Properties

Label 3330.2.a.bj.1.4
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
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 1.4
Root \(-2.67673\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.13277 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +5.13277 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.22069 q^{11} -6.60629 q^{13} -5.13277 q^{14} +1.00000 q^{16} -5.63841 q^{17} -7.51836 q^{19} -1.00000 q^{20} +2.22069 q^{22} +6.10064 q^{23} +1.00000 q^{25} +6.60629 q^{26} +5.13277 q^{28} +7.29767 q^{29} -0.967873 q^{31} -1.00000 q^{32} +5.63841 q^{34} -5.13277 q^{35} -1.00000 q^{37} +7.51836 q^{38} +1.00000 q^{40} +1.03213 q^{41} +5.29767 q^{43} -2.22069 q^{44} -6.10064 q^{46} +5.85911 q^{47} +19.3453 q^{49} -1.00000 q^{50} -6.60629 q^{52} +9.63841 q^{53} +2.22069 q^{55} -5.13277 q^{56} -7.29767 q^{58} +10.2655 q^{59} +6.32134 q^{61} +0.967873 q^{62} +1.00000 q^{64} +6.60629 q^{65} +1.49436 q^{67} -5.63841 q^{68} +5.13277 q^{70} -0.329796 q^{71} +12.8718 q^{73} +1.00000 q^{74} -7.51836 q^{76} -11.3983 q^{77} +4.00000 q^{79} -1.00000 q^{80} -1.03213 q^{82} -9.04767 q^{83} +5.63841 q^{85} -5.29767 q^{86} +2.22069 q^{88} +2.27649 q^{89} -33.9086 q^{91} +6.10064 q^{92} -5.85911 q^{94} +7.51836 q^{95} +17.5982 q^{97} -19.3453 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} - 4q^{5} + 4q^{7} - 4q^{8} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} - 4q^{5} + 4q^{7} - 4q^{8} + 4q^{10} - 2q^{11} - 3q^{13} - 4q^{14} + 4q^{16} - 6q^{17} + 3q^{19} - 4q^{20} + 2q^{22} + q^{23} + 4q^{25} + 3q^{26} + 4q^{28} + 3q^{29} + 3q^{31} - 4q^{32} + 6q^{34} - 4q^{35} - 4q^{37} - 3q^{38} + 4q^{40} + 11q^{41} - 5q^{43} - 2q^{44} - q^{46} + 14q^{49} - 4q^{50} - 3q^{52} + 22q^{53} + 2q^{55} - 4q^{56} - 3q^{58} + 8q^{59} - 5q^{61} - 3q^{62} + 4q^{64} + 3q^{65} + 6q^{67} - 6q^{68} + 4q^{70} + 18q^{71} - 5q^{73} + 4q^{74} + 3q^{76} + 4q^{77} + 16q^{79} - 4q^{80} - 11q^{82} + q^{83} + 6q^{85} + 5q^{86} + 2q^{88} + 5q^{89} - 13q^{91} + q^{92} - 3q^{95} + 7q^{97} - 14q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 5.13277 1.94001 0.970003 0.243094i \(-0.0781625\pi\)
0.970003 + 0.243094i \(0.0781625\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.22069 −0.669564 −0.334782 0.942296i \(-0.608663\pi\)
−0.334782 + 0.942296i \(0.608663\pi\)
\(12\) 0 0
\(13\) −6.60629 −1.83225 −0.916127 0.400888i \(-0.868702\pi\)
−0.916127 + 0.400888i \(0.868702\pi\)
\(14\) −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 \(0.739656\pi\)
\(18\) 0 0
\(19\) −7.51836 −1.72483 −0.862415 0.506201i \(-0.831049\pi\)
−0.862415 + 0.506201i \(0.831049\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.22069 0.473454
\(23\) 6.10064 1.27207 0.636036 0.771659i \(-0.280573\pi\)
0.636036 + 0.771659i \(0.280573\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.60629 1.29560
\(27\) 0 0
\(28\) 5.13277 0.970003
\(29\) 7.29767 1.35514 0.677572 0.735457i \(-0.263032\pi\)
0.677572 + 0.735457i \(0.263032\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 5.63841 0.966980
\(35\) −5.13277 −0.867597
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 7.51836 1.21964
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 1.03213 0.161191 0.0805955 0.996747i \(-0.474318\pi\)
0.0805955 + 0.996747i \(0.474318\pi\)
\(42\) 0 0
\(43\) 5.29767 0.807887 0.403944 0.914784i \(-0.367639\pi\)
0.403944 + 0.914784i \(0.367639\pi\)
\(44\) −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 \(0.359458\pi\)
\(48\) 0 0
\(49\) 19.3453 2.76362
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −6.60629 −0.916127
\(53\) 9.63841 1.32394 0.661969 0.749531i \(-0.269721\pi\)
0.661969 + 0.749531i \(0.269721\pi\)
\(54\) 0 0
\(55\) 2.22069 0.299438
\(56\) −5.13277 −0.685895
\(57\) 0 0
\(58\) −7.29767 −0.958231
\(59\) 10.2655 1.33646 0.668230 0.743955i \(-0.267052\pi\)
0.668230 + 0.743955i \(0.267052\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.60629 0.819409
