LMFDB - Embedded newform 83.13.b.c.82.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [83,13,Mod(82,83)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(83, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("83.82");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$ N $	$=$	$ 83 $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	83.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:	$75.8614868339$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			82.4
Character	$\chi$	$=$	83.82
Dual form			83.13.b.c.82.77

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-115.485i q^{2} -823.731 q^{3} -9240.80 q^{4} -17633.0i q^{5} +95128.7i q^{6} -95956.6 q^{7} +594147. i q^{8} +147092. q^{9} +O(q^{10})$	\(q-115.485i q^{2} -823.731 q^{3} -9240.80 q^{4} -17633.0i q^{5} +95128.7i q^{6} -95956.6 q^{7} +594147. i q^{8} +147092. q^{9} -2.03635e6 q^{10} -495050. q^{11} +7.61194e6 q^{12} +5.35402e6i q^{13} +1.10816e7i q^{14} +1.45249e7i q^{15} +3.07648e7 q^{16} -3.39986e7 q^{17} -1.69870e7i q^{18} -3.22186e7i q^{19} +1.62943e8i q^{20} +7.90425e7 q^{21} +5.71709e7i q^{22} +1.15249e8 q^{23} -4.89418e8i q^{24} -6.67824e7 q^{25} +6.18310e8 q^{26} +3.16600e8 q^{27} +8.86716e8 q^{28} +6.79185e8 q^{29} +1.67740e9 q^{30} -8.71974e8 q^{31} -1.11925e9i q^{32} +4.07788e8 q^{33} +3.92634e9i q^{34} +1.69200e9i q^{35} -1.35925e9 q^{36} +5.58004e8 q^{37} -3.72077e9 q^{38} -4.41028e9i q^{39} +1.04766e10 q^{40} -1.31887e9 q^{41} -9.12823e9i q^{42} -3.26419e9i q^{43} +4.57466e9 q^{44} -2.59368e9i q^{45} -1.33096e10i q^{46} +1.71934e10i q^{47} -2.53420e10 q^{48} -4.63361e9 q^{49} +7.71237e9i q^{50} +2.80057e10 q^{51} -4.94754e10i q^{52} +1.13394e9i q^{53} -3.65626e10i q^{54} +8.72922e9i q^{55} -5.70124e10i q^{56} +2.65395e10i q^{57} -7.84357e10i q^{58} -5.61197e10 q^{59} -1.34221e11i q^{60} -7.19223e10 q^{61} +1.00700e11i q^{62} -1.41145e10 q^{63} -3.24404e9 q^{64} +9.44075e10 q^{65} -4.70934e10i q^{66} -9.09550e10i q^{67} +3.14175e11 q^{68} -9.49346e10 q^{69} +1.95401e11 q^{70} +1.43104e11i q^{71} +8.73945e10i q^{72} +2.69290e9i q^{73} -6.44411e10i q^{74} +5.50107e10 q^{75} +2.97726e11i q^{76} +4.75033e10 q^{77} -5.09321e11 q^{78} +9.84548e10i q^{79} -5.42477e11i q^{80} -3.38964e11 q^{81} +1.52309e11i q^{82} +(-2.91629e11 + 1.47792e11i) q^{83} -7.30416e11 q^{84} +5.99498e11i q^{85} -3.76965e11 q^{86} -5.59466e11 q^{87} -2.94133e11i q^{88} +1.50498e11i q^{89} -2.99531e11 q^{90} -5.13754e11i q^{91} -1.06500e12 q^{92} +7.18273e11 q^{93} +1.98558e12 q^{94} -5.68111e11 q^{95} +9.21962e11i q^{96} +6.29420e11i q^{97} +5.35113e11i q^{98} -7.28180e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9}+O(q^{10}) $ 80 * q + 918 * q^3 - 181150 * q^4 + 103918 * q^7 + 14902142 * q^9	\( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9} + 1177482 q^{10} + 510406 q^{11} - 9260402 q^{12} + 203791850 q^{16} + 5571718 q^{17} + 46565862 q^{21} + 804389464 q^{23} - 4263713272 q^{25} + 18061794 q^{26} - 2326165338 q^{27} - 204811652 q^{28} + 1486960270 q^{29} - 2621683648 q^{30} - 665743010 q^{31} + 135224502 q^{33} - 54936709824 q^{36} - 280627202 q^{37} + 6646145310 q^{38} - 7119722058 q^{40} + 51318109072 q^{41} - 11512674650 q^{44} + 85368259738 q^{48} + 148785395094 q^{49} - 100143389562 q^{51} - 63584241050 q^{59} - 29216180978 q^{61} - 332932206620 q^{63} - 323596534090 q^{64} + 362112989184 q^{65} - 86115426752 q^{68} + 272383417100 q^{69} + 105630718656 q^{70} + 785418808326 q^{75} - 663355117738 q^{77} + 1483841884620 q^{78} + 2430778545148 q^{81} + 837315119192 q^{83} - 3013574788354 q^{84} + 452180651958 q^{86} - 682263689498 q^{87} - 499714512022 q^{90} + 997428187414 q^{92} - 1487992716298 q^{93} + 3169817690580 q^{94} + 1542762610848 q^{95} + 2021347267420 q^{99}+O(q^{100}) \) 80 * q + 918 * q^3 - 181150 * q^4 + 103918 * q^7 + 14902142 * q^9 + 1177482 * q^10 + 510406 * q^11 - 9260402 * q^12 + 203791850 * q^16 + 5571718 * q^17 + 46565862 * q^21 + 804389464 * q^23 - 4263713272 * q^25 + 18061794 * q^26 - 2326165338 * q^27 - 204811652 * q^28 + 1486960270 * q^29 - 2621683648 * q^30 - 665743010 * q^31 + 135224502 * q^33 - 54936709824 * q^36 - 280627202 * q^37 + 6646145310 * q^38 - 7119722058 * q^40 + 51318109072 * q^41 - 11512674650 * q^44 + 85368259738 * q^48 + 148785395094 * q^49 - 100143389562 * q^51 - 63584241050 * q^59 - 29216180978 * q^61 - 332932206620 * q^63 - 323596534090 * q^64 + 362112989184 * q^65 - 86115426752 * q^68 + 272383417100 * q^69 + 105630718656 * q^70 + 785418808326 * q^75 - 663355117738 * q^77 + 1483841884620 * q^78 + 2430778545148 * q^81 + 837315119192 * q^83 - 3013574788354 * q^84 + 452180651958 * q^86 - 682263689498 * q^87 - 499714512022 * q^90 + 997428187414 * q^92 - 1487992716298 * q^93 + 3169817690580 * q^94 + 1542762610848 * q^95 + 2021347267420 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/83\mathbb{Z}\right)^\times$.

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

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 $

$2$		−	115.485i		−	1.80445i	−0.431261	−	0.902227i	$-0.641931\pi$
$2$		−	115.485i		−	1.80445i	0.431261	−	0.902227i	$-0.358069\pi$
$3$	−823.731			−1.12995			−0.564973	−	0.825109i	$-0.691114\pi$
$3$	−823.731			−1.12995			−0.564973	+	0.825109i	$0.691114\pi$
$4$	−9240.80			−2.25605
$5$		−	17633.0i		−	1.12851i	−0.825600	−	0.564256i	$-0.809163\pi$
$5$		−	17633.0i		−	1.12851i	0.825600	−	0.564256i	$-0.190837\pi$
$6$			95128.7i			2.03894i
$7$	−95956.6			−0.815618			−0.407809	−	0.913067i	$-0.633707\pi$
$7$	−95956.6			−0.815618			−0.407809	+	0.913067i	$0.633707\pi$
$8$			594147.i			2.26649i
$9$	147092.			0.276780
$10$	−2.03635e6			−2.03635
$11$	−495050.			−0.279443			−0.139721	−	0.990191i	$-0.544621\pi$
$11$	−495050.			−0.279443			−0.139721	+	0.990191i	$0.544621\pi$
$12$	7.61194e6			2.54922
$13$			5.35402e6i			1.10923i	0.832108	+	0.554613i	$0.187134\pi$
$13$			5.35402e6i			1.10923i	−0.832108	+	0.554613i	$0.812866\pi$
$14$			1.10816e7i			1.47175i
$15$			1.45249e7i			1.27516i
$16$	3.07648e7			1.83373
$17$	−3.39986e7			−1.40854			−0.704268	−	0.709934i	$-0.748725\pi$
$17$	−3.39986e7			−1.40854			−0.704268	+	0.709934i	$0.748725\pi$
$18$		−	1.69870e7i		−	0.499437i
$19$		−	3.22186e7i		−	0.684834i	−0.939548	−	0.342417i	$-0.888754\pi$
$19$		−	3.22186e7i		−	0.684834i	0.939548	−	0.342417i	$-0.111246\pi$
$20$			1.62943e8i			2.54599i
$21$	7.90425e7			0.921605
$22$			5.71709e7i			0.504242i
$23$	1.15249e8			0.778524			0.389262	−	0.921127i	$-0.372730\pi$
$23$	1.15249e8			0.778524			0.389262	+	0.921127i	$0.372730\pi$
$24$		−	4.89418e8i		−	2.56102i
$25$	−6.67824e7			−0.273541
$26$	6.18310e8			2.00155
$27$	3.16600e8			0.817200
$28$	8.86716e8			1.84008
$29$	6.79185e8			1.14183			0.570913	−	0.821010i	$-0.306589\pi$
$29$	6.79185e8			1.14183			0.570913	+	0.821010i	$0.306589\pi$
$30$	1.67740e9			2.30097
$31$	−8.71974e8			−0.982502			−0.491251	−	0.871018i	$-0.663460\pi$
$31$	−8.71974e8			−0.982502			−0.491251	+	0.871018i	$0.663460\pi$
$32$		−	1.11925e9i		−	1.04238i
$33$	4.07788e8			0.315755
$34$			3.92634e9i			2.54164i
$35$			1.69200e9i			0.920435i
$36$	−1.35925e9			−0.624431
$37$	5.58004e8			0.217484			0.108742	−	0.994070i	$-0.465318\pi$
$37$	5.58004e8			0.217484			0.108742	+	0.994070i	$0.465318\pi$
$38$	−3.72077e9			−1.23575
$39$		−	4.41028e9i		−	1.25337i
$40$	1.04766e10			2.55777
