Properties

Label 177.12.a.b.1.7
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
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: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-54.1540 q^{2} +243.000 q^{3} +884.652 q^{4} -9497.19 q^{5} -13159.4 q^{6} -21091.6 q^{7} +62999.9 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-54.1540 q^{2} +243.000 q^{3} +884.652 q^{4} -9497.19 q^{5} -13159.4 q^{6} -21091.6 q^{7} +62999.9 q^{8} +59049.0 q^{9} +514310. q^{10} -843837. q^{11} +214970. q^{12} +2.09397e6 q^{13} +1.14219e6 q^{14} -2.30782e6 q^{15} -5.22346e6 q^{16} -5.90682e6 q^{17} -3.19774e6 q^{18} +4.14828e6 q^{19} -8.40171e6 q^{20} -5.12525e6 q^{21} +4.56971e7 q^{22} -4.05086e7 q^{23} +1.53090e7 q^{24} +4.13685e7 q^{25} -1.13397e8 q^{26} +1.43489e7 q^{27} -1.86587e7 q^{28} +7.14544e7 q^{29} +1.24977e8 q^{30} +2.22524e8 q^{31} +1.53847e8 q^{32} -2.05052e8 q^{33} +3.19878e8 q^{34} +2.00310e8 q^{35} +5.22378e7 q^{36} +5.50364e8 q^{37} -2.24646e8 q^{38} +5.08836e8 q^{39} -5.98322e8 q^{40} -3.58060e8 q^{41} +2.77552e8 q^{42} -8.90997e8 q^{43} -7.46502e8 q^{44} -5.60799e8 q^{45} +2.19370e9 q^{46} +2.37132e9 q^{47} -1.26930e9 q^{48} -1.53247e9 q^{49} -2.24027e9 q^{50} -1.43536e9 q^{51} +1.85244e9 q^{52} +2.90109e9 q^{53} -7.77050e8 q^{54} +8.01408e9 q^{55} -1.32877e9 q^{56} +1.00803e9 q^{57} -3.86954e9 q^{58} -7.14924e8 q^{59} -2.04161e9 q^{60} -7.97897e9 q^{61} -1.20505e10 q^{62} -1.24544e9 q^{63} +2.36620e9 q^{64} -1.98869e10 q^{65} +1.11044e10 q^{66} +9.55219e9 q^{67} -5.22548e9 q^{68} -9.84360e9 q^{69} -1.08476e10 q^{70} -1.49378e9 q^{71} +3.72008e9 q^{72} -4.31992e8 q^{73} -2.98044e10 q^{74} +1.00525e10 q^{75} +3.66979e9 q^{76} +1.77978e10 q^{77} -2.75555e10 q^{78} +4.22383e10 q^{79} +4.96082e10 q^{80} +3.48678e9 q^{81} +1.93904e10 q^{82} +6.44130e9 q^{83} -4.53406e9 q^{84} +5.60982e10 q^{85} +4.82510e10 q^{86} +1.73634e10 q^{87} -5.31617e10 q^{88} +1.02190e11 q^{89} +3.03695e10 q^{90} -4.41651e10 q^{91} -3.58360e10 q^{92} +5.40732e10 q^{93} -1.28416e11 q^{94} -3.93970e10 q^{95} +3.73849e10 q^{96} +1.09002e11 q^{97} +8.29895e10 q^{98} -4.98278e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −54.1540 −1.19664 −0.598322 0.801255i \(-0.704166\pi\)
−0.598322 + 0.801255i \(0.704166\pi\)
\(3\) 243.000 0.577350
\(4\) 884.652 0.431959
\(5\) −9497.19 −1.35913 −0.679563 0.733617i \(-0.737831\pi\)
−0.679563 + 0.733617i \(0.737831\pi\)
\(6\) −13159.4 −0.690883
\(7\) −21091.6 −0.474318 −0.237159 0.971471i \(-0.576216\pi\)
−0.237159 + 0.971471i \(0.576216\pi\)
\(8\) 62999.9 0.679743
\(9\) 59049.0 0.333333
\(10\) 514310. 1.62639
\(11\) −843837. −1.57979 −0.789894 0.613243i \(-0.789865\pi\)
−0.789894 + 0.613243i \(0.789865\pi\)
\(12\) 214970. 0.249392
\(13\) 2.09397e6 1.56417 0.782083 0.623175i \(-0.214157\pi\)
0.782083 + 0.623175i \(0.214157\pi\)
\(14\) 1.14219e6 0.567590
\(15\) −2.30782e6 −0.784692
\(16\) −5.22346e6 −1.24537
\(17\) −5.90682e6 −1.00899 −0.504493 0.863416i \(-0.668320\pi\)
−0.504493 + 0.863416i \(0.668320\pi\)
\(18\) −3.19774e6 −0.398882
\(19\) 4.14828e6 0.384347 0.192173 0.981361i \(-0.438446\pi\)
0.192173 + 0.981361i \(0.438446\pi\)
\(20\) −8.40171e6 −0.587087
\(21\) −5.12525e6 −0.273847
\(22\) 4.56971e7 1.89045
\(23\) −4.05086e7 −1.31233 −0.656167 0.754616i \(-0.727823\pi\)
−0.656167 + 0.754616i \(0.727823\pi\)
\(24\) 1.53090e7 0.392450
\(25\) 4.13685e7 0.847226
\(26\) −1.13397e8 −1.87175
\(27\) 1.43489e7 0.192450
\(28\) −1.86587e7 −0.204886
\(29\) 7.14544e7 0.646904 0.323452 0.946245i \(-0.395157\pi\)
0.323452 + 0.946245i \(0.395157\pi\)
\(30\) 1.24977e8 0.938998
\(31\) 2.22524e8 1.39600 0.698002 0.716096i \(-0.254073\pi\)
0.698002 + 0.716096i \(0.254073\pi\)
\(32\) 1.53847e8 0.810523
\(33\) −2.05052e8 −0.912091
\(34\) 3.19878e8 1.20740
\(35\) 2.00310e8 0.644658
\(36\) 5.22378e7 0.143986
\(37\) 5.50364e8 1.30479 0.652395 0.757879i \(-0.273764\pi\)
0.652395 + 0.757879i \(0.273764\pi\)
\(38\) −2.24646e8 −0.459927
\(39\) 5.08836e8 0.903071
\(40\) −5.98322e8 −0.923858
\(41\) −3.58060e8 −0.482664 −0.241332 0.970443i \(-0.577584\pi\)
−0.241332 + 0.970443i \(0.577584\pi\)
\(42\) 2.77552e8 0.327698
\(43\) −8.90997e8 −0.924272 −0.462136 0.886809i \(-0.652917\pi\)
−0.462136 + 0.886809i \(0.652917\pi\)
\(44\) −7.46502e8 −0.682404
\(45\) −5.60799e8 −0.453042
\(46\) 2.19370e9 1.57040
\(47\) 2.37132e9 1.50818 0.754088 0.656774i \(-0.228079\pi\)
0.754088 + 0.656774i \(0.228079\pi\)
\(48\) −1.26930e9 −0.719015
\(49\) −1.53247e9 −0.775023
\(50\) −2.24027e9 −1.01383
\(51\) −1.43536e9 −0.582538
\(52\) 1.85244e9 0.675655
\(53\) 2.90109e9 0.952891 0.476446 0.879204i \(-0.341925\pi\)
0.476446 + 0.879204i \(0.341925\pi\)
\(54\) −7.77050e8 −0.230294
\(55\) 8.01408e9 2.14713
\(56\) −1.32877e9 −0.322414
\(57\) 1.00803e9 0.221903
\(58\) −3.86954e9 −0.774114
\(59\) −7.14924e8 −0.130189
\(60\) −2.04161e9 −0.338955
\(61\) −7.97897e9 −1.20958 −0.604788 0.796387i \(-0.706742\pi\)
−0.604788 + 0.796387i \(0.706742\pi\)
\(62\) −1.20505e10 −1.67052
\(63\) −1.24544e9 −0.158106
\(64\) 2.36620e9 0.275462
\(65\) −1.98869e10 −2.12590
\(66\) 1.11044e10 1.09145
\(67\) 9.55219e9 0.864354 0.432177 0.901789i \(-0.357746\pi\)
0.432177 + 0.901789i \(0.357746\pi\)
\(68\) −5.22548e9 −0.435840
\(69\) −9.84360e9 −0.757677
\(70\) −1.08476e10 −0.771427
\(71\) −1.49378e9 −0.0982573 −0.0491287 0.998792i \(-0.515644\pi\)
−0.0491287 + 0.998792i \(0.515644\pi\)
\(72\) 3.72008e9 0.226581
\(73\) −4.31992e8 −0.0243893 −0.0121947 0.999926i \(-0.503882\pi\)
