Properties

Label 8034.2.a.ba.1.8
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} - 4126 x^{5} - 22759 x^{4} + 2997 x^{3} + 10504 x^{2} - 506 x - 1492\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.522326\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.522326 q^{5} +1.00000 q^{6} -2.74868 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.522326 q^{5} +1.00000 q^{6} -2.74868 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.522326 q^{10} -0.360369 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.74868 q^{14} -0.522326 q^{15} +1.00000 q^{16} +1.93116 q^{17} -1.00000 q^{18} -0.748297 q^{19} +0.522326 q^{20} +2.74868 q^{21} +0.360369 q^{22} -1.23649 q^{23} +1.00000 q^{24} -4.72718 q^{25} +1.00000 q^{26} -1.00000 q^{27} -2.74868 q^{28} -4.90740 q^{29} +0.522326 q^{30} +1.19890 q^{31} -1.00000 q^{32} +0.360369 q^{33} -1.93116 q^{34} -1.43571 q^{35} +1.00000 q^{36} -2.99837 q^{37} +0.748297 q^{38} +1.00000 q^{39} -0.522326 q^{40} +5.55715 q^{41} -2.74868 q^{42} +5.82398 q^{43} -0.360369 q^{44} +0.522326 q^{45} +1.23649 q^{46} -0.902971 q^{47} -1.00000 q^{48} +0.555246 q^{49} +4.72718 q^{50} -1.93116 q^{51} -1.00000 q^{52} -9.38100 q^{53} +1.00000 q^{54} -0.188230 q^{55} +2.74868 q^{56} +0.748297 q^{57} +4.90740 q^{58} +6.21543 q^{59} -0.522326 q^{60} +6.71192 q^{61} -1.19890 q^{62} -2.74868 q^{63} +1.00000 q^{64} -0.522326 q^{65} -0.360369 q^{66} +2.28386 q^{67} +1.93116 q^{68} +1.23649 q^{69} +1.43571 q^{70} -1.41208 q^{71} -1.00000 q^{72} -15.8684 q^{73} +2.99837 q^{74} +4.72718 q^{75} -0.748297 q^{76} +0.990539 q^{77} -1.00000 q^{78} -7.70563 q^{79} +0.522326 q^{80} +1.00000 q^{81} -5.55715 q^{82} -2.77287 q^{83} +2.74868 q^{84} +1.00870 q^{85} -5.82398 q^{86} +4.90740 q^{87} +0.360369 q^{88} +1.03216 q^{89} -0.522326 q^{90} +2.74868 q^{91} -1.23649 q^{92} -1.19890 q^{93} +0.902971 q^{94} -0.390855 q^{95} +1.00000 q^{96} +7.76875 q^{97} -0.555246 q^{98} -0.360369 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - q^{5} + 14q^{6} + 5q^{7} - 14q^{8} + 14q^{9} + q^{10} - q^{11} - 14q^{12} - 14q^{13} - 5q^{14} + q^{15} + 14q^{16} - 12q^{17} - 14q^{18} + 10q^{19} - q^{20} - 5q^{21} + q^{22} - q^{23} + 14q^{24} + 19q^{25} + 14q^{26} - 14q^{27} + 5q^{28} + 6q^{29} - q^{30} + 20q^{31} - 14q^{32} + q^{33} + 12q^{34} - 16q^{35} + 14q^{36} - 3q^{37} - 10q^{38} + 14q^{39} + q^{40} + q^{41} + 5q^{42} + 6q^{43} - q^{44} - q^{45} + q^{46} - 13q^{47} - 14q^{48} + 9q^{49} - 19q^{50} + 12q^{51} - 14q^{52} - 27q^{53} + 14q^{54} + 10q^{55} - 5q^{56} - 10q^{57} - 6q^{58} - 6q^{59} + q^{60} - 4q^{61} - 20q^{62} + 5q^{63} + 14q^{64} + q^{65} - q^{66} + 13q^{67} - 12q^{68} + q^{69} + 16q^{70} + 18q^{71} - 14q^{72} + 11q^{73} + 3q^{74} - 19q^{75} + 10q^{76} - 15q^{77} - 14q^{78} + 33q^{79} - q^{80} + 14q^{81} - q^{82} - 25q^{83} - 5q^{84} + 25q^{85} - 6q^{86} - 6q^{87} + q^{88} + 3q^{89} + q^{90} - 5q^{91} - q^{92} - 20q^{93} + 13q^{94} + 30q^{95} + 14q^{96} + 11q^{97} - 9q^{98} - q^{99} + O(q^{100}) \)

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
\(5\) 0.522326 0.233591 0.116796 0.993156i \(-0.462738\pi\)
0.116796 + 0.993156i \(0.462738\pi\)
\(6\) 1.00000 0.408248
\(7\) −2.74868 −1.03890 −0.519452 0.854500i \(-0.673864\pi\)
−0.519452 + 0.854500i \(0.673864\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −0.522326 −0.165174
\(11\) −0.360369 −0.108655 −0.0543276 0.998523i \(-0.517302\pi\)
−0.0543276 + 0.998523i \(0.517302\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.74868 0.734616
\(15\) −0.522326 −0.134864
\(16\) 1.00000 0.250000
\(17\) 1.93116 0.468376 0.234188 0.972191i \(-0.424757\pi\)
0.234188 + 0.972191i \(0.424757\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.748297 −0.171671 −0.0858355 0.996309i \(-0.527356\pi\)
−0.0858355 + 0.996309i \(0.527356\pi\)
\(20\) 0.522326 0.116796
\(21\) 2.74868 0.599811
\(22\) 0.360369 0.0768309
\(23\) −1.23649 −0.257827 −0.128913 0.991656i \(-0.541149\pi\)
−0.128913 + 0.991656i \(0.541149\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.72718 −0.945435
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) −2.74868 −0.519452
\(29\) −4.90740 −0.911281 −0.455640 0.890164i \(-0.650590\pi\)
−0.455640 + 0.890164i \(0.650590\pi\)
\(30\) 0.522326 0.0953632
\(31\) 1.19890 0.215330 0.107665 0.994187i \(-0.465663\pi\)
0.107665 + 0.994187i \(0.465663\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.360369 0.0627321
\(34\) −1.93116 −0.331192
\(35\) −1.43571 −0.242679
\(36\) 1.00000 0.166667
\(37\) −2.99837 −0.492929 −0.246465 0.969152i \(-0.579269\pi\)
−0.246465 + 0.969152i \(0.579269\pi\)
\(38\) 0.748297 0.121390
\(39\) 1.00000 0.160128
\(40\) −0.522326 −0.0825869
\(41\) 5.55715 0.867882 0.433941 0.900941i \(-0.357123\pi\)
0.433941 + 0.900941i \(0.357123\pi\)
\(42\) −2.74868 −0.424131
\(43\) 5.82398 0.888148 0.444074 0.895990i \(-0.353533\pi\)
0.444074 + 0.895990i \(0.353533\pi\)
\(44\) −0.360369 −0.0543276
\(45\) 0.522326 0.0778637
\(46\) 1.23649 0.182311
\(47\) −0.902971 −0.131712 −0.0658559 0.997829i \(-0.520978\pi\)
−0.0658559 + 0.997829i \(0.520978\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0.555246 0.0793209
\(50\) 4.72718 0.668524
\(51\) −1.93116 −0.270417
\(52\) −1.00000 −0.138675
\(53\) −9.38100 −1.28858 −0.644290 0.764781i \(-0.722847\pi\)
