Properties

Label 4598.2.a.ca.1.4
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} - 10x^{6} + 16x^{5} + 26x^{4} - 32x^{3} - 16x^{2} + 20x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.488861\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.158731 q^{3} +1.00000 q^{4} +3.07692 q^{5} +0.158731 q^{6} +1.35199 q^{7} +1.00000 q^{8} -2.97480 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.158731 q^{3} +1.00000 q^{4} +3.07692 q^{5} +0.158731 q^{6} +1.35199 q^{7} +1.00000 q^{8} -2.97480 q^{9} +3.07692 q^{10} +0.158731 q^{12} -0.526746 q^{13} +1.35199 q^{14} +0.488401 q^{15} +1.00000 q^{16} +7.23593 q^{17} -2.97480 q^{18} -1.00000 q^{19} +3.07692 q^{20} +0.214602 q^{21} +2.09183 q^{23} +0.158731 q^{24} +4.46742 q^{25} -0.526746 q^{26} -0.948385 q^{27} +1.35199 q^{28} -0.277873 q^{29} +0.488401 q^{30} -1.72447 q^{31} +1.00000 q^{32} +7.23593 q^{34} +4.15995 q^{35} -2.97480 q^{36} +8.91226 q^{37} -1.00000 q^{38} -0.0836108 q^{39} +3.07692 q^{40} -8.37354 q^{41} +0.214602 q^{42} +6.30119 q^{43} -9.15323 q^{45} +2.09183 q^{46} +1.96711 q^{47} +0.158731 q^{48} -5.17213 q^{49} +4.46742 q^{50} +1.14856 q^{51} -0.526746 q^{52} -2.20703 q^{53} -0.948385 q^{54} +1.35199 q^{56} -0.158731 q^{57} -0.277873 q^{58} +8.72295 q^{59} +0.488401 q^{60} +6.38710 q^{61} -1.72447 q^{62} -4.02190 q^{63} +1.00000 q^{64} -1.62075 q^{65} -1.02001 q^{67} +7.23593 q^{68} +0.332037 q^{69} +4.15995 q^{70} -3.71385 q^{71} -2.97480 q^{72} +1.72161 q^{73} +8.91226 q^{74} +0.709116 q^{75} -1.00000 q^{76} -0.0836108 q^{78} +7.53423 q^{79} +3.07692 q^{80} +8.77388 q^{81} -8.37354 q^{82} +6.49693 q^{83} +0.214602 q^{84} +22.2643 q^{85} +6.30119 q^{86} -0.0441070 q^{87} -10.3518 q^{89} -9.15323 q^{90} -0.712155 q^{91} +2.09183 q^{92} -0.273726 q^{93} +1.96711 q^{94} -3.07692 q^{95} +0.158731 q^{96} -10.4421 q^{97} -5.17213 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 2 q^{5} + 8 q^{7} + 8 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 2 q^{5} + 8 q^{7} + 8 q^{8} + 20 q^{9} + 2 q^{10} + 18 q^{13} + 8 q^{14} + 10 q^{15} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 8 q^{19} + 2 q^{20} + 14 q^{21} + 12 q^{23} + 18 q^{26} - 24 q^{27} + 8 q^{28} + 14 q^{29} + 10 q^{30} - 2 q^{31} + 8 q^{32} + 4 q^{34} + 40 q^{35} + 20 q^{36} - 22 q^{37} - 8 q^{38} - 4 q^{39} + 2 q^{40} + 8 q^{41} + 14 q^{42} + 28 q^{43} - 28 q^{45} + 12 q^{46} + 6 q^{47} + 32 q^{49} - 12 q^{51} + 18 q^{52} - 24 q^{53} - 24 q^{54} + 8 q^{56} + 14 q^{58} + 46 q^{59} + 10 q^{60} - 24 q^{61} - 2 q^{62} + 30 q^{63} + 8 q^{64} - 16 q^{65} - 22 q^{67} + 4 q^{68} - 38 q^{69} + 40 q^{70} + 8 q^{71} + 20 q^{72} + 16 q^{73} - 22 q^{74} + 6 q^{75} - 8 q^{76} - 4 q^{78} + 4 q^{79} + 2 q^{80} + 28 q^{81} + 8 q^{82} + 12 q^{83} + 14 q^{84} + 48 q^{85} + 28 q^{86} + 42 q^{87} - 28 q^{89} - 28 q^{90} - 12 q^{91} + 12 q^{92} + 22 q^{93} + 6 q^{94} - 2 q^{95} - 22 q^{97} + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.158731 0.0916432 0.0458216 0.998950i \(-0.485409\pi\)
0.0458216 + 0.998950i \(0.485409\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.07692 1.37604 0.688019 0.725692i \(-0.258480\pi\)
0.688019 + 0.725692i \(0.258480\pi\)
\(6\) 0.158731 0.0648015
\(7\) 1.35199 0.511003 0.255502 0.966809i \(-0.417759\pi\)
0.255502 + 0.966809i \(0.417759\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.97480 −0.991602
\(10\) 3.07692 0.973007
\(11\) 0 0
\(12\) 0.158731 0.0458216
\(13\) −0.526746 −0.146093 −0.0730466 0.997329i \(-0.523272\pi\)
−0.0730466 + 0.997329i \(0.523272\pi\)
\(14\) 1.35199 0.361334
\(15\) 0.488401 0.126105
\(16\) 1.00000 0.250000
\(17\) 7.23593 1.75497 0.877485 0.479604i \(-0.159220\pi\)
0.877485 + 0.479604i \(0.159220\pi\)
\(18\) −2.97480 −0.701168
\(19\) −1.00000 −0.229416
\(20\) 3.07692 0.688019
\(21\) 0.214602 0.0468300
\(22\) 0 0
\(23\) 2.09183 0.436176 0.218088 0.975929i \(-0.430018\pi\)
0.218088 + 0.975929i \(0.430018\pi\)
\(24\) 0.158731 0.0324008
\(25\) 4.46742 0.893483
\(26\) −0.526746 −0.103303
\(27\) −0.948385 −0.182517
\(28\) 1.35199 0.255502
\(29\) −0.277873 −0.0515998 −0.0257999 0.999667i \(-0.508213\pi\)
−0.0257999 + 0.999667i \(0.508213\pi\)
\(30\) 0.488401 0.0891694
\(31\) −1.72447 −0.309724 −0.154862 0.987936i \(-0.549493\pi\)
−0.154862 + 0.987936i \(0.549493\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.23593 1.24095
\(35\) 4.15995 0.703161
\(36\) −2.97480 −0.495801
\(37\) 8.91226 1.46517 0.732584 0.680677i \(-0.238314\pi\)
0.732584 + 0.680677i \(0.238314\pi\)
\(38\) −1.00000 −0.162221
\(39\) −0.0836108 −0.0133884
\(40\) 3.07692 0.486503
\(41\) −8.37354 −1.30773 −0.653864 0.756612i \(-0.726853\pi\)
−0.653864 + 0.756612i \(0.726853\pi\)
\(42\) 0.214602 0.0331138
\(43\) 6.30119 0.960923 0.480461 0.877016i \(-0.340469\pi\)
0.480461 + 0.877016i \(0.340469\pi\)
\(44\) 0 0
\(45\) −9.15323 −1.36448
\(46\) 2.09183 0.308423
\(47\) 1.96711 0.286933 0.143466 0.989655i \(-0.454175\pi\)
0.143466 + 0.989655i \(0.454175\pi\)
\(48\) 0.158731 0.0229108
\(49\) −5.17213 −0.738875
\(50\) 4.46742 0.631788
\(51\) 1.14856 0.160831
\(52\) −0.526746 −0.0730466
\(53\) −2.20703 −0.303158 −0.151579 0.988445i \(-0.548436\pi\)
