Properties

Label 177.12.a.a.1.17
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $26$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+20.9819 q^{2} -243.000 q^{3} -1607.76 q^{4} +5799.05 q^{5} -5098.61 q^{6} +22173.1 q^{7} -76704.8 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+20.9819 q^{2} -243.000 q^{3} -1607.76 q^{4} +5799.05 q^{5} -5098.61 q^{6} +22173.1 q^{7} -76704.8 q^{8} +59049.0 q^{9} +121675. q^{10} +936055. q^{11} +390685. q^{12} -1.11415e6 q^{13} +465235. q^{14} -1.40917e6 q^{15} +1.68328e6 q^{16} -9.92639e6 q^{17} +1.23896e6 q^{18} -1.60881e7 q^{19} -9.32347e6 q^{20} -5.38807e6 q^{21} +1.96402e7 q^{22} -9.19924e6 q^{23} +1.86393e7 q^{24} -1.51992e7 q^{25} -2.33769e7 q^{26} -1.43489e7 q^{27} -3.56490e7 q^{28} +1.93966e8 q^{29} -2.95671e7 q^{30} +2.28248e8 q^{31} +1.92410e8 q^{32} -2.27461e8 q^{33} -2.08275e8 q^{34} +1.28583e8 q^{35} -9.49366e7 q^{36} +6.39964e8 q^{37} -3.37559e8 q^{38} +2.70737e8 q^{39} -4.44815e8 q^{40} +2.59523e8 q^{41} -1.13052e8 q^{42} -5.30335e6 q^{43} -1.50495e9 q^{44} +3.42428e8 q^{45} -1.93018e8 q^{46} +9.16945e8 q^{47} -4.09036e8 q^{48} -1.48568e9 q^{49} -3.18908e8 q^{50} +2.41211e9 q^{51} +1.79128e9 q^{52} -4.20629e9 q^{53} -3.01068e8 q^{54} +5.42823e9 q^{55} -1.70079e9 q^{56} +3.90940e9 q^{57} +4.06977e9 q^{58} +7.14924e8 q^{59} +2.26560e9 q^{60} -8.56840e9 q^{61} +4.78907e9 q^{62} +1.30930e9 q^{63} +5.89780e8 q^{64} -6.46098e9 q^{65} -4.77258e9 q^{66} +1.23134e10 q^{67} +1.59592e10 q^{68} +2.23541e9 q^{69} +2.69792e9 q^{70} -4.22731e9 q^{71} -4.52934e9 q^{72} -6.29909e9 q^{73} +1.34277e10 q^{74} +3.69340e9 q^{75} +2.58658e10 q^{76} +2.07553e10 q^{77} +5.68059e9 q^{78} -5.22111e10 q^{79} +9.76140e9 q^{80} +3.48678e9 q^{81} +5.44529e9 q^{82} -7.62439e9 q^{83} +8.66272e9 q^{84} -5.75636e10 q^{85} -1.11274e8 q^{86} -4.71337e10 q^{87} -7.18000e10 q^{88} -9.11126e10 q^{89} +7.18480e9 q^{90} -2.47041e10 q^{91} +1.47902e10 q^{92} -5.54642e10 q^{93} +1.92393e10 q^{94} -9.32956e10 q^{95} -4.67556e10 q^{96} +1.48369e11 q^{97} -3.11724e10 q^{98} +5.52731e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 20.9819 0.463639 0.231820 0.972759i \(-0.425532\pi\)
0.231820 + 0.972759i \(0.425532\pi\)
\(3\) −243.000 −0.577350
\(4\) −1607.76 −0.785039
\(5\) 5799.05 0.829892 0.414946 0.909846i \(-0.363800\pi\)
0.414946 + 0.909846i \(0.363800\pi\)
\(6\) −5098.61 −0.267682
\(7\) 22173.1 0.498641 0.249320 0.968421i \(-0.419793\pi\)
0.249320 + 0.968421i \(0.419793\pi\)
\(8\) −76704.8 −0.827614
\(9\) 59049.0 0.333333
\(10\) 121675. 0.384771
\(11\) 936055. 1.75243 0.876217 0.481916i \(-0.160059\pi\)
0.876217 + 0.481916i \(0.160059\pi\)
\(12\) 390685. 0.453242
\(13\) −1.11415e6 −0.832249 −0.416125 0.909308i \(-0.636612\pi\)
−0.416125 + 0.909308i \(0.636612\pi\)
\(14\) 465235. 0.231189
\(15\) −1.40917e6 −0.479138
\(16\) 1.68328e6 0.401324
\(17\) −9.92639e6 −1.69559 −0.847797 0.530320i \(-0.822072\pi\)
−0.847797 + 0.530320i \(0.822072\pi\)
\(18\) 1.23896e6 0.154546
\(19\) −1.60881e7 −1.49059 −0.745297 0.666732i \(-0.767692\pi\)
−0.745297 + 0.666732i \(0.767692\pi\)
\(20\) −9.32347e6 −0.651497
\(21\) −5.38807e6 −0.287890
\(22\) 1.96402e7 0.812497
\(23\) −9.19924e6 −0.298022 −0.149011 0.988836i \(-0.547609\pi\)
−0.149011 + 0.988836i \(0.547609\pi\)
\(24\) 1.86393e7 0.477823
\(25\) −1.51992e7 −0.311279
\(26\) −2.33769e7 −0.385863
\(27\) −1.43489e7 −0.192450
\(28\) −3.56490e7 −0.391452
\(29\) 1.93966e8 1.75605 0.878024 0.478617i \(-0.158862\pi\)
0.878024 + 0.478617i \(0.158862\pi\)
\(30\) −2.95671e7 −0.222147
\(31\) 2.28248e8 1.43191 0.715957 0.698144i \(-0.245991\pi\)
0.715957 + 0.698144i \(0.245991\pi\)
\(32\) 1.92410e8 1.01368
\(33\) −2.27461e8 −1.01177
\(34\) −2.08275e8 −0.786144
\(35\) 1.28583e8 0.413818
\(36\) −9.49366e7 −0.261680
\(37\) 6.39964e8 1.51721 0.758605 0.651550i \(-0.225881\pi\)
0.758605 + 0.651550i \(0.225881\pi\)
\(38\) −3.37559e8 −0.691098
\(39\) 2.70737e8 0.480499
\(40\) −4.44815e8 −0.686830
\(41\) 2.59523e8 0.349836 0.174918 0.984583i \(-0.444034\pi\)
0.174918 + 0.984583i \(0.444034\pi\)
\(42\) −1.13052e8 −0.133477
\(43\) −5.30335e6 −0.00550141 −0.00275070 0.999996i \(-0.500876\pi\)
−0.00275070 + 0.999996i \(0.500876\pi\)
\(44\) −1.50495e9 −1.37573
\(45\) 3.42428e8 0.276631
\(46\) −1.93018e8 −0.138175
\(47\) 9.16945e8 0.583183 0.291592 0.956543i \(-0.405815\pi\)
0.291592 + 0.956543i \(0.405815\pi\)
\(48\) −4.09036e8 −0.231705
\(49\) −1.48568e9 −0.751358
\(50\) −3.18908e8 −0.144321
\(51\) 2.41211e9 0.978952
\(52\) 1.79128e9 0.653348
\(53\) −4.20629e9 −1.38160 −0.690800 0.723046i \(-0.742741\pi\)
−0.690800 + 0.723046i \(0.742741\pi\)
\(54\) −3.01068e8 −0.0892274
\(55\) 5.42823e9 1.45433
\(56\) −1.70079e9 −0.412682
\(57\) 3.90940e9 0.860595
\(58\) 4.06977e9 0.814172
\(59\) 7.14924e8 0.130189
\(60\) 2.26560e9 0.376142
\(61\) −8.56840e9 −1.29893 −0.649465 0.760392i \(-0.725007\pi\)
−0.649465 + 0.760392i \(0.725007\pi\)
\(62\) 4.78907e9 0.663892
\(63\) 1.30930e9 0.166214
\(64\) 5.89780e8 0.0686594
\(65\) −6.46098e9 −0.690677
\(66\) −4.77258e9 −0.469096
\(67\) 1.23134e10 1.11421 0.557106 0.830442i \(-0.311912\pi\)
0.557106 + 0.830442i \(0.311912\pi\)
\(68\) 1.59592e10 1.33111
\(69\) 2.23541e9 0.172063
\(70\) 2.69792e9 0.191862
\(71\) −4.22731e9 −0.278063 −0.139031 0.990288i \(-0.544399\pi\)
−0.139031 + 0.990288i \(0.544399\pi\)
\(72\) −4.52934e9 −0.275871
\(73\) −6.29909e9 −0.355633 −0.177817 0.984064i \(-0.556903\pi\)
−0.177817 + 0.984064i \(0.556903\pi\)
