LMFDB - Embedded newform 177.8.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,8,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, 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("177.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	177.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:	$55.2921495107$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 $ x^16 - 6x^15 - 1493x^14 + 8791x^13 + 890490x^12 - 5107725x^11 - 269092298x^10 + 1488374176x^9 + 42885295136x^8 - 226132003872x^7 - 3353576629440x^6 + 16796366777600x^5 + 99470801612800x^4 - 494039551757568x^3 - 493048066650624x^2 + 3193975642099712*x - 2385018853548032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.97136$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.97136 q^{2} -27.0000 q^{3} -124.114 q^{4} -339.775 q^{5} +53.2268 q^{6} +364.700 q^{7} +497.007 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-1.97136 q^{2} -27.0000 q^{3} -124.114 q^{4} -339.775 q^{5} +53.2268 q^{6} +364.700 q^{7} +497.007 q^{8} +729.000 q^{9} +669.820 q^{10} -3588.22 q^{11} +3351.07 q^{12} +857.316 q^{13} -718.956 q^{14} +9173.94 q^{15} +14906.8 q^{16} +3649.35 q^{17} -1437.12 q^{18} -9650.25 q^{19} +42170.8 q^{20} -9846.91 q^{21} +7073.68 q^{22} +116133. q^{23} -13419.2 q^{24} +37322.4 q^{25} -1690.08 q^{26} -19683.0 q^{27} -45264.3 q^{28} +247675. q^{29} -18085.1 q^{30} +114054. q^{31} -93003.6 q^{32} +96882.0 q^{33} -7194.19 q^{34} -123916. q^{35} -90478.9 q^{36} -194589. q^{37} +19024.1 q^{38} -23147.5 q^{39} -168871. q^{40} -594925. q^{41} +19411.8 q^{42} +247788. q^{43} +445348. q^{44} -247696. q^{45} -228939. q^{46} -1.30718e6 q^{47} -402483. q^{48} -690537. q^{49} -73575.9 q^{50} -98532.5 q^{51} -106405. q^{52} +1.20394e6 q^{53} +38802.3 q^{54} +1.21919e6 q^{55} +181259. q^{56} +260557. q^{57} -488257. q^{58} +205379. q^{59} -1.13861e6 q^{60} +428862. q^{61} -224843. q^{62} +265866. q^{63} -1.72472e6 q^{64} -291295. q^{65} -190989. q^{66} -2.01026e6 q^{67} -452934. q^{68} -3.13558e6 q^{69} +244284. q^{70} +1.87858e6 q^{71} +362318. q^{72} +2.35049e6 q^{73} +383605. q^{74} -1.00770e6 q^{75} +1.19773e6 q^{76} -1.30863e6 q^{77} +45632.2 q^{78} -3.74874e6 q^{79} -5.06496e6 q^{80} +531441. q^{81} +1.17281e6 q^{82} +3.21289e6 q^{83} +1.22214e6 q^{84} -1.23996e6 q^{85} -488480. q^{86} -6.68722e6 q^{87} -1.78337e6 q^{88} -544547. q^{89} +488299. q^{90} +312663. q^{91} -1.44136e7 q^{92} -3.07947e6 q^{93} +2.57692e6 q^{94} +3.27892e6 q^{95} +2.51110e6 q^{96} -2.91764e6 q^{97} +1.36130e6 q^{98} -2.61581e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) $ 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9} - 3479 q^{10} + 898 q^{11} - 26298 q^{12} - 8172 q^{13} - 13315 q^{14} + 1836 q^{15} + 3138 q^{16} - 44985 q^{17} - 4374 q^{18} - 40137 q^{19} + 130657 q^{20} + 63261 q^{21} + 109394 q^{22} - 2833 q^{23} - 22113 q^{24} + 285746 q^{25} - 129420 q^{26} - 314928 q^{27} + 112890 q^{28} + 144375 q^{29} + 93933 q^{30} - 141759 q^{31} - 36224 q^{32} - 24246 q^{33} - 341332 q^{34} - 78859 q^{35} + 710046 q^{36} - 297971 q^{37} + 329075 q^{38} + 220644 q^{39} - 203048 q^{40} + 659077 q^{41} + 359505 q^{42} - 1431608 q^{43} + 254916 q^{44} - 49572 q^{45} + 873113 q^{46} - 1574073 q^{47} - 84726 q^{48} + 1893545 q^{49} + 302533 q^{50} + 1214595 q^{51} - 4972548 q^{52} + 587736 q^{53} + 118098 q^{54} - 4624036 q^{55} - 5798506 q^{56} + 1083699 q^{57} - 6991380 q^{58} + 3286064 q^{59} - 3527739 q^{60} - 6117131 q^{61} - 11570258 q^{62} - 1708047 q^{63} - 19063011 q^{64} - 5335514 q^{65} - 2953638 q^{66} - 16518710 q^{67} - 17284669 q^{68} + 76491 q^{69} - 39189486 q^{70} - 10882582 q^{71} + 597051 q^{72} - 21097441 q^{73} - 16717030 q^{74} - 7715142 q^{75} - 40864952 q^{76} - 3404601 q^{77} + 3494340 q^{78} - 3784458 q^{79} - 27466195 q^{80} + 8503056 q^{81} - 24990117 q^{82} - 1951425 q^{83} - 3048030 q^{84} - 23238675 q^{85} - 35910572 q^{86} - 3898125 q^{87} - 27843055 q^{88} + 10499443 q^{89} - 2536191 q^{90} + 699217 q^{91} - 20062766 q^{92} + 3827493 q^{93} - 59358988 q^{94} - 29236333 q^{95} + 978048 q^{96} - 25158976 q^{97} + 2120460 q^{98} + 654642 q^{99}+O(q^{100}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9 - 3479 * q^10 + 898 * q^11 - 26298 * q^12 - 8172 * q^13 - 13315 * q^14 + 1836 * q^15 + 3138 * q^16 - 44985 * q^17 - 4374 * q^18 - 40137 * q^19 + 130657 * q^20 + 63261 * q^21 + 109394 * q^22 - 2833 * q^23 - 22113 * q^24 + 285746 * q^25 - 129420 * q^26 - 314928 * q^27 + 112890 * q^28 + 144375 * q^29 + 93933 * q^30 - 141759 * q^31 - 36224 * q^32 - 24246 * q^33 - 341332 * q^34 - 78859 * q^35 + 710046 * q^36 - 297971 * q^37 + 329075 * q^38 + 220644 * q^39 - 203048 * q^40 + 659077 * q^41 + 359505 * q^42 - 1431608 * q^43 + 254916 * q^44 - 49572 * q^45 + 873113 * q^46 - 1574073 * q^47 - 84726 * q^48 + 1893545 * q^49 + 302533 * q^50 + 1214595 * q^51 - 4972548 * q^52 + 587736 * q^53 + 118098 * q^54 - 4624036 * q^55 - 5798506 * q^56 + 1083699 * q^57 - 6991380 * q^58 + 3286064 * q^59 - 3527739 * q^60 - 6117131 * q^61 - 11570258 * q^62 - 1708047 * q^63 - 19063011 * q^64 - 5335514 * q^65 - 2953638 * q^66 - 16518710 * q^67 - 17284669 * q^68 + 76491 * q^69 - 39189486 * q^70 - 10882582 * q^71 + 597051 * q^72 - 21097441 * q^73 - 16717030 * q^74 - 7715142 * q^75 - 40864952 * q^76 - 3404601 * q^77 + 3494340 * q^78 - 3784458 * q^79 - 27466195 * q^80 + 8503056 * q^81 - 24990117 * q^82 - 1951425 * q^83 - 3048030 * q^84 - 23238675 * q^85 - 35910572 * q^86 - 3898125 * q^87 - 27843055 * q^88 + 10499443 * q^89 - 2536191 * q^90 + 699217 * q^91 - 20062766 * q^92 + 3827493 * q^93 - 59358988 * q^94 - 29236333 * q^95 + 978048 * q^96 - 25158976 * q^97 + 2120460 * q^98 + 654642 * 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 $

$2$	−1.97136	−0.174245	−0.0871227	−	0.996198i	$-0.527767\pi$
$2$	−1.97136	−0.174245	−0.0871227	+	0.996198i	$0.527767\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−124.114	−0.969639
$5$	−339.775	−1.21562	−0.607809	−	0.794083i	$-0.707951\pi$
$5$	−339.775	−1.21562	−0.607809	+	0.794083i	$0.707951\pi$
$6$	53.2268	0.100601
$7$	364.700	0.401877	0.200938	−	0.979604i	$-0.435601\pi$
$7$	364.700	0.401877	0.200938	+	0.979604i	$0.435601\pi$
$8$	497.007	0.343200
$9$	729.000	0.333333
$10$	669.820	0.211816
$11$	−3588.22	−0.812840	−0.406420	−	0.913686i	$-0.633223\pi$
$11$	−3588.22	−0.812840	−0.406420	+	0.913686i	$0.633223\pi$
$12$	3351.07	0.559821
$13$	857.316	0.108228	0.0541140	−	0.998535i	$-0.482767\pi$
$13$	857.316	0.108228	0.0541140	+	0.998535i	$0.482767\pi$
$14$	−718.956	−0.0700252
$15$	9173.94	0.701837
$16$	14906.8	0.909837
$17$	3649.35	0.180154	0.0900771	−	0.995935i	$-0.471289\pi$
$17$	3649.35	0.180154	0.0900771	+	0.995935i	$0.471289\pi$
