LMFDB - Embedded newform 37.8.b.a.36.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,8,Mod(36,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("37.36");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	37.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:	$11.5582459429$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 1702 x^{18} + 1194509 x^{16} + 450999516 x^{14} + 100204783492 x^{12} + 13461378480848 x^{10} + \cdots + 13\!\cdots\!96 $ x^20 + 1702x^18 + 1194509x^16 + 450999516x^14 + 100204783492x^12 + 13461378480848x^10 + 1081011973644416x^8 + 49304995250225664x^6 + 1131877572418003968x^4 + 9402469145336696832*x^2 + 130757963535876096
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			36.7
Root			$-7.77325i$ of defining polynomial
Character	$\chi$	$=$	37.36
Dual form			37.8.b.a.36.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.77325i q^{2} -62.8536 q^{3} +67.5766 q^{4} +184.143i q^{5} +488.577i q^{6} +1071.60 q^{7} -1520.27i q^{8} +1763.58 q^{9} +O(q^{10})$	\(q-7.77325i q^{2} -62.8536 q^{3} +67.5766 q^{4} +184.143i q^{5} +488.577i q^{6} +1071.60 q^{7} -1520.27i q^{8} +1763.58 q^{9} +1431.39 q^{10} -3542.53 q^{11} -4247.44 q^{12} -6313.40i q^{13} -8329.81i q^{14} -11574.1i q^{15} -3167.59 q^{16} -22366.3i q^{17} -13708.8i q^{18} +9601.18i q^{19} +12443.8i q^{20} -67354.0 q^{21} +27537.0i q^{22} -95973.0i q^{23} +95554.2i q^{24} +44216.3 q^{25} -49075.6 q^{26} +26613.4 q^{27} +72415.1 q^{28} -24891.1i q^{29} -89968.1 q^{30} -251973. i q^{31} -169971. i q^{32} +222661. q^{33} -173858. q^{34} +197328. i q^{35} +119177. q^{36} +(-98687.8 + 291878. i) q^{37} +74632.3 q^{38} +396820. i q^{39} +279946. q^{40} +449268. q^{41} +523559. i q^{42} +285878. i q^{43} -239392. q^{44} +324751. i q^{45} -746022. q^{46} -202075. q^{47} +199095. q^{48} +324783. q^{49} -343704. i q^{50} +1.40580e6i q^{51} -426638. i q^{52} -333372. q^{53} -206873. i q^{54} -652333. i q^{55} -1.62912e6i q^{56} -603469. i q^{57} -193484. q^{58} +179819. i q^{59} -782136. i q^{60} -1.60637e6i q^{61} -1.95865e6 q^{62} +1.88985e6 q^{63} -1.72668e6 q^{64} +1.16257e6 q^{65} -1.73080e6i q^{66} +2.14594e6 q^{67} -1.51144e6i q^{68} +6.03225e6i q^{69} +1.53388e6 q^{70} -4.66087e6 q^{71} -2.68111e6i q^{72} +5.02966e6 q^{73} +(2.26884e6 + 767125. i) q^{74} -2.77916e6 q^{75} +648815. i q^{76} -3.79618e6 q^{77} +3.08458e6 q^{78} -3.45009e6i q^{79} -583290. i q^{80} -5.52970e6 q^{81} -3.49227e6i q^{82} -3.38690e6 q^{83} -4.55155e6 q^{84} +4.11859e6 q^{85} +2.22220e6 q^{86} +1.56449e6i q^{87} +5.38559e6i q^{88} +2.64331e6i q^{89} +2.52437e6 q^{90} -6.76544e6i q^{91} -6.48553e6i q^{92} +1.58374e7i q^{93} +1.57078e6i q^{94} -1.76799e6 q^{95} +1.06833e7i q^{96} +8.57697e6i q^{97} -2.52462e6i q^{98} -6.24754e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10}) $ 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9	$ 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9} - 882 q^{10} + 3498 q^{11} - 30374 q^{12} + 36116 q^{16} + 113482 q^{21} - 108112 q^{25} + 49278 q^{26} - 304110 q^{27} - 41192 q^{28} + 429776 q^{30} + 305646 q^{33} - 960356 q^{34} + 484758 q^{36} + 108732 q^{37} + 1049916 q^{38} - 496346 q^{40} - 1577742 q^{41} + 685266 q^{44} - 2906298 q^{46} - 1512786 q^{47} + 1522958 q^{48} + 3269246 q^{49} + 2999358 q^{53} + 405946 q^{58} + 3728310 q^{62} - 11995292 q^{63} - 11109700 q^{64} + 4251792 q^{65} + 3562224 q^{67} + 21605644 q^{70} - 15259086 q^{71} + 11088018 q^{73} - 2036544 q^{74} + 14882062 q^{75} - 2419122 q^{77} - 12178734 q^{78} - 17764972 q^{81} - 12873822 q^{83} + 9944396 q^{84} - 2698920 q^{85} + 15345336 q^{86} - 13219100 q^{90} + 48981192 q^{95} + 43111380 q^{99}+O(q^{100}) $ 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9 - 882 * q^10 + 3498 * q^11 - 30374 * q^12 + 36116 * q^16 + 113482 * q^21 - 108112 * q^25 + 49278 * q^26 - 304110 * q^27 - 41192 * q^28 + 429776 * q^30 + 305646 * q^33 - 960356 * q^34 + 484758 * q^36 + 108732 * q^37 + 1049916 * q^38 - 496346 * q^40 - 1577742 * q^41 + 685266 * q^44 - 2906298 * q^46 - 1512786 * q^47 + 1522958 * q^48 + 3269246 * q^49 + 2999358 * q^53 + 405946 * q^58 + 3728310 * q^62 - 11995292 * q^63 - 11109700 * q^64 + 4251792 * q^65 + 3562224 * q^67 + 21605644 * q^70 - 15259086 * q^71 + 11088018 * q^73 - 2036544 * q^74 + 14882062 * q^75 - 2419122 * q^77 - 12178734 * q^78 - 17764972 * q^81 - 12873822 * q^83 + 9944396 * q^84 - 2698920 * q^85 + 15345336 * q^86 - 13219100 * q^90 + 48981192 * q^95 + 43111380 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		−	7.77325i		−	0.687065i	−0.939141	−	0.343532i	$-0.888377\pi$
$2$		−	7.77325i		−	0.687065i	0.939141	−	0.343532i	$-0.111623\pi$
$3$	−62.8536			−1.34402			−0.672011	−	0.740542i	$-0.734569\pi$
$3$	−62.8536			−1.34402			−0.672011	+	0.740542i	$0.734569\pi$
$4$	67.5766			0.527942
$5$			184.143i			0.658810i	0.944189	+	0.329405i	$0.106848\pi$
$5$			184.143i			0.658810i	−0.944189	+	0.329405i	$0.893152\pi$
$6$			488.577i			0.923429i
$7$	1071.60			1.18084			0.590418	−	0.807098i	$-0.298963\pi$
$7$	1071.60			1.18084			0.590418	+	0.807098i	$0.298963\pi$
$8$		−	1520.27i		−	1.04979i
$9$	1763.58			0.806393
$10$	1431.39			0.452645
$11$	−3542.53			−0.802490			−0.401245	−	0.915971i	$-0.631422\pi$
$11$	−3542.53			−0.802490			−0.401245	+	0.915971i	$0.631422\pi$
$12$	−4247.44			−0.709566
$13$		−	6313.40i		−	0.797006i	−0.917167	−	0.398503i	$-0.869530\pi$
$13$		−	6313.40i		−	0.797006i	0.917167	−	0.398503i	$-0.130470\pi$
$14$		−	8329.81i		−	0.811310i
$15$		−	11574.1i		−	0.885455i
$16$	−3167.59			−0.193335
$17$		−	22366.3i		−	1.10413i	−0.833799	−	0.552067i	$-0.813839\pi$
$17$		−	22366.3i		−	1.10413i	0.833799	−	0.552067i	$-0.186161\pi$
$18$		−	13708.8i		−	0.554044i
$19$			9601.18i			0.321134i	0.987025	+	0.160567i	$0.0513323\pi$
$19$			9601.18i			0.321134i	−0.987025	+	0.160567i	$0.948668\pi$
$20$			12443.8i			0.347814i
$21$	−67354.0			−1.58707
$22$			27537.0i			0.551362i
$23$		−	95973.0i		−	1.64476i	−0.568941	−	0.822378i	$-0.692647\pi$
$23$		−	95973.0i		−	1.64476i	0.568941	−	0.822378i	$-0.307353\pi$
$24$			95554.2i			1.41095i
$25$	44216.3			0.565969
$26$	−49075.6			−0.547595
$27$	26613.4			0.260212
$28$	72415.1			0.623413
$29$		−	24891.1i		−	0.189518i	−0.995500	−	0.0947590i	$-0.969792\pi$
$29$		−	24891.1i		−	0.189518i	0.995500	−	0.0947590i	$-0.0302080\pi$
$30$	−89968.1			−0.608365
$31$		−	251973.i		−	1.51911i	−0.650445	−	0.759553i	$-0.725418\pi$
$31$		−	251973.i		−	1.51911i	0.650445	−	0.759553i	$-0.274582\pi$
$32$		−	169971.i		−	0.916962i
$33$	222661.			1.07856
$34$	−173858.			−0.758612
$35$			197328.i			0.777947i
$36$	119177.			0.425729
$37$	−98687.8	+	291878.i	−0.320300	+	0.947316i
$37$	−98687.8	+	291878.i	−0.320300	+	0.947316i
$38$	74632.3			0.220640
$39$			396820.i			1.07119i
$40$	279946.			0.691616
$41$	449268.			1.01803			0.509017	−	0.860757i	$-0.330009\pi$
$41$	449268.			1.01803			0.509017	+	0.860757i	$0.330009\pi$
$42$			523559.i			1.09042i
$43$			285878.i			0.548330i	0.961683	+	0.274165i	$0.0884014\pi$
$43$			285878.i			0.548330i	−0.961683	+	0.274165i	$0.911599\pi$
$44$	−239392.			−0.423668
