Properties

Label 8004.2.a.i.1.16
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(4.15437\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.15437 q^{5} -4.14427 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.15437 q^{5} -4.14427 q^{7} +1.00000 q^{9} +0.229135 q^{11} -4.10323 q^{13} -4.15437 q^{15} +6.38557 q^{17} -2.89203 q^{19} +4.14427 q^{21} +1.00000 q^{23} +12.2588 q^{25} -1.00000 q^{27} +1.00000 q^{29} +7.43114 q^{31} -0.229135 q^{33} -17.2168 q^{35} -6.31719 q^{37} +4.10323 q^{39} +5.76434 q^{41} +0.993037 q^{43} +4.15437 q^{45} -6.88898 q^{47} +10.1750 q^{49} -6.38557 q^{51} +6.39600 q^{53} +0.951911 q^{55} +2.89203 q^{57} -3.98312 q^{59} -4.98317 q^{61} -4.14427 q^{63} -17.0464 q^{65} +2.09792 q^{67} -1.00000 q^{69} -13.7993 q^{71} -13.8931 q^{73} -12.2588 q^{75} -0.949596 q^{77} +12.1000 q^{79} +1.00000 q^{81} +9.00171 q^{83} +26.5280 q^{85} -1.00000 q^{87} +7.22544 q^{89} +17.0049 q^{91} -7.43114 q^{93} -12.0146 q^{95} +2.29533 q^{97} +0.229135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 4.15437 1.85789 0.928946 0.370216i \(-0.120716\pi\)
0.928946 + 0.370216i \(0.120716\pi\)
\(6\) 0 0
\(7\) −4.14427 −1.56639 −0.783193 0.621778i \(-0.786410\pi\)
−0.783193 + 0.621778i \(0.786410\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.229135 0.0690867 0.0345434 0.999403i \(-0.489002\pi\)
0.0345434 + 0.999403i \(0.489002\pi\)
\(12\) 0 0
\(13\) −4.10323 −1.13803 −0.569016 0.822326i \(-0.692676\pi\)
−0.569016 + 0.822326i \(0.692676\pi\)
\(14\) 0 0
\(15\) −4.15437 −1.07265
\(16\) 0 0
\(17\) 6.38557 1.54873 0.774364 0.632741i \(-0.218070\pi\)
0.774364 + 0.632741i \(0.218070\pi\)
\(18\) 0 0
\(19\) −2.89203 −0.663478 −0.331739 0.943371i \(-0.607635\pi\)
−0.331739 + 0.943371i \(0.607635\pi\)
\(20\) 0 0
\(21\) 4.14427 0.904354
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 12.2588 2.45176
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 7.43114 1.33467 0.667336 0.744757i \(-0.267435\pi\)
0.667336 + 0.744757i \(0.267435\pi\)
\(32\) 0 0
\(33\) −0.229135 −0.0398872
\(34\) 0 0
\(35\) −17.2168 −2.91018
\(36\) 0 0
\(37\) −6.31719 −1.03854 −0.519270 0.854610i \(-0.673796\pi\)
−0.519270 + 0.854610i \(0.673796\pi\)
\(38\) 0 0
\(39\) 4.10323 0.657043
\(40\) 0 0
\(41\) 5.76434 0.900239 0.450119 0.892968i \(-0.351381\pi\)
0.450119 + 0.892968i \(0.351381\pi\)
\(42\) 0 0
\(43\) 0.993037 0.151437 0.0757183 0.997129i \(-0.475875\pi\)
0.0757183 + 0.997129i \(0.475875\pi\)
\(44\) 0 0
\(45\) 4.15437 0.619297
\(46\) 0 0
\(47\) −6.88898 −1.00486 −0.502430 0.864618i \(-0.667561\pi\)
−0.502430 + 0.864618i \(0.667561\pi\)
\(48\) 0 0
\(49\) 10.1750 1.45357
\(50\) 0 0
\(51\) −6.38557 −0.894158
\(52\) 0 0
\(53\) 6.39600 0.878558 0.439279 0.898351i \(-0.355234\pi\)
0.439279 + 0.898351i \(0.355234\pi\)
\(54\) 0 0
\(55\) 0.951911 0.128356
\(56\) 0 0
\(57\) 2.89203 0.383059
\(58\) 0 0
\(59\) −3.98312 −0.518558 −0.259279 0.965802i \(-0.583485\pi\)
−0.259279 + 0.965802i \(0.583485\pi\)
\(60\) 0 0
\(61\) −4.98317 −0.638030 −0.319015 0.947750i \(-0.603352\pi\)
−0.319015 + 0.947750i \(0.603352\pi\)
\(62\) 0 0
\(63\) −4.14427 −0.522129
\(64\) 0 0
\(65\) −17.0464 −2.11434
\(66\) 0 0
\(67\) 2.09792 0.256302 0.128151 0.991755i \(-0.459096\pi\)
0.128151 + 0.991755i \(0.459096\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −13.7993 −1.63768 −0.818840 0.574021i \(-0.805383\pi\)
−0.818840 + 0.574021i \(0.805383\pi\)
\(72\) 0 0
\(73\) −13.8931 −1.62606 −0.813032 0.582219i \(-0.802185\pi\)
−0.813032 + 0.582219i \(0.802185\pi\)
\(74\) 0 0
\(75\) −12.2588 −1.41552
\(76\) 0 0
\(77\) −0.949596 −0.108216
\(78\) 0 0
\(79\) 12.1000 1.36135 0.680677 0.732584i \(-0.261686\pi\)
0.680677 + 0.732584i \(0.261686\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.00171 0.988066 0.494033 0.869443i \(-0.335522\pi\)
0.494033 + 0.869443i \(0.335522\pi\)
\(84\) 0 0
\(85\) 26.5280 2.87737
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 7.22544 0.765895 0.382948 0.923770i \(-0.374909\pi\)
0.382948 + 0.923770i \(0.374909\pi\)
\(90\) 0 0
