Properties

Label 177.12.a.b.1.14
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 1.14
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.91496 q^{2} +243.000 q^{3} -2023.84 q^{4} -1117.68 q^{5} -1194.33 q^{6} -21416.7 q^{7} +20012.9 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-4.91496 q^{2} +243.000 q^{3} -2023.84 q^{4} -1117.68 q^{5} -1194.33 q^{6} -21416.7 q^{7} +20012.9 q^{8} +59049.0 q^{9} +5493.36 q^{10} +110263. q^{11} -491794. q^{12} -34344.1 q^{13} +105262. q^{14} -271597. q^{15} +4.04647e6 q^{16} -8.45888e6 q^{17} -290223. q^{18} +1.50934e7 q^{19} +2.26201e6 q^{20} -5.20426e6 q^{21} -541937. q^{22} +7.31177e6 q^{23} +4.86314e6 q^{24} -4.75789e7 q^{25} +168800. q^{26} +1.43489e7 q^{27} +4.33441e7 q^{28} -3.39103e7 q^{29} +1.33489e6 q^{30} +1.38887e8 q^{31} -6.08747e7 q^{32} +2.67939e7 q^{33} +4.15750e7 q^{34} +2.39371e7 q^{35} -1.19506e8 q^{36} +5.04720e8 q^{37} -7.41832e7 q^{38} -8.34562e6 q^{39} -2.23681e7 q^{40} +6.49327e8 q^{41} +2.55787e7 q^{42} +1.25045e9 q^{43} -2.23155e8 q^{44} -6.59980e7 q^{45} -3.59370e7 q^{46} -1.92452e9 q^{47} +9.83292e8 q^{48} -1.51865e9 q^{49} +2.33848e8 q^{50} -2.05551e9 q^{51} +6.95071e7 q^{52} +5.25997e9 q^{53} -7.05243e7 q^{54} -1.23239e8 q^{55} -4.28611e8 q^{56} +3.66769e9 q^{57} +1.66668e8 q^{58} -7.14924e8 q^{59} +5.49669e8 q^{60} -8.19477e9 q^{61} -6.82622e8 q^{62} -1.26464e9 q^{63} -7.98797e9 q^{64} +3.83858e7 q^{65} -1.31691e8 q^{66} -8.86480e9 q^{67} +1.71194e10 q^{68} +1.77676e9 q^{69} -1.17650e8 q^{70} -3.46434e9 q^{71} +1.18174e9 q^{72} +5.04526e9 q^{73} -2.48068e9 q^{74} -1.15617e10 q^{75} -3.05466e10 q^{76} -2.36147e9 q^{77} +4.10183e7 q^{78} +2.53994e10 q^{79} -4.52266e9 q^{80} +3.48678e9 q^{81} -3.19141e9 q^{82} +2.79697e10 q^{83} +1.05326e10 q^{84} +9.45433e9 q^{85} -6.14593e9 q^{86} -8.24021e9 q^{87} +2.20668e9 q^{88} -9.94834e10 q^{89} +3.24377e8 q^{90} +7.35538e8 q^{91} -1.47979e10 q^{92} +3.37495e10 q^{93} +9.45895e9 q^{94} -1.68696e10 q^{95} -1.47926e10 q^{96} -1.07428e11 q^{97} +7.46410e9 q^{98} +6.51091e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.91496 −0.108606 −0.0543031 0.998524i \(-0.517294\pi\)
−0.0543031 + 0.998524i \(0.517294\pi\)
\(3\) 243.000 0.577350
\(4\) −2023.84 −0.988205
\(5\) −1117.68 −0.159950 −0.0799748 0.996797i \(-0.525484\pi\)
−0.0799748 + 0.996797i \(0.525484\pi\)
\(6\) −1194.33 −0.0627038
\(7\) −21416.7 −0.481630 −0.240815 0.970571i \(-0.577415\pi\)
−0.240815 + 0.970571i \(0.577415\pi\)
\(8\) 20012.9 0.215931
\(9\) 59049.0 0.333333
\(10\) 5493.36 0.0173715
\(11\) 110263. 0.206428 0.103214 0.994659i \(-0.467087\pi\)
0.103214 + 0.994659i \(0.467087\pi\)
\(12\) −491794. −0.570540
\(13\) −34344.1 −0.0256545 −0.0128273 0.999918i \(-0.504083\pi\)
−0.0128273 + 0.999918i \(0.504083\pi\)
\(14\) 105262. 0.0523080
\(15\) −271597. −0.0923469
\(16\) 4.04647e6 0.964753
\(17\) −8.45888e6 −1.44492 −0.722460 0.691413i \(-0.756989\pi\)
−0.722460 + 0.691413i \(0.756989\pi\)
\(18\) −290223. −0.0362021
\(19\) 1.50934e7 1.39843 0.699216 0.714911i \(-0.253533\pi\)
0.699216 + 0.714911i \(0.253533\pi\)
\(20\) 2.26201e6 0.158063
\(21\) −5.20426e6 −0.278069
\(22\) −541937. −0.0224194
\(23\) 7.31177e6 0.236875 0.118438 0.992961i \(-0.462211\pi\)
0.118438 + 0.992961i \(0.462211\pi\)
\(24\) 4.86314e6 0.124668
\(25\) −4.75789e7 −0.974416
\(26\) 168800. 0.00278624
\(27\) 1.43489e7 0.192450
\(28\) 4.33441e7 0.475949
\(29\) −3.39103e7 −0.307003 −0.153502 0.988148i \(-0.549055\pi\)
−0.153502 + 0.988148i \(0.549055\pi\)
\(30\) 1.33489e6 0.0100295
\(31\) 1.38887e8 0.871307 0.435654 0.900114i \(-0.356517\pi\)
0.435654 + 0.900114i \(0.356517\pi\)
\(32\) −6.08747e7 −0.320710
\(33\) 2.67939e7 0.119182
\(34\) 4.15750e7 0.156927
\(35\) 2.39371e7 0.0770365
\(36\) −1.19506e8 −0.329402
\(37\) 5.04720e8 1.19658 0.598289 0.801280i \(-0.295847\pi\)
0.598289 + 0.801280i \(0.295847\pi\)
\(38\) −7.41832e7 −0.151878
\(39\) −8.34562e6 −0.0148116
\(40\) −2.23681e7 −0.0345381
\(41\) 6.49327e8 0.875290 0.437645 0.899148i \(-0.355813\pi\)
0.437645 + 0.899148i \(0.355813\pi\)
\(42\) 2.55787e7 0.0302001
\(43\) 1.25045e9 1.29715 0.648577 0.761149i \(-0.275365\pi\)
0.648577 + 0.761149i \(0.275365\pi\)
\(44\) −2.23155e8 −0.203994
\(45\) −6.59980e7 −0.0533165
\(46\) −3.59370e7 −0.0257261
\(47\) −1.92452e9 −1.22401 −0.612005 0.790854i \(-0.709637\pi\)
−0.612005 + 0.790854i \(0.709637\pi\)
\(48\) 9.83292e8 0.557001
\(49\) −1.51865e9 −0.768032
\(50\) 2.33848e8 0.105828
\(51\) −2.05551e9 −0.834225
\(52\) 6.95071e7 0.0253519
\(53\) 5.25997e9 1.72769 0.863846 0.503756i \(-0.168049\pi\)
0.863846 + 0.503756i \(0.168049\pi\)
\(54\) −7.05243e7 −0.0209013
\(55\) −1.23239e8 −0.0330181
\(56\) −4.28611e8 −0.103999
\(57\) 3.66769e9 0.807385
\(58\) 1.66668e8 0.0333425
\(59\) −7.14924e8 −0.130189
\(60\) 5.49669e8 0.0912576
\(61\) −8.19477e9 −1.24229 −0.621145 0.783696i \(-0.713332\pi\)
−0.621145 + 0.783696i \(0.713332\pi\)
\(62\) −6.82622e8 −0.0946294
\(63\) −1.26464e9 −0.160543
\(64\) −7.98797e9 −0.929922
\(65\) 3.83858e7 0.00410343
\(66\) −1.31691e8 −0.0129439
\(67\) −8.86480e9 −0.802154 −0.401077 0.916044i \(-0.631364\pi\)
−0.401077 + 0.916044i \(0.631364\pi\)
\(68\) 1.71194e10 1.42788
\(69\) 1.77676e9 0.136760
\(70\) −1.17650e8 −0.00836665
\(71\) −3.46434e9 −0.227877 −0.113938 0.993488i \(-0.536347\pi\)
−0.113938 + 0.993488i \(0.536347\pi\)
\(72\) 1.18174e9 0.0719771
\(73\) 5.04526e9 0.284845 0.142422 0.989806i \(-0.454511\pi\)
0.142422 + 0.989806i \(0.454511\pi\)
\(74\) −2.48068e9 −0.129956
\(75\) −1.15617e10 −0.562579