−0.644290 + 0.764781i \(0.722847\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.188230 −0.0253809
\(56\) 2.74868 0.367308
\(57\) 0.748297 0.0991143
\(58\) 4.90740 0.644373
\(59\) 6.21543 0.809181 0.404590 0.914498i \(-0.367414\pi\)
0.404590 + 0.914498i \(0.367414\pi\)
\(60\) −0.522326 −0.0674320
\(61\) 6.71192 0.859374 0.429687 0.902978i \(-0.358624\pi\)
0.429687 + 0.902978i \(0.358624\pi\)
\(62\) −1.19890 −0.152261
\(63\) −2.74868 −0.346301
\(64\) 1.00000 0.125000
\(65\) −0.522326 −0.0647865
\(66\) −0.360369 −0.0443583
\(67\) 2.28386 0.279018 0.139509 0.990221i \(-0.455448\pi\)
0.139509 + 0.990221i \(0.455448\pi\)
\(68\) 1.93116 0.234188
\(69\) 1.23649 0.148856
\(70\) 1.43571 0.171600
\(71\) −1.41208 −0.167583 −0.0837914 0.996483i \(-0.526703\pi\)
−0.0837914 + 0.996483i \(0.526703\pi\)
\(72\) −1.00000 −0.117851
\(73\) −15.8684 −1.85725 −0.928626 0.371017i \(-0.879009\pi\)
−0.928626 + 0.371017i \(0.879009\pi\)
\(74\) 2.99837 0.348554
\(75\) 4.72718 0.545847
\(76\) −0.748297 −0.0858355
\(77\) 0.990539 0.112882
\(78\) −1.00000 −0.113228
\(79\) −7.70563 −0.866951 −0.433475 0.901165i \(-0.642713\pi\)
−0.433475 + 0.901165i \(0.642713\pi\)
\(80\) 0.522326 0.0583978
\(81\) 1.00000 0.111111
\(82\) −5.55715 −0.613685
\(83\) −2.77287 −0.304362 −0.152181 0.988353i \(-0.548630\pi\)
−0.152181 + 0.988353i \(0.548630\pi\)
\(84\) 2.74868 0.299906
\(85\) 1.00870 0.109409
\(86\) −5.82398 −0.628016
\(87\) 4.90740 0.526128
\(88\) 0.360369 0.0384154
\(89\) 1.03216 0.109409 0.0547045 0.998503i \(-0.482578\pi\)
0.0547045 + 0.998503i \(0.482578\pi\)
\(90\) −0.522326 −0.0550580
\(91\) 2.74868 0.288140
\(92\) −1.23649 −0.128913
\(93\) −1.19890 −0.124321
\(94\) 0.902971 0.0931344
\(95\) −0.390855 −0.0401008
\(96\) 1.00000 0.102062
\(97\) 7.76875 0.788797 0.394398 0.918940i \(-0.370953\pi\)
0.394398 + 0.918940i \(0.370953\pi\)
\(98\) −0.555246 −0.0560883
\(99\) −0.360369 −0.0362184
\(100\) −4.72718 −0.472718
\(101\) 8.86370 0.881971 0.440985 0.897514i \(-0.354629\pi\)
0.440985 + 0.897514i \(0.354629\pi\)
\(102\) 1.93116 0.191214
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 1.43571 0.140111
\(106\) 9.38100 0.911164
\(107\) 8.57097 0.828587 0.414293 0.910143i \(-0.364029\pi\)
0.414293 + 0.910143i \(0.364029\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.80669 0.651962 0.325981 0.945376i \(-0.394305\pi\)
0.325981 + 0.945376i \(0.394305\pi\)
\(110\) 0.188230 0.0179470
\(111\) 2.99837 0.284593
\(112\) −2.74868 −0.259726
\(113\) −12.4885 −1.17482 −0.587410 0.809290i \(-0.699852\pi\)
−0.587410 + 0.809290i \(0.699852\pi\)
\(114\) −0.748297 −0.0700844
\(115\) −0.645852 −0.0602260
\(116\) −4.90740 −0.455640
\(117\) −1.00000 −0.0924500
\(118\) −6.21543 −0.572177
\(119\) −5.30816 −0.486598
\(120\) 0.522326 0.0476816
\(121\) −10.8701 −0.988194
\(122\) −6.71192 −0.607669
\(123\) −5.55715 −0.501072
\(124\) 1.19890 0.107665
\(125\) −5.08075 −0.454436
\(126\) 2.74868 0.244872
\(127\) 7.24136 0.642566 0.321283 0.946983i \(-0.395886\pi\)
0.321283 + 0.946983i \(0.395886\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.82398 −0.512773
\(130\) 0.522326 0.0458110
\(131\) 8.82634 0.771161 0.385580 0.922674i \(-0.374001\pi\)
0.385580 + 0.922674i \(0.374001\pi\)
\(132\) 0.360369 0.0313661
\(133\) 2.05683 0.178350
\(134\) −2.28386 −0.197296
\(135\) −0.522326 −0.0449546
\(136\) −1.93116 −0.165596
\(137\) 12.0119 1.02624 0.513122 0.858315i \(-0.328489\pi\)
0.513122 + 0.858315i \(0.328489\pi\)
\(138\) −1.23649 −0.105257
\(139\) −16.0537 −1.36166 −0.680831 0.732441i \(-0.738381\pi\)
−0.680831 + 0.732441i \(0.738381\pi\)
\(140\) −1.43571 −0.121339
\(141\) 0.902971 0.0760439
\(142\) 1.41208 0.118499
\(143\) 0.360369 0.0301355
\(144\) 1.00000 0.0833333
\(145\) −2.56326 −0.212867
\(146\) 15.8684 1.31328
\(147\) −0.555246 −0.0457959
\(148\) −2.99837 −0.246465
\(149\) −19.4381 −1.59243 −0.796216 0.605013i \(-0.793168\pi\)
−0.796216 + 0.605013i \(0.793168\pi\)
\(150\) −4.72718 −0.385972
\(151\) 2.58033 0.209984 0.104992 0.994473i \(-0.466518\pi\)
0.104992 + 0.994473i \(0.466518\pi\)
\(152\) 0.748297 0.0606949
\(153\) 1.93116 0.156125
\(154\) −0.990539 −0.0798199
\(155\) 0.626218 0.0502991
\(156\) 1.00000 0.0800641
\(157\) 1.50247 0.119910 0.0599552 0.998201i \(-0.480904\pi\)
0.0599552 + 0.998201i \(0.480904\pi\)
\(158\) 7.70563 0.613027
\(159\) 9.38100 0.743962
\(160\) −0.522326 −0.0412935
\(161\) 3.39873 0.267857
\(162\) −1.00000 −0.0785674
\(163\) 4.06884 0.318696 0.159348 0.987222i \(-0.449061\pi\)
0.159348 + 0.987222i \(0.449061\pi\)
\(164\) 5.55715 0.433941
\(165\) 0.188230 0.0146537
\(166\) 2.77287 0.215216
\(167\) −18.3623 −1.42091 −0.710457 0.703741i \(-0.751512\pi\)
−0.710457 + 0.703741i \(0.751512\pi\)
\(168\) −2.74868 −0.212065
\(169\) 1.00000 0.0769231
\(170\) −1.00870 −0.0773635
\(171\) −0.748297 −0.0572237
\(172\) 5.82398 0.444074
\(173\) −5.17700 −0.393600 −0.196800 0.980444i \(-0.563055\pi\)
−0.196800 + 0.980444i \(0.563055\pi\)
\(174\) −4.90740 −0.372029
\(175\) 12.9935 0.982216
\(176\) −0.360369 −0.0271638
\(177\) −6.21543 −0.467181
\(178\) −1.03216 −0.0773639
\(179\) −2.12323 −0.158698 −0.0793488 0.996847i \(-0.525284\pi\)
−0.0793488 + 0.996847i \(0.525284\pi\)
\(180\) 0.522326 0.0389319
\(181\) 22.8106 1.69550 0.847750 0.530396i \(-0.177957\pi\)
