Properties

Label 1932.2.a.d.1.1
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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1932.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.618034 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.618034 q^{5} -1.00000 q^{7} +1.00000 q^{9} +0.236068 q^{11} +0.381966 q^{13} +0.618034 q^{15} -2.23607 q^{17} +3.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.61803 q^{25} -1.00000 q^{27} +5.47214 q^{29} -1.76393 q^{31} -0.236068 q^{33} +0.618034 q^{35} -5.47214 q^{37} -0.381966 q^{39} +7.47214 q^{41} -12.5623 q^{43} -0.618034 q^{45} -0.763932 q^{47} +1.00000 q^{49} +2.23607 q^{51} -7.09017 q^{53} -0.145898 q^{55} -3.47214 q^{57} -3.38197 q^{59} +5.32624 q^{61} -1.00000 q^{63} -0.236068 q^{65} -3.38197 q^{67} -1.00000 q^{69} -10.8541 q^{71} +1.94427 q^{73} +4.61803 q^{75} -0.236068 q^{77} -8.23607 q^{79} +1.00000 q^{81} -7.94427 q^{83} +1.38197 q^{85} -5.47214 q^{87} -2.85410 q^{89} -0.381966 q^{91} +1.76393 q^{93} -2.14590 q^{95} -0.236068 q^{97} +0.236068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} + 3 q^{13} - q^{15} - 2 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} - 2 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} - q^{35} - 2 q^{37} - 3 q^{39} + 6 q^{41} - 5 q^{43} + q^{45} - 6 q^{47} + 2 q^{49} - 3 q^{53} - 7 q^{55} + 2 q^{57} - 9 q^{59} - 5 q^{61} - 2 q^{63} + 4 q^{65} - 9 q^{67} - 2 q^{69} - 15 q^{71} - 14 q^{73} + 7 q^{75} + 4 q^{77} - 12 q^{79} + 2 q^{81} + 2 q^{83} + 5 q^{85} - 2 q^{87} + q^{89} - 3 q^{91} + 8 q^{93} - 11 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(5\) −0.618034 −0.276393 −0.138197 0.990405i \(-0.544131\pi\)
−0.138197 + 0.990405i \(0.544131\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.236068 0.0711772 0.0355886 0.999367i \(-0.488669\pi\)
0.0355886 + 0.999367i \(0.488669\pi\)
\(12\) 0 0
\(13\) 0.381966 0.105938 0.0529692 0.998596i \(-0.483131\pi\)
0.0529692 + 0.998596i \(0.483131\pi\)
\(14\) 0 0
\(15\) 0.618034 0.159576
\(16\) 0 0
\(17\) −2.23607 −0.542326 −0.271163 0.962533i \(-0.587408\pi\)
−0.271163 + 0.962533i \(0.587408\pi\)
\(18\) 0 0
\(19\) 3.47214 0.796563 0.398281 0.917263i \(-0.369607\pi\)
0.398281 + 0.917263i \(0.369607\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.61803 −0.923607
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.47214 1.01615 0.508075 0.861313i \(-0.330357\pi\)
0.508075 + 0.861313i \(0.330357\pi\)
\(30\) 0 0
\(31\) −1.76393 −0.316812 −0.158406 0.987374i \(-0.550635\pi\)
−0.158406 + 0.987374i \(0.550635\pi\)
\(32\) 0 0
\(33\) −0.236068 −0.0410942
\(34\) 0 0
\(35\) 0.618034 0.104467
\(36\) 0 0
\(37\) −5.47214 −0.899614 −0.449807 0.893126i \(-0.648507\pi\)
−0.449807 + 0.893126i \(0.648507\pi\)
\(38\) 0 0
\(39\) −0.381966 −0.0611635
\(40\) 0 0
\(41\) 7.47214 1.16695 0.583476 0.812131i \(-0.301692\pi\)
0.583476 + 0.812131i \(0.301692\pi\)
\(42\) 0 0
\(43\) −12.5623 −1.91573 −0.957867 0.287213i \(-0.907271\pi\)
−0.957867 + 0.287213i \(0.907271\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) 0 0
\(47\) −0.763932 −0.111431 −0.0557155 0.998447i \(-0.517744\pi\)
−0.0557155 + 0.998447i \(0.517744\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.23607 0.313112
\(52\) 0 0
\(53\) −7.09017 −0.973910 −0.486955 0.873427i \(-0.661892\pi\)
−0.486955 + 0.873427i \(0.661892\pi\)
\(54\) 0 0
\(55\) −0.145898 −0.0196729
\(56\) 0 0
\(57\) −3.47214 −0.459896
\(58\) 0 0
\(59\) −3.38197 −0.440294 −0.220147 0.975467i \(-0.570654\pi\)
−0.220147 + 0.975467i \(0.570654\pi\)
\(60\) 0 0
\(61\) 5.32624 0.681955 0.340977 0.940071i \(-0.389242\pi\)
0.340977 + 0.940071i \(0.389242\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −0.236068 −0.0292806
\(66\) 0 0
\(67\) −3.38197 −0.413173 −0.206586 0.978428i \(-0.566235\pi\)
−0.206586 + 0.978428i \(0.566235\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −10.8541 −1.28814 −0.644072 0.764964i \(-0.722756\pi\)
−0.644072 + 0.764964i \(0.722756\pi\)
\(72\) 0 0
\(73\) 1.94427 0.227560 0.113780 0.993506i \(-0.463704\pi\)
0.113780 + 0.993506i \(0.463704\pi\)
\(74\) 0 0
\(75\) 4.61803 0.533245
\(76\) 0 0
\(77\) −0.236068 −0.0269024
