Properties

Label 6160.2.a.bd.1.3
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3080)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.86620\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.86620 q^{3} -1.00000 q^{5} +1.00000 q^{7} +0.482696 q^{9} +O(q^{10})\) \(q+1.86620 q^{3} -1.00000 q^{5} +1.00000 q^{7} +0.482696 q^{9} +1.00000 q^{11} +1.86620 q^{13} -1.86620 q^{15} -2.83159 q^{17} -3.03461 q^{19} +1.86620 q^{21} -5.21509 q^{23} +1.00000 q^{25} -4.69779 q^{27} -2.00000 q^{29} -5.08129 q^{31} +1.86620 q^{33} -1.00000 q^{35} +3.21509 q^{37} +3.48270 q^{39} -11.0813 q^{41} +11.9129 q^{43} -0.482696 q^{45} -10.5640 q^{47} +1.00000 q^{49} -5.28431 q^{51} +8.24970 q^{53} -1.00000 q^{55} -5.66318 q^{57} -10.7445 q^{59} -11.4181 q^{61} +0.482696 q^{63} -1.86620 q^{65} +3.55191 q^{67} -9.73240 q^{69} +1.93078 q^{71} -5.16841 q^{73} +1.86620 q^{75} +1.00000 q^{77} +16.2497 q^{79} -10.2151 q^{81} -8.69779 q^{83} +2.83159 q^{85} -3.73240 q^{87} -3.73240 q^{89} +1.86620 q^{91} -9.48270 q^{93} +3.03461 q^{95} -5.80161 q^{97} +0.482696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} + 4 q^{51} + 8 q^{53} - 3 q^{55} - 8 q^{57} - 2 q^{59} - 22 q^{61} + 3 q^{63} + 2 q^{65} + 6 q^{67} - 14 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 3 q^{77} + 32 q^{79} - 17 q^{81} - 14 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 2 q^{91} - 30 q^{93} + 6 q^{95} + 4 q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.86620 1.07745 0.538725 0.842482i \(-0.318906\pi\)
0.538725 + 0.842482i \(0.318906\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0.482696 0.160899
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.86620 0.517590 0.258795 0.965932i \(-0.416675\pi\)
0.258795 + 0.965932i \(0.416675\pi\)
\(14\) 0 0
\(15\) −1.86620 −0.481850
\(16\) 0 0
\(17\) −2.83159 −0.686761 −0.343381 0.939196i \(-0.611572\pi\)
−0.343381 + 0.939196i \(0.611572\pi\)
\(18\) 0 0
\(19\) −3.03461 −0.696187 −0.348093 0.937460i \(-0.613171\pi\)
−0.348093 + 0.937460i \(0.613171\pi\)
\(20\) 0 0
\(21\) 1.86620 0.407238
\(22\) 0 0
\(23\) −5.21509 −1.08742 −0.543711 0.839273i \(-0.682981\pi\)
−0.543711 + 0.839273i \(0.682981\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.69779 −0.904090
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −5.08129 −0.912627 −0.456313 0.889819i \(-0.650830\pi\)
−0.456313 + 0.889819i \(0.650830\pi\)
\(32\) 0 0
\(33\) 1.86620 0.324863
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 3.21509 0.528558 0.264279 0.964446i \(-0.414866\pi\)
0.264279 + 0.964446i \(0.414866\pi\)
\(38\) 0 0
\(39\) 3.48270 0.557678
\(40\) 0 0
\(41\) −11.0813 −1.73061 −0.865303 0.501248i \(-0.832874\pi\)
−0.865303 + 0.501248i \(0.832874\pi\)
\(42\) 0 0
\(43\) 11.9129 1.81670 0.908349 0.418214i \(-0.137344\pi\)
0.908349 + 0.418214i \(0.137344\pi\)
\(44\) 0 0
\(45\) −0.482696 −0.0719561
\(46\) 0 0
\(47\) −10.5640 −1.54092 −0.770458 0.637491i \(-0.779972\pi\)
−0.770458 + 0.637491i \(0.779972\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.28431 −0.739951
\(52\) 0 0
\(53\) 8.24970 1.13318 0.566592 0.823999i \(-0.308262\pi\)
0.566592 + 0.823999i \(0.308262\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −5.66318 −0.750107
\(58\) 0 0
\(59\) −10.7445 −1.39881 −0.699405 0.714725i \(-0.746552\pi\)
−0.699405 + 0.714725i \(0.746552\pi\)
\(60\) 0 0
\(61\) −11.4181 −1.46194 −0.730970 0.682410i \(-0.760932\pi\)
−0.730970 + 0.682410i \(0.760932\pi\)
\(62\) 0 0
\(63\) 0.482696 0.0608140
\(64\) 0 0
\(65\) −1.86620 −0.231473
\(66\) 0 0
\(67\) 3.55191 0.433935 0.216968 0.976179i \(-0.430383\pi\)
0.216968 + 0.976179i \(0.430383\pi\)
\(68\) 0 0
\(69\) −9.73240 −1.17164
\(70\) 0 0
\(71\) 1.93078 0.229142 0.114571 0.993415i \(-0.463451\pi\)
0.114571 + 0.993415i \(0.463451\pi\)
\(72\) 0 0
\(73\) −5.16841 −0.604917 −0.302458 0.953163i \(-0.597807\pi\)
−0.302458 + 0.953163i \(0.597807\pi\)
\(74\) 0 0
\(75\) 1.86620 0.215490
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 16.2497 1.82823 0.914117 0.405450i \(-0.132885\pi\)
0.914117 + 0.405450i \(0.132885\pi\)
\(80\) 0 0
