Properties

Label 8470.2.a.dd.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.745749504.1
Defining polynomial: \(x^{6} - 18 x^{4} - 4 x^{3} + 81 x^{2} + 36 x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.40870\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.40870 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.40870 q^{6} -1.00000 q^{7} +1.00000 q^{8} +8.61924 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.40870 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.40870 q^{6} -1.00000 q^{7} +1.00000 q^{8} +8.61924 q^{9} -1.00000 q^{10} -3.40870 q^{12} -3.69350 q^{13} -1.00000 q^{14} +3.40870 q^{15} +1.00000 q^{16} -3.63060 q^{17} +8.61924 q^{18} +2.01685 q^{19} -1.00000 q^{20} +3.40870 q^{21} -5.90404 q^{23} -3.40870 q^{24} +1.00000 q^{25} -3.69350 q^{26} -19.1543 q^{27} -1.00000 q^{28} -2.77810 q^{29} +3.40870 q^{30} +2.98864 q^{31} +1.00000 q^{32} -3.63060 q^{34} +1.00000 q^{35} +8.61924 q^{36} +3.22190 q^{37} +2.01685 q^{38} +12.5900 q^{39} -1.00000 q^{40} -8.13344 q^{41} +3.40870 q^{42} -3.87280 q^{43} -8.61924 q^{45} -5.90404 q^{46} +9.67464 q^{47} -3.40870 q^{48} +1.00000 q^{49} +1.00000 q^{50} +12.3756 q^{51} -3.69350 q^{52} +9.39390 q^{53} -19.1543 q^{54} -1.00000 q^{56} -6.87485 q^{57} -2.77810 q^{58} +12.8803 q^{59} +3.40870 q^{60} -6.61375 q^{61} +2.98864 q^{62} -8.61924 q^{63} +1.00000 q^{64} +3.69350 q^{65} -3.48599 q^{67} -3.63060 q^{68} +20.1251 q^{69} +1.00000 q^{70} -5.18885 q^{71} +8.61924 q^{72} -1.83909 q^{73} +3.22190 q^{74} -3.40870 q^{75} +2.01685 q^{76} +12.5900 q^{78} +2.95191 q^{79} -1.00000 q^{80} +39.4336 q^{81} -8.13344 q^{82} -8.37746 q^{83} +3.40870 q^{84} +3.63060 q^{85} -3.87280 q^{86} +9.46970 q^{87} +4.85724 q^{89} -8.61924 q^{90} +3.69350 q^{91} -5.90404 q^{92} -10.1874 q^{93} +9.67464 q^{94} -2.01685 q^{95} -3.40870 q^{96} -16.0944 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{7} + 6 q^{8} + 18 q^{9} + O(q^{10}) \) \( 6 q + 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{7} + 6 q^{8} + 18 q^{9} - 6 q^{10} - 6 q^{14} + 6 q^{16} - 6 q^{17} + 18 q^{18} - 6 q^{20} + 6 q^{25} - 12 q^{27} - 6 q^{28} - 12 q^{29} + 6 q^{32} - 6 q^{34} + 6 q^{35} + 18 q^{36} + 24 q^{37} + 24 q^{39} - 6 q^{40} - 12 q^{41} + 18 q^{43} - 18 q^{45} + 24 q^{47} + 6 q^{49} + 6 q^{50} + 12 q^{51} + 36 q^{53} - 12 q^{54} - 6 q^{56} + 12 q^{57} - 12 q^{58} + 30 q^{59} - 36 q^{61} - 18 q^{63} + 6 q^{64} - 12 q^{67} - 6 q^{68} + 6 q^{70} + 6 q^{71} + 18 q^{72} + 6 q^{73} + 24 q^{74} + 24 q^{78} + 24 q^{79} - 6 q^{80} + 54 q^{81} - 12 q^{82} - 24 q^{83} + 6 q^{85} + 18 q^{86} + 24 q^{87} + 36 q^{89} - 18 q^{90} + 24 q^{94} + 6 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.40870 −1.96801 −0.984007 0.178129i \(-0.942995\pi\)
−0.984007 + 0.178129i \(0.942995\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −3.40870 −1.39160
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 8.61924 2.87308
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −3.40870 −0.984007
\(13\) −3.69350 −1.02439 −0.512197 0.858868i \(-0.671168\pi\)
−0.512197 + 0.858868i \(0.671168\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.40870 0.880123
\(16\) 1.00000 0.250000
\(17\) −3.63060 −0.880551 −0.440275 0.897863i \(-0.645119\pi\)
−0.440275 + 0.897863i \(0.645119\pi\)
\(18\) 8.61924 2.03157
\(19\) 2.01685 0.462698 0.231349 0.972871i \(-0.425686\pi\)
0.231349 + 0.972871i \(0.425686\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.40870 0.743839
\(22\) 0 0
\(23\) −5.90404 −1.23108 −0.615539 0.788106i \(-0.711062\pi\)
−0.615539 + 0.788106i \(0.711062\pi\)
\(24\) −3.40870 −0.695798
\(25\) 1.00000 0.200000
\(26\) −3.69350 −0.724356
\(27\) −19.1543 −3.68625
\(28\) −1.00000 −0.188982
\(29\) −2.77810 −0.515880 −0.257940 0.966161i \(-0.583044\pi\)
−0.257940 + 0.966161i \(0.583044\pi\)
\(30\) 3.40870 0.622341
\(31\) 2.98864 0.536775 0.268387 0.963311i \(-0.413509\pi\)
0.268387 + 0.963311i \(0.413509\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.63060 −0.622643
\(35\) 1.00000 0.169031
\(36\) 8.61924 1.43654
\(37\) 3.22190 0.529678 0.264839 0.964293i \(-0.414681\pi\)
0.264839 + 0.964293i \(0.414681\pi\)
\(38\) 2.01685 0.327177
\(39\) 12.5900 2.01602
\(40\) −1.00000 −0.158114
\(41\) −8.13344 −1.27023 −0.635115 0.772417i \(-0.719047\pi\)
−0.635115 + 0.772417i \(0.719047\pi\)
\(42\) 3.40870 0.525974
\(43\) −3.87280 −0.590597 −0.295298 0.955405i \(-0.595419\pi\)
−0.295298 + 0.955405i \(0.595419\pi\)
\(44\) 0 0
\(45\) −8.61924 −1.28488
\(46\) −5.90404 −0.870504
\(47\) 9.67464 1.41119 0.705596 0.708615i \(-0.250680\pi\)
0.705596 + 0.708615i \(0.250680\pi\)
\(48\) −3.40870 −0.492004
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 12.3756 1.73294
\(52\) −3.69350 −0.512197
\(53\) 9.39390 1.29035 0.645175 0.764035i \(-0.276784\pi\)
0.645175 + 0.764035i \(0.276784\pi\)
\(54\) −19.1543 −2.60657
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −6.87485 −0.910596
\(58\) −2.77810 −0.364782
\(59\) 12.8803 1.67687 0.838436 0.545000i \(-0.183470\pi\)
