LMFDB - Embedded newform 471.8.a.c.1.13

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			1.13
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)

$f(q)$	$=$	$q-12.5583 q^{2} -27.0000 q^{3} +29.7115 q^{4} +553.645 q^{5} +339.075 q^{6} -406.725 q^{7} +1234.34 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-12.5583 q^{2} -27.0000 q^{3} +29.7115 q^{4} +553.645 q^{5} +339.075 q^{6} -406.725 q^{7} +1234.34 q^{8} +729.000 q^{9} -6952.86 q^{10} +2658.74 q^{11} -802.209 q^{12} -14814.8 q^{13} +5107.78 q^{14} -14948.4 q^{15} -19304.3 q^{16} -21921.4 q^{17} -9155.02 q^{18} -25078.3 q^{19} +16449.6 q^{20} +10981.6 q^{21} -33389.3 q^{22} -106143. q^{23} -33327.2 q^{24} +228398. q^{25} +186050. q^{26} -19683.0 q^{27} -12084.4 q^{28} -54616.8 q^{29} +187727. q^{30} +121213. q^{31} +84434.2 q^{32} -71786.0 q^{33} +275296. q^{34} -225181. q^{35} +21659.7 q^{36} -92373.1 q^{37} +314942. q^{38} +400001. q^{39} +683386. q^{40} +171664. q^{41} -137910. q^{42} -203663. q^{43} +78995.1 q^{44} +403607. q^{45} +1.33298e6 q^{46} -935476. q^{47} +521216. q^{48} -658118. q^{49} -2.86830e6 q^{50} +591877. q^{51} -440171. q^{52} +724049. q^{53} +247185. q^{54} +1.47200e6 q^{55} -502037. q^{56} +677115. q^{57} +685896. q^{58} +1.54359e6 q^{59} -444140. q^{60} +2.23540e6 q^{61} -1.52223e6 q^{62} -296503. q^{63} +1.41060e6 q^{64} -8.20217e6 q^{65} +901512. q^{66} +1.27008e6 q^{67} -651316. q^{68} +2.86587e6 q^{69} +2.82790e6 q^{70} +538399. q^{71} +899833. q^{72} +5.52700e6 q^{73} +1.16005e6 q^{74} -6.16675e6 q^{75} -745114. q^{76} -1.08138e6 q^{77} -5.02334e6 q^{78} -5.40610e6 q^{79} -1.06877e7 q^{80} +531441. q^{81} -2.15581e6 q^{82} +2.20567e6 q^{83} +326279. q^{84} -1.21367e7 q^{85} +2.55766e6 q^{86} +1.47465e6 q^{87} +3.28179e6 q^{88} -8.77881e6 q^{89} -5.06863e6 q^{90} +6.02557e6 q^{91} -3.15367e6 q^{92} -3.27275e6 q^{93} +1.17480e7 q^{94} -1.38845e7 q^{95} -2.27972e6 q^{96} -6.35053e6 q^{97} +8.26485e6 q^{98} +1.93822e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−12.5583	−1.11001	−0.555005	−	0.831847i	$-0.687284\pi$
$2$	−12.5583	−1.11001	−0.555005	+	0.831847i	$0.687284\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	29.7115	0.232121
$5$	553.645	1.98078	0.990391	−	0.138297i	$-0.0441628\pi$
$5$	553.645	1.98078	0.990391	+	0.138297i	$0.0441628\pi$
$6$	339.075	0.640864
$7$	−406.725	−0.448186	−0.224093	−	0.974568i	$-0.571942\pi$
$7$	−406.725	−0.448186	−0.224093	+	0.974568i	$0.571942\pi$
$8$	1234.34	0.852353
$9$	729.000	0.333333
$10$	−6952.86	−2.19869
$11$	2658.74	0.602284	0.301142	−	0.953579i	$-0.402632\pi$
$11$	2658.74	0.602284	0.301142	+	0.953579i	$0.402632\pi$
$12$	−802.209	−0.134015
$13$	−14814.8	−1.87023	−0.935116	−	0.354342i	$-0.884705\pi$
$13$	−14814.8	−1.87023	−0.935116	+	0.354342i	$0.884705\pi$
$14$	5107.78	0.497490
$15$	−14948.4	−1.14360
$16$	−19304.3	−1.17824
$17$	−21921.4	−1.08217	−0.541087	−	0.840967i	$-0.681987\pi$
$17$	−21921.4	−1.08217	−0.541087	+	0.840967i	$0.681987\pi$
$18$	−9155.02	−0.370003
$19$	−25078.3	−0.838805	−0.419402	−	0.907800i	$-0.637760\pi$
$19$	−25078.3	−0.838805	−0.419402	+	0.907800i	$0.637760\pi$
$20$	16449.6	0.459781
$21$	10981.6	0.258760
$22$	−33389.3	−0.668541
$23$	−106143.	−1.81905	−0.909527	−	0.415646i	$-0.863556\pi$
$23$	−106143.	−1.81905	−0.909527	+	0.415646i	$0.863556\pi$
$24$	−33327.2	−0.492106
$25$	228398.	2.92350
$26$	186050.	2.07597
$27$	−19683.0	−0.192450
$28$	−12084.4	−0.104033
$29$	−54616.8	−0.415847	−0.207923	−	0.978145i	$-0.566670\pi$
$29$	−54616.8	−0.415847	−0.207923	+	0.978145i	$0.566670\pi$
$30$	187727.	1.26941
$31$	121213.	0.730774	0.365387	−	0.930856i	$-0.380937\pi$
$31$	121213.	0.730774	0.365387	+	0.930856i	$0.380937\pi$
$32$	84434.2	0.455505
$33$	−71786.0	−0.347729
$34$	275296.	1.20122
$35$	−225181.	−0.887758
$36$	21659.7	0.0773736
$37$	−92373.1	−0.299806	−0.149903	−	0.988701i	$-0.547896\pi$
$37$	−92373.1	−0.299806	−0.149903	+	0.988701i	$0.547896\pi$
$38$	314942.	0.931081
$39$	400001.	1.07978
$40$	683386.	1.68833
$41$	171664.	0.388987	0.194494	−	0.980904i	$-0.437694\pi$
$41$	171664.	0.388987	0.194494	+	0.980904i	$0.437694\pi$
$42$	−137910.	−0.287226
$43$	−203663.	−0.390635	−0.195318	−	0.980740i	$-0.562574\pi$
$43$	−203663.	−0.390635	−0.195318	+	0.980740i	$0.562574\pi$
$44$	78995.1	0.139803
$45$	403607.	0.660261
$46$	1.33298e6	2.01917
$47$	−935476.	−1.31429	−0.657144	−	0.753765i	$-0.728235\pi$
$47$	−935476.	−1.31429	−0.657144	+	0.753765i	$0.728235\pi$
$48$	521216.	0.680258
$49$	−658118.	−0.799130
$50$	−2.86830e6	−3.24511
$51$	591877.	0.624793
$52$	−440171.	−0.434120
$53$	724049.	0.668040	0.334020	−	0.942566i	$-0.391595\pi$
$53$	724049.	0.668040	0.334020	+	0.942566i	$0.391595\pi$
$54$	247185.	0.213621
$55$	1.47200e6	1.19299
$56$	−502037.	−0.382012
$57$	677115.	0.484284
$58$	685896.	0.461594
$59$	1.54359e6	0.978473	0.489236	−	0.872151i	$-0.337276\pi$
$59$	1.54359e6	0.978473	0.489236	+	0.872151i	$0.337276\pi$
$60$	−444140.	−0.265454
$61$	2.23540e6	1.26096	0.630479	−	0.776206i	$-0.282858\pi$
$61$	2.23540e6	1.26096	0.630479	+	0.776206i	$0.282858\pi$
$62$	−1.52223e6	−0.811166
$63$	−296503.	−0.149395
$64$	1.41060e6	0.672626
$65$	−8.20217e6	−3.70452
$66$	901512.	0.385983
$67$	1.27008e6	0.515905	0.257953	−	0.966158i	$-0.416952\pi$
$67$	1.27008e6	0.515905	0.257953	+	0.966158i	$0.416952\pi$
$68$	−651316.	−0.251195
$69$	2.86587e6	1.05023
$70$	2.82790e6	0.985419
$71$	538399.	0.178525	0.0892626	−	0.996008i	$-0.471549\pi$
$71$	538399.	0.178525	0.0892626	+	0.996008i	$0.471549\pi$
$72$	899833.	0.284118
$73$	5.52700e6	1.66287	0.831437	−	0.555619i	$-0.187519\pi$
$73$	5.52700e6	1.66287	0.831437	+	0.555619i	$0.187519\pi$
$74$	1.16005e6	0.332787
$75$	−6.16675e6	−1.68788
$76$	−745114.	−0.194704
$77$	−1.08138e6	−0.269935
$78$	−5.02334e6	−1.19856
$79$	−5.40610e6	−1.23364	−0.616821	−	0.787103i	$-0.711580\pi$