$18$	−1437.12	−0.0580818
$19$	−9650.25	−0.322776	−0.161388	−	0.986891i	$-0.551597\pi$
$19$	−9650.25	−0.322776	−0.161388	+	0.986891i	$0.551597\pi$
$20$	42170.8	1.17871
$21$	−9846.91	−0.232024
$22$	7073.68	0.141634
$23$	116133.	1.99024	0.995122	−	0.0986484i	$-0.0314519\pi$
$23$	116133.	1.99024	0.995122	+	0.0986484i	$0.0314519\pi$
$24$	−13419.2	−0.198147
$25$	37322.4	0.477727
$26$	−1690.08	−0.0188582
$27$	−19683.0	−0.192450
$28$	−45264.3	−0.389675
$29$	247675.	1.88577	0.942886	−	0.333116i	$-0.108100\pi$
$29$	247675.	1.88577	0.942886	+	0.333116i	$0.108100\pi$
$30$	−18085.1	−0.122292
$31$	114054.	0.687617	0.343808	−	0.939040i	$-0.388283\pi$
$31$	114054.	0.687617	0.343808	+	0.939040i	$0.388283\pi$
$32$	−93003.6	−0.501735
$33$	96882.0	0.469293
$34$	−7194.19	−0.0313910
$35$	−123916.	−0.488529
$36$	−90478.9	−0.323213
$37$	−194589.	−0.631556	−0.315778	−	0.948833i	$-0.602266\pi$
$37$	−194589.	−0.631556	−0.315778	+	0.948833i	$0.602266\pi$
$38$	19024.1	0.0562422
$39$	−23147.5	−0.0624854
$40$	−168871.	−0.417201
$41$	−594925.	−1.34809	−0.674045	−	0.738691i	$-0.735444\pi$
$41$	−594925.	−1.34809	−0.674045	+	0.738691i	$0.735444\pi$
$42$	19411.8	0.0404290
$43$	247788.	0.475270	0.237635	−	0.971354i	$-0.423628\pi$
$43$	247788.	0.475270	0.237635	+	0.971354i	$0.423628\pi$
$44$	445348.	0.788161
$45$	−247696.	−0.405206
$46$	−228939.	−0.346791
$47$	−1.30718e6	−1.83651	−0.918254	−	0.395993i	$-0.870400\pi$
$47$	−1.30718e6	−1.83651	−0.918254	+	0.395993i	$0.870400\pi$
$48$	−402483.	−0.525295
$49$	−690537.	−0.838495
$50$	−73575.9	−0.0832416
$51$	−98532.5	−0.104012
$52$	−106405.	−0.104942
$53$	1.20394e6	1.11081	0.555403	−	0.831581i	$-0.312564\pi$
$53$	1.20394e6	1.11081	0.555403	+	0.831581i	$0.312564\pi$
$54$	38802.3	0.0335335
$55$	1.21919e6	0.988102
$56$	181259.	0.137924
$57$	260557.	0.186355
$58$	−488257.	−0.328587
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	−1.13861e6	−0.680528
$61$	428862.	0.241915	0.120958	−	0.992658i	$-0.461404\pi$
$61$	428862.	0.241915	0.120958	+	0.992658i	$0.461404\pi$
$62$	−224843.	−0.119814
$63$	265866.	0.133959
$64$	−1.72472e6	−0.822412
$65$	−291295.	−0.131564
$66$	−190989.	−0.0817722
$67$	−2.01026e6	−0.816565	−0.408283	−	0.912856i	$-0.633872\pi$
$67$	−2.01026e6	−0.816565	−0.408283	+	0.912856i	$0.633872\pi$
$68$	−452934.	−0.174684
$69$	−3.13558e6	−1.14907
$70$	244284.	0.0851238
$71$	1.87858e6	0.622909	0.311454	−	0.950261i	$-0.399184\pi$
$71$	1.87858e6	0.622909	0.311454	+	0.950261i	$0.399184\pi$
$72$	362318.	0.114400
$73$	2.35049e6	0.707178	0.353589	−	0.935401i	$-0.384961\pi$
$73$	2.35049e6	0.707178	0.353589	+	0.935401i	$0.384961\pi$
$74$	383605.	0.110046
$75$	−1.00770e6	−0.275816
$76$	1.19773e6	0.312976
$77$	−1.30863e6	−0.326661
$78$	45632.2	0.0108878
$79$	−3.74874e6	−0.855442	−0.427721	−	0.903911i	$-0.640683\pi$
$79$	−3.74874e6	−0.855442	−0.427721	+	0.903911i	$0.640683\pi$
$80$	−5.06496e6	−1.10601
$81$	531441.	0.111111
$82$	1.17281e6	0.234898
$83$	3.21289e6	0.616768	0.308384	−	0.951262i	$-0.400212\pi$
$83$	3.21289e6	0.616768	0.308384	+	0.951262i	$0.400212\pi$
$84$	1.22214e6	0.224979
$85$	−1.23996e6	−0.218999
$86$	−488480.	−0.0828137
$87$	−6.68722e6	−1.08875
$88$	−1.78337e6	−0.278967
$89$	−544547.	−0.0818786	−0.0409393	−	0.999162i	$-0.513035\pi$
$89$	−544547.	−0.0818786	−0.0409393	+	0.999162i	$0.513035\pi$
$90$	488299.	0.0706053
$91$	312663.	0.0434943
$92$	−1.44136e7	−1.92982
$93$	−3.07947e6	−0.396996
$94$	2.57692e6	0.320003
$95$	3.27892e6	0.392372
$96$	2.51110e6	0.289677
$97$	−2.91764e6	−0.324586	−0.162293	−	0.986743i	$-0.551889\pi$
$97$	−2.91764e6	−0.324586	−0.162293	+	0.986743i	$0.551889\pi$
$98$	1.36130e6	0.146104
$99$	−2.61581e6	−0.270947
$100$	−4.63222e6	−0.463222
$101$	−1.99790e7	−1.92952	−0.964761	−	0.263129i	$-0.915246\pi$
$101$	−1.99790e7	−1.92952	−0.964761	+	0.263129i	$0.915246\pi$
$102$	194243.	0.0181236
$103$	−1.14245e7	−1.03016	−0.515082	−	0.857141i	$-0.672239\pi$
$103$	−1.14245e7	−1.03016	−0.515082	+	0.857141i	$0.672239\pi$
$104$	426092.	0.0371439
$105$	3.34574e6	0.282052
$106$	−2.37340e6	−0.193553
$107$	2.29986e7	1.81492	0.907460	−	0.420138i	$-0.138018\pi$
$107$	2.29986e7	1.81492	0.907460	+	0.420138i	$0.138018\pi$
$108$	2.44293e6	0.186607
$109$	2.02074e7	1.49457	0.747287	−	0.664502i	$-0.231356\pi$
$109$	2.02074e7	1.49457	0.747287	+	0.664502i	$0.231356\pi$
$110$	−2.40346e6	−0.172172
$111$	5.25390e6	0.364629
$112$	5.43650e6	0.365643
$113$	−1.26444e7	−0.824374	−0.412187	−	0.911099i	$-0.635235\pi$
$113$	−1.26444e7	−0.824374	−0.412187	+	0.911099i	$0.635235\pi$
$114$	−513651.	−0.0324714
$115$	−3.94590e7	−2.41938
$116$	−3.07399e7	−1.82852
$117$	624983.	0.0360760
$118$	−404876.	−0.0226848
$119$	1.33092e6	0.0723998
$120$	4.55951e6	0.240871
$121$	−6.61184e6	−0.339292
$122$	−845441.	−0.0421526
$123$	1.60630e7	0.778320
$124$	−1.41557e7	−0.666740
$125$	1.38637e7	0.634885
$126$	−524119.	−0.0233417
$127$	3.20643e7	1.38902	0.694509	−	0.719484i	$-0.255621\pi$
$127$	3.20643e7	1.38902	0.694509	+	0.719484i	$0.255621\pi$
$128$	1.53045e7	0.645037
$129$	−6.69028e6	−0.274398
$130$	574248.	0.0229244
$131$	−2.38103e7	−0.925369	−0.462684	−	0.886523i	$-0.653114\pi$
$131$	−2.38103e7	−0.925369	−0.462684	+	0.886523i	$0.653114\pi$
$132$	−1.20244e7	−0.455045
$133$	−3.51945e6	−0.129716
$134$	3.96295e6	0.142283
$135$	6.68780e6	0.233946
$136$	1.81375e6	0.0618290
$137$	2.13799e7	0.710370	0.355185	−	0.934796i	$-0.384418\pi$
$137$	2.13799e7	0.710370	0.355185	+	0.934796i	$0.384418\pi$
$138$	6.18136e6	0.200220
$139$	−3.64257e7	−1.15042	−0.575209	−	0.818006i	$-0.695079\pi$
$139$	−3.64257e7	−1.15042	−0.575209	+	0.818006i	$0.695079\pi$
$140$	1.53797e7	0.473696
$141$	3.52938e7	1.06031
$142$	−3.70335e6	−0.108539
$143$	−3.07624e6	−0.0879719
$144$	1.08670e7	0.303279
$145$	−8.41539e7	−2.29238
$146$	−4.63367e6	−0.123222
$147$	1.86445e7	0.484105
$148$	2.41512e7	0.612381
$149$	95229.9	0.00235842	0.00117921	−	0.999999i	$-0.499625\pi$
$149$	95229.9	0.00235842	0.00117921	+	0.999999i	$0.499625\pi$
$150$	1.98655e6	0.0480596
$151$	4.71653e7	1.11482	0.557408	−	0.830238i	$-0.311796\pi$
$151$	4.71653e7	1.11482	0.557408	+	0.830238i	$0.311796\pi$
$152$	−4.79624e6	−0.110777
$153$	2.66038e6	0.0600514
$154$	2.57977e6	0.0569192
$155$	−3.87529e7	−0.835879
$156$	2.87293e6	0.0605883
$157$	4.44571e7	0.916838	0.458419	−	0.888736i	$-0.348416\pi$
$157$	4.44571e7	0.916838	0.458419	+	0.888736i	$0.348416\pi$
$158$	7.39012e6	0.149057
$159$	−3.25063e7	−0.641324
$160$	3.16003e7	0.609918
$161$	4.23536e7	0.799833
$162$	−1.04766e6	−0.0193606
$163$	−9.85994e7	−1.78327	−0.891636	−	0.452753i	$-0.850442\pi$
$163$	−9.85994e7	−1.78327	−0.891636	+	0.452753i	$0.850442\pi$
$164$	7.38384e7	1.30716
$165$	−3.29181e7	−0.570481
$166$	−6.33376e6	−0.107469