$41$	−1.31887e9			−0.277650			−0.138825	−	0.990317i	$-0.544333\pi$
$41$	−1.31887e9			−0.277650			−0.138825	+	0.990317i	$0.544333\pi$
$42$		−	9.12823e9i		−	1.66299i
$43$		−	3.26419e9i		−	0.516374i	−0.966095	−	0.258187i	$-0.916875\pi$
$43$		−	3.26419e9i		−	0.516374i	0.966095	−	0.258187i	$-0.0831251\pi$
$44$	4.57466e9			0.630438
$45$		−	2.59368e9i		−	0.312350i
$46$		−	1.33096e10i		−	1.40481i
$47$			1.71934e10i			1.59505i	0.603287	+	0.797524i	$0.293857\pi$
$47$			1.71934e10i			1.59505i	−0.603287	+	0.797524i	$0.706143\pi$
$48$	−2.53420e10			−2.07201
$49$	−4.63361e9			−0.334767
$50$			7.71237e9i			0.493591i
$51$	2.80057e10			1.59157
$52$		−	4.94754e10i		−	2.50247i
$53$			1.13394e9i			0.0511604i	0.999673	+	0.0255802i	$0.00814333\pi$
$53$			1.13394e9i			0.0511604i	−0.999673	+	0.0255802i	$0.991857\pi$
$54$		−	3.65626e10i		−	1.47460i
$55$			8.72922e9i			0.315355i
$56$		−	5.70124e10i		−	1.84859i
$57$			2.65395e10i			0.773826i
$58$		−	7.84357e10i		−	2.06037i
$59$	−5.61197e10			−1.33046			−0.665232	−	0.746637i	$-0.731667\pi$
$59$	−5.61197e10			−1.33046			−0.665232	+	0.746637i	$0.731667\pi$
$60$		−	1.34221e11i		−	2.87683i
$61$	−7.19223e10			−1.39600			−0.697998	−	0.716099i	$-0.745926\pi$
$61$	−7.19223e10			−1.39600			−0.697998	+	0.716099i	$0.745926\pi$
$62$			1.00700e11i			1.77288i
$63$	−1.41145e10			−0.225747
$64$	−3.24404e9			−0.0472070
$65$	9.44075e10			1.25178
$66$		−	4.70934e10i		−	0.569766i
$67$		−	9.09550e10i		−	1.00549i	−0.864435	−	0.502745i	$-0.832323\pi$
$67$		−	9.09550e10i		−	1.00549i	0.864435	−	0.502745i	$-0.167677\pi$
$68$	3.14175e11			3.17773
$69$	−9.49346e10			−0.879690
$70$	1.95401e11			1.66088
$71$			1.43104e11i			1.11713i	0.829461	+	0.558564i	$0.188648\pi$
$71$			1.43104e11i			1.11713i	−0.829461	+	0.558564i	$0.811352\pi$
$72$			8.73945e10i			0.627320i
$73$			2.69290e9i			0.0177944i	0.999960	+	0.00889720i	$0.00283210\pi$
$73$			2.69290e9i			0.0177944i	−0.999960	+	0.00889720i	$0.997168\pi$
$74$		−	6.44411e10i		−	0.392439i
$75$	5.50107e10			0.309086
$76$			2.97726e11i			1.54502i
$77$	4.75033e10			0.227919
$78$	−5.09321e11			−2.26164
$79$			9.84548e10i			0.405018i	0.979280	+	0.202509i	$0.0649096\pi$
$79$			9.84548e10i			0.405018i	−0.979280	+	0.202509i	$0.935090\pi$
$80$		−	5.42477e11i		−	2.06938i
$81$	−3.38964e11			−1.20017
$82$			1.52309e11i			0.501007i
$83$	−2.91629e11	+	1.47792e11i	−0.891995	+	0.452045i
$83$	−2.91629e11	+	1.47792e11i	−0.891995	+	0.452045i
$84$	−7.30416e11			−2.07919
$85$			5.99498e11i			1.58955i
$86$	−3.76965e11			−0.931773
$87$	−5.59466e11			−1.29020
$88$		−	2.94133e11i		−	0.633355i
$89$			1.50498e11i			0.302825i	0.988471	+	0.151413i	$0.0483822\pi$
$89$			1.50498e11i			0.302825i	−0.988471	+	0.151413i	$0.951618\pi$
$90$	−2.99531e11			−0.563621
$91$		−	5.13754e11i		−	0.904705i
$92$	−1.06500e12			−1.75639
$93$	7.18273e11			1.11018
$94$	1.98558e12			2.87819
$95$	−5.68111e11			−0.772844
$96$			9.21962e11i			1.17784i
$97$			6.29420e11i			0.755632i	0.925881	+	0.377816i	$0.123325\pi$
$97$			6.29420e11i			0.755632i	−0.925881	+	0.377816i	$0.876675\pi$
$98$			5.35113e11i			0.604072i
$99$	−7.28180e10			−0.0773441
$100$	6.17123e11			0.617123
$101$			4.91580e10i			0.0463090i	0.999732	+	0.0231545i	$0.00737097\pi$
$101$			4.91580e10i			0.0463090i	−0.999732	+	0.0231545i	$0.992629\pi$
$102$		−	3.23425e12i		−	2.87192i
$103$			1.11074e12i			0.930223i	0.885252	+	0.465112i	$0.153986\pi$
$103$			1.11074e12i			0.930223i	−0.885252	+	0.465112i	$0.846014\pi$
$104$	−3.18108e12			−2.51405
$105$		−	1.39376e12i		−	1.04004i
$106$	1.30953e11			0.0923167
$107$		−	3.39336e11i		−	0.226114i	−0.993589	−	0.113057i	$-0.963936\pi$
$107$		−	3.39336e11i		−	0.226114i	0.993589	−	0.113057i	$-0.0360643\pi$
$108$	−2.92564e12			−1.84365
$109$	1.28025e12			0.763368			0.381684	−	0.924293i	$-0.375344\pi$
$109$	1.28025e12			0.763368			0.381684	+	0.924293i	$0.375344\pi$
$110$	1.00809e12			0.569043
$111$	−4.59645e11			−0.245745
$112$	−2.95209e12			−1.49562
$113$	2.47592e11			0.118923			0.0594615	−	0.998231i	$-0.481062\pi$
$113$	2.47592e11			0.118923			0.0594615	+	0.998231i	$0.481062\pi$
$114$	3.06491e12			1.39633
$115$		−	2.03219e12i		−	0.878574i
$116$	−6.27621e12			−2.57602
$117$			7.87535e11i			0.307011i
$118$			6.48098e12i			2.40076i
$119$	3.26240e12			1.14883
$120$	−8.62991e12			−2.89014
$121$	−2.89335e12			−0.921912
$122$			8.30595e12i			2.51901i
$123$	1.08639e12			0.313730
$124$	8.05774e12			2.21658
$125$		−	3.12736e12i		−	0.819819i
$126$			1.63001e12i			0.407350i
$127$	6.04894e12			1.44164			0.720820	−	0.693122i	$-0.243765\pi$
$127$	6.04894e12			1.44164			0.720820	+	0.693122i	$0.243765\pi$
$128$		−	4.20982e12i		−	0.957201i
$129$			2.68881e12i			0.583475i
$130$		−	1.09027e13i		−	2.25877i
$131$	−1.27687e12			−0.252651			−0.126325	−	0.991989i	$-0.540318\pi$
$131$	−1.27687e12			−0.252651			−0.126325	+	0.991989i	$0.540318\pi$
$132$	−3.76829e12			−0.712361
$133$			3.09159e12i			0.558563i
$134$	−1.05039e13			−1.81436
$135$		−	5.58261e12i		−	0.922221i
$136$		−	2.02002e13i		−	3.19244i
$137$		−	2.39146e12i		−	0.361692i	−0.983511	−	0.180846i	$-0.942116\pi$
$137$		−	2.39146e12i		−	0.361692i	0.983511	−	0.180846i	$-0.0578836\pi$
$138$			1.09635e13i			1.58736i
$139$		−	5.39070e12i		−	0.747406i	−0.927548	−	0.373703i	$-0.878088\pi$
$139$		−	5.39070e12i		−	0.747406i	0.927548	−	0.373703i	$-0.121912\pi$
$140$		−	1.56355e13i		−	2.07655i
$141$		−	1.41627e13i		−	1.80232i
$142$	1.65264e13			2.01581
$143$		−	2.65051e12i		−	0.309965i
$144$	4.52527e12			0.507539
$145$		−	1.19761e13i		−	1.28857i
$146$	3.10990e11			0.0321092
$147$	3.81685e12			0.378269
$148$	−5.15640e12			−0.490655
$149$		−	1.86128e13i		−	1.70096i	−0.526008	−	0.850479i	$-0.676312\pi$
$149$		−	1.86128e13i		−	1.70096i	0.526008	−	0.850479i	$-0.323688\pi$
$150$		−	6.35292e12i		−	0.557732i
$151$	−1.97636e13			−1.66726			−0.833630	−	0.552323i	$-0.813741\pi$
$151$	−1.97636e13			−1.66726			−0.833630	+	0.552323i	$0.813741\pi$
$152$	1.91426e13			1.55217
$153$	−5.00094e12			−0.389855
$154$		−	5.48592e12i		−	0.411268i
$155$			1.53755e13i			1.10877i
$156$			4.07545e13i			2.82766i
$157$			8.93169e12i			0.596397i	0.954504	+	0.298199i	$0.0963858\pi$
$157$			8.93169e12i			0.596397i	−0.954504	+	0.298199i	$0.903614\pi$
$158$	1.13701e13			0.730837
$159$		−	9.34061e11i		−	0.0578086i
$160$	−1.97358e13			−1.17634
$161$	−1.10590e13			−0.634978
$162$			3.91453e13i			2.16566i
$163$		−	3.01132e13i		−	1.60558i	−0.596264	−	0.802788i	$-0.703349\pi$
$163$		−	3.01132e13i		−	1.60558i	0.596264	−	0.802788i	$-0.296651\pi$
$164$	1.21874e13			0.626393
$165$		−	7.19053e12i		−	0.356334i
$166$	1.70677e13	+	3.36788e13i	0.815694	+	1.60956i
$167$	−6.20942e12			−0.286254			−0.143127	−	0.989704i	$-0.545716\pi$
$167$	−6.20942e12			−0.286254			−0.143127	+	0.989704i	$0.545716\pi$
$168$			4.69629e13i			2.08881i
$169$	−5.36746e12			−0.230382
$170$	6.92331e13			2.86827
$171$		−	4.73911e12i		−	0.189548i
$172$			3.01637e13i			1.16497i
$173$	−4.29866e13			−1.60345			−0.801727	−	0.597691i	$-0.796085\pi$
$173$	−4.29866e13			−1.60345			−0.801727	+	0.597691i	$0.796085\pi$
$174$			6.46099e13i			2.32811i
$175$	6.40821e12			0.223105
$176$	−1.52301e13			−0.512422
$177$	4.62275e13			1.50335
$178$	1.73803e13			0.546434
$179$			1.40492e13i			0.427104i	0.976932	+	0.213552i	$0.0685033\pi$
$179$			1.40492e13i			0.427104i	−0.976932	+	0.213552i	$0.931497\pi$
