Properties

Label 2760.2.a.v.1.1
Level $2760$
Weight $2$
Character 2760.1
Self dual yes
Analytic conductor $22.039$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.67673\) of defining polynomial
Character \(\chi\) \(=\) 2760.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -4.13277 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -4.13277 q^{7} +1.00000 q^{9} +2.92955 q^{11} +2.42391 q^{13} +1.00000 q^{15} -4.15025 q^{17} +3.84163 q^{19} +4.13277 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.64461 q^{29} -3.22069 q^{31} -2.92955 q^{33} +4.13277 q^{35} +5.55049 q^{37} -2.42391 q^{39} -5.06232 q^{41} -7.41772 q^{43} -1.00000 q^{45} -3.84163 q^{47} +10.0798 q^{49} +4.15025 q^{51} +8.48004 q^{53} -2.92955 q^{55} -3.84163 q^{57} +3.64461 q^{59} -1.51183 q^{61} -4.13277 q^{63} -2.42391 q^{65} -13.0448 q^{67} -1.00000 q^{69} -7.06232 q^{71} -1.57609 q^{73} -1.00000 q^{75} -12.1072 q^{77} +0.329796 q^{79} +1.00000 q^{81} -13.5680 q^{83} +4.15025 q^{85} -5.64461 q^{87} +1.08792 q^{89} -10.0175 q^{91} +3.22069 q^{93} -3.84163 q^{95} +0.441388 q^{97} +2.92955 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{9} - 2 q^{13} + 4 q^{15} + 2 q^{17} - 6 q^{19} + 4 q^{23} + 4 q^{25} - 4 q^{27} + 4 q^{29} - 6 q^{31} - 4 q^{37} + 2 q^{39} + 8 q^{41} - 20 q^{43} - 4 q^{45} + 6 q^{47} + 10 q^{49} - 2 q^{51} - 4 q^{53} + 6 q^{57} - 4 q^{59} - 4 q^{61} + 2 q^{65} - 26 q^{67} - 4 q^{69} - 18 q^{73} - 4 q^{75} + 6 q^{77} - 18 q^{79} + 4 q^{81} - 26 q^{83} - 2 q^{85} - 4 q^{87} + 14 q^{89} - 38 q^{91} + 6 q^{93} + 6 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.13277 −1.56204 −0.781020 0.624506i \(-0.785301\pi\)
−0.781020 + 0.624506i \(0.785301\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.92955 0.883294 0.441647 0.897189i \(-0.354394\pi\)
0.441647 + 0.897189i \(0.354394\pi\)
\(12\) 0 0
\(13\) 2.42391 0.672272 0.336136 0.941813i \(-0.390880\pi\)
0.336136 + 0.941813i \(0.390880\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −4.15025 −1.00658 −0.503291 0.864117i \(-0.667878\pi\)
−0.503291 + 0.864117i \(0.667878\pi\)
\(18\) 0 0
\(19\) 3.84163 0.881331 0.440665 0.897672i \(-0.354743\pi\)
0.440665 + 0.897672i \(0.354743\pi\)
\(20\) 0 0
\(21\) 4.13277 0.901845
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.64461 1.04818 0.524089 0.851664i \(-0.324406\pi\)
0.524089 + 0.851664i \(0.324406\pi\)
\(30\) 0 0
\(31\) −3.22069 −0.578454 −0.289227 0.957261i \(-0.593398\pi\)
−0.289227 + 0.957261i \(0.593398\pi\)
\(32\) 0 0
\(33\) −2.92955 −0.509970
\(34\) 0 0
\(35\) 4.13277 0.698566
\(36\) 0 0
\(37\) 5.55049 0.912495 0.456247 0.889853i \(-0.349193\pi\)
0.456247 + 0.889853i \(0.349193\pi\)
\(38\) 0 0
\(39\) −2.42391 −0.388137
\(40\) 0 0
\(41\) −5.06232 −0.790602 −0.395301 0.918552i \(-0.629360\pi\)
−0.395301 + 0.918552i \(0.629360\pi\)
\(42\) 0 0
\(43\) −7.41772 −1.13119 −0.565596 0.824683i \(-0.691354\pi\)
−0.565596 + 0.824683i \(0.691354\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −3.84163 −0.560359 −0.280180 0.959948i \(-0.590394\pi\)
−0.280180 + 0.959948i \(0.590394\pi\)
\(48\) 0 0
\(49\) 10.0798 1.43997
\(50\) 0 0
\(51\) 4.15025 0.581151
\(52\) 0 0
\(53\) 8.48004 1.16482 0.582412 0.812894i \(-0.302109\pi\)
0.582412 + 0.812894i \(0.302109\pi\)
\(54\) 0 0
\(55\) −2.92955 −0.395021
\(56\) 0 0
\(57\) −3.84163 −0.508836
\(58\) 0 0
\(59\) 3.64461 0.474487 0.237244 0.971450i \(-0.423756\pi\)
0.237244 + 0.971450i \(0.423756\pi\)
\(60\) 0 0
\(61\) −1.51183 −0.193571 −0.0967853 0.995305i \(-0.530856\pi\)
−0.0967853 + 0.995305i \(0.530856\pi\)
\(62\) 0 0
\(63\) −4.13277 −0.520680
\(64\) 0 0
\(65\) −2.42391 −0.300649
\(66\) 0 0
\(67\) −13.0448 −1.59368 −0.796841 0.604189i \(-0.793497\pi\)
−0.796841 + 0.604189i \(0.793497\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −7.06232 −0.838144 −0.419072 0.907953i \(-0.637644\pi\)
−0.419072 + 0.907953i \(0.637644\pi\)
\(72\) 0 0
\(73\) −1.57609 −0.184467 −0.0922336 0.995737i \(-0.529401\pi\)
−0.0922336 + 0.995737i \(0.529401\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −12.1072 −1.37974
\(78\) 0 0
\(79\) 0.329796 0.0371049 0.0185525 0.999828i \(-0.494094\pi\)
