LMFDB - Embedded newform 69.8.a.d.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.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:	$21.5545667584$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 757x^{6} - 1170x^{5} + 170343x^{4} + 424132x^{3} - 9973075x^{2} - 5161010x + 130545120 $ x^8 - 757x^6 - 1170x^5 + 170343x^4 + 424132x^3 - 9973075x^2 - 5161010x + 130545120
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-11.6014$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+14.6014 q^{2} +27.0000 q^{3} +85.2018 q^{4} +493.797 q^{5} +394.239 q^{6} -368.336 q^{7} -624.915 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+14.6014 q^{2} +27.0000 q^{3} +85.2018 q^{4} +493.797 q^{5} +394.239 q^{6} -368.336 q^{7} -624.915 q^{8} +729.000 q^{9} +7210.15 q^{10} +7537.54 q^{11} +2300.45 q^{12} -417.322 q^{13} -5378.24 q^{14} +13332.5 q^{15} -20030.5 q^{16} -24880.5 q^{17} +10644.4 q^{18} +20861.4 q^{19} +42072.4 q^{20} -9945.08 q^{21} +110059. q^{22} +12167.0 q^{23} -16872.7 q^{24} +165711. q^{25} -6093.50 q^{26} +19683.0 q^{27} -31382.9 q^{28} -5058.15 q^{29} +194674. q^{30} -228761. q^{31} -212485. q^{32} +203513. q^{33} -363291. q^{34} -181883. q^{35} +62112.1 q^{36} +588441. q^{37} +304606. q^{38} -11267.7 q^{39} -308581. q^{40} +191846. q^{41} -145212. q^{42} +208779. q^{43} +642212. q^{44} +359978. q^{45} +177656. q^{46} -1.36255e6 q^{47} -540823. q^{48} -687871. q^{49} +2.41962e6 q^{50} -671774. q^{51} -35556.6 q^{52} -1.13105e6 q^{53} +287400. q^{54} +3.72202e6 q^{55} +230179. q^{56} +563257. q^{57} -73856.3 q^{58} -1.59555e6 q^{59} +1.13596e6 q^{60} -1.18637e6 q^{61} -3.34024e6 q^{62} -268517. q^{63} -538678. q^{64} -206072. q^{65} +2.97159e6 q^{66} +2.64240e6 q^{67} -2.11986e6 q^{68} +328509. q^{69} -2.65576e6 q^{70} -2.80200e6 q^{71} -455563. q^{72} -5.16484e6 q^{73} +8.59208e6 q^{74} +4.47419e6 q^{75} +1.77743e6 q^{76} -2.77635e6 q^{77} -164524. q^{78} -3.28919e6 q^{79} -9.89100e6 q^{80} +531441. q^{81} +2.80123e6 q^{82} +590082. q^{83} -847338. q^{84} -1.22859e7 q^{85} +3.04847e6 q^{86} -136570. q^{87} -4.71032e6 q^{88} +2.78595e6 q^{89} +5.25620e6 q^{90} +153715. q^{91} +1.03665e6 q^{92} -6.17655e6 q^{93} -1.98952e7 q^{94} +1.03013e7 q^{95} -5.73709e6 q^{96} -289056. q^{97} -1.00439e7 q^{98} +5.49486e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) $ 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9} + 11720 q^{10} + 6932 q^{11} + 15174 q^{12} + 12404 q^{13} + 30222 q^{14} + 10206 q^{15} + 27058 q^{16} + 24434 q^{17} + 17496 q^{18} - 14682 q^{19} - 3760 q^{20} + 3402 q^{21} + 36294 q^{22} + 97336 q^{23} + 113076 q^{24} + 144644 q^{25} + 325840 q^{26} + 157464 q^{27} - 21566 q^{28} + 255356 q^{29} + 316440 q^{30} + 450764 q^{31} + 647588 q^{32} + 187164 q^{33} + 191822 q^{34} + 1022616 q^{35} + 409698 q^{36} + 206240 q^{37} + 737372 q^{38} + 334908 q^{39} + 590028 q^{40} + 1053344 q^{41} + 815994 q^{42} + 1587806 q^{43} + 589366 q^{44} + 275562 q^{45} + 292008 q^{46} + 443336 q^{47} + 730566 q^{48} + 1944828 q^{49} - 1556112 q^{50} + 659718 q^{51} - 614236 q^{52} - 375530 q^{53} + 472392 q^{54} + 407792 q^{55} - 1316922 q^{56} - 396414 q^{57} - 1413384 q^{58} + 624008 q^{59} - 101520 q^{60} - 2005568 q^{61} - 3908272 q^{62} + 91854 q^{63} - 5082310 q^{64} + 646124 q^{65} + 979938 q^{66} - 2712286 q^{67} - 2289698 q^{68} + 2628072 q^{69} - 16499468 q^{70} - 6287176 q^{71} + 3053052 q^{72} - 10358312 q^{73} - 2000150 q^{74} + 3905388 q^{75} - 25107464 q^{76} - 2156840 q^{77} + 8797680 q^{78} - 8800574 q^{79} + 2384344 q^{80} + 4251528 q^{81} - 31799800 q^{82} + 384948 q^{83} - 582282 q^{84} - 17826684 q^{85} - 11563928 q^{86} + 6894612 q^{87} - 25202782 q^{88} - 3445530 q^{89} + 8543880 q^{90} - 16316740 q^{91} + 6837854 q^{92} + 12170628 q^{93} - 24237616 q^{94} + 26164288 q^{95} + 17484876 q^{96} - 28043764 q^{97} - 9998012 q^{98} + 5053428 q^{99}+O(q^{100}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9 + 11720 * q^10 + 6932 * q^11 + 15174 * q^12 + 12404 * q^13 + 30222 * q^14 + 10206 * q^15 + 27058 * q^16 + 24434 * q^17 + 17496 * q^18 - 14682 * q^19 - 3760 * q^20 + 3402 * q^21 + 36294 * q^22 + 97336 * q^23 + 113076 * q^24 + 144644 * q^25 + 325840 * q^26 + 157464 * q^27 - 21566 * q^28 + 255356 * q^29 + 316440 * q^30 + 450764 * q^31 + 647588 * q^32 + 187164 * q^33 + 191822 * q^34 + 1022616 * q^35 + 409698 * q^36 + 206240 * q^37 + 737372 * q^38 + 334908 * q^39 + 590028 * q^40 + 1053344 * q^41 + 815994 * q^42 + 1587806 * q^43 + 589366 * q^44 + 275562 * q^45 + 292008 * q^46 + 443336 * q^47 + 730566 * q^48 + 1944828 * q^49 - 1556112 * q^50 + 659718 * q^51 - 614236 * q^52 - 375530 * q^53 + 472392 * q^54 + 407792 * q^55 - 1316922 * q^56 - 396414 * q^57 - 1413384 * q^58 + 624008 * q^59 - 101520 * q^60 - 2005568 * q^61 - 3908272 * q^62 + 91854 * q^63 - 5082310 * q^64 + 646124 * q^65 + 979938 * q^66 - 2712286 * q^67 - 2289698 * q^68 + 2628072 * q^69 - 16499468 * q^70 - 6287176 * q^71 + 3053052 * q^72 - 10358312 * q^73 - 2000150 * q^74 + 3905388 * q^75 - 25107464 * q^76 - 2156840 * q^77 + 8797680 * q^78 - 8800574 * q^79 + 2384344 * q^80 + 4251528 * q^81 - 31799800 * q^82 + 384948 * q^83 - 582282 * q^84 - 17826684 * q^85 - 11563928 * q^86 + 6894612 * q^87 - 25202782 * q^88 - 3445530 * q^89 + 8543880 * q^90 - 16316740 * q^91 + 6837854 * q^92 + 12170628 * q^93 - 24237616 * q^94 + 26164288 * q^95 + 17484876 * q^96 - 28043764 * q^97 - 9998012 * q^98 + 5053428 * 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 $

$2$	14.6014	1.29060	0.645298	−	0.763931i	$-0.276733\pi$
$2$	14.6014	1.29060	0.645298	+	0.763931i	$0.276733\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	85.2018	0.665639
$5$	493.797	1.76666	0.883332	−	0.468749i	$-0.155295\pi$
$5$	493.797	1.76666	0.883332	+	0.468749i	$0.155295\pi$
$6$	394.239	0.745126
$7$	−368.336	−0.405883	−0.202942	−	0.979191i	$-0.565050\pi$
$7$	−368.336	−0.405883	−0.202942	+	0.979191i	$0.565050\pi$
$8$	−624.915	−0.431525
$9$	729.000	0.333333
$10$	7210.15	2.28005
$11$	7537.54	1.70748	0.853739	−	0.520701i	$-0.174330\pi$
$11$	7537.54	1.70748	0.853739	+	0.520701i	$0.174330\pi$
$12$	2300.45	0.384307
$13$	−417.322	−0.0526829	−0.0263414	−	0.999653i	$-0.508386\pi$
$13$	−417.322	−0.0526829	−0.0263414	+	0.999653i	$0.508386\pi$
$14$	−5378.24	−0.523832
$15$	13332.5	1.01998
$16$	−20030.5	−1.22256
$17$	−24880.5	−1.22825	−0.614127	−	0.789207i	$-0.710492\pi$
$17$	−24880.5	−1.22825	−0.614127	+	0.789207i	$0.710492\pi$
$18$	10644.4	0.430199
$19$	20861.4	0.697759	0.348879	−	0.937168i	$-0.386562\pi$
$19$	20861.4	0.697759	0.348879	+	0.937168i	$0.386562\pi$
$20$	42072.4	1.17596
$21$	−9945.08	−0.234337
$22$	110059.	2.20366
