Properties

Label 6027.2.a.s.1.2
Level 6027
Weight 2
Character 6027.1
Self dual yes
Analytic conductor 48.126
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 123)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\)
Character \(\chi\) = 6027.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} -4.24914 q^{5} +0.470683 q^{6} -1.77846 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.470683 q^{2} +1.00000 q^{3} -1.77846 q^{4} -4.24914 q^{5} +0.470683 q^{6} -1.77846 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.47068 q^{11} -1.77846 q^{12} +0.249141 q^{13} -4.24914 q^{15} +2.71982 q^{16} +5.02760 q^{17} +0.470683 q^{18} +2.24914 q^{19} +7.55691 q^{20} -0.692226 q^{22} -6.24914 q^{23} -1.77846 q^{24} +13.0552 q^{25} +0.117266 q^{26} +1.00000 q^{27} -2.41205 q^{29} -2.00000 q^{30} +2.89572 q^{31} +4.83709 q^{32} -1.47068 q^{33} +2.36641 q^{34} -1.77846 q^{36} +9.71982 q^{37} +1.05863 q^{38} +0.249141 q^{39} +7.55691 q^{40} -1.00000 q^{41} -10.8337 q^{43} +2.61555 q^{44} -4.24914 q^{45} -2.94137 q^{46} +4.08623 q^{47} +2.71982 q^{48} +6.14486 q^{50} +5.02760 q^{51} -0.443086 q^{52} -1.43965 q^{53} +0.470683 q^{54} +6.24914 q^{55} +2.24914 q^{57} -1.13531 q^{58} -2.61555 q^{59} +7.55691 q^{60} +5.71982 q^{61} +1.36297 q^{62} -3.16291 q^{64} -1.05863 q^{65} -0.692226 q^{66} +15.9379 q^{67} -8.94137 q^{68} -6.24914 q^{69} -10.5293 q^{71} -1.77846 q^{72} -9.39400 q^{73} +4.57496 q^{74} +13.0552 q^{75} -4.00000 q^{76} +0.117266 q^{78} -0.560352 q^{79} -11.5569 q^{80} +1.00000 q^{81} -0.470683 q^{82} -3.80605 q^{83} -21.3630 q^{85} -5.09922 q^{86} -2.41205 q^{87} +2.61555 q^{88} -4.11727 q^{89} -2.00000 q^{90} +11.1138 q^{92} +2.89572 q^{93} +1.92332 q^{94} -9.55691 q^{95} +4.83709 q^{96} +7.19051 q^{97} -1.47068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 3q^{3} + 3q^{4} - 4q^{5} + q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} + 3q^{3} + 3q^{4} - 4q^{5} + q^{6} + 3q^{8} + 3q^{9} - 6q^{10} - 4q^{11} + 3q^{12} - 8q^{13} - 4q^{15} - q^{16} - 2q^{17} + q^{18} - 2q^{19} + 6q^{20} - 10q^{22} - 10q^{23} + 3q^{24} + 5q^{25} + 2q^{26} + 3q^{27} - 6q^{29} - 6q^{30} + 2q^{31} + 7q^{32} - 4q^{33} + 3q^{36} + 20q^{37} + 4q^{38} - 8q^{39} + 6q^{40} - 3q^{41} + 10q^{43} - 8q^{44} - 4q^{45} - 8q^{46} - 4q^{47} - q^{48} + 3q^{50} - 2q^{51} - 18q^{52} + 14q^{53} + q^{54} + 10q^{55} - 2q^{57} - 28q^{58} + 8q^{59} + 6q^{60} + 8q^{61} - 38q^{62} - 17q^{64} - 4q^{65} - 10q^{66} + 12q^{67} - 26q^{68} - 10q^{69} - 32q^{71} + 3q^{72} - 4q^{73} + 20q^{74} + 5q^{75} - 12q^{76} + 2q^{78} - 20q^{79} - 18q^{80} + 3q^{81} - q^{82} + 14q^{83} - 22q^{85} + 6q^{86} - 6q^{87} - 8q^{88} - 14q^{89} - 6q^{90} + 2q^{93} - 18q^{94} - 12q^{95} + 7q^{96} + 12q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.470683 0.332823 0.166412 0.986056i \(-0.446782\pi\)
0.166412 + 0.986056i \(0.446782\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.77846 −0.889229
\(5\) −4.24914 −1.90027 −0.950137 0.311834i \(-0.899057\pi\)
−0.950137 + 0.311834i \(0.899057\pi\)
\(6\) 0.470683 0.192156
\(7\) 0 0
\(8\) −1.77846 −0.628780
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −1.47068 −0.443428 −0.221714 0.975112i \(-0.571165\pi\)
−0.221714 + 0.975112i \(0.571165\pi\)
\(12\) −1.77846 −0.513396
\(13\) 0.249141 0.0690992 0.0345496 0.999403i \(-0.489000\pi\)
0.0345496 + 0.999403i \(0.489000\pi\)
\(14\) 0 0
\(15\) −4.24914 −1.09712
\(16\) 2.71982 0.679956
\(17\) 5.02760 1.21937 0.609686 0.792643i \(-0.291296\pi\)
0.609686 + 0.792643i \(0.291296\pi\)
\(18\) 0.470683 0.110941
\(19\) 2.24914 0.515988 0.257994 0.966146i \(-0.416938\pi\)
0.257994 + 0.966146i \(0.416938\pi\)
\(20\) 7.55691 1.68978
\(21\) 0 0
\(22\) −0.692226 −0.147583
\(23\) −6.24914 −1.30304 −0.651518 0.758633i \(-0.725867\pi\)
−0.651518 + 0.758633i \(0.725867\pi\)
\(24\) −1.77846 −0.363026
\(25\) 13.0552 2.61104
\(26\) 0.117266 0.0229978
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.41205 −0.447906 −0.223953 0.974600i \(-0.571896\pi\)
−0.223953 + 0.974600i \(0.571896\pi\)
\(30\) −2.00000 −0.365148
\(31\) 2.89572 0.520087 0.260044 0.965597i \(-0.416263\pi\)
0.260044 + 0.965597i \(0.416263\pi\)
\(32\) 4.83709 0.855085
\(33\) −1.47068 −0.256013
\(34\) 2.36641 0.405835
\(35\) 0 0
\(36\) −1.77846 −0.296410
\(37\) 9.71982 1.59793 0.798965 0.601378i \(-0.205381\pi\)
0.798965 + 0.601378i \(0.205381\pi\)
\(38\) 1.05863 0.171733
\(39\) 0.249141 0.0398944
\(40\) 7.55691 1.19485
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) −10.8337 −1.65212 −0.826058 0.563585i \(-0.809422\pi\)
−0.826058 + 0.563585i \(0.809422\pi\)
\(44\) 2.61555 0.394309
\(45\) −4.24914 −0.633424
\(46\) −2.94137 −0.433681
\(47\) 4.08623 0.596038 0.298019 0.954560i \(-0.403674\pi\)
0.298019 + 0.954560i \(0.403674\pi\)
\(48\) 2.71982 0.392573
\(49\) 0 0
\(50\) 6.14486 0.869015
\(51\) 5.02760 0.704004
\(52\) −0.443086 −0.0614449
\(53\) −1.43965 −0.197751 −0.0988754 0.995100i \(-0.531525\pi\)
−0.0988754 + 0.995100i \(0.531525\pi\)
\(54\) 0.470683 0.0640519
\(55\) 6.24914 0.842634