\(81\) −10.2151 −1.13501
\(82\) 0 0
\(83\) −8.69779 −0.954706 −0.477353 0.878712i \(-0.658404\pi\)
−0.477353 + 0.878712i \(0.658404\pi\)
\(84\) 0 0
\(85\) 2.83159 0.307129
\(86\) 0 0
\(87\) −3.73240 −0.400155
\(88\) 0 0
\(89\) −3.73240 −0.395633 −0.197817 0.980239i \(-0.563385\pi\)
−0.197817 + 0.980239i \(0.563385\pi\)
\(90\) 0 0
\(91\) 1.86620 0.195631
\(92\) 0 0
\(93\) −9.48270 −0.983310
\(94\) 0 0
\(95\) 3.03461 0.311344
\(96\) 0 0
\(97\) −5.80161 −0.589065 −0.294532 0.955642i \(-0.595164\pi\)
−0.294532 + 0.955642i \(0.595164\pi\)
\(98\) 0 0
\(99\) 0.482696 0.0485128
\(100\) 0 0
\(101\) 13.5115 1.34444 0.672221 0.740351i \(-0.265340\pi\)
0.672221 + 0.740351i \(0.265340\pi\)
\(102\) 0 0
\(103\) −3.52938 −0.347760 −0.173880 0.984767i \(-0.555630\pi\)
−0.173880 + 0.984767i \(0.555630\pi\)
\(104\) 0 0
\(105\) −1.86620 −0.182122
\(106\) 0 0
\(107\) −10.4302 −1.00832 −0.504162 0.863609i \(-0.668199\pi\)
−0.504162 + 0.863609i \(0.668199\pi\)
\(108\) 0 0
\(109\) −5.73240 −0.549064 −0.274532 0.961578i \(-0.588523\pi\)
−0.274532 + 0.961578i \(0.588523\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −11.8016 −1.11020 −0.555101 0.831783i \(-0.687320\pi\)
−0.555101 + 0.831783i \(0.687320\pi\)
\(114\) 0 0
\(115\) 5.21509 0.486310
\(116\) 0 0
\(117\) 0.900806 0.0832796
\(118\) 0 0
\(119\) −2.83159 −0.259571
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −20.6799 −1.86464
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.62857 0.765662 0.382831 0.923818i \(-0.374949\pi\)
0.382831 + 0.923818i \(0.374949\pi\)
\(128\) 0 0
\(129\) 22.2318 1.95740
\(130\) 0 0
\(131\) 5.39558 0.471414 0.235707 0.971824i \(-0.424259\pi\)
0.235707 + 0.971824i \(0.424259\pi\)
\(132\) 0 0
\(133\) −3.03461 −0.263134
\(134\) 0 0
\(135\) 4.69779 0.404321
\(136\) 0 0
\(137\) 9.81952 0.838938 0.419469 0.907770i \(-0.362216\pi\)
0.419469 + 0.907770i \(0.362216\pi\)
\(138\) 0 0
\(139\) 15.1972 1.28901 0.644504 0.764601i \(-0.277064\pi\)
0.644504 + 0.764601i \(0.277064\pi\)
\(140\) 0 0
\(141\) −19.7145 −1.66026
\(142\) 0 0
\(143\) 1.86620 0.156059
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 1.86620 0.153921
\(148\) 0 0
\(149\) −24.1626 −1.97948 −0.989738 0.142895i \(-0.954359\pi\)
−0.989738 + 0.142895i \(0.954359\pi\)
\(150\) 0 0
\(151\) −14.0513 −1.14348 −0.571740 0.820435i \(-0.693731\pi\)
−0.571740 + 0.820435i \(0.693731\pi\)
\(152\) 0 0
\(153\) −1.36680 −0.110499
\(154\) 0 0
\(155\) 5.08129 0.408139
\(156\) 0 0
\(157\) 2.06922 0.165141 0.0825707 0.996585i \(-0.473687\pi\)
0.0825707 + 0.996585i \(0.473687\pi\)
\(158\) 0 0
\(159\) 15.3956 1.22095
\(160\) 0 0
\(161\) −5.21509 −0.411007
\(162\) 0 0
\(163\) 8.44809 0.661705 0.330853 0.943682i \(-0.392664\pi\)
0.330853 + 0.943682i \(0.392664\pi\)
\(164\) 0 0
\(165\) −1.86620 −0.145283
\(166\) 0 0
\(167\) 18.8604 1.45946 0.729730 0.683736i \(-0.239646\pi\)
0.729730 + 0.683736i \(0.239646\pi\)
\(168\) 0 0
\(169\) −9.51730 −0.732100
\(170\) 0 0
\(171\) −1.46479 −0.112016
\(172\) 0 0
\(173\) 5.26178 0.400045 0.200023 0.979791i \(-0.435898\pi\)
0.200023 + 0.979791i \(0.435898\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −20.0513 −1.50715
\(178\) 0 0
\(179\) 25.3598 1.89548 0.947739 0.319046i \(-0.103363\pi\)
0.947739 + 0.319046i \(0.103363\pi\)
\(180\) 0 0
\(181\) 7.10382 0.528023 0.264012 0.964520i \(-0.414954\pi\)
0.264012 + 0.964520i \(0.414954\pi\)
\(182\) 0 0
\(183\) −21.3085 −1.57517
\(184\) 0 0
\(185\) −3.21509 −0.236378
\(186\) 0 0
\(187\) −2.83159 −0.207066
\(188\) 0 0
\(189\) −4.69779 −0.341714
\(190\) 0 0
\(191\) 8.26760 0.598223 0.299111 0.954218i \(-0.403310\pi\)
0.299111 + 0.954218i \(0.403310\pi\)
\(192\) 0 0
\(193\) −18.9475 −1.36387 −0.681935 0.731413i \(-0.738861\pi\)
−0.681935 + 0.731413i \(0.738861\pi\)
\(194\) 0 0
\(195\) −3.48270 −0.249401
\(196\) 0 0
\(197\) −10.9475 −0.779976 −0.389988 0.920820i \(-0.627521\pi\)
−0.389988 + 0.920820i \(0.627521\pi\)
\(198\) 0 0
\(199\) 7.70986 0.546538 0.273269 0.961938i \(-0.411895\pi\)
0.273269 + 0.961938i \(0.411895\pi\)
\(200\) 0 0
\(201\) 6.62857 0.467543
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 11.0813 0.773951
