Properties

Label 177.12.a.c.1.16
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 1.16
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.89171 q^{2} -243.000 q^{3} -1985.72 q^{4} -1405.99 q^{5} -1917.69 q^{6} +25105.4 q^{7} -31832.9 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+7.89171 q^{2} -243.000 q^{3} -1985.72 q^{4} -1405.99 q^{5} -1917.69 q^{6} +25105.4 q^{7} -31832.9 q^{8} +59049.0 q^{9} -11095.6 q^{10} +251890. q^{11} +482530. q^{12} +1.97716e6 q^{13} +198125. q^{14} +341655. q^{15} +3.81554e6 q^{16} +384296. q^{17} +465997. q^{18} -1.00048e7 q^{19} +2.79190e6 q^{20} -6.10062e6 q^{21} +1.98784e6 q^{22} +5.12407e7 q^{23} +7.73541e6 q^{24} -4.68513e7 q^{25} +1.56031e7 q^{26} -1.43489e7 q^{27} -4.98524e7 q^{28} -1.79135e8 q^{29} +2.69624e6 q^{30} -1.54653e8 q^{31} +9.53050e7 q^{32} -6.12092e7 q^{33} +3.03275e6 q^{34} -3.52980e7 q^{35} -1.17255e8 q^{36} +6.49876e8 q^{37} -7.89553e7 q^{38} -4.80449e8 q^{39} +4.47567e7 q^{40} -1.77958e8 q^{41} -4.81443e7 q^{42} +1.32372e9 q^{43} -5.00183e8 q^{44} -8.30222e7 q^{45} +4.04377e8 q^{46} -2.57114e9 q^{47} -9.27176e8 q^{48} -1.34704e9 q^{49} -3.69737e8 q^{50} -9.33839e7 q^{51} -3.92608e9 q^{52} -1.18778e9 q^{53} -1.13237e8 q^{54} -3.54154e8 q^{55} -7.99180e8 q^{56} +2.43118e9 q^{57} -1.41368e9 q^{58} -7.14924e8 q^{59} -6.78432e8 q^{60} +8.62224e9 q^{61} -1.22048e9 q^{62} +1.48245e9 q^{63} -7.06211e9 q^{64} -2.77986e9 q^{65} -4.83046e8 q^{66} -9.42876e8 q^{67} -7.63104e8 q^{68} -1.24515e10 q^{69} -2.78561e8 q^{70} +8.96360e9 q^{71} -1.87970e9 q^{72} -9.87272e9 q^{73} +5.12863e9 q^{74} +1.13849e10 q^{75} +1.98668e10 q^{76} +6.32381e9 q^{77} -3.79156e9 q^{78} -1.71048e10 q^{79} -5.36460e9 q^{80} +3.48678e9 q^{81} -1.40440e9 q^{82} +5.86777e10 q^{83} +1.21141e10 q^{84} -5.40315e8 q^{85} +1.04464e10 q^{86} +4.35299e10 q^{87} -8.01840e9 q^{88} -2.87521e10 q^{89} -6.55187e8 q^{90} +4.96374e10 q^{91} -1.01750e11 q^{92} +3.75807e10 q^{93} -2.02907e10 q^{94} +1.40667e10 q^{95} -2.31591e10 q^{96} +8.39510e10 q^{97} -1.06305e10 q^{98} +1.48738e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 46q^{2} - 6561q^{3} + 26142q^{4} - 2442q^{5} + 11178q^{6} + 170093q^{7} - 19341q^{8} + 1594323q^{9} + 140249q^{10} + 256992q^{11} - 6352506q^{12} + 2436978q^{13} + 5233061q^{14} + 593406q^{15} + 28295194q^{16} - 4565351q^{17} - 2716254q^{18} + 33607699q^{19} - 19208463q^{20} - 41332599q^{21} + 79735622q^{22} + 43966161q^{23} + 4699863q^{24} + 406675819q^{25} + 42605404q^{26} - 387420489q^{27} + 635747682q^{28} - 107217773q^{29} - 34080507q^{30} + 570926627q^{31} + 526569236q^{32} - 62449056q^{33} + 129790240q^{34} + 134356079q^{35} + 1543658958q^{36} - 107121371q^{37} + 208302581q^{38} - 592185654q^{39} - 958762162q^{40} - 1935967559q^{41} - 1271633823q^{42} + 1725943824q^{43} + 196885756q^{44} - 144197658q^{45} - 13265966407q^{46} + 1801256065q^{47} - 6875732142q^{48} + 10484289252q^{49} - 10067682271q^{50} + 1109380293q^{51} - 882697024q^{52} - 6214238922q^{53} + 660049722q^{54} + 4460552366q^{55} + 28328012310q^{56} - 8166670857q^{57} + 12220116750q^{58} - 19302956073q^{59} + 4667656509q^{60} + 13167821039q^{61} - 1162130230q^{62} + 10043821557q^{63} - 5337557395q^{64} - 16849896006q^{65} - 19375756146q^{66} - 16856763152q^{67} - 36171071977q^{68} - 10683777123q^{69} - 120177261588q^{70} - 5198545690q^{71} - 1142066709q^{72} - 25075321857q^{73} - 182979651978q^{74} - 98822224017q^{75} - 3501293988q^{76} - 42787697701q^{77} - 10353113172q^{78} + 6850314702q^{79} - 261464428159q^{80} + 94143178827q^{81} - 148881516273q^{82} + 30908370899q^{83} - 154486686726q^{84} - 49419624969q^{85} - 220725475224q^{86} + 26053918839q^{87} - 53091280787q^{88} + 28988060121q^{89} + 8281563201q^{90} + 97120614047q^{91} + 45374597708q^{92} - 138735170361q^{93} + 208966927220q^{94} - 125253904969q^{95} - 127956324348q^{96} + 367722840268q^{97} - 48265639912q^{98} + 15175120608q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.89171 0.174384 0.0871919 0.996192i \(-0.472211\pi\)
0.0871919 + 0.996192i \(0.472211\pi\)
\(3\) −243.000 −0.577350
\(4\) −1985.72 −0.969590
\(5\) −1405.99 −0.201209 −0.100604 0.994927i \(-0.532078\pi\)
−0.100604 + 0.994927i \(0.532078\pi\)
\(6\) −1917.69 −0.100681
\(7\) 25105.4 0.564584 0.282292 0.959329i \(-0.408905\pi\)
0.282292 + 0.959329i \(0.408905\pi\)
\(8\) −31832.9 −0.343465
\(9\) 59049.0 0.333333
\(10\) −11095.6 −0.0350875
\(11\) 251890. 0.471575 0.235788 0.971805i \(-0.424233\pi\)
0.235788 + 0.971805i \(0.424233\pi\)
\(12\) 482530. 0.559793
\(13\) 1.97716e6 1.47691 0.738453 0.674305i \(-0.235557\pi\)
0.738453 + 0.674305i \(0.235557\pi\)
\(14\) 198125. 0.0984543
\(15\) 341655. 0.116168
\(16\) 3.81554e6 0.909696
\(17\) 384296. 0.0656442 0.0328221 0.999461i \(-0.489551\pi\)
0.0328221 + 0.999461i \(0.489551\pi\)
\(18\) 465997. 0.0581279
\(19\) −1.00048e7 −0.926970 −0.463485 0.886105i \(-0.653401\pi\)
−0.463485 + 0.886105i \(0.653401\pi\)
\(20\) 2.79190e6 0.195090
\(21\) −6.10062e6 −0.325963
\(22\) 1.98784e6 0.0822351
\(23\) 5.12407e7 1.66001 0.830007 0.557752i \(-0.188336\pi\)
0.830007 + 0.557752i \(0.188336\pi\)
\(24\) 7.73541e6 0.198299
\(25\) −4.68513e7 −0.959515
\(26\) 1.56031e7 0.257548
\(27\) −1.43489e7 −0.192450
\(28\) −4.98524e7 −0.547415
\(29\) −1.79135e8 −1.62178 −0.810890 0.585198i \(-0.801017\pi\)
−0.810890 + 0.585198i \(0.801017\pi\)
\(30\) 2.69624e6 0.0202578
\(31\) −1.54653e8 −0.970218 −0.485109 0.874454i \(-0.661220\pi\)
−0.485109 + 0.874454i \(0.661220\pi\)
\(32\) 9.53050e7 0.502101
\(33\) −6.12092e7 −0.272264
\(34\) 3.03275e6 0.0114473
\(35\) −3.52980e7 −0.113599
\(36\) −1.17255e8 −0.323197
\(37\) 6.49876e8 1.54071 0.770355 0.637615i \(-0.220079\pi\)
0.770355 + 0.637615i \(0.220079\pi\)
