LMFDB - Embedded newform 471.8.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,8,Mod(1,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, 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("471.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	471.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:	$147.133347003$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.9953 q^{2} -27.0000 q^{3} +271.812 q^{4} -231.869 q^{5} +539.873 q^{6} +342.382 q^{7} -2875.57 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-19.9953 q^{2} -27.0000 q^{3} +271.812 q^{4} -231.869 q^{5} +539.873 q^{6} +342.382 q^{7} -2875.57 q^{8} +729.000 q^{9} +4636.29 q^{10} -1179.20 q^{11} -7338.93 q^{12} -8827.43 q^{13} -6846.04 q^{14} +6260.47 q^{15} +22705.9 q^{16} -924.150 q^{17} -14576.6 q^{18} -41847.9 q^{19} -63024.8 q^{20} -9244.32 q^{21} +23578.5 q^{22} +36377.2 q^{23} +77640.3 q^{24} -24361.7 q^{25} +176507. q^{26} -19683.0 q^{27} +93063.6 q^{28} -144217. q^{29} -125180. q^{30} -90032.4 q^{31} -85938.3 q^{32} +31838.4 q^{33} +18478.7 q^{34} -79387.9 q^{35} +198151. q^{36} -550647. q^{37} +836762. q^{38} +238341. q^{39} +666755. q^{40} +509124. q^{41} +184843. q^{42} +53578.2 q^{43} -320521. q^{44} -169033. q^{45} -727373. q^{46} +1.21335e6 q^{47} -613059. q^{48} -706317. q^{49} +487119. q^{50} +24952.0 q^{51} -2.39940e6 q^{52} -1.60068e6 q^{53} +393568. q^{54} +273420. q^{55} -984543. q^{56} +1.12989e6 q^{57} +2.88366e6 q^{58} +62915.1 q^{59} +1.70167e6 q^{60} -1.02969e6 q^{61} +1.80023e6 q^{62} +249597. q^{63} -1.18799e6 q^{64} +2.04681e6 q^{65} -636618. q^{66} +2.39598e6 q^{67} -251195. q^{68} -982184. q^{69} +1.58738e6 q^{70} +3.52854e6 q^{71} -2.09629e6 q^{72} +915999. q^{73} +1.10104e7 q^{74} +657766. q^{75} -1.13748e7 q^{76} -403737. q^{77} -4.76569e6 q^{78} -3.18874e6 q^{79} -5.26479e6 q^{80} +531441. q^{81} -1.01801e7 q^{82} -5.91707e6 q^{83} -2.51272e6 q^{84} +214282. q^{85} -1.07131e6 q^{86} +3.89386e6 q^{87} +3.39087e6 q^{88} -1.32476e7 q^{89} +3.37986e6 q^{90} -3.02236e6 q^{91} +9.88776e6 q^{92} +2.43087e6 q^{93} -2.42612e7 q^{94} +9.70325e6 q^{95} +2.32033e6 q^{96} -8.60476e6 q^{97} +1.41230e7 q^{98} -859637. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9}+O(q^{10}) $ 48 * q + 2 * q^2 - 1296 * q^3 + 3214 * q^4 + 428 * q^5 - 54 * q^6 - 680 * q^7 + 2355 * q^8 + 34992 * q^9	\( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9} + 6185 q^{10} + 11989 q^{11} - 86778 q^{12} - 2393 q^{13} + 18201 q^{14} - 11556 q^{15} + 208150 q^{16} + 62538 q^{17} + 1458 q^{18} - 39882 q^{19} + 113423 q^{20} + 18360 q^{21} - 93716 q^{22} + 195618 q^{23} - 63585 q^{24} + 886490 q^{25} + 294399 q^{26} - 944784 q^{27} - 60819 q^{28} + 421501 q^{29} - 166995 q^{30} + 392689 q^{31} - 341578 q^{32} - 323703 q^{33} + 50837 q^{34} + 697874 q^{35} + 2343006 q^{36} - 410396 q^{37} + 677216 q^{38} + 64611 q^{39} + 3232376 q^{40} + 3832958 q^{41} - 491427 q^{42} - 1751932 q^{43} + 4888297 q^{44} + 312012 q^{45} + 1163150 q^{46} + 106461 q^{47} - 5620050 q^{48} + 8202048 q^{49} - 2159111 q^{50} - 1688526 q^{51} - 3605030 q^{52} + 1755534 q^{53} - 39366 q^{54} - 1220729 q^{55} - 4430622 q^{56} + 1076814 q^{57} - 10000202 q^{58} - 2037752 q^{59} - 3062421 q^{60} + 1274098 q^{61} + 97748 q^{62} - 495720 q^{63} + 15135201 q^{64} + 6139645 q^{65} + 2530332 q^{66} - 7751257 q^{67} + 1700631 q^{68} - 5281686 q^{69} - 20935703 q^{70} - 12592217 q^{71} + 1716795 q^{72} + 12508355 q^{73} - 14999956 q^{74} - 23935230 q^{75} - 23946874 q^{76} + 1874177 q^{77} - 7948773 q^{78} - 5103480 q^{79} + 3128449 q^{80} + 25509168 q^{81} + 11622426 q^{82} + 3040643 q^{83} + 1642113 q^{84} - 13756076 q^{85} + 964635 q^{86} - 11380527 q^{87} - 29653500 q^{88} + 28462995 q^{89} + 4508865 q^{90} + 3016621 q^{91} + 22938254 q^{92} - 10602603 q^{93} - 10070348 q^{94} - 2579984 q^{95} + 9222606 q^{96} + 16208760 q^{97} + 6323227 q^{98} + 8739981 q^{99}+O(q^{100}) \) 48 * q + 2 * q^2 - 1296 * q^3 + 3214 * q^4 + 428 * q^5 - 54 * q^6 - 680 * q^7 + 2355 * q^8 + 34992 * q^9 + 6185 * q^10 + 11989 * q^11 - 86778 * q^12 - 2393 * q^13 + 18201 * q^14 - 11556 * q^15 + 208150 * q^16 + 62538 * q^17 + 1458 * q^18 - 39882 * q^19 + 113423 * q^20 + 18360 * q^21 - 93716 * q^22 + 195618 * q^23 - 63585 * q^24 + 886490 * q^25 + 294399 * q^26 - 944784 * q^27 - 60819 * q^28 + 421501 * q^29 - 166995 * q^30 + 392689 * q^31 - 341578 * q^32 - 323703 * q^33 + 50837 * q^34 + 697874 * q^35 + 2343006 * q^36 - 410396 * q^37 + 677216 * q^38 + 64611 * q^39 + 3232376 * q^40 + 3832958 * q^41 - 491427 * q^42 - 1751932 * q^43 + 4888297 * q^44 + 312012 * q^45 + 1163150 * q^46 + 106461 * q^47 - 5620050 * q^48 + 8202048 * q^49 - 2159111 * q^50 - 1688526 * q^51 - 3605030 * q^52 + 1755534 * q^53 - 39366 * q^54 - 1220729 * q^55 - 4430622 * q^56 + 1076814 * q^57 - 10000202 * q^58 - 2037752 * q^59 - 3062421 * q^60 + 1274098 * q^61 + 97748 * q^62 - 495720 * q^63 + 15135201 * q^64 + 6139645 * q^65 + 2530332 * q^66 - 7751257 * q^67 + 1700631 * q^68 - 5281686 * q^69 - 20935703 * q^70 - 12592217 * q^71 + 1716795 * q^72 + 12508355 * q^73 - 14999956 * q^74 - 23935230 * q^75 - 23946874 * q^76 + 1874177 * q^77 - 7948773 * q^78 - 5103480 * q^79 + 3128449 * q^80 + 25509168 * q^81 + 11622426 * q^82 + 3040643 * q^83 + 1642113 * q^84 - 13756076 * q^85 + 964635 * q^86 - 11380527 * q^87 - 29653500 * q^88 + 28462995 * q^89 + 4508865 * q^90 + 3016621 * q^91 + 22938254 * q^92 - 10602603 * q^93 - 10070348 * q^94 - 2579984 * q^95 + 9222606 * q^96 + 16208760 * q^97 + 6323227 * q^98 + 8739981 * 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 $

$2$	−19.9953	−1.76735	−0.883676	−	0.468099i	$-0.844939\pi$
$2$	−19.9953	−1.76735	−0.883676	+	0.468099i	$0.844939\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	271.812	2.12353
$5$	−231.869	−0.829560	−0.414780	−	0.909922i	$-0.636142\pi$
$5$	−231.869	−0.829560	−0.414780	+	0.909922i	$0.636142\pi$
$6$	539.873	1.02038
$7$	342.382	0.377284	0.188642	−	0.982046i	$-0.439591\pi$
$7$	342.382	0.377284	0.188642	+	0.982046i	$0.439591\pi$
$8$	−2875.57	−1.98568
$9$	729.000	0.333333
$10$	4636.29	1.46612
$11$	−1179.20	−0.267124	−0.133562	−	0.991040i	$-0.542642\pi$
$11$	−1179.20	−0.267124	−0.133562	+	0.991040i	$0.542642\pi$
$12$	−7338.93	−1.22602
$13$	−8827.43	−1.11438	−0.557189	−	0.830386i	$-0.688120\pi$
$13$	−8827.43	−1.11438	−0.557189	+	0.830386i	$0.688120\pi$
$14$	−6846.04	−0.666793
$15$	6260.47	0.478947
$16$	22705.9	1.38586
$17$	−924.150	−0.0456217	−0.0228108	−	0.999740i	$-0.507262\pi$
$17$	−924.150	−0.0456217	−0.0228108	+	0.999740i	$0.507262\pi$
$18$	−14576.6	−0.589117
$19$	−41847.9	−1.39971	−0.699853	−	0.714287i	$-0.746751\pi$
$19$	−41847.9	−1.39971	−0.699853	+	0.714287i	$0.746751\pi$
