Properties

Label 29.12.a.a.1.9
Level $29$
Weight $12$
Character 29.1
Self dual yes
Analytic conductor $22.282$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2819522362\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + 1279291654973975 x^{2} - 65244901875230266 x - 758324542468966858\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(54.9918\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q+51.9918 q^{2} +135.432 q^{3} +655.142 q^{4} +3682.79 q^{5} +7041.33 q^{6} -83752.5 q^{7} -72417.1 q^{8} -158805. q^{9} +O(q^{10})\) \(q+51.9918 q^{2} +135.432 q^{3} +655.142 q^{4} +3682.79 q^{5} +7041.33 q^{6} -83752.5 q^{7} -72417.1 q^{8} -158805. q^{9} +191474. q^{10} +132980. q^{11} +88727.0 q^{12} +193527. q^{13} -4.35444e6 q^{14} +498766. q^{15} -5.10682e6 q^{16} +2.59769e6 q^{17} -8.25656e6 q^{18} -4.52077e6 q^{19} +2.41275e6 q^{20} -1.13427e7 q^{21} +6.91387e6 q^{22} +7.42631e6 q^{23} -9.80757e6 q^{24} -3.52652e7 q^{25} +1.00618e7 q^{26} -4.54986e7 q^{27} -5.48698e7 q^{28} +2.05111e7 q^{29} +2.59317e7 q^{30} +4.28934e7 q^{31} -1.17202e8 q^{32} +1.80097e7 q^{33} +1.35058e8 q^{34} -3.08443e8 q^{35} -1.04040e8 q^{36} -8.25971e7 q^{37} -2.35043e8 q^{38} +2.62096e7 q^{39} -2.66697e8 q^{40} -7.25750e8 q^{41} -5.89729e8 q^{42} -1.14606e8 q^{43} +8.71209e7 q^{44} -5.84846e8 q^{45} +3.86107e8 q^{46} -3.13358e8 q^{47} -6.91625e8 q^{48} +5.03716e9 q^{49} -1.83350e9 q^{50} +3.51809e8 q^{51} +1.26788e8 q^{52} -5.04903e8 q^{53} -2.36555e9 q^{54} +4.89737e8 q^{55} +6.06512e9 q^{56} -6.12256e8 q^{57} +1.06641e9 q^{58} +8.68166e9 q^{59} +3.26762e8 q^{60} +4.85120e9 q^{61} +2.23010e9 q^{62} +1.33003e10 q^{63} +4.36521e9 q^{64} +7.12717e8 q^{65} +9.36357e8 q^{66} -1.44650e10 q^{67} +1.70186e9 q^{68} +1.00576e9 q^{69} -1.60365e10 q^{70} -2.67681e10 q^{71} +1.15002e10 q^{72} +2.46174e10 q^{73} -4.29437e9 q^{74} -4.77603e9 q^{75} -2.96175e9 q^{76} -1.11374e10 q^{77} +1.36269e9 q^{78} -3.86002e10 q^{79} -1.88073e10 q^{80} +2.19699e10 q^{81} -3.77330e10 q^{82} +1.74415e9 q^{83} -7.43111e9 q^{84} +9.56673e9 q^{85} -5.95859e9 q^{86} +2.77786e9 q^{87} -9.63004e9 q^{88} -2.10441e10 q^{89} -3.04072e10 q^{90} -1.62084e10 q^{91} +4.86529e9 q^{92} +5.80912e9 q^{93} -1.62920e10 q^{94} -1.66490e10 q^{95} -1.58729e10 q^{96} +1.02295e11 q^{97} +2.61891e11 q^{98} -2.11180e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} + O(q^{10}) \) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} - 685834q^{10} - 612246q^{11} + 2578538q^{12} + 1510364q^{13} + 3955400q^{14} - 2462818q^{15} + 3024818q^{16} - 3291098q^{17} - 27885614q^{18} - 44121388q^{19} - 49472662q^{20} - 46916800q^{21} - 43435618q^{22} - 88684076q^{23} - 224700678q^{24} - 44195521q^{25} - 324999762q^{26} - 236304286q^{27} - 391274848q^{28} + 225622639q^{29} - 494910382q^{30} - 292235934q^{31} - 632542514q^{32} - 1079766410q^{33} - 1113307936q^{34} - 1312820120q^{35} - 2236726492q^{36} - 1380429338q^{37} - 1222857284q^{38} - 1186931090q^{39} - 2713154106q^{40} - 1062067494q^{41} + 205598960q^{42} + 74588594q^{43} + 52891466q^{44} + 4527996830q^{45} - 87670324q^{46} - 1821239394q^{47} + 2666035542q^{48} + 4692522003q^{49} + 9494259926q^{50} + 8768158380q^{51} + 3266669866q^{52} + 7818635688q^{53} + 17402728558q^{54} - 191002682q^{55} + 11263587512q^{56} + 15495358340q^{57} - 656356768q^{58} + 1230002712q^{59} + 31834046430q^{60} - 18602654230q^{61} + 22075953162q^{62} - 9964531456q^{63} + 11813658086q^{64} + 32245789334q^{65} + 42677188354q^{66} + 27481284652q^{67} + 29588811820q^{68} - 20565315068q^{69} + 42862666712q^{70} - 20347168516q^{71} + 47061083616q^{72} - 57740010478q^{73} - 2640709564q^{74} - 23544691000q^{75} - 33350650772q^{76} + 871959792q^{77} - 15384525342q^{78} - 120245016462q^{79} - 84319695274q^{80} - 48880047865q^{81} - 111495532412q^{82} - 142463983824q^{83} - 134146226376q^{84} - 181628566552q^{85} + 47870165542q^{86} - 20141948318q^{87} - 180608014462q^{88} - 96700717270q^{89} - 25522461244q^{90} - 355162031176q^{91} - 22429477796q^{92} - 172582115142q^{93} + 172608565078q^{94} - 195922150708q^{95} + 226391047758q^{96} - 303190852014q^{97} - 123776497136q^{98} - 139125462440q^{99} + O(q^{100}) \)

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 (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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 51.9918 1.14887 0.574433 0.818552i \(-0.305223\pi\)
0.574433 + 0.818552i \(0.305223\pi\)
\(3\) 135.432 0.321776 0.160888 0.986973i \(-0.448564\pi\)
0.160888 + 0.986973i \(0.448564\pi\)
\(4\) 655.142 0.319894
\(5\) 3682.79 0.527037 0.263519 0.964654i \(-0.415117\pi\)
0.263519 + 0.964654i \(0.415117\pi\)
\(6\) 7041.33 0.369677
\(7\) −83752.5 −1.88347 −0.941735 0.336355i \(-0.890806\pi\)
−0.941735 + 0.336355i \(0.890806\pi\)
\(8\) −72417.1 −0.781351
\(9\) −158805. −0.896460
\(10\) 191474. 0.605495
\(11\) 132980. 0.248959 0.124479 0.992222i \(-0.460274\pi\)
0.124479 + 0.992222i \(0.460274\pi\)
\(12\) 88727.0 0.102934
\(13\) 193527. 0.144561 0.0722807 0.997384i \(-0.476972\pi\)
0.0722807 + 0.997384i \(0.476972\pi\)
\(14\) −4.35444e6 −2.16386
\(15\) 498766. 0.169588
\(16\) −5.10682e6 −1.21756
\(17\) 2.59769e6 0.443729 0.221865 0.975077i \(-0.428786\pi\)
0.221865 + 0.975077i \(0.428786\pi\)
\(18\) −8.25656e6 −1.02991
\(19\) −4.52077e6 −0.418859 −0.209430 0.977824i \(-0.567161\pi\)
−0.209430 + 0.977824i \(0.567161\pi\)
\(20\) 2.41275e6 0.168596
\(21\) −1.13427e7 −0.606055
\(22\) 6.91387e6 0.286020
\(23\) 7.42631e6 0.240586 0.120293 0.992738i \(-0.461617\pi\)
0.120293 + 0.992738i \(0.461617\pi\)
\(24\) −9.80757e6 −0.251420
\(25\) −3.52652e7 −0.722232
\(26\) 1.00618e7 0.166082
\(27\) −4.54986e7 −0.610235
\(28\) −5.48698e7 −0.602510
\(29\) 2.05111e7 0.185695
\(30\) 2.59317e7 0.194834
