LMFDB - Embedded newform 43.8.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,8,Mod(1,43)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("43.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	43.a (trivial)

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:	yes
Analytic conductor:	$13.4325560958$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + \cdots + 1551032970660 $ x^13 - 3x^12 - 1279x^11 + 3765x^10 + 598742x^9 - 1518614x^8 - 124677082x^7 + 193428526x^6 + 11160446785x^5 + 1754605765x^4 - 349352939351x^3 - 481872751923x^2 + 2098464001560x + 1551032970660
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$17.8435$ of defining polynomial
Character	$\chi$	$=$	43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.8435 q^{2} -90.3374 q^{3} +227.079 q^{4} +70.1157 q^{5} -1702.28 q^{6} +1065.82 q^{7} +1867.00 q^{8} +5973.84 q^{9} +O(q^{10})$	\(q+18.8435 q^{2} -90.3374 q^{3} +227.079 q^{4} +70.1157 q^{5} -1702.28 q^{6} +1065.82 q^{7} +1867.00 q^{8} +5973.84 q^{9} +1321.23 q^{10} -1455.37 q^{11} -20513.7 q^{12} +9741.20 q^{13} +20083.8 q^{14} -6334.07 q^{15} +6114.77 q^{16} +21354.5 q^{17} +112568. q^{18} +45230.6 q^{19} +15921.8 q^{20} -96283.4 q^{21} -27424.2 q^{22} -39774.0 q^{23} -168660. q^{24} -73208.8 q^{25} +183559. q^{26} -342093. q^{27} +242025. q^{28} +106837. q^{29} -119356. q^{30} -195235. q^{31} -123752. q^{32} +131474. q^{33} +402394. q^{34} +74730.8 q^{35} +1.35653e6 q^{36} -34774.3 q^{37} +852305. q^{38} -879994. q^{39} +130906. q^{40} +217638. q^{41} -1.81432e6 q^{42} -79507.0 q^{43} -330483. q^{44} +418860. q^{45} -749483. q^{46} +914679. q^{47} -552392. q^{48} +312430. q^{49} -1.37951e6 q^{50} -1.92911e6 q^{51} +2.21202e6 q^{52} +1.04811e6 q^{53} -6.44624e6 q^{54} -102044. q^{55} +1.98989e6 q^{56} -4.08602e6 q^{57} +2.01318e6 q^{58} +2.62505e6 q^{59} -1.43833e6 q^{60} +305720. q^{61} -3.67892e6 q^{62} +6.36704e6 q^{63} -3.11462e6 q^{64} +683011. q^{65} +2.47743e6 q^{66} -1.51055e6 q^{67} +4.84916e6 q^{68} +3.59308e6 q^{69} +1.40819e6 q^{70} -3.03204e6 q^{71} +1.11532e7 q^{72} -4.88668e6 q^{73} -655271. q^{74} +6.61349e6 q^{75} +1.02709e7 q^{76} -1.55116e6 q^{77} -1.65822e7 q^{78} -2.20917e6 q^{79} +428742. q^{80} +1.78390e7 q^{81} +4.10107e6 q^{82} -8.61841e6 q^{83} -2.18639e7 q^{84} +1.49729e6 q^{85} -1.49819e6 q^{86} -9.65134e6 q^{87} -2.71717e6 q^{88} -4.49534e6 q^{89} +7.89280e6 q^{90} +1.03824e7 q^{91} -9.03184e6 q^{92} +1.76370e7 q^{93} +1.72358e7 q^{94} +3.17138e6 q^{95} +1.11794e7 q^{96} +1.11102e7 q^{97} +5.88729e6 q^{98} -8.69412e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9}+O(q^{10}) $ 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9} + 4667 q^{10} + 1620 q^{11} - 19681 q^{12} + 13550 q^{13} + 44160 q^{14} + 31412 q^{15} + 114026 q^{16} + 110880 q^{17} + 159267 q^{18} + 105058 q^{19} + 167251 q^{20} + 129840 q^{21} + 201504 q^{22} + 160184 q^{23} + 161289 q^{24} + 270149 q^{25} + 272104 q^{26} + 252544 q^{27} + 208172 q^{28} + 285546 q^{29} + 107580 q^{30} - 99616 q^{31} + 200126 q^{32} + 531468 q^{33} - 80941 q^{34} - 187104 q^{35} - 608975 q^{36} + 176038 q^{37} + 652165 q^{38} - 794680 q^{39} - 895387 q^{40} - 410260 q^{41} - 3413218 q^{42} - 1033591 q^{43} - 2177076 q^{44} - 1051178 q^{45} - 3975765 q^{46} - 424556 q^{47} - 2360477 q^{48} - 1561359 q^{49} - 4063801 q^{50} - 2375738 q^{51} - 4172312 q^{52} + 3992458 q^{53} - 10438626 q^{54} + 406960 q^{55} + 1559556 q^{56} - 3116152 q^{57} - 4052005 q^{58} + 2248836 q^{59} - 2911436 q^{60} + 6210394 q^{61} + 885317 q^{62} + 11622368 q^{63} - 3096318 q^{64} + 5600420 q^{65} - 2174604 q^{66} - 1993648 q^{67} + 9327135 q^{68} + 13366240 q^{69} - 1105098 q^{70} + 4978064 q^{71} + 11370663 q^{72} + 8224814 q^{73} - 3613563 q^{74} + 27115592 q^{75} + 10687121 q^{76} + 17261892 q^{77} - 15226630 q^{78} + 6945708 q^{79} + 15822799 q^{80} + 35113185 q^{81} - 508449 q^{82} + 22937328 q^{83} - 14010106 q^{84} - 575532 q^{85} - 1272112 q^{86} + 9081380 q^{87} + 11202656 q^{88} + 9291302 q^{89} + 2841402 q^{90} + 25581108 q^{91} - 14388137 q^{92} + 25930480 q^{93} - 24645805 q^{94} + 30750464 q^{95} - 22461255 q^{96} + 10001852 q^{97} - 32304856 q^{98} + 5055452 q^{99}+O(q^{100}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9 + 4667 * q^10 + 1620 * q^11 - 19681 * q^12 + 13550 * q^13 + 44160 * q^14 + 31412 * q^15 + 114026 * q^16 + 110880 * q^17 + 159267 * q^18 + 105058 * q^19 + 167251 * q^20 + 129840 * q^21 + 201504 * q^22 + 160184 * q^23 + 161289 * q^24 + 270149 * q^25 + 272104 * q^26 + 252544 * q^27 + 208172 * q^28 + 285546 * q^29 + 107580 * q^30 - 99616 * q^31 + 200126 * q^32 + 531468 * q^33 - 80941 * q^34 - 187104 * q^35 - 608975 * q^36 + 176038 * q^37 + 652165 * q^38 - 794680 * q^39 - 895387 * q^40 - 410260 * q^41 - 3413218 * q^42 - 1033591 * q^43 - 2177076 * q^44 - 1051178 * q^45 - 3975765 * q^46 - 424556 * q^47 - 2360477 * q^48 - 1561359 * q^49 - 4063801 * q^50 - 2375738 * q^51 - 4172312 * q^52 + 3992458 * q^53 - 10438626 * q^54 + 406960 * q^55 + 1559556 * q^56 - 3116152 * q^57 - 4052005 * q^58 + 2248836 * q^59 - 2911436 * q^60 + 6210394 * q^61 + 885317 * q^62 + 11622368 * q^63 - 3096318 * q^64 + 5600420 * q^65 - 2174604 * q^66 - 1993648 * q^67 + 9327135 * q^68 + 13366240 * q^69 - 1105098 * q^70 + 4978064 * q^71 + 11370663 * q^72 + 8224814 * q^73 - 3613563 * q^74 + 27115592 * q^75 + 10687121 * q^76 + 17261892 * q^77 - 15226630 * q^78 + 6945708 * q^79 + 15822799 * q^80 + 35113185 * q^81 - 508449 * q^82 + 22937328 * q^83 - 14010106 * q^84 - 575532 * q^85 - 1272112 * q^86 + 9081380 * q^87 + 11202656 * q^88 + 9291302 * q^89 + 2841402 * q^90 + 25581108 * q^91 - 14388137 * q^92 + 25930480 * q^93 - 24645805 * q^94 + 30750464 * q^95 - 22461255 * q^96 + 10001852 * q^97 - 32304856 * q^98 + 5055452 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	18.8435	1.66555	0.832775	−	0.553612i	$-0.186751\pi$
$2$	18.8435	1.66555	0.832775	+	0.553612i	$0.186751\pi$
$3$	−90.3374	−1.93171	−0.965857	−	0.259075i	$-0.916582\pi$
$3$	−90.3374	−1.93171	−0.965857	+	0.259075i	$0.916582\pi$
$4$	227.079	1.77405
$5$	70.1157	0.250854	0.125427	−	0.992103i	$-0.459970\pi$
$5$	70.1157	0.250854	0.125427	+	0.992103i	$0.459970\pi$
$6$	−1702.28	−3.21737
$7$	1065.82	1.17447	0.587234	−	0.809418i	$-0.300217\pi$
$7$	1065.82	1.17447	0.587234	+	0.809418i	$0.300217\pi$
$8$	1867.00	1.28923
$9$	5973.84	2.73152
$10$	1321.23	0.417809
$11$	−1455.37	−0.329684	−0.164842	−	0.986320i	$-0.552711\pi$
$11$	−1455.37	−0.329684	−0.164842	+	0.986320i	$0.552711\pi$
$12$	−20513.7	−3.42697
$13$	9741.20	1.22973	0.614866	−	0.788631i	$-0.289210\pi$
$13$	9741.20	1.22973	0.614866	+	0.788631i	$0.289210\pi$
$14$	20083.8	1.95613
$15$	−6334.07	−0.484578