$20$	−63024.8	−1.76160
$21$	−9244.32	−0.217825
$22$	23578.5	0.472102
$23$	36377.2	0.623421	0.311711	−	0.950177i	$-0.399098\pi$
$23$	36377.2	0.623421	0.311711	+	0.950177i	$0.399098\pi$
$24$	77640.3	1.14643
$25$	−24361.7	−0.311830
$26$	176507.	1.96950
$27$	−19683.0	−0.192450
$28$	93063.6	0.801174
$29$	−144217.	−1.09805	−0.549027	−	0.835805i	$-0.685002\pi$
$29$	−144217.	−1.09805	−0.549027	+	0.835805i	$0.685002\pi$
$30$	−125180.	−0.846468
$31$	−90032.4	−0.542791	−0.271396	−	0.962468i	$-0.587485\pi$
$31$	−90032.4	−0.542791	−0.271396	+	0.962468i	$0.587485\pi$
$32$	−85938.3	−0.463619
$33$	31838.4	0.154224
$34$	18478.7	0.0806295
$35$	−79387.9	−0.312980
$36$	198151.	0.707844
$37$	−550647.	−1.78718	−0.893588	−	0.448889i	$-0.851820\pi$
$37$	−550647.	−1.78718	−0.893588	+	0.448889i	$0.851820\pi$
$38$	836762.	2.47377
$39$	238341.	0.643387
$40$	666755.	1.64724
$41$	509124.	1.15367	0.576833	−	0.816862i	$-0.304288\pi$
$41$	509124.	1.15367	0.576833	+	0.816862i	$0.304288\pi$
$42$	184843.	0.384973
$43$	53578.2	0.102766	0.0513829	−	0.998679i	$-0.483637\pi$
$43$	53578.2	0.102766	0.0513829	+	0.998679i	$0.483637\pi$
$44$	−320521.	−0.567246
$45$	−169033.	−0.276520
$46$	−727373.	−1.10181
$47$	1.21335e6	1.70468	0.852339	−	0.522990i	$-0.175183\pi$
$47$	1.21335e6	1.70468	0.852339	+	0.522990i	$0.175183\pi$
$48$	−613059.	−0.800125
$49$	−706317.	−0.857657
$50$	487119.	0.551113
$51$	24952.0	0.0263397
$52$	−2.39940e6	−2.36642
$53$	−1.60068e6	−1.47685	−0.738427	−	0.674333i	$-0.764431\pi$
$53$	−1.60068e6	−1.47685	−0.738427	+	0.674333i	$0.764431\pi$
$54$	393568.	0.340127
$55$	273420.	0.221595
$56$	−984543.	−0.749163
$57$	1.12989e6	0.808120
$58$	2.88366e6	1.94065
$59$	62915.1	0.0398816	0.0199408	−	0.999801i	$-0.493652\pi$
$59$	62915.1	0.0398816	0.0199408	+	0.999801i	$0.493652\pi$
$60$	1.70167e6	1.01706
$61$	−1.02969e6	−0.580835	−0.290418	−	0.956900i	$-0.593794\pi$
$61$	−1.02969e6	−0.580835	−0.290418	+	0.956900i	$0.593794\pi$
$62$	1.80023e6	0.959303
$63$	249597.	0.125761
$64$	−1.18799e6	−0.566478
$65$	2.04681e6	0.924444
$66$	−636618.	−0.272568
$67$	2.39598e6	0.973245	0.486622	−	0.873612i	$-0.338229\pi$
$67$	2.39598e6	0.973245	0.486622	+	0.873612i	$0.338229\pi$
$68$	−251195.	−0.0968791
$69$	−982184.	−0.359933
$70$	1.58738e6	0.553145
$71$	3.52854e6	1.17001	0.585006	−	0.811029i	$-0.301092\pi$
$71$	3.52854e6	1.17001	0.585006	+	0.811029i	$0.301092\pi$
$72$	−2.09629e6	−0.661892
$73$	915999.	0.275591	0.137795	−	0.990461i	$-0.455998\pi$
$73$	915999.	0.275591	0.137795	+	0.990461i	$0.455998\pi$
$74$	1.10104e7	3.15857
$75$	657766.	0.180035
$76$	−1.13748e7	−2.97232
$77$	−403737.	−0.100782
$78$	−4.76569e6	−1.13709
$79$	−3.18874e6	−0.727652	−0.363826	−	0.931467i	$-0.618530\pi$
$79$	−3.18874e6	−0.727652	−0.363826	+	0.931467i	$0.618530\pi$
$80$	−5.26479e6	−1.14965
$81$	531441.	0.111111
$82$	−1.01801e7	−2.03894
$83$	−5.91707e6	−1.13588	−0.567941	−	0.823069i	$-0.692260\pi$
$83$	−5.91707e6	−1.13588	−0.567941	+	0.823069i	$0.692260\pi$
$84$	−2.51272e6	−0.462558
$85$	214282.	0.0378459
$86$	−1.07131e6	−0.181623
$87$	3.89386e6	0.633962
$88$	3.39087e6	0.530422
$89$	−1.32476e7	−1.99192	−0.995958	−	0.0898222i	$-0.971370\pi$
$89$	−1.32476e7	−1.99192	−0.995958	+	0.0898222i	$0.971370\pi$
$90$	3.37986e6	0.488708
$91$	−3.02236e6	−0.420437
$92$	9.88776e6	1.32386
$93$	2.43087e6	0.313381
$94$	−2.42612e7	−3.01276
$95$	9.70325e6	1.16114
$96$	2.32033e6	0.267671
$97$	−8.60476e6	−0.957278	−0.478639	−	0.878012i	$-0.658870\pi$
$97$	−8.60476e6	−0.957278	−0.478639	+	0.878012i	$0.658870\pi$
$98$	1.41230e7	1.51578
$99$	−859637.	−0.0890413
$100$	−6.62180e6	−0.662180
$101$	−2.70404e6	−0.261149	−0.130575	−	0.991438i	$-0.541682\pi$
$101$	−2.70404e6	−0.261149	−0.130575	+	0.991438i	$0.541682\pi$
$102$	−498924.	−0.0465515
$103$	−6.20952e6	−0.559923	−0.279961	−	0.960011i	$-0.590322\pi$
$103$	−6.20952e6	−0.559923	−0.279961	+	0.960011i	$0.590322\pi$
$104$	2.53839e7	2.21279
$105$	2.14347e6	0.180699
$106$	3.20060e7	2.61012
$107$	−7.10397e6	−0.560606	−0.280303	−	0.959912i	$-0.590435\pi$
$107$	−7.10397e6	−0.560606	−0.280303	+	0.959912i	$0.590435\pi$
$108$	−5.35008e6	−0.408674
$109$	−8.32935e6	−0.616053	−0.308027	−	0.951378i	$-0.599669\pi$
$109$	−8.32935e6	−0.616053	−0.308027	+	0.951378i	$0.599669\pi$
$110$	−5.46712e6	−0.391637
$111$	1.48675e7	1.03183
$112$	7.77409e6	0.522861
$113$	−1.98579e7	−1.29467	−0.647334	−	0.762207i	$-0.724116\pi$
$113$	−1.98579e7	−1.29467	−0.647334	+	0.762207i	$0.724116\pi$
$114$	−2.25926e7	−1.42823
$115$	−8.43475e6	−0.517166
$116$	−3.92000e7	−2.33175
$117$	−6.43520e6	−0.371459
$118$	−1.25801e6	−0.0704849
$119$	−316413.	−0.0172123
$120$	−1.80024e7	−0.951034
$121$	−1.80967e7	−0.928645
$122$	2.05890e7	1.02654
$123$	−1.37464e7	−0.666070
$124$	−2.44719e7	−1.15263
$125$	2.37635e7	1.08824
$126$	−4.99076e6	−0.222264
$127$	−4.56598e7	−1.97798	−0.988988	−	0.147996i	$-0.952718\pi$
$127$	−4.56598e7	−1.97798	−0.988988	+	0.147996i	$0.952718\pi$
$128$	3.47543e7	1.46479
$129$	−1.44661e6	−0.0593319
$130$	−4.09266e7	−1.63382
$131$	−2.12250e6	−0.0824896	−0.0412448	−	0.999149i	$-0.513132\pi$
$131$	−2.12250e6	−0.0824896	−0.0412448	+	0.999149i	$0.513132\pi$
$132$	8.65406e6	0.327500
$133$	−1.43280e7	−0.528086
$134$	−4.79084e7	−1.72007
$135$	4.56388e6	0.159649
$136$	2.65745e6	0.0905899
$137$	−5.87411e7	−1.95173	−0.975866	−	0.218369i	$-0.929926\pi$
$137$	−5.87411e7	−1.95173	−0.975866	+	0.218369i	$0.929926\pi$
$138$	1.96391e7	0.636127
$139$	−2.32626e7	−0.734695	−0.367348	−	0.930084i	$-0.619734\pi$
$139$	−2.32626e7	−0.734695	−0.367348	+	0.930084i	$0.619734\pi$
$140$	−2.15786e7	−0.664622
$141$	−3.27603e7	−0.984196
$142$	−7.05542e7	−2.06782
$143$	1.04093e7	0.297677
$144$	1.65526e7	0.461952
$145$	3.34395e7	0.910902
$146$	−1.83157e7	−0.487066
$147$	1.90706e7	0.495168
$148$	−1.49673e8	−3.79512
$149$	4.87938e7	1.20841	0.604203	−	0.796831i	$-0.293492\pi$
$149$	4.87938e7	1.20841	0.604203	+	0.796831i	$0.293492\pi$
$150$	−1.31522e7	−0.318185
$151$	2.04923e7	0.484364	0.242182	−	0.970231i	$-0.422137\pi$
$151$	2.04923e7	0.484364	0.242182	+	0.970231i	$0.422137\pi$
$152$	1.20337e8	2.77936
$153$	−673705.	−0.0152072
$154$	8.07284e6	0.178116
$155$	2.08757e7	0.450278
$156$	6.47839e7	1.36625
$157$	−3.86989e6	−0.0798087
$157$	−3.86989e6	−0.0798087
$158$	6.37597e7	1.28602
$159$	4.32182e7	0.852662
$160$	1.99264e7	0.384600
$161$	1.24549e7	0.235207
$162$	−1.06263e7	−0.196372
$163$	2.81553e7	0.509218	0.254609	−	0.967044i	$-0.418053\pi$
$163$	2.81553e7	0.509218	0.254609	+	0.967044i	$0.418053\pi$
$164$	1.38386e8	2.44985
$165$	−7.38234e6	−0.127938
$166$	1.18314e8	2.00750
$167$	−9.91460e7	−1.64728	−0.823640	−	0.567113i	$-0.808060\pi$
$167$	−9.91460e7	−1.64728	−0.823640	+	0.567113i	$0.808060\pi$
