Properties

Label 6422.2.a.n.1.1
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.80194 q^{3} +1.00000 q^{4} +2.44504 q^{5} +1.80194 q^{6} +4.49396 q^{7} -1.00000 q^{8} +0.246980 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.80194 q^{3} +1.00000 q^{4} +2.44504 q^{5} +1.80194 q^{6} +4.49396 q^{7} -1.00000 q^{8} +0.246980 q^{9} -2.44504 q^{10} +2.49396 q^{11} -1.80194 q^{12} -4.49396 q^{14} -4.40581 q^{15} +1.00000 q^{16} +7.74094 q^{17} -0.246980 q^{18} -1.00000 q^{19} +2.44504 q^{20} -8.09783 q^{21} -2.49396 q^{22} +4.27413 q^{23} +1.80194 q^{24} +0.978230 q^{25} +4.96077 q^{27} +4.49396 q^{28} -0.713792 q^{29} +4.40581 q^{30} +0.911854 q^{31} -1.00000 q^{32} -4.49396 q^{33} -7.74094 q^{34} +10.9879 q^{35} +0.246980 q^{36} -6.81163 q^{37} +1.00000 q^{38} -2.44504 q^{40} +3.28621 q^{41} +8.09783 q^{42} +7.20775 q^{43} +2.49396 q^{44} +0.603875 q^{45} -4.27413 q^{46} -5.20775 q^{47} -1.80194 q^{48} +13.1957 q^{49} -0.978230 q^{50} -13.9487 q^{51} -8.89008 q^{53} -4.96077 q^{54} +6.09783 q^{55} -4.49396 q^{56} +1.80194 q^{57} +0.713792 q^{58} +11.2567 q^{59} -4.40581 q^{60} +3.43296 q^{61} -0.911854 q^{62} +1.10992 q^{63} +1.00000 q^{64} +4.49396 q^{66} +1.40581 q^{67} +7.74094 q^{68} -7.70171 q^{69} -10.9879 q^{70} -3.86294 q^{71} -0.246980 q^{72} +7.74094 q^{73} +6.81163 q^{74} -1.76271 q^{75} -1.00000 q^{76} +11.2078 q^{77} -1.91723 q^{79} +2.44504 q^{80} -9.67994 q^{81} -3.28621 q^{82} +0.396125 q^{83} -8.09783 q^{84} +18.9269 q^{85} -7.20775 q^{86} +1.28621 q^{87} -2.49396 q^{88} +16.7681 q^{89} -0.603875 q^{90} +4.27413 q^{92} -1.64310 q^{93} +5.20775 q^{94} -2.44504 q^{95} +1.80194 q^{96} -3.16421 q^{97} -13.1957 q^{98} +0.615957 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 7 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} - 4 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + 7 q^{5} + q^{6} + 4 q^{7} - 3 q^{8} - 4 q^{9} - 7 q^{10} - 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} + 4 q^{18} - 3 q^{19} + 7 q^{20} - 6 q^{21} + 2 q^{22} + 2 q^{23} + q^{24} + 6 q^{25} + 2 q^{27} + 4 q^{28} + 6 q^{29} - q^{31} - 3 q^{32} - 4 q^{33} - 9 q^{34} + 14 q^{35} - 4 q^{36} + 6 q^{37} + 3 q^{38} - 7 q^{40} + 18 q^{41} + 6 q^{42} + 4 q^{43} - 2 q^{44} - 7 q^{45} - 2 q^{46} + 2 q^{47} - q^{48} + 3 q^{49} - 6 q^{50} - 10 q^{51} - 26 q^{53} - 2 q^{54} - 4 q^{56} + q^{57} - 6 q^{58} + 7 q^{59} - 9 q^{61} + q^{62} + 4 q^{63} + 3 q^{64} + 4 q^{66} - 9 q^{67} + 9 q^{68} + 4 q^{69} - 14 q^{70} - 17 q^{71} + 4 q^{72} + 9 q^{73} - 6 q^{74} + 12 q^{75} - 3 q^{76} + 16 q^{77} + q^{79} + 7 q^{80} - 5 q^{81} - 18 q^{82} + 10 q^{83} - 6 q^{84} + 28 q^{85} - 4 q^{86} + 12 q^{87} + 2 q^{88} + 30 q^{89} + 7 q^{90} + 2 q^{92} - 9 q^{93} - 2 q^{94} - 7 q^{95} + q^{96} + 2 q^{97} - 3 q^{98} + 12 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.80194 −1.04035 −0.520175 0.854060i \(-0.674133\pi\)
−0.520175 + 0.854060i \(0.674133\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.44504 1.09346 0.546728 0.837310i \(-0.315873\pi\)
0.546728 + 0.837310i \(0.315873\pi\)
\(6\) 1.80194 0.735638
\(7\) 4.49396 1.69856 0.849278 0.527945i \(-0.177037\pi\)
0.849278 + 0.527945i \(0.177037\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0.246980 0.0823265
\(10\) −2.44504 −0.773190
\(11\) 2.49396 0.751957 0.375978 0.926628i \(-0.377307\pi\)
0.375978 + 0.926628i \(0.377307\pi\)
\(12\) −1.80194 −0.520175
\(13\) 0 0
\(14\) −4.49396 −1.20106
\(15\) −4.40581 −1.13758
\(16\) 1.00000 0.250000
\(17\) 7.74094 1.87745 0.938727 0.344662i \(-0.112007\pi\)
0.938727 + 0.344662i \(0.112007\pi\)
\(18\) −0.246980 −0.0582137
\(19\) −1.00000 −0.229416
\(20\) 2.44504 0.546728
\(21\) −8.09783 −1.76709
\(22\) −2.49396 −0.531714
\(23\) 4.27413 0.891217 0.445609 0.895228i \(-0.352987\pi\)
0.445609 + 0.895228i \(0.352987\pi\)
\(24\) 1.80194 0.367819
\(25\) 0.978230 0.195646
\(26\) 0 0
\(27\) 4.96077 0.954701
\(28\) 4.49396 0.849278
\(29\) −0.713792 −0.132548 −0.0662739 0.997801i \(-0.521111\pi\)
−0.0662739 + 0.997801i \(0.521111\pi\)
\(30\) 4.40581 0.804388
\(31\) 0.911854 0.163774 0.0818869 0.996642i \(-0.473905\pi\)
0.0818869 + 0.996642i \(0.473905\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.49396 −0.782298
\(34\) −7.74094 −1.32756
\(35\) 10.9879 1.85730
\(36\) 0.246980 0.0411633
\(37\) −6.81163 −1.11982 −0.559912 0.828552i \(-0.689165\pi\)
−0.559912 + 0.828552i \(0.689165\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −2.44504 −0.386595
\(41\) 3.28621 0.513220 0.256610 0.966515i \(-0.417394\pi\)
0.256610 + 0.966515i \(0.417394\pi\)
\(42\) 8.09783 1.24952
\(43\) 7.20775 1.09917 0.549586 0.835437i \(-0.314786\pi\)
0.549586 + 0.835437i \(0.314786\pi\)
\(44\) 2.49396 0.375978
\(45\) 0.603875 0.0900204
\(46\) −4.27413 −0.630186
\(47\) −5.20775 −0.759629 −0.379814 0.925063i \(-0.624012\pi\)
−0.379814 + 0.925063i \(0.624012\pi\)
\(48\) −1.80194 −0.260087
\(49\) 13.1957 1.88510
\(50\) −0.978230 −0.138343
\(51\) −13.9487 −1.95321
\(52\) 0 0
\(53\) −8.89008 −1.22115 −0.610573 0.791960i \(-0.709061\pi\)
−0.610573 + 0.791960i \(0.709061\pi\)