\(74\) 1.34277e10 0.703439
\(75\) 3.69340e9 0.179717
\(76\) 2.58658e10 1.17017
\(77\) 2.07553e10 0.873835
\(78\) 5.68059e9 0.222778
\(79\) −5.22111e10 −1.90903 −0.954517 0.298155i \(-0.903629\pi\)
−0.954517 + 0.298155i \(0.903629\pi\)
\(80\) 9.76140e9 0.333056
\(81\) 3.48678e9 0.111111
\(82\) 5.44529e9 0.162198
\(83\) −7.62439e9 −0.212459 −0.106230 0.994342i \(-0.533878\pi\)
−0.106230 + 0.994342i \(0.533878\pi\)
\(84\) 8.66272e9 0.226005
\(85\) −5.75636e10 −1.40716
\(86\) −1.11274e8 −0.00255067
\(87\) −4.71337e10 −1.01385
\(88\) −7.18000e10 −1.45034
\(89\) −9.11126e10 −1.72955 −0.864775 0.502159i \(-0.832539\pi\)
−0.864775 + 0.502159i \(0.832539\pi\)
\(90\) 7.18480e9 0.128257
\(91\) −2.47041e10 −0.414993
\(92\) 1.47902e10 0.233959
\(93\) −5.54642e10 −0.826716
\(94\) 1.92393e10 0.270387
\(95\) −9.32956e10 −1.23703
\(96\) −4.67556e10 −0.585251
\(97\) 1.48369e11 1.75428 0.877139 0.480237i \(-0.159449\pi\)
0.877139 + 0.480237i \(0.159449\pi\)
\(98\) −3.11724e10 −0.348359
\(99\) 5.52731e10 0.584145
\(100\) 2.44366e10 0.244366
\(101\) 6.76905e10 0.640855 0.320428 0.947273i \(-0.396173\pi\)
0.320428 + 0.947273i \(0.396173\pi\)
\(102\) 5.06107e10 0.453881
\(103\) −9.24360e10 −0.785664 −0.392832 0.919610i \(-0.628505\pi\)
−0.392832 + 0.919610i \(0.628505\pi\)
\(104\) 8.54603e10 0.688781
\(105\) −3.12457e10 −0.238918
\(106\) −8.82561e10 −0.640564
\(107\) −7.51409e9 −0.0517924 −0.0258962 0.999665i \(-0.508244\pi\)
−0.0258962 + 0.999665i \(0.508244\pi\)
\(108\) 2.30696e10 0.151081
\(109\) −1.87393e11 −1.16656 −0.583281 0.812270i \(-0.698232\pi\)
−0.583281 + 0.812270i \(0.698232\pi\)
\(110\) 1.13895e11 0.674285
\(111\) −1.55511e11 −0.875962
\(112\) 3.73235e10 0.200117
\(113\) −6.22501e10 −0.317840 −0.158920 0.987291i \(-0.550801\pi\)
−0.158920 + 0.987291i \(0.550801\pi\)
\(114\) 8.20268e10 0.399006
\(115\) −5.33468e10 −0.247326
\(116\) −3.11850e11 −1.37856
\(117\) −6.57892e10 −0.277416
\(118\) 1.50005e10 0.0603607
\(119\) −2.20099e11 −0.845492
\(120\) 1.08090e11 0.396542
\(121\) 5.90888e11 2.07103
\(122\) −1.79781e11 −0.602235
\(123\) −6.30641e10 −0.201978
\(124\) −3.66967e11 −1.12411
\(125\) −3.71297e11 −1.08822
\(126\) 2.74716e10 0.0770631
\(127\) 4.50859e11 1.21093 0.605467 0.795870i \(-0.292986\pi\)
0.605467 + 0.795870i \(0.292986\pi\)
\(128\) −3.81681e11 −0.981851
\(129\) 1.28871e9 0.00317624
\(130\) −1.35564e11 −0.320225
\(131\) −4.06406e11 −0.920382 −0.460191 0.887820i \(-0.652219\pi\)
−0.460191 + 0.887820i \(0.652219\pi\)
\(132\) 3.65703e11 0.794277
\(133\) −3.56723e11 −0.743271
\(134\) 2.58359e11 0.516592
\(135\) −8.32100e10 −0.159713
\(136\) 7.61402e11 1.40330
\(137\) −9.22476e11 −1.63302 −0.816511 0.577330i \(-0.804095\pi\)
−0.816511 + 0.577330i \(0.804095\pi\)
\(138\) 4.69033e10 0.0797753
\(139\) 1.60565e11 0.262463 0.131232 0.991352i \(-0.458107\pi\)
0.131232 + 0.991352i \(0.458107\pi\)
\(140\) −2.06730e11 −0.324863
\(141\) −2.22818e11 −0.336701
\(142\) −8.86970e10 −0.128921
\(143\) −1.04290e12 −1.45846
\(144\) 9.93957e10 0.133775
\(145\) 1.12482e12 1.45733
\(146\) −1.32167e11 −0.164886
\(147\) 3.61020e11 0.433796
\(148\) −1.02891e12 −1.19107
\(149\) −4.86839e11 −0.543076 −0.271538 0.962428i \(-0.587532\pi\)
−0.271538 + 0.962428i \(0.587532\pi\)
\(150\) 7.74946e10 0.0833239
\(151\) 1.17088e12 1.21378 0.606891 0.794785i \(-0.292416\pi\)
0.606891 + 0.794785i \(0.292416\pi\)
\(152\) 1.23403e12 1.23364
\(153\) −5.86143e11 −0.565198
\(154\) 4.35485e11 0.405144
\(155\) 1.32362e12 1.18833
\(156\) −4.35280e11 −0.377210
\(157\) −1.01892e11 −0.0852499 −0.0426250 0.999091i \(-0.513572\pi\)
−0.0426250 + 0.999091i \(0.513572\pi\)
\(158\) −1.09549e12 −0.885104
\(159\) 1.02213e12 0.797667
\(160\) 1.11579e12 0.841248
\(161\) −2.03976e11 −0.148606
\(162\) 7.31594e10 0.0515155
\(163\) −3.58023e11 −0.243713 −0.121857 0.992548i \(-0.538885\pi\)
−0.121857 + 0.992548i \(0.538885\pi\)
\(164\) −4.17251e11 −0.274635
\(165\) −1.31906e12 −0.839659
\(166\) −1.59974e11 −0.0985044
\(167\) −3.69568e11 −0.220168 −0.110084 0.993922i \(-0.535112\pi\)
−0.110084 + 0.993922i \(0.535112\pi\)
\(168\) 4.13291e11 0.238262
\(169\) −5.50841e11 −0.307362
\(170\) −1.20779e12 −0.652415
\(171\) −9.49985e11 −0.496865
\(172\) 8.52651e9 0.00431882
\(173\) −6.32932e11 −0.310530 −0.155265 0.987873i \(-0.549623\pi\)
−0.155265 + 0.987873i \(0.549623\pi\)
\(174\) −9.88955e11 −0.470063
\(175\) −3.37013e11 −0.155216
\(176\) 1.57564e12 0.703294
\(177\) −1.73727e11 −0.0751646
\(178\) −1.91172e12 −0.801888
\(179\) 6.95997e11 0.283084 0.141542 0.989932i \(-0.454794\pi\)
0.141542 + 0.989932i \(0.454794\pi\)
\(180\) −5.50542e11 −0.217166
\(181\) −1.21927e12 −0.466517 −0.233258 0.972415i \(-0.574939\pi\)
−0.233258 + 0.972415i \(0.574939\pi\)
\(182\) −5.18339e11 −0.192407
\(183\) 2.08212e12 0.749937
\(184\) 7.05626e11 0.246647
\(185\) 3.71118e12 1.25912
\(186\) −1.16374e12 −0.383298
\(187\) −9.29165e12 −2.97142
\(188\) −1.47423e12 −0.457821
\(189\) −3.18160e11 −0.0959634
\(190\) −1.95752e12 −0.573537
\(191\) −2.61439e12 −0.744195 −0.372097 0.928194i \(-0.621361\pi\)
−0.372097 + 0.928194i \(0.621361\pi\)
\(192\) −1.43317e11 −0.0396405
\(193\) −8.15687e11 −0.219259 −0.109630 0.993972i \(-0.534966\pi\)
−0.109630 + 0.993972i \(0.534966\pi\)
\(194\) 3.11306e12 0.813352
\(195\) 1.57002e12 0.398763
\(196\) 2.38861e12 0.589845
\(197\) −5.44207e12 −1.30677 −0.653386 0.757025i \(-0.726652\pi\)
−0.653386 + 0.757025i \(0.726652\pi\)
\(198\) 1.15974e12 0.270832
\(199\) −2.55159e12 −0.579588 −0.289794 0.957089i \(-0.593587\pi\)