−0.151579 + 0.988445i \(0.548436\pi\)
\(54\) −0.948385 −0.129059
\(55\) 0 0
\(56\) 1.35199 0.180667
\(57\) −0.158731 −0.0210244
\(58\) −0.277873 −0.0364866
\(59\) 8.72295 1.13563 0.567816 0.823156i \(-0.307789\pi\)
0.567816 + 0.823156i \(0.307789\pi\)
\(60\) 0.488401 0.0630523
\(61\) 6.38710 0.817785 0.408892 0.912583i \(-0.365915\pi\)
0.408892 + 0.912583i \(0.365915\pi\)
\(62\) −1.72447 −0.219008
\(63\) −4.02190 −0.506712
\(64\) 1.00000 0.125000
\(65\) −1.62075 −0.201030
\(66\) 0 0
\(67\) −1.02001 −0.124614 −0.0623071 0.998057i \(-0.519846\pi\)
−0.0623071 + 0.998057i \(0.519846\pi\)
\(68\) 7.23593 0.877485
\(69\) 0.332037 0.0399726
\(70\) 4.15995 0.497210
\(71\) −3.71385 −0.440753 −0.220376 0.975415i \(-0.570729\pi\)
−0.220376 + 0.975415i \(0.570729\pi\)
\(72\) −2.97480 −0.350584
\(73\) 1.72161 0.201499 0.100750 0.994912i \(-0.467876\pi\)
0.100750 + 0.994912i \(0.467876\pi\)
\(74\) 8.91226 1.03603
\(75\) 0.709116 0.0818817
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −0.0836108 −0.00946706
\(79\) 7.53423 0.847667 0.423833 0.905740i \(-0.360684\pi\)
0.423833 + 0.905740i \(0.360684\pi\)
\(80\) 3.07692 0.344010
\(81\) 8.77388 0.974875
\(82\) −8.37354 −0.924703
\(83\) 6.49693 0.713131 0.356565 0.934270i \(-0.383948\pi\)
0.356565 + 0.934270i \(0.383948\pi\)
\(84\) 0.214602 0.0234150
\(85\) 22.2643 2.41491
\(86\) 6.30119 0.679475
\(87\) −0.0441070 −0.00472877
\(88\) 0 0
\(89\) −10.3518 −1.09728 −0.548642 0.836057i \(-0.684855\pi\)
−0.548642 + 0.836057i \(0.684855\pi\)
\(90\) −9.15323 −0.964835
\(91\) −0.712155 −0.0746541
\(92\) 2.09183 0.218088
\(93\) −0.273726 −0.0283841
\(94\) 1.96711 0.202892
\(95\) −3.07692 −0.315685
\(96\) 0.158731 0.0162004
\(97\) −10.4421 −1.06023 −0.530116 0.847925i \(-0.677852\pi\)
−0.530116 + 0.847925i \(0.677852\pi\)
\(98\) −5.17213 −0.522464
\(99\) 0 0
\(100\) 4.46742 0.446742
\(101\) 16.6075 1.65251 0.826254 0.563298i \(-0.190468\pi\)
0.826254 + 0.563298i \(0.190468\pi\)
\(102\) 1.14856 0.113725
\(103\) −6.30847 −0.621592 −0.310796 0.950477i \(-0.600596\pi\)
−0.310796 + 0.950477i \(0.600596\pi\)
\(104\) −0.526746 −0.0516517
\(105\) 0.660313 0.0644399
\(106\) −2.20703 −0.214365
\(107\) 13.1999 1.27608 0.638041 0.770002i \(-0.279745\pi\)
0.638041 + 0.770002i \(0.279745\pi\)
\(108\) −0.948385 −0.0912584
\(109\) 4.47355 0.428488 0.214244 0.976780i \(-0.431271\pi\)
0.214244 + 0.976780i \(0.431271\pi\)
\(110\) 0 0
\(111\) 1.41465 0.134273
\(112\) 1.35199 0.127751
\(113\) −12.5601 −1.18156 −0.590778 0.806834i \(-0.701179\pi\)
−0.590778 + 0.806834i \(0.701179\pi\)
\(114\) −0.158731 −0.0148665
\(115\) 6.43637 0.600195
\(116\) −0.277873 −0.0257999
\(117\) 1.56697 0.144866
\(118\) 8.72295 0.803012
\(119\) 9.78288 0.896796
\(120\) 0.488401 0.0445847
\(121\) 0 0
\(122\) 6.38710 0.578261
\(123\) −1.32914 −0.119844
\(124\) −1.72447 −0.154862
\(125\) −1.63872 −0.146571
\(126\) −4.02190 −0.358299
\(127\) −16.4582 −1.46042 −0.730212 0.683220i \(-0.760579\pi\)
−0.730212 + 0.683220i \(0.760579\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.00019 0.0880621
\(130\) −1.62075 −0.142150
\(131\) 15.1571 1.32428 0.662140 0.749380i \(-0.269648\pi\)
0.662140 + 0.749380i \(0.269648\pi\)
\(132\) 0 0
\(133\) −1.35199 −0.117232
\(134\) −1.02001 −0.0881156
\(135\) −2.91810 −0.251150
\(136\) 7.23593 0.620475
\(137\) −5.46038 −0.466512 −0.233256 0.972415i \(-0.574938\pi\)
−0.233256 + 0.972415i \(0.574938\pi\)
\(138\) 0.332037 0.0282649
\(139\) −9.22943 −0.782830 −0.391415 0.920214i \(-0.628014\pi\)
−0.391415 + 0.920214i \(0.628014\pi\)
\(140\) 4.15995 0.351580
\(141\) 0.312241 0.0262954
\(142\) −3.71385 −0.311659
\(143\) 0 0
\(144\) −2.97480 −0.247900
\(145\) −0.854993 −0.0710033
\(146\) 1.72161 0.142481
\(147\) −0.820976 −0.0677129
\(148\) 8.91226 0.732584
\(149\) −12.3302 −1.01013 −0.505065 0.863081i \(-0.668531\pi\)
−0.505065 + 0.863081i \(0.668531\pi\)
\(150\) 0.709116 0.0578991
\(151\) 3.63459 0.295779 0.147889 0.989004i \(-0.452752\pi\)
0.147889 + 0.989004i \(0.452752\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −21.5255 −1.74023
\(154\) 0 0
\(155\) −5.30605 −0.426192
\(156\) −0.0836108 −0.00669422
\(157\) −0.871549 −0.0695572 −0.0347786 0.999395i \(-0.511073\pi\)
−0.0347786 + 0.999395i \(0.511073\pi\)
\(158\) 7.53423 0.599391
\(159\) −0.350323 −0.0277824
\(160\) 3.07692 0.243252
\(161\) 2.82812 0.222887
\(162\) 8.77388 0.689341
\(163\) −7.73571 −0.605908 −0.302954 0.953005i \(-0.597973\pi\)
−0.302954 + 0.953005i \(0.597973\pi\)
\(164\) −8.37354 −0.653864
\(165\) 0 0
\(166\) 6.49693 0.504259
\(167\) 21.8284 1.68913 0.844566 0.535451i \(-0.179858\pi\)
0.844566 + 0.535451i \(0.179858\pi\)
\(168\) 0.214602 0.0165569
\(169\) −12.7225 −0.978657
\(170\) 22.2643 1.70760
\(171\) 2.97480 0.227489
\(172\) 6.30119 0.480461
\(173\) −11.0102 −0.837086 −0.418543 0.908197i \(-0.637459\pi\)
−0.418543 + 0.908197i \(0.637459\pi\)
\(174\) −0.0441070 −0.00334375
\(175\) 6.03989 0.456573
\(176\) 0 0
\(177\) 1.38460 0.104073
\(178\) −10.3518 −0.775897
\(179\) −12.5527 −0.938233 −0.469116 0.883136i \(-0.655428\pi\)
−0.469116 + 0.883136i \(0.655428\pi\)