$167$	1.60639e7	0.266897	0.133449	−	0.991056i	$-0.457395\pi$
$167$	1.60639e7	0.266897	0.133449	+	0.991056i	$0.457395\pi$
$168$	−4.89398e6	−0.0796306
$169$	−6.20135e7	−0.988287
$170$	2.44441e6	0.0381595
$171$	−7.03503e6	−0.107592
$172$	−3.07539e7	−0.460841
$173$	9.63596e7	1.41493	0.707463	−	0.706751i	$-0.249840\pi$
$173$	9.63596e7	1.41493	0.707463	+	0.706751i	$0.249840\pi$
$174$	1.31829e7	0.189710
$175$	1.36115e7	0.191987
$176$	−5.34888e7	−0.739552
$177$	−5.54523e6	−0.0751646
$178$	1.07350e6	0.0142670
$179$	6.96623e7	0.907846	0.453923	−	0.891041i	$-0.350024\pi$
$179$	6.96623e7	0.907846	0.453923	+	0.891041i	$0.350024\pi$
$180$	3.07425e7	0.392903
$181$	−1.04761e8	−1.31319	−0.656593	−	0.754245i	$-0.728003\pi$
$181$	−1.04761e8	−1.31319	−0.656593	+	0.754245i	$0.728003\pi$
$182$	−616372.	−0.00757868
$183$	−1.15793e7	−0.139670
$184$	5.77187e7	0.683053
$185$	6.61165e7	0.767731
$186$	6.07075e6	0.0691747
$187$	−1.30947e7	−0.146436
$188$	1.62239e8	1.78075
$189$	−7.17839e6	−0.0773412
$190$	−6.46393e6	−0.0683690
$191$	3.38716e7	0.351738	0.175869	−	0.984414i	$-0.443727\pi$
$191$	3.38716e7	0.351738	0.175869	+	0.984414i	$0.443727\pi$
$192$	4.65675e7	0.474820
$193$	−5.74536e7	−0.575264	−0.287632	−	0.957741i	$-0.592868\pi$
$193$	−5.74536e7	−0.575264	−0.287632	+	0.957741i	$0.592868\pi$
$194$	5.75172e6	0.0565576
$195$	7.86496e6	0.0759584
$196$	8.57051e7	0.813037
$197$	7.82632e7	0.729333	0.364667	−	0.931138i	$-0.381183\pi$
$197$	7.82632e7	0.729333	0.364667	+	0.931138i	$0.381183\pi$
$198$	5.15671e6	0.0472112
$199$	−1.18386e8	−1.06491	−0.532455	−	0.846458i	$-0.678731\pi$
$199$	−1.18386e8	−1.06491	−0.532455	+	0.846458i	$0.678731\pi$
$200$	1.85495e7	0.163956
$201$	5.42771e7	0.471444
$202$	3.93859e7	0.336210
$203$	9.03271e7	0.757848
$204$	1.22292e7	0.100854
$205$	2.02141e8	1.63876
$206$	2.25218e7	0.179501
$207$	8.46606e7	0.663415
$208$	1.27798e7	0.0984698
$209$	3.46272e7	0.262365
$210$	−6.59566e6	−0.0491463
$211$	4.78501e7	0.350666	0.175333	−	0.984509i	$-0.443900\pi$
$211$	4.78501e7	0.350666	0.175333	+	0.984509i	$0.443900\pi$
$212$	−1.49425e8	−1.07708
$213$	−5.07215e7	−0.359637
$214$	−4.53385e7	−0.316241
$215$	−8.41923e7	−0.577747
$216$	−9.78259e6	−0.0660489
$217$	4.15957e7	0.276337
$218$	−3.98361e7	−0.260423
$219$	−6.34632e7	−0.408289
$220$	−1.51318e8	−0.958102
$221$	3.12865e6	0.0194977
$222$	−1.03573e7	−0.0635349
$223$	1.24666e8	0.752800	0.376400	−	0.926457i	$-0.377162\pi$
$223$	1.24666e8	0.752800	0.376400	+	0.926457i	$0.377162\pi$
$224$	−3.39184e7	−0.201636
$225$	2.72080e7	0.159242
$226$	2.49267e7	0.143643
$227$	7.13606e7	0.404919	0.202460	−	0.979291i	$-0.435107\pi$
$227$	7.13606e7	0.404919	0.202460	+	0.979291i	$0.435107\pi$
$228$	−3.23387e7	−0.180697
$229$	−3.39682e8	−1.86917	−0.934584	−	0.355743i	$-0.884228\pi$
$229$	−3.39682e8	−1.86917	−0.934584	+	0.355743i	$0.884228\pi$
$230$	7.77879e7	0.421565
$231$	3.53329e7	0.188598
$232$	1.23096e8	0.647198
$233$	−3.19085e8	−1.65257	−0.826286	−	0.563250i	$-0.809551\pi$
$233$	−3.19085e8	−1.65257	−0.826286	+	0.563250i	$0.809551\pi$
$234$	−1.23207e6	−0.00628607
$235$	4.44147e8	2.23249
$236$	−2.54904e7	−0.126236
$237$	1.01216e8	0.493890
$238$	−2.62372e6	−0.0126153
$239$	2.48784e8	1.17877	0.589387	−	0.807851i	$-0.299369\pi$
$239$	2.48784e8	1.17877	0.589387	+	0.807851i	$0.299369\pi$
$240$	1.36754e8	0.638558
$241$	−1.02085e8	−0.469788	−0.234894	−	0.972021i	$-0.575474\pi$
$241$	−1.02085e8	−0.469788	−0.234894	+	0.972021i	$0.575474\pi$
$242$	1.30343e7	0.0591200
$243$	−1.43489e7	−0.0641500
$244$	−5.32276e7	−0.234570
$245$	2.34627e8	1.01929
$246$	−3.16659e7	−0.135619
$247$	−8.27331e6	−0.0349333
$248$	5.66859e7	0.235990
$249$	−8.67479e7	−0.356091
$250$	−2.73304e7	−0.110626
$251$	1.69390e8	0.676131	0.338066	−	0.941123i	$-0.390227\pi$
$251$	1.69390e8	0.676131	0.338066	+	0.941123i	$0.390227\pi$
$252$	−3.29977e7	−0.129892
$253$	−4.16709e8	−1.61775
$254$	−6.32102e7	−0.242030
$255$	3.34789e7	0.126439
$256$	1.90594e8	0.710018
$257$	−4.15160e8	−1.52563	−0.762815	−	0.646617i	$-0.776183\pi$
$257$	−4.15160e8	−1.52563	−0.762815	+	0.646617i	$0.776183\pi$
$258$	1.31890e7	0.0478125
$259$	−7.09666e7	−0.253808
$260$	3.61537e7	0.127569
$261$	1.80555e8	0.628591
$262$	4.69387e7	0.161241
$263$	−2.46530e8	−0.835650	−0.417825	−	0.908527i	$-0.637208\pi$
$263$	−2.46530e8	−0.835650	−0.417825	+	0.908527i	$0.637208\pi$
$264$	4.81511e7	0.161062
$265$	−4.09068e8	−1.35032
$266$	6.93810e6	0.0226024
$267$	1.47028e7	0.0472726
$268$	2.49501e8	0.791773
$269$	3.18928e8	0.998987	0.499494	−	0.866318i	$-0.333519\pi$
$269$	3.18928e8	0.998987	0.499494	+	0.866318i	$0.333519\pi$
$270$	−1.31841e7	−0.0407640
$271$	−3.87904e8	−1.18394	−0.591972	−	0.805958i	$-0.701651\pi$
$271$	−3.87904e8	−1.18394	−0.591972	+	0.805958i	$0.701651\pi$
$272$	5.44000e7	0.163911
$273$	−8.44191e6	−0.0251114
$274$	−4.21476e7	−0.123779
$275$	−1.33921e8	−0.388315
$276$	3.89168e8	1.11418
$277$	−7.42290e7	−0.209843	−0.104921	−	0.994481i	$-0.533459\pi$
$277$	−7.42290e7	−0.209843	−0.104921	+	0.994481i	$0.533459\pi$
$278$	7.18082e7	0.200455
$279$	8.31457e7	0.229206
$280$	−6.15873e7	−0.167663
$281$	−6.16298e8	−1.65698	−0.828492	−	0.560001i	$-0.810801\pi$
$281$	−6.16298e8	−1.65698	−0.828492	+	0.560001i	$0.810801\pi$
$282$	−6.95769e7	−0.184754
$283$	7.96612e7	0.208927	0.104463	−	0.994529i	$-0.466687\pi$
$283$	7.96612e7	0.208927	0.104463	+	0.994529i	$0.466687\pi$
$284$	−2.33157e8	−0.603996
$285$	−8.85308e7	−0.226536
$286$	6.06438e6	0.0153287
$287$	−2.16969e8	−0.541766
$288$	−6.77996e7	−0.167245
$289$	−3.97021e8	−0.967544
$290$	1.65898e8	0.399436
$291$	7.87762e7	0.187400
$292$	−2.91728e8	−0.685707
$293$	2.50390e8	0.581542	0.290771	−	0.956793i	$-0.406088\pi$
$293$	2.50390e8	0.581542	0.290771	+	0.956793i	$0.406088\pi$
$294$	−3.67550e7	−0.0843531
$295$	−6.97828e7	−0.158260
$296$	−9.67121e7	−0.216750
$297$	7.06270e7	0.156431
$298$	−187732.	−0.000410944 0
$299$	9.95623e7	0.215400
$300$	1.25070e8	0.267441
$301$	9.03684e7	0.191000
$302$	−9.29799e7	−0.194252
$303$	5.39434e8	1.11401
$304$	−1.43854e8	−0.293673
$305$	−1.45717e8	−0.294076
$306$	−5.24456e6	−0.0104637
$307$	−3.31396e8	−0.653677	−0.326839	−	0.945080i	$-0.605983\pi$
$307$	−3.31396e8	−0.653677	−0.326839	+	0.945080i	$0.605983\pi$
$308$	1.62418e8	0.316743
$309$	3.08461e8	0.594766
$310$	7.63960e7	0.145648
$311$	−9.41919e7	−0.177563	−0.0887815	−	0.996051i	$-0.528297\pi$
$311$	−9.41919e7	−0.177563	−0.0887815	+	0.996051i	$0.528297\pi$
$312$	−1.15045e7	−0.0214450
$313$	−5.39491e8	−0.994441	−0.497220	−	0.867624i	$-0.665646\pi$
$313$	−5.39491e8	−0.994441	−0.497220	+	0.867624i	$0.665646\pi$
$314$	−8.76411e7	−0.159755
$315$	−9.03349e7	−0.162843
$316$	4.65270e8	0.829470
$317$	6.91838e8	1.21982	0.609912	−	0.792469i	$-0.291205\pi$
$317$	6.91838e8	1.21982	0.609912	+	0.792469i	$0.291205\pi$
