LMFDB - Embedded newform 546.8.a.j.1.4

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} - 5672x^{3} - 117684x^{2} + 1695035x + 39011360 $ x^5 - 5672x^3 - 117684x^2 + 1695035*x + 39011360
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-21.9822$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +249.727 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +249.727 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -1997.82 q^{10} +3233.99 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +6742.63 q^{15} +4096.00 q^{16} +9015.75 q^{17} -5832.00 q^{18} -35945.0 q^{19} +15982.5 q^{20} -9261.00 q^{21} -25871.9 q^{22} -38564.1 q^{23} -13824.0 q^{24} -15761.4 q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +68241.3 q^{29} -53941.0 q^{30} +118835. q^{31} -32768.0 q^{32} +87317.8 q^{33} -72126.0 q^{34} -85656.4 q^{35} +46656.0 q^{36} -590446. q^{37} +287560. q^{38} -59319.0 q^{39} -127860. q^{40} +414223. q^{41} +74088.0 q^{42} +47634.0 q^{43} +206975. q^{44} +182051. q^{45} +308513. q^{46} -347512. q^{47} +110592. q^{48} +117649. q^{49} +126091. q^{50} +243425. q^{51} -140608. q^{52} -547089. q^{53} -157464. q^{54} +807615. q^{55} +175616. q^{56} -970514. q^{57} -545931. q^{58} +995629. q^{59} +431528. q^{60} -2.70586e6 q^{61} -950680. q^{62} -250047. q^{63} +262144. q^{64} -548650. q^{65} -698542. q^{66} -3.28358e6 q^{67} +577008. q^{68} -1.04123e6 q^{69} +685251. q^{70} -5.43613e6 q^{71} -373248. q^{72} -618715. q^{73} +4.72357e6 q^{74} -425558. q^{75} -2.30048e6 q^{76} -1.10926e6 q^{77} +474552. q^{78} -6.29147e6 q^{79} +1.02288e6 q^{80} +531441. q^{81} -3.31378e6 q^{82} +5.10103e6 q^{83} -592704. q^{84} +2.25148e6 q^{85} -381072. q^{86} +1.84252e6 q^{87} -1.65580e6 q^{88} +6.51911e6 q^{89} -1.45641e6 q^{90} +753571. q^{91} -2.46810e6 q^{92} +3.20854e6 q^{93} +2.78010e6 q^{94} -8.97643e6 q^{95} -884736. q^{96} +9.09144e6 q^{97} -941192. q^{98} +2.35758e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 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 + 56 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 448 q^{10} - 3679 q^{11} + 8640 q^{12} - 10985 q^{13} + 13720 q^{14} + 1512 q^{15} + 20480 q^{16} + 409 q^{17} - 29160 q^{18} + 33730 q^{19} + 3584 q^{20} - 46305 q^{21} + 29432 q^{22} - 142142 q^{23} - 69120 q^{24} + 153981 q^{25} + 87880 q^{26} + 98415 q^{27} - 109760 q^{28} + 88028 q^{29} - 12096 q^{30} + 244543 q^{31} - 163840 q^{32} - 99333 q^{33} - 3272 q^{34} - 19208 q^{35} + 233280 q^{36} + 730963 q^{37} - 269840 q^{38} - 296595 q^{39} - 28672 q^{40} + 479512 q^{41} + 370440 q^{42} - 406536 q^{43} - 235456 q^{44} + 40824 q^{45} + 1137136 q^{46} + 1138945 q^{47} + 552960 q^{48} + 588245 q^{49} - 1231848 q^{50} + 11043 q^{51} - 703040 q^{52} + 297595 q^{53} - 787320 q^{54} - 1834423 q^{55} + 878080 q^{56} + 910710 q^{57} - 704224 q^{58} + 941652 q^{59} + 96768 q^{60} - 2985259 q^{61} - 1956344 q^{62} - 1250235 q^{63} + 1310720 q^{64} - 123032 q^{65} + 794664 q^{66} - 2333504 q^{67} + 26176 q^{68} - 3837834 q^{69} + 153664 q^{70} - 11322272 q^{71} - 1866240 q^{72} - 6631604 q^{73} - 5847704 q^{74} + 4157487 q^{75} + 2158720 q^{76} + 1261897 q^{77} + 2372760 q^{78} - 10600265 q^{79} + 229376 q^{80} + 2657205 q^{81} - 3836096 q^{82} - 2425229 q^{83} - 2963520 q^{84} - 12267705 q^{85} + 3252288 q^{86} + 2376756 q^{87} + 1883648 q^{88} - 1581837 q^{89} - 326592 q^{90} + 3767855 q^{91} - 9097088 q^{92} + 6602661 q^{93} - 9111560 q^{94} - 11507718 q^{95} - 4423680 q^{96} + 5298407 q^{97} - 4705960 q^{98} - 2681991 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 56 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9 - 448 * q^10 - 3679 * q^11 + 8640 * q^12 - 10985 * q^13 + 13720 * q^14 + 1512 * q^15 + 20480 * q^16 + 409 * q^17 - 29160 * q^18 + 33730 * q^19 + 3584 * q^20 - 46305 * q^21 + 29432 * q^22 - 142142 * q^23 - 69120 * q^24 + 153981 * q^25 + 87880 * q^26 + 98415 * q^27 - 109760 * q^28 + 88028 * q^29 - 12096 * q^30 + 244543 * q^31 - 163840 * q^32 - 99333 * q^33 - 3272 * q^34 - 19208 * q^35 + 233280 * q^36 + 730963 * q^37 - 269840 * q^38 - 296595 * q^39 - 28672 * q^40 + 479512 * q^41 + 370440 * q^42 - 406536 * q^43 - 235456 * q^44 + 40824 * q^45 + 1137136 * q^46 + 1138945 * q^47 + 552960 * q^48 + 588245 * q^49 - 1231848 * q^50 + 11043 * q^51 - 703040 * q^52 + 297595 * q^53 - 787320 * q^54 - 1834423 * q^55 + 878080 * q^56 + 910710 * q^57 - 704224 * q^58 + 941652 * q^59 + 96768 * q^60 - 2985259 * q^61 - 1956344 * q^62 - 1250235 * q^63 + 1310720 * q^64 - 123032 * q^65 + 794664 * q^66 - 2333504 * q^67 + 26176 * q^68 - 3837834 * q^69 + 153664 * q^70 - 11322272 * q^71 - 1866240 * q^72 - 6631604 * q^73 - 5847704 * q^74 + 4157487 * q^75 + 2158720 * q^76 + 1261897 * q^77 + 2372760 * q^78 - 10600265 * q^79 + 229376 * q^80 + 2657205 * q^81 - 3836096 * q^82 - 2425229 * q^83 - 2963520 * q^84 - 12267705 * q^85 + 3252288 * q^86 + 2376756 * q^87 + 1883648 * q^88 - 1581837 * q^89 - 326592 * q^90 + 3767855 * q^91 - 9097088 * q^92 + 6602661 * q^93 - 9111560 * q^94 - 11507718 * q^95 - 4423680 * q^96 + 5298407 * q^97 - 4705960 * q^98 - 2681991 * 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
$5$	249.727	0.893451	0.446725	−	0.894671i	$-0.352590\pi$
$5$	249.727	0.893451	0.446725	+	0.894671i	$0.352590\pi$
$6$	−216.000	−0.408248
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	−1997.82	−0.631765
$11$	3233.99	0.732596	0.366298	−	0.930498i	$-0.380625\pi$
$11$	3233.99	0.732596	0.366298	+	0.930498i	$0.380625\pi$
$12$	1728.00	0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	2744.00	0.267261
$15$	6742.63	0.515834
$16$	4096.00	0.250000
$17$	9015.75	0.445072	0.222536	−	0.974924i	$-0.428566\pi$
$17$	9015.75	0.445072	0.222536	+	0.974924i	$0.428566\pi$
$18$	−5832.00	−0.235702
$19$	−35945.0	−1.20227	−0.601133	−	0.799149i	$-0.705284\pi$
$19$	−35945.0	−1.20227	−0.601133	+	0.799149i	$0.705284\pi$
$20$	15982.5	0.446725
$21$	−9261.00	−0.218218
$22$	−25871.9	−0.518023
$23$	−38564.1	−0.660899	−0.330450	−	0.943824i	$-0.607200\pi$
$23$	−38564.1	−0.660899	−0.330450	+	0.943824i	$0.607200\pi$
$24$	−13824.0	−0.204124
$25$	−15761.4	−0.201746
$26$	17576.0	0.196116
$27$	19683.0	0.192450
$28$	−21952.0	−0.188982
$29$	68241.3	0.519582	0.259791	−	0.965665i	$-0.416346\pi$
$29$	68241.3	0.519582	0.259791	+	0.965665i	$0.416346\pi$
$30$	−53941.0	−0.364750
$31$	118835.	0.716438	0.358219	−	0.933638i	$-0.383384\pi$
$31$	118835.	0.716438	0.358219	+	0.933638i	$0.383384\pi$
$32$	−32768.0	−0.176777
$33$	87317.8	0.422964
$34$	−72126.0	−0.314714
$35$	−85656.4	−0.337693
$36$	46656.0	0.166667
