Properties

Label 177.12.a.b.1.9
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-45.9608 q^{2} +243.000 q^{3} +64.3958 q^{4} -11937.5 q^{5} -11168.5 q^{6} -79874.8 q^{7} +91168.1 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-45.9608 q^{2} +243.000 q^{3} +64.3958 q^{4} -11937.5 q^{5} -11168.5 q^{6} -79874.8 q^{7} +91168.1 q^{8} +59049.0 q^{9} +548659. q^{10} +946726. q^{11} +15648.2 q^{12} -1.76618e6 q^{13} +3.67111e6 q^{14} -2.90082e6 q^{15} -4.32204e6 q^{16} -7.18471e6 q^{17} -2.71394e6 q^{18} -1.18388e7 q^{19} -768727. q^{20} -1.94096e7 q^{21} -4.35123e7 q^{22} +1.07020e7 q^{23} +2.21538e7 q^{24} +9.36768e7 q^{25} +8.11749e7 q^{26} +1.43489e7 q^{27} -5.14360e6 q^{28} +1.10825e6 q^{29} +1.33324e8 q^{30} +1.34986e8 q^{31} +1.19323e7 q^{32} +2.30054e8 q^{33} +3.30215e8 q^{34} +9.53509e8 q^{35} +3.80251e6 q^{36} -4.45212e8 q^{37} +5.44120e8 q^{38} -4.29181e8 q^{39} -1.08832e9 q^{40} +1.34475e9 q^{41} +8.92080e8 q^{42} -5.78738e8 q^{43} +6.09651e7 q^{44} -7.04900e8 q^{45} -4.91872e8 q^{46} +2.46985e9 q^{47} -1.05026e9 q^{48} +4.40266e9 q^{49} -4.30546e9 q^{50} -1.74588e9 q^{51} -1.13734e8 q^{52} +3.35994e9 q^{53} -6.59487e8 q^{54} -1.13016e10 q^{55} -7.28203e9 q^{56} -2.87683e9 q^{57} -5.09360e7 q^{58} -7.14924e8 q^{59} -1.86801e8 q^{60} +2.55522e9 q^{61} -6.20407e9 q^{62} -4.71653e9 q^{63} +8.30312e9 q^{64} +2.10838e10 q^{65} -1.05735e10 q^{66} +1.07691e10 q^{67} -4.62665e8 q^{68} +2.60059e9 q^{69} -4.38241e10 q^{70} +1.60386e10 q^{71} +5.38338e9 q^{72} +2.47907e10 q^{73} +2.04623e10 q^{74} +2.27635e10 q^{75} -7.62368e8 q^{76} -7.56196e10 q^{77} +1.97255e10 q^{78} -3.19630e10 q^{79} +5.15945e10 q^{80} +3.48678e9 q^{81} -6.18057e10 q^{82} -3.85068e10 q^{83} -1.24990e9 q^{84} +8.57677e10 q^{85} +2.65992e10 q^{86} +2.69304e8 q^{87} +8.63111e10 q^{88} -7.10593e10 q^{89} +3.23978e10 q^{90} +1.41073e11 q^{91} +6.89164e8 q^{92} +3.28016e10 q^{93} -1.13516e11 q^{94} +1.41326e11 q^{95} +2.89954e9 q^{96} -6.48923e10 q^{97} -2.02350e11 q^{98} +5.59032e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −45.9608 −1.01560 −0.507800 0.861475i \(-0.669541\pi\)
−0.507800 + 0.861475i \(0.669541\pi\)
\(3\) 243.000 0.577350
\(4\) 64.3958 0.0314433
\(5\) −11937.5 −1.70836 −0.854181 0.519976i \(-0.825941\pi\)
−0.854181 + 0.519976i \(0.825941\pi\)
\(6\) −11168.5 −0.586357
\(7\) −79874.8 −1.79627 −0.898133 0.439723i \(-0.855077\pi\)
−0.898133 + 0.439723i \(0.855077\pi\)
\(8\) 91168.1 0.983666
\(9\) 59049.0 0.333333
\(10\) 548659. 1.73501
\(11\) 946726. 1.77241 0.886205 0.463293i \(-0.153332\pi\)
0.886205 + 0.463293i \(0.153332\pi\)
\(12\) 15648.2 0.0181538
\(13\) −1.76618e6 −1.31931 −0.659653 0.751570i \(-0.729297\pi\)
−0.659653 + 0.751570i \(0.729297\pi\)
\(14\) 3.67111e6 1.82429
\(15\) −2.90082e6 −0.986323
\(16\) −4.32204e6 −1.03045
\(17\) −7.18471e6 −1.22727 −0.613635 0.789590i \(-0.710293\pi\)
−0.613635 + 0.789590i \(0.710293\pi\)
\(18\) −2.71394e6 −0.338533
\(19\) −1.18388e7 −1.09689 −0.548444 0.836187i \(-0.684780\pi\)
−0.548444 + 0.836187i \(0.684780\pi\)
\(20\) −768727. −0.0537165
\(21\) −1.94096e7 −1.03708
\(22\) −4.35123e7 −1.80006
\(23\) 1.07020e7 0.346706 0.173353 0.984860i \(-0.444540\pi\)
0.173353 + 0.984860i \(0.444540\pi\)
\(24\) 2.21538e7 0.567920
\(25\) 9.36768e7 1.91850
\(26\) 8.11749e7 1.33989
\(27\) 1.43489e7 0.192450
\(28\) −5.14360e6 −0.0564805
\(29\) 1.10825e6 0.0100334 0.00501670 0.999987i \(-0.498403\pi\)
0.00501670 + 0.999987i \(0.498403\pi\)
\(30\) 1.33324e8 1.00171
\(31\) 1.34986e8 0.846837 0.423419 0.905934i \(-0.360830\pi\)
0.423419 + 0.905934i \(0.360830\pi\)
\(32\) 1.19323e7 0.0628634
\(33\) 2.30054e8 1.02330
\(34\) 3.30215e8 1.24642
\(35\) 9.53509e8 3.06867
\(36\) 3.80251e6 0.0104811
\(37\) −4.45212e8 −1.05550 −0.527749 0.849400i \(-0.676964\pi\)
−0.527749 + 0.849400i \(0.676964\pi\)
\(38\) 5.44120e8 1.11400
\(39\) −4.29181e8 −0.761702
\(40\) −1.08832e9 −1.68046
\(41\) 1.34475e9 1.81271 0.906357 0.422512i \(-0.138852\pi\)
0.906357 + 0.422512i \(0.138852\pi\)
\(42\) 8.92080e8 1.05325
\(43\) −5.78738e8 −0.600351 −0.300175 0.953884i \(-0.597045\pi\)
−0.300175 + 0.953884i \(0.597045\pi\)
\(44\) 6.09651e7 0.0557304
\(45\) −7.04900e8 −0.569454
\(46\) −4.91872e8 −0.352115
\(47\) 2.46985e9 1.57084 0.785421 0.618962i \(-0.212446\pi\)
0.785421 + 0.618962i \(0.212446\pi\)
\(48\) −1.05026e9 −0.594933
\(49\) 4.40266e9 2.22657
\(50\) −4.30546e9 −1.94843
\(51\) −1.74588e9 −0.708565
\(52\) −1.13734e8 −0.0414833
\(53\) 3.35994e9 1.10361 0.551803 0.833974i \(-0.313940\pi\)
0.551803 + 0.833974i \(0.313940\pi\)
\(54\) −6.59487e8 −0.195452
\(55\) −1.13016e10 −3.02792
\(56\) −7.28203e9 −1.76693
\(57\) −2.87683e9 −0.633289
\(58\) −5.09360e7 −0.0101899
\(59\) −7.14924e8 −0.130189
\(60\) −1.86801e8 −0.0310132
\(61\) 2.55522e9 0.387360 0.193680 0.981065i \(-0.437958\pi\)
0.193680 + 0.981065i \(0.437958\pi\)
\(62\) −6.20407e9 −0.860048
\(63\) −4.71653e9 −0.598756
\(64\) 8.30312e9 0.966610
\(65\) 2.10838e10 2.25385
\(66\) −1.05735e10 −1.03927
\(67\) 1.07691e10 0.974466 0.487233 0.873272i \(-0.338006\pi\)
0.487233 + 0.873272i \(0.338006\pi\)
\(68\) −4.62665e8 −0.0385894
\(69\) 2.60059e9 0.200171
\(70\) −4.38241e10 −3.11654
\(71\) 1.60386e10 1.05498 0.527491 0.849561i \(-0.323133\pi\)
0.527491 + 0.849561i \(0.323133\pi\)
\(72\) 5.38338e9 0.327889
\(73\) 2.47907e10 1.39963 0.699814 0.714326i \(-0.253266\pi\)
0.699814 + 0.714326i \(0.253266\pi\)
\(74\) 2.04623e10 1.07196
\(75\) 2.27635e10 1.10765
