LMFDB - Embedded newform 69.8.a.d.1.3

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			1.3
Root			$5.38924$ 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)

$f(q)$	$=$	$q-2.38924 q^{2} +27.0000 q^{3} -122.292 q^{4} -147.281 q^{5} -64.5095 q^{6} -1223.93 q^{7} +598.006 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-2.38924 q^{2} +27.0000 q^{3} -122.292 q^{4} -147.281 q^{5} -64.5095 q^{6} -1223.93 q^{7} +598.006 q^{8} +729.000 q^{9} +351.889 q^{10} -1637.33 q^{11} -3301.87 q^{12} +12894.7 q^{13} +2924.26 q^{14} -3976.58 q^{15} +14224.5 q^{16} -12052.3 q^{17} -1741.76 q^{18} +27609.1 q^{19} +18011.2 q^{20} -33046.1 q^{21} +3911.97 q^{22} +12167.0 q^{23} +16146.2 q^{24} -56433.4 q^{25} -30808.6 q^{26} +19683.0 q^{27} +149676. q^{28} +240721. q^{29} +9500.99 q^{30} -128196. q^{31} -110531. q^{32} -44207.9 q^{33} +28795.9 q^{34} +180261. q^{35} -89150.5 q^{36} +179159. q^{37} -65964.8 q^{38} +348158. q^{39} -88074.8 q^{40} +473457. q^{41} +78955.1 q^{42} +936296. q^{43} +200232. q^{44} -107368. q^{45} -29069.9 q^{46} +796871. q^{47} +384062. q^{48} +674461. q^{49} +134833. q^{50} -325413. q^{51} -1.57692e6 q^{52} -2.04568e6 q^{53} -47027.4 q^{54} +241147. q^{55} -731918. q^{56} +745447. q^{57} -575140. q^{58} +1.90819e6 q^{59} +486302. q^{60} -3.32363e6 q^{61} +306291. q^{62} -892245. q^{63} -1.55666e6 q^{64} -1.89915e6 q^{65} +105623. q^{66} -285366. q^{67} +1.47390e6 q^{68} +328509. q^{69} -430687. q^{70} +3.91888e6 q^{71} +435947. q^{72} -3.52247e6 q^{73} -428053. q^{74} -1.52370e6 q^{75} -3.37636e6 q^{76} +2.00398e6 q^{77} -831833. q^{78} +1.10676e6 q^{79} -2.09500e6 q^{80} +531441. q^{81} -1.13120e6 q^{82} -3.81576e6 q^{83} +4.04126e6 q^{84} +1.77508e6 q^{85} -2.23703e6 q^{86} +6.49947e6 q^{87} -979134. q^{88} +5.29119e6 q^{89} +256527. q^{90} -1.57823e7 q^{91} -1.48792e6 q^{92} -3.46129e6 q^{93} -1.90392e6 q^{94} -4.06629e6 q^{95} -2.98433e6 q^{96} +6.41098e6 q^{97} -1.61145e6 q^{98} -1.19361e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.38924	−0.211181	−0.105590	−	0.994410i	$-0.533673\pi$
$2$	−2.38924	−0.211181	−0.105590	+	0.994410i	$0.533673\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−122.292	−0.955403
$5$	−147.281	−0.526927	−0.263464	−	0.964669i	$-0.584865\pi$
$5$	−147.281	−0.526927	−0.263464	+	0.964669i	$0.584865\pi$
$6$	−64.5095	−0.121925
$7$	−1223.93	−1.34869	−0.674347	−	0.738415i	$-0.735575\pi$
$7$	−1223.93	−1.34869	−0.674347	+	0.738415i	$0.735575\pi$
$8$	598.006	0.412944
$9$	729.000	0.333333
$10$	351.889	0.111277
$11$	−1637.33	−0.370904	−0.185452	−	0.982653i	$-0.559375\pi$
$11$	−1637.33	−0.370904	−0.185452	+	0.982653i	$0.559375\pi$
$12$	−3301.87	−0.551602
$13$	12894.7	1.62784	0.813919	−	0.580978i	$-0.197330\pi$
$13$	12894.7	1.62784	0.813919	+	0.580978i	$0.197330\pi$
$14$	2924.26	0.284818
$15$	−3976.58	−0.304222
$16$	14224.5	0.868197
$17$	−12052.3	−0.594977	−0.297489	−	0.954725i	$-0.596149\pi$
$17$	−12052.3	−0.594977	−0.297489	+	0.954725i	$0.596149\pi$
$18$	−1741.76	−0.0703936
$19$	27609.1	0.923454	0.461727	−	0.887022i	$-0.347230\pi$
$19$	27609.1	0.923454	0.461727	+	0.887022i	$0.347230\pi$
$20$	18011.2	0.503428
$21$	−33046.1	−0.778669
$22$	3911.97	0.0783279
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	16146.2	0.238413
$25$	−56433.4	−0.722348
$26$	−30808.6	−0.343768
$27$	19683.0	0.192450
$28$	149676.	1.28855
$29$	240721.	1.83283	0.916413	−	0.400235i	$-0.131071\pi$
$29$	240721.	1.83283	0.916413	+	0.400235i	$0.131071\pi$
$30$	9500.99	0.0642458
$31$	−128196.	−0.772874	−0.386437	−	0.922316i	$-0.626294\pi$
$31$	−128196.	−0.772874	−0.386437	+	0.922316i	$0.626294\pi$
$32$	−110531.	−0.596290
$33$	−44207.9	−0.214142
$34$	28795.9	0.125648
$35$	180261.	0.710664
$36$	−89150.5	−0.318468
$37$	179159.	0.581476	0.290738	−	0.956803i	$-0.406099\pi$
$37$	179159.	0.581476	0.290738	+	0.956803i	$0.406099\pi$
$38$	−65964.8	−0.195016
$39$	348158.	0.939833
$40$	−88074.8	−0.217591
$41$	473457.	1.07285	0.536423	−	0.843949i	$-0.319775\pi$
$41$	473457.	1.07285	0.536423	+	0.843949i	$0.319775\pi$
$42$	78955.1	0.164440
$43$	936296.	1.79586	0.897932	−	0.440135i	$-0.145069\pi$
$43$	936296.	1.79586	0.897932	+	0.440135i	$0.145069\pi$
$44$	200232.	0.354363
$45$	−107368.	−0.175642
$46$	−29069.9	−0.0440343
$47$	796871.	1.11956	0.559778	−	0.828643i	$-0.310887\pi$
$47$	796871.	1.11956	0.559778	+	0.828643i	$0.310887\pi$
$48$	384062.	0.501254
$49$	674461.	0.818975
$50$	134833.	0.152546
$51$	−325413.	−0.343510
$52$	−1.57692e6	−1.55524
$53$	−2.04568e6	−1.88743	−0.943717	−	0.330755i	$-0.892697\pi$
$53$	−2.04568e6	−1.88743	−0.943717	+	0.330755i	$0.892697\pi$
$54$	−47027.4	−0.0406418
$55$	241147.	0.195440
$56$	−731918.	−0.556935
$57$	745447.	0.533156
$58$	−575140.	−0.387058
$59$	1.90819e6	1.20959	0.604796	−	0.796380i	$-0.293254\pi$
$59$	1.90819e6	1.20959	0.604796	+	0.796380i	$0.293254\pi$
$60$	486302.	0.290654
$61$	−3.32363e6	−1.87481	−0.937407	−	0.348236i	$-0.886781\pi$
$61$	−3.32363e6	−1.87481	−0.937407	+	0.348236i	$0.886781\pi$
$62$	306291.	0.163216
$63$	−892245.	−0.449565
$64$	−1.55666e6	−0.742272
$65$	−1.89915e6	−0.857752
$66$	105623.	0.0452226
$67$	−285366.	−0.115915	−0.0579576	−	0.998319i	$-0.518459\pi$
$67$	−285366.	−0.115915	−0.0579576	+	0.998319i	$0.518459\pi$
$68$	1.47390e6	0.568443
$69$	328509.	0.120386
$70$	−430687.	−0.150079
$71$	3.91888e6	1.29944	0.649722	−	0.760172i	$-0.274885\pi$
$71$	3.91888e6	1.29944	0.649722	+	0.760172i	$0.274885\pi$
$72$	435947.	0.137648
$73$	−3.52247e6	−1.05978	−0.529891	−	0.848065i	$-0.677767\pi$
$73$	−3.52247e6	−1.05978	−0.529891	+	0.848065i	$0.677767\pi$
$74$	−428053.	−0.122797
$75$	−1.52370e6	−0.417048
$76$	−3.37636e6	−0.882270
$77$	2.00398e6	0.500236
$78$	−831833.	−0.198475
$79$	1.10676e6	0.252556	0.126278	−	0.991995i	$-0.459697\pi$
$79$	1.10676e6	0.252556	0.126278	+	0.991995i	$0.459697\pi$
$80$	−2.09500e6	−0.457477
