LMFDB - Embedded newform 69.6.c.b.68.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,6,Mod(68,69)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	69.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.0664835671$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			68.22
Character	$\chi$	$=$	69.68
Dual form			69.6.c.b.68.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.78145i q^{2} +(-1.49341 - 15.5168i) q^{3} +24.2635 q^{4} +76.0392 q^{5} +(43.1591 - 4.15386i) q^{6} -165.966i q^{7} +156.494i q^{8} +(-238.539 + 46.3458i) q^{9} +O(q^{10})$	\(q+2.78145i q^{2} +(-1.49341 - 15.5168i) q^{3} +24.2635 q^{4} +76.0392 q^{5} +(43.1591 - 4.15386i) q^{6} -165.966i q^{7} +156.494i q^{8} +(-238.539 + 46.3458i) q^{9} +211.500i q^{10} -574.052 q^{11} +(-36.2354 - 376.491i) q^{12} +1110.26 q^{13} +461.627 q^{14} +(-113.558 - 1179.88i) q^{15} +341.150 q^{16} -321.247 q^{17} +(-128.909 - 663.487i) q^{18} -1721.25i q^{19} +1844.98 q^{20} +(-2575.25 + 247.855i) q^{21} -1596.70i q^{22} +(1966.43 - 1602.96i) q^{23} +(2428.29 - 233.710i) q^{24} +2656.97 q^{25} +3088.15i q^{26} +(1075.37 + 3632.15i) q^{27} -4026.92i q^{28} +3174.02i q^{29} +(3281.79 - 315.856i) q^{30} -2461.84 q^{31} +5956.71i q^{32} +(857.295 + 8907.42i) q^{33} -893.534i q^{34} -12619.9i q^{35} +(-5787.80 + 1124.51i) q^{36} -9387.89i q^{37} +4787.57 q^{38} +(-1658.08 - 17227.7i) q^{39} +11899.7i q^{40} +9359.87i q^{41} +(-689.399 - 7162.95i) q^{42} +7969.47i q^{43} -13928.5 q^{44} +(-18138.4 + 3524.10i) q^{45} +(4458.56 + 5469.54i) q^{46} -3859.46i q^{47} +(-509.478 - 5293.55i) q^{48} -10737.7 q^{49} +7390.23i q^{50} +(479.754 + 4984.71i) q^{51} +26938.9 q^{52} +3644.84 q^{53} +(-10102.6 + 2991.10i) q^{54} -43650.5 q^{55} +25972.7 q^{56} +(-26708.2 + 2570.53i) q^{57} -8828.40 q^{58} +11573.5i q^{59} +(-2755.31 - 28628.1i) q^{60} +48496.4i q^{61} -6847.48i q^{62} +(7691.83 + 39589.4i) q^{63} -5651.52 q^{64} +84423.7 q^{65} +(-24775.6 + 2384.53i) q^{66} +7847.72i q^{67} -7794.58 q^{68} +(-27809.4 - 28118.8i) q^{69} +35101.7 q^{70} +40892.6i q^{71} +(-7252.86 - 37330.1i) q^{72} +15152.0 q^{73} +26112.0 q^{74} +(-3967.94 - 41227.5i) q^{75} -41763.5i q^{76} +95273.1i q^{77} +(47918.1 - 4611.88i) q^{78} -18490.6i q^{79} +25940.8 q^{80} +(54753.1 - 22110.6i) q^{81} -26034.1 q^{82} -94071.5 q^{83} +(-62484.7 + 6013.84i) q^{84} -24427.4 q^{85} -22166.7 q^{86} +(49250.5 - 4740.12i) q^{87} -89835.9i q^{88} -32456.7 q^{89} +(-9802.12 - 50451.0i) q^{90} -184266. i q^{91} +(47712.6 - 38893.5i) q^{92} +(3676.53 + 38199.7i) q^{93} +10734.9 q^{94} -130882. i q^{95} +(92428.9 - 8895.82i) q^{96} +174638. i q^{97} -29866.4i q^{98} +(136934. - 26604.9i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) $ 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9	$ 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) $ 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9 - 2484 * q^12 + 520 * q^13 + 4936 * q^16 + 7188 * q^18 + 18660 * q^24 + 36032 * q^25 - 22032 * q^27 + 6544 * q^31 - 33912 * q^36 - 63912 * q^39 + 54328 * q^46 + 88284 * q^48 - 207664 * q^49 + 46296 * q^52 - 38628 * q^54 - 139296 * q^55 - 184144 * q^58 + 486584 * q^64 - 113580 * q^69 + 37176 * q^70 - 15504 * q^72 - 93896 * q^73 + 249840 * q^75 + 368028 * q^78 - 339372 * q^81 - 23512 * q^82 + 259584 * q^85 + 509928 * q^87 + 82740 * q^93 - 562000 * q^94 + 1404 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$			2.78145i			0.491696i	0.969308	+	0.245848i	$0.0790665\pi$
$2$			2.78145i			0.491696i	−0.969308	+	0.245848i	$0.920934\pi$
$3$	−1.49341	−	15.5168i	−0.0958024	−	0.995400i
$3$	−1.49341	−	15.5168i	−0.0958024	−	0.995400i
$4$	24.2635			0.758235
$5$	76.0392			1.36023			0.680116	−	0.733105i	$-0.261930\pi$
$5$	76.0392			1.36023			0.680116	+	0.733105i	$0.261930\pi$
$6$	43.1591	−	4.15386i	0.489435	−	0.0471057i
$7$		−	165.966i		−	1.28019i	−0.768296	−	0.640094i	$-0.778895\pi$
$7$		−	165.966i		−	1.28019i	0.768296	−	0.640094i	$-0.221105\pi$
$8$			156.494i			0.864518i
$9$	−238.539	+	46.3458i	−0.981644	+	0.190723i
$10$			211.500i			0.668821i
$11$	−574.052			−1.43044			−0.715220	−	0.698900i	$-0.753673\pi$
$11$	−574.052			−1.43044			−0.715220	+	0.698900i	$0.753673\pi$
$12$	−36.2354	−	376.491i	−0.0726407	−	0.754747i
$13$	1110.26			1.82208			0.911041	−	0.412315i	$-0.135280\pi$
$13$	1110.26			1.82208			0.911041	+	0.412315i	$0.135280\pi$
$14$	461.627			0.629464
$15$	−113.558	−	1179.88i	−0.130313	−	1.35397i
$16$	341.150			0.333155
$17$	−321.247			−0.269598			−0.134799	−	0.990873i	$-0.543039\pi$
$17$	−321.247			−0.269598			−0.134799	+	0.990873i	$0.543039\pi$
$18$	−128.909	−	663.487i	−0.0937780	−	0.482671i
$19$		−	1721.25i		−	1.09385i	−0.837180	−	0.546927i	$-0.815797\pi$
$19$		−	1721.25i		−	1.09385i	0.837180	−	0.546927i	$-0.184203\pi$
$20$	1844.98			1.03137
$21$	−2575.25	+	247.855i	−1.27430	+	0.122645i
$22$		−	1596.70i		−	0.703342i
$23$	1966.43	−	1602.96i	0.775103	−	0.631834i
$23$	1966.43	−	1602.96i	0.775103	−	0.631834i
$24$	2428.29	−	233.710i	0.860541	−	0.0828228i
$25$	2656.97			0.850229
$26$			3088.15i			0.895911i
$27$	1075.37	+	3632.15i	0.283890	+	0.958857i
$28$		−	4026.92i		−	0.970683i
$29$			3174.02i			0.700834i	0.936594	+	0.350417i	$0.113960\pi$
$29$			3174.02i			0.700834i	−0.936594	+	0.350417i	$0.886040\pi$
$30$	3281.79	−	315.856i	0.665744	−	0.0640746i
$31$	−2461.84			−0.460103			−0.230051	−	0.973178i	$-0.573889\pi$
$31$	−2461.84			−0.460103			−0.230051	+	0.973178i	$0.573889\pi$
$32$			5956.71i			1.02833i
$33$	857.295	+	8907.42i	0.137039	+	1.42386i
$34$		−	893.534i		−	0.132560i
$35$		−	12619.9i		−	1.74135i
$36$	−5787.80	+	1124.51i	−0.744316	+	0.144613i
$37$		−	9387.89i		−	1.12736i	−0.825992	−	0.563682i	$-0.809384\pi$
$37$		−	9387.89i		−	1.12736i	0.825992	−	0.563682i	$-0.190616\pi$
$38$	4787.57			0.537844
$39$	−1658.08	−	17227.7i	−0.174560	−	1.81370i
$40$			11899.7i			1.17594i
$41$			9359.87i			0.869582i	0.900531	+	0.434791i	$0.143178\pi$
$41$			9359.87i			0.869582i	−0.900531	+	0.434791i	$0.856822\pi$
$42$	−689.399	−	7162.95i	−0.0603041	−	0.626569i
$43$			7969.47i			0.657292i	0.944453	+	0.328646i	$0.106592\pi$
$43$			7969.47i			0.657292i	−0.944453	+	0.328646i	$0.893408\pi$
$44$	−13928.5			−1.08461
$45$	−18138.4	+	3524.10i	−1.33526	+	0.259428i
$46$	4458.56	+	5469.54i	0.310671	+	0.381115i
$47$		−	3859.46i		−	0.254849i	−0.991848	−	0.127424i	$-0.959329\pi$
$47$		−	3859.46i		−	0.254849i	0.991848	−	0.127424i	$-0.0406710\pi$
$48$	−509.478	−	5293.55i	−0.0319170	−	0.331622i
$49$	−10737.7			−0.638883
$50$			7390.23i			0.418055i
$51$	479.754	+	4984.71i	0.0258281	+	0.268358i
$52$	26938.9			1.38157
$53$	3644.84			0.178233			0.0891166	−	0.996021i	$-0.471596\pi$
$53$	3644.84			0.178233			0.0891166	+	0.996021i	$0.471596\pi$
$54$	−10102.6	+	2991.10i	−0.471466	+	0.139588i
$55$	−43650.5			−1.94573
$56$	25972.7			1.10675
$57$	−26708.2	+	2570.53i	−1.08882	+	0.104794i
$58$	−8828.40			−0.344597
$59$			11573.5i			0.432848i	0.976299	+	0.216424i	$0.0694394\pi$
$59$			11573.5i			0.432848i	−0.976299	+	0.216424i	$0.930561\pi$
$60$	−2755.31	−	28628.1i	−0.0988081	−	1.02663i
$61$			48496.4i			1.66873i	0.551215	+	0.834364i	$0.314165\pi$
$61$			48496.4i			1.66873i	−0.551215	+	0.834364i	$0.685835\pi$
$62$		−	6847.48i		−	0.226231i
$63$	7691.83	+	39589.4i	0.244162	+	1.25669i
$64$	−5651.52			−0.172471
$65$	84423.7			2.47845
$66$	−24775.6	+	2384.53i	−0.700107	+	0.0673818i
$67$			7847.72i			0.213578i	0.994282	+	0.106789i	$0.0340569\pi$
$67$			7847.72i			0.213578i	−0.994282	+	0.106789i	$0.965943\pi$
$68$	−7794.58			−0.204419
$69$	−27809.4	−	28118.8i	−0.703185	−	0.711007i
$70$	35101.7			0.856217
$71$			40892.6i			0.962716i	0.876524	+	0.481358i	$0.159856\pi$
