LMFDB - Embedded newform 83.13.b.c.82.8

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			82.8
Character	$\chi$	$=$	83.82
Dual form			83.13.b.c.82.73

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-108.230i q^{2} +117.748 q^{3} -7617.70 q^{4} +16840.2i q^{5} -12743.8i q^{6} -114224. q^{7} +381152. i q^{8} -517576. q^{9} +O(q^{10})$	\(q-108.230i q^{2} +117.748 q^{3} -7617.70 q^{4} +16840.2i q^{5} -12743.8i q^{6} -114224. q^{7} +381152. i q^{8} -517576. q^{9} +1.82261e6 q^{10} -1.55230e6 q^{11} -896968. q^{12} -453865. i q^{13} +1.23624e7i q^{14} +1.98290e6i q^{15} +1.00500e7 q^{16} -1.58366e7 q^{17} +5.60172e7i q^{18} +6.65276e7i q^{19} -1.28284e8i q^{20} -1.34496e7 q^{21} +1.68005e8i q^{22} +2.55021e8 q^{23} +4.48799e7i q^{24} -3.94517e7 q^{25} -4.91217e7 q^{26} -1.23520e8 q^{27} +8.70120e8 q^{28} -5.47904e8 q^{29} +2.14609e8 q^{30} -2.30920e8 q^{31} +4.73492e8i q^{32} -1.82780e8 q^{33} +1.71399e9i q^{34} -1.92355e9i q^{35} +3.94274e9 q^{36} -2.52628e9 q^{37} +7.20028e9 q^{38} -5.34416e7i q^{39} -6.41868e9 q^{40} +4.39427e9 q^{41} +1.45565e9i q^{42} -6.98402e9i q^{43} +1.18249e10 q^{44} -8.71609e9i q^{45} -2.76008e10i q^{46} -1.50846e10i q^{47} +1.18337e9 q^{48} -7.94274e8 q^{49} +4.26985e9i q^{50} -1.86473e9 q^{51} +3.45740e9i q^{52} -1.16138e10i q^{53} +1.33685e10i q^{54} -2.61410e10i q^{55} -4.35366e10i q^{56} +7.83350e9i q^{57} +5.92995e10i q^{58} -4.91742e10 q^{59} -1.51051e10i q^{60} +8.52508e10 q^{61} +2.49924e10i q^{62} +5.91194e10 q^{63} +9.24107e10 q^{64} +7.64317e9 q^{65} +1.97822e10i q^{66} -8.86092e10i q^{67} +1.20638e11 q^{68} +3.00281e10 q^{69} -2.08185e11 q^{70} +3.76443e10i q^{71} -1.97276e11i q^{72} +1.21908e11i q^{73} +2.73418e11i q^{74} -4.64536e9 q^{75} -5.06787e11i q^{76} +1.77309e11 q^{77} -5.78398e9 q^{78} +2.98888e10i q^{79} +1.69244e11i q^{80} +2.60517e11 q^{81} -4.75591e11i q^{82} +(-1.84283e11 + 2.70055e11i) q^{83} +1.02455e11 q^{84} -2.66692e11i q^{85} -7.55880e11 q^{86} -6.45146e10 q^{87} -5.91661e11i q^{88} +6.94583e11i q^{89} -9.43341e11 q^{90} +5.18420e10i q^{91} -1.94267e12 q^{92} -2.71904e10 q^{93} -1.63260e12 q^{94} -1.12034e12 q^{95} +5.57528e10i q^{96} -7.59774e11i q^{97} +8.59641e10i q^{98} +8.03432e11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	108.230i		−	1.69109i	−0.533903	−	0.845546i	$-0.679275\pi$
$2$		−	108.230i		−	1.69109i	0.533903	−	0.845546i	$-0.320725\pi$
$3$	117.748			0.161520			0.0807599	−	0.996734i	$-0.474265\pi$
$3$	117.748			0.161520			0.0807599	+	0.996734i	$0.474265\pi$
$4$	−7617.70			−1.85979
$5$			16840.2i			1.07777i	0.842378	+	0.538886i	$0.181155\pi$
$5$			16840.2i			1.07777i	−0.842378	+	0.538886i	$0.818845\pi$
$6$		−	12743.8i		−	0.273145i
$7$	−114224.			−0.970884			−0.485442	−	0.874269i	$-0.661341\pi$
$7$	−114224.			−0.970884			−0.485442	+	0.874269i	$0.661341\pi$
$8$			381152.i			1.45398i
$9$	−517576.			−0.973911
$10$	1.82261e6			1.82261
$11$	−1.55230e6			−0.876231			−0.438115	−	0.898919i	$-0.644354\pi$
$11$	−1.55230e6			−0.876231			−0.438115	+	0.898919i	$0.644354\pi$
$12$	−896968.			−0.300393
$13$		−	453865.i		−	0.0940299i	−0.998894	−	0.0470150i	$-0.985029\pi$
$13$		−	453865.i		−	0.0940299i	0.998894	−	0.0470150i	$-0.0149708\pi$
$14$			1.23624e7i			1.64185i
$15$			1.98290e6i			0.174082i
$16$	1.00500e7			0.599026
$17$	−1.58366e7			−0.656098			−0.328049	−	0.944661i	$-0.606391\pi$
$17$	−1.58366e7			−0.656098			−0.328049	+	0.944661i	$0.606391\pi$
$18$			5.60172e7i			1.64697i
$19$			6.65276e7i			1.41410i	0.707163	+	0.707051i	$0.249975\pi$
$19$			6.65276e7i			1.41410i	−0.707163	+	0.707051i	$0.750025\pi$
$20$		−	1.28284e8i		−	2.00443i
$21$	−1.34496e7			−0.156817
$22$			1.68005e8i			1.48179i
$23$	2.55021e8			1.72269			0.861347	−	0.508017i	$-0.169621\pi$
$23$	2.55021e8			1.72269			0.861347	+	0.508017i	$0.169621\pi$
$24$			4.48799e7i			0.234847i
$25$	−3.94517e7			−0.161594
$26$	−4.91217e7			−0.159013
$27$	−1.23520e8			−0.318826
$28$	8.70120e8			1.80564
$29$	−5.47904e8			−0.921120			−0.460560	−	0.887628i	$-0.652351\pi$
$29$	−5.47904e8			−0.921120			−0.460560	+	0.887628i	$0.652351\pi$
$30$	2.14609e8			0.294388
$31$	−2.30920e8			−0.260191			−0.130095	−	0.991501i	$-0.541528\pi$
$31$	−2.30920e8			−0.260191			−0.130095	+	0.991501i	$0.541528\pi$
$32$			4.73492e8i			0.440974i
$33$	−1.82780e8			−0.141529
$34$			1.71399e9i			1.10952i
$35$		−	1.92355e9i		−	1.04639i
$36$	3.94274e9			1.81127
$37$	−2.52628e9			−0.984624			−0.492312	−	0.870419i	$-0.663848\pi$
$37$	−2.52628e9			−0.984624			−0.492312	+	0.870419i	$0.663848\pi$
$38$	7.20028e9			2.39137
$39$		−	5.34416e7i		−	0.0151877i
$40$	−6.41868e9			−1.56706
$41$	4.39427e9			0.925090			0.462545	−	0.886596i	$-0.346936\pi$
$41$	4.39427e9			0.925090			0.462545	+	0.886596i	$0.346936\pi$
$42$			1.45565e9i			0.265192i
$43$		−	6.98402e9i		−	1.10483i	−0.833570	−	0.552414i	$-0.813707\pi$
$43$		−	6.98402e9i		−	1.10483i	0.833570	−	0.552414i	$-0.186293\pi$
$44$	1.18249e10			1.62960
$45$		−	8.71609e9i		−	1.04966i
$46$		−	2.76008e10i		−	2.91323i
$47$		−	1.50846e10i		−	1.39941i	−0.714431	−	0.699706i	$-0.753315\pi$
$47$		−	1.50846e10i		−	1.39941i	0.714431	−	0.699706i	$-0.246685\pi$
$48$	1.18337e9			0.0967546
$49$	−7.94274e8			−0.0573844
$50$			4.26985e9i			0.273270i
$51$	−1.86473e9			−0.105973
$52$			3.45740e9i			0.174876i
$53$		−	1.16138e10i		−	0.523987i	−0.965070	−	0.261993i	$-0.915620\pi$
$53$		−	1.16138e10i		−	0.523987i	0.965070	−	0.261993i	$-0.0843798\pi$
$54$			1.33685e10i			0.539164i
$55$		−	2.61410e10i		−	0.944377i
$56$		−	4.35366e10i		−	1.41165i
$57$			7.83350e9i			0.228405i
$58$			5.92995e10i			1.55770i
$59$	−4.91742e10			−1.16580			−0.582902	−	0.812542i	$-0.698083\pi$
$59$	−4.91742e10			−1.16580			−0.582902	+	0.812542i	$0.698083\pi$
$60$		−	1.51051e10i		−	0.323755i
$61$	8.52508e10			1.65470			0.827350	−	0.561686i	$-0.189847\pi$
$61$	8.52508e10			1.65470			0.827350	+	0.561686i	$0.189847\pi$
$62$			2.49924e10i			0.440006i
$63$	5.91194e10			0.945555
$64$	9.24107e10			1.34475
$65$	7.64317e9			0.101343
$66$			1.97822e10i			0.239338i
$67$		−	8.86092e10i		−	0.979558i	−0.871847	−	0.489779i	$-0.837077\pi$
$67$		−	8.86092e10i		−	0.979558i	0.871847	−	0.489779i	$-0.162923\pi$
$68$	1.20638e11			1.22020
$69$	3.00281e10			0.278249
$70$	−2.08185e11			−1.76954
$71$			3.76443e10i			0.293866i	0.989146	+	0.146933i	$0.0469401\pi$
$71$			3.76443e10i			0.293866i	−0.989146	+	0.146933i	$0.953060\pi$
$72$		−	1.97276e11i		−	1.41605i
$73$			1.21908e11i			0.805557i	0.915298	+	0.402778i	$0.131955\pi$
$73$			1.21908e11i			0.805557i	−0.915298	+	0.402778i	$0.868045\pi$
$74$			2.73418e11i			1.66509i
$75$	−4.64536e9			−0.0261007
$76$		−	5.06787e11i		−	2.62993i
$77$	1.77309e11			0.850718
$78$	−5.78398e9			−0.0256838