0.838436 + 0.545000i \(0.183470\pi\)
\(60\) 3.40870 0.440061
\(61\) −6.61375 −0.846804 −0.423402 0.905942i \(-0.639164\pi\)
−0.423402 + 0.905942i \(0.639164\pi\)
\(62\) 2.98864 0.379557
\(63\) −8.61924 −1.08592
\(64\) 1.00000 0.125000
\(65\) 3.69350 0.458123
\(66\) 0 0
\(67\) −3.48599 −0.425881 −0.212941 0.977065i \(-0.568304\pi\)
−0.212941 + 0.977065i \(0.568304\pi\)
\(68\) −3.63060 −0.440275
\(69\) 20.1251 2.42278
\(70\) 1.00000 0.119523
\(71\) −5.18885 −0.615803 −0.307901 0.951418i \(-0.599627\pi\)
−0.307901 + 0.951418i \(0.599627\pi\)
\(72\) 8.61924 1.01579
\(73\) −1.83909 −0.215250 −0.107625 0.994192i \(-0.534325\pi\)
−0.107625 + 0.994192i \(0.534325\pi\)
\(74\) 3.22190 0.374539
\(75\) −3.40870 −0.393603
\(76\) 2.01685 0.231349
\(77\) 0 0
\(78\) 12.5900 1.42554
\(79\) 2.95191 0.332115 0.166058 0.986116i \(-0.446896\pi\)
0.166058 + 0.986116i \(0.446896\pi\)
\(80\) −1.00000 −0.111803
\(81\) 39.4336 4.38151
\(82\) −8.13344 −0.898189
\(83\) −8.37746 −0.919546 −0.459773 0.888037i \(-0.652069\pi\)
−0.459773 + 0.888037i \(0.652069\pi\)
\(84\) 3.40870 0.371920
\(85\) 3.63060 0.393794
\(86\) −3.87280 −0.417615
\(87\) 9.46970 1.01526
\(88\) 0 0
\(89\) 4.85724 0.514866 0.257433 0.966296i \(-0.417123\pi\)
0.257433 + 0.966296i \(0.417123\pi\)
\(90\) −8.61924 −0.908548
\(91\) 3.69350 0.387184
\(92\) −5.90404 −0.615539
\(93\) −10.1874 −1.05638
\(94\) 9.67464 0.997863
\(95\) −2.01685 −0.206925
\(96\) −3.40870 −0.347899
\(97\) −16.0944 −1.63414 −0.817071 0.576537i \(-0.804404\pi\)
−0.817071 + 0.576537i \(0.804404\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −18.8577 −1.87641 −0.938207 0.346076i \(-0.887514\pi\)
−0.938207 + 0.346076i \(0.887514\pi\)
\(102\) 12.3756 1.22537
\(103\) −15.0231 −1.48027 −0.740135 0.672458i \(-0.765238\pi\)
−0.740135 + 0.672458i \(0.765238\pi\)
\(104\) −3.69350 −0.362178
\(105\) −3.40870 −0.332655
\(106\) 9.39390 0.912416
\(107\) −3.93323 −0.380240 −0.190120 0.981761i \(-0.560888\pi\)
−0.190120 + 0.981761i \(0.560888\pi\)
\(108\) −19.1543 −1.84312
\(109\) −19.7477 −1.89148 −0.945741 0.324921i \(-0.894662\pi\)
−0.945741 + 0.324921i \(0.894662\pi\)
\(110\) 0 0
\(111\) −10.9825 −1.04241
\(112\) −1.00000 −0.0944911
\(113\) −11.8492 −1.11468 −0.557340 0.830284i \(-0.688178\pi\)
−0.557340 + 0.830284i \(0.688178\pi\)
\(114\) −6.87485 −0.643889
\(115\) 5.90404 0.550555
\(116\) −2.77810 −0.257940
\(117\) −31.8352 −2.94316
\(118\) 12.8803 1.18573
\(119\) 3.63060 0.332817
\(120\) 3.40870 0.311170
\(121\) 0 0
\(122\) −6.61375 −0.598781
\(123\) 27.7245 2.49983
\(124\) 2.98864 0.268387
\(125\) −1.00000 −0.0894427
\(126\) −8.61924 −0.767863
\(127\) 16.7243 1.48404 0.742020 0.670378i \(-0.233868\pi\)
0.742020 + 0.670378i \(0.233868\pi\)
\(128\) 1.00000 0.0883883
\(129\) 13.2012 1.16230
\(130\) 3.69350 0.323942
\(131\) −13.1921 −1.15260 −0.576299 0.817239i \(-0.695504\pi\)
−0.576299 + 0.817239i \(0.695504\pi\)
\(132\) 0 0
\(133\) −2.01685 −0.174883
\(134\) −3.48599 −0.301143
\(135\) 19.1543 1.64854
\(136\) −3.63060 −0.311322
\(137\) 0.0587014 0.00501520 0.00250760 0.999997i \(-0.499202\pi\)
0.00250760 + 0.999997i \(0.499202\pi\)
\(138\) 20.1251 1.71316
\(139\) 20.9129 1.77381 0.886903 0.461956i \(-0.152852\pi\)
0.886903 + 0.461956i \(0.152852\pi\)
\(140\) 1.00000 0.0845154
\(141\) −32.9780 −2.77724
\(142\) −5.18885 −0.435438
\(143\) 0 0
\(144\) 8.61924 0.718270
\(145\) 2.77810 0.230708
\(146\) −1.83909 −0.152205
\(147\) −3.40870 −0.281145
\(148\) 3.22190 0.264839
\(149\) 14.8904 1.21987 0.609935 0.792451i \(-0.291195\pi\)
0.609935 + 0.792451i \(0.291195\pi\)
\(150\) −3.40870 −0.278319
\(151\) −12.3644 −1.00620 −0.503101 0.864227i \(-0.667808\pi\)
−0.503101 + 0.864227i \(0.667808\pi\)
\(152\) 2.01685 0.163588
\(153\) −31.2930 −2.52989
\(154\) 0 0
\(155\) −2.98864 −0.240053
\(156\) 12.5900 1.00801
\(157\) −22.0362 −1.75868 −0.879342 0.476192i \(-0.842017\pi\)
−0.879342 + 0.476192i \(0.842017\pi\)
\(158\) 2.95191 0.234841
\(159\) −32.0210 −2.53943
\(160\) −1.00000 −0.0790569
\(161\) 5.90404 0.465304
\(162\) 39.4336 3.09819
\(163\) 15.4758 1.21216 0.606081 0.795403i \(-0.292741\pi\)
0.606081 + 0.795403i \(0.292741\pi\)
\(164\) −8.13344 −0.635115
\(165\) 0 0
\(166\) −8.37746 −0.650217
\(167\) 17.4387 1.34945 0.674723 0.738071i \(-0.264263\pi\)
0.674723 + 0.738071i \(0.264263\pi\)
\(168\) 3.40870 0.262987
\(169\) 0.641969 0.0493822
\(170\) 3.63060 0.278455
\(171\) 17.3837 1.32937
\(172\) −3.87280 −0.295298
\(173\) −15.2949 −1.16285 −0.581425 0.813600i \(-0.697505\pi\)
−0.581425 + 0.813600i \(0.697505\pi\)
\(174\) 9.46970 0.717896
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −43.9051 −3.30011
\(178\) 4.85724 0.364065
\(179\) 11.6746 0.872604 0.436302 0.899800i \(-0.356288\pi\)
0.436302 + 0.899800i \(0.356288\pi\)
\(180\) −8.61924 −0.642440
\(181\) 9.93097 0.738163 0.369082 0.929397i \(-0.379672\pi\)
0.369082 + 0.929397i \(0.379672\pi\)
\(182\) 3.69350 0.273781
\(183\) 22.5443 1.66652
\(184\) −5.90404 −0.435252
