LMFDB - Embedded newform 177.8.a.d.1.3

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 $ x^18 - 6x^17 - 1798x^16 + 11087x^15 + 1326765x^14 - 8403720x^13 - 518334228x^12 + 3375594921x^11 + 115310342333x^10 - 774932111214x^9 - 14600047830166x^8 + 102185027148481x^7 + 988557475638619x^6 - 7379206238519716x^5 - 30152342836849520x^4 + 260578770749067175x^3 + 182609347488069978x^2 - 3481290425710753600*x + 5164646074739714048
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-17.7235$ 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)

$f(q)$	$=$	$q-16.7235 q^{2} +27.0000 q^{3} +151.676 q^{4} -298.819 q^{5} -451.535 q^{6} +200.095 q^{7} -395.941 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-16.7235 q^{2} +27.0000 q^{3} +151.676 q^{4} -298.819 q^{5} -451.535 q^{6} +200.095 q^{7} -395.941 q^{8} +729.000 q^{9} +4997.30 q^{10} +6819.80 q^{11} +4095.24 q^{12} -7653.89 q^{13} -3346.30 q^{14} -8068.11 q^{15} -12793.0 q^{16} -23868.2 q^{17} -12191.4 q^{18} +41453.1 q^{19} -45323.6 q^{20} +5402.58 q^{21} -114051. q^{22} +87045.3 q^{23} -10690.4 q^{24} +11167.7 q^{25} +128000. q^{26} +19683.0 q^{27} +30349.6 q^{28} -28849.4 q^{29} +134927. q^{30} -136082. q^{31} +264624. q^{32} +184135. q^{33} +399161. q^{34} -59792.3 q^{35} +110572. q^{36} -238332. q^{37} -693241. q^{38} -206655. q^{39} +118315. q^{40} -387202. q^{41} -90350.0 q^{42} -560156. q^{43} +1.03440e6 q^{44} -217839. q^{45} -1.45570e6 q^{46} +343766. q^{47} -345410. q^{48} -783505. q^{49} -186763. q^{50} -644443. q^{51} -1.16091e6 q^{52} +1.90828e6 q^{53} -329169. q^{54} -2.03788e6 q^{55} -79226.0 q^{56} +1.11923e6 q^{57} +482464. q^{58} +205379. q^{59} -1.22374e6 q^{60} +2.52219e6 q^{61} +2.27577e6 q^{62} +145870. q^{63} -2.78794e6 q^{64} +2.28713e6 q^{65} -3.07938e6 q^{66} +1.23640e6 q^{67} -3.62023e6 q^{68} +2.35022e6 q^{69} +999937. q^{70} -5.34185e6 q^{71} -288641. q^{72} -4.51521e6 q^{73} +3.98574e6 q^{74} +301528. q^{75} +6.28743e6 q^{76} +1.36461e6 q^{77} +3.45600e6 q^{78} -1.40570e6 q^{79} +3.82278e6 q^{80} +531441. q^{81} +6.47538e6 q^{82} +8.95946e6 q^{83} +819440. q^{84} +7.13228e6 q^{85} +9.36777e6 q^{86} -778935. q^{87} -2.70024e6 q^{88} -1.30962e6 q^{89} +3.64303e6 q^{90} -1.53151e6 q^{91} +1.32027e7 q^{92} -3.67422e6 q^{93} -5.74897e6 q^{94} -1.23870e7 q^{95} +7.14484e6 q^{96} +1.27503e7 q^{97} +1.31029e7 q^{98} +4.97163e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9}+O(q^{10}) $ 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9} + 3609 q^{10} + 15070 q^{11} + 36666 q^{12} + 13662 q^{13} + 20861 q^{14} + 18306 q^{15} + 60482 q^{16} + 71919 q^{17} + 17496 q^{18} + 56231 q^{19} + 143053 q^{20} + 83187 q^{21} + 274198 q^{22} + 150029 q^{23} + 110889 q^{24} + 399672 q^{25} + 182846 q^{26} + 354294 q^{27} + 434150 q^{28} + 591285 q^{29} + 97443 q^{30} + 426733 q^{31} + 1205630 q^{32} + 406890 q^{33} + 403548 q^{34} + 912879 q^{35} + 989982 q^{36} + 7703 q^{37} - 417859 q^{38} + 368874 q^{39} + 618020 q^{40} + 770959 q^{41} + 563247 q^{42} + 793050 q^{43} + 2591274 q^{44} + 494262 q^{45} - 4068019 q^{46} + 1410373 q^{47} + 1633014 q^{48} + 1637427 q^{49} + 1021549 q^{50} + 1941813 q^{51} - 3749190 q^{52} + 1037934 q^{53} + 472392 q^{54} + 331974 q^{55} - 391748 q^{56} + 1518237 q^{57} + 653724 q^{58} + 3696822 q^{59} + 3862431 q^{60} - 1374623 q^{61} + 5251718 q^{62} + 2246049 q^{63} + 5077197 q^{64} + 3257170 q^{65} + 7403346 q^{66} - 2436904 q^{67} + 14119909 q^{68} + 4050783 q^{69} + 5185580 q^{70} + 14289172 q^{71} + 2994003 q^{72} + 5482515 q^{73} + 14934154 q^{74} + 10791144 q^{75} + 3822912 q^{76} + 23157109 q^{77} + 4936842 q^{78} + 19786414 q^{79} + 31978143 q^{80} + 9565938 q^{81} + 9749509 q^{82} + 30227337 q^{83} + 11722050 q^{84} + 9946981 q^{85} + 44295864 q^{86} + 15964695 q^{87} + 39970897 q^{88} + 31061677 q^{89} + 2630961 q^{90} + 26377785 q^{91} + 4719698 q^{92} + 11521791 q^{93} + 44488296 q^{94} + 15534599 q^{95} + 32552010 q^{96} + 12084118 q^{97} + 42274744 q^{98} + 10986030 q^{99}+O(q^{100}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9 + 3609 * q^10 + 15070 * q^11 + 36666 * q^12 + 13662 * q^13 + 20861 * q^14 + 18306 * q^15 + 60482 * q^16 + 71919 * q^17 + 17496 * q^18 + 56231 * q^19 + 143053 * q^20 + 83187 * q^21 + 274198 * q^22 + 150029 * q^23 + 110889 * q^24 + 399672 * q^25 + 182846 * q^26 + 354294 * q^27 + 434150 * q^28 + 591285 * q^29 + 97443 * q^30 + 426733 * q^31 + 1205630 * q^32 + 406890 * q^33 + 403548 * q^34 + 912879 * q^35 + 989982 * q^36 + 7703 * q^37 - 417859 * q^38 + 368874 * q^39 + 618020 * q^40 + 770959 * q^41 + 563247 * q^42 + 793050 * q^43 + 2591274 * q^44 + 494262 * q^45 - 4068019 * q^46 + 1410373 * q^47 + 1633014 * q^48 + 1637427 * q^49 + 1021549 * q^50 + 1941813 * q^51 - 3749190 * q^52 + 1037934 * q^53 + 472392 * q^54 + 331974 * q^55 - 391748 * q^56 + 1518237 * q^57 + 653724 * q^58 + 3696822 * q^59 + 3862431 * q^60 - 1374623 * q^61 + 5251718 * q^62 + 2246049 * q^63 + 5077197 * q^64 + 3257170 * q^65 + 7403346 * q^66 - 2436904 * q^67 + 14119909 * q^68 + 4050783 * q^69 + 5185580 * q^70 + 14289172 * q^71 + 2994003 * q^72 + 5482515 * q^73 + 14934154 * q^74 + 10791144 * q^75 + 3822912 * q^76 + 23157109 * q^77 + 4936842 * q^78 + 19786414 * q^79 + 31978143 * q^80 + 9565938 * q^81 + 9749509 * q^82 + 30227337 * q^83 + 11722050 * q^84 + 9946981 * q^85 + 44295864 * q^86 + 15964695 * q^87 + 39970897 * q^88 + 31061677 * q^89 + 2630961 * q^90 + 26377785 * q^91 + 4719698 * q^92 + 11521791 * q^93 + 44488296 * q^94 + 15534599 * q^95 + 32552010 * q^96 + 12084118 * q^97 + 42274744 * q^98 + 10986030 * 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 $

$2$	−16.7235	−1.47816	−0.739082	−	0.673616i	$-0.764740\pi$
$2$	−16.7235	−1.47816	−0.739082	+	0.673616i	$0.764740\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	151.676	1.18497
$5$	−298.819	−1.06909	−0.534543	−	0.845141i	$-0.679516\pi$
$5$	−298.819	−1.06909	−0.534543	+	0.845141i	$0.679516\pi$
$6$	−451.535	−0.853418
$7$	200.095	0.220493	0.110246	−	0.993904i	$-0.464836\pi$
$7$	200.095	0.220493	0.110246	+	0.993904i	$0.464836\pi$
$8$	−395.941	−0.273411
$9$	729.000	0.333333
$10$	4997.30	1.58028
$11$	6819.80	1.54489	0.772444	−	0.635082i	$-0.219034\pi$
$11$	6819.80	1.54489	0.772444	+	0.635082i	$0.219034\pi$
$12$	4095.24	0.684141
$13$	−7653.89	−0.966230	−0.483115	−	0.875557i	$-0.660495\pi$
$13$	−7653.89	−0.966230	−0.483115	+	0.875557i	$0.660495\pi$
$14$	−3346.30	−0.325924
$15$	−8068.11	−0.617238
$16$	−12793.0	−0.780821
$17$	−23868.2	−1.17828	−0.589141	−	0.808030i	$-0.700534\pi$
$17$	−23868.2	−1.17828	−0.589141	+	0.808030i	$0.700534\pi$
$18$	−12191.4	−0.492721
$19$	41453.1	1.38650	0.693249	−	0.720698i	$-0.256179\pi$
