Properties

Label 177.12.a.b.1.1
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 1.1
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-88.8229 q^{2} +243.000 q^{3} +5841.51 q^{4} +11395.4 q^{5} -21584.0 q^{6} +23008.9 q^{7} -336951. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-88.8229 q^{2} +243.000 q^{3} +5841.51 q^{4} +11395.4 q^{5} -21584.0 q^{6} +23008.9 q^{7} -336951. q^{8} +59049.0 q^{9} -1.01218e6 q^{10} -240447. q^{11} +1.41949e6 q^{12} -2.32601e6 q^{13} -2.04372e6 q^{14} +2.76909e6 q^{15} +1.79656e7 q^{16} +1.04209e7 q^{17} -5.24491e6 q^{18} -1.05088e7 q^{19} +6.65666e7 q^{20} +5.59117e6 q^{21} +2.13572e7 q^{22} -3.94534e7 q^{23} -8.18791e7 q^{24} +8.10278e7 q^{25} +2.06603e8 q^{26} +1.43489e7 q^{27} +1.34407e8 q^{28} -9.37519e6 q^{29} -2.45959e8 q^{30} -2.38250e7 q^{31} -9.05679e8 q^{32} -5.84287e7 q^{33} -9.25614e8 q^{34} +2.62197e8 q^{35} +3.44936e8 q^{36} +3.44877e8 q^{37} +9.33424e8 q^{38} -5.65220e8 q^{39} -3.83971e9 q^{40} -1.07900e9 q^{41} -4.96624e8 q^{42} +9.52560e8 q^{43} -1.40458e9 q^{44} +6.72889e8 q^{45} +3.50437e9 q^{46} -2.34758e9 q^{47} +4.36563e9 q^{48} -1.44792e9 q^{49} -7.19713e9 q^{50} +2.53228e9 q^{51} -1.35874e10 q^{52} -1.54271e9 q^{53} -1.27451e9 q^{54} -2.74000e9 q^{55} -7.75289e9 q^{56} -2.55364e9 q^{57} +8.32732e8 q^{58} -7.14924e8 q^{59} +1.61757e10 q^{60} +5.48052e9 q^{61} +2.11620e9 q^{62} +1.35865e9 q^{63} +4.36516e10 q^{64} -2.65059e10 q^{65} +5.18981e9 q^{66} -3.63841e9 q^{67} +6.08738e10 q^{68} -9.58717e9 q^{69} -2.32891e10 q^{70} -4.43632e9 q^{71} -1.98966e10 q^{72} -1.08626e10 q^{73} -3.06330e10 q^{74} +1.96898e10 q^{75} -6.13874e10 q^{76} -5.53244e9 q^{77} +5.02045e10 q^{78} -1.30413e10 q^{79} +2.04725e11 q^{80} +3.48678e9 q^{81} +9.58401e10 q^{82} -2.78822e10 q^{83} +3.26609e10 q^{84} +1.18751e11 q^{85} -8.46092e10 q^{86} -2.27817e9 q^{87} +8.10190e10 q^{88} +3.44766e10 q^{89} -5.97680e10 q^{90} -5.35190e10 q^{91} -2.30468e11 q^{92} -5.78946e9 q^{93} +2.08519e11 q^{94} -1.19752e11 q^{95} -2.20080e11 q^{96} +1.26705e11 q^{97} +1.28608e11 q^{98} -1.41982e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −88.8229 −1.96273 −0.981364 0.192157i \(-0.938452\pi\)
−0.981364 + 0.192157i \(0.938452\pi\)
\(3\) 243.000 0.577350
\(4\) 5841.51 2.85230
\(5\) 11395.4 1.63078 0.815391 0.578911i \(-0.196522\pi\)
0.815391 + 0.578911i \(0.196522\pi\)
\(6\) −21584.0 −1.13318
\(7\) 23008.9 0.517437 0.258718 0.965953i \(-0.416700\pi\)
0.258718 + 0.965953i \(0.416700\pi\)
\(8\) −336951. −3.63557
\(9\) 59049.0 0.333333
\(10\) −1.01218e6 −3.20078
\(11\) −240447. −0.450153 −0.225077 0.974341i \(-0.572263\pi\)
−0.225077 + 0.974341i \(0.572263\pi\)
\(12\) 1.41949e6 1.64678
\(13\) −2.32601e6 −1.73749 −0.868747 0.495257i \(-0.835074\pi\)
−0.868747 + 0.495257i \(0.835074\pi\)
\(14\) −2.04372e6 −1.01559
\(15\) 2.76909e6 0.941532
\(16\) 1.79656e7 4.28333
\(17\) 1.04209e7 1.78006 0.890032 0.455897i \(-0.150682\pi\)
0.890032 + 0.455897i \(0.150682\pi\)
\(18\) −5.24491e6 −0.654243
\(19\) −1.05088e7 −0.973663 −0.486832 0.873496i \(-0.661847\pi\)
−0.486832 + 0.873496i \(0.661847\pi\)
\(20\) 6.65666e7 4.65148
\(21\) 5.59117e6 0.298742
\(22\) 2.13572e7 0.883528
\(23\) −3.94534e7 −1.27815 −0.639074 0.769145i \(-0.720682\pi\)
−0.639074 + 0.769145i \(0.720682\pi\)
\(24\) −8.18791e7 −2.09899
\(25\) 8.10278e7 1.65945
\(26\) 2.06603e8 3.41023
\(27\) 1.43489e7 0.192450
\(28\) 1.34407e8 1.47589
\(29\) −9.37519e6 −0.0848773 −0.0424386 0.999099i \(-0.513513\pi\)
−0.0424386 + 0.999099i \(0.513513\pi\)
\(30\) −2.45959e8 −1.84797
\(31\) −2.38250e7 −0.149466 −0.0747331 0.997204i \(-0.523810\pi\)
−0.0747331 + 0.997204i \(0.523810\pi\)
\(32\) −9.05679e8 −4.77144
\(33\) −5.84287e7 −0.259896
\(34\) −9.25614e8 −3.49378
\(35\) 2.62197e8 0.843827
\(36\) 3.44936e8 0.950767
\(37\) 3.44877e8 0.817627 0.408813 0.912618i \(-0.365943\pi\)
0.408813 + 0.912618i \(0.365943\pi\)
\(38\) 9.33424e8 1.91104
\(39\) −5.65220e8 −1.00314
\(40\) −3.83971e9 −5.92882
\(41\) −1.07900e9 −1.45449 −0.727245 0.686378i \(-0.759200\pi\)
−0.727245 + 0.686378i \(0.759200\pi\)
\(42\) −4.96624e8 −0.586350
\(43\) 9.52560e8 0.988134 0.494067 0.869424i \(-0.335510\pi\)
0.494067 + 0.869424i \(0.335510\pi\)
\(44\) −1.40458e9 −1.28397
\(45\) 6.72889e8 0.543594
\(46\) 3.50437e9 2.50866
\(47\) −2.34758e9 −1.49308 −0.746538 0.665342i \(-0.768286\pi\)
−0.746538 + 0.665342i \(0.768286\pi\)
\(48\) 4.36563e9 2.47298
\(49\) −1.44792e9 −0.732259
\(50\) −7.19713e9 −3.25705
\(51\) 2.53228e9 1.02772
\(52\) −1.35874e10 −4.95586
\(53\) −1.54271e9 −0.506718 −0.253359 0.967372i \(-0.581535\pi\)
−0.253359 + 0.967372i \(0.581535\pi\)
\(54\) −1.27451e9 −0.377727
\(55\) −2.74000e9 −0.734102
\(56\) −7.75289e9 −1.88118
\(57\) −2.55364e9 −0.562145
\(58\) 8.32732e8 0.166591
\(59\) −7.14924e8 −0.130189
\(60\) 1.61757e10 2.68554
\(61\) 5.48052e9 0.830822 0.415411 0.909634i \(-0.363638\pi\)
0.415411 + 0.909634i \(0.363638\pi\)
\(62\) 2.11620e9 0.293361
\(63\) 1.35865e9 0.172479
\(64\) 4.36516e10 5.08171
\(65\) −2.65059e10 −2.83347
\(66\) 5.18981e9 0.510105
\(67\) −3.63841e9 −0.329230 −0.164615 0.986358i \(-0.552638\pi\)
−0.164615 + 0.986358i \(0.552638\pi\)
\(68\) 6.08738e10 5.07728
\(69\) −9.58717e9 −0.737939
\(70\) −2.32891e10 −1.65620
\(71\) −4.43632e9 −0.291811 −0.145906 0.989299i \(-0.546610\pi\)
−0.145906 + 0.989299i \(0.546610\pi\)
\(72\) −1.98966e10 −1.21186
\(73\) −1.08626e10 −0.613277 −0.306638 0.951826i \(-0.599204\pi\)