0.0185525 + 0.999828i \(0.494094\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.5680 −1.48928 −0.744639 0.667468i \(-0.767378\pi\)
−0.744639 + 0.667468i \(0.767378\pi\)
\(84\) 0 0
\(85\) 4.15025 0.450157
\(86\) 0 0
\(87\) −5.64461 −0.605165
\(88\) 0 0
\(89\) 1.08792 0.115320 0.0576598 0.998336i \(-0.481636\pi\)
0.0576598 + 0.998336i \(0.481636\pi\)
\(90\) 0 0
\(91\) −10.0175 −1.05012
\(92\) 0 0
\(93\) 3.22069 0.333970
\(94\) 0 0
\(95\) −3.84163 −0.394143
\(96\) 0 0
\(97\) 0.441388 0.0448161 0.0224081 0.999749i \(-0.492867\pi\)
0.0224081 + 0.999749i \(0.492867\pi\)
\(98\) 0 0
\(99\) 2.92955 0.294431
\(100\) 0 0
\(101\) 4.21450 0.419358 0.209679 0.977770i \(-0.432758\pi\)
0.209679 + 0.977770i \(0.432758\pi\)
\(102\) 0 0
\(103\) −1.22882 −0.121079 −0.0605394 0.998166i \(-0.519282\pi\)
−0.0605394 + 0.998166i \(0.519282\pi\)
\(104\) 0 0
\(105\) −4.13277 −0.403317
\(106\) 0 0
\(107\) −0.291141 −0.0281456 −0.0140728 0.999901i \(-0.504480\pi\)
−0.0140728 + 0.999901i \(0.504480\pi\)
\(108\) 0 0
\(109\) −0.929553 −0.0890350 −0.0445175 0.999009i \(-0.514175\pi\)
−0.0445175 + 0.999009i \(0.514175\pi\)
\(110\) 0 0
\(111\) −5.55049 −0.526829
\(112\) 0 0
\(113\) −14.5567 −1.36938 −0.684689 0.728836i \(-0.740062\pi\)
−0.684689 + 0.728836i \(0.740062\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 2.42391 0.224091
\(118\) 0 0
\(119\) 17.1520 1.57232
\(120\) 0 0
\(121\) −2.41772 −0.219793
\(122\) 0 0
\(123\) 5.06232 0.456454
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −14.2481 −1.26431 −0.632156 0.774841i \(-0.717830\pi\)
−0.632156 + 0.774841i \(0.717830\pi\)
\(128\) 0 0
\(129\) 7.41772 0.653094
\(130\) 0 0
\(131\) −20.5311 −1.79381 −0.896905 0.442224i \(-0.854190\pi\)
−0.896905 + 0.442224i \(0.854190\pi\)
\(132\) 0 0
\(133\) −15.8766 −1.37667
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −2.94703 −0.251782 −0.125891 0.992044i \(-0.540179\pi\)
−0.125891 + 0.992044i \(0.540179\pi\)
\(138\) 0 0
\(139\) −14.0561 −1.19223 −0.596113 0.802901i \(-0.703289\pi\)
−0.596113 + 0.802901i \(0.703289\pi\)
\(140\) 0 0
\(141\) 3.84163 0.323524
\(142\) 0 0
\(143\) 7.10098 0.593814
\(144\) 0 0
\(145\) −5.64461 −0.468759
\(146\) 0 0
\(147\) −10.0798 −0.831368
\(148\) 0 0
\(149\) −17.1308 −1.40341 −0.701707 0.712466i \(-0.747578\pi\)
−0.701707 + 0.712466i \(0.747578\pi\)
\(150\) 0 0
\(151\) 9.11337 0.741635 0.370818 0.928706i \(-0.379077\pi\)
0.370818 + 0.928706i \(0.379077\pi\)
\(152\) 0 0
\(153\) −4.15025 −0.335528
\(154\) 0 0
\(155\) 3.22069 0.258692
\(156\) 0 0
\(157\) 10.4333 0.832665 0.416333 0.909212i \(-0.363315\pi\)
0.416333 + 0.909212i \(0.363315\pi\)
\(158\) 0 0
\(159\) −8.48004 −0.672511
\(160\) 0 0
\(161\) −4.13277 −0.325708
\(162\) 0 0
\(163\) −18.4414 −1.44444 −0.722220 0.691663i \(-0.756878\pi\)
−0.722220 + 0.691663i \(0.756878\pi\)
\(164\) 0 0
\(165\) 2.92955 0.228065
\(166\) 0 0
\(167\) 20.3727 1.57649 0.788244 0.615363i \(-0.210990\pi\)
0.788244 + 0.615363i \(0.210990\pi\)
\(168\) 0 0
\(169\) −7.12465 −0.548050
\(170\) 0 0
\(171\) 3.84163 0.293777
\(172\) 0 0
\(173\) −1.73446 −0.131868 −0.0659342 0.997824i \(-0.521003\pi\)
−0.0659342 + 0.997824i \(0.521003\pi\)
\(174\) 0 0
\(175\) −4.13277 −0.312408
\(176\) 0 0
\(177\) −3.64461 −0.273945
\(178\) 0 0
\(179\) 5.24187 0.391796 0.195898 0.980624i \(-0.437238\pi\)
0.195898 + 0.980624i \(0.437238\pi\)
\(180\) 0 0
\(181\) −8.80613 −0.654555 −0.327277 0.944928i \(-0.606131\pi\)
−0.327277 + 0.944928i \(0.606131\pi\)
\(182\) 0 0
\(183\) 1.51183 0.111758
\(184\) 0 0
\(185\) −5.55049 −0.408080
\(186\) 0 0
\(187\) −12.1584 −0.889108
\(188\) 0 0
\(189\) 4.13277 0.300615
\(190\) 0 0
\(191\) 16.7244 1.21014 0.605068 0.796174i \(-0.293146\pi\)
0.605068 + 0.796174i \(0.293146\pi\)
\(192\) 0 0
\(193\) 6.30049 0.453519 0.226760 0.973951i \(-0.427187\pi\)
0.226760 + 0.973951i \(0.427187\pi\)
\(194\) 0 0
\(195\) 2.42391 0.173580
\(196\) 0 0
\(197\) 26.5660 1.89275 0.946376 0.323068i \(-0.104714\pi\)
0.946376 + 0.323068i \(0.104714\pi\)
\(198\) 0 0
\(199\) −7.74751 −0.549207 −0.274603 0.961558i \(-0.588547\pi\)
−0.274603 + 0.961558i \(0.588547\pi\)