\(185\) −3.22190 −0.236879
\(186\) −10.1874 −0.746974
\(187\) 0 0
\(188\) 9.67464 0.705596
\(189\) 19.1543 1.39327
\(190\) −2.01685 −0.146318
\(191\) 14.1868 1.02652 0.513260 0.858233i \(-0.328438\pi\)
0.513260 + 0.858233i \(0.328438\pi\)
\(192\) −3.40870 −0.246002
\(193\) 20.1766 1.45235 0.726173 0.687512i \(-0.241297\pi\)
0.726173 + 0.687512i \(0.241297\pi\)
\(194\) −16.0944 −1.15551
\(195\) −12.5900 −0.901592
\(196\) 1.00000 0.0714286
\(197\) 11.7016 0.833702 0.416851 0.908975i \(-0.363134\pi\)
0.416851 + 0.908975i \(0.363134\pi\)
\(198\) 0 0
\(199\) 8.24220 0.584274 0.292137 0.956376i \(-0.405634\pi\)
0.292137 + 0.956376i \(0.405634\pi\)
\(200\) 1.00000 0.0707107
\(201\) 11.8827 0.838140
\(202\) −18.8577 −1.32682
\(203\) 2.77810 0.194984
\(204\) 12.3756 0.866468
\(205\) 8.13344 0.568064
\(206\) −15.0231 −1.04671
\(207\) −50.8884 −3.53699
\(208\) −3.69350 −0.256098
\(209\) 0 0
\(210\) −3.40870 −0.235223
\(211\) 9.71032 0.668486 0.334243 0.942487i \(-0.391519\pi\)
0.334243 + 0.942487i \(0.391519\pi\)
\(212\) 9.39390 0.645175
\(213\) 17.6872 1.21191
\(214\) −3.93323 −0.268870
\(215\) 3.87280 0.264123
\(216\) −19.1543 −1.30329
\(217\) −2.98864 −0.202882
\(218\) −19.7477 −1.33748
\(219\) 6.26892 0.423615
\(220\) 0 0
\(221\) 13.4096 0.902031
\(222\) −10.9825 −0.737097
\(223\) 2.64541 0.177150 0.0885749 0.996070i \(-0.471769\pi\)
0.0885749 + 0.996070i \(0.471769\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 8.61924 0.574616
\(226\) −11.8492 −0.788198
\(227\) −2.30423 −0.152937 −0.0764686 0.997072i \(-0.524365\pi\)
−0.0764686 + 0.997072i \(0.524365\pi\)
\(228\) −6.87485 −0.455298
\(229\) 3.72474 0.246138 0.123069 0.992398i \(-0.460726\pi\)
0.123069 + 0.992398i \(0.460726\pi\)
\(230\) 5.90404 0.389301
\(231\) 0 0
\(232\) −2.77810 −0.182391
\(233\) 1.72846 0.113235 0.0566177 0.998396i \(-0.481968\pi\)
0.0566177 + 0.998396i \(0.481968\pi\)
\(234\) −31.8352 −2.08113
\(235\) −9.67464 −0.631104
\(236\) 12.8803 0.838436
\(237\) −10.0622 −0.653608
\(238\) 3.63060 0.235337
\(239\) −10.6465 −0.688664 −0.344332 0.938848i \(-0.611895\pi\)
−0.344332 + 0.938848i \(0.611895\pi\)
\(240\) 3.40870 0.220031
\(241\) 22.2210 1.43138 0.715690 0.698418i \(-0.246112\pi\)
0.715690 + 0.698418i \(0.246112\pi\)
\(242\) 0 0
\(243\) −76.9543 −4.93662
\(244\) −6.61375 −0.423402
\(245\) −1.00000 −0.0638877
\(246\) 27.7245 1.76765
\(247\) −7.44926 −0.473985
\(248\) 2.98864 0.189779
\(249\) 28.5563 1.80968
\(250\) −1.00000 −0.0632456
\(251\) 30.4213 1.92017 0.960087 0.279701i \(-0.0902354\pi\)
0.960087 + 0.279701i \(0.0902354\pi\)
\(252\) −8.61924 −0.542961
\(253\) 0 0
\(254\) 16.7243 1.04937
\(255\) −12.3756 −0.774993
\(256\) 1.00000 0.0625000
\(257\) 14.8674 0.927403 0.463701 0.885992i \(-0.346521\pi\)
0.463701 + 0.885992i \(0.346521\pi\)
\(258\) 13.2012 0.821872
\(259\) −3.22190 −0.200199
\(260\) 3.69350 0.229061
\(261\) −23.9451 −1.48216
\(262\) −13.1921 −0.815010
\(263\) 27.6050 1.70220 0.851098 0.525006i \(-0.175937\pi\)
0.851098 + 0.525006i \(0.175937\pi\)
\(264\) 0 0
\(265\) −9.39390 −0.577062
\(266\) −2.01685 −0.123661
\(267\) −16.5569 −1.01326
\(268\) −3.48599 −0.212941
\(269\) −4.63204 −0.282420 −0.141210 0.989980i \(-0.545099\pi\)
−0.141210 + 0.989980i \(0.545099\pi\)
\(270\) 19.1543 1.16569
\(271\) −4.31521 −0.262130 −0.131065 0.991374i \(-0.541840\pi\)
−0.131065 + 0.991374i \(0.541840\pi\)
\(272\) −3.63060 −0.220138
\(273\) −12.5900 −0.761984
\(274\) 0.0587014 0.00354628
\(275\) 0 0
\(276\) 20.1251 1.21139
\(277\) 14.4149 0.866110 0.433055 0.901367i \(-0.357436\pi\)
0.433055 + 0.901367i \(0.357436\pi\)
\(278\) 20.9129 1.25427
\(279\) 25.7598 1.54220
\(280\) 1.00000 0.0597614
\(281\) 29.1262 1.73753 0.868763 0.495229i \(-0.164916\pi\)
0.868763 + 0.495229i \(0.164916\pi\)
\(282\) −32.9780 −1.96381
\(283\) −19.8862 −1.18211 −0.591057 0.806630i \(-0.701289\pi\)
−0.591057 + 0.806630i \(0.701289\pi\)
\(284\) −5.18885 −0.307901
\(285\) 6.87485 0.407231
\(286\) 0 0
\(287\) 8.13344 0.480102
\(288\) 8.61924 0.507894
\(289\) −3.81871 −0.224630
\(290\) 2.77810 0.163135
\(291\) 54.8611 3.21601
\(292\) −1.83909 −0.107625
\(293\) 1.28362 0.0749899 0.0374949 0.999297i \(-0.488062\pi\)
0.0374949 + 0.999297i \(0.488062\pi\)
\(294\) −3.40870 −0.198799
\(295\) −12.8803 −0.749920
\(296\) 3.22190 0.187269
\(297\) 0 0
\(298\) 14.8904 0.862578
\(299\) 21.8066 1.26111
\(300\) −3.40870 −0.196801
\(301\) 3.87280 0.223225
\(302\) −12.3644 −0.711493
\(303\) 64.2803 3.69281
\(304\) 2.01685 0.115675
\(305\) 6.61375 0.378702
\(306\) −31.2930 −1.78890
\(307\) −16.0559 −0.916357 −0.458179 0.888860i \(-0.651498\pi\)
−0.458179 + 0.888860i \(0.651498\pi\)
\(308\) 0 0
\(309\) 51.2093 2.91319
\(310\) −2.98864 −0.169743
\(311\) 5.84180 0.331258 0.165629 0.986188i \(-0.447035\pi\)
0.165629 + 0.986188i \(0.447035\pi\)
\(312\) 12.5900 0.712771
\(313\) 23.0324 1.30187 0.650934 0.759134i \(-0.274377\pi\)
0.650934 + 0.759134i \(0.274377\pi\)
\(314\) −22.0362 −1.24358
\(315\) 8.61924 0.485639
\(316\) 2.95191 0.166058