$168$	2.65827e7	0.432530
$169$	1.51751e7	0.241839
$170$	−4.28463e6	−0.0668871
$171$	−3.05072e7	−0.466568
$172$	1.45632e7	0.218226
$173$	−5.16854e7	−0.758939	−0.379469	−	0.925204i	$-0.623893\pi$
$173$	−5.16854e7	−0.758939	−0.379469	+	0.925204i	$0.623893\pi$
$174$	−7.78589e7	−1.12043
$175$	−8.34101e6	−0.117648
$176$	−2.67748e7	−0.370196
$177$	−1.69871e6	−0.0230257
$178$	2.64889e8	3.52042
$179$	−8.32582e7	−1.08503	−0.542515	−	0.840046i	$-0.682528\pi$
$179$	−8.32582e7	−1.08503	−0.542515	+	0.840046i	$0.682528\pi$
$180$	−4.59451e7	−0.587199
$181$	1.34738e8	1.68895	0.844474	−	0.535597i	$-0.179913\pi$
$181$	1.34738e8	1.68895	0.844474	+	0.535597i	$0.179913\pi$
$182$	6.04329e7	0.743060
$183$	2.78017e7	0.335345
$184$	−1.04605e8	−1.23791
$185$	1.27678e8	1.48257
$186$	−4.86061e7	−0.553854
$187$	1.08976e6	0.0121866
$188$	3.29802e8	3.61994
$189$	−6.73911e6	−0.0726083
$190$	−1.94019e8	−2.05214
$191$	4.88336e7	0.507109	0.253555	−	0.967321i	$-0.418400\pi$
$191$	4.88336e7	0.507109	0.253555	+	0.967321i	$0.418400\pi$
$192$	3.20757e7	0.327056
$193$	8.79116e6	0.0880229	0.0440115	−	0.999031i	$-0.485986\pi$
$193$	8.79116e6	0.0880229	0.0440115	+	0.999031i	$0.485986\pi$
$194$	1.72055e8	1.69185
$195$	−5.52639e7	−0.533728
$196$	−1.91986e8	−1.82126
$197$	−1.66354e8	−1.55025	−0.775123	−	0.631811i	$-0.782312\pi$
$197$	−1.66354e8	−1.55025	−0.775123	+	0.631811i	$0.782312\pi$
$198$	1.71887e7	0.157367
$199$	−4.34472e7	−0.390820	−0.195410	−	0.980722i	$-0.562604\pi$
$199$	−4.34472e7	−0.390820	−0.195410	+	0.980722i	$0.562604\pi$
$200$	7.00537e7	0.619193
$201$	−6.46916e7	−0.561903
$202$	5.40682e7	0.461543
$203$	−4.93774e7	−0.414278
$204$	6.78227e6	0.0559332
$205$	−1.18050e8	−0.957036
$206$	1.24161e8	0.989580
$207$	2.65190e7	0.207807
$208$	−2.00435e8	−1.54437
$209$	4.93471e7	0.373895
$210$	−4.28594e7	−0.319358
$211$	1.43011e8	1.04805	0.524025	−	0.851703i	$-0.324430\pi$
$211$	1.43011e8	1.04805	0.524025	+	0.851703i	$0.324430\pi$
$212$	−4.35083e8	−3.13615
$213$	−9.52705e7	−0.675507
$214$	1.42046e8	0.990788
$215$	−1.24231e7	−0.0852504
$216$	5.65998e7	0.382144
$217$	−3.08255e7	−0.204786
$218$	1.66548e8	1.08878
$219$	−2.47320e7	−0.159113
$220$	7.43189e7	0.470565
$221$	8.15787e6	0.0508398
$222$	−2.97279e8	−1.82360
$223$	2.60493e8	1.57300	0.786500	−	0.617590i	$-0.211891\pi$
$223$	2.60493e8	1.57300	0.786500	+	0.617590i	$0.211891\pi$
$224$	−2.94237e7	−0.174916
$225$	−1.77597e7	−0.103943
$226$	3.97064e8	2.28813
$227$	−2.94237e8	−1.66958	−0.834790	−	0.550568i	$-0.814411\pi$
$227$	−2.94237e8	−1.66958	−0.834790	+	0.550568i	$0.814411\pi$
$228$	3.07119e8	1.71607
$229$	1.00929e8	0.555380	0.277690	−	0.960671i	$-0.410431\pi$
$229$	1.00929e8	0.555380	0.277690	+	0.960671i	$0.410431\pi$
$230$	1.68655e8	0.914014
$231$	1.09009e7	0.0581863
$232$	4.14706e8	2.18038
$233$	3.22668e8	1.67113	0.835565	−	0.549391i	$-0.185140\pi$
$233$	3.22668e8	1.67113	0.835565	+	0.549391i	$0.185140\pi$
$234$	1.28674e8	0.656500
$235$	−2.81338e8	−1.41413
$236$	1.71011e7	0.0846899
$237$	8.60959e7	0.420110
$238$	6.32676e6	0.0304202
$239$	3.32979e8	1.57770	0.788851	−	0.614585i	$-0.210677\pi$
$239$	3.32979e8	1.57770	0.788851	+	0.614585i	$0.210677\pi$
$240$	1.42149e8	0.663752
$241$	4.02266e7	0.185120	0.0925601	−	0.995707i	$-0.470495\pi$
$241$	4.02266e7	0.185120	0.0925601	+	0.995707i	$0.470495\pi$
$242$	3.61848e8	1.64124
$243$	−1.43489e7	−0.0641500
$244$	−2.79883e8	−1.23342
$245$	1.63773e8	0.711478
$246$	2.74863e8	1.17718
$247$	3.69410e8	1.55980
$248$	2.58894e8	1.07781
$249$	1.59761e8	0.655802
$250$	−4.75158e8	−1.92331
$251$	3.52998e8	1.40901	0.704505	−	0.709699i	$-0.251169\pi$
$251$	3.52998e8	1.40901	0.704505	+	0.709699i	$0.251169\pi$
$252$	6.78434e7	0.267058
$253$	−4.28960e7	−0.166531
$254$	9.12982e8	3.49578
$255$	−5.78561e6	−0.0218504
$256$	−5.42860e8	−2.02231
$257$	−1.00380e8	−0.368876	−0.184438	−	0.982844i	$-0.559046\pi$
$257$	−1.00380e8	−0.368876	−0.184438	+	0.982844i	$0.559046\pi$
$258$	2.89254e7	0.104860
$259$	−1.88532e8	−0.674272
$260$	5.56348e8	1.96309
$261$	−1.05134e8	−0.366018
$262$	4.24401e7	0.145788
$263$	−1.16424e8	−0.394638	−0.197319	−	0.980339i	$-0.563223\pi$
$263$	−1.16424e8	−0.394638	−0.197319	+	0.980339i	$0.563223\pi$
$264$	−9.15534e7	−0.306239
$265$	3.71147e8	1.22514
$266$	2.86493e8	0.933314
$267$	3.57684e8	1.15003
$268$	6.51257e8	2.06672
$269$	−3.90454e8	−1.22303	−0.611514	−	0.791234i	$-0.709439\pi$
$269$	−3.90454e8	−1.22303	−0.611514	+	0.791234i	$0.709439\pi$
$270$	−9.12562e7	−0.282156
$271$	−1.80076e8	−0.549622	−0.274811	−	0.961498i	$-0.588615\pi$
$271$	−1.80076e8	−0.549622	−0.274811	+	0.961498i	$0.588615\pi$
$272$	−2.09836e7	−0.0632251
$273$	8.16036e7	0.242739
$274$	1.17455e9	3.44940
$275$	2.87273e7	0.0832972
$276$	−2.66970e8	−0.764328
$277$	−6.77224e8	−1.91449	−0.957245	−	0.289279i	$-0.906585\pi$
$277$	−6.77224e8	−1.91449	−0.957245	+	0.289279i	$0.906585\pi$
$278$	4.65143e8	1.29846
$279$	−6.56336e7	−0.180930
$280$	2.28285e8	0.621476
$281$	3.55754e8	0.956484	0.478242	−	0.878228i	$-0.341274\pi$
$281$	3.55754e8	0.956484	0.478242	+	0.878228i	$0.341274\pi$
$282$	6.55053e8	1.73942
$283$	−1.51881e8	−0.398337	−0.199169	−	0.979965i	$-0.563824\pi$
$283$	−1.51881e8	−0.398337	−0.199169	+	0.979965i	$0.563824\pi$
$284$	9.59099e8	2.48456
$285$	−2.61988e8	−0.670384
$286$	−2.08137e8	−0.526100
$287$	1.74315e8	0.435260
$288$	−6.26490e7	−0.154540
$289$	−4.09485e8	−0.997919
$290$	−6.68633e8	−1.60988
$291$	2.32329e8	0.552684
$292$	2.48980e8	0.585226
$293$	−4.96876e8	−1.15401	−0.577007	−	0.816739i	$-0.695780\pi$
$293$	−4.96876e8	−1.15401	−0.577007	+	0.816739i	$0.695780\pi$
$294$	−3.81322e8	−0.875137
$295$	−1.45881e7	−0.0330842
$296$	1.58342e9	3.54875
$297$	2.32102e7	0.0514080
$298$	−9.75647e8	−2.13568
$299$	−3.21117e8	−0.694727
$300$	1.78789e8	0.382310
$301$	1.83442e7	0.0387719
$302$	−4.09750e8	−0.856041
$303$	7.30092e7	0.150775
$304$	−9.50194e8	−1.93979
$305$	2.38754e8	0.481838
$306$	1.34709e7	0.0268765
$307$	3.37330e8	0.665382	0.332691	−	0.943036i	$-0.392043\pi$
$307$	3.37330e8	0.665382	0.332691	+	0.943036i	$0.392043\pi$
$308$	−1.09741e8	−0.214013
$309$	1.67657e8	0.323271
$310$	−4.17417e8	−0.795800
$311$	−9.19913e7	−0.173414	−0.0867072	−	0.996234i	$-0.527634\pi$
$311$	−9.19913e7	−0.173414	−0.0867072	+	0.996234i	$0.527634\pi$
$312$	−6.85365e8	−1.27756
$313$	−1.12528e8	−0.207423	−0.103712	−	0.994607i	$-0.533072\pi$
$313$	−1.12528e8	−0.207423	−0.103712	+	0.994607i	$0.533072\pi$
$314$	7.73797e7	0.141050
$315$	−5.78738e7	−0.104327
$316$	−8.66737e8	−1.54519
$317$	−8.43351e8	−1.48697	−0.743483	−	0.668755i	$-0.766827\pi$
$317$	−8.43351e8	−1.48697	−0.743483	+	0.668755i	$0.766827\pi$
$318$	−8.64162e8	−1.50695
$319$	1.70061e8	0.293317
$320$	2.75458e8	0.469928
$321$	1.91807e8	0.323666