\(91\) 17.0049 1.78260
\(92\) 0 0
\(93\) −7.43114 −0.770573
\(94\) 0 0
\(95\) −12.0146 −1.23267
\(96\) 0 0
\(97\) 2.29533 0.233055 0.116528 0.993187i \(-0.462824\pi\)
0.116528 + 0.993187i \(0.462824\pi\)
\(98\) 0 0
\(99\) 0.229135 0.0230289
\(100\) 0 0
\(101\) −1.84478 −0.183562 −0.0917812 0.995779i \(-0.529256\pi\)
−0.0917812 + 0.995779i \(0.529256\pi\)
\(102\) 0 0
\(103\) −10.0687 −0.992095 −0.496048 0.868295i \(-0.665216\pi\)
−0.496048 + 0.868295i \(0.665216\pi\)
\(104\) 0 0
\(105\) 17.2168 1.68019
\(106\) 0 0
\(107\) −2.97463 −0.287568 −0.143784 0.989609i \(-0.545927\pi\)
−0.143784 + 0.989609i \(0.545927\pi\)
\(108\) 0 0
\(109\) 17.2466 1.65193 0.825965 0.563722i \(-0.190631\pi\)
0.825965 + 0.563722i \(0.190631\pi\)
\(110\) 0 0
\(111\) 6.31719 0.599602
\(112\) 0 0
\(113\) 14.2563 1.34112 0.670561 0.741854i \(-0.266053\pi\)
0.670561 + 0.741854i \(0.266053\pi\)
\(114\) 0 0
\(115\) 4.15437 0.387397
\(116\) 0 0
\(117\) −4.10323 −0.379344
\(118\) 0 0
\(119\) −26.4635 −2.42591
\(120\) 0 0
\(121\) −10.9475 −0.995227
\(122\) 0 0
\(123\) −5.76434 −0.519753
\(124\) 0 0
\(125\) 30.1558 2.69721
\(126\) 0 0
\(127\) 8.12633 0.721095 0.360548 0.932741i \(-0.382590\pi\)
0.360548 + 0.932741i \(0.382590\pi\)
\(128\) 0 0
\(129\) −0.993037 −0.0874320
\(130\) 0 0
\(131\) −6.66793 −0.582579 −0.291290 0.956635i \(-0.594084\pi\)
−0.291290 + 0.956635i \(0.594084\pi\)
\(132\) 0 0
\(133\) 11.9854 1.03926
\(134\) 0 0
\(135\) −4.15437 −0.357551
\(136\) 0 0
\(137\) 8.92799 0.762770 0.381385 0.924416i \(-0.375447\pi\)
0.381385 + 0.924416i \(0.375447\pi\)
\(138\) 0 0
\(139\) 15.6891 1.33073 0.665365 0.746518i \(-0.268276\pi\)
0.665365 + 0.746518i \(0.268276\pi\)
\(140\) 0 0
\(141\) 6.88898 0.580156
\(142\) 0 0
\(143\) −0.940193 −0.0786229
\(144\) 0 0
\(145\) 4.15437 0.345002
\(146\) 0 0
\(147\) −10.1750 −0.839217
\(148\) 0 0
\(149\) −21.2764 −1.74303 −0.871515 0.490369i \(-0.836862\pi\)
−0.871515 + 0.490369i \(0.836862\pi\)
\(150\) 0 0
\(151\) 16.0301 1.30451 0.652254 0.758000i \(-0.273823\pi\)
0.652254 + 0.758000i \(0.273823\pi\)
\(152\) 0 0
\(153\) 6.38557 0.516242
\(154\) 0 0
\(155\) 30.8717 2.47967
\(156\) 0 0
\(157\) 21.8133 1.74089 0.870447 0.492262i \(-0.163830\pi\)
0.870447 + 0.492262i \(0.163830\pi\)
\(158\) 0 0
\(159\) −6.39600 −0.507236
\(160\) 0 0
\(161\) −4.14427 −0.326614
\(162\) 0 0
\(163\) 20.1201 1.57593 0.787965 0.615719i \(-0.211135\pi\)
0.787965 + 0.615719i \(0.211135\pi\)
\(164\) 0 0
\(165\) −0.951911 −0.0741061
\(166\) 0 0
\(167\) −14.8101 −1.14604 −0.573018 0.819543i \(-0.694228\pi\)
−0.573018 + 0.819543i \(0.694228\pi\)
\(168\) 0 0
\(169\) 3.83652 0.295117
\(170\) 0 0
\(171\) −2.89203 −0.221159
\(172\) 0 0
\(173\) −3.05807 −0.232501 −0.116250 0.993220i \(-0.537087\pi\)
−0.116250 + 0.993220i \(0.537087\pi\)
\(174\) 0 0
\(175\) −50.8038 −3.84040
\(176\) 0 0
\(177\) 3.98312 0.299390
\(178\) 0 0
\(179\) −6.09443 −0.455519 −0.227759 0.973717i \(-0.573140\pi\)
−0.227759 + 0.973717i \(0.573140\pi\)
\(180\) 0 0
\(181\) 24.1648 1.79616 0.898079 0.439834i \(-0.144963\pi\)
0.898079 + 0.439834i \(0.144963\pi\)
\(182\) 0 0
\(183\) 4.98317 0.368367
\(184\) 0 0
\(185\) −26.2440 −1.92950
\(186\) 0 0
\(187\) 1.46315 0.106996
\(188\) 0 0
\(189\) 4.14427 0.301451
\(190\) 0 0
\(191\) 4.15674 0.300771 0.150385 0.988627i \(-0.451949\pi\)
0.150385 + 0.988627i \(0.451949\pi\)
\(192\) 0 0
\(193\) −11.5694 −0.832784 −0.416392 0.909185i \(-0.636706\pi\)
−0.416392 + 0.909185i \(0.636706\pi\)
\(194\) 0 0
\(195\) 17.0464 1.22071
\(196\) 0 0
\(197\) 26.6706 1.90020 0.950100 0.311947i \(-0.100981\pi\)
0.950100 + 0.311947i \(0.100981\pi\)
\(198\) 0 0
\(199\) 2.14916 0.152350 0.0761750 0.997094i \(-0.475729\pi\)
0.0761750 + 0.997094i \(0.475729\pi\)
\(200\) 0 0
\(201\) −2.09792 −0.147976
\(202\) 0 0
\(203\) −4.14427 −0.290871
\(204\) 0 0
\(205\) 23.9472 1.67255
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −0.662665 −0.0458375
\(210\) 0 0
\(211\) 25.3278 1.74364 0.871819 0.489828i \(-0.162940\pi\)
0.871819 + 0.489828i \(0.162940\pi\)
\(212\) 0 0
