Properties

Label 3640.2.a.s.1.2
Level $3640$
Weight $2$
Character 3640.1
Self dual yes
Analytic conductor $29.066$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.0655463357\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.13856\) of defining polynomial
Character \(\chi\) \(=\) 3640.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.13856 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} +O(q^{10})\) \(q-1.13856 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70367 q^{9} -3.23607 q^{11} +1.00000 q^{13} -1.13856 q^{15} +2.98080 q^{17} +1.93974 q^{19} +1.13856 q^{21} -2.17127 q^{23} +1.00000 q^{25} +5.35543 q^{27} +3.58697 q^{29} +7.29517 q^{31} +3.68447 q^{33} -1.00000 q^{35} +2.37463 q^{37} -1.13856 q^{39} +1.76846 q^{41} -5.51320 q^{43} -1.70367 q^{45} -7.53693 q^{47} +1.00000 q^{49} -3.39383 q^{51} -3.40734 q^{53} -3.23607 q^{55} -2.20852 q^{57} -11.6377 q^{59} -5.68447 q^{61} +1.70367 q^{63} +1.00000 q^{65} +2.04559 q^{67} +2.47214 q^{69} -10.9205 q^{71} -7.09181 q^{73} -1.13856 q^{75} +3.23607 q^{77} -14.7949 q^{79} -0.986489 q^{81} +2.61968 q^{83} +2.98080 q^{85} -4.08399 q^{87} +3.92952 q^{89} -1.00000 q^{91} -8.30602 q^{93} +1.93974 q^{95} -1.34192 q^{97} +5.51320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - q^{17} - 7 q^{19} + q^{21} - 6 q^{23} + 4 q^{25} - 4 q^{27} + q^{29} - 11 q^{31} - 4 q^{33} - 4 q^{35} - 3 q^{37} - q^{39} - 5 q^{41} - 6 q^{43} - q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} - 2 q^{53} - 4 q^{55} + 14 q^{57} - q^{59} - 4 q^{61} + q^{63} + 4 q^{65} - 11 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} - q^{75} + 4 q^{77} - 15 q^{79} - 16 q^{81} - 2 q^{83} - q^{85} - 23 q^{87} - 3 q^{89} - 4 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.13856 −0.657350 −0.328675 0.944443i \(-0.606602\pi\)
−0.328675 + 0.944443i \(0.606602\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.70367 −0.567890
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.13856 −0.293976
\(16\) 0 0
\(17\) 2.98080 0.722950 0.361475 0.932382i \(-0.382273\pi\)
0.361475 + 0.932382i \(0.382273\pi\)
\(18\) 0 0
\(19\) 1.93974 0.445007 0.222503 0.974932i \(-0.428577\pi\)
0.222503 + 0.974932i \(0.428577\pi\)
\(20\) 0 0
\(21\) 1.13856 0.248455
\(22\) 0 0
\(23\) −2.17127 −0.452742 −0.226371 0.974041i \(-0.572686\pi\)
−0.226371 + 0.974041i \(0.572686\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.35543 1.03065
\(28\) 0 0
\(29\) 3.58697 0.666083 0.333042 0.942912i \(-0.391925\pi\)
0.333042 + 0.942912i \(0.391925\pi\)
\(30\) 0 0
\(31\) 7.29517 1.31025 0.655126 0.755520i \(-0.272616\pi\)
0.655126 + 0.755520i \(0.272616\pi\)
\(32\) 0 0
\(33\) 3.68447 0.641384
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 2.37463 0.390387 0.195194 0.980765i \(-0.437466\pi\)
0.195194 + 0.980765i \(0.437466\pi\)
\(38\) 0 0
\(39\) −1.13856 −0.182316
\(40\) 0 0
\(41\) 1.76846 0.276188 0.138094 0.990419i \(-0.455902\pi\)
0.138094 + 0.990419i \(0.455902\pi\)
\(42\) 0 0
\(43\) −5.51320 −0.840755 −0.420377 0.907349i \(-0.638102\pi\)
−0.420377 + 0.907349i \(0.638102\pi\)
\(44\) 0 0
\(45\) −1.70367 −0.253968
\(46\) 0 0
\(47\) −7.53693 −1.09937 −0.549687 0.835371i \(-0.685253\pi\)
−0.549687 + 0.835371i \(0.685253\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.39383 −0.475232
\(52\) 0 0
\(53\) −3.40734 −0.468035 −0.234017 0.972232i \(-0.575187\pi\)
−0.234017 + 0.972232i \(0.575187\pi\)
\(54\) 0 0
\(55\) −3.23607 −0.436351
\(56\) 0 0
\(57\) −2.20852 −0.292525
\(58\) 0 0
\(59\) −11.6377 −1.51510 −0.757551 0.652776i \(-0.773604\pi\)
−0.757551 + 0.652776i \(0.773604\pi\)
\(60\) 0 0
\(61\) −5.68447 −0.727822 −0.363911 0.931434i \(-0.618559\pi\)
−0.363911 + 0.931434i \(0.618559\pi\)
\(62\) 0 0
\(63\) 1.70367 0.214642
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 2.04559 0.249909 0.124954 0.992162i \(-0.460122\pi\)
0.124954 + 0.992162i \(0.460122\pi\)
\(68\) 0 0
\(69\) 2.47214 0.297610
\(70\) 0 0
\(71\) −10.9205 −1.29603 −0.648015 0.761628i \(-0.724401\pi\)
−0.648015 + 0.761628i \(0.724401\pi\)
\(72\) 0 0
\(73\) −7.09181 −0.830034 −0.415017 0.909814i \(-0.636224\pi\)
