Properties

Label 177.12.a.c.1.17
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
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: \(0\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+18.0234 q^{2} -243.000 q^{3} -1723.16 q^{4} -3879.65 q^{5} -4379.69 q^{6} +32120.2 q^{7} -67969.1 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+18.0234 q^{2} -243.000 q^{3} -1723.16 q^{4} -3879.65 q^{5} -4379.69 q^{6} +32120.2 q^{7} -67969.1 q^{8} +59049.0 q^{9} -69924.5 q^{10} -362420. q^{11} +418727. q^{12} -286043. q^{13} +578915. q^{14} +942754. q^{15} +2.30399e6 q^{16} +9.02842e6 q^{17} +1.06426e6 q^{18} -2.13529e7 q^{19} +6.68524e6 q^{20} -7.80520e6 q^{21} -6.53205e6 q^{22} -5.38940e7 q^{23} +1.65165e7 q^{24} -3.37765e7 q^{25} -5.15548e6 q^{26} -1.43489e7 q^{27} -5.53480e7 q^{28} +1.58491e8 q^{29} +1.69917e7 q^{30} -1.44016e8 q^{31} +1.80727e8 q^{32} +8.80681e7 q^{33} +1.62723e8 q^{34} -1.24615e8 q^{35} -1.01751e8 q^{36} -1.68394e8 q^{37} -3.84852e8 q^{38} +6.95085e7 q^{39} +2.63696e8 q^{40} -1.21824e9 q^{41} -1.40676e8 q^{42} -6.48935e8 q^{43} +6.24507e8 q^{44} -2.29089e8 q^{45} -9.71354e8 q^{46} +1.87600e8 q^{47} -5.59869e8 q^{48} -9.45622e8 q^{49} -6.08767e8 q^{50} -2.19391e9 q^{51} +4.92897e8 q^{52} -2.62856e9 q^{53} -2.58616e8 q^{54} +1.40606e9 q^{55} -2.18318e9 q^{56} +5.18875e9 q^{57} +2.85655e9 q^{58} -7.14924e8 q^{59} -1.62451e9 q^{60} -3.26106e9 q^{61} -2.59565e9 q^{62} +1.89666e9 q^{63} -1.46126e9 q^{64} +1.10975e9 q^{65} +1.58729e9 q^{66} +1.11768e10 q^{67} -1.55574e10 q^{68} +1.30962e10 q^{69} -2.24599e9 q^{70} -1.81403e10 q^{71} -4.01351e9 q^{72} -4.66168e9 q^{73} -3.03503e9 q^{74} +8.20768e9 q^{75} +3.67943e10 q^{76} -1.16410e10 q^{77} +1.25278e9 q^{78} +3.85674e10 q^{79} -8.93866e9 q^{80} +3.48678e9 q^{81} -2.19569e10 q^{82} +4.48401e10 q^{83} +1.34496e10 q^{84} -3.50271e10 q^{85} -1.16960e10 q^{86} -3.85133e10 q^{87} +2.46334e10 q^{88} +8.91551e10 q^{89} -4.12897e9 q^{90} -9.18775e9 q^{91} +9.28678e10 q^{92} +3.49958e10 q^{93} +3.38120e9 q^{94} +8.28416e10 q^{95} -4.39165e10 q^{96} -1.14808e11 q^{97} -1.70434e10 q^{98} -2.14006e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + 140249q^{10} + 256992q^{11} - 6352506q^{12} + 2436978q^{13} + 5233061q^{14} + 593406q^{15} + 28295194q^{16} - 4565351q^{17} - 2716254q^{18} + 33607699q^{19} - 19208463q^{20} - 41332599q^{21} + 79735622q^{22} + 43966161q^{23} + 4699863q^{24} + 406675819q^{25} + 42605404q^{26} - 387420489q^{27} + 635747682q^{28} - 107217773q^{29} - 34080507q^{30} + 570926627q^{31} + 526569236q^{32} - 62449056q^{33} + 129790240q^{34} + 134356079q^{35} + 1543658958q^{36} - 107121371q^{37} + 208302581q^{38} - 592185654q^{39} - 958762162q^{40} - 1935967559q^{41} - 1271633823q^{42} + 1725943824q^{43} + 196885756q^{44} - 144197658q^{45} - 13265966407q^{46} + 1801256065q^{47} - 6875732142q^{48} + 10484289252q^{49} - 10067682271q^{50} + 1109380293q^{51} - 882697024q^{52} - 6214238922q^{53} + 660049722q^{54} + 4460552366q^{55} + 28328012310q^{56} - 8166670857q^{57} + 12220116750q^{58} - 19302956073q^{59} + 4667656509q^{60} + 13167821039q^{61} - 1162130230q^{62} + 10043821557q^{63} - 5337557395q^{64} - 16849896006q^{65} - 19375756146q^{66} - 16856763152q^{67} - 36171071977q^{68} - 10683777123q^{69} - 120177261588q^{70} - 5198545690q^{71} - 1142066709q^{72} - 25075321857q^{73} - 182979651978q^{74} - 98822224017q^{75} - 3501293988q^{76} - 42787697701q^{77} - 10353113172q^{78} + 6850314702q^{79} - 261464428159q^{80} + 94143178827q^{81} - 148881516273q^{82} + 30908370899q^{83} - 154486686726q^{84} - 49419624969q^{85} - 220725475224q^{86} + 26053918839q^{87} - 53091280787q^{88} + 28988060121q^{89} + 8281563201q^{90} + 97120614047q^{91} + 45374597708q^{92} - 138735170361q^{93} + 208966927220q^{94} - 125253904969q^{95} - 127956324348q^{96} + 367722840268q^{97} - 48265639912q^{98} + 15175120608q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.0234 0.398265 0.199133 0.979973i \(-0.436188\pi\)
0.199133 + 0.979973i \(0.436188\pi\)
\(3\) −243.000 −0.577350
\(4\) −1723.16 −0.841385
\(5\) −3879.65 −0.555210 −0.277605 0.960695i \(-0.589541\pi\)
−0.277605 + 0.960695i \(0.589541\pi\)
\(6\) −4379.69 −0.229938
\(7\) 32120.2 0.722335 0.361167 0.932501i \(-0.382378\pi\)
0.361167 + 0.932501i \(0.382378\pi\)
\(8\) −67969.1 −0.733359
\(9\) 59049.0 0.333333
\(10\) −69924.5 −0.221121
\(11\) −362420. −0.678505 −0.339252 0.940695i \(-0.610174\pi\)
−0.339252 + 0.940695i \(0.610174\pi\)
\(12\) 418727. 0.485774
\(13\) −286043. −0.213670 −0.106835 0.994277i \(-0.534072\pi\)
−0.106835 + 0.994277i \(0.534072\pi\)
\(14\) 578915. 0.287681
\(15\) 942754. 0.320551
\(16\) 2.30399e6 0.549314
\(17\) 9.02842e6 1.54221 0.771104 0.636710i \(-0.219705\pi\)
0.771104 + 0.636710i \(0.219705\pi\)
\(18\) 1.06426e6 0.132755
\(19\) −2.13529e7 −1.97839 −0.989194 0.146615i \(-0.953162\pi\)
−0.989194 + 0.146615i \(0.953162\pi\)
\(20\) 6.68524e6 0.467145
\(21\) −7.80520e6 −0.417040
\(22\) −6.53205e6 −0.270225
\(23\) −5.38940e7 −1.74597 −0.872986 0.487745i \(-0.837820\pi\)
−0.872986 + 0.487745i \(0.837820\pi\)
\(24\) 1.65165e7 0.423405
\(25\) −3.37765e7 −0.691742
\(26\) −5.15548e6 −0.0850972
\(27\) −1.43489e7 −0.192450
\(28\) −5.53480e7 −0.607761
\(29\) 1.58491e8 1.43488 0.717440 0.696621i \(-0.245314\pi\)
0.717440 + 0.696621i \(0.245314\pi\)
\(30\) 1.69917e7 0.127664
\(31\) −1.44016e8 −0.903483 −0.451742 0.892149i \(-0.649197\pi\)
−0.451742 + 0.892149i \(0.649197\pi\)
\(32\) 1.80727e8 0.952132
\(33\) 8.80681e7 0.391735
\(34\) 1.62723e8 0.614207
\(35\) −1.24615e8 −0.401047
\(36\) −1.01751e8 −0.280462
\(37\) −1.68394e8 −0.399224 −0.199612 0.979875i \(-0.563968\pi\)
−0.199612 + 0.979875i \(0.563968\pi\)
