Properties

Label 177.12.a.a.1.2
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $26$
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: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-86.0142 q^{2} -243.000 q^{3} +5350.44 q^{4} +12869.1 q^{5} +20901.4 q^{6} -72810.3 q^{7} -284057. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-86.0142 q^{2} -243.000 q^{3} +5350.44 q^{4} +12869.1 q^{5} +20901.4 q^{6} -72810.3 q^{7} -284057. q^{8} +59049.0 q^{9} -1.10693e6 q^{10} -128428. q^{11} -1.30016e6 q^{12} +1.06672e6 q^{13} +6.26272e6 q^{14} -3.12720e6 q^{15} +1.34752e7 q^{16} -4.24538e6 q^{17} -5.07905e6 q^{18} +4.17284e6 q^{19} +6.88556e7 q^{20} +1.76929e7 q^{21} +1.10466e7 q^{22} -1.73947e7 q^{23} +6.90258e7 q^{24} +1.16787e8 q^{25} -9.17532e7 q^{26} -1.43489e7 q^{27} -3.89567e8 q^{28} +9.35237e7 q^{29} +2.68984e8 q^{30} +5.43406e7 q^{31} -5.77310e8 q^{32} +3.12080e7 q^{33} +3.65163e8 q^{34} -9.37006e8 q^{35} +3.15938e8 q^{36} -5.85281e7 q^{37} -3.58924e8 q^{38} -2.59214e8 q^{39} -3.65557e9 q^{40} -1.22068e9 q^{41} -1.52184e9 q^{42} -1.70364e9 q^{43} -6.87147e8 q^{44} +7.59910e8 q^{45} +1.49619e9 q^{46} +2.74021e9 q^{47} -3.27447e9 q^{48} +3.32401e9 q^{49} -1.00453e10 q^{50} +1.03163e9 q^{51} +5.70743e9 q^{52} +8.82674e8 q^{53} +1.23421e9 q^{54} -1.65276e9 q^{55} +2.06822e10 q^{56} -1.01400e9 q^{57} -8.04436e9 q^{58} +7.14924e8 q^{59} -1.67319e10 q^{60} -2.92979e9 q^{61} -4.67406e9 q^{62} -4.29937e9 q^{63} +2.20597e10 q^{64} +1.37278e10 q^{65} -2.68433e9 q^{66} +1.69588e10 q^{67} -2.27146e10 q^{68} +4.22692e9 q^{69} +8.05958e10 q^{70} -3.58652e9 q^{71} -1.67733e10 q^{72} -1.17507e10 q^{73} +5.03425e9 q^{74} -2.83792e10 q^{75} +2.23265e10 q^{76} +9.35089e9 q^{77} +2.22960e10 q^{78} -4.15998e10 q^{79} +1.73414e11 q^{80} +3.48678e9 q^{81} +1.04996e11 q^{82} -3.90038e10 q^{83} +9.46648e10 q^{84} -5.46344e10 q^{85} +1.46537e11 q^{86} -2.27263e10 q^{87} +3.64809e10 q^{88} +8.36999e10 q^{89} -6.53631e10 q^{90} -7.76684e10 q^{91} -9.30694e10 q^{92} -1.32048e10 q^{93} -2.35697e11 q^{94} +5.37009e10 q^{95} +1.40286e11 q^{96} +2.64875e10 q^{97} -2.85912e11 q^{98} -7.58355e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −86.0142 −1.90066 −0.950331 0.311240i \(-0.899256\pi\)
−0.950331 + 0.311240i \(0.899256\pi\)
\(3\) −243.000 −0.577350
\(4\) 5350.44 2.61252
\(5\) 12869.1 1.84168 0.920841 0.389938i \(-0.127503\pi\)
0.920841 + 0.389938i \(0.127503\pi\)
\(6\) 20901.4 1.09735
\(7\) −72810.3 −1.63740 −0.818698 0.574225i \(-0.805304\pi\)
−0.818698 + 0.574225i \(0.805304\pi\)
\(8\) −284057. −3.06486
\(9\) 59049.0 0.333333
\(10\) −1.10693e6 −3.50042
\(11\) −128428. −0.240437 −0.120218 0.992747i \(-0.538359\pi\)
−0.120218 + 0.992747i \(0.538359\pi\)
\(12\) −1.30016e6 −1.50834
\(13\) 1.06672e6 0.796825 0.398412 0.917206i \(-0.369561\pi\)
0.398412 + 0.917206i \(0.369561\pi\)
\(14\) 6.26272e6 3.11214
\(15\) −3.12720e6 −1.06330
\(16\) 1.34752e7 3.21274
\(17\) −4.24538e6 −0.725182 −0.362591 0.931948i \(-0.618108\pi\)
−0.362591 + 0.931948i \(0.618108\pi\)
\(18\) −5.07905e6 −0.633554
\(19\) 4.17284e6 0.386623 0.193311 0.981137i \(-0.438077\pi\)
0.193311 + 0.981137i \(0.438077\pi\)
\(20\) 6.88556e7 4.81143
\(21\) 1.76929e7 0.945351
\(22\) 1.10466e7 0.456989
\(23\) −1.73947e7 −0.563526 −0.281763 0.959484i \(-0.590919\pi\)
−0.281763 + 0.959484i \(0.590919\pi\)
\(24\) 6.90258e7 1.76950
\(25\) 1.16787e8 2.39179
\(26\) −9.17532e7 −1.51450
\(27\) −1.43489e7 −0.192450
\(28\) −3.89567e8 −4.27773
\(29\) 9.35237e7 0.846706 0.423353 0.905965i \(-0.360853\pi\)
0.423353 + 0.905965i \(0.360853\pi\)
\(30\) 2.68984e8 2.02097
\(31\) 5.43406e7 0.340906 0.170453 0.985366i \(-0.445477\pi\)
0.170453 + 0.985366i \(0.445477\pi\)
\(32\) −5.77310e8 −3.04148
\(33\) 3.12080e7 0.138816
\(34\) 3.65163e8 1.37833
\(35\) −9.37006e8 −3.01556
\(36\) 3.15938e8 0.870840
\(37\) −5.85281e7 −0.138757 −0.0693786 0.997590i \(-0.522102\pi\)
−0.0693786 + 0.997590i \(0.522102\pi\)
\(38\) −3.58924e8 −0.734839
\(39\) −2.59214e8 −0.460047
\(40\) −3.65557e9 −5.64449
\(41\) −1.22068e9 −1.64548 −0.822739 0.568419i \(-0.807555\pi\)
−0.822739 + 0.568419i \(0.807555\pi\)
\(42\) −1.52184e9 −1.79679
\(43\) −1.70364e9 −1.76726 −0.883630 0.468186i \(-0.844908\pi\)
−0.883630 + 0.468186i \(0.844908\pi\)
\(44\) −6.87147e8 −0.628145
\(45\) 7.59910e8 0.613894
\(46\) 1.49619e9 1.07107
\(47\) 2.74021e9 1.74279 0.871395 0.490582i \(-0.163216\pi\)
0.871395 + 0.490582i \(0.163216\pi\)
\(48\) −3.27447e9 −1.85488
\(49\) 3.32401e9 1.68106
\(50\) −1.00453e10 −4.54600
\(51\) 1.03163e9 0.418684
\(52\) 5.70743e9 2.08172
\(53\) 8.82674e8 0.289923 0.144962 0.989437i \(-0.453694\pi\)
0.144962 + 0.989437i \(0.453694\pi\)
\(54\) 1.23421e9 0.365783
\(55\) −1.65276e9 −0.442808
\(56\) 2.06822e10 5.01838
\(57\) −1.01400e9 −0.223217
\(58\) −8.04436e9 −1.60930
\(59\) 7.14924e8 0.130189
\(60\) −1.67319e10 −2.77788
\(61\) −2.92979e9 −0.444142 −0.222071 0.975030i \(-0.571282\pi\)
−0.222071 + 0.975030i \(0.571282\pi\)
\(62\) −4.67406e9 −0.647948
\(63\) −4.29937e9 −0.545798
\(64\) 2.20597e10 2.56808
\(65\) 1.37278e10 1.46750
\(66\) −2.68433e9 −0.263843
\(67\) 1.69588e10 1.53456 0.767281 0.641311i \(-0.221609\pi\)
0.767281 + 0.641311i \(0.221609\pi\)
\(68\) −2.27146e10 −1.89455
\(69\) 4.22692e9 0.325352
\(70\) 8.05958e10 5.73157
\(71\) −3.58652e9 −0.235913 −0.117957 0.993019i \(-0.537634\pi\)
−0.117957 + 0.993019i \(0.537634\pi\)
\(72\) −1.67733e10 −1.02162