$71$			40892.6i			0.962716i	−0.876524	+	0.481358i	$0.840144\pi$
$72$	−7252.86	−	37330.1i	−0.164884	−	0.848648i
$73$	15152.0			0.332784			0.166392	−	0.986060i	$-0.446788\pi$
$73$	15152.0			0.332784			0.166392	+	0.986060i	$0.446788\pi$
$74$	26112.0			0.554320
$75$	−3967.94	−	41227.5i	−0.0814540	−	0.846319i
$76$		−	41763.5i		−	0.829398i
$77$			95273.1i			1.83123i
$78$	47918.1	−	4611.88i	0.891790	−	0.0858304i
$79$		−	18490.6i		−	0.333337i	−0.986013	−	0.166669i	$-0.946699\pi$
$79$		−	18490.6i		−	0.333337i	0.986013	−	0.166669i	$-0.0533010\pi$
$80$	25940.8			0.453167
$81$	54753.1	−	22110.6i	0.927249	−	0.374445i
$82$	−26034.1			−0.427570
$83$	−94071.5			−1.49887			−0.749433	−	0.662080i	$-0.769674\pi$
$83$	−94071.5			−1.49887			−0.749433	+	0.662080i	$0.769674\pi$
$84$	−62484.7	+	6013.84i	−0.966219	+	0.0929938i
$85$	−24427.4			−0.366716
$86$	−22166.7			−0.323188
$87$	49250.5	−	4740.12i	0.697610	−	0.0671415i
$88$		−	89835.9i		−	1.23664i
$89$	−32456.7			−0.434339			−0.217169	−	0.976134i	$-0.569682\pi$
$89$	−32456.7			−0.434339			−0.217169	+	0.976134i	$0.569682\pi$
$90$	−9802.12	−	50451.0i	−0.127560	−	0.656544i
$91$		−	184266.i		−	2.33261i
$92$	47712.6	−	38893.5i	0.587710	−	0.479079i
$93$	3676.53	+	38199.7i	0.0440789	+	0.457986i
$94$	10734.9			0.125308
$95$		−	130882.i		−	1.48789i
$96$	92428.9	−	8895.82i	1.02360	−	0.0985163i
$97$			174638.i			1.88456i	0.334823	+	0.942281i	$0.391323\pi$
$97$			174638.i			1.88456i	−0.334823	+	0.942281i	$0.608677\pi$
$98$		−	29866.4i		−	0.314136i
$99$	136934.	−	26604.9i	1.40418	−	0.272818i
$100$	64467.3			0.644673
$101$		−	46377.2i		−	0.452378i	−0.974083	−	0.226189i	$-0.927373\pi$
$101$		−	46377.2i		−	0.452378i	0.974083	−	0.226189i	$-0.0726266\pi$
$102$	−13864.7	+	1334.41i	−0.131951	+	0.0126996i
$103$			205465.i			1.90829i	0.299345	+	0.954145i	$0.403232\pi$
$103$			205465.i			1.90829i	−0.299345	+	0.954145i	$0.596768\pi$
$104$			173750.i			1.57522i
$105$	−195820.	+	18846.7i	−1.73334	+	0.166826i
$106$			10137.9i			0.0876366i
$107$	92647.1			0.782298			0.391149	−	0.920327i	$-0.372078\pi$
$107$	92647.1			0.782298			0.391149	+	0.920327i	$0.372078\pi$
$108$	26092.3	+	88128.6i	0.215255	+	0.727039i
$109$		−	32257.7i		−	0.260056i	−0.991510	−	0.130028i	$-0.958493\pi$
$109$		−	32257.7i		−	0.260056i	0.991510	−	0.130028i	$-0.0415068\pi$
$110$		−	121412.i		−	0.956707i
$111$	−145670.	+	14020.0i	−1.12218	+	0.108004i
$112$		−	56619.4i		−	0.426501i
$113$	22682.9			0.167110			0.0835550	−	0.996503i	$-0.473373\pi$
$113$	22682.9			0.167110			0.0835550	+	0.996503i	$0.473373\pi$
$114$	−7149.81	−	74287.6i	−0.0515267	−	0.535370i
$115$	149526.	−	121888.i	1.05432	−	0.859441i
$116$			77012.9i			0.531396i
$117$	−264842.	+	51456.1i	−1.78864	+	0.347514i
$118$	−32191.2			−0.212830
$119$			53316.1i			0.345136i
$120$	184645.	−	17771.2i	1.17053	−	0.112658i
$121$	168485.			1.04616
$122$	−134891.			−0.820507
$123$	145235.	−	13978.1i	0.865582	−	0.0833080i
$124$	−59732.8			−0.348866
$125$	−35588.9			−0.203723
$126$	−110116.	+	21394.5i	−0.617909	+	0.120054i
$127$	−186503.			−1.02607			−0.513034	−	0.858368i	$-0.671479\pi$
$127$	−186503.			−1.02607			−0.513034	+	0.858368i	$0.671479\pi$
$128$			174895.i			0.943525i
$129$	123660.	−	11901.7i	0.654269	−	0.0629701i
$130$			234821.i			1.21865i
$131$		−	237832.i		−	1.21086i	−0.795900	−	0.605428i	$-0.793002\pi$
$131$		−	237832.i		−	1.21086i	0.795900	−	0.605428i	$-0.206998\pi$
$132$	20801.0	+	216125.i	0.103908	+	1.07962i
$133$	−285669.			−1.40034
$134$	−21828.1			−0.105015
$135$	81770.6	+	276186.i	0.386156	+	1.30427i
$136$		−	50273.3i		−	0.233072i
$137$	51130.2			0.232743			0.116371	−	0.993206i	$-0.462874\pi$
$137$	51130.2			0.232743			0.116371	+	0.993206i	$0.462874\pi$
$138$	78211.1	−	77350.7i	0.349599	−	0.345753i
$139$	320539.			1.40716			0.703580	−	0.710616i	$-0.251584\pi$
$139$	320539.			1.40716			0.703580	+	0.710616i	$0.251584\pi$
$140$		−	306204.i		−	1.32035i
$141$	−59886.4	+	5763.77i	−0.253677	+	0.0244151i
$142$	−113741.			−0.473364
$143$	−637349.			−2.60638
$144$	−81377.8	+	15810.9i	−0.327039	+	0.0635404i
$145$			241350.i			0.953296i
$146$			42144.6i			0.163629i
$147$	16035.8	+	166614.i	0.0612065	+	0.635944i
$148$		−	227783.i		−	0.854806i
$149$	279356.			1.03084			0.515422	−	0.856936i	$-0.327635\pi$
$149$	279356.			1.03084			0.515422	+	0.856936i	$0.327635\pi$
$150$	114672.	−	11036.7i	0.416132	−	0.0400506i
$151$	−287315.			−1.02545			−0.512726	−	0.858552i	$-0.671364\pi$
$151$	−287315.			−1.02545			−0.512726	+	0.858552i	$0.671364\pi$
$152$	269366.			0.945656
$153$	76630.1	−	14888.4i	0.264649	−	0.0514187i
$154$	−264998.			−0.900410
$155$	−187196.			−0.625846
$156$	−40230.9	−	418005.i	−0.132357	−	1.37521i
$157$		−	144607.i		−	0.468211i	−0.972211	−	0.234105i	$-0.924784\pi$
$157$		−	144607.i		−	0.468211i	0.972211	−	0.234105i	$-0.0752161\pi$
$158$	51430.8			0.163901
$159$	−5443.24	−	56556.1i	−0.0170752	−	0.177413i
$160$			452944.i			1.39876i
$161$	−266037.	−	326361.i	−0.808867	−	0.992278i
$162$	61499.6	+	152293.i	0.184113	+	0.455925i
$163$	−247355.			−0.729207			−0.364604	−	0.931163i	$-0.618796\pi$
$163$	−247355.			−0.729207			−0.364604	+	0.931163i	$0.618796\pi$
$164$			227103.i			0.659347i
$165$	65188.1	+	677314.i	0.186405	+	1.93678i
$166$		−	261656.i		−	0.736987i
$167$			491092.i			1.36261i	0.732000	+	0.681305i	$0.238587\pi$
$167$			491092.i			1.36261i	−0.732000	+	0.681305i	$0.761413\pi$
$168$	−38788.0	−	403013.i	−0.106029	−	1.10165i
$169$	861394.			2.31999
$170$		−	67943.6i		−	0.180313i
$171$	79772.6	+	410586.i	0.208624	+	1.07378i
$172$			193367.i			0.498382i
$173$		−	217004.i		−	0.551254i	−0.961265	−	0.275627i	$-0.911115\pi$
$173$		−	217004.i		−	0.551254i	0.961265	−	0.275627i	$-0.0888854\pi$
$174$	13184.4	+	136988.i	0.0330132	+	0.343012i
$175$		−	440966.i		−	1.08845i
$176$	−195838.			−0.476558
$177$	179584.	−	17284.0i	0.430857	−	0.0414679i
$178$		−	90276.7i		−	0.213563i
$179$		−	213371.i		−	0.497740i	−0.968537	−	0.248870i	$-0.919941\pi$
$179$		−	213371.i		−	0.497740i	0.968537	−	0.248870i	$-0.0800592\pi$
$180$	−440100.	+	85507.0i	−1.01244	+	0.196707i
$181$		−	482595.i		−	1.09493i	−0.836829	−	0.547465i	$-0.815593\pi$
$181$		−	482595.i		−	1.09493i	0.836829	−	0.547465i	$-0.184407\pi$
$182$	512528.			1.14694
$183$	752507.	−	72425.1i	1.66105	−	0.159868i
$184$	250854.	+	307736.i	0.546232	+	0.670090i
$185$		−	713848.i		−	1.53347i
$186$	−106251.	+	10226.1i	−0.225190	+	0.0216734i
$187$	184412.			0.385644
$188$		−	93644.2i		−	0.193235i
$189$	602812.	−	178475.i	1.22752	−	0.363433i
$190$	364043.			0.731592
$191$	−627925.			−1.24545			−0.622723	−	0.782443i	$-0.713974\pi$
$191$	−627925.			−1.24545			−0.622723	+	0.782443i	$0.713974\pi$
$192$	8440.04	+	87693.2i	0.0165231	+	0.171677i
$193$	−392357.			−0.758207			−0.379104	−	0.925354i	$-0.623768\pi$
$193$	−392357.			−0.758207			−0.379104	+	0.925354i	$0.623768\pi$
$194$	−485749.			−0.926632
$195$	−126079.	−	1.30998e6i	−0.237442	−	2.46705i
$196$	−260534.			−0.484423
$197$		−	646521.i		−	1.18691i	−0.804868	−	0.593454i	$-0.797764\pi$
$197$		−	646521.i		−	1.18691i	0.804868	−	0.593454i	$-0.202236\pi$
$198$	74000.3	+	380876.i	0.134144	+	0.690431i
$199$		−	774298.i		−	1.38604i	−0.720919	−	0.693020i	$-0.756280\pi$
$199$		−	774298.i		−	1.38604i	0.720919	−	0.693020i	$-0.243720\pi$
$200$			415800.i			0.735038i
$201$	121771.	−	11719.9i	0.212596	−	0.0204613i
$202$	128996.			0.222432
$203$	526780.			0.897199
$204$	11640.5	+	120947.i	0.0195838	+	0.203478i
$205$			711718.i			1.18283i
$206$	−571491.			−0.938299
$207$	−394781.	+	473505.i	−0.640370	+	0.768067i
$208$	378767.			0.607035
$209$			988085.i			1.56469i
$210$	−52421.3	−	544665.i	−0.0820276	−	0.852278i
$211$	638399.			0.987157			0.493578	−	0.869701i	$-0.335689\pi$
