Properties

Label 138.8.a.h.1.4
Level $138$
Weight $8$
Character 138.1
Self dual yes
Analytic conductor $43.109$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.1091335168\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(-56.1267\) of defining polynomial
Character \(\chi\) \(=\) 138.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +366.876 q^{5} +216.000 q^{6} -561.721 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +366.876 q^{5} +216.000 q^{6} -561.721 q^{7} +512.000 q^{8} +729.000 q^{9} +2935.00 q^{10} -1096.20 q^{11} +1728.00 q^{12} +6925.11 q^{13} -4493.77 q^{14} +9905.64 q^{15} +4096.00 q^{16} +13654.6 q^{17} +5832.00 q^{18} +29540.8 q^{19} +23480.0 q^{20} -15166.5 q^{21} -8769.58 q^{22} -12167.0 q^{23} +13824.0 q^{24} +56472.7 q^{25} +55400.9 q^{26} +19683.0 q^{27} -35950.1 q^{28} +150266. q^{29} +79245.1 q^{30} -204841. q^{31} +32768.0 q^{32} -29597.3 q^{33} +109237. q^{34} -206082. q^{35} +46656.0 q^{36} -492163. q^{37} +236327. q^{38} +186978. q^{39} +187840. q^{40} +416521. q^{41} -121332. q^{42} +371500. q^{43} -70156.6 q^{44} +267452. q^{45} -97336.0 q^{46} +930660. q^{47} +110592. q^{48} -508013. q^{49} +451782. q^{50} +368675. q^{51} +443207. q^{52} +1.01318e6 q^{53} +157464. q^{54} -402168. q^{55} -287601. q^{56} +797602. q^{57} +1.20213e6 q^{58} -2.28957e6 q^{59} +633961. q^{60} +2.95586e6 q^{61} -1.63873e6 q^{62} -409495. q^{63} +262144. q^{64} +2.54066e6 q^{65} -236779. q^{66} -1.46138e6 q^{67} +873896. q^{68} -328509. q^{69} -1.64865e6 q^{70} -1.82588e6 q^{71} +373248. q^{72} -40352.1 q^{73} -3.93731e6 q^{74} +1.52476e6 q^{75} +1.89061e6 q^{76} +615757. q^{77} +1.49582e6 q^{78} -2.36229e6 q^{79} +1.50272e6 q^{80} +531441. q^{81} +3.33217e6 q^{82} +1.27885e6 q^{83} -970654. q^{84} +5.00955e6 q^{85} +2.97200e6 q^{86} +4.05718e6 q^{87} -561253. q^{88} +3.88462e6 q^{89} +2.13962e6 q^{90} -3.88998e6 q^{91} -778688. q^{92} -5.53072e6 q^{93} +7.44528e6 q^{94} +1.08378e7 q^{95} +884736. q^{96} -8.46502e6 q^{97} -4.06410e6 q^{98} -799128. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) 366.876 1.31257 0.656287 0.754511i \(-0.272126\pi\)
0.656287 + 0.754511i \(0.272126\pi\)
\(6\) 216.000 0.408248
\(7\) −561.721 −0.618981 −0.309491 0.950903i \(-0.600159\pi\)
−0.309491 + 0.950903i \(0.600159\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 2935.00 0.928130
\(11\) −1096.20 −0.248322 −0.124161 0.992262i \(-0.539624\pi\)
−0.124161 + 0.992262i \(0.539624\pi\)
\(12\) 1728.00 0.288675
\(13\) 6925.11 0.874229 0.437115 0.899406i \(-0.356000\pi\)
0.437115 + 0.899406i \(0.356000\pi\)
\(14\) −4493.77 −0.437686
\(15\) 9905.64 0.757815
\(16\) 4096.00 0.250000
\(17\) 13654.6 0.674075 0.337038 0.941491i \(-0.390575\pi\)
0.337038 + 0.941491i \(0.390575\pi\)
\(18\) 5832.00 0.235702
\(19\) 29540.8 0.988064 0.494032 0.869444i \(-0.335523\pi\)
0.494032 + 0.869444i \(0.335523\pi\)
\(20\) 23480.0 0.656287
\(21\) −15166.5 −0.357369
\(22\) −8769.58 −0.175590
\(23\) −12167.0 −0.208514
\(24\) 13824.0 0.204124
\(25\) 56472.7 0.722851
\(26\) 55400.9 0.618173
\(27\) 19683.0 0.192450
\(28\) −35950.1 −0.309491
\(29\) 150266. 1.14411 0.572054 0.820216i \(-0.306147\pi\)
0.572054 + 0.820216i \(0.306147\pi\)
\(30\) 79245.1 0.535856
\(31\) −204841. −1.23496 −0.617479 0.786588i \(-0.711846\pi\)
−0.617479 + 0.786588i \(0.711846\pi\)
\(32\) 32768.0 0.176777
\(33\) −29597.3 −0.143368
\(34\) 109237. 0.476643
\(35\) −206082. −0.812459
\(36\) 46656.0 0.166667
\(37\) −492163. −1.59736 −0.798681 0.601755i \(-0.794468\pi\)
−0.798681 + 0.601755i \(0.794468\pi\)
\(38\) 236327. 0.698667
\(39\) 186978. 0.504736
\(40\) 187840. 0.464065
\(41\) 416521. 0.943829 0.471915 0.881644i \(-0.343563\pi\)
0.471915 + 0.881644i \(0.343563\pi\)
\(42\) −121332. −0.252698
\(43\) 371500. 0.712557 0.356279 0.934380i \(-0.384045\pi\)
0.356279 + 0.934380i \(0.384045\pi\)
\(44\) −70156.6 −0.124161
\(45\) 267452. 0.437525
\(46\) −97336.0 −0.147442
\(47\) 930660. 1.30752 0.653760 0.756702i \(-0.273191\pi\)
0.653760 + 0.756702i \(0.273191\pi\)
\(48\) 110592. 0.144338
\(49\) −508013. −0.616862
\(50\) 451782. 0.511133
\(51\) 368675. 0.389177
\(52\) 443207. 0.437115
\(53\) 1.01318e6 0.934804 0.467402 0.884045i \(-0.345190\pi\)
0.467402 + 0.884045i \(0.345190\pi\)
\(54\) 157464. 0.136083
\(55\) −402168. −0.325940
\(56\) −287601. −0.218843
\(57\) 797602. 0.570459
\(58\) 1.20213e6 0.809007
\(59\) −2.28957e6 −1.45135 −0.725676 0.688036i \(-0.758473\pi\)
−0.725676 + 0.688036i \(0.758473\pi\)
\(60\) 633961. 0.378908
\(61\) 2.95586e6 1.66736 0.833681 0.552246i \(-0.186229\pi\)
0.833681 + 0.552246i \(0.186229\pi\)
\(62\) −1.63873e6 −0.873246
\(63\) −409495. −0.206327
\(64\) 262144. 0.125000
\(65\) 2.54066e6 1.14749
\(66\) −236779. −0.101377
\(67\) −1.46138e6 −0.593609 −0.296804 0.954938i \(-0.595921\pi\)
−0.296804 + 0.954938i \(0.595921\pi\)
\(68\) 873896. 0.337038
\(69\) −328509. −0.120386
\(70\) −1.64865e6 −0.574495
\(71\) −1.82588e6 −0.605437 −0.302718 0.953080i \(-0.597894\pi\)
−0.302718 + 0.953080i \(0.597894\pi\)
\(72\) 373248. 0.117851
\(73\) −40352.1 −0.0121405 −0.00607024 0.999982i \(-0.501932\pi\)
−0.00607024 + 0.999982i \(0.501932\pi\)
\(74\) −3.93731e6 −1.12951
\(75\) 1.52476e6 0.417338
\(76\) 1.89061e6 0.494032
\(77\) 615757. 0.153706
\(78\) 1.49582e6 0.356903
\(79\) −2.36229e6 −0.539061 −0.269531 0.962992i \(-0.586869\pi\)