\(317\) 25.6295 1.43950 0.719748 0.694236i \(-0.244257\pi\)
0.719748 + 0.694236i \(0.244257\pi\)
\(318\) −32.0210 −1.79565
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 13.4072 0.748318
\(322\) 5.90404 0.329019
\(323\) −7.32240 −0.407429
\(324\) 39.4336 2.19075
\(325\) −3.69350 −0.204879
\(326\) 15.4758 0.857127
\(327\) 67.3138 3.72246
\(328\) −8.13344 −0.449094
\(329\) −9.67464 −0.533380
\(330\) 0 0
\(331\) −1.61474 −0.0887542 −0.0443771 0.999015i \(-0.514130\pi\)
−0.0443771 + 0.999015i \(0.514130\pi\)
\(332\) −8.37746 −0.459773
\(333\) 27.7704 1.52181
\(334\) 17.4387 0.954203
\(335\) 3.48599 0.190460
\(336\) 3.40870 0.185960
\(337\) 11.2918 0.615105 0.307553 0.951531i \(-0.400490\pi\)
0.307553 + 0.951531i \(0.400490\pi\)
\(338\) 0.641969 0.0349185
\(339\) 40.3904 2.19371
\(340\) 3.63060 0.196897
\(341\) 0 0
\(342\) 17.3837 0.940005
\(343\) −1.00000 −0.0539949
\(344\) −3.87280 −0.208807
\(345\) −20.1251 −1.08350
\(346\) −15.2949 −0.822260
\(347\) −3.57419 −0.191873 −0.0959363 0.995387i \(-0.530585\pi\)
−0.0959363 + 0.995387i \(0.530585\pi\)
\(348\) 9.46970 0.507629
\(349\) 2.69597 0.144312 0.0721560 0.997393i \(-0.477012\pi\)
0.0721560 + 0.997393i \(0.477012\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 70.7465 3.77617
\(352\) 0 0
\(353\) 33.3929 1.77732 0.888662 0.458563i \(-0.151635\pi\)
0.888662 + 0.458563i \(0.151635\pi\)
\(354\) −43.9051 −2.33353
\(355\) 5.18885 0.275395
\(356\) 4.85724 0.257433
\(357\) −12.3756 −0.654988
\(358\) 11.6746 0.617024
\(359\) 10.5951 0.559189 0.279594 0.960118i \(-0.409800\pi\)
0.279594 + 0.960118i \(0.409800\pi\)
\(360\) −8.61924 −0.454274
\(361\) −14.9323 −0.785911
\(362\) 9.93097 0.521960
\(363\) 0 0
\(364\) 3.69350 0.193592
\(365\) 1.83909 0.0962626
\(366\) 22.5443 1.17841
\(367\) 0.872611 0.0455499 0.0227750 0.999741i \(-0.492750\pi\)
0.0227750 + 0.999741i \(0.492750\pi\)
\(368\) −5.90404 −0.307769
\(369\) −70.1041 −3.64947
\(370\) −3.22190 −0.167499
\(371\) −9.39390 −0.487707
\(372\) −10.1874 −0.528190
\(373\) −17.9629 −0.930085 −0.465042 0.885288i \(-0.653961\pi\)
−0.465042 + 0.885288i \(0.653961\pi\)
\(374\) 0 0
\(375\) 3.40870 0.176025
\(376\) 9.67464 0.498931
\(377\) 10.2609 0.528464
\(378\) 19.1543 0.985191
\(379\) −13.6701 −0.702185 −0.351093 0.936341i \(-0.614190\pi\)
−0.351093 + 0.936341i \(0.614190\pi\)
\(380\) −2.01685 −0.103462
\(381\) −57.0080 −2.92061
\(382\) 14.1868 0.725860
\(383\) 35.5188 1.81492 0.907462 0.420134i \(-0.138017\pi\)
0.907462 + 0.420134i \(0.138017\pi\)
\(384\) −3.40870 −0.173950
\(385\) 0 0
\(386\) 20.1766 1.02696
\(387\) −33.3806 −1.69683
\(388\) −16.0944 −0.817071
\(389\) −25.8447 −1.31038 −0.655189 0.755465i \(-0.727411\pi\)
−0.655189 + 0.755465i \(0.727411\pi\)
\(390\) −12.5900 −0.637522
\(391\) 21.4352 1.08403
\(392\) 1.00000 0.0505076
\(393\) 44.9679 2.26833
\(394\) 11.7016 0.589516
\(395\) −2.95191 −0.148526
\(396\) 0 0
\(397\) −38.5967 −1.93711 −0.968557 0.248793i \(-0.919966\pi\)
−0.968557 + 0.248793i \(0.919966\pi\)
\(398\) 8.24220 0.413144
\(399\) 6.87485 0.344173
\(400\) 1.00000 0.0500000
\(401\) −20.0070 −0.999100 −0.499550 0.866285i \(-0.666501\pi\)
−0.499550 + 0.866285i \(0.666501\pi\)
\(402\) 11.8827 0.592654
\(403\) −11.0385 −0.549869
\(404\) −18.8577 −0.938207
\(405\) −39.4336 −1.95947
\(406\) 2.77810 0.137875
\(407\) 0 0
\(408\) 12.3756 0.612686
\(409\) −2.18374 −0.107979 −0.0539894 0.998542i \(-0.517194\pi\)
−0.0539894 + 0.998542i \(0.517194\pi\)
\(410\) 8.13344 0.401682
\(411\) −0.200095 −0.00986998
\(412\) −15.0231 −0.740135
\(413\) −12.8803 −0.633798
\(414\) −50.8884 −2.50103
\(415\) 8.37746 0.411233
\(416\) −3.69350 −0.181089
\(417\) −71.2857 −3.49088
\(418\) 0 0
\(419\) −17.7452 −0.866909 −0.433455 0.901175i \(-0.642706\pi\)
−0.433455 + 0.901175i \(0.642706\pi\)
\(420\) −3.40870 −0.166328
\(421\) 15.0491 0.733447 0.366723 0.930330i \(-0.380480\pi\)
0.366723 + 0.930330i \(0.380480\pi\)
\(422\) 9.71032 0.472691
\(423\) 83.3880 4.05447
\(424\) 9.39390 0.456208
\(425\) −3.63060 −0.176110
\(426\) 17.6872 0.856949
\(427\) 6.61375 0.320062
\(428\) −3.93323 −0.190120
\(429\) 0 0
\(430\) 3.87280 0.186763
\(431\) −9.60354 −0.462586 −0.231293 0.972884i \(-0.574296\pi\)
−0.231293 + 0.972884i \(0.574296\pi\)
\(432\) −19.1543 −0.921562
\(433\) 24.2421 1.16500 0.582501 0.812830i \(-0.302074\pi\)
0.582501 + 0.812830i \(0.302074\pi\)
\(434\) −2.98864 −0.143459
\(435\) −9.46970 −0.454037
\(436\) −19.7477 −0.945741
\(437\) −11.9076 −0.569617
\(438\) 6.26892 0.299541
\(439\) 32.3368 1.54335 0.771677 0.636015i \(-0.219418\pi\)
0.771677 + 0.636015i \(0.219418\pi\)
\(440\) 0 0
\(441\) 8.61924 0.410440
\(442\) 13.4096 0.637832
\(443\) −21.8977 −1.04039 −0.520195 0.854048i \(-0.674141\pi\)
−0.520195 + 0.854048i \(0.674141\pi\)
\(444\) −10.9825 −0.521207
\(445\) −4.85724 −0.230255
\(446\) 2.64541 0.125264
\(447\) −50.7570 −2.40072
\(448\) −1.00000 −0.0472456
\(449\) −12.5764 −0.593518 −0.296759 0.954952i \(-0.595906\pi\)
−0.296759 + 0.954952i \(0.595906\pi\)