−0.415017 + 0.909814i \(0.636224\pi\)
\(74\) 0 0
\(75\) −1.13856 −0.131470
\(76\) 0 0
\(77\) 3.23607 0.368784
\(78\) 0 0
\(79\) −14.7949 −1.66455 −0.832276 0.554362i \(-0.812962\pi\)
−0.832276 + 0.554362i \(0.812962\pi\)
\(80\) 0 0
\(81\) −0.986489 −0.109610
\(82\) 0 0
\(83\) 2.61968 0.287547 0.143774 0.989611i \(-0.454076\pi\)
0.143774 + 0.989611i \(0.454076\pi\)
\(84\) 0 0
\(85\) 2.98080 0.323313
\(86\) 0 0
\(87\) −4.08399 −0.437850
\(88\) 0 0
\(89\) 3.92952 0.416528 0.208264 0.978073i \(-0.433219\pi\)
0.208264 + 0.978073i \(0.433219\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −8.30602 −0.861294
\(94\) 0 0
\(95\) 1.93974 0.199013
\(96\) 0 0
\(97\) −1.34192 −0.136252 −0.0681258 0.997677i \(-0.521702\pi\)
−0.0681258 + 0.997677i \(0.521702\pi\)
\(98\) 0 0
\(99\) 5.51320 0.554097
\(100\) 0 0
\(101\) 7.49853 0.746132 0.373066 0.927805i \(-0.378307\pi\)
0.373066 + 0.927805i \(0.378307\pi\)
\(102\) 0 0
\(103\) 14.5685 1.43548 0.717738 0.696314i \(-0.245178\pi\)
0.717738 + 0.696314i \(0.245178\pi\)
\(104\) 0 0
\(105\) 1.13856 0.111112
\(106\) 0 0
\(107\) −11.7736 −1.13820 −0.569100 0.822269i \(-0.692708\pi\)
−0.569100 + 0.822269i \(0.692708\pi\)
\(108\) 0 0
\(109\) 0.787665 0.0754446 0.0377223 0.999288i \(-0.487990\pi\)
0.0377223 + 0.999288i \(0.487990\pi\)
\(110\) 0 0
\(111\) −2.70367 −0.256621
\(112\) 0 0
\(113\) −7.96160 −0.748964 −0.374482 0.927234i \(-0.622180\pi\)
−0.374482 + 0.927234i \(0.622180\pi\)
\(114\) 0 0
\(115\) −2.17127 −0.202472
\(116\) 0 0
\(117\) −1.70367 −0.157504
\(118\) 0 0
\(119\) −2.98080 −0.273249
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) −2.01351 −0.181552
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −3.23607 −0.287155 −0.143577 0.989639i \(-0.545861\pi\)
−0.143577 + 0.989639i \(0.545861\pi\)
\(128\) 0 0
\(129\) 6.27713 0.552670
\(130\) 0 0
\(131\) −6.42467 −0.561326 −0.280663 0.959806i \(-0.590554\pi\)
−0.280663 + 0.959806i \(0.590554\pi\)
\(132\) 0 0
\(133\) −1.93974 −0.168197
\(134\) 0 0
\(135\) 5.35543 0.460922
\(136\) 0 0
\(137\) 1.83708 0.156952 0.0784760 0.996916i \(-0.474995\pi\)
0.0784760 + 0.996916i \(0.474995\pi\)
\(138\) 0 0
\(139\) 8.62874 0.731880 0.365940 0.930638i \(-0.380747\pi\)
0.365940 + 0.930638i \(0.380747\pi\)
\(140\) 0 0
\(141\) 8.58128 0.722674
\(142\) 0 0
\(143\) −3.23607 −0.270614
\(144\) 0 0
\(145\) 3.58697 0.297881
\(146\) 0 0
\(147\) −1.13856 −0.0939072
\(148\) 0 0
\(149\) −18.8405 −1.54347 −0.771735 0.635944i \(-0.780611\pi\)
−0.771735 + 0.635944i \(0.780611\pi\)
\(150\) 0 0
\(151\) −12.6704 −1.03111 −0.515553 0.856858i \(-0.672413\pi\)
−0.515553 + 0.856858i \(0.672413\pi\)
\(152\) 0 0
\(153\) −5.07830 −0.410557
\(154\) 0 0
\(155\) 7.29517 0.585962
\(156\) 0 0
\(157\) 2.32717 0.185728 0.0928641 0.995679i \(-0.470398\pi\)
0.0928641 + 0.995679i \(0.470398\pi\)
\(158\) 0 0
\(159\) 3.87948 0.307663
\(160\) 0 0
\(161\) 2.17127 0.171120
\(162\) 0 0
\(163\) −11.1623 −0.874299 −0.437149 0.899389i \(-0.644012\pi\)
−0.437149 + 0.899389i \(0.644012\pi\)
\(164\) 0 0
\(165\) 3.68447 0.286836
\(166\) 0 0
\(167\) −8.58128 −0.664039 −0.332020 0.943272i \(-0.607730\pi\)
−0.332020 + 0.943272i \(0.607730\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −3.30468 −0.252715
\(172\) 0 0
\(173\) −13.4010 −1.01886 −0.509431 0.860512i \(-0.670144\pi\)
−0.509431 + 0.860512i \(0.670144\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 13.2503 0.995953
\(178\) 0 0
\(179\) 8.45356 0.631849 0.315925 0.948784i \(-0.397685\pi\)
0.315925 + 0.948784i \(0.397685\pi\)
\(180\) 0 0
\(181\) −21.6989 −1.61286 −0.806432 0.591327i \(-0.798604\pi\)
−0.806432 + 0.591327i \(0.798604\pi\)
\(182\) 0 0
\(183\) 6.47214 0.478434
\(184\) 0 0
\(185\) 2.37463 0.174586
\(186\) 0 0
\(187\) −9.64607 −0.705391
\(188\) 0 0
\(189\) −5.35543 −0.389550
\(190\) 0 0
\(191\) −14.0975 −1.02006 −0.510030 0.860157i \(-0.670366\pi\)
−0.510030 + 0.860157i \(0.670366\pi\)
\(192\) 0 0
\(193\) 14.9077 1.07308 0.536541 0.843874i \(-0.319731\pi\)
0.536541 + 0.843874i \(0.319731\pi\)
\(194\) 0 0
\(195\) −1.13856 −0.0815343
\(196\) 0 0
\(197\) 18.1503 1.29315 0.646577 0.762848i \(-0.276200\pi\)