−0.269531 + 0.962992i \(0.586869\pi\)
\(80\) 1.50272e6 0.328144
\(81\) 531441. 0.111111
\(82\) 3.33217e6 0.667388
\(83\) 1.27885e6 0.245497 0.122749 0.992438i \(-0.460829\pi\)
0.122749 + 0.992438i \(0.460829\pi\)
\(84\) −970654. −0.178684
\(85\) 5.00955e6 0.884774
\(86\) 2.97200e6 0.503854
\(87\) 4.05718e6 0.660551
\(88\) −561253. −0.0877949
\(89\) 3.88462e6 0.584095 0.292047 0.956404i \(-0.405664\pi\)
0.292047 + 0.956404i \(0.405664\pi\)
\(90\) 2.13962e6 0.309377
\(91\) −3.88998e6 −0.541131
\(92\) −778688. −0.104257
\(93\) −5.53072e6 −0.713003
\(94\) 7.44528e6 0.924556
\(95\) 1.08378e7 1.29691
\(96\) 884736. 0.102062
\(97\) −8.46502e6 −0.941731 −0.470865 0.882205i \(-0.656058\pi\)
−0.470865 + 0.882205i \(0.656058\pi\)
\(98\) −4.06410e6 −0.436188
\(99\) −799128. −0.0827738
\(100\) 3.61425e6 0.361425
\(101\) 5.79563e6 0.559727 0.279863 0.960040i \(-0.409711\pi\)
0.279863 + 0.960040i \(0.409711\pi\)
\(102\) 2.94940e6 0.275190
\(103\) 1.34033e6 0.120860 0.0604298 0.998172i \(-0.480753\pi\)
0.0604298 + 0.998172i \(0.480753\pi\)
\(104\) 3.54566e6 0.309087
\(105\) −5.56421e6 −0.469073
\(106\) 8.10543e6 0.661006
\(107\) −1.37815e7 −1.08756 −0.543779 0.839229i \(-0.683007\pi\)
−0.543779 + 0.839229i \(0.683007\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 9.40692e6 0.695752 0.347876 0.937541i \(-0.386903\pi\)
0.347876 + 0.937541i \(0.386903\pi\)
\(110\) −3.21734e6 −0.230475
\(111\) −1.32884e7 −0.922237
\(112\) −2.30081e6 −0.154745
\(113\) −776740. −0.0506409 −0.0253204 0.999679i \(-0.508061\pi\)
−0.0253204 + 0.999679i \(0.508061\pi\)
\(114\) 6.38082e6 0.403375
\(115\) −4.46378e6 −0.273691
\(116\) 9.61701e6 0.572054
\(117\) 5.04841e6 0.291410
\(118\) −1.83166e7 −1.02626
\(119\) −7.67008e6 −0.417240
\(120\) 5.07169e6 0.267928
\(121\) −1.82855e7 −0.938336
\(122\) 2.36469e7 1.17900
\(123\) 1.12461e7 0.544920
\(124\) −1.31099e7 −0.617479
\(125\) −7.94369e6 −0.363779
\(126\) −3.27596e6 −0.145895
\(127\) −2.02779e7 −0.878434 −0.439217 0.898381i \(-0.644744\pi\)
−0.439217 + 0.898381i \(0.644744\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 1.00305e7 0.411395
\(130\) 2.03252e7 0.811398
\(131\) −5.73353e6 −0.222829 −0.111415 0.993774i \(-0.535538\pi\)
−0.111415 + 0.993774i \(0.535538\pi\)
\(132\) −1.89423e6 −0.0716842
\(133\) −1.65937e7 −0.611593
\(134\) −1.16910e7 −0.419745
\(135\) 7.22121e6 0.252605
\(136\) 6.99116e6 0.238322
\(137\) −2.96971e7 −0.986716 −0.493358 0.869826i \(-0.664231\pi\)
−0.493358 + 0.869826i \(0.664231\pi\)
\(138\) −2.62807e6 −0.0851257
\(139\) 2.70200e7 0.853362 0.426681 0.904402i \(-0.359683\pi\)
0.426681 + 0.904402i \(0.359683\pi\)
\(140\) −1.31892e7 −0.406229
\(141\) 2.51278e7 0.754897
\(142\) −1.46071e7 −0.428108
\(143\) −7.59129e6 −0.217090
\(144\) 2.98598e6 0.0833333
\(145\) 5.51289e7 1.50173
\(146\) −322816. −0.00858461
\(147\) −1.37163e7 −0.356146
\(148\) −3.14984e7 −0.798681
\(149\) −1.22268e7 −0.302803 −0.151401 0.988472i \(-0.548379\pi\)
−0.151401 + 0.988472i \(0.548379\pi\)
\(150\) 1.21981e7 0.295103
\(151\) −7.91442e7 −1.87068 −0.935341 0.353747i \(-0.884907\pi\)
−0.935341 + 0.353747i \(0.884907\pi\)
\(152\) 1.51249e7 0.349333
\(153\) 9.95422e6 0.224692
\(154\) 4.92606e6 0.108687
\(155\) −7.51513e7 −1.62097
\(156\) 1.19666e7 0.252368
\(157\) −6.65855e7 −1.37319 −0.686595 0.727040i \(-0.740895\pi\)
−0.686595 + 0.727040i \(0.740895\pi\)
\(158\) −1.88983e7 −0.381174
\(159\) 2.73558e7 0.539709
\(160\) 1.20218e7 0.232033
\(161\) 6.83446e6 0.129066
\(162\) 4.25153e6 0.0785674
\(163\) 5.57898e7 1.00902 0.504508 0.863407i \(-0.331674\pi\)
0.504508 + 0.863407i \(0.331674\pi\)
\(164\) 2.66573e7 0.471915
\(165\) −1.08585e7 −0.188182
\(166\) 1.02308e7 0.173593
\(167\) −2.85952e7 −0.475100 −0.237550 0.971375i \(-0.576344\pi\)
−0.237550 + 0.971375i \(0.576344\pi\)
\(168\) −7.76523e6 −0.126349
\(169\) −1.47913e7 −0.235724
\(170\) 4.00764e7 0.625629
\(171\) 2.15353e7 0.329355
\(172\) 2.37760e7 0.356279
\(173\) −5.94429e7 −0.872847 −0.436424 0.899741i \(-0.643755\pi\)
−0.436424 + 0.899741i \(0.643755\pi\)
\(174\) 3.24574e7 0.467080
\(175\) −3.17219e7 −0.447431
\(176\) −4.49002e6 −0.0620804
\(177\) −6.18185e7 −0.837939
\(178\) 3.10769e7 0.413017
\(179\) −1.09127e8 −1.42216 −0.711079 0.703112i \(-0.751793\pi\)
−0.711079 + 0.703112i \(0.751793\pi\)
\(180\) 1.71169e7 0.218762
\(181\) −9.89431e7 −1.24025 −0.620127 0.784501i \(-0.712919\pi\)
−0.620127 + 0.784501i \(0.712919\pi\)
\(182\) −3.11199e7 −0.382638
\(183\) 7.98083e7 0.962652
\(184\) −6.22950e6 −0.0737210
\(185\) −1.80563e8 −2.09666
\(186\) −4.42457e7 −0.504169
\(187\) −1.49682e7 −0.167387
\(188\) 5.95622e7 0.653760
\(189\) −1.10564e7 −0.119123
\(190\) 8.67025e7 0.917052
\(191\) −6.19114e7 −0.642916 −0.321458 0.946924i \(-0.604173\pi\)
−0.321458 + 0.946924i \(0.604173\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 1.03484e8 1.03615 0.518076 0.855334i \(-0.326648\pi\)
0.518076 + 0.855334i \(0.326648\pi\)
\(194\) −6.77201e7 −0.665904
\(195\) 6.85977e7 0.662504
\(196\) −3.25128e7 −0.308431
\(197\) −1.11986e8 −1.04360 −0.521798 0.853069i \(-0.674739\pi\)
−0.521798 + 0.853069i \(0.674739\pi\)
\(198\) −6.39302e6 −0.0585299
\(199\) −1.09861e7 −0.0988228 −0.0494114 0.998779i \(-0.515735\pi\)
−0.0494114 + 0.998779i \(0.515735\pi\)
\(200\) 2.89140e7 0.255566
\(201\) −3.94572e7 −0.342720
\(202\) 4.63651e7 0.395787