$45$			324751.i			0.531260i
$46$	−746022.			−1.13005
$47$	−202075.			−0.283903			−0.141951	−	0.989874i	$-0.545338\pi$
$47$	−202075.			−0.283903			−0.141951	+	0.989874i	$0.545338\pi$
$48$	199095.			0.259846
$49$	324783.			0.394373
$50$		−	343704.i		−	0.388857i
$51$			1.40580e6i			1.48398i
$52$		−	426638.i		−	0.420773i
$53$	−333372.			−0.307584			−0.153792	−	0.988103i	$-0.549149\pi$
$53$	−333372.			−0.307584			−0.153792	+	0.988103i	$0.549149\pi$
$54$		−	206873.i		−	0.178783i
$55$		−	652333.i		−	0.528688i
$56$		−	1.62912e6i		−	1.23964i
$57$		−	603469.i		−	0.431611i
$58$	−193484.			−0.130211
$59$			179819.i			0.113986i	0.998375	+	0.0569932i	$0.0181513\pi$
$59$			179819.i			0.113986i	−0.998375	+	0.0569932i	$0.981849\pi$
$60$		−	782136.i		−	0.467469i
$61$		−	1.60637e6i		−	0.906131i	−0.891477	−	0.453065i	$-0.850330\pi$
$61$		−	1.60637e6i		−	0.906131i	0.891477	−	0.453065i	$-0.149670\pi$
$62$	−1.95865e6			−1.04372
$63$	1.88985e6			0.952217
$64$	−1.72668e6			−0.823346
$65$	1.16257e6			0.525076
$66$		−	1.73080e6i		−	0.741042i
$67$	2.14594e6			0.871678			0.435839	−	0.900025i	$-0.356452\pi$
$67$	2.14594e6			0.871678			0.435839	+	0.900025i	$0.356452\pi$
$68$		−	1.51144e6i		−	0.582920i
$69$			6.03225e6i			2.21059i
$70$	1.53388e6			0.534500
$71$	−4.66087e6			−1.54548			−0.772740	−	0.634723i	$-0.781114\pi$
$71$	−4.66087e6			−1.54548			−0.772740	+	0.634723i	$0.781114\pi$
$72$		−	2.68111e6i		−	0.846547i
$73$	5.02966e6			1.51324			0.756621	−	0.653854i	$-0.226849\pi$
$73$	5.02966e6			1.51324			0.756621	+	0.653854i	$0.226849\pi$
$74$	2.26884e6	+	767125.i	0.650867	+	0.220067i
$75$	−2.77916e6			−0.760674
$76$			648815.i			0.169540i
$77$	−3.79618e6			−0.947609
$78$	3.08458e6			0.735979
$79$		−	3.45009e6i		−	0.787291i	−0.919262	−	0.393645i	$-0.871214\pi$
$79$		−	3.45009e6i		−	0.787291i	0.919262	−	0.393645i	$-0.128786\pi$
$80$		−	583290.i		−	0.127371i
$81$	−5.52970e6			−1.15612
$82$		−	3.49227e6i		−	0.699455i
$83$	−3.38690e6			−0.650173			−0.325086	−	0.945684i	$-0.605393\pi$
$83$	−3.38690e6			−0.650173			−0.325086	+	0.945684i	$0.605393\pi$
$84$	−4.55155e6			−0.837881
$85$	4.11859e6			0.727416
$86$	2.22220e6			0.376738
$87$			1.56449e6i			0.254716i
$88$			5.38559e6i			0.842450i
$89$			2.64331e6i			0.397451i	0.980055	+	0.198726i	$0.0636802\pi$
$89$			2.64331e6i			0.397451i	−0.980055	+	0.198726i	$0.936320\pi$
$90$	2.52437e6			0.365010
$91$		−	6.76544e6i		−	0.941133i
$92$		−	6.48553e6i		−	0.868337i
$93$			1.58374e7i			2.04171i
$94$			1.57078e6i			0.195060i
$95$	−1.76799e6			−0.211567
$96$			1.06833e7i			1.23242i
$97$			8.57697e6i			0.954185i	0.878853	+	0.477092i	$0.158309\pi$
$97$			8.57697e6i			0.954185i	−0.878853	+	0.477092i	$0.841691\pi$
$98$		−	2.52462e6i		−	0.270960i
$99$	−6.24754e6			−0.647122
$100$	2.98799e6			0.298799
$101$	1.70872e7			1.65023			0.825117	−	0.564961i	$-0.191109\pi$
$101$	1.70872e7			1.65023			0.825117	+	0.564961i	$0.191109\pi$
$102$	1.09276e7			1.01959
$103$			2.22530e6i			0.200659i	0.994954	+	0.100330i	$0.0319897\pi$
$103$			2.22530e6i			0.200659i	−0.994954	+	0.100330i	$0.968010\pi$
$104$	−9.59804e6			−0.836693
$105$		−	1.24028e7i		−	1.04558i
$106$			2.59138e6i			0.211330i
$107$	9.28643e6			0.732834			0.366417	−	0.930451i	$-0.380584\pi$
$107$	9.28643e6			0.732834			0.366417	+	0.930451i	$0.380584\pi$
$108$	1.79845e6			0.137377
$109$			2.12406e7i			1.57099i	0.618867	+	0.785496i	$0.287592\pi$
$109$			2.12406e7i			1.57099i	−0.618867	+	0.785496i	$0.712408\pi$
$110$	−5.07075e6			−0.363243
$111$	6.20289e6	−	1.83456e7i	0.430491	−	1.27321i
$112$	−3.39439e6			−0.228296
$113$			1.85610e7i			1.21011i	0.796182	+	0.605057i	$0.206850\pi$
$113$			1.85610e7i			1.21011i	−0.796182	+	0.605057i	$0.793150\pi$
$114$	−4.69091e6			−0.296545
$115$	1.76728e7			1.08358
$116$		−	1.68205e6i		−	0.100055i
$117$		−	1.11342e7i		−	0.642700i
$118$	1.39778e6			0.0783160
$119$		−	2.39677e7i		−	1.30380i
$120$	−1.75957e7			−0.929546
$121$	−6.93764e6			−0.356010
$122$	−1.24867e7			−0.622570
$123$	−2.82381e7			−1.36826
$124$		−	1.70275e7i		−	0.802000i
$125$			2.25283e7i			1.03168i
$126$		−	1.46903e7i		−	0.654235i
$127$	−2.47678e6			−0.107294			−0.0536468	−	0.998560i	$-0.517085\pi$
$127$	−2.47678e6			−0.107294			−0.0536468	+	0.998560i	$0.517085\pi$
$128$		−	8.33442e6i		−	0.351270i
$129$		−	1.79685e7i		−	0.736966i
$130$		−	9.03694e6i		−	0.360761i
$131$		−	2.87745e7i		−	1.11830i	−0.829066	−	0.559150i	$-0.811127\pi$
$131$		−	2.87745e7i		−	1.11830i	0.829066	−	0.559150i	$-0.188873\pi$
$132$	1.50467e7			0.569419
$133$			1.02886e7i			0.379207i
$134$		−	1.66809e7i		−	0.598899i
$135$			4.90068e6i			0.171431i
$136$	−3.40027e7			−1.15912
$137$	−2.22896e7			−0.740593			−0.370296	−	0.928914i	$-0.620744\pi$
$137$	−2.22896e7			−0.740593			−0.370296	+	0.928914i	$0.620744\pi$
$138$	4.68902e7			1.51882
$139$	1.95351e7			0.616971			0.308485	−	0.951229i	$-0.400178\pi$
$139$	1.95351e7			0.616971			0.308485	+	0.951229i	$0.400178\pi$
$140$			1.33347e7i			0.410711i
$141$	1.27011e7			0.381571
$142$			3.62301e7i			1.06184i
$143$			2.23654e7i			0.639589i
$144$	−5.58631e6			−0.155904
$145$	4.58352e6			0.124856
$146$		−	3.90968e7i		−	1.03970i
$147$	−2.04138e7			−0.530046
$148$	−6.66899e6	+	1.97241e7i	−0.169100	+	0.500128i
$149$	−3.39105e7			−0.839813			−0.419907	−	0.907567i	$-0.637937\pi$
$149$	−3.39105e7			−0.839813			−0.419907	+	0.907567i	$0.637937\pi$
$150$			2.16031e7i			0.522632i
$151$	−5.45454e7			−1.28925			−0.644627	−	0.764497i	$-0.722987\pi$
$151$	−5.45454e7			−1.28925			−0.644627	+	0.764497i	$0.722987\pi$
$152$	1.45963e7			0.337125
$153$		−	3.94447e7i		−	0.890366i
$154$			2.95086e7i			0.651068i
$155$	4.63991e7			1.00080
$156$			2.68158e7i			0.565528i
$157$	9.55167e7			1.96984			0.984919	−	0.173015i	$-0.0553509\pi$
$157$	9.55167e7			1.96984			0.984919	+	0.173015i	$0.0553509\pi$
$158$	−2.68184e7			−0.540920
$159$	2.09536e7			0.413399
$160$	3.12991e7			0.604104
$161$		−	1.02845e8i		−	1.94219i
$162$			4.29837e7i			0.794331i
$163$		−	4.16479e6i		−	0.0753245i	−0.999291	−	0.0376623i	$-0.988009\pi$
$163$		−	4.16479e6i		−	0.0753245i	0.999291	−	0.0376623i	$-0.0119911\pi$
$164$	3.03600e7			0.537463
$165$			4.10015e7i			0.710568i
$166$			2.63272e7i			0.446711i
$167$		−	6.45622e7i		−	1.07268i	−0.844002	−	0.536341i	$-0.819806\pi$
$167$		−	6.45622e7i		−	1.07268i	0.844002	−	0.536341i	$-0.180194\pi$
$168$			1.02396e8i			1.66610i
$169$	2.28895e7			0.364781
$170$		−	3.20148e7i		−	0.499781i
$171$			1.69325e7i			0.258960i
$172$			1.93187e7i			0.289486i
$173$	−1.05385e8			−1.54746			−0.773728	−	0.633518i	$-0.781610\pi$
$173$	−1.05385e8			−1.54746			−0.773728	+	0.633518i	$0.781610\pi$
$174$	1.21612e7			0.175006
$175$	4.73822e7			0.668316
$176$	1.12213e7			0.155149
$177$		−	1.13023e7i		−	0.153200i
$178$	2.05471e7			0.273075
$179$			3.56184e6i			0.0464183i	0.999731	+	0.0232091i	$0.00738836\pi$
$179$			3.56184e6i			0.0464183i	−0.999731	+	0.0232091i	$0.992612\pi$
$180$			2.19456e7i			0.280475i
$181$	−1.04584e8			−1.31096			−0.655481	−	0.755212i	$-0.727534\pi$
$181$	−1.04584e8			−1.31096			−0.655481	+	0.755212i	$0.727534\pi$
$182$	−5.25894e7			−0.646619
$183$			1.00966e8i			1.21786i