\(56\) 0 0
\(57\) 2.24914 0.297906
\(58\) −1.13531 −0.149074
\(59\) −2.61555 −0.340515 −0.170258 0.985400i \(-0.554460\pi\)
−0.170258 + 0.985400i \(0.554460\pi\)
\(60\) 7.55691 0.975593
\(61\) 5.71982 0.732348 0.366174 0.930546i \(-0.380667\pi\)
0.366174 + 0.930546i \(0.380667\pi\)
\(62\) 1.36297 0.173097
\(63\) 0 0
\(64\) −3.16291 −0.395364
\(65\) −1.05863 −0.131307
\(66\) −0.692226 −0.0852072
\(67\) 15.9379 1.94713 0.973564 0.228415i \(-0.0733542\pi\)
0.973564 + 0.228415i \(0.0733542\pi\)
\(68\) −8.94137 −1.08430
\(69\) −6.24914 −0.752308
\(70\) 0 0
\(71\) −10.5293 −1.24960 −0.624800 0.780785i \(-0.714819\pi\)
−0.624800 + 0.780785i \(0.714819\pi\)
\(72\) −1.77846 −0.209593
\(73\) −9.39400 −1.09949 −0.549743 0.835334i \(-0.685274\pi\)
−0.549743 + 0.835334i \(0.685274\pi\)
\(74\) 4.57496 0.531828
\(75\) 13.0552 1.50748
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0.117266 0.0132778
\(79\) −0.560352 −0.0630445 −0.0315223 0.999503i \(-0.510036\pi\)
−0.0315223 + 0.999503i \(0.510036\pi\)
\(80\) −11.5569 −1.29210
\(81\) 1.00000 0.111111
\(82\) −0.470683 −0.0519783
\(83\) −3.80605 −0.417769 −0.208884 0.977940i \(-0.566983\pi\)
−0.208884 + 0.977940i \(0.566983\pi\)
\(84\) 0 0
\(85\) −21.3630 −2.31714
\(86\) −5.09922 −0.549863
\(87\) −2.41205 −0.258599
\(88\) 2.61555 0.278818
\(89\) −4.11727 −0.436429 −0.218215 0.975901i \(-0.570023\pi\)
−0.218215 + 0.975901i \(0.570023\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 11.1138 1.15870
\(93\) 2.89572 0.300273
\(94\) 1.92332 0.198375
\(95\) −9.55691 −0.980519
\(96\) 4.83709 0.493683
\(97\) 7.19051 0.730085 0.365043 0.930991i \(-0.381054\pi\)
0.365043 + 0.930991i \(0.381054\pi\)
\(98\) 0 0
\(99\) −1.47068 −0.147809
\(100\) −23.2181 −2.32181
\(101\) 11.1449 1.10896 0.554478 0.832199i \(-0.312918\pi\)
0.554478 + 0.832199i \(0.312918\pi\)
\(102\) 2.36641 0.234309
\(103\) −13.2767 −1.30820 −0.654098 0.756410i \(-0.726952\pi\)
−0.654098 + 0.756410i \(0.726952\pi\)
\(104\) −0.443086 −0.0434481
\(105\) 0 0
\(106\) −0.677618 −0.0658161
\(107\) 9.32238 0.901229 0.450614 0.892719i \(-0.351205\pi\)
0.450614 + 0.892719i \(0.351205\pi\)
\(108\) −1.77846 −0.171132
\(109\) −0.249141 −0.0238633 −0.0119317 0.999929i \(-0.503798\pi\)
−0.0119317 + 0.999929i \(0.503798\pi\)
\(110\) 2.94137 0.280448
\(111\) 9.71982 0.922565
\(112\) 0 0
\(113\) 9.24570 0.869763 0.434881 0.900488i \(-0.356790\pi\)
0.434881 + 0.900488i \(0.356790\pi\)
\(114\) 1.05863 0.0991501
\(115\) 26.5535 2.47612
\(116\) 4.28973 0.398291
\(117\) 0.249141 0.0230331
\(118\) −1.23109 −0.113331
\(119\) 0 0
\(120\) 7.55691 0.689849
\(121\) −8.83709 −0.803372
\(122\) 2.69223 0.243743
\(123\) −1.00000 −0.0901670
\(124\) −5.14992 −0.462476
\(125\) −34.2277 −3.06141
\(126\) 0 0
\(127\) −0.824101 −0.0731271 −0.0365635 0.999331i \(-0.511641\pi\)
−0.0365635 + 0.999331i \(0.511641\pi\)
\(128\) −11.1629 −0.986671
\(129\) −10.8337 −0.953850
\(130\) −0.498281 −0.0437021
\(131\) −8.13187 −0.710485 −0.355243 0.934774i \(-0.615602\pi\)
−0.355243 + 0.934774i \(0.615602\pi\)
\(132\) 2.61555 0.227654
\(133\) 0 0
\(134\) 7.50172 0.648050
\(135\) −4.24914 −0.365708
\(136\) −8.94137 −0.766716
\(137\) −12.2035 −1.04262 −0.521308 0.853369i \(-0.674556\pi\)
−0.521308 + 0.853369i \(0.674556\pi\)
\(138\) −2.94137 −0.250386
\(139\) −14.2897 −1.21204 −0.606019 0.795450i \(-0.707235\pi\)
−0.606019 + 0.795450i \(0.707235\pi\)
\(140\) 0 0
\(141\) 4.08623 0.344123
\(142\) −4.95597 −0.415896
\(143\) −0.366407 −0.0306405
\(144\) 2.71982 0.226652
\(145\) 10.2491 0.851145
\(146\) −4.42160 −0.365934
\(147\) 0 0
\(148\) −17.2863 −1.42092
\(149\) −22.1104 −1.81135 −0.905677 0.423969i \(-0.860637\pi\)
−0.905677 + 0.423969i \(0.860637\pi\)
\(150\) 6.14486 0.501726
\(151\) 10.1173 0.823331 0.411666 0.911335i \(-0.364947\pi\)
0.411666 + 0.911335i \(0.364947\pi\)
\(152\) −4.00000 −0.324443
\(153\) 5.02760 0.406457
\(154\) 0 0
\(155\) −12.3043 −0.988308
\(156\) −0.443086 −0.0354753
\(157\) −18.7620 −1.49737 −0.748686 0.662924i \(-0.769315\pi\)
−0.748686 + 0.662924i \(0.769315\pi\)
\(158\) −0.263748 −0.0209827
\(159\) −1.43965 −0.114172
\(160\) −20.5535 −1.62489
\(161\) 0 0
\(162\) 0.470683 0.0369804
\(163\) 8.54392 0.669212 0.334606 0.942358i \(-0.391397\pi\)
0.334606 + 0.942358i \(0.391397\pi\)
\(164\) 1.77846 0.138874
\(165\) 6.24914 0.486495
\(166\) −1.79145 −0.139043
\(167\) −5.05863 −0.391449 −0.195724 0.980659i \(-0.562706\pi\)
−0.195724 + 0.980659i \(0.562706\pi\)
\(168\) 0 0
\(169\) −12.9379 −0.995225
\(170\) −10.0552 −0.771198
\(171\) 2.24914 0.171996
\(172\) 19.2672 1.46911
\(173\) −18.8647 −1.43426 −0.717128 0.696942i \(-0.754544\pi\)
−0.717128 + 0.696942i \(0.754544\pi\)
\(174\) −1.13531 −0.0860678
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −2.61555 −0.196597
\(178\) −1.93793 −0.145254
\(179\) −2.85514 −0.213403 −0.106701 0.994291i \(-0.534029\pi\)
−0.106701 + 0.994291i \(0.534029\pi\)
\(180\) 7.55691 0.563259
\(181\) −13.4396 −0.998961 −0.499481 0.866325i \(-0.666476\pi\)
−0.499481 + 0.866325i \(0.666476\pi\)