\(76\) −3.05466e10 −1.38194
\(77\) −2.36147e9 −0.0994222
\(78\) 4.10183e7 0.00160864
\(79\) 2.53994e10 0.928697 0.464348 0.885653i \(-0.346289\pi\)
0.464348 + 0.885653i \(0.346289\pi\)
\(80\) −4.52266e9 −0.154312
\(81\) 3.48678e9 0.111111
\(82\) −3.19141e9 −0.0950620
\(83\) 2.79697e10 0.779398 0.389699 0.920942i \(-0.372579\pi\)
0.389699 + 0.920942i \(0.372579\pi\)
\(84\) 1.05326e10 0.274789
\(85\) 9.45433e9 0.231114
\(86\) −6.14593e9 −0.140879
\(87\) −8.24021e9 −0.177248
\(88\) 2.20668e9 0.0445744
\(89\) −9.94834e10 −1.88845 −0.944226 0.329299i \(-0.893188\pi\)
−0.944226 + 0.329299i \(0.893188\pi\)
\(90\) 3.24377e8 0.00579051
\(91\) 7.35538e8 0.0123560
\(92\) −1.47979e10 −0.234081
\(93\) 3.37495e10 0.503049
\(94\) 9.45895e9 0.132935
\(95\) −1.68696e10 −0.223679
\(96\) −1.47926e10 −0.185162
\(97\) −1.07428e11 −1.27020 −0.635100 0.772430i \(-0.719041\pi\)
−0.635100 + 0.772430i \(0.719041\pi\)
\(98\) 7.46410e9 0.0834131
\(99\) 6.51091e9 0.0688095
\(100\) 9.62923e10 0.962923
\(101\) −1.77126e11 −1.67693 −0.838466 0.544955i \(-0.816547\pi\)
−0.838466 + 0.544955i \(0.816547\pi\)
\(102\) 1.01027e10 0.0906020
\(103\) −8.41645e10 −0.715359 −0.357679 0.933844i \(-0.616432\pi\)
−0.357679 + 0.933844i \(0.616432\pi\)
\(104\) −6.87326e8 −0.00553961
\(105\) 5.81671e9 0.0444771
\(106\) −2.58525e10 −0.187638
\(107\) 1.83896e11 1.26754 0.633771 0.773521i \(-0.281506\pi\)
0.633771 + 0.773521i \(0.281506\pi\)
\(108\) −2.90399e10 −0.190180
\(109\) −3.45814e10 −0.215277 −0.107638 0.994190i \(-0.534329\pi\)
−0.107638 + 0.994190i \(0.534329\pi\)
\(110\) 6.05713e8 0.00358598
\(111\) 1.22647e11 0.690845
\(112\) −8.66621e10 −0.464654
\(113\) 1.64949e11 0.842206 0.421103 0.907013i \(-0.361643\pi\)
0.421103 + 0.907013i \(0.361643\pi\)
\(114\) −1.80265e10 −0.0876870
\(115\) −8.17223e9 −0.0378881
\(116\) 6.86292e10 0.303382
\(117\) −2.02798e9 −0.00855150
\(118\) 3.51382e9 0.0141393
\(119\) 1.81161e11 0.695917
\(120\) −5.43544e9 −0.0199406
\(121\) −2.73154e11 −0.957387
\(122\) 4.02770e10 0.134920
\(123\) 1.57786e11 0.505349
\(124\) −2.81085e11 −0.861030
\(125\) 1.07752e11 0.315807
\(126\) 6.21563e9 0.0174360
\(127\) 4.41527e11 1.18587 0.592934 0.805251i \(-0.297969\pi\)
0.592934 + 0.805251i \(0.297969\pi\)
\(128\) 1.63932e11 0.421705
\(129\) 3.03860e11 0.748912
\(130\) −1.88664e8 −0.000445658 0
\(131\) −1.61559e11 −0.365880 −0.182940 0.983124i \(-0.558561\pi\)
−0.182940 + 0.983124i \(0.558561\pi\)
\(132\) −5.42266e10 −0.117776
\(133\) −3.23250e11 −0.673527
\(134\) 4.35701e10 0.0871189
\(135\) −1.60375e10 −0.0307823
\(136\) −1.69287e11 −0.312004
\(137\) −4.86495e11 −0.861223 −0.430611 0.902538i \(-0.641702\pi\)
−0.430611 + 0.902538i \(0.641702\pi\)
\(138\) −8.73270e9 −0.0148530
\(139\) 4.81449e11 0.786989 0.393495 0.919327i \(-0.371266\pi\)
0.393495 + 0.919327i \(0.371266\pi\)
\(140\) −4.84449e10 −0.0761279
\(141\) −4.67659e11 −0.706683
\(142\) 1.70271e10 0.0247488
\(143\) −3.78688e9 −0.00529582
\(144\) 2.38940e11 0.321584
\(145\) 3.79009e10 0.0491050
\(146\) −2.47973e10 −0.0309359
\(147\) −3.69032e11 −0.443424
\(148\) −1.02147e12 −1.18246
\(149\) 3.55211e10 0.0396243 0.0198121 0.999804i \(-0.493693\pi\)
0.0198121 + 0.999804i \(0.493693\pi\)
\(150\) 5.68251e10 0.0610996
\(151\) 9.14071e11 0.947560 0.473780 0.880643i \(-0.342889\pi\)
0.473780 + 0.880643i \(0.342889\pi\)
\(152\) 3.02063e11 0.301965
\(153\) −4.99488e11 −0.481640
\(154\) 1.16065e10 0.0107979
\(155\) −1.55231e11 −0.139365
\(156\) 1.68902e10 0.0146369
\(157\) −1.23688e12 −1.03485 −0.517426 0.855728i \(-0.673110\pi\)
−0.517426 + 0.855728i \(0.673110\pi\)
\(158\) −1.24837e11 −0.100862
\(159\) 1.27817e12 0.997483
\(160\) 6.80385e10 0.0512974
\(161\) −1.56594e11 −0.114086
\(162\) −1.71374e10 −0.0120674
\(163\) −7.79907e11 −0.530898 −0.265449 0.964125i \(-0.585520\pi\)
−0.265449 + 0.964125i \(0.585520\pi\)
\(164\) −1.31414e12 −0.864966
\(165\) −2.99470e10 −0.0190630
\(166\) −1.37470e11 −0.0846474
\(167\) −2.22097e12 −1.32313 −0.661564 0.749889i \(-0.730107\pi\)
−0.661564 + 0.749889i \(0.730107\pi\)
\(168\) −1.04153e11 −0.0600439
\(169\) −1.79098e12 −0.999342
\(170\) −4.64676e10 −0.0251004
\(171\) 8.91248e11 0.466144
\(172\) −2.53072e12 −1.28185
\(173\) −2.39491e12 −1.17500 −0.587498 0.809226i \(-0.699887\pi\)
−0.587498 + 0.809226i \(0.699887\pi\)
\(174\) 4.05003e10 0.0192503
\(175\) 1.01898e12 0.469308
\(176\) 4.46175e11 0.199153
\(177\) −1.73727e11 −0.0751646
\(178\) 4.88957e11 0.205098
\(179\) −4.47136e12 −1.81865 −0.909323 0.416092i \(-0.863400\pi\)
−0.909323 + 0.416092i \(0.863400\pi\)
\(180\) 1.33570e11 0.0526876
\(181\) −4.87250e12 −1.86432 −0.932158 0.362052i \(-0.882076\pi\)
−0.932158 + 0.362052i \(0.882076\pi\)
\(182\) −3.61514e9 −0.00134194
\(183\) −1.99133e12 −0.717236
\(184\) 1.46330e11 0.0511488
\(185\) −5.64117e11 −0.191392
\(186\) −1.65877e11 −0.0546343
\(187\) −9.32700e11 −0.298272
\(188\) 3.89493e12 1.20957
\(189\) −3.07306e11 −0.0926898
\(190\) 8.29132e10 0.0242929
\(191\) −1.25233e12 −0.356480 −0.178240 0.983987i \(-0.557040\pi\)
−0.178240 + 0.983987i \(0.557040\pi\)
\(192\) −1.94108e12 −0.536891
\(193\) −6.09566e12 −1.63853 −0.819267 0.573412i \(-0.805620\pi\)
−0.819267 + 0.573412i \(0.805620\pi\)
\(194\) 5.28003e11 0.137952
\(195\) 9.32774e9 0.00236911
\(196\) 3.07351e12 0.758973
\(197\) 4.02584e12 0.966700 0.483350 0.875427i \(-0.339420\pi\)
0.483350 + 0.875427i \(0.339420\pi\)
\(198\) −3.20009e10 −0.00747314
\(199\) −4.07863e12 −0.926450 −0.463225 0.886241i \(-0.653308\pi\)
