Properties

Label 8470.2.a.cx.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 \(-0.328832\pi\)
0.512197 + 0.858868i \(0.328832\pi\)
\(14\) −1.00000 −0.267261
\(15\) 3.40870 0.880123
\(16\) 1.00000 0.250000
\(17\) 3.63060 0.880551 0.440275 0.897863i \(-0.354881\pi\)
0.440275 + 0.897863i \(0.354881\pi\)
\(18\) −8.61924 −2.03157
\(19\) −2.01685 −0.462698 −0.231349 0.972871i \(-0.574314\pi\)
−0.231349 + 0.972871i \(0.574314\pi\)
\(20\) −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 \(-0.416956\pi\)
0.257940 + 0.966161i \(0.416956\pi\)
\(30\) −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 \(-0.280953\pi\)
0.635115 + 0.772417i \(0.280953\pi\)
\(42\) 3.40870 0.525974
\(43\) 3.87280 0.590597 0.295298 0.955405i \(-0.404581\pi\)
0.295298 + 0.955405i \(0.404581\pi\)
\(44\) 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 \(-0.360836\pi\)
0.423402 + 0.905942i \(0.360836\pi\)
\(62\) −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 \(-0.465675\pi\)
0.107625 + 0.994192i \(0.465675\pi\)
\(74\) −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 \(-0.553104\pi\)
−0.166058 + 0.986116i \(0.553104\pi\)
\(80\) −1.00000 −0.111803
\(81\) 39.4336 4.38151
\(82\) −8.13344 −0.898189
\(83\) 8.37746 0.919546 0.459773 0.888037i \(-0.347931\pi\)
0.459773 + 0.888037i \(0.347931\pi\)
\(84\) −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 \(-0.112486\pi\)
0.938207 + 0.346076i \(0.112486\pi\)
\(102\) 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 \(-0.439112\pi\)
0.190120 + 0.981761i \(0.439112\pi\)
\(108\) −19.1543 −1.84312
\(109\) 19.7477 1.89148 0.945741 0.324921i \(-0.105338\pi\)
0.945741 + 0.324921i \(0.105338\pi\)
\(110\) 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 \(-0.766132\pi\)
−0.742020 + 0.670378i \(0.766132\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −13.2012 −1.16230
\(130\) 3.69350 0.323942
\(131\) 13.1921 1.15260 0.576299 0.817239i \(-0.304496\pi\)
0.576299 + 0.817239i \(0.304496\pi\)
\(132\) 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 \(-0.847148\pi\)
−0.886903 + 0.461956i \(0.847148\pi\)
\(140\) −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 \(-0.708805\pi\)
−0.609935 + 0.792451i \(0.708805\pi\)
\(150\) 3.40870 0.278319
\(151\) 12.3644 1.00620 0.503101 0.864227i \(-0.332192\pi\)
0.503101 + 0.864227i \(0.332192\pi\)
\(152\) 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 \(-0.735737\pi\)
−0.674723 + 0.738071i \(0.735737\pi\)
\(168\) 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 \(-0.302495\pi\)
0.581425 + 0.813600i \(0.302495\pi\)
\(174\) 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 \(-0.758703\pi\)
−0.726173 + 0.687512i \(0.758703\pi\)
\(194\) 16.0944 1.15551
\(195\) 12.5900 0.901592
\(196\) 1.00000 0.0714286
\(197\) −11.7016 −0.833702 −0.416851 0.908975i \(-0.636866\pi\)
−0.416851 + 0.908975i \(0.636866\pi\)
\(198\) 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 \(-0.608481\pi\)
−0.334243 + 0.942487i \(0.608481\pi\)
\(212\) 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 \(-0.475635\pi\)
0.0764686 + 0.997072i \(0.475635\pi\)
\(228\) 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 \(-0.518032\pi\)
−0.0566177 + 0.998396i \(0.518032\pi\)
\(234\) −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 \(-0.388105\pi\)
0.344332 + 0.938848i \(0.388105\pi\)
\(240\) 3.40870 0.220031
\(241\) −22.2210 −1.43138 −0.715690 0.698418i \(-0.753888\pi\)
−0.715690 + 0.698418i \(0.753888\pi\)
\(242\) 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 \(-0.824063\pi\)