−0.0121947 + 0.999926i \(0.503882\pi\)
\(74\) −2.98044e10 −1.56137
\(75\) 1.00525e10 0.489146
\(76\) 3.66979e9 0.166022
\(77\) 1.77978e10 0.749322
\(78\) −2.75555e10 −1.08066
\(79\) 4.22383e10 1.54439 0.772196 0.635385i \(-0.219159\pi\)
0.772196 + 0.635385i \(0.219159\pi\)
\(80\) 4.96082e10 1.69262
\(81\) 3.48678e9 0.111111
\(82\) 1.93904e10 0.577577
\(83\) 6.44130e9 0.179492 0.0897458 0.995965i \(-0.471395\pi\)
0.0897458 + 0.995965i \(0.471395\pi\)
\(84\) −4.53406e9 −0.118291
\(85\) 5.60982e10 1.37134
\(86\) 4.82510e10 1.10603
\(87\) 1.73634e10 0.373490
\(88\) −5.31617e10 −1.07385
\(89\) 1.02190e11 1.93983 0.969913 0.243451i \(-0.0782795\pi\)
0.969913 + 0.243451i \(0.0782795\pi\)
\(90\) 3.03695e10 0.542131
\(91\) −4.41651e10 −0.741911
\(92\) −3.58360e10 −0.566875
\(93\) 5.40732e10 0.805983
\(94\) −1.28416e11 −1.80475
\(95\) −3.93970e10 −0.522376
\(96\) 3.73849e10 0.467956
\(97\) 1.09002e11 1.28881 0.644407 0.764683i \(-0.277104\pi\)
0.644407 + 0.764683i \(0.277104\pi\)
\(98\) 8.29895e10 0.927427
\(99\) −4.98278e10 −0.526596
\(100\) 3.65967e10 0.365967
\(101\) 1.45203e11 1.37470 0.687350 0.726326i \(-0.258774\pi\)
0.687350 + 0.726326i \(0.258774\pi\)
\(102\) 7.77303e10 0.697091
\(103\) −2.78583e10 −0.236783 −0.118391 0.992967i \(-0.537774\pi\)
−0.118391 + 0.992967i \(0.537774\pi\)
\(104\) 1.31920e11 1.06323
\(105\) 4.86754e10 0.372193
\(106\) −1.57105e11 −1.14027
\(107\) 2.33740e10 0.161110 0.0805549 0.996750i \(-0.474331\pi\)
0.0805549 + 0.996750i \(0.474331\pi\)
\(108\) 1.26938e10 0.0831305
\(109\) −2.25343e11 −1.40281 −0.701404 0.712764i \(-0.747443\pi\)
−0.701404 + 0.712764i \(0.747443\pi\)
\(110\) −4.33994e11 −2.56936
\(111\) 1.33739e11 0.753321
\(112\) 1.10171e11 0.590701
\(113\) −3.49445e11 −1.78422 −0.892109 0.451820i \(-0.850775\pi\)
−0.892109 + 0.451820i \(0.850775\pi\)
\(114\) −5.45889e10 −0.265539
\(115\) 3.84718e11 1.78363
\(116\) 6.32123e10 0.279436
\(117\) 1.23647e11 0.521389
\(118\) 3.87160e10 0.155790
\(119\) 1.24584e11 0.478580
\(120\) −1.45392e11 −0.533389
\(121\) 4.26750e11 1.49573
\(122\) 4.32093e11 1.44743
\(123\) −8.70086e10 −0.278666
\(124\) 1.96856e11 0.603016
\(125\) 7.08458e10 0.207639
\(126\) 6.74452e10 0.189197
\(127\) 3.87569e11 1.04095 0.520473 0.853878i \(-0.325756\pi\)
0.520473 + 0.853878i \(0.325756\pi\)
\(128\) −4.43219e11 −1.14015
\(129\) −2.16512e11 −0.533629
\(130\) 1.07695e12 2.54395
\(131\) −6.16968e10 −0.139724 −0.0698619 0.997557i \(-0.522256\pi\)
−0.0698619 + 0.997557i \(0.522256\pi\)
\(132\) −1.81400e11 −0.393986
\(133\) −8.74937e10 −0.182303
\(134\) −5.17289e11 −1.03432
\(135\) −1.36274e11 −0.261564
\(136\) −3.72129e11 −0.685851
\(137\) −8.92149e11 −1.57934 −0.789668 0.613534i \(-0.789747\pi\)
−0.789668 + 0.613534i \(0.789747\pi\)
\(138\) 5.33070e11 0.906670
\(139\) −7.61455e11 −1.24469 −0.622347 0.782741i \(-0.713821\pi\)
−0.622347 + 0.782741i \(0.713821\pi\)
\(140\) 1.77205e11 0.278466
\(141\) 5.76231e11 0.870745
\(142\) 8.08940e10 0.117579
\(143\) −1.76697e12 −2.47105
\(144\) −3.08440e11 −0.415123
\(145\) −6.78616e11 −0.879225
\(146\) 2.33941e10 0.0291854
\(147\) −3.72391e11 −0.447460
\(148\) 4.86881e11 0.563616
\(149\) −1.13242e12 −1.26323 −0.631614 0.775283i \(-0.717607\pi\)
−0.631614 + 0.775283i \(0.717607\pi\)
\(150\) −5.44385e11 −0.585334
\(151\) 2.38730e11 0.247477 0.123738 0.992315i \(-0.460512\pi\)
0.123738 + 0.992315i \(0.460512\pi\)
\(152\) 2.61341e11 0.261257
\(153\) −3.48792e11 −0.336328
\(154\) −9.63824e11 −0.896672
\(155\) −2.11335e12 −1.89735
\(156\) 4.50142e11 0.390090
\(157\) 4.59169e11 0.384171 0.192086 0.981378i \(-0.438475\pi\)
0.192086 + 0.981378i \(0.438475\pi\)
\(158\) −2.28737e12 −1.84809
\(159\) 7.04964e11 0.550152
\(160\) −1.46112e12 −1.10160
\(161\) 8.54390e11 0.622463
\(162\) −1.88823e11 −0.132961
\(163\) 8.33481e11 0.567367 0.283683 0.958918i \(-0.408444\pi\)
0.283683 + 0.958918i \(0.408444\pi\)
\(164\) −3.16759e11 −0.208491
\(165\) 1.94742e12 1.23965
\(166\) −3.48822e11 −0.214788
\(167\) −2.27175e12 −1.35338 −0.676689 0.736269i \(-0.736586\pi\)
−0.676689 + 0.736269i \(0.736586\pi\)
\(168\) −3.22890e11 −0.186146
\(169\) 2.59256e12 1.44661
\(170\) −3.03794e12 −1.64101
\(171\) 2.44952e11 0.128116
\(172\) −7.88223e11 −0.399248
\(173\) 3.69756e12 1.81410 0.907050 0.421022i \(-0.138329\pi\)
0.907050 + 0.421022i \(0.138329\pi\)
\(174\) −9.40298e11 −0.446935
\(175\) −8.72525e11 −0.401854
\(176\) 4.40775e12 1.96742
\(177\) −1.73727e11 −0.0751646
\(178\) −5.53399e12 −2.32128
\(179\) −3.41963e12 −1.39087 −0.695436 0.718588i \(-0.744789\pi\)
−0.695436 + 0.718588i \(0.744789\pi\)
\(180\) −4.96112e11 −0.195696
\(181\) 8.98229e11 0.343680 0.171840 0.985125i \(-0.445029\pi\)
0.171840 + 0.985125i \(0.445029\pi\)
\(182\) 2.39172e12 0.887805
\(183\) −1.93889e12 −0.698349
\(184\) −2.55204e12 −0.892051
\(185\) −5.22692e12 −1.77338
\(186\) −2.92828e12 −0.964476
\(187\) 4.98440e12 1.59398
\(188\) 2.09779e12 0.651470
\(189\) −3.02641e11 −0.0912825
\(190\) 2.13350e12 0.625099
\(191\) 4.05225e12 1.15349 0.576743 0.816926i \(-0.304323\pi\)
0.576743 + 0.816926i \(0.304323\pi\)
\(192\) 5.74988e11 0.159038
\(193\) −3.96024e12 −1.06453 −0.532264 0.846579i \(-0.678659\pi\)
−0.532264 + 0.846579i \(0.678659\pi\)
\(194\) −5.90289e12 −1.54225
\(195\) −4.83251e12 −1.22739
\(196\) −1.35571e12 −0.334778
\(197\) −4.64358e12 −1.11504 −0.557518 0.830165i \(-0.688246\pi\)
−0.557518 + 0.830165i \(0.688246\pi\)
\(198\) 2.69837e12 0.630149