−0.463225 + 0.886241i \(0.653308\pi\)
\(200\) −9.52194e11 −0.210407
\(201\) −2.15415e12 −0.463124
\(202\) 8.70568e11 0.182125
\(203\) 7.26248e11 0.147862
\(204\) 4.16002e12 0.824385
\(205\) −7.25740e11 −0.140002
\(206\) 4.13665e11 0.0776924
\(207\) 4.31753e11 0.0789584
\(208\) −1.38972e11 −0.0247503
\(209\) 1.66424e12 0.288676
\(210\) −2.85889e10 −0.00483049
\(211\) 3.79950e12 0.625421 0.312710 0.949848i \(-0.398763\pi\)
0.312710 + 0.949848i \(0.398763\pi\)
\(212\) −1.06454e13 −1.70731
\(213\) −8.41835e11 −0.131565
\(214\) −9.03843e11 −0.137663
\(215\) −1.39761e12 −0.207479
\(216\) 2.87164e11 0.0415560
\(217\) −2.97450e12 −0.419648
\(218\) 1.69966e11 0.0233804
\(219\) 1.22600e12 0.164455
\(220\) 2.49416e11 0.0326287
\(221\) 2.90513e11 0.0370687
\(222\) −6.02805e11 −0.0750301
\(223\) 1.60866e12 0.195339 0.0976695 0.995219i \(-0.468861\pi\)
0.0976695 + 0.995219i \(0.468861\pi\)
\(224\) 1.30374e12 0.154463
\(225\) −2.80949e12 −0.324805
\(226\) −8.10718e11 −0.0914688
\(227\) −1.20558e13 −1.32755 −0.663777 0.747931i \(-0.731048\pi\)
−0.663777 + 0.747931i \(0.731048\pi\)
\(228\) −7.42282e12 −0.797862
\(229\) −6.53920e12 −0.686167 −0.343083 0.939305i \(-0.611471\pi\)
−0.343083 + 0.939305i \(0.611471\pi\)
\(230\) 4.01662e10 0.00411488
\(231\) −5.73837e11 −0.0574014
\(232\) −6.78645e11 −0.0662917
\(233\) 9.39440e12 0.896214 0.448107 0.893980i \(-0.352098\pi\)
0.448107 + 0.893980i \(0.352098\pi\)
\(234\) 9.96746e9 0.000928747 0
\(235\) 2.15100e12 0.195780
\(236\) 1.44689e12 0.128653
\(237\) 6.17205e12 0.536183
\(238\) −8.90400e11 −0.0755809
\(239\) 1.81396e12 0.150466 0.0752330 0.997166i \(-0.476030\pi\)
0.0752330 + 0.997166i \(0.476030\pi\)
\(240\) −1.09901e12 −0.0890920
\(241\) 1.38604e12 0.109820 0.0549102 0.998491i \(-0.482513\pi\)
0.0549102 + 0.998491i \(0.482513\pi\)
\(242\) 1.34254e12 0.103978
\(243\) 8.47289e11 0.0641500
\(244\) 1.65849e13 1.22764
\(245\) 1.69737e12 0.122846
\(246\) −7.75513e11 −0.0548841
\(247\) −5.18368e11 −0.0358761
\(248\) 2.77953e12 0.188143
\(249\) 6.79665e12 0.449985
\(250\) −5.29598e11 −0.0342986
\(251\) 5.62633e12 0.356467 0.178234 0.983988i \(-0.442962\pi\)
0.178234 + 0.983988i \(0.442962\pi\)
\(252\) 2.55942e12 0.158650
\(253\) 8.06217e11 0.0488978
\(254\) −2.17008e12 −0.128793
\(255\) 2.29740e12 0.133434
\(256\) 1.55536e13 0.884122
\(257\) −2.81452e13 −1.56593 −0.782964 0.622067i \(-0.786293\pi\)
−0.782964 + 0.622067i \(0.786293\pi\)
\(258\) −1.49346e12 −0.0813365
\(259\) −1.08095e13 −0.576309
\(260\) −7.76868e10 −0.00405503
\(261\) −2.00237e12 −0.102334
\(262\) 7.94056e11 0.0397369
\(263\) −8.05576e12 −0.394775 −0.197388 0.980326i \(-0.563246\pi\)
−0.197388 + 0.980326i \(0.563246\pi\)
\(264\) 5.36224e11 0.0257350
\(265\) −5.87897e12 −0.276343
\(266\) 1.58876e12 0.0731492
\(267\) −2.41745e13 −1.09030
\(268\) 1.79410e13 0.792692
\(269\) −1.30424e13 −0.564571 −0.282286 0.959330i \(-0.591093\pi\)
−0.282286 + 0.959330i \(0.591093\pi\)
\(270\) 7.88237e10 0.00334315
\(271\) −2.56349e13 −1.06537 −0.532686 0.846313i \(-0.678817\pi\)
−0.532686 + 0.846313i \(0.678817\pi\)
\(272\) −3.42286e13 −1.39399
\(273\) 1.78736e11 0.00713373
\(274\) 2.39110e12 0.0935341
\(275\) −5.24619e12 −0.201147
\(276\) −3.59588e12 −0.135147
\(277\) 2.17695e13 0.802066 0.401033 0.916064i \(-0.368651\pi\)
0.401033 + 0.916064i \(0.368651\pi\)
\(278\) −2.36630e12 −0.0854719
\(279\) 8.20112e12 0.290436
\(280\) 4.79051e11 0.0166346
\(281\) 2.32643e13 0.792147 0.396073 0.918219i \(-0.370373\pi\)
0.396073 + 0.918219i \(0.370373\pi\)
\(282\) 2.29853e12 0.0767501
\(283\) 4.41058e13 1.44434 0.722171 0.691715i \(-0.243144\pi\)
0.722171 + 0.691715i \(0.243144\pi\)
\(284\) 7.01128e12 0.225189
\(285\) −4.09931e12 −0.129141
\(286\) 1.86124e10 0.000575159 0
\(287\) −1.39064e13 −0.421566
\(288\) −3.59459e12 −0.106903
\(289\) 3.72807e13 1.08779
\(290\) −1.86282e11 −0.00533311
\(291\) −2.61050e13 −0.733350
\(292\) −1.02108e13 −0.281485
\(293\) 9.13485e12 0.247132 0.123566 0.992336i \(-0.460567\pi\)
0.123566 + 0.992336i \(0.460567\pi\)
\(294\) 1.81378e12 0.0481586
\(295\) 7.99058e11 0.0208237
\(296\) 1.01009e13 0.258379
\(297\) 1.58215e12 0.0397272
\(298\) −1.74585e11 −0.00430345
\(299\) −2.51116e11 −0.00607692
\(300\) 2.33990e13 0.555944
\(301\) −2.67806e13 −0.624748
\(302\) −4.49262e12 −0.102911
\(303\) −4.30417e13 −0.968177
\(304\) 6.10748e13 1.34914
\(305\) 9.15915e12 0.198704
\(306\) 2.45496e12 0.0523091
\(307\) −5.62621e13 −1.17748 −0.588742 0.808321i \(-0.700377\pi\)
−0.588742 + 0.808321i \(0.700377\pi\)
\(308\) 4.77924e12 0.0982495
\(309\) −2.04520e13 −0.413013
\(310\) 7.62954e11 0.0151359
\(311\) 7.74516e13 1.50955 0.754776 0.655983i \(-0.227746\pi\)
0.754776 + 0.655983i \(0.227746\pi\)
\(312\) −1.67020e11 −0.00319830
\(313\) 1.66025e13 0.312378 0.156189 0.987727i \(-0.450079\pi\)
0.156189 + 0.987727i \(0.450079\pi\)
\(314\) 6.07920e12 0.112391
\(315\) 1.41346e12 0.0256788
\(316\) −5.14043e13 −0.917743
\(317\) −1.01258e14 −1.77666 −0.888330 0.459206i \(-0.848134\pi\)
−0.888330 + 0.459206i \(0.848134\pi\)
\(318\) −6.28217e12 −0.108333
\(319\) −3.73905e12 −0.0633742
\(320\) 8.92801e12 0.148741
\(321\) 4.46868e13 0.731816
\(322\) 7.69654e11 0.0123905
\(323\) −1.27673e14 −2.02062
\(324\) −7.05670e12 −0.109801
\(325\) 1.63405e12 0.0249982
\(326\) 3.83321e12 0.0576588
\(327\) −8.40329e12 −0.124290
\(328\) 1.29949e13 0.189003
\(329\) 4.12170e13 0.589520
\(330\) 1.47188e11 0.00207036
\(331\) 7.41087e13 1.02522 0.512608 0.858623i \(-0.328679\pi\)
0.512608 + 0.858623i \(0.328679\pi\)