−0.851098 + 0.525006i \(0.824063\pi\)
\(264\) 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 \(-0.458160\pi\)
0.131065 + 0.991374i \(0.458160\pi\)
\(272\) 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 \(-0.642564\pi\)
−0.433055 + 0.901367i \(0.642564\pi\)
\(278\) 20.9129 1.25427
\(279\) 25.7598 1.54220
\(280\) 1.00000 0.0597614
\(281\) −29.1262 −1.73753 −0.868763 0.495229i \(-0.835084\pi\)
−0.868763 + 0.495229i \(0.835084\pi\)
\(282\) 32.9780 1.96381
\(283\) 19.8862 1.18211 0.591057 0.806630i \(-0.298711\pi\)
0.591057 + 0.806630i \(0.298711\pi\)
\(284\) −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 \(-0.511938\pi\)
−0.0374949 + 0.999297i \(0.511938\pi\)
\(294\) 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 \(-0.348502\pi\)
0.458179 + 0.888860i \(0.348502\pi\)
\(308\) 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 \(-0.599510\pi\)
−0.307553 + 0.951531i \(0.599510\pi\)
\(338\) −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 \(-0.469415\pi\)
0.0959363 + 0.995387i \(0.469415\pi\)
\(348\) −9.46970 −0.507629
\(349\) −2.69597 −0.144312 −0.0721560 0.997393i \(-0.522988\pi\)
−0.0721560 + 0.997393i \(0.522988\pi\)
\(350\) −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 \(-0.590200\pi\)
−0.279594 + 0.960118i \(0.590200\pi\)
\(360\) 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 \(-0.346039\pi\)
0.465042 + 0.885288i \(0.346039\pi\)
\(374\) 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 \(-0.482806\pi\)
0.0539894 + 0.998542i \(0.482806\pi\)
\(410\) 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 \(-0.425704\pi\)
0.231293 + 0.972884i \(0.425704\pi\)
\(432\) −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 \(-0.780582\pi\)
−0.771677 + 0.636015i \(0.780582\pi\)
\(440\) 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 \(-0.245344\pi\)
0.717374 + 0.696688i \(0.245344\pi\)
\(458\) −3.72474 −0.174046
\(459\) −69.5417 −3.24593
\(460\) 5.90404 0.275277
\(461\) −19.5978 −0.912762 −0.456381 0.889784i \(-0.650855\pi\)
−0.456381 + 0.889784i \(0.650855\pi\)
\(462\) 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 \(-0.764748\pi\)
−0.739098 + 0.673598i \(0.764748\pi\)
\(480\) −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 \(-0.580802\pi\)
−0.251129 + 0.967954i \(0.580802\pi\)
\(492\) −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 \(-0.795020\pi\)
−0.799723 + 0.600370i \(0.795020\pi\)
\(504\) −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 \(-0.488132\pi\)
0.0372762 + 0.999305i \(0.488132\pi\)
\(524\) 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 \(-0.267260\pi\)
0.667745 + 0.744390i \(0.267260\pi\)
\(542\) −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 \(-0.579734\pi\)
−0.247880 + 0.968791i \(0.579734\pi\)
\(548\) 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 \(-0.481101\pi\)
0.0593369 + 0.998238i \(0.481101\pi\)
\(558\) −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 \(-0.411859\pi\)
0.273379 + 0.961906i \(0.411859\pi\)
\(564\) −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 \(-0.501626\pi\)
−0.00510832 + 0.999987i \(0.501626\pi\)
\(570\) 6.87485 0.287956
\(571\) −7.03817 −0.294538 −0.147269 0.989096i \(-0.547048\pi\)
−0.147269 + 0.989096i \(0.547048\pi\)
\(572\) 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 \(-0.814864\pi\)
−0.835573 + 0.549380i \(0.814864\pi\)
\(594\) 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 \(-0.693694\pi\)
−0.571643 + 0.820502i \(0.693694\pi\)