$81$	531441.	0.111111
$82$	−1.13120e6	−0.226565
$83$	−3.81576e6	−0.732501	−0.366250	−	0.930516i	$-0.619359\pi$
$83$	−3.81576e6	−0.732501	−0.366250	+	0.930516i	$0.619359\pi$
$84$	4.04126e6	0.743942
$85$	1.77508e6	0.313510
$86$	−2.23703e6	−0.379252
$87$	6.49947e6	1.05818
$88$	−979134.	−0.153163
$89$	5.29119e6	0.795588	0.397794	−	0.917475i	$-0.369776\pi$
$89$	5.29119e6	0.795588	0.397794	+	0.917475i	$0.369776\pi$
$90$	256527.	0.0370923
$91$	−1.57823e7	−2.19546
$92$	−1.48792e6	−0.199215
$93$	−3.46129e6	−0.446219
$94$	−1.90392e6	−0.236429
$95$	−4.06629e6	−0.486593
$96$	−2.98433e6	−0.344268
$97$	6.41098e6	0.713220	0.356610	−	0.934253i	$-0.383932\pi$
$97$	6.41098e6	0.713220	0.356610	+	0.934253i	$0.383932\pi$
$98$	−1.61145e6	−0.172952
$99$	−1.19361e6	−0.123635
$100$	6.90133e6	0.690133
$101$	7.39048e6	0.713752	0.356876	−	0.934152i	$-0.383842\pi$
$101$	7.39048e6	0.713752	0.356876	+	0.934152i	$0.383842\pi$
$102$	777490.	0.0725428
$103$	6.97336e6	0.628799	0.314400	−	0.949291i	$-0.398197\pi$
$103$	6.97336e6	0.628799	0.314400	+	0.949291i	$0.398197\pi$
$104$	7.71114e6	0.672205
$105$	4.86705e6	0.410302
$106$	4.88761e6	0.398590
$107$	−1.70173e7	−1.34291	−0.671456	−	0.741045i	$-0.734331\pi$
$107$	−1.70173e7	−1.34291	−0.671456	+	0.741045i	$0.734331\pi$
$108$	−2.40706e6	−0.183867
$109$	−4.11126e6	−0.304076	−0.152038	−	0.988375i	$-0.548584\pi$
$109$	−4.11126e6	−0.304076	−0.152038	+	0.988375i	$0.548584\pi$
$110$	−576158.	−0.0412731
$111$	4.83729e6	0.335715
$112$	−1.74098e7	−1.17093
$113$	2.09127e7	1.36344	0.681721	−	0.731613i	$-0.261232\pi$
$113$	2.09127e7	1.36344	0.681721	+	0.731613i	$0.261232\pi$
$114$	−1.78105e6	−0.112592
$115$	−1.79196e6	−0.109872
$116$	−2.94381e7	−1.75109
$117$	9.40027e6	0.542613
$118$	−4.55912e6	−0.255443
$119$	1.47512e7	0.802442
$120$	−2.37802e6	−0.125626
$121$	−1.68063e7	−0.862430
$122$	7.94095e6	0.395925
$123$	1.27833e7	0.619408
$124$	1.56773e7	0.738406
$125$	1.98178e7	0.907552
$126$	2.13179e6	0.0949395
$127$	2.31258e7	1.00181	0.500903	−	0.865503i	$-0.333001\pi$
$127$	2.31258e7	1.00181	0.500903	+	0.865503i	$0.333001\pi$
$128$	1.78671e7	0.753044
$129$	2.52800e7	1.03684
$130$	4.53752e6	0.181141
$131$	4.03840e7	1.56950	0.784748	−	0.619815i	$-0.212792\pi$
$131$	4.03840e7	1.56950	0.784748	+	0.619815i	$0.212792\pi$
$132$	5.40625e6	0.204591
$133$	−3.37917e7	−1.24546
$134$	681808.	0.0244791
$135$	−2.89892e6	−0.101407
$136$	−7.20738e6	−0.245692
$137$	4.05545e7	1.34746	0.673732	−	0.738976i	$-0.264690\pi$
$137$	4.05545e7	1.34746	0.673732	+	0.738976i	$0.264690\pi$
$138$	−784887.	−0.0254232
$139$	−2.68652e7	−0.848474	−0.424237	−	0.905551i	$-0.639458\pi$
$139$	−2.68652e7	−0.848474	−0.424237	+	0.905551i	$0.639458\pi$
$140$	−2.20444e7	−0.678970
$141$	2.15155e7	0.646376
$142$	−9.36314e6	−0.274418
$143$	−2.11130e7	−0.603772
$144$	1.03697e7	0.289399
$145$	−3.54536e7	−0.965766
$146$	8.41602e6	0.223806
$147$	1.82105e7	0.472836
$148$	−2.19096e7	−0.555544
$149$	−1.26633e7	−0.313614	−0.156807	−	0.987629i	$-0.550120\pi$
$149$	−1.26633e7	−0.313614	−0.156807	+	0.987629i	$0.550120\pi$
$150$	3.64049e6	0.0880725
$151$	840099.	0.0198569	0.00992844	−	0.999951i	$-0.496840\pi$
$151$	840099.	0.0198569	0.00992844	+	0.999951i	$0.496840\pi$
$152$	1.65104e7	0.381335
$153$	−8.78616e6	−0.198326
$154$	−4.78798e6	−0.105640
$155$	1.88808e7	0.407248
$156$	−4.25768e7	−0.897919
$157$	−960571.	−0.0198098	−0.00990491	−	0.999951i	$-0.503153\pi$
$157$	−960571.	−0.0198098	−0.00990491	+	0.999951i	$0.503153\pi$
$158$	−2.64431e6	−0.0533350
$159$	−5.52333e7	−1.08971
$160$	1.62790e7	0.314202
$161$	−1.48916e7	−0.281222
$162$	−1.26974e6	−0.0234645
$163$	3.15848e7	0.571245	0.285622	−	0.958342i	$-0.407800\pi$
$163$	3.15848e7	0.571245	0.285622	+	0.958342i	$0.407800\pi$
$164$	−5.78998e7	−1.02500
$165$	6.51097e6	0.112837
$166$	9.11677e6	0.154690
$167$	−7.78579e7	−1.29358	−0.646792	−	0.762666i	$-0.723890\pi$
$167$	−7.78579e7	−1.29358	−0.646792	+	0.762666i	$0.723890\pi$
$168$	−1.97618e7	−0.321546
$169$	1.03526e8	1.64986
$170$	−4.24108e6	−0.0662072
$171$	2.01271e7	0.307818
$172$	−1.14501e8	−1.71577
$173$	5.01041e7	0.735719	0.367860	−	0.929881i	$-0.380091\pi$
$173$	5.01041e7	0.735719	0.367860	+	0.929881i	$0.380091\pi$
$174$	−1.55288e7	−0.223468
$175$	6.90705e7	0.974226
$176$	−2.32903e7	−0.322018
$177$	5.15211e7	0.698359
$178$	−1.26419e7	−0.168013
$179$	−3.80878e7	−0.496365	−0.248182	−	0.968713i	$-0.579833\pi$
$179$	−3.80878e7	−0.496365	−0.248182	+	0.968713i	$0.579833\pi$
$180$	1.31301e7	0.167809
$181$	3.63781e7	0.456000	0.228000	−	0.973661i	$-0.426781\pi$
$181$	3.63781e7	0.456000	0.228000	+	0.973661i	$0.426781\pi$
$182$	3.77076e7	0.463638
$183$	−8.97380e7	−1.08242
$184$	7.27594e6	0.0861047
$185$	−2.63866e7	−0.306396
$186$	8.26986e6	0.0942329
$187$	1.97337e7	0.220679
$188$	−9.74506e7	−1.06963
$189$	−2.40906e7	−0.259556
$190$	9.71534e6	0.102759
$191$	9.38592e7	0.974676	0.487338	−	0.873214i	$-0.337968\pi$
$191$	9.38592e7	0.974676	0.487338	+	0.873214i	$0.337968\pi$
$192$	−4.20297e7	−0.428551
$193$	−9.36357e7	−0.937543	−0.468771	−	0.883320i	$-0.655303\pi$
$193$	−9.36357e7	−0.937543	−0.468771	+	0.883320i	$0.655303\pi$
$194$	−1.53174e7	−0.150618
$195$	−5.12770e7	−0.495223
$196$	−8.24809e7	−0.782451
$197$	6.82715e7	0.636220	0.318110	−	0.948054i	$-0.396952\pi$
$197$	6.82715e7	0.636220	0.318110	+	0.948054i	$0.396952\pi$
$198$	2.85183e6	0.0261093
$199$	6.37165e7	0.573147	0.286573	−	0.958058i	$-0.407484\pi$
$199$	6.37165e7	0.573147	0.286573	+	0.958058i	$0.407484\pi$
$200$	−3.37475e7	−0.298289
$201$	−7.70488e6	−0.0669237
$202$	−1.76576e7	−0.150731
$203$	−2.94626e8	−2.47192
$204$	3.97953e7	0.328190
$205$	−6.97311e7	−0.565312
$206$	−1.66610e7	−0.132790
$207$	8.86974e6	0.0695048
$208$	1.83422e8	1.41328
$209$	−4.52053e7	−0.342513
$210$	−1.16286e7	−0.0866479
$211$	−3.67268e7	−0.269150	−0.134575	−	0.990903i	$-0.542967\pi$