\(31\) 4.28934e7 0.269092 0.134546 0.990907i \(-0.457042\pi\)
0.134546 + 0.990907i \(0.457042\pi\)
\(32\) −1.17202e8 −0.617464
\(33\) 1.80097e7 0.0801088
\(34\) 1.35058e8 0.509786
\(35\) −3.08443e8 −0.992659
\(36\) −1.04040e8 −0.286772
\(37\) −8.25971e7 −0.195819 −0.0979096 0.995195i \(-0.531216\pi\)
−0.0979096 + 0.995195i \(0.531216\pi\)
\(38\) −2.35043e8 −0.481213
\(39\) 2.62096e7 0.0465164
\(40\) −2.66697e8 −0.411801
\(41\) −7.25750e8 −0.978309 −0.489154 0.872197i \(-0.662695\pi\)
−0.489154 + 0.872197i \(0.662695\pi\)
\(42\) −5.89729e8 −0.696276
\(43\) −1.14606e8 −0.118887 −0.0594433 0.998232i \(-0.518933\pi\)
−0.0594433 + 0.998232i \(0.518933\pi\)
\(44\) 8.71209e7 0.0796403
\(45\) −5.84846e8 −0.472468
\(46\) 3.86107e8 0.276401
\(47\) −3.13358e8 −0.199298 −0.0996490 0.995023i \(-0.531772\pi\)
−0.0996490 + 0.995023i \(0.531772\pi\)
\(48\) −6.91625e8 −0.391782
\(49\) 5.03716e9 2.54746
\(50\) −1.83350e9 −0.829748
\(51\) 3.51809e8 0.142781
\(52\) 1.26788e8 0.0462443
\(53\) −5.04903e8 −0.165840 −0.0829202 0.996556i \(-0.526425\pi\)
−0.0829202 + 0.996556i \(0.526425\pi\)
\(54\) −2.36555e9 −0.701078
\(55\) 4.89737e8 0.131210
\(56\) 6.06512e9 1.47165
\(57\) −6.12256e8 −0.134779
\(58\) 1.06641e9 0.213339
\(59\) 8.68166e9 1.58095 0.790473 0.612497i \(-0.209835\pi\)
0.790473 + 0.612497i \(0.209835\pi\)
\(60\) 3.26762e8 0.0542501
\(61\) 4.85120e9 0.735420 0.367710 0.929941i \(-0.380142\pi\)
0.367710 + 0.929941i \(0.380142\pi\)
\(62\) 2.23010e9 0.309151
\(63\) 1.33003e10 1.68846
\(64\) 4.36521e9 0.508178
\(65\) 7.12717e8 0.0761893
\(66\) 9.36357e8 0.0920343
\(67\) −1.44650e10 −1.30890 −0.654452 0.756104i \(-0.727101\pi\)
−0.654452 + 0.756104i \(0.727101\pi\)
\(68\) 1.70186e9 0.141946
\(69\) 1.00576e9 0.0774147
\(70\) −1.60365e10 −1.14043
\(71\) −2.67681e10 −1.76075 −0.880374 0.474280i \(-0.842708\pi\)
−0.880374 + 0.474280i \(0.842708\pi\)
\(72\) 1.15002e10 0.700450
\(73\) 2.46174e10 1.38985 0.694924 0.719084i \(-0.255438\pi\)
0.694924 + 0.719084i \(0.255438\pi\)
\(74\) −4.29437e9 −0.224970
\(75\) −4.77603e9 −0.232397
\(76\) −2.96175e9 −0.133990
\(77\) −1.11374e10 −0.468906
\(78\) 1.36269e9 0.0534411
\(79\) −3.86002e10 −1.41137 −0.705685 0.708526i \(-0.749360\pi\)
−0.705685 + 0.708526i \(0.749360\pi\)
\(80\) −1.88073e10 −0.641700
\(81\) 2.19699e10 0.700102
\(82\) −3.77330e10 −1.12395
\(83\) 1.74415e9 0.0486020 0.0243010 0.999705i \(-0.492264\pi\)
0.0243010 + 0.999705i \(0.492264\pi\)
\(84\) −7.43111e9 −0.193873
\(85\) 9.56673e9 0.233862
\(86\) −5.95859e9 −0.136585
\(87\) 2.77786e9 0.0597522
\(88\) −9.63004e9 −0.194524
\(89\) −2.10441e10 −0.399471 −0.199735 0.979850i \(-0.564008\pi\)
−0.199735 + 0.979850i \(0.564008\pi\)
\(90\) −3.04072e10 −0.542803
\(91\) −1.62084e10 −0.272277
\(92\) 4.86529e9 0.0769619
\(93\) 5.80912e9 0.0865873
\(94\) −1.62920e10 −0.228967
\(95\) −1.66490e10 −0.220754
\(96\) −1.58729e10 −0.198685
\(97\) 1.02295e11 1.20951 0.604757 0.796410i \(-0.293270\pi\)
0.604757 + 0.796410i \(0.293270\pi\)
\(98\) 2.61891e11 2.92669
\(99\) −2.11180e10 −0.223182
\(100\) −2.31037e10 −0.231037
\(101\) 2.22830e10 0.210963 0.105481 0.994421i \(-0.466362\pi\)
0.105481 + 0.994421i \(0.466362\pi\)
\(102\) 1.82912e10 0.164037
\(103\) −1.18118e10 −0.100395 −0.0501976 0.998739i \(-0.515985\pi\)
−0.0501976 + 0.998739i \(0.515985\pi\)
\(104\) −1.40146e10 −0.112953
\(105\) −4.17729e10 −0.319414
\(106\) −2.62508e10 −0.190528
\(107\) −2.30403e11 −1.58810 −0.794048 0.607855i \(-0.792030\pi\)
−0.794048 + 0.607855i \(0.792030\pi\)
\(108\) −2.98080e10 −0.195210
\(109\) −7.31425e10 −0.455328 −0.227664 0.973740i \(-0.573109\pi\)
−0.227664 + 0.973740i \(0.573109\pi\)
\(110\) 2.54623e10 0.150743
\(111\) −1.11863e10 −0.0630098
\(112\) 4.27710e11 2.29324
\(113\) −2.78421e11 −1.42158 −0.710788 0.703407i \(-0.751661\pi\)
−0.710788 + 0.703407i \(0.751661\pi\)
\(114\) −3.18322e10 −0.154843
\(115\) 2.73495e10 0.126798
\(116\) 1.34377e10 0.0594028
\(117\) −3.07331e10 −0.129594
\(118\) 4.51375e11 1.81629
\(119\) −2.17563e11 −0.835751
\(120\) −3.61192e10 −0.132508
\(121\) −2.67628e11 −0.938020
\(122\) 2.52223e11 0.844899
\(123\) −9.82895e10 −0.314796
\(124\) 2.81013e10 0.0860808
\(125\) −3.09698e11 −0.907680
\(126\) 6.91508e11 1.93981
\(127\) 6.84159e11 1.83754 0.918769 0.394795i \(-0.129184\pi\)
0.918769 + 0.394795i \(0.129184\pi\)
\(128\) 4.66986e11 1.20129
\(129\) −1.55213e10 −0.0382548
\(130\) 3.70554e10 0.0875313
\(131\) −3.46467e11 −0.784638 −0.392319 0.919829i \(-0.628327\pi\)
−0.392319 + 0.919829i \(0.628327\pi\)
\(132\) 1.17989e10 0.0256263
\(133\) 3.78626e11 0.788909
\(134\) −7.52062e11 −1.50376
\(135\) −1.67561e11 −0.321617
\(136\) −1.88117e11 −0.346708
\(137\) −2.96772e11 −0.525364 −0.262682 0.964883i \(-0.584607\pi\)
−0.262682 + 0.964883i \(0.584607\pi\)
\(138\) 5.22911e10 0.0889391
\(139\) 3.77869e11 0.617674 0.308837 0.951115i \(-0.400060\pi\)
0.308837 + 0.951115i \(0.400060\pi\)
\(140\) −2.02074e11 −0.317545
\(141\) −4.24386e10 −0.0641292
\(142\) −1.39172e12 −2.02286
\(143\) 2.57352e10 0.0359898
\(144\) 8.10991e11 1.09150
\(145\) 7.55382e10 0.0978684
\(146\) 1.27990e12 1.59675
\(147\) 6.82191e11 0.819711
\(148\) −5.41128e10 −0.0626413
\(149\) −6.93233e11 −0.773312 −0.386656 0.922224i \(-0.626370\pi\)
−0.386656 + 0.922224i \(0.626370\pi\)
\(150\) −2.48314e11 −0.266993
\(151\) −7.27671e11 −0.754331 −0.377165 0.926146i \(-0.623101\pi\)
−0.377165 + 0.926146i \(0.623101\pi\)
\(152\) 3.27381e11 0.327276
\(153\) −4.12527e11 −0.397786
\(154\) −5.79054e11 −0.538710
\(155\) 1.57967e11 0.141822
\(156\) 1.71710e10 0.0148803
\(157\) −1.94371e12 −1.62623 −0.813117 0.582101i \(-0.802231\pi\)
−0.813117 + 0.582101i \(0.802231\pi\)
\(158\) −2.00689e12 −1.62147
\(159\) −6.83798e10 −0.0533634
\(160\) −4.31631e11 −0.325427
\(161\) −6.21972e11 −0.453136