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	−16872.7	−0.249141
$25$	165711.	2.12110
$26$	−6093.50	−0.0679923
$27$	19683.0	0.192450
$28$	−31382.9	−0.270172
$29$	−5058.15	−0.0385122	−0.0192561	−	0.999815i	$-0.506130\pi$
$29$	−5058.15	−0.0385122	−0.0192561	+	0.999815i	$0.506130\pi$
$30$	194674.	1.31639
$31$	−228761.	−1.37917	−0.689583	−	0.724207i	$-0.742206\pi$
$31$	−228761.	−1.37917	−0.689583	+	0.724207i	$0.742206\pi$
$32$	−212485.	−1.14631
$33$	203513.	0.985813
$34$	−363291.	−1.58518
$35$	−181883.	−0.717059
$36$	62112.1	0.221880
$37$	588441.	1.90984	0.954920	−	0.296864i	$-0.0959409\pi$
$37$	588441.	1.90984	0.954920	+	0.296864i	$0.0959409\pi$
$38$	304606.	0.900525
$39$	−11267.7	−0.0304165
$40$	−308581.	−0.762359
$41$	191846.	0.434720	0.217360	−	0.976092i	$-0.430255\pi$
$41$	191846.	0.434720	0.217360	+	0.976092i	$0.430255\pi$
$42$	−145212.	−0.302434
$43$	208779.	0.400449	0.200225	−	0.979750i	$-0.435833\pi$
$43$	208779.	0.400449	0.200225	+	0.979750i	$0.435833\pi$
$44$	642212.	1.13656
$45$	359978.	0.588888
$46$	177656.	0.269108
$47$	−1.36255e6	−1.91431	−0.957153	−	0.289584i	$-0.906483\pi$
$47$	−1.36255e6	−1.91431	−0.957153	+	0.289584i	$0.906483\pi$
$48$	−540823.	−0.705847
$49$	−687871.	−0.835259
$50$	2.41962e6	2.73748
$51$	−671774.	−0.709133
$52$	−35556.6	−0.0350678
$53$	−1.13105e6	−1.04355	−0.521777	−	0.853082i	$-0.674731\pi$
$53$	−1.13105e6	−1.04355	−0.521777	+	0.853082i	$0.674731\pi$
$54$	287400.	0.248375
$55$	3.72202e6	3.01654
$56$	230179.	0.175149
$57$	563257.	0.402851
$58$	−73856.3	−0.0497038
$59$	−1.59555e6	−1.01141	−0.505707	−	0.862705i	$-0.668768\pi$
$59$	−1.59555e6	−1.01141	−0.505707	+	0.862705i	$0.668768\pi$
$60$	1.13596e6	0.678941
$61$	−1.18637e6	−0.669217	−0.334608	−	0.942357i	$-0.608604\pi$
$61$	−1.18637e6	−0.669217	−0.334608	+	0.942357i	$0.608604\pi$
$62$	−3.34024e6	−1.77995
$63$	−268517.	−0.135294
$64$	−538678.	−0.256862
$65$	−206072.	−0.0930729
$66$	2.97159e6	1.27229
$67$	2.64240e6	1.07334	0.536669	−	0.843793i	$-0.319682\pi$
$67$	2.64240e6	1.07334	0.536669	+	0.843793i	$0.319682\pi$
$68$	−2.11986e6	−0.817574
$69$	328509.	0.120386
$70$	−2.65576e6	−0.925434
$71$	−2.80200e6	−0.929104	−0.464552	−	0.885546i	$-0.653785\pi$
$71$	−2.80200e6	−0.929104	−0.464552	+	0.885546i	$0.653785\pi$
$72$	−455563.	−0.143842
$73$	−5.16484e6	−1.55391	−0.776956	−	0.629554i	$-0.783237\pi$
$73$	−5.16484e6	−1.55391	−0.776956	+	0.629554i	$0.783237\pi$
$74$	8.59208e6	2.46483
$75$	4.47419e6	1.22462
$76$	1.77743e6	0.464456
$77$	−2.77635e6	−0.693037
$78$	−164524.	−0.0392554
$79$	−3.28919e6	−0.750576	−0.375288	−	0.926908i	$-0.622456\pi$
$79$	−3.28919e6	−0.750576	−0.375288	+	0.926908i	$0.622456\pi$
$80$	−9.89100e6	−2.15986
$81$	531441.	0.111111
$82$	2.80123e6	0.561048
$83$	590082.	0.113276	0.0566382	−	0.998395i	$-0.481962\pi$
$83$	590082.	0.113276	0.0566382	+	0.998395i	$0.481962\pi$
$84$	−847338.	−0.155984
$85$	−1.22859e7	−2.16991
$86$	3.04847e6	0.516818
$87$	−136570.	−0.0222351
$88$	−4.71032e6	−0.736819
$89$	2.78595e6	0.418898	0.209449	−	0.977820i	$-0.432833\pi$
$89$	2.78595e6	0.418898	0.209449	+	0.977820i	$0.432833\pi$
$90$	5.25620e6	0.760016
$91$	153715.	0.0213831
$92$	1.03665e6	0.138795
$93$	−6.17655e6	−0.796262
$94$	−1.98952e7	−2.47060
$95$	1.03013e7	1.23270
$96$	−5.73709e6	−0.661823
$97$	−289056.	−0.0321574	−0.0160787	−	0.999871i	$-0.505118\pi$
$97$	−289056.	−0.0321574	−0.0160787	+	0.999871i	$0.505118\pi$
$98$	−1.00439e7	−1.07798
$99$	5.49486e6	0.569159
$100$	1.41189e7	1.41189
$101$	5.19146e6	0.501377	0.250689	−	0.968068i	$-0.419343\pi$
$101$	5.19146e6	0.501377	0.250689	+	0.968068i	$0.419343\pi$
$102$	−9.80886e6	−0.915204
$103$	−6.83585e6	−0.616399	−0.308200	−	0.951322i	$-0.599726\pi$
$103$	−6.83585e6	−0.616399	−0.308200	+	0.951322i	$0.599726\pi$
$104$	260791.	0.0227340
$105$	−4.91085e6	−0.413994
$106$	−1.65149e7	−1.34681
$107$	5.98559e6	0.472350	0.236175	−	0.971711i	$-0.424106\pi$
$107$	5.98559e6	0.472350	0.236175	+	0.971711i	$0.424106\pi$
$108$	1.67703e6	0.128102
$109$	−1.00111e7	−0.740436	−0.370218	−	0.928945i	$-0.620717\pi$
$109$	−1.00111e7	−0.740436	−0.370218	+	0.928945i	$0.620717\pi$
$110$	5.43468e7	3.89313
$111$	1.58879e7	1.10265
$112$	7.37795e6	0.496218
$113$	2.09806e6	0.136787	0.0683933	−	0.997658i	$-0.478213\pi$
$113$	2.09806e6	0.136787	0.0683933	+	0.997658i	$0.478213\pi$
$114$	8.22436e6	0.519918
$115$	6.00803e6	0.368375
$116$	−430964.	−0.0256353
$117$	−304228.	−0.0175610
$118$	−2.32973e7	−1.30533
$119$	9.16439e6	0.498528
$120$	−8.33169e6	−0.440148
$121$	3.73273e7	1.91548
$122$	−1.73227e7	−0.863689
$123$	5.17984e6	0.250986
$124$	−1.94909e7	−0.918027
$125$	4.32496e7	1.98060
$126$	−3.92073e6	−0.174611
$127$	−9.76439e6	−0.422992	−0.211496	−	0.977379i	$-0.567834\pi$
$127$	−9.76439e6	−0.422992	−0.211496	+	0.977379i	$0.567834\pi$
$128$	1.93326e7	0.814807
$129$	5.63704e6	0.231199
$130$	−3.00895e6	−0.120120
$131$	4.72387e7	1.83590	0.917948	−	0.396700i	$-0.129845\pi$
$131$	4.72387e7	1.83590	0.917948	+	0.396700i	$0.129845\pi$
$132$	1.73397e7	0.656195
$133$	−7.68400e6	−0.283209
$134$	3.85828e7	1.38525
$135$	9.71941e6	0.339994
$136$	1.55482e7	0.530022
$137$	−2.85915e7	−0.949980	−0.474990	−	0.879991i	$-0.657548\pi$
$137$	−2.85915e7	−0.949980	−0.474990	+	0.879991i	$0.657548\pi$
$138$	4.79670e6	0.155370
$139$	1.97562e7	0.623951	0.311976	−	0.950090i	$-0.399009\pi$
$139$	1.97562e7	0.623951	0.311976	+	0.950090i	$0.399009\pi$
$140$	−1.54968e7	−0.477303
$141$	−3.67890e7	−1.10522
$142$	−4.09132e7	−1.19910
$143$	−3.14558e6	−0.0899548
$144$	−1.46022e7	−0.407521
$145$	−2.49770e6	−0.0680382
$146$	−7.54140e7	−2.00547
$147$	−1.85725e7	−0.482237
$148$	5.01362e7	1.27126
$149$	2.13465e7	0.528658	0.264329	−	0.964433i	$-0.414850\pi$
$149$	2.13465e7	0.528658	0.264329	+	0.964433i	$0.414850\pi$
$150$	6.53296e7	1.58049
$151$	6.56691e7	1.55218	0.776090	−	0.630622i	$-0.217200\pi$
$151$	6.56691e7	1.55218	0.776090	+	0.630622i	$0.217200\pi$
$152$	−1.30366e7	−0.301100
$153$	−1.81379e7	−0.409418
$154$	−4.05386e7	−0.894431
$155$	−1.12962e8	−2.43652
$156$	−960027.	−0.0202464
$157$	3.83258e7	0.790392	0.395196	−	0.918597i	$-0.370677\pi$
$157$	3.83258e7	0.790392	0.395196	+	0.918597i	$0.370677\pi$
$158$	−4.80269e7	−0.968691
$159$	−3.05383e7	−0.602496
$160$	−1.04924e8	−2.02515
$161$	−4.48155e6	−0.0846325
$162$	7.75980e6	0.143400
$163$	1.29604e7	0.234403	0.117201	−	0.993108i	$-0.462608\pi$
$163$	1.29604e7	0.234403	0.117201	+	0.993108i	$0.462608\pi$
$164$	1.63456e7	0.289366
$165$	1.00494e8	1.74160
$166$	8.61605e6	0.146194
$167$	9.60453e7	1.59576	0.797881	−	0.602815i	$-0.205954\pi$
$167$	9.60453e7	1.59576	0.797881	+	0.602815i	$0.205954\pi$
$168$	6.21483e6	0.101122
$169$	−6.25744e7	−0.997225
$170$	−1.79392e8	−2.80048
