LMFDB - Embedded newform 208.8.w.b.17.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,8,Mod(17,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.17"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	208.w (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} + 22180 x^{14} + 184473654 x^{12} + 707524481236 x^{10} + \cdots + 18\!\cdots\!76 $ x^16 + 22180x^14 + 184473654x^12 + 707524481236x^10 + 1235305573179121x^8 + 834592841081820432x^6 + 142296444753813668448x^4 + 5907885179463590558976*x^2 + 18970385728293762437376
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{36}\cdot 3^{4}\cdot 13^{2} $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.3
Root			$-32.3527i$ of defining polynomial
Character	$\chi$	$=$	208.17
Dual form			208.8.w.b.49.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-16.1763 + 28.0182i) q^{3} -211.408i q^{5} +(-635.796 + 367.077i) q^{7} +(570.153 + 987.534i) q^{9} +(7445.38 + 4298.59i) q^{11} +(-6928.10 - 3840.57i) q^{13} +(5923.28 + 3419.81i) q^{15} +(-8168.98 - 14149.1i) q^{17} +(-3033.16 + 1751.20i) q^{19} -23751.8i q^{21} +(32638.2 - 56531.0i) q^{23} +33431.5 q^{25} -107647. q^{27} +(56295.7 - 97507.0i) q^{29} +176071. i q^{31} +(-240878. + 139071. i) q^{33} +(77603.1 + 134412. i) q^{35} +(432385. + 249638. i) q^{37} +(219677. - 131987. i) q^{39} +(-250415. - 144577. i) q^{41} +(201010. + 348160. i) q^{43} +(208773. - 120535. i) q^{45} +64966.5i q^{47} +(-142281. + 246438. i) q^{49} +528576. q^{51} +263903. q^{53} +(908757. - 1.57401e6i) q^{55} -113312. i q^{57} +(-2.22659e6 + 1.28552e6i) q^{59} +(269475. + 466745. i) q^{61} +(-725002. - 418580. i) q^{63} +(-811929. + 1.46466e6i) q^{65} +(-771672. - 445525. i) q^{67} +(1.05593e6 + 1.82893e6i) q^{69} +(-3.42530e6 + 1.97760e6i) q^{71} +2.77895e6i q^{73} +(-540800. + 936692. i) q^{75} -6.31165e6 q^{77} -4.53826e6 q^{79} +(494411. - 856345. i) q^{81} +2.61022e6i q^{83} +(-2.99123e6 + 1.72699e6i) q^{85} +(1.82131e6 + 3.15461e6i) q^{87} +(-6.82257e6 - 3.93901e6i) q^{89} +(5.81464e6 - 101323. i) q^{91} +(-4.93320e6 - 2.84818e6i) q^{93} +(370217. + 641235. i) q^{95} +(7.91553e6 - 4.57003e6i) q^{97} +9.80341e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2520 q^{7} - 4684 q^{9} + 8496 q^{11} - 3620 q^{13} - 51648 q^{15} + 41520 q^{17} + 54432 q^{19} + 7560 q^{23} - 273960 q^{25} - 133920 q^{27} - 346056 q^{29} + 486300 q^{33} + 283248 q^{35} - 68280 q^{37}+ \cdots + 83706300 q^{97}+O(q^{100}) $ 16 * q - 2520 * q^7 - 4684 * q^9 + 8496 * q^11 - 3620 * q^13 - 51648 * q^15 + 41520 * q^17 + 54432 * q^19 + 7560 * q^23 - 273960 * q^25 - 133920 * q^27 - 346056 * q^29 + 486300 * q^33 + 283248 * q^35 - 68280 * q^37 + 1122680 * q^39 - 1288008 * q^41 - 285360 * q^43 + 5743980 * q^45 - 2131660 * q^49 - 957280 * q^51 + 3666600 * q^53 + 1091472 * q^55 - 1799280 * q^59 - 4626000 * q^61 + 8609760 * q^63 - 11518020 * q^65 + 7094400 * q^67 + 18027316 * q^69 - 25958520 * q^71 + 10898032 * q^75 - 20577480 * q^77 - 34143488 * q^79 - 15745864 * q^81 + 7123932 * q^85 + 10460440 * q^87 + 10609452 * q^89 + 38683952 * q^91 - 52747080 * q^93 - 33837504 * q^95 + 83706300 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	−16.1763	+	28.0182i	−0.345904	+	0.599123i	−0.985518	−	0.169573i	$-0.945761\pi$
$3$	−16.1763	+	28.0182i	−0.345904	+	0.599123i	0.639614	+	0.768697i	$0.279094\pi$
$4$		0			0
$5$		−	211.408i		−	0.756357i	−0.925733	−	0.378179i	$-0.876551\pi$
$5$		−	211.408i		−	0.756357i	0.925733	−	0.378179i	$-0.123449\pi$
$6$		0			0
$7$	−635.796	+	367.077i	−0.700607	+	0.404496i	−0.807573	−	0.589767i	$-0.799220\pi$
$7$	−635.796	+	367.077i	−0.700607	+	0.404496i	0.106967	+	0.994263i	$0.465886\pi$
$8$		0			0
$9$	570.153	+	987.534i	0.260701	+	0.451547i
$10$		0			0
$11$	7445.38	+	4298.59i	1.68660	+	0.973759i	0.957091	+	0.289789i	$0.0935851\pi$
$11$	7445.38	+	4298.59i	1.68660	+	0.973759i	0.729510	+	0.683970i	$0.239748\pi$
$12$		0			0
$13$	−6928.10	−	3840.57i	−0.874605	−	0.484836i
$13$	−6928.10	−	3840.57i	−0.874605	−	0.484836i
$14$		0			0
$15$	5923.28	+	3419.81i	0.453151	+	0.261627i
$16$		0			0
$17$	−8168.98	−	14149.1i	−0.403271	−	0.698485i	0.590848	−	0.806783i	$-0.298793\pi$
$17$	−8168.98	−	14149.1i	−0.403271	−	0.698485i	−0.994118	+	0.108298i	$0.965460\pi$
$18$		0			0
$19$	−3033.16	+	1751.20i	−0.101451	+	0.0585730i	−0.549867	−	0.835252i	$-0.685322\pi$
$19$	−3033.16	+	1751.20i	−0.101451	+	0.0585730i	0.448416	+	0.893825i	$0.351988\pi$
$20$		0			0
$21$		−	23751.8i		−	0.559666i
$22$		0			0
$23$	32638.2	−	56531.0i	0.559344	−	0.968812i	−0.438207	−	0.898874i	$-0.644386\pi$
$23$	32638.2	−	56531.0i	0.559344	−	0.968812i	0.997551	−	0.0699383i	$-0.0222802\pi$
$24$		0			0
$25$	33431.5			0.427924
$26$		0			0
$27$	−107647.			−1.05252
$28$		0			0
$29$	56295.7	−	97507.0i	0.428630	−	0.742408i	−0.568122	−	0.822944i	$-0.692330\pi$
$29$	56295.7	−	97507.0i	0.428630	−	0.742408i	0.996752	+	0.0805360i	$0.0256632\pi$
$30$		0			0
$31$			176071.i			1.06150i	0.847527	+	0.530752i	$0.178091\pi$
$31$			176071.i			1.06150i	−0.847527	+	0.530752i	$0.821909\pi$
$32$		0			0
$33$	−240878.	+	139071.i	−1.16680	+	0.673654i
$34$		0			0
$35$	77603.1	+	134412.i	0.305943	+	0.529909i
$36$		0			0
$37$	432385.	+	249638.i	1.40335	+	0.810222i	0.994734	−	0.102486i	$-0.0326798\pi$
$37$	432385.	+	249638.i	1.40335	+	0.810222i	0.408611	+	0.912708i	$0.366013\pi$
$38$		0			0
$39$	219677.	−	131987.i	0.593006	−	0.356290i
$40$		0			0
$41$	−250415.	−	144577.i	−0.567436	−	0.327610i	0.188688	−	0.982037i	$-0.439576\pi$
$41$	−250415.	−	144577.i	−0.567436	−	0.327610i	−0.756125	+	0.654427i	$0.772910\pi$
$42$		0			0
$43$	201010.	+	348160.i	0.385548	+	0.667788i	0.991845	−	0.127450i	$-0.0406792\pi$
$43$	201010.	+	348160.i	0.385548	+	0.667788i	−0.606297	+	0.795238i	$0.707346\pi$
$44$		0			0
$45$	208773.	−	120535.i	0.341531	−	0.197183i
$46$		0			0
$47$			64966.5i			0.0912739i	0.998958	+	0.0456370i	$0.0145317\pi$
$47$			64966.5i			0.0912739i	−0.998958	+	0.0456370i	$0.985468\pi$
$48$		0			0
$49$	−142281.	+	246438.i	−0.172767	+	0.299241i
$50$		0			0
$51$	528576.			0.557971
$52$		0			0
$53$	263903.			0.243489			0.121744	−	0.992561i	$-0.461151\pi$
$53$	263903.			0.243489			0.121744	+	0.992561i	$0.461151\pi$
$54$		0			0
$55$	908757.	−	1.57401e6i	0.736510	−	1.27567i
$56$		0			0
$57$		−	113312.i		−	0.0810425i
$58$		0			0
$59$	−2.22659e6	+	1.28552e6i	−1.41143	+	0.814887i	−0.995523	−	0.0945221i	$-0.969868\pi$
$59$	−2.22659e6	+	1.28552e6i	−1.41143	+	0.814887i	−0.415903	+	0.909409i	$0.636534\pi$
$60$		0			0
$61$	269475.	+	466745.i	0.152007	+	0.263285i	0.931965	−	0.362547i	$-0.118093\pi$
$61$	269475.	+	466745.i	0.152007	+	0.263285i	−0.779958	+	0.625832i	$0.784760\pi$
$62$		0			0
$63$	−725002.	−	418580.i	−0.365298	−	0.210905i
$64$		0			0
$65$	−811929.	+	1.46466e6i	−0.366709	+	0.661514i
$66$		0			0
$67$	−771672.	−	445525.i	−0.313452	−	0.180972i	0.335018	−	0.942212i	$-0.391258\pi$