$318$	6.40817e7	0.111748
$319$	−8.88713e8	−1.53283
$320$	5.86019e8	0.999739
$321$	−6.20961e8	−1.04784
$322$	−8.34942e7	−0.139367
$323$	−3.52171e7	−0.0581494
$324$	−6.59591e7	−0.107738
$325$	3.19971e7	0.0517033
$326$	1.94375e8	0.310727
$327$	−5.45599e8	−0.862893
$328$	−2.95682e8	−0.462665
$329$	−4.76729e8	−0.738050
$330$	6.48935e7	0.0994037
$331$	−8.54742e8	−1.29550	−0.647750	−	0.761853i	$-0.724290\pi$
$331$	−8.54742e8	−1.29550	−0.647750	+	0.761853i	$0.724290\pi$
$332$	−3.98763e8	−0.598042
$333$	−1.41855e8	−0.210519
$334$	−3.16678e7	−0.0465056
$335$	6.83038e8	0.992631
$336$	−1.46786e8	−0.211104
$337$	1.27131e8	0.180945	0.0904723	−	0.995899i	$-0.471162\pi$
$337$	1.27131e8	0.180945	0.0904723	+	0.995899i	$0.471162\pi$
$338$	1.22251e8	0.172204
$339$	3.41399e8	0.475952
$340$	1.53896e8	0.212349
$341$	−4.09253e8	−0.558922
$342$	1.38686e7	0.0187474
$343$	−5.52185e8	−0.738848
$344$	1.23152e8	0.163113
$345$	1.06539e9	1.39683
$346$	−1.89960e8	−0.246544
$347$	−6.17139e7	−0.0792920	−0.0396460	−	0.999214i	$-0.512623\pi$
$347$	−6.17139e7	−0.0792920	−0.0396460	+	0.999214i	$0.512623\pi$
$348$	8.29976e8	1.05569
$349$	3.92987e8	0.494868	0.247434	−	0.968905i	$-0.420413\pi$
$349$	3.92987e8	0.494868	0.247434	+	0.968905i	$0.420413\pi$
$350$	−2.68332e7	−0.0334529
$351$	−1.68746e7	−0.0208285
$352$	3.33717e8	0.407830
$353$	−1.45872e9	−1.76506	−0.882531	−	0.470254i	$-0.844162\pi$
$353$	−1.45872e9	−1.76506	−0.882531	+	0.470254i	$0.844162\pi$
$354$	1.09317e7	0.0130971
$355$	−6.38294e8	−0.757219
$356$	6.75857e7	0.0793926
$357$	−3.59348e7	−0.0418000
$358$	−1.37330e8	−0.158188
$359$	−1.00957e9	−1.15162	−0.575808	−	0.817585i	$-0.695313\pi$
$359$	−1.00957e9	−1.15162	−0.575808	+	0.817585i	$0.695313\pi$
$360$	−1.23107e8	−0.139067
$361$	−8.00744e8	−0.895816
$362$	2.06523e8	0.228817
$363$	1.78520e8	0.195890
$364$	−3.88058e7	−0.0421737
$365$	−7.98639e8	−0.859658
$366$	2.28269e7	0.0243368
$367$	6.89693e8	0.728324	0.364162	−	0.931336i	$-0.381355\pi$
$367$	6.89693e8	0.728324	0.364162	+	0.931336i	$0.381355\pi$
$368$	1.73116e9	1.81080
$369$	−4.33700e8	−0.449363
$370$	−1.30340e8	−0.133774
$371$	4.39076e8	0.446407
$372$	3.82205e8	0.384942
$373$	1.52953e7	0.0152608	0.00763041	−	0.999971i	$-0.497571\pi$
$373$	1.52953e7	0.0152608	0.00763041	+	0.999971i	$0.497571\pi$
$374$	2.58143e7	0.0255159
$375$	−3.74321e8	−0.366551
$376$	−6.49678e8	−0.630290
$377$	2.12336e8	0.204093
$378$	1.41512e7	0.0134763
$379$	−1.31066e9	−1.23667	−0.618335	−	0.785915i	$-0.712192\pi$
$379$	−1.31066e9	−1.23667	−0.618335	+	0.785915i	$0.712192\pi$
$380$	−4.06959e8	−0.380459
$381$	−8.65735e8	−0.801950
$382$	−6.67731e7	−0.0612886
$383$	6.67359e8	0.606966	0.303483	−	0.952837i	$-0.401851\pi$
$383$	6.67359e8	0.606966	0.303483	+	0.952837i	$0.401851\pi$
$384$	−4.13222e8	−0.372412
$385$	4.44639e8	0.397095
$386$	1.13262e8	0.100237
$387$	1.80638e8	0.158423
$388$	3.62119e8	0.314731
$389$	2.25715e9	1.94418	0.972089	−	0.234611i	$-0.0753816\pi$
$389$	2.25715e9	1.94418	0.972089	+	0.234611i	$0.0753816\pi$
$390$	−1.55047e7	−0.0132354
$391$	4.23808e8	0.358551
$392$	−3.43202e8	−0.287772
$393$	6.42877e8	0.534262
$394$	−1.54285e8	−0.127083
$395$	1.27373e9	1.03989
$396$	3.24658e8	0.262720
$397$	−9.81086e8	−0.786938	−0.393469	−	0.919338i	$-0.628725\pi$
$397$	−9.81086e8	−0.786938	−0.393469	+	0.919338i	$0.628725\pi$
$398$	2.33381e8	0.185556
$399$	9.50251e7	0.0748916
$400$	5.56357e8	0.434654
$401$	−5.90695e7	−0.0457465	−0.0228732	−	0.999738i	$-0.507281\pi$
$401$	−5.90695e7	−0.0457465	−0.0228732	+	0.999738i	$0.507281\pi$
$402$	−1.07000e8	−0.0821470
$403$	9.77807e7	0.0744193
$404$	2.47967e9	1.87094
$405$	−1.80571e8	−0.135069
$406$	−1.78067e8	−0.132051
$407$	6.98228e8	0.513354
$408$	−4.89713e7	−0.0356970
$409$	−7.74847e8	−0.559995	−0.279998	−	0.960001i	$-0.590334\pi$
$409$	−7.74847e8	−0.559995	−0.279998	+	0.960001i	$0.590334\pi$
$410$	−3.98493e8	−0.285547
$411$	−5.77258e8	−0.410132
$412$	1.41794e9	0.998887
$413$	7.49018e7	0.0523199
$414$	−1.66897e8	−0.115597
$415$	−1.09166e9	−0.749754
$416$	−7.97335e7	−0.0543018
$417$	9.83493e8	0.664194
$418$	−6.82628e7	−0.0457159
$419$	2.71966e8	0.180620	0.0903098	−	0.995914i	$-0.471214\pi$
$419$	2.71966e8	0.180620	0.0903098	+	0.995914i	$0.471214\pi$
$420$	−4.15252e8	−0.273489
$421$	−8.59819e8	−0.561591	−0.280795	−	0.959768i	$-0.590598\pi$
$421$	−8.59819e8	−0.561591	−0.280795	+	0.959768i	$0.590598\pi$
$422$	−9.43298e7	−0.0611020
$423$	−9.52934e8	−0.612169
$424$	5.98366e8	0.381229
$425$	1.36202e8	0.0860644
$426$	9.99905e7	0.0626650
$427$	1.56406e8	0.0972200
$428$	−2.85444e9	−1.75982
$429$	8.30585e7	0.0507906
$430$	1.65973e8	0.100670
$431$	−6.62428e8	−0.398536	−0.199268	−	0.979945i	$-0.563856\pi$
$431$	−6.62428e8	−0.398536	−0.199268	+	0.979945i	$0.563856\pi$
$432$	−2.93410e8	−0.175098
$433$	2.01470e9	1.19262	0.596312	−	0.802753i	$-0.296632\pi$
$433$	2.01470e9	1.19262	0.596312	+	0.802753i	$0.296632\pi$
$434$	−8.20001e7	−0.0481505
$435$	2.27215e9	1.32350
$436$	−2.50801e9	−1.44920
$437$	−1.12071e9	−0.642403
$438$	1.25109e8	0.0711425
$439$	2.41587e9	1.36285	0.681425	−	0.731888i	$-0.261360\pi$
$439$	2.41587e9	1.36285	0.681425	+	0.731888i	$0.261360\pi$
$440$	6.05946e8	0.339117
$441$	−5.03401e8	−0.279498
$442$	−6.16769e6	−0.00339739
$443$	−2.74104e9	−1.49796	−0.748982	−	0.662590i	$-0.769457\pi$
$443$	−2.74104e9	−1.49796	−0.748982	+	0.662590i	$0.769457\pi$
$444$	−6.52081e8	−0.353558
$445$	1.85024e8	0.0995330
$446$	−2.45761e8	−0.131172
$447$	−2.57121e6	−0.00136163
$448$	−6.29007e8	−0.330508
$449$	−1.37537e9	−0.717063	−0.358532	−	0.933518i	$-0.616723\pi$
$449$	−1.37537e9	−0.717063	−0.358532	+	0.933518i	$0.616723\pi$
$450$	−5.36368e7	−0.0277472
$451$	2.13472e9	1.09578
$452$	1.56935e9	0.799344
$453$	−1.27346e9	−0.643640
$454$	−1.40678e8	−0.0705553
$455$	−1.06235e8	−0.0528724
$456$	1.29499e8	0.0639570
$457$	1.14427e9	0.560816	0.280408	−	0.959881i	$-0.409530\pi$
$457$	1.14427e9	0.560816	0.280408	+	0.959881i	$0.409530\pi$
$458$	6.69636e8	0.325694
$459$	−7.18302e7	−0.0346707
$460$	4.89740e9	2.34592
$461$	7.69708e8	0.365909	0.182954	−	0.983121i	$-0.441434\pi$
$461$	7.69708e8	0.365909	0.182954	+	0.983121i	$0.441434\pi$
$462$	−6.96539e7	−0.0328623
$463$	−2.23996e8	−0.104884	−0.0524418	−	0.998624i	$-0.516700\pi$
$463$	−2.23996e8	−0.104884	−0.0524418	+	0.998624i	$0.516700\pi$
$464$	3.69204e9	1.71575
$465$	1.04633e9	0.482595
$466$	6.29032e8	0.287953
$467$	3.61971e8	0.164462	0.0822308	−	0.996613i	$-0.473796\pi$
$467$	3.61971e8	0.164462	0.0822308	+	0.996613i	$0.473796\pi$
$468$	−7.75690e7	−0.0349806
$469$	−7.33143e8	−0.328159
$470$	−8.75575e8	−0.389001
$471$	−1.20034e9	−0.529337
$472$	1.02075e8	0.0446809
$473$	−8.89119e8	−0.386319
$474$	−1.99533e8	−0.0860580
$475$	−3.60170e8	−0.154199