$211$	638399.			0.987157			0.493578	+	0.869701i	$0.335689\pi$
$212$	88436.6			0.135143
$213$	634520.	−	61069.4i	0.958288	−	0.0922305i
$214$			257694.i			0.384653i
$215$			605993.i			0.894069i
$216$	−568410.	+	168290.i	−0.828949	+	0.245428i
$217$			408581.i			0.589018i
$218$	89723.3			0.127869
$219$	−22628.1	−	235110.i	−0.0318815	−	0.331253i
$220$	−1.05911e6			−1.47532
$221$	−356669.			−0.491230
$222$	−38996.0	−	405174.i	−0.0531052	−	0.551771i
$223$	116763.			0.157232			0.0786161	−	0.996905i	$-0.474950\pi$
$223$	116763.			0.157232			0.0786161	+	0.996905i	$0.474950\pi$
$224$	988612.			1.31645
$225$	−633791.	+	123139.i	−0.834622	+	0.162159i
$226$			63091.4i			0.0821674i
$227$	881668.			1.13564			0.567820	−	0.823153i	$-0.307787\pi$
$227$	881668.			1.13564			0.567820	+	0.823153i	$0.307787\pi$
$228$	−648034.	+	62370.1i	−0.825583	+	0.0794583i
$229$			148888.i			0.187616i	0.995590	+	0.0938079i	$0.0299040\pi$
$229$			148888.i			0.187616i	−0.995590	+	0.0938079i	$0.970096\pi$
$230$	339026.	+	415900.i	0.422584	+	0.518405i
$231$	1.47833e6	−	142282.i	1.82281	−	0.175436i
$232$	−496717.			−0.605883
$233$		−	737995.i		−	0.890561i	−0.895391	−	0.445281i	$-0.853104\pi$
$233$		−	737995.i		−	0.890561i	0.895391	−	0.445281i	$-0.146896\pi$
$234$	−143123.	−	736646.i	−0.170871	−	0.879466i
$235$		−	293471.i		−	0.346653i
$236$			280814.i			0.328201i
$237$	−286915.	+	27614.1i	−0.331804	+	0.0319345i
$238$	−148296.			−0.169702
$239$			451607.i			0.511406i	0.966755	+	0.255703i	$0.0823069\pi$
$239$			451607.i			0.511406i	−0.966755	+	0.255703i	$0.917693\pi$
$240$	−38740.3	−	402517.i	−0.0434145	−	0.451083i
$241$			164088.i			0.181984i	0.995852	+	0.0909922i	$0.0290038\pi$
$241$			164088.i			0.181984i	−0.995852	+	0.0909922i	$0.970996\pi$
$242$			468632.i			0.514391i
$243$	−424854.	−	816571.i	−0.461555	−	0.887111i
$244$			1.17669e6i			1.26529i
$245$	−816487.			−0.869028
$246$	38879.6	+	403964.i	0.0409622	+	0.425603i
$247$		−	1.91104e6i		−	1.99309i
$248$		−	385263.i		−	0.397767i
$249$	140487.	+	1.45969e6i	0.143595	+	1.49197i
$250$		−	98988.9i		−	0.100170i
$251$	−51795.9			−0.0518933			−0.0259467	−	0.999663i	$-0.508260\pi$
$251$	−51795.9			−0.0518933			−0.0259467	+	0.999663i	$0.508260\pi$
$252$	186631.	+	960579.i	0.185132	+	0.952865i
$253$	−1.12883e6	+	920183.i	−1.10874	+	0.903801i
$254$		−	518749.i		−	0.504514i
$255$	36480.1	+	379034.i	0.0351322	+	0.365029i
$256$	−667312.			−0.636398
$257$		−	494796.i		−	0.467297i	−0.972321	−	0.233649i	$-0.924933\pi$
$257$		−	494796.i		−	0.467297i	0.972321	−	0.233649i	$-0.0750665\pi$
$258$	33104.0	+	343956.i	0.0309622	+	0.321701i
$259$	−1.55807e6			−1.44324
$260$	2.04841e6			1.87925
$261$	−147103.	−	757129.i	−0.133665	−	0.687969i
$262$	661520.			0.595374
$263$	521212.			0.464649			0.232324	−	0.972638i	$-0.425367\pi$
$263$	521212.			0.464649			0.232324	+	0.972638i	$0.425367\pi$
$264$	−1.39396e6	+	134162.i	−1.23095	+	0.118473i
$265$	277151.			0.242438
$266$		−	794574.i		−	0.688542i
$267$	48471.1	+	503622.i	0.0416107	+	0.432341i
$268$			190413.i			0.161942i
$269$			1.39571e6i			1.17602i	0.808853	+	0.588011i	$0.200089\pi$
$269$			1.39571e6i			1.17602i	−0.808853	+	0.588011i	$0.799911\pi$
$270$	−768198.	+	227441.i	−0.641303	+	0.189872i
$271$	1.91089e6			1.58056			0.790282	−	0.612743i	$-0.209934\pi$
$271$	1.91089e6			1.58056			0.790282	+	0.612743i	$0.209934\pi$
$272$	−109594.			−0.0898178
$273$	−2.85921e6	+	275185.i	−2.32188	+	0.223470i
$274$			142216.i			0.114439i
$275$	−1.52524e6			−1.21620
$276$	−674755.	−	682260.i	−0.533179	−	0.539110i
$277$	−1.45745e6			−1.14128			−0.570641	−	0.821199i	$-0.693305\pi$
$277$	−1.45745e6			−1.14128			−0.570641	+	0.821199i	$0.693305\pi$
$278$			891563.i			0.691895i
$279$	587245.	−	114096.i	0.451657	−	0.0877524i
$280$	1.97495e6			1.50543
$281$	−474943.			−0.358820			−0.179410	−	0.983774i	$-0.557419\pi$
$281$	−474943.			−0.358820			−0.179410	+	0.983774i	$0.557419\pi$
$282$	−16031.7	−	166571.i	−0.0120048	−	0.124732i
$283$			1.67445e6i			1.24281i	0.783489	+	0.621406i	$0.213438\pi$
$283$			1.67445e6i			1.24281i	−0.783489	+	0.621406i	$0.786562\pi$
$284$			992197.i			0.729965i
$285$	−2.03087e6	+	195461.i	−1.48105	+	0.142544i
$286$		−	1.77276e6i		−	1.28155i
$287$	1.55342e6			1.11323
$288$	−276069.	−	1.42091e6i	−0.196126	−	1.00945i
$289$	−1.31666e6			−0.927317
$290$	−671305.			−0.468732
$291$	2.70982e6	−	260807.i	1.87589	−	0.180546i
$292$	367640.			0.252328
$293$	−1.23142e6			−0.837985			−0.418992	−	0.907990i	$-0.637617\pi$
$293$	−1.23142e6			−0.837985			−0.418992	+	0.907990i	$0.637617\pi$
$294$	−463430.	+	44602.9i	−0.312691	+	0.0300950i
$295$			880042.i			0.588774i
$296$	1.46915e6			0.974625
$297$	−617320.	−	2.08504e6i	−0.406087	−	1.37159i
$298$			777017.i			0.506862i
$299$	2.18326e6	−	1.77971e6i	1.41230	−	1.15125i
$300$	−96276.3	−	1.00032e6i	−0.0617612	−	0.641708i
$301$	1.32266e6			0.841458
$302$		−	799153.i		−	0.504211i
$303$	−719624.	+	69260.2i	−0.450297	+	0.0433388i
$304$		−	587204.i		−	0.364423i
$305$			3.68763e6i			2.26985i
$306$	41411.5	+	213143.i	0.0252824	+	0.130127i
$307$	1.81429e6			1.09865			0.549327	−	0.835608i	$-0.314884\pi$
$307$	1.81429e6			1.09865			0.549327	+	0.835608i	$0.314884\pi$
$308$			2.31166e6i			1.38850i
$309$	3.18815e6	−	306843.i	1.89951	−	0.182819i
$310$		−	520677.i		−	0.307726i
$311$		−	2.18699e6i		−	1.28217i	−0.767470	−	0.641085i	$-0.778485\pi$
$311$		−	2.18699e6i		−	1.28217i	0.767470	−	0.641085i	$-0.221515\pi$
$312$	2.69604e6	−	259480.i	1.56798	−	0.150910i
$313$		−	840296.i		−	0.484810i	−0.970175	−	0.242405i	$-0.922064\pi$
$313$		−	840296.i		−	0.484810i	0.970175	−	0.242405i	$-0.0779362\pi$
$314$	402219.			0.230217
$315$	584881.	+	3.01035e6i	0.332117	+	1.70939i
$316$		−	448648.i		−	0.252748i
$317$		−	1.07362e6i		−	0.600070i	−0.953928	−	0.300035i	$-0.903002\pi$
$317$		−	1.07362e6i		−	0.600070i	0.953928	−	0.300035i	$-0.0969984\pi$
$318$	157308.	−	15140.1i	0.0872335	−	0.00839579i
$319$		−	1.82205e6i		−	1.00250i
$320$	−429737.			−0.234600
$321$	−138360.	−	1.43758e6i	−0.0749460	−	0.778700i
$322$	907758.	−	739970.i	0.487900	−	0.397717i
$323$			552946.i			0.294901i
$324$	1.32850e6	−	536481.i	0.703073	−	0.283917i
$325$	2.94994e6			1.54919
$326$		−	688006.i		−	0.358549i
$327$	−500535.	+	48174.0i	−0.258860	+	0.0249140i
$328$	−1.46477e6			−0.751769
$329$	−640540.			−0.326254
$330$	−1.88392e6	+	181318.i	−0.952307	+	0.0916548i
$331$	−65142.0			−0.0326807			−0.0163403	−	0.999866i	$-0.505202\pi$
$331$	−65142.0			−0.0326807			−0.0163403	+	0.999866i	$0.505202\pi$
$332$	−2.28251e6			−1.13649
$333$	435089.	+	2.23938e6i	0.215015	+	1.10667i
$334$	−1.36595e6			−0.669990
$335$			596734.i			0.290515i
$336$	−878549.	+	84556.0i	−0.424539	+	0.0408598i
$337$			1.72274e6i			0.826314i	0.910660	+	0.413157i	$0.135574\pi$
$337$			1.72274e6i			0.826314i	−0.910660	+	0.413157i	$0.864426\pi$
$338$			2.39593e6i			1.14073i
$339$	−33874.9	−	351965.i	−0.0160095	−	0.166341i
$340$	−592694.			−0.278057
$341$	1.41322e6			0.658149
$342$	−1.14202e6	+	221884.i	−0.527971	+	0.102579i
$343$		−	1.00730e6i		−	0.462298i
$344$	−1.24718e6			−0.568240
$345$	−2.11461e6	−	2.13813e6i	−0.956494	−	0.967134i
$346$	603586.			0.271049
$347$		−	2.35098e6i		−	1.04815i	−0.851671	−	0.524077i	$-0.824411\pi$
$347$		−	2.35098e6i		−	1.04815i	0.851671	−	0.524077i	$-0.175589\pi$
$348$	1.19499e6	−	115012.i	0.528952	−	0.0509090i
$349$	−3.31460e6			−1.45669			−0.728346	−	0.685209i	$-0.759711\pi$
$349$	−3.31460e6			−1.45669			−0.728346	+	0.685209i	$0.759711\pi$
$350$	1.22653e6			0.535189
$351$	1.19395e6	+	4.03264e6i	0.517271	+	1.74712i
$352$		−	3.41946e6i		−	1.47096i
$353$		−	4.02537e6i		−	1.71937i	−0.510824	−	0.859685i	$-0.670660\pi$
$353$		−	4.02537e6i		−	1.71937i	0.510824	−	0.859685i	$-0.329340\pi$
