Properties

Label 5586.2.a.bw.1.4
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.29268.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 5 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.83719\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.49248 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.49248 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.49248 q^{10} -3.46193 q^{11} -1.00000 q^{12} -4.98496 q^{13} -2.49248 q^{15} +1.00000 q^{16} +2.24299 q^{17} -1.00000 q^{18} +1.00000 q^{19} +2.49248 q^{20} +3.46193 q^{22} +6.00404 q^{23} +1.00000 q^{24} +1.21245 q^{25} +4.98496 q^{26} -1.00000 q^{27} -5.51156 q^{29} +2.49248 q^{30} -7.28407 q^{31} -1.00000 q^{32} +3.46193 q^{33} -2.24299 q^{34} +1.00000 q^{36} -9.16685 q^{37} -1.00000 q^{38} +4.98496 q^{39} -2.49248 q^{40} +4.64383 q^{41} +9.54211 q^{43} -3.46193 q^{44} +2.49248 q^{45} -6.00404 q^{46} +5.36021 q^{47} -1.00000 q^{48} -1.21245 q^{50} -2.24299 q^{51} -4.98496 q^{52} -5.87423 q^{53} +1.00000 q^{54} -8.62878 q^{55} -1.00000 q^{57} +5.51156 q^{58} -8.67842 q^{59} -2.49248 q^{60} +4.37526 q^{61} +7.28407 q^{62} +1.00000 q^{64} -12.4249 q^{65} -3.46193 q^{66} +15.1583 q^{67} +2.24299 q^{68} -6.00404 q^{69} +12.0907 q^{71} -1.00000 q^{72} +14.4204 q^{73} +9.16685 q^{74} -1.21245 q^{75} +1.00000 q^{76} -4.98496 q^{78} -7.08667 q^{79} +2.49248 q^{80} +1.00000 q^{81} -4.64383 q^{82} -5.59823 q^{83} +5.59061 q^{85} -9.54211 q^{86} +5.51156 q^{87} +3.46193 q^{88} +7.52952 q^{89} -2.49248 q^{90} +6.00404 q^{92} +7.28407 q^{93} -5.36021 q^{94} +2.49248 q^{95} +1.00000 q^{96} +7.00404 q^{97} -3.46193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{8} + 4q^{9} - 2q^{11} - 4q^{12} + 4q^{16} + 10q^{17} - 4q^{18} + 4q^{19} + 2q^{22} - 5q^{23} + 4q^{24} + 4q^{25} - 4q^{27} - 3q^{29} + 9q^{31} - 4q^{32} + 2q^{33} - 10q^{34} + 4q^{36} - 14q^{37} - 4q^{38} + 4q^{41} + 21q^{43} - 2q^{44} + 5q^{46} + 7q^{47} - 4q^{48} - 4q^{50} - 10q^{51} - 7q^{53} + 4q^{54} - 4q^{57} + 3q^{58} + 7q^{59} + 23q^{61} - 9q^{62} + 4q^{64} - 48q^{65} - 2q^{66} + 6q^{67} + 10q^{68} + 5q^{69} + 2q^{71} - 4q^{72} - 5q^{73} + 14q^{74} - 4q^{75} + 4q^{76} - 11q^{79} + 4q^{81} - 4q^{82} + 14q^{83} + 6q^{85} - 21q^{86} + 3q^{87} + 2q^{88} + 10q^{89} - 5q^{92} - 9q^{93} - 7q^{94} + 4q^{96} - q^{97} - 2q^{99} + O(q^{100}) \)

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
\(5\) 2.49248 1.11467 0.557335 0.830288i \(-0.311824\pi\)
0.557335 + 0.830288i \(0.311824\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.49248 −0.788191
\(11\) −3.46193 −1.04381 −0.521906 0.853003i \(-0.674779\pi\)
−0.521906 + 0.853003i \(0.674779\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.98496 −1.38258 −0.691289 0.722578i \(-0.742957\pi\)
−0.691289 + 0.722578i \(0.742957\pi\)
\(14\) 0 0
\(15\) −2.49248 −0.643555
\(16\) 1.00000 0.250000
\(17\) 2.24299 0.544006 0.272003 0.962296i \(-0.412314\pi\)
0.272003 + 0.962296i \(0.412314\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) 2.49248 0.557335
\(21\) 0 0
\(22\) 3.46193 0.738086
\(23\) 6.00404 1.25193 0.625964 0.779852i \(-0.284705\pi\)
0.625964 + 0.779852i \(0.284705\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.21245 0.242489
\(26\) 4.98496 0.977630
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.51156 −1.02347 −0.511736 0.859143i \(-0.670997\pi\)
−0.511736 + 0.859143i \(0.670997\pi\)
\(30\) 2.49248 0.455062
\(31\) −7.28407 −1.30826 −0.654129 0.756383i \(-0.726965\pi\)
−0.654129 + 0.756383i \(0.726965\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.46193 0.602645
\(34\) −2.24299 −0.384670
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −9.16685 −1.50702 −0.753511 0.657436i \(-0.771641\pi\)
−0.753511 + 0.657436i \(0.771641\pi\)
\(38\) −1.00000 −0.162221
\(39\) 4.98496 0.798232
\(40\) −2.49248 −0.394095
\(41\) 4.64383 0.725244 0.362622 0.931936i \(-0.381882\pi\)
0.362622 + 0.931936i \(0.381882\pi\)
\(42\) 0 0
\(43\) 9.54211 1.45516 0.727579 0.686024i \(-0.240645\pi\)
0.727579 + 0.686024i \(0.240645\pi\)
\(44\) −3.46193 −0.521906
\(45\) 2.49248 0.371557
\(46\) −6.00404 −0.885247
\(47\) 5.36021 0.781868 0.390934 0.920419i \(-0.372152\pi\)
0.390934 + 0.920419i \(0.372152\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.21245 −0.171466
\(51\) −2.24299 −0.314082
\(52\) −4.98496 −0.691289
\(53\) −5.87423 −0.806887 −0.403443 0.915005i \(-0.632187\pi\)
−0.403443 + 0.915005i \(0.632187\pi\)
\(54\) 1.00000 0.136083
\(55\) −8.62878 −1.16350
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 5.51156 0.723704
\(59\) −8.67842 −1.12983 −0.564917 0.825148i \(-0.691092\pi\)
−0.564917 + 0.825148i \(0.691092\pi\)
\(60\) −2.49248 −0.321778
\(61\) 4.37526 0.560194 0.280097 0.959972i \(-0.409633\pi\)
0.280097 + 0.959972i \(0.409633\pi\)
\(62\) 7.28407 0.925078
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.4249 −1.54112
\(66\) −3.46193 −0.426134