$476$	−1.65185e8	−0.0702016
$477$	8.77670e8	0.370269
$478$	−4.90444e8	−0.205396
$479$	−1.79221e9	−0.745100	−0.372550	−	0.928012i	$-0.621516\pi$
$479$	−1.79221e9	−0.745100	−0.372550	+	0.928012i	$0.621516\pi$
$480$	−8.53209e8	−0.352137
$481$	−1.66824e8	−0.0683520
$482$	2.01246e8	0.0818583
$483$	−1.14355e9	−0.461784
$484$	8.20620e8	0.328990
$485$	9.91341e8	0.394573
$486$	2.82869e7	0.0111778
$487$	1.52055e9	0.596554	0.298277	−	0.954479i	$-0.403588\pi$
$487$	1.52055e9	0.596554	0.298277	+	0.954479i	$0.403588\pi$
$488$	2.13147e8	0.0830253
$489$	2.66218e9	1.02957
$490$	−4.62536e8	−0.177606
$491$	−3.45739e9	−1.31815	−0.659073	−	0.752079i	$-0.729051\pi$
$491$	−3.45739e9	−1.31815	−0.659073	+	0.752079i	$0.729051\pi$
$492$	−1.99364e9	−0.754689
$493$	9.03853e8	0.339730
$494$	1.63097e7	0.00608697
$495$	8.88789e8	0.329367
$496$	1.70018e9	0.625619
$497$	6.85117e8	0.250333
$498$	1.71011e8	0.0620472
$499$	−5.24675e8	−0.189033	−0.0945167	−	0.995523i	$-0.530131\pi$
$499$	−5.24675e8	−0.189033	−0.0945167	+	0.995523i	$0.530131\pi$
$500$	−1.72068e9	−0.615609
$501$	−4.33726e8	−0.154093
$502$	−3.33930e8	−0.117813
$503$	−2.08499e9	−0.730492	−0.365246	−	0.930911i	$-0.619015\pi$
$503$	−2.08499e9	−0.730492	−0.365246	+	0.930911i	$0.619015\pi$
$504$	1.32138e8	0.0459748
$505$	6.78839e9	2.34556
$506$	8.21485e8	0.281885
$507$	1.67437e9	0.570588
$508$	−3.97961e9	−1.34685
$509$	3.13440e9	1.05352	0.526760	−	0.850014i	$-0.323407\pi$
$509$	3.13440e9	1.05352	0.526760	+	0.850014i	$0.323407\pi$
$510$	−6.59990e7	−0.0220314
$511$	8.57224e8	0.284198
$512$	−2.33471e9	−0.768754
$513$	1.89946e8	0.0621182
$514$	8.18430e8	0.265834
$515$	3.88176e9	1.25229
$516$	8.30355e8	0.266066
$517$	4.69045e9	1.49279
$518$	1.39901e8	0.0442248
$519$	−2.60171e9	−0.816908
$520$	−1.44776e8	−0.0451527
$521$	9.76015e8	0.302360	0.151180	−	0.988506i	$-0.451693\pi$
$521$	9.76015e8	0.302360	0.151180	+	0.988506i	$0.451693\pi$
$522$	−3.55939e8	−0.109529
$523$	−1.99309e9	−0.609216	−0.304608	−	0.952478i	$-0.598526\pi$
$523$	−1.99309e9	−0.609216	−0.304608	+	0.952478i	$0.598526\pi$
$524$	2.95518e9	0.897273
$525$	−3.67510e8	−0.110844
$526$	4.86000e8	0.145608
$527$	4.16225e8	0.123877
$528$	1.44420e9	0.426981
$529$	1.00819e10	2.96107
$530$	8.06422e8	0.235286
$531$	1.49721e8	0.0433963
$532$	4.36812e8	0.125778
$533$	−5.10039e8	−0.145901
$534$	−2.89845e7	−0.00823703
$535$	−7.81435e9	−2.20625
$536$	−9.99115e8	−0.280246
$537$	−1.88088e9	−0.524145
$538$	−6.28723e8	−0.174069
$539$	2.47780e9	0.681562
$540$	−8.30048e8	−0.226843
$541$	3.17125e9	0.861072	0.430536	−	0.902573i	$-0.358325\pi$
$541$	3.17125e9	0.861072	0.430536	+	0.902573i	$0.358325\pi$
$542$	7.64698e8	0.206297
$543$	2.82856e9	0.758168
$544$	−3.39403e8	−0.0903897
$545$	−6.86597e9	−1.81683
$546$	1.66421e7	0.00437555
$547$	2.77827e9	0.725803	0.362901	−	0.931828i	$-0.381786\pi$
$547$	2.77827e9	0.725803	0.362901	+	0.931828i	$0.381786\pi$
$548$	−2.65354e9	−0.688802
$549$	3.12640e8	0.0806383
$550$	2.64007e8	0.0676621
$551$	−2.39012e9	−0.608681
$552$	−1.55841e9	−0.394361
$553$	−1.36717e9	−0.343782
$554$	1.46332e8	0.0365641
$555$	−1.78515e9	−0.443250
$556$	4.52093e9	1.11549
$557$	−7.99807e8	−0.196107	−0.0980533	−	0.995181i	$-0.531262\pi$
$557$	−7.99807e8	−0.196107	−0.0980533	+	0.995181i	$0.531262\pi$
$558$	−1.63910e8	−0.0399380
$559$	2.12433e8	0.0514375
$560$	−1.84719e9	−0.444482
$561$	3.53556e8	0.0845451
$562$	1.21495e9	0.288722
$563$	−1.37529e9	−0.324800	−0.162400	−	0.986725i	$-0.551923\pi$
$563$	−1.37529e9	−0.324800	−0.162400	+	0.986725i	$0.551923\pi$
$564$	−4.38045e9	−1.02812
$565$	4.29626e9	1.00212
$566$	−1.57041e8	−0.0364045
$567$	1.93817e8	0.0446530
$568$	9.33666e8	0.213783
$569$	4.45989e8	0.101492	0.0507459	−	0.998712i	$-0.483840\pi$
$569$	4.45989e8	0.101492	0.0507459	+	0.998712i	$0.483840\pi$
$570$	1.74526e8	0.0394728
$571$	−7.16360e9	−1.61029	−0.805147	−	0.593076i	$-0.797914\pi$
$571$	−7.16360e9	−1.61029	−0.805147	+	0.593076i	$0.797914\pi$
$572$	3.81804e8	0.0853010
$573$	−9.14533e8	−0.203076
$574$	4.27725e8	0.0944002
$575$	4.33434e9	0.950793
$576$	−1.25732e9	−0.274137
$577$	−5.75941e9	−1.24814	−0.624069	−	0.781369i	$-0.714522\pi$
$577$	−5.75941e9	−1.24814	−0.624069	+	0.781369i	$0.714522\pi$
$578$	7.82672e8	0.168590
$579$	1.55125e9	0.332129
$580$	1.04447e10	2.22278
$581$	1.17174e9	0.247865
$582$	−1.55296e8	−0.0326536
$583$	−4.31999e9	−0.902908
$584$	1.16821e9	0.242704
$585$	−2.12354e8	−0.0438546
$586$	−4.93610e8	−0.101331
$587$	−2.89213e9	−0.590180	−0.295090	−	0.955469i	$-0.595350\pi$
$587$	−2.89213e9	−0.590180	−0.295090	+	0.955469i	$0.595350\pi$
$588$	−2.31404e9	−0.469407
$589$	−1.10065e9	−0.221946
$590$	1.37567e8	0.0275761
$591$	−2.11311e9	−0.421081
$592$	−2.90069e9	−0.574613
$593$	−2.52669e9	−0.497577	−0.248789	−	0.968558i	$-0.580032\pi$
$593$	−2.52669e9	−0.497577	−0.248789	+	0.968558i	$0.580032\pi$
$594$	−1.39231e8	−0.0272574
$595$	−4.52214e8	−0.0880104
$596$	−1.18193e7	−0.00228682
$597$	3.19641e9	0.614826
$598$	−1.96273e8	−0.0375325
$599$	−9.87541e9	−1.87742	−0.938709	−	0.344709i	$-0.887977\pi$
$599$	−9.87541e9	−1.87742	−0.938709	+	0.344709i	$0.887977\pi$
$600$	−5.00836e8	−0.0946600
$601$	−1.69668e9	−0.318815	−0.159408	−	0.987213i	$-0.550958\pi$
$601$	−1.69668e9	−0.318815	−0.159408	+	0.987213i	$0.550958\pi$
$602$	−1.78149e8	−0.0332809
$603$	−1.46548e9	−0.272188
$604$	−5.85386e9	−1.08097
$605$	2.24654e9	0.412449
$606$	−1.06342e9	−0.194111
$607$	−9.97951e9	−1.81113	−0.905564	−	0.424211i	$-0.860552\pi$
$607$	−9.97951e9	−1.81113	−0.905564	+	0.424211i	$0.860552\pi$
$608$	8.97507e8	0.161948
$609$	−2.43883e9	−0.437544
$610$	2.87260e8	0.0512414
$611$	−1.12067e9	−0.198761
$612$	−3.30189e8	−0.0582281
$613$	−6.06527e9	−1.06350	−0.531751	−	0.846901i	$-0.678466\pi$
$613$	−6.06527e9	−1.06350	−0.531751	+	0.846901i	$0.678466\pi$
$614$	6.53302e8	0.113900
$615$	−5.45780e9	−0.946139
$616$	−6.50396e8	−0.112110
$617$	−1.40806e9	−0.241336	−0.120668	−	0.992693i	$-0.538504\pi$
$617$	−1.40806e9	−0.241336	−0.120668	+	0.992693i	$0.538504\pi$
$618$	−6.08089e8	−0.103635
$619$	−2.17547e9	−0.368668	−0.184334	−	0.982864i	$-0.559013\pi$
$619$	−2.17547e9	−0.368668	−0.184334	+	0.982864i	$0.559013\pi$
$620$	4.80977e9	0.810501
$621$	−2.28584e9	−0.383023
$622$	1.85686e8	0.0309395
$623$	−1.98596e8	−0.0329051
$624$	−3.45055e8	−0.0568516
$625$	−7.62637e9	−1.24950
$626$	1.06353e9	0.173277
$627$	−9.34935e8	−0.151476
$628$	−5.51774e9	−0.889001
$629$	−7.10123e8	−0.113777
$630$	1.78083e8	0.0283746
$631$	−3.80261e9	−0.602531	−0.301266	−	0.953540i	$-0.597409\pi$
$631$	−3.80261e9	−0.602531	−0.301266	+	0.953540i	$0.597409\pi$
$632$	−1.86315e9	−0.293588
$633$	−1.29195e9	−0.202457
$634$	−1.36386e9	−0.212549
$635$	−1.08946e10	−1.68852
$636$	4.03448e9	0.621853
$637$	−5.92008e8	−0.0907486
$638$	1.75197e9	0.267089
