Properties

Label 8004.2.a.g.1.10
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-2.48147\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.48147 q^{5} -0.0448560 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.48147 q^{5} -0.0448560 q^{7} +1.00000 q^{9} -0.683409 q^{11} -6.72642 q^{13} -2.48147 q^{15} -1.19277 q^{17} +1.53130 q^{19} +0.0448560 q^{21} +1.00000 q^{23} +1.15770 q^{25} -1.00000 q^{27} -1.00000 q^{29} +1.40385 q^{31} +0.683409 q^{33} -0.111309 q^{35} +8.68984 q^{37} +6.72642 q^{39} +11.5300 q^{41} -6.28573 q^{43} +2.48147 q^{45} +5.00128 q^{47} -6.99799 q^{49} +1.19277 q^{51} -1.85106 q^{53} -1.69586 q^{55} -1.53130 q^{57} -2.14821 q^{59} -2.75349 q^{61} -0.0448560 q^{63} -16.6914 q^{65} -0.0568657 q^{67} -1.00000 q^{69} -1.36240 q^{71} -1.19218 q^{73} -1.15770 q^{75} +0.0306550 q^{77} +0.0825347 q^{79} +1.00000 q^{81} +7.83176 q^{83} -2.95982 q^{85} +1.00000 q^{87} -0.473510 q^{89} +0.301720 q^{91} -1.40385 q^{93} +3.79989 q^{95} -5.20158 q^{97} -0.683409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) 2.48147 1.10975 0.554874 0.831934i \(-0.312766\pi\)
0.554874 + 0.831934i \(0.312766\pi\)
\(6\) 0 0
\(7\) −0.0448560 −0.0169540 −0.00847699 0.999964i \(-0.502698\pi\)
−0.00847699 + 0.999964i \(0.502698\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.683409 −0.206056 −0.103028 0.994678i \(-0.532853\pi\)
−0.103028 + 0.994678i \(0.532853\pi\)
\(12\) 0 0
\(13\) −6.72642 −1.86557 −0.932787 0.360429i \(-0.882630\pi\)
−0.932787 + 0.360429i \(0.882630\pi\)
\(14\) 0 0
\(15\) −2.48147 −0.640713
\(16\) 0 0
\(17\) −1.19277 −0.289289 −0.144645 0.989484i \(-0.546204\pi\)
−0.144645 + 0.989484i \(0.546204\pi\)
\(18\) 0 0
\(19\) 1.53130 0.351305 0.175653 0.984452i \(-0.443796\pi\)
0.175653 + 0.984452i \(0.443796\pi\)
\(20\) 0 0
\(21\) 0.0448560 0.00978839
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.15770 0.231541
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 1.40385 0.252139 0.126069 0.992021i \(-0.459764\pi\)
0.126069 + 0.992021i \(0.459764\pi\)
\(32\) 0 0
\(33\) 0.683409 0.118966
\(34\) 0 0
\(35\) −0.111309 −0.0188147
\(36\) 0 0
\(37\) 8.68984 1.42860 0.714301 0.699839i \(-0.246745\pi\)
0.714301 + 0.699839i \(0.246745\pi\)
\(38\) 0 0
\(39\) 6.72642 1.07709
\(40\) 0 0
\(41\) 11.5300 1.80069 0.900345 0.435178i \(-0.143314\pi\)
0.900345 + 0.435178i \(0.143314\pi\)
\(42\) 0 0
\(43\) −6.28573 −0.958565 −0.479283 0.877661i \(-0.659103\pi\)
−0.479283 + 0.877661i \(0.659103\pi\)
\(44\) 0 0
\(45\) 2.48147 0.369916
\(46\) 0 0
\(47\) 5.00128 0.729512 0.364756 0.931103i \(-0.381152\pi\)
0.364756 + 0.931103i \(0.381152\pi\)
\(48\) 0 0
\(49\) −6.99799 −0.999713
\(50\) 0 0
\(51\) 1.19277 0.167021
\(52\) 0 0
\(53\) −1.85106 −0.254263 −0.127131 0.991886i \(-0.540577\pi\)
−0.127131 + 0.991886i \(0.540577\pi\)
\(54\) 0 0
\(55\) −1.69586 −0.228670
\(56\) 0 0
\(57\) −1.53130 −0.202826
\(58\) 0 0
\(59\) −2.14821 −0.279673 −0.139837 0.990175i \(-0.544658\pi\)
−0.139837 + 0.990175i \(0.544658\pi\)
\(60\) 0 0
\(61\) −2.75349 −0.352548 −0.176274 0.984341i \(-0.556404\pi\)
−0.176274 + 0.984341i \(0.556404\pi\)
\(62\) 0 0
\(63\) −0.0448560 −0.00565133
\(64\) 0 0
\(65\) −16.6914 −2.07032
\(66\) 0 0
\(67\) −0.0568657 −0.00694725 −0.00347362 0.999994i \(-0.501106\pi\)
−0.00347362 + 0.999994i \(0.501106\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −1.36240 −0.161687 −0.0808436 0.996727i \(-0.525761\pi\)
−0.0808436 + 0.996727i \(0.525761\pi\)
\(72\) 0 0
\(73\) −1.19218 −0.139534 −0.0697672 0.997563i \(-0.522226\pi\)
−0.0697672 + 0.997563i \(0.522226\pi\)
\(74\) 0 0
\(75\) −1.15770 −0.133680
\(76\) 0 0
\(77\) 0.0306550 0.00349346
\(78\) 0 0
\(79\) 0.0825347 0.00928587 0.00464294 0.999989i \(-0.498522\pi\)
0.00464294 + 0.999989i \(0.498522\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.83176 0.859647 0.429824 0.902913i \(-0.358576\pi\)
0.429824 + 0.902913i \(0.358576\pi\)
\(84\) 0 0
\(85\) −2.95982 −0.321038
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −0.473510 −0.0501920 −0.0250960 0.999685i \(-0.507989\pi\)
−0.0250960 + 0.999685i \(0.507989\pi\)