\(67\) 15.1583 1.85188 0.925940 0.377670i \(-0.123274\pi\)
0.925940 + 0.377670i \(0.123274\pi\)
\(68\) 2.24299 0.272003
\(69\) −6.00404 −0.722802
\(70\) 0 0
\(71\) 12.0907 1.43490 0.717452 0.696608i \(-0.245308\pi\)
0.717452 + 0.696608i \(0.245308\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.4204 1.68778 0.843889 0.536518i \(-0.180261\pi\)
0.843889 + 0.536518i \(0.180261\pi\)
\(74\) 9.16685 1.06563
\(75\) −1.21245 −0.140001
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −4.98496 −0.564435
\(79\) −7.08667 −0.797313 −0.398656 0.917100i \(-0.630523\pi\)
−0.398656 + 0.917100i \(0.630523\pi\)
\(80\) 2.49248 0.278667
\(81\) 1.00000 0.111111
\(82\) −4.64383 −0.512825
\(83\) −5.59823 −0.614486 −0.307243 0.951631i \(-0.599406\pi\)
−0.307243 + 0.951631i \(0.599406\pi\)
\(84\) 0 0
\(85\) 5.59061 0.606387
\(86\) −9.54211 −1.02895
\(87\) 5.51156 0.590902
\(88\) 3.46193 0.369043
\(89\) 7.52952 0.798127 0.399064 0.916923i \(-0.369335\pi\)
0.399064 + 0.916923i \(0.369335\pi\)
\(90\) −2.49248 −0.262730
\(91\) 0 0
\(92\) 6.00404 0.625964
\(93\) 7.28407 0.755323
\(94\) −5.36021 −0.552864
\(95\) 2.49248 0.255723
\(96\) 1.00000 0.102062
\(97\) 7.00404 0.711153 0.355576 0.934647i \(-0.384285\pi\)
0.355576 + 0.934647i \(0.384285\pi\)
\(98\) 0 0
\(99\) −3.46193 −0.347937
\(100\) 1.21245 0.121245
\(101\) 7.09569 0.706047 0.353024 0.935614i \(-0.385154\pi\)
0.353024 + 0.935614i \(0.385154\pi\)
\(102\) 2.24299 0.222089
\(103\) 7.15135 0.704643 0.352322 0.935879i \(-0.385392\pi\)
0.352322 + 0.935879i \(0.385392\pi\)
\(104\) 4.98496 0.488815
\(105\) 0 0
\(106\) 5.87423 0.570555
\(107\) −15.7305 −1.52072 −0.760362 0.649499i \(-0.774979\pi\)
−0.760362 + 0.649499i \(0.774979\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 8.31707 0.796631 0.398315 0.917249i \(-0.369595\pi\)
0.398315 + 0.917249i \(0.369595\pi\)
\(110\) 8.62878 0.822722
\(111\) 9.16685 0.870079
\(112\) 0 0
\(113\) 13.3643 1.25720 0.628602 0.777727i \(-0.283628\pi\)
0.628602 + 0.777727i \(0.283628\pi\)
\(114\) 1.00000 0.0936586
\(115\) 14.9649 1.39549
\(116\) −5.51156 −0.511736
\(117\) −4.98496 −0.460859
\(118\) 8.67842 0.798913
\(119\) 0 0
\(120\) 2.49248 0.227531
\(121\) 0.984956 0.0895414
\(122\) −4.37526 −0.396117
\(123\) −4.64383 −0.418720
\(124\) −7.28407 −0.654129
\(125\) −9.44040 −0.844375
\(126\) 0 0
\(127\) 6.09569 0.540905 0.270452 0.962733i \(-0.412827\pi\)
0.270452 + 0.962733i \(0.412827\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −9.54211 −0.840136
\(130\) 12.4249 1.08974
\(131\) 9.31416 0.813782 0.406891 0.913477i \(-0.366613\pi\)
0.406891 + 0.913477i \(0.366613\pi\)
\(132\) 3.46193 0.301322
\(133\) 0 0
\(134\) −15.1583 −1.30948
\(135\) −2.49248 −0.214518
\(136\) −2.24299 −0.192335
\(137\) −7.78510 −0.665126 −0.332563 0.943081i \(-0.607913\pi\)
−0.332563 + 0.943081i \(0.607913\pi\)
\(138\) 6.00404 0.511098
\(139\) 12.5882 1.06771 0.533857 0.845575i \(-0.320742\pi\)
0.533857 + 0.845575i \(0.320742\pi\)
\(140\) 0 0
\(141\) −5.36021 −0.451411
\(142\) −12.0907 −1.01463
\(143\) 17.2576 1.44315
\(144\) 1.00000 0.0833333
\(145\) −13.7374 −1.14083
\(146\) −14.4204 −1.19344
\(147\) 0 0
\(148\) −9.16685 −0.753511
\(149\) 16.2922 1.33471 0.667353 0.744742i \(-0.267427\pi\)
0.667353 + 0.744742i \(0.267427\pi\)
\(150\) 1.21245 0.0989958
\(151\) 19.9735 1.62542 0.812710 0.582669i \(-0.197991\pi\)
0.812710 + 0.582669i \(0.197991\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 2.24299 0.181335
\(154\) 0 0
\(155\) −18.1554 −1.45828
\(156\) 4.98496 0.399116
\(157\) −0.739053 −0.0589828 −0.0294914 0.999565i \(-0.509389\pi\)
−0.0294914 + 0.999565i \(0.509389\pi\)
\(158\) 7.08667 0.563785
\(159\) 5.87423 0.465856
\(160\) −2.49248 −0.197048
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −5.79808 −0.454141 −0.227071 0.973878i \(-0.572915\pi\)
−0.227071 + 0.973878i \(0.572915\pi\)
\(164\) 4.64383 0.362622
\(165\) 8.62878 0.671750
\(166\) 5.59823 0.434507
\(167\) 17.1058 1.32368 0.661842 0.749644i \(-0.269775\pi\)
0.661842 + 0.749644i \(0.269775\pi\)
\(168\) 0 0
\(169\) 11.8498 0.911522
\(170\) −5.59061 −0.428780
\(171\) 1.00000 0.0764719
\(172\) 9.54211 0.727579
\(173\) −2.58412 −0.196467 −0.0982336 0.995163i \(-0.531319\pi\)
−0.0982336 + 0.995163i \(0.531319\pi\)
\(174\) −5.51156 −0.417830
\(175\) 0 0
\(176\) −3.46193 −0.260953
\(177\) 8.67842 0.652310
\(178\) −7.52952 −0.564361
\(179\) −3.24948 −0.242878 −0.121439 0.992599i \(-0.538751\pi\)
−0.121439 + 0.992599i \(0.538751\pi\)
\(180\) 2.49248 0.185778
\(181\) 12.4559 0.925840 0.462920 0.886400i \(-0.346802\pi\)
0.462920 + 0.886400i \(0.346802\pi\)
\(182\) 0 0
\(183\) −4.37526 −0.323428
\(184\) −6.00404 −0.442624
\(185\) −22.8482 −1.67983
\(186\) −7.28407 −0.534094
\(187\) −7.76509 −0.567839
\(188\) 5.36021 0.390934
\(189\) 0 0
\(190\) −2.49248 −0.180823
\(191\) −16.4444 −1.18988 −0.594939 0.803771i \(-0.702824\pi\)