$180$			2.39677e13i			0.704678i
$181$			9.36663e12i			0.266386i	0.991090	+	0.133193i	$0.0425231\pi$
$181$			9.36663e12i			0.266386i	−0.991090	+	0.133193i	$0.957477\pi$
$182$	−5.93309e13			−1.63250
$183$	5.92446e13			1.57740
$184$			6.84752e13i			1.76452i
$185$		−	9.83928e12i		−	0.245433i
$186$		−	8.29498e13i		−	2.00326i
$187$	1.68310e13			0.393605
$188$		−	1.58880e14i		−	3.59852i
$189$	−3.03799e13			−0.666523
$190$			6.56084e13i			1.39456i
$191$	8.81293e13			1.81518			0.907590	−	0.419857i	$-0.137920\pi$
$191$	8.81293e13			1.81518			0.907590	+	0.419857i	$0.137920\pi$
$192$	2.67222e12			0.0533414
$193$	6.18527e13			1.19678			0.598391	−	0.801204i	$-0.295807\pi$
$193$	6.18527e13			1.19678			0.598391	+	0.801204i	$0.295807\pi$
$194$	7.26887e13			1.36350
$195$	−7.77664e13			−1.41444
$196$	4.28183e13			0.755253
$197$	−8.52405e13			−1.45831			−0.729153	−	0.684351i	$-0.760086\pi$
$197$	−8.52405e13			−1.45831			−0.729153	+	0.684351i	$0.760086\pi$
$198$			8.40939e12i			0.139564i
$199$	7.62507e13			1.22779			0.613897	−	0.789386i	$-0.289601\pi$
$199$	7.62507e13			1.22779			0.613897	+	0.789386i	$0.289601\pi$
$200$		−	3.96786e13i		−	0.619978i
$201$			7.49225e13i			1.13615i
$202$	5.67701e12			0.0835625
$203$	−6.51723e13			−0.931294
$204$	−2.58795e14			−3.59067
$205$			2.32556e13i			0.313331i
$206$	1.28273e14			1.67855
$207$	1.69523e13			0.215480
$208$			1.64716e14i			2.03402i
$209$			1.59498e13i			0.191372i
$210$	−1.60958e14			−1.87671
$211$			1.88127e13i			0.213184i	0.994303	+	0.106592i	$0.0339939\pi$
$211$			1.88127e13i			0.213184i	−0.994303	+	0.106592i	$0.966006\pi$
$212$		−	1.04785e13i		−	0.115421i
$213$		−	1.17880e14i		−	1.26230i
$214$	−3.91883e13			−0.408012
$215$	−5.75575e13			−0.582735
$216$			1.88107e14i			1.85218i
$217$	8.36717e13			0.801346
$218$		−	1.47849e14i		−	1.37746i
$219$		−	2.21823e12i		−	0.0201067i
$220$		−	8.06650e13i		−	0.711457i
$221$		−	1.82029e14i		−	1.56239i
$222$			5.30821e13i			0.443436i
$223$			1.50623e13i			0.122479i	0.998123	+	0.0612395i	$0.0195053\pi$
$223$			1.50623e13i			0.122479i	−0.998123	+	0.0612395i	$0.980495\pi$
$224$			1.07400e14i			0.850187i
$225$	−9.82317e12			−0.0757105
$226$		−	2.85932e13i		−	0.214591i
$227$	9.88408e13			0.722405			0.361203	−	0.932487i	$-0.382366\pi$
$227$	9.88408e13			0.722405			0.361203	+	0.932487i	$0.382366\pi$
$228$		−	2.45246e14i		−	1.74579i
$229$	−2.49498e13			−0.173003			−0.0865015	−	0.996252i	$-0.527569\pi$
$229$	−2.49498e13			−0.173003			−0.0865015	+	0.996252i	$0.527569\pi$
$230$	−2.34688e14			−1.58535
$231$	−3.91300e13			−0.257536
$232$			4.03536e14i			2.58794i
$233$		−	2.27027e14i		−	1.41887i	−0.704772	−	0.709434i	$-0.748951\pi$
$233$		−	2.27027e14i		−	1.41887i	0.704772	−	0.709434i	$-0.251049\pi$
$234$	9.09485e13			0.553988
$235$	3.03171e14			1.80003
$236$	5.18591e14			3.00160
$237$		−	8.11003e13i		−	0.457649i
$238$		−	3.76758e14i		−	2.07301i
$239$		−	1.63205e14i		−	0.875680i	−0.899053	−	0.437840i	$-0.855744\pi$
$239$		−	1.63205e14i		−	0.875680i	0.899053	−	0.437840i	$-0.144256\pi$
$240$			4.46855e14i			2.33829i
$241$	4.69175e13			0.239460			0.119730	−	0.992807i	$-0.461797\pi$
$241$	4.69175e13			0.239460			0.119730	+	0.992807i	$0.461797\pi$
$242$			3.34139e14i			1.66355i
$243$	1.10961e14			0.538931
$244$	6.64619e14			3.14945
$245$			8.17045e13i			0.377789i
$246$		−	1.25462e14i		−	0.566111i
$247$	1.72499e14			0.759636
$248$		−	5.18081e14i		−	2.22683i
$249$	2.40224e14	−	1.21741e14i	1.00791	−	0.510786i
$250$	−3.61163e14			−1.47932
$251$			3.92876e14i			1.57113i	0.618777	+	0.785567i	$0.287628\pi$
$251$			3.92876e14i			1.57113i	−0.618777	+	0.785567i	$0.712372\pi$
$252$	1.30429e14			0.509297
$253$	−5.70542e13			−0.217553
$254$		−	6.98562e14i		−	2.60137i
$255$		−	4.93826e14i		−	1.79611i
$256$	−4.99458e14			−1.77443
$257$		−	5.33535e14i		−	1.85167i	−0.377925	−	0.925836i	$-0.623362\pi$
$257$		−	5.33535e14i		−	1.85167i	0.377925	−	0.925836i	$-0.376638\pi$
$258$	3.10518e14			1.05285
$259$	−5.35441e13			−0.177384
$260$	−8.72401e14			−2.82407
$261$	9.99028e13			0.316035
$262$			1.47460e14i			0.455896i
$263$			5.92899e14i			1.79162i	0.444435	+	0.895811i	$0.353404\pi$
$263$			5.92899e14i			1.79162i	−0.444435	+	0.895811i	$0.646596\pi$
$264$			2.42286e14i			0.715657i
$265$	1.99947e13			0.0577352
$266$	3.57033e14			1.00790
$267$		−	1.23970e14i		−	0.342176i
$268$			8.40497e14i			2.26844i
$269$		−	2.54478e14i		−	0.671641i	−0.941926	−	0.335820i	$-0.890986\pi$
$269$		−	2.54478e14i		−	0.671641i	0.941926	−	0.335820i	$-0.109014\pi$
$270$	−6.44708e14			−1.66410
$271$			3.22893e14i			0.815160i	0.913169	+	0.407580i	$0.133627\pi$
$271$			3.22893e14i			0.815160i	−0.913169	+	0.407580i	$0.866373\pi$
$272$	−1.04596e15			−2.58287
$273$			4.23195e14i			1.02227i
$274$	−2.76177e14			−0.652657
$275$	3.30606e13			0.0764389
$276$	8.77271e14			1.98463
$277$	3.96115e14			0.876885			0.438443	−	0.898759i	$-0.355530\pi$
$277$	3.96115e14			0.876885			0.438443	+	0.898759i	$0.355530\pi$
$278$	−6.22546e14			−1.34866
$279$	−1.28261e14			−0.271937
$280$	−1.00530e15			−2.08616
$281$		−	8.08902e13i		−	0.164308i	−0.996620	−	0.0821539i	$-0.973820\pi$
$281$		−	8.08902e13i		−	0.164308i	0.996620	−	0.0821539i	$-0.0261799\pi$
$282$	−1.63558e15			−3.25220
$283$		−	6.11945e14i		−	1.19123i	−0.803272	−	0.595613i	$-0.796909\pi$
$283$		−	6.11945e14i		−	1.19123i	0.803272	−	0.595613i	$-0.203091\pi$
$284$		−	1.32240e15i		−	2.52030i
$285$	4.67971e14			0.873273
$286$	−3.06094e14			−0.559318
$287$	1.26554e14			0.226456
$288$		−	1.64633e14i		−	0.288511i
$289$	5.73286e14			0.983975
$290$	−1.38306e15			−2.32516
$291$		−	5.18473e14i		−	0.853824i
$292$		−	2.48846e13i		−	0.0401451i
$293$	−7.04659e14			−1.11371			−0.556857	−	0.830609i	$-0.687993\pi$
$293$	−7.04659e14			−1.11371			−0.556857	+	0.830609i	$0.687993\pi$
$294$		−	4.40789e14i		−	0.682570i
$295$			9.89559e14i			1.50144i
$296$			3.31536e14i			0.492925i
$297$	−1.56733e14			−0.228361
$298$	−2.14950e15			−3.06930
$299$			6.17048e14i			0.863559i
$300$	−5.08343e14			−0.697316
$301$			3.13221e14i			0.421164i
$302$			2.28240e15i			3.00849i
$303$		−	4.04930e13i		−	0.0523267i
$304$		−	9.91201e14i		−	1.25580i
$305$			1.26821e15i			1.57540i
$306$			5.77533e14i			0.703475i
$307$		−	7.12029e14i		−	0.850487i	−0.905079	−	0.425243i	$-0.860188\pi$
$307$		−	7.12029e14i		−	0.850487i	0.905079	−	0.425243i	$-0.139812\pi$
$308$	−4.38969e14			−0.514197
$309$		−	9.14947e14i		−	1.05110i
$310$	1.77564e15			2.00072
$311$			1.01672e15i			1.12367i	0.827248	+	0.561837i	$0.189905\pi$
$311$			1.01672e15i			1.12367i	−0.827248	+	0.561837i	$0.810095\pi$
$312$	2.62035e15			2.84075
$313$	1.67140e15			1.77752			0.888762	−	0.458369i	$-0.151566\pi$
$313$	1.67140e15			1.77752			0.888762	+	0.458369i	$0.151566\pi$
$314$	1.03148e15			1.07617
$315$			2.48881e14i			0.254758i
$316$		−	9.09801e14i		−	0.913743i
$317$	3.97857e13			0.0392077			0.0196038	−	0.999808i	$-0.493760\pi$
$317$	3.97857e13			0.0392077			0.0196038	+	0.999808i	$0.493760\pi$
$318$	−1.07870e14			−0.104313
$319$	−3.36230e14			−0.319075
$320$			5.72022e13i			0.0532737i
$321$			2.79522e14i			0.255497i
$322$			1.27714e15i			1.14579i
$323$			1.09539e15i			0.964614i
$324$	3.13230e15			2.70766
$325$		−	3.57554e14i		−	0.303418i
$326$	−3.47762e15			−2.89719
$327$	−1.05458e15			−0.862566
$328$		−	7.83601e14i		−	0.629292i