\(203\) −8.44074e7 −0.708182
\(204\) 2.35952e7 0.194589
\(205\) 1.52811e8 1.23885
\(206\) 1.07226e7 0.0854606
\(207\) −8.86974e6 −0.0695048
\(208\) 2.83653e7 0.218557
\(209\) −3.23826e7 −0.245357
\(210\) −4.45136e7 −0.331685
\(211\) 7.14052e7 0.523289 0.261644 0.965164i \(-0.415735\pi\)
0.261644 + 0.965164i \(0.415735\pi\)
\(212\) 6.48434e7 0.467402
\(213\) −4.92988e7 −0.349549
\(214\) −1.10252e8 −0.769019
\(215\) 1.36294e8 0.935284
\(216\) 1.00777e7 0.0680414
\(217\) 1.15064e8 0.764415
\(218\) 7.52553e7 0.491971
\(219\) −1.08951e6 −0.00700931
\(220\) −2.57388e7 −0.162970
\(221\) 9.45598e7 0.589296
\(222\) −1.06307e8 −0.652120
\(223\) 1.21124e8 0.731415 0.365707 0.930730i \(-0.380827\pi\)
0.365707 + 0.930730i \(0.380827\pi\)
\(224\) −1.84065e7 −0.109421
\(225\) 4.11686e7 0.240950
\(226\) −6.21392e6 −0.0358085
\(227\) 3.23693e7 0.183672 0.0918360 0.995774i \(-0.470726\pi\)
0.0918360 + 0.995774i \(0.470726\pi\)
\(228\) 5.10465e7 0.285229
\(229\) 2.14673e8 1.18128 0.590642 0.806934i \(-0.298875\pi\)
0.590642 + 0.806934i \(0.298875\pi\)
\(230\) −3.57102e7 −0.193528
\(231\) 1.66254e7 0.0887424
\(232\) 7.69361e7 0.404503
\(233\) −2.34580e8 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(234\) 4.03873e7 0.206058
\(235\) 3.41436e8 1.71622
\(236\) −1.46533e8 −0.725676
\(237\) −6.37818e7 −0.311227
\(238\) −6.13607e7 −0.295033
\(239\) −1.73526e8 −0.822189 −0.411095 0.911593i \(-0.634853\pi\)
−0.411095 + 0.911593i \(0.634853\pi\)
\(240\) 4.05735e7 0.189454
\(241\) 1.89649e8 0.872752 0.436376 0.899764i \(-0.356262\pi\)
0.436376 + 0.899764i \(0.356262\pi\)
\(242\) −1.46284e8 −0.663504
\(243\) 1.43489e7 0.0641500
\(244\) 1.89175e8 0.833681
\(245\) −1.86377e8 −0.809678
\(246\) 8.99685e7 0.385317
\(247\) 2.04574e8 0.863794
\(248\) −1.04879e8 −0.436623
\(249\) 3.45290e7 0.141738
\(250\) −6.35495e7 −0.257230
\(251\) 2.71105e8 1.08213 0.541064 0.840981i \(-0.318022\pi\)
0.541064 + 0.840981i \(0.318022\pi\)
\(252\) −2.62076e7 −0.103164
\(253\) 1.33374e7 0.0517786
\(254\) −1.62223e8 −0.621147
\(255\) 1.35258e8 0.510824
\(256\) 1.67772e7 0.0625000
\(257\) 4.36260e8 1.60317 0.801584 0.597882i \(-0.203991\pi\)
0.801584 + 0.597882i \(0.203991\pi\)
\(258\) 8.02441e7 0.290900
\(259\) 2.76458e8 0.988737
\(260\) 1.62602e8 0.573745
\(261\) 1.09544e8 0.381369
\(262\) −4.58682e7 −0.157564
\(263\) 1.27332e8 0.431609 0.215805 0.976437i \(-0.430763\pi\)
0.215805 + 0.976437i \(0.430763\pi\)
\(264\) −1.51538e7 −0.0506884
\(265\) 3.71711e8 1.22700
\(266\) −1.32750e8 −0.432461
\(267\) 1.04885e8 0.337227
\(268\) −9.35281e7 −0.296804
\(269\) 2.50990e8 0.786182 0.393091 0.919500i \(-0.371406\pi\)
0.393091 + 0.919500i \(0.371406\pi\)
\(270\) 5.77697e7 0.178619
\(271\) 1.19209e8 0.363844 0.181922 0.983313i \(-0.441768\pi\)
0.181922 + 0.983313i \(0.441768\pi\)
\(272\) 5.59293e7 0.168519
\(273\) −1.05029e8 −0.312422
\(274\) −2.37577e8 −0.697714
\(275\) −6.19052e7 −0.179499
\(276\) −2.10246e7 −0.0601929
\(277\) −2.70183e8 −0.763799 −0.381899 0.924204i \(-0.624730\pi\)
−0.381899 + 0.924204i \(0.624730\pi\)
\(278\) 2.16160e8 0.603418
\(279\) −1.49329e8 −0.411652
\(280\) −1.05514e8 −0.287248
\(281\) −6.33010e8 −1.70192 −0.850958 0.525233i \(-0.823978\pi\)
−0.850958 + 0.525233i \(0.823978\pi\)
\(282\) 2.01022e8 0.533793
\(283\) 6.68305e8 1.75276 0.876379 0.481622i \(-0.159952\pi\)
0.876379 + 0.481622i \(0.159952\pi\)
\(284\) −1.16857e8 −0.302718
\(285\) 2.92621e8 0.748770
\(286\) −6.07303e7 −0.153506
\(287\) −2.33969e8 −0.584213
\(288\) 2.38879e7 0.0589256
\(289\) −2.23890e8 −0.545623
\(290\) 4.41031e8 1.06188
\(291\) −2.28555e8 −0.543709
\(292\) −2.58253e6 −0.00607024
\(293\) 5.36402e8 1.24582 0.622908 0.782295i \(-0.285951\pi\)
0.622908 + 0.782295i \(0.285951\pi\)
\(294\) −1.09731e8 −0.251833
\(295\) −8.39989e8 −1.90501
\(296\) −2.51988e8 −0.564753
\(297\) −2.15765e7 −0.0477895
\(298\) −9.78141e7 −0.214114
\(299\) −8.42579e7 −0.182289
\(300\) 9.75849e7 0.208669
\(301\) −2.08680e8 −0.441059
\(302\) −6.33154e8 −1.32277
\(303\) 1.56482e8 0.323158
\(304\) 1.20999e8 0.247016
\(305\) 1.08443e9 2.18854
\(306\) 7.96337e7 0.158881
\(307\) −1.07221e8 −0.211493 −0.105746 0.994393i \(-0.533723\pi\)
−0.105746 + 0.994393i \(0.533723\pi\)
\(308\) 3.94084e7 0.0768532
\(309\) 3.61889e7 0.0697783
\(310\) −6.01211e8 −1.14620
\(311\) −9.74611e8 −1.83726 −0.918629 0.395122i \(-0.870702\pi\)
−0.918629 + 0.395122i \(0.870702\pi\)
\(312\) 9.57328e7 0.178451
\(313\) −6.47093e8 −1.19278 −0.596392 0.802693i \(-0.703400\pi\)
−0.596392 + 0.802693i \(0.703400\pi\)
\(314\) −5.32684e8 −0.970992
\(315\) −1.50234e8 −0.270820
\(316\) −1.51186e8 −0.269531
\(317\) 5.22962e8 0.922068 0.461034 0.887382i \(-0.347479\pi\)
0.461034 + 0.887382i \(0.347479\pi\)
\(318\) 2.18847e8 0.381632
\(319\) −1.64721e8 −0.284107
\(320\) 9.61742e7 0.164072
\(321\) −3.72100e8 −0.627902
\(322\) 5.46757e7 0.0912638
\(323\) 4.03369e8 0.666029
\(324\) 3.40122e7 0.0555556
\(325\) 3.91080e8 0.631937
\(326\) 4.46319e8 0.713483
\(327\) 2.53987e8 0.401693
\(328\) 2.13259e8 0.333694
\(329\) −5.22771e8 −0.809330
\(330\) −8.68683e7 −0.133065
\(331\) −1.00533e8 −0.152373 −0.0761867 0.997094i \(-0.524275\pi\)
−0.0761867 + 0.997094i \(0.524275\pi\)
\(332\) 8.18465e7 0.122749
\(333\) −3.58787e8 −0.532454
\(334\) −2.28761e8 −0.335946
\(335\) −5.36144e8 −0.779156
\(336\) −6.21218e7 −0.0893422
