Properties

Label 5550.2.a.cj.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 3 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.67673\) of defining polynomial
Character \(\chi\) \(=\) 5550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.13277 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -5.13277 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.22069 q^{11} -1.00000 q^{12} +6.60629 q^{13} +5.13277 q^{14} +1.00000 q^{16} -5.63841 q^{17} -1.00000 q^{18} -7.51836 q^{19} +5.13277 q^{21} -2.22069 q^{22} +6.10064 q^{23} +1.00000 q^{24} -6.60629 q^{26} -1.00000 q^{27} -5.13277 q^{28} -7.29767 q^{29} -0.967873 q^{31} -1.00000 q^{32} -2.22069 q^{33} +5.63841 q^{34} +1.00000 q^{36} +1.00000 q^{37} +7.51836 q^{38} -6.60629 q^{39} -1.03213 q^{41} -5.13277 q^{42} -5.29767 q^{43} +2.22069 q^{44} -6.10064 q^{46} +5.85911 q^{47} -1.00000 q^{48} +19.3453 q^{49} +5.63841 q^{51} +6.60629 q^{52} +9.63841 q^{53} +1.00000 q^{54} +5.13277 q^{56} +7.51836 q^{57} +7.29767 q^{58} -10.2655 q^{59} +6.32134 q^{61} +0.967873 q^{62} -5.13277 q^{63} +1.00000 q^{64} +2.22069 q^{66} -1.49436 q^{67} -5.63841 q^{68} -6.10064 q^{69} +0.329796 q^{71} -1.00000 q^{72} -12.8718 q^{73} -1.00000 q^{74} -7.51836 q^{76} -11.3983 q^{77} +6.60629 q^{78} +4.00000 q^{79} +1.00000 q^{81} +1.03213 q^{82} -9.04767 q^{83} +5.13277 q^{84} +5.29767 q^{86} +7.29767 q^{87} -2.22069 q^{88} -2.27649 q^{89} -33.9086 q^{91} +6.10064 q^{92} +0.967873 q^{93} -5.85911 q^{94} +1.00000 q^{96} -17.5982 q^{97} -19.3453 q^{98} +2.22069 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{7} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{7} - 4q^{8} + 4q^{9} + 2q^{11} - 4q^{12} + 3q^{13} + 4q^{14} + 4q^{16} - 6q^{17} - 4q^{18} + 3q^{19} + 4q^{21} - 2q^{22} + q^{23} + 4q^{24} - 3q^{26} - 4q^{27} - 4q^{28} - 3q^{29} + 3q^{31} - 4q^{32} - 2q^{33} + 6q^{34} + 4q^{36} + 4q^{37} - 3q^{38} - 3q^{39} - 11q^{41} - 4q^{42} + 5q^{43} + 2q^{44} - q^{46} - 4q^{48} + 14q^{49} + 6q^{51} + 3q^{52} + 22q^{53} + 4q^{54} + 4q^{56} - 3q^{57} + 3q^{58} - 8q^{59} - 5q^{61} - 3q^{62} - 4q^{63} + 4q^{64} + 2q^{66} - 6q^{67} - 6q^{68} - q^{69} - 18q^{71} - 4q^{72} + 5q^{73} - 4q^{74} + 3q^{76} + 4q^{77} + 3q^{78} + 16q^{79} + 4q^{81} + 11q^{82} + q^{83} + 4q^{84} - 5q^{86} + 3q^{87} - 2q^{88} - 5q^{89} - 13q^{91} + q^{92} - 3q^{93} + 4q^{96} - 7q^{97} - 14q^{98} + 2q^{99} + O(q^{100}) \)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −5.13277 −1.94001 −0.970003 0.243094i \(-0.921838\pi\)
−0.970003 + 0.243094i \(0.921838\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.22069 0.669564 0.334782 0.942296i \(-0.391337\pi\)
0.334782 + 0.942296i \(0.391337\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.60629 1.83225 0.916127 0.400888i \(-0.131298\pi\)
0.916127 + 0.400888i \(0.131298\pi\)
\(14\) 5.13277 1.37179
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.63841 −1.36752 −0.683758 0.729709i \(-0.739656\pi\)
−0.683758 + 0.729709i \(0.739656\pi\)
\(18\) −1.00000 −0.235702
\(19\) −7.51836 −1.72483 −0.862415 0.506201i \(-0.831049\pi\)
−0.862415 + 0.506201i \(0.831049\pi\)
\(20\) 0 0
\(21\) 5.13277 1.12006
\(22\) −2.22069 −0.473454
\(23\) 6.10064 1.27207 0.636036 0.771659i \(-0.280573\pi\)
0.636036 + 0.771659i \(0.280573\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −6.60629 −1.29560
\(27\) −1.00000 −0.192450
\(28\) −5.13277 −0.970003
\(29\) −7.29767 −1.35514 −0.677572 0.735457i \(-0.736968\pi\)
−0.677572 + 0.735457i \(0.736968\pi\)
\(30\) 0 0
\(31\) −0.967873 −0.173835 −0.0869176 0.996216i \(-0.527702\pi\)
−0.0869176 + 0.996216i \(0.527702\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.22069 −0.386573
\(34\) 5.63841 0.966980
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 7.51836 1.21964
\(39\) −6.60629 −1.05785
\(40\) 0 0
\(41\) −1.03213 −0.161191 −0.0805955 0.996747i \(-0.525682\pi\)
−0.0805955 + 0.996747i \(0.525682\pi\)
\(42\) −5.13277 −0.792004
\(43\) −5.29767 −0.807887 −0.403944 0.914784i \(-0.632361\pi\)
−0.403944 + 0.914784i \(0.632361\pi\)
\(44\) 2.22069 0.334782
\(45\) 0 0
\(46\) −6.10064 −0.899491
\(47\) 5.85911 0.854638 0.427319 0.904101i \(-0.359458\pi\)
0.427319 + 0.904101i \(0.359458\pi\)
\(48\) −1.00000 −0.144338
\(49\) 19.3453 2.76362
\(50\) 0 0
\(51\) 5.63841 0.789536
\(52\) 6.60629 0.916127
\(53\) 9.63841 1.32394 0.661969 0.749531i \(-0.269721\pi\)
0.661969 + 0.749531i \(0.269721\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 5.13277 0.685895
\(57\) 7.51836 0.995832
\(58\) 7.29767 0.958231
\(59\) −10.2655 −1.33646 −0.668230 0.743955i \(-0.732948\pi\)
−0.668230 + 0.743955i \(0.732948\pi\)
\(60\) 0 0
\(61\) 6.32134 0.809365 0.404682 0.914457i \(-0.367382\pi\)
0.404682 + 0.914457i \(0.367382\pi\)
\(62\) 0.967873 0.122920
\(63\) −5.13277 −0.646668