−0.594939 + 0.803771i \(0.702824\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −18.6854 −1.34500 −0.672501 0.740096i \(-0.734780\pi\)
−0.672501 + 0.740096i \(0.734780\pi\)
\(194\) −7.00404 −0.502861
\(195\) 12.4249 0.889765
\(196\) 0 0
\(197\) −15.4965 −1.10408 −0.552041 0.833817i \(-0.686151\pi\)
−0.552041 + 0.833817i \(0.686151\pi\)
\(198\) 3.46193 0.246029
\(199\) −1.45948 −0.103460 −0.0517299 0.998661i \(-0.516473\pi\)
−0.0517299 + 0.998661i \(0.516473\pi\)
\(200\) −1.21245 −0.0857329
\(201\) −15.1583 −1.06918
\(202\) −7.09569 −0.499251
\(203\) 0 0
\(204\) −2.24299 −0.157041
\(205\) 11.5746 0.808408
\(206\) −7.15135 −0.498258
\(207\) 6.00404 0.417310
\(208\) −4.98496 −0.345644
\(209\) −3.46193 −0.239467
\(210\) 0 0
\(211\) −23.3953 −1.61060 −0.805298 0.592870i \(-0.797995\pi\)
−0.805298 + 0.592870i \(0.797995\pi\)
\(212\) −5.87423 −0.403443
\(213\) −12.0907 −0.828442
\(214\) 15.7305 1.07531
\(215\) 23.7835 1.62202
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −8.31707 −0.563303
\(219\) −14.4204 −0.974439
\(220\) −8.62878 −0.581752
\(221\) −11.1812 −0.752131
\(222\) −9.16685 −0.615239
\(223\) 14.8221 0.992564 0.496282 0.868161i \(-0.334698\pi\)
0.496282 + 0.868161i \(0.334698\pi\)
\(224\) 0 0
\(225\) 1.21245 0.0808297
\(226\) −13.3643 −0.888977
\(227\) 16.2295 1.07719 0.538594 0.842566i \(-0.318956\pi\)
0.538594 + 0.842566i \(0.318956\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −6.39480 −0.422580 −0.211290 0.977423i \(-0.567767\pi\)
−0.211290 + 0.977423i \(0.567767\pi\)
\(230\) −14.9649 −0.986759
\(231\) 0 0
\(232\) 5.51156 0.361852
\(233\) −21.3798 −1.40063 −0.700317 0.713832i \(-0.746958\pi\)
−0.700317 + 0.713832i \(0.746958\pi\)
\(234\) 4.98496 0.325877
\(235\) 13.3602 0.871524
\(236\) −8.67842 −0.564917
\(237\) 7.08667 0.460329
\(238\) 0 0
\(239\) 18.8938 1.22214 0.611068 0.791578i \(-0.290740\pi\)
0.611068 + 0.791578i \(0.290740\pi\)
\(240\) −2.49248 −0.160889
\(241\) 4.05009 0.260889 0.130445 0.991456i \(-0.458359\pi\)
0.130445 + 0.991456i \(0.458359\pi\)
\(242\) −0.984956 −0.0633153
\(243\) −1.00000 −0.0641500
\(244\) 4.37526 0.280097
\(245\) 0 0
\(246\) 4.64383 0.296080
\(247\) −4.98496 −0.317185
\(248\) 7.28407 0.462539
\(249\) 5.59823 0.354774
\(250\) 9.44040 0.597063
\(251\) 6.59917 0.416536 0.208268 0.978072i \(-0.433217\pi\)
0.208268 + 0.978072i \(0.433217\pi\)
\(252\) 0 0
\(253\) −20.7856 −1.30678
\(254\) −6.09569 −0.382477
\(255\) −5.59061 −0.350098
\(256\) 1.00000 0.0625000
\(257\) −2.65125 −0.165380 −0.0826902 0.996575i \(-0.526351\pi\)
−0.0826902 + 0.996575i \(0.526351\pi\)
\(258\) 9.54211 0.594066
\(259\) 0 0
\(260\) −12.4249 −0.770559
\(261\) −5.51156 −0.341157
\(262\) −9.31416 −0.575431
\(263\) −1.45544 −0.0897463 −0.0448731 0.998993i \(-0.514288\pi\)
−0.0448731 + 0.998993i \(0.514288\pi\)
\(264\) −3.46193 −0.213067
\(265\) −14.6414 −0.899413
\(266\) 0 0
\(267\) −7.52952 −0.460799
\(268\) 15.1583 0.925940
\(269\) 14.3232 0.873299 0.436650 0.899632i \(-0.356165\pi\)
0.436650 + 0.899632i \(0.356165\pi\)
\(270\) 2.49248 0.151687
\(271\) 20.6658 1.25536 0.627680 0.778472i \(-0.284005\pi\)
0.627680 + 0.778472i \(0.284005\pi\)
\(272\) 2.24299 0.136001
\(273\) 0 0
\(274\) 7.78510 0.470315
\(275\) −4.19740 −0.253113
\(276\) −6.00404 −0.361401
\(277\) −28.5421 −1.71493 −0.857464 0.514544i \(-0.827961\pi\)
−0.857464 + 0.514544i \(0.827961\pi\)
\(278\) −12.5882 −0.754988
\(279\) −7.28407 −0.436086
\(280\) 0 0
\(281\) −13.6443 −0.813950 −0.406975 0.913439i \(-0.633416\pi\)
−0.406975 + 0.913439i \(0.633416\pi\)
\(282\) 5.36021 0.319196
\(283\) −1.53403 −0.0911886 −0.0455943 0.998960i \(-0.514518\pi\)
−0.0455943 + 0.998960i \(0.514518\pi\)
\(284\) 12.0907 0.717452
\(285\) −2.49248 −0.147642
\(286\) −17.2576 −1.02046
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −11.9690 −0.704058
\(290\) 13.7374 0.806691
\(291\) −7.00404 −0.410584
\(292\) 14.4204 0.843889
\(293\) −33.3773 −1.94992 −0.974962 0.222373i \(-0.928620\pi\)
−0.974962 + 0.222373i \(0.928620\pi\)
\(294\) 0 0
\(295\) −21.6308 −1.25939
\(296\) 9.16685 0.532813
\(297\) 3.46193 0.200882
\(298\) −16.2922 −0.943779
\(299\) −29.9299 −1.73089
\(300\) −1.21245 −0.0700006
\(301\) 0 0
\(302\) −19.9735 −1.14935
\(303\) −7.09569 −0.407636
\(304\) 1.00000 0.0573539
\(305\) 10.9052 0.624432
\(306\) −2.24299 −0.128223
\(307\) 9.52952 0.543878 0.271939 0.962314i \(-0.412335\pi\)
0.271939 + 0.962314i \(0.412335\pi\)
\(308\) 0 0
\(309\) −7.15135 −0.406826
\(310\) 18.1554 1.03116
\(311\) 16.4404 0.932249 0.466125 0.884719i \(-0.345650\pi\)
0.466125 + 0.884719i \(0.345650\pi\)
\(312\) −4.98496 −0.282218
\(313\) 10.9940 0.621416 0.310708 0.950505i \(-0.399434\pi\)
0.310708 + 0.950505i \(0.399434\pi\)
\(314\) 0.739053 0.0417072
\(315\) 0 0
\(316\) −7.08667 −0.398656
\(317\) 6.66291 0.374226 0.187113 0.982338i \(-0.440087\pi\)
0.187113 + 0.982338i \(0.440087\pi\)
\(318\) −5.87423 −0.329410
\(319\) 19.0806 1.06831