0.646577 + 0.762848i \(0.276200\pi\)
\(198\) 0 0
\(199\) −26.6070 −1.88612 −0.943062 0.332618i \(-0.892068\pi\)
−0.943062 + 0.332618i \(0.892068\pi\)
\(200\) 0 0
\(201\) −2.32904 −0.164278
\(202\) 0 0
\(203\) −3.58697 −0.251756
\(204\) 0 0
\(205\) 1.76846 0.123515
\(206\) 0 0
\(207\) 3.69914 0.257108
\(208\) 0 0
\(209\) −6.27713 −0.434198
\(210\) 0 0
\(211\) 6.28432 0.432631 0.216315 0.976324i \(-0.430596\pi\)
0.216315 + 0.976324i \(0.430596\pi\)
\(212\) 0 0
\(213\) 12.4337 0.851946
\(214\) 0 0
\(215\) −5.51320 −0.375997
\(216\) 0 0
\(217\) −7.29517 −0.495229
\(218\) 0 0
\(219\) 8.07449 0.545623
\(220\) 0 0
\(221\) 2.98080 0.200510
\(222\) 0 0
\(223\) 17.8885 1.19791 0.598953 0.800784i \(-0.295584\pi\)
0.598953 + 0.800784i \(0.295584\pi\)
\(224\) 0 0
\(225\) −1.70367 −0.113578
\(226\) 0 0
\(227\) 7.77566 0.516089 0.258044 0.966133i \(-0.416922\pi\)
0.258044 + 0.966133i \(0.416922\pi\)
\(228\) 0 0
\(229\) −5.60048 −0.370090 −0.185045 0.982730i \(-0.559243\pi\)
−0.185045 + 0.982730i \(0.559243\pi\)
\(230\) 0 0
\(231\) −3.68447 −0.242420
\(232\) 0 0
\(233\) 6.88011 0.450731 0.225365 0.974274i \(-0.427642\pi\)
0.225365 + 0.974274i \(0.427642\pi\)
\(234\) 0 0
\(235\) −7.53693 −0.491655
\(236\) 0 0
\(237\) 16.8449 1.09419
\(238\) 0 0
\(239\) 21.2722 1.37598 0.687991 0.725720i \(-0.258493\pi\)
0.687991 + 0.725720i \(0.258493\pi\)
\(240\) 0 0
\(241\) −7.89805 −0.508758 −0.254379 0.967105i \(-0.581871\pi\)
−0.254379 + 0.967105i \(0.581871\pi\)
\(242\) 0 0
\(243\) −14.9431 −0.958601
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 1.93974 0.123423
\(248\) 0 0
\(249\) −2.98267 −0.189019
\(250\) 0 0
\(251\) −29.0792 −1.83546 −0.917731 0.397203i \(-0.869981\pi\)
−0.917731 + 0.397203i \(0.869981\pi\)
\(252\) 0 0
\(253\) 7.02639 0.441746
\(254\) 0 0
\(255\) −3.39383 −0.212530
\(256\) 0 0
\(257\) 8.12959 0.507110 0.253555 0.967321i \(-0.418400\pi\)
0.253555 + 0.967321i \(0.418400\pi\)
\(258\) 0 0
\(259\) −2.37463 −0.147552
\(260\) 0 0
\(261\) −6.11101 −0.378262
\(262\) 0 0
\(263\) 0.0416886 0.00257063 0.00128531 0.999999i \(-0.499591\pi\)
0.00128531 + 0.999999i \(0.499591\pi\)
\(264\) 0 0
\(265\) −3.40734 −0.209311
\(266\) 0 0
\(267\) −4.47401 −0.273805
\(268\) 0 0
\(269\) 24.0068 1.46372 0.731859 0.681456i \(-0.238653\pi\)
0.731859 + 0.681456i \(0.238653\pi\)
\(270\) 0 0
\(271\) −23.8815 −1.45070 −0.725349 0.688381i \(-0.758322\pi\)
−0.725349 + 0.688381i \(0.758322\pi\)
\(272\) 0 0
\(273\) 1.13856 0.0689090
\(274\) 0 0
\(275\) −3.23607 −0.195142
\(276\) 0 0
\(277\) 23.5256 1.41351 0.706757 0.707457i \(-0.250158\pi\)
0.706757 + 0.707457i \(0.250158\pi\)
\(278\) 0 0
\(279\) −12.4286 −0.744079
\(280\) 0 0
\(281\) −18.8405 −1.12393 −0.561964 0.827162i \(-0.689954\pi\)
−0.561964 + 0.827162i \(0.689954\pi\)
\(282\) 0 0
\(283\) −9.11742 −0.541974 −0.270987 0.962583i \(-0.587350\pi\)
−0.270987 + 0.962583i \(0.587350\pi\)
\(284\) 0 0
\(285\) −2.20852 −0.130821
\(286\) 0 0
\(287\) −1.76846 −0.104389
\(288\) 0 0
\(289\) −8.11483 −0.477343
\(290\) 0 0
\(291\) 1.52786 0.0895650
\(292\) 0 0
\(293\) −10.2387 −0.598153 −0.299076 0.954229i \(-0.596679\pi\)
−0.299076 + 0.954229i \(0.596679\pi\)
\(294\) 0 0
\(295\) −11.6377 −0.677574
\(296\) 0 0
\(297\) −17.3305 −1.00562
\(298\) 0 0
\(299\) −2.17127 −0.125568
\(300\) 0 0
\(301\) 5.51320 0.317775
\(302\) 0 0
\(303\) −8.53756 −0.490470
\(304\) 0 0
\(305\) −5.68447 −0.325492
\(306\) 0 0
\(307\) 18.9059 1.07902 0.539508 0.841981i \(-0.318610\pi\)
0.539508 + 0.841981i \(0.318610\pi\)
\(308\) 0 0
\(309\) −16.5872 −0.943610
\(310\) 0 0
\(311\) −17.7499 −1.00650 −0.503252 0.864140i \(-0.667863\pi\)
−0.503252 + 0.864140i \(0.667863\pi\)
\(312\) 0 0
\(313\) 5.53187 0.312680 0.156340 0.987703i \(-0.450031\pi\)
0.156340 + 0.987703i \(0.450031\pi\)
\(314\) 0 0
\(315\) 1.70367 0.0959910
\(316\) 0 0
\(317\) −2.13491 −0.119908 −0.0599542 0.998201i \(-0.519095\pi\)
−0.0599542 + 0.998201i \(0.519095\pi\)
\(318\) 0 0
\(319\) −11.6077 −0.649905
\(320\) 0 0
\(321\) 13.4050 0.748196
\(322\) 0 0
\(323\) 5.78198 0.321718
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) −0.896807 −0.0495935
\(328\) 0 0
\(329\) 7.53693 0.415524
