Properties

Label 8004.2.a.g.1.6
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 1.6
Root \(0.305886\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.305886 q^{5} -0.140026 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.305886 q^{5} -0.140026 q^{7} +1.00000 q^{9} +1.85849 q^{11} -1.94243 q^{13} +0.305886 q^{15} +5.66444 q^{17} -1.04650 q^{19} +0.140026 q^{21} +1.00000 q^{23} -4.90643 q^{25} -1.00000 q^{27} -1.00000 q^{29} -6.72475 q^{31} -1.85849 q^{33} +0.0428320 q^{35} -1.42575 q^{37} +1.94243 q^{39} +0.522747 q^{41} +4.27739 q^{43} -0.305886 q^{45} +7.07300 q^{47} -6.98039 q^{49} -5.66444 q^{51} -11.0449 q^{53} -0.568487 q^{55} +1.04650 q^{57} -10.0473 q^{59} +7.67091 q^{61} -0.140026 q^{63} +0.594162 q^{65} -12.0677 q^{67} -1.00000 q^{69} +8.29946 q^{71} -0.0686045 q^{73} +4.90643 q^{75} -0.260237 q^{77} +8.58428 q^{79} +1.00000 q^{81} +7.65463 q^{83} -1.73267 q^{85} +1.00000 q^{87} +8.93734 q^{89} +0.271991 q^{91} +6.72475 q^{93} +0.320111 q^{95} +4.43366 q^{97} +1.85849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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
\(5\) −0.305886 −0.136796 −0.0683982 0.997658i \(-0.521789\pi\)
−0.0683982 + 0.997658i \(0.521789\pi\)
\(6\) 0 0
\(7\) −0.140026 −0.0529249 −0.0264624 0.999650i \(-0.508424\pi\)
−0.0264624 + 0.999650i \(0.508424\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.85849 0.560356 0.280178 0.959948i \(-0.409606\pi\)
0.280178 + 0.959948i \(0.409606\pi\)
\(12\) 0 0
\(13\) −1.94243 −0.538733 −0.269367 0.963038i \(-0.586814\pi\)
−0.269367 + 0.963038i \(0.586814\pi\)
\(14\) 0 0
\(15\) 0.305886 0.0789795
\(16\) 0 0
\(17\) 5.66444 1.37383 0.686914 0.726738i \(-0.258965\pi\)
0.686914 + 0.726738i \(0.258965\pi\)
\(18\) 0 0
\(19\) −1.04650 −0.240084 −0.120042 0.992769i \(-0.538303\pi\)
−0.120042 + 0.992769i \(0.538303\pi\)
\(20\) 0 0
\(21\) 0.140026 0.0305562
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.90643 −0.981287
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −6.72475 −1.20780 −0.603901 0.797060i \(-0.706388\pi\)
−0.603901 + 0.797060i \(0.706388\pi\)
\(32\) 0 0
\(33\) −1.85849 −0.323522
\(34\) 0 0
\(35\) 0.0428320 0.00723993
\(36\) 0 0
\(37\) −1.42575 −0.234392 −0.117196 0.993109i \(-0.537391\pi\)
−0.117196 + 0.993109i \(0.537391\pi\)
\(38\) 0 0
\(39\) 1.94243 0.311038
\(40\) 0 0
\(41\) 0.522747 0.0816394 0.0408197 0.999167i \(-0.487003\pi\)
0.0408197 + 0.999167i \(0.487003\pi\)
\(42\) 0 0
\(43\) 4.27739 0.652296 0.326148 0.945319i \(-0.394249\pi\)
0.326148 + 0.945319i \(0.394249\pi\)
\(44\) 0 0
\(45\) −0.305886 −0.0455988
\(46\) 0 0
\(47\) 7.07300 1.03170 0.515852 0.856678i \(-0.327476\pi\)
0.515852 + 0.856678i \(0.327476\pi\)
\(48\) 0 0
\(49\) −6.98039 −0.997199
\(50\) 0 0
\(51\) −5.66444 −0.793180
\(52\) 0 0
\(53\) −11.0449 −1.51713 −0.758567 0.651595i \(-0.774100\pi\)
−0.758567 + 0.651595i \(0.774100\pi\)
\(54\) 0 0
\(55\) −0.568487 −0.0766548
\(56\) 0 0
\(57\) 1.04650 0.138613
\(58\) 0 0
\(59\) −10.0473 −1.30804 −0.654021 0.756476i \(-0.726919\pi\)
−0.654021 + 0.756476i \(0.726919\pi\)
\(60\) 0 0
\(61\) 7.67091 0.982159 0.491080 0.871115i \(-0.336602\pi\)
0.491080 + 0.871115i \(0.336602\pi\)
\(62\) 0 0
\(63\) −0.140026 −0.0176416
\(64\) 0 0
\(65\) 0.594162 0.0736968
\(66\) 0 0
\(67\) −12.0677 −1.47430 −0.737149 0.675730i \(-0.763829\pi\)
−0.737149 + 0.675730i \(0.763829\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 8.29946 0.984965 0.492482 0.870322i \(-0.336090\pi\)
0.492482 + 0.870322i \(0.336090\pi\)
\(72\) 0 0
\(73\) −0.0686045 −0.00802955 −0.00401478 0.999992i \(-0.501278\pi\)
−0.00401478 + 0.999992i \(0.501278\pi\)
\(74\) 0 0
\(75\) 4.90643 0.566546
\(76\) 0 0
\(77\) −0.260237 −0.0296568
\(78\) 0 0
\(79\) 8.58428 0.965807 0.482903 0.875674i \(-0.339582\pi\)
0.482903 + 0.875674i \(0.339582\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.65463 0.840205 0.420102 0.907477i \(-0.361994\pi\)
0.420102 + 0.907477i \(0.361994\pi\)
\(84\) 0 0
\(85\) −1.73267 −0.187935
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 8.93734 0.947357 0.473678 0.880698i \(-0.342926\pi\)