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.22069 0.273349
\(67\) −1.49436 −0.182565 −0.0912825 0.995825i \(-0.529097\pi\)
−0.0912825 + 0.995825i \(0.529097\pi\)
\(68\) −5.63841 −0.683758
\(69\) −6.10064 −0.734431
\(70\) 0 0
\(71\) 0.329796 0.0391396 0.0195698 0.999808i \(-0.493770\pi\)
0.0195698 + 0.999808i \(0.493770\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.8718 −1.50653 −0.753267 0.657715i \(-0.771523\pi\)
−0.753267 + 0.657715i \(0.771523\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −7.51836 −0.862415
\(77\) −11.3983 −1.29896
\(78\) 6.60629 0.748015
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.03213 0.113979
\(83\) −9.04767 −0.993111 −0.496556 0.868005i \(-0.665402\pi\)
−0.496556 + 0.868005i \(0.665402\pi\)
\(84\) 5.13277 0.560031
\(85\) 0 0
\(86\) 5.29767 0.571262
\(87\) 7.29767 0.782392
\(88\) −2.22069 −0.236727
\(89\) −2.27649 −0.241307 −0.120654 0.992695i \(-0.538499\pi\)
−0.120654 + 0.992695i \(0.538499\pi\)
\(90\) 0 0
\(91\) −33.9086 −3.55458
\(92\) 6.10064 0.636036
\(93\) 0.967873 0.100364
\(94\) −5.85911 −0.604321
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −17.5982 −1.78682 −0.893411 0.449239i \(-0.851695\pi\)
−0.893411 + 0.449239i \(0.851695\pi\)
\(98\) −19.3453 −1.95417
\(99\) 2.22069 0.223188
\(100\) 0 0
\(101\) 7.61901 0.758120 0.379060 0.925372i \(-0.376247\pi\)
0.379060 + 0.925372i \(0.376247\pi\)
\(102\) −5.63841 −0.558286
\(103\) −5.85911 −0.577315 −0.288657 0.957432i \(-0.593209\pi\)
−0.288657 + 0.957432i \(0.593209\pi\)
\(104\) −6.60629 −0.647800
\(105\) 0 0
\(106\) −9.63841 −0.936165
\(107\) −5.65926 −0.547101 −0.273551 0.961858i \(-0.588198\pi\)
−0.273551 + 0.961858i \(0.588198\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −3.16772 −0.303413 −0.151706 0.988426i \(-0.548477\pi\)
−0.151706 + 0.988426i \(0.548477\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) −5.13277 −0.485001
\(113\) 0.702331 0.0660697 0.0330348 0.999454i \(-0.489483\pi\)
0.0330348 + 0.999454i \(0.489483\pi\)
\(114\) −7.51836 −0.704159
\(115\) 0 0
\(116\) −7.29767 −0.677572
\(117\) 6.60629 0.610751
\(118\) 10.2655 0.945020
\(119\) 28.9407 2.65299
\(120\) 0 0
\(121\) −6.06852 −0.551683
\(122\) −6.32134 −0.572307
\(123\) 1.03213 0.0930637
\(124\) −0.967873 −0.0869176
\(125\) 0 0
\(126\) 5.13277 0.457264
\(127\) 19.3132 1.71377 0.856885 0.515507i \(-0.172396\pi\)
0.856885 + 0.515507i \(0.172396\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.29767 0.466434
\(130\) 0 0
\(131\) −18.5953 −1.62468 −0.812341 0.583183i \(-0.801807\pi\)
−0.812341 + 0.583183i \(0.801807\pi\)
\(132\) −2.22069 −0.193287
\(133\) 38.5900 3.34618
\(134\) 1.49436 0.129093
\(135\) 0 0
\(136\) 5.63841 0.483490
\(137\) 19.5074 1.66663 0.833316 0.552798i \(-0.186440\pi\)
0.833316 + 0.552798i \(0.186440\pi\)
\(138\) 6.10064 0.519321
\(139\) 10.9036 0.924833 0.462417 0.886663i \(-0.346982\pi\)
0.462417 + 0.886663i \(0.346982\pi\)
\(140\) 0 0
\(141\) −5.85911 −0.493426
\(142\) −0.329796 −0.0276759
\(143\) 14.6705 1.22681
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 12.8718 1.06528
\(147\) −19.3453 −1.59558
\(148\) 1.00000 0.0821995
\(149\) −12.6720 −1.03813 −0.519065 0.854735i \(-0.673720\pi\)
−0.519065 + 0.854735i \(0.673720\pi\)
\(150\) 0 0
\(151\) 10.5420 0.857898 0.428949 0.903329i \(-0.358884\pi\)
0.428949 + 0.903329i \(0.358884\pi\)
\(152\) 7.51836 0.609820
\(153\) −5.63841 −0.455839
\(154\) 11.3983 0.918502
\(155\) 0 0
\(156\) −6.60629 −0.528926
\(157\) 15.4990 1.23695 0.618476 0.785804i \(-0.287750\pi\)
0.618476 + 0.785804i \(0.287750\pi\)
\(158\) −4.00000 −0.318223
\(159\) −9.63841 −0.764376
\(160\) 0 0
\(161\) −31.3132 −2.46783
\(162\) −1.00000 −0.0785674
\(163\) 4.12149 0.322820 0.161410 0.986887i \(-0.448396\pi\)
0.161410 + 0.986887i \(0.448396\pi\)
\(164\) −1.03213 −0.0805955
\(165\) 0 0
\(166\) 9.04767 0.702236
\(167\) −21.9474 −1.69834 −0.849169 0.528121i \(-0.822897\pi\)
−0.849169 + 0.528121i \(0.822897\pi\)
\(168\) −5.13277 −0.396002
\(169\) 30.6430 2.35715
\(170\) 0 0
\(171\) −7.51836 −0.574944
\(172\) −5.29767 −0.403944
\(173\) 0.425841 0.0323761 0.0161880 0.999869i \(-0.494847\pi\)
0.0161880 + 0.999869i \(0.494847\pi\)
\(174\) −7.29767 −0.553235
\(175\) 0 0
\(176\) 2.22069 0.167391
\(177\) 10.2655 0.771605
\(178\) 2.27649 0.170630
\(179\) −17.5424 −1.31118 −0.655589 0.755118i \(-0.727580\pi\)
−0.655589 + 0.755118i \(0.727580\pi\)
\(180\) 0 0
\(181\) 18.5311 1.37740 0.688702 0.725044i \(-0.258181\pi\)
0.688702 + 0.725044i \(0.258181\pi\)
\(182\) 33.9086 2.51347
\(183\) −6.32134 −0.467287
\(184\) −6.10064 −0.449746
\(185\) 0 0
\(186\) −0.967873 −0.0709679
\(187\) −12.5212 −0.915640
\(188\) 5.85911 0.427319
\(189\) 5.13277 0.373354
\(190\) 0 0
\(191\) 16.1695 1.16998 0.584992 0.811039i \(-0.301098\pi\)