\(162\) 1.14226e12 0.804323
\(163\) 1.07427e12 0.731277 0.365638 0.930757i \(-0.380851\pi\)
0.365638 + 0.930757i \(0.380851\pi\)
\(164\) −4.75469e11 −0.312955
\(165\) 6.63259e10 0.0422203
\(166\) 9.06813e10 0.0558371
\(167\) −2.05398e12 −1.22365 −0.611824 0.790994i \(-0.709564\pi\)
−0.611824 + 0.790994i \(0.709564\pi\)
\(168\) 8.21409e11 0.473542
\(169\) −1.75471e12 −0.979102
\(170\) 4.97391e11 0.268676
\(171\) 7.17923e11 0.375491
\(172\) −7.50835e10 −0.0380310
\(173\) 5.47972e10 0.0268847 0.0134423 0.999910i \(-0.495721\pi\)
0.0134423 + 0.999910i \(0.495721\pi\)
\(174\) 1.44426e11 0.0686473
\(175\) 2.95355e12 1.36030
\(176\) −6.79106e11 −0.303122
\(177\) 1.17577e12 0.508710
\(178\) −1.09412e12 −0.458938
\(179\) 1.01838e12 0.414209 0.207105 0.978319i \(-0.433596\pi\)
0.207105 + 0.978319i \(0.433596\pi\)
\(180\) −3.83157e11 −0.151140
\(181\) 4.66142e12 1.78355 0.891777 0.452476i \(-0.149459\pi\)
0.891777 + 0.452476i \(0.149459\pi\)
\(182\) −8.42701e11 −0.312810
\(183\) 6.57006e11 0.236640
\(184\) −5.37792e11 −0.187982
\(185\) −3.04187e11 −0.103204
\(186\) 3.02026e11 0.0994772
\(187\) 3.45441e11 0.110470
\(188\) −2.05294e11 −0.0637541
\(189\) 3.81062e12 1.14936
\(190\) −8.65612e11 −0.253617
\(191\) −3.44126e12 −0.979567 −0.489783 0.871844i \(-0.662924\pi\)
−0.489783 + 0.871844i \(0.662924\pi\)
\(192\) 5.91188e11 0.163519
\(193\) 1.88765e12 0.507408 0.253704 0.967282i \(-0.418351\pi\)
0.253704 + 0.967282i \(0.418351\pi\)
\(194\) 5.31851e12 1.38957
\(195\) 9.65245e10 0.0245159
\(196\) 3.30006e12 0.814917
\(197\) 7.01609e12 1.68473 0.842366 0.538906i \(-0.181162\pi\)
0.842366 + 0.538906i \(0.181162\pi\)
\(198\) −1.09796e12 −0.256406
\(199\) 1.98650e12 0.451228 0.225614 0.974217i \(-0.427561\pi\)
0.225614 + 0.974217i \(0.427561\pi\)
\(200\) 2.55381e12 0.564317
\(201\) −1.95902e12 −0.421173
\(202\) 1.15853e12 0.242368
\(203\) −1.71786e12 −0.349752
\(204\) 2.30485e11 0.0456748
\(205\) −2.67278e12 −0.515605
\(206\) −6.14119e11 −0.115341
\(207\) −1.17934e12 −0.215676
\(208\) −9.88307e11 −0.176012
\(209\) −6.01173e11 −0.104279
\(210\) −2.17185e12 −0.366963
\(211\) −6.11632e12 −1.00678 −0.503392 0.864058i \(-0.667915\pi\)
−0.503392 + 0.864058i \(0.667915\pi\)
\(212\) −3.30783e11 −0.0530513
\(213\) −3.62525e12 −0.566566
\(214\) −1.19790e13 −1.82451
\(215\) −4.22071e11 −0.0626576
\(216\) 3.29487e12 0.476808
\(217\) −3.59243e12 −0.506827
\(218\) −3.80281e12 −0.523111
\(219\) 3.33398e12 0.447219
\(220\) 3.20848e11 0.0419734
\(221\) 5.02722e11 0.0641462
\(222\) −5.81593e11 −0.0723899
\(223\) −1.23957e13 −1.50520 −0.752600 0.658478i \(-0.771201\pi\)
−0.752600 + 0.658478i \(0.771201\pi\)
\(224\) 9.81600e12 1.16298
\(225\) 5.60030e12 0.647452
\(226\) −1.44756e13 −1.63320
\(227\) 1.20542e13 1.32739 0.663694 0.748004i \(-0.268988\pi\)
0.663694 + 0.748004i \(0.268988\pi\)
\(228\) −4.01114e11 −0.0431148
\(229\) 9.40324e12 0.986693 0.493347 0.869833i \(-0.335773\pi\)
0.493347 + 0.869833i \(0.335773\pi\)
\(230\) 1.42195e12 0.145674
\(231\) −1.50836e12 −0.150883
\(232\) −1.48536e12 −0.145093
\(233\) −1.26418e13 −1.20601 −0.603003 0.797739i \(-0.706029\pi\)
−0.603003 + 0.797739i \(0.706029\pi\)
\(234\) −1.59787e12 −0.148886
\(235\) −1.15403e12 −0.105037
\(236\) 5.68772e12 0.505734
\(237\) −5.22769e12 −0.454144
\(238\) −1.13115e13 −0.960166
\(239\) −1.22673e13 −1.01756 −0.508781 0.860896i \(-0.669904\pi\)
−0.508781 + 0.860896i \(0.669904\pi\)
\(240\) −2.54711e12 −0.206484
\(241\) −2.89078e12 −0.229046 −0.114523 0.993421i \(-0.536534\pi\)
−0.114523 + 0.993421i \(0.536534\pi\)
\(242\) −1.39144e13 −1.07766
\(243\) 1.10354e13 0.835511
\(244\) 3.17823e12 0.235256
\(245\) 1.85508e13 1.34261
\(246\) −5.11024e12 −0.361658
\(247\) −8.74890e11 −0.0605509
\(248\) −3.10621e12 −0.210255
\(249\) 2.36213e11 0.0156389
\(250\) −1.61017e13 −1.04280
\(251\) 2.68736e13 1.70263 0.851314 0.524657i \(-0.175806\pi\)
0.851314 + 0.524657i \(0.175806\pi\)
\(252\) 8.71362e12 0.540127
\(253\) 9.87552e11 0.0598959
\(254\) 3.55706e13 2.11109
\(255\) 1.29564e12 0.0752511
\(256\) 1.53394e13 0.871947
\(257\) −1.28426e13 −0.714533 −0.357266 0.934003i \(-0.616291\pi\)
−0.357266 + 0.934003i \(0.616291\pi\)
\(258\) −8.06982e11 −0.0439496
\(259\) 6.91771e12 0.368820
\(260\) 4.66931e11 0.0243725
\(261\) −3.25728e12 −0.166469
\(262\) −1.80134e13 −0.901444
\(263\) 1.60237e13 0.785248 0.392624 0.919699i \(-0.371567\pi\)
0.392624 + 0.919699i \(0.371567\pi\)
\(264\) −1.30421e12 −0.0625931
\(265\) −1.85945e12 −0.0874041
\(266\) 1.96854e13 0.906350
\(267\) −2.85003e12 −0.128540
\(268\) −9.47665e12 −0.418710
\(269\) 3.10713e13 1.34500 0.672500 0.740097i \(-0.265220\pi\)
0.672500 + 0.740097i \(0.265220\pi\)
\(270\) −8.71181e12 −0.369494
\(271\) −1.82362e13 −0.757885 −0.378943 0.925420i \(-0.623712\pi\)
−0.378943 + 0.925420i \(0.623712\pi\)
\(272\) −1.32659e13 −0.540268
\(273\) −2.19512e12 −0.0876122
\(274\) −1.54297e13 −0.603573
\(275\) −4.68957e12 −0.179806
\(276\) 6.58914e11 0.0247645
\(277\) −5.73318e12 −0.211230 −0.105615 0.994407i \(-0.533681\pi\)
−0.105615 + 0.994407i \(0.533681\pi\)
\(278\) 1.96461e13 0.709625
\(279\) −6.81169e12 −0.241230
\(280\) 2.23365e13 0.775616
\(281\) −2.17147e13 −0.739383 −0.369692 0.929155i \(-0.620537\pi\)
−0.369692 + 0.929155i \(0.620537\pi\)
\(282\) −2.20646e12 −0.0736759
\(283\) −5.22739e13 −1.71182 −0.855912 0.517121i \(-0.827004\pi\)
−0.855912 + 0.517121i \(0.827004\pi\)
\(284\) −1.75369e13 −0.563252
\(285\) −2.25481e12 −0.0710334
\(286\) 1.33802e12 0.0413475
\(287\) 6.07834e13 1.84262
\(288\) 1.86124e13 0.553532
\(289\) −2.75239e13 −0.803104
\(290\) 3.92736e12 0.112438
\(291\) 1.38540e13 0.389192
\(292\) 1.61279e13 0.444603
\(293\) 4.99463e13 1.35124 0.675619 0.737251i \(-0.263877\pi\)
0.675619 + 0.737251i \(0.263877\pi\)
\(294\) 3.54683e13 0.941738