\(78\) 0 0
\(79\) −8.23607 −0.926630 −0.463315 0.886194i \(-0.653340\pi\)
−0.463315 + 0.886194i \(0.653340\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.94427 −0.871997 −0.435999 0.899947i \(-0.643605\pi\)
−0.435999 + 0.899947i \(0.643605\pi\)
\(84\) 0 0
\(85\) 1.38197 0.149895
\(86\) 0 0
\(87\) −5.47214 −0.586675
\(88\) 0 0
\(89\) −2.85410 −0.302534 −0.151267 0.988493i \(-0.548335\pi\)
−0.151267 + 0.988493i \(0.548335\pi\)
\(90\) 0 0
\(91\) −0.381966 −0.0400409
\(92\) 0 0
\(93\) 1.76393 0.182911
\(94\) 0 0
\(95\) −2.14590 −0.220164
\(96\) 0 0
\(97\) −0.236068 −0.0239691 −0.0119845 0.999928i \(-0.503815\pi\)
−0.0119845 + 0.999928i \(0.503815\pi\)
\(98\) 0 0
\(99\) 0.236068 0.0237257
\(100\) 0 0
\(101\) −1.38197 −0.137511 −0.0687554 0.997634i \(-0.521903\pi\)
−0.0687554 + 0.997634i \(0.521903\pi\)
\(102\) 0 0
\(103\) −8.94427 −0.881305 −0.440653 0.897678i \(-0.645253\pi\)
−0.440653 + 0.897678i \(0.645253\pi\)
\(104\) 0 0
\(105\) −0.618034 −0.0603139
\(106\) 0 0
\(107\) −10.3820 −1.00366 −0.501831 0.864966i \(-0.667340\pi\)
−0.501831 + 0.864966i \(0.667340\pi\)
\(108\) 0 0
\(109\) −6.61803 −0.633893 −0.316946 0.948443i \(-0.602658\pi\)
−0.316946 + 0.948443i \(0.602658\pi\)
\(110\) 0 0
\(111\) 5.47214 0.519392
\(112\) 0 0
\(113\) 7.56231 0.711402 0.355701 0.934600i \(-0.384242\pi\)
0.355701 + 0.934600i \(0.384242\pi\)
\(114\) 0 0
\(115\) −0.618034 −0.0576320
\(116\) 0 0
\(117\) 0.381966 0.0353128
\(118\) 0 0
\(119\) 2.23607 0.204980
\(120\) 0 0
\(121\) −10.9443 −0.994934
\(122\) 0 0
\(123\) −7.47214 −0.673740
\(124\) 0 0
\(125\) 5.94427 0.531672
\(126\) 0 0
\(127\) −7.14590 −0.634096 −0.317048 0.948410i \(-0.602692\pi\)
−0.317048 + 0.948410i \(0.602692\pi\)
\(128\) 0 0
\(129\) 12.5623 1.10605
\(130\) 0 0
\(131\) 0.0557281 0.00486899 0.00243449 0.999997i \(-0.499225\pi\)
0.00243449 + 0.999997i \(0.499225\pi\)
\(132\) 0 0
\(133\) −3.47214 −0.301072
\(134\) 0 0
\(135\) 0.618034 0.0531919
\(136\) 0 0
\(137\) −0.236068 −0.0201686 −0.0100843 0.999949i \(-0.503210\pi\)
−0.0100843 + 0.999949i \(0.503210\pi\)
\(138\) 0 0
\(139\) −12.0344 −1.02075 −0.510374 0.859953i \(-0.670493\pi\)
−0.510374 + 0.859953i \(0.670493\pi\)
\(140\) 0 0
\(141\) 0.763932 0.0643347
\(142\) 0 0
\(143\) 0.0901699 0.00754039
\(144\) 0 0
\(145\) −3.38197 −0.280857
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 14.6525 1.20038 0.600189 0.799858i \(-0.295092\pi\)
0.600189 + 0.799858i \(0.295092\pi\)
\(150\) 0 0
\(151\) −2.29180 −0.186504 −0.0932519 0.995643i \(-0.529726\pi\)
−0.0932519 + 0.995643i \(0.529726\pi\)
\(152\) 0 0
\(153\) −2.23607 −0.180775
\(154\) 0 0
\(155\) 1.09017 0.0875646
\(156\) 0 0
\(157\) −17.2361 −1.37559 −0.687794 0.725906i \(-0.741421\pi\)
−0.687794 + 0.725906i \(0.741421\pi\)
\(158\) 0 0
\(159\) 7.09017 0.562287
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 3.56231 0.279021 0.139511 0.990221i \(-0.455447\pi\)
0.139511 + 0.990221i \(0.455447\pi\)
\(164\) 0 0
\(165\) 0.145898 0.0113581
\(166\) 0 0
\(167\) −14.8885 −1.15211 −0.576055 0.817411i \(-0.695409\pi\)
−0.576055 + 0.817411i \(0.695409\pi\)
\(168\) 0 0
\(169\) −12.8541 −0.988777
\(170\) 0 0
\(171\) 3.47214 0.265521
\(172\) 0 0
\(173\) 19.9443 1.51633 0.758167 0.652060i \(-0.226095\pi\)
0.758167 + 0.652060i \(0.226095\pi\)
\(174\) 0 0
\(175\) 4.61803 0.349091
\(176\) 0 0
\(177\) 3.38197 0.254204
\(178\) 0 0
\(179\) −2.79837 −0.209160 −0.104580 0.994516i \(-0.533350\pi\)
−0.104580 + 0.994516i \(0.533350\pi\)
\(180\) 0 0
\(181\) −4.23607 −0.314864 −0.157432 0.987530i \(-0.550322\pi\)
−0.157432 + 0.987530i \(0.550322\pi\)
\(182\) 0 0
\(183\) −5.32624 −0.393727
\(184\) 0 0
\(185\) 3.38197 0.248647
\(186\) 0 0
\(187\) −0.527864 −0.0386012
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 21.1246 1.52852 0.764262 0.644906i \(-0.223104\pi\)
0.764262 + 0.644906i \(0.223104\pi\)
\(192\) 0 0
\(193\) −4.29180 −0.308930 −0.154465 0.987998i \(-0.549365\pi\)
−0.154465 + 0.987998i \(0.549365\pi\)
\(194\) 0 0
\(195\) 0.236068 0.0169052
\(196\) 0 0
\(197\) −6.38197 −0.454696 −0.227348 0.973814i \(-0.573006\pi\)
−0.227348 + 0.973814i \(0.573006\pi\)
\(198\) 0 0