$184$	−1.45904e8			−1.72666
$185$	−5.37473e7	−	1.81727e7i	−0.624102	−	0.211017i
$186$	1.23108e8			1.40279
$187$			7.92332e7i			0.886057i
$188$	−1.36555e7			−0.149884
$189$	2.85190e7			0.307268
$190$			1.37430e7i			0.145360i
$191$		−	1.27517e8i		−	1.32419i	−0.749419	−	0.662096i	$-0.769667\pi$
$191$		−	1.27517e8i		−	1.32419i	0.749419	−	0.662096i	$-0.230333\pi$
$192$	1.08528e8			1.10659
$193$			1.53071e8i			1.53265i	0.642454	+	0.766324i	$0.277916\pi$
$193$			1.53071e8i			1.53265i	−0.642454	+	0.766324i	$0.722084\pi$
$194$	6.66709e7			0.655587
$195$	−7.30717e7			−0.705713
$196$	2.19478e7			0.208206
$197$	1.49652e8			1.39460			0.697302	−	0.716777i	$-0.254384\pi$
$197$	1.49652e8			1.39460			0.697302	+	0.716777i	$0.254384\pi$
$198$			4.85637e7i			0.444614i
$199$		−	1.20435e8i		−	1.08334i	−0.840590	−	0.541672i	$-0.817792\pi$
$199$		−	1.20435e8i		−	1.08334i	0.840590	−	0.541672i	$-0.182208\pi$
$200$		−	6.72205e7i		−	0.594151i
$201$	−1.34880e8			−1.17155
$202$		−	1.32823e8i		−	1.13382i
$203$		−	2.66732e7i		−	0.223790i
$204$			9.49993e7i			0.783456i
$205$			8.27296e7i			0.670691i
$206$	1.72978e7			0.137866
$207$		−	1.69256e8i		−	1.32632i
$208$			1.99983e7i			0.154089i
$209$		−	3.40125e7i		−	0.257707i
$210$	−9.64098e7			−0.718379
$211$	−1.20633e8			−0.884051			−0.442025	−	0.897003i	$-0.645740\pi$
$211$	−1.20633e8			−0.884051			−0.442025	+	0.897003i	$0.645740\pi$
$212$	−2.25281e7			−0.162386
$213$	2.92953e8			2.07716
$214$		−	7.21857e7i		−	0.503504i
$215$	−5.26425e7			−0.361245
$216$		−	4.04595e7i		−	0.273169i
$217$		−	2.70014e8i		−	1.79381i
$218$	1.65108e8			1.07937
$219$	−3.16132e8			−2.03383
$220$		−	4.40825e7i		−	0.279117i
$221$	−1.41207e8			−0.880002
$222$	−1.42605e8	−	4.82166e7i	−0.874779	−	0.295775i
$223$	−3.05627e7			−0.184555			−0.0922773	−	0.995733i	$-0.529415\pi$
$223$	−3.05627e7			−0.184555			−0.0922773	+	0.995733i	$0.529415\pi$
$224$		−	1.82141e8i		−	1.08278i
$225$	7.79791e7			0.456393
$226$	1.44279e8			0.831426
$227$			6.29729e7i			0.357325i	0.983910	+	0.178662i	$0.0571770\pi$
$227$			6.29729e7i			0.357325i	−0.983910	+	0.178662i	$0.942823\pi$
$228$		−	4.07804e7i		−	0.227866i
$229$	8.90917e7			0.490245			0.245122	−	0.969492i	$-0.421172\pi$
$229$	8.90917e7			0.490245			0.245122	+	0.969492i	$0.421172\pi$
$230$		−	1.37375e8i		−	0.744491i
$231$	2.38604e8			1.27361
$232$	−3.78410e7			−0.198955
$233$	6.60296e7			0.341974			0.170987	−	0.985273i	$-0.445304\pi$
$233$	6.60296e7			0.341974			0.170987	+	0.985273i	$0.445304\pi$
$234$	−8.65488e7			−0.441576
$235$		−	3.72107e7i		−	0.187038i
$236$			1.21515e7i			0.0601783i
$237$			2.16851e8i			1.05814i
$238$	−1.86307e8			−0.895796
$239$			3.87928e8i			1.83806i	0.394192	+	0.919028i	$0.371024\pi$
$239$			3.87928e8i			1.83806i	−0.394192	+	0.919028i	$0.628976\pi$
$240$			3.66619e7i			0.171189i
$241$			3.14017e8i			1.44508i	0.691327	+	0.722542i	$0.257026\pi$
$241$			3.14017e8i			1.44508i	−0.691327	+	0.722542i	$0.742974\pi$
$242$			5.39280e7i			0.244602i
$243$	2.89358e8			1.29364
$244$		−	1.08553e8i		−	0.478385i
$245$			5.98066e7i			0.259817i
$246$			2.19502e8i			0.940082i
$247$	6.06161e7			0.255946
$248$	−3.83066e8			−1.59475
$249$	2.12879e8			0.873846
$250$	1.75118e8			0.708828
$251$			3.98926e7i			0.159233i	0.996826	+	0.0796167i	$0.0253696\pi$
$251$			3.98926e7i			0.159233i	−0.996826	+	0.0796167i	$0.974630\pi$
$252$	1.27710e8			0.502716
$253$			3.39987e8i			1.31990i
$254$			1.92526e7i			0.0737177i
$255$	−2.58869e8			−0.977662
$256$	−2.85801e8			−1.06469
$257$		−	3.94256e8i		−	1.44881i	−0.689372	−	0.724407i	$-0.742114\pi$
$257$		−	3.94256e8i		−	1.44881i	0.689372	−	0.724407i	$-0.257886\pi$
$258$	−1.39674e8			−0.506344
$259$	−1.05754e8	+	3.12776e8i	−0.378222	+	1.11862i
$260$	7.85625e7			0.277210
$261$		−	4.38974e7i		−	0.152826i
$262$	−2.23671e8			−0.768345
$263$	2.66665e8			0.903900			0.451950	−	0.892043i	$-0.350728\pi$
$263$	2.66665e8			0.903900			0.451950	+	0.892043i	$0.350728\pi$
$264$		−	3.38504e8i		−	1.13227i
$265$		−	6.13881e7i		−	0.202639i
$266$	7.99760e7			0.260540
$267$		−	1.66142e8i		−	0.534183i
$268$	1.45015e8			0.460195
$269$	−3.85489e8			−1.20748			−0.603739	−	0.797182i	$-0.706323\pi$
$269$	−3.85489e8			−1.20748			−0.603739	+	0.797182i	$0.706323\pi$
$270$	3.80942e7			0.117784
$271$	5.52563e8			1.68651			0.843256	−	0.537512i	$-0.180636\pi$
$271$	5.52563e8			1.68651			0.843256	+	0.537512i	$0.180636\pi$
$272$			7.08472e7i			0.213467i
$273$			4.25233e8i			1.26490i
$274$			1.73262e8i			0.508835i
$275$	−1.56638e8			−0.454184
$276$			4.07639e8i			1.16706i
$277$			2.07607e8i			0.586898i	0.955975	+	0.293449i	$0.0948030\pi$
$277$			2.07607e8i			0.586898i	−0.955975	+	0.293449i	$0.905197\pi$
$278$		−	1.51851e8i		−	0.423899i
$279$		−	4.44375e8i		−	1.22500i
$280$	2.99991e8			0.816685
$281$			1.72083e8i			0.462665i	0.972875	+	0.231333i	$0.0743086\pi$
$281$			1.72083e8i			0.462665i	−0.972875	+	0.231333i	$0.925691\pi$
$282$		−	9.87291e7i		−	0.262164i
$283$		−	5.46066e8i		−	1.43216i	−0.698016	−	0.716082i	$-0.745934\pi$
$283$		−	5.46066e8i		−	1.43216i	0.698016	−	0.716082i	$-0.254066\pi$
$284$	−3.14966e8			−0.815924
$285$	1.11125e8			0.284350
$286$	1.73852e8			0.439439
$287$	4.81436e8			1.20213
$288$		−	2.99758e8i		−	0.739431i
$289$	−8.99110e7			−0.219114
$290$		−	3.56288e7i		−	0.0857844i
$291$		−	5.39094e8i		−	1.28244i
$292$	3.39887e8			0.798905
$293$	7.10525e7			0.165022			0.0825112	−	0.996590i	$-0.473706\pi$
$293$	7.10525e7			0.165022			0.0825112	+	0.996590i	$0.473706\pi$
$294$			1.58682e8i			0.364176i
$295$	−3.31124e7			−0.0750954
$296$	4.43732e8	+	1.50032e8i	0.994488	+	0.336250i
$297$	−9.42789e7			−0.208818
$298$			2.63595e8i			0.577006i
$299$	−6.05916e8			−1.31088
$300$	−1.87806e8			−0.401592
$301$			3.06347e8i			0.647487i
$302$			4.23995e8i			0.885801i
$303$	−1.07399e9			−2.21795
$304$		−	3.04126e7i		−	0.0620864i
$305$	2.95802e8			0.596968
$306$	−3.06614e8			−0.611739
$307$	2.67756e8			0.528147			0.264074	−	0.964503i	$-0.414934\pi$
$307$	2.67756e8			0.528147			0.264074	+	0.964503i	$0.414934\pi$
$308$	−2.56533e8			−0.500283
$309$		−	1.39869e8i		−	0.269690i
$310$		−	3.60672e8i		−	0.687616i
$311$			9.51878e8i			1.79440i	0.441622	+	0.897201i	$0.354403\pi$
$311$			9.51878e8i			1.79440i	−0.441622	+	0.897201i	$0.645597\pi$
$312$	6.03272e8			1.12453
$313$		−	2.56095e8i		−	0.472058i	−0.971746	−	0.236029i	$-0.924154\pi$
$313$		−	2.56095e8i		−	0.472058i	0.971746	−	0.236029i	$-0.0758460\pi$
$314$		−	7.42475e8i		−	1.35341i
$315$			3.48003e8i			0.627331i
$316$		−	2.33145e8i		−	0.415644i
$317$	4.88498e8			0.861302			0.430651	−	0.902519i	$-0.358284\pi$
$317$	4.88498e8			0.861302			0.430651	+	0.902519i	$0.358284\pi$
$318$		−	1.62878e8i		−	0.284032i
$319$			8.81774e7i			0.152086i
$320$		−	3.17957e8i		−	0.542429i
$321$	−5.83686e8			−0.984944
$322$	−7.99437e8			−1.33441
$323$	2.14742e8			0.354576
$324$	−3.73679e8			−0.610367
$325$		−	2.79155e8i		−	0.451081i
$326$	−3.23739e7			−0.0517528
$327$		−	1.33505e9i		−	2.11145i
$328$		−	6.83007e8i		−	1.06873i
$329$	−2.16543e8			−0.335243
$330$	3.18715e8			0.488206
$331$		−	9.16711e8i		−	1.38942i	−0.719289	−	0.694711i	$-0.755532\pi$