−0.289794 + 0.957089i \(0.593587\pi\)
\(200\) 1.16585e12 0.257619
\(201\) −2.99216e12 −0.643290
\(202\) 1.42028e12 0.297126
\(203\) 4.30083e12 0.875636
\(204\) −3.87809e12 −0.768515
\(205\) 1.50499e12 0.290326
\(206\) −1.93949e12 −0.364264
\(207\) −5.43206e11 −0.0993408
\(208\) −1.87541e12 −0.334002
\(209\) −1.50593e13 −2.61217
\(210\) −6.55594e11 −0.110772
\(211\) 4.77860e12 0.786587 0.393294 0.919413i \(-0.371336\pi\)
0.393294 + 0.919413i \(0.371336\pi\)
\(212\) 6.76270e12 1.08461
\(213\) 1.02724e12 0.160540
\(214\) −1.57660e11 −0.0240130
\(215\) −3.07544e10 −0.00456558
\(216\) 1.10063e12 0.159274
\(217\) 5.06096e12 0.714011
\(218\) −3.93187e12 −0.540864
\(219\) 1.53068e12 0.205325
\(220\) −8.72729e12 −1.14171
\(221\) 1.10594e13 1.41116
\(222\) −3.26292e12 −0.406130
\(223\) 9.75032e12 1.18397 0.591987 0.805948i \(-0.298344\pi\)
0.591987 + 0.805948i \(0.298344\pi\)
\(224\) 4.26633e12 0.505464
\(225\) −8.97496e11 −0.103760
\(226\) −1.30613e12 −0.147363
\(227\) −1.40190e13 −1.54375 −0.771874 0.635776i \(-0.780680\pi\)
−0.771874 + 0.635776i \(0.780680\pi\)
\(228\) −6.28538e12 −0.675600
\(229\) −9.22670e12 −0.968169 −0.484085 0.875021i \(-0.660847\pi\)
−0.484085 + 0.875021i \(0.660847\pi\)
\(230\) −1.11932e12 −0.114670
\(231\) −5.04353e12 −0.504509
\(232\) −1.48781e13 −1.45333
\(233\) 4.25657e12 0.406071 0.203036 0.979171i \(-0.434919\pi\)
0.203036 + 0.979171i \(0.434919\pi\)
\(234\) −1.38038e12 −0.128621
\(235\) 5.31741e12 0.483979
\(236\) −1.14943e12 −0.102203
\(237\) 1.26873e13 1.10218
\(238\) −4.61810e12 −0.392004
\(239\) −3.90272e12 −0.323727 −0.161864 0.986813i \(-0.551750\pi\)
−0.161864 + 0.986813i \(0.551750\pi\)
\(240\) −2.37202e12 −0.192290
\(241\) −1.17046e13 −0.927393 −0.463697 0.885994i \(-0.653477\pi\)
−0.463697 + 0.885994i \(0.653477\pi\)
\(242\) 1.23980e13 0.960209
\(243\) −8.47289e11 −0.0641500
\(244\) 1.37759e13 1.01971
\(245\) −8.61553e12 −0.623546
\(246\) −1.32321e12 −0.0936450
\(247\) 1.79245e13 1.24055
\(248\) −1.75077e13 −1.18507
\(249\) 1.85273e12 0.122663
\(250\) −7.79053e12 −0.504542
\(251\) 1.43145e13 0.906925 0.453462 0.891275i \(-0.350189\pi\)
0.453462 + 0.891275i \(0.350189\pi\)
\(252\) −2.10504e12 −0.130484
\(253\) −8.61100e12 −0.522265
\(254\) 9.45989e12 0.561437
\(255\) 1.39880e13 0.812425
\(256\) −9.21626e12 −0.523884
\(257\) −3.42210e13 −1.90397 −0.951986 0.306143i \(-0.900961\pi\)
−0.951986 + 0.306143i \(0.900961\pi\)
\(258\) 2.70397e10 0.00147263
\(259\) 1.41900e13 0.756543
\(260\) 1.03877e13 0.542208
\(261\) 1.14535e13 0.585349
\(262\) −8.52718e12 −0.426725
\(263\) 1.24549e13 0.610358 0.305179 0.952295i \(-0.401284\pi\)
0.305179 + 0.952295i \(0.401284\pi\)
\(264\) 1.74474e13 0.837354
\(265\) −2.43925e13 −1.14658
\(266\) −7.48473e12 −0.344610
\(267\) 2.21404e13 0.998556
\(268\) −1.97970e13 −0.874699
\(269\) −4.26154e12 −0.184472 −0.0922358 0.995737i \(-0.529401\pi\)
−0.0922358 + 0.995737i \(0.529401\pi\)
\(270\) −1.74591e12 −0.0740491
\(271\) −1.52799e11 −0.00635022 −0.00317511 0.999995i \(-0.501011\pi\)
−0.00317511 + 0.999995i \(0.501011\pi\)
\(272\) −1.67088e13 −0.680483
\(273\) 6.00309e12 0.239596
\(274\) −1.93553e13 −0.757133
\(275\) −1.42273e13 −0.545496
\(276\) −3.59401e12 −0.135076
\(277\) −3.10248e13 −1.14306 −0.571532 0.820579i \(-0.693651\pi\)
−0.571532 + 0.820579i \(0.693651\pi\)
\(278\) 3.36896e12 0.121688
\(279\) 1.34778e13 0.477305
\(280\) −9.86294e12 −0.342482
\(281\) −5.05962e13 −1.72279 −0.861397 0.507932i \(-0.830410\pi\)
−0.861397 + 0.507932i \(0.830410\pi\)
\(282\) −4.67514e12 −0.156108
\(283\) −3.42911e13 −1.12294 −0.561469 0.827498i \(-0.689764\pi\)
−0.561469 + 0.827498i \(0.689764\pi\)
\(284\) 6.79649e12 0.218290
\(285\) 2.26708e13 0.714201
\(286\) −2.18821e13 −0.676200
\(287\) 5.75444e12 0.174443
\(288\) 1.13616e13 0.337895
\(289\) 6.42612e13 1.87504
\(290\) 2.36008e13 0.675675
\(291\) −3.60536e13 −1.01283
\(292\) 1.01274e13 0.279186
\(293\) −9.74901e12 −0.263748 −0.131874 0.991267i \(-0.542099\pi\)
−0.131874 + 0.991267i \(0.542099\pi\)
\(294\) 7.57489e12 0.201125
\(295\) 4.14588e12 0.108043
\(296\) −4.90883e13 −1.25567
\(297\) −1.34314e13 −0.337256
\(298\) −1.02148e13 −0.251791
\(299\) 1.02493e13 0.248029
\(300\) −5.93810e12 −0.141085
\(301\) −1.17592e11 −0.00274323
\(302\) 2.45674e13 0.562757
\(303\) −1.64488e13 −0.369998
\(304\) −2.70807e13 −0.598212
\(305\) −4.96885e13 −1.07797
\(306\) −1.22984e13 −0.262048
\(307\) 3.11117e13 0.651122 0.325561 0.945521i \(-0.394447\pi\)
0.325561 + 0.945521i \(0.394447\pi\)
\(308\) −3.33695e13 −0.685994
\(309\) 2.24620e13 0.453603
\(310\) 2.77721e13 0.550958
\(311\) −6.72271e13 −1.31027 −0.655137 0.755510i \(-0.727389\pi\)
−0.655137 + 0.755510i \(0.727389\pi\)
\(312\) −2.07669e13 −0.397668
\(313\) 1.61471e13 0.303809 0.151905 0.988395i \(-0.451459\pi\)
0.151905 + 0.988395i \(0.451459\pi\)
\(314\) −2.13790e12 −0.0395252
\(315\) 7.59270e12 0.137939
\(316\) 8.39429e13 1.49867
\(317\) 6.58020e13 1.15455 0.577276 0.816549i \(-0.304116\pi\)
0.577276 + 0.816549i \(0.304116\pi\)
\(318\) 2.14462e13 0.369830
\(319\) 1.81563e14 3.07736
\(320\) 3.42016e12 0.0569799
\(321\) 1.82592e12 0.0299023
\(322\) −4.27980e12 −0.0688996
\(323\) 1.59697e14 2.52744
\(324\) −5.60591e12 −0.0872265
\(325\) 1.69341e13 0.259062
\(326\) −7.51201e12 −0.112995
\(327\) 4.55366e13 0.673515
\(328\) −1.99067e13 −0.289529
\(329\) 2.03315e13 0.290799
\(330\) −2.76764e13 −0.389299
\(331\) −2.91682e13 −0.403511 −0.201755 0.979436i \(-0.564665\pi\)