\(199\) −5.67376 −0.402202 −0.201101 0.979570i \(-0.564452\pi\)
−0.201101 + 0.979570i \(0.564452\pi\)
\(200\) 0 0
\(201\) 3.38197 0.238545
\(202\) 0 0
\(203\) −5.47214 −0.384069
\(204\) 0 0
\(205\) −4.61803 −0.322537
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 0.819660 0.0566971
\(210\) 0 0
\(211\) −2.23607 −0.153937 −0.0769686 0.997034i \(-0.524524\pi\)
−0.0769686 + 0.997034i \(0.524524\pi\)
\(212\) 0 0
\(213\) 10.8541 0.743711
\(214\) 0 0
\(215\) 7.76393 0.529496
\(216\) 0 0
\(217\) 1.76393 0.119744
\(218\) 0 0
\(219\) −1.94427 −0.131382
\(220\) 0 0
\(221\) −0.854102 −0.0574531
\(222\) 0 0
\(223\) 26.2148 1.75547 0.877736 0.479145i \(-0.159053\pi\)
0.877736 + 0.479145i \(0.159053\pi\)
\(224\) 0 0
\(225\) −4.61803 −0.307869
\(226\) 0 0
\(227\) 1.09017 0.0723571 0.0361786 0.999345i \(-0.488481\pi\)
0.0361786 + 0.999345i \(0.488481\pi\)
\(228\) 0 0
\(229\) 14.6180 0.965987 0.482993 0.875624i \(-0.339549\pi\)
0.482993 + 0.875624i \(0.339549\pi\)
\(230\) 0 0
\(231\) 0.236068 0.0155321
\(232\) 0 0
\(233\) −9.09017 −0.595517 −0.297758 0.954641i \(-0.596239\pi\)
−0.297758 + 0.954641i \(0.596239\pi\)
\(234\) 0 0
\(235\) 0.472136 0.0307988
\(236\) 0 0
\(237\) 8.23607 0.534990
\(238\) 0 0
\(239\) −22.0344 −1.42529 −0.712645 0.701525i \(-0.752503\pi\)
−0.712645 + 0.701525i \(0.752503\pi\)
\(240\) 0 0
\(241\) −24.7082 −1.59160 −0.795798 0.605563i \(-0.792948\pi\)
−0.795798 + 0.605563i \(0.792948\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.618034 −0.0394847
\(246\) 0 0
\(247\) 1.32624 0.0843865
\(248\) 0 0
\(249\) 7.94427 0.503448
\(250\) 0 0
\(251\) 22.1803 1.40001 0.700005 0.714138i \(-0.253181\pi\)
0.700005 + 0.714138i \(0.253181\pi\)
\(252\) 0 0
\(253\) 0.236068 0.0148415
\(254\) 0 0
\(255\) −1.38197 −0.0865421
\(256\) 0 0
\(257\) 25.7082 1.60363 0.801817 0.597570i \(-0.203867\pi\)
0.801817 + 0.597570i \(0.203867\pi\)
\(258\) 0 0
\(259\) 5.47214 0.340022
\(260\) 0 0
\(261\) 5.47214 0.338717
\(262\) 0 0
\(263\) −7.94427 −0.489865 −0.244932 0.969540i \(-0.578766\pi\)
−0.244932 + 0.969540i \(0.578766\pi\)
\(264\) 0 0
\(265\) 4.38197 0.269182
\(266\) 0 0
\(267\) 2.85410 0.174668
\(268\) 0 0
\(269\) 11.9098 0.726155 0.363078 0.931759i \(-0.381726\pi\)
0.363078 + 0.931759i \(0.381726\pi\)
\(270\) 0 0
\(271\) −13.3607 −0.811603 −0.405802 0.913961i \(-0.633008\pi\)
−0.405802 + 0.913961i \(0.633008\pi\)
\(272\) 0 0
\(273\) 0.381966 0.0231176
\(274\) 0 0
\(275\) −1.09017 −0.0657397
\(276\) 0 0
\(277\) 6.38197 0.383455 0.191728 0.981448i \(-0.438591\pi\)
0.191728 + 0.981448i \(0.438591\pi\)
\(278\) 0 0
\(279\) −1.76393 −0.105604
\(280\) 0 0
\(281\) −8.65248 −0.516163 −0.258082 0.966123i \(-0.583090\pi\)
−0.258082 + 0.966123i \(0.583090\pi\)
\(282\) 0 0
\(283\) −19.0902 −1.13479 −0.567396 0.823445i \(-0.692049\pi\)
−0.567396 + 0.823445i \(0.692049\pi\)
\(284\) 0 0
\(285\) 2.14590 0.127112
\(286\) 0 0
\(287\) −7.47214 −0.441066
\(288\) 0 0
\(289\) −12.0000 −0.705882
\(290\) 0 0
\(291\) 0.236068 0.0138385
\(292\) 0 0
\(293\) 7.05573 0.412200 0.206100 0.978531i \(-0.433923\pi\)
0.206100 + 0.978531i \(0.433923\pi\)
\(294\) 0 0
\(295\) 2.09017 0.121694
\(296\) 0 0
\(297\) −0.236068 −0.0136981
\(298\) 0 0
\(299\) 0.381966 0.0220897
\(300\) 0 0
\(301\) 12.5623 0.724079
\(302\) 0 0
\(303\) 1.38197 0.0793919
\(304\) 0 0
\(305\) −3.29180 −0.188488
\(306\) 0 0
\(307\) 5.76393 0.328965 0.164482 0.986380i \(-0.447405\pi\)
0.164482 + 0.986380i \(0.447405\pi\)
\(308\) 0 0
\(309\) 8.94427 0.508822
\(310\) 0 0
\(311\) −15.9787 −0.906070 −0.453035 0.891493i \(-0.649659\pi\)
−0.453035 + 0.891493i \(0.649659\pi\)
\(312\) 0 0
\(313\) 6.47214 0.365827 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(314\) 0 0
\(315\) 0.618034 0.0348223
\(316\) 0 0
\(317\) 34.0344 1.91156 0.955782 0.294075i \(-0.0950115\pi\)
0.955782 + 0.294075i \(0.0950115\pi\)
\(318\) 0 0
\(319\) 1.29180 0.0723267
\(320\) 0 0
\(321\) 10.3820 0.579465
\(322\) 0 0
\(323\) −7.76393 −0.431997
\(324\) 0 0
\(325\) −1.76393 −0.0978453
\(326\) 0 0
\(327\) 6.61803 0.365978
\(328\) 0 0
\(329\) 0.763932 0.0421169
\(330\) 0 0
\(331\) 1.05573 0.0580281 0.0290140 0.999579i \(-0.490763\pi\)