$639$	1.36948e9	0.207636
$640$	−5.20010e9	−0.784118
$641$	3.59687e9	0.539413	0.269706	−	0.962943i	$-0.413073\pi$
$641$	3.59687e9	0.539413	0.269706	+	0.962943i	$0.413073\pi$
$642$	1.22414e9	0.182582
$643$	4.76649e9	0.707066	0.353533	−	0.935422i	$-0.384980\pi$
$643$	4.76649e9	0.707066	0.353533	+	0.935422i	$0.384980\pi$
$644$	−5.25666e9	−0.775549
$645$	2.27319e9	0.333562
$646$	6.94257e7	0.0101323
$647$	1.17944e9	0.171203	0.0856016	−	0.996329i	$-0.472719\pi$
$647$	1.17944e9	0.171203	0.0856016	+	0.996329i	$0.472719\pi$
$648$	2.64130e8	0.0381334
$649$	−7.36945e8	−0.105823
$650$	−6.30778e7	−0.00900907
$651$	−1.12308e9	−0.159543
$652$	1.22375e10	1.72913
$653$	8.76325e9	1.23160	0.615799	−	0.787903i	$-0.288833\pi$
$653$	8.76325e9	1.23160	0.615799	+	0.787903i	$0.288833\pi$
$654$	1.07557e9	0.150355
$655$	8.09015e9	1.12489
$656$	−8.86841e9	−1.22654
$657$	1.71351e9	0.235726
$658$	9.39804e8	0.128602
$659$	−1.34224e10	−1.82697	−0.913487	−	0.406868i	$-0.866621\pi$
$659$	−1.34224e10	−1.82697	−0.913487	+	0.406868i	$0.866621\pi$
$660$	4.08559e9	0.553161
$661$	−5.67841e9	−0.764754	−0.382377	−	0.924006i	$-0.624894\pi$
$661$	−5.67841e9	−0.764754	−0.382377	+	0.924006i	$0.624894\pi$
$662$	1.68501e9	0.225735
$663$	−8.44735e7	−0.0112570
$664$	1.59683e9	0.211675
$665$	1.19582e9	0.157685
$666$	2.79648e8	0.0366819
$667$	2.87631e10	3.75315
$668$	−1.99375e9	−0.258794
$669$	−3.36597e9	−0.434630
$670$	−1.34651e9	−0.172961
$671$	−1.53885e9	−0.196638
$672$	9.15797e8	0.116414
$673$	2.49548e9	0.315574	0.157787	−	0.987473i	$-0.449564\pi$
$673$	2.49548e9	0.315574	0.157787	+	0.987473i	$0.449564\pi$
$674$	−2.50621e8	−0.0315288
$675$	−7.34617e8	−0.0919385
$676$	7.69673e9	0.958281
$677$	5.54895e9	0.687307	0.343654	−	0.939097i	$-0.388335\pi$
$677$	5.54895e9	0.687307	0.343654	+	0.939097i	$0.388335\pi$
$678$	−6.73021e8	−0.0829325
$679$	−1.06406e9	−0.130444
$680$	−6.16269e8	−0.0751604
$681$	−1.92674e9	−0.233780
$682$	8.06785e8	0.0973896
$683$	4.93572e9	0.592759	0.296379	−	0.955070i	$-0.404221\pi$
$683$	4.93572e9	0.592759	0.296379	+	0.955070i	$0.404221\pi$
$684$	8.73144e8	0.104325
$685$	−7.26438e9	−0.863538
$686$	1.08856e9	0.128741
$687$	9.17141e9	1.07916
$688$	3.69372e9	0.432419
$689$	1.03215e9	0.120220
$690$	−2.10027e9	−0.243391
$691$	−8.63015e9	−0.995050	−0.497525	−	0.867450i	$-0.665758\pi$
$691$	−8.63015e9	−0.995050	−0.497525	+	0.867450i	$0.665758\pi$
$692$	−1.19595e10	−1.37197
$693$	−9.53988e8	−0.108887
$694$	1.21660e8	0.0138163
$695$	1.23766e10	1.39847
$696$	−3.32360e9	−0.373660
$697$	−2.17109e9	−0.242864
$698$	−7.74720e8	−0.0862285
$699$	8.61529e9	0.954113
$700$	−1.68937e9	−0.186158
$701$	8.06267e9	0.884028	0.442014	−	0.897008i	$-0.354264\pi$
$701$	8.06267e9	0.884028	0.442014	+	0.897008i	$0.354264\pi$
$702$	3.32658e7	0.00362926
$703$	1.87783e9	0.203851
$704$	6.18869e9	0.668489
$705$	−1.19920e10	−1.28893
$706$	2.87566e9	0.307554
$707$	−7.28636e9	−0.775430
$708$	6.88240e8	0.0728825
$709$	1.13085e10	1.19163	0.595817	−	0.803120i	$-0.296828\pi$
$709$	1.13085e10	1.19163	0.595817	+	0.803120i	$0.296828\pi$
$710$	1.25831e9	0.131942
$711$	−2.73283e9	−0.285147
$712$	−2.70644e8	−0.0281008
$713$	1.32454e10	1.36853
$714$	7.08405e7	0.00728346
$715$	1.04523e9	0.106940
$716$	−8.64605e9	−0.880283
$717$	−6.71718e9	−0.680565
$718$	1.99024e9	0.200664
$719$	−1.36390e10	−1.36846	−0.684231	−	0.729266i	$-0.739862\pi$
$719$	−1.36390e10	−1.36846	−0.684231	+	0.729266i	$0.739862\pi$
$720$	−3.69235e9	−0.368672
$721$	−4.16652e9	−0.413999
$722$	1.57856e9	0.156092
$723$	2.75629e9	0.271232
$724$	1.30023e10	1.27332
$725$	9.24382e9	0.900883
$726$	−3.51927e8	−0.0341330
$727$	3.69976e9	0.357111	0.178556	−	0.983930i	$-0.442858\pi$
$727$	3.69976e9	0.357111	0.178556	+	0.983930i	$0.442858\pi$
$728$	1.55396e8	0.0149273
$729$	3.87420e8	0.0370370
$730$	1.57441e9	0.149791
$731$	9.04265e8	0.0856219
$732$	1.43715e9	0.135429
$733$	1.64513e10	1.54289	0.771447	−	0.636294i	$-0.219533\pi$
$733$	1.64513e10	1.54289	0.771447	+	0.636294i	$0.219533\pi$
$734$	−1.35963e9	−0.126907
$735$	−6.33494e9	−0.588487
$736$	−1.08007e10	−0.998576
$737$	7.21327e9	0.663737
$738$	8.54980e8	0.0782994
$739$	−4.40695e9	−0.401682	−0.200841	−	0.979624i	$-0.564367\pi$
$739$	−4.40695e9	−0.401682	−0.200841	+	0.979624i	$0.564367\pi$
$740$	−8.20597e9	−0.744421
$741$	2.23379e8	0.0201688
$742$	−8.65578e8	−0.0777844
$743$	2.97820e9	0.266374	0.133187	−	0.991091i	$-0.457479\pi$
$743$	2.97820e9	0.266374	0.133187	+	0.991091i	$0.457479\pi$
$744$	−1.53052e9	−0.136249
$745$	−3.23568e7	−0.00286694
$746$	−3.01526e7	−0.00265913
$747$	2.34219e9	0.205589
$748$	1.62523e9	0.141990
$749$	8.38758e9	0.729374
$750$	7.37921e8	0.0638698
$751$	6.84373e9	0.589594	0.294797	−	0.955560i	$-0.404748\pi$
$751$	6.84373e9	0.589594	0.294797	+	0.955560i	$0.404748\pi$
$752$	−1.94858e10	−1.67092
$753$	−4.57354e9	−0.390364
$754$	−4.18590e8	−0.0355623
$755$	−1.60256e10	−1.35519
$756$	8.90937e8	0.0749930
$757$	−1.28064e10	−1.07298	−0.536491	−	0.843906i	$-0.680250\pi$
$757$	−1.28064e10	−1.07298	−0.536491	+	0.843906i	$0.680250\pi$
$758$	2.58379e9	0.215484
$759$	1.12512e10	0.934008
$760$	1.62965e9	0.134662
$761$	−1.33708e9	−0.109979	−0.0549897	−	0.998487i	$-0.517513\pi$
$761$	−1.33708e9	−0.109979	−0.0549897	+	0.998487i	$0.517513\pi$
$762$	1.70668e9	0.139736
$763$	7.36964e9	0.600634
$764$	−4.20393e9	−0.341058
$765$	−9.03931e8	−0.0729995
$766$	−1.31561e9	−0.105761
$767$	1.76075e8	0.0140901
$768$	−5.14604e9	−0.409929
$769$	1.10898e10	0.879390	0.439695	−	0.898147i	$-0.355086\pi$
$769$	1.10898e10	0.879390	0.439695	+	0.898147i	$0.355086\pi$
$770$	−8.76544e8	−0.0691920
$771$	1.12093e10	0.880823
$772$	7.13078e9	0.557798
$773$	−1.78044e10	−1.38644	−0.693218	−	0.720728i	$-0.743808\pi$
$773$	−1.78044e10	−1.38644	−0.693218	+	0.720728i	$0.743808\pi$
$774$	−3.56102e8	−0.0276046
$775$	4.25679e9	0.328493
$776$	−1.45009e9	−0.111398
$777$	1.91610e9	0.146536
$778$	−4.44965e9	−0.338764
$779$	5.74117e9	0.435130
$780$	−9.76150e8	−0.0736522
$781$	−6.74075e9	−0.506325
$782$	−8.35479e8	−0.0624758
$783$	−4.87499e9	−0.362917
$784$	−1.02937e10	−0.762894
$785$	−1.51054e10	−1.11452
$786$	−1.26734e9	−0.0930927
$787$	1.97486e10	1.44419	0.722095	−	0.691794i	$-0.243179\pi$
$787$	1.97486e10	1.44419	0.722095	+	0.691794i	$0.243179\pi$
$788$	−9.71354e9	−0.707190
$789$	6.65631e9	0.482463
$790$	−2.51098e9	−0.181196
$791$	−4.61142e9	−0.331297
$792$	−1.30008e9	−0.0929890
$793$	3.67670e8	0.0261820
$794$	1.93408e9	0.137120
$795$	1.10448e10	0.779605
$796$	1.46933e10	1.03258
$797$	1.84228e10	1.28900	0.644499	−	0.764605i	$-0.277066\pi$
$797$	1.84228e10	1.28900	0.644499	+	0.764605i	$0.277066\pi$
$798$	−1.87329e8	−0.0130495
$799$	−4.77035e9	−0.330854
$800$	−3.47112e9	−0.239692
$801$	−3.96975e8	−0.0272929