0.584992 + 0.811039i \(0.301098\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 11.8591 0.853637 0.426819 0.904337i \(-0.359634\pi\)
0.426819 + 0.904337i \(0.359634\pi\)
\(194\) 17.5982 1.26347
\(195\) 0 0
\(196\) 19.3453 1.38181
\(197\) −0.100645 −0.00717066 −0.00358533 0.999994i \(-0.501141\pi\)
−0.00358533 + 0.999994i \(0.501141\pi\)
\(198\) −2.22069 −0.157818
\(199\) −2.70693 −0.191889 −0.0959446 0.995387i \(-0.530587\pi\)
−0.0959446 + 0.995387i \(0.530587\pi\)
\(200\) 0 0
\(201\) 1.49436 0.105404
\(202\) −7.61901 −0.536072
\(203\) 37.4573 2.62898
\(204\) 5.63841 0.394768
\(205\) 0 0
\(206\) 5.85911 0.408223
\(207\) 6.10064 0.424024
\(208\) 6.60629 0.458064
\(209\) −16.6960 −1.15489
\(210\) 0 0
\(211\) 11.8033 0.812573 0.406287 0.913746i \(-0.366823\pi\)
0.406287 + 0.913746i \(0.366823\pi\)
\(212\) 9.63841 0.661969
\(213\) −0.329796 −0.0225972
\(214\) 5.65926 0.386859
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 4.96787 0.337241
\(218\) 3.16772 0.214545
\(219\) 12.8718 0.869798
\(220\) 0 0
\(221\) −37.2490 −2.50564
\(222\) 1.00000 0.0671156
\(223\) −13.1692 −0.881872 −0.440936 0.897538i \(-0.645354\pi\)
−0.440936 + 0.897538i \(0.645354\pi\)
\(224\) 5.13277 0.342948
\(225\) 0 0
\(226\) −0.702331 −0.0467183
\(227\) −5.57946 −0.370322 −0.185161 0.982708i \(-0.559281\pi\)
−0.185161 + 0.982708i \(0.559281\pi\)
\(228\) 7.51836 0.497916
\(229\) −1.75990 −0.116298 −0.0581488 0.998308i \(-0.518520\pi\)
−0.0581488 + 0.998308i \(0.518520\pi\)
\(230\) 0 0
\(231\) 11.3983 0.749954
\(232\) 7.29767 0.479115
\(233\) 6.07664 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(234\) −6.60629 −0.431866
\(235\) 0 0
\(236\) −10.2655 −0.668230
\(237\) −4.00000 −0.259828
\(238\) −28.9407 −1.87595
\(239\) 13.9403 0.901726 0.450863 0.892593i \(-0.351116\pi\)
0.450863 + 0.892593i \(0.351116\pi\)
\(240\) 0 0
\(241\) −3.89406 −0.250838 −0.125419 0.992104i \(-0.540028\pi\)
−0.125419 + 0.992104i \(0.540028\pi\)
\(242\) 6.06852 0.390099
\(243\) −1.00000 −0.0641500
\(244\) 6.32134 0.404682
\(245\) 0 0
\(246\) −1.03213 −0.0658060
\(247\) −49.6685 −3.16033
\(248\) 0.967873 0.0614600
\(249\) 9.04767 0.573373
\(250\) 0 0
\(251\) 20.9308 1.32114 0.660570 0.750765i \(-0.270315\pi\)
0.660570 + 0.750765i \(0.270315\pi\)
\(252\) −5.13277 −0.323334
\(253\) 13.5477 0.851734
\(254\) −19.3132 −1.21182
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.43044 0.276363 0.138182 0.990407i \(-0.455874\pi\)
0.138182 + 0.990407i \(0.455874\pi\)
\(258\) −5.29767 −0.329818
\(259\) −5.13277 −0.318935
\(260\) 0 0
\(261\) −7.29767 −0.451714
\(262\) 18.5953 1.14882
\(263\) −4.20975 −0.259584 −0.129792 0.991541i \(-0.541431\pi\)
−0.129792 + 0.991541i \(0.541431\pi\)
\(264\) 2.22069 0.136674
\(265\) 0 0
\(266\) −38.5900 −2.36611
\(267\) 2.27649 0.139319
\(268\) −1.49436 −0.0912825
\(269\) 12.5071 0.762570 0.381285 0.924457i \(-0.375482\pi\)
0.381285 + 0.924457i \(0.375482\pi\)
\(270\) 0 0
\(271\) −6.70693 −0.407417 −0.203709 0.979032i \(-0.565299\pi\)
−0.203709 + 0.979032i \(0.565299\pi\)
\(272\) −5.63841 −0.341879
\(273\) 33.9086 2.05224
\(274\) −19.5074 −1.17849
\(275\) 0 0
\(276\) −6.10064 −0.367216
\(277\) 27.0477 1.62514 0.812569 0.582866i \(-0.198069\pi\)
0.812569 + 0.582866i \(0.198069\pi\)
\(278\) −10.9036 −0.653956
\(279\) −0.967873 −0.0579451
\(280\) 0 0
\(281\) −1.39371 −0.0831420 −0.0415710 0.999136i \(-0.513236\pi\)
−0.0415710 + 0.999136i \(0.513236\pi\)
\(282\) 5.85911 0.348905
\(283\) 15.3132 0.910276 0.455138 0.890421i \(-0.349590\pi\)
0.455138 + 0.890421i \(0.349590\pi\)
\(284\) 0.329796 0.0195698
\(285\) 0 0
\(286\) −14.6705 −0.867487
\(287\) 5.29767 0.312712
\(288\) −1.00000 −0.0589256
\(289\) 14.7917 0.870100
\(290\) 0 0
\(291\) 17.5982 1.03162
\(292\) −12.8718 −0.753267
\(293\) 12.8672 0.751712 0.375856 0.926678i \(-0.377349\pi\)
0.375856 + 0.926678i \(0.377349\pi\)
\(294\) 19.3453 1.12824
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −2.22069 −0.128858
\(298\) 12.6720 0.734068
\(299\) 40.3026 2.33076
\(300\) 0 0
\(301\) 27.1917 1.56731
\(302\) −10.5420 −0.606626
\(303\) −7.61901 −0.437701
\(304\) −7.51836 −0.431208
\(305\) 0 0
\(306\) 5.63841 0.322327
\(307\) 4.08970 0.233411 0.116706 0.993167i \(-0.462767\pi\)
0.116706 + 0.993167i \(0.462767\pi\)
\(308\) −11.3983 −0.649479
\(309\) 5.85911 0.333313
\(310\) 0 0
\(311\) −10.3090 −0.584567 −0.292283 0.956332i \(-0.594415\pi\)
−0.292283 + 0.956332i \(0.594415\pi\)
\(312\) 6.60629 0.374007
\(313\) 2.64653 0.149591 0.0747955 0.997199i \(-0.476170\pi\)
0.0747955 + 0.997199i \(0.476170\pi\)
\(314\) −15.4990 −0.874657
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 8.06885 0.453192 0.226596 0.973989i \(-0.427240\pi\)
0.226596 + 0.973989i \(0.427240\pi\)
\(318\) 9.63841 0.540495
\(319\) −16.2059 −0.907356
\(320\) 0 0