0.847750 + 0.530396i \(0.177957\pi\)
\(182\) −2.74868 −0.203746
\(183\) −6.71192 −0.496160
\(184\) 1.23649 0.0911555
\(185\) −1.56613 −0.115144
\(186\) 1.19890 0.0879079
\(187\) −0.695931 −0.0508915
\(188\) −0.902971 −0.0658559
\(189\) 2.74868 0.199937
\(190\) 0.390855 0.0283556
\(191\) 8.47503 0.613232 0.306616 0.951833i \(-0.400803\pi\)
0.306616 + 0.951833i \(0.400803\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −19.8592 −1.42949 −0.714747 0.699383i \(-0.753458\pi\)
−0.714747 + 0.699383i \(0.753458\pi\)
\(194\) −7.76875 −0.557764
\(195\) 0.522326 0.0374045
\(196\) 0.555246 0.0396604
\(197\) −9.82460 −0.699974 −0.349987 0.936755i \(-0.613814\pi\)
−0.349987 + 0.936755i \(0.613814\pi\)
\(198\) 0.360369 0.0256103
\(199\) 12.1513 0.861381 0.430690 0.902500i \(-0.358270\pi\)
0.430690 + 0.902500i \(0.358270\pi\)
\(200\) 4.72718 0.334262
\(201\) −2.28386 −0.161091
\(202\) −8.86370 −0.623647
\(203\) 13.4889 0.946733
\(204\) −1.93116 −0.135209
\(205\) 2.90264 0.202729
\(206\) −1.00000 −0.0696733
\(207\) −1.23649 −0.0859422
\(208\) −1.00000 −0.0693375
\(209\) 0.269663 0.0186530
\(210\) −1.43571 −0.0990732
\(211\) −2.76064 −0.190050 −0.0950250 0.995475i \(-0.530293\pi\)
−0.0950250 + 0.995475i \(0.530293\pi\)
\(212\) −9.38100 −0.644290
\(213\) 1.41208 0.0967539
\(214\) −8.57097 −0.585899
\(215\) 3.04201 0.207464
\(216\) 1.00000 0.0680414
\(217\) −3.29540 −0.223707
\(218\) −6.80669 −0.461007
\(219\) 15.8684 1.07229
\(220\) −0.188230 −0.0126905
\(221\) −1.93116 −0.129904
\(222\) −2.99837 −0.201238
\(223\) 18.8473 1.26211 0.631055 0.775738i \(-0.282622\pi\)
0.631055 + 0.775738i \(0.282622\pi\)
\(224\) 2.74868 0.183654
\(225\) −4.72718 −0.315145
\(226\) 12.4885 0.830723
\(227\) 9.80664 0.650889 0.325445 0.945561i \(-0.394486\pi\)
0.325445 + 0.945561i \(0.394486\pi\)
\(228\) 0.748297 0.0495572
\(229\) 22.6867 1.49918 0.749590 0.661902i \(-0.230251\pi\)
0.749590 + 0.661902i \(0.230251\pi\)
\(230\) 0.645852 0.0425862
\(231\) −0.990539 −0.0651727
\(232\) 4.90740 0.322186
\(233\) −0.545768 −0.0357544 −0.0178772 0.999840i \(-0.505691\pi\)
−0.0178772 + 0.999840i \(0.505691\pi\)
\(234\) 1.00000 0.0653720
\(235\) −0.471645 −0.0307667
\(236\) 6.21543 0.404590
\(237\) 7.70563 0.500534
\(238\) 5.30816 0.344077
\(239\) 19.9894 1.29300 0.646502 0.762912i \(-0.276231\pi\)
0.646502 + 0.762912i \(0.276231\pi\)
\(240\) −0.522326 −0.0337160
\(241\) 23.8871 1.53870 0.769350 0.638827i \(-0.220580\pi\)
0.769350 + 0.638827i \(0.220580\pi\)
\(242\) 10.8701 0.698759
\(243\) −1.00000 −0.0641500
\(244\) 6.71192 0.429687
\(245\) 0.290019 0.0185287
\(246\) 5.55715 0.354311
\(247\) 0.748297 0.0476130
\(248\) −1.19890 −0.0761305
\(249\) 2.77287 0.175723
\(250\) 5.08075 0.321335
\(251\) −27.8402 −1.75726 −0.878628 0.477508i \(-0.841540\pi\)
−0.878628 + 0.477508i \(0.841540\pi\)
\(252\) −2.74868 −0.173151
\(253\) 0.445594 0.0280142
\(254\) −7.24136 −0.454363
\(255\) −1.00870 −0.0631670
\(256\) 1.00000 0.0625000
\(257\) 0.573865 0.0357967 0.0178983 0.999840i \(-0.494302\pi\)
0.0178983 + 0.999840i \(0.494302\pi\)
\(258\) 5.82398 0.362585
\(259\) 8.24157 0.512106
\(260\) −0.522326 −0.0323933
\(261\) −4.90740 −0.303760
\(262\) −8.82634 −0.545293
\(263\) 31.1711 1.92209 0.961045 0.276391i \(-0.0891385\pi\)
0.961045 + 0.276391i \(0.0891385\pi\)
\(264\) −0.360369 −0.0221792
\(265\) −4.89994 −0.301001
\(266\) −2.05683 −0.126112
\(267\) −1.03216 −0.0631673
\(268\) 2.28386 0.139509
\(269\) 4.14299 0.252602 0.126301 0.991992i \(-0.459689\pi\)
0.126301 + 0.991992i \(0.459689\pi\)
\(270\) 0.522326 0.0317877
\(271\) 25.0410 1.52113 0.760566 0.649261i \(-0.224922\pi\)
0.760566 + 0.649261i \(0.224922\pi\)
\(272\) 1.93116 0.117094
\(273\) −2.74868 −0.166358
\(274\) −12.0119 −0.725665
\(275\) 1.70353 0.102727
\(276\) 1.23649 0.0744282
\(277\) −3.29484 −0.197968 −0.0989839 0.995089i \(-0.531559\pi\)
−0.0989839 + 0.995089i \(0.531559\pi\)
\(278\) 16.0537 0.962840
\(279\) 1.19890 0.0717765
\(280\) 1.43571 0.0857999
\(281\) −20.3656 −1.21491 −0.607456 0.794354i \(-0.707810\pi\)
−0.607456 + 0.794354i \(0.707810\pi\)
\(282\) −0.902971 −0.0537711
\(283\) 13.6604 0.812024 0.406012 0.913868i \(-0.366919\pi\)
0.406012 + 0.913868i \(0.366919\pi\)
\(284\) −1.41208 −0.0837914
\(285\) 0.390855 0.0231522
\(286\) −0.360369 −0.0213091
\(287\) −15.2748 −0.901646
\(288\) −1.00000 −0.0589256
\(289\) −13.2706 −0.780624
\(290\) 2.56326 0.150520
\(291\) −7.76875 −0.455412
\(292\) −15.8684 −0.928626
\(293\) −22.1368 −1.29325 −0.646624 0.762809i \(-0.723820\pi\)
−0.646624 + 0.762809i \(0.723820\pi\)
\(294\) 0.555246 0.0323826
\(295\) 3.24648 0.189017
\(296\) 2.99837 0.174277
\(297\) 0.360369 0.0209107
\(298\) 19.4381 1.12602
\(299\) 1.23649 0.0715083
\(300\) 4.72718 0.272924
\(301\) −16.0083 −0.922701
\(302\) −2.58033 −0.148481
\(303\) −8.86370 −0.509206
\(304\) −0.748297 −0.0429178
\(305\) 3.50581 0.200742
\(306\) −1.93116 −0.110397
\(307\) 13.5244 0.771877 0.385939 0.922524i \(-0.373878\pi\)
0.385939 + 0.922524i \(0.373878\pi\)
\(308\) 0.990539 0.0564412
\(309\) −1.00000 −0.0568880
\(310\) −0.626218 −0.0355668
\(311\) −0.984074 −0.0558017 −0.0279009 0.999611i \(-0.508882\pi\)
−0.0279009 + 0.999611i \(0.508882\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −8.44322 −0.477239 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(314\) −1.50247 −0.0847895
