Properties

Label 8470.2.a.co.1.4
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4400.1
Defining polynomial: \(x^{4} - 7 x^{2} + 11\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.14896\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.67989 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.14896 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.14896 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.67989 q^{9} -1.00000 q^{10} +1.14896 q^{12} +3.79720 q^{13} +1.00000 q^{14} +1.14896 q^{15} +1.00000 q^{16} +6.47709 q^{17} +1.67989 q^{18} +2.67989 q^{19} +1.00000 q^{20} -1.14896 q^{21} +2.65626 q^{23} -1.14896 q^{24} +1.00000 q^{25} -3.79720 q^{26} -5.37701 q^{27} -1.00000 q^{28} +0.386922 q^{29} -1.14896 q^{30} -6.01867 q^{31} -1.00000 q^{32} -6.47709 q^{34} -1.00000 q^{35} -1.67989 q^{36} +2.38692 q^{37} -2.67989 q^{38} +4.36284 q^{39} -1.00000 q^{40} +9.47709 q^{41} +1.14896 q^{42} -4.66428 q^{43} -1.67989 q^{45} -2.65626 q^{46} -4.48205 q^{47} +1.14896 q^{48} +1.00000 q^{49} -1.00000 q^{50} +7.44193 q^{51} +3.79720 q^{52} -5.09819 q^{53} +5.37701 q^{54} +1.00000 q^{56} +3.07909 q^{57} -0.386922 q^{58} +4.74643 q^{59} +1.14896 q^{60} +11.6593 q^{61} +6.01867 q^{62} +1.67989 q^{63} +1.00000 q^{64} +3.79720 q^{65} +5.71316 q^{67} +6.47709 q^{68} +3.05194 q^{69} +1.00000 q^{70} +3.33119 q^{71} +1.67989 q^{72} +15.0127 q^{73} -2.38692 q^{74} +1.14896 q^{75} +2.67989 q^{76} -4.36284 q^{78} -6.03633 q^{79} +1.00000 q^{80} -1.13831 q^{81} -9.47709 q^{82} -11.1551 q^{83} -1.14896 q^{84} +6.47709 q^{85} +4.66428 q^{86} +0.444559 q^{87} -5.87277 q^{89} +1.67989 q^{90} -3.79720 q^{91} +2.65626 q^{92} -6.91522 q^{93} +4.48205 q^{94} +2.67989 q^{95} -1.14896 q^{96} -3.30450 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.14896 0.663353 0.331677 0.943393i \(-0.392386\pi\)
0.331677 + 0.943393i \(0.392386\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.14896 −0.469061
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −1.67989 −0.559963
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.14896 0.331677
\(13\) 3.79720 1.05315 0.526577 0.850127i \(-0.323475\pi\)
0.526577 + 0.850127i \(0.323475\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.14896 0.296660
\(16\) 1.00000 0.250000
\(17\) 6.47709 1.57093 0.785463 0.618909i \(-0.212425\pi\)
0.785463 + 0.618909i \(0.212425\pi\)
\(18\) 1.67989 0.395953
\(19\) 2.67989 0.614809 0.307404 0.951579i \(-0.400540\pi\)
0.307404 + 0.951579i \(0.400540\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.14896 −0.250724
\(22\) 0 0
\(23\) 2.65626 0.553869 0.276934 0.960889i \(-0.410682\pi\)
0.276934 + 0.960889i \(0.410682\pi\)
\(24\) −1.14896 −0.234531
\(25\) 1.00000 0.200000
\(26\) −3.79720 −0.744693
\(27\) −5.37701 −1.03481
\(28\) −1.00000 −0.188982
\(29\) 0.386922 0.0718497 0.0359248 0.999354i \(-0.488562\pi\)
0.0359248 + 0.999354i \(0.488562\pi\)
\(30\) −1.14896 −0.209771
\(31\) −6.01867 −1.08099 −0.540493 0.841349i \(-0.681762\pi\)
−0.540493 + 0.841349i \(0.681762\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.47709 −1.11081
\(35\) −1.00000 −0.169031
\(36\) −1.67989 −0.279981
\(37\) 2.38692 0.392408 0.196204 0.980563i \(-0.437139\pi\)
0.196204 + 0.980563i \(0.437139\pi\)
\(38\) −2.67989 −0.434735
\(39\) 4.36284 0.698613
\(40\) −1.00000 −0.158114
\(41\) 9.47709 1.48007 0.740037 0.672567i \(-0.234808\pi\)
0.740037 + 0.672567i \(0.234808\pi\)
\(42\) 1.14896 0.177289
\(43\) −4.66428 −0.711296 −0.355648 0.934620i \(-0.615740\pi\)
−0.355648 + 0.934620i \(0.615740\pi\)
\(44\) 0 0
\(45\) −1.67989 −0.250423
\(46\) −2.65626 −0.391644
\(47\) −4.48205 −0.653774 −0.326887 0.945063i \(-0.606000\pi\)
−0.326887 + 0.945063i \(0.606000\pi\)
\(48\) 1.14896 0.165838
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 7.44193 1.04208
\(52\) 3.79720 0.526577
\(53\) −5.09819 −0.700290 −0.350145 0.936695i \(-0.613868\pi\)
−0.350145 + 0.936695i \(0.613868\pi\)
\(54\) 5.37701 0.731718
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 3.07909 0.407835
\(58\) −0.386922 −0.0508054
\(59\) 4.74643 0.617933 0.308966 0.951073i \(-0.400017\pi\)
0.308966 + 0.951073i \(0.400017\pi\)
\(60\) 1.14896 0.148330
\(61\) 11.6593 1.49282 0.746412 0.665484i \(-0.231775\pi\)
0.746412 + 0.665484i \(0.231775\pi\)
\(62\) 6.01867 0.764372
\(63\) 1.67989 0.211646
\(64\) 1.00000 0.125000
\(65\) 3.79720 0.470985
\(66\) 0 0
\(67\) 5.71316 0.697974 0.348987 0.937128i \(-0.386526\pi\)
0.348987 + 0.937128i \(0.386526\pi\)
\(68\) 6.47709 0.785463
\(69\) 3.05194 0.367411
\(70\) 1.00000 0.119523
\(71\) 3.33119 0.395340 0.197670 0.980269i \(-0.436663\pi\)
0.197670 + 0.980269i \(0.436663\pi\)
\(72\) 1.67989 0.197977
\(73\) 15.0127 1.75710 0.878552 0.477646i \(-0.158510\pi\)
0.878552 + 0.477646i \(0.158510\pi\)
\(74\) −2.38692 −0.277474
\(75\) 1.14896 0.132671
\(76\) 2.67989 0.307404
\(77\) 0 0
\(78\) −4.36284 −0.493994