\(182\) 0 0
\(183\) 5.71982 0.422822
\(184\) 11.1138 0.819322
\(185\) −41.3009 −3.03650
\(186\) 1.36297 0.0999377
\(187\) −7.39400 −0.540703
\(188\) −7.26719 −0.530014
\(189\) 0 0
\(190\) −4.49828 −0.326340
\(191\) 13.4948 0.976453 0.488226 0.872717i \(-0.337644\pi\)
0.488226 + 0.872717i \(0.337644\pi\)
\(192\) −3.16291 −0.228263
\(193\) −8.01461 −0.576904 −0.288452 0.957494i \(-0.593141\pi\)
−0.288452 + 0.957494i \(0.593141\pi\)
\(194\) 3.38445 0.242990
\(195\) −1.05863 −0.0758103
\(196\) 0 0
\(197\) 13.9785 0.995928 0.497964 0.867198i \(-0.334081\pi\)
0.497964 + 0.867198i \(0.334081\pi\)
\(198\) −0.692226 −0.0491944
\(199\) 4.86469 0.344849 0.172424 0.985023i \(-0.444840\pi\)
0.172424 + 0.985023i \(0.444840\pi\)
\(200\) −23.2181 −1.64177
\(201\) 15.9379 1.12417
\(202\) 5.24570 0.369086
\(203\) 0 0
\(204\) −8.94137 −0.626021
\(205\) 4.24914 0.296773
\(206\) −6.24914 −0.435398
\(207\) −6.24914 −0.434345
\(208\) 0.677618 0.0469844
\(209\) −3.30777 −0.228803
\(210\) 0 0
\(211\) −10.6155 −0.730804 −0.365402 0.930850i \(-0.619069\pi\)
−0.365402 + 0.930850i \(0.619069\pi\)
\(212\) 2.56035 0.175846
\(213\) −10.5293 −0.721457
\(214\) 4.38789 0.299950
\(215\) 46.0337 3.13947
\(216\) −1.77846 −0.121009
\(217\) 0 0
\(218\) −0.117266 −0.00794228
\(219\) −9.39400 −0.634788
\(220\) −11.1138 −0.749294
\(221\) 1.25258 0.0842575
\(222\) 4.57496 0.307051
\(223\) 17.9931 1.20491 0.602454 0.798153i \(-0.294190\pi\)
0.602454 + 0.798153i \(0.294190\pi\)
\(224\) 0 0
\(225\) 13.0552 0.870346
\(226\) 4.35180 0.289477
\(227\) 0.910331 0.0604208 0.0302104 0.999544i \(-0.490382\pi\)
0.0302104 + 0.999544i \(0.490382\pi\)
\(228\) −4.00000 −0.264906
\(229\) 7.42504 0.490660 0.245330 0.969440i \(-0.421104\pi\)
0.245330 + 0.969440i \(0.421104\pi\)
\(230\) 12.4983 0.824112
\(231\) 0 0
\(232\) 4.28973 0.281634
\(233\) 14.7620 0.967093 0.483546 0.875319i \(-0.339348\pi\)
0.483546 + 0.875319i \(0.339348\pi\)
\(234\) 0.117266 0.00766594
\(235\) −17.3630 −1.13264
\(236\) 4.65164 0.302796
\(237\) −0.560352 −0.0363988
\(238\) 0 0
\(239\) −1.38445 −0.0895528 −0.0447764 0.998997i \(-0.514258\pi\)
−0.0447764 + 0.998997i \(0.514258\pi\)
\(240\) −11.5569 −0.745996
\(241\) 5.95436 0.383554 0.191777 0.981439i \(-0.438575\pi\)
0.191777 + 0.981439i \(0.438575\pi\)
\(242\) −4.15947 −0.267381
\(243\) 1.00000 0.0641500
\(244\) −10.1725 −0.651225
\(245\) 0 0
\(246\) −0.470683 −0.0300097
\(247\) 0.560352 0.0356543
\(248\) −5.14992 −0.327020
\(249\) −3.80605 −0.241199
\(250\) −16.1104 −1.01891
\(251\) −2.70683 −0.170854 −0.0854269 0.996344i \(-0.527225\pi\)
−0.0854269 + 0.996344i \(0.527225\pi\)
\(252\) 0 0
\(253\) 9.19051 0.577802
\(254\) −0.387890 −0.0243384
\(255\) −21.3630 −1.33780
\(256\) 1.07162 0.0669764
\(257\) 0.793065 0.0494700 0.0247350 0.999694i \(-0.492126\pi\)
0.0247350 + 0.999694i \(0.492126\pi\)
\(258\) −5.09922 −0.317464
\(259\) 0 0
\(260\) 1.88273 0.116762
\(261\) −2.41205 −0.149302
\(262\) −3.82754 −0.236466
\(263\) −15.5879 −0.961194 −0.480597 0.876942i \(-0.659580\pi\)
−0.480597 + 0.876942i \(0.659580\pi\)
\(264\) 2.61555 0.160976
\(265\) 6.11727 0.375781
\(266\) 0 0
\(267\) −4.11727 −0.251973
\(268\) −28.3449 −1.73144
\(269\) −3.05863 −0.186488 −0.0932441 0.995643i \(-0.529724\pi\)
−0.0932441 + 0.995643i \(0.529724\pi\)
\(270\) −2.00000 −0.121716
\(271\) −12.2802 −0.745968 −0.372984 0.927838i \(-0.621665\pi\)
−0.372984 + 0.927838i \(0.621665\pi\)
\(272\) 13.6742 0.829119
\(273\) 0 0
\(274\) −5.74398 −0.347007
\(275\) −19.2001 −1.15781
\(276\) 11.1138 0.668974
\(277\) −8.56990 −0.514916 −0.257458 0.966290i \(-0.582885\pi\)
−0.257458 + 0.966290i \(0.582885\pi\)
\(278\) −6.72594 −0.403395
\(279\) 2.89572 0.173362
\(280\) 0 0
\(281\) −14.5845 −0.870039 −0.435020 0.900421i \(-0.643259\pi\)
−0.435020 + 0.900421i \(0.643259\pi\)
\(282\) 1.92332 0.114532
\(283\) 20.9509 1.24540 0.622701 0.782460i \(-0.286035\pi\)
0.622701 + 0.782460i \(0.286035\pi\)
\(284\) 18.7259 1.11118
\(285\) −9.55691 −0.566103
\(286\) −0.172462 −0.0101979
\(287\) 0 0
\(288\) 4.83709 0.285028
\(289\) 8.27674 0.486867
\(290\) 4.82410 0.283281
\(291\) 7.19051 0.421515
\(292\) 16.7068 0.977694
\(293\) −31.4086 −1.83491 −0.917455 0.397839i \(-0.869760\pi\)
−0.917455 + 0.397839i \(0.869760\pi\)
\(294\) 0 0
\(295\) 11.1138 0.647072
\(296\) −17.2863 −1.00475
\(297\) −1.47068 −0.0853377
\(298\) −10.4070 −0.602861
\(299\) −1.55691 −0.0900387
\(300\) −23.2181 −1.34050
\(301\) 0 0
\(302\) 4.76203 0.274024
\(303\) 11.1449 0.640256
\(304\) 6.11727 0.350849
\(305\) −24.3043 −1.39166
\(306\) 2.36641 0.135278
\(307\) −8.54392 −0.487628 −0.243814 0.969822i \(-0.578399\pi\)
−0.243814 + 0.969822i \(0.578399\pi\)
\(308\) 0 0
\(309\) −13.2767 −0.755287
\(310\) −5.79145 −0.328932
\(311\) 5.49484 0.311584 0.155792 0.987790i \(-0.450207\pi\)
0.155792 + 0.987790i \(0.450207\pi\)
\(312\) −0.443086 −0.0250848
\(313\) −5.48024 −0.309761 −0.154881 0.987933i \(-0.549499\pi\)
−0.154881 + 0.987933i \(0.549499\pi\)
\(314\) −8.83098 −0.498361
\(315\) 0 0
