Properties

Label 171.2.a.e.1.1
Level $171$
Weight $2$
Character 171.1
Self dual yes
Analytic conductor $1.365$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.04374\) of defining polynomial
Character \(\chi\) \(=\) 171.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.71519 q^{2} +5.37228 q^{4} +3.22060 q^{5} -2.37228 q^{7} -9.15640 q^{8} +O(q^{10})\) \(q-2.71519 q^{2} +5.37228 q^{4} +3.22060 q^{5} -2.37228 q^{7} -9.15640 q^{8} -8.74456 q^{10} +2.20979 q^{11} +2.00000 q^{13} +6.44121 q^{14} +14.1168 q^{16} +3.22060 q^{17} +1.00000 q^{19} +17.3020 q^{20} -6.00000 q^{22} -1.01082 q^{23} +5.37228 q^{25} -5.43039 q^{26} -12.7446 q^{28} -1.01082 q^{29} +4.74456 q^{31} -20.0172 q^{32} -8.74456 q^{34} -7.64018 q^{35} +10.7446 q^{37} -2.71519 q^{38} -29.4891 q^{40} +5.43039 q^{41} -11.1168 q^{43} +11.8716 q^{44} +2.74456 q^{46} +4.23142 q^{47} -1.37228 q^{49} -14.5868 q^{50} +10.7446 q^{52} -9.84996 q^{53} +7.11684 q^{55} +21.7216 q^{56} +2.74456 q^{58} -10.8608 q^{59} -5.11684 q^{61} -12.8824 q^{62} +26.1168 q^{64} +6.44121 q^{65} -4.00000 q^{67} +17.3020 q^{68} +20.7446 q^{70} -2.02163 q^{71} -5.11684 q^{73} -29.1736 q^{74} +5.37228 q^{76} -5.24224 q^{77} -4.00000 q^{79} +45.4647 q^{80} -14.7446 q^{82} +11.8716 q^{83} +10.3723 q^{85} +30.1844 q^{86} -20.2337 q^{88} -9.84996 q^{89} -4.74456 q^{91} -5.43039 q^{92} -11.4891 q^{94} +3.22060 q^{95} +7.48913 q^{97} +3.72601 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55} - 12 q^{58} + 14 q^{61} + 70 q^{64} - 16 q^{67} + 60 q^{70} + 14 q^{73} + 10 q^{76} - 16 q^{79} - 36 q^{82} + 30 q^{85} - 12 q^{88} + 4 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.71519 −1.91993 −0.959966 0.280116i \(-0.909627\pi\)
−0.959966 + 0.280116i \(0.909627\pi\)
\(3\) 0 0
\(4\) 5.37228 2.68614
\(5\) 3.22060 1.44030 0.720149 0.693820i \(-0.244074\pi\)
0.720149 + 0.693820i \(0.244074\pi\)
\(6\) 0 0
\(7\) −2.37228 −0.896638 −0.448319 0.893874i \(-0.647977\pi\)
−0.448319 + 0.893874i \(0.647977\pi\)
\(8\) −9.15640 −3.23728
\(9\) 0 0
\(10\) −8.74456 −2.76527
\(11\) 2.20979 0.666276 0.333138 0.942878i \(-0.391893\pi\)
0.333138 + 0.942878i \(0.391893\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 6.44121 1.72148
\(15\) 0 0
\(16\) 14.1168 3.52921
\(17\) 3.22060 0.781111 0.390555 0.920579i \(-0.372283\pi\)
0.390555 + 0.920579i \(0.372283\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 17.3020 3.86884
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −1.01082 −0.210770 −0.105385 0.994432i \(-0.533607\pi\)
−0.105385 + 0.994432i \(0.533607\pi\)
\(24\) 0 0
\(25\) 5.37228 1.07446
\(26\) −5.43039 −1.06499
\(27\) 0 0
\(28\) −12.7446 −2.40850
\(29\) −1.01082 −0.187704 −0.0938519 0.995586i \(-0.529918\pi\)
−0.0938519 + 0.995586i \(0.529918\pi\)
\(30\) 0 0
\(31\) 4.74456 0.852149 0.426074 0.904688i \(-0.359896\pi\)
0.426074 + 0.904688i \(0.359896\pi\)
\(32\) −20.0172 −3.53857
\(33\) 0 0
\(34\) −8.74456 −1.49968
\(35\) −7.64018 −1.29143
\(36\) 0 0
\(37\) 10.7446 1.76640 0.883198 0.469001i \(-0.155386\pi\)
0.883198 + 0.469001i \(0.155386\pi\)
\(38\) −2.71519 −0.440463
\(39\) 0 0
\(40\) −29.4891 −4.66264
\(41\) 5.43039 0.848084 0.424042 0.905642i \(-0.360611\pi\)
0.424042 + 0.905642i \(0.360611\pi\)
\(42\) 0 0
\(43\) −11.1168 −1.69530 −0.847651 0.530554i \(-0.821984\pi\)
−0.847651 + 0.530554i \(0.821984\pi\)
\(44\) 11.8716 1.78971
\(45\) 0 0
\(46\) 2.74456 0.404664
\(47\) 4.23142 0.617216 0.308608 0.951189i \(-0.400137\pi\)
0.308608 + 0.951189i \(0.400137\pi\)
\(48\) 0 0
\(49\) −1.37228 −0.196040
\(50\) −14.5868 −2.06288
\(51\) 0 0
\(52\) 10.7446 1.49000
\(53\) −9.84996 −1.35300 −0.676498 0.736444i \(-0.736503\pi\)
−0.676498 + 0.736444i \(0.736503\pi\)
\(54\) 0 0
\(55\) 7.11684 0.959635
\(56\) 21.7216 2.90267
\(57\) 0 0
\(58\) 2.74456 0.360379
\(59\) −10.8608 −1.41395 −0.706976 0.707237i \(-0.749941\pi\)
−0.706976 + 0.707237i \(0.749941\pi\)
\(60\) 0 0
\(61\) −5.11684 −0.655145 −0.327572 0.944826i \(-0.606231\pi\)
−0.327572 + 0.944826i \(0.606231\pi\)
\(62\) −12.8824 −1.63607
\(63\) 0 0
\(64\) 26.1168 3.26461
\(65\) 6.44121 0.798933
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 17.3020 2.09817
\(69\) 0 0
\(70\) 20.7446 2.47945
\(71\) −2.02163 −0.239924 −0.119962 0.992779i \(-0.538277\pi\)
−0.119962 + 0.992779i \(0.538277\pi\)
\(72\) 0 0
\(73\) −5.11684 −0.598881 −0.299441 0.954115i \(-0.596800\pi\)