$802$	1.16447e8	0.00797111
$803$	−8.43408e9	−0.574822
$804$	−6.73653e9	−0.457130
$805$	−1.43907e10	−0.972291
$806$	−1.92761e8	−0.0129672
$807$	−8.61106e9	−0.576766
$808$	−9.92972e9	−0.662213
$809$	1.12171e10	0.744838	0.372419	−	0.928065i	$-0.378528\pi$
$809$	1.12171e10	0.744838	0.372419	+	0.928065i	$0.378528\pi$
$810$	3.55970e8	0.0235351
$811$	1.91381e10	1.25987	0.629934	−	0.776649i	$-0.283082\pi$
$811$	1.91381e10	1.25987	0.629934	+	0.776649i	$0.283082\pi$
$812$	−1.12108e10	−0.734838
$813$	1.04734e10	0.683551
$814$	−1.37646e9	−0.0894495
$815$	3.35017e10	2.16778
$816$	−1.46880e9	−0.0946341
$817$	−2.39122e9	−0.153406
$818$	1.52750e9	0.0975766
$819$	2.27932e8	0.0144981
$820$	−2.50885e10	−1.58901
$821$	1.26324e10	0.796685	0.398342	−	0.917237i	$-0.369586\pi$
$821$	1.26324e10	0.796685	0.398342	+	0.917237i	$0.369586\pi$
$822$	1.13799e9	0.0714637
$823$	−5.60518e9	−0.350502	−0.175251	−	0.984524i	$-0.556074\pi$
$823$	−5.60518e9	−0.350502	−0.175251	+	0.984524i	$0.556074\pi$
$824$	−5.67806e9	−0.353553
$825$	3.61587e9	0.224194
$826$	−1.47658e8	−0.00911650
$827$	−9.05725e9	−0.556836	−0.278418	−	0.960460i	$-0.589810\pi$
$827$	−9.05725e9	−0.556836	−0.278418	+	0.960460i	$0.589810\pi$
$828$	−1.05075e10	−0.643273
$829$	1.80271e10	1.09897	0.549483	−	0.835505i	$-0.314825\pi$
$829$	1.80271e10	1.09897	0.549483	+	0.835505i	$0.314825\pi$
$830$	2.15206e9	0.130641
$831$	2.00418e9	0.121153
$832$	−1.47863e9	−0.0890080
$833$	−2.52001e9	−0.151058
$834$	−1.93882e9	−0.115733
$835$	−5.45813e9	−0.324445
$836$	−4.29771e9	−0.254399
$837$	−2.24493e9	−0.132332
$838$	−5.36143e8	−0.0314721
$839$	2.52101e10	1.47370	0.736848	−	0.676059i	$-0.236313\pi$
$839$	2.52101e10	1.47370	0.736848	+	0.676059i	$0.236313\pi$
$840$	1.66286e9	0.0968004
$841$	4.40930e10	2.55613
$842$	1.69501e9	0.0978546
$843$	1.66400e10	0.956660
$844$	−5.93885e9	−0.340020
$845$	2.10707e10	1.20138
$846$	1.87858e9	0.106668
$847$	−2.41134e9	−0.136353
$848$	1.79468e10	1.01065
$849$	−2.15085e9	−0.120624
$850$	−2.68504e8	−0.0149963
$851$	−2.25981e10	−1.25695
$852$	6.29524e9	0.348717
$853$	−4.60485e9	−0.254035	−0.127017	−	0.991900i	$-0.540540\pi$
$853$	−4.60485e9	−0.254035	−0.127017	+	0.991900i	$0.540540\pi$
$854$	−3.08333e8	−0.0169401
$855$	2.39033e9	0.130791
$856$	1.14305e10	0.622881
$857$	2.64820e10	1.43720	0.718600	−	0.695423i	$-0.244783\pi$
$857$	2.64820e10	1.43720	0.718600	+	0.695423i	$0.244783\pi$
$858$	−1.63738e8	−0.00885003
$859$	−3.30189e10	−1.77740	−0.888702	−	0.458485i	$-0.848392\pi$
$859$	−3.30189e10	−1.77740	−0.888702	+	0.458485i	$0.848392\pi$
$860$	1.04494e10	0.560206
$861$	5.85817e9	0.312789
$862$	1.30588e9	0.0694431
$863$	−1.44548e10	−0.765552	−0.382776	−	0.923841i	$-0.625032\pi$
$863$	−1.44548e10	−0.765552	−0.382776	+	0.923841i	$0.625032\pi$
$864$	1.83059e9	0.0965590
$865$	−3.27406e10	−1.72001
$866$	−3.97171e9	−0.207809
$867$	1.07196e10	0.558612
$868$	−5.16260e9	−0.267947
$869$	1.34513e10	0.695337
$870$	−4.47924e9	−0.230615
$871$	−1.72343e9	−0.0883751
$872$	1.00432e10	0.512938
$873$	−2.12696e9	−0.108195
$874$	2.20932e9	0.111936
$875$	5.05610e9	0.255145
$876$	7.87666e9	0.395893
$877$	1.61500e9	0.0808491	0.0404245	−	0.999183i	$-0.487129\pi$
$877$	1.61500e9	0.0808491	0.0404245	+	0.999183i	$0.487129\pi$
$878$	−4.76256e9	−0.237470
$879$	−6.76054e9	−0.335753
$880$	1.81742e10	0.899013
$881$	3.62460e10	1.78585	0.892925	−	0.450206i	$-0.148649\pi$
$881$	3.62460e10	1.78585	0.892925	+	0.450206i	$0.148649\pi$
$882$	9.92386e8	0.0487013
$883$	1.90627e10	0.931796	0.465898	−	0.884838i	$-0.345731\pi$
$883$	1.90627e10	0.931796	0.465898	+	0.884838i	$0.345731\pi$
$884$	−3.88308e8	−0.0189057
$885$	1.88413e9	0.0913714
$886$	5.40357e9	0.261013
$887$	−1.75867e9	−0.0846157	−0.0423079	−	0.999105i	$-0.513471\pi$
$887$	−1.75867e9	−0.0846157	−0.0423079	+	0.999105i	$0.513471\pi$
$888$	2.61123e9	0.125141
$889$	1.16938e10	0.558214
$890$	−3.64748e8	−0.0173432
$891$	−1.90693e9	−0.0903155
$892$	−1.54727e10	−0.729944
$893$	1.26146e10	0.592780
$894$	5.06878e6	0.000237259 0
$895$	−2.36695e10	−1.10359
$896$	5.58156e9	0.259225
$897$	−2.68818e9	−0.124361
$898$	2.71135e9	0.124945
$899$	2.82484e10	1.29669
$900$	−3.37689e9	−0.154407
$901$	4.39359e9	0.200116
$902$	−4.20831e9	−0.190935
$903$	−2.43995e9	−0.110274
$904$	−6.28436e9	−0.282925
$905$	3.55954e10	1.59633
$906$	2.51046e9	0.112151
$907$	−3.74603e10	−1.66704	−0.833520	−	0.552489i	$-0.813678\pi$
$907$	−3.74603e10	−1.66704	−0.833520	+	0.552489i	$0.813678\pi$
$908$	−8.85684e9	−0.392625
$909$	−1.45647e10	−0.643174
$910$	2.09428e8	0.00921277
$911$	8.20611e9	0.359603	0.179801	−	0.983703i	$-0.442455\pi$
$911$	8.20611e9	0.359603	0.179801	+	0.983703i	$0.442455\pi$
$912$	3.88406e9	0.169552
$913$	−1.15285e10	−0.501334
$914$	−2.25576e9	−0.0977196
$915$	3.93435e9	0.169785
$916$	4.21592e10	1.81242
$917$	−8.68361e9	−0.371884
$918$	1.41603e8	0.00604121
$919$	−1.35638e9	−0.0576472	−0.0288236	−	0.999585i	$-0.509176\pi$
$919$	−1.35638e9	−0.0576472	−0.0288236	+	0.999585i	$0.509176\pi$
$920$	−1.96114e10	−0.830331
$921$	8.94770e9	0.377401
$922$	−1.51737e9	−0.0637579
$923$	1.61053e9	0.0674161
$924$	−4.38530e9	−0.182872
$925$	−7.26252e9	−0.301711
$926$	4.41578e8	0.0182755
$927$	−8.32846e9	−0.343388
$928$	−2.30347e10	−0.946158
$929$	6.23174e9	0.255008	0.127504	−	0.991838i	$-0.459303\pi$
$929$	6.23174e9	0.255008	0.127504	+	0.991838i	$0.459303\pi$
$930$	−2.06269e9	−0.0840899
$931$	6.66385e9	0.270646
$932$	3.96028e10	1.60240
$933$	2.54318e9	0.102516
$934$	−7.13575e8	−0.0286567
$935$	4.44925e9	0.178011
$936$	3.10621e8	0.0123813
$937$	−3.18495e10	−1.26478	−0.632389	−	0.774651i	$-0.717926\pi$
$937$	−3.18495e10	−1.26478	−0.632389	+	0.774651i	$0.717926\pi$
$938$	1.44529e9	0.0571801
$939$	1.45663e10	0.574141
$940$	−5.51248e10	−2.16471
$941$	−4.56004e10	−1.78404	−0.892021	−	0.451995i	$-0.850713\pi$
$941$	−4.56004e10	−1.78404	−0.892021	+	0.451995i	$0.850713\pi$
$942$	2.36631e9	0.0922345
$943$	−6.90901e10	−2.68303
$944$	3.06154e9	0.118451
$945$	2.43904e9	0.0940174
$946$	1.75277e9	0.0673142
$947$	1.39598e10	0.534138	0.267069	−	0.963677i	$-0.413945\pi$
$947$	1.39598e10	0.534138	0.267069	+	0.963677i	$0.413945\pi$
$948$	−1.25623e10	−0.478894
$949$	2.01511e9	0.0765363
$950$	7.10026e8	0.0268684
$951$	−1.86796e10	−0.704266
$952$	6.61476e8	0.0248476
$953$	−1.88757e10	−0.706444	−0.353222	−	0.935540i	$-0.614914\pi$
$953$	−1.88757e10	−0.706444	−0.353222	+	0.935540i	$0.614914\pi$
$954$	−1.73021e9	−0.0645176
$955$	−1.15087e10	−0.427578
$956$	−3.08776e10	−1.14298
$957$	2.39952e10	0.884980
$958$	3.53309e9	0.129830
$959$	7.79727e9	0.285481
$960$	−1.58225e10	−0.577200
$961$	−1.45042e10	−0.527183
$962$	3.28871e8	0.0119100
$963$	1.67660e10	0.604973
$964$	1.26701e10	0.455524
$965$	1.95213e10	0.699301
$966$	2.25434e9	0.0804637