$329$		−	1.64982e15i		−	1.30095i
$330$	−8.30399e14			−0.642988
$331$			1.50076e15i			1.14115i	0.821244	+	0.570577i	$0.193280\pi$
$331$			1.50076e15i			1.14115i	−0.821244	+	0.570577i	$0.806720\pi$
$332$	2.69489e15	−	1.36571e15i	2.01239	−	1.01984i
$333$	8.20780e13			0.0601951
$334$			7.17095e14i			0.516533i
$335$	−1.60381e15			−1.13471
$336$	2.43173e15			1.68997
$337$		−	1.48925e15i		−	1.01669i	−0.861154	−	0.508345i	$-0.830257\pi$
$337$		−	1.48925e15i		−	1.01669i	0.861154	−	0.508345i	$-0.169743\pi$
$338$			6.19862e14i			0.415714i
$339$	−2.03949e14			−0.134377
$340$		−	5.53984e15i		−	3.58611i
$341$	4.31671e14			0.274553
$342$	−5.47296e14			−0.342031
$343$	1.77279e15			1.08866
$344$	1.93941e15			1.17036
$345$			1.67398e15i			0.992742i
$346$			4.96431e15i			2.89336i
$347$		−	7.34244e14i		−	0.420594i	−0.977638	−	0.210297i	$-0.932557\pi$
$347$		−	7.34244e14i		−	0.420594i	0.977638	−	0.210297i	$-0.0674432\pi$
$348$	5.16991e15			2.91077
$349$	−2.39054e15			−1.32295			−0.661475	−	0.749967i	$-0.730069\pi$
$349$	−2.39054e15			−1.32295			−0.661475	+	0.749967i	$0.730069\pi$
$350$		−	7.40053e14i		−	0.402582i
$351$			1.69508e15i			0.906460i
$352$			5.54085e14i			0.291287i
$353$	−3.57782e14			−0.184914			−0.0924571	−	0.995717i	$-0.529472\pi$
$353$	−3.57782e14			−0.184914			−0.0924571	+	0.995717i	$0.529472\pi$
$354$		−	5.33859e15i		−	2.71273i
$355$	2.52336e15			1.26069
$356$		−	1.39073e15i		−	0.683190i
$357$	−2.68734e15			−1.29811
$358$	1.62247e15			0.770690
$359$	3.21101e14			0.149995			0.0749973	−	0.997184i	$-0.476105\pi$
$359$	3.21101e14			0.149995			0.0749973	+	0.997184i	$0.476105\pi$
$360$	1.54103e15			0.707938
$361$	1.17528e15			0.531002
$362$	1.08171e15			0.480682
$363$	2.38335e15			1.04171
$364$			4.74750e15i			2.04106i
$365$	4.74839e13			0.0200812
$366$		−	6.84187e15i		−	2.84635i
$367$			4.52046e15i			1.85006i	0.379891	+	0.925031i	$0.375961\pi$
$367$			4.52046e15i			1.85006i	−0.379891	+	0.925031i	$0.624039\pi$
$368$	3.54563e15			1.42760
$369$	−1.93995e14			−0.0768479
$370$	−1.13629e15			−0.442873
$371$		−	1.08809e14i		−	0.0417274i
$372$	−6.63741e15			−2.50462
$373$	−1.90742e15			−0.708261			−0.354130	−	0.935196i	$-0.615223\pi$
$373$	−1.90742e15			−0.708261			−0.354130	+	0.935196i	$0.615223\pi$
$374$		−	1.94373e15i		−	0.710243i
$375$			2.57610e15i			0.926351i
$376$	−1.02154e16			−3.61516
$377$			3.63637e15i			1.26654i
$378$			3.50842e15i			1.20271i
$379$		−	2.76965e15i		−	0.934522i	−0.884119	−	0.467261i	$-0.845241\pi$
$379$		−	2.76965e15i		−	0.934522i	0.884119	−	0.467261i	$-0.154759\pi$
$380$	5.24980e15			1.74358
$381$	−4.98270e15			−1.62898
$382$		−	1.01776e16i		−	3.27541i
$383$	7.47822e14			0.236922			0.118461	−	0.992959i	$-0.462204\pi$
$383$	7.47822e14			0.236922			0.118461	+	0.992959i	$0.462204\pi$
$384$			3.46776e15i			1.08159i
$385$		−	8.37626e14i		−	0.257209i
$386$		−	7.14306e15i		−	2.15954i
$387$		−	4.80137e14i		−	0.142922i
$388$		−	5.81635e15i		−	1.70475i
$389$		−	3.86373e15i		−	1.11509i	−0.830147	−	0.557544i	$-0.811744\pi$
$389$		−	3.86373e15i		−	1.11509i	0.830147	−	0.557544i	$-0.188256\pi$
$390$			8.98086e15i			2.55229i
$391$	−3.91833e15			−1.09658
$392$		−	2.75305e15i		−	0.758748i
$393$	1.05180e15			0.285482
$394$			9.84401e15i			2.63145i
$395$	1.73606e15			0.457068
$396$	6.72896e14			0.174493
$397$	−4.26985e15			−1.09061			−0.545306	−	0.838237i	$-0.683586\pi$
$397$	−4.26985e15			−1.09061			−0.545306	+	0.838237i	$0.683586\pi$
$398$		−	8.80582e15i		−	2.21550i
$399$		−	2.54664e15i		−	0.631147i
$400$	−2.05455e15			−0.501599
$401$	4.30196e15			1.03467			0.517333	−	0.855784i	$-0.326925\pi$
$401$	4.30196e15			1.03467			0.517333	+	0.855784i	$0.326925\pi$
$402$	8.65243e15			2.05013
$403$		−	4.66857e15i		−	1.08982i
$404$		−	4.54259e14i		−	0.104476i
$405$			5.97696e15i			1.35441i
$406$			7.52643e15i			1.68048i
$407$	−2.76240e14			−0.0607742
$408$			1.66395e16i			3.60728i
$409$	2.70822e15			0.578553			0.289277	−	0.957246i	$-0.406585\pi$
$409$	2.70822e15			0.578553			0.289277	+	0.957246i	$0.406585\pi$
$410$	2.68567e15			0.565392
$411$			1.96992e15i			0.408693i
$412$		−	1.02641e16i		−	2.09863i
$413$	5.38505e15			1.08515
$414$		−	1.95774e15i		−	0.388823i
$415$	2.60601e15	+	5.14230e15i	0.510138	+	1.00663i
$416$	5.99250e15			1.15624
$417$			4.44049e15i			0.844529i
$418$	1.84197e15			0.345322
$419$	3.20241e15			0.591825			0.295913	−	0.955215i	$-0.404376\pi$
$419$	3.20241e15			0.591825			0.295913	+	0.955215i	$0.404376\pi$
$420$			1.28794e16i			2.34639i
$421$			6.16662e15i			1.10753i	0.832674	+	0.553764i	$0.186809\pi$
$421$			6.16662e15i			1.10753i	−0.832674	+	0.553764i	$0.813191\pi$
$422$	2.17258e15			0.384682
$423$			2.52901e15i			0.441477i
$424$	−6.73727e14			−0.115955
$425$	2.27051e15			0.385292
$426$	−1.36133e16			−2.27775
$427$	6.90142e15			1.13860
$428$			3.13574e15i			0.510126i
$429$			2.18331e15i			0.350244i
$430$			6.64703e15i			1.05152i
$431$	−7.11655e15			−1.11021			−0.555106	−	0.831780i	$-0.687322\pi$
$431$	−7.11655e15			−1.11021			−0.555106	+	0.831780i	$0.687322\pi$
$432$	9.74015e15			1.49852
$433$		−	1.23301e14i		−	0.0187085i	−0.999956	−	0.00935427i	$-0.997022\pi$
$433$		−	1.23301e14i		−	0.0187085i	0.999956	−	0.00935427i	$-0.00297760\pi$
$434$		−	9.66283e15i		−	1.44599i
$435$			9.86507e15i			1.45601i
$436$	−1.18305e16			−1.72220
$437$		−	3.71318e15i		−	0.533160i
$438$	−2.56172e14			−0.0362816
$439$		−	4.86808e15i		−	0.680097i	−0.940408	−	0.340048i	$-0.889557\pi$
$439$		−	4.86808e15i		−	0.680097i	0.940408	−	0.340048i	$-0.110443\pi$
$440$	−5.18644e15			−0.714749
$441$	−6.81568e14			−0.0926569
$442$	−2.10217e16			−2.81925
$443$	−2.60178e15			−0.344229			−0.172115	−	0.985077i	$-0.555060\pi$
$443$	−2.60178e15			−0.344229			−0.172115	+	0.985077i	$0.555060\pi$
$444$	4.24749e15			0.554414
$445$	2.65374e15			0.341742
$446$	1.73947e15			0.221008
$447$			1.53319e16i			1.92199i
$448$	3.11287e14			0.0385029
$449$		−	4.66956e15i		−	0.569898i	−0.958543	−	0.284949i	$-0.908023\pi$
$449$		−	4.66956e15i		−	0.569898i	0.958543	−	0.284949i	$-0.0919767\pi$
$450$			1.13443e15i			0.136616i
$451$	6.52905e14			0.0775873
$452$	−2.28795e15			−0.268297
$453$	1.62799e16			1.88392
$454$		−	1.14146e16i		−	1.30355i
$455$	−9.05903e15			−1.02097
$456$	−1.57684e16			−1.75387
$457$		−	1.14534e16i		−	1.25729i	−0.777691	−	0.628646i	$-0.783609\pi$
$457$		−	1.14534e16i		−	1.25729i	0.777691	−	0.628646i	$-0.216391\pi$
$458$			2.88133e15i			0.312176i
$459$	−1.07640e16			−1.15106
$460$			1.87791e16i			1.98211i
$461$		−	5.44062e15i		−	0.566817i	−0.958999	−	0.283408i	$-0.908535\pi$
$461$		−	5.44062e15i		−	0.566817i	0.958999	−	0.283408i	$-0.0914652\pi$
$462$			4.51893e15i			0.464712i
$463$	−2.68913e15			−0.272977			−0.136488	−	0.990642i	$-0.543582\pi$
$463$	−2.68913e15			−0.272977			−0.136488	+	0.990642i	$0.543582\pi$
$464$	2.08950e16			2.09380
$465$		−	1.26653e16i		−	1.25285i
$466$	−2.62182e16			−2.56028
$467$		−	6.17995e15i		−	0.595777i	−0.954601	−	0.297888i	$-0.903718\pi$
$467$		−	6.17995e15i		−	0.595777i	0.954601	−	0.297888i	$-0.0962823\pi$
$468$		−	7.27745e15i		−	0.692635i
$469$			8.72774e15i			0.820096i
$470$		−	3.50117e16i		−	3.24807i
$471$		−	7.35731e15i		−	0.673897i
$472$		−	3.33434e16i		−	3.01549i
$473$			1.61594e15i			0.144297i
$474$	−9.36588e15			−0.825807
$475$			2.15164e15i			0.187330i
$476$	−3.01471e16			−2.59182
