# Properties

 Label 570.8.a.b.1.4 Level $570$ Weight $8$ Character 570.1 Self dual yes Analytic conductor $178.059$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 570.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$178.059464526$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 3046 x^{2} + 50476 x + 497070$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-60.6808$$ of defining polynomial Character $$\chi$$ $$=$$ 570.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} +836.803 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$$ $$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} +836.803 q^{7} +512.000 q^{8} +729.000 q^{9} -1000.00 q^{10} -2850.39 q^{11} +1728.00 q^{12} -272.024 q^{13} +6694.43 q^{14} -3375.00 q^{15} +4096.00 q^{16} -18990.8 q^{17} +5832.00 q^{18} +6859.00 q^{19} -8000.00 q^{20} +22593.7 q^{21} -22803.1 q^{22} -83520.5 q^{23} +13824.0 q^{24} +15625.0 q^{25} -2176.19 q^{26} +19683.0 q^{27} +53555.4 q^{28} +39225.6 q^{29} -27000.0 q^{30} -70995.2 q^{31} +32768.0 q^{32} -76960.5 q^{33} -151927. q^{34} -104600. q^{35} +46656.0 q^{36} -342215. q^{37} +54872.0 q^{38} -7344.64 q^{39} -64000.0 q^{40} +12800.5 q^{41} +180749. q^{42} -555589. q^{43} -182425. q^{44} -91125.0 q^{45} -668164. q^{46} +675732. q^{47} +110592. q^{48} -123303. q^{49} +125000. q^{50} -512753. q^{51} -17409.5 q^{52} +597349. q^{53} +157464. q^{54} +356299. q^{55} +428443. q^{56} +185193. q^{57} +313805. q^{58} -2.38800e6 q^{59} -216000. q^{60} -2.70634e6 q^{61} -567961. q^{62} +610030. q^{63} +262144. q^{64} +34003.0 q^{65} -615684. q^{66} +1.27372e6 q^{67} -1.21541e6 q^{68} -2.25505e6 q^{69} -836803. q^{70} +2.54916e6 q^{71} +373248. q^{72} -311362. q^{73} -2.73772e6 q^{74} +421875. q^{75} +438976. q^{76} -2.38521e6 q^{77} -58757.1 q^{78} -1.44553e6 q^{79} -512000. q^{80} +531441. q^{81} +102404. q^{82} +4.27618e6 q^{83} +1.44600e6 q^{84} +2.37386e6 q^{85} -4.44471e6 q^{86} +1.05909e6 q^{87} -1.45940e6 q^{88} +3.07550e6 q^{89} -729000. q^{90} -227630. q^{91} -5.34531e6 q^{92} -1.91687e6 q^{93} +5.40585e6 q^{94} -857375. q^{95} +884736. q^{96} +5.31697e6 q^{97} -986427. q^{98} -2.07793e6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10})$$ $$4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} - 4000q^{10} - 354q^{11} + 6912q^{12} - 6366q^{13} - 5936q^{14} - 13500q^{15} + 16384q^{16} - 16412q^{17} + 23328q^{18} + 27436q^{19} - 32000q^{20} - 20034q^{21} - 2832q^{22} - 68140q^{23} + 55296q^{24} + 62500q^{25} - 50928q^{26} + 78732q^{27} - 47488q^{28} - 120486q^{29} - 108000q^{30} - 223328q^{31} + 131072q^{32} - 9558q^{33} - 131296q^{34} + 92750q^{35} + 186624q^{36} - 409930q^{37} + 219488q^{38} - 171882q^{39} - 256000q^{40} + 209182q^{41} - 160272q^{42} - 983566q^{43} - 22656q^{44} - 364500q^{45} - 545120q^{46} - 371420q^{47} + 442368q^{48} - 832632q^{49} + 500000q^{50} - 443124q^{51} - 407424q^{52} - 1254692q^{53} + 629856q^{54} + 44250q^{55} - 379904q^{56} + 740772q^{57} - 963888q^{58} - 797084q^{59} - 864000q^{60} - 3424652q^{61} - 1786624q^{62} - 540918q^{63} + 1048576q^{64} + 795750q^{65} - 76464q^{66} - 1072972q^{67} - 1050368q^{68} - 1839780q^{69} + 742000q^{70} - 2077240q^{71} + 1492992q^{72} - 257780q^{73} - 3279440q^{74} + 1687500q^{75} + 1755904q^{76} - 2436036q^{77} - 1375056q^{78} - 2112232q^{79} - 2048000q^{80} + 2125764q^{81} + 1673456q^{82} - 8743304q^{83} - 1282176q^{84} + 2051500q^{85} - 7868528q^{86} - 3253122q^{87} - 181248q^{88} - 18352170q^{89} - 2916000q^{90} - 7018432q^{91} - 4360960q^{92} - 6029856q^{93} - 2971360q^{94} - 3429500q^{95} + 3538944q^{96} + 18150q^{97} - 6661056q^{98} - 258066q^{99} + O(q^{100})$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 8.00000 0.707107
$$3$$ 27.0000 0.577350
$$4$$ 64.0000 0.500000
$$5$$ −125.000 −0.447214
$$6$$ 216.000 0.408248
$$7$$ 836.803 0.922105 0.461052 0.887373i $$-0.347472\pi$$
0.461052 + 0.887373i $$0.347472\pi$$
$$8$$ 512.000 0.353553
$$9$$ 729.000 0.333333
$$10$$ −1000.00 −0.316228
$$11$$ −2850.39 −0.645698 −0.322849 0.946450i $$-0.604641\pi$$
−0.322849 + 0.946450i $$0.604641\pi$$
$$12$$ 1728.00 0.288675
$$13$$ −272.024 −0.0343404 −0.0171702 0.999853i $$-0.505466\pi$$
−0.0171702 + 0.999853i $$0.505466\pi$$
$$14$$ 6694.43 0.652026
$$15$$ −3375.00 −0.258199
$$16$$ 4096.00 0.250000
$$17$$ −18990.8 −0.937504 −0.468752 0.883330i $$-0.655296\pi$$
−0.468752 + 0.883330i $$0.655296\pi$$
$$18$$ 5832.00 0.235702
$$19$$ 6859.00 0.229416
$$20$$ −8000.00 −0.223607
$$21$$ 22593.7 0.532377
$$22$$ −22803.1 −0.456578
$$23$$ −83520.5 −1.43135 −0.715675 0.698433i $$-0.753881\pi$$
−0.715675 + 0.698433i $$0.753881\pi$$
$$24$$ 13824.0 0.204124
$$25$$ 15625.0 0.200000
$$26$$ −2176.19 −0.0242823
$$27$$ 19683.0 0.192450
$$28$$ 53555.4 0.461052
$$29$$ 39225.6 0.298659 0.149330 0.988787i $$-0.452288\pi$$
0.149330 + 0.988787i $$0.452288\pi$$
$$30$$ −27000.0 −0.182574
$$31$$ −70995.2 −0.428019 −0.214009 0.976832i $$-0.568652\pi$$
−0.214009 + 0.976832i $$0.568652\pi$$
$$32$$ 32768.0 0.176777
$$33$$ −76960.5 −0.372794
$$34$$ −151927. −0.662915
$$35$$ −104600. −0.412378
$$36$$ 46656.0 0.166667
$$37$$ −342215. −1.11069 −0.555345 0.831620i $$-0.687414\pi$$
−0.555345 + 0.831620i $$0.687414\pi$$
$$38$$ 54872.0 0.162221
$$39$$ −7344.64 −0.0198264
$$40$$ −64000.0 −0.158114
$$41$$ 12800.5 0.0290057 0.0145029 0.999895i $$-0.495383\pi$$
0.0145029 + 0.999895i $$0.495383\pi$$
$$42$$ 180749. 0.376448
$$43$$ −555589. −1.06565 −0.532824 0.846226i $$-0.678869\pi$$
−0.532824 + 0.846226i $$0.678869\pi$$
$$44$$ −182425. −0.322849
$$45$$ −91125.0 −0.149071