−0.306638 + 0.951826i \(0.599204\pi\)
\(74\) −3.06330e10 −1.60478
\(75\) 1.96898e10 0.958084
\(76\) −6.13874e10 −2.77718
\(77\) −5.53244e9 −0.232926
\(78\) 5.02045e10 1.96890
\(79\) −1.30413e10 −0.476840 −0.238420 0.971162i \(-0.576629\pi\)
−0.238420 + 0.971162i \(0.576629\pi\)
\(80\) 2.04725e11 6.98517
\(81\) 3.48678e9 0.111111
\(82\) 9.58401e10 2.85477
\(83\) −2.78822e10 −0.776958 −0.388479 0.921458i \(-0.626999\pi\)
−0.388479 + 0.921458i \(0.626999\pi\)
\(84\) 3.26609e10 0.852103
\(85\) 1.18751e11 2.90290
\(86\) −8.46092e10 −1.93944
\(87\) −2.27817e9 −0.0490039
\(88\) 8.10190e10 1.63656
\(89\) 3.44766e10 0.654454 0.327227 0.944946i \(-0.393886\pi\)
0.327227 + 0.944946i \(0.393886\pi\)
\(90\) −5.97680e10 −1.06693
\(91\) −5.35190e10 −0.899043
\(92\) −2.30468e11 −3.64566
\(93\) −5.78946e9 −0.0862943
\(94\) 2.08519e11 2.93050
\(95\) −1.19752e11 −1.58783
\(96\) −2.20080e11 −2.75479
\(97\) 1.26705e11 1.49812 0.749062 0.662500i \(-0.230504\pi\)
0.749062 + 0.662500i \(0.230504\pi\)
\(98\) 1.28608e11 1.43723
\(99\) −1.41982e10 −0.150051
\(100\) 4.73325e11 4.73325
\(101\) −8.52892e10 −0.807471 −0.403735 0.914876i \(-0.632288\pi\)
−0.403735 + 0.914876i \(0.632288\pi\)
\(102\) −2.24924e11 −2.01714
\(103\) 7.54733e10 0.641488 0.320744 0.947166i \(-0.396067\pi\)
0.320744 + 0.947166i \(0.396067\pi\)
\(104\) 7.83752e11 6.31677
\(105\) 6.37138e10 0.487184
\(106\) 1.37028e11 0.994551
\(107\) −1.82905e11 −1.26071 −0.630355 0.776307i \(-0.717091\pi\)
−0.630355 + 0.776307i \(0.717091\pi\)
\(108\) 8.38194e10 0.548926
\(109\) −1.01699e11 −0.633098 −0.316549 0.948576i \(-0.602524\pi\)
−0.316549 + 0.948576i \(0.602524\pi\)
\(110\) 2.43375e11 1.44084
\(111\) 8.38052e10 0.472057
\(112\) 4.13369e11 2.21635
\(113\) −2.58972e11 −1.32228 −0.661138 0.750264i \(-0.729926\pi\)
−0.661138 + 0.750264i \(0.729926\pi\)
\(114\) 2.26822e11 1.10334
\(115\) −4.49588e11 −2.08438
\(116\) −5.47653e10 −0.242096
\(117\) −1.37349e11 −0.579164
\(118\) 6.35017e10 0.255525
\(119\) 2.39774e11 0.921071
\(120\) −9.33048e11 −3.42300
\(121\) −2.27497e11 −0.797362
\(122\) −4.86796e11 −1.63068
\(123\) −2.62197e11 −0.839751
\(124\) −1.39174e11 −0.426323
\(125\) 3.66930e11 1.07542
\(126\) −1.20680e11 −0.338529
\(127\) −5.59654e11 −1.50314 −0.751570 0.659654i \(-0.770703\pi\)
−0.751570 + 0.659654i \(0.770703\pi\)
\(128\) −2.02243e12 −5.20258
\(129\) 2.31472e11 0.570499
\(130\) 2.35433e12 5.56134
\(131\) 1.30344e11 0.295189 0.147594 0.989048i \(-0.452847\pi\)
0.147594 + 0.989048i \(0.452847\pi\)
\(132\) −3.41312e11 −0.741302
\(133\) −2.41797e11 −0.503809
\(134\) 3.23174e11 0.646189
\(135\) 1.63512e11 0.313844
\(136\) −3.51133e12 −6.47154
\(137\) −6.33227e10 −0.112098 −0.0560488 0.998428i \(-0.517850\pi\)
−0.0560488 + 0.998428i \(0.517850\pi\)
\(138\) 8.51561e11 1.44837
\(139\) −1.36820e11 −0.223650 −0.111825 0.993728i \(-0.535670\pi\)
−0.111825 + 0.993728i \(0.535670\pi\)
\(140\) 1.53163e12 2.40685
\(141\) −5.70462e11 −0.862028
\(142\) 3.94047e11 0.572746
\(143\) 5.59283e11 0.782138
\(144\) 1.06085e12 1.42778
\(145\) −1.06834e11 −0.138416
\(146\) 9.64845e11 1.20370
\(147\) −3.51844e11 −0.422770
\(148\) 2.01461e12 2.33212
\(149\) 1.29089e12 1.44001 0.720006 0.693968i \(-0.244139\pi\)
0.720006 + 0.693968i \(0.244139\pi\)
\(150\) −1.74890e12 −1.88046
\(151\) −1.71851e12 −1.78147 −0.890736 0.454522i \(-0.849810\pi\)
−0.890736 + 0.454522i \(0.849810\pi\)
\(152\) 3.54096e12 3.53982
\(153\) 6.15343e11 0.593355
\(154\) 4.91407e11 0.457170
\(155\) −2.71496e11 −0.243747
\(156\) −3.30174e12 −2.86126
\(157\) 1.17133e12 0.980011 0.490005 0.871719i \(-0.336995\pi\)
0.490005 + 0.871719i \(0.336995\pi\)
\(158\) 1.15837e12 0.935907
\(159\) −3.74878e11 −0.292554
\(160\) −1.03206e13 −7.78118
\(161\) −9.07780e11 −0.661361
\(162\) −3.09706e11 −0.218081
\(163\) 1.09485e11 0.0745289 0.0372644 0.999305i \(-0.488136\pi\)
0.0372644 + 0.999305i \(0.488136\pi\)
\(164\) −6.30301e12 −4.14865
\(165\) −6.65821e11 −0.423834
\(166\) 2.47658e12 1.52496
\(167\) −1.86167e12 −1.10908 −0.554540 0.832157i \(-0.687106\pi\)
−0.554540 + 0.832157i \(0.687106\pi\)
\(168\) −1.88395e12 −1.08610
\(169\) 3.61816e12 2.01888
\(170\) −1.05478e13 −5.69760
\(171\) −6.20535e11 −0.324554
\(172\) 5.56439e12 2.81846
\(173\) −3.00980e11 −0.147667 −0.0738336 0.997271i \(-0.523523\pi\)
−0.0738336 + 0.997271i \(0.523523\pi\)
\(174\) 2.02354e11 0.0961813
\(175\) 1.86436e12 0.858660
\(176\) −4.31977e12 −1.92815
\(177\) −1.73727e11 −0.0751646
\(178\) −3.06231e12 −1.28451
\(179\) −1.70328e12 −0.692780 −0.346390 0.938091i \(-0.612593\pi\)
−0.346390 + 0.938091i \(0.612593\pi\)
\(180\) 3.93069e12 1.55049
\(181\) 2.80974e12 1.07506 0.537531 0.843244i \(-0.319357\pi\)
0.537531 + 0.843244i \(0.319357\pi\)
\(182\) 4.75372e12 1.76458
\(183\) 1.33177e12 0.479675
\(184\) 1.32939e13 4.64679
\(185\) 3.93003e12 1.33337
\(186\) 5.14237e11 0.169372
\(187\) −2.50568e12 −0.801302
\(188\) −1.37134e13 −4.25871
\(189\) 3.30153e11 0.0995808
\(190\) 1.06368e13 3.11648
\(191\) 5.15965e12 1.46871 0.734356 0.678764i \(-0.237484\pi\)
0.734356 + 0.678764i \(0.237484\pi\)
\(192\) 1.06073e13 2.93393
\(193\) −1.39897e11 −0.0376049 −0.0188024 0.999823i \(-0.505985\pi\)
−0.0188024 + 0.999823i \(0.505985\pi\)
\(194\) −1.12543e13 −2.94041
\(195\) −6.44093e12 −1.63591
\(196\) −8.45802e12 −2.08862
\(197\) −5.23182e12 −1.25629 −0.628143 0.778098i \(-0.716185\pi\)
−0.628143 + 0.778098i \(0.716185\pi\)