$37$	−590446.	−1.91635	−0.958174	−	0.286186i	$-0.907612\pi$
$37$	−590446.	−1.91635	−0.958174	+	0.286186i	$0.907612\pi$
$38$	287560.	0.850130
$39$	−59319.0	−0.160128
$40$	−127860.	−0.315882
$41$	414223.	0.938621	0.469311	−	0.883033i	$-0.344503\pi$
$41$	414223.	0.938621	0.469311	+	0.883033i	$0.344503\pi$
$42$	74088.0	0.154303
$43$	47634.0	0.0913645	0.0456822	−	0.998956i	$-0.485454\pi$
$43$	47634.0	0.0913645	0.0456822	+	0.998956i	$0.485454\pi$
$44$	206975.	0.366298
$45$	182051.	0.297817
$46$	308513.	0.467326
$47$	−347512.	−0.488233	−0.244117	−	0.969746i	$-0.578498\pi$
$47$	−347512.	−0.488233	−0.244117	+	0.969746i	$0.578498\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	126091.	0.142656
$51$	243425.	0.256963
$52$	−140608.	−0.138675
$53$	−547089.	−0.504769	−0.252385	−	0.967627i	$-0.581215\pi$
$53$	−547089.	−0.504769	−0.252385	+	0.967627i	$0.581215\pi$
$54$	−157464.	−0.136083
$55$	807615.	0.654538
$56$	175616.	0.133631
$57$	−970514.	−0.694128
$58$	−545931.	−0.367400
$59$	995629.	0.631125	0.315563	−	0.948905i	$-0.397807\pi$
$59$	995629.	0.631125	0.315563	+	0.948905i	$0.397807\pi$
$60$	431528.	0.257917
$61$	−2.70586e6	−1.52634	−0.763168	−	0.646200i	$-0.776357\pi$
$61$	−2.70586e6	−1.52634	−0.763168	+	0.646200i	$0.776357\pi$
$62$	−950680.	−0.506598
$63$	−250047.	−0.125988
$64$	262144.	0.125000
$65$	−548650.	−0.247799
$66$	−698542.	−0.299081
$67$	−3.28358e6	−1.33379	−0.666893	−	0.745154i	$-0.732376\pi$
$67$	−3.28358e6	−1.33379	−0.666893	+	0.745154i	$0.732376\pi$
$68$	577008.	0.222536
$69$	−1.04123e6	−0.381570
$70$	685251.	0.238785
$71$	−5.43613e6	−1.80254	−0.901272	−	0.433253i	$-0.857366\pi$
$71$	−5.43613e6	−1.80254	−0.901272	+	0.433253i	$0.857366\pi$
$72$	−373248.	−0.117851
$73$	−618715.	−0.186149	−0.0930744	−	0.995659i	$-0.529669\pi$
$73$	−618715.	−0.186149	−0.0930744	+	0.995659i	$0.529669\pi$
$74$	4.72357e6	1.35506
$75$	−425558.	−0.116478
$76$	−2.30048e6	−0.601133
$77$	−1.10926e6	−0.276895
$78$	474552.	0.113228
$79$	−6.29147e6	−1.43568	−0.717839	−	0.696209i	$-0.754869\pi$
$79$	−6.29147e6	−1.43568	−0.717839	+	0.696209i	$0.754869\pi$
$80$	1.02288e6	0.223363
$81$	531441.	0.111111
$82$	−3.31378e6	−0.663705
$83$	5.10103e6	0.979230	0.489615	−	0.871939i	$-0.337137\pi$
$83$	5.10103e6	0.979230	0.489615	+	0.871939i	$0.337137\pi$
$84$	−592704.	−0.109109
$85$	2.25148e6	0.397650
$86$	−381072.	−0.0646045
$87$	1.84252e6	0.299981
$88$	−1.65580e6	−0.259012
$89$	6.51911e6	0.980220	0.490110	−	0.871661i	$-0.336957\pi$
$89$	6.51911e6	0.980220	0.490110	+	0.871661i	$0.336957\pi$
$90$	−1.45641e6	−0.210588
$91$	753571.	0.104828
$92$	−2.46810e6	−0.330450
$93$	3.20854e6	0.413635
$94$	2.78010e6	0.345233
$95$	−8.97643e6	−1.07416
$96$	−884736.	−0.102062
$97$	9.09144e6	1.01142	0.505710	−	0.862703i	$-0.331231\pi$
$97$	9.09144e6	1.01142	0.505710	+	0.862703i	$0.331231\pi$
$98$	−941192.	−0.101015
$99$	2.35758e6	0.244199
$100$	−1.00873e6	−0.100873
$101$	−1.64041e7	−1.58426	−0.792130	−	0.610353i	$-0.791028\pi$
$101$	−1.64041e7	−1.58426	−0.792130	+	0.610353i	$0.791028\pi$
$102$	−1.94740e6	−0.181700
$103$	−3.47734e6	−0.313557	−0.156778	−	0.987634i	$-0.550111\pi$
$103$	−3.47734e6	−0.313557	−0.156778	+	0.987634i	$0.550111\pi$
$104$	1.12486e6	0.0980581
$105$	−2.31272e6	−0.194967
$106$	4.37672e6	0.356926
$107$	9.49661e6	0.749420	0.374710	−	0.927142i	$-0.377742\pi$
$107$	9.49661e6	0.749420	0.374710	+	0.927142i	$0.377742\pi$
$108$	1.25971e6	0.0962250
$109$	9.41138e6	0.696082	0.348041	−	0.937479i	$-0.386847\pi$
$109$	9.41138e6	0.696082	0.348041	+	0.937479i	$0.386847\pi$
$110$	−6.46092e6	−0.462828
$111$	−1.59421e7	−1.10640
$112$	−1.40493e6	−0.0944911
$113$	1.22161e7	0.796448	0.398224	−	0.917288i	$-0.369627\pi$
$113$	1.22161e7	0.796448	0.398224	+	0.917288i	$0.369627\pi$
$114$	7.76411e6	0.490823
$115$	−9.63049e6	−0.590481
$116$	4.36744e6	0.259791
$117$	−1.60161e6	−0.0924500
$118$	−7.96503e6	−0.446273
$119$	−3.09240e6	−0.168222
$120$	−3.45223e6	−0.182375
$121$	−9.02847e6	−0.463303
$122$	2.16469e7	1.07928
$123$	1.11840e7	0.541913
$124$	7.60544e6	0.358219
$125$	−2.34460e7	−1.07370
$126$	2.00038e6	0.0890871
$127$	3.04755e7	1.32019	0.660097	−	0.751180i	$-0.270515\pi$
$127$	3.04755e7	1.32019	0.660097	+	0.751180i	$0.270515\pi$
$128$	−2.09715e6	−0.0883883
$129$	1.28612e6	0.0527493
$130$	4.38920e6	0.175220
$131$	1.52623e7	0.593159	0.296580	−	0.955008i	$-0.404154\pi$
$131$	1.52623e7	0.593159	0.296580	+	0.955008i	$0.404154\pi$
$132$	5.58834e6	0.211482
$133$	1.23291e7	0.454414
$134$	2.62687e7	0.943129
$135$	4.91538e6	0.171945
$136$	−4.61606e6	−0.157357
$137$	5.04710e6	0.167695	0.0838475	−	0.996479i	$-0.473279\pi$
$137$	5.04710e6	0.167695	0.0838475	+	0.996479i	$0.473279\pi$
$138$	8.32984e6	0.269811
$139$	−2.25126e7	−0.711007	−0.355504	−	0.934675i	$-0.615691\pi$
$139$	−2.25126e7	−0.711007	−0.355504	+	0.934675i	$0.615691\pi$
$140$	−5.48201e6	−0.168846
$141$	−9.38283e6	−0.281882
$142$	4.34891e7	1.27459
$143$	−7.10508e6	−0.203186
$144$	2.98598e6	0.0833333
$145$	1.70417e7	0.464221
$146$	4.94972e6	0.131627
$147$	3.17652e6	0.0824786
$148$	−3.77886e7	−0.958174
$149$	−3.32626e7	−0.823766	−0.411883	−	0.911237i	$-0.635129\pi$
$149$	−3.32626e7	−0.823766	−0.411883	+	0.911237i	$0.635129\pi$
$150$	3.40446e6	0.0823625
$151$	−3.66037e7	−0.865179	−0.432590	−	0.901591i	$-0.642400\pi$
$151$	−3.66037e7	−0.865179	−0.432590	+	0.901591i	$0.642400\pi$
$152$	1.84038e7	0.425065
$153$	6.57248e6	0.148357
$154$	8.87407e6	0.195794
$155$	2.96763e7	0.640102
$156$	−3.79642e6	−0.0800641
$157$	−4.81360e6	−0.0992708	−0.0496354	−	0.998767i	$-0.515806\pi$
$157$	−4.81360e6	−0.0992708	−0.0496354	+	0.998767i	$0.515806\pi$
$158$	5.03317e7	1.01518
$159$	−1.47714e7	−0.291429
$160$	−8.18306e6	−0.157941
$161$	1.32275e7	0.249796
$162$	−4.25153e6	−0.0785674
$163$	1.76309e7	0.318873	0.159437	−	0.987208i	$-0.449032\pi$
$163$	1.76309e7	0.318873	0.159437	+	0.987208i	$0.449032\pi$
$164$	2.65102e7	0.469311
$165$	2.18056e7	0.377898
$166$	−4.08083e7	−0.692420
$167$	5.12049e7	0.850754	0.425377	−	0.905016i	$-0.360141\pi$
$167$	5.12049e7	0.850754	0.425377	+	0.905016i	$0.360141\pi$
$168$	4.74163e6	0.0771517
$169$	4.82681e6	0.0769231
$170$	−1.80118e7	−0.281181
$171$	−2.62039e7	−0.400755
$172$	3.04858e6	0.0456822
$173$	−6.62195e6	−0.0972355	−0.0486177	−	0.998817i	$-0.515482\pi$
$173$	−6.62195e6	−0.0972355	−0.0486177	+	0.998817i	$0.515482\pi$
$174$	−1.47401e7	−0.212119
$175$	5.40616e6	0.0762528
$176$	1.32464e7	0.183149
$177$	2.68820e7	0.364380
$178$	−5.21529e7	−0.693120
$179$	281431.	0.00366764	0.00183382	−	0.999998i	$-0.499416\pi$
$179$	281431.	0.00366764	0.00183382	+	0.999998i	$0.499416\pi$
$180$	1.16513e7	0.148908