\(450\) 8.61924 0.406315
\(451\) 0 0
\(452\) −11.8492 −0.557340
\(453\) 42.1466 1.98022
\(454\) −2.30423 −0.108143
\(455\) −3.69350 −0.173154
\(456\) −6.87485 −0.321944
\(457\) −30.6714 −1.43475 −0.717374 0.696688i \(-0.754656\pi\)
−0.717374 + 0.696688i \(0.754656\pi\)
\(458\) 3.72474 0.174046
\(459\) 69.5417 3.24593
\(460\) 5.90404 0.275277
\(461\) 19.5978 0.912762 0.456381 0.889784i \(-0.349145\pi\)
0.456381 + 0.889784i \(0.349145\pi\)
\(462\) 0 0
\(463\) −4.29483 −0.199598 −0.0997989 0.995008i \(-0.531820\pi\)
−0.0997989 + 0.995008i \(0.531820\pi\)
\(464\) −2.77810 −0.128970
\(465\) 10.1874 0.472428
\(466\) 1.72846 0.0800695
\(467\) −1.40460 −0.0649973 −0.0324986 0.999472i \(-0.510346\pi\)
−0.0324986 + 0.999472i \(0.510346\pi\)
\(468\) −31.8352 −1.47158
\(469\) 3.48599 0.160968
\(470\) −9.67464 −0.446258
\(471\) 75.1149 3.46111
\(472\) 12.8803 0.592864
\(473\) 0 0
\(474\) −10.0622 −0.462170
\(475\) 2.01685 0.0925396
\(476\) 3.63060 0.166408
\(477\) 80.9682 3.70728
\(478\) −10.6465 −0.486959
\(479\) 32.3519 1.47820 0.739098 0.673598i \(-0.235252\pi\)
0.739098 + 0.673598i \(0.235252\pi\)
\(480\) 3.40870 0.155585
\(481\) −11.9001 −0.542598
\(482\) 22.2210 1.01214
\(483\) −20.1251 −0.915724
\(484\) 0 0
\(485\) 16.0944 0.730810
\(486\) −76.9543 −3.49072
\(487\) −3.75142 −0.169993 −0.0849965 0.996381i \(-0.527088\pi\)
−0.0849965 + 0.996381i \(0.527088\pi\)
\(488\) −6.61375 −0.299390
\(489\) −52.7525 −2.38555
\(490\) −1.00000 −0.0451754
\(491\) 11.1293 0.502257 0.251129 0.967954i \(-0.419198\pi\)
0.251129 + 0.967954i \(0.419198\pi\)
\(492\) 27.7245 1.24992
\(493\) 10.0862 0.454258
\(494\) −7.44926 −0.335158
\(495\) 0 0
\(496\) 2.98864 0.134194
\(497\) 5.18885 0.232752
\(498\) 28.5563 1.27964
\(499\) −23.2827 −1.04228 −0.521139 0.853472i \(-0.674493\pi\)
−0.521139 + 0.853472i \(0.674493\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −59.4433 −2.65573
\(502\) 30.4213 1.35777
\(503\) 35.8718 1.59945 0.799723 0.600370i \(-0.204980\pi\)
0.799723 + 0.600370i \(0.204980\pi\)
\(504\) −8.61924 −0.383931
\(505\) 18.8577 0.839157
\(506\) 0 0
\(507\) −2.18828 −0.0971849
\(508\) 16.7243 0.742020
\(509\) −23.2092 −1.02873 −0.514365 0.857571i \(-0.671972\pi\)
−0.514365 + 0.857571i \(0.671972\pi\)
\(510\) −12.3756 −0.548003
\(511\) 1.83909 0.0813568
\(512\) 1.00000 0.0441942
\(513\) −38.6314 −1.70562
\(514\) 14.8674 0.655773
\(515\) 15.0231 0.661997
\(516\) 13.2012 0.581151
\(517\) 0 0
\(518\) −3.22190 −0.141562
\(519\) 52.1358 2.28851
\(520\) 3.69350 0.161971
\(521\) −9.71728 −0.425722 −0.212861 0.977083i \(-0.568278\pi\)
−0.212861 + 0.977083i \(0.568278\pi\)
\(522\) −23.9451 −1.04805
\(523\) −1.70495 −0.0745523 −0.0372762 0.999305i \(-0.511868\pi\)
−0.0372762 + 0.999305i \(0.511868\pi\)
\(524\) −13.1921 −0.576299
\(525\) 3.40870 0.148768
\(526\) 27.6050 1.20363
\(527\) −10.8506 −0.472657
\(528\) 0 0
\(529\) 11.8577 0.515553
\(530\) −9.39390 −0.408045
\(531\) 111.018 4.81779
\(532\) −2.01685 −0.0874417
\(533\) 30.0409 1.30122
\(534\) −16.5569 −0.716486
\(535\) 3.93323 0.170049
\(536\) −3.48599 −0.150572
\(537\) −39.7954 −1.71730
\(538\) −4.63204 −0.199701
\(539\) 0 0
\(540\) 19.1543 0.824270
\(541\) −31.0627 −1.33549 −0.667745 0.744390i \(-0.732740\pi\)
−0.667745 + 0.744390i \(0.732740\pi\)
\(542\) −4.31521 −0.185354
\(543\) −33.8517 −1.45272
\(544\) −3.63060 −0.155661
\(545\) 19.7477 0.845897
\(546\) −12.5900 −0.538804
\(547\) 11.5948 0.495760 0.247880 0.968791i \(-0.420266\pi\)
0.247880 + 0.968791i \(0.420266\pi\)
\(548\) 0.0587014 0.00250760
\(549\) −57.0055 −2.43294
\(550\) 0 0
\(551\) −5.60301 −0.238696
\(552\) 20.1251 0.856582
\(553\) −2.95191 −0.125528
\(554\) 14.4149 0.612432
\(555\) 10.9825 0.466181
\(556\) 20.9129 0.886903
\(557\) −2.80080 −0.118674 −0.0593369 0.998238i \(-0.518899\pi\)
−0.0593369 + 0.998238i \(0.518899\pi\)
\(558\) 25.7598 1.09050
\(559\) 14.3042 0.605004
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 29.1262 1.22862
\(563\) −12.9733 −0.546758 −0.273379 0.961906i \(-0.588141\pi\)
−0.273379 + 0.961906i \(0.588141\pi\)
\(564\) −32.9780 −1.38862
\(565\) 11.8492 0.498500
\(566\) −19.8862 −0.835881
\(567\) −39.4336 −1.65605
\(568\) −5.18885 −0.217719
\(569\) 0.243705 0.0102166 0.00510832 0.999987i \(-0.498374\pi\)
0.00510832 + 0.999987i \(0.498374\pi\)
\(570\) 6.87485 0.287956
\(571\) 7.03817 0.294538 0.147269 0.989096i \(-0.452952\pi\)
0.147269 + 0.989096i \(0.452952\pi\)
\(572\) 0 0
\(573\) −48.3585 −2.02021
\(574\) 8.13344 0.339483
\(575\) −5.90404 −0.246216
\(576\) 8.61924 0.359135
\(577\) 33.3549 1.38858 0.694291 0.719694i \(-0.255718\pi\)
0.694291 + 0.719694i \(0.255718\pi\)
\(578\) −3.81871 −0.158838
\(579\) −68.7762 −2.85824
\(580\) 2.77810 0.115354
\(581\) 8.37746 0.347556
\(582\) 54.8611 2.27407
\(583\) 0 0
\(584\) −1.83909 −0.0761023
\(585\) 31.8352 1.31622
\(586\) 1.28362 0.0530258
\(587\) −18.3333 −0.756695 −0.378347 0.925664i \(-0.623508\pi\)
−0.378347 + 0.925664i \(0.623508\pi\)
\(588\) −3.40870 −0.140572
\(589\) 6.02764 0.248365