\(54\) −4.96077 −0.675075
\(55\) 6.09783 0.822232
\(56\) −4.49396 −0.600531
\(57\) 1.80194 0.238672
\(58\) 0.713792 0.0937254
\(59\) 11.2567 1.46549 0.732747 0.680501i \(-0.238238\pi\)
0.732747 + 0.680501i \(0.238238\pi\)
\(60\) −4.40581 −0.568788
\(61\) 3.43296 0.439546 0.219773 0.975551i \(-0.429468\pi\)
0.219773 + 0.975551i \(0.429468\pi\)
\(62\) −0.911854 −0.115806
\(63\) 1.10992 0.139836
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.49396 0.553168
\(67\) 1.40581 0.171747 0.0858737 0.996306i \(-0.472632\pi\)
0.0858737 + 0.996306i \(0.472632\pi\)
\(68\) 7.74094 0.938727
\(69\) −7.70171 −0.927177
\(70\) −10.9879 −1.31331
\(71\) −3.86294 −0.458446 −0.229223 0.973374i \(-0.573619\pi\)
−0.229223 + 0.973374i \(0.573619\pi\)
\(72\) −0.246980 −0.0291068
\(73\) 7.74094 0.906008 0.453004 0.891508i \(-0.350352\pi\)
0.453004 + 0.891508i \(0.350352\pi\)
\(74\) 6.81163 0.791835
\(75\) −1.76271 −0.203540
\(76\) −1.00000 −0.114708
\(77\) 11.2078 1.27724
\(78\) 0 0
\(79\) −1.91723 −0.215705 −0.107853 0.994167i \(-0.534397\pi\)
−0.107853 + 0.994167i \(0.534397\pi\)
\(80\) 2.44504 0.273364
\(81\) −9.67994 −1.07555
\(82\) −3.28621 −0.362901
\(83\) 0.396125 0.0434803 0.0217402 0.999764i \(-0.493079\pi\)
0.0217402 + 0.999764i \(0.493079\pi\)
\(84\) −8.09783 −0.883546
\(85\) 18.9269 2.05291
\(86\) −7.20775 −0.777232
\(87\) 1.28621 0.137896
\(88\) −2.49396 −0.265857
\(89\) 16.7681 1.77741 0.888707 0.458476i \(-0.151605\pi\)
0.888707 + 0.458476i \(0.151605\pi\)
\(90\) −0.603875 −0.0636541
\(91\) 0 0
\(92\) 4.27413 0.445609
\(93\) −1.64310 −0.170382
\(94\) 5.20775 0.537138
\(95\) −2.44504 −0.250856
\(96\) 1.80194 0.183910
\(97\) −3.16421 −0.321277 −0.160638 0.987013i \(-0.551355\pi\)
−0.160638 + 0.987013i \(0.551355\pi\)
\(98\) −13.1957 −1.33296
\(99\) 0.615957 0.0619060
\(100\) 0.978230 0.0978230
\(101\) 15.5308 1.54537 0.772686 0.634789i \(-0.218913\pi\)
0.772686 + 0.634789i \(0.218913\pi\)
\(102\) 13.9487 1.38113
\(103\) −3.03684 −0.299228 −0.149614 0.988744i \(-0.547803\pi\)
−0.149614 + 0.988744i \(0.547803\pi\)
\(104\) 0 0
\(105\) −19.7995 −1.93224
\(106\) 8.89008 0.863481
\(107\) −2.53319 −0.244893 −0.122446 0.992475i \(-0.539074\pi\)
−0.122446 + 0.992475i \(0.539074\pi\)
\(108\) 4.96077 0.477350
\(109\) 10.2198 0.978882 0.489441 0.872036i \(-0.337201\pi\)
0.489441 + 0.872036i \(0.337201\pi\)
\(110\) −6.09783 −0.581406
\(111\) 12.2741 1.16501
\(112\) 4.49396 0.424639
\(113\) 7.60388 0.715312 0.357656 0.933853i \(-0.383576\pi\)
0.357656 + 0.933853i \(0.383576\pi\)
\(114\) −1.80194 −0.168767
\(115\) 10.4504 0.974507
\(116\) −0.713792 −0.0662739
\(117\) 0 0
\(118\) −11.2567 −1.03626
\(119\) 34.7875 3.18896
\(120\) 4.40581 0.402194
\(121\) −4.78017 −0.434561
\(122\) −3.43296 −0.310806
\(123\) −5.92154 −0.533928
\(124\) 0.911854 0.0818869
\(125\) −9.83340 −0.879526
\(126\) −1.10992 −0.0988792
\(127\) −18.6920 −1.65865 −0.829324 0.558768i \(-0.811274\pi\)
−0.829324 + 0.558768i \(0.811274\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.9879 −1.14352
\(130\) 0 0
\(131\) −8.49396 −0.742121 −0.371060 0.928609i \(-0.621006\pi\)
−0.371060 + 0.928609i \(0.621006\pi\)
\(132\) −4.49396 −0.391149
\(133\) −4.49396 −0.389676
\(134\) −1.40581 −0.121444
\(135\) 12.1293 1.04392
\(136\) −7.74094 −0.663780
\(137\) −19.7506 −1.68741 −0.843705 0.536807i \(-0.819630\pi\)
−0.843705 + 0.536807i \(0.819630\pi\)
\(138\) 7.70171 0.655613
\(139\) 8.15213 0.691455 0.345727 0.938335i \(-0.387632\pi\)
0.345727 + 0.938335i \(0.387632\pi\)
\(140\) 10.9879 0.928649
\(141\) 9.38404 0.790279
\(142\) 3.86294 0.324170
\(143\) 0 0
\(144\) 0.246980 0.0205816
\(145\) −1.74525 −0.144935
\(146\) −7.74094 −0.640645
\(147\) −23.7778 −1.96116
\(148\) −6.81163 −0.559912
\(149\) −8.11290 −0.664635 −0.332317 0.943168i \(-0.607830\pi\)
−0.332317 + 0.943168i \(0.607830\pi\)
\(150\) 1.76271 0.143925
\(151\) −5.56465 −0.452845 −0.226422 0.974029i \(-0.572703\pi\)
−0.226422 + 0.974029i \(0.572703\pi\)
\(152\) 1.00000 0.0811107
\(153\) 1.91185 0.154564
\(154\) −11.2078 −0.903146
\(155\) 2.22952 0.179079
\(156\) 0 0
\(157\) −6.77240 −0.540496 −0.270248 0.962791i \(-0.587106\pi\)
−0.270248 + 0.962791i \(0.587106\pi\)
\(158\) 1.91723 0.152527
\(159\) 16.0194 1.27042
\(160\) −2.44504 −0.193298
\(161\) 19.2078 1.51378
\(162\) 9.67994 0.760528
\(163\) 8.27413 0.648080 0.324040 0.946043i \(-0.394959\pi\)
0.324040 + 0.946043i \(0.394959\pi\)
\(164\) 3.28621 0.256610
\(165\) −10.9879 −0.855408
\(166\) −0.396125 −0.0307452
\(167\) 6.25236 0.483822 0.241911 0.970298i \(-0.422226\pi\)
0.241911 + 0.970298i \(0.422226\pi\)
\(168\) 8.09783 0.624762
\(169\) 0 0
\(170\) −18.9269 −1.45163
\(171\) −0.246980 −0.0188870
\(172\) 7.20775 0.549586
\(173\) 24.5133 1.86371 0.931857 0.362825i \(-0.118188\pi\)
0.931857 + 0.362825i \(0.118188\pi\)
\(174\) −1.28621 −0.0975072
\(175\) 4.39612 0.332316
\(176\) 2.49396 0.187989
\(177\) −20.2838 −1.52462
\(178\) −16.7681 −1.25682
\(179\) −25.9366 −1.93859 −0.969297 0.245895i \(-0.920918\pi\)
−0.969297 + 0.245895i \(0.920918\pi\)
\(180\) 0.603875 0.0450102