$331$		−	9.16711e8i		−	1.38942i	0.719289	−	0.694711i	$-0.244468\pi$
$332$	−2.28875e8			−0.343254
$333$	−1.74044e8	+	5.14750e8i	−0.258288	+	0.763909i
$334$	−5.01858e8			−0.737001
$335$			3.95160e8i			0.574270i
$336$	2.13350e8			0.306835
$337$	7.08990e8			1.00910			0.504551	−	0.863382i	$-0.331658\pi$
$337$	7.08990e8			1.00910			0.504551	+	0.863382i	$0.331658\pi$
$338$		−	1.77926e8i		−	0.250628i
$339$		−	1.16662e9i		−	1.62642i
$340$	2.78321e8			0.384033
$341$			8.92622e8i			1.21907i
$342$	1.31620e8			0.177923
$343$	−5.34471e8			−0.715146
$344$	4.34611e8			0.575634
$345$	−1.11080e9			−1.45636
$346$			8.19185e8i			1.06320i
$347$			8.52560e8i			1.09540i	0.836676	+	0.547699i	$0.184496\pi$
$347$			8.52560e8i			1.09540i	−0.836676	+	0.547699i	$0.815504\pi$
$348$			1.05723e8i			0.134475i
$349$	6.85056e8			0.862654			0.431327	−	0.902196i	$-0.358045\pi$
$349$	6.85056e8			0.862654			0.431327	+	0.902196i	$0.358045\pi$
$350$		−	3.68314e8i		−	0.459176i
$351$		−	1.68021e8i		−	0.207391i
$352$			6.02130e8i			0.735852i
$353$			9.46593e8i			1.14539i	0.819770	+	0.572693i	$0.194101\pi$
$353$			9.46593e8i			1.14539i	−0.819770	+	0.572693i	$0.805899\pi$
$354$	−8.78553e7			−0.105258
$355$		−	8.58268e8i		−	1.01818i
$356$			1.78626e8i			0.209831i
$357$			1.50646e9i			1.75234i
$358$	2.76871e7			0.0318924
$359$	−3.56084e8			−0.406183			−0.203091	−	0.979160i	$-0.565099\pi$
$359$	−3.56084e8			−0.406183			−0.203091	+	0.979160i	$0.565099\pi$
$360$	4.93708e8			0.557714
$361$	8.01689e8			0.896873
$362$			8.12957e8i			0.900716i
$363$	4.36056e8			0.478486
$364$		−	4.57186e8i		−	0.496864i
$365$			9.26177e8i			0.996940i
$366$	7.84835e8			0.836748
$367$	4.10789e8			0.433798			0.216899	−	0.976194i	$-0.430406\pi$
$367$	4.10789e8			0.433798			0.216899	+	0.976194i	$0.430406\pi$
$368$			3.04003e8i			0.317988i
$369$	7.92321e8			0.820935
$370$	−1.41261e8	+	4.17791e8i	−0.144982	+	0.428798i
$371$	−3.57241e8			−0.363206
$372$			1.07024e9i			1.07791i
$373$	1.08047e9			1.07804			0.539018	−	0.842294i	$-0.318795\pi$
$373$	1.08047e9			1.07804			0.539018	+	0.842294i	$0.318795\pi$
$374$	6.15899e8			0.608778
$375$		−	1.41599e9i		−	1.38660i
$376$			3.07207e8i			0.298040i
$377$	−1.57147e8			−0.151047
$378$		−	2.21685e8i		−	0.211113i
$379$	−5.73572e8			−0.541191			−0.270596	−	0.962693i	$-0.587221\pi$
$379$	−5.73572e8			−0.541191			−0.270596	+	0.962693i	$0.587221\pi$
$380$	−1.19475e8			−0.111695
$381$	1.55675e8			0.144205
$382$	−9.91220e8			−0.909805
$383$		−	2.05358e9i		−	1.86773i	−0.357620	−	0.933867i	$-0.616412\pi$
$383$		−	2.05358e9i		−	1.86773i	0.357620	−	0.933867i	$-0.383588\pi$
$384$			5.23849e8i			0.472114i
$385$		−	6.99040e8i		−	0.624294i
$386$	1.18986e9			1.05303
$387$			5.04170e8i			0.442169i
$388$			5.79602e8i			0.503755i
$389$			7.62806e8i			0.657038i	0.944497	+	0.328519i	$0.106550\pi$
$389$			7.62806e8i			0.657038i	−0.944497	+	0.328519i	$0.893450\pi$
$390$			5.68004e8i			0.484870i
$391$	−2.14656e9			−1.81603
$392$		−	4.93757e8i		−	0.414011i
$393$			1.80858e9i			1.50302i
$394$		−	1.16328e9i		−	0.958183i
$395$	6.35310e8			0.518675
$396$	−4.22188e8			−0.341643
$397$	−1.85548e9			−1.48830			−0.744148	−	0.668015i	$-0.767144\pi$
$397$	−1.85548e9			−1.48830			−0.744148	+	0.668015i	$0.767144\pi$
$398$	−9.36169e8			−0.744327
$399$		−	6.46677e8i		−	0.509662i
$400$	−1.40059e8			−0.109421
$401$			4.58966e7i			0.0355448i	0.999842	+	0.0177724i	$0.00565742\pi$
$401$			4.58966e7i			0.0355448i	−0.999842	+	0.0177724i	$0.994343\pi$
$402$			1.04846e9i			0.804933i
$403$	−1.59081e9			−1.21074
$404$	1.15469e9			0.871229
$405$		−	1.01826e9i		−	0.761666i
$406$	−2.07338e8			−0.153758
$407$	3.49605e8	−	1.03399e9i	0.257038	−	0.760211i
$408$	2.13719e9			1.55788
$409$		−	1.62943e9i		−	1.17761i	−0.808274	−	0.588807i	$-0.799598\pi$
$409$		−	1.62943e9i		−	1.17761i	0.808274	−	0.588807i	$-0.200402\pi$
$410$	6.43078e8			0.460808
$411$	1.40098e9			0.995372
$412$			1.50379e8i			0.105937i
$413$			1.92694e8i			0.134599i
$414$	−1.31567e9			−0.911267
$415$		−	6.23674e8i		−	0.428341i
$416$	−1.07310e9			−0.730824
$417$	−1.22785e9			−0.829222
$418$	−2.64387e8			−0.177061
$419$	2.00862e8			0.133398			0.0666989	−	0.997773i	$-0.478753\pi$
$419$	2.00862e8			0.133398			0.0666989	+	0.997773i	$0.478753\pi$
$420$		−	8.38137e8i		−	0.552004i
$421$			2.03536e9i			1.32940i	0.747112	+	0.664698i	$0.231440\pi$
$421$			2.03536e9i			1.32940i	−0.747112	+	0.664698i	$0.768560\pi$
$422$			9.37710e8i			0.607400i
$423$	−3.56375e8			−0.228937
$424$			5.06814e8i			0.322900i
$425$		−	9.88954e8i		−	0.624906i
$426$		−	2.27720e9i		−	1.42714i
$427$		−	1.72139e9i		−	1.06999i
$428$	6.27545e8			0.386894
$429$		−	1.40575e9i		−	0.859621i
$430$			4.09203e8i			0.248199i
$431$			1.89636e9i			1.14091i	0.821329	+	0.570454i	$0.193233\pi$
$431$			1.89636e9i			1.14091i	−0.821329	+	0.570454i	$0.806767\pi$
$432$	−8.43005e7			−0.0503080
$433$	1.40467e9			0.831511			0.415755	−	0.909476i	$-0.363517\pi$
$433$	1.40467e9			0.831511			0.415755	+	0.909476i	$0.363517\pi$
$434$	−2.09889e9			−1.23247
$435$	−2.88091e8			−0.167810
$436$			1.43537e9i			0.829393i
$437$	9.21454e8			0.528188
$438$			2.45738e9i			1.39737i
$439$		−	2.35723e9i		−	1.32977i	−0.746947	−	0.664883i	$-0.768481\pi$
$439$		−	2.35723e9i		−	1.32977i	0.746947	−	0.664883i	$-0.231519\pi$
$440$	−9.91719e8			−0.555015
$441$	5.72782e8			0.318020
$442$			1.09764e9i			0.604618i
$443$	9.47597e8			0.517858			0.258929	−	0.965896i	$-0.416630\pi$
$443$	9.47597e8			0.517858			0.258929	+	0.965896i	$0.416630\pi$
$444$	4.19170e8	−	1.23973e9i	0.227274	−	0.672183i
$445$	−4.86748e8			−0.261845
$446$			2.37572e8i			0.126801i
$447$	2.13140e9			1.12873
$448$	−1.85031e9			−0.972237
$449$		−	1.91232e9i		−	0.997007i	−0.866888	−	0.498503i	$-0.833883\pi$
$449$		−	1.91232e9i		−	0.997007i	0.866888	−	0.498503i	$-0.166117\pi$
$450$		−	6.06151e8i		−	0.313572i
$451$	−1.59155e9			−0.816962
$452$			1.25429e9i			0.638870i
$453$	3.42838e9			1.73279
$454$	4.89504e8			0.245505
$455$	1.24581e9			0.620028
$456$	−9.17433e8			−0.453104
$457$		−	3.28480e9i		−	1.60991i	−0.593333	−	0.804957i	$-0.702188\pi$
$457$		−	3.28480e9i		−	1.60991i	0.593333	−	0.804957i	$-0.297812\pi$
$458$		−	6.92532e8i		−	0.336830i
$459$		−	5.95243e8i		−	0.287309i
$460$	1.19427e9			0.572069
$461$			2.33847e8i			0.111168i	0.998454	+	0.0555839i	$0.0177020\pi$
$461$			2.33847e8i			0.111168i	−0.998454	+	0.0555839i	$0.982298\pi$
$462$		−	1.85472e9i		−	0.875049i
$463$			2.20388e9i			1.03194i	0.856607	+	0.515969i	$0.172568\pi$
$463$			2.20388e9i			1.03194i	−0.856607	+	0.515969i	$0.827432\pi$
$464$			7.88447e7i			0.0366404i
$465$	−2.91635e9			−1.34510
$466$		−	5.13265e8i		−	0.234958i
$467$			6.99117e8i			0.317644i	0.987307	+	0.158822i	$0.0507696\pi$
$467$			6.99117e8i			0.317644i	−0.987307	+	0.158822i	$0.949230\pi$
$468$		−	7.52411e8i		−	0.339308i
$469$	2.29959e9			1.02931
$470$	−2.89248e8			−0.128507
$471$	−6.00357e9			−2.64750
$472$	2.73372e8			0.119662
$473$		−	1.01273e9i		−	0.440029i
$474$	1.68563e9			0.727007
$475$			4.24529e8i			0.181752i
$476$		−	1.61966e9i		−	0.688332i
$477$	−5.87928e8			−0.248033
$478$	3.01546e9			1.26286