\(73\) −1.17507e10 −0.663420 −0.331710 0.943381i \(-0.607625\pi\)
−0.331710 + 0.943381i \(0.607625\pi\)
\(74\) 5.03425e9 0.263731
\(75\) −2.83792e10 −1.38090
\(76\) 2.23265e10 1.01006
\(77\) 9.35089e9 0.393690
\(78\) 2.22960e10 0.874394
\(79\) −4.15998e10 −1.52104 −0.760522 0.649312i \(-0.775057\pi\)
−0.760522 + 0.649312i \(0.775057\pi\)
\(80\) 1.73414e11 5.91684
\(81\) 3.48678e9 0.111111
\(82\) 1.04996e11 3.12750
\(83\) −3.90038e10 −1.08687 −0.543435 0.839451i \(-0.682877\pi\)
−0.543435 + 0.839451i \(0.682877\pi\)
\(84\) 9.46648e10 2.46975
\(85\) −5.46344e10 −1.33556
\(86\) 1.46537e11 3.35896
\(87\) −2.27263e10 −0.488846
\(88\) 3.64809e10 0.736903
\(89\) 8.36999e10 1.58884 0.794420 0.607369i \(-0.207775\pi\)
0.794420 + 0.607369i \(0.207775\pi\)
\(90\) −6.53631e10 −1.16681
\(91\) −7.76684e10 −1.30472
\(92\) −9.30694e10 −1.47222
\(93\) −1.32048e10 −0.196822
\(94\) −2.35697e11 −3.31246
\(95\) 5.37009e10 0.712036
\(96\) 1.40286e11 1.75600
\(97\) 2.64875e10 0.313182 0.156591 0.987664i \(-0.449950\pi\)
0.156591 + 0.987664i \(0.449950\pi\)
\(98\) −2.85912e11 −3.19513
\(99\) −7.58355e9 −0.0801455
\(100\) 6.24861e11 6.24861
\(101\) 9.15128e10 0.866392 0.433196 0.901300i \(-0.357386\pi\)
0.433196 + 0.901300i \(0.357386\pi\)
\(102\) −8.87345e10 −0.795777
\(103\) −2.16558e11 −1.84064 −0.920320 0.391166i \(-0.872072\pi\)
−0.920320 + 0.391166i \(0.872072\pi\)
\(104\) −3.03010e11 −2.44215
\(105\) 2.27693e11 1.74104
\(106\) −7.59225e10 −0.551046
\(107\) 9.08001e10 0.625858 0.312929 0.949777i \(-0.398690\pi\)
0.312929 + 0.949777i \(0.398690\pi\)
\(108\) −7.67730e10 −0.502780
\(109\) 2.83339e11 1.76385 0.881924 0.471392i \(-0.156248\pi\)
0.881924 + 0.471392i \(0.156248\pi\)
\(110\) 1.42161e11 0.841628
\(111\) 1.42223e10 0.0801115
\(112\) −9.81133e11 −5.26052
\(113\) −2.47545e11 −1.26393 −0.631964 0.774997i \(-0.717751\pi\)
−0.631964 + 0.774997i \(0.717751\pi\)
\(114\) 8.72184e10 0.424260
\(115\) −2.23855e11 −1.03784
\(116\) 5.00393e11 2.21204
\(117\) 6.29889e10 0.265608
\(118\) −6.14936e10 −0.247445
\(119\) 3.09107e11 1.18741
\(120\) 8.88303e11 3.25885
\(121\) −2.68818e11 −0.942190
\(122\) 2.52003e11 0.844165
\(123\) 2.96626e11 0.950017
\(124\) 2.90746e11 0.890624
\(125\) 8.74571e11 2.56324
\(126\) 3.69807e11 1.03738
\(127\) −2.59989e11 −0.698287 −0.349143 0.937069i \(-0.613527\pi\)
−0.349143 + 0.937069i \(0.613527\pi\)
\(128\) −7.15113e11 −1.83958
\(129\) 4.13984e11 1.02033
\(130\) −1.18079e12 −2.78922
\(131\) −6.82477e11 −1.54560 −0.772798 0.634652i \(-0.781143\pi\)
−0.772798 + 0.634652i \(0.781143\pi\)
\(132\) 1.66977e11 0.362660
\(133\) −3.03826e11 −0.633054
\(134\) −1.45870e12 −2.91669
\(135\) −1.84658e11 −0.354432
\(136\) 1.20593e12 2.22258
\(137\) 9.47901e10 0.167803 0.0839016 0.996474i \(-0.473262\pi\)
0.0839016 + 0.996474i \(0.473262\pi\)
\(138\) −3.63575e11 −0.618385
\(139\) 1.07791e12 1.76197 0.880987 0.473140i \(-0.156880\pi\)
0.880987 + 0.473140i \(0.156880\pi\)
\(140\) −5.01340e12 −7.87822
\(141\) −6.65870e11 −1.00620
\(142\) 3.08491e11 0.448391
\(143\) −1.36997e11 −0.191586
\(144\) 7.95697e11 1.07091
\(145\) 1.20357e12 1.55936
\(146\) 1.01073e12 1.26094
\(147\) −8.07735e11 −0.970562
\(148\) −3.13151e11 −0.362506
\(149\) 1.20005e12 1.33867 0.669337 0.742959i \(-0.266578\pi\)
0.669337 + 0.742959i \(0.266578\pi\)
\(150\) 2.44101e12 2.62463
\(151\) 1.89764e11 0.196716 0.0983581 0.995151i \(-0.468641\pi\)
0.0983581 + 0.995151i \(0.468641\pi\)
\(152\) −1.18532e12 −1.18494
\(153\) −2.50685e11 −0.241727
\(154\) −8.04309e11 −0.748271
\(155\) 6.99317e11 0.627841
\(156\) −1.38691e12 −1.20188
\(157\) −1.34772e12 −1.12759 −0.563795 0.825915i \(-0.690659\pi\)
−0.563795 + 0.825915i \(0.690659\pi\)
\(158\) 3.57817e12 2.89099
\(159\) −2.14490e11 −0.167387
\(160\) −7.42949e12 −5.60143
\(161\) 1.26651e12 0.922716
\(162\) −2.99913e11 −0.211185
\(163\) −2.08772e10 −0.0142115 −0.00710577 0.999975i \(-0.502262\pi\)
−0.00710577 + 0.999975i \(0.502262\pi\)
\(164\) −6.53120e12 −4.29884
\(165\) 4.01621e11 0.255655
\(166\) 3.35488e12 2.06577
\(167\) 2.23710e12 1.33274 0.666368 0.745623i \(-0.267848\pi\)
0.666368 + 0.745623i \(0.267848\pi\)
\(168\) −5.02579e12 −2.89736
\(169\) −6.54264e11 −0.365070
\(170\) 4.69933e12 2.53844
\(171\) 2.46402e11 0.128874
\(172\) −9.11520e12 −4.61700
\(173\) 5.09919e11 0.250177 0.125089 0.992146i \(-0.460078\pi\)
0.125089 + 0.992146i \(0.460078\pi\)
\(174\) 1.95478e12 0.929131
\(175\) −8.50328e12 −3.91631
\(176\) −1.73060e12 −0.772460
\(177\) −1.73727e11 −0.0751646
\(178\) −7.19938e12 −3.01985
\(179\) 3.36855e12 1.37010 0.685048 0.728498i \(-0.259781\pi\)
0.685048 + 0.728498i \(0.259781\pi\)
\(180\) 4.06585e12 1.60381
\(181\) 2.30258e12 0.881012 0.440506 0.897750i \(-0.354799\pi\)
0.440506 + 0.897750i \(0.354799\pi\)
\(182\) 6.68058e12 2.47983
\(183\) 7.11938e11 0.256426
\(184\) 4.94109e12 1.72713
\(185\) −7.53207e11 −0.255547
\(186\) 1.13580e12 0.374093
\(187\) 5.45226e11 0.174360
\(188\) 1.46613e13 4.55307
\(189\) 1.04475e12 0.315117
\(190\) −4.61904e12 −1.35334
\(191\) 6.21335e12 1.76865 0.884325 0.466871i \(-0.154619\pi\)
0.884325 + 0.466871i \(0.154619\pi\)
\(192\) −5.36050e12 −1.48268
\(193\) −3.55496e12 −0.955585 −0.477792 0.878473i \(-0.658563\pi\)
−0.477792 + 0.878473i \(0.658563\pi\)
\(194\) −2.27830e12 −0.595254
\(195\) −3.33586e12 −0.847261
\(196\) 1.77849e13 4.39181
\(197\) 2.48699e12 0.597186 0.298593 0.954380i \(-0.403483\pi\)
0.298593 + 0.954380i \(0.403483\pi\)