\(332\) −5.66064e13 −0.770204
\(333\) 2.98032e13 0.398860
\(334\) 1.09160e13 0.143700
\(335\) 9.90802e12 0.128304
\(336\) −2.10589e13 −0.268268
\(337\) 5.34310e13 0.669621 0.334811 0.942285i \(-0.391328\pi\)
0.334811 + 0.942285i \(0.391328\pi\)
\(338\) 8.80259e12 0.108535
\(339\) 4.00826e13 0.486248
\(340\) −1.91341e13 −0.228388
\(341\) 1.53140e13 0.179863
\(342\) −4.38045e12 −0.0506261
\(343\) 7.48724e13 0.851538
\(344\) 2.50253e13 0.280096
\(345\) −1.98585e12 −0.0218747
\(346\) 1.17709e13 0.127612
\(347\) 1.17978e14 1.25889 0.629446 0.777044i \(-0.283282\pi\)
0.629446 + 0.777044i \(0.283282\pi\)
\(348\) 1.66769e13 0.175158
\(349\) 1.24768e14 1.28992 0.644962 0.764214i \(-0.276873\pi\)
0.644962 + 0.764214i \(0.276873\pi\)
\(350\) −5.00826e12 −0.0509698
\(351\) −4.92800e11 −0.00493721
\(352\) −6.71222e12 −0.0662036
\(353\) −2.43773e13 −0.236715 −0.118357 0.992971i \(-0.537763\pi\)
−0.118357 + 0.992971i \(0.537763\pi\)
\(354\) 8.53859e11 0.00816334
\(355\) 3.87203e12 0.0364488
\(356\) 2.01339e14 1.86618
\(357\) 4.40222e13 0.401788
\(358\) 2.19765e13 0.197516
\(359\) 9.88821e12 0.0875182 0.0437591 0.999042i \(-0.486067\pi\)
0.0437591 + 0.999042i \(0.486067\pi\)
\(360\) −1.32081e12 −0.0115127
\(361\) 1.11319e14 0.955611
\(362\) 2.39481e13 0.202476
\(363\) −6.63764e13 −0.552748
\(364\) −1.48861e12 −0.0122102
\(365\) −5.63900e12 −0.0455608
\(366\) 9.78730e12 0.0778963
\(367\) 1.72600e14 1.35325 0.676624 0.736328i \(-0.263442\pi\)
0.676624 + 0.736328i \(0.263442\pi\)
\(368\) 2.95869e13 0.228526
\(369\) 3.83421e13 0.291763
\(370\) 2.77261e12 0.0207864
\(371\) −1.12651e14 −0.832108
\(372\) −6.83036e13 −0.497116
\(373\) −9.71366e13 −0.696601 −0.348301 0.937383i \(-0.613241\pi\)
−0.348301 + 0.937383i \(0.613241\pi\)
\(374\) 4.58418e12 0.0323943
\(375\) 2.61838e13 0.182331
\(376\) −3.85154e13 −0.264302
\(377\) 1.16462e12 0.00787602
\(378\) 1.51040e12 0.0100667
\(379\) −9.40338e13 −0.617687 −0.308843 0.951113i \(-0.599942\pi\)
−0.308843 + 0.951113i \(0.599942\pi\)
\(380\) 3.41414e13 0.221040
\(381\) 1.07291e14 0.684661
\(382\) 6.15514e12 0.0387159
\(383\) 1.42485e14 0.883438 0.441719 0.897153i \(-0.354369\pi\)
0.441719 + 0.897153i \(0.354369\pi\)
\(384\) 3.98355e13 0.243471
\(385\) 2.63937e12 0.0159025
\(386\) 2.99599e13 0.177955
\(387\) 7.38381e13 0.432384
\(388\) 2.17417e14 1.25522
\(389\) 1.74109e14 0.991054 0.495527 0.868593i \(-0.334975\pi\)
0.495527 + 0.868593i \(0.334975\pi\)
\(390\) −4.58454e10 −0.000257301 0
\(391\) −6.18494e13 −0.342266
\(392\) −3.03927e13 −0.165842
\(393\) −3.92588e13 −0.211241
\(394\) −1.97868e13 −0.104990
\(395\) −2.83884e13 −0.148545
\(396\) −1.31771e13 −0.0679979
\(397\) −3.15493e14 −1.60562 −0.802808 0.596238i \(-0.796662\pi\)
−0.802808 + 0.596238i \(0.796662\pi\)
\(398\) 2.00463e13 0.100618
\(399\) −7.85498e13 −0.388861
\(400\) −1.92527e14 −0.940071
\(401\) −2.81606e14 −1.35628 −0.678138 0.734935i \(-0.737213\pi\)
−0.678138 + 0.734935i \(0.737213\pi\)
\(402\) 1.05875e13 0.0502981
\(403\) −4.76994e12 −0.0223530
\(404\) 3.58476e14 1.65715
\(405\) −3.89711e12 −0.0177722
\(406\) −3.56948e12 −0.0160587
\(407\) 5.56519e13 0.247008
\(408\) −4.11367e13 −0.180135
\(409\) 4.50963e14 1.94833 0.974166 0.225831i \(-0.0725097\pi\)
0.974166 + 0.225831i \(0.0725097\pi\)
\(410\) 3.56698e12 0.0152051
\(411\) −1.18218e14 −0.497227
\(412\) 1.70336e14 0.706921
\(413\) 1.53113e13 0.0627029
\(414\) −2.12205e12 −0.00857537
\(415\) −3.12613e13 −0.124664
\(416\) 2.09069e12 0.00822765
\(417\) 1.16992e14 0.454368
\(418\) −8.17966e12 −0.0313520
\(419\) −2.40157e14 −0.908485 −0.454242 0.890878i \(-0.650090\pi\)
−0.454242 + 0.890878i \(0.650090\pi\)
\(420\) −1.17721e13 −0.0439524
\(421\) 2.25985e14 0.832776 0.416388 0.909187i \(-0.363296\pi\)
0.416388 + 0.909187i \(0.363296\pi\)
\(422\) −1.86744e13 −0.0679246
\(423\) −1.13641e14 −0.408003
\(424\) 1.05267e14 0.373063
\(425\) 4.02464e14 1.40795
\(426\) 4.13758e12 0.0142887
\(427\) 1.75505e14 0.598324
\(428\) −3.72177e14 −1.25259
\(429\) −9.20212e11 −0.00305754
\(430\) 6.86919e12 0.0225335
\(431\) 2.23695e13 0.0724487 0.0362244 0.999344i \(-0.488467\pi\)
0.0362244 + 0.999344i \(0.488467\pi\)
\(432\) 5.80624e13 0.185667
\(433\) −5.15388e14 −1.62724 −0.813618 0.581400i \(-0.802505\pi\)
−0.813618 + 0.581400i \(0.802505\pi\)
\(434\) 1.46195e13 0.0455764
\(435\) 9.20993e12 0.0283508
\(436\) 6.99874e13 0.212737
\(437\) 1.10359e14 0.331254
\(438\) −6.02573e12 −0.0178609
\(439\) −4.93570e14 −1.44475 −0.722377 0.691499i \(-0.756951\pi\)
−0.722377 + 0.691499i \(0.756951\pi\)
\(440\) −2.46637e12 −0.00712965
\(441\) −8.96748e13 −0.256011
\(442\) −1.42786e12 −0.00402589
\(443\) 1.66783e14 0.464442 0.232221 0.972663i \(-0.425401\pi\)
0.232221 + 0.972663i \(0.425401\pi\)
\(444\) −2.48218e14 −0.682696
\(445\) 1.11191e14 0.302057
\(446\) −7.90652e12 −0.0212150
\(447\) 8.63162e12 0.0228771
\(448\) 1.71076e14 0.447879
\(449\) −2.74518e14 −0.709932 −0.354966 0.934879i \(-0.615508\pi\)
−0.354966 + 0.934879i \(0.615508\pi\)
\(450\) 1.38085e13 0.0352759
\(451\) 7.15966e13 0.180685
\(452\) −3.33831e14 −0.832272
\(453\) 2.22119e14 0.547074
\(454\) 5.92535e13 0.144181
\(455\) −8.22097e11 −0.00197633
\(456\) 7.34012e13 0.174340
\(457\) −5.18597e14 −1.21700 −0.608501 0.793553i \(-0.708229\pi\)
−0.608501 + 0.793553i \(0.708229\pi\)
\(458\) 3.21399e13 0.0745220
\(459\) −1.21376e14 −0.278075
\(460\) 1.65393e13 0.0374412
\(461\) −7.27089e14 −1.62642 −0.813209 0.581971i \(-0.802282\pi\)
−0.813209 + 0.581971i \(0.802282\pi\)
\(462\) 2.82038e12 0.00623415