\(316\) 0.996562 0.0560610
\(317\) −11.3173 −0.635644 −0.317822 0.948150i \(-0.602951\pi\)
−0.317822 + 0.948150i \(0.602951\pi\)
\(318\) −0.677618 −0.0379990
\(319\) 3.54736 0.198614
\(320\) 13.4396 0.751299
\(321\) 9.32238 0.520325
\(322\) 0 0
\(323\) 11.3078 0.629181
\(324\) −1.77846 −0.0988032
\(325\) 3.25258 0.180421
\(326\) 4.02148 0.222729
\(327\) −0.249141 −0.0137775
\(328\) 1.77846 0.0981989
\(329\) 0 0
\(330\) 2.94137 0.161917
\(331\) 33.8613 1.86118 0.930591 0.366060i \(-0.119293\pi\)
0.930591 + 0.366060i \(0.119293\pi\)
\(332\) 6.76891 0.371492
\(333\) 9.71982 0.532643
\(334\) −2.38101 −0.130283
\(335\) −67.7225 −3.70008
\(336\) 0 0
\(337\) −6.77846 −0.369246 −0.184623 0.982809i \(-0.559106\pi\)
−0.184623 + 0.982809i \(0.559106\pi\)
\(338\) −6.08967 −0.331234
\(339\) 9.24570 0.502158
\(340\) 37.9931 2.06047
\(341\) −4.25869 −0.230621
\(342\) 1.05863 0.0572443
\(343\) 0 0
\(344\) 19.2672 1.03882
\(345\) 26.5535 1.42959
\(346\) −8.87930 −0.477354
\(347\) 19.2001 1.03071 0.515357 0.856976i \(-0.327659\pi\)
0.515357 + 0.856976i \(0.327659\pi\)
\(348\) 4.28973 0.229954
\(349\) 6.54392 0.350288 0.175144 0.984543i \(-0.443961\pi\)
0.175144 + 0.984543i \(0.443961\pi\)
\(350\) 0 0
\(351\) 0.249141 0.0132981
\(352\) −7.11383 −0.379168
\(353\) 15.9931 0.851228 0.425614 0.904905i \(-0.360058\pi\)
0.425614 + 0.904905i \(0.360058\pi\)
\(354\) −1.23109 −0.0654320
\(355\) 44.7405 2.37458
\(356\) 7.32238 0.388085
\(357\) 0 0
\(358\) −1.34387 −0.0710255
\(359\) −12.1319 −0.640296 −0.320148 0.947368i \(-0.603733\pi\)
−0.320148 + 0.947368i \(0.603733\pi\)
\(360\) 7.55691 0.398284
\(361\) −13.9414 −0.733756
\(362\) −6.32582 −0.332478
\(363\) −8.83709 −0.463827
\(364\) 0 0
\(365\) 39.9164 2.08932
\(366\) 2.69223 0.140725
\(367\) −6.39744 −0.333944 −0.166972 0.985962i \(-0.553399\pi\)
−0.166972 + 0.985962i \(0.553399\pi\)
\(368\) −16.9966 −0.886007
\(369\) −1.00000 −0.0520579
\(370\) −19.4396 −1.01062
\(371\) 0 0
\(372\) −5.14992 −0.267011
\(373\) −7.54049 −0.390432 −0.195216 0.980760i \(-0.562541\pi\)
−0.195216 + 0.980760i \(0.562541\pi\)
\(374\) −3.48024 −0.179959
\(375\) −34.2277 −1.76751
\(376\) −7.26719 −0.374777
\(377\) −0.600939 −0.0309500
\(378\) 0 0
\(379\) −0.0620710 −0.00318837 −0.00159419 0.999999i \(-0.500507\pi\)
−0.00159419 + 0.999999i \(0.500507\pi\)
\(380\) 16.9966 0.871905
\(381\) −0.824101 −0.0422199
\(382\) 6.35180 0.324986
\(383\) −23.9329 −1.22291 −0.611456 0.791278i \(-0.709416\pi\)
−0.611456 + 0.791278i \(0.709416\pi\)
\(384\) −11.1629 −0.569655
\(385\) 0 0
\(386\) −3.77234 −0.192007
\(387\) −10.8337 −0.550706
\(388\) −12.7880 −0.649213
\(389\) −23.8613 −1.20981 −0.604907 0.796296i \(-0.706790\pi\)
−0.604907 + 0.796296i \(0.706790\pi\)
\(390\) −0.498281 −0.0252314
\(391\) −31.4182 −1.58888
\(392\) 0 0
\(393\) −8.13187 −0.410199
\(394\) 6.57946 0.331468
\(395\) 2.38101 0.119802
\(396\) 2.61555 0.131436
\(397\) −14.6302 −0.734266 −0.367133 0.930168i \(-0.619661\pi\)
−0.367133 + 0.930168i \(0.619661\pi\)
\(398\) 2.28973 0.114774
\(399\) 0 0
\(400\) 35.5078 1.77539
\(401\) −11.3224 −0.565413 −0.282706 0.959206i \(-0.591232\pi\)
−0.282706 + 0.959206i \(0.591232\pi\)
\(402\) 7.50172 0.374152
\(403\) 0.721442 0.0359376
\(404\) −19.8207 −0.986115
\(405\) −4.24914 −0.211141
\(406\) 0 0
\(407\) −14.2948 −0.708566
\(408\) −8.94137 −0.442664
\(409\) 33.8923 1.67587 0.837933 0.545773i \(-0.183764\pi\)
0.837933 + 0.545773i \(0.183764\pi\)
\(410\) 2.00000 0.0987730
\(411\) −12.2035 −0.601954
\(412\) 23.6121 1.16329
\(413\) 0 0
\(414\) −2.94137 −0.144560
\(415\) 16.1725 0.793875
\(416\) 1.20512 0.0590856
\(417\) −14.2897 −0.699771
\(418\) −1.55691 −0.0761512
\(419\) 1.98539 0.0969928 0.0484964 0.998823i \(-0.484557\pi\)
0.0484964 + 0.998823i \(0.484557\pi\)
\(420\) 0 0
\(421\) −5.67418 −0.276543 −0.138271 0.990394i \(-0.544155\pi\)
−0.138271 + 0.990394i \(0.544155\pi\)
\(422\) −4.99656 −0.243229
\(423\) 4.08623 0.198679
\(424\) 2.56035 0.124342
\(425\) 65.6363 3.18383
\(426\) −4.95597 −0.240118
\(427\) 0 0
\(428\) −16.5795 −0.801398
\(429\) −0.366407 −0.0176903
\(430\) 21.6673 1.04489
\(431\) 1.75086 0.0843359 0.0421680 0.999111i \(-0.486574\pi\)
0.0421680 + 0.999111i \(0.486574\pi\)
\(432\) 2.71982 0.130858
\(433\) 34.0647 1.63705 0.818524 0.574473i \(-0.194793\pi\)
0.818524 + 0.574473i \(0.194793\pi\)
\(434\) 0 0
\(435\) 10.2491 0.491409
\(436\) 0.443086 0.0212200
\(437\) −14.0552 −0.672351
\(438\) −4.42160 −0.211272
\(439\) 4.32582 0.206460 0.103230 0.994658i \(-0.467082\pi\)
0.103230 + 0.994658i \(0.467082\pi\)
\(440\) −11.1138 −0.529831
\(441\) 0 0
\(442\) 0.589568 0.0280429
\(443\) 36.9966 1.75776 0.878880 0.477043i \(-0.158291\pi\)
0.878880 + 0.477043i \(0.158291\pi\)
\(444\) −17.2863 −0.820371
\(445\) 17.4948 0.829335
\(446\) 8.46907 0.401022
\(447\) −22.1104 −1.04579
\(448\) 0 0
\(449\) −15.9448 −0.752482 −0.376241 0.926522i \(-0.622784\pi\)
−0.376241 + 0.926522i \(0.622784\pi\)
\(450\) 6.14486 0.289672
\(451\) 1.47068 0.0692518
\(452\) −16.4431 −0.773418
\(453\) 10.1173 0.475351
