LMFDB - Embedded newform 138.5.b.a.91.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,5,Mod(91,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("138.91");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	138.b (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:	$14.2650549056$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} + \cdots + 274129967370817 $ x^16 + 1428x^14 - 600x^13 + 788282x^12 - 529464x^11 + 213396724x^10 - 175079484x^9 + 29130946113x^8 - 26553654912x^7 + 1790717926120x^6 - 1672125302460x^5 + 32715955056348x^4 - 11255739266544x^3 + 140972261417692x^2 - 70961983286748x + 274129967370817
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.14
Root			$0.707107 - 4.53246i$ of defining polynomial
Character	$\chi$	$=$	138.91
Dual form			138.5.b.a.91.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -7.70519i q^{5} +14.6969 q^{6} +72.1011i q^{7} +22.6274 q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -7.70519i q^{5} +14.6969 q^{6} +72.1011i q^{7} +22.6274 q^{8} +27.0000 q^{9} -21.7936i q^{10} +99.6618i q^{11} +41.5692 q^{12} +228.825 q^{13} +203.933i q^{14} -40.0373i q^{15} +64.0000 q^{16} +208.892i q^{17} +76.3675 q^{18} -517.377i q^{19} -61.6415i q^{20} +374.648i q^{21} +281.886i q^{22} +(445.386 - 285.433i) q^{23} +117.576 q^{24} +565.630 q^{25} +647.214 q^{26} +140.296 q^{27} +576.809i q^{28} -1444.22 q^{29} -113.243i q^{30} -578.381 q^{31} +181.019 q^{32} +517.858i q^{33} +590.836i q^{34} +555.552 q^{35} +216.000 q^{36} +1884.33i q^{37} -1463.36i q^{38} +1189.01 q^{39} -174.349i q^{40} -302.964 q^{41} +1059.66i q^{42} -2300.38i q^{43} +797.294i q^{44} -208.040i q^{45} +(1259.74 - 807.328i) q^{46} +1282.83 q^{47} +332.554 q^{48} -2797.56 q^{49} +1599.84 q^{50} +1085.44i q^{51} +1830.60 q^{52} -979.640i q^{53} +396.817 q^{54} +767.913 q^{55} +1631.46i q^{56} -2688.37i q^{57} -4084.88 q^{58} -3439.18 q^{59} -320.299i q^{60} -2670.93i q^{61} -1635.91 q^{62} +1946.73i q^{63} +512.000 q^{64} -1763.14i q^{65} +1464.72i q^{66} +6724.19i q^{67} +1671.14i q^{68} +(2314.29 - 1483.16i) q^{69} +1571.34 q^{70} -8233.12 q^{71} +610.940 q^{72} +468.443 q^{73} +5329.69i q^{74} +2939.10 q^{75} -4139.01i q^{76} -7185.72 q^{77} +3363.02 q^{78} -11268.8i q^{79} -493.132i q^{80} +729.000 q^{81} -856.911 q^{82} -11253.9i q^{83} +2997.19i q^{84} +1609.55 q^{85} -6506.47i q^{86} -7504.40 q^{87} +2255.09i q^{88} -9357.72i q^{89} -588.426i q^{90} +16498.5i q^{91} +(3563.09 - 2283.47i) q^{92} -3005.36 q^{93} +3628.40 q^{94} -3986.49 q^{95} +940.604 q^{96} +6821.35i q^{97} -7912.71 q^{98} +2690.87i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) $ 16 * q + 128 * q^4 + 432 * q^9	$ 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) $ 16 * q + 128 * q^4 + 432 * q^9 - 208 * q^13 + 1024 * q^16 + 840 * q^23 + 1056 * q^25 + 1920 * q^26 + 3600 * q^29 + 224 * q^31 - 3264 * q^35 + 3456 * q^36 - 2016 * q^39 - 6144 * q^41 + 1280 * q^46 + 8880 * q^47 - 13888 * q^49 + 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 - 18240 * q^59 + 8192 * q^64 + 10584 * q^69 + 19584 * q^70 - 30048 * q^71 + 9536 * q^73 - 4176 * q^75 + 14160 * q^77 + 6912 * q^78 + 11664 * q^81 - 19584 * q^82 - 32496 * q^85 - 8064 * q^87 + 6720 * q^92 - 11952 * q^93 - 21248 * q^94 - 20064 * q^95 + 21504 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.82843			0.707107
$2$	2.82843			0.707107
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	8.00000			0.500000
$5$		−	7.70519i		−	0.308208i	−0.988055	−	0.154104i	$-0.950751\pi$
$5$		−	7.70519i		−	0.308208i	0.988055	−	0.154104i	$-0.0492490\pi$
$6$	14.6969			0.408248
$7$			72.1011i			1.47145i	0.677280	+	0.735725i	$0.263158\pi$
$7$			72.1011i			1.47145i	−0.677280	+	0.735725i	$0.736842\pi$
$8$	22.6274			0.353553
$9$	27.0000			0.333333
$10$		−	21.7936i		−	0.217936i
$11$			99.6618i			0.823651i	0.911263	+	0.411826i	$0.135109\pi$
$11$			99.6618i			0.823651i	−0.911263	+	0.411826i	$0.864891\pi$
$12$	41.5692			0.288675
$13$	228.825			1.35399			0.676996	−	0.735986i	$-0.263281\pi$
$13$	228.825			1.35399			0.676996	+	0.735986i	$0.263281\pi$
$14$			203.933i			1.04047i
$15$		−	40.0373i		−	0.177944i
$16$	64.0000			0.250000
$17$			208.892i			0.722810i	0.932409	+	0.361405i	$0.117703\pi$
$17$			208.892i			0.722810i	−0.932409	+	0.361405i	$0.882297\pi$
$18$	76.3675			0.235702
$19$		−	517.377i		−	1.43318i	−0.697496	−	0.716588i	$-0.745703\pi$
$19$		−	517.377i		−	1.43318i	0.697496	−	0.716588i	$-0.254297\pi$
$20$		−	61.6415i		−	0.154104i
$21$			374.648i			0.849542i
$22$			281.886i			0.582409i
$23$	445.386	−	285.433i	0.841940	−	0.539572i
$23$	445.386	−	285.433i	0.841940	−	0.539572i
$24$	117.576			0.204124
$25$	565.630			0.905008
$26$	647.214			0.957417
$27$	140.296			0.192450
$28$			576.809i			0.735725i
$29$	−1444.22			−1.71727			−0.858634	−	0.512590i	$-0.828686\pi$
$29$	−1444.22			−1.71727			−0.858634	+	0.512590i	$0.828686\pi$
$30$		−	113.243i		−	0.125825i
$31$	−578.381			−0.601853			−0.300927	−	0.953647i	$-0.597296\pi$
$31$	−578.381			−0.601853			−0.300927	+	0.953647i	$0.597296\pi$
$32$	181.019			0.176777
$33$			517.858i			0.475535i
$34$			590.836i			0.511104i
$35$	555.552			0.453512
$36$	216.000			0.166667
$37$			1884.33i			1.37643i	0.725507	+	0.688214i	$0.241605\pi$
$37$			1884.33i			1.37643i	−0.725507	+	0.688214i	$0.758395\pi$
$38$		−	1463.36i		−	1.01341i
$39$	1189.01			0.781728
$40$		−	174.349i		−	0.108968i
$41$	−302.964			−0.180228			−0.0901141	−	0.995931i	$-0.528723\pi$
$41$	−302.964			−0.180228			−0.0901141	+	0.995931i	$0.528723\pi$
$42$			1059.66i			0.600717i
$43$		−	2300.38i		−	1.24412i	−0.782968	−	0.622062i	$-0.786295\pi$
$43$		−	2300.38i		−	1.24412i	0.782968	−	0.622062i	$-0.213705\pi$
$44$			797.294i			0.411826i
$45$		−	208.040i		−	0.102736i
$46$	1259.74	−	807.328i	0.595341	−	0.381535i
$47$	1282.83			0.580731			0.290365	−	0.956916i	$-0.406223\pi$
$47$	1282.83			0.580731			0.290365	+	0.956916i	$0.406223\pi$
$48$	332.554			0.144338
$49$	−2797.56			−1.16517
$50$	1599.84			0.639937
$51$			1085.44i			0.417315i
$52$	1830.60			0.676996
$53$		−	979.640i		−	0.348751i	−0.984679	−	0.174375i	$-0.944209\pi$
$53$		−	979.640i		−	0.348751i	0.984679	−	0.174375i	$-0.0557906\pi$
$54$	396.817			0.136083
$55$	767.913			0.253856
$56$			1631.46i			0.520236i
$57$		−	2688.37i		−	0.827445i
$58$	−4084.88			−1.21429
$59$	−3439.18			−0.987988			−0.493994	−	0.869465i	$-0.664463\pi$
$59$	−3439.18			−0.987988			−0.493994	+	0.869465i	$0.664463\pi$
$60$		−	320.299i		−	0.0889719i
$61$		−	2670.93i		−	0.717800i	−0.933376	−	0.358900i	$-0.883152\pi$