\(200\) 0 0
\(201\) 13.0448 0.920113
\(202\) 0 0
\(203\) −23.3279 −1.63730
\(204\) 0 0
\(205\) 5.06232 0.353568
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 11.2543 0.778474
\(210\) 0 0
\(211\) 19.4700 1.34037 0.670185 0.742194i \(-0.266215\pi\)
0.670185 + 0.742194i \(0.266215\pi\)
\(212\) 0 0
\(213\) 7.06232 0.483903
\(214\) 0 0
\(215\) 7.41772 0.505884
\(216\) 0 0
\(217\) 13.3104 0.903568
\(218\) 0 0
\(219\) 1.57609 0.106502
\(220\) 0 0
\(221\) −10.0598 −0.676698
\(222\) 0 0
\(223\) −18.0897 −1.21138 −0.605688 0.795702i \(-0.707102\pi\)
−0.605688 + 0.795702i \(0.707102\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 5.21257 0.345971 0.172985 0.984924i \(-0.444659\pi\)
0.172985 + 0.984924i \(0.444659\pi\)
\(228\) 0 0
\(229\) −25.3372 −1.67433 −0.837165 0.546950i \(-0.815789\pi\)
−0.837165 + 0.546950i \(0.815789\pi\)
\(230\) 0 0
\(231\) 12.1072 0.796594
\(232\) 0 0
\(233\) −9.38277 −0.614686 −0.307343 0.951599i \(-0.599440\pi\)
−0.307343 + 0.951599i \(0.599440\pi\)
\(234\) 0 0
\(235\) 3.84163 0.250600
\(236\) 0 0
\(237\) −0.329796 −0.0214226
\(238\) 0 0
\(239\) 27.9451 1.80762 0.903809 0.427936i \(-0.140759\pi\)
0.903809 + 0.427936i \(0.140759\pi\)
\(240\) 0 0
\(241\) −25.9787 −1.67343 −0.836717 0.547636i \(-0.815528\pi\)
−0.836717 + 0.547636i \(0.815528\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −10.0798 −0.643975
\(246\) 0 0
\(247\) 9.31178 0.592494
\(248\) 0 0
\(249\) 13.5680 0.859835
\(250\) 0 0
\(251\) −9.75990 −0.616040 −0.308020 0.951380i \(-0.599666\pi\)
−0.308020 + 0.951380i \(0.599666\pi\)
\(252\) 0 0
\(253\) 2.92955 0.184179
\(254\) 0 0
\(255\) −4.15025 −0.259899
\(256\) 0 0
\(257\) 14.1072 0.879981 0.439991 0.898002i \(-0.354982\pi\)
0.439991 + 0.898002i \(0.354982\pi\)
\(258\) 0 0
\(259\) −22.9389 −1.42535
\(260\) 0 0
\(261\) 5.64461 0.349392
\(262\) 0 0
\(263\) 3.60966 0.222581 0.111290 0.993788i \(-0.464502\pi\)
0.111290 + 0.993788i \(0.464502\pi\)
\(264\) 0 0
\(265\) −8.48004 −0.520925
\(266\) 0 0
\(267\) −1.08792 −0.0665798
\(268\) 0 0
\(269\) 20.7456 1.26488 0.632440 0.774609i \(-0.282053\pi\)
0.632440 + 0.774609i \(0.282053\pi\)
\(270\) 0 0
\(271\) −20.7281 −1.25914 −0.629572 0.776943i \(-0.716770\pi\)
−0.629572 + 0.776943i \(0.716770\pi\)
\(272\) 0 0
\(273\) 10.0175 0.606285
\(274\) 0 0
\(275\) 2.92955 0.176659
\(276\) 0 0
\(277\) −10.5311 −0.632752 −0.316376 0.948634i \(-0.602466\pi\)
−0.316376 + 0.948634i \(0.602466\pi\)
\(278\) 0 0
\(279\) −3.22069 −0.192818
\(280\) 0 0
\(281\) 27.5853 1.64560 0.822800 0.568331i \(-0.192411\pi\)
0.822800 + 0.568331i \(0.192411\pi\)
\(282\) 0 0
\(283\) −12.9163 −0.767797 −0.383898 0.923375i \(-0.625419\pi\)
−0.383898 + 0.923375i \(0.625419\pi\)
\(284\) 0 0
\(285\) 3.84163 0.227559
\(286\) 0 0
\(287\) 20.9214 1.23495
\(288\) 0 0
\(289\) 0.224551 0.0132089
\(290\) 0 0
\(291\) −0.441388 −0.0258746
\(292\) 0 0
\(293\) 16.1633 0.944270 0.472135 0.881526i \(-0.343483\pi\)
0.472135 + 0.881526i \(0.343483\pi\)
\(294\) 0 0
\(295\) −3.64461 −0.212197
\(296\) 0 0
\(297\) −2.92955 −0.169990
\(298\) 0 0
\(299\) 2.42391 0.140178
\(300\) 0 0
\(301\) 30.6557 1.76697
\(302\) 0 0
\(303\) −4.21450 −0.242117
\(304\) 0 0
\(305\) 1.51183 0.0865674
\(306\) 0 0
\(307\) −32.4426 −1.85160 −0.925799 0.378016i \(-0.876606\pi\)
−0.925799 + 0.378016i \(0.876606\pi\)
\(308\) 0 0
\(309\) 1.22882 0.0699049
\(310\) 0 0
\(311\) −24.6010 −1.39499 −0.697497 0.716588i \(-0.745703\pi\)
−0.697497 + 0.716588i \(0.745703\pi\)
\(312\) 0 0
\(313\) −17.0922 −0.966108 −0.483054 0.875591i \(-0.660472\pi\)
−0.483054 + 0.875591i \(0.660472\pi\)
\(314\) 0 0
\(315\) 4.13277 0.232855
\(316\) 0 0
\(317\) −2.14212 −0.120314 −0.0601569 0.998189i \(-0.519160\pi\)
−0.0601569 + 0.998189i \(0.519160\pi\)
\(318\) 0 0
\(319\) 16.5362 0.925848
\(320\) 0 0
\(321\) 0.291141 0.0162499
\(322\) 0 0
\(323\) −15.9437 −0.887132
\(324\) 0 0
\(325\) 2.42391 0.134454
\(326\) 0 0
\(327\) 0.929553 0.0514044
\(328\) 0 0
\(329\) 15.8766 0.875304
\(330\) 0 0
\(331\) 13.3453 0.733526 0.366763 0.930314i \(-0.380466\pi\)
0.366763 + 0.930314i \(0.380466\pi\)
\(332\) 0 0