0.473678 + 0.880698i \(0.342926\pi\)
\(90\) 0 0
\(91\) 0.271991 0.0285124
\(92\) 0 0
\(93\) 6.72475 0.697324
\(94\) 0 0
\(95\) 0.320111 0.0328427
\(96\) 0 0
\(97\) 4.43366 0.450170 0.225085 0.974339i \(-0.427734\pi\)
0.225085 + 0.974339i \(0.427734\pi\)
\(98\) 0 0
\(99\) 1.85849 0.186785
\(100\) 0 0
\(101\) 5.01727 0.499237 0.249618 0.968344i \(-0.419695\pi\)
0.249618 + 0.968344i \(0.419695\pi\)
\(102\) 0 0
\(103\) −13.5050 −1.33068 −0.665341 0.746539i \(-0.731714\pi\)
−0.665341 + 0.746539i \(0.731714\pi\)
\(104\) 0 0
\(105\) −0.0428320 −0.00417998
\(106\) 0 0
\(107\) −10.1993 −0.986007 −0.493003 0.870027i \(-0.664101\pi\)
−0.493003 + 0.870027i \(0.664101\pi\)
\(108\) 0 0
\(109\) 0.150542 0.0144193 0.00720965 0.999974i \(-0.497705\pi\)
0.00720965 + 0.999974i \(0.497705\pi\)
\(110\) 0 0
\(111\) 1.42575 0.135327
\(112\) 0 0
\(113\) 11.3408 1.06685 0.533427 0.845846i \(-0.320904\pi\)
0.533427 + 0.845846i \(0.320904\pi\)
\(114\) 0 0
\(115\) −0.305886 −0.0285240
\(116\) 0 0
\(117\) −1.94243 −0.179578
\(118\) 0 0
\(119\) −0.793169 −0.0727097
\(120\) 0 0
\(121\) −7.54601 −0.686001
\(122\) 0 0
\(123\) −0.522747 −0.0471345
\(124\) 0 0
\(125\) 3.03024 0.271033
\(126\) 0 0
\(127\) 4.72184 0.418995 0.209498 0.977809i \(-0.432817\pi\)
0.209498 + 0.977809i \(0.432817\pi\)
\(128\) 0 0
\(129\) −4.27739 −0.376603
\(130\) 0 0
\(131\) −11.9635 −1.04526 −0.522629 0.852560i \(-0.675049\pi\)
−0.522629 + 0.852560i \(0.675049\pi\)
\(132\) 0 0
\(133\) 0.146538 0.0127064
\(134\) 0 0
\(135\) 0.305886 0.0263265
\(136\) 0 0
\(137\) −13.1712 −1.12529 −0.562645 0.826698i \(-0.690216\pi\)
−0.562645 + 0.826698i \(0.690216\pi\)
\(138\) 0 0
\(139\) 11.8454 1.00472 0.502358 0.864660i \(-0.332466\pi\)
0.502358 + 0.864660i \(0.332466\pi\)
\(140\) 0 0
\(141\) −7.07300 −0.595654
\(142\) 0 0
\(143\) −3.60999 −0.301882
\(144\) 0 0
\(145\) 0.305886 0.0254025
\(146\) 0 0
\(147\) 6.98039 0.575733
\(148\) 0 0
\(149\) −15.1656 −1.24241 −0.621207 0.783646i \(-0.713358\pi\)
−0.621207 + 0.783646i \(0.713358\pi\)
\(150\) 0 0
\(151\) 0.571642 0.0465196 0.0232598 0.999729i \(-0.492596\pi\)
0.0232598 + 0.999729i \(0.492596\pi\)
\(152\) 0 0
\(153\) 5.66444 0.457943
\(154\) 0 0
\(155\) 2.05701 0.165223
\(156\) 0 0
\(157\) −17.2615 −1.37762 −0.688810 0.724941i \(-0.741867\pi\)
−0.688810 + 0.724941i \(0.741867\pi\)
\(158\) 0 0
\(159\) 11.0449 0.875918
\(160\) 0 0
\(161\) −0.140026 −0.0110356
\(162\) 0 0
\(163\) 24.3324 1.90586 0.952929 0.303195i \(-0.0980532\pi\)
0.952929 + 0.303195i \(0.0980532\pi\)
\(164\) 0 0
\(165\) 0.568487 0.0442566
\(166\) 0 0
\(167\) 1.90162 0.147152 0.0735758 0.997290i \(-0.476559\pi\)
0.0735758 + 0.997290i \(0.476559\pi\)
\(168\) 0 0
\(169\) −9.22697 −0.709767
\(170\) 0 0
\(171\) −1.04650 −0.0800281
\(172\) 0 0
\(173\) −6.53138 −0.496572 −0.248286 0.968687i \(-0.579867\pi\)
−0.248286 + 0.968687i \(0.579867\pi\)
\(174\) 0 0
\(175\) 0.687028 0.0519345
\(176\) 0 0
\(177\) 10.0473 0.755199
\(178\) 0 0
\(179\) 15.8879 1.18752 0.593760 0.804642i \(-0.297643\pi\)
0.593760 + 0.804642i \(0.297643\pi\)
\(180\) 0 0
\(181\) −7.85906 −0.584160 −0.292080 0.956394i \(-0.594347\pi\)
−0.292080 + 0.956394i \(0.594347\pi\)
\(182\) 0 0
\(183\) −7.67091 −0.567050
\(184\) 0 0
\(185\) 0.436118 0.0320641
\(186\) 0 0
\(187\) 10.5273 0.769834
\(188\) 0 0
\(189\) 0.140026 0.0101854
\(190\) 0 0
\(191\) −3.03406 −0.219537 −0.109769 0.993957i \(-0.535011\pi\)
−0.109769 + 0.993957i \(0.535011\pi\)
\(192\) 0 0
\(193\) −3.82892 −0.275612 −0.137806 0.990459i \(-0.544005\pi\)
−0.137806 + 0.990459i \(0.544005\pi\)
\(194\) 0 0
\(195\) −0.594162 −0.0425489
\(196\) 0 0
\(197\) −6.61415 −0.471238 −0.235619 0.971845i \(-0.575712\pi\)
−0.235619 + 0.971845i \(0.575712\pi\)
\(198\) 0 0
\(199\) 2.10692 0.149356 0.0746779 0.997208i \(-0.476207\pi\)
0.0746779 + 0.997208i \(0.476207\pi\)
\(200\) 0 0
\(201\) 12.0677 0.851187
\(202\) 0 0
\(203\) 0.140026 0.00982790
\(204\) 0 0
\(205\) −0.159901 −0.0111680
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −1.94492 −0.134533
\(210\) 0 0
\(211\) 3.66115 0.252044 0.126022 0.992027i \(-0.459779\pi\)
