Properties

Label 29.12.a.a.1.2
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 1.2
Root \(-65.3340\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q-68.3340 q^{2} +537.644 q^{3} +2621.54 q^{4} +8265.40 q^{5} -36739.4 q^{6} -68659.4 q^{7} -39192.0 q^{8} +111914. q^{9} +O(q^{10})\) \(q-68.3340 q^{2} +537.644 q^{3} +2621.54 q^{4} +8265.40 q^{5} -36739.4 q^{6} -68659.4 q^{7} -39192.0 q^{8} +111914. q^{9} -564808. q^{10} -970140. q^{11} +1.40945e6 q^{12} +1.54127e6 q^{13} +4.69177e6 q^{14} +4.44384e6 q^{15} -2.69076e6 q^{16} -3.34371e6 q^{17} -7.64755e6 q^{18} +5.46951e6 q^{19} +2.16680e7 q^{20} -3.69143e7 q^{21} +6.62936e7 q^{22} -3.32547e7 q^{23} -2.10714e7 q^{24} +1.94887e7 q^{25} -1.05321e8 q^{26} -3.50720e7 q^{27} -1.79993e8 q^{28} +2.05111e7 q^{29} -3.03666e8 q^{30} -8.71995e7 q^{31} +2.64136e8 q^{32} -5.21590e8 q^{33} +2.28489e8 q^{34} -5.67498e8 q^{35} +2.93387e8 q^{36} -1.97918e8 q^{37} -3.73754e8 q^{38} +8.28652e8 q^{39} -3.23938e8 q^{40} +1.10998e9 q^{41} +2.52251e9 q^{42} -9.82013e8 q^{43} -2.54326e9 q^{44} +9.25016e8 q^{45} +2.27243e9 q^{46} -2.38472e9 q^{47} -1.44667e9 q^{48} +2.73679e9 q^{49} -1.33174e9 q^{50} -1.79773e9 q^{51} +4.04048e9 q^{52} +2.86882e9 q^{53} +2.39661e9 q^{54} -8.01859e9 q^{55} +2.69090e9 q^{56} +2.94065e9 q^{57} -1.40161e9 q^{58} -7.54410e9 q^{59} +1.16497e10 q^{60} -1.25957e10 q^{61} +5.95869e9 q^{62} -7.68397e9 q^{63} -1.25388e10 q^{64} +1.27392e10 q^{65} +3.56423e10 q^{66} -2.53077e8 q^{67} -8.76565e9 q^{68} -1.78792e10 q^{69} +3.87794e10 q^{70} -7.66739e9 q^{71} -4.38615e9 q^{72} -1.59232e10 q^{73} +1.35245e10 q^{74} +1.04780e10 q^{75} +1.43385e10 q^{76} +6.66093e10 q^{77} -5.66251e10 q^{78} +1.03493e10 q^{79} -2.22402e10 q^{80} -3.86815e10 q^{81} -7.58492e10 q^{82} +2.23246e10 q^{83} -9.67723e10 q^{84} -2.76371e10 q^{85} +6.71049e10 q^{86} +1.10277e10 q^{87} +3.80218e10 q^{88} +7.10371e10 q^{89} -6.32100e10 q^{90} -1.05822e11 q^{91} -8.71784e10 q^{92} -4.68823e10 q^{93} +1.62957e11 q^{94} +4.52077e10 q^{95} +1.42011e11 q^{96} -5.61657e9 q^{97} -1.87016e11 q^{98} -1.08572e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} + O(q^{10}) \) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} - 685834q^{10} - 612246q^{11} + 2578538q^{12} + 1510364q^{13} + 3955400q^{14} - 2462818q^{15} + 3024818q^{16} - 3291098q^{17} - 27885614q^{18} - 44121388q^{19} - 49472662q^{20} - 46916800q^{21} - 43435618q^{22} - 88684076q^{23} - 224700678q^{24} - 44195521q^{25} - 324999762q^{26} - 236304286q^{27} - 391274848q^{28} + 225622639q^{29} - 494910382q^{30} - 292235934q^{31} - 632542514q^{32} - 1079766410q^{33} - 1113307936q^{34} - 1312820120q^{35} - 2236726492q^{36} - 1380429338q^{37} - 1222857284q^{38} - 1186931090q^{39} - 2713154106q^{40} - 1062067494q^{41} + 205598960q^{42} + 74588594q^{43} + 52891466q^{44} + 4527996830q^{45} - 87670324q^{46} - 1821239394q^{47} + 2666035542q^{48} + 4692522003q^{49} + 9494259926q^{50} + 8768158380q^{51} + 3266669866q^{52} + 7818635688q^{53} + 17402728558q^{54} - 191002682q^{55} + 11263587512q^{56} + 15495358340q^{57} - 656356768q^{58} + 1230002712q^{59} + 31834046430q^{60} - 18602654230q^{61} + 22075953162q^{62} - 9964531456q^{63} + 11813658086q^{64} + 32245789334q^{65} + 42677188354q^{66} + 27481284652q^{67} + 29588811820q^{68} - 20565315068q^{69} + 42862666712q^{70} - 20347168516q^{71} + 47061083616q^{72} - 57740010478q^{73} - 2640709564q^{74} - 23544691000q^{75} - 33350650772q^{76} + 871959792q^{77} - 15384525342q^{78} - 120245016462q^{79} - 84319695274q^{80} - 48880047865q^{81} - 111495532412q^{82} - 142463983824q^{83} - 134146226376q^{84} - 181628566552q^{85} + 47870165542q^{86} - 20141948318q^{87} - 180608014462q^{88} - 96700717270q^{89} - 25522461244q^{90} - 355162031176q^{91} - 22429477796q^{92} - 172582115142q^{93} + 172608565078q^{94} - 195922150708q^{95} + 226391047758q^{96} - 303190852014q^{97} - 123776497136q^{98} - 139125462440q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −68.3340 −1.50998 −0.754991 0.655735i \(-0.772359\pi\)
−0.754991 + 0.655735i \(0.772359\pi\)
\(3\) 537.644 1.27740 0.638702 0.769454i \(-0.279472\pi\)
0.638702 + 0.769454i \(0.279472\pi\)
\(4\) 2621.54 1.28005
\(5\) 8265.40 1.18285 0.591424 0.806361i \(-0.298566\pi\)
0.591424 + 0.806361i \(0.298566\pi\)
\(6\) −36739.4 −1.92886
\(7\) −68659.4 −1.54405 −0.772024 0.635593i \(-0.780756\pi\)
−0.772024 + 0.635593i \(0.780756\pi\)
\(8\) −39192.0 −0.422866
\(9\) 111914. 0.631759
\(10\) −564808. −1.78608
\(11\) −970140. −1.81625 −0.908123 0.418703i \(-0.862485\pi\)
−0.908123 + 0.418703i \(0.862485\pi\)
\(12\) 1.40945e6 1.63514
\(13\) 1.54127e6 1.15130 0.575651 0.817696i \(-0.304749\pi\)
0.575651 + 0.817696i \(0.304749\pi\)
\(14\) 4.69177e6 2.33149
\(15\) 4.44384e6 1.51097
\(16\) −2.69076e6 −0.641527
\(17\) −3.34371e6 −0.571162 −0.285581 0.958355i \(-0.592187\pi\)
−0.285581 + 0.958355i \(0.592187\pi\)
\(18\) −7.64755e6 −0.953945
\(19\) 5.46951e6 0.506762 0.253381 0.967367i \(-0.418457\pi\)
0.253381 + 0.967367i \(0.418457\pi\)
\(20\) 2.16680e7 1.51410
\(21\) −3.69143e7 −1.97237
\(22\) 6.62936e7 2.74250
\(23\) −3.32547e7 −1.07733 −0.538667 0.842519i \(-0.681072\pi\)
−0.538667 + 0.842519i \(0.681072\pi\)
\(24\) −2.10714e7 −0.540171
\(25\) 1.94887e7 0.399128
\(26\) −1.05321e8 −1.73845
\(27\) −3.50720e7 −0.470392
\(28\) −1.79993e8 −1.97646
\(29\) 2.05111e7 0.185695
\(30\) −3.03666e8 −2.28154
\(31\) −8.71995e7 −0.547047 −0.273523 0.961865i \(-0.588189\pi\)
−0.273523 + 0.961865i \(0.588189\pi\)
\(32\) 2.64136e8 1.39156
\(33\) −5.21590e8 −2.32008
\(34\) 2.28489e8 0.862445
\(35\) −5.67498e8 −1.82637
\(36\) 2.93387e8 0.808681
\(37\) −1.97918e8 −0.469220 −0.234610 0.972090i \(-0.575381\pi\)