\(79\) −6.03633 −0.679141 −0.339570 0.940581i \(-0.610282\pi\)
−0.339570 + 0.940581i \(0.610282\pi\)
\(80\) 1.00000 0.111803
\(81\) −1.13831 −0.126479
\(82\) −9.47709 −1.04657
\(83\) −11.1551 −1.22443 −0.612215 0.790691i \(-0.709721\pi\)
−0.612215 + 0.790691i \(0.709721\pi\)
\(84\) −1.14896 −0.125362
\(85\) 6.47709 0.702539
\(86\) 4.66428 0.502962
\(87\) 0.444559 0.0476617
\(88\) 0 0
\(89\) −5.87277 −0.622513 −0.311256 0.950326i \(-0.600750\pi\)
−0.311256 + 0.950326i \(0.600750\pi\)
\(90\) 1.67989 0.177076
\(91\) −3.79720 −0.398055
\(92\) 2.65626 0.276934
\(93\) −6.91522 −0.717075
\(94\) 4.48205 0.462288
\(95\) 2.67989 0.274951
\(96\) −1.14896 −0.117265
\(97\) −3.30450 −0.335522 −0.167761 0.985828i \(-0.553654\pi\)
−0.167761 + 0.985828i \(0.553654\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −5.56420 −0.553658 −0.276829 0.960919i \(-0.589284\pi\)
−0.276829 + 0.960919i \(0.589284\pi\)
\(102\) −7.44193 −0.736861
\(103\) −7.95988 −0.784310 −0.392155 0.919899i \(-0.628270\pi\)
−0.392155 + 0.919899i \(0.628270\pi\)
\(104\) −3.79720 −0.372346
\(105\) −1.14896 −0.112127
\(106\) 5.09819 0.495180
\(107\) −17.4244 −1.68448 −0.842241 0.539100i \(-0.818764\pi\)
−0.842241 + 0.539100i \(0.818764\pi\)
\(108\) −5.37701 −0.517403
\(109\) 4.92398 0.471631 0.235816 0.971798i \(-0.424224\pi\)
0.235816 + 0.971798i \(0.424224\pi\)
\(110\) 0 0
\(111\) 2.74248 0.260305
\(112\) −1.00000 −0.0944911
\(113\) −11.3824 −1.07077 −0.535383 0.844609i \(-0.679833\pi\)
−0.535383 + 0.844609i \(0.679833\pi\)
\(114\) −3.07909 −0.288383
\(115\) 2.65626 0.247698
\(116\) 0.386922 0.0359248
\(117\) −6.37888 −0.589728
\(118\) −4.74643 −0.436944
\(119\) −6.47709 −0.593754
\(120\) −1.14896 −0.104885
\(121\) 0 0
\(122\) −11.6593 −1.05559
\(123\) 10.8888 0.981811
\(124\) −6.01867 −0.540493
\(125\) 1.00000 0.0894427
\(126\) −1.67989 −0.149656
\(127\) −1.43724 −0.127534 −0.0637671 0.997965i \(-0.520311\pi\)
−0.0637671 + 0.997965i \(0.520311\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.35908 −0.471841
\(130\) −3.79720 −0.333037
\(131\) 19.5165 1.70516 0.852582 0.522594i \(-0.175036\pi\)
0.852582 + 0.522594i \(0.175036\pi\)
\(132\) 0 0
\(133\) −2.67989 −0.232376
\(134\) −5.71316 −0.493542
\(135\) −5.37701 −0.462779
\(136\) −6.47709 −0.555406
\(137\) 18.9809 1.62165 0.810823 0.585292i \(-0.199020\pi\)
0.810823 + 0.585292i \(0.199020\pi\)
\(138\) −3.05194 −0.259799
\(139\) 6.58593 0.558611 0.279306 0.960202i \(-0.409896\pi\)
0.279306 + 0.960202i \(0.409896\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −5.14970 −0.433683
\(142\) −3.33119 −0.279548
\(143\) 0 0
\(144\) −1.67989 −0.139991
\(145\) 0.386922 0.0321321
\(146\) −15.0127 −1.24246
\(147\) 1.14896 0.0947647
\(148\) 2.38692 0.196204
\(149\) 10.1138 0.828554 0.414277 0.910151i \(-0.364034\pi\)
0.414277 + 0.910151i \(0.364034\pi\)
\(150\) −1.14896 −0.0938123
\(151\) −11.8335 −0.963000 −0.481500 0.876446i \(-0.659908\pi\)
−0.481500 + 0.876446i \(0.659908\pi\)
\(152\) −2.67989 −0.217368
\(153\) −10.8808 −0.879660
\(154\) 0 0
\(155\) −6.01867 −0.483431
\(156\) 4.36284 0.349307
\(157\) 1.69054 0.134920 0.0674599 0.997722i \(-0.478511\pi\)
0.0674599 + 0.997722i \(0.478511\pi\)
\(158\) 6.03633 0.480225
\(159\) −5.85762 −0.464540
\(160\) −1.00000 −0.0790569
\(161\) −2.65626 −0.209343
\(162\) 1.13831 0.0894341
\(163\) −8.06159 −0.631432 −0.315716 0.948854i \(-0.602245\pi\)
−0.315716 + 0.948854i \(0.602245\pi\)
\(164\) 9.47709 0.740037
\(165\) 0 0
\(166\) 11.1551 0.865803
\(167\) 16.2178 1.25497 0.627487 0.778627i \(-0.284084\pi\)
0.627487 + 0.778627i \(0.284084\pi\)
\(168\) 1.14896 0.0886443
\(169\) 1.41876 0.109135
\(170\) −6.47709 −0.496770
\(171\) −4.50191 −0.344270
\(172\) −4.66428 −0.355648
\(173\) 12.1600 0.924511 0.462255 0.886747i \(-0.347040\pi\)
0.462255 + 0.886747i \(0.347040\pi\)
\(174\) −0.444559 −0.0337019
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 5.45347 0.409908
\(178\) 5.87277 0.440183
\(179\) −23.3033 −1.74177 −0.870886 0.491486i \(-0.836454\pi\)
−0.870886 + 0.491486i \(0.836454\pi\)
\(180\) −1.67989 −0.125211
\(181\) 22.1288 1.64482 0.822411 0.568893i \(-0.192628\pi\)
0.822411 + 0.568893i \(0.192628\pi\)
\(182\) 3.79720 0.281467
\(183\) 13.3961 0.990269
\(184\) −2.65626 −0.195822
\(185\) 2.38692 0.175490
\(186\) 6.91522 0.507048
\(187\) 0 0
\(188\) −4.48205 −0.326887
\(189\) 5.37701 0.391120
\(190\) −2.67989 −0.194420
\(191\) 20.2864 1.46787 0.733936 0.679219i \(-0.237681\pi\)
0.733936 + 0.679219i \(0.237681\pi\)
\(192\) 1.14896 0.0829191
\(193\) −3.84508 −0.276775 −0.138387 0.990378i \(-0.544192\pi\)
−0.138387 + 0.990378i \(0.544192\pi\)
\(194\) 3.30450 0.237250
\(195\) 4.36284 0.312429
\(196\) 1.00000 0.0714286
\(197\) 11.8159 0.841846 0.420923 0.907096i \(-0.361706\pi\)
0.420923 + 0.907096i \(0.361706\pi\)
\(198\) 0 0
\(199\) 12.5496 0.889617 0.444809 0.895626i \(-0.353272\pi\)
0.444809 + 0.895626i \(0.353272\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 6.56420 0.463003