$79$	−5.40610e6	−1.23364	−0.616821	+	0.787103i	$0.711580\pi$
$80$	−1.06877e7	−2.33384
$81$	531441.	0.111111
$82$	−2.15581e6	−0.431779
$83$	2.20567e6	0.423416	0.211708	−	0.977333i	$-0.432097\pi$
$83$	2.20567e6	0.423416	0.211708	+	0.977333i	$0.432097\pi$
$84$	326279.	0.0600636
$85$	−1.21367e7	−2.14355
$86$	2.55766e6	0.433609
$87$	1.47465e6	0.240089
$88$	3.28179e6	0.513359
$89$	−8.77881e6	−1.31999	−0.659995	−	0.751270i	$-0.729442\pi$
$89$	−8.77881e6	−1.31999	−0.659995	+	0.751270i	$0.729442\pi$
$90$	−5.06863e6	−0.732895
$91$	6.02557e6	0.838211
$92$	−3.15367e6	−0.422240
$93$	−3.27275e6	−0.421912
$94$	1.17480e7	1.45887
$95$	−1.38845e7	−1.66149
$96$	−2.27972e6	−0.262986
$97$	−6.35053e6	−0.706495	−0.353247	−	0.935530i	$-0.614923\pi$
$97$	−6.35053e6	−0.706495	−0.353247	+	0.935530i	$0.614923\pi$
$98$	8.26485e6	0.887041
$99$	1.93822e6	0.200761
$100$	6.78604e6	0.678604
$101$	1.39065e7	1.34305	0.671527	−	0.740980i	$-0.265639\pi$
$101$	1.39065e7	1.34305	0.671527	+	0.740980i	$0.265639\pi$
$102$	−7.43299e6	−0.693526
$103$	2.69043e6	0.242601	0.121300	−	0.992616i	$-0.461294\pi$
$103$	2.69043e6	0.242601	0.121300	+	0.992616i	$0.461294\pi$
$104$	−1.82865e7	−1.59410
$105$	6.07990e6	0.512547
$106$	−9.09284e6	−0.741531
$107$	−1.26405e7	−0.997516	−0.498758	−	0.866741i	$-0.666210\pi$
$107$	−1.26405e7	−0.997516	−0.498758	+	0.866741i	$0.666210\pi$
$108$	−584811.	−0.0446717
$109$	1.28930e7	0.953590	0.476795	−	0.879014i	$-0.341798\pi$
$109$	1.28930e7	0.953590	0.476795	+	0.879014i	$0.341798\pi$
$110$	−1.84858e7	−1.32423
$111$	2.49407e6	0.173093
$112$	7.85154e6	0.528070
$113$	1.23910e7	0.807849	0.403925	−	0.914792i	$-0.367646\pi$
$113$	1.23910e7	0.807849	0.403925	+	0.914792i	$0.367646\pi$
$114$	−8.50342e6	−0.537560
$115$	−5.87658e7	−3.60315
$116$	−1.62275e6	−0.0965267
$117$	−1.08000e7	−0.623411
$118$	−1.93848e7	−1.08611
$119$	8.91598e6	0.485014
$120$	−1.84514e7	−0.974755
$121$	−1.24183e7	−0.637254
$122$	−2.80729e7	−1.39968
$123$	−4.63492e6	−0.224582
$124$	3.60141e6	0.169628
$125$	8.31980e7	3.81003
$126$	3.72358e6	0.165830
$127$	−1.46151e7	−0.633124	−0.316562	−	0.948572i	$-0.602529\pi$
$127$	−1.46151e7	−0.633124	−0.316562	+	0.948572i	$0.602529\pi$
$128$	−2.85223e7	−1.20213
$129$	5.49889e6	0.225533
$130$	1.03006e8	4.11205
$131$	3.89252e7	1.51280	0.756399	−	0.654111i	$-0.226957\pi$
$131$	3.89252e7	1.51280	0.756399	+	0.654111i	$0.226957\pi$
$132$	−2.13287e6	−0.0807151
$133$	1.02000e7	0.375940
$134$	−1.59501e7	−0.572660
$135$	−1.08974e7	−0.381202
$136$	−2.70584e7	−0.922394
$137$	−1.87855e7	−0.624166	−0.312083	−	0.950055i	$-0.601027\pi$
$137$	−1.87855e7	−0.624166	−0.312083	+	0.950055i	$0.601027\pi$
$138$	−3.59905e7	−1.16577
$139$	2.59419e7	0.819312	0.409656	−	0.912240i	$-0.365649\pi$
$139$	2.59419e7	0.819312	0.409656	+	0.912240i	$0.365649\pi$
$140$	−6.69047e6	−0.206067
$141$	2.52579e7	0.758804
$142$	−6.76138e6	−0.198165
$143$	−3.93888e7	−1.12641
$144$	−1.40728e7	−0.392747
$145$	−3.02383e7	−0.823702
$146$	−6.94098e7	−1.84581
$147$	1.77692e7	0.461378
$148$	−2.74454e6	−0.0695911
$149$	2.91159e7	0.721072	0.360536	−	0.932745i	$-0.382594\pi$
$149$	2.91159e7	0.721072	0.360536	+	0.932745i	$0.382594\pi$
$150$	7.74440e7	1.87356
$151$	6.80298e7	1.60798	0.803988	−	0.594645i	$-0.202707\pi$
$151$	6.80298e7	1.60798	0.803988	+	0.594645i	$0.202707\pi$
$152$	−3.09552e7	−0.714958
$153$	−1.59807e7	−0.360724
$154$	1.35803e7	0.299631
$155$	6.71089e7	1.44750
$156$	1.18846e7	0.250639
$157$	−3.86989e6	−0.0798087
$157$	−3.86989e6	−0.0798087
$158$	6.78916e7	1.36935
$159$	−1.95493e7	−0.385693
$160$	4.67466e7	0.902256
$161$	4.31712e7	0.815273
$162$	−6.67401e6	−0.123334
$163$	−1.00351e8	−1.81496	−0.907479	−	0.420097i	$-0.861996\pi$
$163$	−1.00351e8	−1.81496	−0.907479	+	0.420097i	$0.861996\pi$
$164$	5.10038e6	0.0902920
$165$	−3.97440e7	−0.688775
$166$	−2.76995e7	−0.469996
$167$	−4.90057e6	−0.0814215	−0.0407107	−	0.999171i	$-0.512962\pi$
$167$	−4.90057e6	−0.0814215	−0.0407107	+	0.999171i	$0.512962\pi$
$168$	1.35550e7	0.220555
$169$	1.56731e8	2.49777
$170$	1.52416e8	2.37936
$171$	−1.82821e7	−0.279602
$172$	−6.05111e6	−0.0906746
$173$	−3.20008e7	−0.469894	−0.234947	−	0.972008i	$-0.575492\pi$
$173$	−3.20008e7	−0.469894	−0.234947	+	0.972008i	$0.575492\pi$
$174$	−1.85192e7	−0.266501
$175$	−9.28952e7	−1.31027
$176$	−5.13251e7	−0.709636
$177$	−4.16768e7	−0.564921
$178$	1.10247e8	1.46520
$179$	6.22652e7	0.811447	0.405723	−	0.913996i	$-0.367020\pi$
$179$	6.22652e7	0.811447	0.405723	+	0.913996i	$0.367020\pi$
$180$	1.19918e7	0.153260
$181$	−4.82363e6	−0.0604642	−0.0302321	−	0.999543i	$-0.509625\pi$
$181$	−4.82363e6	−0.0604642	−0.0302321	+	0.999543i	$0.509625\pi$
$182$	−7.56711e7	−0.930422
$183$	−6.03558e7	−0.728015
$184$	−1.31017e8	−1.55048
$185$	−5.11420e7	−0.593849
$186$	4.11002e7	0.468327
$187$	−5.82833e7	−0.651776
$188$	−2.77944e7	−0.305073
$189$	8.00557e6	0.0862533
$190$	1.74366e8	1.84427
$191$	8.69609e7	0.903040	0.451520	−	0.892261i	$-0.350882\pi$
$191$	8.69609e7	0.903040	0.451520	+	0.892261i	$0.350882\pi$
$192$	−3.80862e7	−0.388341
$193$	1.03298e8	1.03429	0.517145	−	0.855898i	$-0.326995\pi$
$193$	1.03298e8	1.03429	0.517145	+	0.855898i	$0.326995\pi$
$194$	7.97520e7	0.784216
$195$	2.21459e8	2.13881
$196$	−1.95536e7	−0.185495
$197$	1.39859e8	1.30334	0.651670	−	0.758503i	$-0.274069\pi$
$197$	1.39859e8	1.30334	0.651670	+	0.758503i	$0.274069\pi$
$198$	−2.43408e7	−0.222847
$199$	1.16468e8	1.04766	0.523831	−	0.851822i	$-0.324502\pi$
$199$	1.16468e8	1.04766	0.523831	+	0.851822i	$0.324502\pi$
$200$	2.81921e8	2.49185
$201$	−3.42922e7	−0.297858
$202$	−1.74642e8	−1.49080
$203$	2.22140e7	0.186377
$204$	1.75855e7	0.145027
$205$	9.50409e7	0.770499
$206$	−3.37873e7	−0.269289
$207$	−7.73785e7	−0.606351
$208$	2.85990e8	2.20358
$209$	−6.66768e7	−0.505199
$210$	−7.63533e7	−0.568932
$211$	1.51682e8	1.11159	0.555795	−	0.831319i	$-0.312414\pi$
