LMFDB - Embedded newform 43.8.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("43.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.8
Root			$2.37903$ of defining polynomial
Character	$\chi$	$=$	43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37903 q^{2} -66.5036 q^{3} -116.582 q^{4} -241.175 q^{5} -224.718 q^{6} -173.086 q^{7} -826.450 q^{8} +2235.73 q^{9} +O(q^{10})$	\(q+3.37903 q^{2} -66.5036 q^{3} -116.582 q^{4} -241.175 q^{5} -224.718 q^{6} -173.086 q^{7} -826.450 q^{8} +2235.73 q^{9} -814.939 q^{10} +682.590 q^{11} +7753.13 q^{12} +168.040 q^{13} -584.863 q^{14} +16039.0 q^{15} +12129.9 q^{16} +5562.09 q^{17} +7554.59 q^{18} -13825.2 q^{19} +28116.8 q^{20} +11510.8 q^{21} +2306.49 q^{22} -24641.2 q^{23} +54961.9 q^{24} -19959.4 q^{25} +567.813 q^{26} -3240.56 q^{27} +20178.7 q^{28} -33649.4 q^{29} +54196.4 q^{30} +22707.6 q^{31} +146773. q^{32} -45394.7 q^{33} +18794.5 q^{34} +41744.1 q^{35} -260646. q^{36} +181778. q^{37} -46715.6 q^{38} -11175.3 q^{39} +199320. q^{40} +335134. q^{41} +38895.5 q^{42} -79507.0 q^{43} -79577.8 q^{44} -539203. q^{45} -83263.2 q^{46} -743538. q^{47} -806683. q^{48} -793584. q^{49} -67443.4 q^{50} -369899. q^{51} -19590.5 q^{52} -12693.7 q^{53} -10949.9 q^{54} -164624. q^{55} +143047. q^{56} +919423. q^{57} -113702. q^{58} -830705. q^{59} -1.86987e6 q^{60} +2.22282e6 q^{61} +76729.7 q^{62} -386973. q^{63} -1.05668e6 q^{64} -40527.2 q^{65} -153390. q^{66} +3.88681e6 q^{67} -648441. q^{68} +1.63873e6 q^{69} +141055. q^{70} +3.88756e6 q^{71} -1.84772e6 q^{72} +5.22595e6 q^{73} +614232. q^{74} +1.32737e6 q^{75} +1.61177e6 q^{76} -118147. q^{77} -37761.6 q^{78} -2.64353e6 q^{79} -2.92544e6 q^{80} -4.67403e6 q^{81} +1.13243e6 q^{82} +6.76661e6 q^{83} -1.34196e6 q^{84} -1.34144e6 q^{85} -268657. q^{86} +2.23781e6 q^{87} -564127. q^{88} +327679. q^{89} -1.82198e6 q^{90} -29085.4 q^{91} +2.87272e6 q^{92} -1.51014e6 q^{93} -2.51244e6 q^{94} +3.33429e6 q^{95} -9.76093e6 q^{96} +5.05740e6 q^{97} -2.68154e6 q^{98} +1.52608e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9}+O(q^{10}) $ 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9} + 4667 q^{10} + 1620 q^{11} - 19681 q^{12} + 13550 q^{13} + 44160 q^{14} + 31412 q^{15} + 114026 q^{16} + 110880 q^{17} + 159267 q^{18} + 105058 q^{19} + 167251 q^{20} + 129840 q^{21} + 201504 q^{22} + 160184 q^{23} + 161289 q^{24} + 270149 q^{25} + 272104 q^{26} + 252544 q^{27} + 208172 q^{28} + 285546 q^{29} + 107580 q^{30} - 99616 q^{31} + 200126 q^{32} + 531468 q^{33} - 80941 q^{34} - 187104 q^{35} - 608975 q^{36} + 176038 q^{37} + 652165 q^{38} - 794680 q^{39} - 895387 q^{40} - 410260 q^{41} - 3413218 q^{42} - 1033591 q^{43} - 2177076 q^{44} - 1051178 q^{45} - 3975765 q^{46} - 424556 q^{47} - 2360477 q^{48} - 1561359 q^{49} - 4063801 q^{50} - 2375738 q^{51} - 4172312 q^{52} + 3992458 q^{53} - 10438626 q^{54} + 406960 q^{55} + 1559556 q^{56} - 3116152 q^{57} - 4052005 q^{58} + 2248836 q^{59} - 2911436 q^{60} + 6210394 q^{61} + 885317 q^{62} + 11622368 q^{63} - 3096318 q^{64} + 5600420 q^{65} - 2174604 q^{66} - 1993648 q^{67} + 9327135 q^{68} + 13366240 q^{69} - 1105098 q^{70} + 4978064 q^{71} + 11370663 q^{72} + 8224814 q^{73} - 3613563 q^{74} + 27115592 q^{75} + 10687121 q^{76} + 17261892 q^{77} - 15226630 q^{78} + 6945708 q^{79} + 15822799 q^{80} + 35113185 q^{81} - 508449 q^{82} + 22937328 q^{83} - 14010106 q^{84} - 575532 q^{85} - 1272112 q^{86} + 9081380 q^{87} + 11202656 q^{88} + 9291302 q^{89} + 2841402 q^{90} + 25581108 q^{91} - 14388137 q^{92} + 25930480 q^{93} - 24645805 q^{94} + 30750464 q^{95} - 22461255 q^{96} + 10001852 q^{97} - 32304856 q^{98} + 5055452 q^{99}+O(q^{100}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9 + 4667 * q^10 + 1620 * q^11 - 19681 * q^12 + 13550 * q^13 + 44160 * q^14 + 31412 * q^15 + 114026 * q^16 + 110880 * q^17 + 159267 * q^18 + 105058 * q^19 + 167251 * q^20 + 129840 * q^21 + 201504 * q^22 + 160184 * q^23 + 161289 * q^24 + 270149 * q^25 + 272104 * q^26 + 252544 * q^27 + 208172 * q^28 + 285546 * q^29 + 107580 * q^30 - 99616 * q^31 + 200126 * q^32 + 531468 * q^33 - 80941 * q^34 - 187104 * q^35 - 608975 * q^36 + 176038 * q^37 + 652165 * q^38 - 794680 * q^39 - 895387 * q^40 - 410260 * q^41 - 3413218 * q^42 - 1033591 * q^43 - 2177076 * q^44 - 1051178 * q^45 - 3975765 * q^46 - 424556 * q^47 - 2360477 * q^48 - 1561359 * q^49 - 4063801 * q^50 - 2375738 * q^51 - 4172312 * q^52 + 3992458 * q^53 - 10438626 * q^54 + 406960 * q^55 + 1559556 * q^56 - 3116152 * q^57 - 4052005 * q^58 + 2248836 * q^59 - 2911436 * q^60 + 6210394 * q^61 + 885317 * q^62 + 11622368 * q^63 - 3096318 * q^64 + 5600420 * q^65 - 2174604 * q^66 - 1993648 * q^67 + 9327135 * q^68 + 13366240 * q^69 - 1105098 * q^70 + 4978064 * q^71 + 11370663 * q^72 + 8224814 * q^73 - 3613563 * q^74 + 27115592 * q^75 + 10687121 * q^76 + 17261892 * q^77 - 15226630 * q^78 + 6945708 * q^79 + 15822799 * q^80 + 35113185 * q^81 - 508449 * q^82 + 22937328 * q^83 - 14010106 * q^84 - 575532 * q^85 - 1272112 * q^86 + 9081380 * q^87 + 11202656 * q^88 + 9291302 * q^89 + 2841402 * q^90 + 25581108 * q^91 - 14388137 * q^92 + 25930480 * q^93 - 24645805 * q^94 + 30750464 * q^95 - 22461255 * q^96 + 10001852 * q^97 - 32304856 * q^98 + 5055452 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.37903	0.298667	0.149333	−	0.988787i	$-0.452287\pi$
$2$	3.37903	0.298667	0.149333	+	0.988787i	$0.452287\pi$
$3$	−66.5036	−1.42207	−0.711035	−	0.703157i	$-0.751773\pi$
$3$	−66.5036	−1.42207	−0.711035	+	0.703157i	$0.751773\pi$
$4$	−116.582	−0.910798
$5$	−241.175	−0.862856	−0.431428	−	0.902147i	$-0.641990\pi$
$5$	−241.175	−0.862856	−0.431428	+	0.902147i	$0.641990\pi$
$6$	−224.718	−0.424725
$7$	−173.086	−0.190730	−0.0953650	−	0.995442i	$-0.530402\pi$
$7$	−173.086	−0.190730	−0.0953650	+	0.995442i	$0.530402\pi$
$8$	−826.450	−0.570692
$9$	2235.73	1.02228
$10$	−814.939	−0.257706
$11$	682.590	0.154627	0.0773135	−	0.997007i	$-0.475366\pi$
$11$	682.590	0.154627	0.0773135	+	0.997007i	$0.475366\pi$
$12$	7753.13	1.29522
$13$	168.040	0.0212135	0.0106067	−	0.999944i	$-0.496624\pi$
$13$	168.040	0.0212135	0.0106067	+	0.999944i	$0.496624\pi$
$14$	−584.863	−0.0569647
$15$	16039.0	1.22704
$16$	12129.9	0.740351
$17$	5562.09	0.274579	0.137289	−	0.990531i	$-0.456161\pi$
$17$	5562.09	0.274579	0.137289	+	0.990531i	$0.456161\pi$
$18$	7554.59	0.305321
$19$	−13825.2	−0.462416	−0.231208	−	0.972904i	$-0.574268\pi$
$19$	−13825.2	−0.462416	−0.231208	+	0.972904i	$0.574268\pi$
$20$	28116.8	0.785887
$21$	11510.8	0.271231
$22$	2306.49	0.0461820
$23$	−24641.2	−0.422293	−0.211146	−	0.977454i	$-0.567720\pi$
$23$	−24641.2	−0.422293	−0.211146	+	0.977454i	$0.567720\pi$
$24$	54961.9	0.811564