\(66\) 0 0
\(67\) 1.49436 0.182565 0.0912825 0.995825i \(-0.470903\pi\)
0.0912825 + 0.995825i \(0.470903\pi\)
\(68\) −5.63841 −0.683758
\(69\) 0 0
\(70\) 5.13277 0.613484
\(71\) −0.329796 −0.0391396 −0.0195698 0.999808i \(-0.506230\pi\)
−0.0195698 + 0.999808i \(0.506230\pi\)
\(72\) 0 0
\(73\) 12.8718 1.50653 0.753267 0.657715i \(-0.228477\pi\)
0.753267 + 0.657715i \(0.228477\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −7.51836 −0.862415
\(77\) −11.3983 −1.29896
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −1.03213 −0.113979
\(83\) −9.04767 −0.993111 −0.496556 0.868005i \(-0.665402\pi\)
−0.496556 + 0.868005i \(0.665402\pi\)
\(84\) 0 0
\(85\) 5.63841 0.611572
\(86\) −5.29767 −0.571262
\(87\) 0 0
\(88\) 2.22069 0.236727
\(89\) 2.27649 0.241307 0.120654 0.992695i \(-0.461501\pi\)
0.120654 + 0.992695i \(0.461501\pi\)
\(90\) 0 0
\(91\) −33.9086 −3.55458
\(92\) 6.10064 0.636036
\(93\) 0 0
\(94\) −5.85911 −0.604321
\(95\) 7.51836 0.771368
\(96\) 0 0
\(97\) 17.5982 1.78682 0.893411 0.449239i \(-0.148305\pi\)
0.893411 + 0.449239i \(0.148305\pi\)
\(98\) −19.3453 −1.95417
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −7.61901 −0.758120 −0.379060 0.925372i \(-0.623753\pi\)
−0.379060 + 0.925372i \(0.623753\pi\)
\(102\) 0 0
\(103\) 5.85911 0.577315 0.288657 0.957432i \(-0.406791\pi\)
0.288657 + 0.957432i \(0.406791\pi\)
\(104\) 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 \(0.588198\pi\)
\(108\) 0 0
\(109\) −3.16772 −0.303413 −0.151706 0.988426i \(-0.548477\pi\)
−0.151706 + 0.988426i \(0.548477\pi\)
\(110\) −2.22069 −0.211735
\(111\) 0 0
\(112\) 5.13277 0.485001
\(113\) 0.702331 0.0660697 0.0330348 0.999454i \(-0.489483\pi\)
0.0330348 + 0.999454i \(0.489483\pi\)
\(114\) 0 0
\(115\) −6.10064 −0.568888
\(116\) 7.29767 0.677572
\(117\) 0 0
\(118\) −10.2655 −0.945020
\(119\) −28.9407 −2.65299
\(120\) 0 0
\(121\) −6.06852 −0.551683
\(122\) −6.32134 −0.572307
\(123\) 0 0
\(124\) −0.967873 −0.0869176
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −19.3132 −1.71377 −0.856885 0.515507i \(-0.827604\pi\)
−0.856885 + 0.515507i \(0.827604\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.60629 −0.579410
\(131\) 18.5953 1.62468 0.812341 0.583183i \(-0.198193\pi\)
0.812341 + 0.583183i \(0.198193\pi\)
\(132\) 0 0
\(133\) −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 \(0.186440\pi\)
\(138\) 0 0
\(139\) 10.9036 0.924833 0.462417 0.886663i \(-0.346982\pi\)
0.462417 + 0.886663i \(0.346982\pi\)
\(140\) −5.13277 −0.433798
\(141\) 0 0
\(142\) 0.329796 0.0276759
\(143\) 14.6705 1.22681
\(144\) 0 0
\(145\) −7.29767 −0.606038
\(146\) −12.8718 −1.06528
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) 12.6720 1.03813 0.519065 0.854735i \(-0.326280\pi\)
0.519065 + 0.854735i \(0.326280\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 11.3983 0.918502
\(155\) 0.967873 0.0777415
\(156\) 0 0
\(157\) −15.4990 −1.23695 −0.618476 0.785804i \(-0.712250\pi\)
−0.618476 + 0.785804i \(0.712250\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 31.3132 2.46783
\(162\) 0 0
\(163\) −4.12149 −0.322820 −0.161410 0.986887i \(-0.551604\pi\)
−0.161410 + 0.986887i \(0.551604\pi\)
\(164\) 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 \(0.822897\pi\)
\(168\) 0 0
\(169\) 30.6430 2.35715
\(170\) −5.63841 −0.432446
\(171\) 0 0
\(172\) 5.29767 0.403944
\(173\) 0.425841 0.0323761 0.0161880 0.999869i \(-0.494847\pi\)
0.0161880 + 0.999869i \(0.494847\pi\)
\(174\) 0 0
\(175\) 5.13277 0.388001
\(176\) −2.22069 −0.167391
\(177\) 0 0
\(178\) −2.27649 −0.170630
\(179\) 17.5424 1.31118 0.655589 0.755118i \(-0.272420\pi\)
0.655589 + 0.755118i \(0.272420\pi\)
\(180\) 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
\(183\) 0 0
\(184\) −6.10064 −0.449746
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 12.5212 0.915640
\(188\) 5.85911 0.427319
\(189\) 0 0
\(190\) −7.51836 −0.545439
\(191\) −16.1695 −1.16998 −0.584992 0.811039i \(-0.698902\pi\)