$$46$$ −668164. −1.01212
$$47$$ 675732. 0.949362 0.474681 0.880158i $$-0.342563\pi$$
0.474681 + 0.880158i $$0.342563\pi$$
$$48$$ 110592. 0.144338
$$49$$ −123303. −0.149723
$$50$$ 125000. 0.141421
$$51$$ −512753. −0.541268
$$52$$ −17409.5 −0.0171702
$$53$$ 597349. 0.551141 0.275570 0.961281i $$-0.411133\pi$$
0.275570 + 0.961281i $$0.411133\pi$$
$$54$$ 157464. 0.136083
$$55$$ 356299. 0.288765
$$56$$ 428443. 0.326013
$$57$$ 185193. 0.132453
$$58$$ 313805. 0.211184
$$59$$ −2.38800e6 −1.51374 −0.756871 0.653565i $$-0.773273\pi$$
−0.756871 + 0.653565i $$0.773273\pi$$
$$60$$ −216000. −0.129099
$$61$$ −2.70634e6 −1.52661 −0.763306 0.646038i $$-0.776425\pi$$
−0.763306 + 0.646038i $$0.776425\pi$$
$$62$$ −567961. −0.302655
$$63$$ 610030. 0.307368
$$64$$ 262144. 0.125000
$$65$$ 34003.0 0.0153575
$$66$$ −615684. −0.263605
$$67$$ 1.27372e6 0.517385 0.258692 0.965960i $$-0.416708\pi$$
0.258692 + 0.965960i $$0.416708\pi$$
$$68$$ −1.21541e6 −0.468752
$$69$$ −2.25505e6 −0.826390
$$70$$ −836803. −0.291595
$$71$$ 2.54916e6 0.845266 0.422633 0.906301i $$-0.361106\pi$$
0.422633 + 0.906301i $$0.361106\pi$$
$$72$$ 373248. 0.117851
$$73$$ −311362. −0.0936776 −0.0468388 0.998902i $$-0.514915\pi$$
−0.0468388 + 0.998902i $$0.514915\pi$$
$$74$$ −2.73772e6 −0.785376
$$75$$ 421875. 0.115470
$$76$$ 438976. 0.114708
$$77$$ −2.38521e6 −0.595402
$$78$$ −58757.1 −0.0140194
$$79$$ −1.44553e6 −0.329862 −0.164931 0.986305i $$-0.552740\pi$$
−0.164931 + 0.986305i $$0.552740\pi$$
$$80$$ −512000. −0.111803
$$81$$ 531441. 0.111111
$$82$$ 102404. 0.0205101
$$83$$ 4.27618e6 0.820885 0.410443 0.911886i $$-0.365374\pi$$
0.410443 + 0.911886i $$0.365374\pi$$
$$84$$ 1.44600e6 0.266189
$$85$$ 2.37386e6 0.419265
$$86$$ −4.44471e6 −0.753527
$$87$$ 1.05909e6 0.172431
$$88$$ −1.45940e6 −0.228289
$$89$$ 3.07550e6 0.462435 0.231218 0.972902i $$-0.425729\pi$$
0.231218 + 0.972902i $$0.425729\pi$$
$$90$$ −729000. −0.105409
$$91$$ −227630. −0.0316654
$$92$$ −5.34531e6 −0.715675
$$93$$ −1.91687e6 −0.247117
$$94$$ 5.40585e6 0.671300
$$95$$ −857375. −0.102598
$$96$$ 884736. 0.102062
$$97$$ 5.31697e6 0.591512 0.295756 0.955264i $$-0.404429\pi$$
0.295756 + 0.955264i $$0.404429\pi$$
$$98$$ −986427. −0.105870
$$99$$ −2.07793e6 −0.215233
$$100$$ 1.00000e6 0.100000
$$101$$ 1.36841e6 0.132158 0.0660789 0.997814i $$-0.478951\pi$$
0.0660789 + 0.997814i $$0.478951\pi$$
$$102$$ −4.10202e6 −0.382734
$$103$$ −1.59596e7 −1.43910 −0.719551 0.694440i $$-0.755652\pi$$
−0.719551 + 0.694440i $$0.755652\pi$$
$$104$$ −139276. −0.0121412
$$105$$ −2.82421e6 −0.238086
$$106$$ 4.77879e6 0.389715
$$107$$ 2.23193e7 1.76131 0.880656 0.473756i $$-0.157102\pi$$
0.880656 + 0.473756i $$0.157102\pi$$
$$108$$ 1.25971e6 0.0962250
$$109$$ −1.97882e7 −1.46357 −0.731787 0.681534i $$-0.761313\pi$$
−0.731787 + 0.681534i $$0.761313\pi$$
$$110$$ 2.85039e6 0.204188
$$111$$ −9.23980e6 −0.641257
$$112$$ 3.42755e6 0.230526
$$113$$ −5.44310e6 −0.354872 −0.177436 0.984132i $$-0.556780\pi$$
−0.177436 + 0.984132i $$0.556780\pi$$
$$114$$ 1.48154e6 0.0936586
$$115$$ 1.04401e7 0.640119
$$116$$ 2.51044e6 0.149330
$$117$$ −198305. −0.0114468
$$118$$ −1.91040e7 −1.07038
$$119$$ −1.58916e7 −0.864477
$$120$$ −1.72800e6 −0.0912871
$$121$$ −1.13625e7 −0.583073
$$122$$ −2.16507e7 −1.07948
$$123$$ 345613. 0.0167464
$$124$$ −4.54369e6 −0.214009
$$125$$ −1.95312e6 −0.0894427
$$126$$ 4.88024e6 0.217342
$$127$$ −2.66077e7 −1.15264 −0.576321 0.817224i $$-0.695512\pi$$
−0.576321 + 0.817224i $$0.695512\pi$$
$$128$$ 2.09715e6 0.0883883
$$129$$ −1.50009e7 −0.615253
$$130$$ 272024. 0.0108594
$$131$$ −6.56953e6 −0.255320 −0.127660 0.991818i $$-0.540747\pi$$
−0.127660 + 0.991818i $$0.540747\pi$$
$$132$$ −4.92547e6 −0.186397
$$133$$ 5.73963e6 0.211545
$$134$$ 1.01898e7 0.365846
$$135$$ −2.46038e6 −0.0860663
$$136$$ −9.72331e6 −0.331458
$$137$$ −2.40376e7 −0.798675 −0.399337 0.916804i $$-0.630760\pi$$
−0.399337 + 0.916804i $$0.630760\pi$$
$$138$$ −1.80404e7 −0.584346
$$139$$ −1.09363e7 −0.345396 −0.172698 0.984975i $$-0.555249\pi$$
−0.172698 + 0.984975i $$0.555249\pi$$
$$140$$ −6.69443e6 −0.206189
$$141$$ 1.82448e7 0.548114
$$142$$ 2.03933e7 0.597694
$$143$$ 775373. 0.0221735
$$144$$ 2.98598e6 0.0833333
$$145$$ −4.90320e6 −0.133565
$$146$$ −2.49090e6 −0.0662400
$$147$$ −3.32919e6 −0.0864427
$$148$$ −2.19017e7 −0.555345
$$149$$ 5.07363e7 1.25651 0.628256 0.778007i $$-0.283769\pi$$
0.628256 + 0.778007i $$0.283769\pi$$
$$150$$ 3.37500e6 0.0816497
$$151$$ −2.92446e7 −0.691236 −0.345618 0.938375i $$-0.612331\pi$$
−0.345618 + 0.938375i $$0.612331\pi$$
$$152$$ 3.51181e6 0.0811107
$$153$$ −1.38443e7 −0.312501
$$154$$ −1.90817e7 −0.421012
$$155$$ 8.87440e6 0.191416
$$156$$ −470057. −0.00991321
$$157$$ −5.62135e7 −1.15929 −0.579645 0.814869i $$-0.696809\pi$$
−0.579645 + 0.814869i $$0.696809\pi$$
$$158$$ −1.15642e7 −0.233247
$$159$$ 1.61284e7 0.318201
$$160$$ −4.09600e6 −0.0790569
$$161$$ −6.98903e7 −1.31985
$$162$$ 4.25153e6 0.0785674
$$163$$ −9.71809e7 −1.75762 −0.878809 0.477174i $$-0.841661\pi$$
−0.878809 + 0.477174i $$0.841661\pi$$
$$164$$ 819232. 0.0145029
$$165$$ 9.62006e6 0.166719
$$166$$ 3.42094e7 0.580454
$$167$$ 2.50137e7 0.415595 0.207797 0.978172i $$-0.433371\pi$$
0.207797 + 0.978172i $$0.433371\pi$$
$$168$$ 1.15680e7 0.188224
$$169$$ −6.26745e7 −0.998821
$$170$$ 1.89908e7 0.296465
$$171$$ 5.00021e6 0.0764719
$$172$$ −3.55577e7 −0.532824
$$173$$ −7.86742e7 −1.15524 −0.577619 0.816307i $$-0.696018\pi$$
−0.577619 + 0.816307i $$0.696018\pi$$
$$174$$ 8.47272e6 0.121927
$$175$$ 1.30751e7 0.184421
$$176$$ −1.16752e7 −0.161425