$171$	1.52079e7	0.232586
$172$	1.77884e7	0.266555
$173$	1.54094e7	0.226268	0.113134	−	0.993580i	$-0.463911\pi$
$173$	1.54094e7	0.226268	0.113134	+	0.993580i	$0.463911\pi$
$174$	−1.99412e6	−0.0286965
$175$	−6.10373e7	−0.860918
$176$	−1.50980e8	−2.08750
$177$	−4.30799e7	−0.583940
$178$	4.06789e7	0.540629
$179$	8.38575e7	1.09284	0.546419	−	0.837512i	$-0.315990\pi$
$179$	8.38575e7	1.09284	0.546419	+	0.837512i	$0.315990\pi$
$180$	3.06708e7	0.391987
$181$	−1.50898e8	−1.89151	−0.945754	−	0.324883i	$-0.894675\pi$
$181$	−1.50898e8	−1.89151	−0.945754	+	0.324883i	$0.894675\pi$
$182$	2.24445e6	0.0275969
$183$	−3.20321e7	−0.386372
$184$	−7.60334e6	−0.0899792
$185$	2.90571e8	3.37404
$186$	−9.01865e7	−1.02765
$187$	−1.87538e8	−2.09722
$188$	−1.16092e8	−1.27424
$189$	−7.24996e6	−0.0781123
$190$	1.50414e8	1.59092
$191$	2.94854e7	0.306189	0.153095	−	0.988212i	$-0.451076\pi$
$191$	2.94854e7	0.306189	0.153095	+	0.988212i	$0.451076\pi$
$192$	−1.45443e7	−0.148299
$193$	1.42403e8	1.42583	0.712917	−	0.701248i	$-0.247373\pi$
$193$	1.42403e8	1.42583	0.712917	+	0.701248i	$0.247373\pi$
$194$	−4.22063e6	−0.0415022
$195$	−5.56395e6	−0.0537356
$196$	−5.86079e7	−0.555981
$197$	−6.70489e7	−0.624827	−0.312413	−	0.949946i	$-0.601137\pi$
$197$	−6.70489e7	−0.624827	−0.312413	+	0.949946i	$0.601137\pi$
$198$	8.02329e7	0.734555
$199$	1.79632e8	1.61584	0.807918	−	0.589295i	$-0.200594\pi$
$199$	1.79632e8	1.61584	0.807918	+	0.589295i	$0.200594\pi$
$200$	−1.03555e8	−0.915307
$201$	7.13448e7	0.619692
$202$	7.58028e7	0.647076
$203$	1.86310e6	0.0156315
$204$	−5.72363e7	−0.472026
$205$	9.47331e7	0.768003
$206$	−9.98132e7	−0.795523
$207$	8.86974e6	0.0695048
$208$	8.35916e6	0.0644081
$209$	1.57243e8	1.19141
$210$	−7.17055e7	−0.534300
$211$	1.75301e8	1.28468	0.642342	−	0.766418i	$-0.277963\pi$
$211$	1.75301e8	1.28468	0.642342	+	0.766418i	$0.277963\pi$
$212$	−9.63673e7	−0.694631
$213$	−7.56541e7	−0.536418
$214$	8.73982e7	0.609613
$215$	1.03095e8	0.707459
$216$	−1.23002e7	−0.0830470
$217$	8.42610e7	0.559780
$218$	−1.46176e8	−0.955605
$219$	−1.39451e8	−0.897152
$220$	3.17122e8	2.00793
$221$	1.03832e7	0.0647079
$222$	2.31986e8	1.42307
$223$	−2.10447e7	−0.127080	−0.0635399	−	0.997979i	$-0.520239\pi$
$223$	−2.10447e7	−0.127080	−0.0635399	+	0.997979i	$0.520239\pi$
$224$	7.82658e7	0.465269
$225$	1.20803e8	0.707033
$226$	3.06347e7	0.176536
$227$	−3.24371e8	−1.84057	−0.920284	−	0.391251i	$-0.872043\pi$
$227$	−3.24371e8	−1.84057	−0.920284	+	0.391251i	$0.872043\pi$
$228$	4.79905e7	0.268154
$229$	−4.97183e6	−0.0273585	−0.0136792	−	0.999906i	$-0.504354\pi$
$229$	−4.97183e6	−0.0273585	−0.0136792	+	0.999906i	$0.504354\pi$
$230$	8.77259e7	0.475423
$231$	−7.49614e7	−0.400125
$232$	3.16091e6	0.0166190
$233$	−5.34348e7	−0.276744	−0.138372	−	0.990380i	$-0.544187\pi$
$233$	−5.34348e7	−0.276744	−0.138372	+	0.990380i	$0.544187\pi$
$234$	−4.44216e6	−0.0226641
$235$	−6.72826e8	−3.38193
$236$	−1.35944e8	−0.673237
$237$	−8.88082e7	−0.433345
$238$	1.33813e8	0.643398
$239$	−2.65448e8	−1.25773	−0.628864	−	0.777515i	$-0.716480\pi$
$239$	−2.65448e8	−1.25773	−0.628864	+	0.777515i	$0.716480\pi$
$240$	−2.67057e8	−1.24699
$241$	−1.42389e8	−0.655263	−0.327631	−	0.944806i	$-0.606250\pi$
$241$	−1.42389e8	−0.655263	−0.327631	+	0.944806i	$0.606250\pi$
$242$	5.45032e8	2.47211
$243$	1.43489e7	0.0641500
$244$	−1.01081e8	−0.445457
$245$	−3.39669e8	−1.47562
$246$	7.56331e7	0.323921
$247$	−8.70591e6	−0.0367599
$248$	1.42956e8	0.595144
$249$	1.59322e7	0.0654001
$250$	6.31507e8	2.55616
$251$	2.71119e8	1.08219	0.541094	−	0.840962i	$-0.318010\pi$
$251$	2.71119e8	1.08219	0.541094	+	0.840962i	$0.318010\pi$
$252$	−2.28781e7	−0.0900573
$253$	9.17092e7	0.356034
$254$	−1.42574e8	−0.545912
$255$	−3.31720e8	−1.25280
$256$	3.51234e8	1.30845
$257$	−3.70271e8	−1.36067	−0.680336	−	0.732900i	$-0.738166\pi$
$257$	−3.70271e8	−1.36067	−0.680336	+	0.732900i	$0.738166\pi$
$258$	8.23088e7	0.298385
$259$	−2.16744e8	−0.775172
$260$	−1.75577e7	−0.0619529
$261$	−3.68739e6	−0.0128374
$262$	6.89752e8	2.36940
$263$	2.67912e7	0.0908128	0.0454064	−	0.998969i	$-0.485542\pi$
$263$	2.67912e7	0.0908128	0.0454064	+	0.998969i	$0.485542\pi$
$264$	−1.27179e8	−0.425403
$265$	−5.58508e8	−1.84361
$266$	−1.12197e8	−0.365508
$267$	7.52207e7	0.241851
$268$	2.25137e8	0.714456
$269$	−4.19430e7	−0.131379	−0.0656896	−	0.997840i	$-0.520925\pi$
$269$	−4.19430e7	−0.131379	−0.0656896	+	0.997840i	$0.520925\pi$
$270$	1.41917e8	0.438796
$271$	3.59386e8	1.09690	0.548452	−	0.836182i	$-0.315217\pi$
$271$	3.59386e8	1.09690	0.548452	+	0.836182i	$0.315217\pi$
$272$	4.98369e8	1.50162
$273$	4.15030e6	0.0123455
$274$	−4.17476e8	−1.22604
$275$	1.24905e9	3.62173
$276$	2.79896e7	0.0801335
$277$	−5.22440e7	−0.147692	−0.0738460	−	0.997270i	$-0.523527\pi$
$277$	−5.22440e7	−0.147692	−0.0738460	+	0.997270i	$0.523527\pi$
$278$	2.88468e8	0.805269
$279$	−1.66767e8	−0.459722
$280$	1.13662e8	0.309429
$281$	5.82346e8	1.56570	0.782850	−	0.622210i	$-0.213765\pi$
$281$	5.82346e8	1.56570	0.782850	+	0.622210i	$0.213765\pi$
$282$	−5.37171e8	−1.42640
$283$	3.59974e8	0.944102	0.472051	−	0.881571i	$-0.343514\pi$
$283$	3.59974e8	0.944102	0.472051	+	0.881571i	$0.343514\pi$
$284$	−2.38736e8	−0.618448
$285$	2.78135e8	0.711702
$286$	−4.59299e7	−0.116095
$287$	−7.06638e7	−0.176445
$288$	−1.54901e8	−0.382104
$289$	2.08701e8	0.508607
$290$	−3.64700e7	−0.0878098
$291$	−7.80451e6	−0.0185661
$292$	−4.40053e8	−1.03435
$293$	−3.37944e8	−0.784888	−0.392444	−	0.919776i	$-0.628370\pi$
$293$	−3.37944e8	−0.784888	−0.392444	+	0.919776i	$0.628370\pi$
$294$	−2.71186e8	−0.622373
$295$	−7.87879e8	−1.78683
$296$	−3.67725e8	−0.824143
$297$	1.48361e8	0.328604
$298$	3.11690e8	0.682284
$299$	−5.07755e6	−0.0109851
$300$	3.81209e8	0.815153
$301$	−7.69009e7	−0.162536
$302$	9.58863e8	2.00324
$303$	1.40169e8	0.289470
$304$	−4.17863e8	−0.853054
$305$	−5.85828e8	−1.18228
$306$	−2.64839e8	−0.528393
$307$	−3.52294e8	−0.694898	−0.347449	−	0.937699i	$-0.612952\pi$
$307$	−3.52294e8	−0.694898	−0.347449	+	0.937699i	$0.612952\pi$
$308$	−2.36550e8	−0.461312
$309$	−1.84568e8	−0.355878
$310$	−1.64940e9	−3.14457
$311$	−3.08514e8	−0.581585	−0.290792	−	0.956786i	$-0.593919\pi$
$311$	−3.08514e8	−0.581585	−0.290792	+	0.956786i	$0.593919\pi$
$312$	7.04135e6	0.0131255
$313$	2.48147e8	0.457407	0.228704	−	0.973496i	$-0.426551\pi$
$313$	2.48147e8	0.457407	0.228704	+	0.973496i	$0.426551\pi$
$314$	5.59612e8	1.02008
$315$	−1.32593e8	−0.239020
$316$	−2.80245e8	−0.499613
$317$	−3.68196e8	−0.649190	−0.324595	−	0.945853i	$-0.605228\pi$
$317$	−3.68196e8	−0.649190	−0.324595	+	0.945853i	$0.605228\pi$
$318$	−4.45902e8	−0.777580
$319$	−3.81260e7	−0.0657588
$320$	−2.65998e8	−0.453788
$321$	1.61611e8	0.272711
