Properties

Label 1932.2.a.f.1.2
Level $1932$
Weight $2$
Character 1932.1
Self dual yes
Analytic conductor $15.427$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1932.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.4270976705\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1932.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.38197 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.38197 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -6.09017 q^{13} -1.38197 q^{15} -5.00000 q^{17} +5.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.09017 q^{25} +1.00000 q^{27} -1.76393 q^{29} +4.70820 q^{31} -1.00000 q^{33} -1.38197 q^{35} -7.47214 q^{37} -6.09017 q^{39} -10.2361 q^{41} -9.32624 q^{43} -1.38197 q^{45} -9.70820 q^{47} +1.00000 q^{49} -5.00000 q^{51} +2.61803 q^{53} +1.38197 q^{55} +5.47214 q^{57} -1.38197 q^{59} +12.5623 q^{61} +1.00000 q^{63} +8.41641 q^{65} +8.32624 q^{67} +1.00000 q^{69} +11.3262 q^{71} -9.47214 q^{73} -3.09017 q^{75} -1.00000 q^{77} +2.70820 q^{79} +1.00000 q^{81} -13.1803 q^{83} +6.90983 q^{85} -1.76393 q^{87} -14.3820 q^{89} -6.09017 q^{91} +4.70820 q^{93} -7.56231 q^{95} -7.76393 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - q^{13} - 5 q^{15} - 10 q^{17} + 2 q^{19} + 2 q^{21} + 2 q^{23} + 5 q^{25} + 2 q^{27} - 8 q^{29} - 4 q^{31} - 2 q^{33} - 5 q^{35} - 6 q^{37} - q^{39} - 16 q^{41} - 3 q^{43} - 5 q^{45} - 6 q^{47} + 2 q^{49} - 10 q^{51} + 3 q^{53} + 5 q^{55} + 2 q^{57} - 5 q^{59} + 5 q^{61} + 2 q^{63} - 10 q^{65} + q^{67} + 2 q^{69} + 7 q^{71} - 10 q^{73} + 5 q^{75} - 2 q^{77} - 8 q^{79} + 2 q^{81} - 4 q^{83} + 25 q^{85} - 8 q^{87} - 31 q^{89} - q^{91} - 4 q^{93} + 5 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(5\) −1.38197 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −6.09017 −1.68911 −0.844555 0.535469i \(-0.820135\pi\)
−0.844555 + 0.535469i \(0.820135\pi\)
\(14\) 0 0
\(15\) −1.38197 −0.356822
\(16\) 0 0
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 5.47214 1.25539 0.627697 0.778458i \(-0.283998\pi\)
0.627697 + 0.778458i \(0.283998\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.09017 −0.618034
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.76393 −0.327554 −0.163777 0.986497i \(-0.552368\pi\)
−0.163777 + 0.986497i \(0.552368\pi\)
\(30\) 0 0
\(31\) 4.70820 0.845618 0.422809 0.906219i \(-0.361044\pi\)
0.422809 + 0.906219i \(0.361044\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −1.38197 −0.233595
\(36\) 0 0
\(37\) −7.47214 −1.22841 −0.614206 0.789146i \(-0.710524\pi\)
−0.614206 + 0.789146i \(0.710524\pi\)
\(38\) 0 0
\(39\) −6.09017 −0.975208
\(40\) 0 0
\(41\) −10.2361 −1.59861 −0.799303 0.600929i \(-0.794798\pi\)
−0.799303 + 0.600929i \(0.794798\pi\)
\(42\) 0 0
\(43\) −9.32624 −1.42224 −0.711119 0.703072i \(-0.751811\pi\)
−0.711119 + 0.703072i \(0.751811\pi\)
\(44\) 0 0
\(45\) −1.38197 −0.206011
\(46\) 0 0
\(47\) −9.70820 −1.41609 −0.708044 0.706169i \(-0.750422\pi\)
−0.708044 + 0.706169i \(0.750422\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) 2.61803 0.359615 0.179807 0.983702i \(-0.442453\pi\)
0.179807 + 0.983702i \(0.442453\pi\)
\(54\) 0 0
\(55\) 1.38197 0.186344
\(56\) 0 0
\(57\) 5.47214 0.724802
\(58\) 0 0
\(59\) −1.38197 −0.179917 −0.0899583 0.995946i \(-0.528673\pi\)
−0.0899583 + 0.995946i \(0.528673\pi\)
\(60\) 0 0
\(61\) 12.5623 1.60844 0.804219 0.594333i \(-0.202584\pi\)
0.804219 + 0.594333i \(0.202584\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 8.41641 1.04393
\(66\) 0 0
\(67\) 8.32624 1.01721 0.508606 0.860999i \(-0.330161\pi\)
0.508606 + 0.860999i \(0.330161\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 11.3262 1.34418 0.672089 0.740471i \(-0.265397\pi\)
0.672089 + 0.740471i \(0.265397\pi\)
\(72\) 0 0
\(73\) −9.47214 −1.10863 −0.554315 0.832307i \(-0.687020\pi\)
−0.554315 + 0.832307i \(0.687020\pi\)
\(74\) 0 0
\(75\) −3.09017 −0.356822
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 2.70820 0.304697 0.152348 0.988327i \(-0.451316\pi\)
0.152348 + 0.988327i \(0.451316\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.1803 −1.44673 −0.723365 0.690466i \(-0.757406\pi\)