\(602\) −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 \(-0.399853\pi\)
0.309456 + 0.950914i \(0.399853\pi\)
\(608\) 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 \(-0.533487\pi\)
−0.105010 + 0.994471i \(0.533487\pi\)
\(614\) −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 \(-0.408765\pi\)
0.282715 + 0.959204i \(0.408765\pi\)
\(660\) 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 \(-0.561057\pi\)
−0.190642 + 0.981660i \(0.561057\pi\)
\(674\) 11.2918 0.434945
\(675\) −19.1543 −0.737250
\(676\) 0.641969 0.0246911
\(677\) 19.2252 0.738886 0.369443 0.929253i \(-0.379549\pi\)
0.369443 + 0.929253i \(0.379549\pi\)
\(678\) −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 \(-0.480670\pi\)
0.0606899 + 0.998157i \(0.480670\pi\)
\(702\) 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 \(-0.602366\pi\)
−0.316077 + 0.948733i \(0.602366\pi\)
\(734\) −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 \(-0.599357\pi\)
−0.307094 + 0.951679i \(0.599357\pi\)
\(740\) −3.22190 −0.118440
\(741\) 25.3923 0.932809
\(742\) −9.39390 −0.344861
\(743\) 0.812545 0.0298094 0.0149047 0.999889i \(-0.495256\pi\)
0.0149047 + 0.999889i \(0.495256\pi\)
\(744\) 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 \(-0.276793\pi\)
0.645154 + 0.764052i \(0.276793\pi\)
\(762\) −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 \(-0.464552\pi\)
0.111134 + 0.993805i \(0.464552\pi\)
\(770\) 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 \(-0.568727\pi\)
−0.214237 + 0.976782i \(0.568727\pi\)
\(788\) −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 \(-0.345539\pi\)
0.466434 + 0.884556i \(0.345539\pi\)
\(810\) 39.4336 1.38555
\(811\) −2.76840 −0.0972117 −0.0486059 0.998818i \(-0.515478\pi\)
−0.0486059 + 0.998818i \(0.515478\pi\)
\(812\) 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 \(-0.458337\pi\)
0.130515 + 0.991446i \(0.458337\pi\)
\(822\) 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 \(-0.769517\pi\)
−0.749107 + 0.662449i \(0.769517\pi\)
\(828\) −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 \(-0.792160\pi\)
−0.794296 + 0.607531i \(0.792160\pi\)
\(854\) −6.61375 −0.226318
\(855\) 17.3837 0.594512
\(856\) −3.93323 −0.134435
\(857\) 47.0412 1.60690 0.803448 0.595374i \(-0.202996\pi\)
0.803448 + 0.595374i \(0.202996\pi\)
\(858\) 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 \(-0.434673\pi\)
0.203792 + 0.979014i \(0.434673\pi\)
\(878\) 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 \(-0.486375\pi\)
0.0427920 + 0.999084i \(0.486375\pi\)
\(888\) 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 \(-0.873274\pi\)
−0.921791 + 0.387687i \(0.873274\pi\)
\(920\) −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 \(-0.660156\pi\)
−0.482182 + 0.876071i \(0.660156\pi\)
\(938\) 3.48599 0.113821
\(939\) −78.5106 −2.56210
\(940\) −9.67464 −0.315552
\(941\) −5.88530 −0.191855 −0.0959277 0.995388i \(-0.530582\pi\)
−0.0959277 + 0.995388i \(0.530582\pi\)
\(942\) −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 \(-0.613924\pi\)
−0.350311 + 0.936633i \(0.613924\pi\)
\(954\) −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 \(-0.290041\pi\)
0.612806 + 0.790233i \(0.290041\pi\)
\(968\) 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 \(-0.406144\pi\)
0.290603 + 0.956844i \(0.406144\pi\)
\(998\) 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.cx.1.1 6
11.10 odd 2 8470.2.a.dd.1.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cx.1.1 6 1.1 even 1 trivial
8470.2.a.dd.1.1 yes 6 11.10 odd 2