\(202\) 5.56420 0.391496
\(203\) −0.386922 −0.0271566
\(204\) 7.44193 0.521039
\(205\) 9.47709 0.661909
\(206\) 7.95988 0.554591
\(207\) −4.46222 −0.310146
\(208\) 3.79720 0.263289
\(209\) 0 0
\(210\) 1.14896 0.0792859
\(211\) 3.99973 0.275353 0.137676 0.990477i \(-0.456037\pi\)
0.137676 + 0.990477i \(0.456037\pi\)
\(212\) −5.09819 −0.350145
\(213\) 3.82741 0.262250
\(214\) 17.4244 1.19111
\(215\) −4.66428 −0.318101
\(216\) 5.37701 0.365859
\(217\) 6.01867 0.408574
\(218\) −4.92398 −0.333494
\(219\) 17.2490 1.16558
\(220\) 0 0
\(221\) 24.5948 1.65443
\(222\) −2.74248 −0.184063
\(223\) 17.8782 1.19721 0.598605 0.801044i \(-0.295722\pi\)
0.598605 + 0.801044i \(0.295722\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.67989 −0.111993
\(226\) 11.3824 0.757146
\(227\) −5.76204 −0.382440 −0.191220 0.981547i \(-0.561244\pi\)
−0.191220 + 0.981547i \(0.561244\pi\)
\(228\) 3.07909 0.203918
\(229\) −8.57617 −0.566729 −0.283365 0.959012i \(-0.591451\pi\)
−0.283365 + 0.959012i \(0.591451\pi\)
\(230\) −2.65626 −0.175149
\(231\) 0 0
\(232\) −0.386922 −0.0254027
\(233\) 14.4011 0.943445 0.471723 0.881747i \(-0.343632\pi\)
0.471723 + 0.881747i \(0.343632\pi\)
\(234\) 6.37888 0.417000
\(235\) −4.48205 −0.292377
\(236\) 4.74643 0.308966
\(237\) −6.93551 −0.450510
\(238\) 6.47709 0.419848
\(239\) −24.0096 −1.55305 −0.776527 0.630084i \(-0.783020\pi\)
−0.776527 + 0.630084i \(0.783020\pi\)
\(240\) 1.14896 0.0741651
\(241\) −16.7641 −1.07987 −0.539935 0.841707i \(-0.681551\pi\)
−0.539935 + 0.841707i \(0.681551\pi\)
\(242\) 0 0
\(243\) 14.8232 0.950906
\(244\) 11.6593 0.746412
\(245\) 1.00000 0.0638877
\(246\) −10.8888 −0.694245
\(247\) 10.1761 0.647489
\(248\) 6.01867 0.382186
\(249\) −12.8168 −0.812229
\(250\) −1.00000 −0.0632456
\(251\) −2.42488 −0.153057 −0.0765286 0.997067i \(-0.524384\pi\)
−0.0765286 + 0.997067i \(0.524384\pi\)
\(252\) 1.67989 0.105823
\(253\) 0 0
\(254\) 1.43724 0.0901803
\(255\) 7.44193 0.466032
\(256\) 1.00000 0.0625000
\(257\) 5.18367 0.323348 0.161674 0.986844i \(-0.448311\pi\)
0.161674 + 0.986844i \(0.448311\pi\)
\(258\) 5.35908 0.333642
\(259\) −2.38692 −0.148316
\(260\) 3.79720 0.235493
\(261\) −0.649986 −0.0402331
\(262\) −19.5165 −1.20573
\(263\) 20.3770 1.25650 0.628250 0.778011i \(-0.283771\pi\)
0.628250 + 0.778011i \(0.283771\pi\)
\(264\) 0 0
\(265\) −5.09819 −0.313179
\(266\) 2.67989 0.164314
\(267\) −6.74759 −0.412946
\(268\) 5.71316 0.348987
\(269\) −11.4504 −0.698143 −0.349072 0.937096i \(-0.613503\pi\)
−0.349072 + 0.937096i \(0.613503\pi\)
\(270\) 5.37701 0.327234
\(271\) 14.8199 0.900247 0.450124 0.892966i \(-0.351380\pi\)
0.450124 + 0.892966i \(0.351380\pi\)
\(272\) 6.47709 0.392731
\(273\) −4.36284 −0.264051
\(274\) −18.9809 −1.14668
\(275\) 0 0
\(276\) 3.05194 0.183705
\(277\) 24.5158 1.47301 0.736504 0.676433i \(-0.236475\pi\)
0.736504 + 0.676433i \(0.236475\pi\)
\(278\) −6.58593 −0.394998
\(279\) 10.1107 0.605311
\(280\) 1.00000 0.0597614
\(281\) −28.5929 −1.70571 −0.852856 0.522146i \(-0.825132\pi\)
−0.852856 + 0.522146i \(0.825132\pi\)
\(282\) 5.14970 0.306660
\(283\) −5.19494 −0.308807 −0.154404 0.988008i \(-0.549346\pi\)
−0.154404 + 0.988008i \(0.549346\pi\)
\(284\) 3.33119 0.197670
\(285\) 3.07909 0.182389
\(286\) 0 0
\(287\) −9.47709 −0.559415
\(288\) 1.67989 0.0989884
\(289\) 24.9527 1.46781
\(290\) −0.386922 −0.0227209
\(291\) −3.79675 −0.222569
\(292\) 15.0127 0.878552
\(293\) 18.2205 1.06445 0.532225 0.846603i \(-0.321356\pi\)
0.532225 + 0.846603i \(0.321356\pi\)
\(294\) −1.14896 −0.0670088
\(295\) 4.74643 0.276348
\(296\) −2.38692 −0.138737
\(297\) 0 0
\(298\) −10.1138 −0.585876
\(299\) 10.0864 0.583310
\(300\) 1.14896 0.0663353
\(301\) 4.66428 0.268845
\(302\) 11.8335 0.680944
\(303\) −6.39305 −0.367271
\(304\) 2.67989 0.153702
\(305\) 11.6593 0.667611
\(306\) 10.8808 0.622013
\(307\) −10.9710 −0.626146 −0.313073 0.949729i \(-0.601358\pi\)
−0.313073 + 0.949729i \(0.601358\pi\)
\(308\) 0 0
\(309\) −9.14559 −0.520275
\(310\) 6.01867 0.341837
\(311\) 20.8687 1.18335 0.591677 0.806175i \(-0.298466\pi\)
0.591677 + 0.806175i \(0.298466\pi\)
\(312\) −4.36284 −0.246997
\(313\) 8.64805 0.488817 0.244408 0.969672i \(-0.421406\pi\)
0.244408 + 0.969672i \(0.421406\pi\)
\(314\) −1.69054 −0.0954026
\(315\) 1.67989 0.0946510
\(316\) −6.03633 −0.339570
\(317\) 3.53662 0.198636 0.0993182 0.995056i \(-0.468334\pi\)
0.0993182 + 0.995056i \(0.468334\pi\)
\(318\) 5.85762 0.328479
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −20.0200 −1.11741
\(322\) 2.65626 0.148028
\(323\) 17.3579 0.965818
\(324\) −1.13831 −0.0632395
\(325\) 3.79720 0.210631
\(326\) 8.06159 0.446490
\(327\) 5.65746 0.312858
\(328\) −9.47709 −0.523285
\(329\) 4.48205 0.247103
\(330\) 0 0
\(331\) 10.1966 0.560458 0.280229 0.959933i \(-0.409590\pi\)
0.280229 + 0.959933i \(0.409590\pi\)
\(332\) −11.1551 −0.612215
\(333\) −4.00976 −0.219734
\(334\) −16.2178 −0.887400