$25$	−19959.4	−0.255480
$26$	567.813	0.00633576
$27$	−3240.56	−0.0316845
$28$	20178.7	0.173716
$29$	−33649.4	−0.256203	−0.128102	−	0.991761i	$-0.540888\pi$
$29$	−33649.4	−0.256203	−0.128102	+	0.991761i	$0.540888\pi$
$30$	54196.4	0.366476
$31$	22707.6	0.136901	0.0684503	−	0.997655i	$-0.478195\pi$
$31$	22707.6	0.136901	0.0684503	+	0.997655i	$0.478195\pi$
$32$	146773.	0.791811
$33$	−45394.7	−0.219890
$34$	18794.5	0.0820076
$35$	41744.1	0.164572
$36$	−260646.	−0.931091
$37$	181778.	0.589976	0.294988	−	0.955501i	$-0.404684\pi$
$37$	181778.	0.589976	0.294988	+	0.955501i	$0.404684\pi$
$38$	−46715.6	−0.138108
$39$	−11175.3	−0.0301670
$40$	199320.	0.492425
$41$	335134.	0.759407	0.379704	−	0.925108i	$-0.376026\pi$
$41$	335134.	0.759407	0.379704	+	0.925108i	$0.376026\pi$
$42$	38895.5	0.0810078
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	−79577.8	−0.140834
$45$	−539203.	−0.882080
$46$	−83263.2	−0.126125
$47$	−743538.	−1.04463	−0.522313	−	0.852754i	$-0.674931\pi$
$47$	−743538.	−1.04463	−0.522313	+	0.852754i	$0.674931\pi$
$48$	−806683.	−1.05283
$49$	−793584.	−0.963622
$50$	−67443.4	−0.0763035
$51$	−369899.	−0.390470
$52$	−19590.5	−0.0193212
$53$	−12693.7	−0.0117118	−0.00585591	−	0.999983i	$-0.501864\pi$
$53$	−12693.7	−0.0117118	−0.00585591	+	0.999983i	$0.501864\pi$
$54$	−10949.9	−0.00946310
$55$	−164624.	−0.133421
$56$	143047.	0.108848
$57$	919423.	0.657587
$58$	−113702.	−0.0765194
$59$	−830705.	−0.526581	−0.263290	−	0.964717i	$-0.584808\pi$
$59$	−830705.	−0.526581	−0.263290	+	0.964717i	$0.584808\pi$
$60$	−1.86987e6	−1.11759
$61$	2.22282e6	1.25386	0.626931	−	0.779075i	$-0.284311\pi$
$61$	2.22282e6	1.25386	0.626931	+	0.779075i	$0.284311\pi$
$62$	76729.7	0.0408877
$63$	−386973.	−0.194980
$64$	−1.05668e6	−0.503864
$65$	−40527.2	−0.0183042
$66$	−153390.	−0.0656739
$67$	3.88681e6	1.57882	0.789409	−	0.613868i	$-0.210387\pi$
$67$	3.88681e6	1.57882	0.789409	+	0.613868i	$0.210387\pi$
$68$	−648441.	−0.250086
$69$	1.63873e6	0.600530
$70$	141055.	0.0491523
$71$	3.88756e6	1.28906	0.644530	−	0.764579i	$-0.277053\pi$
$71$	3.88756e6	1.28906	0.644530	+	0.764579i	$0.277053\pi$
$72$	−1.84772e6	−0.583407
$73$	5.22595e6	1.57230	0.786150	−	0.618035i	$-0.212071\pi$
$73$	5.22595e6	1.57230	0.786150	+	0.618035i	$0.212071\pi$
$74$	614232.	0.176206
$75$	1.32737e6	0.363311
$76$	1.61177e6	0.421167
$77$	−118147.	−0.0294920
$78$	−37761.6	−0.00900989
$79$	−2.64353e6	−0.603240	−0.301620	−	0.953428i	$-0.597527\pi$
$79$	−2.64353e6	−0.603240	−0.301620	+	0.953428i	$0.597527\pi$
$80$	−2.92544e6	−0.638816
$81$	−4.67403e6	−0.977223
$82$	1.13243e6	0.226810
$83$	6.76661e6	1.29897	0.649483	−	0.760376i	$-0.274985\pi$
$83$	6.76661e6	1.29897	0.649483	+	0.760376i	$0.274985\pi$
$84$	−1.34196e6	−0.247037
$85$	−1.34144e6	−0.236922
$86$	−268657.	−0.0455463
$87$	2.23781e6	0.364339
$88$	−564127.	−0.0882444
$89$	327679.	0.0492701	0.0246350	−	0.999697i	$-0.492158\pi$
$89$	327679.	0.0492701	0.0246350	+	0.999697i	$0.492158\pi$
$90$	−1.82198e6	−0.263448
$91$	−29085.4	−0.00404604
$92$	2.87272e6	0.384624
$93$	−1.51014e6	−0.194682
$94$	−2.51244e6	−0.311995
$95$	3.33429e6	0.398998
$96$	−9.76093e6	−1.12601
$97$	5.05740e6	0.562634	0.281317	−	0.959615i	$-0.409229\pi$
$97$	5.05740e6	0.562634	0.281317	+	0.959615i	$0.409229\pi$
$98$	−2.68154e6	−0.287802
$99$	1.52608e6	0.158072
$100$	2.32691e6	0.232691
$101$	1.31064e7	1.26578	0.632889	−	0.774242i	$-0.281869\pi$
$101$	1.31064e7	1.26578	0.632889	+	0.774242i	$0.281869\pi$
$102$	−1.24990e6	−0.116620
$103$	−1.35656e7	−1.22324	−0.611618	−	0.791153i	$-0.709481\pi$
$103$	−1.35656e7	−1.22324	−0.611618	+	0.791153i	$0.709481\pi$
$104$	−138877.	−0.0121064
$105$	−2.77613e6	−0.234033
$106$	−42892.6	−0.00349793
$107$	−1.06605e7	−0.841269	−0.420635	−	0.907230i	$-0.638193\pi$
$107$	−1.06605e7	−0.841269	−0.420635	+	0.907230i	$0.638193\pi$
$108$	377791.	0.0288581
$109$	9.24482e6	0.683763	0.341881	−	0.939743i	$-0.388936\pi$
$109$	9.24482e6	0.683763	0.341881	+	0.939743i	$0.388936\pi$
$110$	−556269.	−0.0398484
$111$	−1.20889e7	−0.838987
$112$	−2.09952e6	−0.141207
$113$	−4.05844e6	−0.264597	−0.132299	−	0.991210i	$-0.542236\pi$
$113$	−4.05844e6	−0.264597	−0.132299	+	0.991210i	$0.542236\pi$
$114$	3.10676e6	0.196399
$115$	5.94285e6	0.364378
$116$	3.92292e6	0.233349
$117$	375692.	0.0216861
$118$	−2.80698e6	−0.157272
$119$	−962721.	−0.0523704
$120$	−1.32555e7	−0.700262
$121$	−1.90212e7	−0.976090
$122$	7.51098e6	0.374487
$123$	−2.22876e7	−1.07993
$124$	−2.64730e6	−0.124689
$125$	2.36555e7	1.08330
$126$	−1.30759e6	−0.0582339
$127$	−2.45396e7	−1.06305	−0.531525	−	0.847043i	$-0.678381\pi$
$127$	−2.45396e7	−1.06305	−0.531525	+	0.847043i	$0.678381\pi$
$128$	−2.23575e7	−0.942298
$129$	5.28750e6	0.216864
$130$	−136943.	−0.00546684
$131$	−2.67897e7	−1.04116	−0.520581	−	0.853812i	$-0.674285\pi$
$131$	−2.67897e7	−1.04116	−0.520581	+	0.853812i	$0.674285\pi$
$132$	5.29221e6	0.200276
$133$	2.39294e6	0.0881965
$134$	1.31337e7	0.471540
$135$	781543.	0.0273391
$136$	−4.59679e6	−0.156700
$137$	4.80765e7	1.59739	0.798695	−	0.601736i	$-0.205524\pi$
$137$	4.80765e7	1.59739	0.798695	+	0.601736i	$0.205524\pi$
$138$	5.53730e6	0.179358
$139$	−5.94300e6	−0.187696	−0.0938478	−	0.995587i	$-0.529917\pi$
$139$	−5.94300e6	−0.187696	−0.0938478	+	0.995587i	$0.529917\pi$
$140$	−4.86662e6	−0.149892
$141$	4.94480e7	1.48553
$142$	1.31362e7	0.385000
$143$	114703.	0.00328017
$144$	2.71192e7	0.756847
$145$	8.11541e6	0.221066
$146$	1.76587e7	0.469594
$147$	5.27762e7	1.37034
$148$	−2.11920e7	−0.537349
$149$	4.65127e7	1.15191	0.575956	−	0.817481i	$-0.304630\pi$
$149$	4.65127e7	1.15191	0.575956	+	0.817481i	$0.304630\pi$
$150$	4.48523e6	0.108509
$151$	−1.54586e7	−0.365385	−0.182693	−	0.983170i	$-0.558481\pi$
$151$	−1.54586e7	−0.365385	−0.182693	+	0.983170i	$0.558481\pi$
$152$	1.14258e7	0.263897
$153$	1.24353e7	0.280697
$154$	−399221.	−0.00880828
$155$	−5.47652e6	−0.118125
$156$	1.30284e6	0.0274761
$157$	2.72583e6	0.0562147	0.0281074	−	0.999605i	$-0.491052\pi$
$157$	2.72583e6	0.0562147	0.0281074	+	0.999605i	$0.491052\pi$
$158$	−8.93258e6	−0.180168
$159$	844180.	0.0166550
$160$	−3.53980e7	−0.683218
$161$	4.26504e6	0.0805439
$162$	−1.57937e7	−0.291864
$163$	−5.29222e7	−0.957152	−0.478576	−	0.878046i	$-0.658847\pi$
$163$	−5.29222e7	−0.957152	−0.478576	+	0.878046i	$0.658847\pi$
$164$	−3.90706e7	−0.691667
$165$	1.09481e7	0.189734
$166$	2.28646e7	0.387958
$167$	3.70870e7	0.616189	0.308095	−	0.951356i	$-0.400309\pi$
$167$	3.70870e7	0.616189	0.308095	+	0.951356i	$0.400309\pi$
$168$	−9.51314e6	−0.154789
$169$	−6.27203e7	−0.999550
$170$	−4.53277e6	−0.0707607
$171$	−3.09093e7	−0.472719
$172$	9.26910e6	0.138895
$173$	−3.93055e7	−0.577154	−0.288577	−	0.957457i	$-0.593182\pi$