\(90\) 0 0
\(91\) 0.301720 0.0316289
\(92\) 0 0
\(93\) −1.40385 −0.145572
\(94\) 0 0
\(95\) 3.79989 0.389860
\(96\) 0 0
\(97\) −5.20158 −0.528141 −0.264070 0.964503i \(-0.585065\pi\)
−0.264070 + 0.964503i \(0.585065\pi\)
\(98\) 0 0
\(99\) −0.683409 −0.0686852
\(100\) 0 0
\(101\) −13.4401 −1.33734 −0.668669 0.743560i \(-0.733136\pi\)
−0.668669 + 0.743560i \(0.733136\pi\)
\(102\) 0 0
\(103\) 3.12497 0.307912 0.153956 0.988078i \(-0.450799\pi\)
0.153956 + 0.988078i \(0.450799\pi\)
\(104\) 0 0
\(105\) 0.111309 0.0108626
\(106\) 0 0
\(107\) 2.76781 0.267574 0.133787 0.991010i \(-0.457286\pi\)
0.133787 + 0.991010i \(0.457286\pi\)
\(108\) 0 0
\(109\) −20.7736 −1.98975 −0.994877 0.101095i \(-0.967766\pi\)
−0.994877 + 0.101095i \(0.967766\pi\)
\(110\) 0 0
\(111\) −8.68984 −0.824803
\(112\) 0 0
\(113\) −8.18272 −0.769765 −0.384883 0.922965i \(-0.625758\pi\)
−0.384883 + 0.922965i \(0.625758\pi\)
\(114\) 0 0
\(115\) 2.48147 0.231398
\(116\) 0 0
\(117\) −6.72642 −0.621858
\(118\) 0 0
\(119\) 0.0535029 0.00490460
\(120\) 0 0
\(121\) −10.5330 −0.957541
\(122\) 0 0
\(123\) −11.5300 −1.03963
\(124\) 0 0
\(125\) −9.53455 −0.852796
\(126\) 0 0
\(127\) −7.44035 −0.660225 −0.330112 0.943942i \(-0.607087\pi\)
−0.330112 + 0.943942i \(0.607087\pi\)
\(128\) 0 0
\(129\) 6.28573 0.553428
\(130\) 0 0
\(131\) −17.7398 −1.54993 −0.774965 0.632004i \(-0.782233\pi\)
−0.774965 + 0.632004i \(0.782233\pi\)
\(132\) 0 0
\(133\) −0.0686882 −0.00595603
\(134\) 0 0
\(135\) −2.48147 −0.213571
\(136\) 0 0
\(137\) 1.55498 0.132851 0.0664255 0.997791i \(-0.478841\pi\)
0.0664255 + 0.997791i \(0.478841\pi\)
\(138\) 0 0
\(139\) 4.65605 0.394921 0.197461 0.980311i \(-0.436731\pi\)
0.197461 + 0.980311i \(0.436731\pi\)
\(140\) 0 0
\(141\) −5.00128 −0.421184
\(142\) 0 0
\(143\) 4.59690 0.384412
\(144\) 0 0
\(145\) −2.48147 −0.206075
\(146\) 0 0
\(147\) 6.99799 0.577184
\(148\) 0 0
\(149\) 8.44818 0.692102 0.346051 0.938216i \(-0.387522\pi\)
0.346051 + 0.938216i \(0.387522\pi\)
\(150\) 0 0
\(151\) −19.0547 −1.55065 −0.775325 0.631562i \(-0.782414\pi\)
−0.775325 + 0.631562i \(0.782414\pi\)
\(152\) 0 0
\(153\) −1.19277 −0.0964297
\(154\) 0 0
\(155\) 3.48361 0.279810
\(156\) 0 0
\(157\) −2.91226 −0.232423 −0.116212 0.993224i \(-0.537075\pi\)
−0.116212 + 0.993224i \(0.537075\pi\)
\(158\) 0 0
\(159\) 1.85106 0.146799
\(160\) 0 0
\(161\) −0.0448560 −0.00353515
\(162\) 0 0
\(163\) −13.1417 −1.02934 −0.514670 0.857389i \(-0.672085\pi\)
−0.514670 + 0.857389i \(0.672085\pi\)
\(164\) 0 0
\(165\) 1.69586 0.132023
\(166\) 0 0
\(167\) −16.0649 −1.24314 −0.621570 0.783359i \(-0.713505\pi\)
−0.621570 + 0.783359i \(0.713505\pi\)
\(168\) 0 0
\(169\) 32.2447 2.48036
\(170\) 0 0
\(171\) 1.53130 0.117102
\(172\) 0 0
\(173\) −10.3946 −0.790286 −0.395143 0.918620i \(-0.629305\pi\)
−0.395143 + 0.918620i \(0.629305\pi\)
\(174\) 0 0
\(175\) −0.0519300 −0.00392554
\(176\) 0 0
\(177\) 2.14821 0.161469
\(178\) 0 0
\(179\) 0.127749 0.00954841 0.00477421 0.999989i \(-0.498480\pi\)
0.00477421 + 0.999989i \(0.498480\pi\)
\(180\) 0 0
\(181\) −7.28371 −0.541394 −0.270697 0.962665i \(-0.587254\pi\)
−0.270697 + 0.962665i \(0.587254\pi\)
\(182\) 0 0
\(183\) 2.75349 0.203544
\(184\) 0 0
\(185\) 21.5636 1.58539
\(186\) 0 0
\(187\) 0.815150 0.0596097
\(188\) 0 0
\(189\) 0.0448560 0.00326280
\(190\) 0 0
\(191\) 1.58690 0.114824 0.0574119 0.998351i \(-0.481715\pi\)
0.0574119 + 0.998351i \(0.481715\pi\)
\(192\) 0 0
\(193\) 7.87850 0.567107 0.283553 0.958956i \(-0.408487\pi\)
0.283553 + 0.958956i \(0.408487\pi\)
\(194\) 0 0
\(195\) 16.6914 1.19530
\(196\) 0 0
\(197\) 9.56121 0.681208 0.340604 0.940207i \(-0.389368\pi\)
0.340604 + 0.940207i \(0.389368\pi\)
\(198\) 0 0
\(199\) −10.9842 −0.778647 −0.389323 0.921101i \(-0.627291\pi\)
−0.389323 + 0.921101i \(0.627291\pi\)
\(200\) 0 0
\(201\) 0.0568657 0.00401100
\(202\) 0 0
\(203\) 0.0448560 0.00314828
\(204\) 0 0
\(205\) 28.6115 1.99831
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −1.04651 −0.0723885
\(210\) 0 0
\(211\) −8.04284 −0.553692 −0.276846 0.960914i \(-0.589289\pi\)