$79$			2.98888e10i			0.122955i	0.998108	+	0.0614775i	$0.0195813\pi$
$79$			2.98888e10i			0.122955i	−0.998108	+	0.0614775i	$0.980419\pi$
$80$			1.69244e11i			0.645614i
$81$	2.60517e11			0.922415
$82$		−	4.75591e11i		−	1.56441i
$83$	−1.84283e11	+	2.70055e11i	−0.563660	+	0.826007i
$83$	−1.84283e11	+	2.70055e11i	−0.563660	+	0.826007i
$84$	1.02455e11			0.291647
$85$		−	2.66692e11i		−	0.707124i
$86$	−7.55880e11			−1.86837
$87$	−6.45146e10			−0.148779
$88$		−	5.91661e11i		−	1.27402i
$89$			6.94583e11i			1.39760i	0.715315	+	0.698802i	$0.246283\pi$
$89$			6.94583e11i			1.39760i	−0.715315	+	0.698802i	$0.753717\pi$
$90$	−9.43341e11			−1.77506
$91$			5.18420e10i			0.0912921i
$92$	−1.94267e12			−3.20385
$93$	−2.71904e10			−0.0420259
$94$	−1.63260e12			−2.36653
$95$	−1.12034e12			−1.52408
$96$			5.57528e10i			0.0712261i
$97$		−	7.59774e11i		−	0.912124i	−0.889948	−	0.456062i	$-0.849260\pi$
$97$		−	7.59774e11i		−	0.912124i	0.889948	−	0.456062i	$-0.150740\pi$
$98$			8.59641e10i			0.0970423i
$99$	8.03432e11			0.853371
$100$	3.00531e11			0.300531
$101$		−	8.06823e11i		−	0.760064i	−0.924973	−	0.380032i	$-0.875913\pi$
$101$		−	8.06823e11i		−	0.760064i	0.924973	−	0.380032i	$-0.124087\pi$
$102$			2.01819e11i			0.179210i
$103$		−	1.31553e12i		−	1.10174i	−0.834592	−	0.550869i	$-0.814296\pi$
$103$		−	1.31553e12i		−	1.10174i	0.834592	−	0.550869i	$-0.185704\pi$
$104$	1.72992e11			0.136718
$105$		−	2.26494e11i		−	0.169013i
$106$	−1.25696e12			−0.886109
$107$		−	1.22395e12i		−	0.815572i	−0.913078	−	0.407786i	$-0.866301\pi$
$107$		−	1.22395e12i		−	0.815572i	0.913078	−	0.407786i	$-0.133699\pi$
$108$	9.40935e11			0.592949
$109$	2.12246e12			1.26556			0.632778	−	0.774333i	$-0.281915\pi$
$109$	2.12246e12			1.26556			0.632778	+	0.774333i	$0.281915\pi$
$110$	−2.82923e12			−1.59703
$111$	−2.97464e11			−0.159036
$112$	−1.14794e12			−0.581585
$113$	−1.39723e12			−0.671114			−0.335557	−	0.942020i	$-0.608925\pi$
$113$	−1.39723e12			−0.671114			−0.335557	+	0.942020i	$0.608925\pi$
$114$	8.47818e11			0.386254
$115$			4.29460e12i			1.85667i
$116$	4.17376e12			1.71309
$117$			2.34910e11i			0.0915768i
$118$			5.32212e12i			1.97148i
$119$	1.80891e12			0.636995
$120$	−7.55787e11			−0.253112
$121$	−7.28806e11			−0.232220
$122$		−	9.22668e12i		−	2.79825i
$123$	5.17417e11			0.149420
$124$	1.75908e12			0.483899
$125$			3.44700e12i			0.903611i
$126$		−	6.39848e12i		−	1.59902i
$127$	−2.72169e12			−0.648660			−0.324330	−	0.945944i	$-0.605139\pi$
$127$	−2.72169e12			−0.648660			−0.324330	+	0.945944i	$0.605139\pi$
$128$		−	8.06217e12i		−	1.83313i
$129$		−	8.22354e11i		−	0.178452i
$130$		−	8.27219e11i		−	0.171380i
$131$	5.96140e12			1.17956			0.589781	−	0.807564i	$-0.299214\pi$
$131$	5.96140e12			1.17956			0.589781	+	0.807564i	$0.299214\pi$
$132$	1.39236e12			0.263213
$133$		−	7.59902e12i		−	1.37293i
$134$	−9.59016e12			−1.65652
$135$		−	2.08010e12i		−	0.343622i
$136$		−	6.03616e12i		−	0.953954i
$137$			4.24473e12i			0.641988i	0.947081	+	0.320994i	$0.104017\pi$
$137$			4.24473e12i			0.641988i	−0.947081	+	0.320994i	$0.895983\pi$
$138$		−	3.24994e12i		−	0.470545i
$139$			2.72921e12i			0.378397i	0.981939	+	0.189199i	$0.0605890\pi$
$139$			2.72921e12i			0.378397i	−0.981939	+	0.189199i	$0.939411\pi$
$140$			1.46530e13i			1.94607i
$141$		−	1.77618e12i		−	0.226033i
$142$	4.07423e12			0.496954
$143$			7.04532e11i			0.0823919i
$144$	−5.20164e12			−0.583398
$145$		−	9.22681e12i		−	0.992758i
$146$	1.31941e13			1.36227
$147$	−9.35242e10			−0.00926872
$148$	1.92444e13			1.83119
$149$		−	7.66630e12i		−	0.700597i	−0.936638	−	0.350299i	$-0.886080\pi$
$149$		−	7.66630e12i		−	0.700597i	0.936638	−	0.350299i	$-0.113920\pi$
$150$			5.02766e11i			0.0441386i
$151$	1.03194e13			0.870552			0.435276	−	0.900297i	$-0.356651\pi$
$151$	1.03194e13			0.870552			0.435276	+	0.900297i	$0.356651\pi$
$152$	−2.53572e13			−2.05608
$153$	8.19666e12			0.638981
$154$		−	1.91901e13i		−	1.43864i
$155$		−	3.88874e12i		−	0.280426i
$156$			4.07102e11i			0.0282459i
$157$			2.54931e13i			1.70225i	0.524959	+	0.851127i	$0.324080\pi$
$157$			2.54931e13i			1.70225i	−0.524959	+	0.851127i	$0.675920\pi$
$158$	3.23486e12			0.207928
$159$		−	1.36750e12i		−	0.0846342i
$160$	−7.97371e12			−0.475270
$161$	−2.91293e13			−1.67254
$162$		−	2.81957e13i		−	1.55989i
$163$			8.67153e12i			0.462349i	0.972912	+	0.231175i	$0.0742568\pi$
$163$			8.67153e12i			0.462349i	−0.972912	+	0.231175i	$0.925743\pi$
$164$	−3.34742e13			−1.72047
$165$		−	3.07805e12i		−	0.152536i
$166$	2.92280e13	+	1.99449e13i	1.39685	+	0.953200i
$167$	1.60294e13			0.738956			0.369478	−	0.929239i	$-0.379537\pi$
$167$	1.60294e13			0.738956			0.369478	+	0.929239i	$0.379537\pi$
$168$		−	5.12634e12i		−	0.228009i
$169$	2.30921e13			0.991158
$170$	−2.88640e13			−1.19581
$171$		−	3.44331e13i		−	1.37721i
$172$			5.32022e13i			2.05475i
$173$	4.75171e12			0.177245			0.0886224	−	0.996065i	$-0.471754\pi$
$173$	4.75171e12			0.177245			0.0886224	+	0.996065i	$0.471754\pi$
$174$			6.98240e12i			0.251599i
$175$	4.50631e12			0.156889
$176$	−1.56006e13			−0.524885
$177$	−5.79017e12			−0.188300
$178$	7.51746e13			2.36348
$179$			2.96281e13i			0.900713i	0.892849	+	0.450356i	$0.148703\pi$
$179$			2.96281e13i			0.900713i	−0.892849	+	0.450356i	$0.851297\pi$
$180$			6.63965e13i			1.95214i
$181$			4.41774e12i			0.125640i	0.998025	+	0.0628200i	$0.0200094\pi$
$181$			4.41774e12i			0.125640i	−0.998025	+	0.0628200i	$0.979991\pi$
$182$	5.61085e12			0.154383
$183$	1.00381e13			0.267267
$184$			9.72017e13i			2.50476i
$185$		−	4.25430e13i		−	1.06120i
$186$			2.94281e12i			0.0710697i
$187$	2.45831e13			0.574893
$188$			1.14910e14i			2.60261i
$189$	1.41089e13			0.309543
$190$			1.21254e14i			2.57736i
$191$	−4.46277e13			−0.919187			−0.459593	−	0.888129i	$-0.652005\pi$
$191$	−4.46277e13			−0.919187			−0.459593	+	0.888129i	$0.652005\pi$
$192$	1.08812e13			0.217204
$193$	−7.08703e13			−1.37126			−0.685631	−	0.727949i	$-0.740474\pi$
$193$	−7.08703e13			−1.37126			−0.685631	+	0.727949i	$0.740474\pi$
$194$	−8.22302e13			−1.54248
$195$	8.99968e11			0.0163689
$196$	6.05054e12			0.106723
$197$	8.03406e13			1.37448			0.687239	−	0.726431i	$-0.258823\pi$
$197$	8.03406e13			1.37448			0.687239	+	0.726431i	$0.258823\pi$
$198$		−	8.69553e13i		−	1.44313i
$199$	8.83256e13			1.42222			0.711112	−	0.703078i	$-0.248192\pi$
$199$	8.83256e13			1.42222			0.711112	+	0.703078i	$0.248192\pi$
$200$		−	1.50371e13i		−	0.234955i
$201$		−	1.04336e13i		−	0.158218i
$202$	−8.73223e13			−1.28534
$203$	6.25835e13			0.894301
$204$	1.42049e13			0.197087
$205$			7.40004e13i			0.997037i
$206$	−1.42380e14			−1.86314
$207$	−1.31993e14			−1.67775
$208$		−	4.56133e12i		−	0.0563264i
$209$		−	1.03271e14i		−	1.23908i
$210$	−2.45134e13			−0.285817
