Properties

Label 5070.2.a.bt.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.04892 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -6.24698 q^{11} -1.00000 q^{12} +3.04892 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.69202 q^{17} +1.00000 q^{18} -5.82908 q^{19} +1.00000 q^{20} -3.04892 q^{21} -6.24698 q^{22} -5.62565 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.04892 q^{28} -5.14675 q^{29} -1.00000 q^{30} -3.53319 q^{31} +1.00000 q^{32} +6.24698 q^{33} +2.69202 q^{34} +3.04892 q^{35} +1.00000 q^{36} -4.65279 q^{37} -5.82908 q^{38} +1.00000 q^{40} -3.77479 q^{41} -3.04892 q^{42} +2.85086 q^{43} -6.24698 q^{44} +1.00000 q^{45} -5.62565 q^{46} -3.61596 q^{47} -1.00000 q^{48} +2.29590 q^{49} +1.00000 q^{50} -2.69202 q^{51} +0.664874 q^{53} -1.00000 q^{54} -6.24698 q^{55} +3.04892 q^{56} +5.82908 q^{57} -5.14675 q^{58} -12.9487 q^{59} -1.00000 q^{60} -7.91723 q^{61} -3.53319 q^{62} +3.04892 q^{63} +1.00000 q^{64} +6.24698 q^{66} -0.198062 q^{67} +2.69202 q^{68} +5.62565 q^{69} +3.04892 q^{70} -0.374354 q^{71} +1.00000 q^{72} +14.6136 q^{73} -4.65279 q^{74} -1.00000 q^{75} -5.82908 q^{76} -19.0465 q^{77} -11.2567 q^{79} +1.00000 q^{80} +1.00000 q^{81} -3.77479 q^{82} +6.83877 q^{83} -3.04892 q^{84} +2.69202 q^{85} +2.85086 q^{86} +5.14675 q^{87} -6.24698 q^{88} +9.50365 q^{89} +1.00000 q^{90} -5.62565 q^{92} +3.53319 q^{93} -3.61596 q^{94} -5.82908 q^{95} -1.00000 q^{96} -0.335126 q^{97} +2.29590 q^{98} -6.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + 3q^{10} - 14q^{11} - 3q^{12} - 3q^{15} + 3q^{16} + 3q^{17} + 3q^{18} - 7q^{19} + 3q^{20} - 14q^{22} - 5q^{23} - 3q^{24} + 3q^{25} - 3q^{27} + 12q^{29} - 3q^{30} - 14q^{31} + 3q^{32} + 14q^{33} + 3q^{34} + 3q^{36} + 4q^{37} - 7q^{38} + 3q^{40} - 13q^{41} - 5q^{43} - 14q^{44} + 3q^{45} - 5q^{46} - 21q^{47} - 3q^{48} - 7q^{49} + 3q^{50} - 3q^{51} + 3q^{53} - 3q^{54} - 14q^{55} + 7q^{57} + 12q^{58} - 7q^{59} - 3q^{60} - 17q^{61} - 14q^{62} + 3q^{64} + 14q^{66} - 5q^{67} + 3q^{68} + 5q^{69} - 13q^{71} + 3q^{72} + 13q^{73} + 4q^{74} - 3q^{75} - 7q^{76} - 7q^{77} - 7q^{79} + 3q^{80} + 3q^{81} - 13q^{82} - 12q^{83} + 3q^{85} - 5q^{86} - 12q^{87} - 14q^{88} - 3q^{89} + 3q^{90} - 5q^{92} + 14q^{93} - 21q^{94} - 7q^{95} - 3q^{96} - 7q^{98} - 14q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 3.04892 1.15238 0.576191 0.817315i \(-0.304538\pi\)
0.576191 + 0.817315i \(0.304538\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −6.24698 −1.88354 −0.941768 0.336264i \(-0.890836\pi\)
−0.941768 + 0.336264i \(0.890836\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 3.04892 0.814857
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.69202 0.652911 0.326456 0.945213i \(-0.394146\pi\)
0.326456 + 0.945213i \(0.394146\pi\)
\(18\) 1.00000 0.235702
\(19\) −5.82908 −1.33728 −0.668642 0.743585i \(-0.733124\pi\)
−0.668642 + 0.743585i \(0.733124\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.04892 −0.665328
\(22\) −6.24698 −1.33186
\(23\) −5.62565 −1.17303 −0.586514 0.809939i \(-0.699500\pi\)
−0.586514 + 0.809939i \(0.699500\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.04892 0.576191
\(29\) −5.14675 −0.955728 −0.477864 0.878434i \(-0.658589\pi\)
−0.477864 + 0.878434i \(0.658589\pi\)
\(30\) −1.00000 −0.182574
\(31\) −3.53319 −0.634579 −0.317290 0.948329i \(-0.602773\pi\)
−0.317290 + 0.948329i \(0.602773\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.24698 1.08746
\(34\) 2.69202 0.461678
\(35\) 3.04892 0.515361
\(36\) 1.00000 0.166667
\(37\) −4.65279 −0.764914 −0.382457 0.923973i \(-0.624922\pi\)
−0.382457 + 0.923973i \(0.624922\pi\)
\(38\) −5.82908 −0.945602
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −3.77479 −0.589523 −0.294762 0.955571i \(-0.595240\pi\)
−0.294762 + 0.955571i \(0.595240\pi\)
\(42\) −3.04892 −0.470458
\(43\) 2.85086 0.434751 0.217376 0.976088i \(-0.430250\pi\)
0.217376 + 0.976088i \(0.430250\pi\)
\(44\) −6.24698 −0.941768
\(45\) 1.00000 0.149071
\(46\) −5.62565 −0.829456
\(47\) −3.61596 −0.527442 −0.263721 0.964599i \(-0.584950\pi\)
−0.263721 + 0.964599i \(0.584950\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.29590 0.327985
\(50\) 1.00000 0.141421
\(51\) −2.69202 −0.376958
\(52\) 0 0
\(53\) 0.664874 0.0913275 0.0456638 0.998957i \(-0.485460\pi\)
0.0456638 + 0.998957i \(0.485460\pi\)
\(54\) −1.00000 −0.136083
\(55\) −6.24698 −0.842343
\(56\) 3.04892 0.407429
\(57\) 5.82908 0.772081
\(58\) −5.14675 −0.675802
\(59\) −12.9487 −1.68578 −0.842888 0.538089i \(-0.819146\pi\)
−0.842888 + 0.538089i \(0.819146\pi\)
\(60\) −1.00000 −0.129099
\(61\) −7.91723 −1.01370 −0.506849 0.862035i \(-0.669190\pi\)
−0.506849 + 0.862035i \(0.669190\pi\)
\(62\) −3.53319 −0.448715
\(63\) 3.04892 0.384127
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.24698 0.768950
\(67\) −0.198062 −0.0241972 −0.0120986 0.999927i \(-0.503851\pi\)
−0.0120986 + 0.999927i \(0.503851\pi\)
\(68\) 2.69202 0.326456
\(69\) 5.62565 0.677248
\(70\) 3.04892 0.364415
\(71\) −0.374354 −0.0444277 −0.0222138 0.999753i \(-0.507071\pi\)