$67$	−771672.	−	445525.i	−0.313452	−	0.180972i	−0.648470	+	0.761240i	$0.724591\pi$
$68$		0			0
$69$	1.05593e6	+	1.82893e6i	0.386959	+	0.670232i
$70$		0			0
$71$	−3.42530e6	+	1.97760e6i	−1.13578	+	0.655744i	−0.945382	−	0.325963i	$-0.894311\pi$
$71$	−3.42530e6	+	1.97760e6i	−1.13578	+	0.655744i	−0.190399	+	0.981707i	$0.560978\pi$
$72$		0			0
$73$			2.77895e6i			0.836087i	0.908427	+	0.418043i	$0.137284\pi$
$73$			2.77895e6i			0.836087i	−0.908427	+	0.418043i	$0.862716\pi$
$74$		0			0
$75$	−540800.	+	936692.i	−0.148021	+	0.256379i
$76$		0			0
$77$	−6.31165e6			−1.57553
$78$		0			0
$79$	−4.53826e6			−1.03561			−0.517803	−	0.855500i	$-0.673250\pi$
$79$	−4.53826e6			−1.03561			−0.517803	+	0.855500i	$0.673250\pi$
$80$		0			0
$81$	494411.	−	856345.i	0.103369	−	0.179040i
$82$		0			0
$83$			2.61022e6i			0.501076i	0.968107	+	0.250538i	$0.0806074\pi$
$83$			2.61022e6i			0.501076i	−0.968107	+	0.250538i	$0.919393\pi$
$84$		0			0
$85$	−2.99123e6	+	1.72699e6i	−0.528304	+	0.305017i
$86$		0			0
$87$	1.82131e6	+	3.15461e6i	0.296529	+	0.513604i
$88$		0			0
$89$	−6.82257e6	−	3.93901e6i	−1.02585	−	0.592274i	−0.110056	−	0.993925i	$-0.535103\pi$
$89$	−6.82257e6	−	3.93901e6i	−1.02585	−	0.592274i	−0.915793	+	0.401652i	$0.868436\pi$
$90$		0			0
$91$	5.81464e6	−	101323.i	0.808868	−	0.0140949i
$92$		0			0
$93$	−4.93320e6	−	2.84818e6i	−0.635972	−	0.367179i
$94$		0			0
$95$	370217.	+	641235.i	0.0443021	+	0.0767335i
$96$		0			0
$97$	7.91553e6	−	4.57003e6i	0.880600	−	0.508415i	0.00974386	−	0.999953i	$-0.496898\pi$
$97$	7.91553e6	−	4.57003e6i	0.880600	−	0.508415i	0.870856	+	0.491538i	$0.163565\pi$
$98$		0			0
$99$			9.80341e6i			1.01544i
$100$		0			0
$101$	−2.87415e6	+	4.97817e6i	−0.277578	+	0.480779i	−0.970782	−	0.239962i	$-0.922865\pi$
$101$	−2.87415e6	+	4.97817e6i	−0.277578	+	0.480779i	0.693205	+	0.720741i	$0.256198\pi$
$102$		0			0
$103$	9.21760e6			0.831166			0.415583	−	0.909555i	$-0.363578\pi$
$103$	9.21760e6			0.831166			0.415583	+	0.909555i	$0.363578\pi$
$104$		0			0
$105$	−5.02133e6			−0.423308
$106$		0			0
$107$	−8.35748e6	+	1.44756e7i	−0.659526	+	1.14233i	0.321212	+	0.947007i	$0.395910\pi$
$107$	−8.35748e6	+	1.44756e7i	−0.659526	+	1.14233i	−0.980738	+	0.195325i	$0.937424\pi$
$108$		0			0
$109$			1.42886e7i			1.05681i	0.848993	+	0.528403i	$0.177209\pi$
$109$			1.42886e7i			1.05681i	−0.848993	+	0.528403i	$0.822791\pi$
$110$		0			0
$111$	−1.39888e7	+	8.07644e6i	−0.970846	+	0.560518i
$112$		0			0
$113$	1.09827e7	+	1.90225e7i	0.716032	+	1.24020i	0.962560	+	0.271069i	$0.0873770\pi$
$113$	1.09827e7	+	1.90225e7i	0.716032	+	1.24020i	−0.246528	+	0.969136i	$0.579290\pi$
$114$		0			0
$115$	−1.19511e7	−	6.89999e6i	−0.732768	−	0.423064i
$116$		0			0
$117$	−157377.	−	9.03144e6i	−0.00908428	−	0.521323i
$118$		0			0
$119$	1.03876e7	+	5.99728e6i	0.565068	+	0.326242i
$120$		0			0
$121$	2.72122e7	+	4.71328e7i	1.39641	+	2.41866i
$122$		0			0
$123$	8.10159e6	−	4.67746e6i	0.392557	−	0.226643i
$124$		0			0
$125$		−	2.35840e7i		−	1.08002i
$126$		0			0
$127$	1.07006e7	−	1.85339e7i	0.463546	−	0.802886i	−0.535588	−	0.844479i	$-0.679910\pi$
$127$	1.07006e7	−	1.85339e7i	0.463546	−	0.802886i	0.999135	+	0.0415933i	$0.0132434\pi$
$128$		0			0
$129$	−1.30064e7			−0.533450
$130$		0			0
$131$	−2.79550e7			−1.08645			−0.543226	−	0.839587i	$-0.682797\pi$
$131$	−2.79550e7			−1.08645			−0.543226	+	0.839587i	$0.682797\pi$
$132$		0			0
$133$	1.28565e6	−	2.22681e6i	0.0473850	−	0.0820732i
$134$		0			0
$135$			2.27575e7i			0.796079i
$136$		0			0
$137$	−3.87791e7	+	2.23891e7i	−1.28848	+	0.743902i	−0.978382	−	0.206804i	$-0.933694\pi$
$137$	−3.87791e7	+	2.23891e7i	−1.28848	+	0.743902i	−0.310093	+	0.950706i	$0.600360\pi$
$138$		0			0
$139$	1.19723e7	+	2.07367e7i	0.378117	+	0.654918i	0.990788	−	0.135420i	$-0.0432383\pi$
$139$	1.19723e7	+	2.07367e7i	0.378117	+	0.654918i	−0.612671	+	0.790338i	$0.709905\pi$
$140$		0			0
$141$	−1.82024e6	−	1.05092e6i	−0.0546843	−	0.0315720i
$142$		0			0
$143$	−3.50732e7	−	5.83756e7i	−1.00300	−	1.66938i
$144$		0			0
$145$	−2.06138e7	−	1.19014e7i	−0.561526	−	0.324197i
$146$		0			0
$147$	−4.60316e6	−	7.97291e6i	−0.119521	−	0.207017i
$148$		0			0
$149$	3.40494e7	−	1.96584e7i	0.843253	−	0.486852i	−0.0151156	−	0.999886i	$-0.504812\pi$
$149$	3.40494e7	−	1.96584e7i	0.843253	−	0.486852i	0.858369	+	0.513033i	$0.171478\pi$
$150$		0			0
$151$			5.45163e7i			1.28857i	0.764786	+	0.644284i	$0.222845\pi$
$151$			5.45163e7i			1.28857i	−0.764786	+	0.644284i	$0.777155\pi$
$152$		0			0
$153$	9.31513e6	−	1.61343e7i	0.210266	−	0.364191i
$154$		0			0
$155$	3.72229e7			0.802877
$156$		0			0
$157$	−6.58173e7			−1.35735			−0.678674	−	0.734440i	$-0.737445\pi$
$157$	−6.58173e7			−1.35735			−0.678674	+	0.734440i	$0.737445\pi$
$158$		0			0
$159$	−4.26898e6	+	7.39409e6i	−0.0842237	+	0.145880i
$160$		0			0
$161$			4.79229e7i			0.905009i
$162$		0			0
$163$	−5.64786e7	+	3.26079e7i	−1.02147	+	0.589748i	−0.914531	−	0.404517i	$-0.867440\pi$
$163$	−5.64786e7	+	3.26079e7i	−1.02147	+	0.589748i	−0.106944	+	0.994265i	$0.534106\pi$
$164$		0			0
$165$	2.94007e7	+	5.09235e7i	0.509523	+	0.882520i
$166$		0			0
$167$	5.48034e7	+	3.16407e7i	0.910541	+	0.525701i	0.880605	−	0.473851i	$-0.157136\pi$
$167$	5.48034e7	+	3.16407e7i	0.910541	+	0.525701i	0.0299360	+	0.999552i	$0.490470\pi$
$168$		0			0
$169$	3.32485e7	+	5.32157e7i	0.529869	+	0.848079i
$170$		0			0
$171$	−3.45873e6	−	1.99690e6i	−0.0528969	−	0.0305401i
$172$		0			0
$173$	−3.84579e7	−	6.66111e7i	−0.564709	−	0.978104i	−0.997077	−	0.0764069i	$-0.975655\pi$
$173$	−3.84579e7	−	6.66111e7i	−0.564709	−	0.978104i	0.432368	−	0.901697i	$-0.357678\pi$
$174$		0			0
$175$	−2.12556e7	+	1.22719e7i	−0.299806	+	0.173093i
$176$		0			0
$177$		−	8.31801e7i		−	1.12749i
$178$		0			0
$179$	4.32366e7	−	7.48880e7i	0.563464	−	0.975948i	−0.433727	−	0.901044i	$-0.642802\pi$
$179$	4.32366e7	−	7.48880e7i	0.563464	−	0.975948i	0.997191	−	0.0749036i	$-0.0238649\pi$
$180$		0			0
$181$	−8.42468e7			−1.05603			−0.528017	−	0.849234i	$-0.677064\pi$
$181$	−8.42468e7			−1.05603			−0.528017	+	0.849234i	$0.677064\pi$
$182$		0			0
$183$	−1.74365e7			−0.210320
$184$		0			0
$185$	5.27755e7	−	9.14098e7i	0.612817	−	1.06143i
$186$		0			0
$187$		−	1.40460e8i		−	1.57075i
$188$		0			0
$189$	6.84416e7	−	3.95148e7i	0.737401	−	0.425739i
$190$		0			0
$191$	−1.60963e7	−	2.78796e7i	−0.167151	−	0.289514i	0.770266	−	0.637723i	$-0.220123\pi$
$191$	−1.60963e7	−	2.78796e7i	−0.167151	−	0.289514i	−0.937417	+	0.348209i	$0.886790\pi$
$192$		0			0
$193$	7.67729e7	+	4.43249e7i	0.768701	+	0.443810i	0.832411	−	0.554159i	$-0.186960\pi$
$193$	7.67729e7	+	4.43249e7i	0.768701	+	0.443810i	−0.0637099	+	0.997968i	$0.520293\pi$
$194$		0			0
$195$	−2.79030e7	−	4.64416e7i	−0.269482	−	0.448524i
$196$		0			0
$197$	7.69162e6	+	4.44076e6i	0.0716780	+	0.0413833i	0.535411	−	0.844592i	$-0.320157\pi$
$197$	7.69162e6	+	4.44076e6i	0.0716780	+	0.0413833i	−0.463733	+	0.885975i	$0.653490\pi$
$198$		0			0