$181$	1.25701e8	1.57567	0.787835	−	0.615886i	$-0.211202\pi$
$181$	1.25701e8	1.57567	0.787835	+	0.615886i	$0.211202\pi$
$182$	−6.02857e6	−0.0741249
$183$	−7.30581e7	−0.881231
$184$	1.97448e7	0.233663
$185$	−1.47450e8	−1.71216
$186$	−2.56684e7	−0.292484
$187$	2.91569e7	0.326058
$188$	−2.22408e7	−0.244117
$189$	−6.75127e6	−0.0727393
$190$	7.18114e7	0.759549
$191$	−1.56684e7	−0.162708	−0.0813538	−	0.996685i	$-0.525924\pi$
$191$	−1.56684e7	−0.162708	−0.0813538	+	0.996685i	$0.525924\pi$
$192$	7.07789e6	0.0721688
$193$	−9.89310e6	−0.0990563	−0.0495281	−	0.998773i	$-0.515772\pi$
$193$	−9.89310e6	−0.0990563	−0.0495281	+	0.998773i	$0.515772\pi$
$194$	−7.27315e7	−0.715182
$195$	−1.48136e7	−0.143067
$196$	7.52954e6	0.0714286
$197$	−7.26858e7	−0.677357	−0.338679	−	0.940902i	$-0.609980\pi$
$197$	−7.26858e7	−0.677357	−0.338679	+	0.940902i	$0.609980\pi$
$198$	−1.88606e7	−0.172674
$199$	−1.22132e8	−1.09861	−0.549303	−	0.835623i	$-0.685107\pi$
$199$	−1.22132e8	−1.09861	−0.549303	+	0.835623i	$0.685107\pi$
$200$	8.06984e6	0.0713280
$201$	−8.86567e7	−0.770061
$202$	1.31232e8	1.12024
$203$	−2.34068e7	−0.196384
$204$	1.55792e7	0.128481
$205$	1.03443e8	0.838612
$206$	2.78187e7	0.221718
$207$	−2.81132e7	−0.220300
$208$	−8.99891e6	−0.0693375
$209$	−1.16246e8	−0.880775
$210$	1.85018e7	0.137862
$211$	−7.04312e7	−0.516151	−0.258075	−	0.966125i	$-0.583088\pi$
$211$	−7.04312e7	−0.516151	−0.258075	+	0.966125i	$0.583088\pi$
$212$	−3.50137e7	−0.252385
$213$	−1.46776e8	−1.04070
$214$	−7.59729e7	−0.529920
$215$	1.18955e7	0.0816297
$216$	−1.00777e7	−0.0680414
$217$	−4.07604e7	−0.270788
$218$	−7.52910e7	−0.492204
$219$	−1.67053e7	−0.107473
$220$	5.16874e7	0.327269
$221$	−1.98076e7	−0.123441
$222$	1.27536e8	0.782346
$223$	−4.01526e7	−0.242463	−0.121232	−	0.992624i	$-0.538684\pi$
$223$	−4.01526e7	−0.242463	−0.121232	+	0.992624i	$0.538684\pi$
$224$	1.12394e7	0.0668153
$225$	−1.14901e7	−0.0672487
$226$	−9.77287e7	−0.563174
$227$	4.39233e7	0.249232	0.124616	−	0.992205i	$-0.460230\pi$
$227$	4.39233e7	0.249232	0.124616	+	0.992205i	$0.460230\pi$
$228$	−6.21129e7	−0.347064
$229$	7.28347e7	0.400788	0.200394	−	0.979715i	$-0.435778\pi$
$229$	7.28347e7	0.400788	0.200394	+	0.979715i	$0.435778\pi$
$230$	7.70439e7	0.417533
$231$	−2.99500e7	−0.159866
$232$	−3.49396e7	−0.183700
$233$	2.54186e8	1.31646	0.658228	−	0.752819i	$-0.271306\pi$
$233$	2.54186e8	1.31646	0.658228	+	0.752819i	$0.271306\pi$
$234$	1.28129e7	0.0653720
$235$	−8.67832e7	−0.436212
$236$	6.37203e7	0.315563
$237$	−1.69870e8	−0.828889
$238$	2.47392e7	0.118951
$239$	−2.59103e8	−1.22767	−0.613833	−	0.789436i	$-0.710373\pi$
$239$	−2.59103e8	−1.22767	−0.613833	+	0.789436i	$0.710373\pi$
$240$	2.76178e7	0.128958
$241$	−3.73599e8	−1.71928	−0.859639	−	0.510903i	$-0.829311\pi$
$241$	−3.73599e8	−1.71928	−0.859639	+	0.510903i	$0.829311\pi$
$242$	7.22278e7	0.327605
$243$	1.43489e7	0.0641500
$244$	−1.73175e8	−0.763168
$245$	2.93801e7	0.127636
$246$	−8.94721e7	−0.383190
$247$	7.89711e7	0.333448
$248$	−6.08435e7	−0.253299
$249$	1.37728e8	0.565359
$250$	1.87568e8	0.759221
$251$	3.26441e8	1.30301	0.651503	−	0.758646i	$-0.274139\pi$
$251$	3.26441e8	1.30301	0.651503	+	0.758646i	$0.274139\pi$
$252$	−1.60030e7	−0.0629941
$253$	−1.24716e8	−0.484172
$254$	−2.43804e8	−0.933518
$255$	6.07899e7	0.229583
$256$	1.67772e7	0.0625000
$257$	−4.20813e8	−1.54640	−0.773202	−	0.634160i	$-0.781346\pi$
$257$	−4.20813e8	−1.54640	−0.773202	+	0.634160i	$0.781346\pi$
$258$	−1.02889e7	−0.0372994
$259$	2.02523e8	0.724312
$260$	−3.51136e7	−0.123899
$261$	4.97479e7	0.173194
$262$	−1.22099e8	−0.419427
$263$	−4.68224e8	−1.58711	−0.793557	−	0.608496i	$-0.791773\pi$
$263$	−4.68224e8	−1.58711	−0.793557	+	0.608496i	$0.791773\pi$
$264$	−4.47067e7	−0.149541
$265$	−1.36623e8	−0.450986
$266$	−9.86330e7	−0.321319
$267$	1.76016e8	0.565930
$268$	−2.10149e8	−0.666893
$269$	−1.75612e8	−0.550076	−0.275038	−	0.961433i	$-0.588690\pi$
$269$	−1.75612e8	−0.550076	−0.275038	+	0.961433i	$0.588690\pi$
$270$	−3.93230e7	−0.121583
$271$	1.22913e8	0.375150	0.187575	−	0.982250i	$-0.439937\pi$
$271$	1.22913e8	0.375150	0.187575	+	0.982250i	$0.439937\pi$
$272$	3.69285e7	0.111268
$273$	2.03464e7	0.0605228
$274$	−4.03768e7	−0.118578
$275$	−5.09723e7	−0.147798
$276$	−6.66387e7	−0.190785
$277$	3.46024e8	0.978197	0.489099	−	0.872228i	$-0.337326\pi$
$277$	3.46024e8	0.978197	0.489099	+	0.872228i	$0.337326\pi$
$278$	1.80101e8	0.502758
$279$	8.66307e7	0.238813
$280$	4.38561e7	0.119392
$281$	−1.60442e8	−0.431367	−0.215683	−	0.976463i	$-0.569198\pi$
$281$	−1.60442e8	−0.431367	−0.215683	+	0.976463i	$0.569198\pi$
$282$	7.50626e7	0.199320
$283$	−4.63629e8	−1.21596	−0.607979	−	0.793953i	$-0.708019\pi$
$283$	−4.63629e8	−1.21596	−0.607979	+	0.793953i	$0.708019\pi$
$284$	−3.47913e8	−0.901272
$285$	−2.42364e8	−0.620169
$286$	5.68406e7	0.143674
$287$	−1.42078e8	−0.354765
$288$	−2.38879e7	−0.0589256
$289$	−3.29055e8	−0.801911
$290$	−1.36334e8	−0.328254
$291$	2.45469e8	0.583944
$292$	−3.95977e7	−0.0930744
$293$	−7.31152e7	−0.169813	−0.0849066	−	0.996389i	$-0.527059\pi$
$293$	−7.31152e7	−0.169813	−0.0849066	+	0.996389i	$0.527059\pi$
$294$	−2.54122e7	−0.0583212
$295$	2.48635e8	0.563879
$296$	3.02309e8	0.677531
$297$	6.36547e7	0.140988
$298$	2.66101e8	0.582491
$299$	8.47252e7	0.183301
$300$	−2.72357e7	−0.0582391
$301$	−1.63385e7	−0.0345325
$302$	2.92830e8	0.611774
$303$	−4.42909e8	−0.914673
$304$	−1.47231e8	−0.300566
$305$	−6.75726e8	−1.36371
$306$	−5.25798e7	−0.104905
$307$	−7.31387e8	−1.44266	−0.721328	−	0.692594i	$-0.756468\pi$
$307$	−7.31387e8	−1.44266	−0.721328	+	0.692594i	$0.756468\pi$
$308$	−7.09926e7	−0.138448
$309$	−9.38881e7	−0.181032
$310$	−2.37410e8	−0.452620
$311$	−9.69470e8	−1.82757	−0.913783	−	0.406203i	$-0.866853\pi$
$311$	−9.69470e8	−1.82757	−0.913783	+	0.406203i	$0.866853\pi$
$312$	3.03713e7	0.0566139
$313$	7.36168e8	1.35697	0.678487	−	0.734612i	$-0.262636\pi$
$313$	7.36168e8	1.35697	0.678487	+	0.734612i	$0.262636\pi$
$314$	3.85088e7	0.0701950
$315$	−6.24435e7	−0.112564
$316$	−4.02654e8	−0.717839
$317$	−1.27245e8	−0.224355	−0.112177	−	0.993688i	$-0.535782\pi$
$317$	−1.27245e8	−0.224355	−0.112177	+	0.993688i	$0.535782\pi$
$318$	1.18171e8	0.206071
$319$	2.20692e8	0.380644
$320$	6.54644e7	0.111681
$321$	2.56409e8	0.432678
$322$	−1.05820e8	−0.176633
$323$	−3.24071e8	−0.535095
$324$	3.40122e7	0.0555556
$325$	3.46278e7	0.0559543
$326$	−1.41047e8	−0.225477
$327$	2.54107e8	0.401883
$328$	−2.12082e8	−0.331853
$329$	1.19197e8	0.184535
$330$	−1.74445e8	−0.267214
$331$	1.22358e8	0.185452	0.0927262	−	0.995692i	$-0.470442\pi$
$331$	1.22358e8	0.185452	0.0927262	+	0.995692i	$0.470442\pi$