−0.0222138 + 0.999753i \(0.507071\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.6136 1.71039 0.855194 0.518308i \(-0.173438\pi\)
0.855194 + 0.518308i \(0.173438\pi\)
\(74\) −4.65279 −0.540876
\(75\) −1.00000 −0.115470
\(76\) −5.82908 −0.668642
\(77\) −19.0465 −2.17055
\(78\) 0 0
\(79\) −11.2567 −1.26647 −0.633237 0.773958i \(-0.718274\pi\)
−0.633237 + 0.773958i \(0.718274\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −3.77479 −0.416856
\(83\) 6.83877 0.750653 0.375326 0.926893i \(-0.377531\pi\)
0.375326 + 0.926893i \(0.377531\pi\)
\(84\) −3.04892 −0.332664
\(85\) 2.69202 0.291991
\(86\) 2.85086 0.307416
\(87\) 5.14675 0.551790
\(88\) −6.24698 −0.665930
\(89\) 9.50365 1.00738 0.503692 0.863883i \(-0.331975\pi\)
0.503692 + 0.863883i \(0.331975\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −5.62565 −0.586514
\(93\) 3.53319 0.366375
\(94\) −3.61596 −0.372957
\(95\) −5.82908 −0.598051
\(96\) −1.00000 −0.102062
\(97\) −0.335126 −0.0340268 −0.0170134 0.999855i \(-0.505416\pi\)
−0.0170134 + 0.999855i \(0.505416\pi\)
\(98\) 2.29590 0.231921
\(99\) −6.24698 −0.627845
\(100\) 1.00000 0.100000
\(101\) 0.274127 0.0272766 0.0136383 0.999907i \(-0.495659\pi\)
0.0136383 + 0.999907i \(0.495659\pi\)
\(102\) −2.69202 −0.266550
\(103\) −9.00000 −0.886796 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 0 0
\(105\) −3.04892 −0.297544
\(106\) 0.664874 0.0645783
\(107\) −10.8509 −1.04899 −0.524496 0.851413i \(-0.675746\pi\)
−0.524496 + 0.851413i \(0.675746\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −1.02177 −0.0978678 −0.0489339 0.998802i \(-0.515582\pi\)
−0.0489339 + 0.998802i \(0.515582\pi\)
\(110\) −6.24698 −0.595626
\(111\) 4.65279 0.441624
\(112\) 3.04892 0.288096
\(113\) 19.6383 1.84742 0.923709 0.383095i \(-0.125142\pi\)
0.923709 + 0.383095i \(0.125142\pi\)
\(114\) 5.82908 0.545944
\(115\) −5.62565 −0.524594
\(116\) −5.14675 −0.477864
\(117\) 0 0
\(118\) −12.9487 −1.19202
\(119\) 8.20775 0.752403
\(120\) −1.00000 −0.0912871
\(121\) 28.0248 2.54770
\(122\) −7.91723 −0.716792
\(123\) 3.77479 0.340361
\(124\) −3.53319 −0.317290
\(125\) 1.00000 0.0894427
\(126\) 3.04892 0.271619
\(127\) 18.5187 1.64327 0.821635 0.570014i \(-0.193062\pi\)
0.821635 + 0.570014i \(0.193062\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.85086 −0.251004
\(130\) 0 0
\(131\) −12.2349 −1.06897 −0.534484 0.845179i \(-0.679494\pi\)
−0.534484 + 0.845179i \(0.679494\pi\)
\(132\) 6.24698 0.543730
\(133\) −17.7724 −1.54106
\(134\) −0.198062 −0.0171100
\(135\) −1.00000 −0.0860663
\(136\) 2.69202 0.230839
\(137\) −19.3002 −1.64893 −0.824464 0.565914i \(-0.808523\pi\)
−0.824464 + 0.565914i \(0.808523\pi\)
\(138\) 5.62565 0.478887
\(139\) 19.5133 1.65510 0.827550 0.561392i \(-0.189734\pi\)
0.827550 + 0.561392i \(0.189734\pi\)
\(140\) 3.04892 0.257681
\(141\) 3.61596 0.304519
\(142\) −0.374354 −0.0314151
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −5.14675 −0.427414
\(146\) 14.6136 1.20943
\(147\) −2.29590 −0.189362
\(148\) −4.65279 −0.382457
\(149\) −18.0248 −1.47665 −0.738323 0.674448i \(-0.764382\pi\)
−0.738323 + 0.674448i \(0.764382\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 5.30127 0.431412 0.215706 0.976458i \(-0.430795\pi\)
0.215706 + 0.976458i \(0.430795\pi\)
\(152\) −5.82908 −0.472801
\(153\) 2.69202 0.217637
\(154\) −19.0465 −1.53481
\(155\) −3.53319 −0.283792
\(156\) 0 0
\(157\) 7.41789 0.592012 0.296006 0.955186i \(-0.404345\pi\)
0.296006 + 0.955186i \(0.404345\pi\)
\(158\) −11.2567 −0.895532
\(159\) −0.664874 −0.0527280
\(160\) 1.00000 0.0790569
\(161\) −17.1521 −1.35178
\(162\) 1.00000 0.0785674
\(163\) 24.0151 1.88101 0.940503 0.339787i \(-0.110355\pi\)
0.940503 + 0.339787i \(0.110355\pi\)
\(164\) −3.77479 −0.294762
\(165\) 6.24698 0.486327
\(166\) 6.83877 0.530792
\(167\) 11.9922 0.927987 0.463993 0.885839i \(-0.346416\pi\)
0.463993 + 0.885839i \(0.346416\pi\)
\(168\) −3.04892 −0.235229
\(169\) 0 0
\(170\) 2.69202 0.206469
\(171\) −5.82908 −0.445761
\(172\) 2.85086 0.217376
\(173\) 19.8823 1.51162 0.755812 0.654788i \(-0.227242\pi\)
0.755812 + 0.654788i \(0.227242\pi\)
\(174\) 5.14675 0.390174
\(175\) 3.04892 0.230476
\(176\) −6.24698 −0.470884
\(177\) 12.9487 0.973283
\(178\) 9.50365 0.712329
\(179\) 16.7453 1.25160 0.625799 0.779984i \(-0.284773\pi\)
0.625799 + 0.779984i \(0.284773\pi\)
\(180\) 1.00000 0.0745356
\(181\) −3.93900 −0.292784 −0.146392 0.989227i \(-0.546766\pi\)
−0.146392 + 0.989227i \(0.546766\pi\)
\(182\) 0 0
\(183\) 7.91723 0.585259
\(184\) −5.62565 −0.414728
\(185\) −4.65279 −0.342080
\(186\) 3.53319 0.259066
\(187\) −16.8170 −1.22978
\(188\) −3.61596 −0.263721
\(189\) −3.04892 −0.221776
\(190\) −5.82908 −0.422886
\(191\) −27.1540 −1.96480 −0.982399 0.186795i \(-0.940190\pi\)
−0.982399 + 0.186795i \(0.940190\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 15.3937 1.10807 0.554033 0.832495i \(-0.313088\pi\)
0.554033 + 0.832495i \(0.313088\pi\)
\(194\) −0.335126 −0.0240606
\(195\) 0 0