\(38\) −3.84852e8 −0.787923
\(39\) 6.95085e7 0.123362
\(40\) 2.63696e8 0.407168
\(41\) −1.21824e9 −1.64219 −0.821093 0.570795i \(-0.806635\pi\)
−0.821093 + 0.570795i \(0.806635\pi\)
\(42\) −1.40676e8 −0.166092
\(43\) −6.48935e8 −0.673170 −0.336585 0.941653i \(-0.609272\pi\)
−0.336585 + 0.941653i \(0.609272\pi\)
\(44\) 6.24507e8 0.570884
\(45\) −2.29089e8 −0.185070
\(46\) −9.71354e8 −0.695360
\(47\) 1.87600e8 0.119315 0.0596576 0.998219i \(-0.480999\pi\)
0.0596576 + 0.998219i \(0.480999\pi\)
\(48\) −5.59869e8 −0.317146
\(49\) −9.45622e8 −0.478233
\(50\) −6.08767e8 −0.275497
\(51\) −2.19391e9 −0.890394
\(52\) 4.92897e8 0.179779
\(53\) −2.62856e9 −0.863376 −0.431688 0.902023i \(-0.642082\pi\)
−0.431688 + 0.902023i \(0.642082\pi\)
\(54\) −2.58616e8 −0.0766462
\(55\) 1.40606e9 0.376712
\(56\) −2.18318e9 −0.529731
\(57\) 5.18875e9 1.14222
\(58\) 2.85655e9 0.571462
\(59\) −7.14924e8 −0.130189
\(60\) −1.62451e9 −0.269706
\(61\) −3.26106e9 −0.494361 −0.247181 0.968969i \(-0.579504\pi\)
−0.247181 + 0.968969i \(0.579504\pi\)
\(62\) −2.59565e9 −0.359826
\(63\) 1.89666e9 0.240778
\(64\) −1.46126e9 −0.170113
\(65\) 1.10975e9 0.118632
\(66\) 1.58729e9 0.156014
\(67\) 1.11768e10 1.01136 0.505681 0.862721i \(-0.331241\pi\)
0.505681 + 0.862721i \(0.331241\pi\)
\(68\) −1.55574e10 −1.29759
\(69\) 1.30962e10 1.00804
\(70\) −2.24599e9 −0.159723
\(71\) −1.81403e10 −1.19323 −0.596614 0.802528i \(-0.703488\pi\)
−0.596614 + 0.802528i \(0.703488\pi\)
\(72\) −4.01351e9 −0.244453
\(73\) −4.66168e9 −0.263188 −0.131594 0.991304i \(-0.542010\pi\)
−0.131594 + 0.991304i \(0.542010\pi\)
\(74\) −3.03503e9 −0.158997
\(75\) 8.20768e9 0.399377
\(76\) 3.67943e10 1.66459
\(77\) −1.16410e10 −0.490107
\(78\) 1.25278e9 0.0491309
\(79\) 3.85674e10 1.41017 0.705085 0.709122i \(-0.250909\pi\)
0.705085 + 0.709122i \(0.250909\pi\)
\(80\) −8.93866e9 −0.304984
\(81\) 3.48678e9 0.111111
\(82\) −2.19569e10 −0.654025
\(83\) 4.48401e10 1.24950 0.624751 0.780824i \(-0.285201\pi\)
0.624751 + 0.780824i \(0.285201\pi\)
\(84\) 1.34496e10 0.350891
\(85\) −3.50271e10 −0.856249
\(86\) −1.16960e10 −0.268100
\(87\) −3.85133e10 −0.828428
\(88\) 2.46334e10 0.497588
\(89\) 8.91551e10 1.69239 0.846197 0.532871i \(-0.178887\pi\)
0.846197 + 0.532871i \(0.178887\pi\)
\(90\) −4.12897e9 −0.0737069
\(91\) −9.18775e9 −0.154341
\(92\) 9.28678e10 1.46904
\(93\) 3.49958e10 0.521626
\(94\) 3.38120e9 0.0475190
\(95\) 8.28416e10 1.09842
\(96\) −4.39165e10 −0.549713
\(97\) −1.14808e11 −1.35746 −0.678731 0.734387i \(-0.737470\pi\)
−0.678731 + 0.734387i \(0.737470\pi\)
\(98\) −1.70434e10 −0.190463
\(99\) −2.14006e10 −0.226168
\(100\) 5.82021e10 0.582021
\(101\) −1.27258e11 −1.20481 −0.602404 0.798192i \(-0.705790\pi\)
−0.602404 + 0.798192i \(0.705790\pi\)
\(102\) −3.95417e10 −0.354613
\(103\) 1.39054e11 1.18189 0.590945 0.806712i \(-0.298755\pi\)
0.590945 + 0.806712i \(0.298755\pi\)
\(104\) 1.94421e10 0.156697
\(105\) 3.02814e10 0.231545
\(106\) −4.73756e10 −0.343853
\(107\) −2.97788e10 −0.205257 −0.102628 0.994720i \(-0.532725\pi\)
−0.102628 + 0.994720i \(0.532725\pi\)
\(108\) 2.47254e10 0.161925
\(109\) 1.33959e11 0.833924 0.416962 0.908924i \(-0.363095\pi\)
0.416962 + 0.908924i \(0.363095\pi\)
\(110\) 2.53421e10 0.150031
\(111\) 4.09197e10 0.230492
\(112\) 7.40044e10 0.396788
\(113\) −1.64153e11 −0.838140 −0.419070 0.907954i \(-0.637644\pi\)
−0.419070 + 0.907954i \(0.637644\pi\)
\(114\) 9.35189e10 0.454907
\(115\) 2.09090e11 0.969381
\(116\) −2.73105e11 −1.20729
\(117\) −1.68906e10 −0.0712233
\(118\) −1.28854e10 −0.0518497
\(119\) 2.89994e11 1.11399
\(120\) −6.40782e10 −0.235079
\(121\) −1.53963e11 −0.539632
\(122\) −5.87754e10 −0.196887
\(123\) 2.96033e11 0.948116
\(124\) 2.48161e11 0.760177
\(125\) 3.20477e11 0.939272
\(126\) 3.41844e10 0.0958935
\(127\) 2.89349e11 0.777143 0.388571 0.921419i \(-0.372969\pi\)
0.388571 + 0.921419i \(0.372969\pi\)
\(128\) −3.96465e11 −1.01988
\(129\) 1.57691e11 0.388655
\(130\) 2.00014e10 0.0472468
\(131\) 5.85181e11 1.32525 0.662625 0.748951i \(-0.269442\pi\)
0.662625 + 0.748951i \(0.269442\pi\)
\(132\) −1.51755e11 −0.329600
\(133\) −6.85857e11 −1.42906
\(134\) 2.01444e11 0.402790
\(135\) 5.56687e10 0.106850
\(136\) −6.13654e11 −1.13099
\(137\) 2.05345e11 0.363514 0.181757 0.983344i \(-0.441822\pi\)
0.181757 + 0.983344i \(0.441822\pi\)
\(138\) 2.36039e11 0.401466
\(139\) 1.85933e11 0.303930 0.151965 0.988386i \(-0.451440\pi\)
0.151965 + 0.988386i \(0.451440\pi\)
\(140\) 2.14731e11 0.337435
\(141\) −4.55869e10 −0.0688866
\(142\) −3.26950e11 −0.475221
\(143\) 1.03668e11 0.144976
\(144\) 1.36048e11 0.183105
\(145\) −6.14889e11 −0.796659
\(146\) −8.40194e10 −0.104819
\(147\) 2.29786e11 0.276108
\(148\) 2.90169e11 0.335901
\(149\) 1.54005e11 0.171794 0.0858972 0.996304i \(-0.472624\pi\)
0.0858972 + 0.996304i \(0.472624\pi\)
\(150\) 1.47930e11 0.159058
\(151\) 4.71655e11 0.488935 0.244467 0.969658i \(-0.421387\pi\)
0.244467 + 0.969658i \(0.421387\pi\)
\(152\) 1.45134e12 1.45087
\(153\) 5.33119e11 0.514069
\(154\) −2.09811e11 −0.195193
\(155\) 5.58730e11 0.501623
\(156\) −1.19774e11 −0.103795
\(157\) −2.47986e11 −0.207481 −0.103740 0.994604i \(-0.533081\pi\)
−0.103740 + 0.994604i \(0.533081\pi\)
\(158\) 6.95117e11 0.561622
\(159\) 6.38739e11 0.498470
\(160\) −7.01155e11 −0.528633
\(161\) −1.73108e12 −1.26118
\(162\) 6.28438e10 0.0442517
\(163\) 6.84742e11 0.466117 0.233059 0.972463i \(-0.425127\pi\)
0.233059 + 0.972463i \(0.425127\pi\)
\(164\) 2.09922e12 1.38171
\(165\) −3.41673e11 −0.217495
\(166\) 8.08171e11 0.497633
\(167\) 1.79932e12 1.07193 0.535967 0.844239i \(-0.319947\pi\)
0.535967 + 0.844239i \(0.319947\pi\)
\(168\) 5.30513e11 0.305840
\(169\) −1.71034e12 −0.954345