\(76\) −7.62368e8 −0.0344897
\(77\) −7.56196e10 −3.18372
\(78\) 1.97255e10 0.773584
\(79\) −3.19630e10 −1.16869 −0.584343 0.811507i \(-0.698648\pi\)
−0.584343 + 0.811507i \(0.698648\pi\)
\(80\) 5.15945e10 1.76039
\(81\) 3.48678e9 0.111111
\(82\) −6.18057e10 −1.84099
\(83\) −3.85068e10 −1.07302 −0.536509 0.843894i \(-0.680257\pi\)
−0.536509 + 0.843894i \(0.680257\pi\)
\(84\) −1.24990e9 −0.0326090
\(85\) 8.57677e10 2.09662
\(86\) 2.65992e10 0.609716
\(87\) 2.69304e8 0.00579279
\(88\) 8.63111e10 1.74346
\(89\) −7.10593e10 −1.34889 −0.674444 0.738326i \(-0.735617\pi\)
−0.674444 + 0.738326i \(0.735617\pi\)
\(90\) 3.23978e10 0.578337
\(91\) 1.41073e11 2.36983
\(92\) 6.89164e8 0.0109016
\(93\) 3.28016e10 0.488922
\(94\) −1.13516e11 −1.59535
\(95\) 1.41326e11 1.87388
\(96\) 2.89954e9 0.0362942
\(97\) −6.48923e10 −0.767271 −0.383636 0.923485i \(-0.625328\pi\)
−0.383636 + 0.923485i \(0.625328\pi\)
\(98\) −2.02350e11 −2.26131
\(99\) 5.59032e10 0.590804
\(100\) 6.03239e9 0.0603239
\(101\) −8.28856e10 −0.784714 −0.392357 0.919813i \(-0.628340\pi\)
−0.392357 + 0.919813i \(0.628340\pi\)
\(102\) 8.02422e10 0.719618
\(103\) −3.80678e10 −0.323558 −0.161779 0.986827i \(-0.551723\pi\)
−0.161779 + 0.986827i \(0.551723\pi\)
\(104\) −1.61019e11 −1.29776
\(105\) 2.31703e11 1.77170
\(106\) −1.54426e11 −1.12082
\(107\) 1.14863e11 0.791716 0.395858 0.918312i \(-0.370447\pi\)
0.395858 + 0.918312i \(0.370447\pi\)
\(108\) 9.24009e8 0.00605126
\(109\) −1.45645e11 −0.906674 −0.453337 0.891339i \(-0.649767\pi\)
−0.453337 + 0.891339i \(0.649767\pi\)
\(110\) 5.19430e11 3.07515
\(111\) −1.08187e11 −0.609392
\(112\) 3.45222e11 1.85097
\(113\) −5.69768e10 −0.290915 −0.145458 0.989364i \(-0.546465\pi\)
−0.145458 + 0.989364i \(0.546465\pi\)
\(114\) 1.32221e11 0.643168
\(115\) −1.27756e11 −0.592300
\(116\) 7.13665e7 0.000315483 0
\(117\) −1.04291e11 −0.439769
\(118\) 3.28585e10 0.132220
\(119\) 5.73877e11 2.20450
\(120\) −2.64462e11 −0.970213
\(121\) 6.10978e11 2.14144
\(122\) −1.17440e11 −0.393403
\(123\) 3.26774e11 1.04657
\(124\) 8.69254e9 0.0266273
\(125\) −5.35382e11 −1.56913
\(126\) 2.16776e11 0.608096
\(127\) 2.48672e11 0.667892 0.333946 0.942592i \(-0.391620\pi\)
0.333946 + 0.942592i \(0.391620\pi\)
\(128\) −4.06055e11 −1.04455
\(129\) −1.40633e11 −0.346613
\(130\) −9.69029e11 −2.28901
\(131\) 5.44777e10 0.123375 0.0616874 0.998096i \(-0.480352\pi\)
0.0616874 + 0.998096i \(0.480352\pi\)
\(132\) 1.48145e10 0.0321759
\(133\) 9.45621e11 1.97030
\(134\) −4.94955e11 −0.989668
\(135\) −1.71291e11 −0.328774
\(136\) −6.55016e11 −1.20722
\(137\) −2.41952e11 −0.428318 −0.214159 0.976799i \(-0.568701\pi\)
−0.214159 + 0.976799i \(0.568701\pi\)
\(138\) −1.19525e11 −0.203294
\(139\) 1.34844e11 0.220419 0.110210 0.993908i \(-0.464848\pi\)
0.110210 + 0.993908i \(0.464848\pi\)
\(140\) 6.14020e10 0.0964891
\(141\) 6.00174e11 0.906926
\(142\) −7.37146e11 −1.07144
\(143\) −1.67208e12 −2.33835
\(144\) −2.55212e11 −0.343485
\(145\) −1.32298e10 −0.0171407
\(146\) −1.13940e12 −1.42146
\(147\) 1.06985e12 1.28551
\(148\) −2.86698e10 −0.0331883
\(149\) 8.73681e11 0.974604 0.487302 0.873233i \(-0.337981\pi\)
0.487302 + 0.873233i \(0.337981\pi\)
\(150\) −1.04623e12 −1.12493
\(151\) −9.24165e11 −0.958024 −0.479012 0.877808i \(-0.659005\pi\)
−0.479012 + 0.877808i \(0.659005\pi\)
\(152\) −1.07932e12 −1.07897
\(153\) −4.24250e11 −0.409090
\(154\) 3.47554e12 3.23339
\(155\) −1.61140e12 −1.44670
\(156\) −2.76374e10 −0.0239504
\(157\) −1.06220e10 −0.00888706 −0.00444353 0.999990i \(-0.501414\pi\)
−0.00444353 + 0.999990i \(0.501414\pi\)
\(158\) 1.46904e12 1.18692
\(159\) 8.16466e11 0.637168
\(160\) −1.42442e11 −0.107393
\(161\) −8.54820e11 −0.622777
\(162\) −1.60255e11 −0.112844
\(163\) 1.82975e12 1.24555 0.622774 0.782402i \(-0.286006\pi\)
0.622774 + 0.782402i \(0.286006\pi\)
\(164\) 8.65960e10 0.0569976
\(165\) −2.74628e12 −1.74817
\(166\) 1.76980e12 1.08976
\(167\) −1.88729e12 −1.12434 −0.562170 0.827022i \(-0.690033\pi\)
−0.562170 + 0.827022i \(0.690033\pi\)
\(168\) −1.76953e12 −1.02014
\(169\) 1.32722e12 0.740570
\(170\) −3.94195e12 −2.12933
\(171\) −6.99069e11 −0.365629
\(172\) −3.72683e10 −0.0188770
\(173\) −2.29470e12 −1.12583 −0.562915 0.826515i \(-0.690320\pi\)
−0.562915 + 0.826515i \(0.690320\pi\)
\(174\) −1.23774e10 −0.00588315
\(175\) −7.48242e12 −3.44614
\(176\) −4.09179e12 −1.82639
\(177\) −1.73727e11 −0.0751646
\(178\) 3.26594e12 1.36993
\(179\) 2.87738e12 1.17032 0.585162 0.810917i \(-0.301031\pi\)
0.585162 + 0.810917i \(0.301031\pi\)
\(180\) −4.53926e10 −0.0179055
\(181\) −6.19841e11 −0.237164 −0.118582 0.992944i \(-0.537835\pi\)
−0.118582 + 0.992944i \(0.537835\pi\)
\(182\) −6.48383e12 −2.40680
\(183\) 6.20919e11 0.223642
\(184\) 9.75680e11 0.341043
\(185\) 5.31474e12 1.80317
\(186\) −1.50759e12 −0.496549
\(187\) −6.80195e12 −2.17523
\(188\) 1.59048e11 0.0493924
\(189\) −1.14612e12 −0.345692
\(190\) −6.49546e12 −1.90311
\(191\) 2.40903e12 0.685739 0.342870 0.939383i \(-0.388601\pi\)
0.342870 + 0.939383i \(0.388601\pi\)
\(192\) 2.01766e12 0.558073
\(193\) −1.78608e12 −0.480105 −0.240052 0.970760i \(-0.577165\pi\)
−0.240052 + 0.970760i \(0.577165\pi\)
\(194\) 2.98250e12 0.779241
\(195\) 5.12336e12 1.30126
\(196\) 2.83513e11 0.0700107
\(197\) −5.74904e12 −1.38048 −0.690241 0.723579i \(-0.742496\pi\)
−0.690241 + 0.723579i \(0.742496\pi\)
\(198\) −2.56936e12 −0.600020
\(199\) −1.16531e12 −0.264698 −0.132349 0.991203i \(-0.542252\pi\)