$322$	−6.54370e7	−0.109226
$323$	−5.19042e8	−0.857025
$324$	4.52797e7	0.0739599
$325$	−6.91547e7	−0.111746
$326$	1.89241e8	0.302519
$327$	−2.70299e8	−0.427491
$328$	−1.19887e8	−0.187592
$329$	5.01878e8	0.776985
$330$	1.46736e9	2.24770
$331$	−1.67853e8	−0.254409	−0.127204	−	0.991877i	$-0.540600\pi$
$331$	−1.67853e8	−0.254409	−0.127204	+	0.991877i	$0.540600\pi$
$332$	5.02761e7	0.0754012
$333$	4.28973e8	0.636613
$334$	1.40240e9	2.05949
$335$	1.30481e9	1.89623
$336$	1.99205e8	0.286492
$337$	−1.72217e8	−0.245116	−0.122558	−	0.992461i	$-0.539110\pi$
$337$	−1.72217e8	−0.245116	−0.122558	+	0.992461i	$0.539110\pi$
$338$	−9.13675e8	−1.28701
$339$	5.66477e7	0.0789738
$340$	−1.04678e9	−1.44438
$341$	−1.72430e9	−2.35489
$342$	2.22058e8	0.300175
$343$	5.56709e8	0.744901
$344$	−1.30469e8	−0.172804
$345$	1.62217e8	0.212681
$346$	2.24999e8	0.292021
$347$	7.96731e8	1.02367	0.511834	−	0.859085i	$-0.328966\pi$
$347$	7.96731e8	1.02367	0.511834	+	0.859085i	$0.328966\pi$
$348$	−1.16360e7	−0.0148005
$349$	1.09341e9	1.37687	0.688436	−	0.725297i	$-0.258298\pi$
$349$	1.09341e9	1.37687	0.688436	+	0.725297i	$0.258298\pi$
$350$	−8.91232e8	−1.11110
$351$	−8.21414e6	−0.0101388
$352$	−1.60161e9	−1.95730
$353$	6.64482e8	0.804029	0.402015	−	0.915633i	$-0.368310\pi$
$353$	6.64482e8	0.804029	0.402015	+	0.915633i	$0.368310\pi$
$354$	−6.29028e8	−0.753631
$355$	−1.38362e9	−1.64141
$356$	2.37368e8	0.278835
$357$	2.47439e8	0.287825
$358$	1.22444e9	1.41041
$359$	−2.55461e8	−0.291403	−0.145702	−	0.989329i	$-0.546544\pi$
$359$	−2.55461e8	−0.291403	−0.145702	+	0.989329i	$0.546544\pi$
$360$	−2.24956e8	−0.254120
$361$	−4.58675e8	−0.513133
$362$	−2.20333e9	−2.44117
$363$	1.00784e9	1.10590
$364$	1.30968e7	0.0142334
$365$	−2.55038e9	−2.74524
$366$	−4.67714e8	−0.498651
$367$	7.04463e8	0.743921	0.371961	−	0.928249i	$-0.378686\pi$
$367$	7.04463e8	0.743921	0.371961	+	0.928249i	$0.378686\pi$
$368$	−2.43711e8	−0.254922
$369$	1.39856e8	0.144907
$370$	4.24275e9	4.35453
$371$	4.16606e8	0.423561
$372$	−5.26253e8	−0.530023
$373$	−1.53844e9	−1.53497	−0.767484	−	0.641068i	$-0.778491\pi$
$373$	−1.53844e9	−1.53497	−0.767484	+	0.641068i	$0.778491\pi$
$374$	−2.73832e9	−2.70666
$375$	1.16774e9	1.14350
$376$	8.51480e8	0.826070
$377$	2.11088e6	0.00202894
$378$	−1.05860e8	−0.100811
$379$	−2.20215e7	−0.0207782	−0.0103891	−	0.999946i	$-0.503307\pi$
$379$	−2.20215e7	−0.0207782	−0.0103891	+	0.999946i	$0.503307\pi$
$380$	8.77689e8	0.820536
$381$	−2.63638e8	−0.244214
$382$	4.30528e8	0.395166
$383$	1.80851e9	1.64484	0.822421	−	0.568879i	$-0.192623\pi$
$383$	1.80851e9	1.64484	0.822421	+	0.568879i	$0.192623\pi$
$384$	5.21979e8	0.470429
$385$	−1.37095e9	−1.22436
$386$	2.07929e9	1.84018
$387$	1.52200e8	0.133483
$388$	−2.46281e7	−0.0214052
$389$	1.89367e8	0.163110	0.0815549	−	0.996669i	$-0.474011\pi$
$389$	1.89367e8	0.163110	0.0815549	+	0.996669i	$0.474011\pi$
$390$	−8.12417e7	−0.0693510
$391$	−3.02721e8	−0.256109
$392$	4.29861e8	0.360435
$393$	1.27544e9	1.05996
$394$	−9.79010e8	−0.806399
$395$	−1.62420e9	−1.32601
$396$	4.68172e8	0.378855
$397$	1.15077e9	0.923040	0.461520	−	0.887130i	$-0.347304\pi$
$397$	1.15077e9	0.923040	0.461520	+	0.887130i	$0.347304\pi$
$398$	2.62288e9	2.08539
$399$	−2.07468e8	−0.163511
$400$	−3.31927e9	−2.59318
$401$	1.47249e9	1.14037	0.570186	−	0.821516i	$-0.306871\pi$
$401$	1.47249e9	1.14037	0.570186	+	0.821516i	$0.306871\pi$
$402$	1.04174e9	0.799772
$403$	9.54670e7	0.0726584
$404$	4.42322e8	0.333736
$405$	2.62424e8	0.196296
$406$	2.72039e7	0.0201739
$407$	4.43539e9	3.26101
$408$	4.19801e8	0.306008
$409$	−9.24969e8	−0.668491	−0.334246	−	0.942486i	$-0.608482\pi$
$409$	−9.24969e8	−0.668491	−0.334246	+	0.942486i	$0.608482\pi$
$410$	1.38324e9	0.991182
$411$	−7.71969e8	−0.548471
$412$	−5.82427e8	−0.410299
$413$	5.87699e8	0.410516
$414$	1.29511e8	0.0897027
$415$	2.91381e8	0.200121
$416$	8.86745e7	0.0603910
$417$	5.33416e8	0.360238
$418$	2.29598e9	1.53763
$419$	−7.26576e8	−0.482539	−0.241269	−	0.970458i	$-0.577564\pi$
$419$	−7.26576e8	−0.482539	−0.241269	+	0.970458i	$0.577564\pi$
$420$	−4.18413e8	−0.275571
$421$	−2.41347e8	−0.157636	−0.0788178	−	0.996889i	$-0.525115\pi$
$421$	−2.41347e8	−0.157636	−0.0788178	+	0.996889i	$0.525115\pi$
$422$	2.55965e9	1.65801
$423$	−9.93302e8	−0.638102
$424$	7.06808e8	0.450320
$425$	−4.12297e9	−2.60525
$426$	−1.10466e9	−0.692300
$427$	4.36984e8	0.271624
$428$	5.09983e8	0.314415
$429$	−8.49306e7	−0.0519354
$430$	1.50533e9	0.913044
$431$	3.23006e9	1.94330	0.971652	−	0.236417i	$-0.0759732\pi$
$431$	3.23006e9	1.94330	0.971652	+	0.236417i	$0.0759732\pi$
$432$	−3.94260e8	−0.235282
$433$	−2.93355e9	−1.73654	−0.868272	−	0.496088i	$-0.834769\pi$
$433$	−2.93355e9	−1.73654	−0.868272	+	0.496088i	$0.834769\pi$
$434$	1.23033e9	0.722451
$435$	−6.74380e7	−0.0392819
$436$	−8.52961e8	−0.492863
$437$	2.53820e8	0.145493
$438$	−2.03618e9	−1.15786
$439$	−1.39654e9	−0.787819	−0.393910	−	0.919149i	$-0.628878\pi$
$439$	−1.39654e9	−0.787819	−0.393910	+	0.919149i	$0.628878\pi$
$440$	−2.32594e9	−1.30171
$441$	−5.01458e8	−0.278420
$442$	1.51609e8	0.0835118
$443$	9.43130e8	0.515417	0.257708	−	0.966223i	$-0.417033\pi$
$443$	9.43130e8	0.515417	0.257708	+	0.966223i	$0.417033\pi$
$444$	1.35368e9	0.733965
$445$	1.37570e9	0.740052
$446$	−3.07283e8	−0.164009
$447$	5.76356e8	0.305221
$448$	1.98415e8	0.104256
$449$	−3.54094e9	−1.84610	−0.923052	−	0.384676i	$-0.874313\pi$
$449$	−3.54094e9	−1.84610	−0.923052	+	0.384676i	$0.874313\pi$
$450$	1.76390e9	0.912494
$451$	1.44605e9	0.742274
$452$	1.78759e8	0.0910506
$453$	1.77307e9	0.896151
$454$	−4.73629e9	−2.37543
$455$	7.59039e7	0.0377767
$456$	−3.51988e8	−0.173840
$457$	−2.03166e9	−0.995738	−0.497869	−	0.867252i	$-0.665884\pi$
$457$	−2.03166e9	−0.995738	−0.497869	+	0.867252i	$0.665884\pi$
$458$	−7.25958e7	−0.0353087
$459$	−4.89723e8	−0.236378
$460$	5.11895e8	0.245205
$461$	−4.97341e8	−0.236429	−0.118215	−	0.992988i	$-0.537717\pi$
$461$	−4.97341e8	−0.236429	−0.118215	+	0.992988i	$0.537717\pi$
$462$	−1.09454e9	−0.516400
$463$	1.07915e9	0.505301	0.252650	−	0.967558i	$-0.418698\pi$
$463$	1.07915e9	0.505301	0.252650	+	0.967558i	$0.418698\pi$
$464$	1.01317e8	0.0470837
$465$	−3.04997e9	−1.40673
$466$	−7.80225e8	−0.357165
$467$	−1.05677e9	−0.480144	−0.240072	−	0.970755i	$-0.577171\pi$
$467$	−1.05677e9	−0.480144	−0.240072	+	0.970755i	$0.577171\pi$
$468$	−2.59207e7	−0.0116893
$469$	−9.73291e8	−0.435650
$470$	−9.82422e9	−4.36471
$471$	1.03480e9	0.456333
$472$	9.97084e8	0.436450
$473$	1.57368e9	0.683758
$474$	−1.29673e9	−0.559274
$475$	3.45695e9	1.48001
$476$	7.80823e8	0.331840
$477$	−8.24533e8	−0.347851
$478$	−3.87592e9	−1.62322
$479$	−1.21111e8	−0.0503512	−0.0251756	−	0.999683i	$-0.508014\pi$