\(333\) 5.55049 0.304165
\(334\) 0 0
\(335\) 13.0448 0.712716
\(336\) 0 0
\(337\) 0.175845 0.00957888 0.00478944 0.999989i \(-0.498475\pi\)
0.00478944 + 0.999989i \(0.498475\pi\)
\(338\) 0 0
\(339\) 14.5567 0.790611
\(340\) 0 0
\(341\) −9.43519 −0.510944
\(342\) 0 0
\(343\) −12.7281 −0.687253
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) −1.85911 −0.0998021 −0.0499010 0.998754i \(-0.515891\pi\)
−0.0499010 + 0.998754i \(0.515891\pi\)
\(348\) 0 0
\(349\) 7.39654 0.395928 0.197964 0.980209i \(-0.436567\pi\)
0.197964 + 0.980209i \(0.436567\pi\)
\(350\) 0 0
\(351\) −2.42391 −0.129379
\(352\) 0 0
\(353\) 9.16579 0.487846 0.243923 0.969795i \(-0.421566\pi\)
0.243923 + 0.969795i \(0.421566\pi\)
\(354\) 0 0
\(355\) 7.06232 0.374829
\(356\) 0 0
\(357\) −17.1520 −0.907781
\(358\) 0 0
\(359\) −24.0909 −1.27147 −0.635735 0.771907i \(-0.719303\pi\)
−0.635735 + 0.771907i \(0.719303\pi\)
\(360\) 0 0
\(361\) −4.24187 −0.223257
\(362\) 0 0
\(363\) 2.41772 0.126897
\(364\) 0 0
\(365\) 1.57609 0.0824962
\(366\) 0 0
\(367\) −23.9795 −1.25172 −0.625860 0.779936i \(-0.715252\pi\)
−0.625860 + 0.779936i \(0.715252\pi\)
\(368\) 0 0
\(369\) −5.06232 −0.263534
\(370\) 0 0
\(371\) −35.0461 −1.81950
\(372\) 0 0
\(373\) −27.7949 −1.43916 −0.719581 0.694408i \(-0.755666\pi\)
−0.719581 + 0.694408i \(0.755666\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 13.6820 0.704660
\(378\) 0 0
\(379\) 12.8828 0.661744 0.330872 0.943676i \(-0.392657\pi\)
0.330872 + 0.943676i \(0.392657\pi\)
\(380\) 0 0
\(381\) 14.2481 0.729951
\(382\) 0 0
\(383\) −19.0461 −0.973209 −0.486605 0.873622i \(-0.661765\pi\)
−0.486605 + 0.873622i \(0.661765\pi\)
\(384\) 0 0
\(385\) 12.1072 0.617039
\(386\) 0 0
\(387\) −7.41772 −0.377064
\(388\) 0 0
\(389\) −30.2842 −1.53547 −0.767736 0.640766i \(-0.778617\pi\)
−0.767736 + 0.640766i \(0.778617\pi\)
\(390\) 0 0
\(391\) −4.15025 −0.209887
\(392\) 0 0
\(393\) 20.5311 1.03566
\(394\) 0 0
\(395\) −0.329796 −0.0165938
\(396\) 0 0
\(397\) −15.1483 −0.760272 −0.380136 0.924931i \(-0.624123\pi\)
−0.380136 + 0.924931i \(0.624123\pi\)
\(398\) 0 0
\(399\) 15.8766 0.794823
\(400\) 0 0
\(401\) −23.0491 −1.15102 −0.575509 0.817795i \(-0.695196\pi\)
−0.575509 + 0.817795i \(0.695196\pi\)
\(402\) 0 0
\(403\) −7.80668 −0.388878
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 16.2605 0.806001
\(408\) 0 0
\(409\) 11.7906 0.583007 0.291504 0.956570i \(-0.405844\pi\)
0.291504 + 0.956570i \(0.405844\pi\)
\(410\) 0 0
\(411\) 2.94703 0.145366
\(412\) 0 0
\(413\) −15.0623 −0.741169
\(414\) 0 0
\(415\) 13.5680 0.666025
\(416\) 0 0
\(417\) 14.0561 0.688332
\(418\) 0 0
\(419\) −35.9917 −1.75831 −0.879155 0.476535i \(-0.841892\pi\)
−0.879155 + 0.476535i \(0.841892\pi\)
\(420\) 0 0
\(421\) −1.58915 −0.0774502 −0.0387251 0.999250i \(-0.512330\pi\)
−0.0387251 + 0.999250i \(0.512330\pi\)
\(422\) 0 0
\(423\) −3.84163 −0.186786
\(424\) 0 0
\(425\) −4.15025 −0.201317
\(426\) 0 0
\(427\) 6.24807 0.302365
\(428\) 0 0
\(429\) −7.10098 −0.342839
\(430\) 0 0
\(431\) 0.445245 0.0214467 0.0107233 0.999943i \(-0.496587\pi\)
0.0107233 + 0.999943i \(0.496587\pi\)
\(432\) 0 0
\(433\) −9.34920 −0.449294 −0.224647 0.974440i \(-0.572123\pi\)
−0.224647 + 0.974440i \(0.572123\pi\)
\(434\) 0 0
\(435\) 5.64461 0.270638
\(436\) 0 0
\(437\) 3.84163 0.183770
\(438\) 0 0
\(439\) 33.5261 1.60011 0.800057 0.599924i \(-0.204802\pi\)
0.800057 + 0.599924i \(0.204802\pi\)
\(440\) 0 0
\(441\) 10.0798 0.479990
\(442\) 0 0
\(443\) 13.8890 0.659885 0.329942 0.944001i \(-0.392971\pi\)
0.329942 + 0.944001i \(0.392971\pi\)
\(444\) 0 0
\(445\) −1.08792 −0.0515725
\(446\) 0 0
\(447\) 17.1308 0.810261
\(448\) 0 0
\(449\) 21.1644 0.998810 0.499405 0.866369i \(-0.333552\pi\)
0.499405 + 0.866369i \(0.333552\pi\)
\(450\) 0 0
\(451\) −14.8303 −0.698334
\(452\) 0 0
\(453\) −9.11337 −0.428183
\(454\) 0 0
\(455\) 10.0175 0.469626
\(456\) 0 0
\(457\) 31.3104 1.46464 0.732319 0.680962i \(-0.238438\pi\)
0.732319 + 0.680962i \(0.238438\pi\)
\(458\) 0 0
\(459\) 4.15025 0.193717
\(460\) 0 0
\(461\) 20.5187 0.955651 0.477826 0.878455i \(-0.341425\pi\)
0.477826 + 0.878455i \(0.341425\pi\)