−0.276846 + 0.960914i \(0.589289\pi\)
\(212\) 0 0
\(213\) 1.36240 0.0933501
\(214\) 0 0
\(215\) −15.5979 −1.06377
\(216\) 0 0
\(217\) −0.0629711 −0.00427476
\(218\) 0 0
\(219\) 1.19218 0.0805602
\(220\) 0 0
\(221\) 8.02307 0.539690
\(222\) 0 0
\(223\) 26.7333 1.79019 0.895097 0.445871i \(-0.147106\pi\)
0.895097 + 0.445871i \(0.147106\pi\)
\(224\) 0 0
\(225\) 1.15770 0.0771802
\(226\) 0 0
\(227\) 24.0072 1.59342 0.796708 0.604365i \(-0.206573\pi\)
0.796708 + 0.604365i \(0.206573\pi\)
\(228\) 0 0
\(229\) −21.2022 −1.40108 −0.700539 0.713614i \(-0.747057\pi\)
−0.700539 + 0.713614i \(0.747057\pi\)
\(230\) 0 0
\(231\) −0.0306550 −0.00201695
\(232\) 0 0
\(233\) −5.50352 −0.360548 −0.180274 0.983616i \(-0.557698\pi\)
−0.180274 + 0.983616i \(0.557698\pi\)
\(234\) 0 0
\(235\) 12.4105 0.809575
\(236\) 0 0
\(237\) −0.0825347 −0.00536120
\(238\) 0 0
\(239\) 26.0700 1.68633 0.843163 0.537658i \(-0.180691\pi\)
0.843163 + 0.537658i \(0.180691\pi\)
\(240\) 0 0
\(241\) −23.5744 −1.51856 −0.759279 0.650765i \(-0.774448\pi\)
−0.759279 + 0.650765i \(0.774448\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −17.3653 −1.10943
\(246\) 0 0
\(247\) −10.3002 −0.655386
\(248\) 0 0
\(249\) −7.83176 −0.496318
\(250\) 0 0
\(251\) −14.8586 −0.937869 −0.468934 0.883233i \(-0.655362\pi\)
−0.468934 + 0.883233i \(0.655362\pi\)
\(252\) 0 0
\(253\) −0.683409 −0.0429656
\(254\) 0 0
\(255\) 2.95982 0.185351
\(256\) 0 0
\(257\) −6.87765 −0.429016 −0.214508 0.976722i \(-0.568815\pi\)
−0.214508 + 0.976722i \(0.568815\pi\)
\(258\) 0 0
\(259\) −0.389792 −0.0242205
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 21.3741 1.31798 0.658992 0.752150i \(-0.270983\pi\)
0.658992 + 0.752150i \(0.270983\pi\)
\(264\) 0 0
\(265\) −4.59335 −0.282167
\(266\) 0 0
\(267\) 0.473510 0.0289784
\(268\) 0 0
\(269\) −12.5934 −0.767836 −0.383918 0.923367i \(-0.625425\pi\)
−0.383918 + 0.923367i \(0.625425\pi\)
\(270\) 0 0
\(271\) 4.59472 0.279109 0.139555 0.990214i \(-0.455433\pi\)
0.139555 + 0.990214i \(0.455433\pi\)
\(272\) 0 0
\(273\) −0.301720 −0.0182610
\(274\) 0 0
\(275\) −0.791185 −0.0477103
\(276\) 0 0
\(277\) −2.28953 −0.137565 −0.0687823 0.997632i \(-0.521911\pi\)
−0.0687823 + 0.997632i \(0.521911\pi\)
\(278\) 0 0
\(279\) 1.40385 0.0840462
\(280\) 0 0
\(281\) −20.3041 −1.21124 −0.605620 0.795754i \(-0.707075\pi\)
−0.605620 + 0.795754i \(0.707075\pi\)
\(282\) 0 0
\(283\) −2.01310 −0.119666 −0.0598332 0.998208i \(-0.519057\pi\)
−0.0598332 + 0.998208i \(0.519057\pi\)
\(284\) 0 0
\(285\) −3.79989 −0.225086
\(286\) 0 0
\(287\) −0.517192 −0.0305289
\(288\) 0 0
\(289\) −15.5773 −0.916312
\(290\) 0 0
\(291\) 5.20158 0.304922
\(292\) 0 0
\(293\) 14.6955 0.858518 0.429259 0.903181i \(-0.358775\pi\)
0.429259 + 0.903181i \(0.358775\pi\)
\(294\) 0 0
\(295\) −5.33072 −0.310367
\(296\) 0 0
\(297\) 0.683409 0.0396554
\(298\) 0 0
\(299\) −6.72642 −0.388999
\(300\) 0 0
\(301\) 0.281953 0.0162515
\(302\) 0 0
\(303\) 13.4401 0.772112
\(304\) 0 0
\(305\) −6.83271 −0.391240
\(306\) 0 0
\(307\) −24.8416 −1.41779 −0.708893 0.705316i \(-0.750805\pi\)
−0.708893 + 0.705316i \(0.750805\pi\)
\(308\) 0 0
\(309\) −3.12497 −0.177773
\(310\) 0 0
\(311\) −0.957541 −0.0542972 −0.0271486 0.999631i \(-0.508643\pi\)
−0.0271486 + 0.999631i \(0.508643\pi\)
\(312\) 0 0
\(313\) 17.6885 0.999813 0.499907 0.866079i \(-0.333368\pi\)
0.499907 + 0.866079i \(0.333368\pi\)
\(314\) 0 0
\(315\) −0.111309 −0.00627155
\(316\) 0 0
\(317\) −30.3391 −1.70401 −0.852006 0.523533i \(-0.824614\pi\)
−0.852006 + 0.523533i \(0.824614\pi\)
\(318\) 0 0
\(319\) 0.683409 0.0382636
\(320\) 0 0
\(321\) −2.76781 −0.154484
\(322\) 0 0
\(323\) −1.82649 −0.101629
\(324\) 0 0
\(325\) −7.78720 −0.431956
\(326\) 0 0
\(327\) 20.7736 1.14878
\(328\) 0 0
\(329\) −0.224338 −0.0123681
\(330\) 0 0
\(331\) 14.8890 0.818371 0.409186 0.912451i \(-0.365813\pi\)
0.409186 + 0.912451i \(0.365813\pi\)
\(332\) 0 0
\(333\) 8.68984 0.476200
\(334\) 0 0
\(335\) −0.141111 −0.00770970
\(336\) 0 0
\(337\) 18.4693 1.00608 0.503042 0.864262i \(-0.332214\pi\)
0.503042 + 0.864262i \(0.332214\pi\)
\(338\) 0 0
\(339\) 8.18272 0.444424