\(181\) 5.23191 0.388885 0.194443 0.980914i \(-0.437710\pi\)
0.194443 + 0.980914i \(0.437710\pi\)
\(182\) 0 0
\(183\) −6.18598 −0.457281
\(184\) −4.27413 −0.315093
\(185\) −16.6547 −1.22448
\(186\) 1.64310 0.120478
\(187\) 19.3056 1.41176
\(188\) −5.20775 −0.379814
\(189\) 22.2935 1.62161
\(190\) 2.44504 0.177382
\(191\) −17.4819 −1.26494 −0.632472 0.774583i \(-0.717960\pi\)
−0.632472 + 0.774583i \(0.717960\pi\)
\(192\) −1.80194 −0.130044
\(193\) −23.7754 −1.71139 −0.855695 0.517481i \(-0.826870\pi\)
−0.855695 + 0.517481i \(0.826870\pi\)
\(194\) 3.16421 0.227177
\(195\) 0 0
\(196\) 13.1957 0.942548
\(197\) −22.6069 −1.61067 −0.805336 0.592819i \(-0.798015\pi\)
−0.805336 + 0.592819i \(0.798015\pi\)
\(198\) −0.615957 −0.0437742
\(199\) −16.7332 −1.18618 −0.593091 0.805135i \(-0.702093\pi\)
−0.593091 + 0.805135i \(0.702093\pi\)
\(200\) −0.978230 −0.0691713
\(201\) −2.53319 −0.178677
\(202\) −15.5308 −1.09274
\(203\) −3.20775 −0.225140
\(204\) −13.9487 −0.976604
\(205\) 8.03492 0.561183
\(206\) 3.03684 0.211586
\(207\) 1.05562 0.0733708
\(208\) 0 0
\(209\) −2.49396 −0.172511
\(210\) 19.7995 1.36630
\(211\) 15.3817 1.05892 0.529458 0.848336i \(-0.322395\pi\)
0.529458 + 0.848336i \(0.322395\pi\)
\(212\) −8.89008 −0.610573
\(213\) 6.96077 0.476944
\(214\) 2.53319 0.173165
\(215\) 17.6233 1.20190
\(216\) −4.96077 −0.337538
\(217\) 4.09783 0.278179
\(218\) −10.2198 −0.692174
\(219\) −13.9487 −0.942565
\(220\) 6.09783 0.411116
\(221\) 0 0
\(222\) −12.2741 −0.823785
\(223\) 28.4198 1.90313 0.951566 0.307445i \(-0.0994741\pi\)
0.951566 + 0.307445i \(0.0994741\pi\)
\(224\) −4.49396 −0.300265
\(225\) 0.241603 0.0161069
\(226\) −7.60388 −0.505802
\(227\) 10.1981 0.676869 0.338435 0.940990i \(-0.390103\pi\)
0.338435 + 0.940990i \(0.390103\pi\)
\(228\) 1.80194 0.119336
\(229\) 14.5429 0.961020 0.480510 0.876989i \(-0.340452\pi\)
0.480510 + 0.876989i \(0.340452\pi\)
\(230\) −10.4504 −0.689080
\(231\) −20.1957 −1.32878
\(232\) 0.713792 0.0468627
\(233\) −26.0683 −1.70779 −0.853895 0.520445i \(-0.825766\pi\)
−0.853895 + 0.520445i \(0.825766\pi\)
\(234\) 0 0
\(235\) −12.7332 −0.830620
\(236\) 11.2567 0.732747
\(237\) 3.45473 0.224409
\(238\) −34.7875 −2.25494
\(239\) −16.2198 −1.04917 −0.524587 0.851357i \(-0.675780\pi\)
−0.524587 + 0.851357i \(0.675780\pi\)
\(240\) −4.40581 −0.284394
\(241\) −27.3250 −1.76016 −0.880078 0.474830i \(-0.842510\pi\)
−0.880078 + 0.474830i \(0.842510\pi\)
\(242\) 4.78017 0.307281
\(243\) 2.56033 0.164246
\(244\) 3.43296 0.219773
\(245\) 32.2640 2.06127
\(246\) 5.92154 0.377544
\(247\) 0 0
\(248\) −0.911854 −0.0579028
\(249\) −0.713792 −0.0452347
\(250\) 9.83340 0.621919
\(251\) 14.7922 0.933678 0.466839 0.884342i \(-0.345393\pi\)
0.466839 + 0.884342i \(0.345393\pi\)
\(252\) 1.10992 0.0699182
\(253\) 10.6595 0.670157
\(254\) 18.6920 1.17284
\(255\) −34.1051 −2.13575
\(256\) 1.00000 0.0625000
\(257\) −26.5918 −1.65875 −0.829375 0.558692i \(-0.811303\pi\)
−0.829375 + 0.558692i \(0.811303\pi\)
\(258\) 12.9879 0.808592
\(259\) −30.6112 −1.90209
\(260\) 0 0
\(261\) −0.176292 −0.0109122
\(262\) 8.49396 0.524759
\(263\) −24.7332 −1.52511 −0.762556 0.646922i \(-0.776056\pi\)
−0.762556 + 0.646922i \(0.776056\pi\)
\(264\) 4.49396 0.276584
\(265\) −21.7366 −1.33527
\(266\) 4.49396 0.275542
\(267\) −30.2150 −1.84913
\(268\) 1.40581 0.0858737
\(269\) 16.2935 0.993432 0.496716 0.867913i \(-0.334539\pi\)
0.496716 + 0.867913i \(0.334539\pi\)
\(270\) −12.1293 −0.738165
\(271\) −25.7560 −1.56457 −0.782283 0.622923i \(-0.785945\pi\)
−0.782283 + 0.622923i \(0.785945\pi\)
\(272\) 7.74094 0.469363
\(273\) 0 0
\(274\) 19.7506 1.19318
\(275\) 2.43967 0.147117
\(276\) −7.70171 −0.463588
\(277\) 2.19029 0.131602 0.0658010 0.997833i \(-0.479040\pi\)
0.0658010 + 0.997833i \(0.479040\pi\)
\(278\) −8.15213 −0.488932
\(279\) 0.225209 0.0134829
\(280\) −10.9879 −0.656654
\(281\) 24.6353 1.46962 0.734810 0.678273i \(-0.237271\pi\)
0.734810 + 0.678273i \(0.237271\pi\)
\(282\) −9.38404 −0.558812
\(283\) −4.48321 −0.266499 −0.133249 0.991083i \(-0.542541\pi\)
−0.133249 + 0.991083i \(0.542541\pi\)
\(284\) −3.86294 −0.229223
\(285\) 4.40581 0.260978
\(286\) 0 0
\(287\) 14.7681 0.871733
\(288\) −0.246980 −0.0145534
\(289\) 42.9221 2.52483
\(290\) 1.74525 0.102485
\(291\) 5.70171 0.334240
\(292\) 7.74094 0.453004
\(293\) 3.32975 0.194526 0.0972630 0.995259i \(-0.468991\pi\)
0.0972630 + 0.995259i \(0.468991\pi\)
\(294\) 23.7778 1.38675
\(295\) 27.5230 1.60245
\(296\) 6.81163 0.395918
\(297\) 12.3720 0.717894
\(298\) 8.11290 0.469968
\(299\) 0 0
\(300\) −1.76271 −0.101770
\(301\) 32.3913 1.86701
\(302\) 5.56465 0.320209
\(303\) −27.9855 −1.60773
\(304\) −1.00000 −0.0573539
\(305\) 8.39373 0.480624
\(306\) −1.91185 −0.109293
\(307\) 2.74632 0.156741 0.0783703 0.996924i \(-0.475028\pi\)
0.0783703 + 0.996924i \(0.475028\pi\)
\(308\) 11.2078 0.638621
\(309\) 5.47219 0.311302
\(310\) −2.22952 −0.126628
\(311\) −13.6775 −0.775583 −0.387791 0.921747i \(-0.626762\pi\)
−0.387791 + 0.921747i \(0.626762\pi\)
\(312\) 0 0
\(313\) 16.1903 0.915129 0.457565 0.889176i \(-0.348722\pi\)