−0.201755 + 0.979436i \(0.564665\pi\)
\(332\) 1.22582e13 0.166789
\(333\) 3.77892e13 0.505737
\(334\) −7.75425e12 −0.102078
\(335\) 7.14062e13 0.924675
\(336\) −9.06960e12 −0.115537
\(337\) −1.82154e13 −0.228283 −0.114142 0.993464i \(-0.536412\pi\)
−0.114142 + 0.993464i \(0.536412\pi\)
\(338\) −1.15577e13 −0.142505
\(339\) 1.51268e13 0.183505
\(340\) 9.25484e13 1.10468
\(341\) 2.13652e14 2.50934
\(342\) −1.99325e13 −0.230366
\(343\) −7.67857e13 −0.873298
\(344\) 4.06793e11 0.00455304
\(345\) 1.29633e13 0.142794
\(346\) −1.32801e13 −0.143974
\(347\) −6.67404e13 −0.712159 −0.356079 0.934456i \(-0.615887\pi\)
−0.356079 + 0.934456i \(0.615887\pi\)
\(348\) 7.57796e13 0.795915
\(349\) −1.11830e14 −1.15616 −0.578081 0.815979i \(-0.696198\pi\)
−0.578081 + 0.815979i \(0.696198\pi\)
\(350\) −7.07118e12 −0.0719644
\(351\) 1.59868e13 0.160166
\(352\) 1.80106e14 1.77641
\(353\) 1.25238e13 0.121612 0.0608058 0.998150i \(-0.480633\pi\)
0.0608058 + 0.998150i \(0.480633\pi\)
\(354\) −3.64512e12 −0.0348493
\(355\) −2.45143e13 −0.230762
\(356\) 1.46487e14 1.35776
\(357\) 5.34840e13 0.488145
\(358\) 1.46034e13 0.131249
\(359\) 2.07626e14 1.83765 0.918825 0.394665i \(-0.129139\pi\)
0.918825 + 0.394665i \(0.129139\pi\)
\(360\) −2.62659e13 −0.228943
\(361\) 1.42336e14 1.22187
\(362\) −2.55826e13 −0.216295
\(363\) −1.43586e14 −1.19571
\(364\) 3.97182e13 0.325786
\(365\) −3.65288e13 −0.295137
\(366\) 4.36869e13 0.347700
\(367\) −3.41773e13 −0.267963 −0.133981 0.990984i \(-0.542776\pi\)
−0.133981 + 0.990984i \(0.542776\pi\)
\(368\) −1.54849e13 −0.119604
\(369\) 1.53246e13 0.116612
\(370\) 7.78677e13 0.583778
\(371\) −9.32666e13 −0.688921
\(372\) 8.91730e13 0.649004
\(373\) 3.25095e13 0.233137 0.116568 0.993183i \(-0.462811\pi\)
0.116568 + 0.993183i \(0.462811\pi\)
\(374\) −1.94957e14 −1.37767
\(375\) 9.02253e13 0.628284
\(376\) −7.03341e13 −0.482651
\(377\) −2.16106e14 −1.46147
\(378\) −6.67561e12 −0.0444924
\(379\) −1.12920e14 −0.741749 −0.370874 0.928683i \(-0.620942\pi\)
−0.370874 + 0.928683i \(0.620942\pi\)
\(380\) 1.49997e14 0.971118
\(381\) −1.09559e14 −0.699133
\(382\) −5.48549e13 −0.345038
\(383\) −1.21687e14 −0.754488 −0.377244 0.926114i \(-0.623128\pi\)
−0.377244 + 0.926114i \(0.623128\pi\)
\(384\) 9.27484e13 0.566872
\(385\) 1.20361e14 0.725189
\(386\) −1.71147e13 −0.101657
\(387\) −3.13158e11 −0.00183380
\(388\) −2.38541e14 −1.37718
\(389\) 1.67697e14 0.954559 0.477280 0.878751i \(-0.341623\pi\)
0.477280 + 0.878751i \(0.341623\pi\)
\(390\) 3.29420e13 0.184882
\(391\) 9.13152e13 0.505325
\(392\) 1.13959e14 0.621834
\(393\) 9.87566e13 0.531383
\(394\) −1.14185e14 −0.605871
\(395\) −3.02775e14 −1.58429
\(396\) −8.88659e13 −0.458576
\(397\) −1.86310e14 −0.948174 −0.474087 0.880478i \(-0.657222\pi\)
−0.474087 + 0.880478i \(0.657222\pi\)
\(398\) −5.35374e13 −0.268720
\(399\) 8.66837e13 0.429128
\(400\) −2.55844e13 −0.124924
\(401\) −1.10456e14 −0.531981 −0.265990 0.963976i \(-0.585699\pi\)
−0.265990 + 0.963976i \(0.585699\pi\)
\(402\) −6.27813e13 −0.298255
\(403\) −2.54301e14 −1.19171
\(404\) −1.08830e14 −0.503096
\(405\) 2.02200e13 0.0922102
\(406\) 9.02396e13 0.405979
\(407\) 5.99042e14 2.65881
\(408\) −1.85021e14 −0.810195
\(409\) 3.44348e14 1.48771 0.743856 0.668340i \(-0.232995\pi\)
0.743856 + 0.668340i \(0.232995\pi\)
\(410\) 3.15775e13 0.134607
\(411\) 2.24162e14 0.942825
\(412\) 1.48615e14 0.616776
\(413\) 1.58521e13 0.0649175
\(414\) −1.13975e13 −0.0460583
\(415\) −4.42142e13 −0.176318
\(416\) −2.14373e14 −0.843637
\(417\) −3.90172e13 −0.151533
\(418\) −3.15974e14 −1.21110
\(419\) −9.24903e13 −0.349880 −0.174940 0.984579i \(-0.555973\pi\)
−0.174940 + 0.984579i \(0.555973\pi\)
\(420\) 5.02355e13 0.187560
\(421\) 1.47656e14 0.544127 0.272064 0.962279i \(-0.412294\pi\)
0.272064 + 0.962279i \(0.412294\pi\)
\(422\) 1.00264e14 0.364693
\(423\) 5.41447e13 0.194394
\(424\) 3.22643e14 1.14343
\(425\) 1.50873e14 0.527803
\(426\) 2.15534e13 0.0744325
\(427\) −1.89988e14 −0.647699
\(428\) 1.20808e13 0.0406590
\(429\) 2.53425e14 0.842043
\(430\) −6.45286e11 −0.00211678
\(431\) 3.73758e14 1.21050 0.605252 0.796034i \(-0.293072\pi\)
0.605252 + 0.796034i \(0.293072\pi\)
\(432\) −2.41532e13 −0.0772349
\(433\) 3.24128e13 0.102337 0.0511685 0.998690i \(-0.483705\pi\)
0.0511685 + 0.998690i \(0.483705\pi\)
\(434\) 1.06189e14 0.331043
\(435\) −2.73331e14 −0.841390
\(436\) 3.01283e14 0.915797
\(437\) 1.47998e14 0.444230
\(438\) 3.21166e13 0.0951967
\(439\) −1.98839e14 −0.582031 −0.291015 0.956718i \(-0.593993\pi\)
−0.291015 + 0.956718i \(0.593993\pi\)
\(440\) −4.16371e14 −1.20363
\(441\) −8.77279e13 −0.250453
\(442\) 2.32048e14 0.654268
\(443\) −4.47598e14 −1.24643 −0.623214 0.782051i \(-0.714174\pi\)
−0.623214 + 0.782051i \(0.714174\pi\)
\(444\) 2.50025e14 0.687664
\(445\) −5.28366e14 −1.43534
\(446\) 2.04580e14 0.548937
\(447\) 1.18302e14 0.313545
\(448\) 1.30773e13 0.0342364
\(449\) 1.09020e14 0.281937 0.140969 0.990014i \(-0.454978\pi\)
0.140969 + 0.990014i \(0.454978\pi\)
\(450\) −1.88312e13 −0.0481071
\(451\) 2.42928e14 0.613065
\(452\) 1.00083e14 0.249517
\(453\) −2.84525e14 −0.700778
\(454\) −2.94147e14 −0.715742
\(455\) −1.43260e14 −0.344400
\(456\) −2.99870e14 −0.712241
\(457\) −5.84990e14 −1.37281 −0.686403 0.727222i \(-0.740811\pi\)
−0.686403 + 0.727222i \(0.740811\pi\)
\(458\) −1.93594e14 −0.448881
\(459\) 1.42433e14 0.326317
\(460\) 8.57688e13 0.194161
\(461\) 5.81689e14 1.30117 0.650587 0.759432i \(-0.274523\pi\)