\(335\) 5.71316 0.312143
\(336\) −1.14896 −0.0626810
\(337\) −14.1072 −0.768469 −0.384234 0.923236i \(-0.625535\pi\)
−0.384234 + 0.923236i \(0.625535\pi\)
\(338\) −1.41876 −0.0771702
\(339\) −13.0779 −0.710296
\(340\) 6.47709 0.351270
\(341\) 0 0
\(342\) 4.50191 0.243436
\(343\) −1.00000 −0.0539949
\(344\) 4.66428 0.251481
\(345\) 3.05194 0.164311
\(346\) −12.1600 −0.653728
\(347\) 4.16926 0.223817 0.111909 0.993718i \(-0.464304\pi\)
0.111909 + 0.993718i \(0.464304\pi\)
\(348\) 0.444559 0.0238308
\(349\) 8.03734 0.430229 0.215114 0.976589i \(-0.430988\pi\)
0.215114 + 0.976589i \(0.430988\pi\)
\(350\) 1.00000 0.0534522
\(351\) −20.4176 −1.08981
\(352\) 0 0
\(353\) 15.7957 0.840723 0.420361 0.907357i \(-0.361903\pi\)
0.420361 + 0.907357i \(0.361903\pi\)
\(354\) −5.45347 −0.289848
\(355\) 3.33119 0.176801
\(356\) −5.87277 −0.311256
\(357\) −7.44193 −0.393869
\(358\) 23.3033 1.23162
\(359\) 6.64022 0.350458 0.175229 0.984528i \(-0.443933\pi\)
0.175229 + 0.984528i \(0.443933\pi\)
\(360\) 1.67989 0.0885379
\(361\) −11.8182 −0.622010
\(362\) −22.1288 −1.16307
\(363\) 0 0
\(364\) −3.79720 −0.199028
\(365\) 15.0127 0.785801
\(366\) −13.3961 −0.700226
\(367\) 16.6306 0.868108 0.434054 0.900887i \(-0.357083\pi\)
0.434054 + 0.900887i \(0.357083\pi\)
\(368\) 2.65626 0.138467
\(369\) −15.9205 −0.828786
\(370\) −2.38692 −0.124090
\(371\) 5.09819 0.264685
\(372\) −6.91522 −0.358537
\(373\) −7.30477 −0.378227 −0.189113 0.981955i \(-0.560561\pi\)
−0.189113 + 0.981955i \(0.560561\pi\)
\(374\) 0 0
\(375\) 1.14896 0.0593321
\(376\) 4.48205 0.231144
\(377\) 1.46922 0.0756688
\(378\) −5.37701 −0.276564
\(379\) −13.2798 −0.682138 −0.341069 0.940038i \(-0.610789\pi\)
−0.341069 + 0.940038i \(0.610789\pi\)
\(380\) 2.67989 0.137475
\(381\) −1.65133 −0.0846002
\(382\) −20.2864 −1.03794
\(383\) 17.1065 0.874100 0.437050 0.899437i \(-0.356023\pi\)
0.437050 + 0.899437i \(0.356023\pi\)
\(384\) −1.14896 −0.0586327
\(385\) 0 0
\(386\) 3.84508 0.195709
\(387\) 7.83547 0.398299
\(388\) −3.30450 −0.167761
\(389\) 8.54697 0.433348 0.216674 0.976244i \(-0.430479\pi\)
0.216674 + 0.976244i \(0.430479\pi\)
\(390\) −4.36284 −0.220921
\(391\) 17.2049 0.870087
\(392\) −1.00000 −0.0505076
\(393\) 22.4237 1.13113
\(394\) −11.8159 −0.595275
\(395\) −6.03633 −0.303721
\(396\) 0 0
\(397\) 17.0472 0.855572 0.427786 0.903880i \(-0.359294\pi\)
0.427786 + 0.903880i \(0.359294\pi\)
\(398\) −12.5496 −0.629054
\(399\) −3.07909 −0.154147
\(400\) 1.00000 0.0500000
\(401\) −2.71053 −0.135357 −0.0676787 0.997707i \(-0.521559\pi\)
−0.0676787 + 0.997707i \(0.521559\pi\)
\(402\) −6.56420 −0.327392
\(403\) −22.8541 −1.13844
\(404\) −5.56420 −0.276829
\(405\) −1.13831 −0.0565631
\(406\) 0.386922 0.0192026
\(407\) 0 0
\(408\) −7.44193 −0.368430
\(409\) 29.3149 1.44953 0.724763 0.688998i \(-0.241949\pi\)
0.724763 + 0.688998i \(0.241949\pi\)
\(410\) −9.47709 −0.468040
\(411\) 21.8083 1.07572
\(412\) −7.95988 −0.392155
\(413\) −4.74643 −0.233557
\(414\) 4.46222 0.219306
\(415\) −11.1551 −0.547582
\(416\) −3.79720 −0.186173
\(417\) 7.56698 0.370557
\(418\) 0 0
\(419\) 38.3776 1.87487 0.937434 0.348162i \(-0.113194\pi\)
0.937434 + 0.348162i \(0.113194\pi\)
\(420\) −1.14896 −0.0560636
\(421\) −37.1605 −1.81109 −0.905545 0.424249i \(-0.860538\pi\)
−0.905545 + 0.424249i \(0.860538\pi\)
\(422\) −3.99973 −0.194704
\(423\) 7.52934 0.366089
\(424\) 5.09819 0.247590
\(425\) 6.47709 0.314185
\(426\) −3.82741 −0.185439
\(427\) −11.6593 −0.564234
\(428\) −17.4244 −0.842241
\(429\) 0 0
\(430\) 4.66428 0.224932
\(431\) −30.7450 −1.48093 −0.740467 0.672093i \(-0.765395\pi\)
−0.740467 + 0.672093i \(0.765395\pi\)
\(432\) −5.37701 −0.258702
\(433\) −8.55355 −0.411057 −0.205529 0.978651i \(-0.565891\pi\)
−0.205529 + 0.978651i \(0.565891\pi\)
\(434\) −6.01867 −0.288905
\(435\) 0.444559 0.0213150
\(436\) 4.92398 0.235816
\(437\) 7.11849 0.340523
\(438\) −17.2490 −0.824190
\(439\) −24.5405 −1.17126 −0.585628 0.810580i \(-0.699152\pi\)
−0.585628 + 0.810580i \(0.699152\pi\)
\(440\) 0 0
\(441\) −1.67989 −0.0799947
\(442\) −24.5948 −1.16986
\(443\) 26.9683 1.28130 0.640652 0.767831i \(-0.278664\pi\)
0.640652 + 0.767831i \(0.278664\pi\)
\(444\) 2.74248 0.130152
\(445\) −5.87277 −0.278396
\(446\) −17.8782 −0.846555
\(447\) 11.6204 0.549624
\(448\) −1.00000 −0.0472456
\(449\) −35.1803 −1.66026 −0.830131 0.557569i \(-0.811734\pi\)
−0.830131 + 0.557569i \(0.811734\pi\)
\(450\) 1.67989 0.0791907
\(451\) 0 0
\(452\) −11.3824 −0.535383
\(453\) −13.5963 −0.638809
\(454\) 5.76204 0.270426
\(455\) −3.79720 −0.178016
\(456\) −3.07909 −0.144191
\(457\) 20.4364 0.955974 0.477987 0.878367i \(-0.341367\pi\)
0.477987 + 0.878367i \(0.341367\pi\)
\(458\) 8.57617 0.400738
\(459\) −34.8274 −1.62560
\(460\) 2.65626 0.123849
\(461\) 5.59541 0.260604 0.130302 0.991474i \(-0.458405\pi\)
0.130302 + 0.991474i \(0.458405\pi\)
\(462\) 0 0
\(463\) −29.5079 −1.37135 −0.685674 0.727909i \(-0.740492\pi\)