−0.723365 + 0.690466i \(0.757406\pi\)
\(84\) 0 0
\(85\) 6.90983 0.749476
\(86\) 0 0
\(87\) −1.76393 −0.189113
\(88\) 0 0
\(89\) −14.3820 −1.52449 −0.762243 0.647291i \(-0.775902\pi\)
−0.762243 + 0.647291i \(0.775902\pi\)
\(90\) 0 0
\(91\) −6.09017 −0.638423
\(92\) 0 0
\(93\) 4.70820 0.488218
\(94\) 0 0
\(95\) −7.56231 −0.775876
\(96\) 0 0
\(97\) −7.76393 −0.788308 −0.394154 0.919044i \(-0.628962\pi\)
−0.394154 + 0.919044i \(0.628962\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −3.38197 −0.336518 −0.168259 0.985743i \(-0.553815\pi\)
−0.168259 + 0.985743i \(0.553815\pi\)
\(102\) 0 0
\(103\) −6.47214 −0.637719 −0.318859 0.947802i \(-0.603300\pi\)
−0.318859 + 0.947802i \(0.603300\pi\)
\(104\) 0 0
\(105\) −1.38197 −0.134866
\(106\) 0 0
\(107\) −12.3820 −1.19701 −0.598505 0.801119i \(-0.704238\pi\)
−0.598505 + 0.801119i \(0.704238\pi\)
\(108\) 0 0
\(109\) 20.5066 1.96417 0.982087 0.188428i \(-0.0603393\pi\)
0.982087 + 0.188428i \(0.0603393\pi\)
\(110\) 0 0
\(111\) −7.47214 −0.709224
\(112\) 0 0
\(113\) 9.85410 0.926996 0.463498 0.886098i \(-0.346594\pi\)
0.463498 + 0.886098i \(0.346594\pi\)
\(114\) 0 0
\(115\) −1.38197 −0.128869
\(116\) 0 0
\(117\) −6.09017 −0.563036
\(118\) 0 0
\(119\) −5.00000 −0.458349
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −10.2361 −0.922955
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 2.38197 0.211365 0.105683 0.994400i \(-0.466297\pi\)
0.105683 + 0.994400i \(0.466297\pi\)
\(128\) 0 0
\(129\) −9.32624 −0.821129
\(130\) 0 0
\(131\) 13.1803 1.15157 0.575786 0.817601i \(-0.304696\pi\)
0.575786 + 0.817601i \(0.304696\pi\)
\(132\) 0 0
\(133\) 5.47214 0.474494
\(134\) 0 0
\(135\) −1.38197 −0.118941
\(136\) 0 0
\(137\) −13.4721 −1.15100 −0.575501 0.817801i \(-0.695193\pi\)
−0.575501 + 0.817801i \(0.695193\pi\)
\(138\) 0 0
\(139\) 16.3262 1.38477 0.692387 0.721527i \(-0.256559\pi\)
0.692387 + 0.721527i \(0.256559\pi\)
\(140\) 0 0
\(141\) −9.70820 −0.817578
\(142\) 0 0
\(143\) 6.09017 0.509286
\(144\) 0 0
\(145\) 2.43769 0.202439
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −21.1246 −1.73060 −0.865298 0.501258i \(-0.832871\pi\)
−0.865298 + 0.501258i \(0.832871\pi\)
\(150\) 0 0
\(151\) 0.763932 0.0621679 0.0310840 0.999517i \(-0.490104\pi\)
0.0310840 + 0.999517i \(0.490104\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) −6.50658 −0.522621
\(156\) 0 0
\(157\) −16.6525 −1.32901 −0.664506 0.747283i \(-0.731358\pi\)
−0.664506 + 0.747283i \(0.731358\pi\)
\(158\) 0 0
\(159\) 2.61803 0.207624
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −2.90983 −0.227915 −0.113958 0.993486i \(-0.536353\pi\)
−0.113958 + 0.993486i \(0.536353\pi\)
\(164\) 0 0
\(165\) 1.38197 0.107586
\(166\) 0 0
\(167\) 19.1803 1.48422 0.742110 0.670279i \(-0.233825\pi\)
0.742110 + 0.670279i \(0.233825\pi\)
\(168\) 0 0
\(169\) 24.0902 1.85309
\(170\) 0 0
\(171\) 5.47214 0.418465
\(172\) 0 0
\(173\) −13.1803 −1.00208 −0.501041 0.865423i \(-0.667050\pi\)
−0.501041 + 0.865423i \(0.667050\pi\)
\(174\) 0 0
\(175\) −3.09017 −0.233595
\(176\) 0 0
\(177\) −1.38197 −0.103875
\(178\) 0 0
\(179\) −23.2705 −1.73932 −0.869660 0.493652i \(-0.835662\pi\)
−0.869660 + 0.493652i \(0.835662\pi\)
\(180\) 0 0
\(181\) 4.23607 0.314864 0.157432 0.987530i \(-0.449678\pi\)
0.157432 + 0.987530i \(0.449678\pi\)
\(182\) 0 0
\(183\) 12.5623 0.928632
\(184\) 0 0
\(185\) 10.3262 0.759200
\(186\) 0 0
\(187\) 5.00000 0.365636
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −14.6525 −1.06022 −0.530108 0.847930i \(-0.677849\pi\)
−0.530108 + 0.847930i \(0.677849\pi\)
\(192\) 0 0
\(193\) 8.29180 0.596857 0.298428 0.954432i \(-0.403538\pi\)
0.298428 + 0.954432i \(0.403538\pi\)
\(194\) 0 0
\(195\) 8.41641 0.602711
\(196\) 0 0
\(197\) 20.3820 1.45215 0.726077 0.687613i \(-0.241341\pi\)
0.726077 + 0.687613i \(0.241341\pi\)
\(198\) 0 0
\(199\) −20.7984 −1.47436 −0.737179 0.675698i \(-0.763842\pi\)
−0.737179 + 0.675698i \(0.763842\pi\)
\(200\) 0 0
\(201\) 8.32624 0.587288
\(202\) 0 0
\(203\) −1.76393 −0.123804
\(204\) 0 0
\(205\) 14.1459 0.987992