$477$			1.66794e14i			0.0141602i
$478$	−1.88477e16			−1.58012
$479$	1.36357e16			1.12892			0.564461	−	0.825460i	$-0.309084\pi$
$479$	1.36357e16			1.12892			0.564461	+	0.825460i	$0.309084\pi$
$480$	1.62570e16			1.32921
$481$			2.98756e15i			0.241239i
$482$		−	5.41827e15i		−	0.432094i
$483$	9.10960e15			0.717491
$484$	2.67369e16			2.07988
$485$	1.10986e16			0.852740
$486$		−	1.28144e16i		−	0.972477i
$487$			1.95851e16i			1.46808i	0.679104	+	0.734042i	$0.262369\pi$
$487$			1.95851e16i			1.46808i	−0.679104	+	0.734042i	$0.737631\pi$
$488$		−	4.27324e16i		−	3.16402i
$489$			2.48052e16i			1.81422i
$490$	9.43565e15			0.681703
$491$			2.15145e16i			1.53548i	0.640764	+	0.767738i	$0.278618\pi$
$491$			2.15145e16i			1.53548i	−0.640764	+	0.767738i	$0.721382\pi$
$492$	−1.00391e16			−0.707791
$493$	−2.30914e16			−1.60830
$494$		−	1.99211e16i		−	1.37073i
$495$			1.28400e15i			0.0872838i
$496$	−2.68262e16			−1.80164
$497$		−	1.37318e16i		−	0.911150i
$498$	−1.40592e16	−	2.77423e16i	−0.921691	−	1.81872i
$499$	1.67603e16			1.08562			0.542811	−	0.839855i	$-0.317360\pi$
$499$	1.67603e16			1.08562			0.542811	+	0.839855i	$0.317360\pi$
$500$			2.88993e16i			1.84956i
$501$	5.11489e15			0.323452
$502$	4.53713e16			2.83504
$503$		−	2.04185e16i		−	1.26071i	−0.776307	−	0.630355i	$-0.782909\pi$
$503$		−	2.04185e16i		−	1.26071i	0.776307	−	0.630355i	$-0.217091\pi$
$504$		−	8.38608e15i		−	0.511653i
$505$	8.66803e14			0.0522603
$506$			6.58891e15i			0.392564i
$507$	4.42135e15			0.260320
$508$	−5.58971e16			−3.25242
$509$	2.27001e16			1.30534			0.652668	−	0.757644i	$-0.273650\pi$
$509$	2.27001e16			1.30534			0.652668	+	0.757644i	$0.273650\pi$
$510$	−5.70295e16			−3.24099
$511$		−	2.58402e14i		−	0.0145134i
$512$			4.04366e16i			2.24468i
$513$		−	1.02004e16i		−	0.559647i
$514$	−6.16153e16			−3.34126
$515$	1.95856e16			1.04977
$516$		−	2.48468e16i		−	1.31635i
$517$		−	8.51157e15i		−	0.445725i
$518$			6.18355e15i			0.320081i
$519$	3.54094e16			1.81182
$520$			5.60920e16i			2.83714i
$521$	1.54882e16			0.774418			0.387209	−	0.921992i	$-0.373439\pi$
$521$	1.54882e16			0.774418			0.387209	+	0.921992i	$0.373439\pi$
$522$		−	1.15373e16i		−	0.570270i
$523$	1.21071e16			0.591604			0.295802	−	0.955249i	$-0.404413\pi$
$523$	1.21071e16			0.591604			0.295802	+	0.955249i	$0.404413\pi$
$524$	1.17993e16			0.569993
$525$	−5.27864e15			−0.252096
$526$	6.84710e16			3.23290
$527$	2.96459e16			1.38389
$528$	1.25455e16			0.579009
$529$	−8.63219e15			−0.393901
$530$		−	2.30909e15i		−	0.104181i
$531$	−8.25476e15			−0.368246
$532$		−	2.85688e16i		−	1.26015i
$533$		−	7.06124e15i		−	0.307977i
$534$	−1.43167e16			−0.617441
$535$	−5.98352e15			−0.255172
$536$	5.40407e16			2.27894
$537$		−	1.15728e16i		−	0.482605i
$538$	−2.93884e16			−1.21195
$539$	2.29387e15			0.0935483
$540$			5.15878e16i			2.08058i
$541$			1.82078e16i			0.726228i	0.931745	+	0.363114i	$0.118287\pi$
$541$			1.82078e16i			0.726228i	−0.931745	+	0.363114i	$0.881713\pi$
$542$	3.72893e16			1.47092
$543$		−	7.71558e15i		−	0.301002i
$544$			3.80530e16i			1.46824i
$545$		−	2.25746e16i		−	0.861471i
$546$	4.88727e16			1.84464
$547$	3.13296e15			0.116958			0.0584791	−	0.998289i	$-0.481375\pi$
$547$	3.13296e15			0.116958			0.0584791	+	0.998289i	$0.481375\pi$
$548$			2.20990e16i			0.815997i
$549$	−1.05792e16			−0.386384
$550$		−	3.81801e15i		−	0.137931i
$551$		−	2.18824e16i		−	0.781962i
$552$		−	5.64051e16i		−	1.99381i
$553$		−	9.44740e15i		−	0.330340i
$554$		−	4.57454e16i		−	1.58230i
$555$			8.10492e15i			0.277326i
$556$			4.98144e16i			1.68619i
$557$	−3.89202e16			−1.30330			−0.651649	−	0.758521i	$-0.725922\pi$
$557$	−3.89202e16			−1.30330			−0.651649	+	0.758521i	$0.725922\pi$
$558$			1.48122e16i			0.490698i
$559$	1.74765e16			0.572776
$560$			5.20542e16i			1.68783i
$561$	−1.38642e16			−0.444753
$562$	−9.34161e15			−0.296486
$563$	4.26927e16			1.34061			0.670307	−	0.742084i	$-0.266163\pi$
$563$	4.26927e16			1.34061			0.670307	+	0.742084i	$0.266163\pi$
$564$			1.30875e17i			4.06613i
$565$		−	4.36579e15i		−	0.134206i
$566$	−7.06706e16			−2.14951
$567$	3.25259e16			0.978882
$568$	−8.50252e16			−2.53196
$569$			9.95748e15i			0.293411i	0.989180	+	0.146705i	$0.0468669\pi$
$569$			9.95748e15i			0.293411i	−0.989180	+	0.146705i	$0.953133\pi$
$570$		−	5.40437e16i		−	1.57578i
$571$		−	2.82675e16i		−	0.815589i	−0.913074	−	0.407795i	$-0.866298\pi$
$571$		−	2.82675e16i		−	0.815589i	0.913074	−	0.407795i	$-0.133702\pi$
$572$			2.44928e16i			0.699298i
$573$	−7.25948e16			−2.05106
$574$		−	1.46151e16i		−	0.408630i
$575$	−7.69663e15			−0.212958
$576$	−4.77173e14			−0.0130659
$577$		−	5.39911e16i		−	1.46308i	−0.681800	−	0.731538i	$-0.738803\pi$
$577$		−	5.39911e16i		−	1.46308i	0.681800	−	0.731538i	$-0.261197\pi$
$578$		−	6.62059e16i		−	1.77554i
$579$	−5.09500e16			−1.35230
$580$			1.10668e17i			2.90707i
$581$	2.79838e16	−	1.41816e16i	0.727527	−	0.368696i
$582$	−5.98759e16			−1.54069
$583$		−	5.61356e14i		−	0.0142964i
$584$	−1.59998e15			−0.0403309
$585$	1.38866e16			0.346466
$586$			8.13776e16i			2.00964i
$587$		−	8.68029e15i		−	0.212180i	−0.994357	−	0.106090i	$-0.966167\pi$
$587$		−	8.68029e15i		−	0.212180i	0.994357	−	0.106090i	$-0.0338332\pi$
$588$	−3.52707e16			−0.853396
$589$			2.80938e16i			0.672851i
$590$	1.14279e17			2.70929
$591$	7.02153e16			1.64781
$592$	1.71669e16			0.398806
$593$	−7.97236e16			−1.83341			−0.916703	−	0.399570i	$-0.869160\pi$
$593$	−7.97236e16			−1.83341			−0.916703	+	0.399570i	$0.869160\pi$
$594$			1.81003e16i			0.412066i
$595$		−	5.75258e16i		−	1.29647i
$596$			1.71997e17i			3.83746i
$597$	−6.28101e16			−1.38734
$598$	7.12598e16			1.55825
$599$			8.45204e16i			1.82979i	0.403696	+	0.914893i	$0.367725\pi$
$599$			8.45204e16i			1.82979i	−0.403696	+	0.914893i	$0.632275\pi$
$600$			3.26845e16i			0.700542i
$601$			5.97298e16i			1.26749i	0.773543	+	0.633744i	$0.218483\pi$
$601$			5.97298e16i			1.26749i	−0.773543	+	0.633744i	$0.781517\pi$
$602$	3.61723e16			0.759971
$603$		−	1.33788e16i		−	0.278300i
$604$	1.82631e17			3.76143
$605$			5.10185e16i			1.04039i
$606$	−4.67633e15			−0.0944212
$607$	2.30869e15			0.0461565			0.0230783	−	0.999734i	$-0.492653\pi$
$607$	2.30869e15			0.0461565			0.0230783	+	0.999734i	$0.492653\pi$
$608$	−3.60607e16			−0.713860
$609$	5.36845e16			1.05231
$610$	1.46459e17			2.84274
$611$	−9.20537e16			−1.76927
$612$	4.62126e16			0.879533
$613$		−	5.29034e16i		−	0.997059i	−0.866873	−	0.498529i	$-0.833874\pi$
$613$		−	5.29034e16i		−	0.997059i	0.866873	−	0.498529i	$-0.166126\pi$
$614$	−8.22287e16			−1.53466
$615$		−	1.91564e16i		−	0.354048i
$616$			2.82240e16i			0.516576i
$617$	−6.61746e16			−1.19944			−0.599722	−	0.800208i	$-0.704722\pi$
$617$	−6.61746e16			−1.19944			−0.599722	+	0.800208i	$0.704722\pi$
$618$	−1.05663e17			−1.89667
$619$	9.44509e16			1.67905			0.839523	−	0.543325i	$-0.182835\pi$
$619$	9.44509e16			1.67905			0.839523	+	0.543325i	$0.182835\pi$
$620$		−	1.42082e17i		−	2.50144i
$621$	3.64880e16			0.636210
$622$	1.17416e17			2.02762
$623$		−	1.44413e16i		−	0.246990i
$624$		−	1.35681e17i		−	2.29833i
$625$	−7.14490e16			−1.19872
$626$		−	1.93022e17i		−	3.20746i
$627$		−	1.31384e16i		−	0.216240i
$628$		−	8.25360e16i		−	1.34551i
$629$	−1.89714e16			−0.306334
$630$	2.87420e16			0.459699
$631$			1.68011e16i			0.266171i	0.991105	+	0.133086i	$0.0424886\pi$
