378	+	0.449307i	$0.851671\pi$
$752$	− 4.99242e9i	− 0.428103i
$753$	−2.84290e10	−2.42650
$754$	0	0
$755$	−5.70971e9	−0.482836
$756$	1.81219e10i	1.52538i
$757$	2.98409e9	0.250021	0.125010	−	0.992155i	$-0.460104\pi$
$757$	2.98409e9	0.250021	0.125010	+	0.992155i	$0.460104\pi$
$758$	1.33192e10	1.11080
$759$	3.57610e10i	2.96868i
$760$	− 7.83821e8i	− 0.0647694i
$761$	2.11422e10i	1.73901i	0.493920	+	0.869507i	$0.335564\pi$
$761$	2.11422e10i	1.73901i	−0.493920	+	0.869507i	$0.664436\pi$
$762$	− 1.26816e10i	− 1.03832i
$763$	−2.71009e9	−0.220875
$764$	1.28663e10	1.04383
$765$	1.37873e10i	1.11343i
$766$	3.17191e10	2.54989
$767$	0	0
$768$	−1.09120e10	−0.869240
$769$	1.37622e8i	0.0109130i	0.999985	+	0.00545652i	$0.00173687\pi$
$769$	1.37622e8i	0.0109130i	−0.999985	+	0.00545652i	$0.998263\pi$
$770$	−1.89539e10	−1.49617
$771$	−2.91034e10	−2.28693
$772$	6.81399e9i	0.533017i
$773$	− 4.08746e9i	− 0.318292i	−0.987255	−	0.159146i	$-0.949126\pi$
$773$	− 4.08746e9i	− 0.318292i	0.987255	−	0.159146i	$-0.0508740\pi$
$774$	− 7.66966e9i	− 0.594542i
$775$	2.58384e9i	0.199393i
$776$	−4.64495e9	−0.356833
$777$	−1.92118e10	−1.46924
$778$	1.64169e10i	1.24986i
$779$	−4.85683e9	−0.368105
$780$	0	0
$781$	9.85077e9	0.739932
$782$	1.37482e10i	1.02807i
$783$	5.86539e9	0.436648
$784$	−4.75101e9	−0.352111
$785$	9.45295e9i	0.697466i
$786$	− 5.53613e10i	− 4.06656i
$787$	1.83896e10i	1.34481i	0.740184	+	0.672404i	$0.234738\pi$
$787$	1.83896e10i	1.34481i	−0.740184	+	0.672404i	$0.765262\pi$
$788$	1.93662e10i	1.40995i
$789$	1.61651e10	1.17168
$790$	8.00882e9	0.577928
$791$	− 1.40020e10i	− 1.00594i
$792$	−1.39783e10	−0.999805
$793$	0	0
$794$	−1.97576e10	−1.40075
$795$	− 2.71192e10i	− 1.91422i
$796$	−1.47301e10	−1.03516
$797$	1.07541e10	0.752436	0.376218	−	0.926531i	$-0.377224\pi$
$797$	1.07541e10	0.752436	0.376218	+	0.926531i	$0.377224\pi$
$798$	− 7.55908e9i	− 0.526573i
$799$	5.61328e9i	0.389317i
$800$	7.46290e9i	0.515339i
$801$	− 1.75636e9i	− 0.120754i
$802$	1.58089e10	1.08216
$803$	2.02063e10	1.37716
$804$	3.18380e10i	2.16048i
$805$	8.65972e9	0.585084
$806$	0	0
$807$	5.22482e9	0.349956
$808$	− 3.23192e9i	− 0.215537i
$809$	7.93391e9	0.526826	0.263413	−	0.964683i	$-0.415152\pi$
$809$	7.93391e9	0.526826	0.263413	+	0.964683i	$0.415152\pi$
$810$	1.79989e10	1.19000
$811$	− 2.50497e10i	− 1.64903i	−0.565837	−	0.824517i	$-0.691447\pi$
$811$	− 2.50497e10i	− 1.64903i	0.565837	−	0.824517i	$-0.308553\pi$
$812$	3.37379e9i	0.221142i
$813$	3.88500e9i	0.253556i
$814$	− 4.52485e10i	− 2.94049i
$815$	−7.70220e9	−0.498383
$816$	1.43323e10	0.923424
$817$	8.71928e8i	0.0559375i
$818$	−3.30032e10	−2.10824
$819$	0	0
$820$	1.98998e10	1.26038
$821$	5.53755e9i	0.349234i	0.984636	+	0.174617i	$0.0558687\pi$