\(337\) −4.77653e8 −0.679842 −0.339921 0.940454i \(-0.610400\pi\)
−0.339921 + 0.940454i \(0.610400\pi\)
\(338\) −1.18330e8 −0.166682
\(339\) −2.09720e7 −0.0292375
\(340\) 3.20611e8 0.442387
\(341\) 2.24547e8 0.306666
\(342\) 1.72282e8 0.232889
\(343\) 7.47963e8 1.00081
\(344\) 1.90208e8 0.251927
\(345\) −1.20522e8 −0.158015
\(346\) −4.75543e8 −0.617196
\(347\) −9.37582e8 −1.20464 −0.602318 0.798256i \(-0.705756\pi\)
−0.602318 + 0.798256i \(0.705756\pi\)
\(348\) 2.59659e8 0.330276
\(349\) 9.69080e8 1.22031 0.610155 0.792282i \(-0.291107\pi\)
0.610155 + 0.792282i \(0.291107\pi\)
\(350\) −2.53775e8 −0.316382
\(351\) 1.36307e8 0.168245
\(352\) −3.59202e7 −0.0438975
\(353\) −4.61929e8 −0.558938 −0.279469 0.960155i \(-0.590158\pi\)
−0.279469 + 0.960155i \(0.590158\pi\)
\(354\) −4.94548e8 −0.592512
\(355\) −6.69872e8 −0.794680
\(356\) 2.48616e8 0.292047
\(357\) −2.07092e8 −0.240894
\(358\) −8.73019e8 −1.00562
\(359\) −7.75397e8 −0.884491 −0.442245 0.896894i \(-0.645818\pi\)
−0.442245 + 0.896894i \(0.645818\pi\)
\(360\) 1.36936e8 0.154688
\(361\) −2.12115e7 −0.0237299
\(362\) −7.91545e8 −0.876992
\(363\) −4.93709e8 −0.541749
\(364\) −2.48959e8 −0.270566
\(365\) −1.48042e7 −0.0159353
\(366\) 6.38467e8 0.680698
\(367\) −2.89183e8 −0.305381 −0.152690 0.988274i \(-0.548794\pi\)
−0.152690 + 0.988274i \(0.548794\pi\)
\(368\) −4.98360e7 −0.0521286
\(369\) 3.03644e8 0.314610
\(370\) −1.44450e9 −1.48256
\(371\) −5.69124e8 −0.578626
\(372\) −3.53966e8 −0.356501
\(373\) 6.15666e8 0.614277 0.307138 0.951665i \(-0.400629\pi\)
0.307138 + 0.951665i \(0.400629\pi\)
\(374\) −1.19745e8 −0.118361
\(375\) −2.14480e8 −0.210028
\(376\) 4.76498e8 0.462278
\(377\) 1.04061e9 1.00021
\(378\) −8.84508e7 −0.0842327
\(379\) 9.95325e8 0.939134 0.469567 0.882897i \(-0.344410\pi\)
0.469567 + 0.882897i \(0.344410\pi\)
\(380\) 6.93620e8 0.648453
\(381\) −5.47502e8 −0.507164
\(382\) −4.95292e8 −0.454610
\(383\) −2.09760e8 −0.190777 −0.0953887 0.995440i \(-0.530409\pi\)
−0.0953887 + 0.995440i \(0.530409\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 2.25906e8 0.201751
\(386\) 8.27874e8 0.732671
\(387\) 2.70824e8 0.237519
\(388\) −5.41761e8 −0.470865
\(389\) 2.53008e8 0.217927 0.108963 0.994046i \(-0.465247\pi\)
0.108963 + 0.994046i \(0.465247\pi\)
\(390\) 5.48782e8 0.468461
\(391\) −1.66136e8 −0.140554
\(392\) −2.60102e8 −0.218094
\(393\) −1.54805e8 −0.128651
\(394\) −8.95889e8 −0.737934
\(395\) −8.66666e8 −0.707558
\(396\) −5.11442e7 −0.0413869
\(397\) 1.23414e9 0.989914 0.494957 0.868917i \(-0.335184\pi\)
0.494957 + 0.868917i \(0.335184\pi\)
\(398\) −8.78887e7 −0.0698783
\(399\) −4.48030e8 −0.353103
\(400\) 2.31312e8 0.180713
\(401\) 1.41825e9 1.09837 0.549185 0.835701i \(-0.314938\pi\)
0.549185 + 0.835701i \(0.314938\pi\)
\(402\) −3.15657e8 −0.242340
\(403\) −1.41855e9 −1.07964
\(404\) 3.70921e8 0.279863
\(405\) 1.94973e8 0.145842
\(406\) −6.75260e8 −0.500760
\(407\) 5.39508e8 0.396659
\(408\) 1.88761e8 0.137595
\(409\) 8.33868e8 0.602651 0.301325 0.953521i \(-0.402571\pi\)
0.301325 + 0.953521i \(0.402571\pi\)
\(410\) 1.22249e9 0.875996
\(411\) −8.01822e8 −0.569681
\(412\) 8.57810e7 0.0604298
\(413\) 1.28610e9 0.898360
\(414\) −7.09579e7 −0.0491473
\(415\) 4.69180e8 0.322234
\(416\) 2.26922e8 0.154543
\(417\) 7.29540e8 0.492689
\(418\) −2.59061e8 −0.173494
\(419\) −5.56088e8 −0.369313 −0.184656 0.982803i \(-0.559117\pi\)
−0.184656 + 0.982803i \(0.559117\pi\)
\(420\) −3.56109e8 −0.234537
\(421\) 1.76819e9 1.15489 0.577447 0.816428i \(-0.304049\pi\)
0.577447 + 0.816428i \(0.304049\pi\)
\(422\) 5.71242e8 0.370021
\(423\) 6.78451e8 0.435840
\(424\) 5.18747e8 0.330503
\(425\) 7.71113e8 0.487256
\(426\) −3.94391e8 −0.247168
\(427\) −1.66037e9 −1.03207
\(428\) −8.82014e8 −0.543779
\(429\) −2.04965e8 −0.125337
\(430\) 1.09036e9 0.661346
\(431\) 1.86575e9 1.12249 0.561244 0.827650i \(-0.310323\pi\)
0.561244 + 0.827650i \(0.310323\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) −8.86037e8 −0.524498 −0.262249 0.965000i \(-0.584464\pi\)
−0.262249 + 0.965000i \(0.584464\pi\)
\(434\) 9.20510e8 0.540523
\(435\) 1.48848e9 0.867023
\(436\) 6.02043e8 0.347876
\(437\) −3.59423e8 −0.206026
\(438\) −8.71604e6 −0.00495633
\(439\) −1.11162e9 −0.627094 −0.313547 0.949573i \(-0.601517\pi\)
−0.313547 + 0.949573i \(0.601517\pi\)
\(440\) −2.05910e8 −0.115237
\(441\) −3.70341e8 −0.205621
\(442\) 7.56478e8 0.416695
\(443\) 3.79851e8 0.207587 0.103794 0.994599i \(-0.466902\pi\)
0.103794 + 0.994599i \(0.466902\pi\)
\(444\) −8.50458e8 −0.461119
\(445\) 1.42517e9 0.766668
\(446\) 9.68993e8 0.517188
\(447\) −3.30123e8 −0.174823
\(448\) −1.47252e8 −0.0773726
\(449\) 1.04853e9 0.546664 0.273332 0.961920i \(-0.411874\pi\)
0.273332 + 0.961920i \(0.411874\pi\)
\(450\) 3.29349e8 0.170378
\(451\) −4.56589e8 −0.234373
\(452\) −4.97114e7 −0.0253204
\(453\) −2.13689e9 −1.08004
\(454\) 2.58955e8 0.129876
\(455\) −1.42714e9 −0.710275
\(456\) 4.08372e8 0.201688
\(457\) −1.49813e9 −0.734250 −0.367125 0.930172i \(-0.619658\pi\)
−0.367125 + 0.930172i \(0.619658\pi\)
\(458\) 1.71739e9 0.835293
\(459\) 2.68764e8 0.129726
\(460\) −2.85682e8 −0.136845
\(461\) 1.49236e9 0.709447 0.354723 0.934971i \(-0.384575\pi\)
0.354723 + 0.934971i \(0.384575\pi\)
\(462\) 1.33003e8 0.0627503
\(463\) −3.06867e9 −1.43687 −0.718433 0.695596i \(-0.755140\pi\)
−0.718433 + 0.695596i \(0.755140\pi\)
\(464\) 6.15489e8 0.286027