$61$		−	2670.93i		−	0.717800i	0.933376	−	0.358900i	$-0.116848\pi$
$62$	−1635.91			−0.425574
$63$			1946.73i			0.490483i
$64$	512.000			0.125000
$65$		−	1763.14i		−	0.417311i
$66$			1464.72i			0.336254i
$67$			6724.19i			1.49793i	0.662612	+	0.748963i	$0.269448\pi$
$67$			6724.19i			1.49793i	−0.662612	+	0.748963i	$0.730552\pi$
$68$			1671.14i			0.361405i
$69$	2314.29	−	1483.16i	0.486094	−	0.311522i
$70$	1571.34			0.320681
$71$	−8233.12			−1.63323			−0.816616	−	0.577182i	$-0.804152\pi$
$71$	−8233.12			−1.63323			−0.816616	+	0.577182i	$0.804152\pi$
$72$	610.940			0.117851
$73$	468.443			0.0879045			0.0439522	−	0.999034i	$-0.486005\pi$
$73$	468.443			0.0879045			0.0439522	+	0.999034i	$0.486005\pi$
$74$			5329.69i			0.973282i
$75$	2939.10			0.522507
$76$		−	4139.01i		−	0.716588i
$77$	−7185.72			−1.21196
$78$	3363.02			0.552765
$79$		−	11268.8i		−	1.80560i	−0.430058	−	0.902801i	$-0.641507\pi$
$79$		−	11268.8i		−	1.80560i	0.430058	−	0.902801i	$-0.358493\pi$
$80$		−	493.132i		−	0.0770519i
$81$	729.000			0.111111
$82$	−856.911			−0.127441
$83$		−	11253.9i		−	1.63361i	−0.576915	−	0.816804i	$-0.695744\pi$
$83$		−	11253.9i		−	1.63361i	0.576915	−	0.816804i	$-0.304256\pi$
$84$			2997.19i			0.424771i
$85$	1609.55			0.222776
$86$		−	6506.47i		−	0.879728i
$87$	−7504.40			−0.991465
$88$			2255.09i			0.291205i
$89$		−	9357.72i		−	1.18138i	−0.806898	−	0.590691i	$-0.798855\pi$
$89$		−	9357.72i		−	1.18138i	0.806898	−	0.590691i	$-0.201145\pi$
$90$		−	588.426i		−	0.0726452i
$91$			16498.5i			1.99233i
$92$	3563.09	−	2283.47i	0.420970	−	0.269786i
$93$	−3005.36			−0.347480
$94$	3628.40			0.410639
$95$	−3986.49			−0.441716
$96$	940.604			0.102062
$97$			6821.35i			0.724981i	0.931987	+	0.362491i	$0.118073\pi$
$97$			6821.35i			0.724981i	−0.931987	+	0.362491i	$0.881927\pi$
$98$	−7912.71			−0.823897
$99$			2690.87i			0.274550i
$100$	4525.04			0.452504
$101$	−10062.9			−0.986462			−0.493231	−	0.869898i	$-0.664184\pi$
$101$	−10062.9			−0.986462			−0.493231	+	0.869898i	$0.664184\pi$
$102$			3070.08i			0.295086i
$103$		−	13027.9i		−	1.22801i	−0.789304	−	0.614003i	$-0.789558\pi$
$103$		−	13027.9i		−	1.22801i	0.789304	−	0.614003i	$-0.210442\pi$
$104$	5177.71			0.478709
$105$	2886.73			0.261835
$106$		−	2770.84i		−	0.246604i
$107$		−	5449.88i		−	0.476013i	−0.971264	−	0.238007i	$-0.923506\pi$
$107$		−	5449.88i		−	0.476013i	0.971264	−	0.238007i	$-0.0764940\pi$
$108$	1122.37			0.0962250
$109$			13965.6i			1.17545i	0.809059	+	0.587727i	$0.199977\pi$
$109$			13965.6i			1.17545i	−0.809059	+	0.587727i	$0.800023\pi$
$110$	2171.99			0.179503
$111$			9791.27i			0.794681i
$112$			4614.47i			0.367863i
$113$			2414.44i			0.189086i	0.995521	+	0.0945430i	$0.0301390\pi$
$113$			2414.44i			0.189086i	−0.995521	+	0.0945430i	$0.969861\pi$
$114$		−	7603.85i		−	0.585092i
$115$	−2199.32	−	3431.78i	−0.166300	−	0.259492i
$116$	−11553.8			−0.858634
$117$	6178.27			0.451331
$118$	−9727.48			−0.698613
$119$	−15061.3			−1.06358
$120$		−	905.942i		−	0.0629126i
$121$	4708.52			0.321598
$122$		−	7554.54i		−	0.507561i
$123$	−1574.25			−0.104055
$124$	−4627.05			−0.300927
$125$		−	9174.03i		−	0.587138i
$126$			5506.18i			0.346824i
$127$	18498.8			1.14693			0.573463	−	0.819231i	$-0.305600\pi$
$127$	18498.8			1.14693			0.573463	+	0.819231i	$0.305600\pi$
$128$	1448.15			0.0883883
$129$		−	11953.1i		−	0.718295i
$130$		−	4986.91i		−	0.295083i
$131$	−7236.03			−0.421655			−0.210828	−	0.977523i	$-0.567616\pi$
$131$	−7236.03			−0.421655			−0.210828	+	0.977523i	$0.567616\pi$
$132$			4142.86i			0.237768i
$133$	37303.4			2.10885
$134$			19018.9i			1.05919i
$135$		−	1081.01i		−	0.0593146i
$136$			4726.69i			0.255552i
$137$		−	24304.5i		−	1.29493i	−0.762096	−	0.647464i	$-0.775830\pi$
$137$		−	24304.5i		−	1.29493i	0.762096	−	0.647464i	$-0.224170\pi$
$138$	6545.81	−	4195.00i	0.343720	−	0.220279i
$139$	22728.8			1.17638			0.588188	−	0.808724i	$-0.299841\pi$
$139$	22728.8			1.17638			0.588188	+	0.808724i	$0.299841\pi$
$140$	4444.42			0.226756
$141$	6665.80			0.335285
$142$	−23286.8			−1.15487
$143$			22805.1i			1.11522i
$144$	1728.00			0.0833333
$145$			11128.0i			0.529275i
$146$	1324.96			0.0621578
$147$	−14536.6			−0.672709
$148$			15074.6i			0.688214i
$149$			32575.5i			1.46730i	0.679528	+	0.733650i	$0.262185\pi$
$149$			32575.5i			1.46730i	−0.679528	+	0.733650i	$0.737815\pi$
$150$	8313.03			0.369468
$151$	28285.7			1.24055			0.620274	−	0.784385i	$-0.287021\pi$
$151$	28285.7			1.24055			0.620274	+	0.784385i	$0.287021\pi$
$152$		−	11706.9i		−	0.506704i
$153$			5640.09i			0.240937i
$154$	−20324.3			−0.856987
$155$			4456.53i			0.185496i
$156$	9512.07			0.390864
$157$			29018.8i			1.17728i	0.808394	+	0.588641i	$0.200337\pi$
$157$			29018.8i			1.17728i	−0.808394	+	0.588641i	$0.799663\pi$
$158$		−	31872.9i		−	1.27675i
$159$		−	5090.36i		−	0.201351i
$160$		−	1394.79i		−	0.0544839i
$161$	20580.1	+	32112.8i	0.793953	+	1.23887i
$162$	2061.92			0.0785674
$163$	−12511.3			−0.470899			−0.235450	−	0.971887i	$-0.575656\pi$
$163$	−12511.3			−0.470899			−0.235450	+	0.971887i	$0.575656\pi$
$164$	−2423.71			−0.0901141
$165$	3990.19			0.146564
$166$		−	31830.9i		−	1.15514i
$167$	36628.1			1.31335			0.656676	−	0.754173i	$-0.271962\pi$
$167$	36628.1			1.31335			0.656676	+	0.754173i	$0.271962\pi$
$168$			8477.32i			0.300359i
$169$	23799.8			0.833296
$170$	4552.51			0.157526
$171$		−	13969.2i		−	0.477725i
$172$		−	18403.1i		−	0.622062i
$173$	31295.0			1.04564			0.522821	−	0.852442i	$-0.324879\pi$
$173$	31295.0			1.04564			0.522821	+	0.852442i	$0.324879\pi$
$174$	−21225.6			−0.701071
$175$			40782.5i			1.33167i
$176$			6378.36i			0.205913i
$177$	−17870.5			−0.570415
$178$		−	26467.6i		−	0.835363i
$179$	−23052.6			−0.719471			−0.359736	−	0.933054i	$-0.617133\pi$
$179$	−23052.6			−0.719471			−0.359736	+	0.933054i	$0.617133\pi$
$180$		−	1664.32i		−	0.0513679i
$181$			18635.1i			0.568821i	0.958703	+	0.284410i	$0.0917978\pi$
$181$			18635.1i			0.568821i	−0.958703	+	0.284410i	$0.908202\pi$
$182$			46664.8i			1.40879i
$183$		−	13878.6i		−	0.414422i
$184$	10077.9	−	6458.62i	0.297671	−	0.190767i
$185$	14519.1			0.424226
$186$	−8500.43			−0.245706
$187$	−20818.6			−0.595344
$188$	10262.7			0.290365
$189$			10115.5i			0.283181i
$190$	−11275.5			−0.312340
$191$		−	9043.84i		−	0.247905i	−0.992288	−	0.123953i	$-0.960443\pi$
$191$		−	9043.84i		−	0.247905i	0.992288	−	0.123953i	$-0.0395571\pi$
$192$	2660.43			0.0721688
$193$	6623.85			0.177826			0.0889131	−	0.996039i	$-0.471661\pi$
$193$	6623.85			0.177826			0.0889131	+	0.996039i	$0.471661\pi$
$194$			19293.7i			0.512639i
$195$		−	9161.53i		−	0.240934i