$$177$$ −6.44759e7 −0.873959
$$178$$ 2.46040e7 0.326991
$$179$$ 3.16655e7 0.412669 0.206334 0.978482i $$-0.433847\pi$$
0.206334 + 0.978482i $$0.433847\pi$$
$$180$$ −5.83200e6 −0.0745356
$$181$$ 1.43164e8 1.79456 0.897282 0.441458i $$-0.145539\pi$$
0.897282 + 0.441458i $$0.145539\pi$$
$$182$$ −1.82104e6 −0.0223908
$$183$$ −7.30713e7 −0.881389
$$184$$ −4.27625e7 −0.506059
$$185$$ 4.27768e7 0.496715
$$186$$ −1.53350e7 −0.174738
$$187$$ 5.41313e7 0.605345
$$188$$ 4.32468e7 0.474681
$$189$$ 1.64708e7 0.177459
$$190$$ −6.85900e6 −0.0725476
$$191$$ 2.56884e7 0.266760 0.133380 0.991065i $$-0.457417\pi$$
0.133380 + 0.991065i $$0.457417\pi$$
$$192$$ 7.07789e6 0.0721688
$$193$$ 1.15293e6 0.0115439 0.00577196 0.999983i $$-0.498163\pi$$
0.00577196 + 0.999983i $$0.498163\pi$$
$$194$$ 4.25358e7 0.418262
$$195$$ 918080. 0.00886665
$$196$$ −7.89142e6 −0.0748615
$$197$$ 1.25465e8 1.16920 0.584601 0.811321i $$-0.301251\pi$$
0.584601 + 0.811321i $$0.301251\pi$$
$$198$$ −1.66235e7 −0.152193
$$199$$ 7.86052e7 0.707075 0.353537 0.935420i $$-0.384979\pi$$
0.353537 + 0.935420i $$0.384979\pi$$
$$200$$ 8.00000e6 0.0707107
$$201$$ 3.43906e7 0.298712
$$202$$ 1.09473e7 0.0934497
$$203$$ 3.28241e7 0.275395
$$204$$ −3.28162e7 −0.270634
$$205$$ −1.60006e6 −0.0129717
$$206$$ −1.27677e8 −1.01760
$$207$$ −6.08865e7 −0.477117
$$208$$ −1.11421e6 −0.00858509
$$209$$ −1.95508e7 −0.148133
$$210$$ −2.25937e7 −0.168353
$$211$$ −4.92739e7 −0.361100 −0.180550 0.983566i $$-0.557788\pi$$
−0.180550 + 0.983566i $$0.557788\pi$$
$$212$$ 3.82303e7 0.275570
$$213$$ 6.88274e7 0.488015
$$214$$ 1.78554e8 1.24544
$$215$$ 6.94486e7 0.476573
$$216$$ 1.00777e7 0.0680414
$$217$$ −5.94090e7 −0.394678
$$218$$ −1.58306e8 −1.03490
$$219$$ −8.40677e6 −0.0540848
$$220$$ 2.28031e7 0.144383
$$221$$ 5.16596e6 0.0321942
$$222$$ −7.39184e7 −0.453437
$$223$$ −2.10879e8 −1.27340 −0.636702 0.771110i $$-0.719702\pi$$
−0.636702 + 0.771110i $$0.719702\pi$$
$$224$$ 2.74204e7 0.163007
$$225$$ 1.13906e7 0.0666667
$$226$$ −4.35448e7 −0.250932
$$227$$ 1.61754e7 0.0917837 0.0458918 0.998946i $$-0.485387\pi$$
0.0458918 + 0.998946i $$0.485387\pi$$
$$228$$ 1.18524e7 0.0662266
$$229$$ 2.84435e8 1.56516 0.782580 0.622550i $$-0.213903\pi$$
0.782580 + 0.622550i $$0.213903\pi$$
$$230$$ 8.35205e7 0.452633
$$231$$ −6.44008e7 −0.343755
$$232$$ 2.00835e7 0.105592
$$233$$ 1.92691e8 0.997966 0.498983 0.866612i $$-0.333707\pi$$
0.498983 + 0.866612i $$0.333707\pi$$
$$234$$ −1.58644e6 −0.00809410
$$235$$ −8.44664e7 −0.424567
$$236$$ −1.52832e8 −0.756871
$$237$$ −3.90293e7 −0.190446
$$238$$ −1.27133e8 −0.611277
$$239$$ −2.22040e8 −1.05206 −0.526028 0.850468i $$-0.676319\pi$$
−0.526028 + 0.850468i $$0.676319\pi$$
$$240$$ −1.38240e7 −0.0645497
$$241$$ −4.24494e8 −1.95349 −0.976747 0.214395i $$-0.931222\pi$$
−0.976747 + 0.214395i $$0.931222\pi$$
$$242$$ −9.08996e7 −0.412295
$$243$$ 1.43489e7 0.0641500
$$244$$ −1.73206e8 −0.763306
$$245$$ 1.54129e7 0.0669582
$$246$$ 2.76491e6 0.0118415
$$247$$ −1.86581e6 −0.00787822
$$248$$ −3.63495e7 −0.151328
$$249$$ 1.15457e8 0.473938
$$250$$ −1.56250e7 −0.0632456
$$251$$ −3.69876e8 −1.47638 −0.738190 0.674593i $$-0.764319\pi$$
−0.738190 + 0.674593i $$0.764319\pi$$
$$252$$ 3.90419e7 0.153684
$$253$$ 2.38066e8 0.924220
$$254$$ −2.12862e8 −0.815041
$$255$$ 6.40941e7 0.242062
$$256$$ 1.67772e7 0.0625000
$$257$$ −1.34218e8 −0.493223 −0.246612 0.969114i $$-0.579317\pi$$
−0.246612 + 0.969114i $$0.579317\pi$$
$$258$$ −1.20007e8 −0.435049
$$259$$ −2.86366e8 −1.02417
$$260$$ 2.17619e6 0.00767874
$$261$$ 2.85954e7 0.0995532
$$262$$ −5.25563e7 −0.180539
$$263$$ 2.90630e8 0.985133 0.492566 0.870275i $$-0.336059\pi$$
0.492566 + 0.870275i $$0.336059\pi$$
$$264$$ −3.94038e7 −0.131803
$$265$$ −7.46686e7 −0.246478
$$266$$ 4.59171e7 0.149585
$$267$$ 8.30386e7 0.266987
$$268$$ 8.15184e7 0.258692
$$269$$ −2.34158e8 −0.733458 −0.366729 0.930328i $$-0.619522\pi$$
−0.366729 + 0.930328i $$0.619522\pi$$
$$270$$ −1.96830e7 −0.0608581
$$271$$ 4.75402e8 1.45100 0.725502 0.688220i $$-0.241608\pi$$
0.725502 + 0.688220i $$0.241608\pi$$
$$272$$ −7.77865e7 −0.234376
$$273$$ −6.14602e6 −0.0182820
$$274$$ −1.92301e8 −0.564748
$$275$$ −4.45373e7 −0.129140
$$276$$ −1.44323e8 −0.413195
$$277$$ −5.18595e8 −1.46605 −0.733026 0.680201i $$-0.761893\pi$$
−0.733026 + 0.680201i $$0.761893\pi$$
$$278$$ −8.74902e7 −0.244232
$$279$$ −5.17555e7 −0.142673
$$280$$ −5.35554e7 −0.145798
$$281$$ 3.79068e8 1.01917 0.509583 0.860421i $$-0.329800\pi$$
0.509583 + 0.860421i $$0.329800\pi$$
$$282$$ 1.45958e8 0.387575
$$283$$ −1.54943e8 −0.406367 −0.203184 0.979141i $$-0.565129\pi$$
−0.203184 + 0.979141i $$0.565129\pi$$
$$284$$ 1.63146e8 0.422633
$$285$$ −2.31491e7 −0.0592349
$$286$$ 6.20299e6 0.0156791
$$287$$ 1.07115e7 0.0267463
$$288$$ 2.38879e7 0.0589256
$$289$$ −4.96864e7 −0.121086
$$290$$ −3.92256e7 −0.0944444
$$291$$ 1.43558e8 0.341509
$$292$$ −1.99272e7 −0.0468388
$$293$$ −4.92640e8 −1.14418 −0.572089 0.820192i $$-0.693867\pi$$
−0.572089 + 0.820192i $$0.693867\pi$$
$$294$$ −2.66335e7 −0.0611242
$$295$$ 2.98500e8 0.676966
$$296$$ −1.75214e8 −0.392688
$$297$$ −5.61042e7 −0.124265
$$298$$ 4.05890e8 0.888488
$$299$$ 2.27196e7 0.0491531
$$300$$ 2.70000e7 0.0577350
$$301$$ −4.64919e8 −0.982640
$$302$$ −2.33957e8 −0.488778
$$303$$ 3.69472e7 0.0763013
$$304$$ 2.80945e7 0.0573539
$$305$$ 3.38293e8 0.682721
$$306$$ −1.10755e8 −0.220972
$$307$$ 1.42736e8 0.281547 0.140773 0.990042i $$-0.455041\pi$$
0.140773 + 0.990042i $$0.455041\pi$$
$$308$$ −1.52654e8 −0.297701
$$309$$ −4.30909e8 −0.830866
$$310$$ 7.09952e7 0.135351