\(454\) 0.428478 0.0201095
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −29.4396 −1.37713 −0.688564 0.725175i \(-0.741759\pi\)
−0.688564 + 0.725175i \(0.741759\pi\)
\(458\) 3.49484 0.163303
\(459\) 5.02760 0.234668
\(460\) −47.2242 −2.20184
\(461\) −21.8759 −1.01886 −0.509430 0.860512i \(-0.670144\pi\)
−0.509430 + 0.860512i \(0.670144\pi\)
\(462\) 0 0
\(463\) −18.6561 −0.867024 −0.433512 0.901148i \(-0.642726\pi\)
−0.433512 + 0.901148i \(0.642726\pi\)
\(464\) −6.56035 −0.304557
\(465\) −12.3043 −0.570600
\(466\) 6.94824 0.321871
\(467\) −8.67074 −0.401234 −0.200617 0.979670i \(-0.564295\pi\)
−0.200617 + 0.979670i \(0.564295\pi\)
\(468\) −0.443086 −0.0204816
\(469\) 0 0
\(470\) −8.17246 −0.376968
\(471\) −18.7620 −0.864509
\(472\) 4.65164 0.214109
\(473\) 15.9329 0.732594
\(474\) −0.263748 −0.0121144
\(475\) 29.3630 1.34727
\(476\) 0 0
\(477\) −1.43965 −0.0659169
\(478\) −0.651639 −0.0298053
\(479\) 20.4312 0.933523 0.466762 0.884383i \(-0.345421\pi\)
0.466762 + 0.884383i \(0.345421\pi\)
\(480\) −20.5535 −0.938134
\(481\) 2.42160 0.110416
\(482\) 2.80262 0.127656
\(483\) 0 0
\(484\) 15.7164 0.714381
\(485\) −30.5535 −1.38736
\(486\) 0.470683 0.0213506
\(487\) −7.00611 −0.317477 −0.158739 0.987321i \(-0.550743\pi\)
−0.158739 + 0.987321i \(0.550743\pi\)
\(488\) −10.1725 −0.460486
\(489\) 8.54392 0.386370
\(490\) 0 0
\(491\) −15.5715 −0.702733 −0.351366 0.936238i \(-0.614283\pi\)
−0.351366 + 0.936238i \(0.614283\pi\)
\(492\) 1.77846 0.0801790
\(493\) −12.1268 −0.546164
\(494\) 0.263748 0.0118666
\(495\) 6.24914 0.280878
\(496\) 7.87586 0.353636
\(497\) 0 0
\(498\) −1.79145 −0.0802767
\(499\) −1.61899 −0.0724757 −0.0362379 0.999343i \(-0.511537\pi\)
−0.0362379 + 0.999343i \(0.511537\pi\)
\(500\) 60.8724 2.72230
\(501\) −5.05863 −0.226003
\(502\) −1.27406 −0.0568642
\(503\) 39.3725 1.75553 0.877767 0.479088i \(-0.159032\pi\)
0.877767 + 0.479088i \(0.159032\pi\)
\(504\) 0 0
\(505\) −47.3561 −2.10732
\(506\) 4.32582 0.192306
\(507\) −12.9379 −0.574594
\(508\) 1.46563 0.0650267
\(509\) 8.63971 0.382948 0.191474 0.981498i \(-0.438673\pi\)
0.191474 + 0.981498i \(0.438673\pi\)
\(510\) −10.0552 −0.445252
\(511\) 0 0
\(512\) 22.8302 1.00896
\(513\) 2.24914 0.0993020
\(514\) 0.373283 0.0164648
\(515\) 56.4147 2.48593
\(516\) 19.2672 0.848191
\(517\) −6.00955 −0.264300
\(518\) 0 0
\(519\) −18.8647 −0.828068
\(520\) 1.88273 0.0825633
\(521\) −30.7862 −1.34877 −0.674384 0.738381i \(-0.735591\pi\)
−0.674384 + 0.738381i \(0.735591\pi\)
\(522\) −1.13531 −0.0496913
\(523\) −17.4036 −0.761004 −0.380502 0.924780i \(-0.624249\pi\)
−0.380502 + 0.924780i \(0.624249\pi\)
\(524\) 14.4622 0.631784
\(525\) 0 0
\(526\) −7.33699 −0.319908
\(527\) 14.5585 0.634180
\(528\) −4.00000 −0.174078
\(529\) 16.0518 0.697902
\(530\) 2.87930 0.125069
\(531\) −2.61555 −0.113505
\(532\) 0 0
\(533\) −0.249141 −0.0107915
\(534\) −1.93793 −0.0838624
\(535\) −39.6121 −1.71258
\(536\) −28.3449 −1.22431
\(537\) −2.85514 −0.123208
\(538\) −1.43965 −0.0620676
\(539\) 0 0
\(540\) 7.55691 0.325198
\(541\) −9.87586 −0.424596 −0.212298 0.977205i \(-0.568095\pi\)
−0.212298 + 0.977205i \(0.568095\pi\)
\(542\) −5.78008 −0.248275
\(543\) −13.4396 −0.576750
\(544\) 24.3189 1.04267
\(545\) 1.05863 0.0453469
\(546\) 0 0
\(547\) −18.1465 −0.775888 −0.387944 0.921683i \(-0.626815\pi\)
−0.387944 + 0.921683i \(0.626815\pi\)
\(548\) 21.7034 0.927123
\(549\) 5.71982 0.244116
\(550\) −9.03715 −0.385345
\(551\) −5.42504 −0.231114
\(552\) 11.1138 0.473036
\(553\) 0 0
\(554\) −4.03371 −0.171376
\(555\) −41.3009 −1.75313
\(556\) 25.4137 1.07778
\(557\) 37.3725 1.58352 0.791762 0.610829i \(-0.209164\pi\)
0.791762 + 0.610829i \(0.209164\pi\)
\(558\) 1.36297 0.0576991
\(559\) −2.69910 −0.114160
\(560\) 0 0
\(561\) −7.39400 −0.312175
\(562\) −6.86469 −0.289569
\(563\) −22.8742 −0.964034 −0.482017 0.876162i \(-0.660096\pi\)
−0.482017 + 0.876162i \(0.660096\pi\)
\(564\) −7.26719 −0.306004
\(565\) −39.2863 −1.65279
\(566\) 9.86125 0.414499
\(567\) 0 0
\(568\) 18.7259 0.785723
\(569\) 34.4147 1.44274 0.721370 0.692550i \(-0.243513\pi\)
0.721370 + 0.692550i \(0.243513\pi\)
\(570\) −4.49828 −0.188412
\(571\) 36.3449 1.52099 0.760494 0.649345i \(-0.224957\pi\)
0.760494 + 0.649345i \(0.224957\pi\)
\(572\) 0.651639 0.0272464
\(573\) 13.4948 0.563755
\(574\) 0 0
\(575\) −81.5838 −3.40228
\(576\) −3.16291 −0.131788
\(577\) −26.1725 −1.08957 −0.544787 0.838575i \(-0.683389\pi\)
−0.544787 + 0.838575i \(0.683389\pi\)
\(578\) 3.89572 0.162041
\(579\) −8.01461 −0.333076
\(580\) −18.2277 −0.756862
\(581\) 0 0
\(582\) 3.38445 0.140290
\(583\) 2.11727 0.0876882
\(584\) 16.7068 0.691334
\(585\) −1.05863 −0.0437691
\(586\) −14.7835 −0.610701
\(587\) 17.8156 0.735329 0.367664 0.929959i \(-0.380158\pi\)
0.367664 + 0.929959i \(0.380158\pi\)
\(588\) 0 0
\(589\) 6.51289 0.268359
\(590\) 5.23109 0.215361
\(591\) 13.9785 0.574999
\(592\) 26.4362 1.08652
\(593\) 32.7018 1.34290 0.671451 0.741049i \(-0.265672\pi\)
0.671451 + 0.741049i \(0.265672\pi\)
\(594\) −0.692226 −0.0284024