\(315\) −1.43571 −0.0808929
\(316\) −7.70563 −0.433475
\(317\) 2.32174 0.130402 0.0652010 0.997872i \(-0.479231\pi\)
0.0652010 + 0.997872i \(0.479231\pi\)
\(318\) −9.38100 −0.526061
\(319\) 1.76847 0.0990155
\(320\) 0.522326 0.0291989
\(321\) −8.57097 −0.478385
\(322\) −3.39873 −0.189404
\(323\) −1.44508 −0.0804066
\(324\) 1.00000 0.0555556
\(325\) 4.72718 0.262217
\(326\) −4.06884 −0.225352
\(327\) −6.80669 −0.376411
\(328\) −5.55715 −0.306843
\(329\) 2.48198 0.136836
\(330\) −0.188230 −0.0103617
\(331\) −8.39888 −0.461644 −0.230822 0.972996i \(-0.574141\pi\)
−0.230822 + 0.972996i \(0.574141\pi\)
\(332\) −2.77287 −0.152181
\(333\) −2.99837 −0.164310
\(334\) 18.3623 1.00474
\(335\) 1.19292 0.0651762
\(336\) 2.74868 0.149953
\(337\) 30.6516 1.66970 0.834850 0.550477i \(-0.185554\pi\)
0.834850 + 0.550477i \(0.185554\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 12.4885 0.678282
\(340\) 1.00870 0.0547043
\(341\) −0.432048 −0.0233967
\(342\) 0.748297 0.0404633
\(343\) 17.7146 0.956497
\(344\) −5.82398 −0.314008
\(345\) 0.645852 0.0347715
\(346\) 5.17700 0.278317
\(347\) −26.7991 −1.43865 −0.719326 0.694673i \(-0.755549\pi\)
−0.719326 + 0.694673i \(0.755549\pi\)
\(348\) 4.90740 0.263064
\(349\) 36.0370 1.92902 0.964509 0.264051i \(-0.0850587\pi\)
0.964509 + 0.264051i \(0.0850587\pi\)
\(350\) −12.9935 −0.694532
\(351\) 1.00000 0.0533761
\(352\) 0.360369 0.0192077
\(353\) −27.9712 −1.48875 −0.744377 0.667759i \(-0.767254\pi\)
−0.744377 + 0.667759i \(0.767254\pi\)
\(354\) 6.21543 0.330347
\(355\) −0.737564 −0.0391458
\(356\) 1.03216 0.0547045
\(357\) 5.30816 0.280937
\(358\) 2.12323 0.112216
\(359\) 12.1286 0.640122 0.320061 0.947397i \(-0.396297\pi\)
0.320061 + 0.947397i \(0.396297\pi\)
\(360\) −0.522326 −0.0275290
\(361\) −18.4401 −0.970529
\(362\) −22.8106 −1.19890
\(363\) 10.8701 0.570534
\(364\) 2.74868 0.144070
\(365\) −8.28846 −0.433838
\(366\) 6.71192 0.350838
\(367\) −6.87499 −0.358871 −0.179436 0.983770i \(-0.557427\pi\)
−0.179436 + 0.983770i \(0.557427\pi\)
\(368\) −1.23649 −0.0644567
\(369\) 5.55715 0.289294
\(370\) 1.56613 0.0814190
\(371\) 25.7854 1.33871
\(372\) −1.19890 −0.0621603
\(373\) −10.2960 −0.533104 −0.266552 0.963820i \(-0.585884\pi\)
−0.266552 + 0.963820i \(0.585884\pi\)
\(374\) 0.695931 0.0359858
\(375\) 5.08075 0.262369
\(376\) 0.902971 0.0465672
\(377\) 4.90740 0.252744
\(378\) −2.74868 −0.141377
\(379\) 19.9240 1.02342 0.511712 0.859157i \(-0.329011\pi\)
0.511712 + 0.859157i \(0.329011\pi\)
\(380\) −0.390855 −0.0200504
\(381\) −7.24136 −0.370986
\(382\) −8.47503 −0.433620
\(383\) −19.6327 −1.00318 −0.501591 0.865105i \(-0.667252\pi\)
−0.501591 + 0.865105i \(0.667252\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0.517384 0.0263683
\(386\) 19.8592 1.01081
\(387\) 5.82398 0.296049
\(388\) 7.76875 0.394398
\(389\) −4.93870 −0.250402 −0.125201 0.992131i \(-0.539958\pi\)
−0.125201 + 0.992131i \(0.539958\pi\)
\(390\) −0.522326 −0.0264490
\(391\) −2.38787 −0.120760
\(392\) −0.555246 −0.0280442
\(393\) −8.82634 −0.445230
\(394\) 9.82460 0.494956
\(395\) −4.02485 −0.202512
\(396\) −0.360369 −0.0181092
\(397\) 12.0504 0.604791 0.302396 0.953182i \(-0.402214\pi\)
0.302396 + 0.953182i \(0.402214\pi\)
\(398\) −12.1513 −0.609088
\(399\) −2.05683 −0.102970
\(400\) −4.72718 −0.236359
\(401\) 12.3082 0.614640 0.307320 0.951606i \(-0.400568\pi\)
0.307320 + 0.951606i \(0.400568\pi\)
\(402\) 2.28386 0.113909
\(403\) −1.19890 −0.0597217
\(404\) 8.86370 0.440985
\(405\) 0.522326 0.0259546
\(406\) −13.4889 −0.669441
\(407\) 1.08052 0.0535594
\(408\) 1.93116 0.0956069
\(409\) −7.93754 −0.392486 −0.196243 0.980555i \(-0.562874\pi\)
−0.196243 + 0.980555i \(0.562874\pi\)
\(410\) −2.90264 −0.143351
\(411\) −12.0119 −0.592503
\(412\) 1.00000 0.0492665
\(413\) −17.0842 −0.840661
\(414\) 1.23649 0.0607703
\(415\) −1.44834 −0.0710963
\(416\) 1.00000 0.0490290
\(417\) 16.0537 0.786155
\(418\) −0.269663 −0.0131896
\(419\) 8.38895 0.409827 0.204913 0.978780i \(-0.434309\pi\)
0.204913 + 0.978780i \(0.434309\pi\)
\(420\) 1.43571 0.0700553
\(421\) 10.8439 0.528501 0.264250 0.964454i \(-0.414875\pi\)
0.264250 + 0.964454i \(0.414875\pi\)
\(422\) 2.76064 0.134386
\(423\) −0.902971 −0.0439040
\(424\) 9.38100 0.455582
\(425\) −9.12896 −0.442819
\(426\) −1.41208 −0.0684154
\(427\) −18.4489 −0.892807
\(428\) 8.57097 0.414293
\(429\) −0.360369 −0.0173988
\(430\) −3.04201 −0.146699
\(431\) 15.4234 0.742920 0.371460 0.928449i \(-0.378857\pi\)
0.371460 + 0.928449i \(0.378857\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −27.4292 −1.31816 −0.659082 0.752071i \(-0.729055\pi\)
−0.659082 + 0.752071i \(0.729055\pi\)
\(434\) 3.29540 0.158184
\(435\) 2.56326 0.122899
\(436\) 6.80669 0.325981
\(437\) 0.925264 0.0442614
\(438\) −15.8684 −0.758220
\(439\) 9.13157 0.435826 0.217913 0.975968i \(-0.430075\pi\)
0.217913 + 0.975968i \(0.430075\pi\)
\(440\) 0.188230 0.00897351
\(441\) 0.555246 0.0264403
\(442\) 1.93116 0.0918561
\(443\) 11.0611 0.525527 0.262763 0.964860i \(-0.415366\pi\)
0.262763 + 0.964860i \(0.415366\pi\)
\(444\) 2.99837 0.142296
\(445\) 0.539125 0.0255570
\(446\) −18.8473 −0.892446
\(447\) 19.4381 0.919391
\(448\) −2.74868 −0.129863