\(180\) −9.15323 −0.682241
\(181\) −3.45298 −0.256658 −0.128329 0.991732i \(-0.540961\pi\)
−0.128329 + 0.991732i \(0.540961\pi\)
\(182\) −0.712155 −0.0527884
\(183\) 1.01383 0.0749444
\(184\) 2.09183 0.154211
\(185\) 27.4223 2.01613
\(186\) −0.273726 −0.0200706
\(187\) 0 0
\(188\) 1.96711 0.143466
\(189\) −1.28221 −0.0932667
\(190\) −3.07692 −0.223223
\(191\) 21.3218 1.54279 0.771397 0.636354i \(-0.219558\pi\)
0.771397 + 0.636354i \(0.219558\pi\)
\(192\) 0.158731 0.0114554
\(193\) 23.4522 1.68813 0.844063 0.536243i \(-0.180157\pi\)
0.844063 + 0.536243i \(0.180157\pi\)
\(194\) −10.4421 −0.749697
\(195\) −0.257263 −0.0184230
\(196\) −5.17213 −0.369438
\(197\) −7.79605 −0.555445 −0.277723 0.960661i \(-0.589580\pi\)
−0.277723 + 0.960661i \(0.589580\pi\)
\(198\) 0 0
\(199\) −9.48281 −0.672219 −0.336109 0.941823i \(-0.609111\pi\)
−0.336109 + 0.941823i \(0.609111\pi\)
\(200\) 4.46742 0.315894
\(201\) −0.161907 −0.0114200
\(202\) 16.6075 1.16850
\(203\) −0.375681 −0.0263677
\(204\) 1.14856 0.0804155
\(205\) −25.7647 −1.79948
\(206\) −6.30847 −0.439532
\(207\) −6.22277 −0.432513
\(208\) −0.526746 −0.0365233
\(209\) 0 0
\(210\) 0.660313 0.0455659
\(211\) 15.5951 1.07361 0.536805 0.843707i \(-0.319631\pi\)
0.536805 + 0.843707i \(0.319631\pi\)
\(212\) −2.20703 −0.151579
\(213\) −0.589502 −0.0403920
\(214\) 13.1999 0.902327
\(215\) 19.3882 1.32227
\(216\) −0.948385 −0.0645294
\(217\) −2.33146 −0.158270
\(218\) 4.47355 0.302987
\(219\) 0.273272 0.0184660
\(220\) 0 0
\(221\) −3.81150 −0.256389
\(222\) 1.41465 0.0949451
\(223\) −24.4375 −1.63646 −0.818230 0.574892i \(-0.805044\pi\)
−0.818230 + 0.574892i \(0.805044\pi\)
\(224\) 1.35199 0.0903335
\(225\) −13.2897 −0.885979
\(226\) −12.5601 −0.835486
\(227\) −23.2524 −1.54332 −0.771659 0.636037i \(-0.780573\pi\)
−0.771659 + 0.636037i \(0.780573\pi\)
\(228\) −0.158731 −0.0105122
\(229\) −1.94364 −0.128439 −0.0642197 0.997936i \(-0.520456\pi\)
−0.0642197 + 0.997936i \(0.520456\pi\)
\(230\) 6.43637 0.424402
\(231\) 0 0
\(232\) −0.277873 −0.0182433
\(233\) 5.46260 0.357867 0.178933 0.983861i \(-0.442735\pi\)
0.178933 + 0.983861i \(0.442735\pi\)
\(234\) 1.56697 0.102436
\(235\) 6.05264 0.394831
\(236\) 8.72295 0.567816
\(237\) 1.19591 0.0776829
\(238\) 9.78288 0.634130
\(239\) −16.1582 −1.04519 −0.522595 0.852581i \(-0.675036\pi\)
−0.522595 + 0.852581i \(0.675036\pi\)
\(240\) 0.488401 0.0315262
\(241\) −17.9215 −1.15442 −0.577212 0.816594i \(-0.695859\pi\)
−0.577212 + 0.816594i \(0.695859\pi\)
\(242\) 0 0
\(243\) 4.23784 0.271857
\(244\) 6.38710 0.408892
\(245\) −15.9142 −1.01672
\(246\) −1.32914 −0.0847428
\(247\) 0.526746 0.0335161
\(248\) −1.72447 −0.109504
\(249\) 1.03126 0.0653536
\(250\) −1.63872 −0.103641
\(251\) 26.9550 1.70139 0.850693 0.525663i \(-0.176183\pi\)
0.850693 + 0.525663i \(0.176183\pi\)
\(252\) −4.02190 −0.253356
\(253\) 0 0
\(254\) −16.4582 −1.03268
\(255\) 3.53403 0.221310
\(256\) 1.00000 0.0625000
\(257\) −22.2307 −1.38671 −0.693357 0.720594i \(-0.743869\pi\)
−0.693357 + 0.720594i \(0.743869\pi\)
\(258\) 1.00019 0.0622693
\(259\) 12.0493 0.748706
\(260\) −1.62075 −0.100515
\(261\) 0.826619 0.0511664
\(262\) 15.1571 0.936407
\(263\) −0.840832 −0.0518479 −0.0259240 0.999664i \(-0.508253\pi\)
−0.0259240 + 0.999664i \(0.508253\pi\)
\(264\) 0 0
\(265\) −6.79084 −0.417158
\(266\) −1.35199 −0.0828957
\(267\) −1.64314 −0.100559
\(268\) −1.02001 −0.0623071
\(269\) 26.6280 1.62354 0.811770 0.583978i \(-0.198504\pi\)
0.811770 + 0.583978i \(0.198504\pi\)
\(270\) −2.91810 −0.177590
\(271\) −25.3809 −1.54178 −0.770889 0.636970i \(-0.780188\pi\)
−0.770889 + 0.636970i \(0.780188\pi\)
\(272\) 7.23593 0.438742
\(273\) −0.113041 −0.00684154
\(274\) −5.46038 −0.329874
\(275\) 0 0
\(276\) 0.332037 0.0199863
\(277\) −30.1812 −1.81341 −0.906706 0.421763i \(-0.861411\pi\)
−0.906706 + 0.421763i \(0.861411\pi\)
\(278\) −9.22943 −0.553544
\(279\) 5.12996 0.307123
\(280\) 4.15995 0.248605
\(281\) 16.8269 1.00381 0.501904 0.864923i \(-0.332633\pi\)
0.501904 + 0.864923i \(0.332633\pi\)
\(282\) 0.312241 0.0185937
\(283\) −10.4403 −0.620611 −0.310305 0.950637i \(-0.600431\pi\)
−0.310305 + 0.950637i \(0.600431\pi\)
\(284\) −3.71385 −0.220376
\(285\) −0.488401 −0.0289304
\(286\) 0 0
\(287\) −11.3209 −0.668253
\(288\) −2.97480 −0.175292
\(289\) 35.3586 2.07992
\(290\) −0.854993 −0.0502069
\(291\) −1.65748 −0.0971631
\(292\) 1.72161 0.100750
\(293\) 13.4022 0.782965 0.391482 0.920186i \(-0.371962\pi\)
0.391482 + 0.920186i \(0.371962\pi\)
\(294\) −0.820976 −0.0478803
\(295\) 26.8398 1.56267
\(296\) 8.91226 0.518015
\(297\) 0 0
\(298\) −12.3302 −0.714270
\(299\) −1.10186 −0.0637223
\(300\) 0.709116 0.0409408
\(301\) 8.51914 0.491035
\(302\) 3.63459 0.209147
\(303\) 2.63612 0.151441
\(304\) −1.00000 −0.0573539
\(305\) 19.6526 1.12530
\(306\) −21.5255 −1.23053
\(307\) −18.9800 −1.08325 −0.541623 0.840622i \(-0.682190\pi\)
−0.541623 + 0.840622i \(0.682190\pi\)
\(308\) 0 0
\(309\) −1.00135 −0.0569646
\(310\) −5.30605 −0.301363
\(311\) 12.4757 0.707430 0.353715 0.935353i \(-0.384918\pi\)
0.353715 + 0.935353i \(0.384918\pi\)
\(312\) −0.0836108 −0.00473353