$173$	−3.93055e7	−0.577154	−0.288577	+	0.957457i	$0.593182\pi$
$174$	7.56162e6	0.108816
$175$	3.45469e6	0.0487277
$176$	8.27975e6	0.114478
$177$	5.52449e7	0.748834
$178$	1.10724e6	0.0147153
$179$	−4.76706e7	−0.621248	−0.310624	−	0.950533i	$-0.600538\pi$
$179$	−4.76706e7	−0.621248	−0.310624	+	0.950533i	$0.600538\pi$
$180$	6.28614e7	0.803397
$181$	2.04178e7	0.255938	0.127969	−	0.991778i	$-0.459154\pi$
$181$	2.04178e7	0.255938	0.127969	+	0.991778i	$0.459154\pi$
$182$	−98280.5	−0.00120842
$183$	−1.47826e8	−1.78308
$184$	2.03647e7	0.240999
$185$	−4.38403e7	−0.509064
$186$	−5.10280e6	−0.0581451
$187$	3.79663e6	0.0424573
$188$	8.66833e7	0.951444
$189$	560895.	0.00604318
$190$	1.12667e7	0.119167
$191$	7.99910e7	0.830662	0.415331	−	0.909670i	$-0.363666\pi$
$191$	7.99910e7	0.830662	0.415331	+	0.909670i	$0.363666\pi$
$192$	7.02729e7	0.716529
$193$	−3.82721e7	−0.383205	−0.191603	−	0.981473i	$-0.561368\pi$
$193$	−3.82721e7	−0.383205	−0.191603	+	0.981473i	$0.561368\pi$
$194$	1.70891e7	0.168040
$195$	2.69520e6	0.0260298
$196$	9.25178e7	0.877665
$197$	1.23397e8	1.14993	0.574967	−	0.818177i	$-0.305015\pi$
$197$	1.23397e8	1.14993	0.574967	+	0.818177i	$0.305015\pi$
$198$	5.15669e6	0.0472109
$199$	−7.95502e7	−0.715576	−0.357788	−	0.933803i	$-0.616469\pi$
$199$	−7.95502e7	−0.715576	−0.357788	+	0.933803i	$0.616469\pi$
$200$	1.64954e7	0.145801
$201$	−2.58487e8	−2.24519
$202$	4.42868e7	0.378046
$203$	5.82424e6	0.0488656
$204$	4.31236e7	0.355639
$205$	−8.08260e7	−0.655259
$206$	−4.58387e7	−0.365340
$207$	−5.50909e7	−0.431702
$208$	2.03831e6	0.0157054
$209$	−9.43691e6	−0.0715019
$210$	−9.38064e6	−0.0698980
$211$	−2.30074e8	−1.68608	−0.843041	−	0.537849i	$-0.819237\pi$
$211$	−2.30074e8	−1.68608	−0.843041	+	0.537849i	$0.819237\pi$
$212$	1.47986e6	0.0106671
$213$	−2.58537e8	−1.83313
$214$	−3.60222e7	−0.251259
$215$	1.91751e7	0.131584
$216$	2.67816e6	0.0180821
$217$	−3.93037e6	−0.0261110
$218$	3.12385e7	0.204217
$219$	−3.47545e8	−2.23592
$220$	1.91922e7	0.121519
$221$	934655.	0.00582477
$222$	−4.08486e7	−0.250578
$223$	1.94009e8	1.17153	0.585767	−	0.810479i	$-0.300793\pi$
$223$	1.94009e8	1.17153	0.585767	+	0.810479i	$0.300793\pi$
$224$	−2.54044e7	−0.151022
$225$	−4.46238e7	−0.261172
$226$	−1.37136e7	−0.0790264
$227$	2.48770e8	1.41159	0.705793	−	0.708418i	$-0.250591\pi$
$227$	2.48770e8	1.41159	0.705793	+	0.708418i	$0.250591\pi$
$228$	−1.07188e8	−0.598929
$229$	2.61570e8	1.43934	0.719670	−	0.694316i	$-0.244293\pi$
$229$	2.61570e8	1.43934	0.719670	+	0.694316i	$0.244293\pi$
$230$	2.00811e7	0.108828
$231$	7.85718e6	0.0419397
$232$	2.78096e7	0.146213
$233$	1.70926e8	0.885244	0.442622	−	0.896708i	$-0.354048\pi$
$233$	1.70926e8	0.885244	0.442622	+	0.896708i	$0.354048\pi$
$234$	1.26948e6	0.00647692
$235$	1.79323e8	0.901362
$236$	9.68454e7	0.479609
$237$	1.75804e8	0.857848
$238$	−3.25306e6	−0.0156413
$239$	8.66501e7	0.410560	0.205280	−	0.978703i	$-0.434190\pi$
$239$	8.66501e7	0.410560	0.205280	+	0.978703i	$0.434190\pi$
$240$	1.94552e8	0.908441
$241$	−1.19315e8	−0.549079	−0.274540	−	0.961576i	$-0.588525\pi$
$241$	−1.19315e8	−0.549079	−0.274540	+	0.961576i	$0.588525\pi$
$242$	−6.42733e7	−0.291526
$243$	3.17927e8	1.42136
$244$	−2.59141e8	−1.14202
$245$	1.91393e8	0.831467
$246$	−7.53105e7	−0.322539
$247$	−2.32318e6	−0.00980944
$248$	−1.87667e7	−0.0781281
$249$	−4.50004e8	−1.84722
$250$	7.99328e7	0.323545
$251$	4.55167e8	1.81682	0.908412	−	0.418077i	$-0.137296\pi$
$251$	4.55167e8	1.81682	0.908412	+	0.418077i	$0.137296\pi$
$252$	4.51142e7	0.177587
$253$	−1.68198e7	−0.0652979
$254$	−8.29199e7	−0.317498
$255$	8.92106e7	0.336919
$256$	5.97082e7	0.222431
$257$	5.12052e8	1.88169	0.940845	−	0.338837i	$-0.110033\pi$
$257$	5.12052e8	1.88169	0.940845	+	0.338837i	$0.110033\pi$
$258$	1.78666e7	0.0647699
$259$	−3.14632e7	−0.112526
$260$	4.72475e6	0.0166714
$261$	−7.52309e7	−0.261912
$262$	−9.05232e7	−0.310961
$263$	9.38421e7	0.318092	0.159046	−	0.987271i	$-0.449158\pi$
$263$	9.38421e7	0.318092	0.159046	+	0.987271i	$0.449158\pi$
$264$	3.75164e7	0.125490
$265$	3.06142e6	0.0101056
$266$	8.08582e6	0.0263414
$267$	−2.17918e7	−0.0700655
$268$	−4.53133e8	−1.43798
$269$	−5.02612e6	−0.0157434	−0.00787172	−	0.999969i	$-0.502506\pi$
$269$	−5.02612e6	−0.0157434	−0.00787172	+	0.999969i	$0.502506\pi$
$270$	2.64086e6	0.00816529
$271$	−4.90350e8	−1.49663	−0.748314	−	0.663345i	$-0.769136\pi$
$271$	−4.90350e8	−1.49663	−0.748314	+	0.663345i	$0.769136\pi$
$272$	6.74677e7	0.203285
$273$	1.93428e6	0.00575375
$274$	1.62452e8	0.477088
$275$	−1.36241e7	−0.0395041
$276$	−1.91046e8	−0.546961
$277$	5.00692e7	0.141544	0.0707721	−	0.997493i	$-0.477454\pi$
$277$	5.00692e7	0.141544	0.0707721	+	0.997493i	$0.477454\pi$
$278$	−2.00816e7	−0.0560584
$279$	5.07680e7	0.139951
$280$	−3.44994e7	−0.0939202
$281$	5.08518e8	1.36721	0.683604	−	0.729853i	$-0.260412\pi$
$281$	5.08518e8	1.36721	0.683604	+	0.729853i	$0.260412\pi$
$282$	1.67086e8	0.443679
$283$	9.23331e7	0.242161	0.121081	−	0.992643i	$-0.461364\pi$
$283$	9.23331e7	0.242161	0.121081	+	0.992643i	$0.461364\pi$
$284$	−4.53220e8	−1.17407
$285$	−2.21742e8	−0.567403
$286$	387583.	0.000979679 0
$287$	−5.80070e7	−0.144842
$288$	3.28144e8	0.809452
$289$	−3.79402e8	−0.924606
$290$	2.74222e7	0.0660252
$291$	−3.36335e8	−0.800105
$292$	−6.09253e8	−1.43205
$293$	−2.46782e8	−0.573162	−0.286581	−	0.958056i	$-0.592519\pi$
$293$	−2.46782e8	−0.573162	−0.286581	+	0.958056i	$0.592519\pi$
$294$	1.78332e8	0.409274
$295$	2.00346e8	0.454363
$296$	−1.50230e8	−0.336695
$297$	−2.21197e6	−0.00489927
$298$	1.57168e8	0.344038
$299$	−4.14071e6	−0.00895830
$300$	−1.54748e8	−0.330903
$301$	1.37616e7	0.0290860
$302$	−5.22351e7	−0.109128
$303$	−8.71621e8	−1.80002
$304$	−1.67698e8	−0.342350
$305$	−5.36090e8	−1.08190
$306$	4.20193e7	0.0838348
$307$	3.84464e8	0.758352	0.379176	−	0.925325i	$-0.376207\pi$
$307$	3.84464e8	0.758352	0.379176	+	0.925325i	$0.376207\pi$
$308$	1.37738e7	0.0268613
$309$	9.02164e8	1.73953
$310$	−1.85053e7	−0.0352802
$311$	−5.62608e8	−1.06058	−0.530291	−	0.847815i	$-0.677917\pi$
$311$	−5.62608e8	−1.06058	−0.530291	+	0.847815i	$0.677917\pi$
$312$	9.23581e6	0.0172161
$313$	−2.64846e8	−0.488188	−0.244094	−	0.969752i	$-0.578491\pi$
$313$	−2.64846e8	−0.488188	−0.244094	+	0.969752i	$0.578491\pi$
$314$	9.21066e6	0.0167895
$315$	9.33285e7	0.168239
$316$	3.08189e8	0.549429
$317$	9.81092e7	0.172983	0.0864913	−	0.996253i	$-0.472435\pi$
$317$	9.81092e7	0.172983	0.0864913	+	0.996253i	$0.472435\pi$
$318$	2.85251e6	0.00497430
$319$	−2.29687e7	−0.0396159
$320$	2.54845e8	0.434762
$321$	7.08963e8	1.19634
$322$	1.44117e7	0.0240558
$323$	−7.68968e7	−0.126970
$324$	5.44908e8	0.890053
$325$	−3.35398e6	−0.00541962
$326$	−1.78826e8	−0.285870
$327$	−6.14813e8	−0.972358