\(462\) 0 0
\(463\) 29.0323 1.34925 0.674623 0.738163i \(-0.264306\pi\)
0.674623 + 0.738163i \(0.264306\pi\)
\(464\) 0 0
\(465\) −3.22069 −0.149356
\(466\) 0 0
\(467\) 10.6976 0.495025 0.247512 0.968885i \(-0.420387\pi\)
0.247512 + 0.968885i \(0.420387\pi\)
\(468\) 0 0
\(469\) 53.9114 2.48940
\(470\) 0 0
\(471\) −10.4333 −0.480740
\(472\) 0 0
\(473\) −21.7306 −0.999174
\(474\) 0 0
\(475\) 3.84163 0.176266
\(476\) 0 0
\(477\) 8.48004 0.388274
\(478\) 0 0
\(479\) 7.02876 0.321152 0.160576 0.987023i \(-0.448665\pi\)
0.160576 + 0.987023i \(0.448665\pi\)
\(480\) 0 0
\(481\) 13.4539 0.613445
\(482\) 0 0
\(483\) 4.13277 0.188048
\(484\) 0 0
\(485\) −0.441388 −0.0200424
\(486\) 0 0
\(487\) 0.388961 0.0176255 0.00881276 0.999961i \(-0.497195\pi\)
0.00881276 + 0.999961i \(0.497195\pi\)
\(488\) 0 0
\(489\) 18.4414 0.833948
\(490\) 0 0
\(491\) 2.92529 0.132016 0.0660082 0.997819i \(-0.478974\pi\)
0.0660082 + 0.997819i \(0.478974\pi\)
\(492\) 0 0
\(493\) −23.4265 −1.05508
\(494\) 0 0
\(495\) −2.92955 −0.131674
\(496\) 0 0
\(497\) 29.1870 1.30921
\(498\) 0 0
\(499\) 1.23308 0.0552003 0.0276002 0.999619i \(-0.491213\pi\)
0.0276002 + 0.999619i \(0.491213\pi\)
\(500\) 0 0
\(501\) −20.3727 −0.910186
\(502\) 0 0
\(503\) 23.7816 1.06037 0.530186 0.847882i \(-0.322122\pi\)
0.530186 + 0.847882i \(0.322122\pi\)
\(504\) 0 0
\(505\) −4.21450 −0.187543
\(506\) 0 0
\(507\) 7.12465 0.316417
\(508\) 0 0
\(509\) −6.54733 −0.290205 −0.145103 0.989417i \(-0.546351\pi\)
−0.145103 + 0.989417i \(0.546351\pi\)
\(510\) 0 0
\(511\) 6.51361 0.288145
\(512\) 0 0
\(513\) −3.84163 −0.169612
\(514\) 0 0
\(515\) 1.22882 0.0541481
\(516\) 0 0
\(517\) −11.2543 −0.494962
\(518\) 0 0
\(519\) 1.73446 0.0761342
\(520\) 0 0
\(521\) 0.312320 0.0136830 0.00684150 0.999977i \(-0.497822\pi\)
0.00684150 + 0.999977i \(0.497822\pi\)
\(522\) 0 0
\(523\) 33.1907 1.45133 0.725664 0.688050i \(-0.241533\pi\)
0.725664 + 0.688050i \(0.241533\pi\)
\(524\) 0 0
\(525\) 4.13277 0.180369
\(526\) 0 0
\(527\) 13.3667 0.582262
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.64461 0.158162
\(532\) 0 0
\(533\) −12.2706 −0.531500
\(534\) 0 0
\(535\) 0.291141 0.0125871
\(536\) 0 0
\(537\) −5.24187 −0.226204
\(538\) 0 0
\(539\) 29.5293 1.27192
\(540\) 0 0
\(541\) 11.9364 0.513187 0.256593 0.966519i \(-0.417400\pi\)
0.256593 + 0.966519i \(0.417400\pi\)
\(542\) 0 0
\(543\) 8.80613 0.377907
\(544\) 0 0
\(545\) 0.929553 0.0398177
\(546\) 0 0
\(547\) −13.1646 −0.562876 −0.281438 0.959579i \(-0.590811\pi\)
−0.281438 + 0.959579i \(0.590811\pi\)
\(548\) 0 0
\(549\) −1.51183 −0.0645235
\(550\) 0 0
\(551\) 21.6845 0.923790
\(552\) 0 0
\(553\) −1.36297 −0.0579594
\(554\) 0 0
\(555\) 5.55049 0.235605
\(556\) 0 0
\(557\) 7.82045 0.331363 0.165682 0.986179i \(-0.447018\pi\)
0.165682 + 0.986179i \(0.447018\pi\)
\(558\) 0 0
\(559\) −17.9799 −0.760469
\(560\) 0 0
\(561\) 12.1584 0.513327
\(562\) 0 0
\(563\) −1.00193 −0.0422262 −0.0211131 0.999777i \(-0.506721\pi\)
−0.0211131 + 0.999777i \(0.506721\pi\)
\(564\) 0 0
\(565\) 14.5567 0.612404
\(566\) 0 0
\(567\) −4.13277 −0.173560
\(568\) 0 0
\(569\) 5.50675 0.230855 0.115427 0.993316i \(-0.463176\pi\)
0.115427 + 0.993316i \(0.463176\pi\)
\(570\) 0 0
\(571\) −28.7082 −1.20140 −0.600700 0.799475i \(-0.705111\pi\)
−0.600700 + 0.799475i \(0.705111\pi\)
\(572\) 0 0
\(573\) −16.7244 −0.698672
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 14.9499 0.622373 0.311186 0.950349i \(-0.399274\pi\)
0.311186 + 0.950349i \(0.399274\pi\)
\(578\) 0 0
\(579\) −6.30049 −0.261840
\(580\) 0 0
\(581\) 56.0733 2.32631
\(582\) 0 0
\(583\) 24.8427 1.02888
\(584\) 0 0
\(585\) −2.42391 −0.100216
\(586\) 0 0
\(587\) 31.3503 1.29396 0.646982 0.762505i \(-0.276031\pi\)
0.646982 + 0.762505i \(0.276031\pi\)
\(588\) 0 0
\(589\) −12.3727 −0.509809
\(590\) 0 0
\(591\) −26.5660 −1.09278
\(592\) 0 0
\(593\) 8.11956 0.333430 0.166715 0.986005i \(-0.446684\pi\)
0.166715 + 0.986005i \(0.446684\pi\)
\(594\) 0 0
\(595\) −17.1520 −0.703164
\(596\) 0 0
\(597\) 7.74751 0.317085
\(598\) 0 0
\(599\) −17.3789 −0.710083 −0.355042 0.934851i \(-0.615533\pi\)
−0.355042 + 0.934851i \(0.615533\pi\)