\(206\) 0 0
\(207\) −2.51730 −0.174965
\(208\) 0 0
\(209\) −3.03461 −0.209908
\(210\) 0 0
\(211\) −25.4889 −1.75473 −0.877366 0.479823i \(-0.840701\pi\)
−0.877366 + 0.479823i \(0.840701\pi\)
\(212\) 0 0
\(213\) 3.60323 0.246889
\(214\) 0 0
\(215\) −11.9129 −0.812452
\(216\) 0 0
\(217\) −5.08129 −0.344940
\(218\) 0 0
\(219\) −9.64528 −0.651767
\(220\) 0 0
\(221\) −5.28431 −0.355461
\(222\) 0 0
\(223\) 1.33099 0.0891298 0.0445649 0.999006i \(-0.485810\pi\)
0.0445649 + 0.999006i \(0.485810\pi\)
\(224\) 0 0
\(225\) 0.482696 0.0321797
\(226\) 0 0
\(227\) −29.5340 −1.96024 −0.980121 0.198403i \(-0.936425\pi\)
−0.980121 + 0.198403i \(0.936425\pi\)
\(228\) 0 0
\(229\) −21.5582 −1.42460 −0.712302 0.701874i \(-0.752347\pi\)
−0.712302 + 0.701874i \(0.752347\pi\)
\(230\) 0 0
\(231\) 1.86620 0.122787
\(232\) 0 0
\(233\) 3.71449 0.243345 0.121672 0.992570i \(-0.461174\pi\)
0.121672 + 0.992570i \(0.461174\pi\)
\(234\) 0 0
\(235\) 10.5640 0.689119
\(236\) 0 0
\(237\) 30.3252 1.96983
\(238\) 0 0
\(239\) 14.8362 0.959675 0.479838 0.877357i \(-0.340696\pi\)
0.479838 + 0.877357i \(0.340696\pi\)
\(240\) 0 0
\(241\) −10.7203 −0.690557 −0.345278 0.938500i \(-0.612215\pi\)
−0.345278 + 0.938500i \(0.612215\pi\)
\(242\) 0 0
\(243\) −4.97002 −0.318827
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −5.66318 −0.360340
\(248\) 0 0
\(249\) −16.2318 −1.02865
\(250\) 0 0
\(251\) −24.0467 −1.51781 −0.758907 0.651200i \(-0.774266\pi\)
−0.758907 + 0.651200i \(0.774266\pi\)
\(252\) 0 0
\(253\) −5.21509 −0.327870
\(254\) 0 0
\(255\) 5.28431 0.330916
\(256\) 0 0
\(257\) −15.8258 −0.987184 −0.493592 0.869694i \(-0.664316\pi\)
−0.493592 + 0.869694i \(0.664316\pi\)
\(258\) 0 0
\(259\) 3.21509 0.199776
\(260\) 0 0
\(261\) −0.965392 −0.0597563
\(262\) 0 0
\(263\) 27.6966 1.70784 0.853922 0.520400i \(-0.174217\pi\)
0.853922 + 0.520400i \(0.174217\pi\)
\(264\) 0 0
\(265\) −8.24970 −0.506775
\(266\) 0 0
\(267\) −6.96539 −0.426275
\(268\) 0 0
\(269\) −12.1384 −0.740093 −0.370047 0.929013i \(-0.620658\pi\)
−0.370047 + 0.929013i \(0.620658\pi\)
\(270\) 0 0
\(271\) −15.0588 −0.914754 −0.457377 0.889273i \(-0.651211\pi\)
−0.457377 + 0.889273i \(0.651211\pi\)
\(272\) 0 0
\(273\) 3.48270 0.210782
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 11.2151 0.673850 0.336925 0.941532i \(-0.390613\pi\)
0.336925 + 0.941532i \(0.390613\pi\)
\(278\) 0 0
\(279\) −2.45272 −0.146840
\(280\) 0 0
\(281\) −10.3610 −0.618084 −0.309042 0.951048i \(-0.600008\pi\)
−0.309042 + 0.951048i \(0.600008\pi\)
\(282\) 0 0
\(283\) 17.8950 1.06375 0.531873 0.846824i \(-0.321488\pi\)
0.531873 + 0.846824i \(0.321488\pi\)
\(284\) 0 0
\(285\) 5.66318 0.335458
\(286\) 0 0
\(287\) −11.0813 −0.654108
\(288\) 0 0
\(289\) −8.98210 −0.528359
\(290\) 0 0
\(291\) −10.8270 −0.634688
\(292\) 0 0
\(293\) −12.4014 −0.724498 −0.362249 0.932081i \(-0.617991\pi\)
−0.362249 + 0.932081i \(0.617991\pi\)
\(294\) 0 0
\(295\) 10.7445 0.625567
\(296\) 0 0
\(297\) −4.69779 −0.272593
\(298\) 0 0
\(299\) −9.73240 −0.562839
\(300\) 0 0
\(301\) 11.9129 0.686647
\(302\) 0 0
\(303\) 25.2151 1.44857
\(304\) 0 0
\(305\) 11.4181 0.653799
\(306\) 0 0
\(307\) 17.2906 0.986824 0.493412 0.869796i \(-0.335749\pi\)
0.493412 + 0.869796i \(0.335749\pi\)
\(308\) 0 0
\(309\) −6.58652 −0.374694
\(310\) 0 0
\(311\) 11.1747 0.633657 0.316828 0.948483i \(-0.397382\pi\)
0.316828 + 0.948483i \(0.397382\pi\)
\(312\) 0 0
\(313\) 23.3598 1.32037 0.660186 0.751102i \(-0.270477\pi\)
0.660186 + 0.751102i \(0.270477\pi\)
\(314\) 0 0
\(315\) −0.482696 −0.0271968
\(316\) 0 0
\(317\) 15.3956 0.864702 0.432351 0.901705i \(-0.357684\pi\)
0.432351 + 0.901705i \(0.357684\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −19.4648 −1.08642
\(322\) 0 0
\(323\) 8.59277 0.478114
\(324\) 0 0
\(325\) 1.86620 0.103518
\(326\) 0 0
\(327\) −10.6978 −0.591589
\(328\) 0 0
\(329\) −10.5640 −0.582411
\(330\) 0 0
\(331\) −8.23180 −0.452461 −0.226230 0.974074i \(-0.572640\pi\)
−0.226230 + 0.974074i \(0.572640\pi\)
\(332\) 0 0
\(333\) 1.55191 0.0850443
\(334\) 0 0
\(335\) −3.55191 −0.194062