\(199\) −4.70935e12 −1.06972 −0.534859 0.844942i \(-0.679635\pi\)
−0.534859 + 0.844942i \(0.679635\pi\)
\(200\) 2.60621e12 0.575896
\(201\) 2.32118e12 0.499035
\(202\) −7.86332e12 −1.64503
\(203\) −1.50708e12 −0.306838
\(204\) −1.26979e12 −0.251632
\(205\) 3.40057e12 0.656002
\(206\) 1.50864e12 0.283345
\(207\) −2.39199e12 −0.437445
\(208\) −1.09378e13 −1.94797
\(209\) −3.50047e12 −0.607187
\(210\) −2.63597e12 −0.445383
\(211\) −1.17839e13 −1.93971 −0.969853 0.243692i \(-0.921642\pi\)
−0.969853 + 0.243692i \(0.921642\pi\)
\(212\) 2.56645e12 0.411610
\(213\) −3.62988e11 −0.0567289
\(214\) −1.26579e12 −0.192791
\(215\) 8.46197e12 1.25620
\(216\) 9.03980e11 0.130817
\(217\) −4.69337e12 −0.662149
\(218\) 1.22032e13 1.67866
\(219\) −1.04974e11 −0.0140812
\(220\) 7.08967e12 0.927474
\(221\) −1.23687e13 −1.57822
\(222\) −7.24247e12 −0.901458
\(223\) 8.78662e12 1.06695 0.533476 0.845815i \(-0.320885\pi\)
0.533476 + 0.845815i \(0.320885\pi\)
\(224\) −3.24488e12 −0.384445
\(225\) 2.44277e12 0.282409
\(226\) 1.89239e13 2.13508
\(227\) −9.68408e11 −0.106639 −0.0533195 0.998578i \(-0.516980\pi\)
−0.0533195 + 0.998578i \(0.516980\pi\)
\(228\) 8.91758e11 0.0958529
\(229\) 8.09890e12 0.849827 0.424914 0.905234i \(-0.360304\pi\)
0.424914 + 0.905234i \(0.360304\pi\)
\(230\) −2.08340e13 −2.13437
\(231\) 4.32488e12 0.432621
\(232\) 4.50162e12 0.439729
\(233\) 1.38783e13 1.32397 0.661986 0.749516i \(-0.269714\pi\)
0.661986 + 0.749516i \(0.269714\pi\)
\(234\) −6.69598e12 −0.623917
\(235\) −2.25209e13 −2.04980
\(236\) −6.32459e11 −0.0562363
\(237\) 1.02639e13 0.891655
\(238\) −6.74672e12 −0.572690
\(239\) −1.24484e13 −1.03258 −0.516290 0.856414i \(-0.672687\pi\)
−0.516290 + 0.856414i \(0.672687\pi\)
\(240\) 1.20548e13 0.977233
\(241\) −1.18378e13 −0.937943 −0.468972 0.883213i \(-0.655375\pi\)
−0.468972 + 0.883213i \(0.655375\pi\)
\(242\) −2.31102e13 −1.78986
\(243\) 8.47289e11 0.0641500
\(244\) −7.05861e12 −0.522487
\(245\) 1.45542e13 1.05335
\(246\) 4.71186e12 0.333464
\(247\) 8.68639e12 0.601182
\(248\) 1.40190e13 0.948924
\(249\) 1.56524e12 0.103629
\(250\) −3.83658e12 −0.248470
\(251\) −2.56906e12 −0.162768 −0.0813840 0.996683i \(-0.525934\pi\)
−0.0813840 + 0.996683i \(0.525934\pi\)
\(252\) −1.10178e12 −0.0682953
\(253\) 3.41827e13 2.07321
\(254\) −2.09884e13 −1.24564
\(255\) 1.36319e13 0.791743
\(256\) 1.91561e13 1.08890
\(257\) −7.66562e10 −0.00426496 −0.00213248 0.999998i \(-0.500679\pi\)
−0.00213248 + 0.999998i \(0.500679\pi\)
\(258\) 1.17250e13 0.638564
\(259\) −1.16080e13 −0.618885
\(260\) −1.75930e13 −0.918302
\(261\) 4.21931e12 0.215635
\(262\) 3.34113e12 0.167200
\(263\) −2.94684e13 −1.44411 −0.722054 0.691837i \(-0.756802\pi\)
−0.722054 + 0.691837i \(0.756802\pi\)
\(264\) −1.29183e13 −0.619988
\(265\) −2.75522e13 −1.29510
\(266\) 4.73813e12 0.218151
\(267\) 2.48321e13 1.11996
\(268\) 8.45037e12 0.373366
\(269\) 6.96896e12 0.301669 0.150834 0.988559i \(-0.451804\pi\)
0.150834 + 0.988559i \(0.451804\pi\)
\(270\) 7.37979e12 0.312999
\(271\) −4.26561e13 −1.77276 −0.886381 0.462956i \(-0.846789\pi\)
−0.886381 + 0.462956i \(0.846789\pi\)
\(272\) 3.08541e13 1.25656
\(273\) −1.07321e13 −0.428343
\(274\) 4.83134e13 1.88990
\(275\) −3.49083e13 −1.33844
\(276\) −8.70816e12 −0.327285
\(277\) 2.39786e13 0.883455 0.441728 0.897149i \(-0.354366\pi\)
0.441728 + 0.897149i \(0.354366\pi\)
\(278\) 4.12358e13 1.48946
\(279\) 1.31398e13 0.465335
\(280\) 1.26195e13 0.438202
\(281\) −5.17816e13 −1.76316 −0.881578 0.472039i \(-0.843518\pi\)
−0.881578 + 0.472039i \(0.843518\pi\)
\(282\) −3.12052e13 −1.04197
\(283\) −7.35010e11 −0.0240695 −0.0120348 0.999928i \(-0.503831\pi\)
−0.0120348 + 0.999928i \(0.503831\pi\)
\(284\) −1.32147e12 −0.0424431
\(285\) −9.57347e12 −0.301594
\(286\) 9.56886e13 2.95697
\(287\) 7.55205e12 0.228936
\(288\) 9.08453e12 0.270174
\(289\) 6.18651e11 0.0180513
\(290\) 3.67497e13 1.05212
\(291\) 2.64875e13 0.744097
\(292\) −3.82162e11 −0.0105352
\(293\) −4.01628e13 −1.08656 −0.543278 0.839553i \(-0.682817\pi\)
−0.543278 + 0.839553i \(0.682817\pi\)
\(294\) 2.01664e13 0.535450
\(295\) 6.78977e12 0.176943
\(296\) 3.46729e13 0.886923
\(297\) −1.21081e13 −0.304030
\(298\) 6.13248e13 1.51164
\(299\) −8.48240e13 −2.05271
\(300\) 8.89300e12 0.211291
\(301\) 1.87925e13 0.438399
\(302\) −1.29282e13 −0.296142
\(303\) 3.52843e13 0.793684
\(304\) −2.16684e13 −0.478654
\(305\) 7.57778e13 1.64397
\(306\) 1.88885e13 0.402466
\(307\) −5.24893e13 −1.09852 −0.549262 0.835650i \(-0.685091\pi\)
−0.549262 + 0.835650i \(0.685091\pi\)
\(308\) 1.57449e13 0.323676
\(309\) −6.76957e12 −0.136707
\(310\) 1.14446e14 2.27045
\(311\) −2.53969e13 −0.494992 −0.247496 0.968889i \(-0.579608\pi\)
−0.247496 + 0.968889i \(0.579608\pi\)
\(312\) 3.20566e13 0.613857
\(313\) −1.99353e13 −0.375085 −0.187542 0.982256i \(-0.560052\pi\)
−0.187542 + 0.982256i \(0.560052\pi\)
\(314\) −2.48658e13 −0.459716
\(315\) 1.18281e13 0.214886
\(316\) 3.73662e13 0.667114
\(317\) 6.84447e13 1.20092 0.600460 0.799655i \(-0.294984\pi\)
0.600460 + 0.799655i \(0.294984\pi\)
\(318\) −3.81766e13 −0.658337
\(319\) −6.02959e13 −1.02197
\(320\) −2.24723e13 −0.374388
\(321\) 5.67987e12 0.0930167
\(322\) −4.62686e13 −0.744868
\(323\) −2.45032e13 −0.387800
\(324\) 3.08459e12 0.0479954
\(325\) 8.66245e13 1.32520
\(326\) −4.51363e13 −0.678937
\(327\) −5.47583e13 −0.809911
\(328\) −2.25578e13 −0.328088
\(329\) −5.00148e13 −0.715354
\(330\) −1.05461e14 −1.48342