−0.685674 + 0.727909i \(0.740492\pi\)
\(464\) 0.386922 0.0179624
\(465\) −6.91522 −0.320686
\(466\) −14.4011 −0.667117
\(467\) 31.3408 1.45028 0.725140 0.688601i \(-0.241775\pi\)
0.725140 + 0.688601i \(0.241775\pi\)
\(468\) −6.37888 −0.294864
\(469\) −5.71316 −0.263809
\(470\) 4.48205 0.206741
\(471\) 1.94236 0.0894994
\(472\) −4.74643 −0.218472
\(473\) 0 0
\(474\) 6.93551 0.318559
\(475\) 2.67989 0.122962
\(476\) −6.47709 −0.296877
\(477\) 8.56439 0.392136
\(478\) 24.0096 1.09817
\(479\) 6.75139 0.308479 0.154239 0.988034i \(-0.450707\pi\)
0.154239 + 0.988034i \(0.450707\pi\)
\(480\) −1.14896 −0.0524427
\(481\) 9.06363 0.413266
\(482\) 16.7641 0.763584
\(483\) −3.05194 −0.138868
\(484\) 0 0
\(485\) −3.30450 −0.150050
\(486\) −14.8232 −0.672392
\(487\) −25.3561 −1.14900 −0.574498 0.818506i \(-0.694803\pi\)
−0.574498 + 0.818506i \(0.694803\pi\)
\(488\) −11.6593 −0.527793
\(489\) −9.26245 −0.418863
\(490\) −1.00000 −0.0451754
\(491\) 19.7873 0.892987 0.446493 0.894787i \(-0.352673\pi\)
0.446493 + 0.894787i \(0.352673\pi\)
\(492\) 10.8888 0.490905
\(493\) 2.50613 0.112870
\(494\) −10.1761 −0.457844
\(495\) 0 0
\(496\) −6.01867 −0.270246
\(497\) −3.33119 −0.149424
\(498\) 12.8168 0.574333
\(499\) −26.7727 −1.19851 −0.599255 0.800559i \(-0.704536\pi\)
−0.599255 + 0.800559i \(0.704536\pi\)
\(500\) 1.00000 0.0447214
\(501\) 18.6337 0.832490
\(502\) 2.42488 0.108228
\(503\) −0.355087 −0.0158326 −0.00791628 0.999969i \(-0.502520\pi\)
−0.00791628 + 0.999969i \(0.502520\pi\)
\(504\) −1.67989 −0.0748282
\(505\) −5.56420 −0.247604
\(506\) 0 0
\(507\) 1.63010 0.0723951
\(508\) −1.43724 −0.0637671
\(509\) −23.9743 −1.06264 −0.531321 0.847171i \(-0.678304\pi\)
−0.531321 + 0.847171i \(0.678304\pi\)
\(510\) −7.44193 −0.329534
\(511\) −15.0127 −0.664123
\(512\) −1.00000 −0.0441942
\(513\) −14.4098 −0.636208
\(514\) −5.18367 −0.228642
\(515\) −7.95988 −0.350754
\(516\) −5.35908 −0.235920
\(517\) 0 0
\(518\) 2.38692 0.104875
\(519\) 13.9714 0.613277
\(520\) −3.79720 −0.166518
\(521\) −7.25606 −0.317894 −0.158947 0.987287i \(-0.550810\pi\)
−0.158947 + 0.987287i \(0.550810\pi\)
\(522\) 0.649986 0.0284491
\(523\) −17.1723 −0.750893 −0.375447 0.926844i \(-0.622511\pi\)
−0.375447 + 0.926844i \(0.622511\pi\)
\(524\) 19.5165 0.852582
\(525\) −1.14896 −0.0501448
\(526\) −20.3770 −0.888480
\(527\) −38.9835 −1.69815
\(528\) 0 0
\(529\) −15.9443 −0.693229
\(530\) 5.09819 0.221451
\(531\) −7.97348 −0.346019
\(532\) −2.67989 −0.116188
\(533\) 35.9865 1.55875
\(534\) 6.74759 0.291997
\(535\) −17.4244 −0.753324
\(536\) −5.71316 −0.246771
\(537\) −26.7746 −1.15541
\(538\) 11.4504 0.493662
\(539\) 0 0
\(540\) −5.37701 −0.231390
\(541\) −23.7882 −1.02273 −0.511366 0.859363i \(-0.670860\pi\)
−0.511366 + 0.859363i \(0.670860\pi\)
\(542\) −14.8199 −0.636571
\(543\) 25.4252 1.09110
\(544\) −6.47709 −0.277703
\(545\) 4.92398 0.210920
\(546\) 4.36284 0.186712
\(547\) −39.2442 −1.67796 −0.838981 0.544161i \(-0.816848\pi\)
−0.838981 + 0.544161i \(0.816848\pi\)
\(548\) 18.9809 0.810823
\(549\) −19.5864 −0.835926
\(550\) 0 0
\(551\) 1.03691 0.0441738
\(552\) −3.05194 −0.129899
\(553\) 6.03633 0.256691
\(554\) −24.5158 −1.04157
\(555\) 2.74248 0.116412
\(556\) 6.58593 0.279306
\(557\) 43.0812 1.82541 0.912705 0.408618i \(-0.133989\pi\)
0.912705 + 0.408618i \(0.133989\pi\)
\(558\) −10.1107 −0.428020
\(559\) −17.7112 −0.749105
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 28.5929 1.20612
\(563\) 40.8534 1.72176 0.860882 0.508804i \(-0.169912\pi\)
0.860882 + 0.508804i \(0.169912\pi\)
\(564\) −5.14970 −0.216841
\(565\) −11.3824 −0.478861
\(566\) 5.19494 0.218360
\(567\) 1.13831 0.0478045
\(568\) −3.33119 −0.139774
\(569\) −4.46455 −0.187164 −0.0935818 0.995612i \(-0.529832\pi\)
−0.0935818 + 0.995612i \(0.529832\pi\)
\(570\) −3.07909 −0.128969
\(571\) 31.4272 1.31519 0.657593 0.753373i \(-0.271575\pi\)
0.657593 + 0.753373i \(0.271575\pi\)
\(572\) 0 0
\(573\) 23.3083 0.973717
\(574\) 9.47709 0.395566
\(575\) 2.65626 0.110774
\(576\) −1.67989 −0.0699953
\(577\) 8.24246 0.343138 0.171569 0.985172i \(-0.445116\pi\)
0.171569 + 0.985172i \(0.445116\pi\)
\(578\) −24.9527 −1.03790
\(579\) −4.41784 −0.183599
\(580\) 0.386922 0.0160661
\(581\) 11.1551 0.462791
\(582\) 3.79675 0.157380
\(583\) 0 0
\(584\) −15.0127 −0.621230
\(585\) −6.37888 −0.263734
\(586\) −18.2205 −0.752680
\(587\) 1.10956 0.0457966 0.0228983 0.999738i \(-0.492711\pi\)
0.0228983 + 0.999738i \(0.492711\pi\)
\(588\) 1.14896 0.0473824
\(589\) −16.1294 −0.664599
\(590\) −4.74643 −0.195408
\(591\) 13.5760 0.558441
\(592\) 2.38692 0.0981019
\(593\) 46.0484 1.89098 0.945491 0.325648i \(-0.105582\pi\)
0.945491 + 0.325648i \(0.105582\pi\)
\(594\) 0 0
\(595\) −6.47709 −0.265535
\(596\) 10.1138 0.414277
\(597\) 14.4190 0.590130
\(598\) −10.0864 −0.412462
\(599\) −29.5604 −1.20781 −0.603903 0.797058i \(-0.706389\pi\)
−0.603903 + 0.797058i \(0.706389\pi\)