$479$			2.57187e9i			1.06924i	0.845093	+	0.534619i	$0.179545\pi$
$479$			2.57187e9i			1.06924i	−0.845093	+	0.534619i	$0.820455\pi$
$480$	−1.96726e9			−0.811928
$481$	1.84274e9	+	6.23056e8i	0.755017	+	0.255281i
$482$	2.44093e9			0.992866
$483$			6.46416e9i			2.61034i
$484$	−4.68822e8			−0.187953
$485$	−1.57939e9			−0.628627
$486$		−	2.24925e9i		−	0.888816i
$487$		−	1.36755e8i		−	0.0536526i	−0.999640	−	0.0268263i	$-0.991460\pi$
$487$		−	1.36755e8i		−	0.0536526i	0.999640	−	0.0268263i	$-0.00854010\pi$
$488$	−2.44211e9			−0.951252
$489$			2.61772e8i			0.101238i
$490$	4.64892e8			0.178511
$491$	3.76083e9			1.43383			0.716916	−	0.697160i	$-0.245553\pi$
$491$	3.76083e9			1.43383			0.716916	+	0.697160i	$0.245553\pi$
$492$	−1.90824e9			−0.722362
$493$	−5.56720e8			−0.209253
$494$		−	4.71184e8i		−	0.175851i
$495$		−	1.15044e9i		−	0.426331i
$496$			7.98148e8i			0.293696i
$497$	−4.99459e9			−1.82496
$498$		−	1.65476e9i		−	0.600389i
$499$			2.50825e9i			0.903688i	0.892097	+	0.451844i	$0.149234\pi$
$499$			2.50825e9i			0.903688i	−0.892097	+	0.451844i	$0.850766\pi$
$500$			1.52239e9i			0.544666i
$501$			4.05797e9i			1.44171i
$502$	3.10095e8			0.109404
$503$			4.17750e9i			1.46362i	0.681509	+	0.731810i	$0.261324\pi$
$503$			4.17750e9i			1.46362i	−0.681509	+	0.731810i	$0.738676\pi$
$504$		−	2.87308e9i		−	0.999633i
$505$			3.14649e9i			1.08719i
$506$	2.64281e9			0.906856
$507$	−1.43869e9			−0.490274
$508$	−1.67372e8			−0.0566449
$509$	1.10530e9			0.371508			0.185754	−	0.982596i	$-0.440527\pi$
$509$	1.10530e9			0.371508			0.185754	+	0.982596i	$0.440527\pi$
$510$			2.01225e9i			0.671717i
$511$	5.38978e9			1.78689
$512$			1.15480e9i			0.380242i
$513$			2.55520e8i			0.0835631i
$514$	−3.06465e9			−0.995429
$515$	−4.09774e8			−0.132196
$516$		−	1.21425e9i		−	0.389076i
$517$	7.15857e8			0.227829
$518$	2.43129e9	+	8.22051e8i	0.768567	+	0.259863i
$519$	6.62385e9			2.07981
$520$		−	1.76741e9i		−	0.551222i
$521$	4.60719e9			1.42726			0.713631	−	0.700522i	$-0.247049\pi$
$521$	4.60719e9			1.42726			0.713631	+	0.700522i	$0.247049\pi$
$522$	−3.41225e8			−0.105001
$523$			4.53314e9i			1.38562i	0.721122	+	0.692808i	$0.243627\pi$
$523$			4.53314e9i			1.38562i	−0.721122	+	0.692808i	$0.756373\pi$
$524$		−	1.94448e9i		−	0.590398i
$525$	−2.97814e9			−0.898231
$526$		−	2.07285e9i		−	0.621038i
$527$	−5.63569e9			−1.67730
$528$	−7.05300e8			−0.208523
$529$	−5.80599e9			−1.70522
$530$	−4.77185e8			−0.139226
$531$			3.17125e8i			0.0919178i
$532$			6.95270e8i			0.200199i
$533$		−	2.83641e9i		−	0.811379i
$534$	−1.29146e9			−0.367018
$535$			1.71003e9i			0.482799i
$536$		−	3.26240e9i		−	0.915083i
$537$		−	2.23875e8i		−	0.0623872i
$538$			2.99651e9i			0.829616i
$539$	−1.15056e9			−0.316481
$540$			3.31171e8i			0.0905054i
$541$		−	1.63791e9i		−	0.444733i	−0.974963	−	0.222366i	$-0.928622\pi$
$541$		−	1.63791e9i		−	0.444733i	0.974963	−	0.222366i	$-0.0713781\pi$
$542$		−	4.29521e9i		−	1.15874i
$543$	6.57349e9			1.76196
$544$	−3.80163e9			−1.01245
$545$	−3.91131e9			−1.03499
$546$	3.30544e9			0.869070
$547$		−	1.51913e9i		−	0.396861i	−0.980115	−	0.198431i	$-0.936416\pi$
$547$		−	1.51913e9i		−	0.396861i	0.980115	−	0.198431i	$-0.0635845\pi$
$548$	−1.50625e9			−0.390990
$549$		−	2.83296e9i		−	0.730697i
$550$			1.21758e9i			0.312054i
$551$	2.38983e8			0.0608607
$552$	9.17063e9			2.32066
$553$		−	3.69711e9i		−	0.929661i
$554$	1.61378e9			0.403237
$555$	3.37821e9	+	1.14222e9i	0.838806	+	0.283612i
$556$	1.32012e9			0.325725
$557$		−	1.87650e9i		−	0.460104i	−0.973178	−	0.230052i	$-0.926110\pi$
$557$		−	1.87650e9i		−	0.460104i	0.973178	−	0.230052i	$-0.0738897\pi$
$558$	−3.45423e9			−0.841651
$559$	1.80486e9			0.437022
$560$		−	6.25054e8i		−	0.150404i
$561$		−	4.98010e9i		−	1.19088i
$562$	1.33765e9			0.317881
$563$		−	7.96840e8i		−	0.188188i	−0.995563	−	0.0940939i	$-0.970005\pi$
$563$		−	7.96840e8i		−	0.188188i	0.995563	−	0.0940939i	$-0.0299954\pi$
$564$	8.58300e8			0.201448
$565$	−3.41787e9			−0.797235
$566$	−4.24471e9			−0.983989
$567$	−5.92563e9			−1.36519
$568$			7.08577e9i			1.62244i
$569$		−	3.29667e9i		−	0.750210i	−0.926982	−	0.375105i	$-0.877607\pi$
$569$		−	3.29667e9i		−	0.750210i	0.926982	−	0.375105i	$-0.122393\pi$
$570$		−	8.63799e8i		−	0.195367i
$571$	2.75252e9			0.618735			0.309367	−	0.950943i	$-0.399883\pi$
$571$	2.75252e9			0.618735			0.309367	+	0.950943i	$0.399883\pi$
$572$			1.51138e9i			0.337666i
$573$			8.01490e9i			1.77974i
$574$		−	3.74232e9i		−	0.825941i
$575$		−	4.24357e9i		−	0.930881i
$576$	−3.04514e9			−0.663941
$577$		−	6.48727e9i		−	1.40587i	−0.711252	−	0.702937i	$-0.751872\pi$
$577$		−	6.48727e9i		−	1.40587i	0.711252	−	0.702937i	$-0.248128\pi$
$578$			6.98900e8i			0.150546i
$579$		−	9.62108e9i		−	2.05991i
$580$	3.09738e8			0.0659170
$581$	−3.62940e9			−0.767747
$582$	−4.19051e9			−0.881122
$583$	1.18098e9			0.246833
$584$		−	7.64642e9i		−	1.58859i
$585$	2.05028e9			0.423417
$586$		−	5.52309e8i		−	0.113381i
$587$			5.72083e9i			1.16742i	0.811964	+	0.583708i	$0.198399\pi$
$587$			5.72083e9i			1.16742i	−0.811964	+	0.583708i	$0.801601\pi$
$588$	−1.37950e9			−0.279834
$589$	2.41924e9			0.487837
$590$			2.57391e8i			0.0515954i
$591$	−9.40618e9			−1.87438
$592$	3.12603e8	−	9.24550e8i	0.0619251	−	0.183149i
$593$	9.68426e9			1.90711			0.953553	−	0.301225i	$-0.0973955\pi$
$593$	9.68426e9			1.90711			0.953553	+	0.301225i	$0.0973955\pi$
$594$			7.32854e8i			0.143471i
$595$	4.41348e9			0.858958
$596$	−2.29156e9			−0.443373
$597$			7.56976e9i			1.45604i
$598$			4.70993e9i			0.900660i
$599$	−7.04695e9			−1.33970			−0.669850	−	0.742497i	$-0.733641\pi$
$599$	−7.04695e9			−1.33970			−0.669850	+	0.742497i	$0.733641\pi$
$600$			4.22506e9i			0.798552i
$601$	1.81048e9			0.340199			0.170100	−	0.985427i	$-0.445591\pi$
$601$	1.81048e9			0.340199			0.170100	+	0.985427i	$0.445591\pi$
$602$	2.38131e9			0.444865
$603$	3.78454e9			0.702914
$604$	−3.68599e9			−0.680652
$605$		−	1.27752e9i		−	0.234543i
$606$			8.34841e9i			1.52388i
$607$			9.44627e9i			1.71435i	0.515024	+	0.857176i	$0.327783\pi$
$607$			9.44627e9i			1.71435i	−0.515024	+	0.857176i	$0.672217\pi$
$608$	1.63193e9			0.294468
$609$			1.67651e9i			0.300778i
$610$		−	2.29934e9i		−	0.410156i
$611$			1.27578e9i			0.226272i
$612$		−	2.66554e9i		−	0.470062i
$613$	−5.38264e9			−0.943809			−0.471904	−	0.881650i	$-0.656433\pi$
$613$	−5.38264e9			−0.943809			−0.471904	+	0.881650i	$0.656433\pi$
$614$		−	2.08133e9i		−	0.362871i
$615$		−	5.19986e9i		−	0.901423i
$616$			5.77120e9i			0.994795i
$617$	−7.47778e9			−1.28167			−0.640833	−	0.767680i	$-0.721411\pi$
$617$	−7.47778e9			−1.28167			−0.640833	+	0.767680i	$0.721411\pi$
$618$	−1.08723e9			−0.185295
$619$	−3.23165e9			−0.547655			−0.273828	−	0.961779i	$-0.588290\pi$
$619$	−3.23165e9			−0.547655			−0.273828	+	0.961779i	$0.588290\pi$
$620$	3.13549e9			0.528366
$621$		−	2.55417e9i		−	0.427986i
$622$	7.39918e9			1.23287
$623$			2.83258e9i			0.469325i
$624$		−	1.25696e9i		−	0.207099i
$625$	−6.94033e8			−0.113710
$626$	−1.99069e9			−0.324334
$627$			2.13781e9i			0.346364i
$628$	6.45470e9			1.03996
$629$	6.52821e9	+	2.20728e9i	1.04596	+	0.353655i