−0.299441 + 0.954115i \(0.596800\pi\)
\(74\) −29.1736 −3.39136
\(75\) 0 0
\(76\) 5.37228 0.616243
\(77\) −5.24224 −0.597408
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 45.4647 5.08311
\(81\) 0 0
\(82\) −14.7446 −1.62826
\(83\) 11.8716 1.30308 0.651538 0.758616i \(-0.274124\pi\)
0.651538 + 0.758616i \(0.274124\pi\)
\(84\) 0 0
\(85\) 10.3723 1.12503
\(86\) 30.1844 3.25487
\(87\) 0 0
\(88\) −20.2337 −2.15692
\(89\) −9.84996 −1.04409 −0.522047 0.852917i \(-0.674831\pi\)
−0.522047 + 0.852917i \(0.674831\pi\)
\(90\) 0 0
\(91\) −4.74456 −0.497365
\(92\) −5.43039 −0.566157
\(93\) 0 0
\(94\) −11.4891 −1.18501
\(95\) 3.22060 0.330427
\(96\) 0 0
\(97\) 7.48913 0.760405 0.380203 0.924903i \(-0.375854\pi\)
0.380203 + 0.924903i \(0.375854\pi\)
\(98\) 3.72601 0.376384
\(99\) 0 0
\(100\) 28.8614 2.88614
\(101\) −12.8824 −1.28185 −0.640924 0.767604i \(-0.721449\pi\)
−0.640924 + 0.767604i \(0.721449\pi\)
\(102\) 0 0
\(103\) −7.25544 −0.714899 −0.357450 0.933932i \(-0.616354\pi\)
−0.357450 + 0.933932i \(0.616354\pi\)
\(104\) −18.3128 −1.79572
\(105\) 0 0
\(106\) 26.7446 2.59766
\(107\) −4.41957 −0.427256 −0.213628 0.976915i \(-0.568528\pi\)
−0.213628 + 0.976915i \(0.568528\pi\)
\(108\) 0 0
\(109\) 5.25544 0.503380 0.251690 0.967808i \(-0.419014\pi\)
0.251690 + 0.967808i \(0.419014\pi\)
\(110\) −19.3236 −1.84243
\(111\) 0 0
\(112\) −33.4891 −3.16442
\(113\) 11.8716 1.11679 0.558393 0.829577i \(-0.311418\pi\)
0.558393 + 0.829577i \(0.311418\pi\)
\(114\) 0 0
\(115\) −3.25544 −0.303571
\(116\) −5.43039 −0.504199
\(117\) 0 0
\(118\) 29.4891 2.71469
\(119\) −7.64018 −0.700374
\(120\) 0 0
\(121\) −6.11684 −0.556077
\(122\) 13.8932 1.25783
\(123\) 0 0
\(124\) 25.4891 2.28899
\(125\) 1.19897 0.107239
\(126\) 0 0
\(127\) −12.7446 −1.13090 −0.565449 0.824784i \(-0.691297\pi\)
−0.565449 + 0.824784i \(0.691297\pi\)
\(128\) −30.8780 −2.72925
\(129\) 0 0
\(130\) −17.4891 −1.53390
\(131\) 15.0922 1.31861 0.659306 0.751875i \(-0.270850\pi\)
0.659306 + 0.751875i \(0.270850\pi\)
\(132\) 0 0
\(133\) −2.37228 −0.205703
\(134\) 10.8608 0.938228
\(135\) 0 0
\(136\) −29.4891 −2.52867
\(137\) −12.0597 −1.03033 −0.515167 0.857090i \(-0.672270\pi\)
−0.515167 + 0.857090i \(0.672270\pi\)
\(138\) 0 0
\(139\) 3.11684 0.264367 0.132184 0.991225i \(-0.457801\pi\)
0.132184 + 0.991225i \(0.457801\pi\)
\(140\) −41.0452 −3.46895
\(141\) 0 0
\(142\) 5.48913 0.460637
\(143\) 4.41957 0.369583
\(144\) 0 0
\(145\) −3.25544 −0.270349
\(146\) 13.8932 1.14981
\(147\) 0 0
\(148\) 57.7228 4.74479
\(149\) −20.5226 −1.68128 −0.840638 0.541598i \(-0.817820\pi\)
−0.840638 + 0.541598i \(0.817820\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −9.15640 −0.742682
\(153\) 0 0
\(154\) 14.2337 1.14698
\(155\) 15.2804 1.22735
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 10.8608 0.864037
\(159\) 0 0
\(160\) −64.4674 −5.09659
\(161\) 2.39794 0.188984
\(162\) 0 0
\(163\) −21.4891 −1.68316 −0.841579 0.540134i \(-0.818374\pi\)
−0.841579 + 0.540134i \(0.818374\pi\)
\(164\) 29.1736 2.27807
\(165\) 0 0
\(166\) −32.2337 −2.50182
\(167\) 6.44121 0.498435 0.249218 0.968447i \(-0.419827\pi\)
0.249218 + 0.968447i \(0.419827\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −28.1628 −2.15999
\(171\) 0 0
\(172\) −59.7228 −4.55382
\(173\) 1.01082 0.0768509 0.0384255 0.999261i \(-0.487766\pi\)
0.0384255 + 0.999261i \(0.487766\pi\)
\(174\) 0 0
\(175\) −12.7446 −0.963398
\(176\) 31.1952 2.35143
\(177\) 0 0
\(178\) 26.7446 2.00459
\(179\) 4.41957 0.330334 0.165167 0.986266i \(-0.447184\pi\)
0.165167 + 0.986266i \(0.447184\pi\)
\(180\) 0 0
\(181\) 19.4891 1.44862 0.724308 0.689477i \(-0.242160\pi\)
0.724308 + 0.689477i \(0.242160\pi\)
\(182\) 12.8824 0.954908
\(183\) 0 0
\(184\) 9.25544 0.682320
\(185\) 34.6040 2.54413
\(186\) 0 0
\(187\) 7.11684 0.520435
\(188\) 22.7324 1.65793
\(189\) 0 0
\(190\) −8.74456 −0.634397
\(191\) 21.5334 1.55810 0.779051 0.626960i \(-0.215701\pi\)
0.779051 + 0.626960i \(0.215701\pi\)
\(192\) 0 0
\(193\) 16.2337 1.16853 0.584263 0.811564i \(-0.301384\pi\)
0.584263 + 0.811564i \(0.301384\pi\)
\(194\) −20.3344 −1.45993
\(195\) 0 0
\(196\) −7.37228 −0.526592
\(197\) 23.7432 1.69163 0.845816 0.533475i \(-0.179114\pi\)
0.845816 + 0.533475i \(0.179114\pi\)
\(198\) 0 0
\(199\) 0.883156 0.0626053 0.0313026 0.999510i \(-0.490034\pi\)
0.0313026 + 0.999510i \(0.490034\pi\)
\(200\) −49.1908 −3.47831