$328$	−2.76971e8	−0.433388
$329$	1.28696e8	0.199241
$330$	3.69939e7	0.0566671
$331$	6.54036e8	0.991296	0.495648	−	0.868523i	$-0.334931\pi$
$331$	6.54036e8	0.991296	0.495648	+	0.868523i	$0.334931\pi$
$332$	−7.88866e8	−1.18310
$333$	4.06405e8	0.603121
$334$	1.25318e8	0.184035
$335$	−9.37404e8	−1.36229
$336$	1.39626e8	0.200806
$337$	6.62977e8	0.943613	0.471806	−	0.881702i	$-0.343602\pi$
$337$	6.62977e8	0.943613	0.471806	+	0.881702i	$0.343602\pi$
$338$	−2.11934e8	−0.298532
$339$	2.69901e8	0.376275
$340$	1.56388e8	0.215788
$341$	1.55000e7	0.0211685
$342$	−1.04443e8	−0.141185
$343$	2.79902e8	0.374522
$344$	6.57086e7	0.0870297
$345$	−3.95221e8	−0.518170
$346$	−1.32814e8	−0.172377
$347$	−5.98992e8	−0.769605	−0.384802	−	0.922999i	$-0.625730\pi$
$347$	−5.98992e8	−0.769605	−0.384802	+	0.922999i	$0.625730\pi$
$348$	−2.60888e8	−0.331839
$349$	1.09709e9	1.38151	0.690753	−	0.723091i	$-0.257279\pi$
$349$	1.09709e9	1.38151	0.690753	+	0.723091i	$0.257279\pi$
$350$	1.16735e7	0.0145534
$351$	−544544.	−0.000672137 0
$352$	1.00186e8	0.122435
$353$	1.20482e9	1.45784	0.728921	−	0.684598i	$-0.240022\pi$
$353$	1.20482e9	1.45784	0.728921	+	0.684598i	$0.240022\pi$
$354$	1.86674e8	0.223652
$355$	−9.37585e8	−1.11227
$356$	−3.82015e7	−0.0448751
$357$	6.40244e7	0.0744743
$358$	−1.61080e8	−0.185546
$359$	−5.38564e8	−0.614338	−0.307169	−	0.951655i	$-0.599382\pi$
$359$	−5.38564e8	−0.614338	−0.307169	+	0.951655i	$0.599382\pi$
$360$	4.45624e8	0.503396
$361$	−7.02737e8	−0.786172
$362$	6.89924e7	0.0764401
$363$	1.26498e9	1.38807
$364$	3.39084e6	0.00368513
$365$	−1.26037e9	−1.35667
$366$	−4.99507e8	−0.532547
$367$	1.10179e9	1.16350	0.581750	−	0.813368i	$-0.302368\pi$
$367$	1.10179e9	1.16350	0.581750	+	0.813368i	$0.302368\pi$
$368$	−2.98895e8	−0.312645
$369$	7.49268e8	0.776327
$370$	−1.48138e8	−0.152041
$371$	2.19711e6	0.00223379
$372$	1.76055e8	0.177316
$373$	1.71818e9	1.71431	0.857154	−	0.515060i	$-0.172231\pi$
$373$	1.71818e9	1.71431	0.857154	+	0.515060i	$0.172231\pi$
$374$	1.28289e7	0.0126806
$375$	−1.57318e9	−1.54052
$376$	6.14498e8	0.596160
$377$	−5.65446e6	−0.00543496
$378$	1.89528e6	0.00180490
$379$	1.07347e8	0.101287	0.0506433	−	0.998717i	$-0.483873\pi$
$379$	1.07347e8	0.101287	0.0506433	+	0.998717i	$0.483873\pi$
$380$	−3.88719e8	−0.363407
$381$	1.63197e9	1.51173
$382$	2.70292e8	0.248091
$383$	−6.46792e8	−0.588259	−0.294130	−	0.955766i	$-0.595030\pi$
$383$	−6.46792e8	−0.588259	−0.294130	+	0.955766i	$0.595030\pi$
$384$	1.48685e9	1.34001
$385$	2.84941e7	0.0254473
$386$	−1.29322e8	−0.114451
$387$	−1.77756e8	−0.155896
$388$	−5.89603e8	−0.512446
$389$	−5.05273e7	−0.0435214	−0.0217607	−	0.999763i	$-0.506927\pi$
$389$	−5.05273e7	−0.0435214	−0.0217607	+	0.999763i	$0.506927\pi$
$390$	9.10717e6	0.00777423
$391$	−1.37056e8	−0.115953
$392$	6.55858e8	0.549932
$393$	1.78161e9	1.48060
$394$	4.16962e8	0.343447
$395$	6.37555e8	0.520509
$396$	−1.77914e8	−0.143972
$397$	−1.12623e9	−0.903359	−0.451679	−	0.892180i	$-0.649175\pi$
$397$	−1.12623e9	−0.903359	−0.451679	+	0.892180i	$0.649175\pi$
$398$	−2.68803e8	−0.213719
$399$	−1.59139e8	−0.125422
$400$	−2.42106e8	−0.189145
$401$	−90676.9	−7.02250e−5 0	−3.51125e−5	−	1.00000i	$-0.500011\pi$
$401$	−90676.9	−7.02250e−5 0	−3.51125e−5		1.00000i	$0.500011\pi$
$402$	−8.73435e8	−0.670563
$403$	3.81579e6	0.00290413
$404$	−1.52797e9	−1.15287
$405$	1.12726e9	0.843202
$406$	1.96803e7	0.0145945
$407$	1.24080e8	0.0912262
$408$	3.05703e8	0.222838
$409$	−3.28251e8	−0.237233	−0.118616	−	0.992940i	$-0.537846\pi$
$409$	−3.28251e8	−0.237233	−0.118616	+	0.992940i	$0.537846\pi$
$410$	−2.73114e8	−0.195704
$411$	−3.19726e9	−2.27160
$412$	1.58151e9	1.11412
$413$	1.43783e8	0.100435
$414$	−1.86154e8	−0.128935
$415$	−1.63194e9	−1.12082
$416$	2.46638e7	0.0167970
$417$	3.95231e8	0.266916
$418$	−3.18876e7	−0.0213553
$419$	2.00896e9	1.33421	0.667103	−	0.744965i	$-0.267534\pi$
$419$	2.00896e9	1.33421	0.667103	+	0.744965i	$0.267534\pi$
$420$	3.23648e8	0.213157
$421$	−1.17015e9	−0.764283	−0.382141	−	0.924104i	$-0.624813\pi$
$421$	−1.17015e9	−0.764283	−0.382141	+	0.924104i	$0.624813\pi$
$422$	−7.77427e8	−0.503577
$423$	−1.66235e9	−1.06790
$424$	1.04908e7	0.00668384
$425$	−1.11016e8	−0.0701495
$426$	−8.73604e8	−0.547496
$427$	−3.84739e8	−0.239149
$428$	1.24283e9	0.766227
$429$	−7.62813e6	−0.00466463
$430$	6.47934e7	0.0392999
$431$	−2.84974e9	−1.71449	−0.857245	−	0.514909i	$-0.827826\pi$
$431$	−2.84974e9	−1.71449	−0.857245	+	0.514909i	$0.827826\pi$
$432$	−3.93077e7	−0.0234576
$433$	1.92069e9	1.13697	0.568486	−	0.822693i	$-0.307529\pi$
$433$	1.92069e9	1.13697	0.568486	+	0.822693i	$0.307529\pi$
$434$	−1.32808e7	−0.00779850
$435$	−5.39704e8	−0.314372
$436$	−1.07778e9	−0.622770
$437$	3.40668e8	0.195275
$438$	−1.17436e9	−0.667795
$439$	7.86877e8	0.443896	0.221948	−	0.975059i	$-0.428759\pi$
$439$	7.86877e8	0.443896	0.221948	+	0.975059i	$0.428759\pi$
$440$	1.36053e8	0.0761422
$441$	−1.77424e9	−0.985092
$442$	3.15823e6	0.00173966
$443$	−1.97285e9	−1.07816	−0.539078	−	0.842256i	$-0.681227\pi$
$443$	−1.97285e9	−1.07816	−0.539078	+	0.842256i	$0.681227\pi$
$444$	1.40935e9	0.764148
$445$	−7.90281e7	−0.0425130
$446$	6.55562e8	0.349898
$447$	−3.09326e9	−1.63810
$448$	1.82896e8	0.0961019
$449$	5.76219e7	0.0300417	0.0150209	−	0.999887i	$-0.495219\pi$
$449$	5.76219e7	0.0300417	0.0150209	+	0.999887i	$0.495219\pi$
$450$	−1.50785e8	−0.0780036
$451$	2.28759e8	0.117425
$452$	4.73142e8	0.240995
$453$	1.02805e9	0.519603
$454$	8.40601e8	0.421594
$455$	7.01469e6	0.00349115
$456$	−7.59857e8	−0.375280
$457$	4.53691e8	0.222358	0.111179	−	0.993800i	$-0.464537\pi$
$457$	4.53691e8	0.222358	0.111179	+	0.993800i	$0.464537\pi$
$458$	8.83852e8	0.429883
$459$	−1.80243e7	−0.00869988
$460$	−6.92830e8	−0.331875
$461$	2.09843e9	0.997563	0.498781	−	0.866728i	$-0.333781\pi$
$461$	2.09843e9	0.997563	0.498781	+	0.866728i	$0.333781\pi$
$462$	2.65497e7	0.0125260
$463$	−2.12353e8	−0.0994318	−0.0497159	−	0.998763i	$-0.515832\pi$
$463$	−2.12353e8	−0.0994318	−0.0497159	+	0.998763i	$0.515832\pi$
$464$	−4.08165e8	−0.189680
$465$	3.64208e8	0.167983
$466$	5.77565e8	0.264393
$467$	1.04433e9	0.474491	0.237246	−	0.971450i	$-0.423755\pi$
$467$	1.04433e9	0.474491	0.237246	+	0.971450i	$0.423755\pi$
$468$	−4.37990e7	−0.0197517
$469$	−6.72753e8	−0.301128
$470$	6.05939e8	0.269207
$471$	−1.81277e8	−0.0799412
$472$	6.86537e8	0.300515
$473$	−5.42707e7	−0.0235804
$474$	5.94048e8	0.256211
$475$	2.75942e8	0.118138
$476$	1.12236e8	0.0476989
$477$	−2.83798e7	−0.0119728
$478$	2.92793e8	0.122621
$479$	1.22092e9	0.507589	0.253794	−	0.967258i	$-0.418321\pi$
$479$	1.22092e9	0.507589	0.253794	+	0.967258i	$0.418321\pi$
$480$	2.35410e9	0.971583
$481$	3.05460e7	0.0125154
$482$	−4.03168e8	−0.163992