$199$	3.95228e7	+	6.84555e7i	0.355519	+	0.615776i	0.987207	−	0.159447i	$-0.0509710\pi$
$199$	3.95228e7	+	6.84555e7i	0.355519	+	0.615776i	−0.631688	+	0.775223i	$0.717638\pi$
$200$		0			0
$201$	2.49656e7	−	1.44139e7i	0.216848	−	0.125198i
$202$		0			0
$203$			8.26594e7i			0.693515i
$204$		0			0
$205$	−3.05648e7	+	5.29398e7i	−0.247790	+	0.429185i
$206$		0			0
$207$	7.44351e7			0.583286
$208$		0			0
$209$	−3.01107e7			−0.228144
$210$		0			0
$211$	−1.08770e8	+	1.88395e8i	−0.797114	+	1.38064i	0.124373	+	0.992235i	$0.460308\pi$
$211$	−1.08770e8	+	1.88395e8i	−0.797114	+	1.38064i	−0.921488	+	0.388407i	$0.873025\pi$
$212$		0			0
$213$		−	1.27961e8i		−	0.907297i
$214$		0			0
$215$	7.36038e7	−	4.24952e7i	0.505086	−	0.291612i
$216$		0			0
$217$	−6.46316e7	−	1.11945e8i	−0.429374	−	0.743698i
$218$		0			0
$219$	−7.78614e7	−	4.49533e7i	−0.500919	−	0.289206i
$220$		0			0
$221$	2.25485e6	+	1.29400e8i	0.0140522	+	0.806419i
$222$		0			0
$223$	1.08999e8	+	6.29308e7i	0.658198	+	0.380011i	0.791590	−	0.611052i	$-0.209254\pi$
$223$	1.08999e8	+	6.29308e7i	0.658198	+	0.380011i	−0.133392	+	0.991063i	$0.542587\pi$
$224$		0			0
$225$	1.90611e7	+	3.30148e7i	0.111560	+	0.193228i
$226$		0			0
$227$	2.28753e8	−	1.32070e8i	1.29800	−	0.749402i	0.317943	−	0.948110i	$-0.397008\pi$
$227$	2.28753e8	−	1.32070e8i	1.29800	−	0.749402i	0.980059	+	0.198708i	$0.0636744\pi$
$228$		0			0
$229$			6.38112e7i			0.351134i	0.984467	+	0.175567i	$0.0561759\pi$
$229$			6.38112e7i			0.351134i	−0.984467	+	0.175567i	$0.943824\pi$
$230$		0			0
$231$	1.02099e8	−	1.76841e8i	0.544980	−	0.943934i
$232$		0			0
$233$	−1.58319e8			−0.819947			−0.409974	−	0.912097i	$-0.634462\pi$
$233$	−1.58319e8			−0.819947			−0.409974	+	0.912097i	$0.634462\pi$
$234$		0			0
$235$	1.37344e7			0.0690357
$236$		0			0
$237$	7.34124e7	−	1.27154e8i	0.358220	−	0.620456i
$238$		0			0
$239$		−	2.79618e8i		−	1.32487i	−0.749120	−	0.662434i	$-0.769523\pi$
$239$		−	2.79618e8i		−	1.32487i	0.749120	−	0.662434i	$-0.230477\pi$
$240$		0			0
$241$	1.95982e8	−	1.13150e8i	0.901895	−	0.520710i	0.0240807	−	0.999710i	$-0.492334\pi$
$241$	1.95982e8	−	1.13150e8i	0.901895	−	0.520710i	0.877815	+	0.479000i	$0.159001\pi$
$242$		0			0
$243$	−1.01717e8	−	1.76178e8i	−0.454747	−	0.787646i
$244$		0			0
$245$	5.20989e7	+	3.00793e7i	0.226333	+	0.130673i
$246$		0			0
$247$	2.77396e7	−	483375.i	0.117128	−	0.00204101i
$248$		0			0
$249$	−7.31336e7	−	4.22237e7i	−0.300206	−	0.173324i
$250$		0			0
$251$	−1.08219e8	−	1.87441e8i	−0.431962	−	0.748181i	0.565080	−	0.825036i	$-0.308845\pi$
$251$	−1.08219e8	−	1.87441e8i	−0.431962	−	0.748181i	−0.997042	+	0.0768554i	$0.975512\pi$
$252$		0			0
$253$	4.86008e8	−	2.80597e8i	1.88678	−	1.08933i
$254$		0			0
$255$		−	1.11745e8i		−	0.422026i
$256$		0			0
$257$	1.21788e8	−	2.10943e8i	0.447547	−	0.775174i	−0.550679	−	0.834717i	$-0.685631\pi$
$257$	1.21788e8	−	2.10943e8i	0.447547	−	0.775174i	0.998226	+	0.0595432i	$0.0189644\pi$
$258$		0			0
$259$	−3.66545e8			−1.31093
$260$		0			0
$261$	1.28389e8			0.446977
$262$		0			0
$263$	−2.52252e6	+	4.36913e6i	−0.00855045	+	0.0148098i	−0.870269	−	0.492577i	$-0.836055\pi$
$263$	−2.52252e6	+	4.36913e6i	−0.00855045	+	0.0148098i	0.861719	+	0.507387i	$0.169388\pi$
$264$		0			0
$265$		−	5.57913e7i		−	0.184164i
$266$		0			0
$267$	2.20728e8	−	1.27438e8i	0.709690	−	0.409740i
$268$		0			0
$269$	2.32684e8	+	4.03021e8i	0.728844	+	1.26239i	0.957372	+	0.288857i	$0.0932752\pi$
$269$	2.32684e8	+	4.03021e8i	0.728844	+	1.26239i	−0.228529	+	0.973537i	$0.573391\pi$
$270$		0			0
$271$	2.87325e8	+	1.65887e8i	0.876962	+	0.506314i	0.869655	−	0.493659i	$-0.164341\pi$
$271$	2.87325e8	+	1.65887e8i	0.876962	+	0.506314i	0.00730633	+	0.999973i	$0.497674\pi$
$272$		0			0
$273$	−9.12206e7	+	1.64555e8i	−0.271346	+	0.489487i
$274$		0			0
$275$	2.48910e8	+	1.43708e8i	0.721736	+	0.416695i
$276$		0			0
$277$	−2.04267e8	−	3.53801e8i	−0.577456	−	1.00018i	−0.995770	−	0.0918807i	$-0.970712\pi$
$277$	−2.04267e8	−	3.53801e8i	−0.577456	−	1.00018i	0.418314	−	0.908302i	$-0.362621\pi$
$278$		0			0
$279$	−1.73876e8	+	1.00387e8i	−0.479320	+	0.276735i
$280$		0			0
$281$			6.26412e8i			1.68418i	0.539339	+	0.842089i	$0.318674\pi$
$281$			6.26412e8i			1.68418i	−0.539339	+	0.842089i	$0.681326\pi$
$282$		0			0
$283$	1.37435e7	−	2.38045e7i	0.0360450	−	0.0624318i	−0.847440	−	0.530891i	$-0.821857\pi$
$283$	1.37435e7	−	2.38045e7i	0.0360450	−	0.0624318i	0.883485	+	0.468459i	$0.155191\pi$
$284$		0			0
$285$	−2.39550e7			−0.0612971
$286$		0			0
$287$	2.12284e8			0.530066
$288$		0			0
$289$	7.17049e7	−	1.24197e8i	0.174746	−	0.302669i
$290$		0			0
$291$			2.95705e8i			0.703451i
$292$		0			0
$293$	−1.60266e8	+	9.25295e7i	−0.372224	+	0.214904i	−0.674430	−	0.738339i	$-0.735610\pi$
$293$	−1.60266e8	+	9.25295e7i	−0.372224	+	0.214904i	0.302206	+	0.953243i	$0.402277\pi$
$294$		0			0
$295$	2.71770e8	+	4.70719e8i	0.616346	+	1.06754i
$296$		0			0
$297$	−8.01474e8	−	4.62731e8i	−1.77518	−	1.02490i
$298$		0			0
$299$	−4.43232e8	+	2.66303e8i	−0.958920	+	0.576139i
$300$		0			0
$301$	−2.55603e8	−	1.47572e8i	−0.540235	−	0.311905i
$302$		0			0
$303$	−9.29864e7	−	1.61057e8i	−0.192030	−	0.332606i
$304$		0			0
$305$	9.86738e7	−	5.69693e7i	0.199137	−	0.114972i
$306$		0			0
$307$			2.08464e7i			0.0411194i	0.999789	+	0.0205597i	$0.00654481\pi$
$307$			2.08464e7i			0.0411194i	−0.999789	+	0.0205597i	$0.993455\pi$
$308$		0			0
$309$	−1.49107e8	+	2.58261e8i	−0.287504	+	0.497971i
$310$		0			0
$311$	−7.13088e7			−0.134426			−0.0672128	−	0.997739i	$-0.521411\pi$
$311$	−7.13088e7			−0.134426			−0.0672128	+	0.997739i	$0.521411\pi$
$312$		0			0
$313$	3.19037e8			0.588080			0.294040	−	0.955793i	$-0.405000\pi$
$313$	3.19037e8			0.588080			0.294040	+	0.955793i	$0.405000\pi$
$314$		0			0
$315$	−8.84912e7	+	1.53271e8i	−0.159519	+	0.276296i
$316$		0			0
$317$		−	3.53993e8i		−	0.624147i	−0.950058	−	0.312074i	$-0.898976\pi$
$317$		−	3.53993e8i		−	0.624147i	0.950058	−	0.312074i	$-0.101024\pi$
$318$		0			0
$319$	8.38285e8	−	4.83984e8i	1.44585	−	0.834764i
$320$		0			0
$321$	−2.70387e8	−	4.68323e8i	−0.456265	−	0.790275i
$322$		0			0
$323$	4.95556e7	+	2.86110e7i	0.0818247	+	0.0472415i
$324$		0			0
$325$	−2.31617e8	−	1.28396e8i	−0.374264	−	0.207473i
$326$		0			0
$327$	−4.00340e8	−	2.31136e8i	−0.633157	−	0.365554i
$328$		0			0
$329$	−2.38477e7	−	4.13054e7i	−0.0369199	−	0.0639471i
$330$		0			0
$331$	−8.73685e8	+	5.04422e8i	−1.32421	+	0.764533i	−0.984397	−	0.175960i	$-0.943697\pi$
$331$	−8.73685e8	+	5.04422e8i	−1.32421	+	0.764533i	−0.339813	+	0.940493i	$0.610364\pi$
$332$		0			0
$333$			5.69327e8i			0.844903i
$334$		0			0
$335$	−9.41877e7	+	1.63138e8i	−0.136879	+	0.237082i
$336$		0			0
$337$	6.85528e8			0.975709			0.487855	−	0.872925i	$-0.337780\pi$
$337$	6.85528e8			0.975709			0.487855	+	0.872925i	$0.337780\pi$
$338$		0			0
$339$	−7.10636e8			−0.990714
$340$		0			0
$341$	−7.56857e8	+	1.31091e9i	−1.03365	+	1.79033i
$342$		0			0
$343$		−	8.13519e8i		−	1.08852i