\(201\) 0 0
\(202\) 34.9783 2.46106
\(203\) 2.39794 0.168302
\(204\) 0 0
\(205\) 17.4891 1.22149
\(206\) 19.6999 1.37256
\(207\) 0 0
\(208\) 28.2337 1.95765
\(209\) 2.20979 0.152854
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −52.9168 −3.63434
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −35.8029 −2.44174
\(216\) 0 0
\(217\) −11.2554 −0.764069
\(218\) −14.2695 −0.966455
\(219\) 0 0
\(220\) 38.2337 2.57771
\(221\) 6.44121 0.433282
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 47.4864 3.17282
\(225\) 0 0
\(226\) −32.2337 −2.14415
\(227\) −19.3236 −1.28255 −0.641277 0.767310i \(-0.721595\pi\)
−0.641277 + 0.767310i \(0.721595\pi\)
\(228\) 0 0
\(229\) 15.6277 1.03271 0.516354 0.856375i \(-0.327289\pi\)
0.516354 + 0.856375i \(0.327289\pi\)
\(230\) 8.83915 0.582836
\(231\) 0 0
\(232\) 9.25544 0.607649
\(233\) 7.64018 0.500525 0.250262 0.968178i \(-0.419483\pi\)
0.250262 + 0.968178i \(0.419483\pi\)
\(234\) 0 0
\(235\) 13.6277 0.888974
\(236\) −58.3472 −3.79808
\(237\) 0 0
\(238\) 20.7446 1.34467
\(239\) 12.6943 0.821123 0.410562 0.911833i \(-0.365333\pi\)
0.410562 + 0.911833i \(0.365333\pi\)
\(240\) 0 0
\(241\) −24.2337 −1.56103 −0.780515 0.625138i \(-0.785043\pi\)
−0.780515 + 0.625138i \(0.785043\pi\)
\(242\) 16.6084 1.06763
\(243\) 0 0
\(244\) −27.4891 −1.75981
\(245\) −4.41957 −0.282356
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) −43.4431 −2.75864
\(249\) 0 0
\(250\) −3.25544 −0.205892
\(251\) −23.9313 −1.51053 −0.755266 0.655418i \(-0.772493\pi\)
−0.755266 + 0.655418i \(0.772493\pi\)
\(252\) 0 0
\(253\) −2.23369 −0.140431
\(254\) 34.6040 2.17125
\(255\) 0 0
\(256\) 31.6060 1.97537
\(257\) −5.43039 −0.338738 −0.169369 0.985553i \(-0.554173\pi\)
−0.169369 + 0.985553i \(0.554173\pi\)
\(258\) 0 0
\(259\) −25.4891 −1.58382
\(260\) 34.6040 2.14605
\(261\) 0 0
\(262\) −40.9783 −2.53164
\(263\) −13.0706 −0.805966 −0.402983 0.915208i \(-0.632027\pi\)
−0.402983 + 0.915208i \(0.632027\pi\)
\(264\) 0 0
\(265\) −31.7228 −1.94872
\(266\) 6.44121 0.394936
\(267\) 0 0
\(268\) −21.4891 −1.31266
\(269\) −7.82833 −0.477302 −0.238651 0.971105i \(-0.576705\pi\)
−0.238651 + 0.971105i \(0.576705\pi\)
\(270\) 0 0
\(271\) 25.4891 1.54835 0.774177 0.632969i \(-0.218164\pi\)
0.774177 + 0.632969i \(0.218164\pi\)
\(272\) 45.4647 2.75671
\(273\) 0 0
\(274\) 32.7446 1.97817
\(275\) 11.8716 0.715884
\(276\) 0 0
\(277\) 9.11684 0.547778 0.273889 0.961761i \(-0.411690\pi\)
0.273889 + 0.961761i \(0.411690\pi\)
\(278\) −8.46284 −0.507567
\(279\) 0 0
\(280\) 69.9565 4.18070
\(281\) −24.7540 −1.47670 −0.738350 0.674418i \(-0.764395\pi\)
−0.738350 + 0.674418i \(0.764395\pi\)
\(282\) 0 0
\(283\) 6.37228 0.378793 0.189396 0.981901i \(-0.439347\pi\)
0.189396 + 0.981901i \(0.439347\pi\)
\(284\) −10.8608 −0.644469
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −12.8824 −0.760425
\(288\) 0 0
\(289\) −6.62772 −0.389866
\(290\) 8.83915 0.519053
\(291\) 0 0
\(292\) −27.4891 −1.60868
\(293\) 25.1303 1.46813 0.734064 0.679080i \(-0.237621\pi\)
0.734064 + 0.679080i \(0.237621\pi\)
\(294\) 0 0
\(295\) −34.9783 −2.03651
\(296\) −98.3815 −5.71831
\(297\) 0 0
\(298\) 55.7228 3.22794
\(299\) −2.02163 −0.116914
\(300\) 0 0
\(301\) 26.3723 1.52007
\(302\) 10.8608 0.624968
\(303\) 0 0
\(304\) 14.1168 0.809657
\(305\) −16.4793 −0.943603
\(306\) 0 0
\(307\) 25.4891 1.45474 0.727371 0.686245i \(-0.240742\pi\)
0.727371 + 0.686245i \(0.240742\pi\)
\(308\) −28.1628 −1.60472
\(309\) 0 0
\(310\) −41.4891 −2.35642
\(311\) −12.6943 −0.719825 −0.359913 0.932986i \(-0.617194\pi\)
−0.359913 + 0.932986i \(0.617194\pi\)
\(312\) 0 0
\(313\) 7.48913 0.423310 0.211655 0.977344i \(-0.432115\pi\)
0.211655 + 0.977344i \(0.432115\pi\)
\(314\) −5.43039 −0.306455
\(315\) 0 0
\(316\) −21.4891 −1.20886
\(317\) 12.2479 0.687911 0.343955 0.938986i \(-0.388233\pi\)
0.343955 + 0.938986i \(0.388233\pi\)
\(318\) 0 0
\(319\) −2.23369 −0.125063
\(320\) 84.1120 4.70200
\(321\) 0 0
\(322\) −6.51087 −0.362837
\(323\) 3.22060 0.179199
\(324\) 0 0
\(325\) 10.7446 0.596001
\(326\) 58.3472 3.23155
\(327\) 0 0
\(328\) −49.7228 −2.74548
\(329\) −10.0381 −0.553419
\(330\) 0 0
\(331\) 18.9783 1.04314 0.521569 0.853209i \(-0.325347\pi\)
0.521569 + 0.853209i \(0.325347\pi\)
\(332\) 63.7775 3.50025
\(333\) 0 0
\(334\) −17.4891 −0.956962
\(335\) −12.8824 −0.703841
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 24.4368 1.32918