\(590\) −12.8803 −0.530274
\(591\) −39.8871 −1.64074
\(592\) 3.22190 0.132419
\(593\) 40.6951 1.67115 0.835573 0.549380i \(-0.185136\pi\)
0.835573 + 0.549380i \(0.185136\pi\)
\(594\) 0 0
\(595\) −3.63060 −0.148840
\(596\) 14.8904 0.609935
\(597\) −28.0952 −1.14986
\(598\) 21.8066 0.891738
\(599\) 27.7826 1.13516 0.567582 0.823317i \(-0.307879\pi\)
0.567582 + 0.823317i \(0.307879\pi\)
\(600\) −3.40870 −0.139160
\(601\) 28.0280 1.14329 0.571643 0.820502i \(-0.306306\pi\)
0.571643 + 0.820502i \(0.306306\pi\)
\(602\) 3.87280 0.157844
\(603\) −30.0466 −1.22359
\(604\) −12.3644 −0.503101
\(605\) 0 0
\(606\) 64.2803 2.61121
\(607\) −15.2484 −0.618912 −0.309456 0.950914i \(-0.600147\pi\)
−0.309456 + 0.950914i \(0.600147\pi\)
\(608\) 2.01685 0.0817942
\(609\) −9.46970 −0.383732
\(610\) 6.61375 0.267783
\(611\) −35.7333 −1.44562
\(612\) −31.2930 −1.26495
\(613\) 5.19985 0.210020 0.105010 0.994471i \(-0.466513\pi\)
0.105010 + 0.994471i \(0.466513\pi\)
\(614\) −16.0559 −0.647963
\(615\) −27.7245 −1.11796
\(616\) 0 0
\(617\) 5.07263 0.204217 0.102108 0.994773i \(-0.467441\pi\)
0.102108 + 0.994773i \(0.467441\pi\)
\(618\) 51.2093 2.05994
\(619\) 18.1421 0.729194 0.364597 0.931165i \(-0.381207\pi\)
0.364597 + 0.931165i \(0.381207\pi\)
\(620\) −2.98864 −0.120026
\(621\) 113.088 4.53806
\(622\) 5.84180 0.234235
\(623\) −4.85724 −0.194601
\(624\) 12.5900 0.504005
\(625\) 1.00000 0.0400000
\(626\) 23.0324 0.920560
\(627\) 0 0
\(628\) −22.0362 −0.879342
\(629\) −11.6975 −0.466408
\(630\) 8.61924 0.343399
\(631\) −23.7253 −0.944490 −0.472245 0.881467i \(-0.656556\pi\)
−0.472245 + 0.881467i \(0.656556\pi\)
\(632\) 2.95191 0.117421
\(633\) −33.0996 −1.31559
\(634\) 25.6295 1.01788
\(635\) −16.7243 −0.663683
\(636\) −32.0210 −1.26971
\(637\) −3.69350 −0.146342
\(638\) 0 0
\(639\) −44.7239 −1.76925
\(640\) −1.00000 −0.0395285
\(641\) 25.0709 0.990240 0.495120 0.868825i \(-0.335124\pi\)
0.495120 + 0.868825i \(0.335124\pi\)
\(642\) 13.4072 0.529141
\(643\) 39.6473 1.56354 0.781768 0.623569i \(-0.214318\pi\)
0.781768 + 0.623569i \(0.214318\pi\)
\(644\) 5.90404 0.232652
\(645\) −13.2012 −0.519798
\(646\) −7.32240 −0.288096
\(647\) 33.8481 1.33070 0.665352 0.746530i \(-0.268281\pi\)
0.665352 + 0.746530i \(0.268281\pi\)
\(648\) 39.4336 1.54910
\(649\) 0 0
\(650\) −3.69350 −0.144871
\(651\) 10.1874 0.399274
\(652\) 15.4758 0.606081
\(653\) 2.39913 0.0938852 0.0469426 0.998898i \(-0.485052\pi\)
0.0469426 + 0.998898i \(0.485052\pi\)
\(654\) 67.3138 2.63218
\(655\) 13.1921 0.515458
\(656\) −8.13344 −0.317558
\(657\) −15.8516 −0.618430
\(658\) −9.67464 −0.377157
\(659\) −14.5152 −0.565430 −0.282715 0.959204i \(-0.591235\pi\)
−0.282715 + 0.959204i \(0.591235\pi\)
\(660\) 0 0
\(661\) 10.1729 0.395680 0.197840 0.980234i \(-0.436607\pi\)
0.197840 + 0.980234i \(0.436607\pi\)
\(662\) −1.61474 −0.0627587
\(663\) −45.7095 −1.77521
\(664\) −8.37746 −0.325109
\(665\) 2.01685 0.0782102
\(666\) 27.7704 1.07608
\(667\) 16.4020 0.635088
\(668\) 17.4387 0.674723
\(669\) −9.01741 −0.348633
\(670\) 3.48599 0.134675
\(671\) 0 0
\(672\) 3.40870 0.131493
\(673\) 9.89134 0.381283 0.190642 0.981660i \(-0.438943\pi\)
0.190642 + 0.981660i \(0.438943\pi\)
\(674\) 11.2918 0.434945
\(675\) −19.1543 −0.737250
\(676\) 0.641969 0.0246911
\(677\) −19.2252 −0.738886 −0.369443 0.929253i \(-0.620451\pi\)
−0.369443 + 0.929253i \(0.620451\pi\)
\(678\) 40.3904 1.55118
\(679\) 16.0944 0.617647
\(680\) 3.63060 0.139227
\(681\) 7.85444 0.300983
\(682\) 0 0
\(683\) 37.5641 1.43735 0.718674 0.695347i \(-0.244749\pi\)
0.718674 + 0.695347i \(0.244749\pi\)
\(684\) 17.3837 0.664684
\(685\) −0.0587014 −0.00224286
\(686\) −1.00000 −0.0381802
\(687\) −12.6965 −0.484403
\(688\) −3.87280 −0.147649
\(689\) −34.6964 −1.32183
\(690\) −20.1251 −0.766150
\(691\) 50.4772 1.92024 0.960121 0.279585i \(-0.0901970\pi\)
0.960121 + 0.279585i \(0.0901970\pi\)
\(692\) −15.2949 −0.581425
\(693\) 0 0
\(694\) −3.57419 −0.135674
\(695\) −20.9129 −0.793270
\(696\) 9.46970 0.358948
\(697\) 29.5293 1.11850
\(698\) 2.69597 0.102044
\(699\) −5.89181 −0.222849
\(700\) −1.00000 −0.0377964
\(701\) −3.21370 −0.121380 −0.0606899 0.998157i \(-0.519330\pi\)
−0.0606899 + 0.998157i \(0.519330\pi\)
\(702\) 70.7465 2.67015
\(703\) 6.49811 0.245081
\(704\) 0 0
\(705\) 32.9780 1.24202
\(706\) 33.3929 1.25676
\(707\) 18.8577 0.709217
\(708\) −43.9051 −1.65005
\(709\) −23.5837 −0.885706 −0.442853 0.896594i \(-0.646034\pi\)
−0.442853 + 0.896594i \(0.646034\pi\)
\(710\) 5.18885 0.194734
\(711\) 25.4432 0.954194
\(712\) 4.85724 0.182033
\(713\) −17.6450 −0.660812
\(714\) −12.3756 −0.463147
\(715\) 0 0
\(716\) 11.6746 0.436302
\(717\) 36.2907 1.35530
\(718\) 10.5951 0.395406
\(719\) 31.6922 1.18192 0.590959 0.806702i \(-0.298749\pi\)
0.590959 + 0.806702i \(0.298749\pi\)
\(720\) −8.61924 −0.321220
\(721\) 15.0231 0.559490
\(722\) −14.9323 −0.555723
\(723\) −75.7447 −2.81697
\(724\) 9.93097 0.369082
\(725\) −2.77810 −0.103176
\(726\) 0 0
\(727\) 8.59020 0.318593 0.159296 0.987231i \(-0.449077\pi\)