$631$			1.68011e16i			0.266171i	−0.991105	+	0.133086i	$0.957511\pi$
$632$	−5.84967e16			−0.917971
$633$		−	1.54966e16i		−	0.240887i
$634$		−	4.59465e15i		−	0.0707485i
$635$		−	1.06661e17i		−	1.62691i
$636$			8.63147e15i			0.130419i
$637$		−	2.48085e16i		−	0.371333i
$638$			3.88296e16i			0.575756i
$639$			2.10496e16i			0.309199i
$640$	−7.42317e16			−1.08021
$641$			3.84203e15i			0.0553876i	0.999616	+	0.0276938i	$0.00881634\pi$
$641$			3.84203e15i			0.0553876i	−0.999616	+	0.0276938i	$0.991184\pi$
$642$	3.22806e16			0.461032
$643$		−	4.33825e16i		−	0.613831i	−0.951737	−	0.306916i	$-0.900703\pi$
$643$		−	4.33825e16i		−	0.613831i	0.951737	−	0.306916i	$-0.0992970\pi$
$644$	1.02194e17			1.43254
$645$	4.74119e16			0.658459
$646$	1.26501e17			1.74060
$647$		−	1.39735e17i		−	1.90493i	−0.304651	−	0.952464i	$-0.598540\pi$
$647$		−	1.39735e17i		−	1.90493i	0.304651	−	0.952464i	$-0.401460\pi$
$648$		−	2.01395e17i		−	2.72018i
$649$	2.77820e16			0.371788
$650$	−4.12922e16			−0.547504
$651$	−6.89230e16			−0.905479
$652$			2.78270e17i			3.62227i
$653$			1.33222e17i			1.71830i	0.511726	+	0.859148i	$0.329006\pi$
$653$			1.33222e17i			1.71830i	−0.511726	+	0.859148i	$0.670994\pi$
$654$			1.21788e17i			1.55646i
$655$			2.25151e16i			0.285119i
$656$	−4.05747e16			−0.509134
$657$			3.96105e14i			0.00492513i
$658$	−1.90529e17			−2.34750
$659$	1.48973e17			1.81884			0.909419	−	0.415881i	$-0.136527\pi$
$659$	1.48973e17			1.81884			0.909419	+	0.415881i	$0.136527\pi$
$660$			6.64462e16i			0.803909i
$661$			1.01860e17i			1.22122i	0.791930	+	0.610612i	$0.209077\pi$
$661$			1.01860e17i			1.22122i	−0.791930	+	0.610612i	$0.790923\pi$
$662$	1.73316e17			2.05916
$663$			1.49943e17i			1.76541i
$664$	−8.78100e16	−	1.73271e17i	−1.02456	−	2.02170i
$665$	5.45140e16			0.630345
$666$		−	9.47878e15i		−	0.108619i
$667$	7.82757e16			0.888939
$668$	5.73800e16			0.645806
$669$		−	1.24073e16i		−	0.138395i
$670$			1.85216e17i			2.04753i
$671$	3.56051e16			0.390101
$672$		−	8.84684e16i		−	0.960666i
$673$	−4.82614e16			−0.519410			−0.259705	−	0.965688i	$-0.583625\pi$
$673$	−4.82614e16			−0.519410			−0.259705	+	0.965688i	$0.583625\pi$
$674$	−1.71986e17			−1.83457
$675$	−2.11433e16			−0.223537
$676$	4.95997e16			0.519755
$677$		−	1.19575e17i		−	1.24196i	−0.783827	−	0.620979i	$-0.786735\pi$
$677$		−	1.19575e17i		−	1.24196i	0.783827	−	0.620979i	$-0.213265\pi$
$678$			2.35531e16i			0.242476i
$679$		−	6.03971e16i		−	0.616307i
$680$	−3.56190e17			−3.60271
$681$	−8.14182e16			−0.816280
$682$		−	4.98515e16i		−	0.495418i
$683$		−	4.39955e16i		−	0.433395i	−0.976239	−	0.216698i	$-0.930471\pi$
$683$		−	4.39955e16i		−	0.433395i	0.976239	−	0.216698i	$-0.0695285\pi$
$684$			4.37931e16i			0.427631i
$685$	−4.21686e16			−0.408174
$686$		−	2.04731e17i		−	1.96444i
$687$	2.05519e16			0.195484
$688$		−	1.00422e17i		−	0.946889i
$689$	−6.07113e15			−0.0567485
$690$	1.93320e17			1.79136
$691$	−1.85060e17			−1.69998			−0.849990	−	0.526799i	$-0.823392\pi$
$691$	−1.85060e17			−1.69998			−0.849990	+	0.526799i	$0.823392\pi$
$692$	3.97230e17			3.61748
$693$	6.98737e15			0.0630833
$694$	−8.47942e16			−0.758943
$695$	−9.50543e16			−0.843457
$696$		−	3.32405e17i		−	2.92424i
$697$	4.48397e16			0.391080
$698$			2.76072e17i			2.38720i
$699$			1.87009e17i			1.60325i
$700$	−5.92170e16			−0.503336
$701$	−4.67259e16			−0.393777			−0.196888	−	0.980426i	$-0.563084\pi$
$701$	−4.67259e16			−0.393777			−0.196888	+	0.980426i	$0.563084\pi$
$702$	1.95757e17			1.63566
$703$		−	1.79781e16i		−	0.148940i
$704$	1.60596e15			0.0131916
$705$	−2.49731e17			−2.03394
$706$			4.13185e16i			0.333669i
$707$		−	4.71703e15i		−	0.0377705i
$708$	−4.27179e17			−3.39165
$709$		−	1.76848e17i		−	1.39227i	−0.717912	−	0.696134i	$-0.754902\pi$
$709$		−	1.76848e17i		−	1.39227i	0.717912	−	0.696134i	$-0.245098\pi$
$710$		−	2.91411e17i		−	2.27486i
$711$			1.44819e16i			0.112101i
$712$	−8.94182e16			−0.686351
$713$	−1.00495e17			−0.764901
$714$			3.10347e17i			2.34239i
$715$	−4.67364e16			−0.349800
$716$		−	1.29826e17i		−	0.963570i
$717$			1.34437e17i			0.989472i
$718$		−	3.70824e16i		−	0.270658i
$719$			2.70170e17i			1.95552i	0.209718	+	0.977762i	$0.432745\pi$
$719$			2.70170e17i			1.95552i	−0.209718	+	0.977762i	$0.567255\pi$
$720$		−	7.97941e16i		−	0.572764i
$721$		−	1.06582e17i		−	0.758707i
$722$		−	1.35727e17i		−	0.958169i
$723$	−3.86474e16			−0.270577
$724$		−	8.65551e16i		−	0.600982i
$725$	−4.53576e16			−0.312336
$726$		−	2.75241e17i		−	1.87972i
$727$	7.10314e16			0.481109			0.240555	−	0.970636i	$-0.422671\pi$
$727$	7.10314e16			0.481109			0.240555	+	0.970636i	$0.422671\pi$
$728$	3.05246e17			2.05051
$729$	8.87373e16			0.591209
$730$		−	5.48369e15i		−	0.0362356i
$731$			1.10978e17i			0.727332i
$732$	−5.47468e17			−3.55871
$733$	1.71824e17			1.10780			0.553899	−	0.832584i	$-0.313139\pi$
$733$	1.71824e17			1.10780			0.553899	+	0.832584i	$0.313139\pi$
$734$	5.22046e17			3.33835
$735$		−	6.73026e16i		−	0.426882i
$736$		−	1.28993e17i		−	0.811521i
$737$			4.50273e16i			0.280977i
$738$			2.24035e16i			0.138669i
$739$	2.77892e17			1.70612			0.853058	−	0.521816i	$-0.174745\pi$
$739$	2.77892e17			1.70612			0.853058	+	0.521816i	$0.174745\pi$
$740$			9.09228e16i			0.553710i
$741$	−1.42093e17			−0.858348
$742$	−1.25658e16			−0.0752951
$743$			1.13012e16i			0.0671724i	0.999436	+	0.0335862i	$0.0106928\pi$
$743$			1.13012e16i			0.0671724i	−0.999436	+	0.0335862i	$0.989307\pi$
$744$			4.26760e17i			2.51620i
$745$	−3.28199e17			−1.91955
$746$			2.20278e17i			1.27802i
$747$	−4.28964e16	+	2.17390e16i	−0.246886	+	0.125117i
$748$	−1.55532e17			−0.887995
$749$			3.25616e16i			0.184423i
$750$	2.97502e17			1.67156
$751$	−2.84402e17			−1.58523			−0.792617	−	0.609720i	$-0.791282\pi$
$751$	−2.84402e17			−1.58523			−0.792617	+	0.609720i	$0.791282\pi$
$752$			5.28951e17i			2.92488i
$753$		−	3.23624e17i		−	1.77530i
$754$	4.19946e17			2.28542
$755$			3.48491e17i			1.88152i
$756$	2.80734e17			1.50371
$757$	−7.15984e16			−0.380476			−0.190238	−	0.981738i	$-0.560926\pi$
$757$	−7.15984e16			−0.380476			−0.190238	+	0.981738i	$0.560926\pi$
$758$	−3.19853e17			−1.68630
$759$	4.69974e16			0.245823
$760$		−	3.37542e17i		−	1.75165i
$761$			2.93036e15i			0.0150873i	0.999972	+	0.00754367i	$0.00240125\pi$
$761$			2.93036e15i			0.0150873i	−0.999972	+	0.00754367i	$0.997599\pi$
$762$			5.75428e17i			2.93942i
$763$	−1.22848e17			−0.622617
$764$	−8.14385e17			−4.09515
$765$			8.81815e16i			0.439956i
$766$		−	8.63622e16i		−	0.427515i
$767$		−	3.00466e17i		−	1.47578i
$768$	4.11420e17			2.00501
$769$			1.87792e17i			0.908069i	0.890984	+	0.454034i	$0.150016\pi$
$769$			1.87792e17i			0.908069i	−0.890984	+	0.454034i	$0.849984\pi$
$770$	−9.67333e16			−0.464122
$771$			4.39490e17i			2.09229i
$772$	−5.71568e17			−2.70000
$773$	3.15199e17			1.47743			0.738716	−	0.674016i	$-0.235432\pi$
$773$	3.15199e17			1.47743			0.738716	+	0.674016i	$0.235432\pi$
$774$	−5.54486e16			−0.257896
$775$	5.82325e16			0.268754
$776$	−3.73969e17			−1.71263
$777$	4.41060e16			0.200434
$778$	−4.46203e17			−2.01213
$779$			4.24921e16i			0.190144i
$780$	7.18624e17			3.19105
$781$		−	7.08438e16i		−	0.312173i
$782$			4.52508e17i			1.97873i
$783$	2.15030e17			0.933101
$784$	−1.42552e17			−0.613872
$785$	1.57493e17			0.673042
$786$		−	1.21467e17i		−	0.515139i
$787$	−2.14610e17			−0.903238			−0.451619	−	0.892211i	$-0.649153\pi$