$322$	−2.49040e8	−0.415693
$323$	3.86738e7	0.0638569
$324$	1.44452e8	0.235948
$325$	2.15051e8	0.347496
$326$	−5.62974e8	−0.899967
$327$	2.24892e8	0.355678
$328$	−1.46402e9	−2.29081
$329$	4.15428e8	0.643147
$330$	1.47612e8	0.226112
$331$	−5.45465e8	−0.826740	−0.413370	−	0.910563i	$-0.635648\pi$
$331$	−5.45465e8	−0.826740	−0.413370	+	0.910563i	$0.635648\pi$
$332$	−1.60833e9	−2.41208
$333$	−4.01422e8	−0.595725
$334$	1.98245e9	2.91132
$335$	−5.55555e8	−0.807365
$336$	−2.09900e8	−0.301874
$337$	3.51028e8	0.499617	0.249809	−	0.968295i	$-0.419632\pi$
$337$	3.51028e8	0.499617	0.249809	+	0.968295i	$0.419632\pi$
$338$	−3.03430e8	−0.427415
$339$	5.36163e8	0.747477
$340$	5.82444e7	0.0803670
$341$	1.06166e8	0.144993
$342$	6.10000e8	0.824590
$343$	−5.23797e8	−0.700864
$344$	−1.54068e8	−0.204060
$345$	2.27738e8	0.298586
$346$	1.03347e9	1.34131
$347$	8.75888e8	1.12537	0.562685	−	0.826671i	$-0.309768\pi$
$347$	8.75888e8	1.12537	0.562685	+	0.826671i	$0.309768\pi$
$348$	1.05840e9	1.34624
$349$	−1.33144e9	−1.67661	−0.838303	−	0.545204i	$-0.816452\pi$
$349$	−1.33144e9	−1.67661	−0.838303	+	0.545204i	$0.816452\pi$
$350$	1.66781e8	0.207926
$351$	1.73750e8	0.214462
$352$	1.01338e8	0.123844
$353$	9.93225e8	1.20181	0.600905	−	0.799320i	$-0.294807\pi$
$353$	9.93225e8	1.20181	0.600905	+	0.799320i	$0.294807\pi$
$354$	3.39662e7	0.0406945
$355$	−8.18159e8	−0.970596
$356$	−3.60085e9	−4.22990
$357$	8.54314e6	0.00993753
$358$	1.66477e9	1.91763
$359$	1.00039e9	1.14114	0.570571	−	0.821248i	$-0.306722\pi$
$359$	1.00039e9	1.14114	0.570571	+	0.821248i	$0.306722\pi$
$360$	4.86065e8	0.549079
$361$	8.57379e8	0.959174
$362$	−2.69413e9	−2.98496
$363$	4.88610e8	0.536153
$364$	−8.21513e8	−0.892811
$365$	−2.12392e8	−0.228619
$366$	−5.55903e8	−0.592673
$367$	6.76033e8	0.713899	0.356949	−	0.934124i	$-0.383817\pi$
$367$	6.76033e8	0.713899	0.356949	+	0.934124i	$0.383817\pi$
$368$	8.25976e8	0.863973
$369$	3.71152e8	0.384556
$370$	−2.55296e9	−2.62022
$371$	−5.48043e8	−0.557193
$372$	6.60741e8	0.665474
$373$	2.29644e8	0.229125	0.114563	−	0.993416i	$-0.463453\pi$
$373$	2.29644e8	0.229125	0.114563	+	0.993416i	$0.463453\pi$
$374$	−2.17900e7	−0.0215381
$375$	−6.41615e8	−0.628297
$376$	−3.48906e9	−3.38494
$377$	1.27307e9	1.22365
$378$	1.34751e8	0.128324
$379$	−9.24411e8	−0.872224	−0.436112	−	0.899892i	$-0.643645\pi$
$379$	−9.24411e8	−0.872224	−0.436112	+	0.899892i	$0.643645\pi$
$380$	2.63746e9	2.46572
$381$	1.23281e9	1.14198
$382$	−9.76442e8	−0.896241
$383$	−1.35960e9	−1.23656	−0.618281	−	0.785957i	$-0.712171\pi$
$383$	−1.35960e9	−1.23656	−0.618281	+	0.785957i	$0.712171\pi$
$384$	−9.38367e8	−0.845694
$385$	9.36142e7	0.0836044
$386$	−1.75782e8	−0.155567
$387$	3.90585e7	0.0342553
$388$	−2.33888e9	−2.03281
$389$	−1.06062e9	−0.913556	−0.456778	−	0.889581i	$-0.650997\pi$
$389$	−1.06062e9	−0.913556	−0.456778	+	0.889581i	$0.650997\pi$
$390$	1.10502e9	0.943285
$391$	−3.36180e7	−0.0284415
$392$	2.03106e9	1.70303
$393$	5.73076e7	0.0476254
$394$	3.32629e9	2.73983
$395$	7.39369e8	0.603631
$396$	−2.33660e8	−0.189082
$397$	5.09419e8	0.408610	0.204305	−	0.978907i	$-0.434507\pi$
$397$	5.09419e8	0.408610	0.204305	+	0.978907i	$0.434507\pi$
$398$	8.68741e8	0.690716
$399$	3.86856e8	0.304891
$400$	−5.53154e8	−0.432151
$401$	8.98086e7	0.0695525	0.0347762	−	0.999395i	$-0.488928\pi$
$401$	8.98086e7	0.0695525	0.0347762	+	0.999395i	$0.488928\pi$
$402$	1.29353e9	0.993081
$403$	7.94755e8	0.604875
$404$	−7.34992e8	−0.554559
$405$	−1.23225e8	−0.0921734
$406$	9.87316e8	0.732175
$407$	6.49323e8	0.477397
$408$	−7.17513e7	−0.0523021
$409$	−4.55927e8	−0.329506	−0.164753	−	0.986335i	$-0.552683\pi$
$409$	−4.55927e8	−0.329506	−0.164753	+	0.986335i	$0.552683\pi$
$410$	2.36045e9	1.69142
$411$	1.58601e9	1.12683
$412$	−1.68782e9	−1.18901
$413$	2.15410e7	0.0150467
$414$	−5.30255e8	−0.367268
$415$	1.37199e9	0.942283
$416$	7.58614e8	0.516648
$417$	6.28091e8	0.424176
$418$	−9.86710e8	−0.660804
$419$	1.14549e9	0.760752	0.380376	−	0.924832i	$-0.375795\pi$
$419$	1.14549e9	0.760752	0.380376	+	0.924832i	$0.375795\pi$
$420$	5.82622e8	0.383720
$421$	6.43340e7	0.0420197	0.0210099	−	0.999779i	$-0.493312\pi$
$421$	6.43340e7	0.0420197	0.0210099	+	0.999779i	$0.493312\pi$
$422$	−2.85956e9	−1.85227
$423$	8.84529e8	0.568226
$424$	4.60285e9	2.93255
$425$	2.25139e7	0.0142262
$426$	1.90496e9	1.19386
$427$	−3.52548e8	−0.219140
$428$	−1.93094e9	−1.19046
$429$	−2.81051e8	−0.171864
$430$	2.48404e8	0.150667
$431$	2.01447e9	1.21197	0.605984	−	0.795477i	$-0.292779\pi$
$431$	2.01447e9	1.21197	0.605984	+	0.795477i	$0.292779\pi$
$432$	−4.46920e8	−0.266708
$433$	−2.47860e9	−1.46723	−0.733616	−	0.679564i	$-0.762169\pi$
$433$	−2.47860e9	−1.46723	−0.733616	+	0.679564i	$0.762169\pi$
$434$	6.16365e8	0.361929
$435$	−9.02866e8	−0.525910
$436$	−2.26402e9	−1.30821
$437$	−1.52231e9	−0.872606
$438$	4.94523e8	0.281208
$439$	2.37627e9	1.34051	0.670255	−	0.742131i	$-0.266185\pi$
$439$	2.37627e9	1.34051	0.670255	+	0.742131i	$0.266185\pi$
$440$	−7.86238e8	−0.440017
$441$	−5.14905e8	−0.285886
$442$	−1.63119e8	−0.0898518
$443$	1.78958e9	0.977996	0.488998	−	0.872285i	$-0.337363\pi$
$443$	1.78958e9	0.977996	0.488998	+	0.872285i	$0.337363\pi$
$444$	4.04116e9	2.19112
$445$	3.07170e9	1.65241
$446$	−5.20863e9	−2.78004
$447$	−1.31743e9	−0.697673
$448$	−4.06747e8	−0.213723
$449$	1.38305e9	0.721068	0.360534	−	0.932746i	$-0.382594\pi$
$449$	1.38305e9	0.721068	0.360534	+	0.932746i	$0.382594\pi$
$450$	3.55110e8	0.183704
$451$	−6.00359e8	−0.308172
$452$	−5.39761e9	−2.74927
$453$	−5.53292e8	−0.279647
$454$	5.88337e9	2.95074
$455$	7.00791e8	0.348778
$456$	−3.24909e9	−1.60467
$457$	1.14169e9	0.559553	0.279776	−	0.960065i	$-0.409740\pi$
$457$	1.14169e9	0.559553	0.279776	+	0.960065i	$0.409740\pi$
$458$	−2.01810e9	−0.981551
$459$	1.81900e7	0.00877989
$460$	−2.29267e9	−1.09822
$461$	1.81586e8	0.0863234	0.0431617	−	0.999068i	$-0.486257\pi$
$461$	1.81586e8	0.0863234	0.0431617	+	0.999068i	$0.486257\pi$
$462$	−2.17967e8	−0.102836
$463$	2.97724e9	1.39406	0.697028	−	0.717044i	$-0.254505\pi$
$463$	2.97724e9	1.39406	0.697028	+	0.717044i	$0.254505\pi$
$464$	−3.27458e9	−1.52175
$465$	−5.63645e8	−0.259968
$466$	−6.45185e9	−2.95348
$467$	−2.08365e9	−0.946708	−0.473354	−	0.880872i	$-0.656957\pi$
$467$	−2.08365e9	−0.946708	−0.473354	+	0.880872i	$0.656957\pi$
$468$	−1.74916e9	−0.788806
$469$	8.20342e8	0.367189
$470$	5.62543e9	2.49927
$471$	1.04487e8	0.0460776
$472$	−1.80917e8	−0.0791920
$473$	−6.31794e7	−0.0274512
$474$	−1.72151e9	−0.742482
$475$	1.01949e9	0.436470
$476$	−8.60048e7	−0.0365509
$477$	−1.16689e9	−0.492285
$478$	−6.65802e9	−2.78835
$479$	8.17179e7	0.0339737	0.0169868	−	0.999856i	$-0.494593\pi$
$479$	8.17179e7	0.0339737	0.0169868	+	0.999856i	$0.494593\pi$