$344$		0			0
$345$	3.86651e8	−	2.23233e8i	0.506935	−	0.292679i
$346$		0			0
$347$	−2.49148e8	−	4.31537e8i	−0.320114	−	0.554453i	0.660398	−	0.750916i	$-0.270388\pi$
$347$	−2.49148e8	−	4.31537e8i	−0.320114	−	0.554453i	−0.980511	+	0.196463i	$0.937054\pi$
$348$		0			0
$349$	−2.43339e8	−	1.40492e8i	−0.306424	−	0.176914i	0.338901	−	0.940822i	$-0.389945\pi$
$349$	−2.43339e8	−	1.40492e8i	−0.306424	−	0.176914i	−0.645325	+	0.763908i	$0.723278\pi$
$350$		0			0
$351$	7.45790e8	+	4.13427e8i	0.920538	+	0.510298i
$352$		0			0
$353$	5.71431e8	+	3.29916e8i	0.691437	+	0.399201i	0.804150	−	0.594426i	$-0.202621\pi$
$353$	5.71431e8	+	3.29916e8i	0.691437	+	0.399201i	−0.112713	+	0.993628i	$0.535954\pi$
$354$		0			0
$355$	4.18081e8	+	7.24137e8i	0.495976	+	0.859056i
$356$		0			0
$357$	−3.36066e8	+	1.94028e8i	−0.390919	+	0.225697i
$358$		0			0
$359$			3.72973e7i			0.0425448i	0.999774	+	0.0212724i	$0.00677172\pi$
$359$			3.72973e7i			0.0425448i	−0.999774	+	0.0212724i	$0.993228\pi$
$360$		0			0
$361$	−4.40802e8	+	7.63492e8i	−0.493138	+	0.854141i
$362$		0			0
$363$	−1.76077e9			−1.93210
$364$		0			0
$365$	5.87494e8			0.632380
$366$		0			0
$367$	−3.60243e8	+	6.23959e8i	−0.380421	+	0.658908i	−0.991122	−	0.132953i	$-0.957554\pi$
$367$	−3.60243e8	+	6.23959e8i	−0.380421	+	0.658908i	0.610702	+	0.791861i	$0.290887\pi$
$368$		0			0
$369$		−	3.29725e8i		−	0.341632i
$370$		0			0
$371$	−1.67788e8	+	9.68727e7i	−0.170590	+	0.0984901i
$372$		0			0
$373$	1.45502e8	+	2.52017e8i	0.145174	+	0.251449i	0.929438	−	0.368979i	$-0.120292\pi$
$373$	1.45502e8	+	2.52017e8i	0.145174	+	0.251449i	−0.784264	+	0.620427i	$0.786959\pi$
$374$		0			0
$375$	6.60781e8	+	3.81502e8i	0.647065	+	0.373583i
$376$		0			0
$377$	−7.64505e8	+	4.59330e8i	−0.734828	+	0.441499i
$378$		0			0
$379$	−9.12613e8	−	5.26897e8i	−0.861092	−	0.497152i	0.00328585	−	0.999995i	$-0.498954\pi$
$379$	−9.12613e8	−	5.26897e8i	−0.861092	−	0.497152i	−0.864378	+	0.502843i	$0.832287\pi$
$380$		0			0
$381$	3.46191e8	+	5.99621e8i	0.320685	+	0.555443i
$382$		0			0
$383$	1.61663e9	−	9.33363e8i	1.47033	−	0.848896i	0.470886	−	0.882194i	$-0.343934\pi$
$383$	1.61663e9	−	9.33363e8i	1.47033	−	0.848896i	0.999445	+	0.0332976i	$0.0106009\pi$
$384$		0			0
$385$			1.33433e9i			1.19166i
$386$		0			0
$387$	−2.29213e8	+	3.97008e8i	−0.201025	+	0.348186i
$388$		0			0
$389$	1.47943e9			1.27430			0.637151	−	0.770739i	$-0.280113\pi$
$389$	1.47943e9			1.27430			0.637151	+	0.770739i	$0.280113\pi$
$390$		0			0
$391$	−1.06648e9			−0.902268
$392$		0			0
$393$	4.52210e8	−	7.83250e8i	0.375808	−	0.650919i
$394$		0			0
$395$			9.59426e8i			0.783289i
$396$		0			0
$397$	−1.94946e9	+	1.12552e9i	−1.56368	+	0.902789i	−0.566796	+	0.823858i	$0.691817\pi$
$397$	−1.94946e9	+	1.12552e9i	−1.56368	+	0.902789i	−0.996880	+	0.0789312i	$0.974849\pi$
$398$		0			0
$399$	4.15941e7	+	7.20431e7i	0.0327813	+	0.0567789i
$400$		0			0
$401$	−1.62628e9	−	9.38935e8i	−1.25948	−	0.727161i	−0.286506	−	0.958078i	$-0.592494\pi$
$401$	−1.62628e9	−	9.38935e8i	−1.25948	−	0.727161i	−0.972973	+	0.230918i	$0.925827\pi$
$402$		0			0
$403$	6.76214e8	−	1.21984e9i	0.514655	−	0.928398i
$404$		0			0
$405$	−1.81038e8	−	1.04523e8i	−0.135419	−	0.0781839i
$406$		0			0
$407$	2.14618e9	+	3.71729e9i	1.57792	+	2.73304i
$408$		0			0
$409$	−5.51529e8	+	3.18426e8i	−0.398600	+	0.230132i	−0.685880	−	0.727715i	$-0.740582\pi$
$409$	−5.51529e8	+	3.18426e8i	−0.398600	+	0.230132i	0.287280	+	0.957847i	$0.407249\pi$
$410$		0			0
$411$		−	1.44870e9i		−	1.02927i
$412$		0			0
$413$	9.43770e8	−	1.63466e9i	0.659236	−	1.14183i
$414$		0			0
$415$	5.51821e8			0.378992
$416$		0			0
$417$	−7.74673e8			−0.523169
$418$		0			0
$419$	−7.77149e8	+	1.34606e9i	−0.516125	+	0.893955i	0.483699	+	0.875234i	$0.339293\pi$
$419$	−7.77149e8	+	1.34606e9i	−0.516125	+	0.893955i	−0.999825	+	0.0187211i	$0.994041\pi$
$420$		0			0
$421$		−	9.46533e8i		−	0.618228i	−0.951025	−	0.309114i	$-0.899968\pi$
$421$		−	9.46533e8i		−	0.618228i	0.951025	−	0.309114i	$-0.100032\pi$
$422$		0			0
$423$	−6.41566e7	+	3.70408e7i	−0.0412145	+	0.0237952i
$424$		0			0
$425$	−2.73102e8	−	4.73026e8i	−0.172569	−	0.298898i
$426$		0			0
$427$	−3.42663e8	−	1.97836e8i	−0.212995	−	0.122973i
$428$		0			0
$429$	2.20293e9	−	3.83871e7i	1.34710	−	0.0234739i
$430$		0			0
$431$	4.49874e7	+	2.59735e7i	0.0270658	+	0.0156264i	0.513472	−	0.858106i	$-0.328359\pi$
$431$	4.49874e7	+	2.59735e7i	0.0270658	+	0.0156264i	−0.486406	+	0.873733i	$0.661692\pi$
$432$		0			0
$433$	7.84073e8	+	1.35806e9i	0.464140	+	0.803914i	0.999162	−	0.0409237i	$-0.0130301\pi$
$433$	7.84073e8	+	1.35806e9i	0.464140	+	0.803914i	−0.535022	+	0.844838i	$0.679697\pi$
$434$		0			0
$435$	6.66911e8	−	3.85041e8i	0.388468	−	0.224282i
$436$		0			0
$437$			2.28624e8i			0.131050i
$438$		0			0
$439$	−4.29729e8	+	7.44312e8i	−0.242420	+	0.419884i	−0.961403	−	0.275144i	$-0.911275\pi$
$439$	−4.29729e8	+	7.44312e8i	−0.242420	+	0.419884i	0.718983	+	0.695028i	$0.244608\pi$
$440$		0			0
$441$	−3.24487e8			−0.180162
$442$		0			0
$443$	−3.13741e8			−0.171458			−0.0857292	−	0.996318i	$-0.527322\pi$
$443$	−3.13741e8			−0.171458			−0.0857292	+	0.996318i	$0.527322\pi$
$444$		0			0
$445$	−8.32740e8	+	1.44235e9i	−0.447971	+	0.775908i
$446$		0			0
$447$			1.27201e9i			0.673617i
$448$		0			0
$449$	−7.07915e7	+	4.08715e7i	−0.0369079	+	0.0213088i	−0.518340	−	0.855174i	$-0.673450\pi$
$449$	−7.07915e7	+	4.08715e7i	−0.0369079	+	0.0213088i	0.481433	+	0.876483i	$0.340117\pi$
$450$		0			0
$451$	−1.24296e9	−	2.15286e9i	−0.638026	−	1.10509i
$452$		0			0
$453$	−1.52745e9	−	8.81874e8i	−0.772011	−	0.445721i
$454$		0			0
$455$	−2.14204e7	−	1.22926e9i	−0.0106608	−	0.611793i
$456$		0			0
$457$	1.68781e9	+	9.74458e8i	0.827213	+	0.477592i	0.852897	−	0.522079i	$-0.174843\pi$
$457$	1.68781e9	+	9.74458e8i	0.827213	+	0.477592i	−0.0256846	+	0.999670i	$0.508177\pi$
$458$		0			0
$459$	8.79367e8	+	1.52311e9i	0.424449	+	0.735168i
$460$		0			0
$461$	2.99997e9	−	1.73204e9i	1.42615	−	0.823386i	0.429333	−	0.903146i	$-0.358749\pi$
$461$	2.99997e9	−	1.73204e9i	1.42615	−	0.823386i	0.996814	+	0.0797602i	$0.0254155\pi$
$462$		0			0
$463$			1.13332e9i			0.530665i	0.964157	+	0.265332i	$0.0854817\pi$
$463$			1.13332e9i			0.530665i	−0.964157	+	0.265332i	$0.914518\pi$
$464$		0			0
$465$	−6.02129e8	+	1.04292e9i	−0.277718	+	0.481022i
$466$		0			0
$467$	−2.08755e9			−0.948479			−0.474239	−	0.880396i	$-0.657277\pi$
$467$	−2.08755e9			−0.948479			−0.474239	+	0.880396i	$0.657277\pi$
$468$		0			0
$469$	6.54167e8			0.292809
$470$		0			0
$471$	1.06468e9	−	1.84408e9i	0.469512	−	0.813218i
$472$		0			0
$473$			3.45624e9i			1.50172i
$474$		0			0
$475$	−1.01403e8	+	5.85452e7i	−0.0434134	+	0.0250648i
$476$		0			0
$477$	1.50465e8	+	2.60613e8i	0.0634777	+	0.109947i
$478$		0			0
$479$	1.60014e8	+	9.23844e7i	0.0665250	+	0.0384082i	0.532894	−	0.846182i	$-0.321105\pi$
$479$	1.60014e8	+	9.23844e7i	0.0665250	+	0.0384082i	−0.466369	+	0.884590i	$0.654438\pi$