$630$	2.70512e9			0.431017
$631$		−	1.11428e10i		−	1.76560i	−0.469746	−	0.882801i	$-0.655655\pi$
$631$		−	1.11428e10i		−	1.76560i	0.469746	−	0.882801i	$-0.344345\pi$
$632$	−5.24505e9			−0.826494
$633$	7.58222e9			1.18818
$634$		−	3.79721e9i		−	0.591770i
$635$		−	4.56082e8i		−	0.0706862i
$636$	1.41598e9			0.218251
$637$		−	2.05049e9i		−	0.314318i
$638$	6.85424e8			0.104493
$639$	−8.21983e9			−1.24626
$640$	1.53473e9			0.231420
$641$	−1.02739e9			−0.154075			−0.0770377	−	0.997028i	$-0.524546\pi$
$641$	−1.02739e9			−0.154075			−0.0770377	+	0.997028i	$0.524546\pi$
$642$			4.53714e9i			0.676720i
$643$		−	2.21119e9i		−	0.328010i	−0.986459	−	0.164005i	$-0.947559\pi$
$643$		−	2.21119e9i		−	0.328010i	0.986459	−	0.164005i	$-0.0524414\pi$
$644$		−	6.94989e9i		−	1.02536i
$645$	3.30877e9			0.485521
$646$		−	1.66925e9i		−	0.243616i
$647$		−	4.26115e9i		−	0.618531i	−0.950976	−	0.309266i	$-0.899917\pi$
$647$		−	4.26115e9i		−	0.618531i	0.950976	−	0.309266i	$-0.100083\pi$
$648$			8.40662e9i			1.21369i
$649$		−	6.37014e8i		−	0.0914729i
$650$	−2.16994e9			−0.309921
$651$			1.69714e10i			2.41092i
$652$		−	2.81442e8i		−	0.0397670i
$653$		−	1.44976e9i		−	0.203751i	−0.994797	−	0.101875i	$-0.967516\pi$
$653$		−	1.44976e9i		−	0.203751i	0.994797	−	0.101875i	$-0.0324843\pi$
$654$	−1.03777e10			−1.45070
$655$	5.29863e9			0.736748
$656$	−1.42310e9			−0.196821
$657$	8.87021e9			1.22027
$658$			1.68325e9i			0.230333i
$659$	1.16735e10			1.58892			0.794459	−	0.607318i	$-0.207755\pi$
$659$	1.16735e10			1.58892			0.794459	+	0.607318i	$0.207755\pi$
$660$			2.77074e9i			0.375139i
$661$		−	2.45761e9i		−	0.330984i	−0.986211	−	0.165492i	$-0.947079\pi$
$661$		−	2.45761e9i		−	0.330984i	0.986211	−	0.165492i	$-0.0529213\pi$
$662$	−7.12582e9			−0.954623
$663$	8.87538e9			1.18274
$664$			5.14898e9i			0.682548i
$665$	−1.89458e9			−0.249826
$666$	4.00128e9	+	1.35289e9i	0.524855	+	0.177460i
$667$	−2.38887e9			−0.311711
$668$		−	4.36290e9i		−	0.566314i
$669$	1.92098e9			0.248045
$670$	3.07168e9			0.394561
$671$			5.69062e9i			0.727161i
$672$			1.14483e10i			1.45528i
$673$	−7.64447e9			−0.966707			−0.483353	−	0.875425i	$-0.660581\pi$
$673$	−7.64447e9			−0.966707			−0.483353	+	0.875425i	$0.660581\pi$
$674$		−	5.51115e9i		−	0.693318i
$675$	1.17675e9			0.147272
$676$	1.54679e9			0.192584
$677$	7.84124e8			0.0971235			0.0485618	−	0.998820i	$-0.484536\pi$
$677$	7.84124e8			0.0971235			0.0485618	+	0.998820i	$0.484536\pi$
$678$	−9.06846e9			−1.11745
$679$			9.19108e9i			1.12674i
$680$		−	6.26135e9i		−	0.763637i
$681$		−	3.95807e9i		−	0.480252i
$682$	6.93858e9			0.837577
$683$		−	2.48885e9i		−	0.298901i	−0.988769	−	0.149450i	$-0.952250\pi$
$683$		−	2.48885e9i		−	0.298901i	0.988769	−	0.149450i	$-0.0477504\pi$
$684$			1.14424e9i			0.136716i
$685$		−	4.10447e9i		−	0.487910i
$686$			4.15457e9i			0.491351i
$687$	−5.59974e9			−0.658899
$688$		−	9.05546e8i		−	0.106011i
$689$			2.10471e9i			0.245146i
$690$			8.63451e9i			1.00061i
$691$	−6.77193e9			−0.780800			−0.390400	−	0.920645i	$-0.627663\pi$
$691$	−6.77193e9			−0.780800			−0.390400	+	0.920645i	$0.627663\pi$
$692$	−7.12158e9			−0.816968
$693$	−6.69487e9			−0.764145
$694$	6.62716e9			0.752609
$695$			3.59726e9i			0.406467i
$696$	2.37845e9			0.267400
$697$		−	1.00485e10i		−	1.12405i
$698$		−	5.32511e9i		−	0.592699i
$699$	−4.15020e9			−0.459620
$700$	3.20193e9			0.352833
$701$			2.74369e9i			0.300831i	0.988623	+	0.150415i	$0.0480611\pi$
$701$			2.74369e9i			0.300831i	−0.988623	+	0.150415i	$0.951939\pi$
$702$	−1.30607e9			−0.142491
$703$	−2.80237e9	−	9.47519e8i	−0.304216	−	0.102859i
$704$	6.11683e9			0.660727
$705$			2.33883e9i			0.251383i
$706$	7.35810e9			0.786954
$707$	1.83106e10			1.94866
$708$		−	7.63769e8i		−	0.0808808i
$709$			8.68705e9i			0.915399i	0.889107	+	0.457699i	$0.151326\pi$
$709$			8.68705e9i			0.915399i	−0.889107	+	0.457699i	$0.848674\pi$
$710$	−6.67153e9			−0.699554
$711$		−	6.08451e9i		−	0.634866i
$712$	4.01854e9			0.417242
$713$	−2.41826e10			−2.49856
$714$	1.17101e10			1.20397
$715$	−4.11844e9			−0.421368
$716$			2.40697e8i			0.0245062i
$717$		−	2.43827e10i		−	2.47039i
$718$			2.76793e9i			0.279074i
$719$	2.98122e8			0.0299119			0.0149559	−	0.999888i	$-0.495239\pi$
$719$	2.98122e8			0.0299119			0.0149559	+	0.999888i	$0.495239\pi$
$720$		−	1.02868e9i		−	0.102711i
$721$			2.38464e9i			0.236946i
$722$		−	6.23173e9i		−	0.616209i
$723$		−	1.97371e10i		−	1.94222i
$724$	−7.06743e9			−0.692113
$725$		−	1.10059e9i		−	0.107261i
$726$		−	3.38957e9i		−	0.328750i
$727$			5.12917e9i			0.495082i	0.968877	+	0.247541i	$0.0796224\pi$
$727$			5.12917e9i			0.495082i	−0.968877	+	0.247541i	$0.920378\pi$
$728$	−1.02853e10			−0.987997
$729$	−6.09377e9			−0.582559
$730$	7.19940e9			0.684962
$731$	6.39403e9			0.605430
$732$			6.82295e9i			0.642959i
$733$	−4.77269e9			−0.447610			−0.223805	−	0.974634i	$-0.571848\pi$
$733$	−4.77269e9			−0.447610			−0.223805	+	0.974634i	$0.571848\pi$
$734$		−	3.19316e9i		−	0.298047i
$735$		−	3.75906e9i		−	0.349200i
$736$	−1.63127e10			−1.50818
$737$	−7.60206e9			−0.699512
$738$		−	6.15891e9i		−	0.564035i
$739$	−1.66773e10			−1.52010			−0.760048	−	0.649867i	$-0.774825\pi$
$739$	−1.66773e10			−1.52010			−0.760048	+	0.649867i	$0.774825\pi$
$740$	−3.63206e9	−	1.22805e9i	−0.329490	−	0.111405i
$741$	−3.80994e9			−0.343997
$742$			2.77692e9i			0.249546i
$743$	6.41020e9			0.573338			0.286669	−	0.958030i	$-0.407452\pi$
$743$	6.41020e9			0.573338			0.286669	+	0.958030i	$0.407452\pi$
$744$	2.40771e10			2.14338
$745$		−	6.24439e9i		−	0.553278i
$746$		−	8.39879e9i		−	0.740680i
$747$	−5.97307e9			−0.524295
$748$			5.35431e9i			0.467787i
$749$	9.95134e9			0.865356
$750$	−1.10068e10			−0.952680
$751$	−1.09108e10			−0.939973			−0.469987	−	0.882674i	$-0.655741\pi$
$751$	−1.09108e10			−0.939973			−0.469987	+	0.882674i	$0.655741\pi$
$752$	6.40091e8			0.0548882
$753$		−	2.50739e9i		−	0.214013i
$754$			1.22154e9i			0.103779i
$755$		−	1.00442e10i		−	0.849374i
$756$	1.92721e9			0.162220
$757$		−	1.61426e10i		−	1.35250i	−0.736673	−	0.676250i	$-0.763604\pi$
$757$		−	1.61426e10i		−	1.35250i	0.736673	−	0.676250i	$-0.236396\pi$
$758$			4.45852e9i			0.371833i
$759$		−	2.13695e10i		−	1.77397i
$760$			2.68781e9i			0.222102i
$761$	2.09317e10			1.72170			0.860852	−	0.508855i	$-0.169931\pi$
$761$	2.09317e10			1.72170			0.860852	+	0.508855i	$0.169931\pi$
$762$		−	1.21010e9i		−	0.0990781i
$763$			2.27614e10i			1.85508i
$764$		−	8.61715e9i		−	0.699097i
$765$	7.26347e9			0.586583
$766$	−1.59630e10			−1.28325
$767$	1.13527e9			0.0908479
$768$	1.79636e10			1.43097
$769$			2.14671e10i			1.70228i	0.524936	+	0.851142i	$0.324089\pi$
$769$			2.14671e10i			1.70228i	−0.524936	+	0.851142i	$0.675911\pi$
$770$	−5.43381e9			−0.428930
$771$			2.47804e10i			1.94724i
$772$			1.03440e10i			0.809150i
$773$	2.18462e10			1.70117			0.850583	−	0.525840i	$-0.176249\pi$
$773$	2.18462e10			1.70117			0.850583	+	0.525840i	$0.176249\pi$
$774$	3.91904e9			0.303799
$775$		−	1.11413e10i		−	0.859767i
$776$	1.30393e10			1.00170
$777$	6.64702e9	−	1.96591e10i	0.508339	−	1.50346i
$778$	5.92948e9			0.451428
$779$			4.31350e9i			0.326926i