0.0290140 + 0.999579i \(0.490763\pi\)
\(332\) 0 0
\(333\) −5.47214 −0.299871
\(334\) 0 0
\(335\) 2.09017 0.114198
\(336\) 0 0
\(337\) −14.5623 −0.793259 −0.396630 0.917979i \(-0.629820\pi\)
−0.396630 + 0.917979i \(0.629820\pi\)
\(338\) 0 0
\(339\) −7.56231 −0.410728
\(340\) 0 0
\(341\) −0.416408 −0.0225498
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0.618034 0.0332738
\(346\) 0 0
\(347\) 19.1803 1.02965 0.514827 0.857294i \(-0.327856\pi\)
0.514827 + 0.857294i \(0.327856\pi\)
\(348\) 0 0
\(349\) 23.0902 1.23599 0.617994 0.786183i \(-0.287945\pi\)
0.617994 + 0.786183i \(0.287945\pi\)
\(350\) 0 0
\(351\) −0.381966 −0.0203878
\(352\) 0 0
\(353\) 8.05573 0.428763 0.214382 0.976750i \(-0.431226\pi\)
0.214382 + 0.976750i \(0.431226\pi\)
\(354\) 0 0
\(355\) 6.70820 0.356034
\(356\) 0 0
\(357\) −2.23607 −0.118345
\(358\) 0 0
\(359\) 8.32624 0.439442 0.219721 0.975563i \(-0.429485\pi\)
0.219721 + 0.975563i \(0.429485\pi\)
\(360\) 0 0
\(361\) −6.94427 −0.365488
\(362\) 0 0
\(363\) 10.9443 0.574425
\(364\) 0 0
\(365\) −1.20163 −0.0628960
\(366\) 0 0
\(367\) −11.7984 −0.615870 −0.307935 0.951407i \(-0.599638\pi\)
−0.307935 + 0.951407i \(0.599638\pi\)
\(368\) 0 0
\(369\) 7.47214 0.388984
\(370\) 0 0
\(371\) 7.09017 0.368103
\(372\) 0 0
\(373\) 25.9443 1.34334 0.671672 0.740849i \(-0.265577\pi\)
0.671672 + 0.740849i \(0.265577\pi\)
\(374\) 0 0
\(375\) −5.94427 −0.306961
\(376\) 0 0
\(377\) 2.09017 0.107649
\(378\) 0 0
\(379\) 26.8328 1.37831 0.689155 0.724614i \(-0.257982\pi\)
0.689155 + 0.724614i \(0.257982\pi\)
\(380\) 0 0
\(381\) 7.14590 0.366095
\(382\) 0 0
\(383\) 0.527864 0.0269726 0.0134863 0.999909i \(-0.495707\pi\)
0.0134863 + 0.999909i \(0.495707\pi\)
\(384\) 0 0
\(385\) 0.145898 0.00743565
\(386\) 0 0
\(387\) −12.5623 −0.638578
\(388\) 0 0
\(389\) −0.416408 −0.0211127 −0.0105564 0.999944i \(-0.503360\pi\)
−0.0105564 + 0.999944i \(0.503360\pi\)
\(390\) 0 0
\(391\) −2.23607 −0.113083
\(392\) 0 0
\(393\) −0.0557281 −0.00281111
\(394\) 0 0
\(395\) 5.09017 0.256114
\(396\) 0 0
\(397\) 28.1803 1.41433 0.707165 0.707048i \(-0.249974\pi\)
0.707165 + 0.707048i \(0.249974\pi\)
\(398\) 0 0
\(399\) 3.47214 0.173824
\(400\) 0 0
\(401\) −11.3607 −0.567325 −0.283663 0.958924i \(-0.591550\pi\)
−0.283663 + 0.958924i \(0.591550\pi\)
\(402\) 0 0
\(403\) −0.673762 −0.0335625
\(404\) 0 0
\(405\) −0.618034 −0.0307104
\(406\) 0 0
\(407\) −1.29180 −0.0640320
\(408\) 0 0
\(409\) −4.23607 −0.209460 −0.104730 0.994501i \(-0.533398\pi\)
−0.104730 + 0.994501i \(0.533398\pi\)
\(410\) 0 0
\(411\) 0.236068 0.0116444
\(412\) 0 0
\(413\) 3.38197 0.166416
\(414\) 0 0
\(415\) 4.90983 0.241014
\(416\) 0 0
\(417\) 12.0344 0.589329
\(418\) 0 0
\(419\) 40.2148 1.96462 0.982310 0.187260i \(-0.0599608\pi\)
0.982310 + 0.187260i \(0.0599608\pi\)
\(420\) 0 0
\(421\) 17.5066 0.853218 0.426609 0.904436i \(-0.359708\pi\)
0.426609 + 0.904436i \(0.359708\pi\)
\(422\) 0 0
\(423\) −0.763932 −0.0371436
\(424\) 0 0
\(425\) 10.3262 0.500896
\(426\) 0 0
\(427\) −5.32624 −0.257755
\(428\) 0 0
\(429\) −0.0901699 −0.00435345
\(430\) 0 0
\(431\) −22.5623 −1.08679 −0.543394 0.839478i \(-0.682861\pi\)
−0.543394 + 0.839478i \(0.682861\pi\)
\(432\) 0 0
\(433\) −23.7082 −1.13934 −0.569672 0.821872i \(-0.692930\pi\)
−0.569672 + 0.821872i \(0.692930\pi\)
\(434\) 0 0
\(435\) 3.38197 0.162153
\(436\) 0 0
\(437\) 3.47214 0.166095
\(438\) 0 0
\(439\) 6.81966 0.325485 0.162742 0.986669i \(-0.447966\pi\)
0.162742 + 0.986669i \(0.447966\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −13.8885 −0.659865 −0.329932 0.944005i \(-0.607026\pi\)
−0.329932 + 0.944005i \(0.607026\pi\)
\(444\) 0 0
\(445\) 1.76393 0.0836184
\(446\) 0 0
\(447\) −14.6525 −0.693038
\(448\) 0 0
\(449\) 9.09017 0.428992 0.214496 0.976725i \(-0.431189\pi\)
0.214496 + 0.976725i \(0.431189\pi\)
\(450\) 0 0
\(451\) 1.76393 0.0830603
\(452\) 0 0
\(453\) 2.29180 0.107678
\(454\) 0 0
\(455\) 0.236068 0.0110670
\(456\) 0 0
\(457\) −3.85410 −0.180287 −0.0901436 0.995929i \(-0.528733\pi\)
−0.0901436 + 0.995929i \(0.528733\pi\)
\(458\) 0 0
\(459\) 2.23607 0.104371
\(460\) 0 0
\(461\) 15.5623 0.724809 0.362404 0.932021i \(-0.381956\pi\)