$211$		−	1.45164e14i		−	1.64500i	−0.568766	−	0.822500i	$-0.692579\pi$
$211$		−	1.45164e14i		−	1.64500i	0.568766	−	0.822500i	$-0.307421\pi$
$212$			8.84706e13i			0.974504i
$213$			4.43254e12i			0.0474652i
$214$	−1.32468e14			−1.37921
$215$	1.17612e14			1.19075
$216$		−	4.70798e13i		−	0.463567i
$217$	2.63765e13			0.252615
$218$		−	2.29714e14i		−	2.14017i
$219$			1.43545e13i			0.130113i
$220$			1.99134e14i			1.75634i
$221$			7.18767e12i			0.0616928i
$222$			3.21945e13i			0.268945i
$223$		−	6.20732e13i		−	0.504749i	−0.967630	−	0.252374i	$-0.918789\pi$
$223$		−	6.20732e13i		−	0.504749i	0.967630	−	0.252374i	$-0.0812114\pi$
$224$		−	5.40840e13i		−	0.428135i
$225$	2.04193e13			0.157378
$226$			1.51222e14i			1.13492i
$227$	−2.19417e13			−0.160367			−0.0801836	−	0.996780i	$-0.525551\pi$
$227$	−2.19417e13			−0.160367			−0.0801836	+	0.996780i	$0.525551\pi$
$228$		−	5.96732e13i		−	0.424786i
$229$	2.63340e14			1.82601			0.913006	−	0.407946i	$-0.133755\pi$
$229$	2.63340e14			1.82601			0.913006	+	0.407946i	$0.133755\pi$
$230$	4.64803e14			3.13980
$231$	2.08777e13			0.137408
$232$		−	2.08835e14i		−	1.33929i
$233$			6.61821e13i			0.413623i	0.978381	+	0.206812i	$0.0663087\pi$
$233$			6.61821e13i			0.413623i	−0.978381	+	0.206812i	$0.933691\pi$
$234$	2.54242e13			0.154865
$235$	2.54027e14			1.50825
$236$	3.74594e14			2.16815
$237$			3.51935e12i			0.0198597i
$238$		−	1.95778e14i		−	1.07722i
$239$			8.82557e12i			0.0473538i	0.999720	+	0.0236769i	$0.00753730\pi$
$239$			8.82557e12i			0.0473538i	−0.999720	+	0.0236769i	$0.992463\pi$
$240$			1.99281e13i			0.104279i
$241$	2.02269e14			1.03235			0.516174	−	0.856484i	$-0.327356\pi$
$241$	2.02269e14			1.03235			0.516174	+	0.856484i	$0.327356\pi$
$242$			7.88785e13i			0.392705i
$243$	9.63188e13			0.467814
$244$	−6.49415e14			−3.07739
$245$		−	1.33757e13i		−	0.0618474i
$246$		−	5.59999e13i		−	0.252683i
$247$	3.01945e13			0.132968
$248$		−	8.80157e13i		−	0.378312i
$249$	−2.16990e13	+	3.17984e13i	−0.0910422	+	0.133417i
$250$	3.73068e14			1.52809
$251$			2.44959e14i			0.979607i	0.871833	+	0.489803i	$0.162931\pi$
$251$			2.44959e14i			0.979607i	−0.871833	+	0.489803i	$0.837069\pi$
$252$	−4.50354e14			−1.75853
$253$	−3.95867e14			−1.50948
$254$			2.94569e14i			1.09694i
$255$		−	3.14024e13i		−	0.114215i
$256$	−4.94053e14			−1.75523
$257$		−	2.71294e14i		−	0.941547i	−0.882254	−	0.470773i	$-0.843975\pi$
$257$		−	2.71294e14i		−	0.941547i	0.882254	−	0.470773i	$-0.156025\pi$
$258$	−8.90033e13			−0.301778
$259$	2.88560e14			0.955955
$260$	−5.82233e13			−0.188476
$261$	2.83582e14			0.897090
$262$		−	6.45201e14i		−	1.99475i
$263$		−	3.60328e14i		−	1.08884i	−0.838813	−	0.544419i	$-0.816750\pi$
$263$		−	3.60328e14i		−	1.08884i	0.838813	−	0.544419i	$-0.183250\pi$
$264$		−	6.96669e13i		−	0.205780i
$265$	1.95579e14			0.564739
$266$	−8.22441e14			−2.32175
$267$			8.17857e13i			0.225741i
$268$			6.74998e14i			1.82177i
$269$		−	4.79450e14i		−	1.26540i	−0.774395	−	0.632702i	$-0.781946\pi$
$269$		−	4.79450e14i		−	1.26540i	0.774395	−	0.632702i	$-0.218054\pi$
$270$	−2.25128e14			−0.581096
$271$			2.76761e13i			0.0698698i	0.999390	+	0.0349349i	$0.0111224\pi$
$271$			2.76761e13i			0.0698698i	−0.999390	+	0.0349349i	$0.988878\pi$
$272$	−1.59158e14			−0.393020
$273$			6.10429e12i			0.0147455i
$274$	4.59407e14			1.08566
$275$	6.12407e13			0.141594
$276$	−2.28745e14			−0.517485
$277$	−6.12760e14			−1.35647			−0.678237	−	0.734843i	$-0.737256\pi$
$277$	−6.12760e14			−1.35647			−0.678237	+	0.734843i	$0.737256\pi$
$278$	2.95382e14			0.639904
$279$	1.19519e14			0.253403
$280$	7.33165e14			1.52143
$281$			1.23440e14i			0.250736i	0.992110	+	0.125368i	$0.0400111\pi$
$281$			1.23440e14i			0.250736i	−0.992110	+	0.125368i	$0.959989\pi$
$282$	−1.92235e14			−0.382242
$283$			6.62958e14i			1.29053i	0.763959	+	0.645264i	$0.223253\pi$
$283$			6.62958e14i			1.29053i	−0.763959	+	0.645264i	$0.776747\pi$
$284$		−	2.86763e14i		−	0.546528i
$285$	−1.31918e14			−0.246169
$286$	7.62514e13			0.139332
$287$	−5.01929e14			−0.898155
$288$		−	2.45068e14i		−	0.429470i
$289$	−3.31824e14			−0.569536
$290$	−9.98616e14			−1.67884
$291$		−	8.94618e13i		−	0.147326i
$292$		−	9.28660e14i		−	1.49817i
$293$	−7.18472e14			−1.13554			−0.567772	−	0.823186i	$-0.692195\pi$
$293$	−7.18472e14			−1.13554			−0.567772	+	0.823186i	$0.692195\pi$
$294$			1.01221e13i			0.0156743i
$295$		−	8.28104e14i		−	1.25647i
$296$		−	9.62896e14i		−	1.43162i
$297$	1.91739e14			0.279365
$298$	−8.29723e14			−1.18477
$299$		−	1.15745e14i		−	0.161985i
$300$	3.53869e13			0.0485417
$301$			7.97740e14i			1.07266i
$302$		−	1.11687e15i		−	1.47218i
$303$		−	9.50018e13i		−	0.122765i
$304$			6.68602e14i			0.847083i
$305$			1.43564e15i			1.78339i
$306$		−	8.87123e14i		−	1.08058i
$307$			6.39261e14i			0.763569i	0.924251	+	0.381784i	$0.124690\pi$
$307$			6.39261e14i			0.763569i	−0.924251	+	0.381784i	$0.875310\pi$
$308$	−1.35068e15			−1.58216
$309$		−	1.54901e14i		−	0.177953i
$310$	−4.20878e14			−0.474226
$311$		−	5.98708e14i		−	0.661688i	−0.943686	−	0.330844i	$-0.892667\pi$
$311$		−	5.98708e14i		−	0.661688i	0.943686	−	0.330844i	$-0.107333\pi$
$312$	2.03694e13			0.0220826
$313$	−6.55936e13			−0.0697583			−0.0348791	−	0.999392i	$-0.511105\pi$
$313$	−6.55936e13			−0.0697583			−0.0348791	+	0.999392i	$0.511105\pi$
$314$	2.75911e15			2.87867
$315$			9.95583e14i			1.01909i
$316$		−	2.27684e14i		−	0.228670i
$317$	−1.37166e15			−1.35174			−0.675868	−	0.737022i	$-0.736231\pi$
$317$	−1.37166e15			−1.35174			−0.675868	+	0.737022i	$0.736231\pi$
$318$	−1.48005e14			−0.143124
$319$	8.50509e14			0.807114
$320$			1.55622e15i			1.44934i
$321$		−	1.44118e14i		−	0.131731i
$322$			3.15266e15i			2.82841i
$323$		−	1.05357e15i		−	0.927789i
$324$	−1.98454e15			−1.71550
$325$			1.79057e13i			0.0151947i
$326$	9.38518e14			0.781874
$327$	2.49916e14			0.204412
$328$			1.67489e15i			1.34506i
$329$			1.72301e15i			1.35867i
$330$	−3.33136e14			−0.257952
$331$			6.39829e14i			0.486515i	0.969962	+	0.243257i	$0.0782160\pi$
$331$			6.39829e14i			0.486515i	−0.969962	+	0.243257i	$0.921784\pi$
$332$	1.40381e15	−	2.05720e15i	1.04829	−	1.53620i
$333$	1.30754e15			0.958936
$334$		−	1.73486e15i		−	1.24964i
$335$	1.49220e15			1.05574
$336$	−1.35168e14			−0.0939374
$337$		−	3.41335e14i		−	0.233024i	−0.993189	−	0.116512i	$-0.962829\pi$
$337$		−	3.41335e14i		−	0.233024i	0.993189	−	0.116512i	$-0.0371714\pi$
$338$		−	2.49925e15i		−	1.67614i
$339$	−1.64521e14			−0.108398
$340$			2.03158e15i			1.31510i
$341$	3.58456e14			0.227987
$342$	−3.72669e15			−2.32899
$343$	1.67173e15			1.02660
$344$	2.66198e15			1.60640
$345$			5.05680e14i			0.299889i
$346$		−	5.14277e14i		−	0.299737i
$347$			2.85705e15i			1.63659i	0.574796	+	0.818297i	$0.305081\pi$
$347$			2.85705e15i			1.63659i	−0.574796	+	0.818297i	$0.694919\pi$