\(313\) −18.2742 −1.03292 −0.516459 0.856312i \(-0.672750\pi\)
−0.516459 + 0.856312i \(0.672750\pi\)
\(314\) −0.871549 −0.0491844
\(315\) −12.3751 −0.697255
\(316\) 7.53423 0.423833
\(317\) −13.1830 −0.740432 −0.370216 0.928946i \(-0.620716\pi\)
−0.370216 + 0.928946i \(0.620716\pi\)
\(318\) −0.350323 −0.0196451
\(319\) 0 0
\(320\) 3.07692 0.172005
\(321\) 2.09523 0.116944
\(322\) 2.82812 0.157605
\(323\) −7.23593 −0.402618
\(324\) 8.77388 0.487438
\(325\) −2.35319 −0.130532
\(326\) −7.73571 −0.428441
\(327\) 0.710090 0.0392681
\(328\) −8.37354 −0.462352
\(329\) 2.65951 0.146624
\(330\) 0 0
\(331\) 3.69222 0.202943 0.101471 0.994838i \(-0.467645\pi\)
0.101471 + 0.994838i \(0.467645\pi\)
\(332\) 6.49693 0.356565
\(333\) −26.5122 −1.45286
\(334\) 21.8284 1.19440
\(335\) −3.13849 −0.171474
\(336\) 0.214602 0.0117075
\(337\) 26.2983 1.43256 0.716279 0.697814i \(-0.245844\pi\)
0.716279 + 0.697814i \(0.245844\pi\)
\(338\) −12.7225 −0.692015
\(339\) −1.99368 −0.108282
\(340\) 22.2643 1.20745
\(341\) 0 0
\(342\) 2.97480 0.160859
\(343\) −16.4566 −0.888571
\(344\) 6.30119 0.339738
\(345\) 1.02165 0.0550038
\(346\) −11.0102 −0.591909
\(347\) −5.33343 −0.286313 −0.143157 0.989700i \(-0.545725\pi\)
−0.143157 + 0.989700i \(0.545725\pi\)
\(348\) −0.0441070 −0.00236439
\(349\) 19.4450 1.04087 0.520433 0.853903i \(-0.325771\pi\)
0.520433 + 0.853903i \(0.325771\pi\)
\(350\) 6.03989 0.322846
\(351\) 0.499558 0.0266644
\(352\) 0 0
\(353\) −27.8728 −1.48352 −0.741761 0.670665i \(-0.766009\pi\)
−0.741761 + 0.670665i \(0.766009\pi\)
\(354\) 1.38460 0.0735906
\(355\) −11.4272 −0.606493
\(356\) −10.3518 −0.548642
\(357\) 1.55284 0.0821852
\(358\) −12.5527 −0.663431
\(359\) 35.7338 1.88596 0.942978 0.332856i \(-0.108012\pi\)
0.942978 + 0.332856i \(0.108012\pi\)
\(360\) −9.15323 −0.482417
\(361\) 1.00000 0.0526316
\(362\) −3.45298 −0.181485
\(363\) 0 0
\(364\) −0.712155 −0.0373270
\(365\) 5.29725 0.277271
\(366\) 1.01383 0.0529937
\(367\) 37.8147 1.97391 0.986956 0.160991i \(-0.0514691\pi\)
0.986956 + 0.160991i \(0.0514691\pi\)
\(368\) 2.09183 0.109044
\(369\) 24.9097 1.29675
\(370\) 27.4223 1.42562
\(371\) −2.98387 −0.154915
\(372\) −0.273726 −0.0141920
\(373\) −20.1806 −1.04491 −0.522456 0.852667i \(-0.674984\pi\)
−0.522456 + 0.852667i \(0.674984\pi\)
\(374\) 0 0
\(375\) −0.260114 −0.0134322
\(376\) 1.96711 0.101446
\(377\) 0.146369 0.00753837
\(378\) −1.28221 −0.0659495
\(379\) −32.7152 −1.68047 −0.840233 0.542225i \(-0.817582\pi\)
−0.840233 + 0.542225i \(0.817582\pi\)
\(380\) −3.07692 −0.157842
\(381\) −2.61241 −0.133838
\(382\) 21.3218 1.09092
\(383\) −25.6817 −1.31227 −0.656136 0.754643i \(-0.727810\pi\)
−0.656136 + 0.754643i \(0.727810\pi\)
\(384\) 0.158731 0.00810019
\(385\) 0 0
\(386\) 23.4522 1.19369
\(387\) −18.7448 −0.952853
\(388\) −10.4421 −0.530116
\(389\) −4.46845 −0.226559 −0.113280 0.993563i \(-0.536136\pi\)
−0.113280 + 0.993563i \(0.536136\pi\)
\(390\) −0.257263 −0.0130270
\(391\) 15.1363 0.765475
\(392\) −5.17213 −0.261232
\(393\) 2.40589 0.121361
\(394\) −7.79605 −0.392759
\(395\) 23.1822 1.16642
\(396\) 0 0
\(397\) 28.8810 1.44950 0.724748 0.689014i \(-0.241956\pi\)
0.724748 + 0.689014i \(0.241956\pi\)
\(398\) −9.48281 −0.475330
\(399\) −0.214602 −0.0107435
\(400\) 4.46742 0.223371
\(401\) 0.797439 0.0398222 0.0199111 0.999802i \(-0.493662\pi\)
0.0199111 + 0.999802i \(0.493662\pi\)
\(402\) −0.161907 −0.00807519
\(403\) 0.908358 0.0452485
\(404\) 16.6075 0.826254
\(405\) 26.9965 1.34147
\(406\) −0.375681 −0.0186448
\(407\) 0 0
\(408\) 1.14856 0.0568624
\(409\) 22.7240 1.12363 0.561814 0.827263i \(-0.310103\pi\)
0.561814 + 0.827263i \(0.310103\pi\)
\(410\) −25.7647 −1.27243
\(411\) −0.866730 −0.0427527
\(412\) −6.30847 −0.310796
\(413\) 11.7933 0.580311
\(414\) −6.22277 −0.305833
\(415\) 19.9905 0.981296
\(416\) −0.526746 −0.0258259
\(417\) −1.46499 −0.0717410
\(418\) 0 0
\(419\) −0.427754 −0.0208971 −0.0104486 0.999945i \(-0.503326\pi\)
−0.0104486 + 0.999945i \(0.503326\pi\)
\(420\) 0.660313 0.0322200
\(421\) −9.69726 −0.472615 −0.236308 0.971678i \(-0.575937\pi\)
−0.236308 + 0.971678i \(0.575937\pi\)
\(422\) 15.5951 0.759156
\(423\) −5.85177 −0.284523
\(424\) −2.20703 −0.107183
\(425\) 32.3259 1.56804
\(426\) −0.589502 −0.0285615
\(427\) 8.63529 0.417891
\(428\) 13.1999 0.638041
\(429\) 0 0
\(430\) 19.3882 0.934984
\(431\) −6.70352 −0.322897 −0.161449 0.986881i \(-0.551617\pi\)
−0.161449 + 0.986881i \(0.551617\pi\)
\(432\) −0.948385 −0.0456292
\(433\) −15.3773 −0.738988 −0.369494 0.929233i \(-0.620469\pi\)
−0.369494 + 0.929233i \(0.620469\pi\)
\(434\) −2.33146 −0.111914
\(435\) −0.135714 −0.00650697
\(436\) 4.47355 0.214244
\(437\) −2.09183 −0.100066
\(438\) 0.273272 0.0130575
\(439\) −2.15073 −0.102649 −0.0513244 0.998682i \(-0.516344\pi\)
−0.0513244 + 0.998682i \(0.516344\pi\)
\(440\) 0 0
\(441\) 15.3861 0.732670
\(442\) −3.81150 −0.181294
\(443\) 30.3841 1.44359 0.721797 0.692105i \(-0.243317\pi\)
0.721797 + 0.692105i \(0.243317\pi\)
\(444\) 1.41465 0.0671363
\(445\) −31.8515 −1.50991
\(446\) −24.4375 −1.15715
\(447\) −1.95718 −0.0925715
\(448\) 1.35199 0.0638754