$16$	6114.77	0.373216
$17$	21354.5	1.05419	0.527094	−	0.849807i	$-0.323281\pi$
$17$	21354.5	1.05419	0.527094	+	0.849807i	$0.323281\pi$
$18$	112568.	4.54948
$19$	45230.6	1.51285	0.756424	−	0.654082i	$-0.226945\pi$
$19$	45230.6	1.51285	0.756424	+	0.654082i	$0.226945\pi$
$20$	15921.8	0.445028
$21$	−96283.4	−2.26874
$22$	−27424.2	−0.549105
$23$	−39774.0	−0.681635	−0.340817	−	0.940130i	$-0.610704\pi$
$23$	−39774.0	−0.681635	−0.340817	+	0.940130i	$0.610704\pi$
$24$	−168660.	−2.49042
$25$	−73208.8	−0.937072
$26$	183559.	2.04818
$27$	−342093.	−3.34481
$28$	242025.	2.08357
$29$	106837.	0.813443	0.406721	−	0.913552i	$-0.366672\pi$
$29$	106837.	0.813443	0.406721	+	0.913552i	$0.366672\pi$
$30$	−119356.	−0.807088
$31$	−195235.	−1.17704	−0.588521	−	0.808482i	$-0.700290\pi$
$31$	−195235.	−1.17704	−0.588521	+	0.808482i	$0.700290\pi$
$32$	−123752.	−0.667617
$33$	131474.	0.636855
$34$	402394.	1.75580
$35$	74730.8	0.294619
$36$	1.35653e6	4.84587
$37$	−34774.3	−0.112863	−0.0564316	−	0.998406i	$-0.517972\pi$
$37$	−34774.3	−0.112863	−0.0564316	+	0.998406i	$0.517972\pi$
$38$	852305.	2.51972
$39$	−879994.	−2.37549
$40$	130906.	0.323407
$41$	217638.	0.493164	0.246582	−	0.969122i	$-0.420692\pi$
$41$	217638.	0.493164	0.246582	+	0.969122i	$0.420692\pi$
$42$	−1.81432e6	−3.77869
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	−330483.	−0.584877
$45$	418860.	0.685212
$46$	−749483.	−1.13530
$47$	914679.	1.28507	0.642534	−	0.766257i	$-0.277883\pi$
$47$	914679.	1.28507	0.642534	+	0.766257i	$0.277883\pi$
$48$	−552392.	−0.720947
$49$	312430.	0.379373
$50$	−1.37951e6	−1.56074
$51$	−1.92911e6	−2.03639
$52$	2.21202e6	2.18161
$53$	1.04811e6	0.967031	0.483515	−	0.875336i	$-0.339360\pi$
$53$	1.04811e6	0.967031	0.483515	+	0.875336i	$0.339360\pi$
$54$	−6.44624e6	−5.57094
$55$	−102044.	−0.0827024
$56$	1.98989e6	1.51415
$57$	−4.08602e6	−2.92239
$58$	2.01318e6	1.35483
$59$	2.62505e6	1.66401	0.832005	−	0.554768i	$-0.187193\pi$
$59$	2.62505e6	1.66401	0.832005	+	0.554768i	$0.187193\pi$
$60$	−1.43833e6	−0.859667
$61$	305720.	0.172452	0.0862262	−	0.996276i	$-0.472519\pi$
$61$	305720.	0.172452	0.0862262	+	0.996276i	$0.472519\pi$
$62$	−3.67892e6	−1.96042
$63$	6.36704e6	3.20808
$64$	−3.11462e6	−1.48517
$65$	683011.	0.308483
$66$	2.47743e6	1.06071
$67$	−1.51055e6	−0.613583	−0.306791	−	0.951777i	$-0.599255\pi$
$67$	−1.51055e6	−0.613583	−0.306791	+	0.951777i	$0.599255\pi$
$68$	4.84916e6	1.87019
$69$	3.59308e6	1.31672
$70$	1.40819e6	0.490703
$71$	−3.03204e6	−1.00538	−0.502690	−	0.864467i	$-0.667656\pi$
$71$	−3.03204e6	−1.00538	−0.502690	+	0.864467i	$0.667656\pi$
$72$	1.11532e7	3.52155
$73$	−4.88668e6	−1.47023	−0.735113	−	0.677945i	$-0.762871\pi$
$73$	−4.88668e6	−1.47023	−0.735113	+	0.677945i	$0.762871\pi$
$74$	−655271.	−0.187979
$75$	6.61349e6	1.81016
$76$	1.02709e7	2.68387
$77$	−1.55116e6	−0.387203
$78$	−1.65822e7	−3.95650
$79$	−2.20917e6	−0.504120	−0.252060	−	0.967712i	$-0.581108\pi$
$79$	−2.20917e6	−0.504120	−0.252060	+	0.967712i	$0.581108\pi$
$80$	428742.	0.0936226
$81$	1.78390e7	3.72969
$82$	4.10107e6	0.821389
$83$	−8.61841e6	−1.65445	−0.827225	−	0.561871i	$-0.810082\pi$
$83$	−8.61841e6	−1.65445	−0.827225	+	0.561871i	$0.810082\pi$
$84$	−2.18639e7	−4.02486
$85$	1.49729e6	0.264447
$86$	−1.49819e6	−0.253994
$87$	−9.65134e6	−1.57134
$88$	−2.71717e6	−0.425037
$89$	−4.49534e6	−0.675923	−0.337962	−	0.941160i	$-0.609737\pi$
$89$	−4.49534e6	−0.675923	−0.337962	+	0.941160i	$0.609737\pi$
$90$	7.89280e6	1.14125
$91$	1.03824e7	1.44428
$92$	−9.03184e6	−1.20926
$93$	1.76370e7	2.27371
$94$	1.72358e7	2.14034
$95$	3.17138e6	0.379503
$96$	1.11794e7	1.28965
$97$	1.11102e7	1.23601	0.618003	−	0.786175i	$-0.287942\pi$
$97$	1.11102e7	1.23601	0.618003	+	0.786175i	$0.287942\pi$
$98$	5.88729e6	0.631865
$99$	−8.69412e6	−0.900539
$100$	−1.66242e7	−1.66242
$101$	−4.64538e6	−0.448638	−0.224319	−	0.974516i	$-0.572016\pi$
$101$	−4.64538e6	−0.448638	−0.224319	+	0.974516i	$0.572016\pi$
$102$	−3.63513e7	−3.39171
$103$	−1.71624e6	−0.154756	−0.0773782	−	0.997002i	$-0.524655\pi$
$103$	−1.71624e6	−0.154756	−0.0773782	+	0.997002i	$0.524655\pi$
$104$	1.81868e7	1.58540
$105$	−6.75098e6	−0.569120
$106$	1.97501e7	1.61064
$107$	−7.01098e6	−0.553268	−0.276634	−	0.960975i	$-0.589219\pi$
$107$	−7.01098e6	−0.553268	−0.276634	+	0.960975i	$0.589219\pi$
$108$	−7.76821e7	−5.93387
$109$	9.93632e6	0.734908	0.367454	−	0.930042i	$-0.380230\pi$
$109$	9.93632e6	0.734908	0.367454	+	0.930042i	$0.380230\pi$
$110$	−1.92287e6	−0.137745
$111$	3.14142e6	0.218020
$112$	6.51725e6	0.438330
$113$	−1.02922e7	−0.671017	−0.335508	−	0.942037i	$-0.608908\pi$
$113$	−1.02922e7	−0.671017	−0.335508	+	0.942037i	$0.608908\pi$
$114$	−7.69950e7	−4.86738
$115$	−2.78878e6	−0.170990
$116$	2.42604e7	1.44309
$117$	5.81923e7	3.35904
$118$	4.94653e7	2.77149
$119$	2.27601e7	1.23811
$120$	−1.18257e7	−0.624730
$121$	−1.73691e7	−0.891308
$122$	5.76085e6	0.287228
$123$	−1.96609e7	−0.952653
$124$	−4.43338e7	−2.08814
$125$	−1.06109e7	−0.485922
$126$	1.19978e8	5.34322
$127$	−2.57945e7	−1.11741	−0.558706	−	0.829365i	$-0.688702\pi$
$127$	−2.57945e7	−1.11741	−0.558706	+	0.829365i	$0.688702\pi$
$128$	−4.28502e7	−1.80600
$129$	7.18245e6	0.294584
$130$	1.28703e7	0.513793
$131$	−218805.	−0.00850371	−0.00425185	−	0.999991i	$-0.501353\pi$
$131$	−218805.	−0.00850371	−0.00425185	+	0.999991i	$0.501353\pi$
$132$	2.98550e7	1.12982
$133$	4.82077e7	1.77679
$134$	−2.84641e7	−1.02195
$135$	−2.39861e7	−0.839057
$136$	3.98689e7	1.35909
$137$	1.69620e7	0.563578	0.281789	−	0.959476i	$-0.409072\pi$
$137$	1.69620e7	0.563578	0.281789	+	0.959476i	$0.409072\pi$
$138$	6.77063e7	2.19307
$139$	−4.70595e7	−1.48626	−0.743131	−	0.669146i	$-0.766660\pi$
$139$	−4.70595e7	−1.48626	−0.743131	+	0.669146i	$0.766660\pi$
$140$	1.69698e7	0.522671
$141$	−8.26297e7	−2.48238
$142$	−5.71343e7	−1.67451
$143$	−1.41770e7	−0.405423
$144$	3.65287e7	1.01945
$145$	7.49092e6	0.204055
$146$	−9.20824e7	−2.44873
$147$	−2.82241e7	−0.732841
$148$	−7.89652e6	−0.200226
$149$	3.38648e7	0.838681	0.419341	−	0.907829i	$-0.362261\pi$
$149$	3.38648e7	0.838681	0.419341	+	0.907829i	$0.362261\pi$
$150$	1.24622e8	3.01491
$151$	−5.68824e7	−1.34449	−0.672247	−	0.740327i	$-0.734671\pi$
$151$	−5.68824e7	−1.34449	−0.672247	+	0.740327i	$0.734671\pi$
$152$	8.44456e7	1.95040
$153$	1.27568e8	2.87954
$154$	−2.92293e7	−0.644906
$155$	−1.36891e7	−0.295265
$156$	−1.99828e8	−4.21425
$157$	9.38899e7	1.93629	0.968144	−	0.250392i	$-0.0805596\pi$
$157$	9.38899e7	1.93629	0.968144	+	0.250392i	$0.0805596\pi$
$158$	−4.16285e7	−0.839636
$159$	−9.46833e7	−1.86803
$160$	−8.67696e6	−0.167474
$161$	−4.23919e7	−0.800557
$162$	3.36150e8	6.21198
$163$	4.02174e7	0.727373	0.363686	−	0.931521i	$-0.381518\pi$
$163$	4.02174e7	0.727373	0.363686	+	0.931521i	$0.381518\pi$