\(38\) −7.89553e7 −0.161648
\(39\) −4.80449e8 −0.852692
\(40\) 4.47567e7 0.0691080
\(41\) −1.77958e8 −0.239887 −0.119944 0.992781i \(-0.538271\pi\)
−0.119944 + 0.992781i \(0.538271\pi\)
\(42\) −4.81443e7 −0.0568426
\(43\) 1.32372e9 1.37315 0.686577 0.727057i \(-0.259113\pi\)
0.686577 + 0.727057i \(0.259113\pi\)
\(44\) −5.00183e8 −0.457235
\(45\) −8.30222e7 −0.0670695
\(46\) 4.04377e8 0.289480
\(47\) −2.57114e9 −1.63526 −0.817632 0.575741i \(-0.804714\pi\)
−0.817632 + 0.575741i \(0.804714\pi\)
\(48\) −9.27176e8 −0.525213
\(49\) −1.34704e9 −0.681245
\(50\) −3.69737e8 −0.167324
\(51\) −9.33839e7 −0.0378997
\(52\) −3.92608e9 −1.43199
\(53\) −1.18778e9 −0.390138 −0.195069 0.980790i \(-0.562493\pi\)
−0.195069 + 0.980790i \(0.562493\pi\)
\(54\) −1.13237e8 −0.0335602
\(55\) −3.54154e8 −0.0948850
\(56\) −7.99180e8 −0.193915
\(57\) 2.43118e9 0.535186
\(58\) −1.41368e9 −0.282812
\(59\) −7.14924e8 −0.130189
\(60\) −6.78432e8 −0.112635
\(61\) 8.62224e9 1.30709 0.653545 0.756887i \(-0.273281\pi\)
0.653545 + 0.756887i \(0.273281\pi\)
\(62\) −1.22048e9 −0.169190
\(63\) 1.48245e9 0.188195
\(64\) −7.06211e9 −0.822137
\(65\) −2.77986e9 −0.297166
\(66\) −4.83046e8 −0.0474784
\(67\) −9.42876e8 −0.0853185 −0.0426592 0.999090i \(-0.513583\pi\)
−0.0426592 + 0.999090i \(0.513583\pi\)
\(68\) −7.63104e8 −0.0636480
\(69\) −1.24515e10 −0.958410
\(70\) −2.78561e8 −0.0198099
\(71\) 8.96360e9 0.589606 0.294803 0.955558i \(-0.404746\pi\)
0.294803 + 0.955558i \(0.404746\pi\)
\(72\) −1.87970e9 −0.114488
\(73\) −9.87272e9 −0.557393 −0.278696 0.960379i \(-0.589902\pi\)
−0.278696 + 0.960379i \(0.589902\pi\)
\(74\) 5.12863e9 0.268675
\(75\) 1.13849e10 0.553976
\(76\) 1.98668e10 0.898781
\(77\) 6.32381e9 0.266244
\(78\) −3.79156e9 −0.148696
\(79\) −1.71048e10 −0.625416 −0.312708 0.949849i \(-0.601236\pi\)
−0.312708 + 0.949849i \(0.601236\pi\)
\(80\) −5.36460e9 −0.183039
\(81\) 3.48678e9 0.111111
\(82\) −1.40440e9 −0.0418325
\(83\) 5.86777e10 1.63510 0.817548 0.575860i \(-0.195333\pi\)
0.817548 + 0.575860i \(0.195333\pi\)
\(84\) 1.21141e10 0.316050
\(85\) −5.40315e8 −0.0132082
\(86\) 1.04464e10 0.239456
\(87\) 4.35299e10 0.936336
\(88\) −8.01840e9 −0.161969
\(89\) −2.87521e10 −0.545788 −0.272894 0.962044i \(-0.587981\pi\)
−0.272894 + 0.962044i \(0.587981\pi\)
\(90\) −6.55187e8 −0.0116958
\(91\) 4.96374e10 0.833837
\(92\) −1.01750e11 −1.60953
\(93\) 3.75807e10 0.560156
\(94\) −2.02907e10 −0.285164
\(95\) 1.40667e10 0.186514
\(96\) −2.31591e10 −0.289888
\(97\) 8.39510e10 0.992617 0.496308 0.868146i \(-0.334689\pi\)
0.496308 + 0.868146i \(0.334689\pi\)
\(98\) −1.06305e10 −0.118798
\(99\) 1.48738e10 0.157192
\(100\) 9.30337e10 0.930337
\(101\) 9.28263e10 0.878827 0.439413 0.898285i \(-0.355186\pi\)
0.439413 + 0.898285i \(0.355186\pi\)
\(102\) −7.36958e8 −0.00660910
\(103\) −1.89238e11 −1.60844 −0.804219 0.594333i \(-0.797416\pi\)
−0.804219 + 0.594333i \(0.797416\pi\)
\(104\) −6.29387e10 −0.507265
\(105\) 8.57740e9 0.0655865
\(106\) −9.37360e9 −0.0680337
\(107\) −1.43004e11 −0.985680 −0.492840 0.870120i \(-0.664041\pi\)
−0.492840 + 0.870120i \(0.664041\pi\)
\(108\) 2.84929e10 0.186598
\(109\) 2.89853e11 1.80440 0.902199 0.431321i \(-0.141952\pi\)
0.902199 + 0.431321i \(0.141952\pi\)
\(110\) −2.79488e9 −0.0165464
\(111\) −1.57920e11 −0.889530
\(112\) 9.57908e10 0.513600
\(113\) 1.90809e11 0.974243 0.487122 0.873334i \(-0.338047\pi\)
0.487122 + 0.873334i \(0.338047\pi\)
\(114\) 1.91861e10 0.0933278
\(115\) −7.20438e10 −0.334009
\(116\) 3.55713e11 1.57246
\(117\) 1.16749e11 0.492302
\(118\) −5.64197e9 −0.0227028
\(119\) 9.64792e9 0.0370617
\(120\) −1.08759e10 −0.0398995
\(121\) −2.21863e11 −0.777617
\(122\) 6.80442e10 0.227935
\(123\) 4.32439e10 0.138499
\(124\) 3.07098e11 0.940714
\(125\) 1.34524e11 0.394271
\(126\) 1.16991e10 0.0328181
\(127\) −1.82107e11 −0.489108 −0.244554 0.969636i \(-0.578642\pi\)
−0.244554 + 0.969636i \(0.578642\pi\)
\(128\) −2.50917e11 −0.645468
\(129\) −3.21664e11 −0.792791
\(130\) −2.19378e10 −0.0518209
\(131\) 6.09391e11 1.38008 0.690039 0.723772i \(-0.257593\pi\)
0.690039 + 0.723772i \(0.257593\pi\)
\(132\) 1.21544e11 0.263985
\(133\) −2.51176e11 −0.523352
\(134\) −7.44090e9 −0.0148782
\(135\) 2.01744e10 0.0387226
\(136\) −1.22333e10 −0.0225465
\(137\) −7.08371e11 −1.25400 −0.627000 0.779019i \(-0.715718\pi\)
−0.627000 + 0.779019i \(0.715718\pi\)
\(138\) −9.82635e10 −0.167131
\(139\) 3.39953e10 0.0555696 0.0277848 0.999614i \(-0.491155\pi\)
0.0277848 + 0.999614i \(0.491155\pi\)
\(140\) 7.00919e10 0.110145
\(141\) 6.24788e11 0.944120
\(142\) 7.07381e10 0.102818
\(143\) 4.98026e11 0.696472
\(144\) 2.25304e11 0.303232
\(145\) 2.51862e11 0.326316
\(146\) −7.79127e10 −0.0972002
\(147\) 3.27332e11 0.393317
\(148\) −1.29047e12 −1.49386
\(149\) 1.33224e12 1.48613 0.743066 0.669219i \(-0.233371\pi\)
0.743066 + 0.669219i \(0.233371\pi\)
\(150\) 8.98461e10 0.0966045
\(151\) −3.50958e11 −0.363816 −0.181908 0.983316i \(-0.558227\pi\)
−0.181908 + 0.983316i \(0.558227\pi\)
\(152\) 3.18484e11 0.318381
\(153\) 2.26923e10 0.0218814
\(154\) 4.99057e10 0.0464286
\(155\) 2.17441e11 0.195216
\(156\) 9.54038e11 0.826762
\(157\) −8.04739e11 −0.673298 −0.336649 0.941630i \(-0.609294\pi\)
−0.336649 + 0.941630i \(0.609294\pi\)
\(158\) −1.34986e11 −0.109062
\(159\) 2.88630e11 0.225246
\(160\) −1.33998e11 −0.101027
\(161\) 1.28642e12 0.937218
\(162\) 2.75167e10 0.0193760
\(163\) 1.19198e12 0.811405 0.405702 0.914005i \(-0.367027\pi\)
0.405702 + 0.914005i \(0.367027\pi\)
\(164\) 3.53376e11 0.232592
\(165\) 8.60595e10 0.0547819
\(166\) 4.63067e11 0.285134
\(167\) −8.90159e11 −0.530306 −0.265153 0.964206i \(-0.585423\pi\)
−0.265153 + 0.964206i \(0.585423\pi\)
\(168\) 1.94201e11 0.111957
\(169\) 2.11699e12 1.18125