\(340\) 0 0
\(341\) −0.959404 −0.0519546
\(342\) 0 0
\(343\) 0.627894 0.0339031
\(344\) 0 0
\(345\) −2.48147 −0.133598
\(346\) 0 0
\(347\) 36.2946 1.94840 0.974198 0.225697i \(-0.0724658\pi\)
0.974198 + 0.225697i \(0.0724658\pi\)
\(348\) 0 0
\(349\) 5.14729 0.275528 0.137764 0.990465i \(-0.456008\pi\)
0.137764 + 0.990465i \(0.456008\pi\)
\(350\) 0 0
\(351\) 6.72642 0.359030
\(352\) 0 0
\(353\) −23.6393 −1.25819 −0.629096 0.777327i \(-0.716575\pi\)
−0.629096 + 0.777327i \(0.716575\pi\)
\(354\) 0 0
\(355\) −3.38076 −0.179432
\(356\) 0 0
\(357\) −0.0535029 −0.00283167
\(358\) 0 0
\(359\) 24.7411 1.30578 0.652891 0.757451i \(-0.273556\pi\)
0.652891 + 0.757451i \(0.273556\pi\)
\(360\) 0 0
\(361\) −16.6551 −0.876585
\(362\) 0 0
\(363\) 10.5330 0.552837
\(364\) 0 0
\(365\) −2.95837 −0.154848
\(366\) 0 0
\(367\) −12.6328 −0.659426 −0.329713 0.944081i \(-0.606952\pi\)
−0.329713 + 0.944081i \(0.606952\pi\)
\(368\) 0 0
\(369\) 11.5300 0.600230
\(370\) 0 0
\(371\) 0.0830312 0.00431076
\(372\) 0 0
\(373\) −11.6850 −0.605024 −0.302512 0.953146i \(-0.597825\pi\)
−0.302512 + 0.953146i \(0.597825\pi\)
\(374\) 0 0
\(375\) 9.53455 0.492362
\(376\) 0 0
\(377\) 6.72642 0.346428
\(378\) 0 0
\(379\) 18.8719 0.969384 0.484692 0.874685i \(-0.338932\pi\)
0.484692 + 0.874685i \(0.338932\pi\)
\(380\) 0 0
\(381\) 7.44035 0.381181
\(382\) 0 0
\(383\) 7.17604 0.366678 0.183339 0.983050i \(-0.441309\pi\)
0.183339 + 0.983050i \(0.441309\pi\)
\(384\) 0 0
\(385\) 0.0760696 0.00387687
\(386\) 0 0
\(387\) −6.28573 −0.319522
\(388\) 0 0
\(389\) −23.5320 −1.19312 −0.596560 0.802568i \(-0.703466\pi\)
−0.596560 + 0.802568i \(0.703466\pi\)
\(390\) 0 0
\(391\) −1.19277 −0.0603210
\(392\) 0 0
\(393\) 17.7398 0.894853
\(394\) 0 0
\(395\) 0.204807 0.0103050
\(396\) 0 0
\(397\) −5.90830 −0.296529 −0.148264 0.988948i \(-0.547369\pi\)
−0.148264 + 0.988948i \(0.547369\pi\)
\(398\) 0 0
\(399\) 0.0686882 0.00343871
\(400\) 0 0
\(401\) −33.3654 −1.66619 −0.833095 0.553130i \(-0.813433\pi\)
−0.833095 + 0.553130i \(0.813433\pi\)
\(402\) 0 0
\(403\) −9.44288 −0.470383
\(404\) 0 0
\(405\) 2.48147 0.123305
\(406\) 0 0
\(407\) −5.93872 −0.294371
\(408\) 0 0
\(409\) 16.6673 0.824143 0.412071 0.911152i \(-0.364805\pi\)
0.412071 + 0.911152i \(0.364805\pi\)
\(410\) 0 0
\(411\) −1.55498 −0.0767016
\(412\) 0 0
\(413\) 0.0963602 0.00474157
\(414\) 0 0
\(415\) 19.4343 0.953992
\(416\) 0 0
\(417\) −4.65605 −0.228008
\(418\) 0 0
\(419\) −30.9304 −1.51105 −0.755523 0.655122i \(-0.772617\pi\)
−0.755523 + 0.655122i \(0.772617\pi\)
\(420\) 0 0
\(421\) 28.5383 1.39087 0.695435 0.718589i \(-0.255212\pi\)
0.695435 + 0.718589i \(0.255212\pi\)
\(422\) 0 0
\(423\) 5.00128 0.243171
\(424\) 0 0
\(425\) −1.38087 −0.0669822
\(426\) 0 0
\(427\) 0.123511 0.00597710
\(428\) 0 0
\(429\) −4.59690 −0.221940
\(430\) 0 0
\(431\) −30.6926 −1.47841 −0.739206 0.673480i \(-0.764799\pi\)
−0.739206 + 0.673480i \(0.764799\pi\)
\(432\) 0 0
\(433\) −18.3757 −0.883078 −0.441539 0.897242i \(-0.645567\pi\)
−0.441539 + 0.897242i \(0.645567\pi\)
\(434\) 0 0
\(435\) 2.48147 0.118977
\(436\) 0 0
\(437\) 1.53130 0.0732522
\(438\) 0 0
\(439\) 39.5696 1.88855 0.944277 0.329151i \(-0.106762\pi\)
0.944277 + 0.329151i \(0.106762\pi\)
\(440\) 0 0
\(441\) −6.99799 −0.333238
\(442\) 0 0
\(443\) −30.7869 −1.46273 −0.731365 0.681986i \(-0.761116\pi\)
−0.731365 + 0.681986i \(0.761116\pi\)
\(444\) 0 0
\(445\) −1.17500 −0.0557005
\(446\) 0 0
\(447\) −8.44818 −0.399585
\(448\) 0 0
\(449\) −15.8707 −0.748986 −0.374493 0.927230i \(-0.622183\pi\)
−0.374493 + 0.927230i \(0.622183\pi\)
\(450\) 0 0
\(451\) −7.87973 −0.371042
\(452\) 0 0
\(453\) 19.0547 0.895269
\(454\) 0 0
\(455\) 0.748711 0.0351001
\(456\) 0 0
\(457\) 11.4522 0.535710 0.267855 0.963459i \(-0.413685\pi\)
0.267855 + 0.963459i \(0.413685\pi\)
\(458\) 0 0
\(459\) 1.19277 0.0556737
\(460\) 0 0
\(461\) −0.768140 −0.0357759 −0.0178879 0.999840i \(-0.505694\pi\)
−0.0178879 + 0.999840i \(0.505694\pi\)
\(462\) 0 0
\(463\) 4.41811 0.205327 0.102663 0.994716i \(-0.467264\pi\)
0.102663 + 0.994716i \(0.467264\pi\)
\(464\) 0 0