−0.584992 + 0.811039i \(0.698902\pi\)
\(192\) 0 0
\(193\) −11.8591 −0.853637 −0.426819 0.904337i \(-0.640366\pi\)
−0.426819 + 0.904337i \(0.640366\pi\)
\(194\) −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 \(0.501141\pi\)
\(198\) 0 0
\(199\) −2.70693 −0.191889 −0.0959446 0.995387i \(-0.530587\pi\)
−0.0959446 + 0.995387i \(0.530587\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 7.61901 0.536072
\(203\) 37.4573 2.62898
\(204\) 0 0
\(205\) −1.03213 −0.0720868
\(206\) −5.85911 −0.408223
\(207\) 0 0
\(208\) −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 0
\(214\) 5.65926 0.386859
\(215\) −5.29767 −0.361298
\(216\) 0 0
\(217\) −4.96787 −0.337241
\(218\) 3.16772 0.214545
\(219\) 0 0
\(220\) 2.22069 0.149719
\(221\) 37.2490 2.50564
\(222\) 0 0
\(223\) 13.1692 0.881872 0.440936 0.897538i \(-0.354646\pi\)
0.440936 + 0.897538i \(0.354646\pi\)
\(224\) −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 \(0.559281\pi\)
\(228\) 0 0
\(229\) −1.75990 −0.116298 −0.0581488 0.998308i \(-0.518520\pi\)
−0.0581488 + 0.998308i \(0.518520\pi\)
\(230\) 6.10064 0.402265
\(231\) 0 0
\(232\) −7.29767 −0.479115
\(233\) 6.07664 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(234\) 0 0
\(235\) −5.85911 −0.382206
\(236\) 10.2655 0.668230
\(237\) 0 0
\(238\) 28.9407 1.87595
\(239\) −13.9403 −0.901726 −0.450863 0.892593i \(-0.648884\pi\)
−0.450863 + 0.892593i \(0.648884\pi\)
\(240\) 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
\(243\) 0 0
\(244\) 6.32134 0.404682
\(245\) −19.3453 −1.23593
\(246\) 0 0
\(247\) 49.6685 3.16033
\(248\) 0.967873 0.0614600
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −20.9308 −1.32114 −0.660570 0.750765i \(-0.729685\pi\)
−0.660570 + 0.750765i \(0.729685\pi\)
\(252\) 0 0
\(253\) −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 \(0.455874\pi\)
\(258\) 0 0
\(259\) −5.13277 −0.318935
\(260\) 6.60629 0.409704
\(261\) 0 0
\(262\) −18.5953 −1.14882
\(263\) −4.20975 −0.259584 −0.129792 0.991541i \(-0.541431\pi\)
−0.129792 + 0.991541i \(0.541431\pi\)
\(264\) 0 0
\(265\) −9.63841 −0.592083
\(266\) 38.5900 2.36611
\(267\) 0 0
\(268\) 1.49436 0.0912825
\(269\) −12.5071 −0.762570 −0.381285 0.924457i \(-0.624518\pi\)
−0.381285 + 0.924457i \(0.624518\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −19.5074 −1.17849
\(275\) −2.22069 −0.133913
\(276\) 0 0
\(277\) −27.0477 −1.62514 −0.812569 0.582866i \(-0.801931\pi\)
−0.812569 + 0.582866i \(0.801931\pi\)
\(278\) −10.9036 −0.653956
\(279\) 0 0
\(280\) 5.13277 0.306742
\(281\) 1.39371 0.0831420 0.0415710 0.999136i \(-0.486764\pi\)
0.0415710 + 0.999136i \(0.486764\pi\)
\(282\) 0 0
\(283\) −15.3132 −0.910276 −0.455138 0.890421i \(-0.650410\pi\)
−0.455138 + 0.890421i \(0.650410\pi\)
\(284\) −0.329796 −0.0195698
\(285\) 0 0
\(286\) −14.6705 −0.867487
\(287\) 5.29767 0.312712
\(288\) 0 0
\(289\) 14.7917 0.870100
\(290\) 7.29767 0.428534
\(291\) 0 0
\(292\) 12.8718 0.753267
\(293\) 12.8672 0.751712 0.375856 0.926678i \(-0.377349\pi\)
0.375856 + 0.926678i \(0.377349\pi\)
\(294\) 0 0
\(295\) −10.2655 −0.597683
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −12.6720 −0.734068
\(299\) −40.3026 −2.33076
\(300\) 0 0
\(301\) 27.1917 1.56731
\(302\) −10.5420 −0.606626
\(303\) 0 0
\(304\) −7.51836 −0.431208
\(305\) −6.32134 −0.361959
\(306\) 0 0
\(307\) −4.08970 −0.233411 −0.116706 0.993167i \(-0.537233\pi\)
−0.116706 + 0.993167i \(0.537233\pi\)
\(308\) −11.3983 −0.649479
\(309\) 0 0
\(310\) −0.967873 −0.0549715
\(311\) 10.3090 0.584567 0.292283 0.956332i \(-0.405585\pi\)
0.292283 + 0.956332i \(0.405585\pi\)
\(312\) 0 0
\(313\) −2.64653 −0.149591 −0.0747955 0.997199i \(-0.523830\pi\)
−0.0747955 + 0.997199i \(0.523830\pi\)
\(314\) 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 \(0.427240\pi\)
\(318\) 0 0
\(319\) −16.2059 −0.907356
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −31.3132 −1.74502
\(323\) 42.3916 2.35873