$211$	1.51682e8	1.11159	0.555795	+	0.831319i	$0.312414\pi$
$212$	2.15126e7	0.155066
$213$	−1.45368e7	−0.103072
$214$	1.58743e8	1.10725
$215$	−1.12757e8	−0.773763
$216$	−2.42955e7	−0.164035
$217$	−4.93003e7	−0.327522
$218$	−1.61915e8	−1.05849
$219$	−1.49229e8	−0.960061
$220$	4.37353e7	0.276919
$221$	3.24762e8	2.02391
$222$	−3.13214e7	−0.192135
$223$	1.09792e8	0.662983	0.331491	−	0.943458i	$-0.392448\pi$
$223$	1.09792e8	0.662983	0.331491	+	0.943458i	$0.392448\pi$
$224$	−3.43415e7	−0.204151
$225$	1.66502e8	0.974499
$226$	−1.55610e8	−0.896720
$227$	−2.15709e8	−1.22399	−0.611995	−	0.790862i	$-0.709633\pi$
$227$	−2.15709e8	−1.22399	−0.611995	+	0.790862i	$0.709633\pi$
$228$	2.01181e7	0.112412
$229$	1.86163e8	1.02440	0.512200	−	0.858866i	$-0.328831\pi$
$229$	1.86163e8	1.02440	0.512200	+	0.858866i	$0.328831\pi$
$230$	7.38000e8	3.99953
$231$	2.91972e7	0.155847
$232$	−6.74157e7	−0.354448
$233$	−1.90525e8	−0.986748	−0.493374	−	0.869817i	$-0.664237\pi$
$233$	−1.90525e8	−0.986748	−0.493374	+	0.869817i	$0.664237\pi$
$234$	1.35630e8	0.691992
$235$	−5.17922e8	−2.60332
$236$	4.58622e7	0.227124
$237$	1.45965e8	0.712244
$238$	−1.11970e8	−0.538371
$239$	−6.41208e7	−0.303813	−0.151906	−	0.988395i	$-0.548541\pi$
$239$	−6.41208e7	−0.303813	−0.151906	+	0.988395i	$0.548541\pi$
$240$	2.88569e8	1.34744
$241$	−7.28919e7	−0.335444	−0.167722	−	0.985834i	$-0.553641\pi$
$241$	−7.28919e7	−0.335444	−0.167722	+	0.985834i	$0.553641\pi$
$242$	1.55953e8	0.707357
$243$	−1.43489e7	−0.0641500
$244$	6.64170e7	0.292695
$245$	−3.64364e8	−1.58290
$246$	5.82069e7	0.249288
$247$	3.71531e8	1.56876
$248$	1.49618e8	0.622877
$249$	−5.95531e7	−0.244459
$250$	−1.04483e9	−4.22916
$251$	−4.55611e8	−1.81860	−0.909298	−	0.416146i	$-0.863381\pi$
$251$	−4.55611e8	−1.81860	−0.909298	+	0.416146i	$0.863381\pi$
$252$	−8.80953e6	−0.0346777
$253$	−2.82208e8	−1.09559
$254$	1.83541e8	0.702774
$255$	3.27690e8	1.23758
$256$	1.77636e8	0.661745
$257$	−1.90354e8	−0.699513	−0.349757	−	0.936841i	$-0.613736\pi$
$257$	−1.90354e8	−0.699513	−0.349757	+	0.936841i	$0.613736\pi$
$258$	−6.90568e7	−0.250344
$259$	3.75705e7	0.134369
$260$	−2.43698e8	−0.859896
$261$	−3.98157e7	−0.138616
$262$	−4.88835e8	−1.67922
$263$	4.01692e8	1.36160	0.680799	−	0.732471i	$-0.261633\pi$
$263$	4.01692e8	1.36160	0.680799	+	0.732471i	$0.261633\pi$
$264$	−8.86083e7	−0.296388
$265$	4.00866e8	1.32324
$266$	−1.28095e8	−0.417297
$267$	2.37028e8	0.762096
$268$	3.77360e7	0.119752
$269$	3.61524e8	1.13241	0.566205	−	0.824264i	$-0.308411\pi$
$269$	3.61524e8	1.13241	0.566205	+	0.824264i	$0.308411\pi$
$270$	1.36853e8	0.423137
$271$	−4.06650e8	−1.24116	−0.620581	−	0.784142i	$-0.713103\pi$
$271$	−4.06650e8	−1.24116	−0.620581	+	0.784142i	$0.713103\pi$
$272$	4.23177e8	1.27506
$273$	−1.62690e8	−0.483941
$274$	2.35914e8	0.692830
$275$	6.07251e8	1.76078
$276$	8.51492e7	0.243780
$277$	4.39647e8	1.24287	0.621433	−	0.783467i	$-0.286551\pi$
$277$	4.39647e8	1.24287	0.621433	+	0.783467i	$0.286551\pi$
$278$	−3.25786e8	−0.909444
$279$	8.83642e7	0.243591
$280$	−2.77950e8	−0.756683
$281$	−6.03539e8	−1.62268	−0.811340	−	0.584574i	$-0.801262\pi$
$281$	−6.03539e8	−1.62268	−0.811340	+	0.584574i	$0.801262\pi$
$282$	−3.17196e8	−0.842280
$283$	1.37655e8	0.361027	0.180513	−	0.983573i	$-0.442224\pi$
$283$	1.37655e8	0.361027	0.180513	+	0.983573i	$0.442224\pi$
$284$	1.59966e7	0.0414394
$285$	3.74881e8	0.959261
$286$	4.94658e8	1.25033
$287$	−6.98200e7	−0.174338
$288$	6.15525e7	0.151835
$289$	7.02085e7	0.171099
$290$	3.79743e8	0.914317
$291$	1.71464e8	0.407895
$292$	1.64215e8	0.385988
$293$	3.59633e8	0.835262	0.417631	−	0.908617i	$-0.362860\pi$
$293$	3.59633e8	0.835262	0.417631	+	0.908617i	$0.362860\pi$
$294$	−2.23151e8	−0.512134
$295$	8.54599e8	1.93814
$296$	−1.14020e8	−0.255540
$297$	−5.23320e7	−0.115910
$298$	−3.65647e8	−0.800397
$299$	1.57250e9	3.40205
$300$	−1.83223e8	−0.391792
$301$	8.28347e7	0.175077
$302$	−8.54340e8	−1.78487
$303$	−3.75476e8	−0.775413
$304$	4.84119e8	0.988314
$305$	1.23762e9	2.49768
$306$	2.00691e8	0.400407
$307$	5.42932e8	1.07093	0.535465	−	0.844557i	$-0.320136\pi$
$307$	5.42932e8	1.07093	0.535465	+	0.844557i	$0.320136\pi$
$308$	−3.21293e7	−0.0626576
$309$	−7.26417e7	−0.140066
$310$	−8.42776e8	−1.60674
$311$	−8.37396e8	−1.57859	−0.789295	−	0.614014i	$-0.789554\pi$
$311$	−8.37396e8	−1.57859	−0.789295	+	0.614014i	$0.789554\pi$
$312$	4.93737e8	0.920353
$313$	−3.01975e8	−0.556630	−0.278315	−	0.960490i	$-0.589776\pi$
$313$	−3.01975e8	−0.556630	−0.278315	+	0.960490i	$0.589776\pi$
$314$	4.85994e7	0.0885884
$315$	−1.64157e8	−0.295919
$316$	−1.60623e8	−0.286354
$317$	−3.10546e8	−0.547544	−0.273772	−	0.961795i	$-0.588271\pi$
$317$	−3.10546e8	−0.547544	−0.273772	+	0.961795i	$0.588271\pi$
$318$	2.45507e8	0.428123
$319$	−1.45212e8	−0.250458
$320$	7.80971e8	1.33232
$321$	3.41293e8	0.575916
$322$	−5.42157e8	−0.904961
$323$	5.49752e8	0.907732
$324$	1.57899e7	0.0257912
$325$	−3.38368e9	−5.46762
$326$	1.26024e9	2.01462
$327$	−3.48111e8	−0.550556
$328$	2.11891e8	0.331554
$329$	3.80482e8	0.589044
$330$	4.99118e8	0.764547
$331$	5.46149e8	0.827777	0.413889	−	0.910328i	$-0.364170\pi$
$331$	5.46149e8	0.827777	0.413889	+	0.910328i	$0.364170\pi$
$332$	6.55337e7	0.0982837
$333$	−6.73400e7	−0.0999352
$334$	6.15430e7	0.0903786
$335$	7.03175e8	1.02190
$336$	−2.11992e8	−0.304882
$337$	−2.71456e8	−0.386362	−0.193181	−	0.981163i	$-0.561880\pi$
$337$	−2.71456e8	−0.386362	−0.193181	+	0.981163i	$0.561880\pi$
$338$	−1.96828e9	−2.77255
$339$	−3.34556e8	−0.466412
$340$	−3.60598e8	−0.497562
$341$	3.22274e8	0.440134
$342$	2.29592e8	0.310360
$343$	6.02629e8	0.806344
$344$	−2.51389e8	−0.332959
$345$	1.58668e9	2.08028
$346$	4.01877e8	0.521587
$347$	1.37777e9	1.77021	0.885105	−	0.465392i	$-0.154087\pi$
$347$	1.37777e9	1.77021	0.885105	+	0.465392i	$0.154087\pi$
$348$	4.38141e7	0.0557297
$349$	−2.60530e8	−0.328072	−0.164036	−	0.986454i	$-0.552451\pi$