\(465\) −3.48361 −0.161549
\(466\) 0 0
\(467\) −14.1128 −0.653060 −0.326530 0.945187i \(-0.605879\pi\)
−0.326530 + 0.945187i \(0.605879\pi\)
\(468\) 0 0
\(469\) 0.00255077 0.000117784 0
\(470\) 0 0
\(471\) 2.91226 0.134190
\(472\) 0 0
\(473\) 4.29573 0.197518
\(474\) 0 0
\(475\) 1.77280 0.0813415
\(476\) 0 0
\(477\) −1.85106 −0.0847542
\(478\) 0 0
\(479\) −11.8727 −0.542480 −0.271240 0.962512i \(-0.587434\pi\)
−0.271240 + 0.962512i \(0.587434\pi\)
\(480\) 0 0
\(481\) −58.4515 −2.66516
\(482\) 0 0
\(483\) 0.0448560 0.00204102
\(484\) 0 0
\(485\) −12.9076 −0.586103
\(486\) 0 0
\(487\) 17.7138 0.802687 0.401343 0.915928i \(-0.368543\pi\)
0.401343 + 0.915928i \(0.368543\pi\)
\(488\) 0 0
\(489\) 13.1417 0.594289
\(490\) 0 0
\(491\) 39.0560 1.76257 0.881287 0.472581i \(-0.156678\pi\)
0.881287 + 0.472581i \(0.156678\pi\)
\(492\) 0 0
\(493\) 1.19277 0.0537196
\(494\) 0 0
\(495\) −1.69586 −0.0762233
\(496\) 0 0
\(497\) 0.0611119 0.00274124
\(498\) 0 0
\(499\) −35.9240 −1.60818 −0.804091 0.594507i \(-0.797347\pi\)
−0.804091 + 0.594507i \(0.797347\pi\)
\(500\) 0 0
\(501\) 16.0649 0.717727
\(502\) 0 0
\(503\) 33.3600 1.48745 0.743724 0.668487i \(-0.233058\pi\)
0.743724 + 0.668487i \(0.233058\pi\)
\(504\) 0 0
\(505\) −33.3512 −1.48411
\(506\) 0 0
\(507\) −32.2447 −1.43204
\(508\) 0 0
\(509\) −20.7899 −0.921497 −0.460748 0.887531i \(-0.652419\pi\)
−0.460748 + 0.887531i \(0.652419\pi\)
\(510\) 0 0
\(511\) 0.0534766 0.00236566
\(512\) 0 0
\(513\) −1.53130 −0.0676087
\(514\) 0 0
\(515\) 7.75452 0.341705
\(516\) 0 0
\(517\) −3.41792 −0.150320
\(518\) 0 0
\(519\) 10.3946 0.456272
\(520\) 0 0
\(521\) −27.7417 −1.21539 −0.607693 0.794172i \(-0.707905\pi\)
−0.607693 + 0.794172i \(0.707905\pi\)
\(522\) 0 0
\(523\) 24.1577 1.05634 0.528170 0.849139i \(-0.322878\pi\)
0.528170 + 0.849139i \(0.322878\pi\)
\(524\) 0 0
\(525\) 0.0519300 0.00226641
\(526\) 0 0
\(527\) −1.67447 −0.0729410
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.14821 −0.0932244
\(532\) 0 0
\(533\) −77.5559 −3.35932
\(534\) 0 0
\(535\) 6.86823 0.296940
\(536\) 0 0
\(537\) −0.127749 −0.00551278
\(538\) 0 0
\(539\) 4.78249 0.205996
\(540\) 0 0
\(541\) −27.5006 −1.18234 −0.591171 0.806546i \(-0.701334\pi\)
−0.591171 + 0.806546i \(0.701334\pi\)
\(542\) 0 0
\(543\) 7.28371 0.312574
\(544\) 0 0
\(545\) −51.5492 −2.20813
\(546\) 0 0
\(547\) 29.6516 1.26781 0.633905 0.773411i \(-0.281451\pi\)
0.633905 + 0.773411i \(0.281451\pi\)
\(548\) 0 0
\(549\) −2.75349 −0.117516
\(550\) 0 0
\(551\) −1.53130 −0.0652358
\(552\) 0 0
\(553\) −0.00370218 −0.000157433 0
\(554\) 0 0
\(555\) −21.5636 −0.915324
\(556\) 0 0
\(557\) 14.8913 0.630965 0.315483 0.948931i \(-0.397834\pi\)
0.315483 + 0.948931i \(0.397834\pi\)
\(558\) 0 0
\(559\) 42.2805 1.78827
\(560\) 0 0
\(561\) −0.815150 −0.0344157
\(562\) 0 0
\(563\) −33.4014 −1.40770 −0.703850 0.710348i \(-0.748537\pi\)
−0.703850 + 0.710348i \(0.748537\pi\)
\(564\) 0 0
\(565\) −20.3052 −0.854246
\(566\) 0 0
\(567\) −0.0448560 −0.00188378
\(568\) 0 0
\(569\) −33.3289 −1.39722 −0.698611 0.715502i \(-0.746198\pi\)
−0.698611 + 0.715502i \(0.746198\pi\)
\(570\) 0 0
\(571\) −2.86311 −0.119818 −0.0599088 0.998204i \(-0.519081\pi\)
−0.0599088 + 0.998204i \(0.519081\pi\)
\(572\) 0 0
\(573\) −1.58690 −0.0662936
\(574\) 0 0
\(575\) 1.15770 0.0482796
\(576\) 0 0
\(577\) 35.6396 1.48370 0.741848 0.670568i \(-0.233949\pi\)
0.741848 + 0.670568i \(0.233949\pi\)
\(578\) 0 0
\(579\) −7.87850 −0.327419
\(580\) 0 0
\(581\) −0.351302 −0.0145744
\(582\) 0 0
\(583\) 1.26503 0.0523922
\(584\) 0 0
\(585\) −16.6914 −0.690105
\(586\) 0 0
\(587\) 6.66341 0.275029 0.137514 0.990500i \(-0.456089\pi\)
0.137514 + 0.990500i \(0.456089\pi\)
\(588\) 0 0
\(589\) 2.14972 0.0885777
\(590\) 0 0
\(591\) −9.56121 −0.393296
\(592\) 0 0
\(593\) −35.6118 −1.46240 −0.731201 0.682162i \(-0.761040\pi\)
−0.731201 + 0.682162i \(0.761040\pi\)
\(594\) 0 0
\(595\) 0.132766 0.00544287
\(596\) 0 0
\(597\) 10.9842 0.449552
\(598\) 0 0
\(599\) −9.38122 −0.383306 −0.191653 0.981463i \(-0.561385\pi\)
−0.191653 + 0.981463i \(0.561385\pi\)
\(600\) 0 0