\(170\) −4.26401e9 −0.00230329
\(171\) −5.90776e11 −0.308990
\(172\) −2.62854e12 −1.33140
\(173\) 9.49944e11 0.466063 0.233032 0.972469i \(-0.425135\pi\)
0.233032 + 0.972469i \(0.425135\pi\)
\(174\) 3.43525e11 0.163282
\(175\) −1.17622e12 −0.541727
\(176\) 9.61096e11 0.428990
\(177\) 1.73727e11 0.0751646
\(178\) −2.26903e11 −0.0951766
\(179\) −1.13637e12 −0.462197 −0.231099 0.972930i \(-0.574232\pi\)
−0.231099 + 0.972930i \(0.574232\pi\)
\(180\) 1.64859e11 0.0650300
\(181\) 1.56925e12 0.600425 0.300213 0.953872i \(-0.402942\pi\)
0.300213 + 0.953872i \(0.402942\pi\)
\(182\) 3.91724e11 0.145408
\(183\) −2.09520e12 −0.754649
\(184\) −1.63114e12 −0.570156
\(185\) −9.13718e11 −0.310004
\(186\) 2.96576e11 0.0976821
\(187\) 9.68002e10 0.0309562
\(188\) 5.10557e12 1.58554
\(189\) −3.60236e11 −0.108654
\(190\) 1.11010e11 0.0325251
\(191\) 6.19963e12 1.76475 0.882374 0.470549i \(-0.155944\pi\)
0.882374 + 0.470549i \(0.155944\pi\)
\(192\) 1.71609e12 0.474661
\(193\) −1.65838e12 −0.445778 −0.222889 0.974844i \(-0.571549\pi\)
−0.222889 + 0.974844i \(0.571549\pi\)
\(194\) 6.62517e11 0.173096
\(195\) 6.75506e11 0.171569
\(196\) 2.67485e12 0.660528
\(197\) −3.41122e12 −0.819117 −0.409559 0.912284i \(-0.634317\pi\)
−0.409559 + 0.912284i \(0.634317\pi\)
\(198\) 1.17380e11 0.0274117
\(199\) 7.79320e12 1.77021 0.885104 0.465394i \(-0.154087\pi\)
0.885104 + 0.465394i \(0.154087\pi\)
\(200\) 1.49142e12 0.329559
\(201\) 2.29119e11 0.0492586
\(202\) 7.32558e11 0.153253
\(203\) −4.49727e12 −0.915632
\(204\) 1.85434e11 0.0367472
\(205\) 2.50207e11 0.0482674
\(206\) −1.49341e12 −0.280485
\(207\) 3.02571e12 0.553338
\(208\) 7.54392e12 1.34353
\(209\) −2.52012e12 −0.437136
\(210\) 6.76904e10 0.0114372
\(211\) 2.52623e12 0.415834 0.207917 0.978146i \(-0.433332\pi\)
0.207917 + 0.978146i \(0.433332\pi\)
\(212\) 2.35860e12 0.378274
\(213\) −2.17816e12 −0.340409
\(214\) −1.12854e12 −0.171887
\(215\) −1.86113e12 −0.276290
\(216\) 4.56768e11 0.0660998
\(217\) −3.88264e12 −0.547770
\(218\) 2.28744e12 0.314658
\(219\) 2.39907e12 0.321811
\(220\) 7.03251e11 0.0919996
\(221\) 7.59813e11 0.0969503
\(222\) −1.24626e12 −0.155120
\(223\) 1.16037e13 1.40903 0.704513 0.709691i \(-0.251166\pi\)
0.704513 + 0.709691i \(0.251166\pi\)
\(224\) 2.39267e12 0.283478
\(225\) −2.76652e12 −0.319838
\(226\) 1.50581e12 0.169892
\(227\) 1.77498e13 1.95457 0.977283 0.211939i \(-0.0679778\pi\)
0.977283 + 0.211939i \(0.0679778\pi\)
\(228\) −4.82764e12 −0.518911
\(229\) 6.45345e12 0.677169 0.338584 0.940936i \(-0.390052\pi\)
0.338584 + 0.940936i \(0.390052\pi\)
\(230\) −5.68549e11 −0.0582458
\(231\) −1.53669e12 −0.153716
\(232\) 5.70241e12 0.557024
\(233\) −1.56906e13 −1.49686 −0.748432 0.663212i \(-0.769193\pi\)
−0.748432 + 0.663212i \(0.769193\pi\)
\(234\) 9.21350e11 0.0858494
\(235\) 3.61500e12 0.329029
\(236\) 1.41964e12 0.126230
\(237\) 4.15646e12 0.361084
\(238\) 7.61385e10 0.00646296
\(239\) −8.80380e12 −0.730267 −0.365134 0.930955i \(-0.618977\pi\)
−0.365134 + 0.930955i \(0.618977\pi\)
\(240\) 1.30360e12 0.105677
\(241\) −1.06050e13 −0.840264 −0.420132 0.907463i \(-0.638016\pi\)
−0.420132 + 0.907463i \(0.638016\pi\)
\(242\) −1.75088e12 −0.135604
\(243\) −8.47289e11 −0.0641500
\(244\) −1.71214e13 −1.26734
\(245\) 1.89393e12 0.137072
\(246\) 3.41268e11 0.0241520
\(247\) −1.97811e13 −1.36905
\(248\) 4.92307e12 0.333236
\(249\) −1.42587e13 −0.944023
\(250\) 1.06163e12 0.0687545
\(251\) 2.79673e13 1.77192 0.885962 0.463758i \(-0.153499\pi\)
0.885962 + 0.463758i \(0.153499\pi\)
\(252\) −2.94373e12 −0.182472
\(253\) 1.29070e13 0.782822
\(254\) −1.43713e12 −0.0852926
\(255\) 1.31297e11 0.00762575
\(256\) 1.24830e13 0.709578
\(257\) 2.75903e13 1.53506 0.767528 0.641015i \(-0.221487\pi\)
0.767528 + 0.641015i \(0.221487\pi\)
\(258\) −2.53848e12 −0.138250
\(259\) 1.63154e13 0.869861
\(260\) 5.52002e12 0.288129
\(261\) −1.05778e13 −0.540594
\(262\) 4.80913e12 0.240663
\(263\) −4.32633e12 −0.212013 −0.106007 0.994365i \(-0.533806\pi\)
−0.106007 + 0.994365i \(0.533806\pi\)
\(264\) 1.94847e12 0.0935131
\(265\) 1.67000e12 0.0784991
\(266\) −1.98221e12 −0.0912642
\(267\) 6.98676e12 0.315111
\(268\) 1.87229e12 0.0827240
\(269\) −1.99206e13 −0.862315 −0.431157 0.902277i \(-0.641895\pi\)
−0.431157 + 0.902277i \(0.641895\pi\)
\(270\) 1.59210e11 0.00675260
\(271\) 4.12737e13 1.71531 0.857654 0.514227i \(-0.171921\pi\)
0.857654 + 0.514227i \(0.171921\pi\)
\(272\) 1.46630e12 0.0597163
\(273\) −1.20619e13 −0.481416
\(274\) −5.59026e12 −0.218677
\(275\) −1.18014e13 −0.452484
\(276\) 2.47252e13 0.929265
\(277\) 4.39141e13 1.61795 0.808976 0.587842i \(-0.200022\pi\)
0.808976 + 0.587842i \(0.200022\pi\)
\(278\) 2.68281e11 0.00969043
\(279\) −9.13211e12 −0.323406
\(280\) 1.12364e12 0.0390173
\(281\) 9.21235e12 0.313679 0.156840 0.987624i \(-0.449869\pi\)
0.156840 + 0.987624i \(0.449869\pi\)
\(282\) 4.93064e12 0.164639
\(283\) −2.81171e13 −0.920758 −0.460379 0.887723i \(-0.652286\pi\)
−0.460379 + 0.887723i \(0.652286\pi\)
\(284\) −1.77992e13 −0.571676
\(285\) −3.41821e12 −0.107684
\(286\) 3.93027e12 0.121453
\(287\) −4.46773e12 −0.135437
\(288\) 5.62767e12 0.167367
\(289\) −3.41242e13 −0.995691
\(290\) 1.98762e12 0.0569043
\(291\) −2.04001e13 −0.573087
\(292\) 1.96045e13 0.540443
\(293\) −4.90947e13 −1.32820 −0.664098 0.747645i \(-0.731184\pi\)
−0.664098 + 0.747645i \(0.731184\pi\)
\(294\) 2.58321e12 0.0685881
\(295\) 1.00518e12 0.0261951
\(296\) −2.06875e13 −0.529180
\(297\) −3.61434e12 −0.0907547
\(298\) 1.05136e13 0.259157
\(299\) 1.01311e14 2.45168
\(300\) −2.26072e13 −0.537130
\(301\) 3.32325e13 0.775261
\(302\) −2.76966e12 −0.0634436
\(303\) −2.25568e13 −0.507391
\(304\) −3.81739e13 −0.843260
\(305\) −1.21228e13 −0.262998