0.126022 + 0.992027i \(0.459779\pi\)
\(212\) 0 0
\(213\) −8.29946 −0.568670
\(214\) 0 0
\(215\) −1.30840 −0.0892318
\(216\) 0 0
\(217\) 0.941641 0.0639227
\(218\) 0 0
\(219\) 0.0686045 0.00463586
\(220\) 0 0
\(221\) −11.0028 −0.740127
\(222\) 0 0
\(223\) 8.80133 0.589381 0.294690 0.955593i \(-0.404784\pi\)
0.294690 + 0.955593i \(0.404784\pi\)
\(224\) 0 0
\(225\) −4.90643 −0.327096
\(226\) 0 0
\(227\) −21.2093 −1.40771 −0.703854 0.710344i \(-0.748539\pi\)
−0.703854 + 0.710344i \(0.748539\pi\)
\(228\) 0 0
\(229\) 15.9425 1.05351 0.526754 0.850018i \(-0.323409\pi\)
0.526754 + 0.850018i \(0.323409\pi\)
\(230\) 0 0
\(231\) 0.260237 0.0171223
\(232\) 0 0
\(233\) 6.42481 0.420903 0.210452 0.977604i \(-0.432507\pi\)
0.210452 + 0.977604i \(0.432507\pi\)
\(234\) 0 0
\(235\) −2.16353 −0.141133
\(236\) 0 0
\(237\) −8.58428 −0.557609
\(238\) 0 0
\(239\) −7.49563 −0.484852 −0.242426 0.970170i \(-0.577943\pi\)
−0.242426 + 0.970170i \(0.577943\pi\)
\(240\) 0 0
\(241\) 0.687449 0.0442825 0.0221412 0.999755i \(-0.492952\pi\)
0.0221412 + 0.999755i \(0.492952\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.13521 0.136413
\(246\) 0 0
\(247\) 2.03276 0.129341
\(248\) 0 0
\(249\) −7.65463 −0.485092
\(250\) 0 0
\(251\) −17.9404 −1.13239 −0.566193 0.824273i \(-0.691584\pi\)
−0.566193 + 0.824273i \(0.691584\pi\)
\(252\) 0 0
\(253\) 1.85849 0.116842
\(254\) 0 0
\(255\) 1.73267 0.108504
\(256\) 0 0
\(257\) −21.1889 −1.32172 −0.660862 0.750507i \(-0.729809\pi\)
−0.660862 + 0.750507i \(0.729809\pi\)
\(258\) 0 0
\(259\) 0.199643 0.0124052
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 0.533746 0.0329122 0.0164561 0.999865i \(-0.494762\pi\)
0.0164561 + 0.999865i \(0.494762\pi\)
\(264\) 0 0
\(265\) 3.37848 0.207539
\(266\) 0 0
\(267\) −8.93734 −0.546957
\(268\) 0 0
\(269\) −30.3949 −1.85321 −0.926605 0.376035i \(-0.877287\pi\)
−0.926605 + 0.376035i \(0.877287\pi\)
\(270\) 0 0
\(271\) −19.7790 −1.20149 −0.600743 0.799442i \(-0.705128\pi\)
−0.600743 + 0.799442i \(0.705128\pi\)
\(272\) 0 0
\(273\) −0.271991 −0.0164616
\(274\) 0 0
\(275\) −9.11856 −0.549870
\(276\) 0 0
\(277\) 32.4269 1.94835 0.974173 0.225802i \(-0.0725002\pi\)
0.974173 + 0.225802i \(0.0725002\pi\)
\(278\) 0 0
\(279\) −6.72475 −0.402600
\(280\) 0 0
\(281\) −25.0927 −1.49691 −0.748454 0.663187i \(-0.769203\pi\)
−0.748454 + 0.663187i \(0.769203\pi\)
\(282\) 0 0
\(283\) −26.0505 −1.54854 −0.774270 0.632856i \(-0.781883\pi\)
−0.774270 + 0.632856i \(0.781883\pi\)
\(284\) 0 0
\(285\) −0.320111 −0.0189617
\(286\) 0 0
\(287\) −0.0731982 −0.00432075
\(288\) 0 0
\(289\) 15.0859 0.887406
\(290\) 0 0
\(291\) −4.43366 −0.259906
\(292\) 0 0
\(293\) −26.2399 −1.53295 −0.766474 0.642275i \(-0.777991\pi\)
−0.766474 + 0.642275i \(0.777991\pi\)
\(294\) 0 0
\(295\) 3.07332 0.178936
\(296\) 0 0
\(297\) −1.85849 −0.107841
\(298\) 0 0
\(299\) −1.94243 −0.112334
\(300\) 0 0
\(301\) −0.598946 −0.0345227
\(302\) 0 0
\(303\) −5.01727 −0.288234
\(304\) 0 0
\(305\) −2.34642 −0.134356
\(306\) 0 0
\(307\) 0.970427 0.0553852 0.0276926 0.999616i \(-0.491184\pi\)
0.0276926 + 0.999616i \(0.491184\pi\)
\(308\) 0 0
\(309\) 13.5050 0.768270
\(310\) 0 0
\(311\) 10.0703 0.571036 0.285518 0.958373i \(-0.407834\pi\)
0.285518 + 0.958373i \(0.407834\pi\)
\(312\) 0 0
\(313\) −29.9323 −1.69188 −0.845938 0.533281i \(-0.820959\pi\)
−0.845938 + 0.533281i \(0.820959\pi\)
\(314\) 0 0
\(315\) 0.0428320 0.00241331
\(316\) 0 0
\(317\) 20.1859 1.13375 0.566876 0.823803i \(-0.308152\pi\)
0.566876 + 0.823803i \(0.308152\pi\)
\(318\) 0 0
\(319\) −1.85849 −0.104056
\(320\) 0 0
\(321\) 10.1993 0.569271
\(322\) 0 0
\(323\) −5.92786 −0.329835
\(324\) 0 0
\(325\) 9.53040 0.528652
\(326\) 0 0
\(327\) −0.150542 −0.00832498
\(328\) 0 0
\(329\) −0.990404 −0.0546028
\(330\) 0 0
\(331\) −28.2007 −1.55005 −0.775026 0.631930i \(-0.782263\pi\)
−0.775026 + 0.631930i \(0.782263\pi\)
\(332\) 0 0
\(333\) −1.42575 −0.0781308
\(334\) 0 0
\(335\) 3.69133 0.201679
\(336\) 0 0
\(337\) 1.55048 0.0844602 0.0422301 0.999108i \(-0.486554\pi\)
0.0422301 + 0.999108i \(0.486554\pi\)
\(338\) 0 0