\(595\) 0 0
\(596\) 39.3224 1.61071
\(597\) 4.86469 0.199098
\(598\) −0.732814 −0.0299670
\(599\) 10.2784 0.419962 0.209981 0.977705i \(-0.432660\pi\)
0.209981 + 0.977705i \(0.432660\pi\)
\(600\) −23.2181 −0.947875
\(601\) −31.8398 −1.29877 −0.649386 0.760459i \(-0.724974\pi\)
−0.649386 + 0.760459i \(0.724974\pi\)
\(602\) 0 0
\(603\) 15.9379 0.649043
\(604\) −17.9931 −0.732130
\(605\) 37.5500 1.52663
\(606\) 5.24570 0.213092
\(607\) −10.2086 −0.414352 −0.207176 0.978304i \(-0.566427\pi\)
−0.207176 + 0.978304i \(0.566427\pi\)
\(608\) 10.8793 0.441214
\(609\) 0 0
\(610\) −11.4396 −0.463178
\(611\) 1.01805 0.0411857
\(612\) −8.94137 −0.361433
\(613\) −4.94137 −0.199580 −0.0997900 0.995009i \(-0.531817\pi\)
−0.0997900 + 0.995009i \(0.531817\pi\)
\(614\) −4.02148 −0.162294
\(615\) 4.24914 0.171342
\(616\) 0 0
\(617\) 15.2526 0.614046 0.307023 0.951702i \(-0.400667\pi\)
0.307023 + 0.951702i \(0.400667\pi\)
\(618\) −6.24914 −0.251377
\(619\) 20.6543 0.830167 0.415084 0.909783i \(-0.363752\pi\)
0.415084 + 0.909783i \(0.363752\pi\)
\(620\) 21.8827 0.878832
\(621\) −6.24914 −0.250769
\(622\) 2.58633 0.103702
\(623\) 0 0
\(624\) 0.677618 0.0271264
\(625\) 80.1621 3.20649
\(626\) −2.57946 −0.103096
\(627\) −3.30777 −0.132100
\(628\) 33.3675 1.33151
\(629\) 48.8674 1.94847
\(630\) 0 0
\(631\) 5.83709 0.232371 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(632\) 0.996562 0.0396411
\(633\) −10.6155 −0.421930
\(634\) −5.32688 −0.211557
\(635\) 3.50172 0.138961
\(636\) 2.56035 0.101525
\(637\) 0 0
\(638\) 1.66968 0.0661034
\(639\) −10.5293 −0.416533
\(640\) 47.4328 1.87494
\(641\) 42.6328 1.68390 0.841948 0.539559i \(-0.181409\pi\)
0.841948 + 0.539559i \(0.181409\pi\)
\(642\) 4.38789 0.173176
\(643\) −30.7259 −1.21171 −0.605856 0.795574i \(-0.707169\pi\)
−0.605856 + 0.795574i \(0.707169\pi\)
\(644\) 0 0
\(645\) 46.0337 1.81258
\(646\) 5.32238 0.209406
\(647\) −18.3595 −0.721788 −0.360894 0.932607i \(-0.617528\pi\)
−0.360894 + 0.932607i \(0.617528\pi\)
\(648\) −1.77846 −0.0698644
\(649\) 3.84664 0.150994
\(650\) 1.53093 0.0600482
\(651\) 0 0
\(652\) −15.1950 −0.595082
\(653\) −35.5811 −1.39240 −0.696198 0.717850i \(-0.745126\pi\)
−0.696198 + 0.717850i \(0.745126\pi\)
\(654\) −0.117266 −0.00458548
\(655\) 34.5535 1.35012
\(656\) −2.71982 −0.106191
\(657\) −9.39400 −0.366495
\(658\) 0 0
\(659\) −6.38101 −0.248569 −0.124285 0.992247i \(-0.539664\pi\)
−0.124285 + 0.992247i \(0.539664\pi\)
\(660\) −11.1138 −0.432605
\(661\) −30.9053 −1.20208 −0.601038 0.799220i \(-0.705246\pi\)
−0.601038 + 0.799220i \(0.705246\pi\)
\(662\) 15.9379 0.619445
\(663\) 1.25258 0.0486461
\(664\) 6.76891 0.262684
\(665\) 0 0
\(666\) 4.57496 0.177276
\(667\) 15.0732 0.583638
\(668\) 8.99656 0.348087
\(669\) 17.9931 0.695654
\(670\) −31.8759 −1.23147
\(671\) −8.41205 −0.324744
\(672\) 0 0
\(673\) 36.7259 1.41568 0.707840 0.706372i \(-0.249670\pi\)
0.707840 + 0.706372i \(0.249670\pi\)
\(674\) −3.19051 −0.122894
\(675\) 13.0552 0.502495
\(676\) 23.0096 0.884983
\(677\) 5.66281 0.217639 0.108820 0.994062i \(-0.465293\pi\)
0.108820 + 0.994062i \(0.465293\pi\)
\(678\) 4.35180 0.167130
\(679\) 0 0
\(680\) 37.9931 1.45697
\(681\) 0.910331 0.0348840
\(682\) −2.00450 −0.0767561
\(683\) −29.1690 −1.11612 −0.558061 0.829800i \(-0.688454\pi\)
−0.558061 + 0.829800i \(0.688454\pi\)
\(684\) −4.00000 −0.152944
\(685\) 51.8544 1.98125
\(686\) 0 0
\(687\) 7.42504 0.283283
\(688\) −29.4656 −1.12337
\(689\) −0.358675 −0.0136644
\(690\) 12.4983 0.475801
\(691\) −1.68879 −0.0642445 −0.0321223 0.999484i \(-0.510227\pi\)
−0.0321223 + 0.999484i \(0.510227\pi\)
\(692\) 33.5500 1.27538
\(693\) 0 0
\(694\) 9.03715 0.343046
\(695\) 60.7191 2.30321
\(696\) 4.28973 0.162602
\(697\) −5.02760 −0.190434
\(698\) 3.08012 0.116584
\(699\) 14.7620 0.558351
\(700\) 0 0
\(701\) −20.3810 −0.769780 −0.384890 0.922962i \(-0.625761\pi\)
−0.384890 + 0.922962i \(0.625761\pi\)
\(702\) 0.117266 0.00442593
\(703\) 21.8613 0.824513
\(704\) 4.65164 0.175315
\(705\) −17.3630 −0.653927
\(706\) 7.52770 0.283309
\(707\) 0 0
\(708\) 4.65164 0.174819
\(709\) −40.8363 −1.53364 −0.766820 0.641862i \(-0.778162\pi\)
−0.766820 + 0.641862i \(0.778162\pi\)
\(710\) 21.0586 0.790316
\(711\) −0.560352 −0.0210148
\(712\) 7.32238 0.274418
\(713\) −18.0958 −0.677692
\(714\) 0 0
\(715\) 1.55691 0.0582253
\(716\) 5.07774 0.189764
\(717\) −1.38445 −0.0517033
\(718\) −5.71027 −0.213105
\(719\) −9.14486 −0.341046 −0.170523 0.985354i \(-0.554546\pi\)
−0.170523 + 0.985354i \(0.554546\pi\)
\(720\) −11.5569 −0.430701
\(721\) 0 0
\(722\) −6.56197 −0.244211
\(723\) 5.95436 0.221445
\(724\) 23.9018 0.888305
\(725\) −31.4898 −1.16950
\(726\) −4.15947 −0.154372
\(727\) 2.13875 0.0793218 0.0396609 0.999213i \(-0.487372\pi\)
0.0396609 + 0.999213i \(0.487372\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 18.7880 0.695375
\(731\) −54.4672 −2.01454
\(732\) −10.1725 −0.375985
\(733\) −15.3388 −0.566552 −0.283276 0.959038i \(-0.591421\pi\)
−0.283276 + 0.959038i \(0.591421\pi\)
\(734\) −3.01117 −0.111144