$19$	41453.1	1.38650	0.693249	+	0.720698i	$0.256179\pi$
$20$	−45323.6	−1.26683
$21$	5402.58	0.127301
$22$	−114051.	−2.28360
$23$	87045.3	1.49176	0.745878	−	0.666082i	$-0.232030\pi$
$23$	87045.3	1.49176	0.745878	+	0.666082i	$0.232030\pi$
$24$	−10690.4	−0.157854
$25$	11167.7	0.142947
$26$	128000.	1.42825
$27$	19683.0	0.192450
$28$	30349.6	0.261276
$29$	−28849.4	−0.219657	−0.109828	−	0.993951i	$-0.535030\pi$
$29$	−28849.4	−0.219657	−0.109828	+	0.993951i	$0.535030\pi$
$30$	134927.	0.912378
$31$	−136082.	−0.820419	−0.410209	−	0.911991i	$-0.634544\pi$
$31$	−136082.	−0.820419	−0.410209	+	0.911991i	$0.634544\pi$
$32$	264624.	1.42759
$33$	184135.	0.891942
$34$	399161.	1.74169
$35$	−59792.3	−0.235726
$36$	110572.	0.394989
$37$	−238332.	−0.773527	−0.386764	−	0.922179i	$-0.626407\pi$
$37$	−238332.	−0.773527	−0.386764	+	0.922179i	$0.626407\pi$
$38$	−693241.	−2.04947
$39$	−206655.	−0.557853
$40$	118315.	0.292300
$41$	−387202.	−0.877394	−0.438697	−	0.898635i	$-0.644560\pi$
$41$	−387202.	−0.877394	−0.438697	+	0.898635i	$0.644560\pi$
$42$	−90350.0	−0.188172
$43$	−560156.	−1.07441	−0.537204	−	0.843452i	$-0.680519\pi$
$43$	−560156.	−1.07441	−0.537204	+	0.843452i	$0.680519\pi$
$44$	1.03440e6	1.83064
$45$	−217839.	−0.356362
$46$	−1.45570e6	−2.20506
$47$	343766.	0.482970	0.241485	−	0.970405i	$-0.422365\pi$
$47$	343766.	0.482970	0.241485	+	0.970405i	$0.422365\pi$
$48$	−345410.	−0.450807
$49$	−783505.	−0.951383
$50$	−186763.	−0.211298
$51$	−644443.	−0.680281
$52$	−1.16091e6	−1.14495
$53$	1.90828e6	1.76066	0.880332	−	0.474358i	$-0.157320\pi$
$53$	1.90828e6	1.76066	0.880332	+	0.474358i	$0.157320\pi$
$54$	−329169.	−0.284473
$55$	−2.03788e6	−1.65162
$56$	−79226.0	−0.0602851
$57$	1.11923e6	0.800495
$58$	482464.	0.324689
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	−1.22374e6	−0.731406
$61$	2.52219e6	1.42273	0.711366	−	0.702822i	$-0.248077\pi$
$61$	2.52219e6	1.42273	0.711366	+	0.702822i	$0.248077\pi$
$62$	2.27577e6	1.21271
$63$	145870.	0.0734975
$64$	−2.78794e6	−1.32939
$65$	2.28713e6	1.03298
$66$	−3.07938e6	−1.31844
$67$	1.23640e6	0.502224	0.251112	−	0.967958i	$-0.419204\pi$
$67$	1.23640e6	0.502224	0.251112	+	0.967958i	$0.419204\pi$
$68$	−3.62023e6	−1.39622
$69$	2.35022e6	0.861266
$70$	999937.	0.348441
$71$	−5.34185e6	−1.77128	−0.885640	−	0.464373i	$-0.846280\pi$
$71$	−5.34185e6	−1.77128	−0.885640	+	0.464373i	$0.846280\pi$
$72$	−288641.	−0.0911369
$73$	−4.51521e6	−1.35846	−0.679232	−	0.733923i	$-0.737687\pi$
$73$	−4.51521e6	−1.35846	−0.679232	+	0.733923i	$0.737687\pi$
$74$	3.98574e6	1.14340
$75$	301528.	0.0825303
$76$	6.28743e6	1.64295
$77$	1.36461e6	0.340637
$78$	3.45600e6	0.824598
$79$	−1.40570e6	−0.320772	−0.160386	−	0.987054i	$-0.551274\pi$
$79$	−1.40570e6	−0.320772	−0.160386	+	0.987054i	$0.551274\pi$
$80$	3.82278e6	0.834765
$81$	531441.	0.111111
$82$	6.47538e6	1.29693
$83$	8.95946e6	1.71992	0.859961	−	0.510361i	$-0.170488\pi$
$83$	8.95946e6	1.71992	0.859961	+	0.510361i	$0.170488\pi$
$84$	819440.	0.150848
$85$	7.13228e6	1.25969
$86$	9.36777e6	1.58815
$87$	−778935.	−0.126819
$88$	−2.70024e6	−0.422389
$89$	−1.30962e6	−0.196916	−0.0984581	−	0.995141i	$-0.531391\pi$
$89$	−1.30962e6	−0.196916	−0.0984581	+	0.995141i	$0.531391\pi$
$90$	3.64303e6	0.526762
$91$	−1.53151e6	−0.213047
$92$	1.32027e7	1.76768
$93$	−3.67422e6	−0.473669
$94$	−5.74897e6	−0.713909
$95$	−1.23870e7	−1.48229
$96$	7.14484e6	0.824220
$97$	1.27503e7	1.41847	0.709236	−	0.704971i	$-0.249040\pi$
$97$	1.27503e7	1.41847	0.709236	+	0.704971i	$0.249040\pi$
$98$	1.31029e7	1.40630
$99$	4.97163e6	0.514963
$100$	1.69387e6	0.169387
$101$	1.88135e7	1.81696	0.908478	−	0.417933i	$-0.137245\pi$
$101$	1.88135e7	1.81696	0.908478	+	0.417933i	$0.137245\pi$
$102$	1.07773e7	1.00557
$103$	1.54829e7	1.39611	0.698057	−	0.716042i	$-0.254048\pi$
$103$	1.54829e7	1.39611	0.698057	+	0.716042i	$0.254048\pi$
$104$	3.03049e6	0.264178
$105$	−1.61439e6	−0.136096
$106$	−3.19131e7	−2.60255
$107$	2.71022e6	0.213876	0.106938	−	0.994266i	$-0.465895\pi$
$107$	2.71022e6	0.213876	0.106938	+	0.994266i	$0.465895\pi$
$108$	2.98543e6	0.228047
$109$	4.25363e6	0.314606	0.157303	−	0.987550i	$-0.449720\pi$
$109$	4.25363e6	0.314606	0.157303	+	0.987550i	$0.449720\pi$
$110$	3.40806e7	2.44136
$111$	−6.43495e6	−0.446596
$112$	−2.55981e6	−0.172165
$113$	1.36740e7	0.891501	0.445750	−	0.895157i	$-0.352937\pi$
$113$	1.36740e7	0.891501	0.445750	+	0.895157i	$0.352937\pi$
$114$	−1.87175e7	−1.18326
$115$	−2.60108e7	−1.59482
$116$	−4.37576e6	−0.260286
$117$	−5.57969e6	−0.322077
$118$	−3.43466e6	−0.192440
$119$	−4.77593e6	−0.259802
$120$	3.19450e6	0.168759
$121$	2.70225e7	1.38668
$122$	−4.21798e7	−2.10303
$123$	−1.04545e7	−0.506563
$124$	−2.06404e7	−0.972169
$125$	2.00081e7	0.916264
$126$	−2.43945e6	−0.108641
$127$	3.13310e7	1.35726	0.678628	−	0.734482i	$-0.262575\pi$
$127$	3.13310e7	1.35726	0.678628	+	0.734482i	$0.262575\pi$
$128$	1.27523e7	0.537468
$129$	−1.51242e7	−0.620309
$130$	−3.82488e7	−1.52692
$131$	−2.66364e6	−0.103520	−0.0517602	−	0.998660i	$-0.516483\pi$
$131$	−2.66364e6	−0.103520	−0.0517602	+	0.998660i	$0.516483\pi$
$132$	2.79287e7	1.05692
$133$	8.29457e6	0.305713
$134$	−2.06770e7	−0.742369
$135$	−5.88165e6	−0.205746
$136$	9.45042e6	0.322155
$137$	2.83551e6	0.0942126	0.0471063	−	0.998890i	$-0.485000\pi$
$137$	2.83551e6	0.0942126	0.0471063	+	0.998890i	$0.485000\pi$
$138$	−3.93040e7	−1.27309
$139$	3.75740e7	1.18669	0.593343	−	0.804950i	$-0.297808\pi$
$139$	3.75740e7	1.18669	0.593343	+	0.804950i	$0.297808\pi$
$140$	−9.06904e6	−0.279327
$141$	9.28168e6	0.278843
$142$	8.93344e7	2.61824
$143$	−5.21980e7	−1.49272
$144$	−9.32607e6	−0.260274
$145$	8.62076e6	0.234832
$146$	7.55102e7	2.00803
$147$	−2.11546e7	−0.549281
$148$	−3.61491e7	−0.916604
$149$	2.26234e7	0.560280	0.280140	−	0.959959i	$-0.409619\pi$
$149$	2.26234e7	0.560280	0.280140	+	0.959959i	$0.409619\pi$
$150$	−5.04261e6	−0.121993
$151$	−2.07436e7	−0.490303	−0.245152	−	0.969485i	$-0.578838\pi$
$151$	−2.07436e7	−0.490303	−0.245152	+	0.969485i	$0.578838\pi$
$152$	−1.64130e7	−0.379084
$153$	−1.74000e7	−0.392761
$154$	−2.28211e7	−0.503516
$155$	4.06639e7	0.877099
$156$	−3.13446e7	−0.661037
$157$	−5.37534e7	−1.10855	−0.554277	−	0.832332i	$-0.687005\pi$
$157$	−5.37534e7	−1.10855	−0.554277	+	0.832332i	$0.687005\pi$
$158$	2.35082e7	0.474154
$159$	5.15236e7	1.01652
$160$	−7.90746e7	−1.52622
$161$	1.74174e7	0.328921
$162$	−8.88756e6	−0.164240
$163$	2.39837e7	0.433769	0.216885	−	0.976197i	$-0.430410\pi$
$163$	2.39837e7	0.433769	0.216885	+	0.976197i	$0.430410\pi$
$164$	−5.87292e7	−1.03968
$165$	−5.50229e7	−0.953563
$166$	−1.49834e8	−2.54232
$167$	7.25545e7	1.20547	0.602736	−	0.797941i	$-0.294077\pi$
$167$	7.25545e7	1.20547	0.602736	+	0.797941i	$0.294077\pi$