$$311$$ 8.89524e7 0.167686 0.0838429 0.996479i $$-0.473281\pi$$
0.0838429 + 0.996479i $$0.473281\pi$$
$$312$$ −3.76046e6 −0.00700970
$$313$$ 8.07590e8 1.48863 0.744314 0.667830i $$-0.232777\pi$$
0.744314 + 0.667830i $$0.232777\pi$$
$$314$$ −4.49708e8 −0.819742
$$315$$ −7.62537e7 −0.137459
$$316$$ −9.25138e7 −0.164931
$$317$$ −6.88675e8 −1.21425 −0.607123 0.794608i $$-0.707677\pi$$
−0.607123 + 0.794608i $$0.707677\pi$$
$$318$$ 1.29027e8 0.225002
$$319$$ −1.11808e8 −0.192844
$$320$$ −3.27680e7 −0.0559017
$$321$$ 6.02620e8 1.01689
$$322$$ −5.59122e8 −0.933278
$$323$$ −1.30258e8 −0.215078
$$324$$ 3.40122e7 0.0555556
$$325$$ −4.25037e6 −0.00686807
$$326$$ −7.77447e8 −1.24282
$$327$$ −5.34283e8 −0.844995
$$328$$ 6.55386e6 0.0102551
$$329$$ 5.65454e8 0.875411
$$330$$ 7.69605e7 0.117888
$$331$$ 1.48744e8 0.225445 0.112723 0.993626i $$-0.464043\pi$$
0.112723 + 0.993626i $$0.464043\pi$$
$$332$$ 2.73675e8 0.410443
$$333$$ −2.49474e8 −0.370230
$$334$$ 2.00110e8 0.293870
$$335$$ −1.59216e8 −0.231382
$$336$$ 9.25437e7 0.133094
$$337$$ −3.35266e6 −0.00477183 −0.00238592 0.999997i $$-0.500759\pi$$
−0.00238592 + 0.999997i $$0.500759\pi$$
$$338$$ −5.01396e8 −0.706273
$$339$$ −1.46964e8 −0.204885
$$340$$ 1.51927e8 0.209632
$$341$$ 2.02364e8 0.276371
$$342$$ 4.00017e7 0.0540738
$$343$$ −7.92324e8 −1.06016
$$344$$ −2.84462e8 −0.376764
$$345$$ 2.81882e8 0.369573
$$346$$ −6.29394e8 −0.816876
$$347$$ 8.54721e8 1.09817 0.549087 0.835765i $$-0.314976\pi$$
0.549087 + 0.835765i $$0.314976\pi$$
$$348$$ 6.77818e7 0.0862156
$$349$$ −9.51858e7 −0.119862 −0.0599312 0.998203i $$-0.519088\pi$$
−0.0599312 + 0.998203i $$0.519088\pi$$
$$350$$ 1.04600e8 0.130405
$$351$$ −5.35424e6 −0.00660881
$$352$$ −9.34016e7 −0.114144
$$353$$ −9.20946e8 −1.11435 −0.557176 0.830394i $$-0.688115\pi$$
−0.557176 + 0.830394i $$0.688115\pi$$
$$354$$ −5.15807e8 −0.617982
$$355$$ −3.18646e8 −0.378015
$$356$$ 1.96832e8 0.231218
$$357$$ −4.29073e8 −0.499106
$$358$$ 2.53324e8 0.291801
$$359$$ −2.97767e8 −0.339661 −0.169830 0.985473i $$-0.554322\pi$$
−0.169830 + 0.985473i $$0.554322\pi$$
$$360$$ −4.66560e7 −0.0527046
$$361$$ 4.70459e7 0.0526316
$$362$$ 1.14531e9 1.26895
$$363$$ −3.06786e8 −0.336638
$$364$$ −1.45683e7 −0.0158327
$$365$$ 3.89203e7 0.0418939
$$366$$ −5.84570e8 −0.623236
$$367$$ 3.30475e7 0.0348986 0.0174493 0.999848i $$-0.494445\pi$$
0.0174493 + 0.999848i $$0.494445\pi$$
$$368$$ −3.42100e8 −0.357837
$$369$$ 9.33156e6 0.00966857
$$370$$ 3.42215e8 0.351231
$$371$$ 4.99864e8 0.508210
$$372$$ −1.22680e8 −0.123558
$$373$$ 4.57835e8 0.456802 0.228401 0.973567i $$-0.426650\pi$$
0.228401 + 0.973567i $$0.426650\pi$$
$$374$$ 4.33050e8 0.428043
$$375$$ −5.27344e7 −0.0516398
$$376$$ 3.45975e8 0.335650
$$377$$ −1.06703e7 −0.0102561
$$378$$ 1.31766e8 0.125483
$$379$$ −1.10898e9 −1.04637 −0.523185 0.852219i $$-0.675256\pi$$
−0.523185 + 0.852219i $$0.675256\pi$$
$$380$$ −5.48720e7 −0.0512989
$$381$$ −7.18408e8 −0.665478
$$382$$ 2.05507e8 0.188628
$$383$$ −1.36304e9 −1.23969 −0.619843 0.784726i $$-0.712804\pi$$
−0.619843 + 0.784726i $$0.712804\pi$$
$$384$$ 5.66231e7 0.0510310
$$385$$ 2.98152e8 0.266272
$$386$$ 9.22345e6 0.00816278
$$387$$ −4.05024e8 −0.355216
$$388$$ 3.40286e8 0.295756
$$389$$ 1.22100e9 1.05170 0.525850 0.850577i $$-0.323747\pi$$
0.525850 + 0.850577i $$0.323747\pi$$
$$390$$ 7.34464e6 0.00626967
$$391$$ 1.58613e9 1.34190
$$392$$ −6.31313e7 −0.0529351
$$393$$ −1.77377e8 −0.147409
$$394$$ 1.00372e9 0.826751
$$395$$ 1.80691e8 0.147519
$$396$$ −1.32988e8 −0.107616
$$397$$ 1.95045e9 1.56447 0.782236 0.622982i $$-0.214079\pi$$
0.782236 + 0.622982i $$0.214079\pi$$
$$398$$ 6.28842e8 0.499977
$$399$$ 1.54970e8 0.122136
$$400$$ 6.40000e7 0.0500000
$$401$$ −3.28954e8 −0.254759 −0.127380 0.991854i $$-0.540657\pi$$
−0.127380 + 0.991854i $$0.540657\pi$$
$$402$$ 2.75124e8 0.211221
$$403$$ 1.93124e7 0.0146983
$$404$$ 8.75785e7 0.0660789
$$405$$ −6.64301e7 −0.0496904
$$406$$ 2.62593e8 0.194734
$$407$$ 9.75445e8 0.717170
$$408$$ −2.62529e8 −0.191367
$$409$$ 8.61975e8 0.622964 0.311482 0.950252i $$-0.399175\pi$$
0.311482 + 0.950252i $$0.399175\pi$$
$$410$$ −1.28005e7 −0.00917241
$$411$$ −6.49016e8 −0.461115
$$412$$ −1.02141e9 −0.719551
$$413$$ −1.99828e9 −1.39583
$$414$$ −4.87092e8 −0.337372
$$415$$ −5.34522e8 −0.367111
$$416$$ −8.91367e6 −0.00607058
$$417$$ −2.95279e8 −0.199415
$$418$$ −1.56407e8 −0.104746
$$419$$ −1.76742e9 −1.17379 −0.586894 0.809664i $$-0.699649\pi$$
−0.586894 + 0.809664i $$0.699649\pi$$
$$420$$ −1.80749e8 −0.119043
$$421$$ −2.80577e8 −0.183259 −0.0916295 0.995793i $$-0.529208\pi$$
−0.0916295 + 0.995793i $$0.529208\pi$$
$$422$$ −3.94191e8 −0.255337
$$423$$ 4.92608e8 0.316454
$$424$$ 3.05843e8 0.194858
$$425$$ −2.96732e8 −0.187501
$$426$$ 5.50619e8 0.345079
$$427$$ −2.26468e9 −1.40770
$$428$$ 1.42843e9 0.880656
$$429$$ 2.09351e7 0.0128019
$$430$$ 5.55589e8 0.336988
$$431$$ 8.32322e8 0.500750 0.250375 0.968149i $$-0.419446\pi$$
0.250375 + 0.968149i $$0.419446\pi$$
$$432$$ 8.06216e7 0.0481125
$$433$$ 2.14254e9 1.26830 0.634148 0.773211i $$-0.281351\pi$$
0.634148 + 0.773211i $$0.281351\pi$$
$$434$$ −4.75272e8 −0.279080
$$435$$ −1.32386e8 −0.0771135
$$436$$ −1.26645e9 −0.731787
$$437$$ −5.72867e8 −0.328374
$$438$$ −6.72542e7 −0.0382437
$$439$$ −2.21507e9 −1.24957 −0.624785 0.780797i $$-0.714813\pi$$
−0.624785 + 0.780797i $$0.714813\pi$$
$$440$$ 1.82425e8 0.102094
$$441$$ −8.98882e7 −0.0499077
$$442$$ 4.13277e7 0.0227648
$$443$$ −1.40738e9 −0.769128 −0.384564 0.923098i $$-0.625648\pi$$
−0.384564 + 0.923098i $$0.625648\pi$$
$$444$$ −5.91347e8 −0.320628