$164$	4.94211e7	0.874900
$165$	9.21839e6	0.159757
$166$	−1.62401e8	−2.75557
$167$	1.22868e7	0.204141	0.102071	−	0.994777i	$-0.467453\pi$
$167$	1.22868e7	0.204141	0.102071	+	0.994777i	$0.467453\pi$
$168$	−1.79761e8	−2.92491
$169$	3.21424e7	0.512242
$170$	2.82142e7	0.440450
$171$	2.70200e8	4.13237
$172$	−1.80544e7	−0.270541
$173$	−5.25459e6	−0.0771574	−0.0385787	−	0.999256i	$-0.512283\pi$
$173$	−5.25459e6	−0.0771574	−0.0385787	+	0.999256i	$0.512283\pi$
$174$	−1.81865e8	−2.61714
$175$	−7.80274e7	−1.10056
$176$	−8.89923e6	−0.123043
$177$	−2.37140e8	−3.21439
$178$	−8.47081e7	−1.12578
$179$	5.70630e7	0.743651	0.371826	−	0.928303i	$-0.378732\pi$
$179$	5.70630e7	0.743651	0.371826	+	0.928303i	$0.378732\pi$
$180$	9.51143e7	1.21560
$181$	−6.26864e7	−0.785775	−0.392888	−	0.919586i	$-0.628524\pi$
$181$	−6.26864e7	−0.785775	−0.392888	+	0.919586i	$0.628524\pi$
$182$	1.95641e8	2.40552
$183$	−2.76179e7	−0.333129
$184$	−7.42580e7	−0.878782
$185$	−2.43823e6	−0.0283121
$186$	3.32344e8	3.78698
$187$	−3.10786e7	−0.347549
$188$	2.07704e8	2.27978
$189$	−3.64610e8	−3.92836
$190$	5.97600e7	0.632081
$191$	−7.15929e7	−0.743453	−0.371726	−	0.928342i	$-0.621234\pi$
$191$	−7.15929e7	−0.743453	−0.371726	+	0.928342i	$0.621234\pi$
$192$	2.81366e8	2.86892
$193$	−6.80335e7	−0.681196	−0.340598	−	0.940209i	$-0.610630\pi$
$193$	−6.80335e7	−0.681196	−0.340598	+	0.940209i	$0.610630\pi$
$194$	2.09356e8	2.05863
$195$	−6.17014e7	−0.595901
$196$	7.09463e7	0.673029
$197$	1.26363e8	1.17758	0.588788	−	0.808288i	$-0.299605\pi$
$197$	1.26363e8	1.17758	0.588788	+	0.808288i	$0.299605\pi$
$198$	−1.63828e8	−1.49989
$199$	−2.86067e7	−0.257325	−0.128662	−	0.991688i	$-0.541068\pi$
$199$	−2.86067e7	−0.257325	−0.128662	+	0.991688i	$0.541068\pi$
$200$	−1.36681e8	−1.20810
$201$	1.36459e8	1.18527
$202$	−8.75354e7	−0.747229
$203$	1.13869e8	0.955362
$204$	−4.38060e8	−3.61267
$205$	1.52599e7	0.123712
$206$	−3.23401e7	−0.257754
$207$	−2.37603e8	−1.86190
$208$	5.95652e7	0.458956
$209$	−6.58271e7	−0.498761
$210$	−1.27212e8	−0.947898
$211$	−1.14904e8	−0.842066	−0.421033	−	0.907045i	$-0.638332\pi$
$211$	−1.14904e8	−0.842066	−0.421033	+	0.907045i	$0.638332\pi$
$212$	2.38003e8	1.71557
$213$	2.73906e8	1.94211
$214$	−1.32112e8	−0.921495
$215$	−5.57469e6	−0.0382548
$216$	−6.38687e8	−4.31221
$217$	−2.08086e8	−1.38240
$218$	1.87235e8	1.22402
$219$	4.41450e8	2.84006
$220$	−2.31721e7	−0.146719
$221$	2.08018e8	1.29637
$222$	5.91955e7	0.363122
$223$	−8.16498e7	−0.493047	−0.246523	−	0.969137i	$-0.579288\pi$
$223$	−8.16498e7	−0.493047	−0.246523	+	0.969137i	$0.579288\pi$
$224$	−1.31897e8	−0.784094
$225$	−4.37337e8	−2.55963
$226$	−1.93941e8	−1.11761
$227$	3.41280e8	1.93651	0.968257	−	0.249958i	$-0.0804169\pi$
$227$	3.41280e8	1.93651	0.968257	+	0.249958i	$0.0804169\pi$
$228$	−9.27848e8	−5.18448
$229$	−1.15657e8	−0.636427	−0.318213	−	0.948019i	$-0.603083\pi$
$229$	−1.15657e8	−0.636427	−0.318213	+	0.948019i	$0.603083\pi$
$230$	−5.25505e7	−0.284793
$231$	1.40128e8	0.747966
$232$	1.99464e8	1.04871
$233$	−1.82084e8	−0.943029	−0.471514	−	0.881858i	$-0.656292\pi$
$233$	−1.82084e8	−0.943029	−0.471514	+	0.881858i	$0.656292\pi$
$234$	1.09655e9	5.59465
$235$	6.41334e7	0.322364
$236$	5.96094e8	2.95204
$237$	1.99570e8	0.973815
$238$	4.28880e8	2.06213
$239$	−3.58475e6	−0.0169850	−0.00849252	−	0.999964i	$-0.502703\pi$
$239$	−3.58475e6	−0.0169850	−0.00849252	+	0.999964i	$0.502703\pi$
$240$	−3.87314e7	−0.180852
$241$	1.50219e8	0.691299	0.345650	−	0.938364i	$-0.387659\pi$
$241$	1.50219e8	0.691299	0.345650	+	0.938364i	$0.387659\pi$
$242$	−3.27295e8	−1.48452
$243$	−8.63369e8	−3.85989
$244$	6.94226e7	0.305940
$245$	2.19063e7	0.0951671
$246$	−3.70480e8	−1.58669
$247$	4.40601e8	1.86040
$248$	−3.64504e8	−1.51748
$249$	7.78564e8	3.19593
$250$	−1.99946e8	−0.809327
$251$	−2.63281e8	−1.05090	−0.525450	−	0.850824i	$-0.676103\pi$
$251$	−2.63281e8	−1.05090	−0.525450	+	0.850824i	$0.676103\pi$
$252$	1.44582e9	5.69131
$253$	5.78857e7	0.224724
$254$	−4.86059e8	−1.86111
$255$	−1.35261e8	−0.510836
$256$	−4.08778e8	−1.52282
$257$	2.87010e8	1.05470	0.527352	−	0.849647i	$-0.323185\pi$
$257$	2.87010e8	1.05470	0.527352	+	0.849647i	$0.323185\pi$
$258$	1.35343e8	0.490644
$259$	−3.70632e7	−0.132554
$260$	1.55098e8	0.547266
$261$	6.38224e8	2.22194
$262$	−4.12307e6	−0.0141633
$263$	4.14761e8	1.40589	0.702947	−	0.711242i	$-0.251867\pi$
$263$	4.14761e8	1.40589	0.702947	+	0.711242i	$0.251867\pi$
$264$	2.45462e8	0.821051
$265$	7.34888e7	0.242583
$266$	9.08404e8	2.95933
$267$	4.06097e8	1.30569
$268$	−3.43014e8	−1.08853
$269$	−1.11250e8	−0.348470	−0.174235	−	0.984704i	$-0.555745\pi$
$269$	−1.11250e8	−0.348470	−0.174235	+	0.984704i	$0.555745\pi$
$270$	−4.51983e8	−1.39749
$271$	1.01854e8	0.310874	0.155437	−	0.987846i	$-0.450321\pi$
$271$	1.01854e8	0.310874	0.155437	+	0.987846i	$0.450321\pi$
$272$	1.30578e8	0.393440
$273$	−9.37916e8	−2.78994
$274$	3.19623e8	0.938667
$275$	1.06546e8	0.308938
$276$	8.15912e8	2.33594
$277$	3.01221e8	0.851543	0.425772	−	0.904831i	$-0.360003\pi$
$277$	3.01221e8	0.851543	0.425772	+	0.904831i	$0.360003\pi$
$278$	−8.86767e8	−2.47544
$279$	−1.16630e9	−3.21512
$280$	1.39522e8	0.379831
$281$	539783.	0.00145127	0.000725633	−	1.00000i	$-0.499769\pi$
$281$	539783.	0.00145127	0.000725633		1.00000i	$0.499769\pi$
$282$	−1.55704e9	−4.13453
$283$	7.75243e7	0.203322	0.101661	−	0.994819i	$-0.467584\pi$
$283$	7.75243e7	0.203322	0.101661	+	0.994819i	$0.467584\pi$
$284$	−6.88512e8	−1.78360
$285$	−2.86494e8	−0.733092
$286$	−2.67145e8	−0.675252
$287$	2.31963e8	0.579205
$288$	−7.39275e8	−1.82361
$289$	4.56762e7	0.111313
$290$	1.41156e8	0.339864
$291$	−1.00367e9	−2.38761
$292$	−1.10966e9	−2.60826
$293$	−3.93510e8	−0.913943	−0.456971	−	0.889481i	$-0.651066\pi$
$293$	−3.93510e8	−0.913943	−0.456971	+	0.889481i	$0.651066\pi$
$294$	−5.31842e8	−1.22058
$295$	1.84057e8	0.417423
$296$	−6.49236e7	−0.145506
$297$	4.97870e8	1.10273
$298$	6.38133e8	1.39687
$299$	−3.87446e8	−0.838228
$300$	1.50178e9	3.21132
$301$	−8.47402e7	−0.179105
$302$	−1.07187e9	−2.23932
$303$	4.19651e8	0.866641
$304$	2.76575e8	0.564619
$305$	2.14358e7	0.0432603
$306$	2.40384e9	4.79601
$307$	5.00982e8	0.988184	0.494092	−	0.869409i	$-0.335501\pi$
$307$	5.00982e8	0.988184	0.494092	+	0.869409i	$0.335501\pi$
$308$	−3.52236e8	−0.686919
$309$	1.55041e8	0.298945
$310$	−2.57950e8	−0.491779
$311$	−4.05811e8	−0.765002	−0.382501	−	0.923955i	$-0.624937\pi$
$311$	−4.05811e8	−0.765002	−0.382501	+	0.923955i	$0.624937\pi$
$312$	−1.64295e9	−3.06255
$313$	−8.16758e7	−0.150553	−0.0752763	−	0.997163i	$-0.523984\pi$
$313$	−8.16758e7	−0.150553	−0.0752763	+	0.997163i	$0.523984\pi$
$314$	1.76922e9	3.22499
$315$	4.46429e8	0.804759
$316$	−5.01655e8	−0.894336
$317$	−3.74278e8	−0.659913	−0.329956	−	0.943996i	$-0.607034\pi$