\(198\) 6.52293e11 0.152330
\(199\) −9.47524e11 −0.215228 −0.107614 0.994193i \(-0.534321\pi\)
−0.107614 + 0.994193i \(0.534321\pi\)
\(200\) −3.31741e13 −7.33051
\(201\) −4.12100e12 −0.885980
\(202\) −7.87140e12 −1.64672
\(203\) −6.80949e12 −1.38639
\(204\) 5.51966e12 1.09382
\(205\) −1.57092e13 −3.03045
\(206\) 1.86270e13 3.49844
\(207\) −1.02714e12 −0.187842
\(208\) 1.43743e13 2.55999
\(209\) −5.35910e11 −0.0929582
\(210\) −1.95848e13 −3.30912
\(211\) −8.10742e12 −1.33453 −0.667266 0.744819i \(-0.732536\pi\)
−0.667266 + 0.744819i \(0.732536\pi\)
\(212\) 4.72269e12 0.757430
\(213\) 8.71524e11 0.136205
\(214\) −7.81010e12 −1.18954
\(215\) −2.19243e13 −3.25473
\(216\) 4.07590e12 0.589832
\(217\) −3.95655e12 −0.558198
\(218\) −2.43712e13 −3.35248
\(219\) 2.85542e12 0.383026
\(220\) −8.84300e12 −1.15684
\(221\) −4.52864e12 −0.577843
\(222\) −1.22332e12 −0.152265
\(223\) −4.22592e12 −0.513150 −0.256575 0.966524i \(-0.582594\pi\)
−0.256575 + 0.966524i \(0.582594\pi\)
\(224\) 4.20341e13 4.98010
\(225\) 6.89615e12 0.797265
\(226\) 2.12924e13 2.40230
\(227\) −6.14055e12 −0.676184 −0.338092 0.941113i \(-0.609782\pi\)
−0.338092 + 0.941113i \(0.609782\pi\)
\(228\) −5.42535e12 −0.583158
\(229\) −7.87370e12 −0.826197 −0.413099 0.910686i \(-0.635553\pi\)
−0.413099 + 0.910686i \(0.635553\pi\)
\(230\) 1.92547e13 1.97258
\(231\) −2.27227e12 −0.227297
\(232\) −2.65660e13 −2.59503
\(233\) −9.24605e12 −0.882062 −0.441031 0.897492i \(-0.645387\pi\)
−0.441031 + 0.897492i \(0.645387\pi\)
\(234\) −5.41794e12 −0.504832
\(235\) 3.52641e13 3.20967
\(236\) 3.82516e12 0.340121
\(237\) 1.01087e13 0.878176
\(238\) −2.65876e13 −2.25687
\(239\) 7.24389e12 0.600874 0.300437 0.953802i \(-0.402868\pi\)
0.300437 + 0.953802i \(0.402868\pi\)
\(240\) −4.21397e13 −3.41609
\(241\) −9.26650e12 −0.734213 −0.367106 0.930179i \(-0.619652\pi\)
−0.367106 + 0.930179i \(0.619652\pi\)
\(242\) 2.31222e13 1.79079
\(243\) −8.47289e11 −0.0641500
\(244\) −1.56757e13 −1.16033
\(245\) 4.27772e13 3.09599
\(246\) −2.55141e13 −1.80566
\(247\) 4.45126e12 0.308070
\(248\) −1.54358e13 −1.04483
\(249\) 9.47792e12 0.627505
\(250\) −7.52255e13 −4.87186
\(251\) −2.37605e13 −1.50539 −0.752697 0.658367i \(-0.771248\pi\)
−0.752697 + 0.658367i \(0.771248\pi\)
\(252\) −2.30035e13 −1.42591
\(253\) 2.23397e12 0.135492
\(254\) 2.23627e13 1.32721
\(255\) 1.32762e13 0.771083
\(256\) 1.63316e13 0.928346
\(257\) 8.98996e11 0.0500179 0.0250089 0.999687i \(-0.492039\pi\)
0.0250089 + 0.999687i \(0.492039\pi\)
\(258\) −3.56085e13 −1.93930
\(259\) 4.26145e12 0.227200
\(260\) 7.34498e13 3.83387
\(261\) 5.52248e12 0.282235
\(262\) 5.87027e13 2.93766
\(263\) −5.82370e12 −0.285392 −0.142696 0.989767i \(-0.545577\pi\)
−0.142696 + 0.989767i \(0.545577\pi\)
\(264\) −8.86485e12 −0.425451
\(265\) 1.13593e13 0.533947
\(266\) 2.61333e13 1.20322
\(267\) −2.03391e13 −0.917317
\(268\) 9.07372e13 4.00907
\(269\) 5.34952e12 0.231567 0.115784 0.993274i \(-0.463062\pi\)
0.115784 + 0.993274i \(0.463062\pi\)
\(270\) 1.58832e13 0.673656
\(271\) 8.76617e11 0.0364316 0.0182158 0.999834i \(-0.494201\pi\)
0.0182158 + 0.999834i \(0.494201\pi\)
\(272\) −5.72073e13 −2.32982
\(273\) 1.88734e13 0.753279
\(274\) −8.15330e12 −0.318937
\(275\) −1.49987e13 −0.575075
\(276\) 2.26159e13 0.849989
\(277\) −7.56380e11 −0.0278677 −0.0139339 0.999903i \(-0.504435\pi\)
−0.0139339 + 0.999903i \(0.504435\pi\)
\(278\) −9.27152e13 −3.34892
\(279\) 3.20876e12 0.113635
\(280\) 2.66163e14 9.24226
\(281\) 7.34680e12 0.250157 0.125079 0.992147i \(-0.460082\pi\)
0.125079 + 0.992147i \(0.460082\pi\)
\(282\) 5.72743e13 1.91245
\(283\) 2.26371e13 0.741304 0.370652 0.928772i \(-0.379134\pi\)
0.370652 + 0.928772i \(0.379134\pi\)
\(284\) −1.91894e13 −0.616328
\(285\) −1.30493e13 −0.411094
\(286\) 1.17837e13 0.364140
\(287\) 8.88784e13 2.69430
\(288\) −3.40896e13 −1.01383
\(289\) −1.62487e13 −0.474111
\(290\) −1.03524e14 −2.96382
\(291\) −6.43647e12 −0.180816
\(292\) −6.28715e13 −1.73320
\(293\) 3.79170e13 1.02580 0.512899 0.858449i \(-0.328572\pi\)
0.512899 + 0.858449i \(0.328572\pi\)
\(294\) 6.94767e13 1.84471
\(295\) 9.20047e12 0.239767
\(296\) 1.66253e13 0.425271
\(297\) 1.84280e12 0.0462720
\(298\) −1.03221e14 −2.54437
\(299\) −1.85553e13 −0.449032
\(300\) −1.51841e14 −3.60764
\(301\) 1.24042e14 2.89370
\(302\) −1.63224e13 −0.373891
\(303\) −2.22376e13 −0.500211
\(304\) 5.62299e13 1.24212
\(305\) −3.77039e13 −0.817969
\(306\) 2.15625e13 0.459442
\(307\) −7.27579e13 −1.52272 −0.761358 0.648332i \(-0.775467\pi\)
−0.761358 + 0.648332i \(0.775467\pi\)
\(308\) 5.00314e13 1.02852
\(309\) 5.26235e13 1.06269
\(310\) −6.01512e13 −1.19331
\(311\) −9.34073e13 −1.82053 −0.910266 0.414023i \(-0.864123\pi\)
−0.910266 + 0.414023i \(0.864123\pi\)
\(312\) 7.36313e13 1.40998
\(313\) 5.52589e13 1.03970 0.519850 0.854258i \(-0.325988\pi\)
0.519850 + 0.854258i \(0.325988\pi\)
\(314\) 1.15923e14 2.14317
\(315\) −5.53293e13 −1.00519
\(316\) −2.22577e14 −3.97376
\(317\) −2.21602e13 −0.388820 −0.194410 0.980920i \(-0.562279\pi\)
−0.194410 + 0.980920i \(0.562279\pi\)
\(318\) 1.84492e13 0.318147
\(319\) −1.20111e13 −0.203579
\(320\) 2.83889e14 4.72959
\(321\) −2.20644e13 −0.361339
\(322\) −1.08938e14 −1.75377
\(323\) −1.77153e13 −0.280372
\(324\) 1.86558e13 0.290280
\(325\) 1.24579e14 1.90584
\(326\) 1.79574e12 0.0270114
\(327\) −6.88514e13 −1.01836
\(328\) 3.46744e14 5.04315
\(329\) −1.99515e14 −2.85364
\(330\) −3.45451e13 −0.485914