\(170\) −6.31308e11 −0.341014
\(171\) −1.26087e12 −0.659462
\(172\) 1.11822e12 0.566395
\(173\) 2.05890e11 0.101014 0.0505070 0.998724i \(-0.483916\pi\)
0.0505070 + 0.998724i \(0.483916\pi\)
\(174\) −6.94141e11 −0.329934
\(175\) −1.08491e12 −0.499669
\(176\) −8.35012e11 −0.372712
\(177\) 1.73727e11 0.0751646
\(178\) 1.60688e12 0.674021
\(179\) −3.45127e12 −1.40374 −0.701872 0.712304i \(-0.747652\pi\)
−0.701872 + 0.712304i \(0.747652\pi\)
\(180\) 3.94757e11 0.155715
\(181\) −3.37062e12 −1.28967 −0.644834 0.764323i \(-0.723073\pi\)
−0.644834 + 0.764323i \(0.723073\pi\)
\(182\) −1.65595e11 −0.0614687
\(183\) 7.92437e11 0.285420
\(184\) 3.66313e12 1.28043
\(185\) 6.53308e11 0.221653
\(186\) 6.30744e11 0.207746
\(187\) −3.27208e12 −1.04639
\(188\) −3.23265e11 −0.100390
\(189\) −4.60889e11 −0.139013
\(190\) 1.49309e12 0.437462
\(191\) 2.38809e12 0.679778 0.339889 0.940465i \(-0.389610\pi\)
0.339889 + 0.940465i \(0.389610\pi\)
\(192\) 3.55085e11 0.0982146
\(193\) 5.57651e12 1.49899 0.749493 0.662012i \(-0.230297\pi\)
0.749493 + 0.662012i \(0.230297\pi\)
\(194\) −2.06923e12 −0.540629
\(195\) −2.69668e11 −0.0684920
\(196\) 1.62946e12 0.402378
\(197\) −8.64153e11 −0.207504 −0.103752 0.994603i \(-0.533085\pi\)
−0.103752 + 0.994603i \(0.533085\pi\)
\(198\) −3.85711e11 −0.0900749
\(199\) 3.45541e11 0.0784889 0.0392445 0.999230i \(-0.487505\pi\)
0.0392445 + 0.999230i \(0.487505\pi\)
\(200\) 2.29576e12 0.507295
\(201\) −2.71596e12 −0.583910
\(202\) −2.29363e12 −0.479833
\(203\) 5.09075e12 1.03646
\(204\) 3.78044e12 0.749164
\(205\) 4.72635e12 0.911758
\(206\) 2.50622e12 0.470706
\(207\) −3.18239e12 −0.581991
\(208\) −6.59040e11 −0.117372
\(209\) 7.73871e12 1.34234
\(210\) 5.45775e11 0.0922162
\(211\) 6.76550e12 1.11364 0.556822 0.830632i \(-0.312021\pi\)
0.556822 + 0.830632i \(0.312021\pi\)
\(212\) 4.52941e12 0.726432
\(213\) 4.40809e12 0.688911
\(214\) −5.36717e11 −0.0817465
\(215\) 2.51764e12 0.373751
\(216\) 9.75283e11 0.141135
\(217\) −4.62580e12 −0.652617
\(218\) 2.41440e12 0.332123
\(219\) 1.13279e12 0.151952
\(220\) −2.42287e12 −0.316960
\(221\) −2.58252e12 −0.329523
\(222\) 7.37512e11 0.0917968
\(223\) 3.60649e12 0.437933 0.218967 0.975732i \(-0.429731\pi\)
0.218967 + 0.975732i \(0.429731\pi\)
\(224\) 5.80496e12 0.687758
\(225\) −1.99447e12 −0.230581
\(226\) −2.95859e12 −0.333802
\(227\) −2.38826e12 −0.262990 −0.131495 0.991317i \(-0.541978\pi\)
−0.131495 + 0.991317i \(0.541978\pi\)
\(228\) −8.94102e12 −0.961049
\(229\) 5.83285e12 0.612048 0.306024 0.952024i \(-0.401001\pi\)
0.306024 + 0.952024i \(0.401001\pi\)
\(230\) 3.76851e12 0.386071
\(231\) 2.82876e12 0.282964
\(232\) −1.07725e13 −1.05228
\(233\) −6.07236e12 −0.579295 −0.289648 0.957133i \(-0.593538\pi\)
−0.289648 + 0.957133i \(0.593538\pi\)
\(234\) −3.04426e11 −0.0283657
\(235\) −7.27823e11 −0.0662449
\(236\) 1.23193e12 0.109539
\(237\) −9.37189e12 −0.814163
\(238\) 5.22669e12 0.443663
\(239\) −5.24155e12 −0.434782 −0.217391 0.976085i \(-0.569755\pi\)
−0.217391 + 0.976085i \(0.569755\pi\)
\(240\) 2.17209e12 0.176083
\(241\) 1.50319e13 1.19103 0.595513 0.803346i \(-0.296949\pi\)
0.595513 + 0.803346i \(0.296949\pi\)
\(242\) −2.77494e12 −0.214916
\(243\) −8.47289e11 −0.0641500
\(244\) 5.61931e12 0.415948
\(245\) 3.66868e12 0.265520
\(246\) 5.33552e12 0.377602
\(247\) 6.10784e12 0.422722
\(248\) 9.78861e12 0.662578
\(249\) −1.08961e13 −0.721400
\(250\) 5.77609e12 0.374079
\(251\) 2.00908e13 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(252\) −3.26825e12 −0.202587
\(253\) 1.95323e13 1.18465
\(254\) 5.21505e12 0.309509
\(255\) 8.51158e12 0.494356
\(256\) −4.15300e12 −0.236071
\(257\) −1.74585e13 −0.971349 −0.485674 0.874140i \(-0.661426\pi\)
−0.485674 + 0.874140i \(0.661426\pi\)
\(258\) 2.84214e12 0.154788
\(259\) −5.40883e12 −0.288373
\(260\) −1.91227e12 −0.0998148
\(261\) 9.35873e12 0.478293
\(262\) 1.05470e13 0.527801
\(263\) −1.19269e13 −0.584482 −0.292241 0.956345i \(-0.594401\pi\)
−0.292241 + 0.956345i \(0.594401\pi\)
\(264\) −5.98592e12 −0.287282
\(265\) 1.01979e13 0.479355
\(266\) −1.23615e13 −0.569144
\(267\) −2.16647e13 −0.977104
\(268\) −1.92594e13 −0.850944
\(269\) −1.82284e13 −0.789063 −0.394531 0.918882i \(-0.629093\pi\)
−0.394531 + 0.918882i \(0.629093\pi\)
\(270\) 1.00334e12 0.0425547
\(271\) 5.45181e12 0.226574 0.113287 0.993562i \(-0.463862\pi\)
0.113287 + 0.993562i \(0.463862\pi\)
\(272\) 2.08014e13 0.847155
\(273\) 2.23262e12 0.0891089
\(274\) 3.70102e12 0.144775
\(275\) 1.22413e13 0.469350
\(276\) −2.25669e13 −0.848148
\(277\) −1.60314e13 −0.590652 −0.295326 0.955397i \(-0.595428\pi\)
−0.295326 + 0.955397i \(0.595428\pi\)
\(278\) 3.35114e12 0.121045
\(279\) −8.50397e12 −0.301161
\(280\) 8.46996e12 0.294112
\(281\) 3.31008e13 1.12708 0.563538 0.826090i \(-0.309440\pi\)
0.563538 + 0.826090i \(0.309440\pi\)
\(282\) −8.21632e11 −0.0274351
\(283\) 4.22069e13 1.38216 0.691079 0.722779i \(-0.257136\pi\)
0.691079 + 0.722779i \(0.257136\pi\)
\(284\) 3.12586e13 1.00396
\(285\) −2.01305e13 −0.634173
\(286\) 1.86845e12 0.0577389
\(287\) −3.91301e13 −1.18621
\(288\) 1.06717e13 0.317377
\(289\) 4.72405e13 1.37840
\(290\) −1.10824e13 −0.317282
\(291\) 2.78983e13 0.783730
\(292\) 8.03280e12 0.221443
\(293\) 2.36378e13 0.639492 0.319746 0.947503i \(-0.396402\pi\)
0.319746 + 0.947503i \(0.396402\pi\)
\(294\) 4.14153e12 0.109964
\(295\) 2.77365e12 0.0722822
\(296\) 1.14456e13 0.292774
\(297\) 5.20034e12 0.130578
\(298\) 2.77569e12 0.0684197
\(299\) 1.54160e13 0.373062
\(300\) −1.41431e13 −0.336030
\(301\) −2.08439e13 −0.486254
\(302\) 8.50083e12 0.194726
\(303\) 3.09237e13 0.695596
\(304\) −4.91967e13 −1.08675
\(305\) 1.26518e13 0.274474
\(306\) 9.60863e12 0.204736