\(213\) 13.7993 0.945515
\(214\) 0 0
\(215\) 4.12544 0.281353
\(216\) 0 0
\(217\) −30.7966 −2.09061
\(218\) 0 0
\(219\) 13.8931 0.938809
\(220\) 0 0
\(221\) −26.2015 −1.76250
\(222\) 0 0
\(223\) −2.20969 −0.147972 −0.0739858 0.997259i \(-0.523572\pi\)
−0.0739858 + 0.997259i \(0.523572\pi\)
\(224\) 0 0
\(225\) 12.2588 0.817253
\(226\) 0 0
\(227\) 2.88544 0.191514 0.0957568 0.995405i \(-0.469473\pi\)
0.0957568 + 0.995405i \(0.469473\pi\)
\(228\) 0 0
\(229\) −21.1310 −1.39638 −0.698188 0.715915i \(-0.746010\pi\)
−0.698188 + 0.715915i \(0.746010\pi\)
\(230\) 0 0
\(231\) 0.949596 0.0624788
\(232\) 0 0
\(233\) −14.1662 −0.928057 −0.464029 0.885820i \(-0.653597\pi\)
−0.464029 + 0.885820i \(0.653597\pi\)
\(234\) 0 0
\(235\) −28.6194 −1.86692
\(236\) 0 0
\(237\) −12.1000 −0.785978
\(238\) 0 0
\(239\) 19.0504 1.23227 0.616133 0.787642i \(-0.288698\pi\)
0.616133 + 0.787642i \(0.288698\pi\)
\(240\) 0 0
\(241\) 28.9548 1.86514 0.932570 0.360989i \(-0.117561\pi\)
0.932570 + 0.360989i \(0.117561\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 42.2706 2.70057
\(246\) 0 0
\(247\) 11.8667 0.755059
\(248\) 0 0
\(249\) −9.00171 −0.570460
\(250\) 0 0
\(251\) 13.5507 0.855313 0.427656 0.903941i \(-0.359339\pi\)
0.427656 + 0.903941i \(0.359339\pi\)
\(252\) 0 0
\(253\) 0.229135 0.0144056
\(254\) 0 0
\(255\) −26.5280 −1.66125
\(256\) 0 0
\(257\) 1.47343 0.0919100 0.0459550 0.998944i \(-0.485367\pi\)
0.0459550 + 0.998944i \(0.485367\pi\)
\(258\) 0 0
\(259\) 26.1802 1.62676
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 22.8721 1.41035 0.705176 0.709032i \(-0.250868\pi\)
0.705176 + 0.709032i \(0.250868\pi\)
\(264\) 0 0
\(265\) 26.5714 1.63227
\(266\) 0 0
\(267\) −7.22544 −0.442190
\(268\) 0 0
\(269\) 25.0211 1.52556 0.762781 0.646657i \(-0.223833\pi\)
0.762781 + 0.646657i \(0.223833\pi\)
\(270\) 0 0
\(271\) −2.80847 −0.170602 −0.0853012 0.996355i \(-0.527185\pi\)
−0.0853012 + 0.996355i \(0.527185\pi\)
\(272\) 0 0
\(273\) −17.0049 −1.02918
\(274\) 0 0
\(275\) 2.80892 0.169384
\(276\) 0 0
\(277\) −3.13857 −0.188578 −0.0942892 0.995545i \(-0.530058\pi\)
−0.0942892 + 0.995545i \(0.530058\pi\)
\(278\) 0 0
\(279\) 7.43114 0.444890
\(280\) 0 0
\(281\) −7.87615 −0.469852 −0.234926 0.972013i \(-0.575485\pi\)
−0.234926 + 0.972013i \(0.575485\pi\)
\(282\) 0 0
\(283\) −13.1634 −0.782484 −0.391242 0.920288i \(-0.627954\pi\)
−0.391242 + 0.920288i \(0.627954\pi\)
\(284\) 0 0
\(285\) 12.0146 0.711683
\(286\) 0 0
\(287\) −23.8890 −1.41012
\(288\) 0 0
\(289\) 23.7754 1.39856
\(290\) 0 0
\(291\) −2.29533 −0.134554
\(292\) 0 0
\(293\) −32.9068 −1.92243 −0.961216 0.275795i \(-0.911059\pi\)
−0.961216 + 0.275795i \(0.911059\pi\)
\(294\) 0 0
\(295\) −16.5474 −0.963425
\(296\) 0 0
\(297\) −0.229135 −0.0132957
\(298\) 0 0
\(299\) −4.10323 −0.237296
\(300\) 0 0
\(301\) −4.11541 −0.237208
\(302\) 0 0
\(303\) 1.84478 0.105980
\(304\) 0 0
\(305\) −20.7019 −1.18539
\(306\) 0 0
\(307\) 3.01583 0.172123 0.0860613 0.996290i \(-0.472572\pi\)
0.0860613 + 0.996290i \(0.472572\pi\)
\(308\) 0 0
\(309\) 10.0687 0.572786
\(310\) 0 0
\(311\) −7.45318 −0.422631 −0.211315 0.977418i \(-0.567775\pi\)
−0.211315 + 0.977418i \(0.567775\pi\)
\(312\) 0 0
\(313\) −3.43534 −0.194177 −0.0970886 0.995276i \(-0.530953\pi\)
−0.0970886 + 0.995276i \(0.530953\pi\)
\(314\) 0 0
\(315\) −17.2168 −0.970059
\(316\) 0 0
\(317\) 0.318803 0.0179057 0.00895287 0.999960i \(-0.497150\pi\)
0.00895287 + 0.999960i \(0.497150\pi\)
\(318\) 0 0
\(319\) 0.229135 0.0128291
\(320\) 0 0
\(321\) 2.97463 0.166028
\(322\) 0 0
\(323\) −18.4673 −1.02755
\(324\) 0 0
\(325\) −50.3007 −2.79018
\(326\) 0 0
\(327\) −17.2466 −0.953742
\(328\) 0 0
\(329\) 28.5498 1.57400
\(330\) 0 0
\(331\) 26.0762 1.43328 0.716638 0.697446i \(-0.245680\pi\)
0.716638 + 0.697446i \(0.245680\pi\)
\(332\) 0 0
\(333\) −6.31719 −0.346180
\(334\) 0 0
\(335\) 8.71556 0.476182
\(336\) 0 0
\(337\) −15.0255 −0.818489 −0.409244 0.912425i \(-0.634208\pi\)
−0.409244 + 0.912425i \(0.634208\pi\)
\(338\) 0 0
\(339\) −14.2563 −0.774298
\(340\) 0 0