−0.132349 + 0.991203i \(0.542252\pi\)
\(200\) 8.54033e12 1.88716
\(201\) 2.61688e12 0.562608
\(202\) 3.80949e12 0.796955
\(203\) −8.85212e10 −0.0180227
\(204\) −1.12428e11 −0.0222796
\(205\) −1.60530e13 −3.09677
\(206\) 1.74963e12 0.328606
\(207\) 6.31942e11 0.115569
\(208\) 7.63349e12 1.35949
\(209\) −1.12081e13 −1.94414
\(210\) −1.06492e13 −1.79934
\(211\) 5.68606e12 0.935961 0.467980 0.883739i \(-0.344982\pi\)
0.467980 + 0.883739i \(0.344982\pi\)
\(212\) 2.16366e11 0.0347010
\(213\) 3.89738e12 0.609094
\(214\) −5.27919e12 −0.804066
\(215\) 6.90870e12 1.02562
\(216\) 1.30816e12 0.189307
\(217\) −1.07820e13 −1.52115
\(218\) 6.69398e12 0.920818
\(219\) 6.02413e12 0.808075
\(220\) −7.27774e11 −0.0952076
\(221\) 1.26895e13 1.61914
\(222\) 4.97234e12 0.618899
\(223\) −7.10973e12 −0.863329 −0.431665 0.902034i \(-0.642074\pi\)
−0.431665 + 0.902034i \(0.642074\pi\)
\(224\) −9.53088e11 −0.112920
\(225\) 5.53152e12 0.639500
\(226\) 2.61870e12 0.295453
\(227\) 1.49744e13 1.64895 0.824474 0.565900i \(-0.191471\pi\)
0.824474 + 0.565900i \(0.191471\pi\)
\(228\) −1.85255e11 −0.0199127
\(229\) −1.02108e13 −1.07143 −0.535715 0.844399i \(-0.679958\pi\)
−0.535715 + 0.844399i \(0.679958\pi\)
\(230\) 5.87175e12 0.601540
\(231\) −1.83756e13 −1.83812
\(232\) 1.01037e11 0.00986951
\(233\) −6.44340e12 −0.614692 −0.307346 0.951598i \(-0.599441\pi\)
−0.307346 + 0.951598i \(0.599441\pi\)
\(234\) 4.79330e12 0.446629
\(235\) −2.94840e13 −2.68357
\(236\) −4.60381e10 −0.00409356
\(237\) −7.76700e12 −0.674741
\(238\) −2.63759e13 −2.23889
\(239\) −2.10460e13 −1.74575 −0.872873 0.487948i \(-0.837746\pi\)
−0.872873 + 0.487948i \(0.837746\pi\)
\(240\) 1.25375e13 1.01636
\(241\) 2.15329e13 1.70612 0.853059 0.521815i \(-0.174745\pi\)
0.853059 + 0.521815i \(0.174745\pi\)
\(242\) −2.80810e13 −2.17485
\(243\) 8.47289e11 0.0641500
\(244\) 1.64546e11 0.0121799
\(245\) −5.25570e13 −3.80379
\(246\) −1.50188e13 −1.06290
\(247\) 2.09094e13 1.44713
\(248\) 1.23064e13 0.833005
\(249\) −9.35714e12 −0.619508
\(250\) 2.46066e13 1.59361
\(251\) 2.15058e13 1.36254 0.681270 0.732032i \(-0.261428\pi\)
0.681270 + 0.732032i \(0.261428\pi\)
\(252\) −3.03725e11 −0.0188268
\(253\) 1.01319e13 0.614506
\(254\) −1.14292e13 −0.678311
\(255\) 2.08416e13 1.21048
\(256\) 1.65784e12 0.0942375
\(257\) −1.16136e13 −0.646154 −0.323077 0.946373i \(-0.604717\pi\)
−0.323077 + 0.946373i \(0.604717\pi\)
\(258\) 6.46362e12 0.352020
\(259\) 3.55613e13 1.89596
\(260\) 1.35771e12 0.0708685
\(261\) 6.54410e10 0.00334447
\(262\) −2.50384e12 −0.125299
\(263\) 1.27802e12 0.0626297 0.0313148 0.999510i \(-0.490031\pi\)
0.0313148 + 0.999510i \(0.490031\pi\)
\(264\) 2.09736e13 1.00659
\(265\) −4.01094e13 −1.88536
\(266\) −4.34615e13 −2.00104
\(267\) −1.72674e13 −0.778781
\(268\) 6.93483e11 0.0306404
\(269\) 1.33643e13 0.578505 0.289253 0.957253i \(-0.406593\pi\)
0.289253 + 0.957253i \(0.406593\pi\)
\(270\) 7.87266e12 0.333903
\(271\) 8.16318e12 0.339257 0.169628 0.985508i \(-0.445743\pi\)
0.169628 + 0.985508i \(0.445743\pi\)
\(272\) 3.10526e13 1.26465
\(273\) 3.42808e13 1.36822
\(274\) 1.11203e13 0.434999
\(275\) 8.86862e13 3.40037
\(276\) 1.67467e11 0.00629403
\(277\) 2.11796e13 0.780331 0.390166 0.920745i \(-0.372418\pi\)
0.390166 + 0.920745i \(0.372418\pi\)
\(278\) −6.19753e12 −0.223858
\(279\) 7.97080e12 0.282279
\(280\) 8.69296e13 3.01855
\(281\) 2.20969e13 0.752396 0.376198 0.926539i \(-0.377231\pi\)
0.376198 + 0.926539i \(0.377231\pi\)
\(282\) −2.75845e13 −0.921074
\(283\) −2.65237e13 −0.868578 −0.434289 0.900774i \(-0.643000\pi\)
−0.434289 + 0.900774i \(0.643000\pi\)
\(284\) 1.03282e12 0.0331721
\(285\) 3.43422e13 1.08189
\(286\) 7.68504e13 2.37483
\(287\) −1.07411e14 −3.25612
\(288\) 7.04589e11 0.0209545
\(289\) 1.73481e13 0.506191
\(290\) 6.08050e11 0.0174081
\(291\) −1.57688e13 −0.442984
\(292\) 1.59641e12 0.0440088
\(293\) 6.41386e12 0.173519 0.0867596 0.996229i \(-0.472349\pi\)
0.0867596 + 0.996229i \(0.472349\pi\)
\(294\) −4.91711e13 −1.30557
\(295\) 8.53444e12 0.222410
\(296\) −4.05891e13 −1.03826
\(297\) 1.35845e13 0.341101
\(298\) −4.01551e13 −0.989808
\(299\) −1.89016e13 −0.457412
\(300\) 1.46587e12 0.0348280
\(301\) 4.62266e13 1.07839
\(302\) 4.24754e13 0.972969
\(303\) −2.01412e13 −0.453055
\(304\) 5.11677e13 1.13029
\(305\) −3.05031e13 −0.661751
\(306\) 1.94989e13 0.415472
\(307\) 1.23817e13 0.259132 0.129566 0.991571i \(-0.458642\pi\)
0.129566 + 0.991571i \(0.458642\pi\)
\(308\) −4.86958e12 −0.100107
\(309\) −9.25047e12 −0.186807
\(310\) 7.40614e13 1.46927
\(311\) −3.10921e13 −0.605994 −0.302997 0.952991i \(-0.597987\pi\)
−0.302997 + 0.952991i \(0.597987\pi\)
\(312\) −3.91276e13 −0.749260
\(313\) −2.14700e13 −0.403960 −0.201980 0.979390i \(-0.564738\pi\)
−0.201980 + 0.979390i \(0.564738\pi\)
\(314\) 4.88196e11 0.00902570
\(315\) 5.63038e13 1.02289
\(316\) −2.05828e12 −0.0367473
\(317\) −8.12504e13 −1.42561 −0.712803 0.701364i \(-0.752575\pi\)
−0.712803 + 0.701364i \(0.752575\pi\)
\(318\) −3.75254e13 −0.647107
\(319\) 1.04921e12 0.0177833
\(320\) −9.91188e13 −1.65132
\(321\) 2.79117e13 0.457097
\(322\) 3.92882e13 0.632492
\(323\) 8.50582e13 1.34618
\(324\) 2.24534e11 0.00349370
\(325\) −1.65450e14 −2.53109
\(326\) −8.40968e13 −1.26498
\(327\) −3.53918e13 −0.523468
\(328\) 1.22598e14 1.78311
\(329\) −1.97279e14 −2.82165
\(330\) 1.26221e14 1.77544
\(331\) 5.57980e13 0.771907 0.385953 0.922518i \(-0.373873\pi\)
0.385953 + 0.922518i \(0.373873\pi\)
\(332\) −2.47967e12 −0.0337392