$211$	−3.67268e7	−0.269150	−0.134575	+	0.990903i	$0.542967\pi$
$212$	2.50169e8	1.80326
$213$	1.05810e8	0.750234
$214$	4.06584e7	0.283597
$215$	−1.37898e8	−0.946289
$216$	1.17706e7	0.0794711
$217$	1.56903e8	1.04237
$218$	9.82278e6	0.0642150
$219$	−9.51066e7	−0.611866
$220$	−2.94902e7	−0.186723
$221$	−1.55412e8	−0.968526
$222$	−1.15574e7	−0.0708967
$223$	1.94555e8	1.17483	0.587415	−	0.809286i	$-0.300146\pi$
$223$	1.94555e8	1.17483	0.587415	+	0.809286i	$0.300146\pi$
$224$	1.35282e8	0.804213
$225$	−4.11400e7	−0.240783
$226$	−4.99655e7	−0.287933
$227$	2.74851e8	1.55958	0.779789	−	0.626043i	$-0.215326\pi$
$227$	2.74851e8	1.55958	0.779789	+	0.626043i	$0.215326\pi$
$228$	−9.11618e7	−0.509379
$229$	−1.43628e8	−0.790344	−0.395172	−	0.918607i	$-0.629315\pi$
$229$	−1.43628e8	−0.790344	−0.395172	+	0.918607i	$0.629315\pi$
$230$	4.28143e6	0.0232029
$231$	5.41074e7	0.288812
$232$	1.43953e8	0.756854
$233$	3.20512e8	1.65996	0.829982	−	0.557790i	$-0.188351\pi$
$233$	3.20512e8	1.65996	0.829982	+	0.557790i	$0.188351\pi$
$234$	−2.24595e7	−0.114589
$235$	−1.17364e8	−0.589924
$236$	−2.33355e8	−1.15565
$237$	2.98825e7	0.145813
$238$	−3.52442e7	−0.169460
$239$	−4.41622e7	−0.209247	−0.104623	−	0.994512i	$-0.533364\pi$
$239$	−4.41622e7	−0.209247	−0.104623	+	0.994512i	$0.533364\pi$
$240$	−5.65650e7	−0.264124
$241$	7.53980e7	0.346977	0.173488	−	0.984836i	$-0.444496\pi$
$241$	7.53980e7	0.346977	0.173488	+	0.984836i	$0.444496\pi$
$242$	4.01543e7	0.182129
$243$	1.43489e7	0.0641500
$244$	4.06452e8	1.79120
$245$	−9.93351e7	−0.431540
$246$	−3.05425e7	−0.130807
$247$	3.56013e8	1.50323
$248$	−7.66620e7	−0.319153
$249$	−1.03026e8	−0.422910
$250$	−4.73496e7	−0.191658
$251$	7.67520e7	0.306360	0.153180	−	0.988198i	$-0.451049\pi$
$251$	7.67520e7	0.306360	0.153180	+	0.988198i	$0.451049\pi$
$252$	1.09114e8	0.429515
$253$	−1.99214e7	−0.0773389
$254$	−5.52530e7	−0.211562
$255$	4.79271e7	0.181005
$256$	1.56563e8	0.583243
$257$	8.08367e6	0.0297059	0.0148529	−	0.999890i	$-0.495272\pi$
$257$	8.08367e6	0.0297059	0.0148529	+	0.999890i	$0.495272\pi$
$258$	−6.03999e7	−0.218961
$259$	−2.19278e8	−0.784234
$260$	2.32250e8	0.819499
$261$	1.75486e8	0.610942
$262$	−9.64871e7	−0.331448
$263$	2.78979e8	0.945640	0.472820	−	0.881159i	$-0.343236\pi$
$263$	2.78979e8	0.945640	0.472820	+	0.881159i	$0.343236\pi$
$264$	−2.64366e7	−0.0884284
$265$	3.01289e8	0.994540
$266$	8.07364e7	0.263017
$267$	1.42862e8	0.459333
$268$	3.48978e7	0.110746
$269$	−3.28037e8	−1.02752	−0.513760	−	0.857934i	$-0.671748\pi$
$269$	−3.28037e8	−1.02752	−0.513760	+	0.857934i	$0.671748\pi$
$270$	6.92622e6	0.0214153
$271$	−1.91549e8	−0.584639	−0.292319	−	0.956321i	$-0.594427\pi$
$271$	−1.91549e8	−0.584639	−0.292319	+	0.956321i	$0.594427\pi$
$272$	−1.71439e8	−0.516557
$273$	−4.26121e8	−1.26755
$274$	−9.68944e7	−0.284558
$275$	9.24001e7	0.267922
$276$	−4.01739e7	−0.115017
$277$	1.80215e8	0.509463	0.254731	−	0.967012i	$-0.418013\pi$
$277$	1.80215e8	0.509463	0.254731	+	0.967012i	$0.418013\pi$
$278$	6.41874e7	0.179181
$279$	−9.34549e7	−0.257625
$280$	1.07797e8	0.293464
$281$	−1.57503e8	−0.423465	−0.211733	−	0.977328i	$-0.567911\pi$
$281$	−1.57503e8	−0.423465	−0.211733	+	0.977328i	$0.567911\pi$
$282$	−5.14057e7	−0.136502
$283$	6.94865e8	1.82242	0.911208	−	0.411946i	$-0.135151\pi$
$283$	6.94865e8	1.82242	0.911208	+	0.411946i	$0.135151\pi$
$284$	−4.79246e8	−1.24149
$285$	−1.09790e8	−0.280935
$286$	5.04439e7	0.127505
$287$	−5.79479e8	−1.44694
$288$	−8.05768e7	−0.198763
$289$	−2.65080e8	−0.646002
$290$	8.47070e7	0.203951
$291$	1.73097e8	0.411778
$292$	4.30768e8	1.01252
$293$	−3.61524e8	−0.839653	−0.419827	−	0.907604i	$-0.637909\pi$
$293$	−3.61524e8	−0.839653	−0.419827	+	0.907604i	$0.637909\pi$
$294$	−4.35091e7	−0.0998539
$295$	−2.81039e8	−0.637367
$296$	1.07138e8	0.240117
$297$	−3.22276e7	−0.0713805
$298$	3.02557e7	0.0662293
$299$	1.56890e8	0.339428
$300$	1.86336e8	0.398448
$301$	−1.14596e9	−2.42207
$302$	−2.00720e6	−0.00419339
$303$	1.99543e8	0.412085
$304$	3.92727e8	0.801740
$305$	4.89506e8	0.987891
$306$	2.09922e7	0.0418826
$307$	−3.33653e8	−0.658129	−0.329065	−	0.944307i	$-0.606733\pi$
$307$	−3.33653e8	−0.658129	−0.329065	+	0.944307i	$0.606733\pi$
$308$	−2.45069e8	−0.477927
$309$	1.88281e8	0.363037
$310$	−4.51107e7	−0.0860031
$311$	2.96226e8	0.558421	0.279211	−	0.960230i	$-0.409927\pi$
$311$	2.96226e8	0.558421	0.279211	+	0.960230i	$0.409927\pi$
$312$	2.08201e8	0.388098
$313$	−3.47401e8	−0.640363	−0.320181	−	0.947356i	$-0.603744\pi$
$313$	−3.47401e8	−0.640363	−0.320181	+	0.947356i	$0.603744\pi$
$314$	2.29503e6	0.00418346
$315$	1.31410e8	0.236888
$316$	−1.35347e8	−0.241293
$317$	−2.12080e8	−0.373931	−0.186966	−	0.982366i	$-0.559865\pi$
$317$	−2.12080e8	−0.373931	−0.186966	+	0.982366i	$0.559865\pi$
$318$	1.31966e8	0.230126
$319$	−3.94140e8	−0.679803
$320$	2.29265e8	0.391123
$321$	−4.59467e8	−0.775330
$322$	3.55795e7	0.0593887
$323$	−3.32755e8	−0.549434
$324$	−6.49907e7	−0.106156
$325$	−7.27695e8	−1.17586
$326$	−7.54638e7	−0.120636
$327$	−1.11004e8	−0.175558
$328$	2.83130e8	0.443025
$329$	−9.75314e8	−1.50994
$330$	−1.55563e7	−0.0238290
$331$	3.92104e8	0.594296	0.297148	−	0.954831i	$-0.403964\pi$
$331$	3.92104e8	0.594296	0.297148	+	0.954831i	$0.403964\pi$
$332$	4.66636e8	0.699833
$333$	1.30607e8	0.193825
$334$	1.86021e8	0.273180
$335$	4.20289e7	0.0610789
$336$	−4.70066e8	−0.676038
$337$	−9.07353e8	−1.29143	−0.645716	−	0.763578i	$-0.723441\pi$
$337$	−9.07353e8	−1.29143	−0.645716	+	0.763578i	$0.723441\pi$
$338$	−2.47348e8	−0.348418
$339$	5.64644e8	0.787183
$340$	−2.17077e8	−0.299528
$341$	2.09899e8	0.286662
$342$	−4.80884e7	−0.0650053
$343$	1.82465e8	0.244147
$344$	5.59911e8	0.741591
$345$	−4.83830e7	−0.0634346
$346$	−1.19711e8	−0.155370
$347$	−3.73740e8	−0.480194	−0.240097	−	0.970749i	$-0.577179\pi$
$347$	−3.73740e8	−0.480194	−0.240097	+	0.970749i	$0.577179\pi$
$348$	−7.94830e8	−1.01099