$$445$$ −3.84438e8 −0.206807
$$446$$ −1.68703e9 −0.900433
$$447$$ 1.36988e9 0.725447
$$448$$ 2.19363e8 0.115263
$$449$$ 1.42630e9 0.743618 0.371809 0.928309i $$-0.378738\pi$$
0.371809 + 0.928309i $$0.378738\pi$$
$$450$$ 9.11250e7 0.0471405
$$451$$ −3.64864e7 −0.0187289
$$452$$ −3.48358e8 −0.177436
$$453$$ −7.89605e8 −0.399086
$$454$$ 1.29404e8 0.0649009
$$455$$ 2.84538e7 0.0141612
$$456$$ 9.48188e7 0.0468293
$$457$$ 2.56687e9 1.25805 0.629024 0.777386i $$-0.283455\pi$$
0.629024 + 0.777386i $$0.283455\pi$$
$$458$$ 2.27548e9 1.10673
$$459$$ −3.73797e8 −0.180423
$$460$$ 6.68164e8 0.320060
$$461$$ 1.12083e9 0.532826 0.266413 0.963859i $$-0.414161\pi$$
0.266413 + 0.963859i $$0.414161\pi$$
$$462$$ −5.15206e8 −0.243072
$$463$$ 1.77884e9 0.832922 0.416461 0.909153i $$-0.363270\pi$$
0.416461 + 0.909153i $$0.363270\pi$$
$$464$$ 1.60668e8 0.0746649
$$465$$ 2.39609e8 0.110514
$$466$$ 1.54153e9 0.705669
$$467$$ 1.47455e9 0.669962 0.334981 0.942225i $$-0.391270\pi$$
0.334981 + 0.942225i $$0.391270\pi$$
$$468$$ −1.26915e7 −0.00572340
$$469$$ 1.06586e9 0.477083
$$470$$ −6.75732e8 −0.300215
$$471$$ −1.51777e9 −0.669317
$$472$$ −1.22265e9 −0.535188
$$473$$ 1.58364e9 0.688088
$$474$$ −3.12234e8 −0.134665
$$475$$ 1.07172e8 0.0458831
$$476$$ −1.01706e9 −0.432238
$$477$$ 4.35467e8 0.183714
$$478$$ −1.77632e9 −0.743915
$$479$$ −2.31447e9 −0.962226 −0.481113 0.876658i $$-0.659767\pi$$
−0.481113 + 0.876658i $$0.659767\pi$$
$$480$$ −1.10592e8 −0.0456435
$$481$$ 9.30905e7 0.0381415
$$482$$ −3.39595e9 −1.38133
$$483$$ −1.88704e9 −0.762018
$$484$$ −7.27197e8 −0.291537
$$485$$ −6.64622e8 −0.264532
$$486$$ 1.14791e8 0.0453609
$$487$$ 2.23123e9 0.875372 0.437686 0.899128i $$-0.355798\pi$$
0.437686 + 0.899128i $$0.355798\pi$$
$$488$$ −1.38565e9 −0.539739
$$489$$ −2.62388e9 −1.01476
$$490$$ 1.23303e8 0.0473466
$$491$$ 5.31217e7 0.0202529 0.0101264 0.999949i $$-0.496777\pi$$
0.0101264 + 0.999949i $$0.496777\pi$$
$$492$$ 2.21193e7 0.00837322
$$493$$ −7.44927e8 −0.279994
$$494$$ −1.49265e7 −0.00557074
$$495$$ 2.59742e8 0.0962550
$$496$$ −2.90796e8 −0.107005
$$497$$ 2.13315e9 0.779424
$$498$$ 9.23654e8 0.335125
$$499$$ 3.92892e9 1.41554 0.707769 0.706444i $$-0.249702\pi$$
0.707769 + 0.706444i $$0.249702\pi$$
$$500$$ −1.25000e8 −0.0447214
$$501$$ 6.75370e8 0.239944
$$502$$ −2.95901e9 −1.04396
$$503$$ −3.54976e9 −1.24369 −0.621843 0.783142i $$-0.713616\pi$$
−0.621843 + 0.783142i $$0.713616\pi$$
$$504$$ 3.12335e8 0.108671
$$505$$ −1.71052e8 −0.0591028
$$506$$ 1.90453e9 0.653523
$$507$$ −1.69221e9 −0.576669
$$508$$ −1.70289e9 −0.576321
$$509$$ 5.26540e9 1.76978 0.884890 0.465800i $$-0.154233\pi$$
0.884890 + 0.465800i $$0.154233\pi$$
$$510$$ 5.12753e8 0.171164
$$511$$ −2.60549e8 −0.0863805
$$512$$ 1.34218e8 0.0441942
$$513$$ 1.35006e8 0.0441511
$$514$$ −1.07374e9 −0.348762
$$515$$ 1.99495e9 0.643586
$$516$$ −9.60058e8 −0.307626
$$517$$ −1.92610e9 −0.613001
$$518$$ −2.29093e9 −0.724199
$$519$$ −2.12420e9 −0.666977
$$520$$ 1.74095e7 0.00542969
$$521$$ 2.83042e9 0.876836 0.438418 0.898771i $$-0.355539\pi$$
0.438418 + 0.898771i $$0.355539\pi$$
$$522$$ 2.28764e8 0.0703947
$$523$$ −2.22245e9 −0.679322 −0.339661 0.940548i $$-0.610312\pi$$
−0.339661 + 0.940548i $$0.610312\pi$$
$$524$$ −4.20450e8 −0.127660
$$525$$ 3.53026e8 0.106475
$$526$$ 2.32504e9 0.696594
$$527$$ 1.34826e9 0.401269
$$528$$ −3.15230e8 −0.0931985
$$529$$ 3.57085e9 1.04876
$$530$$ −5.97349e8 −0.174286
$$531$$ −1.74085e9 −0.504580
$$532$$ 3.67337e8 0.105773
$$533$$ −3.48204e6 −0.000996067 0
$$534$$ 6.64309e8 0.188788
$$535$$ −2.78991e9 −0.787683
$$536$$ 6.52147e8 0.182923
$$537$$ 8.54970e8 0.238254
$$538$$ −1.87326e9 −0.518633
$$539$$ 3.51463e8 0.0966760
$$540$$ −1.57464e8 −0.0430331
$$541$$ 4.33154e9 1.17612 0.588060 0.808817i $$-0.299892\pi$$
0.588060 + 0.808817i $$0.299892\pi$$
$$542$$ 3.80322e9 1.02601
$$543$$ 3.86543e9 1.03609
$$544$$ −6.22292e8 −0.165729
$$545$$ 2.47353e9 0.654530
$$546$$ −4.91681e7 −0.0129274
$$547$$ −4.41256e9 −1.15275 −0.576375 0.817186i $$-0.695533\pi$$
−0.576375 + 0.817186i $$0.695533\pi$$
$$548$$ −1.53841e9 −0.399337
$$549$$ −1.97292e9 −0.508870
$$550$$ −3.56299e8 −0.0913156
$$551$$ 2.69048e8 0.0685172
$$552$$ −1.15459e9 −0.292173
$$553$$ −1.20962e9 −0.304167
$$554$$ −4.14876e9 −1.03666
$$555$$ 1.15497e9 0.286779
$$556$$ −6.99922e8 −0.172698
$$557$$ −3.89753e9 −0.955644 −0.477822 0.878457i $$-0.658574\pi$$
−0.477822 + 0.878457i $$0.658574\pi$$
$$558$$ −4.14044e8 −0.100885
$$559$$ 1.51133e8 0.0365948
$$560$$ −4.28443e8 −0.103094
$$561$$ 1.46155e9 0.349496
$$562$$ 3.03255e9 0.720660
$$563$$ 7.98467e8 0.188572 0.0942860 0.995545i $$-0.469943\pi$$
0.0942860 + 0.995545i $$0.469943\pi$$
$$564$$ 1.16766e9 0.274057
$$565$$ 6.80387e8 0.158704
$$566$$ −1.23954e9 −0.287345
$$567$$ 4.44712e8 0.102456
$$568$$ 1.30517e9 0.298847
$$569$$ −8.96244e8 −0.203955 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$570$$ −1.85193e8 −0.0418854
$$571$$ 2.09621e9 0.471204 0.235602 0.971850i $$-0.424294\pi$$
0.235602 + 0.971850i $$0.424294\pi$$
$$572$$ 4.96239e7 0.0110868
$$573$$ 6.93587e8 0.154014
$$574$$ 8.56920e7 0.0189125
$$575$$ −1.30501e9 −0.286270
$$576$$ 1.91103e8 0.0416667
$$577$$ −6.21510e9 −1.34689 −0.673446 0.739237i $$-0.735187\pi$$
−0.673446 + 0.739237i $$0.735187\pi$$
$$578$$ −3.97491e8 −0.0856209
$$579$$ 3.11292e7 0.00666488
$$580$$ −3.13805e8 −0.0667823
$$581$$ 3.57832e9 0.756942
$$582$$ 1.14847e9 0.241484
$$583$$ −1.70268e9 −0.355871
$$584$$ −1.59417e8 −0.0331200
$$585$$ 2.47882e7 0.00511916
$$586$$ −3.94112e9 −0.809056