\(306\) 1.79081e11 0.00381576
\(307\) −5.09295e13 −1.06588 −0.532940 0.846153i \(-0.678913\pi\)
−0.532940 + 0.846153i \(0.678913\pi\)
\(308\) −1.25573e13 −0.258148
\(309\) 4.59849e13 0.928632
\(310\) 1.71598e12 0.0340426
\(311\) −1.84718e13 −0.360021 −0.180010 0.983665i \(-0.557613\pi\)
−0.180010 + 0.983665i \(0.557613\pi\)
\(312\) 1.52941e13 0.292869
\(313\) 2.10290e13 0.395663 0.197831 0.980236i \(-0.436610\pi\)
0.197831 + 0.980236i \(0.436610\pi\)
\(314\) −6.35077e12 −0.117412
\(315\) −2.08431e12 −0.0378664
\(316\) 3.39653e13 0.606397
\(317\) 2.81419e13 0.493774 0.246887 0.969044i \(-0.420592\pi\)
0.246887 + 0.969044i \(0.420592\pi\)
\(318\) 2.27778e12 0.0392793
\(319\) −4.51224e13 −0.764792
\(320\) 9.92924e12 0.165421
\(321\) 3.47499e13 0.569083
\(322\) 1.01521e13 0.163436
\(323\) −3.84482e12 −0.0608502
\(324\) −6.92378e12 −0.107732
\(325\) −9.26324e13 −1.41711
\(326\) 9.40677e12 0.141496
\(327\) −7.04343e13 −1.04177
\(328\) 5.66494e12 0.0823928
\(329\) −6.45497e13 −0.923245
\(330\) 6.79156e11 0.00955307
\(331\) −1.08032e14 −1.49451 −0.747253 0.664540i \(-0.768628\pi\)
−0.747253 + 0.664540i \(0.768628\pi\)
\(332\) −1.16517e14 −1.58537
\(333\) 3.83745e13 0.513570
\(334\) −7.02487e12 −0.0924768
\(335\) 1.32567e12 0.0171668
\(336\) −2.32772e13 −0.296527
\(337\) 1.05771e14 1.32556 0.662782 0.748812i \(-0.269376\pi\)
0.662782 + 0.748812i \(0.269376\pi\)
\(338\) 1.67066e13 0.205991
\(339\) −4.63666e13 −0.562480
\(340\) 1.07292e12 0.0128065
\(341\) −3.89556e13 −0.457531
\(342\) −4.66223e12 −0.0538828
\(343\) −8.34598e13 −0.949204
\(344\) −4.21379e13 −0.471630
\(345\) 1.75066e13 0.192840
\(346\) 7.49668e12 0.0812738
\(347\) −1.77689e14 −1.89605 −0.948025 0.318197i \(-0.896923\pi\)
−0.948025 + 0.318197i \(0.896923\pi\)
\(348\) −8.64382e13 −0.907862
\(349\) 9.19114e13 0.950232 0.475116 0.879923i \(-0.342406\pi\)
0.475116 + 0.879923i \(0.342406\pi\)
\(350\) −9.28241e12 −0.0944684
\(351\) −2.83700e13 −0.284231
\(352\) 2.40064e13 0.236778
\(353\) −1.12600e14 −1.09340 −0.546699 0.837329i \(-0.684116\pi\)
−0.546699 + 0.837329i \(0.684116\pi\)
\(354\) 1.37100e12 0.0131075
\(355\) −1.26027e13 −0.118634
\(356\) 5.70936e13 0.529191
\(357\) −2.34444e12 −0.0213976
\(358\) −8.96788e12 −0.0805997
\(359\) −1.66558e13 −0.147416 −0.0737081 0.997280i \(-0.523483\pi\)
−0.0737081 + 0.997280i \(0.523483\pi\)
\(360\) 2.64284e12 0.0230360
\(361\) −1.63933e13 −0.140727
\(362\) 1.23840e13 0.104704
\(363\) 5.39127e13 0.448957
\(364\) −9.85660e13 −0.808481
\(365\) 1.38809e13 0.112152
\(366\) −1.65347e13 −0.131599
\(367\) −1.46428e14 −1.14805 −0.574027 0.818836i \(-0.694620\pi\)
−0.574027 + 0.818836i \(0.694620\pi\)
\(368\) 1.95511e14 1.51011
\(369\) −1.05083e13 −0.0799625
\(370\) −7.21080e12 −0.0540597
\(371\) −2.98197e13 −0.220266
\(372\) −7.46248e13 −0.543122
\(373\) 2.08804e14 1.49740 0.748702 0.662906i \(-0.230677\pi\)
0.748702 + 0.662906i \(0.230677\pi\)
\(374\) 7.63919e11 0.00539826
\(375\) −3.26894e13 −0.227633
\(376\) 8.18471e13 0.561655
\(377\) −3.54179e14 −2.39522
\(378\) −2.84287e12 −0.0189475
\(379\) 8.51228e13 0.559153 0.279576 0.960123i \(-0.409806\pi\)
0.279576 + 0.960123i \(0.409806\pi\)
\(380\) −2.79325e13 −0.180842
\(381\) 4.42519e13 0.282387
\(382\) 4.89257e13 0.307743
\(383\) 2.25630e14 1.39895 0.699476 0.714656i \(-0.253417\pi\)
0.699476 + 0.714656i \(0.253417\pi\)
\(384\) 6.09728e13 0.372661
\(385\) −8.89120e12 −0.0535706
\(386\) −1.30875e13 −0.0777365
\(387\) 7.81643e13 0.457718
\(388\) −1.66703e14 −0.962431
\(389\) 1.19250e14 0.678789 0.339394 0.940644i \(-0.389778\pi\)
0.339394 + 0.940644i \(0.389778\pi\)
\(390\) 5.33089e12 0.0299188
\(391\) 1.96916e13 0.108970
\(392\) 4.28804e13 0.233983
\(393\) −1.48082e14 −0.796788
\(394\) −2.69204e13 −0.142841
\(395\) 2.40491e13 0.125839
\(396\) −2.95353e13 −0.152412
\(397\) −2.31280e14 −1.17704 −0.588518 0.808484i \(-0.700288\pi\)
−0.588518 + 0.808484i \(0.700288\pi\)
\(398\) 6.15017e13 0.308695
\(399\) 6.10358e13 0.302158
\(400\) −1.78763e14 −0.872867
\(401\) −5.79434e12 −0.0279068 −0.0139534 0.999903i \(-0.504442\pi\)
−0.0139534 + 0.999903i \(0.504442\pi\)
\(402\) 1.80814e12 0.00858991
\(403\) −3.05774e14 −1.43292
\(404\) −1.84327e14 −0.852102
\(405\) −4.90238e12 −0.0223565
\(406\) −3.54912e13 −0.159671
\(407\) 1.63697e14 0.726561
\(408\) 2.97268e12 0.0130172
\(409\) 2.40534e14 1.03920 0.519600 0.854410i \(-0.326081\pi\)
0.519600 + 0.854410i \(0.326081\pi\)
\(410\) 1.97456e12 0.00841705
\(411\) 1.72134e14 0.723998
\(412\) 3.75774e14 1.55953
\(413\) −1.79485e13 −0.0735026
\(414\) 2.38780e13 0.0964932
\(415\) −8.25001e13 −0.328996
\(416\) 1.88433e14 0.741555
\(417\) −8.26085e12 −0.0320831
\(418\) −1.98880e13 −0.0762294
\(419\) −2.03602e13 −0.0770203 −0.0385101 0.999258i \(-0.512261\pi\)
−0.0385101 + 0.999258i \(0.512261\pi\)
\(420\) −1.70323e13 −0.0635921
\(421\) −3.06582e13 −0.112978 −0.0564891 0.998403i \(-0.517991\pi\)
−0.0564891 + 0.998403i \(0.517991\pi\)
\(422\) 1.99363e13 0.0725147
\(423\) −1.51823e14 −0.545088
\(424\) 3.78105e13 0.133999
\(425\) −1.80048e13 −0.0629866
\(426\) −1.71894e13 −0.0593618
\(427\) 2.16465e14 0.737963
\(428\) 2.83965e14 0.955706
\(429\) −1.21020e14 −0.402108
\(430\) −1.46875e13 −0.0481806
\(431\) −3.65677e14 −1.18433 −0.592165 0.805816i \(-0.701727\pi\)
−0.592165 + 0.805816i \(0.701727\pi\)
\(432\) −5.47488e13 −0.175071
\(433\) 4.13063e14 1.30417 0.652083 0.758148i \(-0.273895\pi\)
0.652083 + 0.758148i \(0.273895\pi\)
\(434\) −3.06406e13 −0.0955222
\(435\) −6.12025e13 −0.188399
\(436\) −5.75567e14 −1.74953
\(437\) −5.12655e14 −1.53878
\(438\) 1.89328e13 0.0561186
\(439\) −6.75685e13 −0.197783 −0.0988916 0.995098i \(-0.531530\pi\)
−0.0988916 + 0.995098i \(0.531530\pi\)
\(440\) 1.12738e13 0.0325896