0.650587 + 0.759432i \(0.274523\pi\)
\(462\) −1.05823e14 −0.233910
\(463\) −4.75087e14 −1.03771 −0.518857 0.854861i \(-0.673642\pi\)
−0.518857 + 0.854861i \(0.673642\pi\)
\(464\) 3.26498e14 0.704744
\(465\) −3.21639e14 −0.686085
\(466\) 8.93111e13 0.188271
\(467\) −7.16934e14 −1.49361 −0.746803 0.665045i \(-0.768412\pi\)
−0.746803 + 0.665045i \(0.768412\pi\)
\(468\) 1.05773e14 0.217783
\(469\) 2.73027e14 0.555591
\(470\) 1.11569e14 0.224392
\(471\) 2.47599e13 0.0492191
\(472\) −5.48382e13 −0.107746
\(473\) −4.96423e12 −0.00964086
\(474\) 2.66204e14 0.511015
\(475\) 2.44526e14 0.463991
\(476\) 3.53866e14 0.663744
\(477\) −2.48377e14 −0.460533
\(478\) −8.18866e13 −0.150093
\(479\) 7.87256e14 1.42650 0.713248 0.700912i \(-0.247223\pi\)
0.713248 + 0.700912i \(0.247223\pi\)
\(480\) −2.71138e14 −0.485695
\(481\) −7.13013e14 −1.26270
\(482\) −2.45586e14 −0.429976
\(483\) 4.95661e13 0.0857977
\(484\) −9.50005e14 −1.62584
\(485\) 8.60398e14 1.45586
\(486\) −1.77777e13 −0.0297425
\(487\) 3.00862e14 0.497689 0.248844 0.968543i \(-0.419949\pi\)
0.248844 + 0.968543i \(0.419949\pi\)
\(488\) 6.57238e14 1.07501
\(489\) 8.69995e13 0.140708
\(490\) −1.80770e14 −0.289100
\(491\) 1.05712e14 0.167177 0.0835887 0.996500i \(-0.473362\pi\)
0.0835887 + 0.996500i \(0.473362\pi\)
\(492\) 1.01392e14 0.158561
\(493\) −1.92538e15 −2.97754
\(494\) 3.76090e14 0.575166
\(495\) 3.20532e14 0.484777
\(496\) 3.84204e14 0.574662
\(497\) −9.37326e13 −0.138653
\(498\) 3.88737e13 0.0568715
\(499\) −1.27645e15 −1.84694 −0.923469 0.383673i \(-0.874659\pi\)
−0.923469 + 0.383673i \(0.874659\pi\)
\(500\) 5.96957e14 0.854295
\(501\) 8.98051e13 0.127114
\(502\) 3.00346e14 0.420486
\(503\) −6.89981e14 −0.955462 −0.477731 0.878506i \(-0.658541\pi\)
−0.477731 + 0.878506i \(0.658541\pi\)
\(504\) −1.00430e14 −0.137561
\(505\) 3.92540e14 0.531841
\(506\) −1.80675e14 −0.242142
\(507\) 1.33854e14 0.177455
\(508\) −7.24873e14 −0.950630
\(509\) −1.24066e14 −0.160955 −0.0804776 0.996756i \(-0.525645\pi\)
−0.0804776 + 0.996756i \(0.525645\pi\)
\(510\) 2.93494e14 0.376672
\(511\) −1.39671e14 −0.177333
\(512\) 5.88307e14 0.738957
\(513\) 2.30846e14 0.286865
\(514\) −7.18022e14 −0.882756
\(515\) −5.36041e14 −0.652016
\(516\) −2.07194e12 −0.00249347
\(517\) 8.58311e14 1.02199
\(518\) 2.97733e14 0.350763
\(519\) 1.53803e14 0.179285
\(520\) 4.95588e14 0.571614
\(521\) 1.85354e14 0.211541 0.105771 0.994391i \(-0.466269\pi\)
0.105771 + 0.994391i \(0.466269\pi\)
\(522\) 2.40316e14 0.271391
\(523\) −4.44489e14 −0.496708 −0.248354 0.968669i \(-0.579890\pi\)
−0.248354 + 0.968669i \(0.579890\pi\)
\(524\) 6.53403e14 0.722535
\(525\) 8.18942e13 0.0896142
\(526\) 2.61328e14 0.282986
\(527\) −2.26567e15 −2.42795
\(528\) −3.82880e14 −0.406047
\(529\) −8.68184e14 −0.911183
\(530\) −5.11801e14 −0.531599
\(531\) 4.22156e13 0.0433963
\(532\) 5.73525e14 0.583496
\(533\) −2.89146e14 −0.291151
\(534\) 4.64547e14 0.462970
\(535\) −4.35746e13 −0.0429821
\(536\) −9.44500e14 −0.922137
\(537\) −1.69127e14 −0.163439
\(538\) −8.94153e13 −0.0855282
\(539\) −1.39068e15 −1.31670
\(540\) 1.33782e14 0.125381
\(541\) 2.10768e15 1.95533 0.977665 0.210169i \(-0.0674014\pi\)
0.977665 + 0.210169i \(0.0674014\pi\)
\(542\) −3.20601e12 −0.00294421
\(543\) 2.96282e14 0.269343
\(544\) −1.90993e15 −1.71880
\(545\) −1.08670e15 −0.968121
\(546\) 1.25956e14 0.111086
\(547\) −1.26350e14 −0.110317 −0.0551587 0.998478i \(-0.517566\pi\)
−0.0551587 + 0.998478i \(0.517566\pi\)
\(548\) 1.48312e15 1.28198
\(549\) −5.05955e14 −0.432976
\(550\) −2.98515e14 −0.252913
\(551\) −3.12054e15 −2.61755
\(552\) −1.71467e14 −0.142402
\(553\) −1.15768e15 −0.951922
\(554\) −6.50961e14 −0.529970
\(555\) −9.01817e14 −0.726954
\(556\) −2.58149e14 −0.206044
\(557\) 1.08838e15 0.860152 0.430076 0.902793i \(-0.358487\pi\)
0.430076 + 0.902793i \(0.358487\pi\)
\(558\) 2.82790e14 0.221297
\(559\) 5.90870e12 0.00457854
\(560\) 2.16441e14 0.166075
\(561\) 2.25787e15 1.71555
\(562\) −1.06161e15 −0.798755
\(563\) 1.44529e15 1.07686 0.538429 0.842671i \(-0.319018\pi\)
0.538429 + 0.842671i \(0.319018\pi\)
\(564\) 3.58237e14 0.264323
\(565\) −3.60991e14 −0.263773
\(566\) −7.19493e14 −0.520638
\(567\) 7.73129e13 0.0554045
\(568\) 3.24255e14 0.230129
\(569\) 1.80278e15 1.26714 0.633571 0.773685i \(-0.281588\pi\)
0.633571 + 0.773685i \(0.281588\pi\)
\(570\) 4.75677e14 0.331132
\(571\) −2.82982e13 −0.0195101 −0.00975506 0.999952i \(-0.503105\pi\)
−0.00975506 + 0.999952i \(0.503105\pi\)
\(572\) 1.67673e15 1.14495
\(573\) 6.35296e14 0.429661
\(574\) 1.20739e14 0.0808784
\(575\) 1.39821e14 0.0927681
\(576\) 3.48259e13 0.0228865
\(577\) −9.32447e14 −0.606956 −0.303478 0.952838i \(-0.598148\pi\)
−0.303478 + 0.952838i \(0.598148\pi\)
\(578\) 1.34832e15 0.869343
\(579\) 1.98212e14 0.126589
\(580\) −1.80843e15 −1.14406
\(581\) −1.69056e14 −0.105941
\(582\) −7.56475e14 −0.469589
\(583\) −3.93732e15 −2.42116
\(584\) 4.83171e14 0.294327
\(585\) −3.81514e14 −0.230226
\(586\) −2.04553e14 −0.122284
\(587\) −2.28177e15 −1.35134 −0.675668 0.737206i \(-0.736145\pi\)
−0.675668 + 0.737206i \(0.736145\pi\)
\(588\) −5.80433e14 −0.340547
\(589\) −3.67207e15 −2.13440
\(590\) 8.69885e13 0.0500929
\(591\) 1.32242e15 0.754465
\(592\) 1.07724e15 0.608893
\(593\) 3.18237e15 1.78217 0.891085 0.453835i \(-0.149945\pi\)
0.891085 + 0.453835i \(0.149945\pi\)
\(594\) −2.81816e14 −0.156365
\(595\) −1.27636e15 −0.701668
\(596\) 7.82719e14 0.426336
\(597\) 6.20037e14 0.334626