$348$	4.91452e14			0.276698
$349$	2.33430e15			1.29182			0.645912	−	0.763412i	$-0.276477\pi$
$349$	2.33430e15			1.29182			0.645912	+	0.763412i	$0.276477\pi$
$350$		−	4.87717e14i		−	0.265314i
$351$			5.60612e13i			0.0299792i
$352$		−	7.35000e14i		−	0.386395i
$353$	−5.94747e13			−0.0307386			−0.0153693	−	0.999882i	$-0.504892\pi$
$353$	−5.94747e13			−0.0307386			−0.0153693	+	0.999882i	$0.504892\pi$
$354$			6.26669e14i			0.318433i
$355$	−6.33937e14			−0.316721
$356$		−	5.29112e15i		−	2.59925i
$357$	2.12996e14			0.102887
$358$	3.20665e15			1.52319
$359$	3.07499e15			1.43640			0.718202	−	0.695835i	$-0.244965\pi$
$359$	3.07499e15			1.43640			0.718202	+	0.695835i	$0.244965\pi$
$360$	3.32216e15			1.52618
$361$	−2.21261e15			−0.999683
$362$	4.78131e14			0.212469
$363$	−8.58154e13			−0.0375081
$364$		−	3.94917e14i		−	0.169784i
$365$	−2.05296e15			−0.868207
$366$		−	1.08642e15i		−	0.451973i
$367$		−	2.08440e15i		−	0.853068i	−0.904471	−	0.426534i	$-0.859734\pi$
$367$		−	2.08440e15i		−	0.853068i	0.904471	−	0.426534i	$-0.140266\pi$
$368$	2.56295e15			1.03194
$369$	−2.27437e15			−0.900955
$370$	−4.60442e15			−1.79459
$371$			1.32657e15i			0.508730i
$372$	2.07128e14			0.0781594
$373$	−2.43918e15			−0.905714			−0.452857	−	0.891583i	$-0.649595\pi$
$373$	−2.43918e15			−0.905714			−0.452857	+	0.891583i	$0.649595\pi$
$374$		−	2.66062e15i		−	0.972196i
$375$			4.05878e14i			0.145951i
$376$	5.74952e15			2.03472
$377$			2.48674e14i			0.0866129i
$378$		−	1.52700e15i		−	0.523465i
$379$			1.30116e15i			0.439032i	0.975609	+	0.219516i	$0.0704478\pi$
$379$			1.30116e15i			0.439032i	−0.975609	+	0.219516i	$0.929552\pi$
$380$	8.53440e15			2.83447
$381$	−3.20474e14			−0.104771
$382$			4.83004e15i			1.55443i
$383$	−1.32809e15			−0.420759			−0.210380	−	0.977620i	$-0.567470\pi$
$383$	−1.32809e15			−0.420759			−0.210380	+	0.977620i	$0.567470\pi$
$384$		−	9.49304e14i		−	0.296086i
$385$			2.98591e15i			0.916881i
$386$			7.67028e15i			2.31893i
$387$			3.61477e15i			1.07601i
$388$			5.78772e15i			1.69636i
$389$		−	1.45438e15i		−	0.419741i	−0.977729	−	0.209870i	$-0.932696\pi$
$389$		−	1.45438e15i		−	0.419741i	0.977729	−	0.209870i	$-0.0673042\pi$
$390$		−	9.74033e13i		−	0.0276813i
$391$	−4.03866e15			−1.13026
$392$		−	3.02740e14i		−	0.0834359i
$393$	7.01943e14			0.190523
$394$		−	8.69525e15i		−	2.32437i
$395$	−5.03334e14			−0.132518
$396$	−6.12030e15			−1.58709
$397$	−3.33429e15			−0.851649			−0.425825	−	0.904806i	$-0.640016\pi$
$397$	−3.33429e15			−0.851649			−0.425825	+	0.904806i	$0.640016\pi$
$398$		−	9.55947e15i		−	2.40511i
$399$		−	8.94769e14i		−	0.221755i
$400$	−3.96489e14			−0.0967991
$401$	4.06761e15			0.978302			0.489151	−	0.872199i	$-0.337307\pi$
$401$	4.06761e15			0.978302			0.489151	+	0.872199i	$0.337307\pi$
$402$	−1.12922e15			−0.267561
$403$			1.04806e14i			0.0244657i
$404$			6.14613e15i			1.41356i
$405$			4.38716e15i			0.994153i
$406$		−	6.77340e15i		−	1.51234i
$407$	3.92153e15			0.862758
$408$		−	7.10746e14i		−	0.154083i
$409$	−3.47911e14			−0.0743238			−0.0371619	−	0.999309i	$-0.511832\pi$
$409$	−3.47911e14			−0.0743238			−0.0371619	+	0.999309i	$0.511832\pi$
$410$	8.00905e15			1.68608
$411$			4.99809e14i			0.103694i
$412$			1.00213e16i			2.04900i
$413$	5.61685e15			1.13186
$414$			1.42855e16i			2.83723i
$415$	−4.54778e15	−	3.10336e15i	−0.890248	−	0.607497i
$416$	2.14901e14			0.0414648
$417$			3.21359e14i			0.0611187i
$418$	−1.11770e16			−2.09540
$419$	3.59696e15			0.664740			0.332370	−	0.943149i	$-0.392152\pi$
$419$	3.59696e15			0.664740			0.332370	+	0.943149i	$0.392152\pi$
$420$			1.72536e15i			0.314329i
$421$		−	7.98998e15i		−	1.43500i	−0.696556	−	0.717502i	$-0.745285\pi$
$421$		−	7.98998e15i		−	1.43500i	0.696556	−	0.717502i	$-0.254715\pi$
$422$	−1.57111e16			−2.78184
$423$			7.80741e15i			1.36290i
$424$	4.42664e15			0.761867
$425$	6.24781e14			0.106022
$426$	4.79733e14			0.0802679
$427$	−9.73765e15			−1.60652
$428$			9.32370e15i			1.51679i
$429$			8.29572e13i			0.0133079i
$430$		−	1.27292e16i		−	2.01367i
$431$	4.58333e15			0.715019			0.357510	−	0.933909i	$-0.383626\pi$
$431$	4.58333e15			0.715019			0.357510	+	0.933909i	$0.383626\pi$
$432$	−1.24137e15			−0.190985
$433$			2.84170e15i			0.431173i	0.976485	+	0.215586i	$0.0691663\pi$
$433$			2.84170e15i			0.431173i	−0.976485	+	0.215586i	$0.930834\pi$
$434$		−	2.85472e15i		−	0.427195i
$435$		−	1.08644e15i		−	0.160350i
$436$	−1.61683e16			−2.35367
$437$			1.69659e16i			2.43606i
$438$	1.55358e15			0.220034
$439$			4.11386e15i			0.574728i	0.957821	+	0.287364i	$0.0927790\pi$
$439$			4.11386e15i			0.574728i	−0.957821	+	0.287364i	$0.907221\pi$
$440$	9.96370e15			1.37311
$441$	4.11098e14			0.0558873
$442$	7.77921e14			0.104328
$443$	1.44646e16			1.91374			0.956872	−	0.290509i	$-0.0938246\pi$
$443$	1.44646e16			1.91374			0.956872	+	0.290509i	$0.0938246\pi$
$444$	2.26599e15			0.295774
$445$	−1.16969e16			−1.50630
$446$	−6.71817e15			−0.853576
$447$		−	9.02692e14i		−	0.113160i
$448$	−1.05555e16			−1.30560
$449$			1.18579e16i			1.44721i	0.690216	+	0.723604i	$0.257516\pi$
$449$			1.18579e16i			1.44721i	−0.690216	+	0.723604i	$0.742484\pi$
$450$		−	2.20997e15i		−	0.266141i
$451$	−6.82121e15			−0.810592
$452$	1.06437e16			1.24813
$453$	1.21509e15			0.140611
$454$			2.37475e15i			0.271195i
$455$	−8.73030e14			−0.0983922
$456$	−2.98576e15			−0.332097
$457$			2.92103e15i			0.320655i	0.987064	+	0.160328i	$0.0512551\pi$
$457$			2.92103e15i			0.320655i	−0.987064	+	0.160328i	$0.948745\pi$
$458$		−	2.85012e16i		−	3.08795i
$459$	1.95613e15			0.209181
$460$		−	3.27149e16i		−	3.45302i
$461$		−	1.25856e16i		−	1.31120i	−0.755110	−	0.655598i	$-0.772416\pi$
$461$		−	1.25856e16i		−	1.31120i	0.755110	−	0.655598i	$-0.227584\pi$
$462$		−	2.25959e15i		−	0.232369i
$463$	−2.39831e15			−0.243456			−0.121728	−	0.992564i	$-0.538843\pi$
$463$	−2.39831e15			−0.243456			−0.121728	+	0.992564i	$0.538843\pi$
$464$	−5.50643e15			−0.551775
$465$		−	4.57891e14i		−	0.0452944i
$466$	7.16287e15			0.699474
$467$		−	1.82357e16i		−	1.75801i	−0.476810	−	0.879006i	$-0.658207\pi$
$467$		−	1.82357e16i		−	1.75801i	0.476810	−	0.879006i	$-0.341793\pi$
$468$		−	1.78947e15i		−	0.170314i
$469$			1.01213e16i			0.951037i
$470$		−	2.74933e16i		−	2.55058i
$471$			3.00176e15i			0.274948i
$472$		−	1.87429e16i		−	1.69506i
$473$			1.08413e16i			0.968085i
$474$	3.80899e14			0.0335845
$475$		−	2.62463e15i		−	0.228511i
$476$	−1.37797e16			−1.18468
$477$			6.01104e15i			0.510316i
$478$	9.55190e14			0.0800797
$479$	−8.20174e15			−0.679036			−0.339518	−	0.940599i	$-0.610264\pi$
$479$	−8.20174e15			−0.679036			−0.339518	+	0.940599i	$0.610264\pi$
$480$	−9.38888e14			−0.0767655
$481$			1.14659e15i			0.0925841i
$482$		−	2.18915e16i		−	1.74579i