$196$	−22380.5			−0.582583
$197$	2868.82			0.0739214			0.0369607	−	0.999317i	$-0.488232\pi$
$197$	2868.82			0.0739214			0.0369607	+	0.999317i	$0.488232\pi$
$198$			7610.93i			0.194136i
$199$		−	39601.4i		−	1.00001i	−0.866022	−	0.500005i	$-0.833331\pi$
$199$		−	39601.4i		−	1.00001i	0.866022	−	0.500005i	$-0.166669\pi$
$200$	12798.7			0.319969
$201$			34939.9i			0.864828i
$202$	−28462.2			−0.697534
$203$		−	104130.i		−	2.52687i
$204$			8683.48i			0.208657i
$205$			2334.39i			0.0555477i
$206$		−	36848.5i		−	0.868331i
$207$	12025.4	−	7706.70i	0.280647	−	0.179857i
$208$	14644.8			0.338498
$209$	51562.7			1.18044
$210$	8164.92			0.185146
$211$	−71119.7			−1.59744			−0.798720	−	0.601703i	$-0.794489\pi$
$211$	−71119.7			−1.59744			−0.798720	+	0.601703i	$0.794489\pi$
$212$		−	7837.12i		−	0.174375i
$213$	−42780.6			−0.942947
$214$		−	15414.6i		−	0.336592i
$215$	−17724.9			−0.383448
$216$	3174.54			0.0680414
$217$		−	41701.9i		−	0.885597i
$218$			39500.6i			0.831171i
$219$	2434.10			0.0507517
$220$	6143.30			0.126928
$221$			47799.7i			0.978680i
$222$			27693.9i			0.561925i
$223$	−49668.1			−0.998775			−0.499388	−	0.866379i	$-0.666442\pi$
$223$	−49668.1			−0.998775			−0.499388	+	0.866379i	$0.666442\pi$
$224$			13051.7i			0.260118i
$225$	15272.0			0.301669
$226$			6829.06i			0.133704i
$227$			75466.9i			1.46455i	0.681008	+	0.732276i	$0.261542\pi$
$227$			75466.9i			1.46455i	−0.681008	+	0.732276i	$0.738458\pi$
$228$		−	21506.9i		−	0.413722i
$229$			17702.8i			0.337576i	0.985652	+	0.168788i	$0.0539853\pi$
$229$			17702.8i			0.337576i	−0.985652	+	0.168788i	$0.946015\pi$
$230$	−6220.61	−	9706.55i	−0.117592	−	0.183489i
$231$	−37338.1			−0.699727
$232$	−32679.0			−0.607146
$233$	−25351.6			−0.466974			−0.233487	−	0.972360i	$-0.575014\pi$
$233$	−25351.6			−0.466974			−0.233487	+	0.972360i	$0.575014\pi$
$234$	17474.8			0.319139
$235$		−	9884.48i		−	0.178986i
$236$	−27513.5			−0.493994
$237$		−	58554.2i		−	1.04246i
$238$	−42599.9			−0.752064
$239$	−63489.9			−1.11150			−0.555750	−	0.831350i	$-0.687569\pi$
$239$	−63489.9			−1.11150			−0.555750	+	0.831350i	$0.687569\pi$
$240$		−	2562.39i		−	0.0444859i
$241$		−	63390.5i		−	1.09142i	−0.837976	−	0.545708i	$-0.816261\pi$
$241$		−	63390.5i		−	1.09142i	0.837976	−	0.545708i	$-0.183739\pi$
$242$	13317.7			0.227404
$243$	3788.00			0.0641500
$244$		−	21367.5i		−	0.358900i
$245$			21555.8i			0.359113i
$246$	−4452.64			−0.0735779
$247$		−	118389.i		−	1.94051i
$248$	−13087.3			−0.212787
$249$		−	58477.1i		−	0.943164i
$250$		−	25948.1i		−	0.415169i
$251$		−	87203.8i		−	1.38417i	−0.721818	−	0.692083i	$-0.756693\pi$
$251$		−	87203.8i		−	1.38417i	0.721818	−	0.692083i	$-0.243307\pi$
$252$			15573.8i			0.245242i
$253$	28446.8	+	44388.0i	0.444419	+	0.693465i
$254$	52322.4			0.810999
$255$	8363.49			0.128620
$256$	4096.00			0.0625000
$257$	−2032.18			−0.0307678			−0.0153839	−	0.999882i	$-0.504897\pi$
$257$	−2032.18			−0.0307678			−0.0153839	+	0.999882i	$0.504897\pi$
$258$		−	33808.6i		−	0.507911i
$259$	−135862.			−2.02535
$260$		−	14105.1i		−	0.208655i
$261$	−38994.0			−0.572422
$262$	−20466.6			−0.298155
$263$			79775.8i			1.15335i	0.816975	+	0.576673i	$0.195649\pi$
$263$			79775.8i			1.15335i	−0.816975	+	0.576673i	$0.804351\pi$
$264$			11717.8i			0.168127i
$265$	−7548.31			−0.107488
$266$	105510.			1.49118
$267$		−	48624.1i		−	0.682071i
$268$			53793.5i			0.748963i
$269$	−31306.7			−0.432646			−0.216323	−	0.976322i	$-0.569407\pi$
$269$	−31306.7			−0.432646			−0.216323	+	0.976322i	$0.569407\pi$
$270$		−	3057.55i		−	0.0419417i
$271$	−19525.3			−0.265863			−0.132932	−	0.991125i	$-0.542439\pi$
$271$	−19525.3			−0.265863			−0.132932	+	0.991125i	$0.542439\pi$
$272$			13369.1i			0.180703i
$273$			85728.8i			1.15027i
$274$		−	68743.5i		−	0.915652i
$275$			56371.7i			0.745411i
$276$	18514.4	−	11865.2i	0.243047	−	0.155761i
$277$	37926.8			0.494295			0.247148	−	0.968978i	$-0.420507\pi$
$277$	37926.8			0.494295			0.247148	+	0.968978i	$0.420507\pi$
$278$	64286.7			0.831824
$279$	−15616.3			−0.200618
$280$	12570.7			0.160341
$281$		−	87235.2i		−	1.10479i	−0.833583	−	0.552394i	$-0.813714\pi$
$281$		−	87235.2i		−	1.10479i	0.833583	−	0.552394i	$-0.186286\pi$
$282$	18853.7			0.237082
$283$			108189.i			1.35085i	0.737427	+	0.675427i	$0.236041\pi$
$283$			108189.i			1.35085i	−0.737427	+	0.675427i	$0.763959\pi$
$284$	−65865.0			−0.816616
$285$	−20714.4			−0.255025
$286$			64502.5i			0.788578i
$287$		−	21844.0i		−	0.265197i
$288$	4887.52			0.0589256
$289$	39885.1			0.477545
$290$			31474.7i			0.374254i
$291$			35444.8i			0.418568i
$292$	3747.54			0.0439522
$293$			63410.0i			0.738622i	0.929306	+	0.369311i	$0.120406\pi$
$293$			63410.0i			0.738622i	−0.929306	+	0.369311i	$0.879594\pi$
$294$	−41115.6			−0.475677
$295$			26499.6i			0.304505i
$296$			42637.5i			0.486641i
$297$			13982.2i			0.158512i
$298$			92137.4i			1.03754i
$299$	101915.	−	65314.2i	1.13998	−	0.730576i
$300$	23512.8			0.261253
$301$	165860.			1.83067
$302$	80004.1			0.877200
$303$	−52288.4			−0.569534
$304$		−	33112.1i		−	0.358294i
$305$	−20580.0			−0.221231
$306$			15952.6i			0.170368i
$307$	7067.17			0.0749840			0.0374920	−	0.999297i	$-0.488063\pi$
$307$	7067.17			0.0749840			0.0374920	+	0.999297i	$0.488063\pi$
$308$	−57485.8			−0.605981
$309$		−	67695.0i		−	0.708989i
$310$			12605.0i			0.131165i
$311$	6189.13			0.0639895			0.0319948	−	0.999488i	$-0.489814\pi$
$311$	6189.13			0.0639895			0.0319948	+	0.999488i	$0.489814\pi$
$312$	26904.2			0.276383
$313$			75353.4i			0.769156i	0.923093	+	0.384578i	$0.125653\pi$
$313$			75353.4i			0.769156i	−0.923093	+	0.384578i	$0.874347\pi$
$314$			82077.6i			0.832464i
$315$	14999.9			0.151171
$316$		−	90150.1i		−	0.902801i
$317$	141581.			1.40892			0.704460	−	0.709744i	$-0.251189\pi$
$317$	141581.			1.40892			0.704460	+	0.709744i	$0.251189\pi$
$318$		−	14397.7i		−	0.142377i
$319$		−	143934.i		−	1.41443i
$320$		−	3945.06i		−	0.0385259i
$321$		−	28318.4i		−	0.274826i
$322$	58209.2	+	90828.7i	0.561410	+	0.876015i
$323$	108076.			1.03591
$324$	5832.00			0.0555556
$325$	129430.			1.22537
$326$	−35387.4			−0.332976
$327$			72567.2i			0.678648i
$328$	−6855.29			−0.0637203
$329$			92493.7i			0.854517i
$330$	11286.0			0.103636
$331$	−103794.			−0.947361			−0.473680	−	0.880697i	$-0.657075\pi$
$331$	−103794.			−0.947361			−0.473680	+	0.880697i	$0.657075\pi$
$332$		−	90031.4i		−	0.816804i
$333$			50876.9i			0.458810i
$334$	103600.			0.928680
$335$	51811.2			0.461672
$336$			23977.5i			0.212386i
$337$			53507.5i			0.471146i	0.971857	+	0.235573i	$0.0756966\pi$