\(336\) 0 0
\(337\) −3.12173 −0.170051 −0.0850257 0.996379i \(-0.527097\pi\)
−0.0850257 + 0.996379i \(0.527097\pi\)
\(338\) 0 0
\(339\) −22.0241 −1.19619
\(340\) 0 0
\(341\) −5.08129 −0.275167
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 9.73240 0.523975
\(346\) 0 0
\(347\) −13.4827 −0.723789 −0.361895 0.932219i \(-0.617870\pi\)
−0.361895 + 0.932219i \(0.617870\pi\)
\(348\) 0 0
\(349\) −8.31429 −0.445054 −0.222527 0.974927i \(-0.571430\pi\)
−0.222527 + 0.974927i \(0.571430\pi\)
\(350\) 0 0
\(351\) −8.76700 −0.467948
\(352\) 0 0
\(353\) 27.8950 1.48470 0.742350 0.670012i \(-0.233711\pi\)
0.742350 + 0.670012i \(0.233711\pi\)
\(354\) 0 0
\(355\) −1.93078 −0.102475
\(356\) 0 0
\(357\) −5.28431 −0.279675
\(358\) 0 0
\(359\) 6.54145 0.345245 0.172622 0.984988i \(-0.444776\pi\)
0.172622 + 0.984988i \(0.444776\pi\)
\(360\) 0 0
\(361\) −9.79115 −0.515324
\(362\) 0 0
\(363\) 1.86620 0.0979500
\(364\) 0 0
\(365\) 5.16841 0.270527
\(366\) 0 0
\(367\) −25.5294 −1.33262 −0.666311 0.745674i \(-0.732128\pi\)
−0.666311 + 0.745674i \(0.732128\pi\)
\(368\) 0 0
\(369\) −5.34889 −0.278452
\(370\) 0 0
\(371\) 8.24970 0.428303
\(372\) 0 0
\(373\) 2.41228 0.124903 0.0624516 0.998048i \(-0.480108\pi\)
0.0624516 + 0.998048i \(0.480108\pi\)
\(374\) 0 0
\(375\) −1.86620 −0.0963701
\(376\) 0 0
\(377\) −3.73240 −0.192228
\(378\) 0 0
\(379\) −11.4648 −0.588907 −0.294453 0.955666i \(-0.595138\pi\)
−0.294453 + 0.955666i \(0.595138\pi\)
\(380\) 0 0
\(381\) 16.1026 0.824963
\(382\) 0 0
\(383\) −26.5040 −1.35429 −0.677146 0.735848i \(-0.736784\pi\)
−0.677146 + 0.735848i \(0.736784\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) 5.75030 0.292304
\(388\) 0 0
\(389\) 26.0576 1.32117 0.660585 0.750751i \(-0.270308\pi\)
0.660585 + 0.750751i \(0.270308\pi\)
\(390\) 0 0
\(391\) 14.7670 0.746800
\(392\) 0 0
\(393\) 10.0692 0.507925
\(394\) 0 0
\(395\) −16.2497 −0.817611
\(396\) 0 0
\(397\) 4.49940 0.225818 0.112909 0.993605i \(-0.463983\pi\)
0.112909 + 0.993605i \(0.463983\pi\)
\(398\) 0 0
\(399\) −5.66318 −0.283514
\(400\) 0 0
\(401\) −32.9296 −1.64443 −0.822213 0.569181i \(-0.807261\pi\)
−0.822213 + 0.569181i \(0.807261\pi\)
\(402\) 0 0
\(403\) −9.48270 −0.472367
\(404\) 0 0
\(405\) 10.2151 0.507592
\(406\) 0 0
\(407\) 3.21509 0.159366
\(408\) 0 0
\(409\) −9.44226 −0.466890 −0.233445 0.972370i \(-0.575000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(410\) 0 0
\(411\) 18.3252 0.903914
\(412\) 0 0
\(413\) −10.7445 −0.528701
\(414\) 0 0
\(415\) 8.69779 0.426958
\(416\) 0 0
\(417\) 28.3610 1.38884
\(418\) 0 0
\(419\) 38.4169 1.87679 0.938394 0.345566i \(-0.112313\pi\)
0.938394 + 0.345566i \(0.112313\pi\)
\(420\) 0 0
\(421\) −18.2497 −0.889436 −0.444718 0.895671i \(-0.646696\pi\)
−0.444718 + 0.895671i \(0.646696\pi\)
\(422\) 0 0
\(423\) −5.09919 −0.247931
\(424\) 0 0
\(425\) −2.83159 −0.137352
\(426\) 0 0
\(427\) −11.4181 −0.552561
\(428\) 0 0
\(429\) 3.48270 0.168146
\(430\) 0 0
\(431\) 8.92334 0.429822 0.214911 0.976634i \(-0.431054\pi\)
0.214911 + 0.976634i \(0.431054\pi\)
\(432\) 0 0
\(433\) −0.872027 −0.0419069 −0.0209535 0.999780i \(-0.506670\pi\)
−0.0209535 + 0.999780i \(0.506670\pi\)
\(434\) 0 0
\(435\) 3.73240 0.178955
\(436\) 0 0
\(437\) 15.8258 0.757049
\(438\) 0 0
\(439\) 24.9054 1.18867 0.594336 0.804217i \(-0.297415\pi\)
0.594336 + 0.804217i \(0.297415\pi\)
\(440\) 0 0
\(441\) 0.482696 0.0229855
\(442\) 0 0
\(443\) −13.2151 −0.627868 −0.313934 0.949445i \(-0.601647\pi\)
−0.313934 + 0.949445i \(0.601647\pi\)
\(444\) 0 0
\(445\) 3.73240 0.176933
\(446\) 0 0
\(447\) −45.0922 −2.13279
\(448\) 0 0
\(449\) −3.64528 −0.172031 −0.0860156 0.996294i \(-0.527414\pi\)
−0.0860156 + 0.996294i \(0.527414\pi\)
\(450\) 0 0
\(451\) −11.0813 −0.521798
\(452\) 0 0
\(453\) −26.2225 −1.23204
\(454\) 0 0
\(455\) −1.86620 −0.0874887
\(456\) 0 0
\(457\) −30.4123 −1.42263 −0.711313 0.702875i \(-0.751899\pi\)
−0.711313 + 0.702875i \(0.751899\pi\)
\(458\) 0 0
\(459\) 13.3022 0.620894
\(460\) 0 0
\(461\) 11.7099 0.545383 0.272691 0.962102i \(-0.412086\pi\)
0.272691 + 0.962102i \(0.412086\pi\)
\(462\) 0 0