\(196\) 2.29590 0.163993
\(197\) −23.5036 −1.67457 −0.837283 0.546770i \(-0.815857\pi\)
−0.837283 + 0.546770i \(0.815857\pi\)
\(198\) −6.24698 −0.443954
\(199\) −20.8442 −1.47760 −0.738801 0.673923i \(-0.764608\pi\)
−0.738801 + 0.673923i \(0.764608\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0.198062 0.0139702
\(202\) 0.274127 0.0192875
\(203\) −15.6920 −1.10136
\(204\) −2.69202 −0.188479
\(205\) −3.77479 −0.263643
\(206\) −9.00000 −0.627060
\(207\) −5.62565 −0.391009
\(208\) 0 0
\(209\) 36.4142 2.51882
\(210\) −3.04892 −0.210395
\(211\) −19.0073 −1.30852 −0.654258 0.756271i \(-0.727019\pi\)
−0.654258 + 0.756271i \(0.727019\pi\)
\(212\) 0.664874 0.0456638
\(213\) 0.374354 0.0256503
\(214\) −10.8509 −0.741749
\(215\) 2.85086 0.194427
\(216\) −1.00000 −0.0680414
\(217\) −10.7724 −0.731278
\(218\) −1.02177 −0.0692030
\(219\) −14.6136 −0.987493
\(220\) −6.24698 −0.421171
\(221\) 0 0
\(222\) 4.65279 0.312275
\(223\) −20.9433 −1.40247 −0.701234 0.712931i \(-0.747367\pi\)
−0.701234 + 0.712931i \(0.747367\pi\)
\(224\) 3.04892 0.203714
\(225\) 1.00000 0.0666667
\(226\) 19.6383 1.30632
\(227\) −19.0978 −1.26757 −0.633784 0.773510i \(-0.718499\pi\)
−0.633784 + 0.773510i \(0.718499\pi\)
\(228\) 5.82908 0.386041
\(229\) −9.33513 −0.616882 −0.308441 0.951243i \(-0.599807\pi\)
−0.308441 + 0.951243i \(0.599807\pi\)
\(230\) −5.62565 −0.370944
\(231\) 19.0465 1.25317
\(232\) −5.14675 −0.337901
\(233\) −14.3599 −0.940747 −0.470374 0.882467i \(-0.655881\pi\)
−0.470374 + 0.882467i \(0.655881\pi\)
\(234\) 0 0
\(235\) −3.61596 −0.235879
\(236\) −12.9487 −0.842888
\(237\) 11.2567 0.731199
\(238\) 8.20775 0.532029
\(239\) −27.7482 −1.79488 −0.897442 0.441132i \(-0.854577\pi\)
−0.897442 + 0.441132i \(0.854577\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 14.5579 0.937759 0.468880 0.883262i \(-0.344658\pi\)
0.468880 + 0.883262i \(0.344658\pi\)
\(242\) 28.0248 1.80150
\(243\) −1.00000 −0.0641500
\(244\) −7.91723 −0.506849
\(245\) 2.29590 0.146679
\(246\) 3.77479 0.240672
\(247\) 0 0
\(248\) −3.53319 −0.224358
\(249\) −6.83877 −0.433390
\(250\) 1.00000 0.0632456
\(251\) 27.8756 1.75949 0.879746 0.475443i \(-0.157712\pi\)
0.879746 + 0.475443i \(0.157712\pi\)
\(252\) 3.04892 0.192064
\(253\) 35.1433 2.20944
\(254\) 18.5187 1.16197
\(255\) −2.69202 −0.168581
\(256\) 1.00000 0.0625000
\(257\) −5.64848 −0.352343 −0.176171 0.984360i \(-0.556371\pi\)
−0.176171 + 0.984360i \(0.556371\pi\)
\(258\) −2.85086 −0.177486
\(259\) −14.1860 −0.881474
\(260\) 0 0
\(261\) −5.14675 −0.318576
\(262\) −12.2349 −0.755875
\(263\) 6.06100 0.373737 0.186869 0.982385i \(-0.440166\pi\)
0.186869 + 0.982385i \(0.440166\pi\)
\(264\) 6.24698 0.384475
\(265\) 0.664874 0.0408429
\(266\) −17.7724 −1.08970
\(267\) −9.50365 −0.581614
\(268\) −0.198062 −0.0120986
\(269\) 8.19136 0.499436 0.249718 0.968319i \(-0.419662\pi\)
0.249718 + 0.968319i \(0.419662\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 12.8006 0.777582 0.388791 0.921326i \(-0.372893\pi\)
0.388791 + 0.921326i \(0.372893\pi\)
\(272\) 2.69202 0.163228
\(273\) 0 0
\(274\) −19.3002 −1.16597
\(275\) −6.24698 −0.376707
\(276\) 5.62565 0.338624
\(277\) −7.88338 −0.473666 −0.236833 0.971550i \(-0.576109\pi\)
−0.236833 + 0.971550i \(0.576109\pi\)
\(278\) 19.5133 1.17033
\(279\) −3.53319 −0.211526
\(280\) 3.04892 0.182208
\(281\) 7.16182 0.427238 0.213619 0.976917i \(-0.431475\pi\)
0.213619 + 0.976917i \(0.431475\pi\)
\(282\) 3.61596 0.215327
\(283\) 11.7342 0.697528 0.348764 0.937211i \(-0.386602\pi\)
0.348764 + 0.937211i \(0.386602\pi\)
\(284\) −0.374354 −0.0222138
\(285\) 5.82908 0.345285
\(286\) 0 0
\(287\) −11.5090 −0.679356
\(288\) 1.00000 0.0589256
\(289\) −9.75302 −0.573707
\(290\) −5.14675 −0.302228
\(291\) 0.335126 0.0196454
\(292\) 14.6136 0.855194
\(293\) −26.4456 −1.54497 −0.772485 0.635033i \(-0.780987\pi\)
−0.772485 + 0.635033i \(0.780987\pi\)
\(294\) −2.29590 −0.133899
\(295\) −12.9487 −0.753902
\(296\) −4.65279 −0.270438
\(297\) 6.24698 0.362487
\(298\) −18.0248 −1.04415
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 8.69202 0.501000
\(302\) 5.30127 0.305054
\(303\) −0.274127 −0.0157482
\(304\) −5.82908 −0.334321
\(305\) −7.91723 −0.453339
\(306\) 2.69202 0.153893
\(307\) −10.8562 −0.619598 −0.309799 0.950802i \(-0.600262\pi\)
−0.309799 + 0.950802i \(0.600262\pi\)
\(308\) −19.0465 −1.08528
\(309\) 9.00000 0.511992
\(310\) −3.53319 −0.200672
\(311\) 11.5453 0.654672 0.327336 0.944908i \(-0.393849\pi\)
0.327336 + 0.944908i \(0.393849\pi\)
\(312\) 0 0
\(313\) 18.4862 1.04490 0.522451 0.852670i \(-0.325018\pi\)
0.522451 + 0.852670i \(0.325018\pi\)
\(314\) 7.41789 0.418616
\(315\) 3.04892 0.171787
\(316\) −11.2567 −0.633237
\(317\) −3.05131 −0.171379 −0.0856893 0.996322i \(-0.527309\pi\)
−0.0856893 + 0.996322i \(0.527309\pi\)
\(318\) −0.664874 −0.0372843
\(319\) 32.1517 1.80015
\(320\) 1.00000 0.0559017
\(321\) 10.8509 0.605636
\(322\) −17.1521 −0.955851
\(323\) −15.6920 −0.873127
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 24.0151 1.33007