\(324\) 0 0
\(325\) −6.60629 −0.366451
\(326\) 4.12149 0.228268
\(327\) 0 0
\(328\) −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
\(333\) 0 0
\(334\) 21.9474 1.20091
\(335\) −1.49436 −0.0816456
\(336\) 0 0
\(337\) −23.4891 −1.27953 −0.639765 0.768570i \(-0.720968\pi\)
−0.639765 + 0.768570i \(0.720968\pi\)
\(338\) −30.6430 −1.66676
\(339\) 0 0
\(340\) 5.63841 0.305786
\(341\) 2.14935 0.116394
\(342\) 0 0
\(343\) 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 \(0.667447\pi\)
\(348\) 0 0
\(349\) 16.1123 0.862470 0.431235 0.902240i \(-0.358078\pi\)
0.431235 + 0.902240i \(0.358078\pi\)
\(350\) −5.13277 −0.274358
\(351\) 0 0
\(352\) 2.22069 0.118363
\(353\) −2.37320 −0.126313 −0.0631565 0.998004i \(-0.520117\pi\)
−0.0631565 + 0.998004i \(0.520117\pi\)
\(354\) 0 0
\(355\) 0.329796 0.0175038
\(356\) 2.27649 0.120654
\(357\) 0 0
\(358\) −17.5424 −0.927143
\(359\) 18.5953 0.981424 0.490712 0.871322i \(-0.336737\pi\)
0.490712 + 0.871322i \(0.336737\pi\)
\(360\) 0 0
\(361\) 37.5258 1.97504
\(362\) −18.5311 −0.973972
\(363\) 0 0
\(364\) −33.9086 −1.77729
\(365\) −12.8718 −0.673742
\(366\) 0 0
\(367\) −29.9520 −1.56348 −0.781740 0.623605i \(-0.785668\pi\)
−0.781740 + 0.623605i \(0.785668\pi\)
\(368\) 6.10064 0.318018
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) 49.4718 2.56845
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) −12.5212 −0.647455
\(375\) 0 0
\(376\) −5.85911 −0.302160
\(377\) −48.2105 −2.48297
\(378\) 0 0
\(379\) 23.3185 1.19779 0.598896 0.800827i \(-0.295606\pi\)
0.598896 + 0.800827i \(0.295606\pi\)
\(380\) 7.51836 0.385684
\(381\) 0 0
\(382\) 16.1695 0.827304
\(383\) 16.6063 0.848542 0.424271 0.905535i \(-0.360530\pi\)
0.424271 + 0.905535i \(0.360530\pi\)
\(384\) 0 0
\(385\) 11.3983 0.580912
\(386\) 11.8591 0.603613
\(387\) 0 0
\(388\) 17.5982 0.893411
\(389\) −15.5795 −0.789910 −0.394955 0.918701i \(-0.629240\pi\)
−0.394955 + 0.918701i \(0.629240\pi\)
\(390\) 0 0
\(391\) −34.3980 −1.73958
\(392\) −19.3453 −0.977087
\(393\) 0 0
\(394\) 0.100645 0.00507042
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −23.2126 −1.16501 −0.582503 0.812829i \(-0.697926\pi\)
−0.582503 + 0.812829i \(0.697926\pi\)
\(398\) 2.70693 0.135686
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −5.72351 −0.285818 −0.142909 0.989736i \(-0.545646\pi\)
−0.142909 + 0.989736i \(0.545646\pi\)
\(402\) 0 0
\(403\) 6.39405 0.318510
\(404\) −7.61901 −0.379060
\(405\) 0 0
\(406\) −37.4573 −1.85897
\(407\) 2.22069 0.110076
\(408\) 0 0
\(409\) 28.0254 1.38577 0.692885 0.721049i \(-0.256340\pi\)
0.692885 + 0.721049i \(0.256340\pi\)
\(410\) 1.03213 0.0509731
\(411\) 0 0
\(412\) 5.85911 0.288657
\(413\) 52.6907 2.59274
\(414\) 0 0
\(415\) 9.04767 0.444133
\(416\) 6.60629 0.323900
\(417\) 0 0
\(418\) −16.6960 −0.816627
\(419\) −16.2415 −0.793451 −0.396726 0.917937i \(-0.629854\pi\)
−0.396726 + 0.917937i \(0.629854\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −9.63841 −0.468083
\(425\) −5.63841 −0.273503
\(426\) 0 0
\(427\) 32.4460 1.57017
\(428\) −5.65926 −0.273551
\(429\) 0 0
\(430\) 5.29767 0.255476
\(431\) 33.2924 1.60364 0.801819 0.597568i \(-0.203866\pi\)
0.801819 + 0.597568i \(0.203866\pi\)
\(432\) 0 0
\(433\) 8.67054 0.416680 0.208340 0.978057i \(-0.433194\pi\)
0.208340 + 0.978057i \(0.433194\pi\)
\(434\) 4.96787 0.238466
\(435\) 0 0
\(436\) −3.16772 −0.151706
\(437\) −45.8669 −2.19411
\(438\) 0 0
\(439\) 32.8874 1.56963 0.784814 0.619731i \(-0.212758\pi\)
0.784814 + 0.619731i \(0.212758\pi\)
\(440\) −2.22069 −0.105867
\(441\) 0 0
\(442\) −37.2490 −1.77175
\(443\) −14.8185 −0.704049 −0.352025 0.935991i \(-0.614507\pi\)
−0.352025 + 0.935991i \(0.614507\pi\)
\(444\) 0 0
\(445\) −2.27649 −0.107916
\(446\) −13.1692 −0.623578
\(447\) 0 0
\(448\) 5.13277 0.242501
\(449\) −9.22882 −0.435535 −0.217767 0.976001i \(-0.569877\pi\)
−0.217767 + 0.976001i \(0.569877\pi\)
\(450\) 0 0
\(451\) −2.29204 −0.107928