\(341\) 1.70273 0.0922080
\(342\) 0 0
\(343\) −13.1579 −0.710460
\(344\) 0 0
\(345\) −4.15437 −0.223664
\(346\) 0 0
\(347\) −10.0828 −0.541274 −0.270637 0.962681i \(-0.587234\pi\)
−0.270637 + 0.962681i \(0.587234\pi\)
\(348\) 0 0
\(349\) −4.11851 −0.220459 −0.110229 0.993906i \(-0.535159\pi\)
−0.110229 + 0.993906i \(0.535159\pi\)
\(350\) 0 0
\(351\) 4.10323 0.219014
\(352\) 0 0
\(353\) 27.9036 1.48516 0.742579 0.669759i \(-0.233603\pi\)
0.742579 + 0.669759i \(0.233603\pi\)
\(354\) 0 0
\(355\) −57.3276 −3.04263
\(356\) 0 0
\(357\) 26.4635 1.40060
\(358\) 0 0
\(359\) −17.4428 −0.920598 −0.460299 0.887764i \(-0.652258\pi\)
−0.460299 + 0.887764i \(0.652258\pi\)
\(360\) 0 0
\(361\) −10.6361 −0.559797
\(362\) 0 0
\(363\) 10.9475 0.574595
\(364\) 0 0
\(365\) −57.7171 −3.02105
\(366\) 0 0
\(367\) −33.8999 −1.76956 −0.884781 0.466006i \(-0.845692\pi\)
−0.884781 + 0.466006i \(0.845692\pi\)
\(368\) 0 0
\(369\) 5.76434 0.300080
\(370\) 0 0
\(371\) −26.5067 −1.37616
\(372\) 0 0
\(373\) −18.2447 −0.944676 −0.472338 0.881417i \(-0.656590\pi\)
−0.472338 + 0.881417i \(0.656590\pi\)
\(374\) 0 0
\(375\) −30.1558 −1.55724
\(376\) 0 0
\(377\) −4.10323 −0.211327
\(378\) 0 0
\(379\) −12.1061 −0.621848 −0.310924 0.950435i \(-0.600638\pi\)
−0.310924 + 0.950435i \(0.600638\pi\)
\(380\) 0 0
\(381\) −8.12633 −0.416325
\(382\) 0 0
\(383\) 22.0811 1.12829 0.564147 0.825675i \(-0.309205\pi\)
0.564147 + 0.825675i \(0.309205\pi\)
\(384\) 0 0
\(385\) −3.94497 −0.201054
\(386\) 0 0
\(387\) 0.993037 0.0504789
\(388\) 0 0
\(389\) 31.1346 1.57859 0.789293 0.614016i \(-0.210447\pi\)
0.789293 + 0.614016i \(0.210447\pi\)
\(390\) 0 0
\(391\) 6.38557 0.322932
\(392\) 0 0
\(393\) 6.66793 0.336352
\(394\) 0 0
\(395\) 50.2678 2.52925
\(396\) 0 0
\(397\) 6.74900 0.338723 0.169361 0.985554i \(-0.445830\pi\)
0.169361 + 0.985554i \(0.445830\pi\)
\(398\) 0 0
\(399\) −11.9854 −0.600019
\(400\) 0 0
\(401\) 21.2099 1.05917 0.529586 0.848256i \(-0.322347\pi\)
0.529586 + 0.848256i \(0.322347\pi\)
\(402\) 0 0
\(403\) −30.4917 −1.51890
\(404\) 0 0
\(405\) 4.15437 0.206432
\(406\) 0 0
\(407\) −1.44749 −0.0717493
\(408\) 0 0
\(409\) 15.9041 0.786406 0.393203 0.919452i \(-0.371367\pi\)
0.393203 + 0.919452i \(0.371367\pi\)
\(410\) 0 0
\(411\) −8.92799 −0.440385
\(412\) 0 0
\(413\) 16.5071 0.812262
\(414\) 0 0
\(415\) 37.3964 1.83572
\(416\) 0 0
\(417\) −15.6891 −0.768297
\(418\) 0 0
\(419\) 31.7059 1.54894 0.774468 0.632613i \(-0.218018\pi\)
0.774468 + 0.632613i \(0.218018\pi\)
\(420\) 0 0
\(421\) 21.4102 1.04347 0.521735 0.853108i \(-0.325285\pi\)
0.521735 + 0.853108i \(0.325285\pi\)
\(422\) 0 0
\(423\) −6.88898 −0.334953
\(424\) 0 0
\(425\) 78.2794 3.79711
\(426\) 0 0
\(427\) 20.6516 0.999401
\(428\) 0 0
\(429\) 0.940193 0.0453929
\(430\) 0 0
\(431\) −2.77251 −0.133547 −0.0667735 0.997768i \(-0.521270\pi\)
−0.0667735 + 0.997768i \(0.521270\pi\)
\(432\) 0 0
\(433\) 1.68490 0.0809711 0.0404856 0.999180i \(-0.487110\pi\)
0.0404856 + 0.999180i \(0.487110\pi\)
\(434\) 0 0
\(435\) −4.15437 −0.199187
\(436\) 0 0
\(437\) −2.89203 −0.138345
\(438\) 0 0
\(439\) −14.6829 −0.700777 −0.350389 0.936604i \(-0.613950\pi\)
−0.350389 + 0.936604i \(0.613950\pi\)
\(440\) 0 0
\(441\) 10.1750 0.484522
\(442\) 0 0
\(443\) 18.2376 0.866494 0.433247 0.901275i \(-0.357368\pi\)
0.433247 + 0.901275i \(0.357368\pi\)
\(444\) 0 0
\(445\) 30.0172 1.42295
\(446\) 0 0
\(447\) 21.2764 1.00634
\(448\) 0 0
\(449\) 6.91352 0.326269 0.163135 0.986604i \(-0.447840\pi\)
0.163135 + 0.986604i \(0.447840\pi\)
\(450\) 0 0
\(451\) 1.32081 0.0621945
\(452\) 0 0
\(453\) −16.0301 −0.753158
\(454\) 0 0
\(455\) 70.6447 3.31187
\(456\) 0 0
\(457\) 1.39169 0.0651005 0.0325502 0.999470i \(-0.489637\pi\)
0.0325502 + 0.999470i \(0.489637\pi\)
\(458\) 0 0
\(459\) −6.38557 −0.298053
\(460\) 0 0
\(461\) 17.4374 0.812140 0.406070 0.913842i \(-0.366899\pi\)
0.406070 + 0.913842i \(0.366899\pi\)
\(462\) 0 0
\(463\) −13.7841 −0.640599 −0.320300 0.947316i \(-0.603784\pi\)
−0.320300 + 0.947316i \(0.603784\pi\)
\(464\) 0 0
\(465\) −30.8717 −1.43164
