LMFDB - Embedded newform 546.8.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,8,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("546.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	546.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:	$170.562223914$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 86504x^{3} - 9117228x^{2} + 89606664x + 21810067776 $ x^5 - 2x^4 - 86504x^3 - 9117228x^2 + 89606664x + 21810067776
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5\cdot 13 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-65.0947$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -428.929 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -428.929 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +3431.43 q^{10} -5641.82 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -11581.1 q^{15} +4096.00 q^{16} +5030.39 q^{17} -5832.00 q^{18} -7398.89 q^{19} -27451.5 q^{20} +9261.00 q^{21} +45134.5 q^{22} +72906.4 q^{23} -13824.0 q^{24} +105855. q^{25} -17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} -59340.1 q^{29} +92648.7 q^{30} -113966. q^{31} -32768.0 q^{32} -152329. q^{33} -40243.1 q^{34} -147123. q^{35} +46656.0 q^{36} +112781. q^{37} +59191.2 q^{38} +59319.0 q^{39} +219612. q^{40} +224080. q^{41} -74088.0 q^{42} +509419. q^{43} -361076. q^{44} -312689. q^{45} -583251. q^{46} -454823. q^{47} +110592. q^{48} +117649. q^{49} -846841. q^{50} +135820. q^{51} +140608. q^{52} +869626. q^{53} -157464. q^{54} +2.41994e6 q^{55} -175616. q^{56} -199770. q^{57} +474721. q^{58} +1.63913e6 q^{59} -741189. q^{60} +2.14907e6 q^{61} +911729. q^{62} +250047. q^{63} +262144. q^{64} -942357. q^{65} +1.21863e6 q^{66} +55040.3 q^{67} +321945. q^{68} +1.96847e6 q^{69} +1.17698e6 q^{70} -4.22051e6 q^{71} -373248. q^{72} +1.72232e6 q^{73} -902247. q^{74} +2.85809e6 q^{75} -473529. q^{76} -1.93514e6 q^{77} -474552. q^{78} -3.59169e6 q^{79} -1.75689e6 q^{80} +531441. q^{81} -1.79264e6 q^{82} -8.13749e6 q^{83} +592704. q^{84} -2.15768e6 q^{85} -4.07536e6 q^{86} -1.60218e6 q^{87} +2.88861e6 q^{88} -4.86198e6 q^{89} +2.50151e6 q^{90} +753571. q^{91} +4.66601e6 q^{92} -3.07708e6 q^{93} +3.63859e6 q^{94} +3.17360e6 q^{95} -884736. q^{96} +6.88065e6 q^{97} -941192. q^{98} -4.11288e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) $ 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 - 340 * q^5 - 1080 * q^6 + 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + 2720 q^{10} - 1303 q^{11} + 8640 q^{12} + 10985 q^{13} - 13720 q^{14} - 9180 q^{15} + 20480 q^{16} - 4247 q^{17} - 29160 q^{18} - 16984 q^{19} - 21760 q^{20} + 46305 q^{21} + 10424 q^{22} - 78072 q^{23} - 69120 q^{24} - 79555 q^{25} - 87880 q^{26} + 98415 q^{27} + 109760 q^{28} - 213142 q^{29} + 73440 q^{30} - 186027 q^{31} - 163840 q^{32} - 35181 q^{33} + 33976 q^{34} - 116620 q^{35} + 233280 q^{36} + 101025 q^{37} + 135872 q^{38} + 296595 q^{39} + 174080 q^{40} - 23976 q^{41} - 370440 q^{42} - 55528 q^{43} - 83392 q^{44} - 247860 q^{45} + 624576 q^{46} - 985981 q^{47} + 552960 q^{48} + 588245 q^{49} + 636440 q^{50} - 114669 q^{51} + 703040 q^{52} - 1891657 q^{53} - 787320 q^{54} + 1746955 q^{55} - 878080 q^{56} - 458568 q^{57} + 1705136 q^{58} - 2802208 q^{59} - 587520 q^{60} + 1140591 q^{61} + 1488216 q^{62} + 1250235 q^{63} + 1310720 q^{64} - 746980 q^{65} + 281448 q^{66} + 265168 q^{67} - 271808 q^{68} - 2107944 q^{69} + 932960 q^{70} - 4483276 q^{71} - 1866240 q^{72} - 2350578 q^{73} - 808200 q^{74} - 2147985 q^{75} - 1086976 q^{76} - 446929 q^{77} - 2372760 q^{78} - 4079889 q^{79} - 1392640 q^{80} + 2657205 q^{81} + 191808 q^{82} - 8731571 q^{83} + 2963520 q^{84} - 1715895 q^{85} + 444224 q^{86} - 5754834 q^{87} + 667136 q^{88} - 20077879 q^{89} + 1982880 q^{90} + 3767855 q^{91} - 4996608 q^{92} - 5022729 q^{93} + 7887848 q^{94} - 11580740 q^{95} - 4423680 q^{96} + 3780209 q^{97} - 4705960 q^{98} - 949887 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 - 340 * q^5 - 1080 * q^6 + 1715 * q^7 - 2560 * q^8 + 3645 * q^9 + 2720 * q^10 - 1303 * q^11 + 8640 * q^12 + 10985 * q^13 - 13720 * q^14 - 9180 * q^15 + 20480 * q^16 - 4247 * q^17 - 29160 * q^18 - 16984 * q^19 - 21760 * q^20 + 46305 * q^21 + 10424 * q^22 - 78072 * q^23 - 69120 * q^24 - 79555 * q^25 - 87880 * q^26 + 98415 * q^27 + 109760 * q^28 - 213142 * q^29 + 73440 * q^30 - 186027 * q^31 - 163840 * q^32 - 35181 * q^33 + 33976 * q^34 - 116620 * q^35 + 233280 * q^36 + 101025 * q^37 + 135872 * q^38 + 296595 * q^39 + 174080 * q^40 - 23976 * q^41 - 370440 * q^42 - 55528 * q^43 - 83392 * q^44 - 247860 * q^45 + 624576 * q^46 - 985981 * q^47 + 552960 * q^48 + 588245 * q^49 + 636440 * q^50 - 114669 * q^51 + 703040 * q^52 - 1891657 * q^53 - 787320 * q^54 + 1746955 * q^55 - 878080 * q^56 - 458568 * q^57 + 1705136 * q^58 - 2802208 * q^59 - 587520 * q^60 + 1140591 * q^61 + 1488216 * q^62 + 1250235 * q^63 + 1310720 * q^64 - 746980 * q^65 + 281448 * q^66 + 265168 * q^67 - 271808 * q^68 - 2107944 * q^69 + 932960 * q^70 - 4483276 * q^71 - 1866240 * q^72 - 2350578 * q^73 - 808200 * q^74 - 2147985 * q^75 - 1086976 * q^76 - 446929 * q^77 - 2372760 * q^78 - 4079889 * q^79 - 1392640 * q^80 + 2657205 * q^81 + 191808 * q^82 - 8731571 * q^83 + 2963520 * q^84 - 1715895 * q^85 + 444224 * q^86 - 5754834 * q^87 + 667136 * q^88 - 20077879 * q^89 + 1982880 * q^90 + 3767855 * q^91 - 4996608 * q^92 - 5022729 * q^93 + 7887848 * q^94 - 11580740 * q^95 - 4423680 * q^96 + 3780209 * q^97 - 4705960 * q^98 - 949887 * 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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	−428.929	−1.53458	−0.767292	−	0.641298i	$-0.778396\pi$
$5$	−428.929	−1.53458	−0.767292	+	0.641298i	$0.778396\pi$
$6$	−216.000	−0.408248
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	3431.43	1.08511
$11$	−5641.82	−1.27804	−0.639020	−	0.769190i	$-0.720660\pi$
$11$	−5641.82	−1.27804	−0.639020	+	0.769190i	$0.720660\pi$
$12$	1728.00	0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	−2744.00	−0.267261
$15$	−11581.1	−0.885992
$16$	4096.00	0.250000
$17$	5030.39	0.248331	0.124165	−	0.992262i	$-0.460375\pi$
$17$	5030.39	0.248331	0.124165	+	0.992262i	$0.460375\pi$
$18$	−5832.00	−0.235702
$19$	−7398.89	−0.247474	−0.123737	−	0.992315i	$-0.539488\pi$
$19$	−7398.89	−0.247474	−0.123737	+	0.992315i	$0.539488\pi$
$20$	−27451.5	−0.767292
$21$	9261.00	0.218218
$22$	45134.5	0.903711
$23$	72906.4	1.24945	0.624724	−	0.780846i	$-0.285211\pi$
$23$	72906.4	1.24945	0.624724	+	0.780846i	$0.285211\pi$
$24$	−13824.0	−0.204124
$25$	105855.	1.35495
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	21952.0	0.188982