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −5.47214 −0.378516
\(210\) 0 0
\(211\) 13.7639 0.947548 0.473774 0.880646i \(-0.342891\pi\)
0.473774 + 0.880646i \(0.342891\pi\)
\(212\) 0 0
\(213\) 11.3262 0.776061
\(214\) 0 0
\(215\) 12.8885 0.878991
\(216\) 0 0
\(217\) 4.70820 0.319614
\(218\) 0 0
\(219\) −9.47214 −0.640068
\(220\) 0 0
\(221\) 30.4508 2.04835
\(222\) 0 0
\(223\) −9.56231 −0.640339 −0.320170 0.947360i \(-0.603740\pi\)
−0.320170 + 0.947360i \(0.603740\pi\)
\(224\) 0 0
\(225\) −3.09017 −0.206011
\(226\) 0 0
\(227\) −10.1459 −0.673407 −0.336703 0.941611i \(-0.609312\pi\)
−0.336703 + 0.941611i \(0.609312\pi\)
\(228\) 0 0
\(229\) 16.7984 1.11007 0.555034 0.831828i \(-0.312705\pi\)
0.555034 + 0.831828i \(0.312705\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) −1.56231 −0.102350 −0.0511750 0.998690i \(-0.516297\pi\)
−0.0511750 + 0.998690i \(0.516297\pi\)
\(234\) 0 0
\(235\) 13.4164 0.875190
\(236\) 0 0
\(237\) 2.70820 0.175917
\(238\) 0 0
\(239\) 28.3262 1.83227 0.916136 0.400868i \(-0.131291\pi\)
0.916136 + 0.400868i \(0.131291\pi\)
\(240\) 0 0
\(241\) 5.65248 0.364108 0.182054 0.983289i \(-0.441725\pi\)
0.182054 + 0.983289i \(0.441725\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.38197 −0.0882906
\(246\) 0 0
\(247\) −33.3262 −2.12050
\(248\) 0 0
\(249\) −13.1803 −0.835270
\(250\) 0 0
\(251\) 26.1803 1.65249 0.826244 0.563312i \(-0.190473\pi\)
0.826244 + 0.563312i \(0.190473\pi\)
\(252\) 0 0
\(253\) −1.00000 −0.0628695
\(254\) 0 0
\(255\) 6.90983 0.432710
\(256\) 0 0
\(257\) −13.7082 −0.855094 −0.427547 0.903993i \(-0.640622\pi\)
−0.427547 + 0.903993i \(0.640622\pi\)
\(258\) 0 0
\(259\) −7.47214 −0.464296
\(260\) 0 0
\(261\) −1.76393 −0.109185
\(262\) 0 0
\(263\) −4.81966 −0.297193 −0.148596 0.988898i \(-0.547476\pi\)
−0.148596 + 0.988898i \(0.547476\pi\)
\(264\) 0 0
\(265\) −3.61803 −0.222254
\(266\) 0 0
\(267\) −14.3820 −0.880162
\(268\) 0 0
\(269\) 13.1459 0.801520 0.400760 0.916183i \(-0.368746\pi\)
0.400760 + 0.916183i \(0.368746\pi\)
\(270\) 0 0
\(271\) −15.4721 −0.939865 −0.469933 0.882702i \(-0.655722\pi\)
−0.469933 + 0.882702i \(0.655722\pi\)
\(272\) 0 0
\(273\) −6.09017 −0.368594
\(274\) 0 0
\(275\) 3.09017 0.186344
\(276\) 0 0
\(277\) 19.9098 1.19627 0.598133 0.801397i \(-0.295909\pi\)
0.598133 + 0.801397i \(0.295909\pi\)
\(278\) 0 0
\(279\) 4.70820 0.281873
\(280\) 0 0
\(281\) −14.1803 −0.845928 −0.422964 0.906146i \(-0.639010\pi\)
−0.422964 + 0.906146i \(0.639010\pi\)
\(282\) 0 0
\(283\) 32.0344 1.90425 0.952125 0.305709i \(-0.0988935\pi\)
0.952125 + 0.305709i \(0.0988935\pi\)
\(284\) 0 0
\(285\) −7.56231 −0.447952
\(286\) 0 0
\(287\) −10.2361 −0.604216
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −7.76393 −0.455130
\(292\) 0 0
\(293\) −6.47214 −0.378106 −0.189053 0.981967i \(-0.560542\pi\)
−0.189053 + 0.981967i \(0.560542\pi\)
\(294\) 0 0
\(295\) 1.90983 0.111195
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −6.09017 −0.352204
\(300\) 0 0
\(301\) −9.32624 −0.537555
\(302\) 0 0
\(303\) −3.38197 −0.194289
\(304\) 0 0
\(305\) −17.3607 −0.994070
\(306\) 0 0
\(307\) −29.0689 −1.65905 −0.829524 0.558470i \(-0.811388\pi\)
−0.829524 + 0.558470i \(0.811388\pi\)
\(308\) 0 0
\(309\) −6.47214 −0.368187
\(310\) 0 0
\(311\) −3.90983 −0.221706 −0.110853 0.993837i \(-0.535358\pi\)
−0.110853 + 0.993837i \(0.535358\pi\)
\(312\) 0 0
\(313\) −27.4164 −1.54967 −0.774833 0.632165i \(-0.782166\pi\)
−0.774833 + 0.632165i \(0.782166\pi\)
\(314\) 0 0
\(315\) −1.38197 −0.0778650
\(316\) 0 0
\(317\) 28.6180 1.60735 0.803674 0.595069i \(-0.202875\pi\)
0.803674 + 0.595069i \(0.202875\pi\)
\(318\) 0 0
\(319\) 1.76393 0.0987612
\(320\) 0 0
\(321\) −12.3820 −0.691094
\(322\) 0 0
\(323\) −27.3607 −1.52239
\(324\) 0 0
\(325\) 18.8197 1.04393
\(326\) 0 0
\(327\) 20.5066 1.13402
\(328\) 0 0
\(329\) −9.70820 −0.535231
\(330\) 0 0
\(331\) −6.58359 −0.361867 −0.180933 0.983495i \(-0.557912\pi\)
−0.180933 + 0.983495i \(0.557912\pi\)
\(332\) 0 0
\(333\) −7.47214 −0.409471
\(334\) 0 0
\(335\) −11.5066 −0.628672
\(336\) 0 0
\(337\) 33.9787 1.85094 0.925469 0.378823i \(-0.123671\pi\)