$337$			53507.5i			0.471146i	−0.971857	+	0.235573i	$0.924303\pi$
$338$	67315.9			0.589229
$339$			12545.8i			0.109169i
$340$	12876.4			0.111388
$341$		−	57642.5i		−	0.495717i
$342$		−	39510.8i		−	0.337803i
$343$		−	28592.7i		−	0.243034i
$344$		−	52051.7i		−	0.439864i
$345$	−11428.0	−	17832.1i	−0.0960134	−	0.149818i
$346$	88515.8			0.739381
$347$	−198614.			−1.64949			−0.824747	−	0.565501i	$-0.808683\pi$
$347$	−198614.			−1.64949			−0.824747	+	0.565501i	$0.808683\pi$
$348$	−60035.2			−0.495732
$349$	−66271.3			−0.544095			−0.272047	−	0.962284i	$-0.587701\pi$
$349$	−66271.3			−0.544095			−0.272047	+	0.962284i	$0.587701\pi$
$350$			115350.i			0.941636i
$351$	32103.2			0.260576
$352$			18040.7i			0.145602i
$353$	64814.4			0.520143			0.260071	−	0.965589i	$-0.416254\pi$
$353$	64814.4			0.520143			0.260071	+	0.965589i	$0.416254\pi$
$354$	−50545.5			−0.403344
$355$			63437.8i			0.503374i
$356$		−	74861.8i		−	0.590691i
$357$	−78261.1			−0.614058
$358$	−65202.5			−0.508743
$359$			26201.2i			0.203298i	0.994820	+	0.101649i	$0.0324118\pi$
$359$			26201.2i			0.203298i	−0.994820	+	0.101649i	$0.967588\pi$
$360$		−	4707.41i		−	0.0363226i
$361$	−137358.			−1.05399
$362$			52708.1i			0.402217i
$363$	24466.2			0.185675
$364$			131988.i			0.996167i
$365$		−	3609.44i		−	0.0270928i
$366$		−	39254.5i		−	0.293040i
$367$			50116.5i			0.372091i	0.982541	+	0.186045i	$0.0595671\pi$
$367$			50116.5i			0.372091i	−0.982541	+	0.186045i	$0.940433\pi$
$368$	28504.7	−	18267.7i	0.210485	−	0.134893i
$369$	−8180.02			−0.0600761
$370$	41066.3			0.299973
$371$	70633.1			0.513169
$372$	−24042.8			−0.173740
$373$		−	116958.i		−	0.840645i	−0.907375	−	0.420323i	$-0.861917\pi$
$373$		−	116958.i		−	0.840645i	0.907375	−	0.420323i	$-0.138083\pi$
$374$	−58883.8			−0.420972
$375$		−	47669.7i		−	0.338984i
$376$	29027.2			0.205319
$377$	−330474.			−2.32517
$378$			28611.0i			0.200239i
$379$		−	68538.9i		−	0.477154i	−0.971124	−	0.238577i	$-0.923319\pi$
$379$		−	68538.9i		−	0.477154i	0.971124	−	0.238577i	$-0.0766810\pi$
$380$	−31891.9			−0.220858
$381$	96122.4			0.662178
$382$		−	25579.8i		−	0.175296i
$383$		−	130722.i		−	0.891150i	−0.895245	−	0.445575i	$-0.852999\pi$
$383$		−	130722.i		−	0.891150i	0.895245	−	0.445575i	$-0.147001\pi$
$384$	7524.83			0.0510310
$385$			55367.4i			0.373536i
$386$	18735.1			0.125742
$387$		−	62110.4i		−	0.414708i
$388$			54570.8i			0.362491i
$389$			67786.0i			0.447962i	0.974594	+	0.223981i	$0.0719053\pi$
$389$			67786.0i			0.447962i	−0.974594	+	0.223981i	$0.928095\pi$
$390$		−	25912.7i		−	0.170366i
$391$	59624.8	+	93037.7i	0.390008	+	0.608563i
$392$	−63301.6			−0.411948
$393$	−37599.5			−0.243443
$394$	8114.24			0.0522703
$395$	−86828.0			−0.556500
$396$			21527.0i			0.137275i
$397$	305065.			1.93558			0.967791	−	0.251757i	$-0.0810083\pi$
$397$	305065.			1.93558			0.967791	+	0.251757i	$0.0810083\pi$
$398$		−	112010.i		−	0.707114i
$399$	193834.			1.21754
$400$	36200.3			0.226252
$401$			134850.i			0.838616i	0.907844	+	0.419308i	$0.137727\pi$
$401$			134850.i			0.838616i	−0.907844	+	0.419308i	$0.862273\pi$
$402$			98825.0i			0.611526i
$403$	−132348.			−0.814905
$404$	−80503.2			−0.493231
$405$		−	5617.08i		−	0.0342453i
$406$		−	294524.i		−	1.78677i
$407$	−187796.			−1.13370
$408$			24560.6i			0.147543i
$409$	−122501.			−0.732306			−0.366153	−	0.930555i	$-0.619325\pi$
$409$	−122501.			−0.732306			−0.366153	+	0.930555i	$0.619325\pi$
$410$			6602.66i			0.0392782i
$411$		−	126290.i		−	0.747627i
$412$		−	104223.i		−	0.614003i
$413$		−	247969.i		−	1.45377i
$414$	34013.0	−	21797.8i	0.198447	−	0.127178i
$415$	−86713.6			−0.503490
$416$	41421.7			0.239354
$417$	118102.			0.679181
$418$	145841.			0.834695
$419$			23357.6i			0.133046i	0.997785	+	0.0665228i	$0.0211905\pi$
$419$			23357.6i			0.133046i	−0.997785	+	0.0665228i	$0.978809\pi$
$420$	23093.9			0.130918
$421$			13625.1i			0.0768733i	0.999261	+	0.0384367i	$0.0122378\pi$
$421$			13625.1i			0.0768733i	−0.999261	+	0.0384367i	$0.987762\pi$
$422$	−201157.			−1.12956
$423$	34636.5			0.193577
$424$		−	22166.7i		−	0.123302i
$425$			118156.i			0.654149i
$426$	−121002.			−0.666764
$427$	192577.			1.05621
$428$		−	43599.0i		−	0.238007i
$429$			118499.i			0.643871i
$430$	−50133.6			−0.271139
$431$		−	82366.5i		−	0.443400i	−0.975115	−	0.221700i	$-0.928839\pi$
$431$		−	82366.5i		−	0.443400i	0.975115	−	0.221700i	$-0.0711606\pi$
$432$	8978.95			0.0481125
$433$			133392.i			0.711465i	0.934588	+	0.355732i	$0.115769\pi$
$433$			133392.i			0.711465i	−0.934588	+	0.355732i	$0.884231\pi$
$434$		−	117951.i		−	0.626212i
$435$			57822.8i			0.305577i
$436$			111725.i			0.587727i
$437$	−147677.	−	230432.i	−0.773302	−	1.20665i
$438$	6884.68			0.0358868
$439$	123925.			0.643029			0.321515	−	0.946905i	$-0.395808\pi$
$439$	123925.			0.643029			0.321515	+	0.946905i	$0.395808\pi$
$440$	17375.9			0.0897515
$441$	−75534.2			−0.388389
$442$			135198.i			0.692031i
$443$	78337.7			0.399175			0.199588	−	0.979880i	$-0.436040\pi$
$443$	78337.7			0.399175			0.199588	+	0.979880i	$0.436040\pi$
$444$			78330.2i			0.397341i
$445$	−72103.0			−0.364111
$446$	−140483.			−0.706241
$447$			169267.i			0.847145i
$448$			36915.7i			0.183931i
$449$	−147331.			−0.730804			−0.365402	−	0.930850i	$-0.619068\pi$
$449$	−147331.			−0.730804			−0.365402	+	0.930850i	$0.619068\pi$
$450$	43195.8			0.213312
$451$		−	30193.9i		−	0.148445i
$452$			19315.5i			0.0945430i
$453$	146977.			0.716231
$454$			213453.i			1.03559i
$455$	127124.			0.614052
$456$		−	60830.8i		−	0.292546i
$457$		−	131155.i		−	0.627990i	−0.949425	−	0.313995i	$-0.898333\pi$
$457$		−	131155.i		−	0.627990i	0.949425	−	0.313995i	$-0.101667\pi$
$458$			50071.1i			0.238702i
$459$			29306.8i			0.139105i
$460$	−17594.5	−	27454.3i	−0.0831500	−	0.129746i
$461$	121914.			0.573656			0.286828	−	0.957982i	$-0.407399\pi$
$461$	121914.			0.573656			0.286828	+	0.957982i	$0.407399\pi$
$462$	−105608.			−0.494781
$463$	394501.			1.84029			0.920144	−	0.391579i	$-0.128071\pi$
$463$	394501.			1.84029			0.920144	+	0.391579i	$0.128071\pi$
$464$	−92430.2			−0.429317
$465$			23156.8i			0.107096i
$466$	−71705.1			−0.330201
$467$		−	37769.5i		−	0.173184i	−0.996244	−	0.0865921i	$-0.972402\pi$
$467$		−	37769.5i		−	0.173184i	0.996244	−	0.0865921i	$-0.0275977\pi$
$468$	49426.1			0.225665
$469$	−484821.			−2.20412
$470$		−	27957.5i		−	0.126562i
$471$			150786.i			0.679704i
$472$	−77819.9			−0.349306
$473$	229260.			1.02472
$474$		−	165616.i		−	0.737134i
$475$		−	292644.i		−	1.29704i
$476$	−120491.			−0.531790
$477$		−	26450.3i		−	0.116250i
$478$	−179577.			−0.785949