\(333\) −2.62893e13 −0.351833
\(334\) 8.67413e13 1.14188
\(335\) −1.28556e14 −1.66474
\(336\) 8.38890e13 1.06866
\(337\) 4.36993e12 0.0547659 0.0273830 0.999625i \(-0.491283\pi\)
0.0273830 + 0.999625i \(0.491283\pi\)
\(338\) −6.10001e13 −0.752122
\(339\) −1.38454e13 −0.167960
\(340\) 5.52308e12 0.0659246
\(341\) 1.27795e14 1.50094
\(342\) 3.21298e13 0.371333
\(343\) −1.93723e14 −2.20325
\(344\) −5.27624e13 −0.590545
\(345\) −3.10446e13 −0.341964
\(346\) 1.05466e14 1.14339
\(347\) −1.11128e13 −0.118580 −0.0592900 0.998241i \(-0.518884\pi\)
−0.0592900 + 0.998241i \(0.518884\pi\)
\(348\) 1.73421e10 0.000182144 0
\(349\) −1.11101e13 −0.114863 −0.0574313 0.998349i \(-0.518291\pi\)
−0.0574313 + 0.998349i \(0.518291\pi\)
\(350\) 3.43898e14 3.49990
\(351\) −2.53427e13 −0.253901
\(352\) 1.12966e13 0.111420
\(353\) −6.20785e13 −0.602810 −0.301405 0.953496i \(-0.597456\pi\)
−0.301405 + 0.953496i \(0.597456\pi\)
\(354\) 7.98461e12 0.0763372
\(355\) −1.91461e14 −1.80229
\(356\) −4.57592e12 −0.0424134
\(357\) 1.39452e14 1.27277
\(358\) −1.32247e14 −1.18858
\(359\) 5.26471e13 0.465967 0.232983 0.972481i \(-0.425151\pi\)
0.232983 + 0.972481i \(0.425151\pi\)
\(360\) −6.42643e13 −0.560153
\(361\) 2.36666e13 0.203164
\(362\) 2.84884e13 0.240863
\(363\) 1.48468e14 1.23636
\(364\) 9.08451e12 0.0745151
\(365\) −2.95940e14 −2.39107
\(366\) −2.85380e13 −0.227131
\(367\) −7.12029e12 −0.0558257 −0.0279129 0.999610i \(-0.508886\pi\)
−0.0279129 + 0.999610i \(0.508886\pi\)
\(368\) −4.62545e13 −0.357265
\(369\) 7.94060e13 0.604238
\(370\) −2.44270e14 −1.83130
\(371\) −2.68375e14 −1.98237
\(372\) 2.11229e12 0.0153733
\(373\) 1.28958e14 0.924801 0.462401 0.886671i \(-0.346988\pi\)
0.462401 + 0.886671i \(0.346988\pi\)
\(374\) 3.12623e14 2.20916
\(375\) −1.30098e14 −0.905938
\(376\) 2.25172e14 1.54518
\(377\) −1.95736e12 −0.0132371
\(378\) 5.26764e13 0.351084
\(379\) 1.49484e14 0.981928 0.490964 0.871180i \(-0.336645\pi\)
0.490964 + 0.871180i \(0.336645\pi\)
\(380\) 9.10080e12 0.0589210
\(381\) 6.04272e13 0.385607
\(382\) −1.10721e14 −0.696437
\(383\) 1.40469e14 0.870940 0.435470 0.900203i \(-0.356582\pi\)
0.435470 + 0.900203i \(0.356582\pi\)
\(384\) −9.86715e13 −0.603073
\(385\) 9.02712e14 5.43895
\(386\) 8.20897e13 0.487594
\(387\) −3.41739e13 −0.200117
\(388\) −4.17879e12 −0.0241255
\(389\) 1.27503e14 0.725767 0.362884 0.931834i \(-0.381792\pi\)
0.362884 + 0.931834i \(0.381792\pi\)
\(390\) −2.35474e14 −1.32156
\(391\) −7.68907e13 −0.425502
\(392\) 4.01382e14 2.19021
\(393\) 1.32381e13 0.0712305
\(394\) 2.64230e14 1.40202
\(395\) 3.81559e14 1.99654
\(396\) 3.59993e12 0.0185768
\(397\) −1.77885e14 −0.905299 −0.452650 0.891688i \(-0.649521\pi\)
−0.452650 + 0.891688i \(0.649521\pi\)
\(398\) 5.35587e13 0.268827
\(399\) 2.29786e14 1.13756
\(400\) −4.04875e14 −1.97693
\(401\) 8.12140e13 0.391144 0.195572 0.980689i \(-0.437344\pi\)
0.195572 + 0.980689i \(0.437344\pi\)
\(402\) −1.20274e14 −0.571385
\(403\) −2.38409e14 −1.11724
\(404\) −5.33748e12 −0.0246740
\(405\) −4.16236e13 −0.189818
\(406\) 4.06850e12 0.0183038
\(407\) −4.21494e14 −1.87078
\(408\) −1.59169e14 −0.696991
\(409\) −2.01695e14 −0.871398 −0.435699 0.900093i \(-0.643499\pi\)
−0.435699 + 0.900093i \(0.643499\pi\)
\(410\) 7.37808e14 3.14508
\(411\) −5.87943e13 −0.247289
\(412\) −2.45140e12 −0.0101737
\(413\) 5.71045e13 0.233854
\(414\) −2.90446e13 −0.117372
\(415\) 4.59676e14 1.83310
\(416\) −2.10745e13 −0.0829361
\(417\) 3.27671e13 0.127259
\(418\) 5.15133e14 1.97446
\(419\) −2.81412e14 −1.06455 −0.532274 0.846572i \(-0.678663\pi\)
−0.532274 + 0.846572i \(0.678663\pi\)
\(420\) 1.49207e13 0.0557080
\(421\) 1.37748e14 0.507615 0.253808 0.967255i \(-0.418317\pi\)
0.253808 + 0.967255i \(0.418317\pi\)
\(422\) −2.61336e14 −0.950562
\(423\) 1.45842e14 0.523614
\(424\) 3.06319e14 1.08558
\(425\) −6.73040e14 −2.35452
\(426\) −1.79127e14 −0.618596
\(427\) −2.04098e14 −0.695802
\(428\) 7.39669e12 0.0248941
\(429\) −4.06317e14 −1.35005
\(430\) −3.17530e14 −1.04162
\(431\) 1.15761e14 0.374919 0.187460 0.982272i \(-0.439975\pi\)
0.187460 + 0.982272i \(0.439975\pi\)
\(432\) −6.20165e13 −0.198311
\(433\) −3.86740e14 −1.22106 −0.610528 0.791995i \(-0.709043\pi\)
−0.610528 + 0.791995i \(0.709043\pi\)
\(434\) 4.95549e14 1.54488
\(435\) −3.21483e12 −0.00989617
\(436\) −9.37895e12 −0.0285088
\(437\) −1.26699e14 −0.380298
\(438\) −2.76874e14 −0.820681
\(439\) 3.30433e14 0.967228 0.483614 0.875281i \(-0.339324\pi\)
0.483614 + 0.875281i \(0.339324\pi\)
\(440\) −1.03034e15 −2.97846
\(441\) 2.59973e14 0.742191
\(442\) −5.83218e14 −1.64440
\(443\) 5.19386e14 1.44634 0.723169 0.690672i \(-0.242685\pi\)
0.723169 + 0.690672i \(0.242685\pi\)
\(444\) −6.96676e12 −0.0191613
\(445\) 8.48273e14 2.30439
\(446\) 3.26769e14 0.876797
\(447\) 2.12304e14 0.562688
\(448\) −6.63210e14 −1.73629
\(449\) 7.61057e14 1.96817 0.984085 0.177699i \(-0.0568654\pi\)
0.984085 + 0.177699i \(0.0568654\pi\)
\(450\) −2.54233e14 −0.649476
\(451\) 1.27311e15 3.21287
\(452\) −3.66906e12 −0.00914732
\(453\) −2.24572e14 −0.553115
\(454\) −6.88235e14 −1.67467
\(455\) −1.68407e15 −4.04852
\(456\) −2.62275e14 −0.622945
\(457\) −7.47629e13 −0.175448 −0.0877238 0.996145i \(-0.527959\pi\)
−0.0877238 + 0.996145i \(0.527959\pi\)
\(458\) 4.69296e14 1.08814
\(459\) −1.03093e14 −0.236188
\(460\) −8.22692e12 −0.0186238
\(461\) −3.26176e14 −0.729621 −0.364810 0.931082i \(-0.618866\pi\)
−0.364810 + 0.931082i \(0.618866\pi\)