$332$	3.26466e8	0.489615
$333$	−4.30435e8	−0.638783
$334$	−4.09640e8	−0.601574
$335$	−8.19999e8	−1.19167
$336$	−3.79331e7	−0.0545545
$337$	−1.06041e8	−0.150928	−0.0754639	−	0.997149i	$-0.524044\pi$
$337$	−1.06041e8	−0.150928	−0.0754639	+	0.997149i	$0.524044\pi$
$338$	−3.86145e7	−0.0543928
$339$	3.29834e8	0.459830
$340$	1.44094e8	0.198825
$341$	3.84311e8	0.524859
$342$	2.09631e8	0.283377
$343$	−4.03536e7	−0.0539949
$344$	−2.43886e7	−0.0323022
$345$	−2.60023e8	−0.340914
$346$	5.29756e7	0.0687559
$347$	−1.27113e9	−1.63319	−0.816596	−	0.577210i	$-0.804141\pi$
$347$	−1.27113e9	−1.63319	−0.816596	+	0.577210i	$0.804141\pi$
$348$	1.17921e8	0.149991
$349$	1.17229e9	1.47621	0.738103	−	0.674688i	$-0.235722\pi$
$349$	1.17229e9	1.47621	0.738103	+	0.674688i	$0.235722\pi$
$350$	−4.32493e7	−0.0539189
$351$	−4.32436e7	−0.0533761
$352$	−1.05971e8	−0.129506
$353$	9.12868e8	1.10458	0.552289	−	0.833653i	$-0.313754\pi$
$353$	9.12868e8	1.10458	0.552289	+	0.833653i	$0.313754\pi$
$354$	−2.15056e8	−0.257656
$355$	−1.35755e9	−1.61048
$356$	4.17223e8	0.490110
$357$	−8.34948e7	−0.0971227
$358$	−2.25145e6	−0.00259341
$359$	−9.38327e8	−1.07035	−0.535173	−	0.844743i	$-0.679753\pi$
$359$	−9.38327e8	−1.07035	−0.535173	+	0.844743i	$0.679753\pi$
$360$	−9.32101e7	−0.105294
$361$	3.98168e8	0.445442
$362$	−1.00561e9	−1.11417
$363$	−2.43769e8	−0.267488
$364$	4.82285e7	0.0524142
$365$	−1.54510e8	−0.166315
$366$	5.84465e8	0.623124
$367$	1.09083e8	0.115193	0.0575967	−	0.998340i	$-0.481656\pi$
$367$	1.09083e8	0.115193	0.0575967	+	0.998340i	$0.481656\pi$
$368$	−1.57958e8	−0.165225
$369$	3.01968e8	0.312874
$370$	1.17960e9	1.21068
$371$	1.87652e8	0.190785
$372$	2.05347e8	0.206818
$373$	−6.53066e8	−0.651593	−0.325796	−	0.945440i	$-0.605632\pi$
$373$	−6.53066e8	−0.651593	−0.325796	+	0.945440i	$0.605632\pi$
$374$	−2.33255e8	−0.230558
$375$	−6.33041e8	−0.619901
$376$	1.77926e8	0.172617
$377$	−1.49926e8	−0.144106
$378$	5.40102e7	0.0514344
$379$	−2.65598e8	−0.250604	−0.125302	−	0.992119i	$-0.539990\pi$
$379$	−2.65598e8	−0.250604	−0.125302	+	0.992119i	$0.539990\pi$
$380$	−5.74491e8	−0.537082
$381$	8.22839e8	0.762215
$382$	1.25347e8	0.115052
$383$	−1.11095e9	−1.01041	−0.505205	−	0.863000i	$-0.668583\pi$
$383$	−1.11095e9	−1.01041	−0.505205	+	0.863000i	$0.668583\pi$
$384$	−5.66231e7	−0.0510310
$385$	−2.77012e8	−0.247392
$386$	7.91448e7	0.0700434
$387$	3.47252e7	0.0304548
$388$	5.81852e8	0.505710
$389$	−1.13929e9	−0.981323	−0.490661	−	0.871350i	$-0.663245\pi$
$389$	−1.13929e9	−0.981323	−0.490661	+	0.871350i	$0.663245\pi$
$390$	1.18508e8	0.101163
$391$	−3.47684e8	−0.294148
$392$	−6.02363e7	−0.0505076
$393$	4.12083e8	0.342461
$394$	5.81487e8	0.478964
$395$	−1.57115e9	−1.28271
$396$	1.50885e8	0.122099
$397$	−1.23511e9	−0.990690	−0.495345	−	0.868696i	$-0.664958\pi$
$397$	−1.23511e9	−0.990690	−0.495345	+	0.868696i	$0.664958\pi$
$398$	9.77053e8	0.776832
$399$	3.32886e8	0.262356
$400$	−6.45587e7	−0.0504365
$401$	−1.44292e9	−1.11747	−0.558736	−	0.829346i	$-0.688713\pi$
$401$	−1.44292e9	−1.11747	−0.558736	+	0.829346i	$0.688713\pi$
$402$	7.09254e8	0.544516
$403$	−2.61080e8	−0.198704
$404$	−1.04986e9	−0.792130
$405$	1.32715e8	0.0992723
$406$	1.87254e8	0.138864
$407$	−1.90950e9	−1.40391
$408$	−1.24634e8	−0.0908500
$409$	−1.83879e9	−1.32892	−0.664462	−	0.747322i	$-0.731339\pi$
$409$	−1.83879e9	−1.32892	−0.664462	+	0.747322i	$0.731339\pi$
$410$	−8.27541e8	−0.592988
$411$	1.36272e8	0.0968188
$412$	−2.22550e8	−0.156778
$413$	−3.41501e8	−0.238543
$414$	2.24906e8	0.155775
$415$	1.27387e9	0.874894
$416$	7.19913e7	0.0490290
$417$	−6.07840e8	−0.410500
$418$	9.29965e8	0.622802
$419$	−1.01820e9	−0.676215	−0.338107	−	0.941108i	$-0.609787\pi$
$419$	−1.01820e9	−0.676215	−0.338107	+	0.941108i	$0.609787\pi$
$420$	−1.48014e8	−0.0974835
$421$	−5.82161e8	−0.380238	−0.190119	−	0.981761i	$-0.560887\pi$
$421$	−5.82161e8	−0.380238	−0.190119	+	0.981761i	$0.560887\pi$
$422$	5.63450e8	0.364974
$423$	−2.53336e8	−0.162744
$424$	2.80110e8	0.178463
$425$	−1.42101e8	−0.0897916
$426$	1.17421e9	0.735886
$427$	9.28109e8	0.576901
$428$	6.07783e8	0.374710
$429$	−1.91837e8	−0.117309
$430$	−9.51640e7	−0.0577209
$431$	1.89524e8	0.114023	0.0570117	−	0.998374i	$-0.481843\pi$
$431$	1.89524e8	0.114023	0.0570117	+	0.998374i	$0.481843\pi$
$432$	8.06216e7	0.0481125
$433$	2.17028e9	1.28472	0.642359	−	0.766404i	$-0.277956\pi$
$433$	2.17028e9	1.28472	0.642359	+	0.766404i	$0.277956\pi$
$434$	3.26083e8	0.191476
$435$	4.60126e8	0.268018
$436$	6.02328e8	0.348041
$437$	1.38618e9	0.794577
$438$	1.33642e8	0.0759950
$439$	5.12061e8	0.288866	0.144433	−	0.989515i	$-0.453864\pi$
$439$	5.12061e8	0.288866	0.144433	+	0.989515i	$0.453864\pi$
$440$	−4.13499e8	−0.231414
$441$	8.57661e7	0.0476190
$442$	1.58461e8	0.0872859
$443$	9.82509e8	0.536937	0.268469	−	0.963288i	$-0.413482\pi$
$443$	9.82509e8	0.536937	0.268469	+	0.963288i	$0.413482\pi$
$444$	−1.02029e9	−0.553202
$445$	1.62800e9	0.875778
$446$	3.21221e8	0.171448
$447$	−8.98090e8	−0.475602
$448$	−8.99154e7	−0.0472456
$449$	−2.04698e9	−1.06721	−0.533607	−	0.845732i	$-0.679164\pi$
$449$	−2.04698e9	−1.06721	−0.533607	+	0.845732i	$0.679164\pi$
$450$	9.19205e7	0.0475520
$451$	1.33959e9	0.687630
$452$	7.81830e8	0.398224
$453$	−9.88301e8	−0.499511
$454$	−3.51386e8	−0.176234
$455$	1.88187e8	0.0936591
$456$	4.96903e8	0.245411
$457$	2.14052e9	1.04909	0.524544	−	0.851383i	$-0.324236\pi$
$457$	2.14052e9	1.04909	0.524544	+	0.851383i	$0.324236\pi$
$458$	−5.82678e8	−0.283400
$459$	1.77457e8	0.0856542
$460$	−6.16351e8	−0.295240
$461$	1.84663e9	0.877862	0.438931	−	0.898521i	$-0.355357\pi$
$461$	1.84663e9	0.877862	0.438931	+	0.898521i	$0.355357\pi$
$462$	2.39600e8	0.113042
$463$	2.49284e9	1.16724	0.583622	−	0.812026i	$-0.301635\pi$
$463$	2.49284e9	1.16724	0.583622	+	0.812026i	$0.301635\pi$
$464$	2.79516e8	0.129896
$465$	8.01260e8	0.369563
$466$	−2.03349e9	−0.930874
$467$	−5.56531e8	−0.252860	−0.126430	−	0.991976i	$-0.540352\pi$
$467$	−5.56531e8	−0.252860	−0.126430	+	0.991976i	$0.540352\pi$
$468$	−1.02503e8	−0.0462250
$469$	1.12627e9	0.504124
$470$	6.94265e8	0.308449
$471$	−1.29967e8	−0.0573140
$472$	−5.09762e8	−0.223136
$473$	1.54048e8	0.0669333
$474$	1.35896e9	0.586113
$475$	5.66543e8	0.242552
$476$	−1.97914e8	−0.0841108
$477$	−3.98828e8	−0.168256
$478$	2.07283e9	0.868091
$479$	4.38428e8	0.182274	0.0911368	−	0.995838i	$-0.470950\pi$
$479$	4.38428e8	0.182274	0.0911368	+	0.995838i	$0.470950\pi$
$480$	−2.20943e8	−0.0911874
$481$	1.29721e9	0.531499
$482$	2.98879e9	1.21571
$483$	3.57142e8	0.144220
$484$	−5.77822e8	−0.231652
$485$	2.27038e9	0.903654
$486$	−1.14791e8	−0.0453609