$479$			346948.i			1.51215i	0.654487	+	0.756073i	$0.272884\pi$
$479$			346948.i			1.51215i	−0.654487	+	0.756073i	$0.727116\pi$
$480$		−	7247.53i		−	0.0314563i
$481$			431182.i			1.86367i
$482$		−	179295.i		−	0.771747i
$483$	106937.	+	166863.i	0.458389	+	0.715263i
$484$	37668.2			0.160799
$485$	52559.8			0.223445
$486$	10714.1			0.0453609
$487$	−428565.			−1.80700			−0.903502	−	0.428585i	$-0.859012\pi$
$487$	−428565.			−1.80700			−0.903502	+	0.428585i	$0.859012\pi$
$488$		−	60436.3i		−	0.253781i
$489$	−65010.8			−0.271874
$490$			60968.9i			0.253931i
$491$	130900.			0.542973			0.271486	−	0.962442i	$-0.412485\pi$
$491$	130900.			0.542973			0.271486	+	0.962442i	$0.412485\pi$
$492$	−12594.0			−0.0520274
$493$		−	301687.i		−	1.24126i
$494$		−	334854.i		−	1.37215i
$495$	20733.7			0.0846185
$496$	−37016.4			−0.150463
$497$		−	593617.i		−	2.40322i
$498$		−	165398.i		−	0.666918i
$499$	−148926.			−0.598093			−0.299047	−	0.954239i	$-0.596669\pi$
$499$	−148926.			−0.598093			−0.299047	+	0.954239i	$0.596669\pi$
$500$		−	73392.2i		−	0.293569i
$501$	190325.			0.758264
$502$		−	246650.i		−	0.978752i
$503$			380434.i			1.50364i	0.659369	+	0.751820i	$0.270824\pi$
$503$			380434.i			1.50364i	−0.659369	+	0.751820i	$0.729176\pi$
$504$			44049.4i			0.173412i
$505$			77536.5i			0.304035i
$506$	80459.7	+	125548.i	0.314252	+	0.490354i
$507$	123667.			0.481104
$508$	147990.			0.573463
$509$	−299931.			−1.15767			−0.578836	−	0.815444i	$-0.696493\pi$
$509$	−299931.			−1.15767			−0.578836	+	0.815444i	$0.696493\pi$
$510$	23655.5			0.0909478
$511$			33775.2i			0.129347i
$512$	11585.2			0.0441942
$513$		−	72585.9i		−	0.275815i
$514$	−5747.88			−0.0217561
$515$	−100383.			−0.378481
$516$		−	95625.2i		−	0.359147i
$517$			127850.i			0.478320i
$518$	−384277.			−1.43214
$519$	162614.			0.603702
$520$		−	39895.3i		−	0.147542i
$521$		−	115006.i		−	0.423686i	−0.977304	−	0.211843i	$-0.932053\pi$
$521$		−	115006.i		−	0.423686i	0.977304	−	0.211843i	$-0.0679466\pi$
$522$	−110292.			−0.404764
$523$		−	211108.i		−	0.771795i	−0.922542	−	0.385897i	$-0.873892\pi$
$523$		−	211108.i		−	0.771795i	0.922542	−	0.385897i	$-0.126108\pi$
$524$	−57888.2			−0.210828
$525$			211912.i			0.768843i
$526$			225640.i			0.815539i
$527$		−	120819.i		−	0.435026i
$528$			33142.9i			0.118884i
$529$	116896.	−	254256.i	0.417725	−	0.908574i
$530$	−21349.9			−0.0760052
$531$	−92858.0			−0.329329
$532$	298427.			1.05442
$533$	−69325.6			−0.244028
$534$		−	137530.i		−	0.482297i
$535$	−41992.3			−0.146711
$536$			152151.i			0.529597i
$537$	−119785.			−0.415387
$538$	−88548.8			−0.305927
$539$		−	278810.i		−	0.959691i
$540$		−	8648.06i		−	0.0296573i
$541$	26983.2			0.0921931			0.0460966	−	0.998937i	$-0.485322\pi$
$541$	26983.2			0.0921931			0.0460966	+	0.998937i	$0.485322\pi$
$542$	−55225.8			−0.187994
$543$			96831.0i			0.328409i
$544$			37813.5i			0.127776i
$545$	107607.			0.362284
$546$			242478.i			0.813367i
$547$	−458870.			−1.53361			−0.766805	−	0.641881i	$-0.778154\pi$
$547$	−458870.			−1.53361			−0.766805	+	0.641881i	$0.778154\pi$
$548$		−	194436.i		−	0.647464i
$549$		−	72115.2i		−	0.239267i
$550$			159443.i			0.527085i
$551$			747207.i			2.46115i
$552$	52366.5	−	33560.0i	0.171860	−	0.110140i
$553$	812490.			2.65685
$554$	107273.			0.349520
$555$	75443.6			0.244927
$556$	181830.			0.588188
$557$			158747.i			0.511677i	0.966720	+	0.255839i	$0.0823516\pi$
$557$			158747.i			0.511677i	−0.966720	+	0.255839i	$0.917648\pi$
$558$	−44169.5			−0.141858
$559$		−	526385.i		−	1.68453i
$560$	35555.4			0.113378
$561$	−108176.			−0.343722
$562$		−	246738.i		−	0.781204i
$563$			285531.i			0.900816i	0.892823	+	0.450408i	$0.148721\pi$
$563$			285531.i			0.900816i	−0.892823	+	0.450408i	$0.851279\pi$
$564$	53326.4			0.167643
$565$	18603.7			0.0582777
$566$			306004.i			0.955199i
$567$			52561.7i			0.163494i
$568$	−186294.			−0.577435
$569$			273788.i			0.845650i	0.906211	+	0.422825i	$0.138961\pi$
$569$			273788.i			0.845650i	−0.906211	+	0.422825i	$0.861039\pi$
$570$	−58589.1			−0.180330
$571$		−	76315.0i		−	0.234066i	−0.993128	−	0.117033i	$-0.962662\pi$
$571$		−	76315.0i		−	0.234066i	0.993128	−	0.117033i	$-0.0373383\pi$
$572$			182441.i			0.557609i
$573$		−	46993.2i		−	0.143128i
$574$		−	61784.2i		−	0.187523i
$575$	251924.	−	161450.i	0.761962	−	0.488317i
$576$	13824.0			0.0416667
$577$	−441103.			−1.32492			−0.662458	−	0.749099i	$-0.730487\pi$
$577$	−441103.			−1.32492			−0.662458	+	0.749099i	$0.730487\pi$
$578$	112812.			0.337676
$579$	34418.5			0.102668
$580$			89024.0i			0.264637i
$581$	811420.			2.40377
$582$			100253.i			0.295972i
$583$	97632.7			0.287249
$584$	10599.7			0.0310789
$585$		−	47604.7i		−	0.139104i
$586$			179351.i			0.522285i
$587$	−565485.			−1.64114			−0.820569	−	0.571547i	$-0.806343\pi$
$587$	−565485.			−1.64114			−0.820569	+	0.571547i	$0.806343\pi$
$588$	−116293.			−0.336354
$589$			299241.i			0.862562i
$590$			74952.1i			0.215318i
$591$	14906.8			0.0426785
$592$			120597.i			0.344107i
$593$	−334994.			−0.952636			−0.476318	−	0.879273i	$-0.658029\pi$
$593$	−334994.			−0.952636			−0.476318	+	0.879273i	$0.658029\pi$
$594$			39547.5i			0.112085i
$595$			116051.i			0.327803i
$596$			260604.i			0.733650i
$597$		−	205775.i		−	0.577356i
$598$	288260.	−	184737.i	0.806088	−	0.516595i
$599$	461073.			1.28504			0.642519	−	0.766270i	$-0.277889\pi$
$599$	461073.			1.28504			0.642519	+	0.766270i	$0.277889\pi$
$600$	66504.2			0.184734
$601$	23595.8			0.0653261			0.0326630	−	0.999466i	$-0.489601\pi$
$601$	23595.8			0.0653261			0.0326630	+	0.999466i	$0.489601\pi$
$602$	469123.			1.29448
$603$			181553.i			0.499309i
$604$	226286.			0.620274
$605$		−	36280.1i		−	0.0991191i
$606$	−147894.			−0.402721
$607$	289866.			0.786720			0.393360	−	0.919385i	$-0.371313\pi$
$607$	289866.			0.786720			0.393360	+	0.919385i	$0.371313\pi$
$608$		−	93655.2i		−	0.253352i
$609$		−	541075.i		−	1.45889i
$610$	−58209.1			−0.156434
$611$	293544.			0.786305
$612$			45120.7i			0.120468i
$613$			666644.i			1.77408i	0.461693	+	0.887040i	$0.347242\pi$
$613$			666644.i			1.77408i	−0.461693	+	0.887040i	$0.652758\pi$
$614$	19989.0			0.0530217
$615$			12129.9i			0.0320705i
$616$	−162594.			−0.428493
$617$			170341.i			0.447453i	0.974652	+	0.223727i	$0.0718223\pi$
$617$			170341.i			0.447453i	−0.974652	+	0.223727i	$0.928178\pi$
$618$		−	191470.i		−	0.501331i
$619$			24409.2i			0.0637047i	0.999493	+	0.0318524i	$0.0101406\pi$
$619$			24409.2i			0.0637047i	−0.999493	+	0.0318524i	$0.989859\pi$
$620$			35652.3i			0.0927479i
$621$	62485.9	−	40045.2i	0.162031	−	0.103841i
$622$	17505.5			0.0452474
$623$	674702.			1.73834