$483$	−3.42992e15			−0.270148
$484$	5.55182e15			0.431880
$485$	1.27947e16			0.983062
$486$		−	1.04246e16i		−	0.791116i
$487$		−	1.40709e16i		−	1.05474i	−0.849635	−	0.527371i	$-0.823178\pi$
$487$		−	1.40709e16i		−	1.05474i	0.849635	−	0.527371i	$-0.176822\pi$
$488$			3.24936e16i			2.40590i
$489$			1.02105e15i			0.0746786i
$490$	−1.44765e15			−0.104590
$491$		−	8.22183e15i		−	0.586785i	−0.955992	−	0.293393i	$-0.905216\pi$
$491$		−	8.22183e15i		−	0.586785i	0.955992	−	0.293393i	$-0.0947844\pi$
$492$	−3.94152e15			−0.277890
$493$	8.67694e15			0.604345
$494$		−	3.26795e15i		−	0.224861i
$495$			1.35300e16i			0.919740i
$496$	−2.32074e15			−0.155861
$497$		−	4.29986e15i		−	0.285310i
$498$	3.44154e15	+	2.34847e15i	0.225620	+	0.153961i
$499$	2.92191e15			0.189262			0.0946309	−	0.995512i	$-0.469833\pi$
$499$	2.92191e15			0.189262			0.0946309	+	0.995512i	$0.469833\pi$
$500$		−	2.62582e16i		−	1.68053i
$501$	1.88743e15			0.119356
$502$	2.65119e16			1.65660
$503$			1.24826e16i			0.770723i	0.922766	+	0.385361i	$0.125923\pi$
$503$			1.24826e16i			0.770723i	−0.922766	+	0.385361i	$0.874077\pi$
$504$			2.25335e16i			1.37482i
$505$	1.35871e16			0.819176
$506$			4.28447e16i			2.55266i
$507$	2.71905e15			0.160092
$508$	2.07330e16			1.20637
$509$	6.94480e15			0.399350			0.199675	−	0.979862i	$-0.436011\pi$
$509$	6.94480e15			0.399350			0.199675	+	0.979862i	$0.436011\pi$
$510$	−3.39868e15			−0.193147
$511$		−	1.39248e16i		−	0.782102i
$512$			2.04486e16i			1.13513i
$513$		−	8.21747e15i		−	0.450852i
$514$	−2.93621e16			−1.59224
$515$	2.21538e16			1.18742
$516$			6.26445e15i			0.331883i
$517$			2.34157e16i			1.22621i
$518$		−	3.12308e16i		−	1.61661i
$519$	5.59505e14			0.0286286
$520$			2.91321e15i			0.147351i
$521$	1.26141e16			0.630712			0.315356	−	0.948973i	$-0.397876\pi$
$521$	1.26141e16			0.630712			0.315356	+	0.948973i	$0.397876\pi$
$522$		−	3.06920e16i		−	1.51706i
$523$	−1.87510e16			−0.916251			−0.458125	−	0.888888i	$-0.651479\pi$
$523$	−1.87510e16			−0.916251			−0.458125	+	0.888888i	$0.651479\pi$
$524$	−4.54121e16			−2.19373
$525$	5.30609e14			0.0253407
$526$	−3.89982e16			−1.84133
$527$	3.65699e15			0.170710
$528$	−1.83693e15			−0.0847793
$529$	4.31208e16			1.96767
$530$		−	2.11675e16i		−	0.955024i
$531$	2.54514e16			1.13539
$532$			5.78870e16i			2.55336i
$533$		−	1.99440e15i		−	0.0869861i
$534$	8.85166e15			0.381748
$535$	2.06116e16			0.879001
$536$	3.37736e16			1.42426
$537$			3.48865e15i			0.145483i
$538$	−5.18908e16			−2.13991
$539$	1.23295e15			0.0502820
$540$			1.58455e16i			0.639064i
$541$		−	2.99318e16i		−	1.19385i	−0.802298	−	0.596924i	$-0.796389\pi$
$541$		−	2.99318e16i		−	1.19385i	0.802298	−	0.596924i	$-0.203611\pi$
$542$	2.99538e15			0.118156
$543$			5.20179e14i			0.0202934i
$544$		−	7.49851e15i		−	0.289322i
$545$			3.57427e16i			1.36398i
$546$	6.60666e14			0.0249360
$547$	−1.87124e16			−0.698562			−0.349281	−	0.937018i	$-0.613574\pi$
$547$	−1.87124e16			−0.698562			−0.349281	+	0.937018i	$0.613574\pi$
$548$		−	3.23351e16i		−	1.19396i
$549$	−4.41238e16			−1.61153
$550$		−	6.62807e15i		−	0.239448i
$551$		−	3.64508e16i		−	1.30256i
$552$			1.14453e16i			0.404569i
$553$		−	3.41401e15i		−	0.119375i
$554$			6.63189e16i			2.29392i
$555$		−	5.00935e15i		−	0.171405i
$556$		−	2.07903e16i		−	0.703739i
$557$	4.09178e16			1.37019			0.685096	−	0.728453i	$-0.259760\pi$
$557$	4.09178e16			1.37019			0.685096	+	0.728453i	$0.259760\pi$
$558$		−	1.29355e16i		−	0.428527i
$559$	−3.16980e15			−0.103887
$560$		−	1.93316e16i		−	0.626816i
$561$	2.89461e15			0.0928566
$562$	1.33598e16			0.424017
$563$	−3.00625e16			−0.944005			−0.472002	−	0.881597i	$-0.656469\pi$
$563$	−3.00625e16			−0.944005			−0.472002	+	0.881597i	$0.656469\pi$
$564$			1.35304e16i			0.420373i
$565$		−	2.35296e16i		−	0.723309i
$566$	7.17519e16			2.18240
$567$	−2.97572e16			−0.895558
$568$	−1.43482e16			−0.427275
$569$		−	9.90392e15i		−	0.291832i	−0.989297	−	0.145916i	$-0.953387\pi$
$569$		−	9.90392e15i		−	0.291832i	0.989297	−	0.145916i	$-0.0466130\pi$
$570$			1.42774e16i			0.416294i
$571$			4.93668e16i			1.42435i	0.702000	+	0.712177i	$0.252291\pi$
$571$			4.93668e16i			1.42435i	−0.702000	+	0.712177i	$0.747709\pi$
$572$		−	5.36691e15i		−	0.153232i
$573$	−5.25482e15			−0.148467
$574$			5.43237e16i			1.51886i
$575$	−1.00610e16			−0.278377
$576$	−4.78296e16			−1.30967
$577$		−	4.70900e16i		−	1.27607i	−0.770009	−	0.638033i	$-0.779748\pi$
$577$		−	4.70900e16i		−	1.27607i	0.770009	−	0.638033i	$-0.220252\pi$
$578$			3.59133e16i			0.963136i
$579$	−8.34483e15			−0.221486
$580$			7.02870e16i			1.84632i
$581$	2.10495e16	−	3.08466e16i	0.547248	−	0.801957i
$582$	−9.68244e15			−0.249142
$583$			1.80281e16i			0.459133i
$584$	−4.64657e16			−1.17126
$585$	−3.95592e15			−0.0986990
$586$			7.77601e16i			1.92031i
$587$		−	6.42550e16i		−	1.57065i	−0.619086	−	0.785323i	$-0.712497\pi$
$587$		−	6.42550e16i		−	1.57065i	0.619086	−	0.785323i	$-0.287503\pi$
$588$	7.12439e14			0.0172379
$589$		−	1.53626e16i		−	0.367936i
$590$	−8.96256e16			−2.12481
$591$	9.45994e15			0.222005
$592$	−2.53890e16			−0.589815
$593$	−6.56769e15			−0.151037			−0.0755187	−	0.997144i	$-0.524061\pi$
$593$	−6.56769e15			−0.151037			−0.0755187	+	0.997144i	$0.524061\pi$
$594$		−	2.07519e16i		−	0.472432i
$595$			3.04625e16i			0.686536i
$596$			5.83996e16i			1.30296i
$597$	1.04002e16			0.229718
$598$	−1.25270e16			−0.273931
$599$			5.94643e16i			1.28735i	0.765300	+	0.643674i	$0.222591\pi$
$599$			5.94643e16i			1.28735i	−0.765300	+	0.643674i	$0.777409\pi$
$600$		−	1.77059e15i		−	0.0379499i
$601$			2.41527e16i			0.512530i	0.966607	+	0.256265i	$0.0824920\pi$
$601$			2.41527e16i			0.512530i	−0.966607	+	0.256265i	$0.917508\pi$
$602$	8.63392e16			1.81397
$603$			4.58620e16i			0.954003i
$604$	−7.86104e16			−1.61904
$605$		−	1.22732e16i		−	0.250280i
$606$	−1.02820e16			−0.207607
$607$	5.94605e16			1.18877			0.594383	−	0.804182i	$-0.297396\pi$
$607$	5.94605e16			1.18877			0.594383	+	0.804182i	$0.297396\pi$
$608$	−3.15003e16			−0.623582
$609$	7.36908e15			0.144447
$610$	1.55379e17			3.01588
$611$	−6.84634e15			−0.131587
$612$	−6.24396e16			−1.18837
$613$		−	1.10353e16i		−	0.207979i	−0.994578	−	0.103990i	$-0.966839\pi$
$613$		−	1.10353e16i		−	0.207979i	0.994578	−	0.103990i	$-0.0331609\pi$
$614$	6.91871e16			1.29126
$615$			8.71340e15i			0.161041i
$616$			6.75816e16i			1.23693i
$617$	3.56432e16			0.646050			0.323025	−	0.946390i	$-0.395300\pi$
$617$	3.56432e16			0.646050			0.323025	+	0.946390i	$0.395300\pi$
$618$	−1.67649e16			−0.300934
$619$	8.56588e16			1.52275			0.761374	−	0.648313i	$-0.224525\pi$
$619$	8.56588e16			1.52275			0.761374	+	0.648313i	$0.224525\pi$
$620$			2.96232e16i			0.521534i
$621$	−3.15001e16			−0.549239
