Properties

Label 8112.2.a.cp.1.1
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.7746461197\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 8112.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.04892 q^{5} -0.554958 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.04892 q^{5} -0.554958 q^{7} +1.00000 q^{9} -2.91185 q^{11} -1.04892 q^{15} -4.85086 q^{17} +0.753020 q^{19} -0.554958 q^{21} -5.76271 q^{23} -3.89977 q^{25} +1.00000 q^{27} -1.91185 q^{29} +9.51573 q^{31} -2.91185 q^{33} +0.582105 q^{35} +5.75302 q^{37} +4.91185 q^{41} +11.0978 q^{43} -1.04892 q^{45} -0.753020 q^{47} -6.69202 q^{49} -4.85086 q^{51} -7.58211 q^{53} +3.05429 q^{55} +0.753020 q^{57} -4.09783 q^{59} -3.42327 q^{61} -0.554958 q^{63} +1.87263 q^{67} -5.76271 q^{69} +10.5036 q^{71} +10.4765 q^{73} -3.89977 q^{75} +1.61596 q^{77} -1.33513 q^{79} +1.00000 q^{81} -2.64310 q^{83} +5.08815 q^{85} -1.91185 q^{87} +9.92692 q^{89} +9.51573 q^{93} -0.789856 q^{95} +17.0737 q^{97} -2.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 6q^{5} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 6q^{5} - 2q^{7} + 3q^{9} - 5q^{11} + 6q^{15} - q^{17} + 7q^{19} - 2q^{21} + 11q^{25} + 3q^{27} - 2q^{29} + 16q^{31} - 5q^{33} - 4q^{35} + 22q^{37} + 11q^{41} + 15q^{43} + 6q^{45} - 7q^{47} - 15q^{49} - q^{51} - 17q^{53} - 3q^{55} + 7q^{57} + 6q^{59} - 13q^{61} - 2q^{63} - 11q^{67} + 6q^{73} + 11q^{75} + 15q^{77} - 3q^{79} + 3q^{81} - 12q^{83} + 19q^{85} - 2q^{87} + q^{89} + 16q^{93} + 21q^{95} - 5q^{97} - 5q^{99} + O(q^{100}) \)

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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.04892 −0.469090 −0.234545 0.972105i \(-0.575360\pi\)
−0.234545 + 0.972105i \(0.575360\pi\)
\(6\) 0 0
\(7\) −0.554958 −0.209754 −0.104877 0.994485i \(-0.533445\pi\)
−0.104877 + 0.994485i \(0.533445\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.91185 −0.877957 −0.438979 0.898498i \(-0.644660\pi\)
−0.438979 + 0.898498i \(0.644660\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −1.04892 −0.270829
\(16\) 0 0
\(17\) −4.85086 −1.17651 −0.588253 0.808677i \(-0.700184\pi\)
−0.588253 + 0.808677i \(0.700184\pi\)
\(18\) 0 0
\(19\) 0.753020 0.172755 0.0863774 0.996262i \(-0.472471\pi\)
0.0863774 + 0.996262i \(0.472471\pi\)
\(20\) 0 0
\(21\) −0.554958 −0.121102
\(22\) 0 0
\(23\) −5.76271 −1.20161 −0.600804 0.799396i \(-0.705153\pi\)
−0.600804 + 0.799396i \(0.705153\pi\)
\(24\) 0 0
\(25\) −3.89977 −0.779954
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.91185 −0.355022 −0.177511 0.984119i \(-0.556805\pi\)
−0.177511 + 0.984119i \(0.556805\pi\)
\(30\) 0 0
\(31\) 9.51573 1.70908 0.854538 0.519389i \(-0.173841\pi\)
0.854538 + 0.519389i \(0.173841\pi\)
\(32\) 0 0
\(33\) −2.91185 −0.506889
\(34\) 0 0
\(35\) 0.582105 0.0983937
\(36\) 0 0
\(37\) 5.75302 0.945791 0.472895 0.881119i \(-0.343209\pi\)
0.472895 + 0.881119i \(0.343209\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.91185 0.767103 0.383551 0.923520i \(-0.374701\pi\)
0.383551 + 0.923520i \(0.374701\pi\)
\(42\) 0 0
\(43\) 11.0978 1.69240 0.846202 0.532862i \(-0.178884\pi\)
0.846202 + 0.532862i \(0.178884\pi\)
\(44\) 0 0
\(45\) −1.04892 −0.156363
\(46\) 0 0
\(47\) −0.753020 −0.109839 −0.0549197 0.998491i \(-0.517490\pi\)
−0.0549197 + 0.998491i \(0.517490\pi\)
\(48\) 0 0
\(49\) −6.69202 −0.956003
\(50\) 0 0
\(51\) −4.85086 −0.679256
\(52\) 0 0
\(53\) −7.58211 −1.04148 −0.520741 0.853715i \(-0.674344\pi\)
−0.520741 + 0.853715i \(0.674344\pi\)
\(54\) 0 0
\(55\) 3.05429 0.411841
\(56\) 0 0
\(57\) 0.753020 0.0997400
\(58\) 0 0
\(59\) −4.09783 −0.533493 −0.266746 0.963767i \(-0.585949\pi\)
−0.266746 + 0.963767i \(0.585949\pi\)
\(60\) 0 0
\(61\) −3.42327 −0.438305 −0.219153 0.975691i \(-0.570329\pi\)
−0.219153 + 0.975691i \(0.570329\pi\)
\(62\) 0 0
\(63\) −0.554958 −0.0699182
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.87263 0.228778 0.114389 0.993436i \(-0.463509\pi\)
0.114389 + 0.993436i \(0.463509\pi\)
\(68\) 0 0
\(69\) −5.76271 −0.693749
\(70\) 0 0
\(71\) 10.5036 1.24655 0.623277 0.782001i \(-0.285801\pi\)
0.623277 + 0.782001i \(0.285801\pi\)
\(72\) 0 0
\(73\) 10.4765 1.22618 0.613091 0.790012i \(-0.289926\pi\)
0.613091 + 0.790012i \(0.289926\pi\)