\(600\) −1.14896 −0.0469061
\(601\) 5.68051 0.231713 0.115856 0.993266i \(-0.463039\pi\)
0.115856 + 0.993266i \(0.463039\pi\)
\(602\) −4.66428 −0.190102
\(603\) −9.59747 −0.390839
\(604\) −11.8335 −0.481500
\(605\) 0 0
\(606\) 6.39305 0.259700
\(607\) 24.3166 0.986980 0.493490 0.869751i \(-0.335721\pi\)
0.493490 + 0.869751i \(0.335721\pi\)
\(608\) −2.67989 −0.108684
\(609\) −0.444559 −0.0180144
\(610\) −11.6593 −0.472072
\(611\) −17.0193 −0.688525
\(612\) −10.8808 −0.439830
\(613\) 30.4381 1.22938 0.614692 0.788767i \(-0.289280\pi\)
0.614692 + 0.788767i \(0.289280\pi\)
\(614\) 10.9710 0.442752
\(615\) 10.8888 0.439079
\(616\) 0 0
\(617\) 33.4297 1.34583 0.672914 0.739721i \(-0.265043\pi\)
0.672914 + 0.739721i \(0.265043\pi\)
\(618\) 9.14559 0.367890
\(619\) −4.33525 −0.174248 −0.0871241 0.996197i \(-0.527768\pi\)
−0.0871241 + 0.996197i \(0.527768\pi\)
\(620\) −6.01867 −0.241716
\(621\) −14.2827 −0.573147
\(622\) −20.8687 −0.836758
\(623\) 5.87277 0.235288
\(624\) 4.36284 0.174653
\(625\) 1.00000 0.0400000
\(626\) −8.64805 −0.345646
\(627\) 0 0
\(628\) 1.69054 0.0674599
\(629\) 15.4603 0.616443
\(630\) −1.67989 −0.0669284
\(631\) −15.4234 −0.613998 −0.306999 0.951710i \(-0.599325\pi\)
−0.306999 + 0.951710i \(0.599325\pi\)
\(632\) 6.03633 0.240113
\(633\) 4.59554 0.182656
\(634\) −3.53662 −0.140457
\(635\) −1.43724 −0.0570350
\(636\) −5.85762 −0.232270
\(637\) 3.79720 0.150451
\(638\) 0 0
\(639\) −5.59603 −0.221376
\(640\) −1.00000 −0.0395285
\(641\) 35.5278 1.40327 0.701633 0.712539i \(-0.252455\pi\)
0.701633 + 0.712539i \(0.252455\pi\)
\(642\) 20.0200 0.790126
\(643\) 49.2065 1.94051 0.970257 0.242079i \(-0.0778292\pi\)
0.970257 + 0.242079i \(0.0778292\pi\)
\(644\) −2.65626 −0.104671
\(645\) −5.35908 −0.211013
\(646\) −17.3579 −0.682937
\(647\) 11.9820 0.471063 0.235531 0.971867i \(-0.424317\pi\)
0.235531 + 0.971867i \(0.424317\pi\)
\(648\) 1.13831 0.0447171
\(649\) 0 0
\(650\) −3.79720 −0.148939
\(651\) 6.91522 0.271029
\(652\) −8.06159 −0.315716
\(653\) −50.4638 −1.97480 −0.987401 0.158241i \(-0.949418\pi\)
−0.987401 + 0.158241i \(0.949418\pi\)
\(654\) −5.65746 −0.221224
\(655\) 19.5165 0.762572
\(656\) 9.47709 0.370018
\(657\) −25.2197 −0.983913
\(658\) −4.48205 −0.174728
\(659\) 29.2967 1.14124 0.570618 0.821216i \(-0.306704\pi\)
0.570618 + 0.821216i \(0.306704\pi\)
\(660\) 0 0
\(661\) 35.0445 1.36307 0.681537 0.731784i \(-0.261312\pi\)
0.681537 + 0.731784i \(0.261312\pi\)
\(662\) −10.1966 −0.396304
\(663\) 28.2585 1.09747
\(664\) 11.1551 0.432901
\(665\) −2.67989 −0.103922
\(666\) 4.00976 0.155375
\(667\) 1.02777 0.0397953
\(668\) 16.2178 0.627487
\(669\) 20.5413 0.794173
\(670\) −5.71316 −0.220719
\(671\) 0 0
\(672\) 1.14896 0.0443221
\(673\) 37.4744 1.44453 0.722265 0.691616i \(-0.243101\pi\)
0.722265 + 0.691616i \(0.243101\pi\)
\(674\) 14.1072 0.543390
\(675\) −5.37701 −0.206961
\(676\) 1.41876 0.0545676
\(677\) 8.42279 0.323714 0.161857 0.986814i \(-0.448252\pi\)
0.161857 + 0.986814i \(0.448252\pi\)
\(678\) 13.0779 0.502255
\(679\) 3.30450 0.126815
\(680\) −6.47709 −0.248385
\(681\) −6.62036 −0.253693
\(682\) 0 0
\(683\) 14.9986 0.573904 0.286952 0.957945i \(-0.407358\pi\)
0.286952 + 0.957945i \(0.407358\pi\)
\(684\) −4.50191 −0.172135
\(685\) 18.9809 0.725222
\(686\) 1.00000 0.0381802
\(687\) −9.85369 −0.375942
\(688\) −4.66428 −0.177824
\(689\) −19.3589 −0.737514
\(690\) −3.05194 −0.116185
\(691\) 43.8937 1.66980 0.834898 0.550404i \(-0.185526\pi\)
0.834898 + 0.550404i \(0.185526\pi\)
\(692\) 12.1600 0.462255
\(693\) 0 0
\(694\) −4.16926 −0.158263
\(695\) 6.58593 0.249819
\(696\) −0.444559 −0.0168510
\(697\) 61.3840 2.32508
\(698\) −8.03734 −0.304218
\(699\) 16.5463 0.625837
\(700\) −1.00000 −0.0377964
\(701\) −43.3836 −1.63857 −0.819287 0.573383i \(-0.805631\pi\)
−0.819287 + 0.573383i \(0.805631\pi\)
\(702\) 20.4176 0.770613
\(703\) 6.39668 0.241256
\(704\) 0 0
\(705\) −5.14970 −0.193949
\(706\) −15.7957 −0.594481
\(707\) 5.56420 0.209263
\(708\) 5.45347 0.204954
\(709\) 6.61088 0.248277 0.124138 0.992265i \(-0.460383\pi\)
0.124138 + 0.992265i \(0.460383\pi\)
\(710\) −3.33119 −0.125017
\(711\) 10.1404 0.380294
\(712\) 5.87277 0.220091
\(713\) −15.9872 −0.598724
\(714\) 7.44193 0.278507
\(715\) 0 0
\(716\) −23.3033 −0.870886
\(717\) −27.5861 −1.03022
\(718\) −6.64022 −0.247811
\(719\) −27.5111 −1.02599 −0.512995 0.858392i \(-0.671464\pi\)
−0.512995 + 0.858392i \(0.671464\pi\)
\(720\) −1.67989 −0.0626057
\(721\) 7.95988 0.296441
\(722\) 11.8182 0.439828
\(723\) −19.2613 −0.716335
\(724\) 22.1288 0.822411
\(725\) 0.386922 0.0143699
\(726\) 0 0
\(727\) −13.3895 −0.496590 −0.248295 0.968685i \(-0.579870\pi\)
−0.248295 + 0.968685i \(0.579870\pi\)
\(728\) 3.79720 0.140734
\(729\) 20.4462 0.757265
\(730\) −15.0127 −0.555645
\(731\) −30.2110 −1.11739
\(732\) 13.3961 0.495135
\(733\) −45.9046 −1.69552 −0.847762 0.530376i \(-0.822051\pi\)
−0.847762 + 0.530376i \(0.822051\pi\)