\(600\) 0 0
\(601\) 3.13100 0.127716 0.0638580 0.997959i \(-0.479660\pi\)
0.0638580 + 0.997959i \(0.479660\pi\)
\(602\) 0 0
\(603\) −13.0448 −0.531227
\(604\) 0 0
\(605\) 2.41772 0.0982942
\(606\) 0 0
\(607\) −10.8141 −0.438931 −0.219465 0.975620i \(-0.570431\pi\)
−0.219465 + 0.975620i \(0.570431\pi\)
\(608\) 0 0
\(609\) 23.3279 0.945293
\(610\) 0 0
\(611\) −9.31178 −0.376714
\(612\) 0 0
\(613\) 4.22385 0.170600 0.0852999 0.996355i \(-0.472815\pi\)
0.0852999 + 0.996355i \(0.472815\pi\)
\(614\) 0 0
\(615\) −5.06232 −0.204133
\(616\) 0 0
\(617\) 9.72757 0.391617 0.195809 0.980642i \(-0.437267\pi\)
0.195809 + 0.980642i \(0.437267\pi\)
\(618\) 0 0
\(619\) 29.3440 1.17943 0.589717 0.807610i \(-0.299239\pi\)
0.589717 + 0.807610i \(0.299239\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −4.49613 −0.180134
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −11.2543 −0.449452
\(628\) 0 0
\(629\) −23.0359 −0.918502
\(630\) 0 0
\(631\) 20.8236 0.828975 0.414487 0.910055i \(-0.363961\pi\)
0.414487 + 0.910055i \(0.363961\pi\)
\(632\) 0 0
\(633\) −19.4700 −0.773863
\(634\) 0 0
\(635\) 14.2481 0.565417
\(636\) 0 0
\(637\) 24.4326 0.968053
\(638\) 0 0
\(639\) −7.06232 −0.279381
\(640\) 0 0
\(641\) −37.1277 −1.46645 −0.733227 0.679984i \(-0.761987\pi\)
−0.733227 + 0.679984i \(0.761987\pi\)
\(642\) 0 0
\(643\) 18.0049 0.710045 0.355023 0.934858i \(-0.384473\pi\)
0.355023 + 0.934858i \(0.384473\pi\)
\(644\) 0 0
\(645\) −7.41772 −0.292072
\(646\) 0 0
\(647\) 6.56481 0.258089 0.129045 0.991639i \(-0.458809\pi\)
0.129045 + 0.991639i \(0.458809\pi\)
\(648\) 0 0
\(649\) 10.6771 0.419112
\(650\) 0 0
\(651\) −13.3104 −0.521675
\(652\) 0 0
\(653\) −39.8215 −1.55834 −0.779168 0.626815i \(-0.784358\pi\)
−0.779168 + 0.626815i \(0.784358\pi\)
\(654\) 0 0
\(655\) 20.5311 0.802216
\(656\) 0 0
\(657\) −1.57609 −0.0614890
\(658\) 0 0
\(659\) 44.3186 1.72641 0.863205 0.504854i \(-0.168454\pi\)
0.863205 + 0.504854i \(0.168454\pi\)
\(660\) 0 0
\(661\) 10.3895 0.404106 0.202053 0.979375i \(-0.435239\pi\)
0.202053 + 0.979375i \(0.435239\pi\)
\(662\) 0 0
\(663\) 10.0598 0.390692
\(664\) 0 0
\(665\) 15.8766 0.615667
\(666\) 0 0
\(667\) 5.64461 0.218560
\(668\) 0 0
\(669\) 18.0897 0.699388
\(670\) 0 0
\(671\) −4.42900 −0.170980
\(672\) 0 0
\(673\) −11.5373 −0.444729 −0.222365 0.974964i \(-0.571378\pi\)
−0.222365 + 0.974964i \(0.571378\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 15.9677 0.613687 0.306844 0.951760i \(-0.400727\pi\)
0.306844 + 0.951760i \(0.400727\pi\)
\(678\) 0 0
\(679\) −1.82416 −0.0700046
\(680\) 0 0
\(681\) −5.21257 −0.199746
\(682\) 0 0
\(683\) 32.1845 1.23151 0.615753 0.787940i \(-0.288852\pi\)
0.615753 + 0.787940i \(0.288852\pi\)
\(684\) 0 0
\(685\) 2.94703 0.112600
\(686\) 0 0
\(687\) 25.3372 0.966675
\(688\) 0 0
\(689\) 20.5549 0.783079
\(690\) 0 0
\(691\) −40.9601 −1.55820 −0.779098 0.626903i \(-0.784322\pi\)
−0.779098 + 0.626903i \(0.784322\pi\)
\(692\) 0 0
\(693\) −12.1072 −0.459913
\(694\) 0 0
\(695\) 14.0561 0.533179
\(696\) 0 0
\(697\) 21.0099 0.795807
\(698\) 0 0
\(699\) 9.38277 0.354889
\(700\) 0 0
\(701\) −25.4737 −0.962128 −0.481064 0.876685i \(-0.659750\pi\)
−0.481064 + 0.876685i \(0.659750\pi\)
\(702\) 0 0
\(703\) 21.3229 0.804210
\(704\) 0 0
\(705\) −3.84163 −0.144684
\(706\) 0 0
\(707\) −17.4176 −0.655055
\(708\) 0 0
\(709\) 4.46560 0.167709 0.0838546 0.996478i \(-0.473277\pi\)
0.0838546 + 0.996478i \(0.473277\pi\)
\(710\) 0 0
\(711\) 0.329796 0.0123683
\(712\) 0 0
\(713\) −3.22069 −0.120616
\(714\) 0 0
\(715\) −7.10098 −0.265562
\(716\) 0 0
\(717\) −27.9451 −1.04363
\(718\) 0 0
\(719\) 4.90272 0.182841 0.0914204 0.995812i \(-0.470859\pi\)
0.0914204 + 0.995812i \(0.470859\pi\)
\(720\) 0 0
\(721\) 5.07842 0.189130
\(722\) 0 0
\(723\) 25.9787 0.966157
\(724\) 0 0
\(725\) 5.64461 0.209635
\(726\) 0 0
\(727\) −35.4220 −1.31373 −0.656864 0.754009i \(-0.728118\pi\)
−0.656864 + 0.754009i \(0.728118\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.7854 1.13864
\(732\) 0 0
\(733\) −51.9244 −1.91787 −0.958936 0.283621i \(-0.908464\pi\)
−0.958936 + 0.283621i \(0.908464\pi\)
\(734\) 0 0