$168$	−2.13910e6	−0.0348056
$169$	−4.16645e6	−0.0663992
$170$	−1.19277e8	−1.86202
$171$	3.02193e7	0.462166
$172$	−8.49620e7	−1.27314
$173$	8.99082e7	1.32020	0.660098	−	0.751180i	$-0.270515\pi$
$173$	8.99082e7	1.32020	0.660098	+	0.751180i	$0.270515\pi$
$174$	1.30265e7	0.187459
$175$	2.23461e6	0.0315187
$176$	−8.72455e7	−1.20628
$177$	5.54523e6	0.0751646
$178$	2.19015e7	0.291074
$179$	2.66685e7	0.347547	0.173774	−	0.984786i	$-0.444404\pi$
$179$	2.66685e7	0.347547	0.173774	+	0.984786i	$0.444404\pi$
$180$	−3.30409e7	−0.422277
$181$	−1.27888e8	−1.60308	−0.801539	−	0.597943i	$-0.795985\pi$
$181$	−1.27888e8	−1.60308	−0.801539	+	0.597943i	$0.795985\pi$
$182$	2.56122e7	0.314918
$183$	6.80991e7	0.821414
$184$	−3.44648e7	−0.407862
$185$	7.12180e7	0.826968
$186$	6.14459e7	0.700160
$187$	−1.62777e8	−1.82031
$188$	5.21410e7	0.572304
$189$	3.93848e6	0.0424338
$190$	2.07153e8	2.19106
$191$	−3.33882e7	−0.346718	−0.173359	−	0.984859i	$-0.555462\pi$
$191$	−3.33882e7	−0.346718	−0.173359	+	0.984859i	$0.555462\pi$
$192$	−7.52743e7	−0.767525
$193$	6.61151e7	0.661988	0.330994	−	0.943633i	$-0.392616\pi$
$193$	6.61151e7	0.661988	0.330994	+	0.943633i	$0.392616\pi$
$194$	−2.13230e8	−2.09673
$195$	6.17524e7	0.596394
$196$	−1.18839e8	−1.12736
$197$	−7.50492e6	−0.0699381	−0.0349691	−	0.999388i	$-0.511133\pi$
$197$	−7.50492e6	−0.0699381	−0.0349691	+	0.999388i	$0.511133\pi$
$198$	−8.31432e7	−0.761199
$199$	2.88579e7	0.259584	0.129792	−	0.991541i	$-0.458569\pi$
$199$	2.88579e7	0.259584	0.129792	+	0.991541i	$0.458569\pi$
$200$	−4.42175e6	−0.0390832
$201$	3.33828e7	0.289959
$202$	−3.14627e8	−2.68576
$203$	−5.77264e6	−0.0484327
$204$	−9.77463e7	−0.806111
$205$	1.15703e8	0.938010
$206$	−2.58928e8	−2.06368
$207$	6.34560e7	0.497252
$208$	9.79160e7	0.754453
$209$	2.82702e8	2.14199
$210$	2.69983e7	0.201173
$211$	−1.61521e8	−1.18370	−0.591849	−	0.806049i	$-0.701602\pi$
$211$	−1.61521e8	−1.18370	−0.591849	+	0.806049i	$0.701602\pi$
$212$	2.89440e8	2.08633
$213$	−1.44230e8	−1.02265
$214$	−4.53245e7	−0.316144
$215$	1.67385e8	1.14863
$216$	−7.79331e6	−0.0526179
$217$	−2.72294e7	−0.180896
$218$	−7.11356e7	−0.465039
$219$	−1.21911e8	−0.784310
$220$	−3.09098e8	−1.95711
$221$	1.82685e8	1.13849
$222$	1.07615e8	0.660142
$223$	1.37272e8	0.828924	0.414462	−	0.910067i	$-0.363970\pi$
$223$	1.37272e8	0.828924	0.414462	+	0.910067i	$0.363970\pi$
$224$	5.29500e7	0.314773
$225$	8.14126e6	0.0476489
$226$	−2.28678e8	−1.31778
$227$	−5.03590e7	−0.285750	−0.142875	−	0.989741i	$-0.545635\pi$
$227$	−5.03590e7	−0.285750	−0.142875	+	0.989741i	$0.545635\pi$
$228$	1.69761e8	0.948560
$229$	3.29634e8	1.81388	0.906938	−	0.421264i	$-0.138413\pi$
$229$	3.29634e8	1.81388	0.906938	+	0.421264i	$0.138413\pi$
$230$	4.34991e8	2.35740
$231$	3.68445e7	0.196667
$232$	1.14227e7	0.0600565
$233$	4.35459e7	0.225529	0.112764	−	0.993622i	$-0.464029\pi$
$233$	4.35459e7	0.225529	0.112764	+	0.993622i	$0.464029\pi$
$234$	9.33119e7	0.476082
$235$	−1.02724e8	−0.516337
$236$	3.11510e7	0.154270
$237$	−3.79538e7	−0.185198
$238$	7.98703e7	0.384030
$239$	−1.16895e8	−0.553865	−0.276933	−	0.960889i	$-0.589318\pi$
$239$	−1.16895e8	−0.553865	−0.276933	+	0.960889i	$0.589318\pi$
$240$	1.03215e8	0.481952
$241$	−3.76459e8	−1.73244	−0.866219	−	0.499664i	$-0.833457\pi$
$241$	−3.76459e8	−1.73244	−0.866219	+	0.499664i	$0.833457\pi$
$242$	−4.51911e8	−2.04974
$243$	1.43489e7	0.0641500
$244$	3.82555e8	1.68589
$245$	2.34126e8	1.01711
$246$	1.74835e8	0.748783
$247$	−3.17277e8	−1.33968
$248$	5.38806e7	0.224311
$249$	2.41905e8	0.992997
$250$	−3.34606e8	−1.35439
$251$	3.72328e8	1.48617	0.743084	−	0.669198i	$-0.233362\pi$
$251$	3.72328e8	1.48617	0.743084	+	0.669198i	$0.233362\pi$
$252$	2.21249e7	0.0870921
$253$	5.93631e8	2.30460
$254$	−5.23965e8	−2.00625
$255$	1.92572e8	0.727280
$256$	1.43594e8	0.534928
$257$	−4.73110e8	−1.73859	−0.869294	−	0.494296i	$-0.835426\pi$
$257$	−4.73110e8	−1.73859	−0.869294	+	0.494296i	$0.835426\pi$
$258$	2.52930e8	0.916919
$259$	−4.76891e7	−0.170557
$260$	3.46902e8	1.22405
$261$	−2.10312e7	−0.0732189
$262$	4.45454e7	0.153020
$263$	−4.08150e8	−1.38349	−0.691743	−	0.722144i	$-0.743157\pi$
$263$	−4.08150e8	−1.38349	−0.691743	+	0.722144i	$0.743157\pi$
$264$	−7.29065e7	−0.243867
$265$	−5.70230e8	−1.88230
$266$	−1.38714e8	−0.451893
$267$	−3.53598e7	−0.113690
$268$	1.87532e8	0.595119
$269$	5.16986e8	1.61937	0.809684	−	0.586865i	$-0.199638\pi$
$269$	5.16986e8	1.61937	0.809684	+	0.586865i	$0.199638\pi$
$270$	9.83618e7	0.304126
$271$	−5.66936e7	−0.173038	−0.0865189	−	0.996250i	$-0.527574\pi$
$271$	−5.66936e7	−0.173038	−0.0865189	+	0.996250i	$0.527574\pi$
$272$	3.05346e8	0.920027
$273$	−4.13507e7	−0.123003
$274$	−4.74196e7	−0.139262
$275$	7.61615e7	0.220837
$276$	3.56472e8	1.02057
$277$	2.31128e8	0.653392	0.326696	−	0.945129i	$-0.394065\pi$
$277$	2.31128e8	0.653392	0.326696	+	0.945129i	$0.394065\pi$
$278$	−6.28370e8	−1.75412
$279$	−9.92040e7	−0.273473
$280$	2.36742e7	0.0644500
$281$	3.68378e8	0.990426	0.495213	−	0.868772i	$-0.335090\pi$
$281$	3.68378e8	0.990426	0.495213	+	0.868772i	$0.335090\pi$
$282$	−1.55222e8	−0.412176
$283$	5.19507e8	1.36251	0.681254	−	0.732047i	$-0.261435\pi$
$283$	5.19507e8	1.36251	0.681254	+	0.732047i	$0.261435\pi$
$284$	−8.10228e8	−2.09891
$285$	−3.34448e8	−0.855799
$286$	8.72934e8	2.20648
$287$	−7.74774e7	−0.193459
$288$	1.92911e8	0.475864
$289$	1.59354e8	0.388349
$290$	−1.44169e8	−0.347120
$291$	3.44259e8	0.818955
$292$	−6.84848e8	−1.60974
$293$	−8.25048e8	−1.91621	−0.958104	−	0.286421i	$-0.907534\pi$
$293$	−8.25048e8	−1.91621	−0.958104	+	0.286421i	$0.907534\pi$
$294$	3.53780e8	0.811927
$295$	−6.13711e7	−0.139183
$296$	9.43653e7	0.211491
$297$	1.34234e8	0.297314
$298$	−3.78342e8	−0.828186
$299$	−6.66235e8	−1.44138
$300$	4.57345e7	0.0977956
$301$	−1.12085e8	−0.236899
$302$	3.46906e8	0.724748
$303$	5.07964e8	1.04902
$304$	−5.30308e8	−1.08261
$305$	−7.53677e8	−1.52102
$306$	2.90988e8	0.580564
$307$	−2.01480e8	−0.397418	−0.198709	−	0.980059i	$-0.563675\pi$
$307$	−2.01480e8	−0.397418	−0.198709	+	0.980059i	$0.563675\pi$
$308$	2.06978e8	0.403643
$309$	4.18037e8	0.806047
$310$	−6.80044e8	−1.29650
$311$	1.01602e8	0.191533	0.0957663	−	0.995404i	$-0.469470\pi$
$311$	1.01602e8	0.191533	0.0957663	+	0.995404i	$0.469470\pi$
$312$	8.18233e7	0.152523
$313$	−5.09494e8	−0.939148	−0.469574	−	0.882893i	$-0.655592\pi$
$313$	−5.09494e8	−0.939148	−0.469574	+	0.882893i	$0.655592\pi$
$314$	8.98946e8	1.63863
$315$	−4.35886e7	−0.0785752
$316$	−2.13210e8	−0.380104
$317$	5.25634e8	0.926779	0.463389	−	0.886155i	$-0.346633\pi$
$317$	5.25634e8	0.926779	0.463389	+	0.886155i	$0.346633\pi$
$318$	−8.61655e8	−1.50258
$319$	−1.96747e8	−0.339345
$320$	8.33088e8	1.42124
