Properties

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

Related objects

Downloads

Learn more

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: \(1\)
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

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

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 \(-0.676525\pi\)
−0.526577 + 0.850127i \(0.676525\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.14896 0.296660
\(16\) 1.00000 0.250000
\(17\) −6.47709 −1.57093 −0.785463 0.618909i \(-0.787575\pi\)
−0.785463 + 0.618909i \(0.787575\pi\)
\(18\) −1.67989 −0.395953
\(19\) −2.67989 −0.614809 −0.307404 0.951579i \(-0.599460\pi\)
−0.307404 + 0.951579i \(0.599460\pi\)
\(20\) 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 \(-0.511438\pi\)
−0.0359248 + 0.999354i \(0.511438\pi\)
\(30\) 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 \(-0.765192\pi\)
−0.740037 + 0.672567i \(0.765192\pi\)
\(42\) 1.14896 0.177289
\(43\) 4.66428 0.711296 0.355648 0.934620i \(-0.384260\pi\)
0.355648 + 0.934620i \(0.384260\pi\)
\(44\) 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 \(-0.768225\pi\)
−0.746412 + 0.665484i \(0.768225\pi\)
\(62\) −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 \(-0.841490\pi\)
−0.878552 + 0.477646i \(0.841490\pi\)
\(74\) 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 \(-0.389718\pi\)
0.339570 + 0.940581i \(0.389718\pi\)
\(80\) 1.00000 0.111803
\(81\) −1.13831 −0.126479
\(82\) −9.47709 −1.04657
\(83\) 11.1551 1.22443 0.612215 0.790691i \(-0.290279\pi\)
0.612215 + 0.790691i \(0.290279\pi\)
\(84\) 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 \(-0.410716\pi\)
0.276829 + 0.960919i \(0.410716\pi\)
\(102\) −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 \(-0.181236\pi\)
0.842241 + 0.539100i \(0.181236\pi\)
\(108\) −5.37701 −0.517403
\(109\) −4.92398 −0.471631 −0.235816 0.971798i \(-0.575776\pi\)
−0.235816 + 0.971798i \(0.575776\pi\)
\(110\) 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 \(-0.479689\pi\)
0.0637671 + 0.997965i \(0.479689\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.35908 0.471841
\(130\) −3.79720 −0.333037
\(131\) −19.5165 −1.70516 −0.852582 0.522594i \(-0.824964\pi\)
−0.852582 + 0.522594i \(0.824964\pi\)
\(132\) 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 \(-0.590104\pi\)
−0.279306 + 0.960202i \(0.590104\pi\)
\(140\) 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 \(-0.635966\pi\)
−0.414277 + 0.910151i \(0.635966\pi\)
\(150\) 1.14896 0.0938123
\(151\) 11.8335 0.963000 0.481500 0.876446i \(-0.340092\pi\)
0.481500 + 0.876446i \(0.340092\pi\)
\(152\) −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 \(-0.715916\pi\)
−0.627487 + 0.778627i \(0.715916\pi\)
\(168\) 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 \(-0.652960\pi\)
−0.462255 + 0.886747i \(0.652960\pi\)
\(174\) −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 \(-0.455808\pi\)
0.138387 + 0.990378i \(0.455808\pi\)
\(194\) −3.30450 −0.237250
\(195\) −4.36284 −0.312429
\(196\) 1.00000 0.0714286
\(197\) −11.8159 −0.841846 −0.420923 0.907096i \(-0.638294\pi\)
−0.420923 + 0.907096i \(0.638294\pi\)
\(198\) 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 \(-0.543963\pi\)
−0.137676 + 0.990477i \(0.543963\pi\)
\(212\) −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 \(-0.438756\pi\)
0.191220 + 0.981547i \(0.438756\pi\)
\(228\) −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 \(-0.656368\pi\)
−0.471723 + 0.881747i \(0.656368\pi\)
\(234\) 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 \(-0.216980\pi\)
0.776527 + 0.630084i \(0.216980\pi\)
\(240\) 1.14896 0.0741651