\(463\) 6.19864e14 1.35395 0.676973 0.736008i \(-0.263291\pi\)
0.676973 + 0.736008i \(0.263291\pi\)
\(464\) −1.37217e14 −0.296182
\(465\) −3.77211e13 −0.0804625
\(466\) −4.61731e13 −0.0973344
\(467\) −7.39953e14 −1.54156 −0.770782 0.637099i \(-0.780134\pi\)
−0.770782 + 0.637099i \(0.780134\pi\)
\(468\) 4.10432e12 0.00845064
\(469\) 1.89855e14 0.386341
\(470\) −1.05721e13 −0.0212629
\(471\) −3.00561e14 −0.597473
\(472\) −1.43077e13 −0.0281119
\(473\) 1.37879e14 0.267769
\(474\) −3.03354e13 −0.0582329
\(475\) −7.18126e14 −1.36265
\(476\) −3.66642e14 −0.687708
\(477\) 3.10596e14 0.575897
\(478\) −8.91551e12 −0.0163415
\(479\) 3.46919e14 0.628611 0.314306 0.949322i \(-0.398228\pi\)
0.314306 + 0.949322i \(0.398228\pi\)
\(480\) 1.65334e13 0.0296165
\(481\) −1.73342e13 −0.0306976
\(482\) −6.81234e12 −0.0119272
\(483\) −3.80524e13 −0.0658677
\(484\) 5.52820e14 0.946095
\(485\) 1.20070e14 0.203168
\(486\) −4.16439e12 −0.00696709
\(487\) 9.31866e14 1.54150 0.770751 0.637136i \(-0.219881\pi\)
0.770751 + 0.637136i \(0.219881\pi\)
\(488\) −1.64001e14 −0.268249
\(489\) −1.89517e14 −0.306514
\(490\) −8.34249e12 −0.0133419
\(491\) 7.29686e14 1.15395 0.576976 0.816761i \(-0.304232\pi\)
0.576976 + 0.816761i \(0.304232\pi\)
\(492\) −3.19335e14 −0.499388
\(493\) 2.86843e14 0.443595
\(494\) 2.54776e12 0.00389637
\(495\) −7.27713e12 −0.0110060
\(496\) 5.62000e14 0.840596
\(497\) 7.41948e13 0.109752
\(498\) −3.34052e13 −0.0488712
\(499\) 1.14051e15 1.65023 0.825117 0.564962i \(-0.191109\pi\)
0.825117 + 0.564962i \(0.191109\pi\)
\(500\) −2.18074e14 −0.312082
\(501\) −5.39695e14 −0.763908
\(502\) −2.76532e13 −0.0387146
\(503\) 1.51279e14 0.209486 0.104743 0.994499i \(-0.466598\pi\)
0.104743 + 0.994499i \(0.466598\pi\)
\(504\) −2.53091e13 −0.0346664
\(505\) 1.97971e14 0.268224
\(506\) −3.96252e12 −0.00531060
\(507\) −4.35208e14 −0.576970
\(508\) −8.93581e14 −1.17188
\(509\) −1.04534e15 −1.35616 −0.678079 0.734989i \(-0.737187\pi\)
−0.678079 + 0.734989i \(0.737187\pi\)
\(510\) −1.12916e13 −0.0144917
\(511\) −1.08053e14 −0.137190
\(512\) −4.12178e14 −0.517726
\(513\) 2.16573e14 0.269128
\(514\) 1.38332e14 0.170070
\(515\) 9.40690e13 0.114421
\(516\) −6.14966e14 −0.740078
\(517\) −2.12204e14 −0.252671
\(518\) 5.31280e13 0.0625907
\(519\) −5.81964e14 −0.678384
\(520\) 7.68212e11 0.000886059 0
\(521\) −9.86353e14 −1.12571 −0.562853 0.826557i \(-0.690296\pi\)
−0.562853 + 0.826557i \(0.690296\pi\)
\(522\) 9.84157e12 0.0111142
\(523\) 1.59830e15 1.78607 0.893033 0.449990i \(-0.148573\pi\)
0.893033 + 0.449990i \(0.148573\pi\)
\(524\) 3.26970e14 0.361565
\(525\) 2.47613e14 0.270955
\(526\) 3.95937e13 0.0428750
\(527\) −1.17483e15 −1.25897
\(528\) 1.08421e14 0.114981
\(529\) −8.99348e14 −0.943890
\(530\) 2.88949e13 0.0300126
\(531\) −4.22156e13 −0.0433963
\(532\) 6.54208e14 0.665582
\(533\) −2.23005e13 −0.0224551
\(534\) 1.18817e14 0.118413
\(535\) −2.05538e14 −0.202743
\(536\) −1.77411e14 −0.173210
\(537\) −1.08654e15 −1.05000
\(538\) 6.41027e13 0.0613160
\(539\) −1.67451e14 −0.158544
\(540\) 3.24574e13 0.0304192
\(541\) −5.76950e14 −0.535246 −0.267623 0.963524i \(-0.586238\pi\)
−0.267623 + 0.963524i \(0.586238\pi\)
\(542\) 1.25995e14 0.115706
\(543\) −1.18402e15 −1.07636
\(544\) 5.14932e14 0.463400
\(545\) 3.86510e13 0.0344334
\(546\) −8.78478e11 −0.000774768 0
\(547\) 1.07123e15 0.935300 0.467650 0.883914i \(-0.345101\pi\)
0.467650 + 0.883914i \(0.345101\pi\)
\(548\) 9.84590e14 0.851064
\(549\) −4.83893e14 −0.414097
\(550\) 2.57848e13 0.0218458
\(551\) −5.11821e14 −0.429323
\(552\) 3.55582e13 0.0295308
\(553\) −5.43971e14 −0.447288
\(554\) −1.06996e14 −0.0871094
\(555\) −1.37080e14 −0.110500
\(556\) −9.74377e14 −0.777706
\(557\) 1.18124e15 0.933546 0.466773 0.884377i \(-0.345417\pi\)
0.466773 + 0.884377i \(0.345417\pi\)
\(558\) −4.03082e13 −0.0315431
\(559\) −4.29457e13 −0.0332778
\(560\) 9.68606e13 0.0743212
\(561\) −2.26646e14 −0.172208
\(562\) −1.14343e14 −0.0860321
\(563\) −1.70145e15 −1.26772 −0.633858 0.773449i \(-0.718530\pi\)
−0.633858 + 0.773449i \(0.718530\pi\)
\(564\) 9.46469e14 0.698347
\(565\) −1.84361e14 −0.134710
\(566\) −2.16778e14 −0.156865
\(567\) −7.46755e13 −0.0535145
\(568\) −6.93316e13 −0.0492057
\(569\) −1.64736e15 −1.15790 −0.578951 0.815363i \(-0.696538\pi\)
−0.578951 + 0.815363i \(0.696538\pi\)
\(570\) 2.01479e13 0.0140255
\(571\) 1.91521e14 0.132044 0.0660219 0.997818i \(-0.478969\pi\)
0.0660219 + 0.997818i \(0.478969\pi\)
\(572\) 7.66405e12 0.00523336
\(573\) −3.04316e14 −0.205814
\(574\) 6.83496e13 0.0457847
\(575\) −3.47886e14 −0.230815
\(576\) −4.71682e14 −0.309974
\(577\) 1.78676e15 1.16305 0.581526 0.813528i \(-0.302456\pi\)
0.581526 + 0.813528i \(0.302456\pi\)
\(578\) −1.83233e14 −0.118141
\(579\) −1.48125e15 −0.946008
\(580\) −7.67056e13 −0.0485258
\(581\) −5.99020e14 −0.375381
\(582\) 1.28305e14 0.0796464
\(583\) 5.79980e14 0.356645
\(584\) 1.00971e14 0.0615069
\(585\) 2.26664e12 0.00136781
\(586\) −4.48974e13 −0.0268401
\(587\) −2.87784e15 −1.70434 −0.852170 0.523264i \(-0.824714\pi\)
−0.852170 + 0.523264i \(0.824714\pi\)
\(588\) 7.46863e14 0.438193
\(589\) 2.09627e15 1.21846
\(590\) −3.92733e12 −0.00226158
\(591\) 9.78278e14 0.558125
\(592\) 2.04233e15 1.15440
\(593\) −8.09682e14 −0.453434 −0.226717 0.973961i \(-0.572799\pi\)
−0.226717 + 0.973961i \(0.572799\pi\)
\(594\) −7.77621e12 −0.00431462
\(595\) −2.02481e14 −0.111312
\(596\) −7.18891e13 −0.0391569
\(597\) −9.91107e14 −0.534886
\(598\) 1.23423e12 0.000659991 0