\(466\) 0 0
\(467\) −10.2314 −0.473454 −0.236727 0.971576i \(-0.576075\pi\)
−0.236727 + 0.971576i \(0.576075\pi\)
\(468\) 0 0
\(469\) −8.69436 −0.401468
\(470\) 0 0
\(471\) −21.8133 −1.00511
\(472\) 0 0
\(473\) 0.227539 0.0104623
\(474\) 0 0
\(475\) −35.4529 −1.62669
\(476\) 0 0
\(477\) 6.39600 0.292853
\(478\) 0 0
\(479\) −23.0206 −1.05184 −0.525920 0.850534i \(-0.676279\pi\)
−0.525920 + 0.850534i \(0.676279\pi\)
\(480\) 0 0
\(481\) 25.9209 1.18189
\(482\) 0 0
\(483\) 4.14427 0.188571
\(484\) 0 0
\(485\) 9.53564 0.432991
\(486\) 0 0
\(487\) −22.2168 −1.00674 −0.503370 0.864071i \(-0.667907\pi\)
−0.503370 + 0.864071i \(0.667907\pi\)
\(488\) 0 0
\(489\) −20.1201 −0.909864
\(490\) 0 0
\(491\) 25.0400 1.13004 0.565020 0.825077i \(-0.308868\pi\)
0.565020 + 0.825077i \(0.308868\pi\)
\(492\) 0 0
\(493\) 6.38557 0.287591
\(494\) 0 0
\(495\) 0.951911 0.0427852
\(496\) 0 0
\(497\) 57.1882 2.56524
\(498\) 0 0
\(499\) 23.6498 1.05871 0.529356 0.848400i \(-0.322434\pi\)
0.529356 + 0.848400i \(0.322434\pi\)
\(500\) 0 0
\(501\) 14.8101 0.661664
\(502\) 0 0
\(503\) −20.9246 −0.932980 −0.466490 0.884526i \(-0.654482\pi\)
−0.466490 + 0.884526i \(0.654482\pi\)
\(504\) 0 0
\(505\) −7.66390 −0.341039
\(506\) 0 0
\(507\) −3.83652 −0.170386
\(508\) 0 0
\(509\) −27.1188 −1.20202 −0.601009 0.799242i \(-0.705235\pi\)
−0.601009 + 0.799242i \(0.705235\pi\)
\(510\) 0 0
\(511\) 57.5767 2.54704
\(512\) 0 0
\(513\) 2.89203 0.127686
\(514\) 0 0
\(515\) −41.8290 −1.84320
\(516\) 0 0
\(517\) −1.57850 −0.0694225
\(518\) 0 0
\(519\) 3.05807 0.134234
\(520\) 0 0
\(521\) 38.8200 1.70073 0.850367 0.526190i \(-0.176380\pi\)
0.850367 + 0.526190i \(0.176380\pi\)
\(522\) 0 0
\(523\) 39.7447 1.73791 0.868957 0.494887i \(-0.164791\pi\)
0.868957 + 0.494887i \(0.164791\pi\)
\(524\) 0 0
\(525\) 50.8038 2.21726
\(526\) 0 0
\(527\) 47.4520 2.06704
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.98312 −0.172853
\(532\) 0 0
\(533\) −23.6524 −1.02450
\(534\) 0 0
\(535\) −12.3577 −0.534270
\(536\) 0 0
\(537\) 6.09443 0.262994
\(538\) 0 0
\(539\) 2.33144 0.100422
\(540\) 0 0
\(541\) −35.9805 −1.54692 −0.773462 0.633843i \(-0.781477\pi\)
−0.773462 + 0.633843i \(0.781477\pi\)
\(542\) 0 0
\(543\) −24.1648 −1.03701
\(544\) 0 0
\(545\) 71.6490 3.06911
\(546\) 0 0
\(547\) 16.3011 0.696986 0.348493 0.937311i \(-0.386693\pi\)
0.348493 + 0.937311i \(0.386693\pi\)
\(548\) 0 0
\(549\) −4.98317 −0.212677
\(550\) 0 0
\(551\) −2.89203 −0.123205
\(552\) 0 0
\(553\) −50.1456 −2.13241
\(554\) 0 0
\(555\) 26.2440 1.11399
\(556\) 0 0
\(557\) 4.14809 0.175760 0.0878800 0.996131i \(-0.471991\pi\)
0.0878800 + 0.996131i \(0.471991\pi\)
\(558\) 0 0
\(559\) −4.07466 −0.172340
\(560\) 0 0
\(561\) −1.46315 −0.0617744
\(562\) 0 0
\(563\) −25.5993 −1.07888 −0.539440 0.842024i \(-0.681364\pi\)
−0.539440 + 0.842024i \(0.681364\pi\)
\(564\) 0 0
\(565\) 59.2261 2.49166
\(566\) 0 0
\(567\) −4.14427 −0.174043
\(568\) 0 0
\(569\) −19.4642 −0.815984 −0.407992 0.912986i \(-0.633771\pi\)
−0.407992 + 0.912986i \(0.633771\pi\)
\(570\) 0 0
\(571\) 37.7938 1.58162 0.790810 0.612062i \(-0.209660\pi\)
0.790810 + 0.612062i \(0.209660\pi\)
\(572\) 0 0
\(573\) −4.15674 −0.173650
\(574\) 0 0
\(575\) 12.2588 0.511227
\(576\) 0 0
\(577\) 8.28292 0.344822 0.172411 0.985025i \(-0.444844\pi\)
0.172411 + 0.985025i \(0.444844\pi\)
\(578\) 0 0
\(579\) 11.5694 0.480808
\(580\) 0 0
\(581\) −37.3055 −1.54769
\(582\) 0 0
\(583\) 1.46555 0.0606967
\(584\) 0 0
\(585\) −17.0464 −0.704780
\(586\) 0 0
\(587\) 25.5240 1.05349 0.526744 0.850024i \(-0.323413\pi\)
0.526744 + 0.850024i \(0.323413\pi\)
\(588\) 0 0
\(589\) −21.4911 −0.885525
\(590\) 0 0
\(591\) −26.6706 −1.09708
\(592\) 0 0
\(593\) 35.2203 1.44633 0.723163 0.690677i \(-0.242688\pi\)
0.723163 + 0.690677i \(0.242688\pi\)
\(594\) 0 0
\(595\) −109.939 −4.50707
\(596\) 0 0
\(597\) −2.14916 −0.0879593
\(598\) 0 0
\(599\) −26.8228 −1.09595 −0.547974 0.836495i \(-0.684601\pi\)
−0.547974 + 0.836495i \(0.684601\pi\)
\(600\) 0 0
\(601\) −32.5755 −1.32878 −0.664390 0.747386i \(-0.731309\pi\)