\(241\) 16.7641 1.07987 0.539935 0.841707i \(-0.318449\pi\)
0.539935 + 0.841707i \(0.318449\pi\)
\(242\) 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 \(-0.716229\pi\)
−0.628250 + 0.778011i \(0.716229\pi\)
\(264\) 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 \(-0.648620\pi\)
−0.450124 + 0.892966i \(0.648620\pi\)
\(272\) −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 \(-0.763525\pi\)
−0.736504 + 0.676433i \(0.763525\pi\)
\(278\) −6.58593 −0.394998
\(279\) 10.1107 0.605311
\(280\) 1.00000 0.0597614
\(281\) 28.5929 1.70571 0.852856 0.522146i \(-0.174868\pi\)
0.852856 + 0.522146i \(0.174868\pi\)
\(282\) −5.14970 −0.306660
\(283\) 5.19494 0.308807 0.154404 0.988008i \(-0.450654\pi\)
0.154404 + 0.988008i \(0.450654\pi\)
\(284\) 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 \(-0.678644\pi\)
−0.532225 + 0.846603i \(0.678644\pi\)
\(294\) 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 \(-0.398642\pi\)
0.313073 + 0.949729i \(0.398642\pi\)
\(308\) 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 \(-0.374465\pi\)
0.384234 + 0.923236i \(0.374465\pi\)
\(338\) 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 \(-0.535696\pi\)
−0.111909 + 0.993718i \(0.535696\pi\)
\(348\) −0.444559 −0.0238308
\(349\) −8.03734 −0.430229 −0.215114 0.976589i \(-0.569012\pi\)
−0.215114 + 0.976589i \(0.569012\pi\)
\(350\) 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 \(-0.556067\pi\)
−0.175229 + 0.984528i \(0.556067\pi\)
\(360\) −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 \(-0.439439\pi\)
0.189113 + 0.981955i \(0.439439\pi\)
\(374\) 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 \(-0.758051\pi\)
−0.724763 + 0.688998i \(0.758051\pi\)
\(410\) −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 \(-0.234605\pi\)
0.740467 + 0.672093i \(0.234605\pi\)
\(432\) −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 \(-0.300848\pi\)
0.585628 + 0.810580i \(0.300848\pi\)
\(440\) 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 \(-0.658633\pi\)
−0.477987 + 0.878367i \(0.658633\pi\)
\(458\) −8.57617 −0.400738
\(459\) 34.8274 1.62560
\(460\) 2.65626 0.123849
\(461\) −5.59541 −0.260604 −0.130302 0.991474i \(-0.541595\pi\)
−0.130302 + 0.991474i \(0.541595\pi\)
\(462\) 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 \(-0.549293\pi\)
−0.154239 + 0.988034i \(0.549293\pi\)
\(480\) 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 \(-0.647327\pi\)
−0.446493 + 0.894787i \(0.647327\pi\)
\(492\) −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 \(-0.497480\pi\)
0.00791628 + 0.999969i \(0.497480\pi\)
\(504\) −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 \(-0.377489\pi\)
0.375447 + 0.926844i \(0.377489\pi\)
\(524\) −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 \(-0.329140\pi\)
0.511366 + 0.859363i \(0.329140\pi\)
\(542\) −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 \(-0.183152\pi\)
0.838981 + 0.544161i \(0.183152\pi\)
\(548\) 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 \(-0.866011\pi\)
−0.912705 + 0.408618i \(0.866011\pi\)
\(558\) 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 \(-0.830088\pi\)
−0.860882 + 0.508804i \(0.830088\pi\)
\(564\) −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 \(-0.470168\pi\)
0.0935818 + 0.995612i \(0.470168\pi\)
\(570\) −3.07909 −0.128969
\(571\) −31.4272 −1.31519 −0.657593 0.753373i \(-0.728425\pi\)
−0.657593 + 0.753373i \(0.728425\pi\)
\(572\) 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 \(-0.894418\pi\)
−0.945491 + 0.325648i \(0.894418\pi\)
\(594\) 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 \(-0.536961\pi\)
−0.115856 + 0.993266i \(0.536961\pi\)