\(734\) −16.6306 −0.613845
\(735\) 1.14896 0.0423801
\(736\) −2.65626 −0.0979111
\(737\) 0 0
\(738\) 15.9205 0.586040
\(739\) −15.5601 −0.572388 −0.286194 0.958172i \(-0.592390\pi\)
−0.286194 + 0.958172i \(0.592390\pi\)
\(740\) 2.38692 0.0877450
\(741\) 11.6919 0.429514
\(742\) −5.09819 −0.187160
\(743\) 31.9688 1.17282 0.586411 0.810014i \(-0.300541\pi\)
0.586411 + 0.810014i \(0.300541\pi\)
\(744\) 6.91522 0.253524
\(745\) 10.1138 0.370541
\(746\) 7.30477 0.267447
\(747\) 18.7393 0.685635
\(748\) 0 0
\(749\) 17.4244 0.636675
\(750\) −1.14896 −0.0419541
\(751\) 31.7056 1.15695 0.578476 0.815699i \(-0.303647\pi\)
0.578476 + 0.815699i \(0.303647\pi\)
\(752\) −4.48205 −0.163443
\(753\) −2.78610 −0.101531
\(754\) −1.46922 −0.0535059
\(755\) −11.8335 −0.430667
\(756\) 5.37701 0.195560
\(757\) −22.3306 −0.811620 −0.405810 0.913957i \(-0.633011\pi\)
−0.405810 + 0.913957i \(0.633011\pi\)
\(758\) 13.2798 0.482344
\(759\) 0 0
\(760\) −2.67989 −0.0972098
\(761\) 17.3657 0.629508 0.314754 0.949173i \(-0.398078\pi\)
0.314754 + 0.949173i \(0.398078\pi\)
\(762\) 1.65133 0.0598214
\(763\) −4.92398 −0.178260
\(764\) 20.2864 0.733936
\(765\) −10.8808 −0.393396
\(766\) −17.1065 −0.618082
\(767\) 18.0232 0.650779
\(768\) 1.14896 0.0414596
\(769\) −36.3673 −1.31144 −0.655720 0.755004i \(-0.727635\pi\)
−0.655720 + 0.755004i \(0.727635\pi\)
\(770\) 0 0
\(771\) 5.95584 0.214494
\(772\) −3.84508 −0.138387
\(773\) −53.0722 −1.90887 −0.954437 0.298413i \(-0.903543\pi\)
−0.954437 + 0.298413i \(0.903543\pi\)
\(774\) −7.83547 −0.281640
\(775\) −6.01867 −0.216197
\(776\) 3.30450 0.118625
\(777\) −2.74248 −0.0983860
\(778\) −8.54697 −0.306424
\(779\) 25.3975 0.909962
\(780\) 4.36284 0.156215
\(781\) 0 0
\(782\) −17.2049 −0.615244
\(783\) −2.08048 −0.0743505
\(784\) 1.00000 0.0357143
\(785\) 1.69054 0.0603379
\(786\) −22.4237 −0.799826
\(787\) −38.3366 −1.36655 −0.683276 0.730160i \(-0.739445\pi\)
−0.683276 + 0.730160i \(0.739445\pi\)
\(788\) 11.8159 0.420923
\(789\) 23.4124 0.833503
\(790\) 6.03633 0.214763
\(791\) 11.3824 0.404711
\(792\) 0 0
\(793\) 44.2728 1.57217
\(794\) −17.0472 −0.604981
\(795\) −5.85762 −0.207748
\(796\) 12.5496 0.444809
\(797\) −30.5021 −1.08044 −0.540219 0.841524i \(-0.681659\pi\)
−0.540219 + 0.841524i \(0.681659\pi\)
\(798\) 3.07909 0.108999
\(799\) −29.0306 −1.02703
\(800\) −1.00000 −0.0353553
\(801\) 9.86560 0.348584
\(802\) 2.71053 0.0957121
\(803\) 0 0
\(804\) 6.56420 0.231501
\(805\) −2.65626 −0.0936209
\(806\) 22.8541 0.805002
\(807\) −13.1561 −0.463116
\(808\) 5.56420 0.195748
\(809\) −33.8990 −1.19183 −0.595913 0.803049i \(-0.703210\pi\)
−0.595913 + 0.803049i \(0.703210\pi\)
\(810\) 1.13831 0.0399962
\(811\) 42.9037 1.50655 0.753276 0.657705i \(-0.228473\pi\)
0.753276 + 0.657705i \(0.228473\pi\)
\(812\) −0.386922 −0.0135783
\(813\) 17.0275 0.597182
\(814\) 0 0
\(815\) −8.06159 −0.282385
\(816\) 7.44193 0.260520
\(817\) −12.4998 −0.437311
\(818\) −29.3149 −1.02497
\(819\) 6.37888 0.222896
\(820\) 9.47709 0.330954
\(821\) 50.2053 1.75218 0.876088 0.482151i \(-0.160144\pi\)
0.876088 + 0.482151i \(0.160144\pi\)
\(822\) −21.8083 −0.760651
\(823\) −19.9384 −0.695010 −0.347505 0.937678i \(-0.612971\pi\)
−0.347505 + 0.937678i \(0.612971\pi\)
\(824\) 7.95988 0.277296
\(825\) 0 0
\(826\) 4.74643 0.165149
\(827\) −34.0394 −1.18366 −0.591832 0.806061i \(-0.701595\pi\)
−0.591832 + 0.806061i \(0.701595\pi\)
\(828\) −4.46222 −0.155073
\(829\) −12.9629 −0.450222 −0.225111 0.974333i \(-0.572274\pi\)
−0.225111 + 0.974333i \(0.572274\pi\)
\(830\) 11.1551 0.387199
\(831\) 28.1676 0.977125
\(832\) 3.79720 0.131644
\(833\) 6.47709 0.224418
\(834\) −7.56698 −0.262023
\(835\) 16.2178 0.561241
\(836\) 0 0
\(837\) 32.3624 1.11861
\(838\) −38.3776 −1.32573
\(839\) −14.4663 −0.499430 −0.249715 0.968319i \(-0.580337\pi\)
−0.249715 + 0.968319i \(0.580337\pi\)
\(840\) 1.14896 0.0396429
\(841\) −28.8503 −0.994838
\(842\) 37.1605 1.28063
\(843\) −32.8522 −1.13149
\(844\) 3.99973 0.137676
\(845\) 1.41876 0.0488067
\(846\) −7.52934 −0.258864
\(847\) 0 0
\(848\) −5.09819 −0.175073
\(849\) −5.96879 −0.204848
\(850\) −6.47709 −0.222162
\(851\) 6.34029 0.217342
\(852\) 3.82741 0.131125
\(853\) −51.1805 −1.75239 −0.876193 0.481960i \(-0.839925\pi\)
−0.876193 + 0.481960i \(0.839925\pi\)
\(854\) 11.6593 0.398974
\(855\) −4.50191 −0.153962
\(856\) 17.4244 0.595555
\(857\) −50.8960 −1.73857 −0.869287 0.494308i \(-0.835422\pi\)
−0.869287 + 0.494308i \(0.835422\pi\)
\(858\) 0 0
\(859\) −20.8508 −0.711422 −0.355711 0.934596i \(-0.615761\pi\)
−0.355711 + 0.934596i \(0.615761\pi\)
\(860\) −4.66428 −0.159051
\(861\) −10.8888 −0.371090
\(862\) 30.7450 1.04718
\(863\) 54.0762 1.84077 0.920387 0.391009i \(-0.127874\pi\)
0.920387 + 0.391009i \(0.127874\pi\)
\(864\) 5.37701 0.182930
\(865\) 12.1600 0.413454
\(866\) 8.55355 0.290661
\(867\) 28.6697 0.973674
\(868\) 6.01867 0.204287
\(869\) 0 0
\(870\) −0.444559 −0.0150719