0.457565 + 0.889176i \(0.348722\pi\)
\(314\) 6.77240 0.382189
\(315\) 2.71379 0.152905
\(316\) −1.91723 −0.107853
\(317\) 19.8538 1.11510 0.557551 0.830142i \(-0.311741\pi\)
0.557551 + 0.830142i \(0.311741\pi\)
\(318\) −16.0194 −0.898322
\(319\) −1.78017 −0.0996702
\(320\) 2.44504 0.136682
\(321\) 4.56465 0.254774
\(322\) −19.2078 −1.07041
\(323\) −7.74094 −0.430717
\(324\) −9.67994 −0.537774
\(325\) 0 0
\(326\) −8.27413 −0.458261
\(327\) −18.4155 −1.01838
\(328\) −3.28621 −0.181450
\(329\) −23.4034 −1.29027
\(330\) 10.9879 0.604865
\(331\) −19.2295 −1.05695 −0.528475 0.848949i \(-0.677236\pi\)
−0.528475 + 0.848949i \(0.677236\pi\)
\(332\) 0.396125 0.0217402
\(333\) −1.68233 −0.0921913
\(334\) −6.25236 −0.342114
\(335\) 3.43727 0.187798
\(336\) −8.09783 −0.441773
\(337\) −10.8465 −0.590849 −0.295424 0.955366i \(-0.595461\pi\)
−0.295424 + 0.955366i \(0.595461\pi\)
\(338\) 0 0
\(339\) −13.7017 −0.744175
\(340\) 18.9269 1.02646
\(341\) 2.27413 0.123151
\(342\) 0.246980 0.0133551
\(343\) 27.8431 1.50339
\(344\) −7.20775 −0.388616
\(345\) −18.8310 −1.01383
\(346\) −24.5133 −1.31785
\(347\) 13.8974 0.746050 0.373025 0.927821i \(-0.378320\pi\)
0.373025 + 0.927821i \(0.378320\pi\)
\(348\) 1.28621 0.0689480
\(349\) −1.89977 −0.101692 −0.0508462 0.998706i \(-0.516192\pi\)
−0.0508462 + 0.998706i \(0.516192\pi\)
\(350\) −4.39612 −0.234983
\(351\) 0 0
\(352\) −2.49396 −0.132928
\(353\) −17.1618 −0.913431 −0.456716 0.889613i \(-0.650974\pi\)
−0.456716 + 0.889613i \(0.650974\pi\)
\(354\) 20.2838 1.07807
\(355\) −9.44504 −0.501291
\(356\) 16.7681 0.888707
\(357\) −62.6848 −3.31763
\(358\) 25.9366 1.37079
\(359\) −18.5181 −0.977349 −0.488675 0.872466i \(-0.662519\pi\)
−0.488675 + 0.872466i \(0.662519\pi\)
\(360\) −0.603875 −0.0318270
\(361\) 1.00000 0.0526316
\(362\) −5.23191 −0.274983
\(363\) 8.61356 0.452095
\(364\) 0 0
\(365\) 18.9269 0.990680
\(366\) 6.18598 0.323346
\(367\) 8.39612 0.438274 0.219137 0.975694i \(-0.429676\pi\)
0.219137 + 0.975694i \(0.429676\pi\)
\(368\) 4.27413 0.222804
\(369\) 0.811626 0.0422516
\(370\) 16.6547 0.865837
\(371\) −39.9517 −2.07419
\(372\) −1.64310 −0.0851910
\(373\) 29.5991 1.53258 0.766291 0.642493i \(-0.222100\pi\)
0.766291 + 0.642493i \(0.222100\pi\)
\(374\) −19.3056 −0.998268
\(375\) 17.7192 0.915014
\(376\) 5.20775 0.268569
\(377\) 0 0
\(378\) −22.2935 −1.14665
\(379\) −18.9638 −0.974103 −0.487051 0.873373i \(-0.661928\pi\)
−0.487051 + 0.873373i \(0.661928\pi\)
\(380\) −2.44504 −0.125428
\(381\) 33.6819 1.72557
\(382\) 17.4819 0.894451
\(383\) 28.4644 1.45446 0.727232 0.686392i \(-0.240806\pi\)
0.727232 + 0.686392i \(0.240806\pi\)
\(384\) 1.80194 0.0919548
\(385\) 27.4034 1.39661
\(386\) 23.7754 1.21014
\(387\) 1.78017 0.0904910
\(388\) −3.16421 −0.160638
\(389\) 6.90084 0.349886 0.174943 0.984579i \(-0.444026\pi\)
0.174943 + 0.984579i \(0.444026\pi\)
\(390\) 0 0
\(391\) 33.0858 1.67322
\(392\) −13.1957 −0.666482
\(393\) 15.3056 0.772065
\(394\) 22.6069 1.13892
\(395\) −4.68771 −0.235864
\(396\) 0.615957 0.0309530
\(397\) −27.8726 −1.39889 −0.699443 0.714688i \(-0.746569\pi\)
−0.699443 + 0.714688i \(0.746569\pi\)
\(398\) 16.7332 0.838758
\(399\) 8.09783 0.405399
\(400\) 0.978230 0.0489115
\(401\) 27.5120 1.37388 0.686942 0.726712i \(-0.258953\pi\)
0.686942 + 0.726712i \(0.258953\pi\)
\(402\) 2.53319 0.126344
\(403\) 0 0
\(404\) 15.5308 0.772686
\(405\) −23.6679 −1.17607
\(406\) 3.20775 0.159198
\(407\) −16.9879 −0.842060
\(408\) 13.9487 0.690563
\(409\) 10.8358 0.535795 0.267898 0.963447i \(-0.413671\pi\)
0.267898 + 0.963447i \(0.413671\pi\)
\(410\) −8.03492 −0.396816
\(411\) 35.5894 1.75550
\(412\) −3.03684 −0.149614
\(413\) 50.5870 2.48922
\(414\) −1.05562 −0.0518810
\(415\) 0.968541 0.0475438
\(416\) 0 0
\(417\) −14.6896 −0.719354
\(418\) 2.49396 0.121984
\(419\) −28.4892 −1.39179 −0.695894 0.718145i \(-0.744991\pi\)
−0.695894 + 0.718145i \(0.744991\pi\)
\(420\) −19.7995 −0.966119
\(421\) 28.1957 1.37417 0.687086 0.726576i \(-0.258889\pi\)
0.687086 + 0.726576i \(0.258889\pi\)
\(422\) −15.3817 −0.748767
\(423\) −1.28621 −0.0625376
\(424\) 8.89008 0.431741
\(425\) 7.57242 0.367316
\(426\) −6.96077 −0.337250
\(427\) 15.4276 0.746593
\(428\) −2.53319 −0.122446
\(429\) 0 0
\(430\) −17.6233 −0.849869
\(431\) −20.4722 −0.986111 −0.493055 0.869998i \(-0.664120\pi\)
−0.493055 + 0.869998i \(0.664120\pi\)
\(432\) 4.96077 0.238675
\(433\) 9.69096 0.465718 0.232859 0.972511i \(-0.425192\pi\)
0.232859 + 0.972511i \(0.425192\pi\)
\(434\) −4.09783 −0.196702
\(435\) 3.14483 0.150783
\(436\) 10.2198 0.489441
\(437\) −4.27413 −0.204459
\(438\) 13.9487 0.666494
\(439\) −7.67994 −0.366544 −0.183272 0.983062i \(-0.558669\pi\)
−0.183272 + 0.983062i \(0.558669\pi\)
\(440\) −6.09783 −0.290703
\(441\) 3.25906 0.155193
\(442\) 0 0
\(443\) −11.5168 −0.547179 −0.273590 0.961846i \(-0.588211\pi\)
−0.273590 + 0.961846i \(0.588211\pi\)
\(444\) 12.2741 0.582504
\(445\) 40.9987 1.94352
\(446\) −28.4198 −1.34572
\(447\) 14.6189 0.691452
\(448\) 4.49396 0.212320
\(449\) −1.16421 −0.0549425 −0.0274712 0.999623i \(-0.508745\pi\)