\(598\) 2.15050e14 0.114996
\(599\) 1.65920e15 0.879127 0.439564 0.898211i \(-0.355133\pi\)
0.439564 + 0.898211i \(0.355133\pi\)
\(600\) −2.83302e14 −0.148736
\(601\) −1.92039e15 −0.999034 −0.499517 0.866304i \(-0.666489\pi\)
−0.499517 + 0.866304i \(0.666489\pi\)
\(602\) −2.46730e12 −0.00127187
\(603\) 7.27096e14 0.371404
\(604\) −1.88250e15 −0.952866
\(605\) 3.42659e15 1.71873
\(606\) −3.45127e14 −0.171546
\(607\) 3.30391e15 1.62738 0.813692 0.581296i \(-0.197454\pi\)
0.813692 + 0.581296i \(0.197454\pi\)
\(608\) −3.09551e15 −1.51099
\(609\) −1.04510e15 −0.505549
\(610\) −1.04256e15 −0.499790
\(611\) −1.02161e15 −0.485354
\(612\) 9.42377e14 0.443702
\(613\) −1.00720e15 −0.469984 −0.234992 0.971997i \(-0.575506\pi\)
−0.234992 + 0.971997i \(0.575506\pi\)
\(614\) 6.52782e14 0.301886
\(615\) −3.65712e14 −0.167620
\(616\) −1.59203e15 −0.723198
\(617\) 1.41632e15 0.637666 0.318833 0.947811i \(-0.396709\pi\)
0.318833 + 0.947811i \(0.396709\pi\)
\(618\) 4.71295e14 0.210308
\(619\) 2.38868e15 1.05648 0.528238 0.849096i \(-0.322853\pi\)
0.528238 + 0.849096i \(0.322853\pi\)
\(620\) −2.12806e15 −0.932888
\(621\) 1.31999e14 0.0573544
\(622\) −1.41055e15 −0.607494
\(623\) −2.02025e15 −0.862424
\(624\) 4.55725e14 0.192836
\(625\) −1.41102e15 −0.591826
\(626\) 3.38798e14 0.140858
\(627\) 3.65942e15 1.50814
\(628\) 1.63819e14 0.0669245
\(629\) −6.35253e15 −2.57258
\(630\) 1.59309e14 0.0639541
\(631\) 3.09629e15 1.23220 0.616099 0.787669i \(-0.288712\pi\)
0.616099 + 0.787669i \(0.288712\pi\)
\(632\) 4.00484e15 1.57994
\(633\) −1.16120e15 −0.454136
\(634\) 1.38065e15 0.535296
\(635\) 2.61455e15 1.00494
\(636\) −1.64334e15 −0.626199
\(637\) 1.65526e15 0.625317
\(638\) 3.80953e15 1.42678
\(639\) −2.49618e14 −0.0926876
\(640\) −2.21338e15 −0.814830
\(641\) 2.23695e15 0.816463 0.408232 0.912878i \(-0.366146\pi\)
0.408232 + 0.912878i \(0.366146\pi\)
\(642\) 3.83114e13 0.0138639
\(643\) 3.70986e15 1.33106 0.665529 0.746372i \(-0.268206\pi\)
0.665529 + 0.746372i \(0.268206\pi\)
\(644\) 3.27944e14 0.116661
\(645\) 7.47332e12 0.00263594
\(646\) 3.35074e15 1.17182
\(647\) 3.78140e15 1.31123 0.655615 0.755096i \(-0.272410\pi\)
0.655615 + 0.755096i \(0.272410\pi\)
\(648\) −2.67453e14 −0.0919571
\(649\) 6.69209e14 0.228148
\(650\) 3.55310e14 0.120111
\(651\) −1.22981e15 −0.412234
\(652\) 5.75614e14 0.191324
\(653\) 4.33846e15 1.42992 0.714962 0.699163i \(-0.246444\pi\)
0.714962 + 0.699163i \(0.246444\pi\)
\(654\) 9.55444e14 0.312268
\(655\) −2.35677e15 −0.763817
\(656\) 4.36849e14 0.140398
\(657\) −3.71955e14 −0.118544
\(658\) 4.26595e14 0.134826
\(659\) 1.70685e15 0.534965 0.267482 0.963563i \(-0.413808\pi\)
0.267482 + 0.963563i \(0.413808\pi\)
\(660\) 2.12073e15 0.659164
\(661\) 2.39059e15 0.736880 0.368440 0.929651i \(-0.379892\pi\)
0.368440 + 0.929651i \(0.379892\pi\)
\(662\) −6.12004e14 −0.187084
\(663\) −2.68744e15 −0.814732
\(664\) 5.84827e14 0.175834
\(665\) −2.06865e15 −0.616835
\(666\) 7.92891e14 0.234480
\(667\) −1.78434e15 −0.523341
\(668\) 5.94177e14 0.172840
\(669\) −2.36933e15 −0.683567
\(670\) 1.49824e15 0.428716
\(671\) −8.02049e15 −2.27629
\(672\) −1.03672e15 −0.291830
\(673\) −4.28194e15 −1.19552 −0.597761 0.801674i \(-0.703943\pi\)
−0.597761 + 0.801674i \(0.703943\pi\)
\(674\) −3.82194e14 −0.105841
\(675\) 2.18092e14 0.0599057
\(676\) 8.85620e14 0.241291
\(677\) 2.18531e14 0.0590576 0.0295288 0.999564i \(-0.490599\pi\)
0.0295288 + 0.999564i \(0.490599\pi\)
\(678\) 3.17389e14 0.0850802
\(679\) 3.28980e15 0.874754
\(680\) 4.41541e15 1.16459
\(681\) 3.40663e15 0.891283
\(682\) 4.48284e15 1.16343
\(683\) −4.04892e15 −1.04238 −0.521189 0.853441i \(-0.674511\pi\)
−0.521189 + 0.853441i \(0.674511\pi\)
\(684\) 1.52735e15 0.390058
\(685\) −5.34948e15 −1.35523
\(686\) −1.61111e15 −0.404895
\(687\) 2.24209e15 0.558973
\(688\) −8.92700e12 −0.00220785
\(689\) 4.68642e15 1.14983
\(690\) 2.71994e14 0.0662049
\(691\) −1.81826e15 −0.439062 −0.219531 0.975606i \(-0.570453\pi\)
−0.219531 + 0.975606i \(0.570453\pi\)
\(692\) 1.01760e15 0.243778
\(693\) 1.22558e15 0.291278
\(694\) −1.40034e15 −0.330185
\(695\) 9.31122e14 0.217816
\(696\) 3.61538e15 0.839080
\(697\) −2.57613e15 −0.593181
\(698\) −2.34641e15 −0.536042
\(699\) −1.03435e15 −0.234445
\(700\) 5.41836e14 0.121851
\(701\) 2.38322e15 0.531759 0.265879 0.964006i \(-0.414338\pi\)
0.265879 + 0.964006i \(0.414338\pi\)
\(702\) 3.35433e14 0.0742594
\(703\) −1.02958e16 −2.26155
\(704\) 5.52067e14 0.120321
\(705\) −1.29213e15 −0.279426
\(706\) 2.62773e14 0.0563839
\(707\) 1.50091e15 0.319556
\(708\) 2.79311e14 0.0590071
\(709\) 2.96484e15 0.621509 0.310754 0.950490i \(-0.399418\pi\)
0.310754 + 0.950490i \(0.399418\pi\)
\(710\) −5.14358e14 −0.106990
\(711\) −3.08301e15 −0.636345
\(712\) 6.98878e15 1.43140
\(713\) −2.09970e15 −0.426742
\(714\) 1.12220e15 0.226323
\(715\) −6.04784e15 −1.21037
\(716\) −1.11900e15 −0.222232
\(717\) 9.48362e14 0.186904
\(718\) 4.35640e15 0.852007
\(719\) −4.68185e15 −0.908675 −0.454338 0.890830i \(-0.650124\pi\)
−0.454338 + 0.890830i \(0.650124\pi\)
\(720\) 5.76401e14 0.111019
\(721\) −2.04960e15 −0.391764
\(722\) 2.98649e15 0.566508
\(723\) 2.84423e15 0.535431
\(724\) 1.96029e15 0.366233
\(725\) −2.94812e15 −0.546621
\(726\) −3.01271e15 −0.554377
\(727\) −6.06976e15 −1.10849 −0.554246 0.832353i \(-0.686993\pi\)
−0.554246 + 0.832353i \(0.686993\pi\)
\(728\) 1.89492e15 0.343454
\(729\) 2.05891e14 0.0370370
\(730\) −7.66443e14 −0.136837
\(731\) 5.26431e13 0.00932816