\(330\) 0 0
\(331\) 12.8528 0.706454 0.353227 0.935538i \(-0.385084\pi\)
0.353227 + 0.935538i \(0.385084\pi\)
\(332\) 0 0
\(333\) −4.04559 −0.221697
\(334\) 0 0
\(335\) 2.04559 0.111763
\(336\) 0 0
\(337\) 20.8147 1.13385 0.566924 0.823770i \(-0.308133\pi\)
0.566924 + 0.823770i \(0.308133\pi\)
\(338\) 0 0
\(339\) 9.06479 0.492332
\(340\) 0 0
\(341\) −23.6077 −1.27843
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 2.47214 0.133095
\(346\) 0 0
\(347\) 17.2889 0.928114 0.464057 0.885805i \(-0.346393\pi\)
0.464057 + 0.885805i \(0.346393\pi\)
\(348\) 0 0
\(349\) −28.9920 −1.55191 −0.775953 0.630791i \(-0.782731\pi\)
−0.775953 + 0.630791i \(0.782731\pi\)
\(350\) 0 0
\(351\) 5.35543 0.285852
\(352\) 0 0
\(353\) −30.9149 −1.64544 −0.822718 0.568450i \(-0.807543\pi\)
−0.822718 + 0.568450i \(0.807543\pi\)
\(354\) 0 0
\(355\) −10.9205 −0.579602
\(356\) 0 0
\(357\) 3.39383 0.179621
\(358\) 0 0
\(359\) −25.6584 −1.35420 −0.677100 0.735891i \(-0.736764\pi\)
−0.677100 + 0.735891i \(0.736764\pi\)
\(360\) 0 0
\(361\) −15.2374 −0.801969
\(362\) 0 0
\(363\) 0.601007 0.0315447
\(364\) 0 0
\(365\) −7.09181 −0.371203
\(366\) 0 0
\(367\) −12.4317 −0.648930 −0.324465 0.945898i \(-0.605184\pi\)
−0.324465 + 0.945898i \(0.605184\pi\)
\(368\) 0 0
\(369\) −3.01288 −0.156844
\(370\) 0 0
\(371\) 3.40734 0.176900
\(372\) 0 0
\(373\) −29.1613 −1.50991 −0.754957 0.655774i \(-0.772343\pi\)
−0.754957 + 0.655774i \(0.772343\pi\)
\(374\) 0 0
\(375\) −1.13856 −0.0587952
\(376\) 0 0
\(377\) 3.58697 0.184738
\(378\) 0 0
\(379\) 12.4921 0.641677 0.320839 0.947134i \(-0.396035\pi\)
0.320839 + 0.947134i \(0.396035\pi\)
\(380\) 0 0
\(381\) 3.68447 0.188761
\(382\) 0 0
\(383\) −11.7012 −0.597902 −0.298951 0.954268i \(-0.596637\pi\)
−0.298951 + 0.954268i \(0.596637\pi\)
\(384\) 0 0
\(385\) 3.23607 0.164925
\(386\) 0 0
\(387\) 9.39268 0.477457
\(388\) 0 0
\(389\) 15.1758 0.769444 0.384722 0.923032i \(-0.374297\pi\)
0.384722 + 0.923032i \(0.374297\pi\)
\(390\) 0 0
\(391\) −6.47214 −0.327310
\(392\) 0 0
\(393\) 7.31490 0.368988
\(394\) 0 0
\(395\) −14.7949 −0.744410
\(396\) 0 0
\(397\) 35.0150 1.75735 0.878677 0.477417i \(-0.158427\pi\)
0.878677 + 0.477417i \(0.158427\pi\)
\(398\) 0 0
\(399\) 2.20852 0.110564
\(400\) 0 0
\(401\) −17.0059 −0.849237 −0.424618 0.905372i \(-0.639592\pi\)
−0.424618 + 0.905372i \(0.639592\pi\)
\(402\) 0 0
\(403\) 7.29517 0.363398
\(404\) 0 0
\(405\) −0.986489 −0.0490191
\(406\) 0 0
\(407\) −7.68447 −0.380905
\(408\) 0 0
\(409\) 4.05279 0.200397 0.100199 0.994967i \(-0.468052\pi\)
0.100199 + 0.994967i \(0.468052\pi\)
\(410\) 0 0
\(411\) −2.09163 −0.103172
\(412\) 0 0
\(413\) 11.6377 0.572655
\(414\) 0 0
\(415\) 2.61968 0.128595
\(416\) 0 0
\(417\) −9.82438 −0.481102
\(418\) 0 0
\(419\) −8.73256 −0.426614 −0.213307 0.976985i \(-0.568423\pi\)
−0.213307 + 0.976985i \(0.568423\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 0 0
\(423\) 12.8405 0.624324
\(424\) 0 0
\(425\) 2.98080 0.144590
\(426\) 0 0
\(427\) 5.68447 0.275091
\(428\) 0 0
\(429\) 3.68447 0.177888
\(430\) 0 0
\(431\) 3.20530 0.154394 0.0771970 0.997016i \(-0.475403\pi\)
0.0771970 + 0.997016i \(0.475403\pi\)
\(432\) 0 0
\(433\) −33.4199 −1.60606 −0.803028 0.595941i \(-0.796779\pi\)
−0.803028 + 0.595941i \(0.796779\pi\)
\(434\) 0 0
\(435\) −4.08399 −0.195812
\(436\) 0 0
\(437\) −4.21171 −0.201473
\(438\) 0 0
\(439\) 23.6551 1.12900 0.564499 0.825434i \(-0.309069\pi\)
0.564499 + 0.825434i \(0.309069\pi\)
\(440\) 0 0
\(441\) −1.70367 −0.0811272
\(442\) 0 0
\(443\) 6.15864 0.292606 0.146303 0.989240i \(-0.453263\pi\)
0.146303 + 0.989240i \(0.453263\pi\)
\(444\) 0 0
\(445\) 3.92952 0.186277
\(446\) 0 0
\(447\) 21.4511 1.01460
\(448\) 0 0
\(449\) 15.4073 0.727117 0.363559 0.931571i \(-0.381562\pi\)
0.363559 + 0.931571i \(0.381562\pi\)
\(450\) 0 0
\(451\) −5.72287 −0.269479
\(452\) 0 0
\(453\) 14.4261 0.677797
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) 12.6518 0.591824 0.295912 0.955215i \(-0.404376\pi\)
0.295912 + 0.955215i \(0.404376\pi\)
\(458\) 0 0
\(459\) 15.9635 0.745111
\(460\) 0 0
\(461\) 7.19838 0.335262 0.167631 0.985850i \(-0.446388\pi\)