\(320\) 2.49248 0.139334
\(321\) 15.7305 0.877991
\(322\) 0 0
\(323\) 2.24299 0.124804
\(324\) 1.00000 0.0555556
\(325\) −6.04399 −0.335260
\(326\) 5.79808 0.321126
\(327\) −8.31707 −0.459935
\(328\) −4.64383 −0.256412
\(329\) 0 0
\(330\) −8.62878 −0.474999
\(331\) 25.8102 1.41866 0.709329 0.704878i \(-0.248998\pi\)
0.709329 + 0.704878i \(0.248998\pi\)
\(332\) −5.59823 −0.307243
\(333\) −9.16685 −0.502340
\(334\) −17.1058 −0.935985
\(335\) 37.7817 2.06424
\(336\) 0 0
\(337\) 10.0932 0.549810 0.274905 0.961471i \(-0.411354\pi\)
0.274905 + 0.961471i \(0.411354\pi\)
\(338\) −11.8498 −0.644543
\(339\) −13.3643 −0.725847
\(340\) 5.59061 0.303194
\(341\) 25.2169 1.36557
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −9.54211 −0.514476
\(345\) −14.9649 −0.805685
\(346\) 2.58412 0.138923
\(347\) −0.369228 −0.0198212 −0.00991059 0.999951i \(-0.503155\pi\)
−0.00991059 + 0.999951i \(0.503155\pi\)
\(348\) 5.51156 0.295451
\(349\) 31.3643 1.67889 0.839445 0.543445i \(-0.182880\pi\)
0.839445 + 0.543445i \(0.182880\pi\)
\(350\) 0 0
\(351\) 4.98496 0.266077
\(352\) 3.46193 0.184521
\(353\) −22.8192 −1.21454 −0.607272 0.794494i \(-0.707736\pi\)
−0.607272 + 0.794494i \(0.707736\pi\)
\(354\) −8.67842 −0.461253
\(355\) 30.1358 1.59944
\(356\) 7.52952 0.399064
\(357\) 0 0
\(358\) 3.24948 0.171741
\(359\) 24.0276 1.26813 0.634065 0.773280i \(-0.281385\pi\)
0.634065 + 0.773280i \(0.281385\pi\)
\(360\) −2.49248 −0.131365
\(361\) 1.00000 0.0526316
\(362\) −12.4559 −0.654668
\(363\) −0.984956 −0.0516968
\(364\) 0 0
\(365\) 35.9425 1.88131
\(366\) 4.37526 0.228698
\(367\) 15.8096 0.825256 0.412628 0.910900i \(-0.364611\pi\)
0.412628 + 0.910900i \(0.364611\pi\)
\(368\) 6.00404 0.312982
\(369\) 4.64383 0.241748
\(370\) 22.8482 1.18782
\(371\) 0 0
\(372\) 7.28407 0.377662
\(373\) 30.8507 1.59739 0.798694 0.601737i \(-0.205524\pi\)
0.798694 + 0.601737i \(0.205524\pi\)
\(374\) 7.76509 0.401523
\(375\) 9.44040 0.487500
\(376\) −5.36021 −0.276432
\(377\) 27.4749 1.41503
\(378\) 0 0
\(379\) 23.5450 1.20943 0.604713 0.796443i \(-0.293288\pi\)
0.604713 + 0.796443i \(0.293288\pi\)
\(380\) 2.49248 0.127861
\(381\) −6.09569 −0.312291
\(382\) 16.4444 0.841371
\(383\) 4.87271 0.248984 0.124492 0.992221i \(-0.460270\pi\)
0.124492 + 0.992221i \(0.460270\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 18.6854 0.951060
\(387\) 9.54211 0.485053
\(388\) 7.00404 0.355576
\(389\) −28.2856 −1.43414 −0.717068 0.697003i \(-0.754516\pi\)
−0.717068 + 0.697003i \(0.754516\pi\)
\(390\) −12.4249 −0.629159
\(391\) 13.4670 0.681057
\(392\) 0 0
\(393\) −9.31416 −0.469837
\(394\) 15.4965 0.780703
\(395\) −17.6634 −0.888741
\(396\) −3.46193 −0.173969
\(397\) 2.65125 0.133062 0.0665312 0.997784i \(-0.478807\pi\)
0.0665312 + 0.997784i \(0.478807\pi\)
\(398\) 1.45948 0.0731571
\(399\) 0 0
\(400\) 1.21245 0.0606223
\(401\) −7.45292 −0.372181 −0.186090 0.982533i \(-0.559582\pi\)
−0.186090 + 0.982533i \(0.559582\pi\)
\(402\) 15.1583 0.756027
\(403\) 36.3108 1.80877
\(404\) 7.09569 0.353024
\(405\) 2.49248 0.123852
\(406\) 0 0
\(407\) 31.7350 1.57305
\(408\) 2.24299 0.111045
\(409\) 17.4037 0.860560 0.430280 0.902696i \(-0.358415\pi\)
0.430280 + 0.902696i \(0.358415\pi\)
\(410\) −11.5746 −0.571630
\(411\) 7.78510 0.384011
\(412\) 7.15135 0.352322
\(413\) 0 0
\(414\) −6.00404 −0.295082
\(415\) −13.9535 −0.684949
\(416\) 4.98496 0.244408
\(417\) −12.5882 −0.616445
\(418\) 3.46193 0.169329
\(419\) −3.76105 −0.183739 −0.0918696 0.995771i \(-0.529284\pi\)
−0.0918696 + 0.995771i \(0.529284\pi\)
\(420\) 0 0
\(421\) 31.6343 1.54176 0.770882 0.636978i \(-0.219816\pi\)
0.770882 + 0.636978i \(0.219816\pi\)
\(422\) 23.3953 1.13886
\(423\) 5.36021 0.260623
\(424\) 5.87423 0.285278
\(425\) 2.71951 0.131916
\(426\) 12.0907 0.585797
\(427\) 0 0
\(428\) −15.7305 −0.760362
\(429\) −17.2576 −0.833203
\(430\) −23.7835 −1.14694
\(431\) −1.66582 −0.0802398 −0.0401199 0.999195i \(-0.512774\pi\)
−0.0401199 + 0.999195i \(0.512774\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 16.1062 0.774015 0.387008 0.922076i \(-0.373509\pi\)
0.387008 + 0.922076i \(0.373509\pi\)
\(434\) 0 0
\(435\) 13.7374 0.658660
\(436\) 8.31707 0.398315
\(437\) 6.00404 0.287212
\(438\) 14.4204 0.689032
\(439\) 21.0867 1.00641 0.503206 0.864167i \(-0.332154\pi\)
0.503206 + 0.864167i \(0.332154\pi\)
\(440\) 8.62878 0.411361
\(441\) 0 0
\(442\) 11.1812 0.531837
\(443\) −18.5958 −0.883512 −0.441756 0.897135i \(-0.645644\pi\)
−0.441756 + 0.897135i \(0.645644\pi\)
\(444\) 9.16685 0.435040
\(445\) 18.7672 0.889648
\(446\) −14.8221 −0.701849
\(447\) −16.2922 −0.770592
\(448\) 0 0
\(449\) 19.7410 0.931637 0.465818 0.884880i \(-0.345760\pi\)
0.465818 + 0.884880i \(0.345760\pi\)
\(450\) −1.21245 −0.0571552
\(451\) −16.0766 −0.757018
\(452\) 13.3643 0.628602
\(453\) −19.9735 −0.938437
\(454\) −16.2295 −0.761687
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 0.468893 0.0219339 0.0109669 0.999940i \(-0.496509\pi\)