\(327\) 1.02177 0.0565040
\(328\) −3.77479 −0.208428
\(329\) −11.0248 −0.607814
\(330\) 6.24698 0.343885
\(331\) −12.1575 −0.668237 −0.334118 0.942531i \(-0.608439\pi\)
−0.334118 + 0.942531i \(0.608439\pi\)
\(332\) 6.83877 0.375326
\(333\) −4.65279 −0.254971
\(334\) 11.9922 0.656186
\(335\) −0.198062 −0.0108213
\(336\) −3.04892 −0.166332
\(337\) 6.48188 0.353090 0.176545 0.984293i \(-0.443508\pi\)
0.176545 + 0.984293i \(0.443508\pi\)
\(338\) 0 0
\(339\) −19.6383 −1.06661
\(340\) 2.69202 0.145995
\(341\) 22.0718 1.19525
\(342\) −5.82908 −0.315201
\(343\) −14.3424 −0.774418
\(344\) 2.85086 0.153708
\(345\) 5.62565 0.302875
\(346\) 19.8823 1.06888
\(347\) 19.1933 1.03035 0.515175 0.857085i \(-0.327727\pi\)
0.515175 + 0.857085i \(0.327727\pi\)
\(348\) 5.14675 0.275895
\(349\) −11.5767 −0.619688 −0.309844 0.950787i \(-0.600277\pi\)
−0.309844 + 0.950787i \(0.600277\pi\)
\(350\) 3.04892 0.162971
\(351\) 0 0
\(352\) −6.24698 −0.332965
\(353\) −21.6963 −1.15478 −0.577390 0.816469i \(-0.695929\pi\)
−0.577390 + 0.816469i \(0.695929\pi\)
\(354\) 12.9487 0.688215
\(355\) −0.374354 −0.0198687
\(356\) 9.50365 0.503692
\(357\) −8.20775 −0.434400
\(358\) 16.7453 0.885014
\(359\) 13.0476 0.688625 0.344313 0.938855i \(-0.388112\pi\)
0.344313 + 0.938855i \(0.388112\pi\)
\(360\) 1.00000 0.0527046
\(361\) 14.9782 0.788328
\(362\) −3.93900 −0.207029
\(363\) −28.0248 −1.47092
\(364\) 0 0
\(365\) 14.6136 0.764909
\(366\) 7.91723 0.413840
\(367\) 1.76377 0.0920683 0.0460341 0.998940i \(-0.485342\pi\)
0.0460341 + 0.998940i \(0.485342\pi\)
\(368\) −5.62565 −0.293257
\(369\) −3.77479 −0.196508
\(370\) −4.65279 −0.241887
\(371\) 2.02715 0.105244
\(372\) 3.53319 0.183187
\(373\) 15.5013 0.802625 0.401312 0.915941i \(-0.368554\pi\)
0.401312 + 0.915941i \(0.368554\pi\)
\(374\) −16.8170 −0.869587
\(375\) −1.00000 −0.0516398
\(376\) −3.61596 −0.186479
\(377\) 0 0
\(378\) −3.04892 −0.156819
\(379\) 8.62863 0.443223 0.221611 0.975135i \(-0.428868\pi\)
0.221611 + 0.975135i \(0.428868\pi\)
\(380\) −5.82908 −0.299026
\(381\) −18.5187 −0.948742
\(382\) −27.1540 −1.38932
\(383\) 31.1008 1.58918 0.794589 0.607148i \(-0.207686\pi\)
0.794589 + 0.607148i \(0.207686\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −19.0465 −0.970701
\(386\) 15.3937 0.783520
\(387\) 2.85086 0.144917
\(388\) −0.335126 −0.0170134
\(389\) 7.07069 0.358498 0.179249 0.983804i \(-0.442633\pi\)
0.179249 + 0.983804i \(0.442633\pi\)
\(390\) 0 0
\(391\) −15.1444 −0.765883
\(392\) 2.29590 0.115960
\(393\) 12.2349 0.617169
\(394\) −23.5036 −1.18410
\(395\) −11.2567 −0.566384
\(396\) −6.24698 −0.313923
\(397\) −28.5948 −1.43513 −0.717565 0.696491i \(-0.754744\pi\)
−0.717565 + 0.696491i \(0.754744\pi\)
\(398\) −20.8442 −1.04482
\(399\) 17.7724 0.889733
\(400\) 1.00000 0.0500000
\(401\) −11.8267 −0.590597 −0.295298 0.955405i \(-0.595419\pi\)
−0.295298 + 0.955405i \(0.595419\pi\)
\(402\) 0.198062 0.00987845
\(403\) 0 0
\(404\) 0.274127 0.0136383
\(405\) 1.00000 0.0496904
\(406\) −15.6920 −0.778782
\(407\) 29.0659 1.44074
\(408\) −2.69202 −0.133275
\(409\) −9.00000 −0.445021 −0.222511 0.974930i \(-0.571425\pi\)
−0.222511 + 0.974930i \(0.571425\pi\)
\(410\) −3.77479 −0.186424
\(411\) 19.3002 0.952009
\(412\) −9.00000 −0.443398
\(413\) −39.4795 −1.94266
\(414\) −5.62565 −0.276485
\(415\) 6.83877 0.335702
\(416\) 0 0
\(417\) −19.5133 −0.955572
\(418\) 36.4142 1.78108
\(419\) 19.7735 0.965997 0.482998 0.875621i \(-0.339548\pi\)
0.482998 + 0.875621i \(0.339548\pi\)
\(420\) −3.04892 −0.148772
\(421\) −21.0127 −1.02409 −0.512047 0.858957i \(-0.671113\pi\)
−0.512047 + 0.858957i \(0.671113\pi\)
\(422\) −19.0073 −0.925261
\(423\) −3.61596 −0.175814
\(424\) 0.664874 0.0322892
\(425\) 2.69202 0.130582
\(426\) 0.374354 0.0181375
\(427\) −24.1390 −1.16817
\(428\) −10.8509 −0.524496
\(429\) 0 0
\(430\) 2.85086 0.137480
\(431\) −26.5633 −1.27951 −0.639755 0.768579i \(-0.720964\pi\)
−0.639755 + 0.768579i \(0.720964\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.7047 −0.706662 −0.353331 0.935498i \(-0.614951\pi\)
−0.353331 + 0.935498i \(0.614951\pi\)
\(434\) −10.7724 −0.517092
\(435\) 5.14675 0.246768
\(436\) −1.02177 −0.0489339
\(437\) 32.7924 1.56867
\(438\) −14.6136 −0.698263
\(439\) −28.2640 −1.34897 −0.674483 0.738291i \(-0.735633\pi\)
−0.674483 + 0.738291i \(0.735633\pi\)
\(440\) −6.24698 −0.297813
\(441\) 2.29590 0.109328
\(442\) 0 0
\(443\) 20.5109 0.974504 0.487252 0.873261i \(-0.337999\pi\)
0.487252 + 0.873261i \(0.337999\pi\)
\(444\) 4.65279 0.220812
\(445\) 9.50365 0.450516
\(446\) −20.9433 −0.991695
\(447\) 18.0248 0.852542
\(448\) 3.04892 0.144048
\(449\) −3.19806 −0.150926 −0.0754629 0.997149i \(-0.524043\pi\)
−0.0754629 + 0.997149i \(0.524043\pi\)
\(450\) 1.00000 0.0471405
\(451\) 23.5810 1.11039
\(452\) 19.6383 0.923709
\(453\) −5.30127 −0.249076
\(454\) −19.0978 −0.896306
\(455\) 0 0
\(456\) 5.82908 0.272972
\(457\) 8.65710 0.404962 0.202481 0.979286i \(-0.435100\pi\)
0.202481 + 0.979286i \(0.435100\pi\)
\(458\) −9.33513 −0.436202
\(459\) −2.69202 −0.125653