−0.234610 + 0.972090i \(0.575381\pi\)
\(38\) −3.73754e8 −0.765201
\(39\) 8.28652e8 1.47068
\(40\) −3.23938e8 −0.500186
\(41\) 1.10998e9 1.49625 0.748123 0.663560i \(-0.230955\pi\)
0.748123 + 0.663560i \(0.230955\pi\)
\(42\) 2.52251e9 2.97825
\(43\) −9.82013e8 −1.01869 −0.509344 0.860563i \(-0.670112\pi\)
−0.509344 + 0.860563i \(0.670112\pi\)
\(44\) −2.54326e9 −2.32488
\(45\) 9.25016e8 0.747275
\(46\) 2.27243e9 1.62675
\(47\) −2.38472e9 −1.51670 −0.758348 0.651850i \(-0.773993\pi\)
−0.758348 + 0.651850i \(0.773993\pi\)
\(48\) −1.44667e9 −0.819488
\(49\) 2.73679e9 1.38409
\(50\) −1.33174e9 −0.602676
\(51\) −1.79773e9 −0.729604
\(52\) 4.04048e9 1.47372
\(53\) 2.86882e9 0.942293 0.471147 0.882055i \(-0.343840\pi\)
0.471147 + 0.882055i \(0.343840\pi\)
\(54\) 2.39661e9 0.710284
\(55\) −8.01859e9 −2.14834
\(56\) 2.69090e9 0.652926
\(57\) 2.94065e9 0.647339
\(58\) −1.40161e9 −0.280397
\(59\) −7.54410e9 −1.37379 −0.686896 0.726755i \(-0.741027\pi\)
−0.686896 + 0.726755i \(0.741027\pi\)
\(60\) 1.16497e10 1.93412
\(61\) −1.25957e10 −1.90945 −0.954723 0.297496i \(-0.903849\pi\)
−0.954723 + 0.297496i \(0.903849\pi\)
\(62\) 5.95869e9 0.826031
\(63\) −7.68397e9 −0.975467
\(64\) −1.25388e10 −1.45970
\(65\) 1.27392e10 1.36181
\(66\) 3.56423e10 3.50328
\(67\) −2.53077e8 −0.0229003 −0.0114502 0.999934i \(-0.503645\pi\)
−0.0114502 + 0.999934i \(0.503645\pi\)
\(68\) −8.76565e9 −0.731114
\(69\) −1.78792e10 −1.37619
\(70\) 3.87794e10 2.75779
\(71\) −7.66739e9 −0.504344 −0.252172 0.967682i \(-0.581145\pi\)
−0.252172 + 0.967682i \(0.581145\pi\)
\(72\) −4.38615e9 −0.267150
\(73\) −1.59232e10 −0.898991 −0.449495 0.893283i \(-0.648396\pi\)
−0.449495 + 0.893283i \(0.648396\pi\)
\(74\) 1.35245e10 0.708513
\(75\) 1.04780e10 0.509848
\(76\) 1.43385e10 0.648679
\(77\) 6.66093e10 2.80437
\(78\) −5.66251e10 −2.22070
\(79\) 1.03493e10 0.378411 0.189205 0.981938i \(-0.439409\pi\)
0.189205 + 0.981938i \(0.439409\pi\)
\(80\) −2.22402e10 −0.758828
\(81\) −3.86815e10 −1.23264
\(82\) −7.58492e10 −2.25930
\(83\) 2.23246e10 0.622091 0.311046 0.950395i \(-0.399321\pi\)
0.311046 + 0.950395i \(0.399321\pi\)
\(84\) −9.67723e10 −2.52473
\(85\) −2.76371e10 −0.675598
\(86\) 6.71049e10 1.53820
\(87\) 1.10277e10 0.237208
\(88\) 3.80218e10 0.768029
\(89\) 7.10371e10 1.34847 0.674233 0.738518i \(-0.264474\pi\)
0.674233 + 0.738518i \(0.264474\pi\)
\(90\) −6.32100e10 −1.12837
\(91\) −1.05822e11 −1.77767
\(92\) −8.71784e10 −1.37904
\(93\) −4.68823e10 −0.698800
\(94\) 1.62957e11 2.29018
\(95\) 4.52077e10 0.599422
\(96\) 1.42011e11 1.77758
\(97\) −5.61657e9 −0.0664090 −0.0332045 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(98\) −1.87016e11 −2.08995
\(99\) −1.08572e11 −1.14743
\(100\) 5.10903e10 0.510903
\(101\) 1.50942e11 1.42904 0.714518 0.699617i \(-0.246646\pi\)
0.714518 + 0.699617i \(0.246646\pi\)
\(102\) 1.22846e11 1.10169
\(103\) −4.18049e10 −0.355322 −0.177661 0.984092i \(-0.556853\pi\)
−0.177661 + 0.984092i \(0.556853\pi\)
\(104\) −6.04054e10 −0.486847
\(105\) −3.05112e11 −2.33302
\(106\) −1.96038e11 −1.42285
\(107\) 3.91482e10 0.269837 0.134918 0.990857i \(-0.456923\pi\)
0.134918 + 0.990857i \(0.456923\pi\)
\(108\) −9.19426e10 −0.602124
\(109\) −1.25846e11 −0.783421 −0.391710 0.920089i \(-0.628116\pi\)
−0.391710 + 0.920089i \(0.628116\pi\)
\(110\) 5.47943e11 3.24396
\(111\) −1.06410e11 −0.599383
\(112\) 1.84746e11 0.990548
\(113\) 8.94509e10 0.456724 0.228362 0.973576i \(-0.426663\pi\)
0.228362 + 0.973576i \(0.426663\pi\)
\(114\) −2.00947e11 −0.977471
\(115\) −2.74863e11 −1.27432
\(116\) 5.37707e10 0.237699
\(117\) 1.72490e11 0.727345
\(118\) 5.15518e11 2.07440
\(119\) 2.29577e11 0.881902
\(120\) −1.74163e11 −0.638940
\(121\) 6.55860e11 2.29875
\(122\) 8.60713e11 2.88323
\(123\) 5.96773e11 1.91131
\(124\) −2.28597e11 −0.700246
\(125\) −2.42502e11 −0.710740
\(126\) 5.25076e11 1.47294
\(127\) 1.50285e11 0.403642 0.201821 0.979422i \(-0.435314\pi\)
0.201821 + 0.979422i \(0.435314\pi\)
\(128\) 3.15875e11 0.812569
\(129\) −5.27974e11 −1.30127
\(130\) −8.70519e11 −2.05632
\(131\) −2.82004e11 −0.638651 −0.319326 0.947645i \(-0.603456\pi\)
−0.319326 + 0.947645i \(0.603456\pi\)
\(132\) −1.36737e12 −2.96981
\(133\) −3.75534e11 −0.782465
\(134\) 1.72938e10 0.0345791
\(135\) −2.89884e11 −0.556402
\(136\) 1.31047e11 0.241525
\(137\) 3.90301e11 0.690933 0.345467 0.938431i \(-0.387721\pi\)
0.345467 + 0.938431i \(0.387721\pi\)
\(138\) 1.22176e12 2.07802
\(139\) −4.80262e10 −0.0785049 −0.0392524 0.999229i \(-0.512498\pi\)
−0.0392524 + 0.999229i \(0.512498\pi\)
\(140\) −1.48772e12 −2.33785
\(141\) −1.28213e12 −1.93743
\(142\) 5.23943e11 0.761550
\(143\) −1.49524e12 −2.09105
\(144\) −3.01134e11 −0.405290
\(145\) 1.69533e11 0.219649
\(146\) 1.08810e12 1.35746
\(147\) 1.47142e12 1.76804
\(148\) −5.18850e11 −0.600623
\(149\) −7.91388e11 −0.882805 −0.441402 0.897309i \(-0.645519\pi\)
−0.441402 + 0.897309i \(0.645519\pi\)
\(150\) −7.16002e11 −0.769861
\(151\) 1.31177e12 1.35983 0.679915 0.733291i \(-0.262016\pi\)
0.679915 + 0.733291i \(0.262016\pi\)
\(152\) −2.14361e11 −0.214292
\(153\) −3.74209e11 −0.360837
\(154\) −4.55168e12 −4.23455
\(155\) −7.20738e11 −0.647073
\(156\) 2.17234e12 1.88253
\(157\) −6.76641e11 −0.566122 −0.283061 0.959102i \(-0.591350\pi\)
−0.283061 + 0.959102i \(0.591350\pi\)
\(158\) −7.07211e11 −0.571394
\(159\) 1.54240e12 1.20369
\(160\) 2.18319e12 1.64600
\(161\) 2.28325e12 1.66346
\(162\) 2.64326e12 1.86126
\(163\) 8.90844e11 0.606415 0.303207 0.952925i \(-0.401943\pi\)
0.303207 + 0.952925i \(0.401943\pi\)
\(164\) 2.90985e12 1.91527
\(165\) −4.31115e12 −2.74430
\(166\) −1.52553e12 −0.939347