\(735\) 10.0798 0.371799
\(736\) 0 0
\(737\) −38.2156 −1.40769
\(738\) 0 0
\(739\) −13.7394 −0.505412 −0.252706 0.967543i \(-0.581320\pi\)
−0.252706 + 0.967543i \(0.581320\pi\)
\(740\) 0 0
\(741\) −9.31178 −0.342077
\(742\) 0 0
\(743\) 15.8016 0.579704 0.289852 0.957071i \(-0.406394\pi\)
0.289852 + 0.957071i \(0.406394\pi\)
\(744\) 0 0
\(745\) 17.1308 0.627626
\(746\) 0 0
\(747\) −13.5680 −0.496426
\(748\) 0 0
\(749\) 1.20322 0.0439646
\(750\) 0 0
\(751\) −26.5616 −0.969247 −0.484624 0.874723i \(-0.661043\pi\)
−0.484624 + 0.874723i \(0.661043\pi\)
\(752\) 0 0
\(753\) 9.75990 0.355671
\(754\) 0 0
\(755\) −9.11337 −0.331669
\(756\) 0 0
\(757\) −24.9682 −0.907485 −0.453742 0.891133i \(-0.649911\pi\)
−0.453742 + 0.891133i \(0.649911\pi\)
\(758\) 0 0
\(759\) −2.92955 −0.106336
\(760\) 0 0
\(761\) −29.3279 −1.06313 −0.531567 0.847016i \(-0.678397\pi\)
−0.531567 + 0.847016i \(0.678397\pi\)
\(762\) 0 0
\(763\) 3.84163 0.139076
\(764\) 0 0
\(765\) 4.15025 0.150052
\(766\) 0 0
\(767\) 8.83421 0.318985
\(768\) 0 0
\(769\) 3.09975 0.111780 0.0558899 0.998437i \(-0.482200\pi\)
0.0558899 + 0.998437i \(0.482200\pi\)
\(770\) 0 0
\(771\) −14.1072 −0.508057
\(772\) 0 0
\(773\) −7.50742 −0.270023 −0.135012 0.990844i \(-0.543107\pi\)
−0.135012 + 0.990844i \(0.543107\pi\)
\(774\) 0 0
\(775\) −3.22069 −0.115691
\(776\) 0 0
\(777\) 22.9389 0.822929
\(778\) 0 0
\(779\) −19.4476 −0.696782
\(780\) 0 0
\(781\) −20.6895 −0.740327
\(782\) 0 0
\(783\) −5.64461 −0.201722
\(784\) 0 0
\(785\) −10.4333 −0.372379
\(786\) 0 0
\(787\) −43.5635 −1.55287 −0.776436 0.630196i \(-0.782975\pi\)
−0.776436 + 0.630196i \(0.782975\pi\)
\(788\) 0 0
\(789\) −3.60966 −0.128507
\(790\) 0 0
\(791\) 60.1594 2.13902
\(792\) 0 0
\(793\) −3.66455 −0.130132
\(794\) 0 0
\(795\) 8.48004 0.300756
\(796\) 0 0
\(797\) −16.3268 −0.578324 −0.289162 0.957280i \(-0.593377\pi\)
−0.289162 + 0.957280i \(0.593377\pi\)
\(798\) 0 0
\(799\) 15.9437 0.564048
\(800\) 0 0
\(801\) 1.08792 0.0384398
\(802\) 0 0
\(803\) −4.61723 −0.162939
\(804\) 0 0
\(805\) 4.13277 0.145661
\(806\) 0 0
\(807\) −20.7456 −0.730279
\(808\) 0 0
\(809\) 22.0736 0.776067 0.388033 0.921645i \(-0.373155\pi\)
0.388033 + 0.921645i \(0.373155\pi\)
\(810\) 0 0
\(811\) 42.8365 1.50419 0.752097 0.659053i \(-0.229043\pi\)
0.752097 + 0.659053i \(0.229043\pi\)
\(812\) 0 0
\(813\) 20.7281 0.726967
\(814\) 0 0
\(815\) 18.4414 0.645974
\(816\) 0 0
\(817\) −28.4961 −0.996954
\(818\) 0 0
\(819\) −10.0175 −0.350039
\(820\) 0 0
\(821\) 29.0424 1.01359 0.506793 0.862068i \(-0.330831\pi\)
0.506793 + 0.862068i \(0.330831\pi\)
\(822\) 0 0
\(823\) 44.9089 1.56543 0.782713 0.622383i \(-0.213835\pi\)
0.782713 + 0.622383i \(0.213835\pi\)
\(824\) 0 0
\(825\) −2.92955 −0.101994
\(826\) 0 0
\(827\) −11.6739 −0.405942 −0.202971 0.979185i \(-0.565060\pi\)
−0.202971 + 0.979185i \(0.565060\pi\)
\(828\) 0 0
\(829\) 27.3291 0.949179 0.474589 0.880207i \(-0.342597\pi\)
0.474589 + 0.880207i \(0.342597\pi\)
\(830\) 0 0
\(831\) 10.5311 0.365319
\(832\) 0 0
\(833\) −41.8337 −1.44945
\(834\) 0 0
\(835\) −20.3727 −0.705027
\(836\) 0 0
\(837\) 3.22069 0.111323
\(838\) 0 0
\(839\) −15.1010 −0.521344 −0.260672 0.965427i \(-0.583944\pi\)
−0.260672 + 0.965427i \(0.583944\pi\)
\(840\) 0 0
\(841\) 2.86158 0.0986751
\(842\) 0 0
\(843\) −27.5853 −0.950088
\(844\) 0 0
\(845\) 7.12465 0.245095
\(846\) 0 0
\(847\) 9.99188 0.343325
\(848\) 0 0
\(849\) 12.9163 0.443288
\(850\) 0 0
\(851\) 5.55049 0.190268
\(852\) 0 0
\(853\) −14.3676 −0.491938 −0.245969 0.969278i \(-0.579106\pi\)
−0.245969 + 0.969278i \(0.579106\pi\)
\(854\) 0 0
\(855\) −3.84163 −0.131381
\(856\) 0 0
\(857\) 25.5649 0.873281 0.436641 0.899636i \(-0.356168\pi\)
0.436641 + 0.899636i \(0.356168\pi\)
\(858\) 0 0
\(859\) 50.3054 1.71640 0.858200 0.513316i \(-0.171583\pi\)
0.858200 + 0.513316i \(0.171583\pi\)
\(860\) 0 0
\(861\) −20.9214 −0.713000
\(862\) 0 0
\(863\) 17.8429 0.607378 0.303689 0.952771i \(-0.401782\pi\)
0.303689 + 0.952771i \(0.401782\pi\)
\(864\) 0 0
\(865\) 1.73446 0.0589733
\(866\) 0 0
\(867\) −0.224551 −0.00762614
\(868\) 0 0
\(869\) 0.966155 0.0327746
\(870\) 0 0