\(449\) 3.34155 0.157698 0.0788488 0.996887i \(-0.474876\pi\)
0.0788488 + 0.996887i \(0.474876\pi\)
\(450\) −13.2897 −0.626482
\(451\) 0 0
\(452\) −12.5601 −0.590778
\(453\) 0.576921 0.0271061
\(454\) −23.2524 −1.09129
\(455\) −2.19124 −0.102727
\(456\) −0.158731 −0.00743325
\(457\) 29.3880 1.37471 0.687356 0.726321i \(-0.258771\pi\)
0.687356 + 0.726321i \(0.258771\pi\)
\(458\) −1.94364 −0.0908204
\(459\) −6.86244 −0.320311
\(460\) 6.43637 0.300097
\(461\) −1.74308 −0.0811833 −0.0405917 0.999176i \(-0.512924\pi\)
−0.0405917 + 0.999176i \(0.512924\pi\)
\(462\) 0 0
\(463\) −31.2015 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(464\) −0.277873 −0.0128999
\(465\) −0.842233 −0.0390576
\(466\) 5.46260 0.253050
\(467\) 26.0733 1.20653 0.603264 0.797542i \(-0.293867\pi\)
0.603264 + 0.797542i \(0.293867\pi\)
\(468\) 1.56697 0.0724331
\(469\) −1.37904 −0.0636783
\(470\) 6.05264 0.279187
\(471\) −0.138342 −0.00637444
\(472\) 8.72295 0.401506
\(473\) 0 0
\(474\) 1.19591 0.0549301
\(475\) −4.46742 −0.204979
\(476\) 9.78288 0.448398
\(477\) 6.56547 0.300612
\(478\) −16.1582 −0.739061
\(479\) −29.3946 −1.34307 −0.671536 0.740972i \(-0.734365\pi\)
−0.671536 + 0.740972i \(0.734365\pi\)
\(480\) 0.488401 0.0222924
\(481\) −4.69450 −0.214051
\(482\) −17.9215 −0.816301
\(483\) 0.448910 0.0204261
\(484\) 0 0
\(485\) −32.1294 −1.45892
\(486\) 4.23784 0.192232
\(487\) 43.0919 1.95268 0.976340 0.216242i \(-0.0693801\pi\)
0.976340 + 0.216242i \(0.0693801\pi\)
\(488\) 6.38710 0.289131
\(489\) −1.22790 −0.0555273
\(490\) −15.9142 −0.718931
\(491\) −29.5401 −1.33313 −0.666563 0.745449i \(-0.732235\pi\)
−0.666563 + 0.745449i \(0.732235\pi\)
\(492\) −1.32914 −0.0599222
\(493\) −2.01067 −0.0905561
\(494\) 0.526746 0.0236994
\(495\) 0 0
\(496\) −1.72447 −0.0774309
\(497\) −5.02108 −0.225226
\(498\) 1.03126 0.0462120
\(499\) −27.8915 −1.24859 −0.624297 0.781187i \(-0.714615\pi\)
−0.624297 + 0.781187i \(0.714615\pi\)
\(500\) −1.63872 −0.0732856
\(501\) 3.46484 0.154798
\(502\) 26.9550 1.20306
\(503\) 0.0484888 0.00216201 0.00108100 0.999999i \(-0.499656\pi\)
0.00108100 + 0.999999i \(0.499656\pi\)
\(504\) −4.02190 −0.179150
\(505\) 51.0999 2.27392
\(506\) 0 0
\(507\) −2.01946 −0.0896873
\(508\) −16.4582 −0.730212
\(509\) 9.75291 0.432290 0.216145 0.976361i \(-0.430652\pi\)
0.216145 + 0.976361i \(0.430652\pi\)
\(510\) 3.53403 0.156490
\(511\) 2.32759 0.102967
\(512\) 1.00000 0.0441942
\(513\) 0.948385 0.0418722
\(514\) −22.2307 −0.980555
\(515\) −19.4106 −0.855334
\(516\) 1.00019 0.0440310
\(517\) 0 0
\(518\) 12.0493 0.529415
\(519\) −1.74765 −0.0767133
\(520\) −1.62075 −0.0710748
\(521\) −40.4077 −1.77029 −0.885147 0.465312i \(-0.845942\pi\)
−0.885147 + 0.465312i \(0.845942\pi\)
\(522\) 0.826619 0.0361801
\(523\) 22.7718 0.995742 0.497871 0.867251i \(-0.334115\pi\)
0.497871 + 0.867251i \(0.334115\pi\)
\(524\) 15.1571 0.662140
\(525\) 0.958717 0.0418418
\(526\) −0.840832 −0.0366620
\(527\) −12.4781 −0.543556
\(528\) 0 0
\(529\) −18.6243 −0.809751
\(530\) −6.79084 −0.294975
\(531\) −25.9491 −1.12609
\(532\) −1.35199 −0.0586161
\(533\) 4.41073 0.191050
\(534\) −1.64314 −0.0711057
\(535\) 40.6150 1.75594
\(536\) −1.02001 −0.0440578
\(537\) −1.99250 −0.0859827
\(538\) 26.6280 1.14802
\(539\) 0 0
\(540\) −2.91810 −0.125575
\(541\) −7.41552 −0.318818 −0.159409 0.987213i \(-0.550959\pi\)
−0.159409 + 0.987213i \(0.550959\pi\)
\(542\) −25.3809 −1.09020
\(543\) −0.548094 −0.0235210
\(544\) 7.23593 0.310238
\(545\) 13.7647 0.589617
\(546\) −0.113041 −0.00483770
\(547\) 2.41910 0.103433 0.0517167 0.998662i \(-0.483531\pi\)
0.0517167 + 0.998662i \(0.483531\pi\)
\(548\) −5.46038 −0.233256
\(549\) −19.0004 −0.810917
\(550\) 0 0
\(551\) 0.277873 0.0118378
\(552\) 0.332037 0.0141324
\(553\) 10.1862 0.433161
\(554\) −30.1812 −1.28228
\(555\) 4.35276 0.184764
\(556\) −9.22943 −0.391415
\(557\) 0.412176 0.0174645 0.00873224 0.999962i \(-0.497220\pi\)
0.00873224 + 0.999962i \(0.497220\pi\)
\(558\) 5.12996 0.217168
\(559\) −3.31913 −0.140384
\(560\) 4.15995 0.175790
\(561\) 0 0
\(562\) 16.8269 0.709800
\(563\) 4.49045 0.189250 0.0946249 0.995513i \(-0.469835\pi\)
0.0946249 + 0.995513i \(0.469835\pi\)
\(564\) 0.312241 0.0131477
\(565\) −38.6464 −1.62587
\(566\) −10.4403 −0.438838
\(567\) 11.8622 0.498165
\(568\) −3.71385 −0.155830
\(569\) −32.6319 −1.36800 −0.684000 0.729482i \(-0.739761\pi\)
−0.684000 + 0.729482i \(0.739761\pi\)
\(570\) −0.488401 −0.0204569
\(571\) 18.1484 0.759486 0.379743 0.925092i \(-0.376012\pi\)
0.379743 + 0.925092i \(0.376012\pi\)
\(572\) 0 0
\(573\) 3.38443 0.141387
\(574\) −11.3209 −0.472527
\(575\) 9.34506 0.389716
\(576\) −2.97480 −0.123950
\(577\) −21.9198 −0.912532 −0.456266 0.889843i \(-0.650814\pi\)
−0.456266 + 0.889843i \(0.650814\pi\)
\(578\) 35.3586 1.47072
\(579\) 3.72259 0.154705
\(580\) −0.854993 −0.0355017
\(581\) 8.78377 0.364412
\(582\) −1.65748 −0.0687047
\(583\) 0 0
\(584\) 1.72161 0.0712407
\(585\) 4.82143 0.199341
\(586\) 13.4022 0.553640
\(587\) 16.9253 0.698581 0.349291 0.937014i \(-0.386423\pi\)
0.349291 + 0.937014i \(0.386423\pi\)
\(588\) −0.820976 −0.0338565
\(589\) 1.72447 0.0710555