\(462\) 8.44555e14 1.86680
\(463\) −2.70438e14 −0.590707 −0.295353 0.955388i \(-0.595437\pi\)
−0.295353 + 0.955388i \(0.595437\pi\)
\(464\) −4.78989e12 −0.0103390
\(465\) −3.91571e14 −0.835255
\(466\) 2.96144e14 0.624281
\(467\) 5.44006e14 1.13334 0.566671 0.823944i \(-0.308231\pi\)
0.566671 + 0.823944i \(0.308231\pi\)
\(468\) −6.71590e12 −0.0138278
\(469\) −8.60178e14 −1.75040
\(470\) 1.35511e15 2.72543
\(471\) −2.58115e12 −0.00513095
\(472\) −6.51783e13 −0.128062
\(473\) −5.47906e14 −1.06407
\(474\) 3.56977e14 0.685267
\(475\) −1.10902e15 −2.10438
\(476\) 3.69553e13 0.0693168
\(477\) 1.98401e14 0.367869
\(478\) 9.67291e14 1.77298
\(479\) −1.44509e14 −0.261847 −0.130924 0.991392i \(-0.541794\pi\)
−0.130924 + 0.991392i \(0.541794\pi\)
\(480\) −3.46134e13 −0.0620037
\(481\) 7.86323e14 1.39253
\(482\) −9.89670e14 −1.73273
\(483\) −2.07721e14 −0.359560
\(484\) 3.93444e13 0.0673338
\(485\) 7.74655e14 1.31078
\(486\) −3.89421e13 −0.0651508
\(487\) 4.92506e14 0.814709 0.407354 0.913270i \(-0.366451\pi\)
0.407354 + 0.913270i \(0.366451\pi\)
\(488\) 2.32955e14 0.381033
\(489\) 4.44630e14 0.719117
\(490\) 2.41556e15 3.86313
\(491\) 3.57330e14 0.565095 0.282548 0.959253i \(-0.408820\pi\)
0.282548 + 0.959253i \(0.408820\pi\)
\(492\) 2.10428e13 0.0329076
\(493\) −7.96244e12 −0.0123137
\(494\) −9.61012e14 −1.46971
\(495\) −6.67347e14 −1.00931
\(496\) −5.83416e14 −0.872627
\(497\) −1.28108e15 −1.89503
\(498\) 4.30062e14 0.629172
\(499\) 1.02704e14 0.148606 0.0743029 0.997236i \(-0.476327\pi\)
0.0743029 + 0.997236i \(0.476327\pi\)
\(500\) −3.44764e13 −0.0493386
\(501\) −4.58611e14 −0.649138
\(502\) −9.88422e14 −1.38380
\(503\) −1.08451e15 −1.50179 −0.750893 0.660424i \(-0.770376\pi\)
−0.750893 + 0.660424i \(0.770376\pi\)
\(504\) −4.29997e14 −0.588976
\(505\) 9.89450e14 1.34058
\(506\) −4.65668e14 −0.624092
\(507\) 3.22514e14 0.427568
\(508\) 1.60134e13 0.0210007
\(509\) 1.09362e15 1.41879 0.709395 0.704811i \(-0.248968\pi\)
0.709395 + 0.704811i \(0.248968\pi\)
\(510\) −9.57895e14 −1.22937
\(511\) −1.98015e15 −2.51410
\(512\) 7.55406e14 0.948845
\(513\) −1.69874e14 −0.211096
\(514\) 5.33772e14 0.656234
\(515\) 4.54436e14 0.552755
\(516\) −9.05619e12 −0.0108986
\(517\) 2.33827e15 2.78418
\(518\) −1.63442e15 −1.92553
\(519\) −5.57613e14 −0.649999
\(520\) 1.92217e15 2.21704
\(521\) −5.41160e14 −0.617616 −0.308808 0.951124i \(-0.599930\pi\)
−0.308808 + 0.951124i \(0.599930\pi\)
\(522\) −3.00772e12 −0.00339664
\(523\) −1.71054e15 −1.91150 −0.955748 0.294185i \(-0.904952\pi\)
−0.955748 + 0.294185i \(0.904952\pi\)
\(524\) 3.50813e12 0.00387931
\(525\) −1.81823e15 −1.98963
\(526\) −5.87387e13 −0.0636067
\(527\) −9.69836e14 −1.03930
\(528\) −9.94304e14 −1.05447
\(529\) −8.38277e14 −0.879795
\(530\) 1.84346e15 1.91477
\(531\) −4.22156e13 −0.0433963
\(532\) 6.08940e13 0.0619528
\(533\) −2.37506e15 −2.39153
\(534\) 7.93624e14 0.790930
\(535\) −1.37118e15 −1.35254
\(536\) 9.81795e14 0.958549
\(537\) 6.99204e14 0.675687
\(538\) −6.14232e14 −0.587530
\(539\) 4.16811e15 3.94640
\(540\) −1.10304e13 −0.0103377
\(541\) 7.76317e14 0.720202 0.360101 0.932913i \(-0.382742\pi\)
0.360101 + 0.932913i \(0.382742\pi\)
\(542\) −3.75186e14 −0.344549
\(543\) −1.50621e14 −0.136927
\(544\) −8.57299e13 −0.0771504
\(545\) 1.73865e15 1.54893
\(546\) −1.57557e15 −1.38956
\(547\) −2.24867e15 −1.96334 −0.981668 0.190598i \(-0.938957\pi\)
−0.981668 + 0.190598i \(0.938957\pi\)
\(548\) −1.55807e13 −0.0134677
\(549\) 1.50883e14 0.129120
\(550\) −4.07609e15 −3.45342
\(551\) −1.31203e13 −0.0110055
\(552\) 2.37090e14 0.196901
\(553\) 2.55304e15 2.09927
\(554\) −9.73431e14 −0.792504
\(555\) 1.29148e15 1.04106
\(556\) 8.68338e12 0.00693070
\(557\) −9.31655e14 −0.736295 −0.368147 0.929767i \(-0.620008\pi\)
−0.368147 + 0.929767i \(0.620008\pi\)
\(558\) −3.66344e14 −0.286683
\(559\) 1.02215e15 0.792047
\(560\) −4.12110e15 −3.16213
\(561\) −1.65287e15 −1.25587
\(562\) −1.01559e15 −0.764133
\(563\) 8.93930e14 0.666051 0.333025 0.942918i \(-0.391930\pi\)
0.333025 + 0.942918i \(0.391930\pi\)
\(564\) 3.86487e13 0.0285167
\(565\) 6.80163e14 0.496988
\(566\) 1.21905e15 0.882127
\(567\) −2.78506e14 −0.199585
\(568\) 1.46221e15 1.03775
\(569\) −2.06733e15 −1.45309 −0.726544 0.687120i \(-0.758875\pi\)
−0.726544 + 0.687120i \(0.758875\pi\)
\(570\) −1.57840e15 −1.09876
\(571\) −2.02285e14 −0.139465 −0.0697326 0.997566i \(-0.522215\pi\)
−0.0697326 + 0.997566i \(0.522215\pi\)
\(572\) −1.07675e14 −0.0735254
\(573\) 5.85395e14 0.395912
\(574\) 4.93672e15 3.30691
\(575\) 1.00253e15 0.665156
\(576\) 4.90291e14 0.322203
\(577\) −1.99205e14 −0.129668 −0.0648341 0.997896i \(-0.520652\pi\)
−0.0648341 + 0.997896i \(0.520652\pi\)
\(578\) −7.97334e14 −0.514088
\(579\) −4.34018e14 −0.277189
\(580\) −8.51941e11 −0.000538959 0
\(581\) 3.07572e15 1.92743
\(582\) 7.24748e14 0.449895
\(583\) 3.18094e15 1.95604
\(584\) 2.26012e15 1.37677
\(585\) 1.24498e15 0.751284
\(586\) −2.94786e14 −0.176226
\(587\) −1.43802e15 −0.851639 −0.425819 0.904808i \(-0.640014\pi\)
−0.425819 + 0.904808i \(0.640014\pi\)
\(588\) 6.88937e13 0.0404207
\(589\) −1.59807e15 −0.928886
\(590\) −3.92250e14 −0.225879
\(591\) −1.39702e15 −0.797022
\(592\) 1.92422e15 1.08764
\(593\) 3.94642e14 0.221005 0.110503 0.993876i \(-0.464754\pi\)
0.110503 + 0.993876i \(0.464754\pi\)
\(594\) −6.24354e14 −0.346422
\(595\) −6.85069e15 −3.76609
\(596\) 5.62614e13 0.0306447
\(597\) −2.83171e14 −0.152823
\(598\) 8.68734e14 0.464547