\(339\) −11.3408 −0.615949
\(340\) 0 0
\(341\) −12.4979 −0.676799
\(342\) 0 0
\(343\) 1.95762 0.105701
\(344\) 0 0
\(345\) 0.305886 0.0164684
\(346\) 0 0
\(347\) −19.2286 −1.03225 −0.516123 0.856515i \(-0.672625\pi\)
−0.516123 + 0.856515i \(0.672625\pi\)
\(348\) 0 0
\(349\) 6.23325 0.333658 0.166829 0.985986i \(-0.446647\pi\)
0.166829 + 0.985986i \(0.446647\pi\)
\(350\) 0 0
\(351\) 1.94243 0.103679
\(352\) 0 0
\(353\) −28.2992 −1.50621 −0.753106 0.657899i \(-0.771445\pi\)
−0.753106 + 0.657899i \(0.771445\pi\)
\(354\) 0 0
\(355\) −2.53869 −0.134740
\(356\) 0 0
\(357\) 0.793169 0.0419790
\(358\) 0 0
\(359\) 32.8506 1.73379 0.866895 0.498491i \(-0.166112\pi\)
0.866895 + 0.498491i \(0.166112\pi\)
\(360\) 0 0
\(361\) −17.9048 −0.942359
\(362\) 0 0
\(363\) 7.54601 0.396063
\(364\) 0 0
\(365\) 0.0209852 0.00109841
\(366\) 0 0
\(367\) 7.67381 0.400570 0.200285 0.979738i \(-0.435813\pi\)
0.200285 + 0.979738i \(0.435813\pi\)
\(368\) 0 0
\(369\) 0.522747 0.0272131
\(370\) 0 0
\(371\) 1.54657 0.0802941
\(372\) 0 0
\(373\) −7.94179 −0.411210 −0.205605 0.978635i \(-0.565916\pi\)
−0.205605 + 0.978635i \(0.565916\pi\)
\(374\) 0 0
\(375\) −3.03024 −0.156481
\(376\) 0 0
\(377\) 1.94243 0.100040
\(378\) 0 0
\(379\) −13.9632 −0.717243 −0.358621 0.933483i \(-0.616753\pi\)
−0.358621 + 0.933483i \(0.616753\pi\)
\(380\) 0 0
\(381\) −4.72184 −0.241907
\(382\) 0 0
\(383\) 1.11226 0.0568337 0.0284169 0.999596i \(-0.490953\pi\)
0.0284169 + 0.999596i \(0.490953\pi\)
\(384\) 0 0
\(385\) 0.0796030 0.00405694
\(386\) 0 0
\(387\) 4.27739 0.217432
\(388\) 0 0
\(389\) 32.0187 1.62341 0.811707 0.584065i \(-0.198539\pi\)
0.811707 + 0.584065i \(0.198539\pi\)
\(390\) 0 0
\(391\) 5.66444 0.286463
\(392\) 0 0
\(393\) 11.9635 0.603480
\(394\) 0 0
\(395\) −2.62581 −0.132119
\(396\) 0 0
\(397\) 25.6356 1.28661 0.643306 0.765609i \(-0.277562\pi\)
0.643306 + 0.765609i \(0.277562\pi\)
\(398\) 0 0
\(399\) −0.146538 −0.00733606
\(400\) 0 0
\(401\) 10.0646 0.502603 0.251301 0.967909i \(-0.419141\pi\)
0.251301 + 0.967909i \(0.419141\pi\)
\(402\) 0 0
\(403\) 13.0624 0.650683
\(404\) 0 0
\(405\) −0.305886 −0.0151996
\(406\) 0 0
\(407\) −2.64975 −0.131343
\(408\) 0 0
\(409\) −16.3638 −0.809140 −0.404570 0.914507i \(-0.632579\pi\)
−0.404570 + 0.914507i \(0.632579\pi\)
\(410\) 0 0
\(411\) 13.1712 0.649687
\(412\) 0 0
\(413\) 1.40688 0.0692280
\(414\) 0 0
\(415\) −2.34145 −0.114937
\(416\) 0 0
\(417\) −11.8454 −0.580073
\(418\) 0 0
\(419\) 32.4761 1.58656 0.793282 0.608855i \(-0.208371\pi\)
0.793282 + 0.608855i \(0.208371\pi\)
\(420\) 0 0
\(421\) −0.972317 −0.0473878 −0.0236939 0.999719i \(-0.507543\pi\)
−0.0236939 + 0.999719i \(0.507543\pi\)
\(422\) 0 0
\(423\) 7.07300 0.343901
\(424\) 0 0
\(425\) −27.7922 −1.34812
\(426\) 0 0
\(427\) −1.07413 −0.0519806
\(428\) 0 0
\(429\) 3.60999 0.174292
\(430\) 0 0
\(431\) −23.2012 −1.11756 −0.558781 0.829315i \(-0.688731\pi\)
−0.558781 + 0.829315i \(0.688731\pi\)
\(432\) 0 0
\(433\) −7.39839 −0.355544 −0.177772 0.984072i \(-0.556889\pi\)
−0.177772 + 0.984072i \(0.556889\pi\)
\(434\) 0 0
\(435\) −0.305886 −0.0146661
\(436\) 0 0
\(437\) −1.04650 −0.0500611
\(438\) 0 0
\(439\) 30.8260 1.47124 0.735622 0.677392i \(-0.236890\pi\)
0.735622 + 0.677392i \(0.236890\pi\)
\(440\) 0 0
\(441\) −6.98039 −0.332400
\(442\) 0 0
\(443\) 11.1200 0.528328 0.264164 0.964478i \(-0.414904\pi\)
0.264164 + 0.964478i \(0.414904\pi\)
\(444\) 0 0
\(445\) −2.73381 −0.129595
\(446\) 0 0
\(447\) 15.1656 0.717308
\(448\) 0 0
\(449\) −5.32321 −0.251218 −0.125609 0.992080i \(-0.540088\pi\)
−0.125609 + 0.992080i \(0.540088\pi\)
\(450\) 0 0
\(451\) 0.971521 0.0457471
\(452\) 0 0
\(453\) −0.571642 −0.0268581
\(454\) 0 0
\(455\) −0.0831982 −0.00390039
\(456\) 0 0
\(457\) −25.1085 −1.17453 −0.587264 0.809396i \(-0.699795\pi\)
−0.587264 + 0.809396i \(0.699795\pi\)
\(458\) 0 0
\(459\) −5.66444 −0.264393
\(460\) 0 0
\(461\) −25.0268 −1.16562 −0.582808 0.812610i \(-0.698046\pi\)
−0.582808 + 0.812610i \(0.698046\pi\)
\(462\) 0 0
\(463\) 12.1410 0.564242 0.282121 0.959379i \(-0.408962\pi\)
0.282121 + 0.959379i \(0.408962\pi\)
\(464\) 0 0