$349$	−1.88713e8	−0.237636	−0.118818	−	0.992916i	$-0.537910\pi$
$349$	−1.88713e8	−0.237636	−0.118818	+	0.992916i	$0.537910\pi$
$350$	−1.65026e8	−0.205738
$351$	2.53807e8	0.313278
$352$	1.80975e8	0.221167
$353$	1.22544e9	1.48279	0.741395	−	0.671069i	$-0.234164\pi$
$353$	1.22544e9	1.48279	0.741395	+	0.671069i	$0.234164\pi$
$354$	−1.23096e8	−0.147480
$355$	−5.77175e8	−0.684712
$356$	−6.47067e8	−0.760107
$357$	3.98283e8	0.463290
$358$	9.10010e7	0.104823
$359$	−4.27509e8	−0.487657	−0.243829	−	0.969818i	$-0.578403\pi$
$359$	−4.27509e8	−0.487657	−0.243829	+	0.969818i	$0.578403\pi$
$360$	−6.42065e7	−0.0725304
$361$	−1.31607e8	−0.147232
$362$	−8.69159e7	−0.0962984
$363$	−4.53771e8	−0.497924
$364$	1.93004e9	2.09754
$365$	5.18791e8	0.558428
$366$	2.14406e8	0.228587
$367$	−1.58489e9	−1.67366	−0.836831	−	0.547462i	$-0.815594\pi$
$367$	−1.58489e9	−1.67366	−0.836831	+	0.547462i	$0.815594\pi$
$368$	1.73070e8	0.181032
$369$	3.45150e8	0.357615
$370$	6.30439e7	0.0647049
$371$	2.50377e9	2.54557
$372$	4.23287e8	0.426319
$373$	−1.23598e9	−1.23320	−0.616598	−	0.787278i	$-0.711490\pi$
$373$	−1.23598e9	−1.23320	−0.616598	+	0.787278i	$0.711490\pi$
$374$	−4.71484e7	−0.0466033
$375$	5.35082e8	0.523975
$376$	4.76534e8	0.462313
$377$	3.10404e9	2.98354
$378$	5.75582e7	0.0548133
$379$	6.22899e8	0.587733	0.293867	−	0.955846i	$-0.405058\pi$
$379$	6.22899e8	0.587733	0.293867	+	0.955846i	$0.405058\pi$
$380$	4.97273e8	0.464892
$381$	6.24396e8	0.578393
$382$	−2.24252e8	−0.205833
$383$	−1.18281e9	−1.07577	−0.537884	−	0.843019i	$-0.680776\pi$
$383$	−1.18281e9	−1.07577	−0.537884	+	0.843019i	$0.680776\pi$
$384$	4.82413e8	0.434770
$385$	−2.95147e8	−0.263588
$386$	2.23718e8	0.197991
$387$	6.82559e8	0.598621
$388$	−7.84009e8	−0.681412
$389$	7.89553e8	0.680077	0.340038	−	0.940412i	$-0.389560\pi$
$389$	7.89553e8	0.680077	0.340038	+	0.940412i	$0.389560\pi$
$390$	1.22513e8	0.104582
$391$	−1.46641e8	−0.124061
$392$	4.03332e8	0.338191
$393$	1.09037e9	0.906149
$394$	−1.63117e8	−0.134358
$395$	−1.63004e8	−0.133079
$396$	1.45969e8	0.118121
$397$	−1.58576e9	−1.27195	−0.635977	−	0.771708i	$-0.719403\pi$
$397$	−1.58576e9	−1.27195	−0.635977	+	0.771708i	$0.719403\pi$
$398$	−1.52234e8	−0.121038
$399$	−9.12375e8	−0.719065
$400$	−8.02739e8	−0.627140
$401$	−9.63924e8	−0.746513	−0.373256	−	0.927728i	$-0.621759\pi$
$401$	−9.63924e8	−0.746513	−0.373256	+	0.927728i	$0.621759\pi$
$402$	1.84088e7	0.0141330
$403$	−1.65306e9	−1.25811
$404$	−9.03793e8	−0.681921
$405$	−7.82710e7	−0.0585475
$406$	7.03931e8	0.522022
$407$	−2.93342e8	−0.215672
$408$	−1.94599e8	−0.141850
$409$	1.37917e9	0.996752	0.498376	−	0.866961i	$-0.333930\pi$
$409$	1.37917e9	0.996752	0.498376	+	0.866961i	$0.333930\pi$
$410$	1.66604e8	0.119383
$411$	1.09497e9	0.777958
$412$	−8.52783e8	−0.600756
$413$	−2.33549e9	−1.63137
$414$	−2.11919e7	−0.0146781
$415$	5.61988e8	0.385975
$416$	−1.42526e9	−0.970664
$417$	−7.25361e8	−0.489867
$418$	1.08006e8	0.0723322
$419$	2.43158e9	1.61487	0.807437	−	0.589953i	$-0.200854\pi$
$419$	2.43158e9	1.61487	0.807437	+	0.589953i	$0.200854\pi$
$420$	−5.95199e8	−0.392003
$421$	2.63767e7	0.0172279	0.00861396	−	0.999963i	$-0.497258\pi$
$421$	2.63767e7	0.0172279	0.00861396	+	0.999963i	$0.497258\pi$
$422$	8.77491e7	0.0568393
$423$	5.80919e8	0.373185
$424$	−1.22333e9	−0.779404
$425$	6.80155e8	0.429780
$426$	−2.52805e8	−0.158435
$427$	4.06789e9	2.52855
$428$	2.08107e9	1.28302
$429$	−5.70050e8	−0.348588
$430$	3.29472e8	0.199838
$431$	−2.18762e9	−1.31614	−0.658070	−	0.752957i	$-0.728627\pi$
$431$	−2.18762e9	−1.31614	−0.658070	+	0.752957i	$0.728627\pi$
$432$	2.79982e8	0.167085
$433$	−1.05899e6	−0.000626880 0	−0.000313440	−	1.00000i	$-0.500100\pi$
$433$	−1.05899e6	−0.000626880 0	−0.000313440		1.00000i	$0.500100\pi$
$434$	−3.74879e8	−0.220129
$435$	−9.57246e8	−0.557585
$436$	5.02772e8	0.290515
$437$	3.35920e8	0.192553
$438$	2.27232e8	0.129214
$439$	2.62942e9	1.48332	0.741660	−	0.670776i	$-0.234039\pi$
$439$	2.62942e9	1.48332	0.741660	+	0.670776i	$0.234039\pi$
$440$	1.44207e8	0.0807055
$441$	4.91682e8	0.272992
$442$	3.71316e8	0.204534
$443$	6.92479e8	0.378437	0.189219	−	0.981935i	$-0.439405\pi$
$443$	6.92479e8	0.378437	0.189219	+	0.981935i	$0.439405\pi$
$444$	−5.91559e8	−0.320743
$445$	−7.79289e8	−0.419217
$446$	−4.64838e8	−0.248102
$447$	−3.41909e8	−0.181065
$448$	1.90524e9	1.00110
$449$	−1.34070e9	−0.698990	−0.349495	−	0.936938i	$-0.613647\pi$
$449$	−1.34070e9	−0.698990	−0.349495	+	0.936938i	$0.613647\pi$
$450$	9.82932e7	0.0508487
$451$	−7.75206e8	−0.397923
$452$	−2.55745e9	−1.30264
$453$	2.26827e7	0.0114644
$454$	−6.56685e8	−0.329353
$455$	2.32442e9	1.15685
$456$	4.45782e8	0.220164
$457$	1.96222e8	0.0961701	0.0480851	−	0.998843i	$-0.484688\pi$
$457$	1.96222e8	0.0961701	0.0480851	+	0.998843i	$0.484688\pi$
$458$	3.43162e8	0.166905
$459$	−2.37226e8	−0.114503
$460$	2.19142e8	0.104972
$461$	−2.83515e9	−1.34779	−0.673895	−	0.738827i	$-0.735380\pi$
$461$	−2.83515e9	−1.34779	−0.673895	+	0.738827i	$0.735380\pi$
$462$	−1.29275e8	−0.0609915
$463$	−2.19801e9	−1.02919	−0.514595	−	0.857433i	$-0.672058\pi$
$463$	−2.19801e9	−1.02919	−0.514595	+	0.857433i	$0.672058\pi$
$464$	3.42415e9	1.59125
$465$	5.09781e8	0.235125
$466$	−7.65780e8	−0.350553
$467$	1.42423e9	0.647100	0.323550	−	0.946211i	$-0.395124\pi$
$467$	1.42423e9	0.647100	0.323550	+	0.946211i	$0.395124\pi$
$468$	−1.14957e9	−0.518414
$469$	3.49268e8	0.156334
$470$	2.80410e8	0.124581
$471$	−2.59354e7	−0.0114372
$472$	1.14111e9	0.499494
$473$	−1.53302e9	−0.666093
$474$	−7.13964e7	−0.0307930
$475$	−1.55808e9	−0.667055
$476$	−1.80395e9	−0.766655
$477$	−1.49130e9	−0.629144
$478$	1.05514e8	0.0441889
$479$	3.08517e9	1.28264	0.641320	−	0.767274i	$-0.278387\pi$
$479$	3.08517e9	1.28264	0.641320	+	0.767274i	$0.278387\pi$
$480$	4.39534e8	0.181404
$481$	2.31021e9	0.946549
$482$	−1.80144e8	−0.0732749
$483$	−4.02072e8	−0.162364