−0.0274712 + 0.999623i \(0.508745\pi\)
\(450\) −0.241603 −0.0113893
\(451\) 8.19567 0.385919
\(452\) 7.60388 0.357656
\(453\) 10.0271 0.471116
\(454\) −10.1981 −0.478619
\(455\) 0 0
\(456\) −1.80194 −0.0843835
\(457\) 30.3230 1.41845 0.709226 0.704981i \(-0.249045\pi\)
0.709226 + 0.704981i \(0.249045\pi\)
\(458\) −14.5429 −0.679544
\(459\) 38.4010 1.79241
\(460\) 10.4504 0.487253
\(461\) −0.992230 −0.0462128 −0.0231064 0.999733i \(-0.507356\pi\)
−0.0231064 + 0.999733i \(0.507356\pi\)
\(462\) 20.1957 0.939588
\(463\) 28.8767 1.34201 0.671006 0.741452i \(-0.265862\pi\)
0.671006 + 0.741452i \(0.265862\pi\)
\(464\) −0.713792 −0.0331369
\(465\) −4.01746 −0.186305
\(466\) 26.0683 1.20759
\(467\) −19.8189 −0.917110 −0.458555 0.888666i \(-0.651633\pi\)
−0.458555 + 0.888666i \(0.651633\pi\)
\(468\) 0 0
\(469\) 6.31767 0.291723
\(470\) 12.7332 0.587337
\(471\) 12.2034 0.562305
\(472\) −11.2567 −0.518130
\(473\) 17.9758 0.826530
\(474\) −3.45473 −0.158681
\(475\) −0.978230 −0.0448843
\(476\) 34.7875 1.59448
\(477\) −2.19567 −0.100533
\(478\) 16.2198 0.741878
\(479\) −21.9758 −1.00410 −0.502051 0.864838i \(-0.667421\pi\)
−0.502051 + 0.864838i \(0.667421\pi\)
\(480\) 4.40581 0.201097
\(481\) 0 0
\(482\) 27.3250 1.24462
\(483\) −34.6112 −1.57486
\(484\) −4.78017 −0.217280
\(485\) −7.73663 −0.351302
\(486\) −2.56033 −0.116139
\(487\) 36.1148 1.63652 0.818259 0.574849i \(-0.194939\pi\)
0.818259 + 0.574849i \(0.194939\pi\)
\(488\) −3.43296 −0.155403
\(489\) −14.9095 −0.674229
\(490\) −32.2640 −1.45754
\(491\) −21.2814 −0.960417 −0.480209 0.877154i \(-0.659439\pi\)
−0.480209 + 0.877154i \(0.659439\pi\)
\(492\) −5.92154 −0.266964
\(493\) −5.52542 −0.248852
\(494\) 0 0
\(495\) 1.50604 0.0676915
\(496\) 0.911854 0.0409435
\(497\) −17.3599 −0.778697
\(498\) 0.713792 0.0319858
\(499\) 5.00730 0.224157 0.112079 0.993699i \(-0.464249\pi\)
0.112079 + 0.993699i \(0.464249\pi\)
\(500\) −9.83340 −0.439763
\(501\) −11.2664 −0.503344
\(502\) −14.7922 −0.660210
\(503\) 44.0930 1.96601 0.983006 0.183574i \(-0.0587666\pi\)
0.983006 + 0.183574i \(0.0587666\pi\)
\(504\) −1.10992 −0.0494396
\(505\) 37.9734 1.68980
\(506\) −10.6595 −0.473872
\(507\) 0 0
\(508\) −18.6920 −0.829324
\(509\) −24.8901 −1.10323 −0.551617 0.834098i \(-0.685989\pi\)
−0.551617 + 0.834098i \(0.685989\pi\)
\(510\) 34.1051 1.51020
\(511\) 34.7875 1.53891
\(512\) −1.00000 −0.0441942
\(513\) −4.96077 −0.219023
\(514\) 26.5918 1.17291
\(515\) −7.42519 −0.327193
\(516\) −12.9879 −0.571761
\(517\) −12.9879 −0.571208
\(518\) 30.6112 1.34498
\(519\) −44.1715 −1.93891
\(520\) 0 0
\(521\) −13.8974 −0.608855 −0.304428 0.952535i \(-0.598465\pi\)
−0.304428 + 0.952535i \(0.598465\pi\)
\(522\) 0.176292 0.00771609
\(523\) 1.78017 0.0778413 0.0389206 0.999242i \(-0.487608\pi\)
0.0389206 + 0.999242i \(0.487608\pi\)
\(524\) −8.49396 −0.371060
\(525\) −7.92154 −0.345724
\(526\) 24.7332 1.07842
\(527\) 7.05861 0.307478
\(528\) −4.49396 −0.195574
\(529\) −4.73184 −0.205732
\(530\) 21.7366 0.944179
\(531\) 2.78017 0.120649
\(532\) −4.49396 −0.194838
\(533\) 0 0
\(534\) 30.2150 1.30753
\(535\) −6.19375 −0.267779
\(536\) −1.40581 −0.0607219
\(537\) 46.7362 2.01681
\(538\) −16.2935 −0.702463
\(539\) 32.9095 1.41751
\(540\) 12.1293 0.521962
\(541\) 38.7308 1.66517 0.832583 0.553900i \(-0.186861\pi\)
0.832583 + 0.553900i \(0.186861\pi\)
\(542\) 25.7560 1.10632
\(543\) −9.42758 −0.404576
\(544\) −7.74094 −0.331890
\(545\) 24.9879 1.07036
\(546\) 0 0
\(547\) −6.25906 −0.267618 −0.133809 0.991007i \(-0.542721\pi\)
−0.133809 + 0.991007i \(0.542721\pi\)
\(548\) −19.7506 −0.843705
\(549\) 0.847871 0.0361863
\(550\) −2.43967 −0.104028
\(551\) 0.713792 0.0304086
\(552\) 7.70171 0.327807
\(553\) −8.61596 −0.366388
\(554\) −2.19029 −0.0930566
\(555\) 30.0108 1.27389
\(556\) 8.15213 0.345727
\(557\) 25.3980 1.07615 0.538075 0.842897i \(-0.319152\pi\)
0.538075 + 0.842897i \(0.319152\pi\)
\(558\) −0.225209 −0.00953387
\(559\) 0 0
\(560\) 10.9879 0.464324
\(561\) −34.7875 −1.46873
\(562\) −24.6353 −1.03918
\(563\) 8.61835 0.363220 0.181610 0.983371i \(-0.441869\pi\)
0.181610 + 0.983371i \(0.441869\pi\)
\(564\) 9.38404 0.395139
\(565\) 18.5918 0.782163
\(566\) 4.48321 0.188443
\(567\) −43.5013 −1.82688
\(568\) 3.86294 0.162085
\(569\) −3.10992 −0.130374 −0.0651872 0.997873i \(-0.520764\pi\)
−0.0651872 + 0.997873i \(0.520764\pi\)
\(570\) −4.40581 −0.184539
\(571\) −44.6741 −1.86955 −0.934776 0.355237i \(-0.884400\pi\)
−0.934776 + 0.355237i \(0.884400\pi\)
\(572\) 0 0
\(573\) 31.5013 1.31598
\(574\) −14.7681 −0.616408
\(575\) 4.18108 0.174363
\(576\) 0.246980 0.0102908
\(577\) 35.3817 1.47296 0.736479 0.676461i \(-0.236487\pi\)
0.736479 + 0.676461i \(0.236487\pi\)
\(578\) −42.9221 −1.78533
\(579\) 42.8418 1.78044
\(580\) −1.74525 −0.0724676
\(581\) 1.78017 0.0738538
\(582\) −5.70171 −0.236343
\(583\) −22.1715 −0.918250
\(584\) −7.74094 −0.320322
\(585\) 0 0
\(586\) −3.32975 −0.137551
\(587\) 19.4711 0.803659 0.401830 0.915714i \(-0.368374\pi\)
0.401830 + 0.915714i \(0.368374\pi\)
\(588\) −23.7778 −0.980579