\(463\) 32.5507 1.51276 0.756380 0.654132i \(-0.226966\pi\)
0.756380 + 0.654132i \(0.226966\pi\)
\(464\) 0 0
\(465\) 9.48270 0.439749
\(466\) 0 0
\(467\) −23.9354 −1.10760 −0.553799 0.832650i \(-0.686823\pi\)
−0.553799 + 0.832650i \(0.686823\pi\)
\(468\) 0 0
\(469\) 3.55191 0.164012
\(470\) 0 0
\(471\) 3.86157 0.177932
\(472\) 0 0
\(473\) 11.9129 0.547755
\(474\) 0 0
\(475\) −3.03461 −0.139237
\(476\) 0 0
\(477\) 3.98210 0.182328
\(478\) 0 0
\(479\) 2.53521 0.115837 0.0579183 0.998321i \(-0.481554\pi\)
0.0579183 + 0.998321i \(0.481554\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) −9.73240 −0.442839
\(484\) 0 0
\(485\) 5.80161 0.263438
\(486\) 0 0
\(487\) 42.5749 1.92925 0.964626 0.263624i \(-0.0849177\pi\)
0.964626 + 0.263624i \(0.0849177\pi\)
\(488\) 0 0
\(489\) 15.7658 0.712954
\(490\) 0 0
\(491\) −42.6441 −1.92450 −0.962250 0.272166i \(-0.912260\pi\)
−0.962250 + 0.272166i \(0.912260\pi\)
\(492\) 0 0
\(493\) 5.66318 0.255057
\(494\) 0 0
\(495\) −0.482696 −0.0216956
\(496\) 0 0
\(497\) 1.93078 0.0866075
\(498\) 0 0
\(499\) −11.9642 −0.535591 −0.267795 0.963476i \(-0.586295\pi\)
−0.267795 + 0.963476i \(0.586295\pi\)
\(500\) 0 0
\(501\) 35.1972 1.57249
\(502\) 0 0
\(503\) −21.2330 −0.946732 −0.473366 0.880866i \(-0.656961\pi\)
−0.473366 + 0.880866i \(0.656961\pi\)
\(504\) 0 0
\(505\) −13.5115 −0.601253
\(506\) 0 0
\(507\) −17.7612 −0.788802
\(508\) 0 0
\(509\) 13.8708 0.614814 0.307407 0.951578i \(-0.400539\pi\)
0.307407 + 0.951578i \(0.400539\pi\)
\(510\) 0 0
\(511\) −5.16841 −0.228637
\(512\) 0 0
\(513\) 14.2559 0.629415
\(514\) 0 0
\(515\) 3.52938 0.155523
\(516\) 0 0
\(517\) −10.5640 −0.464604
\(518\) 0 0
\(519\) 9.81952 0.431029
\(520\) 0 0
\(521\) 32.6861 1.43201 0.716003 0.698097i \(-0.245970\pi\)
0.716003 + 0.698097i \(0.245970\pi\)
\(522\) 0 0
\(523\) −19.3598 −0.846544 −0.423272 0.906003i \(-0.639118\pi\)
−0.423272 + 0.906003i \(0.639118\pi\)
\(524\) 0 0
\(525\) 1.86620 0.0814476
\(526\) 0 0
\(527\) 14.3881 0.626757
\(528\) 0 0
\(529\) 4.19719 0.182487
\(530\) 0 0
\(531\) −5.18631 −0.225067
\(532\) 0 0
\(533\) −20.6799 −0.895745
\(534\) 0 0
\(535\) 10.4302 0.450936
\(536\) 0 0
\(537\) 47.3264 2.04228
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 4.69779 0.201974 0.100987 0.994888i \(-0.467800\pi\)
0.100987 + 0.994888i \(0.467800\pi\)
\(542\) 0 0
\(543\) 13.2571 0.568919
\(544\) 0 0
\(545\) 5.73240 0.245549
\(546\) 0 0
\(547\) −25.8016 −1.10320 −0.551599 0.834110i \(-0.685982\pi\)
−0.551599 + 0.834110i \(0.685982\pi\)
\(548\) 0 0
\(549\) −5.51148 −0.235224
\(550\) 0 0
\(551\) 6.06922 0.258557
\(552\) 0 0
\(553\) 16.2497 0.691008
\(554\) 0 0
\(555\) −6.00000 −0.254686
\(556\) 0 0
\(557\) −5.28431 −0.223903 −0.111952 0.993714i \(-0.535710\pi\)
−0.111952 + 0.993714i \(0.535710\pi\)
\(558\) 0 0
\(559\) 22.2318 0.940305
\(560\) 0 0
\(561\) −5.28431 −0.223104
\(562\) 0 0
\(563\) −35.6966 −1.50443 −0.752216 0.658917i \(-0.771015\pi\)
−0.752216 + 0.658917i \(0.771015\pi\)
\(564\) 0 0
\(565\) 11.8016 0.496498
\(566\) 0 0
\(567\) −10.2151 −0.428994
\(568\) 0 0
\(569\) 7.12797 0.298820 0.149410 0.988775i \(-0.452263\pi\)
0.149410 + 0.988775i \(0.452263\pi\)
\(570\) 0 0
\(571\) 23.4197 0.980085 0.490043 0.871699i \(-0.336981\pi\)
0.490043 + 0.871699i \(0.336981\pi\)
\(572\) 0 0
\(573\) 15.4290 0.644555
\(574\) 0 0
\(575\) −5.21509 −0.217484
\(576\) 0 0
\(577\) 34.1626 1.42221 0.711103 0.703087i \(-0.248196\pi\)
0.711103 + 0.703087i \(0.248196\pi\)
\(578\) 0 0
\(579\) −35.3598 −1.46950
\(580\) 0 0
\(581\) −8.69779 −0.360845
\(582\) 0 0
\(583\) 8.24970 0.341668
\(584\) 0 0
\(585\) −0.900806 −0.0372438
\(586\) 0 0
\(587\) 27.4936 1.13478 0.567391 0.823449i \(-0.307953\pi\)
0.567391 + 0.823449i \(0.307953\pi\)
\(588\) 0 0
\(589\) 15.4197 0.635359
\(590\) 0 0
\(591\) −20.4302 −0.840386
\(592\) 0 0
\(593\) −9.59859 −0.394167 −0.197084 0.980387i \(-0.563147\pi\)
−0.197084 + 0.980387i \(0.563147\pi\)
\(594\) 0 0
\(595\) 2.83159 0.116084
\(596\) 0 0
\(597\) 14.3881 0.588867
\(598\) 0 0
\(599\) −6.96539 −0.284598 −0.142299 0.989824i \(-0.545449\pi\)
−0.142299 + 0.989824i \(0.545449\pi\)