\(452\) 0.702331 0.0330348
\(453\) 0 0
\(454\) 5.57946 0.261857
\(455\) 33.9086 1.58966
\(456\) 0 0
\(457\) −36.4753 −1.70624 −0.853121 0.521713i \(-0.825293\pi\)
−0.853121 + 0.521713i \(0.825293\pi\)
\(458\) 1.75990 0.0822348
\(459\) 0 0
\(460\) −6.10064 −0.284444
\(461\) −26.6642 −1.24188 −0.620938 0.783860i \(-0.713248\pi\)
−0.620938 + 0.783860i \(0.713248\pi\)
\(462\) 0 0
\(463\) 6.25249 0.290578 0.145289 0.989389i \(-0.453589\pi\)
0.145289 + 0.989389i \(0.453589\pi\)
\(464\) 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 \(0.622096\pi\)
\(468\) 0 0
\(469\) 7.67020 0.354177
\(470\) 5.85911 0.270260
\(471\) 0 0
\(472\) −10.2655 −0.472510
\(473\) −11.7645 −0.540932
\(474\) 0 0
\(475\) −7.51836 −0.344966
\(476\) −28.9407 −1.32649
\(477\) 0 0
\(478\) 13.9403 0.637617
\(479\) −7.72351 −0.352896 −0.176448 0.984310i \(-0.556461\pi\)
−0.176448 + 0.984310i \(0.556461\pi\)
\(480\) 0 0
\(481\) 6.60629 0.301221
\(482\) 3.89406 0.177369
\(483\) 0 0
\(484\) −6.06852 −0.275842
\(485\) −17.5982 −0.799091
\(486\) 0 0
\(487\) 35.8429 1.62420 0.812098 0.583522i \(-0.198326\pi\)
0.812098 + 0.583522i \(0.198326\pi\)
\(488\) −6.32134 −0.286154
\(489\) 0 0
\(490\) 19.3453 0.873934
\(491\) −2.74718 −0.123978 −0.0619892 0.998077i \(-0.519744\pi\)
−0.0619892 + 0.998077i \(0.519744\pi\)
\(492\) 0 0
\(493\) −41.1473 −1.85318
\(494\) −49.6685 −2.23469
\(495\) 0 0
\(496\) −0.967873 −0.0434588
\(497\) −1.69277 −0.0759310
\(498\) 0 0
\(499\) −3.07698 −0.137744 −0.0688722 0.997625i \(-0.521940\pi\)
−0.0688722 + 0.997625i \(0.521940\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 20.9308 0.934187
\(503\) −8.88278 −0.396063 −0.198032 0.980196i \(-0.563455\pi\)
−0.198032 + 0.980196i \(0.563455\pi\)
\(504\) 0 0
\(505\) 7.61901 0.339041
\(506\) 13.5477 0.602267
\(507\) 0 0
\(508\) −19.3132 −0.856885
\(509\) −2.32836 −0.103203 −0.0516013 0.998668i \(-0.516432\pi\)
−0.0516013 + 0.998668i \(0.516432\pi\)
\(510\) 0 0
\(511\) 66.0682 2.92268
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −4.43044 −0.195418
\(515\) −5.85911 −0.258183
\(516\) 0 0
\(517\) −13.0113 −0.572236
\(518\) 5.13277 0.225521
\(519\) 0 0
\(520\) −6.60629 −0.289705
\(521\) 26.3344 1.15373 0.576865 0.816839i \(-0.304276\pi\)
0.576865 + 0.816839i \(0.304276\pi\)
\(522\) 0 0
\(523\) 10.9887 0.480503 0.240252 0.970711i \(-0.422770\pi\)
0.240252 + 0.970711i \(0.422770\pi\)
\(524\) 18.5953 0.812341
\(525\) 0 0
\(526\) 4.20975 0.183554
\(527\) 5.45727 0.237722
\(528\) 0 0
\(529\) 14.2179 0.618168
\(530\) 9.63841 0.418666
\(531\) 0 0
\(532\) −38.5900 −1.67309
\(533\) −6.81852 −0.295343
\(534\) 0 0
\(535\) 5.65926 0.244671
\(536\) −1.49436 −0.0645465
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) 5.63841 0.241745
\(545\) 3.16772 0.135690
\(546\) 0 0
\(547\) −23.1801 −0.991110 −0.495555 0.868577i \(-0.665035\pi\)
−0.495555 + 0.868577i \(0.665035\pi\)
\(548\) 19.5074 0.833316
\(549\) 0 0
\(550\) 2.22069 0.0946907
\(551\) −54.8665 −2.33739
\(552\) 0 0
\(553\) 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 0
\(559\) −34.9979 −1.48025
\(560\) −5.13277 −0.216899
\(561\) 0 0
\(562\) −1.39371 −0.0587903
\(563\) −33.3705 −1.40640 −0.703198 0.710994i \(-0.748245\pi\)
−0.703198 + 0.710994i \(0.748245\pi\)
\(564\) 0 0
\(565\) −0.702331 −0.0295473
\(566\) 15.3132 0.643663
\(567\) 0 0
\(568\) 0.329796 0.0138379
\(569\) 0.500343 0.0209755 0.0104877 0.999945i \(-0.496662\pi\)
0.0104877 + 0.999945i \(0.496662\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −5.29767 −0.221120
\(575\) 6.10064 0.254414
\(576\) 0 0
\(577\) 12.3902 0.515810 0.257905 0.966170i \(-0.416968\pi\)
0.257905 + 0.966170i \(0.416968\pi\)
\(578\) −14.7917 −0.615253
\(579\) 0 0
\(580\) −7.29767 −0.303019
\(581\) −46.4396 −1.92664
\(582\) 0 0
\(583\) −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 \(0.505310\pi\)
\(588\) 0 0
\(589\) 7.27682 0.299836
\(590\) 10.2655 0.422626