$29$	−59340.1	−0.451809	−0.225905	−	0.974149i	$-0.572534\pi$
$29$	−59340.1	−0.451809	−0.225905	+	0.974149i	$0.572534\pi$
$30$	92648.7	0.626491
$31$	−113966.	−0.687084	−0.343542	−	0.939137i	$-0.611627\pi$
$31$	−113966.	−0.687084	−0.343542	+	0.939137i	$0.611627\pi$
$32$	−32768.0	−0.176777
$33$	−152329.	−0.737877
$34$	−40243.1	−0.175596
$35$	−147123.	−0.580018
$36$	46656.0	0.166667
$37$	112781.	0.366041	0.183020	−	0.983109i	$-0.441413\pi$
$37$	112781.	0.366041	0.183020	+	0.983109i	$0.441413\pi$
$38$	59191.2	0.174990
$39$	59319.0	0.160128
$40$	219612.	0.542557
$41$	224080.	0.507762	0.253881	−	0.967235i	$-0.418293\pi$
$41$	224080.	0.507762	0.253881	+	0.967235i	$0.418293\pi$
$42$	−74088.0	−0.154303
$43$	509419.	0.977093	0.488546	−	0.872538i	$-0.337527\pi$
$43$	509419.	0.977093	0.488546	+	0.872538i	$0.337527\pi$
$44$	−361076.	−0.639020
$45$	−312689.	−0.511528
$46$	−583251.	−0.883493
$47$	−454823.	−0.638999	−0.319500	−	0.947586i	$-0.603515\pi$
$47$	−454823.	−0.638999	−0.319500	+	0.947586i	$0.603515\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	−846841.	−0.958091
$51$	135820.	0.143374
$52$	140608.	0.138675
$53$	869626.	0.802356	0.401178	−	0.916000i	$-0.368601\pi$
$53$	869626.	0.802356	0.401178	+	0.916000i	$0.368601\pi$
$54$	−157464.	−0.136083
$55$	2.41994e6	1.96126
$56$	−175616.	−0.133631
$57$	−199770.	−0.142879
$58$	474721.	0.319477
$59$	1.63913e6	1.03904	0.519520	−	0.854458i	$-0.326111\pi$
$59$	1.63913e6	1.03904	0.519520	+	0.854458i	$0.326111\pi$
$60$	−741189.	−0.442996
$61$	2.14907e6	1.21226	0.606130	−	0.795366i	$-0.292721\pi$
$61$	2.14907e6	1.21226	0.606130	+	0.795366i	$0.292721\pi$
$62$	911729.	0.485842
$63$	250047.	0.125988
$64$	262144.	0.125000
$65$	−942357.	−0.425617
$66$	1.21863e6	0.521758
$67$	55040.3	0.0223573	0.0111786	−	0.999938i	$-0.496442\pi$
$67$	55040.3	0.0223573	0.0111786	+	0.999938i	$0.496442\pi$
$68$	321945.	0.124165
$69$	1.96847e6	0.721369
$70$	1.17698e6	0.410135
$71$	−4.22051e6	−1.39946	−0.699730	−	0.714408i	$-0.746696\pi$
$71$	−4.22051e6	−1.39946	−0.699730	+	0.714408i	$0.746696\pi$
$72$	−373248.	−0.117851
$73$	1.72232e6	0.518182	0.259091	−	0.965853i	$-0.416577\pi$
$73$	1.72232e6	0.518182	0.259091	+	0.965853i	$0.416577\pi$
$74$	−902247.	−0.258830
$75$	2.85809e6	0.782278
$76$	−473529.	−0.123737
$77$	−1.93514e6	−0.483054
$78$	−474552.	−0.113228
$79$	−3.59169e6	−0.819603	−0.409802	−	0.912175i	$-0.634402\pi$
$79$	−3.59169e6	−0.819603	−0.409802	+	0.912175i	$0.634402\pi$
$80$	−1.75689e6	−0.383646
$81$	531441.	0.111111
$82$	−1.79264e6	−0.359042
$83$	−8.13749e6	−1.56213	−0.781065	−	0.624450i	$-0.785323\pi$
$83$	−8.13749e6	−1.56213	−0.781065	+	0.624450i	$0.785323\pi$
$84$	592704.	0.109109
$85$	−2.15768e6	−0.381084
$86$	−4.07536e6	−0.690909
$87$	−1.60218e6	−0.260852
$88$	2.88861e6	0.451856
$89$	−4.86198e6	−0.731051	−0.365526	−	0.930801i	$-0.619111\pi$
$89$	−4.86198e6	−0.731051	−0.365526	+	0.930801i	$0.619111\pi$
$90$	2.50151e6	0.361705
$91$	753571.	0.104828
$92$	4.66601e6	0.624724
$93$	−3.07708e6	−0.396688
$94$	3.63859e6	0.451841
$95$	3.17360e6	0.379769
$96$	−884736.	−0.102062
$97$	6.88065e6	0.765470	0.382735	−	0.923858i	$-0.374982\pi$
$97$	6.88065e6	0.765470	0.382735	+	0.923858i	$0.374982\pi$
$98$	−941192.	−0.101015
$99$	−4.11288e6	−0.426014
$100$	6.77473e6	0.677473
$101$	7.61335e6	0.735277	0.367638	−	0.929969i	$-0.380166\pi$
$101$	7.61335e6	0.735277	0.367638	+	0.929969i	$0.380166\pi$
$102$	−1.08656e6	−0.101381
$103$	6.69446e6	0.603651	0.301825	−	0.953363i	$-0.402404\pi$
$103$	6.69446e6	0.603651	0.301825	+	0.953363i	$0.402404\pi$
$104$	−1.12486e6	−0.0980581
$105$	−3.97231e6	−0.334874
$106$	−6.95701e6	−0.567351
$107$	−1.40813e7	−1.11122	−0.555608	−	0.831445i	$-0.687514\pi$
$107$	−1.40813e7	−1.11122	−0.555608	+	0.831445i	$0.687514\pi$
$108$	1.25971e6	0.0962250
$109$	1.67215e7	1.23675	0.618376	−	0.785882i	$-0.287791\pi$
$109$	1.67215e7	1.23675	0.618376	+	0.785882i	$0.287791\pi$
$110$	−1.93595e7	−1.38682
$111$	3.04508e6	0.211334
$112$	1.40493e6	0.0944911
$113$	−1.98555e7	−1.29451	−0.647255	−	0.762274i	$-0.724083\pi$
$113$	−1.98555e7	−1.29451	−0.647255	+	0.762274i	$0.724083\pi$
$114$	1.59816e6	0.101031
$115$	−3.12717e7	−1.91738
$116$	−3.79776e6	−0.225905
$117$	1.60161e6	0.0924500
$118$	−1.31131e7	−0.734712
$119$	1.72542e6	0.0938601
$120$	5.92952e6	0.313245
$121$	1.23429e7	0.633388
$122$	−1.71925e7	−0.857197
$123$	6.05017e6	0.293156
$124$	−7.29383e6	−0.343542
$125$	−1.18943e7	−0.544694
$126$	−2.00038e6	−0.0890871
$127$	1.12422e7	0.487009	0.243504	−	0.969900i	$-0.421703\pi$
$127$	1.12422e7	0.487009	0.243504	+	0.969900i	$0.421703\pi$
$128$	−2.09715e6	−0.0883883
$129$	1.37543e7	0.564125
$130$	7.53886e6	0.300957
$131$	−1.49026e7	−0.579178	−0.289589	−	0.957151i	$-0.593519\pi$
$131$	−1.49026e7	−0.579178	−0.289589	+	0.957151i	$0.593519\pi$
$132$	−9.74906e6	−0.368939
$133$	−2.53782e6	−0.0935363
$134$	−440322.	−0.0158090
$135$	−8.44261e6	−0.295331
$136$	−2.57556e6	−0.0877981
$137$	2.84703e7	0.945953	0.472976	−	0.881075i	$-0.343180\pi$
$137$	2.84703e7	0.945953	0.472976	+	0.881075i	$0.343180\pi$
$138$	−1.57478e7	−0.510085
$139$	3.57651e7	1.12956	0.564778	−	0.825243i	$-0.308962\pi$
$139$	3.57651e7	1.12956	0.564778	+	0.825243i	$0.308962\pi$
$140$	−9.41585e6	−0.290009
$141$	−1.22802e7	−0.368926
$142$	3.37640e7	0.989567
$143$	−1.23951e7	−0.354465
$144$	2.98598e6	0.0833333
$145$	2.54527e7	0.693339
$146$	−1.37785e7	−0.366410
$147$	3.17652e6	0.0824786
$148$	7.21797e6	0.183020
$149$	869193.	0.0215261	0.0107630	−	0.999942i	$-0.496574\pi$
$149$	869193.	0.0215261	0.0107630	+	0.999942i	$0.496574\pi$
$150$	−2.28647e7	−0.553154
$151$	−1.96266e7	−0.463901	−0.231950	−	0.972728i	$-0.574511\pi$
$151$	−1.96266e7	−0.463901	−0.231950	+	0.972728i	$0.574511\pi$
$152$	3.78823e6	0.0874952
$153$	3.66715e6	0.0827768
$154$	1.54811e7	0.341571
$155$	4.88834e7	1.05439
$156$	3.79642e6	0.0800641
$157$	−6.86346e7	−1.41545	−0.707725	−	0.706488i	$-0.750278\pi$
$157$	−6.86346e7	−1.41545	−0.707725	+	0.706488i	$0.750278\pi$
$158$	2.87335e7	0.579547
$159$	2.34799e7	0.463241
$160$	1.40551e7	0.271279
$161$	2.50069e7	0.472247
$162$	−4.25153e6	−0.0785674
$163$	−6.13655e7	−1.10986	−0.554929	−	0.831898i	$-0.687255\pi$
$163$	−6.13655e7	−1.10986	−0.554929	+	0.831898i	$0.687255\pi$
$164$	1.43411e7	0.253881
$165$	6.53384e7	1.13233
$166$	6.50999e7	1.10459
$167$	−5.21552e7	−0.866543	−0.433272	−	0.901263i	$-0.642641\pi$
$167$	−5.21552e7	−0.866543	−0.433272	+	0.901263i	$0.642641\pi$
$168$	−4.74163e6	−0.0771517
$169$	4.82681e6	0.0769231
$170$	1.72614e7	0.269467
$171$	−5.39379e6	−0.0824913
$172$	3.26028e7	0.488546
$173$	−1.06934e7	−0.157020	−0.0785098	−	0.996913i	$-0.525016\pi$