\(441\) −7.95416e13 −0.227082
\(442\) 5.99622e12 0.0169066
\(443\) −6.24507e13 −0.173907 −0.0869535 0.996212i \(-0.527713\pi\)
−0.0869535 + 0.996212i \(0.527713\pi\)
\(444\) 3.13585e14 0.862479
\(445\) 4.04251e13 0.109817
\(446\) 9.15728e13 0.245711
\(447\) −3.23734e14 −0.858018
\(448\) −1.77297e14 −0.464166
\(449\) 4.13909e14 1.07041 0.535205 0.844722i \(-0.320234\pi\)
0.535205 + 0.844722i \(0.320234\pi\)
\(450\) −2.18326e13 −0.0557746
\(451\) −4.48259e13 −0.113125
\(452\) −3.78893e14 −0.944617
\(453\) 8.52827e13 0.210049
\(454\) 1.40076e14 0.340845
\(455\) −6.97896e13 −0.167775
\(456\) −7.73915e13 −0.183818
\(457\) 4.01500e14 0.942207 0.471104 0.882078i \(-0.343856\pi\)
0.471104 + 0.882078i \(0.343856\pi\)
\(458\) 5.09288e13 0.118087
\(459\) −5.51422e12 −0.0126332
\(460\) 1.43059e14 0.323852
\(461\) −4.86101e13 −0.108735 −0.0543677 0.998521i \(-0.517314\pi\)
−0.0543677 + 0.998521i \(0.517314\pi\)
\(462\) −1.21271e13 −0.0268056
\(463\) 7.54487e14 1.64800 0.823998 0.566593i \(-0.191739\pi\)
0.823998 + 0.566593i \(0.191739\pi\)
\(464\) −6.83498e14 −1.47533
\(465\) −5.28380e13 −0.112708
\(466\) −1.23826e14 −0.261029
\(467\) 9.12970e14 1.90201 0.951007 0.309170i \(-0.100051\pi\)
0.951007 + 0.309170i \(0.100051\pi\)
\(468\) −2.31831e14 −0.477331
\(469\) −2.36713e13 −0.0481695
\(470\) 2.85285e13 0.0573774
\(471\) 1.95552e14 0.388729
\(472\) 2.27581e13 0.0447153
\(473\) 3.33431e14 0.647545
\(474\) 3.28016e13 0.0629672
\(475\) 4.68740e14 0.889441
\(476\) −1.91581e13 −0.0359347
\(477\) −7.01371e13 −0.130046
\(478\) −6.94770e13 −0.127347
\(479\) 7.41424e13 0.134345 0.0671724 0.997741i \(-0.478602\pi\)
0.0671724 + 0.997741i \(0.478602\pi\)
\(480\) 3.25614e13 0.0583280
\(481\) 1.28491e15 2.27548
\(482\) −8.36913e13 −0.146528
\(483\) −3.12600e14 −0.541103
\(484\) 4.40558e14 0.753970
\(485\) −1.18034e14 −0.199723
\(486\) −6.68655e12 −0.0111867
\(487\) 9.63385e14 1.59364 0.796820 0.604216i \(-0.206514\pi\)
0.796820 + 0.604216i \(0.206514\pi\)
\(488\) −2.74471e14 −0.448939
\(489\) −2.89651e14 −0.468465
\(490\) 1.49463e13 0.0239032
\(491\) −5.31814e14 −0.841030 −0.420515 0.907286i \(-0.638151\pi\)
−0.420515 + 0.907286i \(0.638151\pi\)
\(492\) −8.58703e13 −0.134287
\(493\) −6.88409e13 −0.106461
\(494\) −1.56107e14 −0.238739
\(495\) −2.09125e13 −0.0316283
\(496\) −5.90085e14 −0.882603
\(497\) 2.25035e14 0.332882
\(498\) −1.12525e14 −0.164622
\(499\) −7.62146e14 −1.10277 −0.551385 0.834251i \(-0.685901\pi\)
−0.551385 + 0.834251i \(0.685901\pi\)
\(500\) −2.67127e14 −0.382282
\(501\) 2.16309e14 0.306172
\(502\) 2.20710e14 0.308995
\(503\) −3.50610e14 −0.485513 −0.242756 0.970087i \(-0.578052\pi\)
−0.242756 + 0.970087i \(0.578052\pi\)
\(504\) −4.71908e13 −0.0646382
\(505\) −1.30513e14 −0.176828
\(506\) 1.01858e14 0.136511
\(507\) −5.14428e14 −0.681994
\(508\) 3.61613e14 0.474235
\(509\) −5.65144e14 −0.733181 −0.366591 0.930382i \(-0.619475\pi\)
−0.366591 + 0.930382i \(0.619475\pi\)
\(510\) 1.03615e12 0.00132981
\(511\) −2.47859e14 −0.314695
\(512\) 6.12390e14 0.769207
\(513\) 1.43559e14 0.178395
\(514\) 2.17735e14 0.267689
\(515\) 2.66067e14 0.323632
\(516\) 6.38734e14 0.768682
\(517\) −6.47645e14 −0.771150
\(518\) 1.28757e14 0.151690
\(519\) −2.30837e14 −0.269082
\(520\) 8.84911e13 0.102066
\(521\) −2.17536e14 −0.248270 −0.124135 0.992265i \(-0.539616\pi\)
−0.124135 + 0.992265i \(0.539616\pi\)
\(522\) −8.34766e13 −0.0942708
\(523\) 5.34086e14 0.596832 0.298416 0.954436i \(-0.403542\pi\)
0.298416 + 0.954436i \(0.403542\pi\)
\(524\) −1.21008e15 −1.33811
\(525\) 2.85822e14 0.312766
\(526\) −3.41421e13 −0.0369717
\(527\) −5.94326e13 −0.0636892
\(528\) −2.33546e14 −0.247677
\(529\) 1.67280e15 1.75565
\(530\) 1.31792e13 0.0136890
\(531\) −4.22156e13 −0.0433963
\(532\) 4.98766e14 0.507437
\(533\) −3.51852e14 −0.354291
\(534\) 5.51374e13 0.0549503
\(535\) 2.01061e14 0.198327
\(536\) 3.00145e13 0.0293039
\(537\) 2.76137e14 0.266850
\(538\) −1.57208e14 −0.150374
\(539\) −3.39307e14 −0.321258
\(540\) −4.00607e13 −0.0375451
\(541\) 1.01852e15 0.944898 0.472449 0.881358i \(-0.343370\pi\)
0.472449 + 0.881358i \(0.343370\pi\)
\(542\) 3.25720e14 0.299122
\(543\) −3.81327e14 −0.346656
\(544\) 3.66253e13 0.0329600
\(545\) −4.07530e14 −0.363060
\(546\) −9.51889e13 −0.0839512
\(547\) 5.00644e14 0.437118 0.218559 0.975824i \(-0.429864\pi\)
0.218559 + 0.975824i \(0.429864\pi\)
\(548\) 1.40663e15 1.21587
\(549\) 5.09134e14 0.435697
\(550\) −9.31330e13 −0.0789058
\(551\) 1.79222e15 1.50334
\(552\) 3.96368e14 0.329180
\(553\) −4.29423e14 −0.353100
\(554\) 3.46557e14 0.282144
\(555\) 2.22033e14 0.178981
\(556\) −6.75051e13 −0.0538797
\(557\) −1.12979e15 −0.892886 −0.446443 0.894812i \(-0.647309\pi\)
−0.446443 + 0.894812i \(0.647309\pi\)
\(558\) −7.20680e13 −0.0563968
\(559\) 2.61720e15 2.02802
\(560\) −1.34681e14 −0.103341
\(561\) −2.35225e13 −0.0178726
\(562\) 7.27012e13 0.0547006
\(563\) 1.26181e15 0.940152 0.470076 0.882626i \(-0.344226\pi\)
0.470076 + 0.882626i \(0.344226\pi\)
\(564\) −1.24065e15 −0.915410
\(565\) −2.68275e14 −0.196026
\(566\) −2.21892e14 −0.160565
\(567\) 8.75373e13 0.0627316
\(568\) −2.85338e14 −0.202509
\(569\) 2.42444e15 1.70410 0.852048 0.523464i \(-0.175361\pi\)
0.852048 + 0.523464i \(0.175361\pi\)
\(570\) −2.69755e13 −0.0187784
\(571\) 2.53107e15 1.74504 0.872521 0.488576i \(-0.162484\pi\)
0.872521 + 0.488576i \(0.162484\pi\)
\(572\) −9.88940e14 −0.675292
\(573\) −1.50651e15 −1.01888
\(574\) −3.52580e13 −0.0236179
\(575\) −2.40069e15 −1.59281
\(576\) −4.17010e14 −0.274046
\(577\) 1.93944e15 1.26244 0.631219 0.775604i \(-0.282555\pi\)
0.631219 + 0.775604i \(0.282555\pi\)
\(578\) −2.69298e14 −0.173632
\(579\) 4.02986e14 0.257370
\(580\) −5.00128e14 −0.316393
\(581\) 1.47313e15 0.923150