\(465\) −2.05701 −0.0953915
\(466\) 0 0
\(467\) −31.1080 −1.43951 −0.719754 0.694229i \(-0.755746\pi\)
−0.719754 + 0.694229i \(0.755746\pi\)
\(468\) 0 0
\(469\) 1.68979 0.0780271
\(470\) 0 0
\(471\) 17.2615 0.795370
\(472\) 0 0
\(473\) 7.94950 0.365518
\(474\) 0 0
\(475\) 5.13460 0.235592
\(476\) 0 0
\(477\) −11.0449 −0.505711
\(478\) 0 0
\(479\) 0.963078 0.0440042 0.0220021 0.999758i \(-0.492996\pi\)
0.0220021 + 0.999758i \(0.492996\pi\)
\(480\) 0 0
\(481\) 2.76943 0.126275
\(482\) 0 0
\(483\) 0.140026 0.00637140
\(484\) 0 0
\(485\) −1.35619 −0.0615816
\(486\) 0 0
\(487\) −17.3396 −0.785734 −0.392867 0.919595i \(-0.628517\pi\)
−0.392867 + 0.919595i \(0.628517\pi\)
\(488\) 0 0
\(489\) −24.3324 −1.10035
\(490\) 0 0
\(491\) −29.5883 −1.33530 −0.667650 0.744475i \(-0.732700\pi\)
−0.667650 + 0.744475i \(0.732700\pi\)
\(492\) 0 0
\(493\) −5.66444 −0.255114
\(494\) 0 0
\(495\) −0.568487 −0.0255516
\(496\) 0 0
\(497\) −1.16214 −0.0521291
\(498\) 0 0
\(499\) 15.6338 0.699864 0.349932 0.936775i \(-0.386205\pi\)
0.349932 + 0.936775i \(0.386205\pi\)
\(500\) 0 0
\(501\) −1.90162 −0.0849580
\(502\) 0 0
\(503\) −1.19850 −0.0534386 −0.0267193 0.999643i \(-0.508506\pi\)
−0.0267193 + 0.999643i \(0.508506\pi\)
\(504\) 0 0
\(505\) −1.53471 −0.0682938
\(506\) 0 0
\(507\) 9.22697 0.409784
\(508\) 0 0
\(509\) −10.8016 −0.478770 −0.239385 0.970925i \(-0.576946\pi\)
−0.239385 + 0.970925i \(0.576946\pi\)
\(510\) 0 0
\(511\) 0.00960642 0.000424963 0
\(512\) 0 0
\(513\) 1.04650 0.0462043
\(514\) 0 0
\(515\) 4.13098 0.182033
\(516\) 0 0
\(517\) 13.1451 0.578121
\(518\) 0 0
\(519\) 6.53138 0.286696
\(520\) 0 0
\(521\) 23.6412 1.03574 0.517869 0.855460i \(-0.326725\pi\)
0.517869 + 0.855460i \(0.326725\pi\)
\(522\) 0 0
\(523\) −14.1143 −0.617175 −0.308587 0.951196i \(-0.599856\pi\)
−0.308587 + 0.951196i \(0.599856\pi\)
\(524\) 0 0
\(525\) −0.687028 −0.0299844
\(526\) 0 0
\(527\) −38.0920 −1.65931
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −10.0473 −0.436014
\(532\) 0 0
\(533\) −1.01540 −0.0439818
\(534\) 0 0
\(535\) 3.11984 0.134882
\(536\) 0 0
\(537\) −15.8879 −0.685615
\(538\) 0 0
\(539\) −12.9730 −0.558787
\(540\) 0 0
\(541\) −13.4317 −0.577473 −0.288736 0.957409i \(-0.593235\pi\)
−0.288736 + 0.957409i \(0.593235\pi\)
\(542\) 0 0
\(543\) 7.85906 0.337265
\(544\) 0 0
\(545\) −0.0460487 −0.00197251
\(546\) 0 0
\(547\) −9.51664 −0.406902 −0.203451 0.979085i \(-0.565216\pi\)
−0.203451 + 0.979085i \(0.565216\pi\)
\(548\) 0 0
\(549\) 7.67091 0.327386
\(550\) 0 0
\(551\) 1.04650 0.0445826
\(552\) 0 0
\(553\) −1.20202 −0.0511152
\(554\) 0 0
\(555\) −0.436118 −0.0185122
\(556\) 0 0
\(557\) −19.2946 −0.817538 −0.408769 0.912638i \(-0.634042\pi\)
−0.408769 + 0.912638i \(0.634042\pi\)
\(558\) 0 0
\(559\) −8.30854 −0.351414
\(560\) 0 0
\(561\) −10.5273 −0.444464
\(562\) 0 0
\(563\) −26.4275 −1.11379 −0.556893 0.830584i \(-0.688007\pi\)
−0.556893 + 0.830584i \(0.688007\pi\)
\(564\) 0 0
\(565\) −3.46900 −0.145942
\(566\) 0 0
\(567\) −0.140026 −0.00588054
\(568\) 0 0
\(569\) −20.2718 −0.849839 −0.424919 0.905231i \(-0.639698\pi\)
−0.424919 + 0.905231i \(0.639698\pi\)
\(570\) 0 0
\(571\) 20.4605 0.856244 0.428122 0.903721i \(-0.359175\pi\)
0.428122 + 0.903721i \(0.359175\pi\)
\(572\) 0 0
\(573\) 3.03406 0.126750
\(574\) 0 0
\(575\) −4.90643 −0.204612
\(576\) 0 0
\(577\) 1.16293 0.0484132 0.0242066 0.999707i \(-0.492294\pi\)
0.0242066 + 0.999707i \(0.492294\pi\)
\(578\) 0 0
\(579\) 3.82892 0.159125
\(580\) 0 0
\(581\) −1.07185 −0.0444677
\(582\) 0 0
\(583\) −20.5269 −0.850135
\(584\) 0 0
\(585\) 0.594162 0.0245656
\(586\) 0 0
\(587\) 39.1994 1.61793 0.808965 0.587856i \(-0.200028\pi\)
0.808965 + 0.587856i \(0.200028\pi\)
\(588\) 0 0
\(589\) 7.03748 0.289974
\(590\) 0 0
\(591\) 6.61415 0.272070
\(592\) 0 0
\(593\) −16.1639 −0.663772 −0.331886 0.943320i \(-0.607685\pi\)
−0.331886 + 0.943320i \(0.607685\pi\)
\(594\) 0 0
\(595\) 0.242620 0.00994643
\(596\) 0 0
\(597\) −2.10692 −0.0862306
\(598\) 0 0
\(599\) −28.5186 −1.16524 −0.582619 0.812745i \(-0.697972\pi\)
−0.582619 + 0.812745i \(0.697972\pi\)
\(600\) 0 0