$173$	−1.06934e7	−0.157020	−0.0785098	+	0.996913i	$0.525016\pi$
$174$	1.28175e7	0.184450
$175$	3.63083e7	0.512121
$176$	−2.31089e7	−0.319510
$177$	4.42566e7	0.599890
$178$	3.88958e7	0.516931
$179$	2.83238e7	0.369119	0.184559	−	0.982821i	$-0.440914\pi$
$179$	2.83238e7	0.369119	0.184559	+	0.982821i	$0.440914\pi$
$180$	−2.00121e7	−0.255764
$181$	5.39796e7	0.676635	0.338318	−	0.941032i	$-0.390142\pi$
$181$	5.39796e7	0.676635	0.338318	+	0.941032i	$0.390142\pi$
$182$	−6.02857e6	−0.0741249
$183$	5.80248e7	0.699899
$184$	−3.73281e7	−0.441747
$185$	−4.83750e7	−0.561720
$186$	2.46167e7	0.280501
$187$	−2.83805e7	−0.317376
$188$	−2.91087e7	−0.319500
$189$	6.75127e6	0.0727393
$190$	−2.53888e7	−0.268537
$191$	3.80423e7	0.395048	0.197524	−	0.980298i	$-0.436710\pi$
$191$	3.80423e7	0.395048	0.197524	+	0.980298i	$0.436710\pi$
$192$	7.07789e6	0.0721688
$193$	8.81233e7	0.882348	0.441174	−	0.897422i	$-0.354562\pi$
$193$	8.81233e7	0.882348	0.441174	+	0.897422i	$0.354562\pi$
$194$	−5.50452e7	−0.541269
$195$	−2.54436e7	−0.245730
$196$	7.52954e6	0.0714286
$197$	8.53734e7	0.795593	0.397797	−	0.917474i	$-0.369775\pi$
$197$	8.53734e7	0.795593	0.397797	+	0.917474i	$0.369775\pi$
$198$	3.29031e7	0.301237
$199$	−4.12660e7	−0.371199	−0.185600	−	0.982625i	$-0.559423\pi$
$199$	−4.12660e7	−0.371199	−0.185600	+	0.982625i	$0.559423\pi$
$200$	−5.41978e7	−0.479046
$201$	1.48609e6	0.0129080
$202$	−6.09068e7	−0.519919
$203$	−2.03536e7	−0.170768
$204$	8.69251e6	0.0716868
$205$	−9.61145e7	−0.779203
$206$	−5.35557e7	−0.426845
$207$	5.31488e7	0.416483
$208$	8.99891e6	0.0693375
$209$	4.17432e7	0.316282
$210$	3.17785e7	0.236791
$211$	−9.56908e7	−0.701264	−0.350632	−	0.936513i	$-0.614033\pi$
$211$	−9.56908e7	−0.701264	−0.350632	+	0.936513i	$0.614033\pi$
$212$	5.56561e7	0.401178
$213$	−1.13954e8	−0.807978
$214$	1.12650e8	0.785748
$215$	−2.18505e8	−1.49943
$216$	−1.00777e7	−0.0680414
$217$	−3.90904e7	−0.259693
$218$	−1.33772e8	−0.874516
$219$	4.65025e7	0.299173
$220$	1.54876e8	0.980630
$221$	1.10518e7	0.0688745
$222$	−2.43607e7	−0.149435
$223$	−1.26066e8	−0.761254	−0.380627	−	0.924729i	$-0.624292\pi$
$223$	−1.26066e8	−0.761254	−0.380627	+	0.924729i	$0.624292\pi$
$224$	−1.12394e7	−0.0668153
$225$	7.71684e7	0.451649
$226$	1.58844e8	0.915356
$227$	−2.67097e8	−1.51558	−0.757790	−	0.652499i	$-0.773721\pi$
$227$	−2.67097e8	−1.51558	−0.757790	+	0.652499i	$0.773721\pi$
$228$	−1.27853e7	−0.0714395
$229$	1.13200e8	0.622903	0.311452	−	0.950262i	$-0.399185\pi$
$229$	1.13200e8	0.622903	0.311452	+	0.950262i	$0.399185\pi$
$230$	2.50173e8	1.35579
$231$	−5.22489e7	−0.278891
$232$	3.03821e7	0.159739
$233$	4.56825e7	0.236594	0.118297	−	0.992978i	$-0.462256\pi$
$233$	4.56825e7	0.236594	0.118297	+	0.992978i	$0.462256\pi$
$234$	−1.28129e7	−0.0653720
$235$	1.95087e8	0.980597
$236$	1.04905e8	0.519520
$237$	−9.69755e7	−0.473198
$238$	−1.38034e7	−0.0663691
$239$	−3.73712e8	−1.77070	−0.885349	−	0.464927i	$-0.846081\pi$
$239$	−3.73712e8	−1.77070	−0.885349	+	0.464927i	$0.846081\pi$
$240$	−4.74361e7	−0.221498
$241$	−1.57552e8	−0.725045	−0.362522	−	0.931975i	$-0.618084\pi$
$241$	−1.57552e8	−0.725045	−0.362522	+	0.931975i	$0.618084\pi$
$242$	−9.87435e7	−0.447873
$243$	1.43489e7	0.0641500
$244$	1.37540e8	0.606130
$245$	−5.04631e7	−0.219226
$246$	−4.84013e7	−0.207293
$247$	−1.62554e7	−0.0686369
$248$	5.83506e7	0.242921
$249$	−2.19712e8	−0.901896
$250$	9.51541e7	0.385157
$251$	8.41091e7	0.335726	0.167863	−	0.985810i	$-0.446313\pi$
$251$	8.41091e7	0.335726	0.167863	+	0.985810i	$0.446313\pi$
$252$	1.60030e7	0.0629941
$253$	−4.11325e8	−1.59685
$254$	−8.99373e7	−0.344367
$255$	−5.82573e7	−0.220019
$256$	1.67772e7	0.0625000
$257$	−2.77967e8	−1.02147	−0.510737	−	0.859737i	$-0.670627\pi$
$257$	−2.77967e8	−1.02147	−0.510737	+	0.859737i	$0.670627\pi$
$258$	−1.10035e8	−0.398897
$259$	3.86838e7	0.138350
$260$	−6.03109e7	−0.212808
$261$	−4.32589e7	−0.150603
$262$	1.19221e8	0.409541
$263$	−3.24918e8	−1.10136	−0.550679	−	0.834717i	$-0.685631\pi$
$263$	−3.24918e8	−1.10136	−0.550679	+	0.834717i	$0.685631\pi$
$264$	7.79925e7	0.260879
$265$	−3.73008e8	−1.23128
$266$	2.03026e7	0.0661402
$267$	−1.31273e8	−0.422073
$268$	3.52258e6	0.0111786
$269$	−7.17429e7	−0.224722	−0.112361	−	0.993667i	$-0.535841\pi$
$269$	−7.17429e7	−0.224722	−0.112361	+	0.993667i	$0.535841\pi$
$270$	6.75409e7	0.208830
$271$	2.13653e8	0.652102	0.326051	−	0.945352i	$-0.394282\pi$
$271$	2.13653e8	0.652102	0.326051	+	0.945352i	$0.394282\pi$
$272$	2.06045e7	0.0620826
$273$	2.03464e7	0.0605228
$274$	−2.27762e8	−0.668889
$275$	−5.97215e8	−1.73168
$276$	1.25982e8	0.360685
$277$	−5.22223e8	−1.47631	−0.738153	−	0.674633i	$-0.764302\pi$
$277$	−5.22223e8	−1.47631	−0.738153	+	0.674633i	$0.764302\pi$
$278$	−2.86121e8	−0.798717
$279$	−8.30813e7	−0.229028
$280$	7.53268e7	0.205067
$281$	−3.80140e8	−1.02205	−0.511025	−	0.859566i	$-0.670734\pi$
$281$	−3.80140e8	−1.02205	−0.511025	+	0.859566i	$0.670734\pi$
$282$	9.82418e7	0.260870
$283$	5.95810e7	0.156263	0.0781314	−	0.996943i	$-0.475105\pi$
$283$	5.95810e7	0.156263	0.0781314	+	0.996943i	$0.475105\pi$
$284$	−2.70112e8	−0.699730
$285$	8.56872e7	0.219260
$286$	9.91606e7	0.250644
$287$	7.68595e7	0.191916
$288$	−2.38879e7	−0.0589256
$289$	−3.85034e8	−0.938332
$290$	−2.03621e8	−0.490265
$291$	1.85777e8	0.441944
$292$	1.10228e8	0.259091
$293$	−3.72478e8	−0.865095	−0.432548	−	0.901611i	$-0.642385\pi$
$293$	−3.72478e8	−0.865095	−0.432548	+	0.901611i	$0.642385\pi$
$294$	−2.54122e7	−0.0583212
$295$	−7.03072e8	−1.59449
$296$	−5.77438e7	−0.129415
$297$	−1.11048e8	−0.245959
$298$	−6.95355e6	−0.0152212
$299$	1.60175e8	0.346535
$300$	1.82918e8	0.391139
$301$	1.74731e8	0.369306
$302$	1.57013e8	0.328027
$303$	2.05560e8	0.424512
$304$	−3.03059e7	−0.0618685
$305$	−9.21798e8	−1.86031
$306$	−2.93372e7	−0.0585321
$307$	−4.41815e8	−0.871477	−0.435739	−	0.900073i	$-0.643513\pi$
$307$	−4.41815e8	−0.871477	−0.435739	+	0.900073i	$0.643513\pi$
$308$	−1.23849e8	−0.241527
$309$	1.80751e8	0.348518
$310$	−3.91067e8	−0.745565
$311$	−1.19881e8	−0.225989	−0.112995	−	0.993596i	$-0.536044\pi$
$311$	−1.19881e8	−0.225989	−0.112995	+	0.993596i	$0.536044\pi$
$312$	−3.03713e7	−0.0566139
$313$	−1.18663e8	−0.218732	−0.109366	−	0.994002i	$-0.534882\pi$
$313$	−1.18663e8	−0.218732	−0.109366	+	0.994002i	$0.534882\pi$
$314$	5.49077e8	1.00087
$315$	−1.07252e8	−0.193339
$316$	−2.29868e8	−0.409802
$317$	8.52854e7	0.150372	0.0751860	−	0.997170i	$-0.476045\pi$
$317$	8.52854e7	0.150372	0.0751860	+	0.997170i	$0.476045\pi$
$318$	−1.87839e8	−0.327561
$319$	3.34786e8	0.577430
$320$	−1.12441e8	−0.191823
$321$	−3.80194e8	−0.641561
$322$	−2.00055e8	−0.333929
$323$	−3.72193e7	−0.0614553
$324$	3.40122e7	0.0555556