$484$	2.05527e9	0.823968
$485$	−9.44214e8	−0.375815
$486$	−3.42830e7	−0.0135473
$487$	−2.32140e9	−0.910748	−0.455374	−	0.890300i	$-0.650494\pi$
$487$	−2.32140e9	−0.910748	−0.455374	+	0.890300i	$0.650494\pi$
$488$	−1.98755e9	−0.774193
$489$	8.52791e8	0.329808
$490$	2.37335e8	0.0911331
$491$	−3.13714e8	−0.119605	−0.0598024	−	0.998210i	$-0.519047\pi$
$491$	−3.13714e8	−0.119605	−0.0598024	+	0.998210i	$0.519047\pi$
$492$	−1.56329e9	−0.591784
$493$	−2.90125e9	−1.09049
$494$	−8.50600e8	−0.317454
$495$	1.75796e8	0.0651465
$496$	−1.82353e9	−0.671007
$497$	−4.79643e9	−1.75255
$498$	2.46153e8	0.0893104
$499$	5.75078e8	0.207193	0.103596	−	0.994619i	$-0.466965\pi$
$499$	5.75078e8	0.207193	0.103596	+	0.994619i	$0.466965\pi$
$500$	−2.42356e9	−0.867077
$501$	−2.10216e9	−0.746852
$502$	−1.83379e8	−0.0646973
$503$	−1.87631e9	−0.657379	−0.328690	−	0.944438i	$-0.606607\pi$
$503$	−1.87631e9	−0.657379	−0.328690	+	0.944438i	$0.606607\pi$
$504$	−5.33568e8	−0.185645
$505$	−1.08847e9	−0.376096
$506$	4.75970e7	0.0163325
$507$	2.79520e9	0.952545
$508$	−2.82809e9	−0.957128
$509$	4.24575e9	1.42706	0.713529	−	0.700625i	$-0.247096\pi$
$509$	4.24575e9	1.42706	0.713529	+	0.700625i	$0.247096\pi$
$510$	−1.14509e8	−0.0382248
$511$	4.31125e9	1.42932
$512$	−2.66106e9	−0.876214
$513$	5.43431e8	0.177719
$514$	−1.93138e7	−0.00627331
$515$	−1.02704e9	−0.331331
$516$	−3.09153e9	−0.990602
$517$	−1.30474e9	−0.415248
$518$	5.23907e8	0.165615
$519$	1.35281e9	0.424768
$520$	−1.13570e9	−0.354203
$521$	2.59048e9	0.802506	0.401253	−	0.915967i	$-0.368575\pi$
$521$	2.59048e9	0.802506	0.401253	+	0.915967i	$0.368575\pi$
$522$	−4.19277e8	−0.129019
$523$	6.54783e8	0.200144	0.100072	−	0.994980i	$-0.468093\pi$
$523$	6.54783e8	0.200144	0.100072	+	0.994980i	$0.468093\pi$
$524$	−4.93863e9	−1.49950
$525$	1.86490e9	0.562470
$526$	−6.66547e8	−0.199701
$527$	1.54506e9	0.459842
$528$	−6.28837e8	−0.185917
$529$	1.48036e8	0.0434783
$530$	−7.19851e8	−0.210028
$531$	1.39107e9	0.403198
$532$	4.13243e9	1.18991
$533$	6.10511e9	1.74642
$534$	−3.41332e8	−0.0970023
$535$	2.50632e9	0.707617
$536$	−1.70651e8	−0.0478665
$537$	−1.02837e9	−0.286576
$538$	7.83759e8	0.216992
$539$	−1.10432e9	−0.303761
$540$	3.54514e8	0.0968847
$541$	−4.31638e9	−1.17200	−0.586002	−	0.810310i	$-0.699299\pi$
$541$	−4.31638e9	−1.17200	−0.586002	+	0.810310i	$0.699299\pi$
$542$	4.57657e8	0.123465
$543$	9.82208e8	0.263272
$544$	1.33215e9	0.354779
$545$	6.05509e8	0.160226
$546$	1.01811e9	0.267682
$547$	−5.33812e9	−1.39455	−0.697273	−	0.716806i	$-0.745603\pi$
$547$	−5.33812e9	−1.39455	−0.697273	+	0.716806i	$0.745603\pi$
$548$	−4.95947e9	−1.28737
$549$	−2.42293e9	−0.624938
$550$	−2.20766e8	−0.0565800
$551$	6.64610e9	1.69253
$552$	1.96450e8	0.0497126
$553$	−1.35459e9	−0.340621
$554$	−4.30577e8	−0.107589
$555$	−7.12439e8	−0.176898
$556$	3.28539e9	0.810634
$557$	−5.69967e9	−1.39751	−0.698757	−	0.715359i	$-0.746263\pi$
$557$	−5.69967e9	−1.39751	−0.698757	+	0.715359i	$0.746263\pi$
$558$	2.23286e8	0.0544054
$559$	1.20733e10	2.92337
$560$	2.56413e9	0.616996
$561$	5.32809e8	0.127409
$562$	3.76313e8	0.0894278
$563$	4.19873e9	0.991605	0.495802	−	0.868435i	$-0.334874\pi$
$563$	4.19873e9	0.991605	0.495802	+	0.868435i	$0.334874\pi$
$564$	−2.63117e9	−0.617549
$565$	−3.08004e9	−0.718434
$566$	−1.66020e9	−0.384860
$567$	−6.50447e8	−0.149855
$568$	2.34351e9	0.536597
$569$	−1.69971e9	−0.386795	−0.193398	−	0.981120i	$-0.561951\pi$
$569$	−1.69971e9	−0.386795	−0.193398	+	0.981120i	$0.561951\pi$
$570$	2.62314e8	0.0593280
$571$	6.47824e9	1.45623	0.728116	−	0.685454i	$-0.240396\pi$
$571$	6.47824e9	1.45623	0.728116	+	0.685454i	$0.240396\pi$
$572$	2.58194e9	0.576845
$573$	2.53420e9	0.562729
$574$	1.38451e9	0.305566
$575$	−6.86625e8	−0.150620
$576$	−1.13480e9	−0.247424
$577$	3.45826e9	0.749450	0.374725	−	0.927136i	$-0.377737\pi$
$577$	3.45826e9	0.749450	0.374725	+	0.927136i	$0.377737\pi$
$578$	6.33339e8	0.136423
$579$	−2.52817e9	−0.541291
$580$	4.33567e9	0.922695
$581$	4.67023e9	0.987920
$582$	−4.13569e8	−0.0869596
$583$	3.34945e9	0.700057
$584$	−2.10646e9	−0.437631
$585$	−1.38448e9	−0.285917
$586$	8.63766e8	0.177319
$587$	1.31138e9	0.267606	0.133803	−	0.991008i	$-0.457281\pi$
$587$	1.31138e9	0.267606	0.133803	+	0.991008i	$0.457281\pi$
$588$	−2.22699e9	−0.451748
$589$	−3.53938e9	−0.713714
$590$	6.71470e8	0.134600
$591$	1.84333e9	0.367322
$592$	2.54845e9	0.504836
$593$	−9.08126e9	−1.78836	−0.894179	−	0.447709i	$-0.852240\pi$
$593$	−9.08126e9	−1.78836	−0.894179	+	0.447709i	$0.852240\pi$
$594$	7.69994e7	0.0150742
$595$	−2.17257e9	−0.422828
$596$	1.54862e9	0.299628
$597$	1.72034e9	0.330906
$598$	−3.74849e8	−0.0716806
$599$	−6.26416e8	−0.119088	−0.0595441	−	0.998226i	$-0.518965\pi$
$599$	−6.26416e8	−0.119088	−0.0595441	+	0.998226i	$0.518965\pi$
$600$	−9.11184e8	−0.172217
$601$	9.92718e9	1.86537	0.932686	−	0.360690i	$-0.117459\pi$
$601$	9.92718e9	1.86537	0.932686	+	0.360690i	$0.117459\pi$
$602$	2.73797e9	0.511495
$603$	−2.08032e8	−0.0386384
$604$	−1.02737e8	−0.0189713
$605$	2.47525e9	0.454438
$606$	−4.76756e8	−0.0870245
$607$	1.65614e9	0.300563	0.150282	−	0.988643i	$-0.451982\pi$
$607$	1.65614e9	0.300563	0.150282	+	0.988643i	$0.451982\pi$
$608$	−3.05166e9	−0.550647
$609$	−7.95489e9	−1.42716
$610$	−1.16955e9	−0.208624
$611$	1.02755e10	1.82245
$612$	1.07447e9	0.189481
$613$	4.45864e8	0.0781791	0.0390895	−	0.999236i	$-0.487554\pi$
$613$	4.45864e8	0.0781791	0.0390895	+	0.999236i	$0.487554\pi$
$614$	7.97178e8	0.138984
$615$	−1.88274e9	−0.326383
$616$	1.19839e9	0.206569
$617$	−6.91965e9	−1.18600	−0.593002	−	0.805201i	$-0.702057\pi$
$617$	−6.91965e9	−1.18600	−0.593002	+	0.805201i	$0.702057\pi$
$618$	−4.49848e8	−0.0766666
$619$	−6.02751e9	−1.02146	−0.510729	−	0.859742i	$-0.670624\pi$
$619$	−6.02751e9	−1.02146	−0.510729	+	0.859742i	$0.670624\pi$
$620$	−2.30896e9	−0.389086
$621$	2.39483e8	0.0401286