\(599\) −1.61368e15 −0.855008 −0.427504 0.904013i \(-0.640607\pi\)
−0.427504 + 0.904013i \(0.640607\pi\)
\(600\) 2.07530e15 1.08955
\(601\) −3.82239e15 −1.98850 −0.994248 0.107099i \(-0.965844\pi\)
−0.994248 + 0.107099i \(0.965844\pi\)
\(602\) −2.12461e15 −1.09521
\(603\) 6.35903e14 0.324822
\(604\) −5.95123e13 −0.0301234
\(605\) −7.29357e15 −3.65835
\(606\) 9.25705e14 0.460122
\(607\) 3.49037e15 1.71923 0.859613 0.510945i \(-0.170705\pi\)
0.859613 + 0.510945i \(0.170705\pi\)
\(608\) −1.41264e14 −0.0689542
\(609\) −2.15106e13 −0.0104054
\(610\) 1.40195e15 0.672074
\(611\) −4.36219e15 −2.07242
\(612\) −2.73199e13 −0.0128631
\(613\) −3.05426e14 −0.142519 −0.0712596 0.997458i \(-0.522702\pi\)
−0.0712596 + 0.997458i \(0.522702\pi\)
\(614\) −5.69075e14 −0.263174
\(615\) −3.90087e15 −1.78792
\(616\) −6.89409e15 −3.13172
\(617\) −1.80306e15 −0.811786 −0.405893 0.913921i \(-0.633039\pi\)
−0.405893 + 0.913921i \(0.633039\pi\)
\(618\) 4.25159e14 0.189721
\(619\) 2.65088e15 1.17244 0.586222 0.810151i \(-0.300615\pi\)
0.586222 + 0.810151i \(0.300615\pi\)
\(620\) −1.03768e14 −0.0454891
\(621\) 1.53562e14 0.0667237
\(622\) 1.42902e15 0.615448
\(623\) 5.67585e15 2.42296
\(624\) 1.85494e15 0.784899
\(625\) 1.81709e15 0.762143
\(626\) 9.86779e14 0.410262
\(627\) −2.72356e15 −1.12245
\(628\) −6.84012e11 −0.000279438 0
\(629\) 3.19872e15 1.29538
\(630\) −2.58777e15 −1.03885
\(631\) 3.66420e15 1.45820 0.729102 0.684405i \(-0.239938\pi\)
0.729102 + 0.684405i \(0.239938\pi\)
\(632\) −2.91400e15 −1.14960
\(633\) 1.38171e15 0.540377
\(634\) 3.73434e15 1.44785
\(635\) −2.96853e15 −1.14100
\(636\) 5.25769e13 0.0200346
\(637\) −7.77588e15 −2.93753
\(638\) −4.82224e13 −0.0180607
\(639\) 9.47063e14 0.351661
\(640\) 4.84730e15 1.78447
\(641\) −1.21408e15 −0.443129 −0.221564 0.975146i \(-0.571116\pi\)
−0.221564 + 0.975146i \(0.571116\pi\)
\(642\) −1.28284e15 −0.464228
\(643\) 3.44096e15 1.23458 0.617290 0.786736i \(-0.288231\pi\)
0.617290 + 0.786736i \(0.288231\pi\)
\(644\) −5.50468e13 −0.0195821
\(645\) 1.67881e15 0.592140
\(646\) −3.90934e15 −1.36718
\(647\) 2.32690e15 0.806873 0.403436 0.915008i \(-0.367816\pi\)
0.403436 + 0.915008i \(0.367816\pi\)
\(648\) 3.17883e14 0.109296
\(649\) −6.76837e14 −0.230748
\(650\) 7.60420e15 2.57057
\(651\) −2.62003e15 −0.878234
\(652\) 1.17828e14 0.0391641
\(653\) 5.76213e15 1.89916 0.949578 0.313531i \(-0.101512\pi\)
0.949578 + 0.313531i \(0.101512\pi\)
\(654\) 1.62664e15 0.531634
\(655\) −6.50330e14 −0.210769
\(656\) −5.81205e15 −1.86792
\(657\) 1.46386e15 0.466542
\(658\) 9.06710e15 2.86567
\(659\) −5.53554e15 −1.73496 −0.867481 0.497470i \(-0.834263\pi\)
−0.867481 + 0.497470i \(0.834263\pi\)
\(660\) −1.76849e14 −0.0549681
\(661\) −6.17895e14 −0.190461 −0.0952307 0.995455i \(-0.530359\pi\)
−0.0952307 + 0.995455i \(0.530359\pi\)
\(662\) −2.56452e15 −0.783948
\(663\) 3.08354e15 0.934814
\(664\) −3.51059e15 −1.05549
\(665\) −1.12884e16 −3.36599
\(666\) 1.20828e15 0.357321
\(667\) 1.18605e13 0.00347864
\(668\) −1.21533e14 −0.0353529
\(669\) −1.72766e15 −0.498443
\(670\) 5.90855e15 1.69071
\(671\) 2.41910e15 0.686561
\(672\) −2.31600e14 −0.0651941
\(673\) −2.86734e15 −0.800565 −0.400283 0.916392i \(-0.631088\pi\)
−0.400283 + 0.916392i \(0.631088\pi\)
\(674\) −2.00846e14 −0.0556203
\(675\) 1.34416e15 0.369216
\(676\) 8.54674e13 0.0232859
\(677\) 1.79243e15 0.484400 0.242200 0.970226i \(-0.422131\pi\)
0.242200 + 0.970226i \(0.422131\pi\)
\(678\) 6.36344e14 0.170580
\(679\) 5.18327e15 1.37822
\(680\) 7.81928e15 2.06237
\(681\) 3.63878e15 0.952021
\(682\) −5.87355e15 −1.52436
\(683\) 7.45091e15 1.91821 0.959104 0.283055i \(-0.0913478\pi\)
0.959104 + 0.283055i \(0.0913478\pi\)
\(684\) −4.50171e13 −0.0114966
\(685\) 2.88831e15 0.731721
\(686\) 8.90369e15 2.23762
\(687\) −2.48122e15 −0.618591
\(688\) 2.50133e15 0.618634
\(689\) −5.93425e15 −1.45600
\(690\) 1.42683e15 0.347299
\(691\) 2.90089e15 0.700491 0.350246 0.936658i \(-0.386098\pi\)
0.350246 + 0.936658i \(0.386098\pi\)
\(692\) −1.47769e14 −0.0353998
\(693\) −4.46526e15 −1.06124
\(694\) 5.10754e14 0.120430
\(695\) −1.60970e15 −0.376556
\(696\) 2.45520e13 0.00569817
\(697\) −9.66161e15 −2.22469
\(698\) 5.10630e14 0.116655
\(699\) −1.56575e15 −0.354893
\(700\) −4.81836e14 −0.108358
\(701\) −4.72763e15 −1.05486 −0.527430 0.849599i \(-0.676844\pi\)
−0.527430 + 0.849599i \(0.676844\pi\)
\(702\) 1.16477e15 0.257861
\(703\) 5.27077e15 1.15776
\(704\) 7.86078e15 1.71323
\(705\) −7.16460e15 −1.54936
\(706\) 2.85318e15 0.612213
\(707\) 6.62047e15 1.40956
\(708\) −1.11873e13 −0.00236342
\(709\) −6.66263e15 −1.39666 −0.698331 0.715775i \(-0.746074\pi\)
−0.698331 + 0.715775i \(0.746074\pi\)
\(710\) 8.79972e15 1.83041
\(711\) −1.88738e15 −0.389562
\(712\) −6.47834e15 −1.32686
\(713\) 1.44462e15 0.293604
\(714\) −6.40934e15 −1.29263
\(715\) 1.99606e16 3.99475
\(716\) 1.85291e14 0.0367988
\(717\) −5.11418e15 −1.00791
\(718\) −2.41970e15 −0.473236
\(719\) 9.90807e15 1.92300 0.961502 0.274798i \(-0.0886110\pi\)
0.961502 + 0.274798i \(0.0886110\pi\)
\(720\) 3.04661e15 0.586796
\(721\) 3.04066e15 0.581197
\(722\) −1.08774e15 −0.206333
\(723\) 5.23250e15 0.985027
\(724\) −3.99152e13 −0.00745720
\(725\) 1.03817e14 0.0192491
\(726\) −6.82369e15 −1.25565
\(727\) 2.85309e15 0.521047 0.260523 0.965468i \(-0.416105\pi\)
0.260523 + 0.965468i \(0.416105\pi\)
\(728\) 1.28614e16 2.33112
\(729\) 2.05891e14 0.0370370
\(730\) 1.36016e16 2.42837
\(731\) 4.15806e15 0.736792
\(732\) 3.99846e13 0.00703205