\(295\) 3.19727e13 0.833217
\(296\) 5.98144e12 0.153004
\(297\) −6.05041e12 −0.151923
\(298\) −3.60424e13 −0.888432
\(299\) 1.43719e12 0.0347794
\(300\) −3.12898e12 −0.0743422
\(301\) 9.59859e12 0.223919
\(302\) −3.78329e13 −0.866625
\(303\) 3.01782e12 0.0678827
\(304\) 2.30868e13 0.509987
\(305\) 1.78659e13 0.387594
\(306\) −2.14480e13 −0.457003
\(307\) −5.28549e13 −1.10618 −0.553088 0.833123i \(-0.686551\pi\)
−0.553088 + 0.833123i \(0.686551\pi\)
\(308\) −7.29660e12 −0.150000
\(309\) −1.59970e12 −0.0323047
\(310\) 8.21298e12 0.162934
\(311\) −7.83474e13 −1.52701 −0.763506 0.645800i \(-0.776524\pi\)
−0.763506 + 0.645800i \(0.776524\pi\)
\(312\) −1.89803e12 −0.0363456
\(313\) 8.71210e13 1.63919 0.819594 0.572944i \(-0.194199\pi\)
0.819594 + 0.572944i \(0.194199\pi\)
\(314\) −1.01057e14 −1.86832
\(315\) 4.89823e13 0.889880
\(316\) −2.52886e13 −0.451488
\(317\) 7.95836e13 1.39636 0.698180 0.715922i \(-0.253993\pi\)
0.698180 + 0.715922i \(0.253993\pi\)
\(318\) −3.55519e12 −0.0613074
\(319\) 2.72758e12 0.0462304
\(320\) 1.60761e13 0.267829
\(321\) −3.12038e13 −0.511011
\(322\) −3.23374e13 −0.520593
\(323\) −1.17436e13 −0.185860
\(324\) 1.43934e13 0.223958
\(325\) −6.82476e12 −0.104407
\(326\) 5.58532e13 0.840139
\(327\) −9.90581e12 −0.146513
\(328\) 5.25567e13 0.764403
\(329\) 2.62445e13 0.375372
\(330\) 3.44840e12 0.0485055
\(331\) −5.59223e13 −0.773626 −0.386813 0.922158i \(-0.626424\pi\)
−0.386813 + 0.922158i \(0.626424\pi\)
\(332\) 1.14266e12 0.0155475
\(333\) 1.31168e13 0.175544
\(334\) −1.06790e14 −1.40581
\(335\) −5.32716e13 −0.689841
\(336\) 5.79254e13 0.737909
\(337\) −8.02614e13 −1.00587 −0.502935 0.864324i \(-0.667747\pi\)
−0.502935 + 0.864324i \(0.667747\pi\)
\(338\) −9.12303e13 −1.12486
\(339\) −3.77069e13 −0.457428
\(340\) 6.26757e12 0.0748109
\(341\) 5.70397e12 0.0669928
\(342\) 3.73260e13 0.431388
\(343\) −2.56269e14 −2.91460
\(344\) 8.29947e12 0.0928922
\(345\) 3.70399e12 0.0408004
\(346\) 2.84900e12 0.0308869
\(347\) 1.59883e13 0.170605 0.0853023 0.996355i \(-0.472814\pi\)
0.0853023 + 0.996355i \(0.472814\pi\)
\(348\) 1.81989e12 0.0191144
\(349\) −2.67236e13 −0.276284 −0.138142 0.990412i \(-0.544113\pi\)
−0.138142 + 0.990412i \(0.544113\pi\)
\(350\) 1.53560e14 1.56281
\(351\) −8.80519e12 −0.0882164
\(352\) −1.55856e13 −0.153723
\(353\) −9.23737e13 −0.896989 −0.448495 0.893786i \(-0.648040\pi\)
−0.448495 + 0.893786i \(0.648040\pi\)
\(354\) 6.11304e13 0.584439
\(355\) −9.85813e13 −0.927980
\(356\) −1.37869e13 −0.127788
\(357\) −2.94649e13 −0.268924
\(358\) 5.29475e13 0.475871
\(359\) 5.10662e13 0.451975 0.225987 0.974130i \(-0.427439\pi\)
0.225987 + 0.974130i \(0.427439\pi\)
\(360\) 4.23528e13 0.369164
\(361\) −9.60529e13 −0.824557
\(362\) 2.42355e14 2.04906
\(363\) −3.62453e13 −0.301832
\(364\) −1.06188e13 −0.0870997
\(365\) 9.06607e13 0.732501
\(366\) 3.41589e13 0.271868
\(367\) 5.30370e13 0.415830 0.207915 0.978147i \(-0.433332\pi\)
0.207915 + 0.978147i \(0.433332\pi\)
\(368\) −3.79249e13 −0.292928
\(369\) 1.15253e14 0.877015
\(370\) −1.58152e13 −0.118568
\(371\) 4.22869e13 0.312356
\(372\) 3.80580e12 0.0276987
\(373\) 2.07858e14 1.49062 0.745312 0.666716i \(-0.232301\pi\)
0.745312 + 0.666716i \(0.232301\pi\)
\(374\) 1.79601e13 0.126916
\(375\) −4.19429e13 −0.292069
\(376\) 2.26925e13 0.155722
\(377\) 3.96946e12 0.0268444
\(378\) 1.98121e14 1.32046
\(379\) 1.89936e14 1.24765 0.623823 0.781566i \(-0.285579\pi\)
0.623823 + 0.781566i \(0.285579\pi\)
\(380\) −1.09075e13 −0.0706179
\(381\) 9.26567e13 0.591275
\(382\) −1.78917e14 −1.12539
\(383\) −2.53366e14 −1.57092 −0.785462 0.618910i \(-0.787574\pi\)
−0.785462 + 0.618910i \(0.787574\pi\)
\(384\) 6.32446e13 0.386547
\(385\) −4.10168e13 −0.247131
\(386\) 9.81425e13 0.582944
\(387\) 1.82001e13 0.106577
\(388\) 6.70179e13 0.386916
\(389\) 2.50859e14 1.42793 0.713964 0.700182i \(-0.246898\pi\)
0.713964 + 0.700182i \(0.246898\pi\)
\(390\) 5.01848e12 0.0281654
\(391\) 1.92912e13 0.106755
\(392\) −3.64777e14 −1.99046
\(393\) −4.69225e13 −0.252477
\(394\) 3.64779e14 1.93553
\(395\) −1.42156e14 −0.743844
\(396\) −1.38353e13 −0.0713943
\(397\) −8.33694e13 −0.424286 −0.212143 0.977239i \(-0.568044\pi\)
−0.212143 + 0.977239i \(0.568044\pi\)
\(398\) 1.03281e14 0.518401
\(399\) 5.12780e13 0.253852
\(400\) 1.80093e14 0.879362
\(401\) −8.55764e13 −0.412154 −0.206077 0.978536i \(-0.566070\pi\)
−0.206077 + 0.978536i \(0.566070\pi\)
\(402\) −1.01853e14 −0.483872
\(403\) 8.30101e12 0.0389003
\(404\) 1.45985e13 0.0674857
\(405\) 8.09105e13 0.368980
\(406\) −8.93146e13 −0.401818
\(407\) −1.09838e13 −0.0487508
\(408\) −2.54770e13 −0.111562
\(409\) 2.89966e14 1.25276 0.626382 0.779516i \(-0.284535\pi\)
0.626382 + 0.779516i \(0.284535\pi\)
\(410\) −1.38963e14 −0.592361
\(411\) −4.01923e13 −0.169049
\(412\) −7.73844e12 −0.0321158
\(413\) −7.27111e14 −2.97766
\(414\) −6.13158e13 −0.247782
\(415\) 6.42332e12 0.0256150
\(416\) −2.26818e13 −0.0892615
\(417\) 5.11754e13 0.198753
\(418\) −3.12560e13 −0.119802
\(419\) 1.15977e14 0.438727 0.219364 0.975643i \(-0.429602\pi\)
0.219364 + 0.975643i \(0.429602\pi\)
\(420\) −2.73672e13 −0.102178
\(421\) −1.29586e14 −0.477535 −0.238767 0.971077i \(-0.576743\pi\)
−0.238767 + 0.971077i \(0.576743\pi\)
\(422\) −3.17998e14 −1.15666
\(423\) 4.97629e13 0.178663
\(424\) 3.65636e13 0.129580
\(425\) −9.16081e13 −0.320475
\(426\) −1.88483e14 −0.650908
\(427\) −4.06301e14 −1.38514
\(428\) −1.50947e14 −0.508022
\(429\) 3.48536e12 0.0115806
\(430\) −2.19442e13 −0.0719853
\(431\) 8.77012e13 0.284041 0.142020 0.989864i \(-0.454640\pi\)
0.142020 + 0.989864i \(0.454640\pi\)
\(432\) 2.32353e14 0.742999
\(433\) 3.21721e14 1.01577 0.507886 0.861424i \(-0.330427\pi\)
0.507886 + 0.861424i \(0.330427\pi\)
\(434\) −1.86777e14 −0.582276
\(435\) 1.02303e13 0.0314917