\(307\) −7.37980e12 −0.154449 −0.0772243 0.997014i \(-0.524606\pi\)
−0.0772243 + 0.997014i \(0.524606\pi\)
\(308\) 2.00593e13 0.412369
\(309\) −3.37900e13 −0.682365
\(310\) 1.00702e13 0.199779
\(311\) −2.70914e13 −0.528019 −0.264010 0.964520i \(-0.585045\pi\)
−0.264010 + 0.964520i \(0.585045\pi\)
\(312\) −4.72443e12 −0.0904689
\(313\) −6.62339e13 −1.24620 −0.623098 0.782144i \(-0.714126\pi\)
−0.623098 + 0.782144i \(0.714126\pi\)
\(314\) −4.46955e12 −0.0826324
\(315\) −7.35838e12 −0.133682
\(316\) −6.64577e13 −1.18650
\(317\) −2.57294e13 −0.451445 −0.225722 0.974192i \(-0.572474\pi\)
−0.225722 + 0.974192i \(0.572474\pi\)
\(318\) 1.15123e13 0.198523
\(319\) −5.74403e13 −0.973572
\(320\) 5.66916e12 0.0944483
\(321\) 7.23626e12 0.118505
\(322\) −3.12001e13 −0.502283
\(323\) −1.92783e14 −3.05108
\(324\) −6.00827e12 −0.0934872
\(325\) 9.66153e12 0.147804
\(326\) 1.23414e13 0.185638
\(327\) −3.25521e13 −0.481466
\(328\) 8.28028e13 1.20431
\(329\) 6.02575e12 0.0861854
\(330\) −6.15812e12 −0.0866207
\(331\) −5.48873e13 −0.759308 −0.379654 0.925129i \(-0.623957\pi\)
−0.379654 + 0.925129i \(0.623957\pi\)
\(332\) −7.72664e13 −1.05131
\(333\) −9.94348e12 −0.133075
\(334\) 3.24299e13 0.426914
\(335\) −4.33621e13 −0.561518
\(336\) −1.79831e13 −0.229086
\(337\) −9.79581e13 −1.22765 −0.613827 0.789441i \(-0.710371\pi\)
−0.613827 + 0.789441i \(0.710371\pi\)
\(338\) −3.08262e13 −0.380082
\(339\) 3.98891e13 0.483901
\(340\) 6.03572e13 0.720435
\(341\) 5.21942e13 0.613017
\(342\) −2.27251e13 −0.262641
\(343\) −9.38856e13 −1.06778
\(344\) 4.41076e13 0.493676
\(345\) −5.08088e13 −0.559673
\(346\) 3.71084e12 0.0402303
\(347\) 7.13794e12 0.0761659 0.0380829 0.999275i \(-0.487875\pi\)
0.0380829 + 0.999275i \(0.487875\pi\)
\(348\) 6.63644e13 0.697027
\(349\) −1.62450e14 −1.67950 −0.839748 0.542976i \(-0.817298\pi\)
−0.839748 + 0.542976i \(0.817298\pi\)
\(350\) −1.95537e13 −0.199001
\(351\) 4.10441e12 0.0411208
\(352\) −6.54990e13 −0.646026
\(353\) 1.51005e14 1.46632 0.733162 0.680054i \(-0.238044\pi\)
0.733162 + 0.680054i \(0.238044\pi\)
\(354\) 3.13115e12 0.0299354
\(355\) 7.03780e13 0.662492
\(356\) −1.53628e14 −1.42395
\(357\) −7.04686e13 −0.643162
\(358\) −6.22037e13 −0.559062
\(359\) 2.50228e13 0.221471 0.110736 0.993850i \(-0.464679\pi\)
0.110736 + 0.993850i \(0.464679\pi\)
\(360\) 1.55710e13 0.135723
\(361\) 3.39455e14 2.91402
\(362\) −6.07501e13 −0.513630
\(363\) 3.74131e13 0.311556
\(364\) 1.58319e13 0.129860
\(365\) 1.80857e13 0.146125
\(366\) 1.42824e13 0.113673
\(367\) 1.47297e14 1.15487 0.577433 0.816438i \(-0.304055\pi\)
0.577433 + 0.816438i \(0.304055\pi\)
\(368\) −1.24171e14 −0.959086
\(369\) −7.19359e13 −0.547395
\(370\) 1.17748e13 0.0882766
\(371\) −8.44296e13 −0.623646
\(372\) −6.03032e13 −0.438888
\(373\) −7.52002e13 −0.539288 −0.269644 0.962960i \(-0.586906\pi\)
−0.269644 + 0.962960i \(0.586906\pi\)
\(374\) −5.89741e13 −0.416743
\(375\) −7.78758e13 −0.542289
\(376\) −1.27510e13 −0.0875009
\(377\) −4.53352e13 −0.306590
\(378\) −8.30680e12 −0.0553642
\(379\) 1.75453e14 1.15251 0.576257 0.817269i \(-0.304513\pi\)
0.576257 + 0.817269i \(0.304513\pi\)
\(380\) −1.42749e14 −0.924194
\(381\) −7.03117e13 −0.448684
\(382\) 4.30416e13 0.270732
\(383\) −2.55697e14 −1.58537 −0.792687 0.609629i \(-0.791319\pi\)
−0.792687 + 0.609629i \(0.791319\pi\)
\(384\) 9.63409e13 0.588829
\(385\) 4.51630e13 0.272112
\(386\) 1.00508e14 0.596994
\(387\) −3.83190e13 −0.224390
\(388\) 1.97832e14 1.14215
\(389\) 2.60045e14 1.48022 0.740108 0.672488i \(-0.234774\pi\)
0.740108 + 0.672488i \(0.234774\pi\)
\(390\) −4.86035e12 −0.0272780
\(391\) −4.86578e14 −2.69265
\(392\) 6.42731e13 0.350716
\(393\) −1.42199e14 −0.765134
\(394\) −1.55750e13 −0.0826415
\(395\) −1.49628e14 −0.782941
\(396\) 3.68765e13 0.190295
\(397\) −1.25874e14 −0.640602 −0.320301 0.947316i \(-0.603784\pi\)
−0.320301 + 0.947316i \(0.603784\pi\)
\(398\) 6.22784e12 0.0312594
\(399\) 1.66663e14 0.825067
\(400\) −7.78206e13 −0.379983
\(401\) −3.18523e13 −0.153407 −0.0767037 0.997054i \(-0.524440\pi\)
−0.0767037 + 0.997054i \(0.524440\pi\)
\(402\) −4.89510e13 −0.232551
\(403\) 4.11947e13 0.193047
\(404\) 2.19285e14 1.01371
\(405\) −1.35275e13 −0.0616900
\(406\) 9.17528e13 0.412787
\(407\) 6.10293e13 0.270875
\(408\) 1.49118e14 0.652979
\(409\) −2.50325e14 −1.08150 −0.540749 0.841184i \(-0.681859\pi\)
−0.540749 + 0.841184i \(0.681859\pi\)
\(410\) 8.51849e13 0.363121
\(411\) −4.98988e13 −0.209875
\(412\) −2.39611e14 −0.994425
\(413\) −2.29635e13 −0.0940399
\(414\) −5.73575e13 −0.231787
\(415\) −1.73964e14 −0.693735
\(416\) −5.16956e13 −0.203442
\(417\) −4.51816e13 −0.175474
\(418\) 1.39478e14 0.534609
\(419\) 2.94744e13 0.111498 0.0557491 0.998445i \(-0.482245\pi\)
0.0557491 + 0.998445i \(0.482245\pi\)
\(420\) −5.21796e13 −0.194818
\(421\) 4.40544e14 1.62344 0.811722 0.584044i \(-0.198530\pi\)
0.811722 + 0.584044i \(0.198530\pi\)
\(422\) 1.21937e14 0.443525
\(423\) 1.10776e13 0.0397717
\(424\) 1.78661e14 0.633165
\(425\) −3.04948e14 −1.06681
\(426\) 7.94489e13 0.274369
\(427\) −1.04746e14 −0.357094
\(428\) 5.13136e13 0.172700
\(429\) −2.51913e13 −0.0837019
\(430\) 4.53765e13 0.148852
\(431\) −1.08455e14 −0.351256 −0.175628 0.984457i \(-0.556196\pi\)
−0.175628 + 0.984457i \(0.556196\pi\)
\(432\) −3.30597e13 −0.105715
\(433\) −2.49683e14 −0.788326 −0.394163 0.919041i \(-0.628965\pi\)
−0.394163 + 0.919041i \(0.628965\pi\)
\(434\) −8.33728e13 −0.259915
\(435\) 1.49418e14 0.459951
\(436\) −2.30833e14 −0.701651
\(437\) 1.15079e15 3.45421
\(438\) 2.04167e13 0.0605171
\(439\) 5.41840e14 1.58605 0.793024 0.609190i \(-0.208505\pi\)
0.793024 + 0.609190i \(0.208505\pi\)
\(440\) −9.55689e13 −0.276266
\(441\) −5.58381e13 −0.159411