$624$	76096.5			0.195432
$625$	282831.			0.724048
$626$			213132.i			0.543875i
$627$	267928.			0.681526
$628$			232151.i			0.588641i
$629$	−393622.			−0.994897
$630$	42426.2			0.106894
$631$			114143.i			0.286676i	0.989674	+	0.143338i	$0.0457837\pi$
$631$			114143.i			0.286676i	−0.989674	+	0.143338i	$0.954216\pi$
$632$		−	254983.i		−	0.638377i
$633$	−369549.			−0.922283
$634$	400451.			0.996257
$635$		−	142537.i		−	0.353491i
$636$		−	40722.9i		−	0.100676i
$637$	−640152.			−1.57763
$638$		−	407106.i		−	1.00015i
$639$	−222294.			−0.544411
$640$		−	11158.3i		−	0.0272420i
$641$		−	695865.i		−	1.69359i	−0.531917	−	0.846797i	$-0.678528\pi$
$641$		−	695865.i		−	1.69359i	0.531917	−	0.846797i	$-0.321472\pi$
$642$		−	80096.5i		−	0.194332i
$643$		−	400685.i		−	0.969128i	−0.874756	−	0.484564i	$-0.838978\pi$
$643$		−	400685.i		−	0.969128i	0.874756	−	0.484564i	$-0.161022\pi$
$644$	164640.	+	256902.i	0.396977	+	0.619436i
$645$	−92101.2			−0.221384
$646$	305685.			0.732502
$647$	307949.			0.735647			0.367824	−	0.929896i	$-0.380103\pi$
$647$	307949.			0.735647			0.367824	+	0.929896i	$0.380103\pi$
$648$	16495.4			0.0392837
$649$		−	342755.i		−	0.813757i
$650$	366084.			0.866471
$651$		−	216689.i		−	0.511300i
$652$	−100091.			−0.235450
$653$	−751786.			−1.76306			−0.881532	−	0.472124i	$-0.843487\pi$
$653$	−751786.			−1.76306			−0.881532	+	0.472124i	$0.843487\pi$
$654$			205251.i			0.479877i
$655$			55755.0i			0.129957i
$656$	−19389.7			−0.0450571
$657$	12648.0			0.0293015
$658$			261612.i			0.604235i
$659$			181668.i			0.418318i	0.977882	+	0.209159i	$0.0670726\pi$
$659$			181668.i			0.418318i	−0.977882	+	0.209159i	$0.932927\pi$
$660$	31921.5			0.0732818
$661$			457771.i			1.04772i	0.851804	+	0.523860i	$0.175509\pi$
$661$			457771.i			1.04772i	−0.851804	+	0.523860i	$0.824491\pi$
$662$	−293573.			−0.669885
$663$			248375.i			0.565041i
$664$		−	254647.i		−	0.577568i
$665$		−	287430.i		−	0.649963i
$666$			143902.i			0.324427i
$667$	−643236.	+	412229.i	−1.44584	+	0.926589i
$668$	293025.			0.656676
$669$	−258083.			−0.576643
$670$	146544.			0.326452
$671$	266190.			0.591217
$672$			67818.6i			0.150179i
$673$	−49642.8			−0.109604			−0.0548019	−	0.998497i	$-0.517453\pi$
$673$	−49642.8			−0.109604			−0.0548019	+	0.998497i	$0.517453\pi$
$674$			151342.i			0.333150i
$675$	79355.7			0.174169
$676$	190398.			0.416648
$677$			680350.i			1.48441i	0.670171	+	0.742207i	$0.266221\pi$
$677$			680350.i			1.48441i	−0.670171	+	0.742207i	$0.733779\pi$
$678$			35484.9i			0.0771940i
$679$	−491826.			−1.06677
$680$	36420.0			0.0787631
$681$			392137.i			0.845559i
$682$		−	163038.i		−	0.350525i
$683$	−217234.			−0.465680			−0.232840	−	0.972515i	$-0.574802\pi$
$683$	−217234.			−0.465680			−0.232840	+	0.972515i	$0.574802\pi$
$684$		−	111753.i		−	0.238863i
$685$	−187271.			−0.399106
$686$		−	80872.3i		−	0.171851i
$687$			91986.5i			0.194900i
$688$		−	147225.i		−	0.311031i
$689$		−	224166.i		−	0.472206i
$690$	−32323.3	−	50436.7i	−0.0678917	−	0.105937i
$691$	−363510.			−0.761307			−0.380654	−	0.924718i	$-0.624301\pi$
$691$	−363510.			−0.761307			−0.380654	+	0.924718i	$0.624301\pi$
$692$	250360.			0.522821
$693$	−194015.			−0.403987
$694$	−561765.			−1.16637
$695$		−	175129.i		−	0.362568i
$696$	−169805.			−0.350536
$697$		−	63286.7i		−	0.130271i
$698$	−187444.			−0.384733
$699$	−131731.			−0.269608
$700$			326260.i			0.665837i
$701$			125589.i			0.255573i	0.991802	+	0.127786i	$0.0407872\pi$
$701$			125589.i			0.255573i	−0.991802	+	0.127786i	$0.959213\pi$
$702$	90801.6			0.184255
$703$	974909.			1.97267
$704$			51026.8i			0.102956i
$705$		−	51361.3i		−	0.103337i
$706$	183323.			0.367796
$707$		−	725546.i		−	1.45153i
$708$	−142964.			−0.285207
$709$		−	674978.i		−	1.34276i	−0.741115	−	0.671378i	$-0.765703\pi$
$709$		−	674978.i		−	1.34276i	0.741115	−	0.671378i	$-0.234297\pi$
$710$			179429.i			0.355939i
$711$		−	304257.i		−	0.601867i
$712$		−	211741.i		−	0.417681i
$713$	−257603.	+	165089.i	−0.506724	+	0.324743i
$714$	−221356.			−0.434205
$715$	175718.			0.343719
$716$	−184421.			−0.359736
$717$	−329903.			−0.641724
$718$			74108.3i			0.143753i
$719$	661813.			1.28020			0.640100	−	0.768292i	$-0.278893\pi$
$719$	661813.			1.28020			0.640100	+	0.768292i	$0.278893\pi$
$720$		−	13314.6i		−	0.0256840i
$721$	939326.			1.80695
$722$	−388506.			−0.745287
$723$		−	329387.i		−	0.630129i
$724$			149081.i			0.284410i
$725$	−816895.			−1.55414
$726$	69200.9			0.131292
$727$		−	692426.i		−	1.31010i	−0.755585	−	0.655050i	$-0.772647\pi$
$727$		−	692426.i		−	1.31010i	0.755585	−	0.655050i	$-0.227353\pi$
$728$			373319.i			0.704396i
$729$	19683.0			0.0370370
$730$		−	10209.0i		−	0.0191575i
$731$	480532.			0.899265
$732$		−	111029.i		−	0.207211i
$733$		−	340091.i		−	0.632976i	−0.948597	−	0.316488i	$-0.897496\pi$
$733$		−	340091.i		−	0.632976i	0.948597	−	0.316488i	$-0.102504\pi$
$734$			141751.i			0.263108i
$735$			112007.i			0.207334i
$736$	80623.5	−	51669.0i	0.148835	−	0.0953837i
$737$	−670145.			−1.23377
$738$	−23136.6			−0.0424802
$739$	−717291.			−1.31343			−0.656714	−	0.754140i	$-0.728054\pi$
$739$	−717291.			−1.31343			−0.656714	+	0.754140i	$0.728054\pi$
$740$	116153.			0.212113
$741$		−	615165.i		−	1.12035i
$742$	199781.			0.362865
$743$			598272.i			1.08373i	0.840466	+	0.541865i	$0.182281\pi$
$743$			598272.i			1.08373i	−0.840466	+	0.541865i	$0.817719\pi$
$744$	−68003.4			−0.122853
$745$	251000.			0.452233
$746$		−	330808.i		−	0.594426i
$747$		−	303856.i		−	0.544536i
$748$	−166549.			−0.297672
$749$	392942.			0.700430
$750$		−	134830.i		−	0.239698i
$751$			572205.i			1.01455i	0.861786	+	0.507273i	$0.169346\pi$
$751$			572205.i			1.01455i	−0.861786	+	0.507273i	$0.830654\pi$
$752$	82101.4			0.145183
$753$		−	453124.i		−	0.799148i
$754$	−934721.			−1.64414
$755$		−	217947.i		−	0.382346i
$756$			80924.0i			0.141590i
$757$		−	831178.i		−	1.45045i	−0.688513	−	0.725224i	$-0.741736\pi$
$757$		−	831178.i		−	1.45045i	0.688513	−	0.725224i	$-0.258264\pi$
$758$		−	193857.i		−	0.337399i
$759$	147814.	+	230647.i	0.256585	+	0.400372i
$760$	−90203.9			−0.156170
$761$	685233.			1.18323			0.591614	−	0.806221i	$-0.298491\pi$
$761$	685233.			1.18323			0.591614	+	0.806221i	$0.298491\pi$
$762$	271875.			0.468231
$763$	−1.00693e6			−1.72962
$764$		−	72350.7i		−	0.123953i
$765$	43457.9			0.0742585
$766$		−	369737.i		−	0.630138i
$767$	−786971.			−1.33773
$768$	21283.4			0.0360844
$769$			1.04368e6i			1.76488i	0.470420	+	0.882442i	$0.344102\pi$
$769$			1.04368e6i			1.76488i	−0.470420	+	0.882442i	$0.655898\pi$
$770$			156603.i			0.264130i