$483$	−2.83641e8	−0.114539
$484$	2.21754e9	0.889021
$485$	−1.21972e9	−0.485472
$486$	1.07428e9	0.424514
$487$	−8.85642e8	−0.347462	−0.173731	−	0.984793i	$-0.555582\pi$
$487$	−8.85642e8	−0.347462	−0.173731	+	0.984793i	$0.555582\pi$
$488$	−1.83705e9	−0.715569
$489$	3.51951e9	1.36114
$490$	6.46723e8	0.248332
$491$	−1.70646e9	−0.650597	−0.325298	−	0.945611i	$-0.605465\pi$
$491$	−1.70646e9	−0.650597	−0.325298	+	0.945611i	$0.605465\pi$
$492$	2.59834e9	0.983598
$493$	−1.87161e8	−0.0703480
$494$	−7.85010e6	−0.00292975
$495$	−3.68054e8	−0.136393
$496$	2.75441e8	0.101355
$497$	−6.72883e8	−0.245862
$498$	−1.52058e9	−0.551703
$499$	−1.84137e9	−0.663422	−0.331711	−	0.943381i	$-0.607626\pi$
$499$	−1.84137e9	−0.663422	−0.331711	+	0.943381i	$0.607626\pi$
$500$	−2.75781e9	−0.986666
$501$	−2.46642e9	−0.876264
$502$	1.53802e9	0.542625
$503$	−1.71136e9	−0.599587	−0.299794	−	0.954004i	$-0.596918\pi$
$503$	−1.71136e9	−0.599587	−0.299794	+	0.954004i	$0.596918\pi$
$504$	3.19814e8	0.111273
$505$	−3.16093e9	−1.09218
$506$	−5.68346e7	−0.0195023
$507$	4.17112e9	1.42143
$508$	2.86087e9	0.968224
$509$	2.79743e8	0.0940257	0.0470128	−	0.998894i	$-0.485030\pi$
$509$	2.79743e8	0.0940257	0.0470128	+	0.998894i	$0.485030\pi$
$510$	3.01445e8	0.100627
$511$	−9.04540e8	−0.299885
$512$	3.06352e9	1.00873
$513$	4.48012e7	0.0146514
$514$	1.73024e9	0.561999
$515$	3.27170e9	1.05548
$516$	−6.16428e8	−0.197519
$517$	−5.07532e8	−0.161527
$518$	−1.06315e8	−0.0336078
$519$	2.61395e9	0.820753
$520$	3.34937e7	0.0104460
$521$	−3.76777e9	−1.16722	−0.583609	−	0.812035i	$-0.698360\pi$
$521$	−3.76777e9	−1.16722	−0.583609	+	0.812035i	$0.698360\pi$
$522$	−2.54208e8	−0.0782243
$523$	8.71991e8	0.266536	0.133268	−	0.991080i	$-0.457453\pi$
$523$	8.71991e8	0.266536	0.133268	+	0.991080i	$0.457453\pi$
$524$	3.12320e9	0.948288
$525$	−2.29749e8	−0.0692942
$526$	3.17095e8	0.0950035
$527$	1.26302e8	0.0375900
$528$	−5.50633e8	−0.162796
$529$	−2.79764e9	−0.821669
$530$	1.03446e7	0.00301821
$531$	−1.85723e9	−0.538313
$532$	−2.78974e8	−0.0803292
$533$	5.63159e7	0.0161097
$534$	−7.36352e7	−0.0209262
$535$	2.57106e9	0.725894
$536$	−3.21226e9	−0.901018
$537$	3.17026e9	0.883457
$538$	−1.69834e7	−0.00470205
$539$	−5.41692e8	−0.149002
$540$	−9.11139e7	−0.0249004
$541$	−1.47997e9	−0.401848	−0.200924	−	0.979607i	$-0.564394\pi$
$541$	−1.47997e9	−0.401848	−0.200924	+	0.979607i	$0.564394\pi$
$542$	−1.65691e9	−0.446993
$543$	−1.35786e9	−0.363961
$544$	8.16365e8	0.217414
$545$	−2.22962e9	−0.589989
$546$	6.53601e6	0.00171846
$547$	6.46090e9	1.68786	0.843932	−	0.536450i	$-0.180235\pi$
$547$	6.46090e9	1.68786	0.843932	+	0.536450i	$0.180235\pi$
$548$	−5.60486e9	−1.45490
$549$	4.96962e9	1.28180
$550$	−4.60362e7	−0.0117986
$551$	4.65209e8	0.118472
$552$	−1.35433e9	−0.342718
$553$	4.57559e8	0.115056
$554$	1.69185e8	0.0422745
$555$	2.91554e9	0.723925
$556$	6.92848e8	0.170953
$557$	3.65569e9	0.896347	0.448174	−	0.893946i	$-0.352075\pi$
$557$	3.65569e9	0.896347	0.448174	+	0.893946i	$0.352075\pi$
$558$	1.71547e8	0.0417987
$559$	−1.33604e7	−0.00323502
$560$	5.06352e8	0.121841
$561$	−2.52489e8	−0.0603772
$562$	1.71830e9	0.408340
$563$	7.39782e9	1.74713	0.873564	−	0.486710i	$-0.161803\pi$
$563$	7.39782e9	1.74713	0.873564	+	0.486710i	$0.161803\pi$
$564$	−5.76475e9	−1.35302
$565$	9.78797e8	0.228309
$566$	3.11996e8	0.0723256
$567$	8.09009e8	0.186386
$568$	−3.21288e9	−0.735657
$569$	−1.65447e9	−0.376502	−0.188251	−	0.982121i	$-0.560282\pi$
$569$	−1.65447e9	−0.376502	−0.188251	+	0.982121i	$0.560282\pi$
$570$	−7.49274e8	−0.169464
$571$	−1.49420e9	−0.335878	−0.167939	−	0.985797i	$-0.553711\pi$
$571$	−1.49420e9	−0.335878	−0.167939	+	0.985797i	$0.553711\pi$
$572$	−1.33723e7	−0.00298758
$573$	−5.31969e9	−1.18126
$574$	−1.96007e8	−0.0432594
$575$	4.91823e8	0.107888
$576$	−2.36245e9	−0.515090
$577$	−5.15169e9	−1.11644	−0.558219	−	0.829694i	$-0.688515\pi$
$577$	−5.15169e9	−1.11644	−0.558219	+	0.829694i	$0.688515\pi$
$578$	−1.28201e9	−0.276149
$579$	2.54523e9	0.544944
$580$	−9.46113e8	−0.201347
$581$	−1.17121e9	−0.247752
$582$	−1.13649e9	−0.238965
$583$	−8.66462e6	−0.00181096
$584$	−4.31899e9	−0.897300
$585$	−9.06077e7	−0.0187120
$586$	−8.33885e8	−0.171184
$587$	−7.65712e8	−0.156254	−0.0781272	−	0.996943i	$-0.524894\pi$
$587$	−7.65712e8	−0.156254	−0.0781272	+	0.996943i	$0.524894\pi$
$588$	−6.15276e9	−1.24810
$589$	−3.13936e8	−0.0633050
$590$	6.76974e8	0.135703
$591$	−8.20634e9	−1.63528
$592$	2.20495e9	0.436790
$593$	1.51945e9	0.299222	0.149611	−	0.988745i	$-0.452198\pi$
$593$	1.51945e9	0.299222	0.149611	+	0.988745i	$0.452198\pi$
$594$	−7.47431e6	−0.00146325
$595$	2.32185e8	0.0451881
$596$	−5.42255e9	−1.04916
$597$	5.29038e9	1.01760
$598$	−1.39916e7	−0.00267555
$599$	−8.16357e9	−1.55198	−0.775991	−	0.630745i	$-0.782750\pi$
$599$	−8.16357e9	−1.55198	−0.775991	+	0.630745i	$0.782750\pi$
$600$	−1.09701e9	−0.207338
$601$	−8.29079e9	−1.55788	−0.778942	−	0.627096i	$-0.784243\pi$
$601$	−8.29079e9	−1.55788	−0.778942	+	0.627096i	$0.784243\pi$
$602$	4.65007e7	0.00868704
$603$	8.68985e9	1.61399
$604$	1.80220e9	0.332792
$605$	4.58746e9	0.842225
$606$	−2.94523e9	−0.537608
$607$	−5.48624e9	−0.995668	−0.497834	−	0.867272i	$-0.665871\pi$
$607$	−5.48624e9	−0.995668	−0.497834	+	0.867272i	$0.665871\pi$
$608$	−2.02916e9	−0.366146
$609$	−3.87333e8	−0.0694903
$610$	−1.81146e9	−0.323128
$611$	−1.24944e8	−0.0221601
$612$	−1.44974e9	−0.255658
$613$	−5.20382e8	−0.0912453	−0.0456227	−	0.998959i	$-0.514527\pi$
$613$	−5.20382e8	−0.0912453	−0.0456227	+	0.998959i	$0.514527\pi$
$614$	1.29911e9	0.226495
$615$	5.37522e9	0.931823
$616$	9.76424e7	0.0168309
$617$	1.65572e9	0.283784	0.141892	−	0.989882i	$-0.454681\pi$
$617$	1.65572e9	0.283784	0.141892	+	0.989882i	$0.454681\pi$
$618$	3.04844e9	0.519539
$619$	−4.63680e8	−0.0785781	−0.0392890	−	0.999228i	$-0.512509\pi$
$619$	−4.63680e8	−0.0785781	−0.0392890	+	0.999228i	$0.512509\pi$
$620$	6.38464e8	0.107588
$621$	7.98511e7	0.0133801
$622$	−1.90107e9	−0.316761
$623$	−5.67166e7	−0.00939728
$624$	−1.35555e8	−0.0223342
$625$	−4.14581e9	−0.679250
$626$	−8.94921e8	−0.145806
$627$	6.27588e8	0.101681
$628$	−3.17783e8	−0.0512003
$629$	1.01106e9	0.161995
$630$	3.15360e8	0.0502475
$631$	−5.00414e8	−0.0792915	−0.0396458	−	0.999214i	$-0.512623\pi$
$631$	−5.00414e8	−0.0792915	−0.0396458	+	0.999214i	$0.512623\pi$
$632$	2.18475e9	0.344264
$633$	1.53007e10	2.39773
$634$	3.31514e8	0.0516642
$635$	5.91834e9	0.917258
$636$	−9.84163e7	−0.0151694
$637$	−1.33354e8	−0.0204418
$638$	−7.76121e7	−0.0118320
$639$	8.69153e9	1.31778
$640$	5.39208e9	0.813067
$641$	6.72118e9	1.00796	0.503979	−	0.863716i	$-0.331869\pi$
$641$	6.72118e9	1.00796	0.503979	+	0.863716i	$0.331869\pi$
$642$	2.39561e9	0.357308