$317$	−3.74278e8	−0.659913	−0.329956	+	0.943996i	$0.607034\pi$
$318$	−1.78417e9	−3.11129
$319$	−1.55486e8	−0.268179
$320$	−2.18384e8	−0.372559
$321$	6.33353e8	1.06876
$322$	−7.98814e8	−1.33337
$323$	9.65878e8	1.59483
$324$	4.05086e9	6.61667
$325$	−7.13141e8	−1.15235
$326$	7.57838e8	1.21148
$327$	−8.97621e8	−1.41963
$328$	4.06330e8	0.635801
$329$	9.74883e8	1.50927
$330$	1.73707e8	0.266084
$331$	8.80018e7	0.133381	0.0666904	−	0.997774i	$-0.478756\pi$
$331$	8.80018e7	0.133381	0.0666904	+	0.997774i	$0.478756\pi$
$332$	−1.95706e9	−2.93509
$333$	−2.07736e8	−0.308288
$334$	2.31527e8	0.340007
$335$	−1.05913e8	−0.153919
$336$	−5.88751e8	−0.846728
$337$	−2.56628e8	−0.365257	−0.182629	−	0.983182i	$-0.558461\pi$
$337$	−2.56628e8	−0.365257	−0.182629	+	0.983182i	$0.558461\pi$
$338$	6.05677e8	0.853165
$339$	9.29769e8	1.29621
$340$	3.40002e8	0.469144
$341$	2.84139e8	0.388052
$342$	5.09153e9	6.88267
$343$	−5.44755e8	−0.728906
$344$	−1.48440e8	−0.196605
$345$	2.51931e8	0.330305
$346$	−9.90151e7	−0.128510
$347$	−3.07698e8	−0.395341	−0.197670	−	0.980269i	$-0.563338\pi$
$347$	−3.07698e8	−0.395341	−0.197670	+	0.980269i	$0.563338\pi$
$348$	−2.19162e9	−2.78764
$349$	−1.28877e9	−1.62288	−0.811439	−	0.584437i	$-0.801316\pi$
$349$	−1.28877e9	−1.62288	−0.811439	+	0.584437i	$0.801316\pi$
$350$	−1.47031e9	−1.83304
$351$	−3.33239e9	−4.11322
$352$	1.80104e8	0.220103
$353$	1.44420e9	1.74749	0.873747	−	0.486380i	$-0.161683\pi$
$353$	1.44420e9	1.74749	0.873747	+	0.486380i	$0.161683\pi$
$354$	−4.46856e9	−5.35373
$355$	−2.12593e8	−0.252203
$356$	−1.02080e9	−1.19912
$357$	−2.05608e9	−2.39167
$358$	1.07527e9	1.23859
$359$	8.83030e8	1.00727	0.503634	−	0.863917i	$-0.331996\pi$
$359$	8.83030e8	1.00727	0.503634	+	0.863917i	$0.331996\pi$
$360$	7.82011e8	0.883394
$361$	1.15194e9	1.28871
$362$	−1.18123e9	−1.30875
$363$	1.56908e9	1.72175
$364$	2.35762e9	2.56223
$365$	−3.42633e8	−0.368811
$366$	−5.20420e8	−0.554843
$367$	−1.60796e8	−0.169803	−0.0849015	−	0.996389i	$-0.527058\pi$
$367$	−1.60796e8	−0.169803	−0.0849015	+	0.996389i	$0.527058\pi$
$368$	−2.43209e8	−0.254397
$369$	1.30014e9	1.34709
$370$	−4.59448e7	−0.0471553
$371$	1.11709e9	1.13575
$372$	4.00500e9	4.03369
$373$	1.15384e8	0.115124	0.0575620	−	0.998342i	$-0.481667\pi$
$373$	1.15384e8	0.115124	0.0575620	+	0.998342i	$0.481667\pi$
$374$	−5.85631e8	−0.578860
$375$	9.58559e8	0.938662
$376$	1.70770e9	1.65674
$377$	1.04072e9	1.00032
$378$	−6.87054e9	−6.54288
$379$	1.76476e9	1.66513	0.832565	−	0.553928i	$-0.186872\pi$
$379$	1.76476e9	1.66513	0.832565	+	0.553928i	$0.186872\pi$
$380$	7.20153e8	0.673259
$381$	2.33020e9	2.15852
$382$	−1.34906e9	−1.23826
$383$	2.52128e8	0.229311	0.114656	−	0.993405i	$-0.463424\pi$
$383$	2.52128e8	0.229311	0.114656	+	0.993405i	$0.463424\pi$
$384$	3.87097e9	3.48867
$385$	−1.08761e8	−0.0971313
$386$	−1.28199e9	−1.13457
$387$	−4.74962e8	−0.416553
$388$	2.52289e9	2.19274
$389$	−1.17424e9	−1.01142	−0.505710	−	0.862703i	$-0.668770\pi$
$389$	−1.17424e9	−1.01142	−0.505710	+	0.862703i	$0.668770\pi$
$390$	−1.16267e9	−0.992502
$391$	−8.49354e8	−0.718571
$392$	5.83307e8	0.489098
$393$	1.97663e7	0.0164267
$394$	2.38113e9	1.96131
$395$	−1.54897e8	−0.126460
$396$	−1.97425e9	−1.59761
$397$	−1.20026e9	−0.962738	−0.481369	−	0.876518i	$-0.659860\pi$
$397$	−1.20026e9	−0.962738	−0.481369	+	0.876518i	$0.659860\pi$
$398$	−5.39051e8	−0.428587
$399$	−4.35496e9	−3.43225
$400$	−4.47655e8	−0.349730
$401$	−1.92289e8	−0.148919	−0.0744593	−	0.997224i	$-0.523723\pi$
$401$	−1.92289e8	−0.148919	−0.0744593	+	0.997224i	$0.523723\pi$
$402$	2.57137e9	1.97412
$403$	−1.90183e9	−1.44745
$404$	−1.05487e9	−0.795909
$405$	1.25079e9	0.935606
$406$	2.14569e9	1.59120
$407$	5.06093e7	0.0372092
$408$	−3.60165e9	−2.62537
$409$	4.65392e8	0.336347	0.168173	−	0.985757i	$-0.446213\pi$
$409$	4.65392e8	0.336347	0.168173	+	0.985757i	$0.446213\pi$
$410$	2.87550e8	0.206049
$411$	−1.53230e9	−1.08867
$412$	−3.89723e8	−0.274546
$413$	2.79783e9	1.95432
$414$	−4.47729e9	−3.10109
$415$	−6.04286e8	−0.415025
$416$	−1.20549e9	−0.820990
$417$	4.25123e9	2.87103
$418$	−1.24042e9	−0.830712
$419$	−2.24682e9	−1.49217	−0.746086	−	0.665850i	$-0.768069\pi$
$419$	−2.24682e9	−1.49217	−0.746086	+	0.665850i	$0.768069\pi$
$420$	−1.53301e9	−1.00965
$421$	6.62749e8	0.432874	0.216437	−	0.976297i	$-0.430556\pi$
$421$	6.62749e8	0.432874	0.216437	+	0.976297i	$0.430556\pi$
$422$	−2.16520e9	−1.40250
$423$	5.46414e9	3.51019
$424$	1.95682e9	1.24672
$425$	−1.56334e9	−0.987851
$426$	5.16136e9	3.23467
$427$	3.25843e8	0.202540
$428$	−1.59205e9	−0.981528
$429$	1.28071e9	0.783162
$430$	−1.05047e8	−0.0637153
$431$	−4.38748e8	−0.263964	−0.131982	−	0.991252i	$-0.542134\pi$
$431$	−4.38748e8	−0.263964	−0.131982	+	0.991252i	$0.542134\pi$
$432$	−2.09182e9	−1.24833
$433$	1.08042e8	0.0639567	0.0319784	−	0.999489i	$-0.489819\pi$
$433$	1.08042e8	0.0639567	0.0319784	+	0.999489i	$0.489819\pi$
$434$	−3.92107e9	−2.30245
$435$	−6.76710e8	−0.394176
$436$	2.25633e9	1.30377
$437$	−1.79900e9	−1.03121
$438$	8.31848e9	4.73025
$439$	−2.59454e9	−1.46364	−0.731820	−	0.681498i	$-0.761329\pi$
$439$	−2.59454e9	−1.46364	−0.731820	+	0.681498i	$0.761329\pi$
$440$	−1.90516e8	−0.106622
$441$	1.86641e9	1.03627
$442$	3.91980e9	2.15917
$443$	1.32379e9	0.723446	0.361723	−	0.932286i	$-0.382189\pi$
$443$	1.32379e9	0.723446	0.361723	+	0.932286i	$0.382189\pi$
$444$	7.13350e8	0.386779
$445$	−3.15194e8	−0.169558
$446$	−1.53857e9	−0.821194
$447$	−3.05926e9	−1.62009
$448$	−3.31962e9	−1.74428
$449$	1.73729e9	0.905753	0.452876	−	0.891573i	$-0.350398\pi$
$449$	1.73729e9	0.905753	0.452876	+	0.891573i	$0.350398\pi$
$450$	−8.24098e9	−4.26320
$451$	−3.16743e8	−0.162588
$452$	−2.33714e9	−1.19042
$453$	5.13861e9	2.59718
$454$	6.43093e9	3.22536
$455$	7.27967e8	0.362303
$456$	−7.62859e9	−3.76762
$457$	−1.37475e9	−0.673776	−0.336888	−	0.941545i	$-0.609374\pi$
$457$	−1.37475e9	−0.673776	−0.336888	+	0.941545i	$0.609374\pi$
$458$	−2.17939e9	−1.06000
$459$	−7.30522e9	−3.52606
$460$	−6.33274e8	−0.303347
$461$	−1.34864e9	−0.641124	−0.320562	−	0.947228i	$-0.603872\pi$
$461$	−1.34864e9	−0.641124	−0.320562	+	0.947228i	$0.603872\pi$
$462$	2.64050e9	1.24577
$463$	−1.82505e9	−0.854557	−0.427279	−	0.904120i	$-0.640528\pi$
$463$	−1.82505e9	−0.854557	−0.427279	+	0.904120i	$0.640528\pi$
$464$	6.53281e8	0.303590
$465$	1.23663e9	0.570369
$466$	−3.43110e9	−1.57066
$467$	1.36048e9	0.618135	0.309067	−	0.951040i	$-0.399983\pi$
$467$	1.36048e9	0.618135	0.309067	+	0.951040i	$0.399983\pi$
$468$	1.32143e10	5.95912
$469$	−1.60997e9	−0.720633
$470$	1.20850e9	0.536913
$471$	−8.48177e9	−3.74036
$472$	4.90097e9	2.14529
$473$	1.15712e8	0.0502763
$474$	3.76061e9	1.62194
$475$	−3.31128e9	−1.41765
$476$	5.16833e9	2.19647
$477$	6.26122e9	2.64147