$967$	−4.38222e10	−1.55848	−0.779241	−	0.626724i	$-0.784395\pi$
$967$	−4.38222e10	−1.55848	−0.779241	+	0.626724i	$0.784395\pi$
$968$	−3.28613e9	−0.116445
$969$	9.50862e8	0.0335726
$970$	−1.95429e9	−0.0687525
$971$	7.68907e8	0.0269529	0.0134765	−	0.999909i	$-0.495710\pi$
$971$	7.68907e8	0.0269529	0.0134765	+	0.999909i	$0.495710\pi$
$972$	1.78090e9	0.0622023
$973$	−1.32845e10	−0.462326
$974$	−2.99756e9	−0.103947
$975$	−8.63921e8	−0.0298509
$976$	6.39295e9	0.220103
$977$	−3.25858e10	−1.11788	−0.558942	−	0.829206i	$-0.688793\pi$
$977$	−3.25858e10	−1.11788	−0.558942	+	0.829206i	$0.688793\pi$
$978$	−5.24813e9	−0.179398
$979$	1.95395e9	0.0665541
$980$	−2.91205e10	−0.988342
$981$	1.47312e10	0.498191
$982$	6.81577e9	0.229681
$983$	−3.29965e10	−1.10798	−0.553988	−	0.832525i	$-0.686894\pi$
$983$	−3.29965e10	−1.10798	−0.553988	+	0.832525i	$0.686894\pi$
$984$	7.98341e9	0.267120
$985$	−2.65919e10	−0.886590
$986$	−1.78182e9	−0.0591963
$987$	1.28717e10	0.426113
$988$	1.02683e9	0.0338727
$989$	2.87763e10	0.945904
$990$	−1.75213e9	−0.0573908
$991$	2.11125e10	0.689100	0.344550	−	0.938768i	$-0.388031\pi$
$991$	2.11125e10	0.689100	0.344550	+	0.938768i	$0.388031\pi$
$992$	−1.06075e10	−0.345002
$993$	2.30780e10	0.747957
$994$	−1.35061e9	−0.0436193
$995$	4.02245e10	1.29452
$996$	1.07666e10	0.345280
$997$	−2.81284e10	−0.898902	−0.449451	−	0.893305i	$-0.648380\pi$
$997$	−2.81284e10	−0.898902	−0.449451	+	0.893305i	$0.648380\pi$
$998$	1.03432e9	0.0329382
$999$	3.83009e9	0.121543

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.a.1.8	✓	16
3.2	odd	2		531.8.a.b.1.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.a.1.8	✓	16	1.1	even	1	trivial
531.8.a.b.1.9		16	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-1.97136 q^{2} -27.0000 q^{3} -124.114 q^{4} -339.775 q^{5} +53.2268 q^{6} +364.700 q^{7} +497.007 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-1.97136 q^{2} -27.0000 q^{3} -124.114 q^{4} -339.775 q^{5} +53.2268 q^{6} +364.700 q^{7} +497.007 q^{8} +729.000 q^{9} +669.820 q^{10} -3588.22 q^{11} +3351.07 q^{12} +857.316 q^{13} -718.956 q^{14} +9173.94 q^{15} +14906.8 q^{16} +3649.35 q^{17} -1437.12 q^{18} -9650.25 q^{19} +42170.8 q^{20} -9846.91 q^{21} +7073.68 q^{22} +116133. q^{23} -13419.2 q^{24} +37322.4 q^{25} -1690.08 q^{26} -19683.0 q^{27} -45264.3 q^{28} +247675. q^{29} -18085.1 q^{30} +114054. q^{31} -93003.6 q^{32} +96882.0 q^{33} -7194.19 q^{34} -123916. q^{35} -90478.9 q^{36} -194589. q^{37} +19024.1 q^{38} -23147.5 q^{39} -168871. q^{40} -594925. q^{41} +19411.8 q^{42} +247788. q^{43} +445348. q^{44} -247696. q^{45} -228939. q^{46} -1.30718e6 q^{47} -402483. q^{48} -690537. q^{49} -73575.9 q^{50} -98532.5 q^{51} -106405. q^{52} +1.20394e6 q^{53} +38802.3 q^{54} +1.21919e6 q^{55} +181259. q^{56} +260557. q^{57} -488257. q^{58} +205379. q^{59} -1.13861e6 q^{60} +428862. q^{61} -224843. q^{62} +265866. q^{63} -1.72472e6 q^{64} -291295. q^{65} -190989. q^{66} -2.01026e6 q^{67} -452934. q^{68} -3.13558e6 q^{69} +244284. q^{70} +1.87858e6 q^{71} +362318. q^{72} +2.35049e6 q^{73} +383605. q^{74} -1.00770e6 q^{75} +1.19773e6 q^{76} -1.30863e6 q^{77} +45632.2 q^{78} -3.74874e6 q^{79} -5.06496e6 q^{80} +531441. q^{81} +1.17281e6 q^{82} +3.21289e6 q^{83} +1.22214e6 q^{84} -1.23996e6 q^{85} -488480. q^{86} -6.68722e6 q^{87} -1.78337e6 q^{88} -544547. q^{89} +488299. q^{90} +312663. q^{91} -1.44136e7 q^{92} -3.07947e6 q^{93} +2.57692e6 q^{94} +3.27892e6 q^{95} +2.51110e6 q^{96} -2.91764e6 q^{97} +1.36130e6 q^{98} -2.61581e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9} - 3479 q^{10} + 898 q^{11} - 26298 q^{12} - 8172 q^{13} - 13315 q^{14} + 1836 q^{15} + 3138 q^{16} - 44985 q^{17} - 4374 q^{18} - 40137 q^{19} + 130657 q^{20} + 63261 q^{21} + 109394 q^{22} - 2833 q^{23} - 22113 q^{24} + 285746 q^{25} - 129420 q^{26} - 314928 q^{27} + 112890 q^{28} + 144375 q^{29} + 93933 q^{30} - 141759 q^{31} - 36224 q^{32} - 24246 q^{33} - 341332 q^{34} - 78859 q^{35} + 710046 q^{36} - 297971 q^{37} + 329075 q^{38} + 220644 q^{39} - 203048 q^{40} + 659077 q^{41} + 359505 q^{42} - 1431608 q^{43} + 254916 q^{44} - 49572 q^{45} + 873113 q^{46} - 1574073 q^{47} - 84726 q^{48} + 1893545 q^{49} + 302533 q^{50} + 1214595 q^{51} - 4972548 q^{52} + 587736 q^{53} + 118098 q^{54} - 4624036 q^{55} - 5798506 q^{56} + 1083699 q^{57} - 6991380 q^{58} + 3286064 q^{59} - 3527739 q^{60} - 6117131 q^{61} - 11570258 q^{62} - 1708047 q^{63} - 19063011 q^{64} - 5335514 q^{65} - 2953638 q^{66} - 16518710 q^{67} - 17284669 q^{68} + 76491 q^{69} - 39189486 q^{70} - 10882582 q^{71} + 597051 q^{72} - 21097441 q^{73} - 16717030 q^{74} - 7715142 q^{75} - 40864952 q^{76} - 3404601 q^{77} + 3494340 q^{78} - 3784458 q^{79} - 27466195 q^{80} + 8503056 q^{81} - 24990117 q^{82} - 1951425 q^{83} - 3048030 q^{84} - 23238675 q^{85} - 35910572 q^{86} - 3898125 q^{87} - 27843055 q^{88} + 10499443 q^{89} - 2536191 q^{90} + 699217 q^{91} - 20062766 q^{92} + 3827493 q^{93} - 59358988 q^{94} - 29236333 q^{95} + 978048 q^{96} - 25158976 q^{97} + 2120460 q^{98} + 654642 q^{99}+O(q^{100}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9 - 3479 * q^10 + 898 * q^11 - 26298 * q^12 - 8172 * q^13 - 13315 * q^14 + 1836 * q^15 + 3138 * q^16 - 44985 * q^17 - 4374 * q^18 - 40137 * q^19 + 130657 * q^20 + 63261 * q^21 + 109394 * q^22 - 2833 * q^23 - 22113 * q^24 + 285746 * q^25 - 129420 * q^26 - 314928 * q^27 + 112890 * q^28 + 144375 * q^29 + 93933 * q^30 - 141759 * q^31 - 36224 * q^32 - 24246 * q^33 - 341332 * q^34 - 78859 * q^35 + 710046 * q^36 - 297971 * q^37 + 329075 * q^38 + 220644 * q^39 - 203048 * q^40 + 659077 * q^41 + 359505 * q^42 - 1431608 * q^43 + 254916 * q^44 - 49572 * q^45 + 873113 * q^46 - 1574073 * q^47 - 84726 * q^48 + 1893545 * q^49 + 302533 * q^50 + 1214595 * q^51 - 4972548 * q^52 + 587736 * q^53 + 118098 * q^54 - 4624036 * q^55 - 5798506 * q^56 + 1083699 * q^57 - 6991380 * q^58 + 3286064 * q^59 - 3527739 * q^60 - 6117131 * q^61 - 11570258 * q^62 - 1708047 * q^63 - 19063011 * q^64 - 5335514 * q^65 - 2953638 * q^66 - 16518710 * q^67 - 17284669 * q^68 + 76491 * q^69 - 39189486 * q^70 - 10882582 * q^71 + 597051 * q^72 - 21097441 * q^73 - 16717030 * q^74 - 7715142 * q^75 - 40864952 * q^76 - 3404601 * q^77 + 3494340 * q^78 - 3784458 * q^79 - 27466195 * q^80 + 8503056 * q^81 - 24990117 * q^82 - 1951425 * q^83 - 3048030 * q^84 - 23238675 * q^85 - 35910572 * q^86 - 3898125 * q^87 - 27843055 * q^88 + 10499443 * q^89 - 2536191 * q^90 + 699217 * q^91 - 20062766 * q^92 + 3827493 * q^93 - 59358988 * q^94 - 29236333 * q^95 + 978048 * q^96 - 25158976 * q^97 + 2120460 * q^98 + 654642 * q^99

Introduction

L-functions

Modular forms

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