Properties

Label 5070.2.a.bk.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
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: \(0\)
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 1.1
Root \(1.80194\) 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 \(-0.695462\pi\)
−0.576191 + 0.817315i \(0.695462\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 6.24698 1.88354 0.941768 0.336264i \(-0.109164\pi\)
0.941768 + 0.336264i \(0.109164\pi\)
\(12\) −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 \(-0.266876\pi\)
0.668642 + 0.743585i \(0.266876\pi\)
\(20\) −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 \(-0.397227\pi\)
0.317290 + 0.948329i \(0.397227\pi\)
\(32\) −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 \(-0.375078\pi\)
0.382457 + 0.923973i \(0.375078\pi\)
\(38\) −5.82908 −0.945602
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.77479 0.589523 0.294762 0.955571i \(-0.404760\pi\)
0.294762 + 0.955571i \(0.404760\pi\)
\(42\) −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 \(-0.415050\pi\)
0.263721 + 0.964599i \(0.415050\pi\)
\(48\) −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 \(-0.180854\pi\)
0.842888 + 0.538089i \(0.180854\pi\)
\(60\) 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 \(-0.496149\pi\)
0.0120986 + 0.999927i \(0.496149\pi\)
\(68\) 2.69202 0.326456
\(69\) 5.62565 0.677248
\(70\) −3.04892 −0.364415
\(71\) 0.374354 0.0444277 0.0222138 0.999753i \(-0.492929\pi\)
0.0222138 + 0.999753i \(0.492929\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.6136 −1.71039 −0.855194 0.518308i \(-0.826562\pi\)
−0.855194 + 0.518308i \(0.826562\pi\)
\(74\) −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 \(-0.622469\pi\)
−0.375326 + 0.926893i \(0.622469\pi\)
\(84\) 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 \(-0.668025\pi\)
−0.503692 + 0.863883i \(0.668025\pi\)
\(90\) 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 \(-0.494584\pi\)
0.0170134 + 0.999855i \(0.494584\pi\)
\(98\) −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 \(-0.484418\pi\)
0.0489339 + 0.998802i \(0.484418\pi\)
\(110\) 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 \(-0.191477\pi\)
0.824464 + 0.565914i \(0.191477\pi\)
\(138\) −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 \(-0.235618\pi\)
0.738323 + 0.674448i \(0.235618\pi\)
\(150\) 1.00000 0.0816497
\(151\) −5.30127 −0.431412 −0.215706 0.976458i \(-0.569205\pi\)
−0.215706 + 0.976458i \(0.569205\pi\)
\(152\) −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 \(-0.889645\pi\)
−0.940503 + 0.339787i \(0.889645\pi\)
\(164\) 3.77479 0.294762
\(165\) 6.24698 0.486327
\(166\) 6.83877 0.530792
\(167\) −11.9922 −0.927987 −0.463993 0.885839i \(-0.653584\pi\)
−0.463993 + 0.885839i \(0.653584\pi\)
\(168\) −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 \(-0.686912\pi\)
−0.554033 + 0.832495i \(0.686912\pi\)
\(194\) −0.335126 −0.0240606
\(195\) 0 0
\(196\) 2.29590 0.163993
\(197\) 23.5036 1.67457 0.837283 0.546770i \(-0.184143\pi\)
0.837283 + 0.546770i \(0.184143\pi\)
\(198\) −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 \(-0.252633\pi\)
0.701234 + 0.712931i \(0.252633\pi\)
\(224\) 3.04892 0.203714
\(225\) 1.00000 0.0666667
\(226\) −19.6383 −1.30632
\(227\) 19.0978 1.26757 0.633784 0.773510i \(-0.281501\pi\)
0.633784 + 0.773510i \(0.281501\pi\)
\(228\) −5.82908 −0.386041
\(229\) 9.33513 0.616882 0.308441 0.951243i \(-0.400193\pi\)
0.308441 + 0.951243i \(0.400193\pi\)
\(230\) −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 \(-0.145423\pi\)
0.897442 + 0.441132i \(0.145423\pi\)
\(240\) 1.00000 0.0645497
\(241\) −14.5579 −0.937759 −0.468880 0.883262i \(-0.655342\pi\)
−0.468880 + 0.883262i \(0.655342\pi\)
\(242\) −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 \(-0.627107\pi\)