\(436\) −4.79188e13 −0.145656
\(437\) −3.35727e13 −0.100772
\(438\) 1.73339e14 0.513795
\(439\) 4.76314e14 1.39424 0.697121 0.716954i \(-0.254464\pi\)
0.697121 + 0.716954i \(0.254464\pi\)
\(440\) −3.54654e13 −0.102521
\(441\) −7.99928e14 −2.28370
\(442\) 2.61374e13 0.0736954
\(443\) 6.68817e13 0.186246 0.0931230 0.995655i \(-0.470315\pi\)
0.0931230 + 0.995655i \(0.470315\pi\)
\(444\) −7.32859e12 −0.0201564
\(445\) −7.75008e13 −0.210536
\(446\) −6.44474e14 −1.72927
\(447\) −9.38856e13 −0.248833
\(448\) −3.65598e14 −0.957138
\(449\) 3.99733e14 1.03375 0.516874 0.856062i \(-0.327096\pi\)
0.516874 + 0.856062i \(0.327096\pi\)
\(450\) 2.91170e14 0.743836
\(451\) −9.65104e13 −0.243558
\(452\) −1.82405e14 −0.454753
\(453\) −9.85497e13 −0.242725
\(454\) 6.26721e14 1.52499
\(455\) −5.96919e13 −0.143500
\(456\) 4.43378e13 0.105309
\(457\) 4.42060e14 1.03739 0.518695 0.854959i \(-0.326418\pi\)
0.518695 + 0.854959i \(0.326418\pi\)
\(458\) 4.88891e14 1.13358
\(459\) −1.18191e14 −0.270779
\(460\) 1.79178e13 0.0405618
\(461\) 6.61295e14 1.47925 0.739623 0.673022i \(-0.235004\pi\)
0.739623 + 0.673022i \(0.235004\pi\)
\(462\) −7.84223e13 −0.173344
\(463\) −1.41874e14 −0.309890 −0.154945 0.987923i \(-0.549520\pi\)
−0.154945 + 0.987923i \(0.549520\pi\)
\(464\) −1.04747e14 −0.226096
\(465\) 2.13937e13 0.0456347
\(466\) −6.57267e14 −1.38554
\(467\) 3.99126e14 0.831509 0.415755 0.909477i \(-0.363518\pi\)
0.415755 + 0.909477i \(0.363518\pi\)
\(468\) −2.01345e13 −0.0414562
\(469\) 1.21148e15 2.46528
\(470\) −6.00001e13 −0.120674
\(471\) −2.63240e14 −0.523282
\(472\) −6.28701e14 −1.23527
\(473\) −1.52404e13 −0.0295978
\(474\) −2.71797e14 −0.521751
\(475\) 1.59426e14 0.302513
\(476\) −1.42535e14 −0.267351
\(477\) 8.01812e13 0.148669
\(478\) −6.37799e14 −1.16904
\(479\) 3.65975e14 0.663141 0.331570 0.943431i \(-0.392422\pi\)
0.331570 + 0.943431i \(0.392422\pi\)
\(480\) −5.84566e13 −0.104714
\(481\) −1.59847e13 −0.0283079
\(482\) −1.50297e14 −0.263143
\(483\) −8.42347e13 −0.145808
\(484\) −1.75334e14 −0.300066
\(485\) 3.76731e14 0.637459
\(486\) 5.73748e14 0.959890
\(487\) 5.80639e14 0.960499 0.480249 0.877132i \(-0.340546\pi\)
0.480249 + 0.877132i \(0.340546\pi\)
\(488\) −3.51310e14 −0.574621
\(489\) 1.45490e14 0.235307
\(490\) 9.64488e14 1.54248
\(491\) 6.75492e14 1.06825 0.534124 0.845406i \(-0.320642\pi\)
0.534124 + 0.845406i \(0.320642\pi\)
\(492\) −6.43936e13 −0.100701
\(493\) 5.32816e13 0.0823985
\(494\) −4.54871e13 −0.0695648
\(495\) −7.77729e13 −0.117625
\(496\) −2.19049e14 −0.327636
\(497\) 2.24190e15 3.31632
\(498\) 1.22811e13 0.0179670
\(499\) −5.89501e14 −0.852965 −0.426483 0.904496i \(-0.640247\pi\)
−0.426483 + 0.904496i \(0.640247\pi\)
\(500\) −2.02896e14 −0.290361
\(501\) −2.78174e14 −0.393740
\(502\) 1.39720e15 1.95609
\(503\) −9.33902e14 −1.29324 −0.646618 0.762814i \(-0.723817\pi\)
−0.646618 + 0.762814i \(0.723817\pi\)
\(504\) −9.63173e14 −1.31928
\(505\) 8.20635e13 0.111185
\(506\) 5.13446e13 0.0688124
\(507\) −2.37643e14 −0.315051
\(508\) 4.48221e14 0.587817
\(509\) −5.80951e14 −0.753688 −0.376844 0.926277i \(-0.622991\pi\)
−0.376844 + 0.926277i \(0.622991\pi\)
\(510\) 6.73625e13 0.0864534
\(511\) −2.06177e15 −2.61774
\(512\) −1.58862e14 −0.199543
\(513\) 2.05689e14 0.255602
\(514\) −6.67711e14 −0.820902
\(515\) −4.35005e13 −0.0529120
\(516\) −1.01687e13 −0.0122375
\(517\) −4.16704e13 −0.0496169
\(518\) 3.59664e14 0.423724
\(519\) 7.42128e12 0.00865084
\(520\) −5.16129e13 −0.0595306
\(521\) 6.16566e14 0.703675 0.351837 0.936061i \(-0.385557\pi\)
0.351837 + 0.936061i \(0.385557\pi\)
\(522\) −1.69352e14 −0.191250
\(523\) −9.25923e14 −1.03470 −0.517351 0.855773i \(-0.673082\pi\)
−0.517351 + 0.855773i \(0.673082\pi\)
\(524\) −2.26985e14 −0.251001
\(525\) 4.00004e14 0.437712
\(526\) 8.33101e14 0.902145
\(527\) 1.11424e14 0.119404
\(528\) −9.19725e13 −0.0975374
\(529\) −8.97660e14 −0.942118
\(530\) −9.66760e13 −0.100416
\(531\) −1.37869e15 −1.41725
\(532\) 2.48054e14 0.252367
\(533\) −1.40452e14 −0.141426
\(534\) −1.48178e14 −0.147675
\(535\) −8.48524e14 −0.836986
\(536\) 1.04752e15 1.02271
\(537\) 1.37921e14 0.133282
\(538\) 1.61545e15 1.54523
\(539\) 6.69843e14 0.634212
\(540\) −1.09777e14 −0.102883
\(541\) 1.71163e15 1.58790 0.793952 0.607980i \(-0.208020\pi\)
0.793952 + 0.607980i \(0.208020\pi\)
\(542\) −9.48133e14 −0.870709
\(543\) 6.31304e14 0.573904
\(544\) −3.04456e14 −0.273987
\(545\) −2.69368e14 −0.239975
\(546\) −1.14128e14 −0.100655
\(547\) 3.09884e14 0.270563 0.135282 0.990807i \(-0.456806\pi\)
0.135282 + 0.990807i \(0.456806\pi\)
\(548\) −1.94428e14 −0.168060
\(549\) −7.70397e14 −0.659275
\(550\) −2.43819e14 −0.206573
\(551\) −9.27262e13 −0.0777802
\(552\) −7.28341e13 −0.0604880
\(553\) 3.23287e15 2.65827
\(554\) −2.98078e14 −0.242676
\(555\) −4.11966e13 −0.0332085
\(556\) 2.47558e14 0.197590
\(557\) −1.41441e15 −1.11782 −0.558911 0.829227i \(-0.688781\pi\)
−0.558911 + 0.829227i \(0.688781\pi\)
\(558\) −3.54152e14 −0.277141
\(559\) −2.21794e13 −0.0171864
\(560\) 1.57516e15 1.20862
\(561\) 4.67837e13 0.0355466
\(562\) −1.12899e15 −0.849452
\(563\) 1.58073e14 0.117778 0.0588888 0.998265i \(-0.481244\pi\)
0.0588888 + 0.998265i \(0.481244\pi\)
\(564\) −2.78033e13 −0.0205145
\(565\) −1.02536e15 −0.749223
\(566\) −2.71781e15 −1.96666
\(567\) −1.84004e15 −1.31862
\(568\) 1.93847e15 1.37576
\(569\) −7.94297e14 −0.558297 −0.279149 0.960248i \(-0.590052\pi\)
−0.279149 + 0.960248i \(0.590052\pi\)
\(570\) −1.17231e14 −0.0816078
\(571\) −2.46506e15 −1.69953 −0.849764 0.527164i \(-0.823255\pi\)
−0.849764 + 0.527164i \(0.823255\pi\)
\(572\) 1.68602e13 0.0115129
\(573\) −4.66055e14 −0.315201
\(574\) 3.16024e15 2.11692
\(575\) −2.61890e14 −0.173759
\(576\) −6.93219e14 −0.455561
\(577\) −7.32129e14 −0.476563 −0.238282 0.971196i \(-0.576584\pi\)