$478$	−6.75494e7	−0.0282894
$479$	3.05531e9	1.27023	0.635113	−	0.772419i	$-0.280954\pi$
$479$	3.05531e9	1.27023	0.635113	+	0.772419i	$0.280954\pi$
$480$	7.83854e8	0.323512
$481$	−3.38743e8	−0.138792
$482$	2.83066e9	1.15139
$483$	3.82957e9	1.54645
$484$	−3.94415e9	−1.58123
$485$	7.79000e8	0.310057
$486$	−1.62689e10	−6.42883
$487$	−3.84229e9	−1.50743	−0.753717	−	0.657199i	$-0.771741\pi$
$487$	−3.84229e9	−1.50743	−0.753717	+	0.657199i	$0.771741\pi$
$488$	5.70779e8	0.222330
$489$	−3.63313e9	−1.40508
$490$	4.12792e8	0.158506
$491$	−9.60708e7	−0.0366274	−0.0183137	−	0.999832i	$-0.505830\pi$
$491$	−9.60708e7	−0.0366274	−0.0183137	+	0.999832i	$0.505830\pi$
$492$	−4.46457e9	−1.69006
$493$	2.28144e9	0.857522
$494$	8.30247e9	3.09858
$495$	−6.09594e8	−0.225903
$496$	−1.19382e9	−0.439291
$497$	−3.23160e9	−1.18079
$498$	1.46709e10	5.32297
$499$	4.64015e8	0.167178	0.0835892	−	0.996500i	$-0.473362\pi$
$499$	4.64015e8	0.167178	0.0835892	+	0.996500i	$0.473362\pi$
$500$	−2.40951e9	−0.862052
$501$	−1.10996e9	−0.394343
$502$	−4.96115e9	−1.75033
$503$	3.05552e9	1.07052	0.535262	−	0.844686i	$-0.320213\pi$
$503$	3.05552e9	1.07052	0.535262	+	0.844686i	$0.320213\pi$
$504$	1.18873e10	4.13595
$505$	−3.25714e8	−0.112543
$506$	1.09077e9	0.374289
$507$	−2.90366e9	−0.989506
$508$	−5.85738e9	−1.98235
$509$	−2.74667e9	−0.923198	−0.461599	−	0.887089i	$-0.652724\pi$
$509$	−2.74667e9	−0.923198	−0.461599	+	0.887089i	$0.652724\pi$
$510$	−2.54879e9	−0.850823
$511$	−5.20832e9	−1.72673
$512$	−2.21800e9	−0.730326
$513$	−1.54731e10	−5.06018
$514$	5.40828e9	1.75666
$515$	−1.20336e8	−0.0388212
$516$	1.63098e9	0.522608
$517$	−1.33119e9	−0.423666
$518$	−6.98401e8	−0.220775
$519$	4.74686e8	0.149046
$520$	1.27518e9	0.397704
$521$	−3.04048e9	−0.941912	−0.470956	−	0.882157i	$-0.656091\pi$
$521$	−3.04048e9	−0.941912	−0.470956	+	0.882157i	$0.656091\pi$
$522$	1.20264e10	3.70075
$523$	3.17164e9	0.969457	0.484729	−	0.874665i	$-0.338918\pi$
$523$	3.17164e9	0.969457	0.484729	+	0.874665i	$0.338918\pi$
$524$	−4.96861e7	−0.0150860
$525$	7.04879e9	2.12597
$526$	7.81557e9	2.34159
$527$	−4.16915e9	−1.24082
$528$	8.03933e8	0.237685
$529$	−1.82286e9	−0.535374
$530$	1.38479e9	0.404034
$531$	1.56816e10	4.54528
$532$	1.09470e10	3.15212
$533$	2.12006e9	0.606460
$534$	7.65230e9	2.17469
$535$	−4.91580e8	−0.138789
$536$	−2.82020e9	−0.791047
$537$	−5.15492e9	−1.43652
$538$	−2.09634e9	−0.580394
$539$	−4.54700e8	−0.125073
$540$	−5.44674e9	−1.48853
$541$	4.49965e9	1.22177	0.610884	−	0.791720i	$-0.290814\pi$
$541$	4.49965e9	1.22177	0.610884	+	0.791720i	$0.290814\pi$
$542$	1.91929e9	0.517777
$543$	5.66292e9	1.51789
$544$	−2.64266e9	−0.703794
$545$	6.96692e8	0.184354
$546$	−1.76737e10	−4.64678
$547$	4.37317e8	0.114246	0.0571229	−	0.998367i	$-0.481807\pi$
$547$	4.37317e8	0.114246	0.0571229	+	0.998367i	$0.481807\pi$
$548$	3.85170e9	0.999818
$549$	1.82632e9	0.471058
$550$	2.00770e9	0.514551
$551$	4.83229e9	1.23061
$552$	6.70827e9	1.69756
$553$	−2.35457e9	−0.592072
$554$	5.67608e9	1.41829
$555$	2.20263e8	0.0546910
$556$	−1.06862e10	−2.63671
$557$	−7.67789e9	−1.88256	−0.941280	−	0.337627i	$-0.890376\pi$
$557$	−7.67789e9	−1.88256	−0.941280	+	0.337627i	$0.890376\pi$
$558$	−2.19773e10	−5.35494
$559$	−7.74493e8	−0.187532
$560$	4.56962e8	0.109957
$561$	2.80756e9	0.671366
$562$	1.01714e7	0.00241716
$563$	7.00323e8	0.165394	0.0826968	−	0.996575i	$-0.473647\pi$
$563$	7.00323e8	0.165394	0.0826968	+	0.996575i	$0.473647\pi$
$564$	−1.87635e10	−4.40389
$565$	−7.21644e8	−0.168327
$566$	1.46083e9	0.338643
$567$	1.90132e10	4.38040
$568$	−5.66081e9	−1.29616
$569$	1.86146e9	0.423605	0.211803	−	0.977312i	$-0.432067\pi$
$569$	1.86146e9	0.423605	0.211803	+	0.977312i	$0.432067\pi$
$570$	−5.39856e9	−1.22100
$571$	6.78752e9	1.52575	0.762877	−	0.646544i	$-0.223786\pi$
$571$	6.78752e9	1.52575	0.762877	+	0.646544i	$0.223786\pi$
$572$	−3.21930e9	−0.719243
$573$	6.46752e9	1.43614
$574$	4.37101e9	0.964695
$575$	2.91180e9	0.638741
$576$	−1.86062e10	−4.05676
$577$	−4.99855e8	−0.108325	−0.0541625	−	0.998532i	$-0.517249\pi$
$577$	−4.99855e8	−0.108325	−0.0541625	+	0.998532i	$0.517249\pi$
$578$	8.60702e8	0.185398
$579$	6.14597e9	1.31588
$580$	1.70103e9	0.362005
$581$	−9.18567e9	−1.94310
$582$	−1.89126e10	−3.97669
$583$	−1.52538e9	−0.318814
$584$	−9.12343e9	−1.89545
$585$	4.08020e9	0.842628
$586$	−7.41512e9	−1.52222
$587$	7.26067e9	1.48164	0.740821	−	0.671703i	$-0.234437\pi$
$587$	7.26067e9	1.48164	0.740821	+	0.671703i	$0.234437\pi$
$588$	−6.40910e9	−1.30010
$589$	−8.83061e9	−1.78069
$590$	3.46829e9	0.695238
$591$	−1.14153e10	−2.27474
$592$	−2.12637e8	−0.0421224
$593$	6.06525e9	1.19442	0.597210	−	0.802085i	$-0.296276\pi$
$593$	6.06525e9	1.19442	0.597210	+	0.802085i	$0.296276\pi$
$594$	9.38164e9	1.83665
$595$	1.59584e9	0.310584
$596$	7.68999e9	1.48787
$597$	2.58425e9	0.497078
$598$	−7.30086e9	−1.39611
$599$	−5.97141e8	−0.113523	−0.0567614	−	0.998388i	$-0.518077\pi$
$599$	−5.97141e8	−0.113523	−0.0567614	+	0.998388i	$0.518077\pi$
$600$	1.23474e10	2.33370
$601$	2.66025e8	0.0499876	0.0249938	−	0.999688i	$-0.492043\pi$
$601$	2.66025e8	0.0499876	0.0249938	+	0.999688i	$0.492043\pi$
$602$	−1.59681e9	−0.298308
$603$	−9.02378e9	−1.67601
$604$	−1.29168e10	−2.38521
$605$	−1.21785e9	−0.223588
$606$	7.90771e9	1.44343
$607$	5.53315e9	1.00418	0.502090	−	0.864815i	$-0.332564\pi$
$607$	5.53315e9	1.00418	0.502090	+	0.864815i	$0.332564\pi$
$608$	−5.59738e9	−1.01000
$609$	−1.02866e10	−1.84549
$610$	4.03926e8	0.0720522
$611$	8.91007e9	1.58029
$612$	2.89681e10	5.10846
$613$	3.58911e9	0.629325	0.314662	−	0.949204i	$-0.398109\pi$
$613$	3.58911e9	0.629325	0.314662	+	0.949204i	$0.398109\pi$
$614$	9.44028e9	1.64587
$615$	−1.37854e9	−0.238976
$616$	−2.89601e9	−0.499192
$617$	−3.27338e9	−0.561046	−0.280523	−	0.959847i	$-0.590508\pi$
$617$	−3.27338e9	−0.561046	−0.280523	+	0.959847i	$0.590508\pi$
$618$	2.92152e9	0.497908
$619$	−9.05095e9	−1.53383	−0.766914	−	0.641749i	$-0.778209\pi$
$619$	−9.05095e9	−1.53383	−0.766914	+	0.641749i	$0.778209\pi$
$620$	−3.10850e9	−0.523817
$621$	1.36064e10	2.27993
$622$	−7.64691e9	−1.27415
$623$	−4.79122e9	−0.793850
$624$	−5.38096e9	−0.886572
$625$	4.97545e9	0.815177
$626$	−1.53906e9	−0.250753
$627$	5.94665e9	0.963465
$628$	2.13204e10	3.43508
$629$	−7.42588e8	−0.118979
$630$	8.41231e9	1.34037
$631$	−4.70226e9	−0.745082	−0.372541	−	0.928016i	$-0.621513\pi$
$631$	−4.70226e9	−0.745082	−0.372541	+	0.928016i	$0.621513\pi$
$632$	−4.12451e9	−0.649925
$633$	1.03801e10	1.62663
$634$	−7.05271e9	−1.09912
$635$	−1.80860e9	−0.280307
$636$	−2.15006e10	−3.31398
$637$	3.04344e9	0.466528
$638$	−2.92991e9	−0.446665
$639$	−1.81129e10	−2.74622
$640$	−3.00447e9	−0.453041
$641$	8.63472e9	1.29493	0.647463	−	0.762097i	$-0.275830\pi$