$325$	2.32564e8	0.375794
$326$	4.90924e8	0.784788
$327$	4.51481e8	0.714040
$328$	−1.14729e8	−0.179521
$329$	−1.56004e8	−0.241519
$330$	−5.22707e8	−0.800681
$331$	2.94895e8	0.446961	0.223481	−	0.974708i	$-0.428258\pi$
$331$	2.94895e8	0.446961	0.223481	+	0.974708i	$0.428258\pi$
$332$	−5.20799e8	−0.781065
$333$	8.22172e7	0.122014
$334$	4.17242e8	0.612739
$335$	−2.36084e7	−0.0343091
$336$	3.79331e7	0.0545545
$337$	−7.07623e8	−1.00716	−0.503579	−	0.863949i	$-0.667984\pi$
$337$	−7.07623e8	−1.00716	−0.503579	+	0.863949i	$0.667984\pi$
$338$	−3.86145e7	−0.0543928
$339$	−5.36097e8	−0.747385
$340$	−1.38091e8	−0.190542
$341$	6.42976e8	0.878121
$342$	4.31504e7	0.0583301
$343$	4.03536e7	0.0539949
$344$	−2.60823e8	−0.345455
$345$	−8.44335e8	−1.10700
$346$	8.55471e7	0.111030
$347$	−4.77862e8	−0.613974	−0.306987	−	0.951714i	$-0.599321\pi$
$347$	−4.77862e8	−0.613974	−0.306987	+	0.951714i	$0.599321\pi$
$348$	−1.02540e8	−0.130426
$349$	−7.46777e8	−0.940377	−0.470189	−	0.882566i	$-0.655814\pi$
$349$	−7.46777e8	−0.940377	−0.470189	+	0.882566i	$0.655814\pi$
$350$	−2.90467e8	−0.362124
$351$	4.32436e7	0.0533761
$352$	1.84871e8	0.225928
$353$	7.56646e8	0.915548	0.457774	−	0.889069i	$-0.348647\pi$
$353$	7.56646e8	0.915548	0.457774	+	0.889069i	$0.348647\pi$
$354$	−3.54053e8	−0.424186
$355$	1.81030e9	2.14759
$356$	−3.11166e8	−0.365526
$357$	4.65864e7	0.0541902
$358$	−2.26590e8	−0.261006
$359$	−1.01218e8	−0.115458	−0.0577292	−	0.998332i	$-0.518386\pi$
$359$	−1.01218e8	−0.115458	−0.0577292	+	0.998332i	$0.518386\pi$
$360$	1.60097e8	0.180852
$361$	−8.39128e8	−0.938757
$362$	−4.31837e8	−0.478453
$363$	3.33259e8	0.365687
$364$	4.82285e7	0.0524142
$365$	−7.38751e8	−0.795194
$366$	−4.64199e8	−0.494903
$367$	−1.22591e9	−1.29457	−0.647286	−	0.762248i	$-0.724096\pi$
$367$	−1.22591e9	−1.29457	−0.647286	+	0.762248i	$0.724096\pi$
$368$	2.98625e8	0.312362
$369$	1.63354e8	0.169254
$370$	3.87000e8	0.397196
$371$	2.98282e8	0.303262
$372$	−1.96933e8	−0.198344
$373$	−5.25342e8	−0.524157	−0.262079	−	0.965047i	$-0.584408\pi$
$373$	−5.25342e8	−0.524157	−0.262079	+	0.965047i	$0.584408\pi$
$374$	2.27044e8	0.224419
$375$	−3.21145e8	−0.314479
$376$	2.32870e8	0.225920
$377$	−1.30370e8	−0.125309
$378$	−5.40102e7	−0.0514344
$379$	5.47041e8	0.516158	0.258079	−	0.966124i	$-0.416911\pi$
$379$	5.47041e8	0.516158	0.258079	+	0.966124i	$0.416911\pi$
$380$	2.03110e8	0.189885
$381$	3.03538e8	0.281175
$382$	−3.04338e8	−0.279341
$383$	4.51036e8	0.410219	0.205110	−	0.978739i	$-0.434245\pi$
$383$	4.51036e8	0.410219	0.205110	+	0.978739i	$0.434245\pi$
$384$	−5.66231e7	−0.0510310
$385$	8.30039e8	0.741286
$386$	−7.04986e8	−0.623914
$387$	3.71367e8	0.325698
$388$	4.40361e8	0.382735
$389$	−3.75167e8	−0.323148	−0.161574	−	0.986861i	$-0.551657\pi$
$389$	−3.75167e8	−0.323148	−0.161574	+	0.986861i	$0.551657\pi$
$390$	2.03549e8	0.173757
$391$	3.66747e8	0.310276
$392$	−6.02363e7	−0.0505076
$393$	−4.02370e8	−0.334389
$394$	−6.82987e8	−0.562569
$395$	1.54058e9	1.25775
$396$	−2.63225e8	−0.213007
$397$	−8.62998e8	−0.692218	−0.346109	−	0.938194i	$-0.612497\pi$
$397$	−8.62998e8	−0.692218	−0.346109	+	0.938194i	$0.612497\pi$
$398$	3.30128e8	0.262478
$399$	−6.85212e7	−0.0540032
$400$	4.33583e8	0.338736
$401$	1.21875e9	0.943862	0.471931	−	0.881635i	$-0.343557\pi$
$401$	1.21875e9	0.943862	0.471931	+	0.881635i	$0.343557\pi$
$402$	−1.18887e7	−0.00912732
$403$	−2.50384e8	−0.190563
$404$	4.87254e8	0.367638
$405$	−2.27950e8	−0.170509
$406$	1.62829e8	0.120751
$407$	−6.36289e8	−0.467815
$408$	−6.95401e7	−0.0506903
$409$	6.28426e8	0.454174	0.227087	−	0.973874i	$-0.427080\pi$
$409$	6.28426e8	0.454174	0.227087	+	0.973874i	$0.427080\pi$
$410$	7.68916e8	0.550980
$411$	7.68697e8	0.546146
$412$	4.28446e8	0.301825
$413$	5.62223e8	0.392720
$414$	−4.25190e8	−0.294498
$415$	3.49041e9	2.39722
$416$	−7.19913e7	−0.0490290
$417$	9.65658e8	0.652150
$418$	−3.33946e8	−0.223645
$419$	2.28168e9	1.51532	0.757661	−	0.652648i	$-0.226342\pi$
$419$	2.28168e9	1.51532	0.757661	+	0.652648i	$0.226342\pi$
$420$	−2.54228e8	−0.167437
$421$	−2.96806e9	−1.93859	−0.969294	−	0.245906i	$-0.920915\pi$
$421$	−2.96806e9	−1.93859	−0.969294	+	0.245906i	$0.920915\pi$
$422$	7.65526e8	0.495869
$423$	−3.31566e8	−0.213000
$424$	−4.45249e8	−0.283676
$425$	5.32492e8	0.336474
$426$	9.11629e8	0.571327
$427$	7.37130e8	0.458191
$428$	−9.01201e8	−0.555608
$429$	−3.34667e8	−0.204650
$430$	1.74804e9	1.06026
$431$	9.96218e8	0.599355	0.299677	−	0.954041i	$-0.403121\pi$
$431$	9.96218e8	0.599355	0.299677	+	0.954041i	$0.403121\pi$
$432$	8.06216e7	0.0481125
$433$	−2.75006e9	−1.62793	−0.813963	−	0.580917i	$-0.802694\pi$
$433$	−2.75006e9	−1.62793	−0.813963	+	0.580917i	$0.802694\pi$
$434$	3.12723e8	0.183631
$435$	6.87222e8	0.400299
$436$	1.07018e9	0.618376
$437$	−5.39427e8	−0.309206
$438$	−3.72020e8	−0.211547
$439$	−4.24947e8	−0.239723	−0.119861	−	0.992791i	$-0.538245\pi$
$439$	−4.24947e8	−0.239723	−0.119861	+	0.992791i	$0.538245\pi$
$440$	−1.23901e9	−0.693410
$441$	8.57661e7	0.0476190
$442$	−8.84141e7	−0.0487016
$443$	1.92764e9	1.05345	0.526724	−	0.850036i	$-0.323420\pi$
$443$	1.92764e9	1.05345	0.526724	+	0.850036i	$0.323420\pi$
$444$	1.94885e8	0.105667
$445$	2.08544e9	1.12186
$446$	1.00853e9	0.538288
$447$	2.34682e7	0.0124281
$448$	8.99154e7	0.0472456
$449$	−1.97858e9	−1.03155	−0.515776	−	0.856724i	$-0.672496\pi$
$449$	−1.97858e9	−1.03155	−0.515776	+	0.856724i	$0.672496\pi$
$450$	−6.17347e8	−0.319364
$451$	−1.26422e9	−0.648940
$452$	−1.27075e9	−0.647255
$453$	−5.29917e8	−0.267833
$454$	2.13678e9	1.07168
$455$	−3.23229e8	−0.160868
$456$	1.02282e8	0.0505154
$457$	2.66925e9	1.30822	0.654112	−	0.756398i	$-0.273043\pi$
$457$	2.66925e9	1.30822	0.654112	+	0.756398i	$0.273043\pi$
$458$	−9.05596e8	−0.440459
$459$	9.90131e7	0.0477912
$460$	−2.00139e9	−0.958691
$461$	−2.19410e9	−1.04304	−0.521522	−	0.853238i	$-0.674635\pi$
$461$	−2.19410e9	−1.04304	−0.521522	+	0.853238i	$0.674635\pi$
$462$	4.17991e8	0.197206
$463$	4.35755e8	0.204037	0.102019	−	0.994782i	$-0.467470\pi$
$463$	4.35755e8	0.204037	0.102019	+	0.994782i	$0.467470\pi$
$464$	−2.43057e8	−0.112952
$465$	1.31985e9	0.608751
$466$	−3.65460e8	−0.167298
$467$	3.76838e8	0.171216	0.0856082	−	0.996329i	$-0.472717\pi$
$467$	3.76838e8	0.171216	0.0856082	+	0.996329i	$0.472717\pi$
$468$	1.02503e8	0.0462250
$469$	1.88788e7	0.00845025
$470$	−1.56070e9	−0.693387
$471$	−1.85313e9	−0.817210
$472$	−8.39236e8	−0.367356
$473$	−2.87405e9	−1.24876
$474$	7.75804e8	0.334602
$475$	−7.83211e8	−0.335314
$476$	1.10427e8	0.0469301
$477$	6.33958e8	0.267452
$478$	2.98970e9	1.25207
$479$	−1.75733e8	−0.0730600	−0.0365300	−	0.999333i	$-0.511630\pi$
$479$	−1.75733e8	−0.0730600	−0.0365300	+	0.999333i	$0.511630\pi$