0.0109669 + 0.999940i \(0.496509\pi\)
\(458\) 6.39480 0.298809
\(459\) −2.24299 −0.104694
\(460\) 14.9649 0.697744
\(461\) −19.0556 −0.887510 −0.443755 0.896148i \(-0.646354\pi\)
−0.443755 + 0.896148i \(0.646354\pi\)
\(462\) 0 0
\(463\) −9.65575 −0.448741 −0.224370 0.974504i \(-0.572033\pi\)
−0.224370 + 0.974504i \(0.572033\pi\)
\(464\) −5.51156 −0.255868
\(465\) 18.1554 0.841936
\(466\) 21.3798 0.990398
\(467\) −9.16281 −0.424004 −0.212002 0.977269i \(-0.567998\pi\)
−0.212002 + 0.977269i \(0.567998\pi\)
\(468\) −4.98496 −0.230430
\(469\) 0 0
\(470\) −13.3602 −0.616261
\(471\) 0.739053 0.0340537
\(472\) 8.67842 0.399456
\(473\) −33.0341 −1.51891
\(474\) −7.08667 −0.325502
\(475\) 1.21245 0.0556308
\(476\) 0 0
\(477\) −5.87423 −0.268962
\(478\) −18.8938 −0.864181
\(479\) 42.8212 1.95655 0.978276 0.207309i \(-0.0664706\pi\)
0.978276 + 0.207309i \(0.0664706\pi\)
\(480\) 2.49248 0.113766
\(481\) 45.6964 2.08357
\(482\) −4.05009 −0.184477
\(483\) 0 0
\(484\) 0.984956 0.0447707
\(485\) 17.4574 0.792700
\(486\) 1.00000 0.0453609
\(487\) −26.3153 −1.19246 −0.596230 0.802814i \(-0.703335\pi\)
−0.596230 + 0.802814i \(0.703335\pi\)
\(488\) −4.37526 −0.198059
\(489\) 5.79808 0.262198
\(490\) 0 0
\(491\) −14.2064 −0.641127 −0.320563 0.947227i \(-0.603872\pi\)
−0.320563 + 0.947227i \(0.603872\pi\)
\(492\) −4.64383 −0.209360
\(493\) −12.3624 −0.556775
\(494\) 4.98496 0.224284
\(495\) −8.62878 −0.387835
\(496\) −7.28407 −0.327065
\(497\) 0 0
\(498\) −5.59823 −0.250863
\(499\) 34.0325 1.52351 0.761753 0.647868i \(-0.224339\pi\)
0.761753 + 0.647868i \(0.224339\pi\)
\(500\) −9.44040 −0.422187
\(501\) −17.1058 −0.764229
\(502\) −6.59917 −0.294535
\(503\) 18.2003 0.811512 0.405756 0.913981i \(-0.367008\pi\)
0.405756 + 0.913981i \(0.367008\pi\)
\(504\) 0 0
\(505\) 17.6858 0.787009
\(506\) 20.7856 0.924031
\(507\) −11.8498 −0.526267
\(508\) 6.09569 0.270452
\(509\) −4.56478 −0.202330 −0.101165 0.994870i \(-0.532257\pi\)
−0.101165 + 0.994870i \(0.532257\pi\)
\(510\) 5.59061 0.247556
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 2.65125 0.116942
\(515\) 17.8246 0.785445
\(516\) −9.54211 −0.420068
\(517\) −18.5567 −0.816122
\(518\) 0 0
\(519\) 2.58412 0.113430
\(520\) 12.4249 0.544868
\(521\) −40.5160 −1.77504 −0.887520 0.460769i \(-0.847574\pi\)
−0.887520 + 0.460769i \(0.847574\pi\)
\(522\) 5.51156 0.241235
\(523\) −29.4644 −1.28839 −0.644195 0.764861i \(-0.722807\pi\)
−0.644195 + 0.764861i \(0.722807\pi\)
\(524\) 9.31416 0.406891
\(525\) 0 0
\(526\) 1.45544 0.0634602
\(527\) −16.3381 −0.711700
\(528\) 3.46193 0.150661
\(529\) 13.0485 0.567326
\(530\) 14.6414 0.635981
\(531\) −8.67842 −0.376611
\(532\) 0 0
\(533\) −23.1493 −1.00271
\(534\) 7.52952 0.325834
\(535\) −39.2079 −1.69511
\(536\) −15.1583 −0.654739
\(537\) 3.24948 0.140226
\(538\) −14.3232 −0.617516
\(539\) 0 0
\(540\) −2.49248 −0.107259
\(541\) −5.52800 −0.237667 −0.118834 0.992914i \(-0.537916\pi\)
−0.118834 + 0.992914i \(0.537916\pi\)
\(542\) −20.6658 −0.887673
\(543\) −12.4559 −0.534534
\(544\) −2.24299 −0.0961676
\(545\) 20.7301 0.887980
\(546\) 0 0
\(547\) −13.7329 −0.587178 −0.293589 0.955932i \(-0.594850\pi\)
−0.293589 + 0.955932i \(0.594850\pi\)
\(548\) −7.78510 −0.332563
\(549\) 4.37526 0.186731
\(550\) 4.19740 0.178978
\(551\) −5.51156 −0.234800
\(552\) 6.00404 0.255549
\(553\) 0 0
\(554\) 28.5421 1.21264
\(555\) 22.8482 0.969851
\(556\) 12.5882 0.533857
\(557\) −0.399242 −0.0169164 −0.00845821 0.999964i \(-0.502692\pi\)
−0.00845821 + 0.999964i \(0.502692\pi\)
\(558\) 7.28407 0.308359
\(559\) −47.5670 −2.01187
\(560\) 0 0
\(561\) 7.76509 0.327842
\(562\) 13.6443 0.575549
\(563\) 26.1242 1.10100 0.550501 0.834834i \(-0.314437\pi\)
0.550501 + 0.834834i \(0.314437\pi\)
\(564\) −5.36021 −0.225706
\(565\) 33.3101 1.40137
\(566\) 1.53403 0.0644801
\(567\) 0 0
\(568\) −12.0907 −0.507315
\(569\) 3.06316 0.128414 0.0642072 0.997937i \(-0.479548\pi\)
0.0642072 + 0.997937i \(0.479548\pi\)
\(570\) 2.49248 0.104398
\(571\) 5.89066 0.246517 0.123258 0.992375i \(-0.460666\pi\)
0.123258 + 0.992375i \(0.460666\pi\)
\(572\) 17.2576 0.721575
\(573\) 16.4444 0.686976
\(574\) 0 0
\(575\) 7.27957 0.303579
\(576\) 1.00000 0.0416667
\(577\) −25.7896 −1.07364 −0.536818 0.843698i \(-0.680374\pi\)
−0.536818 + 0.843698i \(0.680374\pi\)
\(578\) 11.9690 0.497844
\(579\) 18.6854 0.776537
\(580\) −13.7374 −0.570416
\(581\) 0 0
\(582\) 7.00404 0.290327
\(583\) 20.3362 0.842237
\(584\) −14.4204 −0.596719
\(585\) −12.4249 −0.513706
\(586\) 33.3773 1.37880
\(587\) 0.822143 0.0339335 0.0169667 0.999856i \(-0.494599\pi\)
0.0169667 + 0.999856i \(0.494599\pi\)
\(588\) 0 0
\(589\) −7.28407 −0.300135
\(590\) 21.6308 0.890524
\(591\) 15.4965 0.637442
\(592\) −9.16685 −0.376755
\(593\) −16.6407 −0.683352 −0.341676 0.939818i \(-0.610995\pi\)
−0.341676 + 0.939818i \(0.610995\pi\)
\(594\) −3.46193 −0.142045