$641$	8.63472e9	1.29493	0.647463	+	0.762097i	$0.275830\pi$
$642$	1.19346e10	1.78007
$643$	1.85229e9	0.274770	0.137385	−	0.990518i	$-0.456130\pi$
$643$	1.85229e9	0.274770	0.137385	+	0.990518i	$0.456130\pi$
$644$	−9.62632e9	−1.42023
$645$	5.03603e8	0.0738974
$646$	1.82006e10	2.65626
$647$	4.30999e9	0.625622	0.312811	−	0.949815i	$-0.398729\pi$
$647$	4.30999e9	0.625622	0.312811	+	0.949815i	$0.398729\pi$
$648$	3.33054e10	4.80841
$649$	−3.82041e9	−0.548597
$650$	−1.34381e10	−1.91929
$651$	1.87979e10	2.67040
$652$	9.13252e9	1.29040
$653$	−5.51577e9	−0.775193	−0.387597	−	0.921829i	$-0.626695\pi$
$653$	−5.51577e9	−0.775193	−0.387597	+	0.921829i	$0.626695\pi$
$654$	−1.69144e10	−2.36447
$655$	−1.53417e7	−0.00213319
$656$	1.33081e9	0.184057
$657$	−2.91922e10	−4.01595
$658$	1.83703e10	2.51376
$659$	1.13199e10	1.54079	0.770397	−	0.637564i	$-0.220058\pi$
$659$	1.13199e10	1.54079	0.770397	+	0.637564i	$0.220058\pi$
$660$	2.09330e9	0.283419
$661$	5.19183e9	0.699222	0.349611	−	0.936895i	$-0.386314\pi$
$661$	5.19183e9	0.699222	0.349611	+	0.936895i	$0.386314\pi$
$662$	1.65826e9	0.222152
$663$	−1.87918e10	−2.50422
$664$	−1.60906e10	−2.13296
$665$	3.38012e9	0.445714
$666$	−3.91448e9	−0.513469
$667$	−4.24932e9	−0.554471
$668$	2.79007e9	0.362158
$669$	7.37603e9	0.952426
$670$	−1.99578e9	−0.256360
$671$	−4.44934e8	−0.0568548
$672$	1.19153e10	1.51465
$673$	−7.58270e9	−0.958895	−0.479447	−	0.877571i	$-0.659163\pi$
$673$	−7.58270e9	−0.958895	−0.479447	+	0.877571i	$0.659163\pi$
$674$	−4.83577e9	−0.608354
$675$	2.50442e10	3.13432
$676$	7.29887e9	0.908746
$677$	4.96130e9	0.614519	0.307259	−	0.951626i	$-0.400588\pi$
$677$	4.96130e9	0.614519	0.307259	+	0.951626i	$0.400588\pi$
$678$	1.75201e10	2.15891
$679$	1.18415e10	1.45165
$680$	2.79543e9	0.340932
$681$	−3.08303e10	−3.74079
$682$	5.35418e9	0.646320
$683$	−1.05886e10	−1.27165	−0.635823	−	0.771835i	$-0.719339\pi$
$683$	−1.05886e10	−1.27165	−0.635823	+	0.771835i	$0.719339\pi$
$684$	6.13569e10	7.33106
$685$	1.18930e9	0.141376
$686$	−1.02651e10	−1.21403
$687$	1.04482e10	1.22939
$688$	−4.86167e8	−0.0569149
$689$	1.02098e10	1.18919
$690$	4.74727e9	0.550139
$691$	−1.26917e10	−1.46335	−0.731674	−	0.681655i	$-0.761261\pi$
$691$	−1.26917e10	−1.46335	−0.731674	+	0.681655i	$0.761261\pi$
$692$	−1.19321e9	−0.136882
$693$	−9.26637e9	−1.05765
$694$	−5.79812e9	−0.658460
$695$	−3.29961e9	−0.372834
$696$	−1.80190e10	−2.02581
$697$	4.64756e9	0.519888
$698$	−2.42850e10	−2.70298
$699$	1.64489e10	1.82166
$700$	−1.77184e10	−1.95246
$701$	−1.15011e10	−1.26104	−0.630519	−	0.776174i	$-0.717158\pi$
$701$	−1.15011e10	−1.26104	−0.630519	+	0.776174i	$0.717158\pi$
$702$	−6.27941e10	−6.85076
$703$	−1.57286e9	−0.170745
$704$	4.53291e9	0.489635
$705$	−5.79364e9	−0.622715
$706$	2.72138e10	2.91054
$707$	−4.95114e9	−0.526911
$708$	−5.38496e10	−5.70251
$709$	−1.09458e9	−0.115342	−0.0576709	−	0.998336i	$-0.518367\pi$
$709$	−1.09458e9	−0.115342	−0.0576709	+	0.998336i	$0.518367\pi$
$710$	−4.00601e9	−0.420057
$711$	−1.31972e10	−1.37701
$712$	−8.39280e9	−0.871418
$713$	7.76528e9	0.802313
$714$	−3.87439e10	−3.98345
$715$	−9.94031e8	−0.101702
$716$	1.29578e10	1.31928
$717$	3.23837e8	0.0328102
$718$	1.66394e10	1.67765
$719$	−6.22726e9	−0.624807	−0.312403	−	0.949950i	$-0.601134\pi$
$719$	−6.22726e9	−0.624807	−0.312403	+	0.949950i	$0.601134\pi$
$720$	2.56123e9	0.255732
$721$	−1.82921e9	−0.181756
$722$	2.17066e10	2.14640
$723$	−1.35704e10	−1.33539
$724$	−1.42348e10	−1.39401
$725$	−7.82138e9	−0.762255
$726$	2.95670e10	2.86767
$727$	1.91377e10	1.84722	0.923611	−	0.383332i	$-0.125223\pi$
$727$	1.91377e10	1.84722	0.923611	+	0.383332i	$0.125223\pi$
$728$	1.93839e10	1.86201
$729$	3.89806e10	3.72651
$730$	−6.45642e9	−0.614273
$731$	−1.69783e9	−0.160762
$732$	−6.27145e9	−0.590989
$733$	9.03821e9	0.847654	0.423827	−	0.905743i	$-0.360686\pi$
$733$	9.03821e9	0.847654	0.423827	+	0.905743i	$0.360686\pi$
$734$	−3.02997e9	−0.282815
$735$	−1.97895e9	−0.183836
$736$	4.92211e9	0.455071
$737$	2.19840e9	0.202288
$738$	2.44991e10	2.24364
$739$	−1.24567e9	−0.113540	−0.0567698	−	0.998387i	$-0.518080\pi$
$739$	−1.24567e9	−0.113540	−0.0567698	+	0.998387i	$0.518080\pi$
$740$	−5.53670e8	−0.0502273
$741$	−3.98027e10	−3.59376
$742$	2.10500e10	1.89164
$743$	1.95885e10	1.75203	0.876013	−	0.482287i	$-0.160194\pi$
$743$	1.95885e10	1.75203	0.876013	+	0.482287i	$0.160194\pi$
$744$	3.29283e10	2.93133
$745$	2.37446e9	0.210386
$746$	2.17425e9	0.191745
$747$	−5.14850e10	−4.51917
$748$	−7.05730e9	−0.616571
$749$	−7.47244e9	−0.649795
$750$	1.80626e10	1.56339
$751$	1.10559e10	0.952473	0.476236	−	0.879317i	$-0.342001\pi$
$751$	1.10559e10	0.952473	0.476236	+	0.879317i	$0.342001\pi$
$752$	5.59305e9	0.479608
$753$	2.37841e10	2.03004
$754$	1.96108e10	1.66608
$755$	−3.98835e9	−0.337271
$756$	−8.27952e10	−6.96913
$757$	−1.11431e10	−0.933617	−0.466808	−	0.884358i	$-0.654596\pi$
$757$	−1.11431e10	−0.933617	−0.466808	+	0.884358i	$0.654596\pi$
$758$	3.32543e10	2.77335
$759$	−5.22924e9	−0.434103
$760$	5.92096e9	0.489266
$761$	−1.64914e10	−1.35647	−0.678236	−	0.734844i	$-0.737255\pi$
$761$	−1.64914e10	−1.35647	−0.678236	+	0.734844i	$0.737255\pi$
$762$	4.39093e10	3.59513
$763$	1.05903e10	0.863125
$764$	−1.62573e10	−1.31893
$765$	8.94455e9	0.722343
$766$	4.75099e9	0.381929
$767$	2.55712e10	2.04629
$768$	3.69279e10	2.94165
$769$	−2.19764e10	−1.74266	−0.871332	−	0.490694i	$-0.836743\pi$
$769$	−2.19764e10	−1.74266	−0.871332	+	0.490694i	$0.836743\pi$
$770$	−2.04943e9	−0.161777
$771$	−2.59277e10	−2.03739
$772$	−1.54490e10	−1.20848
$773$	1.65022e10	1.28503	0.642517	−	0.766271i	$-0.277890\pi$
$773$	1.65022e10	1.28503	0.642517	+	0.766271i	$0.277890\pi$
$774$	−8.94996e9	−0.693790
$775$	1.42929e10	1.10297
$776$	2.07427e10	1.59349
$777$	3.34819e9	0.256057
$778$	−2.21268e10	−1.68457
$779$	9.84391e9	0.746082
$780$	−1.40111e10	−1.05716
$781$	4.41272e9	0.331457
$782$	−1.60048e10	−1.19682
$783$	−3.65480e10	−2.72081
$784$	1.91044e9	0.141588
$785$	6.58316e9	0.485725
$786$	3.72467e8	0.0273595
$787$	1.33119e10	0.973483	0.486741	−	0.873546i	$-0.338185\pi$
$787$	1.33119e10	0.973483	0.486741	+	0.873546i	$0.338185\pi$
$788$	2.86944e10	2.08908
$789$	−3.74684e10	−2.71579
$790$	−2.91881e9	−0.210626
$791$	−1.09696e10	−0.788087
$792$	−1.62319e10	−1.16100
$793$	2.97808e9	0.212070
$794$	−2.26171e10	−1.60349
$795$	−6.63878e9	−0.468601
$796$	−6.49597e9	−0.456508
$797$	6.86024e9	0.479993	0.239997	−	0.970774i	$-0.422854\pi$
$797$	6.86024e9	0.479993	0.239997	+	0.970774i	$0.422854\pi$
$798$	−8.20628e10	−5.71658
$799$	1.95325e10	1.35470
$800$	9.05974e9	0.625606
$801$	−2.68544e10	−1.84630
$802$	−3.62340e9	−0.248031
$803$	7.11191e9	0.484710
$804$	3.09870e10	2.10273
$805$	−2.97234e9	−0.200823
$806$	−3.58371e10	−2.41080
$807$	1.00500e10	0.673145