$622$	−7.07755e8	−0.117928
$623$	−6.47604e9	−1.07300
$624$	4.95239e9	0.815960
$625$	1.49008e9	0.244134
$626$	8.30025e8	0.135232
$627$	−1.22054e9	−0.197750
$628$	1.17470e8	0.0189264
$629$	−2.15928e9	−0.345965
$630$	−3.13971e8	−0.0500262
$631$	−8.87449e9	−1.40618	−0.703089	−	0.711102i	$-0.748197\pi$
$631$	−8.87449e9	−1.40618	−0.703089	+	0.711102i	$0.748197\pi$
$632$	6.61848e8	0.104291
$633$	−9.91623e8	−0.155394
$634$	5.06709e8	0.0789671
$635$	−3.40598e9	−0.527879
$636$	6.75456e9	1.04111
$637$	8.69701e9	1.33316
$638$	9.41694e8	0.143561
$639$	2.85686e9	0.433148
$640$	−2.63148e9	−0.396799
$641$	6.81517e9	1.02205	0.511027	−	0.859565i	$-0.329265\pi$
$641$	6.81517e9	1.02205	0.511027	+	0.859565i	$0.329265\pi$
$642$	1.09778e9	0.163735
$643$	−8.35764e9	−1.23978	−0.619891	−	0.784688i	$-0.712823\pi$
$643$	−8.35764e9	−1.23978	−0.619891	+	0.784688i	$0.712823\pi$
$644$	1.82111e9	0.268680
$645$	−3.72325e9	−0.546340
$646$	7.95031e8	0.116030
$647$	3.48454e9	0.505802	0.252901	−	0.967492i	$-0.418615\pi$
$647$	3.48454e9	0.505802	0.252901	+	0.967492i	$0.418615\pi$
$648$	3.17805e8	0.0458826
$649$	−3.12433e9	−0.448643
$650$	1.73864e9	0.248320
$651$	4.23638e9	0.601813
$652$	−3.86256e9	−0.545769
$653$	−1.41407e10	−1.98736	−0.993678	−	0.112267i	$-0.964189\pi$
$653$	−1.41407e10	−1.98736	−0.993678	+	0.112267i	$0.964189\pi$
$654$	2.65215e8	0.0370746
$655$	−5.94779e9	−0.827010
$656$	6.73471e9	0.931441
$657$	−2.56788e9	−0.353261
$658$	2.33026e9	0.318870
$659$	4.91129e9	0.668492	0.334246	−	0.942486i	$-0.391518\pi$
$659$	4.91129e9	0.668492	0.334246	+	0.942486i	$0.391518\pi$
$660$	−7.96236e8	−0.107805
$661$	6.95658e8	0.0936894	0.0468447	−	0.998902i	$-0.485083\pi$
$661$	6.95658e8	0.0936894	0.0468447	+	0.998902i	$0.485083\pi$
$662$	−9.36830e8	−0.125504
$663$	−4.19612e9	−0.559179
$664$	−2.28185e9	−0.302482
$665$	4.97686e9	0.656265
$666$	−3.12051e8	−0.0409322
$667$	2.92885e9	0.382171
$668$	9.52136e9	1.23589
$669$	5.25298e9	0.678288
$670$	−1.00417e8	−0.0128987
$671$	5.44188e9	0.695376
$672$	3.65261e9	0.464313
$673$	−2.86105e9	−0.361804	−0.180902	−	0.983501i	$-0.557902\pi$
$673$	−2.86105e9	−0.361804	−0.180902	+	0.983501i	$0.557902\pi$
$674$	2.16788e9	0.272726
$675$	−1.11078e9	−0.139016
$676$	−1.26604e10	−1.57628
$677$	−5.03345e9	−0.623455	−0.311728	−	0.950171i	$-0.600908\pi$
$677$	−5.03345e9	−0.623455	−0.311728	+	0.950171i	$0.600908\pi$
$678$	−1.34907e9	−0.166238
$679$	−7.84659e9	−0.961916
$680$	1.06151e9	0.129462
$681$	7.42098e9	0.900423
$682$	−5.01499e8	−0.0605376
$683$	7.71371e9	0.926383	0.463192	−	0.886258i	$-0.346704\pi$
$683$	7.71371e9	0.926383	0.463192	+	0.886258i	$0.346704\pi$
$684$	−2.46137e9	−0.294090
$685$	−5.97289e9	−0.710015
$686$	−4.35953e8	−0.0515591
$687$	−3.87796e9	−0.456305
$688$	1.33184e10	1.55916
$689$	−2.63785e10	−3.07244
$690$	1.15599e8	0.0133962
$691$	−1.62102e10	−1.86903	−0.934515	−	0.355923i	$-0.884167\pi$
$691$	−1.62102e10	−1.86903	−0.934515	+	0.355923i	$0.884167\pi$
$692$	−6.12731e9	−0.702908
$693$	1.46090e9	0.166745
$694$	8.92954e8	0.101408
$695$	3.95672e9	0.447084
$696$	3.88672e9	0.436970
$697$	−5.70627e9	−0.638319
$698$	4.50879e8	0.0501841
$699$	8.65383e9	0.958381
$700$	−8.44674e9	−0.930778
$701$	1.56994e10	1.72135	0.860674	−	0.509156i	$-0.170042\pi$
$701$	1.56994e10	1.72135	0.860674	+	0.509156i	$0.170042\pi$
$702$	−6.06406e8	−0.0661582
$703$	4.94642e9	0.536967
$704$	2.54876e9	0.275312
$705$	−3.16882e9	−0.340593
$706$	−2.92786e9	−0.313137
$707$	−9.04543e9	−0.962634
$708$	−6.30059e9	−0.667214
$709$	8.60054e6	0.000906283 0	0.000453141	−	1.00000i	$-0.499856\pi$
$709$	8.60054e6	0.000906283 0	0.000453141		1.00000i	$0.499856\pi$
$710$	1.37901e9	0.144598
$711$	8.06827e8	0.0841854
$712$	3.16416e9	0.328533
$713$	−1.55976e9	−0.161155
$714$	−9.51593e8	−0.0978380
$715$	3.10953e9	0.318144
$716$	4.65782e9	0.474228
$717$	−1.19238e9	−0.120809
$718$	1.02142e9	0.102984
$719$	−1.69098e10	−1.69663	−0.848314	−	0.529493i	$-0.822382\pi$
$719$	−1.69098e10	−1.69663	−0.848314	+	0.529493i	$0.822382\pi$
$720$	−1.52725e9	−0.152492
$721$	−8.53491e9	−0.848058
$722$	3.14441e8	0.0310927
$723$	2.03575e9	0.200327
$724$	−4.44873e9	−0.435663
$725$	−1.35847e10	−1.32394
$726$	1.08417e9	0.105152
$727$	1.35812e10	1.31089	0.655447	−	0.755242i	$-0.272480\pi$
$727$	1.35812e10	1.31089	0.655447	+	0.755242i	$0.272480\pi$
$728$	−9.43790e9	−0.906599
$729$	3.87420e8	0.0370370
$730$	−1.23952e9	−0.117929
$731$	−1.12846e10	−1.06850
$732$	1.09742e10	1.03415
$733$	−7.43282e9	−0.697092	−0.348546	−	0.937292i	$-0.613324\pi$
$733$	−7.43282e9	−0.697092	−0.348546	+	0.937292i	$0.613324\pi$
$734$	3.78668e9	0.353445
$735$	−2.68205e9	−0.249150
$736$	−1.34483e9	−0.124335
$737$	4.67238e8	0.0429934
$738$	−8.24647e8	−0.0755215
$739$	5.64451e9	0.514483	0.257242	−	0.966347i	$-0.417186\pi$
$739$	5.64451e9	0.514483	0.257242	+	0.966347i	$0.417186\pi$
$740$	3.22686e9	0.292731
$741$	9.61235e9	0.867892
$742$	−5.98210e9	−0.537576
$743$	−9.78408e9	−0.875103	−0.437552	−	0.899193i	$-0.644154\pi$
$743$	−9.78408e9	−0.875103	−0.437552	+	0.899193i	$0.644154\pi$
$744$	−2.06988e9	−0.184263
$745$	1.86506e9	0.165252
$746$	2.95306e9	0.260428
$747$	−2.78169e9	−0.244167
$748$	−2.41326e9	−0.210838
$749$	2.08280e10	1.81118
$750$	−1.27844e9	−0.110654
$751$	−1.20646e10	−1.03937	−0.519687	−	0.854356i	$-0.673952\pi$
$751$	−1.20646e10	−1.03937	−0.519687	+	0.854356i	$0.673952\pi$
$752$	1.13351e10	0.971994
$753$	2.07230e9	0.176877
$754$	−7.41629e9	−0.630067
$755$	−1.23730e8	−0.0104631
$756$	2.94608e9	0.247981
$757$	1.24335e10	1.04174	0.520869	−	0.853637i	$-0.325608\pi$
$757$	1.24335e10	1.04174	0.520869	+	0.853637i	$0.325608\pi$
$758$	−1.48825e9	−0.124118
$759$	−5.37878e8	−0.0446516
$760$	−2.43167e9	−0.200936
$761$	1.74989e10	1.43935	0.719673	−	0.694313i	$-0.244292\pi$
$761$	1.74989e10	1.43935	0.719673	+	0.694313i	$0.244292\pi$
$762$	−1.49183e9	−0.122146
$763$	5.03189e9	0.410105