\(74\) 0 0
\(75\) −3.89977 −0.450307
\(76\) 0 0
\(77\) 1.61596 0.184155
\(78\) 0 0
\(79\) −1.33513 −0.150213 −0.0751067 0.997176i \(-0.523930\pi\)
−0.0751067 + 0.997176i \(0.523930\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.64310 −0.290118 −0.145059 0.989423i \(-0.546337\pi\)
−0.145059 + 0.989423i \(0.546337\pi\)
\(84\) 0 0
\(85\) 5.08815 0.551887
\(86\) 0 0
\(87\) −1.91185 −0.204972
\(88\) 0 0
\(89\) 9.92692 1.05225 0.526126 0.850407i \(-0.323644\pi\)
0.526126 + 0.850407i \(0.323644\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 9.51573 0.986735
\(94\) 0 0
\(95\) −0.789856 −0.0810375
\(96\) 0 0
\(97\) 17.0737 1.73357 0.866784 0.498683i \(-0.166183\pi\)
0.866784 + 0.498683i \(0.166183\pi\)
\(98\) 0 0
\(99\) −2.91185 −0.292652
\(100\) 0 0
\(101\) 7.32304 0.728670 0.364335 0.931268i \(-0.381296\pi\)
0.364335 + 0.931268i \(0.381296\pi\)
\(102\) 0 0
\(103\) 4.21983 0.415792 0.207896 0.978151i \(-0.433338\pi\)
0.207896 + 0.978151i \(0.433338\pi\)
\(104\) 0 0
\(105\) 0.582105 0.0568077
\(106\) 0 0
\(107\) −6.39373 −0.618105 −0.309053 0.951045i \(-0.600012\pi\)
−0.309053 + 0.951045i \(0.600012\pi\)
\(108\) 0 0
\(109\) −3.46011 −0.331418 −0.165709 0.986175i \(-0.552991\pi\)
−0.165709 + 0.986175i \(0.552991\pi\)
\(110\) 0 0
\(111\) 5.75302 0.546053
\(112\) 0 0
\(113\) 9.35690 0.880223 0.440111 0.897943i \(-0.354939\pi\)
0.440111 + 0.897943i \(0.354939\pi\)
\(114\) 0 0
\(115\) 6.04461 0.563662
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.69202 0.246777
\(120\) 0 0
\(121\) −2.52111 −0.229191
\(122\) 0 0
\(123\) 4.91185 0.442887
\(124\) 0 0
\(125\) 9.33513 0.834959
\(126\) 0 0
\(127\) 4.48188 0.397702 0.198851 0.980030i \(-0.436279\pi\)
0.198851 + 0.980030i \(0.436279\pi\)
\(128\) 0 0
\(129\) 11.0978 0.977110
\(130\) 0 0
\(131\) 9.21744 0.805331 0.402666 0.915347i \(-0.368084\pi\)
0.402666 + 0.915347i \(0.368084\pi\)
\(132\) 0 0
\(133\) −0.417895 −0.0362361
\(134\) 0 0
\(135\) −1.04892 −0.0902764
\(136\) 0 0
\(137\) −7.46980 −0.638188 −0.319094 0.947723i \(-0.603379\pi\)
−0.319094 + 0.947723i \(0.603379\pi\)
\(138\) 0 0
\(139\) 17.9976 1.52654 0.763269 0.646081i \(-0.223593\pi\)
0.763269 + 0.646081i \(0.223593\pi\)
\(140\) 0 0
\(141\) −0.753020 −0.0634158
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.00538 0.166537
\(146\) 0 0
\(147\) −6.69202 −0.551949
\(148\) 0 0
\(149\) 15.3351 1.25630 0.628151 0.778091i \(-0.283812\pi\)
0.628151 + 0.778091i \(0.283812\pi\)
\(150\) 0 0
\(151\) −2.53079 −0.205953 −0.102977 0.994684i \(-0.532837\pi\)
−0.102977 + 0.994684i \(0.532837\pi\)
\(152\) 0 0
\(153\) −4.85086 −0.392168
\(154\) 0 0
\(155\) −9.98121 −0.801710
\(156\) 0 0
\(157\) 17.2392 1.37584 0.687919 0.725787i \(-0.258524\pi\)
0.687919 + 0.725787i \(0.258524\pi\)
\(158\) 0 0
\(159\) −7.58211 −0.601300
\(160\) 0 0
\(161\) 3.19806 0.252043
\(162\) 0 0
\(163\) −15.7071 −1.23027 −0.615137 0.788420i \(-0.710899\pi\)
−0.615137 + 0.788420i \(0.710899\pi\)
\(164\) 0 0
\(165\) 3.05429 0.237776
\(166\) 0 0
\(167\) −5.39612 −0.417565 −0.208782 0.977962i \(-0.566950\pi\)
−0.208782 + 0.977962i \(0.566950\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0.753020 0.0575849
\(172\) 0 0
\(173\) 23.9420 1.82028 0.910138 0.414306i \(-0.135976\pi\)
0.910138 + 0.414306i \(0.135976\pi\)
\(174\) 0 0
\(175\) 2.16421 0.163599
\(176\) 0 0
\(177\) −4.09783 −0.308012
\(178\) 0 0
\(179\) −18.4088 −1.37594 −0.687969 0.725740i \(-0.741498\pi\)
−0.687969 + 0.725740i \(0.741498\pi\)
\(180\) 0 0
\(181\) −3.63342 −0.270070 −0.135035 0.990841i \(-0.543115\pi\)
−0.135035 + 0.990841i \(0.543115\pi\)
\(182\) 0 0
\(183\) −3.42327 −0.253056
\(184\) 0 0
\(185\) −6.03444 −0.443661
\(186\) 0 0
\(187\) 14.1250 1.03292
\(188\) 0 0
\(189\) −0.554958 −0.0403673
\(190\) 0 0
\(191\) 21.1782 1.53240 0.766201 0.642601i \(-0.222145\pi\)
0.766201 + 0.642601i \(0.222145\pi\)
\(192\) 0 0
\(193\) −17.6112 −1.26768 −0.633840 0.773464i \(-0.718522\pi\)
−0.633840 + 0.773464i \(0.718522\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.66248 −0.332188 −0.166094 0.986110i \(-0.553116\pi\)
−0.166094 + 0.986110i \(0.553116\pi\)
\(198\) 0 0
\(199\) −15.0368 −1.06593 −0.532967 0.846136i \(-0.678923\pi\)
−0.532967 + 0.846136i \(0.678923\pi\)
\(200\) 0 0