\(590\) 26.8398 1.10498
\(591\) −1.23747 −0.0509028
\(592\) 8.91226 0.366292
\(593\) 9.83301 0.403793 0.201897 0.979407i \(-0.435290\pi\)
0.201897 + 0.979407i \(0.435290\pi\)
\(594\) 0 0
\(595\) 30.1011 1.23403
\(596\) −12.3302 −0.505065
\(597\) −1.50521 −0.0616043
\(598\) −1.10186 −0.0450585
\(599\) 43.2136 1.76566 0.882829 0.469694i \(-0.155636\pi\)
0.882829 + 0.469694i \(0.155636\pi\)
\(600\) 0.709116 0.0289495
\(601\) −6.60416 −0.269389 −0.134695 0.990887i \(-0.543005\pi\)
−0.134695 + 0.990887i \(0.543005\pi\)
\(602\) 8.51914 0.347214
\(603\) 3.03433 0.123568
\(604\) 3.63459 0.147889
\(605\) 0 0
\(606\) 2.63612 0.107085
\(607\) 19.6695 0.798359 0.399180 0.916873i \(-0.369295\pi\)
0.399180 + 0.916873i \(0.369295\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −0.0596322 −0.00241642
\(610\) 19.6526 0.795710
\(611\) −1.03617 −0.0419189
\(612\) −21.5255 −0.870115
\(613\) −1.55586 −0.0628407 −0.0314203 0.999506i \(-0.510003\pi\)
−0.0314203 + 0.999506i \(0.510003\pi\)
\(614\) −18.9800 −0.765970
\(615\) −4.08965 −0.164911
\(616\) 0 0
\(617\) −43.2521 −1.74126 −0.870632 0.491935i \(-0.836290\pi\)
−0.870632 + 0.491935i \(0.836290\pi\)
\(618\) −1.00135 −0.0402801
\(619\) 3.26969 0.131420 0.0657100 0.997839i \(-0.479069\pi\)
0.0657100 + 0.997839i \(0.479069\pi\)
\(620\) −5.30605 −0.213096
\(621\) −1.98386 −0.0796094
\(622\) 12.4757 0.500228
\(623\) −13.9955 −0.560716
\(624\) −0.0836108 −0.00334711
\(625\) −27.3793 −1.09517
\(626\) −18.2742 −0.730383
\(627\) 0 0
\(628\) −0.871549 −0.0347786
\(629\) 64.4885 2.57132
\(630\) −12.3751 −0.493034
\(631\) 27.6628 1.10124 0.550620 0.834756i \(-0.314391\pi\)
0.550620 + 0.834756i \(0.314391\pi\)
\(632\) 7.53423 0.299695
\(633\) 2.47542 0.0983890
\(634\) −13.1830 −0.523565
\(635\) −50.6404 −2.00960
\(636\) −0.350323 −0.0138912
\(637\) 2.72440 0.107945
\(638\) 0 0
\(639\) 11.0480 0.437051
\(640\) 3.07692 0.121626
\(641\) −40.1504 −1.58585 −0.792924 0.609321i \(-0.791442\pi\)
−0.792924 + 0.609321i \(0.791442\pi\)
\(642\) 2.09523 0.0826921
\(643\) −29.3892 −1.15900 −0.579499 0.814973i \(-0.696752\pi\)
−0.579499 + 0.814973i \(0.696752\pi\)
\(644\) 2.82812 0.111444
\(645\) 3.07751 0.121177
\(646\) −7.23593 −0.284694
\(647\) 3.63135 0.142763 0.0713816 0.997449i \(-0.477259\pi\)
0.0713816 + 0.997449i \(0.477259\pi\)
\(648\) 8.77388 0.344670
\(649\) 0 0
\(650\) −2.35319 −0.0922999
\(651\) −0.370075 −0.0145044
\(652\) −7.73571 −0.302954
\(653\) −25.9076 −1.01384 −0.506921 0.861992i \(-0.669217\pi\)
−0.506921 + 0.861992i \(0.669217\pi\)
\(654\) 0.710090 0.0277667
\(655\) 46.6371 1.82226
\(656\) −8.37354 −0.326932
\(657\) −5.12145 −0.199807
\(658\) 2.65951 0.103679
\(659\) −22.9664 −0.894644 −0.447322 0.894373i \(-0.647622\pi\)
−0.447322 + 0.894373i \(0.647622\pi\)
\(660\) 0 0
\(661\) 43.4685 1.69073 0.845365 0.534189i \(-0.179383\pi\)
0.845365 + 0.534189i \(0.179383\pi\)
\(662\) 3.69222 0.143502
\(663\) −0.605002 −0.0234963
\(664\) 6.49693 0.252130
\(665\) −4.15995 −0.161316
\(666\) −26.5122 −1.02733
\(667\) −0.581263 −0.0225066
\(668\) 21.8284 0.844566
\(669\) −3.87899 −0.149970
\(670\) −3.13849 −0.121250
\(671\) 0 0
\(672\) 0.214602 0.00827845
\(673\) −29.2936 −1.12918 −0.564592 0.825370i \(-0.690967\pi\)
−0.564592 + 0.825370i \(0.690967\pi\)
\(674\) 26.2983 1.01297
\(675\) −4.23683 −0.163076
\(676\) −12.7225 −0.489328
\(677\) −38.9988 −1.49884 −0.749422 0.662093i \(-0.769669\pi\)
−0.749422 + 0.662093i \(0.769669\pi\)
\(678\) −1.99368 −0.0765667
\(679\) −14.1176 −0.541782
\(680\) 22.2643 0.853798
\(681\) −3.69087 −0.141435
\(682\) 0 0
\(683\) −13.6716 −0.523130 −0.261565 0.965186i \(-0.584239\pi\)
−0.261565 + 0.965186i \(0.584239\pi\)
\(684\) 2.97480 0.113744
\(685\) −16.8011 −0.641939
\(686\) −16.4566 −0.628315
\(687\) −0.308515 −0.0117706
\(688\) 6.30119 0.240231
\(689\) 1.16254 0.0442894
\(690\) 1.02165 0.0388936
\(691\) 8.06092 0.306652 0.153326 0.988176i \(-0.451002\pi\)
0.153326 + 0.988176i \(0.451002\pi\)
\(692\) −11.0102 −0.418543
\(693\) 0 0
\(694\) −5.33343 −0.202454
\(695\) −28.3982 −1.07720
\(696\) −0.0441070 −0.00167187
\(697\) −60.5903 −2.29502
\(698\) 19.4450 0.736003
\(699\) 0.867082 0.0327961
\(700\) 6.03989 0.228287
\(701\) −11.0066 −0.415713 −0.207857 0.978159i \(-0.566649\pi\)
−0.207857 + 0.978159i \(0.566649\pi\)
\(702\) 0.499558 0.0188546
\(703\) −8.91226 −0.336132
\(704\) 0 0
\(705\) 0.960740 0.0361836
\(706\) −27.8728 −1.04901
\(707\) 22.4531 0.844437
\(708\) 1.38460 0.0520364
\(709\) 3.81971 0.143452 0.0717260 0.997424i \(-0.477149\pi\)
0.0717260 + 0.997424i \(0.477149\pi\)
\(710\) −11.4272 −0.428855
\(711\) −22.4129 −0.840548
\(712\) −10.3518 −0.387948
\(713\) −3.60729 −0.135094
\(714\) 1.55284 0.0581137
\(715\) 0 0
\(716\) −12.5527 −0.469116
\(717\) −2.56481 −0.0957846
\(718\) 35.7338 1.33357
\(719\) 12.4943 0.465958 0.232979 0.972482i \(-0.425153\pi\)
0.232979 + 0.972482i \(0.425153\pi\)
\(720\) −9.15323 −0.341121
\(721\) −8.52897 −0.317635
\(722\) 1.00000 0.0372161
\(723\) −2.84469 −0.105795
\(724\) −3.45298 −0.128329
\(725\) −1.24138 −0.0461036
\(726\) 0 0
\(727\) 21.9023 0.812311 0.406155 0.913804i \(-0.366869\pi\)