\(733\) 1.97703e15 0.345097 0.172548 0.985001i \(-0.444800\pi\)
0.172548 + 0.985001i \(0.444800\pi\)
\(734\) 3.27254e14 0.0566966
\(735\) −1.27713e16 −2.19612
\(736\) 1.27699e14 0.0217951
\(737\) 1.01954e16 1.72715
\(738\) −3.64956e15 −0.613664
\(739\) 4.92807e15 0.822493 0.411247 0.911524i \(-0.365093\pi\)
0.411247 + 0.911524i \(0.365093\pi\)
\(740\) 3.42247e14 0.0566976
\(741\) 5.08098e15 0.835502
\(742\) 1.23347e16 2.01330
\(743\) −3.34590e14 −0.0542094 −0.0271047 0.999633i \(-0.508629\pi\)
−0.0271047 + 0.999633i \(0.508629\pi\)
\(744\) 2.99046e15 0.480936
\(745\) −1.04296e16 −1.66498
\(746\) −5.92700e15 −0.939228
\(747\) −2.27379e15 −0.357673
\(748\) −4.38017e14 −0.0683962
\(749\) −9.17466e15 −1.42213
\(750\) 5.97941e15 0.920071
\(751\) 1.47597e15 0.225454 0.112727 0.993626i \(-0.464041\pi\)
0.112727 + 0.993626i \(0.464041\pi\)
\(752\) −1.06748e16 −1.61868
\(753\) 5.22590e15 0.786663
\(754\) 8.99619e13 0.0134436
\(755\) 1.10323e16 1.63665
\(756\) −7.38051e13 −0.0108697
\(757\) 1.34026e16 1.95958 0.979789 0.200036i \(-0.0641059\pi\)
0.979789 + 0.200036i \(0.0641059\pi\)
\(758\) −6.87042e15 −0.997246
\(759\) 2.46204e15 0.354785
\(760\) 1.28844e16 1.84327
\(761\) −3.37274e15 −0.479035 −0.239517 0.970892i \(-0.576989\pi\)
−0.239517 + 0.970892i \(0.576989\pi\)
\(762\) −2.77728e15 −0.391623
\(763\) 1.16334e16 1.62863
\(764\) 1.55131e14 0.0215619
\(765\) 5.06450e15 0.698874
\(766\) −6.45609e15 −0.884527
\(767\) 1.26268e15 0.171759
\(768\) 4.02856e14 0.0544080
\(769\) −9.79858e15 −1.31392 −0.656959 0.753926i \(-0.728158\pi\)
−0.656959 + 0.753926i \(0.728158\pi\)
\(770\) −4.14894e16 −5.52380
\(771\) −2.82211e15 −0.373057
\(772\) −1.15016e14 −0.0150961
\(773\) −1.36395e16 −1.77751 −0.888754 0.458384i \(-0.848429\pi\)
−0.888754 + 0.458384i \(0.848429\pi\)
\(774\) 1.57066e15 0.203239
\(775\) 1.26451e16 1.62466
\(776\) −5.91611e15 −0.754739
\(777\) 8.64138e15 1.09463
\(778\) −5.86014e15 −0.737089
\(779\) −1.59202e16 −1.98835
\(780\) 3.29923e14 0.0409159
\(781\) 1.51841e16 1.86986
\(782\) 3.53396e15 0.432140
\(783\) 1.59022e13 0.00193093
\(784\) −1.90285e16 −2.29438
\(785\) 1.26801e14 0.0151823
\(786\) −6.08433e14 −0.0723417
\(787\) −3.22872e15 −0.381214 −0.190607 0.981666i \(-0.561046\pi\)
−0.190607 + 0.981666i \(0.561046\pi\)
\(788\) −3.70214e14 −0.0434069
\(789\) 3.10558e14 0.0361593
\(790\) −1.75368e16 −2.02768
\(791\) 4.55101e15 0.522561
\(792\) 5.09659e15 0.581153
\(793\) −4.51298e15 −0.511047
\(794\) 8.17575e15 0.919422
\(795\) −9.74659e15 −1.08851
\(796\) −7.50412e13 −0.00832296
\(797\) −8.17049e15 −0.899968 −0.449984 0.893037i \(-0.648570\pi\)
−0.449984 + 0.893037i \(0.648570\pi\)
\(798\) −1.05611e16 −1.15530
\(799\) −1.77452e16 −1.92785
\(800\) 1.11778e15 0.120604
\(801\) −4.19598e15 −0.449629
\(802\) −3.73266e15 −0.397246
\(803\) 2.34700e16 2.48071
\(804\) 1.68516e14 0.0176902
\(805\) 1.02045e16 1.06393
\(806\) 1.09575e16 1.13467
\(807\) 3.24751e15 0.334000
\(808\) −7.55652e15 −0.771896
\(809\) 1.38353e16 1.40369 0.701846 0.712329i \(-0.252360\pi\)
0.701846 + 0.712329i \(0.252360\pi\)
\(810\) 1.91306e15 0.192779
\(811\) −9.02019e15 −0.902820 −0.451410 0.892317i \(-0.649079\pi\)
−0.451410 + 0.892317i \(0.649079\pi\)
\(812\) −5.70039e12 −0.000566691 0
\(813\) 1.98365e15 0.195870
\(814\) 1.93722e16 1.89996
\(815\) −2.18427e16 −2.12785
\(816\) 7.54578e15 0.730144
\(817\) 6.85155e15 0.658518
\(818\) 9.27005e15 0.884991
\(819\) 8.33022e15 0.789942
\(820\) −1.03374e15 −0.0973726
\(821\) 1.23694e15 0.115734 0.0578670 0.998324i \(-0.481570\pi\)
0.0578670 + 0.998324i \(0.481570\pi\)
\(822\) 2.70223e15 0.251147
\(823\) 9.81837e15 0.906442 0.453221 0.891398i \(-0.350275\pi\)
0.453221 + 0.891398i \(0.350275\pi\)
\(824\) −3.47056e15 −0.318274
\(825\) 2.15507e16 1.96320
\(826\) −2.62457e15 −0.237502
\(827\) −7.96875e15 −0.716325 −0.358162 0.933659i \(-0.616597\pi\)
−0.358162 + 0.933659i \(0.616597\pi\)
\(828\) 4.06944e13 0.00363386
\(829\) −1.14116e16 −1.01227 −0.506134 0.862455i \(-0.668926\pi\)
−0.506134 + 0.862455i \(0.668926\pi\)
\(830\) −2.11271e16 −1.86170
\(831\) 5.14664e15 0.450524
\(832\) −1.46648e16 −1.27526
\(833\) −3.16319e16 −2.73261
\(834\) −1.50600e15 −0.129244
\(835\) 2.25296e16 1.92078
\(836\) −7.21753e14 −0.0611300
\(837\) 1.93690e15 0.162974
\(838\) 1.29339e16 1.08116
\(839\) 2.51139e15 0.208556 0.104278 0.994548i \(-0.466747\pi\)
0.104278 + 0.994548i \(0.466747\pi\)
\(840\) 2.11239e16 1.74276
\(841\) −1.21993e16 −0.999899
\(842\) −6.33102e15 −0.515534
\(843\) 5.36955e15 0.434396
\(844\) 3.66158e14 0.0294297
\(845\) −1.58437e16 −1.26516
\(846\) −6.70303e15 −0.531783
\(847\) −4.88017e16 −3.84660
\(848\) −1.45218e16 −1.13722
\(849\) −6.44526e15 −0.501474
\(850\) 3.09335e16 2.39125
\(851\) −4.76466e15 −0.365948
\(852\) 2.50975e14 0.0191519
\(853\) 1.25539e16 0.951828 0.475914 0.879492i \(-0.342117\pi\)
0.475914 + 0.879492i \(0.342117\pi\)
\(854\) 9.38051e15 0.706656
\(855\) 8.34516e15 0.624627
\(856\) 1.04718e16 0.778784
\(857\) 1.60910e16 1.18902 0.594508 0.804090i \(-0.297347\pi\)
0.594508 + 0.804090i \(0.297347\pi\)
\(858\) 1.86746e16 1.37111
\(859\) 1.83850e16 1.34122 0.670610 0.741810i \(-0.266032\pi\)
0.670610 + 0.741810i \(0.266032\pi\)
\(860\) 4.44891e14 0.0322487
\(861\) −2.61010e16 −1.87992
\(862\) −5.32047e15 −0.380768
\(863\) 1.15024e16 0.817956 0.408978 0.912544i \(-0.365885\pi\)
0.408978 + 0.912544i \(0.365885\pi\)
\(864\) 1.71215e14 0.0120981
\(865\) 2.73931e16 1.92333