\(321\) 5.65926 0.315869
\(322\) 31.3132 1.74502
\(323\) 42.3916 2.35873
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.12149 −0.228268
\(327\) 3.16772 0.175175
\(328\) 1.03213 0.0569897
\(329\) −30.0735 −1.65800
\(330\) 0 0
\(331\) 19.6833 1.08189 0.540945 0.841058i \(-0.318067\pi\)
0.540945 + 0.841058i \(0.318067\pi\)
\(332\) −9.04767 −0.496556
\(333\) 1.00000 0.0547997
\(334\) 21.9474 1.20091
\(335\) 0 0
\(336\) 5.13277 0.280016
\(337\) 23.4891 1.27953 0.639765 0.768570i \(-0.279032\pi\)
0.639765 + 0.768570i \(0.279032\pi\)
\(338\) −30.6430 −1.66676
\(339\) −0.702331 −0.0381454
\(340\) 0 0
\(341\) −2.14935 −0.116394
\(342\) 7.51836 0.406547
\(343\) −63.3658 −3.42143
\(344\) 5.29767 0.285631
\(345\) 0 0
\(346\) −0.425841 −0.0228933
\(347\) −18.7069 −1.00424 −0.502120 0.864798i \(-0.667447\pi\)
−0.502120 + 0.864798i \(0.667447\pi\)
\(348\) 7.29767 0.391196
\(349\) 16.1123 0.862470 0.431235 0.902240i \(-0.358078\pi\)
0.431235 + 0.902240i \(0.358078\pi\)
\(350\) 0 0
\(351\) −6.60629 −0.352617
\(352\) −2.22069 −0.118363
\(353\) −2.37320 −0.126313 −0.0631565 0.998004i \(-0.520117\pi\)
−0.0631565 + 0.998004i \(0.520117\pi\)
\(354\) −10.2655 −0.545607
\(355\) 0 0
\(356\) −2.27649 −0.120654
\(357\) −28.9407 −1.53170
\(358\) 17.5424 0.927143
\(359\) −18.5953 −0.981424 −0.490712 0.871322i \(-0.663263\pi\)
−0.490712 + 0.871322i \(0.663263\pi\)
\(360\) 0 0
\(361\) 37.5258 1.97504
\(362\) −18.5311 −0.973972
\(363\) 6.06852 0.318515
\(364\) −33.9086 −1.77729
\(365\) 0 0
\(366\) 6.32134 0.330422
\(367\) 29.9520 1.56348 0.781740 0.623605i \(-0.214332\pi\)
0.781740 + 0.623605i \(0.214332\pi\)
\(368\) 6.10064 0.318018
\(369\) −1.03213 −0.0537304
\(370\) 0 0
\(371\) −49.4718 −2.56845
\(372\) 0.967873 0.0501819
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 12.5212 0.647455
\(375\) 0 0
\(376\) −5.85911 −0.302160
\(377\) −48.2105 −2.48297
\(378\) −5.13277 −0.264001
\(379\) 23.3185 1.19779 0.598896 0.800827i \(-0.295606\pi\)
0.598896 + 0.800827i \(0.295606\pi\)
\(380\) 0 0
\(381\) −19.3132 −0.989446
\(382\) −16.1695 −0.827304
\(383\) 16.6063 0.848542 0.424271 0.905535i \(-0.360530\pi\)
0.424271 + 0.905535i \(0.360530\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −11.8591 −0.603613
\(387\) −5.29767 −0.269296
\(388\) −17.5982 −0.893411
\(389\) 15.5795 0.789910 0.394955 0.918701i \(-0.370760\pi\)
0.394955 + 0.918701i \(0.370760\pi\)
\(390\) 0 0
\(391\) −34.3980 −1.73958
\(392\) −19.3453 −0.977087
\(393\) 18.5953 0.938011
\(394\) 0.100645 0.00507042
\(395\) 0 0
\(396\) 2.22069 0.111594
\(397\) 23.2126 1.16501 0.582503 0.812829i \(-0.302074\pi\)
0.582503 + 0.812829i \(0.302074\pi\)
\(398\) 2.70693 0.135686
\(399\) −38.5900 −1.93192
\(400\) 0 0
\(401\) 5.72351 0.285818 0.142909 0.989736i \(-0.454354\pi\)
0.142909 + 0.989736i \(0.454354\pi\)
\(402\) −1.49436 −0.0745319
\(403\) −6.39405 −0.318510
\(404\) 7.61901 0.379060
\(405\) 0 0
\(406\) −37.4573 −1.85897
\(407\) 2.22069 0.110076
\(408\) −5.63841 −0.279143
\(409\) 28.0254 1.38577 0.692885 0.721049i \(-0.256340\pi\)
0.692885 + 0.721049i \(0.256340\pi\)
\(410\) 0 0
\(411\) −19.5074 −0.962230
\(412\) −5.85911 −0.288657
\(413\) 52.6907 2.59274
\(414\) −6.10064 −0.299830
\(415\) 0 0
\(416\) −6.60629 −0.323900
\(417\) −10.9036 −0.533953
\(418\) 16.6960 0.816627
\(419\) 16.2415 0.793451 0.396726 0.917937i \(-0.370146\pi\)
0.396726 + 0.917937i \(0.370146\pi\)
\(420\) 0 0
\(421\) 6.03495 0.294126 0.147063 0.989127i \(-0.453018\pi\)
0.147063 + 0.989127i \(0.453018\pi\)
\(422\) −11.8033 −0.574576
\(423\) 5.85911 0.284879
\(424\) −9.63841 −0.468083
\(425\) 0 0
\(426\) 0.329796 0.0159787
\(427\) −32.4460 −1.57017
\(428\) −5.65926 −0.273551
\(429\) −14.6705 −0.708300
\(430\) 0 0
\(431\) −33.2924 −1.60364 −0.801819 0.597568i \(-0.796134\pi\)
−0.801819 + 0.597568i \(0.796134\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −8.67054 −0.416680 −0.208340 0.978057i \(-0.566806\pi\)
−0.208340 + 0.978057i \(0.566806\pi\)
\(434\) −4.96787 −0.238466
\(435\) 0 0
\(436\) −3.16772 −0.151706
\(437\) −45.8669 −2.19411
\(438\) −12.8718 −0.615040
\(439\) 32.8874 1.56963 0.784814 0.619731i \(-0.212758\pi\)
0.784814 + 0.619731i \(0.212758\pi\)
\(440\) 0 0
\(441\) 19.3453 0.921207
\(442\) 37.2490 1.77175
\(443\) −14.8185 −0.704049 −0.352025 0.935991i \(-0.614507\pi\)
−0.352025 + 0.935991i \(0.614507\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) 13.1692 0.623578
\(447\) 12.6720 0.599364
\(448\) −5.13277 −0.242501
\(449\) 9.22882 0.435535 0.217767 0.976001i \(-0.430123\pi\)
0.217767 + 0.976001i \(0.430123\pi\)
\(450\) 0 0
\(451\) −2.29204 −0.107928
\(452\) 0.702331 0.0330348
\(453\) −10.5420 −0.495308
\(454\) 5.57946 0.261857
\(455\) 0 0
\(456\) −7.51836 −0.352080
\(457\) 36.4753 1.70624 0.853121 0.521713i \(-0.174707\pi\)