$487$	−1.05882e9	−0.415402	−0.207701	−	0.978192i	$-0.566598\pi$
$487$	−1.05882e9	−0.415402	−0.207701	+	0.978192i	$0.566598\pi$
$488$	1.38540e9	0.539642
$489$	4.76034e8	0.184101
$490$	−2.35041e8	−0.0902521
$491$	5.19176e8	0.197938	0.0989690	−	0.995091i	$-0.468446\pi$
$491$	5.19176e8	0.197938	0.0989690	+	0.995091i	$0.468446\pi$
$492$	7.15777e8	0.270957
$493$	6.15247e8	0.231252
$494$	−6.31769e8	−0.235784
$495$	5.88751e8	0.218179
$496$	4.86748e8	0.179109
$497$	1.86459e9	0.681298
$498$	−1.10182e9	−0.399769
$499$	1.25064e8	0.0450590	0.0225295	−	0.999746i	$-0.492828\pi$
$499$	1.25064e8	0.0450590	0.0225295	+	0.999746i	$0.492828\pi$
$500$	−1.50054e9	−0.536850
$501$	1.38253e9	0.491183
$502$	−2.61153e9	−0.921364
$503$	2.84960e9	0.998380	0.499190	−	0.866492i	$-0.333631\pi$
$503$	2.84960e9	0.998380	0.499190	+	0.866492i	$0.333631\pi$
$504$	1.28024e8	0.0445435
$505$	−4.09654e9	−1.41546
$506$	9.97727e8	0.342361
$507$	1.30324e8	0.0444116
$508$	1.95043e9	0.660097
$509$	4.26498e9	1.43352	0.716761	−	0.697319i	$-0.245624\pi$
$509$	4.26498e9	1.43352	0.716761	+	0.697319i	$0.245624\pi$
$510$	−4.86319e8	−0.162340
$511$	2.12219e8	0.0703577
$512$	−1.34218e8	−0.0441942
$513$	−7.07505e8	−0.231376
$514$	3.36650e9	1.09347
$515$	−8.68385e8	−0.280148
$516$	8.23116e7	0.0263747
$517$	−1.12385e9	−0.357678
$518$	−1.62018e9	−0.512166
$519$	−1.78793e8	−0.0561389
$520$	2.80909e8	0.0876100
$521$	−6.08449e8	−0.188492	−0.0942458	−	0.995549i	$-0.530044\pi$
$521$	−6.08449e8	−0.188492	−0.0942458	+	0.995549i	$0.530044\pi$
$522$	−3.97983e8	−0.122467
$523$	−5.94178e9	−1.81619	−0.908093	−	0.418768i	$-0.862462\pi$
$523$	−5.94178e9	−1.81619	−0.908093	+	0.418768i	$0.862462\pi$
$524$	9.76789e8	0.296580
$525$	1.45966e8	0.0440246
$526$	3.74579e9	1.12226
$527$	1.07139e9	0.318867
$528$	3.57654e8	0.105741
$529$	−1.91764e9	−0.563212
$530$	1.09298e9	0.318895
$531$	7.25813e8	0.210375
$532$	7.89064e8	0.227207
$533$	−9.10047e8	−0.260327
$534$	−1.40813e9	−0.400173
$535$	2.37156e9	0.669570
$536$	1.68119e9	0.471564
$537$	7.59864e6	0.00211751
$538$	1.40490e9	0.388962
$539$	3.80476e8	0.104657
$540$	3.14584e8	0.0859723
$541$	−1.89699e9	−0.515081	−0.257541	−	0.966267i	$-0.582912\pi$
$541$	−1.89699e9	−0.515081	−0.257541	+	0.966267i	$0.582912\pi$
$542$	−9.83303e8	−0.265271
$543$	3.39394e9	0.909714
$544$	−2.95428e8	−0.0786784
$545$	2.35027e9	0.621915
$546$	−1.62771e8	−0.0427960
$547$	3.82199e9	0.998468	0.499234	−	0.866467i	$-0.333615\pi$
$547$	3.82199e9	0.998468	0.499234	+	0.866467i	$0.333615\pi$
$548$	3.23015e8	0.0838475
$549$	−1.97257e9	−0.508779
$550$	4.07778e8	0.104509
$551$	−2.45293e9	−0.624676
$552$	5.33110e8	0.134906
$553$	2.15797e9	0.542635
$554$	−2.76819e9	−0.691690
$555$	−3.98116e9	−0.988517
$556$	−1.44081e9	−0.355504
$557$	5.09004e9	1.24804	0.624019	−	0.781409i	$-0.285499\pi$
$557$	5.09004e9	1.24804	0.624019	+	0.781409i	$0.285499\pi$
$558$	−6.93046e8	−0.168866
$559$	−1.04652e8	−0.0253400
$560$	−3.50849e8	−0.0844231
$561$	7.87235e8	0.188250
$562$	1.28354e9	0.305022
$563$	−6.69908e9	−1.58211	−0.791053	−	0.611748i	$-0.790467\pi$
$563$	−6.69908e9	−1.58211	−0.791053	+	0.611748i	$0.790467\pi$
$564$	−6.00501e8	−0.140941
$565$	3.05069e9	0.711587
$566$	3.70904e9	0.859812
$567$	−1.82284e8	−0.0419961
$568$	2.78330e9	0.637296
$569$	−1.92335e9	−0.437689	−0.218845	−	0.975760i	$-0.570229\pi$
$569$	−1.92335e9	−0.437689	−0.218845	+	0.975760i	$0.570229\pi$
$570$	1.93891e9	0.438526
$571$	4.72517e9	1.06216	0.531081	−	0.847321i	$-0.321786\pi$
$571$	4.72517e9	1.06216	0.531081	+	0.847321i	$0.321786\pi$
$572$	−4.54725e8	−0.101593
$573$	−4.23047e8	−0.0939393
$574$	1.13663e9	0.250857
$575$	6.07824e8	0.133334
$576$	1.91103e8	0.0416667
$577$	7.32990e9	1.58848	0.794241	−	0.607602i	$-0.207869\pi$
$577$	7.32990e9	1.58848	0.794241	+	0.607602i	$0.207869\pi$
$578$	2.63244e9	0.567036
$579$	−2.67114e8	−0.0571902
$580$	1.09067e9	0.232111
$581$	−1.74965e9	−0.370114
$582$	−1.96375e9	−0.412911
$583$	−1.76928e9	−0.369792
$584$	3.16782e8	0.0658136
$585$	−3.99966e8	−0.0825995
$586$	5.84922e8	0.120076
$587$	6.87279e9	1.40249	0.701245	−	0.712921i	$-0.252628\pi$
$587$	6.87279e9	1.40249	0.701245	+	0.712921i	$0.252628\pi$
$588$	2.03297e8	0.0412393
$589$	−4.27152e9	−0.861348
$590$	−1.98908e9	−0.398723
$591$	−1.96252e9	−0.391073
$592$	−2.41847e9	−0.479087
$593$	−4.80630e9	−0.946497	−0.473249	−	0.880929i	$-0.656919\pi$
$593$	−4.80630e9	−0.946497	−0.473249	+	0.880929i	$0.656919\pi$
$594$	−5.09237e8	−0.0996937
$595$	−7.72256e8	−0.150298
$596$	−2.12881e9	−0.411883
$597$	−3.29755e9	−0.634281
$598$	−6.77802e8	−0.129613
$599$	−1.59452e9	−0.303134	−0.151567	−	0.988447i	$-0.548432\pi$
$599$	−1.59452e9	−0.303134	−0.151567	+	0.988447i	$0.548432\pi$
$600$	2.17886e8	0.0411812
$601$	6.26679e9	1.17756	0.588782	−	0.808292i	$-0.299608\pi$
$601$	6.26679e9	1.17756	0.588782	+	0.808292i	$0.299608\pi$
$602$	1.30708e8	0.0244182
$603$	−2.39373e9	−0.444595
$604$	−2.34264e9	−0.432590
$605$	−2.25465e9	−0.413939
$606$	3.54327e9	0.646771
$607$	2.80434e9	0.508944	0.254472	−	0.967080i	$-0.418098\pi$
$607$	2.80434e9	0.508944	0.254472	+	0.967080i	$0.418098\pi$
$608$	1.17784e9	0.212533
$609$	−6.31983e8	−0.113382
$610$	5.40580e9	0.964286
$611$	7.63484e8	0.135412
$612$	4.20639e8	0.0741787
$613$	3.19476e9	0.560178	0.280089	−	0.959974i	$-0.409636\pi$
$613$	3.19476e9	0.560178	0.280089	+	0.959974i	$0.409636\pi$
$614$	5.85109e9	1.02011
$615$	2.79295e9	0.484173
$616$	5.67941e8	0.0978972
$617$	−4.24149e9	−0.726977	−0.363488	−	0.931599i	$-0.618414\pi$
$617$	−4.24149e9	−0.726977	−0.363488	+	0.931599i	$0.618414\pi$
$618$	7.51105e8	0.128009
$619$	5.76481e9	0.976939	0.488470	−	0.872581i	$-0.337555\pi$
$619$	5.76481e9	0.976939	0.488470	+	0.872581i	$0.337555\pi$
$620$	1.89928e9	0.320051
$621$	−7.59056e8	−0.127190
$622$	7.75576e9	1.29228
$623$	−2.23606e9	−0.370488
$624$	−2.42971e8	−0.0400320
$625$	−4.62373e9	−0.757553
$626$	−5.88934e9	−0.959526
$627$	−3.13863e9	−0.508516
$628$	−3.08071e8	−0.0496354
$629$	−5.32332e9	−0.852913
$630$	4.99548e8	0.0795949
$631$	−2.32202e9	−0.367929	−0.183964	−	0.982933i	$-0.558893\pi$
$631$	−2.32202e9	−0.367929	−0.183964	+	0.982933i	$0.558893\pi$
$632$	3.22123e9	0.507589
$633$	−1.90164e9	−0.298000
$634$	1.01796e9	0.158643
$635$	7.61056e9	1.17953
$636$	−9.45371e8	−0.145714
$637$	−2.58475e8	−0.0396214
$638$	−1.76553e9	−0.269156
$639$	−3.96294e9	−0.600848
$640$	−5.23716e8	−0.0789706
$641$	−1.21454e10	−1.82141	−0.910706	−	0.413055i	$-0.864462\pi$
$641$	−1.21454e10	−1.82141	−0.910706	+	0.413055i	$0.864462\pi$
$642$	−2.05127e9	−0.305950
$643$	4.30565e9	0.638705	0.319353	−	0.947636i	$-0.396535\pi$
$643$	4.30565e9	0.638705	0.319353	+	0.947636i	$0.396535\pi$