0.362404 + 0.932021i \(0.381956\pi\)
\(462\) 0 0
\(463\) −36.8885 −1.71436 −0.857178 0.515020i \(-0.827784\pi\)
−0.857178 + 0.515020i \(0.827784\pi\)
\(464\) 0 0
\(465\) −1.09017 −0.0505554
\(466\) 0 0
\(467\) 14.4164 0.667112 0.333556 0.942730i \(-0.391751\pi\)
0.333556 + 0.942730i \(0.391751\pi\)
\(468\) 0 0
\(469\) 3.38197 0.156165
\(470\) 0 0
\(471\) 17.2361 0.794196
\(472\) 0 0
\(473\) −2.96556 −0.136357
\(474\) 0 0
\(475\) −16.0344 −0.735711
\(476\) 0 0
\(477\) −7.09017 −0.324637
\(478\) 0 0
\(479\) 7.18034 0.328078 0.164039 0.986454i \(-0.447548\pi\)
0.164039 + 0.986454i \(0.447548\pi\)
\(480\) 0 0
\(481\) −2.09017 −0.0953035
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 0.145898 0.00662489
\(486\) 0 0
\(487\) 20.5279 0.930206 0.465103 0.885256i \(-0.346017\pi\)
0.465103 + 0.885256i \(0.346017\pi\)
\(488\) 0 0
\(489\) −3.56231 −0.161093
\(490\) 0 0
\(491\) −0.854102 −0.0385451 −0.0192725 0.999814i \(-0.506135\pi\)
−0.0192725 + 0.999814i \(0.506135\pi\)
\(492\) 0 0
\(493\) −12.2361 −0.551085
\(494\) 0 0
\(495\) −0.145898 −0.00655763
\(496\) 0 0
\(497\) 10.8541 0.486873
\(498\) 0 0
\(499\) −1.27051 −0.0568758 −0.0284379 0.999596i \(-0.509053\pi\)
−0.0284379 + 0.999596i \(0.509053\pi\)
\(500\) 0 0
\(501\) 14.8885 0.665171
\(502\) 0 0
\(503\) −4.43769 −0.197867 −0.0989335 0.995094i \(-0.531543\pi\)
−0.0989335 + 0.995094i \(0.531543\pi\)
\(504\) 0 0
\(505\) 0.854102 0.0380070
\(506\) 0 0
\(507\) 12.8541 0.570871
\(508\) 0 0
\(509\) 2.29180 0.101582 0.0507910 0.998709i \(-0.483826\pi\)
0.0507910 + 0.998709i \(0.483826\pi\)
\(510\) 0 0
\(511\) −1.94427 −0.0860095
\(512\) 0 0
\(513\) −3.47214 −0.153299
\(514\) 0 0
\(515\) 5.52786 0.243587
\(516\) 0 0
\(517\) −0.180340 −0.00793134
\(518\) 0 0
\(519\) −19.9443 −0.875456
\(520\) 0 0
\(521\) 8.47214 0.371171 0.185586 0.982628i \(-0.440582\pi\)
0.185586 + 0.982628i \(0.440582\pi\)
\(522\) 0 0
\(523\) −17.4164 −0.761566 −0.380783 0.924664i \(-0.624346\pi\)
−0.380783 + 0.924664i \(0.624346\pi\)
\(524\) 0 0
\(525\) −4.61803 −0.201548
\(526\) 0 0
\(527\) 3.94427 0.171815
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.38197 −0.146765
\(532\) 0 0
\(533\) 2.85410 0.123625
\(534\) 0 0
\(535\) 6.41641 0.277406
\(536\) 0 0
\(537\) 2.79837 0.120759
\(538\) 0 0
\(539\) 0.236068 0.0101682
\(540\) 0 0
\(541\) −2.29180 −0.0985320 −0.0492660 0.998786i \(-0.515688\pi\)
−0.0492660 + 0.998786i \(0.515688\pi\)
\(542\) 0 0
\(543\) 4.23607 0.181787
\(544\) 0 0
\(545\) 4.09017 0.175204
\(546\) 0 0
\(547\) 3.09017 0.132126 0.0660631 0.997815i \(-0.478956\pi\)
0.0660631 + 0.997815i \(0.478956\pi\)
\(548\) 0 0
\(549\) 5.32624 0.227318
\(550\) 0 0
\(551\) 19.0000 0.809427
\(552\) 0 0
\(553\) 8.23607 0.350233
\(554\) 0 0
\(555\) −3.38197 −0.143556
\(556\) 0 0
\(557\) 4.65248 0.197132 0.0985659 0.995131i \(-0.468574\pi\)
0.0985659 + 0.995131i \(0.468574\pi\)
\(558\) 0 0
\(559\) −4.79837 −0.202950
\(560\) 0 0
\(561\) 0.527864 0.0222864
\(562\) 0 0
\(563\) −37.5066 −1.58071 −0.790357 0.612647i \(-0.790105\pi\)
−0.790357 + 0.612647i \(0.790105\pi\)
\(564\) 0 0
\(565\) −4.67376 −0.196627
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 26.4721 1.10977 0.554885 0.831927i \(-0.312762\pi\)
0.554885 + 0.831927i \(0.312762\pi\)
\(570\) 0 0
\(571\) 3.29180 0.137757 0.0688787 0.997625i \(-0.478058\pi\)
0.0688787 + 0.997625i \(0.478058\pi\)
\(572\) 0 0
\(573\) −21.1246 −0.882493
\(574\) 0 0
\(575\) −4.61803 −0.192585
\(576\) 0 0
\(577\) −30.9443 −1.28823 −0.644113 0.764930i \(-0.722774\pi\)
−0.644113 + 0.764930i \(0.722774\pi\)
\(578\) 0 0
\(579\) 4.29180 0.178361
\(580\) 0 0
\(581\) 7.94427 0.329584
\(582\) 0 0
\(583\) −1.67376 −0.0693201
\(584\) 0 0
\(585\) −0.236068 −0.00976021
\(586\) 0 0
\(587\) −17.0344 −0.703087 −0.351543 0.936172i \(-0.614343\pi\)
−0.351543 + 0.936172i \(0.614343\pi\)
\(588\) 0 0
\(589\) −6.12461 −0.252360
\(590\) 0 0
\(591\) 6.38197 0.262519
\(592\) 0 0
\(593\) 2.76393 0.113501 0.0567505 0.998388i \(-0.481926\pi\)
0.0567505 + 0.998388i \(0.481926\pi\)
\(594\) 0 0
\(595\) −1.38197 −0.0566551
\(596\) 0 0
\(597\) 5.67376 0.232212
\(598\) 0 0
\(599\) 37.3951 1.52792 0.763962 0.645262i \(-0.223252\pi\)