$354$	48074.8	+	499504.i	0.0203896	+	0.211851i
$355$			3.10944e6i			1.30952i
$356$	−787512.			−0.329331
$357$	827292.	−	79622.8i	0.343549	−	0.0330649i
$358$	593481.			0.244737
$359$	−1.83829e6			−0.752798			−0.376399	−	0.926458i	$-0.622838\pi$
$359$	−1.83829e6			−0.752798			−0.376399	+	0.926458i	$0.622838\pi$
$360$	−551502.	−	2.83855e6i	−0.224280	−	1.15436i
$361$	−486595.			−0.196517
$362$	1.34232e6			0.538373
$363$	−251617.	−	2.61433e6i	−0.100224	−	1.04134i
$364$		−	4.47094e6i		−	1.76867i
$365$	1.15215e6			0.452663
$366$	201447.	+	2.09307e6i	0.0786065	+	0.816733i
$367$		−	2.12310e6i		−	0.822821i	−0.911450	−	0.411411i	$-0.865036\pi$
$367$		−	2.12310e6i		−	0.822821i	0.911450	−	0.411411i	$-0.134964\pi$
$368$	670850.	−	546851.i	0.258229	−	0.210499i
$369$	−433791.	−	2.23270e6i	−0.165850	−	0.853620i
$370$	1.98554e6			0.754004
$371$		−	604919.i		−	0.228172i
$372$	89205.6	+	926859.i	0.0334222	+	0.347261i
$373$			2.60971e6i			0.971225i	0.874174	+	0.485612i	$0.161403\pi$
$373$			2.60971e6i			0.971225i	−0.874174	+	0.485612i	$0.838597\pi$
$374$			512935.i			0.189620i
$375$	53148.9	+	552224.i	0.0195171	+	0.202786i
$376$	603985.			0.220321
$377$			3.52400e6i			1.27698i
$378$	496421.	+	1.67670e6i	0.178699	+	0.603566i
$379$		−	1.09232e6i		−	0.390619i	−0.980742	−	0.195309i	$-0.937429\pi$
$379$		−	1.09232e6i		−	0.390619i	0.980742	−	0.195309i	$-0.0625711\pi$
$380$		−	3.17567e6i		−	1.12817i
$381$	278526.	+	2.89392e6i	0.0982998	+	1.02135i
$382$		−	1.74655e6i		−	0.612381i
$383$	−4.66060e6			−1.62347			−0.811736	−	0.584025i	$-0.801477\pi$
$383$	−4.66060e6			−1.62347			−0.811736	+	0.584025i	$0.801477\pi$
$384$	2.71381e6	−	261191.i	0.939185	−	0.0903920i
$385$			7.24449e6i			2.49090i
$386$		−	1.09132e6i		−	0.372808i
$387$	−369351.	−	1.90103e6i	−0.125361	−	0.645227i
$388$			4.23734e6i			1.42894i
$389$	1.29105e6			0.432582			0.216291	−	0.976329i	$-0.430604\pi$
$389$	1.29105e6			0.432582			0.216291	+	0.976329i	$0.430604\pi$
$390$	3.64365e6	−	350684.i	1.21304	−	0.116749i
$391$	−631711.	+	514946.i	−0.208966	+	0.170341i
$392$		−	1.68039e6i		−	0.552325i
$393$	−3.69039e6	+	355181.i	−1.20529	+	0.116003i
$394$	1.79827e6			0.583598
$395$		−	1.40601e6i		−	0.453416i
$396$	3.32250e6	−	645528.i	1.06470	−	0.206860i
$397$	−4.97421e6			−1.58397			−0.791986	−	0.610539i	$-0.790953\pi$
$397$	−4.97421e6			−1.58397			−0.791986	+	0.610539i	$0.790953\pi$
$398$	2.15368e6			0.681511
$399$	426621.	+	4.43265e6i	0.134156	+	1.39390i
$400$	906425.			0.283258
$401$	−4.21550e6			−1.30914			−0.654572	−	0.755999i	$-0.727151\pi$
$401$	−4.21550e6			−1.30914			−0.654572	+	0.755999i	$0.727151\pi$
$402$	32598.3	+	338701.i	0.0100607	+	0.104532i
$403$	−2.73329e6			−0.838345
$404$		−	1.12527e6i		−	0.343008i
$405$	4.16339e6	−	1.68127e6i	1.26127	−	0.509332i
$406$			1.46521e6i			0.441149i
$407$			5.38914e6i			1.61262i
$408$	−780079.	+	75078.8i	−0.232000	+	0.0223289i
$409$	−15950.8			−0.00471491			−0.00235745	−	0.999997i	$-0.500750\pi$
$409$	−15950.8			−0.00471491			−0.00235745	+	0.999997i	$0.500750\pi$
$410$	−1.97961e6			−0.581594
$411$	−76358.5	−	793375.i	−0.0222973	−	0.231672i
$412$			4.98530e6i			1.44693i
$413$	1.92081e6			0.554127
$414$	−1.31703e6	−	1.09807e6i	−0.377656	−	0.314867i
$415$	−7.15313e6			−2.03881
$416$			6.61353e6i			1.87370i
$417$	−478696.	−	4.97372e6i	−0.134809	−	1.40069i
$418$	−2.74831e6			−0.769353
$419$	−3.60482e6			−1.00311			−0.501555	−	0.865126i	$-0.667238\pi$
$419$	−3.60482e6			−1.00311			−0.501555	+	0.865126i	$0.667238\pi$
$420$	−4.75129e6	+	457288.i	−1.31428	+	0.126493i
$421$		−	1.62475e6i		−	0.446766i	−0.974731	−	0.223383i	$-0.928290\pi$
$421$		−	1.62475e6i		−	0.446766i	0.974731	−	0.223383i	$-0.0717101\pi$
$422$			1.77568e6i			0.485381i
$423$	178870.	+	920635.i	0.0486056	+	0.250171i
$424$			570397.i			0.154086i
$425$	−853542.			−0.229220
$426$	169862.	+	1.76489e6i	0.0453494	+	0.471187i
$427$	8.04876e6			2.13629
$428$	2.24794e6			0.593166
$429$	951825.	+	9.88959e6i	0.249697	+	2.59439i
$430$	−1.68554e6			−0.439610
$431$	2.38418e6			0.618224			0.309112	−	0.951026i	$-0.399968\pi$
$431$	2.38418e6			0.618224			0.309112	+	0.951026i	$0.399968\pi$
$432$	366864.	+	1.23911e6i	0.0945793	+	0.319448i
$433$			3.23730e6i			0.829781i	0.909871	+	0.414890i	$0.136180\pi$
$433$			3.23730e6i			0.829781i	−0.909871	+	0.414890i	$0.863820\pi$
$434$	−1.13645e6			−0.289618
$435$	3.74497e6	−	360435.i	0.948911	−	0.0913280i
$436$		−	782685.i		−	0.197184i
$437$	−2.75909e6	−	3.38472e6i	−0.691135	−	0.847850i
$438$	653947.	−	62939.2i	0.162876	−	0.0156760i
$439$	5.19931e6			1.28761			0.643805	−	0.765190i	$-0.277355\pi$
$439$	5.19931e6			1.28761			0.643805	+	0.765190i	$0.277355\pi$
$440$		−	6.83105e6i		−	1.68212i
$441$	2.56137e6	−	497647.i	0.627155	−	0.121850i
$442$		−	992059.i		−	0.241536i
$443$		−	1.58760e6i		−	0.384355i	−0.981360	−	0.192177i	$-0.938445\pi$
$443$		−	1.58760e6i		−	0.384355i	0.981360	−	0.192177i	$-0.0615549\pi$
$444$	−3.53446e6	+	340174.i	−0.850874	+	0.0818924i
$445$	−2.46798e6			−0.590801
$446$			324770.i			0.0773105i
$447$	−417194.	−	4.33471e6i	−0.0987573	−	1.02610i
$448$			937959.i			0.220795i
$449$			1.24574e6i			0.291617i	0.989313	+	0.145809i	$0.0465784\pi$
$449$			1.24574e6i			0.291617i	−0.989313	+	0.145809i	$0.953422\pi$
$450$	−342506.	−	1.76286e6i	−0.0797328	−	0.410381i
$451$		−	5.37305e6i		−	1.24388i
$452$	550367.			0.126709
$453$	429079.	+	4.45819e6i	0.0982408	+	1.02074i
$454$			2.45232e6i			0.558390i
$455$		−	1.40115e7i		−	3.17289i
$456$	−402274.	−	4.17968e6i	−0.0905961	−	0.941306i
$457$			1.71273e6i			0.383618i	0.981432	+	0.191809i	$0.0614354\pi$
$457$			1.71273e6i			0.383618i	−0.981432	+	0.191809i	$0.938565\pi$
$458$	−414124.			−0.0922500
$459$	−345461.	−	1.16682e6i	−0.0765362	−	0.258506i
$460$	3.62803e6	−	2.95743e6i	0.799422	−	0.651658i
$461$		−	4.35189e6i		−	0.953731i	−0.878976	−	0.476866i	$-0.841773\pi$
$461$		−	4.35189e6i		−	0.953731i	0.878976	−	0.476866i	$-0.158227\pi$
$462$	395751.	+	4.11190e6i	0.0862614	+	0.896268i
$463$	3.48868e6			0.756326			0.378163	−	0.925739i	$-0.376556\pi$
$463$	3.48868e6			0.756326			0.378163	+	0.925739i	$0.376556\pi$
$464$			1.08282e6i			0.233486i
$465$	279561.	+	2.90468e6i	0.0599575	+	0.622967i
$466$	2.05270e6			0.437886
$467$	−7.88255e6			−1.67253			−0.836266	−	0.548324i	$-0.815266\pi$
$467$	−7.88255e6			−1.67253			−0.836266	+	0.548324i	$0.815266\pi$
$468$	−6.42599e6	+	1.24851e6i	−1.35621	+	0.263497i
$469$	1.30245e6			0.273420
$470$	816276.			0.170448
$471$	−2.24384e6	+	215958.i	−0.466057	+	0.0448557i
$472$	−1.81119e6			−0.374205
$473$		−	4.57489e6i		−	0.940216i
$474$	−76807.4	−	798040.i	−0.0157021	−	0.163147i
$475$		−	4.57330e6i		−	0.930027i
$476$			1.29363e6i			0.261694i
$477$	−869438.	+	168923.i	−0.174961	+	0.0339932i
$478$	−1.25612e6			−0.251456
$479$	768753.			0.153091			0.0765453	−	0.997066i	$-0.475611\pi$
$479$	768753.			0.153091			0.0765453	+	0.997066i	$0.475611\pi$
$480$	7.02822e6	−	676432.i	1.39233	−	0.134005i
$481$		−	1.04230e7i		−	2.05415i
$482$	−456403.			−0.0894811
$483$	−4.66676e6	+	4.61542e6i	−0.910223	+	0.900209i
$484$	4.08803e6			0.793232
$485$			1.32794e7i			2.56344i
$486$	2.27125e6	−	1.18171e6i	0.436189	−	0.226945i
$487$	−3.02694e6			−0.578338			−0.289169	−	0.957278i	$-0.593379\pi$
$487$	−3.02694e6			−0.578338			−0.289169	+	0.957278i	$0.593379\pi$
$488$	−7.58942e6			−1.44264
$489$	369402.	+	3.83814e6i	0.0698598	+	0.725853i
$490$		−	2.27102e6i		−	0.427298i
$491$			9.30511e6i			1.74188i	0.491392	+	0.870939i	$0.336488\pi$
$491$			9.30511e6i			1.74188i	−0.491392	+	0.870939i	$0.663512\pi$
$492$	3.52391e6	−	339159.i	0.656314	−	0.0631670i
$493$		−	1.01964e6i		−	0.188943i
$494$	5.31547e6			0.979996
$495$	1.04124e7	−	2.02302e6i	1.91001	−	0.371096i