\(331\) −9.22756e13 −1.27653 −0.638267 0.769815i \(-0.720349\pi\)
−0.638267 + 0.769815i \(0.720349\pi\)
\(332\) 5.69831e12 0.0775330
\(333\) 3.24985e13 0.434930
\(334\) 1.23024e14 1.61951
\(335\) −9.07190e13 −1.17477
\(336\) 2.67715e13 0.341042
\(337\) 5.13142e13 0.643092 0.321546 0.946894i \(-0.395797\pi\)
0.321546 + 0.946894i \(0.395797\pi\)
\(338\) −1.40398e14 −1.73108
\(339\) −8.49152e13 −1.03012
\(340\) 4.96274e13 0.592362
\(341\) −1.87774e14 −2.20539
\(342\) −1.32651e13 −0.153309
\(343\) 7.40271e13 0.841925
\(344\) −5.61327e13 −0.628268
\(345\) 9.34865e13 1.02978
\(346\) −2.00237e14 −2.17083
\(347\) −8.79861e13 −0.938862 −0.469431 0.882969i \(-0.655541\pi\)
−0.469431 + 0.882969i \(0.655541\pi\)
\(348\) 1.53606e13 0.161332
\(349\) 6.86479e13 0.709720 0.354860 0.934919i \(-0.384528\pi\)
0.354860 + 0.934919i \(0.384528\pi\)
\(350\) 4.72507e13 0.480877
\(351\) 3.00462e13 0.301024
\(352\) −1.29822e14 −1.28045
\(353\) 3.14822e13 0.305706 0.152853 0.988249i \(-0.451154\pi\)
0.152853 + 0.988249i \(0.451154\pi\)
\(354\) 9.40798e12 0.0899453
\(355\) 1.41867e13 0.133544
\(356\) 9.04025e13 0.837926
\(357\) 3.02739e13 0.276308
\(358\) 1.85186e14 1.66438
\(359\) 1.59050e14 1.40772 0.703858 0.710340i \(-0.251459\pi\)
0.703858 + 0.710340i \(0.251459\pi\)
\(360\) −3.53303e13 −0.307953
\(361\) −9.92820e13 −0.852277
\(362\) −4.86426e13 −0.411263
\(363\) 1.03700e14 0.863561
\(364\) −3.90708e13 −0.320475
\(365\) 4.10271e12 0.0331482
\(366\) 1.04999e14 0.835675
\(367\) −2.18228e14 −1.71099 −0.855495 0.517811i \(-0.826747\pi\)
−0.855495 + 0.517811i \(0.826747\pi\)
\(368\) 2.11595e14 1.63434
\(369\) −2.11431e13 −0.160888
\(370\) 2.83058e14 2.12210
\(371\) −6.11884e13 −0.451973
\(372\) 4.78360e13 0.348152
\(373\) −1.49568e13 −0.107261 −0.0536303 0.998561i \(-0.517079\pi\)
−0.0536303 + 0.998561i \(0.517079\pi\)
\(374\) −2.69925e14 −1.90743
\(375\) 1.72155e13 0.119880
\(376\) 1.49393e14 1.02517
\(377\) 1.49624e14 1.01186
\(378\) 1.63892e13 0.109233
\(379\) 1.40600e14 0.923568 0.461784 0.886992i \(-0.347210\pi\)
0.461784 + 0.886992i \(0.347210\pi\)
\(380\) −3.48526e13 −0.225645
\(381\) 9.41792e13 0.600990
\(382\) −2.19445e14 −1.38031
\(383\) 1.75255e14 1.08662 0.543310 0.839532i \(-0.317171\pi\)
0.543310 + 0.839532i \(0.317171\pi\)
\(384\) −1.07702e14 −0.658268
\(385\) −1.69029e14 −1.01842
\(386\) 2.14463e14 1.27386
\(387\) −5.26125e13 −0.308091
\(388\) 9.64289e13 0.556715
\(389\) −6.98850e13 −0.397797 −0.198898 0.980020i \(-0.563736\pi\)
−0.198898 + 0.980020i \(0.563736\pi\)
\(390\) 2.61699e14 1.46875
\(391\) 2.39277e14 1.32413
\(392\) −9.65457e13 −0.526817
\(393\) −1.49923e13 −0.0806696
\(394\) 2.51468e14 1.33430
\(395\) −4.01145e14 −2.09902
\(396\) −4.40802e13 −0.227468
\(397\) 1.34492e14 0.684463 0.342231 0.939616i \(-0.388817\pi\)
0.342231 + 0.939616i \(0.388817\pi\)
\(398\) 2.55030e14 1.28007
\(399\) −2.12610e13 −0.105252
\(400\) −2.16087e14 −1.05511
\(401\) 1.81027e14 0.871863 0.435932 0.899980i \(-0.356419\pi\)
0.435932 + 0.899980i \(0.356419\pi\)
\(402\) −1.25701e14 −0.597168
\(403\) 4.65958e14 2.18358
\(404\) 1.28454e14 0.593814
\(405\) −3.31146e13 −0.151014
\(406\) 8.16146e13 0.367176
\(407\) −4.64418e14 −2.06129
\(408\) −9.04274e13 −0.395976
\(409\) −1.82091e14 −0.786704 −0.393352 0.919388i \(-0.628685\pi\)
−0.393352 + 0.919388i \(0.628685\pi\)
\(410\) −1.84154e14 −0.785001
\(411\) −2.16792e14 −0.911830
\(412\) −2.46449e13 −0.102280
\(413\) 1.50789e13 0.0617509
\(414\) 1.29536e14 0.523466
\(415\) −6.11742e13 −0.243952
\(416\) 3.22152e14 1.26779
\(417\) −1.85034e14 −0.718625
\(418\) 1.89565e14 0.726587
\(419\) 5.36966e13 0.203128 0.101564 0.994829i \(-0.467615\pi\)
0.101564 + 0.994829i \(0.467615\pi\)
\(420\) 4.30608e13 0.160772
\(421\) 1.97113e14 0.726380 0.363190 0.931715i \(-0.381688\pi\)
0.363190 + 0.931715i \(0.381688\pi\)
\(422\) 6.38145e14 2.32114
\(423\) 1.40024e14 0.502725
\(424\) 1.82768e14 0.647722
\(425\) −2.44356e14 −0.854839
\(426\) 1.96572e13 0.0678843
\(427\) 1.68289e14 0.573723
\(428\) 2.06778e13 0.0695928
\(429\) −4.29374e14 −1.42666
\(430\) −4.58249e14 −1.50323
\(431\) 1.11050e14 0.359660 0.179830 0.983698i \(-0.442445\pi\)
0.179830 + 0.983698i \(0.442445\pi\)
\(432\) −7.49510e13 −0.239672
\(433\) 3.97259e14 1.25427 0.627133 0.778912i \(-0.284228\pi\)
0.627133 + 0.778912i \(0.284228\pi\)
\(434\) 2.54164e14 0.792358
\(435\) −1.64904e14 −0.507621
\(436\) −1.99350e14 −0.605955
\(437\) −1.68041e14 −0.504392
\(438\) 5.68476e12 0.0168502
\(439\) 2.43660e14 0.713229 0.356615 0.934252i \(-0.383931\pi\)
0.356615 + 0.934252i \(0.383931\pi\)
\(440\) 5.04886e14 1.45950
\(441\) −9.04910e13 −0.258341
\(442\) 6.69816e14 1.88857
\(443\) −2.28088e14 −0.635159 −0.317580 0.948232i \(-0.602870\pi\)
−0.317580 + 0.948232i \(0.602870\pi\)
\(444\) 1.18312e14 0.325404
\(445\) −9.70517e14 −2.63647
\(446\) −4.75830e14 −1.27676
\(447\) −2.75177e14 −0.729325
\(448\) −4.99069e13 −0.130657
\(449\) −4.97053e14 −1.28543 −0.642714 0.766107i \(-0.722191\pi\)
−0.642714 + 0.766107i \(0.722191\pi\)
\(450\) −1.32285e14 −0.337943
\(451\) 3.02145e14 0.762507
\(452\) −3.09138e14 −0.770709
\(453\) 5.80114e13 0.142881
\(454\) 5.24431e13 0.127609
\(455\) 4.19445e14 1.00835
\(456\) 6.35059e13 0.150837
\(457\) −5.40036e13 −0.126731 −0.0633656 0.997990i \(-0.520183\pi\)
−0.0633656 + 0.997990i \(0.520183\pi\)
\(458\) −4.38587e14 −1.01694
\(459\) −8.47564e13 −0.194179
\(460\) 3.40342e14 0.770455
\(461\) −1.30960e14 −0.292943 −0.146472 0.989215i \(-0.546792\pi\)