\(331\) −8.20321e13 −1.13483 −0.567414 0.823433i \(-0.692056\pi\)
−0.567414 + 0.823433i \(0.692056\pi\)
\(332\) −2.08688e14 −2.83947
\(333\) −3.45603e12 −0.0462524
\(334\) −1.92422e14 −2.53308
\(335\) 2.18246e14 2.82618
\(336\) 2.38415e14 3.03716
\(337\) −5.07694e12 −0.0636265 −0.0318132 0.999494i \(-0.510128\pi\)
−0.0318132 + 0.999494i \(0.510128\pi\)
\(338\) 5.62760e13 0.693875
\(339\) 6.01534e13 0.729730
\(340\) −2.92318e14 −3.48916
\(341\) −6.97886e12 −0.0819663
\(342\) −2.11941e13 −0.244946
\(343\) −9.80525e13 −1.11517
\(344\) 4.83929e14 5.41640
\(345\) 5.43968e13 0.599195
\(346\) −4.38603e13 −0.475502
\(347\) −4.22761e13 −0.451110 −0.225555 0.974230i \(-0.572420\pi\)
−0.225555 + 0.974230i \(0.572420\pi\)
\(348\) −1.21595e14 −1.27712
\(349\) 1.42158e14 1.46971 0.734856 0.678224i \(-0.237250\pi\)
0.734856 + 0.678224i \(0.237250\pi\)
\(350\) 7.31403e14 7.44359
\(351\) −1.53063e13 −0.153349
\(352\) 7.41429e13 0.731282
\(353\) −9.55970e13 −0.928290 −0.464145 0.885759i \(-0.653638\pi\)
−0.464145 + 0.885759i \(0.653638\pi\)
\(354\) 1.49430e13 0.142863
\(355\) −4.61554e13 −0.434477
\(356\) 4.47831e14 4.15087
\(357\) −7.51130e13 −0.685551
\(358\) −2.89743e14 −2.60409
\(359\) −1.34153e14 −1.18736 −0.593678 0.804703i \(-0.702325\pi\)
−0.593678 + 0.804703i \(0.702325\pi\)
\(360\) −2.15858e14 −1.88150
\(361\) −9.90776e13 −0.850523
\(362\) −1.98054e14 −1.67451
\(363\) 6.53227e13 0.543974
\(364\) −4.15560e14 −3.40860
\(365\) −1.51222e14 −1.22181
\(366\) −6.12368e13 −0.487379
\(367\) −1.11235e14 −0.872121 −0.436061 0.899917i \(-0.643627\pi\)
−0.436061 + 0.899917i \(0.643627\pi\)
\(368\) −2.34397e14 −1.81046
\(369\) −7.20802e13 −0.548493
\(370\) 6.47865e13 0.485708
\(371\) −6.42677e13 −0.474719
\(372\) −7.06513e13 −0.514202
\(373\) −1.25235e14 −0.898105 −0.449053 0.893505i \(-0.648238\pi\)
−0.449053 + 0.893505i \(0.648238\pi\)
\(374\) −4.68972e13 −0.331400
\(375\) −2.12521e14 −1.47989
\(376\) −7.78374e14 −5.34140
\(377\) 9.97638e13 0.674676
\(378\) −8.98632e13 −0.598931
\(379\) 4.79622e13 0.315053 0.157526 0.987515i \(-0.449648\pi\)
0.157526 + 0.987515i \(0.449648\pi\)
\(380\) 2.87324e14 1.86021
\(381\) 6.31772e13 0.403156
\(382\) −5.34436e14 −3.36161
\(383\) −2.67637e14 −1.65941 −0.829703 0.558206i \(-0.811490\pi\)
−0.829703 + 0.558206i \(0.811490\pi\)
\(384\) 1.73772e14 1.06208
\(385\) 1.20338e14 0.725051
\(386\) 3.05777e14 1.81624
\(387\) −1.00598e14 −0.589087
\(388\) 1.41720e14 0.818195
\(389\) 3.46706e14 1.97351 0.986754 0.162225i \(-0.0518670\pi\)
0.986754 + 0.162225i \(0.0518670\pi\)
\(390\) 2.86931e14 1.61036
\(391\) 7.38471e13 0.408659
\(392\) −9.44208e14 −5.15222
\(393\) 1.65842e14 0.892350
\(394\) −2.13916e14 −1.13505
\(395\) −5.35354e14 −2.80128
\(396\) −4.05753e13 −0.209382
\(397\) 5.58638e13 0.284304 0.142152 0.989845i \(-0.454598\pi\)
0.142152 + 0.989845i \(0.454598\pi\)
\(398\) 8.15005e13 0.409076
\(399\) 7.38297e13 0.365494
\(400\) 1.57373e15 7.68421
\(401\) −2.73514e14 −1.31730 −0.658651 0.752449i \(-0.728872\pi\)
−0.658651 + 0.752449i \(0.728872\pi\)
\(402\) 3.54464e14 1.68395
\(403\) 5.79663e13 0.271643
\(404\) 4.89634e14 2.26346
\(405\) 4.48719e13 0.204631
\(406\) 5.85712e14 2.63506
\(407\) 7.51666e12 0.0333623
\(408\) −2.93040e14 −1.28321
\(409\) 6.33500e13 0.273696 0.136848 0.990592i \(-0.456303\pi\)
0.136848 + 0.990592i \(0.456303\pi\)
\(410\) 1.35121e15 5.75986
\(411\) −2.30340e13 −0.0968812
\(412\) −1.15868e15 −4.80871
\(413\) −5.20538e13 −0.213171
\(414\) 8.83487e13 0.357025
\(415\) −5.01946e14 −2.00167
\(416\) −6.15830e14 −2.42352
\(417\) −2.61931e14 −1.01728
\(418\) 4.60959e13 0.176682
\(419\) −5.18193e13 −0.196026 −0.0980131 0.995185i \(-0.531249\pi\)
−0.0980131 + 0.995185i \(0.531249\pi\)
\(420\) 1.21826e15 4.54849
\(421\) −2.44339e14 −0.900411 −0.450206 0.892925i \(-0.648649\pi\)
−0.450206 + 0.892925i \(0.648649\pi\)
\(422\) 6.97353e14 2.53650
\(423\) 1.61806e14 0.580930
\(424\) −2.50729e14 −0.888573
\(425\) −4.95804e14 −1.73449
\(426\) −7.49634e13 −0.258879
\(427\) 2.13319e14 0.727236
\(428\) 4.85820e14 1.63507
\(429\) 3.32903e13 0.110612
\(430\) 1.88580e15 6.18615
\(431\) −3.38705e14 −1.09698 −0.548488 0.836159i \(-0.684796\pi\)
−0.548488 + 0.836159i \(0.684796\pi\)
\(432\) −1.93354e14 −0.618292
\(433\) −2.42661e14 −0.766155 −0.383078 0.923716i \(-0.625136\pi\)
−0.383078 + 0.923716i \(0.625136\pi\)
\(434\) 3.40320e14 1.06095
\(435\) −2.92467e14 −0.900299
\(436\) 1.51599e15 4.60809
\(437\) −7.25854e13 −0.217872
\(438\) −2.45607e14 −0.728002
\(439\) −2.47248e14 −0.723731 −0.361865 0.932230i \(-0.617860\pi\)
−0.361865 + 0.932230i \(0.617860\pi\)
\(440\) 4.69478e14 1.35714
\(441\) 1.96280e14 0.560354
\(442\) 3.89527e14 1.09829
\(443\) −1.88537e13 −0.0525019 −0.0262510 0.999655i \(-0.508357\pi\)
−0.0262510 + 0.999655i \(0.508357\pi\)
\(444\) 7.60958e13 0.209293
\(445\) 1.07715e15 2.92614
\(446\) 3.63489e14 0.975326
\(447\) −2.91612e14 −0.772883
\(448\) −1.60617e15 −4.20497
\(449\) 2.27295e14 0.587808 0.293904 0.955835i \(-0.405045\pi\)
0.293904 + 0.955835i \(0.405045\pi\)
\(450\) −5.93166e14 −1.51533
\(451\) 1.56770e14 0.395633
\(452\) −1.32447e15 −3.30204
\(453\) −4.61126e13 −0.113574
\(454\) 5.28174e14 1.28520
\(455\) −9.99526e14 −2.40288
\(456\) 2.88034e14 0.684127
\(457\) −1.45117e14 −0.340549 −0.170275 0.985397i \(-0.554465\pi\)
−0.170275 + 0.985397i \(0.554465\pi\)
\(458\) 6.77250e14 1.57032
\(459\) 6.09165e13 0.139561
\(460\) −1.19772e15 −2.71137