\(732\) −3.34755e15 −0.588730
\(733\) 6.27486e15 1.09530 0.547649 0.836708i \(-0.315523\pi\)
0.547649 + 0.836708i \(0.315523\pi\)
\(734\) −7.17105e14 −0.124238
\(735\) 2.09357e15 0.360004
\(736\) −1.77002e15 −0.302100
\(737\) 1.15261e16 1.95258
\(738\) 3.21539e14 0.0540659
\(739\) 5.07444e15 0.846923 0.423461 0.905914i \(-0.360815\pi\)
0.423461 + 0.905914i \(0.360815\pi\)
\(740\) −5.96668e15 −0.988459
\(741\) −4.35564e15 −0.716229
\(742\) −1.95691e15 −0.319411
\(743\) 5.18914e15 0.840731 0.420365 0.907355i \(-0.361902\pi\)
0.420365 + 0.907355i \(0.361902\pi\)
\(744\) 4.25437e15 0.684202
\(745\) −2.82320e15 −0.450694
\(746\) 6.82111e14 0.108091
\(747\) −4.50212e14 −0.0708197
\(748\) 1.49387e16 2.33268
\(749\) −1.66611e14 −0.0258258
\(750\) 1.89310e15 0.291297
\(751\) 1.07342e16 1.63964 0.819822 0.572619i \(-0.194072\pi\)
0.819822 + 0.572619i \(0.194072\pi\)
\(752\) 1.54347e15 0.234046
\(753\) −3.47843e15 −0.523613
\(754\) −4.53432e15 −0.677594
\(755\) 6.79002e15 1.00731
\(756\) 5.11525e14 0.0753350
\(757\) −3.18630e15 −0.465865 −0.232932 0.972493i \(-0.574832\pi\)
−0.232932 + 0.972493i \(0.574832\pi\)
\(758\) −2.36929e15 −0.343904
\(759\) 2.09247e15 0.301530
\(760\) 7.15622e15 1.02379
\(761\) −5.24754e15 −0.745315 −0.372657 0.927969i \(-0.621553\pi\)
−0.372657 + 0.927969i \(0.621553\pi\)
\(762\) −2.29875e15 −0.324146
\(763\) −4.15509e15 −0.581696
\(764\) 4.20331e15 0.584221
\(765\) −3.39907e15 −0.469054
\(766\) −2.55323e15 −0.349810
\(767\) −7.96529e14 −0.108350
\(768\) 2.23955e15 0.302465
\(769\) −2.97443e15 −0.398850 −0.199425 0.979913i \(-0.563907\pi\)
−0.199425 + 0.979913i \(0.563907\pi\)
\(770\) 2.52540e15 0.336226
\(771\) 8.31570e15 1.09926
\(772\) 1.31143e15 0.172127
\(773\) 4.08767e15 0.532707 0.266353 0.963875i \(-0.414181\pi\)
0.266353 + 0.963875i \(0.414181\pi\)
\(774\) −6.57065e12 −0.000850223 0
\(775\) −3.46918e15 −0.445725
\(776\) −1.13806e16 −1.45187
\(777\) −3.44817e15 −0.436790
\(778\) 3.51861e15 0.442571
\(779\) −4.17523e15 −0.521464
\(780\) −2.52421e15 −0.313044
\(781\) −3.95699e15 −0.487287
\(782\) 1.91597e15 0.234289
\(783\) −2.78320e15 −0.337951
\(784\) −2.50081e15 −0.301538
\(785\) −5.90879e14 −0.0707482
\(786\) 2.07210e15 0.246370
\(787\) −1.59650e16 −1.88498 −0.942491 0.334233i \(-0.891523\pi\)
−0.942491 + 0.334233i \(0.891523\pi\)
\(788\) 8.74954e15 1.02587
\(789\) −3.02655e15 −0.352390
\(790\) −6.35279e15 −0.734540
\(791\) −1.38028e15 −0.158488
\(792\) −4.23972e15 −0.483446
\(793\) 9.54644e15 1.08103
\(794\) −3.90914e15 −0.439611
\(795\) 5.92737e15 0.661977
\(796\) 4.10235e15 0.454999
\(797\) −1.32433e15 −0.145873 −0.0729365 0.997337i \(-0.523237\pi\)
−0.0729365 + 0.997337i \(0.523237\pi\)
\(798\) 1.81879e15 0.198960
\(799\) −9.10195e15 −0.988843
\(800\) −2.92447e15 −0.315538
\(801\) −5.38011e15 −0.576517
\(802\) −2.31758e15 −0.246647
\(803\) −5.89630e15 −0.623224
\(804\) 4.81068e15 0.505008
\(805\) −1.18287e15 −0.123327
\(806\) −5.33572e15 −0.552523
\(807\) 1.03555e15 0.106505
\(808\) −5.19219e15 −0.530381
\(809\) 1.26213e16 1.28052 0.640262 0.768157i \(-0.278826\pi\)
0.640262 + 0.768157i \(0.278826\pi\)
\(810\) 4.24255e14 0.0427523
\(811\) −5.72424e15 −0.572932 −0.286466 0.958090i \(-0.592481\pi\)
−0.286466 + 0.958090i \(0.592481\pi\)
\(812\) −6.91469e15 −0.687408
\(813\) 3.71301e13 0.00366630
\(814\) 1.25690e16 1.23273
\(815\) −2.07619e15 −0.202256
\(816\) 4.06025e15 0.392877
\(817\) 8.53208e13 0.00820037
\(818\) 7.22508e15 0.689762
\(819\) −1.45875e15 −0.138331
\(820\) −2.41966e15 −0.227917
\(821\) 4.67650e15 0.437556 0.218778 0.975775i \(-0.429793\pi\)
0.218778 + 0.975775i \(0.429793\pi\)
\(822\) 4.70334e15 0.437131
\(823\) 1.27239e16 1.17468 0.587340 0.809340i \(-0.300175\pi\)
0.587340 + 0.809340i \(0.300175\pi\)
\(824\) 7.09029e15 0.650226
\(825\) 3.45723e15 0.314942
\(826\) 3.32608e14 0.0300983
\(827\) −1.69863e16 −1.52693 −0.763463 0.645851i \(-0.776503\pi\)
−0.763463 + 0.645851i \(0.776503\pi\)
\(828\) 8.73344e14 0.0779863
\(829\) 1.04792e16 0.929564 0.464782 0.885425i \(-0.346133\pi\)
0.464782 + 0.885425i \(0.346133\pi\)
\(830\) −9.27698e14 −0.0817480
\(831\) 7.53904e15 0.659949
\(832\) −6.57100e14 −0.0571417
\(833\) 1.47474e16 1.27400
\(834\) −8.18656e14 −0.0702568
\(835\) −2.14314e15 −0.182716
\(836\) 2.42118e16 2.05065
\(837\) −3.27510e15 −0.275572
\(838\) −1.94062e15 −0.162218
\(839\) −1.14027e15 −0.0946927 −0.0473464 0.998879i \(-0.515076\pi\)
−0.0473464 + 0.998879i \(0.515076\pi\)
\(840\) 2.39669e15 0.197732
\(841\) 2.54222e16 2.08370
\(842\) 3.09811e15 0.252279
\(843\) 1.22949e16 0.994656
\(844\) −7.68283e15 −0.617501
\(845\) −3.19435e15 −0.255077
\(846\) 1.13606e15 0.0901289
\(847\) 1.31018e16 1.03270
\(848\) −7.08035e15 −0.554469
\(849\) 8.33274e15 0.648329
\(850\) 3.16560e15 0.244710
\(851\) −5.88718e15 −0.452163
\(852\) −1.65155e15 −0.126030
\(853\) −1.95747e16 −1.48414 −0.742072 0.670320i \(-0.766157\pi\)
−0.742072 + 0.670320i \(0.766157\pi\)
\(854\) −3.98632e15 −0.300299
\(855\) −5.50901e15 −0.412344
\(856\) 5.76367e14 0.0428641
\(857\) 1.25651e16 0.928478 0.464239 0.885710i \(-0.346328\pi\)
0.464239 + 0.885710i \(0.346328\pi\)
\(858\) 5.31734e15 0.390404
\(859\) −1.22364e16 −0.892674 −0.446337 0.894865i \(-0.647272\pi\)
−0.446337 + 0.894865i \(0.647272\pi\)
\(860\) 4.94456e13 0.00358415
\(861\) −1.39833e15 −0.100714
\(862\) 7.84217e15 0.561237
\(863\) −1.15780e16 −0.823330 −0.411665 0.911335i \(-0.635053\pi\)
−0.411665 + 0.911335i \(0.635053\pi\)
\(864\) −2.76087e15 −0.195084