\(449\) 15.6357 0.737892 0.368946 0.929451i \(-0.379719\pi\)
0.368946 + 0.929451i \(0.379719\pi\)
\(450\) 4.72718 0.222841
\(451\) −2.00262 −0.0942999
\(452\) −12.4885 −0.587410
\(453\) −2.58033 −0.121235
\(454\) −9.80664 −0.460248
\(455\) 1.43571 0.0673070
\(456\) −0.748297 −0.0350422
\(457\) 16.3840 0.766412 0.383206 0.923663i \(-0.374820\pi\)
0.383206 + 0.923663i \(0.374820\pi\)
\(458\) −22.6867 −1.06008
\(459\) −1.93116 −0.0901390
\(460\) −0.645852 −0.0301130
\(461\) −10.9385 −0.509457 −0.254728 0.967013i \(-0.581986\pi\)
−0.254728 + 0.967013i \(0.581986\pi\)
\(462\) 0.990539 0.0460840
\(463\) −9.38112 −0.435977 −0.217989 0.975951i \(-0.569950\pi\)
−0.217989 + 0.975951i \(0.569950\pi\)
\(464\) −4.90740 −0.227820
\(465\) −0.626218 −0.0290402
\(466\) 0.545768 0.0252822
\(467\) 29.8882 1.38306 0.691532 0.722346i \(-0.256936\pi\)
0.691532 + 0.722346i \(0.256936\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −6.27762 −0.289873
\(470\) 0.471645 0.0217554
\(471\) −1.50247 −0.0692303
\(472\) −6.21543 −0.286089
\(473\) −2.09878 −0.0965020
\(474\) −7.70563 −0.353931
\(475\) 3.53733 0.162304
\(476\) −5.30816 −0.243299
\(477\) −9.38100 −0.429527
\(478\) −19.9894 −0.914292
\(479\) −1.85725 −0.0848597 −0.0424298 0.999099i \(-0.513510\pi\)
−0.0424298 + 0.999099i \(0.513510\pi\)
\(480\) 0.522326 0.0238408
\(481\) 2.99837 0.136714
\(482\) −23.8871 −1.08803
\(483\) −3.39873 −0.154647
\(484\) −10.8701 −0.494097
\(485\) 4.05782 0.184256
\(486\) 1.00000 0.0453609
\(487\) 22.4287 1.01634 0.508171 0.861256i \(-0.330322\pi\)
0.508171 + 0.861256i \(0.330322\pi\)
\(488\) −6.71192 −0.303835
\(489\) −4.06884 −0.183999
\(490\) −0.290019 −0.0131017
\(491\) −42.4005 −1.91351 −0.956753 0.290900i \(-0.906045\pi\)
−0.956753 + 0.290900i \(0.906045\pi\)
\(492\) −5.55715 −0.250536
\(493\) −9.47699 −0.426822
\(494\) −0.748297 −0.0336675
\(495\) −0.188230 −0.00846030
\(496\) 1.19890 0.0538324
\(497\) 3.88135 0.174102
\(498\) −2.77287 −0.124255
\(499\) 5.80163 0.259717 0.129858 0.991533i \(-0.458548\pi\)
0.129858 + 0.991533i \(0.458548\pi\)
\(500\) −5.08075 −0.227218
\(501\) 18.3623 0.820365
\(502\) 27.8402 1.24257
\(503\) 24.9902 1.11426 0.557129 0.830426i \(-0.311903\pi\)
0.557129 + 0.830426i \(0.311903\pi\)
\(504\) 2.74868 0.122436
\(505\) 4.62974 0.206021
\(506\) −0.445594 −0.0198091
\(507\) −1.00000 −0.0444116
\(508\) 7.24136 0.321283
\(509\) 3.33723 0.147920 0.0739601 0.997261i \(-0.476436\pi\)
0.0739601 + 0.997261i \(0.476436\pi\)
\(510\) 1.00870 0.0446658
\(511\) 43.6171 1.92951
\(512\) −1.00000 −0.0441942
\(513\) 0.748297 0.0330381
\(514\) −0.573865 −0.0253121
\(515\) 0.522326 0.0230164
\(516\) −5.82398 −0.256386
\(517\) 0.325403 0.0143112
\(518\) −8.24157 −0.362114
\(519\) 5.17700 0.227245
\(520\) 0.522326 0.0229055
\(521\) −31.5427 −1.38191 −0.690956 0.722897i \(-0.742810\pi\)
−0.690956 + 0.722897i \(0.742810\pi\)
\(522\) 4.90740 0.214791
\(523\) 15.5475 0.679843 0.339921 0.940454i \(-0.389600\pi\)
0.339921 + 0.940454i \(0.389600\pi\)
\(524\) 8.82634 0.385580
\(525\) −12.9935 −0.567083
\(526\) −31.1711 −1.35912
\(527\) 2.31528 0.100855
\(528\) 0.360369 0.0156830
\(529\) −21.4711 −0.933525
\(530\) 4.89994 0.212840
\(531\) 6.21543 0.269727
\(532\) 2.05683 0.0891748
\(533\) −5.55715 −0.240707
\(534\) 1.03216 0.0446660
\(535\) 4.47684 0.193550
\(536\) −2.28386 −0.0986479
\(537\) 2.12323 0.0916241
\(538\) −4.14299 −0.178617
\(539\) −0.200093 −0.00861863
\(540\) −0.522326 −0.0224773
\(541\) 20.5163 0.882066 0.441033 0.897491i \(-0.354612\pi\)
0.441033 + 0.897491i \(0.354612\pi\)
\(542\) −25.0410 −1.07560
\(543\) −22.8106 −0.978897
\(544\) −1.93116 −0.0827980
\(545\) 3.55531 0.152293
\(546\) 2.74868 0.117633
\(547\) −19.0763 −0.815645 −0.407823 0.913061i \(-0.633712\pi\)
−0.407823 + 0.913061i \(0.633712\pi\)
\(548\) 12.0119 0.513122
\(549\) 6.71192 0.286458
\(550\) −1.70353 −0.0726386
\(551\) 3.67219 0.156441
\(552\) −1.23649 −0.0526287
\(553\) 21.1803 0.900678
\(554\) 3.29484 0.139984
\(555\) 1.56613 0.0664784
\(556\) −16.0537 −0.680831
\(557\) 7.58335 0.321317 0.160659 0.987010i \(-0.448638\pi\)
0.160659 + 0.987010i \(0.448638\pi\)
\(558\) −1.19890 −0.0507537
\(559\) −5.82398 −0.246328
\(560\) −1.43571 −0.0606697
\(561\) 0.695931 0.0293822
\(562\) 20.3656 0.859072
\(563\) 22.8927 0.964814 0.482407 0.875947i \(-0.339763\pi\)
0.482407 + 0.875947i \(0.339763\pi\)
\(564\) 0.902971 0.0380219
\(565\) −6.52306 −0.274427
\(566\) −13.6604 −0.574188
\(567\) −2.74868 −0.115434
\(568\) 1.41208 0.0592494
\(569\) 46.6115 1.95406 0.977029 0.213108i \(-0.0683586\pi\)
0.977029 + 0.213108i \(0.0683586\pi\)
\(570\) −0.390855 −0.0163711
\(571\) −1.60969 −0.0673636 −0.0336818 0.999433i \(-0.510723\pi\)
−0.0336818 + 0.999433i \(0.510723\pi\)
\(572\) 0.360369 0.0150678
\(573\) −8.47503 −0.354050
\(574\) 15.2748 0.637560
\(575\) 5.84512 0.243758
\(576\) 1.00000 0.0416667
\(577\) 5.73655 0.238816 0.119408 0.992845i \(-0.461900\pi\)
0.119408 + 0.992845i \(0.461900\pi\)
\(578\) 13.2706 0.551984
\(579\) 19.8592 0.825319
\(580\) −2.56326 −0.106434
\(581\) 7.62173 0.316203
\(582\) 7.76875 0.322025
\(583\) 3.38062 0.140011
\(584\) 15.8684 0.656638
\(585\) −0.522326 −0.0215955
\(586\) 22.1368 0.914464
\(587\) −5.36219 −0.221321 −0.110661 0.993858i \(-0.535297\pi\)