\(201\) 1.87263 0.132085
\(202\) 0 0
\(203\) 1.06100 0.0744675
\(204\) 0 0
\(205\) −5.15213 −0.359840
\(206\) 0 0
\(207\) −5.76271 −0.400536
\(208\) 0 0
\(209\) −2.19269 −0.151671
\(210\) 0 0
\(211\) 0.460107 0.0316751 0.0158375 0.999875i \(-0.494959\pi\)
0.0158375 + 0.999875i \(0.494959\pi\)
\(212\) 0 0
\(213\) 10.5036 0.719698
\(214\) 0 0
\(215\) −11.6407 −0.793890
\(216\) 0 0
\(217\) −5.28083 −0.358486
\(218\) 0 0
\(219\) 10.4765 0.707936
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.3502 1.09489 0.547445 0.836842i \(-0.315601\pi\)
0.547445 + 0.836842i \(0.315601\pi\)
\(224\) 0 0
\(225\) −3.89977 −0.259985
\(226\) 0 0
\(227\) −6.56033 −0.435425 −0.217712 0.976013i \(-0.569859\pi\)
−0.217712 + 0.976013i \(0.569859\pi\)
\(228\) 0 0
\(229\) −3.95539 −0.261380 −0.130690 0.991423i \(-0.541719\pi\)
−0.130690 + 0.991423i \(0.541719\pi\)
\(230\) 0 0
\(231\) 1.61596 0.106322
\(232\) 0 0
\(233\) 8.35690 0.547478 0.273739 0.961804i \(-0.411739\pi\)
0.273739 + 0.961804i \(0.411739\pi\)
\(234\) 0 0
\(235\) 0.789856 0.0515245
\(236\) 0 0
\(237\) −1.33513 −0.0867257
\(238\) 0 0
\(239\) −20.1008 −1.30021 −0.650107 0.759843i \(-0.725276\pi\)
−0.650107 + 0.759843i \(0.725276\pi\)
\(240\) 0 0
\(241\) −19.0127 −1.22471 −0.612357 0.790581i \(-0.709778\pi\)
−0.612357 + 0.790581i \(0.709778\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 7.01938 0.448452
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −2.64310 −0.167500
\(250\) 0 0
\(251\) 0.763774 0.0482090 0.0241045 0.999709i \(-0.492327\pi\)
0.0241045 + 0.999709i \(0.492327\pi\)
\(252\) 0 0
\(253\) 16.7802 1.05496
\(254\) 0 0
\(255\) 5.08815 0.318632
\(256\) 0 0
\(257\) −13.0911 −0.816602 −0.408301 0.912847i \(-0.633879\pi\)
−0.408301 + 0.912847i \(0.633879\pi\)
\(258\) 0 0
\(259\) −3.19269 −0.198384
\(260\) 0 0
\(261\) −1.91185 −0.118341
\(262\) 0 0
\(263\) −18.3773 −1.13320 −0.566598 0.823995i \(-0.691741\pi\)
−0.566598 + 0.823995i \(0.691741\pi\)
\(264\) 0 0
\(265\) 7.95300 0.488549
\(266\) 0 0
\(267\) 9.92692 0.607518
\(268\) 0 0
\(269\) 23.6625 1.44273 0.721363 0.692557i \(-0.243516\pi\)
0.721363 + 0.692557i \(0.243516\pi\)
\(270\) 0 0
\(271\) 19.7530 1.19991 0.599955 0.800034i \(-0.295185\pi\)
0.599955 + 0.800034i \(0.295185\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11.3556 0.684767
\(276\) 0 0
\(277\) 1.77777 0.106816 0.0534081 0.998573i \(-0.482992\pi\)
0.0534081 + 0.998573i \(0.482992\pi\)
\(278\) 0 0
\(279\) 9.51573 0.569692
\(280\) 0 0
\(281\) 1.62133 0.0967207 0.0483603 0.998830i \(-0.484600\pi\)
0.0483603 + 0.998830i \(0.484600\pi\)
\(282\) 0 0
\(283\) −4.98361 −0.296245 −0.148122 0.988969i \(-0.547323\pi\)
−0.148122 + 0.988969i \(0.547323\pi\)
\(284\) 0 0
\(285\) −0.789856 −0.0467870
\(286\) 0 0
\(287\) −2.72587 −0.160903
\(288\) 0 0
\(289\) 6.53079 0.384164
\(290\) 0 0
\(291\) 17.0737 1.00088
\(292\) 0 0
\(293\) −0.0717525 −0.00419183 −0.00209591 0.999998i \(-0.500667\pi\)
−0.00209591 + 0.999998i \(0.500667\pi\)
\(294\) 0 0
\(295\) 4.29829 0.250256
\(296\) 0 0
\(297\) −2.91185 −0.168963
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −6.15883 −0.354989
\(302\) 0 0
\(303\) 7.32304 0.420698
\(304\) 0 0
\(305\) 3.59073 0.205605
\(306\) 0 0
\(307\) 5.19806 0.296669 0.148335 0.988937i \(-0.452609\pi\)
0.148335 + 0.988937i \(0.452609\pi\)
\(308\) 0 0
\(309\) 4.21983 0.240058
\(310\) 0 0
\(311\) 22.5429 1.27829 0.639145 0.769087i \(-0.279289\pi\)
0.639145 + 0.769087i \(0.279289\pi\)
\(312\) 0 0
\(313\) 22.6612 1.28088 0.640442 0.768006i \(-0.278751\pi\)
0.640442 + 0.768006i \(0.278751\pi\)
\(314\) 0 0
\(315\) 0.582105 0.0327979
\(316\) 0 0
\(317\) 26.3424 1.47954 0.739769 0.672861i \(-0.234935\pi\)
0.739769 + 0.672861i \(0.234935\pi\)
\(318\) 0 0
\(319\) 5.56704 0.311694
\(320\) 0 0
\(321\) −6.39373 −0.356863
\(322\) 0 0
\(323\) −3.65279 −0.203247
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.46011 −0.191344
\(328\) 0 0
\(329\) 0.417895 0.0230393
\(330\) 0 0
\(331\) 11.2295 0.617230 0.308615 0.951187i \(-0.400134\pi\)
0.308615 + 0.951187i \(0.400134\pi\)
\(332\) 0 0
\(333\) 5.75302 0.315264
\(334\) 0 0
\(335\) −1.96423 −0.107317
\(336\) 0 0
\(337\) 2.30798 0.125724 0.0628618 0.998022i \(-0.479977\pi\)
0.0628618 + 0.998022i \(0.479977\pi\)