$480$	3.79489e8	0.156623
$481$	2.47779e8	0.101521
$482$	1.26042e9	0.512684
$483$	6.75186e8	0.272652
$484$	7.89948e8	0.316694
$485$	−2.95131e9	−1.17468
$486$	−1.14791e8	−0.0453609
$487$	−3.91286e9	−1.53512	−0.767560	−	0.640977i	$-0.778529\pi$
$487$	−3.91286e9	−1.53512	−0.767560	+	0.640977i	$0.778529\pi$
$488$	−1.10032e9	−0.428599
$489$	−1.65687e9	−0.640777
$490$	4.03705e8	0.155016
$491$	−3.92505e8	−0.149644	−0.0748221	−	0.997197i	$-0.523839\pi$
$491$	−3.92505e8	−0.149644	−0.0748221	+	0.997197i	$0.523839\pi$
$492$	3.87211e8	0.146578
$493$	−2.98503e8	−0.112198
$494$	1.30043e8	0.0485336
$495$	1.76414e9	0.653753
$496$	−4.66805e8	−0.171771
$497$	−1.44763e9	−0.528946
$498$	1.75770e9	0.637737
$499$	−2.32523e9	−0.837748	−0.418874	−	0.908044i	$-0.637575\pi$
$499$	−2.32523e9	−0.837748	−0.418874	+	0.908044i	$0.637575\pi$
$500$	−7.61233e8	−0.272347
$501$	−1.40819e9	−0.500299
$502$	−6.72873e8	−0.237394
$503$	1.41640e9	0.496246	0.248123	−	0.968729i	$-0.420186\pi$
$503$	1.41640e9	0.496246	0.248123	+	0.968729i	$0.420186\pi$
$504$	−1.28024e8	−0.0445435
$505$	−3.26559e9	−1.12834
$506$	3.29060e9	1.12914
$507$	1.30324e8	0.0444116
$508$	7.19498e8	0.243504
$509$	1.15453e9	0.388053	0.194027	−	0.980996i	$-0.437845\pi$
$509$	1.15453e9	0.388053	0.194027	+	0.980996i	$0.437845\pi$
$510$	4.66059e8	0.155577
$511$	5.90754e8	0.195855
$512$	−1.34218e8	−0.0441942
$513$	−1.45632e8	−0.0476264
$514$	2.22374e9	0.722291
$515$	−2.87145e9	−0.926352
$516$	8.80277e8	0.282062
$517$	2.56603e9	0.816667
$518$	−3.09471e8	−0.0978285
$519$	−2.88721e8	−0.0906553
$520$	4.82487e8	0.150478
$521$	−4.96724e9	−1.53880	−0.769401	−	0.638766i	$-0.779445\pi$
$521$	−4.96724e9	−1.53880	−0.769401	+	0.638766i	$0.779445\pi$
$522$	3.46071e8	0.106492
$523$	−4.20708e8	−0.128595	−0.0642976	−	0.997931i	$-0.520481\pi$
$523$	−4.20708e8	−0.128595	−0.0642976	+	0.997931i	$0.520481\pi$
$524$	−9.53766e8	−0.289589
$525$	9.80324e8	0.295673
$526$	2.59934e9	0.778778
$527$	−5.73294e8	−0.170624
$528$	−6.23940e8	−0.184469
$529$	1.91052e9	0.561121
$530$	2.98406e9	0.870648
$531$	1.19493e9	0.346347
$532$	−1.62421e8	−0.0467682
$533$	4.92304e8	0.140828
$534$	1.05019e9	0.298450
$535$	6.03986e9	1.70525
$536$	−2.81806e7	−0.00790449
$537$	7.64743e8	0.213111
$538$	5.73943e8	0.158903
$539$	−6.63754e8	−0.182577
$540$	−5.40327e8	−0.147665
$541$	5.73067e9	1.55602	0.778010	−	0.628252i	$-0.216229\pi$
$541$	5.73067e9	1.55602	0.778010	+	0.628252i	$0.216229\pi$
$542$	−1.70922e9	−0.461106
$543$	1.45745e9	0.390656
$544$	−1.64836e8	−0.0438990
$545$	−7.17234e9	−1.89790
$546$	−1.62771e8	−0.0427960
$547$	−1.35334e9	−0.353549	−0.176775	−	0.984251i	$-0.556566\pi$
$547$	−1.35334e9	−0.353549	−0.176775	+	0.984251i	$0.556566\pi$
$548$	1.82210e9	0.472976
$549$	1.56667e9	0.404087
$550$	4.77772e9	1.22448
$551$	4.39051e8	0.111811
$552$	−1.00786e9	−0.255043
$553$	−1.23195e9	−0.309781
$554$	4.17778e9	1.04391
$555$	−1.30612e9	−0.324309
$556$	2.28897e9	0.564778
$557$	−3.92720e9	−0.962919	−0.481459	−	0.876468i	$-0.659893\pi$
$557$	−3.92720e9	−0.962919	−0.481459	+	0.876468i	$0.659893\pi$
$558$	6.64650e8	0.161947
$559$	1.11919e9	0.270997
$560$	−6.02614e8	−0.145004
$561$	−7.66274e8	−0.183237
$562$	3.04112e9	0.722698
$563$	3.44486e9	0.813564	0.406782	−	0.913525i	$-0.366651\pi$
$563$	3.44486e9	0.813564	0.406782	+	0.913525i	$0.366651\pi$
$564$	−7.85935e8	−0.184463
$565$	8.51658e9	1.98653
$566$	−4.76648e8	−0.110494
$567$	1.82284e8	0.0419961
$568$	2.16090e9	0.494784
$569$	5.20185e9	1.18376	0.591882	−	0.806024i	$-0.298385\pi$
$569$	5.20185e9	1.18376	0.591882	+	0.806024i	$0.298385\pi$
$570$	−6.85498e8	−0.155040
$571$	4.94153e9	1.11080	0.555398	−	0.831584i	$-0.312566\pi$
$571$	4.94153e9	1.11080	0.555398	+	0.831584i	$0.312566\pi$
$572$	−7.93285e8	−0.177232
$573$	1.02714e9	0.228081
$574$	−6.14876e8	−0.135705
$575$	7.71752e9	1.69293
$576$	1.91103e8	0.0416667
$577$	−1.20233e8	−0.0260561	−0.0130280	−	0.999915i	$-0.504147\pi$
$577$	−1.20233e8	−0.0260561	−0.0130280	+	0.999915i	$0.504147\pi$
$578$	3.08027e9	0.663501
$579$	2.37933e9	0.509424
$580$	1.62897e9	0.346669
$581$	−2.79116e9	−0.590429
$582$	−1.48622e9	−0.312502
$583$	−4.90627e9	−1.02544
$584$	−8.81825e8	−0.183205
$585$	−6.86978e8	−0.141872
$586$	2.97982e9	0.611715
$587$	−1.35701e9	−0.276917	−0.138459	−	0.990368i	$-0.544215\pi$
$587$	−1.35701e9	−0.276917	−0.138459	+	0.990368i	$0.544215\pi$
$588$	2.03297e8	0.0412393
$589$	8.43223e8	0.170035
$590$	5.62458e9	1.12748
$591$	2.30508e9	0.459336
$592$	4.61950e8	0.0915101
$593$	8.65668e8	0.170475	0.0852373	−	0.996361i	$-0.472835\pi$
$593$	8.65668e8	0.170475	0.0852373	+	0.996361i	$0.472835\pi$
$594$	8.88383e8	0.173919
$595$	−7.40084e8	−0.144036
$596$	5.56284e7	0.0107630
$597$	−1.11418e9	−0.214312
$598$	−1.28140e9	−0.245037
$599$	2.71350e9	0.515865	0.257932	−	0.966163i	$-0.416959\pi$
$599$	2.71350e9	0.515865	0.257932	+	0.966163i	$0.416959\pi$
$600$	−1.46334e9	−0.276577
$601$	7.29865e9	1.37146	0.685728	−	0.727858i	$-0.259484\pi$
$601$	7.29865e9	1.37146	0.685728	+	0.727858i	$0.259484\pi$
$602$	−1.39785e9	−0.261139
$603$	4.01244e7	0.00745242
$604$	−1.25610e9	−0.231950
$605$	−5.29424e9	−0.971986
$606$	−1.64448e9	−0.300176
$607$	7.24336e9	1.31456	0.657279	−	0.753647i	$-0.271707\pi$
$607$	7.24336e9	1.31456	0.657279	+	0.753647i	$0.271707\pi$
$608$	2.42447e8	0.0437476
$609$	−5.49548e8	−0.0985928
$610$	7.37438e9	1.31544
$611$	−9.99247e8	−0.177226
$612$	2.34698e8	0.0413884
$613$	5.90927e9	1.03615	0.518075	−	0.855335i	$-0.326649\pi$
$613$	5.90927e9	1.03615	0.518075	+	0.855335i	$0.326649\pi$
$614$	3.53452e9	0.616227
$615$	−2.59509e9	−0.449873
$616$	9.90793e8	0.170785
$617$	−6.28936e9	−1.07797	−0.538987	−	0.842314i	$-0.681193\pi$
$617$	−6.28936e9	−1.07797	−0.538987	+	0.842314i	$0.681193\pi$
$618$	−1.44600e9	−0.246439
$619$	−3.11689e9	−0.528207	−0.264103	−	0.964494i	$-0.585076\pi$
$619$	−3.11689e9	−0.528207	−0.264103	+	0.964494i	$0.585076\pi$
$620$	3.12854e9	0.527194
$621$	1.43502e9	0.240456
$622$	9.59045e8	0.159799
$623$	−1.66766e9	−0.276311
$624$	2.42971e8	0.0400320
$625$	−3.16814e9	−0.519068
$626$	9.49308e8	0.154667
$627$	1.12707e9	0.182605
$628$	−4.39262e9	−0.707725
$629$	5.67331e8	0.0908990
$630$	8.58019e8	0.136712
$631$	1.46856e9	0.232695	0.116348	−	0.993209i	$-0.462881\pi$
$631$	1.46856e9	0.232695	0.116348	+	0.993209i	$0.462881\pi$
$632$	1.83894e9	0.289773
$633$	−2.58365e9	−0.404875
$634$	−6.82283e8	−0.106329
$635$	−4.82209e9	−0.747355
$636$	1.50271e9	0.231620
$637$	2.58475e8	0.0396214
$638$	−2.67829e9	−0.408305
$639$	−3.07675e9	−0.466486
$640$	8.99529e8	0.135639
$641$	−6.63645e9	−0.995250	−0.497625	−	0.867392i	$-0.665795\pi$
$641$	−6.63645e9	−0.995250	−0.497625	+	0.867392i	$0.665795\pi$