\(599\) −6.86915e14 −0.363961 −0.181981 0.983302i \(-0.558251\pi\)
−0.181981 + 0.983302i \(0.558251\pi\)
\(600\) −2.31383e14 −0.121479
\(601\) 2.86881e15 1.49243 0.746213 0.665707i \(-0.231870\pi\)
0.746213 + 0.665707i \(0.231870\pi\)
\(602\) 1.31626e14 0.0678516
\(603\) −5.23458e14 −0.267385
\(604\) −1.84994e15 −0.936383
\(605\) 3.05299e14 0.153134
\(606\) 2.11548e14 0.105150
\(607\) −3.35448e15 −1.65229 −0.826147 0.563454i \(-0.809472\pi\)
−0.826147 + 0.563454i \(0.809472\pi\)
\(608\) −9.18804e14 −0.448491
\(609\) 1.76478e14 0.0853682
\(610\) −4.50168e13 −0.0215805
\(611\) 6.60960e13 0.0314014
\(612\) 1.01089e15 0.475959
\(613\) −1.76110e15 −0.821771 −0.410886 0.911687i \(-0.634780\pi\)
−0.410886 + 0.911687i \(0.634780\pi\)
\(614\) 2.76526e14 0.127882
\(615\) −1.76355e14 −0.0808303
\(616\) −4.72599e13 −0.0214684
\(617\) 2.20526e15 0.992868 0.496434 0.868074i \(-0.334642\pi\)
0.496434 + 0.868074i \(0.334642\pi\)
\(618\) 1.00521e14 0.0448557
\(619\) −3.35617e15 −1.48438 −0.742190 0.670189i \(-0.766213\pi\)
−0.742190 + 0.670189i \(0.766213\pi\)
\(620\) 3.14163e14 0.137721
\(621\) 1.04916e14 0.0455866
\(622\) −3.80671e14 −0.163947
\(623\) 2.13061e15 0.909535
\(624\) −3.37703e13 −0.0142896
\(625\) 2.20276e15 0.923903
\(626\) −8.16008e13 −0.0339262
\(627\) 4.04410e14 0.166667
\(628\) 2.50325e15 1.02265
\(629\) −4.26937e15 −1.72896
\(630\) −6.94709e12 −0.00278888
\(631\) 3.89824e15 1.55134 0.775669 0.631140i \(-0.217412\pi\)
0.775669 + 0.631140i \(0.217412\pi\)
\(632\) 5.08316e14 0.200535
\(633\) 9.23277e14 0.361087
\(634\) 4.97680e14 0.192956
\(635\) −4.93486e14 −0.189679
\(636\) −2.58682e15 −0.985718
\(637\) 5.21567e13 0.0197035
\(638\) 1.83773e13 0.00688284
\(639\) −2.04566e14 −0.0759589
\(640\) −1.83224e14 −0.0674515
\(641\) 2.92809e15 1.06872 0.534362 0.845256i \(-0.320552\pi\)
0.534362 + 0.845256i \(0.320552\pi\)
\(642\) −2.19634e14 −0.0794798
\(643\) −8.03768e14 −0.288384 −0.144192 0.989550i \(-0.546058\pi\)
−0.144192 + 0.989550i \(0.546058\pi\)
\(644\) 3.16922e14 0.112741
\(645\) −3.39619e14 −0.119788
\(646\) 6.27507e14 0.219452
\(647\) 3.77555e15 1.30920 0.654601 0.755975i \(-0.272837\pi\)
0.654601 + 0.755975i \(0.272837\pi\)
\(648\) 6.97808e13 0.0239924
\(649\) −7.88296e13 −0.0268747
\(650\) −8.03131e12 −0.00271496
\(651\) −7.22803e14 −0.242284
\(652\) 1.57841e15 0.524636
\(653\) −4.89684e15 −1.61396 −0.806982 0.590576i \(-0.798900\pi\)
−0.806982 + 0.590576i \(0.798900\pi\)
\(654\) 4.13018e13 0.0134987
\(655\) 1.80571e14 0.0585224
\(656\) 2.62748e15 0.844439
\(657\) 2.97918e14 0.0949482
\(658\) −2.02580e14 −0.0640256
\(659\) −4.06346e15 −1.27358 −0.636789 0.771038i \(-0.719738\pi\)
−0.636789 + 0.771038i \(0.719738\pi\)
\(660\) 6.06081e13 0.0188382
\(661\) 6.20815e15 1.91361 0.956807 0.290724i \(-0.0938961\pi\)
0.956807 + 0.290724i \(0.0938961\pi\)
\(662\) −3.64241e14 −0.111345
\(663\) 7.05945e13 0.0214016
\(664\) 5.59757e14 0.168296
\(665\) 3.61291e14 0.107730
\(666\) −1.46482e14 −0.0433186
\(667\) −2.47945e14 −0.0727215
\(668\) 4.49489e15 1.30752
\(669\) 3.90906e14 0.112779
\(670\) −4.86975e13 −0.0139346
\(671\) −9.03579e14 −0.256444
\(672\) 3.16808e14 0.0891795
\(673\) −1.49379e15 −0.417068 −0.208534 0.978015i \(-0.566869\pi\)
−0.208534 + 0.978015i \(0.566869\pi\)
\(674\) −2.62611e14 −0.0727250
\(675\) −6.82705e14 −0.187526
\(676\) 3.62466e15 0.987554
\(677\) −7.20599e15 −1.94740 −0.973702 0.227824i \(-0.926839\pi\)
−0.973702 + 0.227824i \(0.926839\pi\)
\(678\) −1.97004e14 −0.0528096
\(679\) 2.30075e15 0.611767
\(680\) 1.89209e14 0.0499048
\(681\) −2.92955e15 −0.766463
\(682\) −7.52679e13 −0.0195342
\(683\) 5.08063e15 1.30799 0.653993 0.756500i \(-0.273092\pi\)
0.653993 + 0.756500i \(0.273092\pi\)
\(684\) −1.80375e15 −0.460646
\(685\) 5.43747e14 0.137752
\(686\) −3.67994e14 −0.0924823
\(687\) −1.58903e15 −0.396158
\(688\) 5.05992e15 1.25143
\(689\) −1.80649e14 −0.0443231
\(690\) 9.76038e12 0.00237573
\(691\) −5.57864e15 −1.34710 −0.673549 0.739143i \(-0.735231\pi\)
−0.673549 + 0.739143i \(0.735231\pi\)
\(692\) 4.84693e15 1.16114
\(693\) −1.39442e14 −0.0331407
\(694\) −5.79856e14 −0.136723
\(695\) −5.38106e14 −0.125879
\(696\) −1.64911e14 −0.0382735
\(697\) −5.49257e15 −1.26472
\(698\) −6.13231e14 −0.140094
\(699\) 2.28284e15 0.517429
\(700\) −2.06226e15 −0.463773
\(701\) −1.09283e15 −0.243839 −0.121920 0.992540i \(-0.538905\pi\)
−0.121920 + 0.992540i \(0.538905\pi\)
\(702\) 2.42209e12 0.000536212 0
\(703\) 7.61793e15 1.67333
\(704\) −8.80777e14 −0.191962
\(705\) 5.22694e14 0.113034
\(706\) 1.19813e14 0.0257087
\(707\) 3.79346e15 0.807661
\(708\) 3.51595e14 0.0742780
\(709\) 4.80995e15 1.00829 0.504146 0.863618i \(-0.331807\pi\)
0.504146 + 0.863618i \(0.331807\pi\)
\(710\) −1.90309e13 −0.00395856
\(711\) 1.49981e15 0.309566
\(712\) −1.99096e15 −0.407776
\(713\) 1.01551e15 0.206391
\(714\) −2.16367e14 −0.0436367
\(715\) 4.23252e12 0.000847064 0
\(716\) 9.04933e15 1.79719
\(717\) 4.40791e14 0.0868716
\(718\) −4.86001e13 −0.00950502
\(719\) 6.79376e15 1.31856 0.659282 0.751895i \(-0.270860\pi\)
0.659282 + 0.751895i \(0.270860\pi\)
\(720\) −2.67059e14 −0.0514373
\(721\) 1.80253e15 0.344538
\(722\) −5.47130e14 −0.103785
\(723\) 3.36808e14 0.0634048
\(724\) 9.86117e15 1.84233
\(725\) 1.61342e15 0.299149
\(726\) 3.26237e14 0.0600319
\(727\) −6.19146e15 −1.13072 −0.565359 0.824845i \(-0.691262\pi\)
−0.565359 + 0.824845i \(0.691262\pi\)
\(728\) 1.47203e13 0.00266805
\(729\) 2.05891e14 0.0370370
\(730\) 2.77154e13 0.00494819
\(731\) −1.05774e16 −1.87428
\(732\) 4.03014e15 0.708776