\(465\) −2.02909e9 −0.935869
\(466\) −1.87664e9 −0.859074
\(467\) 1.68526e9 0.765698 0.382849 0.923811i \(-0.374943\pi\)
0.382849 + 0.923811i \(0.374943\pi\)
\(468\) 3.23098e8 0.145705
\(469\) 8.20886e8 0.367433
\(470\) 2.73149e9 1.21355
\(471\) −1.79781e9 −0.792812
\(472\) −1.17226e9 −0.513131
\(473\) −4.07238e8 −0.176943
\(474\) −5.10254e8 −0.220071
\(475\) 1.66825e9 0.714223
\(476\) −4.90885e8 −0.208620
\(477\) 7.38607e8 0.311601
\(478\) −1.38821e9 −0.581376
\(479\) −2.69509e9 −1.12047 −0.560233 0.828335i \(-0.689288\pi\)
−0.560233 + 0.828335i \(0.689288\pi\)
\(480\) 3.24588e8 0.133964
\(481\) −3.40829e9 −1.39646
\(482\) 1.51719e9 0.617129
\(483\) 1.84530e8 0.0745166
\(484\) −1.17027e9 −0.469168
\(485\) −3.10561e9 −1.23609
\(486\) 1.14791e8 0.0453609
\(487\) −2.88969e9 −1.13371 −0.566853 0.823819i \(-0.691839\pi\)
−0.566853 + 0.823819i \(0.691839\pi\)
\(488\) 1.51340e9 0.589502
\(489\) 1.50633e9 0.582556
\(490\) −1.49102e9 −0.572528
\(491\) −2.94886e9 −1.12427 −0.562133 0.827047i \(-0.690019\pi\)
−0.562133 + 0.827047i \(0.690019\pi\)
\(492\) 7.19748e8 0.272460
\(493\) 2.05182e9 0.771215
\(494\) 1.63659e9 0.610795
\(495\) −2.93181e8 −0.108647
\(496\) −8.39030e8 −0.308739
\(497\) 1.02564e9 0.374754
\(498\) 2.76232e8 0.100224
\(499\) 3.10068e9 1.11713 0.558567 0.829459i \(-0.311351\pi\)
0.558567 + 0.829459i \(0.311351\pi\)
\(500\) −5.08396e8 −0.181889
\(501\) −7.72070e8 −0.274299
\(502\) 2.16884e9 0.765181
\(503\) −6.25596e8 −0.219183 −0.109591 0.993977i \(-0.534954\pi\)
−0.109591 + 0.993977i \(0.534954\pi\)
\(504\) −2.09661e8 −0.0729476
\(505\) 2.12628e9 0.734683
\(506\) 1.06699e8 0.0366130
\(507\) −3.99365e8 −0.136095
\(508\) −1.29778e9 −0.439217
\(509\) 4.35887e9 1.46508 0.732541 0.680723i \(-0.238334\pi\)
0.732541 + 0.680723i \(0.238334\pi\)
\(510\) 1.08206e9 0.361207
\(511\) 2.26666e7 0.00751472
\(512\) 1.34218e8 0.0441942
\(513\) 5.81452e8 0.190153
\(514\) 3.49008e9 1.13361
\(515\) 4.91734e8 0.158637
\(516\) 6.41953e8 0.205698
\(517\) −1.02019e9 −0.324685
\(518\) 2.21167e9 0.699142
\(519\) −1.60496e9 −0.503939
\(520\) 1.30082e9 0.405699
\(521\) 1.86006e9 0.576229 0.288115 0.957596i \(-0.406972\pi\)
0.288115 + 0.957596i \(0.406972\pi\)
\(522\) 8.76350e8 0.269669
\(523\) 1.89981e9 0.580703 0.290352 0.956920i \(-0.406228\pi\)
0.290352 + 0.956920i \(0.406228\pi\)
\(524\) −3.66946e8 −0.111415
\(525\) −8.56491e8 −0.258324
\(526\) 1.01865e9 0.305194
\(527\) −2.79703e9 −0.832454
\(528\) −1.21231e8 −0.0358421
\(529\) 1.48036e8 0.0434783
\(530\) 2.97368e9 0.867620
\(531\) −1.66910e9 −0.483784
\(532\) −1.06200e9 −0.305796
\(533\) 2.88446e9 0.825123
\(534\) 8.39077e8 0.238456
\(535\) −5.05608e9 −1.42750
\(536\) −7.48225e8 −0.209872
\(537\) −2.94644e9 −0.821083
\(538\) 2.00792e9 0.555914
\(539\) 5.56882e8 0.153180
\(540\) 4.62158e8 0.126303
\(541\) 6.17189e9 1.67582 0.837910 0.545808i \(-0.183777\pi\)
0.837910 + 0.545808i \(0.183777\pi\)
\(542\) 9.53670e8 0.257277
\(543\) −2.67147e9 −0.716061
\(544\) 4.47435e8 0.119161
\(545\) 3.45117e9 0.913226
\(546\) −8.40236e8 −0.220916
\(547\) 4.97696e9 1.30020 0.650098 0.759851i \(-0.274728\pi\)
0.650098 + 0.759851i \(0.274728\pi\)
\(548\) −1.90061e9 −0.493358
\(549\) 2.15483e9 0.555788
\(550\) −4.95242e8 −0.126925
\(551\) 4.43898e9 1.13045
\(552\) −1.68197e8 −0.0425628
\(553\) 1.32695e9 0.333669
\(554\) −2.16146e9 −0.540087
\(555\) −4.87519e9 −1.21050
\(556\) 1.72928e9 0.426681
\(557\) −2.16771e9 −0.531506 −0.265753 0.964041i \(-0.585621\pi\)
−0.265753 + 0.964041i \(0.585621\pi\)
\(558\) −1.19464e9 −0.291082
\(559\) 2.57268e9 0.622938
\(560\) −8.44111e8 −0.203115
\(561\) −4.04140e8 −0.0966411
\(562\) −5.06408e9 −1.20344
\(563\) 4.33727e9 1.02432 0.512162 0.858889i \(-0.328845\pi\)
0.512162 + 0.858889i \(0.328845\pi\)
\(564\) 1.60818e9 0.377449
\(565\) −2.84967e8 −0.0664699
\(566\) 5.34644e9 1.23939
\(567\) −2.98522e8 −0.0687757
\(568\) −9.34852e8 −0.214054
\(569\) −1.76759e9 −0.402243 −0.201122 0.979566i \(-0.564459\pi\)
−0.201122 + 0.979566i \(0.564459\pi\)
\(570\) 2.34097e9 0.529460
\(571\) 4.35096e9 0.978046 0.489023 0.872271i \(-0.337353\pi\)
0.489023 + 0.872271i \(0.337353\pi\)
\(572\) −4.85843e8 −0.108545
\(573\) −1.67161e9 −0.371188
\(574\) −1.87175e9 −0.413101
\(575\) −6.87104e8 −0.150725
\(576\) 1.91103e8 0.0416667
\(577\) 7.46172e9 1.61705 0.808525 0.588462i \(-0.200266\pi\)
0.808525 + 0.588462i \(0.200266\pi\)
\(578\) −1.79112e9 −0.385813
\(579\) 2.79407e9 0.598223
\(580\) 3.52825e9 0.750864
\(581\) −7.18358e8 −0.151958
\(582\) −1.82844e9 −0.384460
\(583\) −1.11064e9 −0.232132
\(584\) −2.06603e7 −0.00429231
\(585\) 1.85214e9 0.382497
\(586\) 4.29122e9 0.880925
\(587\) −3.27341e9 −0.667985 −0.333992 0.942576i \(-0.608396\pi\)
−0.333992 + 0.942576i \(0.608396\pi\)
\(588\) −8.77846e8 −0.178073
\(589\) −6.05118e9 −1.22022
\(590\) −6.71991e9 −1.34704
\(591\) −3.02363e9 −0.602521
\(592\) −2.01590e9 −0.399340
\(593\) −5.07631e9 −0.999670 −0.499835 0.866121i \(-0.666606\pi\)
−0.499835 + 0.866121i \(0.666606\pi\)
\(594\) −1.72612e8 −0.0337923
\(595\) −2.81397e9 −0.547658
\(596\) −7.82513e8 −0.151401
\(597\) −2.96624e8 −0.0570554
\(598\) −6.74063e8 −0.128898
\(599\) −9.57665e9 −1.82062 −0.910311 0.413926i \(-0.864157\pi\)
−0.910311 + 0.413926i \(0.864157\pi\)
\(600\) 7.80679e8 0.147551
\(601\) −1.02812e9 −0.193189 −0.0965945 0.995324i \(-0.530795\pi\)