$480$		0			0
$481$	−2.03685e9	−	3.39012e9i	−0.834549	−	1.38902i
$482$		0			0
$483$	−1.34271e9	−	7.75217e8i	−0.542212	−	0.313046i
$484$		0			0
$485$	−9.66142e8	−	1.67341e9i	−0.384543	−	0.666048i
$486$		0			0
$487$	1.97464e9	−	1.14006e9i	0.774704	−	0.447275i	−0.0598463	−	0.998208i	$-0.519061\pi$
$487$	1.97464e9	−	1.14006e9i	0.774704	−	0.447275i	0.834550	+	0.550932i	$0.185728\pi$
$488$		0			0
$489$		−	2.10991e9i		−	0.815985i
$490$		0			0
$491$	−9.75730e8	+	1.69001e9i	−0.372001	+	0.644325i	−0.989873	−	0.141953i	$-0.954662\pi$
$491$	−9.75730e8	+	1.69001e9i	−0.372001	+	0.644325i	0.617872	+	0.786279i	$0.287995\pi$
$492$		0			0
$493$	−1.83951e9			−0.691415
$494$		0			0
$495$	2.07252e9			0.768035
$496$		0			0
$497$	1.45186e9	−	2.51470e9i	0.530491	−	0.918837i
$498$		0			0
$499$		−	3.18870e9i		−	1.14885i	−0.818559	−	0.574423i	$-0.805227\pi$
$499$		−	3.18870e9i		−	1.14885i	0.818559	−	0.574423i	$-0.194773\pi$
$500$		0			0
$501$	−1.77303e9	+	1.02366e9i	−0.629920	+	0.363684i
$502$		0			0
$503$	−8.88123e8	−	1.53827e9i	−0.311161	−	0.538947i	0.667453	−	0.744652i	$-0.267384\pi$
$503$	−8.88123e8	−	1.53827e9i	−0.311161	−	0.538947i	−0.978614	+	0.205705i	$0.934051\pi$
$504$		0			0
$505$	1.05243e9	+	6.07619e8i	0.363640	+	0.209948i
$506$		0			0
$507$	−2.02885e9	+	7.07286e7i	−0.691388	+	0.0241028i
$508$		0			0
$509$	−5.11200e9	−	2.95142e9i	−1.71822	−	0.992015i	−0.922169	−	0.386787i	$-0.873585\pi$
$509$	−5.11200e9	−	2.95142e9i	−1.71822	−	0.992015i	−0.796052	−	0.605228i	$-0.793082\pi$
$510$		0			0
$511$	−1.02009e9	−	1.76685e9i	−0.338193	−	0.585768i
$512$		0			0
$513$	3.26511e8	−	1.88511e8i	0.106779	−	0.0616491i
$514$		0			0
$515$		−	1.94868e9i		−	0.628658i
$516$		0			0
$517$	−2.79264e8	+	4.83700e8i	−0.0888788	+	0.153943i
$518$		0			0
$519$	2.48843e9			0.781340
$520$		0			0
$521$	2.84267e9			0.880633			0.440317	−	0.897843i	$-0.354866\pi$
$521$	2.84267e9			0.880633			0.440317	+	0.897843i	$0.354866\pi$
$522$		0			0
$523$	−1.53452e9	+	2.65787e9i	−0.469048	+	0.812415i	−0.999374	−	0.0353788i	$-0.988736\pi$
$523$	−1.53452e9	+	2.65787e9i	−0.469048	+	0.812415i	0.530326	+	0.847794i	$0.322070\pi$
$524$		0			0
$525$		−	7.94060e8i		−	0.239495i
$526$		0			0
$527$	2.49124e9	−	1.43832e9i	0.741445	−	0.428074i
$528$		0			0
$529$	−4.28094e8	−	7.41480e8i	−0.125731	−	0.217773i
$530$		0			0
$531$	−2.53899e9	−	1.46589e9i	−0.735920	−	0.424884i
$532$		0			0
$533$	1.17964e9	+	1.96338e9i	0.337446	+	0.561642i
$534$		0			0
$535$	3.06026e9	+	1.76684e9i	0.864012	+	0.498837i
$536$		0			0
$537$	1.39882e9	+	2.42283e9i	0.389809	+	0.675168i
$538$		0			0
$539$	−2.11867e9	+	1.22321e9i	−0.582777	+	0.336466i
$540$		0			0
$541$		−	2.24439e9i		−	0.609407i	−0.952447	−	0.304703i	$-0.901443\pi$
$541$		−	2.24439e9i		−	0.609407i	0.952447	−	0.304703i	$-0.0985573\pi$
$542$		0			0
$543$	1.36280e9	−	2.36044e9i	0.365287	−	0.632695i
$544$		0			0
$545$	3.02072e9			0.799323
$546$		0			0
$547$	3.05066e9			0.796963			0.398481	−	0.917176i	$-0.369537\pi$
$547$	3.05066e9			0.796963			0.398481	+	0.917176i	$0.369537\pi$
$548$		0			0
$549$	−3.07284e8	+	5.32232e8i	−0.0792569	+	0.137277i
$550$		0			0
$551$			3.94339e8i			0.100424i
$552$		0			0
$553$	2.88541e9	−	1.66589e9i	0.725553	−	0.418898i
$554$		0			0
$555$	1.70743e9	+	2.95735e9i	0.423952	+	0.734306i
$556$		0			0
$557$	1.71586e9	+	9.90653e8i	0.420716	+	0.242901i	0.695384	−	0.718639i	$-0.255234\pi$
$557$	1.71586e9	+	9.90653e8i	0.420716	+	0.242901i	−0.274668	+	0.961539i	$0.588568\pi$
$558$		0			0
$559$	−5.54839e7	−	3.18408e9i	−0.0134346	−	0.770978i
$560$		0			0
$561$	3.93545e9	+	2.27213e9i	0.941075	+	0.543330i
$562$		0			0
$563$	−3.19492e9	−	5.53376e9i	−0.754537	−	1.30690i	−0.945604	−	0.325320i	$-0.894528\pi$
$563$	−3.19492e9	−	5.53376e9i	−0.754537	−	1.30690i	0.191067	−	0.981577i	$-0.438805\pi$
$564$		0			0
$565$	4.02152e9	−	2.32182e9i	0.938037	−	0.541576i
$566$		0			0
$567$			7.25947e8i			0.167249i
$568$		0			0
$569$	−1.97808e9	+	3.42613e9i	−0.450143	+	0.779671i	−0.998395	−	0.0566429i	$-0.981960\pi$
$569$	−1.97808e9	+	3.42613e9i	−0.450143	+	0.779671i	0.548251	+	0.836314i	$0.315294\pi$
$570$		0			0
$571$	1.72462e9			0.387675			0.193837	−	0.981034i	$-0.437907\pi$
$571$	1.72462e9			0.387675			0.193837	+	0.981034i	$0.437907\pi$
$572$		0			0
$573$	1.04152e9			0.231273
$574$		0			0
$575$	1.09115e9	−	1.88992e9i	0.239357	−	0.414578i
$576$		0			0
$577$			1.44292e9i			0.312698i	0.987702	+	0.156349i	$0.0499725\pi$
$577$			1.44292e9i			0.312698i	−0.987702	+	0.156349i	$0.950027\pi$
$578$		0			0
$579$	−2.48381e9	+	1.43403e9i	−0.531794	+	0.307031i
$580$		0			0
$581$	−9.58150e8	−	1.65956e9i	−0.202683	−	0.351057i
$582$		0			0
$583$	1.96486e9	+	1.13441e9i	0.410668	+	0.237099i
$584$		0			0
$585$	−1.90932e9	+	3.32708e7i	−0.394306	+	0.00687096i
$586$		0			0
$587$	4.68617e9	+	2.70556e9i	0.956280	+	0.552108i	0.895026	−	0.446014i	$-0.147157\pi$
$587$	4.68617e9	+	2.70556e9i	0.956280	+	0.552108i	0.0612537	+	0.998122i	$0.480490\pi$
$588$		0			0
$589$	−3.08335e8	−	5.34052e8i	−0.0621755	−	0.107691i
$590$		0			0
$591$	−2.48844e8	+	1.43670e8i	−0.0495874	+	0.0286293i
$592$		0			0
$593$			1.30921e9i			0.257820i	0.991656	+	0.128910i	$0.0411478\pi$
$593$			1.30921e9i			0.257820i	−0.991656	+	0.128910i	$0.958852\pi$
$594$		0			0
$595$	1.26788e9	−	2.19602e9i	0.246756	−	0.427393i
$596$		0			0
$597$	−2.55734e9			−0.491901
$598$		0			0
$599$	5.82086e9			1.10661			0.553303	−	0.832980i	$-0.313367\pi$
$599$	5.82086e9			1.10661			0.553303	+	0.832980i	$0.313367\pi$
$600$		0			0
$601$	4.65256e9	−	8.05847e9i	0.874241	−	1.51423i	0.0166722	−	0.999861i	$-0.494693\pi$
$601$	4.65256e9	−	8.05847e9i	0.874241	−	1.51423i	0.857569	−	0.514369i	$-0.171974\pi$
$602$		0			0
$603$		−	1.01607e9i		−	0.188718i
$604$		0			0
$605$	9.96427e9	−	5.75287e9i	1.82937	−	1.05619i
$606$		0			0
$607$	−1.77466e9	−	3.07379e9i	−0.322072	−	0.557846i	0.658843	−	0.752280i	$-0.271046\pi$
$607$	−1.77466e9	−	3.07379e9i	−0.322072	−	0.557846i	−0.980915	+	0.194435i	$0.937713\pi$
$608$		0			0
$609$	−2.31597e9	−	1.33712e9i	−0.415501	−	0.239890i
$610$		0			0
$611$	2.49509e8	−	4.50094e8i	0.0442528	−	0.0798287i
$612$		0			0
$613$	−5.38574e9	−	3.10946e9i	−0.944352	−	0.545222i	−0.0530301	−	0.998593i	$-0.516888\pi$
$613$	−5.38574e9	−	3.10946e9i	−0.944352	−	0.545222i	−0.891322	+	0.453371i	$0.850221\pi$
$614$		0			0
$615$	−9.88853e8	−	1.71274e9i	−0.171423	−	0.296913i
$616$		0			0
$617$	2.81896e9	−	1.62753e9i	0.483159	−	0.278952i	−0.238573	−	0.971125i	$-0.576680\pi$
$617$	2.81896e9	−	1.62753e9i	0.483159	−	0.278952i	0.721732	+	0.692172i	$0.243346\pi$
$618$		0			0
$619$			8.80728e9i			1.49254i	0.665646	+	0.746268i	$0.268156\pi$
$619$			8.80728e9i			1.49254i	−0.665646	+	0.746268i	$0.731844\pi$
$620$		0			0
$621$	−3.51341e9	+	6.08541e9i	−0.588720	+	1.01969i
$622$		0			0
$623$	5.78368e9			0.958289
$624$		0			0
$625$	−2.37401e9			−0.388957
$626$		0			0
$627$	4.87080e8	−	8.43648e8i	0.0789158	−	0.136686i
$628$		0			0