$496$	−839856.			−0.153285
$497$	6.78677e6			1.23246
$498$	−4.06005e6	+	390760.i	−0.733597	+	0.0706051i
$499$	858866.			0.154409			0.0772047	−	0.997015i	$-0.475400\pi$
$499$	858866.			0.154409			0.0772047	+	0.997015i	$0.475400\pi$
$500$	−863512.			−0.154470
$501$	7.62015e6	−	733402.i	1.35634	−	0.130541i
$502$		−	144068.i		−	0.0255158i
$503$	4.24325e6			0.747788			0.373894	−	0.927472i	$-0.378022\pi$
$503$	4.24325e6			0.747788			0.373894	+	0.927472i	$0.378022\pi$
$504$	−6.19552e6	+	1.20373e6i	−1.08643	+	0.211082i
$505$		−	3.52649e6i		−	0.615338i
$506$	−2.55945e6	−	3.13980e6i	−0.444396	−	0.545162i
$507$	−1.28642e6	−	1.33660e7i	−0.222260	−	2.30931i
$508$	−4.52522e6			−0.778001
$509$		−	5.31532e6i		−	0.909359i	−0.890655	−	0.454679i	$-0.849754\pi$
$509$		−	5.31532e6i		−	0.909359i	0.890655	−	0.454679i	$-0.150246\pi$
$510$	−1.05426e6	+	101468.i	−0.179483	+	0.0172744i
$511$		−	2.51471e6i		−	0.426026i
$512$			3.74056e6i			0.630611i
$513$	6.25182e6	−	1.85099e6i	1.04885	−	0.310534i
$514$	1.37625e6			0.229768
$515$			1.56234e7i			2.59572i
$516$	3.00043e6	−	288777.i	0.496089	−	0.0477461i
$517$			2.21553e6i			0.364546i
$518$		−	4.33370e6i		−	0.709635i
$519$	−3.36719e6	+	324076.i	−0.548718	+	0.0528114i
$520$			1.32118e7i			2.14267i
$521$	1.16287e7			1.87689			0.938443	−	0.345435i	$-0.112269\pi$
$521$	1.16287e7			1.87689			0.938443	+	0.345435i	$0.112269\pi$
$522$	2.10592e6	−	409159.i	0.338272	−	0.0657228i
$523$		−	4.14177e6i		−	0.662112i	−0.943611	−	0.331056i	$-0.892595\pi$
$523$		−	4.14177e6i		−	0.662112i	0.943611	−	0.331056i	$-0.107405\pi$
$524$		−	5.77065e6i		−	0.918113i
$525$	−6.84236e6	+	658544.i	−1.08345	+	0.104276i
$526$			1.44973e6i			0.228466i
$527$	790857.			0.124043
$528$	292467.	+	3.03877e6i	0.0456553	+	0.474366i
$529$	1.29738e6	−	6.30423e6i	0.201570	−	0.979474i
$530$			770882.i			0.119206i
$531$	−536384.	−	2.76074e6i	−0.0825543	−	0.424903i
$532$	−6.93132e6			−1.06179
$533$			1.03919e7i			1.58445i
$534$	−1.40080e6	+	134820.i	−0.212581	+	0.0204598i
$535$	7.04481e6			1.06411
$536$	−1.22812e6			−0.184642
$537$	−3.31082e6	+	318650.i	−0.495450	+	0.0476847i
$538$	−3.88211e6			−0.578245
$539$	6.16400e6			0.913883
$540$	1.98404e6	+	6.70123e6i	0.292797	+	0.988941i
$541$	3.52579e6			0.517921			0.258961	−	0.965888i	$-0.416620\pi$
$541$	3.52579e6			0.517921			0.258961	+	0.965888i	$0.416620\pi$
$542$			5.31505e6i			0.777157i
$543$	−7.48831e6	+	720713.i	−1.08989	+	0.104897i
$544$		−	1.91358e6i		−	0.277235i
$545$		−	2.45285e6i		−	0.353737i
$546$	−765415.	−	7.95277e6i	−0.109879	−	1.14166i
$547$	8.53774e6			1.22004			0.610021	−	0.792385i	$-0.291161\pi$
$547$	8.53774e6			1.22004			0.610021	+	0.792385i	$0.291161\pi$
$548$	1.24060e6			0.176474
$549$	−2.24761e6	−	1.15683e7i	−0.318265	−	1.63810i
$550$		−	4.24238e6i		−	0.598002i
$551$	5.46328e6			0.766610
$552$	4.40043e6	−	4.35202e6i	0.614678	−	0.607916i
$553$	−3.06882e6			−0.426735
$554$		−	4.05382e6i		−	0.561164i
$555$	−1.10766e7	+	1.06607e6i	−1.52642	+	0.146911i
$556$	7.77739e6			1.06696
$557$	1.04184e7			1.42287			0.711433	−	0.702754i	$-0.248046\pi$
$557$	1.04184e7			1.42287			0.711433	+	0.702754i	$0.248046\pi$
$558$	317352.	+	1.63339e6i	0.0431475	+	0.222078i
$559$			8.84822e6i			1.19764i
$560$		−	4.30529e6i		−	0.580140i
$561$	−275404.	−	2.86148e6i	−0.0369456	−	0.383870i
$562$		−	1.32103e6i		−	0.176430i
$563$	−5.22857e6			−0.695204			−0.347602	−	0.937642i	$-0.613004\pi$
$563$	−5.22857e6			−0.695204			−0.347602	+	0.937642i	$0.613004\pi$
$564$	−1.45305e6	+	139849.i	−0.192346	+	0.0185124i
$565$	1.72479e6			0.227308
$566$	−4.65740e6			−0.611086
$567$	−3.66961e6	−	9.08716e6i	−0.479360	−	1.18705i
$568$	−6.39946e6			−0.832285
$569$	−7.84404e6			−1.01569			−0.507843	−	0.861450i	$-0.669557\pi$
$569$	−7.84404e6			−1.01569			−0.507843	+	0.861450i	$0.669557\pi$
$570$	−543666.	−	5.64877e6i	−0.0700883	−	0.728227i
$571$			8.89226e6i			1.14136i	0.821173	+	0.570679i	$0.193320\pi$
$571$			8.89226e6i			1.14136i	−0.821173	+	0.570679i	$0.806680\pi$
$572$	−1.54643e7			−1.97625
$573$	937751.	+	9.74337e6i	0.119317	+	1.23972i
$574$			4.32077e6i			0.547370i
$575$	5.22475e6	−	4.25901e6i	0.659016	−	0.537204i
$576$	1.34811e6	−	261924.i	0.169305	−	0.0328942i
$577$	−1.59062e6			−0.198896			−0.0994479	−	0.995043i	$-0.531708\pi$
$577$	−1.59062e6			−0.198896			−0.0994479	+	0.995043i	$0.531708\pi$
$578$		−	3.66222e6i		−	0.455958i
$579$	585950.	+	6.08810e6i	0.0726380	+	0.754720i
$580$			5.85600e6i			0.722822i
$581$			1.56127e7i			1.91883i
$582$	725423.	+	7.53724e6i	0.0887736	+	0.922370i
$583$	−2.09233e6			−0.254952
$584$			2.37120e6i			0.287698i
$585$	−2.01384e7	+	3.91268e6i	−2.43296	+	0.472699i
$586$		−	3.42513e6i		−	0.412034i
$587$		−	537172.i		−	0.0643454i	−0.999482	−	0.0321727i	$-0.989757\pi$
$587$		−	537172.i		−	0.0643454i	0.999482	−	0.0321727i	$-0.0102427\pi$
$588$	389085.	+	4.04265e6i	0.0464089	+	0.482195i
$589$			4.23743e6i			0.503285i
$590$	−2.44780e6			−0.289498
$591$	−1.00319e7	+	965522.i	−1.18145	+	0.113709i
$592$		−	3.20268e6i		−	0.375586i
$593$		−	8.95599e6i		−	1.04587i	−0.852373	−	0.522934i	$-0.824837\pi$
$593$		−	8.95599e6i		−	1.04587i	0.852373	−	0.522934i	$-0.175163\pi$
$594$	5.79944e6	−	1.71705e6i	0.674404	−	0.199672i
$595$			4.05411e6i			0.469465i
$596$	6.77817e6			0.781622
$597$	−1.20146e7	+	1.15635e6i	−1.37966	+	0.132786i
$598$	4.95018e6	+	6.07264e6i	0.566068	+	0.694424i
$599$		−	6.89748e6i		−	0.785459i	−0.919654	−	0.392729i	$-0.871531\pi$
$599$		−	6.89748e6i		−	0.785459i	0.919654	−	0.392729i	$-0.128469\pi$
$600$	6.45187e6	−	620961.i	0.731657	−	0.0704184i
$601$	−2.34695e6			−0.265043			−0.132522	−	0.991180i	$-0.542307\pi$
$601$	−2.34695e6			−0.265043			−0.132522	+	0.991180i	$0.542307\pi$
$602$			3.67892e6i			0.413742i
$603$	−363709.	−	1.87199e6i	−0.0407343	−	0.209657i
$604$	−6.97127e6			−0.777534
$605$	1.28114e7			1.42301
$606$	−192644.	−	2.00160e6i	−0.0213095	−	0.221409i
$607$	3.35665e6			0.369773			0.184886	−	0.982760i	$-0.440808\pi$
$607$	3.35665e6			0.369773			0.184886	+	0.982760i	$0.440808\pi$
$608$	1.02530e7			1.12484
$609$	−786699.	−	8.17391e6i	−0.0859538	−	0.893072i
$610$	−1.02570e7			−1.11608
$611$		−	4.28503e6i		−	0.464355i
$612$	1.85931e6	−	361246.i	0.200666	−	0.0389874i
$613$		−	2.71153e6i		−	0.291449i	−0.989325	−	0.145724i	$-0.953449\pi$
$613$		−	2.71153e6i		−	0.291449i	0.989325	−	0.145724i	$-0.0465513\pi$
$614$			5.04637e6i			0.540204i
$615$	1.10436e7	−	1.06289e6i	1.17739	−	0.113318i
$616$	−1.49097e7			−1.58313
$617$	4.96146e6			0.524683			0.262341	−	0.964975i	$-0.415505\pi$
$617$	4.96146e6			0.524683			0.262341	+	0.964975i	$0.415505\pi$
$618$	853471.	+	8.86768e6i	0.0898913	+	0.933983i
$619$			2.40949e6i			0.252755i	0.991982	+	0.126377i	$0.0403350\pi$
$619$			2.40949e6i			0.252755i	−0.991982	+	0.126377i	$0.959665\pi$
$620$	−4.54203e6			−0.474538
$621$	7.93684e6	+	5.41859e6i	0.825883	+	0.563842i
$622$	6.08301e6			0.630438
$623$			5.38670e6i			0.556036i
$624$	−565655.	−	5.87724e6i	−0.0581554	−	0.604243i
$625$	−1.10092e7			−1.12734
$626$	2.33724e6			0.238379
$627$	1.53319e7	−	1.47562e6i	1.55749	−	0.149901i
$628$		−	3.50868e6i		−	0.355014i
$629$			3.01583e6i			0.303935i
$630$	−8.37315e6	+	1.62682e6i	−0.840500	+	0.163301i
$631$			3.04372e6i			0.304320i	0.988356	+	0.152160i	$0.0486229\pi$
$631$			3.04372e6i			0.304320i	−0.988356	+	0.152160i	$0.951377\pi$
$632$	2.89368e6			0.288176
$633$	−953392.	−	9.90588e6i	−0.0945719	−	0.982616i
$634$	2.98622e6			0.295052
$635$	−1.41815e7			−1.39569
$636$	−132072.	−	1.37225e6i	−0.0129470	−	0.134521i
$637$	−1.19217e7			−1.16410
$638$	5.06796e6			0.492925
$639$	−1.89520e6	−	9.75449e6i	−0.183613	−	0.945045i
$640$			1.32989e7i			1.28341i
$641$	−463097.			−0.0445171			−0.0222586	−	0.999752i	$-0.507086\pi$