0.925469 + 0.378823i \(0.123671\pi\)
\(338\) 0 0
\(339\) 9.85410 0.535201
\(340\) 0 0
\(341\) −4.70820 −0.254964
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −1.38197 −0.0744025
\(346\) 0 0
\(347\) 28.3050 1.51949 0.759745 0.650221i \(-0.225324\pi\)
0.759745 + 0.650221i \(0.225324\pi\)
\(348\) 0 0
\(349\) 32.0344 1.71476 0.857382 0.514680i \(-0.172089\pi\)
0.857382 + 0.514680i \(0.172089\pi\)
\(350\) 0 0
\(351\) −6.09017 −0.325069
\(352\) 0 0
\(353\) 15.6525 0.833097 0.416549 0.909113i \(-0.363240\pi\)
0.416549 + 0.909113i \(0.363240\pi\)
\(354\) 0 0
\(355\) −15.6525 −0.830747
\(356\) 0 0
\(357\) −5.00000 −0.264628
\(358\) 0 0
\(359\) 3.56231 0.188011 0.0940057 0.995572i \(-0.470033\pi\)
0.0940057 + 0.995572i \(0.470033\pi\)
\(360\) 0 0
\(361\) 10.9443 0.576014
\(362\) 0 0
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 13.0902 0.685171
\(366\) 0 0
\(367\) 6.27051 0.327318 0.163659 0.986517i \(-0.447670\pi\)
0.163659 + 0.986517i \(0.447670\pi\)
\(368\) 0 0
\(369\) −10.2361 −0.532868
\(370\) 0 0
\(371\) 2.61803 0.135922
\(372\) 0 0
\(373\) −2.88854 −0.149563 −0.0747816 0.997200i \(-0.523826\pi\)
−0.0747816 + 0.997200i \(0.523826\pi\)
\(374\) 0 0
\(375\) 11.1803 0.577350
\(376\) 0 0
\(377\) 10.7426 0.553274
\(378\) 0 0
\(379\) 17.8885 0.918873 0.459436 0.888211i \(-0.348051\pi\)
0.459436 + 0.888211i \(0.348051\pi\)
\(380\) 0 0
\(381\) 2.38197 0.122032
\(382\) 0 0
\(383\) −11.7639 −0.601109 −0.300554 0.953765i \(-0.597172\pi\)
−0.300554 + 0.953765i \(0.597172\pi\)
\(384\) 0 0
\(385\) 1.38197 0.0704315
\(386\) 0 0
\(387\) −9.32624 −0.474079
\(388\) 0 0
\(389\) −20.1246 −1.02036 −0.510179 0.860068i \(-0.670421\pi\)
−0.510179 + 0.860068i \(0.670421\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) 0 0
\(393\) 13.1803 0.664860
\(394\) 0 0
\(395\) −3.74265 −0.188313
\(396\) 0 0
\(397\) 20.1803 1.01282 0.506411 0.862292i \(-0.330972\pi\)
0.506411 + 0.862292i \(0.330972\pi\)
\(398\) 0 0
\(399\) 5.47214 0.273949
\(400\) 0 0
\(401\) 12.7082 0.634617 0.317309 0.948322i \(-0.397221\pi\)
0.317309 + 0.948322i \(0.397221\pi\)
\(402\) 0 0
\(403\) −28.6738 −1.42834
\(404\) 0 0
\(405\) −1.38197 −0.0686704
\(406\) 0 0
\(407\) 7.47214 0.370380
\(408\) 0 0
\(409\) 2.81966 0.139423 0.0697116 0.997567i \(-0.477792\pi\)
0.0697116 + 0.997567i \(0.477792\pi\)
\(410\) 0 0
\(411\) −13.4721 −0.664531
\(412\) 0 0
\(413\) −1.38197 −0.0680021
\(414\) 0 0
\(415\) 18.2148 0.894128
\(416\) 0 0
\(417\) 16.3262 0.799499
\(418\) 0 0
\(419\) −9.85410 −0.481404 −0.240702 0.970599i \(-0.577378\pi\)
−0.240702 + 0.970599i \(0.577378\pi\)
\(420\) 0 0
\(421\) 8.96556 0.436955 0.218477 0.975842i \(-0.429891\pi\)
0.218477 + 0.975842i \(0.429891\pi\)
\(422\) 0 0
\(423\) −9.70820 −0.472029
\(424\) 0 0
\(425\) 15.4508 0.749476
\(426\) 0 0
\(427\) 12.5623 0.607933
\(428\) 0 0
\(429\) 6.09017 0.294036
\(430\) 0 0
\(431\) −11.0344 −0.531510 −0.265755 0.964041i \(-0.585621\pi\)
−0.265755 + 0.964041i \(0.585621\pi\)
\(432\) 0 0
\(433\) 7.70820 0.370433 0.185216 0.982698i \(-0.440701\pi\)
0.185216 + 0.982698i \(0.440701\pi\)
\(434\) 0 0
\(435\) 2.43769 0.116878
\(436\) 0 0
\(437\) 5.47214 0.261768
\(438\) 0 0
\(439\) −15.2918 −0.729838 −0.364919 0.931039i \(-0.618903\pi\)
−0.364919 + 0.931039i \(0.618903\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 8.94427 0.424955 0.212478 0.977166i \(-0.431847\pi\)
0.212478 + 0.977166i \(0.431847\pi\)
\(444\) 0 0
\(445\) 19.8754 0.942184
\(446\) 0 0
\(447\) −21.1246 −0.999160
\(448\) 0 0
\(449\) −25.8541 −1.22013 −0.610065 0.792351i \(-0.708857\pi\)
−0.610065 + 0.792351i \(0.708857\pi\)
\(450\) 0 0
\(451\) 10.2361 0.481998
\(452\) 0 0
\(453\) 0.763932 0.0358927
\(454\) 0 0
\(455\) 8.41641 0.394567
\(456\) 0 0
\(457\) 27.2705 1.27566 0.637830 0.770177i \(-0.279832\pi\)
0.637830 + 0.770177i \(0.279832\pi\)
\(458\) 0 0
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) −37.2705 −1.73586 −0.867930 0.496686i \(-0.834550\pi\)
−0.867930 + 0.496686i \(0.834550\pi\)
\(462\) 0 0
\(463\) 31.4721 1.46263 0.731317 0.682038i \(-0.238906\pi\)
0.731317 + 0.682038i \(0.238906\pi\)