0.159296 + 0.987231i \(0.449077\pi\)
\(728\) 3.69350 0.136890
\(729\) 144.014 5.33383
\(730\) 1.83909 0.0680679
\(731\) 14.0606 0.520051
\(732\) 22.5443 0.833261
\(733\) 17.1149 0.632155 0.316077 0.948733i \(-0.397634\pi\)
0.316077 + 0.948733i \(0.397634\pi\)
\(734\) 0.872611 0.0322086
\(735\) 3.40870 0.125732
\(736\) −5.90404 −0.217626
\(737\) 0 0
\(738\) −70.1041 −2.58057
\(739\) 16.6965 0.614189 0.307094 0.951679i \(-0.400643\pi\)
0.307094 + 0.951679i \(0.400643\pi\)
\(740\) −3.22190 −0.118440
\(741\) 25.3923 0.932809
\(742\) −9.39390 −0.344861
\(743\) −0.812545 −0.0298094 −0.0149047 0.999889i \(-0.504744\pi\)
−0.0149047 + 0.999889i \(0.504744\pi\)
\(744\) −10.1874 −0.373487
\(745\) −14.8904 −0.545543
\(746\) −17.9629 −0.657669
\(747\) −72.2073 −2.64193
\(748\) 0 0
\(749\) 3.93323 0.143717
\(750\) 3.40870 0.124468
\(751\) 3.97989 0.145228 0.0726140 0.997360i \(-0.476866\pi\)
0.0726140 + 0.997360i \(0.476866\pi\)
\(752\) 9.67464 0.352798
\(753\) −103.697 −3.77893
\(754\) 10.2609 0.373680
\(755\) 12.3644 0.449988
\(756\) 19.1543 0.696635
\(757\) 49.2340 1.78944 0.894720 0.446627i \(-0.147375\pi\)
0.894720 + 0.446627i \(0.147375\pi\)
\(758\) −13.6701 −0.496520
\(759\) 0 0
\(760\) −2.01685 −0.0731590
\(761\) −35.5947 −1.29031 −0.645154 0.764052i \(-0.723207\pi\)
−0.645154 + 0.764052i \(0.723207\pi\)
\(762\) −57.0080 −2.06518
\(763\) 19.7477 0.714913
\(764\) 14.1868 0.513260
\(765\) 31.2930 1.13140
\(766\) 35.5188 1.28335
\(767\) −47.5734 −1.71778
\(768\) −3.40870 −0.123001
\(769\) −6.16367 −0.222268 −0.111134 0.993805i \(-0.535448\pi\)
−0.111134 + 0.993805i \(0.535448\pi\)
\(770\) 0 0
\(771\) −50.6785 −1.82514
\(772\) 20.1766 0.726173
\(773\) −18.3589 −0.660323 −0.330162 0.943924i \(-0.607103\pi\)
−0.330162 + 0.943924i \(0.607103\pi\)
\(774\) −33.3806 −1.19984
\(775\) 2.98864 0.107355
\(776\) −16.0944 −0.577756
\(777\) 10.9825 0.393995
\(778\) −25.8447 −0.926577
\(779\) −16.4040 −0.587733
\(780\) −12.5900 −0.450796
\(781\) 0 0
\(782\) 21.4352 0.766523
\(783\) 53.2125 1.90166
\(784\) 1.00000 0.0357143
\(785\) 22.0362 0.786507
\(786\) 44.9679 1.60395
\(787\) 12.0202 0.428475 0.214237 0.976782i \(-0.431273\pi\)
0.214237 + 0.976782i \(0.431273\pi\)
\(788\) 11.7016 0.416851
\(789\) −94.0972 −3.34995
\(790\) −2.95191 −0.105024
\(791\) 11.8492 0.421309
\(792\) 0 0
\(793\) 24.4279 0.867461
\(794\) −38.5967 −1.36975
\(795\) 32.0210 1.13567
\(796\) 8.24220 0.292137
\(797\) 15.7618 0.558313 0.279156 0.960246i \(-0.409945\pi\)
0.279156 + 0.960246i \(0.409945\pi\)
\(798\) 6.87485 0.243367
\(799\) −35.1248 −1.24263
\(800\) 1.00000 0.0353553
\(801\) 41.8657 1.47925
\(802\) −20.0070 −0.706470
\(803\) 0 0
\(804\) 11.8827 0.419070
\(805\) −5.90404 −0.208090
\(806\) −11.0385 −0.388816
\(807\) 15.7892 0.555807
\(808\) −18.8577 −0.663412
\(809\) −26.5335 −0.932867 −0.466434 0.884556i \(-0.654461\pi\)
−0.466434 + 0.884556i \(0.654461\pi\)
\(810\) −39.4336 −1.38555
\(811\) 2.76840 0.0972117 0.0486059 0.998818i \(-0.484522\pi\)
0.0486059 + 0.998818i \(0.484522\pi\)
\(812\) 2.77810 0.0974921
\(813\) 14.7093 0.515876
\(814\) 0 0
\(815\) −15.4758 −0.542095
\(816\) 12.3756 0.433234
\(817\) −7.81088 −0.273268
\(818\) −2.18374 −0.0763525
\(819\) 31.8352 1.11241
\(820\) 8.13344 0.284032
\(821\) −7.47931 −0.261030 −0.130515 0.991446i \(-0.541663\pi\)
−0.130515 + 0.991446i \(0.541663\pi\)
\(822\) −0.200095 −0.00697913
\(823\) 28.1564 0.981472 0.490736 0.871308i \(-0.336728\pi\)
0.490736 + 0.871308i \(0.336728\pi\)
\(824\) −15.0231 −0.523355
\(825\) 0 0
\(826\) −12.8803 −0.448163
\(827\) 43.0851 1.49821 0.749107 0.662449i \(-0.230483\pi\)
0.749107 + 0.662449i \(0.230483\pi\)
\(828\) −50.8884 −1.76849
\(829\) −6.22008 −0.216032 −0.108016 0.994149i \(-0.534450\pi\)
−0.108016 + 0.994149i \(0.534450\pi\)
\(830\) 8.37746 0.290786
\(831\) −49.1362 −1.70452
\(832\) −3.69350 −0.128049
\(833\) −3.63060 −0.125793
\(834\) −71.2857 −2.46842
\(835\) −17.4387 −0.603491
\(836\) 0 0
\(837\) −57.2452 −1.97868
\(838\) −17.7452 −0.612997
\(839\) 14.5476 0.502240 0.251120 0.967956i \(-0.419201\pi\)
0.251120 + 0.967956i \(0.419201\pi\)
\(840\) −3.40870 −0.117611
\(841\) −21.2822 −0.733868
\(842\) 15.0491 0.518625
\(843\) −99.2826 −3.41947
\(844\) 9.71032 0.334243
\(845\) −0.641969 −0.0220844
\(846\) 83.3880 2.86694
\(847\) 0 0
\(848\) 9.39390 0.322588
\(849\) 67.7863 2.32642
\(850\) −3.63060 −0.124529
\(851\) −19.0223 −0.652075
\(852\) 17.6872 0.605954
\(853\) 46.3967 1.58859 0.794296 0.607531i \(-0.207840\pi\)
0.794296 + 0.607531i \(0.207840\pi\)
\(854\) 6.61375 0.226318
\(855\) −17.3837 −0.594512
\(856\) −3.93323 −0.134435
\(857\) −47.0412 −1.60690 −0.803448 0.595374i \(-0.797004\pi\)
−0.803448 + 0.595374i \(0.797004\pi\)
\(858\) 0 0
\(859\) −3.80378 −0.129783 −0.0648917 0.997892i \(-0.520670\pi\)
−0.0648917 + 0.997892i \(0.520670\pi\)
\(860\) 3.87280 0.132061
\(861\) −27.7245 −0.944848
\(862\) −9.60354 −0.327098
\(863\) −30.7524 −1.04682 −0.523412 0.852080i \(-0.675341\pi\)