\(167\) 7.38410e11 0.439903 0.219952 0.975511i \(-0.429410\pi\)
0.219952 + 0.975511i \(0.429410\pi\)
\(168\) 1.44675e12 0.834050
\(169\) 5.83339e11 0.325495
\(170\) 1.88855e12 1.02014
\(171\) 6.12116e11 0.320151
\(172\) −2.57438e12 −1.30397
\(173\) −1.13157e12 −0.555170 −0.277585 0.960701i \(-0.589534\pi\)
−0.277585 + 0.960701i \(0.589534\pi\)
\(174\) −7.53567e11 −0.358180
\(175\) −1.33808e12 −0.616273
\(176\) 2.61041e12 1.16517
\(177\) −4.05604e12 −1.75489
\(178\) −4.85425e12 −2.03616
\(179\) 4.03407e12 1.64079 0.820393 0.571799i \(-0.193754\pi\)
0.820393 + 0.571799i \(0.193754\pi\)
\(180\) 2.42496e12 0.956547
\(181\) −3.88701e12 −1.48725 −0.743625 0.668597i \(-0.766895\pi\)
−0.743625 + 0.668597i \(0.766895\pi\)
\(182\) 7.23127e12 2.68424
\(183\) −6.77199e12 −2.43913
\(184\) 1.30332e12 0.455568
\(185\) −1.63587e12 −0.555015
\(186\) 3.20366e12 1.05518
\(187\) 3.24387e12 1.03737
\(188\) −6.25162e12 −1.94144
\(189\) 2.40802e12 0.726308
\(190\) −3.08922e12 −0.905117
\(191\) −2.12510e12 −0.604916 −0.302458 0.953163i \(-0.597807\pi\)
−0.302458 + 0.953163i \(0.597807\pi\)
\(192\) −6.74140e12 −1.86463
\(193\) 1.38130e12 0.371298 0.185649 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(194\) 3.83803e11 0.100276
\(195\) 6.84914e12 1.73959
\(196\) 7.17460e12 1.77170
\(197\) 4.09906e12 0.984283 0.492142 0.870515i \(-0.336214\pi\)
0.492142 + 0.870515i \(0.336214\pi\)
\(198\) 7.41919e12 1.73260
\(199\) 2.67780e12 0.608255 0.304127 0.952631i \(-0.401635\pi\)
0.304127 + 0.952631i \(0.401635\pi\)
\(200\) −7.63801e11 −0.168778
\(201\) −1.36065e11 −0.0292529
\(202\) −1.03145e13 −2.15782
\(203\) −1.40828e12 −0.286723
\(204\) −4.71280e12 −0.933928
\(205\) 9.17440e12 1.76983
\(206\) 2.85669e12 0.536530
\(207\) −3.72168e12 −0.680615
\(208\) −4.14717e12 −0.738590
\(209\) −5.30619e12 −0.920404
\(210\) 2.08495e13 3.52281
\(211\) −4.90054e12 −0.806660 −0.403330 0.915055i \(-0.632147\pi\)
−0.403330 + 0.915055i \(0.632147\pi\)
\(212\) 7.52072e12 1.20618
\(213\) −4.12233e12 −0.644250
\(214\) −2.67515e12 −0.407449
\(215\) −8.11673e12 −1.20495
\(216\) 1.37454e12 0.198913
\(217\) 5.98707e12 0.844667
\(218\) 8.59959e12 1.18295
\(219\) −8.56103e12 −1.14837
\(220\) −2.10210e13 −2.74998
\(221\) −5.15354e12 −0.657580
\(222\) 7.27139e12 0.905057
\(223\) 1.59341e13 1.93487 0.967435 0.253120i \(-0.0814566\pi\)
0.967435 + 0.253120i \(0.0814566\pi\)
\(224\) −1.81354e13 −2.14864
\(225\) 2.18106e12 0.252153
\(226\) −6.11254e12 −0.689645
\(227\) 1.88453e12 0.207520 0.103760 0.994602i \(-0.466913\pi\)
0.103760 + 0.994602i \(0.466913\pi\)
\(228\) 7.70903e12 0.828625
\(229\) −4.25535e12 −0.446519 −0.223259 0.974759i \(-0.571670\pi\)
−0.223259 + 0.974759i \(0.571670\pi\)
\(230\) 1.87825e13 1.92420
\(231\) 3.58121e13 3.58232
\(232\) −8.03874e11 −0.0785243
\(233\) 6.22697e12 0.594045 0.297022 0.954871i \(-0.404006\pi\)
0.297022 + 0.954871i \(0.404006\pi\)
\(234\) −1.17869e13 −1.09828
\(235\) −1.97106e13 −1.79402
\(236\) −1.97771e13 −1.75852
\(237\) 5.56426e12 0.483383
\(238\) −1.56879e13 −1.33166
\(239\) 1.79781e13 1.49127 0.745633 0.666357i \(-0.232147\pi\)
0.745633 + 0.666357i \(0.232147\pi\)
\(240\) −1.19573e13 −0.969329
\(241\) −1.00728e13 −0.798097 −0.399048 0.916930i \(-0.630659\pi\)
−0.399048 + 0.916930i \(0.630659\pi\)
\(242\) −4.48176e13 −3.47107
\(243\) −1.45840e13 −1.10419
\(244\) −3.30200e13 −2.44418
\(245\) 2.26207e13 1.63716
\(246\) −4.07799e13 −2.88604
\(247\) 8.42997e12 0.583436
\(248\) 3.41753e12 0.231328
\(249\) 1.20027e13 0.794661
\(250\) 1.65711e13 1.07320
\(251\) 1.15166e13 0.729654 0.364827 0.931075i \(-0.381128\pi\)
0.364827 + 0.931075i \(0.381128\pi\)
\(252\) −2.01438e13 −1.24864
\(253\) 3.22617e13 1.95670
\(254\) −1.02696e13 −0.609493
\(255\) −1.48589e13 −0.863011
\(256\) 4.09441e12 0.232740
\(257\) −3.13280e13 −1.74301 −0.871506 0.490384i \(-0.836857\pi\)
−0.871506 + 0.490384i \(0.836857\pi\)
\(258\) 3.60786e13 1.96490
\(259\) 1.35890e13 0.724498
\(260\) 3.33962e13 1.74319
\(261\) 2.29549e12 0.117315
\(262\) 1.92705e13 0.964352
\(263\) 9.17362e12 0.449556 0.224778 0.974410i \(-0.427834\pi\)
0.224778 + 0.974410i \(0.427834\pi\)
\(264\) 2.04422e13 0.981083
\(265\) 2.37119e13 1.11459
\(266\) 2.56617e13 1.18151
\(267\) 3.81927e13 1.72254
\(268\) −6.63450e11 −0.0293135
\(269\) −2.94832e13 −1.27625 −0.638127 0.769931i \(-0.720291\pi\)
−0.638127 + 0.769931i \(0.720291\pi\)
\(270\) 1.98089e13 0.840157
\(271\) −8.20600e12 −0.341036 −0.170518 0.985355i \(-0.554544\pi\)
−0.170518 + 0.985355i \(0.554544\pi\)
\(272\) 8.99711e12 0.366416
\(273\) −5.68948e13 −2.27080
\(274\) −2.66708e13 −1.04330
\(275\) −1.89067e13 −0.724915
\(276\) −4.68710e13 −1.76159
\(277\) 2.31768e13 0.853915 0.426958 0.904272i \(-0.359585\pi\)
0.426958 + 0.904272i \(0.359585\pi\)
\(278\) 3.28182e12 0.118541
\(279\) −9.75886e12 −0.345602
\(280\) 2.22414e13 0.772312
\(281\) 4.19264e13 1.42759 0.713793 0.700357i \(-0.246976\pi\)
0.713793 + 0.700357i \(0.246976\pi\)
\(282\) 8.76130e13 2.92549
\(283\) −1.71390e13 −0.561254 −0.280627 0.959817i \(-0.590542\pi\)
−0.280627 + 0.959817i \(0.590542\pi\)
\(284\) −2.01003e13 −0.645584
\(285\) 2.43057e13 0.765703
\(286\) 1.02176e14 3.15744
\(287\) −7.62104e13 −2.31028
\(288\) 2.95605e13 0.879131
\(289\) −2.30915e13 −0.673774
\(290\) −1.15849e13 −0.331667
\(291\) −3.01972e12 −0.0848310
\(292\) −4.17433e13 −1.15075
\(293\) 6.88609e13 1.86295 0.931474 0.363807i \(-0.118523\pi\)
0.931474 + 0.363807i \(0.118523\pi\)
\(294\) −1.00548e14 −2.66971
\(295\) −6.23550e13 −1.62499
\(296\) 7.75682e12 0.198417
\(297\) 3.40248e13 0.854348
\(298\) 5.40787e13 1.33302
\(299\) −5.12543e13 −1.24034
\(300\) 2.74684e13 0.652629
\(301\) 6.74245e13 1.57290
\(302\) −8.96386e13 −2.05332