\(461\) −4.87579e14 −1.09066 −0.545330 0.838221i \(-0.683596\pi\)
−0.545330 + 0.838221i \(0.683596\pi\)
\(462\) 1.95447e14 0.432015
\(463\) 1.64119e14 0.358479 0.179240 0.983805i \(-0.442636\pi\)
0.179240 + 0.983805i \(0.442636\pi\)
\(464\) 1.26025e15 2.72024
\(465\) −1.69934e14 −0.362484
\(466\) 7.95292e14 1.67650
\(467\) −2.77889e14 −0.578934 −0.289467 0.957188i \(-0.593478\pi\)
−0.289467 + 0.957188i \(0.593478\pi\)
\(468\) 3.37018e14 0.693907
\(469\) −1.23478e15 −2.51269
\(470\) −3.03321e15 −6.10049
\(471\) 3.27495e14 0.651014
\(472\) −2.03079e14 −0.399010
\(473\) 2.18795e14 0.424914
\(474\) −8.69495e14 −1.66912
\(475\) 4.87333e14 0.924722
\(476\) 1.65386e15 3.10213
\(477\) 5.21210e13 0.0966411
\(478\) −6.23077e14 −1.14206
\(479\) 8.00287e14 1.45011 0.725054 0.688692i \(-0.241815\pi\)
0.725054 + 0.688692i \(0.241815\pi\)
\(480\) 1.80537e15 3.23399
\(481\) −6.24333e13 −0.110565
\(482\) 7.97050e14 1.39549
\(483\) −3.07763e14 −0.532730
\(484\) −1.43829e15 −2.46149
\(485\) 3.40872e14 0.576782
\(486\) 7.28788e13 0.121928
\(487\) 4.29990e14 0.711294 0.355647 0.934620i \(-0.384261\pi\)
0.355647 + 0.934620i \(0.384261\pi\)
\(488\) 8.32226e14 1.36123
\(489\) 5.07317e12 0.00820504
\(490\) −3.67945e15 −5.88442
\(491\) −1.99682e14 −0.315784 −0.157892 0.987456i \(-0.550470\pi\)
−0.157892 + 0.987456i \(0.550470\pi\)
\(492\) 1.58708e15 2.48194
\(493\) −3.97043e14 −0.614016
\(494\) −3.82872e14 −0.585538
\(495\) −9.75939e13 −0.147603
\(496\) 7.32250e14 1.09524
\(497\) 2.61135e14 0.386283
\(498\) −8.15236e14 −1.19267
\(499\) −1.23903e15 −1.79278 −0.896392 0.443261i \(-0.853821\pi\)
−0.896392 + 0.443261i \(0.853821\pi\)
\(500\) 4.67934e15 6.69652
\(501\) −5.43615e14 −0.769456
\(502\) 2.04374e15 2.86125
\(503\) −1.56087e14 −0.216143 −0.108072 0.994143i \(-0.534468\pi\)
−0.108072 + 0.994143i \(0.534468\pi\)
\(504\) 1.22127e15 1.67279
\(505\) 1.17769e15 1.59562
\(506\) −1.92153e14 −0.257525
\(507\) 1.58986e14 0.210773
\(508\) −1.39105e15 −1.82429
\(509\) −5.82804e14 −0.756092 −0.378046 0.925787i \(-0.623404\pi\)
−0.378046 + 0.925787i \(0.623404\pi\)
\(510\) −1.14194e15 −1.46557
\(511\) 8.55573e14 1.08628
\(512\) 5.97989e13 0.0751118
\(513\) −5.98757e13 −0.0744055
\(514\) −7.73264e13 −0.0950671
\(515\) −2.78691e15 −3.38987
\(516\) 2.21499e15 2.66563
\(517\) −3.51920e14 −0.419030
\(518\) −3.66545e14 −0.431831
\(519\) −1.23910e14 −0.144440
\(520\) −3.89947e15 −4.49767
\(521\) 7.16798e14 0.818067 0.409034 0.912519i \(-0.365866\pi\)
0.409034 + 0.912519i \(0.365866\pi\)
\(522\) −4.75012e14 −0.536434
\(523\) −2.01108e13 −0.0224735 −0.0112368 0.999937i \(-0.503577\pi\)
−0.0112368 + 0.999937i \(0.503577\pi\)
\(524\) −3.65155e15 −4.03790
\(525\) 2.06630e15 2.26108
\(526\) 5.00921e14 0.542434
\(527\) −2.30696e14 −0.247219
\(528\) 4.20535e14 0.445980
\(529\) −6.50234e14 −0.682438
\(530\) −9.77058e14 −1.01485
\(531\) 4.22156e13 0.0433963
\(532\) −1.62560e15 −1.65387
\(533\) −1.30213e15 −1.31116
\(534\) 1.74945e15 1.74351
\(535\) 1.16852e15 1.15263
\(536\) −4.81727e15 −4.70321
\(537\) −8.18557e14 −0.791026
\(538\) −4.60135e14 −0.440132
\(539\) −4.26897e14 −0.404189
\(540\) −9.88003e14 −0.925960
\(541\) 4.54140e14 0.421313 0.210657 0.977560i \(-0.432440\pi\)
0.210657 + 0.977560i \(0.432440\pi\)
\(542\) −7.54015e13 −0.0692442
\(543\) −5.59526e14 −0.508653
\(544\) 2.45090e15 2.20562
\(545\) 3.64633e15 3.24845
\(546\) −1.62338e15 −1.43173
\(547\) 7.23349e14 0.631564 0.315782 0.948832i \(-0.397733\pi\)
0.315782 + 0.948832i \(0.397733\pi\)
\(548\) 5.07169e14 0.438389
\(549\) −1.73001e14 −0.148047
\(550\) 1.29010e15 1.09302
\(551\) 3.90259e14 0.327356
\(552\) −1.20068e15 −0.997157
\(553\) 3.02889e15 2.49055
\(554\) 6.50594e13 0.0529671
\(555\) 1.83029e14 0.147540
\(556\) 5.76727e15 4.60319
\(557\) −1.59614e14 −0.126144 −0.0630719 0.998009i \(-0.520090\pi\)
−0.0630719 + 0.998009i \(0.520090\pi\)
\(558\) −2.75999e14 −0.215983
\(559\) −1.81731e15 −1.40820
\(560\) −1.26263e16 −9.68821
\(561\) −1.32490e14 −0.100667
\(562\) −6.31929e14 −0.475465
\(563\) −1.01573e14 −0.0756799 −0.0378399 0.999284i \(-0.512048\pi\)
−0.0378399 + 0.999284i \(0.512048\pi\)
\(564\) −3.56270e15 −2.62872
\(565\) −3.18569e15 −2.32776
\(566\) −1.94712e15 −1.40897
\(567\) −2.53874e14 −0.181933
\(568\) 1.01877e15 0.723040
\(569\) 2.30804e15 1.62228 0.811139 0.584853i \(-0.198848\pi\)
0.811139 + 0.584853i \(0.198848\pi\)
\(570\) 1.12243e15 0.781351
\(571\) 4.18920e14 0.288823 0.144412 0.989518i \(-0.453871\pi\)
0.144412 + 0.989518i \(0.453871\pi\)
\(572\) −7.32995e14 −0.500522
\(573\) −1.50984e15 −1.02113
\(574\) −7.64480e15 −5.12095
\(575\) −2.03147e15 −1.34784
\(576\) 1.30260e15 0.856028
\(577\) 7.42714e14 0.483454 0.241727 0.970344i \(-0.422286\pi\)
0.241727 + 0.970344i \(0.422286\pi\)
\(578\) 1.39762e15 0.901125
\(579\) 8.63854e14 0.551707
\(580\) 6.43963e15 4.07387
\(581\) 2.83988e15 1.77964
\(582\) 5.53628e14 0.343670
\(583\) −1.13360e14 −0.0697081
\(584\) 3.33787e15 2.03329
\(585\) 8.10613e14 0.489166
\(586\) −3.26140e15 −1.94970
\(587\) −2.92445e15 −1.73195 −0.865973 0.500091i \(-0.833300\pi\)
−0.865973 + 0.500091i \(0.833300\pi\)
\(588\) −4.32174e15 −2.53561
\(589\) 2.26755e14 0.131802
\(590\) −7.91371e14 −0.455716
\(591\) −6.04339e14 −0.344786
\(592\) −7.88679e14 −0.445790
\(593\) −5.58726e14 −0.312895 −0.156447 0.987686i \(-0.550004\pi\)
−0.156447 + 0.987686i \(0.550004\pi\)
\(594\) −1.58507e14 −0.0879475
\(595\) 3.97794e15 2.18683
\(596\) 6.42079e15 3.49731