\(601\) 33.4516 1.36452 0.682259 0.731111i \(-0.260998\pi\)
0.682259 + 0.731111i \(0.260998\pi\)
\(602\) 0 0
\(603\) −0.0568657 −0.00231575
\(604\) 0 0
\(605\) −26.1372 −1.06263
\(606\) 0 0
\(607\) −15.7731 −0.640208 −0.320104 0.947382i \(-0.603718\pi\)
−0.320104 + 0.947382i \(0.603718\pi\)
\(608\) 0 0
\(609\) −0.0448560 −0.00181766
\(610\) 0 0
\(611\) −33.6407 −1.36096
\(612\) 0 0
\(613\) −4.26903 −0.172424 −0.0862122 0.996277i \(-0.527476\pi\)
−0.0862122 + 0.996277i \(0.527476\pi\)
\(614\) 0 0
\(615\) −28.6115 −1.15373
\(616\) 0 0
\(617\) 24.0080 0.966527 0.483263 0.875475i \(-0.339451\pi\)
0.483263 + 0.875475i \(0.339451\pi\)
\(618\) 0 0
\(619\) −30.9394 −1.24356 −0.621780 0.783192i \(-0.713590\pi\)
−0.621780 + 0.783192i \(0.713590\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0.0212398 0.000850954 0
\(624\) 0 0
\(625\) −29.4482 −1.17793
\(626\) 0 0
\(627\) 1.04651 0.0417935
\(628\) 0 0
\(629\) −10.3650 −0.413279
\(630\) 0 0
\(631\) −23.3669 −0.930222 −0.465111 0.885252i \(-0.653986\pi\)
−0.465111 + 0.885252i \(0.653986\pi\)
\(632\) 0 0
\(633\) 8.04284 0.319674
\(634\) 0 0
\(635\) −18.4630 −0.732683
\(636\) 0 0
\(637\) 47.0714 1.86504
\(638\) 0 0
\(639\) −1.36240 −0.0538957
\(640\) 0 0
\(641\) −17.1059 −0.675641 −0.337820 0.941211i \(-0.609690\pi\)
−0.337820 + 0.941211i \(0.609690\pi\)
\(642\) 0 0
\(643\) −12.0493 −0.475179 −0.237589 0.971366i \(-0.576357\pi\)
−0.237589 + 0.971366i \(0.576357\pi\)
\(644\) 0 0
\(645\) 15.5979 0.614166
\(646\) 0 0
\(647\) 18.7704 0.737940 0.368970 0.929441i \(-0.379711\pi\)
0.368970 + 0.929441i \(0.379711\pi\)
\(648\) 0 0
\(649\) 1.46811 0.0576282
\(650\) 0 0
\(651\) 0.0629711 0.00246803
\(652\) 0 0
\(653\) −15.6034 −0.610607 −0.305303 0.952255i \(-0.598758\pi\)
−0.305303 + 0.952255i \(0.598758\pi\)
\(654\) 0 0
\(655\) −44.0207 −1.72003
\(656\) 0 0
\(657\) −1.19218 −0.0465115
\(658\) 0 0
\(659\) 14.8795 0.579623 0.289812 0.957084i \(-0.406407\pi\)
0.289812 + 0.957084i \(0.406407\pi\)
\(660\) 0 0
\(661\) −4.34822 −0.169126 −0.0845630 0.996418i \(-0.526949\pi\)
−0.0845630 + 0.996418i \(0.526949\pi\)
\(662\) 0 0
\(663\) −8.02307 −0.311590
\(664\) 0 0
\(665\) −0.170448 −0.00660969
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −26.7333 −1.03357
\(670\) 0 0
\(671\) 1.88176 0.0726445
\(672\) 0 0
\(673\) 21.5468 0.830567 0.415284 0.909692i \(-0.363682\pi\)
0.415284 + 0.909692i \(0.363682\pi\)
\(674\) 0 0
\(675\) −1.15770 −0.0445600
\(676\) 0 0
\(677\) 32.0704 1.23257 0.616283 0.787525i \(-0.288638\pi\)
0.616283 + 0.787525i \(0.288638\pi\)
\(678\) 0 0
\(679\) 0.233322 0.00895409
\(680\) 0 0
\(681\) −24.0072 −0.919959
\(682\) 0 0
\(683\) 18.8385 0.720836 0.360418 0.932791i \(-0.382634\pi\)
0.360418 + 0.932791i \(0.382634\pi\)
\(684\) 0 0
\(685\) 3.85864 0.147431
\(686\) 0 0
\(687\) 21.2022 0.808913
\(688\) 0 0
\(689\) 12.4510 0.474345
\(690\) 0 0
\(691\) −11.6594 −0.443544 −0.221772 0.975099i \(-0.571184\pi\)
−0.221772 + 0.975099i \(0.571184\pi\)
\(692\) 0 0
\(693\) 0.0306550 0.00116449
\(694\) 0 0
\(695\) 11.5539 0.438263
\(696\) 0 0
\(697\) −13.7527 −0.520920
\(698\) 0 0
\(699\) 5.50352 0.208162
\(700\) 0 0
\(701\) 29.2820 1.10597 0.552983 0.833192i \(-0.313489\pi\)
0.552983 + 0.833192i \(0.313489\pi\)
\(702\) 0 0
\(703\) 13.3068 0.501875
\(704\) 0 0
\(705\) −12.4105 −0.467408
\(706\) 0 0
\(707\) 0.602869 0.0226732
\(708\) 0 0
\(709\) −48.8523 −1.83469 −0.917343 0.398097i \(-0.869671\pi\)
−0.917343 + 0.398097i \(0.869671\pi\)
\(710\) 0 0
\(711\) 0.0825347 0.00309529
\(712\) 0 0
\(713\) 1.40385 0.0525746
\(714\) 0 0
\(715\) 11.4071 0.426600
\(716\) 0 0
\(717\) −26.0700 −0.973601
\(718\) 0 0
\(719\) −21.3601 −0.796597 −0.398299 0.917256i \(-0.630399\pi\)
−0.398299 + 0.917256i \(0.630399\pi\)
\(720\) 0 0
\(721\) −0.140174 −0.00522034
\(722\) 0 0
\(723\) 23.5744 0.876740
\(724\) 0 0
\(725\) −1.15770 −0.0429960
\(726\) 0 0
\(727\) 18.2219 0.675814 0.337907 0.941180i \(-0.390281\pi\)
0.337907 + 0.941180i \(0.390281\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.49743 0.277303
\(732\) 0 0