−0.238282 + 0.971196i \(0.576584\pi\)
\(578\) −1.43102e15 −0.922659
\(579\) 2.55648e14 0.163272
\(580\) 4.94882e13 0.0313075
\(581\) −1.46077e14 −0.0915403
\(582\) 7.20294e14 0.447130
\(583\) −6.71421e13 −0.0412874
\(584\) −1.78272e15 −1.08596
\(585\) −1.13183e14 −0.0683007
\(586\) 2.59680e15 1.55239
\(587\) −1.85488e15 −1.09851 −0.549256 0.835654i \(-0.685089\pi\)
−0.549256 + 0.835654i \(0.685089\pi\)
\(588\) 4.46932e14 0.262220
\(589\) −1.93911e14 −0.112712
\(590\) 1.66232e15 0.957255
\(591\) 9.50200e14 0.542106
\(592\) 4.21809e14 0.238422
\(593\) 2.47471e15 1.38587 0.692936 0.720999i \(-0.256317\pi\)
0.692936 + 0.720999i \(0.256317\pi\)
\(594\) −3.14571e14 −0.174539
\(595\) −8.01238e14 −0.440472
\(596\) −4.54166e14 −0.247377
\(597\) 2.69035e14 0.145194
\(598\) 7.47220e13 0.0399569
\(599\) −1.41526e15 −0.749877 −0.374938 0.927050i \(-0.622336\pi\)
−0.374938 + 0.927050i \(0.622336\pi\)
\(600\) 3.45866e14 0.181583
\(601\) −3.03423e15 −1.57848 −0.789240 0.614084i \(-0.789525\pi\)
−0.789240 + 0.614084i \(0.789525\pi\)
\(602\) 4.99047e14 0.257253
\(603\) 2.29712e15 1.17338
\(604\) −4.76728e14 −0.241306
\(605\) −9.85616e14 −0.494371
\(606\) 1.56902e14 0.0779882
\(607\) 3.26634e14 0.160888 0.0804440 0.996759i \(-0.474366\pi\)
0.0804440 + 0.996759i \(0.474366\pi\)
\(608\) 5.29846e14 0.258630
\(609\) −2.32653e14 −0.112542
\(610\) 9.28882e14 0.445293
\(611\) −6.06432e13 −0.0288108
\(612\) −2.70264e14 −0.127249
\(613\) 1.34079e15 0.625645 0.312822 0.949812i \(-0.398726\pi\)
0.312822 + 0.949812i \(0.398726\pi\)
\(614\) −2.74802e15 −1.27085
\(615\) −3.61979e14 −0.165909
\(616\) 8.06540e14 0.366380
\(617\) 1.57502e15 0.709117 0.354558 0.935034i \(-0.384631\pi\)
0.354558 + 0.935034i \(0.384631\pi\)
\(618\) −8.31711e13 −0.0371138
\(619\) 1.74802e15 0.773121 0.386561 0.922264i \(-0.373663\pi\)
0.386561 + 0.922264i \(0.373663\pi\)
\(620\) 1.03491e14 0.0453678
\(621\) −3.37886e14 −0.146814
\(622\) −4.07342e15 −1.75433
\(623\) 1.76250e15 0.752391
\(624\) −1.33848e14 −0.0566365
\(625\) 5.81384e14 0.243850
\(626\) 4.52957e15 1.88321
\(627\) −8.14179e13 −0.0335543
\(628\) −1.27340e15 −0.520222
\(629\) −2.14561e14 −0.0868907
\(630\) 2.54668e15 1.02235
\(631\) −4.49331e15 −1.78815 −0.894077 0.447913i \(-0.852167\pi\)
−0.894077 + 0.447913i \(0.852167\pi\)
\(632\) 2.79532e15 1.10278
\(633\) −8.28343e14 −0.323959
\(634\) 4.13769e15 1.60423
\(635\) 2.51961e15 0.968451
\(636\) −4.47985e13 −0.0170706
\(637\) 9.74826e14 0.368265
\(638\) 1.41811e14 0.0531126
\(639\) 4.25092e15 1.57844
\(640\) 1.71981e15 0.633126
\(641\) −1.93652e15 −0.706809 −0.353405 0.935471i \(-0.614976\pi\)
−0.353405 + 0.935471i \(0.614976\pi\)
\(642\) −1.62234e15 −0.587083
\(643\) 1.70043e15 0.610095 0.305048 0.952337i \(-0.401328\pi\)
0.305048 + 0.952337i \(0.401328\pi\)
\(644\) −4.07480e14 −0.144955
\(645\) −5.71618e13 −0.0201617
\(646\) −6.10568e14 −0.213528
\(647\) 2.04196e14 0.0708068 0.0354034 0.999373i \(-0.488728\pi\)
0.0354034 + 0.999373i \(0.488728\pi\)
\(648\) −1.59100e15 −0.547025
\(649\) 1.15449e15 0.393590
\(650\) −3.54831e14 −0.119950
\(651\) −4.86529e14 −0.163085
\(652\) 7.03800e14 0.233931
\(653\) 2.57221e15 0.847783 0.423891 0.905713i \(-0.360664\pi\)
0.423891 + 0.905713i \(0.360664\pi\)
\(654\) −5.15021e14 −0.168324
\(655\) −1.27596e15 −0.413533
\(656\) 3.70628e15 1.19115
\(657\) −3.90938e15 −1.24594
\(658\) 1.36450e15 0.431252
\(659\) −1.82638e14 −0.0572429 −0.0286215 0.999590i \(-0.509112\pi\)
−0.0286215 + 0.999590i \(0.509112\pi\)
\(660\) 4.34529e13 0.0135060
\(661\) 1.53812e15 0.474114 0.237057 0.971496i \(-0.423817\pi\)
0.237057 + 0.971496i \(0.423817\pi\)
\(662\) −2.90750e15 −0.888793
\(663\) 6.80845e13 0.0206407
\(664\) −1.26306e14 −0.0379752
\(665\) 1.39440e15 0.415784
\(666\) 6.81968e14 0.201677
\(667\) 1.52322e14 0.0446757
\(668\) −1.34565e15 −0.391437
\(669\) −1.67877e15 −0.484337
\(670\) −2.76968e15 −0.792535
\(671\) 6.45114e14 0.183089
\(672\) 1.32940e15 0.374217
\(673\) 5.36855e15 1.49890 0.749452 0.662058i \(-0.230317\pi\)
0.749452 + 0.662058i \(0.230317\pi\)
\(674\) −4.17293e15 −1.15561
\(675\) 1.60452e15 0.440731
\(676\) −1.14958e15 −0.313208
\(677\) 1.47849e15 0.399558 0.199779 0.979841i \(-0.435978\pi\)
0.199779 + 0.979841i \(0.435978\pi\)
\(678\) −1.96045e15 −0.525524
\(679\) −8.56749e15 −2.27808
\(680\) −6.92795e14 −0.182728
\(681\) 1.63253e15 0.427121
\(682\) 2.96559e14 0.0769657
\(683\) −5.89943e15 −1.51879 −0.759393 0.650632i \(-0.774504\pi\)
−0.759393 + 0.650632i \(0.774504\pi\)
\(684\) 4.70341e14 0.120117
\(685\) −1.09295e15 −0.276886
\(686\) −1.33239e16 −3.34848
\(687\) 1.27350e15 0.317494
\(688\) 5.85275e14 0.144752
\(689\) −9.77122e13 −0.0239741
\(690\) 1.92577e14 0.0468742
\(691\) −2.68896e15 −0.649314 −0.324657 0.945832i \(-0.605249\pi\)
−0.324657 + 0.945832i \(0.605249\pi\)
\(692\) 3.59000e13 0.00860024
\(693\) 1.76868e15 0.420356
\(694\) 8.31260e14 0.196002
\(695\) 1.39161e15 0.325537
\(696\) −2.01165e14 −0.0466875
\(697\) −1.88527e15 −0.434104
\(698\) −1.38941e15 −0.317413
\(699\) −1.71209e15 −0.388064
\(700\) 1.93500e15 0.435152
\(701\) 1.81744e14 0.0405520 0.0202760 0.999794i \(-0.493546\pi\)
0.0202760 + 0.999794i \(0.493546\pi\)
\(702\) −4.57797e14 −0.101349
\(703\) 3.73403e14 0.0820206
\(704\) 5.80487e14 0.126515
\(705\) −1.56292e14 −0.0337985
\(706\) −4.80267e15 −1.03052
\(707\) −1.86626e15 −0.397342
\(708\) 7.70298e14 0.162733
\(709\) 1.92897e15 0.404362 0.202181 0.979348i \(-0.435197\pi\)
0.202181 + 0.979348i \(0.435197\pi\)
\(710\) −5.12541e15 −1.06612
\(711\) 6.12992e15 1.26524
\(712\) 1.52395e15 0.312127
\(713\) 3.18539e14 0.0647397
\(714\) −1.53193e15 −0.308958
\(715\) 9.47773e13 0.0189680
\(716\) 6.67186e14 0.132503
\(717\) −1.66138e15 −0.327427
\(718\) 2.65502e15 0.519258