$622$	−6.47981e16			−1.11897
$623$		−	7.93377e16i		−	1.35691i
$624$		−	5.37088e14i		−	0.00909782i
$625$	−6.76800e16			−1.13548
$626$			7.09919e15i			0.117968i
$627$		−	1.21599e16i		−	0.200136i
$628$		−	1.94199e17i		−	3.16583i
$629$	4.00076e16			0.646010
$630$	1.07752e17			1.72338
$631$			1.17845e17i			1.86697i	0.358619	+	0.933484i	$0.383248\pi$
$631$			1.17845e17i			1.86697i	−0.358619	+	0.933484i	$0.616752\pi$
$632$	−1.13922e16			−0.178774
$633$		−	1.70928e16i		−	0.265700i
$634$			1.48455e17i			2.28591i
$635$		−	4.58339e16i		−	0.699108i
$636$			1.04172e16i			0.157402i
$637$			3.60493e14i			0.00539585i
$638$		−	9.20504e16i		−	1.36490i
$639$		−	1.94838e16i		−	0.286199i
$640$	1.35769e17			1.97569
$641$		−	3.99161e16i		−	0.575439i	−0.957715	−	0.287720i	$-0.907103\pi$
$641$		−	3.99161e16i		−	0.575439i	0.957715	−	0.287720i	$-0.0928972\pi$
$642$	−1.55979e16			−0.222769
$643$			1.47500e16i			0.208701i	0.994541	+	0.104351i	$0.0332764\pi$
$643$			1.47500e16i			0.208701i	−0.994541	+	0.104351i	$0.966724\pi$
$644$	2.21898e17			3.11056
$645$	1.38486e16			0.192330
$646$	−1.14028e17			−1.56898
$647$			3.34592e16i			0.456131i	0.973646	+	0.228066i	$0.0732401\pi$
$647$			3.34592e16i			0.456131i	−0.973646	+	0.228066i	$0.926760\pi$
$648$			9.92967e16i			1.34117i
$649$	7.63330e16			1.02151
$650$	1.93793e15			0.0256956
$651$	3.10578e15			0.0408023
$652$		−	6.60571e16i		−	0.859872i
$653$			4.85731e16i			0.626493i	0.949672	+	0.313247i	$0.101417\pi$
$653$			4.85731e16i			0.626493i	−0.949672	+	0.313247i	$0.898583\pi$
$654$		−	2.70483e16i		−	0.345680i
$655$			1.00391e17i			1.27130i
$656$	4.41624e16			0.554153
$657$		−	6.30969e16i		−	0.784541i
$658$	1.86481e17			2.29763
$659$	2.47859e16			0.302617			0.151308	−	0.988487i	$-0.451651\pi$
$659$	2.47859e16			0.302617			0.151308	+	0.988487i	$0.451651\pi$
$660$			2.34476e16i			0.283684i
$661$		−	1.38403e17i		−	1.65935i	−0.558248	−	0.829674i	$-0.688526\pi$
$661$		−	1.38403e17i		−	1.65935i	0.558248	−	0.829674i	$-0.311474\pi$
$662$	6.92486e16			0.822741
$663$			8.46334e14i			0.00996462i
$664$	−1.02932e17	−	7.02400e16i	−1.20100	−	0.819551i
$665$	1.27969e17			1.47970
$666$		−	1.41515e17i		−	1.62165i
$667$	−1.39727e17			−1.58681
$668$	−1.22107e17			−1.37430
$669$		−	7.30900e15i		−	0.0815269i
$670$		−	1.61500e17i		−	1.78535i
$671$	−1.32334e17			−1.44990
$672$		−	6.36828e15i		−	0.0691522i
$673$	−1.45610e17			−1.56712			−0.783559	−	0.621317i	$-0.786598\pi$
$673$	−1.45610e17			−1.56712			−0.783559	+	0.621317i	$0.786598\pi$
$674$	−3.69426e16			−0.394065
$675$	4.87306e15			0.0515204
$676$	−1.75909e17			−1.84335
$677$		−	3.07331e16i		−	0.319208i	−0.987181	−	0.159604i	$-0.948978\pi$
$677$		−	3.07331e16i		−	0.319208i	0.987181	−	0.159604i	$-0.0510217\pi$
$678$			1.78060e16i			0.183311i
$679$			8.67840e16i			0.885566i
$680$	1.01650e17			1.02815
$681$	−2.58359e15			−0.0259025
$682$		−	3.87957e16i		−	0.385547i
$683$		−	3.46043e16i		−	0.340883i	−0.985368	−	0.170442i	$-0.945481\pi$
$683$		−	3.46043e16i		−	0.340883i	0.985368	−	0.170442i	$-0.0545194\pi$
$684$			2.62301e17i			2.56132i
$685$	−7.14821e16			−0.691917
$686$		−	1.80931e17i		−	1.73607i
$687$	3.10077e16			0.294937
$688$		−	7.01893e16i		−	0.661821i
$689$	−5.27110e15			−0.0492704
$690$	5.47297e16			0.507140
$691$	1.31138e17			1.20465			0.602325	−	0.798251i	$-0.294241\pi$
$691$	1.31138e17			1.20465			0.602325	+	0.798251i	$0.294241\pi$
$692$	−3.61971e16			−0.329638
$693$	−9.17708e16			−0.828524
$694$	3.09218e17			2.76763
$695$	−4.59604e16			−0.407826
$696$		−	2.45899e16i		−	0.216322i
$697$	−6.95904e16			−0.606950
$698$		−	2.52640e17i		−	2.18459i
$699$			7.79280e15i			0.0668083i
$700$	−3.43277e16			−0.291781
$701$	−1.42082e17			−1.19737			−0.598687	−	0.800983i	$-0.704311\pi$
$701$	−1.42082e17			−1.19737			−0.598687	+	0.800983i	$0.704311\pi$
$702$	6.06749e15			0.0506975
$703$		−	1.68067e17i		−	1.39236i
$704$	−1.43449e17			−1.17831
$705$	2.99112e16			0.243612
$706$			6.43693e15i			0.0519817i
$707$			9.21582e16i			0.737934i
$708$	4.41077e16			0.350199
$709$			5.87244e15i			0.0462318i	0.999733	+	0.0231159i	$0.00735868\pi$
$709$			5.87244e15i			0.0462318i	−0.999733	+	0.0231159i	$0.992641\pi$
$710$			6.86109e16i			0.535603i
$711$		−	1.54698e16i		−	0.119747i
$712$	−2.64742e17			−2.03209
$713$	−5.88894e16			−0.448229
$714$		−	2.30525e16i		−	0.173992i
$715$	−1.18645e16			−0.0887997
$716$		−	2.25698e17i		−	1.67514i
$717$			1.03919e15i			0.00764858i
$718$		−	3.32805e17i		−	2.42909i
$719$		−	9.72782e16i		−	0.704113i	−0.935979	−	0.352056i	$-0.885483\pi$
$719$		−	9.72782e16i		−	0.704113i	0.935979	−	0.352056i	$-0.114517\pi$
$720$		−	8.75966e16i		−	0.628771i
$721$			1.50265e17i			1.06966i
$722$			2.39471e17i			1.69055i
$723$	2.38167e16			0.166745
$724$		−	3.36530e16i		−	0.233664i
$725$	2.16157e16			0.148848
$726$			9.28779e15i			0.0634297i
$727$	−2.45163e17			−1.66054			−0.830268	−	0.557364i	$-0.811813\pi$
$727$	−2.45163e17			−1.66054			−0.830268	+	0.557364i	$0.811813\pi$
$728$	−1.97597e16			−0.132737
$729$	−1.27108e17			−0.846853
$730$			2.22192e17i			1.46822i
$731$			1.10603e17i			0.724876i
$732$	−7.64673e16			−0.497060
$733$	5.64648e16			0.364044			0.182022	−	0.983294i	$-0.441736\pi$
$733$	5.64648e16			0.364044			0.182022	+	0.983294i	$0.441736\pi$
$734$	−2.25594e17			−1.44262
$735$		−	1.57497e15i		−	0.00998957i
$736$			1.20750e17i			0.759663i
$737$			1.37548e17i			0.858319i
$738$			2.46155e17i			1.52360i
$739$	−3.48064e16			−0.213694			−0.106847	−	0.994275i	$-0.534076\pi$
$739$	−3.48064e16			−0.213694			−0.106847	+	0.994275i	$0.534076\pi$
$740$			3.24079e17i			1.97361i
$741$	3.55535e15			0.0214769
$742$	1.43575e17			0.860309
$743$		−	2.92987e17i		−	1.74147i	−0.491757	−	0.870733i	$-0.663645\pi$
$743$		−	2.92987e17i		−	1.74147i	0.491757	−	0.870733i	$-0.336355\pi$
$744$		−	1.03637e16i		−	0.0611049i
$745$	1.29102e17			0.755085
$746$			2.63992e17i			1.53165i
$747$	9.53806e16	−	1.39774e17i	0.548955	−	0.804458i
$748$	−1.87267e17			−1.06918
$749$			1.39804e17i			0.791826i
$750$	4.39281e16			0.246817
$751$	−2.26981e17			−1.26517			−0.632587	−	0.774489i	$-0.718007\pi$
$751$	−2.26981e17			−1.26517			−0.632587	+	0.774489i	$0.718007\pi$
$752$		−	1.51600e17i		−	0.838283i
$753$			2.88435e16i			0.158226i
$754$	2.69140e16			0.146470
$755$			1.73781e17i			0.938257i
$756$	−1.07477e17			−0.575684
$757$	−1.52671e17			−0.811301			−0.405651	−	0.914028i	$-0.632955\pi$
$757$	−1.52671e17			−0.811301			−0.405651	+	0.914028i	$0.632955\pi$
$758$	1.40824e17			0.742443
$759$	−4.66126e16			−0.243811
$760$		−	4.27020e17i		−	2.21598i
$761$		−	1.46189e17i		−	0.752672i	−0.926483	−	0.376336i	$-0.877184\pi$
$761$		−	1.46189e17i		−	0.752672i	0.926483	−	0.376336i	$-0.122816\pi$
$762$			3.46848e16i			0.177178i