\(600\) 0 0
\(601\) 14.8137 0.604263 0.302131 0.953266i \(-0.402302\pi\)
0.302131 + 0.953266i \(0.402302\pi\)
\(602\) 0 0
\(603\) 1.71449 0.0698196
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 9.86157 0.400269 0.200134 0.979768i \(-0.435862\pi\)
0.200134 + 0.979768i \(0.435862\pi\)
\(608\) 0 0
\(609\) −3.73240 −0.151244
\(610\) 0 0
\(611\) −19.7145 −0.797563
\(612\) 0 0
\(613\) 39.0409 1.57685 0.788423 0.615134i \(-0.210898\pi\)
0.788423 + 0.615134i \(0.210898\pi\)
\(614\) 0 0
\(615\) 20.6799 0.833893
\(616\) 0 0
\(617\) −9.64528 −0.388304 −0.194152 0.980971i \(-0.562196\pi\)
−0.194152 + 0.980971i \(0.562196\pi\)
\(618\) 0 0
\(619\) −21.9175 −0.880939 −0.440470 0.897768i \(-0.645188\pi\)
−0.440470 + 0.897768i \(0.645188\pi\)
\(620\) 0 0
\(621\) 24.4994 0.983127
\(622\) 0 0
\(623\) −3.73240 −0.149535
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.66318 −0.226166
\(628\) 0 0
\(629\) −9.10382 −0.362993
\(630\) 0 0
\(631\) 10.7429 0.427666 0.213833 0.976870i \(-0.431405\pi\)
0.213833 + 0.976870i \(0.431405\pi\)
\(632\) 0 0
\(633\) −47.5674 −1.89064
\(634\) 0 0
\(635\) −8.62857 −0.342414
\(636\) 0 0
\(637\) 1.86620 0.0739415
\(638\) 0 0
\(639\) 0.931982 0.0368686
\(640\) 0 0
\(641\) 1.71449 0.0677184 0.0338592 0.999427i \(-0.489220\pi\)
0.0338592 + 0.999427i \(0.489220\pi\)
\(642\) 0 0
\(643\) 38.3897 1.51394 0.756972 0.653447i \(-0.226678\pi\)
0.756972 + 0.653447i \(0.226678\pi\)
\(644\) 0 0
\(645\) −22.2318 −0.875376
\(646\) 0 0
\(647\) 8.99417 0.353597 0.176799 0.984247i \(-0.443426\pi\)
0.176799 + 0.984247i \(0.443426\pi\)
\(648\) 0 0
\(649\) −10.7445 −0.421757
\(650\) 0 0
\(651\) −9.48270 −0.371656
\(652\) 0 0
\(653\) −30.5928 −1.19719 −0.598594 0.801053i \(-0.704274\pi\)
−0.598594 + 0.801053i \(0.704274\pi\)
\(654\) 0 0
\(655\) −5.39558 −0.210823
\(656\) 0 0
\(657\) −2.49477 −0.0973303
\(658\) 0 0
\(659\) 45.7028 1.78033 0.890165 0.455639i \(-0.150589\pi\)
0.890165 + 0.455639i \(0.150589\pi\)
\(660\) 0 0
\(661\) −28.9537 −1.12617 −0.563085 0.826399i \(-0.690386\pi\)
−0.563085 + 0.826399i \(0.690386\pi\)
\(662\) 0 0
\(663\) −9.86157 −0.382992
\(664\) 0 0
\(665\) 3.03461 0.117677
\(666\) 0 0
\(667\) 10.4302 0.403858
\(668\) 0 0
\(669\) 2.48389 0.0960329
\(670\) 0 0
\(671\) −11.4181 −0.440791
\(672\) 0 0
\(673\) −15.7052 −0.605392 −0.302696 0.953087i \(-0.597887\pi\)
−0.302696 + 0.953087i \(0.597887\pi\)
\(674\) 0 0
\(675\) −4.69779 −0.180818
\(676\) 0 0
\(677\) −5.42436 −0.208475 −0.104237 0.994552i \(-0.533240\pi\)
−0.104237 + 0.994552i \(0.533240\pi\)
\(678\) 0 0
\(679\) −5.80161 −0.222645
\(680\) 0 0
\(681\) −55.1163 −2.11206
\(682\) 0 0
\(683\) 28.7374 1.09961 0.549804 0.835294i \(-0.314702\pi\)
0.549804 + 0.835294i \(0.314702\pi\)
\(684\) 0 0
\(685\) −9.81952 −0.375184
\(686\) 0 0
\(687\) −40.2318 −1.53494
\(688\) 0 0
\(689\) 15.3956 0.586525
\(690\) 0 0
\(691\) 27.3131 1.03904 0.519519 0.854459i \(-0.326111\pi\)
0.519519 + 0.854459i \(0.326111\pi\)
\(692\) 0 0
\(693\) 0.482696 0.0183361
\(694\) 0 0
\(695\) −15.1972 −0.576462
\(696\) 0 0
\(697\) 31.3777 1.18851
\(698\) 0 0
\(699\) 6.93198 0.262192
\(700\) 0 0
\(701\) 42.2318 1.59507 0.797536 0.603271i \(-0.206136\pi\)
0.797536 + 0.603271i \(0.206136\pi\)
\(702\) 0 0
\(703\) −9.75655 −0.367975
\(704\) 0 0
\(705\) 19.7145 0.742491
\(706\) 0 0
\(707\) 13.5115 0.508151
\(708\) 0 0
\(709\) 47.5823 1.78699 0.893496 0.449072i \(-0.148245\pi\)
0.893496 + 0.449072i \(0.148245\pi\)
\(710\) 0 0
\(711\) 7.84366 0.294160
\(712\) 0 0
\(713\) 26.4994 0.992410
\(714\) 0 0
\(715\) −1.86620 −0.0697919
\(716\) 0 0
\(717\) 27.6873 1.03400
\(718\) 0 0
\(719\) −9.55654 −0.356399 −0.178199 0.983994i \(-0.557027\pi\)
−0.178199 + 0.983994i \(0.557027\pi\)
\(720\) 0 0
\(721\) −3.52938 −0.131441
\(722\) 0 0
\(723\) −20.0062 −0.744040
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 4.66901 0.173164 0.0865820 0.996245i \(-0.472406\pi\)
0.0865820 + 0.996245i \(0.472406\pi\)
\(728\) 0 0
\(729\) 21.3702 0.791490
\(730\) 0 0
\(731\) −33.7324 −1.24764
\(732\) 0 0
\(733\) 21.8903 0.808538 0.404269 0.914640i \(-0.367526\pi\)