\(733\) −9.15432e15 −1.59792 −0.798958 0.601387i \(-0.794615\pi\)
−0.798958 + 0.601387i \(0.794615\pi\)
\(734\) −8.48322e14 −0.146971
\(735\) 4.12460e14 0.0709254
\(736\) −4.45102e14 −0.0759682
\(737\) −9.77459e14 −0.165587
\(738\) −1.88450e14 −0.0316873
\(739\) 8.83935e15 1.47529 0.737643 0.675191i \(-0.235939\pi\)
0.737643 + 0.675191i \(0.235939\pi\)
\(740\) 1.14168e15 0.189135
\(741\) −1.25963e14 −0.0207131
\(742\) 5.53677e14 0.0903722
\(743\) 8.60916e15 1.39483 0.697417 0.716666i \(-0.254333\pi\)
0.697417 + 0.716666i \(0.254333\pi\)
\(744\) 6.75426e14 0.108624
\(745\) −3.97012e13 −0.00633789
\(746\) 4.77422e14 0.0756553
\(747\) 1.65159e15 0.259799
\(748\) 1.88764e15 0.294754
\(749\) −3.93846e15 −0.610487
\(750\) −1.28692e14 −0.0198023
\(751\) −3.67452e15 −0.561281 −0.280641 0.959813i \(-0.590547\pi\)
−0.280641 + 0.959813i \(0.590547\pi\)
\(752\) −7.78752e15 −1.18087
\(753\) 1.36720e15 0.205806
\(754\) −5.72406e12 −0.000855385 0
\(755\) −1.02164e15 −0.151562
\(756\) 6.21940e14 0.0915965
\(757\) −5.82737e15 −0.852011 −0.426006 0.904721i \(-0.640080\pi\)
−0.426006 + 0.904721i \(0.640080\pi\)
\(758\) 4.62172e14 0.0670846
\(759\) 1.95911e14 0.0282311
\(760\) −3.37610e14 −0.0482992
\(761\) 8.24910e14 0.117163 0.0585816 0.998283i \(-0.481342\pi\)
0.0585816 + 0.998283i \(0.481342\pi\)
\(762\) −5.27330e14 −0.0743585
\(763\) 7.40621e14 0.103684
\(764\) 2.53452e15 0.352275
\(765\) 5.58269e14 0.0770381
\(766\) −7.00308e14 −0.0959469
\(767\) 2.45534e13 0.00333993
\(768\) 3.77954e15 0.510448
\(769\) 1.40962e16 1.89020 0.945099 0.326783i \(-0.105965\pi\)
0.945099 + 0.326783i \(0.105965\pi\)
\(770\) −1.29724e13 −0.00172711
\(771\) −6.83928e15 −0.904089
\(772\) 1.23367e16 1.61921
\(773\) −7.51145e14 −0.0978896 −0.0489448 0.998801i \(-0.515586\pi\)
−0.0489448 + 0.998801i \(0.515586\pi\)
\(774\) −3.62911e14 −0.0469596
\(775\) −6.60808e15 −0.849016
\(776\) −2.14995e15 −0.274276
\(777\) −2.62670e15 −0.332732
\(778\) −8.55736e14 −0.107635
\(779\) 9.80052e15 1.22403
\(780\) −1.88779e13 −0.00234117
\(781\) −3.81988e14 −0.0470402
\(782\) 3.03987e14 0.0371722
\(783\) −4.86576e14 −0.0590828
\(784\) −6.14517e15 −0.740962
\(785\) 1.38244e15 0.165524
\(786\) 1.92955e14 0.0229421
\(787\) −4.03518e15 −0.476433 −0.238216 0.971212i \(-0.576563\pi\)
−0.238216 + 0.971212i \(0.576563\pi\)
\(788\) −8.14766e15 −0.955298
\(789\) −1.95755e15 −0.227923
\(790\) 1.39528e14 0.0161329
\(791\) −3.53267e15 −0.405632
\(792\) 1.30302e14 0.0148581
\(793\) 2.81442e14 0.0318703
\(794\) 1.55063e15 0.174380
\(795\) −1.42859e15 −0.159547
\(796\) 8.25450e15 0.915523
\(797\) −1.52361e16 −1.67824 −0.839119 0.543949i \(-0.816929\pi\)
−0.839119 + 0.543949i \(0.816929\pi\)
\(798\) 3.86069e14 0.0422327
\(799\) 1.62793e16 1.76860
\(800\) 2.89635e15 0.312505
\(801\) −5.87440e15 −0.629484
\(802\) 1.38408e15 0.147300
\(803\) 5.56305e14 0.0588001
\(804\) 4.35966e15 0.457661
\(805\) 1.75022e14 0.0182480
\(806\) 2.34440e13 0.00242767
\(807\) −3.16929e15 −0.325955
\(808\) −3.54481e15 −0.362102
\(809\) −1.31662e16 −1.33581 −0.667905 0.744246i \(-0.732809\pi\)
−0.667905 + 0.744246i \(0.732809\pi\)
\(810\) 1.91541e13 0.00193017
\(811\) 2.59337e15 0.259567 0.129783 0.991542i \(-0.458572\pi\)
0.129783 + 0.991542i \(0.458572\pi\)
\(812\) −1.46981e15 −0.146118
\(813\) −6.22929e15 −0.615093
\(814\) −2.73527e14 −0.0268266
\(815\) 8.71687e14 0.0849168
\(816\) −8.31754e15 −0.804821
\(817\) 1.88736e16 1.81398
\(818\) −2.21647e15 −0.211601
\(819\) 4.34328e13 0.00411866
\(820\) 1.46878e15 0.138351
\(821\) 1.40402e16 1.31366 0.656832 0.754037i \(-0.271896\pi\)
0.656832 + 0.754037i \(0.271896\pi\)
\(822\) 5.81038e14 0.0540020
\(823\) −1.93117e15 −0.178287 −0.0891437 0.996019i \(-0.528413\pi\)
−0.0891437 + 0.996019i \(0.528413\pi\)
\(824\) −1.68438e15 −0.154468
\(825\) −1.27482e15 −0.116132
\(826\) −7.52545e13 −0.00680993
\(827\) 7.18683e15 0.646037 0.323018 0.946393i \(-0.395302\pi\)
0.323018 + 0.946393i \(0.395302\pi\)
\(828\) −8.73800e14 −0.0780270
\(829\) −2.90585e15 −0.257765 −0.128882 0.991660i \(-0.541139\pi\)
−0.128882 + 0.991660i \(0.541139\pi\)
\(830\) 1.53648e14 0.0135393
\(831\) 5.29000e15 0.463073
\(832\) 2.74340e14 0.0238567
\(833\) 1.28461e16 1.10974
\(834\) −5.75011e14 −0.0493472
\(835\) 2.48233e15 0.211634
\(836\) −3.36816e15 −0.285271
\(837\) 1.99287e15 0.167683
\(838\) 1.18036e15 0.0986671
\(839\) 3.48645e15 0.289530 0.144765 0.989466i \(-0.453757\pi\)
0.144765 + 0.989466i \(0.453757\pi\)
\(840\) 1.16409e14 0.00960400
\(841\) −1.10506e16 −0.905749
\(842\) −1.11071e15 −0.0904446
\(843\) 5.65323e15 0.457346
\(844\) −7.68958e15 −0.618044
\(845\) 2.00175e15 0.159844
\(846\) 5.58542e14 0.0443117
\(847\) 5.85006e15 0.461107
\(848\) 2.12843e16 1.66680
\(849\) 1.07177e16 0.833891
\(850\) −1.97809e15 −0.152912
\(851\) 3.69040e15 0.283440
\(852\) 1.70374e15 0.130013
\(853\) −1.56899e16 −1.18960 −0.594798 0.803875i \(-0.702768\pi\)
−0.594798 + 0.803875i \(0.702768\pi\)
\(854\) −8.62600e14 −0.0649817
\(855\) −9.96131e14 −0.0745595
\(856\) 3.68031e15 0.273702
\(857\) −2.24133e16 −1.65619 −0.828096 0.560586i \(-0.810576\pi\)
−0.828096 + 0.560586i \(0.810576\pi\)
\(858\) 4.52280e12 0.000332068 0
\(859\) −1.61591e16 −1.17884 −0.589419 0.807827i \(-0.700643\pi\)
−0.589419 + 0.807827i \(0.700643\pi\)
\(860\) 2.82854e15 0.205032
\(861\) −3.37927e15 −0.243391
\(862\) −1.09945e14 −0.00786839
\(863\) −2.44408e16 −1.73802 −0.869012 0.494791i \(-0.835245\pi\)
−0.869012 + 0.494791i \(0.835245\pi\)
\(864\) −8.73486e14 −0.0617206