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) 24.4121 1.00248 0.501242 0.865307i \(-0.332877\pi\)
0.501242 + 0.865307i \(0.332877\pi\)
\(594\) 0 0
\(595\) 28.9407 1.18645
\(596\) 12.6720 0.519065
\(597\) 0 0
\(598\) 40.3026 1.64810
\(599\) 29.3440 1.19896 0.599481 0.800389i \(-0.295374\pi\)
0.599481 + 0.800389i \(0.295374\pi\)
\(600\) 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
\(603\) 0 0
\(604\) 10.5420 0.428949
\(605\) 6.06852 0.246720
\(606\) 0 0
\(607\) −31.4912 −1.27819 −0.639094 0.769129i \(-0.720690\pi\)
−0.639094 + 0.769129i \(0.720690\pi\)
\(608\) 7.51836 0.304910
\(609\) 0 0
\(610\) 6.32134 0.255944
\(611\) −38.7069 −1.56591
\(612\) 0 0
\(613\) 6.41489 0.259095 0.129548 0.991573i \(-0.458648\pi\)
0.129548 + 0.991573i \(0.458648\pi\)
\(614\) 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 \(0.214545\pi\)
\(618\) 0 0
\(619\) −0.00460032 −0.000184902 0 −9.24512e−5 1.00000i \(-0.500029\pi\)
−9.24512e−5 1.00000i \(0.500029\pi\)
\(620\) 0.967873 0.0388707
\(621\) 0 0
\(622\) −10.3090 −0.413351
\(623\) 11.6847 0.468138
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.64653 0.105777
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) −8.06885 −0.320455
\(635\) 19.3132 0.766422
\(636\) 0 0
\(637\) −127.801 −5.06365
\(638\) 16.2059 0.641597
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 40.8711 1.61431 0.807156 0.590338i \(-0.201005\pi\)
0.807156 + 0.590338i \(0.201005\pi\)
\(642\) 0 0
\(643\) 11.4788 0.452680 0.226340 0.974048i \(-0.427324\pi\)
0.226340 + 0.974048i \(0.427324\pi\)
\(644\) 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 \(0.386558\pi\)
\(648\) 0 0
\(649\) −22.7966 −0.894846
\(650\) 6.60629 0.259120
\(651\) 0 0
\(652\) −4.12149 −0.161410
\(653\) −21.2768 −0.832626 −0.416313 0.909221i \(-0.636678\pi\)
−0.416313 + 0.909221i \(0.636678\pi\)
\(654\) 0 0
\(655\) −18.5953 −0.726580
\(656\) 1.03213 0.0402978
\(657\) 0 0
\(658\) −30.0735 −1.17239
\(659\) −23.5548 −0.917563 −0.458781 0.888549i \(-0.651714\pi\)
−0.458781 + 0.888549i \(0.651714\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 9.04767 0.351118
\(665\) 38.5900 1.49646
\(666\) 0 0
\(667\) 44.5205 1.72384
\(668\) −21.9474 −0.849169
\(669\) 0 0
\(670\) 1.49436 0.0577321
\(671\) −14.0378 −0.541922
\(672\) 0 0
\(673\) 17.5562 0.676742 0.338371 0.941013i \(-0.390124\pi\)
0.338371 + 0.941013i \(0.390124\pi\)
\(674\) 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 0
\(679\) 90.3274 3.46645
\(680\) −5.63841 −0.216223
\(681\) 0 0
\(682\) −2.14935 −0.0823029
\(683\) 34.7532 1.32980 0.664898 0.746935i \(-0.268475\pi\)
0.664898 + 0.746935i \(0.268475\pi\)
\(684\) 0 0
\(685\) −19.5074 −0.745340
\(686\) −63.3658 −2.41932
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) 18.7069 0.710105
\(695\) −10.9036 −0.413598
\(696\) 0 0
\(697\) −5.81955 −0.220431
\(698\) −16.1123 −0.609858
\(699\) 0 0
\(700\) 5.13277 0.194001
\(701\) 8.81354 0.332883 0.166441 0.986051i \(-0.446772\pi\)
0.166441 + 0.986051i \(0.446772\pi\)
\(702\) 0 0
\(703\) 7.51836 0.283560
\(704\) −2.22069 −0.0836956
\(705\) 0 0
\(706\) 2.37320 0.0893167
\(707\) −39.1066 −1.47076
\(708\) 0 0
\(709\) −9.07238 −0.340720 −0.170360 0.985382i \(-0.554493\pi\)
−0.170360 + 0.985382i \(0.554493\pi\)
\(710\) −0.329796 −0.0123770
\(711\) 0 0
\(712\) −2.27649 −0.0853151
\(713\) −5.90465 −0.221131
\(714\) 0 0
\(715\) −14.6705 −0.548647
\(716\) 17.5424 0.655589
\(717\) 0 0
\(718\) −18.5953 −0.693972
\(719\) −22.8665 −0.852778 −0.426389 0.904540i \(-0.640214\pi\)
−0.426389 + 0.904540i \(0.640214\pi\)
\(720\) 0 0
\(721\) 30.0735 1.11999
\(722\) −37.5258 −1.39657
\(723\) 0 0
\(724\) 18.5311 0.688702
\(725\) 7.29767 0.271029
\(726\) 0 0
\(727\) −2.86407 −0.106222 −0.0531112 0.998589i \(-0.516914\pi\)
−0.0531112 + 0.998589i \(0.516914\pi\)
\(728\) 33.9086 1.25673
\(729\) 0 0
\(730\) 12.8718 0.476408
\(731\) −29.8704 −1.10480
\(732\) 0 0
\(733\) −21.6360 −0.799144 −0.399572 0.916702i \(-0.630841\pi\)