\(733\) −11.7818 −0.435169 −0.217584 0.976042i \(-0.569818\pi\)
−0.217584 + 0.976042i \(0.569818\pi\)
\(734\) 0 0
\(735\) 17.3653 0.640529
\(736\) 0 0
\(737\) 0.0388625 0.00143152
\(738\) 0 0
\(739\) 22.3717 0.822958 0.411479 0.911419i \(-0.365012\pi\)
0.411479 + 0.911419i \(0.365012\pi\)
\(740\) 0 0
\(741\) 10.3002 0.378387
\(742\) 0 0
\(743\) −39.4351 −1.44673 −0.723367 0.690464i \(-0.757407\pi\)
−0.723367 + 0.690464i \(0.757407\pi\)
\(744\) 0 0
\(745\) 20.9639 0.768059
\(746\) 0 0
\(747\) 7.83176 0.286549
\(748\) 0 0
\(749\) −0.124153 −0.00453644
\(750\) 0 0
\(751\) −27.4160 −1.00042 −0.500211 0.865903i \(-0.666744\pi\)
−0.500211 + 0.865903i \(0.666744\pi\)
\(752\) 0 0
\(753\) 14.8586 0.541479
\(754\) 0 0
\(755\) −47.2837 −1.72083
\(756\) 0 0
\(757\) −22.2228 −0.807700 −0.403850 0.914825i \(-0.632328\pi\)
−0.403850 + 0.914825i \(0.632328\pi\)
\(758\) 0 0
\(759\) 0.683409 0.0248062
\(760\) 0 0
\(761\) 38.5659 1.39801 0.699006 0.715116i \(-0.253626\pi\)
0.699006 + 0.715116i \(0.253626\pi\)
\(762\) 0 0
\(763\) 0.931823 0.0337343
\(764\) 0 0
\(765\) −2.95982 −0.107013
\(766\) 0 0
\(767\) 14.4498 0.521751
\(768\) 0 0
\(769\) −10.9131 −0.393535 −0.196768 0.980450i \(-0.563044\pi\)
−0.196768 + 0.980450i \(0.563044\pi\)
\(770\) 0 0
\(771\) 6.87765 0.247692
\(772\) 0 0
\(773\) −11.0493 −0.397416 −0.198708 0.980059i \(-0.563675\pi\)
−0.198708 + 0.980059i \(0.563675\pi\)
\(774\) 0 0
\(775\) 1.62524 0.0583804
\(776\) 0 0
\(777\) 0.389792 0.0139837
\(778\) 0 0
\(779\) 17.6560 0.632592
\(780\) 0 0
\(781\) 0.931077 0.0333166
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −7.22668 −0.257931
\(786\) 0 0
\(787\) 43.2832 1.54288 0.771440 0.636301i \(-0.219537\pi\)
0.771440 + 0.636301i \(0.219537\pi\)
\(788\) 0 0
\(789\) −21.3741 −0.760939
\(790\) 0 0
\(791\) 0.367044 0.0130506
\(792\) 0 0
\(793\) 18.5211 0.657704
\(794\) 0 0
\(795\) 4.59335 0.162909
\(796\) 0 0
\(797\) −12.8858 −0.456437 −0.228218 0.973610i \(-0.573290\pi\)
−0.228218 + 0.973610i \(0.573290\pi\)
\(798\) 0 0
\(799\) −5.96538 −0.211040
\(800\) 0 0
\(801\) −0.473510 −0.0167307
\(802\) 0 0
\(803\) 0.814748 0.0287518
\(804\) 0 0
\(805\) −0.111309 −0.00392313
\(806\) 0 0
\(807\) 12.5934 0.443310
\(808\) 0 0
\(809\) −3.36974 −0.118474 −0.0592368 0.998244i \(-0.518867\pi\)
−0.0592368 + 0.998244i \(0.518867\pi\)
\(810\) 0 0
\(811\) −12.5650 −0.441218 −0.220609 0.975362i \(-0.570805\pi\)
−0.220609 + 0.975362i \(0.570805\pi\)
\(812\) 0 0
\(813\) −4.59472 −0.161144
\(814\) 0 0
\(815\) −32.6108 −1.14231
\(816\) 0 0
\(817\) −9.62537 −0.336749
\(818\) 0 0
\(819\) 0.301720 0.0105430
\(820\) 0 0
\(821\) 36.1970 1.26328 0.631642 0.775261i \(-0.282381\pi\)
0.631642 + 0.775261i \(0.282381\pi\)
\(822\) 0 0
\(823\) 12.1322 0.422903 0.211452 0.977388i \(-0.432181\pi\)
0.211452 + 0.977388i \(0.432181\pi\)
\(824\) 0 0
\(825\) 0.791185 0.0275455
\(826\) 0 0
\(827\) −0.718101 −0.0249708 −0.0124854 0.999922i \(-0.503974\pi\)
−0.0124854 + 0.999922i \(0.503974\pi\)
\(828\) 0 0
\(829\) 12.3297 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(830\) 0 0
\(831\) 2.28953 0.0794230
\(832\) 0 0
\(833\) 8.34699 0.289206
\(834\) 0 0
\(835\) −39.8646 −1.37957
\(836\) 0 0
\(837\) −1.40385 −0.0485241
\(838\) 0 0
\(839\) 29.8107 1.02918 0.514590 0.857437i \(-0.327944\pi\)
0.514590 + 0.857437i \(0.327944\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 20.3041 0.699310
\(844\) 0 0
\(845\) 80.0144 2.75258
\(846\) 0 0
\(847\) 0.472466 0.0162341
\(848\) 0 0
\(849\) 2.01310 0.0690894
\(850\) 0 0
\(851\) 8.68984 0.297884
\(852\) 0 0
\(853\) 16.8950 0.578474 0.289237 0.957258i \(-0.406598\pi\)
0.289237 + 0.957258i \(0.406598\pi\)
\(854\) 0 0
\(855\) 3.79989 0.129953
\(856\) 0 0
\(857\) 13.3776 0.456970 0.228485 0.973547i \(-0.426623\pi\)
0.228485 + 0.973547i \(0.426623\pi\)
\(858\) 0 0
\(859\) 29.9155 1.02070 0.510352 0.859966i \(-0.329515\pi\)
0.510352 + 0.859966i \(0.329515\pi\)
\(860\) 0 0
\(861\) 0.517192 0.0176258
\(862\) 0 0
\(863\) −23.9042 −0.813709 −0.406854 0.913493i \(-0.633374\pi\)
−0.406854 + 0.913493i \(0.633374\pi\)
\(864\) 0 0
\(865\) −25.7939 −0.877018
\(866\) 0 0