$643$	−2.00229e9	−0.297021	−0.148511	−	0.988911i	$-0.547448\pi$
$643$	−2.00229e9	−0.297021	−0.148511	+	0.988911i	$0.547448\pi$
$644$	−4.97228e8	−0.0733592
$645$	−1.27522e9	−0.187122
$646$	−2.59837e8	−0.0379216
$647$	1.06244e10	1.54219	0.771095	−	0.636720i	$-0.219709\pi$
$647$	1.06244e10	1.54219	0.771095	+	0.636720i	$0.219709\pi$
$648$	3.86285e9	0.557693
$649$	−5.67031e8	−0.0814236
$650$	−1.13332e7	−0.00161866
$651$	2.61384e8	0.0371317
$652$	6.16978e9	0.871772
$653$	−2.49361e9	−0.350456	−0.175228	−	0.984528i	$-0.556066\pi$
$653$	−2.49361e9	−0.350456	−0.175228	+	0.984528i	$0.556066\pi$
$654$	−2.07747e9	−0.290411
$655$	6.46102e9	0.898372
$656$	4.06514e9	0.562228
$657$	1.16838e10	1.60733
$658$	4.34868e8	0.0595068
$659$	−7.14359e9	−0.972339	−0.486170	−	0.873864i	$-0.661606\pi$
$659$	−7.14359e9	−0.972339	−0.486170	+	0.873864i	$0.661606\pi$
$660$	−1.27635e9	−0.172809
$661$	−1.06348e10	−1.43227	−0.716134	−	0.697963i	$-0.754090\pi$
$661$	−1.06348e10	−1.43227	−0.716134	+	0.697963i	$0.754090\pi$
$662$	2.21001e9	0.296067
$663$	−6.21579e7	−0.00828322
$664$	−5.59227e9	−0.741310
$665$	−5.77119e8	−0.0761009
$666$	1.37326e9	0.180132
$667$	8.29161e8	0.108193
$668$	−4.32368e9	−0.561224
$669$	−1.29023e10	−1.66600
$670$	−3.16752e9	−0.406871
$671$	1.51727e9	0.193881
$672$	1.68948e9	0.214764
$673$	6.11664e9	0.773500	0.386750	−	0.922185i	$-0.373598\pi$
$673$	6.11664e9	0.773500	0.386750	+	0.922185i	$0.373598\pi$
$674$	2.24022e9	0.281826
$675$	6.46795e7	0.00809475
$676$	7.31207e9	0.910388
$677$	9.09572e9	1.12662	0.563309	−	0.826246i	$-0.309528\pi$
$677$	9.09572e9	1.12662	0.563309	+	0.826246i	$0.309528\pi$
$678$	9.12004e8	0.112381
$679$	−8.75366e8	−0.107311
$680$	1.10863e9	0.135209
$681$	−1.65441e10	−2.00737
$682$	5.23749e7	0.00632234
$683$	7.39960e9	0.888660	0.444330	−	0.895863i	$-0.353442\pi$
$683$	7.39960e9	0.888660	0.444330	+	0.895863i	$0.353442\pi$
$684$	3.60347e9	0.430551
$685$	−1.15949e10	−1.37832
$686$	9.45798e8	0.111857
$687$	−1.73953e10	−2.04684
$688$	−9.64413e8	−0.112903
$689$	−2.13306e6	−0.000248448 0
$690$	−1.33546e9	−0.154760
$691$	7.73691e9	0.892061	0.446030	−	0.895018i	$-0.352837\pi$
$691$	7.73691e9	0.892061	0.446030	+	0.895018i	$0.352837\pi$
$692$	4.58232e9	0.525671
$693$	−2.64144e8	−0.0301491
$694$	−2.02401e9	−0.229855
$695$	1.43331e9	0.161954
$696$	−1.84944e9	−0.207925
$697$	1.86405e9	0.208517
$698$	3.70710e9	0.412610
$699$	−1.13672e10	−1.25888
$700$	−4.02755e8	−0.0443811
$701$	9.43506e9	1.03450	0.517251	−	0.855833i	$-0.326955\pi$
$701$	9.43506e9	1.03450	0.517251	+	0.855833i	$0.326955\pi$
$702$	−1.84003e6	−0.000200745 0
$703$	−2.51311e9	−0.272814
$704$	−7.21278e8	−0.0779109
$705$	−1.19256e10	−1.28180
$706$	4.07112e9	0.435409
$707$	−2.26853e9	−0.241422
$708$	−6.44057e9	−0.682037
$709$	−1.29011e10	−1.35945	−0.679726	−	0.733466i	$-0.737901\pi$
$709$	−1.29011e10	−1.35945	−0.679726	+	0.733466i	$0.737901\pi$
$710$	−3.16813e9	−0.332199
$711$	−5.91022e9	−0.616680
$712$	−2.70810e8	−0.0281180
$713$	−5.59542e8	−0.0578122
$714$	2.16340e8	0.0222430
$715$	−2.76634e7	−0.00283032
$716$	5.55754e9	0.565831
$717$	−5.76254e9	−0.583844
$718$	−1.81983e9	−0.183482
$719$	1.56677e10	1.57201	0.786004	−	0.618222i	$-0.212147\pi$
$719$	1.56677e10	1.57201	0.786004	+	0.618222i	$0.212147\pi$
$720$	−6.54048e9	−0.653049
$721$	2.34802e9	0.233308
$722$	−2.37457e9	−0.234803
$723$	7.93487e9	0.780828
$724$	−2.38035e9	−0.233108
$725$	6.71622e8	0.0654549
$726$	4.27441e9	0.414570
$727$	1.29468e10	1.24966	0.624830	−	0.780761i	$-0.285168\pi$
$727$	1.29468e10	1.24966	0.624830	+	0.780761i	$0.285168\pi$
$728$	2.40377e7	0.00230904
$729$	−1.09212e10	−1.04405
$730$	−4.25884e9	−0.405192
$731$	−4.42225e8	−0.0418729
$732$	1.72338e10	1.62403
$733$	1.96470e10	1.84261	0.921305	−	0.388842i	$-0.127125\pi$
$733$	1.96470e10	1.84261	0.921305	+	0.388842i	$0.127125\pi$
$734$	3.72297e9	0.347499
$735$	−1.27283e10	−1.18240
$736$	−3.61666e9	−0.334376
$737$	2.65310e9	0.244128
$738$	2.53180e9	0.231863
$739$	5.46414e9	0.498042	0.249021	−	0.968498i	$-0.419891\pi$
$739$	5.46414e9	0.498042	0.249021	+	0.968498i	$0.419891\pi$
$740$	5.11100e9	0.463655
$741$	1.54500e8	0.0139497
$742$	7.42410e6	0.000667160 0
$743$	1.10547e10	0.988750	0.494375	−	0.869249i	$-0.335397\pi$
$743$	1.10547e10	0.988750	0.494375	+	0.869249i	$0.335397\pi$
$744$	1.24805e9	0.111104
$745$	−1.12177e10	−0.993934
$746$	5.80580e9	0.512007
$747$	1.51283e10	1.32791
$748$	−4.42619e8	−0.0386700
$749$	1.84519e9	0.160455
$750$	−5.31582e9	−0.460104
$751$	−1.34448e10	−1.15828	−0.579141	−	0.815227i	$-0.696612\pi$
$751$	−1.34448e10	−1.15828	−0.579141	+	0.815227i	$0.696612\pi$
$752$	−9.01906e9	−0.773390
$753$	−3.02702e10	−2.58365
$754$	−1.91066e7	−0.00162324
$755$	3.72824e9	0.315275
$756$	−6.53903e7	−0.00550411
$757$	4.89813e9	0.410388	0.205194	−	0.978721i	$-0.434217\pi$
$757$	4.89813e9	0.410388	0.205194	+	0.978721i	$0.434217\pi$
$758$	3.62728e8	0.0302509
$759$	1.11858e9	0.0928581
$760$	−2.75562e9	−0.227705
$761$	1.08629e10	0.893512	0.446756	−	0.894656i	$-0.352579\pi$
$761$	1.08629e10	0.893512	0.446756	+	0.894656i	$0.352579\pi$
$762$	5.51447e9	0.451504
$763$	−1.60015e9	−0.130414
$764$	−9.32552e9	−0.756565
$765$	−2.99910e9	−0.242201
$766$	−2.18553e9	−0.175694
$767$	−1.39592e8	−0.0111706
$768$	−3.97081e9	−0.316312
$769$	1.74439e9	0.138325	0.0691627	−	0.997605i	$-0.477967\pi$
$769$	1.74439e9	0.138325	0.0691627	+	0.997605i	$0.477967\pi$
$770$	9.62824e7	0.00760028
$771$	−3.40533e10	−2.67589
$772$	4.46184e9	0.349022
$773$	−2.29156e10	−1.78444	−0.892221	−	0.451598i	$-0.850854\pi$
$773$	−2.29156e10	−1.78444	−0.892221	+	0.451598i	$0.850854\pi$
$774$	−6.00643e8	−0.0465611
$775$	−4.53230e8	−0.0349754
$776$	−4.17969e9	−0.321091
$777$	2.09241e9	0.160020
$778$	−1.70733e8	−0.0129984
$779$	−4.63328e9	−0.351162
$780$	−3.14213e8	−0.0237079
$781$	2.65361e9	0.199324
$782$	−4.63118e8	−0.0346312
$783$	1.09043e8	0.00811766
$784$	−9.62611e9	−0.713419
$785$	−6.57403e8	−0.0485052
$786$	6.02012e9	0.442207
$787$	1.46509e10	1.07140	0.535701	−	0.844407i	$-0.320047\pi$
$787$	1.46509e10	1.07140	0.535701	+	0.844407i	$0.320047\pi$
$788$	−1.43859e10	−1.04736
$789$	−6.24084e9	−0.452349
$790$	2.15432e9	0.155459
$791$	7.02460e8	0.0504666
$792$	−1.26123e9	−0.0902105
$793$	3.73523e8	0.0265988
$794$	−3.80556e9	−0.269803
$795$	−2.03595e8	−0.0143709
$796$	9.27414e9	0.651745
$797$	−1.25977e10	−0.881426	−0.440713	−	0.897648i	$-0.645274\pi$
$797$	−1.25977e10	−0.881426	−0.440713	+	0.897648i	$0.645274\pi$
$798$	−5.37736e8	−0.0374593
$799$	−4.13563e9	−0.286832
$800$	−2.92950e9	−0.202292
$801$	7.32600e8	0.0503678
$802$	−306400.	−2.09739e−5 0
$803$	3.56718e9	0.243120
$804$	3.01350e10	2.04491
$805$	−1.02862e9	−0.0694978
$806$	1.28937e7	0.000867369 0
$807$	3.34255e8	0.0223883