\(442\) −4.65458e13 −0.131238
\(443\) −3.54094e14 −0.986048 −0.493024 0.870016i \(-0.664109\pi\)
−0.493024 + 0.870016i \(0.664109\pi\)
\(444\) −7.05110e13 −0.193932
\(445\) −3.45890e14 −0.939633
\(446\) 6.50013e13 0.174413
\(447\) −3.74231e13 −0.0991856
\(448\) −4.69358e13 −0.122878
\(449\) −4.79011e14 −1.23877 −0.619385 0.785087i \(-0.712618\pi\)
−0.619385 + 0.785087i \(0.712618\pi\)
\(450\) −3.59471e13 −0.0918322
\(451\) 4.41515e14 1.11423
\(452\) 2.82861e14 0.705199
\(453\) −1.14612e14 −0.282287
\(454\) −4.30446e13 −0.104740
\(455\) 3.56452e13 0.0856917
\(456\) −3.52675e14 −0.837659
\(457\) 1.22388e14 0.287210 0.143605 0.989635i \(-0.454131\pi\)
0.143605 + 0.989635i \(0.454131\pi\)
\(458\) 1.05128e14 0.243757
\(459\) −1.29548e14 −0.296798
\(460\) −3.60294e14 −0.815623
\(461\) −8.45619e14 −1.89156 −0.945779 0.324812i \(-0.894699\pi\)
−0.945779 + 0.324812i \(0.894699\pi\)
\(462\) 5.09840e13 0.112695
\(463\) 7.32000e14 1.59888 0.799439 0.600747i \(-0.205130\pi\)
0.799439 + 0.600747i \(0.205130\pi\)
\(464\) 3.65161e14 0.788199
\(465\) −1.35771e14 −0.289612
\(466\) −1.09445e14 −0.230713
\(467\) −1.14805e14 −0.239175 −0.119588 0.992824i \(-0.538157\pi\)
−0.119588 + 0.992824i \(0.538157\pi\)
\(468\) 2.91051e13 0.0599262
\(469\) 3.59001e14 0.730541
\(470\) −1.31179e13 −0.0263830
\(471\) 6.02605e13 0.119789
\(472\) 4.85928e13 0.0954753
\(473\) 2.35187e14 0.456749
\(474\) −1.68913e14 −0.324253
\(475\) 7.21224e14 1.36853
\(476\) −4.99706e14 −0.937294
\(477\) −1.55214e14 −0.287792
\(478\) −9.44707e13 −0.173158
\(479\) 9.95247e14 1.80337 0.901686 0.432391i \(-0.142330\pi\)
0.901686 + 0.432391i \(0.142330\pi\)
\(480\) 1.70381e14 0.305206
\(481\) 4.81679e13 0.0853020
\(482\) 2.70927e14 0.474344
\(483\) 4.20653e14 0.728141
\(484\) 2.65303e14 0.454038
\(485\) 4.45414e14 0.753676
\(486\) −1.52710e13 −0.0255487
\(487\) −5.98697e13 −0.0990371 −0.0495186 0.998773i \(-0.515769\pi\)
−0.0495186 + 0.998773i \(0.515769\pi\)
\(488\) 2.21651e14 0.362545
\(489\) −1.66392e14 −0.269113
\(490\) 6.61222e13 0.105747
\(491\) −5.55539e13 −0.0878550 −0.0439275 0.999035i \(-0.513987\pi\)
−0.0439275 + 0.999035i \(0.513987\pi\)
\(492\) −5.10111e14 −0.797731
\(493\) 1.43092e15 2.21288
\(494\) 1.10084e14 0.168355
\(495\) 8.30266e13 0.125571
\(496\) −3.31810e14 −0.496296
\(497\) −5.82669e14 −0.861910
\(498\) −1.96386e14 −0.287308
\(499\) 1.10932e14 0.160510 0.0802550 0.996774i \(-0.474427\pi\)
0.0802550 + 0.996774i \(0.474427\pi\)
\(500\) −5.52231e14 −0.790289
\(501\) −4.37235e14 −0.618881
\(502\) 3.62105e14 0.506949
\(503\) −6.51056e14 −0.901560 −0.450780 0.892635i \(-0.648854\pi\)
−0.450780 + 0.892635i \(0.648854\pi\)
\(504\) −1.28915e14 −0.176577
\(505\) 4.93716e14 0.668921
\(506\) 3.52039e14 0.471805
\(507\) 4.15613e14 0.550991
\(508\) −4.98593e14 −0.653876
\(509\) −2.39761e13 −0.0311050 −0.0155525 0.999879i \(-0.504951\pi\)
−0.0155525 + 0.999879i \(0.504951\pi\)
\(510\) 1.53408e14 0.196885
\(511\) −1.49734e14 −0.190110
\(512\) 7.37109e14 0.925863
\(513\) 3.06390e14 0.380741
\(514\) −3.14662e14 −0.386854
\(515\) −5.39479e14 −0.656197
\(516\) −2.71727e14 −0.327008
\(517\) −6.79902e13 −0.0809559
\(518\) −9.74856e13 −0.114849
\(519\) −5.00312e13 −0.0583204
\(520\) −7.54285e13 −0.0869996
\(521\) −2.07535e14 −0.236856 −0.118428 0.992963i \(-0.537785\pi\)
−0.118428 + 0.992963i \(0.537785\pi\)
\(522\) 1.68676e14 0.190487
\(523\) 2.09577e14 0.234199 0.117099 0.993120i \(-0.462640\pi\)
0.117099 + 0.993120i \(0.462640\pi\)
\(524\) −1.00836e15 −1.11505
\(525\) 2.63632e14 0.288484
\(526\) −2.14964e14 −0.232779
\(527\) −1.30023e15 −1.39336
\(528\) 2.02908e14 0.215185
\(529\) 1.95176e15 2.04842
\(530\) 1.83801e14 0.190910
\(531\) −4.22156e13 −0.0433963
\(532\) 1.18184e15 1.20239
\(533\) 3.48470e14 0.350885
\(534\) −3.90472e14 −0.389146
\(535\) 1.15531e14 0.113960
\(536\) −7.59678e14 −0.741691
\(537\) 8.38659e14 0.810451
\(538\) −3.28538e14 −0.314256
\(539\) 3.42713e14 0.324483
\(540\) −9.59259e13 −0.0899021
\(541\) −4.31551e12 −0.00400357 −0.00200178 0.999998i \(-0.500637\pi\)
−0.00200178 + 0.999998i \(0.500637\pi\)
\(542\) 9.82602e13 0.0902364
\(543\) 8.19061e14 0.744590
\(544\) 1.63168e15 1.46838
\(545\) −5.19714e14 −0.463003
\(546\) 4.02395e13 0.0354890
\(547\) 1.27963e15 1.11726 0.558630 0.829417i \(-0.311327\pi\)
0.558630 + 0.829417i \(0.311327\pi\)
\(548\) −3.53841e14 −0.305855
\(549\) −1.92562e14 −0.164787
\(550\) 2.20630e14 0.186926
\(551\) −3.38423e15 −2.83875
\(552\) −8.90141e14 −0.739254
\(553\) 1.23879e15 1.01862
\(554\) −2.88940e14 −0.235236
\(555\) −1.58754e14 −0.127971
\(556\) −3.20391e14 −0.255722
\(557\) −2.47444e14 −0.195557 −0.0977785 0.995208i \(-0.531174\pi\)
−0.0977785 + 0.995208i \(0.531174\pi\)
\(558\) −1.53271e14 −0.119942
\(559\) 1.85624e14 0.143836
\(560\) −2.87111e14 −0.220301
\(561\) 7.95116e14 0.604136
\(562\) 5.96589e14 0.448875
\(563\) −7.22104e14 −0.538027 −0.269013 0.963136i \(-0.586698\pi\)
−0.269013 + 0.963136i \(0.586698\pi\)
\(564\) 7.85533e13 0.0579602
\(565\) 6.36855e14 0.465344
\(566\) 7.60712e14 0.550465
\(567\) 1.11996e14 0.0802594
\(568\) 1.23298e15 0.875065
\(569\) 9.44684e14 0.664002 0.332001 0.943279i \(-0.392276\pi\)
0.332001 + 0.943279i \(0.392276\pi\)
\(570\) −3.62821e14 −0.252569
\(571\) −9.77473e13 −0.0673916 −0.0336958 0.999432i \(-0.510728\pi\)
−0.0336958 + 0.999432i \(0.510728\pi\)
\(572\) −1.78636e14 −0.121981
\(573\) −5.80306e14 −0.392470
\(574\) −7.05258e14 −0.472425
\(575\) 1.82035e15 1.20776
\(576\) −8.62858e13 −0.0567042
\(577\) 1.50543e14 0.0979924 0.0489962 0.998799i \(-0.484398\pi\)
0.0489962 + 0.998799i \(0.484398\pi\)
\(578\) 8.51436e14 0.548970
\(579\) −1.35509e15 −0.865440
\(580\) 1.05955e15 0.670297