\(597\) 2.30248e14 0.124262
\(598\) 1.59602e15 0.853458
\(599\) −3.05362e15 −1.61796 −0.808981 0.587835i \(-0.799980\pi\)
−0.808981 + 0.587835i \(0.799980\pi\)
\(600\) 8.06130e15 4.23227
\(601\) −2.62270e15 −1.36439 −0.682195 0.731170i \(-0.738975\pi\)
−0.682195 + 0.731170i \(0.738975\pi\)
\(602\) −1.06694e16 −5.49995
\(603\) 1.00140e15 0.511521
\(604\) 1.01532e15 0.513925
\(605\) −3.45946e15 −1.73522
\(606\) 1.91275e15 0.950733
\(607\) −9.29433e14 −0.457805 −0.228902 0.973449i \(-0.573514\pi\)
−0.228902 + 0.973449i \(0.573514\pi\)
\(608\) −2.40902e15 −1.17590
\(609\) 1.65470e15 0.800434
\(610\) 3.24307e15 1.55468
\(611\) 2.92304e15 1.38870
\(612\) −1.34128e15 −0.631518
\(613\) 1.23597e15 0.576735 0.288367 0.957520i \(-0.406888\pi\)
0.288367 + 0.957520i \(0.406888\pi\)
\(614\) 6.25821e15 2.89417
\(615\) 3.81733e15 1.74963
\(616\) −2.65618e15 −1.20660
\(617\) 3.02427e15 1.36161 0.680804 0.732466i \(-0.261631\pi\)
0.680804 + 0.732466i \(0.261631\pi\)
\(618\) −4.52637e15 −2.01982
\(619\) −2.25456e14 −0.0997156 −0.0498578 0.998756i \(-0.515877\pi\)
−0.0498578 + 0.998756i \(0.515877\pi\)
\(620\) 3.74165e15 1.64025
\(621\) 2.49595e14 0.108451
\(622\) 8.03435e15 3.46022
\(623\) −6.09422e15 −2.60156
\(624\) −3.49295e15 −1.47801
\(625\) 5.55250e15 2.32889
\(626\) −4.75305e15 −1.97612
\(627\) 1.30226e14 0.0536694
\(628\) −7.21088e15 −2.94585
\(629\) 2.48474e14 0.100624
\(630\) 4.75910e15 1.91052
\(631\) −3.97438e15 −1.58164 −0.790820 0.612049i \(-0.790346\pi\)
−0.790820 + 0.612049i \(0.790346\pi\)
\(632\) 1.18167e16 4.66178
\(633\) 1.97010e15 0.770493
\(634\) 1.90610e15 0.739016
\(635\) −3.34583e15 −1.28602
\(636\) −1.14761e15 −0.437302
\(637\) 3.54580e15 1.33951
\(638\) 1.03312e15 0.386935
\(639\) −2.11780e14 −0.0786377
\(640\) −9.20289e15 −3.38793
\(641\) 1.18795e15 0.433590 0.216795 0.976217i \(-0.430440\pi\)
0.216795 + 0.976217i \(0.430440\pi\)
\(642\) 1.89785e15 0.686784
\(643\) −3.97154e15 −1.42495 −0.712473 0.701699i \(-0.752425\pi\)
−0.712473 + 0.701699i \(0.752425\pi\)
\(644\) 6.77641e15 2.41061
\(645\) 5.32762e15 1.87912
\(646\) 1.52377e15 0.532892
\(647\) −6.53729e14 −0.226686 −0.113343 0.993556i \(-0.536156\pi\)
−0.113343 + 0.993556i \(0.536156\pi\)
\(648\) −9.90444e14 −0.340540
\(649\) −9.18164e13 −0.0313022
\(650\) −1.07156e16 −3.62236
\(651\) 9.61442e14 0.322276
\(652\) −1.11702e14 −0.0371279
\(653\) 2.17325e15 0.716286 0.358143 0.933667i \(-0.383410\pi\)
0.358143 + 0.933667i \(0.383410\pi\)
\(654\) 5.92220e15 1.93555
\(655\) −8.78290e15 −2.84650
\(656\) −1.64490e16 −5.28649
\(657\) −6.93868e14 −0.221140
\(658\) 1.71611e16 5.42380
\(659\) −1.49091e15 −0.467285 −0.233643 0.972323i \(-0.575065\pi\)
−0.233643 + 0.972323i \(0.575065\pi\)
\(660\) 2.14885e15 0.667904
\(661\) −3.78616e15 −1.16705 −0.583527 0.812094i \(-0.698328\pi\)
−0.583527 + 0.812094i \(0.698328\pi\)
\(662\) 7.05593e15 2.15693
\(663\) 1.10046e15 0.333618
\(664\) 1.10793e16 3.33110
\(665\) −3.90998e15 −1.16588
\(666\) 2.97267e14 0.0879102
\(667\) −1.62682e15 −0.477141
\(668\) 1.19695e16 3.48180
\(669\) 1.02690e15 0.296268
\(670\) −1.87722e16 −5.37161
\(671\) 3.76267e14 0.106788
\(672\) −1.02143e16 −2.87526
\(673\) −2.04085e15 −0.569809 −0.284904 0.958556i \(-0.591962\pi\)
−0.284904 + 0.958556i \(0.591962\pi\)
\(674\) 4.36689e14 0.120932
\(675\) −1.67576e15 −0.460301
\(676\) −3.50060e15 −0.953753
\(677\) 3.81365e15 1.03063 0.515316 0.857000i \(-0.327675\pi\)
0.515316 + 0.857000i \(0.327675\pi\)
\(678\) −5.17405e15 −1.38697
\(679\) −1.92857e15 −0.512803
\(680\) 1.55193e16 4.09328
\(681\) 1.49215e15 0.390395
\(682\) 6.00281e14 0.155790
\(683\) −3.34357e15 −0.860789 −0.430394 0.902641i \(-0.641626\pi\)
−0.430394 + 0.902641i \(0.641626\pi\)
\(684\) 1.31836e15 0.336686
\(685\) 1.21987e15 0.309040
\(686\) 8.43391e15 2.11956
\(687\) 1.91331e15 0.477005
\(688\) −2.29568e16 −5.67774
\(689\) 9.41568e14 0.231018
\(690\) −4.67890e15 −1.13887
\(691\) −6.30069e14 −0.152145 −0.0760727 0.997102i \(-0.524238\pi\)
−0.0760727 + 0.997102i \(0.524238\pi\)
\(692\) 2.72829e15 0.653593
\(693\) 5.52161e14 0.131230
\(694\) 3.63634e15 0.857409
\(695\) 1.38717e16 3.24500
\(696\) 6.45554e15 1.49824
\(697\) 5.18226e15 1.19327
\(698\) −1.22276e16 −2.79343
\(699\) 2.24679e15 0.509258
\(700\) −4.54963e16 −10.2314
\(701\) −2.69170e15 −0.600589 −0.300295 0.953846i \(-0.597085\pi\)
−0.300295 + 0.953846i \(0.597085\pi\)
\(702\) 1.31656e15 0.291465
\(703\) −2.44229e14 −0.0536466
\(704\) −2.83308e15 −0.617461
\(705\) −8.56918e15 −1.85310
\(706\) 8.22270e15 1.76437
\(707\) −6.66307e15 −1.41863
\(708\) −9.29514e14 −0.196369
\(709\) −9.45615e15 −1.98226 −0.991129 0.132904i \(-0.957570\pi\)
−0.991129 + 0.132904i \(0.957570\pi\)
\(710\) 3.97002e15 0.825794
\(711\) −2.45642e15 −0.507015
\(712\) −2.37755e16 −4.86956
\(713\) −9.45239e14 −0.192110
\(714\) 6.46079e15 1.30300
\(715\) −1.76304e15 −0.352840
\(716\) 1.80232e16 3.57940
\(717\) −1.76026e15 −0.346915
\(718\) 1.15391e16 2.25676
\(719\) 9.04031e15 1.75459 0.877293 0.479956i \(-0.159347\pi\)
0.877293 + 0.479956i \(0.159347\pi\)
\(720\) 1.02399e16 1.97228
\(721\) 1.57676e16 3.01386
\(722\) 8.52208e15 1.61656
\(723\) 2.25176e15 0.423898
\(724\) 1.23198e16 2.30166
\(725\) 1.09223e16 2.02515
\(726\) −5.61868e15 −1.03391
\(727\) 4.79773e15 0.876187 0.438093 0.898929i \(-0.355654\pi\)
0.438093 + 0.898929i \(0.355654\pi\)
\(728\) 2.20622e16 3.99877
\(729\) 2.05891e14 0.0370370
\(730\) 1.30072e16 2.32225