\(303\) 8.11532e13 1.82545
\(304\) −1.47171e13 −0.325101
\(305\) −1.04108e14 −2.25858
\(306\) 2.55712e13 0.544857
\(307\) −3.43334e13 −0.718547 −0.359273 0.933232i \(-0.616975\pi\)
−0.359273 + 0.933232i \(0.616975\pi\)
\(308\) 1.74619e14 3.58973
\(309\) −2.24761e13 −0.453890
\(310\) 4.92509e13 0.977069
\(311\) −7.11998e13 −1.38770 −0.693852 0.720118i \(-0.744088\pi\)
−0.693852 + 0.720118i \(0.744088\pi\)
\(312\) −3.24766e13 −0.621899
\(313\) −8.48846e13 −1.59711 −0.798555 0.601922i \(-0.794402\pi\)
−0.798555 + 0.601922i \(0.794402\pi\)
\(314\) 4.62376e13 0.854834
\(315\) −6.35111e13 −1.15383
\(316\) 2.71312e13 0.484384
\(317\) 6.68644e13 1.17319 0.586595 0.809880i \(-0.300468\pi\)
0.586595 + 0.809880i \(0.300468\pi\)
\(318\) −1.05399e14 −1.81755
\(319\) −1.98987e13 −0.337268
\(320\) −1.03638e14 −1.72661
\(321\) 2.10478e13 0.344690
\(322\) −1.56024e14 −2.51179
\(323\) −1.82885e13 −0.289443
\(324\) −1.01405e14 −1.57784
\(325\) 3.00372e13 0.459517
\(326\) −6.08749e13 −0.915675
\(327\) −6.76606e13 −1.00074
\(328\) −4.35023e13 −0.632712
\(329\) 1.63733e14 2.34185
\(330\) 2.94598e14 4.14384
\(331\) −1.01165e13 −0.139951 −0.0699755 0.997549i \(-0.522292\pi\)
−0.0699755 + 0.997549i \(0.522292\pi\)
\(332\) 5.85247e13 0.796306
\(333\) −2.21499e13 −0.296434
\(334\) −5.04585e13 −0.664246
\(335\) −2.09178e12 −0.0270876
\(336\) 9.93275e13 1.26533
\(337\) −1.36391e14 −1.70931 −0.854653 0.519199i \(-0.826230\pi\)
−0.854653 + 0.519199i \(0.826230\pi\)
\(338\) −3.98619e13 −0.491492
\(339\) 4.80928e13 0.583420
\(340\) −7.24516e13 −0.864797
\(341\) 8.45957e13 0.993572
\(342\) −4.18284e13 −0.483423
\(343\) −5.21445e13 −0.593049
\(344\) 3.84871e13 0.430768
\(345\) −1.47779e14 −1.62782
\(346\) 7.73244e13 0.838297
\(347\) 1.51964e14 1.62154 0.810769 0.585366i \(-0.199049\pi\)
0.810769 + 0.585366i \(0.199049\pi\)
\(348\) 2.89095e13 0.303637
\(349\) −3.81552e13 −0.394470 −0.197235 0.980356i \(-0.563196\pi\)
−0.197235 + 0.980356i \(0.563196\pi\)
\(350\) 9.14365e13 0.930562
\(351\) −5.40553e13 −0.541563
\(352\) −2.56248e14 −2.52742
\(353\) 5.20789e13 0.505710 0.252855 0.967504i \(-0.418631\pi\)
0.252855 + 0.967504i \(0.418631\pi\)
\(354\) 2.77165e14 2.64985
\(355\) −6.33740e13 −0.596562
\(356\) 1.86226e14 1.72610
\(357\) 1.23431e14 1.12654
\(358\) −2.75664e14 −2.47756
\(359\) −1.56134e14 −1.38190 −0.690950 0.722902i \(-0.742808\pi\)
−0.690950 + 0.722902i \(0.742808\pi\)
\(360\) −3.62533e13 −0.315997
\(361\) −8.65747e13 −0.743192
\(362\) 2.65615e14 2.24572
\(363\) 3.52619e14 2.93643
\(364\) −2.77417e14 −2.27550
\(365\) −1.31612e14 −1.06337
\(366\) 4.62757e14 3.68305
\(367\) −1.58571e14 −1.24326 −0.621629 0.783312i \(-0.713529\pi\)
−0.621629 + 0.783312i \(0.713529\pi\)
\(368\) 8.94803e13 0.691138
\(369\) 1.24222e14 0.945267
\(370\) 1.11786e14 0.838063
\(371\) −1.96972e14 −1.45495
\(372\) −1.22904e14 −0.894496
\(373\) 1.84813e14 1.32536 0.662682 0.748901i \(-0.269418\pi\)
0.662682 + 0.748901i \(0.269418\pi\)
\(374\) −2.21666e14 −1.56641
\(375\) −1.30380e14 −0.907901
\(376\) 9.34619e13 0.641360
\(377\) 3.16131e13 0.213791
\(378\) −1.64550e14 −1.09671
\(379\) −5.17503e13 −0.339936 −0.169968 0.985450i \(-0.554366\pi\)
−0.169968 + 0.985450i \(0.554366\pi\)
\(380\) 1.18514e14 0.767288
\(381\) 8.08001e13 0.515614
\(382\) 1.45216e14 0.913413
\(383\) 1.91406e14 1.18676 0.593379 0.804923i \(-0.297793\pi\)
0.593379 + 0.804923i \(0.297793\pi\)
\(384\) 1.69828e14 1.03798
\(385\) 5.50552e14 3.31715
\(386\) −9.43897e13 −0.560653
\(387\) −1.09901e14 −0.643565
\(388\) −1.47240e13 −0.0850066
\(389\) 7.30496e12 0.0415810 0.0207905 0.999784i \(-0.493382\pi\)
0.0207905 + 0.999784i \(0.493382\pi\)
\(390\) −4.68029e14 −2.62674
\(391\) 1.11194e14 0.615332
\(392\) −1.07260e14 −0.585284
\(393\) −1.51618e14 −0.815815
\(394\) −2.80105e14 −1.48625
\(395\) 8.55414e13 0.447602
\(396\) −2.84627e14 −1.46876
\(397\) 2.82104e13 0.143569 0.0717846 0.997420i \(-0.477131\pi\)
0.0717846 + 0.997420i \(0.477131\pi\)
\(398\) −1.82985e14 −0.918454
\(399\) −2.01904e14 −0.999523
\(400\) −5.24393e13 −0.256051
\(401\) −1.73909e14 −0.837585 −0.418793 0.908082i \(-0.637547\pi\)
−0.418793 + 0.908082i \(0.637547\pi\)
\(402\) 9.29789e12 0.0441714
\(403\) −1.34398e14 −0.629816
\(404\) 3.95700e14 1.82923
\(405\) −3.19718e14 −1.45802
\(406\) 9.62337e13 0.432946
\(407\) 1.92008e14 0.852218
\(408\) 7.04565e13 0.308525
\(409\) −1.51365e14 −0.653953 −0.326977 0.945032i \(-0.606030\pi\)
−0.326977 + 0.945032i \(0.606030\pi\)
\(410\) −6.26924e14 −2.67241
\(411\) 2.09843e14 0.882600
\(412\) −1.09593e14 −0.454829
\(413\) 5.17974e14 2.12120
\(414\) 2.54317e14 1.02772
\(415\) 1.84522e14 0.735839
\(416\) 4.07103e14 1.60211
\(417\) −2.58210e13 −0.100282
\(418\) 3.62594e14 1.38979
\(419\) 1.96755e14 0.744303 0.372151 0.928172i \(-0.378620\pi\)
0.372151 + 0.928172i \(0.378620\pi\)
\(420\) −7.99862e14 −2.98637
\(421\) 1.30439e14 0.480681 0.240340 0.970689i \(-0.422741\pi\)
0.240340 + 0.970689i \(0.422741\pi\)
\(422\) 3.34874e14 1.21804
\(423\) −2.66884e14 −0.958187
\(424\) −1.12435e14 −0.398464
\(425\) −6.51645e13 −0.227967
\(426\) 2.81695e14 0.972806
\(427\) 8.64812e14 2.94828
\(428\) 1.02628e14 0.345404
\(429\) −8.03909e14 −2.67111
\(430\) 5.54649e14 1.81946
\(431\) −3.01242e14 −0.975642 −0.487821 0.872944i \(-0.662208\pi\)
−0.487821 + 0.872944i \(0.662208\pi\)
\(432\) 9.43703e13 0.301769
\(433\) −4.01236e14 −1.26682 −0.633412 0.773815i \(-0.718346\pi\)
−0.633412 + 0.773815i \(0.718346\pi\)
\(434\) −4.09120e14 −1.27543
\(435\) 9.11483e13 0.280581
\(436\) −3.29911e14 −1.00282
\(437\) −1.81887e14 −0.545951
\(438\) 5.85009e14 1.73402
\(439\) 1.67628e14 0.490673 0.245337 0.969438i \(-0.421102\pi\)