0.853121 + 0.521713i \(0.174707\pi\)
\(458\) 1.75990 0.0822348
\(459\) 5.63841 0.263179
\(460\) 0 0
\(461\) 26.6642 1.24188 0.620938 0.783860i \(-0.286752\pi\)
0.620938 + 0.783860i \(0.286752\pi\)
\(462\) −11.3983 −0.530298
\(463\) −6.25249 −0.290578 −0.145289 0.989389i \(-0.546411\pi\)
−0.145289 + 0.989389i \(0.546411\pi\)
\(464\) −7.29767 −0.338786
\(465\) 0 0
\(466\) −6.07664 −0.281495
\(467\) −16.1748 −0.748480 −0.374240 0.927332i \(-0.622096\pi\)
−0.374240 + 0.927332i \(0.622096\pi\)
\(468\) 6.60629 0.305376
\(469\) 7.67020 0.354177
\(470\) 0 0
\(471\) −15.4990 −0.714154
\(472\) 10.2655 0.472510
\(473\) −11.7645 −0.540932
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 28.9407 1.32649
\(477\) 9.63841 0.441313
\(478\) −13.9403 −0.637617
\(479\) 7.72351 0.352896 0.176448 0.984310i \(-0.443539\pi\)
0.176448 + 0.984310i \(0.443539\pi\)
\(480\) 0 0
\(481\) 6.60629 0.301221
\(482\) 3.89406 0.177369
\(483\) 31.3132 1.42480
\(484\) −6.06852 −0.275842
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −35.8429 −1.62420 −0.812098 0.583522i \(-0.801674\pi\)
−0.812098 + 0.583522i \(0.801674\pi\)
\(488\) −6.32134 −0.286154
\(489\) −4.12149 −0.186380
\(490\) 0 0
\(491\) 2.74718 0.123978 0.0619892 0.998077i \(-0.480256\pi\)
0.0619892 + 0.998077i \(0.480256\pi\)
\(492\) 1.03213 0.0465319
\(493\) 41.1473 1.85318
\(494\) 49.6685 2.23469
\(495\) 0 0
\(496\) −0.967873 −0.0434588
\(497\) −1.69277 −0.0759310
\(498\) −9.04767 −0.405436
\(499\) −3.07698 −0.137744 −0.0688722 0.997625i \(-0.521940\pi\)
−0.0688722 + 0.997625i \(0.521940\pi\)
\(500\) 0 0
\(501\) 21.9474 0.980536
\(502\) −20.9308 −0.934187
\(503\) −8.88278 −0.396063 −0.198032 0.980196i \(-0.563455\pi\)
−0.198032 + 0.980196i \(0.563455\pi\)
\(504\) 5.13277 0.228632
\(505\) 0 0
\(506\) −13.5477 −0.602267
\(507\) −30.6430 −1.36090
\(508\) 19.3132 0.856885
\(509\) 2.32836 0.103203 0.0516013 0.998668i \(-0.483568\pi\)
0.0516013 + 0.998668i \(0.483568\pi\)
\(510\) 0 0
\(511\) 66.0682 2.92268
\(512\) −1.00000 −0.0441942
\(513\) 7.51836 0.331944
\(514\) −4.43044 −0.195418
\(515\) 0 0
\(516\) 5.29767 0.233217
\(517\) 13.0113 0.572236
\(518\) 5.13277 0.225521
\(519\) −0.425841 −0.0186923
\(520\) 0 0
\(521\) −26.3344 −1.15373 −0.576865 0.816839i \(-0.695724\pi\)
−0.576865 + 0.816839i \(0.695724\pi\)
\(522\) 7.29767 0.319410
\(523\) −10.9887 −0.480503 −0.240252 0.970711i \(-0.577230\pi\)
−0.240252 + 0.970711i \(0.577230\pi\)
\(524\) −18.5953 −0.812341
\(525\) 0 0
\(526\) 4.20975 0.183554
\(527\) 5.45727 0.237722
\(528\) −2.22069 −0.0966433
\(529\) 14.2179 0.618168
\(530\) 0 0
\(531\) −10.2655 −0.445487
\(532\) 38.5900 1.67309
\(533\) −6.81852 −0.295343
\(534\) −2.27649 −0.0985134
\(535\) 0 0
\(536\) 1.49436 0.0645465
\(537\) 17.5424 0.757009
\(538\) −12.5071 −0.539219
\(539\) 42.9601 1.85042
\(540\) 0 0
\(541\) −6.61934 −0.284588 −0.142294 0.989824i \(-0.545448\pi\)
−0.142294 + 0.989824i \(0.545448\pi\)
\(542\) 6.70693 0.288087
\(543\) −18.5311 −0.795245
\(544\) 5.63841 0.241745
\(545\) 0 0
\(546\) −33.9086 −1.45115
\(547\) 23.1801 0.991110 0.495555 0.868577i \(-0.334965\pi\)
0.495555 + 0.868577i \(0.334965\pi\)
\(548\) 19.5074 0.833316
\(549\) 6.32134 0.269788
\(550\) 0 0
\(551\) 54.8665 2.33739
\(552\) 6.10064 0.259661
\(553\) −20.5311 −0.873071
\(554\) −27.0477 −1.14915
\(555\) 0 0
\(556\) 10.9036 0.462417
\(557\) −11.3411 −0.480537 −0.240268 0.970706i \(-0.577235\pi\)
−0.240268 + 0.970706i \(0.577235\pi\)
\(558\) 0.967873 0.0409734
\(559\) −34.9979 −1.48025
\(560\) 0 0
\(561\) 12.5212 0.528645
\(562\) 1.39371 0.0587903
\(563\) −33.3705 −1.40640 −0.703198 0.710994i \(-0.748245\pi\)
−0.703198 + 0.710994i \(0.748245\pi\)
\(564\) −5.85911 −0.246713
\(565\) 0 0
\(566\) −15.3132 −0.643663
\(567\) −5.13277 −0.215556
\(568\) −0.329796 −0.0138379
\(569\) −0.500343 −0.0209755 −0.0104877 0.999945i \(-0.503338\pi\)
−0.0104877 + 0.999945i \(0.503338\pi\)
\(570\) 0 0
\(571\) 27.5463 1.15278 0.576388 0.817176i \(-0.304462\pi\)
0.576388 + 0.817176i \(0.304462\pi\)
\(572\) 14.6705 0.613406
\(573\) −16.1695 −0.675490
\(574\) −5.29767 −0.221120
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −12.3902 −0.515810 −0.257905 0.966170i \(-0.583032\pi\)
−0.257905 + 0.966170i \(0.583032\pi\)
\(578\) −14.7917 −0.615253
\(579\) −11.8591 −0.492848
\(580\) 0 0
\(581\) 46.4396 1.92664
\(582\) −17.5982 −0.729467
\(583\) 21.4040 0.886462
\(584\) 12.8718 0.532640
\(585\) 0 0
\(586\) −12.8672 −0.531540
\(587\) −0.808273 −0.0333610 −0.0166805 0.999861i \(-0.505310\pi\)
−0.0166805 + 0.999861i \(0.505310\pi\)
\(588\) −19.3453 −0.797789
\(589\) 7.27682 0.299836
\(590\) 0 0
\(591\) 0.100645 0.00413998
\(592\) 1.00000 0.0410997
\(593\) 24.4121 1.00248 0.501242 0.865307i \(-0.332877\pi\)
0.501242 + 0.865307i \(0.332877\pi\)
\(594\) 2.22069 0.0911162