−0.523412 + 0.852080i \(0.675341\pi\)
\(864\) −19.1543 −0.651643
\(865\) 15.2949 0.520043
\(866\) 24.2421 0.823780
\(867\) 13.0168 0.442075
\(868\) −2.98864 −0.101441
\(869\) 0 0
\(870\) −9.46970 −0.321053
\(871\) 12.8755 0.436270
\(872\) −19.7477 −0.668740
\(873\) −138.722 −4.69502
\(874\) −11.9076 −0.402780
\(875\) 1.00000 0.0338062
\(876\) 6.26892 0.211807
\(877\) −12.0703 −0.407583 −0.203792 0.979014i \(-0.565327\pi\)
−0.203792 + 0.979014i \(0.565327\pi\)
\(878\) 32.3368 1.09132
\(879\) −4.37548 −0.147581
\(880\) 0 0
\(881\) 45.7309 1.54071 0.770357 0.637613i \(-0.220078\pi\)
0.770357 + 0.637613i \(0.220078\pi\)
\(882\) 8.61924 0.290225
\(883\) −12.6357 −0.425225 −0.212612 0.977137i \(-0.568197\pi\)
−0.212612 + 0.977137i \(0.568197\pi\)
\(884\) 13.4096 0.451015
\(885\) 43.9051 1.47585
\(886\) −21.8977 −0.735667
\(887\) −2.54891 −0.0855840 −0.0427920 0.999084i \(-0.513625\pi\)
−0.0427920 + 0.999084i \(0.513625\pi\)
\(888\) −10.9825 −0.368549
\(889\) −16.7243 −0.560914
\(890\) −4.85724 −0.162815
\(891\) 0 0
\(892\) 2.64541 0.0885749
\(893\) 19.5123 0.652955
\(894\) −50.7570 −1.69757
\(895\) −11.6746 −0.390240
\(896\) −1.00000 −0.0334077
\(897\) −74.3322 −2.48188
\(898\) −12.5764 −0.419681
\(899\) −8.30272 −0.276911
\(900\) 8.61924 0.287308
\(901\) −34.1055 −1.13622
\(902\) 0 0
\(903\) −13.2012 −0.439309
\(904\) −11.8492 −0.394099
\(905\) −9.93097 −0.330117
\(906\) 42.1466 1.40023
\(907\) 16.6544 0.553001 0.276500 0.961014i \(-0.410825\pi\)
0.276500 + 0.961014i \(0.410825\pi\)
\(908\) −2.30423 −0.0764686
\(909\) −162.539 −5.39108
\(910\) −3.69350 −0.122438
\(911\) −26.6728 −0.883709 −0.441855 0.897087i \(-0.645679\pi\)
−0.441855 + 0.897087i \(0.645679\pi\)
\(912\) −6.87485 −0.227649
\(913\) 0 0
\(914\) −30.6714 −1.01452
\(915\) −22.5443 −0.745291
\(916\) 3.72474 0.123069
\(917\) 13.1921 0.435641
\(918\) 69.5417 2.29522
\(919\) 55.8882 1.84358 0.921791 0.387687i \(-0.126726\pi\)
0.921791 + 0.387687i \(0.126726\pi\)
\(920\) 5.90404 0.194651
\(921\) 54.7297 1.80340
\(922\) 19.5978 0.645420
\(923\) 19.1650 0.630824
\(924\) 0 0
\(925\) 3.22190 0.105936
\(926\) −4.29483 −0.141137
\(927\) −129.488 −4.25293
\(928\) −2.77810 −0.0911955
\(929\) −45.2542 −1.48474 −0.742371 0.669989i \(-0.766299\pi\)
−0.742371 + 0.669989i \(0.766299\pi\)
\(930\) 10.1874 0.334057
\(931\) 2.01685 0.0660997
\(932\) 1.72846 0.0566177
\(933\) −19.9129 −0.651921
\(934\) −1.40460 −0.0459600
\(935\) 0 0
\(936\) −31.8352 −1.04057
\(937\) 29.5196 0.964364 0.482182 0.876071i \(-0.339844\pi\)
0.482182 + 0.876071i \(0.339844\pi\)
\(938\) 3.48599 0.113821
\(939\) −78.5106 −2.56210
\(940\) −9.67464 −0.315552
\(941\) 5.88530 0.191855 0.0959277 0.995388i \(-0.469418\pi\)
0.0959277 + 0.995388i \(0.469418\pi\)
\(942\) 75.1149 2.44738
\(943\) 48.0202 1.56375
\(944\) 12.8803 0.419218
\(945\) −19.1543 −0.623090
\(946\) 0 0
\(947\) −10.3432 −0.336109 −0.168055 0.985778i \(-0.553749\pi\)
−0.168055 + 0.985778i \(0.553749\pi\)
\(948\) −10.0622 −0.326804
\(949\) 6.79270 0.220500
\(950\) 2.01685 0.0654354
\(951\) −87.3633 −2.83295
\(952\) 3.63060 0.117669
\(953\) 21.6287 0.700623 0.350311 0.936633i \(-0.386076\pi\)
0.350311 + 0.936633i \(0.386076\pi\)
\(954\) 80.9682 2.62144
\(955\) −14.1868 −0.459074
\(956\) −10.6465 −0.344332
\(957\) 0 0
\(958\) 32.3519 1.04524
\(959\) −0.0587014 −0.00189557
\(960\) 3.40870 0.110015
\(961\) −22.0681 −0.711873
\(962\) −11.9001 −0.383675
\(963\) −33.9015 −1.09246
\(964\) 22.2210 0.715690
\(965\) −20.1766 −0.649509
\(966\) −20.1251 −0.647515
\(967\) −38.1124 −1.22561 −0.612806 0.790233i \(-0.709959\pi\)
−0.612806 + 0.790233i \(0.709959\pi\)
\(968\) 0 0
\(969\) 24.9599 0.801826
\(970\) 16.0944 0.516761
\(971\) −16.2741 −0.522261 −0.261130 0.965304i \(-0.584095\pi\)
−0.261130 + 0.965304i \(0.584095\pi\)
\(972\) −76.9543 −2.46831
\(973\) −20.9129 −0.670436
\(974\) −3.75142 −0.120203
\(975\) 12.5900 0.403204
\(976\) −6.61375 −0.211701
\(977\) 33.2995 1.06535 0.532673 0.846321i \(-0.321187\pi\)
0.532673 + 0.846321i \(0.321187\pi\)
\(978\) −52.7525 −1.68684
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −170.210 −5.43438
\(982\) 11.1293 0.355150
\(983\) −28.8235 −0.919326 −0.459663 0.888094i \(-0.652030\pi\)
−0.459663 + 0.888094i \(0.652030\pi\)
\(984\) 27.7245 0.883824
\(985\) −11.7016 −0.372843
\(986\) 10.0862 0.321209
\(987\) 32.9780 1.04970
\(988\) −7.44926 −0.236992
\(989\) 22.8652 0.727071
\(990\) 0 0
\(991\) 19.6269 0.623469 0.311734 0.950169i \(-0.399090\pi\)
0.311734 + 0.950169i \(0.399090\pi\)
\(992\) 2.98864 0.0948893
\(993\) 5.50417 0.174670
\(994\) 5.18885 0.164580
\(995\) −8.24220 −0.261295
\(996\) 28.5563 0.904840
\(997\) −18.3518 −0.581207 −0.290603 0.956844i \(-0.593856\pi\)
−0.290603 + 0.956844i \(0.593856\pi\)
\(998\) −23.2827 −0.737002
\(999\) −61.7133 −1.95252
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 8470.2.a.dd.1.1 yes 6
11.10 odd 2 8470.2.a.cx.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.1 6 11.10 odd 2
8470.2.a.dd.1.1 yes 6 1.1 even 1 trivial