\(735\) 0 0
\(736\) −30.2277 −1.11421
\(737\) −23.4396 −0.863411
\(738\) −0.470683 −0.0173261
\(739\) −15.8111 −0.581621 −0.290811 0.956781i \(-0.593925\pi\)
−0.290811 + 0.956781i \(0.593925\pi\)
\(740\) 73.4519 2.70014
\(741\) 0.560352 0.0205850
\(742\) 0 0
\(743\) −20.2017 −0.741128 −0.370564 0.928807i \(-0.620836\pi\)
−0.370564 + 0.928807i \(0.620836\pi\)
\(744\) −5.14992 −0.188805
\(745\) 93.9502 3.44207
\(746\) −3.54918 −0.129945
\(747\) −3.80605 −0.139256
\(748\) 13.1499 0.480809
\(749\) 0 0
\(750\) −16.1104 −0.588268
\(751\) 32.9053 1.20073 0.600365 0.799726i \(-0.295022\pi\)
0.600365 + 0.799726i \(0.295022\pi\)
\(752\) 11.1138 0.405280
\(753\) −2.70683 −0.0986425
\(754\) −0.282852 −0.0103009
\(755\) −42.9897 −1.56455
\(756\) 0 0
\(757\) 11.3009 0.410738 0.205369 0.978685i \(-0.434161\pi\)
0.205369 + 0.978685i \(0.434161\pi\)
\(758\) −0.0292158 −0.00106117
\(759\) 9.19051 0.333594
\(760\) 16.9966 0.616530
\(761\) 2.93449 0.106375 0.0531876 0.998585i \(-0.483062\pi\)
0.0531876 + 0.998585i \(0.483062\pi\)
\(762\) −0.387890 −0.0140518
\(763\) 0 0
\(764\) −24.0000 −0.868290
\(765\) −21.3630 −0.772380
\(766\) −11.2648 −0.407014
\(767\) −0.651639 −0.0235293
\(768\) 1.07162 0.0386689
\(769\) 22.7785 0.821412 0.410706 0.911768i \(-0.365282\pi\)
0.410706 + 0.911768i \(0.365282\pi\)
\(770\) 0 0
\(771\) 0.793065 0.0285615
\(772\) 14.2536 0.513000
\(773\) −10.5553 −0.379648 −0.189824 0.981818i \(-0.560792\pi\)
−0.189824 + 0.981818i \(0.560792\pi\)
\(774\) −5.09922 −0.183288
\(775\) 37.8042 1.35797
\(776\) −12.7880 −0.459063
\(777\) 0 0
\(778\) −11.2311 −0.402654
\(779\) −2.24914 −0.0805838
\(780\) 1.88273 0.0674127
\(781\) 15.4853 0.554107
\(782\) −14.7880 −0.528818
\(783\) −2.41205 −0.0861996
\(784\) 0 0
\(785\) 79.7225 2.84542
\(786\) −3.82754 −0.136524
\(787\) 23.0682 0.822292 0.411146 0.911570i \(-0.365129\pi\)
0.411146 + 0.911570i \(0.365129\pi\)
\(788\) −24.8602 −0.885608
\(789\) −15.5879 −0.554946
\(790\) 1.12070 0.0398729
\(791\) 0 0
\(792\) 2.61555 0.0929394
\(793\) 1.42504 0.0506047
\(794\) −6.88617 −0.244381
\(795\) 6.11727 0.216957
\(796\) −8.65164 −0.306649
\(797\) −0.0360915 −0.00127843 −0.000639213 1.00000i \(-0.500203\pi\)
−0.000639213 1.00000i \(0.500203\pi\)
\(798\) 0 0
\(799\) 20.5439 0.726792
\(800\) 63.1492 2.23266
\(801\) −4.11727 −0.145476
\(802\) −5.32926 −0.188183
\(803\) 13.8156 0.487542
\(804\) −28.3449 −0.999648
\(805\) 0 0
\(806\) 0.339571 0.0119609
\(807\) −3.05863 −0.107669
\(808\) −19.8207 −0.697288
\(809\) 33.1430 1.16525 0.582624 0.812742i \(-0.302026\pi\)
0.582624 + 0.812742i \(0.302026\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 1.10428 0.0387764 0.0193882 0.999812i \(-0.493828\pi\)
0.0193882 + 0.999812i \(0.493828\pi\)
\(812\) 0 0
\(813\) −12.2802 −0.430685
\(814\) −6.72832 −0.235827
\(815\) −36.3043 −1.27169
\(816\) 13.6742 0.478692
\(817\) −24.3664 −0.852473
\(818\) 15.9525 0.557767
\(819\) 0 0
\(820\) −7.55691 −0.263899
\(821\) −7.42504 −0.259136 −0.129568 0.991571i \(-0.541359\pi\)
−0.129568 + 0.991571i \(0.541359\pi\)
\(822\) −5.74398 −0.200344
\(823\) 42.1656 1.46980 0.734900 0.678176i \(-0.237229\pi\)
0.734900 + 0.678176i \(0.237229\pi\)
\(824\) 23.6121 0.822567
\(825\) −19.2001 −0.668460
\(826\) 0 0
\(827\) −54.4863 −1.89468 −0.947338 0.320235i \(-0.896238\pi\)
−0.947338 + 0.320235i \(0.896238\pi\)
\(828\) 11.1138 0.386232
\(829\) −24.0388 −0.834901 −0.417450 0.908700i \(-0.637076\pi\)
−0.417450 + 0.908700i \(0.637076\pi\)
\(830\) 7.61211 0.264220
\(831\) −8.56990 −0.297287
\(832\) −0.788009 −0.0273193
\(833\) 0 0
\(834\) −6.72594 −0.232900
\(835\) 21.4948 0.743860
\(836\) 5.88273 0.203459
\(837\) 2.89572 0.100091
\(838\) 0.934491 0.0322815
\(839\) 13.6190 0.470180 0.235090 0.971974i \(-0.424462\pi\)
0.235090 + 0.971974i \(0.424462\pi\)
\(840\) 0 0
\(841\) −23.1820 −0.799380
\(842\) −2.67074 −0.0920399
\(843\) −14.5845 −0.502317
\(844\) 18.8793 0.649852
\(845\) 54.9751 1.89120
\(846\) 1.92332 0.0661251
\(847\) 0 0
\(848\) −3.91559 −0.134462
\(849\) 20.9509 0.719034
\(850\) 30.8939 1.05965
\(851\) −60.7405 −2.08216
\(852\) 18.7259 0.641540
\(853\) 6.93449 0.237432 0.118716 0.992928i \(-0.462122\pi\)
0.118716 + 0.992928i \(0.462122\pi\)
\(854\) 0 0
\(855\) −9.55691 −0.326840
\(856\) −16.5795 −0.566674
\(857\) 34.9345 1.19334 0.596670 0.802487i \(-0.296490\pi\)
0.596670 + 0.802487i \(0.296490\pi\)
\(858\) −0.172462 −0.00588774
\(859\) −4.39057 −0.149804 −0.0749021 0.997191i \(-0.523864\pi\)
−0.0749021 + 0.997191i \(0.523864\pi\)
\(860\) −81.8690 −2.79171
\(861\) 0 0
\(862\) 0.824101 0.0280690
\(863\) 15.4611 0.526303 0.263152 0.964755i \(-0.415238\pi\)
0.263152 + 0.964755i \(0.415238\pi\)
\(864\) 4.83709 0.164561
\(865\) 80.1587 2.72548
\(866\) 16.0337 0.544848
\(867\) 8.27674 0.281093
\(868\) 0 0
\(869\) 0.824101 0.0279557
\(870\) 4.82410 0.163552
\(871\) 3.97078 0.134545
\(872\) 0.443086 0.0150048
\(873\) 7.19051 0.243362
\(874\) −6.61555 −0.223774
\(875\) 0 0
\(876\) 16.7068 0.564472
\(877\) −55.2338 −1.86511 −0.932556 0.361025i \(-0.882427\pi\)