\(719\) −5.16510e15 −1.00247 −0.501234 0.865312i \(-0.667120\pi\)
−0.501234 + 0.865312i \(0.667120\pi\)
\(720\) 2.98670e15 0.575259
\(721\) 9.89272e14 0.189091
\(722\) −4.99396e15 −0.947306
\(723\) −3.91504e14 −0.0737013
\(724\) 3.05389e15 0.570547
\(725\) −7.23330e14 −0.134115
\(726\) −1.88446e15 −0.346764
\(727\) −1.46638e15 −0.267797 −0.133899 0.990995i \(-0.542750\pi\)
−0.133899 + 0.990995i \(0.542750\pi\)
\(728\) 1.17376e15 0.212744
\(729\) −2.39737e15 −0.431255
\(730\) 4.71361e15 0.841546
\(731\) −2.97712e14 −0.0527534
\(732\) 4.30433e14 0.0756997
\(733\) 5.01638e15 0.875626 0.437813 0.899066i \(-0.355753\pi\)
0.437813 + 0.899066i \(0.355753\pi\)
\(734\) 2.75749e15 0.477733
\(735\) 2.51236e15 0.432018
\(736\) −8.70382e14 −0.148553
\(737\) −1.92356e15 −0.325863
\(738\) 5.99220e15 1.00757
\(739\) −6.41227e15 −1.07021 −0.535103 0.844787i \(-0.679727\pi\)
−0.535103 + 0.844787i \(0.679727\pi\)
\(740\) −1.99286e14 −0.0330143
\(741\) −1.18488e14 −0.0194838
\(742\) 2.19857e15 0.358855
\(743\) 9.01527e15 1.46063 0.730315 0.683110i \(-0.239373\pi\)
0.730315 + 0.683110i \(0.239373\pi\)
\(744\) −4.20680e14 −0.0676551
\(745\) −2.55303e15 −0.407564
\(746\) 1.08069e16 1.71253
\(747\) −2.76980e14 −0.0435697
\(748\) 2.26313e14 0.0353387
\(749\) 1.92968e16 2.99113
\(750\) −2.18068e15 −0.335549
\(751\) −4.51349e15 −0.689434 −0.344717 0.938707i \(-0.612025\pi\)
−0.344717 + 0.938707i \(0.612025\pi\)
\(752\) 1.60027e15 0.242658
\(753\) 3.63953e15 0.547864
\(754\) 2.06379e14 0.0308406
\(755\) −2.67986e15 −0.397560
\(756\) 2.49650e15 0.367673
\(757\) −1.27132e15 −0.185878 −0.0929389 0.995672i \(-0.529626\pi\)
−0.0929389 + 0.995672i \(0.529626\pi\)
\(758\) 9.87509e15 1.43338
\(759\) 1.33746e14 0.0192730
\(760\) 1.20568e15 0.172487
\(761\) −8.49793e15 −1.20697 −0.603486 0.797373i \(-0.706222\pi\)
−0.603486 + 0.797373i \(0.706222\pi\)
\(762\) 4.81739e15 0.679296
\(763\) 6.12587e15 0.857597
\(764\) −2.25451e15 −0.313357
\(765\) −1.51925e15 −0.209648
\(766\) −1.31729e16 −1.80478
\(767\) 1.68013e15 0.228544
\(768\) 2.07745e15 0.280571
\(769\) −3.91757e15 −0.525317 −0.262659 0.964889i \(-0.584599\pi\)
−0.262659 + 0.964889i \(0.584599\pi\)
\(770\) −2.13253e15 −0.283920
\(771\) −1.73930e15 −0.229919
\(772\) 1.23668e15 0.162317
\(773\) 1.11302e16 1.45049 0.725244 0.688492i \(-0.241727\pi\)
0.725244 + 0.688492i \(0.241727\pi\)
\(774\) 9.46256e14 0.122443
\(775\) −1.51264e15 −0.194347
\(776\) −7.40793e15 −0.945055
\(777\) 9.36877e14 0.118677
\(778\) 1.30426e16 1.64050
\(779\) 3.28095e15 0.409773
\(780\) 6.32372e13 0.00784247
\(781\) −3.55963e15 −0.438353
\(782\) 1.00299e15 0.122647
\(783\) −9.33228e14 −0.113318
\(784\) −2.57239e16 −3.10169
\(785\) −7.15826e15 −0.857086
\(786\) −2.43958e15 −0.290063
\(787\) −2.14922e15 −0.253758 −0.126879 0.991918i \(-0.540496\pi\)
−0.126879 + 0.991918i \(0.540496\pi\)
\(788\) 4.59654e15 0.538935
\(789\) 2.17012e15 0.252674
\(790\) −7.39095e15 −0.854578
\(791\) 2.33184e16 2.67749
\(792\) 1.52930e15 0.174383
\(793\) 9.38838e14 0.106313
\(794\) −4.33452e15 −0.487448
\(795\) −2.51828e14 −0.0281245
\(796\) 1.30144e15 0.144345
\(797\) −1.56355e16 −1.72223 −0.861116 0.508409i \(-0.830234\pi\)
−0.861116 + 0.508409i \(0.830234\pi\)
\(798\) 2.66603e15 0.291642
\(799\) −8.14007e14 −0.0884343
\(800\) 4.13317e15 0.445952
\(801\) 3.34191e15 0.358110
\(802\) −4.44927e15 −0.473510
\(803\) 3.27363e15 0.346014
\(804\) −1.28344e15 −0.134731
\(805\) −2.29059e15 −0.238820
\(806\) 4.31584e14 0.0446913
\(807\) 4.20804e15 0.432789
\(808\) −1.61367e15 −0.164836
\(809\) −1.58636e16 −1.60948 −0.804740 0.593628i \(-0.797695\pi\)
−0.804740 + 0.593628i \(0.797695\pi\)
\(810\) 4.20668e15 0.423908
\(811\) 1.03213e16 1.03305 0.516524 0.856273i \(-0.327226\pi\)
0.516524 + 0.856273i \(0.327226\pi\)
\(812\) −1.12544e15 −0.111883
\(813\) −2.46976e15 −0.243869
\(814\) −5.71065e14 −0.0560082
\(815\) 3.95631e15 0.385410
\(816\) −1.79663e15 −0.173845
\(817\) 5.18110e14 0.0497967
\(818\) 1.50759e16 1.43926
\(819\) 2.57397e15 0.244086
\(820\) −1.75105e15 −0.164939
\(821\) 1.30461e16 1.22065 0.610327 0.792149i \(-0.291038\pi\)
0.610327 + 0.792149i \(0.291038\pi\)
\(822\) −2.08967e15 −0.194215
\(823\) −9.40594e15 −0.868366 −0.434183 0.900825i \(-0.642963\pi\)
−0.434183 + 0.900825i \(0.642963\pi\)
\(824\) 8.55380e14 0.0784440
\(825\) −6.35117e14 −0.0578571
\(826\) −3.78038e16 −3.42094
\(827\) −5.58618e15 −0.502151 −0.251076 0.967967i \(-0.580784\pi\)
−0.251076 + 0.967967i \(0.580784\pi\)
\(828\) −7.72634e14 −0.0689933
\(829\) 1.46565e15 0.130011 0.0650054 0.997885i \(-0.479294\pi\)
0.0650054 + 0.997885i \(0.479294\pi\)
\(830\) 3.33960e14 0.0294283
\(831\) −7.76454e14 −0.0679688
\(832\) 8.44786e14 0.0734629
\(833\) 1.30850e16 1.13038
\(834\) 2.66070e15 0.228340
\(835\) −7.56438e15 −0.644908
\(836\) −3.93854e14 −0.0333580
\(837\) −1.95159e15 −0.164209
\(838\) 6.02984e15 0.504039
\(839\) −2.31777e16 −1.92477 −0.962387 0.271684i \(-0.912420\pi\)
−0.962387 + 0.271684i \(0.912420\pi\)
\(840\) 3.02507e15 0.249574
\(841\) 4.20707e14 0.0344828
\(842\) −6.73738e15 −0.548623
\(843\) −2.94086e15 −0.237915
\(844\) −4.00706e15 −0.322064
\(845\) −6.46221e15 −0.516023
\(846\) 2.58726e15 0.205260
\(847\) 2.24145e16 1.76673
\(848\) 2.57845e15 0.201921
\(849\) −7.07953e15 −0.550823
\(850\) −4.76286e15 −0.368183
\(851\) −6.13391e14 −0.0471113
\(852\) −2.37506e15 −0.181241
\(853\) −1.62059e16 −1.22872 −0.614359 0.789027i \(-0.710585\pi\)
−0.614359 + 0.789027i \(0.710585\pi\)
\(854\) −2.11243e16 −1.59134
\(855\) 2.64395e15 0.197898
\(856\) 1.66851e16 1.24086
\(857\) −1.76845e15 −0.130677 −0.0653383 0.997863i \(-0.520813\pi\)
−0.0653383 + 0.997863i \(0.520813\pi\)
\(858\) 1.81210e14 0.0133046
\(859\) 2.15096e16 1.56917 0.784586 0.620020i \(-0.212875\pi\)