\(582\) −1.60992e14 −0.0999371
\(583\) −2.99189e14 −0.183979
\(584\) 3.14278e14 0.191445
\(585\) −1.64148e14 −0.0990554
\(586\) −3.87441e14 −0.231616
\(587\) −1.27549e15 −0.755383 −0.377691 0.925932i \(-0.623282\pi\)
−0.377691 + 0.925932i \(0.623282\pi\)
\(588\) −6.49989e14 −0.381356
\(589\) 1.54728e15 0.899363
\(590\) 7.93255e12 0.00456801
\(591\) 8.28928e14 0.472918
\(592\) 2.47963e15 1.40158
\(593\) 2.98297e15 1.67051 0.835253 0.549866i \(-0.185321\pi\)
0.835253 + 0.549866i \(0.185321\pi\)
\(594\) −2.85234e13 −0.0158261
\(595\) −1.35649e13 −0.00745713
\(596\) −2.64545e15 −1.44094
\(597\) −1.89375e15 −1.02203
\(598\) 7.99516e14 0.427534
\(599\) 6.46227e14 0.342403 0.171202 0.985236i \(-0.445235\pi\)
0.171202 + 0.985236i \(0.445235\pi\)
\(600\) −3.62414e14 −0.190271
\(601\) 3.46992e15 1.80513 0.902567 0.430549i \(-0.141680\pi\)
0.902567 + 0.430549i \(0.141680\pi\)
\(602\) 2.62262e14 0.135193
\(603\) −5.56759e13 −0.0284395
\(604\) 6.96904e14 0.352752
\(605\) 3.11937e14 0.156463
\(606\) −1.78012e14 −0.0884808
\(607\) 2.95267e14 0.145438 0.0727188 0.997352i \(-0.476832\pi\)
0.0727188 + 0.997352i \(0.476832\pi\)
\(608\) −9.53512e14 −0.465432
\(609\) 1.09284e15 0.528640
\(610\) −9.56693e13 −0.0458626
\(611\) −5.08355e15 −2.41513
\(612\) −4.50605e13 −0.0212160
\(613\) −9.01113e14 −0.420481 −0.210241 0.977650i \(-0.567425\pi\)
−0.210241 + 0.977650i \(0.567425\pi\)
\(614\) −4.01921e14 −0.185872
\(615\) −6.08004e13 −0.0278672
\(616\) −2.01305e14 −0.0914454
\(617\) −3.98663e14 −0.179489 −0.0897444 0.995965i \(-0.528605\pi\)
−0.0897444 + 0.995965i \(0.528605\pi\)
\(618\) 3.62899e14 0.161938
\(619\) 2.59898e15 1.14949 0.574744 0.818333i \(-0.305101\pi\)
0.574744 + 0.818333i \(0.305101\pi\)
\(620\) −4.31776e14 −0.189280
\(621\) −7.35248e14 −0.319470
\(622\) −1.45774e14 −0.0627818
\(623\) −7.21834e14 −0.308143
\(624\) −1.83317e15 −0.775690
\(625\) 2.09852e15 0.880184
\(626\) 1.65955e14 0.0689972
\(627\) 6.12389e14 0.252381
\(628\) 1.59799e15 0.652823
\(629\) 2.49745e14 0.101139
\(630\) −1.64488e13 −0.00660329
\(631\) −6.97014e13 −0.0277383 −0.0138691 0.999904i \(-0.504415\pi\)
−0.0138691 + 0.999904i \(0.504415\pi\)
\(632\) 5.44496e14 0.214808
\(633\) −6.13875e14 −0.240082
\(634\) 2.22088e14 0.0861062
\(635\) 2.56040e14 0.0984128
\(636\) −5.73139e14 −0.218397
\(637\) −2.66332e15 −1.00613
\(638\) −3.56093e14 −0.133367
\(639\) 5.29292e14 0.196535
\(640\) 3.52786e14 0.129874
\(641\) −3.70395e15 −1.35190 −0.675952 0.736946i \(-0.736267\pi\)
−0.675952 + 0.736946i \(0.736267\pi\)
\(642\) 2.74236e14 0.0992388
\(643\) 3.57318e14 0.128202 0.0641009 0.997943i \(-0.479582\pi\)
0.0641009 + 0.997943i \(0.479582\pi\)
\(644\) −2.55447e15 −0.908718
\(645\) 4.52255e14 0.159516
\(646\) −3.03422e13 −0.0106113
\(647\) 1.55248e14 0.0538333 0.0269167 0.999638i \(-0.491431\pi\)
0.0269167 + 0.999638i \(0.491431\pi\)
\(648\) −1.10995e14 −0.0381627
\(649\) −1.80082e14 −0.0613939
\(650\) −7.31028e14 −0.247121
\(651\) 9.43481e14 0.316255
\(652\) −2.36694e15 −0.786730
\(653\) −2.43695e15 −0.803200 −0.401600 0.915815i \(-0.631546\pi\)
−0.401600 + 0.915815i \(0.631546\pi\)
\(654\) −5.55847e14 −0.181668
\(655\) −8.56796e14 −0.277684
\(656\) −6.79008e14 −0.218224
\(657\) −5.82974e14 −0.185798
\(658\) −5.09407e14 −0.160999
\(659\) 4.86047e14 0.152338 0.0761690 0.997095i \(-0.475731\pi\)
0.0761690 + 0.997095i \(0.475731\pi\)
\(660\) −1.70890e14 −0.0531160
\(661\) −3.09073e15 −0.952694 −0.476347 0.879257i \(-0.658039\pi\)
−0.476347 + 0.879257i \(0.658039\pi\)
\(662\) −8.52556e14 −0.260618
\(663\) −1.84635e14 −0.0559743
\(664\) −1.86788e15 −0.561598
\(665\) 3.53151e14 0.105303
\(666\) 3.02841e14 0.0895583
\(667\) −9.17902e15 −2.69218
\(668\) 1.76761e15 0.514180
\(669\) −2.81969e15 −0.813501
\(670\) 1.04618e13 0.00299361
\(671\) 2.17185e15 0.616392
\(672\) −5.81420e14 −0.163666
\(673\) 5.15556e15 1.43944 0.719719 0.694266i \(-0.244271\pi\)
0.719719 + 0.694266i \(0.244271\pi\)
\(674\) 8.34711e14 0.231157
\(675\) 6.72265e14 0.184659
\(676\) −4.20375e15 −1.14533
\(677\) −1.78330e15 −0.481934 −0.240967 0.970533i \(-0.577464\pi\)
−0.240967 + 0.970533i \(0.577464\pi\)
\(678\) −3.65912e14 −0.0980873
\(679\) 2.10763e15 0.560416
\(680\) 1.71998e13 0.00453654
\(681\) −4.31319e15 −1.12847
\(682\) −3.07426e14 −0.0797860
\(683\) 6.38541e15 1.64390 0.821949 0.569561i \(-0.192887\pi\)
0.821949 + 0.569561i \(0.192887\pi\)
\(684\) 1.17312e15 0.299594
\(685\) 9.95961e14 0.252316
\(686\) −6.58640e14 −0.165526
\(687\) −1.56819e15 −0.390964
\(688\) 5.05070e15 1.24915
\(689\) −2.34842e15 −0.576197
\(690\) 1.38157e14 0.0336282
\(691\) −3.04808e15 −0.736033 −0.368016 0.929819i \(-0.619963\pi\)
−0.368016 + 0.929819i \(0.619963\pi\)
\(692\) −1.88632e15 −0.451890
\(693\) 3.73415e14 0.0887480
\(694\) −1.40227e15 −0.330640
\(695\) −4.77969e13 −0.0111811
\(696\) −1.38568e15 −0.321598
\(697\) −6.83887e13 −0.0157472
\(698\) 7.25338e14 0.165705
\(699\) 3.81282e15 0.864215
\(700\) 2.33565e15 0.525253
\(701\) −1.60759e15 −0.358696 −0.179348 0.983786i \(-0.557399\pi\)
−0.179348 + 0.983786i \(0.557399\pi\)
\(702\) −2.23888e14 −0.0495652
\(703\) −6.50191e15 −1.42819
\(704\) −1.77887e15 −0.387700
\(705\) −8.78444e14 −0.189965
\(706\) −8.88607e14 −0.190671
\(707\) 2.33044e15 0.496172
\(708\) −3.44973e14 −0.0728789
\(709\) 1.47197e15 0.308564 0.154282 0.988027i \(-0.450694\pi\)
0.154282 + 0.988027i \(0.450694\pi\)
\(710\) −9.94570e13 −0.0206878
\(711\) −1.01002e15 −0.208472
\(712\) 9.15264e14 0.187459
\(713\) −7.92454e15 −1.61058
\(714\) −1.85017e13 −0.00373139
\(715\) −7.00218e14 −0.140136
\(716\) 2.25651e15 0.448142
\(717\) 2.13932e15 0.421620
\(718\) −1.31442e14 −0.0257070
\(719\) −2.97725e14 −0.0577838 −0.0288919 0.999583i \(-0.509198\pi\)