0.406155 + 0.913804i \(0.366869\pi\)
\(728\) −0.712155 −0.0263942
\(729\) −25.6490 −0.949961
\(730\) 5.29725 0.196060
\(731\) 45.5950 1.68639
\(732\) 1.01383 0.0374722
\(733\) 27.3319 1.00953 0.504764 0.863257i \(-0.331579\pi\)
0.504764 + 0.863257i \(0.331579\pi\)
\(734\) 37.8147 1.39577
\(735\) −2.52607 −0.0931756
\(736\) 2.09183 0.0771057
\(737\) 0 0
\(738\) 24.9097 0.916937
\(739\) −6.87044 −0.252733 −0.126367 0.991984i \(-0.540332\pi\)
−0.126367 + 0.991984i \(0.540332\pi\)
\(740\) 27.4223 1.00806
\(741\) 0.0836108 0.00307152
\(742\) −2.98387 −0.109541
\(743\) −38.5332 −1.41365 −0.706824 0.707390i \(-0.749873\pi\)
−0.706824 + 0.707390i \(0.749873\pi\)
\(744\) −0.273726 −0.0100353
\(745\) −37.9390 −1.38998
\(746\) −20.1806 −0.738864
\(747\) −19.3271 −0.707141
\(748\) 0 0
\(749\) 17.8461 0.652083
\(750\) −0.260114 −0.00949803
\(751\) 19.0301 0.694418 0.347209 0.937788i \(-0.387130\pi\)
0.347209 + 0.937788i \(0.387130\pi\)
\(752\) 1.96711 0.0717332
\(753\) 4.27859 0.155920
\(754\) 0.146369 0.00533044
\(755\) 11.1833 0.407003
\(756\) −1.28221 −0.0466333
\(757\) −42.0004 −1.52653 −0.763266 0.646084i \(-0.776405\pi\)
−0.763266 + 0.646084i \(0.776405\pi\)
\(758\) −32.7152 −1.18827
\(759\) 0 0
\(760\) −3.07692 −0.111612
\(761\) −21.9298 −0.794955 −0.397477 0.917612i \(-0.630114\pi\)
−0.397477 + 0.917612i \(0.630114\pi\)
\(762\) −2.61241 −0.0946378
\(763\) 6.04819 0.218959
\(764\) 21.3218 0.771397
\(765\) −66.2321 −2.39463
\(766\) −25.6817 −0.927917
\(767\) −4.59478 −0.165908
\(768\) 0.158731 0.00572770
\(769\) −20.9969 −0.757167 −0.378584 0.925567i \(-0.623589\pi\)
−0.378584 + 0.925567i \(0.623589\pi\)
\(770\) 0 0
\(771\) −3.52870 −0.127083
\(772\) 23.4522 0.844063
\(773\) 38.3380 1.37892 0.689460 0.724323i \(-0.257848\pi\)
0.689460 + 0.724323i \(0.257848\pi\)
\(774\) −18.7448 −0.673769
\(775\) −7.70392 −0.276733
\(776\) −10.4421 −0.374849
\(777\) 1.91259 0.0686138
\(778\) −4.46845 −0.160202
\(779\) 8.37354 0.300013
\(780\) −0.257263 −0.00921151
\(781\) 0 0
\(782\) 15.1363 0.541273
\(783\) 0.263531 0.00941783
\(784\) −5.17213 −0.184719
\(785\) −2.68168 −0.0957134
\(786\) 2.40589 0.0858154
\(787\) 3.85555 0.137435 0.0687177 0.997636i \(-0.478109\pi\)
0.0687177 + 0.997636i \(0.478109\pi\)
\(788\) −7.79605 −0.277723
\(789\) −0.133466 −0.00475151
\(790\) 23.1822 0.824785
\(791\) −16.9811 −0.603779
\(792\) 0 0
\(793\) −3.36438 −0.119473
\(794\) 28.8810 1.02495
\(795\) −1.07791 −0.0382297
\(796\) −9.48281 −0.336109
\(797\) 7.72685 0.273699 0.136850 0.990592i \(-0.456302\pi\)
0.136850 + 0.990592i \(0.456302\pi\)
\(798\) −0.214602 −0.00759683
\(799\) 14.2339 0.503558
\(800\) 4.46742 0.157947
\(801\) 30.7945 1.08807
\(802\) 0.797439 0.0281586
\(803\) 0 0
\(804\) −0.161907 −0.00571002
\(805\) 8.70190 0.306702
\(806\) 0.908358 0.0319955
\(807\) 4.22669 0.148786
\(808\) 16.6075 0.584250
\(809\) −45.8483 −1.61194 −0.805970 0.591957i \(-0.798356\pi\)
−0.805970 + 0.591957i \(0.798356\pi\)
\(810\) 26.9965 0.948560
\(811\) −11.9071 −0.418115 −0.209058 0.977903i \(-0.567040\pi\)
−0.209058 + 0.977903i \(0.567040\pi\)
\(812\) −0.375681 −0.0131838
\(813\) −4.02872 −0.141293
\(814\) 0 0
\(815\) −23.8021 −0.833753
\(816\) 1.14856 0.0402078
\(817\) −6.30119 −0.220451
\(818\) 22.7240 0.794526
\(819\) 2.11852 0.0740271
\(820\) −25.7647 −0.899742
\(821\) −25.0389 −0.873865 −0.436932 0.899494i \(-0.643935\pi\)
−0.436932 + 0.899494i \(0.643935\pi\)
\(822\) −0.866730 −0.0302307
\(823\) −4.14621 −0.144528 −0.0722639 0.997386i \(-0.523022\pi\)
−0.0722639 + 0.997386i \(0.523022\pi\)
\(824\) −6.30847 −0.219766
\(825\) 0 0
\(826\) 11.7933 0.410342
\(827\) −47.6070 −1.65546 −0.827729 0.561127i \(-0.810368\pi\)
−0.827729 + 0.561127i \(0.810368\pi\)
\(828\) −6.22277 −0.216256
\(829\) 5.19254 0.180344 0.0901722 0.995926i \(-0.471258\pi\)
0.0901722 + 0.995926i \(0.471258\pi\)
\(830\) 19.9905 0.693881
\(831\) −4.79068 −0.166187
\(832\) −0.526746 −0.0182616
\(833\) −37.4251 −1.29670
\(834\) −1.46499 −0.0507286
\(835\) 67.1642 2.32431
\(836\) 0 0
\(837\) 1.63546 0.0565298
\(838\) −0.427754 −0.0147765
\(839\) −43.4560 −1.50027 −0.750133 0.661287i \(-0.770011\pi\)
−0.750133 + 0.661287i \(0.770011\pi\)
\(840\) 0.660313 0.0227829
\(841\) −28.9228 −0.997337
\(842\) −9.69726 −0.334190
\(843\) 2.67094 0.0919922
\(844\) 15.5951 0.536805
\(845\) −39.1462 −1.34667
\(846\) −5.85177 −0.201188
\(847\) 0 0
\(848\) −2.20703 −0.0757896
\(849\) −1.65719 −0.0568748
\(850\) 32.3259 1.10877
\(851\) 18.6429 0.639071
\(852\) −0.589502 −0.0201960
\(853\) −29.3178 −1.00382 −0.501911 0.864919i \(-0.667369\pi\)
−0.501911 + 0.864919i \(0.667369\pi\)
\(854\) 8.63529 0.295493
\(855\) 9.15323 0.313034
\(856\) 13.1999 0.451163
\(857\) 3.67262 0.125454 0.0627272 0.998031i \(-0.480020\pi\)
0.0627272 + 0.998031i \(0.480020\pi\)
\(858\) 0 0
\(859\) −19.7758 −0.674742 −0.337371 0.941372i \(-0.609538\pi\)
−0.337371 + 0.941372i \(0.609538\pi\)
\(860\) 19.3882 0.661134
\(861\) −1.79698 −0.0612409
\(862\) −6.70352 −0.228323
\(863\) −33.2555 −1.13203 −0.566015 0.824395i \(-0.691516\pi\)
−0.566015 + 0.824395i \(0.691516\pi\)
\(864\) −0.948385 −0.0322647
\(865\) −33.8773 −1.15186