$780$	−4.93794e9			−0.372576
$781$	1.65113e10			1.24023
$782$			1.66857e10i			1.24773i
$783$		−	6.62436e8i		−	0.0493149i
$784$	−1.02878e9			−0.0762460
$785$			1.75887e10i			1.29775i
$786$	1.40586e10			1.03267
$787$	−1.05205e10			−0.769348			−0.384674	−	0.923053i	$-0.625686\pi$
$787$	−1.05205e10			−0.769348			−0.384674	+	0.923053i	$0.625686\pi$
$788$	1.01130e10			0.736271
$789$	−1.67609e10			−1.21486
$790$		−	4.93842e9i		−	0.356363i
$791$			1.98899e10i			1.42895i
$792$			9.49792e9i			0.679345i
$793$	−1.01417e10			−0.722192
$794$			1.44231e10i			1.02256i
$795$			3.85847e9i			0.272352i
$796$		−	8.13857e9i		−	0.571943i
$797$			5.41224e9i			0.378680i	0.981912	+	0.189340i	$0.0606349\pi$
$797$			5.41224e9i			0.378680i	−0.981912	+	0.189340i	$0.939365\pi$
$798$	−5.02678e9			−0.350171
$799$			4.51966e9i			0.313467i
$800$		−	7.51551e9i		−	0.518972i
$801$			4.66170e9i			0.320502i
$802$	3.56766e8			0.0244215
$803$	−1.78177e10			−1.21436
$804$	−9.11475e9			−0.618512
$805$	1.89381e10			1.27953
$806$			1.23657e10i			0.831854i
$807$	2.42294e10			1.62288
$808$		−	2.59771e10i		−	1.73241i
$809$			2.71407e10i			1.80219i	0.433620	+	0.901096i	$0.357236\pi$
$809$			2.71407e10i			1.80219i	−0.433620	+	0.901096i	$0.642764\pi$
$810$	−7.91516e9			−0.523314
$811$	−1.08718e10			−0.715698			−0.357849	−	0.933779i	$-0.616490\pi$
$811$	−1.08718e10			−0.715698			−0.357849	+	0.933779i	$0.616490\pi$
$812$		−	1.80249e9i		−	0.118148i
$813$	−3.47306e10			−2.26671
$814$	−8.03743e9	−	2.71756e9i	−0.522314	−	0.176602i
$815$	7.66917e8			0.0496246
$816$		−	4.45301e9i		−	0.286905i
$817$	−2.74477e9			−0.176087
$818$	−1.26659e10			−0.809097
$819$		−	1.19314e10i		−	0.758923i
$820$			5.59059e9i			0.354086i
$821$	−2.38172e10			−1.50207			−0.751036	−	0.660262i	$-0.770445\pi$
$821$	−2.38172e10			−1.50207			−0.751036	+	0.660262i	$0.770445\pi$
$822$		−	1.08902e10i		−	0.683885i
$823$	2.55428e10			1.59724			0.798620	−	0.601836i	$-0.205564\pi$
$823$	2.55428e10			1.59724			0.798620	+	0.601836i	$0.205564\pi$
$824$	3.38305e9			0.210651
$825$	9.84525e9			0.610433
$826$	1.49786e9			0.0924784
$827$			7.04244e9i			0.432966i	0.976286	+	0.216483i	$0.0694586\pi$
$827$			7.04244e9i			0.432966i	−0.976286	+	0.216483i	$0.930541\pi$
$828$		−	1.14378e10i		−	0.700220i
$829$		−	7.32740e9i		−	0.446693i	−0.974739	−	0.223346i	$-0.928302\pi$
$829$		−	7.32740e9i		−	0.446693i	0.974739	−	0.223346i	$-0.0716981\pi$
$830$	−4.84797e9			−0.294298
$831$		−	1.30488e10i		−	0.788803i
$832$			1.09012e10i			0.656212i
$833$		−	7.26419e9i		−	0.435441i
$834$			9.54442e9i			0.569729i
$835$	1.18887e10			0.706693
$836$		−	2.29845e9i		−	0.136054i
$837$		−	6.70587e9i		−	0.395290i
$838$		−	1.56135e9i		−	0.0916529i
$839$	3.41089e9			0.199389			0.0996945	−	0.995018i	$-0.468213\pi$
$839$	3.41089e9			0.199389			0.0996945	+	0.995018i	$0.468213\pi$
$840$	−1.88555e10			−1.09764
$841$	1.66303e10			0.964083
$842$	1.58214e10			0.913381
$843$		−	1.08161e10i		−	0.621832i
$844$	−8.15196e9			−0.466728
$845$			4.21494e9i			0.240322i
$846$			2.77019e9i			0.157295i
$847$	−7.43437e9			−0.420390
$848$	1.05599e9			0.0594666
$849$			3.43223e10i			1.92486i
$850$	−7.68738e9			−0.429351
$851$	2.80124e10	+	9.47137e9i	1.55810	+	0.526816i
$852$	1.97968e10			1.09662
$853$		−	2.05251e10i		−	1.13231i	−0.824300	−	0.566153i	$-0.808431\pi$
$853$		−	2.05251e10i		−	1.13231i	0.824300	−	0.566153i	$-0.191569\pi$
$854$	−1.33808e10			−0.735153
$855$	−3.11799e9			−0.170606
$856$		−	1.41178e10i		−	0.769325i
$857$			2.12242e10i			1.15186i	0.817500	+	0.575929i	$0.195360\pi$
$857$			2.12242e10i			1.15186i	−0.817500	+	0.575929i	$0.804640\pi$
$858$	−1.09272e10			−0.590615
$859$			2.74545e10i			1.47788i	0.673773	+	0.738938i	$0.264672\pi$
$859$			2.74545e10i			1.47788i	−0.673773	+	0.738938i	$0.735328\pi$
$860$	−3.55740e9			−0.190717
$861$	−3.02600e10			−1.61569
$862$	1.47409e10			0.783878
$863$	2.34906e10			1.24410			0.622052	−	0.782976i	$-0.286299\pi$
$863$	2.34906e10			1.24410			0.622052	+	0.782976i	$0.286299\pi$
$864$		−	4.52352e9i		−	0.238605i
$865$		−	1.94060e10i		−	1.01948i
$866$		−	1.09189e10i		−	0.571302i
$867$	5.65123e9			0.294494
$868$		−	1.82466e10i		−	0.947031i
$869$			1.22220e10i			0.631793i
$870$			2.23940e9i			0.115296i
$871$		−	1.35482e10i		−	0.694732i
$872$	3.22913e10			1.64922
$873$			1.51262e10i			0.769448i
$874$		−	7.16269e9i		−	0.362899i
$875$			2.41413e10i			1.21824i
$876$	−2.13632e10			−1.07374
$877$	−2.98579e10			−1.49472			−0.747362	−	0.664417i	$-0.768680\pi$
$877$	−2.98579e10			−1.49472			−0.747362	+	0.664417i	$0.768680\pi$
$878$	−1.83233e10			−0.913635
$879$	−4.46591e9			−0.221793
$880$			2.06633e9i			0.102214i
$881$	2.66422e10			1.31267			0.656334	−	0.754470i	$-0.272106\pi$
$881$	2.66422e10			1.31267			0.656334	+	0.754470i	$0.272106\pi$
$882$		−	4.45238e9i		−	0.218500i
$883$			1.29867e10i			0.634798i	0.948292	+	0.317399i	$0.102810\pi$
$883$			1.29867e10i			0.634798i	−0.948292	+	0.317399i	$0.897190\pi$
$884$	−9.54230e9			−0.464590
$885$	2.08123e9			0.100930
$886$		−	7.36591e9i		−	0.355802i
$887$	−1.03028e10			−0.495704			−0.247852	−	0.968798i	$-0.579725\pi$
$887$	−1.03028e10			−0.495704			−0.247852	+	0.968798i	$0.579725\pi$
$888$	−2.78901e10	−	9.43004e9i	−1.33661	−	0.451927i
$889$	−2.65412e9			−0.126696
$890$			3.78361e9i			0.179904i
$891$	1.95892e10			0.927777
$892$	−2.06532e9			−0.0974342
$893$		−	1.94016e9i		−	0.0911710i
$894$		−	1.65679e10i		−	0.775508i
$895$	−6.55889e8			−0.0305808
$896$		−	8.93117e9i		−	0.414792i
$897$	3.80840e10			1.76185
$898$	−1.48649e10			−0.685008
$899$	−6.27187e9			−0.287898
$900$	5.26956e9			0.240949
$901$			7.45628e9i			0.339614i
$902$			1.23715e10i			0.561305i
$903$		−	1.92550e10i		−	0.870236i
$904$	2.82176e10			1.27037
$905$		−	1.92584e10i		−	0.863676i
$906$		−	2.66496e10i		−	1.19054i
$907$			1.44372e10i			0.642479i	0.946998	+	0.321239i	$0.104099\pi$
$907$			1.44372e10i			0.642479i	−0.946998	+	0.321239i	$0.895901\pi$
$908$			4.25549e9i			0.188647i
$909$	3.01346e10			1.33074
$910$		−	9.68398e9i		−	0.425999i
$911$		−	1.74011e10i		−	0.762541i	−0.924463	−	0.381271i	$-0.875487\pi$
$911$		−	1.74011e10i		−	0.762541i	0.924463	−	0.381271i	$-0.124513\pi$
$912$			1.91154e9i			0.0834454i
$913$	1.19982e10			0.521757
$914$	−2.55336e10			−1.10611
$915$	−1.85922e10			−0.802338
$916$	6.02052e9			0.258821
$917$		−	3.08348e10i		−	1.32053i
$918$	−4.62697e9			−0.197400
$919$		−	2.21315e10i		−	0.940604i	−0.882506	−	0.470302i	$-0.844145\pi$
$919$		−	2.21315e10i		−	0.940604i	0.882506	−	0.470302i	$-0.155855\pi$
$920$		−	2.68673e10i		−	1.13754i
$921$	−1.68294e10			−0.709841
$922$	1.81775e9			0.0763794
$923$			2.94260e10i			1.23176i
$924$	1.61240e10			0.672390
$925$	−4.36361e9	+	1.29058e10i	−0.181280	+	0.536151i
$926$	1.71313e10			0.709008
$927$			3.92450e9i			0.161810i
$928$	−4.23077e9			−0.173781
$929$	−1.88336e10			−0.770687			−0.385343	−	0.922773i	$-0.625917\pi$
$929$	−1.88336e10			−0.770687			−0.385343	+	0.922773i	$0.625917\pi$
$930$			2.26695e10i			0.924170i
$931$			3.11830e9i			0.126647i
$932$	4.46206e9			0.180543
$933$		−	5.98290e10i		−	2.41171i
$934$	5.43441e9			0.218242
$935$	−1.45902e10			−0.583743