$808$	−1.08318e10	−0.722370
$809$	2.20916e10	1.46692	0.733462	−	0.679731i	$-0.237903\pi$
$809$	2.20916e10	1.46692	0.733462	+	0.679731i	$0.237903\pi$
$810$	3.80905e9	0.251837
$811$	−7.33733e9	−0.483020	−0.241510	−	0.970398i	$-0.577643\pi$
$811$	−7.33733e9	−0.483020	−0.241510	+	0.970398i	$0.577643\pi$
$812$	−6.79003e8	−0.0445067
$813$	3.26100e10	2.12831
$814$	4.19269e8	0.0272463
$815$	1.27635e10	0.825884
$816$	−4.48685e9	−0.289085
$817$	1.09920e9	0.0705177
$818$	−1.10917e9	−0.0708535
$819$	−6.50271e7	−0.00413619
$820$	9.42287e9	0.596808
$821$	−1.04735e10	−0.660528	−0.330264	−	0.943889i	$-0.607138\pi$
$821$	−1.04735e10	−0.660528	−0.330264	+	0.943889i	$0.607138\pi$
$822$	−1.08036e10	−0.678451
$823$	2.13220e10	1.33330	0.666650	−	0.745371i	$-0.267727\pi$
$823$	2.13220e10	1.33330	0.666650	+	0.745371i	$0.267727\pi$
$824$	1.12113e10	0.698091
$825$	9.06050e8	0.0561776
$826$	4.85849e8	0.0299965
$827$	4.07306e9	0.250410	0.125205	−	0.992131i	$-0.460041\pi$
$827$	4.07306e9	0.250410	0.125205	+	0.992131i	$0.460041\pi$
$828$	6.42262e9	0.393193
$829$	−4.97505e9	−0.303289	−0.151644	−	0.988435i	$-0.548457\pi$
$829$	−4.97505e9	−0.303289	−0.151644	+	0.988435i	$0.548457\pi$
$830$	−5.51438e9	−0.334752
$831$	−3.32978e9	−0.201286
$832$	−1.77565e8	−0.0106887
$833$	−4.41399e9	−0.264590
$834$	1.33550e9	0.0797190
$835$	−8.94448e9	−0.531682
$836$	1.10018e9	0.0651238
$837$	−7.35852e7	−0.00433762
$838$	6.78835e9	0.398483
$839$	5.31627e9	0.310771	0.155386	−	0.987854i	$-0.450338\pi$
$839$	5.31627e9	0.310771	0.155386	+	0.987854i	$0.450338\pi$
$840$	2.29434e9	0.133561
$841$	−1.61176e10	−0.934360
$842$	−3.95397e9	−0.228266
$843$	−3.38183e10	−1.94426
$844$	2.68225e10	1.53568
$845$	1.51266e10	0.862467
$846$	−5.61713e9	−0.318947
$847$	3.29231e9	0.186170
$848$	−1.53974e8	−0.00867086
$849$	−6.14048e9	−0.344370
$850$	−3.75126e8	−0.0209513
$851$	−4.47921e9	−0.249143
$852$	3.01408e10	1.66961
$853$	1.45978e10	0.805315	0.402658	−	0.915351i	$-0.368087\pi$
$853$	1.45978e10	0.805315	0.402658	+	0.915351i	$0.368087\pi$
$854$	−1.30005e9	−0.0714259
$855$	7.45456e9	0.407888
$856$	8.81039e9	0.480106
$857$	−1.57329e10	−0.853839	−0.426919	−	0.904290i	$-0.640401\pi$
$857$	−1.57329e10	−0.853839	−0.426919	+	0.904290i	$0.640401\pi$
$858$	−2.57757e7	−0.00139317
$859$	1.26238e10	0.679540	0.339770	−	0.940509i	$-0.389651\pi$
$859$	1.26238e10	0.679540	0.339770	+	0.940509i	$0.389651\pi$
$860$	−2.23548e9	−0.119847
$861$	3.85767e9	0.205975
$862$	−9.62936e9	−0.512061
$863$	5.25485e9	0.278306	0.139153	−	0.990271i	$-0.455562\pi$
$863$	5.25485e9	0.278306	0.139153	+	0.990271i	$0.455562\pi$
$864$	−4.75626e8	−0.0250881
$865$	9.47952e9	0.498000
$866$	6.49008e9	0.339576
$867$	2.52316e10	1.31485
$868$	4.58211e8	0.0237819
$869$	−1.80445e9	−0.0932771
$870$	−1.82368e9	−0.0938924
$871$	6.53141e8	0.0334922
$872$	−7.64038e9	−0.390218
$873$	1.13070e10	0.575170
$874$	1.15113e9	0.0583221
$875$	−4.09445e9	−0.206617
$876$	4.05175e10	2.03647
$877$	−2.81828e10	−1.41086	−0.705432	−	0.708778i	$-0.749247\pi$
$877$	−2.81828e10	−1.41086	−0.705432	+	0.708778i	$0.749247\pi$
$878$	2.65888e9	0.132577
$879$	1.64119e10	0.815076
$880$	−1.99687e9	−0.0987782
$881$	7.05789e9	0.347744	0.173872	−	0.984768i	$-0.444372\pi$
$881$	7.05789e9	0.347744	0.173872	+	0.984768i	$0.444372\pi$
$882$	−5.99520e9	−0.294214
$883$	−3.54924e10	−1.73489	−0.867447	−	0.497530i	$-0.834240\pi$
$883$	−3.54924e10	−1.73489	−0.867447	+	0.497530i	$0.834240\pi$
$884$	−1.08964e8	−0.00530519
$885$	−1.33237e10	−0.646136
$886$	−6.66633e9	−0.322010
$887$	−2.92725e10	−1.40840	−0.704202	−	0.709999i	$-0.748695\pi$
$887$	−2.92725e10	−1.40840	−0.704202	+	0.709999i	$0.748695\pi$
$888$	9.99085e9	0.478803
$889$	4.24745e9	0.202755
$890$	−2.67038e8	−0.0126972
$891$	−3.19044e9	−0.151105
$892$	−2.26180e10	−1.06703
$893$	1.02795e10	0.483051
$894$	−1.04522e10	−0.489246
$895$	1.14970e10	0.536047
$896$	3.86977e9	0.179724
$897$	2.75372e8	0.0127393
$898$	1.94706e8	0.00897248
$899$	−7.64098e8	−0.0350744
$900$	5.20233e9	0.237875
$901$	−7.06038e7	−0.00321582
$902$	7.72983e8	0.0350709
$903$	−9.15193e8	−0.0413624
$904$	3.35410e9	0.151003
$905$	−4.92428e9	−0.220837
$906$	3.47382e9	0.155188
$907$	−3.76454e10	−1.67528	−0.837639	−	0.546224i	$-0.816065\pi$
$907$	−3.76454e10	−1.67528	−0.837639	+	0.546224i	$0.816065\pi$
$908$	−2.90021e10	−1.28567
$909$	2.93023e10	1.29398
$910$	2.37028e7	0.00104269
$911$	6.61567e9	0.289908	0.144954	−	0.989438i	$-0.453697\pi$
$911$	6.61567e9	0.289908	0.144954	+	0.989438i	$0.453697\pi$
$912$	1.11525e10	0.486845
$913$	4.61882e9	0.200855
$914$	1.53303e9	0.0664110
$915$	3.56519e10	1.53854
$916$	−3.04944e10	−1.31095
$917$	4.63692e9	0.198581
$918$	−6.09046e7	−0.00259837
$919$	2.38719e10	1.01457	0.507285	−	0.861778i	$-0.330649\pi$
$919$	2.38719e10	1.01457	0.507285	+	0.861778i	$0.330649\pi$
$920$	−4.91147e9	−0.207948
$921$	−2.55682e10	−1.07843
$922$	7.09064e9	0.297939
$923$	6.53267e8	0.0273454
$924$	−9.16007e8	−0.0381986
$925$	−3.62817e9	−0.150727
$926$	−7.17548e8	−0.0296970
$927$	−3.03291e10	−1.25049
$928$	−4.93883e9	−0.202864
$929$	−2.31224e10	−0.946190	−0.473095	−	0.881012i	$-0.656863\pi$
$929$	−2.31224e10	−0.946190	−0.473095	+	0.881012i	$0.656863\pi$
$930$	1.23067e9	0.0501708
$931$	1.09714e10	0.445594
$932$	−1.99270e10	−0.806279
$933$	3.74155e10	1.50822
$934$	3.52882e9	0.141715
$935$	−9.15653e8	−0.0366345
$936$	−3.10491e8	−0.0123761
$937$	−3.07105e10	−1.21954	−0.609772	−	0.792577i	$-0.708739\pi$
$937$	−3.07105e10	−1.21954	−0.609772	+	0.792577i	$0.708739\pi$
$938$	−2.27325e9	−0.0899369
$939$	1.76132e10	0.694238
$940$	−2.09059e10	−0.820958
$941$	4.74404e8	0.0185603	0.00928015	−	0.999957i	$-0.497046\pi$
$941$	4.74404e8	0.0185603	0.00928015	+	0.999957i	$0.497046\pi$
$942$	−6.12542e8	−0.0238758
$943$	−8.25809e9	−0.320692
$944$	−1.00764e10	−0.389855
$945$	−1.35274e8	−0.00521439
$946$	−1.83382e8	−0.00704268
$947$	−3.03487e10	−1.16122	−0.580611	−	0.814181i	$-0.697186\pi$
$947$	−3.03487e10	−1.16122	−0.580611	+	0.814181i	$0.697186\pi$
$948$	−2.04957e10	−0.781327
$949$	8.78171e8	0.0333539
$950$	9.32416e8	0.0352839
$951$	−6.52461e9	−0.245993
$952$	7.95641e8	0.0298874
$953$	1.34857e10	0.504717	0.252359	−	0.967634i	$-0.418794\pi$
$953$	1.34857e10	0.504717	0.252359	+	0.967634i	$0.418794\pi$
$954$	−9.58961e7	−0.00357587
$955$	−1.92919e10	−0.716741
$956$	−1.01019e10	−0.373937
$957$	1.52750e9	0.0563366
$958$	4.12552e9	0.151600
$959$	−8.32137e9	−0.304670
$960$	−1.69481e10	−0.618261
$961$	−2.69970e10	−0.981258
$962$	1.03216e8	0.00373795
$963$	−2.38340e10	−0.860013
$964$	1.39100e10	0.500100
$965$	9.23028e9	0.330651
$966$	−9.58430e8	−0.0342090
$967$	5.31442e10	1.89001	0.945004	−	0.327060i	$-0.106058\pi$
$967$	5.31442e10	1.89001	0.945004	+	0.327060i	$0.106058\pi$