0.245337 + 0.969438i \(0.421102\pi\)
\(440\) 3.14265e14 0.908461
\(441\) 3.06286e14 0.874410
\(442\) 3.52162e14 0.992934
\(443\) 2.50447e14 0.697421 0.348710 0.937231i \(-0.386620\pi\)
0.348710 + 0.937231i \(0.386620\pi\)
\(444\) −2.78957e14 −0.767238
\(445\) 5.87150e14 1.59503
\(446\) −1.08884e15 −2.92162
\(447\) −4.25485e14 −1.12770
\(448\) 8.60905e14 2.25386
\(449\) −5.39495e14 −1.39519 −0.697593 0.716494i \(-0.745746\pi\)
−0.697593 + 0.716494i \(0.745746\pi\)
\(450\) −1.49041e14 −0.380746
\(451\) −1.07683e15 −2.71755
\(452\) 2.34499e14 0.584628
\(453\) 7.05266e14 1.73705
\(454\) −1.28777e14 −0.313352
\(455\) −8.74665e14 −2.10271
\(456\) −1.15250e14 −0.273738
\(457\) 1.34613e14 0.315899 0.157949 0.987447i \(-0.449512\pi\)
0.157949 + 0.987447i \(0.449512\pi\)
\(458\) 2.90785e14 0.674236
\(459\) 1.17271e14 0.268670
\(460\) −7.20564e14 −1.63119
\(461\) −7.83557e14 −1.75273 −0.876367 0.481645i \(-0.840040\pi\)
−0.876367 + 0.481645i \(0.840040\pi\)
\(462\) −2.44718e15 −5.40923
\(463\) −1.86516e14 −0.407401 −0.203700 0.979033i \(-0.565297\pi\)
−0.203700 + 0.979033i \(0.565297\pi\)
\(464\) −5.51905e13 −0.119128
\(465\) −3.87501e14 −0.826573
\(466\) −4.25514e14 −0.896997
\(467\) −6.06343e14 −1.26321 −0.631605 0.775290i \(-0.717604\pi\)
−0.631605 + 0.775290i \(0.717604\pi\)
\(468\) 4.52188e14 0.931036
\(469\) 1.73761e13 0.0353592
\(470\) 1.34691e15 2.70894
\(471\) −3.63792e14 −0.723166
\(472\) 2.95669e14 0.580931
\(473\) 9.52690e14 1.85019
\(474\) −3.80228e14 −0.729900
\(475\) 1.06594e14 0.202263
\(476\) 6.01845e14 1.12888
\(477\) 3.21062e14 0.595302
\(478\) −1.22851e15 −2.25179
\(479\) 6.70378e14 1.21471 0.607357 0.794429i \(-0.292230\pi\)
0.607357 + 0.794429i \(0.292230\pi\)
\(480\) 1.17378e15 2.10261
\(481\) −3.05044e14 −0.540213
\(482\) 6.88313e14 1.20511
\(483\) 1.22758e15 2.12490
\(484\) 1.71936e15 2.94251
\(485\) −4.64232e13 −0.0785517
\(486\) 9.96583e14 1.66730
\(487\) 3.36789e14 0.557120 0.278560 0.960419i \(-0.410143\pi\)
0.278560 + 0.960419i \(0.410143\pi\)
\(488\) 4.93650e14 0.807440
\(489\) 4.78957e14 0.774636
\(490\) −1.54576e15 −2.47209
\(491\) 3.28879e14 0.520102 0.260051 0.965595i \(-0.416261\pi\)
0.260051 + 0.965595i \(0.416261\pi\)
\(492\) 1.56446e15 2.44657
\(493\) −6.85833e13 −0.106062
\(494\) −5.76054e14 −0.880978
\(495\) −8.97395e14 −1.35723
\(496\) 2.34633e14 0.350945
\(497\) 5.26439e14 0.778731
\(498\) −8.20191e14 −1.19992
\(499\) −1.89240e14 −0.273817 −0.136908 0.990584i \(-0.543717\pi\)
−0.136908 + 0.990584i \(0.543717\pi\)
\(500\) −6.35728e14 −0.909780
\(501\) 3.97002e14 0.561934
\(502\) −7.86972e14 −1.10176
\(503\) 1.64969e14 0.228444 0.114222 0.993455i \(-0.463563\pi\)
0.114222 + 0.993455i \(0.463563\pi\)
\(504\) 3.01150e14 0.412492
\(505\) 1.24760e15 1.69033
\(506\) −2.20457e15 −2.95459
\(507\) 3.13629e14 0.415788
\(508\) 3.93979e14 0.516681
\(509\) 7.09880e14 0.920952 0.460476 0.887672i \(-0.347679\pi\)
0.460476 + 0.887672i \(0.347679\pi\)
\(510\) 1.01537e15 1.30313
\(511\) 1.09328e15 1.38809
\(512\) −9.26699e14 −1.16400
\(513\) −1.91827e14 −0.238377
\(514\) 2.14077e15 2.63192
\(515\) −3.45534e14 −0.420292
\(516\) −1.38410e15 −1.66569
\(517\) 2.31351e15 2.75469
\(518\) −9.28587e14 −1.09398
\(519\) −6.08379e14 −0.709176
\(520\) −4.99274e14 −0.575865
\(521\) −2.38844e14 −0.272588 −0.136294 0.990668i \(-0.543519\pi\)
−0.136294 + 0.990668i \(0.543519\pi\)
\(522\) −1.56860e14 −0.177143
\(523\) −5.08191e13 −0.0567895 −0.0283948 0.999597i \(-0.509040\pi\)
−0.0283948 + 0.999597i \(0.509040\pi\)
\(524\) −7.39285e14 −0.817504
\(525\) −7.19412e14 −0.787230
\(526\) −6.26870e14 −0.678822
\(527\) 2.91570e14 0.312452
\(528\) 1.40347e15 1.48839
\(529\) 1.53066e14 0.160647
\(530\) −1.62033e15 −1.68301
\(531\) −8.44292e14 −0.867906
\(532\) −9.84475e14 −1.00159
\(533\) 1.71077e15 1.72263
\(534\) −2.60986e15 −2.60100
\(535\) 3.23575e14 0.319176
\(536\) 9.91860e12 0.00968376
\(537\) 2.16890e15 2.09595
\(538\) 2.01471e15 1.92712
\(539\) −2.65507e15 −2.51384
\(540\) −7.59942e14 −0.712221
\(541\) −1.83247e15 −1.70001 −0.850007 0.526771i \(-0.823402\pi\)
−0.850007 + 0.526771i \(0.823402\pi\)
\(542\) 5.60749e14 0.514959
\(543\) −2.08983e15 −1.89982
\(544\) −8.83192e14 −0.794806
\(545\) −1.04017e15 −0.926667
\(546\) 3.88785e15 3.42886
\(547\) 3.76524e14 0.328747 0.164374 0.986398i \(-0.447440\pi\)
0.164374 + 0.986398i \(0.447440\pi\)
\(548\) 1.02319e15 0.884427
\(549\) −1.40964e15 −1.20631
\(550\) 1.29197e15 1.09461
\(551\) 1.12186e14 0.0941033
\(552\) 7.00722e14 0.581944
\(553\) −7.10580e14 −0.584285
\(554\) −1.58376e15 −1.28940
\(555\) −8.79517e14 −0.708978
\(556\) −1.25902e14 −0.100490
\(557\) −7.28798e14 −0.575975 −0.287987 0.957634i \(-0.592986\pi\)
−0.287987 + 0.957634i \(0.592986\pi\)
\(558\) 6.66862e14 0.521853
\(559\) −1.51354e15 −1.17282
\(560\) 1.52700e15 1.17167
\(561\) 1.74405e15 1.32514
\(562\) −2.86500e15 −2.15563
\(563\) −2.20017e15 −1.63930 −0.819652 0.572862i \(-0.805833\pi\)
−0.819652 + 0.572862i \(0.805833\pi\)
\(564\) −3.36115e15 −2.48001
\(565\) 7.39348e14 0.540234
\(566\) 1.17117e15 0.847483
\(567\) 2.65585e15 1.90326
\(568\) 3.00501e14 0.213270
\(569\) −1.15616e15 −0.812643 −0.406321 0.913730i \(-0.633189\pi\)
−0.406321 + 0.913730i \(0.633189\pi\)
\(570\) −1.66090e15 −1.15620
\(571\) 1.61701e15 1.11485 0.557423 0.830229i \(-0.311790\pi\)
0.557423 + 0.830229i \(0.311790\pi\)
\(572\) −3.91984e15 −2.67664
\(573\) −1.14255e15 −0.772722
\(574\) 5.20776e15 3.48848
\(575\) −6.48090e14 −0.429994
\(576\) −1.40327e15 −0.922182
\(577\) 1.33066e15 0.866163 0.433081 0.901355i \(-0.357426\pi\)
0.433081 + 0.901355i \(0.357426\pi\)
\(578\) 1.57794e15 1.01739
\(579\) 7.42647e14 0.474297