\(338\) 0 0
\(339\) 9.35690 0.508197
\(340\) 0 0
\(341\) −27.7084 −1.50049
\(342\) 0 0
\(343\) 7.59850 0.410280
\(344\) 0 0
\(345\) 6.04461 0.325431
\(346\) 0 0
\(347\) 9.13706 0.490503 0.245252 0.969459i \(-0.421129\pi\)
0.245252 + 0.969459i \(0.421129\pi\)
\(348\) 0 0
\(349\) 23.9758 1.28340 0.641699 0.766957i \(-0.278230\pi\)
0.641699 + 0.766957i \(0.278230\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −27.1239 −1.44366 −0.721830 0.692070i \(-0.756699\pi\)
−0.721830 + 0.692070i \(0.756699\pi\)
\(354\) 0 0
\(355\) −11.0175 −0.584746
\(356\) 0 0
\(357\) 2.69202 0.142477
\(358\) 0 0
\(359\) 26.0790 1.37640 0.688200 0.725521i \(-0.258401\pi\)
0.688200 + 0.725521i \(0.258401\pi\)
\(360\) 0 0
\(361\) −18.4330 −0.970156
\(362\) 0 0
\(363\) −2.52111 −0.132324
\(364\) 0 0
\(365\) −10.9890 −0.575190
\(366\) 0 0
\(367\) −9.57434 −0.499776 −0.249888 0.968275i \(-0.580394\pi\)
−0.249888 + 0.968275i \(0.580394\pi\)
\(368\) 0 0
\(369\) 4.91185 0.255701
\(370\) 0 0
\(371\) 4.20775 0.218456
\(372\) 0 0
\(373\) 28.1497 1.45754 0.728769 0.684760i \(-0.240093\pi\)
0.728769 + 0.684760i \(0.240093\pi\)
\(374\) 0 0
\(375\) 9.33513 0.482064
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −16.0465 −0.824255 −0.412127 0.911126i \(-0.635214\pi\)
−0.412127 + 0.911126i \(0.635214\pi\)
\(380\) 0 0
\(381\) 4.48188 0.229614
\(382\) 0 0
\(383\) 24.6165 1.25785 0.628923 0.777467i \(-0.283496\pi\)
0.628923 + 0.777467i \(0.283496\pi\)
\(384\) 0 0
\(385\) −1.69501 −0.0863855
\(386\) 0 0
\(387\) 11.0978 0.564135
\(388\) 0 0
\(389\) −17.2198 −0.873080 −0.436540 0.899685i \(-0.643796\pi\)
−0.436540 + 0.899685i \(0.643796\pi\)
\(390\) 0 0
\(391\) 27.9541 1.41370
\(392\) 0 0
\(393\) 9.21744 0.464958
\(394\) 0 0
\(395\) 1.40044 0.0704636
\(396\) 0 0
\(397\) −2.03923 −0.102346 −0.0511730 0.998690i \(-0.516296\pi\)
−0.0511730 + 0.998690i \(0.516296\pi\)
\(398\) 0 0
\(399\) −0.417895 −0.0209209
\(400\) 0 0
\(401\) −1.46144 −0.0729806 −0.0364903 0.999334i \(-0.511618\pi\)
−0.0364903 + 0.999334i \(0.511618\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.04892 −0.0521211
\(406\) 0 0
\(407\) −16.7520 −0.830364
\(408\) 0 0
\(409\) −29.9390 −1.48039 −0.740194 0.672393i \(-0.765266\pi\)
−0.740194 + 0.672393i \(0.765266\pi\)
\(410\) 0 0
\(411\) −7.46980 −0.368458
\(412\) 0 0
\(413\) 2.27413 0.111902
\(414\) 0 0
\(415\) 2.77240 0.136092
\(416\) 0 0
\(417\) 17.9976 0.881347
\(418\) 0 0
\(419\) −6.64742 −0.324748 −0.162374 0.986729i \(-0.551915\pi\)
−0.162374 + 0.986729i \(0.551915\pi\)
\(420\) 0 0
\(421\) 13.5646 0.661100 0.330550 0.943788i \(-0.392766\pi\)
0.330550 + 0.943788i \(0.392766\pi\)
\(422\) 0 0
\(423\) −0.753020 −0.0366131
\(424\) 0 0
\(425\) 18.9172 0.917620
\(426\) 0 0
\(427\) 1.89977 0.0919364
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −35.9463 −1.73147 −0.865736 0.500501i \(-0.833149\pi\)
−0.865736 + 0.500501i \(0.833149\pi\)
\(432\) 0 0
\(433\) −32.4741 −1.56061 −0.780303 0.625402i \(-0.784935\pi\)
−0.780303 + 0.625402i \(0.784935\pi\)
\(434\) 0 0
\(435\) 2.00538 0.0961505
\(436\) 0 0
\(437\) −4.33944 −0.207583
\(438\) 0 0
\(439\) −12.8321 −0.612441 −0.306221 0.951961i \(-0.599065\pi\)
−0.306221 + 0.951961i \(0.599065\pi\)
\(440\) 0 0
\(441\) −6.69202 −0.318668
\(442\) 0 0
\(443\) −11.9608 −0.568273 −0.284137 0.958784i \(-0.591707\pi\)
−0.284137 + 0.958784i \(0.591707\pi\)
\(444\) 0 0
\(445\) −10.4125 −0.493601
\(446\) 0 0
\(447\) 15.3351 0.725327
\(448\) 0 0
\(449\) −12.4789 −0.588915 −0.294458 0.955665i \(-0.595139\pi\)
−0.294458 + 0.955665i \(0.595139\pi\)
\(450\) 0 0
\(451\) −14.3026 −0.673483
\(452\) 0 0
\(453\) −2.53079 −0.118907
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 32.4523 1.51806 0.759028 0.651058i \(-0.225674\pi\)
0.759028 + 0.651058i \(0.225674\pi\)
\(458\) 0 0
\(459\) −4.85086 −0.226419
\(460\) 0 0
\(461\) 24.4034 1.13658 0.568290 0.822828i \(-0.307605\pi\)
0.568290 + 0.822828i \(0.307605\pi\)
\(462\) 0 0
\(463\) 33.1836 1.54217 0.771086 0.636731i \(-0.219714\pi\)
0.771086 + 0.636731i \(0.219714\pi\)
\(464\) 0 0
\(465\) −9.98121 −0.462868
\(466\) 0 0
\(467\) 38.5206 1.78252 0.891261 0.453490i \(-0.149821\pi\)
0.891261 + 0.453490i \(0.149821\pi\)
\(468\) 0 0
\(469\) −1.03923 −0.0479871
\(470\) 0 0