−0.146472 + 0.989215i \(0.546792\pi\)
\(462\) −2.34209e14 −0.517694
\(463\) 7.29383e14 1.59316 0.796581 0.604532i \(-0.206640\pi\)
0.796581 + 0.604532i \(0.206640\pi\)
\(464\) −3.73239e14 −0.805635
\(465\) −5.13544e14 −1.09543
\(466\) −7.51565e14 −1.58432
\(467\) 6.78306e14 1.41313 0.706566 0.707647i \(-0.250243\pi\)
0.706566 + 0.707647i \(0.250243\pi\)
\(468\) 1.09385e14 0.225218
\(469\) −2.01471e14 −0.409978
\(470\) 1.21959e15 2.45288
\(471\) 1.11578e14 0.221801
\(472\) −4.50402e13 −0.0884950
\(473\) 7.51857e14 1.46015
\(474\) −5.55831e14 −1.06699
\(475\) 1.71608e14 0.325629
\(476\) 1.10214e14 0.206727
\(477\) 1.71306e14 0.317630
\(478\) 6.74128e14 1.23563
\(479\) 5.16339e14 0.935599 0.467800 0.883835i \(-0.345047\pi\)
0.467800 + 0.883835i \(0.345047\pi\)
\(480\) −3.55052e14 −0.636011
\(481\) 1.15245e15 2.04091
\(482\) 6.41063e14 1.12239
\(483\) 2.07617e14 0.359379
\(484\) 3.77525e14 0.646095
\(485\) −1.03521e15 −1.75166
\(486\) −4.58840e13 −0.0767648
\(487\) 3.93458e14 0.650862 0.325431 0.945566i \(-0.394491\pi\)
0.325431 + 0.945566i \(0.394491\pi\)
\(488\) −5.02675e14 −0.822201
\(489\) 2.02536e14 0.327569
\(490\) −7.88167e14 −1.26049
\(491\) −3.95822e14 −0.625968 −0.312984 0.949758i \(-0.601329\pi\)
−0.312984 + 0.949758i \(0.601329\pi\)
\(492\) −7.69724e13 −0.120372
\(493\) −4.22068e14 −0.652717
\(494\) −4.70402e14 −0.719402
\(495\) 4.73224e14 0.715711
\(496\) −1.16234e15 −1.73854
\(497\) 3.15061e13 0.0466052
\(498\) −8.47637e13 −0.124008
\(499\) −4.03104e14 −0.583262 −0.291631 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(500\) 6.26739e13 0.0896916
\(501\) −5.52034e14 −0.781373
\(502\) 1.39125e14 0.194776
\(503\) 5.09614e14 0.705695 0.352848 0.935681i \(-0.385213\pi\)
0.352848 + 0.935681i \(0.385213\pi\)
\(504\) −7.84623e13 −0.107471
\(505\) −1.37902e15 −1.86839
\(506\) −1.85113e15 −2.48090
\(507\) 6.29993e14 0.835203
\(508\) 3.42863e14 0.449646
\(509\) −9.09009e14 −1.17929 −0.589645 0.807663i \(-0.700732\pi\)
−0.589645 + 0.807663i \(0.700732\pi\)
\(510\) −7.38219e14 −0.947435
\(511\) 9.11138e12 0.0115683
\(512\) −1.29665e14 −0.162869
\(513\) 5.95233e13 0.0739676
\(514\) 4.15123e12 0.00510364
\(515\) 2.64576e14 0.321818
\(516\) −1.91538e14 −0.230506
\(517\) −2.00101e15 −2.38260
\(518\) 6.28621e14 0.740586
\(519\) 8.98507e14 1.04737
\(520\) −1.25287e15 −1.44507
\(521\) −4.16939e14 −0.475844 −0.237922 0.971284i \(-0.576466\pi\)
−0.237922 + 0.971284i \(0.576466\pi\)
\(522\) −2.28492e14 −0.258038
\(523\) 7.53757e12 0.00842310 0.00421155 0.999991i \(-0.498659\pi\)
0.00421155 + 0.999991i \(0.498659\pi\)
\(524\) −5.45802e13 −0.0603550
\(525\) −2.12024e14 −0.232011
\(526\) 1.59583e15 1.72808
\(527\) −1.31441e15 −1.40855
\(528\) 1.07108e15 1.13589
\(529\) 6.88140e14 0.722221
\(530\) 1.49206e15 1.54978
\(531\) −4.22156e13 −0.0433963
\(532\) −7.74015e13 −0.0787472
\(533\) −7.49769e14 −0.754966
\(534\) −1.34476e15 −1.34019
\(535\) −2.21987e14 −0.218969
\(536\) 6.01787e14 0.587539
\(537\) −8.30970e14 −0.803021
\(538\) −3.77397e14 −0.360990
\(539\) 1.29316e15 1.22437
\(540\) −1.20555e14 −0.112985
\(541\) −5.57911e14 −0.517583 −0.258791 0.965933i \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(542\) 2.31000e15 2.12137
\(543\) 2.18270e14 0.198424
\(544\) −9.08749e14 −0.817806
\(545\) 2.14012e15 1.90659
\(546\) 5.81187e14 0.512574
\(547\) 9.47126e14 0.826947 0.413473 0.910516i \(-0.364315\pi\)
0.413473 + 0.910516i \(0.364315\pi\)
\(548\) −7.89242e14 −0.682208
\(549\) −4.71150e14 −0.403192
\(550\) 1.89042e15 1.60164
\(551\) 2.96413e14 0.248636
\(552\) −6.20146e14 −0.515026
\(553\) −8.90871e14 −0.732532
\(554\) −1.29853e15 −1.05718
\(555\) −1.27014e15 −1.02386
\(556\) −6.73623e14 −0.537657
\(557\) −1.55868e14 −0.123184 −0.0615919 0.998101i \(-0.519618\pi\)
−0.0615919 + 0.998101i \(0.519618\pi\)
\(558\) −7.11572e14 −0.556840
\(559\) −1.86572e15 −1.44571
\(560\) −1.04631e15 −0.802838
\(561\) 1.21121e15 0.920287
\(562\) 2.80418e15 2.10987
\(563\) −1.20905e15 −0.900837 −0.450419 0.892817i \(-0.648725\pi\)
−0.450419 + 0.892817i \(0.648725\pi\)
\(564\) 5.09764e14 0.376126
\(565\) 3.31875e15 2.42498
\(566\) 3.98037e13 0.0288027
\(567\) −7.35417e13 −0.0527020
\(568\) −9.41078e13 −0.0667898
\(569\) −2.21496e15 −1.55686 −0.778428 0.627734i \(-0.783983\pi\)
−0.778428 + 0.627734i \(0.783983\pi\)
\(570\) 5.18441e14 0.360901
\(571\) 1.13013e14 0.0779164 0.0389582 0.999241i \(-0.487596\pi\)
0.0389582 + 0.999241i \(0.487596\pi\)
\(572\) −1.56316e15 −1.06739
\(573\) 9.84696e14 0.665966
\(574\) −4.08973e14 −0.273955
\(575\) −1.67578e15 −1.11184
\(576\) 1.39722e14 0.0918208
\(577\) 2.63239e15 1.71350 0.856750 0.515732i \(-0.172480\pi\)
0.856750 + 0.515732i \(0.172480\pi\)
\(578\) −3.35024e13 −0.0216009
\(579\) −9.62340e14 −0.614605
\(580\) −6.00339e14 −0.379789
\(581\) −1.35857e14 −0.0851360
\(582\) −1.43440e15 −0.890420
\(583\) −2.44805e15 −1.50537
\(584\) −2.72154e13 −0.0165785
\(585\) −1.17430e15 −0.708633
\(586\) 2.17498e15 1.30022
\(587\) −7.18576e14 −0.425563 −0.212781 0.977100i \(-0.568252\pi\)
−0.212781 + 0.977100i \(0.568252\pi\)
\(588\) −3.29436e14 −0.193284
\(589\) 9.23090e14 0.536550
\(590\) −3.67693e14 −0.211738
\(591\) −1.12839e15 −0.643766
\(592\) −2.87481e15 −1.62495
\(593\) −7.56098e13 −0.0423426 −0.0211713 0.999776i \(-0.506740\pi\)
−0.0211713 + 0.999776i \(0.506740\pi\)
\(594\) 6.55704e14 0.363817
\(595\) −1.18320e15 −0.650450
\(596\) −1.00179e15 −0.545663