\(595\) 0 0
\(596\) −12.6720 −0.519065
\(597\) 2.70693 0.110787
\(598\) −40.3026 −1.64810
\(599\) −29.3440 −1.19896 −0.599481 0.800389i \(-0.704626\pi\)
−0.599481 + 0.800389i \(0.704626\pi\)
\(600\) 0 0
\(601\) −9.55791 −0.389875 −0.194938 0.980816i \(-0.562450\pi\)
−0.194938 + 0.980816i \(0.562450\pi\)
\(602\) −27.1917 −1.10825
\(603\) −1.49436 −0.0608550
\(604\) 10.5420 0.428949
\(605\) 0 0
\(606\) 7.61901 0.309501
\(607\) 31.4912 1.27819 0.639094 0.769129i \(-0.279310\pi\)
0.639094 + 0.769129i \(0.279310\pi\)
\(608\) 7.51836 0.304910
\(609\) −37.4573 −1.51785
\(610\) 0 0
\(611\) 38.7069 1.56591
\(612\) −5.63841 −0.227919
\(613\) −6.41489 −0.259095 −0.129548 0.991573i \(-0.541352\pi\)
−0.129548 + 0.991573i \(0.541352\pi\)
\(614\) −4.08970 −0.165047
\(615\) 0 0
\(616\) 11.3983 0.459251
\(617\) 38.8153 1.56265 0.781323 0.624127i \(-0.214545\pi\)
0.781323 + 0.624127i \(0.214545\pi\)
\(618\) −5.85911 −0.235688
\(619\) −0.00460032 −0.000184902 0 −9.24512e−5 1.00000i \(-0.500029\pi\)
−9.24512e−5 1.00000i \(0.500029\pi\)
\(620\) 0 0
\(621\) −6.10064 −0.244810
\(622\) 10.3090 0.413351
\(623\) 11.6847 0.468138
\(624\) −6.60629 −0.264463
\(625\) 0 0
\(626\) −2.64653 −0.105777
\(627\) 16.6960 0.666773
\(628\) 15.4990 0.618476
\(629\) −5.63841 −0.224818
\(630\) 0 0
\(631\) 34.5999 1.37740 0.688701 0.725046i \(-0.258181\pi\)
0.688701 + 0.725046i \(0.258181\pi\)
\(632\) −4.00000 −0.159111
\(633\) −11.8033 −0.469139
\(634\) −8.06885 −0.320455
\(635\) 0 0
\(636\) −9.63841 −0.382188
\(637\) 127.801 5.06365
\(638\) 16.2059 0.641597
\(639\) 0.329796 0.0130465
\(640\) 0 0
\(641\) −40.8711 −1.61431 −0.807156 0.590338i \(-0.798995\pi\)
−0.807156 + 0.590338i \(0.798995\pi\)
\(642\) −5.65926 −0.223353
\(643\) −11.4788 −0.452680 −0.226340 0.974048i \(-0.572676\pi\)
−0.226340 + 0.974048i \(0.572676\pi\)
\(644\) −31.3132 −1.23391
\(645\) 0 0
\(646\) −42.3916 −1.66788
\(647\) 17.7490 0.697783 0.348892 0.937163i \(-0.386558\pi\)
0.348892 + 0.937163i \(0.386558\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −22.7966 −0.894846
\(650\) 0 0
\(651\) −4.96787 −0.194706
\(652\) 4.12149 0.161410
\(653\) −21.2768 −0.832626 −0.416313 0.909221i \(-0.636678\pi\)
−0.416313 + 0.909221i \(0.636678\pi\)
\(654\) −3.16772 −0.123868
\(655\) 0 0
\(656\) −1.03213 −0.0402978
\(657\) −12.8718 −0.502178
\(658\) 30.0735 1.17239
\(659\) 23.5548 0.917563 0.458781 0.888549i \(-0.348286\pi\)
0.458781 + 0.888549i \(0.348286\pi\)
\(660\) 0 0
\(661\) 1.34357 0.0522587 0.0261294 0.999659i \(-0.491682\pi\)
0.0261294 + 0.999659i \(0.491682\pi\)
\(662\) −19.6833 −0.765012
\(663\) 37.2490 1.44663
\(664\) 9.04767 0.351118
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) −44.5205 −1.72384
\(668\) −21.9474 −0.849169
\(669\) 13.1692 0.509149
\(670\) 0 0
\(671\) 14.0378 0.541922
\(672\) −5.13277 −0.198001
\(673\) −17.5562 −0.676742 −0.338371 0.941013i \(-0.609876\pi\)
−0.338371 + 0.941013i \(0.609876\pi\)
\(674\) −23.4891 −0.904765
\(675\) 0 0
\(676\) 30.6430 1.17858
\(677\) 19.2073 0.738196 0.369098 0.929391i \(-0.379667\pi\)
0.369098 + 0.929391i \(0.379667\pi\)
\(678\) 0.702331 0.0269728
\(679\) 90.3274 3.46645
\(680\) 0 0
\(681\) 5.57946 0.213805
\(682\) 2.14935 0.0823029
\(683\) 34.7532 1.32980 0.664898 0.746935i \(-0.268475\pi\)
0.664898 + 0.746935i \(0.268475\pi\)
\(684\) −7.51836 −0.287472
\(685\) 0 0
\(686\) 63.3658 2.41932
\(687\) 1.75990 0.0671445
\(688\) −5.29767 −0.201972
\(689\) 63.6741 2.42579
\(690\) 0 0
\(691\) 24.3344 0.925724 0.462862 0.886430i \(-0.346823\pi\)
0.462862 + 0.886430i \(0.346823\pi\)
\(692\) 0.425841 0.0161880
\(693\) −11.3983 −0.432986
\(694\) 18.7069 0.710105
\(695\) 0 0
\(696\) −7.29767 −0.276617
\(697\) 5.81955 0.220431
\(698\) −16.1123 −0.609858
\(699\) −6.07664 −0.229840
\(700\) 0 0
\(701\) −8.81354 −0.332883 −0.166441 0.986051i \(-0.553228\pi\)
−0.166441 + 0.986051i \(0.553228\pi\)
\(702\) 6.60629 0.249338
\(703\) −7.51836 −0.283560
\(704\) 2.22069 0.0836956
\(705\) 0 0
\(706\) 2.37320 0.0893167
\(707\) −39.1066 −1.47076
\(708\) 10.2655 0.385803
\(709\) −9.07238 −0.340720 −0.170360 0.985382i \(-0.554493\pi\)
−0.170360 + 0.985382i \(0.554493\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 2.27649 0.0853151
\(713\) −5.90465 −0.221131
\(714\) 28.9407 1.08308
\(715\) 0 0
\(716\) −17.5424 −0.655589
\(717\) −13.9403 −0.520612
\(718\) 18.5953 0.693972
\(719\) 22.8665 0.852778 0.426389 0.904540i \(-0.359786\pi\)
0.426389 + 0.904540i \(0.359786\pi\)
\(720\) 0 0
\(721\) 30.0735 1.11999
\(722\) −37.5258 −1.39657
\(723\) 3.89406 0.144822
\(724\) 18.5311 0.688702
\(725\) 0 0
\(726\) −6.06852 −0.225224
\(727\) 2.86407 0.106222 0.0531112 0.998589i \(-0.483086\pi\)
0.0531112 + 0.998589i \(0.483086\pi\)
\(728\) 33.9086 1.25673
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 29.8704 1.10480