0.167631 + 0.985850i \(0.446388\pi\)
\(462\) 0 0
\(463\) 19.2849 0.896248 0.448124 0.893972i \(-0.352092\pi\)
0.448124 + 0.893972i \(0.352092\pi\)
\(464\) 0 0
\(465\) −8.30602 −0.385183
\(466\) 0 0
\(467\) 2.59407 0.120039 0.0600197 0.998197i \(-0.480884\pi\)
0.0600197 + 0.998197i \(0.480884\pi\)
\(468\) 0 0
\(469\) −2.04559 −0.0944567
\(470\) 0 0
\(471\) −2.64963 −0.122088
\(472\) 0 0
\(473\) 17.8411 0.820334
\(474\) 0 0
\(475\) 1.93974 0.0890013
\(476\) 0 0
\(477\) 5.80499 0.265792
\(478\) 0 0
\(479\) −18.6808 −0.853548 −0.426774 0.904358i \(-0.640350\pi\)
−0.426774 + 0.904358i \(0.640350\pi\)
\(480\) 0 0
\(481\) 2.37463 0.108274
\(482\) 0 0
\(483\) −2.47214 −0.112486
\(484\) 0 0
\(485\) −1.34192 −0.0609335
\(486\) 0 0
\(487\) −8.27081 −0.374786 −0.187393 0.982285i \(-0.560004\pi\)
−0.187393 + 0.982285i \(0.560004\pi\)
\(488\) 0 0
\(489\) 12.7090 0.574720
\(490\) 0 0
\(491\) 1.87448 0.0845940 0.0422970 0.999105i \(-0.486532\pi\)
0.0422970 + 0.999105i \(0.486532\pi\)
\(492\) 0 0
\(493\) 10.6920 0.481545
\(494\) 0 0
\(495\) 5.51320 0.247800
\(496\) 0 0
\(497\) 10.9205 0.489853
\(498\) 0 0
\(499\) −36.2217 −1.62151 −0.810754 0.585387i \(-0.800943\pi\)
−0.810754 + 0.585387i \(0.800943\pi\)
\(500\) 0 0
\(501\) 9.77034 0.436506
\(502\) 0 0
\(503\) −1.53364 −0.0683817 −0.0341908 0.999415i \(-0.510885\pi\)
−0.0341908 + 0.999415i \(0.510885\pi\)
\(504\) 0 0
\(505\) 7.49853 0.333680
\(506\) 0 0
\(507\) −1.13856 −0.0505654
\(508\) 0 0
\(509\) 8.06229 0.357355 0.178677 0.983908i \(-0.442818\pi\)
0.178677 + 0.983908i \(0.442818\pi\)
\(510\) 0 0
\(511\) 7.09181 0.313723
\(512\) 0 0
\(513\) 10.3881 0.458648
\(514\) 0 0
\(515\) 14.5685 0.641964
\(516\) 0 0
\(517\) 24.3900 1.07267
\(518\) 0 0
\(519\) 15.2579 0.669749
\(520\) 0 0
\(521\) −10.4835 −0.459291 −0.229646 0.973274i \(-0.573757\pi\)
−0.229646 + 0.973274i \(0.573757\pi\)
\(522\) 0 0
\(523\) −25.2517 −1.10418 −0.552090 0.833784i \(-0.686170\pi\)
−0.552090 + 0.833784i \(0.686170\pi\)
\(524\) 0 0
\(525\) 1.13856 0.0496910
\(526\) 0 0
\(527\) 21.7454 0.947247
\(528\) 0 0
\(529\) −18.2856 −0.795025
\(530\) 0 0
\(531\) 19.8269 0.860412
\(532\) 0 0
\(533\) 1.76846 0.0766007
\(534\) 0 0
\(535\) −11.7736 −0.509018
\(536\) 0 0
\(537\) −9.62493 −0.415346
\(538\) 0 0
\(539\) −3.23607 −0.139387
\(540\) 0 0
\(541\) −15.3947 −0.661870 −0.330935 0.943654i \(-0.607364\pi\)
−0.330935 + 0.943654i \(0.607364\pi\)
\(542\) 0 0
\(543\) 24.7055 1.06022
\(544\) 0 0
\(545\) 0.787665 0.0337398
\(546\) 0 0
\(547\) 37.9613 1.62311 0.811554 0.584277i \(-0.198622\pi\)
0.811554 + 0.584277i \(0.198622\pi\)
\(548\) 0 0
\(549\) 9.68447 0.413323
\(550\) 0 0
\(551\) 6.95778 0.296411
\(552\) 0 0
\(553\) 14.7949 0.629141
\(554\) 0 0
\(555\) −2.70367 −0.114764
\(556\) 0 0
\(557\) −43.5779 −1.84645 −0.923227 0.384254i \(-0.874459\pi\)
−0.923227 + 0.384254i \(0.874459\pi\)
\(558\) 0 0
\(559\) −5.51320 −0.233183
\(560\) 0 0
\(561\) 10.9827 0.463689
\(562\) 0 0
\(563\) 21.6423 0.912116 0.456058 0.889950i \(-0.349261\pi\)
0.456058 + 0.889950i \(0.349261\pi\)
\(564\) 0 0
\(565\) −7.96160 −0.334947
\(566\) 0 0
\(567\) 0.986489 0.0414287
\(568\) 0 0
\(569\) −2.47845 −0.103902 −0.0519511 0.998650i \(-0.516544\pi\)
−0.0519511 + 0.998650i \(0.516544\pi\)
\(570\) 0 0
\(571\) −10.0040 −0.418655 −0.209327 0.977846i \(-0.567127\pi\)
−0.209327 + 0.977846i \(0.567127\pi\)
\(572\) 0 0
\(573\) 16.0509 0.670537
\(574\) 0 0
\(575\) −2.17127 −0.0905484
\(576\) 0 0
\(577\) 28.2953 1.17795 0.588974 0.808152i \(-0.299532\pi\)
0.588974 + 0.808152i \(0.299532\pi\)
\(578\) 0 0
\(579\) −16.9734 −0.705391
\(580\) 0 0
\(581\) −2.61968 −0.108683
\(582\) 0 0
\(583\) 11.0264 0.456667
\(584\) 0 0
\(585\) −1.70367 −0.0704381
\(586\) 0 0
\(587\) 0.614357 0.0253572 0.0126786 0.999920i \(-0.495964\pi\)
0.0126786 + 0.999920i \(0.495964\pi\)
\(588\) 0 0
\(589\) 14.1507 0.583071
\(590\) 0 0
\(591\) −20.6653 −0.850056
\(592\) 0 0
\(593\) −34.6227 −1.42178 −0.710892 0.703302i \(-0.751708\pi\)
−0.710892 + 0.703302i \(0.751708\pi\)
\(594\) 0 0
\(595\) −2.98080 −0.122201
\(596\) 0 0
\(597\) 30.2938 1.23984
\(598\) 0 0
\(599\) 38.2311 1.56208 0.781040 0.624481i \(-0.214689\pi\)