$480$	−5.38014e8	−0.222049
$481$	4.86080e9	1.99159
$482$	−8.04343e8	−0.327172
$483$	−3.36282e8	−0.135797
$484$	−4.91889e9	−1.97201
$485$	1.99518e9	0.794120
$486$	2.86911e8	0.113376
$487$	−2.10921e8	−0.0827501	−0.0413750	−	0.999144i	$-0.513174\pi$
$487$	−2.10921e8	−0.0827501	−0.0413750	+	0.999144i	$0.513174\pi$
$488$	2.96095e9	1.15335
$489$	−7.60193e8	−0.293997
$490$	−3.27470e9	−1.25743
$491$	−5.98742e8	−0.228273	−0.114136	−	0.993465i	$-0.536410\pi$
$491$	−5.98742e8	−0.228273	−0.114136	+	0.993465i	$0.536410\pi$
$492$	−3.73643e9	−1.41442
$493$	1.33278e8	0.0500951
$494$	−7.38646e9	−2.75672
$495$	1.99323e8	0.0738652
$496$	−2.04426e9	−0.752231
$497$	1.20811e9	0.441427
$498$	−3.19447e9	−1.15903
$499$	−3.93744e9	−1.41861	−0.709303	−	0.704903i	$-0.750990\pi$
$499$	−3.93744e9	−1.41861	−0.709303	+	0.704903i	$0.750990\pi$
$500$	6.45921e9	2.31092
$501$	2.67694e9	0.951057
$502$	−7.05830e9	−2.49022
$503$	3.54146e9	1.24078	0.620390	−	0.784293i	$-0.286974\pi$
$503$	3.54146e9	1.24078	0.620390	+	0.784293i	$0.286974\pi$
$504$	−7.17732e8	−0.249721
$505$	6.26984e8	0.216639
$506$	8.57718e8	0.294319
$507$	−4.09726e8	−0.139626
$508$	−1.24109e10	−4.20030
$509$	−1.44434e9	−0.485465	−0.242733	−	0.970093i	$-0.578044\pi$
$509$	−1.44434e9	−0.485465	−0.242733	+	0.970093i	$0.578044\pi$
$510$	1.15685e8	0.0386173
$511$	3.13622e8	0.103976
$512$	6.40610e9	2.10935
$513$	8.23693e8	0.269373
$514$	2.00712e9	0.651933
$515$	1.43980e9	0.464490
$516$	−3.93207e8	−0.125993
$517$	−1.43078e9	−0.455360
$518$	3.76975e9	1.19168
$519$	1.39551e9	0.438174
$520$	−5.88574e9	−1.83565
$521$	−1.14128e9	−0.353557	−0.176779	−	0.984251i	$-0.556568\pi$
$521$	−1.14128e9	−0.353557	−0.176779	+	0.984251i	$0.556568\pi$
$522$	2.10219e9	0.646883
$523$	−3.14566e9	−0.961515	−0.480758	−	0.876854i	$-0.659638\pi$
$523$	−3.14566e9	−0.961515	−0.480758	+	0.876854i	$0.659638\pi$
$524$	−5.76923e8	−0.175169
$525$	2.25207e8	0.0679243
$526$	2.32794e9	0.697464
$527$	8.32034e7	0.0247630
$528$	7.22918e8	0.213733
$529$	−2.08153e9	−0.611346
$530$	−7.42120e9	−2.16525
$531$	4.58651e7	0.0132939
$532$	−3.89452e9	−1.12141
$533$	−4.49426e9	−1.28562
$534$	−7.15200e9	−2.03251
$535$	1.64719e9	0.465056
$536$	−6.88981e9	−1.93255
$537$	2.24797e9	0.626442
$538$	7.80724e9	2.16152
$539$	8.32889e8	0.229101
$540$	1.24052e9	0.339020
$541$	−1.90873e9	−0.518267	−0.259133	−	0.965842i	$-0.583437\pi$
$541$	−1.90873e9	−0.518267	−0.259133	+	0.965842i	$0.583437\pi$
$542$	3.60068e9	0.971375
$543$	−3.63794e9	−0.975114
$544$	7.94198e7	0.0211511
$545$	1.93132e9	0.511053
$546$	−1.63169e9	−0.429006
$547$	3.10216e9	0.810417	0.405209	−	0.914224i	$-0.367199\pi$
$547$	3.10216e9	0.810417	0.405209	+	0.914224i	$0.367199\pi$
$548$	−1.59665e10	−4.14457
$549$	−7.50646e8	−0.193612
$550$	−5.74411e8	−0.147215
$551$	6.03519e9	1.53695
$552$	2.82434e9	0.714710
$553$	−1.09177e9	−0.274531
$554$	1.35413e10	3.38358
$555$	−3.44731e9	−0.855962
$556$	−6.32307e9	−1.56015
$557$	2.45986e8	0.0603139	0.0301569	−	0.999545i	$-0.490399\pi$
$557$	2.45986e8	0.0603139	0.0301569	+	0.999545i	$0.490399\pi$
$558$	1.31236e9	0.319768
$559$	−4.72958e8	−0.114520
$560$	−1.80257e9	−0.433745
$561$	−2.94234e7	−0.00703596
$562$	−7.11341e9	−1.69044
$563$	3.15888e9	0.746026	0.373013	−	0.927826i	$-0.378325\pi$
$563$	3.15888e9	0.746026	0.373013	+	0.927826i	$0.378325\pi$
$564$	−8.90466e9	−2.08997
$565$	4.60443e9	1.07401
$566$	3.03691e9	0.704002
$567$	1.81956e8	0.0419204
$568$	−1.01465e10	−2.32327
$569$	5.92367e9	1.34802	0.674012	−	0.738720i	$-0.264569\pi$
$569$	5.92367e9	1.34802	0.674012	+	0.738720i	$0.264569\pi$
$570$	5.23852e9	1.18481
$571$	2.39172e8	0.0537630	0.0268815	−	0.999639i	$-0.491442\pi$
$571$	2.39172e8	0.0537630	0.0268815	+	0.999639i	$0.491442\pi$
$572$	2.82938e9	0.632127
$573$	−1.31851e9	−0.292780
$574$	−3.48548e9	−0.769257
$575$	−8.86210e8	−0.194401
$576$	−8.66045e8	−0.188826
$577$	−7.90673e9	−1.71349	−0.856745	−	0.515740i	$-0.827517\pi$
$577$	−7.90673e9	−1.71349	−0.856745	+	0.515740i	$0.827517\pi$
$578$	8.18777e9	1.76367
$579$	−2.37361e8	−0.0508201
$580$	9.08926e9	1.93433
$581$	−2.02590e9	−0.428550
$582$	−4.64548e9	−0.976788
$583$	1.88752e9	0.394503
$584$	−2.63402e9	−0.547234
$585$	1.49212e9	0.308148
$586$	9.93518e9	2.03955
$587$	1.56494e9	0.319349	0.159674	−	0.987170i	$-0.448956\pi$
$587$	1.56494e9	0.319349	0.159674	+	0.987170i	$0.448956\pi$
$588$	5.18361e9	1.05151
$589$	3.76767e9	0.759748
$590$	2.91693e8	0.0584715
$591$	4.49155e9	0.895035
$592$	−1.25029e10	−2.47677
$593$	−7.22873e9	−1.42354	−0.711772	−	0.702411i	$-0.752107\pi$
$593$	−7.22873e9	−1.42354	−0.711772	+	0.702411i	$0.752107\pi$
$594$	−4.64095e8	−0.0908561
$595$	7.33663e7	0.0142787
$596$	1.32627e10	2.56609
$597$	1.17308e9	0.225640
$598$	6.42084e9	1.22783
$599$	4.56573e9	0.867993	0.433997	−	0.900914i	$-0.357103\pi$
$599$	4.56573e9	0.867993	0.433997	+	0.900914i	$0.357103\pi$
$600$	−1.89145e9	−0.357491
$601$	−4.43141e9	−0.832687	−0.416343	−	0.909207i	$-0.636689\pi$
$601$	−4.43141e9	−0.832687	−0.416343	+	0.909207i	$0.636689\pi$
$602$	−3.66798e8	−0.0685235
$603$	1.74667e9	0.324415
$604$	5.57006e9	1.02856
$605$	4.19606e9	0.770367
$606$	−1.45984e9	−0.266472
$607$	−5.85610e9	−1.06279	−0.531396	−	0.847124i	$-0.678332\pi$
$607$	−5.85610e9	−1.06279	−0.531396	+	0.847124i	$0.678332\pi$
$608$	3.59634e9	0.648931
$609$	1.33319e9	0.239184
$610$	−4.77396e9	−0.851577
$611$	−1.07107e10	−1.89966
$612$	−1.83121e8	−0.0322930
$613$	5.18599e8	0.0909327	0.0454664	−	0.998966i	$-0.485523\pi$
$613$	5.18599e8	0.0909327	0.0454664	+	0.998966i	$0.485523\pi$
$614$	−6.74502e9	−1.17596
$615$	3.18736e9	0.552545
$616$	1.16097e9	0.200120
$617$	3.01722e9	0.517141	0.258570	−	0.965992i	$-0.416749\pi$
$617$	3.01722e9	0.517141	0.258570	+	0.965992i	$0.416749\pi$
$618$	−3.35236e9	−0.571334
$619$	3.76322e9	0.637738	0.318869	−	0.947799i	$-0.396697\pi$
$619$	3.76322e9	0.637738	0.318869	+	0.947799i	$0.396697\pi$
$620$	5.67428e9	0.956180
$621$	−7.16012e8	−0.119978
$622$	1.83939e9	0.306484
$623$	−4.53573e9	−0.751517
$624$	5.41173e9	0.891642
$625$	−3.60677e9	−0.590933
$626$	2.25004e9	0.366590
$627$	−1.33237e9	−0.215868
$628$	−1.05188e9	−0.169476
$629$	5.08880e8	0.0815339
$630$	1.15720e9	0.184382
$631$	−1.30220e9	−0.206337	−0.103168	−	0.994664i	$-0.532898\pi$
$631$	−1.30220e9	−0.206337	−0.103168	+	0.994664i	$0.532898\pi$
$632$	9.16942e9	1.44488
$633$	−3.86131e9	−0.605092
$634$	1.68631e10	2.62799
$635$	1.05871e10	1.64085
$636$	1.17472e10	1.81066
$637$	6.23497e9	0.955754
$638$	−3.40042e9	−0.518394
$639$	2.57230e9	0.390004
$640$	−8.05845e9	−1.21513
$641$	7.58612e9	1.13767	0.568835	−	0.822452i	$-0.307394\pi$
$641$	7.58612e9	1.13767	0.568835	+	0.822452i	$0.307394\pi$
$642$	−3.83524e9	−0.572032
$643$	−6.43058e9	−0.953919	−0.476959	−	0.878925i	$-0.658261\pi$