\(460\) −5.62565 −0.262297
\(461\) −11.1239 −0.518092 −0.259046 0.965865i \(-0.583408\pi\)
−0.259046 + 0.965865i \(0.583408\pi\)
\(462\) 19.0465 0.886125
\(463\) −33.0659 −1.53670 −0.768351 0.640028i \(-0.778923\pi\)
−0.768351 + 0.640028i \(0.778923\pi\)
\(464\) −5.14675 −0.238932
\(465\) 3.53319 0.163848
\(466\) −14.3599 −0.665209
\(467\) 15.3937 0.712337 0.356168 0.934422i \(-0.384083\pi\)
0.356168 + 0.934422i \(0.384083\pi\)
\(468\) 0 0
\(469\) −0.603875 −0.0278844
\(470\) −3.61596 −0.166792
\(471\) −7.41789 −0.341799
\(472\) −12.9487 −0.596012
\(473\) −17.8092 −0.818869
\(474\) 11.2567 0.517036
\(475\) −5.82908 −0.267457
\(476\) 8.20775 0.376202
\(477\) 0.664874 0.0304425
\(478\) −27.7482 −1.26917
\(479\) 42.2137 1.92879 0.964397 0.264459i \(-0.0851933\pi\)
0.964397 + 0.264459i \(0.0851933\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 14.5579 0.663096
\(483\) 17.1521 0.780449
\(484\) 28.0248 1.27385
\(485\) −0.335126 −0.0152173
\(486\) −1.00000 −0.0453609
\(487\) 14.9903 0.679276 0.339638 0.940556i \(-0.389695\pi\)
0.339638 + 0.940556i \(0.389695\pi\)
\(488\) −7.91723 −0.358396
\(489\) −24.0151 −1.08600
\(490\) 2.29590 0.103718
\(491\) −21.3706 −0.964443 −0.482222 0.876049i \(-0.660170\pi\)
−0.482222 + 0.876049i \(0.660170\pi\)
\(492\) 3.77479 0.170181
\(493\) −13.8552 −0.624005
\(494\) 0 0
\(495\) −6.24698 −0.280781
\(496\) −3.53319 −0.158645
\(497\) −1.14138 −0.0511977
\(498\) −6.83877 −0.306453
\(499\) −26.6160 −1.19149 −0.595747 0.803172i \(-0.703144\pi\)
−0.595747 + 0.803172i \(0.703144\pi\)
\(500\) 1.00000 0.0447214
\(501\) −11.9922 −0.535773
\(502\) 27.8756 1.24415
\(503\) −25.2218 −1.12458 −0.562291 0.826939i \(-0.690080\pi\)
−0.562291 + 0.826939i \(0.690080\pi\)
\(504\) 3.04892 0.135810
\(505\) 0.274127 0.0121985
\(506\) 35.1433 1.56231
\(507\) 0 0
\(508\) 18.5187 0.821635
\(509\) −17.3797 −0.770343 −0.385172 0.922845i \(-0.625858\pi\)
−0.385172 + 0.922845i \(0.625858\pi\)
\(510\) −2.69202 −0.119205
\(511\) 44.5555 1.97102
\(512\) 1.00000 0.0441942
\(513\) 5.82908 0.257360
\(514\) −5.64848 −0.249144
\(515\) −9.00000 −0.396587
\(516\) −2.85086 −0.125502
\(517\) 22.5888 0.993455
\(518\) −14.1860 −0.623296
\(519\) −19.8823 −0.872737
\(520\) 0 0
\(521\) 8.27114 0.362365 0.181183 0.983449i \(-0.442007\pi\)
0.181183 + 0.983449i \(0.442007\pi\)
\(522\) −5.14675 −0.225267
\(523\) −40.9154 −1.78911 −0.894553 0.446961i \(-0.852506\pi\)
−0.894553 + 0.446961i \(0.852506\pi\)
\(524\) −12.2349 −0.534484
\(525\) −3.04892 −0.133066
\(526\) 6.06100 0.264272
\(527\) −9.51142 −0.414324
\(528\) 6.24698 0.271865
\(529\) 8.64789 0.375995
\(530\) 0.664874 0.0288803
\(531\) −12.9487 −0.561925
\(532\) −17.7724 −0.770531
\(533\) 0 0
\(534\) −9.50365 −0.411263
\(535\) −10.8509 −0.469123
\(536\) −0.198062 −0.00855499
\(537\) −16.7453 −0.722611
\(538\) 8.19136 0.353154
\(539\) −14.3424 −0.617772
\(540\) −1.00000 −0.0430331
\(541\) 27.2252 1.17050 0.585252 0.810852i \(-0.300996\pi\)
0.585252 + 0.810852i \(0.300996\pi\)
\(542\) 12.8006 0.549833
\(543\) 3.93900 0.169039
\(544\) 2.69202 0.115419
\(545\) −1.02177 −0.0437678
\(546\) 0 0
\(547\) −32.7023 −1.39825 −0.699125 0.715000i \(-0.746427\pi\)
−0.699125 + 0.715000i \(0.746427\pi\)
\(548\) −19.3002 −0.824464
\(549\) −7.91723 −0.337899
\(550\) −6.24698 −0.266372
\(551\) 30.0009 1.27808
\(552\) 5.62565 0.239443
\(553\) −34.3207 −1.45946
\(554\) −7.88338 −0.334933
\(555\) 4.65279 0.197500
\(556\) 19.5133 0.827550
\(557\) 0.225209 0.00954243 0.00477121 0.999989i \(-0.498481\pi\)
0.00477121 + 0.999989i \(0.498481\pi\)
\(558\) −3.53319 −0.149572
\(559\) 0 0
\(560\) 3.04892 0.128840
\(561\) 16.8170 0.710014
\(562\) 7.16182 0.302103
\(563\) −9.55735 −0.402794 −0.201397 0.979510i \(-0.564548\pi\)
−0.201397 + 0.979510i \(0.564548\pi\)
\(564\) 3.61596 0.152259
\(565\) 19.6383 0.826190
\(566\) 11.7342 0.493227
\(567\) 3.04892 0.128042
\(568\) −0.374354 −0.0157076
\(569\) 34.8455 1.46080 0.730399 0.683020i \(-0.239334\pi\)
0.730399 + 0.683020i \(0.239334\pi\)
\(570\) 5.82908 0.244153
\(571\) 36.5472 1.52945 0.764726 0.644355i \(-0.222874\pi\)
0.764726 + 0.644355i \(0.222874\pi\)
\(572\) 0 0
\(573\) 27.1540 1.13438
\(574\) −11.5090 −0.480377
\(575\) −5.62565 −0.234606
\(576\) 1.00000 0.0416667
\(577\) 18.3284 0.763022 0.381511 0.924364i \(-0.375404\pi\)
0.381511 + 0.924364i \(0.375404\pi\)
\(578\) −9.75302 −0.405672
\(579\) −15.3937 −0.639742
\(580\) −5.14675 −0.213707
\(581\) 20.8509 0.865039
\(582\) 0.335126 0.0138914
\(583\) −4.15346 −0.172019
\(584\) 14.6136 0.604714
\(585\) 0 0
\(586\) −26.4456 −1.09246
\(587\) −17.3937 −0.717916 −0.358958 0.933354i \(-0.616868\pi\)
−0.358958 + 0.933354i \(0.616868\pi\)
\(588\) −2.29590 −0.0946812
\(589\) 20.5953 0.848613
\(590\) −12.9487 −0.533089
\(591\) 23.5036 0.966811
\(592\) −4.65279 −0.191229
\(593\) −19.5362 −0.802254 −0.401127 0.916022i \(-0.631381\pi\)
−0.401127 + 0.916022i \(0.631381\pi\)
\(594\) 6.24698 0.256317
\(595\) 8.20775 0.336485
\(596\) −18.0248 −0.738323
\(597\) 20.8442 0.853094