$821$	5.53755e9i	0.349234i	−0.984636	+	0.174617i	$0.944131\pi$
$822$	6.61868e10	4.15643
$823$	−2.94660e10	−1.84256	−0.921282	−	0.388896i	$-0.872856\pi$
$823$	−2.94660e10	−1.84256	−0.921282	+	0.388896i	$0.872856\pi$
$824$	2.04362e8i	0.0127249i
$825$	− 1.73770e10i	− 1.07742i
$826$	− 3.69682e9i	− 0.228243i
$827$	2.47966e10i	1.52448i	0.647294	+	0.762241i	$0.275901\pi$
$827$	2.47966e10i	1.52448i	−0.647294	+	0.762241i	$0.724099\pi$
$828$	3.89679e10	2.38562
$829$	1.08636e10	0.662266	0.331133	−	0.943584i	$-0.392569\pi$
$829$	1.08636e10	0.662266	0.331133	+	0.943584i	$0.392569\pi$
$830$	6.10880e9i	0.370837i
$831$	4.80480e10	2.90450
$832$	0	0
$833$	5.34185e9	0.320210
$834$	− 7.23889e10i	− 4.32106i
$835$	−1.78868e9	−0.106324
$836$	9.69632e9	0.573965
$837$	− 1.62321e10i	− 0.956834i
$838$	3.28060e10i	1.92575i
$839$	− 1.43527e10i	− 0.839007i	−0.907754	−	0.419504i	$-0.862204\pi$
$839$	− 1.43527e10i	− 0.839007i	0.907754	−	0.419504i	$-0.137796\pi$
$840$	5.07594e9i	0.295488i
$841$	−1.61579e10	−0.936697
$842$	−4.90568e9	−0.283209
$843$	− 1.41594e10i	− 0.814044i
$844$	7.70857e9	0.441342
$845$	0	0
$846$	2.92131e10	1.65875
$847$	− 2.54312e10i	− 1.43805i
$848$	−1.87995e10	−1.05867
$849$	5.35656e10	3.00406
$850$	− 6.68053e9i	− 0.373117i
$851$	2.06733e10i	1.14989i
$852$	− 1.60967e10i	− 0.891658i
$853$	2.42734e10i	1.33909i	0.742774	+	0.669543i	$0.233510\pi$
$853$	2.42734e10i	1.33909i	−0.742774	+	0.669543i	$0.766490\pi$
$854$	−1.61314e10	−0.886280
$855$	−8.15725e9	−0.446336
$856$	− 2.94346e7i	− 0.00160398i
$857$	9.63070e9	0.522667	0.261334	−	0.965249i	$-0.415838\pi$
$857$	9.63070e9	0.522667	0.261334	+	0.965249i	$0.415838\pi$
$858$	0	0
$859$	2.19480e10	1.18146	0.590729	−	0.806870i	$-0.298840\pi$
$859$	2.19480e10	1.18146	0.590729	+	0.806870i	$0.298840\pi$
$860$	− 3.57254e9i	− 0.191528i
$861$	3.14523e10	1.67935
$862$	−1.63905e9	−0.0871599
$863$	1.90983e10i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$863$	1.90983e10i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$864$	− 4.68833e10i	− 2.47298i
$865$	− 8.33290e9i	− 0.437764i
$866$	− 1.09527e10i	− 0.573070i
$867$	1.71321e10	0.892778
$868$	9.33676e9	0.484593
$869$	1.62371e10i	0.839345i
$870$	1.00244e10	0.516110
$871$	0	0
$872$	−1.70938e9	−0.0873031
$873$	4.83401e10i	2.45899i
$874$	−8.13412e9	−0.412117
$875$	−1.58433e10	−0.799498
$876$	− 3.30182e10i	− 1.65955i
$877$	2.78380e10i	1.39360i	0.717263	+	0.696802i	$0.245394\pi$
$877$	2.78380e10i	1.39360i	−0.717263	+	0.696802i	$0.754606\pi$
$878$	2.64169e10i	1.31720i
$879$	− 4.34885e10i	− 2.15980i
$880$	2.12624e10	1.05178
$881$	−2.24473e10	−1.10598	−0.552992	−	0.833186i	$-0.686514\pi$
$881$	−2.24473e10	−1.10598	−0.552992	+	0.833186i	$0.686514\pi$
$882$	− 2.78005e10i	− 1.36431i
$883$	1.41508e10	0.691700	0.345850	−	0.938290i	$-0.387591\pi$