\(597\) −1.14437e15 −0.617602
\(598\) 4.59356e15 2.45636
\(599\) 1.94727e14 0.103176 0.0515879 0.998668i \(-0.483572\pi\)
0.0515879 + 0.998668i \(0.483572\pi\)
\(600\) 6.33309e14 0.332494
\(601\) 1.51012e15 0.785603 0.392802 0.919623i \(-0.371506\pi\)
0.392802 + 0.919623i \(0.371506\pi\)
\(602\) −1.01769e15 −0.524608
\(603\) 5.64048e14 0.288118
\(604\) 2.11193e14 0.106900
\(605\) −4.05292e15 −2.03289
\(606\) −1.91079e15 −0.949758
\(607\) 2.27582e15 1.12099 0.560493 0.828159i \(-0.310612\pi\)
0.560493 + 0.828159i \(0.310612\pi\)
\(608\) 6.38202e14 0.311522
\(609\) −3.66221e14 −0.177153
\(610\) −4.10367e15 −1.96724
\(611\) 4.96548e15 2.35904
\(612\) −3.08560e14 −0.145280
\(613\) 7.28934e14 0.340138 0.170069 0.985432i \(-0.445601\pi\)
0.170069 + 0.985432i \(0.445601\pi\)
\(614\) 2.84250e15 1.31454
\(615\) 8.26337e14 0.378743
\(616\) 1.12126e15 0.509346
\(617\) −3.28279e15 −1.47800 −0.739001 0.673704i \(-0.764702\pi\)
−0.739001 + 0.673704i \(0.764702\pi\)
\(618\) 3.66599e14 0.163589
\(619\) 3.79250e14 0.167736 0.0838682 0.996477i \(-0.473273\pi\)
0.0838682 + 0.996477i \(0.473273\pi\)
\(620\) −1.86958e15 −0.819576
\(621\) −5.81255e14 −0.252559
\(622\) 1.37534e15 0.592330
\(623\) −2.15534e15 −0.920094
\(624\) −2.65788e15 −1.12466
\(625\) −2.69278e15 −1.12943
\(626\) 1.07958e15 0.448843
\(627\) −8.50615e14 −0.350559
\(628\) 4.06205e14 0.165946
\(629\) −3.25090e15 −1.31651
\(630\) −6.40540e14 −0.257142
\(631\) −3.80252e15 −1.51325 −0.756623 0.653851i \(-0.773152\pi\)
−0.756623 + 0.653851i \(0.773152\pi\)
\(632\) 2.66101e15 1.04979
\(633\) −2.86349e15 −1.11989
\(634\) −3.70655e15 −1.43708
\(635\) −3.68081e15 −1.41478
\(636\) 6.23648e14 0.237643
\(637\) −3.20896e15 −1.21226
\(638\) 3.26526e15 1.22294
\(639\) −8.82061e13 −0.0327524
\(640\) 4.20933e15 1.54961
\(641\) 3.00206e15 1.09572 0.547861 0.836570i \(-0.315442\pi\)
0.547861 + 0.836570i \(0.315442\pi\)
\(642\) −3.07588e14 −0.111308
\(643\) −1.62443e15 −0.582829 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(644\) 7.55838e14 0.268879
\(645\) 2.05626e15 0.725269
\(646\) 1.32694e15 0.464059
\(647\) 1.51869e15 0.526618 0.263309 0.964712i \(-0.415186\pi\)
0.263309 + 0.964712i \(0.415186\pi\)
\(648\) 2.19667e14 0.0755270
\(649\) 6.03280e14 0.205671
\(650\) −4.69106e15 −1.58580
\(651\) −1.14049e15 −0.382292
\(652\) 7.37341e14 0.245079
\(653\) 2.96946e14 0.0978711 0.0489356 0.998802i \(-0.484417\pi\)
0.0489356 + 0.998802i \(0.484417\pi\)
\(654\) 2.96538e15 0.969176
\(655\) 5.85946e14 0.189902
\(656\) 1.87031e15 0.601095
\(657\) −2.55087e13 −0.00812977
\(658\) 2.70850e15 0.856025
\(659\) 9.64420e13 0.0302271 0.0151136 0.999886i \(-0.495189\pi\)
0.0151136 + 0.999886i \(0.495189\pi\)
\(660\) 1.72279e15 0.535477
\(661\) −2.32501e15 −0.716665 −0.358333 0.933594i \(-0.616655\pi\)
−0.358333 + 0.933594i \(0.616655\pi\)
\(662\) 4.99709e15 1.52756
\(663\) −3.00560e15 −0.911186
\(664\) 4.05801e14 0.122008
\(665\) 8.30944e14 0.247772
\(666\) −1.75992e15 −0.520457
\(667\) −2.89452e15 −0.848954
\(668\) −2.00970e15 −0.584604
\(669\) 2.13515e15 0.616005
\(670\) 4.91279e15 1.40578
\(671\) 6.73296e15 1.91087
\(672\) −7.88506e14 −0.221960
\(673\) −8.70992e14 −0.243182 −0.121591 0.992580i \(-0.538800\pi\)
−0.121591 + 0.992580i \(0.538800\pi\)
\(674\) −2.77887e15 −0.769553
\(675\) 5.93592e14 0.163049
\(676\) 2.29352e15 0.624878
\(677\) −6.51500e14 −0.176067 −0.0880333 0.996118i \(-0.528058\pi\)
−0.0880333 + 0.996118i \(0.528058\pi\)
\(678\) 4.59850e15 1.23269
\(679\) −2.29902e15 −0.611307
\(680\) 3.53418e15 0.932159
\(681\) −2.35323e14 −0.0615681
\(682\) 1.01687e16 2.63907
\(683\) 1.63952e15 0.422087 0.211043 0.977477i \(-0.432314\pi\)
0.211043 + 0.977477i \(0.432314\pi\)
\(684\) 2.16697e14 0.0553407
\(685\) 8.47291e15 2.14652
\(686\) −4.00886e15 −1.00748
\(687\) 1.96803e15 0.490648
\(688\) 4.65409e15 1.15106
\(689\) 6.07480e15 1.49048
\(690\) −5.06266e15 −1.23228
\(691\) −7.87017e15 −1.90044 −0.950221 0.311576i \(-0.899143\pi\)
−0.950221 + 0.311576i \(0.899143\pi\)
\(692\) 3.27105e15 0.783617
\(693\) 1.05094e15 0.249774
\(694\) 4.76480e15 1.12348
\(695\) 7.23168e15 1.69170
\(696\) 1.09389e15 0.253877
\(697\) 2.11500e15 0.487001
\(698\) −3.71755e15 −0.849283
\(699\) 3.37243e15 0.764396
\(700\) −7.71881e14 −0.173585
\(701\) 1.03894e15 0.231816 0.115908 0.993260i \(-0.463022\pi\)
0.115908 + 0.993260i \(0.463022\pi\)
\(702\) −1.62712e15 −0.360219
\(703\) 2.28307e15 0.501492
\(704\) −1.99669e15 −0.435172
\(705\) −5.47257e15 −1.18345
\(706\) −1.70488e15 −0.365821
\(707\) −3.06256e15 −0.652045
\(708\) −1.53688e14 −0.0324680
\(709\) −5.34928e15 −1.12135 −0.560675 0.828036i \(-0.689458\pi\)
−0.560675 + 0.828036i \(0.689458\pi\)
\(710\) −7.68265e14 −0.159805
\(711\) 2.49413e15 0.514797
\(712\) 6.43795e15 1.31858
\(713\) −9.01412e15 −1.83202
\(714\) −1.63945e15 −0.330643
\(715\) 1.67813e16 3.35847
\(716\) −3.02518e15 −0.600800
\(717\) −3.02495e15 −0.596161
\(718\) −8.61321e15 −1.68454
\(719\) 5.92269e15 1.14950 0.574751 0.818328i \(-0.305099\pi\)
0.574751 + 0.818328i \(0.305099\pi\)
\(720\) 2.92931e15 0.564206
\(721\) 5.87575e14 0.112310
\(722\) 5.37652e15 1.01987
\(723\) −2.87658e15 −0.541522
\(724\) 7.94620e14 0.148456
\(725\) 2.95596e15 0.548074
\(726\) −5.61578e15 −1.03338
\(727\) −3.27751e15 −0.598556 −0.299278 0.954166i \(-0.596746\pi\)
−0.299278 + 0.954166i \(0.596746\pi\)
\(728\) −2.78240e15 −0.504309
\(729\) 2.05891e14 0.0370370
\(730\) −2.22178e14 −0.0396666