−0.664390 + 0.747386i \(0.731309\pi\)
\(602\) 0 0
\(603\) 2.09792 0.0854341
\(604\) 0 0
\(605\) −45.4800 −1.84902
\(606\) 0 0
\(607\) 15.2479 0.618892 0.309446 0.950917i \(-0.399856\pi\)
0.309446 + 0.950917i \(0.399856\pi\)
\(608\) 0 0
\(609\) 4.14427 0.167934
\(610\) 0 0
\(611\) 28.2671 1.14356
\(612\) 0 0
\(613\) −10.1136 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(614\) 0 0
\(615\) −23.9472 −0.965645
\(616\) 0 0
\(617\) 8.26929 0.332909 0.166454 0.986049i \(-0.446768\pi\)
0.166454 + 0.986049i \(0.446768\pi\)
\(618\) 0 0
\(619\) 0.388463 0.0156137 0.00780683 0.999970i \(-0.497515\pi\)
0.00780683 + 0.999970i \(0.497515\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −29.9442 −1.19969
\(624\) 0 0
\(625\) 63.9842 2.55937
\(626\) 0 0
\(627\) 0.662665 0.0264643
\(628\) 0 0
\(629\) −40.3389 −1.60842
\(630\) 0 0
\(631\) −11.3853 −0.453241 −0.226621 0.973983i \(-0.572768\pi\)
−0.226621 + 0.973983i \(0.572768\pi\)
\(632\) 0 0
\(633\) −25.3278 −1.00669
\(634\) 0 0
\(635\) 33.7598 1.33972
\(636\) 0 0
\(637\) −41.7502 −1.65420
\(638\) 0 0
\(639\) −13.7993 −0.545894
\(640\) 0 0
\(641\) −16.0469 −0.633815 −0.316907 0.948457i \(-0.602644\pi\)
−0.316907 + 0.948457i \(0.602644\pi\)
\(642\) 0 0
\(643\) 8.54739 0.337076 0.168538 0.985695i \(-0.446095\pi\)
0.168538 + 0.985695i \(0.446095\pi\)
\(644\) 0 0
\(645\) −4.12544 −0.162439
\(646\) 0 0
\(647\) 20.7023 0.813891 0.406945 0.913453i \(-0.366594\pi\)
0.406945 + 0.913453i \(0.366594\pi\)
\(648\) 0 0
\(649\) −0.912671 −0.0358255
\(650\) 0 0
\(651\) 30.7966 1.20701
\(652\) 0 0
\(653\) −31.2691 −1.22366 −0.611828 0.790991i \(-0.709565\pi\)
−0.611828 + 0.790991i \(0.709565\pi\)
\(654\) 0 0
\(655\) −27.7010 −1.08237
\(656\) 0 0
\(657\) −13.8931 −0.542021
\(658\) 0 0
\(659\) −2.05556 −0.0800734 −0.0400367 0.999198i \(-0.512747\pi\)
−0.0400367 + 0.999198i \(0.512747\pi\)
\(660\) 0 0
\(661\) 23.8207 0.926517 0.463258 0.886223i \(-0.346680\pi\)
0.463258 + 0.886223i \(0.346680\pi\)
\(662\) 0 0
\(663\) 26.2015 1.01758
\(664\) 0 0
\(665\) 49.7917 1.93084
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 2.20969 0.0854314
\(670\) 0 0
\(671\) −1.14182 −0.0440794
\(672\) 0 0
\(673\) −3.62405 −0.139697 −0.0698483 0.997558i \(-0.522252\pi\)
−0.0698483 + 0.997558i \(0.522252\pi\)
\(674\) 0 0
\(675\) −12.2588 −0.471841
\(676\) 0 0
\(677\) 8.74392 0.336056 0.168028 0.985782i \(-0.446260\pi\)
0.168028 + 0.985782i \(0.446260\pi\)
\(678\) 0 0
\(679\) −9.51245 −0.365054
\(680\) 0 0
\(681\) −2.88544 −0.110570
\(682\) 0 0
\(683\) −10.7533 −0.411464 −0.205732 0.978608i \(-0.565958\pi\)
−0.205732 + 0.978608i \(0.565958\pi\)
\(684\) 0 0
\(685\) 37.0902 1.41714
\(686\) 0 0
\(687\) 21.1310 0.806198
\(688\) 0 0
\(689\) −26.2443 −0.999827
\(690\) 0 0
\(691\) −29.9770 −1.14038 −0.570190 0.821513i \(-0.693130\pi\)
−0.570190 + 0.821513i \(0.693130\pi\)
\(692\) 0 0
\(693\) −0.949596 −0.0360722
\(694\) 0 0
\(695\) 65.1782 2.47235
\(696\) 0 0
\(697\) 36.8086 1.39422
\(698\) 0 0
\(699\) 14.1662 0.535814
\(700\) 0 0
\(701\) −12.1979 −0.460710 −0.230355 0.973107i \(-0.573989\pi\)
−0.230355 + 0.973107i \(0.573989\pi\)
\(702\) 0 0
\(703\) 18.2695 0.689049
\(704\) 0 0
\(705\) 28.6194 1.07787
\(706\) 0 0
\(707\) 7.64526 0.287530
\(708\) 0 0
\(709\) 15.6535 0.587878 0.293939 0.955824i \(-0.405034\pi\)
0.293939 + 0.955824i \(0.405034\pi\)
\(710\) 0 0
\(711\) 12.1000 0.453785
\(712\) 0 0
\(713\) 7.43114 0.278298
\(714\) 0 0
\(715\) −3.90591 −0.146073
\(716\) 0 0
\(717\) −19.0504 −0.711449
\(718\) 0 0
\(719\) −0.652811 −0.0243457 −0.0121729 0.999926i \(-0.503875\pi\)
−0.0121729 + 0.999926i \(0.503875\pi\)
\(720\) 0 0
\(721\) 41.7273 1.55400
\(722\) 0 0
\(723\) −28.9548 −1.07684
\(724\) 0 0
\(725\) 12.2588 0.455280
\(726\) 0 0
\(727\) 27.4190 1.01692 0.508458 0.861087i \(-0.330216\pi\)
0.508458 + 0.861087i \(0.330216\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.34110 0.234534
\(732\) 0 0
\(733\) 40.6326 1.50080 0.750400 0.660984i \(-0.229861\pi\)
0.750400 + 0.660984i \(0.229861\pi\)