−0.0288919 + 0.999583i \(0.509198\pi\)
\(720\) −3.16775e14 −0.0610129
\(721\) −4.75091e15 −0.908098
\(722\) −1.29371e14 −0.0245405
\(723\) 2.57701e15 0.485126
\(724\) −3.11609e15 −0.582167
\(725\) 8.39273e15 1.55612
\(726\) 4.25464e14 0.0782909
\(727\) −1.05253e16 −1.92219 −0.961097 0.276212i \(-0.910921\pi\)
−0.961097 + 0.276212i \(0.910921\pi\)
\(728\) −1.58010e15 −0.286394
\(729\) 2.05891e14 0.0370370
\(730\) 1.09544e14 0.0195575
\(731\) 5.08700e14 0.0901396
\(732\) 4.16049e15 0.731701
\(733\) −1.44108e15 −0.251545 −0.125773 0.992059i \(-0.540141\pi\)
−0.125773 + 0.992059i \(0.540141\pi\)
\(734\) −1.15557e15 −0.200202
\(735\) −4.60224e14 −0.0791387
\(736\) 4.88350e15 0.833495
\(737\) −2.37501e14 −0.0402341
\(738\) −8.29282e13 −0.0139442
\(739\) 5.43782e15 0.907571 0.453786 0.891111i \(-0.350073\pi\)
0.453786 + 0.891111i \(0.350073\pi\)
\(740\) 1.81439e15 0.300577
\(741\) 4.80682e15 0.790419
\(742\) −2.35328e14 −0.0384108
\(743\) 5.32566e15 0.862849 0.431425 0.902149i \(-0.358011\pi\)
0.431425 + 0.902149i \(0.358011\pi\)
\(744\) −1.19631e15 −0.192394
\(745\) −1.87311e15 −0.299022
\(746\) 1.64782e15 0.261123
\(747\) 3.46486e15 0.545032
\(748\) −1.92218e14 −0.0300148
\(749\) −3.59017e15 −0.556499
\(750\) −2.57975e14 −0.0396954
\(751\) 3.43705e15 0.525008 0.262504 0.964931i \(-0.415452\pi\)
0.262504 + 0.964931i \(0.415452\pi\)
\(752\) −9.81030e15 −1.48759
\(753\) −6.79605e15 −1.02302
\(754\) −2.79507e15 −0.417687
\(755\) 4.93443e14 0.0732029
\(756\) 7.15328e14 0.105350
\(757\) 1.01041e16 1.47731 0.738654 0.674085i \(-0.235462\pi\)
0.738654 + 0.674085i \(0.235462\pi\)
\(758\) 6.71764e14 0.0975071
\(759\) −3.13640e15 −0.451962
\(760\) −4.47784e14 −0.0640611
\(761\) 7.59905e15 1.07930 0.539652 0.841888i \(-0.318556\pi\)
0.539652 + 0.841888i \(0.318556\pi\)
\(762\) 3.49223e14 0.0492437
\(763\) 7.27689e15 1.01873
\(764\) −1.23107e16 −1.71108
\(765\) −3.19051e13 −0.00440273
\(766\) 1.78060e15 0.243955
\(767\) −1.41352e15 −0.192277
\(768\) −3.03338e15 −0.409675
\(769\) 1.12501e16 1.50856 0.754281 0.656551i \(-0.227985\pi\)
0.754281 + 0.656551i \(0.227985\pi\)
\(770\) −7.01668e13 −0.00934184
\(771\) −6.70444e15 −0.886265
\(772\) 3.29308e15 0.432222
\(773\) −4.90363e15 −0.639044 −0.319522 0.947579i \(-0.603522\pi\)
−0.319522 + 0.947579i \(0.603522\pi\)
\(774\) 6.16850e14 0.0798186
\(775\) 7.24571e15 0.930939
\(776\) −2.67241e15 −0.340929
\(777\) −3.96465e15 −0.502214
\(778\) 9.41084e14 0.118370
\(779\) 1.78045e15 0.222368
\(780\) −1.34137e15 −0.166352
\(781\) 2.25784e15 0.278043
\(782\) 1.55400e14 0.0190027
\(783\) 2.57040e15 0.312112
\(784\) −5.13970e15 −0.619725
\(785\) 1.13145e15 0.135473
\(786\) −1.16862e15 −0.138947
\(787\) −9.33119e15 −1.10173 −0.550866 0.834594i \(-0.685702\pi\)
−0.550866 + 0.834594i \(0.685702\pi\)
\(788\) 6.77374e15 0.794208
\(789\) 1.05130e15 0.122406
\(790\) 1.89789e14 0.0219443
\(791\) 4.79034e15 0.550042
\(792\) −4.73478e14 −0.0539898
\(793\) 1.70475e16 1.93045
\(794\) −1.82519e15 −0.205256
\(795\) −4.05810e14 −0.0453215
\(796\) −1.54751e16 −1.71638
\(797\) 9.30024e15 1.02441 0.512204 0.858864i \(-0.328829\pi\)
0.512204 + 0.858864i \(0.328829\pi\)
\(798\) 4.81677e14 0.0526914
\(799\) −9.88080e14 −0.107346
\(800\) −4.46517e15 −0.481773
\(801\) −1.69778e15 −0.181929
\(802\) −4.57272e13 −0.00486649
\(803\) −2.48684e15 −0.262853
\(804\) −4.54966e14 −0.0477607
\(805\) −1.80869e15 −0.188576
\(806\) −2.41308e15 −0.249878
\(807\) 4.84072e15 0.497858
\(808\) −2.95493e15 −0.301846
\(809\) −1.14715e16 −1.16387 −0.581935 0.813235i \(-0.697704\pi\)
−0.581935 + 0.813235i \(0.697704\pi\)
\(810\) −3.86881e13 −0.00389861
\(811\) 1.30920e16 1.31036 0.655179 0.755473i \(-0.272593\pi\)
0.655179 + 0.755473i \(0.272593\pi\)
\(812\) 8.93033e15 0.887788
\(813\) −1.00295e16 −0.990334
\(814\) 1.29185e15 0.126700
\(815\) −1.67591e15 −0.163262
\(816\) −3.56310e14 −0.0344772
\(817\) −1.32436e16 −1.27287
\(818\) 1.89823e15 0.181220
\(819\) 2.93104e15 0.277946
\(820\) −4.96842e14 −0.0467996
\(821\) −9.77412e15 −0.914514 −0.457257 0.889335i \(-0.651168\pi\)
−0.457257 + 0.889335i \(0.651168\pi\)
\(822\) 1.35843e15 0.126253
\(823\) 9.15167e15 0.844892 0.422446 0.906388i \(-0.361172\pi\)
0.422446 + 0.906388i \(0.361172\pi\)
\(824\) 6.02401e15 0.552441
\(825\) 2.86773e15 0.261242
\(826\) −1.41644e14 −0.0128177
\(827\) −1.75245e16 −1.57531 −0.787654 0.616117i \(-0.788705\pi\)
−0.787654 + 0.616117i \(0.788705\pi\)
\(828\) −6.00822e15 −0.536511
\(829\) −1.97295e16 −1.75011 −0.875056 0.484022i \(-0.839175\pi\)
−0.875056 + 0.484022i \(0.839175\pi\)
\(830\) −6.51067e14 −0.0573715
\(831\) −1.06711e16 −0.934125
\(832\) −1.39629e16 −1.21422
\(833\) −5.17663e14 −0.0447198
\(834\) −6.51922e13 −0.00559477
\(835\) 1.25155e15 0.106702
\(836\) 5.00425e15 0.423843
\(837\) 2.21910e15 0.186719
\(838\) −1.60677e14 −0.0134311
\(839\) 5.08692e15 0.422439 0.211219 0.977439i \(-0.432257\pi\)
0.211219 + 0.977439i \(0.432257\pi\)
\(840\) −2.73044e14 −0.0225267
\(841\) 1.98889e16 1.63017
\(842\) −2.41945e14 −0.0197016
\(843\) −2.23860e15 −0.181103
\(844\) −5.01639e15 −0.403189
\(845\) −2.97646e15 −0.237678
\(846\) −1.19815e15 −0.0950545
\(847\) −5.56997e15 −0.439030
\(848\) −4.53202e15 −0.354907
\(849\) 6.83246e15 0.531600
\(850\) −1.42088e14 −0.0109838
\(851\) 3.33001e16 2.55760
\(852\) 4.32521e15 0.330057
\(853\) 1.72061e16 1.30455 0.652277 0.757981i \(-0.273814\pi\)
0.652277 + 0.757981i \(0.273814\pi\)
\(854\) 1.70828e15 0.128689
\(855\) 8.30624e14 0.0621714
\(856\) 4.55222e15 0.338546
\(857\) −1.38966e16 −1.02687 −0.513434 0.858129i \(-0.671627\pi\)
−0.513434 + 0.858129i \(0.671627\pi\)
\(858\) −9.55057e14 −0.0701212
\(859\) −1.08043e15 −0.0788199 −0.0394100 0.999223i \(-0.512548\pi\)