\(589\) −0.911854 −0.0375723
\(590\) −27.5230 −1.13311
\(591\) 40.7362 1.67566
\(592\) −6.81163 −0.279956
\(593\) −5.31634 −0.218316 −0.109158 0.994024i \(-0.534815\pi\)
−0.109158 + 0.994024i \(0.534815\pi\)
\(594\) −12.3720 −0.507628
\(595\) 85.0568 3.48699
\(596\) −8.11290 −0.332317
\(597\) 30.1521 1.23404
\(598\) 0 0
\(599\) −13.2644 −0.541970 −0.270985 0.962584i \(-0.587349\pi\)
−0.270985 + 0.962584i \(0.587349\pi\)
\(600\) 1.76271 0.0719623
\(601\) 32.0388 1.30689 0.653444 0.756975i \(-0.273323\pi\)
0.653444 + 0.756975i \(0.273323\pi\)
\(602\) −32.3913 −1.32017
\(603\) 0.347207 0.0141394
\(604\) −5.56465 −0.226422
\(605\) −11.6877 −0.475173
\(606\) 27.9855 1.13683
\(607\) −24.0355 −0.975571 −0.487786 0.872963i \(-0.662195\pi\)
−0.487786 + 0.872963i \(0.662195\pi\)
\(608\) 1.00000 0.0405554
\(609\) 5.78017 0.234224
\(610\) −8.39373 −0.339852
\(611\) 0 0
\(612\) 1.91185 0.0772821
\(613\) 41.6122 1.68070 0.840351 0.542042i \(-0.182349\pi\)
0.840351 + 0.542042i \(0.182349\pi\)
\(614\) −2.74632 −0.110832
\(615\) −14.4784 −0.583826
\(616\) −11.2078 −0.451573
\(617\) 2.20477 0.0887606 0.0443803 0.999015i \(-0.485869\pi\)
0.0443803 + 0.999015i \(0.485869\pi\)
\(618\) −5.47219 −0.220124
\(619\) −19.1207 −0.768525 −0.384262 0.923224i \(-0.625544\pi\)
−0.384262 + 0.923224i \(0.625544\pi\)
\(620\) 2.22952 0.0895397
\(621\) 21.2030 0.850846
\(622\) 13.6775 0.548420
\(623\) 75.3551 3.01904
\(624\) 0 0
\(625\) −28.9342 −1.15737
\(626\) −16.1903 −0.647094
\(627\) 4.49396 0.179471
\(628\) −6.77240 −0.270248
\(629\) −52.7284 −2.10242
\(630\) −2.71379 −0.108120
\(631\) 40.7332 1.62156 0.810781 0.585350i \(-0.199043\pi\)
0.810781 + 0.585350i \(0.199043\pi\)
\(632\) 1.91723 0.0762633
\(633\) −27.7168 −1.10164
\(634\) −19.8538 −0.788497
\(635\) −45.7028 −1.81366
\(636\) 16.0194 0.635210
\(637\) 0 0
\(638\) 1.78017 0.0704775
\(639\) −0.954067 −0.0377423
\(640\) −2.44504 −0.0966488
\(641\) 11.3491 0.448264 0.224132 0.974559i \(-0.428045\pi\)
0.224132 + 0.974559i \(0.428045\pi\)
\(642\) −4.56465 −0.180152
\(643\) 35.9409 1.41737 0.708686 0.705524i \(-0.249288\pi\)
0.708686 + 0.705524i \(0.249288\pi\)
\(644\) 19.2078 0.756891
\(645\) −31.7560 −1.25039
\(646\) 7.74094 0.304563
\(647\) −22.7788 −0.895529 −0.447764 0.894152i \(-0.647780\pi\)
−0.447764 + 0.894152i \(0.647780\pi\)
\(648\) 9.67994 0.380264
\(649\) 28.0737 1.10199
\(650\) 0 0
\(651\) −7.38404 −0.289403
\(652\) 8.27413 0.324040
\(653\) 22.2543 0.870877 0.435439 0.900218i \(-0.356593\pi\)
0.435439 + 0.900218i \(0.356593\pi\)
\(654\) 18.4155 0.720103
\(655\) −20.7681 −0.811476
\(656\) 3.28621 0.128305
\(657\) 1.91185 0.0745885
\(658\) 23.4034 0.912360
\(659\) −46.3666 −1.80619 −0.903093 0.429445i \(-0.858709\pi\)
−0.903093 + 0.429445i \(0.858709\pi\)
\(660\) −10.9879 −0.427704
\(661\) 31.0315 1.20698 0.603492 0.797369i \(-0.293776\pi\)
0.603492 + 0.797369i \(0.293776\pi\)
\(662\) 19.2295 0.747377
\(663\) 0 0
\(664\) −0.396125 −0.0153726
\(665\) −10.9879 −0.426093
\(666\) 1.68233 0.0651891
\(667\) −3.05084 −0.118129
\(668\) 6.25236 0.241911
\(669\) −51.2107 −1.97992
\(670\) −3.43727 −0.132793
\(671\) 8.56166 0.330519
\(672\) 8.09783 0.312381
\(673\) −22.6461 −0.872943 −0.436471 0.899718i \(-0.643772\pi\)
−0.436471 + 0.899718i \(0.643772\pi\)
\(674\) 10.8465 0.417793
\(675\) 4.85277 0.186783
\(676\) 0 0
\(677\) 24.7138 0.949828 0.474914 0.880032i \(-0.342479\pi\)
0.474914 + 0.880032i \(0.342479\pi\)
\(678\) 13.7017 0.526211
\(679\) −14.2198 −0.545707
\(680\) −18.9269 −0.725814
\(681\) −18.3763 −0.704180
\(682\) −2.27413 −0.0870808
\(683\) 30.9536 1.18441 0.592203 0.805789i \(-0.298258\pi\)
0.592203 + 0.805789i \(0.298258\pi\)
\(684\) −0.246980 −0.00944350
\(685\) −48.2911 −1.84511
\(686\) −27.8431 −1.06305
\(687\) −26.2054 −0.999797
\(688\) 7.20775 0.274793
\(689\) 0 0
\(690\) 18.8310 0.716884
\(691\) −5.94092 −0.226003 −0.113002 0.993595i \(-0.536047\pi\)
−0.113002 + 0.993595i \(0.536047\pi\)
\(692\) 24.5133 0.931857
\(693\) 2.76809 0.105151
\(694\) −13.8974 −0.527537
\(695\) 19.9323 0.756075
\(696\) −1.28621 −0.0487536
\(697\) 25.4383 0.963546
\(698\) 1.89977 0.0719074
\(699\) 46.9734 1.77670
\(700\) 4.39612 0.166158
\(701\) −7.48427 −0.282677 −0.141338 0.989961i \(-0.545141\pi\)
−0.141338 + 0.989961i \(0.545141\pi\)
\(702\) 0 0
\(703\) 6.81163 0.256905
\(704\) 2.49396 0.0939946
\(705\) 22.9444 0.864135
\(706\) 17.1618 0.645894
\(707\) 69.7948 2.62490
\(708\) −20.2838 −0.762312
\(709\) −20.7966 −0.781031 −0.390516 0.920596i \(-0.627703\pi\)
−0.390516 + 0.920596i \(0.627703\pi\)
\(710\) 9.44504 0.354466
\(711\) −0.473517 −0.0177583
\(712\) −16.7681 −0.628411
\(713\) 3.89738 0.145958
\(714\) 62.6848 2.34592
\(715\) 0 0
\(716\) −25.9366 −0.969297
\(717\) 29.2271 1.09151
\(718\) 18.5181 0.691090
\(719\) −36.3913 −1.35717 −0.678584 0.734523i \(-0.737406\pi\)
−0.678584 + 0.734523i \(0.737406\pi\)
\(720\) 0.603875 0.0225051
\(721\) −13.6474 −0.508256
\(722\) −1.00000 −0.0372161
\(723\) 49.2379 1.83118
\(724\) 5.23191 0.194443
\(725\) −0.698252 −0.0259324
\(726\) −8.61356 −0.319679