\(464\) 0 0
\(465\) −6.50658 −0.301735
\(466\) 0 0
\(467\) −14.5967 −0.675457 −0.337728 0.941244i \(-0.609659\pi\)
−0.337728 + 0.941244i \(0.609659\pi\)
\(468\) 0 0
\(469\) 8.32624 0.384470
\(470\) 0 0
\(471\) −16.6525 −0.767306
\(472\) 0 0
\(473\) 9.32624 0.428821
\(474\) 0 0
\(475\) −16.9098 −0.775876
\(476\) 0 0
\(477\) 2.61803 0.119872
\(478\) 0 0
\(479\) 10.8885 0.497510 0.248755 0.968566i \(-0.419979\pi\)
0.248755 + 0.968566i \(0.419979\pi\)
\(480\) 0 0
\(481\) 45.5066 2.07492
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 10.7295 0.487201
\(486\) 0 0
\(487\) −28.7771 −1.30401 −0.652007 0.758213i \(-0.726073\pi\)
−0.652007 + 0.758213i \(0.726073\pi\)
\(488\) 0 0
\(489\) −2.90983 −0.131587
\(490\) 0 0
\(491\) −6.09017 −0.274846 −0.137423 0.990512i \(-0.543882\pi\)
−0.137423 + 0.990512i \(0.543882\pi\)
\(492\) 0 0
\(493\) 8.81966 0.397218
\(494\) 0 0
\(495\) 1.38197 0.0621148
\(496\) 0 0
\(497\) 11.3262 0.508051
\(498\) 0 0
\(499\) 8.03444 0.359671 0.179836 0.983697i \(-0.442443\pi\)
0.179836 + 0.983697i \(0.442443\pi\)
\(500\) 0 0
\(501\) 19.1803 0.856914
\(502\) 0 0
\(503\) −3.67376 −0.163805 −0.0819025 0.996640i \(-0.526100\pi\)
−0.0819025 + 0.996640i \(0.526100\pi\)
\(504\) 0 0
\(505\) 4.67376 0.207980
\(506\) 0 0
\(507\) 24.0902 1.06988
\(508\) 0 0
\(509\) 11.2361 0.498030 0.249015 0.968500i \(-0.419893\pi\)
0.249015 + 0.968500i \(0.419893\pi\)
\(510\) 0 0
\(511\) −9.47214 −0.419023
\(512\) 0 0
\(513\) 5.47214 0.241601
\(514\) 0 0
\(515\) 8.94427 0.394132
\(516\) 0 0
\(517\) 9.70820 0.426966
\(518\) 0 0
\(519\) −13.1803 −0.578553
\(520\) 0 0
\(521\) −31.3050 −1.37149 −0.685747 0.727840i \(-0.740525\pi\)
−0.685747 + 0.727840i \(0.740525\pi\)
\(522\) 0 0
\(523\) 0.472136 0.0206451 0.0103225 0.999947i \(-0.496714\pi\)
0.0103225 + 0.999947i \(0.496714\pi\)
\(524\) 0 0
\(525\) −3.09017 −0.134866
\(526\) 0 0
\(527\) −23.5410 −1.02546
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −1.38197 −0.0599722
\(532\) 0 0
\(533\) 62.3394 2.70022
\(534\) 0 0
\(535\) 17.1115 0.739793
\(536\) 0 0
\(537\) −23.2705 −1.00420
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −0.763932 −0.0328440 −0.0164220 0.999865i \(-0.505228\pi\)
−0.0164220 + 0.999865i \(0.505228\pi\)
\(542\) 0 0
\(543\) 4.23607 0.181787
\(544\) 0 0
\(545\) −28.3394 −1.21393
\(546\) 0 0
\(547\) 23.4508 1.00269 0.501343 0.865249i \(-0.332840\pi\)
0.501343 + 0.865249i \(0.332840\pi\)
\(548\) 0 0
\(549\) 12.5623 0.536146
\(550\) 0 0
\(551\) −9.65248 −0.411209
\(552\) 0 0
\(553\) 2.70820 0.115165
\(554\) 0 0
\(555\) 10.3262 0.438324
\(556\) 0 0
\(557\) −1.81966 −0.0771015 −0.0385507 0.999257i \(-0.512274\pi\)
−0.0385507 + 0.999257i \(0.512274\pi\)
\(558\) 0 0
\(559\) 56.7984 2.40232
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) 0 0
\(563\) 17.4377 0.734911 0.367456 0.930041i \(-0.380229\pi\)
0.367456 + 0.930041i \(0.380229\pi\)
\(564\) 0 0
\(565\) −13.6180 −0.572915
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.11146 −0.256206 −0.128103 0.991761i \(-0.540889\pi\)
−0.128103 + 0.991761i \(0.540889\pi\)
\(570\) 0 0
\(571\) −1.18034 −0.0493957 −0.0246978 0.999695i \(-0.507862\pi\)
−0.0246978 + 0.999695i \(0.507862\pi\)
\(572\) 0 0
\(573\) −14.6525 −0.612116
\(574\) 0 0
\(575\) −3.09017 −0.128869
\(576\) 0 0
\(577\) 21.4164 0.891577 0.445788 0.895138i \(-0.352923\pi\)
0.445788 + 0.895138i \(0.352923\pi\)
\(578\) 0 0
\(579\) 8.29180 0.344595
\(580\) 0 0
\(581\) −13.1803 −0.546813
\(582\) 0 0
\(583\) −2.61803 −0.108428
\(584\) 0 0
\(585\) 8.41641 0.347976
\(586\) 0 0
\(587\) −40.3820 −1.66674 −0.833371 0.552714i \(-0.813592\pi\)
−0.833371 + 0.552714i \(0.813592\pi\)
\(588\) 0 0
\(589\) 25.7639 1.06158
\(590\) 0 0
\(591\) 20.3820 0.838402
\(592\) 0 0
\(593\) 41.0132 1.68421 0.842104 0.539315i \(-0.181317\pi\)
0.842104 + 0.539315i \(0.181317\pi\)
\(594\) 0 0
\(595\) 6.90983 0.283275
\(596\) 0 0
\(597\) −20.7984 −0.851221
\(598\) 0 0
\(599\) 19.7984 0.808940 0.404470 0.914551i \(-0.367456\pi\)
0.404470 + 0.914551i \(0.367456\pi\)
\(600\) 0 0
\(601\) −41.0902 −1.67610 −0.838051 0.545591i \(-0.816305\pi\)