\(471\) 17.2392 0.794341
\(472\) 0 0
\(473\) −32.3153 −1.48586
\(474\) 0 0
\(475\) −2.93661 −0.134741
\(476\) 0 0
\(477\) −7.58211 −0.347161
\(478\) 0 0
\(479\) 8.34481 0.381284 0.190642 0.981660i \(-0.438943\pi\)
0.190642 + 0.981660i \(0.438943\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 3.19806 0.145517
\(484\) 0 0
\(485\) −17.9089 −0.813200
\(486\) 0 0
\(487\) 14.8586 0.673309 0.336654 0.941628i \(-0.390705\pi\)
0.336654 + 0.941628i \(0.390705\pi\)
\(488\) 0 0
\(489\) −15.7071 −0.710299
\(490\) 0 0
\(491\) −4.99894 −0.225599 −0.112799 0.993618i \(-0.535982\pi\)
−0.112799 + 0.993618i \(0.535982\pi\)
\(492\) 0 0
\(493\) 9.27413 0.417686
\(494\) 0 0
\(495\) 3.05429 0.137280
\(496\) 0 0
\(497\) −5.82908 −0.261470
\(498\) 0 0
\(499\) 0.385371 0.0172516 0.00862579 0.999963i \(-0.497254\pi\)
0.00862579 + 0.999963i \(0.497254\pi\)
\(500\) 0 0
\(501\) −5.39612 −0.241081
\(502\) 0 0
\(503\) −22.6179 −1.00848 −0.504241 0.863563i \(-0.668228\pi\)
−0.504241 + 0.863563i \(0.668228\pi\)
\(504\) 0 0
\(505\) −7.68127 −0.341812
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11.6039 −0.514333 −0.257166 0.966367i \(-0.582789\pi\)
−0.257166 + 0.966367i \(0.582789\pi\)
\(510\) 0 0
\(511\) −5.81402 −0.257197
\(512\) 0 0
\(513\) 0.753020 0.0332467
\(514\) 0 0
\(515\) −4.42626 −0.195044
\(516\) 0 0
\(517\) 2.19269 0.0964342
\(518\) 0 0
\(519\) 23.9420 1.05094
\(520\) 0 0
\(521\) −1.62671 −0.0712675 −0.0356337 0.999365i \(-0.511345\pi\)
−0.0356337 + 0.999365i \(0.511345\pi\)
\(522\) 0 0
\(523\) −10.0718 −0.440407 −0.220203 0.975454i \(-0.570672\pi\)
−0.220203 + 0.975454i \(0.570672\pi\)
\(524\) 0 0
\(525\) 2.16421 0.0944539
\(526\) 0 0
\(527\) −46.1594 −2.01074
\(528\) 0 0
\(529\) 10.2088 0.443862
\(530\) 0 0
\(531\) −4.09783 −0.177831
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.70650 0.289947
\(536\) 0 0
\(537\) −18.4088 −0.794398
\(538\) 0 0
\(539\) 19.4862 0.839330
\(540\) 0 0
\(541\) −20.4674 −0.879962 −0.439981 0.898007i \(-0.645015\pi\)
−0.439981 + 0.898007i \(0.645015\pi\)
\(542\) 0 0
\(543\) −3.63342 −0.155925
\(544\) 0 0
\(545\) 3.62937 0.155465
\(546\) 0 0
\(547\) 27.5478 1.17786 0.588929 0.808185i \(-0.299550\pi\)
0.588929 + 0.808185i \(0.299550\pi\)
\(548\) 0 0
\(549\) −3.42327 −0.146102
\(550\) 0 0
\(551\) −1.43967 −0.0613318
\(552\) 0 0
\(553\) 0.740939 0.0315079
\(554\) 0 0
\(555\) −6.03444 −0.256148
\(556\) 0 0
\(557\) 37.9855 1.60950 0.804749 0.593615i \(-0.202300\pi\)
0.804749 + 0.593615i \(0.202300\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 14.1250 0.596357
\(562\) 0 0
\(563\) 29.7724 1.25476 0.627378 0.778714i \(-0.284128\pi\)
0.627378 + 0.778714i \(0.284128\pi\)
\(564\) 0 0
\(565\) −9.81461 −0.412904
\(566\) 0 0
\(567\) −0.554958 −0.0233061
\(568\) 0 0
\(569\) −21.9541 −0.920362 −0.460181 0.887825i \(-0.652216\pi\)
−0.460181 + 0.887825i \(0.652216\pi\)
\(570\) 0 0
\(571\) −2.46575 −0.103188 −0.0515942 0.998668i \(-0.516430\pi\)
−0.0515942 + 0.998668i \(0.516430\pi\)
\(572\) 0 0
\(573\) 21.1782 0.884732
\(574\) 0 0
\(575\) 22.4733 0.937199
\(576\) 0 0
\(577\) 17.4547 0.726650 0.363325 0.931662i \(-0.381641\pi\)
0.363325 + 0.931662i \(0.381641\pi\)
\(578\) 0 0
\(579\) −17.6112 −0.731895
\(580\) 0 0
\(581\) 1.46681 0.0608536
\(582\) 0 0
\(583\) 22.0780 0.914377
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.26337 −0.258517 −0.129259 0.991611i \(-0.541260\pi\)
−0.129259 + 0.991611i \(0.541260\pi\)
\(588\) 0 0
\(589\) 7.16554 0.295251
\(590\) 0 0
\(591\) −4.66248 −0.191789
\(592\) 0 0
\(593\) 22.8745 0.939345 0.469672 0.882841i \(-0.344372\pi\)
0.469672 + 0.882841i \(0.344372\pi\)
\(594\) 0 0
\(595\) −2.82371 −0.115761
\(596\) 0 0
\(597\) −15.0368 −0.615417
\(598\) 0 0
\(599\) 1.05621 0.0431557 0.0215778 0.999767i \(-0.493131\pi\)
0.0215778 + 0.999767i \(0.493131\pi\)
\(600\) 0 0
\(601\) −33.3236 −1.35930 −0.679650 0.733537i \(-0.737868\pi\)
−0.679650 + 0.733537i \(0.737868\pi\)
\(602\) 0 0
\(603\) 1.87263 0.0762592
\(604\) 0 0
\(605\) 2.64443 0.107511
\(606\) 0 0
\(607\) −16.2403 −0.659172 −0.329586 0.944125i \(-0.606909\pi\)
−0.329586 + 0.944125i \(0.606909\pi\)
\(608\) 0 0
\(609\) 1.06100 0.0429938
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 11.0479 0.446219 0.223109 0.974793i \(-0.428379\pi\)