\(580\) 4.44436e14 0.281161
\(581\) −1.53279e15 −0.960539
\(582\) 2.06349e14 0.128093
\(583\) −2.78316e15 −1.71144
\(584\) 6.24064e14 0.380153
\(585\) 1.42569e15 0.860338
\(586\) −4.70554e15 −2.81302
\(587\) 2.02127e15 1.19706 0.598530 0.801101i \(-0.295752\pi\)
0.598530 + 0.801101i \(0.295752\pi\)
\(588\) 3.85738e15 2.26317
\(589\) −4.76939e14 −0.277222
\(590\) 4.26097e15 2.45370
\(591\) 2.20384e15 1.25733
\(592\) 5.32550e14 0.301017
\(593\) 1.76241e15 0.986973 0.493486 0.869754i \(-0.335722\pi\)
0.493486 + 0.869754i \(0.335722\pi\)
\(594\) −2.32505e15 −1.29005
\(595\) 1.89755e15 1.04316
\(596\) −2.07465e15 −1.13003
\(597\) 1.43970e15 0.776987
\(598\) 3.50241e15 1.87288
\(599\) −3.33607e15 −1.76761 −0.883807 0.467851i \(-0.845028\pi\)
−0.883807 + 0.467851i \(0.845028\pi\)
\(600\) −4.10653e14 −0.215597
\(601\) −9.06258e14 −0.471457 −0.235729 0.971819i \(-0.575748\pi\)
−0.235729 + 0.971819i \(0.575748\pi\)
\(602\) −4.60738e15 −2.37506
\(603\) −2.83229e13 −0.0144675
\(604\) 3.43886e15 1.74065
\(605\) 5.42095e15 2.71907
\(606\) −5.54552e15 −2.75640
\(607\) 1.31379e15 0.647125 0.323562 0.946207i \(-0.395119\pi\)
0.323562 + 0.946207i \(0.395119\pi\)
\(608\) 1.44469e15 0.705189
\(609\) −7.57156e14 −0.366260
\(610\) 7.11414e15 3.41042
\(611\) −3.67548e15 −1.74617
\(612\) −9.81001e14 −0.461888
\(613\) −1.67634e15 −0.782219 −0.391110 0.920344i \(-0.627909\pi\)
−0.391110 + 0.920344i \(0.627909\pi\)
\(614\) 2.34614e15 1.08499
\(615\) 4.93257e15 2.26079
\(616\) −2.61055e15 −1.18587
\(617\) −1.29504e15 −0.583063 −0.291532 0.956561i \(-0.594165\pi\)
−0.291532 + 0.956561i \(0.594165\pi\)
\(618\) 1.53589e15 0.685365
\(619\) −2.90800e15 −1.28616 −0.643082 0.765797i \(-0.722345\pi\)
−0.643082 + 0.765797i \(0.722345\pi\)
\(620\) −1.88944e15 −0.828284
\(621\) 1.16631e15 0.506769
\(622\) 4.86537e15 2.09541
\(623\) −4.87737e15 −2.08210
\(624\) −2.22970e15 −0.943478
\(625\) −2.95597e15 −1.23982
\(626\) 5.80050e15 2.41161
\(627\) −2.85284e15 −1.17573
\(628\) −1.77384e15 −0.724663
\(629\) 6.61781e14 0.268000
\(630\) 4.33997e15 1.74226
\(631\) −7.31735e14 −0.291201 −0.145600 0.989343i \(-0.546511\pi\)
−0.145600 + 0.989343i \(0.546511\pi\)
\(632\) −4.05612e14 −0.160017
\(633\) −2.63475e15 −1.03043
\(634\) −4.56911e15 −1.77150
\(635\) 1.24217e15 0.477447
\(636\) 4.04347e15 1.54078
\(637\) 4.21812e15 1.59350
\(638\) 1.35976e15 0.509269
\(639\) −8.58090e14 −0.318624
\(640\) 2.61083e15 0.961145
\(641\) −4.90606e15 −1.79066 −0.895331 0.445401i \(-0.853061\pi\)
−0.895331 + 0.445401i \(0.853061\pi\)
\(642\) −1.43828e15 −0.520476
\(643\) −1.88025e15 −0.674612 −0.337306 0.941395i \(-0.609516\pi\)
−0.337306 + 0.941395i \(0.609516\pi\)
\(644\) 5.98562e15 2.12930
\(645\) −4.36391e15 −1.53921
\(646\) 1.24972e15 0.437054
\(647\) −1.81526e15 −0.629457 −0.314728 0.949182i \(-0.601913\pi\)
−0.314728 + 0.949182i \(0.601913\pi\)
\(648\) 1.51601e15 0.521242
\(649\) 7.31883e15 2.49515
\(650\) −2.05256e15 −0.693862
\(651\) 3.21891e15 1.07898
\(652\) 2.33538e15 0.776239
\(653\) 1.09915e15 0.362271 0.181135 0.983458i \(-0.442023\pi\)
0.181135 + 0.983458i \(0.442023\pi\)
\(654\) 4.62352e15 1.51111
\(655\) −2.33088e15 −0.755427
\(656\) −2.98668e15 −0.959881
\(657\) −1.78204e15 −0.567946
\(658\) −1.11886e16 −3.53616
\(659\) 4.07550e14 0.127735 0.0638677 0.997958i \(-0.479656\pi\)
0.0638677 + 0.997958i \(0.479656\pi\)
\(660\) −1.13018e16 −3.51283
\(661\) −1.78308e15 −0.549620 −0.274810 0.961499i \(-0.588615\pi\)
−0.274810 + 0.961499i \(0.588615\pi\)
\(662\) 6.91300e14 0.211324
\(663\) −2.77077e15 −0.839995
\(664\) −8.74946e14 −0.263061
\(665\) −3.10394e15 −0.925537
\(666\) 1.51359e15 0.447610
\(667\) −6.82092e14 −0.200056
\(668\) 1.93577e15 0.563097
\(669\) 8.56690e15 2.47161
\(670\) 1.42940e14 0.0409017
\(671\) 1.22196e16 3.46802
\(672\) −9.75039e15 −2.74468
\(673\) −2.92949e15 −0.817917 −0.408958 0.912553i \(-0.634108\pi\)
−0.408958 + 0.912553i \(0.634108\pi\)
\(674\) 9.32012e15 2.58102
\(675\) −6.83507e14 −0.187747
\(676\) 1.52925e15 0.416649
\(677\) −5.65022e15 −1.52696 −0.763480 0.645831i \(-0.776511\pi\)
−0.763480 + 0.645831i \(0.776511\pi\)
\(678\) −3.28637e15 −0.880954
\(679\) 3.85631e14 0.102539
\(680\) 1.08315e15 0.285687
\(681\) 1.01320e15 0.265087
\(682\) −5.78076e15 −1.50028
\(683\) −1.87327e15 −0.482266 −0.241133 0.970492i \(-0.577519\pi\)
−0.241133 + 0.970492i \(0.577519\pi\)
\(684\) 1.60469e15 0.409809
\(685\) 3.22599e15 0.817268
\(686\) 3.56324e15 0.895494
\(687\) −2.28786e15 −0.570385
\(688\) 2.64236e15 0.653515
\(689\) 4.42161e15 1.08486
\(690\) 1.00983e16 2.45798
\(691\) −4.71553e15 −1.13868 −0.569339 0.822103i \(-0.692801\pi\)
−0.569339 + 0.822103i \(0.692801\pi\)
\(692\) −2.96644e15 −0.710644
\(693\) 7.45453e15 1.77169
\(694\) −1.03843e16 −2.44849
\(695\) −3.96956e14 −0.0928593
\(696\) −4.32198e14 −0.100307
\(697\) −3.71144e15 −0.854599
\(698\) 2.60730e15 0.595643
\(699\) 3.34789e15 0.758835
\(700\) −3.50783e15 −0.788859
\(701\) −2.35508e15 −0.525480 −0.262740 0.964867i \(-0.584626\pi\)
−0.262740 + 0.964867i \(0.584626\pi\)
\(702\) 3.69381e15 0.817751
\(703\) −1.08252e15 −0.237783
\(704\) 1.21644e16 2.65118
\(705\) −1.05973e16 −2.29169
\(706\) −3.55876e15 −0.763613
\(707\) −1.03636e16 −2.20650
\(708\) −1.06331e16 −2.24634
\(709\) 7.95473e15 1.66752 0.833760 0.552126i \(-0.186183\pi\)
0.833760 + 0.552126i \(0.186183\pi\)
\(710\) 4.33060e15 0.900797
\(711\) 1.15824e15 0.239064
\(712\) −2.78409e15 −0.570221
\(713\) 2.89979e15 0.589352
\(714\) −8.43452e15 −1.70106
\(715\) −1.23588e16 −2.47339
\(716\) 1.05755e16 2.10028
\(717\) 9.66581e15 1.90495
\(718\) 1.06692e16 2.08665
\(719\) −1.98037e15 −0.384360 −0.192180 0.981360i \(-0.561556\pi\)