\(871\) −31.6196 −1.07139
\(872\) 0 0
\(873\) 0.441388 0.0149387
\(874\) 0 0
\(875\) 4.13277 0.139713
\(876\) 0 0
\(877\) −51.3464 −1.73385 −0.866923 0.498443i \(-0.833905\pi\)
−0.866923 + 0.498443i \(0.833905\pi\)
\(878\) 0 0
\(879\) −16.1633 −0.545175
\(880\) 0 0
\(881\) −15.6365 −0.526807 −0.263403 0.964686i \(-0.584845\pi\)
−0.263403 + 0.964686i \(0.584845\pi\)
\(882\) 0 0
\(883\) 20.5874 0.692820 0.346410 0.938083i \(-0.387401\pi\)
0.346410 + 0.938083i \(0.387401\pi\)
\(884\) 0 0
\(885\) 3.64461 0.122512
\(886\) 0 0
\(887\) −6.05475 −0.203299 −0.101649 0.994820i \(-0.532412\pi\)
−0.101649 + 0.994820i \(0.532412\pi\)
\(888\) 0 0
\(889\) 58.8840 1.97491
\(890\) 0 0
\(891\) 2.92955 0.0981437
\(892\) 0 0
\(893\) −14.7581 −0.493862
\(894\) 0 0
\(895\) −5.24187 −0.175217
\(896\) 0 0
\(897\) −2.42391 −0.0809321
\(898\) 0 0
\(899\) −18.1795 −0.606322
\(900\) 0 0
\(901\) −35.1943 −1.17249
\(902\) 0 0
\(903\) −30.6557 −1.02016
\(904\) 0 0
\(905\) 8.80613 0.292726
\(906\) 0 0
\(907\) 2.19317 0.0728229 0.0364115 0.999337i \(-0.488407\pi\)
0.0364115 + 0.999337i \(0.488407\pi\)
\(908\) 0 0
\(909\) 4.21450 0.139786
\(910\) 0 0
\(911\) 20.9089 0.692742 0.346371 0.938098i \(-0.387414\pi\)
0.346371 + 0.938098i \(0.387414\pi\)
\(912\) 0 0
\(913\) −39.7481 −1.31547
\(914\) 0 0
\(915\) −1.51183 −0.0499797
\(916\) 0 0
\(917\) 84.8503 2.80200
\(918\) 0 0
\(919\) 4.04801 0.133531 0.0667657 0.997769i \(-0.478732\pi\)
0.0667657 + 0.997769i \(0.478732\pi\)
\(920\) 0 0
\(921\) 32.4426 1.06902
\(922\) 0 0
\(923\) −17.1185 −0.563461
\(924\) 0 0
\(925\) 5.55049 0.182499
\(926\) 0 0
\(927\) −1.22882 −0.0403596
\(928\) 0 0
\(929\) −12.8155 −0.420463 −0.210231 0.977652i \(-0.567422\pi\)
−0.210231 + 0.977652i \(0.567422\pi\)
\(930\) 0 0
\(931\) 38.7229 1.26909
\(932\) 0 0
\(933\) 24.6010 0.805400
\(934\) 0 0
\(935\) 12.1584 0.397621
\(936\) 0 0
\(937\) 22.9499 0.749741 0.374870 0.927077i \(-0.377687\pi\)
0.374870 + 0.927077i \(0.377687\pi\)
\(938\) 0 0
\(939\) 17.0922 0.557782
\(940\) 0 0
\(941\) −44.5323 −1.45171 −0.725856 0.687847i \(-0.758556\pi\)
−0.725856 + 0.687847i \(0.758556\pi\)
\(942\) 0 0
\(943\) −5.06232 −0.164852
\(944\) 0 0
\(945\) −4.13277 −0.134439
\(946\) 0 0
\(947\) −52.1557 −1.69483 −0.847417 0.530928i \(-0.821843\pi\)
−0.847417 + 0.530928i \(0.821843\pi\)
\(948\) 0 0
\(949\) −3.82030 −0.124012
\(950\) 0 0
\(951\) 2.14212 0.0694632
\(952\) 0 0
\(953\) −42.9556 −1.39147 −0.695734 0.718300i \(-0.744921\pi\)
−0.695734 + 0.718300i \(0.744921\pi\)
\(954\) 0 0
\(955\) −16.7244 −0.541189
\(956\) 0 0
\(957\) −16.5362 −0.534539
\(958\) 0 0
\(959\) 12.1794 0.393293
\(960\) 0 0
\(961\) −20.6271 −0.665391
\(962\) 0 0
\(963\) −0.291141 −0.00938188
\(964\) 0 0
\(965\) −6.30049 −0.202820
\(966\) 0 0
\(967\) −14.1196 −0.454054 −0.227027 0.973888i \(-0.572901\pi\)
−0.227027 + 0.973888i \(0.572901\pi\)
\(968\) 0 0
\(969\) 15.9437 0.512186
\(970\) 0 0
\(971\) 2.21368 0.0710403 0.0355201 0.999369i \(-0.488691\pi\)
0.0355201 + 0.999369i \(0.488691\pi\)
\(972\) 0 0
\(973\) 58.0908 1.86230
\(974\) 0 0
\(975\) −2.42391 −0.0776273
\(976\) 0 0
\(977\) −34.4034 −1.10066 −0.550331 0.834946i \(-0.685499\pi\)
−0.550331 + 0.834946i \(0.685499\pi\)
\(978\) 0 0
\(979\) 3.18713 0.101861
\(980\) 0 0
\(981\) −0.929553 −0.0296783
\(982\) 0 0
\(983\) −2.05104 −0.0654181 −0.0327091 0.999465i \(-0.510413\pi\)
−0.0327091 + 0.999465i \(0.510413\pi\)
\(984\) 0 0
\(985\) −26.5660 −0.846464
\(986\) 0 0
\(987\) −15.8766 −0.505357
\(988\) 0 0
\(989\) −7.41772 −0.235870
\(990\) 0 0
\(991\) −47.5371 −1.51007 −0.755033 0.655686i \(-0.772379\pi\)
−0.755033 + 0.655686i \(0.772379\pi\)
\(992\) 0 0
\(993\) −13.3453 −0.423502
\(994\) 0 0
\(995\) 7.74751 0.245613
\(996\) 0 0
\(997\) −57.1868 −1.81112 −0.905562 0.424213i \(-0.860551\pi\)
−0.905562 + 0.424213i \(0.860551\pi\)
\(998\) 0 0
\(999\) −5.55049 −0.175610
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 2760.2.a.v.1.1 4
3.2 odd 2 8280.2.a.bq.1.1 4
4.3 odd 2 5520.2.a.cb.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.v.1.1 4 1.1 even 1 trivial
5520.2.a.cb.1.4 4 4.3 odd 2
8280.2.a.bq.1.1 4 3.2 odd 2