\(595\) 0 0
\(596\) 16.2922 0.667353
\(597\) 1.45948 0.0597325
\(598\) 29.9299 1.22392
\(599\) 8.57670 0.350434 0.175217 0.984530i \(-0.443937\pi\)
0.175217 + 0.984530i \(0.443937\pi\)
\(600\) 1.21245 0.0494979
\(601\) 13.6993 0.558805 0.279403 0.960174i \(-0.409864\pi\)
0.279403 + 0.960174i \(0.409864\pi\)
\(602\) 0 0
\(603\) 15.1583 0.617294
\(604\) 19.9735 0.812710
\(605\) 2.45498 0.0998091
\(606\) 7.09569 0.288243
\(607\) 42.5405 1.72667 0.863333 0.504635i \(-0.168373\pi\)
0.863333 + 0.504635i \(0.168373\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −10.9052 −0.441540
\(611\) −26.7204 −1.08099
\(612\) 2.24299 0.0906677
\(613\) 14.5096 0.586036 0.293018 0.956107i \(-0.405340\pi\)
0.293018 + 0.956107i \(0.405340\pi\)
\(614\) −9.52952 −0.384580
\(615\) −11.5746 −0.466734
\(616\) 0 0
\(617\) −25.3468 −1.02042 −0.510211 0.860049i \(-0.670433\pi\)
−0.510211 + 0.860049i \(0.670433\pi\)
\(618\) 7.15135 0.287669
\(619\) −15.9944 −0.642867 −0.321434 0.946932i \(-0.604165\pi\)
−0.321434 + 0.946932i \(0.604165\pi\)
\(620\) −18.1554 −0.729138
\(621\) −6.00404 −0.240934
\(622\) −16.4404 −0.659200
\(623\) 0 0
\(624\) 4.98496 0.199558
\(625\) −29.5922 −1.18369
\(626\) −10.9940 −0.439407
\(627\) 3.46193 0.138256
\(628\) −0.739053 −0.0294914
\(629\) −20.5612 −0.819829
\(630\) 0 0
\(631\) −25.6970 −1.02298 −0.511491 0.859288i \(-0.670907\pi\)
−0.511491 + 0.859288i \(0.670907\pi\)
\(632\) 7.08667 0.281893
\(633\) 23.3953 0.929878
\(634\) −6.66291 −0.264618
\(635\) 15.1934 0.602930
\(636\) 5.87423 0.232928
\(637\) 0 0
\(638\) −19.0806 −0.755410
\(639\) 12.0907 0.478301
\(640\) −2.49248 −0.0985238
\(641\) 44.0747 1.74085 0.870424 0.492303i \(-0.163845\pi\)
0.870424 + 0.492303i \(0.163845\pi\)
\(642\) −15.7305 −0.620833
\(643\) 22.7730 0.898078 0.449039 0.893512i \(-0.351766\pi\)
0.449039 + 0.893512i \(0.351766\pi\)
\(644\) 0 0
\(645\) −23.7835 −0.936474
\(646\) −2.24299 −0.0882494
\(647\) 30.3678 1.19388 0.596941 0.802285i \(-0.296382\pi\)
0.596941 + 0.802285i \(0.296382\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 30.0441 1.17933
\(650\) 6.04399 0.237065
\(651\) 0 0
\(652\) −5.79808 −0.227071
\(653\) −4.63535 −0.181395 −0.0906976 0.995878i \(-0.528910\pi\)
−0.0906976 + 0.995878i \(0.528910\pi\)
\(654\) 8.31707 0.325223
\(655\) 23.2153 0.907098
\(656\) 4.64383 0.181311
\(657\) 14.4204 0.562592
\(658\) 0 0
\(659\) −18.8998 −0.736232 −0.368116 0.929780i \(-0.619997\pi\)
−0.368116 + 0.929780i \(0.619997\pi\)
\(660\) 8.62878 0.335875
\(661\) 25.8042 1.00367 0.501833 0.864964i \(-0.332659\pi\)
0.501833 + 0.864964i \(0.332659\pi\)
\(662\) −25.8102 −1.00314
\(663\) 11.1812 0.434243
\(664\) 5.59823 0.217254
\(665\) 0 0
\(666\) 9.16685 0.355208
\(667\) −33.0916 −1.28131
\(668\) 17.1058 0.661842
\(669\) −14.8221 −0.573057
\(670\) −37.7817 −1.45964
\(671\) −15.1468 −0.584737
\(672\) 0 0
\(673\) 15.7844 0.608445 0.304223 0.952601i \(-0.401603\pi\)
0.304223 + 0.952601i \(0.401603\pi\)
\(674\) −10.0932 −0.388774
\(675\) −1.21245 −0.0466671
\(676\) 11.8498 0.455761
\(677\) −24.0786 −0.925415 −0.462707 0.886511i \(-0.653122\pi\)
−0.462707 + 0.886511i \(0.653122\pi\)
\(678\) 13.3643 0.513251
\(679\) 0 0
\(680\) −5.59061 −0.214390
\(681\) −16.2295 −0.621914
\(682\) −25.2169 −0.965607
\(683\) 14.8506 0.568243 0.284122 0.958788i \(-0.408298\pi\)
0.284122 + 0.958788i \(0.408298\pi\)
\(684\) 1.00000 0.0382360
\(685\) −19.4042 −0.741396
\(686\) 0 0
\(687\) 6.39480 0.243977
\(688\) 9.54211 0.363790
\(689\) 29.2828 1.11558
\(690\) 14.9649 0.569705
\(691\) 38.2941 1.45678 0.728389 0.685164i \(-0.240269\pi\)
0.728389 + 0.685164i \(0.240269\pi\)
\(692\) −2.58412 −0.0982336
\(693\) 0 0
\(694\) 0.369228 0.0140157
\(695\) 31.3757 1.19015
\(696\) −5.51156 −0.208915
\(697\) 10.4161 0.394537
\(698\) −31.3643 −1.18715
\(699\) 21.3798 0.808657
\(700\) 0 0
\(701\) −24.6068 −0.929385 −0.464693 0.885472i \(-0.653835\pi\)
−0.464693 + 0.885472i \(0.653835\pi\)
\(702\) −4.98496 −0.188145
\(703\) −9.16685 −0.345734
\(704\) −3.46193 −0.130476
\(705\) −13.3602 −0.503175
\(706\) 22.8192 0.858813
\(707\) 0 0
\(708\) 8.67842 0.326155
\(709\) 17.2665 0.648457 0.324229 0.945979i \(-0.394895\pi\)
0.324229 + 0.945979i \(0.394895\pi\)
\(710\) −30.1358 −1.13098
\(711\) −7.08667 −0.265771
\(712\) −7.52952 −0.282181
\(713\) −43.7339 −1.63785
\(714\) 0 0
\(715\) 43.0141 1.60864
\(716\) −3.24948 −0.121439
\(717\) −18.8938 −0.705601
\(718\) −24.0276 −0.896703
\(719\) 43.2527 1.61305 0.806526 0.591198i \(-0.201345\pi\)
0.806526 + 0.591198i \(0.201345\pi\)
\(720\) 2.49248 0.0928892
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) −4.05009 −0.150625
\(724\) 12.4559 0.462920
\(725\) −6.68247 −0.248181
\(726\) 0.984956 0.0365551
\(727\) −26.0766 −0.967127 −0.483564 0.875309i \(-0.660658\pi\)
−0.483564 + 0.875309i \(0.660658\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −35.9425 −1.33029
\(731\) 21.4029 0.791615