−0.110661 + 0.993858i \(0.535297\pi\)
\(588\) −0.555246 −0.0228980
\(589\) −0.897136 −0.0369658
\(590\) −3.24648 −0.133656
\(591\) 9.82460 0.404130
\(592\) −2.99837 −0.123232
\(593\) 0.564040 0.0231623 0.0115812 0.999933i \(-0.496314\pi\)
0.0115812 + 0.999933i \(0.496314\pi\)
\(594\) −0.360369 −0.0147861
\(595\) −2.77259 −0.113665
\(596\) −19.4381 −0.796216
\(597\) −12.1513 −0.497319
\(598\) −1.23649 −0.0505640
\(599\) 12.6688 0.517633 0.258817 0.965927i \(-0.416668\pi\)
0.258817 + 0.965927i \(0.416668\pi\)
\(600\) −4.72718 −0.192986
\(601\) 24.1607 0.985534 0.492767 0.870161i \(-0.335985\pi\)
0.492767 + 0.870161i \(0.335985\pi\)
\(602\) 16.0083 0.652448
\(603\) 2.28386 0.0930062
\(604\) 2.58033 0.104992
\(605\) −5.67775 −0.230833
\(606\) 8.86370 0.360063
\(607\) 28.1195 1.14133 0.570667 0.821182i \(-0.306685\pi\)
0.570667 + 0.821182i \(0.306685\pi\)
\(608\) 0.748297 0.0303474
\(609\) −13.4889 −0.546597
\(610\) −3.50581 −0.141946
\(611\) 0.902971 0.0365303
\(612\) 1.93116 0.0780627
\(613\) 2.51093 0.101415 0.0507077 0.998714i \(-0.483852\pi\)
0.0507077 + 0.998714i \(0.483852\pi\)
\(614\) −13.5244 −0.545800
\(615\) −2.90264 −0.117046
\(616\) −0.990539 −0.0399099
\(617\) 9.97891 0.401736 0.200868 0.979618i \(-0.435624\pi\)
0.200868 + 0.979618i \(0.435624\pi\)
\(618\) 1.00000 0.0402259
\(619\) 18.2603 0.733944 0.366972 0.930232i \(-0.380395\pi\)
0.366972 + 0.930232i \(0.380395\pi\)
\(620\) 0.626218 0.0251495
\(621\) 1.23649 0.0496188
\(622\) 0.984074 0.0394578
\(623\) −2.83709 −0.113665
\(624\) 1.00000 0.0400320
\(625\) 20.9821 0.839283
\(626\) 8.44322 0.337459
\(627\) −0.269663 −0.0107693
\(628\) 1.50247 0.0599552
\(629\) −5.79035 −0.230876
\(630\) 1.43571 0.0571999
\(631\) 28.8571 1.14878 0.574391 0.818581i \(-0.305239\pi\)
0.574391 + 0.818581i \(0.305239\pi\)
\(632\) 7.70563 0.306513
\(633\) 2.76064 0.109725
\(634\) −2.32174 −0.0922081
\(635\) 3.78235 0.150098
\(636\) 9.38100 0.371981
\(637\) −0.555246 −0.0219997
\(638\) −1.76847 −0.0700145
\(639\) −1.41208 −0.0558609
\(640\) −0.522326 −0.0206467
\(641\) 17.7597 0.701467 0.350734 0.936475i \(-0.385932\pi\)
0.350734 + 0.936475i \(0.385932\pi\)
\(642\) 8.57097 0.338269
\(643\) 44.8959 1.77052 0.885260 0.465096i \(-0.153980\pi\)
0.885260 + 0.465096i \(0.153980\pi\)
\(644\) 3.39873 0.133929
\(645\) −3.04201 −0.119779
\(646\) 1.44508 0.0568561
\(647\) 32.9478 1.29531 0.647655 0.761934i \(-0.275750\pi\)
0.647655 + 0.761934i \(0.275750\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −2.23985 −0.0879217
\(650\) −4.72718 −0.185415
\(651\) 3.29540 0.129157
\(652\) 4.06884 0.159348
\(653\) 3.03062 0.118597 0.0592987 0.998240i \(-0.481114\pi\)
0.0592987 + 0.998240i \(0.481114\pi\)
\(654\) 6.80669 0.266163
\(655\) 4.61022 0.180136
\(656\) 5.55715 0.216970
\(657\) −15.8684 −0.619084
\(658\) −2.48198 −0.0967576
\(659\) −44.6671 −1.73998 −0.869991 0.493068i \(-0.835875\pi\)
−0.869991 + 0.493068i \(0.835875\pi\)
\(660\) 0.188230 0.00732684
\(661\) 28.3968 1.10451 0.552254 0.833676i \(-0.313768\pi\)
0.552254 + 0.833676i \(0.313768\pi\)
\(662\) 8.39888 0.326432
\(663\) 1.93116 0.0750002
\(664\) 2.77287 0.107608
\(665\) 1.07433 0.0416609
\(666\) 2.99837 0.116185
\(667\) 6.06797 0.234953
\(668\) −18.3623 −0.710457
\(669\) −18.8473 −0.728679
\(670\) −1.19292 −0.0460866
\(671\) −2.41877 −0.0933755
\(672\) −2.74868 −0.106033
\(673\) 1.87967 0.0724559 0.0362279 0.999344i \(-0.488466\pi\)
0.0362279 + 0.999344i \(0.488466\pi\)
\(674\) −30.6516 −1.18066
\(675\) 4.72718 0.181949
\(676\) 1.00000 0.0384615
\(677\) −11.0758 −0.425679 −0.212840 0.977087i \(-0.568271\pi\)
−0.212840 + 0.977087i \(0.568271\pi\)
\(678\) −12.4885 −0.479618
\(679\) −21.3538 −0.819484
\(680\) −1.00870 −0.0386818
\(681\) −9.80664 −0.375791
\(682\) 0.432048 0.0165440
\(683\) −0.794064 −0.0303840 −0.0151920 0.999885i \(-0.504836\pi\)
−0.0151920 + 0.999885i \(0.504836\pi\)
\(684\) −0.748297 −0.0286118
\(685\) 6.27412 0.239722
\(686\) −17.7146 −0.676345
\(687\) −22.6867 −0.865552
\(688\) 5.82398 0.222037
\(689\) 9.38100 0.357388
\(690\) −0.645852 −0.0245872
\(691\) 43.5172 1.65547 0.827736 0.561117i \(-0.189628\pi\)
0.827736 + 0.561117i \(0.189628\pi\)
\(692\) −5.17700 −0.196800
\(693\) 0.990539 0.0376275
\(694\) 26.7991 1.01728
\(695\) −8.38529 −0.318072
\(696\) −4.90740 −0.186014
\(697\) 10.7318 0.406495
\(698\) −36.0370 −1.36402
\(699\) 0.545768 0.0206428
\(700\) 12.9935 0.491108
\(701\) 15.7089 0.593318 0.296659 0.954983i \(-0.404127\pi\)
0.296659 + 0.954983i \(0.404127\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 2.24367 0.0846217
\(704\) −0.360369 −0.0135819
\(705\) 0.471645 0.0177632
\(706\) 27.9712 1.05271
\(707\) −24.3635 −0.916283
\(708\) −6.21543 −0.233590
\(709\) −1.72192 −0.0646682 −0.0323341 0.999477i \(-0.510294\pi\)
−0.0323341 + 0.999477i \(0.510294\pi\)
\(710\) 0.737564 0.0276803
\(711\) −7.70563 −0.288984
\(712\) −1.03216 −0.0386819
\(713\) −1.48244 −0.0555177
\(714\) −5.30816 −0.198653
\(715\) 0.188230 0.00703940
\(716\) −2.12323 −0.0793488
\(717\) −19.9894 −0.746517
\(718\) −12.1286 −0.452634
\(719\) −11.1623 −0.416285 −0.208142 0.978099i \(-0.566742\pi\)
−0.208142 + 0.978099i \(0.566742\pi\)
\(720\) 0.522326 0.0194659
\(721\) −2.74868 −0.102366
\(722\) 18.4401 0.686268
\(723\) −23.8871 −0.888369