$629$		−	8.15714e9i		−	1.30695i
$630$		0			0
$631$	−4.23686e9	+	2.44615e9i	−0.671338	+	0.387597i	−0.796583	−	0.604529i	$-0.793361\pi$
$631$	−4.23686e9	+	2.44615e9i	−0.671338	+	0.387597i	0.125246	+	0.992126i	$0.460028\pi$
$632$		0			0
$633$	−3.51900e9	−	6.09508e9i	−0.551450	−	0.955139i
$634$		0			0
$635$	−3.91822e9	−	2.26219e9i	−0.607269	−	0.350607i
$636$		0			0
$637$	1.93220e9	−	1.16090e9i	0.296185	−	0.177954i
$638$		0			0
$639$	−3.90589e9	−	2.25507e9i	−0.592199	−	0.341906i
$640$		0			0
$641$	1.45291e9	+	2.51651e9i	0.217889	+	0.377395i	0.954162	−	0.299289i	$-0.0967496\pi$
$641$	1.45291e9	+	2.51651e9i	0.217889	+	0.377395i	−0.736273	+	0.676684i	$0.763416\pi$
$642$		0			0
$643$	3.26011e8	−	1.88222e8i	0.0483608	−	0.0279211i	−0.475625	−	0.879648i	$-0.657778\pi$
$643$	3.26011e8	−	1.88222e8i	0.0483608	−	0.0279211i	0.523985	+	0.851727i	$0.324445\pi$
$644$		0			0
$645$			2.74966e9i			0.403479i
$646$		0			0
$647$	−1.69765e9	+	2.94042e9i	−0.246425	+	0.426820i	−0.962531	−	0.271171i	$-0.912589\pi$
$647$	−1.69765e9	+	2.94042e9i	−0.246425	+	0.426820i	0.716106	+	0.697991i	$0.245923\pi$
$648$		0			0
$649$	−2.21037e10			−3.17401
$650$		0			0
$651$	4.18201e9			0.594089
$652$		0			0
$653$	−6.46268e9	+	1.11937e10i	−0.908273	+	1.57317i	−0.0918103	+	0.995777i	$0.529265\pi$
$653$	−6.46268e9	+	1.11937e10i	−0.908273	+	1.57317i	−0.816463	+	0.577398i	$0.804068\pi$
$654$		0			0
$655$			5.90993e9i			0.821746i
$656$		0			0
$657$	−2.74431e9	+	1.58443e9i	−0.377533	+	0.217969i
$658$		0			0
$659$	2.50007e9	+	4.33024e9i	0.340293	+	0.589405i	0.984487	−	0.175457i	$-0.0561405\pi$
$659$	2.50007e9	+	4.33024e9i	0.340293	+	0.589405i	−0.644194	+	0.764862i	$0.722807\pi$
$660$		0			0
$661$	1.10755e10	+	6.39445e9i	1.49162	+	0.861189i	0.999954	−	0.00959383i	$-0.00305386\pi$
$661$	1.10755e10	+	6.39445e9i	1.49162	+	0.861189i	0.491668	+	0.870782i	$0.336387\pi$
$662$		0			0
$663$	−3.66203e9	−	2.03004e9i	−0.488005	−	0.270524i
$664$		0			0
$665$	−4.70765e8	−	2.71796e8i	−0.0620767	−	0.0358400i
$666$		0			0
$667$	−3.67478e9	−	6.36491e9i	−0.479503	−	0.830523i
$668$		0			0
$669$	−3.52642e9	+	2.03598e9i	−0.455347	+	0.262895i
$670$		0			0
$671$			4.63346e9i			0.592074i
$672$		0			0
$673$	1.96087e9	−	3.39633e9i	0.247969	−	0.429494i	−0.714993	−	0.699131i	$-0.753570\pi$
$673$	1.96087e9	−	3.39633e9i	0.247969	−	0.429494i	0.962962	+	0.269637i	$0.0869037\pi$
$674$		0			0
$675$	−3.59881e9			−0.450397
$676$		0			0
$677$	1.10745e10			1.37171			0.685855	−	0.727739i	$-0.259429\pi$
$677$	1.10745e10			1.37171			0.685855	+	0.727739i	$0.259429\pi$
$678$		0			0
$679$	−3.35510e9	+	5.81121e9i	−0.411303	+	0.712398i
$680$		0			0
$681$			8.54565e9i			1.03688i
$682$		0			0
$683$	−2.30202e9	+	1.32907e9i	−0.276463	+	0.159616i	−0.631821	−	0.775114i	$-0.717692\pi$
$683$	−2.30202e9	+	1.32907e9i	−0.276463	+	0.159616i	0.355358	+	0.934730i	$0.384359\pi$
$684$		0			0
$685$	4.73325e9	+	8.19823e9i	0.562655	+	0.974548i
$686$		0			0
$687$	−1.78788e9	−	1.03223e9i	−0.210373	−	0.121459i
$688$		0			0
$689$	−1.82835e9	−	1.01354e9i	−0.212957	−	0.118052i
$690$		0			0
$691$	−2.79991e9	−	1.61653e9i	−0.322828	−	0.186385i	0.329824	−	0.944042i	$-0.393011\pi$
$691$	−2.79991e9	−	1.61653e9i	−0.322828	−	0.186385i	−0.652653	+	0.757657i	$0.726344\pi$
$692$		0			0
$693$	−3.59861e9	−	6.23297e9i	−0.410741	−	0.711424i
$694$		0			0
$695$	4.38390e9	−	2.53105e9i	0.495352	−	0.285992i
$696$		0			0
$697$			4.72419e9i			0.528461i
$698$		0			0
$699$	2.56101e9	−	4.43580e9i	0.283623	−	0.491249i
$700$		0			0
$701$	−1.01439e10			−1.11223			−0.556114	−	0.831106i	$-0.687708\pi$
$701$	−1.01439e10			−1.11223			−0.556114	+	0.831106i	$0.687708\pi$
$702$		0			0
$703$	−1.74866e9			−0.189828
$704$		0			0
$705$	−2.22173e8	+	3.84815e8i	−0.0238797	+	0.0413609i
$706$		0			0
$707$		−	4.22013e9i		−	0.449116i
$708$		0			0
$709$	−2.29529e8	+	1.32519e8i	−0.0241867	+	0.0139642i	−0.512045	−	0.858959i	$-0.671112\pi$
$709$	−2.29529e8	+	1.32519e8i	−0.0241867	+	0.0139642i	0.487858	+	0.872923i	$0.337778\pi$
$710$		0			0
$711$	−2.58750e9	−	4.48169e9i	−0.269984	−	0.467625i
$712$		0			0
$713$	9.95348e9	+	5.74664e9i	1.02840	+	0.593746i
$714$		0			0
$715$	−1.23411e10	+	7.41477e9i	−1.26265	+	0.758624i
$716$		0			0
$717$	7.83440e9	+	4.52319e9i	0.793759	+	0.458277i
$718$		0			0
$719$	−1.30879e9	−	2.26689e9i	−0.131317	−	0.227447i	0.792868	−	0.609394i	$-0.208587\pi$
$719$	−1.30879e9	−	2.26689e9i	−0.131317	−	0.227447i	−0.924184	+	0.381947i	$0.875254\pi$
$720$		0			0
$721$	−5.86051e9	+	3.38357e9i	−0.582321	+	0.336203i
$722$		0			0
$723$			7.32142e9i			0.720462i
$724$		0			0
$725$	1.88205e9	−	3.25981e9i	0.183421	−	0.317694i
$726$		0			0
$727$	−1.98815e10			−1.91902			−0.959509	−	0.281676i	$-0.909110\pi$
$727$	−1.98815e10			−1.91902			−0.959509	+	0.281676i	$0.909110\pi$
$728$		0			0
$729$	8.74416e9			0.835934
$730$		0			0
$731$	3.28409e9	−	5.68822e9i	0.310960	−	0.538599i
$732$		0			0
$733$			1.77576e10i			1.66541i	0.553720	+	0.832703i	$0.313208\pi$
$733$			1.77576e10i			1.66541i	−0.553720	+	0.832703i	$0.686792\pi$
$734$		0			0
$735$	−1.68554e9	+	9.73146e8i	−0.156579	+	0.0904008i
$736$		0			0
$737$	−3.83026e9	−	6.63420e9i	−0.352445	−	0.610453i
$738$		0			0
$739$	9.78135e9	+	5.64727e9i	0.891545	+	0.514734i	0.874448	−	0.485120i	$-0.161224\pi$
$739$	9.78135e9	+	5.64727e9i	0.891545	+	0.514734i	0.0170975	+	0.999854i	$0.494557\pi$
$740$		0			0
$741$	−4.35182e8	+	7.85034e8i	−0.0392923	+	0.0708802i
$742$		0			0
$743$	6.78319e9	+	3.91628e9i	0.606699	+	0.350278i	0.771672	−	0.636020i	$-0.219421\pi$
$743$	6.78319e9	+	3.91628e9i	0.606699	+	0.350278i	−0.164974	+	0.986298i	$0.552754\pi$
$744$		0			0
$745$	−4.15596e9	−	7.19833e9i	−0.368234	−	0.637800i
$746$		0			0
$747$	−2.57768e9	+	1.48822e9i	−0.226259	+	0.130631i
$748$		0			0
$749$		−	1.22713e10i		−	1.06710i
$750$		0			0
$751$	1.17937e9	−	2.04273e9i	0.101604	−	0.175983i	−0.810742	−	0.585404i	$-0.800936\pi$
$751$	1.17937e9	−	2.04273e9i	0.101604	−	0.175983i	0.912346	+	0.409421i	$0.134269\pi$
$752$		0			0
$753$	7.00235e9			0.597670
$754$		0			0
$755$	1.15252e10			0.974618
$756$		0			0
$757$	−6.42271e9	+	1.11245e10i	−0.538125	+	0.932059i	0.460880	+	0.887462i	$0.347534\pi$
$757$	−6.42271e9	+	1.11245e10i	−0.538125	+	0.932059i	−0.999005	+	0.0445970i	$0.985800\pi$
$758$		0			0
$759$			1.81561e10i			1.50722i
$760$		0			0
$761$	8.56209e8	−	4.94332e8i	0.0704261	−	0.0406605i	−0.464373	−	0.885639i	$-0.653720\pi$
$761$	8.56209e8	−	4.94332e8i	0.0704261	−	0.0406605i	0.534800	+	0.844979i	$0.320387\pi$
$762$		0			0
$763$	−5.24500e9	−	9.08460e9i	−0.427474	−	0.740406i
$764$		0			0
$765$	−3.41092e9	−	1.96930e9i	−0.275459	−	0.159036i
$766$		0			0
$767$	2.03632e10	−	3.54837e8i	1.62953	−	0.0283952i
$768$		0			0
$769$	−1.15938e10	−	6.69371e9i	−0.919359	−	0.530792i	−0.0359285	−	0.999354i	$-0.511439\pi$
$769$	−1.15938e10	−	6.69371e9i	−0.919359	−	0.530792i	−0.883431	+	0.468562i	$0.844772\pi$
$770$		0			0
$771$	3.94016e9	+	6.82457e9i	0.309616	+	0.536271i