$883$	1.41508e10	0.691700	0.345850	+	0.938290i	$0.387591\pi$
$884$	0	0
$885$	−5.98234e9	−0.290115
$886$	− 3.97346e10i	− 1.91933i
$887$	3.27385e10	1.57516	0.787582	−	0.616210i	$-0.211333\pi$
$887$	3.27385e10	1.57516	0.787582	+	0.616210i	$0.211333\pi$
$888$	−1.21178e10	−0.580733
$889$	− 6.22603e9i	− 0.297204i
$890$	− 1.50215e9i	− 0.0714248i
$891$	3.64911e10i	1.72828i
$892$	3.75316e10i	1.77060i
$893$	−3.32110e9	−0.156064
$894$	−4.92274e10	−2.30423
$895$	1.53019e10i	0.713452i
$896$	9.01594e9	0.418729
$897$	0	0
$898$	−3.44548e9	−0.158775
$899$	− 3.02197e9i	− 0.138718i
$900$	−1.89353e10	−0.865814
$901$	2.11375e10	0.962756
$902$	7.40780e10i	3.36098i
$903$	− 5.64651e9i	− 0.255195i
$904$	− 8.83171e9i	− 0.397608i
$905$	− 2.69902e10i	− 1.21042i
$906$	−3.47313e10	−1.55157
$907$	−4.64948e9	−0.206909	−0.103454	−	0.994634i	$-0.532990\pi$
$907$	−4.64948e9	−0.206909	−0.103454	+	0.994634i	$0.532990\pi$
$908$	− 2.92239e8i	− 0.0129550i
$909$	−3.36347e10	−1.48530
$910$	0	0
$911$	−1.91286e9	−0.0838242	−0.0419121	−	0.999121i	$-0.513345\pi$
$911$	−1.91286e9	−0.0838242	−0.0419121	+	0.999121i	$0.513345\pi$
$912$	8.47974e9i	0.370170i
$913$	−1.23850e10	−0.538579
$914$	−4.52200e10	−1.95893
$915$	2.61046e10i	1.12653i
$916$	3.17519e10i	1.36501i
$917$	− 2.71795e10i	− 1.16399i
$918$	4.19683e10i	1.79049i
$919$	−4.51305e10	−1.91808	−0.959039	−	0.283275i	$-0.908579\pi$
$919$	−4.51305e10	−1.91808	−0.959039	+	0.283275i	$0.908579\pi$
$920$	5.46208e9	0.231260
$921$	− 2.90171e10i	− 1.22390i
$922$	−3.22925e10	−1.35689
$923$	0	0
$924$	−6.27923e10	−2.61851
$925$	− 1.00456e10i	− 0.417330i
$926$	−3.93619e10	−1.62906
$927$	2.12680e9	0.0876897
$928$	− 8.72837e9i	− 0.358522i
$929$	6.56214e9i	0.268529i	0.990946	+	0.134264i	$0.0428671\pi$
$929$	6.56214e9i	0.268529i	−0.990946	+	0.134264i	$0.957133\pi$
$930$	− 2.77420e10i	− 1.13096i
$931$	3.16051e9i	0.128361i
$932$	−5.25029e10	−2.12436
$933$	−4.01139e10	−1.61700
$934$	2.87267e10i	1.15364i
$935$	−2.39067e10	−0.956485
$936$	0	0
$937$	−1.55199e10	−0.616314	−0.308157	−	0.951336i	$-0.599712\pi$
$937$	−1.55199e10	−0.616314	−0.308157	+	0.951336i	$0.599712\pi$
$938$	2.86999e10i	1.13546i
$939$	2.55057e10	1.00533
$940$	1.36075e10	0.534357
$941$	1.49766e10i	0.585933i	0.956123	+	0.292967i	$0.0946425\pi$
$941$	1.49766e10i	0.585933i	−0.956123	+	0.292967i	$0.905358\pi$
$942$	5.75007e10i	2.24128i
$943$	− 3.38449e10i	− 1.31432i
$944$	4.14707e9i	0.160450i
$945$	2.64351e10	1.01899
$946$	1.32989e10	0.510738
$947$	1.19869e10i	0.458651i	0.973350	+	0.229325i	$0.0736519\pi$
$947$	1.19869e10i	0.458651i	−0.973350	+	0.229325i	$0.926348\pi$
$948$	2.65324e10	1.01146
$949$	0	0
$950$	3.95254e9	0.149570
$951$	5.05726e10i	1.90671i
$952$	−3.95633e9	−0.148615