−0.838051 + 0.545591i \(0.816305\pi\)
\(602\) 0 0
\(603\) 8.32624 0.339071
\(604\) 0 0
\(605\) 13.8197 0.561849
\(606\) 0 0
\(607\) −46.3394 −1.88086 −0.940429 0.339990i \(-0.889576\pi\)
−0.940429 + 0.339990i \(0.889576\pi\)
\(608\) 0 0
\(609\) −1.76393 −0.0714781
\(610\) 0 0
\(611\) 59.1246 2.39193
\(612\) 0 0
\(613\) −21.0689 −0.850964 −0.425482 0.904967i \(-0.639895\pi\)
−0.425482 + 0.904967i \(0.639895\pi\)
\(614\) 0 0
\(615\) 14.1459 0.570418
\(616\) 0 0
\(617\) −0.965558 −0.0388719 −0.0194360 0.999811i \(-0.506187\pi\)
−0.0194360 + 0.999811i \(0.506187\pi\)
\(618\) 0 0
\(619\) −44.9230 −1.80561 −0.902804 0.430053i \(-0.858495\pi\)
−0.902804 + 0.430053i \(0.858495\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −14.3820 −0.576201
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.47214 −0.218536
\(628\) 0 0
\(629\) 37.3607 1.48967
\(630\) 0 0
\(631\) 9.47214 0.377080 0.188540 0.982066i \(-0.439625\pi\)
0.188540 + 0.982066i \(0.439625\pi\)
\(632\) 0 0
\(633\) 13.7639 0.547067
\(634\) 0 0
\(635\) −3.29180 −0.130631
\(636\) 0 0
\(637\) −6.09017 −0.241301
\(638\) 0 0
\(639\) 11.3262 0.448059
\(640\) 0 0
\(641\) −14.3820 −0.568054 −0.284027 0.958816i \(-0.591670\pi\)
−0.284027 + 0.958816i \(0.591670\pi\)
\(642\) 0 0
\(643\) −22.0902 −0.871151 −0.435576 0.900152i \(-0.643455\pi\)
−0.435576 + 0.900152i \(0.643455\pi\)
\(644\) 0 0
\(645\) 12.8885 0.507486
\(646\) 0 0
\(647\) 27.0902 1.06502 0.532512 0.846422i \(-0.321248\pi\)
0.532512 + 0.846422i \(0.321248\pi\)
\(648\) 0 0
\(649\) 1.38197 0.0542469
\(650\) 0 0
\(651\) 4.70820 0.184529
\(652\) 0 0
\(653\) 14.4377 0.564991 0.282495 0.959269i \(-0.408838\pi\)
0.282495 + 0.959269i \(0.408838\pi\)
\(654\) 0 0
\(655\) −18.2148 −0.711710
\(656\) 0 0
\(657\) −9.47214 −0.369543
\(658\) 0 0
\(659\) −1.00000 −0.0389545 −0.0194772 0.999810i \(-0.506200\pi\)
−0.0194772 + 0.999810i \(0.506200\pi\)
\(660\) 0 0
\(661\) −25.2918 −0.983737 −0.491868 0.870670i \(-0.663686\pi\)
−0.491868 + 0.870670i \(0.663686\pi\)
\(662\) 0 0
\(663\) 30.4508 1.18261
\(664\) 0 0
\(665\) −7.56231 −0.293254
\(666\) 0 0
\(667\) −1.76393 −0.0682997
\(668\) 0 0
\(669\) −9.56231 −0.369700
\(670\) 0 0
\(671\) −12.5623 −0.484962
\(672\) 0 0
\(673\) 12.0557 0.464714 0.232357 0.972631i \(-0.425356\pi\)
0.232357 + 0.972631i \(0.425356\pi\)
\(674\) 0 0
\(675\) −3.09017 −0.118941
\(676\) 0 0
\(677\) −20.5623 −0.790274 −0.395137 0.918622i \(-0.629303\pi\)
−0.395137 + 0.918622i \(0.629303\pi\)
\(678\) 0 0
\(679\) −7.76393 −0.297952
\(680\) 0 0
\(681\) −10.1459 −0.388792
\(682\) 0 0
\(683\) 1.94427 0.0743955 0.0371977 0.999308i \(-0.488157\pi\)
0.0371977 + 0.999308i \(0.488157\pi\)
\(684\) 0 0
\(685\) 18.6180 0.711359
\(686\) 0 0
\(687\) 16.7984 0.640898
\(688\) 0 0
\(689\) −15.9443 −0.607428
\(690\) 0 0
\(691\) −37.2148 −1.41572 −0.707859 0.706354i \(-0.750339\pi\)
−0.707859 + 0.706354i \(0.750339\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −22.5623 −0.855837
\(696\) 0 0
\(697\) 51.1803 1.93859
\(698\) 0 0
\(699\) −1.56231 −0.0590918
\(700\) 0 0
\(701\) −37.0902 −1.40088 −0.700438 0.713713i \(-0.747012\pi\)
−0.700438 + 0.713713i \(0.747012\pi\)
\(702\) 0 0
\(703\) −40.8885 −1.54214
\(704\) 0 0
\(705\) 13.4164 0.505291
\(706\) 0 0
\(707\) −3.38197 −0.127192
\(708\) 0 0
\(709\) −42.1591 −1.58332 −0.791658 0.610964i \(-0.790782\pi\)
−0.791658 + 0.610964i \(0.790782\pi\)
\(710\) 0 0
\(711\) 2.70820 0.101566
\(712\) 0 0
\(713\) 4.70820 0.176324
\(714\) 0 0
\(715\) −8.41641 −0.314756
\(716\) 0 0
\(717\) 28.3262 1.05786
\(718\) 0 0
\(719\) 50.7771 1.89367 0.946833 0.321726i \(-0.104263\pi\)
0.946833 + 0.321726i \(0.104263\pi\)
\(720\) 0 0
\(721\) −6.47214 −0.241035
\(722\) 0 0
\(723\) 5.65248 0.210218
\(724\) 0 0
\(725\) 5.45085 0.202439
\(726\) 0 0
\(727\) 43.2492 1.60402 0.802012 0.597307i \(-0.203763\pi\)
0.802012 + 0.597307i \(0.203763\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 46.6312 1.72472
\(732\) 0 0
\(733\) 15.7082 0.580196 0.290098 0.956997i \(-0.406312\pi\)
0.290098 + 0.956997i \(0.406312\pi\)
\(734\) 0 0
\(735\) −1.38197 −0.0509746