$321$	7.31761e7	0.123481
$322$	−2.91279e8	−0.486199
$323$	−9.89412e8	−1.63369
$324$	8.06067e7	0.131663
$325$	−8.54764e7	−0.138119
$326$	−4.01091e8	−0.641182
$327$	1.14848e8	0.181638
$328$	1.53309e8	0.239889
$329$	6.87860e7	0.106491
$330$	9.20176e8	1.40952
$331$	2.47062e8	0.374463	0.187231	−	0.982316i	$-0.440049\pi$
$331$	2.47062e8	0.374463	0.187231	+	0.982316i	$0.440049\pi$
$332$	1.35893e9	2.03805
$333$	−1.73744e8	−0.257842
$334$	−1.21337e9	−1.78188
$335$	−3.69460e8	−0.536921
$336$	−6.91150e7	−0.0993996
$337$	1.28783e9	1.83296	0.916482	−	0.400076i	$-0.131016\pi$
$337$	1.28783e9	1.83296	0.916482	+	0.400076i	$0.131016\pi$
$338$	6.96777e7	0.0981488
$339$	3.69199e8	0.514708
$340$	1.08179e9	1.49269
$341$	−9.28054e8	−1.26746
$342$	−5.05373e8	−0.683157
$343$	−3.21563e8	−0.430266
$344$	2.21789e8	0.293755
$345$	−7.02291e8	−0.920768
$346$	−1.50358e9	−1.95146
$347$	−1.16047e9	−1.49101	−0.745504	−	0.666502i	$-0.767791\pi$
$347$	−1.16047e9	−1.49101	−0.745504	+	0.666502i	$0.767791\pi$
$348$	−1.18146e8	−0.150276
$349$	1.00669e9	1.26767	0.633836	−	0.773467i	$-0.281479\pi$
$349$	1.00669e9	1.26767	0.633836	+	0.773467i	$0.281479\pi$
$350$	−3.73705e7	−0.0465897
$351$	−1.50652e8	−0.185951
$352$	1.80468e9	2.20547
$353$	2.36378e8	0.286019	0.143010	−	0.989721i	$-0.454322\pi$
$353$	2.36378e8	0.286019	0.143010	+	0.989721i	$0.454322\pi$
$354$	−9.27357e7	−0.111106
$355$	1.59624e9	1.89365
$356$	−1.98638e8	−0.233339
$357$	−1.28950e8	−0.149997
$358$	−4.45992e8	−0.513731
$359$	−4.28699e8	−0.489015	−0.244507	−	0.969647i	$-0.578626\pi$
$359$	−4.28699e8	−0.489015	−0.244507	+	0.969647i	$0.578626\pi$
$360$	8.62514e7	0.0974333
$361$	8.24486e8	0.922377
$362$	2.13873e9	2.36961
$363$	7.29608e8	0.800601
$364$	−2.32293e8	−0.252453
$365$	1.34923e9	1.45232
$366$	−1.13886e9	−1.21418
$367$	1.21528e9	1.28335	0.641677	−	0.766975i	$-0.278239\pi$
$367$	1.21528e9	1.28335	0.641677	+	0.766975i	$0.278239\pi$
$368$	−1.11357e9	−1.16479
$369$	−2.82270e8	−0.292465
$370$	−1.19101e9	−1.22239
$371$	3.81838e8	0.388213
$372$	−5.57290e8	−0.561282
$373$	9.41218e8	0.939094	0.469547	−	0.882907i	$-0.344417\pi$
$373$	9.41218e8	0.939094	0.469547	+	0.882907i	$0.344417\pi$
$374$	2.72220e9	2.69072
$375$	5.40219e8	0.529006
$376$	−1.36111e8	−0.132049
$377$	2.20811e8	0.212239
$378$	−6.58652e7	−0.0627241
$379$	2.27326e8	0.214492	0.107246	−	0.994233i	$-0.465797\pi$
$379$	2.27326e8	0.214492	0.107246	+	0.994233i	$0.465797\pi$
$380$	−1.87880e9	−1.75646
$381$	8.45938e8	0.783612
$382$	5.58368e8	0.512506
$383$	1.40852e9	1.28105	0.640525	−	0.767938i	$-0.278717\pi$
$383$	1.40852e9	1.28105	0.640525	+	0.767938i	$0.278717\pi$
$384$	3.44311e8	0.310307
$385$	−4.07771e8	−0.364170
$386$	−1.10568e9	−0.978527
$387$	−4.08353e8	−0.358136
$388$	1.93392e9	1.68084
$389$	1.22433e9	1.05457	0.527285	−	0.849689i	$-0.323210\pi$
$389$	1.22433e9	1.05457	0.527285	+	0.849689i	$0.323210\pi$
$390$	−1.03272e9	−0.881567
$391$	−2.07762e9	−1.75771
$392$	3.10222e8	0.260118
$393$	−7.19183e7	−0.0597676
$394$	1.25509e8	0.103380
$395$	4.20049e8	0.342933
$396$	7.54076e8	0.610214
$397$	−3.38844e8	−0.271790	−0.135895	−	0.990723i	$-0.543391\pi$
$397$	−3.38844e8	−0.271790	−0.135895	+	0.990723i	$0.543391\pi$
$398$	−4.82605e8	−0.383708
$399$	2.23953e8	0.176503
$400$	−1.42868e8	−0.111616
$401$	1.32434e8	0.102564	0.0512820	−	0.998684i	$-0.483669\pi$
$401$	1.32434e8	0.102564	0.0512820	+	0.998684i	$0.483669\pi$
$402$	−5.58278e8	−0.428607
$403$	1.04156e9	0.792713
$404$	2.85355e9	2.15303
$405$	−1.58805e8	−0.118787
$406$	9.65388e7	0.0715914
$407$	−1.62537e9	−1.19501
$408$	2.55161e8	0.185996
$409$	1.81103e9	1.30886	0.654431	−	0.756122i	$-0.272908\pi$
$409$	1.81103e9	1.30886	0.654431	+	0.756122i	$0.272908\pi$
$410$	−1.93497e9	−1.38653
$411$	7.65587e7	0.0543936
$412$	2.34837e9	1.65435
$413$	4.10954e7	0.0287057
$414$	−1.06121e9	−0.735020
$415$	−2.67726e9	−1.83874
$416$	−2.02540e9	−1.37938
$417$	1.01450e9	0.685134
$418$	−4.72776e9	−3.16620
$419$	−2.75900e9	−1.83233	−0.916164	−	0.400805i	$-0.868731\pi$
$419$	−2.75900e9	−1.83233	−0.916164	+	0.400805i	$0.868731\pi$
$420$	−2.44864e8	−0.161270
$421$	−6.24044e8	−0.407594	−0.203797	−	0.979013i	$-0.565328\pi$
$421$	−6.24044e8	−0.407594	−0.203797	+	0.979013i	$0.565328\pi$
$422$	2.70120e9	1.74970
$423$	2.50605e8	0.160990
$424$	−7.55566e8	−0.481385
$425$	−2.66554e8	−0.168431
$426$	2.41203e9	1.51164
$427$	5.04678e8	0.313702
$428$	4.11075e8	0.253436
$429$	−1.40935e9	−0.861821
$430$	−2.79927e9	−1.69787
$431$	2.11891e9	1.27480	0.637401	−	0.770532i	$-0.280010\pi$
$431$	2.11891e9	1.27480	0.637401	+	0.770532i	$0.280010\pi$
$432$	−2.51804e8	−0.150269
$433$	−1.12211e9	−0.664244	−0.332122	−	0.943236i	$-0.607765\pi$
$433$	−1.12211e9	−0.664244	−0.332122	+	0.943236i	$0.607765\pi$
$434$	4.55372e8	0.267394
$435$	2.32760e8	0.135580
$436$	6.45172e8	0.372797
$437$	3.60830e9	2.06832
$438$	2.03878e9	1.15934
$439$	−1.19818e9	−0.675922	−0.337961	−	0.941160i	$-0.609737\pi$
$439$	−1.19818e9	−0.675922	−0.337961	+	0.941160i	$0.609737\pi$
$440$	8.06882e8	0.451571
$441$	−5.71175e8	−0.317128
$442$	−3.05513e9	−1.68288
$443$	−5.64409e8	−0.308447	−0.154224	−	0.988036i	$-0.549288\pi$
$443$	−5.64409e8	−0.308447	−0.154224	+	0.988036i	$0.549288\pi$
$444$	−9.76026e8	−0.529201
$445$	3.91340e8	0.210521
$446$	−2.29567e9	−1.22529
$447$	6.10831e8	0.323478
$448$	−5.57854e8	−0.293121
$449$	−1.48373e9	−0.773558	−0.386779	−	0.922172i	$-0.626412\pi$
$449$	−1.48373e9	−0.773558	−0.386779	+	0.922172i	$0.626412\pi$
$450$	−1.36150e8	−0.0704328
$451$	−2.64064e9	−1.35548
$452$	2.07402e9	1.05640
$453$	−5.60077e8	−0.283077
$454$	8.42179e8	0.422386
$455$	4.57644e8	0.227765
$456$	−4.43150e8	−0.218864
$457$	−1.51906e9	−0.744505	−0.372253	−	0.928131i	$-0.621415\pi$
$457$	−1.51906e9	−0.744505	−0.372253	+	0.928131i	$0.621415\pi$
$458$	−5.51264e9	−2.68121
$459$	−4.69799e8	−0.226760
$460$	−3.94520e9	−1.88980
$461$	8.02602e8	0.381546	0.190773	−	0.981634i	$-0.438901\pi$
$461$	8.02602e8	0.381546	0.190773	+	0.981634i	$0.438901\pi$
$462$	−6.16169e8	−0.290705
$463$	9.47165e8	0.443499	0.221749	−	0.975104i	$-0.428823\pi$
$463$	9.47165e8	0.443499	0.221749	+	0.975104i	$0.428823\pi$
$464$	3.69070e8	0.171513
$465$	1.09793e9	0.506393
$466$	−7.28241e8	−0.333368
$467$	1.58449e9	0.719914	0.359957	−	0.932969i	$-0.382791\pi$
$467$	1.58449e9	0.719914	0.359957	+	0.932969i	$0.382791\pi$
$468$	−8.46303e8	−0.381650
$469$	2.47398e8	0.110737
$470$	1.71790e9	0.763231
$471$	−1.45134e9	−0.640025
$472$	−8.13180e7	−0.0355951
$473$	−3.82015e9	−1.65984
$474$	6.34721e8	0.273753
$475$	4.62936e8	0.198195
$476$	−7.24392e8	−0.307857
$477$	1.39114e9	0.586888
$478$	1.95490e9	0.818703
$479$	−2.85017e6	−0.00118494	−0.000592471	−	1.00000i	$-0.500189\pi$