$349$	−2.60530e8	−0.328072	−0.164036	+	0.986454i	$0.552451\pi$
$350$	1.16661e9	1.45441
$351$	2.91601e8	0.359926
$352$	2.24489e8	0.274344
$353$	−3.07556e7	−0.0372146	−0.0186073	−	0.999827i	$-0.505923\pi$
$353$	−3.07556e7	−0.0372146	−0.0186073	+	0.999827i	$0.505923\pi$
$354$	5.23391e8	0.627068
$355$	2.98082e8	0.353620
$356$	−2.60831e8	−0.306397
$357$	−2.40731e8	−0.280023
$358$	−7.81947e8	−0.900714
$359$	5.62387e8	0.641512	0.320756	−	0.947162i	$-0.396063\pi$
$359$	5.62387e8	0.641512	0.320756	+	0.947162i	$0.396063\pi$
$360$	4.98188e8	0.562775
$361$	−2.64950e8	−0.296407
$362$	6.05766e7	0.0671159
$363$	3.35293e8	0.367918
$364$	1.79029e8	0.194566
$365$	3.06000e9	3.29379
$366$	7.57967e8	0.808103
$367$	3.87540e8	0.409246	0.204623	−	0.978841i	$-0.434403\pi$
$367$	3.87540e8	0.409246	0.204623	+	0.978841i	$0.434403\pi$
$368$	2.04902e9	2.14328
$369$	1.25143e8	0.129662
$370$	6.42257e8	0.659178
$371$	−2.94489e8	−0.299406
$372$	−9.72381e7	−0.0979346
$373$	1.33496e9	1.33194	0.665972	−	0.745977i	$-0.268017\pi$
$373$	1.33496e9	1.33194	0.665972	+	0.745977i	$0.268017\pi$
$374$	7.31940e8	0.723477
$375$	−2.24635e9	−2.19972
$376$	−1.15469e9	−1.12024
$377$	8.09140e8	0.777730
$378$	−1.00537e8	−0.0957420
$379$	−8.81517e8	−0.831752	−0.415876	−	0.909421i	$-0.636525\pi$
$379$	−8.81517e8	−0.831752	−0.415876	+	0.909421i	$0.636525\pi$
$380$	−4.12529e8	−0.385666
$381$	3.94608e8	0.365535
$382$	−1.09208e9	−1.00238
$383$	−8.46372e8	−0.769778	−0.384889	−	0.922963i	$-0.625760\pi$
$383$	−8.46372e8	−0.769778	−0.384889	+	0.922963i	$0.625760\pi$
$384$	7.70103e8	0.694048
$385$	−5.98699e8	−0.534683
$386$	−1.29725e9	−1.14807
$387$	−1.48470e8	−0.130212
$388$	−1.88684e8	−0.163992
$389$	−1.70602e9	−1.46947	−0.734734	−	0.678355i	$-0.762693\pi$
$389$	−1.70602e9	−1.46947	−0.734734	+	0.678355i	$0.762693\pi$
$390$	−2.78115e9	−2.37409
$391$	2.32681e9	1.96853
$392$	−8.12340e8	−0.681141
$393$	−1.05098e9	−0.873414
$394$	−1.75639e9	−1.44672
$395$	−2.99306e9	−2.44358
$396$	5.75874e7	0.0466009
$397$	2.94647e8	0.236339	0.118170	−	0.992993i	$-0.462297\pi$
$397$	2.94647e8	0.236339	0.118170	+	0.992993i	$0.462297\pi$
$398$	−1.46264e9	−1.16291
$399$	−2.75400e8	−0.217049
$400$	−4.40906e9	−3.44458
$401$	−2.05695e8	−0.159301	−0.0796503	−	0.996823i	$-0.525380\pi$
$401$	−2.05695e8	−0.159301	−0.0796503	+	0.996823i	$0.525380\pi$
$402$	4.30653e8	0.330625
$403$	−1.79575e9	−1.36672
$404$	4.13183e8	0.311751
$405$	2.94230e8	0.220087
$406$	−2.78971e8	−0.206880
$407$	−2.45596e8	−0.180568
$408$	7.30577e8	0.532544
$409$	−1.91589e9	−1.38465	−0.692323	−	0.721587i	$-0.743413\pi$
$409$	−1.91589e9	−1.38465	−0.692323	+	0.721587i	$0.743413\pi$
$410$	−1.19355e9	−0.855261
$411$	5.07208e8	0.360363
$412$	7.99367e7	0.0563127
$413$	−6.27815e8	−0.438537
$414$	9.71744e8	0.673055
$415$	1.22116e9	0.838695
$416$	−1.25088e9	−0.851900
$417$	−7.00430e8	−0.473030
$418$	8.37348e8	0.560776
$419$	−4.37661e8	−0.290662	−0.145331	−	0.989383i	$-0.546425\pi$
$419$	−4.37661e8	−0.290662	−0.145331	+	0.989383i	$0.546425\pi$
$420$	1.80643e8	0.118973
$421$	−2.26258e8	−0.147780	−0.0738901	−	0.997266i	$-0.523541\pi$
$421$	−2.26258e8	−0.147780	−0.0738901	+	0.997266i	$0.523541\pi$
$422$	−1.90487e9	−1.23388
$423$	−6.81962e8	−0.438096
$424$	8.93722e8	0.569406
$425$	−5.00680e9	−3.16373
$426$	1.82557e8	0.114410
$427$	−9.09193e8	−0.565143
$428$	−3.75567e8	−0.231544
$429$	1.06350e9	0.650334
$430$	1.41604e9	0.858885
$431$	2.14998e9	1.29349	0.646745	−	0.762706i	$-0.276130\pi$
$431$	2.14998e9	1.29349	0.646745	+	0.762706i	$0.276130\pi$
$432$	3.79966e8	0.226753
$433$	1.48451e9	0.878769	0.439385	−	0.898299i	$-0.355197\pi$
$433$	1.48451e9	0.878769	0.439385	+	0.898299i	$0.355197\pi$
$434$	6.19129e8	0.363553
$435$	8.16435e8	0.475564
$436$	3.83070e8	0.221348
$437$	2.66190e9	1.52583
$438$	1.87407e9	1.06568
$439$	−1.76020e8	−0.0992967	−0.0496484	−	0.998767i	$-0.515810\pi$
$439$	−1.76020e8	−0.0992967	−0.0496484	+	0.998767i	$0.515810\pi$
$440$	1.81695e9	1.01685
$441$	−4.79768e8	−0.266377
$442$	−4.07847e9	−2.24656
$443$	1.14525e9	0.625876	0.312938	−	0.949773i	$-0.398687\pi$
$443$	1.14525e9	0.625876	0.312938	+	0.949773i	$0.398687\pi$
$444$	7.41026e7	0.0401784
$445$	−4.86035e9	−2.61461
$446$	−1.37880e9	−0.735917
$447$	−7.86130e8	−0.416311
$448$	−5.73726e8	−0.301461
$449$	−8.95733e8	−0.466999	−0.233500	−	0.972357i	$-0.575018\pi$
$449$	−8.95733e8	−0.466999	−0.233500	+	0.972357i	$0.575018\pi$
$450$	−2.09099e9	−1.08170
$451$	4.56410e8	0.234281
$452$	3.68153e8	0.187519
$453$	−1.83680e9	−0.928366
$454$	2.70894e9	1.35864
$455$	3.33603e9	1.66031
$456$	8.35789e8	0.412781
$457$	−2.63392e9	−1.29091	−0.645455	−	0.763798i	$-0.723332\pi$
$457$	−2.63392e9	−1.29091	−0.645455	+	0.763798i	$0.723332\pi$
$458$	−2.33790e9	−1.13709
$459$	4.31479e8	0.208264
$460$	−1.74602e9	−0.836365
$461$	−9.33971e6	−0.00443997	−0.00221999	−	0.999998i	$-0.500707\pi$
$461$	−9.33971e6	−0.00443997	−0.00221999	+	0.999998i	$0.500707\pi$
$462$	−3.66667e8	−0.172992
$463$	2.25955e9	1.05801	0.529004	−	0.848619i	$-0.322566\pi$
$463$	2.25955e9	1.05801	0.529004	+	0.848619i	$0.322566\pi$
$464$	1.05434e9	0.489968
$465$	−1.81194e9	−0.835716
$466$	2.39267e9	1.09530
$467$	2.00141e9	0.909342	0.454671	−	0.890659i	$-0.349757\pi$
$467$	2.00141e9	0.909342	0.454671	+	0.890659i	$0.349757\pi$
$468$	−3.20885e8	−0.144707
$469$	−5.16574e8	−0.231221
$470$	6.50423e9	2.88970
$471$	1.04487e8	0.0460776
$472$	1.90531e9	0.834004
$473$	−5.41486e8	−0.235274
$474$	−1.83307e9	−0.790597
$475$	−5.72784e9	−2.45224
$476$	2.64907e8	0.112582
$477$	5.27832e8	0.222680
$478$	8.05249e8	0.337235
$479$	−3.16592e8	−0.131621	−0.0658106	−	0.997832i	$-0.520963\pi$
$479$	−3.16592e8	−0.131621	−0.0658106	+	0.997832i	$0.520963\pi$
$480$	−1.26216e9	−0.520918
$481$	1.36849e9	0.560706
$482$	9.15400e8	0.372346
$483$	−1.16562e9	−0.470698
$484$	−3.68965e8	−0.147920