\(581\) 1.44027e15 0.902558
\(582\) 5.02823e14 0.312132
\(583\) 9.52642e14 0.585805
\(584\) 3.16850e14 0.193012
\(585\) 6.55294e13 0.0395439
\(586\) 4.26034e14 0.254687
\(587\) −6.10101e14 −0.361320 −0.180660 0.983546i \(-0.557823\pi\)
−0.180660 + 0.983546i \(0.557823\pi\)
\(588\) −3.95958e14 −0.232313
\(589\) 3.07514e15 1.78744
\(590\) 4.99907e13 0.0287875
\(591\) 2.09989e14 0.119802
\(592\) −3.87977e14 −0.219299
\(593\) −7.60782e13 −0.0426049 −0.0213024 0.999773i \(-0.506781\pi\)
−0.0213024 + 0.999773i \(0.506781\pi\)
\(594\) 9.37278e13 0.0520048
\(595\) −1.12508e15 −0.618498
\(596\) −2.65374e14 −0.144545
\(597\) −8.39666e13 −0.0453156
\(598\) 2.77849e14 0.148577
\(599\) 3.07445e15 1.62899 0.814497 0.580168i \(-0.197013\pi\)
0.814497 + 0.580168i \(0.197013\pi\)
\(600\) −5.57869e14 −0.292887
\(601\) 2.51066e15 1.30611 0.653053 0.757312i \(-0.273488\pi\)
0.653053 + 0.757312i \(0.273488\pi\)
\(602\) −3.75678e14 −0.193658
\(603\) 6.59979e14 0.337121
\(604\) −8.12735e14 −0.411382
\(605\) 5.97323e14 0.299609
\(606\) 5.57351e14 0.277032
\(607\) 3.18922e15 1.57089 0.785446 0.618931i \(-0.212434\pi\)
0.785446 + 0.618931i \(0.212434\pi\)
\(608\) −3.85903e15 −1.88369
\(609\) −1.23705e15 −0.598402
\(610\) 2.28028e14 0.109314
\(611\) −5.36618e13 −0.0254940
\(612\) −9.18648e14 −0.432530
\(613\) 1.38627e15 0.646868 0.323434 0.946251i \(-0.395163\pi\)
0.323434 + 0.946251i \(0.395163\pi\)
\(614\) −1.33009e14 −0.0615115
\(615\) −1.14850e15 −0.526403
\(616\) 7.91228e14 0.359425
\(617\) 1.52685e15 0.687429 0.343715 0.939074i \(-0.388315\pi\)
0.343715 + 0.939074i \(0.388315\pi\)
\(618\) −6.09011e14 −0.271762
\(619\) −4.10703e15 −1.81648 −0.908238 0.418455i \(-0.862572\pi\)
−0.908238 + 0.418455i \(0.862572\pi\)
\(620\) −9.62778e14 −0.422058
\(621\) 7.73320e14 0.336013
\(622\) −4.88280e14 −0.210292
\(623\) 2.86368e15 1.22247
\(624\) 1.60147e14 0.0677646
\(625\) 4.05905e14 0.170249
\(626\) −1.19376e15 −0.496316
\(627\) −1.88051e15 −0.775003
\(628\) 4.27318e14 0.174571
\(629\) −1.52033e15 −0.615686
\(630\) −1.32623e14 −0.0532410
\(631\) −4.75010e14 −0.189035 −0.0945173 0.995523i \(-0.530131\pi\)
−0.0945173 + 0.995523i \(0.530131\pi\)
\(632\) −2.62140e15 −1.03416
\(633\) −1.64402e15 −0.642963
\(634\) −4.63732e14 −0.179795
\(635\) −1.12257e15 −0.431477
\(636\) −1.10065e15 −0.419406
\(637\) 2.70489e14 0.102184
\(638\) −1.03527e15 −0.387740
\(639\) −1.07117e15 −0.397743
\(640\) 1.53814e15 0.566248
\(641\) 3.96744e15 1.44808 0.724038 0.689760i \(-0.242284\pi\)
0.724038 + 0.689760i \(0.242284\pi\)
\(642\) 1.30422e14 0.0471964
\(643\) 5.00082e15 1.79424 0.897120 0.441787i \(-0.145655\pi\)
0.897120 + 0.441787i \(0.145655\pi\)
\(644\) 2.98293e15 1.06113
\(645\) −6.11786e14 −0.215785
\(646\) −3.47460e15 −1.21514
\(647\) −5.11962e14 −0.177527 −0.0887634 0.996053i \(-0.528292\pi\)
−0.0887634 + 0.996053i \(0.528292\pi\)
\(648\) −2.36994e14 −0.0814844
\(649\) 2.59103e14 0.0883338
\(650\) 1.74134e14 0.0588653
\(651\) 1.12407e15 0.376789
\(652\) −1.17992e15 −0.392184
\(653\) −5.33380e15 −1.75798 −0.878990 0.476840i \(-0.841782\pi\)
−0.878990 + 0.476840i \(0.841782\pi\)
\(654\) −5.86700e14 −0.191751
\(655\) −2.27030e15 −0.735792
\(656\) −2.80681e15 −0.902075
\(657\) −2.75267e14 −0.0877294
\(658\) 1.08605e14 0.0343247
\(659\) −3.31774e15 −1.03985 −0.519927 0.854211i \(-0.674041\pi\)
−0.519927 + 0.854211i \(0.674041\pi\)
\(660\) 5.88757e14 0.182997
\(661\) −2.53460e15 −0.781270 −0.390635 0.920546i \(-0.627745\pi\)
−0.390635 + 0.920546i \(0.627745\pi\)
\(662\) −9.89257e14 −0.302406
\(663\) 6.27552e14 0.190250
\(664\) −3.04774e15 −0.916333
\(665\) 2.66088e15 0.793427
\(666\) −1.79215e14 −0.0529989
\(667\) −8.54171e15 −2.50526
\(668\) −3.10051e15 −0.901909
\(669\) −8.76377e14 −0.252841
\(670\) −7.81533e14 −0.223633
\(671\) 1.18187e15 0.335426
\(672\) −1.41061e15 −0.397077
\(673\) 5.86295e15 1.63694 0.818471 0.574548i \(-0.194822\pi\)
0.818471 + 0.574548i \(0.194822\pi\)
\(674\) −1.76554e15 −0.488932
\(675\) 4.84655e14 0.133126
\(676\) 2.94718e15 0.802972
\(677\) −1.87975e13 −0.00507999 −0.00254000 0.999997i \(-0.500809\pi\)
−0.00254000 + 0.999997i \(0.500809\pi\)
\(678\) 7.18938e14 0.192721
\(679\) −3.68765e15 −0.980541
\(680\) 2.38076e15 0.627938
\(681\) 5.80347e14 0.151837
\(682\) 9.40717e14 0.244143
\(683\) −6.07603e15 −1.56425 −0.782125 0.623121i \(-0.785864\pi\)
−0.782125 + 0.623121i \(0.785864\pi\)
\(684\) 2.17267e15 0.554862
\(685\) −7.96665e14 −0.201826
\(686\) −1.69214e15 −0.425259
\(687\) −1.41738e15 −0.353366
\(688\) −1.49514e15 −0.369781
\(689\) 7.51881e14 0.184477
\(690\) −9.15749e14 −0.222898
\(691\) 1.75688e15 0.424242 0.212121 0.977243i \(-0.431963\pi\)
0.212121 + 0.977243i \(0.431963\pi\)
\(692\) −3.54780e14 −0.0849916
\(693\) −6.87389e14 −0.163369
\(694\) 1.28650e14 0.0303342
\(695\) −7.21352e14 −0.168745
\(696\) 2.61772e15 0.607535
\(697\) −1.09988e16 −2.53259
\(698\) −2.92790e15 −0.668885
\(699\) 1.47558e15 0.334456
\(700\) 1.86946e15 0.420414
\(701\) −3.30315e15 −0.737020 −0.368510 0.929624i \(-0.620132\pi\)
−0.368510 + 0.929624i \(0.620132\pi\)
\(702\) 7.39755e13 0.0163770
\(703\) 3.59569e15 0.789819
\(704\) 5.29589e14 0.115422
\(705\) 1.76861e14 0.0382465
\(706\) 2.72162e15 0.583986
\(707\) −4.08755e15 −0.870274
\(708\) −2.99358e14 −0.0632424
\(709\) 4.88990e15 1.02505 0.512526 0.858672i \(-0.328710\pi\)
0.512526 + 0.858672i \(0.328710\pi\)
\(710\) 1.26845e15 0.263848
\(711\) 2.27737e15 0.470057
\(712\) −6.05980e15 −1.24113
\(713\) 7.76158e15 1.57746
\(714\) −1.27009e15 −0.256149
\(715\) −4.02195e14 −0.0804921
\(716\) 5.94708e15 1.18109
\(717\) 1.27370e15 0.251021
\(718\) 4.50997e14 0.0882042
\(719\) 7.32137e15 1.42097 0.710483 0.703715i \(-0.248477\pi\)
