Properties

Label 8016.2.a.q.1.2
Level 8016
Weight 2
Character 8016.1
Self dual yes
Analytic conductor 64.008
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.161121.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56399\) of \(x^{5} - x^{4} - 6 x^{3} + 3 x^{2} + 5 x + 1\)
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.85181 q^{5} -2.41580 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.85181 q^{5} -2.41580 q^{7} +1.00000 q^{9} +2.49614 q^{11} +0.116176 q^{13} +1.85181 q^{15} -6.12798 q^{17} +3.34227 q^{19} +2.41580 q^{21} +8.19583 q^{23} -1.57081 q^{25} -1.00000 q^{27} +0.0936338 q^{29} -1.44840 q^{31} -2.49614 q^{33} +4.47360 q^{35} -0.230021 q^{37} -0.116176 q^{39} -4.49763 q^{41} -5.35233 q^{43} -1.85181 q^{45} -7.01021 q^{47} -1.16392 q^{49} +6.12798 q^{51} +3.67773 q^{53} -4.62237 q^{55} -3.34227 q^{57} +1.76972 q^{59} +8.66144 q^{61} -2.41580 q^{63} -0.215135 q^{65} +15.1723 q^{67} -8.19583 q^{69} +8.51567 q^{71} +4.74977 q^{73} +1.57081 q^{75} -6.03017 q^{77} +3.24808 q^{79} +1.00000 q^{81} +0.437500 q^{83} +11.3478 q^{85} -0.0936338 q^{87} +11.9663 q^{89} -0.280657 q^{91} +1.44840 q^{93} -6.18925 q^{95} +5.72224 q^{97} +2.49614 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} - 7q^{5} + 2q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} - 7q^{5} + 2q^{7} + 5q^{9} + 5q^{11} - 8q^{13} + 7q^{15} - 7q^{17} - 2q^{19} - 2q^{21} + 13q^{23} + 2q^{25} - 5q^{27} - 11q^{29} + 12q^{31} - 5q^{33} + 12q^{35} - 7q^{37} + 8q^{39} - 12q^{41} - 7q^{45} + 19q^{47} - 9q^{49} + 7q^{51} - 21q^{53} + q^{55} + 2q^{57} + 7q^{59} - 6q^{61} + 2q^{63} + 14q^{65} - 10q^{67} - 13q^{69} + 35q^{71} - 8q^{73} - 2q^{75} - 6q^{77} + 5q^{81} + 11q^{83} + 5q^{85} + 11q^{87} - 32q^{89} - 5q^{91} - 12q^{93} + 19q^{95} + 11q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) −1.85181 −0.828154 −0.414077 0.910242i \(-0.635896\pi\)
−0.414077 + 0.910242i \(0.635896\pi\)
\(6\) 0 0
\(7\) −2.41580 −0.913086 −0.456543 0.889701i \(-0.650913\pi\)
−0.456543 + 0.889701i \(0.650913\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.49614 0.752614 0.376307 0.926495i \(-0.377194\pi\)
0.376307 + 0.926495i \(0.377194\pi\)
\(12\) 0 0
\(13\) 0.116176 0.0322213 0.0161107 0.999870i \(-0.494872\pi\)
0.0161107 + 0.999870i \(0.494872\pi\)
\(14\) 0 0
\(15\) 1.85181 0.478135
\(16\) 0 0
\(17\) −6.12798 −1.48625 −0.743127 0.669151i \(-0.766658\pi\)
−0.743127 + 0.669151i \(0.766658\pi\)
\(18\) 0 0
\(19\) 3.34227 0.766770 0.383385 0.923589i \(-0.374758\pi\)
0.383385 + 0.923589i \(0.374758\pi\)
\(20\) 0 0
\(21\) 2.41580 0.527170
\(22\) 0 0
\(23\) 8.19583 1.70895 0.854475 0.519493i \(-0.173879\pi\)
0.854475 + 0.519493i \(0.173879\pi\)
\(24\) 0 0
\(25\) −1.57081 −0.314161
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.0936338 0.0173874 0.00869368 0.999962i \(-0.497233\pi\)
0.00869368 + 0.999962i \(0.497233\pi\)
\(30\) 0 0
\(31\) −1.44840 −0.260140 −0.130070 0.991505i \(-0.541520\pi\)
−0.130070 + 0.991505i \(0.541520\pi\)
\(32\) 0 0
\(33\) −2.49614 −0.434522
\(34\) 0 0
\(35\) 4.47360 0.756176
\(36\) 0 0
\(37\) −0.230021 −0.0378152 −0.0189076 0.999821i \(-0.506019\pi\)
−0.0189076 + 0.999821i \(0.506019\pi\)
\(38\) 0 0
\(39\) −0.116176 −0.0186030
\(40\) 0 0
\(41\) −4.49763 −0.702411 −0.351206 0.936298i \(-0.614228\pi\)
−0.351206 + 0.936298i \(0.614228\pi\)
\(42\) 0 0
\(43\) −5.35233 −0.816223 −0.408111 0.912932i \(-0.633812\pi\)
−0.408111 + 0.912932i \(0.633812\pi\)
\(44\) 0 0
\(45\) −1.85181 −0.276051
\(46\) 0 0
\(47\) −7.01021 −1.02254 −0.511272 0.859419i \(-0.670826\pi\)
−0.511272 + 0.859419i \(0.670826\pi\)
\(48\) 0 0
\(49\) −1.16392 −0.166274