\(736\) 0 0
\(737\) −8.32624 −0.306701
\(738\) 0 0
\(739\) −15.7639 −0.579886 −0.289943 0.957044i \(-0.593636\pi\)
−0.289943 + 0.957044i \(0.593636\pi\)
\(740\) 0 0
\(741\) −33.3262 −1.22427
\(742\) 0 0
\(743\) 0.145898 0.00535248 0.00267624 0.999996i \(-0.499148\pi\)
0.00267624 + 0.999996i \(0.499148\pi\)
\(744\) 0 0
\(745\) 29.1935 1.06957
\(746\) 0 0
\(747\) −13.1803 −0.482243
\(748\) 0 0
\(749\) −12.3820 −0.452427
\(750\) 0 0
\(751\) −38.3262 −1.39854 −0.699272 0.714856i \(-0.746492\pi\)
−0.699272 + 0.714856i \(0.746492\pi\)
\(752\) 0 0
\(753\) 26.1803 0.954065
\(754\) 0 0
\(755\) −1.05573 −0.0384219
\(756\) 0 0
\(757\) −10.4164 −0.378591 −0.189295 0.981920i \(-0.560620\pi\)
−0.189295 + 0.981920i \(0.560620\pi\)
\(758\) 0 0
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) −24.9443 −0.904229 −0.452115 0.891960i \(-0.649330\pi\)
−0.452115 + 0.891960i \(0.649330\pi\)
\(762\) 0 0
\(763\) 20.5066 0.742388
\(764\) 0 0
\(765\) 6.90983 0.249825
\(766\) 0 0
\(767\) 8.41641 0.303899
\(768\) 0 0
\(769\) −44.6525 −1.61021 −0.805105 0.593133i \(-0.797891\pi\)
−0.805105 + 0.593133i \(0.797891\pi\)
\(770\) 0 0
\(771\) −13.7082 −0.493689
\(772\) 0 0
\(773\) −14.7082 −0.529017 −0.264509 0.964383i \(-0.585210\pi\)
−0.264509 + 0.964383i \(0.585210\pi\)
\(774\) 0 0
\(775\) −14.5492 −0.522621
\(776\) 0 0
\(777\) −7.47214 −0.268061
\(778\) 0 0
\(779\) −56.0132 −2.00688
\(780\) 0 0
\(781\) −11.3262 −0.405285
\(782\) 0 0
\(783\) −1.76393 −0.0630378
\(784\) 0 0
\(785\) 23.0132 0.821375
\(786\) 0 0
\(787\) −17.7426 −0.632457 −0.316229 0.948683i \(-0.602417\pi\)
−0.316229 + 0.948683i \(0.602417\pi\)
\(788\) 0 0
\(789\) −4.81966 −0.171584
\(790\) 0 0
\(791\) 9.85410 0.350372
\(792\) 0 0
\(793\) −76.5066 −2.71683
\(794\) 0 0
\(795\) −3.61803 −0.128318
\(796\) 0 0
\(797\) −18.0689 −0.640033 −0.320016 0.947412i \(-0.603688\pi\)
−0.320016 + 0.947412i \(0.603688\pi\)
\(798\) 0 0
\(799\) 48.5410 1.71726
\(800\) 0 0
\(801\) −14.3820 −0.508162
\(802\) 0 0
\(803\) 9.47214 0.334264
\(804\) 0 0
\(805\) −1.38197 −0.0487079
\(806\) 0 0
\(807\) 13.1459 0.462758
\(808\) 0 0
\(809\) 49.6180 1.74448 0.872239 0.489081i \(-0.162668\pi\)
0.872239 + 0.489081i \(0.162668\pi\)
\(810\) 0 0
\(811\) −2.63932 −0.0926791 −0.0463395 0.998926i \(-0.514756\pi\)
−0.0463395 + 0.998926i \(0.514756\pi\)
\(812\) 0 0
\(813\) −15.4721 −0.542631
\(814\) 0 0
\(815\) 4.02129 0.140860
\(816\) 0 0
\(817\) −51.0344 −1.78547
\(818\) 0 0
\(819\) −6.09017 −0.212808
\(820\) 0 0
\(821\) 38.6525 1.34898 0.674490 0.738284i \(-0.264363\pi\)
0.674490 + 0.738284i \(0.264363\pi\)
\(822\) 0 0
\(823\) 10.7984 0.376408 0.188204 0.982130i \(-0.439733\pi\)
0.188204 + 0.982130i \(0.439733\pi\)
\(824\) 0 0
\(825\) 3.09017 0.107586
\(826\) 0 0
\(827\) 29.2016 1.01544 0.507720 0.861522i \(-0.330488\pi\)
0.507720 + 0.861522i \(0.330488\pi\)
\(828\) 0 0
\(829\) 0.236068 0.00819898 0.00409949 0.999992i \(-0.498695\pi\)
0.00409949 + 0.999992i \(0.498695\pi\)
\(830\) 0 0
\(831\) 19.9098 0.690664
\(832\) 0 0
\(833\) −5.00000 −0.173240
\(834\) 0 0
\(835\) −26.5066 −0.917298
\(836\) 0 0
\(837\) 4.70820 0.162739
\(838\) 0 0
\(839\) 26.2705 0.906959 0.453479 0.891267i \(-0.350183\pi\)
0.453479 + 0.891267i \(0.350183\pi\)
\(840\) 0 0
\(841\) −25.8885 −0.892708
\(842\) 0 0
\(843\) −14.1803 −0.488397
\(844\) 0 0
\(845\) −33.2918 −1.14527
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 32.0344 1.09942
\(850\) 0 0
\(851\) −7.47214 −0.256142
\(852\) 0 0
\(853\) 22.5410 0.771790 0.385895 0.922543i \(-0.373893\pi\)
0.385895 + 0.922543i \(0.373893\pi\)
\(854\) 0 0
\(855\) −7.56231 −0.258625
\(856\) 0 0
\(857\) −8.65248 −0.295563 −0.147781 0.989020i \(-0.547213\pi\)
−0.147781 + 0.989020i \(0.547213\pi\)
\(858\) 0 0
\(859\) −11.8885 −0.405632 −0.202816 0.979217i \(-0.565009\pi\)
−0.202816 + 0.979217i \(0.565009\pi\)
\(860\) 0 0
\(861\) −10.2361 −0.348844
\(862\) 0 0
\(863\) 31.4853 1.07177 0.535886 0.844290i \(-0.319978\pi\)
0.535886 + 0.844290i \(0.319978\pi\)
\(864\) 0 0
\(865\) 18.2148 0.619321
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −2.70820 −0.0918695
\(870\) 0 0