\(198\) 1.26112e12 0.294509
\(199\) 3.51930e12 0.799399 0.399700 0.916646i \(-0.369114\pi\)
0.399700 + 0.916646i \(0.369114\pi\)
\(200\) −2.73024e13 −6.03304
\(201\) −8.84132e11 −0.190081
\(202\) 7.57564e12 1.58485
\(203\) −2.15713e11 −0.0439186
\(204\) 1.47923e13 2.93137
\(205\) −1.22957e13 −2.37196
\(206\) −6.70376e12 −1.25907
\(207\) −2.32968e12 −0.426049
\(208\) −4.17881e13 −7.44225
\(209\) 2.52682e12 0.438298
\(210\) −5.65925e12 −0.956209
\(211\) 6.22111e12 1.02403 0.512017 0.858975i \(-0.328898\pi\)
0.512017 + 0.858975i \(0.328898\pi\)
\(212\) −9.01176e12 −1.44531
\(213\) −1.07803e12 −0.168477
\(214\) 1.62462e13 2.47443
\(215\) 1.08548e13 1.61143
\(216\) −4.83488e12 −0.699665
\(217\) −5.48187e11 −0.0773393
\(218\) 9.03321e12 1.24260
\(219\) −2.63960e12 −0.354076
\(220\) −1.60058e13 −2.09388
\(221\) −2.42391e13 −3.09285
\(222\) −7.44382e12 −0.926520
\(223\) 8.46778e12 1.02824 0.514118 0.857720i \(-0.328119\pi\)
0.514118 + 0.857720i \(0.328119\pi\)
\(224\) −2.08387e13 −2.46892
\(225\) 4.78461e12 0.553150
\(226\) 2.30027e13 2.59527
\(227\) −7.90846e12 −0.870863 −0.435432 0.900222i \(-0.643404\pi\)
−0.435432 + 0.900222i \(0.643404\pi\)
\(228\) −1.49171e13 −1.60341
\(229\) −1.50654e13 −1.58083 −0.790413 0.612574i \(-0.790134\pi\)
−0.790413 + 0.612574i \(0.790134\pi\)
\(230\) 3.99338e13 4.09107
\(231\) −1.34438e12 −0.134480
\(232\) 3.15898e12 0.308577
\(233\) 6.84441e12 0.652948 0.326474 0.945206i \(-0.394139\pi\)
0.326474 + 0.945206i \(0.394139\pi\)
\(234\) 1.21997e13 1.13674
\(235\) −2.67517e13 −2.43488
\(236\) −4.17624e12 −0.371338
\(237\) −3.16904e12 −0.275304
\(238\) −2.12974e13 −1.80781
\(239\) −1.26859e13 −1.05229 −0.526143 0.850396i \(-0.676362\pi\)
−0.526143 + 0.850396i \(0.676362\pi\)
\(240\) 4.97483e13 4.03289
\(241\) 1.47735e13 1.17055 0.585273 0.810836i \(-0.300987\pi\)
0.585273 + 0.810836i \(0.300987\pi\)
\(242\) 2.02069e13 1.56501
\(243\) 8.47289e11 0.0641500
\(244\) 3.20146e13 2.36976
\(245\) −1.64996e13 −1.19416
\(246\) 2.32891e13 1.64820
\(247\) 2.44436e13 1.69173
\(248\) 8.02785e12 0.543394
\(249\) −6.77538e12 −0.448577
\(250\) −3.25918e13 −2.11076
\(251\) 9.62805e11 0.0610004 0.0305002 0.999535i \(-0.490290\pi\)
0.0305002 + 0.999535i \(0.490290\pi\)
\(252\) 7.93660e12 0.491962
\(253\) 9.48646e12 0.575362
\(254\) 4.97101e13 2.95025
\(255\) 2.88564e13 1.67599
\(256\) 9.02398e13 5.12954
\(257\) −6.84840e12 −0.381028 −0.190514 0.981684i \(-0.561015\pi\)
−0.190514 + 0.981684i \(0.561015\pi\)
\(258\) −2.05600e13 −1.11973
\(259\) 7.93526e12 0.423070
\(260\) −1.54835e14 −8.08192
\(261\) −5.53596e11 −0.0282924
\(262\) −1.15776e13 −0.579376
\(263\) −9.94610e12 −0.487412 −0.243706 0.969849i \(-0.578363\pi\)
−0.243706 + 0.969849i \(0.578363\pi\)
\(264\) 1.96876e13 0.944869
\(265\) −1.75798e13 −0.826347
\(266\) 2.14771e13 0.988841
\(267\) 8.37780e12 0.377849
\(268\) −2.12538e13 −0.939064
\(269\) 2.96879e13 1.28511 0.642557 0.766238i \(-0.277874\pi\)
0.642557 + 0.766238i \(0.277874\pi\)
\(270\) −1.45236e13 −0.615991
\(271\) 1.68982e13 0.702277 0.351139 0.936323i \(-0.385795\pi\)
0.351139 + 0.936323i \(0.385795\pi\)
\(272\) 1.87217e14 7.62460
\(273\) −1.30051e13 −0.519063
\(274\) 5.62451e12 0.220017
\(275\) −1.94829e13 −0.747007
\(276\) −5.60036e13 −2.10483
\(277\) 2.76547e13 1.01890 0.509449 0.860501i \(-0.329849\pi\)
0.509449 + 0.860501i \(0.329849\pi\)
\(278\) 1.21528e13 0.438964
\(279\) −1.40684e12 −0.0498220
\(280\) −8.83475e13 −3.06779
\(281\) −6.84594e12 −0.233103 −0.116552 0.993185i \(-0.537184\pi\)
−0.116552 + 0.993185i \(0.537184\pi\)
\(282\) 5.06701e13 1.69193
\(283\) 2.56673e13 0.840533 0.420267 0.907401i \(-0.361937\pi\)
0.420267 + 0.907401i \(0.361937\pi\)
\(284\) −2.59148e13 −0.832333
\(285\) −2.90999e13 −0.916736
\(286\) −4.96772e13 −1.53512
\(287\) −2.48267e13 −0.752607
\(288\) −5.34794e13 −1.59048
\(289\) 7.43231e13 2.16863
\(290\) 9.48935e12 0.271674
\(291\) 3.07892e13 0.864943
\(292\) −6.34538e13 −1.74925
\(293\) −6.78053e13 −1.83439 −0.917195 0.398439i \(-0.869552\pi\)
−0.917195 + 0.398439i \(0.869552\pi\)
\(294\) 3.12518e13 0.829783
\(295\) −8.14687e12 −0.212310
\(296\) −1.16207e14 −2.97254
\(297\) −3.45016e12 −0.0866320
\(298\) −1.14661e14 −2.82635
\(299\) 9.17689e13 2.22077
\(300\) 1.15018e14 2.73274
\(301\) 2.19174e13 0.511297
\(302\) 1.52643e14 3.49654
\(303\) −2.07253e13 −0.466193
\(304\) −1.88797e14 −4.17052
\(305\) 6.24530e13 1.35489
\(306\) −5.46566e13 −1.16459
\(307\) 4.73650e13 0.991280 0.495640 0.868528i \(-0.334933\pi\)
0.495640 + 0.868528i \(0.334933\pi\)
\(308\) −3.23178e13 −0.664375
\(309\) 1.83400e13 0.370363
\(310\) 2.41151e13 0.478408
\(311\) 8.09067e13 1.57689 0.788447 0.615103i \(-0.210885\pi\)
0.788447 + 0.615103i \(0.210885\pi\)
\(312\) 1.90452e14 3.64699
\(313\) −7.41330e13 −1.39482 −0.697410 0.716673i \(-0.745664\pi\)
−0.697410 + 0.716673i \(0.745664\pi\)
\(314\) −1.04041e14 −1.92350
\(315\) 1.54825e13 0.281276
\(316\) −7.61811e13 −1.36009
\(317\) −8.33862e13 −1.46308 −0.731540 0.681799i \(-0.761198\pi\)
−0.731540 + 0.681799i \(0.761198\pi\)
\(318\) 3.32978e13 0.574204
\(319\) 2.25424e12 0.0382078
\(320\) 4.97429e14 8.28716
\(321\) −4.44459e13 −0.727871
\(322\) 8.06317e13 1.29807
\(323\) −1.09511e14 −1.73318
\(324\) 2.03681e13 0.316922
\(325\) −1.88472e14 −2.88328
\(326\) −9.72482e12 −0.146280
\(327\) −2.47129e13 −0.365520
\(328\) 3.63571e14 5.28790
\(329\) −5.40153e13 −0.772573
\(330\) 5.91401e13 0.831870