$643$	−6.43058e9	−0.953919	−0.476959	+	0.878925i	$0.658261\pi$
$644$	3.38539e9	0.499469
$645$	3.35425e8	0.0492194
$646$	−7.73294e8	−0.112858
$647$	−1.97099e9	−0.286102	−0.143051	−	0.989715i	$-0.545691\pi$
$647$	−1.97099e9	−0.286102	−0.143051	+	0.989715i	$0.545691\pi$
$648$	−1.52819e9	−0.220631
$649$	−7.41895e7	−0.0106533
$650$	−4.30001e9	−0.614148
$651$	8.32288e8	0.118233
$652$	7.65295e9	1.08134
$653$	−4.04663e9	−0.568719	−0.284359	−	0.958718i	$-0.591781\pi$
$653$	−4.04663e9	−0.568719	−0.284359	+	0.958718i	$0.591781\pi$
$654$	−4.49679e9	−0.628609
$655$	4.92143e8	0.0684301
$656$	1.15601e10	1.59882
$657$	6.67763e8	0.0918637
$658$	−8.30661e9	−1.13667
$659$	7.33454e9	0.998329	0.499164	−	0.866507i	$-0.333640\pi$
$659$	7.33454e9	0.998329	0.499164	+	0.866507i	$0.333640\pi$
$660$	−2.00661e9	−0.271681
$661$	−3.95852e9	−0.533124	−0.266562	−	0.963818i	$-0.585888\pi$
$661$	−3.95852e9	−0.533124	−0.266562	+	0.963818i	$0.585888\pi$
$662$	1.09067e10	1.46114
$663$	−2.20263e8	−0.0293524
$664$	1.70149e10	2.25549
$665$	3.32222e9	0.438079
$666$	8.02655e9	1.05286
$667$	−5.24621e9	−0.684551
$668$	−2.69491e10	−3.49805
$669$	−7.03331e9	−0.908172
$670$	1.11085e10	1.42690
$671$	1.21421e9	0.155155
$672$	7.94441e8	0.100988
$673$	1.42864e10	1.80663	0.903315	−	0.428979i	$-0.141127\pi$
$673$	1.42864e10	1.80663	0.903315	+	0.428979i	$0.141127\pi$
$674$	−7.01892e9	−0.883000
$675$	4.79511e8	0.0600117
$676$	4.12476e9	0.513553
$677$	−4.20363e9	−0.520672	−0.260336	−	0.965518i	$-0.583833\pi$
$677$	−4.20363e9	−0.520672	−0.260336	+	0.965518i	$0.583833\pi$
$678$	−1.07207e10	−1.32105
$679$	−2.94612e9	−0.361165
$680$	−6.16182e8	−0.0751498
$681$	7.94441e9	0.963933
$682$	−2.12282e9	−0.256253
$683$	−1.40588e9	−0.168840	−0.0844201	−	0.996430i	$-0.526904\pi$
$683$	−1.40588e9	−0.168840	−0.0844201	+	0.996430i	$0.526904\pi$
$684$	−8.29221e9	−0.990773
$685$	1.36203e10	1.61908
$686$	1.04735e10	1.23867
$687$	−2.72507e9	−0.320649
$688$	1.21654e9	0.142419
$689$	1.41299e10	1.64577
$690$	−4.55369e9	−0.527706
$691$	1.56591e10	1.80548	0.902741	−	0.430185i	$-0.141552\pi$
$691$	1.56591e10	1.80548	0.902741	+	0.430185i	$0.141552\pi$
$692$	−1.40487e10	−1.61163
$693$	−2.94324e8	−0.0335939
$694$	−1.75136e10	−1.98892
$695$	5.39389e9	0.609474
$696$	−1.11971e10	−1.25884
$697$	−4.70507e8	−0.0526322
$698$	2.66225e10	2.96315
$699$	−8.71204e9	−0.964828
$700$	−2.26719e9	−0.249830
$701$	1.43698e10	1.57557	0.787787	−	0.615947i	$-0.211227\pi$
$701$	1.43698e10	1.57557	0.787787	+	0.615947i	$0.211227\pi$
$702$	−3.47419e9	−0.379030
$703$	2.30434e10	2.50152
$704$	1.40088e9	0.151320
$705$	7.59611e9	0.816450
$706$	−1.98598e10	−2.12402
$707$	−9.25817e8	−0.0985274
$708$	−4.61729e8	−0.0488958
$709$	1.40381e9	0.147926	0.0739632	−	0.997261i	$-0.476435\pi$
$709$	1.40381e9	0.147926	0.0739632	+	0.997261i	$0.476435\pi$
$710$	1.63593e10	1.71538
$711$	−2.32459e9	−0.242551
$712$	3.80942e10	3.95530
$713$	−3.27513e9	−0.338388
$714$	−1.70823e8	−0.0175631
$715$	−2.41360e9	−0.246941
$716$	−2.26306e10	−2.30409
$717$	−8.99044e9	−0.910886
$718$	−2.00031e10	−2.01680
$719$	1.61359e10	1.61898	0.809491	−	0.587133i	$-0.199743\pi$
$719$	1.61359e10	1.61898	0.809491	+	0.587133i	$0.199743\pi$
$720$	−3.83803e9	−0.383217
$721$	−2.12603e9	−0.211250
$722$	−1.71436e10	−1.69520
$723$	−1.08612e9	−0.106879
$724$	3.66235e10	3.58653
$725$	3.51337e9	0.342406
$726$	−9.76990e9	−0.947571
$727$	−1.40134e10	−1.35261	−0.676307	−	0.736620i	$-0.736421\pi$
$727$	−1.40134e10	−1.35261	−0.676307	+	0.736620i	$0.736421\pi$
$728$	8.69099e9	0.834852
$729$	3.87420e8	0.0370370
$730$	4.24684e9	0.404051
$731$	−4.95143e7	−0.00468835
$732$	7.55684e9	0.712117
$733$	5.27511e9	0.494729	0.247365	−	0.968922i	$-0.420435\pi$
$733$	5.27511e9	0.494729	0.247365	+	0.968922i	$0.420435\pi$
$734$	−1.35175e10	−1.26171
$735$	−4.42188e9	−0.410772
$736$	−3.12619e9	−0.289030
$737$	−2.82534e9	−0.259977
$738$	−7.42129e9	−0.679645
$739$	7.80183e9	0.711117	0.355558	−	0.934654i	$-0.384291\pi$
$739$	7.80183e9	0.711117	0.355558	+	0.934654i	$0.384291\pi$
$740$	3.47044e10	3.14828
$741$	−9.97407e9	−0.900552
$742$	1.09583e10	0.984756
$743$	1.17555e10	1.05143	0.525715	−	0.850661i	$-0.323798\pi$
$743$	1.17555e10	1.05143	0.525715	+	0.850661i	$0.323798\pi$
$744$	−6.99014e9	−0.622273
$745$	−1.13138e10	−1.00245
$746$	−4.59179e9	−0.404945
$747$	−4.31354e9	−0.378627
$748$	2.96209e8	0.0258787
$749$	−2.43227e9	−0.211507
$750$	1.28293e10	1.11042
$751$	−1.27368e10	−1.09729	−0.548644	−	0.836056i	$-0.684856\pi$
$751$	−1.27368e10	−1.09729	−0.548644	+	0.836056i	$0.684856\pi$
$752$	2.75501e10	2.36244
$753$	−9.53094e9	−0.813492
$754$	−2.54554e10	−2.16262
$755$	−4.75153e9	−0.401809
$756$	−1.83177e9	−0.154186
$757$	1.82827e10	1.53181	0.765906	−	0.642952i	$-0.222291\pi$
$757$	1.82827e10	1.53181	0.765906	+	0.642952i	$0.222291\pi$
$758$	1.84839e10	1.54153
$759$	1.15819e9	0.0961466
$760$	−2.79023e10	−2.30565
$761$	−7.62266e9	−0.626989	−0.313495	−	0.949590i	$-0.601500\pi$
$761$	−7.62266e9	−0.626989	−0.313495	+	0.949590i	$0.601500\pi$
$762$	−2.46505e10	−2.01829
$763$	−2.85182e9	−0.232427
$764$	1.32736e10	1.07686
$765$	1.56211e8	0.0126153
$766$	2.71857e10	2.18544
$767$	−5.55379e8	−0.0444432
$768$	1.46572e10	1.16758
$769$	8.63633e8	0.0684837	0.0342419	−	0.999414i	$-0.489098\pi$
$769$	8.63633e8	0.0684837	0.0342419	+	0.999414i	$0.489098\pi$
$770$	−1.87184e9	−0.147758
$771$	2.71025e9	0.212971
$772$	2.38954e9	0.186920
$773$	−4.87081e9	−0.379292	−0.189646	−	0.981853i	$-0.560734\pi$
$773$	−4.87081e9	−0.379292	−0.189646	+	0.981853i	$0.560734\pi$
$774$	−7.80987e8	−0.0605411
$775$	2.19334e9	0.169258
$776$	2.47436e10	1.90084
$777$	5.09036e9	0.389291
$778$	2.12074e10	1.61458
$779$	−2.13058e10	−1.61479
$780$	−1.50214e10	−1.13339
$781$	−4.16085e9	−0.312538
$782$	6.72202e8	0.0502662
$783$	2.83863e9	0.211321
$784$	−1.60376e10	−1.18859
$785$	8.97309e8	0.0662061
$786$	−1.14588e9	−0.0841708
$787$	−1.20865e9	−0.0883874	−0.0441937	−	0.999023i	$-0.514072\pi$
$787$	−1.20865e9	−0.0883874	−0.0441937	+	0.999023i	$0.514072\pi$
$788$	−4.52169e10	−3.29200
$789$	3.14346e9	0.227844
$790$	−1.47839e10	−1.06683
$791$	−6.79899e9	−0.488457
$792$	2.47194e9	0.176807
$793$	9.08954e9	0.647270
$794$	−1.01860e10	−0.722157
$795$	−1.00210e10	−0.707335
$796$	−1.18095e10	−0.829918
$797$	1.01022e10	0.706824	0.353412	−	0.935468i	$-0.385021\pi$
$797$	1.01022e10	0.706824	0.353412	+	0.935468i	$0.385021\pi$
$798$	−7.73530e9	−0.538849
$799$	−1.12131e9	−0.0777702
$800$	2.09360e9	0.144570
$801$	−9.65747e9	−0.663972
$802$	−1.79575e9	−0.122924
$803$	−1.08015e9	−0.0736170
$804$	−1.75840e10	−1.19322
$805$	−2.88791e9	−0.195118
$806$	−1.58914e10	−1.06903
$807$	1.05422e10	0.706116
$808$	7.77566e9	0.518558