$642$	3.04155e9	0.453652
$643$	−3.68719e8	−0.0546961	−0.0273481	−	0.999626i	$-0.508706\pi$
$643$	−3.68719e8	−0.0546961	−0.0273481	+	0.999626i	$0.508706\pi$
$644$	1.60044e9	0.236124
$645$	−5.89963e9	−0.865697
$646$	2.97754e8	0.0434555
$647$	−4.62229e9	−0.670953	−0.335476	−	0.942049i	$-0.608897\pi$
$647$	−4.62229e9	−0.670953	−0.335476	+	0.942049i	$0.608897\pi$
$648$	−2.72098e8	−0.0392837
$649$	−9.24769e9	−1.32794
$650$	−1.86051e9	−0.265727
$651$	−1.05544e9	−0.149934
$652$	−3.92739e9	−0.554929
$653$	6.70504e9	0.942335	0.471168	−	0.882044i	$-0.343833\pi$
$653$	6.70504e9	0.942335	0.471168	+	0.882044i	$0.343833\pi$
$654$	−3.61185e9	−0.504902
$655$	6.39216e9	0.888797
$656$	9.17833e8	0.126940
$657$	1.25557e9	0.172727
$658$	1.24804e9	0.170780
$659$	−2.22581e9	−0.302963	−0.151482	−	0.988460i	$-0.548404\pi$
$659$	−2.22581e9	−0.302963	−0.151482	+	0.988460i	$0.548404\pi$
$660$	4.18166e9	0.566167
$661$	5.49601e8	0.0740189	0.0370094	−	0.999315i	$-0.488217\pi$
$661$	5.49601e8	0.0740189	0.0370094	+	0.999315i	$0.488217\pi$
$662$	−2.35916e9	−0.316049
$663$	2.98397e8	0.0397647
$664$	4.16639e9	0.552296
$665$	1.08855e9	0.143539
$666$	−6.57738e8	−0.0862766
$667$	−4.32627e9	−0.564512
$668$	−3.33794e9	−0.433272
$669$	−3.40377e9	−0.439510
$670$	1.88867e8	0.0242602
$671$	−1.21247e10	−1.54932
$672$	−3.03464e8	−0.0385758
$673$	1.49304e10	1.88807	0.944036	−	0.329844i	$-0.106996\pi$
$673$	1.49304e10	1.88807	0.944036	+	0.329844i	$0.106996\pi$
$674$	5.66098e9	0.712168
$675$	2.08355e9	0.260759
$676$	3.08916e8	0.0384615
$677$	1.16981e10	1.44895	0.724476	−	0.689300i	$-0.242082\pi$
$677$	1.16981e10	1.44895	0.724476	+	0.689300i	$0.242082\pi$
$678$	4.28878e9	0.528481
$679$	2.36006e9	0.289320
$680$	1.10473e9	0.134733
$681$	−7.21162e9	−0.875020
$682$	−5.14381e9	−0.620925
$683$	3.79059e9	0.455234	0.227617	−	0.973751i	$-0.426907\pi$
$683$	3.79059e9	0.455234	0.227617	+	0.973751i	$0.426907\pi$
$684$	−3.45203e8	−0.0412456
$685$	−1.22117e10	−1.45164
$686$	−3.22829e8	−0.0381802
$687$	3.05639e9	0.359633
$688$	2.08658e9	0.244273
$689$	1.91057e9	0.222534
$690$	6.75468e9	0.782768
$691$	1.05651e10	1.21815	0.609077	−	0.793111i	$-0.291540\pi$
$691$	1.05651e10	1.21815	0.609077	+	0.793111i	$0.291540\pi$
$692$	−6.84377e8	−0.0785098
$693$	−1.41072e9	−0.161018
$694$	3.82290e9	0.434145
$695$	−1.53407e10	−1.73340
$696$	8.20317e8	0.0922252
$697$	1.12721e9	0.126093
$698$	5.97422e9	0.664947
$699$	1.23343e9	0.136598
$700$	2.32373e9	0.256061
$701$	−1.41529e10	−1.55179	−0.775896	−	0.630860i	$-0.782702\pi$
$701$	−1.41529e10	−1.55179	−0.775896	+	0.630860i	$0.782702\pi$
$702$	−3.45948e8	−0.0377426
$703$	−8.34453e8	−0.0905855
$704$	−1.47897e9	−0.159755
$705$	5.26735e9	0.566148
$706$	−6.05317e9	−0.647390
$707$	2.61138e9	0.277909
$708$	2.83242e9	0.299945
$709$	−6.21578e9	−0.654989	−0.327494	−	0.944853i	$-0.606204\pi$
$709$	−6.21578e9	−0.654989	−0.327494	+	0.944853i	$0.606204\pi$
$710$	−1.44824e10	−1.51857
$711$	−2.61834e9	−0.273201
$712$	2.48933e9	0.258466
$713$	−8.30886e9	−0.858476
$714$	−3.72691e8	−0.0383182
$715$	5.31661e9	0.543956
$716$	1.81272e9	0.184559
$717$	−1.00902e10	−1.02231
$718$	8.09740e8	0.0816414
$719$	−1.18293e10	−1.18688	−0.593439	−	0.804879i	$-0.702230\pi$
$719$	−1.18293e10	−1.18688	−0.593439	+	0.804879i	$0.702230\pi$
$720$	−1.28078e9	−0.127882
$721$	2.29620e9	0.228158
$722$	6.71302e9	0.663801
$723$	−4.25391e9	−0.418605
$724$	3.45469e9	0.338318
$725$	−6.28145e9	−0.612177
$726$	−2.66607e9	−0.258579
$727$	−1.83987e10	−1.77589	−0.887947	−	0.459945i	$-0.847869\pi$
$727$	−1.83987e10	−1.77589	−0.887947	+	0.459945i	$0.847869\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	5.91001e9	0.562287
$731$	2.56258e9	0.242642
$732$	3.71359e9	0.349949
$733$	1.19751e10	1.12309	0.561544	−	0.827447i	$-0.310207\pi$
$733$	1.19751e10	1.12309	0.561544	+	0.827447i	$0.310207\pi$
$734$	9.80725e9	0.915400
$735$	−1.36250e9	−0.126570
$736$	−2.38900e9	−0.220873
$737$	−3.10527e8	−0.0285735
$738$	−1.30684e9	−0.119681
$739$	1.22577e10	1.11726	0.558631	−	0.829416i	$-0.311327\pi$
$739$	1.22577e10	1.11726	0.558631	+	0.829416i	$0.311327\pi$
$740$	−3.09600e9	−0.280860
$741$	−4.38895e8	−0.0396275
$742$	−2.38625e9	−0.214439
$743$	1.69423e10	1.51534	0.757671	−	0.652636i	$-0.226337\pi$
$743$	1.69423e10	1.51534	0.757671	+	0.652636i	$0.226337\pi$
$744$	1.57547e9	0.140250
$745$	−3.72822e8	−0.0330335
$746$	4.20274e9	0.370635
$747$	−5.93223e9	−0.520710
$748$	−1.81635e9	−0.158688
$749$	−4.82987e9	−0.420000
$750$	2.56916e9	0.222370
$751$	−1.50990e9	−0.130080	−0.0650399	−	0.997883i	$-0.520717\pi$
$751$	−1.50990e9	−0.130080	−0.0650399	+	0.997883i	$0.520717\pi$
$752$	−1.86296e9	−0.159750
$753$	2.27094e9	0.193831
$754$	1.04296e9	0.0886071
$755$	8.41841e9	0.711894
$756$	4.32081e8	0.0363696
$757$	1.06891e10	0.895586	0.447793	−	0.894137i	$-0.352210\pi$
$757$	1.06891e10	0.895586	0.447793	+	0.894137i	$0.352210\pi$
$758$	−4.37633e9	−0.364979
$759$	−1.11058e10	−0.921939
$760$	−1.62488e9	−0.134269
$761$	7.92640e9	0.651973	0.325987	−	0.945374i	$-0.394304\pi$
$761$	7.92640e9	0.651973	0.325987	+	0.945374i	$0.394304\pi$
$762$	−2.42831e9	−0.198820
$763$	5.73548e9	0.467449
$764$	2.43471e9	0.197524
$765$	−1.57295e9	−0.127028
$766$	−3.60829e9	−0.290069
$767$	3.60118e9	0.288178
$768$	4.52985e8	0.0360844
$769$	−5.60784e9	−0.444686	−0.222343	−	0.974969i	$-0.571370\pi$
$769$	−5.60784e9	−0.444686	−0.222343	+	0.974969i	$0.571370\pi$
$770$	−6.64031e9	−0.524169
$771$	−7.50511e9	−0.589748
$772$	5.63989e9	0.441174
$773$	7.85923e9	0.612000	0.306000	−	0.952031i	$-0.401009\pi$
$773$	7.85923e9	0.612000	0.306000	+	0.952031i	$0.401009\pi$
$774$	−2.97093e9	−0.230303
$775$	−1.20639e10	−0.930961
$776$	−3.52289e9	−0.270634
$777$	1.04446e9	0.0798766
$778$	3.00134e9	0.228500
$779$	−1.65795e9	−0.125658
$780$	−1.62839e9	−0.122865
$781$	2.38113e10	1.78857
$782$	−2.93398e9	−0.219398
$783$	−1.16799e9	−0.0869507
$784$	4.81890e8	0.0357143
$785$	2.94394e10	2.17213
$786$	3.21896e9	0.236449
$787$	−8.00008e9	−0.585036	−0.292518	−	0.956260i	$-0.594493\pi$
$787$	−8.00008e9	−0.585036	−0.292518	+	0.956260i	$0.594493\pi$
$788$	5.46390e9	0.397797
$789$	−8.77279e9	−0.635870
$790$	−1.23246e10	−0.889363
$791$	−6.81042e9	−0.489279
$792$	2.10580e9	0.150619
$793$	4.72150e9	0.336220
$794$	6.90398e9	0.489472
$795$	−1.00712e10	−0.710881
$796$	−2.64103e9	−0.185600
$797$	−1.46243e10	−1.02323	−0.511614	−	0.859215i	$-0.670952\pi$
$797$	−1.46243e10	−1.02323	−0.511614	+	0.859215i	$0.670952\pi$
$798$	5.48169e8	0.0381860
$799$	−2.28794e9	−0.158683
$800$	−3.46866e9	−0.239523
$801$	−3.54438e9	−0.243684
$802$	−9.74998e9	−0.667411
$803$	−9.71699e9	−0.662258
$804$	9.51096e7	0.00645399