$772$		0			0
$773$	−9.24800e9	+	5.33934e9i	−0.720145	+	0.415776i	−0.814806	−	0.579734i	$-0.803157\pi$
$773$	−9.24800e9	+	5.33934e9i	−0.720145	+	0.415776i	0.0946613	+	0.995510i	$0.469823\pi$
$774$		0			0
$775$			5.88633e9i			0.454243i
$776$		0			0
$777$	5.92935e9	−	1.02699e10i	0.453454	−	0.785406i
$778$		0			0
$779$	1.01273e9			0.0767562
$780$		0			0
$781$	−3.40036e10			−2.55415
$782$		0			0
$783$	−6.06007e9	+	1.04964e10i	−0.451140	+	0.781398i
$784$		0			0
$785$			1.39143e10i			1.02664i
$786$		0			0
$787$	1.33997e10	−	7.73632e9i	0.979903	−	0.565748i	0.0776624	−	0.996980i	$-0.475254\pi$
$787$	1.33997e10	−	7.73632e9i	0.979903	−	0.565748i	0.902241	+	0.431232i	$0.141921\pi$
$788$		0			0
$789$	−8.16101e7	−	1.41353e8i	−0.00591527	−	0.0102455i
$790$		0			0
$791$	−1.39654e10	−	8.06295e9i	−1.00331	−	0.579264i
$792$		0			0
$793$	−7.43821e7	−	4.26859e9i	−0.00529679	−	0.303969i
$794$		0			0
$795$	1.56317e9	+	9.02498e8i	0.110337	+	0.0637032i
$796$		0			0
$797$	9.15716e9	+	1.58607e10i	0.640703	+	1.10973i	0.985276	+	0.170971i	$0.0546904\pi$
$797$	9.15716e9	+	1.58607e10i	0.640703	+	1.10973i	−0.344573	+	0.938760i	$0.611976\pi$
$798$		0			0
$799$	9.19216e8	−	5.30710e8i	0.0637535	−	0.0368081i
$800$		0			0
$801$		−	8.98336e9i		−	0.617625i
$802$		0			0
$803$	−1.19456e10	+	2.06904e10i	−0.814147	+	1.41014i
$804$		0			0
$805$	1.01313e10			0.684510
$806$		0			0
$807$	−1.50559e10			−1.00844
$808$		0			0
$809$	−3.03106e9	+	5.24995e9i	−0.201268	+	0.348606i	−0.948937	−	0.315465i	$-0.897840\pi$
$809$	−3.03106e9	+	5.24995e9i	−0.201268	+	0.348606i	0.747669	+	0.664071i	$0.231173\pi$
$810$		0			0
$811$			2.18462e10i			1.43815i	0.694933	+	0.719074i	$0.255434\pi$
$811$			2.18462e10i			1.43815i	−0.694933	+	0.719074i	$0.744566\pi$
$812$		0			0
$813$	−9.29572e9	+	5.36689e9i	−0.606689	+	0.350272i
$814$		0			0
$815$	6.89359e9	+	1.19400e10i	0.446060	+	0.772599i
$816$		0			0
$817$	−1.21939e9	−	7.04016e8i	−0.0782287	−	0.0451653i
$818$		0			0
$819$	3.41529e9	+	5.68438e9i	0.217237	+	0.361568i
$820$		0			0
$821$	2.26398e10	+	1.30711e10i	1.42782	+	0.824350i	0.996948	−	0.0780644i	$-0.0248740\pi$
$821$	2.26398e10	+	1.30711e10i	1.42782	+	0.824350i	0.430868	+	0.902415i	$0.358207\pi$
$822$		0			0
$823$	−4.08115e9	−	7.06876e9i	−0.255202	−	0.442022i	0.709749	−	0.704455i	$-0.248809\pi$
$823$	−4.08115e9	−	7.06876e9i	−0.255202	−	0.442022i	−0.964950	+	0.262433i	$0.915475\pi$
$824$		0			0
$825$	−8.05291e9	+	4.64935e9i	−0.499303	+	0.288273i
$826$		0			0
$827$		−	2.10648e10i		−	1.29505i	−0.762042	−	0.647527i	$-0.775803\pi$
$827$		−	2.10648e10i		−	1.29505i	0.762042	−	0.647527i	$-0.224197\pi$
$828$		0			0
$829$	1.44410e10	−	2.50126e10i	0.880353	−	1.52482i	0.0294044	−	0.999568i	$-0.490639\pi$
$829$	1.44410e10	−	2.50126e10i	0.880353	−	1.52482i	0.850949	−	0.525249i	$-0.176028\pi$
$830$		0			0
$831$	1.32172e10			0.798977
$832$		0			0
$833$	4.64915e9			0.278687
$834$		0			0
$835$	6.68911e9	−	1.15859e10i	0.397618	−	0.688695i
$836$		0			0
$837$		−	1.89535e10i		−	1.11725i
$838$		0			0
$839$	−1.02477e10	+	5.91652e9i	−0.599046	+	0.345859i	−0.768666	−	0.639650i	$-0.779079\pi$
$839$	−1.02477e10	+	5.91652e9i	−0.599046	+	0.345859i	0.169620	+	0.985509i	$0.445746\pi$
$840$		0			0
$841$	2.28653e9	+	3.96038e9i	0.132553	+	0.229589i
$842$		0			0
$843$	−1.75509e10	−	1.01330e10i	−1.00903	−	0.582564i
$844$		0			0
$845$	1.12502e10	−	7.02901e9i	0.641451	−	0.400770i
$846$		0			0
$847$	−3.46027e10	−	1.99779e10i	−1.95667	−	1.12969i
$848$		0			0
$849$	4.44639e8	+	7.70138e8i	0.0249362	+	0.0431908i
$850$		0			0
$851$	2.82246e10	−	1.62955e10i	1.56991	−	0.906386i
$852$		0			0
$853$			4.54152e9i			0.250542i	0.992123	+	0.125271i	$0.0399800\pi$
$853$			4.54152e9i			0.250542i	−0.992123	+	0.125271i	$0.960020\pi$
$854$		0			0
$855$	−4.22161e8	+	7.31204e8i	−0.0230992	+	0.0400090i
$856$		0			0
$857$	2.71958e10			1.47594			0.737971	−	0.674833i	$-0.235784\pi$
$857$	2.71958e10			1.47594			0.737971	+	0.674833i	$0.235784\pi$
$858$		0			0
$859$	1.24708e10			0.671304			0.335652	−	0.941986i	$-0.391043\pi$
$859$	1.24708e10			0.671304			0.335652	+	0.941986i	$0.391043\pi$
$860$		0			0
$861$	−3.43397e9	+	5.94781e9i	−0.183352	+	0.317575i
$862$		0			0
$863$		−	3.88024e9i		−	0.205504i	−0.994707	−	0.102752i	$-0.967235\pi$
$863$		−	3.88024e9i		−	0.205504i	0.994707	−	0.102752i	$-0.0327648\pi$
$864$		0			0
$865$	−1.40821e10	+	8.13032e9i	−0.739796	+	0.427121i
$866$		0			0
$867$	2.31984e9	+	4.01809e9i	0.120890	+	0.209388i
$868$		0			0
$869$	−3.37891e10	−	1.95081e10i	−1.74665	−	1.00843i
$870$		0			0
$871$	3.63514e9	+	6.05030e9i	0.186405	+	0.310251i
$872$		0			0
$873$	9.02612e9	+	5.21123e9i	0.459147	+	0.265088i
$874$		0			0
$875$	8.65713e9	+	1.49946e10i	0.436863	+	0.756670i
$876$		0			0
$877$	2.70830e10	−	1.56364e10i	1.35581	−	0.782777i	0.366754	−	0.930318i	$-0.380469\pi$
$877$	2.70830e10	−	1.56364e10i	1.35581	−	0.782777i	0.989056	+	0.147541i	$0.0471358\pi$
$878$		0			0
$879$		−	5.98715e9i		−	0.297344i
$880$		0			0
$881$	4.00303e9	−	6.93345e9i	0.197230	−	0.341612i	−0.750399	−	0.660985i	$-0.770139\pi$
$881$	4.00303e9	−	6.93345e9i	0.197230	−	0.341612i	0.947629	+	0.319372i	$0.103472\pi$
$882$		0			0
$883$	−5.19038e9			−0.253709			−0.126855	−	0.991921i	$-0.540488\pi$
$883$	−5.19038e9			−0.253709			−0.126855	+	0.991921i	$0.540488\pi$
$884$		0			0
$885$	−1.75850e10			−0.852785
$886$		0			0
$887$	1.30177e10	−	2.25474e10i	0.626329	−	1.08483i	−0.361954	−	0.932196i	$-0.617890\pi$
$887$	1.30177e10	−	2.25474e10i	0.626329	−	1.08483i	0.988282	−	0.152637i	$-0.0487765\pi$
$888$		0			0
$889$			1.57117e10i			0.750010i
$890$		0			0
$891$	7.36215e9	−	4.25054e9i	0.348685	−	0.201313i
$892$		0			0
$893$	−1.13769e8	−	1.97054e8i	−0.00534618	−	0.00925986i
$894$		0			0
$895$	−1.58319e10	−	9.14058e9i	−0.738165	−	0.426180i
$896$		0			0
$897$	−2.91465e8	−	1.67264e10i	−0.0134838	−	0.773800i
$898$		0			0
$899$	1.71682e10	+	9.91204e9i	0.788070	+	0.454992i
$900$		0			0
$901$	−2.15582e9	−	3.73399e9i	−0.0981918	−	0.170073i
$902$		0			0
$903$	8.26942e9	−	4.77435e9i	0.373739	−	0.215778i
$904$		0			0
$905$			1.78105e10i			0.798740i
$906$		0			0
$907$	1.83101e10	−	3.17140e10i	0.814826	−	1.41132i	−0.0946274	−	0.995513i	$-0.530166\pi$
$907$	1.83101e10	−	3.17140e10i	0.814826	−	1.41132i	0.909453	−	0.415807i	$-0.136501\pi$
$908$		0			0
$909$	−6.55482e9			−0.289459
$910$		0			0
$911$	−3.91136e10			−1.71401			−0.857006	−	0.515306i	$-0.827678\pi$
$911$	−3.91136e10			−1.71401			−0.857006	+	0.515306i	$0.827678\pi$
$912$		0			0
$913$	−1.12202e10	+	1.94340e10i	−0.487927	+	0.845114i
$914$		0			0
$915$			3.68622e9i			0.159077i
$916$		0			0
$917$	1.77737e10	−	1.02616e10i	0.761176	−	0.439465i
$918$		0			0
$919$	6.48209e9	+	1.12273e10i	0.275493	+	0.477168i	0.970259	−	0.242068i	$-0.0778256\pi$
$919$	6.48209e9	+	1.12273e10i	0.275493	+	0.477168i	−0.694766	+	0.719235i	$0.744492\pi$
$920$		0			0
$921$	−5.84078e8	−	3.37218e8i	−0.0246356	−	0.0142233i