−0.0965945 + 0.995324i \(0.530795\pi\)
\(602\) −1.66944e9 −0.311876
\(603\) −1.06534e9 −0.197870
\(604\) −5.06523e9 −0.935341
\(605\) −6.70851e9 −1.23164
\(606\) 1.25186e9 0.228507
\(607\) 2.45045e9 0.444719 0.222359 0.974965i \(-0.428624\pi\)
0.222359 + 0.974965i \(0.428624\pi\)
\(608\) 9.67994e8 0.174667
\(609\) −2.27900e9 −0.408869
\(610\) 8.67548e9 1.54753
\(611\) 6.44492e9 1.14307
\(612\) 6.37070e8 0.112346
\(613\) 8.90460e8 0.156136 0.0780679 0.996948i \(-0.475125\pi\)
0.0780679 + 0.996948i \(0.475125\pi\)
\(614\) −8.57767e8 −0.149548
\(615\) 4.12591e9 0.715248
\(616\) 3.15268e8 0.0543434
\(617\) 3.73214e9 0.639675 0.319838 0.947472i \(-0.396372\pi\)
0.319838 + 0.947472i \(0.396372\pi\)
\(618\) 2.89511e8 0.0493407
\(619\) −9.73916e9 −1.65046 −0.825228 0.564799i \(-0.808954\pi\)
−0.825228 + 0.564799i \(0.808954\pi\)
\(620\) −4.80968e9 −0.810486
\(621\) −2.39483e8 −0.0401286
\(622\) −7.79689e9 −1.29914
\(623\) −2.18207e9 −0.361544
\(624\) 7.65862e8 0.126184
\(625\) −7.32628e9 −1.20034
\(626\) −5.17675e9 −0.843426
\(627\) −8.74329e8 −0.141657
\(628\) −4.26147e9 −0.686595
\(629\) −6.72030e9 −1.07674
\(630\) −1.20187e9 −0.191498
\(631\) −8.10602e8 −0.128441 −0.0642207 0.997936i \(-0.520456\pi\)
−0.0642207 + 0.997936i \(0.520456\pi\)
\(632\) −1.20949e9 −0.190587
\(633\) 1.92794e9 0.302121
\(634\) 4.18370e9 0.652001
\(635\) −7.43946e9 −1.15301
\(636\) 1.75077e9 0.269855
\(637\) −3.51805e9 −0.539279
\(638\) −1.31777e9 −0.200894
\(639\) −1.33107e9 −0.201812
\(640\) 7.69394e8 0.116016
\(641\) −1.53566e9 −0.230299 −0.115150 0.993348i \(-0.536735\pi\)
−0.115150 + 0.993348i \(0.536735\pi\)
\(642\) −2.97680e9 −0.443993
\(643\) 9.03556e9 1.34035 0.670173 0.742205i \(-0.266220\pi\)
0.670173 + 0.742205i \(0.266220\pi\)
\(644\) 4.37405e8 0.0645332
\(645\) 3.67995e9 0.539986
\(646\) 3.22695e9 0.470954
\(647\) −6.44225e8 −0.0935132 −0.0467566 0.998906i \(-0.514889\pi\)
−0.0467566 + 0.998906i \(0.514889\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 2.50983e9 0.360402
\(650\) 3.12864e9 0.446847
\(651\) 3.10672e9 0.441335
\(652\) 3.57055e9 0.504508
\(653\) −3.14760e9 −0.442368 −0.221184 0.975232i \(-0.570992\pi\)
−0.221184 + 0.975232i \(0.570992\pi\)
\(654\) 2.03189e9 0.284040
\(655\) −2.10349e9 −0.292480
\(656\) 1.70607e9 0.235957
\(657\) −2.94166e7 −0.00404682
\(658\) −4.18217e9 −0.572283
\(659\) 1.35636e10 1.84619 0.923093 0.384577i \(-0.125653\pi\)
0.923093 + 0.384577i \(0.125653\pi\)
\(660\) −6.94946e8 −0.0940909
\(661\) 1.38238e10 1.86176 0.930880 0.365325i \(-0.119042\pi\)
0.930880 + 0.365325i \(0.119042\pi\)
\(662\) −8.04261e8 −0.107744
\(663\) 2.55311e9 0.340230
\(664\) 6.54772e8 0.0867964
\(665\) −6.08782e9 −0.802761
\(666\) −2.87030e9 −0.376502
\(667\) −1.82828e9 −0.238563
\(668\) −1.83009e9 −0.237550
\(669\) 3.27035e9 0.422283
\(670\) −4.28915e9 −0.550946
\(671\) −3.24021e9 −0.414042
\(672\) −4.96975e8 −0.0631745
\(673\) −3.45394e9 −0.436780 −0.218390 0.975862i \(-0.570080\pi\)
−0.218390 + 0.975862i \(0.570080\pi\)
\(674\) −3.82123e9 −0.480721
\(675\) 1.11155e9 0.139113
\(676\) −9.46644e8 −0.117862
\(677\) 6.53708e9 0.809699 0.404849 0.914383i \(-0.367324\pi\)
0.404849 + 0.914383i \(0.367324\pi\)
\(678\) −1.67776e8 −0.0206741
\(679\) 4.75498e9 0.582914
\(680\) 2.56489e9 0.312815
\(681\) 8.73971e8 0.106043
\(682\) 1.79637e9 0.216846
\(683\) −1.53942e10 −1.84877 −0.924386 0.381457i \(-0.875422\pi\)
−0.924386 + 0.381457i \(0.875422\pi\)
\(684\) 1.37826e9 0.164677
\(685\) −1.08951e10 −1.29514
\(686\) 5.98370e9 0.707678
\(687\) 5.79618e9 0.682014
\(688\) 1.52167e9 0.178139
\(689\) 7.01638e9 0.817233
\(690\) −9.64176e8 −0.111734
\(691\) −6.43606e9 −0.742074 −0.371037 0.928618i \(-0.620998\pi\)
−0.371037 + 0.928618i \(0.620998\pi\)
\(692\) −3.80434e9 −0.436424
\(693\) 4.48887e8 0.0512354
\(694\) −7.50066e9 −0.851807
\(695\) 9.91297e9 1.12010
\(696\) 2.07727e9 0.233540
\(697\) 5.68744e9 0.636212
\(698\) 7.75264e9 0.862890
\(699\) −6.33366e9 −0.701431
\(700\) −2.03020e9 −0.223716
\(701\) 3.29232e9 0.360985 0.180492 0.983576i \(-0.442231\pi\)
0.180492 + 0.983576i \(0.442231\pi\)
\(702\) 1.09046e9 0.118968
\(703\) −1.45389e10 −1.57829
\(704\) −2.87362e8 −0.0310402
\(705\) 9.21878e9 0.990858
\(706\) −3.69543e9 −0.395229
\(707\) −3.25553e9 −0.346460
\(708\) −3.95639e9 −0.418969
\(709\) −1.33790e10 −1.40981 −0.704907 0.709300i \(-0.749011\pi\)
−0.704907 + 0.709300i \(0.749011\pi\)
\(710\) −5.35898e9 −0.561924
\(711\) −1.72211e9 −0.179687
\(712\) 1.98892e9 0.206509
\(713\) 2.49231e9 0.257506
\(714\) −1.65674e9 −0.170337
\(715\) −2.78506e9 −0.284947
\(716\) −6.98415e9 −0.711079
\(717\) −4.68520e9 −0.474691
\(718\) −6.20317e9 −0.625430
\(719\) 9.99931e9 1.00327 0.501636 0.865079i \(-0.332732\pi\)
0.501636 + 0.865079i \(0.332732\pi\)
\(720\) 1.09548e9 0.109381
\(721\) −7.52890e8 −0.0748098
\(722\) −1.69692e8 −0.0167796
\(723\) 5.12052e9 0.503884
\(724\) −6.33236e9 −0.620127
\(725\) 8.48592e9 0.827020
\(726\) −3.94967e9 −0.383074
\(727\) 3.87514e9 0.374039 0.187019 0.982356i \(-0.440117\pi\)
0.187019 + 0.982356i \(0.440117\pi\)
\(728\) −1.99167e9 −0.191319
\(729\) 3.87420e8 0.0370370
\(730\) −1.18433e8 −0.0112679
\(731\) 5.07270e9 0.480317
\(732\) 5.10773e9 0.481326
\(733\) 2.03185e10 1.90559 0.952793 0.303620i \(-0.0981954\pi\)
0.952793 + 0.303620i \(0.0981954\pi\)
\(734\) −2.31346e9 −0.215937