$479$	−2.85017e6	−0.00118494	−0.000592471		1.00000i	$0.500189\pi$
$480$	−2.13501e9	−0.881163
$481$	1.82416e9	0.747405
$482$	6.29571e9	2.56083
$483$	4.70269e8	0.189903
$484$	4.09866e9	1.64317
$485$	−3.81004e9	−1.51647
$486$	−2.39964e8	−0.0948242
$487$	1.28087e9	0.502521	0.251260	−	0.967920i	$-0.419155\pi$
$487$	1.28087e9	0.502521	0.251260	+	0.967920i	$0.419155\pi$
$488$	−9.98638e8	−0.388990
$489$	6.47559e8	0.250437
$490$	−3.91541e9	−1.50346
$491$	−1.37484e8	−0.0524164	−0.0262082	−	0.999657i	$-0.508343\pi$
$491$	−1.37484e8	−0.0524164	−0.0262082	+	0.999657i	$0.508343\pi$
$492$	−1.58569e9	−0.600261
$493$	6.88586e8	0.258818
$494$	5.30599e9	1.98026
$495$	−1.48562e9	−0.550540
$496$	1.74090e9	0.640600
$497$	−1.06888e9	−0.390554
$498$	−4.04551e9	−1.46781
$499$	1.97156e8	0.0710327	0.0355163	−	0.999369i	$-0.488692\pi$
$499$	1.97156e8	0.0710327	0.0355163	+	0.999369i	$0.488692\pi$
$500$	3.03474e9	1.08574
$501$	1.95897e9	0.695979
$502$	−6.22663e9	−2.19680
$503$	−2.00636e9	−0.702946	−0.351473	−	0.936198i	$-0.614319\pi$
$503$	−2.00636e9	−0.702946	−0.351473	+	0.936198i	$0.614319\pi$
$504$	−5.77558e7	−0.0200950
$505$	−5.62182e9	−1.94248
$506$	−9.92760e9	−3.40657
$507$	−1.12494e8	−0.0383356
$508$	4.75216e9	1.60830
$509$	4.88057e9	1.64043	0.820216	−	0.572053i	$-0.193853\pi$
$509$	4.88057e9	1.64043	0.820216	+	0.572053i	$0.193853\pi$
$510$	−3.22047e9	−1.07504
$511$	−9.03474e8	−0.299531
$512$	−4.03368e9	−1.32818
$513$	8.15921e8	0.266832
$514$	7.91206e9	2.56992
$515$	−4.62657e9	−1.49257
$516$	−2.29397e9	−0.735046
$517$	2.34442e9	0.746135
$518$	7.97528e8	0.252111
$519$	2.42752e9	0.762215
$520$	−9.05568e8	−0.282429
$521$	3.48111e9	1.07842	0.539208	−	0.842173i	$-0.318724\pi$
$521$	3.48111e9	1.07842	0.539208	+	0.842173i	$0.318724\pi$
$522$	3.51716e8	0.108230
$523$	−8.79645e8	−0.268876	−0.134438	−	0.990922i	$-0.542923\pi$
$523$	−8.79645e8	−0.268876	−0.134438	+	0.990922i	$0.542923\pi$
$524$	−4.04010e8	−0.122668
$525$	6.03344e7	0.0181973
$526$	6.82570e9	2.04502
$527$	3.24804e9	0.966685
$528$	−2.35563e9	−0.696447
$529$	4.17206e9	1.22534
$530$	9.53625e9	2.78235
$531$	1.49721e8	0.0433963
$532$	1.25809e9	0.362259
$533$	2.96360e9	0.847764
$534$	5.91341e8	0.168052
$535$	−8.09866e8	−0.228652
$536$	−4.89542e8	−0.137314
$537$	7.20051e8	0.200656
$538$	−8.64582e9	−2.39369
$539$	−5.34335e9	−1.46978
$540$	−8.92104e8	−0.243802
$541$	−5.31588e9	−1.44339	−0.721697	−	0.692209i	$-0.756638\pi$
$541$	−5.31588e9	−1.44339	−0.721697	+	0.692209i	$0.756638\pi$
$542$	9.48115e8	0.255778
$543$	−3.45297e9	−0.925537
$544$	−6.31611e9	−1.68211
$545$	−1.27106e9	−0.336341
$546$	6.91529e8	0.181818
$547$	4.68079e9	1.22282	0.611412	−	0.791313i	$-0.290602\pi$
$547$	4.68079e9	1.22282	0.611412	+	0.791313i	$0.290602\pi$
$548$	4.30078e8	0.111639
$549$	1.83867e9	0.474244
$550$	−1.27369e9	−0.326433
$551$	−1.19590e9	−0.304554
$552$	−9.30550e8	−0.235479
$553$	−2.81274e8	−0.0707279
$554$	−3.86527e9	−0.965820
$555$	1.92288e9	0.477450
$556$	5.69907e9	1.40618
$557$	9.19162e8	0.225371	0.112686	−	0.993631i	$-0.464055\pi$
$557$	9.19162e8	0.225371	0.112686	+	0.993631i	$0.464055\pi$
$558$	1.65904e9	0.404238
$559$	4.28737e9	1.03813
$560$	7.64921e8	0.184060
$561$	−4.39497e9	−1.05096
$562$	−6.16058e9	−1.46401
$563$	2.14197e9	0.505865	0.252933	−	0.967484i	$-0.418605\pi$
$563$	2.14197e9	0.505865	0.252933	+	0.967484i	$0.418605\pi$
$564$	1.40781e9	0.330420
$565$	−4.08606e9	−0.953092
$566$	−8.68798e9	−2.01401
$567$	1.06339e8	0.0244992
$568$	2.11506e9	0.484287
$569$	−6.07580e9	−1.38265	−0.691323	−	0.722546i	$-0.742972\pi$
$569$	−6.07580e9	−1.38265	−0.691323	+	0.722546i	$0.742972\pi$
$570$	5.59314e9	1.26501
$571$	−6.78352e9	−1.52486	−0.762428	−	0.647073i	$-0.775993\pi$
$571$	−6.78352e9	−1.52486	−0.762428	+	0.647073i	$0.775993\pi$
$572$	−7.91717e9	−1.76882
$573$	−9.01481e8	−0.200178
$574$	1.29569e9	0.285964
$575$	9.72096e8	0.213242
$576$	−2.03241e9	−0.443131
$577$	−2.20797e9	−0.478495	−0.239248	−	0.970959i	$-0.576901\pi$
$577$	−2.20797e9	−0.478495	−0.239248	+	0.970959i	$0.576901\pi$
$578$	−2.66497e9	−0.574043
$579$	1.78511e9	0.382199
$580$	1.30756e9	0.278268
$581$	1.79275e9	0.379230
$582$	−5.75722e9	−1.21055
$583$	1.30141e10	2.72003
$584$	1.78776e9	0.371419
$585$	1.66732e9	0.344328
$586$	1.37977e10	2.83247
$587$	−3.43972e9	−0.701924	−0.350962	−	0.936390i	$-0.614145\pi$
$587$	−3.43972e9	−0.701924	−0.350962	+	0.936390i	$0.614145\pi$
$588$	−3.20864e9	−0.650880
$589$	−5.64103e9	−1.13751
$590$	1.02634e9	0.205736
$591$	−2.02633e8	−0.0403788
$592$	3.04897e9	0.603986
$593$	−3.62804e9	−0.714464	−0.357232	−	0.934016i	$-0.616279\pi$
$593$	−3.62804e9	−0.714464	−0.357232	+	0.934016i	$0.616279\pi$
$594$	−2.24487e9	−0.439479
$595$	1.42714e9	0.277751
$596$	3.43142e9	0.663914
$597$	7.79162e8	0.149871
$598$	1.11418e10	2.13059
$599$	3.45165e9	0.656194	0.328097	−	0.944644i	$-0.393593\pi$
$599$	3.45165e9	0.656194	0.328097	+	0.944644i	$0.393593\pi$
$600$	−1.19387e8	−0.0225647
$601$	7.18642e9	1.35037	0.675184	−	0.737649i	$-0.264064\pi$
$601$	7.18642e9	1.35037	0.675184	+	0.737649i	$0.264064\pi$
$602$	1.87445e9	0.350175
$603$	9.01337e8	0.167408
$604$	−3.14630e9	−0.580993
$605$	−8.07483e9	−1.48248
$606$	−8.49494e9	−1.55062
$607$	7.76353e9	1.40896	0.704480	−	0.709724i	$-0.251180\pi$
$607$	7.76353e9	1.40896	0.704480	+	0.709724i	$0.251180\pi$
$608$	1.09695e10	1.97935
$609$	−1.55861e8	−0.0279626
$610$	1.26041e10	2.24832
$611$	−2.63115e9	−0.466660
$612$	−2.63915e9	−0.465408
$613$	2.87864e9	0.504750	0.252375	−	0.967630i	$-0.418788\pi$
$613$	2.87864e9	0.504750	0.252375	+	0.967630i	$0.418788\pi$
$614$	3.36945e9	0.587449
$615$	3.12399e9	0.541560
$616$	−5.40306e8	−0.0931337
$617$	9.59531e9	1.64460	0.822301	−	0.569053i	$-0.192690\pi$
$617$	9.59531e9	1.64460	0.822301	+	0.569053i	$0.192690\pi$
$618$	−6.99105e9	−1.19147
$619$	6.93284e8	0.117488	0.0587441	−	0.998273i	$-0.481290\pi$
$619$	6.93284e8	0.117488	0.0587441	+	0.998273i	$0.481290\pi$
$620$	6.16773e9	1.03933
$621$	1.71331e9	0.287089
$622$	−1.69915e9	−0.283116
$623$	−2.62050e8	−0.0434186
$624$	2.64373e9	0.435583
$625$	−6.85127e9	−1.12251
$626$	8.52053e9	1.38821
$627$	7.63295e9	1.23668
$628$	−8.15309e9	−1.31360
$629$	5.68856e9	0.911433
$630$	7.28954e8	0.116147
$631$	4.90432e9	0.777099	0.388549	−	0.921428i	$-0.372976\pi$
$631$	4.90432e9	0.777099	0.388549	+	0.921428i	$0.372976\pi$
$632$	5.56573e8	0.0877026
$633$	−4.36107e9	−0.683408
$634$	−8.79044e9	−1.36993
$635$	−9.36231e9	−1.45102
$636$	7.81487e9	1.20454
$637$	5.99686e9	0.919255
$638$	3.29031e9	0.501608
$639$	−3.89421e9	−0.590427
$640$	−3.81062e9	−0.574599
$641$	−9.16838e9	−1.37496	−0.687479	−	0.726204i	$-0.741283\pi$
$641$	−9.16838e9	−1.37496	−0.687479	+	0.726204i	$0.741283\pi$
$642$	−1.22376e9	−0.182526