\(331\) −1.37867e14 −1.90725 −0.953624 0.301000i \(-0.902680\pi\)
−0.953624 + 0.301000i \(0.902680\pi\)
\(332\) −1.62874e14 −2.21612
\(333\) 2.03647e13 0.272542
\(334\) 1.65359e14 2.17682
\(335\) −4.14612e13 −0.536903
\(336\) 1.00449e14 1.27961
\(337\) 8.41270e13 1.05432 0.527158 0.849767i \(-0.323258\pi\)
0.527158 + 0.849767i \(0.323258\pi\)
\(338\) −3.21376e14 −3.96252
\(339\) −6.29303e13 −0.763416
\(340\) 6.93683e14 8.27994
\(341\) 5.72865e12 0.0672826
\(342\) 5.51177e13 0.637012
\(343\) −7.88112e13 −0.896335
\(344\) −3.20966e14 −3.59242
\(345\) −1.09250e14 −1.20342
\(346\) 2.67339e13 0.289831
\(347\) −1.54945e13 −0.165335 −0.0826676 0.996577i \(-0.526344\pi\)
−0.0826676 + 0.996577i \(0.526344\pi\)
\(348\) −1.33080e13 −0.139774
\(349\) 1.51619e14 1.56752 0.783762 0.621061i \(-0.213298\pi\)
0.783762 + 0.621061i \(0.213298\pi\)
\(350\) −1.65598e14 −1.68532
\(351\) −3.33757e13 −0.334381
\(352\) 2.17768e14 2.14788
\(353\) 1.58801e14 1.54203 0.771014 0.636818i \(-0.219750\pi\)
0.771014 + 0.636818i \(0.219750\pi\)
\(354\) 1.54309e13 0.147528
\(355\) −5.05538e13 −0.475880
\(356\) 2.01395e14 1.86670
\(357\) 5.82650e13 0.531781
\(358\) 1.51291e14 1.35974
\(359\) −1.05114e14 −0.930341 −0.465171 0.885221i \(-0.654007\pi\)
−0.465171 + 0.885221i \(0.654007\pi\)
\(360\) −2.26731e14 −1.97627
\(361\) −6.05513e12 −0.0519798
\(362\) −2.49569e14 −2.11005
\(363\) −5.52817e13 −0.460357
\(364\) −3.12632e14 −2.56434
\(365\) −1.23784e14 −1.00012
\(366\) −1.18292e14 −0.941472
\(367\) −1.98072e14 −1.55295 −0.776477 0.630145i \(-0.782995\pi\)
−0.776477 + 0.630145i \(0.782995\pi\)
\(368\) −7.08802e14 −5.47472
\(369\) −6.37140e13 −0.484830
\(370\) −3.49077e14 −2.61705
\(371\) −3.54961e13 −0.262195
\(372\) −3.38192e13 −0.246137
\(373\) −1.78362e14 −1.27910 −0.639548 0.768751i \(-0.720879\pi\)
−0.639548 + 0.768751i \(0.720879\pi\)
\(374\) 2.22561e14 1.57274
\(375\) 8.91639e13 0.620894
\(376\) 7.91020e14 5.42818
\(377\) 2.18068e13 0.147474
\(378\) −2.93252e13 −0.195450
\(379\) −1.86888e14 −1.22762 −0.613811 0.789453i \(-0.710364\pi\)
−0.613811 + 0.789453i \(0.710364\pi\)
\(380\) −6.99536e14 −4.52898
\(381\) −1.35996e14 −0.867838
\(382\) −4.58295e14 −2.88268
\(383\) −1.15566e14 −0.716533 −0.358266 0.933619i \(-0.616632\pi\)
−0.358266 + 0.933619i \(0.616632\pi\)
\(384\) −4.91451e14 −3.00371
\(385\) −6.30445e13 −0.379851
\(386\) 1.24261e13 0.0738082
\(387\) 5.62477e13 0.329378
\(388\) 7.40147e14 4.27310
\(389\) 5.04611e13 0.287233 0.143616 0.989633i \(-0.454127\pi\)
0.143616 + 0.989633i \(0.454127\pi\)
\(390\) 5.72103e14 3.21084
\(391\) −4.11139e14 −2.27519
\(392\) 4.87877e14 2.66218
\(393\) 3.16737e13 0.170427
\(394\) 4.64706e14 2.46575
\(395\) −1.48612e14 −0.777622
\(396\) −8.29389e13 −0.427991
\(397\) −2.93044e14 −1.49137 −0.745685 0.666299i \(-0.767877\pi\)
−0.745685 + 0.666299i \(0.767877\pi\)
\(398\) −3.12594e14 −1.56900
\(399\) −5.87566e13 −0.290874
\(400\) 1.45571e15 7.10796
\(401\) −2.54837e14 −1.22735 −0.613676 0.789558i \(-0.710310\pi\)
−0.613676 + 0.789558i \(0.710310\pi\)
\(402\) 7.85312e13 0.373078
\(403\) 5.54171e13 0.259696
\(404\) −4.98218e14 −2.30315
\(405\) 3.97334e13 0.181198
\(406\) 1.91603e13 0.0862003
\(407\) −8.29248e13 −0.368057
\(408\) −8.53254e14 −3.73635
\(409\) 3.55207e14 1.53463 0.767314 0.641271i \(-0.221593\pi\)
0.767314 + 0.641271i \(0.221593\pi\)
\(410\) 1.09214e15 4.65551
\(411\) −1.53874e13 −0.0647196
\(412\) 4.40879e14 1.82972
\(413\) −1.64496e13 −0.0673645
\(414\) 2.06929e14 0.836219
\(415\) −3.17730e14 −1.26705
\(416\) 2.10662e15 8.29034
\(417\) −3.32473e13 −0.129124
\(418\) −2.24439e14 −0.860259
\(419\) 1.32237e14 0.500236 0.250118 0.968215i \(-0.419531\pi\)
0.250118 + 0.968215i \(0.419531\pi\)
\(420\) 3.72185e14 1.38959
\(421\) −3.26349e14 −1.20263 −0.601314 0.799013i \(-0.705356\pi\)
−0.601314 + 0.799013i \(0.705356\pi\)
\(422\) −5.52578e14 −2.00990
\(423\) −1.38622e14 −0.497692
\(424\) 5.19818e14 1.84221
\(425\) 8.44382e14 2.95393
\(426\) 9.57534e13 0.330675
\(427\) 1.26101e14 0.429898
\(428\) −1.06844e15 −3.59593
\(429\) 1.35906e14 0.451568
\(430\) −9.64158e14 −3.16280
\(431\) 3.23418e14 1.04747 0.523733 0.851883i \(-0.324539\pi\)
0.523733 + 0.851883i \(0.324539\pi\)
\(432\) 2.57786e14 0.824326
\(433\) 2.43979e14 0.770316 0.385158 0.922851i \(-0.374147\pi\)
0.385158 + 0.922851i \(0.374147\pi\)
\(434\) 4.86916e13 0.151796
\(435\) −2.59608e13 −0.0799147
\(436\) −5.94077e14 −1.80579
\(437\) 4.14608e14 1.24449
\(438\) 2.34457e14 0.694954
\(439\) −5.42131e14 −1.58690 −0.793449 0.608637i \(-0.791717\pi\)
−0.793449 + 0.608637i \(0.791717\pi\)
\(440\) 9.23247e14 2.66887
\(441\) −8.54980e13 −0.244086
\(442\) 2.15299e15 6.07042
\(443\) 1.28812e14 0.358703 0.179352 0.983785i \(-0.442600\pi\)
0.179352 + 0.983785i \(0.442600\pi\)
\(444\) 4.89549e14 1.34645
\(445\) 3.92875e14 1.06727
\(446\) −7.52133e14 −2.01815
\(447\) 3.13687e14 0.831391
\(448\) 1.00438e15 2.62946
\(449\) −6.51432e13 −0.168467 −0.0842334 0.996446i \(-0.526844\pi\)
−0.0842334 + 0.996446i \(0.526844\pi\)
\(450\) −4.24983e14 −1.08568
\(451\) 2.59443e14 0.654744
\(452\) −1.51279e15 −3.77153
\(453\) −4.17598e14 −1.02853
\(454\) 7.02453e14 1.70927
\(455\) −6.09872e14 −1.46614
\(456\) 8.60452e14 2.04371
\(457\) −4.29850e14 −1.00874 −0.504369 0.863488i \(-0.668275\pi\)
−0.504369 + 0.863488i \(0.668275\pi\)
\(458\) 1.33815e15 3.10273
\(459\) 1.49528e14 0.342574
\(460\) −2.62628e15 −5.94528