$644$	8.46558e8	0.124898
$645$	3.21178e8	0.0471289
$646$	2.59257e9	0.378369
$647$	5.95403e9	0.864263	0.432131	−	0.901811i	$-0.357762\pi$
$647$	5.95403e9	0.864263	0.432131	+	0.901811i	$0.357762\pi$
$648$	−2.72098e8	−0.0392837
$649$	3.21986e9	0.462360
$650$	−2.77022e8	−0.0395656
$651$	−1.10053e9	−0.156339
$652$	1.12838e9	0.159437
$653$	2.52831e9	0.355332	0.177666	−	0.984091i	$-0.443145\pi$
$653$	2.52831e9	0.355332	0.177666	+	0.984091i	$0.443145\pi$
$654$	−2.03286e9	−0.284174
$655$	3.81142e9	0.529958
$656$	1.69666e9	0.234655
$657$	−4.51043e8	−0.0620496
$658$	−9.53573e8	−0.130486
$659$	1.19368e10	1.62475	0.812376	−	0.583134i	$-0.198174\pi$
$659$	1.19368e10	1.62475	0.812376	+	0.583134i	$0.198174\pi$
$660$	1.39556e9	0.188949
$661$	−1.39313e10	−1.87624	−0.938119	−	0.346313i	$-0.887433\pi$
$661$	−1.39313e10	−1.87624	−0.938119	+	0.346313i	$0.887433\pi$
$662$	−9.78860e8	−0.131135
$663$	−5.34805e8	−0.0712686
$664$	−2.61173e9	−0.346210
$665$	3.07891e9	0.405996
$666$	3.44348e9	0.451688
$667$	−2.63166e9	−0.343392
$668$	3.27712e9	0.425377
$669$	−1.08412e9	−0.139986
$670$	6.55999e9	0.842639
$671$	−8.75072e9	−1.11819
$672$	3.03464e8	0.0385758
$673$	−6.16910e9	−0.780134	−0.390067	−	0.920787i	$-0.627548\pi$
$673$	−6.16910e9	−0.780134	−0.390067	+	0.920787i	$0.627548\pi$
$674$	8.48328e8	0.106722
$675$	−3.10232e8	−0.0388260
$676$	3.08916e8	0.0384615
$677$	−1.17438e10	−1.45461	−0.727306	−	0.686314i	$-0.759228\pi$
$677$	−1.17438e10	−1.45461	−0.727306	+	0.686314i	$0.759228\pi$
$678$	−2.63867e9	−0.325149
$679$	−3.11836e9	−0.382281
$680$	−1.15276e9	−0.140591
$681$	1.18593e9	0.143894
$682$	−3.07449e9	−0.371131
$683$	4.19103e9	0.503325	0.251662	−	0.967815i	$-0.419023\pi$
$683$	4.19103e9	0.503325	0.251662	+	0.967815i	$0.419023\pi$
$684$	−1.67705e9	−0.200378
$685$	1.26040e9	0.149827
$686$	3.22829e8	0.0381802
$687$	1.96654e9	0.231395
$688$	1.95109e8	0.0228411
$689$	1.20196e9	0.139998
$690$	2.08019e9	0.241063
$691$	−4.69866e9	−0.541752	−0.270876	−	0.962614i	$-0.587313\pi$
$691$	−4.69866e9	−0.541752	−0.270876	+	0.962614i	$0.587313\pi$
$692$	−4.23805e8	−0.0486177
$693$	−8.08650e8	−0.0922984
$694$	1.01690e10	1.15484
$695$	−5.62201e9	−0.635250
$696$	−9.43368e8	−0.106059
$697$	3.73453e9	0.417754
$698$	−9.37834e9	−1.04384
$699$	6.86303e9	0.760056
$700$	3.45994e8	0.0381264
$701$	3.88240e9	0.425683	0.212842	−	0.977087i	$-0.431728\pi$
$701$	3.88240e9	0.425683	0.212842	+	0.977087i	$0.431728\pi$
$702$	3.45948e8	0.0377426
$703$	2.12236e10	2.30396
$704$	8.47771e8	0.0915745
$705$	−2.34315e9	−0.251847
$706$	−7.30294e9	−0.781055
$707$	5.62659e9	0.598794
$708$	1.72045e9	0.182190
$709$	9.11818e9	0.960830	0.480415	−	0.877041i	$-0.340486\pi$
$709$	9.11818e9	0.960830	0.480415	+	0.877041i	$0.340486\pi$
$710$	1.08604e10	1.13878
$711$	−4.58648e9	−0.478560
$712$	−3.33779e9	−0.346560
$713$	−4.58276e9	−0.473493
$714$	6.67959e8	0.0686761
$715$	−1.77433e9	−0.181536
$716$	1.80116e7	0.00183382
$717$	−6.99578e9	−0.708793
$718$	7.50662e9	0.756848
$719$	−1.22833e10	−1.23243	−0.616216	−	0.787577i	$-0.711335\pi$
$719$	−1.22833e10	−1.23243	−0.616216	+	0.787577i	$0.711335\pi$
$720$	7.45681e8	0.0744542
$721$	1.19273e9	0.118513
$722$	−3.18535e9	−0.314975
$723$	−1.00872e10	−0.992625
$724$	8.04489e9	0.787835
$725$	−1.07558e9	−0.104824
$726$	1.95015e9	0.189143
$727$	4.84128e8	0.0467293	0.0233647	−	0.999727i	$-0.492562\pi$
$727$	4.84128e8	0.0467293	0.0233647	+	0.999727i	$0.492562\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	1.23608e9	0.117602
$731$	4.29456e8	0.0406638
$732$	−4.67572e9	−0.440615
$733$	−7.90496e9	−0.741371	−0.370686	−	0.928758i	$-0.620877\pi$
$733$	−7.90496e9	−0.741371	−0.370686	+	0.928758i	$0.620877\pi$
$734$	−8.72667e8	−0.0814540
$735$	7.93264e8	0.0736906
$736$	1.26367e9	0.116832
$737$	−1.06191e10	−0.977126
$738$	−2.41575e9	−0.221235
$739$	4.35082e9	0.396566	0.198283	−	0.980145i	$-0.436463\pi$
$739$	4.35082e9	0.396566	0.198283	+	0.980145i	$0.436463\pi$
$740$	−9.43683e9	−0.856081
$741$	2.13222e9	0.192517
$742$	−1.50121e9	−0.134905
$743$	2.89756e9	0.259162	0.129581	−	0.991569i	$-0.458637\pi$
$743$	2.89756e9	0.259162	0.129581	+	0.991569i	$0.458637\pi$
$744$	−1.64277e9	−0.146242
$745$	−8.30657e9	−0.735995
$746$	5.22453e9	0.460746
$747$	3.71865e9	0.326410
$748$	1.86604e9	0.163029
$749$	−3.25734e9	−0.283254
$750$	5.06433e9	0.438336
$751$	−2.15345e10	−1.85522	−0.927609	−	0.373554i	$-0.878139\pi$
$751$	−2.15345e10	−1.85522	−0.927609	+	0.373554i	$0.878139\pi$
$752$	−1.42341e9	−0.122058
$753$	8.81390e9	0.752290
$754$	1.19941e9	0.101898
$755$	−9.14094e9	−0.772995
$756$	−4.32081e8	−0.0363696
$757$	1.64305e10	1.37663	0.688313	−	0.725414i	$-0.258351\pi$
$757$	1.64305e10	1.37663	0.688313	+	0.725414i	$0.258351\pi$
$758$	2.12479e9	0.177204
$759$	−3.36733e9	−0.279537
$760$	4.59593e9	0.379775
$761$	1.79037e10	1.47264	0.736319	−	0.676635i	$-0.236562\pi$
$761$	1.79037e10	1.47264	0.736319	+	0.676635i	$0.236562\pi$
$762$	−6.58271e9	−0.538967
$763$	−3.22810e9	−0.263094
$764$	−1.00278e9	−0.0813538
$765$	1.64133e9	0.132550
$766$	8.88757e9	0.714467
$767$	−2.18740e9	−0.175043
$768$	4.52985e8	0.0360844
$769$	5.11934e9	0.405949	0.202974	−	0.979184i	$-0.434939\pi$
$769$	5.11934e9	0.405949	0.202974	+	0.979184i	$0.434939\pi$
$770$	2.21610e9	0.174933
$771$	−1.13619e10	−0.892817
$772$	−6.33159e8	−0.0495281
$773$	1.08379e10	0.843954	0.421977	−	0.906606i	$-0.361336\pi$
$773$	1.08379e10	0.843954	0.421977	+	0.906606i	$0.361336\pi$
$774$	−2.77802e8	−0.0215348
$775$	−1.87301e9	−0.144538
$776$	−4.65482e9	−0.357591
$777$	5.46812e9	0.418181
$778$	9.11434e9	0.693900
$779$	−1.48892e10	−1.12847
$780$	−9.48068e8	−0.0715333
$781$	−1.75804e10	−1.32054
$782$	2.78147e9	0.207994
$783$	1.34319e9	0.0999937
$784$	4.81890e8	0.0357143
$785$	−1.20209e9	−0.0886935
$786$	−3.29666e9	−0.242156
$787$	−3.56535e8	−0.0260730	−0.0130365	−	0.999915i	$-0.504150\pi$
$787$	−3.56535e8	−0.0260730	−0.0130365	+	0.999915i	$0.504150\pi$
$788$	−4.65189e9	−0.338679
$789$	−1.26420e10	−0.916321
$790$	1.25692e10	0.907011
$791$	−4.19012e9	−0.301029
$792$	−1.20708e9	−0.0863372
$793$	5.94477e9	0.423330
$794$	9.88085e9	0.700524
$795$	−3.68882e9	−0.260377
$796$	−7.81642e9	−0.549303
$797$	−1.99101e10	−1.39306	−0.696528	−	0.717530i	$-0.745273\pi$
$797$	−1.99101e10	−1.39306	−0.696528	+	0.717530i	$0.745273\pi$
$798$	−2.66309e9	−0.185514
$799$	−3.13308e9	−0.217299
$800$	5.16470e8	0.0356640
$801$	4.75243e9	0.326740
$802$	1.15433e10	0.790172
$803$	−2.00092e9	−0.136372
$804$	−5.67403e9	−0.385031
$805$	3.30326e9	0.223181
$806$	2.08864e9	0.140505
$807$	−4.74154e9	−0.317586
$808$	8.39887e9	0.560120