$643$	4.70285e9	0.697626	0.348813	−	0.937192i	$-0.386585\pi$
$643$	4.70285e9	0.697626	0.348813	+	0.937192i	$0.386585\pi$
$644$	2.64179e9	0.389761
$645$	4.51940e9	0.663165
$646$	1.65464e10	2.41485
$647$	5.90195e9	0.856703	0.428352	−	0.903612i	$-0.359094\pi$
$647$	5.90195e9	0.856703	0.428352	+	0.903612i	$0.359094\pi$
$648$	−2.10419e8	−0.0303790
$649$	1.40064e9	0.201127
$650$	1.42947e9	0.204163
$651$	−7.35195e8	−0.104441
$652$	3.63774e9	0.514002
$653$	7.53884e9	1.05952	0.529759	−	0.848148i	$-0.322282\pi$
$653$	7.53884e9	1.05952	0.529759	+	0.848148i	$0.322282\pi$
$654$	−1.92066e9	−0.268490
$655$	7.95947e8	0.110672
$656$	4.95347e9	0.685087
$657$	−3.29159e9	−0.452822
$658$	−1.15034e9	−0.157412
$659$	−8.90395e9	−1.21195	−0.605974	−	0.795484i	$-0.707216\pi$
$659$	−8.90395e9	−1.21195	−0.605974	+	0.795484i	$0.707216\pi$
$660$	−8.34564e9	−1.12994
$661$	−2.51361e9	−0.338527	−0.169264	−	0.985571i	$-0.554139\pi$
$661$	−2.51361e9	−0.338527	−0.169264	+	0.985571i	$0.554139\pi$
$662$	−4.13175e9	−0.553517
$663$	4.93249e9	0.657309
$664$	−3.54742e9	−0.470245
$665$	−2.47857e9	−0.326833
$666$	2.90560e9	0.381133
$667$	−2.51121e9	−0.327674
$668$	1.10048e10	1.42844
$669$	3.70634e9	0.478580
$670$	6.17867e9	0.793657
$671$	1.72008e10	2.19796
$672$	1.42965e9	0.181734
$673$	1.80021e9	0.227651	0.113826	−	0.993501i	$-0.463690\pi$
$673$	1.80021e9	0.227651	0.113826	+	0.993501i	$0.463690\pi$
$674$	−2.15370e10	−2.70942
$675$	2.19814e8	0.0275101
$676$	−6.31949e8	−0.0786808
$677$	7.89862e9	0.978343	0.489171	−	0.872188i	$-0.337299\pi$
$677$	7.89862e9	0.978343	0.489171	+	0.872188i	$0.337299\pi$
$678$	−6.17430e9	−0.760823
$679$	2.55128e9	0.312762
$680$	−2.82396e9	−0.344412
$681$	−1.35969e9	−0.164978
$682$	1.55203e10	1.87351
$683$	3.31446e9	0.398053	0.199026	−	0.979994i	$-0.436222\pi$
$683$	3.31446e9	0.398053	0.199026	+	0.979994i	$0.436222\pi$
$684$	4.58353e9	0.547651
$685$	−8.47303e8	−0.100721
$686$	5.37766e9	0.636003
$687$	8.90012e9	1.04724
$688$	7.16605e9	0.838920
$689$	−1.46058e10	−1.70121
$690$	1.17448e10	1.36105
$691$	−5.78938e9	−0.667512	−0.333756	−	0.942659i	$-0.608316\pi$
$691$	−5.78938e9	−0.667512	−0.333756	+	0.942659i	$0.608316\pi$
$692$	1.36369e10	1.56439
$693$	9.94801e8	0.113546
$694$	1.94071e10	2.20395
$695$	−1.12278e10	−1.26867
$696$	3.08412e8	0.0346737
$697$	9.24184e9	1.03382
$698$	−1.68354e10	−1.87383
$699$	1.17574e9	0.130209
$700$	3.38936e8	0.0373486
$701$	−1.14194e10	−1.25207	−0.626036	−	0.779794i	$-0.715324\pi$
$701$	−1.14194e10	−1.25207	−0.626036	+	0.779794i	$0.715324\pi$
$702$	2.51942e9	0.274866
$703$	−9.87958e9	−1.07249
$704$	−1.90132e10	−2.05376
$705$	−2.77354e9	−0.298107
$706$	−3.95307e9	−0.422783
$707$	3.76449e9	0.400625
$708$	8.41077e8	0.0890675
$709$	−1.23857e10	−1.30515	−0.652575	−	0.757725i	$-0.726311\pi$
$709$	−1.23857e10	−1.30515	−0.652575	+	0.757725i	$0.726311\pi$
$710$	−2.66948e10	−2.79913
$711$	−1.02475e9	−0.106924
$712$	5.18534e8	0.0538390
$713$	−1.18453e10	−1.22386
$714$	2.15650e9	0.221720
$715$	1.55978e10	1.59585
$716$	4.04497e9	0.411832
$717$	−3.15617e9	−0.319774
$718$	7.16935e9	0.722844
$719$	−1.16266e10	−1.16655	−0.583274	−	0.812275i	$-0.698229\pi$
$719$	−1.16266e10	−1.16655	−0.583274	+	0.812275i	$0.698229\pi$
$720$	2.78681e9	0.278255
$721$	3.09805e9	0.307833
$722$	−1.37883e10	−1.36342
$723$	−1.01644e10	−1.00022
$724$	−1.93975e10	−1.89959
$725$	−3.22182e8	−0.0313992
$726$	−1.22016e10	−1.18342
$727$	8.72736e9	0.842389	0.421194	−	0.906970i	$-0.361611\pi$
$727$	8.72736e9	0.842389	0.421194	+	0.906970i	$0.361611\pi$
$728$	6.06387e8	0.0582492
$729$	3.87420e8	0.0370370
$730$	−2.25639e10	−2.14676
$731$	1.33699e10	1.26596
$732$	1.03290e10	0.973349
$733$	2.02816e10	1.90212	0.951062	−	0.309000i	$-0.0999943\pi$
$733$	2.02816e10	1.90212	0.951062	+	0.309000i	$0.0999943\pi$
$734$	−2.03238e10	−1.89701
$735$	6.32140e9	0.587229
$736$	2.30343e10	2.12962
$737$	8.43201e9	0.775881
$738$	4.72055e9	0.432310
$739$	−3.02731e9	−0.275932	−0.137966	−	0.990437i	$-0.544056\pi$
$739$	−3.02731e9	−0.275932	−0.137966	+	0.990437i	$0.544056\pi$
$740$	1.08020e10	0.979929
$741$	−8.56649e9	−0.773462
$742$	−6.38567e9	−0.573843
$743$	1.35696e10	1.21369	0.606843	−	0.794822i	$-0.292436\pi$
$743$	1.35696e10	1.21369	0.606843	+	0.794822i	$0.292436\pi$
$744$	1.45478e9	0.129506
$745$	−6.76029e9	−0.598988
$746$	−1.57405e10	−1.38813
$747$	6.53145e9	0.573307
$748$	−2.46893e10	−2.15701
$749$	5.42304e8	0.0471581
$750$	−9.03435e9	−0.781957
$751$	−1.38238e10	−1.19094	−0.595469	−	0.803378i	$-0.703034\pi$
$751$	−1.38238e10	−1.19094	−0.595469	+	0.803378i	$0.703034\pi$
$752$	−4.39779e9	−0.377113
$753$	1.00529e10	0.858039
$754$	−3.69273e9	−0.313724
$755$	6.19858e9	0.524177
$756$	5.97372e8	0.0502827
$757$	−1.53342e9	−0.128477	−0.0642386	−	0.997935i	$-0.520462\pi$
$757$	−1.53342e9	−0.128477	−0.0642386	+	0.997935i	$0.520462\pi$
$758$	−3.80168e9	−0.317054
$759$	1.60280e10	1.33056
$760$	4.90451e9	0.405273
$761$	−8.31351e9	−0.683814	−0.341907	−	0.939734i	$-0.611073\pi$
$761$	−8.31351e9	−0.683814	−0.341907	+	0.939734i	$0.611073\pi$
$762$	−1.41471e10	−1.15831
$763$	8.51131e8	0.0693682
$764$	−5.06418e9	−0.410849
$765$	5.19943e9	0.419895
$766$	−2.35553e10	−1.89360
$767$	−1.57195e9	−0.125792
$768$	3.87703e9	0.308841
$769$	1.21279e10	0.961710	0.480855	−	0.876800i	$-0.340326\pi$
$769$	1.21279e10	0.961710	0.480855	+	0.876800i	$0.340326\pi$
$770$	6.81937e9	0.538303
$771$	−1.27740e10	−1.00377
$772$	1.00281e10	0.784434
$773$	−8.47150e9	−0.659678	−0.329839	−	0.944037i	$-0.606994\pi$
$773$	−8.47150e9	−0.659678	−0.329839	+	0.944037i	$0.606994\pi$
$774$	6.82910e9	0.529383
$775$	−1.51973e9	−0.117276
$776$	−5.04838e9	−0.387825
$777$	−1.28760e9	−0.0984711
$778$	−2.04751e10	−1.55883
$779$	−1.60507e10	−1.21650
$780$	9.36635e9	0.706706
$781$	−3.64303e10	−2.73643
$782$	3.47451e10	2.59818
$783$	−5.67844e8	−0.0422730
$784$	1.00234e10	0.742860
$785$	1.60625e10	1.18514
$786$	1.20273e9	0.0883463
$787$	−1.35820e10	−0.993233	−0.496617	−	0.867970i	$-0.665424\pi$
$787$	−1.35820e10	−0.993233	−0.496617	+	0.867970i	$0.665424\pi$
$788$	−1.13831e9	−0.0828744
$789$	−1.10200e10	−0.798756
$790$	−7.02469e9	−0.506912
$791$	2.73611e9	0.196569
$792$	−1.96847e9	−0.140796
$793$	−1.93045e10	−1.37469
$794$	5.66666e9	0.401750
$795$	−1.53962e10	−1.08675
$796$	4.37704e9	0.307599
$797$	−2.11682e10	−1.48108	−0.740542	−	0.672010i	$-0.765431\pi$
$797$	−2.11682e10	−1.48108	−0.740542	+	0.672010i	$0.765431\pi$
$798$	−3.74529e9	−0.260901
$799$	−8.20509e9	−0.569075
$800$	2.95524e9	0.204069
$801$	−9.54716e8	−0.0656388
$802$	−2.21477e9	−0.151606
$803$	−3.07929e10	−2.09868
$804$	5.06337e9	0.343592
$805$	−5.20464e9	−0.351645
$806$	−1.74185e10	−1.17176
$807$	1.39586e10	0.934943
$808$	−7.44903e9	−0.496775