$953$	6.04429e9	0.226214	0.113107	−	0.993583i	$-0.463920\pi$
$953$	6.04429e9	0.226214	0.113107	+	0.993583i	$0.463920\pi$
$954$	− 1.10006e11i	− 4.10200i
$955$	− 1.87686e10i	− 0.697302i
$956$	6.04648e10i	2.23820i
$957$	2.03236e10i	0.749565i
$958$	1.91028e9	0.0701969
$959$	3.24943e10	1.18971
$960$	− 5.10776e10i	− 1.86329i
$961$	1.91495e10	0.696025
$962$	0	0
$963$	−3.06327e8	−0.0110533
$964$	1.19008e10i	0.427865i
$965$	9.93983e9	0.356068
$966$	5.26757e10	1.88014
$967$	− 4.09241e9i	− 0.145541i	−0.997349	−	0.0727706i	$-0.976816\pi$
$967$	− 4.09241e9i	− 0.145541i	0.997349	−	0.0727706i	$-0.0231841\pi$
$968$	− 1.60406e10i	− 0.568404i
$969$	− 9.53430e9i	− 0.336632i
$970$	4.13435e10i	1.45447i
$971$	4.94435e10	1.73317	0.866587	−	0.499026i	$-0.166309\pi$
$971$	4.94435e10	1.73317	0.866587	+	0.499026i	$0.166309\pi$
$972$	2.01119e8	0.00702460
$973$	− 3.55392e10i	− 1.23684i
$974$	−1.33498e10	−0.462935
$975$	0	0
$976$	1.80962e10	0.623035
$977$	4.89243e10i	1.67839i	0.543828	+	0.839197i	$0.316974\pi$
$977$	4.89243e10i	1.67839i	−0.543828	+	0.839197i	$0.683026\pi$
$978$	−4.68512e10	−1.60153
$979$	3.04548e9	0.103733
$980$	− 1.29495e10i	− 0.439504i
$981$	1.77895e10i	0.601620i
$982$	6.12991e10i	2.06568i
$983$	− 1.74370e10i	− 0.585511i	−0.956187	−	0.292755i	$-0.905428\pi$
$983$	− 1.74370e10i	− 0.585511i	0.956187	−	0.292755i	$-0.0945722\pi$
$984$	1.98384e10	0.663779
$985$	2.82503e10	0.941881
$986$	7.81333e9i	0.259577i
$987$	2.15071e10	0.711987
$988$	0	0
$989$	−6.07605e9	−0.199726
$990$	1.24417e11i	4.07528i
$991$	−4.84851e10	−1.58252	−0.791262	−	0.611477i	$-0.790576\pi$
$991$	−4.84851e10	−1.58252	−0.791262	+	0.611477i	$0.790576\pi$
$992$	−2.41553e10	−0.785635
$993$	− 4.08193e10i	− 1.32295i
$994$	− 1.45101e10i	− 0.468618i
$995$	2.14873e10i	0.691515i
$996$	2.02378e10i	0.649017i
$997$	2.19592e10	0.701752	0.350876	−	0.936422i	$-0.385884\pi$
$997$	2.19592e10	0.701752	0.350876	+	0.936422i	$0.385884\pi$
$998$	−1.25094e10	−0.398365
$999$	6.31082e10i	2.00266i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.d.168.2		14
13.5	odd	4		169.8.a.g.1.2		14
13.8	odd	4		169.8.a.g.1.13		14
13.9	even	3		13.8.e.a.10.1	yes	14
13.10	even	6		13.8.e.a.4.1	✓	14
13.12	even	2	inner	169.8.b.d.168.13		14
39.23	odd	6		117.8.q.b.82.7		14
39.35	odd	6		117.8.q.b.10.7		14
52.23	odd	6		208.8.w.a.17.1		14
52.35	odd	6		208.8.w.a.49.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.1	✓	14	13.10	even	6
13.8.e.a.10.1	yes	14	13.9	even	3
117.8.q.b.10.7		14	39.35	odd	6
117.8.q.b.82.7		14	39.23	odd	6
169.8.a.g.1.2		14	13.5	odd	4
169.8.a.g.1.13		14	13.8	odd	4
169.8.b.d.168.2		14	1.1	even	1	trivial
169.8.b.d.168.13		14	13.12	even	2	inner
208.8.w.a.17.1		14	52.23	odd	6