0.404269 + 0.914640i \(0.367526\pi\)
\(734\) 0 0
\(735\) −1.86620 −0.0688358
\(736\) 0 0
\(737\) 3.55191 0.130836
\(738\) 0 0
\(739\) 46.2559 1.70155 0.850776 0.525528i \(-0.176132\pi\)
0.850776 + 0.525528i \(0.176132\pi\)
\(740\) 0 0
\(741\) −10.5686 −0.388248
\(742\) 0 0
\(743\) −20.8962 −0.766606 −0.383303 0.923623i \(-0.625214\pi\)
−0.383303 + 0.923623i \(0.625214\pi\)
\(744\) 0 0
\(745\) 24.1626 0.885248
\(746\) 0 0
\(747\) −4.19839 −0.153611
\(748\) 0 0
\(749\) −10.4302 −0.381111
\(750\) 0 0
\(751\) 13.9400 0.508679 0.254340 0.967115i \(-0.418142\pi\)
0.254340 + 0.967115i \(0.418142\pi\)
\(752\) 0 0
\(753\) −44.8759 −1.63537
\(754\) 0 0
\(755\) 14.0513 0.511380
\(756\) 0 0
\(757\) 42.0934 1.52991 0.764955 0.644084i \(-0.222761\pi\)
0.764955 + 0.644084i \(0.222761\pi\)
\(758\) 0 0
\(759\) −9.73240 −0.353264
\(760\) 0 0
\(761\) 31.1054 1.12757 0.563786 0.825921i \(-0.309344\pi\)
0.563786 + 0.825921i \(0.309344\pi\)
\(762\) 0 0
\(763\) −5.73240 −0.207527
\(764\) 0 0
\(765\) 1.36680 0.0494167
\(766\) 0 0
\(767\) −20.0513 −0.724011
\(768\) 0 0
\(769\) −40.7445 −1.46928 −0.734642 0.678455i \(-0.762650\pi\)
−0.734642 + 0.678455i \(0.762650\pi\)
\(770\) 0 0
\(771\) −29.5340 −1.06364
\(772\) 0 0
\(773\) −19.3356 −0.695454 −0.347727 0.937596i \(-0.613046\pi\)
−0.347727 + 0.937596i \(0.613046\pi\)
\(774\) 0 0
\(775\) −5.08129 −0.182525
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) 33.6274 1.20483
\(780\) 0 0
\(781\) 1.93078 0.0690889
\(782\) 0 0
\(783\) 9.39558 0.335771
\(784\) 0 0
\(785\) −2.06922 −0.0738535
\(786\) 0 0
\(787\) −19.4197 −0.692238 −0.346119 0.938191i \(-0.612501\pi\)
−0.346119 + 0.938191i \(0.612501\pi\)
\(788\) 0 0
\(789\) 51.6873 1.84012
\(790\) 0 0
\(791\) −11.8016 −0.419617
\(792\) 0 0
\(793\) −21.3085 −0.756686
\(794\) 0 0
\(795\) −15.3956 −0.546025
\(796\) 0 0
\(797\) −18.0692 −0.640044 −0.320022 0.947410i \(-0.603690\pi\)
−0.320022 + 0.947410i \(0.603690\pi\)
\(798\) 0 0
\(799\) 29.9129 1.05824
\(800\) 0 0
\(801\) −1.80161 −0.0636569
\(802\) 0 0
\(803\) −5.16841 −0.182389
\(804\) 0 0
\(805\) 5.21509 0.183808
\(806\) 0 0
\(807\) −22.6527 −0.797414
\(808\) 0 0
\(809\) 28.5686 1.00442 0.502210 0.864746i \(-0.332521\pi\)
0.502210 + 0.864746i \(0.332521\pi\)
\(810\) 0 0
\(811\) 5.73240 0.201292 0.100646 0.994922i \(-0.467909\pi\)
0.100646 + 0.994922i \(0.467909\pi\)
\(812\) 0 0
\(813\) −28.1026 −0.985602
\(814\) 0 0
\(815\) −8.44809 −0.295924
\(816\) 0 0
\(817\) −36.1509 −1.26476
\(818\) 0 0
\(819\) 0.900806 0.0314767
\(820\) 0 0
\(821\) −56.5236 −1.97269 −0.986343 0.164706i \(-0.947333\pi\)
−0.986343 + 0.164706i \(0.947333\pi\)
\(822\) 0 0
\(823\) −29.1342 −1.01556 −0.507778 0.861488i \(-0.669533\pi\)
−0.507778 + 0.861488i \(0.669533\pi\)
\(824\) 0 0
\(825\) 1.86620 0.0649727
\(826\) 0 0
\(827\) −21.5853 −0.750595 −0.375298 0.926904i \(-0.622459\pi\)
−0.375298 + 0.926904i \(0.622459\pi\)
\(828\) 0 0
\(829\) 13.6032 0.472460 0.236230 0.971697i \(-0.424088\pi\)
0.236230 + 0.971697i \(0.424088\pi\)
\(830\) 0 0
\(831\) 20.9296 0.726039
\(832\) 0 0
\(833\) −2.83159 −0.0981088
\(834\) 0 0
\(835\) −18.8604 −0.652690
\(836\) 0 0
\(837\) 23.8708 0.825097
\(838\) 0 0
\(839\) 23.7191 0.818875 0.409438 0.912338i \(-0.365725\pi\)
0.409438 + 0.912338i \(0.365725\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −19.3356 −0.665954
\(844\) 0 0
\(845\) 9.51730 0.327405
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 33.3956 1.14613
\(850\) 0 0
\(851\) −16.7670 −0.574766
\(852\) 0 0
\(853\) −28.5282 −0.976786 −0.488393 0.872624i \(-0.662417\pi\)
−0.488393 + 0.872624i \(0.662417\pi\)
\(854\) 0 0
\(855\) 1.46479 0.0500949
\(856\) 0 0
\(857\) −49.1809 −1.67999 −0.839994 0.542596i \(-0.817441\pi\)
−0.839994 + 0.542596i \(0.817441\pi\)
\(858\) 0 0
\(859\) −38.8020 −1.32391 −0.661954 0.749544i \(-0.730273\pi\)
−0.661954 + 0.749544i \(0.730273\pi\)
\(860\) 0 0
\(861\) −20.6799 −0.704769
\(862\) 0 0
\(863\) 17.7145 0.603008 0.301504 0.953465i \(-0.402511\pi\)
0.301504 + 0.953465i \(0.402511\pi\)
\(864\) 0 0
\(865\) −5.26178 −0.178906
\(866\) 0 0
\(867\) −16.7624 −0.569280