$809$	−2.43083e10	−1.61411	−0.807057	−	0.590474i	$-0.798941\pi$
$809$	−2.43083e10	−1.61411	−0.807057	+	0.590474i	$0.798941\pi$
$810$	−1.06172e9	−0.0701961
$811$	6.52755e9	0.429712	0.214856	−	0.976646i	$-0.431072\pi$
$811$	6.52755e9	0.429712	0.214856	+	0.976646i	$0.431072\pi$
$812$	−1.49803e9	−0.0981918
$813$	3.31865e9	0.216593
$814$	1.52760e10	0.992713
$815$	4.40291e9	0.284897
$816$	9.97070e8	0.0642407
$817$	−1.71220e9	−0.109844
$818$	1.47103e10	0.939691
$819$	5.49353e8	0.0349428
$820$	6.62033e9	0.419306
$821$	−5.20602e9	−0.328326	−0.164163	−	0.986433i	$-0.552492\pi$
$821$	−5.20602e9	−0.328326	−0.164163	+	0.986433i	$0.552492\pi$
$822$	−1.09017e9	−0.0684612
$823$	1.61413e10	1.00934	0.504671	−	0.863311i	$-0.331614\pi$
$823$	1.61413e10	1.00934	0.504671	+	0.863311i	$0.331614\pi$
$824$	1.78040e9	0.110859
$825$	−1.37625e9	−0.0853314
$826$	2.73201e9	0.168675
$827$	1.20070e10	0.738182	0.369091	−	0.929393i	$-0.379669\pi$
$827$	1.20070e10	0.738182	0.369091	+	0.929393i	$0.379669\pi$
$828$	−1.79924e9	−0.110150
$829$	1.61936e10	0.987192	0.493596	−	0.869691i	$-0.335682\pi$
$829$	1.61936e10	0.987192	0.493596	+	0.869691i	$0.335682\pi$
$830$	−1.01909e10	−0.618643
$831$	9.34264e9	0.564763
$832$	−5.75930e8	−0.0346688
$833$	1.06069e9	0.0635818
$834$	4.86272e9	0.290267
$835$	1.27873e10	0.760107
$836$	−7.43972e9	−0.440387
$837$	2.33903e9	0.137878
$838$	8.14561e9	0.478156
$839$	1.91601e10	1.12003	0.560017	−	0.828481i	$-0.310795\pi$
$839$	1.91601e10	1.12003	0.560017	+	0.828481i	$0.310795\pi$
$840$	1.18411e9	0.0689312
$841$	−1.25930e10	−0.730034
$842$	4.65729e9	0.268869
$843$	−4.33194e9	−0.249050
$844$	−4.50760e9	−0.258075
$845$	1.20538e9	0.0687270
$846$	2.02669e9	0.115078
$847$	3.09677e9	0.175112
$848$	−2.24088e9	−0.126192
$849$	−1.25180e10	−0.702033
$850$	1.13681e9	0.0634922
$851$	2.27700e10	1.26651
$852$	−9.39364e9	−0.520350
$853$	−8.97603e9	−0.495179	−0.247590	−	0.968865i	$-0.579639\pi$
$853$	−8.97603e9	−0.495179	−0.247590	+	0.968865i	$0.579639\pi$
$854$	−7.42487e9	−0.407931
$855$	−6.54382e9	−0.358055
$856$	−4.86227e9	−0.264960
$857$	1.85639e10	1.00748	0.503740	−	0.863855i	$-0.331957\pi$
$857$	1.85639e10	1.00748	0.503740	+	0.863855i	$0.331957\pi$
$858$	1.53470e9	0.0829501
$859$	1.30357e10	0.701709	0.350855	−	0.936430i	$-0.385891\pi$
$859$	1.30357e10	0.701709	0.350855	+	0.936430i	$0.385891\pi$
$860$	7.61312e8	0.0408148
$861$	−3.83612e9	−0.204824
$862$	−1.51619e9	−0.0806268
$863$	4.32991e8	0.0229320	0.0114660	−	0.999934i	$-0.496350\pi$
$863$	4.32991e8	0.0229320	0.0114660	+	0.999934i	$0.496350\pi$
$864$	−6.44973e8	−0.0340207
$865$	−1.65368e9	−0.0868751
$866$	−1.73622e10	−0.908433
$867$	−8.88448e9	−0.462983
$868$	−2.60867e9	−0.135394
$869$	−2.03466e10	−1.05177
$870$	−3.68101e9	−0.189518
$871$	7.21403e9	0.369926
$872$	−4.81862e9	−0.246102
$873$	6.62766e9	0.337140
$874$	−1.10895e10	−0.561850
$875$	8.04197e9	0.405821
$876$	−1.06914e9	−0.0537365
$877$	1.50599e10	0.753915	0.376957	−	0.926231i	$-0.376970\pi$
$877$	1.50599e10	0.753915	0.376957	+	0.926231i	$0.376970\pi$
$878$	−4.09649e9	−0.204259
$879$	−1.97411e9	−0.0980417
$880$	3.30799e9	0.163635
$881$	3.03610e10	1.49589	0.747947	−	0.663758i	$-0.231040\pi$
$881$	3.03610e10	1.49589	0.747947	+	0.663758i	$0.231040\pi$
$882$	−6.86129e8	−0.0336718
$883$	−2.74514e10	−1.34184	−0.670921	−	0.741529i	$-0.734101\pi$
$883$	−2.74514e10	−1.34184	−0.670921	+	0.741529i	$0.734101\pi$
$884$	−1.26769e9	−0.0617204
$885$	6.71316e9	0.325556
$886$	−7.86007e9	−0.379672
$887$	−2.30605e10	−1.10952	−0.554762	−	0.832009i	$-0.687191\pi$
$887$	−2.30605e10	−1.10952	−0.554762	+	0.832009i	$0.687191\pi$
$888$	8.16233e9	0.391173
$889$	−1.04531e10	−0.498987
$890$	−1.30240e10	−0.619268
$891$	1.71868e9	0.0813995
$892$	−2.56976e9	−0.121232
$893$	1.24913e10	0.586986
$894$	7.18472e9	0.336301
$895$	7.02809e7	0.00327685
$896$	7.19323e8	0.0334077
$897$	2.28758e9	0.105829
$898$	1.63759e10	0.754635
$899$	8.10946e9	0.372248
$900$	−7.35364e8	−0.0336243
$901$	−4.93242e9	−0.224659
$902$	−1.07167e10	−0.486228
$903$	−4.41138e8	−0.0199374
$904$	−6.25464e9	−0.281587
$905$	3.13911e10	1.40778
$906$	7.90641e9	0.353208
$907$	1.74751e10	0.777667	0.388834	−	0.921308i	$-0.372878\pi$
$907$	1.74751e10	0.777667	0.388834	+	0.921308i	$0.372878\pi$
$908$	2.81109e9	0.124616
$909$	−1.19586e10	−0.528086
$910$	−1.50550e9	−0.0662270
$911$	−3.45096e10	−1.51226	−0.756128	−	0.654424i	$-0.772911\pi$
$911$	−3.45096e10	−1.51226	−0.756128	+	0.654424i	$0.772911\pi$
$912$	−3.97522e9	−0.173532
$913$	1.64967e10	0.717380
$914$	−1.71241e10	−0.741817
$915$	−1.82446e10	−0.787336
$916$	4.66142e9	0.200394
$917$	−5.23498e9	−0.224193
$918$	−1.41966e9	−0.0605667
$919$	−2.03987e10	−0.866957	−0.433478	−	0.901164i	$-0.642714\pi$
$919$	−2.03987e10	−0.866957	−0.433478	+	0.901164i	$0.642714\pi$
$920$	4.93081e9	0.208767
$921$	−1.97474e10	−0.832918
$922$	−1.47730e10	−0.620742
$923$	1.19432e10	0.499936
$924$	−1.91680e9	−0.0799328
$925$	9.30627e9	0.386616
$926$	−1.99427e10	−0.825366
$927$	−2.53498e9	−0.104519
$928$	−2.23613e9	−0.0918501
$929$	−4.32372e10	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$929$	−4.32372e10	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$930$	−6.41008e9	−0.261320
$931$	−4.22889e9	−0.171752
$932$	1.62679e10	0.658228
$933$	−2.61757e10	−1.05515
$934$	4.45225e9	0.178799
$935$	7.28125e9	0.291317
$936$	8.20026e8	0.0326860
$937$	1.71234e10	0.679989	0.339995	−	0.940427i	$-0.389575\pi$
$937$	1.71234e10	0.679989	0.339995	+	0.940427i	$0.389575\pi$
$938$	−9.01015e9	−0.356469
$939$	1.98765e10	0.783450
$940$	−5.55412e9	−0.218106
$941$	1.40113e10	0.548169	0.274085	−	0.961706i	$-0.411625\pi$
$941$	1.40113e10	0.548169	0.274085	+	0.961706i	$0.411625\pi$
$942$	1.03974e9	0.0405271
$943$	−1.59741e10	−0.620334
$944$	4.07810e9	0.157781
$945$	−1.68597e9	−0.0649890
$946$	−1.23238e9	−0.0473290
$947$	−1.27534e10	−0.487978	−0.243989	−	0.969778i	$-0.578456\pi$
$947$	−1.27534e10	−0.487978	−0.243989	+	0.969778i	$0.578456\pi$
$948$	−1.08717e10	−0.414445
$949$	1.35932e9	0.0516284
$950$	−4.53234e9	−0.171510
$951$	−3.43563e9	−0.129531
$952$	1.58331e9	0.0594753
$953$	−2.20228e10	−0.824227	−0.412114	−	0.911132i	$-0.635209\pi$
$953$	−2.20228e10	−0.824227	−0.412114	+	0.911132i	$0.635209\pi$
$954$	3.19063e9	0.118975
$955$	−3.91282e9	−0.145371
$956$	−1.65826e10	−0.613833
$957$	5.95868e9	0.219765
$958$	−3.50742e9	−0.128887
$959$	−1.73116e9	−0.0633828
$960$	1.76754e9	0.0644792
$961$	−1.33909e10	−0.486717
$962$	−1.03777e10	−0.375827
$963$	6.92303e9	0.249807
$964$	−2.39103e10	−0.859639
$965$	−2.47058e9	−0.0885019
$966$	−2.85713e9	−0.101979
$967$	−1.58385e10	−0.563275	−0.281637	−	0.959521i	$-0.590878\pi$