\(735\) −5.03219e9 −0.467468
\(736\) −3.98688e8 −0.0368605
\(737\) 1.60196e9 0.147406
\(738\) 2.42915e9 0.222463
\(739\) 6.01270e9 0.548042 0.274021 0.961724i \(-0.411646\pi\)
0.274021 + 0.961724i \(0.411646\pi\)
\(740\) −1.15560e10 −1.04833
\(741\) 5.52349e9 0.498712
\(742\) −4.55299e9 −0.409150
\(743\) 7.10581e9 0.635554 0.317777 0.948165i \(-0.397064\pi\)
0.317777 + 0.948165i \(0.397064\pi\)
\(744\) −2.83173e9 −0.252085
\(745\) −4.48570e9 −0.397451
\(746\) 4.92533e9 0.434359
\(747\) 9.32283e8 0.0818325
\(748\) −9.57962e8 −0.0836937
\(749\) 7.74134e9 0.673178
\(750\) −1.71584e9 −0.148512
\(751\) −2.66674e9 −0.229743 −0.114871 0.993380i \(-0.536646\pi\)
−0.114871 + 0.993380i \(0.536646\pi\)
\(752\) 3.81198e9 0.326880
\(753\) 7.31983e9 0.624767
\(754\) 8.32486e9 0.707257
\(755\) −2.90361e10 −2.45541
\(756\) −7.07607e8 −0.0595615
\(757\) 1.21053e10 1.01424 0.507119 0.861876i \(-0.330711\pi\)
0.507119 + 0.861876i \(0.330711\pi\)
\(758\) 7.96260e9 0.664068
\(759\) 3.60111e8 0.0298944
\(760\) 5.54896e9 0.458526
\(761\) −2.97000e9 −0.244293 −0.122146 0.992512i \(-0.538978\pi\)
−0.122146 + 0.992512i \(0.538978\pi\)
\(762\) −4.38002e9 −0.358619
\(763\) −5.28406e9 −0.430657
\(764\) −3.96233e9 −0.321458
\(765\) 3.65196e9 0.294925
\(766\) −1.67808e9 −0.134900
\(767\) −1.58556e10 −1.26881
\(768\) 4.52985e8 0.0360844
\(769\) 1.95706e10 1.55190 0.775948 0.630797i \(-0.217272\pi\)
0.775948 + 0.630797i \(0.217272\pi\)
\(770\) 1.80725e9 0.142659
\(771\) 1.17790e10 0.925589
\(772\) 6.62299e9 0.518076
\(773\) −1.54223e9 −0.120094 −0.0600468 0.998196i \(-0.519125\pi\)
−0.0600468 + 0.998196i \(0.519125\pi\)
\(774\) 2.16659e9 0.167951
\(775\) −1.15680e10 −0.892690
\(776\) −4.33409e9 −0.332952
\(777\) 7.46438e9 0.570847
\(778\) 2.02406e9 0.154098
\(779\) 1.23044e10 0.932564
\(780\) 4.39025e9 0.331252
\(781\) 2.00153e9 0.150343
\(782\) −1.32909e9 −0.0993870
\(783\) 2.95768e9 0.220184
\(784\) −2.08082e9 −0.154216
\(785\) −2.44286e10 −1.80241
\(786\) −1.23844e9 −0.0909697
\(787\) −1.76348e10 −1.28961 −0.644805 0.764347i \(-0.723061\pi\)
−0.644805 + 0.764347i \(0.723061\pi\)
\(788\) −7.16712e9 −0.521798
\(789\) 3.43795e9 0.249190
\(790\) −6.93333e9 −0.500319
\(791\) 4.36311e8 0.0313458
\(792\) −4.09153e8 −0.0292650
\(793\) 2.04697e10 1.45766
\(794\) 9.87312e9 0.699975
\(795\) 1.00362e10 0.708408
\(796\) −7.03110e8 −0.0494114
\(797\) 2.10641e10 1.47380 0.736901 0.676000i \(-0.236288\pi\)
0.736901 + 0.676000i \(0.236288\pi\)
\(798\) −3.58424e9 −0.249682
\(799\) 1.27078e10 0.881367
\(800\) 1.85050e9 0.127783
\(801\) 2.83189e9 0.194698
\(802\) 1.13460e10 0.776664
\(803\) 4.42338e7 0.00301474
\(804\) −2.52526e9 −0.171360
\(805\) 2.50740e9 0.169409
\(806\) −1.13484e10 −0.763417
\(807\) 6.77672e9 0.453902
\(808\) 2.96736e9 0.197893
\(809\) −2.12308e10 −1.40977 −0.704884 0.709323i \(-0.749001\pi\)
−0.704884 + 0.709323i \(0.749001\pi\)
\(810\) 1.55978e9 0.103126
\(811\) 9.31477e9 0.613196 0.306598 0.951839i \(-0.400809\pi\)
0.306598 + 0.951839i \(0.400809\pi\)
\(812\) −5.40208e9 −0.354091
\(813\) 3.21864e9 0.210066
\(814\) 4.31606e9 0.280480
\(815\) 2.04679e10 1.32441
\(816\) 1.51009e9 0.0972944
\(817\) 1.09744e10 0.704052
\(818\) 6.67094e9 0.426138
\(819\) −2.83580e9 −0.180377
\(820\) 9.77993e9 0.619423
\(821\) −1.51846e10 −0.957642 −0.478821 0.877913i \(-0.658936\pi\)
−0.478821 + 0.877913i \(0.658936\pi\)
\(822\) −6.41458e9 −0.402825
\(823\) −9.38917e9 −0.587121 −0.293561 0.955940i \(-0.594840\pi\)
−0.293561 + 0.955940i \(0.594840\pi\)
\(824\) 6.86248e8 0.0427303
\(825\) −1.67144e9 −0.103634
\(826\) 1.02888e10 0.635236
\(827\) 1.40878e10 0.866114 0.433057 0.901366i \(-0.357435\pi\)
0.433057 + 0.901366i \(0.357435\pi\)
\(828\) −5.67664e8 −0.0347524
\(829\) 1.99910e10 1.21869 0.609345 0.792905i \(-0.291432\pi\)
0.609345 + 0.792905i \(0.291432\pi\)
\(830\) 3.75344e9 0.227854
\(831\) −7.29494e9 −0.440979
\(832\) 1.81538e9 0.109279
\(833\) −6.93672e9 −0.415812
\(834\) 5.83632e9 0.348384
\(835\) −1.04909e10 −0.623604
\(836\) −2.07248e9 −0.122679
\(837\) −4.03189e9 −0.237668
\(838\) −4.44870e9 −0.261143
\(839\) 2.49729e10 1.45983 0.729915 0.683538i \(-0.239560\pi\)
0.729915 + 0.683538i \(0.239560\pi\)
\(840\) −2.84887e9 −0.165842
\(841\) 5.32994e9 0.308984
\(842\) 1.41455e10 0.816633
\(843\) −1.70913e10 −0.982602
\(844\) 4.56993e9 0.261644
\(845\) −5.42657e9 −0.309405
\(846\) 5.42761e9 0.308185
\(847\) 1.02714e10 0.580813
\(848\) 4.14998e9 0.233701
\(849\) 1.80442e10 1.01196
\(850\) 6.16891e9 0.344542
\(851\) 5.98815e9 0.333073
\(852\) −3.15513e9 −0.174775
\(853\) 2.15537e10 1.18905 0.594526 0.804077i \(-0.297340\pi\)
0.594526 + 0.804077i \(0.297340\pi\)
\(854\) −1.32830e10 −0.729781
\(855\) 7.90076e9 0.432302
\(856\) −7.05611e9 −0.384510
\(857\) −1.83252e9 −0.0994524 −0.0497262 0.998763i \(-0.515835\pi\)
−0.0497262 + 0.998763i \(0.515835\pi\)
\(858\) −1.63972e9 −0.0886266
\(859\) 2.88323e10 1.55204 0.776020 0.630708i \(-0.217235\pi\)
0.776020 + 0.630708i \(0.217235\pi\)
\(860\) 8.72284e9 0.467642
\(861\) −6.31715e9 −0.337295
\(862\) 1.49260e10 0.793719
\(863\) −2.80605e10 −1.48613 −0.743067 0.669217i \(-0.766630\pi\)
−0.743067 + 0.669217i \(0.766630\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −2.18081e10 −1.14568
\(866\) −7.08829e9 −0.370876
\(867\) −6.04503e9 −0.315015
\(868\) 7.36408e9 0.382208
\(869\) 2.58953e9 0.133860