\(731\) 7.23258e15 1.28159
\(732\) 3.80918e15 0.669917
\(733\) −7.50700e15 −1.31037 −0.655186 0.755467i \(-0.727410\pi\)
−0.655186 + 0.755467i \(0.727410\pi\)
\(734\) 9.56776e15 1.65761
\(735\) −1.03949e16 −1.78747
\(736\) 1.00422e16 1.71395
\(737\) −2.17799e15 −0.368965
\(738\) 6.19992e15 1.04250
\(739\) 9.01567e15 1.50471 0.752356 0.658756i \(-0.228917\pi\)
0.752356 + 0.658756i \(0.228917\pi\)
\(740\) −4.02999e15 −0.667620
\(741\) −1.08166e15 −0.177865
\(742\) 5.52794e15 0.902281
\(743\) 6.14048e15 0.994864 0.497432 0.867503i \(-0.334276\pi\)
0.497432 + 0.867503i \(0.334276\pi\)
\(744\) 3.75090e15 0.603232
\(745\) 1.54436e16 2.46541
\(746\) 1.07720e16 1.70700
\(747\) −2.30314e15 −0.362290
\(748\) 2.91720e15 0.455520
\(749\) −6.61118e15 −1.02478
\(750\) 1.82798e16 2.81277
\(751\) 7.14670e15 1.09166 0.545828 0.837897i \(-0.316215\pi\)
0.545828 + 0.837897i \(0.316215\pi\)
\(752\) 3.69248e16 5.59913
\(753\) 5.77380e15 0.869140
\(754\) −8.58110e15 −1.28233
\(755\) 2.44210e15 0.362289
\(756\) 5.58986e15 0.823249
\(757\) −5.55750e14 −0.0812553 −0.0406277 0.999174i \(-0.512936\pi\)
−0.0406277 + 0.999174i \(0.512936\pi\)
\(758\) −4.12543e15 −0.598809
\(759\) −5.42855e14 −0.0782265
\(760\) −1.52541e16 −2.18229
\(761\) −6.98562e15 −0.992177 −0.496089 0.868272i \(-0.665231\pi\)
−0.496089 + 0.868272i \(0.665231\pi\)
\(762\) −5.43414e15 −0.766264
\(763\) −2.06300e16 −2.88812
\(764\) 3.32441e16 4.62063
\(765\) −3.22611e15 −0.445185
\(766\) 2.30206e16 3.15397
\(767\) 7.62626e14 0.103738
\(768\) −3.96859e15 −0.535981
\(769\) −1.23209e16 −1.65215 −0.826073 0.563563i \(-0.809430\pi\)
−0.826073 + 0.563563i \(0.809430\pi\)
\(770\) −1.03508e16 −1.37808
\(771\) −2.18456e14 −0.0288778
\(772\) −1.90206e16 −2.49648
\(773\) −6.39007e15 −0.832757 −0.416378 0.909191i \(-0.636701\pi\)
−0.416378 + 0.909191i \(0.636701\pi\)
\(774\) 8.65286e15 1.11965
\(775\) 6.34627e15 0.815378
\(776\) −7.52396e15 −0.959858
\(777\) −1.03553e15 −0.131174
\(778\) −2.98217e16 −3.75097
\(779\) −5.09372e15 −0.636179
\(780\) −1.78483e16 −2.21348
\(781\) 4.60610e14 0.0567221
\(782\) −6.35190e15 −0.776724
\(783\) −1.34196e15 −0.162949
\(784\) 4.47917e16 5.40082
\(785\) −1.73440e16 −2.07666
\(786\) −1.42648e16 −1.69606
\(787\) −1.64082e16 −1.93731 −0.968657 0.248403i \(-0.920094\pi\)
−0.968657 + 0.248403i \(0.920094\pi\)
\(788\) 1.33065e16 1.56016
\(789\) 1.41516e15 0.164771
\(790\) 4.60480e16 5.32429
\(791\) 1.80238e16 2.06955
\(792\) 2.15416e15 0.245634
\(793\) −3.12527e15 −0.353904
\(794\) −4.80508e15 −0.540366
\(795\) −2.76030e15 −0.308274
\(796\) −5.06967e15 −0.562287
\(797\) −8.89142e15 −0.979378 −0.489689 0.871897i \(-0.662890\pi\)
−0.489689 + 0.871897i \(0.662890\pi\)
\(798\) −6.35040e15 −0.694681
\(799\) −1.16332e16 −1.26384
\(800\) −6.74223e16 −7.27459
\(801\) 4.94240e15 0.529613
\(802\) 2.35261e16 2.50375
\(803\) 1.50912e15 0.159510
\(804\) −2.20491e16 −2.31464
\(805\) 1.62990e16 1.69935
\(806\) −4.98592e15 −0.516301
\(807\) −1.29993e15 −0.133696
\(808\) −2.59948e16 −2.65537
\(809\) −3.54139e15 −0.359300 −0.179650 0.983731i \(-0.557497\pi\)
−0.179650 + 0.983731i \(0.557497\pi\)
\(810\) −3.85962e15 −0.388935
\(811\) −3.04072e15 −0.304342 −0.152171 0.988354i \(-0.548626\pi\)
−0.152171 + 0.988354i \(0.548626\pi\)
\(812\) −3.64337e16 −3.62198
\(813\) −2.13018e14 −0.0210338
\(814\) −6.46540e14 −0.0634105
\(815\) −2.68672e14 −0.0261732
\(816\) 1.39014e16 1.34512
\(817\) −7.10901e15 −0.683262
\(818\) −5.44900e15 −0.520204
\(819\) −4.58624e15 −0.434906
\(820\) −8.40510e16 −7.91711
\(821\) 1.85223e16 1.73304 0.866518 0.499146i \(-0.166353\pi\)
0.866518 + 0.499146i \(0.166353\pi\)
\(822\) 1.98125e15 0.184138
\(823\) 8.11072e15 0.748791 0.374395 0.927269i \(-0.377850\pi\)
0.374395 + 0.927269i \(0.377850\pi\)
\(824\) 6.15147e16 5.64130
\(825\) 3.64469e15 0.332020
\(826\) 4.47737e15 0.405166
\(827\) 2.56159e15 0.230266 0.115133 0.993350i \(-0.463271\pi\)
0.115133 + 0.993350i \(0.463271\pi\)
\(828\) −5.49565e15 −0.490741
\(829\) −1.05268e16 −0.933788 −0.466894 0.884313i \(-0.654627\pi\)
−0.466894 + 0.884313i \(0.654627\pi\)
\(830\) 4.31745e16 3.80450
\(831\) 1.83800e14 0.0160894
\(832\) 2.35315e16 2.04631
\(833\) −1.41117e16 −1.21908
\(834\) 2.25298e16 1.93350
\(835\) 2.87895e16 2.45448
\(836\) −2.86736e15 −0.242855
\(837\) −7.79728e14 −0.0656074
\(838\) 4.45719e15 0.372580
\(839\) −1.03348e16 −0.858242 −0.429121 0.903247i \(-0.641177\pi\)
−0.429121 + 0.903247i \(0.641177\pi\)
\(840\) −6.46776e16 −5.33602
\(841\) −3.45383e15 −0.283089
\(842\) 2.10166e16 1.71138
\(843\) −1.78527e15 −0.144428
\(844\) −4.33783e16 −3.48649
\(845\) −8.41982e15 −0.672343
\(846\) −1.39176e16 −1.10415
\(847\) 1.95727e16 1.54274
\(848\) 1.18942e16 0.931447
\(849\) −5.50083e15 −0.427992
\(850\) 4.26462e16 3.29667
\(851\) 1.01808e15 0.0781933
\(852\) 4.66304e15 0.355837
\(853\) −1.89957e16 −1.44024 −0.720122 0.693848i \(-0.755914\pi\)
−0.720122 + 0.693848i \(0.755914\pi\)
\(854\) −1.83484e16 −1.38223
\(855\) 3.17099e15 0.237345
\(856\) −2.57924e16 −1.91816
\(857\) −1.71475e16 −1.26709 −0.633543 0.773708i \(-0.718400\pi\)
−0.633543 + 0.773708i \(0.718400\pi\)
\(858\) −2.86344e15 −0.210236
\(859\) −1.16779e16 −0.851930 −0.425965 0.904740i \(-0.640065\pi\)
−0.425965 + 0.904740i \(0.640065\pi\)
\(860\) −1.17305e17 −8.50305
\(861\) −2.15974e16 −1.55555
\(862\) 2.91335e16 2.08498
\(863\) 4.65167e15 0.330788 0.165394 0.986228i \(-0.447110\pi\)