\(865\) −3.67040e15 −0.257706
\(866\) 6.80083e14 0.0474475
\(867\) −1.56155e16 −1.08256
\(868\) −8.13681e15 −0.560526
\(869\) −4.88725e16 −3.34546
\(870\) −5.73500e15 −0.390101
\(871\) −1.37189e16 −0.927301
\(872\) 1.43740e16 0.965464
\(873\) 8.76104e15 0.584759
\(874\) 3.10528e15 0.205963
\(875\) −8.23282e15 −0.542631
\(876\) −2.46096e15 −0.161188
\(877\) 9.00687e14 0.0586241 0.0293121 0.999570i \(-0.490668\pi\)
0.0293121 + 0.999570i \(0.490668\pi\)
\(878\) −4.17202e15 −0.269852
\(879\) 2.36901e15 0.152275
\(880\) 9.13721e15 0.583658
\(881\) −2.87968e16 −1.82800 −0.914001 0.405712i \(-0.867024\pi\)
−0.914001 + 0.405712i \(0.867024\pi\)
\(882\) −1.84070e15 −0.116120
\(883\) −1.92900e16 −1.20934 −0.604671 0.796475i \(-0.706695\pi\)
−0.604671 + 0.796475i \(0.706695\pi\)
\(884\) −1.77809e16 −1.10781
\(885\) −1.00745e15 −0.0623785
\(886\) −9.39146e15 −0.577893
\(887\) 1.91234e16 1.16946 0.584730 0.811228i \(-0.301201\pi\)
0.584730 + 0.811228i \(0.301201\pi\)
\(888\) 1.19285e16 0.724959
\(889\) 9.99696e15 0.603821
\(890\) −1.10861e16 −0.665480
\(891\) 3.26382e15 0.194715
\(892\) −1.56762e16 −0.929465
\(893\) −1.47519e16 −0.869290
\(894\) 2.48220e15 0.145372
\(895\) 4.03612e15 0.234929
\(896\) −8.46305e15 −0.489591
\(897\) −2.49058e15 −0.143199
\(898\) 2.28746e15 0.130717
\(899\) 4.42722e16 2.51451
\(900\) 1.44296e15 0.0814554
\(901\) 4.17533e16 2.34263
\(902\) 5.09710e15 0.284241
\(903\) 2.85748e13 0.00158380
\(904\) 4.77489e15 0.263049
\(905\) −7.07059e15 −0.387158
\(906\) −5.96988e15 −0.324908
\(907\) −2.57494e16 −1.39292 −0.696461 0.717595i \(-0.745243\pi\)
−0.696461 + 0.717595i \(0.745243\pi\)
\(908\) 2.25393e16 1.21190
\(909\) 3.99705e15 0.213618
\(910\) −3.00587e15 −0.159677
\(911\) 1.39563e16 0.736917 0.368459 0.929644i \(-0.379886\pi\)
0.368459 + 0.929644i \(0.379886\pi\)
\(912\) 6.58061e15 0.345378
\(913\) −7.13685e15 −0.372321
\(914\) −1.22742e16 −0.636487
\(915\) 1.20743e16 0.622367
\(916\) 1.48343e16 0.760050
\(917\) −9.01129e15 −0.458940
\(918\) 2.98851e15 0.151294
\(919\) −3.54732e16 −1.78511 −0.892556 0.450936i \(-0.851090\pi\)
−0.892556 + 0.450936i \(0.851090\pi\)
\(920\) 4.09196e15 0.204691
\(921\) −7.56013e15 −0.375925
\(922\) 1.22049e16 0.603276
\(923\) 4.70983e15 0.231417
\(924\) 8.10878e15 0.396059
\(925\) −9.72692e15 −0.472276
\(926\) −9.96823e15 −0.481125
\(927\) −5.45826e15 −0.261888
\(928\) 3.73209e16 1.78008
\(929\) 1.56899e16 0.743934 0.371967 0.928246i \(-0.378683\pi\)
0.371967 + 0.928246i \(0.378683\pi\)
\(930\) −6.74861e15 −0.318096
\(931\) 2.39017e16 1.11997
\(932\) −6.84354e15 −0.318782
\(933\) 1.63362e16 0.756487
\(934\) −1.50426e16 −0.692495
\(935\) −5.38827e16 −2.46596
\(936\) 5.04635e15 0.229594
\(937\) −2.75211e16 −1.24480 −0.622399 0.782700i \(-0.713842\pi\)
−0.622399 + 0.782700i \(0.713842\pi\)
\(938\) 5.72863e15 0.257594
\(939\) −3.92375e15 −0.175404
\(940\) −8.54911e15 −0.379942
\(941\) 1.84209e16 0.813893 0.406946 0.913452i \(-0.366594\pi\)
0.406946 + 0.913452i \(0.366594\pi\)
\(942\) 5.19510e14 0.0228199
\(943\) −2.38741e15 −0.104259
\(944\) 1.20341e15 0.0522480
\(945\) −1.84503e15 −0.0796393
\(946\) −1.04159e14 −0.00446988
\(947\) −1.07792e16 −0.459899 −0.229950 0.973203i \(-0.573856\pi\)
−0.229950 + 0.973203i \(0.573856\pi\)
\(948\) −2.03981e16 −0.865255
\(949\) 7.01811e15 0.295975
\(950\) 5.13062e15 0.215124
\(951\) −1.59899e16 −0.666581
\(952\) 1.68827e16 0.699741
\(953\) −1.89737e16 −0.781881 −0.390940 0.920416i \(-0.627850\pi\)
−0.390940 + 0.920416i \(0.627850\pi\)
\(954\) −5.21143e15 −0.213521
\(955\) −1.51610e16 −0.617601
\(956\) 6.27464e15 0.254138
\(957\) −4.41197e16 −1.77671
\(958\) 1.65181e16 0.661380
\(959\) −2.04542e16 −0.814291
\(960\) −8.31099e14 −0.0328974
\(961\) 2.66885e16 1.05038
\(962\) −1.49604e16 −0.585436
\(963\) −4.43700e14 −0.0172641
\(964\) 1.88182e16 0.728040
\(965\) −4.73021e15 −0.181962
\(966\) 1.03999e15 0.0397792
\(967\) 3.78791e16 1.44064 0.720318 0.693644i \(-0.243996\pi\)
0.720318 + 0.693644i \(0.243996\pi\)
\(968\) −4.53240e16 −1.71401
\(969\) −3.88063e16 −1.45922
\(970\) 1.80528e16 0.674995
\(971\) 4.82137e16 1.79252 0.896262 0.443525i \(-0.146272\pi\)
0.896262 + 0.443525i \(0.146272\pi\)
\(972\) 1.36224e15 0.0503602
\(973\) 3.56022e15 0.130875
\(974\) 6.31266e15 0.230748
\(975\) −4.11498e15 −0.149569
\(976\) −1.44230e16 −0.521292
\(977\) 1.43458e16 0.515590 0.257795 0.966200i \(-0.417004\pi\)
0.257795 + 0.966200i \(0.417004\pi\)
\(978\) 1.82542e15 0.0652377
\(979\) −8.52864e16 −3.03092
\(980\) 1.38517e16 0.489507
\(981\) −1.10654e16 −0.388854
\(982\) 2.21805e15 0.0775100
\(983\) −1.40114e16 −0.486898 −0.243449 0.969914i \(-0.578279\pi\)
−0.243449 + 0.969914i \(0.578279\pi\)
\(984\) 4.83732e15 0.167160
\(985\) −3.15588e16 −1.08448
\(986\) −4.03982e16 −1.38051
\(987\) −4.94056e15 −0.167893
\(988\) −2.88182e16 −0.973876
\(989\) 4.87868e13 0.00163954
\(990\) 6.72537e15 0.224762
\(991\) −2.49514e16 −0.829259 −0.414630 0.909990i \(-0.636089\pi\)
−0.414630 + 0.909990i \(0.636089\pi\)
\(992\) 4.39171e16 1.45151
\(993\) 7.08787e15 0.232967
\(994\) −1.96669e15 −0.0642851
\(995\) −1.47968e16 −0.480996
\(996\) −2.97874e15 −0.0962954
\(997\) −1.67167e16 −0.537438 −0.268719 0.963219i \(-0.586600\pi\)
−0.268719 + 0.963219i \(0.586600\pi\)
\(998\) −2.67825e16 −0.856313
\(999\) −9.18278e15 −0.291987
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 177.12.a.a.1.17 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.a.1.17 26 1.1 even 1 trivial