\(724\) 22.8106 0.847750
\(725\) 23.1981 0.861557
\(726\) −10.8701 −0.403429
\(727\) −0.284917 −0.0105670 −0.00528349 0.999986i \(-0.501682\pi\)
−0.00528349 + 0.999986i \(0.501682\pi\)
\(728\) −2.74868 −0.101873
\(729\) 1.00000 0.0370370
\(730\) 8.28846 0.306770
\(731\) 11.2471 0.415988
\(732\) −6.71192 −0.248080
\(733\) 3.90487 0.144230 0.0721148 0.997396i \(-0.477025\pi\)
0.0721148 + 0.997396i \(0.477025\pi\)
\(734\) 6.87499 0.253760
\(735\) −0.290019 −0.0106975
\(736\) 1.23649 0.0455778
\(737\) −0.823034 −0.0303168
\(738\) −5.55715 −0.204562
\(739\) 10.5454 0.387918 0.193959 0.981010i \(-0.437867\pi\)
0.193959 + 0.981010i \(0.437867\pi\)
\(740\) −1.56613 −0.0575720
\(741\) −0.748297 −0.0274894
\(742\) −25.7854 −0.946611
\(743\) 34.6724 1.27201 0.636004 0.771686i \(-0.280586\pi\)
0.636004 + 0.771686i \(0.280586\pi\)
\(744\) 1.19890 0.0439540
\(745\) −10.1530 −0.371978
\(746\) 10.2960 0.376962
\(747\) −2.77287 −0.101454
\(748\) −0.695931 −0.0254458
\(749\) −23.5589 −0.860822
\(750\) −5.08075 −0.185523
\(751\) 28.7573 1.04937 0.524684 0.851297i \(-0.324184\pi\)
0.524684 + 0.851297i \(0.324184\pi\)
\(752\) −0.902971 −0.0329280
\(753\) 27.8402 1.01455
\(754\) −4.90740 −0.178717
\(755\) 1.34777 0.0490505
\(756\) 2.74868 0.0999686
\(757\) −19.2177 −0.698478 −0.349239 0.937034i \(-0.613560\pi\)
−0.349239 + 0.937034i \(0.613560\pi\)
\(758\) −19.9240 −0.723670
\(759\) −0.445594 −0.0161740
\(760\) 0.390855 0.0141778
\(761\) −20.4907 −0.742787 −0.371394 0.928476i \(-0.621120\pi\)
−0.371394 + 0.928476i \(0.621120\pi\)
\(762\) 7.24136 0.262327
\(763\) −18.7094 −0.677326
\(764\) 8.47503 0.306616
\(765\) 1.00870 0.0364695
\(766\) 19.6327 0.709357
\(767\) −6.21543 −0.224426
\(768\) −1.00000 −0.0360844
\(769\) 44.3911 1.60078 0.800392 0.599476i \(-0.204624\pi\)
0.800392 + 0.599476i \(0.204624\pi\)
\(770\) −0.517384 −0.0186452
\(771\) −0.573865 −0.0206672
\(772\) −19.8592 −0.714747
\(773\) 6.45905 0.232316 0.116158 0.993231i \(-0.462942\pi\)
0.116158 + 0.993231i \(0.462942\pi\)
\(774\) −5.82398 −0.209339
\(775\) −5.66743 −0.203580
\(776\) −7.76875 −0.278882
\(777\) −8.24157 −0.295665
\(778\) 4.93870 0.177061
\(779\) −4.15840 −0.148990
\(780\) 0.522326 0.0187023
\(781\) 0.508868 0.0182087
\(782\) 2.38787 0.0853902
\(783\) 4.90740 0.175376
\(784\) 0.555246 0.0198302
\(785\) 0.784780 0.0280100
\(786\) 8.82634 0.314825
\(787\) 19.5577 0.697157 0.348579 0.937280i \(-0.386664\pi\)
0.348579 + 0.937280i \(0.386664\pi\)
\(788\) −9.82460 −0.349987
\(789\) −31.1711 −1.10972
\(790\) 4.02485 0.143198
\(791\) 34.3269 1.22052
\(792\) 0.360369 0.0128051
\(793\) −6.71192 −0.238347
\(794\) −12.0504 −0.427652
\(795\) 4.89994 0.173783
\(796\) 12.1513 0.430690
\(797\) 22.0929 0.782571 0.391285 0.920269i \(-0.372031\pi\)
0.391285 + 0.920269i \(0.372031\pi\)
\(798\) 2.05683 0.0728110
\(799\) −1.74379 −0.0616907
\(800\) 4.72718 0.167131
\(801\) 1.03216 0.0364697
\(802\) −12.3082 −0.434616
\(803\) 5.71846 0.201800
\(804\) −2.28386 −0.0805457
\(805\) 1.77524 0.0625691
\(806\) 1.19890 0.0422296
\(807\) −4.14299 −0.145840
\(808\) −8.86370 −0.311824
\(809\) 48.2741 1.69723 0.848614 0.529013i \(-0.177438\pi\)
0.848614 + 0.529013i \(0.177438\pi\)
\(810\) −0.522326 −0.0183527
\(811\) 11.3494 0.398531 0.199266 0.979946i \(-0.436144\pi\)
0.199266 + 0.979946i \(0.436144\pi\)
\(812\) 13.4889 0.473367
\(813\) −25.0410 −0.878226
\(814\) −1.08052 −0.0378722
\(815\) 2.12526 0.0744447
\(816\) −1.93116 −0.0676043
\(817\) −4.35806 −0.152469
\(818\) 7.93754 0.277530
\(819\) 2.74868 0.0960467
\(820\) 2.90264 0.101365
\(821\) 20.2922 0.708204 0.354102 0.935207i \(-0.384787\pi\)
0.354102 + 0.935207i \(0.384787\pi\)
\(822\) 12.0119 0.418963
\(823\) −31.7986 −1.10843 −0.554215 0.832374i \(-0.686981\pi\)
−0.554215 + 0.832374i \(0.686981\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −1.70353 −0.0593092
\(826\) 17.0842 0.594437
\(827\) −21.1750 −0.736326 −0.368163 0.929761i \(-0.620013\pi\)
−0.368163 + 0.929761i \(0.620013\pi\)
\(828\) −1.23649 −0.0429711
\(829\) 55.0767 1.91289 0.956446 0.291908i \(-0.0942901\pi\)
0.956446 + 0.291908i \(0.0942901\pi\)
\(830\) 1.44834 0.0502726
\(831\) 3.29484 0.114297
\(832\) −1.00000 −0.0346688
\(833\) 1.07227 0.0371520
\(834\) −16.0537 −0.555896
\(835\) −9.59107 −0.331913
\(836\) 0.269663 0.00932648
\(837\) −1.19890 −0.0414402
\(838\) −8.38895 −0.289791
\(839\) 20.3412 0.702255 0.351127 0.936328i \(-0.385798\pi\)
0.351127 + 0.936328i \(0.385798\pi\)
\(840\) −1.43571 −0.0495366
\(841\) −4.91744 −0.169567
\(842\) −10.8439 −0.373707
\(843\) 20.3656 0.701429
\(844\) −2.76064 −0.0950250
\(845\) 0.522326 0.0179685
\(846\) 0.902971 0.0310448
\(847\) 29.8785 1.02664
\(848\) −9.38100 −0.322145
\(849\) −13.6604 −0.468822
\(850\) 9.12896 0.313121
\(851\) 3.70747 0.127090
\(852\) 1.41208 0.0483770
\(853\) −2.80668 −0.0960988 −0.0480494 0.998845i \(-0.515300\pi\)
−0.0480494 + 0.998845i \(0.515300\pi\)
\(854\) 18.4489 0.631310
\(855\) −0.390855 −0.0133669
\(856\) −8.57097 −0.292950
\(857\) −47.3635 −1.61791 −0.808953 0.587873i \(-0.799965\pi\)
−0.808953 + 0.587873i \(0.799965\pi\)
\(858\) 0.360369 0.0123028
\(859\) −11.5391 −0.393710 −0.196855 0.980433i \(-0.563073\pi\)
−0.196855 + 0.980433i \(0.563073\pi\)
\(860\) 3.04201 0.103732
\(861\) 15.2748 0.520565