\(601\) −7.67695 −0.313149 −0.156575 0.987666i \(-0.550045\pi\)
−0.156575 + 0.987666i \(0.550045\pi\)
\(602\) 0 0
\(603\) −12.0677 −0.491433
\(604\) 0 0
\(605\) 2.30822 0.0938425
\(606\) 0 0
\(607\) 23.1764 0.940703 0.470351 0.882479i \(-0.344127\pi\)
0.470351 + 0.882479i \(0.344127\pi\)
\(608\) 0 0
\(609\) −0.140026 −0.00567414
\(610\) 0 0
\(611\) −13.7388 −0.555813
\(612\) 0 0
\(613\) −41.7207 −1.68508 −0.842542 0.538631i \(-0.818942\pi\)
−0.842542 + 0.538631i \(0.818942\pi\)
\(614\) 0 0
\(615\) 0.159901 0.00644783
\(616\) 0 0
\(617\) 12.8239 0.516270 0.258135 0.966109i \(-0.416892\pi\)
0.258135 + 0.966109i \(0.416892\pi\)
\(618\) 0 0
\(619\) −21.1322 −0.849373 −0.424687 0.905340i \(-0.639616\pi\)
−0.424687 + 0.905340i \(0.639616\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −1.25146 −0.0501387
\(624\) 0 0
\(625\) 23.6053 0.944210
\(626\) 0 0
\(627\) 1.94492 0.0776726
\(628\) 0 0
\(629\) −8.07610 −0.322015
\(630\) 0 0
\(631\) −3.19211 −0.127076 −0.0635380 0.997979i \(-0.520238\pi\)
−0.0635380 + 0.997979i \(0.520238\pi\)
\(632\) 0 0
\(633\) −3.66115 −0.145518
\(634\) 0 0
\(635\) −1.44435 −0.0573171
\(636\) 0 0
\(637\) 13.5589 0.537224
\(638\) 0 0
\(639\) 8.29946 0.328322
\(640\) 0 0
\(641\) 19.7436 0.779825 0.389912 0.920852i \(-0.372505\pi\)
0.389912 + 0.920852i \(0.372505\pi\)
\(642\) 0 0
\(643\) 8.39805 0.331187 0.165593 0.986194i \(-0.447046\pi\)
0.165593 + 0.986194i \(0.447046\pi\)
\(644\) 0 0
\(645\) 1.30840 0.0515180
\(646\) 0 0
\(647\) 0.653272 0.0256828 0.0128414 0.999918i \(-0.495912\pi\)
0.0128414 + 0.999918i \(0.495912\pi\)
\(648\) 0 0
\(649\) −18.6728 −0.732970
\(650\) 0 0
\(651\) −0.941641 −0.0369058
\(652\) 0 0
\(653\) 11.5695 0.452749 0.226374 0.974040i \(-0.427313\pi\)
0.226374 + 0.974040i \(0.427313\pi\)
\(654\) 0 0
\(655\) 3.65948 0.142988
\(656\) 0 0
\(657\) −0.0686045 −0.00267652
\(658\) 0 0
\(659\) −24.8891 −0.969543 −0.484771 0.874641i \(-0.661097\pi\)
−0.484771 + 0.874641i \(0.661097\pi\)
\(660\) 0 0
\(661\) −25.3623 −0.986480 −0.493240 0.869893i \(-0.664188\pi\)
−0.493240 + 0.869893i \(0.664188\pi\)
\(662\) 0 0
\(663\) 11.0028 0.427313
\(664\) 0 0
\(665\) −0.0448239 −0.00173820
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −8.80133 −0.340279
\(670\) 0 0
\(671\) 14.2563 0.550359
\(672\) 0 0
\(673\) −17.9669 −0.692572 −0.346286 0.938129i \(-0.612557\pi\)
−0.346286 + 0.938129i \(0.612557\pi\)
\(674\) 0 0
\(675\) 4.90643 0.188849
\(676\) 0 0
\(677\) −30.9240 −1.18850 −0.594252 0.804279i \(-0.702552\pi\)
−0.594252 + 0.804279i \(0.702552\pi\)
\(678\) 0 0
\(679\) −0.620827 −0.0238252
\(680\) 0 0
\(681\) 21.2093 0.812741
\(682\) 0 0
\(683\) −8.21255 −0.314244 −0.157122 0.987579i \(-0.550222\pi\)
−0.157122 + 0.987579i \(0.550222\pi\)
\(684\) 0 0
\(685\) 4.02888 0.153936
\(686\) 0 0
\(687\) −15.9425 −0.608243
\(688\) 0 0
\(689\) 21.4539 0.817330
\(690\) 0 0
\(691\) 15.9772 0.607801 0.303901 0.952704i \(-0.401711\pi\)
0.303901 + 0.952704i \(0.401711\pi\)
\(692\) 0 0
\(693\) −0.260237 −0.00988559
\(694\) 0 0
\(695\) −3.62335 −0.137442
\(696\) 0 0
\(697\) 2.96107 0.112158
\(698\) 0 0
\(699\) −6.42481 −0.243009
\(700\) 0 0
\(701\) −50.8260 −1.91967 −0.959835 0.280565i \(-0.909478\pi\)
−0.959835 + 0.280565i \(0.909478\pi\)
\(702\) 0 0
\(703\) 1.49206 0.0562740
\(704\) 0 0
\(705\) 2.16353 0.0814834
\(706\) 0 0
\(707\) −0.702548 −0.0264220
\(708\) 0 0
\(709\) −27.8310 −1.04522 −0.522608 0.852573i \(-0.675041\pi\)
−0.522608 + 0.852573i \(0.675041\pi\)
\(710\) 0 0
\(711\) 8.58428 0.321936
\(712\) 0 0
\(713\) −6.72475 −0.251844
\(714\) 0 0
\(715\) 1.10425 0.0412965
\(716\) 0 0
\(717\) 7.49563 0.279929
\(718\) 0 0
\(719\) 0.498348 0.0185852 0.00929262 0.999957i \(-0.497042\pi\)
0.00929262 + 0.999957i \(0.497042\pi\)
\(720\) 0 0
\(721\) 1.89105 0.0704262
\(722\) 0 0
\(723\) −0.687449 −0.0255665
\(724\) 0 0
\(725\) 4.90643 0.182220
\(726\) 0 0
\(727\) 48.4281 1.79610 0.898049 0.439895i \(-0.144985\pi\)
0.898049 + 0.439895i \(0.144985\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.2290 0.896144
\(732\) 0 0
\(733\) 34.3149 1.26745 0.633725 0.773558i \(-0.281525\pi\)