$485$	−3.51594e9	−1.39941
$486$	1.80198e8	0.0712071
$487$	−1.04649e9	−0.410566	−0.205283	−	0.978703i	$-0.565811\pi$
$487$	−1.04649e9	−0.410566	−0.205283	+	0.978703i	$0.565811\pi$
$488$	2.75924e9	1.07478
$489$	2.70949e9	1.04787
$490$	4.57580e9	1.75704
$491$	−2.41177e9	−0.919498	−0.459749	−	0.888049i	$-0.652061\pi$
$491$	−2.41177e9	−0.919498	−0.459749	+	0.888049i	$0.652061\pi$
$492$	−1.37710e8	−0.0521301
$493$	1.19728e9	0.450018
$494$	−4.66581e9	−1.74134
$495$	1.07309e9	0.397665
$496$	−2.33993e9	−0.861027
$497$	−2.18980e8	−0.0800124
$498$	7.47887e8	0.271352
$499$	−4.07577e9	−1.46844	−0.734222	−	0.678910i	$-0.762453\pi$
$499$	−4.07577e9	−1.46844	−0.734222	+	0.678910i	$0.762453\pi$
$500$	2.47193e9	0.884386
$501$	1.32315e8	0.0470087
$502$	5.72171e9	2.01866
$503$	−1.60149e9	−0.561094	−0.280547	−	0.959840i	$-0.590516\pi$
$503$	−1.60149e9	−0.561094	−0.280547	+	0.959840i	$0.590516\pi$
$504$	−3.65985e8	−0.127337
$505$	7.69928e9	2.66030
$506$	3.54406e9	1.21611
$507$	−4.23174e9	−1.44209
$508$	−4.34236e8	−0.146961
$509$	−2.70133e9	−0.907956	−0.453978	−	0.891013i	$-0.649996\pi$
$509$	−2.70133e9	−0.907956	−0.453978	+	0.891013i	$0.649996\pi$
$510$	−4.11524e9	−1.37372
$511$	−2.24797e9	−0.745276
$512$	1.42005e9	0.467582
$513$	4.93617e8	0.161428
$514$	2.39053e9	0.776466
$515$	1.48955e9	0.480539
$516$	1.63380e8	0.0523510
$517$	−2.48719e9	−0.791575
$518$	−4.71822e8	−0.149150
$519$	8.64023e8	0.271294
$520$	−1.01243e10	−3.15756
$521$	4.07843e9	1.26346	0.631729	−	0.775190i	$-0.282346\pi$
$521$	4.07843e9	1.26346	0.631729	+	0.775190i	$0.282346\pi$
$522$	5.00018e8	0.153865
$523$	−3.73517e9	−1.14171	−0.570853	−	0.821053i	$-0.693387\pi$
$523$	−3.73517e9	−1.14171	−0.570853	+	0.821053i	$0.693387\pi$
$524$	1.15652e9	0.351152
$525$	2.50817e9	0.756484
$526$	−5.04458e9	−1.51139
$527$	−2.65715e9	−0.790824
$528$	1.38578e9	0.409709
$529$	7.86159e9	2.30895
$530$	−5.03421e9	−1.46881
$531$	1.12527e9	0.326158
$532$	3.03056e8	0.0872635
$533$	−2.54317e9	−0.727496
$534$	−2.97667e9	−0.845934
$535$	−6.99833e9	−1.97586
$536$	1.56771e9	0.439734
$537$	−1.68116e9	−0.468489
$538$	−4.54013e9	−1.25699
$539$	−1.74976e9	−0.481303
$540$	−3.23778e8	−0.0884848
$541$	2.71503e9	0.737198	0.368599	−	0.929588i	$-0.379838\pi$
$541$	2.71503e9	0.737198	0.368599	+	0.929588i	$0.379838\pi$
$542$	5.10685e9	1.37770
$543$	1.30238e8	0.0349090
$544$	−1.85091e9	−0.492935
$545$	7.13816e9	1.88885
$546$	2.04312e9	0.537179
$547$	−4.64278e8	−0.121289	−0.0606447	−	0.998159i	$-0.519316\pi$
$547$	−4.64278e8	−0.121289	−0.0606447	+	0.998159i	$0.519316\pi$
$548$	−5.58144e8	−0.144882
$549$	1.62961e9	0.420319
$550$	−7.62606e9	−1.95448
$551$	1.36970e9	0.348814
$552$	3.53746e9	0.895168
$553$	2.19880e9	0.552901
$554$	−5.52122e9	−1.37959
$555$	1.38083e9	0.342859
$556$	7.70771e8	0.190179
$557$	2.09116e8	0.0512735	0.0256368	−	0.999671i	$-0.491839\pi$
$557$	2.09116e8	0.0512735	0.0256368	+	0.999671i	$0.491839\pi$
$558$	−1.10971e9	−0.270389
$559$	3.01723e9	0.730579
$560$	4.34697e9	1.04599
$561$	1.57365e9	0.376303
$562$	7.57943e9	1.80119
$563$	1.32152e9	0.312100	0.156050	−	0.987749i	$-0.450124\pi$
$563$	1.32152e9	0.312100	0.156050	+	0.987749i	$0.450124\pi$
$564$	7.50448e8	0.176134
$565$	6.86019e9	1.60017
$566$	−1.72872e9	−0.400743
$567$	−2.16150e8	−0.0497984
$568$	6.64567e8	0.152167
$569$	4.06408e9	0.924847	0.462423	−	0.886659i	$-0.346980\pi$
$569$	4.06408e9	0.924847	0.462423	+	0.886659i	$0.346980\pi$
$570$	−4.70788e9	−1.06479
$571$	−2.07817e9	−0.467149	−0.233575	−	0.972339i	$-0.575042\pi$
$571$	−2.07817e9	−0.467149	−0.233575	+	0.972339i	$0.575042\pi$
$572$	−1.17030e9	−0.261464
$573$	−2.34794e9	−0.521370
$574$	8.76822e8	0.193517
$575$	−2.42429e10	−5.31799
$576$	1.02833e9	0.224209
$577$	7.97259e9	1.72776	0.863882	−	0.503695i	$-0.168026\pi$
$577$	7.97259e9	1.72776	0.863882	+	0.503695i	$0.168026\pi$
$578$	−8.81701e8	−0.189921
$579$	−2.78905e9	−0.597148
$580$	−8.98425e8	−0.191198
$581$	−8.97102e8	−0.189769
$582$	−2.15330e9	−0.452767
$583$	1.92506e9	0.402350
$584$	6.82219e9	1.41736
$585$	−5.97938e9	−1.23484
$586$	−4.51639e9	−0.927149
$587$	−5.36376e9	−1.09455	−0.547275	−	0.836953i	$-0.684335\pi$
$587$	−5.36376e9	−1.09455	−0.547275	+	0.836953i	$0.684335\pi$
$588$	5.27948e8	0.107095
$589$	−3.03982e9	−0.612976
$590$	−1.07323e10	−2.15135
$591$	−3.77619e9	−0.752484
$592$	1.78320e9	0.353243
$593$	7.44323e9	1.46578	0.732892	−	0.680345i	$-0.238170\pi$
$593$	7.44323e9	1.46578	0.732892	+	0.680345i	$0.238170\pi$
$594$	6.57202e8	0.128661
$595$	4.93629e9	0.960708
$596$	8.65077e8	0.167376
$597$	−3.14464e9	−0.604868
$598$	−1.97479e10	−3.77631
$599$	−3.99053e9	−0.758641	−0.379321	−	0.925265i	$-0.623842\pi$
$599$	−3.99053e9	−0.758641	−0.379321	+	0.925265i	$0.623842\pi$
$600$	−7.61186e9	−1.43867
$601$	6.75434e9	1.26918	0.634588	−	0.772850i	$-0.281170\pi$
$601$	6.75434e9	1.26918	0.634588	+	0.772850i	$0.281170\pi$
$602$	−1.04026e9	−0.194337
$603$	9.25890e8	0.171968
$604$	2.02126e9	0.373245
$605$	−6.87532e9	−1.26226
$606$	4.71535e9	0.860715
$607$	−3.43073e9	−0.622624	−0.311312	−	0.950308i	$-0.600768\pi$
$607$	−3.43073e9	−0.622624	−0.311312	+	0.950308i	$0.600768\pi$
$608$	−2.11747e9	−0.382080
$609$	−5.99779e8	−0.107605
$610$	−1.55424e10	−2.77245
$611$	1.38589e10	2.45802
$612$	−4.74810e8	−0.0837316
$613$	3.18350e9	0.558204	0.279102	−	0.960261i	$-0.409963\pi$
$613$	3.18350e9	0.558204	0.279102	+	0.960261i	$0.409963\pi$
$614$	−6.81832e9	−1.18874
$615$	−2.56610e9	−0.444848
$616$	−1.33479e9	−0.230080
$617$	7.63861e9	1.30923	0.654616	−	0.755962i	$-0.272830\pi$
$617$	7.63861e9	1.30923	0.654616	+	0.755962i	$0.272830\pi$
$618$	9.12258e8	0.155474
$619$	4.68087e9	0.793248	0.396624	−	0.917981i	$-0.370182\pi$
$619$	4.68087e9	0.793248	0.396624	+	0.917981i	$0.370182\pi$
$620$	1.99390e9	0.335996
$621$	2.08922e9	0.350077
$622$	1.05163e10	1.75225
$623$	3.57056e9	0.591600
$624$	−7.72174e9	−1.27224