$808$	−8.67292e9	−0.578396
$809$	7.51249e9	0.498843	0.249422	−	0.968395i	$-0.419760\pi$
$809$	7.51249e9	0.498843	0.249422	+	0.968395i	$0.419760\pi$
$810$	2.35694e10	1.55830
$811$	1.36176e10	0.896453	0.448226	−	0.893920i	$-0.352056\pi$
$811$	1.36176e10	0.896453	0.448226	+	0.893920i	$0.352056\pi$
$812$	2.58572e10	1.69486
$813$	−9.20121e9	−0.600521
$814$	9.53659e8	0.0619737
$815$	2.81987e9	0.182464
$816$	−1.17961e10	−0.760014
$817$	−3.59615e9	−0.230707
$818$	8.76964e9	0.560202
$819$	6.20226e10	3.94508
$820$	3.46519e9	0.219472
$821$	−8.07643e8	−0.0509353	−0.0254676	−	0.999676i	$-0.508107\pi$
$821$	−8.07643e8	−0.0509353	−0.0254676	+	0.999676i	$0.508107\pi$
$822$	−2.88739e10	−1.81324
$823$	−2.23299e10	−1.39633	−0.698163	−	0.715938i	$-0.745999\pi$
$823$	−2.23299e10	−1.39633	−0.698163	+	0.715938i	$0.745999\pi$
$824$	−3.20423e9	−0.199516
$825$	−9.62504e9	−0.596780
$826$	5.27211e10	3.25502
$827$	1.07538e9	0.0661136	0.0330568	−	0.999453i	$-0.489476\pi$
$827$	1.07538e9	0.0661136	0.0330568	+	0.999453i	$0.489476\pi$
$828$	−5.39547e10	−3.30311
$829$	1.30676e10	0.796628	0.398314	−	0.917249i	$-0.369595\pi$
$829$	1.30676e10	0.796628	0.398314	+	0.917249i	$0.369595\pi$
$830$	−1.13869e10	−0.691244
$831$	−2.72115e10	−1.64494
$832$	−3.03401e10	−1.82636
$833$	6.67179e9	0.399931
$834$	8.01082e10	4.78185
$835$	8.61497e8	0.0512096
$836$	−1.49480e10	−0.884830
$837$	6.67886e10	3.93698
$838$	−4.23380e10	−2.48529
$839$	−7.30739e9	−0.427165	−0.213582	−	0.976925i	$-0.568513\pi$
$839$	−7.30739e9	−0.427165	−0.213582	+	0.976925i	$0.568513\pi$
$840$	−1.26041e10	−0.733725
$841$	−5.83582e9	−0.338311
$842$	1.24885e10	0.720974
$843$	−4.87626e7	−0.00280343
$844$	−2.60923e10	−1.49387
$845$	2.25369e9	0.128498
$846$	1.02964e11	5.84640
$847$	−1.85123e10	−1.04681
$848$	6.40894e9	0.360911
$849$	−7.00334e9	−0.392761
$850$	−2.94588e10	−1.64531
$851$	1.38311e9	0.0769315
$852$	6.21983e10	3.44540
$853$	−2.19746e10	−1.21227	−0.606134	−	0.795363i	$-0.707280\pi$
$853$	−2.19746e10	−1.21227	−0.606134	+	0.795363i	$0.707280\pi$
$854$	6.14003e9	0.337340
$855$	1.89453e10	1.03662
$856$	−1.30895e10	−0.713288
$857$	−4.09566e9	−0.222275	−0.111138	−	0.993805i	$-0.535449\pi$
$857$	−4.09566e9	−0.222275	−0.111138	+	0.993805i	$0.535449\pi$
$858$	2.41332e10	1.30439
$859$	−2.48200e10	−1.33606	−0.668031	−	0.744134i	$-0.732863\pi$
$859$	−2.48200e10	−1.33606	−0.668031	+	0.744134i	$0.732863\pi$
$860$	−1.26590e9	−0.0678661
$861$	−2.09549e10	−1.11886
$862$	−8.26757e9	−0.439645
$863$	−9.32141e9	−0.493678	−0.246839	−	0.969056i	$-0.579392\pi$
$863$	−9.32141e9	−0.493678	−0.246839	+	0.969056i	$0.579392\pi$
$864$	4.23347e10	2.23305
$865$	−3.68430e8	−0.0193552
$866$	2.03590e9	0.106523
$867$	−4.12627e9	−0.215026
$868$	−4.72519e10	−2.45245
$869$	3.21515e9	0.166200
$870$	−1.27516e10	−0.656520
$871$	−1.47146e10	−0.754543
$872$	1.85511e10	0.947463
$873$	6.63705e10	3.37618
$874$	−3.38996e10	−1.71753
$875$	−1.13093e10	−0.570699
$876$	1.00244e11	5.03841
$877$	−1.06318e10	−0.532239	−0.266120	−	0.963940i	$-0.585742\pi$
$877$	−1.06318e10	−0.532239	−0.266120	+	0.963940i	$0.585742\pi$
$878$	−4.88903e10	−2.43777
$879$	3.55486e10	1.76548
$880$	−6.23976e8	−0.0308659
$881$	8.99181e9	0.443028	0.221514	−	0.975157i	$-0.428900\pi$
$881$	8.99181e9	0.443028	0.221514	+	0.975157i	$0.428900\pi$
$882$	3.51697e10	1.72595
$883$	7.28785e9	0.356235	0.178118	−	0.984009i	$-0.442999\pi$
$883$	7.28785e9	0.356235	0.178118	+	0.984009i	$0.442999\pi$
$884$	4.72366e10	2.29983
$885$	−1.66273e10	−0.806342
$886$	2.49449e10	1.20494
$887$	2.86600e10	1.37893	0.689467	−	0.724317i	$-0.257845\pi$
$887$	2.86600e10	1.37893	0.689467	+	0.724317i	$0.257845\pi$
$888$	5.86503e9	0.281077
$889$	−2.74923e10	−1.31236
$890$	−5.93937e9	−0.282407
$891$	−2.59622e10	−1.22962
$892$	−1.85410e10	−0.874692
$893$	4.13715e10	1.94411
$894$	−5.76473e10	−2.69834
$895$	4.00102e9	0.186548
$896$	−4.56706e10	−2.12109
$897$	3.50009e10	1.61922
$898$	3.27367e10	1.50858
$899$	−2.08583e10	−0.957457
$900$	−9.93102e10	−4.54093
$901$	2.23818e10	1.01943
$902$	−5.96856e9	−0.270799
$903$	7.65520e9	0.345979
$904$	−1.92155e10	−0.865092
$905$	−4.39530e9	−0.197115
$906$	9.68296e10	4.32573
$907$	2.38033e9	0.105928	0.0529642	−	0.998596i	$-0.483133\pi$
$907$	2.38033e9	0.105928	0.0529642	+	0.998596i	$0.483133\pi$
$908$	7.74976e10	3.43548
$909$	−2.77507e10	−1.22546
$910$	1.37175e10	0.603434
$911$	−3.27857e10	−1.43671	−0.718356	−	0.695676i	$-0.755105\pi$
$911$	−3.27857e10	−1.43671	−0.718356	+	0.695676i	$0.755105\pi$
$912$	−2.49851e10	−1.09068
$913$	1.25429e10	0.545446
$914$	−2.59051e10	−1.12221
$915$	−1.93645e9	−0.0835666
$916$	−2.62633e10	−1.12906
$917$	−2.33207e8	−0.00998732
$918$	−1.37656e11	−5.87282
$919$	1.71738e10	0.729899	0.364950	−	0.931027i	$-0.381086\pi$
$919$	1.71738e10	0.729899	0.364950	+	0.931027i	$0.381086\pi$
$920$	−5.20665e9	−0.220446
$921$	−4.52574e10	−1.90889
$922$	−2.54131e10	−1.06782
$923$	−2.95357e10	−1.23635
$924$	3.18200e10	1.32693
$925$	2.54578e9	0.105761
$926$	−3.43904e10	−1.42331
$927$	−1.02526e10	−0.422721
$928$	−1.32212e10	−0.543068
$929$	−4.09833e9	−0.167707	−0.0838536	−	0.996478i	$-0.526723\pi$
$929$	−4.09833e9	−0.167707	−0.0838536	+	0.996478i	$0.526723\pi$
$930$	2.33025e10	0.949977
$931$	1.41314e10	0.573934
$932$	−4.13474e10	−1.67299
$933$	3.66599e10	1.47776
$934$	2.56363e10	1.02953
$935$	−2.17910e9	−0.0871839
$936$	1.08645e11	4.33057
$937$	3.03563e10	1.20548	0.602740	−	0.797938i	$-0.294076\pi$
$937$	3.03563e10	1.20548	0.602740	+	0.797938i	$0.294076\pi$
$938$	−3.03376e10	−1.20025
$939$	7.37838e9	0.290825
$940$	1.45633e10	0.571891
$941$	−3.15855e10	−1.23573	−0.617867	−	0.786283i	$-0.712003\pi$
$941$	−3.15855e10	−1.23573	−0.617867	+	0.786283i	$0.712003\pi$
$942$	−1.59827e11	−6.22975
$943$	−8.65634e9	−0.336158
$944$	1.60516e10	0.621035
$945$	−2.55649e10	−0.985444
$946$	2.18042e9	0.0837377
$947$	−1.44552e10	−0.553095	−0.276548	−	0.961000i	$-0.589190\pi$
$947$	−1.44552e10	−0.553095	−0.276548	+	0.961000i	$0.589190\pi$
$948$	4.53182e10	1.72760
$949$	−4.76021e10	−1.80798
$950$	−6.23962e10	−2.36116
$951$	3.38112e10	1.27476
$952$	4.24930e10	1.59620
$953$	−8.95669e9	−0.335214	−0.167607	−	0.985854i	$-0.553604\pi$
$953$	−8.95669e9	−0.335214	−0.167607	+	0.985854i	$0.553604\pi$
$954$	1.17984e11	4.39949
$955$	−5.01979e9	−0.186498
$956$	−8.14022e8	−0.0301324
$957$	1.40462e10	0.518045
$958$	5.75729e10	2.11562
$959$	1.80784e10	0.661904
$960$	1.97282e10	0.719678
$961$	1.06042e10	0.385430
$962$	−6.38313e9	−0.231164
$963$	−4.18824e10	−1.51126
$964$	3.41117e10	1.22640
$965$	−4.77022e9	−0.170881
$966$	7.21627e10	2.57569
$967$	3.78882e10	1.34745	0.673723	−	0.738984i	$-0.264694\pi$
$967$	3.78882e10	1.34745	0.673723	+	0.738984i	$0.264694\pi$
$968$	−3.24281e10	−1.14910
$969$	−8.72548e10	−3.08075