$787$	−2.14610e17			−0.903238			−0.451619	+	0.892211i	$0.649153\pi$
$788$	7.87690e17			3.29002
$789$		−	4.88390e17i		−	2.02444i
$790$		−	2.00488e17i		−	0.824758i
$791$	−2.37581e16			−0.0969957
$792$		−	4.32646e16i		−	0.175300i
$793$		−	3.85073e17i		−	1.54848i
$794$			4.93104e17i			1.96796i
$795$	−1.64703e16			−0.0652377
$796$	−7.04618e17			−2.76997
$797$			4.16361e17i			1.62450i	0.583308	+	0.812251i	$0.301758\pi$
$797$			4.16361e17i			1.62450i	−0.583308	+	0.812251i	$0.698242\pi$
$798$	−2.94099e17			−1.13887
$799$		−	5.84551e17i		−	2.24668i
$800$			7.47463e16i			0.285134i
$801$			2.21371e16i			0.0838159i
$802$		−	4.96812e17i		−	1.86701i
$803$		−	1.33312e15i		−	0.00497251i
$804$		−	6.92344e17i		−	2.56322i
$805$			1.95003e17i			0.716581i
$806$	−5.39150e17			−1.96652
$807$			2.09622e17i			0.758919i
$808$	−2.92071e16			−0.104959
$809$			3.62699e17i			1.29376i	0.762590	+	0.646882i	$0.223927\pi$
$809$			3.62699e17i			1.29376i	−0.762590	+	0.646882i	$0.776073\pi$
$810$	6.90250e17			2.44397
$811$	1.51249e17			0.531579			0.265789	−	0.964031i	$-0.414367\pi$
$811$	1.51249e17			0.531579			0.265789	+	0.964031i	$0.414367\pi$
$812$	6.02244e17			2.10105
$813$		−	2.65977e17i		−	0.921088i
$814$			3.19015e16i			0.109664i
$815$	−5.30986e17			−1.81191
$816$	8.61592e17			2.91851
$817$	−1.05168e17			−0.353631
$818$		−	3.12758e17i		−	1.04397i
$819$		−	7.55692e16i		−	0.250404i
$820$		−	2.14900e17i		−	0.706893i
$821$		−	1.45287e17i		−	0.474424i	−0.971458	−	0.237212i	$-0.923766\pi$
$821$		−	1.45287e17i		−	0.474424i	0.971458	−	0.237212i	$-0.0762336\pi$
$822$	2.27496e17			0.737467
$823$		−	1.61793e17i		−	0.520668i	−0.965519	−	0.260334i	$-0.916167\pi$
$823$		−	1.61793e17i		−	0.520668i	0.965519	−	0.260334i	$-0.0838328\pi$
$824$	−6.59941e17			−2.10834
$825$	−2.72331e16			−0.0863719
$826$		−	6.21893e17i		−	1.95810i
$827$			1.20273e17i			0.375956i	0.982173	+	0.187978i	$0.0601933\pi$
$827$			1.20273e17i			0.375956i	−0.982173	+	0.187978i	$0.939807\pi$
$828$	−1.56653e17			−0.486134
$829$		−	7.70910e16i		−	0.237507i	−0.992924	−	0.118753i	$-0.962110\pi$
$829$		−	7.70910e16i		−	0.237507i	0.992924	−	0.118753i	$-0.0378898\pi$
$830$	5.93859e17	−	3.00955e17i	1.81641	−	0.920521i
$831$	−3.26292e17			−0.990834
$832$		−	1.73687e16i		−	0.0523632i
$833$	1.57536e17			0.471532
$834$	5.12810e17			1.52391
$835$			1.09491e17i			0.323042i
$836$		−	1.47389e17i		−	0.431745i
$837$	−2.76067e17			−0.802901
$838$		−	3.69831e17i		−	1.06792i
$839$	3.63699e17			1.04273			0.521363	−	0.853335i	$-0.325424\pi$
$839$	3.63699e17			1.04273			0.521363	+	0.853335i	$0.325424\pi$
$840$	8.28097e17			2.35725
$841$	1.07477e17			0.303767
$842$	7.12153e17			1.99848
$843$			6.66318e16i			0.185659i
$844$		−	1.73844e17i		−	0.480956i
$845$			9.46445e16i			0.259989i
$846$	2.92063e17			0.796626
$847$	2.77637e17			0.751928
$848$			3.48854e16i			0.0938143i
$849$			5.04079e17i			1.34602i
$850$		−	2.62210e17i		−	0.695242i
$851$	6.43096e16			0.169316
$852$			1.08930e18i			2.84781i
$853$	2.46078e17			0.638820			0.319410	−	0.947617i	$-0.396515\pi$
$853$	2.46078e17			0.638820			0.319410	+	0.947617i	$0.396515\pi$
$854$		−	7.97011e17i		−	2.05455i
$855$	−8.35647e16			−0.213908
$856$	2.01616e17			0.512486
$857$	−5.77286e17			−1.45716			−0.728578	−	0.684963i	$-0.759818\pi$
$857$	−5.77286e17			−1.45716			−0.728578	+	0.684963i	$0.759818\pi$
$858$	2.52139e17			0.631999
$859$	−9.60936e16			−0.239186			−0.119593	−	0.992823i	$-0.538159\pi$
$859$	−9.60936e16			−0.239186			−0.119593	+	0.992823i	$0.538159\pi$
$860$	5.31877e17			1.31468
$861$	−1.04246e17			−0.255884
$862$			8.21855e17i			2.00333i
$863$	−3.27889e17			−0.793710			−0.396855	−	0.917881i	$-0.629898\pi$
$863$	−3.27889e17			−0.793710			−0.396855	+	0.917881i	$0.629898\pi$
$864$		−	3.54355e17i		−	0.851837i
$865$			7.57983e17i			1.80952i
$866$	−1.42395e16			−0.0337587
$867$	−4.72233e17			−1.11184
$868$	−7.73194e17			−1.80788
$869$		−	4.87401e16i		−	0.113179i
$870$	1.13927e18			2.62730
$871$	4.86975e17			1.11532
$872$			7.60654e17i			1.73017i
$873$			9.25828e16i			0.209144i
$874$	−4.28817e17			−0.962062
$875$			3.00091e17i			0.668659i
$876$			2.04982e16i			0.0453618i
$877$			2.09783e17i			0.461077i	0.973063	+	0.230538i	$0.0740487\pi$
$877$			2.09783e17i			0.461077i	−0.973063	+	0.230538i	$0.925951\pi$
$878$	−5.62191e17			−1.22720
$879$	5.80450e17			1.25844
$880$			2.68553e17i			0.578274i
$881$	3.66084e17			0.782934			0.391467	−	0.920192i	$-0.371968\pi$
$881$	3.66084e17			0.782934			0.391467	+	0.920192i	$0.371968\pi$
$882$			7.87109e16i			0.167195i
$883$		−	4.22533e17i		−	0.891449i	−0.895170	−	0.445724i	$-0.852946\pi$
$883$		−	4.22533e17i		−	0.891449i	0.895170	−	0.445724i	$-0.147054\pi$
$884$			1.68210e18i			3.52483i
$885$		−	8.15130e17i		−	1.69655i
$886$			3.00466e17i			0.621146i
$887$			8.31380e16i			0.170710i	0.996351	+	0.0853548i	$0.0272024\pi$
$887$			8.31380e16i			0.170710i	−0.996351	+	0.0853548i	$0.972798\pi$
$888$		−	2.73097e17i		−	0.556979i
$889$	−5.80436e17			−1.17583
$890$		−	3.06467e17i		−	0.616657i
$891$	1.67804e17			0.335380
$892$		−	1.39188e17i		−	0.276319i
$893$	5.53947e17			1.09234
$894$	1.77061e18			3.46815
$895$	2.47730e17			0.481992
$896$			4.03960e17i			0.780711i
$897$		−	5.08282e17i		−	0.975776i
$898$	−5.39264e17			−1.02836
$899$	−5.92232e17			−1.12185
$900$	9.07739e16			0.170807
$901$		−	3.85524e16i		−	0.0720613i
$902$		−	7.54007e16i		−	0.140003i
$903$		−	2.58010e17i		−	0.475893i
$904$			1.47106e17i			0.269538i
$905$	1.65162e17			0.300620
$906$		−	1.88008e18i		−	3.39944i
$907$	1.28531e17			0.230869			0.115435	−	0.993315i	$-0.463174\pi$
$907$	1.28531e17			0.230869			0.115435	+	0.993315i	$0.463174\pi$
$908$	−9.13368e17			−1.62979
$909$			7.23075e15i			0.0128174i
$910$			1.04618e18i			1.84229i
$911$	−4.99109e17			−0.873143			−0.436571	−	0.899670i	$-0.643807\pi$
$911$	−4.99109e17			−0.873143			−0.436571	+	0.899670i	$0.643807\pi$
$912$			8.16483e17i			1.41899i
$913$	1.44371e17	−	7.31642e16i	0.249262	−	0.126321i
$914$	−1.32269e18			−2.26873
$915$		−	1.04466e18i		−	1.78012i
$916$	2.30556e17			0.390304
$917$	1.22525e17			0.206066
$918$			1.24308e18i			2.07703i
$919$			1.27807e17i			0.212160i	0.994358	+	0.106080i	$0.0338299\pi$
$919$			1.27807e17i			0.212160i	−0.994358	+	0.106080i	$0.966170\pi$
$920$	1.20742e18			1.99128
$921$			5.86521e17i			0.961005i
$922$	−6.28310e17			−1.02279
$923$	−7.66184e17			−1.23915
$924$	3.61592e17			0.581015
$925$	−3.72648e16			−0.0594906
$926$			3.10554e17i			0.492574i
$927$			1.63380e17i			0.257467i
$928$		−	7.60179e17i		−	1.19022i
$929$	4.47635e17			0.696354			0.348177	−	0.937429i	$-0.386801\pi$
$929$	4.47635e17			0.696354			0.348177	+	0.937429i	$0.386801\pi$
$930$	−1.46265e18			−2.26070
$931$			1.49289e17i			0.229260i
$932$			2.09791e18i			3.20104i
$933$		−	8.37507e17i		−	1.26969i
$934$	−7.13692e17			−1.07505
$935$		−	2.96782e17i		−	0.444188i
$936$	−4.67912e17			−0.695839
$937$		−	1.21274e18i		−	1.79197i	−0.444083	−	0.895986i	$-0.646471\pi$
$937$		−	1.21274e18i		−	1.79197i	0.444083	−	0.895986i	$-0.353529\pi$
$938$	1.00792e18			1.47983
$939$	−1.37679e18			−2.00851
$940$	−2.80154e18			−4.06097
$941$	8.26182e17			1.18998			0.594988	−	0.803735i	$-0.297157\pi$
$941$	8.26182e17			1.18998			0.594988	+	0.803735i	$0.297157\pi$
$942$	−8.49660e17			−1.21602
$943$	−1.51999e17			−0.216157