−0.0394100 + 0.999223i \(0.512548\pi\)
\(860\) 3.69569e15 0.267889
\(861\) 1.08566e15 0.0781944
\(862\) −2.88582e15 −0.206528
\(863\) −2.25406e16 −1.60290 −0.801448 0.598064i \(-0.795937\pi\)
−0.801448 + 0.598064i \(0.795937\pi\)
\(864\) −1.36752e15 −0.0966293
\(865\) −1.33561e15 −0.0937759
\(866\) 3.25977e15 0.227425
\(867\) 8.29218e15 0.574862
\(868\) 7.70983e15 0.531112
\(869\) −4.30852e15 −0.294931
\(870\) −4.82992e14 −0.0328537
\(871\) −1.86421e15 −0.126007
\(872\) −9.22688e15 −0.619747
\(873\) 4.95722e15 0.330872
\(874\) −4.04573e15 −0.268339
\(875\) 3.37729e15 0.222599
\(876\) −4.76389e15 −0.312025
\(877\) 2.44292e15 0.159006 0.0795028 0.996835i \(-0.474667\pi\)
0.0795028 + 0.996835i \(0.474667\pi\)
\(878\) −5.33231e14 −0.0344902
\(879\) 1.19300e16 0.766834
\(880\) −1.35129e15 −0.0863165
\(881\) −2.77508e16 −1.76160 −0.880801 0.473487i \(-0.842995\pi\)
−0.880801 + 0.473487i \(0.842995\pi\)
\(882\) −6.27719e14 −0.0395993
\(883\) 2.43119e16 1.52417 0.762086 0.647475i \(-0.224175\pi\)
0.762086 + 0.647475i \(0.224175\pi\)
\(884\) −1.50878e15 −0.0940021
\(885\) −2.44258e14 −0.0151238
\(886\) −4.92843e14 −0.0303265
\(887\) 1.62559e16 0.994103 0.497051 0.867721i \(-0.334416\pi\)
0.497051 + 0.867721i \(0.334416\pi\)
\(888\) 5.02706e15 0.305522
\(889\) −4.57187e15 −0.276143
\(890\) 3.19023e14 0.0191504
\(891\) 8.78286e14 0.0523973
\(892\) −2.30417e16 −1.36618
\(893\) 2.57239e16 1.51584
\(894\) −2.55481e15 −0.149624
\(895\) 1.59772e15 0.0929980
\(896\) −6.29938e15 −0.364421
\(897\) −2.46185e16 −1.41548
\(898\) 3.26645e15 0.186662
\(899\) 2.77038e16 1.57348
\(900\) 5.49354e15 0.310112
\(901\) −4.56458e14 −0.0256103
\(902\) −3.53753e14 −0.0197272
\(903\) −8.07551e15 −0.447597
\(904\) −6.07401e15 −0.334618
\(905\) −2.20634e15 −0.120811
\(906\) 6.73027e14 0.0366292
\(907\) 1.95882e16 1.05963 0.529815 0.848113i \(-0.322261\pi\)
0.529815 + 0.848113i \(0.322261\pi\)
\(908\) −3.52461e16 −1.89513
\(909\) 5.48130e15 0.292942
\(910\) −5.50759e14 −0.0292573
\(911\) −6.40605e15 −0.338251 −0.169125 0.985595i \(-0.554094\pi\)
−0.169125 + 0.985595i \(0.554094\pi\)
\(912\) 9.27625e15 0.486857
\(913\) 1.47803e16 0.771071
\(914\) 3.16852e15 0.164306
\(915\) 2.94583e15 0.151842
\(916\) −1.28148e16 −0.656576
\(917\) 1.52990e16 0.779170
\(918\) −4.35167e13 −0.00220303
\(919\) 9.03519e15 0.454676 0.227338 0.973816i \(-0.426998\pi\)
0.227338 + 0.973816i \(0.426998\pi\)
\(920\) 2.29337e15 0.114720
\(921\) 1.23759e16 0.615386
\(922\) −3.83617e14 −0.0189617
\(923\) 1.77224e16 0.870792
\(924\) 3.05143e15 0.149042
\(925\) −3.04476e16 −1.47834
\(926\) 5.95419e15 0.287384
\(927\) −1.11743e16 −0.536146
\(928\) −1.70725e16 −0.814297
\(929\) 2.26269e16 1.07285 0.536424 0.843949i \(-0.319775\pi\)
0.536424 + 0.843949i \(0.319775\pi\)
\(930\) −4.16982e14 −0.0196545
\(931\) 1.34770e16 0.631493
\(932\) 3.11572e16 1.45134
\(933\) 4.48865e15 0.207858
\(934\) 7.20489e15 0.331680
\(935\) −1.36100e14 −0.00622865
\(936\) −3.71647e15 −0.169088
\(937\) 3.99367e16 1.80636 0.903180 0.429262i \(-0.141226\pi\)
0.903180 + 0.429262i \(0.141226\pi\)
\(938\) −1.86807e14 −0.00839997
\(939\) −5.11005e15 −0.228436
\(940\) −7.17837e15 −0.319024
\(941\) 4.19219e16 1.85224 0.926122 0.377223i \(-0.123121\pi\)
0.926122 + 0.377223i \(0.123121\pi\)
\(942\) 1.54324e15 0.0677880
\(943\) −9.11871e15 −0.398217
\(944\) −2.72782e15 −0.118432
\(945\) 5.06487e14 0.0218622
\(946\) 2.63134e15 0.112921
\(947\) −1.38333e16 −0.590201 −0.295100 0.955466i \(-0.595353\pi\)
−0.295100 + 0.955466i \(0.595353\pi\)
\(948\) −8.25358e15 −0.350103
\(949\) −1.95199e16 −0.823216
\(950\) 3.69916e15 0.155104
\(951\) −6.83849e15 −0.285081
\(952\) −3.07122e14 −0.0127294
\(953\) −8.04510e15 −0.331528 −0.165764 0.986165i \(-0.553009\pi\)
−0.165764 + 0.986165i \(0.553009\pi\)
\(954\) −5.53502e14 −0.0226779
\(955\) −8.71661e15 −0.355082
\(956\) 1.74819e16 0.708060
\(957\) 1.09647e16 0.441553
\(958\) 5.85110e14 0.0234276
\(959\) −1.77840e16 −0.707989
\(960\) −2.41280e15 −0.0955059
\(961\) −1.49088e15 −0.0586763
\(962\) 1.01401e16 0.396807
\(963\) −8.44421e15 −0.328560
\(964\) 2.10585e16 0.814712
\(965\) 2.33166e15 0.0896944
\(966\) −2.46695e15 −0.0943596
\(967\) 9.74385e14 0.0370582 0.0185291 0.999828i \(-0.494102\pi\)
0.0185291 + 0.999828i \(0.494102\pi\)
\(968\) 7.06256e15 0.267084
\(969\) 9.34291e14 0.0351319
\(970\) −9.31491e14 −0.0348285
\(971\) 3.42108e16 1.27191 0.635957 0.771725i \(-0.280606\pi\)
0.635957 + 0.771725i \(0.280606\pi\)
\(972\) 1.68248e15 0.0621992
\(973\) 8.53466e14 0.0313737
\(974\) 7.60275e15 0.277905
\(975\) 2.25097e16 0.818170
\(976\) 3.28985e16 1.18905
\(977\) 7.60864e14 0.0273456 0.0136728 0.999907i \(-0.495648\pi\)
0.0136728 + 0.999907i \(0.495648\pi\)
\(978\) −2.28584e15 −0.0816926
\(979\) −7.24236e15 −0.257380
\(980\) −3.76081e15 −0.132904
\(981\) 1.71155e16 0.601466
\(982\) −4.19692e15 −0.146662
\(983\) −3.75600e15 −0.130521 −0.0652607 0.997868i \(-0.520788\pi\)
−0.0652607 + 0.997868i \(0.520788\pi\)
\(984\) −1.37658e15 −0.0475695
\(985\) 4.79614e15 0.164813
\(986\) −5.43273e14 −0.0185650
\(987\) 1.56856e16 0.533035
\(988\) 3.92798e16 1.32741
\(989\) 6.78283e16 2.27946
\(990\) −1.65035e14 −0.00551547
\(991\) 1.43031e16 0.475362 0.237681 0.971343i \(-0.423613\pi\)
0.237681 + 0.971343i \(0.423613\pi\)
\(992\) −1.47392e16 −0.487147
\(993\) 2.62517e16 0.862853
\(994\) 1.77591e15 0.0580492
\(995\) −1.09572e16 −0.356181
\(996\) 2.83137e16 0.915316
\(997\) −1.74274e15 −0.0560287 −0.0280143 0.999608i \(-0.508918\pi\)
−0.0280143 + 0.999608i \(0.508918\pi\)
\(998\) −6.01463e15 −0.192305
\(999\) −9.32501e15 −0.296510
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.1.16 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.c.1.16 27 1.1 even 1 trivial