0.710483 + 0.703715i \(0.248477\pi\)
\(720\) −5.27819e14 −0.101661
\(721\) 4.46642e15 0.853720
\(722\) 6.11813e15 1.16055
\(723\) −3.65276e15 −0.687639
\(724\) 5.80811e15 1.08511
\(725\) −5.35326e15 −0.992566
\(726\) 6.74311e14 0.124082
\(727\) 8.09516e15 1.47838 0.739190 0.673497i \(-0.235209\pi\)
0.739190 + 0.673497i \(0.235209\pi\)
\(728\) 6.24483e14 0.113187
\(729\) 2.05891e14 0.0370370
\(730\) 3.25966e14 0.0581964
\(731\) −5.85886e15 −1.03817
\(732\) −1.36549e15 −0.240148
\(733\) −3.29018e15 −0.574313 −0.287156 0.957884i \(-0.592710\pi\)
−0.287156 + 0.957884i \(0.592710\pi\)
\(734\) 2.65480e15 0.459942
\(735\) −8.91490e14 −0.153298
\(736\) −9.74008e15 −1.66240
\(737\) −4.05070e15 −0.686213
\(738\) −1.29653e15 −0.218008
\(739\) 2.17098e15 0.362336 0.181168 0.983452i \(-0.442012\pi\)
0.181168 + 0.983452i \(0.442012\pi\)
\(740\) −1.12575e15 −0.186495
\(741\) −1.48421e15 −0.244058
\(742\) −1.52171e15 −0.248377
\(743\) 4.01946e15 0.651222 0.325611 0.945504i \(-0.394430\pi\)
0.325611 + 0.945504i \(0.394430\pi\)
\(744\) −2.37863e15 −0.382539
\(745\) −5.97483e14 −0.0953820
\(746\) −1.35537e15 −0.214780
\(747\) 2.64776e15 0.416500
\(748\) 5.63831e15 0.880421
\(749\) −9.56501e14 −0.148264
\(750\) −1.40359e15 −0.215975
\(751\) 1.10261e16 1.68423 0.842116 0.539296i \(-0.181310\pi\)
0.842116 + 0.539296i \(0.181310\pi\)
\(752\) 4.32229e14 0.0655414
\(753\) −4.88206e15 −0.734905
\(754\) −8.17096e14 −0.122104
\(755\) −1.82985e15 −0.271461
\(756\) 7.94184e14 0.116964
\(757\) −1.19658e16 −1.74949 −0.874747 0.484579i \(-0.838973\pi\)
−0.874747 + 0.484579i \(0.838973\pi\)
\(758\) 3.16227e15 0.459006
\(759\) −4.74635e15 −0.683958
\(760\) −5.63067e15 −0.805537
\(761\) 4.06587e15 0.577481 0.288740 0.957407i \(-0.406764\pi\)
0.288740 + 0.957407i \(0.406764\pi\)
\(762\) −1.26726e15 −0.178695
\(763\) 4.30279e15 0.602372
\(764\) −4.11505e15 −0.571955
\(765\) −2.06831e15 −0.285416
\(766\) −4.60853e15 −0.631399
\(767\) 2.04499e14 0.0278174
\(768\) 1.00918e15 0.136295
\(769\) −1.12489e16 −1.50839 −0.754195 0.656650i \(-0.771973\pi\)
−0.754195 + 0.656650i \(0.771973\pi\)
\(770\) 8.13991e14 0.108373
\(771\) 4.24242e15 0.560808
\(772\) −9.60921e15 −1.26122
\(773\) −1.25407e16 −1.63431 −0.817156 0.576417i \(-0.804451\pi\)
−0.817156 + 0.576417i \(0.804451\pi\)
\(774\) −6.90639e14 −0.0893667
\(775\) 4.86434e15 0.624977
\(776\) 7.80340e15 0.995507
\(777\) 1.31435e15 0.166492
\(778\) 4.68689e15 0.589518
\(779\) 2.60129e16 3.24888
\(780\) 4.64681e14 0.0576281
\(781\) 6.57441e15 0.809611
\(782\) −8.76980e15 −1.07239
\(783\) −2.27417e15 −0.276143
\(784\) −2.17870e15 −0.262700
\(785\) 9.62097e14 0.115196
\(786\) −2.56291e15 −0.304726
\(787\) 1.32562e16 1.56516 0.782579 0.622551i \(-0.213904\pi\)
0.782579 + 0.622551i \(0.213904\pi\)
\(788\) 1.48907e15 0.174591
\(789\) 2.89824e15 0.337451
\(790\) −2.69681e15 −0.311818
\(791\) −5.27261e15 −0.605418
\(792\) 1.45458e15 0.165863
\(793\) 9.32804e14 0.105630
\(794\) −2.26868e15 −0.255130
\(795\) −2.47808e15 −0.276756
\(796\) −5.95422e14 −0.0660394
\(797\) 1.25975e15 0.138759 0.0693797 0.997590i \(-0.477898\pi\)
0.0693797 + 0.997590i \(0.477898\pi\)
\(798\) 3.00384e15 0.328595
\(799\) 1.69374e15 0.184009
\(800\) −6.10430e15 −0.658629
\(801\) 5.26452e15 0.564131
\(802\) −5.74087e14 −0.0610968
\(803\) 1.68949e15 0.178574
\(804\) 4.68003e15 0.491293
\(805\) 6.71600e15 0.700218
\(806\) 7.42469e14 0.0768839
\(807\) 4.42951e15 0.455566
\(808\) 8.64962e15 0.883557
\(809\) 2.69532e15 0.273460 0.136730 0.990608i \(-0.456341\pi\)
0.136730 + 0.990608i \(0.456341\pi\)
\(810\) −2.43812e14 −0.0245690
\(811\) −7.63973e15 −0.764651 −0.382326 0.924028i \(-0.624877\pi\)
−0.382326 + 0.924028i \(0.624877\pi\)
\(812\) −8.77216e15 −0.872064
\(813\) −1.32479e15 −0.130812
\(814\) 1.09996e15 0.107880
\(815\) −2.65656e15 −0.258793
\(816\) −5.05473e15 −0.489105
\(817\) 1.38566e16 1.33179
\(818\) −4.51171e15 −0.430723
\(819\) −5.42528e14 −0.0514470
\(820\) −8.14423e15 −0.767139
\(821\) 2.91872e15 0.273089 0.136545 0.990634i \(-0.456400\pi\)
0.136545 + 0.990634i \(0.456400\pi\)
\(822\) −8.99347e14 −0.0835857
\(823\) −6.18239e15 −0.570765 −0.285382 0.958414i \(-0.592121\pi\)
−0.285382 + 0.958414i \(0.592121\pi\)
\(824\) −9.45135e15 −0.866750
\(825\) −2.97463e15 −0.270979
\(826\) −4.13880e14 −0.0374528
\(827\) −5.31928e15 −0.478160 −0.239080 0.971000i \(-0.576846\pi\)
−0.239080 + 0.971000i \(0.576846\pi\)
\(828\) 5.48375e15 0.489678
\(829\) −7.02110e15 −0.622809 −0.311405 0.950277i \(-0.600799\pi\)
−0.311405 + 0.950277i \(0.600799\pi\)
\(830\) −3.13542e15 −0.276291
\(831\) 3.89562e15 0.341013
\(832\) 4.17983e14 0.0363480
\(833\) −8.53748e15 −0.737534
\(834\) −8.14327e14 −0.0698852
\(835\) −6.98073e15 −0.595148
\(836\) −1.33350e16 −1.12943
\(837\) 2.06647e15 0.173875
\(838\) 5.31229e14 0.0444058
\(839\) −1.16618e16 −0.968443 −0.484222 0.874945i \(-0.660897\pi\)
−0.484222 + 0.874945i \(0.660897\pi\)
\(840\) −2.05820e15 −0.169806
\(841\) 1.29189e16 1.05888
\(842\) 7.94010e15 0.646561
\(843\) −8.04348e15 −0.650718
\(844\) −1.16580e16 −0.937003
\(845\) 6.63551e15 0.529862
\(846\) 1.99657e14 0.0158397
\(847\) −4.94532e15 −0.389795
\(848\) −6.05616e15 −0.474264
\(849\) −1.02563e16 −0.797989
\(850\) −5.49621e15 −0.424873
\(851\) 9.07541e15 0.697033
\(852\) −7.59583e15 −0.579639
\(853\) −8.18494e15 −0.620577 −0.310289 0.950642i \(-0.600426\pi\)
−0.310289 + 0.950642i \(0.600426\pi\)
\(854\) −1.88788e15 −0.142218
\(855\) 4.89171e15 0.366140
\(856\) 2.02404e15 0.150527
\(857\) 1.48309e16 1.09590 0.547952 0.836510i \(-0.315408\pi\)
0.547952 + 0.836510i \(0.315408\pi\)
\(858\) −4.54033e14 −0.0333355
\(859\) 1.84576e16 1.34652 0.673262 0.739404i \(-0.264893\pi\)