$763$	−2.42435e17			−1.22871
$764$	3.39960e17			1.70949
$765$			1.38033e17i			0.688677i
$766$			1.43739e17i			0.711542i
$767$			2.23184e16i			0.109620i
$768$	−5.81738e16			−0.283504
$769$		−	9.96906e16i		−	0.482055i	−0.970518	−	0.241027i	$-0.922516\pi$
$769$		−	9.96906e16i		−	0.482055i	0.970518	−	0.241027i	$-0.0774843\pi$
$770$	3.23165e17			1.55053
$771$		−	3.19443e16i		−	0.152078i
$772$	5.39868e17			2.55026
$773$	−1.75739e17			−0.823741			−0.411870	−	0.911242i	$-0.635124\pi$
$773$	−1.75739e17			−0.823741			−0.411870	+	0.911242i	$0.635124\pi$
$774$	3.91225e17			1.81962
$775$	9.11019e15			0.0420453
$776$	2.89590e17			1.32621
$777$	3.39774e16			0.154406
$778$	−1.57408e17			−0.709820
$779$			2.92341e17i			1.30817i
$780$	−6.85568e15			−0.0304427
$781$		−	5.84351e16i		−	0.257494i
$782$			4.37104e17i			1.91137i
$783$	6.76769e16			0.293677
$784$	−7.98244e15			−0.0343747
$785$	−4.29309e17			−1.83464
$786$		−	7.59711e16i		−	0.322191i
$787$	3.11269e17			1.31005			0.655025	−	0.755607i	$-0.272658\pi$
$787$	3.11269e17			1.31005			0.655025	+	0.755607i	$0.272658\pi$
$788$	−6.12010e17			−2.55624
$789$		−	4.24279e16i		−	0.175869i
$790$			5.44757e16i			0.224099i
$791$	1.59596e17			0.651574
$792$			3.06230e17i			1.24079i
$793$		−	3.86923e16i		−	0.155591i
$794$			3.60870e17i			1.44022i
$795$	2.30291e16			0.0912165
$796$	−6.72838e17			−2.64504
$797$		−	3.32614e17i		−	1.29775i	−0.760896	−	0.648874i	$-0.775240\pi$
$797$		−	3.32614e17i		−	1.29775i	0.760896	−	0.648874i	$-0.224760\pi$
$798$	−9.68407e16			−0.375008
$799$			2.38888e17i			0.918151i
$800$		−	1.86801e16i		−	0.0712589i
$801$		−	3.59500e17i		−	1.36114i
$802$		−	4.40237e17i		−	1.65440i
$803$		−	1.89238e17i		−	0.705853i
$804$			7.94797e16i			0.294252i
$805$		−	4.90544e17i		−	1.80261i
$806$	1.13432e16			0.0413737
$807$		−	5.64542e16i		−	0.204388i
$808$	3.07523e17			1.10512
$809$			3.15708e17i			1.12615i	0.826407	+	0.563073i	$0.190381\pi$
$809$			3.15708e17i			1.12615i	−0.826407	+	0.563073i	$0.809619\pi$
$810$	4.74822e17			1.68120
$811$	−3.65167e16			−0.128341			−0.0641707	−	0.997939i	$-0.520440\pi$
$811$	−3.65167e16			−0.128341			−0.0641707	+	0.997939i	$0.520440\pi$
$812$	−4.76742e17			−1.66321
$813$			3.25881e15i			0.0112854i
$814$		−	4.24426e17i		−	1.45900i
$815$	−1.46030e17			−0.498307
$816$	−1.87405e16			−0.0634805
$817$	4.64631e17			1.56234
$818$			3.76543e16i			0.125688i
$819$		−	2.68322e16i		−	0.0889105i
$820$		−	5.63713e17i		−	1.85428i
$821$			4.14664e17i			1.35406i	0.735956	+	0.677029i	$0.236733\pi$
$821$			4.14664e17i			1.35406i	−0.735956	+	0.677029i	$0.763267\pi$
$822$	5.40942e16			0.175356
$823$		−	2.30525e17i		−	0.741854i	−0.928662	−	0.370927i	$-0.879040\pi$
$823$		−	2.30525e17i		−	0.741854i	0.928662	−	0.370927i	$-0.120960\pi$
$824$	5.01419e17			1.60191
$825$	7.21097e15			0.0228702
$826$		−	6.07911e17i		−	1.91408i
$827$		−	5.89525e17i		−	1.84276i	−0.388660	−	0.921381i	$-0.627062\pi$
$827$		−	5.89525e17i		−	1.84276i	0.388660	−	0.921381i	$-0.372938\pi$
$828$	1.00548e18			3.12026
$829$		−	3.04984e17i		−	0.939615i	−0.882769	−	0.469807i	$-0.844323\pi$
$829$		−	3.04984e17i		−	0.939615i	0.882769	−	0.469807i	$-0.155677\pi$
$830$	−3.35877e17	+	4.92206e17i	−1.02733	+	1.50549i
$831$	−7.21512e16			−0.219098
$832$		−	4.19420e16i		−	0.126447i
$833$	1.25786e16			0.0376498
$834$	3.47806e16			0.103357
$835$			2.69938e17i			0.796427i
$836$			7.86684e17i			2.30443i
$837$	2.85232e16			0.0829555
$838$		−	3.89299e17i		−	1.12414i
$839$	1.37791e16			0.0395046			0.0197523	−	0.999805i	$-0.493712\pi$
$839$	1.37791e16			0.0395046			0.0197523	+	0.999805i	$0.493712\pi$
$840$	8.63286e16			0.245742
$841$	−5.36161e16			−0.151537
$842$	−8.64754e17			−2.42672
$843$			1.45348e16i			0.0404988i
$844$			1.10582e18i			3.05935i
$845$			3.88875e17i			1.06824i
$846$	8.44995e17			2.30479
$847$	8.32468e16			0.225459
$848$		−	1.16719e17i		−	0.313881i
$849$			7.80620e16i			0.208446i
$850$		−	6.76200e16i		−	0.179292i
$851$	−6.44252e17			−1.69621
$852$		−	3.37657e16i		−	0.0882752i
$853$	−6.26268e17			−1.62580			−0.812898	−	0.582406i	$-0.802111\pi$
$853$	−6.26268e17			−1.62580			−0.812898	+	0.582406i	$0.802111\pi$
$854$			1.05390e18i			2.71678i
$855$	5.79861e17			1.48432
$856$	4.66513e17			1.18583
$857$	5.36100e17			1.35320			0.676598	−	0.736352i	$-0.263453\pi$
$857$	5.36100e17			1.35320			0.676598	+	0.736352i	$0.263453\pi$
$858$	8.97844e15			0.0225049
$859$	6.51468e15			0.0162156			0.00810782	−	0.999967i	$-0.497419\pi$
$859$	6.51468e15			0.0162156			0.00810782	+	0.999967i	$0.497419\pi$
$860$	−8.95935e17			−2.21455
$861$	−5.91012e16			−0.145070
$862$		−	4.96053e17i		−	1.20916i
$863$	4.29083e17			1.03867			0.519333	−	0.854572i	$-0.326180\pi$
$863$	4.29083e17			1.03867			0.519333	+	0.854572i	$0.326180\pi$
$864$		−	5.84856e16i		−	0.140594i
$865$			8.00198e16i			0.191030i
$866$	3.07557e17			0.729152
$867$	−3.90716e16			−0.0919913
$868$	−2.00928e17			−0.469810
$869$		−	4.63963e16i		−	0.107737i
$870$	−1.17585e17			−0.271167
$871$	−4.02166e16			−0.0921078
$872$			8.08982e17i			1.84009i
$873$			3.93241e17i			0.888328i
$874$	1.83622e18			4.11961
$875$		−	3.93729e17i		−	0.877301i
$876$		−	1.09348e17i		−	0.241983i
$877$			7.46896e17i			1.64158i	0.571228	+	0.820791i	$0.306467\pi$
$877$			7.46896e17i			1.64158i	−0.571228	+	0.820791i	$0.693533\pi$
$878$	4.45243e17			0.971918
$879$	−8.45987e16			−0.183413
$880$		−	2.62716e17i		−	0.565706i
$881$	2.52786e17			0.540626			0.270313	−	0.962773i	$-0.412873\pi$
$881$	2.52786e17			0.540626			0.270313	+	0.962773i	$0.412873\pi$
$882$		−	4.44930e16i		−	0.0945106i
$883$		−	7.80657e17i		−	1.64701i	−0.567310	−	0.823504i	$-0.692016\pi$
$883$		−	7.80657e17i		−	1.64701i	0.567310	−	0.823504i	$-0.307984\pi$
$884$		−	5.47535e16i		−	0.114736i
$885$		−	9.75076e16i		−	0.202945i
$886$		−	1.56550e18i		−	3.23632i
$887$		−	3.61609e17i		−	0.742502i	−0.928533	−	0.371251i	$-0.878929\pi$
$887$		−	3.61609e17i		−	0.742502i	0.928533	−	0.371251i	$-0.121071\pi$
$888$		−	1.13379e17i		−	0.231236i
$889$	3.10882e17			0.629774
$890$			1.26596e18i			2.54729i
$891$	−4.04400e17			−0.808248
$892$			4.72855e17i			0.938726i
$893$	1.00354e18			1.97891
$894$	−9.76982e16			−0.191364
$895$	−4.98944e17			−0.970764
$896$			9.20890e17i			1.77975i
$897$		−	1.36287e16i		−	0.0261638i
$898$	1.28338e18			2.44736
$899$	1.26522e17			0.239667
$900$	−1.55548e17			−0.292691
$901$			1.83924e17i			0.343787i
$902$			7.38259e17i			1.37078i
$903$			9.39322e16i			0.173256i
$904$		−	5.32557e17i		−	0.975788i
$905$	−7.43955e16			−0.135411
$906$		−	1.31509e17i		−	0.237787i
$907$	−3.83604e17			−0.689033			−0.344516	−	0.938780i	$-0.611957\pi$
$907$	−3.83604e17			−0.689033			−0.344516	+	0.938780i	$0.611957\pi$
$908$	1.67145e17			0.298249
$909$			4.17593e17i			0.740235i