$771$	−10559.5			−0.0177638
$772$	52990.8			0.0889131
$773$			1.10190e6i			1.84409i	0.387078	+	0.922047i	$0.373484\pi$
$773$			1.10190e6i			1.84409i	−0.387078	+	0.922047i	$0.626516\pi$
$774$		−	175675.i		−	0.293243i
$775$	−327150.			−0.544682
$776$			154349.i			0.256320i
$777$	−705961.			−1.16933
$778$			191728.i			0.316757i
$779$			156746.i			0.258299i
$780$		−	73292.3i		−	0.120467i
$781$		−	820528.i		−	1.34521i
$782$	168644.	+	263150.i	0.275777	+	0.430319i
$783$	−202619.			−0.330488
$784$	−179044.			−0.291292
$785$	223596.			0.362847
$786$	−106347.			−0.172140
$787$			451034.i			0.728215i	0.931357	+	0.364107i	$0.118626\pi$
$787$			451034.i			0.728215i	−0.931357	+	0.364107i	$0.881374\pi$
$788$	22950.5			0.0369607
$789$			414527.i			0.665885i
$790$	−245587.			−0.393505
$791$	−174084.			−0.278231
$792$			60887.4i			0.0970682i
$793$		−	611176.i		−	0.971896i
$794$	862854.			1.36866
$795$	−39222.2			−0.0620580
$796$		−	316811.i		−	0.500005i
$797$		−	772691.i		−	1.21644i	−0.793770	−	0.608218i	$-0.791885\pi$
$797$		−	772691.i		−	1.21644i	0.793770	−	0.608218i	$-0.208115\pi$
$798$	548246.			0.860934
$799$			267974.i			0.419758i
$800$	102390.			0.159984
$801$		−	252658.i		−	0.393794i
$802$			381414.i			0.592991i
$803$			46685.9i			0.0724026i
$804$			279519.i			0.432414i
$805$	247435.	−	158573.i	0.381830	−	0.244702i
$806$	−374336.			−0.576225
$807$	−162675.			−0.249789
$808$	−227697.			−0.348767
$809$	−783092.			−1.19651			−0.598254	−	0.801307i	$-0.704139\pi$
$809$	−783092.			−1.19651			−0.598254	+	0.801307i	$0.704139\pi$
$810$		−	15887.5i		−	0.0242151i
$811$	1.08801e6			1.65421			0.827105	−	0.562048i	$-0.189986\pi$
$811$	1.08801e6			1.65421			0.827105	+	0.562048i	$0.189986\pi$
$812$		−	833039.i		−	1.26344i
$813$	−101456.			−0.153496
$814$	−531167.			−0.801645
$815$			96402.1i			0.145135i
$816$			69467.9i			0.104329i
$817$	−1.19017e6			−1.78305
$818$	−346485.			−0.517819
$819$			445460.i			0.664111i
$820$			18675.1i			0.0277739i
$821$	−681755.			−1.01144			−0.505722	−	0.862696i	$-0.668774\pi$
$821$	−681755.			−1.01144			−0.505722	+	0.862696i	$0.668774\pi$
$822$		−	357202.i		−	0.528652i
$823$	−226095.			−0.333805			−0.166902	−	0.985973i	$-0.553376\pi$
$823$	−226095.			−0.333805			−0.166902	+	0.985973i	$0.553376\pi$
$824$		−	294788.i		−	0.434166i
$825$			292916.i			0.430363i
$826$		−	701362.i		−	1.02797i
$827$			519090.i			0.758982i	0.925195	+	0.379491i	$0.123901\pi$
$827$			519090.i			0.758982i	−0.925195	+	0.379491i	$0.876099\pi$
$828$	96203.4	−	61653.6i	0.140323	−	0.0899286i
$829$	435193.			0.633246			0.316623	−	0.948551i	$-0.397451\pi$
$829$	435193.			0.633246			0.316623	+	0.948551i	$0.397451\pi$
$830$	−245263.			−0.356021
$831$	197073.			0.285382
$832$	117158.			0.169249
$833$		−	584389.i		−	0.842194i
$834$	334043.			0.480254
$835$		−	282226.i		−	0.404785i
$836$	412502.			0.590219
$837$	−81144.6			−0.115827
$838$			66065.3i			0.0940775i
$839$		−	624392.i		−	0.887020i	−0.896270	−	0.443510i	$-0.853733\pi$
$839$		−	624392.i		−	0.887020i	0.896270	−	0.443510i	$-0.146267\pi$
$840$	65319.3			0.0925728
$841$	1.37849e6			1.94901
$842$			38537.6i			0.0543576i
$843$		−	453288.i		−	0.637850i
$844$	−568957.			−0.798720
$845$		−	183382.i		−	0.256828i
$846$	97966.9			0.136880
$847$			339490.i			0.473216i
$848$		−	62697.0i		−	0.0871877i
$849$			562164.i			0.779916i
$850$			334195.i			0.462553i
$851$	537851.	+	839255.i	0.742682	+	1.15887i
$852$	−342244.			−0.471473
$853$	688720.			0.946552			0.473276	−	0.880914i	$-0.343071\pi$
$853$	688720.			0.946552			0.473276	+	0.880914i	$0.343071\pi$
$854$	544690.			0.746851
$855$	−107635.			−0.147239
$856$		−	123317.i		−	0.168296i
$857$	467260.			0.636205			0.318102	−	0.948056i	$-0.396954\pi$
$857$	467260.			0.636205			0.318102	+	0.948056i	$0.396954\pi$
$858$			335165.i			0.455286i
$859$	771074.			1.04498			0.522492	−	0.852644i	$-0.325002\pi$
$859$	771074.			1.04498			0.522492	+	0.852644i	$0.325002\pi$
$860$	−141799.			−0.191724
$861$		−	113505.i		−	0.153111i
$862$		−	232968.i		−	0.313531i
$863$	−112834.			−0.151501			−0.0757507	−	0.997127i	$-0.524135\pi$
$863$	−112834.			−0.151501			−0.0757507	+	0.997127i	$0.524135\pi$
$864$	25396.3			0.0340207
$865$		−	241134.i		−	0.322275i
$866$			377289.i			0.503082i
$867$	207249.			0.275711
$868$		−	333615.i		−	0.442799i
$869$	1.12307e6			1.48719
$870$			163548.i			0.216075i
$871$			1.53866e6i			2.02818i
$872$			316005.i			0.415586i
$873$			184176.i			0.241660i
$874$	−417693.	−	651761.i	−0.546807	−	0.853229i
$875$	661457.			0.863944
$876$	19472.8			0.0253758
$877$	−785321.			−1.02105			−0.510526	−	0.859862i	$-0.670549\pi$
$877$	−785321.			−1.02105			−0.510526	+	0.859862i	$0.670549\pi$
$878$	350514.			0.454690
$879$			329488.i			0.426444i
$880$	49146.4			0.0634639
$881$			955717.i			1.23134i	0.788004	+	0.615669i	$0.211114\pi$
$881$			955717.i			1.23134i	−0.788004	+	0.615669i	$0.788886\pi$
$882$	−213643.			−0.274632
$883$	−581246.			−0.745484			−0.372742	−	0.927935i	$-0.621582\pi$
$883$	−581246.			−0.745484			−0.372742	+	0.927935i	$0.621582\pi$
$884$			382398.i			0.489340i
$885$			137696.i			0.175806i
$886$	221572.			0.282259
$887$	721565.			0.917124			0.458562	−	0.888662i	$-0.348365\pi$
$887$	721565.			0.917124			0.458562	+	0.888662i	$0.348365\pi$
$888$			221551.i			0.280962i
$889$			1.33378e6i			1.68764i
$890$	−203938.			−0.257465
$891$			72653.5i			0.0915168i
$892$	−397345.			−0.499388
$893$		−	663709.i		−	0.832290i
$894$			478760.i			0.599022i
$895$			177624.i			0.221746i
$896$			104413.i			0.130059i
$897$	529568.	−	339383.i	0.658168	−	0.421798i
$898$	−416714.			−0.516756
$899$	835310.			1.03354
$900$	122176.			0.150835
$901$	204639.			0.252081
$902$		−	85401.3i		−	0.104967i
$903$	861835.			1.05694
$904$			54632.5i			0.0668520i
$905$	143587.			0.175315
$906$	415714.			0.506452
$907$		−	1.41735e6i		−	1.72291i	−0.507834	−	0.861455i	$-0.669554\pi$
$907$		−	1.41735e6i		−	1.72291i	0.507834	−	0.861455i	$-0.330446\pi$
$908$			603735.i			0.732276i
$909$	−271698.			−0.328821
$910$	359561.			0.434200
$911$		−	230045.i		−	0.277189i	−0.990349	−	0.138594i	$-0.955742\pi$
$911$		−	230045.i		−	0.277189i	0.990349	−	0.138594i	$-0.0442584\pi$
$912$		−	172056.i		−	0.206861i
$913$	1.12159e6			1.34552
$914$		−	370962.i		−	0.444056i
$915$	−106937.			−0.127728
$916$			141623.i			0.168788i
$917$		−	521725.i		−	0.620445i
$918$			82892.0i			0.0983620i
$919$		−	757945.i		−	0.897442i	−0.893672	−	0.448721i	$-0.851880\pi$
$919$		−	757945.i		−	0.897442i	0.893672	−	0.448721i	$-0.148120\pi$
$920$	−49764.9	−	77652.4i	−0.0587960	−	0.0917443i