−0.932556 + 0.361025i \(0.882427\pi\)
\(878\) 2.03609 0.0687148
\(879\) −31.4086 −1.05939
\(880\) 16.9966 0.572954
\(881\) 28.4431 0.958272 0.479136 0.877741i \(-0.340950\pi\)
0.479136 + 0.877741i \(0.340950\pi\)
\(882\) 0 0
\(883\) −24.1319 −0.812102 −0.406051 0.913850i \(-0.633095\pi\)
−0.406051 + 0.913850i \(0.633095\pi\)
\(884\) −2.22766 −0.0749242
\(885\) 11.1138 0.373587
\(886\) 17.4137 0.585024
\(887\) 2.06025 0.0691765 0.0345882 0.999402i \(-0.488988\pi\)
0.0345882 + 0.999402i \(0.488988\pi\)
\(888\) −17.2863 −0.580090
\(889\) 0 0
\(890\) 8.23453 0.276022
\(891\) −1.47068 −0.0492697
\(892\) −32.0000 −1.07144
\(893\) 9.19051 0.307549
\(894\) −10.4070 −0.348062
\(895\) 12.1319 0.405524
\(896\) 0 0
\(897\) −1.55691 −0.0519839
\(898\) −7.50496 −0.250444
\(899\) −6.98463 −0.232950
\(900\) −23.2181 −0.773937
\(901\) −7.23797 −0.241132
\(902\) 0.692226 0.0230486
\(903\) 0 0
\(904\) −16.4431 −0.546889
\(905\) 57.1070 1.89830
\(906\) 4.76203 0.158208
\(907\) −19.5208 −0.648178 −0.324089 0.946027i \(-0.605058\pi\)
−0.324089 + 0.946027i \(0.605058\pi\)
\(908\) −1.61899 −0.0537279
\(909\) 11.1449 0.369652
\(910\) 0 0
\(911\) −39.9785 −1.32455 −0.662274 0.749262i \(-0.730408\pi\)
−0.662274 + 0.749262i \(0.730408\pi\)
\(912\) 6.11727 0.202563
\(913\) 5.59750 0.185250
\(914\) −13.8568 −0.458341
\(915\) −24.3043 −0.803477
\(916\) −13.2051 −0.436309
\(917\) 0 0
\(918\) 2.36641 0.0781031
\(919\) −26.4508 −0.872532 −0.436266 0.899818i \(-0.643699\pi\)
−0.436266 + 0.899818i \(0.643699\pi\)
\(920\) −47.2242 −1.55694
\(921\) −8.54392 −0.281532
\(922\) −10.2966 −0.339101
\(923\) −2.62328 −0.0863463
\(924\) 0 0
\(925\) 126.894 4.17226
\(926\) −8.78113 −0.288566
\(927\) −13.2767 −0.436065
\(928\) −11.6673 −0.382998
\(929\) −33.2913 −1.09225 −0.546127 0.837703i \(-0.683898\pi\)
−0.546127 + 0.837703i \(0.683898\pi\)
\(930\) −5.79145 −0.189909
\(931\) 0 0
\(932\) −26.2536 −0.859966
\(933\) 5.49484 0.179893
\(934\) −4.08117 −0.133540
\(935\) 31.4182 1.02748
\(936\) −0.443086 −0.0144827
\(937\) 38.7811 1.26692 0.633462 0.773774i \(-0.281633\pi\)
0.633462 + 0.773774i \(0.281633\pi\)
\(938\) 0 0
\(939\) −5.48024 −0.178841
\(940\) 30.8793 1.00717
\(941\) −15.6482 −0.510117 −0.255058 0.966926i \(-0.582095\pi\)
−0.255058 + 0.966926i \(0.582095\pi\)
\(942\) −8.83098 −0.287729
\(943\) 6.24914 0.203500
\(944\) −7.11383 −0.231535
\(945\) 0 0
\(946\) 7.49934 0.243825
\(947\) −47.5500 −1.54517 −0.772584 0.634912i \(-0.781036\pi\)
−0.772584 + 0.634912i \(0.781036\pi\)
\(948\) 0.996562 0.0323668
\(949\) −2.34043 −0.0759735
\(950\) 13.8207 0.448402
\(951\) −11.3173 −0.366989
\(952\) 0 0
\(953\) −29.6819 −0.961491 −0.480746 0.876860i \(-0.659634\pi\)
−0.480746 + 0.876860i \(0.659634\pi\)
\(954\) −0.677618 −0.0219387
\(955\) −57.3415 −1.85553
\(956\) 2.46219 0.0796329
\(957\) 3.54736 0.114670
\(958\) 9.61661 0.310698
\(959\) 0 0
\(960\) 13.4396 0.433763
\(961\) −22.6148 −0.729509
\(962\) 1.13981 0.0367489
\(963\) 9.32238 0.300410
\(964\) −10.5896 −0.341067
\(965\) 34.0552 1.09628
\(966\) 0 0
\(967\) −1.21199 −0.0389750 −0.0194875 0.999810i \(-0.506203\pi\)
−0.0194875 + 0.999810i \(0.506203\pi\)
\(968\) 15.7164 0.505144
\(969\) 11.3078 0.363258
\(970\) −14.3810 −0.461747
\(971\) −12.5362 −0.402306 −0.201153 0.979560i \(-0.564469\pi\)
−0.201153 + 0.979560i \(0.564469\pi\)
\(972\) −1.77846 −0.0570440
\(973\) 0 0
\(974\) −3.29766 −0.105664
\(975\) 3.25258 0.104166
\(976\) 15.5569 0.497965
\(977\) 15.2069 0.486513 0.243256 0.969962i \(-0.421784\pi\)
0.243256 + 0.969962i \(0.421784\pi\)
\(978\) 4.02148 0.128593
\(979\) 6.05520 0.193525
\(980\) 0 0
\(981\) −0.249141 −0.00795445
\(982\) −7.32926 −0.233886
\(983\) −39.0303 −1.24487 −0.622436 0.782671i \(-0.713857\pi\)
−0.622436 + 0.782671i \(0.713857\pi\)
\(984\) 1.77846 0.0566951
\(985\) −59.3967 −1.89254
\(986\) −5.70789 −0.181776
\(987\) 0 0
\(988\) −0.996562 −0.0317049
\(989\) 67.7010 2.15277
\(990\) 2.94137 0.0934828
\(991\) −8.91539 −0.283207 −0.141603 0.989923i \(-0.545226\pi\)
−0.141603 + 0.989923i \(0.545226\pi\)
\(992\) 14.0069 0.444719
\(993\) 33.8613 1.07455
\(994\) 0 0
\(995\) −20.6707 −0.655307
\(996\) 6.76891 0.214481
\(997\) 33.6336 1.06519 0.532593 0.846371i \(-0.321218\pi\)
0.532593 + 0.846371i \(0.321218\pi\)
\(998\) −0.762030 −0.0241216
\(999\) 9.71982 0.307522
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 6027.2.a.s.1.2 3
7.6 odd 2 123.2.a.d.1.2 3
21.20 even 2 369.2.a.e.1.2 3
28.27 even 2 1968.2.a.w.1.3 3
35.34 odd 2 3075.2.a.t.1.2 3
56.13 odd 2 7872.2.a.bx.1.1 3
56.27 even 2 7872.2.a.bs.1.1 3
84.83 odd 2 5904.2.a.bd.1.1 3
105.104 even 2 9225.2.a.bx.1.2 3
287.286 odd 2 5043.2.a.n.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
123.2.a.d.1.2 3 7.6 odd 2
369.2.a.e.1.2 3 21.20 even 2
1968.2.a.w.1.3 3 28.27 even 2
3075.2.a.t.1.2 3 35.34 odd 2
5043.2.a.n.1.2 3 287.286 odd 2
5904.2.a.bd.1.1 3 84.83 odd 2
6027.2.a.s.1.2 3 1.1 even 1 trivial
7872.2.a.bs.1.1 3 56.27 even 2
7872.2.a.bx.1.1 3 56.13 odd 2
9225.2.a.bx.1.2 3 105.104 even 2