$$587$$ −3.38000e9 −0.689736 −0.344868 0.938651i $$-0.612076\pi$$
−0.344868 + 0.938651i $$0.612076\pi$$
$$588$$ −2.13068e8 −0.0432213
$$589$$ −4.86956e8 −0.0981943
$$590$$ 2.38800e9 0.478687
$$591$$ 3.38755e9 0.675039
$$592$$ −1.40171e9 −0.277672
$$593$$ 7.87465e9 1.55074 0.775371 0.631506i $$-0.217563\pi$$
0.775371 + 0.631506i $$0.217563\pi$$
$$594$$ −4.48834e8 −0.0878684
$$595$$ 1.98645e9 0.386606
$$596$$ 3.24712e9 0.628256
$$597$$ 2.12234e9 0.408230
$$598$$ 1.81757e8 0.0347565
$$599$$ 7.17992e9 1.36498 0.682489 0.730895i $$-0.260897\pi$$
0.682489 + 0.730895i $$0.260897\pi$$
$$600$$ 2.16000e8 0.0408248
$$601$$ −2.45321e9 −0.460971 −0.230486 0.973076i $$-0.574031\pi$$
−0.230486 + 0.973076i $$0.574031\pi$$
$$602$$ −3.71935e9 −0.694831
$$603$$ 9.28545e8 0.172462
$$604$$ −1.87166e9 −0.345618
$$605$$ 1.42031e9 0.260758
$$606$$ 2.95577e8 0.0539532
$$607$$ 7.61815e9 1.38258 0.691288 0.722579i $$-0.257044\pi$$
0.691288 + 0.722579i $$0.257044\pi$$
$$608$$ 2.24756e8 0.0405554
$$609$$ 8.86250e8 0.159000
$$610$$ 2.70634e9 0.482757
$$611$$ −1.83815e8 −0.0326014
$$612$$ −8.86037e8 −0.156251
$$613$$ 6.97473e9 1.22297 0.611485 0.791256i $$-0.290572\pi$$
0.611485 + 0.791256i $$0.290572\pi$$
$$614$$ 1.14189e9 0.199083
$$615$$ −4.32017e7 −0.00748924
$$616$$ −1.22123e9 −0.210506
$$617$$ 5.04278e9 0.864315 0.432157 0.901798i $$-0.357753\pi$$
0.432157 + 0.901798i $$0.357753\pi$$
$$618$$ −3.44727e9 −0.587511
$$619$$ 8.53979e8 0.144720 0.0723602 0.997379i $$-0.476947\pi$$
0.0723602 + 0.997379i $$0.476947\pi$$
$$620$$ 5.67961e8 0.0957079
$$621$$ −1.64393e9 −0.275463
$$622$$ 7.11619e8 0.118572
$$623$$ 2.57359e9 0.426414
$$624$$ −3.00836e7 −0.00495661
$$625$$ 2.44141e8 0.0400000
$$626$$ 6.46072e9 1.05262
$$627$$ −5.27872e8 −0.0855249
$$628$$ −3.59767e9 −0.579645
$$629$$ 6.49895e9 1.04128
$$630$$ −6.10030e8 −0.0971984
$$631$$ 5.87967e8 0.0931645 0.0465823 0.998914i $$-0.485167\pi$$
0.0465823 + 0.998914i $$0.485167\pi$$
$$632$$ −7.40111e8 −0.116624
$$633$$ −1.33039e9 −0.208481
$$634$$ −5.50940e9 −0.858602
$$635$$ 3.32596e9 0.515477
$$636$$ 1.03222e9 0.159101
$$637$$ 3.35414e7 0.00514155
$$638$$ −8.94465e8 −0.136361
$$639$$ 1.85834e9 0.281755
$$640$$ −2.62144e8 −0.0395285
$$641$$ −1.94543e9 −0.291751 −0.145876 0.989303i $$-0.546600\pi$$
−0.145876 + 0.989303i $$0.546600\pi$$
$$642$$ 4.82096e9 0.719053
$$643$$ 9.67317e9 1.43493 0.717465 0.696595i $$-0.245303\pi$$
0.717465 + 0.696595i $$0.245303\pi$$
$$644$$ −4.47298e9 −0.659927
$$645$$ 1.87511e9 0.275149
$$646$$ −1.04207e9 −0.152083
$$647$$ −6.66124e9 −0.966919 −0.483459 0.875367i $$-0.660620\pi$$
−0.483459 + 0.875367i $$0.660620\pi$$
$$648$$ 2.72098e8 0.0392837
$$649$$ 6.80672e9 0.977420
$$650$$ −3.40030e7 −0.00485646
$$651$$ −1.60404e9 −0.227868
$$652$$ −6.21958e9 −0.878809
$$653$$ −1.14296e9 −0.160633 −0.0803167 0.996769i $$-0.525593\pi$$
−0.0803167 + 0.996769i $$0.525593\pi$$
$$654$$ −4.27426e9 −0.597501
$$655$$ 8.21192e8 0.114183
$$656$$ 5.24308e7 0.00725143
$$657$$ −2.26983e8 −0.0312259
$$658$$ 4.52363e9 0.619009
$$659$$ 6.76019e9 0.920152 0.460076 0.887879i $$-0.347822\pi$$
0.460076 + 0.887879i $$0.347822\pi$$
$$660$$ 6.15684e8 0.0833593
$$661$$ 1.05365e10 1.41903 0.709515 0.704691i $$-0.248914\pi$$
0.709515 + 0.704691i $$0.248914\pi$$
$$662$$ 1.18995e9 0.159414
$$663$$ 1.39481e8 0.0185874
$$664$$ 2.18940e9 0.290227
$$665$$ −7.17454e8 −0.0946059
$$666$$ −1.99580e9 −0.261792
$$667$$ −3.27614e9 −0.427486
$$668$$ 1.60088e9 0.207797
$$669$$ −5.69373e9 −0.735200
$$670$$ −1.27372e9 −0.163611
$$671$$ 7.71413e9 0.985731
$$672$$ 7.40350e8 0.0941119
$$673$$ −8.07302e9 −1.02090 −0.510450 0.859907i $$-0.670521\pi$$
−0.510450 + 0.859907i $$0.670521\pi$$
$$674$$ −2.68213e7 −0.00337419
$$675$$ 3.07547e8 0.0384900
$$676$$ −4.01117e9 −0.499410
$$677$$ 7.42536e9 0.919724 0.459862 0.887991i $$-0.347899\pi$$
0.459862 + 0.887991i $$0.347899\pi$$
$$678$$ −1.17571e9 −0.144876
$$679$$ 4.44926e9 0.545436
$$680$$ 1.21541e9 0.148232
$$681$$ 4.36737e8 0.0529913
$$682$$ 1.61891e9 0.195424
$$683$$ 8.93289e9 1.07280 0.536401 0.843963i $$-0.319783\pi$$
0.536401 + 0.843963i $$0.319783\pi$$
$$684$$ 3.20014e8 0.0382360
$$685$$ 3.00471e9 0.357178
$$686$$ −6.33859e9 −0.749650
$$687$$ 7.67974e9 0.903645
$$688$$ −2.27569e9 −0.266412
$$689$$ −1.62493e8 −0.0189264
$$690$$ 2.25505e9 0.261328
$$691$$ 4.10109e9 0.472853 0.236426 0.971649i $$-0.424024\pi$$
0.236426 + 0.971649i $$0.424024\pi$$
$$692$$ −5.03515e9 −0.577619
$$693$$ −1.73882e9 −0.198467
$$694$$ 6.83776e9 0.776526
$$695$$ 1.36703e9 0.154466
$$696$$ 5.42254e8 0.0609636
$$697$$ −2.43092e8 −0.0271930
$$698$$ −7.61487e8 −0.0847556
$$699$$ 5.20266e9 0.576176
$$700$$ 8.36803e8 0.0922105
$$701$$ −1.01608e10 −1.11408 −0.557040 0.830486i $$-0.688063\pi$$
−0.557040 + 0.830486i $$0.688063\pi$$
$$702$$ −4.28339e7 −0.00467313
$$703$$ −2.34725e9 −0.254810
$$704$$ −7.47212e8 −0.0807123
$$705$$ −2.28059e9 −0.245124
$$706$$ −7.36757e9 −0.787967
$$707$$ 1.14509e9 0.121863
$$708$$ −4.12646e9 −0.436979
$$709$$ 1.15232e10 1.21426 0.607129 0.794603i $$-0.292321\pi$$
0.607129 + 0.794603i $$0.292321\pi$$
$$710$$ −2.54916e9 −0.267297
$$711$$ −1.05379e9 −0.109954
$$712$$ 1.57466e9 0.163496
$$713$$ 5.92956e9 0.612645
$$714$$ −3.43259e9 −0.352921
$$715$$ −9.69217e7 −0.00991630
$$716$$ 2.02660e9 0.206334
$$717$$ −5.99508e9 −0.607404
$$718$$ −2.38213e9 −0.240177
$$719$$ 4.00451e9 0.401789 0.200895 0.979613i $$-0.435615\pi$$
0.200895 + 0.979613i $$0.435615\pi$$
$$720$$ −3.73248e8 −0.0372678
$$721$$ −1.33550e10 −1.32700
$$722$$ 3.76367e8 0.0372161
$$723$$ −1.14613e10 −1.12785
$$724$$ 9.16250e9 0.897282