\(461\) 1.02578e14 0.229456 0.114728 0.993397i \(-0.463400\pi\)
0.114728 + 0.993397i \(0.463400\pi\)
\(462\) 1.19412e14 0.263947
\(463\) 1.89958e14 0.414919 0.207459 0.978244i \(-0.433481\pi\)
0.207459 + 0.978244i \(0.433481\pi\)
\(464\) −1.68431e14 −0.363557
\(465\) −6.59735e13 −0.140727
\(466\) −6.07941e14 −1.28156
\(467\) −7.59612e14 −1.58252 −0.791260 0.611480i \(-0.790575\pi\)
−0.791260 + 0.611480i \(0.790575\pi\)
\(468\) −8.02324e14 −1.65195
\(469\) −8.37158e13 −0.170356
\(470\) 2.37616e15 4.77901
\(471\) 2.84633e14 0.565810
\(472\) 2.40895e14 0.473310
\(473\) −2.29040e14 −0.444811
\(474\) 2.81484e14 0.540346
\(475\) −8.51506e14 −1.61575
\(476\) 1.40064e15 2.62717
\(477\) −9.10954e13 −0.168906
\(478\) 1.12680e15 2.06535
\(479\) 3.77438e14 0.683912 0.341956 0.939716i \(-0.388911\pi\)
0.341956 + 0.939716i \(0.388911\pi\)
\(480\) −2.50791e15 −4.49246
\(481\) −8.02188e14 −1.42062
\(482\) −1.31222e15 −2.29746
\(483\) −2.20591e14 −0.381837
\(484\) −1.32893e15 −2.27432
\(485\) 1.44385e15 2.44311
\(486\) −7.52587e13 −0.125909
\(487\) 6.56935e14 1.08671 0.543354 0.839503i \(-0.317154\pi\)
0.543354 + 0.839503i \(0.317154\pi\)
\(488\) −1.84667e15 −3.02051
\(489\) 2.66050e13 0.0430293
\(490\) 1.46555e15 2.34380
\(491\) −4.51043e14 −0.713297 −0.356648 0.934239i \(-0.616081\pi\)
−0.356648 + 0.934239i \(0.616081\pi\)
\(492\) −1.53163e15 −2.39522
\(493\) −9.76979e13 −0.151087
\(494\) −2.17115e15 −3.32041
\(495\) −1.61794e14 −0.244701
\(496\) −4.28029e14 −0.640212
\(497\) −1.02075e14 −0.150994
\(498\) 6.01809e14 0.880435
\(499\) 7.01153e13 0.101452 0.0507259 0.998713i \(-0.483847\pi\)
0.0507259 + 0.998713i \(0.483847\pi\)
\(500\) 2.14343e15 3.06742
\(501\) −4.52387e14 −0.640328
\(502\) −8.55192e13 −0.119727
\(503\) −1.06161e15 −1.47009 −0.735043 0.678021i \(-0.762838\pi\)
−0.735043 + 0.678021i \(0.762838\pi\)
\(504\) −4.57800e14 −0.627058
\(505\) −9.71908e14 −1.31681
\(506\) −8.42615e14 −1.12928
\(507\) 8.79213e14 1.16560
\(508\) −3.26923e15 −4.28741
\(509\) 7.42466e14 0.963227 0.481614 0.876384i \(-0.340051\pi\)
0.481614 + 0.876384i \(0.340051\pi\)
\(510\) −2.56311e15 −3.28951
\(511\) −2.49936e14 −0.317332
\(512\) −3.87343e15 −4.86531
\(513\) −1.50790e14 −0.187382
\(514\) 6.08295e14 0.747854
\(515\) 8.60051e14 1.04613
\(516\) 1.35215e15 1.62724
\(517\) 5.64469e14 0.672113
\(518\) −7.04833e14 −0.830372
\(519\) −7.31381e13 −0.0852557
\(520\) 8.93119e15 10.3013
\(521\) 1.13056e15 1.29029 0.645145 0.764061i \(-0.276797\pi\)
0.645145 + 0.764061i \(0.276797\pi\)
\(522\) 4.91720e13 0.0555303
\(523\) 6.85320e14 0.765834 0.382917 0.923783i \(-0.374920\pi\)
0.382917 + 0.923783i \(0.374920\pi\)
\(524\) 7.61408e14 0.841968
\(525\) 4.53040e14 0.495748
\(526\) 8.83442e14 0.956658
\(527\) −2.48277e14 −0.266059
\(528\) −1.04970e15 −1.11322
\(529\) 6.03759e14 0.633662
\(530\) 1.56149e15 1.62190
\(531\) −4.22156e13 −0.0433963
\(532\) −1.41246e15 −1.43702
\(533\) 2.50977e15 2.52717
\(534\) −7.44141e14 −0.741615
\(535\) −2.08428e15 −2.05594
\(536\) 1.22596e15 1.19694
\(537\) −4.13898e14 −0.399977
\(538\) −2.63696e15 −2.52233
\(539\) 3.48147e14 0.329629
\(540\) 9.55158e14 0.895178
\(541\) −1.17281e15 −1.08803 −0.544017 0.839074i \(-0.683097\pi\)
−0.544017 + 0.839074i \(0.683097\pi\)
\(542\) −1.50095e15 −1.37838
\(543\) 6.82766e14 0.620687
\(544\) −9.43798e15 −8.49347
\(545\) −1.15891e15 −1.03245
\(546\) 1.15515e15 1.01878
\(547\) 1.75059e15 1.52846 0.764228 0.644946i \(-0.223120\pi\)
0.764228 + 0.644946i \(0.223120\pi\)
\(548\) −3.69900e14 −0.319736
\(549\) 3.23620e14 0.276941
\(550\) 1.73053e15 1.46617
\(551\) 9.85222e13 0.0826419
\(552\) 3.23041e15 2.68283
\(553\) −3.00067e14 −0.246735
\(554\) −2.45638e15 −1.99982
\(555\) 9.54997e14 0.769822
\(556\) −7.99237e14 −0.637917
\(557\) 5.14429e14 0.406557 0.203279 0.979121i \(-0.434840\pi\)
0.203279 + 0.979121i \(0.434840\pi\)
\(558\) 1.24960e14 0.0977871
\(559\) −2.21566e15 −1.71688
\(560\) 4.71052e15 3.61438
\(561\) −6.08879e14 −0.462632
\(562\) 6.08077e14 0.457518
\(563\) −5.38082e14 −0.400915 −0.200458 0.979702i \(-0.564243\pi\)
−0.200458 + 0.979702i \(0.564243\pi\)
\(564\) −3.33236e15 −2.45877
\(565\) −2.95110e15 −2.15634
\(566\) −2.27985e15 −1.64974
\(567\) 8.02272e13 0.0574930
\(568\) 1.49482e15 1.06090
\(569\) −2.65655e14 −0.186724 −0.0933620 0.995632i \(-0.529761\pi\)
−0.0933620 + 0.995632i \(0.529761\pi\)
\(570\) 2.58473e15 1.79930
\(571\) −1.72754e15 −1.19105 −0.595525 0.803337i \(-0.703056\pi\)
−0.595525 + 0.803337i \(0.703056\pi\)
\(572\) 3.26706e15 2.23089
\(573\) 1.25380e15 0.847962
\(574\) 2.20518e15 1.47716
\(575\) −3.19682e15 −2.12102
\(576\) 2.57758e15 1.69390
\(577\) −6.32164e13 −0.0411494 −0.0205747 0.999788i \(-0.506550\pi\)
−0.0205747 + 0.999788i \(0.506550\pi\)
\(578\) −6.60160e15 −4.25643
\(579\) −3.39951e13 −0.0217112
\(580\) −6.24075e14 −0.394805
\(581\) −6.41540e14 −0.402027
\(582\) −2.73479e15 −1.69765
\(583\) 3.70940e14 0.228101
\(584\) 3.66015e15 2.22961
\(585\) −1.56515e15 −0.944491
\(586\) 6.02266e15 3.60041
\(587\) 5.95482e14 0.352663 0.176331 0.984331i \(-0.443577\pi\)
0.176331 + 0.984331i \(0.443577\pi\)
\(588\) −2.05530e15 −1.20587
\(589\) 2.50372e14 0.145530
\(590\) 7.23629e14 0.416706
\(591\) −1.27133e15 −0.725317
\(592\) 6.19592e15 3.50216
\(593\) 1.66761e15 0.933885 0.466943 0.884288i \(-0.345355\pi\)
0.466943 + 0.884288i \(0.345355\pi\)
\(594\) 3.06453e14 0.170035
\(595\) 2.73232e15 1.50207
\(596\) 7.54078e15 4.10735
\(597\) 8.55189e14 0.461533