0.633725 + 0.773558i \(0.281525\pi\)
\(734\) 0 0
\(735\) −2.13521 −0.0787583
\(736\) 0 0
\(737\) −22.4276 −0.826133
\(738\) 0 0
\(739\) −17.0461 −0.627050 −0.313525 0.949580i \(-0.601510\pi\)
−0.313525 + 0.949580i \(0.601510\pi\)
\(740\) 0 0
\(741\) −2.03276 −0.0746753
\(742\) 0 0
\(743\) 26.9392 0.988304 0.494152 0.869375i \(-0.335479\pi\)
0.494152 + 0.869375i \(0.335479\pi\)
\(744\) 0 0
\(745\) 4.63895 0.169958
\(746\) 0 0
\(747\) 7.65463 0.280068
\(748\) 0 0
\(749\) 1.42817 0.0521843
\(750\) 0 0
\(751\) −36.4238 −1.32912 −0.664562 0.747233i \(-0.731382\pi\)
−0.664562 + 0.747233i \(0.731382\pi\)
\(752\) 0 0
\(753\) 17.9404 0.653783
\(754\) 0 0
\(755\) −0.174857 −0.00636371
\(756\) 0 0
\(757\) −3.92322 −0.142592 −0.0712959 0.997455i \(-0.522713\pi\)
−0.0712959 + 0.997455i \(0.522713\pi\)
\(758\) 0 0
\(759\) −1.85849 −0.0674590
\(760\) 0 0
\(761\) 8.71135 0.315786 0.157893 0.987456i \(-0.449530\pi\)
0.157893 + 0.987456i \(0.449530\pi\)
\(762\) 0 0
\(763\) −0.0210798 −0.000763139 0
\(764\) 0 0
\(765\) −1.73267 −0.0626450
\(766\) 0 0
\(767\) 19.5161 0.704686
\(768\) 0 0
\(769\) −40.4343 −1.45810 −0.729049 0.684461i \(-0.760037\pi\)
−0.729049 + 0.684461i \(0.760037\pi\)
\(770\) 0 0
\(771\) 21.1889 0.763098
\(772\) 0 0
\(773\) 6.95931 0.250309 0.125155 0.992137i \(-0.460057\pi\)
0.125155 + 0.992137i \(0.460057\pi\)
\(774\) 0 0
\(775\) 32.9946 1.18520
\(776\) 0 0
\(777\) −0.199643 −0.00716214
\(778\) 0 0
\(779\) −0.547057 −0.0196003
\(780\) 0 0
\(781\) 15.4245 0.551931
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 5.28007 0.188454
\(786\) 0 0
\(787\) 16.8140 0.599355 0.299677 0.954041i \(-0.403121\pi\)
0.299677 + 0.954041i \(0.403121\pi\)
\(788\) 0 0
\(789\) −0.533746 −0.0190019
\(790\) 0 0
\(791\) −1.58801 −0.0564631
\(792\) 0 0
\(793\) −14.9002 −0.529122
\(794\) 0 0
\(795\) −3.37848 −0.119822
\(796\) 0 0
\(797\) −5.59612 −0.198225 −0.0991123 0.995076i \(-0.531600\pi\)
−0.0991123 + 0.995076i \(0.531600\pi\)
\(798\) 0 0
\(799\) 40.0646 1.41738
\(800\) 0 0
\(801\) 8.93734 0.315786
\(802\) 0 0
\(803\) −0.127501 −0.00449941
\(804\) 0 0
\(805\) 0.0428320 0.00150963
\(806\) 0 0
\(807\) 30.3949 1.06995
\(808\) 0 0
\(809\) −4.38279 −0.154091 −0.0770454 0.997028i \(-0.524549\pi\)
−0.0770454 + 0.997028i \(0.524549\pi\)
\(810\) 0 0
\(811\) 40.9682 1.43859 0.719294 0.694706i \(-0.244466\pi\)
0.719294 + 0.694706i \(0.244466\pi\)
\(812\) 0 0
\(813\) 19.7790 0.693678
\(814\) 0 0
\(815\) −7.44293 −0.260715
\(816\) 0 0
\(817\) −4.47631 −0.156606
\(818\) 0 0
\(819\) 0.271991 0.00950412
\(820\) 0 0
\(821\) −27.8672 −0.972573 −0.486286 0.873800i \(-0.661649\pi\)
−0.486286 + 0.873800i \(0.661649\pi\)
\(822\) 0 0
\(823\) −14.8442 −0.517436 −0.258718 0.965953i \(-0.583300\pi\)
−0.258718 + 0.965953i \(0.583300\pi\)
\(824\) 0 0
\(825\) 9.11856 0.317468
\(826\) 0 0
\(827\) −18.9515 −0.659009 −0.329504 0.944154i \(-0.606882\pi\)
−0.329504 + 0.944154i \(0.606882\pi\)
\(828\) 0 0
\(829\) −30.5368 −1.06059 −0.530294 0.847814i \(-0.677918\pi\)
−0.530294 + 0.847814i \(0.677918\pi\)
\(830\) 0 0
\(831\) −32.4269 −1.12488
\(832\) 0 0
\(833\) −39.5400 −1.36998
\(834\) 0 0
\(835\) −0.581679 −0.0201298
\(836\) 0 0
\(837\) 6.72475 0.232441
\(838\) 0 0
\(839\) 28.8921 0.997466 0.498733 0.866756i \(-0.333799\pi\)
0.498733 + 0.866756i \(0.333799\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 25.0927 0.864240
\(844\) 0 0
\(845\) 2.82240 0.0970936
\(846\) 0 0
\(847\) 1.05664 0.0363065
\(848\) 0 0
\(849\) 26.0505 0.894049
\(850\) 0 0
\(851\) −1.42575 −0.0488742
\(852\) 0 0
\(853\) −7.03925 −0.241019 −0.120510 0.992712i \(-0.538453\pi\)
−0.120510 + 0.992712i \(0.538453\pi\)
\(854\) 0 0
\(855\) 0.320111 0.0109476
\(856\) 0 0
\(857\) −0.221845 −0.00757809 −0.00378904 0.999993i \(-0.501206\pi\)
−0.00378904 + 0.999993i \(0.501206\pi\)
\(858\) 0 0
\(859\) 36.2549 1.23700 0.618500 0.785785i \(-0.287741\pi\)
0.618500 + 0.785785i \(0.287741\pi\)
\(860\) 0 0
\(861\) 0.0731982 0.00249459
\(862\) 0 0
\(863\) 0.0417367 0.00142074 0.000710368 1.00000i \(-0.499774\pi\)
0.000710368 1.00000i \(0.499774\pi\)
\(864\) 0 0