\(598\) 0 0
\(599\) −14.6364 −0.598027 −0.299014 0.954249i \(-0.596658\pi\)
−0.299014 + 0.954249i \(0.596658\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 33.3980 1.36233 0.681167 0.732128i \(-0.261473\pi\)
0.681167 + 0.732128i \(0.261473\pi\)
\(602\) 8.69202 0.354260
\(603\) −0.198062 −0.00806572
\(604\) 5.30127 0.215706
\(605\) 28.0248 1.13937
\(606\) −0.274127 −0.0111356
\(607\) 29.9694 1.21642 0.608210 0.793776i \(-0.291888\pi\)
0.608210 + 0.793776i \(0.291888\pi\)
\(608\) −5.82908 −0.236401
\(609\) 15.6920 0.635873
\(610\) −7.91723 −0.320559
\(611\) 0 0
\(612\) 2.69202 0.108819
\(613\) 1.58748 0.0641178 0.0320589 0.999486i \(-0.489794\pi\)
0.0320589 + 0.999486i \(0.489794\pi\)
\(614\) −10.8562 −0.438122
\(615\) 3.77479 0.152214
\(616\) −19.0465 −0.767406
\(617\) −12.9511 −0.521391 −0.260695 0.965421i \(-0.583952\pi\)
−0.260695 + 0.965421i \(0.583952\pi\)
\(618\) 9.00000 0.362033
\(619\) 23.9554 0.962849 0.481424 0.876488i \(-0.340120\pi\)
0.481424 + 0.876488i \(0.340120\pi\)
\(620\) −3.53319 −0.141896
\(621\) 5.62565 0.225749
\(622\) 11.5453 0.462923
\(623\) 28.9758 1.16089
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 18.4862 0.738857
\(627\) −36.4142 −1.45424
\(628\) 7.41789 0.296006
\(629\) −12.5254 −0.499421
\(630\) 3.04892 0.121472
\(631\) 35.4601 1.41164 0.705822 0.708389i \(-0.250578\pi\)
0.705822 + 0.708389i \(0.250578\pi\)
\(632\) −11.2567 −0.447766
\(633\) 19.0073 0.755472
\(634\) −3.05131 −0.121183
\(635\) 18.5187 0.734893
\(636\) −0.664874 −0.0263640
\(637\) 0 0
\(638\) 32.1517 1.27290
\(639\) −0.374354 −0.0148092
\(640\) 1.00000 0.0395285
\(641\) 16.8358 0.664974 0.332487 0.943108i \(-0.392112\pi\)
0.332487 + 0.943108i \(0.392112\pi\)
\(642\) 10.8509 0.428249
\(643\) 18.4838 0.728930 0.364465 0.931217i \(-0.381252\pi\)
0.364465 + 0.931217i \(0.381252\pi\)
\(644\) −17.1521 −0.675889
\(645\) −2.85086 −0.112252
\(646\) −15.6920 −0.617394
\(647\) −1.20237 −0.0472702 −0.0236351 0.999721i \(-0.507524\pi\)
−0.0236351 + 0.999721i \(0.507524\pi\)
\(648\) 1.00000 0.0392837
\(649\) 80.8902 3.17522
\(650\) 0 0
\(651\) 10.7724 0.422204
\(652\) 24.0151 0.940503
\(653\) −36.9530 −1.44608 −0.723041 0.690805i \(-0.757256\pi\)
−0.723041 + 0.690805i \(0.757256\pi\)
\(654\) 1.02177 0.0399544
\(655\) −12.2349 −0.478057
\(656\) −3.77479 −0.147381
\(657\) 14.6136 0.570129
\(658\) −11.0248 −0.429790
\(659\) −26.4494 −1.03032 −0.515160 0.857094i \(-0.672268\pi\)
−0.515160 + 0.857094i \(0.672268\pi\)
\(660\) 6.24698 0.243163
\(661\) −2.21073 −0.0859876 −0.0429938 0.999075i \(-0.513690\pi\)
−0.0429938 + 0.999075i \(0.513690\pi\)
\(662\) −12.1575 −0.472515
\(663\) 0 0
\(664\) 6.83877 0.265396
\(665\) −17.7724 −0.689184
\(666\) −4.65279 −0.180292
\(667\) 28.9538 1.12110
\(668\) 11.9922 0.463993
\(669\) 20.9433 0.809715
\(670\) −0.198062 −0.00765181
\(671\) 49.4588 1.90934
\(672\) −3.04892 −0.117615
\(673\) 16.5555 0.638170 0.319085 0.947726i \(-0.396625\pi\)
0.319085 + 0.947726i \(0.396625\pi\)
\(674\) 6.48188 0.249673
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −8.53617 −0.328072 −0.164036 0.986454i \(-0.552451\pi\)
−0.164036 + 0.986454i \(0.552451\pi\)
\(678\) −19.6383 −0.754205
\(679\) −1.02177 −0.0392119
\(680\) 2.69202 0.103234
\(681\) 19.0978 0.731831
\(682\) 22.0718 0.845171
\(683\) 0.204767 0.00783519 0.00391760 0.999992i \(-0.498753\pi\)
0.00391760 + 0.999992i \(0.498753\pi\)
\(684\) −5.82908 −0.222881
\(685\) −19.3002 −0.737423
\(686\) −14.3424 −0.547596
\(687\) 9.33513 0.356157
\(688\) 2.85086 0.108688
\(689\) 0 0
\(690\) 5.62565 0.214165
\(691\) 33.5187 1.27511 0.637556 0.770404i \(-0.279945\pi\)
0.637556 + 0.770404i \(0.279945\pi\)
\(692\) 19.8823 0.755812
\(693\) −19.0465 −0.723518
\(694\) 19.1933 0.728567
\(695\) 19.5133 0.740183
\(696\) 5.14675 0.195087
\(697\) −10.1618 −0.384906
\(698\) −11.5767 −0.438186
\(699\) 14.3599 0.543141
\(700\) 3.04892 0.115238
\(701\) 19.0271 0.718645 0.359323 0.933213i \(-0.383008\pi\)
0.359323 + 0.933213i \(0.383008\pi\)
\(702\) 0 0
\(703\) 27.1215 1.02291
\(704\) −6.24698 −0.235442
\(705\) 3.61596 0.136185
\(706\) −21.6963 −0.816552
\(707\) 0.835790 0.0314331
\(708\) 12.9487 0.486642
\(709\) 42.5760 1.59897 0.799487 0.600683i \(-0.205105\pi\)
0.799487 + 0.600683i \(0.205105\pi\)
\(710\) −0.374354 −0.0140493
\(711\) −11.2567 −0.422158
\(712\) 9.50365 0.356164
\(713\) 19.8765 0.744379
\(714\) −8.20775 −0.307167
\(715\) 0 0
\(716\) 16.7453 0.625799
\(717\) 27.7482 1.03628
\(718\) 13.0476 0.486932
\(719\) 7.51632 0.280311 0.140156 0.990129i \(-0.455240\pi\)
0.140156 + 0.990129i \(0.455240\pi\)
\(720\) 1.00000 0.0372678
\(721\) −27.4403 −1.02193
\(722\) 14.9782 0.557432
\(723\) −14.5579 −0.541416
\(724\) −3.93900 −0.146392
\(725\) −5.14675 −0.191146
\(726\) −28.0248 −1.04010
\(727\) 39.6558 1.47075 0.735376 0.677660i \(-0.237006\pi\)
0.735376 + 0.677660i \(0.237006\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.6136 0.540872
\(731\) 7.67456 0.283854
\(732\) 7.91723 0.292629
\(733\) 24.4886 0.904506 0.452253 0.891890i \(-0.350620\pi\)