\(731\) 5.26296e15 0.932577
\(732\) −1.71524e15 −0.301658
\(733\) 8.20340e15 1.43193 0.715966 0.698135i \(-0.245987\pi\)
0.715966 + 0.698135i \(0.245987\pi\)
\(734\) 1.18179e16 2.04745
\(735\) 3.53667e15 0.608154
\(736\) −6.23215e15 −1.06368
\(737\) −8.06050e15 −1.36550
\(738\) 1.14498e15 0.192526
\(739\) 1.02218e16 1.70601 0.853006 0.521900i \(-0.174777\pi\)
0.853006 + 0.521900i \(0.174777\pi\)
\(740\) −4.62400e15 −0.766026
\(741\) 2.11079e15 0.347093
\(742\) 3.31360e15 0.540852
\(743\) −4.91377e15 −0.796116 −0.398058 0.917360i \(-0.630316\pi\)
−0.398058 + 0.917360i \(0.630316\pi\)
\(744\) 3.40661e15 0.547862
\(745\) 1.07548e16 1.71689
\(746\) 8.09970e14 0.128353
\(747\) 3.80352e14 0.0598305
\(748\) 4.40946e15 0.688536
\(749\) −4.92993e14 −0.0764172
\(750\) −9.32289e14 −0.143454
\(751\) 3.36651e15 0.514234 0.257117 0.966380i \(-0.417227\pi\)
0.257117 + 0.966380i \(0.417227\pi\)
\(752\) −1.23865e16 −1.87824
\(753\) −6.24282e14 −0.0939742
\(754\) −8.10271e15 −1.21084
\(755\) −2.26727e15 −0.336352
\(756\) −2.67732e14 −0.0394303
\(757\) −9.37741e14 −0.137106 −0.0685529 0.997647i \(-0.521838\pi\)
−0.0685529 + 0.997647i \(0.521838\pi\)
\(758\) −7.61403e15 −1.10518
\(759\) 8.30640e15 1.19697
\(760\) −2.48201e15 −0.355082
\(761\) 1.09329e16 1.55281 0.776407 0.630232i \(-0.217040\pi\)
0.776407 + 0.630232i \(0.217040\pi\)
\(762\) −5.10018e15 −0.719172
\(763\) 4.75283e15 0.665376
\(764\) 3.58483e15 0.498259
\(765\) 3.31254e15 0.457113
\(766\) −9.49076e15 −1.30030
\(767\) −1.49703e15 −0.203637
\(768\) 4.65492e15 0.628675
\(769\) 7.90499e15 1.06000 0.530001 0.847997i \(-0.322192\pi\)
0.530001 + 0.847997i \(0.322192\pi\)
\(770\) 9.15361e15 1.21869
\(771\) −1.86274e13 −0.00246238
\(772\) −3.50344e15 −0.459832
\(773\) 5.18761e15 0.676051 0.338026 0.941137i \(-0.390241\pi\)
0.338026 + 0.941137i \(0.390241\pi\)
\(774\) 2.84918e15 0.368675
\(775\) 9.20546e15 1.18273
\(776\) 6.86712e15 0.876062
\(777\) −2.82075e15 −0.357314
\(778\) 3.78455e15 0.476021
\(779\) −1.48533e15 −0.185510
\(780\) −4.27509e15 −0.530182
\(781\) 1.26051e15 0.155226
\(782\) −1.29578e16 −1.58451
\(783\) 1.02529e15 0.124497
\(784\) 8.00482e15 0.965190
\(785\) −4.36082e15 −0.522137
\(786\) 8.11894e14 0.0965329
\(787\) −3.95929e15 −0.467473 −0.233736 0.972300i \(-0.575095\pi\)
−0.233736 + 0.972300i \(0.575095\pi\)
\(788\) −4.10795e15 −0.481650
\(789\) −7.16082e15 −0.833756
\(790\) 2.17236e16 2.51179
\(791\) 7.37035e15 0.846286
\(792\) −3.13914e15 −0.357950
\(793\) −1.67078e16 −1.89198
\(794\) −7.28329e15 −0.819059
\(795\) −6.69518e15 −0.747727
\(796\) −4.16614e15 −0.462074
\(797\) −5.07604e15 −0.559119 −0.279560 0.960128i \(-0.590188\pi\)
−0.279560 + 0.960128i \(0.590188\pi\)
\(798\) 1.15137e15 0.125950
\(799\) −1.40070e16 −1.52173
\(800\) 6.36443e15 0.686696
\(801\) 6.03421e15 0.646609
\(802\) −9.80331e15 −1.04331
\(803\) 3.64531e14 0.0385300
\(804\) 2.05344e15 0.215563
\(805\) −8.11430e15 −0.846007
\(806\) −2.52335e16 −2.61297
\(807\) 1.69346e15 0.174168
\(808\) 9.14778e15 0.934444
\(809\) −7.66663e15 −0.777836 −0.388918 0.921272i \(-0.627151\pi\)
−0.388918 + 0.921272i \(0.627151\pi\)
\(810\) 1.79329e15 0.180710
\(811\) 4.76464e14 0.0476886 0.0238443 0.999716i \(-0.492409\pi\)
0.0238443 + 0.999716i \(0.492409\pi\)
\(812\) −1.33324e15 −0.132541
\(813\) −1.03654e16 −1.02350
\(814\) 2.51501e16 2.46664
\(815\) −7.91573e15 −0.771124
\(816\) 7.49754e15 0.725475
\(817\) −3.69611e15 −0.355241
\(818\) 9.86097e15 0.941405
\(819\) −2.60791e15 −0.247304
\(820\) 3.00832e15 0.283366
\(821\) 1.12022e16 1.04814 0.524068 0.851676i \(-0.324414\pi\)
0.524068 + 0.851676i \(0.324414\pi\)
\(822\) 1.17402e16 1.09114
\(823\) 6.54641e14 0.0604372 0.0302186 0.999543i \(-0.490380\pi\)
0.0302186 + 0.999543i \(0.490380\pi\)
\(824\) −1.75507e15 −0.160952
\(825\) −8.48271e15 −0.772748
\(826\) −8.16580e14 −0.0738939
\(827\) −1.97462e16 −1.77502 −0.887508 0.460792i \(-0.847565\pi\)
−0.887508 + 0.460792i \(0.847565\pi\)
\(828\) −2.11608e15 −0.188958
\(829\) −8.65393e15 −0.767650 −0.383825 0.923406i \(-0.625393\pi\)
−0.383825 + 0.923406i \(0.625393\pi\)
\(830\) 3.31283e15 0.291924
\(831\) 5.82679e15 0.510063
\(832\) 4.95477e15 0.430869
\(833\) 9.05205e15 0.781987
\(834\) 1.00203e16 0.859939
\(835\) 2.15752e16 1.83941
\(836\) −3.09670e15 −0.262280
\(837\) 3.19297e15 0.268661
\(838\) −2.90788e15 −0.243072
\(839\) 3.61570e15 0.300263 0.150131 0.988666i \(-0.452030\pi\)
0.150131 + 0.988666i \(0.452030\pi\)
\(840\) 3.06655e15 0.252996
\(841\) −7.09478e15 −0.581515
\(842\) −1.06745e16 −0.869219
\(843\) −1.25829e16 −1.01796
\(844\) −1.04247e16 −0.837873
\(845\) −2.46221e16 −1.96613
\(846\) −7.58286e15 −0.601583
\(847\) −9.00082e15 −0.709452
\(848\) −1.51537e16 −1.18670
\(849\) −1.78607e14 −0.0138966
\(850\) 1.32329e16 1.02294
\(851\) −2.22945e16 −1.71232
\(852\) −3.21118e14 −0.0245046
\(853\) 5.30884e15 0.402513 0.201257 0.979539i \(-0.435498\pi\)
0.201257 + 0.979539i \(0.435498\pi\)
\(854\) −9.11351e15 −0.686543
\(855\) −2.32635e15 −0.174125
\(856\) 1.47256e15 0.109513
\(857\) 2.14371e16 1.58406 0.792028 0.610484i \(-0.209025\pi\)
0.792028 + 0.610484i \(0.209025\pi\)
\(858\) 2.32523e16 1.70721
\(859\) −6.17747e15 −0.450659 −0.225330 0.974283i \(-0.572346\pi\)
−0.225330 + 0.974283i \(0.572346\pi\)
\(860\) 7.48590e15 0.542628
\(861\) 1.83515e15 0.132176
\(862\) −6.01378e15 −0.430385
\(863\) 2.52467e16 1.79534 0.897668 0.440673i \(-0.145260\pi\)
0.897668 + 0.440673i \(0.145260\pi\)