\(867\) 15.5773 0.529033
\(868\) 0 0
\(869\) −0.0564050 −0.00191341
\(870\) 0 0
\(871\) 0.382502 0.0129606
\(872\) 0 0
\(873\) −5.20158 −0.176047
\(874\) 0 0
\(875\) 0.427682 0.0144583
\(876\) 0 0
\(877\) 22.7542 0.768356 0.384178 0.923259i \(-0.374485\pi\)
0.384178 + 0.923259i \(0.374485\pi\)
\(878\) 0 0
\(879\) −14.6955 −0.495666
\(880\) 0 0
\(881\) 35.2527 1.18769 0.593846 0.804579i \(-0.297609\pi\)
0.593846 + 0.804579i \(0.297609\pi\)
\(882\) 0 0
\(883\) 20.1303 0.677437 0.338719 0.940888i \(-0.390007\pi\)
0.338719 + 0.940888i \(0.390007\pi\)
\(884\) 0 0
\(885\) 5.33072 0.179190
\(886\) 0 0
\(887\) 28.5084 0.957219 0.478610 0.878028i \(-0.341141\pi\)
0.478610 + 0.878028i \(0.341141\pi\)
\(888\) 0 0
\(889\) 0.333745 0.0111934
\(890\) 0 0
\(891\) −0.683409 −0.0228951
\(892\) 0 0
\(893\) 7.65849 0.256282
\(894\) 0 0
\(895\) 0.317006 0.0105963
\(896\) 0 0
\(897\) 6.72642 0.224589
\(898\) 0 0
\(899\) −1.40385 −0.0468210
\(900\) 0 0
\(901\) 2.20789 0.0735554
\(902\) 0 0
\(903\) −0.281953 −0.00938281
\(904\) 0 0
\(905\) −18.0743 −0.600811
\(906\) 0 0
\(907\) −3.24488 −0.107744 −0.0538722 0.998548i \(-0.517156\pi\)
−0.0538722 + 0.998548i \(0.517156\pi\)
\(908\) 0 0
\(909\) −13.4401 −0.445779
\(910\) 0 0
\(911\) −50.9154 −1.68690 −0.843451 0.537207i \(-0.819479\pi\)
−0.843451 + 0.537207i \(0.819479\pi\)
\(912\) 0 0
\(913\) −5.35230 −0.177135
\(914\) 0 0
\(915\) 6.83271 0.225882
\(916\) 0 0
\(917\) 0.795735 0.0262775
\(918\) 0 0
\(919\) −25.6095 −0.844778 −0.422389 0.906415i \(-0.638808\pi\)
−0.422389 + 0.906415i \(0.638808\pi\)
\(920\) 0 0
\(921\) 24.8416 0.818559
\(922\) 0 0
\(923\) 9.16407 0.301639
\(924\) 0 0
\(925\) 10.0603 0.330779
\(926\) 0 0
\(927\) 3.12497 0.102637
\(928\) 0 0
\(929\) −4.01347 −0.131678 −0.0658388 0.997830i \(-0.520972\pi\)
−0.0658388 + 0.997830i \(0.520972\pi\)
\(930\) 0 0
\(931\) −10.7161 −0.351204
\(932\) 0 0
\(933\) 0.957541 0.0313485
\(934\) 0 0
\(935\) 2.02277 0.0661517
\(936\) 0 0
\(937\) 46.3819 1.51523 0.757615 0.652702i \(-0.226365\pi\)
0.757615 + 0.652702i \(0.226365\pi\)
\(938\) 0 0
\(939\) −17.6885 −0.577242
\(940\) 0 0
\(941\) 11.1442 0.363292 0.181646 0.983364i \(-0.441858\pi\)
0.181646 + 0.983364i \(0.441858\pi\)
\(942\) 0 0
\(943\) 11.5300 0.375470
\(944\) 0 0
\(945\) 0.111309 0.00362088
\(946\) 0 0
\(947\) 16.6491 0.541022 0.270511 0.962717i \(-0.412807\pi\)
0.270511 + 0.962717i \(0.412807\pi\)
\(948\) 0 0
\(949\) 8.01912 0.260312
\(950\) 0 0
\(951\) 30.3391 0.983811
\(952\) 0 0
\(953\) −47.7628 −1.54719 −0.773594 0.633681i \(-0.781543\pi\)
−0.773594 + 0.633681i \(0.781543\pi\)
\(954\) 0 0
\(955\) 3.93784 0.127426
\(956\) 0 0
\(957\) −0.683409 −0.0220915
\(958\) 0 0
\(959\) −0.0697503 −0.00225235
\(960\) 0 0
\(961\) −29.0292 −0.936426
\(962\) 0 0
\(963\) 2.76781 0.0891913
\(964\) 0 0
\(965\) 19.5503 0.629345
\(966\) 0 0
\(967\) −45.5564 −1.46499 −0.732497 0.680771i \(-0.761645\pi\)
−0.732497 + 0.680771i \(0.761645\pi\)
\(968\) 0 0
\(969\) 1.82649 0.0586754
\(970\) 0 0
\(971\) −41.9958 −1.34771 −0.673854 0.738864i \(-0.735362\pi\)
−0.673854 + 0.738864i \(0.735362\pi\)
\(972\) 0 0
\(973\) −0.208852 −0.00669549
\(974\) 0 0
\(975\) 7.78720 0.249390
\(976\) 0 0
\(977\) −44.1202 −1.41153 −0.705765 0.708446i \(-0.749397\pi\)
−0.705765 + 0.708446i \(0.749397\pi\)
\(978\) 0 0
\(979\) 0.323601 0.0103423
\(980\) 0 0
\(981\) −20.7736 −0.663251
\(982\) 0 0
\(983\) −16.0584 −0.512185 −0.256092 0.966652i \(-0.582435\pi\)
−0.256092 + 0.966652i \(0.582435\pi\)
\(984\) 0 0
\(985\) 23.7259 0.755969
\(986\) 0 0
\(987\) 0.224338 0.00714075
\(988\) 0 0
\(989\) −6.28573 −0.199875
\(990\) 0 0
\(991\) 50.4668 1.60313 0.801565 0.597907i \(-0.204001\pi\)
0.801565 + 0.597907i \(0.204001\pi\)
\(992\) 0 0
\(993\) −14.8890 −0.472487
\(994\) 0 0
\(995\) −27.2569 −0.864102
\(996\) 0 0
\(997\) 19.0699 0.603950 0.301975 0.953316i \(-0.402354\pi\)
0.301975 + 0.953316i \(0.402354\pi\)
\(998\) 0 0
\(999\) −8.68984 −0.274934
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.g.1.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.10 12 1.1 even 1 trivial