−0.192180 + 0.981360i \(0.561556\pi\)
\(720\) −2.48899e15 −0.479397
\(721\) 2.87030e15 0.548635
\(722\) 5.91600e15 1.12221
\(723\) −5.41557e15 −1.01949
\(724\) −1.01899e16 −1.90375
\(725\) 3.99735e14 0.0741162
\(726\) −2.40959e16 −4.43396
\(727\) 3.46705e15 0.633171 0.316585 0.948564i \(-0.397464\pi\)
0.316585 + 0.948564i \(0.397464\pi\)
\(728\) 4.14740e15 0.751715
\(729\) −9.88684e14 −0.177851
\(730\) 8.99356e15 1.60567
\(731\) 3.28357e15 0.581835
\(732\) −1.77530e16 −3.12220
\(733\) 4.26101e15 0.743774 0.371887 0.928278i \(-0.378711\pi\)
0.371887 + 0.928278i \(0.378711\pi\)
\(734\) 1.08358e16 1.87730
\(735\) 1.21619e16 2.09132
\(736\) −8.78375e15 −1.49917
\(737\) 2.45520e14 0.0415926
\(738\) −8.48861e15 −1.42734
\(739\) 4.74592e15 0.792093 0.396046 0.918230i \(-0.370382\pi\)
0.396046 + 0.918230i \(0.370382\pi\)
\(740\) −4.28850e15 −0.710446
\(741\) 4.53233e15 0.745283
\(742\) 1.34599e16 2.19694
\(743\) 4.32897e15 0.701369 0.350685 0.936494i \(-0.385949\pi\)
0.350685 + 0.936494i \(0.385949\pi\)
\(744\) 1.83741e15 0.295499
\(745\) −6.54113e15 −1.04422
\(746\) −1.26290e16 −2.00128
\(747\) 2.49844e15 0.393012
\(748\) 8.50391e15 1.32788
\(749\) −2.68789e15 −0.416641
\(750\) 8.90938e15 1.37092
\(751\) 1.89924e15 0.290108 0.145054 0.989424i \(-0.453664\pi\)
0.145054 + 0.989424i \(0.453664\pi\)
\(752\) 6.41669e15 0.973001
\(753\) 6.19181e15 0.932063
\(754\) −2.16025e15 −0.322821
\(755\) 1.08423e16 1.60847
\(756\) 6.31273e15 0.929709
\(757\) −6.04958e15 −0.884500 −0.442250 0.896892i \(-0.645820\pi\)
−0.442250 + 0.896892i \(0.645820\pi\)
\(758\) 3.53631e15 0.513298
\(759\) 1.73453e16 2.49950
\(760\) −1.77178e15 −0.253475
\(761\) −9.10355e15 −1.29299 −0.646495 0.762918i \(-0.723766\pi\)
−0.646495 + 0.762918i \(0.723766\pi\)
\(762\) −5.52139e15 −0.778568
\(763\) 8.64054e15 1.20964
\(764\) −5.57102e15 −0.774321
\(765\) −3.09298e15 −0.426815
\(766\) −1.30795e16 −1.79198
\(767\) −1.16275e16 −1.58165
\(768\) 2.20134e15 0.297303
\(769\) −5.47973e15 −0.734792 −0.367396 0.930065i \(-0.619751\pi\)
−0.367396 + 0.930065i \(0.619751\pi\)
\(770\) −3.76214e16 −5.00883
\(771\) −1.68433e16 −2.22653
\(772\) 3.62113e15 0.475279
\(773\) 7.09370e15 0.924455 0.462228 0.886761i \(-0.347050\pi\)
0.462228 + 0.886761i \(0.347050\pi\)
\(774\) 7.50999e15 0.971772
\(775\) −1.69940e15 −0.218342
\(776\) 2.20125e14 0.0280821
\(777\) 7.30602e15 0.925476
\(778\) −4.99177e14 −0.0627866
\(779\) 6.07104e15 0.758240
\(780\) 1.79553e16 2.22675
\(781\) 7.43844e15 0.916012
\(782\) −7.59834e15 −0.929140
\(783\) −7.19367e14 −0.0873496
\(784\) −7.36404e15 −0.887928
\(785\) −5.59271e15 −0.669636
\(786\) 1.03607e16 1.23187
\(787\) −2.33437e15 −0.275619 −0.137810 0.990459i \(-0.544006\pi\)
−0.137810 + 0.990459i \(0.544006\pi\)
\(788\) 1.07458e16 1.25993
\(789\) 4.93214e15 0.574265
\(790\) −5.84538e15 −0.675871
\(791\) −6.14165e15 −0.705204
\(792\) 4.25518e15 0.485209
\(793\) −1.94133e16 −2.19835
\(794\) −1.92773e15 −0.216787
\(795\) 1.27486e16 1.42378
\(796\) 7.01994e15 0.778595
\(797\) −5.40712e13 −0.00595587 −0.00297793 0.999996i \(-0.500948\pi\)
−0.00297793 + 0.999996i \(0.500948\pi\)
\(798\) 1.37969e16 1.50926
\(799\) 7.97380e15 0.866279
\(800\) 5.14765e15 0.555411
\(801\) 7.95006e15 0.851906
\(802\) 1.18839e16 1.26474
\(803\) 1.54478e16 1.63279
\(804\) −3.56700e14 −0.0374451
\(805\) 1.88720e16 1.96761
\(806\) 9.18392e15 0.951011
\(807\) −1.58515e16 −1.63029
\(808\) −5.91573e15 −0.604291
\(809\) 4.45684e15 0.452179 0.226090 0.974107i \(-0.427406\pi\)
0.226090 + 0.974107i \(0.427406\pi\)
\(810\) 2.18476e16 2.20159
\(811\) −5.02280e15 −0.502726 −0.251363 0.967893i \(-0.580879\pi\)
−0.251363 + 0.967893i \(0.580879\pi\)
\(812\) −3.69187e15 −0.367019
\(813\) −4.41191e15 −0.435641
\(814\) −1.31207e16 −1.28683
\(815\) 7.36318e15 0.717296
\(816\) 4.83724e15 0.468060
\(817\) −5.37113e15 −0.516232
\(818\) 1.03434e16 0.987458
\(819\) −1.18430e16 −1.12306
\(820\) 2.40510e16 2.26547
\(821\) 1.69234e15 0.158343 0.0791716 0.996861i \(-0.474772\pi\)
0.0791716 + 0.996861i \(0.474772\pi\)
\(822\) −1.43394e16 −1.33271
\(823\) −9.20529e15 −0.849842 −0.424921 0.905230i \(-0.639698\pi\)
−0.424921 + 0.905230i \(0.639698\pi\)
\(824\) 1.63842e15 0.150254
\(825\) −1.01651e16 −0.926009
\(826\) −3.53952e16 −3.20298
\(827\) 2.50113e15 0.224831 0.112415 0.993661i \(-0.464141\pi\)
0.112415 + 0.993661i \(0.464141\pi\)
\(828\) −9.75651e15 −0.871219
\(829\) 4.39695e15 0.390033 0.195017 0.980800i \(-0.437524\pi\)
0.195017 + 0.980800i \(0.437524\pi\)
\(830\) −1.26091e16 −1.11110
\(831\) 1.24609e16 1.09079
\(832\) −1.93256e16 −1.68056
\(833\) −9.15104e15 −0.790538
\(834\) 1.76445e15 0.151425
\(835\) 6.10326e15 0.520339
\(836\) −1.39104e16 −1.17816
\(837\) 3.05826e15 0.257327
\(838\) −1.34451e16 −1.12388
\(839\) 9.17137e15 0.761629 0.380814 0.924652i \(-0.375644\pi\)
0.380814 + 0.924652i \(0.375644\pi\)
\(840\) 1.19580e16 0.986554
\(841\) 4.20707e14 0.0344828
\(842\) −8.91344e15 −0.725820
\(843\) 2.25415e16 1.82360
\(844\) −1.28469e16 −1.03256
\(845\) 4.82153e15 0.385011
\(846\) 1.82372e16 1.44685
\(847\) −4.50310e16 −3.54938
\(848\) −7.71930e15 −0.604506
\(849\) −9.21467e15 −0.716947
\(850\) 4.45295e15 0.344226
\(851\) 6.58171e15 0.505506
\(852\) −1.08068e16 −0.824671
\(853\) 7.56207e15 0.573352 0.286676 0.958028i \(-0.407450\pi\)
0.286676 + 0.958028i \(0.407450\pi\)
\(854\) −5.90961e16 −4.45185
\(855\) 5.05939e15 0.378690
\(856\) −1.53430e15 −0.114105
\(857\) −9.34890e15 −0.690821 −0.345411 0.938452i \(-0.612260\pi\)
−0.345411 + 0.938452i \(0.612260\pi\)
\(858\) 5.49343e16 4.03333
\(859\) −1.20867e16 −0.881749 −0.440875 0.897569i \(-0.645332\pi\)
−0.440875 + 0.897569i \(0.645332\pi\)