−0.399572 + 0.916702i \(0.630841\pi\)
\(734\) 29.9520 1.10555
\(735\) 0 0
\(736\) −6.10064 −0.224873
\(737\) −3.31851 −0.122239
\(738\) 0 0
\(739\) 16.7278 0.615341 0.307671 0.951493i \(-0.400451\pi\)
0.307671 + 0.951493i \(0.400451\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) −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 0
\(745\) −12.6720 −0.464265
\(746\) 6.00000 0.219676
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 48.2105 1.75572
\(755\) −10.5420 −0.383664
\(756\) 0 0
\(757\) 1.97281 0.0717030 0.0358515 0.999357i \(-0.488586\pi\)
0.0358515 + 0.999357i \(0.488586\pi\)
\(758\) −23.3185 −0.846967
\(759\) 0 0
\(760\) −7.51836 −0.272720
\(761\) −25.4735 −0.923414 −0.461707 0.887032i \(-0.652763\pi\)
−0.461707 + 0.887032i \(0.652763\pi\)
\(762\) 0 0
\(763\) −16.2592 −0.588622
\(764\) −16.1695 −0.584992
\(765\) 0 0
\(766\) −16.6063 −0.600009
\(767\) −67.8171 −2.44873
\(768\) 0 0
\(769\) −41.9675 −1.51339 −0.756694 0.653770i \(-0.773187\pi\)
−0.756694 + 0.653770i \(0.773187\pi\)
\(770\) −11.3983 −0.410767
\(771\) 0 0
\(772\) −11.8591 −0.426819
\(773\) −46.0410 −1.65598 −0.827990 0.560743i \(-0.810515\pi\)
−0.827990 + 0.560743i \(0.810515\pi\)
\(774\) 0 0
\(775\) −0.967873 −0.0347670
\(776\) −17.5982 −0.631737
\(777\) 0 0
\(778\) 15.5795 0.558551
\(779\) −7.75990 −0.278027
\(780\) 0 0
\(781\) 0.732376 0.0262065
\(782\) 34.3980 1.23007
\(783\) 0 0
\(784\) 19.3453 0.690905
\(785\) 15.4990 0.553182
\(786\) 0 0
\(787\) −8.04801 −0.286881 −0.143440 0.989659i \(-0.545816\pi\)
−0.143440 + 0.989659i \(0.545816\pi\)
\(788\) −0.100645 −0.00358533
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) 3.60490 0.128176
\(792\) 0 0
\(793\) −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 \(0.345983\pi\)
\(798\) 0 0
\(799\) −33.0361 −1.16873
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 5.72351 0.202104
\(803\) −28.5844 −1.00872
\(804\) 0 0
\(805\) −31.3132 −1.10365
\(806\) −6.39405 −0.225221
\(807\) 0 0
\(808\) 7.61901 0.268036
\(809\) −13.3301 −0.468662 −0.234331 0.972157i \(-0.575290\pi\)
−0.234331 + 0.972157i \(0.575290\pi\)
\(810\) 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
\(813\) 0 0
\(814\) −2.22069 −0.0778353
\(815\) 4.12149 0.144369
\(816\) 0 0
\(817\) −39.8298 −1.39347
\(818\) −28.0254 −0.979887
\(819\) 0 0
\(820\) −1.03213 −0.0360434
\(821\) 26.2895 0.917512 0.458756 0.888562i \(-0.348295\pi\)
0.458756 + 0.888562i \(0.348295\pi\)
\(822\) 0 0
\(823\) 24.4050 0.850705 0.425352 0.905028i \(-0.360150\pi\)
0.425352 + 0.905028i \(0.360150\pi\)
\(824\) −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 \(0.273282\pi\)
\(828\) 0 0
\(829\) 36.4481 1.26589 0.632947 0.774195i \(-0.281845\pi\)
0.632947 + 0.774195i \(0.281845\pi\)
\(830\) −9.04767 −0.314049
\(831\) 0 0
\(832\) −6.60629 −0.229032
\(833\) −109.077 −3.77929
\(834\) 0 0
\(835\) 21.9474 0.759520
\(836\) 16.6960 0.577443
\(837\) 0 0
\(838\) 16.2415 0.561055
\(839\) 43.3277 1.49584 0.747919 0.663790i \(-0.231053\pi\)
0.747919 + 0.663790i \(0.231053\pi\)
\(840\) 0 0
\(841\) 24.2560 0.836413
\(842\) −6.03495 −0.207978
\(843\) 0 0
\(844\) 11.8033 0.406287
\(845\) −30.6430 −1.05415
\(846\) 0 0
\(847\) −31.1483 −1.07027
\(848\) 9.63841 0.330984
\(849\) 0 0
\(850\) 5.63841 0.193396
\(851\) −6.10064 −0.209127
\(852\) 0 0
\(853\) 17.6430 0.604085 0.302043 0.953294i \(-0.402332\pi\)
0.302043 + 0.953294i \(0.402332\pi\)
\(854\) −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 \(0.708194\pi\)
\(858\) 0 0
\(859\) −5.49292 −0.187416 −0.0937080 0.995600i \(-0.529872\pi\)
−0.0937080 + 0.995600i \(0.529872\pi\)
\(860\) −5.29767 −0.180649
\(861\) 0 0
\(862\) −33.2924 −1.13394
\(863\) 1.12501 0.0382958 0.0191479 0.999817i \(-0.493905\pi\)
0.0191479 + 0.999817i \(0.493905\pi\)
\(864\) 0 0
\(865\) −0.425841 −0.0144790
\(866\) −8.67054 −0.294637
\(867\) 0 0
\(868\) −4.96787 −0.168621
\(869\) −8.88278 −0.301328
\(870\) 0 0
\(871\) −9.87216 −0.334506