$970$	1.46791e10	0.516415
$971$	3.53468e10	1.23903	0.619516	−	0.784984i	$-0.287329\pi$
$971$	3.53468e10	1.23903	0.619516	+	0.784984i	$0.287329\pi$
$972$	−1.96053e11	−6.84765
$973$	−5.01570e10	−1.74557
$974$	−7.24023e10	−2.51071
$975$	6.44233e10	2.22601
$976$	1.86941e9	0.0643620
$977$	4.28246e10	1.46914	0.734568	−	0.678535i	$-0.237385\pi$
$977$	4.28246e10	1.46914	0.734568	+	0.678535i	$0.237385\pi$
$978$	−6.84610e10	−2.34022
$979$	6.54236e9	0.222841
$980$	4.97445e9	0.168832
$981$	5.93579e10	2.00742
$982$	−1.81031e9	−0.0610048
$983$	1.30981e10	0.439815	0.219908	−	0.975521i	$-0.429424\pi$
$983$	1.30981e10	0.439815	0.219908	+	0.975521i	$0.429424\pi$
$984$	−3.67068e10	−1.22819
$985$	8.86005e9	0.295399
$986$	4.29905e10	1.42825
$987$	−8.80684e10	−2.91548
$988$	1.00051e11	3.30045
$989$	3.16231e9	0.103948
$990$	−1.14869e10	−0.376253
$991$	−2.48042e10	−0.809593	−0.404796	−	0.914407i	$-0.632658\pi$
$991$	−2.48042e10	−0.809593	−0.404796	+	0.914407i	$0.632658\pi$
$992$	2.41608e10	0.785814
$993$	−7.94985e9	−0.257654
$994$	−6.08949e10	−1.96666
$995$	−2.00578e9	−0.0645508
$996$	1.76796e11	5.66975
$997$	4.36888e10	1.39617	0.698083	−	0.716017i	$-0.254037\pi$
$997$	4.36888e10	1.39617	0.698083	+	0.716017i	$0.254037\pi$
$998$	8.74368e9	0.278444
$999$	1.18960e10	0.377506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.a.b.1.12	✓	13
3.2	odd	2		387.8.a.d.1.2		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.b.1.12	✓	13	1.1	even	1	trivial
387.8.a.d.1.2		13	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.a (trivial)

\(f(q)\)	\(=\)	\(q+18.8435 q^{2} -90.3374 q^{3} +227.079 q^{4} +70.1157 q^{5} -1702.28 q^{6} +1065.82 q^{7} +1867.00 q^{8} +5973.84 q^{9} +O(q^{10})\)	\(q+18.8435 q^{2} -90.3374 q^{3} +227.079 q^{4} +70.1157 q^{5} -1702.28 q^{6} +1065.82 q^{7} +1867.00 q^{8} +5973.84 q^{9} +1321.23 q^{10} -1455.37 q^{11} -20513.7 q^{12} +9741.20 q^{13} +20083.8 q^{14} -6334.07 q^{15} +6114.77 q^{16} +21354.5 q^{17} +112568. q^{18} +45230.6 q^{19} +15921.8 q^{20} -96283.4 q^{21} -27424.2 q^{22} -39774.0 q^{23} -168660. q^{24} -73208.8 q^{25} +183559. q^{26} -342093. q^{27} +242025. q^{28} +106837. q^{29} -119356. q^{30} -195235. q^{31} -123752. q^{32} +131474. q^{33} +402394. q^{34} +74730.8 q^{35} +1.35653e6 q^{36} -34774.3 q^{37} +852305. q^{38} -879994. q^{39} +130906. q^{40} +217638. q^{41} -1.81432e6 q^{42} -79507.0 q^{43} -330483. q^{44} +418860. q^{45} -749483. q^{46} +914679. q^{47} -552392. q^{48} +312430. q^{49} -1.37951e6 q^{50} -1.92911e6 q^{51} +2.21202e6 q^{52} +1.04811e6 q^{53} -6.44624e6 q^{54} -102044. q^{55} +1.98989e6 q^{56} -4.08602e6 q^{57} +2.01318e6 q^{58} +2.62505e6 q^{59} -1.43833e6 q^{60} +305720. q^{61} -3.67892e6 q^{62} +6.36704e6 q^{63} -3.11462e6 q^{64} +683011. q^{65} +2.47743e6 q^{66} -1.51055e6 q^{67} +4.84916e6 q^{68} +3.59308e6 q^{69} +1.40819e6 q^{70} -3.03204e6 q^{71} +1.11532e7 q^{72} -4.88668e6 q^{73} -655271. q^{74} +6.61349e6 q^{75} +1.02709e7 q^{76} -1.55116e6 q^{77} -1.65822e7 q^{78} -2.20917e6 q^{79} +428742. q^{80} +1.78390e7 q^{81} +4.10107e6 q^{82} -8.61841e6 q^{83} -2.18639e7 q^{84} +1.49729e6 q^{85} -1.49819e6 q^{86} -9.65134e6 q^{87} -2.71717e6 q^{88} -4.49534e6 q^{89} +7.89280e6 q^{90} +1.03824e7 q^{91} -9.03184e6 q^{92} +1.76370e7 q^{93} +1.72358e7 q^{94} +3.17138e6 q^{95} +1.11794e7 q^{96} +1.11102e7 q^{97} +5.88729e6 q^{98} -8.69412e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9}+O(q^{10}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9} + 4667 q^{10} + 1620 q^{11} - 19681 q^{12} + 13550 q^{13} + 44160 q^{14} + 31412 q^{15} + 114026 q^{16} + 110880 q^{17} + 159267 q^{18} + 105058 q^{19} + 167251 q^{20} + 129840 q^{21} + 201504 q^{22} + 160184 q^{23} + 161289 q^{24} + 270149 q^{25} + 272104 q^{26} + 252544 q^{27} + 208172 q^{28} + 285546 q^{29} + 107580 q^{30} - 99616 q^{31} + 200126 q^{32} + 531468 q^{33} - 80941 q^{34} - 187104 q^{35} - 608975 q^{36} + 176038 q^{37} + 652165 q^{38} - 794680 q^{39} - 895387 q^{40} - 410260 q^{41} - 3413218 q^{42} - 1033591 q^{43} - 2177076 q^{44} - 1051178 q^{45} - 3975765 q^{46} - 424556 q^{47} - 2360477 q^{48} - 1561359 q^{49} - 4063801 q^{50} - 2375738 q^{51} - 4172312 q^{52} + 3992458 q^{53} - 10438626 q^{54} + 406960 q^{55} + 1559556 q^{56} - 3116152 q^{57} - 4052005 q^{58} + 2248836 q^{59} - 2911436 q^{60} + 6210394 q^{61} + 885317 q^{62} + 11622368 q^{63} - 3096318 q^{64} + 5600420 q^{65} - 2174604 q^{66} - 1993648 q^{67} + 9327135 q^{68} + 13366240 q^{69} - 1105098 q^{70} + 4978064 q^{71} + 11370663 q^{72} + 8224814 q^{73} - 3613563 q^{74} + 27115592 q^{75} + 10687121 q^{76} + 17261892 q^{77} - 15226630 q^{78} + 6945708 q^{79} + 15822799 q^{80} + 35113185 q^{81} - 508449 q^{82} + 22937328 q^{83} - 14010106 q^{84} - 575532 q^{85} - 1272112 q^{86} + 9081380 q^{87} + 11202656 q^{88} + 9291302 q^{89} + 2841402 q^{90} + 25581108 q^{91} - 14388137 q^{92} + 25930480 q^{93} - 24645805 q^{94} + 30750464 q^{95} - 22461255 q^{96} + 10001852 q^{97} - 32304856 q^{98} + 5055452 q^{99}+O(q^{100}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9 + 4667 * q^10 + 1620 * q^11 - 19681 * q^12 + 13550 * q^13 + 44160 * q^14 + 31412 * q^15 + 114026 * q^16 + 110880 * q^17 + 159267 * q^18 + 105058 * q^19 + 167251 * q^20 + 129840 * q^21 + 201504 * q^22 + 160184 * q^23 + 161289 * q^24 + 270149 * q^25 + 272104 * q^26 + 252544 * q^27 + 208172 * q^28 + 285546 * q^29 + 107580 * q^30 - 99616 * q^31 + 200126 * q^32 + 531468 * q^33 - 80941 * q^34 - 187104 * q^35 - 608975 * q^36 + 176038 * q^37 + 652165 * q^38 - 794680 * q^39 - 895387 * q^40 - 410260 * q^41 - 3413218 * q^42 - 1033591 * q^43 - 2177076 * q^44 - 1051178 * q^45 - 3975765 * q^46 - 424556 * q^47 - 2360477 * q^48 - 1561359 * q^49 - 4063801 * q^50 - 2375738 * q^51 - 4172312 * q^52 + 3992458 * q^53 - 10438626 * q^54 + 406960 * q^55 + 1559556 * q^56 - 3116152 * q^57 - 4052005 * q^58 + 2248836 * q^59 - 2911436 * q^60 + 6210394 * q^61 + 885317 * q^62 + 11622368 * q^63 - 3096318 * q^64 + 5600420 * q^65 - 2174604 * q^66 - 1993648 * q^67 + 9327135 * q^68 + 13366240 * q^69 - 1105098 * q^70 + 4978064 * q^71 + 11370663 * q^72 + 8224814 * q^73 - 3613563 * q^74 + 27115592 * q^75 + 10687121 * q^76 + 17261892 * q^77 - 15226630 * q^78 + 6945708 * q^79 + 15822799 * q^80 + 35113185 * q^81 - 508449 * q^82 + 22937328 * q^83 - 14010106 * q^84 - 575532 * q^85 - 1272112 * q^86 + 9081380 * q^87 + 11202656 * q^88 + 9291302 * q^89 + 2841402 * q^90 + 25581108 * q^91 - 14388137 * q^92 + 25930480 * q^93 - 24645805 * q^94 + 30750464 * q^95 - 22461255 * q^96 + 10001852 * q^97 - 32304856 * q^98 + 5055452 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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