0.784586 + 0.620020i \(0.212875\pi\)
\(860\) −2.76517e14 −0.0200438
\(861\) 8.23200e15 0.592909
\(862\) 4.55974e15 0.326325
\(863\) 2.05957e16 1.46459 0.732297 0.680986i \(-0.238448\pi\)
0.732297 + 0.680986i \(0.238448\pi\)
\(864\) 5.33254e15 0.376798
\(865\) 2.01806e14 0.0141692
\(866\) 1.67269e16 1.16699
\(867\) −3.72761e15 −0.258419
\(868\) −2.35355e15 −0.162131
\(869\) −5.13306e15 −0.351373
\(870\) 5.31889e14 0.0361797
\(871\) −2.79937e15 −0.189217
\(872\) 5.29677e15 0.355771
\(873\) −1.62450e16 −1.08428
\(874\) −1.74550e15 −0.115773
\(875\) 2.59380e16 1.70959
\(876\) 2.18423e15 0.143062
\(877\) 5.19470e15 0.338114 0.169057 0.985606i \(-0.445928\pi\)
0.169057 + 0.985606i \(0.445928\pi\)
\(878\) 2.47644e16 1.60180
\(879\) 6.76431e15 0.434795
\(880\) −2.50100e15 −0.159757
\(881\) −1.55161e16 −0.984952 −0.492476 0.870326i \(-0.663908\pi\)
−0.492476 + 0.870326i \(0.663908\pi\)
\(882\) −4.15897e16 −2.62366
\(883\) −1.33984e16 −0.839977 −0.419989 0.907529i \(-0.637966\pi\)
−0.419989 + 0.907529i \(0.637966\pi\)
\(884\) 3.29355e14 0.0205199
\(885\) 4.33012e15 0.268109
\(886\) 3.47730e15 0.213972
\(887\) 4.71681e15 0.288449 0.144224 0.989545i \(-0.453931\pi\)
0.144224 + 0.989545i \(0.453931\pi\)
\(888\) 8.10076e14 0.0492328
\(889\) −5.73000e16 −3.46095
\(890\) −4.02940e15 −0.241878
\(891\) 2.92157e15 0.174296
\(892\) −8.12095e15 −0.481504
\(893\) 1.41662e15 0.0834777
\(894\) −4.88128e15 −0.285876
\(895\) 3.75049e15 0.218304
\(896\) −3.91112e16 −2.26260
\(897\) 1.94641e14 0.0111912
\(898\) 2.07828e16 1.18764
\(899\) 8.79792e14 0.0499691
\(900\) 3.66899e15 0.207116
\(901\) −1.31158e15 −0.0735883
\(902\) −5.01774e15 −0.279816
\(903\) 1.29995e15 0.0720518
\(904\) 2.01624e16 1.11075
\(905\) 1.71670e16 0.939999
\(906\) −5.12377e15 −0.278859
\(907\) −4.70220e15 −0.254367 −0.127183 0.991879i \(-0.540594\pi\)
−0.127183 + 0.991879i \(0.540594\pi\)
\(908\) 7.89724e15 0.424623
\(909\) −3.53866e15 −0.189120
\(910\) −3.10349e15 −0.164863
\(911\) 7.70359e15 0.406763 0.203382 0.979100i \(-0.434807\pi\)
0.203382 + 0.979100i \(0.434807\pi\)
\(912\) 3.12668e15 0.164101
\(913\) 2.31937e14 0.0120999
\(914\) 2.29835e16 1.19182
\(915\) 2.41961e15 0.124718
\(916\) 6.16046e15 0.315637
\(917\) 2.90175e16 1.47784
\(918\) −6.14496e15 −0.311089
\(919\) 2.83879e16 1.42856 0.714280 0.699860i \(-0.246754\pi\)
0.714280 + 0.699860i \(0.246754\pi\)
\(920\) −1.98057e15 −0.0990735
\(921\) −7.15823e15 −0.355941
\(922\) 3.43819e16 1.69945
\(923\) −5.18035e15 −0.254536
\(924\) −9.88190e14 −0.0482664
\(925\) 2.91280e15 0.141427
\(926\) −7.37627e15 −0.356022
\(927\) 1.87578e15 0.0900004
\(928\) −2.40396e15 −0.114660
\(929\) −2.43302e13 −0.00115361 −0.000576804 1.00000i \(-0.500184\pi\)
−0.000576804 1.00000i \(0.500184\pi\)
\(930\) 1.11230e15 0.0524282
\(931\) −2.27719e16 −1.06703
\(932\) −8.28214e15 −0.385794
\(933\) −1.06107e16 −0.491356
\(934\) 2.07512e16 0.955293
\(935\) 1.27219e15 0.0582219
\(936\) 2.22560e15 0.101258
\(937\) −6.94518e15 −0.314135 −0.157067 0.987588i \(-0.550204\pi\)
−0.157067 + 0.987588i \(0.550204\pi\)
\(938\) 6.29871e16 2.83228
\(939\) 1.17989e16 0.527451
\(940\) −7.56054e14 −0.0336008
\(941\) −3.11414e16 −1.37593 −0.687963 0.725745i \(-0.741495\pi\)
−0.687963 + 0.725745i \(0.741495\pi\)
\(942\) −1.36863e16 −0.601181
\(943\) −5.38965e15 −0.235367
\(944\) −4.43357e16 −1.92490
\(945\) 1.40337e16 0.605755
\(946\) −7.92375e14 −0.0340039
\(947\) −3.66620e16 −1.56420 −0.782098 0.623155i \(-0.785851\pi\)
−0.782098 + 0.623155i \(0.785851\pi\)
\(948\) −3.42488e15 −0.145278
\(949\) 4.76413e15 0.200918
\(950\) 8.28884e15 0.347547
\(951\) 1.07781e16 0.449315
\(952\) 1.57553e16 0.653015
\(953\) 1.01542e16 0.418440 0.209220 0.977869i \(-0.432908\pi\)
0.209220 + 0.977869i \(0.432908\pi\)
\(954\) 4.16876e15 0.170801
\(955\) −1.26734e16 −0.516268
\(956\) −8.03683e15 −0.325512
\(957\) 3.69400e14 0.0148758
\(958\) 1.90277e16 0.761860
\(959\) 2.48554e16 0.989507
\(960\) 2.17722e15 0.0861808
\(961\) −2.35686e16 −0.927589
\(962\) −8.31074e14 −0.0325220
\(963\) 3.65892e16 1.42367
\(964\) −1.89388e15 −0.0732702
\(965\) 6.95183e15 0.267423
\(966\) −4.37951e15 −0.167514
\(967\) −3.28531e16 −1.24949 −0.624743 0.780830i \(-0.714796\pi\)
−0.624743 + 0.780830i \(0.714796\pi\)
\(968\) 1.93808e16 0.732923
\(969\) −1.59045e15 −0.0598052
\(970\) 1.95869e16 0.732355
\(971\) −2.39850e16 −0.891732 −0.445866 0.895100i \(-0.647104\pi\)
−0.445866 + 0.895100i \(0.647104\pi\)
\(972\) 7.22973e15 0.267274
\(973\) −3.16475e16 −1.16337
\(974\) 3.01884e16 1.10348
\(975\) −9.24289e14 −0.0335956
\(976\) −2.47742e16 −0.895419
\(977\) −1.64069e16 −0.589667 −0.294834 0.955549i \(-0.595264\pi\)
−0.294834 + 0.955549i \(0.595264\pi\)
\(978\) 7.56429e15 0.270336
\(979\) −2.79845e15 −0.0994517
\(980\) 1.21534e16 0.429491
\(981\) 1.16154e16 0.408183
\(982\) 3.51200e16 1.22727
\(983\) −4.35578e16 −1.51364 −0.756818 0.653626i \(-0.773247\pi\)
−0.756818 + 0.653626i \(0.773247\pi\)
\(984\) 7.11784e15 0.245966
\(985\) 2.58387e16 0.887917
\(986\) 2.77020e15 0.0946648
\(987\) 3.55434e15 0.120786
\(988\) −5.73178e14 −0.0193698
\(989\) −8.51103e14 −0.0286024
\(990\) −4.04355e15 −0.135135
\(991\) −2.49086e16 −0.827835 −0.413918 0.910314i \(-0.635840\pi\)
−0.413918 + 0.910314i \(0.635840\pi\)
\(992\) −5.02721e15 −0.166155
\(993\) −7.57365e15 −0.248934
\(994\) 1.16560e17 3.81000
\(995\) 7.31584e15 0.237814
\(996\) 1.54753e14 0.00500279
\(997\) −2.41171e16 −0.775358 −0.387679 0.921795i \(-0.626723\pi\)
−0.387679 + 0.921795i \(0.626723\pi\)
\(998\) −3.06492e16 −0.979943
\(999\) 3.75805e15 0.119496
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 29.12.a.a.1.9 11
3.2 odd 2 261.12.a.a.1.3 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.12.a.a.1.9 11 1.1 even 1 trivial
261.12.a.a.1.3 11 3.2 odd 2