\(727\) 34.4698 1.27841 0.639207 0.769035i \(-0.279263\pi\)
0.639207 + 0.769035i \(0.279263\pi\)
\(728\) 0 0
\(729\) 24.4263 0.904676
\(730\) −18.9269 −0.700517
\(731\) 55.7948 2.06364
\(732\) −6.18598 −0.228640
\(733\) 15.6732 0.578904 0.289452 0.957193i \(-0.406527\pi\)
0.289452 + 0.957193i \(0.406527\pi\)
\(734\) −8.39612 −0.309907
\(735\) −58.1377 −2.14444
\(736\) −4.27413 −0.157546
\(737\) 3.50604 0.129147
\(738\) −0.811626 −0.0298764
\(739\) 31.7861 1.16927 0.584636 0.811296i \(-0.301237\pi\)
0.584636 + 0.811296i \(0.301237\pi\)
\(740\) −16.6547 −0.612239
\(741\) 0 0
\(742\) 39.9517 1.46667
\(743\) −26.7308 −0.980657 −0.490329 0.871538i \(-0.663123\pi\)
−0.490329 + 0.871538i \(0.663123\pi\)
\(744\) 1.64310 0.0602391
\(745\) −19.8364 −0.726749
\(746\) −29.5991 −1.08370
\(747\) 0.0978347 0.00357958
\(748\) 19.3056 0.705882
\(749\) −11.3840 −0.415964
\(750\) −17.7192 −0.647013
\(751\) 3.09916 0.113090 0.0565450 0.998400i \(-0.481992\pi\)
0.0565450 + 0.998400i \(0.481992\pi\)
\(752\) −5.20775 −0.189907
\(753\) −26.6547 −0.971352
\(754\) 0 0
\(755\) −13.6058 −0.495166
\(756\) 22.2935 0.810807
\(757\) −25.0670 −0.911074 −0.455537 0.890217i \(-0.650553\pi\)
−0.455537 + 0.890217i \(0.650553\pi\)
\(758\) 18.9638 0.688795
\(759\) −19.2078 −0.697197
\(760\) 2.44504 0.0886910
\(761\) −32.7375 −1.18673 −0.593366 0.804933i \(-0.702201\pi\)
−0.593366 + 0.804933i \(0.702201\pi\)
\(762\) −33.6819 −1.22017
\(763\) 45.9275 1.66269
\(764\) −17.4819 −0.632472
\(765\) 4.67456 0.169009
\(766\) −28.4644 −1.02846
\(767\) 0 0
\(768\) −1.80194 −0.0650218
\(769\) −0.836381 −0.0301607 −0.0150803 0.999886i \(-0.504800\pi\)
−0.0150803 + 0.999886i \(0.504800\pi\)
\(770\) −27.4034 −0.987551
\(771\) 47.9168 1.72568
\(772\) −23.7754 −0.855695
\(773\) 9.52542 0.342605 0.171303 0.985218i \(-0.445202\pi\)
0.171303 + 0.985218i \(0.445202\pi\)
\(774\) −1.78017 −0.0639868
\(775\) 0.892003 0.0320417
\(776\) 3.16421 0.113589
\(777\) 55.1594 1.97883
\(778\) −6.90084 −0.247407
\(779\) −3.28621 −0.117741
\(780\) 0 0
\(781\) −9.63401 −0.344732
\(782\) −33.0858 −1.18314
\(783\) −3.54096 −0.126544
\(784\) 13.1957 0.471274
\(785\) −16.5588 −0.591009
\(786\) −15.3056 −0.545932
\(787\) −27.4040 −0.976848 −0.488424 0.872607i \(-0.662428\pi\)
−0.488424 + 0.872607i \(0.662428\pi\)
\(788\) −22.6069 −0.805336
\(789\) 44.5676 1.58665
\(790\) 4.68771 0.166781
\(791\) 34.1715 1.21500
\(792\) −0.615957 −0.0218871
\(793\) 0 0
\(794\) 27.8726 0.989162
\(795\) 39.1680 1.38915
\(796\) −16.7332 −0.593091
\(797\) −28.4241 −1.00683 −0.503417 0.864044i \(-0.667924\pi\)
−0.503417 + 0.864044i \(0.667924\pi\)
\(798\) −8.09783 −0.286660
\(799\) −40.3129 −1.42617
\(800\) −0.978230 −0.0345856
\(801\) 4.14138 0.146328
\(802\) −27.5120 −0.971483
\(803\) 19.3056 0.681279
\(804\) −2.53319 −0.0893386
\(805\) 46.9638 1.65525
\(806\) 0 0
\(807\) −29.3599 −1.03352
\(808\) −15.5308 −0.546371
\(809\) −34.9855 −1.23003 −0.615013 0.788517i \(-0.710849\pi\)
−0.615013 + 0.788517i \(0.710849\pi\)
\(810\) 23.6679 0.831604
\(811\) 6.08383 0.213632 0.106816 0.994279i \(-0.465934\pi\)
0.106816 + 0.994279i \(0.465934\pi\)
\(812\) −3.20775 −0.112570
\(813\) 46.4107 1.62769
\(814\) 16.9879 0.595426
\(815\) 20.2306 0.708647
\(816\) −13.9487 −0.488302
\(817\) −7.20775 −0.252167
\(818\) −10.8358 −0.378864
\(819\) 0 0
\(820\) 8.03492 0.280591
\(821\) −15.0804 −0.526309 −0.263154 0.964754i \(-0.584763\pi\)
−0.263154 + 0.964754i \(0.584763\pi\)
\(822\) −35.5894 −1.24132
\(823\) 9.20297 0.320795 0.160398 0.987052i \(-0.448722\pi\)
0.160398 + 0.987052i \(0.448722\pi\)
\(824\) 3.03684 0.105793
\(825\) −4.39612 −0.153053
\(826\) −50.5870 −1.76015
\(827\) −38.4849 −1.33825 −0.669125 0.743150i \(-0.733331\pi\)
−0.669125 + 0.743150i \(0.733331\pi\)
\(828\) 1.05562 0.0366854
\(829\) 46.1366 1.60239 0.801195 0.598403i \(-0.204198\pi\)
0.801195 + 0.598403i \(0.204198\pi\)
\(830\) −0.968541 −0.0336186
\(831\) −3.94677 −0.136912
\(832\) 0 0
\(833\) 102.147 3.53918
\(834\) 14.6896 0.508660
\(835\) 15.2873 0.529038
\(836\) −2.49396 −0.0862554
\(837\) 4.52350 0.156355
\(838\) 28.4892 0.984142
\(839\) 27.7808 0.959098 0.479549 0.877515i \(-0.340800\pi\)
0.479549 + 0.877515i \(0.340800\pi\)
\(840\) 19.7995 0.683149
\(841\) −28.4905 −0.982431
\(842\) −28.1957 −0.971687
\(843\) −44.3913 −1.52892
\(844\) 15.3817 0.529458
\(845\) 0 0
\(846\) 1.28621 0.0442208
\(847\) −21.4819 −0.738126
\(848\) −8.89008 −0.305287
\(849\) 8.07846 0.277252
\(850\) −7.57242 −0.259732
\(851\) −29.1138 −0.998007
\(852\) 6.96077 0.238472
\(853\) 7.16746 0.245409 0.122705 0.992443i \(-0.460843\pi\)
0.122705 + 0.992443i \(0.460843\pi\)
\(854\) −15.4276 −0.527921
\(855\) −0.603875 −0.0206521
\(856\) 2.53319 0.0865826
\(857\) 6.47458 0.221168 0.110584 0.993867i \(-0.464728\pi\)
0.110584 + 0.993867i \(0.464728\pi\)
\(858\) 0 0
\(859\) 4.64874 0.158613 0.0793065 0.996850i \(-0.474729\pi\)
0.0793065 + 0.996850i \(0.474729\pi\)
\(860\) 17.6233 0.600948
\(861\) −26.6112 −0.906906
\(862\) 20.4722 0.697286
\(863\) −3.16900 −0.107874 −0.0539369 0.998544i \(-0.517177\pi\)