0.781040 + 0.624481i \(0.214689\pi\)
\(600\) 0 0
\(601\) −36.3390 −1.48230 −0.741149 0.671341i \(-0.765719\pi\)
−0.741149 + 0.671341i \(0.765719\pi\)
\(602\) 0 0
\(603\) −3.48502 −0.141921
\(604\) 0 0
\(605\) −0.527864 −0.0214607
\(606\) 0 0
\(607\) −2.03537 −0.0826132 −0.0413066 0.999147i \(-0.513152\pi\)
−0.0413066 + 0.999147i \(0.513152\pi\)
\(608\) 0 0
\(609\) 4.08399 0.165492
\(610\) 0 0
\(611\) −7.53693 −0.304912
\(612\) 0 0
\(613\) 29.8464 1.20548 0.602742 0.797936i \(-0.294075\pi\)
0.602742 + 0.797936i \(0.294075\pi\)
\(614\) 0 0
\(615\) −2.01351 −0.0811926
\(616\) 0 0
\(617\) 2.72163 0.109569 0.0547843 0.998498i \(-0.482553\pi\)
0.0547843 + 0.998498i \(0.482553\pi\)
\(618\) 0 0
\(619\) −8.52795 −0.342767 −0.171384 0.985204i \(-0.554824\pi\)
−0.171384 + 0.985204i \(0.554824\pi\)
\(620\) 0 0
\(621\) −11.6281 −0.466620
\(622\) 0 0
\(623\) −3.92952 −0.157433
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.14691 0.285420
\(628\) 0 0
\(629\) 7.07830 0.282230
\(630\) 0 0
\(631\) −1.39268 −0.0554415 −0.0277208 0.999616i \(-0.508825\pi\)
−0.0277208 + 0.999616i \(0.508825\pi\)
\(632\) 0 0
\(633\) −7.15510 −0.284390
\(634\) 0 0
\(635\) −3.23607 −0.128419
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 18.6050 0.736003
\(640\) 0 0
\(641\) 3.19838 0.126329 0.0631643 0.998003i \(-0.479881\pi\)
0.0631643 + 0.998003i \(0.479881\pi\)
\(642\) 0 0
\(643\) −25.2845 −0.997124 −0.498562 0.866854i \(-0.666138\pi\)
−0.498562 + 0.866854i \(0.666138\pi\)
\(644\) 0 0
\(645\) 6.27713 0.247162
\(646\) 0 0
\(647\) 0.414007 0.0162763 0.00813814 0.999967i \(-0.497410\pi\)
0.00813814 + 0.999967i \(0.497410\pi\)
\(648\) 0 0
\(649\) 37.6605 1.47830
\(650\) 0 0
\(651\) 8.30602 0.325539
\(652\) 0 0
\(653\) 22.4170 0.877246 0.438623 0.898671i \(-0.355466\pi\)
0.438623 + 0.898671i \(0.355466\pi\)
\(654\) 0 0
\(655\) −6.42467 −0.251033
\(656\) 0 0
\(657\) 12.0821 0.471368
\(658\) 0 0
\(659\) 18.8679 0.734990 0.367495 0.930026i \(-0.380216\pi\)
0.367495 + 0.930026i \(0.380216\pi\)
\(660\) 0 0
\(661\) 14.8398 0.577203 0.288601 0.957449i \(-0.406810\pi\)
0.288601 + 0.957449i \(0.406810\pi\)
\(662\) 0 0
\(663\) −3.39383 −0.131806
\(664\) 0 0
\(665\) −1.93974 −0.0752199
\(666\) 0 0
\(667\) −7.78829 −0.301564
\(668\) 0 0
\(669\) −20.3673 −0.787444
\(670\) 0 0
\(671\) 18.3953 0.710144
\(672\) 0 0
\(673\) −0.488836 −0.0188432 −0.00942162 0.999956i \(-0.502999\pi\)
−0.00942162 + 0.999956i \(0.502999\pi\)
\(674\) 0 0
\(675\) 5.35543 0.206131
\(676\) 0 0
\(677\) 24.5736 0.944442 0.472221 0.881480i \(-0.343452\pi\)
0.472221 + 0.881480i \(0.343452\pi\)
\(678\) 0 0
\(679\) 1.34192 0.0514982
\(680\) 0 0
\(681\) −8.85309 −0.339251
\(682\) 0 0
\(683\) 19.8392 0.759126 0.379563 0.925166i \(-0.376074\pi\)
0.379563 + 0.925166i \(0.376074\pi\)
\(684\) 0 0
\(685\) 1.83708 0.0701910
\(686\) 0 0
\(687\) 6.37650 0.243279
\(688\) 0 0
\(689\) −3.40734 −0.129809
\(690\) 0 0
\(691\) 25.4788 0.969259 0.484630 0.874719i \(-0.338954\pi\)
0.484630 + 0.874719i \(0.338954\pi\)
\(692\) 0 0
\(693\) −5.51320 −0.209429
\(694\) 0 0
\(695\) 8.62874 0.327307
\(696\) 0 0
\(697\) 5.27144 0.199670
\(698\) 0 0
\(699\) −7.83344 −0.296288
\(700\) 0 0
\(701\) 27.8981 1.05369 0.526847 0.849960i \(-0.323374\pi\)
0.526847 + 0.849960i \(0.323374\pi\)
\(702\) 0 0
\(703\) 4.60617 0.173725
\(704\) 0 0
\(705\) 8.58128 0.323190
\(706\) 0 0
\(707\) −7.49853 −0.282011
\(708\) 0 0
\(709\) 21.7860 0.818190 0.409095 0.912492i \(-0.365845\pi\)
0.409095 + 0.912492i \(0.365845\pi\)
\(710\) 0 0
\(711\) 25.2056 0.945283
\(712\) 0 0
\(713\) −15.8398 −0.593206
\(714\) 0 0
\(715\) −3.23607 −0.121022
\(716\) 0 0
\(717\) −24.2197 −0.904502
\(718\) 0 0
\(719\) −2.76064 −0.102955 −0.0514773 0.998674i \(-0.516393\pi\)
−0.0514773 + 0.998674i \(0.516393\pi\)
\(720\) 0 0
\(721\) −14.5685 −0.542559
\(722\) 0 0
\(723\) 8.99244 0.334432
\(724\) 0 0
\(725\) 3.58697 0.133217
\(726\) 0 0
\(727\) 5.00366 0.185575 0.0927877 0.995686i \(-0.470422\pi\)
0.0927877 + 0.995686i \(0.470422\pi\)
\(728\) 0 0
\(729\) 19.9732 0.739747
\(730\) 0 0
\(731\) −16.4337 −0.607824
\(732\) 0 0