\(860\) −2.12783e16 −1.54239
\(861\) −4.09741e16 −2.95115
\(862\) 2.05851e16 1.47320
\(863\) −5.66080e15 −0.402549 −0.201275 0.979535i \(-0.564508\pi\)
−0.201275 + 0.979535i \(0.564508\pi\)
\(864\) −9.26377e15 −0.654579
\(865\) −9.35284e15 −0.656681
\(866\) 2.74181e16 1.91288
\(867\) −1.24150e16 −0.860681
\(868\) 1.56953e16 1.08121
\(869\) −1.00403e16 −0.687287
\(870\) −6.22853e15 −0.423672
\(871\) −3.90059e14 −0.0263651
\(872\) 4.93218e15 0.331282
\(873\) −6.28574e14 −0.0419545
\(874\) 1.24291e16 0.824377
\(875\) 1.66501e16 1.09742
\(876\) −2.24430e16 −1.46997
\(877\) −8.61237e15 −0.560564 −0.280282 0.959918i \(-0.590428\pi\)
−0.280282 + 0.959918i \(0.590428\pi\)
\(878\) −1.14547e16 −0.740908
\(879\) 3.70227e16 2.37974
\(880\) 2.15761e16 1.37822
\(881\) 7.82338e15 0.496623 0.248312 0.968680i \(-0.420124\pi\)
0.248312 + 0.968680i \(0.420124\pi\)
\(882\) −2.09298e16 −1.32034
\(883\) 2.57988e16 1.61739 0.808696 0.588226i \(-0.200174\pi\)
0.808696 + 0.588226i \(0.200174\pi\)
\(884\) −1.35102e16 −0.841733
\(885\) −3.35248e16 −2.07576
\(886\) −1.71140e16 −1.05309
\(887\) 1.53064e16 0.936035 0.468017 0.883719i \(-0.344968\pi\)
0.468017 + 0.883719i \(0.344968\pi\)
\(888\) 4.17041e15 0.253459
\(889\) −1.03185e16 −0.623243
\(890\) −4.01223e16 −2.40847
\(891\) 3.75265e16 2.23878
\(892\) 4.17719e16 2.47672
\(893\) −1.30432e16 −0.768604
\(894\) 2.90751e16 1.70280
\(895\) 3.33432e16 1.94080
\(896\) −2.16878e16 −1.25465
\(897\) −2.75566e16 −1.58441
\(898\) 3.68658e16 2.10671
\(899\) −1.78856e15 −0.101584
\(900\) 5.71773e15 0.322767
\(901\) −9.59250e15 −0.538202
\(902\) 7.35844e16 4.10345
\(903\) 3.62504e16 2.00923
\(904\) −3.50577e15 −0.193133
\(905\) −3.21277e16 −1.75919
\(906\) −4.81937e16 −2.62292
\(907\) 3.15987e15 0.170934 0.0854672 0.996341i \(-0.472762\pi\)
0.0854672 + 0.996341i \(0.472762\pi\)
\(908\) 4.94036e15 0.265635
\(909\) 1.68926e16 0.902806
\(910\) 5.97693e16 3.17505
\(911\) −2.95195e16 −1.55868 −0.779342 0.626599i \(-0.784446\pi\)
−0.779342 + 0.626599i \(0.784446\pi\)
\(912\) −7.91258e15 −0.415285
\(913\) −2.16580e16 −1.12987
\(914\) −9.19864e15 −0.477001
\(915\) −5.59732e16 −2.88512
\(916\) −1.11555e16 −0.571565
\(917\) 1.93623e16 0.986109
\(918\) −8.01357e15 −0.405687
\(919\) 2.89215e16 1.45541 0.727704 0.685891i \(-0.240587\pi\)
0.727704 + 0.685891i \(0.240587\pi\)
\(920\) 1.07725e16 0.538867
\(921\) −1.84591e16 −0.917874
\(922\) 5.35436e16 2.64660
\(923\) −1.18175e16 −0.580652
\(924\) 9.38827e16 4.58553
\(925\) −3.85716e15 −0.187279
\(926\) 1.27454e16 0.615168
\(927\) −4.67856e15 −0.224478
\(928\) 5.41772e15 0.258406
\(929\) 1.99050e16 0.943792 0.471896 0.881654i \(-0.343570\pi\)
0.471896 + 0.881654i \(0.343570\pi\)
\(930\) 2.64795e16 1.24811
\(931\) 1.49689e16 0.701402
\(932\) 1.63242e16 0.760405
\(933\) −3.82802e16 −1.77266
\(934\) 4.14339e16 1.90743
\(935\) 2.68118e16 1.22705
\(936\) −6.76022e15 −0.307570
\(937\) 2.81833e16 1.27475 0.637373 0.770556i \(-0.280021\pi\)
0.637373 + 0.770556i \(0.280021\pi\)
\(938\) −1.18738e15 −0.0533917
\(939\) −4.56377e16 −2.04015
\(940\) −5.16722e16 −2.29643
\(941\) 1.92625e16 0.851079 0.425540 0.904940i \(-0.360084\pi\)
0.425540 + 0.904940i \(0.360084\pi\)
\(942\) 2.48594e16 1.09197
\(943\) −3.69120e16 −1.61196
\(944\) 2.02993e16 0.881325
\(945\) 1.99033e16 0.859112
\(946\) −6.51012e16 −2.79375
\(947\) −1.32271e16 −0.564339 −0.282170 0.959365i \(-0.591054\pi\)
−0.282170 + 0.959365i \(0.591054\pi\)
\(948\) 1.45869e16 0.618753
\(949\) −2.45419e16 −1.03501
\(950\) −7.28397e15 −0.305413
\(951\) 3.59492e16 1.49864
\(952\) −8.99760e15 −0.372927
\(953\) −7.28946e15 −0.300389 −0.150195 0.988656i \(-0.547990\pi\)
−0.150195 + 0.988656i \(0.547990\pi\)
\(954\) −2.19394e16 −0.898896
\(955\) −1.75648e16 −0.715523
\(956\) 4.71302e16 1.90889
\(957\) −1.06984e16 −0.430828
\(958\) −4.58096e16 −1.83420
\(959\) −2.67978e16 −1.06683
\(960\) −5.57203e16 −2.20557
\(961\) −1.78047e16 −0.700740
\(962\) 2.08449e16 0.815713
\(963\) 4.38124e15 0.170472
\(964\) −2.64062e16 −1.02160
\(965\) 1.14170e16 0.439189
\(966\) −8.38852e16 −3.20857
\(967\) −1.90917e16 −0.726104 −0.363052 0.931769i \(-0.618265\pi\)
−0.363052 + 0.931769i \(0.618265\pi\)
\(968\) −2.57045e16 −0.972064
\(969\) −9.83268e15 −0.369736
\(970\) 3.17228e15 0.118612
\(971\) 1.19393e16 0.443888 0.221944 0.975059i \(-0.428760\pi\)
0.221944 + 0.975059i \(0.428760\pi\)
\(972\) −3.82325e16 −1.41341
\(973\) 3.29745e15 0.121215
\(974\) −2.30141e16 −0.841241
\(975\) 1.61493e16 0.586988
\(976\) 3.38919e16 1.22496
\(977\) 4.01267e16 1.44216 0.721080 0.692852i \(-0.243646\pi\)
0.721080 + 0.692852i \(0.243646\pi\)
\(978\) −3.27290e16 −1.16969
\(979\) −6.89160e16 −2.44915
\(980\) 5.93009e16 2.09565
\(981\) −1.40840e16 −0.494933
\(982\) −2.24736e16 −0.785344
\(983\) −3.80717e16 −1.32299 −0.661497 0.749948i \(-0.730078\pi\)
−0.661497 + 0.749948i \(0.730078\pi\)
\(984\) −2.33887e16 −0.808228
\(985\) 3.38804e16 1.16426
\(986\) 4.68657e15 0.160152
\(987\) 8.80303e16 2.99149
\(988\) 2.20995e16 0.746825
\(989\) 3.26566e16 1.09747
\(990\) 6.13226e16 2.04940
\(991\) −2.32664e16 −0.773257 −0.386629 0.922236i \(-0.626360\pi\)
−0.386629 + 0.922236i \(0.626360\pi\)
\(992\) −2.30325e16 −0.761249
\(993\) −5.43907e15 −0.178774
\(994\) −3.59737e16 −1.17587
\(995\) 2.21331e16 0.719473
\(996\) 3.14655e16 1.01720
\(997\) 6.76473e14 0.0217484 0.0108742 0.999941i \(-0.496539\pi\)
0.0108742 + 0.999941i \(0.496539\pi\)
\(998\) 1.29315e16 0.413459
\(999\) 6.94139e15 0.220717
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.1.2 11
3.2 odd 2 261.12.a.a.1.10 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.12.a.a.1.2 11 1.1 even 1 trivial
261.12.a.a.1.10 11 3.2 odd 2