\(339\) 0 0
\(340\) 55.7228 3.02199
\(341\) 10.4845 0.567766
\(342\) 0 0
\(343\) 19.8614 1.07242
\(344\) 101.790 5.48816
\(345\) 0 0
\(346\) −2.74456 −0.147549
\(347\) −32.3942 −1.73901 −0.869505 0.493924i \(-0.835562\pi\)
−0.869505 + 0.493924i \(0.835562\pi\)
\(348\) 0 0
\(349\) 17.8614 0.956099 0.478050 0.878333i \(-0.341344\pi\)
0.478050 + 0.878333i \(0.341344\pi\)
\(350\) 34.6040 1.84966
\(351\) 0 0
\(352\) −44.2337 −2.35766
\(353\) −14.9040 −0.793262 −0.396631 0.917978i \(-0.629821\pi\)
−0.396631 + 0.917978i \(0.629821\pi\)
\(354\) 0 0
\(355\) −6.51087 −0.345561
\(356\) −52.9168 −2.80458
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 4.60773 0.243187 0.121593 0.992580i \(-0.461200\pi\)
0.121593 + 0.992580i \(0.461200\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −52.9168 −2.78124
\(363\) 0 0
\(364\) −25.4891 −1.33599
\(365\) −16.4793 −0.862567
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −14.2695 −0.743851
\(369\) 0 0
\(370\) −93.9565 −4.88457
\(371\) 23.3669 1.21315
\(372\) 0 0
\(373\) −24.2337 −1.25477 −0.627386 0.778708i \(-0.715875\pi\)
−0.627386 + 0.778708i \(0.715875\pi\)
\(374\) −19.3236 −0.999200
\(375\) 0 0
\(376\) −38.7446 −1.99810
\(377\) −2.02163 −0.104119
\(378\) 0 0
\(379\) 28.7446 1.47651 0.738255 0.674522i \(-0.235650\pi\)
0.738255 + 0.674522i \(0.235650\pi\)
\(380\) 17.3020 0.887573
\(381\) 0 0
\(382\) −58.4674 −2.99145
\(383\) 17.3020 0.884090 0.442045 0.896993i \(-0.354253\pi\)
0.442045 + 0.896993i \(0.354253\pi\)
\(384\) 0 0
\(385\) −16.8832 −0.860445
\(386\) −44.0776 −2.24349
\(387\) 0 0
\(388\) 40.2337 2.04256
\(389\) −12.0597 −0.611454 −0.305727 0.952119i \(-0.598899\pi\)
−0.305727 + 0.952119i \(0.598899\pi\)
\(390\) 0 0
\(391\) −3.25544 −0.164635
\(392\) 12.5652 0.634636
\(393\) 0 0
\(394\) −64.4674 −3.24782
\(395\) −12.8824 −0.648184
\(396\) 0 0
\(397\) −17.1168 −0.859070 −0.429535 0.903050i \(-0.641322\pi\)
−0.429535 + 0.903050i \(0.641322\pi\)
\(398\) −2.39794 −0.120198
\(399\) 0 0
\(400\) 75.8397 3.79198
\(401\) −3.40876 −0.170225 −0.0851126 0.996371i \(-0.527125\pi\)
−0.0851126 + 0.996371i \(0.527125\pi\)
\(402\) 0 0
\(403\) 9.48913 0.472687
\(404\) −69.2079 −3.44322
\(405\) 0 0
\(406\) −6.51087 −0.323129
\(407\) 23.7432 1.17691
\(408\) 0 0
\(409\) 7.48913 0.370313 0.185157 0.982709i \(-0.440721\pi\)
0.185157 + 0.982709i \(0.440721\pi\)
\(410\) −47.4864 −2.34519
\(411\) 0 0
\(412\) −38.9783 −1.92032
\(413\) 25.7648 1.26780
\(414\) 0 0
\(415\) 38.2337 1.87682
\(416\) −40.0344 −1.96285
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) −11.8716 −0.579965 −0.289983 0.957032i \(-0.593650\pi\)
−0.289983 + 0.957032i \(0.593650\pi\)
\(420\) 0 0
\(421\) −8.97825 −0.437573 −0.218787 0.975773i \(-0.570210\pi\)
−0.218787 + 0.975773i \(0.570210\pi\)
\(422\) 10.8608 0.528694
\(423\) 0 0
\(424\) 90.1902 4.38002
\(425\) 17.3020 0.839269
\(426\) 0 0
\(427\) 12.1386 0.587428
\(428\) −23.7432 −1.14767
\(429\) 0 0
\(430\) 97.2119 4.68798
\(431\) 30.1844 1.45393 0.726966 0.686674i \(-0.240930\pi\)
0.726966 + 0.686674i \(0.240930\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 30.5607 1.46696
\(435\) 0 0
\(436\) 28.2337 1.35215
\(437\) −1.01082 −0.0483539
\(438\) 0 0
\(439\) −18.2337 −0.870246 −0.435123 0.900371i \(-0.643295\pi\)
−0.435123 + 0.900371i \(0.643295\pi\)
\(440\) −65.1647 −3.10660
\(441\) 0 0
\(442\) −17.4891 −0.831873
\(443\) 27.9746 1.32911 0.664557 0.747238i \(-0.268620\pi\)
0.664557 + 0.747238i \(0.268620\pi\)
\(444\) 0 0
\(445\) −31.7228 −1.50381
\(446\) 10.8608 0.514273
\(447\) 0 0
\(448\) −61.9565 −2.92717
\(449\) −22.3561 −1.05505 −0.527524 0.849540i \(-0.676880\pi\)
−0.527524 + 0.849540i \(0.676880\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 63.7775 2.99984
\(453\) 0 0
\(454\) 52.4674 2.46242
\(455\) −15.2804 −0.716354
\(456\) 0 0
\(457\) −8.37228 −0.391639 −0.195819 0.980640i \(-0.562737\pi\)
−0.195819 + 0.980640i \(0.562737\pi\)
\(458\) −42.4323 −1.98273
\(459\) 0 0
\(460\) −17.4891 −0.815435
\(461\) 26.9638 1.25583 0.627914 0.778282i \(-0.283909\pi\)
0.627914 + 0.778282i \(0.283909\pi\)
\(462\) 0 0
\(463\) −2.37228 −0.110249 −0.0551246 0.998479i \(-0.517556\pi\)
−0.0551246 + 0.998479i \(0.517556\pi\)
\(464\) −14.2695 −0.662447
\(465\) 0 0
\(466\) −20.7446 −0.960973
\(467\) 36.4374 1.68612 0.843062 0.537816i \(-0.180751\pi\)
0.843062 + 0.537816i \(0.180751\pi\)