$910$			9.44879e16i			0.166390i
$911$	3.96178e17			0.693074			0.346537	−	0.938036i	$-0.387357\pi$
$911$	3.96178e17			0.693074			0.346537	+	0.938036i	$0.387357\pi$
$912$			7.87265e16i			0.136821i
$913$	2.86062e17	−	4.19205e17i	0.493896	−	0.723773i
$914$	3.16142e17			0.542257
$915$			1.69044e17i			0.288053i
$916$	−2.00604e18			−3.39600
$917$	−6.80932e17			−1.14522
$918$		−	2.11712e17i		−	0.353744i
$919$		−	4.06569e17i		−	0.674903i	−0.941343	−	0.337451i	$-0.890435\pi$
$919$		−	4.06569e17i		−	0.674903i	0.941343	−	0.337451i	$-0.109565\pi$
$920$	−1.63690e18			−2.69957
$921$			7.52717e16i			0.123332i
$922$	−1.36214e18			−2.21735
$923$	1.70854e16			0.0276322
$924$	−1.59040e17			−0.255550
$925$	9.96659e16			0.159109
$926$			2.59569e17i			0.411706i
$927$			6.80889e17i			1.07300i
$928$		−	2.59428e17i		−	0.406190i
$929$	−4.99059e16			−0.0776350			−0.0388175	−	0.999246i	$-0.512359\pi$
$929$	−4.99059e16			−0.0776350			−0.0388175	+	0.999246i	$0.512359\pi$
$930$	−4.95575e16			−0.0765970
$931$		−	5.28412e16i		−	0.0811474i
$932$		−	5.04155e17i		−	0.769251i
$933$		−	7.04967e16i		−	0.106876i
$934$	−1.97365e18			−2.97296
$935$			4.13984e17i			0.619604i
$936$	−8.95364e16			−0.133151
$937$			4.30531e17i			0.636160i	0.948064	+	0.318080i	$0.103038\pi$
$937$			4.30531e17i			0.636160i	−0.948064	+	0.318080i	$0.896962\pi$
$938$	1.09542e18			1.60829
$939$	−7.72352e15			−0.0112673
$940$	−1.93510e18			−2.80502
$941$	−7.40817e17			−1.06702			−0.533511	−	0.845793i	$-0.679128\pi$
$941$	−7.40817e17			−1.06702			−0.533511	+	0.845793i	$0.679128\pi$
$942$	3.24880e17			0.464962
$943$	1.12063e18			1.59365
$944$	−4.94200e17			−0.698347
$945$			2.37596e17i			0.333617i
$946$	1.17335e18			1.63712
$947$		−	1.35357e18i		−	1.87664i	−0.345769	−	0.938320i	$-0.612382\pi$
$947$		−	1.35357e18i		−	1.87664i	0.345769	−	0.938320i	$-0.387618\pi$
$948$		−	2.68093e16i		−	0.0369348i
$949$	5.53299e16			0.0757464
$950$	−2.84063e17			−0.386432
$951$	−1.61511e17			−0.218332
$952$			6.89472e17i			0.926179i
$953$	−9.21381e17			−1.22993			−0.614967	−	0.788553i	$-0.710831\pi$
$953$	−9.21381e17			−1.22993			−0.614967	+	0.788553i	$0.710831\pi$
$954$	6.50574e17			0.862992
$955$		−	7.51539e17i		−	0.990675i
$956$		−	6.72305e16i		−	0.0880681i
$957$	1.00146e17			0.130365
$958$			8.87673e17i			1.14831i
$959$		−	4.84848e17i		−	0.623296i
$960$			1.83241e17i			0.234097i
$961$	−7.34339e17			−0.932301
$962$	1.24095e17			0.156568
$963$			6.33489e17i			0.794295i
$964$	−1.54082e18			−1.91995
$965$		−	1.19347e18i		−	1.47791i
$966$			3.71220e17i			0.456844i
$967$			2.13934e17i			0.261650i	0.991405	+	0.130825i	$0.0417626\pi$
$967$			2.13934e17i			0.261650i	−0.991405	+	0.130825i	$0.958237\pi$
$968$		−	2.77786e17i		−	0.337644i
$969$		−	1.24056e17i		−	0.149856i
$970$		−	1.38477e18i		−	1.66245i
$971$			7.17898e17i			0.856539i	0.903651	+	0.428270i	$0.140877\pi$
$971$			7.17898e17i			0.856539i	−0.903651	+	0.428270i	$0.859123\pi$
$972$	−7.33727e17			−0.870035
$973$		−	3.11740e17i		−	0.367380i
$974$	−1.52289e18			−1.78367
$975$			2.10836e15i			0.00245424i
$976$	8.56769e17			0.991209
$977$	5.71628e17			0.657274			0.328637	−	0.944456i	$-0.393411\pi$
$977$	5.71628e17			0.657274			0.328637	+	0.944456i	$0.393411\pi$
$978$	1.10509e17			0.126288
$979$		−	1.07820e18i		−	1.22462i
$980$			1.01892e17i			0.115023i
$981$	−1.09854e18			−1.23254
$982$	−8.89847e17			−0.992308
$983$	1.49353e18			1.65536			0.827681	−	0.561200i	$-0.189660\pi$
$983$	1.49353e18			1.65536			0.827681	+	0.561200i	$0.189660\pi$
$984$			1.97215e17i			0.217254i
$985$			1.35295e18i			1.48138i
$986$		−	9.39104e17i		−	1.02200i
$987$			2.02881e17i			0.219451i
$988$	−2.30013e17			−0.247292
$989$		−	1.78107e18i		−	1.90328i
$990$	1.46434e18			1.55536
$991$	1.29876e18			1.37116			0.685579	−	0.727999i	$-0.259549\pi$
$991$	1.29876e18			1.37116			0.685579	+	0.727999i	$0.259549\pi$
$992$		−	1.09339e17i		−	0.114737i
$993$			7.53386e16i			0.0785818i
$994$	−4.65373e17			−0.482484
$995$			1.48742e18i			1.53284i
$996$	1.65296e17	−	2.42231e17i	0.169319	−	0.248127i
$997$	3.38260e17			0.344413			0.172207	−	0.985061i	$-0.444910\pi$
$997$	3.38260e17			0.344413			0.172207	+	0.985061i	$0.444910\pi$
$998$		−	3.16238e17i		−	0.320059i
$999$	3.12045e17			0.313924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	83.13.b.c.82.8	✓	80
83.82	odd	2	inner	83.13.b.c.82.73	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
83.13.b.c.82.8	✓	80	1.1	even	1	trivial
83.13.b.c.82.73	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)

\(f(q)\)	\(=\)	\(q-108.230i q^{2} +117.748 q^{3} -7617.70 q^{4} +16840.2i q^{5} -12743.8i q^{6} -114224. q^{7} +381152. i q^{8} -517576. q^{9} +O(q^{10})\)	\(q-108.230i q^{2} +117.748 q^{3} -7617.70 q^{4} +16840.2i q^{5} -12743.8i q^{6} -114224. q^{7} +381152. i q^{8} -517576. q^{9} +1.82261e6 q^{10} -1.55230e6 q^{11} -896968. q^{12} -453865. i q^{13} +1.23624e7i q^{14} +1.98290e6i q^{15} +1.00500e7 q^{16} -1.58366e7 q^{17} +5.60172e7i q^{18} +6.65276e7i q^{19} -1.28284e8i q^{20} -1.34496e7 q^{21} +1.68005e8i q^{22} +2.55021e8 q^{23} +4.48799e7i q^{24} -3.94517e7 q^{25} -4.91217e7 q^{26} -1.23520e8 q^{27} +8.70120e8 q^{28} -5.47904e8 q^{29} +2.14609e8 q^{30} -2.30920e8 q^{31} +4.73492e8i q^{32} -1.82780e8 q^{33} +1.71399e9i q^{34} -1.92355e9i q^{35} +3.94274e9 q^{36} -2.52628e9 q^{37} +7.20028e9 q^{38} -5.34416e7i q^{39} -6.41868e9 q^{40} +4.39427e9 q^{41} +1.45565e9i q^{42} -6.98402e9i q^{43} +1.18249e10 q^{44} -8.71609e9i q^{45} -2.76008e10i q^{46} -1.50846e10i q^{47} +1.18337e9 q^{48} -7.94274e8 q^{49} +4.26985e9i q^{50} -1.86473e9 q^{51} +3.45740e9i q^{52} -1.16138e10i q^{53} +1.33685e10i q^{54} -2.61410e10i q^{55} -4.35366e10i q^{56} +7.83350e9i q^{57} +5.92995e10i q^{58} -4.91742e10 q^{59} -1.51051e10i q^{60} +8.52508e10 q^{61} +2.49924e10i q^{62} +5.91194e10 q^{63} +9.24107e10 q^{64} +7.64317e9 q^{65} +1.97822e10i q^{66} -8.86092e10i q^{67} +1.20638e11 q^{68} +3.00281e10 q^{69} -2.08185e11 q^{70} +3.76443e10i q^{71} -1.97276e11i q^{72} +1.21908e11i q^{73} +2.73418e11i q^{74} -4.64536e9 q^{75} -5.06787e11i q^{76} +1.77309e11 q^{77} -5.78398e9 q^{78} +2.98888e10i q^{79} +1.69244e11i q^{80} +2.60517e11 q^{81} -4.75591e11i q^{82} +(-1.84283e11 + 2.70055e11i) q^{83} +1.02455e11 q^{84} -2.66692e11i q^{85} -7.55880e11 q^{86} -6.45146e10 q^{87} -5.91661e11i q^{88} +6.94583e11i q^{89} -9.43341e11 q^{90} +5.18420e10i q^{91} -1.94267e12 q^{92} -2.71904e10 q^{93} -1.63260e12 q^{94} -1.12034e12 q^{95} +5.57528e10i q^{96} -7.59774e11i q^{97} +8.59641e10i q^{98} +8.03432e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9}+O(q^{10}) \) 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

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