−0.388791 + 0.921326i \(0.627107\pi\)
\(272\) 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 \(-0.568525\pi\)
−0.213619 + 0.976917i \(0.568525\pi\)
\(282\) 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 \(-0.219013\pi\)
0.772485 + 0.635033i \(0.219013\pi\)
\(294\) 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 \(-0.399738\pi\)
0.309799 + 0.950802i \(0.399738\pi\)
\(308\) −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 \(-0.472691\pi\)
0.0856893 + 0.996322i \(0.472691\pi\)
\(318\) 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 \(-0.391561\pi\)
0.334118 + 0.942531i \(0.391561\pi\)
\(332\) −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 \(-0.399723\pi\)
0.309844 + 0.950787i \(0.399723\pi\)
\(350\) 3.04892 0.162971
\(351\) 0 0
\(352\) −6.24698 −0.332965
\(353\) 21.6963 1.15478 0.577390 0.816469i \(-0.304071\pi\)
0.577390 + 0.816469i \(0.304071\pi\)
\(354\) 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 \(-0.611888\pi\)
−0.344313 + 0.938855i \(0.611888\pi\)
\(360\) 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 \(-0.571132\pi\)
−0.221611 + 0.975135i \(0.571132\pi\)
\(380\) −5.82908 −0.299026
\(381\) −18.5187 −0.948742
\(382\) 27.1540 1.38932
\(383\) −31.1008 −1.58918 −0.794589 0.607148i \(-0.792314\pi\)
−0.794589 + 0.607148i \(0.792314\pi\)
\(384\) 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 \(-0.245256\pi\)
0.717565 + 0.696491i \(0.245256\pi\)
\(398\) 20.8442 1.04482
\(399\) 17.7724 0.889733
\(400\) 1.00000 0.0500000
\(401\) 11.8267 0.590597 0.295298 0.955405i \(-0.404581\pi\)
0.295298 + 0.955405i \(0.404581\pi\)
\(402\) 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 \(-0.428575\pi\)
0.222511 + 0.974930i \(0.428575\pi\)
\(410\) 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 \(-0.328887\pi\)
0.512047 + 0.858957i \(0.328887\pi\)
\(422\) 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 \(-0.279036\pi\)
0.639755 + 0.768579i \(0.279036\pi\)
\(432\) −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 \(-0.475957\pi\)
0.0754629 + 0.997149i \(0.475957\pi\)
\(450\) −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 \(-0.564900\pi\)
−0.202481 + 0.979286i \(0.564900\pi\)
\(458\) −9.33513 −0.436202
\(459\) −2.69202 −0.125653
\(460\) 5.62565 0.262297
\(461\) 11.1239 0.518092 0.259046 0.965865i \(-0.416592\pi\)
0.259046 + 0.965865i \(0.416592\pi\)
\(462\) −19.0465 −0.886125
\(463\) 33.0659 1.53670 0.768351 0.640028i \(-0.221077\pi\)
0.768351 + 0.640028i \(0.221077\pi\)
\(464\) −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 \(-0.914807\pi\)
−0.964397 + 0.264459i \(0.914807\pi\)
\(480\) −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 \(-0.610305\pi\)
−0.339638 + 0.940556i \(0.610305\pi\)
\(488\) 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 \(-0.296856\pi\)
0.595747 + 0.803172i \(0.296856\pi\)
\(500\) −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 \(-0.374142\pi\)
0.385172 + 0.922845i \(0.374142\pi\)
\(510\) −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 \(-0.699004\pi\)
−0.585252 + 0.810852i \(0.699004\pi\)
\(542\) 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 \(-0.501519\pi\)
−0.00477121 + 0.999989i \(0.501519\pi\)
\(558\) −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 \(-0.624596\pi\)
−0.381511 + 0.924364i \(0.624596\pi\)
\(578\) 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 \(-0.383132\pi\)
0.358958 + 0.933354i \(0.383132\pi\)
\(588\) −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 \(-0.368619\pi\)
0.401127 + 0.916022i \(0.368619\pi\)
\(594\) 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 \(-0.510206\pi\)
−0.0320589 + 0.999486i \(0.510206\pi\)
\(614\) −10.8562 −0.438122
\(615\) 3.77479 0.152214