\(871\) −50.7082 −1.71818
\(872\) 0 0
\(873\) −7.76393 −0.262769
\(874\) 0 0
\(875\) 11.1803 0.377964
\(876\) 0 0
\(877\) 5.30495 0.179135 0.0895677 0.995981i \(-0.471451\pi\)
0.0895677 + 0.995981i \(0.471451\pi\)
\(878\) 0 0
\(879\) −6.47214 −0.218300
\(880\) 0 0
\(881\) −51.1246 −1.72243 −0.861216 0.508239i \(-0.830297\pi\)
−0.861216 + 0.508239i \(0.830297\pi\)
\(882\) 0 0
\(883\) 10.6738 0.359201 0.179600 0.983740i \(-0.442520\pi\)
0.179600 + 0.983740i \(0.442520\pi\)
\(884\) 0 0
\(885\) 1.90983 0.0641982
\(886\) 0 0
\(887\) −48.2705 −1.62077 −0.810383 0.585901i \(-0.800741\pi\)
−0.810383 + 0.585901i \(0.800741\pi\)
\(888\) 0 0
\(889\) 2.38197 0.0798886
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −53.1246 −1.77775
\(894\) 0 0
\(895\) 32.1591 1.07496
\(896\) 0 0
\(897\) −6.09017 −0.203345
\(898\) 0 0
\(899\) −8.30495 −0.276986
\(900\) 0 0
\(901\) −13.0902 −0.436097
\(902\) 0 0
\(903\) −9.32624 −0.310358
\(904\) 0 0
\(905\) −5.85410 −0.194597
\(906\) 0 0
\(907\) 15.9230 0.528714 0.264357 0.964425i \(-0.414840\pi\)
0.264357 + 0.964425i \(0.414840\pi\)
\(908\) 0 0
\(909\) −3.38197 −0.112173
\(910\) 0 0
\(911\) −24.5410 −0.813080 −0.406540 0.913633i \(-0.633265\pi\)
−0.406540 + 0.913633i \(0.633265\pi\)
\(912\) 0 0
\(913\) 13.1803 0.436206
\(914\) 0 0
\(915\) −17.3607 −0.573926
\(916\) 0 0
\(917\) 13.1803 0.435253
\(918\) 0 0
\(919\) 19.0000 0.626752 0.313376 0.949629i \(-0.398540\pi\)
0.313376 + 0.949629i \(0.398540\pi\)
\(920\) 0 0
\(921\) −29.0689 −0.957852
\(922\) 0 0
\(923\) −68.9787 −2.27046
\(924\) 0 0
\(925\) 23.0902 0.759200
\(926\) 0 0
\(927\) −6.47214 −0.212573
\(928\) 0 0
\(929\) 9.96556 0.326959 0.163480 0.986547i \(-0.447728\pi\)
0.163480 + 0.986547i \(0.447728\pi\)
\(930\) 0 0
\(931\) 5.47214 0.179342
\(932\) 0 0
\(933\) −3.90983 −0.128002
\(934\) 0 0
\(935\) −6.90983 −0.225976
\(936\) 0 0
\(937\) −13.0000 −0.424691 −0.212346 0.977195i \(-0.568110\pi\)
−0.212346 + 0.977195i \(0.568110\pi\)
\(938\) 0 0
\(939\) −27.4164 −0.894701
\(940\) 0 0
\(941\) 42.8328 1.39631 0.698155 0.715947i \(-0.254005\pi\)
0.698155 + 0.715947i \(0.254005\pi\)
\(942\) 0 0
\(943\) −10.2361 −0.333332
\(944\) 0 0
\(945\) −1.38197 −0.0449554
\(946\) 0 0
\(947\) −42.3607 −1.37654 −0.688269 0.725456i \(-0.741629\pi\)
−0.688269 + 0.725456i \(0.741629\pi\)
\(948\) 0 0
\(949\) 57.6869 1.87260
\(950\) 0 0
\(951\) 28.6180 0.928003
\(952\) 0 0
\(953\) −54.6312 −1.76968 −0.884839 0.465897i \(-0.845732\pi\)
−0.884839 + 0.465897i \(0.845732\pi\)
\(954\) 0 0
\(955\) 20.2492 0.655249
\(956\) 0 0
\(957\) 1.76393 0.0570198
\(958\) 0 0
\(959\) −13.4721 −0.435038
\(960\) 0 0
\(961\) −8.83282 −0.284930
\(962\) 0 0
\(963\) −12.3820 −0.399003
\(964\) 0 0
\(965\) −11.4590 −0.368878
\(966\) 0 0
\(967\) −29.0557 −0.934369 −0.467185 0.884160i \(-0.654732\pi\)
−0.467185 + 0.884160i \(0.654732\pi\)
\(968\) 0 0
\(969\) −27.3607 −0.878952
\(970\) 0 0
\(971\) −26.7426 −0.858212 −0.429106 0.903254i \(-0.641171\pi\)
−0.429106 + 0.903254i \(0.641171\pi\)
\(972\) 0 0
\(973\) 16.3262 0.523395
\(974\) 0 0
\(975\) 18.8197 0.602711
\(976\) 0 0
\(977\) 17.2574 0.552112 0.276056 0.961142i \(-0.410973\pi\)
0.276056 + 0.961142i \(0.410973\pi\)
\(978\) 0 0
\(979\) 14.3820 0.459650
\(980\) 0 0
\(981\) 20.5066 0.654725
\(982\) 0 0
\(983\) −4.70820 −0.150168 −0.0750842 0.997177i \(-0.523923\pi\)
−0.0750842 + 0.997177i \(0.523923\pi\)
\(984\) 0 0
\(985\) −28.1672 −0.897481
\(986\) 0 0
\(987\) −9.70820 −0.309016
\(988\) 0 0
\(989\) −9.32624 −0.296557
\(990\) 0 0
\(991\) −44.2705 −1.40630 −0.703150 0.711042i \(-0.748224\pi\)
−0.703150 + 0.711042i \(0.748224\pi\)
\(992\) 0 0
\(993\) −6.58359 −0.208924
\(994\) 0 0
\(995\) 28.7426 0.911203
\(996\) 0 0
\(997\) −22.2361 −0.704223 −0.352112 0.935958i \(-0.614536\pi\)
−0.352112 + 0.935958i \(0.614536\pi\)
\(998\) 0 0
\(999\) −7.47214 −0.236408
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 1932.2.a.f.1.2 2
3.2 odd 2 5796.2.a.n.1.1 2
4.3 odd 2 7728.2.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.f.1.2 2 1.1 even 1 trivial
5796.2.a.n.1.1 2 3.2 odd 2
7728.2.a.w.1.2 2 4.3 odd 2