$764$	−1.14782e10	−0.931208
$765$	1.29403e9	0.104503
$766$	2.82601e9	0.227182
$767$	2.46056e10	1.96902
$768$	4.22720e9	0.336736
$769$	−8.61237e9	−0.682937	−0.341469	−	0.939893i	$-0.610924\pi$
$769$	−8.61237e9	−0.682937	−0.341469	+	0.939893i	$0.610924\pi$
$770$	7.05177e8	0.0556648
$771$	2.18259e8	0.0171507
$772$	1.14509e10	0.895731
$773$	−3.27403e9	−0.254950	−0.127475	−	0.991842i	$-0.540687\pi$
$773$	−3.27403e9	−0.254950	−0.127475	+	0.991842i	$0.540687\pi$
$774$	−1.63080e9	−0.126417
$775$	7.23454e9	0.558284
$776$	3.83381e9	0.294520
$777$	−5.92050e9	−0.452777
$778$	−1.88643e9	−0.143619
$779$	1.30717e10	0.990724
$780$	6.27074e9	0.473138
$781$	−6.41649e9	−0.481969
$782$	3.50360e8	0.0261994
$783$	4.73811e9	0.352727
$784$	9.59390e9	0.711032
$785$	1.41473e8	0.0104383
$786$	−2.60515e9	−0.191361
$787$	1.90391e10	1.39231	0.696154	−	0.717893i	$-0.254893\pi$
$787$	1.90391e10	1.39231	0.696154	+	0.717893i	$0.254893\pi$
$788$	−8.34902e9	−0.607846
$789$	7.53243e9	0.545966
$790$	3.89456e8	0.0281037
$791$	−2.55957e10	−1.83887
$792$	−7.13788e8	−0.0510542
$793$	−4.28574e10	−3.05189
$794$	3.78876e9	0.268612
$795$	8.13479e9	0.574198
$796$	−7.79198e9	−0.547586
$797$	−1.09654e10	−0.767222	−0.383611	−	0.923495i	$-0.625320\pi$
$797$	−1.09654e10	−0.767222	−0.383611	+	0.923495i	$0.625320\pi$
$798$	2.17988e9	0.151853
$799$	−9.60416e9	−0.666110
$800$	6.23762e9	0.430729
$801$	3.85728e9	0.265196
$802$	2.30304e9	0.157649
$803$	5.76744e9	0.393078
$804$	9.42242e8	0.0639391
$805$	2.19324e9	0.148184
$806$	3.94955e9	0.265690
$807$	−8.85700e9	−0.593238
$808$	4.41955e9	0.294740
$809$	2.54612e10	1.69067	0.845336	−	0.534235i	$-0.179400\pi$
$809$	2.54612e10	1.69067	0.845336	+	0.534235i	$0.179400\pi$
$810$	1.87008e8	0.0123641
$811$	1.38636e10	0.912645	0.456323	−	0.889814i	$-0.349166\pi$
$811$	1.38636e10	0.912645	0.456323	+	0.889814i	$0.349166\pi$
$812$	3.60302e10	2.36168
$813$	−5.17183e9	−0.337541
$814$	7.00864e8	0.0455458
$815$	−4.65184e9	−0.301004
$816$	−4.62885e9	−0.298234
$817$	2.58503e10	1.65840
$818$	−3.29517e9	−0.210495
$819$	−1.15053e10	−0.731818
$820$	8.52752e9	0.540100
$821$	1.57708e10	0.994607	0.497304	−	0.867577i	$-0.334323\pi$
$821$	1.57708e10	0.994607	0.497304	+	0.867577i	$0.334323\pi$
$822$	−2.61615e9	−0.164290
$823$	−5.42493e7	−0.00339231	−0.00169615	−	0.999999i	$-0.500540\pi$
$823$	−5.42493e7	−0.00339231	−0.00169615	+	0.999999i	$0.500540\pi$
$824$	4.17011e9	0.259659
$825$	2.49480e9	0.154685
$826$	5.58004e9	0.344514
$827$	−2.41236e10	−1.48311	−0.741554	−	0.670893i	$-0.765911\pi$
$827$	−2.41236e10	−1.48311	−0.741554	+	0.670893i	$0.765911\pi$
$828$	−1.08469e9	−0.0664051
$829$	3.13296e7	0.00190991	0.000954956	−	1.00000i	$-0.499696\pi$
$829$	3.13296e7	0.00190991	0.000954956		1.00000i	$0.499696\pi$
$830$	−1.34272e9	−0.0815105
$831$	4.86581e9	0.294138
$832$	−2.00727e10	−1.20830
$833$	−8.12884e9	−0.487272
$834$	1.73306e9	0.103450
$835$	1.14670e10	0.681625
$836$	5.52822e9	0.327238
$837$	−2.52328e9	−0.148740
$838$	−5.80962e9	−0.341031
$839$	−2.29764e8	−0.0134312	−0.00671559	−	0.999977i	$-0.502138\pi$
$839$	−2.29764e8	−0.0134312	−0.00671559	+	0.999977i	$0.502138\pi$
$840$	2.91053e9	0.169432
$841$	4.06968e10	2.35925
$842$	−6.30202e7	−0.00363821
$843$	−4.25259e9	−0.244488
$844$	4.49137e9	0.257147
$845$	−1.52474e10	−0.869354
$846$	−1.38795e9	−0.0788096
$847$	2.05698e10	1.16315
$848$	−2.90988e10	−1.63866
$849$	1.87613e10	1.05217
$850$	−1.62505e9	−0.0907614
$851$	2.17982e9	0.121246
$852$	−1.29396e10	−0.716776
$853$	−3.52100e10	−1.94243	−0.971213	−	0.238213i	$-0.923439\pi$
$853$	−3.52100e10	−1.94243	−0.971213	+	0.238213i	$0.923439\pi$
$854$	−9.71916e9	−0.533982
$855$	−2.96433e9	−0.162198
$856$	−1.01765e10	−0.554547
$857$	2.17887e10	1.18249	0.591246	−	0.806491i	$-0.298636\pi$
$857$	2.17887e10	1.18249	0.591246	+	0.806491i	$0.298636\pi$
$858$	1.36199e9	0.0736151
$859$	2.04776e10	1.10231	0.551155	−	0.834403i	$-0.314187\pi$
$859$	2.04776e10	1.10231	0.551155	+	0.834403i	$0.314187\pi$
$860$	1.68638e10	0.904087
$861$	−1.56459e10	−0.835392
$862$	5.22676e9	0.277944
$863$	2.39053e9	0.126606	0.0633032	−	0.997994i	$-0.479836\pi$
$863$	2.39053e9	0.126606	0.0633032	+	0.997994i	$0.479836\pi$
$864$	−2.17557e9	−0.114756
$865$	−7.37937e9	−0.387670
$866$	2.53018e6	0.000132385 0
$867$	−7.15715e9	−0.372970
$868$	−1.91879e10	−0.995883
$869$	−1.81213e9	−0.0936741
$870$	2.28709e9	0.117751
$871$	−3.67972e9	−0.188691
$872$	−2.45856e9	−0.125566
$873$	4.67361e9	0.237740
$874$	−8.02594e8	−0.0406636
$875$	−2.42557e10	−1.22401
$876$	1.16307e10	0.584578
$877$	−5.33337e9	−0.266995	−0.133497	−	0.991049i	$-0.542621\pi$
$877$	−5.33337e9	−0.266995	−0.133497	+	0.991049i	$0.542621\pi$
$878$	−6.28232e9	−0.313249
$879$	−9.76114e9	−0.484774
$880$	3.43020e9	0.169680
$881$	−7.74528e9	−0.381612	−0.190806	−	0.981628i	$-0.561110\pi$
$881$	−7.74528e9	−0.381612	−0.190806	+	0.981628i	$0.561110\pi$
$882$	−1.17475e9	−0.0576507
$883$	−7.35743e9	−0.359636	−0.179818	−	0.983700i	$-0.557551\pi$
$883$	−7.35743e9	−0.359636	−0.179818	+	0.983700i	$0.557551\pi$
$884$	1.90056e10	0.925332
$885$	−7.58806e9	−0.367984
$886$	−1.65450e9	−0.0799187
$887$	2.53512e10	1.21974	0.609869	−	0.792502i	$-0.291222\pi$
$887$	2.53512e10	1.21974	0.609869	+	0.792502i	$0.291222\pi$
$888$	2.89273e9	0.138632
$889$	−2.83043e10	−1.35113
$890$	1.86191e9	0.0885306
$891$	−8.70144e8	−0.0412116
$892$	−2.37924e10	−1.12244
$893$	2.20009e10	1.03386
$894$	8.16903e8	0.0382375
$895$	5.60960e9	0.261548
$896$	−2.18681e10	−1.01563
$897$	4.23604e9	0.195969
$898$	3.20326e9	0.147613
$899$	−3.08595e10	−1.41654
$900$	5.03107e9	0.230044
$901$	2.46552e10	1.12298
$902$	1.85215e9	0.0840337
$903$	−3.09409e10	−1.39838
$904$	1.25060e10	0.563025
$905$	−5.35778e9	−0.240279
$906$	−5.41943e7	−0.00242106
$907$	1.17598e10	0.523329	0.261665	−	0.965159i	$-0.415729\pi$
$907$	1.17598e10	0.523329	0.261665	+	0.965159i	$0.415729\pi$