0.673262 + 0.739404i \(0.264893\pi\)
\(860\) −4.33829e15 −0.314468
\(861\) 9.50862e15 0.684857
\(862\) −1.95473e15 −0.139893
\(863\) 1.57578e16 1.12056 0.560281 0.828303i \(-0.310693\pi\)
0.560281 + 0.828303i \(0.310693\pi\)
\(864\) −2.59323e15 −0.183238
\(865\) −7.98780e14 −0.0560840
\(866\) −4.50014e15 −0.313963
\(867\) −1.14794e16 −0.795822
\(868\) 7.97098e15 0.549102
\(869\) −1.39776e16 −0.956807
\(870\) 2.69302e15 0.183183
\(871\) −3.19705e15 −0.216097
\(872\) −9.10509e15 −0.611566
\(873\) −6.77929e15 −0.452487
\(874\) 2.07412e16 1.37569
\(875\) 1.02938e16 0.678469
\(876\) −1.95197e15 −0.127850
\(877\) −1.76366e16 −1.14794 −0.573968 0.818878i \(-0.694597\pi\)
−0.573968 + 0.818878i \(0.694597\pi\)
\(878\) 9.76582e15 0.631668
\(879\) −5.74398e15 −0.369211
\(880\) 3.23955e15 0.206933
\(881\) 2.32278e16 1.47449 0.737243 0.675628i \(-0.236127\pi\)
0.737243 + 0.675628i \(0.236127\pi\)
\(882\) −1.00639e15 −0.0634878
\(883\) 2.21922e16 1.39128 0.695642 0.718389i \(-0.255120\pi\)
0.695642 + 0.718389i \(0.255120\pi\)
\(884\) 4.45008e15 0.277256
\(885\) −6.73998e14 −0.0417321
\(886\) −6.38199e15 −0.392709
\(887\) 1.93898e16 1.18575 0.592875 0.805295i \(-0.297993\pi\)
0.592875 + 0.805295i \(0.297993\pi\)
\(888\) −2.78127e15 −0.169033
\(889\) 9.29392e15 0.561357
\(890\) −6.23413e15 −0.374223
\(891\) −1.26368e15 −0.0753894
\(892\) −6.21454e15 −0.368470
\(893\) −4.00581e15 −0.236052
\(894\) −6.74492e14 −0.0395021
\(895\) 1.33897e16 0.779372
\(896\) −1.27345e16 −0.736696
\(897\) −3.74609e15 −0.215387
\(898\) −8.63342e15 −0.493359
\(899\) −2.28252e16 −1.29639
\(900\) 3.43678e15 0.194007
\(901\) −2.37317e16 −1.33151
\(902\) 7.95762e15 0.443759
\(903\) 5.06507e15 0.280739
\(904\) 1.11573e16 0.614658
\(905\) 1.30768e16 0.716036
\(906\) −2.06570e15 −0.112425
\(907\) −2.54420e16 −1.37629 −0.688146 0.725572i \(-0.741575\pi\)
−0.688146 + 0.725572i \(0.741575\pi\)
\(908\) 4.11534e15 0.221276
\(909\) −7.51446e15 −0.401603
\(910\) 6.42449e14 0.0341280
\(911\) −9.76594e15 −0.515659 −0.257830 0.966190i \(-0.583007\pi\)
−0.257830 + 0.966190i \(0.583007\pi\)
\(912\) 1.19548e16 0.627438
\(913\) −1.62509e16 −0.847792
\(914\) 2.20585e15 0.114386
\(915\) −3.07438e15 −0.158468
\(916\) −1.00509e16 −0.514968
\(917\) 1.87961e16 0.957274
\(918\) −2.33490e15 −0.118204
\(919\) 3.66644e16 1.84505 0.922527 0.385933i \(-0.126121\pi\)
0.922527 + 0.385933i \(0.126121\pi\)
\(920\) −1.42117e16 −0.710905
\(921\) 1.79329e15 0.0891709
\(922\) −1.52409e16 −0.753341
\(923\) 5.18891e15 0.254957
\(924\) −4.87440e15 −0.238081
\(925\) 5.68774e15 0.276160
\(926\) 1.31931e16 0.636778
\(927\) 8.21097e15 0.393964
\(928\) 2.86435e16 1.36619
\(929\) 1.21481e16 0.576001 0.288001 0.957630i \(-0.407009\pi\)
0.288001 + 0.957630i \(0.407009\pi\)
\(930\) −2.44706e15 −0.115342
\(931\) 2.01917e16 0.946130
\(932\) 1.04636e16 0.487410
\(933\) 6.58321e15 0.304852
\(934\) −2.06917e15 −0.0952552
\(935\) 1.26945e16 0.580969
\(936\) 1.14804e15 0.0522322
\(937\) −1.51708e16 −0.686183 −0.343091 0.939302i \(-0.611474\pi\)
−0.343091 + 0.939302i \(0.611474\pi\)
\(938\) 6.47042e15 0.290949
\(939\) 1.60948e16 0.719491
\(940\) 1.25415e15 0.0557375
\(941\) −3.17507e16 −1.40285 −0.701423 0.712745i \(-0.747451\pi\)
−0.701423 + 0.712745i \(0.747451\pi\)
\(942\) 1.08610e15 0.0477079
\(943\) 6.56559e16 2.86721
\(944\) −1.64718e15 −0.0715145
\(945\) 1.78809e15 0.0771816
\(946\) 4.23888e15 0.181907
\(947\) 2.85074e16 1.21628 0.608138 0.793831i \(-0.291917\pi\)
0.608138 + 0.793831i \(0.291917\pi\)
\(948\) 1.61492e16 0.685024
\(949\) 1.33344e15 0.0562354
\(950\) 1.29989e16 0.545039
\(951\) 6.25225e15 0.260642
\(952\) −1.97107e16 −0.816955
\(953\) −7.54001e15 −0.310714 −0.155357 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(954\) −2.79748e15 −0.114618
\(955\) −9.26495e15 −0.377420
\(956\) 9.03201e15 0.365819
\(957\) 1.39580e16 0.562092
\(958\) 1.79378e16 0.718220
\(959\) 6.59571e15 0.262578
\(960\) −1.37761e15 −0.0545297
\(961\) −4.66800e15 −0.183718
\(962\) 8.68149e14 0.0339728
\(963\) −1.75841e15 −0.0684188
\(964\) −2.59024e16 −1.00211
\(965\) −2.16349e16 −0.832252
\(966\) 7.58161e15 0.289993
\(967\) 4.65003e16 1.76852 0.884260 0.466995i \(-0.154663\pi\)
0.884260 + 0.466995i \(0.154663\pi\)
\(968\) 1.04647e16 0.395744
\(969\) 4.68462e16 1.76154
\(970\) 8.02789e15 0.300163
\(971\) 2.80464e15 0.104273 0.0521365 0.998640i \(-0.483397\pi\)
0.0521365 + 0.998640i \(0.483397\pi\)
\(972\) 1.46001e15 0.0539749
\(973\) 5.97218e15 0.219539
\(974\) −1.07906e15 −0.0394430
\(975\) −2.34775e15 −0.0853349
\(976\) −7.51344e15 −0.271559
\(977\) 3.02212e16 1.08615 0.543077 0.839683i \(-0.317259\pi\)
0.543077 + 0.839683i \(0.317259\pi\)
\(978\) −2.99896e15 −0.107178
\(979\) −3.23116e16 −1.14830
\(980\) −6.32171e15 −0.223404
\(981\) 7.91015e15 0.277975
\(982\) −1.00127e15 −0.0349896
\(983\) −6.55268e15 −0.227706 −0.113853 0.993498i \(-0.536319\pi\)
−0.113853 + 0.993498i \(0.536319\pi\)
\(984\) −2.01211e16 −0.695310
\(985\) 3.35261e15 0.115208
\(986\) 2.57901e16 0.881313
\(987\) −1.46426e15 −0.0497592
\(988\) −1.05248e16 −0.355672
\(989\) 3.49737e16 1.17534
\(990\) 1.49642e15 0.0500105
\(991\) −4.07952e16 −1.35583 −0.677913 0.735143i \(-0.737115\pi\)
−0.677913 + 0.735143i \(0.737115\pi\)
\(992\) −2.60274e16 −0.860235
\(993\) 1.33376e16 0.438387
\(994\) −1.05017e16 −0.343269
\(995\) −1.34058e15 −0.0435778
\(996\) 1.87757e16 0.606975
\(997\) −1.46734e16 −0.471746 −0.235873 0.971784i \(-0.575795\pi\)
−0.235873 + 0.971784i \(0.575795\pi\)
\(998\) 1.99937e15 0.0639255
\(999\) 2.41626e15 0.0768306
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.c.1.17 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.c.1.17 27 1.1 even 1 trivial