\(732\) −4.37526 −0.161714
\(733\) −8.21894 −0.303573 −0.151787 0.988413i \(-0.548503\pi\)
−0.151787 + 0.988413i \(0.548503\pi\)
\(734\) −15.8096 −0.583544
\(735\) 0 0
\(736\) −6.00404 −0.221312
\(737\) −52.4770 −1.93301
\(738\) −4.64383 −0.170942
\(739\) 38.5978 1.41984 0.709921 0.704282i \(-0.248731\pi\)
0.709921 + 0.704282i \(0.248731\pi\)
\(740\) −22.8482 −0.839916
\(741\) 4.98496 0.183127
\(742\) 0 0
\(743\) −5.51309 −0.202256 −0.101128 0.994873i \(-0.532245\pi\)
−0.101128 + 0.994873i \(0.532245\pi\)
\(744\) −7.28407 −0.267047
\(745\) 40.6078 1.48776
\(746\) −30.8507 −1.12952
\(747\) −5.59823 −0.204829
\(748\) −7.76509 −0.283920
\(749\) 0 0
\(750\) −9.44040 −0.344714
\(751\) 19.2600 0.702808 0.351404 0.936224i \(-0.385704\pi\)
0.351404 + 0.936224i \(0.385704\pi\)
\(752\) 5.36021 0.195467
\(753\) −6.59917 −0.240487
\(754\) −27.4749 −1.00058
\(755\) 49.7835 1.81181
\(756\) 0 0
\(757\) −0.556024 −0.0202090 −0.0101045 0.999949i \(-0.503216\pi\)
−0.0101045 + 0.999949i \(0.503216\pi\)
\(758\) −23.5450 −0.855194
\(759\) 20.7856 0.754468
\(760\) −2.49248 −0.0904117
\(761\) 9.17692 0.332663 0.166332 0.986070i \(-0.446808\pi\)
0.166332 + 0.986070i \(0.446808\pi\)
\(762\) 6.09569 0.220823
\(763\) 0 0
\(764\) −16.4444 −0.594939
\(765\) 5.59061 0.202129
\(766\) −4.87271 −0.176058
\(767\) 43.2615 1.56208
\(768\) −1.00000 −0.0360844
\(769\) 10.2515 0.369680 0.184840 0.982769i \(-0.440823\pi\)
0.184840 + 0.982769i \(0.440823\pi\)
\(770\) 0 0
\(771\) 2.65125 0.0954824
\(772\) −18.6854 −0.672501
\(773\) −25.3728 −0.912596 −0.456298 0.889827i \(-0.650825\pi\)
−0.456298 + 0.889827i \(0.650825\pi\)
\(774\) −9.54211 −0.342984
\(775\) −8.83154 −0.317238
\(776\) −7.00404 −0.251430
\(777\) 0 0
\(778\) 28.2856 1.01409
\(779\) 4.64383 0.166382
\(780\) 12.4249 0.444882
\(781\) −41.8572 −1.49777
\(782\) −13.4670 −0.481580
\(783\) 5.51156 0.196967
\(784\) 0 0
\(785\) −1.84207 −0.0657464
\(786\) 9.31416 0.332225
\(787\) 21.2886 0.758856 0.379428 0.925221i \(-0.376121\pi\)
0.379428 + 0.925221i \(0.376121\pi\)
\(788\) −15.4965 −0.552041
\(789\) 1.45544 0.0518150
\(790\) 17.6634 0.628435
\(791\) 0 0
\(792\) 3.46193 0.123014
\(793\) −21.8105 −0.774512
\(794\) −2.65125 −0.0940893
\(795\) 14.6414 0.519276
\(796\) −1.45948 −0.0517299
\(797\) 41.1949 1.45920 0.729600 0.683874i \(-0.239706\pi\)
0.729600 + 0.683874i \(0.239706\pi\)
\(798\) 0 0
\(799\) 12.0229 0.425341
\(800\) −1.21245 −0.0428664
\(801\) 7.52952 0.266042
\(802\) 7.45292 0.263172
\(803\) −49.9223 −1.76172
\(804\) −15.1583 −0.534592
\(805\) 0 0
\(806\) −36.3108 −1.27899
\(807\) −14.3232 −0.504200
\(808\) −7.09569 −0.249625
\(809\) −18.3968 −0.646797 −0.323398 0.946263i \(-0.604825\pi\)
−0.323398 + 0.946263i \(0.604825\pi\)
\(810\) −2.49248 −0.0875767
\(811\) −18.8662 −0.662482 −0.331241 0.943546i \(-0.607467\pi\)
−0.331241 + 0.943546i \(0.607467\pi\)
\(812\) 0 0
\(813\) −20.6658 −0.724782
\(814\) −31.7350 −1.11231
\(815\) −14.4516 −0.506217
\(816\) −2.24299 −0.0785205
\(817\) 9.54211 0.333836
\(818\) −17.4037 −0.608508
\(819\) 0 0
\(820\) 11.5746 0.404204
\(821\) 16.6517 0.581149 0.290574 0.956852i \(-0.406154\pi\)
0.290574 + 0.956852i \(0.406154\pi\)
\(822\) −7.78510 −0.271537
\(823\) −35.3894 −1.23360 −0.616798 0.787122i \(-0.711570\pi\)
−0.616798 + 0.787122i \(0.711570\pi\)
\(824\) −7.15135 −0.249129
\(825\) 4.19740 0.146135
\(826\) 0 0
\(827\) −2.71432 −0.0943862 −0.0471931 0.998886i \(-0.515028\pi\)
−0.0471931 + 0.998886i \(0.515028\pi\)
\(828\) 6.00404 0.208655
\(829\) 12.5291 0.435152 0.217576 0.976043i \(-0.430185\pi\)
0.217576 + 0.976043i \(0.430185\pi\)
\(830\) 13.9535 0.484332
\(831\) 28.5421 0.990114
\(832\) −4.98496 −0.172822
\(833\) 0 0
\(834\) 12.5882 0.435892
\(835\) 42.6357 1.47547
\(836\) −3.46193 −0.119733
\(837\) 7.28407 0.251774
\(838\) 3.76105 0.129923
\(839\) 5.50546 0.190070 0.0950348 0.995474i \(-0.469704\pi\)
0.0950348 + 0.995474i \(0.469704\pi\)
\(840\) 0 0
\(841\) 1.37732 0.0474939
\(842\) −31.6343 −1.09019
\(843\) 13.6443 0.469934
\(844\) −23.3953 −0.805298
\(845\) 29.5353 1.01605
\(846\) −5.36021 −0.184288
\(847\) 0 0
\(848\) −5.87423 −0.201722
\(849\) 1.53403 0.0526478
\(850\) −2.71951 −0.0932784
\(851\) −55.0382 −1.88668
\(852\) −12.0907 −0.414221
\(853\) −47.3276 −1.62047 −0.810233 0.586108i \(-0.800660\pi\)
−0.810233 + 0.586108i \(0.800660\pi\)
\(854\) 0 0
\(855\) 2.49248 0.0852409
\(856\) 15.7305 0.537657
\(857\) 54.1773 1.85066 0.925331 0.379161i \(-0.123787\pi\)
0.925331 + 0.379161i \(0.123787\pi\)
\(858\) 17.2576 0.589164
\(859\) −51.7842 −1.76685 −0.883426 0.468570i \(-0.844769\pi\)
−0.883426 + 0.468570i \(0.844769\pi\)
\(860\) 23.7835 0.811011
\(861\) 0 0
\(862\) 1.66582 0.0567381
\(863\) −26.6679 −0.907785 −0.453893 0.891056i \(-0.649965\pi\)
−0.453893 + 0.891056i \(0.649965\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.44087 −0.218996
\(866\) −16.1062 −0.547311
\(867\) 11.9690 0.406488