\(865\) 1.99786 0.0679292
\(866\) 0 0
\(867\) −15.0859 −0.512344
\(868\) 0 0
\(869\) 15.9538 0.541196
\(870\) 0 0
\(871\) 23.4406 0.794254
\(872\) 0 0
\(873\) 4.43366 0.150057
\(874\) 0 0
\(875\) −0.424313 −0.0143444
\(876\) 0 0
\(877\) −44.8890 −1.51579 −0.757897 0.652374i \(-0.773773\pi\)
−0.757897 + 0.652374i \(0.773773\pi\)
\(878\) 0 0
\(879\) 26.2399 0.885048
\(880\) 0 0
\(881\) −4.54343 −0.153072 −0.0765360 0.997067i \(-0.524386\pi\)
−0.0765360 + 0.997067i \(0.524386\pi\)
\(882\) 0 0
\(883\) −23.5745 −0.793345 −0.396672 0.917960i \(-0.629835\pi\)
−0.396672 + 0.917960i \(0.629835\pi\)
\(884\) 0 0
\(885\) −3.07332 −0.103309
\(886\) 0 0
\(887\) 28.4093 0.953891 0.476945 0.878933i \(-0.341744\pi\)
0.476945 + 0.878933i \(0.341744\pi\)
\(888\) 0 0
\(889\) −0.661180 −0.0221753
\(890\) 0 0
\(891\) 1.85849 0.0622618
\(892\) 0 0
\(893\) −7.40192 −0.247696
\(894\) 0 0
\(895\) −4.85990 −0.162449
\(896\) 0 0
\(897\) 1.94243 0.0648558
\(898\) 0 0
\(899\) 6.72475 0.224283
\(900\) 0 0
\(901\) −62.5632 −2.08428
\(902\) 0 0
\(903\) 0.598946 0.0199317
\(904\) 0 0
\(905\) 2.40398 0.0799110
\(906\) 0 0
\(907\) 27.4950 0.912956 0.456478 0.889735i \(-0.349111\pi\)
0.456478 + 0.889735i \(0.349111\pi\)
\(908\) 0 0
\(909\) 5.01727 0.166412
\(910\) 0 0
\(911\) −10.1181 −0.335227 −0.167613 0.985853i \(-0.553606\pi\)
−0.167613 + 0.985853i \(0.553606\pi\)
\(912\) 0 0
\(913\) 14.2261 0.470814
\(914\) 0 0
\(915\) 2.34642 0.0775704
\(916\) 0 0
\(917\) 1.67521 0.0553202
\(918\) 0 0
\(919\) 22.5438 0.743651 0.371826 0.928303i \(-0.378732\pi\)
0.371826 + 0.928303i \(0.378732\pi\)
\(920\) 0 0
\(921\) −0.970427 −0.0319766
\(922\) 0 0
\(923\) −16.1211 −0.530633
\(924\) 0 0
\(925\) 6.99536 0.230006
\(926\) 0 0
\(927\) −13.5050 −0.443561
\(928\) 0 0
\(929\) −38.5125 −1.26355 −0.631777 0.775150i \(-0.717674\pi\)
−0.631777 + 0.775150i \(0.717674\pi\)
\(930\) 0 0
\(931\) 7.30501 0.239412
\(932\) 0 0
\(933\) −10.0703 −0.329688
\(934\) 0 0
\(935\) −3.22016 −0.105311
\(936\) 0 0
\(937\) −9.71092 −0.317242 −0.158621 0.987340i \(-0.550705\pi\)
−0.158621 + 0.987340i \(0.550705\pi\)
\(938\) 0 0
\(939\) 29.9323 0.976805
\(940\) 0 0
\(941\) −29.5549 −0.963462 −0.481731 0.876319i \(-0.659992\pi\)
−0.481731 + 0.876319i \(0.659992\pi\)
\(942\) 0 0
\(943\) 0.522747 0.0170230
\(944\) 0 0
\(945\) −0.0428320 −0.00139333
\(946\) 0 0
\(947\) −48.0594 −1.56172 −0.780861 0.624705i \(-0.785219\pi\)
−0.780861 + 0.624705i \(0.785219\pi\)
\(948\) 0 0
\(949\) 0.133259 0.00432578
\(950\) 0 0
\(951\) −20.1859 −0.654572
\(952\) 0 0
\(953\) 50.2867 1.62895 0.814473 0.580201i \(-0.197026\pi\)
0.814473 + 0.580201i \(0.197026\pi\)
\(954\) 0 0
\(955\) 0.928078 0.0300319
\(956\) 0 0
\(957\) 1.85849 0.0600765
\(958\) 0 0
\(959\) 1.84431 0.0595558
\(960\) 0 0
\(961\) 14.2223 0.458784
\(962\) 0 0
\(963\) −10.1993 −0.328669
\(964\) 0 0
\(965\) 1.17122 0.0377027
\(966\) 0 0
\(967\) 41.3154 1.32861 0.664306 0.747461i \(-0.268727\pi\)
0.664306 + 0.747461i \(0.268727\pi\)
\(968\) 0 0
\(969\) 5.92786 0.190430
\(970\) 0 0
\(971\) −45.3278 −1.45464 −0.727319 0.686299i \(-0.759234\pi\)
−0.727319 + 0.686299i \(0.759234\pi\)
\(972\) 0 0
\(973\) −1.65867 −0.0531745
\(974\) 0 0
\(975\) −9.53040 −0.305217
\(976\) 0 0
\(977\) 43.0997 1.37888 0.689441 0.724342i \(-0.257856\pi\)
0.689441 + 0.724342i \(0.257856\pi\)
\(978\) 0 0
\(979\) 16.6100 0.530857
\(980\) 0 0
\(981\) 0.150542 0.00480643
\(982\) 0 0
\(983\) 17.4186 0.555567 0.277783 0.960644i \(-0.410400\pi\)
0.277783 + 0.960644i \(0.410400\pi\)
\(984\) 0 0
\(985\) 2.02318 0.0644638
\(986\) 0 0
\(987\) 0.990404 0.0315249
\(988\) 0 0
\(989\) 4.27739 0.136013
\(990\) 0 0
\(991\) −31.7050 −1.00714 −0.503571 0.863954i \(-0.667981\pi\)
−0.503571 + 0.863954i \(0.667981\pi\)
\(992\) 0 0
\(993\) 28.2007 0.894922
\(994\) 0 0
\(995\) −0.644479 −0.0204313
\(996\) 0 0
\(997\) 33.2432 1.05282 0.526412 0.850230i \(-0.323537\pi\)
0.526412 + 0.850230i \(0.323537\pi\)
\(998\) 0 0
\(999\) 1.42575 0.0451088
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.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.6 12 1.1 even 1 trivial