$936$	−1.69269e10			−0.674703
$937$	3.11973e9			0.123888			0.0619439	−	0.998080i	$-0.480270\pi$
$937$	3.11973e9			0.123888			0.0619439	+	0.998080i	$0.480270\pi$
$938$		−	1.78753e10i		−	0.707201i
$939$			1.60965e10i			0.634456i
$940$		−	2.51457e9i		−	0.0987454i
$941$	−1.98041e10			−0.774803			−0.387402	−	0.921911i	$-0.626627\pi$
$941$	−1.98041e10			−0.774803			−0.387402	+	0.921911i	$0.626627\pi$
$942$			4.66673e10i			1.81901i
$943$		−	4.31176e10i		−	1.67442i
$944$		−	5.69593e8i		−	0.0220375i
$945$			5.25157e9i			0.202431i
$946$	−7.87223e9			−0.302328
$947$		−	9.16582e9i		−	0.350709i	−0.984505	−	0.175354i	$-0.943893\pi$
$947$		−	9.16582e9i		−	0.350709i	0.984505	−	0.175354i	$-0.0561071\pi$
$948$			1.46540e10i			0.558635i
$949$		−	3.17542e10i		−	1.20606i
$950$	3.29997e9			0.124875
$951$	−3.07039e10			−1.15761
$952$	−3.64372e10			−1.36873
$953$	2.84259e10			1.06387			0.531936	−	0.846785i	$-0.321465\pi$
$953$	2.84259e10			1.06387			0.531936	+	0.846785i	$0.321465\pi$
$954$			4.57011e9i			0.170415i
$955$	2.34813e10			0.872391
$956$			2.62149e10i			0.970388i
$957$		−	5.54227e9i		−	0.204407i
$958$	1.99918e10			0.734635
$959$	−2.38855e10			−0.874518
$960$			1.99847e10i			0.729036i
$961$	−3.59778e10			−1.30768
$962$	4.84317e9	−	1.43241e10i	0.175395	−	0.518745i
$963$	1.63774e10			0.590952
$964$			2.12202e10i			0.762921i
$965$	−2.81870e10			−1.00972
$966$	5.02475e10			1.79347
$967$			4.67990e10i			1.66435i	0.554515	+	0.832174i	$0.312904\pi$
$967$			4.67990e10i			1.66435i	−0.554515	+	0.832174i	$0.687096\pi$
$968$			1.05470e10i			0.373738i
$969$	−1.34973e10			−0.476557
$970$			1.22770e10i			0.431907i
$971$	−1.22368e10			−0.428944			−0.214472	−	0.976730i	$-0.568803\pi$
$971$	−1.22368e10			−0.428944			−0.214472	+	0.976730i	$0.568803\pi$
$972$	1.95539e10			0.682968
$973$	2.09338e10			0.728541
$974$	−1.06303e9			−0.0368628
$975$			1.75459e10i			0.606262i
$976$			5.08832e9i			0.175186i
$977$			1.63789e9i			0.0561893i	0.999605	+	0.0280946i	$0.00894398\pi$
$977$			1.63789e9i			0.0561893i	−0.999605	+	0.0280946i	$0.991056\pi$
$978$	2.03482e9			0.0695569
$979$		−	9.36403e9i		−	0.318950i
$980$			4.04153e9i			0.137169i
$981$			3.74595e10i			1.26684i
$982$		−	2.92338e10i		−	0.985135i
$983$	−3.00060e10			−1.00756			−0.503780	−	0.863832i	$-0.668058\pi$
$983$	−3.00060e10			−1.00756			−0.503780	+	0.863832i	$0.668058\pi$
$984$			4.29295e10i			1.43639i
$985$			2.75574e10i			0.918780i
$986$			4.32752e9i			0.143771i
$987$	1.36105e10			0.450573
$988$	4.09623e9			0.135125
$989$	2.74366e10			0.901869
$990$	−8.94267e9			−0.292917
$991$		−	3.44246e10i		−	1.12360i	−0.827274	−	0.561799i	$-0.810110\pi$
$991$		−	3.44246e10i		−	1.12360i	0.827274	−	0.561799i	$-0.189890\pi$
$992$	−4.28282e10			−1.39296
$993$			5.76186e10i			1.86741i
$994$			3.88242e10i			1.25386i
$995$	2.21772e10			0.713718
$996$	1.43856e10			0.461340
$997$		−	5.97739e10i		−	1.91020i	−0.296286	−	0.955099i	$-0.595748\pi$
$997$		−	5.97739e10i		−	1.91020i	0.296286	−	0.955099i	$-0.404252\pi$
$998$	1.94972e10			0.620892
$999$	−2.62642e9	+	7.76787e9i	−0.0833461	+	0.246503i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.b.a.36.7	✓	20
37.36	even	2	inner	37.8.b.a.36.14	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.b.a.36.7	✓	20	1.1	even	1	trivial
37.8.b.a.36.14	yes	20	37.36	even	2	inner

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:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.77325i q^{2} -62.8536 q^{3} +67.5766 q^{4} +184.143i q^{5} +488.577i q^{6} +1071.60 q^{7} -1520.27i q^{8} +1763.58 q^{9} +O(q^{10})\)	\(q-7.77325i q^{2} -62.8536 q^{3} +67.5766 q^{4} +184.143i q^{5} +488.577i q^{6} +1071.60 q^{7} -1520.27i q^{8} +1763.58 q^{9} +1431.39 q^{10} -3542.53 q^{11} -4247.44 q^{12} -6313.40i q^{13} -8329.81i q^{14} -11574.1i q^{15} -3167.59 q^{16} -22366.3i q^{17} -13708.8i q^{18} +9601.18i q^{19} +12443.8i q^{20} -67354.0 q^{21} +27537.0i q^{22} -95973.0i q^{23} +95554.2i q^{24} +44216.3 q^{25} -49075.6 q^{26} +26613.4 q^{27} +72415.1 q^{28} -24891.1i q^{29} -89968.1 q^{30} -251973. i q^{31} -169971. i q^{32} +222661. q^{33} -173858. q^{34} +197328. i q^{35} +119177. q^{36} +(-98687.8 + 291878. i) q^{37} +74632.3 q^{38} +396820. i q^{39} +279946. q^{40} +449268. q^{41} +523559. i q^{42} +285878. i q^{43} -239392. q^{44} +324751. i q^{45} -746022. q^{46} -202075. q^{47} +199095. q^{48} +324783. q^{49} -343704. i q^{50} +1.40580e6i q^{51} -426638. i q^{52} -333372. q^{53} -206873. i q^{54} -652333. i q^{55} -1.62912e6i q^{56} -603469. i q^{57} -193484. q^{58} +179819. i q^{59} -782136. i q^{60} -1.60637e6i q^{61} -1.95865e6 q^{62} +1.88985e6 q^{63} -1.72668e6 q^{64} +1.16257e6 q^{65} -1.73080e6i q^{66} +2.14594e6 q^{67} -1.51144e6i q^{68} +6.03225e6i q^{69} +1.53388e6 q^{70} -4.66087e6 q^{71} -2.68111e6i q^{72} +5.02966e6 q^{73} +(2.26884e6 + 767125. i) q^{74} -2.77916e6 q^{75} +648815. i q^{76} -3.79618e6 q^{77} +3.08458e6 q^{78} -3.45009e6i q^{79} -583290. i q^{80} -5.52970e6 q^{81} -3.49227e6i q^{82} -3.38690e6 q^{83} -4.55155e6 q^{84} +4.11859e6 q^{85} +2.22220e6 q^{86} +1.56449e6i q^{87} +5.38559e6i q^{88} +2.64331e6i q^{89} +2.52437e6 q^{90} -6.76544e6i q^{91} -6.48553e6i q^{92} +1.58374e7i q^{93} +1.57078e6i q^{94} -1.76799e6 q^{95} +1.06833e7i q^{96} +8.57697e6i q^{97} -2.52462e6i q^{98} -6.24754e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9}+O(q^{10}) \) 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9	\( 20 q + 78 q^{3} - 844 q^{4} - 1746 q^{7} + 12362 q^{9} - 882 q^{10} + 3498 q^{11} - 30374 q^{12} + 36116 q^{16} + 113482 q^{21} - 108112 q^{25} + 49278 q^{26} - 304110 q^{27} - 41192 q^{28} + 429776 q^{30} + 305646 q^{33} - 960356 q^{34} + 484758 q^{36} + 108732 q^{37} + 1049916 q^{38} - 496346 q^{40} - 1577742 q^{41} + 685266 q^{44} - 2906298 q^{46} - 1512786 q^{47} + 1522958 q^{48} + 3269246 q^{49} + 2999358 q^{53} + 405946 q^{58} + 3728310 q^{62} - 11995292 q^{63} - 11109700 q^{64} + 4251792 q^{65} + 3562224 q^{67} + 21605644 q^{70} - 15259086 q^{71} + 11088018 q^{73} - 2036544 q^{74} + 14882062 q^{75} - 2419122 q^{77} - 12178734 q^{78} - 17764972 q^{81} - 12873822 q^{83} + 9944396 q^{84} - 2698920 q^{85} + 15345336 q^{86} - 13219100 q^{90} + 48981192 q^{95} + 43111380 q^{99}+O(q^{100}) \) 20 * q + 78 * q^3 - 844 * q^4 - 1746 * q^7 + 12362 * q^9 - 882 * q^10 + 3498 * q^11 - 30374 * q^12 + 36116 * q^16 + 113482 * q^21 - 108112 * q^25 + 49278 * q^26 - 304110 * q^27 - 41192 * q^28 + 429776 * q^30 + 305646 * q^33 - 960356 * q^34 + 484758 * q^36 + 108732 * q^37 + 1049916 * q^38 - 496346 * q^40 - 1577742 * q^41 + 685266 * q^44 - 2906298 * q^46 - 1512786 * q^47 + 1522958 * q^48 + 3269246 * q^49 + 2999358 * q^53 + 405946 * q^58 + 3728310 * q^62 - 11995292 * q^63 - 11109700 * q^64 + 4251792 * q^65 + 3562224 * q^67 + 21605644 * q^70 - 15259086 * q^71 + 11088018 * q^73 - 2036544 * q^74 + 14882062 * q^75 - 2419122 * q^77 - 12178734 * q^78 - 17764972 * q^81 - 12873822 * q^83 + 9944396 * q^84 - 2698920 * q^85 + 15345336 * q^86 - 13219100 * q^90 + 48981192 * q^95 + 43111380 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more