\(868\) 0 0
\(869\) 16.2497 0.551233
\(870\) 0 0
\(871\) 6.62857 0.224601
\(872\) 0 0
\(873\) −2.80041 −0.0947797
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −4.33380 −0.146342 −0.0731711 0.997319i \(-0.523312\pi\)
−0.0731711 + 0.997319i \(0.523312\pi\)
\(878\) 0 0
\(879\) −23.1435 −0.780610
\(880\) 0 0
\(881\) −20.3610 −0.685978 −0.342989 0.939339i \(-0.611439\pi\)
−0.342989 + 0.939339i \(0.611439\pi\)
\(882\) 0 0
\(883\) 31.5161 1.06060 0.530301 0.847810i \(-0.322079\pi\)
0.530301 + 0.847810i \(0.322079\pi\)
\(884\) 0 0
\(885\) 20.0513 0.674018
\(886\) 0 0
\(887\) −29.2664 −0.982670 −0.491335 0.870971i \(-0.663491\pi\)
−0.491335 + 0.870971i \(0.663491\pi\)
\(888\) 0 0
\(889\) 8.62857 0.289393
\(890\) 0 0
\(891\) −10.2151 −0.342218
\(892\) 0 0
\(893\) 32.0576 1.07277
\(894\) 0 0
\(895\) −25.3598 −0.847684
\(896\) 0 0
\(897\) −18.1626 −0.606431
\(898\) 0 0
\(899\) 10.1626 0.338941
\(900\) 0 0
\(901\) −23.3598 −0.778227
\(902\) 0 0
\(903\) 22.2318 0.739828
\(904\) 0 0
\(905\) −7.10382 −0.236139
\(906\) 0 0
\(907\) 29.1793 0.968882 0.484441 0.874824i \(-0.339023\pi\)
0.484441 + 0.874824i \(0.339023\pi\)
\(908\) 0 0
\(909\) 6.52193 0.216319
\(910\) 0 0
\(911\) −46.7195 −1.54789 −0.773944 0.633254i \(-0.781719\pi\)
−0.773944 + 0.633254i \(0.781719\pi\)
\(912\) 0 0
\(913\) −8.69779 −0.287855
\(914\) 0 0
\(915\) 21.3085 0.704436
\(916\) 0 0
\(917\) 5.39558 0.178178
\(918\) 0 0
\(919\) 38.8362 1.28109 0.640544 0.767921i \(-0.278709\pi\)
0.640544 + 0.767921i \(0.278709\pi\)
\(920\) 0 0
\(921\) 32.2676 1.06325
\(922\) 0 0
\(923\) 3.60323 0.118602
\(924\) 0 0
\(925\) 3.21509 0.105712
\(926\) 0 0
\(927\) −1.70362 −0.0559541
\(928\) 0 0
\(929\) 46.0817 1.51189 0.755946 0.654634i \(-0.227177\pi\)
0.755946 + 0.654634i \(0.227177\pi\)
\(930\) 0 0
\(931\) −3.03461 −0.0994553
\(932\) 0 0
\(933\) 20.8541 0.682733
\(934\) 0 0
\(935\) 2.83159 0.0926029
\(936\) 0 0
\(937\) 9.05413 0.295785 0.147893 0.989003i \(-0.452751\pi\)
0.147893 + 0.989003i \(0.452751\pi\)
\(938\) 0 0
\(939\) 43.5940 1.42264
\(940\) 0 0
\(941\) −13.3489 −0.435162 −0.217581 0.976042i \(-0.569817\pi\)
−0.217581 + 0.976042i \(0.569817\pi\)
\(942\) 0 0
\(943\) 57.7900 1.88190
\(944\) 0 0
\(945\) 4.69779 0.152819
\(946\) 0 0
\(947\) 33.4018 1.08541 0.542707 0.839922i \(-0.317400\pi\)
0.542707 + 0.839922i \(0.317400\pi\)
\(948\) 0 0
\(949\) −9.64528 −0.313099
\(950\) 0 0
\(951\) 28.7312 0.931673
\(952\) 0 0
\(953\) −22.3672 −0.724545 −0.362273 0.932072i \(-0.617999\pi\)
−0.362273 + 0.932072i \(0.617999\pi\)
\(954\) 0 0
\(955\) −8.26760 −0.267533
\(956\) 0 0
\(957\) −3.73240 −0.120651
\(958\) 0 0
\(959\) 9.81952 0.317089
\(960\) 0 0
\(961\) −5.18048 −0.167112
\(962\) 0 0
\(963\) −5.03461 −0.162238
\(964\) 0 0
\(965\) 18.9475 0.609941
\(966\) 0 0
\(967\) −20.9475 −0.673626 −0.336813 0.941572i \(-0.609349\pi\)
−0.336813 + 0.941572i \(0.609349\pi\)
\(968\) 0 0
\(969\) 16.0358 0.515144
\(970\) 0 0
\(971\) −31.3489 −1.00603 −0.503017 0.864277i \(-0.667777\pi\)
−0.503017 + 0.864277i \(0.667777\pi\)
\(972\) 0 0
\(973\) 15.1972 0.487200
\(974\) 0 0
\(975\) 3.48270 0.111536
\(976\) 0 0
\(977\) −9.91913 −0.317341 −0.158670 0.987332i \(-0.550721\pi\)
−0.158670 + 0.987332i \(0.550721\pi\)
\(978\) 0 0
\(979\) −3.73240 −0.119288
\(980\) 0 0
\(981\) −2.76700 −0.0883437
\(982\) 0 0
\(983\) 46.2248 1.47434 0.737171 0.675707i \(-0.236161\pi\)
0.737171 + 0.675707i \(0.236161\pi\)
\(984\) 0 0
\(985\) 10.9475 0.348816
\(986\) 0 0
\(987\) −19.7145 −0.627519
\(988\) 0 0
\(989\) −62.1268 −1.97552
\(990\) 0 0
\(991\) 15.7682 0.500893 0.250447 0.968130i \(-0.419423\pi\)
0.250447 + 0.968130i \(0.419423\pi\)
\(992\) 0 0
\(993\) −15.3622 −0.487504
\(994\) 0 0
\(995\) −7.70986 −0.244419
\(996\) 0 0
\(997\) −14.4256 −0.456862 −0.228431 0.973560i \(-0.573360\pi\)
−0.228431 + 0.973560i \(0.573360\pi\)
\(998\) 0 0
\(999\) −15.1038 −0.477864
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 6160.2.a.bd.1.3 3
4.3 odd 2 3080.2.a.m.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3080.2.a.m.1.1 3 4.3 odd 2
6160.2.a.bd.1.3 3 1.1 even 1 trivial