\(598\) −8.15119e15 −4.35877
\(599\) −2.02609e15 −1.07352 −0.536761 0.843734i \(-0.680353\pi\)
−0.536761 + 0.843734i \(0.680353\pi\)
\(600\) −6.63449e15 −3.48318
\(601\) −1.52237e15 −0.791975 −0.395988 0.918256i \(-0.629598\pi\)
−0.395988 + 0.918256i \(0.629598\pi\)
\(602\) −1.94677e15 −1.00354
\(603\) −2.14844e14 −0.109743
\(604\) −1.00387e16 −5.08129
\(605\) −2.59242e15 −1.30032
\(606\) 1.84088e15 0.915011
\(607\) 2.77705e15 1.36788 0.683938 0.729540i \(-0.260266\pi\)
0.683938 + 0.729540i \(0.260266\pi\)
\(608\) 9.51761e15 4.64577
\(609\) −5.24183e13 −0.0253564
\(610\) −5.54726e15 −2.65928
\(611\) 5.46049e15 2.59421
\(612\) 3.59454e15 1.69243
\(613\) −4.11696e15 −1.92107 −0.960537 0.278153i \(-0.910278\pi\)
−0.960537 + 0.278153i \(0.910278\pi\)
\(614\) −4.20710e15 −1.94561
\(615\) −2.98785e15 −1.36945
\(616\) 1.86416e15 0.846817
\(617\) 8.99302e14 0.404890 0.202445 0.979294i \(-0.435111\pi\)
0.202445 + 0.979294i \(0.435111\pi\)
\(618\) −1.62901e15 −0.726923
\(619\) 3.53814e15 1.56486 0.782431 0.622737i \(-0.213980\pi\)
0.782431 + 0.622737i \(0.213980\pi\)
\(620\) −1.58595e15 −0.695239
\(621\) −5.66113e14 −0.245980
\(622\) −7.18637e15 −3.09501
\(623\) 7.93269e14 0.338638
\(624\) −1.01545e16 −4.29678
\(625\) 2.24887e14 0.0943244
\(626\) 6.58471e15 2.73765
\(627\) 6.14016e14 0.253051
\(628\) 6.84234e15 2.79529
\(629\) 3.59393e15 1.45543
\(630\) −1.37520e15 −0.552067
\(631\) 1.08963e15 0.433630 0.216815 0.976213i \(-0.430433\pi\)
0.216815 + 0.976213i \(0.430433\pi\)
\(632\) 4.39429e15 1.73358
\(633\) 1.51173e15 0.591227
\(634\) 7.40660e15 2.87163
\(635\) −6.37750e15 −2.45129
\(636\) −2.18986e15 −0.834453
\(637\) 3.36787e15 1.27230
\(638\) −2.00228e14 −0.0749914
\(639\) −2.61960e14 −0.0972704
\(640\) −2.30465e16 −8.48427
\(641\) 2.66708e15 0.973457 0.486728 0.873553i \(-0.338190\pi\)
0.486728 + 0.873553i \(0.338190\pi\)
\(642\) 3.94782e15 1.42861
\(643\) 1.95524e14 0.0701518 0.0350759 0.999385i \(-0.488833\pi\)
0.0350759 + 0.999385i \(0.488833\pi\)
\(644\) −5.30281e15 −1.88640
\(645\) 2.63772e15 0.930360
\(646\) 9.72711e15 3.40177
\(647\) 3.70613e14 0.128513 0.0642565 0.997933i \(-0.479532\pi\)
0.0642565 + 0.997933i \(0.479532\pi\)
\(648\) −1.17488e15 −0.403952
\(649\) 1.71902e14 0.0586049
\(650\) 1.67406e16 5.65910
\(651\) −1.33209e14 −0.0446518
\(652\) 6.39561e14 0.212579
\(653\) 5.59755e14 0.184491 0.0922456 0.995736i \(-0.470596\pi\)
0.0922456 + 0.995736i \(0.470596\pi\)
\(654\) 2.19507e15 0.717416
\(655\) 1.48533e15 0.481389
\(656\) −1.93849e16 −6.23006
\(657\) −6.41424e14 −0.204426
\(658\) 4.79780e15 1.51635
\(659\) 2.30893e15 0.723670 0.361835 0.932242i \(-0.382150\pi\)
0.361835 + 0.932242i \(0.382150\pi\)
\(660\) −3.88940e15 −1.20890
\(661\) 1.89110e15 0.582916 0.291458 0.956584i \(-0.405860\pi\)
0.291458 + 0.956584i \(0.405860\pi\)
\(662\) 1.22458e16 3.74341
\(663\) −5.89010e15 −1.78566
\(664\) 9.39494e15 2.82468
\(665\) −2.75538e15 −0.821603
\(666\) −1.80885e15 −0.534926
\(667\) 3.69883e14 0.108486
\(668\) −1.08750e16 −3.16343
\(669\) 2.05767e15 0.593652
\(670\) 3.68271e15 1.05379
\(671\) −1.31778e15 −0.373997
\(672\) −5.06380e15 −1.42543
\(673\) 2.18803e15 0.610900 0.305450 0.952208i \(-0.401193\pi\)
0.305450 + 0.952208i \(0.401193\pi\)
\(674\) −7.47241e15 −2.06934
\(675\) 1.16266e15 0.319361
\(676\) 2.11355e16 5.75846
\(677\) −4.32264e15 −1.16819 −0.584093 0.811687i \(-0.698550\pi\)
−0.584093 + 0.811687i \(0.698550\pi\)
\(678\) 5.58965e15 1.49838
\(679\) 2.91534e15 0.775185
\(680\) −4.00132e16 −10.5537
\(681\) −1.92176e15 −0.502793
\(682\) −5.08835e14 −0.132058
\(683\) 2.53936e15 0.653747 0.326873 0.945068i \(-0.394005\pi\)
0.326873 + 0.945068i \(0.394005\pi\)
\(684\) −3.62486e15 −0.925727
\(685\) −7.21590e14 −0.182807
\(686\) 7.00024e15 1.75926
\(687\) −3.66088e15 −0.912691
\(688\) 1.71133e16 4.23250
\(689\) 3.58836e15 0.880420
\(690\) 9.70391e15 2.36198
\(691\) −2.88128e15 −0.695756 −0.347878 0.937540i \(-0.613098\pi\)
−0.347878 + 0.937540i \(0.613098\pi\)
\(692\) −1.75818e15 −0.421191
\(693\) −3.26685e14 −0.0776419
\(694\) 1.37627e15 0.324508
\(695\) −1.55913e15 −0.364724
\(696\) 7.67633e14 0.178157
\(697\) −1.12442e16 −2.58909
\(698\) −1.34673e16 −3.07662
\(699\) 1.66319e15 0.376980
\(700\) 1.08907e16 2.44916
\(701\) −2.44431e15 −0.545389 −0.272695 0.962101i \(-0.587915\pi\)
−0.272695 + 0.962101i \(0.587915\pi\)
\(702\) 2.96453e15 0.656298
\(703\) −3.62425e15 −0.796093
\(704\) −1.04959e16 −2.28755
\(705\) −6.50066e15 −1.40578
\(706\) −1.41052e16 −3.02658
\(707\) −1.96241e15 −0.417815
\(708\) −1.01483e15 −0.214392
\(709\) 7.69783e15 1.61367 0.806834 0.590779i \(-0.201179\pi\)
0.806834 + 0.590779i \(0.201179\pi\)
\(710\) 4.49034e15 0.934024
\(711\) −7.70077e14 −0.158947
\(712\) −1.16169e16 −2.37931
\(713\) 9.39975e14 0.191040
\(714\) −5.17527e15 −1.04374
\(715\) 6.37327e15 1.27550
\(716\) −9.94976e15 −1.97602
\(717\) −3.08268e15 −0.607538
\(718\) 9.33656e15 1.82601
\(719\) 5.87465e15 1.14018 0.570090 0.821583i \(-0.306908\pi\)
0.570090 + 0.821583i \(0.306908\pi\)
\(720\) 1.20888e16 2.32839
\(721\) 1.73656e15 0.331930
\(722\) 5.37835e14 0.102022
\(723\) 3.58995e15 0.675815
\(724\) 1.64131e16 3.06640
\(725\) −7.59652e14 −0.140850
\(726\) 4.91028e15 0.903556
\(727\) −1.37032e15 −0.250255 −0.125128 0.992141i \(-0.539934\pi\)
−0.125128 + 0.992141i \(0.539934\pi\)
\(728\) 1.80333e16 3.26853
\(729\) 2.05891e14 0.0370370
\(730\) 1.09948e16 1.96297