\(870\) 1.19078e10 0.613078
\(871\) −1.01202e10 −0.518950
\(872\) 4.81634e9 0.245986
\(873\) −6.17100e9 −0.313910
\(874\) −2.87539e9 −0.145682
\(875\) 4.46214e9 0.225172
\(876\) −6.97283e7 −0.00350465
\(877\) −1.07928e10 −0.540299 −0.270149 0.962818i \(-0.587073\pi\)
−0.270149 + 0.962818i \(0.587073\pi\)
\(878\) −8.89300e9 −0.443422
\(879\) 1.44829e10 0.719272
\(880\) −1.64728e9 −0.0814851
\(881\) −2.86597e10 −1.41207 −0.706034 0.708178i \(-0.749517\pi\)
−0.706034 + 0.708178i \(0.749517\pi\)
\(882\) −2.96273e9 −0.145396
\(883\) −3.47774e10 −1.69994 −0.849972 0.526828i \(-0.823381\pi\)
−0.849972 + 0.526828i \(0.823381\pi\)
\(884\) 6.05183e9 0.294648
\(885\) −2.26797e10 −1.09986
\(886\) 3.03881e9 0.146786
\(887\) 6.42202e9 0.308986 0.154493 0.987994i \(-0.450626\pi\)
0.154493 + 0.987994i \(0.450626\pi\)
\(888\) −6.80367e9 −0.326060
\(889\) 1.13905e10 0.543734
\(890\) 1.14014e10 0.542116
\(891\) −5.82564e8 −0.0275913
\(892\) 7.75195e9 0.365707
\(893\) 2.74925e10 1.29191
\(894\) −2.64098e9 −0.123619
\(895\) −4.00362e10 −1.86669
\(896\) −1.17801e9 −0.0547107
\(897\) −2.27496e9 −0.105245
\(898\) 8.38827e9 0.386550
\(899\) −3.07807e10 −1.41292
\(900\) 2.63479e9 0.120475
\(901\) 1.38346e10 0.630128
\(902\) −3.65271e9 −0.165727
\(903\) −5.63435e9 −0.254646
\(904\) −3.97691e8 −0.0179043
\(905\) −3.62998e10 −1.62793
\(906\) −1.70952e10 −0.763703
\(907\) −3.96724e10 −1.76548 −0.882740 0.469861i \(-0.844304\pi\)
−0.882740 + 0.469861i \(0.844304\pi\)
\(908\) 2.07164e9 0.0918360
\(909\) 4.22502e9 0.186576
\(910\) −1.14171e10 −0.502240
\(911\) 3.39607e10 1.48820 0.744102 0.668066i \(-0.232878\pi\)
0.744102 + 0.668066i \(0.232878\pi\)
\(912\) 3.26698e9 0.142615
\(913\) −1.40187e9 −0.0609623
\(914\) −1.19851e10 −0.519193
\(915\) 2.92797e10 1.26355
\(916\) 1.37391e10 0.590642
\(917\) 3.22064e9 0.137927
\(918\) 2.15011e9 0.0917300
\(919\) 1.98186e10 0.842304 0.421152 0.906990i \(-0.361626\pi\)
0.421152 + 0.906990i \(0.361626\pi\)
\(920\) −2.28545e9 −0.0967642
\(921\) −2.89496e9 −0.122105
\(922\) 1.19389e10 0.501655
\(923\) −1.26444e10 −0.529290
\(924\) 1.06403e9 0.0443712
\(925\) −2.77938e10 −1.15465
\(926\) −2.45493e10 −1.01602
\(927\) 9.77099e8 0.0402865
\(928\) 4.92391e9 0.202252
\(929\) −2.08980e10 −0.855164 −0.427582 0.903976i \(-0.640635\pi\)
−0.427582 + 0.903976i \(0.640635\pi\)
\(930\) −1.62327e10 −0.661759
\(931\) −1.50071e10 −0.609499
\(932\) −1.50131e10 −0.607457
\(933\) −2.63145e10 −1.06074
\(934\) 1.34821e10 0.541430
\(935\) −5.49145e9 −0.219708
\(936\) 2.58479e9 0.103029
\(937\) −4.61325e9 −0.183197 −0.0915986 0.995796i \(-0.529198\pi\)
−0.0915986 + 0.995796i \(0.529198\pi\)
\(938\) 6.56709e9 0.259814
\(939\) −1.74715e10 −0.688654
\(940\) 2.18519e10 0.858109
\(941\) 2.61949e10 1.02483 0.512417 0.858737i \(-0.328750\pi\)
0.512417 + 0.858737i \(0.328750\pi\)
\(942\) −1.43825e10 −0.560603
\(943\) −5.06781e9 −0.196802
\(944\) −9.37810e9 −0.362838
\(945\) −4.05631e9 −0.156358
\(946\) −3.25790e9 −0.125118
\(947\) −1.01216e10 −0.387278 −0.193639 0.981073i \(-0.562029\pi\)
−0.193639 + 0.981073i \(0.562029\pi\)
\(948\) −4.08203e9 −0.155614
\(949\) −2.79443e8 −0.0106136
\(950\) 1.33460e10 0.505032
\(951\) 1.41200e10 0.532356
\(952\) −3.92708e9 −0.147517
\(953\) −5.16517e9 −0.193312 −0.0966562 0.995318i \(-0.530815\pi\)
−0.0966562 + 0.995318i \(0.530815\pi\)
\(954\) 5.90886e9 0.220335
\(955\) −2.27138e10 −0.843875
\(956\) −1.11057e10 −0.411095
\(957\) −4.44747e9 −0.164029
\(958\) −2.15607e10 −0.792289
\(959\) 1.66815e10 0.610759
\(960\) 2.59670e9 0.0947269
\(961\) 1.44474e10 0.525119
\(962\) −2.72663e10 −0.987446
\(963\) −1.00467e10 −0.362519
\(964\) 1.21375e10 0.436376
\(965\) 3.79659e10 1.36003
\(966\) 1.47624e9 0.0526912
\(967\) 2.33871e10 0.831734 0.415867 0.909425i \(-0.363478\pi\)
0.415867 + 0.909425i \(0.363478\pi\)
\(968\) −9.36219e9 −0.331752
\(969\) 1.08910e10 0.384532
\(970\) −2.48449e10 −0.874049
\(971\) −4.05447e9 −0.142124 −0.0710619 0.997472i \(-0.522639\pi\)
−0.0710619 + 0.997472i \(0.522639\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −1.51777e10 −0.528215
\(974\) −2.31176e10 −0.801651
\(975\) 1.05592e10 0.364849
\(976\) 1.21072e10 0.416841
\(977\) 5.68903e10 1.95167 0.975837 0.218499i \(-0.0701160\pi\)
0.975837 + 0.218499i \(0.0701160\pi\)
\(978\) 1.20506e10 0.411929
\(979\) −4.25831e9 −0.145043
\(980\) −1.19282e10 −0.404839
\(981\) 6.85764e9 0.231917
\(982\) −2.35909e10 −0.794977
\(983\) 3.74559e10 1.25772 0.628859 0.777519i \(-0.283522\pi\)
0.628859 + 0.777519i \(0.283522\pi\)
\(984\) 5.75799e9 0.192658
\(985\) −4.10850e10 −1.36980
\(986\) 1.64146e10 0.545331
\(987\) −1.41148e10 −0.467267
\(988\) 1.30927e10 0.431897
\(989\) −4.52004e9 −0.148578
\(990\) −2.34544e9 −0.0768249
\(991\) 1.00112e8 0.00326759 0.00163379 0.999999i \(-0.499480\pi\)
0.00163379 + 0.999999i \(0.499480\pi\)
\(992\) −6.71224e9 −0.218312
\(993\) −2.71438e9 −0.0879729
\(994\) 8.20509e9 0.264991
\(995\) −4.03053e9 −0.129712
\(996\) 2.20986e9 0.0708690
\(997\) −3.72236e10 −1.18956 −0.594779 0.803889i \(-0.702760\pi\)
−0.594779 + 0.803889i \(0.702760\pi\)
\(998\) 2.48055e10 0.789934
\(999\) −9.68725e9 −0.307412
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 138.8.a.h.1.4 4
3.2 odd 2 414.8.a.i.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.8.a.h.1.4 4 1.1 even 1 trivial
414.8.a.i.1.1 4 3.2 odd 2