$944$	−1.72651e18			−2.43971
$945$			5.35689e17i			0.752180i
$946$	1.86616e17			0.260377
$947$		−	1.15950e18i		−	1.60757i	−0.594921	−	0.803785i	$-0.702816\pi$
$947$		−	1.15950e18i		−	1.60757i	0.594921	−	0.803785i	$-0.297184\pi$
$948$			7.49432e17i			1.03248i
$949$	−1.44178e16			−0.0197380
$950$	2.48482e17			0.338028
$951$	−3.27727e16			−0.0443026
$952$			1.93834e18i			2.60381i
$953$	6.19464e17			0.826910			0.413455	−	0.910525i	$-0.364322\pi$
$953$	6.19464e17			0.826910			0.413455	+	0.910525i	$0.364322\pi$
$954$	1.92622e16			0.0255514
$955$		−	1.55398e18i		−	2.04845i
$956$			1.50814e18i			1.97558i
$957$	2.76963e17			0.360538
$958$		−	1.57472e18i		−	2.03709i
$959$			2.29476e17i			0.295002i
$960$		−	4.71192e16i		−	0.0601964i
$961$	−2.73235e16			−0.0346893
$962$	3.45019e17			0.435304
$963$		−	4.99137e16i		−	0.0625838i
$964$	−4.33555e17			−0.540234
$965$		−	1.09065e18i		−	1.35058i
$966$		−	1.05202e18i		−	1.29468i
$967$			9.88646e17i			1.20916i	0.796546	+	0.604578i	$0.206658\pi$
$967$			9.88646e17i			1.20916i	−0.796546	+	0.604578i	$0.793342\pi$
$968$		−	1.71908e18i		−	2.08951i
$969$		−	9.02307e17i		−	1.08996i
$970$		−	1.28172e18i		−	1.53873i
$971$		−	6.17588e17i		−	0.736858i	−0.929656	−	0.368429i	$-0.879896\pi$
$971$		−	6.17588e17i		−	0.736858i	0.929656	−	0.368429i	$-0.120104\pi$
$972$	−1.02537e18			−1.21586
$973$			5.17274e17i			0.609598i
$974$	2.26178e18			2.64909
$975$			2.94529e17i			0.342847i
$976$	−2.21268e18			−2.55988
$977$	−8.46447e17			−0.973268			−0.486634	−	0.873606i	$-0.661775\pi$
$977$	−8.46447e17			−0.973268			−0.486634	+	0.873606i	$0.661775\pi$
$978$	2.86463e18			3.27367
$979$		−	7.45042e16i		−	0.0846223i
$980$		−	7.55015e17i		−	0.852313i
$981$	1.88314e17			0.211285
$982$	2.48461e18			2.77070
$983$	−1.38980e18			−1.54039			−0.770196	−	0.637808i	$-0.779841\pi$
$983$	−1.38980e18			−1.54039			−0.770196	+	0.637808i	$0.779841\pi$
$984$			6.45477e17i			0.711066i
$985$			1.50305e18i			1.64572i
$986$			2.66671e18i			2.90211i
$987$			1.35901e18i			1.47000i
$988$	−1.59403e18			−1.71378
$989$		−	3.76196e17i		−	0.402010i
$990$	1.48283e17			0.157500
$991$	1.77493e18			1.87387			0.936933	−	0.349508i	$-0.113651\pi$
$991$	1.77493e18			1.87387			0.936933	+	0.349508i	$0.113651\pi$
$992$			9.75959e17i			1.02414i
$993$		−	1.23622e18i		−	1.28944i
$994$	−1.58582e18			−1.64413
$995$		−	1.34453e18i		−	1.38558i
$996$	−2.21986e18	+	1.12498e18i	−2.27389	+	1.15236i
$997$	−4.50462e17			−0.458656			−0.229328	−	0.973349i	$-0.573653\pi$
$997$	−4.50462e17			−0.458656			−0.229328	+	0.973349i	$0.573653\pi$
$998$		−	1.93556e18i		−	1.95895i
$999$	1.76664e17			0.177728

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	83.13.b.c.82.4	✓	80
83.82	odd	2	inner	83.13.b.c.82.77	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
83.13.b.c.82.4	✓	80	1.1	even	1	trivial
83.13.b.c.82.77	yes	80	83.82	odd	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:	\( N \)	\(=\)	\( 83 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	83.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-115.485i q^{2} -823.731 q^{3} -9240.80 q^{4} -17633.0i q^{5} +95128.7i q^{6} -95956.6 q^{7} +594147. i q^{8} +147092. q^{9} +O(q^{10})\)	\(q-115.485i q^{2} -823.731 q^{3} -9240.80 q^{4} -17633.0i q^{5} +95128.7i q^{6} -95956.6 q^{7} +594147. i q^{8} +147092. q^{9} -2.03635e6 q^{10} -495050. q^{11} +7.61194e6 q^{12} +5.35402e6i q^{13} +1.10816e7i q^{14} +1.45249e7i q^{15} +3.07648e7 q^{16} -3.39986e7 q^{17} -1.69870e7i q^{18} -3.22186e7i q^{19} +1.62943e8i q^{20} +7.90425e7 q^{21} +5.71709e7i q^{22} +1.15249e8 q^{23} -4.89418e8i q^{24} -6.67824e7 q^{25} +6.18310e8 q^{26} +3.16600e8 q^{27} +8.86716e8 q^{28} +6.79185e8 q^{29} +1.67740e9 q^{30} -8.71974e8 q^{31} -1.11925e9i q^{32} +4.07788e8 q^{33} +3.92634e9i q^{34} +1.69200e9i q^{35} -1.35925e9 q^{36} +5.58004e8 q^{37} -3.72077e9 q^{38} -4.41028e9i q^{39} +1.04766e10 q^{40} -1.31887e9 q^{41} -9.12823e9i q^{42} -3.26419e9i q^{43} +4.57466e9 q^{44} -2.59368e9i q^{45} -1.33096e10i q^{46} +1.71934e10i q^{47} -2.53420e10 q^{48} -4.63361e9 q^{49} +7.71237e9i q^{50} +2.80057e10 q^{51} -4.94754e10i q^{52} +1.13394e9i q^{53} -3.65626e10i q^{54} +8.72922e9i q^{55} -5.70124e10i q^{56} +2.65395e10i q^{57} -7.84357e10i q^{58} -5.61197e10 q^{59} -1.34221e11i q^{60} -7.19223e10 q^{61} +1.00700e11i q^{62} -1.41145e10 q^{63} -3.24404e9 q^{64} +9.44075e10 q^{65} -4.70934e10i q^{66} -9.09550e10i q^{67} +3.14175e11 q^{68} -9.49346e10 q^{69} +1.95401e11 q^{70} +1.43104e11i q^{71} +8.73945e10i q^{72} +2.69290e9i q^{73} -6.44411e10i q^{74} +5.50107e10 q^{75} +2.97726e11i q^{76} +4.75033e10 q^{77} -5.09321e11 q^{78} +9.84548e10i q^{79} -5.42477e11i q^{80} -3.38964e11 q^{81} +1.52309e11i q^{82} +(-2.91629e11 + 1.47792e11i) q^{83} -7.30416e11 q^{84} +5.99498e11i q^{85} -3.76965e11 q^{86} -5.59466e11 q^{87} -2.94133e11i q^{88} +1.50498e11i q^{89} -2.99531e11 q^{90} -5.13754e11i q^{91} -1.06500e12 q^{92} +7.18273e11 q^{93} +1.98558e12 q^{94} -5.68111e11 q^{95} +9.21962e11i q^{96} +6.29420e11i q^{97} +5.35113e11i q^{98} -7.28180e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9}+O(q^{10}) \) 80 * q + 918 * q^3 - 181150 * q^4 + 103918 * q^7 + 14902142 * q^9	\( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9} + 1177482 q^{10} + 510406 q^{11} - 9260402 q^{12} + 203791850 q^{16} + 5571718 q^{17} + 46565862 q^{21} + 804389464 q^{23} - 4263713272 q^{25} + 18061794 q^{26} - 2326165338 q^{27} - 204811652 q^{28} + 1486960270 q^{29} - 2621683648 q^{30} - 665743010 q^{31} + 135224502 q^{33} - 54936709824 q^{36} - 280627202 q^{37} + 6646145310 q^{38} - 7119722058 q^{40} + 51318109072 q^{41} - 11512674650 q^{44} + 85368259738 q^{48} + 148785395094 q^{49} - 100143389562 q^{51} - 63584241050 q^{59} - 29216180978 q^{61} - 332932206620 q^{63} - 323596534090 q^{64} + 362112989184 q^{65} - 86115426752 q^{68} + 272383417100 q^{69} + 105630718656 q^{70} + 785418808326 q^{75} - 663355117738 q^{77} + 1483841884620 q^{78} + 2430778545148 q^{81} + 837315119192 q^{83} - 3013574788354 q^{84} + 452180651958 q^{86} - 682263689498 q^{87} - 499714512022 q^{90} + 997428187414 q^{92} - 1487992716298 q^{93} + 3169817690580 q^{94} + 1542762610848 q^{95} + 2021347267420 q^{99}+O(q^{100}) \) 80 * q + 918 * q^3 - 181150 * q^4 + 103918 * q^7 + 14902142 * q^9 + 1177482 * q^10 + 510406 * q^11 - 9260402 * q^12 + 203791850 * q^16 + 5571718 * q^17 + 46565862 * q^21 + 804389464 * q^23 - 4263713272 * q^25 + 18061794 * q^26 - 2326165338 * q^27 - 204811652 * q^28 + 1486960270 * q^29 - 2621683648 * q^30 - 665743010 * q^31 + 135224502 * q^33 - 54936709824 * q^36 - 280627202 * q^37 + 6646145310 * q^38 - 7119722058 * q^40 + 51318109072 * q^41 - 11512674650 * q^44 + 85368259738 * q^48 + 148785395094 * q^49 - 100143389562 * q^51 - 63584241050 * q^59 - 29216180978 * q^61 - 332932206620 * q^63 - 323596534090 * q^64 + 362112989184 * q^65 - 86115426752 * q^68 + 272383417100 * q^69 + 105630718656 * q^70 + 785418808326 * q^75 - 663355117738 * q^77 + 1483841884620 * q^78 + 2430778545148 * q^81 + 837315119192 * q^83 - 3013574788354 * q^84 + 452180651958 * q^86 - 682263689498 * q^87 - 499714512022 * q^90 + 997428187414 * q^92 - 1487992716298 * q^93 + 3169817690580 * q^94 + 1542762610848 * q^95 + 2021347267420 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more