$625$	2.82186e10	4.62333
$626$	3.79230e9	0.617864
$627$	1.80027e9	0.291677
$628$	−1.14980e8	−0.0185253
$629$	2.02495e9	0.324442
$630$	2.06154e9	0.328473
$631$	−1.17450e10	−1.86102	−0.930510	−	0.366266i	$-0.880636\pi$
$631$	−1.17450e10	−1.86102	−0.930510	+	0.366266i	$0.880636\pi$
$632$	−6.67296e9	−1.05150
$633$	−4.09541e9	−0.641777
$634$	3.89994e9	0.607779
$635$	−8.09159e9	−1.25408
$636$	−5.80839e8	−0.0895274
$637$	9.74991e9	1.49456
$638$	1.82362e9	0.278011
$639$	3.92493e8	0.0595084
$640$	−1.57912e10	−2.38115
$641$	1.22273e10	1.83370	0.916848	−	0.399238i	$-0.130725\pi$
$641$	1.22273e10	1.83370	0.916848	+	0.399238i	$0.130725\pi$
$642$	−4.28606e9	−0.639272
$643$	−1.03024e10	−1.52827	−0.764134	−	0.645058i	$-0.776833\pi$
$643$	−1.03024e10	−1.52827	−0.764134	+	0.645058i	$0.776833\pi$
$644$	1.28268e9	0.189242
$645$	3.04443e9	0.446733
$646$	−6.90396e9	−1.00759
$647$	1.14849e10	1.66710	0.833550	−	0.552444i	$-0.186305\pi$
$647$	1.14849e10	1.66710	0.833550	+	0.552444i	$0.186305\pi$
$648$	6.55978e8	0.0947059
$649$	4.10399e9	0.589319
$650$	4.24934e10	6.06910
$651$	1.33111e9	0.189095
$652$	−2.98159e9	−0.421290
$653$	9.47100e9	1.33107	0.665533	−	0.746368i	$-0.268204\pi$
$653$	9.47100e9	1.33107	0.665533	+	0.746368i	$0.268204\pi$
$654$	4.37170e9	0.611122
$655$	2.15507e10	2.99652
$656$	−3.31385e9	−0.458321
$657$	4.02918e9	0.554291
$658$	−4.77821e9	−0.653845
$659$	7.03312e9	0.957303	0.478651	−	0.878005i	$-0.341126\pi$
$659$	7.03312e9	0.957303	0.478651	+	0.878005i	$0.341126\pi$
$660$	−1.18085e9	−0.159879
$661$	8.85871e9	1.19307	0.596535	−	0.802587i	$-0.296544\pi$
$661$	8.85871e9	1.19307	0.596535	+	0.802587i	$0.296544\pi$
$662$	−6.85872e9	−0.918840
$663$	−8.76857e9	−1.16851
$664$	2.72255e9	0.360900
$665$	5.64717e9	0.744655
$666$	8.45678e8	0.110929
$667$	5.79721e9	0.756447
$668$	−1.45603e8	−0.0188996
$669$	−2.96437e9	−0.382773
$670$	−8.83070e9	−1.13431
$671$	5.94335e9	0.759455
$672$	9.27220e8	0.117867
$673$	1.19441e10	1.51044	0.755218	−	0.655474i	$-0.227531\pi$
$673$	1.19441e10	1.51044	0.755218	+	0.655474i	$0.227531\pi$
$674$	3.40903e9	0.428866
$675$	−4.49556e9	−0.562627
$676$	4.65671e9	0.579784
$677$	5.03496e9	0.623642	0.311821	−	0.950141i	$-0.399061\pi$
$677$	5.03496e9	0.623642	0.311821	+	0.950141i	$0.399061\pi$
$678$	4.20146e9	0.517722
$679$	2.58292e9	0.316641
$680$	−1.49808e10	−1.82706
$681$	5.82415e9	0.706671
$682$	−4.04722e9	−0.488552
$683$	8.90050e9	1.06891	0.534456	−	0.845196i	$-0.320517\pi$
$683$	8.90050e9	1.06891	0.534456	+	0.845196i	$0.320517\pi$
$684$	−5.43188e8	−0.0649013
$685$	−1.04005e10	−1.23634
$686$	−7.56800e9	−0.895049
$687$	−5.02641e9	−0.591438
$688$	3.93156e9	0.460263
$689$	−1.07267e10	−1.24939
$690$	−1.99260e10	−2.30913
$691$	−4.31339e9	−0.497331	−0.248666	−	0.968589i	$-0.579992\pi$
$691$	−4.31339e9	−0.497331	−0.248666	+	0.968589i	$0.579992\pi$
$692$	−9.50792e8	−0.109072
$693$	−7.88324e8	−0.0899784
$694$	−1.73025e10	−1.96495
$695$	1.43626e10	1.62288
$696$	1.82022e9	0.204641
$697$	−3.76311e9	−0.420952
$698$	3.27182e9	0.364163
$699$	5.14417e9	0.569699
$700$	−2.76005e9	−0.304141
$701$	5.57650e8	0.0611432	0.0305716	−	0.999533i	$-0.490267\pi$
$701$	5.57650e8	0.0611432	0.0305716	+	0.999533i	$0.490267\pi$
$702$	−3.66201e9	−0.399522
$703$	2.31656e9	0.251478
$704$	3.75042e9	0.405112
$705$	1.39839e10	1.50303
$706$	3.86239e8	0.0413085
$707$	−5.65613e9	−0.601937
$708$	−1.23828e9	−0.131130
$709$	4.93207e9	0.519718	0.259859	−	0.965647i	$-0.416324\pi$
$709$	4.93207e9	0.519718	0.259859	+	0.965647i	$0.416324\pi$
$710$	−3.74341e9	−0.392521
$711$	−3.94105e9	−0.411214
$712$	−1.08360e10	−1.12510
$713$	−1.28659e10	−1.32932
$714$	3.02318e9	0.310828
$715$	−2.18074e10	−2.23118
$716$	1.84999e9	0.188354
$717$	1.73126e9	0.175406
$718$	−7.06264e9	−0.712085
$719$	9.07009e9	0.910040	0.455020	−	0.890481i	$-0.349632\pi$
$719$	9.07009e9	0.910040	0.455020	+	0.890481i	$0.349632\pi$
$720$	−7.79136e9	−0.777946
$721$	−1.09427e9	−0.108730
$722$	3.32732e9	0.329014
$723$	1.96808e9	0.193669
$724$	−1.43317e8	−0.0140350
$725$	−1.24744e10	−1.21573
$726$	−4.21072e9	−0.408393
$727$	−1.33049e10	−1.28422	−0.642112	−	0.766611i	$-0.721942\pi$
$727$	−1.33049e10	−1.28422	−0.642112	+	0.766611i	$0.721942\pi$
$728$	7.43760e9	0.714452
$729$	3.87420e8	0.0370370
$730$	−3.84284e10	−3.65614
$731$	4.46457e9	0.422735
$732$	−1.79326e9	−0.168987
$733$	9.36494e9	0.878296	0.439148	−	0.898415i	$-0.355280\pi$
$733$	9.36494e9	0.878296	0.439148	+	0.898415i	$0.355280\pi$
$734$	−4.86685e9	−0.454267
$735$	9.83782e9	0.913889
$736$	−8.96213e9	−0.828588
$737$	3.37682e9	0.310722
$738$	−1.57159e9	−0.143926
$739$	1.46403e10	1.33442	0.667211	−	0.744869i	$-0.267488\pi$
$739$	1.46403e10	1.33442	0.667211	+	0.744869i	$0.267488\pi$
$740$	−1.51950e9	−0.137845
$741$	−1.00314e10	−0.905724
$742$	3.69829e9	0.332343
$743$	−8.82056e9	−0.788924	−0.394462	−	0.918912i	$-0.629069\pi$
$743$	−8.82056e9	−0.788924	−0.394462	+	0.918912i	$0.629069\pi$
$744$	−4.03968e9	−0.359618
$745$	1.61199e10	1.42829
$746$	−1.67648e10	−1.47847
$747$	1.60793e9	0.141139
$748$	−1.73168e9	−0.151291
$749$	5.14119e9	0.447072
$750$	2.82103e10	2.44171
$751$	−7.99931e9	−0.689149	−0.344574	−	0.938759i	$-0.611977\pi$
$751$	−7.99931e9	−0.689149	−0.344574	+	0.938759i	$0.611977\pi$
$752$	1.80587e10	1.54855
$753$	1.23015e10	1.04997
$754$	−1.01614e10	−0.863288
$755$	3.76644e10	3.18505
$756$	2.37857e8	0.0200212
$757$	−6.11589e9	−0.512417	−0.256209	−	0.966621i	$-0.582473\pi$
$757$	−6.11589e9	−0.512417	−0.256209	+	0.966621i	$0.582473\pi$
$758$	1.10704e10	0.923252
$759$	7.61961e9	0.632538
$760$	−1.71382e10	−1.41618
$761$	2.03009e10	1.66982	0.834910	−	0.550386i	$-0.185520\pi$
$761$	2.03009e10	1.66982	0.834910	+	0.550386i	$0.185520\pi$
$762$	−4.95561e9	−0.405747
$763$	−5.24391e9	−0.427385
$764$	2.58373e9	0.209614
$765$	−8.84763e9	−0.714516
$766$	1.06290e10	0.854461
$767$	−2.28680e10	−1.82997