$$725$$ 6.12900e8 0.0597319
$$726$$ −2.45429e9 −0.238039
$$727$$ 9.31051e9 0.898676 0.449338 0.893362i $$-0.351660\pi$$
0.449338 + 0.893362i $$0.351660\pi$$
$$728$$ −1.16547e8 −0.0111954
$$729$$ 3.87420e8 0.0370370
$$730$$ 3.11362e8 0.0296234
$$731$$ 1.05511e10 0.999050
$$732$$ −4.67656e9 −0.440695
$$733$$ 1.75464e10 1.64560 0.822798 0.568334i $$-0.192412\pi$$
0.822798 + 0.568334i $$0.192412\pi$$
$$734$$ 2.64380e8 0.0246770
$$735$$ 4.16149e8 0.0386583
$$736$$ −2.73680e9 −0.253029
$$737$$ −3.63061e9 −0.334075
$$738$$ 7.46525e7 0.00683671
$$739$$ 9.97408e9 0.909112 0.454556 0.890718i $$-0.349798\pi$$
0.454556 + 0.890718i $$0.349798\pi$$
$$740$$ 2.73772e9 0.248358
$$741$$ −5.03769e7 −0.00454849
$$742$$ 3.99891e9 0.359358
$$743$$ 1.46948e10 1.31433 0.657163 0.753749i $$-0.271756\pi$$
0.657163 + 0.753749i $$0.271756\pi$$
$$744$$ −9.81437e8 −0.0873690
$$745$$ −6.34203e9 −0.561929
$$746$$ 3.66268e9 0.323008
$$747$$ 3.11733e9 0.273628
$$748$$ 3.46440e9 0.302672
$$749$$ 1.86768e10 1.62411
$$750$$ −4.21875e8 −0.0365148
$$751$$ −5.42860e9 −0.467679 −0.233839 0.972275i $$-0.575129\pi$$
−0.233839 + 0.972275i $$0.575129\pi$$
$$752$$ 2.76780e9 0.237340
$$753$$ −9.98665e9 −0.852388
$$754$$ −8.53623e7 −0.00725214
$$755$$ 3.65558e9 0.309130
$$756$$ 1.05413e9 0.0887296
$$757$$ −1.31707e10 −1.10350 −0.551751 0.834009i $$-0.686040\pi$$
−0.551751 + 0.834009i $$0.686040\pi$$
$$758$$ −8.87182e9 −0.739896
$$759$$ 6.42778e9 0.533599
$$760$$ −4.38976e8 −0.0362738
$$761$$ −1.00210e10 −0.824260 −0.412130 0.911125i $$-0.635215\pi$$
−0.412130 + 0.911125i $$0.635215\pi$$
$$762$$ −5.74726e9 −0.470564
$$763$$ −1.65589e10 −1.34957
$$764$$ 1.64406e9 0.133380
$$765$$ 1.73054e9 0.139755
$$766$$ −1.09043e10 −0.876590
$$767$$ 6.49592e8 0.0519824
$$768$$ 4.52985e8 0.0360844
$$769$$ 9.00079e9 0.713737 0.356869 0.934155i $$-0.383844\pi$$
0.356869 + 0.934155i $$0.383844\pi$$
$$770$$ 2.38521e9 0.188282
$$771$$ −3.62388e9 −0.284763
$$772$$ 7.37876e7 0.00577196
$$773$$ −5.57639e9 −0.434235 −0.217118 0.976145i $$-0.569666\pi$$
−0.217118 + 0.976145i $$0.569666\pi$$
$$774$$ −3.24020e9 −0.251176
$$775$$ −1.10930e9 −0.0856038
$$776$$ 2.72229e9 0.209131
$$777$$ −7.73189e9 −0.591306
$$778$$ 9.76800e9 0.743664
$$779$$ 8.77986e7 0.00665436
$$780$$ 5.87571e7 0.00443332
$$781$$ −7.26611e9 −0.545787
$$782$$ 1.26890e10 0.948864
$$783$$ 7.72077e8 0.0574770
$$784$$ −5.05051e8 −0.0374308
$$785$$ 7.02669e9 0.518450
$$786$$ −1.41902e9 −0.104234
$$787$$ 3.98455e7 0.00291385 0.00145693 0.999999i $$-0.499536\pi$$
0.00145693 + 0.999999i $$0.499536\pi$$
$$788$$ 8.02974e9 0.584601
$$789$$ 7.84700e9 0.568767
$$790$$ 1.44553e9 0.104311
$$791$$ −4.55480e9 −0.327229
$$792$$ −1.06390e9 −0.0760963
$$793$$ 7.36190e8 0.0524244
$$794$$ 1.56036e10 1.10625
$$795$$ −2.01605e9 −0.142304
$$796$$ 5.03073e9 0.353537
$$797$$ −5.19745e9 −0.363652 −0.181826 0.983331i $$-0.558201\pi$$
−0.181826 + 0.983331i $$0.558201\pi$$
$$798$$ 1.23976e9 0.0863630
$$799$$ −1.28327e10 −0.890030
$$800$$ 5.12000e8 0.0353553
$$801$$ 2.24204e9 0.154145
$$802$$ −2.63163e9 −0.180142
$$803$$ 8.87503e8 0.0604875
$$804$$ 2.20100e9 0.149356
$$805$$ 8.73628e9 0.590257
$$806$$ 1.54499e8 0.0103933
$$807$$ −6.32225e9 −0.423462
$$808$$ 7.00628e8 0.0467248
$$809$$ 2.93360e9 0.194796 0.0973981 0.995246i $$-0.468948\pi$$
0.0973981 + 0.995246i $$0.468948\pi$$
$$810$$ −5.31441e8 −0.0351364
$$811$$ −2.05022e10 −1.34967 −0.674833 0.737970i $$-0.735784\pi$$
−0.674833 + 0.737970i $$0.735784\pi$$
$$812$$ 2.10074e9 0.137698
$$813$$ 1.28359e10 0.837738
$$814$$ 7.80356e9 0.507116
$$815$$ 1.21476e10 0.786030
$$816$$ −2.10024e9 −0.135317
$$817$$ −3.81079e9 −0.244477
$$818$$ 6.89580e9 0.440502
$$819$$ −1.65942e8 −0.0105551
$$820$$ −1.02404e8 −0.00648587
$$821$$ 4.09630e9 0.258339 0.129170 0.991623i $$-0.458769\pi$$
0.129170 + 0.991623i $$0.458769\pi$$
$$822$$ −5.19213e9 −0.326058
$$823$$ −2.13738e10 −1.33654 −0.668270 0.743919i $$-0.732965\pi$$
−0.668270 + 0.743919i $$0.732965\pi$$
$$824$$ −8.17131e9 −0.508799
$$825$$ −1.20251e9 −0.0745588
$$826$$ −1.59863e10 −0.986999
$$827$$ 1.24488e10 0.765346 0.382673 0.923884i $$-0.375004\pi$$
0.382673 + 0.923884i $$0.375004\pi$$
$$828$$ −3.89673e9 −0.238558
$$829$$ 2.19927e10 1.34072 0.670358 0.742038i $$-0.266141\pi$$
0.670358 + 0.742038i $$0.266141\pi$$
$$830$$ −4.27618e9 −0.259587
$$831$$ −1.40021e10 −0.846426
$$832$$ −7.13094e7 −0.00429255
$$833$$ 2.34164e9 0.140366
$$834$$ −2.36224e9 −0.141007
$$835$$ −3.12671e9 −0.185860
$$836$$ −1.25125e9 −0.0740667
$$837$$ −1.39740e9 −0.0823723
$$838$$ −1.41393e10 −0.829993
$$839$$ 2.85756e10 1.67043 0.835215 0.549923i $$-0.185343\pi$$
0.835215 + 0.549923i $$0.185343\pi$$
$$840$$ −1.44600e9 −0.0841763
$$841$$ −1.57112e10 −0.910803
$$842$$ −2.24462e9 −0.129584
$$843$$ 1.02348e10 0.588416
$$844$$ −3.15353e9 −0.180550
$$845$$ 7.83432e9 0.446686
$$846$$ 3.94087e9 0.223767
$$847$$ −9.50814e9 −0.537655
$$848$$ 2.44674e9 0.137785
$$849$$ −4.18345e9 −0.234616
$$850$$ −2.37386e9 −0.132583
$$851$$ 2.85819e10 1.58978
$$852$$ 4.40496e9 0.244007
$$853$$ −4.93353e9 −0.272168 −0.136084 0.990697i $$-0.543452\pi$$
−0.136084 + 0.990697i $$0.543452\pi$$
$$854$$ −1.81174e10 −0.995391
$$855$$ −6.25026e8 −0.0341993
$$856$$ 1.14275e10 0.622718
$$857$$ −2.34984e9 −0.127528 −0.0637641 0.997965i $$-0.520311\pi$$
−0.0637641 + 0.997965i $$0.520311\pi$$
$$858$$ 1.67481e8 0.00905230
$$859$$ −9.50673e9 −0.511747 −0.255873 0.966710i $$-0.582363\pi$$
−0.255873 + 0.966710i $$0.582363\pi$$
$$860$$ 4.44471e9 0.238286
$$861$$ 2.89210e8 0.0154420
$$862$$ 6.65858e9 0.354084