\(732\) −6.32134 −0.233643
\(733\) 21.6360 0.799144 0.399572 0.916702i \(-0.369159\pi\)
0.399572 + 0.916702i \(0.369159\pi\)
\(734\) −29.9520 −1.10555
\(735\) 0 0
\(736\) −6.10064 −0.224873
\(737\) −3.31851 −0.122239
\(738\) 1.03213 0.0379931
\(739\) 16.7278 0.615341 0.307671 0.951493i \(-0.400451\pi\)
0.307671 + 0.951493i \(0.400451\pi\)
\(740\) 0 0
\(741\) 49.6685 1.82462
\(742\) 49.4718 1.81617
\(743\) 21.6624 0.794717 0.397358 0.917663i \(-0.369927\pi\)
0.397358 + 0.917663i \(0.369927\pi\)
\(744\) −0.967873 −0.0354840
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) −9.04767 −0.331037
\(748\) −12.5212 −0.457820
\(749\) 29.0477 1.06138
\(750\) 0 0
\(751\) 24.8129 0.905435 0.452717 0.891654i \(-0.350455\pi\)
0.452717 + 0.891654i \(0.350455\pi\)
\(752\) 5.85911 0.213660
\(753\) −20.9308 −0.762760
\(754\) 48.2105 1.75572
\(755\) 0 0
\(756\) 5.13277 0.186677
\(757\) −1.97281 −0.0717030 −0.0358515 0.999357i \(-0.511414\pi\)
−0.0358515 + 0.999357i \(0.511414\pi\)
\(758\) −23.3185 −0.846967
\(759\) −13.5477 −0.491749
\(760\) 0 0
\(761\) 25.4735 0.923414 0.461707 0.887032i \(-0.347237\pi\)
0.461707 + 0.887032i \(0.347237\pi\)
\(762\) 19.3132 0.699644
\(763\) 16.2592 0.588622
\(764\) 16.1695 0.584992
\(765\) 0 0
\(766\) −16.6063 −0.600009
\(767\) −67.8171 −2.44873
\(768\) −1.00000 −0.0360844
\(769\) −41.9675 −1.51339 −0.756694 0.653770i \(-0.773187\pi\)
−0.756694 + 0.653770i \(0.773187\pi\)
\(770\) 0 0
\(771\) −4.43044 −0.159558
\(772\) 11.8591 0.426819
\(773\) −46.0410 −1.65598 −0.827990 0.560743i \(-0.810515\pi\)
−0.827990 + 0.560743i \(0.810515\pi\)
\(774\) 5.29767 0.190421
\(775\) 0 0
\(776\) 17.5982 0.631737
\(777\) 5.13277 0.184137
\(778\) −15.5795 −0.558551
\(779\) 7.75990 0.278027
\(780\) 0 0
\(781\) 0.732376 0.0262065
\(782\) 34.3980 1.23007
\(783\) 7.29767 0.260797
\(784\) 19.3453 0.690905
\(785\) 0 0
\(786\) −18.5953 −0.663274
\(787\) 8.04801 0.286881 0.143440 0.989659i \(-0.454184\pi\)
0.143440 + 0.989659i \(0.454184\pi\)
\(788\) −0.100645 −0.00358533
\(789\) 4.20975 0.149871
\(790\) 0 0
\(791\) −3.60490 −0.128176
\(792\) −2.22069 −0.0789089
\(793\) 41.7606 1.48296
\(794\) −23.2126 −0.823783
\(795\) 0 0
\(796\) −2.70693 −0.0959446
\(797\) 26.2662 0.930397 0.465198 0.885206i \(-0.345983\pi\)
0.465198 + 0.885206i \(0.345983\pi\)
\(798\) 38.5900 1.36607
\(799\) −33.0361 −1.16873
\(800\) 0 0
\(801\) −2.27649 −0.0804358
\(802\) −5.72351 −0.202104
\(803\) −28.5844 −1.00872
\(804\) 1.49436 0.0527020
\(805\) 0 0
\(806\) 6.39405 0.225221
\(807\) −12.5071 −0.440270
\(808\) −7.61901 −0.268036
\(809\) 13.3301 0.468662 0.234331 0.972157i \(-0.424710\pi\)
0.234331 + 0.972157i \(0.424710\pi\)
\(810\) 0 0
\(811\) −30.4251 −1.06837 −0.534186 0.845367i \(-0.679382\pi\)
−0.534186 + 0.845367i \(0.679382\pi\)
\(812\) 37.4573 1.31449
\(813\) 6.70693 0.235222
\(814\) −2.22069 −0.0778353
\(815\) 0 0
\(816\) 5.63841 0.197384
\(817\) 39.8298 1.39347
\(818\) −28.0254 −0.979887
\(819\) −33.9086 −1.18486
\(820\) 0 0
\(821\) −26.2895 −0.917512 −0.458756 0.888562i \(-0.651705\pi\)
−0.458756 + 0.888562i \(0.651705\pi\)
\(822\) 19.5074 0.680399
\(823\) −24.4050 −0.850705 −0.425352 0.905028i \(-0.639850\pi\)
−0.425352 + 0.905028i \(0.639850\pi\)
\(824\) 5.85911 0.204112
\(825\) 0 0
\(826\) −52.6907 −1.83334
\(827\) 37.5887 1.30709 0.653543 0.756890i \(-0.273282\pi\)
0.653543 + 0.756890i \(0.273282\pi\)
\(828\) 6.10064 0.212012
\(829\) 36.4481 1.26589 0.632947 0.774195i \(-0.281845\pi\)
0.632947 + 0.774195i \(0.281845\pi\)
\(830\) 0 0
\(831\) −27.0477 −0.938273
\(832\) 6.60629 0.229032
\(833\) −109.077 −3.77929
\(834\) 10.9036 0.377561
\(835\) 0 0
\(836\) −16.6960 −0.577443
\(837\) 0.967873 0.0334546
\(838\) −16.2415 −0.561055
\(839\) −43.3277 −1.49584 −0.747919 0.663790i \(-0.768947\pi\)
−0.747919 + 0.663790i \(0.768947\pi\)
\(840\) 0 0
\(841\) 24.2560 0.836413
\(842\) −6.03495 −0.207978
\(843\) 1.39371 0.0480021
\(844\) 11.8033 0.406287
\(845\) 0 0
\(846\) −5.85911 −0.201440
\(847\) 31.1483 1.07027
\(848\) 9.63841 0.330984
\(849\) −15.3132 −0.525548
\(850\) 0 0
\(851\) 6.10064 0.209127
\(852\) −0.329796 −0.0112986
\(853\) −17.6430 −0.604085 −0.302043 0.953294i \(-0.597668\pi\)
−0.302043 + 0.953294i \(0.597668\pi\)
\(854\) 32.4460 1.11028
\(855\) 0 0
\(856\) 5.65926 0.193429
\(857\) −35.6222 −1.21683 −0.608415 0.793619i \(-0.708194\pi\)
−0.608415 + 0.793619i \(0.708194\pi\)
\(858\) 14.6705 0.500844
\(859\) −5.49292 −0.187416 −0.0937080 0.995600i \(-0.529872\pi\)
−0.0937080 + 0.995600i \(0.529872\pi\)
\(860\) 0 0
\(861\) −5.29767 −0.180544
\(862\) 33.2924 1.13394
\(863\) 1.12501 0.0382958 0.0191479 0.999817i \(-0.493905\pi\)
0.0191479 + 0.999817i \(0.493905\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 8.67054 0.294637
\(867\) −14.7917 −0.502352