0.452253 + 0.891890i \(0.350620\pi\)
\(734\) 1.76377 0.0651021
\(735\) −2.29590 −0.0846854
\(736\) −5.62565 −0.207364
\(737\) 1.23729 0.0455762
\(738\) −3.77479 −0.138952
\(739\) −41.9842 −1.54441 −0.772207 0.635371i \(-0.780847\pi\)
−0.772207 + 0.635371i \(0.780847\pi\)
\(740\) −4.65279 −0.171040
\(741\) 0 0
\(742\) 2.02715 0.0744189
\(743\) 29.7711 1.09219 0.546097 0.837722i \(-0.316113\pi\)
0.546097 + 0.837722i \(0.316113\pi\)
\(744\) 3.53319 0.129533
\(745\) −18.0248 −0.660376
\(746\) 15.5013 0.567541
\(747\) 6.83877 0.250218
\(748\) −16.8170 −0.614891
\(749\) −33.0834 −1.20884
\(750\) −1.00000 −0.0365148
\(751\) 35.4999 1.29541 0.647705 0.761891i \(-0.275729\pi\)
0.647705 + 0.761891i \(0.275729\pi\)
\(752\) −3.61596 −0.131860
\(753\) −27.8756 −1.01584
\(754\) 0 0
\(755\) 5.30127 0.192933
\(756\) −3.04892 −0.110888
\(757\) 2.73258 0.0993172 0.0496586 0.998766i \(-0.484187\pi\)
0.0496586 + 0.998766i \(0.484187\pi\)
\(758\) 8.62863 0.313406
\(759\) −35.1433 −1.27562
\(760\) −5.82908 −0.211443
\(761\) −30.2349 −1.09601 −0.548007 0.836474i \(-0.684613\pi\)
−0.548007 + 0.836474i \(0.684613\pi\)
\(762\) −18.5187 −0.670862
\(763\) −3.11529 −0.112781
\(764\) −27.1540 −0.982399
\(765\) 2.69202 0.0973302
\(766\) 31.1008 1.12372
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −35.3183 −1.27361 −0.636804 0.771025i \(-0.719744\pi\)
−0.636804 + 0.771025i \(0.719744\pi\)
\(770\) −19.0465 −0.686389
\(771\) 5.64848 0.203425
\(772\) 15.3937 0.554033
\(773\) −32.0551 −1.15294 −0.576472 0.817117i \(-0.695571\pi\)
−0.576472 + 0.817117i \(0.695571\pi\)
\(774\) 2.85086 0.102472
\(775\) −3.53319 −0.126916
\(776\) −0.335126 −0.0120303
\(777\) 14.1860 0.508919
\(778\) 7.07069 0.253496
\(779\) 22.0036 0.788360
\(780\) 0 0
\(781\) 2.33858 0.0836811
\(782\) −15.1444 −0.541561
\(783\) 5.14675 0.183930
\(784\) 2.29590 0.0819963
\(785\) 7.41789 0.264756
\(786\) 12.2349 0.436404
\(787\) −43.9235 −1.56570 −0.782851 0.622209i \(-0.786235\pi\)
−0.782851 + 0.622209i \(0.786235\pi\)
\(788\) −23.5036 −0.837283
\(789\) −6.06100 −0.215777
\(790\) −11.2567 −0.400494
\(791\) 59.8756 2.12893
\(792\) −6.24698 −0.221977
\(793\) 0 0
\(794\) −28.5948 −1.01479
\(795\) −0.664874 −0.0235807
\(796\) −20.8442 −0.738801
\(797\) −26.4523 −0.936990 −0.468495 0.883466i \(-0.655204\pi\)
−0.468495 + 0.883466i \(0.655204\pi\)
\(798\) 17.7724 0.629136
\(799\) −9.73423 −0.344372
\(800\) 1.00000 0.0353553
\(801\) 9.50365 0.335795
\(802\) −11.8267 −0.417615
\(803\) −91.2906 −3.22158
\(804\) 0.198062 0.00698512
\(805\) −17.1521 −0.604533
\(806\) 0 0
\(807\) −8.19136 −0.288349
\(808\) 0.274127 0.00964374
\(809\) 41.4634 1.45777 0.728887 0.684634i \(-0.240038\pi\)
0.728887 + 0.684634i \(0.240038\pi\)
\(810\) 1.00000 0.0351364
\(811\) 24.9095 0.874689 0.437345 0.899294i \(-0.355919\pi\)
0.437345 + 0.899294i \(0.355919\pi\)
\(812\) −15.6920 −0.550682
\(813\) −12.8006 −0.448937
\(814\) 29.0659 1.01876
\(815\) 24.0151 0.841211
\(816\) −2.69202 −0.0942396
\(817\) −16.6179 −0.581386
\(818\) −9.00000 −0.314678
\(819\) 0 0
\(820\) −3.77479 −0.131821
\(821\) 3.11901 0.108854 0.0544272 0.998518i \(-0.482667\pi\)
0.0544272 + 0.998518i \(0.482667\pi\)
\(822\) 19.3002 0.673172
\(823\) 36.4077 1.26909 0.634547 0.772884i \(-0.281187\pi\)
0.634547 + 0.772884i \(0.281187\pi\)
\(824\) −9.00000 −0.313530
\(825\) 6.24698 0.217492
\(826\) −39.4795 −1.37367
\(827\) −8.04785 −0.279851 −0.139926 0.990162i \(-0.544686\pi\)
−0.139926 + 0.990162i \(0.544686\pi\)
\(828\) −5.62565 −0.195505
\(829\) 28.7006 0.996815 0.498407 0.866943i \(-0.333918\pi\)
0.498407 + 0.866943i \(0.333918\pi\)
\(830\) 6.83877 0.237377
\(831\) 7.88338 0.273471
\(832\) 0 0
\(833\) 6.18060 0.214145
\(834\) −19.5133 −0.675692
\(835\) 11.9922 0.415008
\(836\) 36.4142 1.25941
\(837\) 3.53319 0.122125
\(838\) 19.7735 0.683063
\(839\) −34.8491 −1.20312 −0.601561 0.798827i \(-0.705455\pi\)
−0.601561 + 0.798827i \(0.705455\pi\)
\(840\) −3.04892 −0.105198
\(841\) −2.51094 −0.0865843
\(842\) −21.0127 −0.724145
\(843\) −7.16182 −0.246666
\(844\) −19.0073 −0.654258
\(845\) 0 0
\(846\) −3.61596 −0.124319
\(847\) 85.4452 2.93593
\(848\) 0.664874 0.0228319
\(849\) −11.7342 −0.402718
\(850\) 2.69202 0.0923356
\(851\) 26.1750 0.897266
\(852\) 0.374354 0.0128252
\(853\) 0.0779834 0.00267010 0.00133505 0.999999i \(-0.499575\pi\)
0.00133505 + 0.999999i \(0.499575\pi\)
\(854\) −24.1390 −0.826019
\(855\) −5.82908 −0.199350
\(856\) −10.8509 −0.370875
\(857\) 18.3811 0.627885 0.313943 0.949442i \(-0.398350\pi\)
0.313943 + 0.949442i \(0.398350\pi\)
\(858\) 0 0
\(859\) 0.871297 0.0297283 0.0148641 0.999890i \(-0.495268\pi\)
0.0148641 + 0.999890i \(0.495268\pi\)
\(860\) 2.85086 0.0972134
\(861\) 11.5090 0.392227
\(862\) −26.5633 −0.904750
\(863\) 1.88172 0.0640546 0.0320273 0.999487i \(-0.489804\pi\)
0.0320273 + 0.999487i \(0.489804\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 19.8823 0.676019
\(866\) −14.7047 −0.499686
\(867\) 9.75302 0.331230
\(868\) −10.7724 −0.365639
\(869\) 70.3202 2.38545