$768$	−4.79617e9	−0.382059
$769$	−1.07347e10	−0.851229	−0.425614	−	0.904905i	$-0.639942\pi$
$769$	−1.07347e10	−0.851229	−0.425614	+	0.904905i	$0.639942\pi$
$770$	7.51866e9	0.593503
$771$	5.13955e9	0.403864
$772$	3.06914e9	0.240080
$773$	−2.01375e10	−1.56812	−0.784058	−	0.620687i	$-0.786854\pi$
$773$	−2.01375e10	−1.56812	−0.784058	+	0.620687i	$0.786854\pi$
$774$	1.86453e9	0.144536
$775$	2.76848e10	2.13641
$776$	−7.83871e9	−0.602183
$777$	−1.01440e9	−0.0775777
$778$	2.14247e10	1.63112
$779$	−4.30504e9	−0.326284
$780$	6.57986e9	0.496461
$781$	1.43146e9	0.107523
$782$	−2.92208e10	−2.18509
$783$	1.07502e9	0.0800298
$784$	1.27045e10	0.941567
$785$	−2.14255e9	−0.158084
$786$	1.31985e10	0.969498
$787$	6.97938e9	0.510393	0.255197	−	0.966889i	$-0.417860\pi$
$787$	6.97938e9	0.510393	0.255197	+	0.966889i	$0.417860\pi$
$788$	4.15541e9	0.302532
$789$	−1.08457e10	−0.786118
$790$	3.75878e10	2.71239
$791$	−5.03971e9	−0.362066
$792$	2.39242e9	0.171120
$793$	−3.31171e10	−2.35828
$794$	−3.70028e9	−0.262339
$795$	−1.08234e10	−0.763974
$796$	3.46044e9	0.243184
$797$	7.75045e9	0.542279	0.271140	−	0.962540i	$-0.412600\pi$
$797$	7.75045e9	0.542279	0.271140	+	0.962540i	$0.412600\pi$
$798$	3.45856e9	0.240927
$799$	2.05069e10	1.42229
$800$	1.92846e10	1.33167
$801$	−6.39975e9	−0.439997
$802$	2.58318e9	0.176825
$803$	1.46949e10	1.00152
$804$	−1.01887e9	−0.0691391
$805$	2.39015e10	1.61488
$806$	2.25516e10	1.51707
$807$	−9.76114e9	−0.653797
$808$	1.71654e10	1.14476
$809$	2.60767e10	1.73154	0.865770	−	0.500442i	$-0.166829\pi$
$809$	2.60767e10	1.73154	0.865770	+	0.500442i	$0.166829\pi$
$810$	−3.69503e9	−0.244298
$811$	2.33510e9	0.153721	0.0768604	−	0.997042i	$-0.475510\pi$
$811$	2.33510e9	0.153721	0.0768604	+	0.997042i	$0.475510\pi$
$812$	6.60011e8	0.0432619
$813$	1.09796e10	0.716586
$814$	3.08428e9	0.200432
$815$	−5.55591e10	−3.59504
$816$	−1.14258e10	−0.736157
$817$	5.10752e9	0.327667
$818$	2.40604e10	1.53697
$819$	4.39264e9	0.279404
$820$	2.82380e9	0.178849
$821$	1.89146e9	0.119288	0.0596438	−	0.998220i	$-0.481004\pi$
$821$	1.89146e9	0.119288	0.0596438	+	0.998220i	$0.481004\pi$
$822$	−6.36968e9	−0.400006
$823$	−1.13187e10	−0.707777	−0.353888	−	0.935288i	$-0.615141\pi$
$823$	−1.13187e10	−0.707777	−0.353888	+	0.935288i	$0.615141\pi$
$824$	3.32091e9	0.206781
$825$	−1.63958e10	−1.01658
$826$	7.88430e9	0.486781
$827$	−2.95827e10	−1.81873	−0.909365	−	0.416000i	$-0.863432\pi$
$827$	−2.95827e10	−1.81873	−0.909365	+	0.416000i	$0.863432\pi$
$828$	−2.29903e9	−0.140747
$829$	−1.72656e10	−1.05254	−0.526272	−	0.850316i	$-0.676411\pi$
$829$	−1.72656e10	−1.05254	−0.526272	+	0.850316i	$0.676411\pi$
$830$	−1.53357e10	−0.930959
$831$	−1.18705e10	−0.717569
$832$	−2.08978e10	−1.25797
$833$	1.44269e10	0.864797
$834$	8.79623e9	0.525068
$835$	−2.71318e9	−0.161278
$836$	−1.98106e9	−0.117267
$837$	−2.38583e9	−0.140637
$838$	5.49629e9	0.322638
$839$	−2.61914e10	−1.53106	−0.765529	−	0.643402i	$-0.777522\pi$
$839$	−2.61914e10	−1.53106	−0.765529	+	0.643402i	$0.777522\pi$
$840$	7.50466e9	0.436871
$841$	−1.42669e10	−0.827071
$842$	2.84142e9	0.164037
$843$	1.62955e10	0.936855
$844$	4.50669e9	0.258023
$845$	8.67735e10	4.94753
$846$	8.56430e9	0.486290
$847$	5.05082e9	0.285608
$848$	−1.39773e10	−0.787112
$849$	−3.71668e9	−0.208439
$850$	6.28770e10	3.51177
$851$	9.80480e9	0.545362
$852$	−4.31908e8	−0.0239251
$853$	3.20697e10	1.76919	0.884593	−	0.466364i	$-0.154436\pi$
$853$	3.20697e10	1.76919	0.884593	+	0.466364i	$0.154436\pi$
$854$	1.14179e10	0.627314
$855$	−1.01218e10	−0.553830
$856$	−1.56026e10	−0.850236
$857$	−1.94131e10	−1.05356	−0.526782	−	0.850000i	$-0.676602\pi$
$857$	−1.94131e10	−1.05356	−0.526782	+	0.850000i	$0.676602\pi$
$858$	−1.33558e10	−0.721877
$859$	1.52337e10	0.820028	0.410014	−	0.912079i	$-0.365524\pi$
$859$	1.52337e10	0.820028	0.410014	+	0.912079i	$0.365524\pi$
$860$	−3.35017e9	−0.179607
$861$	1.88514e9	0.100654
$862$	−2.70001e10	−1.43579
$863$	3.37211e9	0.178593	0.0892965	−	0.996005i	$-0.471538\pi$
$863$	3.37211e9	0.178593	0.0892965	+	0.996005i	$0.471538\pi$
$864$	−1.66192e9	−0.0876620
$865$	−1.77171e10	−0.930758
$866$	−1.86429e10	−0.975442
$867$	−1.89563e9	−0.0987840
$868$	−1.46478e9	−0.0760247
$869$	−1.43734e10	−0.743004
$870$	−1.02531e10	−0.527881
$871$	−1.88161e10	−0.964863
$872$	1.59144e10	0.812796
$873$	−4.62954e9	−0.235498
$874$	−3.34290e10	−1.69369
$875$	−3.38387e10	−1.70760
$876$	−4.43381e9	−0.222850
$877$	2.44223e10	1.22261	0.611305	−	0.791395i	$-0.290645\pi$
$877$	2.44223e10	1.22261	0.611305	+	0.791395i	$0.290645\pi$
$878$	2.21051e9	0.110220
$879$	−9.71009e9	−0.482239
$880$	−2.84159e10	−1.40563
$881$	−2.87447e9	−0.141626	−0.0708129	−	0.997490i	$-0.522559\pi$
$881$	−2.87447e9	−0.141626	−0.0708129	+	0.997490i	$0.522559\pi$
$882$	6.02508e9	0.295680
$883$	1.80247e10	0.881060	0.440530	−	0.897738i	$-0.354791\pi$
$883$	1.80247e10	0.881060	0.440530	+	0.897738i	$0.354791\pi$
$884$	9.64915e9	0.469793
$885$	−2.30742e10	−1.11899
$886$	−1.43825e10	−0.694729
$887$	−2.95511e10	−1.42181	−0.710904	−	0.703289i	$-0.751714\pi$
$887$	−2.95511e10	−1.42181	−0.710904	+	0.703289i	$0.751714\pi$
$888$	3.07853e9	0.147536
$889$	5.94433e9	0.283757
$890$	6.10378e10	2.90224
$891$	1.41296e9	0.0669205
$892$	3.26207e9	0.153892
$893$	2.34602e10	1.10243
$894$	9.87247e9	0.462109
$895$	3.44729e10	1.60730
$896$	1.16007e10	0.538775
$897$	−4.24574e10	−1.96418
$898$	1.12489e10	0.518374
$899$	−6.62026e9	−0.303890
$900$	4.94702e9	0.226201
$901$	−1.58722e10	−0.722935
$902$	−5.73174e9	−0.260054
$903$	−2.23654e9	−0.101081
$904$	1.52946e10	0.688573
$905$	−2.67058e9	−0.119766
$906$	2.30672e10	1.03049
$907$	1.75270e10	0.779979	0.389990	−	0.920819i	$-0.372479\pi$
$907$	1.75270e10	0.779979	0.389990	+	0.920819i	$0.372479\pi$
$908$	−6.40903e9	−0.284114
$909$	1.01378e10	0.447685
$910$	−4.18949e10	−1.84296
$911$	4.00967e10	1.75709	0.878545	−	0.477660i	$-0.158515\pi$