\(731\) 9.92652e15 1.75894
\(732\) 7.77954e15 1.36818
\(733\) −2.97135e15 −0.518659 −0.259330 0.965789i \(-0.583502\pi\)
−0.259330 + 0.965789i \(0.583502\pi\)
\(734\) 1.75933e16 3.04803
\(735\) −4.00941e15 −0.689446
\(736\) 3.57321e16 6.09860
\(737\) 8.74845e14 0.148204
\(738\) 5.65926e15 0.951590
\(739\) 4.90213e14 0.0818164 0.0409082 0.999163i \(-0.486975\pi\)
0.0409082 + 0.999163i \(0.486975\pi\)
\(740\) 2.29573e16 3.80318
\(741\) 5.93979e15 0.976723
\(742\) 3.15287e15 0.514617
\(743\) −1.22261e16 −1.98084 −0.990419 0.138094i \(-0.955902\pi\)
−0.990419 + 0.138094i \(0.955902\pi\)
\(744\) 1.95077e15 0.313729
\(745\) 1.47103e16 2.34835
\(746\) 1.58426e16 2.51052
\(747\) −1.64642e15 −0.258986
\(748\) −1.46369e16 −2.28555
\(749\) −4.20845e15 −0.652338
\(750\) −7.91980e15 −1.21865
\(751\) 1.46728e15 0.224126 0.112063 0.993701i \(-0.464254\pi\)
0.112063 + 0.993701i \(0.464254\pi\)
\(752\) −4.21756e16 −6.39533
\(753\) 2.33962e14 0.0352186
\(754\) −1.93694e15 −0.289451
\(755\) −1.95832e16 −2.90519
\(756\) 1.92859e15 0.284034
\(757\) −6.98997e15 −1.02199 −0.510996 0.859583i \(-0.670723\pi\)
−0.510996 + 0.859583i \(0.670723\pi\)
\(758\) 1.65999e16 2.40949
\(759\) 2.30521e15 0.332186
\(760\) 4.03507e16 5.77267
\(761\) 2.23176e15 0.316980 0.158490 0.987361i \(-0.449337\pi\)
0.158490 + 0.987361i \(0.449337\pi\)
\(762\) 1.20796e16 1.70333
\(763\) −2.33999e15 −0.327588
\(764\) 3.01402e16 4.18921
\(765\) 7.01210e15 0.967633
\(766\) 1.02649e16 1.40636
\(767\) 1.66292e15 0.226202
\(768\) 2.19283e16 2.96154
\(769\) −5.22092e15 −0.700087 −0.350044 0.936733i \(-0.613833\pi\)
−0.350044 + 0.936733i \(0.613833\pi\)
\(770\) 5.59980e15 0.745545
\(771\) −1.66416e15 −0.219987
\(772\) −8.17212e14 −0.107261
\(773\) −1.43588e15 −0.187124 −0.0935622 0.995613i \(-0.529825\pi\)
−0.0935622 + 0.995613i \(0.529825\pi\)
\(774\) −4.99609e15 −0.646479
\(775\) −1.93048e15 −0.248032
\(776\) −4.26933e16 −5.44653
\(777\) 1.92827e15 0.244260
\(778\) −4.48211e15 −0.563760
\(779\) 1.13390e16 1.41618
\(780\) −3.76248e16 −4.66610
\(781\) 1.06670e15 0.131360
\(782\) 3.65186e16 4.46557
\(783\) −1.34524e14 −0.0163346
\(784\) −2.60126e16 −3.13650
\(785\) 1.33478e16 1.59818
\(786\) −2.81335e15 −0.334503
\(787\) 2.85594e15 0.337200 0.168600 0.985685i \(-0.446075\pi\)
0.168600 + 0.985685i \(0.446075\pi\)
\(788\) −3.05618e16 −3.58331
\(789\) −2.41690e15 −0.281408
\(790\) 1.32001e16 1.52626
\(791\) −5.95868e15 −0.684194
\(792\) 4.78409e15 0.545520
\(793\) −1.27478e16 −1.44355
\(794\) 2.60290e16 2.92715
\(795\) −4.27190e15 −0.477092
\(796\) 2.05580e16 2.28013
\(797\) 1.02947e15 0.113395 0.0566974 0.998391i \(-0.481943\pi\)
0.0566974 + 0.998391i \(0.481943\pi\)
\(798\) 5.21893e15 0.570907
\(799\) −2.44639e16 −2.65777
\(800\) −7.33852e16 −7.91796
\(801\) 2.03581e15 0.218151
\(802\) 2.26354e16 2.40896
\(803\) 2.61187e15 0.276069
\(804\) −5.16467e15 −0.542169
\(805\) −1.03446e16 −1.07854
\(806\) −4.92231e15 −0.509713
\(807\) 7.21415e15 0.741960
\(808\) 2.87383e16 2.93561
\(809\) −4.09115e15 −0.415077 −0.207538 0.978227i \(-0.566545\pi\)
−0.207538 + 0.978227i \(0.566545\pi\)
\(810\) −3.52924e15 −0.355642
\(811\) −4.62714e15 −0.463124 −0.231562 0.972820i \(-0.574384\pi\)
−0.231562 + 0.972820i \(0.574384\pi\)
\(812\) −1.26009e15 −0.125269
\(813\) 4.10626e15 0.405460
\(814\) 7.36563e15 0.722396
\(815\) 1.24763e15 0.121540
\(816\) 4.54938e16 4.40206
\(817\) −1.00103e16 −0.962109
\(818\) −3.15505e16 −3.01206
\(819\) −3.16024e15 −0.299681
\(820\) −7.18255e16 −6.76554
\(821\) −3.20316e15 −0.299703 −0.149851 0.988709i \(-0.547880\pi\)
−0.149851 + 0.988709i \(0.547880\pi\)
\(822\) 1.36676e15 0.127027
\(823\) 5.02055e15 0.463503 0.231751 0.972775i \(-0.425554\pi\)
0.231751 + 0.972775i \(0.425554\pi\)
\(824\) −2.54308e16 −2.33217
\(825\) −4.73435e15 −0.431284
\(826\) 1.46111e15 0.132218
\(827\) −5.59661e15 −0.503089 −0.251545 0.967846i \(-0.580939\pi\)
−0.251545 + 0.967846i \(0.580939\pi\)
\(828\) −1.36089e16 −1.21522
\(829\) 1.80300e16 1.59935 0.799677 0.600430i \(-0.205004\pi\)
0.799677 + 0.600430i \(0.205004\pi\)
\(830\) 2.82217e16 2.48687
\(831\) 6.72010e15 0.588261
\(832\) −1.01534e17 −8.82944
\(833\) −1.50886e16 −1.30347
\(834\) 2.95312e15 0.253436
\(835\) −2.12146e16 −1.80867
\(836\) 1.47604e16 1.25016
\(837\) −3.41862e14 −0.0287648
\(838\) −1.17457e16 −0.981827
\(839\) 1.29962e16 1.07926 0.539628 0.841903i \(-0.318565\pi\)
0.539628 + 0.841903i \(0.318565\pi\)
\(840\) −2.14684e16 −1.77119
\(841\) −1.21126e16 −0.992796
\(842\) 2.89873e16 2.36043
\(843\) −1.66356e15 −0.134582
\(844\) 3.63407e16 2.92086
\(845\) 4.12305e16 3.29236
\(846\) 1.23128e16 0.976835
\(847\) −5.23446e15 −0.412585
\(848\) −2.77156e16 −2.17044
\(849\) 6.23715e15 0.485282
\(850\) −7.50005e16 −5.79776
\(851\) −1.36066e16 −1.04505
\(852\) −6.29730e15 −0.480548
\(853\) 8.73059e15 0.661948 0.330974 0.943640i \(-0.392623\pi\)
0.330974 + 0.943640i \(0.392623\pi\)
\(854\) −1.12007e16 −0.843773
\(855\) −7.07126e15 −0.529278
\(856\) 6.16301e16 4.58339
\(857\) 3.42182e15 0.252850 0.126425 0.991976i \(-0.459650\pi\)
0.126425 + 0.991976i \(0.459650\pi\)
\(858\) −1.20715e16 −0.886304
\(859\) 1.24428e15 0.0907730 0.0453865 0.998970i \(-0.485548\pi\)
0.0453865 + 0.998970i \(0.485548\pi\)
\(860\) 6.34087e16 4.59629
\(861\) −6.03288e15 −0.434518
\(862\) −2.87270e16 −2.05589
\(863\) −8.73742e15 −0.621332 −0.310666 0.950519i \(-0.600552\pi\)
−0.310666 + 0.950519i \(0.600552\pi\)