$$863$$ 1.22499e10 0.648778 0.324389 0.945924i $$-0.394841\pi$$
0.324389 + 0.945924i $$0.394841\pi$$
$$864$$ 6.44973e8 0.0340207
$$865$$ 9.83428e9 0.516638
$$866$$ 1.71403e10 0.896821
$$867$$ −1.34153e9 −0.0699092
$$868$$ −3.80218e9 −0.197339
$$869$$ 4.12032e9 0.212991
$$870$$ −1.05909e9 −0.0545275
$$871$$ −3.46483e8 −0.0177672
$$872$$ −1.01316e10 −0.517451
$$873$$ 3.87607e9 0.197171
$$874$$ −4.58294e9 −0.232196
$$875$$ −1.63438e9 −0.0824755
$$876$$ −5.38034e8 −0.0270424
$$877$$ −2.77100e10 −1.38719 −0.693597 0.720363i $$-0.743975\pi$$
−0.693597 + 0.720363i $$0.743975\pi$$
$$878$$ −1.77205e10 −0.883580
$$879$$ −1.33013e10 −0.660591
$$880$$ 1.45940e9 0.0721913
$$881$$ 1.72189e10 0.848379 0.424190 0.905573i $$-0.360559\pi$$
0.424190 + 0.905573i $$0.360559\pi$$
$$882$$ −7.19105e8 −0.0352901
$$883$$ 2.21121e10 1.08085 0.540427 0.841391i $$-0.318263\pi$$
0.540427 + 0.841391i $$0.318263\pi$$
$$884$$ 3.30621e8 0.0160971
$$885$$ 8.05949e9 0.390846
$$886$$ −1.12591e10 −0.543856
$$887$$ 3.13319e10 1.50749 0.753745 0.657167i $$-0.228245\pi$$
0.753745 + 0.657167i $$0.228245\pi$$
$$888$$ −4.73078e9 −0.226718
$$889$$ −2.22654e10 −1.06286
$$890$$ −3.07550e9 −0.146235
$$891$$ −1.51481e9 −0.0717443
$$892$$ −1.34963e10 −0.636702
$$893$$ 4.63484e9 0.217799
$$894$$ 1.09590e10 0.512969
$$895$$ −3.95819e9 −0.184551
$$896$$ 1.75490e9 0.0815033
$$897$$ 6.13428e8 0.0283786
$$898$$ 1.14104e10 0.525817
$$899$$ −2.78483e9 −0.127832
$$900$$ 7.29000e8 0.0333333
$$901$$ −1.13442e10 −0.516697
$$902$$ −2.91891e8 −0.0132434
$$903$$ −1.25528e10 −0.567327
$$904$$ −2.78687e9 −0.125466
$$905$$ −1.78955e10 −0.802553
$$906$$ −6.31684e9 −0.282196
$$907$$ −2.96071e10 −1.31756 −0.658781 0.752335i $$-0.728927\pi$$
−0.658781 + 0.752335i $$0.728927\pi$$
$$908$$ 1.03523e9 0.0458918
$$909$$ 9.97574e8 0.0440526
$$910$$ 2.27630e8 0.0100135
$$911$$ −8.84001e9 −0.387381 −0.193690 0.981063i $$-0.562046\pi$$
−0.193690 + 0.981063i $$0.562046\pi$$
$$912$$ 7.58551e8 0.0331133
$$913$$ −1.21888e10 −0.530044
$$914$$ 2.05350e10 0.889575
$$915$$ 9.13391e9 0.394169
$$916$$ 1.82038e10 0.782580
$$917$$ −5.49741e9 −0.235432
$$918$$ −2.99037e9 −0.127578
$$919$$ −1.09414e10 −0.465018 −0.232509 0.972594i $$-0.574693\pi$$
−0.232509 + 0.972594i $$0.574693\pi$$
$$920$$ 5.34531e9 0.226316
$$921$$ 3.85388e9 0.162551
$$922$$ 8.96662e9 0.376765
$$923$$ −6.93433e8 −0.0290268
$$924$$ −4.12165e9 −0.171878
$$925$$ −5.34710e9 −0.222138
$$926$$ 1.42308e10 0.588965
$$927$$ −1.16345e10 −0.479701
$$928$$ 1.28534e9 0.0527960
$$929$$ 1.20285e10 0.492216 0.246108 0.969242i $$-0.420848\pi$$
0.246108 + 0.969242i $$0.420848\pi$$
$$930$$ 1.91687e9 0.0781452
$$931$$ −8.45738e8 −0.0343488
$$932$$ 1.23322e10 0.498983
$$933$$ 2.40171e9 0.0968134
$$934$$ 1.17964e10 0.473735
$$935$$ −6.76641e9 −0.270718
$$936$$ −1.01532e8 −0.00404705
$$937$$ −1.05850e10 −0.420341 −0.210171 0.977665i $$-0.567402\pi$$
−0.210171 + 0.977665i $$0.567402\pi$$
$$938$$ 8.52685e9 0.337349
$$939$$ 2.18049e10 0.859459
$$940$$ −5.40585e9 −0.212284
$$941$$ −2.76549e10 −1.08195 −0.540977 0.841037i $$-0.681945\pi$$
−0.540977 + 0.841037i $$0.681945\pi$$
$$942$$ −1.21421e10 −0.473278
$$943$$ −1.06910e9 −0.0415173
$$944$$ −9.78123e9 −0.378435
$$945$$ −2.05885e9 −0.0793621
$$946$$ 1.26692e10 0.486552
$$947$$ 4.89171e7 0.00187170 0.000935849 1.00000i $$-0.499702\pi$$
0.000935849 1.00000i $$0.499702\pi$$
$$948$$ −2.49787e9 −0.0952229
$$949$$ 8.46979e7 0.00321692
$$950$$ 8.57375e8 0.0324443
$$951$$ −1.85942e10 −0.701046
$$952$$ −8.13650e9 −0.305639
$$953$$ 2.42208e10 0.906489 0.453245 0.891386i $$-0.350266\pi$$
0.453245 + 0.891386i $$0.350266\pi$$
$$954$$ 3.48374e9 0.129905
$$955$$ −3.21105e9 −0.119299
$$956$$ −1.42106e10 −0.526028
$$957$$ −3.01882e9 −0.111339
$$958$$ −1.85158e10 −0.680397
$$959$$ −2.01148e10 −0.736462
$$960$$ −8.84736e8 −0.0322749
$$961$$ −2.24723e10 −0.816800
$$962$$ 7.44724e8 0.0269701
$$963$$ 1.62707e10 0.587104
$$964$$ −2.71676e10 −0.976747
$$965$$ −1.44116e8 −0.00516259
$$966$$ −1.50963e10 −0.538828
$$967$$ −4.40304e10 −1.56589 −0.782944 0.622093i $$-0.786283\pi$$
−0.782944 + 0.622093i $$0.786283\pi$$
$$968$$ −5.81758e9 −0.206148
$$969$$ −3.51697e9 −0.124175
$$970$$ −5.31697e9 −0.187052
$$971$$ −2.09837e10 −0.735554 −0.367777 0.929914i $$-0.619881\pi$$
−0.367777 + 0.929914i $$0.619881\pi$$
$$972$$ 9.18330e8 0.0320750
$$973$$ −9.15151e9 −0.318492
$$974$$ 1.78498e10 0.618981
$$975$$ −1.14760e8 −0.00396528
$$976$$ −1.10852e10 −0.381653
$$977$$ −3.82831e10 −1.31334 −0.656669 0.754179i $$-0.728035\pi$$
−0.656669 + 0.754179i $$0.728035\pi$$
$$978$$ −2.09911e10 −0.717544
$$979$$ −8.76638e9 −0.298594
$$980$$ 9.86427e8 0.0334791
$$981$$ −1.44256e10 −0.487858
$$982$$ 4.24974e8 0.0143210
$$983$$ 2.18431e9 0.0733460 0.0366730 0.999327i $$-0.488324\pi$$
0.0366730 + 0.999327i $$0.488324\pi$$
$$984$$ 1.76954e8 0.00592076
$$985$$ −1.56831e10 −0.522883
$$986$$ −5.95941e9 −0.197986
$$987$$ 1.52673e10 0.505419
$$988$$ −1.19412e8 −0.00393911
$$989$$ 4.64031e10 1.52532
$$990$$ 2.07793e9 0.0680626
$$991$$ −3.30368e10 −1.07830 −0.539151 0.842209i $$-0.681255\pi$$
−0.539151 + 0.842209i $$0.681255\pi$$
$$992$$ −2.32637e9 −0.0756638
$$993$$ 4.01609e9 0.130161
$$994$$ 1.70652e10 0.551136
$$995$$ −9.82565e9 −0.316214
$$996$$ 7.38924e9 0.236969
$$997$$ 4.25291e10 1.35911 0.679553 0.733627i $$-0.262174\pi$$
0.679553 + 0.733627i $$0.262174\pi$$
$$998$$ 3.14314e10 1.00094
$$999$$ −6.73581e9 −0.213752
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 570.8.a.b.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.b.1.4 4 1.1 even 1 trivial