$908$	−3.36120e10	−1.49002
$909$	5.38766e9	0.237917
$910$	−5.55360e9	−0.244304
$911$	1.28379e10	0.562572	0.281286	−	0.959624i	$-0.409239\pi$
$911$	1.28379e10	0.562572	0.281286	+	0.959624i	$0.409239\pi$
$912$	1.06036e10	0.462885
$913$	6.24766e9	0.271688
$914$	−4.68820e8	−0.0203093
$915$	1.32167e10	0.570359
$916$	1.75645e10	0.755096
$917$	−4.94272e10	−2.11677
$918$	5.66790e8	0.0241809
$919$	2.54812e10	1.08297	0.541485	−	0.840711i	$-0.317862\pi$
$919$	2.54812e10	1.08297	0.541485	+	0.840711i	$0.317862\pi$
$920$	−1.07161e9	−0.0453709
$921$	−9.00864e9	−0.379971
$922$	6.77384e9	0.284627
$923$	5.05329e10	2.11528
$924$	−6.61687e9	−0.275931
$925$	−1.01105e10	−0.420028
$926$	5.25156e9	0.217345
$927$	5.08358e9	0.209600
$928$	−2.66071e10	−1.09290
$929$	−4.93247e9	−0.201841	−0.100921	−	0.994894i	$-0.532179\pi$
$929$	−4.93247e9	−0.201841	−0.100921	+	0.994894i	$0.532179\pi$
$930$	−1.21799e9	−0.0496539
$931$	1.86213e10	0.756286
$932$	−3.91959e10	−1.58593
$933$	7.99810e9	0.322405
$934$	−3.40283e9	−0.136655
$935$	−2.90639e9	−0.116282
$936$	5.62142e9	0.224068
$937$	8.62092e9	0.342346	0.171173	−	0.985241i	$-0.445244\pi$
$937$	8.62092e9	0.342346	0.171173	+	0.985241i	$0.445244\pi$
$938$	−8.34485e8	−0.0330148
$939$	−9.37983e9	−0.369714
$940$	1.43526e10	0.563615
$941$	−2.08832e10	−0.817022	−0.408511	−	0.912753i	$-0.633952\pi$
$941$	−2.08832e10	−0.817022	−0.408511	+	0.912753i	$0.633952\pi$
$942$	6.19659e7	0.00241532
$943$	5.76055e9	0.223704
$944$	2.71431e10	1.05016
$945$	3.54808e9	0.136767
$946$	3.66276e9	0.140666
$947$	4.19780e10	1.60619	0.803095	−	0.595851i	$-0.203185\pi$
$947$	4.19780e10	1.60619	0.803095	+	0.595851i	$0.203185\pi$
$948$	−3.65437e9	−0.139310
$949$	−4.54213e10	−1.72515
$950$	3.72262e9	0.140869
$951$	−5.72615e9	−0.215889
$952$	8.82132e9	0.331363
$953$	−3.26688e10	−1.22267	−0.611333	−	0.791373i	$-0.709366\pi$
$953$	−3.26688e10	−1.22267	−0.611333	+	0.791373i	$0.709366\pi$
$954$	3.56307e9	0.132863
$955$	−1.38236e10	−0.513583
$956$	5.40067e9	0.199915
$957$	−1.06418e10	−0.392484
$958$	−7.37121e9	−0.270869
$959$	−4.96358e10	−1.81732
$960$	6.19016e9	0.225815
$961$	−1.10784e10	−0.402666
$962$	−5.51964e9	−0.199893
$963$	−1.24056e10	−0.447637
$964$	−9.22054e9	−0.331502
$965$	1.37907e10	0.494017
$966$	9.60646e8	0.0342881
$967$	5.39963e10	1.92031	0.960155	−	0.279467i	$-0.0901576\pi$
$967$	5.39963e10	1.92031	0.960155	+	0.279467i	$0.0901576\pi$
$968$	−1.00503e10	−0.356135
$969$	−8.98438e9	−0.317216
$970$	2.25595e9	0.0793650
$971$	1.25581e10	0.440207	0.220104	−	0.975476i	$-0.429360\pi$
$971$	1.25581e10	0.440207	0.220104	+	0.975476i	$0.429360\pi$
$972$	−1.75475e9	−0.0612891
$973$	3.28811e10	1.14433
$974$	5.54638e9	0.192333
$975$	−1.96478e10	−0.678886
$976$	−4.72771e10	−1.62771
$977$	−1.93161e10	−0.662657	−0.331329	−	0.943515i	$-0.607497\pi$
$977$	−1.93161e10	−0.662657	−0.331329	+	0.943515i	$0.607497\pi$
$978$	−2.03752e9	−0.0696492
$979$	−8.66342e9	−0.295087
$980$	1.21478e10	0.412295
$981$	−2.99711e9	−0.101359
$982$	7.49538e8	0.0252583
$983$	−3.38434e10	−1.13642	−0.568208	−	0.822885i	$-0.692363\pi$
$983$	−3.38434e10	−1.13642	−0.568208	+	0.822885i	$0.692363\pi$
$984$	7.64452e9	0.255781
$985$	−1.00551e10	−0.335242
$986$	6.93179e9	0.230290
$987$	−2.63335e10	−0.871763
$988$	−4.35374e10	−1.43619
$989$	1.13919e10	0.374463
$990$	−4.20019e8	−0.0137577
$991$	2.92468e10	0.954599	0.477300	−	0.878741i	$-0.341616\pi$
$991$	2.92468e10	0.954599	0.477300	+	0.878741i	$0.341616\pi$
$992$	1.41696e10	0.460857
$993$	1.05868e10	0.343117
$994$	1.14598e10	0.370105
$995$	−9.38420e9	−0.302007
$996$	1.25992e10	0.404049
$997$	−2.05938e9	−0.0658117	−0.0329059	−	0.999458i	$-0.510476\pi$
$997$	−2.05938e9	−0.0658117	−0.0329059	+	0.999458i	$0.510476\pi$
$998$	−1.37400e9	−0.0437552
$999$	3.52638e9	0.111905

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.d.1.3	✓	8	1.1	even	1	trivial
207.8.a.e.1.6		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)

\(f(q)\)	\(=\)	\(q-2.38924 q^{2} +27.0000 q^{3} -122.292 q^{4} -147.281 q^{5} -64.5095 q^{6} -1223.93 q^{7} +598.006 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-2.38924 q^{2} +27.0000 q^{3} -122.292 q^{4} -147.281 q^{5} -64.5095 q^{6} -1223.93 q^{7} +598.006 q^{8} +729.000 q^{9} +351.889 q^{10} -1637.33 q^{11} -3301.87 q^{12} +12894.7 q^{13} +2924.26 q^{14} -3976.58 q^{15} +14224.5 q^{16} -12052.3 q^{17} -1741.76 q^{18} +27609.1 q^{19} +18011.2 q^{20} -33046.1 q^{21} +3911.97 q^{22} +12167.0 q^{23} +16146.2 q^{24} -56433.4 q^{25} -30808.6 q^{26} +19683.0 q^{27} +149676. q^{28} +240721. q^{29} +9500.99 q^{30} -128196. q^{31} -110531. q^{32} -44207.9 q^{33} +28795.9 q^{34} +180261. q^{35} -89150.5 q^{36} +179159. q^{37} -65964.8 q^{38} +348158. q^{39} -88074.8 q^{40} +473457. q^{41} +78955.1 q^{42} +936296. q^{43} +200232. q^{44} -107368. q^{45} -29069.9 q^{46} +796871. q^{47} +384062. q^{48} +674461. q^{49} +134833. q^{50} -325413. q^{51} -1.57692e6 q^{52} -2.04568e6 q^{53} -47027.4 q^{54} +241147. q^{55} -731918. q^{56} +745447. q^{57} -575140. q^{58} +1.90819e6 q^{59} +486302. q^{60} -3.32363e6 q^{61} +306291. q^{62} -892245. q^{63} -1.55666e6 q^{64} -1.89915e6 q^{65} +105623. q^{66} -285366. q^{67} +1.47390e6 q^{68} +328509. q^{69} -430687. q^{70} +3.91888e6 q^{71} +435947. q^{72} -3.52247e6 q^{73} -428053. q^{74} -1.52370e6 q^{75} -3.37636e6 q^{76} +2.00398e6 q^{77} -831833. q^{78} +1.10676e6 q^{79} -2.09500e6 q^{80} +531441. q^{81} -1.13120e6 q^{82} -3.81576e6 q^{83} +4.04126e6 q^{84} +1.77508e6 q^{85} -2.23703e6 q^{86} +6.49947e6 q^{87} -979134. q^{88} +5.29119e6 q^{89} +256527. q^{90} -1.57823e7 q^{91} -1.48792e6 q^{92} -3.46129e6 q^{93} -1.90392e6 q^{94} -4.06629e6 q^{95} -2.98433e6 q^{96} +6.41098e6 q^{97} -1.61145e6 q^{98} -1.19361e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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