$809$	1.12069e10	0.744160	0.372080	−	0.928201i	$-0.378645\pi$
$809$	1.12069e10	0.744160	0.372080	+	0.928201i	$0.378645\pi$
$810$	2.46392e9	0.162903
$811$	2.51143e10	1.65328	0.826642	−	0.562729i	$-0.190248\pi$
$811$	2.51143e10	1.65328	0.826642	+	0.562729i	$0.190248\pi$
$812$	−1.34214e10	−0.879733
$813$	4.86206e9	0.317324
$814$	−1.29834e10	−0.843729
$815$	−6.52834e9	−0.422427
$816$	5.66558e8	0.0365030
$817$	−2.24214e9	−0.143842
$818$	9.11640e9	0.582354
$819$	−2.20330e9	−0.140146
$820$	−3.20875e10	−2.03230
$821$	−1.45342e10	−0.916623	−0.458312	−	0.888792i	$-0.651546\pi$
$821$	−1.45342e10	−0.916623	−0.458312	+	0.888792i	$0.651546\pi$
$822$	−3.17128e10	−1.99151
$823$	−1.81214e10	−1.13316	−0.566582	−	0.824005i	$-0.691735\pi$
$823$	−1.81214e10	−1.13316	−0.566582	+	0.824005i	$0.691735\pi$
$824$	1.78559e10	1.11183
$825$	−7.75637e8	−0.0480917
$826$	−4.30719e8	−0.0265928
$827$	−2.10420e10	−1.29365	−0.646827	−	0.762637i	$-0.723904\pi$
$827$	−2.10420e10	−1.29365	−0.646827	+	0.762637i	$0.723904\pi$
$828$	7.20818e9	0.441285
$829$	1.89577e10	1.15570	0.577849	−	0.816144i	$-0.303892\pi$
$829$	1.89577e10	1.15570	0.577849	+	0.816144i	$0.303892\pi$
$830$	−2.74333e10	−1.66534
$831$	1.82850e10	1.10533
$832$	1.04869e10	0.631271
$833$	6.52743e8	0.0391277
$834$	−1.25589e10	−0.749669
$835$	2.29889e10	1.36652
$836$	1.34131e10	0.793978
$837$	1.77211e9	0.104460
$838$	−2.29045e10	−1.34452
$839$	1.79258e10	1.04788	0.523940	−	0.851755i	$-0.324462\pi$
$839$	1.79258e10	1.04788	0.523940	+	0.851755i	$0.324462\pi$
$840$	−6.16370e9	−0.358809
$841$	3.54870e9	0.205723
$842$	−1.28638e9	−0.0742637
$843$	−9.60536e9	−0.552226
$844$	3.88722e10	2.22557
$845$	−3.51863e9	−0.200620
$846$	−1.76864e10	−1.00425
$847$	−6.19597e9	−0.350363
$848$	−3.63447e10	−2.04671
$849$	4.10079e9	0.229980
$850$	−4.50171e8	−0.0251427
$851$	−2.00310e10	−1.11416
$852$	−2.58957e10	−1.43446
$853$	−1.77620e10	−0.979875	−0.489937	−	0.871758i	$-0.662980\pi$
$853$	−1.77620e10	−0.979875	−0.489937	+	0.871758i	$0.662980\pi$
$854$	7.04931e9	0.387297
$855$	7.07367e9	0.387047
$856$	2.04279e10	1.11318
$857$	−1.55746e10	−0.845250	−0.422625	−	0.906305i	$-0.638891\pi$
$857$	−1.55746e10	−0.845250	−0.422625	+	0.906305i	$0.638891\pi$
$858$	5.61970e9	0.303744
$859$	1.29423e10	0.696681	0.348341	−	0.937368i	$-0.386745\pi$
$859$	1.29423e10	0.696681	0.348341	+	0.937368i	$0.386745\pi$
$860$	−3.37676e9	−0.181032
$861$	−4.70651e9	−0.251297
$862$	−4.02800e10	−2.14197
$863$	−3.18503e10	−1.68685	−0.843424	−	0.537249i	$-0.819463\pi$
$863$	−3.18503e10	−1.68685	−0.843424	+	0.537249i	$0.819463\pi$
$864$	1.69152e9	0.0892236
$865$	1.19843e10	0.629586
$866$	4.95604e10	2.59312
$867$	1.10561e10	0.576149
$868$	−8.37874e9	−0.434870
$869$	3.76016e9	0.194373
$870$	1.80531e10	0.929467
$871$	−2.11504e10	−1.08456
$872$	2.39516e10	1.22328
$873$	−6.27287e9	−0.319093
$874$	3.04391e10	1.54220
$875$	8.13620e9	0.410576
$876$	−6.72245e9	−0.337881
$877$	−1.65008e10	−0.826050	−0.413025	−	0.910720i	$-0.635528\pi$
$877$	−1.65008e10	−0.826050	−0.413025	+	0.910720i	$0.635528\pi$
$878$	−4.75143e10	−2.36915
$879$	1.34156e10	0.666271
$880$	6.20824e9	0.307100
$881$	1.62477e10	0.800528	0.400264	−	0.916400i	$-0.368918\pi$
$881$	1.62477e10	0.800528	0.400264	+	0.916400i	$0.368918\pi$
$882$	1.02957e10	0.505261
$883$	2.64434e9	0.129257	0.0646286	−	0.997909i	$-0.479414\pi$
$883$	2.64434e9	0.129257	0.0646286	+	0.997909i	$0.479414\pi$
$884$	2.21741e9	0.107960
$885$	3.93878e8	0.0191012
$886$	−3.57831e10	−1.72846
$887$	3.17508e10	1.52765	0.763823	−	0.645426i	$-0.223320\pi$
$887$	3.17508e10	1.52765	0.763823	+	0.645426i	$0.223320\pi$
$888$	−4.27524e10	−2.04887
$889$	−1.56331e10	−0.746258
$890$	−6.14196e10	−2.92040
$891$	−6.26675e8	−0.0296804
$892$	7.08051e10	3.34032
$893$	−5.07761e10	−2.38605
$894$	2.63425e10	1.23303
$895$	1.93050e10	0.900097
$896$	1.18993e10	0.552640
$897$	8.67016e9	0.401101
$898$	−2.76546e10	−1.27438
$899$	1.29842e10	0.596014
$900$	−4.82729e9	−0.220727
$901$	1.47926e9	0.0673766
$902$	1.20044e10	0.544649
$903$	−4.95294e8	−0.0223849
$904$	5.71027e10	2.57079
$905$	−3.12417e10	−1.40108
$906$	1.10632e10	0.494235
$907$	2.85053e10	1.26853	0.634263	−	0.773117i	$-0.281304\pi$
$907$	2.85053e10	1.26853	0.634263	+	0.773117i	$0.281304\pi$
$908$	−7.99773e10	−3.54541
$909$	−1.97125e9	−0.0870498
$910$	−1.40125e10	−0.616413
$911$	−1.16016e10	−0.508396	−0.254198	−	0.967152i	$-0.581812\pi$
$911$	−1.16016e10	−0.508396	−0.254198	+	0.967152i	$0.581812\pi$
$912$	2.56552e10	1.11994
$913$	6.97740e9	0.303421
$914$	−2.28284e10	−0.988926
$915$	−6.44636e9	−0.278189
$916$	2.74336e10	1.17937
$917$	−7.26708e8	−0.0311220
$918$	−3.63715e8	−0.0155172
$919$	1.83222e10	0.778705	0.389353	−	0.921089i	$-0.372699\pi$
$919$	1.83222e10	0.778705	0.389353	+	0.921089i	$0.372699\pi$
$920$	2.42547e10	1.02692
$921$	−9.10792e9	−0.384158
$922$	−3.63086e9	−0.152564
$923$	−3.11479e10	−1.30384
$924$	2.96300e9	0.123560
$925$	1.34147e10	0.557294
$926$	−5.95308e10	−2.46379
$927$	−4.52674e9	−0.186641
$928$	1.23938e10	0.509079
$929$	1.68494e10	0.689491	0.344745	−	0.938696i	$-0.387965\pi$
$929$	1.68494e10	0.689491	0.344745	+	0.938696i	$0.387965\pi$
$930$	1.12702e10	0.459455
$931$	2.95579e10	1.20047
$932$	8.77051e10	3.54870
$933$	2.48377e9	0.100121
$934$	4.16632e10	1.67317
$935$	−2.52681e8	−0.0101096
$936$	1.85048e10	0.737598
$937$	1.89386e10	0.752072	0.376036	−	0.926605i	$-0.377287\pi$
$937$	1.89386e10	0.752072	0.376036	+	0.926605i	$0.377287\pi$
$938$	−1.64030e10	−0.648953
$939$	3.03827e9	0.119756
$940$	−7.64710e10	−3.00296
$941$	−2.44551e10	−0.956765	−0.478382	−	0.878152i	$-0.658777\pi$
$941$	−2.44551e10	−0.956765	−0.478382	+	0.878152i	$0.658777\pi$
$942$	−2.08925e9	−0.0814353
$943$	1.85205e10	0.719221
$944$	1.42854e9	0.0552702
$945$	1.56259e9	0.0602330
$946$	1.26329e9	0.0485159
$947$	2.23693e10	0.855908	0.427954	−	0.903801i	$-0.359235\pi$
$947$	2.23693e10	0.855908	0.427954	+	0.903801i	$0.359235\pi$
$948$	2.34019e10	0.892117
$949$	−8.08592e9	−0.307113
$950$	−2.03849e10	−0.771395
$951$	2.27705e10	0.858500
$952$	9.09865e8	0.0341781
$953$	4.85182e10	1.81585	0.907924	−	0.419134i	$-0.137666\pi$
$953$	4.85182e10	1.81585	0.907924	+	0.419134i	$0.137666\pi$
$954$	2.33324e10	0.870040
$955$	−1.13230e10	−0.420678
$956$	9.05078e10	3.35030
$957$	−4.59164e9	−0.169346
$958$	−1.63397e9	−0.0600435
$959$	−2.01119e10	−0.736357
$960$	−7.43737e9	−0.271313
$961$	−1.94068e10	−0.705378
$962$	−9.71931e10	−3.51984
$963$	−5.17879e9	−0.186869
$964$	1.09341e10	0.393108
$965$	−2.03840e9	−0.0730203
$966$	6.72407e9	0.240001
$967$	−3.41671e10	−1.21511	−0.607555	−	0.794278i	$-0.707850\pi$
$967$	−3.41671e10	−1.21511	−0.607555	+	0.794278i	$0.707850\pi$
$968$	5.20381e10	1.84399
$969$	−1.04419e9	−0.0368678
$970$	−3.98942e10	−1.40349