\(468\) 0 0
\(469\) 9.48913 0.438167
\(470\) −37.0019 −1.70677
\(471\) 0 0
\(472\) 99.4456 4.57736
\(473\) −24.5659 −1.12954
\(474\) 0 0
\(475\) 5.37228 0.246497
\(476\) −41.0452 −1.88130
\(477\) 0 0
\(478\) −34.4674 −1.57650
\(479\) −33.5932 −1.53491 −0.767455 0.641103i \(-0.778477\pi\)
−0.767455 + 0.641103i \(0.778477\pi\)
\(480\) 0 0
\(481\) 21.4891 0.979820
\(482\) 65.7992 2.99707
\(483\) 0 0
\(484\) −32.8614 −1.49370
\(485\) 24.1195 1.09521
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 46.8519 2.12088
\(489\) 0 0
\(490\) 12.0000 0.542105
\(491\) 1.01082 0.0456175 0.0228087 0.999740i \(-0.492739\pi\)
0.0228087 + 0.999740i \(0.492739\pi\)
\(492\) 0 0
\(493\) −3.25544 −0.146618
\(494\) −5.43039 −0.244325
\(495\) 0 0
\(496\) 66.9783 3.00741
\(497\) 4.79588 0.215125
\(498\) 0 0
\(499\) −11.1168 −0.497658 −0.248829 0.968547i \(-0.580046\pi\)
−0.248829 + 0.968547i \(0.580046\pi\)
\(500\) 6.44121 0.288059
\(501\) 0 0
\(502\) 64.9783 2.90012
\(503\) 31.5715 1.40770 0.703852 0.710346i \(-0.251462\pi\)
0.703852 + 0.710346i \(0.251462\pi\)
\(504\) 0 0
\(505\) −41.4891 −1.84624
\(506\) 6.06490 0.269618
\(507\) 0 0
\(508\) −68.4674 −3.03775
\(509\) −24.7540 −1.09720 −0.548601 0.836084i \(-0.684839\pi\)
−0.548601 + 0.836084i \(0.684839\pi\)
\(510\) 0 0
\(511\) 12.1386 0.536980
\(512\) −24.0604 −1.06333
\(513\) 0 0
\(514\) 14.7446 0.650355
\(515\) −23.3669 −1.02967
\(516\) 0 0
\(517\) 9.35053 0.411236
\(518\) 69.2079 3.04082
\(519\) 0 0
\(520\) −58.9783 −2.58637
\(521\) −16.2912 −0.713729 −0.356864 0.934156i \(-0.616154\pi\)
−0.356864 + 0.934156i \(0.616154\pi\)
\(522\) 0 0
\(523\) −18.2337 −0.797304 −0.398652 0.917102i \(-0.630522\pi\)
−0.398652 + 0.917102i \(0.630522\pi\)
\(524\) 81.0795 3.54198
\(525\) 0 0
\(526\) 35.4891 1.54740
\(527\) 15.2804 0.665623
\(528\) 0 0
\(529\) −21.9783 −0.955576
\(530\) 86.1336 3.74140
\(531\) 0 0
\(532\) −12.7446 −0.552547
\(533\) 10.8608 0.470433
\(534\) 0 0
\(535\) −14.2337 −0.615376
\(536\) 36.6256 1.58198
\(537\) 0 0
\(538\) 21.2554 0.916387
\(539\) −3.03245 −0.130617
\(540\) 0 0
\(541\) 21.1168 0.907884 0.453942 0.891031i \(-0.350017\pi\)
0.453942 + 0.891031i \(0.350017\pi\)
\(542\) −69.2079 −2.97274
\(543\) 0 0
\(544\) −64.4674 −2.76402
\(545\) 16.9257 0.725016
\(546\) 0 0
\(547\) 22.2337 0.950644 0.475322 0.879812i \(-0.342332\pi\)
0.475322 + 0.879812i \(0.342332\pi\)
\(548\) −64.7884 −2.76762
\(549\) 0 0
\(550\) −32.2337 −1.37445
\(551\) −1.01082 −0.0430622
\(552\) 0 0
\(553\) 9.48913 0.403519
\(554\) −24.7540 −1.05170
\(555\) 0 0
\(556\) 16.7446 0.710128
\(557\) 20.5226 0.869570 0.434785 0.900534i \(-0.356824\pi\)
0.434785 + 0.900534i \(0.356824\pi\)
\(558\) 0 0
\(559\) −22.2337 −0.940385
\(560\) −107.855 −4.55771
\(561\) 0 0
\(562\) 67.2119 2.83516
\(563\) 38.6472 1.62879 0.814393 0.580313i \(-0.197070\pi\)
0.814393 + 0.580313i \(0.197070\pi\)
\(564\) 0 0
\(565\) 38.2337 1.60850
\(566\) −17.3020 −0.727257
\(567\) 0 0
\(568\) 18.5109 0.776699
\(569\) −3.03245 −0.127127 −0.0635634 0.997978i \(-0.520247\pi\)
−0.0635634 + 0.997978i \(0.520247\pi\)
\(570\) 0 0
\(571\) 2.51087 0.105077 0.0525384 0.998619i \(-0.483269\pi\)
0.0525384 + 0.998619i \(0.483269\pi\)
\(572\) 23.7432 0.992753
\(573\) 0 0
\(574\) 34.9783 1.45996
\(575\) −5.43039 −0.226463
\(576\) 0 0
\(577\) −41.1168 −1.71172 −0.855858 0.517210i \(-0.826970\pi\)
−0.855858 + 0.517210i \(0.826970\pi\)
\(578\) 17.9955 0.748516
\(579\) 0 0
\(580\) −17.4891 −0.726196
\(581\) −28.1628 −1.16839
\(582\) 0 0
\(583\) −21.7663 −0.901469
\(584\) 46.8519 1.93874
\(585\) 0 0
\(586\) −68.2337 −2.81871
\(587\) 1.83348 0.0756758 0.0378379 0.999284i \(-0.487953\pi\)
0.0378379 + 0.999284i \(0.487953\pi\)
\(588\) 0 0
\(589\) 4.74456 0.195496
\(590\) 94.9728 3.90997
\(591\) 0 0
\(592\) 151.679 6.23398
\(593\) −4.04326 −0.166037 −0.0830185 0.996548i \(-0.526456\pi\)
−0.0830185 + 0.996548i \(0.526456\pi\)
\(594\) 0 0
\(595\) −24.6060 −1.00875
\(596\) −110.253 −4.51614
\(597\) 0 0
\(598\) 5.48913 0.224467
\(599\) −12.8824 −0.526361 −0.263181 0.964747i \(-0.584771\pi\)
−0.263181 + 0.964747i \(0.584771\pi\)
\(600\) 0 0
\(601\) −25.2554 −1.03019 −0.515095 0.857133i \(-0.672244\pi\)
−0.515095 + 0.857133i \(0.672244\pi\)
\(602\) −71.6059 −2.91844
\(603\) 0 0
\(604\) −21.4891 −0.874380
\(605\) −19.6999 −0.800916
\(606\) 0 0
\(607\) 28.7446 1.16671 0.583353 0.812219i \(-0.301740\pi\)