$805$	−1.07262e10	−0.724702
$806$	2.00307e9	0.134748
$807$	−1.93706e9	−0.129743
$808$	−3.89803e9	−0.259960
$809$	−8.61170e9	−0.571833	−0.285916	−	0.958255i	$-0.592298\pi$
$809$	−8.61170e9	−0.571833	−0.285916	+	0.958255i	$0.592298\pi$
$810$	1.82360e9	0.120568
$811$	1.28149e10	0.843609	0.421804	−	0.906687i	$-0.361397\pi$
$811$	1.28149e10	0.843609	0.421804	+	0.906687i	$0.361397\pi$
$812$	−1.30263e9	−0.0853839
$813$	5.76862e9	0.376491
$814$	5.09031e9	0.330795
$815$	2.63214e10	1.70317
$816$	5.56320e8	0.0358434
$817$	−3.76914e9	−0.241805
$818$	−5.02741e9	−0.321150
$819$	5.49353e8	0.0349428
$820$	−6.15133e9	−0.389601
$821$	1.55861e10	0.982964	0.491482	−	0.870888i	$-0.336455\pi$
$821$	1.55861e10	0.982964	0.491482	+	0.870888i	$0.336455\pi$
$822$	−6.14957e9	−0.386184
$823$	1.84906e10	1.15625	0.578125	−	0.815948i	$-0.303785\pi$
$823$	1.84906e10	1.15625	0.578125	+	0.815948i	$0.303785\pi$
$824$	−3.42757e9	−0.213423
$825$	−1.61248e10	−0.999783
$826$	−4.49778e9	−0.277695
$827$	−3.22049e10	−1.97994	−0.989970	−	0.141275i	$-0.954880\pi$
$827$	−3.22049e10	−1.97994	−0.989970	+	0.141275i	$0.954880\pi$
$828$	3.40152e9	0.208241
$829$	2.14528e10	1.30781	0.653903	−	0.756579i	$-0.273131\pi$
$829$	2.14528e10	1.30781	0.653903	+	0.756579i	$0.273131\pi$
$830$	−2.79232e10	−1.69509
$831$	−1.41000e10	−0.852346
$832$	5.75930e8	0.0346688
$833$	5.91820e8	0.0354758
$834$	−7.72527e9	−0.461139
$835$	2.23709e10	1.32978
$836$	2.67157e9	0.158141
$837$	−2.24319e9	−0.132229
$838$	−1.82534e10	−1.07149
$839$	1.62577e9	0.0950372	0.0475186	−	0.998870i	$-0.484869\pi$
$839$	1.62577e9	0.0950372	0.0475186	+	0.998870i	$0.484869\pi$
$840$	2.03382e9	0.118396
$841$	−1.37286e10	−0.795868
$842$	2.37445e10	1.37079
$843$	−1.02638e10	−0.590080
$844$	−6.12421e9	−0.350632
$845$	−2.07036e9	−0.118045
$846$	2.65253e9	0.150614
$847$	4.23363e9	0.239398
$848$	3.56199e9	0.200589
$849$	1.60869e9	0.0902183
$850$	−4.25994e9	−0.237923
$851$	8.22244e9	0.457349
$852$	−7.29303e9	−0.403989
$853$	−2.46400e10	−1.35931	−0.679657	−	0.733531i	$-0.737871\pi$
$853$	−2.46400e10	−1.35931	−0.679657	+	0.733531i	$0.737871\pi$
$854$	−5.89704e9	−0.323990
$855$	2.31356e9	0.126590
$856$	7.20961e9	0.392874
$857$	1.99504e10	1.08273	0.541364	−	0.840788i	$-0.317908\pi$
$857$	1.99504e10	1.08273	0.541364	+	0.840788i	$0.317908\pi$
$858$	2.67734e9	0.144710
$859$	2.30206e10	1.23920	0.619598	−	0.784919i	$-0.287296\pi$
$859$	2.30206e10	1.23920	0.619598	+	0.784919i	$0.287296\pi$
$860$	−1.39843e10	−0.749715
$861$	2.07521e9	0.110803
$862$	−7.96974e9	−0.423808
$863$	2.84131e10	1.50481	0.752403	−	0.658704i	$-0.228895\pi$
$863$	2.84131e10	1.50481	0.752403	+	0.658704i	$0.228895\pi$
$864$	−6.44973e8	−0.0340207
$865$	4.58670e9	0.240960
$866$	2.20005e10	1.15112
$867$	−1.03959e10	−0.541746
$868$	−2.50178e9	−0.129847
$869$	2.02636e10	1.04749
$870$	−5.49778e9	−0.283054
$871$	1.20923e8	0.00620079
$872$	−8.56142e9	−0.437258
$873$	5.01599e9	0.255157
$874$	4.31541e9	0.218641
$875$	−4.07973e9	−0.205875
$876$	2.97616e9	0.149586
$877$	−2.43482e10	−1.21890	−0.609450	−	0.792825i	$-0.708610\pi$
$877$	−2.43482e10	−1.21890	−0.609450	+	0.792825i	$0.708610\pi$
$878$	3.39958e9	0.169509
$879$	−1.00569e10	−0.499463
$880$	9.91207e9	0.490315
$881$	−2.41724e10	−1.19098	−0.595490	−	0.803363i	$-0.703042\pi$
$881$	−2.41724e10	−1.19098	−0.595490	+	0.803363i	$0.703042\pi$
$882$	−6.86129e8	−0.0336718
$883$	−1.97895e10	−0.967324	−0.483662	−	0.875255i	$-0.660694\pi$
$883$	−1.97895e10	−0.967324	−0.483662	+	0.875255i	$0.660694\pi$
$884$	7.07313e8	0.0344372
$885$	−1.89829e10	−0.920581
$886$	−1.54211e10	−0.744901
$887$	−3.75324e10	−1.80582	−0.902908	−	0.429833i	$-0.858572\pi$
$887$	−3.75324e10	−1.80582	−0.902908	+	0.429833i	$0.858572\pi$
$888$	−1.55908e9	−0.0747177
$889$	3.85606e9	0.184072
$890$	−1.66835e10	−0.793274
$891$	−2.99829e9	−0.142005
$892$	−8.06820e9	−0.380627
$893$	3.36519e9	0.158136
$894$	−1.87746e8	−0.00878798
$895$	−1.21489e10	−0.566443
$896$	−7.19323e8	−0.0334077
$897$	4.32473e9	0.200072
$898$	1.58286e10	0.729417
$899$	6.76276e9	0.310431
$900$	4.93878e9	0.225824
$901$	4.37456e9	0.199250
$902$	1.01138e10	0.458870
$903$	4.71773e9	0.213219
$904$	1.01660e10	0.457678
$905$	−2.31534e10	−1.03835
$906$	4.23934e9	0.189387
$907$	3.67145e10	1.63385	0.816924	−	0.576745i	$-0.195678\pi$
$907$	3.67145e10	1.63385	0.816924	+	0.576745i	$0.195678\pi$
$908$	−1.70942e10	−0.757790
$909$	5.55013e9	0.245092
$910$	2.58583e9	0.113751
$911$	3.24643e10	1.42263	0.711315	−	0.702874i	$-0.248100\pi$
$911$	3.24643e10	1.42263	0.711315	+	0.702874i	$0.248100\pi$
$912$	−8.18259e8	−0.0357198
$913$	4.59102e10	1.99646
$914$	−2.13540e10	−0.925054
$915$	−2.48885e10	−1.07405
$916$	7.24477e9	0.311452
$917$	−5.11159e9	−0.218909
$918$	−7.92105e8	−0.0337935
$919$	−2.65751e10	−1.12946	−0.564730	−	0.825276i	$-0.691020\pi$
$919$	−2.65751e10	−1.12946	−0.564730	+	0.825276i	$0.691020\pi$
$920$	1.60111e10	0.677897
$921$	−1.19290e10	−0.503148
$922$	1.75528e10	0.737543
$923$	−9.27245e9	−0.388140
$924$	−3.34393e9	−0.139446
$925$	1.19384e10	0.495965
$926$	−3.48604e9	−0.144276
$927$	4.88026e9	0.201217
$928$	1.94446e9	0.0798693
$929$	3.05820e10	1.25144	0.625722	−	0.780046i	$-0.284805\pi$
$929$	3.05820e10	1.25144	0.625722	+	0.780046i	$0.284805\pi$
$930$	−1.05588e10	−0.430452
$931$	−8.70473e8	−0.0353534
$932$	2.92368e9	0.118297
$933$	−3.23678e9	−0.130475
$934$	−3.01470e9	−0.121068
$935$	1.21732e10	0.487041
$936$	−8.20026e8	−0.0326860
$937$	2.98201e10	1.18419	0.592095	−	0.805868i	$-0.298301\pi$
$937$	2.98201e10	1.18419	0.592095	+	0.805868i	$0.298301\pi$
$938$	−1.51031e8	−0.00597523
$939$	−3.20391e9	−0.126285
$940$	1.24856e10	0.490299
$941$	3.92420e10	1.53528	0.767639	−	0.640882i	$-0.221431\pi$
$941$	3.92420e10	1.53528	0.767639	+	0.640882i	$0.221431\pi$
$942$	1.48251e10	0.577855
$943$	1.63369e10	0.634422
$944$	6.71389e9	0.259760
$945$	−2.89582e9	−0.111625
$946$	2.29924e10	0.883010
$947$	−3.38049e10	−1.29346	−0.646732	−	0.762718i	$-0.723865\pi$
$947$	−3.38049e10	−1.29346	−0.646732	+	0.762718i	$0.723865\pi$
$948$	−6.20643e9	−0.236599
$949$	3.78393e9	0.143718
$950$	6.26569e9	0.237103
$951$	2.30270e9	0.0868173
$952$	−8.83416e8	−0.0331846
$953$	−8.95517e9	−0.335157	−0.167579	−	0.985859i	$-0.553595\pi$
$953$	−8.95517e9	−0.335157	−0.167579	+	0.985859i	$0.553595\pi$
$954$	−5.07166e9	−0.189117
$955$	−1.63174e10	−0.606234
$956$	−2.39176e10	−0.885349
$957$	9.03922e9	0.333380
$958$	1.40587e9	0.0516612
$959$	9.76530e9	0.357536
$960$	−3.03591e9	−0.110749
$961$	−1.45243e10	−0.527916
$962$	−1.98224e9	−0.0717865
$963$	−1.02652e10	−0.370405
$964$	−1.00833e10	−0.362522
$965$	−3.77986e10	−1.35404
$966$	−5.40149e9	−0.192794
$967$	2.29472e10	0.816089	0.408044	−	0.912962i	$-0.366211\pi$