\(866\) −15.3773 −0.522544
\(867\) 5.61250 0.190610
\(868\) −2.33146 −0.0791350
\(869\) 0 0
\(870\) −0.135714 −0.00460112
\(871\) 0.537287 0.0182053
\(872\) 4.47355 0.151494
\(873\) 31.0631 1.05133
\(874\) −2.09183 −0.0707571
\(875\) −2.21552 −0.0748984
\(876\) 0.273272 0.00923301
\(877\) −14.7200 −0.497060 −0.248530 0.968624i \(-0.579948\pi\)
−0.248530 + 0.968624i \(0.579948\pi\)
\(878\) −2.15073 −0.0725837
\(879\) 2.12734 0.0717534
\(880\) 0 0
\(881\) −2.16151 −0.0728230 −0.0364115 0.999337i \(-0.511593\pi\)
−0.0364115 + 0.999337i \(0.511593\pi\)
\(882\) 15.3861 0.518076
\(883\) −11.6171 −0.390947 −0.195473 0.980709i \(-0.562624\pi\)
−0.195473 + 0.980709i \(0.562624\pi\)
\(884\) −3.81150 −0.128194
\(885\) 4.26030 0.143208
\(886\) 30.3841 1.02077
\(887\) 22.5487 0.757111 0.378555 0.925579i \(-0.376421\pi\)
0.378555 + 0.925579i \(0.376421\pi\)
\(888\) 1.41465 0.0474725
\(889\) −22.2512 −0.746282
\(890\) −31.8515 −1.06766
\(891\) 0 0
\(892\) −24.4375 −0.818230
\(893\) −1.96711 −0.0658269
\(894\) −1.95718 −0.0654580
\(895\) −38.6236 −1.29104
\(896\) 1.35199 0.0451668
\(897\) −0.174899 −0.00583971
\(898\) 3.34155 0.111509
\(899\) 0.479184 0.0159817
\(900\) −13.2897 −0.442990
\(901\) −15.9699 −0.532034
\(902\) 0 0
\(903\) 1.35225 0.0450000
\(904\) −12.5601 −0.417743
\(905\) −10.6245 −0.353171
\(906\) 0.576921 0.0191669
\(907\) 18.3913 0.610672 0.305336 0.952245i \(-0.401231\pi\)
0.305336 + 0.952245i \(0.401231\pi\)
\(908\) −23.2524 −0.771659
\(909\) −49.4041 −1.63863
\(910\) −2.19124 −0.0726389
\(911\) 47.2453 1.56531 0.782654 0.622458i \(-0.213866\pi\)
0.782654 + 0.622458i \(0.213866\pi\)
\(912\) −0.158731 −0.00525610
\(913\) 0 0
\(914\) 29.3880 0.972068
\(915\) 3.11947 0.103126
\(916\) −1.94364 −0.0642197
\(917\) 20.4922 0.676712
\(918\) −6.86244 −0.226494
\(919\) −9.79492 −0.323105 −0.161552 0.986864i \(-0.551650\pi\)
−0.161552 + 0.986864i \(0.551650\pi\)
\(920\) 6.43637 0.212201
\(921\) −3.01271 −0.0992721
\(922\) −1.74308 −0.0574053
\(923\) 1.95626 0.0643910
\(924\) 0 0
\(925\) 39.8148 1.30910
\(926\) −31.2015 −1.02535
\(927\) 18.7665 0.616371
\(928\) −0.277873 −0.00912164
\(929\) 22.9921 0.754346 0.377173 0.926143i \(-0.376896\pi\)
0.377173 + 0.926143i \(0.376896\pi\)
\(930\) −0.842233 −0.0276179
\(931\) 5.17213 0.169510
\(932\) 5.46260 0.178933
\(933\) 1.98027 0.0648311
\(934\) 26.0733 0.853144
\(935\) 0 0
\(936\) 1.56697 0.0512179
\(937\) 36.6455 1.19716 0.598578 0.801065i \(-0.295733\pi\)
0.598578 + 0.801065i \(0.295733\pi\)
\(938\) −1.37904 −0.0450274
\(939\) −2.90067 −0.0946599
\(940\) 6.05264 0.197415
\(941\) −14.2908 −0.465868 −0.232934 0.972493i \(-0.574833\pi\)
−0.232934 + 0.972493i \(0.574833\pi\)
\(942\) −0.138342 −0.00450741
\(943\) −17.5160 −0.570399
\(944\) 8.72295 0.283908
\(945\) −3.94524 −0.128339
\(946\) 0 0
\(947\) 49.0452 1.59375 0.796877 0.604142i \(-0.206484\pi\)
0.796877 + 0.604142i \(0.206484\pi\)
\(948\) 1.19591 0.0388414
\(949\) −0.906851 −0.0294376
\(950\) −4.46742 −0.144942
\(951\) −2.09255 −0.0678556
\(952\) 9.78288 0.317065
\(953\) −9.95579 −0.322500 −0.161250 0.986914i \(-0.551553\pi\)
−0.161250 + 0.986914i \(0.551553\pi\)
\(954\) 6.56547 0.212565
\(955\) 65.6055 2.12294
\(956\) −16.1582 −0.522595
\(957\) 0 0
\(958\) −29.3946 −0.949695
\(959\) −7.38237 −0.238389
\(960\) 0.488401 0.0157631
\(961\) −28.0262 −0.904071
\(962\) −4.69450 −0.151357
\(963\) −39.2671 −1.26537
\(964\) −17.9215 −0.577212
\(965\) 72.1605 2.32293
\(966\) 0.448910 0.0144434
\(967\) −12.0203 −0.386546 −0.193273 0.981145i \(-0.561910\pi\)
−0.193273 + 0.981145i \(0.561910\pi\)
\(968\) 0 0
\(969\) −1.14856 −0.0368972
\(970\) −32.1294 −1.03161
\(971\) 51.9013 1.66559 0.832796 0.553580i \(-0.186739\pi\)
0.832796 + 0.553580i \(0.186739\pi\)
\(972\) 4.23784 0.135929
\(973\) −12.4781 −0.400029
\(974\) 43.0919 1.38075
\(975\) −0.373524 −0.0119624
\(976\) 6.38710 0.204446
\(977\) −28.7379 −0.919407 −0.459703 0.888073i \(-0.652044\pi\)
−0.459703 + 0.888073i \(0.652044\pi\)
\(978\) −1.22790 −0.0392638
\(979\) 0 0
\(980\) −15.9142 −0.508361
\(981\) −13.3079 −0.424890
\(982\) −29.5401 −0.942662
\(983\) 36.7856 1.17328 0.586639 0.809849i \(-0.300451\pi\)
0.586639 + 0.809849i \(0.300451\pi\)
\(984\) −1.32914 −0.0423714
\(985\) −23.9878 −0.764315
\(986\) −2.01067 −0.0640328
\(987\) 0.422146 0.0134371
\(988\) 0.526746 0.0167580
\(989\) 13.1810 0.419131
\(990\) 0 0
\(991\) −59.5095 −1.89038 −0.945191 0.326517i \(-0.894125\pi\)
−0.945191 + 0.326517i \(0.894125\pi\)
\(992\) −1.72447 −0.0547519
\(993\) 0.586069 0.0185983
\(994\) −5.02108 −0.159259
\(995\) −29.1778 −0.924999
\(996\) 1.03126 0.0326768
\(997\) −27.5241 −0.871697 −0.435848 0.900020i \(-0.643552\pi\)
−0.435848 + 0.900020i \(0.643552\pi\)
\(998\) −27.8915 −0.882889
\(999\) −8.45226 −0.267418
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 4598.2.a.ca.1.4 8
11.3 even 5 418.2.f.f.229.3 yes 16
11.4 even 5 418.2.f.f.115.3 16
11.10 odd 2 4598.2.a.bx.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.f.f.115.3 16 11.4 even 5
418.2.f.f.229.3 yes 16 11.3 even 5
4598.2.a.bx.1.4 8 11.10 odd 2
4598.2.a.ca.1.4 8 1.1 even 1 trivial