$479$	−1.21111e8	−0.0503512	−0.0251756	+	0.999683i	$0.508014\pi$
$480$	−2.83296e9	−1.16922
$481$	−2.45569e8	−0.100616
$482$	−2.07908e9	−0.845680
$483$	−1.21002e8	−0.0488626
$484$	3.18035e9	1.27502
$485$	−1.42735e8	−0.0568113
$486$	2.09515e8	0.0827918
$487$	4.13714e9	1.62311	0.811556	−	0.584274i	$-0.198621\pi$
$487$	4.13714e9	1.62311	0.811556	+	0.584274i	$0.198621\pi$
$488$	7.41382e8	0.288784
$489$	3.49931e8	0.135332
$490$	−4.95966e9	−1.90443
$491$	−3.63021e8	−0.138403	−0.0692016	−	0.997603i	$-0.522045\pi$
$491$	−3.63021e8	−0.138403	−0.0692016	+	0.997603i	$0.522045\pi$
$492$	4.41332e8	0.167066
$493$	1.25849e8	0.0473028
$494$	−1.27119e8	−0.0474422
$495$	2.71335e9	1.00551
$496$	4.58220e9	1.68612
$497$	1.03208e9	0.377108
$498$	2.32633e8	0.0844052
$499$	−4.68516e9	−1.68800	−0.844001	−	0.536342i	$-0.819806\pi$
$499$	−4.68516e9	−1.68800	−0.844001	+	0.536342i	$0.819806\pi$
$500$	3.68495e9	1.31837
$501$	2.59322e9	0.921314
$502$	3.95873e9	1.39667
$503$	3.24026e9	1.13525	0.567626	−	0.823286i	$-0.307862\pi$
$503$	3.24026e9	1.13525	0.567626	+	0.823286i	$0.307862\pi$
$504$	1.67800e8	0.0583829
$505$	2.56353e9	0.885765
$506$	1.33909e9	0.459496
$507$	−1.68951e9	−0.575748
$508$	−8.31943e8	−0.281560
$509$	−2.41699e9	−0.812388	−0.406194	−	0.913787i	$-0.633144\pi$
$509$	−2.41699e9	−0.812388	−0.406194	+	0.913787i	$0.633144\pi$
$510$	−4.84359e9	−1.61686
$511$	1.90240e9	0.630707
$512$	2.65395e9	0.873872
$513$	4.10614e8	0.134284
$514$	−5.40649e9	−1.75608
$515$	−3.37552e9	−1.08897
$516$	4.80286e8	0.153895
$517$	−1.02703e10	−3.26863
$518$	−3.16477e9	−1.00043
$519$	4.16053e8	0.130636
$520$	1.28778e8	0.0401633
$521$	5.28397e9	1.63692	0.818461	−	0.574562i	$-0.194827\pi$
$521$	5.28397e9	1.63692	0.818461	+	0.574562i	$0.194827\pi$
$522$	−5.38412e7	−0.0165679
$523$	−1.32356e9	−0.404566	−0.202283	−	0.979327i	$-0.564836\pi$
$523$	−1.32356e9	−0.404566	−0.202283	+	0.979327i	$0.564836\pi$
$524$	4.02482e9	1.22204
$525$	−1.64801e9	−0.497052
$526$	3.91190e8	0.117203
$527$	5.69170e9	1.69397
$528$	−4.07647e9	−1.20522
$529$	1.48036e8	0.0434783
$530$	−8.15502e9	−2.37936
$531$	−1.16316e9	−0.337138
$532$	−6.54690e8	−0.188515
$533$	−8.00615e7	−0.0229023
$534$	1.09833e9	0.312132
$535$	2.95567e9	0.834483
$536$	−1.65127e9	−0.463172
$537$	2.26415e9	0.630951
$538$	−6.12428e8	−0.169558
$539$	−5.18486e9	−1.42619
$540$	8.28112e8	0.226314
$541$	6.76224e9	1.83612	0.918058	−	0.396445i	$-0.129756\pi$
$541$	6.76224e9	1.83612	0.918058	+	0.396445i	$0.129756\pi$
$542$	5.24755e9	1.41566
$543$	−4.07424e9	−1.09206
$544$	5.28673e9	1.40796
$545$	−4.94344e9	−1.30810
$546$	6.06003e7	0.0159331
$547$	8.50953e7	0.0222305	0.0111153	−	0.999938i	$-0.496462\pi$
$547$	8.50953e7	0.0222305	0.0111153	+	0.999938i	$0.496462\pi$
$548$	−2.43604e9	−0.632344
$549$	−8.64866e8	−0.223072
$550$	1.82379e10	4.67419
$551$	−1.05520e8	−0.0268723
$552$	−2.05290e8	−0.0519495
$553$	1.21153e9	0.304646
$554$	−7.62837e8	−0.190611
$555$	7.84541e9	1.94800
$556$	1.68326e9	0.415326
$557$	−1.99497e9	−0.489150	−0.244575	−	0.969630i	$-0.578648\pi$
$557$	−1.99497e9	−0.489150	−0.244575	+	0.969630i	$0.578648\pi$
$558$	−2.43504e9	−0.593315
$559$	−8.71281e7	−0.0210968
$560$	3.64321e9	0.876650
$561$	−5.06352e9	−1.21083
$562$	8.50308e9	2.02069
$563$	−3.64076e9	−0.859829	−0.429915	−	0.902870i	$-0.641456\pi$
$563$	−3.64076e9	−0.859829	−0.429915	+	0.902870i	$0.641456\pi$
$564$	−3.13449e9	−0.735681
$565$	1.03602e9	0.241656
$566$	5.25614e9	1.21845
$567$	−1.95749e8	−0.0450981
$568$	1.75101e9	0.400931
$569$	2.21399e9	0.503828	0.251914	−	0.967750i	$-0.418940\pi$
$569$	2.21399e9	0.503828	0.251914	+	0.967750i	$0.418940\pi$
$570$	4.06117e9	0.918520
$571$	1.63533e9	0.367602	0.183801	−	0.982963i	$-0.441160\pi$
$571$	1.63533e9	0.367602	0.183801	+	0.982963i	$0.441160\pi$
$572$	−2.68009e8	−0.0598774
$573$	7.96105e8	0.176778
$574$	−1.03179e9	−0.227720
$575$	2.01620e9	0.442280
$576$	−3.92696e8	−0.0856206
$577$	4.85985e9	1.05319	0.526596	−	0.850115i	$-0.323468\pi$
$577$	4.85985e9	1.05319	0.526596	+	0.850115i	$0.323468\pi$
$578$	3.04733e9	0.656406
$579$	3.84489e9	0.823206
$580$	−2.12809e8	−0.0452889
$581$	−2.17349e8	−0.0459770
$582$	−1.13957e8	−0.0239613
$583$	−8.52531e9	−1.78185
$584$	3.22758e9	0.670552
$585$	−1.50227e8	−0.0310243
$586$	−4.93446e9	−1.01297
$587$	−6.39733e9	−1.30547	−0.652733	−	0.757588i	$-0.726377\pi$
$587$	−6.39733e9	−1.30547	−0.652733	+	0.757588i	$0.726377\pi$
$588$	−1.58241e9	−0.320996
$589$	−4.77227e9	−0.962325
$590$	−1.15042e10	−2.30607
$591$	−1.81032e9	−0.360744
$592$	−1.17868e10	−2.33490
$593$	−2.73477e9	−0.538554	−0.269277	−	0.963063i	$-0.586785\pi$
$593$	−2.73477e9	−0.538554	−0.269277	+	0.963063i	$0.586785\pi$
$594$	2.16629e9	0.424095
$595$	4.52535e9	0.880730
$596$	1.81876e9	0.351895
$597$	4.85006e9	0.932903
$598$	−7.41396e7	−0.0141774
$599$	−4.75928e8	−0.0904790	−0.0452395	−	0.998976i	$-0.514405\pi$
$599$	−4.75928e8	−0.0904790	−0.0452395	+	0.998976i	$0.514405\pi$
$600$	−2.79599e9	−0.528453
$601$	4.25604e9	0.799733	0.399866	−	0.916573i	$-0.369057\pi$
$601$	4.25604e9	0.799733	0.399866	+	0.916573i	$0.369057\pi$
$602$	−1.12286e9	−0.209768
$603$	1.92631e9	0.357779
$604$	5.59513e9	1.03319
$605$	1.84321e10	3.38401
$606$	2.04667e9	0.373589
$607$	−6.01074e9	−1.09086	−0.545428	−	0.838157i	$-0.683633\pi$
$607$	−6.01074e9	−1.09086	−0.545428	+	0.838157i	$0.683633\pi$
$608$	−4.43272e9	−0.799849
$609$	5.03037e7	0.00902484
$610$	−8.55392e9	−1.52585
$611$	5.68623e8	0.100851
$612$	−1.54538e9	−0.272525
$613$	5.65645e8	0.0991819	0.0495910	−	0.998770i	$-0.484208\pi$
$613$	5.65645e8	0.0991819	0.0495910	+	0.998770i	$0.484208\pi$
$614$	−5.14400e9	−0.896833
$615$	2.55779e9	0.443407
$616$	1.73498e9	0.299063
$617$	7.85121e9	1.34567	0.672834	−	0.739793i	$-0.265077\pi$
$617$	7.85121e9	1.34567	0.672834	+	0.739793i	$0.265077\pi$
$618$	−2.69496e9	−0.459295
$619$	3.13703e9	0.531620	0.265810	−	0.964025i	$-0.414361\pi$
$619$	3.13703e9	0.531620	0.265810	+	0.964025i	$0.414361\pi$
$620$	−9.62454e9	−1.62184
$621$	2.39483e8	0.0401286
$622$	−4.50474e9	−0.750591
$623$	−1.02617e9	−0.170024
$624$	2.25697e8	0.0371861
$625$	8.41040e9	1.37796
$626$	3.62329e9	0.590328
$627$	4.24557e9	0.687859
$628$	3.26543e9	0.526116
$629$	−1.46407e10	−2.34577
$630$	−1.93605e9	−0.308478
$631$	−2.25037e9	−0.356576	−0.178288	−	0.983978i	$-0.557056\pi$
$631$	−2.25037e9	−0.356576	−0.178288	+	0.983978i	$0.557056\pi$
$632$	2.05547e9	0.323892
$633$	4.73313e9	0.741712
$634$	−5.37619e9	−0.837842
$635$	−4.82163e9	−0.747284
$636$	−2.60192e9	−0.401045
$637$	2.87064e8	0.0440038
$638$	−5.56694e8	−0.0848681
$639$	−2.04266e9	−0.309701
$640$	9.54637e9	1.43949
$641$	−2.64631e9	−0.396860	−0.198430	−	0.980115i	$-0.563584\pi$
$641$	−2.64631e9	−0.396860	−0.198430	+	0.980115i	$0.563584\pi$