\(864\) −1.29955e16 −0.918264
\(865\) −3.42980e15 −0.240813
\(866\) −2.16709e16 −1.51192
\(867\) 1.80605e16 1.25206
\(868\) −3.20224e15 −0.220595
\(869\) 3.13575e15 0.214651
\(870\) 2.30591e15 0.156851
\(871\) 8.46297e15 0.572035
\(872\) 3.42676e16 2.30167
\(873\) 7.48178e15 0.499375
\(874\) −3.68267e16 −2.44259
\(875\) 8.44266e15 0.556461
\(876\) −1.54193e16 −1.00993
\(877\) −5.54899e15 −0.361174 −0.180587 0.983559i \(-0.557800\pi\)
−0.180587 + 0.983559i \(0.557800\pi\)
\(878\) 4.81536e16 3.11465
\(879\) −1.64767e16 −1.05909
\(880\) −4.92257e16 −3.14440
\(881\) 3.77264e14 0.0239485 0.0119742 0.999928i \(-0.496188\pi\)
0.0119742 + 0.999928i \(0.496188\pi\)
\(882\) 7.59418e15 0.479075
\(883\) −1.26882e16 −0.795455 −0.397728 0.917504i \(-0.630201\pi\)
−0.397728 + 0.917504i \(0.630201\pi\)
\(884\) −1.41593e17 −8.82174
\(885\) −1.97969e15 −0.122577
\(886\) −1.14414e16 −0.704037
\(887\) −1.05798e16 −0.646989 −0.323494 0.946230i \(-0.604858\pi\)
−0.323494 + 0.946230i \(0.604858\pi\)
\(888\) −2.82383e16 −1.71619
\(889\) −1.28770e16 −0.777780
\(890\) −3.48963e16 −2.09476
\(891\) −8.38388e14 −0.0500170
\(892\) 4.94647e16 2.93284
\(893\) 2.46703e16 1.45375
\(894\) −2.78626e16 −1.63180
\(895\) −1.94097e16 −1.12977
\(896\) −4.65340e16 −2.69201
\(897\) 2.22999e16 1.28216
\(898\) 5.78621e15 0.330655
\(899\) 2.23364e14 0.0126863
\(900\) 2.79494e16 1.57775
\(901\) −1.60764e16 −0.901992
\(902\) −2.30445e16 −1.28508
\(903\) 5.32592e15 0.295197
\(904\) 8.72611e16 4.80722
\(905\) 3.20182e16 1.75319
\(906\) 3.70923e16 2.01873
\(907\) 2.55994e16 1.38481 0.692404 0.721510i \(-0.256552\pi\)
0.692404 + 0.721510i \(0.256552\pi\)
\(908\) −4.61974e16 −2.48396
\(909\) −5.03624e15 −0.269157
\(910\) 5.41707e16 2.87764
\(911\) −2.75529e16 −1.45485 −0.727423 0.686189i \(-0.759282\pi\)
−0.727423 + 0.686189i \(0.759282\pi\)
\(912\) −4.58776e16 −2.40785
\(913\) 6.70420e15 0.349750
\(914\) 3.81806e16 1.97988
\(915\) 1.51761e16 0.782246
\(916\) −8.80045e16 −4.50899
\(917\) 2.99908e15 0.152742
\(918\) −1.32816e16 −0.672379
\(919\) 5.89532e15 0.296669 0.148335 0.988937i \(-0.452609\pi\)
0.148335 + 0.988937i \(0.452609\pi\)
\(920\) 1.51489e17 7.57790
\(921\) 1.15097e16 0.572316
\(922\) −9.11128e15 −0.450359
\(923\) 1.03189e16 0.507020
\(924\) −7.85323e15 −0.383577
\(925\) 2.79447e16 1.35681
\(926\) −1.68727e16 −0.814373
\(927\) 4.45662e15 0.213829
\(928\) 8.49091e15 0.404987
\(929\) 8.28063e15 0.392624 0.196312 0.980541i \(-0.437103\pi\)
0.196312 + 0.980541i \(0.437103\pi\)
\(930\) 5.85996e15 0.276209
\(931\) 1.52159e16 0.712974
\(932\) 3.99818e16 1.86241
\(933\) 1.96603e16 0.910420
\(934\) 6.74710e16 3.10606
\(935\) −2.85533e16 −1.30675
\(936\) 4.62798e16 2.10559
\(937\) −2.43077e16 −1.09945 −0.549725 0.835346i \(-0.685268\pi\)
−0.549725 + 0.835346i \(0.685268\pi\)
\(938\) 7.43589e15 0.334362
\(939\) −1.80143e16 −0.805299
\(940\) −1.56270e17 −6.94502
\(941\) 1.84021e16 0.813063 0.406531 0.913637i \(-0.366738\pi\)
0.406531 + 0.913637i \(0.366738\pi\)
\(942\) −2.52819e16 −1.11053
\(943\) 4.25703e16 1.85905
\(944\) −1.28440e16 −0.557641
\(945\) 3.76224e15 0.162395
\(946\) 2.03440e16 0.873044
\(947\) −3.43633e16 −1.46612 −0.733061 0.680163i \(-0.761909\pi\)
−0.733061 + 0.680163i \(0.761909\pi\)
\(948\) −1.85120e16 −0.785249
\(949\) 2.52664e16 1.06556
\(950\) 7.56333e16 3.17127
\(951\) −2.02628e16 −0.844709
\(952\) −8.07920e16 −3.34861
\(953\) −8.61832e15 −0.355150 −0.177575 0.984107i \(-0.556825\pi\)
−0.177575 + 0.984107i \(0.556825\pi\)
\(954\) 8.09136e15 0.331517
\(955\) 5.87965e16 2.39515
\(956\) −7.41051e16 −3.00144
\(957\) 5.47781e14 0.0220593
\(958\) −3.35251e16 −1.34233
\(959\) −1.45699e15 −0.0580034
\(960\) 1.20875e17 4.78460
\(961\) −2.48408e16 −0.977660
\(962\) 7.12527e16 2.78829
\(963\) −1.08004e16 −0.420237
\(964\) 8.62994e16 3.33875
\(965\) −1.59419e15 −0.0613254
\(966\) 1.95935e16 0.749442
\(967\) −4.27912e15 −0.162745 −0.0813727 0.996684i \(-0.525930\pi\)
−0.0813727 + 0.996684i \(0.525930\pi\)
\(968\) 7.66553e16 2.89886
\(969\) −2.66112e16 −1.00065
\(970\) −1.28247e17 −4.79517
\(971\) −1.14410e16 −0.425362 −0.212681 0.977122i \(-0.568220\pi\)
−0.212681 + 0.977122i \(0.568220\pi\)
\(972\) 4.94945e15 0.182975
\(973\) −3.14809e15 −0.115725
\(974\) −5.83509e16 −2.13291
\(975\) −4.57986e16 −1.66466
\(976\) 9.84607e16 3.55868
\(977\) −1.57282e16 −0.565273 −0.282636 0.959227i \(-0.591209\pi\)
−0.282636 + 0.959227i \(0.591209\pi\)
\(978\) −2.36313e15 −0.0844548
\(979\) −8.28980e15 −0.294604
\(980\) −9.63828e16 −3.40609
\(981\) −6.00523e15 −0.211033
\(982\) 4.00630e16 1.40001
\(983\) 4.00337e16 1.39117 0.695587 0.718442i \(-0.255145\pi\)
0.695587 + 0.718442i \(0.255145\pi\)
\(984\) 8.83477e16 3.05297
\(985\) −5.96189e16 −2.04873
\(986\) 8.67781e15 0.296543
\(987\) −1.31257e16 −0.446045
\(988\) 1.42788e17 4.82533
\(989\) −3.75817e16 −1.26298
\(990\) 1.43711e16 0.480281
\(991\) 2.40351e15 0.0798806 0.0399403 0.999202i \(-0.487283\pi\)
0.0399403 + 0.999202i \(0.487283\pi\)
\(992\) 2.15778e16 0.713168
\(993\) −3.35018e16 −1.10115
\(994\) 9.06660e15 0.296360
\(995\) 4.01039e16 1.30365
\(996\) −3.95785e16 −1.27948
\(997\) −3.36628e16 −1.08225 −0.541124 0.840943i \(-0.682001\pi\)
−0.541124 + 0.840943i \(0.682001\pi\)
\(998\) −6.22785e15 −0.199122
\(999\) 4.94861e15 0.157352
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.1.1 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.1 27 1.1 even 1 trivial