0.583353 + 0.812219i \(0.301740\pi\)
\(608\) −20.0172 −0.811804
\(609\) 0 0
\(610\) 44.7446 1.81165
\(611\) 8.46284 0.342370
\(612\) 0 0
\(613\) 2.60597 0.105254 0.0526271 0.998614i \(-0.483241\pi\)
0.0526271 + 0.998614i \(0.483241\pi\)
\(614\) −69.2079 −2.79300
\(615\) 0 0
\(616\) 48.0000 1.93398
\(617\) −3.22060 −0.129657 −0.0648283 0.997896i \(-0.520650\pi\)
−0.0648283 + 0.997896i \(0.520650\pi\)
\(618\) 0 0
\(619\) 6.97825 0.280480 0.140240 0.990118i \(-0.455213\pi\)
0.140240 + 0.990118i \(0.455213\pi\)
\(620\) 82.0903 3.29683
\(621\) 0 0
\(622\) 34.4674 1.38202
\(623\) 23.3669 0.936174
\(624\) 0 0
\(625\) −23.0000 −0.920000
\(626\) −20.3344 −0.812727
\(627\) 0 0
\(628\) 10.7446 0.428755
\(629\) 34.6040 1.37975
\(630\) 0 0
\(631\) 15.1168 0.601792 0.300896 0.953657i \(-0.402714\pi\)
0.300896 + 0.953657i \(0.402714\pi\)
\(632\) 36.6256 1.45689
\(633\) 0 0
\(634\) −33.2554 −1.32074
\(635\) −41.0452 −1.62883
\(636\) 0 0
\(637\) −2.74456 −0.108744
\(638\) 6.06490 0.240112
\(639\) 0 0
\(640\) −99.4456 −3.93093
\(641\) 24.7540 0.977724 0.488862 0.872361i \(-0.337412\pi\)
0.488862 + 0.872361i \(0.337412\pi\)
\(642\) 0 0
\(643\) −35.1168 −1.38487 −0.692437 0.721479i \(-0.743463\pi\)
−0.692437 + 0.721479i \(0.743463\pi\)
\(644\) 12.8824 0.507638
\(645\) 0 0
\(646\) −8.74456 −0.344050
\(647\) 17.1138 0.672814 0.336407 0.941717i \(-0.390788\pi\)
0.336407 + 0.941717i \(0.390788\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) −29.1736 −1.14428
\(651\) 0 0
\(652\) −115.446 −4.52120
\(653\) −14.0814 −0.551047 −0.275524 0.961294i \(-0.588851\pi\)
−0.275524 + 0.961294i \(0.588851\pi\)
\(654\) 0 0
\(655\) 48.6060 1.89919
\(656\) 76.6600 2.99307
\(657\) 0 0
\(658\) 27.2554 1.06253
\(659\) −11.2371 −0.437735 −0.218867 0.975755i \(-0.570236\pi\)
−0.218867 + 0.975755i \(0.570236\pi\)
\(660\) 0 0
\(661\) −6.74456 −0.262333 −0.131167 0.991360i \(-0.541872\pi\)
−0.131167 + 0.991360i \(0.541872\pi\)
\(662\) −51.5296 −2.00276
\(663\) 0 0
\(664\) −108.701 −4.21842
\(665\) −7.64018 −0.296273
\(666\) 0 0
\(667\) 1.02175 0.0395623
\(668\) 34.6040 1.33887
\(669\) 0 0
\(670\) 34.9783 1.35133
\(671\) −11.3071 −0.436507
\(672\) 0 0
\(673\) −25.2554 −0.973526 −0.486763 0.873534i \(-0.661822\pi\)
−0.486763 + 0.873534i \(0.661822\pi\)
\(674\) −38.0127 −1.46420
\(675\) 0 0
\(676\) −48.3505 −1.85964
\(677\) −44.4539 −1.70850 −0.854252 0.519860i \(-0.825984\pi\)
−0.854252 + 0.519860i \(0.825984\pi\)
\(678\) 0 0
\(679\) −17.7663 −0.681808
\(680\) −94.9728 −3.64204
\(681\) 0 0
\(682\) −28.4674 −1.09007
\(683\) −34.2277 −1.30968 −0.654842 0.755765i \(-0.727265\pi\)
−0.654842 + 0.755765i \(0.727265\pi\)
\(684\) 0 0
\(685\) −38.8397 −1.48399
\(686\) −53.9276 −2.05896
\(687\) 0 0
\(688\) −156.935 −5.98308
\(689\) −19.6999 −0.750507
\(690\) 0 0
\(691\) 27.1168 1.03157 0.515787 0.856717i \(-0.327500\pi\)
0.515787 + 0.856717i \(0.327500\pi\)
\(692\) 5.43039 0.206432
\(693\) 0 0
\(694\) 87.9565 3.33878
\(695\) 10.0381 0.380767
\(696\) 0 0
\(697\) 17.4891 0.662448
\(698\) −48.4972 −1.83565
\(699\) 0 0
\(700\) −68.4674 −2.58782
\(701\) 30.5607 1.15426 0.577131 0.816652i \(-0.304172\pi\)
0.577131 + 0.816652i \(0.304172\pi\)
\(702\) 0 0
\(703\) 10.7446 0.405239
\(704\) 57.7126 2.17513
\(705\) 0 0
\(706\) 40.4674 1.52301
\(707\) 30.5607 1.14935
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 17.6783 0.663454
\(711\) 0 0
\(712\) 90.1902 3.38002
\(713\) −4.79588 −0.179607
\(714\) 0 0
\(715\) 14.2337 0.532310
\(716\) 23.7432 0.887325
\(717\) 0 0
\(718\) −12.5109 −0.466902
\(719\) 38.8354 1.44832 0.724158 0.689634i \(-0.242229\pi\)
0.724158 + 0.689634i \(0.242229\pi\)
\(720\) 0 0
\(721\) 17.2119 0.641006
\(722\) −2.71519 −0.101049
\(723\) 0 0
\(724\) 104.701 3.89118
\(725\) −5.43039 −0.201680
\(726\) 0 0
\(727\) −37.3505 −1.38525 −0.692627 0.721296i \(-0.743547\pi\)
−0.692627 + 0.721296i \(0.743547\pi\)
\(728\) 43.4431 1.61011
\(729\) 0 0
\(730\) 44.7446 1.65607
\(731\) −35.8029 −1.32422
\(732\) 0 0
\(733\) 18.4674 0.682108 0.341054 0.940044i \(-0.389216\pi\)
0.341054 + 0.940044i \(0.389216\pi\)
\(734\) −21.7216 −0.801757
\(735\) 0 0
\(736\) 20.2337 0.745824
\(737\) −8.83915 −0.325594
\(738\) 0 0
\(739\) 27.1168 0.997509 0.498755 0.866743i \(-0.333791\pi\)
0.498755 + 0.866743i \(0.333791\pi\)
\(740\) 185.902 6.83390
\(741\) 0 0
\(742\) −63.4456 −2.32916