\(734\) 0 0
\(735\) −42.2706 −1.55917
\(736\) 0 0
\(737\) 0.480707 0.0177071
\(738\) 0 0
\(739\) 28.6218 1.05287 0.526435 0.850215i \(-0.323528\pi\)
0.526435 + 0.850215i \(0.323528\pi\)
\(740\) 0 0
\(741\) −11.8667 −0.435934
\(742\) 0 0
\(743\) −7.39927 −0.271453 −0.135726 0.990746i \(-0.543337\pi\)
−0.135726 + 0.990746i \(0.543337\pi\)
\(744\) 0 0
\(745\) −88.3900 −3.23836
\(746\) 0 0
\(747\) 9.00171 0.329355
\(748\) 0 0
\(749\) 12.3277 0.450443
\(750\) 0 0
\(751\) 19.6285 0.716253 0.358126 0.933673i \(-0.383416\pi\)
0.358126 + 0.933673i \(0.383416\pi\)
\(752\) 0 0
\(753\) −13.5507 −0.493815
\(754\) 0 0
\(755\) 66.5949 2.42364
\(756\) 0 0
\(757\) 27.8155 1.01097 0.505486 0.862835i \(-0.331313\pi\)
0.505486 + 0.862835i \(0.331313\pi\)
\(758\) 0 0
\(759\) −0.229135 −0.00831706
\(760\) 0 0
\(761\) −23.6768 −0.858285 −0.429143 0.903237i \(-0.641184\pi\)
−0.429143 + 0.903237i \(0.641184\pi\)
\(762\) 0 0
\(763\) −71.4747 −2.58756
\(764\) 0 0
\(765\) 26.5280 0.959122
\(766\) 0 0
\(767\) 16.3437 0.590136
\(768\) 0 0
\(769\) −15.4026 −0.555433 −0.277716 0.960663i \(-0.589578\pi\)
−0.277716 + 0.960663i \(0.589578\pi\)
\(770\) 0 0
\(771\) −1.47343 −0.0530643
\(772\) 0 0
\(773\) −37.7016 −1.35603 −0.678016 0.735047i \(-0.737160\pi\)
−0.678016 + 0.735047i \(0.737160\pi\)
\(774\) 0 0
\(775\) 91.0968 3.27229
\(776\) 0 0
\(777\) −26.1802 −0.939208
\(778\) 0 0
\(779\) −16.6707 −0.597289
\(780\) 0 0
\(781\) −3.16191 −0.113142
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 90.6207 3.23439
\(786\) 0 0
\(787\) −30.1332 −1.07413 −0.537067 0.843540i \(-0.680468\pi\)
−0.537067 + 0.843540i \(0.680468\pi\)
\(788\) 0 0
\(789\) −22.8721 −0.814268
\(790\) 0 0
\(791\) −59.0821 −2.10072
\(792\) 0 0
\(793\) 20.4471 0.726098
\(794\) 0 0
\(795\) −26.5714 −0.942389
\(796\) 0 0
\(797\) −33.8038 −1.19739 −0.598696 0.800976i \(-0.704314\pi\)
−0.598696 + 0.800976i \(0.704314\pi\)
\(798\) 0 0
\(799\) −43.9900 −1.55625
\(800\) 0 0
\(801\) 7.22544 0.255298
\(802\) 0 0
\(803\) −3.18339 −0.112339
\(804\) 0 0
\(805\) −17.2168 −0.606814
\(806\) 0 0
\(807\) −25.0211 −0.880784
\(808\) 0 0
\(809\) 22.2495 0.782252 0.391126 0.920337i \(-0.372086\pi\)
0.391126 + 0.920337i \(0.372086\pi\)
\(810\) 0 0
\(811\) −32.0653 −1.12596 −0.562982 0.826469i \(-0.690346\pi\)
−0.562982 + 0.826469i \(0.690346\pi\)
\(812\) 0 0
\(813\) 2.80847 0.0984973
\(814\) 0 0
\(815\) 83.5865 2.92791
\(816\) 0 0
\(817\) −2.87190 −0.100475
\(818\) 0 0
\(819\) 17.0049 0.594199
\(820\) 0 0
\(821\) −32.1049 −1.12047 −0.560234 0.828334i \(-0.689289\pi\)
−0.560234 + 0.828334i \(0.689289\pi\)
\(822\) 0 0
\(823\) −24.7750 −0.863603 −0.431801 0.901969i \(-0.642122\pi\)
−0.431801 + 0.901969i \(0.642122\pi\)
\(824\) 0 0
\(825\) −2.80892 −0.0977939
\(826\) 0 0
\(827\) −37.7080 −1.31124 −0.655618 0.755093i \(-0.727592\pi\)
−0.655618 + 0.755093i \(0.727592\pi\)
\(828\) 0 0
\(829\) 33.3634 1.15876 0.579379 0.815059i \(-0.303295\pi\)
0.579379 + 0.815059i \(0.303295\pi\)
\(830\) 0 0
\(831\) 3.13857 0.108876
\(832\) 0 0
\(833\) 64.9729 2.25118
\(834\) 0 0
\(835\) −61.5265 −2.12921
\(836\) 0 0
\(837\) −7.43114 −0.256858
\(838\) 0 0
\(839\) 15.9819 0.551756 0.275878 0.961193i \(-0.411031\pi\)
0.275878 + 0.961193i \(0.411031\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 7.87615 0.271269
\(844\) 0 0
\(845\) 15.9383 0.548295
\(846\) 0 0
\(847\) 45.3694 1.55891
\(848\) 0 0
\(849\) 13.1634 0.451767
\(850\) 0 0
\(851\) −6.31719 −0.216551
\(852\) 0 0
\(853\) −3.22370 −0.110378 −0.0551888 0.998476i \(-0.517576\pi\)
−0.0551888 + 0.998476i \(0.517576\pi\)
\(854\) 0 0
\(855\) −12.0146 −0.410890
\(856\) 0 0
\(857\) 35.1231 1.19978 0.599891 0.800082i \(-0.295211\pi\)
0.599891 + 0.800082i \(0.295211\pi\)
\(858\) 0 0
\(859\) −51.7428 −1.76544 −0.882720 0.469899i \(-0.844290\pi\)
−0.882720 + 0.469899i \(0.844290\pi\)
\(860\) 0 0
\(861\) 23.8890 0.814134
\(862\) 0 0
\(863\) 13.7214 0.467080 0.233540 0.972347i \(-0.424969\pi\)
0.233540 + 0.972347i \(0.424969\pi\)
\(864\) 0 0
\(865\) −12.7044 −0.431961
\(866\) 0 0