$641$	−463097.			−0.0445171			−0.0222586	+	0.999752i	$0.507086\pi$
$642$	3.99857e6	−	384843.i	0.382884	−	0.0368507i
$643$			1.44142e7i			1.37487i	0.726244	+	0.687437i	$0.241264\pi$
$643$			1.44142e7i			1.37487i	−0.726244	+	0.687437i	$0.758736\pi$
$644$	−6.45499e6	−	7.91866e6i	−0.613311	−	0.752380i
$645$	9.40304e6	−	904996.i	0.889957	−	0.0856540i
$646$	−1.53799e6			−0.145002
$647$			2.84721e6i			0.267398i	0.991022	+	0.133699i	$0.0426855\pi$
$647$			2.84721e6i			0.267398i	−0.991022	+	0.133699i	$0.957314\pi$
$648$	3.46018e6	+	8.56856e6i	0.323714	+	0.801623i
$649$		−	6.64380e6i		−	0.619163i
$650$			8.20511e6i			0.761730i
$651$	6.33985e6	−	610179.i	0.586309	−	0.0564293i
$652$	−6.00169e6			−0.552910
$653$			6.90890e6i			0.634053i	0.948417	+	0.317027i	$0.102684\pi$
$653$			6.90890e6i			0.634053i	−0.948417	+	0.317027i	$0.897316\pi$
$654$	−133994.	−	1.39222e6i	−0.0122501	−	0.127281i
$655$		−	1.80846e7i		−	1.64704i
$656$			3.19313e6i			0.289705i
$657$	−3.61435e6	+	702231.i	−0.326675	+	0.0634697i
$658$		−	1.78163e6i		−	0.160418i
$659$	−5.77029e6			−0.517588			−0.258794	−	0.965933i	$-0.583325\pi$
$659$	−5.77029e6			−0.517588			−0.258794	+	0.965933i	$0.583325\pi$
$660$	1.58169e6	+	1.64340e7i	0.141339	+	1.46853i
$661$		−	9.91098e6i		−	0.882293i	−0.897435	−	0.441146i	$-0.854572\pi$
$661$		−	9.91098e6i		−	0.882293i	0.897435	−	0.441146i	$-0.145428\pi$
$662$		−	181189.i		−	0.0160690i
$663$	532654.	+	5.53435e6i	0.0470610	+	0.488970i
$664$		−	1.47217e7i		−	1.29580i
$665$	−2.17220e7			−1.90479
$666$	−6.22874e6	+	1.21018e6i	−0.544145	+	0.105722i
$667$	5.08783e6	+	6.24150e6i	0.442811	+	0.543218i
$668$			1.19156e7i			1.03318i
$669$	−174374.	−	1.81178e6i	−0.0150632	−	0.156509i
$670$	−1.65979e6			−0.142845
$671$		−	2.78395e7i		−	2.38701i
$672$	−1.47640e6	−	1.53400e7i	−0.126119	−	1.31040i
$673$	1.02840e7			0.875236			0.437618	−	0.899161i	$-0.355822\pi$
$673$	1.02840e7			0.875236			0.437618	+	0.899161i	$0.355822\pi$
$674$	−4.79173e6			−0.406296
$675$	2.85723e6	+	9.65049e6i	0.241372	+	0.815248i
$676$	2.09004e7			1.75909
$677$	1.50848e7			1.26493			0.632467	−	0.774587i	$-0.282042\pi$
$677$	1.50848e7			1.26493			0.632467	+	0.774587i	$0.282042\pi$
$678$	978974.	−	94221.5i	0.0817894	−	0.00787183i
$679$	2.89840e7			2.41259
$680$		−	3.82275e6i		−	0.317032i
$681$	−1.31669e6	−	1.36806e7i	−0.108797	−	1.13042i
$682$			3.93081e6i			0.323609i
$683$			1.54100e7i			1.26401i	0.774965	+	0.632004i	$0.217768\pi$
$683$			1.54100e7i			1.26401i	−0.774965	+	0.632004i	$0.782232\pi$
$684$	1.93556e6	+	9.96225e6i	0.158186	+	0.814174i
$685$	3.88790e6			0.316584
$686$	2.80175e6			0.227310
$687$	2.31025e6	−	222350.i	0.186753	−	0.0179740i
$688$			2.71879e6i			0.218980i
$689$	4.04673e6			0.324756
$690$	5.94711e6	−	5.88169e6i	0.475536	−	0.470305i
$691$	−8.16882e6			−0.650825			−0.325412	−	0.945572i	$-0.605503\pi$
$691$	−8.16882e6			−0.650825			−0.325412	+	0.945572i	$0.605503\pi$
$692$		−	5.26527e6i		−	0.417980i
$693$	−4.41551e6	−	2.27264e7i	−0.349259	−	1.79762i
$694$	6.53914e6			0.515373
$695$	2.43735e7			1.91406
$696$	741802.	+	7.70743e6i	0.0580450	+	0.603096i
$697$		−	3.00683e6i		−	0.234437i
$698$		−	9.21941e6i		−	0.716250i
$699$	−1.14513e7	+	1.10213e6i	−0.886465	+	0.0853179i
$700$		−	1.06994e7i		−	0.825304i
$701$	−7.44999e6			−0.572612			−0.286306	−	0.958138i	$-0.592427\pi$
$701$	−7.44999e6			−0.572612			−0.286306	+	0.958138i	$0.592427\pi$
$702$	−1.12166e7	+	3.32092e6i	−0.859051	+	0.254340i
$703$	−1.61589e7			−1.23317
$704$	3.24426e6			0.246709
$705$	−4.55371e6	+	438273.i	−0.345059	+	0.0332102i
$706$	1.11964e7			0.845408
$707$	−7.69704e6			−0.579129
$708$	4.35733e6	−	419371.i	0.326691	−	0.0314424i
$709$		−	1.69220e7i		−	1.26426i	−0.774864	−	0.632128i	$-0.782181\pi$
$709$		−	1.69220e7i		−	1.26426i	0.774864	−	0.632128i	$-0.217819\pi$
$710$	−8.64876e6			−0.643885
$711$	856963.	+	4.41075e6i	0.0635753	+	0.327219i
$712$		−	5.07928e6i		−	0.375494i
$713$	−4.84103e6	+	3.94623e6i	−0.356627	+	0.290709i
$714$	221467.	+	2.30108e6i	0.0162579	+	0.168922i
$715$	−4.84636e7			−3.54528
$716$		−	5.17713e6i		−	0.377404i
$717$	7.00747e6	−	674435.i	0.509054	−	0.0489939i
$718$		−	5.11312e6i		−	0.370148i
$719$			2.86571e6i			0.206733i	0.994643	+	0.103366i	$0.0329614\pi$
$719$			2.86571e6i			0.206733i	−0.994643	+	0.103366i	$0.967039\pi$
$720$	−6.18791e6	+	1.20225e6i	−0.444849	+	0.0864297i
$721$	3.41002e7			2.44297
$722$		−	1.35344e6i		−	0.0966266i
$723$	2.54611e6	−	245051.i	0.181147	−	0.0174345i
$724$		−	1.17094e7i		−	0.830214i
$725$			8.43327e6i			0.595869i
$726$	7.27165e6	−	699860.i	0.512025	−	0.0492799i
$727$			9.36513e6i			0.657170i	0.944474	+	0.328585i	$0.106572\pi$
$727$			9.36513e6i			0.657170i	−0.944474	+	0.328585i	$0.893428\pi$
$728$	2.88366e7			2.01658
$729$	−1.20360e7	+	7.81183e6i	−0.838813	+	0.544420i
$730$			3.20464e6i			0.222573i
$731$		−	2.56017e6i		−	0.177205i
$732$	1.82585e7	−	1.75729e6i	1.25947	−	0.121217i
$733$			9.17191e6i			0.630521i	0.949005	+	0.315261i	$0.102092\pi$
$733$			9.17191e6i			0.630521i	−0.949005	+	0.315261i	$0.897908\pi$
$734$	5.90531e6			0.404578
$735$	1.21935e6	+	1.26692e7i	0.0832550	+	0.865031i
$736$	9.54838e6	+	1.17135e7i	0.649733	+	0.797061i
$737$		−	4.50500e6i		−	0.305510i
$738$	6.21015e6	−	1.20657e6i	0.419722	−	0.0815476i
$739$	−2.54254e6			−0.171260			−0.0856301	−	0.996327i	$-0.527290\pi$
$739$	−2.54254e6			−0.171260			−0.0856301	+	0.996327i	$0.527290\pi$
$740$		−	1.73205e7i		−	1.16273i
$741$	−2.96531e7	+	2.85397e6i	−1.98393	+	0.190943i
$742$	1.68255e6			0.112191
$743$	−1.42849e7			−0.949303			−0.474652	−	0.880174i	$-0.657426\pi$
$743$	−1.42849e7			−0.949303			−0.474652	+	0.880174i	$0.657426\pi$
$744$	−5.97804e6	+	575357.i	−0.395937	+	0.0381070i
$745$	2.12421e7			1.40219
$746$	−7.25878e6			−0.477548
$747$	2.24398e7	−	4.35982e6i	1.47135	−	0.285869i
$748$	4.47449e6			0.292408
$749$		−	1.53763e7i		−	1.00149i
$750$	−1.53599e6	+	147831.i	−0.0997090	+	0.00959650i
$751$		−	1.34957e7i		−	0.873162i	−0.899665	−	0.436581i	$-0.856189\pi$
$751$		−	1.34957e7i		−	0.873162i	0.899665	−	0.436581i	$-0.143811\pi$
$752$		−	1.31666e6i		−	0.0849041i
$753$	77352.7	+	803705.i	0.00497150	+	0.0516546i
$754$	−9.80185e6			−0.627885
$755$	−2.18472e7			−1.39485
$756$	1.46263e7	−	4.33044e6i	0.930746	−	0.275567i
$757$			1.46194e7i			0.927234i	0.886036	+	0.463617i	$0.153449\pi$
$757$			1.46194e7i			0.927234i	−0.886036	+	0.463617i	$0.846551\pi$
$758$	3.03825e6			0.192066
$759$	1.59641e7	+	1.61416e7i	1.00586	+	1.01705i
$760$	2.04824e7			1.28631
$761$			2.00908e7i			1.25758i	0.777575	+	0.628790i	$0.216450\pi$
$761$			2.00908e7i			1.25758i	−0.777575	+	0.628790i	$0.783550\pi$
$762$	−8.04931e6	+	774706.i	−0.502194	+	0.0483337i
$763$	−5.35368e6			−0.332921
$764$	−1.52357e7			−0.944340
$765$	5.82689e6	−	1.13211e6i	0.359984	−	0.0699413i
$766$		−	1.29632e7i		−	0.798255i
$767$			1.28497e7i			0.788685i
$768$	996571.	+	1.03545e7i	0.0609685	+	0.633471i
$769$			1.58291e7i			0.965253i	0.875826	+	0.482627i	$0.160317\pi$
$769$			1.58291e7i			0.965253i	−0.875826	+	0.482627i	$0.839683\pi$
$770$	−2.01502e7			−1.22477
$771$	−7.67762e6	+	738933.i	−0.465148	+	0.0447682i
$772$	−9.51995e6			−0.574899
$773$	2.60813e7			1.56993			0.784966	−	0.619539i	$-0.212680\pi$
$773$	2.60813e7			1.56993			0.784966	+	0.619539i	$0.212680\pi$
$774$	5.28764e6	−	1.02733e6i	0.317256	−	0.0616395i
$775$	−6.54101e6			−0.391193
$776$	−2.73299e7			−1.62924
$777$	2.32684e6	+	2.41762e7i	0.138266	+	1.43660i
$778$			3.59099e6i			0.212699i
$779$	1.61107e7			0.951196
$780$	−3.05913e6	−	3.17848e7i	−0.180037	−	1.87061i
$781$		−	2.34744e7i		−	1.37711i
$782$	−1.43230e6	−	1.75707e6i	−0.0837562	−	0.102748i
$783$	−1.15285e7	+	3.41326e6i	−0.671999	+	0.198960i
$784$	−3.66317e6			−0.212847