\(868\) 4.96787 0.168621
\(869\) 8.88278 0.301328
\(870\) 0 0
\(871\) −9.87216 −0.334506
\(872\) 3.16772 0.107273
\(873\) −17.5982 −0.595608
\(874\) 45.8669 1.55147
\(875\) 0 0
\(876\) 12.8718 0.434899
\(877\) −31.6367 −1.06829 −0.534147 0.845392i \(-0.679367\pi\)
−0.534147 + 0.845392i \(0.679367\pi\)
\(878\) −32.8874 −1.10990
\(879\) −12.8672 −0.434001
\(880\) 0 0
\(881\) −24.8012 −0.835575 −0.417787 0.908545i \(-0.637194\pi\)
−0.417787 + 0.908545i \(0.637194\pi\)
\(882\) −19.3453 −0.651392
\(883\) 27.7980 0.935478 0.467739 0.883867i \(-0.345069\pi\)
0.467739 + 0.883867i \(0.345069\pi\)
\(884\) −37.2490 −1.25282
\(885\) 0 0
\(886\) 14.8185 0.497838
\(887\) 28.5226 0.957696 0.478848 0.877898i \(-0.341054\pi\)
0.478848 + 0.877898i \(0.341054\pi\)
\(888\) 1.00000 0.0335578
\(889\) −99.1303 −3.32472
\(890\) 0 0
\(891\) 2.22069 0.0743960
\(892\) −13.1692 −0.440936
\(893\) −44.0509 −1.47411
\(894\) −12.6720 −0.423814
\(895\) 0 0
\(896\) 5.13277 0.171474
\(897\) −40.3026 −1.34566
\(898\) −9.22882 −0.307970
\(899\) 7.06322 0.235572
\(900\) 0 0
\(901\) −54.3453 −1.81051
\(902\) 2.29204 0.0763165
\(903\) −27.1917 −0.904884
\(904\) −0.702331 −0.0233592
\(905\) 0 0
\(906\) 10.5420 0.350236
\(907\) −41.1854 −1.36754 −0.683769 0.729698i \(-0.739660\pi\)
−0.683769 + 0.729698i \(0.739660\pi\)
\(908\) −5.57946 −0.185161
\(909\) 7.61901 0.252707
\(910\) 0 0
\(911\) 21.3440 0.707157 0.353578 0.935405i \(-0.384965\pi\)
0.353578 + 0.935405i \(0.384965\pi\)
\(912\) 7.51836 0.248958
\(913\) −20.0921 −0.664952
\(914\) −36.4753 −1.20650
\(915\) 0 0
\(916\) −1.75990 −0.0581488
\(917\) 95.4456 3.15189
\(918\) −5.63841 −0.186095
\(919\) 36.9089 1.21751 0.608756 0.793358i \(-0.291669\pi\)
0.608756 + 0.793358i \(0.291669\pi\)
\(920\) 0 0
\(921\) −4.08970 −0.134760
\(922\) −26.6642 −0.878138
\(923\) 2.17873 0.0717137
\(924\) 11.3983 0.374977
\(925\) 0 0
\(926\) 6.25249 0.205469
\(927\) −5.85911 −0.192438
\(928\) 7.29767 0.239558
\(929\) 21.0350 0.690136 0.345068 0.938578i \(-0.387856\pi\)
0.345068 + 0.938578i \(0.387856\pi\)
\(930\) 0 0
\(931\) −145.445 −4.76678
\(932\) 6.07664 0.199047
\(933\) 10.3090 0.337500
\(934\) 16.1748 0.529255
\(935\) 0 0
\(936\) −6.60629 −0.215933
\(937\) −9.67652 −0.316118 −0.158059 0.987430i \(-0.550524\pi\)
−0.158059 + 0.987430i \(0.550524\pi\)
\(938\) −7.67020 −0.250441
\(939\) −2.64653 −0.0863664
\(940\) 0 0
\(941\) −48.1332 −1.56910 −0.784548 0.620068i \(-0.787105\pi\)
−0.784548 + 0.620068i \(0.787105\pi\)
\(942\) 15.4990 0.504983
\(943\) −6.29664 −0.205047
\(944\) −10.2655 −0.334115
\(945\) 0 0
\(946\) 11.7645 0.382497
\(947\) 17.7871 0.578002 0.289001 0.957329i \(-0.406677\pi\)
0.289001 + 0.957329i \(0.406677\pi\)
\(948\) −4.00000 −0.129914
\(949\) −85.0350 −2.76035
\(950\) 0 0
\(951\) −8.06885 −0.261650
\(952\) −28.9407 −0.937973
\(953\) −30.0442 −0.973226 −0.486613 0.873618i \(-0.661768\pi\)
−0.486613 + 0.873618i \(0.661768\pi\)
\(954\) −9.63841 −0.312055
\(955\) 0 0
\(956\) 13.9403 0.450863
\(957\) 16.2059 0.523862
\(958\) −7.72351 −0.249535
\(959\) −100.127 −3.23327
\(960\) 0 0
\(961\) −30.0632 −0.969781
\(962\) −6.60629 −0.212995
\(963\) −5.65926 −0.182367
\(964\) −3.89406 −0.125419
\(965\) 0 0
\(966\) −31.3132 −1.00749
\(967\) −15.7052 −0.505044 −0.252522 0.967591i \(-0.581260\pi\)
−0.252522 + 0.967591i \(0.581260\pi\)
\(968\) 6.06852 0.195050
\(969\) −42.3916 −1.36182
\(970\) 0 0
\(971\) 36.8693 1.18319 0.591597 0.806234i \(-0.298498\pi\)
0.591597 + 0.806234i \(0.298498\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −55.9658 −1.79418
\(974\) 35.8429 1.14848
\(975\) 0 0
\(976\) 6.32134 0.202341
\(977\) −31.8404 −1.01866 −0.509332 0.860570i \(-0.670107\pi\)
−0.509332 + 0.860570i \(0.670107\pi\)
\(978\) 4.12149 0.131791
\(979\) −5.05539 −0.161571
\(980\) 0 0
\(981\) −3.16772 −0.101138
\(982\) −2.74718 −0.0876660
\(983\) −11.8708 −0.378618 −0.189309 0.981918i \(-0.560625\pi\)
−0.189309 + 0.981918i \(0.560625\pi\)
\(984\) −1.03213 −0.0329030
\(985\) 0 0
\(986\) −41.1473 −1.31040
\(987\) 30.0735 0.957249
\(988\) −49.6685 −1.58016
\(989\) −32.3192 −1.02769
\(990\) 0 0
\(991\) −13.9990 −0.444691 −0.222346 0.974968i \(-0.571371\pi\)
−0.222346 + 0.974968i \(0.571371\pi\)
\(992\) 0.967873 0.0307300
\(993\) −19.6833 −0.624629
\(994\) 1.69277 0.0536913
\(995\) 0 0
\(996\) 9.04767 0.286687
\(997\) 26.4050 0.836255 0.418127 0.908388i \(-0.362687\pi\)
0.418127 + 0.908388i \(0.362687\pi\)
\(998\) 3.07698 0.0974000
\(999\) −1.00000 −0.0316386
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 5550.2.a.cj.1.1 4
5.4 even 2 1110.2.a.s.1.4 4
15.14 odd 2 3330.2.a.bj.1.4 4
20.19 odd 2 8880.2.a.cg.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.s.1.4 4 5.4 even 2
3330.2.a.bj.1.4 4 15.14 odd 2
5550.2.a.cj.1.1 4 1.1 even 1 trivial
8880.2.a.cg.1.1 4 20.19 odd 2