$922$		0			0
$923$	3.13259e10	−	5.45868e8i	1.31129	−	0.0228498i
$924$		0			0
$925$	1.44553e10	+	8.34577e9i	0.600525	+	0.346713i
$926$		0			0
$927$	5.25544e9	+	9.10270e9i	0.216686	+	0.375311i
$928$		0			0
$929$	−5.10464e9	+	2.94716e9i	−0.208886	+	0.120601i	−0.600794	−	0.799404i	$-0.705149\pi$
$929$	−5.10464e9	+	2.94716e9i	−0.208886	+	0.120601i	0.391907	+	0.920005i	$0.371815\pi$
$930$		0			0
$931$		−	9.96646e8i		−	0.0404778i
$932$		0			0
$933$	1.15351e9	−	1.99795e9i	0.0464983	−	0.0805374i
$934$		0			0
$935$	−2.96945e10			−1.18805
$936$		0			0
$937$	2.40381e10			0.954577			0.477288	−	0.878747i	$-0.341620\pi$
$937$	2.40381e10			0.954577			0.477288	+	0.878747i	$0.341620\pi$
$938$		0			0
$939$	−5.16085e9	+	8.93886e9i	−0.203419	+	0.352332i
$940$		0			0
$941$			5.30161e9i			0.207417i	0.994608	+	0.103708i	$0.0330709\pi$
$941$			5.30161e9i			0.207417i	−0.994608	+	0.103708i	$0.966929\pi$
$942$		0			0
$943$	−1.63462e10	+	9.43749e9i	−0.634784	+	0.366493i
$944$		0			0
$945$	−8.35375e9	−	1.44691e10i	−0.322011	−	0.557739i
$946$		0			0
$947$	2.49066e10	+	1.43798e10i	0.952991	+	0.550210i	0.894009	−	0.448049i	$-0.147881\pi$
$947$	2.49066e10	+	1.43798e10i	0.952991	+	0.550210i	0.0589823	+	0.998259i	$0.481214\pi$
$948$		0			0
$949$	1.06728e10	−	1.92529e10i	0.405365	−	0.731246i
$950$		0			0
$951$	9.91825e9	+	5.72630e9i	0.373941	+	0.215895i
$952$		0			0
$953$	−1.73485e10	−	3.00485e10i	−0.649288	−	1.12460i	−0.983293	−	0.182028i	$-0.941734\pi$
$953$	−1.73485e10	−	3.00485e10i	−0.649288	−	1.12460i	0.334006	−	0.942571i	$-0.391600\pi$
$954$		0			0
$955$	−5.89398e9	+	3.40289e9i	−0.218976	+	0.126426i
$956$		0			0
$957$			3.13163e10i			1.15499i
$958$		0			0
$959$	1.64371e10	−	2.84698e10i	0.601810	−	1.04237i
$960$		0			0
$961$	−3.48839e9			−0.126792
$962$		0			0
$963$	−1.90602e10			−0.687756
$964$		0			0
$965$	9.37064e9	−	1.62304e10i	0.335679	−	0.581413i
$966$		0			0
$967$			2.32799e10i			0.827921i	0.910295	+	0.413960i	$0.135855\pi$
$967$			2.32799e10i			0.827921i	−0.910295	+	0.413960i	$0.864145\pi$
$968$		0			0
$969$	−1.60326e9	+	9.25641e8i	−0.0566070	+	0.0326820i
$970$		0			0
$971$	−1.04009e9	−	1.80150e9i	−0.0364591	−	0.0631489i	0.847220	−	0.531242i	$-0.178275\pi$
$971$	−1.04009e9	−	1.80150e9i	−0.0364591	−	0.0631489i	−0.883679	+	0.468093i	$0.844941\pi$
$972$		0			0
$973$	−1.52239e10	−	8.78952e9i	−0.529823	−	0.305894i
$974$		0			0
$975$	7.34415e9	−	4.41251e9i	0.253761	−	0.152465i
$976$		0			0
$977$	−8.50985e9	−	4.91316e9i	−0.291938	−	0.168551i	0.346877	−	0.937910i	$-0.387242\pi$
$977$	−8.50985e9	−	4.91316e9i	−0.291938	−	0.168551i	−0.638816	+	0.769360i	$0.720575\pi$
$978$		0			0
$979$	−3.38644e10	−	5.86549e10i	−1.15346	−	1.99786i
$980$		0			0
$981$	−1.41104e10	+	8.14666e9i	−0.477198	+	0.275511i
$982$		0			0
$983$		−	3.81722e10i		−	1.28177i	−0.767637	−	0.640885i	$-0.778568\pi$
$983$		−	3.81722e10i		−	1.28177i	0.767637	−	0.640885i	$-0.221432\pi$
$984$		0			0
$985$	9.38813e8	−	1.62607e9i	0.0313006	−	0.0542142i
$986$		0			0
$987$	1.54307e9			0.0510830
$988$		0			0
$989$	2.62424e10			0.862615
$990$		0			0
$991$	6.66158e9	−	1.15382e10i	0.217430	−	0.376600i	−0.736592	−	0.676338i	$-0.763566\pi$
$991$	6.66158e9	−	1.15382e10i	0.217430	−	0.376600i	0.954022	+	0.299738i	$0.0968993\pi$
$992$		0			0
$993$		−	3.26388e10i		−	1.05782i
$994$		0			0
$995$	1.44721e10	−	8.35545e9i	0.465747	−	0.268899i
$996$		0			0
$997$	−1.91796e10	−	3.32200e10i	−0.612923	−	1.06161i	−0.990745	−	0.135736i	$-0.956660\pi$
$997$	−1.91796e10	−	3.32200e10i	−0.612923	−	1.06161i	0.377822	−	0.925878i	$-0.376673\pi$
$998$		0			0
$999$	−4.65450e10	−	2.68728e10i	−1.47705	−	0.852773i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.w.b.17.3		16
4.3	odd	2		26.8.e.a.17.3	✓	16
12.11	even	2		234.8.l.c.199.7		16
13.10	even	6	inner	208.8.w.b.49.3		16
52.7	even	12		338.8.a.m.1.3		8
52.19	even	12		338.8.a.n.1.3		8
52.23	odd	6		26.8.e.a.23.3	yes	16
52.35	odd	6		338.8.b.i.337.11		16
52.43	odd	6		338.8.b.i.337.3		16
156.23	even	6		234.8.l.c.127.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.e.a.17.3	✓	16	4.3	odd	2
26.8.e.a.23.3	yes	16	52.23	odd	6
208.8.w.b.17.3		16	1.1	even	1	trivial
208.8.w.b.49.3		16	13.10	even	6	inner
234.8.l.c.127.6		16	156.23	even	6
234.8.l.c.199.7		16	12.11	even	2
338.8.a.m.1.3		8	52.7	even	12
338.8.a.n.1.3		8	52.19	even	12
338.8.b.i.337.3		16	52.43	odd	6
338.8.b.i.337.11		16	52.35	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	208.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-16.1763 + 28.0182i) q^{3} -211.408i q^{5} +(-635.796 + 367.077i) q^{7} +(570.153 + 987.534i) q^{9} +(7445.38 + 4298.59i) q^{11} +(-6928.10 - 3840.57i) q^{13} +(5923.28 + 3419.81i) q^{15} +(-8168.98 - 14149.1i) q^{17} +(-3033.16 + 1751.20i) q^{19} -23751.8i q^{21} +(32638.2 - 56531.0i) q^{23} +33431.5 q^{25} -107647. q^{27} +(56295.7 - 97507.0i) q^{29} +176071. i q^{31} +(-240878. + 139071. i) q^{33} +(77603.1 + 134412. i) q^{35} +(432385. + 249638. i) q^{37} +(219677. - 131987. i) q^{39} +(-250415. - 144577. i) q^{41} +(201010. + 348160. i) q^{43} +(208773. - 120535. i) q^{45} +64966.5i q^{47} +(-142281. + 246438. i) q^{49} +528576. q^{51} +263903. q^{53} +(908757. - 1.57401e6i) q^{55} -113312. i q^{57} +(-2.22659e6 + 1.28552e6i) q^{59} +(269475. + 466745. i) q^{61} +(-725002. - 418580. i) q^{63} +(-811929. + 1.46466e6i) q^{65} +(-771672. - 445525. i) q^{67} +(1.05593e6 + 1.82893e6i) q^{69} +(-3.42530e6 + 1.97760e6i) q^{71} +2.77895e6i q^{73} +(-540800. + 936692. i) q^{75} -6.31165e6 q^{77} -4.53826e6 q^{79} +(494411. - 856345. i) q^{81} +2.61022e6i q^{83} +(-2.99123e6 + 1.72699e6i) q^{85} +(1.82131e6 + 3.15461e6i) q^{87} +(-6.82257e6 - 3.93901e6i) q^{89} +(5.81464e6 - 101323. i) q^{91} +(-4.93320e6 - 2.84818e6i) q^{93} +(370217. + 641235. i) q^{95} +(7.91553e6 - 4.57003e6i) q^{97} +9.80341e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2520 q^{7} - 4684 q^{9} + 8496 q^{11} - 3620 q^{13} - 51648 q^{15} + 41520 q^{17} + 54432 q^{19} + 7560 q^{23} - 273960 q^{25} - 133920 q^{27} - 346056 q^{29} + 486300 q^{33} + 283248 q^{35} - 68280 q^{37}+ \cdots + 83706300 q^{97}+O(q^{100}) \) 16 * q - 2520 * q^7 - 4684 * q^9 + 8496 * q^11 - 3620 * q^13 - 51648 * q^15 + 41520 * q^17 + 54432 * q^19 + 7560 * q^23 - 273960 * q^25 - 133920 * q^27 - 346056 * q^29 + 486300 * q^33 + 283248 * q^35 - 68280 * q^37 + 1122680 * q^39 - 1288008 * q^41 - 285360 * q^43 + 5743980 * q^45 - 2131660 * q^49 - 957280 * q^51 + 3666600 * q^53 + 1091472 * q^55 - 1799280 * q^59 - 4626000 * q^61 + 8609760 * q^63 - 11518020 * q^65 + 7094400 * q^67 + 18027316 * q^69 - 25958520 * q^71 + 10898032 * q^75 - 20577480 * q^77 - 34143488 * q^79 - 15745864 * q^81 + 7123932 * q^85 + 10460440 * q^87 + 10609452 * q^89 + 38683952 * q^91 - 52747080 * q^93 - 33837504 * q^95 + 83706300 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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