$809$	−5.28973e9	−0.351248	−0.175624	−	0.984457i	$-0.556194\pi$
$809$	−5.28973e9	−0.351248	−0.175624	+	0.984457i	$0.556194\pi$
$810$	2.65577e9	0.175587
$811$	2.57240e10	1.69342	0.846710	−	0.532054i	$-0.178580\pi$
$811$	2.57240e10	1.69342	0.846710	+	0.532054i	$0.178580\pi$
$812$	−8.75570e8	−0.0573911
$813$	−1.53073e9	−0.0999035
$814$	2.71819e10	1.76642
$815$	−7.16677e9	−0.463737
$816$	8.24433e9	0.531178
$817$	−2.32202e10	−1.48966
$818$	−3.02868e10	−1.93471
$819$	−1.11647e9	−0.0710155
$820$	1.75494e10	1.11151
$821$	−3.97061e8	−0.0250413	−0.0125206	−	0.999922i	$-0.503986\pi$
$821$	−3.97061e8	−0.0250413	−0.0125206	+	0.999922i	$0.503986\pi$
$822$	−1.28033e9	−0.0804027
$823$	−2.17465e10	−1.35985	−0.679923	−	0.733283i	$-0.737987\pi$
$823$	−2.17465e10	−1.35985	−0.679923	+	0.733283i	$0.737987\pi$
$824$	−6.13030e9	−0.381713
$825$	2.05636e9	0.127500
$826$	−6.87259e8	−0.0424317
$827$	1.11227e10	0.683821	0.341910	−	0.939733i	$-0.388926\pi$
$827$	1.11227e10	0.683821	0.341910	+	0.939733i	$0.388926\pi$
$828$	9.62474e9	0.589227
$829$	−9.06605e9	−0.552684	−0.276342	−	0.961059i	$-0.589122\pi$
$829$	−9.06605e9	−0.552684	−0.276342	+	0.961059i	$0.589122\pi$
$830$	4.47731e10	2.71797
$831$	6.24046e9	0.377236
$832$	2.13386e10	1.28450
$833$	1.87009e10	1.12100
$834$	−1.69660e10	−1.01274
$835$	−2.16807e10	−1.28875
$836$	4.28790e10	2.53818
$837$	−2.67851e9	−0.157890
$838$	4.61402e10	2.70848
$839$	5.43132e9	0.317496	0.158748	−	0.987319i	$-0.449254\pi$
$839$	5.43132e9	0.317496	0.158748	+	0.987319i	$0.449254\pi$
$840$	6.39204e8	0.0372102
$841$	−1.64176e10	−0.951751
$842$	1.04362e10	0.602491
$843$	9.94622e9	0.571823
$844$	−2.44988e10	−1.40264
$845$	1.24501e9	0.0709865
$846$	−4.19100e9	−0.237970
$847$	5.40708e9	0.305753
$848$	−2.44126e10	−1.37476
$849$	1.40267e10	0.786644
$850$	4.45771e9	0.248969
$851$	−2.07456e10	−1.15391
$852$	−2.18762e10	−1.21180
$853$	−6.82993e9	−0.376786	−0.188393	−	0.982094i	$-0.560328\pi$
$853$	−6.82993e9	−0.376786	−0.188393	+	0.982094i	$0.560328\pi$
$854$	−8.43999e9	−0.463702
$855$	−9.03010e9	−0.494096
$856$	−1.07309e9	−0.0584760
$857$	3.60369e10	1.95576	0.977878	−	0.209178i	$-0.0670788\pi$
$857$	3.60369e10	1.95576	0.977878	+	0.209178i	$0.0670788\pi$
$858$	2.35692e10	1.27391
$859$	1.90837e10	1.02727	0.513637	−	0.858008i	$-0.328298\pi$
$859$	1.90837e10	1.02727	0.513637	+	0.858008i	$0.328298\pi$
$860$	2.53882e10	1.36109
$861$	−2.09189e9	−0.111693
$862$	−3.54357e10	−1.88437
$863$	−3.33278e10	−1.76510	−0.882550	−	0.470218i	$-0.844175\pi$
$863$	−3.33278e10	−1.76510	−0.882550	+	0.470218i	$0.844175\pi$
$864$	5.20859e9	0.274740
$865$	−2.68663e10	−1.41140
$866$	1.87656e10	0.981861
$867$	4.30257e9	0.224213
$868$	−4.13004e9	−0.214356
$869$	−9.58657e9	−0.495558
$870$	−3.89257e9	−0.200410
$871$	−9.46328e9	−0.485264
$872$	−1.68419e9	−0.0860166
$873$	9.29500e9	0.472824
$874$	−6.03434e10	−3.05731
$875$	4.00353e9	0.202030
$876$	−1.84909e10	−0.929381
$877$	2.92873e10	1.46616	0.733079	−	0.680143i	$-0.238082\pi$
$877$	2.92873e10	1.46616	0.733079	+	0.680143i	$0.238082\pi$
$878$	2.00378e10	0.999124
$879$	−2.22763e10	−1.10632
$880$	2.60706e10	1.28962
$881$	−7.06390e9	−0.348040	−0.174020	−	0.984742i	$-0.555676\pi$
$881$	−7.06390e9	−0.348040	−0.174020	+	0.984742i	$0.555676\pi$
$882$	9.55205e9	0.468766
$883$	−2.11700e10	−1.03480	−0.517401	−	0.855743i	$-0.673100\pi$
$883$	−2.11700e10	−1.03480	−0.517401	+	0.855743i	$0.673100\pi$
$884$	2.77089e10	1.34907
$885$	−1.65702e9	−0.0803575
$886$	9.43890e9	0.455935
$887$	3.92843e9	0.189011	0.0945053	−	0.995524i	$-0.469873\pi$
$887$	3.92843e9	0.189011	0.0945053	+	0.995524i	$0.469873\pi$
$888$	2.54786e9	0.122104
$889$	6.26920e9	0.299265
$890$	−6.54458e9	−0.311184
$891$	3.62432e9	0.171654
$892$	2.08208e10	0.982248
$893$	1.42502e10	0.669637
$894$	−1.02152e10	−0.478153
$895$	−7.96906e9	−0.371558
$896$	2.55167e9	0.118508
$897$	−1.79884e10	−0.832181
$898$	2.48132e10	1.14345
$899$	3.92590e9	0.180211
$900$	1.23483e9	0.0564623
$901$	−4.55473e10	−2.07456
$902$	4.41608e10	2.00361
$903$	−3.02628e9	−0.136774
$904$	−5.41411e9	−0.243746
$905$	3.82153e10	1.71383
$906$	9.36645e9	0.418434
$907$	1.73810e10	0.773482	0.386741	−	0.922188i	$-0.373601\pi$
$907$	1.73810e10	0.773482	0.386741	+	0.922188i	$0.373601\pi$
$908$	−7.63824e9	−0.338605
$909$	1.37150e10	0.605652
$910$	−7.65341e9	−0.336674
$911$	3.94515e9	0.172882	0.0864409	−	0.996257i	$-0.472451\pi$
$911$	3.94515e9	0.172882	0.0864409	+	0.996257i	$0.472451\pi$
$912$	−1.43183e10	−0.625043
$913$	6.11017e10	2.65709
$914$	2.54040e10	1.10050
$915$	−2.03493e10	−0.878163
$916$	4.99975e10	2.14938
$917$	−5.32983e8	−0.0228255
$918$	7.85668e9	0.335189
$919$	−1.68687e10	−0.716930	−0.358465	−	0.933543i	$-0.616700\pi$
$919$	−1.68687e10	−0.716930	−0.358465	+	0.933543i	$0.616700\pi$
$920$	1.02987e10	0.436040
$921$	−5.43996e9	−0.229449
$922$	−1.34223e10	−0.563987
$923$	4.08859e10	1.71146
$924$	5.58841e9	0.233043
$925$	−2.66162e9	−0.110573
$926$	−1.58399e10	−0.655564
$927$	1.12870e10	0.465371
$928$	−7.63425e9	−0.313580
$929$	5.86634e9	0.240056	0.120028	−	0.992771i	$-0.461702\pi$
$929$	5.86634e9	0.240056	0.120028	+	0.992771i	$0.461702\pi$
$930$	−1.83612e10	−0.748532
$931$	−3.24787e10	−1.31909
$932$	6.60486e9	0.267244
$933$	2.74326e9	0.110581
$934$	−2.64983e10	−1.06415
$935$	4.86407e10	1.94607
$936$	2.20923e9	0.0880593
$937$	2.42022e10	0.961093	0.480546	−	0.876969i	$-0.340438\pi$
$937$	2.42022e10	0.961093	0.480546	+	0.876969i	$0.340438\pi$
$938$	−4.13737e9	−0.163687
$939$	−1.37563e10	−0.542217
$940$	−1.55807e10	−0.611842
$941$	2.22320e10	0.869792	0.434896	−	0.900481i	$-0.356785\pi$
$941$	2.22320e10	0.869792	0.434896	+	0.900481i	$0.356785\pi$
$942$	2.42715e10	0.946061
$943$	−3.37041e10	−1.30886
$944$	−2.62741e9	−0.101654
$945$	−1.17689e9	−0.0453654
$946$	6.38863e10	2.45351
$947$	2.37539e9	0.0908889	0.0454444	−	0.998967i	$-0.485530\pi$
$947$	2.37539e9	0.0908889	0.0454444	+	0.998967i	$0.485530\pi$
$948$	−5.75667e9	−0.219453
$949$	3.45590e10	1.31259
$950$	−7.74191e9	−0.292965
$951$	1.41921e10	0.535076
$952$	1.89099e9	0.0710328
$953$	−2.11402e10	−0.791197	−0.395599	−	0.918423i	$-0.629463\pi$
$953$	−2.11402e10	−0.791197	−0.395599	+	0.918423i	$0.629463\pi$
$954$	−2.32647e10	−0.867516
$955$	9.97702e9	0.370671
$956$	−1.77302e10	−0.656312
$957$	−5.31218e9	−0.195921
$958$	4.76649e7	0.00175154
$959$	5.67372e8	0.0207732
$960$	2.24934e10	0.820551
$961$	−8.99423e9	−0.326913
$962$	−3.05064e10	−1.10479
$963$	1.97575e9	0.0712920
$964$	−5.70996e10	−2.05288
$965$	−1.97564e10	−0.707723
$966$	−7.86454e9	−0.280707
$967$	−9.04540e9	−0.321688	−0.160844	−	0.986980i	$-0.551422\pi$
$967$	−9.04540e9	−0.321688	−0.160844	+	0.986980i	$0.551422\pi$
$968$	−1.06993e10	−0.379134
$969$	−2.67141e10	−0.943209
$970$	6.37173e10	2.24159