\(864\) 2.20754e15 0.155985
\(865\) −3.51164e16 −2.46559
\(866\) −2.15131e16 −1.50091
\(867\) 1.50332e14 0.0104219
\(868\) −4.15200e15 −0.286021
\(869\) −3.56422e16 −2.43981
\(870\) 8.93018e15 0.607442
\(871\) 2.00020e16 1.35199
\(872\) −1.41966e16 −0.953549
\(873\) 6.43646e15 0.429605
\(874\) 9.10010e15 0.603578
\(875\) −1.49425e15 −0.0984869
\(876\) −9.28655e13 −0.00608249
\(877\) −2.49885e16 −1.62646 −0.813228 0.581945i \(-0.802292\pi\)
−0.813228 + 0.581945i \(0.802292\pi\)
\(878\) −1.31951e16 −0.853482
\(879\) −9.75957e15 −0.627324
\(880\) −4.18613e16 −2.67398
\(881\) 2.60473e16 1.65346 0.826731 0.562597i \(-0.190198\pi\)
0.826731 + 0.562597i \(0.190198\pi\)
\(882\) 4.90045e15 0.309142
\(883\) −2.06710e16 −1.29592 −0.647961 0.761674i \(-0.724378\pi\)
−0.647961 + 0.761674i \(0.724378\pi\)
\(884\) −1.09420e16 −0.681726
\(885\) 1.64991e15 0.102158
\(886\) 1.23519e16 0.760060
\(887\) 1.33866e16 0.818637 0.409319 0.912392i \(-0.365766\pi\)
0.409319 + 0.912392i \(0.365766\pi\)
\(888\) 8.42552e15 0.512065
\(889\) −8.17442e15 −0.493739
\(890\) 5.25573e16 3.15492
\(891\) −2.94228e15 −0.175532
\(892\) 7.77310e15 0.460880
\(893\) 9.83690e15 0.579662
\(894\) 1.49019e16 0.872743
\(895\) 3.24769e16 1.89037
\(896\) 9.34817e15 0.540795
\(897\) −2.06122e16 −1.18513
\(898\) 2.69174e16 1.53820
\(899\) 1.59003e16 0.903080
\(900\) 2.16100e15 0.121989
\(901\) −1.71362e16 −0.961453
\(902\) −1.63623e16 −0.912450
\(903\) 4.56658e15 0.253110
\(904\) −2.20150e16 −1.21281
\(905\) −8.53065e15 −0.467105
\(906\) −3.14155e15 −0.170977
\(907\) −2.11356e16 −1.14334 −0.571668 0.820485i \(-0.693704\pi\)
−0.571668 + 0.820485i \(0.693704\pi\)
\(908\) −8.56704e14 −0.0460637
\(909\) 8.57409e15 0.458234
\(910\) −2.27146e16 −1.20664
\(911\) −2.91385e16 −1.53857 −0.769283 0.638908i \(-0.779387\pi\)
−0.769283 + 0.638908i \(0.779387\pi\)
\(912\) −5.26542e15 −0.276351
\(913\) −5.43541e15 −0.283559
\(914\) 2.92451e15 0.151652
\(915\) 1.84140e16 0.949144
\(916\) 7.16471e15 0.367091
\(917\) 1.30128e15 0.0662735
\(918\) 4.58990e15 0.232364
\(919\) −2.56566e16 −1.29111 −0.645557 0.763712i \(-0.723375\pi\)
−0.645557 + 0.763712i \(0.723375\pi\)
\(920\) 2.42372e16 1.21241
\(921\) −1.27549e16 −0.634233
\(922\) 7.09200e15 0.350549
\(923\) −3.12793e15 −0.153691
\(924\) 3.82601e15 0.186875
\(925\) 2.27677e16 1.10545
\(926\) −3.94990e16 −1.90645
\(927\) −1.64501e15 −0.0789276
\(928\) 1.09931e16 0.524330
\(929\) −9.20273e15 −0.436345 −0.218173 0.975910i \(-0.570010\pi\)
−0.218173 + 0.975910i \(0.570010\pi\)
\(930\) 2.78104e16 1.31084
\(931\) −6.35713e15 −0.297878
\(932\) 1.22775e16 0.571902
\(933\) −6.17144e15 −0.285784
\(934\) −3.67330e16 −1.69102
\(935\) −4.73378e16 −2.16643
\(936\) 7.78975e15 0.354410
\(937\) −2.05016e16 −0.927299 −0.463650 0.886019i \(-0.653460\pi\)
−0.463650 + 0.886019i \(0.653460\pi\)
\(938\) 1.09104e16 0.490599
\(939\) −4.84429e15 −0.216555
\(940\) −1.99231e16 −0.885430
\(941\) −1.85084e16 −0.817762 −0.408881 0.912588i \(-0.634081\pi\)
−0.408881 + 0.912588i \(0.634081\pi\)
\(942\) −6.04240e15 −0.265417
\(943\) 1.45045e16 0.633416
\(944\) 3.73438e15 0.162133
\(945\) 2.87424e15 0.124064
\(946\) −4.07160e16 −1.74729
\(947\) −5.79326e15 −0.247171 −0.123586 0.992334i \(-0.539439\pi\)
−0.123586 + 0.992334i \(0.539439\pi\)
\(948\) 9.07998e15 0.385158
\(949\) −9.04579e14 −0.0381489
\(950\) −9.29325e15 −0.389662
\(951\) 1.66321e16 0.693352
\(952\) 7.84878e15 0.325311
\(953\) −2.10008e16 −0.865418 −0.432709 0.901534i \(-0.642442\pi\)
−0.432709 + 0.901534i \(0.642442\pi\)
\(954\) −9.27691e15 −0.380091
\(955\) −3.84850e16 −1.56773
\(956\) −1.10125e16 −0.446032
\(957\) −1.46519e16 −0.590036
\(958\) −2.79618e16 −1.11958
\(959\) 1.88168e16 0.749107
\(960\) −5.46077e15 −0.216153
\(961\) 2.41082e16 0.948827
\(962\) −6.24097e16 −2.44224
\(963\) 1.38021e15 0.0537032
\(964\) −1.04723e16 −0.405153
\(965\) 3.76112e16 1.44683
\(966\) −1.12433e16 −0.430050
\(967\) 3.25364e16 1.23744 0.618720 0.785612i \(-0.287652\pi\)
0.618720 + 0.785612i \(0.287652\pi\)
\(968\) 2.68852e16 1.01671
\(969\) −5.95427e15 −0.223897
\(970\) 5.60609e16 2.09612
\(971\) 8.82887e14 0.0328246 0.0164123 0.999865i \(-0.494776\pi\)
0.0164123 + 0.999865i \(0.494776\pi\)
\(972\) 7.49556e14 0.0277102
\(973\) 1.60603e16 0.590381
\(974\) −2.13073e16 −0.778851
\(975\) 2.10497e16 0.765106
\(976\) 4.16779e16 1.50637
\(977\) 2.39526e16 0.860859 0.430429 0.902624i \(-0.358362\pi\)
0.430429 + 0.902624i \(0.358362\pi\)
\(978\) −1.09681e16 −0.391984
\(979\) −8.62316e16 −3.06452
\(980\) 1.28754e16 0.455006
\(981\) −1.33063e16 −0.467602
\(982\) 2.14353e16 0.749061
\(983\) −9.42036e15 −0.327358 −0.163679 0.986514i \(-0.552336\pi\)
−0.163679 + 0.986514i \(0.552336\pi\)
\(984\) −5.48154e15 −0.189421
\(985\) 4.41010e16 1.51548
\(986\) 2.28567e16 0.781070
\(987\) −1.21536e16 −0.413010
\(988\) 7.68443e15 0.259686
\(989\) 3.60931e16 1.21295
\(990\) −2.56269e16 −0.856452
\(991\) −1.26830e16 −0.421520 −0.210760 0.977538i \(-0.567594\pi\)
−0.210760 + 0.977538i \(0.567594\pi\)
\(992\) 3.42347e16 1.13149
\(993\) −2.24230e16 −0.737008
\(994\) −1.70618e15 −0.0557699
\(995\) 4.47256e16 1.45388
\(996\) 1.38469e15 0.0447637
\(997\) −3.73904e16 −1.20209 −0.601045 0.799215i \(-0.705249\pi\)
−0.601045 + 0.799215i \(0.705249\pi\)
\(998\) 2.18297e16 0.697958
\(999\) 7.89713e15 0.251107
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.b.1.7 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.7 27 1.1 even 1 trivial