0.763962 + 0.645262i \(0.223252\pi\)
\(600\) 0 0
\(601\) 34.6869 1.41491 0.707454 0.706759i \(-0.249843\pi\)
0.707454 + 0.706759i \(0.249843\pi\)
\(602\) 0 0
\(603\) −3.38197 −0.137724
\(604\) 0 0
\(605\) 6.76393 0.274993
\(606\) 0 0
\(607\) 40.6312 1.64917 0.824585 0.565739i \(-0.191409\pi\)
0.824585 + 0.565739i \(0.191409\pi\)
\(608\) 0 0
\(609\) 5.47214 0.221742
\(610\) 0 0
\(611\) −0.291796 −0.0118048
\(612\) 0 0
\(613\) −17.7639 −0.717478 −0.358739 0.933438i \(-0.616793\pi\)
−0.358739 + 0.933438i \(0.616793\pi\)
\(614\) 0 0
\(615\) 4.61803 0.186217
\(616\) 0 0
\(617\) −15.9098 −0.640506 −0.320253 0.947332i \(-0.603768\pi\)
−0.320253 + 0.947332i \(0.603768\pi\)
\(618\) 0 0
\(619\) 7.03444 0.282738 0.141369 0.989957i \(-0.454850\pi\)
0.141369 + 0.989957i \(0.454850\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 2.85410 0.114347
\(624\) 0 0
\(625\) 19.4164 0.776656
\(626\) 0 0
\(627\) −0.819660 −0.0327341
\(628\) 0 0
\(629\) 12.2361 0.487884
\(630\) 0 0
\(631\) −9.47214 −0.377080 −0.188540 0.982066i \(-0.560375\pi\)
−0.188540 + 0.982066i \(0.560375\pi\)
\(632\) 0 0
\(633\) 2.23607 0.0888757
\(634\) 0 0
\(635\) 4.41641 0.175260
\(636\) 0 0
\(637\) 0.381966 0.0151340
\(638\) 0 0
\(639\) −10.8541 −0.429382
\(640\) 0 0
\(641\) −34.8541 −1.37665 −0.688327 0.725400i \(-0.741655\pi\)
−0.688327 + 0.725400i \(0.741655\pi\)
\(642\) 0 0
\(643\) 19.9787 0.787884 0.393942 0.919135i \(-0.371111\pi\)
0.393942 + 0.919135i \(0.371111\pi\)
\(644\) 0 0
\(645\) −7.76393 −0.305705
\(646\) 0 0
\(647\) −30.9098 −1.21519 −0.607595 0.794247i \(-0.707866\pi\)
−0.607595 + 0.794247i \(0.707866\pi\)
\(648\) 0 0
\(649\) −0.798374 −0.0313389
\(650\) 0 0
\(651\) −1.76393 −0.0691339
\(652\) 0 0
\(653\) −10.0344 −0.392678 −0.196339 0.980536i \(-0.562905\pi\)
−0.196339 + 0.980536i \(0.562905\pi\)
\(654\) 0 0
\(655\) −0.0344419 −0.00134575
\(656\) 0 0
\(657\) 1.94427 0.0758533
\(658\) 0 0
\(659\) −41.0689 −1.59982 −0.799908 0.600122i \(-0.795119\pi\)
−0.799908 + 0.600122i \(0.795119\pi\)
\(660\) 0 0
\(661\) −8.81966 −0.343045 −0.171523 0.985180i \(-0.554869\pi\)
−0.171523 + 0.985180i \(0.554869\pi\)
\(662\) 0 0
\(663\) 0.854102 0.0331706
\(664\) 0 0
\(665\) 2.14590 0.0832144
\(666\) 0 0
\(667\) 5.47214 0.211882
\(668\) 0 0
\(669\) −26.2148 −1.01352
\(670\) 0 0
\(671\) 1.25735 0.0485396
\(672\) 0 0
\(673\) −34.4164 −1.32666 −0.663328 0.748329i \(-0.730856\pi\)
−0.663328 + 0.748329i \(0.730856\pi\)
\(674\) 0 0
\(675\) 4.61803 0.177748
\(676\) 0 0
\(677\) −21.9787 −0.844711 −0.422355 0.906430i \(-0.638797\pi\)
−0.422355 + 0.906430i \(0.638797\pi\)
\(678\) 0 0
\(679\) 0.236068 0.00905946
\(680\) 0 0
\(681\) −1.09017 −0.0417754
\(682\) 0 0
\(683\) −15.7639 −0.603190 −0.301595 0.953436i \(-0.597519\pi\)
−0.301595 + 0.953436i \(0.597519\pi\)
\(684\) 0 0
\(685\) 0.145898 0.00557448
\(686\) 0 0
\(687\) −14.6180 −0.557713
\(688\) 0 0
\(689\) −2.70820 −0.103174
\(690\) 0 0
\(691\) 2.56231 0.0974747 0.0487374 0.998812i \(-0.484480\pi\)
0.0487374 + 0.998812i \(0.484480\pi\)
\(692\) 0 0
\(693\) −0.236068 −0.00896748
\(694\) 0 0
\(695\) 7.43769 0.282128
\(696\) 0 0
\(697\) −16.7082 −0.632868
\(698\) 0 0
\(699\) 9.09017 0.343822
\(700\) 0 0
\(701\) −46.2148 −1.74551 −0.872754 0.488160i \(-0.837668\pi\)
−0.872754 + 0.488160i \(0.837668\pi\)
\(702\) 0 0
\(703\) −19.0000 −0.716599
\(704\) 0 0
\(705\) −0.472136 −0.0177817
\(706\) 0 0
\(707\) 1.38197 0.0519742
\(708\) 0 0
\(709\) 8.27051 0.310606 0.155303 0.987867i \(-0.450365\pi\)
0.155303 + 0.987867i \(0.450365\pi\)
\(710\) 0 0
\(711\) −8.23607 −0.308877
\(712\) 0 0
\(713\) −1.76393 −0.0660598
\(714\) 0 0
\(715\) −0.0557281 −0.00208411
\(716\) 0 0
\(717\) 22.0344 0.822891
\(718\) 0 0
\(719\) −1.76393 −0.0657836 −0.0328918 0.999459i \(-0.510472\pi\)
−0.0328918 + 0.999459i \(0.510472\pi\)
\(720\) 0 0
\(721\) 8.94427 0.333102
\(722\) 0 0
\(723\) 24.7082 0.918908
\(724\) 0 0
\(725\) −25.2705 −0.938523
\(726\) 0 0
\(727\) 45.4721 1.68647 0.843234 0.537547i \(-0.180649\pi\)
0.843234 + 0.537547i \(0.180649\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.0902 1.03895
\(732\) 0 0