\(862\) −15.4234 −0.525324
\(863\) 4.17586 0.142148 0.0710740 0.997471i \(-0.477357\pi\)
0.0710740 + 0.997471i \(0.477357\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.70408 −0.0919415
\(866\) 27.4292 0.932083
\(867\) 13.2706 0.450693
\(868\) −3.29540 −0.111853
\(869\) 2.77687 0.0941987
\(870\) −2.56326 −0.0869026
\(871\) −2.28386 −0.0773858
\(872\) −6.80669 −0.230504
\(873\) 7.76875 0.262932
\(874\) −0.925264 −0.0312975
\(875\) 13.9654 0.472116
\(876\) 15.8684 0.536143
\(877\) −48.9089 −1.65154 −0.825769 0.564009i \(-0.809258\pi\)
−0.825769 + 0.564009i \(0.809258\pi\)
\(878\) −9.13157 −0.308175
\(879\) 22.1368 0.746657
\(880\) −0.188230 −0.00634523
\(881\) −11.0560 −0.372485 −0.186243 0.982504i \(-0.559631\pi\)
−0.186243 + 0.982504i \(0.559631\pi\)
\(882\) −0.555246 −0.0186961
\(883\) −23.3797 −0.786789 −0.393394 0.919370i \(-0.628699\pi\)
−0.393394 + 0.919370i \(0.628699\pi\)
\(884\) −1.93116 −0.0649521
\(885\) −3.24648 −0.109129
\(886\) −11.0611 −0.371603
\(887\) 19.2982 0.647970 0.323985 0.946062i \(-0.394977\pi\)
0.323985 + 0.946062i \(0.394977\pi\)
\(888\) −2.99837 −0.100619
\(889\) −19.9042 −0.667565
\(890\) −0.539125 −0.0180715
\(891\) −0.360369 −0.0120728
\(892\) 18.8473 0.631055
\(893\) 0.675690 0.0226111
\(894\) −19.4381 −0.650107
\(895\) −1.10902 −0.0370704
\(896\) 2.74868 0.0918270
\(897\) −1.23649 −0.0412853
\(898\) −15.6357 −0.521769
\(899\) −5.88350 −0.196226
\(900\) −4.72718 −0.157573
\(901\) −18.1163 −0.603540
\(902\) 2.00262 0.0666801
\(903\) 16.0083 0.532721
\(904\) 12.4885 0.415361
\(905\) 11.9146 0.396054
\(906\) 2.58033 0.0857257
\(907\) 20.6555 0.685854 0.342927 0.939362i \(-0.388582\pi\)
0.342927 + 0.939362i \(0.388582\pi\)
\(908\) 9.80664 0.325445
\(909\) 8.86370 0.293990
\(910\) −1.43571 −0.0475932
\(911\) −30.7241 −1.01793 −0.508967 0.860786i \(-0.669972\pi\)
−0.508967 + 0.860786i \(0.669972\pi\)
\(912\) 0.748297 0.0247786
\(913\) 0.999256 0.0330705
\(914\) −16.3840 −0.541935
\(915\) −3.50581 −0.115898
\(916\) 22.6867 0.749590
\(917\) −24.2608 −0.801162
\(918\) 1.93116 0.0637379
\(919\) −8.56740 −0.282612 −0.141306 0.989966i \(-0.545130\pi\)
−0.141306 + 0.989966i \(0.545130\pi\)
\(920\) 0.645852 0.0212931
\(921\) −13.5244 −0.445644
\(922\) 10.9385 0.360240
\(923\) 1.41208 0.0464791
\(924\) −0.990539 −0.0325863
\(925\) 14.1738 0.466033
\(926\) 9.38112 0.308283
\(927\) 1.00000 0.0328443
\(928\) 4.90740 0.161093
\(929\) 32.8528 1.07786 0.538932 0.842349i \(-0.318828\pi\)
0.538932 + 0.842349i \(0.318828\pi\)
\(930\) 0.626218 0.0205345
\(931\) −0.415489 −0.0136171
\(932\) −0.545768 −0.0178772
\(933\) 0.984074 0.0322171
\(934\) −29.8882 −0.977973
\(935\) −0.363503 −0.0118878
\(936\) 1.00000 0.0326860
\(937\) 26.5121 0.866113 0.433057 0.901367i \(-0.357435\pi\)
0.433057 + 0.901367i \(0.357435\pi\)
\(938\) 6.27762 0.204971
\(939\) 8.44322 0.275534
\(940\) −0.471645 −0.0153834
\(941\) 21.1200 0.688493 0.344246 0.938879i \(-0.388134\pi\)
0.344246 + 0.938879i \(0.388134\pi\)
\(942\) 1.50247 0.0489532
\(943\) −6.87139 −0.223763
\(944\) 6.21543 0.202295
\(945\) 1.43571 0.0467035
\(946\) 2.09878 0.0682372
\(947\) 31.2612 1.01585 0.507926 0.861401i \(-0.330412\pi\)
0.507926 + 0.861401i \(0.330412\pi\)
\(948\) 7.70563 0.250267
\(949\) 15.8684 0.515109
\(950\) −3.53733 −0.114766
\(951\) −2.32174 −0.0752876
\(952\) 5.30816 0.172038
\(953\) 27.6348 0.895180 0.447590 0.894239i \(-0.352282\pi\)
0.447590 + 0.894239i \(0.352282\pi\)
\(954\) 9.38100 0.303721
\(955\) 4.42673 0.143246
\(956\) 19.9894 0.646502
\(957\) −1.76847 −0.0571666
\(958\) 1.85725 0.0600049
\(959\) −33.0168 −1.06617
\(960\) −0.522326 −0.0168580
\(961\) −29.5626 −0.953633
\(962\) −2.99837 −0.0966714
\(963\) 8.57097 0.276196
\(964\) 23.8871 0.769350
\(965\) −10.3730 −0.333917
\(966\) 3.39873 0.109352
\(967\) 42.4643 1.36556 0.682780 0.730624i \(-0.260771\pi\)
0.682780 + 0.730624i \(0.260771\pi\)
\(968\) 10.8701 0.349379
\(969\) 1.44508 0.0464228
\(970\) −4.05782 −0.130289
\(971\) −41.4928 −1.33157 −0.665784 0.746144i \(-0.731903\pi\)
−0.665784 + 0.746144i \(0.731903\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 44.1266 1.41463
\(974\) −22.4287 −0.718662
\(975\) −4.72718 −0.151391
\(976\) 6.71192 0.214843
\(977\) −8.75460 −0.280084 −0.140042 0.990146i \(-0.544724\pi\)
−0.140042 + 0.990146i \(0.544724\pi\)
\(978\) 4.06884 0.130107
\(979\) −0.371959 −0.0118879
\(980\) 0.290019 0.00926433
\(981\) 6.80669 0.217321
\(982\) 42.4005 1.35305
\(983\) 21.7911 0.695029 0.347515 0.937675i \(-0.387026\pi\)
0.347515 + 0.937675i \(0.387026\pi\)
\(984\) 5.55715 0.177156
\(985\) −5.13164 −0.163508
\(986\) 9.47699 0.301809
\(987\) −2.48198 −0.0790023
\(988\) 0.748297 0.0238065
\(989\) −7.20131 −0.228988
\(990\) 0.188230 0.00598234
\(991\) −26.0086 −0.826191 −0.413096 0.910688i \(-0.635552\pi\)
−0.413096 + 0.910688i \(0.635552\pi\)
\(992\) −1.19890 −0.0380652
\(993\) 8.39888 0.266530
\(994\) −3.88135 −0.123109
\(995\) 6.34692 0.201211
\(996\) 2.77287 0.0878617
\(997\) −31.0258 −0.982597 −0.491299 0.870991i \(-0.663478\pi\)
−0.491299 + 0.870991i \(0.663478\pi\)
\(998\) −5.80163 −0.183647
\(999\) 2.99837 0.0948643
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 8034.2.a.ba.1.8 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.ba.1.8 14 1.1 even 1 trivial