$971$	−2.42258e10	−0.849201	−0.424601	−	0.905381i	$-0.639586\pi$
$971$	−2.42258e10	−0.849201	−0.424601	+	0.905381i	$0.639586\pi$
$972$	−3.90021e9	−0.136225
$973$	−7.96471e9	−0.277189
$974$	4.21743e9	0.146248
$975$	−5.80638e9	−0.200627
$976$	−2.33801e10	−0.804955
$977$	−2.15583e10	−0.739576	−0.369788	−	0.929116i	$-0.620570\pi$
$977$	−2.15583e10	−0.739576	−0.369788	+	0.929116i	$0.620570\pi$
$978$	1.52003e10	0.519596
$979$	1.56215e10	0.532089
$980$	4.45155e10	1.51085
$981$	−6.07209e9	−0.205351
$982$	1.19720e10	0.403439
$983$	−1.23761e10	−0.415573	−0.207786	−	0.978174i	$-0.566626\pi$
$983$	−1.23761e10	−0.415573	−0.207786	+	0.978174i	$0.566626\pi$
$984$	3.95286e10	1.32260
$985$	3.85723e10	1.28602
$986$	−2.66494e9	−0.0885356
$987$	−1.12166e10	−0.371321
$988$	1.00410e11	3.31229
$989$	1.94902e9	0.0640664
$990$	−3.98553e9	−0.130546
$991$	4.43757e10	1.44840	0.724198	−	0.689592i	$-0.242210\pi$
$991$	4.43757e10	1.44840	0.724198	+	0.689592i	$0.242210\pi$
$992$	7.73723e9	0.251649
$993$	1.47276e10	0.477318
$994$	−2.41565e10	−0.780156
$995$	1.00741e10	0.324209
$996$	4.34249e10	1.39262
$997$	5.62080e10	1.79624	0.898121	−	0.439748i	$-0.144932\pi$
$997$	5.62080e10	1.79624	0.898121	+	0.439748i	$0.144932\pi$
$998$	7.87303e10	2.50718
$999$	1.08384e10	0.343942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.8.a.c.1.4	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.8.a.c.1.4	✓	48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q-19.9953 q^{2} -27.0000 q^{3} +271.812 q^{4} -231.869 q^{5} +539.873 q^{6} +342.382 q^{7} -2875.57 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-19.9953 q^{2} -27.0000 q^{3} +271.812 q^{4} -231.869 q^{5} +539.873 q^{6} +342.382 q^{7} -2875.57 q^{8} +729.000 q^{9} +4636.29 q^{10} -1179.20 q^{11} -7338.93 q^{12} -8827.43 q^{13} -6846.04 q^{14} +6260.47 q^{15} +22705.9 q^{16} -924.150 q^{17} -14576.6 q^{18} -41847.9 q^{19} -63024.8 q^{20} -9244.32 q^{21} +23578.5 q^{22} +36377.2 q^{23} +77640.3 q^{24} -24361.7 q^{25} +176507. q^{26} -19683.0 q^{27} +93063.6 q^{28} -144217. q^{29} -125180. q^{30} -90032.4 q^{31} -85938.3 q^{32} +31838.4 q^{33} +18478.7 q^{34} -79387.9 q^{35} +198151. q^{36} -550647. q^{37} +836762. q^{38} +238341. q^{39} +666755. q^{40} +509124. q^{41} +184843. q^{42} +53578.2 q^{43} -320521. q^{44} -169033. q^{45} -727373. q^{46} +1.21335e6 q^{47} -613059. q^{48} -706317. q^{49} +487119. q^{50} +24952.0 q^{51} -2.39940e6 q^{52} -1.60068e6 q^{53} +393568. q^{54} +273420. q^{55} -984543. q^{56} +1.12989e6 q^{57} +2.88366e6 q^{58} +62915.1 q^{59} +1.70167e6 q^{60} -1.02969e6 q^{61} +1.80023e6 q^{62} +249597. q^{63} -1.18799e6 q^{64} +2.04681e6 q^{65} -636618. q^{66} +2.39598e6 q^{67} -251195. q^{68} -982184. q^{69} +1.58738e6 q^{70} +3.52854e6 q^{71} -2.09629e6 q^{72} +915999. q^{73} +1.10104e7 q^{74} +657766. q^{75} -1.13748e7 q^{76} -403737. q^{77} -4.76569e6 q^{78} -3.18874e6 q^{79} -5.26479e6 q^{80} +531441. q^{81} -1.01801e7 q^{82} -5.91707e6 q^{83} -2.51272e6 q^{84} +214282. q^{85} -1.07131e6 q^{86} +3.89386e6 q^{87} +3.39087e6 q^{88} -1.32476e7 q^{89} +3.37986e6 q^{90} -3.02236e6 q^{91} +9.88776e6 q^{92} +2.43087e6 q^{93} -2.42612e7 q^{94} +9.70325e6 q^{95} +2.32033e6 q^{96} -8.60476e6 q^{97} +1.41230e7 q^{98} -859637. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9}+O(q^{10}) \) 48 * q + 2 * q^2 - 1296 * q^3 + 3214 * q^4 + 428 * q^5 - 54 * q^6 - 680 * q^7 + 2355 * q^8 + 34992 * q^9	\( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9} + 6185 q^{10} + 11989 q^{11} - 86778 q^{12} - 2393 q^{13} + 18201 q^{14} - 11556 q^{15} + 208150 q^{16} + 62538 q^{17} + 1458 q^{18} - 39882 q^{19} + 113423 q^{20} + 18360 q^{21} - 93716 q^{22} + 195618 q^{23} - 63585 q^{24} + 886490 q^{25} + 294399 q^{26} - 944784 q^{27} - 60819 q^{28} + 421501 q^{29} - 166995 q^{30} + 392689 q^{31} - 341578 q^{32} - 323703 q^{33} + 50837 q^{34} + 697874 q^{35} + 2343006 q^{36} - 410396 q^{37} + 677216 q^{38} + 64611 q^{39} + 3232376 q^{40} + 3832958 q^{41} - 491427 q^{42} - 1751932 q^{43} + 4888297 q^{44} + 312012 q^{45} + 1163150 q^{46} + 106461 q^{47} - 5620050 q^{48} + 8202048 q^{49} - 2159111 q^{50} - 1688526 q^{51} - 3605030 q^{52} + 1755534 q^{53} - 39366 q^{54} - 1220729 q^{55} - 4430622 q^{56} + 1076814 q^{57} - 10000202 q^{58} - 2037752 q^{59} - 3062421 q^{60} + 1274098 q^{61} + 97748 q^{62} - 495720 q^{63} + 15135201 q^{64} + 6139645 q^{65} + 2530332 q^{66} - 7751257 q^{67} + 1700631 q^{68} - 5281686 q^{69} - 20935703 q^{70} - 12592217 q^{71} + 1716795 q^{72} + 12508355 q^{73} - 14999956 q^{74} - 23935230 q^{75} - 23946874 q^{76} + 1874177 q^{77} - 7948773 q^{78} - 5103480 q^{79} + 3128449 q^{80} + 25509168 q^{81} + 11622426 q^{82} + 3040643 q^{83} + 1642113 q^{84} - 13756076 q^{85} + 964635 q^{86} - 11380527 q^{87} - 29653500 q^{88} + 28462995 q^{89} + 4508865 q^{90} + 3016621 q^{91} + 22938254 q^{92} - 10602603 q^{93} - 10070348 q^{94} - 2579984 q^{95} + 9222606 q^{96} + 16208760 q^{97} + 6323227 q^{98} + 8739981 q^{99}+O(q^{100}) \) 48 * q + 2 * q^2 - 1296 * q^3 + 3214 * q^4 + 428 * q^5 - 54 * q^6 - 680 * q^7 + 2355 * q^8 + 34992 * q^9 + 6185 * q^10 + 11989 * q^11 - 86778 * q^12 - 2393 * q^13 + 18201 * q^14 - 11556 * q^15 + 208150 * q^16 + 62538 * q^17 + 1458 * q^18 - 39882 * q^19 + 113423 * q^20 + 18360 * q^21 - 93716 * q^22 + 195618 * q^23 - 63585 * q^24 + 886490 * q^25 + 294399 * q^26 - 944784 * q^27 - 60819 * q^28 + 421501 * q^29 - 166995 * q^30 + 392689 * q^31 - 341578 * q^32 - 323703 * q^33 + 50837 * q^34 + 697874 * q^35 + 2343006 * q^36 - 410396 * q^37 + 677216 * q^38 + 64611 * q^39 + 3232376 * q^40 + 3832958 * q^41 - 491427 * q^42 - 1751932 * q^43 + 4888297 * q^44 + 312012 * q^45 + 1163150 * q^46 + 106461 * q^47 - 5620050 * q^48 + 8202048 * q^49 - 2159111 * q^50 - 1688526 * q^51 - 3605030 * q^52 + 1755534 * q^53 - 39366 * q^54 - 1220729 * q^55 - 4430622 * q^56 + 1076814 * q^57 - 10000202 * q^58 - 2037752 * q^59 - 3062421 * q^60 + 1274098 * q^61 + 97748 * q^62 - 495720 * q^63 + 15135201 * q^64 + 6139645 * q^65 + 2530332 * q^66 - 7751257 * q^67 + 1700631 * q^68 - 5281686 * q^69 - 20935703 * q^70 - 12592217 * q^71 + 1716795 * q^72 + 12508355 * q^73 - 14999956 * q^74 - 23935230 * q^75 - 23946874 * q^76 + 1874177 * q^77 - 7948773 * q^78 - 5103480 * q^79 + 3128449 * q^80 + 25509168 * q^81 + 11622426 * q^82 + 3040643 * q^83 + 1642113 * q^84 - 13756076 * q^85 + 964635 * q^86 - 11380527 * q^87 - 29653500 * q^88 + 28462995 * q^89 + 4508865 * q^90 + 3016621 * q^91 + 22938254 * q^92 - 10602603 * q^93 - 10070348 * q^94 - 2579984 * q^95 + 9222606 * q^96 + 16208760 * q^97 + 6323227 * q^98 + 8739981 * q^99

Introduction

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