$642$	2.35975e9	0.351960
$643$	−8.30752e9	−1.23235	−0.616173	−	0.787611i	$-0.711318\pi$
$643$	−8.30752e9	−1.23235	−0.616173	+	0.787611i	$0.711318\pi$
$644$	−3.81836e8	−0.0563347
$645$	2.78355e9	0.408452
$646$	−7.57875e9	−1.10607
$647$	−4.09897e9	−0.594990	−0.297495	−	0.954723i	$-0.596151\pi$
$647$	−4.09897e9	−0.594990	−0.297495	+	0.954723i	$0.596151\pi$
$648$	−3.32105e8	−0.0479472
$649$	−1.20265e10	−1.72697
$650$	−1.00976e9	−0.144218
$651$	2.27505e9	0.323189
$652$	1.10425e9	0.156027
$653$	−5.29390e9	−0.744011	−0.372006	−	0.928230i	$-0.621330\pi$
$653$	−5.29390e9	−0.744011	−0.372006	+	0.928230i	$0.621330\pi$
$654$	−3.94675e9	−0.551719
$655$	2.33263e10	3.24341
$656$	−3.84277e9	−0.531473
$657$	−3.76517e9	−0.517971
$658$	7.32814e9	1.00277
$659$	9.66411e9	1.31542	0.657708	−	0.753273i	$-0.271526\pi$
$659$	9.66411e9	1.31542	0.657708	+	0.753273i	$0.271526\pi$
$660$	8.56231e9	1.15928
$661$	4.20815e9	0.566743	0.283372	−	0.959010i	$-0.408547\pi$
$661$	4.20815e9	0.566743	0.283372	+	0.959010i	$0.408547\pi$
$662$	−2.45090e9	−0.328339
$663$	2.80346e8	0.0373591
$664$	−3.68751e8	−0.0488816
$665$	−3.79434e9	−0.500334
$666$	6.26363e9	0.821611
$667$	−6.15425e7	−0.00803036
$668$	8.18323e9	1.06220
$669$	−5.68208e8	−0.0733696
$670$	1.90521e10	2.44726
$671$	−8.94233e9	−1.14267
$672$	2.11318e9	0.268623
$673$	−5.81522e9	−0.735383	−0.367691	−	0.929948i	$-0.619852\pi$
$673$	−5.81522e9	−0.735383	−0.367691	+	0.929948i	$0.619852\pi$
$674$	−2.51462e9	−0.316346
$675$	3.26169e9	0.408206
$676$	−5.33145e9	−0.663792
$677$	4.45305e9	0.551565	0.275783	−	0.961220i	$-0.411063\pi$
$677$	4.45305e9	0.551565	0.275783	+	0.961220i	$0.411063\pi$
$678$	8.27137e8	0.101923
$679$	1.06470e8	0.0130521
$680$	7.67766e9	0.936370
$681$	−8.75802e9	−1.06265
$682$	−2.51772e10	−3.03922
$683$	5.09118e9	0.611428	0.305714	−	0.952123i	$-0.401105\pi$
$683$	5.09118e9	0.611428	0.305714	+	0.952123i	$0.401105\pi$
$684$	1.29574e9	0.154819
$685$	−1.41184e10	−1.67829
$686$	8.12874e9	0.961366
$687$	−1.34239e8	−0.0157954
$688$	−4.18195e9	−0.489575
$689$	4.72011e8	0.0549774
$690$	2.36860e9	0.274486
$691$	3.95531e9	0.456045	0.228022	−	0.973656i	$-0.426774\pi$
$691$	3.95531e9	0.456045	0.228022	+	0.973656i	$0.426774\pi$
$692$	1.31291e9	0.150613
$693$	−2.02396e9	−0.231012
$694$	1.16334e10	1.32114
$695$	9.75554e9	1.10231
$696$	8.53447e7	0.00959498
$697$	−4.77323e9	−0.533946
$698$	1.59653e10	1.77699
$699$	−1.44274e9	−0.159778
$700$	−5.20049e9	−0.573061
$701$	1.26904e10	1.39143	0.695716	−	0.718317i	$-0.255087\pi$
$701$	1.26904e10	1.39143	0.695716	+	0.718317i	$0.255087\pi$
$702$	−1.19938e8	−0.0130851
$703$	1.22757e10	1.33261
$704$	−4.06031e9	−0.438586
$705$	−1.81663e10	−1.95256
$706$	9.70239e9	1.03768
$707$	−1.91220e9	−0.203501
$708$	−3.67049e9	−0.388693
$709$	−1.25513e10	−1.32260	−0.661300	−	0.750122i	$-0.729995\pi$
$709$	−1.25513e10	−1.32260	−0.661300	+	0.750122i	$0.729995\pi$
$710$	−2.02028e10	−2.11840
$711$	−2.39782e9	−0.250192
$712$	−1.74098e9	−0.180765
$713$	−2.78334e9	−0.287576
$714$	3.61296e9	0.371466
$715$	−1.55328e9	−0.158920
$716$	7.14481e9	0.727436
$717$	−7.16709e9	−0.726149
$718$	−3.73010e9	−0.376084
$719$	−5.65642e9	−0.567532	−0.283766	−	0.958894i	$-0.591584\pi$
$719$	−5.65642e9	−0.567532	−0.283766	+	0.958894i	$0.591584\pi$
$720$	−7.21054e9	−0.719953
$721$	2.51789e9	0.250186
$722$	−6.69731e9	−0.662247
$723$	−3.84449e9	−0.378316
$724$	−1.28568e10	−1.25906
$725$	−8.38190e8	−0.0816883
$726$	1.47159e10	1.42727
$727$	−1.59955e10	−1.54393	−0.771963	−	0.635667i	$-0.780725\pi$
$727$	−1.59955e10	−1.54393	−0.771963	+	0.635667i	$0.780725\pi$
$728$	−9.60586e7	−0.00922734
$729$	3.87420e8	0.0370370
$730$	−3.72392e10	−3.54300
$731$	−5.19453e9	−0.491853
$732$	−2.72919e9	−0.257185
$733$	−1.22763e10	−1.15134	−0.575670	−	0.817682i	$-0.695259\pi$
$733$	−1.22763e10	−1.15134	−0.575670	+	0.817682i	$0.695259\pi$
$734$	1.02862e10	0.960102
$735$	−9.17107e9	−0.851950
$736$	−2.58530e9	−0.239022
$737$	1.99172e10	1.83270
$738$	2.04209e9	0.187016
$739$	−2.44144e9	−0.222531	−0.111265	−	0.993791i	$-0.535490\pi$
$739$	−2.44144e9	−0.222531	−0.111265	+	0.993791i	$0.535490\pi$
$740$	2.47571e10	2.24589
$741$	−2.35059e8	−0.0212234
$742$	6.08304e9	0.546647
$743$	2.03182e9	0.181729	0.0908646	−	0.995863i	$-0.471037\pi$
$743$	2.03182e9	0.181729	0.0908646	+	0.995863i	$0.471037\pi$
$744$	3.85982e9	0.343607
$745$	1.05408e10	0.933961
$746$	−2.24634e10	−1.98102
$747$	4.30170e8	0.0377588
$748$	−1.59786e10	−1.39599
$749$	−2.20471e9	−0.191719
$750$	1.70507e10	1.47580
$751$	1.99854e10	1.72176	0.860880	−	0.508809i	$-0.169914\pi$
$751$	1.99854e10	1.72176	0.860880	+	0.508809i	$0.169914\pi$
$752$	2.72926e10	2.34036
$753$	7.32022e9	0.624801
$754$	3.08218e7	0.00261854
$755$	3.24272e10	2.74218
$756$	−6.17710e8	−0.0519946
$757$	−1.68290e10	−1.41001	−0.705006	−	0.709202i	$-0.749056\pi$
$757$	−1.68290e10	−1.41001	−0.705006	+	0.709202i	$0.749056\pi$
$758$	−3.21545e8	−0.0268163
$759$	2.47615e9	0.205556
$760$	−6.43743e9	−0.531943
$761$	−2.47157e9	−0.203295	−0.101648	−	0.994820i	$-0.532411\pi$
$761$	−2.47157e9	−0.203295	−0.101648	+	0.994820i	$0.532411\pi$
$762$	−3.84950e9	−0.315182
$763$	3.68744e9	0.300531
$764$	2.51221e9	0.203811
$765$	−8.95644e9	−0.723303
$766$	2.64068e10	2.12283
$767$	6.65859e8	0.0532842
$768$	9.48331e9	0.755433
$769$	−2.24237e10	−1.77813	−0.889067	−	0.457777i	$-0.848646\pi$
$769$	−2.24237e10	−1.77813	−0.889067	+	0.457777i	$0.848646\pi$
$770$	−2.00179e10	−1.58016
$771$	−9.99732e9	−0.785585
$772$	1.21330e10	0.949091
$773$	−1.92307e10	−1.49750	−0.748748	−	0.662854i	$-0.769345\pi$
$773$	−1.92307e10	−1.49750	−0.748748	+	0.662854i	$0.769345\pi$
$774$	2.22234e9	0.172273
$775$	−3.79082e10	−2.92535
$776$	1.80635e8	0.0138767
$777$	−5.85209e9	−0.447546
$778$	2.76503e9	0.210509
$779$	4.00217e9	0.303329
$780$	−4.74059e8	−0.0357685
$781$	−2.11202e10	−1.58642
$782$	−4.42016e9	−0.330533
$783$	−9.95596e7	−0.00741169
$784$	1.37784e10	1.02116
$785$	1.89252e10	1.39636
$786$	1.86233e10	1.36797
$787$	8.73845e9	0.639032	0.319516	−	0.947581i	$-0.396480\pi$
$787$	8.73845e9	0.639032	0.319516	+	0.947581i	$0.396480\pi$
$788$	−5.71269e9	−0.415909
$789$	7.23362e8	0.0524308
$790$	−2.37156e10	−1.71135
$791$	−7.72792e8	−0.0555194
$792$	−3.43382e9	−0.245606
$793$	4.95099e8	0.0352563
$794$	1.68028e10	1.19127
$795$	−1.50797e10	−1.06441
$796$	1.53049e10	1.07556
$797$	−1.61543e10	−1.13027	−0.565137	−	0.824997i	$-0.691177\pi$
$797$	−1.61543e10	−1.13027	−0.565137	+	0.824997i	$0.691177\pi$
$798$	−3.02933e9	−0.211026
$799$	3.39010e10	2.35125
$800$	−3.52110e10	−2.43144
$801$	2.03096e9	0.139633
$802$	2.15004e10	1.47176
$803$	−3.89301e10	−2.65327
$804$	6.07870e9	0.412491
$805$	−2.21298e9	−0.149517