\(868\) 0 0
\(869\) 24.5336 0.832244
\(870\) −13.7374 −0.465743
\(871\) −75.5634 −2.56037
\(872\) −8.31707 −0.281652
\(873\) 7.00404 0.237051
\(874\) −6.00404 −0.203090
\(875\) 0 0
\(876\) −14.4204 −0.487219
\(877\) −11.7318 −0.396155 −0.198078 0.980186i \(-0.563470\pi\)
−0.198078 + 0.980186i \(0.563470\pi\)
\(878\) −21.0867 −0.711641
\(879\) 33.3773 1.12579
\(880\) −8.62878 −0.290876
\(881\) −11.8406 −0.398918 −0.199459 0.979906i \(-0.563919\pi\)
−0.199459 + 0.979906i \(0.563919\pi\)
\(882\) 0 0
\(883\) 35.5167 1.19523 0.597616 0.801782i \(-0.296115\pi\)
0.597616 + 0.801782i \(0.296115\pi\)
\(884\) −11.1812 −0.376065
\(885\) 21.6308 0.727110
\(886\) 18.5958 0.624738
\(887\) −38.9544 −1.30796 −0.653980 0.756512i \(-0.726902\pi\)
−0.653980 + 0.756512i \(0.726902\pi\)
\(888\) −9.16685 −0.307619
\(889\) 0 0
\(890\) −18.7672 −0.629076
\(891\) −3.46193 −0.115979
\(892\) 14.8221 0.496282
\(893\) 5.36021 0.179373
\(894\) 16.2922 0.544891
\(895\) −8.09927 −0.270729
\(896\) 0 0
\(897\) 29.9299 0.999329
\(898\) −19.7410 −0.658767
\(899\) 40.1466 1.33896
\(900\) 1.21245 0.0404149
\(901\) −13.1759 −0.438951
\(902\) 16.0766 0.535292
\(903\) 0 0
\(904\) −13.3643 −0.444488
\(905\) 31.0461 1.03201
\(906\) 19.9735 0.663575
\(907\) 55.1428 1.83099 0.915493 0.402333i \(-0.131801\pi\)
0.915493 + 0.402333i \(0.131801\pi\)
\(908\) 16.2295 0.538594
\(909\) 7.09569 0.235349
\(910\) 0 0
\(911\) 11.9618 0.396313 0.198157 0.980170i \(-0.436505\pi\)
0.198157 + 0.980170i \(0.436505\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 19.3807 0.641407
\(914\) −0.468893 −0.0155096
\(915\) −10.9052 −0.360516
\(916\) −6.39480 −0.211290
\(917\) 0 0
\(918\) 2.24299 0.0740298
\(919\) 36.9365 1.21842 0.609211 0.793008i \(-0.291486\pi\)
0.609211 + 0.793008i \(0.291486\pi\)
\(920\) −14.9649 −0.493379
\(921\) −9.52952 −0.314008
\(922\) 19.0556 0.627564
\(923\) −60.2717 −1.98387
\(924\) 0 0
\(925\) −11.1143 −0.365436
\(926\) 9.65575 0.317308
\(927\) 7.15135 0.234881
\(928\) 5.51156 0.180926
\(929\) −24.2806 −0.796620 −0.398310 0.917251i \(-0.630403\pi\)
−0.398310 + 0.917251i \(0.630403\pi\)
\(930\) −18.1554 −0.595339
\(931\) 0 0
\(932\) −21.3798 −0.700317
\(933\) −16.4404 −0.538234
\(934\) 9.16281 0.299816
\(935\) −19.3543 −0.632954
\(936\) 4.98496 0.162938
\(937\) −24.5823 −0.803068 −0.401534 0.915844i \(-0.631523\pi\)
−0.401534 + 0.915844i \(0.631523\pi\)
\(938\) 0 0
\(939\) −10.9940 −0.358775
\(940\) 13.3602 0.435762
\(941\) −7.35723 −0.239839 −0.119919 0.992784i \(-0.538264\pi\)
−0.119919 + 0.992784i \(0.538264\pi\)
\(942\) −0.739053 −0.0240796
\(943\) 27.8817 0.907954
\(944\) −8.67842 −0.282458
\(945\) 0 0
\(946\) 33.0341 1.07403
\(947\) −29.0462 −0.943876 −0.471938 0.881632i \(-0.656445\pi\)
−0.471938 + 0.881632i \(0.656445\pi\)
\(948\) 7.08667 0.230164
\(949\) −71.8849 −2.33348
\(950\) −1.21245 −0.0393369
\(951\) −6.66291 −0.216060
\(952\) 0 0
\(953\) −5.84772 −0.189426 −0.0947131 0.995505i \(-0.530193\pi\)
−0.0947131 + 0.995505i \(0.530193\pi\)
\(954\) 5.87423 0.190185
\(955\) −40.9874 −1.32632
\(956\) 18.8938 0.611068
\(957\) −19.0806 −0.616790
\(958\) −42.8212 −1.38349
\(959\) 0 0
\(960\) −2.49248 −0.0804444
\(961\) 22.0577 0.711539
\(962\) −45.6964 −1.47331
\(963\) −15.7305 −0.506908
\(964\) 4.05009 0.130445
\(965\) −46.5729 −1.49923
\(966\) 0 0
\(967\) 11.7775 0.378738 0.189369 0.981906i \(-0.439356\pi\)
0.189369 + 0.981906i \(0.439356\pi\)
\(968\) −0.984956 −0.0316577
\(969\) −2.24299 −0.0720553
\(970\) −17.4574 −0.560524
\(971\) −33.4810 −1.07446 −0.537228 0.843437i \(-0.680529\pi\)
−0.537228 + 0.843437i \(0.680529\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 26.3153 0.843196
\(975\) 6.04399 0.193563
\(976\) 4.37526 0.140049
\(977\) −17.3969 −0.556575 −0.278288 0.960498i \(-0.589767\pi\)
−0.278288 + 0.960498i \(0.589767\pi\)
\(978\) −5.79808 −0.185402
\(979\) −26.0667 −0.833094
\(980\) 0 0
\(981\) 8.31707 0.265544
\(982\) 14.2064 0.453345
\(983\) 13.9218 0.444036 0.222018 0.975043i \(-0.428736\pi\)
0.222018 + 0.975043i \(0.428736\pi\)
\(984\) 4.64383 0.148040
\(985\) −38.6247 −1.23069
\(986\) 12.3624 0.393699
\(987\) 0 0
\(988\) −4.98496 −0.158593
\(989\) 57.2912 1.82175
\(990\) 8.62878 0.274241
\(991\) −39.9694 −1.26967 −0.634834 0.772648i \(-0.718932\pi\)
−0.634834 + 0.772648i \(0.718932\pi\)
\(992\) 7.28407 0.231270
\(993\) −25.8102 −0.819062
\(994\) 0 0
\(995\) −3.63772 −0.115324
\(996\) 5.59823 0.177387
\(997\) 35.3277 1.11884 0.559419 0.828885i \(-0.311024\pi\)
0.559419 + 0.828885i \(0.311024\pi\)
\(998\) −34.0325 −1.07728
\(999\) 9.16685 0.290026
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 5586.2.a.bw.1.4 4
7.2 even 3 798.2.j.l.571.1 yes 8
7.4 even 3 798.2.j.l.457.1 8
7.6 odd 2 5586.2.a.bz.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.l.457.1 8 7.4 even 3
798.2.j.l.571.1 yes 8 7.2 even 3
5586.2.a.bw.1.4 4 1.1 even 1 trivial
5586.2.a.bz.1.1 4 7.6 odd 2