\(870\) 5.14675 0.174491
\(871\) 0 0
\(872\) −1.02177 −0.0346015
\(873\) −0.335126 −0.0113423
\(874\) 32.7924 1.10922
\(875\) 3.04892 0.103072
\(876\) −14.6136 −0.493747
\(877\) 46.6859 1.57647 0.788236 0.615374i \(-0.210995\pi\)
0.788236 + 0.615374i \(0.210995\pi\)
\(878\) −28.2640 −0.953863
\(879\) 26.4456 0.891989
\(880\) −6.24698 −0.210586
\(881\) 51.9821 1.75132 0.875660 0.482928i \(-0.160427\pi\)
0.875660 + 0.482928i \(0.160427\pi\)
\(882\) 2.29590 0.0773069
\(883\) 22.3889 0.753448 0.376724 0.926326i \(-0.377051\pi\)
0.376724 + 0.926326i \(0.377051\pi\)
\(884\) 0 0
\(885\) 12.9487 0.435265
\(886\) 20.5109 0.689079
\(887\) −20.7332 −0.696152 −0.348076 0.937466i \(-0.613165\pi\)
−0.348076 + 0.937466i \(0.613165\pi\)
\(888\) 4.65279 0.156138
\(889\) 56.4620 1.89368
\(890\) 9.50365 0.318563
\(891\) −6.24698 −0.209282
\(892\) −20.9433 −0.701234
\(893\) 21.0777 0.705339
\(894\) 18.0248 0.602838
\(895\) 16.7453 0.559732
\(896\) 3.04892 0.101857
\(897\) 0 0
\(898\) −3.19806 −0.106721
\(899\) 18.1844 0.606485
\(900\) 1.00000 0.0333333
\(901\) 1.78986 0.0596288
\(902\) 23.5810 0.785163
\(903\) −8.69202 −0.289252
\(904\) 19.6383 0.653161
\(905\) −3.93900 −0.130937
\(906\) −5.30127 −0.176123
\(907\) −9.27365 −0.307927 −0.153963 0.988077i \(-0.549204\pi\)
−0.153963 + 0.988077i \(0.549204\pi\)
\(908\) −19.0978 −0.633784
\(909\) 0.274127 0.00909221
\(910\) 0 0
\(911\) −3.87071 −0.128242 −0.0641211 0.997942i \(-0.520424\pi\)
−0.0641211 + 0.997942i \(0.520424\pi\)
\(912\) 5.82908 0.193020
\(913\) −42.7217 −1.41388
\(914\) 8.65710 0.286352
\(915\) 7.91723 0.261736
\(916\) −9.33513 −0.308441
\(917\) −37.3032 −1.23186
\(918\) −2.69202 −0.0888499
\(919\) −33.2644 −1.09729 −0.548646 0.836055i \(-0.684857\pi\)
−0.548646 + 0.836055i \(0.684857\pi\)
\(920\) −5.62565 −0.185472
\(921\) 10.8562 0.357725
\(922\) −11.1239 −0.366347
\(923\) 0 0
\(924\) 19.0465 0.626585
\(925\) −4.65279 −0.152983
\(926\) −33.0659 −1.08661
\(927\) −9.00000 −0.295599
\(928\) −5.14675 −0.168950
\(929\) −1.92095 −0.0630244 −0.0315122 0.999503i \(-0.510032\pi\)
−0.0315122 + 0.999503i \(0.510032\pi\)
\(930\) 3.53319 0.115858
\(931\) −13.3830 −0.438609
\(932\) −14.3599 −0.470374
\(933\) −11.5453 −0.377975
\(934\) 15.3937 0.503698
\(935\) −16.8170 −0.549975
\(936\) 0 0
\(937\) 26.6886 0.871877 0.435939 0.899976i \(-0.356416\pi\)
0.435939 + 0.899976i \(0.356416\pi\)
\(938\) −0.603875 −0.0197172
\(939\) −18.4862 −0.603274
\(940\) −3.61596 −0.117940
\(941\) 2.69309 0.0877921 0.0438961 0.999036i \(-0.486023\pi\)
0.0438961 + 0.999036i \(0.486023\pi\)
\(942\) −7.41789 −0.241688
\(943\) 21.2356 0.691527
\(944\) −12.9487 −0.421444
\(945\) −3.04892 −0.0991813
\(946\) −17.8092 −0.579028
\(947\) −23.1226 −0.751383 −0.375692 0.926745i \(-0.622595\pi\)
−0.375692 + 0.926745i \(0.622595\pi\)
\(948\) 11.2567 0.365600
\(949\) 0 0
\(950\) −5.82908 −0.189120
\(951\) 3.05131 0.0989455
\(952\) 8.20775 0.266015
\(953\) −31.7313 −1.02788 −0.513938 0.857827i \(-0.671814\pi\)
−0.513938 + 0.857827i \(0.671814\pi\)
\(954\) 0.664874 0.0215261
\(955\) −27.1540 −0.878684
\(956\) −27.7482 −0.897442
\(957\) −32.1517 −1.03932
\(958\) 42.2137 1.36386
\(959\) −58.8447 −1.90020
\(960\) −1.00000 −0.0322749
\(961\) −18.5166 −0.597309
\(962\) 0 0
\(963\) −10.8509 −0.349664
\(964\) 14.5579 0.468880
\(965\) 15.3937 0.495542
\(966\) 17.1521 0.551861
\(967\) −29.7095 −0.955392 −0.477696 0.878525i \(-0.658528\pi\)
−0.477696 + 0.878525i \(0.658528\pi\)
\(968\) 28.0248 0.900750
\(969\) 15.6920 0.504100
\(970\) −0.335126 −0.0107602
\(971\) −50.0428 −1.60595 −0.802975 0.596013i \(-0.796751\pi\)
−0.802975 + 0.596013i \(0.796751\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 59.4946 1.90731
\(974\) 14.9903 0.480321
\(975\) 0 0
\(976\) −7.91723 −0.253424
\(977\) −15.4800 −0.495248 −0.247624 0.968856i \(-0.579650\pi\)
−0.247624 + 0.968856i \(0.579650\pi\)
\(978\) −24.0151 −0.767917
\(979\) −59.3691 −1.89744
\(980\) 2.29590 0.0733397
\(981\) −1.02177 −0.0326226
\(982\) −21.3706 −0.681964
\(983\) 24.3080 0.775304 0.387652 0.921806i \(-0.373286\pi\)
0.387652 + 0.921806i \(0.373286\pi\)
\(984\) 3.77479 0.120336
\(985\) −23.5036 −0.748888
\(986\) −13.8552 −0.441238
\(987\) 11.0248 0.350922
\(988\) 0 0
\(989\) −16.0379 −0.509976
\(990\) −6.24698 −0.198542
\(991\) 41.1196 1.30621 0.653104 0.757269i \(-0.273467\pi\)
0.653104 + 0.757269i \(0.273467\pi\)
\(992\) −3.53319 −0.112179
\(993\) 12.1575 0.385807
\(994\) −1.14138 −0.0362022
\(995\) −20.8442 −0.660804
\(996\) −6.83877 −0.216695
\(997\) 8.75063 0.277135 0.138568 0.990353i \(-0.455750\pi\)
0.138568 + 0.990353i \(0.455750\pi\)
\(998\) −26.6160 −0.842513
\(999\) 4.65279 0.147208
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 5070.2.a.bt.1.3 yes 3
13.5 odd 4 5070.2.b.s.1351.3 6
13.8 odd 4 5070.2.b.s.1351.4 6
13.12 even 2 5070.2.a.bk.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bk.1.1 3 13.12 even 2
5070.2.a.bt.1.3 yes 3 1.1 even 1 trivial
5070.2.b.s.1351.3 6 13.5 odd 4
5070.2.b.s.1351.4 6 13.8 odd 4