\(871\) 21.6940 0.735074
\(872\) −4.92398 −0.166747
\(873\) 5.55120 0.187880
\(874\) −7.11849 −0.240786
\(875\) −1.00000 −0.0338062
\(876\) 17.2490 0.582790
\(877\) 2.58531 0.0872997 0.0436499 0.999047i \(-0.486101\pi\)
0.0436499 + 0.999047i \(0.486101\pi\)
\(878\) 24.5405 0.828203
\(879\) 20.9346 0.706107
\(880\) 0 0
\(881\) 42.8631 1.44410 0.722048 0.691843i \(-0.243201\pi\)
0.722048 + 0.691843i \(0.243201\pi\)
\(882\) 1.67989 0.0565648
\(883\) −14.4297 −0.485596 −0.242798 0.970077i \(-0.578065\pi\)
−0.242798 + 0.970077i \(0.578065\pi\)
\(884\) 24.5948 0.827214
\(885\) 5.45347 0.183316
\(886\) −26.9683 −0.906019
\(887\) −8.49359 −0.285187 −0.142593 0.989781i \(-0.545544\pi\)
−0.142593 + 0.989781i \(0.545544\pi\)
\(888\) −2.74248 −0.0920316
\(889\) 1.43724 0.0482034
\(890\) 5.87277 0.196856
\(891\) 0 0
\(892\) 17.8782 0.598605
\(893\) −12.0114 −0.401946
\(894\) −11.6204 −0.388643
\(895\) −23.3033 −0.778944
\(896\) 1.00000 0.0334077
\(897\) 11.5888 0.386940
\(898\) 35.1803 1.17398
\(899\) −2.32876 −0.0776684
\(900\) −1.67989 −0.0559963
\(901\) −33.0214 −1.10010
\(902\) 0 0
\(903\) 5.35908 0.178339
\(904\) 11.3824 0.378573
\(905\) 22.1288 0.735587
\(906\) 13.5963 0.451706
\(907\) −47.4123 −1.57430 −0.787150 0.616762i \(-0.788444\pi\)
−0.787150 + 0.616762i \(0.788444\pi\)
\(908\) −5.76204 −0.191220
\(909\) 9.34723 0.310028
\(910\) 3.79720 0.125876
\(911\) 14.7684 0.489300 0.244650 0.969611i \(-0.421327\pi\)
0.244650 + 0.969611i \(0.421327\pi\)
\(912\) 3.07909 0.101959
\(913\) 0 0
\(914\) −20.4364 −0.675976
\(915\) 13.3961 0.442862
\(916\) −8.57617 −0.283365
\(917\) −19.5165 −0.644491
\(918\) 34.8274 1.14948
\(919\) 29.6648 0.978552 0.489276 0.872129i \(-0.337261\pi\)
0.489276 + 0.872129i \(0.337261\pi\)
\(920\) −2.65626 −0.0875744
\(921\) −12.6052 −0.415356
\(922\) −5.59541 −0.184275
\(923\) 12.6492 0.416354
\(924\) 0 0
\(925\) 2.38692 0.0784815
\(926\) 29.5079 0.969690
\(927\) 13.3717 0.439184
\(928\) −0.386922 −0.0127013
\(929\) −14.6996 −0.482277 −0.241139 0.970491i \(-0.577521\pi\)
−0.241139 + 0.970491i \(0.577521\pi\)
\(930\) 6.91522 0.226759
\(931\) 2.67989 0.0878298
\(932\) 14.4011 0.471723
\(933\) 23.9773 0.784982
\(934\) −31.3408 −1.02550
\(935\) 0 0
\(936\) 6.37888 0.208500
\(937\) 54.2518 1.77233 0.886164 0.463371i \(-0.153360\pi\)
0.886164 + 0.463371i \(0.153360\pi\)
\(938\) 5.71316 0.186541
\(939\) 9.93628 0.324258
\(940\) −4.48205 −0.146188
\(941\) 31.8477 1.03820 0.519102 0.854712i \(-0.326266\pi\)
0.519102 + 0.854712i \(0.326266\pi\)
\(942\) −1.94236 −0.0632856
\(943\) 25.1736 0.819767
\(944\) 4.74643 0.154483
\(945\) 5.37701 0.174914
\(946\) 0 0
\(947\) −44.7035 −1.45267 −0.726335 0.687341i \(-0.758778\pi\)
−0.726335 + 0.687341i \(0.758778\pi\)
\(948\) −6.93551 −0.225255
\(949\) 57.0063 1.85050
\(950\) −2.67989 −0.0869471
\(951\) 4.06344 0.131766
\(952\) 6.47709 0.209924
\(953\) −45.4259 −1.47149 −0.735745 0.677259i \(-0.763168\pi\)
−0.735745 + 0.677259i \(0.763168\pi\)
\(954\) −8.56439 −0.277282
\(955\) 20.2864 0.656452
\(956\) −24.0096 −0.776527
\(957\) 0 0
\(958\) −6.75139 −0.218127
\(959\) −18.9809 −0.612924
\(960\) 1.14896 0.0370826
\(961\) 5.22439 0.168529
\(962\) −9.06363 −0.292223
\(963\) 29.2711 0.943248
\(964\) −16.7641 −0.539935
\(965\) −3.84508 −0.123777
\(966\) 3.05194 0.0981946
\(967\) −34.6596 −1.11458 −0.557289 0.830319i \(-0.688158\pi\)
−0.557289 + 0.830319i \(0.688158\pi\)
\(968\) 0 0
\(969\) 19.9435 0.640679
\(970\) 3.30450 0.106101
\(971\) 4.29180 0.137730 0.0688651 0.997626i \(-0.478062\pi\)
0.0688651 + 0.997626i \(0.478062\pi\)
\(972\) 14.8232 0.475453
\(973\) −6.58593 −0.211135
\(974\) 25.3561 0.812463
\(975\) 4.36284 0.139723
\(976\) 11.6593 0.373206
\(977\) −37.8036 −1.20944 −0.604722 0.796437i \(-0.706716\pi\)
−0.604722 + 0.796437i \(0.706716\pi\)
\(978\) 9.26245 0.296181
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) −8.27173 −0.264096
\(982\) −19.7873 −0.631437
\(983\) −14.4139 −0.459731 −0.229866 0.973222i \(-0.573829\pi\)
−0.229866 + 0.973222i \(0.573829\pi\)
\(984\) −10.8888 −0.347123
\(985\) 11.8159 0.376485
\(986\) −2.50613 −0.0798115
\(987\) 5.14970 0.163917
\(988\) 10.1761 0.323744
\(989\) −12.3896 −0.393965
\(990\) 0 0
\(991\) −24.3859 −0.774644 −0.387322 0.921944i \(-0.626600\pi\)
−0.387322 + 0.921944i \(0.626600\pi\)
\(992\) 6.01867 0.191093
\(993\) 11.7155 0.371782
\(994\) 3.33119 0.105659
\(995\) 12.5496 0.397849
\(996\) −12.8168 −0.406115
\(997\) 15.5791 0.493395 0.246697 0.969093i \(-0.420655\pi\)
0.246697 + 0.969093i \(0.420655\pi\)
\(998\) 26.7727 0.847474
\(999\) −12.8345 −0.406066
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 8470.2.a.co.1.4 4
11.3 even 5 770.2.n.f.141.1 yes 8
11.4 even 5 770.2.n.f.71.1 8
11.10 odd 2 8470.2.a.cs.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.f.71.1 8 11.4 even 5
770.2.n.f.141.1 yes 8 11.3 even 5
8470.2.a.co.1.4 4 1.1 even 1 trivial
8470.2.a.cs.1.4 4 11.10 odd 2