0.165394 + 0.986228i \(0.447110\pi\)
\(864\) 8.28377e15 0.585332
\(865\) 6.56222e15 0.460747
\(866\) 2.08723e16 1.45620
\(867\) 3.94843e15 0.273728
\(868\) −2.11693e16 −1.45830
\(869\) 5.34258e15 0.365715
\(870\) 2.51564e16 1.71116
\(871\) 1.80904e16 1.22278
\(872\) −8.04844e16 −5.40594
\(873\) 1.56406e15 0.104394
\(874\) 6.24338e15 0.414101
\(875\) −6.36778e16 −4.19704
\(876\) 1.52778e16 1.00066
\(877\) 1.09513e15 0.0712798 0.0356399 0.999365i \(-0.488653\pi\)
0.0356399 + 0.999365i \(0.488653\pi\)
\(878\) 2.12668e16 1.37557
\(879\) −9.21382e15 −0.592245
\(880\) −2.22713e16 −1.42263
\(881\) 3.96514e15 0.251705 0.125852 0.992049i \(-0.459833\pi\)
0.125852 + 0.992049i \(0.459833\pi\)
\(882\) −1.68828e16 −1.06504
\(883\) −7.13057e15 −0.447034 −0.223517 0.974700i \(-0.571754\pi\)
−0.223517 + 0.974700i \(0.571754\pi\)
\(884\) −2.42302e16 −1.50963
\(885\) −2.23571e15 −0.138429
\(886\) 1.62168e15 0.0997885
\(887\) −1.38631e16 −0.847772 −0.423886 0.905716i \(-0.639334\pi\)
−0.423886 + 0.905716i \(0.639334\pi\)
\(888\) −4.03995e15 −0.245530
\(889\) 1.89298e16 1.14337
\(890\) −9.26499e16 −5.56160
\(891\) −4.47801e14 −0.0267152
\(892\) −2.26105e16 −1.34062
\(893\) 1.14344e16 0.673802
\(894\) 2.50828e16 1.46899
\(895\) 4.33504e16 2.52328
\(896\) 5.20676e16 3.01213
\(897\) 4.50895e15 0.259249
\(898\) −1.95506e16 −1.11722
\(899\) 5.08213e15 0.288647
\(900\) 3.68974e16 2.08287
\(901\) −3.74728e15 −0.210247
\(902\) −1.34845e16 −0.751965
\(903\) −3.01423e16 −1.67068
\(904\) 7.03168e16 3.87376
\(905\) 2.96322e16 1.62254
\(906\) 3.96634e15 0.215866
\(907\) −2.73671e16 −1.48043 −0.740216 0.672369i \(-0.765277\pi\)
−0.740216 + 0.672369i \(0.765277\pi\)
\(908\) −3.28546e16 −1.76654
\(909\) 5.40374e15 0.288797
\(910\) 8.59734e16 4.56706
\(911\) 8.41323e15 0.444234 0.222117 0.975020i \(-0.428703\pi\)
0.222117 + 0.975020i \(0.428703\pi\)
\(912\) −1.36639e16 −0.717137
\(913\) 5.00919e15 0.261323
\(914\) 1.24821e16 0.647269
\(915\) 9.16204e15 0.472255
\(916\) −4.21277e16 −2.15846
\(917\) 4.96914e16 2.53075
\(918\) −5.23968e15 −0.265259
\(919\) 8.20052e15 0.412673 0.206337 0.978481i \(-0.433846\pi\)
0.206337 + 0.978481i \(0.433846\pi\)
\(920\) 6.35876e16 3.18082
\(921\) 1.76802e16 0.879140
\(922\) 4.19387e16 2.07298
\(923\) −3.82582e15 −0.187981
\(924\) −1.21576e16 −0.593817
\(925\) −6.83532e15 −0.331879
\(926\) −1.41166e16 −0.681349
\(927\) −1.27875e16 −0.613547
\(928\) −5.39922e16 −2.57524
\(929\) 1.47434e16 0.699057 0.349528 0.936926i \(-0.386342\pi\)
0.349528 + 0.936926i \(0.386342\pi\)
\(930\) 1.46167e16 0.688960
\(931\) 1.38706e16 0.649937
\(932\) −4.94705e16 −2.30440
\(933\) 2.26980e16 1.05109
\(934\) 2.39024e16 1.10036
\(935\) 7.01659e15 0.321116
\(936\) −1.78924e16 −0.814051
\(937\) 1.63627e16 0.740094 0.370047 0.929013i \(-0.379342\pi\)
0.370047 + 0.929013i \(0.379342\pi\)
\(938\) 1.06208e17 4.77577
\(939\) −1.34279e16 −0.600271
\(940\) 1.88679e17 8.38531
\(941\) −1.85372e16 −0.819033 −0.409517 0.912303i \(-0.634303\pi\)
−0.409517 + 0.912303i \(0.634303\pi\)
\(942\) −2.81693e16 −1.23736
\(943\) 2.12335e16 0.927271
\(944\) 9.63375e15 0.418263
\(945\) 1.34450e16 0.580345
\(946\) −1.88195e16 −0.807618
\(947\) −4.50910e16 −1.92382 −0.961912 0.273360i \(-0.911865\pi\)
−0.961912 + 0.273360i \(0.911865\pi\)
\(948\) 5.40862e16 2.29425
\(949\) −1.25347e16 −0.528629
\(950\) −4.19176e16 −1.75758
\(951\) 5.38494e15 0.224485
\(952\) −8.78039e16 −3.63924
\(953\) −4.10630e15 −0.169215 −0.0846077 0.996414i \(-0.526964\pi\)
−0.0846077 + 0.996414i \(0.526964\pi\)
\(954\) −4.48315e15 −0.183682
\(955\) 7.99605e16 3.25729
\(956\) 3.87580e16 1.56979
\(957\) 2.91869e15 0.117536
\(958\) −6.88361e16 −2.75617
\(959\) −6.90170e15 −0.274760
\(960\) −6.89851e16 −2.73063
\(961\) −2.24556e16 −0.883783
\(962\) 5.37015e15 0.210147
\(963\) 5.36166e15 0.208619
\(964\) −4.95798e16 −1.91815
\(965\) −4.57493e16 −1.75988
\(966\) 2.64720e16 1.01254
\(967\) −1.49354e16 −0.568031 −0.284016 0.958820i \(-0.591667\pi\)
−0.284016 + 0.958820i \(0.591667\pi\)
\(968\) 7.63595e16 2.88768
\(969\) 4.30481e15 0.161873
\(970\) −2.93198e16 −1.09627
\(971\) 2.32789e16 0.865478 0.432739 0.901519i \(-0.357547\pi\)
0.432739 + 0.901519i \(0.357547\pi\)
\(972\) −4.53337e15 −0.167593
\(973\) −7.84827e16 −2.88505
\(974\) −3.69852e16 −1.35193
\(975\) −3.02727e16 −1.10034
\(976\) −3.94795e16 −1.42691
\(977\) −8.28859e15 −0.297893 −0.148947 0.988845i \(-0.547588\pi\)
−0.148947 + 0.988845i \(0.547588\pi\)
\(978\) −4.36365e14 −0.0155950
\(979\) −1.07494e16 −0.382015
\(980\) 2.28877e17 8.08832
\(981\) 1.67309e16 0.587949
\(982\) 1.71755e16 0.600199
\(983\) 3.73103e16 1.29653 0.648267 0.761413i \(-0.275494\pi\)
0.648267 + 0.761413i \(0.275494\pi\)
\(984\) −8.42587e16 −2.91167
\(985\) 3.20055e16 1.09983
\(986\) 3.41513e16 1.16704
\(987\) 4.84822e16 1.64755
\(988\) 2.38162e16 0.804840
\(989\) 2.96343e16 0.995898
\(990\) 8.39446e15 0.280543
\(991\) 4.50911e15 0.149860 0.0749300 0.997189i \(-0.476127\pi\)
0.0749300 + 0.997189i \(0.476127\pi\)
\(992\) −3.13714e16 −1.03686
\(993\) 1.99338e16 0.655193
\(994\) −2.24613e16 −0.734194
\(995\) −1.21938e16 −0.396381
\(996\) 5.07111e16 1.63937
\(997\) 3.16876e16 1.01875 0.509373 0.860546i \(-0.329877\pi\)
0.509373 + 0.860546i \(0.329877\pi\)
\(998\) 1.06574e17 3.40748
\(999\) 8.39815e14 0.0267038
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.a.1.2 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.a.1.2 26 1.1 even 1 trivial