\(733\) −1.26512 −0.0467283 −0.0233642 0.999727i \(-0.507438\pi\)
−0.0233642 + 0.999727i \(0.507438\pi\)
\(734\) 0 0
\(735\) −1.13856 −0.0419966
\(736\) 0 0
\(737\) −6.61968 −0.243839
\(738\) 0 0
\(739\) 28.1432 1.03526 0.517632 0.855603i \(-0.326814\pi\)
0.517632 + 0.855603i \(0.326814\pi\)
\(740\) 0 0
\(741\) −2.20852 −0.0811319
\(742\) 0 0
\(743\) −13.1298 −0.481685 −0.240842 0.970564i \(-0.577424\pi\)
−0.240842 + 0.970564i \(0.577424\pi\)
\(744\) 0 0
\(745\) −18.8405 −0.690261
\(746\) 0 0
\(747\) −4.46307 −0.163295
\(748\) 0 0
\(749\) 11.7736 0.430199
\(750\) 0 0
\(751\) 27.3409 0.997685 0.498842 0.866693i \(-0.333759\pi\)
0.498842 + 0.866693i \(0.333759\pi\)
\(752\) 0 0
\(753\) 33.1085 1.20654
\(754\) 0 0
\(755\) −12.6704 −0.461124
\(756\) 0 0
\(757\) 29.5640 1.07452 0.537260 0.843417i \(-0.319459\pi\)
0.537260 + 0.843417i \(0.319459\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −17.3639 −0.629440 −0.314720 0.949185i \(-0.601911\pi\)
−0.314720 + 0.949185i \(0.601911\pi\)
\(762\) 0 0
\(763\) −0.787665 −0.0285154
\(764\) 0 0
\(765\) −5.07830 −0.183606
\(766\) 0 0
\(767\) −11.6377 −0.420214
\(768\) 0 0
\(769\) 10.9443 0.394661 0.197330 0.980337i \(-0.436773\pi\)
0.197330 + 0.980337i \(0.436773\pi\)
\(770\) 0 0
\(771\) −9.25606 −0.333349
\(772\) 0 0
\(773\) −38.4003 −1.38116 −0.690582 0.723254i \(-0.742646\pi\)
−0.690582 + 0.723254i \(0.742646\pi\)
\(774\) 0 0
\(775\) 7.29517 0.262050
\(776\) 0 0
\(777\) 2.70367 0.0969937
\(778\) 0 0
\(779\) 3.43036 0.122905
\(780\) 0 0
\(781\) 35.3396 1.26455
\(782\) 0 0
\(783\) 19.2098 0.686501
\(784\) 0 0
\(785\) 2.32717 0.0830602
\(786\) 0 0
\(787\) 22.1836 0.790761 0.395380 0.918517i \(-0.370613\pi\)
0.395380 + 0.918517i \(0.370613\pi\)
\(788\) 0 0
\(789\) −0.0474651 −0.00168980
\(790\) 0 0
\(791\) 7.96160 0.283082
\(792\) 0 0
\(793\) −5.68447 −0.201861
\(794\) 0 0
\(795\) 3.87948 0.137591
\(796\) 0 0
\(797\) 21.2368 0.752245 0.376123 0.926570i \(-0.377257\pi\)
0.376123 + 0.926570i \(0.377257\pi\)
\(798\) 0 0
\(799\) −22.4661 −0.794793
\(800\) 0 0
\(801\) −6.69461 −0.236542
\(802\) 0 0
\(803\) 22.9496 0.809874
\(804\) 0 0
\(805\) 2.17127 0.0765274
\(806\) 0 0
\(807\) −27.3332 −0.962175
\(808\) 0 0
\(809\) −20.3677 −0.716090 −0.358045 0.933704i \(-0.616557\pi\)
−0.358045 + 0.933704i \(0.616557\pi\)
\(810\) 0 0
\(811\) 41.3800 1.45305 0.726525 0.687140i \(-0.241134\pi\)
0.726525 + 0.687140i \(0.241134\pi\)
\(812\) 0 0
\(813\) 27.1906 0.953617
\(814\) 0 0
\(815\) −11.1623 −0.390998
\(816\) 0 0
\(817\) −10.6942 −0.374141
\(818\) 0 0
\(819\) 1.70367 0.0595311
\(820\) 0 0
\(821\) −28.8328 −1.00627 −0.503136 0.864207i \(-0.667821\pi\)
−0.503136 + 0.864207i \(0.667821\pi\)
\(822\) 0 0
\(823\) −11.1527 −0.388758 −0.194379 0.980926i \(-0.562269\pi\)
−0.194379 + 0.980926i \(0.562269\pi\)
\(824\) 0 0
\(825\) 3.68447 0.128277
\(826\) 0 0
\(827\) −27.0330 −0.940028 −0.470014 0.882659i \(-0.655751\pi\)
−0.470014 + 0.882659i \(0.655751\pi\)
\(828\) 0 0
\(829\) 22.2928 0.774260 0.387130 0.922025i \(-0.373466\pi\)
0.387130 + 0.922025i \(0.373466\pi\)
\(830\) 0 0
\(831\) −26.7854 −0.929174
\(832\) 0 0
\(833\) 2.98080 0.103279
\(834\) 0 0
\(835\) −8.58128 −0.296967
\(836\) 0 0
\(837\) 39.0688 1.35042
\(838\) 0 0
\(839\) 50.3602 1.73863 0.869314 0.494260i \(-0.164561\pi\)
0.869314 + 0.494260i \(0.164561\pi\)
\(840\) 0 0
\(841\) −16.1337 −0.556333
\(842\) 0 0
\(843\) 21.4511 0.738814
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 0.527864 0.0181376
\(848\) 0 0
\(849\) 10.3808 0.356267
\(850\) 0 0
\(851\) −5.15598 −0.176745
\(852\) 0 0
\(853\) 35.8989 1.22915 0.614577 0.788857i \(-0.289327\pi\)
0.614577 + 0.788857i \(0.289327\pi\)
\(854\) 0 0
\(855\) −3.30468 −0.113018
\(856\) 0 0
\(857\) −8.48752 −0.289928 −0.144964 0.989437i \(-0.546307\pi\)
−0.144964 + 0.989437i \(0.546307\pi\)
\(858\) 0 0
\(859\) 16.8200 0.573891 0.286946 0.957947i \(-0.407360\pi\)
0.286946 + 0.957947i \(0.407360\pi\)
\(860\) 0 0
\(861\) 2.01351 0.0686203
\(862\) 0 0
\(863\) −28.5718 −0.972594 −0.486297 0.873793i \(-0.661653\pi\)
−0.486297 + 0.873793i \(0.661653\pi\)
\(864\) 0 0
\(865\) −13.4010 −0.455649
\(866\) 0 0
\(867\) 9.23926 0.313782