$806$	1.39396e9	0.0937727
$807$	−1.13246e9	−0.0758519
$808$	−3.24422e9	−0.216357
$809$	2.10541e10	1.39803	0.699015	−	0.715107i	$-0.253622\pi$
$809$	2.10541e10	1.39803	0.699015	+	0.715107i	$0.253622\pi$
$810$	3.83177e9	0.253339
$811$	−6.03214e9	−0.397098	−0.198549	−	0.980091i	$-0.563623\pi$
$811$	−6.03214e9	−0.397098	−0.198549	+	0.980091i	$0.563623\pi$
$812$	1.58740e8	0.0104049
$813$	9.70343e9	0.633298
$814$	6.47631e10	4.20865
$815$	6.39982e9	0.414110
$816$	1.34560e10	0.866960
$817$	4.35542e9	0.279417
$818$	−1.35059e10	−0.862753
$819$	1.12058e8	0.00712770
$820$	8.07143e9	0.511213
$821$	2.34470e10	1.47872	0.739361	−	0.673309i	$-0.235128\pi$
$821$	2.34470e10	1.47872	0.739361	+	0.673309i	$0.235128\pi$
$822$	−1.12719e10	−0.707855
$823$	−2.27370e10	−1.42179	−0.710893	−	0.703300i	$-0.751709\pi$
$823$	−2.27370e10	−1.42179	−0.710893	+	0.703300i	$0.751709\pi$
$824$	4.27182e9	0.265992
$825$	3.37244e10	2.09101
$826$	8.58125e9	0.529810
$827$	1.91123e10	1.17501	0.587507	−	0.809219i	$-0.300110\pi$
$827$	1.91123e10	1.17501	0.587507	+	0.809219i	$0.300110\pi$
$828$	7.55718e8	0.0462651
$829$	−1.83939e8	−0.0112133	−0.00560663	−	0.999984i	$-0.501785\pi$
$829$	−1.83939e8	−0.0112133	−0.00560663	+	0.999984i	$0.501785\pi$
$830$	4.25458e9	0.258276
$831$	−1.41059e9	−0.0852701
$832$	2.24802e8	0.0135322
$833$	1.71146e10	1.02591
$834$	7.78864e9	0.464922
$835$	4.74269e10	2.81917
$836$	1.33974e10	0.793047
$837$	−4.50271e9	−0.265421
$838$	−1.06091e10	−0.622763
$839$	9.03520e8	0.0528166	0.0264083	−	0.999651i	$-0.491593\pi$
$839$	9.03520e8	0.0528166	0.0264083	+	0.999651i	$0.491593\pi$
$840$	3.06886e9	0.178649
$841$	−1.72243e10	−0.998517
$842$	−3.52401e9	−0.203444
$843$	1.57233e10	0.903958
$844$	1.49360e10	0.855136
$845$	−3.08991e10	−1.76176
$846$	−1.45036e10	−0.823532
$847$	−1.37490e10	−0.777461
$848$	2.26554e10	1.27581
$849$	9.71930e9	0.545077
$850$	−6.02013e10	−3.36232
$851$	7.15956e9	0.398229
$852$	−6.44586e9	−0.357061
$853$	−1.51293e10	−0.834635	−0.417317	−	0.908761i	$-0.637030\pi$
$853$	−1.51293e10	−0.834635	−0.417317	+	0.908761i	$0.637030\pi$
$854$	6.38059e9	0.350557
$855$	7.50964e9	0.410902
$856$	−3.74049e9	−0.203831
$857$	−4.11557e9	−0.223356	−0.111678	−	0.993744i	$-0.535622\pi$
$857$	−4.11557e9	−0.223356	−0.111678	+	0.993744i	$0.535622\pi$
$858$	−1.24011e9	−0.0670277
$859$	1.02304e10	0.550703	0.275351	−	0.961344i	$-0.411206\pi$
$859$	1.02304e10	0.550703	0.275351	+	0.961344i	$0.411206\pi$
$860$	8.78384e9	0.470912
$861$	−1.90792e9	−0.101871
$862$	4.71635e10	2.50802
$863$	−1.78056e9	−0.0943015	−0.0471508	−	0.998888i	$-0.515014\pi$
$863$	−1.78056e9	−0.0943015	−0.0471508	+	0.998888i	$0.515014\pi$
$864$	−4.18234e9	−0.220608
$865$	7.60911e9	0.399740
$866$	−4.28340e10	−2.24118
$867$	5.63493e9	0.293644
$868$	7.17919e9	0.372612
$869$	−2.47924e10	−1.28159
$870$	−9.84691e8	−0.0506970
$871$	−1.10273e9	−0.0565465
$872$	6.25607e9	0.319517
$873$	−2.10722e8	−0.0107191
$874$	3.70614e9	0.187772
$875$	−1.59304e10	−0.803894
$876$	−1.18814e10	−0.597179
$877$	3.47384e10	1.73905	0.869523	−	0.493893i	$-0.164427\pi$
$877$	3.47384e10	1.73905	0.869523	+	0.493893i	$0.164427\pi$
$878$	−2.03914e10	−1.01676
$879$	−9.12448e9	−0.453156
$880$	−7.45538e10	−3.68791
$881$	2.81250e10	1.38572	0.692862	−	0.721070i	$-0.256349\pi$
$881$	2.81250e10	1.38572	0.692862	+	0.721070i	$0.256349\pi$
$882$	−7.32201e9	−0.359327
$883$	2.00161e10	0.978399	0.489200	−	0.872172i	$-0.337289\pi$
$883$	2.00161e10	0.978399	0.489200	+	0.872172i	$0.337289\pi$
$884$	8.84666e8	0.0430721
$885$	−2.12727e10	−1.03163
$886$	1.37711e10	0.665195
$887$	1.97644e10	0.950936	0.475468	−	0.879733i	$-0.342279\pi$
$887$	1.97644e10	0.950936	0.475468	+	0.879733i	$0.342279\pi$
$888$	−9.92859e9	−0.475819
$889$	3.59658e9	0.171685
$890$	2.00871e10	0.955109
$891$	4.00576e9	0.189720
$892$	−1.79305e9	−0.0845893
$893$	−2.84247e10	−1.33572
$894$	8.41562e9	0.393917
$895$	4.14086e10	1.93068
$896$	−7.12088e9	−0.330716
$897$	−1.37094e8	−0.00634227
$898$	−5.17027e10	−2.38257
$899$	1.15711e9	0.0531148
$900$	1.02926e10	0.470629
$901$	2.81410e10	1.28175
$902$	2.11143e10	0.957976
$903$	−2.07632e9	−0.0938400
$904$	−1.31111e9	−0.0590269
$905$	−7.45130e10	−3.34166
$906$	2.58893e10	1.15657
$907$	7.73861e8	0.0344380	0.0172190	−	0.999852i	$-0.494519\pi$
$907$	7.73861e8	0.0344380	0.0172190	+	0.999852i	$0.494519\pi$
$908$	−2.76370e10	−1.22515
$909$	3.78458e9	0.167126
$910$	1.10831e9	0.0487545
$911$	4.47493e9	0.196097	0.0980487	−	0.995182i	$-0.468740\pi$
$911$	4.47493e9	0.196097	0.0980487	+	0.995182i	$0.468740\pi$
$912$	−1.12823e10	−0.492511
$913$	4.44777e9	0.193417
$914$	−2.96652e10	−1.28510
$915$	−1.58174e10	−0.682590
$916$	−4.23608e8	−0.0182109
$917$	−1.73997e10	−0.745160
$918$	−7.15066e9	−0.305068
$919$	−2.87268e10	−1.22091	−0.610454	−	0.792052i	$-0.709013\pi$
$919$	−2.87268e10	−1.22091	−0.610454	+	0.792052i	$0.709013\pi$
$920$	−3.75451e9	−0.158963
$921$	−9.51194e9	−0.401200
$922$	−7.26189e9	−0.305135
$923$	1.16934e9	0.0489478
$924$	−6.38684e9	−0.266339
$925$	9.75110e10	4.05096
$926$	1.57572e10	0.652139
$927$	−4.98333e9	−0.205466
$928$	1.07478e9	0.0441470
$929$	2.38092e10	0.974294	0.487147	−	0.873320i	$-0.338038\pi$
$929$	2.38092e10	0.974294	0.487147	+	0.873320i	$0.338038\pi$
$930$	−4.45339e10	−1.81552
$931$	−1.43499e10	−0.582809
$932$	−4.55274e9	−0.184212
$933$	−8.32987e9	−0.335778
$934$	−1.54304e10	−0.619672
$935$	−9.26056e10	−3.70507
$936$	1.90116e8	0.00757799
$937$	−2.55610e10	−1.01505	−0.507527	−	0.861636i	$-0.669440\pi$
$937$	−2.55610e10	−1.01505	−0.507527	+	0.861636i	$0.669440\pi$
$938$	−1.42114e10	−0.562248
$939$	6.69996e9	0.264084
$940$	−5.73260e10	−2.25115
$941$	3.51742e9	0.137613	0.0688067	−	0.997630i	$-0.478081\pi$
$941$	3.51742e9	0.137613	0.0688067	+	0.997630i	$0.478081\pi$
$942$	1.51095e10	0.588942
$943$	2.33419e9	0.0906453
$944$	3.19597e10	1.23652
$945$	−3.58001e9	−0.137998
$946$	2.29780e10	0.882456
$947$	−4.81870e10	−1.84376	−0.921882	−	0.387471i	$-0.873349\pi$
$947$	−4.81870e10	−1.84376	−0.921882	+	0.387471i	$0.873349\pi$
$948$	−7.56662e9	−0.288452
$949$	2.15540e9	0.0818646
$950$	5.04765e10	1.91010
$951$	−9.94129e9	−0.374810
$952$	−5.72696e9	−0.215127
$953$	−4.14619e10	−1.55176	−0.775879	−	0.630882i	$-0.782693\pi$
$953$	−4.14619e10	−1.55176	−0.775879	+	0.630882i	$0.782693\pi$
$954$	−1.20394e10	−0.448936
$955$	1.45598e10	0.540933
$956$	−2.26166e10	−0.837193
$957$	−1.02940e9	−0.0379659
$958$	−1.76840e9	−0.0649830
$959$	1.05313e10	0.385581
$960$	−7.18194e9	−0.261995
$961$	2.48191e10	0.902098
$962$	−3.58566e9	−0.129854
$963$	4.36350e9	0.157450
$964$	−1.21318e10	−0.436168
$965$	7.03183e10	2.51897
$966$	−1.76680e9	−0.0630619
$967$	1.23269e10	0.438389	0.219195	−	0.975681i	$-0.429657\pi$