$785$		−	1.09958e7i		−	0.636875i
$786$	−987921.	−	1.02646e7i	−0.0570382	−	0.592635i
$787$			1.23067e7i			0.708281i	0.935192	+	0.354141i	$0.115227\pi$
$787$			1.23067e7i			0.708281i	−0.935192	+	0.354141i	$0.884773\pi$
$788$		−	1.56869e7i		−	0.899955i
$789$	−778384.	−	8.08752e6i	−0.0445145	−	0.462512i
$790$	3.91076e6			0.222943
$791$		−	3.76459e6i		−	0.213932i
$792$	4.16352e6	+	2.14294e7i	0.235856	+	1.21394i
$793$			5.38439e7i			3.04056i
$794$		−	1.38355e7i		−	0.778833i
$795$	−413900.	−	4.30048e6i	−0.0232262	−	0.241323i
$796$		−	1.87872e7i		−	1.05094i
$797$	−2.86235e7			−1.59616			−0.798082	−	0.602549i	$-0.794152\pi$
$797$	−2.86235e7			−1.59616			−0.798082	+	0.602549i	$0.794152\pi$
$798$	−1.23292e7	+	1.18663e6i	−0.685375	+	0.0659639i
$799$			1.23984e6i			0.0687067i
$800$			1.58268e7i			0.874315i
$801$	7.74219e6	−	1.50423e6i	0.426366	−	0.0828386i
$802$		−	1.17252e7i		−	0.643702i
$803$	−8.69803e6			−0.476027
$804$	2.95459e6	−	284365.i	0.161197	−	0.0155144i
$805$	−2.02292e7	−	2.48162e7i	−1.10025	−	1.34973i
$806$		−	7.60252e6i		−	0.412211i
$807$	2.16569e7	−	2.08437e6i	1.17061	−	0.112666i
$808$	7.25777e6			0.391088
$809$		−	3.93621e6i		−	0.211450i	−0.994395	−	0.105725i	$-0.966284\pi$
$809$		−	3.93621e6i		−	0.211450i	0.994395	−	0.105725i	$-0.0337163\pi$
$810$	4.67639e6	+	1.15803e7i	0.250437	+	0.620163i
$811$	−2.29805e7			−1.22689			−0.613447	−	0.789736i	$-0.710218\pi$
$811$	−2.29805e7			−1.22689			−0.613447	+	0.789736i	$0.710218\pi$
$812$	1.27815e7			0.680288
$813$	−2.85374e6	−	2.96508e7i	−0.151422	−	1.57329i
$814$	−1.49896e7			−0.792922
$815$	−1.88087e7			−0.991891
$816$	163668.	+	1.70054e6i	0.00860476	+	0.0894047i
$817$	1.37174e7			0.718982
$818$		−	44366.3i		−	0.00231830i
$819$	8.53996e6	+	4.39547e7i	0.444883	+	2.28979i
$820$			1.72688e7i			0.896865i
$821$		−	2.91696e7i		−	1.51033i	−0.655533	−	0.755166i	$-0.727556\pi$
$821$		−	2.91696e7i		−	1.51033i	0.655533	−	0.755166i	$-0.272444\pi$
$822$	2.20674e6	−	212388.i	0.113912	−	0.0109635i
$823$	−2.31304e7			−1.19037			−0.595186	−	0.803588i	$-0.702922\pi$
$823$	−2.31304e7			−1.19037			−0.595186	+	0.803588i	$0.702922\pi$
$824$	−3.21541e7			−1.64975
$825$	2.27781e6	+	2.36667e7i	0.116515	+	1.21061i
$826$			5.34265e6i			0.272462i
$827$	1.79438e7			0.912328			0.456164	−	0.889896i	$-0.349223\pi$
$827$	1.79438e7			0.912328			0.456164	+	0.889896i	$0.349223\pi$
$828$	−9.57878e6	+	1.14889e7i	−0.485551	+	0.582375i
$829$	−1.45260e7			−0.734106			−0.367053	−	0.930200i	$-0.619633\pi$
$829$	−1.45260e7			−0.734106			−0.367053	+	0.930200i	$0.619633\pi$
$830$		−	1.98961e7i		−	1.00247i
$831$	2.17657e6	+	2.26148e7i	0.109338	+	1.13603i
$832$	−6.27468e6			−0.314256
$833$	3.44945e6			0.172242
$834$	1.38342e7	−	1.33147e6i	0.688713	−	0.0662852i
$835$			3.73422e7i			1.85346i
$836$			2.39744e7i			1.18640i
$837$	−2.64739e6	−	8.94174e6i	−0.130619	−	0.441173i
$838$		−	1.00266e7i		−	0.493225i
$839$	−1.57775e7			−0.773809			−0.386905	−	0.922120i	$-0.626456\pi$
$839$	−1.57775e7			−0.773809			−0.386905	+	0.922120i	$0.626456\pi$
$840$	−2.94941e6	−	3.06448e7i	−0.144224	−	1.49851i
$841$	1.04367e7			0.508832
$842$	4.51916e6			0.219673
$843$	709286.	+	7.36958e6i	0.0343758	+	0.357169i
$844$	1.54898e7			0.748496
$845$	6.54998e7			3.15572
$846$	−2.56070e6	+	497519.i	−0.123008	+	0.0238992i
$847$		−	2.79627e7i		−	1.33928i
$848$	1.24344e6			0.0593792
$849$	2.59820e7	−	2.50064e6i	1.23710	−	0.119064i
$850$		−	2.37409e6i		−	0.112707i
$851$	−1.50484e7	−	1.84607e7i	−0.712307	−	0.873823i
$852$	1.53957e7	−	1.48176e6i	0.726608	−	0.0699324i
$853$	−1.68869e7			−0.794655			−0.397328	−	0.917677i	$-0.630062\pi$
$853$	−1.68869e7			−0.794655			−0.397328	+	0.917677i	$0.630062\pi$
$854$			2.23873e7i			1.05040i
$855$	6.06585e6	+	3.12206e7i	0.283776	+	1.46058i
$856$			1.44987e7i			0.676310i
$857$			3.38112e7i			1.57257i	0.617867	+	0.786283i	$0.287997\pi$
$857$			3.38112e7i			1.57257i	−0.617867	+	0.786283i	$0.712003\pi$
$858$	−2.75075e7	+	2.64746e6i	−1.27565	+	0.122775i
$859$	−1.12780e7			−0.521493			−0.260747	−	0.965407i	$-0.583969\pi$
$859$	−1.12780e7			−0.521493			−0.260747	+	0.965407i	$0.583969\pi$
$860$			1.47035e7i			0.677914i
$861$	−2.31990e6	−	2.41041e7i	−0.106650	−	1.10811i
$862$			6.63149e6i			0.303978i
$863$		−	9.53233e6i		−	0.435684i	−0.975984	−	0.217842i	$-0.930098\pi$
$863$		−	9.53233e6i		−	0.435684i	0.975984	−	0.217842i	$-0.0699018\pi$
$864$	−2.16357e7	+	6.40570e6i	−0.986020	+	0.291932i
$865$		−	1.65008e7i		−	0.749833i
$866$	−9.00440e6			−0.408000
$867$	1.96631e6	+	2.04303e7i	0.0888392	+	0.923052i
$868$			9.91361e6i			0.446614i
$869$			1.06146e7i			0.476819i
$870$	1.00253e6	+	1.04165e7i	0.0449056	+	0.466576i
$871$			8.71304e6i			0.389157i
$872$	5.04815e6			0.224823
$873$	−8.09376e6	−	4.16581e7i	−0.359430	−	1.84997i
$874$	9.41444e6	−	7.67429e6i	0.416885	−	0.339828i
$875$			5.90655e6i			0.260804i
$876$	−549038.	−	5.70459e6i	−0.0241737	−	0.251168i
$877$	1.05897e7			0.464926			0.232463	−	0.972605i	$-0.425321\pi$
$877$	1.05897e7			0.464926			0.232463	+	0.972605i	$0.425321\pi$
$878$			1.44616e7i			0.633113i
$879$	1.83901e6	+	1.91076e7i	0.0802809	+	0.834130i
$880$	−1.48914e7			−0.648229
$881$	1.26015e7			0.546993			0.273497	−	0.961873i	$-0.411820\pi$
$881$	1.26015e7			0.546993			0.273497	+	0.961873i	$0.411820\pi$
$882$	1.38418e6	+	7.12432e6i	0.0599132	+	0.308370i
$883$	−6.35190e6			−0.274159			−0.137079	−	0.990560i	$-0.543772\pi$
$883$	−6.35190e6			−0.274159			−0.137079	+	0.990560i	$0.543772\pi$
$884$	−8.65404e6			−0.372468
$885$	1.36554e7	−	1.31426e6i	0.586066	−	0.0564059i
$886$	4.41584e6			0.188986
$887$		−	1.84499e7i		−	0.787380i	−0.919243	−	0.393690i	$-0.871198\pi$
$887$		−	1.84499e7i		−	0.787380i	0.919243	−	0.393690i	$-0.128802\pi$
$888$	−2.19405e6	−	2.27965e7i	−0.0933714	−	0.970142i
$889$			3.09531e7i			1.31356i
$890$		−	6.86457e6i		−	0.290495i
$891$	−3.14311e7	+	1.26926e7i	−1.32637	+	0.535621i
$892$	2.83307e6			0.119219
$893$	−6.64310e6			−0.278767
$894$	1.20568e7	−	1.16041e6i	0.504531	−	0.0485586i
$895$		−	1.62246e7i		−	0.677041i
$896$	2.90267e7			1.20789
$897$	−3.08758e7	−	3.12193e7i	−1.28126	−	1.29551i
$898$	−3.46498e6			−0.143387
$899$		−	7.81392e6i		−	0.322455i
$900$	−1.53780e7	+	2.98779e6i	−0.632840	+	0.122954i
$901$	−1.17089e6			−0.0480513
$902$	1.49449e7			0.611613
$903$	−1.97528e6	−	2.05234e7i	−0.0806136	−	0.837587i
$904$			3.54975e6i			0.144470i
$905$		−	3.66961e7i		−	1.48936i
$906$	−1.24003e7	+	1.19346e6i	−0.501892	+	0.0483046i
$907$		−	3.15877e7i		−	1.27497i	−0.770463	−	0.637485i	$-0.779975\pi$
$907$		−	3.15877e7i		−	1.27497i	0.770463	−	0.637485i	$-0.220025\pi$
$908$	2.13924e7			0.861082
$909$	2.14939e6	+	1.10628e7i	0.0862790	+	0.444074i
$910$	3.89722e7			1.56010
$911$	1.53170e6			0.0611472			0.0305736	−	0.999533i	$-0.490267\pi$
$911$	1.53170e6			0.0611472			0.0305736	+	0.999533i	$0.490267\pi$
$912$	−9.11151e6	+	876938.i	−0.362746	+	0.0349126i
$913$	5.40019e7			2.14404
$914$	−4.76388e6			−0.188623
$915$	5.72201e7	−	5.50715e6i	2.25941	−	0.217457i
$916$			3.61253e6i			0.142257i
$917$	−3.94721e7			−1.55012
$918$	3.24544e6	−	960883.i	0.127106	−	0.0376326i
$919$			3.63199e7i			1.41859i	0.704913	+	0.709294i	$0.250986\pi$
$919$			3.63199e7i			1.41859i	−0.704913	+	0.709294i	$0.749014\pi$
$920$	1.90748e7	+	2.34000e7i	0.743002	+	0.911478i
$921$	−2.70948e6	−	2.81519e7i	−0.105254	−	1.09360i
$922$	1.21046e7			0.468946
$923$			4.54016e7i			1.75415i
$924$	3.58695e7	−	3.45226e6i	1.38212	−	0.133022i
$925$		−	2.49433e7i		−	0.958517i
$926$			9.70361e6i			0.371883i
$927$	−9.52243e6	−	4.90115e7i	−0.363956	−	1.87326i
$928$	−1.89067e7			−0.720687
$929$		−	4.64683e6i		−	0.176651i	−0.996092	−	0.0883257i	$-0.971848\pi$
$929$		−	4.64683e6i		−	0.176651i	0.996092	−	0.0883257i	$-0.0281516\pi$