\(867\) −23.7754 −0.807456
\(868\) 0 0
\(869\) 2.77252 0.0940515
\(870\) 0 0
\(871\) −8.60827 −0.291680
\(872\) 0 0
\(873\) 2.29533 0.0776850
\(874\) 0 0
\(875\) −124.974 −4.22488
\(876\) 0 0
\(877\) 29.5106 0.996502 0.498251 0.867033i \(-0.333976\pi\)
0.498251 + 0.867033i \(0.333976\pi\)
\(878\) 0 0
\(879\) 32.9068 1.10992
\(880\) 0 0
\(881\) 23.4896 0.791386 0.395693 0.918383i \(-0.370504\pi\)
0.395693 + 0.918383i \(0.370504\pi\)
\(882\) 0 0
\(883\) 44.0606 1.48276 0.741378 0.671088i \(-0.234173\pi\)
0.741378 + 0.671088i \(0.234173\pi\)
\(884\) 0 0
\(885\) 16.5474 0.556234
\(886\) 0 0
\(887\) −59.3172 −1.99168 −0.995839 0.0911296i \(-0.970952\pi\)
−0.995839 + 0.0911296i \(0.970952\pi\)
\(888\) 0 0
\(889\) −33.6777 −1.12951
\(890\) 0 0
\(891\) 0.229135 0.00767630
\(892\) 0 0
\(893\) 19.9232 0.666703
\(894\) 0 0
\(895\) −25.3185 −0.846305
\(896\) 0 0
\(897\) 4.10323 0.137003
\(898\) 0 0
\(899\) 7.43114 0.247842
\(900\) 0 0
\(901\) 40.8421 1.36065
\(902\) 0 0
\(903\) 4.11541 0.136952
\(904\) 0 0
\(905\) 100.390 3.33707
\(906\) 0 0
\(907\) 32.9700 1.09475 0.547375 0.836887i \(-0.315627\pi\)
0.547375 + 0.836887i \(0.315627\pi\)
\(908\) 0 0
\(909\) −1.84478 −0.0611874
\(910\) 0 0
\(911\) 49.0127 1.62386 0.811931 0.583753i \(-0.198416\pi\)
0.811931 + 0.583753i \(0.198416\pi\)
\(912\) 0 0
\(913\) 2.06260 0.0682622
\(914\) 0 0
\(915\) 20.7019 0.684385
\(916\) 0 0
\(917\) 27.6337 0.912544
\(918\) 0 0
\(919\) 24.1928 0.798048 0.399024 0.916940i \(-0.369349\pi\)
0.399024 + 0.916940i \(0.369349\pi\)
\(920\) 0 0
\(921\) −3.01583 −0.0993751
\(922\) 0 0
\(923\) 56.6219 1.86373
\(924\) 0 0
\(925\) −77.4412 −2.54625
\(926\) 0 0
\(927\) −10.0687 −0.330698
\(928\) 0 0
\(929\) −37.1169 −1.21777 −0.608883 0.793260i \(-0.708382\pi\)
−0.608883 + 0.793260i \(0.708382\pi\)
\(930\) 0 0
\(931\) −29.4263 −0.964409
\(932\) 0 0
\(933\) 7.45318 0.244006
\(934\) 0 0
\(935\) 6.07849 0.198788
\(936\) 0 0
\(937\) −37.2959 −1.21840 −0.609202 0.793015i \(-0.708510\pi\)
−0.609202 + 0.793015i \(0.708510\pi\)
\(938\) 0 0
\(939\) 3.43534 0.112108
\(940\) 0 0
\(941\) 48.0014 1.56480 0.782401 0.622775i \(-0.213995\pi\)
0.782401 + 0.622775i \(0.213995\pi\)
\(942\) 0 0
\(943\) 5.76434 0.187713
\(944\) 0 0
\(945\) 17.2168 0.560064
\(946\) 0 0
\(947\) −32.5481 −1.05767 −0.528835 0.848725i \(-0.677371\pi\)
−0.528835 + 0.848725i \(0.677371\pi\)
\(948\) 0 0
\(949\) 57.0066 1.85051
\(950\) 0 0
\(951\) −0.318803 −0.0103379
\(952\) 0 0
\(953\) −54.1571 −1.75432 −0.877161 0.480196i \(-0.840565\pi\)
−0.877161 + 0.480196i \(0.840565\pi\)
\(954\) 0 0
\(955\) 17.2686 0.558800
\(956\) 0 0
\(957\) −0.229135 −0.00740687
\(958\) 0 0
\(959\) −37.0000 −1.19479
\(960\) 0 0
\(961\) 24.2218 0.781348
\(962\) 0 0
\(963\) −2.97463 −0.0958560
\(964\) 0 0
\(965\) −48.0636 −1.54722
\(966\) 0 0
\(967\) 55.4514 1.78320 0.891598 0.452828i \(-0.149585\pi\)
0.891598 + 0.452828i \(0.149585\pi\)
\(968\) 0 0
\(969\) 18.4673 0.593254
\(970\) 0 0
\(971\) 22.7413 0.729803 0.364902 0.931046i \(-0.381103\pi\)
0.364902 + 0.931046i \(0.381103\pi\)
\(972\) 0 0
\(973\) −65.0197 −2.08444
\(974\) 0 0
\(975\) 50.3007 1.61091
\(976\) 0 0
\(977\) −0.809033 −0.0258833 −0.0129416 0.999916i \(-0.504120\pi\)
−0.0129416 + 0.999916i \(0.504120\pi\)
\(978\) 0 0
\(979\) 1.65560 0.0529132
\(980\) 0 0
\(981\) 17.2466 0.550643
\(982\) 0 0
\(983\) 3.46754 0.110597 0.0552986 0.998470i \(-0.482389\pi\)
0.0552986 + 0.998470i \(0.482389\pi\)
\(984\) 0 0
\(985\) 110.799 3.53036
\(986\) 0 0
\(987\) −28.5498 −0.908749
\(988\) 0 0
\(989\) 0.993037 0.0315767
\(990\) 0 0
\(991\) −34.3123 −1.08997 −0.544984 0.838447i \(-0.683464\pi\)
−0.544984 + 0.838447i \(0.683464\pi\)
\(992\) 0 0
\(993\) −26.0762 −0.827502
\(994\) 0 0
\(995\) 8.92841 0.283050
\(996\) 0 0
\(997\) −19.7495 −0.625473 −0.312736 0.949840i \(-0.601246\pi\)
−0.312736 + 0.949840i \(0.601246\pi\)
\(998\) 0 0
\(999\) 6.31719 0.199867
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 8004.2.a.i.1.16 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.16 16 1.1 even 1 trivial