\(866\) 1.77749e16 1.24010
\(867\) 4.21559e15 0.292249
\(868\) −6.94315e14 −0.0478298
\(869\) −3.02601e16 −2.07139
\(870\) 1.47756e14 0.0100506
\(871\) −1.90201e16 −1.28562
\(872\) −1.32782e16 −0.891864
\(873\) −3.83183e15 −0.255757
\(874\) 5.82317e15 0.386231
\(875\) 4.27636e16 2.81858
\(876\) 3.87929e14 0.0254085
\(877\) −1.34056e16 −0.872549 −0.436274 0.899814i \(-0.643702\pi\)
−0.436274 + 0.899814i \(0.643702\pi\)
\(878\) −1.51870e16 −0.982317
\(879\) 1.55857e15 0.100181
\(880\) 4.88459e16 3.12013
\(881\) −7.65716e14 −0.0486072 −0.0243036 0.999705i \(-0.507737\pi\)
−0.0243036 + 0.999705i \(0.507737\pi\)
\(882\) −1.19486e16 −0.753769
\(883\) −5.83349e15 −0.365717 −0.182858 0.983139i \(-0.558535\pi\)
−0.182858 + 0.983139i \(0.558535\pi\)
\(884\) 8.17148e14 0.0509112
\(885\) 2.07387e15 0.128408
\(886\) −2.38714e16 −1.46890
\(887\) −7.79861e15 −0.476911 −0.238455 0.971153i \(-0.576641\pi\)
−0.238455 + 0.971153i \(0.576641\pi\)
\(888\) −9.86316e15 −0.599439
\(889\) −1.98626e16 −1.19971
\(890\) −3.89873e16 −2.34034
\(891\) 3.30103e15 0.196935
\(892\) −4.57837e14 −0.0271459
\(893\) −2.92401e16 −1.72304
\(894\) −9.75769e15 −0.571466
\(895\) −3.43489e16 −1.99934
\(896\) 3.24336e16 1.87630
\(897\) −4.59309e15 −0.264087
\(898\) −3.49788e16 −1.99887
\(899\) 1.49598e14 0.00849665
\(900\) 3.56207e14 0.0201080
\(901\) −2.41402e16 −1.35442
\(902\) −5.85130e16 −3.26299
\(903\) 1.12331e16 0.622609
\(904\) −5.19446e15 −0.286163
\(905\) 7.39938e15 0.405161
\(906\) 1.03215e16 0.561744
\(907\) 1.16614e15 0.0630828 0.0315414 0.999502i \(-0.489958\pi\)
0.0315414 + 0.999502i \(0.489958\pi\)
\(908\) 9.64288e14 0.0518483
\(909\) −4.89431e15 −0.261571
\(910\) 7.74010e16 4.11168
\(911\) 1.68621e16 0.890349 0.445174 0.895444i \(-0.353142\pi\)
0.445174 + 0.895444i \(0.353142\pi\)
\(912\) 1.24338e16 0.652575
\(913\) −3.64553e16 −1.90183
\(914\) 3.43617e15 0.178185
\(915\) −7.41225e15 −0.382062
\(916\) −6.57532e14 −0.0336893
\(917\) −4.35140e15 −0.221614
\(918\) 4.73822e15 0.239873
\(919\) −1.31495e16 −0.661717 −0.330859 0.943680i \(-0.607338\pi\)
−0.330859 + 0.943680i \(0.607338\pi\)
\(920\) −1.16472e16 −0.582625
\(921\) 3.00876e15 0.149610
\(922\) 1.49913e16 0.741003
\(923\) −2.83270e16 −1.39185
\(924\) −1.18331e15 −0.0577966
\(925\) −4.17060e16 −2.02497
\(926\) 1.24295e16 0.599922
\(927\) −2.24786e15 −0.107853
\(928\) 1.32239e13 0.000630734 0
\(929\) −1.08228e16 −0.513162 −0.256581 0.966523i \(-0.582596\pi\)
−0.256581 + 0.966523i \(0.582596\pi\)
\(930\) 1.79969e16 0.848285
\(931\) −5.21222e16 −2.44230
\(932\) −4.14928e14 −0.0193279
\(933\) −7.55539e15 −0.349871
\(934\) −2.50029e16 −1.15102
\(935\) 8.11985e16 3.71607
\(936\) −9.50800e15 −0.432586
\(937\) −6.72798e15 −0.304311 −0.152155 0.988357i \(-0.548621\pi\)
−0.152155 + 0.988357i \(0.548621\pi\)
\(938\) 3.95345e16 1.77771
\(939\) −5.21721e15 −0.233226
\(940\) −1.89864e15 −0.0843801
\(941\) −3.78738e16 −1.67338 −0.836692 0.547674i \(-0.815513\pi\)
−0.836692 + 0.547674i \(0.815513\pi\)
\(942\) 1.18632e14 0.00521099
\(943\) 1.43915e16 0.628479
\(944\) 3.08993e15 0.134154
\(945\) 1.36818e16 0.590566
\(946\) 2.51822e16 1.08067
\(947\) −3.31537e16 −1.41451 −0.707257 0.706957i \(-0.750068\pi\)
−0.707257 + 0.706957i \(0.750068\pi\)
\(948\) −5.00162e14 −0.0212161
\(949\) −4.37847e16 −1.84654
\(950\) 5.09714e16 2.13721
\(951\) −1.97439e16 −0.823075
\(952\) 5.23193e16 2.16850
\(953\) 3.02516e16 1.24663 0.623314 0.781971i \(-0.285786\pi\)
0.623314 + 0.781971i \(0.285786\pi\)
\(954\) −9.11868e15 −0.373608
\(955\) −2.87579e16 −1.17149
\(956\) −1.35527e15 −0.0548919
\(957\) 2.54957e14 0.0102672
\(958\) 6.64173e15 0.265932
\(959\) 1.93259e16 0.769373
\(960\) −2.40859e16 −0.953390
\(961\) −7.18722e15 −0.282867
\(962\) −3.61401e16 −1.41425
\(963\) 6.78254e15 0.263905
\(964\) 1.38663e15 0.0536459
\(965\) 2.13214e16 0.820192
\(966\) 9.54704e15 0.365170
\(967\) −3.09551e16 −1.17730 −0.588649 0.808389i \(-0.700340\pi\)
−0.588649 + 0.808389i \(0.700340\pi\)
\(968\) 5.57016e16 2.10646
\(969\) 2.06691e16 0.777216
\(970\) −3.56038e16 −1.33122
\(971\) −1.20450e16 −0.447819 −0.223910 0.974610i \(-0.571882\pi\)
−0.223910 + 0.974610i \(0.571882\pi\)
\(972\) 5.45618e13 0.00201709
\(973\) −1.07706e16 −0.395932
\(974\) −2.26360e16 −0.827418
\(975\) −4.02043e16 −1.46133
\(976\) −1.10438e16 −0.399157
\(977\) −4.83305e15 −0.173701 −0.0868504 0.996221i \(-0.527680\pi\)
−0.0868504 + 0.996221i \(0.527680\pi\)
\(978\) −2.04355e16 −0.730335
\(979\) −6.72737e16 −2.39078
\(980\) −3.38445e15 −0.119604
\(981\) −8.60021e15 −0.302225
\(982\) −1.64232e16 −0.573911
\(983\) 3.08059e15 0.107051 0.0535254 0.998566i \(-0.482954\pi\)
0.0535254 + 0.998566i \(0.482954\pi\)
\(984\) 2.97913e16 1.02948
\(985\) 6.86294e16 2.35836
\(986\) 3.65960e14 0.0125058
\(987\) −4.79388e16 −1.62908
\(988\) 1.34648e15 0.0455025
\(989\) −6.19365e15 −0.208145
\(990\) 3.06718e16 1.02505
\(991\) −2.17402e15 −0.0722535 −0.0361268 0.999347i \(-0.511502\pi\)
−0.0361268 + 0.999347i \(0.511502\pi\)
\(992\) 1.61069e15 0.0532351
\(993\) 1.35589e16 0.445661
\(994\) 5.88795e16 1.92459
\(995\) 1.39110e16 0.452200
\(996\) −6.02560e14 −0.0194793
\(997\) 9.08856e15 0.292194 0.146097 0.989270i \(-0.453329\pi\)
0.146097 + 0.989270i \(0.453329\pi\)
\(998\) −4.72037e15 −0.150924
\(999\) −6.38831e15 −0.203131
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.12.a.b.1.9 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.9 27 1.1 even 1 trivial