−0.0539369 + 0.998544i \(0.517177\pi\)
\(864\) −4.96077 −0.168769
\(865\) 59.9361 2.03789
\(866\) −9.69096 −0.329312
\(867\) −77.3430 −2.62671
\(868\) 4.09783 0.139090
\(869\) −4.78150 −0.162201
\(870\) −3.14483 −0.106620
\(871\) 0 0
\(872\) −10.2198 −0.346087
\(873\) −0.781495 −0.0264496
\(874\) 4.27413 0.144574
\(875\) −44.1909 −1.49392
\(876\) −13.9487 −0.471283
\(877\) 12.1763 0.411164 0.205582 0.978640i \(-0.434091\pi\)
0.205582 + 0.978640i \(0.434091\pi\)
\(878\) 7.67994 0.259185
\(879\) −6.00000 −0.202375
\(880\) 6.09783 0.205558
\(881\) −21.2704 −0.716618 −0.358309 0.933603i \(-0.616647\pi\)
−0.358309 + 0.933603i \(0.616647\pi\)
\(882\) −3.25906 −0.109738
\(883\) −36.0978 −1.21479 −0.607394 0.794400i \(-0.707785\pi\)
−0.607394 + 0.794400i \(0.707785\pi\)
\(884\) 0 0
\(885\) −49.5948 −1.66711
\(886\) 11.5168 0.386914
\(887\) −2.97956 −0.100044 −0.0500219 0.998748i \(-0.515929\pi\)
−0.0500219 + 0.998748i \(0.515929\pi\)
\(888\) −12.2741 −0.411893
\(889\) −84.0012 −2.81731
\(890\) −40.9987 −1.37428
\(891\) −24.1414 −0.808766
\(892\) 28.4198 0.951566
\(893\) 5.20775 0.174271
\(894\) −14.6189 −0.488931
\(895\) −63.4161 −2.11977
\(896\) −4.49396 −0.150133
\(897\) 0 0
\(898\) 1.16421 0.0388502
\(899\) −0.650874 −0.0217079
\(900\) 0.241603 0.00805343
\(901\) −68.8176 −2.29265
\(902\) −8.19567 −0.272886
\(903\) −58.3672 −1.94234
\(904\) −7.60388 −0.252901
\(905\) 12.7922 0.425229
\(906\) −10.0271 −0.333130
\(907\) −11.1594 −0.370543 −0.185271 0.982687i \(-0.559316\pi\)
−0.185271 + 0.982687i \(0.559316\pi\)
\(908\) 10.1981 0.338435
\(909\) 3.83579 0.127225
\(910\) 0 0
\(911\) −19.3787 −0.642044 −0.321022 0.947072i \(-0.604026\pi\)
−0.321022 + 0.947072i \(0.604026\pi\)
\(912\) 1.80194 0.0596681
\(913\) 0.987918 0.0326953
\(914\) −30.3230 −1.00300
\(915\) −15.1250 −0.500016
\(916\) 14.5429 0.480510
\(917\) −38.1715 −1.26053
\(918\) −38.4010 −1.26742
\(919\) 16.1763 0.533607 0.266803 0.963751i \(-0.414033\pi\)
0.266803 + 0.963751i \(0.414033\pi\)
\(920\) −10.4504 −0.344540
\(921\) −4.94869 −0.163065
\(922\) 0.992230 0.0326774
\(923\) 0 0
\(924\) −20.1957 −0.664389
\(925\) −6.66334 −0.219089
\(926\) −28.8767 −0.948946
\(927\) −0.750036 −0.0246344
\(928\) 0.713792 0.0234314
\(929\) 20.2127 0.663156 0.331578 0.943428i \(-0.392419\pi\)
0.331578 + 0.943428i \(0.392419\pi\)
\(930\) 4.01746 0.131738
\(931\) −13.1957 −0.432471
\(932\) −26.0683 −0.853895
\(933\) 24.6461 0.806877
\(934\) 19.8189 0.648495
\(935\) 47.2030 1.54370
\(936\) 0 0
\(937\) 30.8769 1.00871 0.504353 0.863498i \(-0.331731\pi\)
0.504353 + 0.863498i \(0.331731\pi\)
\(938\) −6.31767 −0.206279
\(939\) −29.1739 −0.952054
\(940\) −12.7332 −0.415310
\(941\) 17.5749 0.572926 0.286463 0.958091i \(-0.407520\pi\)
0.286463 + 0.958091i \(0.407520\pi\)
\(942\) −12.2034 −0.397610
\(943\) 14.0457 0.457390
\(944\) 11.2567 0.366373
\(945\) 54.5086 1.77316
\(946\) −17.9758 −0.584445
\(947\) 20.0435 0.651328 0.325664 0.945486i \(-0.394412\pi\)
0.325664 + 0.945486i \(0.394412\pi\)
\(948\) 3.45473 0.112204
\(949\) 0 0
\(950\) 0.978230 0.0317380
\(951\) −35.7754 −1.16010
\(952\) −34.7875 −1.12747
\(953\) −11.3685 −0.368262 −0.184131 0.982902i \(-0.558947\pi\)
−0.184131 + 0.982902i \(0.558947\pi\)
\(954\) 2.19567 0.0710874
\(955\) −42.7439 −1.38316
\(956\) −16.2198 −0.524587
\(957\) 3.20775 0.103692
\(958\) 21.9758 0.710007
\(959\) −88.7585 −2.86616
\(960\) −4.40581 −0.142197
\(961\) −30.1685 −0.973178
\(962\) 0 0
\(963\) −0.625646 −0.0201612
\(964\) −27.3250 −0.880078
\(965\) −58.1318 −1.87133
\(966\) 34.6112 1.11360
\(967\) −20.2741 −0.651972 −0.325986 0.945375i \(-0.605696\pi\)
−0.325986 + 0.945375i \(0.605696\pi\)
\(968\) 4.78017 0.153640
\(969\) 13.9487 0.448096
\(970\) 7.73663 0.248408
\(971\) 20.8528 0.669197 0.334599 0.942361i \(-0.391399\pi\)
0.334599 + 0.942361i \(0.391399\pi\)
\(972\) 2.56033 0.0821228
\(973\) 36.6353 1.17447
\(974\) −36.1148 −1.15719
\(975\) 0 0
\(976\) 3.43296 0.109886
\(977\) 16.2452 0.519729 0.259865 0.965645i \(-0.416322\pi\)
0.259865 + 0.965645i \(0.416322\pi\)
\(978\) 14.9095 0.476752
\(979\) 41.8189 1.33654
\(980\) 32.2640 1.03063
\(981\) 2.52409 0.0805880
\(982\) 21.2814 0.679117
\(983\) −6.02582 −0.192194 −0.0960969 0.995372i \(-0.530636\pi\)
−0.0960969 + 0.995372i \(0.530636\pi\)
\(984\) 5.92154 0.188772
\(985\) −55.2747 −1.76120
\(986\) 5.52542 0.175965
\(987\) 42.1715 1.34233
\(988\) 0 0
\(989\) 30.8068 0.979601
\(990\) −1.50604 −0.0478651
\(991\) −39.6334 −1.25900 −0.629498 0.777002i \(-0.716740\pi\)
−0.629498 + 0.777002i \(0.716740\pi\)
\(992\) −0.911854 −0.0289514
\(993\) 34.6504 1.09960
\(994\) 17.3599 0.550622
\(995\) −40.9133 −1.29704
\(996\) −0.713792 −0.0226174
\(997\) −15.9627 −0.505543 −0.252772 0.967526i \(-0.581342\pi\)
−0.252772 + 0.967526i \(0.581342\pi\)
\(998\) −5.00730 −0.158503
\(999\) −33.7909 −1.06910
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 6422.2.a.n.1.1 3
13.12 even 2 6422.2.a.v.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6422.2.a.n.1.1 3 1.1 even 1 trivial
6422.2.a.v.1.1 yes 3 13.12 even 2