\(865\) 2.67675e15 0.187940
\(866\) 2.53311e15 0.176728
\(867\) 9.05921e15 0.628037
\(868\) 6.01992e15 0.414698
\(869\) 2.80061e15 0.191709
\(870\) −4.52664e13 −0.00307907
\(871\) 3.04454e14 0.0205789
\(872\) −6.92076e14 −0.0464850
\(873\) −6.34350e15 −0.423400
\(874\) −5.42411e14 −0.0359762
\(875\) −2.30770e15 −0.152102
\(876\) −2.48123e15 −0.162515
\(877\) 6.14296e15 0.399835 0.199917 0.979813i \(-0.435933\pi\)
0.199917 + 0.979813i \(0.435933\pi\)
\(878\) 2.42588e15 0.156909
\(879\) 2.21977e15 0.142682
\(880\) −4.98682e14 −0.0318544
\(881\) −7.46371e15 −0.473791 −0.236896 0.971535i \(-0.576130\pi\)
−0.236896 + 0.971535i \(0.576130\pi\)
\(882\) 4.40748e14 0.0278044
\(883\) −4.58663e15 −0.287548 −0.143774 0.989611i \(-0.545924\pi\)
−0.143774 + 0.989611i \(0.545924\pi\)
\(884\) −5.87952e14 −0.0366315
\(885\) 1.94171e14 0.0120225
\(886\) −8.19731e14 −0.0504413
\(887\) −9.08615e15 −0.555649 −0.277824 0.960632i \(-0.589613\pi\)
−0.277824 + 0.960632i \(0.589613\pi\)
\(888\) 2.45453e15 0.149175
\(889\) −9.45605e15 −0.571150
\(890\) −5.46498e14 −0.0328053
\(891\) 3.84463e14 0.0229365
\(892\) −3.25569e15 −0.193035
\(893\) −2.90475e16 −1.71169
\(894\) −4.24240e13 −0.00248460
\(895\) 4.99756e15 0.290891
\(896\) −3.51088e15 −0.203106
\(897\) −6.10212e13 −0.00350851
\(898\) 1.34925e15 0.0771030
\(899\) −4.70969e15 −0.267494
\(900\) 5.68596e15 0.320974
\(901\) −4.44935e16 −2.49637
\(902\) −3.51894e14 −0.0196235
\(903\) −6.50769e15 −0.360699
\(904\) 3.30112e15 0.181859
\(905\) 5.44590e15 0.298196
\(906\) −1.09171e15 −0.0594156
\(907\) 5.09917e14 0.0275842 0.0137921 0.999905i \(-0.495610\pi\)
0.0137921 + 0.999905i \(0.495610\pi\)
\(908\) 2.43990e16 1.31189
\(909\) −1.04591e16 −0.558977
\(910\) 4.04057e12 0.000214642 0
\(911\) −2.44976e16 −1.29352 −0.646759 0.762695i \(-0.723876\pi\)
−0.646759 + 0.762695i \(0.723876\pi\)
\(912\) 1.48412e16 0.778927
\(913\) 3.08402e15 0.160890
\(914\) 2.54888e15 0.132174
\(915\) 2.22567e15 0.114722
\(916\) 1.32343e16 0.678073
\(917\) 3.46006e15 0.176219
\(918\) 5.96556e14 0.0302007
\(919\) −6.89140e15 −0.346795 −0.173397 0.984852i \(-0.555474\pi\)
−0.173397 + 0.984852i \(0.555474\pi\)
\(920\) −1.63550e14 −0.00818123
\(921\) −1.36717e16 −0.679821
\(922\) 3.57361e15 0.176639
\(923\) 1.18980e14 0.00584606
\(924\) 1.16136e15 0.0567244
\(925\) −2.40140e16 −1.16597
\(926\) −3.04661e15 −0.147047
\(927\) −4.96983e15 −0.238453
\(928\) 2.06428e15 0.0984589
\(929\) 3.98005e15 0.188713 0.0943566 0.995538i \(-0.469921\pi\)
0.0943566 + 0.995538i \(0.469921\pi\)
\(930\) 1.85398e14 0.00873873
\(931\) −2.29216e16 −1.07404
\(932\) −1.90128e16 −0.885642
\(933\) 1.88207e16 0.871540
\(934\) 3.63684e15 0.167423
\(935\) 1.04246e15 0.0477085
\(936\) −4.05859e13 −0.00184654
\(937\) 1.17505e16 0.531483 0.265741 0.964044i \(-0.414383\pi\)
0.265741 + 0.964044i \(0.414383\pi\)
\(938\) −9.33129e14 −0.0419591
\(939\) 4.03442e15 0.180352
\(940\) −4.35330e15 −0.193471
\(941\) 4.44968e15 0.196601 0.0983005 0.995157i \(-0.468659\pi\)
0.0983005 + 0.995157i \(0.468659\pi\)
\(942\) 1.47725e15 0.0648893
\(943\) 4.74773e15 0.207334
\(944\) −2.89292e15 −0.125600
\(945\) 3.43471e14 0.0148257
\(946\) −6.77668e14 −0.0290814
\(947\) 7.19250e15 0.306870 0.153435 0.988159i \(-0.450966\pi\)
0.153435 + 0.988159i \(0.450966\pi\)
\(948\) −1.24913e16 −0.529859
\(949\) −1.73275e14 −0.00730755
\(950\) 3.52956e15 0.147993
\(951\) −2.46057e16 −1.02575
\(952\) 3.62557e15 0.150270
\(953\) −4.08844e16 −1.68479 −0.842397 0.538858i \(-0.818856\pi\)
−0.842397 + 0.538858i \(0.818856\pi\)
\(954\) −1.52657e15 −0.0625460
\(955\) 1.39970e15 0.0570187
\(956\) −3.67116e15 −0.148691
\(957\) −9.08589e14 −0.0365891
\(958\) −1.70509e15 −0.0682711
\(959\) 1.04191e16 0.414791
\(960\) 2.16951e15 0.0858754
\(961\) −6.11897e15 −0.240824
\(962\) 8.51967e13 0.00333396
\(963\) 1.08589e16 0.422514
\(964\) −2.80513e15 −0.108525
\(965\) 6.81301e15 0.262083
\(966\) 1.87026e14 0.00715364
\(967\) −1.05658e16 −0.401842 −0.200921 0.979607i \(-0.564393\pi\)
−0.200921 + 0.979607i \(0.564393\pi\)
\(968\) −5.46661e15 −0.206730
\(969\) −3.10245e16 −1.16661
\(970\) −5.90139e14 −0.0220653
\(971\) −3.89633e15 −0.144860 −0.0724302 0.997373i \(-0.523075\pi\)
−0.0724302 + 0.997373i \(0.523075\pi\)
\(972\) −1.71478e15 −0.0633934
\(973\) −1.03111e16 −0.379038
\(974\) −4.58008e15 −0.167417
\(975\) 3.97075e14 0.0144327
\(976\) −3.31599e16 −1.19850
\(977\) 1.92641e16 0.692355 0.346177 0.938169i \(-0.387480\pi\)
0.346177 + 0.938169i \(0.387480\pi\)
\(978\) 9.31470e14 0.0332893
\(979\) −1.09693e16 −0.389830
\(980\) −3.43521e15 −0.121397
\(981\) −2.04200e15 −0.0717589
\(982\) −3.58638e15 −0.125326
\(983\) −1.44449e16 −0.501961 −0.250981 0.967992i \(-0.580753\pi\)
−0.250981 + 0.967992i \(0.580753\pi\)
\(984\) 3.15777e15 0.109121
\(985\) −4.49960e15 −0.154623
\(986\) −1.40982e15 −0.0481772
\(987\) 1.00157e16 0.340360
\(988\) 1.04910e15 0.0354529
\(989\) 9.14304e15 0.307263
\(990\) 3.57668e13 0.00119533
\(991\) −1.69598e16 −0.563659 −0.281830 0.959464i \(-0.590941\pi\)
−0.281830 + 0.959464i \(0.590941\pi\)
\(992\) −8.45469e15 −0.279437
\(993\) 1.80084e16 0.591909
\(994\) −3.64664e14 −0.0119198
\(995\) 4.55861e15 0.148185
\(996\) −1.37554e16 −0.444678
\(997\) −1.62198e16 −0.521461 −0.260731 0.965412i \(-0.583963\pi\)
−0.260731 + 0.965412i \(0.583963\pi\)
\(998\) −5.60555e15 −0.179226
\(999\) 7.24219e15 0.230282
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.1.14 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.14 27 1.1 even 1 trivial