0.223109 + 0.974793i \(0.428379\pi\)
\(614\) 0 0
\(615\) −5.15213 −0.207754
\(616\) 0 0
\(617\) −4.65950 −0.187584 −0.0937922 0.995592i \(-0.529899\pi\)
−0.0937922 + 0.995592i \(0.529899\pi\)
\(618\) 0 0
\(619\) −31.9259 −1.28321 −0.641604 0.767036i \(-0.721731\pi\)
−0.641604 + 0.767036i \(0.721731\pi\)
\(620\) 0 0
\(621\) −5.76271 −0.231250
\(622\) 0 0
\(623\) −5.50902 −0.220714
\(624\) 0 0
\(625\) 9.70709 0.388283
\(626\) 0 0
\(627\) −2.19269 −0.0875674
\(628\) 0 0
\(629\) −27.9071 −1.11273
\(630\) 0 0
\(631\) 39.4413 1.57013 0.785067 0.619411i \(-0.212628\pi\)
0.785067 + 0.619411i \(0.212628\pi\)
\(632\) 0 0
\(633\) 0.460107 0.0182876
\(634\) 0 0
\(635\) −4.70112 −0.186558
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 10.5036 0.415518
\(640\) 0 0
\(641\) 45.4510 1.79521 0.897603 0.440804i \(-0.145307\pi\)
0.897603 + 0.440804i \(0.145307\pi\)
\(642\) 0 0
\(643\) 29.7469 1.17310 0.586552 0.809912i \(-0.300485\pi\)
0.586552 + 0.809912i \(0.300485\pi\)
\(644\) 0 0
\(645\) −11.6407 −0.458353
\(646\) 0 0
\(647\) 16.9312 0.665635 0.332818 0.942991i \(-0.392001\pi\)
0.332818 + 0.942991i \(0.392001\pi\)
\(648\) 0 0
\(649\) 11.9323 0.468384
\(650\) 0 0
\(651\) −5.28083 −0.206972
\(652\) 0 0
\(653\) −33.1976 −1.29912 −0.649561 0.760309i \(-0.725047\pi\)
−0.649561 + 0.760309i \(0.725047\pi\)
\(654\) 0 0
\(655\) −9.66833 −0.377773
\(656\) 0 0
\(657\) 10.4765 0.408727
\(658\) 0 0
\(659\) 42.6571 1.66168 0.830842 0.556508i \(-0.187859\pi\)
0.830842 + 0.556508i \(0.187859\pi\)
\(660\) 0 0
\(661\) 38.6902 1.50488 0.752438 0.658664i \(-0.228878\pi\)
0.752438 + 0.658664i \(0.228878\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.438337 0.0169980
\(666\) 0 0
\(667\) 11.0175 0.426598
\(668\) 0 0
\(669\) 16.3502 0.632135
\(670\) 0 0
\(671\) 9.96807 0.384813
\(672\) 0 0
\(673\) −2.59419 −0.0999986 −0.0499993 0.998749i \(-0.515922\pi\)
−0.0499993 + 0.998749i \(0.515922\pi\)
\(674\) 0 0
\(675\) −3.89977 −0.150102
\(676\) 0 0
\(677\) 1.75302 0.0673740 0.0336870 0.999432i \(-0.489275\pi\)
0.0336870 + 0.999432i \(0.489275\pi\)
\(678\) 0 0
\(679\) −9.47517 −0.363624
\(680\) 0 0
\(681\) −6.56033 −0.251393
\(682\) 0 0
\(683\) −16.3351 −0.625046 −0.312523 0.949910i \(-0.601174\pi\)
−0.312523 + 0.949910i \(0.601174\pi\)
\(684\) 0 0
\(685\) 7.83520 0.299368
\(686\) 0 0
\(687\) −3.95539 −0.150908
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −15.9105 −0.605265 −0.302632 0.953107i \(-0.597865\pi\)
−0.302632 + 0.953107i \(0.597865\pi\)
\(692\) 0 0
\(693\) 1.61596 0.0613851
\(694\) 0 0
\(695\) −18.8780 −0.716083
\(696\) 0 0
\(697\) −23.8267 −0.902500
\(698\) 0 0
\(699\) 8.35690 0.316087
\(700\) 0 0
\(701\) −20.8635 −0.788005 −0.394002 0.919109i \(-0.628910\pi\)
−0.394002 + 0.919109i \(0.628910\pi\)
\(702\) 0 0
\(703\) 4.33214 0.163390
\(704\) 0 0
\(705\) 0.789856 0.0297477
\(706\) 0 0
\(707\) −4.06398 −0.152842
\(708\) 0 0
\(709\) 15.6485 0.587691 0.293846 0.955853i \(-0.405065\pi\)
0.293846 + 0.955853i \(0.405065\pi\)
\(710\) 0 0
\(711\) −1.33513 −0.0500711
\(712\) 0 0
\(713\) −54.8364 −2.05364
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −20.1008 −0.750679
\(718\) 0 0
\(719\) 27.1594 1.01288 0.506438 0.862276i \(-0.330962\pi\)
0.506438 + 0.862276i \(0.330962\pi\)
\(720\) 0 0
\(721\) −2.34183 −0.0872143
\(722\) 0 0
\(723\) −19.0127 −0.707089
\(724\) 0 0
\(725\) 7.45580 0.276901
\(726\) 0 0
\(727\) −31.7784 −1.17859 −0.589297 0.807916i \(-0.700595\pi\)
−0.589297 + 0.807916i \(0.700595\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −53.8340 −1.99112
\(732\) 0 0
\(733\) 46.8907 1.73195 0.865973 0.500090i \(-0.166700\pi\)
0.865973 + 0.500090i \(0.166700\pi\)
\(734\) 0 0
\(735\) 7.01938 0.258914
\(736\) 0 0
\(737\) −5.45281 −0.200857
\(738\) 0 0
\(739\) −17.0479 −0.627115 −0.313558 0.949569i \(-0.601521\pi\)
−0.313558 + 0.949569i \(0.601521\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 11.6324 0.426750 0.213375 0.976970i \(-0.431554\pi\)
0.213375 + 0.976970i \(0.431554\pi\)
\(744\) 0 0
\(745\) −16.0853 −0.589319
\(746\) 0 0
\(747\) −2.64310 −0.0967061
\(748\) 0 0
\(749\) 3.54825 0.129650
\(750\) 0 0
\(751\) 13.0295 0.475455 0.237727 0.971332i \(-0.423598\pi\)
0.237727 + 0.971332i \(0.423598\pi\)