$911$	4.00967e10	1.75709	0.878545	+	0.477660i	$0.158515\pi$
$912$	−1.30712e10	−0.570603
$913$	5.86431e9	0.255017
$914$	3.30776e10	1.43292
$915$	−3.34157e10	−1.44204
$916$	5.53118e9	0.237785
$917$	−1.58318e10	−0.678014
$918$	−5.41865e9	−0.231175
$919$	−1.02527e10	−0.435745	−0.217873	−	0.975977i	$-0.569912\pi$
$919$	−1.02527e10	−0.435745	−0.217873	+	0.975977i	$0.569912\pi$
$920$	−7.25369e10	−3.07115
$921$	−1.46592e10	−0.618302
$922$	1.17291e8	0.00492841
$923$	−7.97629e9	−0.333884
$924$	8.67491e8	0.0361754
$925$	−2.10978e10	−0.876480
$926$	−2.83762e10	−1.17440
$927$	1.96133e9	0.0808669
$928$	−4.61152e9	−0.189420
$929$	3.70669e9	0.151681	0.0758404	−	0.997120i	$-0.475836\pi$
$929$	3.70669e9	0.151681	0.0758404	+	0.997120i	$0.475836\pi$
$930$	2.27549e10	0.927653
$931$	1.65045e10	0.670314
$932$	−5.66077e9	−0.229045
$933$	2.26097e10	0.911399
$934$	−2.51344e10	−1.00938
$935$	−3.22683e10	−1.29103
$936$	−1.33309e10	−0.531366
$937$	1.60218e10	0.636242	0.318121	−	0.948050i	$-0.396948\pi$
$937$	1.60218e10	0.636242	0.318121	+	0.948050i	$0.396948\pi$
$938$	6.48731e9	0.256658
$939$	8.15334e9	0.321370
$940$	−1.53882e10	−0.604284
$941$	3.65751e10	1.43094	0.715472	−	0.698642i	$-0.246212\pi$
$941$	3.65751e10	1.43094	0.715472	+	0.698642i	$0.246212\pi$
$942$	−1.31218e9	−0.0511465
$943$	−1.82210e10	−0.707588
$944$	−2.97978e10	−1.15288
$945$	4.43225e9	0.170849
$946$	6.80016e9	0.261156
$947$	−4.35149e10	−1.66499	−0.832497	−	0.554029i	$-0.813090\pi$
$947$	−4.35149e10	−1.66499	−0.832497	+	0.554029i	$0.813090\pi$
$948$	4.33682e9	0.165327
$949$	−8.18816e10	−3.10996
$950$	7.19321e10	2.72201
$951$	8.38475e9	0.316125
$952$	1.10053e10	0.413404
$953$	1.21468e10	0.454606	0.227303	−	0.973824i	$-0.427009\pi$
$953$	1.21468e10	0.454606	0.227303	+	0.973824i	$0.427009\pi$
$954$	−6.62868e9	−0.247177
$955$	4.81455e10	1.78873
$956$	−1.90512e9	−0.0705213
$957$	3.92072e9	0.144602
$958$	3.97587e9	0.146101
$959$	7.64053e9	0.279742
$960$	−2.10862e10	−0.769218
$961$	−1.28201e10	−0.465970
$962$	−1.71860e10	−0.622389
$963$	−9.21490e9	−0.332505
$964$	−2.16573e9	−0.0778635
$965$	5.71906e10	2.04870
$966$	1.46383e10	0.522480
$967$	−2.25311e10	−0.801292	−0.400646	−	0.916233i	$-0.631214\pi$
$967$	−2.25311e10	−0.801292	−0.400646	+	0.916233i	$0.631214\pi$
$968$	−1.53284e10	−0.543165
$969$	−1.48433e10	−0.524079
$970$	4.41543e10	1.55336
$971$	−6.44406e9	−0.225888	−0.112944	−	0.993601i	$-0.536028\pi$
$971$	−6.44406e9	−0.225888	−0.112944	+	0.993601i	$0.536028\pi$
$972$	−4.26327e8	−0.0148906
$973$	−1.05512e10	−0.367204
$974$	1.31421e10	0.455732
$975$	9.13594e10	3.15673
$976$	−4.31528e10	−1.48571
$977$	−3.72903e10	−1.27928	−0.639640	−	0.768675i	$-0.720916\pi$
$977$	−3.72903e10	−1.27928	−0.639640	+	0.768675i	$0.720916\pi$
$978$	−3.40266e10	−1.16314
$979$	−2.33406e10	−0.795009
$980$	−1.08258e10	−0.367424
$981$	9.39901e9	0.317863
$982$	3.02878e10	1.02065
$983$	−3.33480e10	−1.11978	−0.559890	−	0.828567i	$-0.689156\pi$
$983$	−3.33480e10	−1.11978	−0.559890	+	0.828567i	$0.689156\pi$
$984$	−5.72107e9	−0.191423
$985$	7.74321e10	2.58163
$986$	−1.50358e10	−0.499524
$987$	−1.02730e10	−0.340085
$988$	1.10387e10	0.364142
$989$	2.16174e10	0.710587
$990$	−1.34762e10	−0.441411
$991$	−3.23461e10	−1.05576	−0.527879	−	0.849320i	$-0.677012\pi$
$991$	−3.23461e10	−1.05576	−0.527879	+	0.849320i	$0.677012\pi$
$992$	1.02345e10	0.332871
$993$	−1.47460e10	−0.477917
$994$	2.75002e9	0.0888146
$995$	6.44820e10	2.07519
$996$	−1.76941e9	−0.0567441
$997$	−2.75371e10	−0.880005	−0.440003	−	0.897996i	$-0.645022\pi$
$997$	−2.75371e10	−0.880005	−0.440003	+	0.897996i	$0.645022\pi$
$998$	5.11848e10	1.62999
$999$	1.81818e9	0.0576976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.8.a.c.1.13	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.8.a.c.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-12.5583 q^{2} -27.0000 q^{3} +29.7115 q^{4} +553.645 q^{5} +339.075 q^{6} -406.725 q^{7} +1234.34 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-12.5583 q^{2} -27.0000 q^{3} +29.7115 q^{4} +553.645 q^{5} +339.075 q^{6} -406.725 q^{7} +1234.34 q^{8} +729.000 q^{9} -6952.86 q^{10} +2658.74 q^{11} -802.209 q^{12} -14814.8 q^{13} +5107.78 q^{14} -14948.4 q^{15} -19304.3 q^{16} -21921.4 q^{17} -9155.02 q^{18} -25078.3 q^{19} +16449.6 q^{20} +10981.6 q^{21} -33389.3 q^{22} -106143. q^{23} -33327.2 q^{24} +228398. q^{25} +186050. q^{26} -19683.0 q^{27} -12084.4 q^{28} -54616.8 q^{29} +187727. q^{30} +121213. q^{31} +84434.2 q^{32} -71786.0 q^{33} +275296. q^{34} -225181. q^{35} +21659.7 q^{36} -92373.1 q^{37} +314942. q^{38} +400001. q^{39} +683386. q^{40} +171664. q^{41} -137910. q^{42} -203663. q^{43} +78995.1 q^{44} +403607. q^{45} +1.33298e6 q^{46} -935476. q^{47} +521216. q^{48} -658118. q^{49} -2.86830e6 q^{50} +591877. q^{51} -440171. q^{52} +724049. q^{53} +247185. q^{54} +1.47200e6 q^{55} -502037. q^{56} +677115. q^{57} +685896. q^{58} +1.54359e6 q^{59} -444140. q^{60} +2.23540e6 q^{61} -1.52223e6 q^{62} -296503. q^{63} +1.41060e6 q^{64} -8.20217e6 q^{65} +901512. q^{66} +1.27008e6 q^{67} -651316. q^{68} +2.86587e6 q^{69} +2.82790e6 q^{70} +538399. q^{71} +899833. q^{72} +5.52700e6 q^{73} +1.16005e6 q^{74} -6.16675e6 q^{75} -745114. q^{76} -1.08138e6 q^{77} -5.02334e6 q^{78} -5.40610e6 q^{79} -1.06877e7 q^{80} +531441. q^{81} -2.15581e6 q^{82} +2.20567e6 q^{83} +326279. q^{84} -1.21367e7 q^{85} +2.55766e6 q^{86} +1.47465e6 q^{87} +3.28179e6 q^{88} -8.77881e6 q^{89} -5.06863e6 q^{90} +6.02557e6 q^{91} -3.15367e6 q^{92} -3.27275e6 q^{93} +1.17480e7 q^{94} -1.38845e7 q^{95} -2.27972e6 q^{96} -6.35053e6 q^{97} +8.26485e6 q^{98} +1.93822e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 2 q^{2} - 1296 q^{3} + 3214 q^{4} + 428 q^{5} - 54 q^{6} - 680 q^{7} + 2355 q^{8} + 34992 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more