\(868\) 0 0
\(869\) 47.8772 1.62412
\(870\) 0 0
\(871\) 2.04559 0.0693123
\(872\) 0 0
\(873\) 2.28619 0.0773759
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 16.7638 0.566072 0.283036 0.959109i \(-0.408658\pi\)
0.283036 + 0.959109i \(0.408658\pi\)
\(878\) 0 0
\(879\) 11.6575 0.393196
\(880\) 0 0
\(881\) −29.5933 −0.997023 −0.498512 0.866883i \(-0.666120\pi\)
−0.498512 + 0.866883i \(0.666120\pi\)
\(882\) 0 0
\(883\) 3.18203 0.107084 0.0535418 0.998566i \(-0.482949\pi\)
0.0535418 + 0.998566i \(0.482949\pi\)
\(884\) 0 0
\(885\) 13.2503 0.445404
\(886\) 0 0
\(887\) −4.31856 −0.145003 −0.0725015 0.997368i \(-0.523098\pi\)
−0.0725015 + 0.997368i \(0.523098\pi\)
\(888\) 0 0
\(889\) 3.23607 0.108534
\(890\) 0 0
\(891\) 3.19235 0.106948
\(892\) 0 0
\(893\) −14.6197 −0.489229
\(894\) 0 0
\(895\) 8.45356 0.282571
\(896\) 0 0
\(897\) 2.47214 0.0825422
\(898\) 0 0
\(899\) 26.1675 0.872736
\(900\) 0 0
\(901\) −10.1566 −0.338366
\(902\) 0 0
\(903\) −6.27713 −0.208890
\(904\) 0 0
\(905\) −21.6989 −0.721294
\(906\) 0 0
\(907\) −8.46653 −0.281127 −0.140563 0.990072i \(-0.544891\pi\)
−0.140563 + 0.990072i \(0.544891\pi\)
\(908\) 0 0
\(909\) −12.7750 −0.423721
\(910\) 0 0
\(911\) 4.45356 0.147553 0.0737766 0.997275i \(-0.476495\pi\)
0.0737766 + 0.997275i \(0.476495\pi\)
\(912\) 0 0
\(913\) −8.47746 −0.280563
\(914\) 0 0
\(915\) 6.47214 0.213962
\(916\) 0 0
\(917\) 6.42467 0.212161
\(918\) 0 0
\(919\) −0.310467 −0.0102414 −0.00512068 0.999987i \(-0.501630\pi\)
−0.00512068 + 0.999987i \(0.501630\pi\)
\(920\) 0 0
\(921\) −21.5256 −0.709291
\(922\) 0 0
\(923\) −10.9205 −0.359454
\(924\) 0 0
\(925\) 2.37463 0.0780774
\(926\) 0 0
\(927\) −24.8199 −0.815193
\(928\) 0 0
\(929\) −0.886424 −0.0290826 −0.0145413 0.999894i \(-0.504629\pi\)
−0.0145413 + 0.999894i \(0.504629\pi\)
\(930\) 0 0
\(931\) 1.93974 0.0635724
\(932\) 0 0
\(933\) 20.2094 0.661626
\(934\) 0 0
\(935\) −9.64607 −0.315460
\(936\) 0 0
\(937\) −22.3307 −0.729513 −0.364757 0.931103i \(-0.618848\pi\)
−0.364757 + 0.931103i \(0.618848\pi\)
\(938\) 0 0
\(939\) −6.29839 −0.205540
\(940\) 0 0
\(941\) −10.0673 −0.328184 −0.164092 0.986445i \(-0.552469\pi\)
−0.164092 + 0.986445i \(0.552469\pi\)
\(942\) 0 0
\(943\) −3.83982 −0.125042
\(944\) 0 0
\(945\) −5.35543 −0.174212
\(946\) 0 0
\(947\) 26.0559 0.846703 0.423352 0.905965i \(-0.360853\pi\)
0.423352 + 0.905965i \(0.360853\pi\)
\(948\) 0 0
\(949\) −7.09181 −0.230210
\(950\) 0 0
\(951\) 2.43073 0.0788218
\(952\) 0 0
\(953\) 11.7206 0.379666 0.189833 0.981816i \(-0.439205\pi\)
0.189833 + 0.981816i \(0.439205\pi\)
\(954\) 0 0
\(955\) −14.0975 −0.456185
\(956\) 0 0
\(957\) 13.2161 0.427215
\(958\) 0 0
\(959\) −1.83708 −0.0593222
\(960\) 0 0
\(961\) 22.2195 0.716759
\(962\) 0 0
\(963\) 20.0584 0.646373
\(964\) 0 0
\(965\) 14.9077 0.479897
\(966\) 0 0
\(967\) −12.0758 −0.388332 −0.194166 0.980969i \(-0.562200\pi\)
−0.194166 + 0.980969i \(0.562200\pi\)
\(968\) 0 0
\(969\) −6.58315 −0.211481
\(970\) 0 0
\(971\) −60.2174 −1.93247 −0.966234 0.257666i \(-0.917047\pi\)
−0.966234 + 0.257666i \(0.917047\pi\)
\(972\) 0 0
\(973\) −8.62874 −0.276625
\(974\) 0 0
\(975\) −1.13856 −0.0364632
\(976\) 0 0
\(977\) −27.1157 −0.867508 −0.433754 0.901031i \(-0.642811\pi\)
−0.433754 + 0.901031i \(0.642811\pi\)
\(978\) 0 0
\(979\) −12.7162 −0.406411
\(980\) 0 0
\(981\) −1.34192 −0.0428443
\(982\) 0 0
\(983\) −28.4274 −0.906692 −0.453346 0.891335i \(-0.649770\pi\)
−0.453346 + 0.891335i \(0.649770\pi\)
\(984\) 0 0
\(985\) 18.1503 0.578316
\(986\) 0 0
\(987\) −8.58128 −0.273145
\(988\) 0 0
\(989\) 11.9707 0.380645
\(990\) 0 0
\(991\) 25.3582 0.805529 0.402765 0.915304i \(-0.368049\pi\)
0.402765 + 0.915304i \(0.368049\pi\)
\(992\) 0 0
\(993\) −14.6337 −0.464388
\(994\) 0 0
\(995\) −26.6070 −0.843500
\(996\) 0 0
\(997\) −40.7554 −1.29074 −0.645368 0.763872i \(-0.723296\pi\)
−0.645368 + 0.763872i \(0.723296\pi\)
\(998\) 0 0
\(999\) 12.7172 0.402354
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 3640.2.a.s.1.2 4
4.3 odd 2 7280.2.a.ca.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3640.2.a.s.1.2 4 1.1 even 1 trivial
7280.2.a.ca.1.3 4 4.3 odd 2