\(752\) 0 0
\(753\) 0.763774 0.0278335
\(754\) 0 0
\(755\) 2.65459 0.0966106
\(756\) 0 0
\(757\) 22.7899 0.828311 0.414156 0.910206i \(-0.364077\pi\)
0.414156 + 0.910206i \(0.364077\pi\)
\(758\) 0 0
\(759\) 16.7802 0.609081
\(760\) 0 0
\(761\) −38.3424 −1.38991 −0.694956 0.719052i \(-0.744576\pi\)
−0.694956 + 0.719052i \(0.744576\pi\)
\(762\) 0 0
\(763\) 1.92021 0.0695164
\(764\) 0 0
\(765\) 5.08815 0.183962
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −3.63879 −0.131218 −0.0656091 0.997845i \(-0.520899\pi\)
−0.0656091 + 0.997845i \(0.520899\pi\)
\(770\) 0 0
\(771\) −13.0911 −0.471466
\(772\) 0 0
\(773\) −39.3424 −1.41505 −0.707524 0.706689i \(-0.750188\pi\)
−0.707524 + 0.706689i \(0.750188\pi\)
\(774\) 0 0
\(775\) −37.1092 −1.33300
\(776\) 0 0
\(777\) −3.19269 −0.114537
\(778\) 0 0
\(779\) 3.69873 0.132521
\(780\) 0 0
\(781\) −30.5851 −1.09442
\(782\) 0 0
\(783\) −1.91185 −0.0683241
\(784\) 0 0
\(785\) −18.0825 −0.645392
\(786\) 0 0
\(787\) 25.4252 0.906310 0.453155 0.891432i \(-0.350298\pi\)
0.453155 + 0.891432i \(0.350298\pi\)
\(788\) 0 0
\(789\) −18.3773 −0.654251
\(790\) 0 0
\(791\) −5.19269 −0.184631
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 7.95300 0.282064
\(796\) 0 0
\(797\) 20.7138 0.733720 0.366860 0.930276i \(-0.380433\pi\)
0.366860 + 0.930276i \(0.380433\pi\)
\(798\) 0 0
\(799\) 3.65279 0.129227
\(800\) 0 0
\(801\) 9.92692 0.350750
\(802\) 0 0
\(803\) −30.5060 −1.07653
\(804\) 0 0
\(805\) −3.35450 −0.118231
\(806\) 0 0
\(807\) 23.6625 0.832959
\(808\) 0 0
\(809\) 11.0978 0.390179 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(810\) 0 0
\(811\) 4.84223 0.170034 0.0850169 0.996380i \(-0.472906\pi\)
0.0850169 + 0.996380i \(0.472906\pi\)
\(812\) 0 0
\(813\) 19.7530 0.692769
\(814\) 0 0
\(815\) 16.4754 0.577109
\(816\) 0 0
\(817\) 8.35690 0.292371
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −43.7381 −1.52647 −0.763235 0.646121i \(-0.776390\pi\)
−0.763235 + 0.646121i \(0.776390\pi\)
\(822\) 0 0
\(823\) −12.0998 −0.421771 −0.210885 0.977511i \(-0.567635\pi\)
−0.210885 + 0.977511i \(0.567635\pi\)
\(824\) 0 0
\(825\) 11.3556 0.395350
\(826\) 0 0
\(827\) 35.3212 1.22824 0.614120 0.789213i \(-0.289511\pi\)
0.614120 + 0.789213i \(0.289511\pi\)
\(828\) 0 0
\(829\) 53.4946 1.85794 0.928971 0.370152i \(-0.120694\pi\)
0.928971 + 0.370152i \(0.120694\pi\)
\(830\) 0 0
\(831\) 1.77777 0.0616703
\(832\) 0 0
\(833\) 32.4620 1.12474
\(834\) 0 0
\(835\) 5.66009 0.195875
\(836\) 0 0
\(837\) 9.51573 0.328912
\(838\) 0 0
\(839\) −23.1521 −0.799300 −0.399650 0.916668i \(-0.630868\pi\)
−0.399650 + 0.916668i \(0.630868\pi\)
\(840\) 0 0
\(841\) −25.3448 −0.873959
\(842\) 0 0
\(843\) 1.62133 0.0558417
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.39911 0.0480739
\(848\) 0 0
\(849\) −4.98361 −0.171037
\(850\) 0 0
\(851\) −33.1530 −1.13647
\(852\) 0 0
\(853\) 26.7265 0.915097 0.457548 0.889185i \(-0.348728\pi\)
0.457548 + 0.889185i \(0.348728\pi\)
\(854\) 0 0
\(855\) −0.789856 −0.0270125
\(856\) 0 0
\(857\) 42.6064 1.45541 0.727703 0.685892i \(-0.240588\pi\)
0.727703 + 0.685892i \(0.240588\pi\)
\(858\) 0 0
\(859\) −33.6079 −1.14669 −0.573344 0.819315i \(-0.694354\pi\)
−0.573344 + 0.819315i \(0.694354\pi\)
\(860\) 0 0
\(861\) −2.72587 −0.0928975
\(862\) 0 0
\(863\) −18.7047 −0.636715 −0.318358 0.947971i \(-0.603131\pi\)
−0.318358 + 0.947971i \(0.603131\pi\)
\(864\) 0 0
\(865\) −25.1132 −0.853873
\(866\) 0 0
\(867\) 6.53079 0.221797
\(868\) 0 0
\(869\) 3.88769 0.131881
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 17.0737 0.577856
\(874\) 0 0
\(875\) −5.18060 −0.175136
\(876\) 0 0
\(877\) 36.8237 1.24345 0.621724 0.783236i \(-0.286433\pi\)
0.621724 + 0.783236i \(0.286433\pi\)
\(878\) 0 0
\(879\) −0.0717525 −0.00242015
\(880\) 0 0
\(881\) −41.1250 −1.38554 −0.692768 0.721161i \(-0.743609\pi\)
−0.692768 + 0.721161i \(0.743609\pi\)
\(882\) 0 0
\(883\) −30.7482 −1.03476 −0.517380 0.855756i \(-0.673093\pi\)
−0.517380 + 0.855756i \(0.673093\pi\)
\(884\) 0 0
\(885\) 4.29829 0.144485
\(886\) 0 0
\(887\) −7.58940 −0.254827 −0.127414 0.991850i \(-0.540668\pi\)
−0.127414 + 0.991850i \(0.540668\pi\)
\(888\) 0 0
\(889\) −2.48725 −0.0834198
\(890\) 0 0
\(891\) −2.91185 −0.0975508
\(892\) 0 0