$921$	36722.1			0.0432920
$922$	344825.			0.405636
$923$	−1.88394e6			−2.21138
$924$	−298705.			−0.349863
$925$			1.06583e6i			1.24568i
$926$	1.11582e6			1.30128
$927$		−	351754.i		−	0.409335i
$928$	−261432.			−0.303573
$929$	384242.			0.445219			0.222609	−	0.974908i	$-0.428543\pi$
$929$	384242.			0.445219			0.222609	+	0.974908i	$0.428543\pi$
$930$			65497.4i			0.0757283i
$931$			1.44739e6i			1.66989i
$932$	−202813.			−0.233487
$933$	32159.7			0.0369444
$934$		−	106828.i		−	0.122460i
$935$			160411.i			0.183489i
$936$	139798.			0.159570
$937$			524670.i			0.597595i	0.954317	+	0.298797i	$0.0965855\pi$
$937$			524670.i			0.597595i	−0.954317	+	0.298797i	$0.903415\pi$
$938$	−1.37128e6			−1.55855
$939$			391548.i			0.444072i
$940$		−	79075.9i		−	0.0894928i
$941$			720611.i			0.813807i	0.913471	+	0.406904i	$0.133392\pi$
$941$			720611.i			0.813807i	−0.913471	+	0.406904i	$0.866608\pi$
$942$			426488.i			0.480623i
$943$	−134936.	+	86476.0i	−0.151741	+	0.0972461i
$944$	−220108.			−0.246997
$945$	77941.8			0.0872784
$946$	648446.			0.724589
$947$	−828915.			−0.924294			−0.462147	−	0.886803i	$-0.652921\pi$
$947$	−828915.			−0.924294			−0.462147	+	0.886803i	$0.652921\pi$
$948$		−	468434.i		−	0.521232i
$949$	107191.			0.119022
$950$		−	827722.i		−	0.917143i
$951$	735676.			0.813440
$952$	−340799.			−0.376032
$953$		−	1.25385e6i		−	1.38057i	−0.723537	−	0.690285i	$-0.757485\pi$
$953$		−	1.25385e6i		−	1.38057i	0.723537	−	0.690285i	$-0.242515\pi$
$954$		−	74812.7i		−	0.0822013i
$955$	−69684.5			−0.0764063
$956$	−507920.			−0.555750
$957$		−	747902.i		−	0.816621i
$958$			981318.i			1.06925i
$959$	1.75238e6			1.90542
$960$		−	20499.1i		−	0.0222430i
$961$	−588997.			−0.637773
$962$			1.21957e6i			1.31782i
$963$		−	147147.i		−	0.158671i
$964$		−	507124.i		−	0.545708i
$965$		−	51038.0i		−	0.0548074i
$966$	302464.	+	471960.i	0.324130	+	0.505768i
$967$	1.38784e6			1.48418			0.742092	−	0.670298i	$-0.233834\pi$
$967$	1.38784e6			1.48418			0.742092	+	0.670298i	$0.233834\pi$
$968$	106542.			0.113702
$969$	561579.			0.598086
$970$	148661.			0.157999
$971$		−	571092.i		−	0.605714i	−0.953036	−	0.302857i	$-0.902060\pi$
$971$		−	571092.i		−	0.605714i	0.953036	−	0.302857i	$-0.0979404\pi$
$972$	30304.0			0.0320750
$973$			1.63877e6i			1.73098i
$974$	−1.21217e6			−1.27774
$975$	672539.			0.707470
$976$		−	170940.i		−	0.179450i
$977$			291602.i			0.305493i	0.988265	+	0.152746i	$0.0488117\pi$
$977$			291602.i			0.305493i	−0.988265	+	0.152746i	$0.951188\pi$
$978$	−183878.			−0.192244
$979$	932607.			0.973046
$980$			172446.i			0.179557i
$981$			377070.i			0.391818i
$982$	370242.			0.383940
$983$			1.04146e6i			1.07780i	0.842370	+	0.538899i	$0.181160\pi$
$983$			1.04146e6i			1.07780i	−0.842370	+	0.538899i	$0.818840\pi$
$984$	−35621.1			−0.0367889
$985$		−	22104.8i		−	0.0227831i
$986$		−	853299.i		−	0.877702i
$987$			480612.i			0.493355i
$988$		−	947109.i		−	0.970255i
$989$	−656607.	−	1.02456e6i	−0.671294	−	1.04748i
$990$	58643.6			0.0598343
$991$	885848.			0.902011			0.451006	−	0.892521i	$-0.351065\pi$
$991$	885848.			0.902011			0.451006	+	0.892521i	$0.351065\pi$
$992$	−104698.			−0.106394
$993$	−539328.			−0.546959
$994$		−	1.67900e6i		−	1.69933i
$995$	−305136.			−0.308211
$996$		−	467817.i		−	0.471582i
$997$	334176.			0.336190			0.168095	−	0.985771i	$-0.446238\pi$
$997$	334176.			0.336190			0.168095	+	0.985771i	$0.446238\pi$
$998$	−421226.			−0.422916
$999$			264364.i			0.264894i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.b.a.91.14	✓	16
3.2	odd	2		414.5.b.b.91.6		16
4.3	odd	2		1104.5.c.a.1057.4		16
23.22	odd	2	inner	138.5.b.a.91.15	yes	16
69.68	even	2		414.5.b.b.91.3		16
92.91	even	2		1104.5.c.a.1057.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.b.a.91.14	✓	16	1.1	even	1	trivial
138.5.b.a.91.15	yes	16	23.22	odd	2	inner
414.5.b.b.91.3		16	69.68	even	2
414.5.b.b.91.6		16	3.2	odd	2
1104.5.c.a.1057.4		16	4.3	odd	2
1104.5.c.a.1057.5		16	92.91	even	2

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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -7.70519i q^{5} +14.6969 q^{6} +72.1011i q^{7} +22.6274 q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} -7.70519i q^{5} +14.6969 q^{6} +72.1011i q^{7} +22.6274 q^{8} +27.0000 q^{9} -21.7936i q^{10} +99.6618i q^{11} +41.5692 q^{12} +228.825 q^{13} +203.933i q^{14} -40.0373i q^{15} +64.0000 q^{16} +208.892i q^{17} +76.3675 q^{18} -517.377i q^{19} -61.6415i q^{20} +374.648i q^{21} +281.886i q^{22} +(445.386 - 285.433i) q^{23} +117.576 q^{24} +565.630 q^{25} +647.214 q^{26} +140.296 q^{27} +576.809i q^{28} -1444.22 q^{29} -113.243i q^{30} -578.381 q^{31} +181.019 q^{32} +517.858i q^{33} +590.836i q^{34} +555.552 q^{35} +216.000 q^{36} +1884.33i q^{37} -1463.36i q^{38} +1189.01 q^{39} -174.349i q^{40} -302.964 q^{41} +1059.66i q^{42} -2300.38i q^{43} +797.294i q^{44} -208.040i q^{45} +(1259.74 - 807.328i) q^{46} +1282.83 q^{47} +332.554 q^{48} -2797.56 q^{49} +1599.84 q^{50} +1085.44i q^{51} +1830.60 q^{52} -979.640i q^{53} +396.817 q^{54} +767.913 q^{55} +1631.46i q^{56} -2688.37i q^{57} -4084.88 q^{58} -3439.18 q^{59} -320.299i q^{60} -2670.93i q^{61} -1635.91 q^{62} +1946.73i q^{63} +512.000 q^{64} -1763.14i q^{65} +1464.72i q^{66} +6724.19i q^{67} +1671.14i q^{68} +(2314.29 - 1483.16i) q^{69} +1571.34 q^{70} -8233.12 q^{71} +610.940 q^{72} +468.443 q^{73} +5329.69i q^{74} +2939.10 q^{75} -4139.01i q^{76} -7185.72 q^{77} +3363.02 q^{78} -11268.8i q^{79} -493.132i q^{80} +729.000 q^{81} -856.911 q^{82} -11253.9i q^{83} +2997.19i q^{84} +1609.55 q^{85} -6506.47i q^{86} -7504.40 q^{87} +2255.09i q^{88} -9357.72i q^{89} -588.426i q^{90} +16498.5i q^{91} +(3563.09 - 2283.47i) q^{92} -3005.36 q^{93} +3628.40 q^{94} -3986.49 q^{95} +940.604 q^{96} +6821.35i q^{97} -7912.71 q^{98} +2690.87i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) \) 16 * q + 128 * q^4 + 432 * q^9	\( 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) \) 16 * q + 128 * q^4 + 432 * q^9 - 208 * q^13 + 1024 * q^16 + 840 * q^23 + 1056 * q^25 + 1920 * q^26 + 3600 * q^29 + 224 * q^31 - 3264 * q^35 + 3456 * q^36 - 2016 * q^39 - 6144 * q^41 + 1280 * q^46 + 8880 * q^47 - 13888 * q^49 + 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 - 18240 * q^59 + 8192 * q^64 + 10584 * q^69 + 19584 * q^70 - 30048 * q^71 + 9536 * q^73 - 4176 * q^75 + 14160 * q^77 + 6912 * q^78 + 11664 * q^81 - 19584 * q^82 - 32496 * q^85 - 8064 * q^87 + 6720 * q^92 - 11952 * q^93 - 21248 * q^94 - 20064 * q^95 + 21504 * q^98

Introduction

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