\(872\) 3.16772 0.107273
\(873\) 0 0
\(874\) 45.8669 1.55147
\(875\) −5.13277 −0.173519
\(876\) 0 0
\(877\) 31.6367 1.06829 0.534147 0.845392i \(-0.320633\pi\)
0.534147 + 0.845392i \(0.320633\pi\)
\(878\) −32.8874 −1.10990
\(879\) 0 0
\(880\) 2.22069 0.0748596
\(881\) 24.8012 0.835575 0.417787 0.908545i \(-0.362806\pi\)
0.417787 + 0.908545i \(0.362806\pi\)
\(882\) 0 0
\(883\) −27.7980 −0.935478 −0.467739 0.883867i \(-0.654931\pi\)
−0.467739 + 0.883867i \(0.654931\pi\)
\(884\) 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 \(0.341054\pi\)
\(888\) 0 0
\(889\) −99.1303 −3.32472
\(890\) 2.27649 0.0763081
\(891\) 0 0
\(892\) 13.1692 0.440936
\(893\) −44.0509 −1.47411
\(894\) 0 0
\(895\) −17.5424 −0.586377
\(896\) −5.13277 −0.171474
\(897\) 0 0
\(898\) 9.22882 0.307970
\(899\) −7.06322 −0.235572
\(900\) 0 0
\(901\) −54.3453 −1.81051
\(902\) 2.29204 0.0763165
\(903\) 0 0
\(904\) −0.702331 −0.0233592
\(905\) −18.5311 −0.615994
\(906\) 0 0
\(907\) 41.1854 1.36754 0.683769 0.729698i \(-0.260340\pi\)
0.683769 + 0.729698i \(0.260340\pi\)
\(908\) −5.57946 −0.185161
\(909\) 0 0
\(910\) −33.9086 −1.12406
\(911\) −21.3440 −0.707157 −0.353578 0.935405i \(-0.615035\pi\)
−0.353578 + 0.935405i \(0.615035\pi\)
\(912\) 0 0
\(913\) 20.0921 0.664952
\(914\) 36.4753 1.20650
\(915\) 0 0
\(916\) −1.75990 −0.0581488
\(917\) 95.4456 3.15189
\(918\) 0 0
\(919\) 36.9089 1.21751 0.608756 0.793358i \(-0.291669\pi\)
0.608756 + 0.793358i \(0.291669\pi\)
\(920\) 6.10064 0.201132
\(921\) 0 0
\(922\) 26.6642 0.878138
\(923\) 2.17873 0.0717137
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) −6.25249 −0.205469
\(927\) 0 0
\(928\) −7.29767 −0.239558
\(929\) −21.0350 −0.690136 −0.345068 0.938578i \(-0.612144\pi\)
−0.345068 + 0.938578i \(0.612144\pi\)
\(930\) 0 0
\(931\) −145.445 −4.76678
\(932\) 6.07664 0.199047
\(933\) 0 0
\(934\) 16.1748 0.529255
\(935\) −12.5212 −0.409487
\(936\) 0 0
\(937\) 9.67652 0.316118 0.158059 0.987430i \(-0.449476\pi\)
0.158059 + 0.987430i \(0.449476\pi\)
\(938\) −7.67020 −0.250441
\(939\) 0 0
\(940\) −5.85911 −0.191103
\(941\) 48.1332 1.56910 0.784548 0.620068i \(-0.212895\pi\)
0.784548 + 0.620068i \(0.212895\pi\)
\(942\) 0 0
\(943\) 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 \(0.406677\pi\)
\(948\) 0 0
\(949\) −85.0350 −2.76035
\(950\) 7.51836 0.243928
\(951\) 0 0
\(952\) 28.9407 0.937973
\(953\) −30.0442 −0.973226 −0.486613 0.873618i \(-0.661768\pi\)
−0.486613 + 0.873618i \(0.661768\pi\)
\(954\) 0 0
\(955\) 16.1695 0.523233
\(956\) −13.9403 −0.450863
\(957\) 0 0
\(958\) 7.72351 0.249535
\(959\) 100.127 3.23327
\(960\) 0 0
\(961\) −30.0632 −0.969781
\(962\) −6.60629 −0.212995
\(963\) 0 0
\(964\) −3.89406 −0.125419
\(965\) 11.8591 0.381758
\(966\) 0 0
\(967\) 15.7052 0.505044 0.252522 0.967591i \(-0.418740\pi\)
0.252522 + 0.967591i \(0.418740\pi\)
\(968\) 6.06852 0.195050
\(969\) 0 0
\(970\) 17.5982 0.565043
\(971\) −36.8693 −1.18319 −0.591597 0.806234i \(-0.701502\pi\)
−0.591597 + 0.806234i \(0.701502\pi\)
\(972\) 0 0
\(973\) 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 \(0.670107\pi\)
\(978\) 0 0
\(979\) −5.05539 −0.161571
\(980\) −19.3453 −0.617964
\(981\) 0 0
\(982\) 2.74718 0.0876660
\(983\) −11.8708 −0.378618 −0.189309 0.981918i \(-0.560625\pi\)
−0.189309 + 0.981918i \(0.560625\pi\)
\(984\) 0 0
\(985\) 0.100645 0.00320682
\(986\) 41.1473 1.31040
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) 1.69277 0.0536913
\(995\) 2.70693 0.0858155
\(996\) 0 0
\(997\) −26.4050 −0.836255 −0.418127 0.908388i \(-0.637313\pi\)
−0.418127 + 0.908388i \(0.637313\pi\)
\(998\) 3.07698 0.0974000
\(999\) 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 3330.2.a.bj.1.4 4
3.2 odd 2 1110.2.a.s.1.4 4
12.11 even 2 8880.2.a.cg.1.1 4
15.14 odd 2 5550.2.a.cj.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.s.1.4 4 3.2 odd 2
3330.2.a.bj.1.4 4 1.1 even 1 trivial
5550.2.a.cj.1.1 4 15.14 odd 2
8880.2.a.cg.1.1 4 12.11 even 2