\(616\) 19.0465 0.767406
\(617\) 12.9511 0.521391 0.260695 0.965421i \(-0.416048\pi\)
0.260695 + 0.965421i \(0.416048\pi\)
\(618\) −9.00000 −0.362033
\(619\) −23.9554 −0.962849 −0.481424 0.876488i \(-0.659880\pi\)
−0.481424 + 0.876488i \(0.659880\pi\)
\(620\) −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 \(-0.749422\pi\)
−0.705822 + 0.708389i \(0.749422\pi\)
\(632\) 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 \(-0.618748\pi\)
−0.364465 + 0.931217i \(0.618748\pi\)
\(644\) 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 \(-0.486310\pi\)
0.0429938 + 0.999075i \(0.486310\pi\)
\(662\) −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 \(-0.501247\pi\)
−0.00391760 + 0.999992i \(0.501247\pi\)
\(684\) 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 \(-0.720055\pi\)
−0.637556 + 0.770404i \(0.720055\pi\)
\(692\) 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 \(-0.794895\pi\)
−0.799487 + 0.600683i \(0.794895\pi\)
\(710\) 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 \(-0.649380\pi\)
−0.452253 + 0.891890i \(0.649380\pi\)
\(734\) −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 \(-0.219153\pi\)
0.772207 + 0.635371i \(0.219153\pi\)
\(740\) −4.65279 −0.171040
\(741\) 0 0
\(742\) 2.02715 0.0744189
\(743\) −29.7711 −1.09219 −0.546097 0.837722i \(-0.683887\pi\)
−0.546097 + 0.837722i \(0.683887\pi\)
\(744\) 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 \(-0.315387\pi\)
0.548007 + 0.836474i \(0.315387\pi\)
\(762\) 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 \(-0.280256\pi\)
0.636804 + 0.771025i \(0.280256\pi\)
\(770\) −19.0465 −0.686389
\(771\) 5.64848 0.203425
\(772\) −15.3937 −0.554033
\(773\) 32.0551 1.15294 0.576472 0.817117i \(-0.304429\pi\)
0.576472 + 0.817117i \(0.304429\pi\)
\(774\) −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 \(-0.213765\pi\)
0.782851 + 0.622209i \(0.213765\pi\)
\(788\) 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 \(-0.644081\pi\)
−0.437345 + 0.899294i \(0.644081\pi\)
\(812\) 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 \(-0.517333\pi\)
−0.0544272 + 0.998518i \(0.517333\pi\)
\(822\) 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 \(-0.455314\pi\)
0.139926 + 0.990162i \(0.455314\pi\)
\(828\) −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 \(-0.294545\pi\)
0.601561 + 0.798827i \(0.294545\pi\)
\(840\) 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 \(-0.500425\pi\)
−0.00133505 + 0.999999i \(0.500425\pi\)
\(854\) −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 \(-0.510196\pi\)
−0.0320273 + 0.999487i \(0.510196\pi\)
\(864\) 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 \(-0.789005\pi\)
−0.788236 + 0.615374i \(0.789005\pi\)
\(878\) 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 \(-0.489968\pi\)
0.0315122 + 0.999503i \(0.489968\pi\)
\(930\) −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 \(-0.513977\pi\)
−0.0438961 + 0.999036i \(0.513977\pi\)
\(942\) 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 \(-0.377405\pi\)
0.375692 + 0.926745i \(0.377405\pi\)
\(948\) 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 \(-0.341472\pi\)
0.477696 + 0.878525i \(0.341472\pi\)
\(968\) −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 \(-0.420350\pi\)
0.247624 + 0.968856i \(0.420350\pi\)
\(978\) −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 \(-0.626714\pi\)
−0.387652 + 0.921806i \(0.626714\pi\)
\(984\) 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.bk.1.1 3
13.5 odd 4 5070.2.b.s.1351.4 6
13.8 odd 4 5070.2.b.s.1351.3 6
13.12 even 2 5070.2.a.bt.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bk.1.1 3 1.1 even 1 trivial
5070.2.a.bt.1.3 yes 3 13.12 even 2
5070.2.b.s.1351.3 6 13.8 odd 4
5070.2.b.s.1351.4 6 13.5 odd 4