$967$	1.23269e10	0.438389	0.219195	+	0.975681i	$0.429657\pi$
$968$	−2.33264e10	−0.826577
$969$	−1.40141e10	−0.494803
$970$	−2.08414e9	−0.0733204
$971$	−6.05848e9	−0.212372	−0.106186	−	0.994346i	$-0.533864\pi$
$971$	−6.05848e9	−0.212372	−0.106186	+	0.994346i	$0.533864\pi$
$972$	1.22255e9	0.0427008
$973$	−7.27690e9	−0.253251
$974$	6.04081e10	2.09478
$975$	−1.86718e9	−0.0645163
$976$	2.37636e10	0.818160
$977$	3.09327e10	1.06118	0.530588	−	0.847630i	$-0.321971\pi$
$977$	3.09327e10	1.06118	0.530588	+	0.847630i	$0.321971\pi$
$978$	5.10950e9	0.174659
$979$	2.09992e10	0.715259
$980$	−2.89404e10	−0.982231
$981$	−7.29807e9	−0.246812
$982$	−5.30062e9	−0.178623
$983$	1.95459e9	0.0656325	0.0328162	−	0.999461i	$-0.489552\pi$
$983$	1.95459e9	0.0656325	0.0328162	+	0.999461i	$0.489552\pi$
$984$	−3.23696e9	−0.108307
$985$	−3.31086e10	−1.10386
$986$	1.83758e9	0.0610488
$987$	1.35507e10	0.448592
$988$	−7.41759e8	−0.0244688
$989$	2.54022e9	0.0834994
$990$	3.96188e10	1.29771
$991$	−2.87355e10	−0.937911	−0.468956	−	0.883222i	$-0.655370\pi$
$991$	−2.87355e10	−0.937911	−0.468956	+	0.883222i	$0.655370\pi$
$992$	4.86082e10	1.58095
$993$	−4.53204e9	−0.146883
$994$	1.50698e10	0.486694
$995$	8.87017e10	2.85464
$996$	1.35745e9	0.0435329
$997$	−4.10733e9	−0.131258	−0.0656291	−	0.997844i	$-0.520905\pi$
$997$	−4.10733e9	−0.131258	−0.0656291	+	0.997844i	$0.520905\pi$
$998$	−6.84101e10	−2.17853
$999$	1.15823e10	0.367549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.d.1.6	✓	8
3.2	odd	2		207.8.a.e.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.d.1.6	✓	8	1.1	even	1	trivial
207.8.a.e.1.3		8	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+14.6014 q^{2} +27.0000 q^{3} +85.2018 q^{4} +493.797 q^{5} +394.239 q^{6} -368.336 q^{7} -624.915 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+14.6014 q^{2} +27.0000 q^{3} +85.2018 q^{4} +493.797 q^{5} +394.239 q^{6} -368.336 q^{7} -624.915 q^{8} +729.000 q^{9} +7210.15 q^{10} +7537.54 q^{11} +2300.45 q^{12} -417.322 q^{13} -5378.24 q^{14} +13332.5 q^{15} -20030.5 q^{16} -24880.5 q^{17} +10644.4 q^{18} +20861.4 q^{19} +42072.4 q^{20} -9945.08 q^{21} +110059. q^{22} +12167.0 q^{23} -16872.7 q^{24} +165711. q^{25} -6093.50 q^{26} +19683.0 q^{27} -31382.9 q^{28} -5058.15 q^{29} +194674. q^{30} -228761. q^{31} -212485. q^{32} +203513. q^{33} -363291. q^{34} -181883. q^{35} +62112.1 q^{36} +588441. q^{37} +304606. q^{38} -11267.7 q^{39} -308581. q^{40} +191846. q^{41} -145212. q^{42} +208779. q^{43} +642212. q^{44} +359978. q^{45} +177656. q^{46} -1.36255e6 q^{47} -540823. q^{48} -687871. q^{49} +2.41962e6 q^{50} -671774. q^{51} -35556.6 q^{52} -1.13105e6 q^{53} +287400. q^{54} +3.72202e6 q^{55} +230179. q^{56} +563257. q^{57} -73856.3 q^{58} -1.59555e6 q^{59} +1.13596e6 q^{60} -1.18637e6 q^{61} -3.34024e6 q^{62} -268517. q^{63} -538678. q^{64} -206072. q^{65} +2.97159e6 q^{66} +2.64240e6 q^{67} -2.11986e6 q^{68} +328509. q^{69} -2.65576e6 q^{70} -2.80200e6 q^{71} -455563. q^{72} -5.16484e6 q^{73} +8.59208e6 q^{74} +4.47419e6 q^{75} +1.77743e6 q^{76} -2.77635e6 q^{77} -164524. q^{78} -3.28919e6 q^{79} -9.89100e6 q^{80} +531441. q^{81} +2.80123e6 q^{82} +590082. q^{83} -847338. q^{84} -1.22859e7 q^{85} +3.04847e6 q^{86} -136570. q^{87} -4.71032e6 q^{88} +2.78595e6 q^{89} +5.25620e6 q^{90} +153715. q^{91} +1.03665e6 q^{92} -6.17655e6 q^{93} -1.98952e7 q^{94} +1.03013e7 q^{95} -5.73709e6 q^{96} -289056. q^{97} -1.00439e7 q^{98} +5.49486e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9} + 11720 q^{10} + 6932 q^{11} + 15174 q^{12} + 12404 q^{13} + 30222 q^{14} + 10206 q^{15} + 27058 q^{16} + 24434 q^{17} + 17496 q^{18} - 14682 q^{19} - 3760 q^{20} + 3402 q^{21} + 36294 q^{22} + 97336 q^{23} + 113076 q^{24} + 144644 q^{25} + 325840 q^{26} + 157464 q^{27} - 21566 q^{28} + 255356 q^{29} + 316440 q^{30} + 450764 q^{31} + 647588 q^{32} + 187164 q^{33} + 191822 q^{34} + 1022616 q^{35} + 409698 q^{36} + 206240 q^{37} + 737372 q^{38} + 334908 q^{39} + 590028 q^{40} + 1053344 q^{41} + 815994 q^{42} + 1587806 q^{43} + 589366 q^{44} + 275562 q^{45} + 292008 q^{46} + 443336 q^{47} + 730566 q^{48} + 1944828 q^{49} - 1556112 q^{50} + 659718 q^{51} - 614236 q^{52} - 375530 q^{53} + 472392 q^{54} + 407792 q^{55} - 1316922 q^{56} - 396414 q^{57} - 1413384 q^{58} + 624008 q^{59} - 101520 q^{60} - 2005568 q^{61} - 3908272 q^{62} + 91854 q^{63} - 5082310 q^{64} + 646124 q^{65} + 979938 q^{66} - 2712286 q^{67} - 2289698 q^{68} + 2628072 q^{69} - 16499468 q^{70} - 6287176 q^{71} + 3053052 q^{72} - 10358312 q^{73} - 2000150 q^{74} + 3905388 q^{75} - 25107464 q^{76} - 2156840 q^{77} + 8797680 q^{78} - 8800574 q^{79} + 2384344 q^{80} + 4251528 q^{81} - 31799800 q^{82} + 384948 q^{83} - 582282 q^{84} - 17826684 q^{85} - 11563928 q^{86} + 6894612 q^{87} - 25202782 q^{88} - 3445530 q^{89} + 8543880 q^{90} - 16316740 q^{91} + 6837854 q^{92} + 12170628 q^{93} - 24237616 q^{94} + 26164288 q^{95} + 17484876 q^{96} - 28043764 q^{97} - 9998012 q^{98} + 5053428 q^{99}+O(q^{100}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9 + 11720 * q^10 + 6932 * q^11 + 15174 * q^12 + 12404 * q^13 + 30222 * q^14 + 10206 * q^15 + 27058 * q^16 + 24434 * q^17 + 17496 * q^18 - 14682 * q^19 - 3760 * q^20 + 3402 * q^21 + 36294 * q^22 + 97336 * q^23 + 113076 * q^24 + 144644 * q^25 + 325840 * q^26 + 157464 * q^27 - 21566 * q^28 + 255356 * q^29 + 316440 * q^30 + 450764 * q^31 + 647588 * q^32 + 187164 * q^33 + 191822 * q^34 + 1022616 * q^35 + 409698 * q^36 + 206240 * q^37 + 737372 * q^38 + 334908 * q^39 + 590028 * q^40 + 1053344 * q^41 + 815994 * q^42 + 1587806 * q^43 + 589366 * q^44 + 275562 * q^45 + 292008 * q^46 + 443336 * q^47 + 730566 * q^48 + 1944828 * q^49 - 1556112 * q^50 + 659718 * q^51 - 614236 * q^52 - 375530 * q^53 + 472392 * q^54 + 407792 * q^55 - 1316922 * q^56 - 396414 * q^57 - 1413384 * q^58 + 624008 * q^59 - 101520 * q^60 - 2005568 * q^61 - 3908272 * q^62 + 91854 * q^63 - 5082310 * q^64 + 646124 * q^65 + 979938 * q^66 - 2712286 * q^67 - 2289698 * q^68 + 2628072 * q^69 - 16499468 * q^70 - 6287176 * q^71 + 3053052 * q^72 - 10358312 * q^73 - 2000150 * q^74 + 3905388 * q^75 - 25107464 * q^76 - 2156840 * q^77 + 8797680 * q^78 - 8800574 * q^79 + 2384344 * q^80 + 4251528 * q^81 - 31799800 * q^82 + 384948 * q^83 - 582282 * q^84 - 17826684 * q^85 - 11563928 * q^86 + 6894612 * q^87 - 25202782 * q^88 - 3445530 * q^89 + 8543880 * q^90 - 16316740 * q^91 + 6837854 * q^92 + 12170628 * q^93 - 24237616 * q^94 + 26164288 * q^95 + 17484876 * q^96 - 28043764 * q^97 - 9998012 * q^98 + 5053428 * q^99

Introduction

L-functions

Modular forms

Varieties

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