\(602\) 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 \(-0.664279\pi\)
−0.493490 + 0.869751i \(0.664279\pi\)
\(608\) −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 \(-0.710720\pi\)
−0.614692 + 0.788767i \(0.710720\pi\)
\(614\) 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 \(-0.693296\pi\)
−0.570618 + 0.821216i \(0.693296\pi\)
\(660\) 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 \(-0.756899\pi\)
−0.722265 + 0.691616i \(0.756899\pi\)
\(674\) 14.1072 0.543390
\(675\) −5.37701 −0.206961
\(676\) 1.41876 0.0545676
\(677\) −8.42279 −0.323714 −0.161857 0.986814i \(-0.551748\pi\)
−0.161857 + 0.986814i \(0.551748\pi\)
\(678\) −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 \(-0.194369\pi\)
0.819287 + 0.573383i \(0.194369\pi\)
\(702\) 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 \(-0.177949\pi\)
0.847762 + 0.530376i \(0.177949\pi\)
\(734\) 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 \(-0.407610\pi\)
0.286194 + 0.958172i \(0.407610\pi\)
\(740\) 2.38692 0.0877450
\(741\) 11.6919 0.429514
\(742\) −5.09819 −0.187160
\(743\) −31.9688 −1.17282 −0.586411 0.810014i \(-0.699459\pi\)
−0.586411 + 0.810014i \(0.699459\pi\)
\(744\) −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 \(-0.601922\pi\)
−0.314754 + 0.949173i \(0.601922\pi\)
\(762\) 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 \(-0.272365\pi\)
0.655720 + 0.755004i \(0.272365\pi\)
\(770\) 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 \(-0.260555\pi\)
0.683276 + 0.730160i \(0.260555\pi\)
\(788\) −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 \(-0.296790\pi\)
0.595913 + 0.803049i \(0.296790\pi\)
\(810\) −1.13831 −0.0399962
\(811\) −42.9037 −1.50655 −0.753276 0.657705i \(-0.771527\pi\)
−0.753276 + 0.657705i \(0.771527\pi\)
\(812\) −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 \(-0.839856\pi\)
−0.876088 + 0.482151i \(0.839856\pi\)
\(822\) 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 \(-0.298405\pi\)
0.591832 + 0.806061i \(0.298405\pi\)
\(828\) −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 \(-0.160075\pi\)
0.876193 + 0.481960i \(0.160075\pi\)
\(854\) −11.6593 −0.398974
\(855\) 4.50191 0.153962
\(856\) 17.4244 0.595555
\(857\) 50.8960 1.73857 0.869287 0.494308i \(-0.164578\pi\)
0.869287 + 0.494308i \(0.164578\pi\)
\(858\) 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 \(-0.513899\pi\)
−0.0436499 + 0.999047i \(0.513899\pi\)
\(878\) 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 \(-0.454456\pi\)
0.142593 + 0.989781i \(0.454456\pi\)
\(888\) 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 \(-0.662739\pi\)
−0.489276 + 0.872129i \(0.662739\pi\)
\(920\) 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 \(-0.846640\pi\)
−0.886164 + 0.463371i \(0.846640\pi\)
\(938\) 5.71316 0.186541
\(939\) 9.93628 0.324258
\(940\) −4.48205 −0.146188
\(941\) −31.8477 −1.03820 −0.519102 0.854712i \(-0.673734\pi\)
−0.519102 + 0.854712i \(0.673734\pi\)
\(942\) 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 \(-0.236832\pi\)
0.735745 + 0.677259i \(0.236832\pi\)
\(954\) 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 \(-0.311842\pi\)
0.557289 + 0.830319i \(0.311842\pi\)
\(968\) 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 \(-0.579345\pi\)
−0.246697 + 0.969093i \(0.579345\pi\)
\(998\) −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.cs.1.4 4
11.7 odd 10 770.2.n.f.71.1 8
11.8 odd 10 770.2.n.f.141.1 yes 8
11.10 odd 2 8470.2.a.co.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.f.71.1 8 11.7 odd 10
770.2.n.f.141.1 yes 8 11.8 odd 10
8470.2.a.co.1.4 4 11.10 odd 2
8470.2.a.cs.1.4 4 1.1 even 1 trivial