$971$	−4.75592e9	−0.166712	−0.0833560	−	0.996520i	$-0.526564\pi$
$971$	−4.75592e9	−0.166712	−0.0833560	+	0.996520i	$0.526564\pi$
$972$	2.17638e9	0.0760156
$973$	7.51839e9	0.261656
$974$	−2.14207e10	−0.742808
$975$	−2.30786e9	−0.0797432
$976$	−3.22663e10	−1.11090
$977$	1.97475e9	0.0677456	0.0338728	−	0.999426i	$-0.489216\pi$
$977$	1.97475e9	0.0677456	0.0338728	+	0.999426i	$0.489216\pi$
$978$	−1.08295e10	−0.370187
$979$	−8.93137e9	−0.304214
$980$	3.55112e10	1.20524
$981$	3.10089e9	0.104869
$982$	2.29922e9	0.0774801
$983$	1.84818e10	0.620592	0.310296	−	0.950640i	$-0.399572\pi$
$983$	1.84818e10	0.620592	0.310296	+	0.950640i	$0.399572\pi$
$984$	4.13935e9	0.138500
$985$	2.24261e9	0.0747699
$986$	−1.15156e10	−0.382575
$987$	1.85722e9	0.0614828
$988$	−4.81233e10	−1.58747
$989$	−4.87589e10	−1.60275
$990$	2.48447e10	0.813788
$991$	1.09246e9	0.0356573	0.0178286	−	0.999841i	$-0.494325\pi$
$991$	1.09246e9	0.0356573	0.0178286	+	0.999841i	$0.494325\pi$
$992$	−3.60106e10	−1.17122
$993$	6.67069e9	0.216196
$994$	1.78754e10	0.577303
$995$	−8.62327e9	−0.277518
$996$	3.66912e10	1.17667
$997$	−7.48711e9	−0.239266	−0.119633	−	0.992818i	$-0.538172\pi$
$997$	−7.48711e9	−0.239266	−0.119633	+	0.992818i	$0.538172\pi$
$998$	−3.29714e9	−0.104998
$999$	−4.69108e9	−0.148865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.d.1.3	✓	18
3.2	odd	2		531.8.a.e.1.16		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.3	✓	18	1.1	even	1	trivial
531.8.a.e.1.16		18	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)

\(f(q)\)	\(=\)	\(q-16.7235 q^{2} +27.0000 q^{3} +151.676 q^{4} -298.819 q^{5} -451.535 q^{6} +200.095 q^{7} -395.941 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-16.7235 q^{2} +27.0000 q^{3} +151.676 q^{4} -298.819 q^{5} -451.535 q^{6} +200.095 q^{7} -395.941 q^{8} +729.000 q^{9} +4997.30 q^{10} +6819.80 q^{11} +4095.24 q^{12} -7653.89 q^{13} -3346.30 q^{14} -8068.11 q^{15} -12793.0 q^{16} -23868.2 q^{17} -12191.4 q^{18} +41453.1 q^{19} -45323.6 q^{20} +5402.58 q^{21} -114051. q^{22} +87045.3 q^{23} -10690.4 q^{24} +11167.7 q^{25} +128000. q^{26} +19683.0 q^{27} +30349.6 q^{28} -28849.4 q^{29} +134927. q^{30} -136082. q^{31} +264624. q^{32} +184135. q^{33} +399161. q^{34} -59792.3 q^{35} +110572. q^{36} -238332. q^{37} -693241. q^{38} -206655. q^{39} +118315. q^{40} -387202. q^{41} -90350.0 q^{42} -560156. q^{43} +1.03440e6 q^{44} -217839. q^{45} -1.45570e6 q^{46} +343766. q^{47} -345410. q^{48} -783505. q^{49} -186763. q^{50} -644443. q^{51} -1.16091e6 q^{52} +1.90828e6 q^{53} -329169. q^{54} -2.03788e6 q^{55} -79226.0 q^{56} +1.11923e6 q^{57} +482464. q^{58} +205379. q^{59} -1.22374e6 q^{60} +2.52219e6 q^{61} +2.27577e6 q^{62} +145870. q^{63} -2.78794e6 q^{64} +2.28713e6 q^{65} -3.07938e6 q^{66} +1.23640e6 q^{67} -3.62023e6 q^{68} +2.35022e6 q^{69} +999937. q^{70} -5.34185e6 q^{71} -288641. q^{72} -4.51521e6 q^{73} +3.98574e6 q^{74} +301528. q^{75} +6.28743e6 q^{76} +1.36461e6 q^{77} +3.45600e6 q^{78} -1.40570e6 q^{79} +3.82278e6 q^{80} +531441. q^{81} +6.47538e6 q^{82} +8.95946e6 q^{83} +819440. q^{84} +7.13228e6 q^{85} +9.36777e6 q^{86} -778935. q^{87} -2.70024e6 q^{88} -1.30962e6 q^{89} +3.64303e6 q^{90} -1.53151e6 q^{91} +1.32027e7 q^{92} -3.67422e6 q^{93} -5.74897e6 q^{94} -1.23870e7 q^{95} +7.14484e6 q^{96} +1.27503e7 q^{97} +1.31029e7 q^{98} +4.97163e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9}+O(q^{10}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9} + 3609 q^{10} + 15070 q^{11} + 36666 q^{12} + 13662 q^{13} + 20861 q^{14} + 18306 q^{15} + 60482 q^{16} + 71919 q^{17} + 17496 q^{18} + 56231 q^{19} + 143053 q^{20} + 83187 q^{21} + 274198 q^{22} + 150029 q^{23} + 110889 q^{24} + 399672 q^{25} + 182846 q^{26} + 354294 q^{27} + 434150 q^{28} + 591285 q^{29} + 97443 q^{30} + 426733 q^{31} + 1205630 q^{32} + 406890 q^{33} + 403548 q^{34} + 912879 q^{35} + 989982 q^{36} + 7703 q^{37} - 417859 q^{38} + 368874 q^{39} + 618020 q^{40} + 770959 q^{41} + 563247 q^{42} + 793050 q^{43} + 2591274 q^{44} + 494262 q^{45} - 4068019 q^{46} + 1410373 q^{47} + 1633014 q^{48} + 1637427 q^{49} + 1021549 q^{50} + 1941813 q^{51} - 3749190 q^{52} + 1037934 q^{53} + 472392 q^{54} + 331974 q^{55} - 391748 q^{56} + 1518237 q^{57} + 653724 q^{58} + 3696822 q^{59} + 3862431 q^{60} - 1374623 q^{61} + 5251718 q^{62} + 2246049 q^{63} + 5077197 q^{64} + 3257170 q^{65} + 7403346 q^{66} - 2436904 q^{67} + 14119909 q^{68} + 4050783 q^{69} + 5185580 q^{70} + 14289172 q^{71} + 2994003 q^{72} + 5482515 q^{73} + 14934154 q^{74} + 10791144 q^{75} + 3822912 q^{76} + 23157109 q^{77} + 4936842 q^{78} + 19786414 q^{79} + 31978143 q^{80} + 9565938 q^{81} + 9749509 q^{82} + 30227337 q^{83} + 11722050 q^{84} + 9946981 q^{85} + 44295864 q^{86} + 15964695 q^{87} + 39970897 q^{88} + 31061677 q^{89} + 2630961 q^{90} + 26377785 q^{91} + 4719698 q^{92} + 11521791 q^{93} + 44488296 q^{94} + 15534599 q^{95} + 32552010 q^{96} + 12084118 q^{97} + 42274744 q^{98} + 10986030 q^{99}+O(q^{100}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9 + 3609 * q^10 + 15070 * q^11 + 36666 * q^12 + 13662 * q^13 + 20861 * q^14 + 18306 * q^15 + 60482 * q^16 + 71919 * q^17 + 17496 * q^18 + 56231 * q^19 + 143053 * q^20 + 83187 * q^21 + 274198 * q^22 + 150029 * q^23 + 110889 * q^24 + 399672 * q^25 + 182846 * q^26 + 354294 * q^27 + 434150 * q^28 + 591285 * q^29 + 97443 * q^30 + 426733 * q^31 + 1205630 * q^32 + 406890 * q^33 + 403548 * q^34 + 912879 * q^35 + 989982 * q^36 + 7703 * q^37 - 417859 * q^38 + 368874 * q^39 + 618020 * q^40 + 770959 * q^41 + 563247 * q^42 + 793050 * q^43 + 2591274 * q^44 + 494262 * q^45 - 4068019 * q^46 + 1410373 * q^47 + 1633014 * q^48 + 1637427 * q^49 + 1021549 * q^50 + 1941813 * q^51 - 3749190 * q^52 + 1037934 * q^53 + 472392 * q^54 + 331974 * q^55 - 391748 * q^56 + 1518237 * q^57 + 653724 * q^58 + 3696822 * q^59 + 3862431 * q^60 - 1374623 * q^61 + 5251718 * q^62 + 2246049 * q^63 + 5077197 * q^64 + 3257170 * q^65 + 7403346 * q^66 - 2436904 * q^67 + 14119909 * q^68 + 4050783 * q^69 + 5185580 * q^70 + 14289172 * q^71 + 2994003 * q^72 + 5482515 * q^73 + 14934154 * q^74 + 10791144 * q^75 + 3822912 * q^76 + 23157109 * q^77 + 4936842 * q^78 + 19786414 * q^79 + 31978143 * q^80 + 9565938 * q^81 + 9749509 * q^82 + 30227337 * q^83 + 11722050 * q^84 + 9946981 * q^85 + 44295864 * q^86 + 15964695 * q^87 + 39970897 * q^88 + 31061677 * q^89 + 2630961 * q^90 + 26377785 * q^91 + 4719698 * q^92 + 11521791 * q^93 + 44488296 * q^94 + 15534599 * q^95 + 32552010 * q^96 + 12084118 * q^97 + 42274744 * q^98 + 10986030 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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