\(733\) −41.5967 −1.53641 −0.768205 0.640203i \(-0.778850\pi\)
−0.768205 + 0.640203i \(0.778850\pi\)
\(734\) 0 0
\(735\) 0.618034 0.0227965
\(736\) 0 0
\(737\) −0.798374 −0.0294085
\(738\) 0 0
\(739\) 24.5967 0.904806 0.452403 0.891814i \(-0.350567\pi\)
0.452403 + 0.891814i \(0.350567\pi\)
\(740\) 0 0
\(741\) −1.32624 −0.0487206
\(742\) 0 0
\(743\) −29.2016 −1.07130 −0.535652 0.844439i \(-0.679934\pi\)
−0.535652 + 0.844439i \(0.679934\pi\)
\(744\) 0 0
\(745\) −9.05573 −0.331776
\(746\) 0 0
\(747\) −7.94427 −0.290666
\(748\) 0 0
\(749\) 10.3820 0.379349
\(750\) 0 0
\(751\) −38.0344 −1.38790 −0.693948 0.720025i \(-0.744130\pi\)
−0.693948 + 0.720025i \(0.744130\pi\)
\(752\) 0 0
\(753\) −22.1803 −0.808297
\(754\) 0 0
\(755\) 1.41641 0.0515484
\(756\) 0 0
\(757\) 35.0000 1.27210 0.636048 0.771649i \(-0.280568\pi\)
0.636048 + 0.771649i \(0.280568\pi\)
\(758\) 0 0
\(759\) −0.236068 −0.00856872
\(760\) 0 0
\(761\) 10.8328 0.392689 0.196345 0.980535i \(-0.437093\pi\)
0.196345 + 0.980535i \(0.437093\pi\)
\(762\) 0 0
\(763\) 6.61803 0.239589
\(764\) 0 0
\(765\) 1.38197 0.0499651
\(766\) 0 0
\(767\) −1.29180 −0.0466441
\(768\) 0 0
\(769\) 30.1803 1.08833 0.544165 0.838978i \(-0.316846\pi\)
0.544165 + 0.838978i \(0.316846\pi\)
\(770\) 0 0
\(771\) −25.7082 −0.925858
\(772\) 0 0
\(773\) 6.52786 0.234791 0.117395 0.993085i \(-0.462545\pi\)
0.117395 + 0.993085i \(0.462545\pi\)
\(774\) 0 0
\(775\) 8.14590 0.292609
\(776\) 0 0
\(777\) −5.47214 −0.196312
\(778\) 0 0
\(779\) 25.9443 0.929550
\(780\) 0 0
\(781\) −2.56231 −0.0916865
\(782\) 0 0
\(783\) −5.47214 −0.195558
\(784\) 0 0
\(785\) 10.6525 0.380203
\(786\) 0 0
\(787\) −17.4508 −0.622056 −0.311028 0.950401i \(-0.600673\pi\)
−0.311028 + 0.950401i \(0.600673\pi\)
\(788\) 0 0
\(789\) 7.94427 0.282824
\(790\) 0 0
\(791\) −7.56231 −0.268885
\(792\) 0 0
\(793\) 2.03444 0.0722451
\(794\) 0 0
\(795\) −4.38197 −0.155412
\(796\) 0 0
\(797\) 28.5410 1.01097 0.505487 0.862834i \(-0.331313\pi\)
0.505487 + 0.862834i \(0.331313\pi\)
\(798\) 0 0
\(799\) 1.70820 0.0604319
\(800\) 0 0
\(801\) −2.85410 −0.100845
\(802\) 0 0
\(803\) 0.458980 0.0161971
\(804\) 0 0
\(805\) 0.618034 0.0217828
\(806\) 0 0
\(807\) −11.9098 −0.419246
\(808\) 0 0
\(809\) 16.3820 0.575959 0.287980 0.957637i \(-0.407016\pi\)
0.287980 + 0.957637i \(0.407016\pi\)
\(810\) 0 0
\(811\) 5.58359 0.196066 0.0980332 0.995183i \(-0.468745\pi\)
0.0980332 + 0.995183i \(0.468745\pi\)
\(812\) 0 0
\(813\) 13.3607 0.468579
\(814\) 0 0
\(815\) −2.20163 −0.0771196
\(816\) 0 0
\(817\) −43.6180 −1.52600
\(818\) 0 0
\(819\) −0.381966 −0.0133470
\(820\) 0 0
\(821\) 41.1246 1.43526 0.717629 0.696425i \(-0.245227\pi\)
0.717629 + 0.696425i \(0.245227\pi\)
\(822\) 0 0
\(823\) 49.2705 1.71746 0.858731 0.512427i \(-0.171253\pi\)
0.858731 + 0.512427i \(0.171253\pi\)
\(824\) 0 0
\(825\) 1.09017 0.0379548
\(826\) 0 0
\(827\) −23.5623 −0.819342 −0.409671 0.912233i \(-0.634356\pi\)
−0.409671 + 0.912233i \(0.634356\pi\)
\(828\) 0 0
\(829\) −20.1246 −0.698957 −0.349478 0.936944i \(-0.613641\pi\)
−0.349478 + 0.936944i \(0.613641\pi\)
\(830\) 0 0
\(831\) −6.38197 −0.221388
\(832\) 0 0
\(833\) −2.23607 −0.0774752
\(834\) 0 0
\(835\) 9.20163 0.318435
\(836\) 0 0
\(837\) 1.76393 0.0609704
\(838\) 0 0
\(839\) −34.9230 −1.20568 −0.602838 0.797864i \(-0.705963\pi\)
−0.602838 + 0.797864i \(0.705963\pi\)
\(840\) 0 0
\(841\) 0.944272 0.0325611
\(842\) 0 0
\(843\) 8.65248 0.298007
\(844\) 0 0
\(845\) 7.94427 0.273291
\(846\) 0 0
\(847\) 10.9443 0.376050
\(848\) 0 0
\(849\) 19.0902 0.655173
\(850\) 0 0
\(851\) −5.47214 −0.187582
\(852\) 0 0
\(853\) −28.6525 −0.981042 −0.490521 0.871429i \(-0.663194\pi\)
−0.490521 + 0.871429i \(0.663194\pi\)
\(854\) 0 0
\(855\) −2.14590 −0.0733882
\(856\) 0 0
\(857\) 0.875388 0.0299027 0.0149513 0.999888i \(-0.495241\pi\)
0.0149513 + 0.999888i \(0.495241\pi\)
\(858\) 0 0
\(859\) −14.9443 −0.509892 −0.254946 0.966955i \(-0.582058\pi\)
−0.254946 + 0.966955i \(0.582058\pi\)
\(860\) 0 0
\(861\) 7.47214 0.254650
\(862\) 0 0
\(863\) 3.12461 0.106363 0.0531815 0.998585i \(-0.483064\pi\)
0.0531815 + 0.998585i \(0.483064\pi\)
\(864\) 0 0
\(865\) −12.3262 −0.419105
\(866\) 0 0