$967$	2.29472e10	0.816089	0.408044	+	0.912962i	$0.366211\pi$
$968$	−6.31958e9	−0.223936
$969$	−1.00492e9	−0.0354812
$970$	2.36105e10	0.830622
$971$	3.46493e9	0.121458	0.0607292	−	0.998154i	$-0.480657\pi$
$971$	3.46493e9	0.121458	0.0607292	+	0.998154i	$0.480657\pi$
$972$	9.18330e8	0.0320750
$973$	1.22674e10	0.426932
$974$	3.13028e10	1.08549
$975$	6.27922e9	0.216965
$976$	8.80258e9	0.303065
$977$	−2.44027e9	−0.0837158	−0.0418579	−	0.999124i	$-0.513328\pi$
$977$	−2.44027e9	−0.0837158	−0.0418579	+	0.999124i	$0.513328\pi$
$978$	1.32549e10	0.453098
$979$	2.74304e10	0.934313
$980$	−3.22964e9	−0.109613
$981$	1.21900e10	0.412251
$982$	3.14004e9	0.105814
$983$	−1.38431e10	−0.464833	−0.232417	−	0.972616i	$-0.574663\pi$
$983$	−1.38431e10	−0.464833	−0.232417	+	0.972616i	$0.574663\pi$
$984$	−3.09768e9	−0.103646
$985$	−3.66191e10	−1.22090
$986$	2.38803e9	0.0793360
$987$	−4.21212e9	−0.139441
$988$	−1.04034e9	−0.0343184
$989$	3.71399e10	1.22083
$990$	−1.41131e10	−0.462273
$991$	−4.50663e10	−1.47094	−0.735469	−	0.677558i	$-0.763038\pi$
$991$	−4.50663e10	−1.47094	−0.735469	+	0.677558i	$0.763038\pi$
$992$	3.73444e9	0.121460
$993$	7.96218e9	0.258053
$994$	1.15811e10	0.374021
$995$	1.77002e10	0.569636
$996$	−1.40616e10	−0.450948
$997$	−2.19579e10	−0.701710	−0.350855	−	0.936430i	$-0.614109\pi$
$997$	−2.19579e10	−0.701710	−0.350855	+	0.936430i	$0.614109\pi$
$998$	1.86018e10	0.592377
$999$	2.21986e9	0.0704445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.i.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.i.1.1	✓	5	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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -428.929 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -428.929 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +3431.43 q^{10} -5641.82 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -11581.1 q^{15} +4096.00 q^{16} +5030.39 q^{17} -5832.00 q^{18} -7398.89 q^{19} -27451.5 q^{20} +9261.00 q^{21} +45134.5 q^{22} +72906.4 q^{23} -13824.0 q^{24} +105855. q^{25} -17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} -59340.1 q^{29} +92648.7 q^{30} -113966. q^{31} -32768.0 q^{32} -152329. q^{33} -40243.1 q^{34} -147123. q^{35} +46656.0 q^{36} +112781. q^{37} +59191.2 q^{38} +59319.0 q^{39} +219612. q^{40} +224080. q^{41} -74088.0 q^{42} +509419. q^{43} -361076. q^{44} -312689. q^{45} -583251. q^{46} -454823. q^{47} +110592. q^{48} +117649. q^{49} -846841. q^{50} +135820. q^{51} +140608. q^{52} +869626. q^{53} -157464. q^{54} +2.41994e6 q^{55} -175616. q^{56} -199770. q^{57} +474721. q^{58} +1.63913e6 q^{59} -741189. q^{60} +2.14907e6 q^{61} +911729. q^{62} +250047. q^{63} +262144. q^{64} -942357. q^{65} +1.21863e6 q^{66} +55040.3 q^{67} +321945. q^{68} +1.96847e6 q^{69} +1.17698e6 q^{70} -4.22051e6 q^{71} -373248. q^{72} +1.72232e6 q^{73} -902247. q^{74} +2.85809e6 q^{75} -473529. q^{76} -1.93514e6 q^{77} -474552. q^{78} -3.59169e6 q^{79} -1.75689e6 q^{80} +531441. q^{81} -1.79264e6 q^{82} -8.13749e6 q^{83} +592704. q^{84} -2.15768e6 q^{85} -4.07536e6 q^{86} -1.60218e6 q^{87} +2.88861e6 q^{88} -4.86198e6 q^{89} +2.50151e6 q^{90} +753571. q^{91} +4.66601e6 q^{92} -3.07708e6 q^{93} +3.63859e6 q^{94} +3.17360e6 q^{95} -884736. q^{96} +6.88065e6 q^{97} -941192. q^{98} -4.11288e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 - 340 * q^5 - 1080 * q^6 + 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} - 340 q^{5} - 1080 q^{6} + 1715 q^{7} - 2560 q^{8} + 3645 q^{9} + 2720 q^{10} - 1303 q^{11} + 8640 q^{12} + 10985 q^{13} - 13720 q^{14} - 9180 q^{15} + 20480 q^{16} - 4247 q^{17} - 29160 q^{18} - 16984 q^{19} - 21760 q^{20} + 46305 q^{21} + 10424 q^{22} - 78072 q^{23} - 69120 q^{24} - 79555 q^{25} - 87880 q^{26} + 98415 q^{27} + 109760 q^{28} - 213142 q^{29} + 73440 q^{30} - 186027 q^{31} - 163840 q^{32} - 35181 q^{33} + 33976 q^{34} - 116620 q^{35} + 233280 q^{36} + 101025 q^{37} + 135872 q^{38} + 296595 q^{39} + 174080 q^{40} - 23976 q^{41} - 370440 q^{42} - 55528 q^{43} - 83392 q^{44} - 247860 q^{45} + 624576 q^{46} - 985981 q^{47} + 552960 q^{48} + 588245 q^{49} + 636440 q^{50} - 114669 q^{51} + 703040 q^{52} - 1891657 q^{53} - 787320 q^{54} + 1746955 q^{55} - 878080 q^{56} - 458568 q^{57} + 1705136 q^{58} - 2802208 q^{59} - 587520 q^{60} + 1140591 q^{61} + 1488216 q^{62} + 1250235 q^{63} + 1310720 q^{64} - 746980 q^{65} + 281448 q^{66} + 265168 q^{67} - 271808 q^{68} - 2107944 q^{69} + 932960 q^{70} - 4483276 q^{71} - 1866240 q^{72} - 2350578 q^{73} - 808200 q^{74} - 2147985 q^{75} - 1086976 q^{76} - 446929 q^{77} - 2372760 q^{78} - 4079889 q^{79} - 1392640 q^{80} + 2657205 q^{81} + 191808 q^{82} - 8731571 q^{83} + 2963520 q^{84} - 1715895 q^{85} + 444224 q^{86} - 5754834 q^{87} + 667136 q^{88} - 20077879 q^{89} + 1982880 q^{90} + 3767855 q^{91} - 4996608 q^{92} - 5022729 q^{93} + 7887848 q^{94} - 11580740 q^{95} - 4423680 q^{96} + 3780209 q^{97} - 4705960 q^{98} - 949887 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 - 340 * q^5 - 1080 * q^6 + 1715 * q^7 - 2560 * q^8 + 3645 * q^9 + 2720 * q^10 - 1303 * q^11 + 8640 * q^12 + 10985 * q^13 - 13720 * q^14 - 9180 * q^15 + 20480 * q^16 - 4247 * q^17 - 29160 * q^18 - 16984 * q^19 - 21760 * q^20 + 46305 * q^21 + 10424 * q^22 - 78072 * q^23 - 69120 * q^24 - 79555 * q^25 - 87880 * q^26 + 98415 * q^27 + 109760 * q^28 - 213142 * q^29 + 73440 * q^30 - 186027 * q^31 - 163840 * q^32 - 35181 * q^33 + 33976 * q^34 - 116620 * q^35 + 233280 * q^36 + 101025 * q^37 + 135872 * q^38 + 296595 * q^39 + 174080 * q^40 - 23976 * q^41 - 370440 * q^42 - 55528 * q^43 - 83392 * q^44 - 247860 * q^45 + 624576 * q^46 - 985981 * q^47 + 552960 * q^48 + 588245 * q^49 + 636440 * q^50 - 114669 * q^51 + 703040 * q^52 - 1891657 * q^53 - 787320 * q^54 + 1746955 * q^55 - 878080 * q^56 - 458568 * q^57 + 1705136 * q^58 - 2802208 * q^59 - 587520 * q^60 + 1140591 * q^61 + 1488216 * q^62 + 1250235 * q^63 + 1310720 * q^64 - 746980 * q^65 + 281448 * q^66 + 265168 * q^67 - 271808 * q^68 - 2107944 * q^69 + 932960 * q^70 - 4483276 * q^71 - 1866240 * q^72 - 2350578 * q^73 - 808200 * q^74 - 2147985 * q^75 - 1086976 * q^76 - 446929 * q^77 - 2372760 * q^78 - 4079889 * q^79 - 1392640 * q^80 + 2657205 * q^81 + 191808 * q^82 - 8731571 * q^83 + 2963520 * q^84 - 1715895 * q^85 + 444224 * q^86 - 5754834 * q^87 + 667136 * q^88 - 20077879 * q^89 + 1982880 * q^90 + 3767855 * q^91 - 4996608 * q^92 - 5022729 * q^93 + 7887848 * q^94 - 11580740 * q^95 - 4423680 * q^96 + 3780209 * q^97 - 4705960 * q^98 - 949887 * q^99

Introduction

L-functions

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