\(893\) −0.567040 −0.0189753
\(894\) 0 0
\(895\) 19.3093 0.645439
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −18.1927 −0.606760
\(900\) 0 0
\(901\) 36.7797 1.22531
\(902\) 0 0
\(903\) −6.15883 −0.204953
\(904\) 0 0
\(905\) 3.81115 0.126687
\(906\) 0 0
\(907\) −19.3333 −0.641952 −0.320976 0.947087i \(-0.604011\pi\)
−0.320976 + 0.947087i \(0.604011\pi\)
\(908\) 0 0
\(909\) 7.32304 0.242890
\(910\) 0 0
\(911\) −22.7149 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(912\) 0 0
\(913\) 7.69633 0.254711
\(914\) 0 0
\(915\) 3.59073 0.118706
\(916\) 0 0
\(917\) −5.11529 −0.168922
\(918\) 0 0
\(919\) −14.2911 −0.471420 −0.235710 0.971823i \(-0.575742\pi\)
−0.235710 + 0.971823i \(0.575742\pi\)
\(920\) 0 0
\(921\) 5.19806 0.171282
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −22.4355 −0.737674
\(926\) 0 0
\(927\) 4.21983 0.138597
\(928\) 0 0
\(929\) 32.7211 1.07354 0.536772 0.843727i \(-0.319644\pi\)
0.536772 + 0.843727i \(0.319644\pi\)
\(930\) 0 0
\(931\) −5.03923 −0.165154
\(932\) 0 0
\(933\) 22.5429 0.738021
\(934\) 0 0
\(935\) −14.8159 −0.484533
\(936\) 0 0
\(937\) −4.01400 −0.131132 −0.0655658 0.997848i \(-0.520885\pi\)
−0.0655658 + 0.997848i \(0.520885\pi\)
\(938\) 0 0
\(939\) 22.6612 0.739519
\(940\) 0 0
\(941\) −28.2669 −0.921476 −0.460738 0.887536i \(-0.652415\pi\)
−0.460738 + 0.887536i \(0.652415\pi\)
\(942\) 0 0
\(943\) −28.3056 −0.921757
\(944\) 0 0
\(945\) 0.582105 0.0189359
\(946\) 0 0
\(947\) 37.8455 1.22981 0.614906 0.788600i \(-0.289194\pi\)
0.614906 + 0.788600i \(0.289194\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 26.3424 0.854212
\(952\) 0 0
\(953\) −40.8256 −1.32247 −0.661236 0.750178i \(-0.729968\pi\)
−0.661236 + 0.750178i \(0.729968\pi\)
\(954\) 0 0
\(955\) −22.2142 −0.718834
\(956\) 0 0
\(957\) 5.56704 0.179957
\(958\) 0 0
\(959\) 4.14542 0.133863
\(960\) 0 0
\(961\) 59.5491 1.92094
\(962\) 0 0
\(963\) −6.39373 −0.206035
\(964\) 0 0
\(965\) 18.4727 0.594656
\(966\) 0 0
\(967\) −10.2798 −0.330575 −0.165288 0.986245i \(-0.552855\pi\)
−0.165288 + 0.986245i \(0.552855\pi\)
\(968\) 0 0
\(969\) −3.65279 −0.117345
\(970\) 0 0
\(971\) 19.4805 0.625161 0.312580 0.949891i \(-0.398807\pi\)
0.312580 + 0.949891i \(0.398807\pi\)
\(972\) 0 0
\(973\) −9.98792 −0.320198
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 47.0538 1.50539 0.752693 0.658372i \(-0.228755\pi\)
0.752693 + 0.658372i \(0.228755\pi\)
\(978\) 0 0
\(979\) −28.9057 −0.923831
\(980\) 0 0
\(981\) −3.46011 −0.110473
\(982\) 0 0
\(983\) −34.4295 −1.09813 −0.549065 0.835779i \(-0.685016\pi\)
−0.549065 + 0.835779i \(0.685016\pi\)
\(984\) 0 0
\(985\) 4.89056 0.155826
\(986\) 0 0
\(987\) 0.417895 0.0133017
\(988\) 0 0
\(989\) −63.9536 −2.03361
\(990\) 0 0
\(991\) −7.30798 −0.232146 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(992\) 0 0
\(993\) 11.2295 0.356358
\(994\) 0 0
\(995\) 15.7724 0.500019
\(996\) 0 0
\(997\) −35.6915 −1.13036 −0.565181 0.824967i \(-0.691194\pi\)
−0.565181 + 0.824967i \(0.691194\pi\)
\(998\) 0 0
\(999\) 5.75302 0.182018
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 8112.2.a.cp.1.1 3
4.3 odd 2 507.2.a.l.1.3 yes 3
12.11 even 2 1521.2.a.n.1.1 3
13.12 even 2 8112.2.a.cg.1.3 3
52.3 odd 6 507.2.e.i.22.1 6
52.7 even 12 507.2.j.i.361.6 12
52.11 even 12 507.2.j.i.316.1 12
52.15 even 12 507.2.j.i.316.6 12
52.19 even 12 507.2.j.i.361.1 12
52.23 odd 6 507.2.e.l.22.3 6
52.31 even 4 507.2.b.f.337.1 6
52.35 odd 6 507.2.e.i.484.1 6
52.43 odd 6 507.2.e.l.484.3 6
52.47 even 4 507.2.b.f.337.6 6
52.51 odd 2 507.2.a.i.1.1 3
156.47 odd 4 1521.2.b.k.1351.1 6
156.83 odd 4 1521.2.b.k.1351.6 6
156.155 even 2 1521.2.a.s.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.1 3 52.51 odd 2
507.2.a.l.1.3 yes 3 4.3 odd 2
507.2.b.f.337.1 6 52.31 even 4
507.2.b.f.337.6 6 52.47 even 4
507.2.e.i.22.1 6 52.3 odd 6
507.2.e.i.484.1 6 52.35 odd 6
507.2.e.l.22.3 6 52.23 odd 6
507.2.e.l.484.3 6 52.43 odd 6
507.2.j.i.316.1 12 52.11 even 12
507.2.j.i.316.6 12 52.15 even 12
507.2.j.i.361.1 12 52.19 even 12
507.2.j.i.361.6 12 52.7 even 12
1521.2.a.n.1.1 3 12.11 even 2
1521.2.a.s.1.3 3 156.155 even 2
1521.2.b.k.1351.1 6 156.47 odd 4
1521.2.b.k.1351.6 6 156.83 odd 4
8112.2.a.cg.1.3 3 13.12 even 2
8112.2.a.cp.1.1 3 1.1 even 1 trivial