\(867\) 12.0000 0.407541
\(868\) 0 0
\(869\) −1.94427 −0.0659549
\(870\) 0 0
\(871\) −1.29180 −0.0437708
\(872\) 0 0
\(873\) −0.236068 −0.00798969
\(874\) 0 0
\(875\) −5.94427 −0.200953
\(876\) 0 0
\(877\) −47.1935 −1.59361 −0.796806 0.604236i \(-0.793478\pi\)
−0.796806 + 0.604236i \(0.793478\pi\)
\(878\) 0 0
\(879\) −7.05573 −0.237984
\(880\) 0 0
\(881\) −35.7082 −1.20304 −0.601520 0.798858i \(-0.705438\pi\)
−0.601520 + 0.798858i \(0.705438\pi\)
\(882\) 0 0
\(883\) −32.7426 −1.10188 −0.550939 0.834546i \(-0.685730\pi\)
−0.550939 + 0.834546i \(0.685730\pi\)
\(884\) 0 0
\(885\) −2.09017 −0.0702603
\(886\) 0 0
\(887\) −20.5623 −0.690415 −0.345207 0.938526i \(-0.612191\pi\)
−0.345207 + 0.938526i \(0.612191\pi\)
\(888\) 0 0
\(889\) 7.14590 0.239666
\(890\) 0 0
\(891\) 0.236068 0.00790857
\(892\) 0 0
\(893\) −2.65248 −0.0887617
\(894\) 0 0
\(895\) 1.72949 0.0578105
\(896\) 0 0
\(897\) −0.381966 −0.0127535
\(898\) 0 0
\(899\) −9.65248 −0.321928
\(900\) 0 0
\(901\) 15.8541 0.528177
\(902\) 0 0
\(903\) −12.5623 −0.418047
\(904\) 0 0
\(905\) 2.61803 0.0870264
\(906\) 0 0
\(907\) 17.9656 0.596537 0.298268 0.954482i \(-0.403591\pi\)
0.298268 + 0.954482i \(0.403591\pi\)
\(908\) 0 0
\(909\) −1.38197 −0.0458369
\(910\) 0 0
\(911\) 32.5410 1.07813 0.539066 0.842264i \(-0.318777\pi\)
0.539066 + 0.842264i \(0.318777\pi\)
\(912\) 0 0
\(913\) −1.87539 −0.0620663
\(914\) 0 0
\(915\) 3.29180 0.108823
\(916\) 0 0
\(917\) −0.0557281 −0.00184030
\(918\) 0 0
\(919\) −6.41641 −0.211658 −0.105829 0.994384i \(-0.533750\pi\)
−0.105829 + 0.994384i \(0.533750\pi\)
\(920\) 0 0
\(921\) −5.76393 −0.189928
\(922\) 0 0
\(923\) −4.14590 −0.136464
\(924\) 0 0
\(925\) 25.2705 0.830889
\(926\) 0 0
\(927\) −8.94427 −0.293768
\(928\) 0 0
\(929\) −30.5066 −1.00089 −0.500444 0.865769i \(-0.666830\pi\)
−0.500444 + 0.865769i \(0.666830\pi\)
\(930\) 0 0
\(931\) 3.47214 0.113795
\(932\) 0 0
\(933\) 15.9787 0.523120
\(934\) 0 0
\(935\) 0.326238 0.0106691
\(936\) 0 0
\(937\) 31.4721 1.02815 0.514075 0.857745i \(-0.328135\pi\)
0.514075 + 0.857745i \(0.328135\pi\)
\(938\) 0 0
\(939\) −6.47214 −0.211210
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 0 0
\(943\) 7.47214 0.243326
\(944\) 0 0
\(945\) −0.618034 −0.0201046
\(946\) 0 0
\(947\) 24.1115 0.783517 0.391759 0.920068i \(-0.371867\pi\)
0.391759 + 0.920068i \(0.371867\pi\)
\(948\) 0 0
\(949\) 0.742646 0.0241073
\(950\) 0 0
\(951\) −34.0344 −1.10364
\(952\) 0 0
\(953\) −13.4377 −0.435290 −0.217645 0.976028i \(-0.569837\pi\)
−0.217645 + 0.976028i \(0.569837\pi\)
\(954\) 0 0
\(955\) −13.0557 −0.422473
\(956\) 0 0
\(957\) −1.29180 −0.0417578
\(958\) 0 0
\(959\) 0.236068 0.00762303
\(960\) 0 0
\(961\) −27.8885 −0.899630
\(962\) 0 0
\(963\) −10.3820 −0.334554
\(964\) 0 0
\(965\) 2.65248 0.0853862
\(966\) 0 0
\(967\) 16.4721 0.529708 0.264854 0.964288i \(-0.414676\pi\)
0.264854 + 0.964288i \(0.414676\pi\)
\(968\) 0 0
\(969\) 7.76393 0.249413
\(970\) 0 0
\(971\) −29.3262 −0.941124 −0.470562 0.882367i \(-0.655949\pi\)
−0.470562 + 0.882367i \(0.655949\pi\)
\(972\) 0 0
\(973\) 12.0344 0.385806
\(974\) 0 0
\(975\) 1.76393 0.0564910
\(976\) 0 0
\(977\) −45.9098 −1.46879 −0.734393 0.678725i \(-0.762533\pi\)
−0.734393 + 0.678725i \(0.762533\pi\)
\(978\) 0 0
\(979\) −0.673762 −0.0215335
\(980\) 0 0
\(981\) −6.61803 −0.211298
\(982\) 0 0
\(983\) 22.7771 0.726476 0.363238 0.931696i \(-0.381671\pi\)
0.363238 + 0.931696i \(0.381671\pi\)
\(984\) 0 0
\(985\) 3.94427 0.125675
\(986\) 0 0
\(987\) −0.763932 −0.0243162
\(988\) 0 0
\(989\) −12.5623 −0.399458
\(990\) 0 0
\(991\) −10.1591 −0.322713 −0.161356 0.986896i \(-0.551587\pi\)
−0.161356 + 0.986896i \(0.551587\pi\)
\(992\) 0 0
\(993\) −1.05573 −0.0335025
\(994\) 0 0
\(995\) 3.50658 0.111166
\(996\) 0 0
\(997\) −12.1246 −0.383990 −0.191995 0.981396i \(-0.561496\pi\)
−0.191995 + 0.981396i \(0.561496\pi\)
\(998\) 0 0
\(999\) 5.47214 0.173131
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.d.1.1 2
3.2 odd 2 5796.2.a.i.1.2 2
4.3 odd 2 7728.2.a.bo.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1932.2.a.d.1.1 2 1.1 even 1 trivial
5796.2.a.i.1.2 2 3.2 odd 2
7728.2.a.bo.1.1 2 4.3 odd 2