$968$	1.57201e10	0.557047
$969$	5.11391e9	0.180559
$970$	−4.12147e9	−0.144995
$971$	1.88877e10	0.662083	0.331042	−	0.943616i	$-0.392600\pi$
$971$	1.88877e10	0.662083	0.331042	+	0.943616i	$0.392600\pi$
$972$	−3.70646e10	−1.29457
$973$	1.02865e9	0.0357992
$974$	−2.99261e9	−0.103775
$975$	2.23052e8	0.00770707
$976$	2.69626e10	0.928299
$977$	−4.88964e9	−0.167744	−0.0838718	−	0.996477i	$-0.526729\pi$
$977$	−4.88964e9	−0.167744	−0.0838718	+	0.996477i	$0.526729\pi$
$978$	1.18925e10	0.406526
$979$	2.23670e8	0.00761848
$980$	−2.23130e10	−0.757298
$981$	2.06689e10	0.698998
$982$	−5.76619e9	−0.194312
$983$	4.22030e10	1.41712	0.708560	−	0.705651i	$-0.249345\pi$
$983$	4.22030e10	1.41712	0.708560	+	0.705651i	$0.249345\pi$
$984$	1.84196e10	0.616307
$985$	−2.97603e10	−0.992227
$986$	−6.32423e8	−0.0210106
$987$	−8.55875e9	−0.283335
$988$	2.70842e8	0.00893442
$989$	1.95915e9	0.0643991
$990$	−1.24367e9	−0.0407362
$991$	4.76499e10	1.55527	0.777633	−	0.628719i	$-0.216420\pi$
$991$	4.76499e10	1.55527	0.777633	+	0.628719i	$0.216420\pi$
$992$	3.33286e9	0.108399
$993$	−4.34957e10	−1.40969
$994$	−2.27369e9	−0.0734310
$995$	1.91856e10	0.617439
$996$	5.24624e10	1.68244
$997$	−6.81917e8	−0.0217921	−0.0108960	−	0.999941i	$-0.503468\pi$
$997$	−6.81917e8	−0.0217921	−0.0108960	+	0.999941i	$0.503468\pi$
$998$	−6.22205e9	−0.198142
$999$	−5.89061e8	−0.0186931

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.a.b.1.8	✓	13
3.2	odd	2		387.8.a.d.1.6		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.b.1.8	✓	13	1.1	even	1	trivial
387.8.a.d.1.6		13	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.37903 q^{2} -66.5036 q^{3} -116.582 q^{4} -241.175 q^{5} -224.718 q^{6} -173.086 q^{7} -826.450 q^{8} +2235.73 q^{9} +O(q^{10})\)	\(q+3.37903 q^{2} -66.5036 q^{3} -116.582 q^{4} -241.175 q^{5} -224.718 q^{6} -173.086 q^{7} -826.450 q^{8} +2235.73 q^{9} -814.939 q^{10} +682.590 q^{11} +7753.13 q^{12} +168.040 q^{13} -584.863 q^{14} +16039.0 q^{15} +12129.9 q^{16} +5562.09 q^{17} +7554.59 q^{18} -13825.2 q^{19} +28116.8 q^{20} +11510.8 q^{21} +2306.49 q^{22} -24641.2 q^{23} +54961.9 q^{24} -19959.4 q^{25} +567.813 q^{26} -3240.56 q^{27} +20178.7 q^{28} -33649.4 q^{29} +54196.4 q^{30} +22707.6 q^{31} +146773. q^{32} -45394.7 q^{33} +18794.5 q^{34} +41744.1 q^{35} -260646. q^{36} +181778. q^{37} -46715.6 q^{38} -11175.3 q^{39} +199320. q^{40} +335134. q^{41} +38895.5 q^{42} -79507.0 q^{43} -79577.8 q^{44} -539203. q^{45} -83263.2 q^{46} -743538. q^{47} -806683. q^{48} -793584. q^{49} -67443.4 q^{50} -369899. q^{51} -19590.5 q^{52} -12693.7 q^{53} -10949.9 q^{54} -164624. q^{55} +143047. q^{56} +919423. q^{57} -113702. q^{58} -830705. q^{59} -1.86987e6 q^{60} +2.22282e6 q^{61} +76729.7 q^{62} -386973. q^{63} -1.05668e6 q^{64} -40527.2 q^{65} -153390. q^{66} +3.88681e6 q^{67} -648441. q^{68} +1.63873e6 q^{69} +141055. q^{70} +3.88756e6 q^{71} -1.84772e6 q^{72} +5.22595e6 q^{73} +614232. q^{74} +1.32737e6 q^{75} +1.61177e6 q^{76} -118147. q^{77} -37761.6 q^{78} -2.64353e6 q^{79} -2.92544e6 q^{80} -4.67403e6 q^{81} +1.13243e6 q^{82} +6.76661e6 q^{83} -1.34196e6 q^{84} -1.34144e6 q^{85} -268657. q^{86} +2.23781e6 q^{87} -564127. q^{88} +327679. q^{89} -1.82198e6 q^{90} -29085.4 q^{91} +2.87272e6 q^{92} -1.51014e6 q^{93} -2.51244e6 q^{94} +3.33429e6 q^{95} -9.76093e6 q^{96} +5.05740e6 q^{97} -2.68154e6 q^{98} +1.52608e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9}+O(q^{10}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9} + 4667 q^{10} + 1620 q^{11} - 19681 q^{12} + 13550 q^{13} + 44160 q^{14} + 31412 q^{15} + 114026 q^{16} + 110880 q^{17} + 159267 q^{18} + 105058 q^{19} + 167251 q^{20} + 129840 q^{21} + 201504 q^{22} + 160184 q^{23} + 161289 q^{24} + 270149 q^{25} + 272104 q^{26} + 252544 q^{27} + 208172 q^{28} + 285546 q^{29} + 107580 q^{30} - 99616 q^{31} + 200126 q^{32} + 531468 q^{33} - 80941 q^{34} - 187104 q^{35} - 608975 q^{36} + 176038 q^{37} + 652165 q^{38} - 794680 q^{39} - 895387 q^{40} - 410260 q^{41} - 3413218 q^{42} - 1033591 q^{43} - 2177076 q^{44} - 1051178 q^{45} - 3975765 q^{46} - 424556 q^{47} - 2360477 q^{48} - 1561359 q^{49} - 4063801 q^{50} - 2375738 q^{51} - 4172312 q^{52} + 3992458 q^{53} - 10438626 q^{54} + 406960 q^{55} + 1559556 q^{56} - 3116152 q^{57} - 4052005 q^{58} + 2248836 q^{59} - 2911436 q^{60} + 6210394 q^{61} + 885317 q^{62} + 11622368 q^{63} - 3096318 q^{64} + 5600420 q^{65} - 2174604 q^{66} - 1993648 q^{67} + 9327135 q^{68} + 13366240 q^{69} - 1105098 q^{70} + 4978064 q^{71} + 11370663 q^{72} + 8224814 q^{73} - 3613563 q^{74} + 27115592 q^{75} + 10687121 q^{76} + 17261892 q^{77} - 15226630 q^{78} + 6945708 q^{79} + 15822799 q^{80} + 35113185 q^{81} - 508449 q^{82} + 22937328 q^{83} - 14010106 q^{84} - 575532 q^{85} - 1272112 q^{86} + 9081380 q^{87} + 11202656 q^{88} + 9291302 q^{89} + 2841402 q^{90} + 25581108 q^{91} - 14388137 q^{92} + 25930480 q^{93} - 24645805 q^{94} + 30750464 q^{95} - 22461255 q^{96} + 10001852 q^{97} - 32304856 q^{98} + 5055452 q^{99}+O(q^{100}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9 + 4667 * q^10 + 1620 * q^11 - 19681 * q^12 + 13550 * q^13 + 44160 * q^14 + 31412 * q^15 + 114026 * q^16 + 110880 * q^17 + 159267 * q^18 + 105058 * q^19 + 167251 * q^20 + 129840 * q^21 + 201504 * q^22 + 160184 * q^23 + 161289 * q^24 + 270149 * q^25 + 272104 * q^26 + 252544 * q^27 + 208172 * q^28 + 285546 * q^29 + 107580 * q^30 - 99616 * q^31 + 200126 * q^32 + 531468 * q^33 - 80941 * q^34 - 187104 * q^35 - 608975 * q^36 + 176038 * q^37 + 652165 * q^38 - 794680 * q^39 - 895387 * q^40 - 410260 * q^41 - 3413218 * q^42 - 1033591 * q^43 - 2177076 * q^44 - 1051178 * q^45 - 3975765 * q^46 - 424556 * q^47 - 2360477 * q^48 - 1561359 * q^49 - 4063801 * q^50 - 2375738 * q^51 - 4172312 * q^52 + 3992458 * q^53 - 10438626 * q^54 + 406960 * q^55 + 1559556 * q^56 - 3116152 * q^57 - 4052005 * q^58 + 2248836 * q^59 - 2911436 * q^60 + 6210394 * q^61 + 885317 * q^62 + 11622368 * q^63 - 3096318 * q^64 + 5600420 * q^65 - 2174604 * q^66 - 1993648 * q^67 + 9327135 * q^68 + 13366240 * q^69 - 1105098 * q^70 + 4978064 * q^71 + 11370663 * q^72 + 8224814 * q^73 - 3613563 * q^74 + 27115592 * q^75 + 10687121 * q^76 + 17261892 * q^77 - 15226630 * q^78 + 6945708 * q^79 + 15822799 * q^80 + 35113185 * q^81 - 508449 * q^82 + 22937328 * q^83 - 14010106 * q^84 - 575532 * q^85 - 1272112 * q^86 + 9081380 * q^87 + 11202656 * q^88 + 9291302 * q^89 + 2841402 * q^90 + 25581108 * q^91 - 14388137 * q^92 + 25930480 * q^93 - 24645805 * q^94 + 30750464 * q^95 - 22461255 * q^96 + 10001852 * q^97 - 32304856 * q^98 + 5055452 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more