\(743\) −28.5391 −1.04700 −0.523498 0.852027i \(-0.675373\pi\)
−0.523498 + 0.852027i \(0.675373\pi\)
\(744\) 0 0
\(745\) −66.0951 −2.42154
\(746\) 65.7992 2.40908
\(747\) 0 0
\(748\) 38.2337 1.39796
\(749\) 10.4845 0.383094
\(750\) 0 0
\(751\) 24.4674 0.892827 0.446414 0.894827i \(-0.352701\pi\)
0.446414 + 0.894827i \(0.352701\pi\)
\(752\) 59.7343 2.17829
\(753\) 0 0
\(754\) 5.48913 0.199902
\(755\) −12.8824 −0.468839
\(756\) 0 0
\(757\) −5.11684 −0.185975 −0.0929874 0.995667i \(-0.529642\pi\)
−0.0929874 + 0.995667i \(0.529642\pi\)
\(758\) −78.0471 −2.83480
\(759\) 0 0
\(760\) −29.4891 −1.06968
\(761\) −0.822662 −0.0298215 −0.0149107 0.999889i \(-0.504746\pi\)
−0.0149107 + 0.999889i \(0.504746\pi\)
\(762\) 0 0
\(763\) −12.4674 −0.451349
\(764\) 115.683 4.18528
\(765\) 0 0
\(766\) −46.9783 −1.69739
\(767\) −21.7216 −0.784320
\(768\) 0 0
\(769\) 6.88316 0.248213 0.124106 0.992269i \(-0.460394\pi\)
0.124106 + 0.992269i \(0.460394\pi\)
\(770\) 45.8411 1.65200
\(771\) 0 0
\(772\) 87.2119 3.13883
\(773\) −5.43039 −0.195318 −0.0976588 0.995220i \(-0.531135\pi\)
−0.0976588 + 0.995220i \(0.531135\pi\)
\(774\) 0 0
\(775\) 25.4891 0.915596
\(776\) −68.5734 −2.46164
\(777\) 0 0
\(778\) 32.7446 1.17395
\(779\) 5.43039 0.194564
\(780\) 0 0
\(781\) −4.46738 −0.159855
\(782\) 8.83915 0.316087
\(783\) 0 0
\(784\) −19.3723 −0.691867
\(785\) 6.44121 0.229896
\(786\) 0 0
\(787\) −49.9565 −1.78076 −0.890378 0.455221i \(-0.849560\pi\)
−0.890378 + 0.455221i \(0.849560\pi\)
\(788\) 127.555 4.54396
\(789\) 0 0
\(790\) 34.9783 1.24447
\(791\) −28.1628 −1.00135
\(792\) 0 0
\(793\) −10.2337 −0.363409
\(794\) 46.4756 1.64936
\(795\) 0 0
\(796\) 4.74456 0.168167
\(797\) 7.45202 0.263964 0.131982 0.991252i \(-0.457866\pi\)
0.131982 + 0.991252i \(0.457866\pi\)
\(798\) 0 0
\(799\) 13.6277 0.482114
\(800\) −107.538 −3.80204
\(801\) 0 0
\(802\) 9.25544 0.326821
\(803\) −11.3071 −0.399020
\(804\) 0 0
\(805\) 7.72281 0.272193
\(806\) −25.7648 −0.907527
\(807\) 0 0
\(808\) 117.957 4.14970
\(809\) 3.59691 0.126461 0.0632303 0.997999i \(-0.479860\pi\)
0.0632303 + 0.997999i \(0.479860\pi\)
\(810\) 0 0
\(811\) −36.7446 −1.29028 −0.645138 0.764066i \(-0.723200\pi\)
−0.645138 + 0.764066i \(0.723200\pi\)
\(812\) 12.8824 0.452084
\(813\) 0 0
\(814\) −64.4674 −2.25958
\(815\) −69.2079 −2.42425
\(816\) 0 0
\(817\) −11.1168 −0.388929
\(818\) −20.3344 −0.710977
\(819\) 0 0
\(820\) 93.9565 3.28110
\(821\) 29.3617 1.02473 0.512366 0.858767i \(-0.328769\pi\)
0.512366 + 0.858767i \(0.328769\pi\)
\(822\) 0 0
\(823\) 8.60597 0.299985 0.149993 0.988687i \(-0.452075\pi\)
0.149993 + 0.988687i \(0.452075\pi\)
\(824\) 66.4337 2.31433
\(825\) 0 0
\(826\) −69.9565 −2.43410
\(827\) 28.5391 0.992401 0.496200 0.868208i \(-0.334728\pi\)
0.496200 + 0.868208i \(0.334728\pi\)
\(828\) 0 0
\(829\) −1.25544 −0.0436031 −0.0218016 0.999762i \(-0.506940\pi\)
−0.0218016 + 0.999762i \(0.506940\pi\)
\(830\) −103.812 −3.60336
\(831\) 0 0
\(832\) 52.2337 1.81088
\(833\) −4.41957 −0.153129
\(834\) 0 0
\(835\) 20.7446 0.717895
\(836\) 11.8716 0.410588
\(837\) 0 0
\(838\) 32.2337 1.11349
\(839\) −30.1844 −1.04208 −0.521041 0.853532i \(-0.674456\pi\)
−0.521041 + 0.853532i \(0.674456\pi\)
\(840\) 0 0
\(841\) −27.9783 −0.964767
\(842\) 24.3777 0.840111
\(843\) 0 0
\(844\) −21.4891 −0.739686
\(845\) −28.9854 −0.997129
\(846\) 0 0
\(847\) 14.5109 0.498600
\(848\) −139.050 −4.77501
\(849\) 0 0
\(850\) −46.9783 −1.61134
\(851\) −10.8608 −0.372303
\(852\) 0 0
\(853\) −38.4674 −1.31710 −0.658549 0.752538i \(-0.728829\pi\)
−0.658549 + 0.752538i \(0.728829\pi\)
\(854\) −32.9586 −1.12782
\(855\) 0 0
\(856\) 40.4674 1.38315
\(857\) 35.2385 1.20372 0.601862 0.798600i \(-0.294426\pi\)
0.601862 + 0.798600i \(0.294426\pi\)
\(858\) 0 0
\(859\) 3.11684 0.106345 0.0531727 0.998585i \(-0.483067\pi\)
0.0531727 + 0.998585i \(0.483067\pi\)
\(860\) −192.343 −6.55886
\(861\) 0 0
\(862\) −81.9565 −2.79145
\(863\) 14.9040 0.507340 0.253670 0.967291i \(-0.418362\pi\)
0.253670 + 0.967291i \(0.418362\pi\)
\(864\) 0 0
\(865\) 3.25544 0.110688
\(866\) 59.7343 2.02985
\(867\) 0 0
\(868\) −60.4674 −2.05240
\(869\) −8.83915 −0.299847
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −48.1209 −1.62958
\(873\) 0 0
\(874\) 2.74456 0.0928362
\(875\) −2.84429 −0.0961547
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 49.5080 1.67081
\(879\) 0 0
\(880\) 100.467 3.38675