$967$	−1.58385e10	−0.563275	−0.281637	+	0.959521i	$0.590878\pi$
$968$	4.62258e9	0.163802
$969$	−8.74991e9	−0.308937
$970$	−1.81630e10	−0.638980
$971$	5.83241e9	0.204447	0.102223	−	0.994761i	$-0.467404\pi$
$971$	5.83241e9	0.204447	0.102223	+	0.994761i	$0.467404\pi$
$972$	9.18330e8	0.0320750
$973$	7.72182e9	0.268735
$974$	8.47052e9	0.293734
$975$	9.34951e8	0.0323052
$976$	−1.10832e10	−0.381584
$977$	2.30405e10	0.790426	0.395213	−	0.918589i	$-0.370671\pi$
$977$	2.30405e10	0.790426	0.395213	+	0.918589i	$0.370671\pi$
$978$	−3.80827e9	−0.130179
$979$	2.10828e10	0.718105
$980$	1.88033e9	0.0638179
$981$	6.86089e9	0.232027
$982$	−4.15340e9	−0.139963
$983$	1.46813e10	0.492979	0.246489	−	0.969145i	$-0.420723\pi$
$983$	1.46813e10	0.492979	0.246489	+	0.969145i	$0.420723\pi$
$984$	−5.72621e9	−0.191595
$985$	−1.81516e10	−0.605185
$986$	−4.92197e9	−0.163520
$987$	3.21831e9	0.106541
$988$	5.05415e9	0.166724
$989$	−1.83696e9	−0.0603827
$990$	−4.71001e9	−0.154276
$991$	3.27312e10	1.06833	0.534164	−	0.845381i	$-0.320626\pi$
$991$	3.27312e10	1.06833	0.534164	+	0.845381i	$0.320626\pi$
$992$	−3.89398e9	−0.126649
$993$	3.30365e9	0.107071
$994$	−1.49168e10	−0.481750
$995$	−3.04996e10	−0.981551
$996$	8.81458e9	0.282679
$997$	−5.28138e10	−1.68778	−0.843888	−	0.536520i	$-0.819739\pi$
$997$	−5.28138e10	−1.68778	−0.843888	+	0.536520i	$0.819739\pi$
$998$	−1.00051e9	−0.0318615
$999$	−1.16218e10	−0.368801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.j.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.j.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +249.727 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +249.727 q^{5} -216.000 q^{6} -343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -1997.82 q^{10} +3233.99 q^{11} +1728.00 q^{12} -2197.00 q^{13} +2744.00 q^{14} +6742.63 q^{15} +4096.00 q^{16} +9015.75 q^{17} -5832.00 q^{18} -35945.0 q^{19} +15982.5 q^{20} -9261.00 q^{21} -25871.9 q^{22} -38564.1 q^{23} -13824.0 q^{24} -15761.4 q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +68241.3 q^{29} -53941.0 q^{30} +118835. q^{31} -32768.0 q^{32} +87317.8 q^{33} -72126.0 q^{34} -85656.4 q^{35} +46656.0 q^{36} -590446. q^{37} +287560. q^{38} -59319.0 q^{39} -127860. q^{40} +414223. q^{41} +74088.0 q^{42} +47634.0 q^{43} +206975. q^{44} +182051. q^{45} +308513. q^{46} -347512. q^{47} +110592. q^{48} +117649. q^{49} +126091. q^{50} +243425. q^{51} -140608. q^{52} -547089. q^{53} -157464. q^{54} +807615. q^{55} +175616. q^{56} -970514. q^{57} -545931. q^{58} +995629. q^{59} +431528. q^{60} -2.70586e6 q^{61} -950680. q^{62} -250047. q^{63} +262144. q^{64} -548650. q^{65} -698542. q^{66} -3.28358e6 q^{67} +577008. q^{68} -1.04123e6 q^{69} +685251. q^{70} -5.43613e6 q^{71} -373248. q^{72} -618715. q^{73} +4.72357e6 q^{74} -425558. q^{75} -2.30048e6 q^{76} -1.10926e6 q^{77} +474552. q^{78} -6.29147e6 q^{79} +1.02288e6 q^{80} +531441. q^{81} -3.31378e6 q^{82} +5.10103e6 q^{83} -592704. q^{84} +2.25148e6 q^{85} -381072. q^{86} +1.84252e6 q^{87} -1.65580e6 q^{88} +6.51911e6 q^{89} -1.45641e6 q^{90} +753571. q^{91} -2.46810e6 q^{92} +3.20854e6 q^{93} +2.78010e6 q^{94} -8.97643e6 q^{95} -884736. q^{96} +9.09144e6 q^{97} -941192. q^{98} +2.35758e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 56 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9	\( 5 q - 40 q^{2} + 135 q^{3} + 320 q^{4} + 56 q^{5} - 1080 q^{6} - 1715 q^{7} - 2560 q^{8} + 3645 q^{9} - 448 q^{10} - 3679 q^{11} + 8640 q^{12} - 10985 q^{13} + 13720 q^{14} + 1512 q^{15} + 20480 q^{16} + 409 q^{17} - 29160 q^{18} + 33730 q^{19} + 3584 q^{20} - 46305 q^{21} + 29432 q^{22} - 142142 q^{23} - 69120 q^{24} + 153981 q^{25} + 87880 q^{26} + 98415 q^{27} - 109760 q^{28} + 88028 q^{29} - 12096 q^{30} + 244543 q^{31} - 163840 q^{32} - 99333 q^{33} - 3272 q^{34} - 19208 q^{35} + 233280 q^{36} + 730963 q^{37} - 269840 q^{38} - 296595 q^{39} - 28672 q^{40} + 479512 q^{41} + 370440 q^{42} - 406536 q^{43} - 235456 q^{44} + 40824 q^{45} + 1137136 q^{46} + 1138945 q^{47} + 552960 q^{48} + 588245 q^{49} - 1231848 q^{50} + 11043 q^{51} - 703040 q^{52} + 297595 q^{53} - 787320 q^{54} - 1834423 q^{55} + 878080 q^{56} + 910710 q^{57} - 704224 q^{58} + 941652 q^{59} + 96768 q^{60} - 2985259 q^{61} - 1956344 q^{62} - 1250235 q^{63} + 1310720 q^{64} - 123032 q^{65} + 794664 q^{66} - 2333504 q^{67} + 26176 q^{68} - 3837834 q^{69} + 153664 q^{70} - 11322272 q^{71} - 1866240 q^{72} - 6631604 q^{73} - 5847704 q^{74} + 4157487 q^{75} + 2158720 q^{76} + 1261897 q^{77} + 2372760 q^{78} - 10600265 q^{79} + 229376 q^{80} + 2657205 q^{81} - 3836096 q^{82} - 2425229 q^{83} - 2963520 q^{84} - 12267705 q^{85} + 3252288 q^{86} + 2376756 q^{87} + 1883648 q^{88} - 1581837 q^{89} - 326592 q^{90} + 3767855 q^{91} - 9097088 q^{92} + 6602661 q^{93} - 9111560 q^{94} - 11507718 q^{95} - 4423680 q^{96} + 5298407 q^{97} - 4705960 q^{98} - 2681991 q^{99}+O(q^{100}) \) 5 * q - 40 * q^2 + 135 * q^3 + 320 * q^4 + 56 * q^5 - 1080 * q^6 - 1715 * q^7 - 2560 * q^8 + 3645 * q^9 - 448 * q^10 - 3679 * q^11 + 8640 * q^12 - 10985 * q^13 + 13720 * q^14 + 1512 * q^15 + 20480 * q^16 + 409 * q^17 - 29160 * q^18 + 33730 * q^19 + 3584 * q^20 - 46305 * q^21 + 29432 * q^22 - 142142 * q^23 - 69120 * q^24 + 153981 * q^25 + 87880 * q^26 + 98415 * q^27 - 109760 * q^28 + 88028 * q^29 - 12096 * q^30 + 244543 * q^31 - 163840 * q^32 - 99333 * q^33 - 3272 * q^34 - 19208 * q^35 + 233280 * q^36 + 730963 * q^37 - 269840 * q^38 - 296595 * q^39 - 28672 * q^40 + 479512 * q^41 + 370440 * q^42 - 406536 * q^43 - 235456 * q^44 + 40824 * q^45 + 1137136 * q^46 + 1138945 * q^47 + 552960 * q^48 + 588245 * q^49 - 1231848 * q^50 + 11043 * q^51 - 703040 * q^52 + 297595 * q^53 - 787320 * q^54 - 1834423 * q^55 + 878080 * q^56 + 910710 * q^57 - 704224 * q^58 + 941652 * q^59 + 96768 * q^60 - 2985259 * q^61 - 1956344 * q^62 - 1250235 * q^63 + 1310720 * q^64 - 123032 * q^65 + 794664 * q^66 - 2333504 * q^67 + 26176 * q^68 - 3837834 * q^69 + 153664 * q^70 - 11322272 * q^71 - 1866240 * q^72 - 6631604 * q^73 - 5847704 * q^74 + 4157487 * q^75 + 2158720 * q^76 + 1261897 * q^77 + 2372760 * q^78 - 10600265 * q^79 + 229376 * q^80 + 2657205 * q^81 - 3836096 * q^82 - 2425229 * q^83 - 2963520 * q^84 - 12267705 * q^85 + 3252288 * q^86 + 2376756 * q^87 + 1883648 * q^88 - 1581837 * q^89 - 326592 * q^90 + 3767855 * q^91 - 9097088 * q^92 + 6602661 * q^93 - 9111560 * q^94 - 11507718 * q^95 - 4423680 * q^96 + 5298407 * q^97 - 4705960 * q^98 - 2681991 * q^99

Introduction

L-functions

Modular forms

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