$930$	−8.07922e6	+	777585.i	−0.306311	+	0.0294809i
$931$			1.84822e7i			0.698845i
$932$		−	1.79064e7i		−	0.675254i
$933$	−3.39350e7	+	3.26607e6i	−1.27627	+	0.122835i
$934$		−	2.19249e7i		−	0.822378i
$935$	1.40226e7			0.524564
$936$	−8.05259e6	−	4.14463e7i	−0.300432	−	1.54631i
$937$			963927.i			0.0358670i	0.999839	+	0.0179335i	$0.00570872\pi$
$937$			963927.i			0.0358670i	−0.999839	+	0.0179335i	$0.994291\pi$
$938$			3.62272e6i			0.134440i
$939$	−1.30387e7	+	1.25491e6i	−0.482580	+	0.0464459i
$940$		−	7.12063e6i		−	0.262845i
$941$	2.59553e7			0.955548			0.477774	−	0.878483i	$-0.341444\pi$
$941$	2.59553e7			0.955548			0.477774	+	0.878483i	$0.341444\pi$
$942$	−600678.	−	6.24113e6i	−0.0220554	−	0.229159i
$943$	1.50035e7	+	1.84056e7i	0.549432	+	0.674016i
$944$			3.94831e6i			0.144205i
$945$	4.58374e7	−	1.35711e7i	1.66971	−	0.494353i
$946$	1.27248e7			0.462301
$947$		−	2.30958e6i		−	0.0836869i	−0.999124	−	0.0418435i	$-0.986677\pi$
$947$		−	2.30958e6i		−	0.0836869i	0.999124	−	0.0418435i	$-0.0133231\pi$
$948$	−6.96156e6	+	670016.i	−0.251585	+	0.0242139i
$949$	1.68227e7			0.606360
$950$	1.27204e7			0.457291
$951$	−1.66591e7	+	1.60335e6i	−0.597310	+	0.0574881i
$952$	−8.34366e6			−0.298376
$953$	−5.98904e6			−0.213612			−0.106806	−	0.994280i	$-0.534062\pi$
$953$	−5.98904e6			−0.213612			−0.106806	+	0.994280i	$0.534062\pi$
$954$	−469851.	−	2.41830e6i	−0.0167144	−	0.0860279i
$955$	−4.77470e7			−1.69409
$956$			1.09576e7i			0.387766i
$957$	−2.82724e7	+	2.72107e6i	−0.997889	+	0.0960419i
$958$			2.13825e6i			0.0752740i
$959$		−	8.48588e6i		−	0.297955i
$960$	641774.	+	6.66812e6i	0.0224752	+	0.233521i
$961$	−2.25685e7			−0.788306
$962$	2.89912e7			1.01002
$963$	−2.21000e7	+	4.29380e6i	−0.767938	+	0.149203i
$964$			3.98135e6i			0.137987i
$965$	−2.98345e7			−1.03134
$966$	−1.28376e7	−	1.29804e7i	−0.442630	−	0.447553i
$967$	3.61195e7			1.24216			0.621078	−	0.783749i	$-0.286695\pi$
$967$	3.61195e7			1.24216			0.621078	+	0.783749i	$0.286695\pi$
$968$			2.63669e7i			0.904420i
$969$	8.57992e6	−	825775.i	0.293544	−	0.0282522i
$970$	−3.69360e7			−1.26043
$971$	−7.79287e6			−0.265246			−0.132623	−	0.991167i	$-0.542340\pi$
$971$	−7.79287e6			−0.265246			−0.132623	+	0.991167i	$0.542340\pi$
$972$	−1.03084e7	−	1.98129e7i	−0.349967	−	0.672639i
$973$		−	5.31985e7i		−	1.80143i
$974$		−	8.41931e6i		−	0.284367i
$975$	−4.40547e6	−	4.57734e7i	−0.148416	−	1.54206i
$976$			1.65446e7i			0.555944i
$977$	964694.			0.0323335			0.0161668	−	0.999869i	$-0.494854\pi$
$977$	964694.			0.0323335			0.0161668	+	0.999869i	$0.494854\pi$
$978$	−1.06756e7	+	1.02748e6i	−0.356899	+	0.0343498i
$979$	1.86318e7			0.621295
$980$	−1.98108e7			−0.658927
$981$	1.49501e6	+	7.69474e6i	0.0495988	+	0.255283i
$982$	−2.58817e7			−0.856475
$983$	−3.47266e7			−1.14625			−0.573123	−	0.819469i	$-0.694268\pi$
$983$	−3.47266e7			−1.14625			−0.573123	+	0.819469i	$0.694268\pi$
$984$	2.18750e6	+	2.27284e7i	0.0720212	+	0.748311i
$985$		−	4.91610e7i		−	1.61447i
$986$	2.83610e6			0.0929027
$987$	956589.	+	9.93910e6i	0.0312559	+	0.324754i
$988$		−	4.63686e7i		−	1.51123i
$989$	1.27747e7	+	1.56714e7i	0.415300	+	0.509469i
$990$	5.62693e6	+	2.89615e7i	0.182467	+	0.939146i
$991$	1.77874e7			0.575345			0.287672	−	0.957729i	$-0.407119\pi$
$991$	1.77874e7			0.575345			0.287672	+	0.957729i	$0.407119\pi$
$992$		−	1.46644e7i		−	0.473137i
$993$	97283.8	+	1.01079e6i	0.00313089	+	0.0325303i
$994$			1.88771e7i			0.605995i
$995$		−	5.88771e7i		−	1.88533i
$996$	3.40872e6	+	3.54171e7i	0.108879	+	1.13127i
$997$	5.73042e7			1.82578			0.912890	−	0.408206i	$-0.133846\pi$
$997$	5.73042e7			1.82578			0.912890	+	0.408206i	$0.133846\pi$
$998$			2.38890e6i			0.0759226i
$999$	3.40982e7	−	1.00955e7i	1.08098	−	0.320047i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.c.b.68.22	yes	32
3.2	odd	2	inner	69.6.c.b.68.11	✓	32
23.22	odd	2	inner	69.6.c.b.68.21	yes	32
69.68	even	2	inner	69.6.c.b.68.12	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.c.b.68.11	✓	32	3.2	odd	2	inner
69.6.c.b.68.12	yes	32	69.68	even	2	inner
69.6.c.b.68.21	yes	32	23.22	odd	2	inner
69.6.c.b.68.22	yes	32	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.78145i q^{2} +(-1.49341 - 15.5168i) q^{3} +24.2635 q^{4} +76.0392 q^{5} +(43.1591 - 4.15386i) q^{6} -165.966i q^{7} +156.494i q^{8} +(-238.539 + 46.3458i) q^{9} +O(q^{10})\)	\(q+2.78145i q^{2} +(-1.49341 - 15.5168i) q^{3} +24.2635 q^{4} +76.0392 q^{5} +(43.1591 - 4.15386i) q^{6} -165.966i q^{7} +156.494i q^{8} +(-238.539 + 46.3458i) q^{9} +211.500i q^{10} -574.052 q^{11} +(-36.2354 - 376.491i) q^{12} +1110.26 q^{13} +461.627 q^{14} +(-113.558 - 1179.88i) q^{15} +341.150 q^{16} -321.247 q^{17} +(-128.909 - 663.487i) q^{18} -1721.25i q^{19} +1844.98 q^{20} +(-2575.25 + 247.855i) q^{21} -1596.70i q^{22} +(1966.43 - 1602.96i) q^{23} +(2428.29 - 233.710i) q^{24} +2656.97 q^{25} +3088.15i q^{26} +(1075.37 + 3632.15i) q^{27} -4026.92i q^{28} +3174.02i q^{29} +(3281.79 - 315.856i) q^{30} -2461.84 q^{31} +5956.71i q^{32} +(857.295 + 8907.42i) q^{33} -893.534i q^{34} -12619.9i q^{35} +(-5787.80 + 1124.51i) q^{36} -9387.89i q^{37} +4787.57 q^{38} +(-1658.08 - 17227.7i) q^{39} +11899.7i q^{40} +9359.87i q^{41} +(-689.399 - 7162.95i) q^{42} +7969.47i q^{43} -13928.5 q^{44} +(-18138.4 + 3524.10i) q^{45} +(4458.56 + 5469.54i) q^{46} -3859.46i q^{47} +(-509.478 - 5293.55i) q^{48} -10737.7 q^{49} +7390.23i q^{50} +(479.754 + 4984.71i) q^{51} +26938.9 q^{52} +3644.84 q^{53} +(-10102.6 + 2991.10i) q^{54} -43650.5 q^{55} +25972.7 q^{56} +(-26708.2 + 2570.53i) q^{57} -8828.40 q^{58} +11573.5i q^{59} +(-2755.31 - 28628.1i) q^{60} +48496.4i q^{61} -6847.48i q^{62} +(7691.83 + 39589.4i) q^{63} -5651.52 q^{64} +84423.7 q^{65} +(-24775.6 + 2384.53i) q^{66} +7847.72i q^{67} -7794.58 q^{68} +(-27809.4 - 28118.8i) q^{69} +35101.7 q^{70} +40892.6i q^{71} +(-7252.86 - 37330.1i) q^{72} +15152.0 q^{73} +26112.0 q^{74} +(-3967.94 - 41227.5i) q^{75} -41763.5i q^{76} +95273.1i q^{77} +(47918.1 - 4611.88i) q^{78} -18490.6i q^{79} +25940.8 q^{80} +(54753.1 - 22110.6i) q^{81} -26034.1 q^{82} -94071.5 q^{83} +(-62484.7 + 6013.84i) q^{84} -24427.4 q^{85} -22166.7 q^{86} +(49250.5 - 4740.12i) q^{87} -89835.9i q^{88} -32456.7 q^{89} +(-9802.12 - 50451.0i) q^{90} -184266. i q^{91} +(47712.6 - 38893.5i) q^{92} +(3676.53 + 38199.7i) q^{93} +10734.9 q^{94} -130882. i q^{95} +(92428.9 - 8895.82i) q^{96} +174638. i q^{97} -29866.4i q^{98} +(136934. - 26604.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9}+O(q^{10}) \) 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9	\( 32 q - 408 q^{4} - 528 q^{6} - 444 q^{9} - 2484 q^{12} + 520 q^{13} + 4936 q^{16} + 7188 q^{18} + 18660 q^{24} + 36032 q^{25} - 22032 q^{27} + 6544 q^{31} - 33912 q^{36} - 63912 q^{39} + 54328 q^{46} + 88284 q^{48} - 207664 q^{49} + 46296 q^{52} - 38628 q^{54} - 139296 q^{55} - 184144 q^{58} + 486584 q^{64} - 113580 q^{69} + 37176 q^{70} - 15504 q^{72} - 93896 q^{73} + 249840 q^{75} + 368028 q^{78} - 339372 q^{81} - 23512 q^{82} + 259584 q^{85} + 509928 q^{87} + 82740 q^{93} - 562000 q^{94} + 1404 q^{96}+O(q^{100}) \) 32 * q - 408 * q^4 - 528 * q^6 - 444 * q^9 - 2484 * q^12 + 520 * q^13 + 4936 * q^16 + 7188 * q^18 + 18660 * q^24 + 36032 * q^25 - 22032 * q^27 + 6544 * q^31 - 33912 * q^36 - 63912 * q^39 + 54328 * q^46 + 88284 * q^48 - 207664 * q^49 + 46296 * q^52 - 38628 * q^54 - 139296 * q^55 - 184144 * q^58 + 486584 * q^64 - 113580 * q^69 + 37176 * q^70 - 15504 * q^72 - 93896 * q^73 + 249840 * q^75 + 368028 * q^78 - 339372 * q^81 - 23512 * q^82 + 259584 * q^85 + 509928 * q^87 + 82740 * q^93 - 562000 * q^94 + 1404 * q^96

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