208.8.w.a.49.1		14	52.35	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-16.7657i q^{2} -81.0229 q^{3} -153.090 q^{4} +223.318i q^{5} +1358.41i q^{6} +666.907i q^{7} +420.649i q^{8} +4377.70 q^{9} +3744.09 q^{10} +7590.80i q^{11} +12403.8 q^{12} +11181.2 q^{14} -18093.9i q^{15} -12543.0 q^{16} +14102.9 q^{17} -73395.4i q^{18} +8343.98i q^{19} -34187.7i q^{20} -54034.7i q^{21} +127265. q^{22} -58145.3 q^{23} -34082.2i q^{24} +28254.0 q^{25} -177497. q^{27} -102097. i q^{28} -33045.0 q^{29} -303357. q^{30} +91450.2i q^{31} +264136. i q^{32} -615028. i q^{33} -236445. i q^{34} -148932. q^{35} -670182. q^{36} -355545. i q^{37} +139893. q^{38} -93938.5 q^{40} +582075. i q^{41} -905932. q^{42} +104498. q^{43} -1.16207e6i q^{44} +977620. i q^{45} +974848. i q^{46} +398024. i q^{47} +1.01627e6 q^{48} +378778. q^{49} -473700. i q^{50} -1.14266e6 q^{51} +1.49881e6 q^{53} +2.97587e6i q^{54} -1.69516e6 q^{55} -280534. q^{56} -676053. i q^{57} +554024. i q^{58} -330628. i q^{59} +2.76999e6i q^{60} -1.44273e6 q^{61} +1.53323e6 q^{62} +2.91952e6i q^{63} +2.82292e6 q^{64} -1.03114e7 q^{66} +2.56680e6i q^{67} -2.15901e6 q^{68} +4.71110e6 q^{69} +2.49696e6i q^{70} -1.29773e6i q^{71} +1.84148e6i q^{72} -2.66195e6i q^{73} -5.96097e6 q^{74} -2.28922e6 q^{75} -1.27738e6i q^{76} -5.06236e6 q^{77} +2.13906e6 q^{79} -2.80108e6i q^{80} +4.80728e6 q^{81} +9.75892e6 q^{82} +1.63158e6i q^{83} +8.27217e6i q^{84} +3.14943e6i q^{85} -1.75198e6i q^{86} +2.67740e6 q^{87} -3.19306e6 q^{88} -401206. i q^{89} +1.63905e7 q^{90} +8.90145e6 q^{92} -7.40955e6i q^{93} +6.67316e6 q^{94} -1.86336e6 q^{95} -2.14010e7i q^{96} +1.10423e7i q^{97} -6.35048e6i q^{98} +3.32303e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 52 q^{3} - 766 q^{4} + 6982 q^{9} + 1018 q^{10} + 38380 q^{12} - 47916 q^{14} + 1266 q^{16} + 76806 q^{17} + 251764 q^{22} + 137100 q^{23} + 39380 q^{25} - 432400 q^{27} - 443166 q^{29} + 315780 q^{30}+ \cdots + 86840772 q^{95}+O(q^{100}) \) 14 * q - 52 * q^3 - 766 * q^4 + 6982 * q^9 + 1018 * q^10 + 38380 * q^12 - 47916 * q^14 + 1266 * q^16 + 76806 * q^17 + 251764 * q^22 + 137100 * q^23 + 39380 * q^25 - 432400 * q^27 - 443166 * q^29 + 315780 * q^30 - 1319232 * q^35 - 1271350 * q^36 + 953640 * q^38 - 77182 * q^40 - 412812 * q^42 + 1765384 * q^43 - 2267260 * q^48 - 5307958 * q^49 - 38964 * q^51 - 1791210 * q^53 - 4907072 * q^55 + 14915904 * q^56 - 7717050 * q^61 - 3808284 * q^62 + 14570922 * q^64 - 31956396 * q^66 - 19139238 * q^68 + 27487680 * q^69 - 17280210 * q^74 - 20632304 * q^75 + 19599336 * q^77 - 30688440 * q^79 - 27772442 * q^81 + 74719438 * q^82 - 25539372 * q^87 - 54091072 * q^88 + 41193546 * q^90 - 47685588 * q^92 - 38277620 * q^94 + 86840772 * q^95

Introduction

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