\(881\) 0.822662 0.0277162 0.0138581 0.999904i \(-0.495589\pi\)
0.0138581 + 0.999904i \(0.495589\pi\)
\(882\) 0 0
\(883\) 3.11684 0.104890 0.0524451 0.998624i \(-0.483299\pi\)
0.0524451 + 0.998624i \(0.483299\pi\)
\(884\) 34.6040 1.16386
\(885\) 0 0
\(886\) −75.9565 −2.55181
\(887\) −21.7216 −0.729338 −0.364669 0.931137i \(-0.618818\pi\)
−0.364669 + 0.931137i \(0.618818\pi\)
\(888\) 0 0
\(889\) 30.2337 1.01401
\(890\) 86.1336 2.88721
\(891\) 0 0
\(892\) −21.4891 −0.719509
\(893\) 4.23142 0.141599
\(894\) 0 0
\(895\) 14.2337 0.475780
\(896\) 73.2512 2.44715
\(897\) 0 0
\(898\) 60.7011 2.02562
\(899\) −4.79588 −0.159952
\(900\) 0 0
\(901\) −31.7228 −1.05684
\(902\) −32.5823 −1.08487
\(903\) 0 0
\(904\) −108.701 −3.61534
\(905\) 62.7667 2.08644
\(906\) 0 0
\(907\) 23.2554 0.772184 0.386092 0.922460i \(-0.373825\pi\)
0.386092 + 0.922460i \(0.373825\pi\)
\(908\) −103.812 −3.44512
\(909\) 0 0
\(910\) 41.4891 1.37535
\(911\) 37.0019 1.22593 0.612964 0.790111i \(-0.289977\pi\)
0.612964 + 0.790111i \(0.289977\pi\)
\(912\) 0 0
\(913\) 26.2337 0.868208
\(914\) 22.7324 0.751920
\(915\) 0 0
\(916\) 83.9565 2.77400
\(917\) −35.8029 −1.18232
\(918\) 0 0
\(919\) 18.9783 0.626035 0.313017 0.949747i \(-0.398660\pi\)
0.313017 + 0.949747i \(0.398660\pi\)
\(920\) 29.8081 0.982743
\(921\) 0 0
\(922\) −73.2119 −2.41111
\(923\) −4.04326 −0.133086
\(924\) 0 0
\(925\) 57.7228 1.89791
\(926\) 6.44121 0.211671
\(927\) 0 0
\(928\) 20.2337 0.664203
\(929\) −4.79588 −0.157348 −0.0786739 0.996900i \(-0.525069\pi\)
−0.0786739 + 0.996900i \(0.525069\pi\)
\(930\) 0 0
\(931\) −1.37228 −0.0449747
\(932\) 41.0452 1.34448
\(933\) 0 0
\(934\) −98.9348 −3.23724
\(935\) 22.9205 0.749581
\(936\) 0 0
\(937\) −5.11684 −0.167160 −0.0835800 0.996501i \(-0.526635\pi\)
−0.0835800 + 0.996501i \(0.526635\pi\)
\(938\) −25.7648 −0.841251
\(939\) 0 0
\(940\) 73.2119 2.38791
\(941\) −24.7540 −0.806957 −0.403479 0.914989i \(-0.632199\pi\)
−0.403479 + 0.914989i \(0.632199\pi\)
\(942\) 0 0
\(943\) −5.48913 −0.178751
\(944\) −153.320 −4.99014
\(945\) 0 0
\(946\) 66.7011 2.16864
\(947\) 9.84996 0.320081 0.160040 0.987110i \(-0.448838\pi\)
0.160040 + 0.987110i \(0.448838\pi\)
\(948\) 0 0
\(949\) −10.2337 −0.332200
\(950\) −14.5868 −0.473258
\(951\) 0 0
\(952\) 69.9565 2.26730
\(953\) −9.47365 −0.306882 −0.153441 0.988158i \(-0.549035\pi\)
−0.153441 + 0.988158i \(0.549035\pi\)
\(954\) 0 0
\(955\) 69.3505 2.24413
\(956\) 68.1971 2.20565
\(957\) 0 0
\(958\) 91.2119 2.94692
\(959\) 28.6091 0.923837
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) −58.3472 −1.88119
\(963\) 0 0
\(964\) −130.190 −4.19314
\(965\) 52.2823 1.68303
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 56.0083 1.80017
\(969\) 0 0
\(970\) −65.4891 −2.10273
\(971\) 51.5296 1.65366 0.826832 0.562448i \(-0.190140\pi\)
0.826832 + 0.562448i \(0.190140\pi\)
\(972\) 0 0
\(973\) −7.39403 −0.237042
\(974\) −21.7216 −0.696004
\(975\) 0 0
\(976\) −72.2337 −2.31214
\(977\) 20.7107 0.662595 0.331298 0.943526i \(-0.392514\pi\)
0.331298 + 0.943526i \(0.392514\pi\)
\(978\) 0 0
\(979\) −21.7663 −0.695654
\(980\) −23.7432 −0.758448
\(981\) 0 0
\(982\) −2.74456 −0.0875825
\(983\) −52.2823 −1.66755 −0.833773 0.552108i \(-0.813824\pi\)
−0.833773 + 0.552108i \(0.813824\pi\)
\(984\) 0 0
\(985\) 76.4674 2.43645
\(986\) 8.83915 0.281496
\(987\) 0 0
\(988\) 10.7446 0.341830
\(989\) 11.2371 0.357319
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −94.9728 −3.01539
\(993\) 0 0
\(994\) −13.0217 −0.413025
\(995\) 2.84429 0.0901702
\(996\) 0 0
\(997\) −19.3505 −0.612837 −0.306419 0.951897i \(-0.599131\pi\)
−0.306419 + 0.951897i \(0.599131\pi\)
\(998\) 30.1844 0.955470
\(999\) 0 0
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 171.2.a.e.1.1 4
3.2 odd 2 inner 171.2.a.e.1.4 yes 4
4.3 odd 2 2736.2.a.bf.1.4 4
5.4 even 2 4275.2.a.bp.1.4 4
7.6 odd 2 8379.2.a.bw.1.1 4
12.11 even 2 2736.2.a.bf.1.1 4
15.14 odd 2 4275.2.a.bp.1.1 4
19.18 odd 2 3249.2.a.bf.1.4 4
21.20 even 2 8379.2.a.bw.1.4 4
57.56 even 2 3249.2.a.bf.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.1 4 1.1 even 1 trivial
171.2.a.e.1.4 yes 4 3.2 odd 2 inner
2736.2.a.bf.1.1 4 12.11 even 2
2736.2.a.bf.1.4 4 4.3 odd 2
3249.2.a.bf.1.1 4 57.56 even 2
3249.2.a.bf.1.4 4 19.18 odd 2
4275.2.a.bp.1.1 4 15.14 odd 2
4275.2.a.bp.1.4 4 5.4 even 2
8379.2.a.bw.1.1 4 7.6 odd 2
8379.2.a.bw.1.4 4 21.20 even 2