\(50\) 0 0
\(51\) 6.12798 0.858089
\(52\) 0 0
\(53\) 3.67773 0.505176 0.252588 0.967574i \(-0.418718\pi\)
0.252588 + 0.967574i \(0.418718\pi\)
\(54\) 0 0
\(55\) −4.62237 −0.623280
\(56\) 0 0
\(57\) −3.34227 −0.442695
\(58\) 0 0
\(59\) 1.76972 0.230398 0.115199 0.993342i \(-0.463249\pi\)
0.115199 + 0.993342i \(0.463249\pi\)
\(60\) 0 0
\(61\) 8.66144 1.10898 0.554492 0.832189i \(-0.312913\pi\)
0.554492 + 0.832189i \(0.312913\pi\)
\(62\) 0 0
\(63\) −2.41580 −0.304362
\(64\) 0 0
\(65\) −0.215135 −0.0266842
\(66\) 0 0
\(67\) 15.1723 1.85359 0.926795 0.375569i \(-0.122553\pi\)
0.926795 + 0.375569i \(0.122553\pi\)
\(68\) 0 0
\(69\) −8.19583 −0.986662
\(70\) 0 0
\(71\) 8.51567 1.01062 0.505312 0.862937i \(-0.331377\pi\)
0.505312 + 0.862937i \(0.331377\pi\)
\(72\) 0 0
\(73\) 4.74977 0.555918 0.277959 0.960593i \(-0.410342\pi\)
0.277959 + 0.960593i \(0.410342\pi\)
\(74\) 0 0
\(75\) 1.57081 0.181381
\(76\) 0 0
\(77\) −6.03017 −0.687201
\(78\) 0 0
\(79\) 3.24808 0.365437 0.182719 0.983165i \(-0.441510\pi\)
0.182719 + 0.983165i \(0.441510\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.437500 0.0480219 0.0240109 0.999712i \(-0.492356\pi\)
0.0240109 + 0.999712i \(0.492356\pi\)
\(84\) 0 0
\(85\) 11.3478 1.23085
\(86\) 0 0
\(87\) −0.0936338 −0.0100386
\(88\) 0 0
\(89\) 11.9663 1.26842 0.634212 0.773159i \(-0.281325\pi\)
0.634212 + 0.773159i \(0.281325\pi\)
\(90\) 0 0
\(91\) −0.280657 −0.0294209
\(92\) 0 0
\(93\) 1.44840 0.150192
\(94\) 0 0
\(95\) −6.18925 −0.635004
\(96\) 0 0
\(97\) 5.72224 0.581005 0.290503 0.956874i \(-0.406178\pi\)
0.290503 + 0.956874i \(0.406178\pi\)
\(98\) 0 0
\(99\) 2.49614 0.250871
\(100\) 0 0
\(101\) −18.5958 −1.85035 −0.925176 0.379539i \(-0.876083\pi\)
−0.925176 + 0.379539i \(0.876083\pi\)
\(102\) 0 0
\(103\) 19.3000 1.90169 0.950843 0.309674i \(-0.100220\pi\)
0.950843 + 0.309674i \(0.100220\pi\)
\(104\) 0 0
\(105\) −4.47360 −0.436578
\(106\) 0 0
\(107\) 18.2861 1.76779 0.883893 0.467689i \(-0.154913\pi\)
0.883893 + 0.467689i \(0.154913\pi\)
\(108\) 0 0
\(109\) −16.0245 −1.53487 −0.767435 0.641127i \(-0.778467\pi\)
−0.767435 + 0.641127i \(0.778467\pi\)
\(110\) 0 0
\(111\) 0.230021 0.0218326
\(112\) 0 0
\(113\) −9.02419 −0.848924 −0.424462 0.905446i \(-0.639537\pi\)
−0.424462 + 0.905446i \(0.639537\pi\)
\(114\) 0 0
\(115\) −15.1771 −1.41527
\(116\) 0 0
\(117\) 0.116176 0.0107404
\(118\) 0 0
\(119\) 14.8040 1.35708
\(120\) 0 0
\(121\) −4.76930 −0.433572
\(122\) 0 0
\(123\) 4.49763 0.405537
\(124\) 0 0
\(125\) 12.1679 1.08833
\(126\) 0 0
\(127\) 0.925332 0.0821099 0.0410550 0.999157i \(-0.486928\pi\)
0.0410550 + 0.999157i \(0.486928\pi\)
\(128\) 0 0
\(129\) 5.35233 0.471246
\(130\) 0 0
\(131\) −2.88590 −0.252142 −0.126071 0.992021i \(-0.540237\pi\)
−0.126071 + 0.992021i \(0.540237\pi\)
\(132\) 0 0
\(133\) −8.07426 −0.700127
\(134\) 0 0
\(135\) 1.85181 0.159378
\(136\) 0 0
\(137\) 18.1131 1.54750 0.773752 0.633488i \(-0.218377\pi\)
0.773752 + 0.633488i \(0.218377\pi\)
\(138\) 0 0
\(139\) −14.8150 −1.25659 −0.628294 0.777976i \(-0.716247\pi\)
−0.628294 + 0.777976i \(0.716247\pi\)
\(140\) 0 0
\(141\) 7.01021 0.590366
\(142\) 0 0
\(143\) 0.289991 0.0242502
\(144\) 0 0
\(145\) −0.173392 −0.0143994
\(146\) 0 0
\(147\) 1.16392 0.0959983
\(148\) 0 0
\(149\) −0.577645 −0.0473225 −0.0236613 0.999720i \(-0.507532\pi\)
−0.0236613 + 0.999720i \(0.507532\pi\)
\(150\) 0 0
\(151\) 1.65587 0.134753 0.0673766 0.997728i \(-0.478537\pi\)
0.0673766 + 0.997728i \(0.478537\pi\)
\(152\) 0 0
\(153\) −6.12798 −0.495418
\(154\) 0 0
\(155\) 2.68215 0.215436
\(156\) 0 0
\(157\) −4.20297 −0.335434 −0.167717 0.985835i \(-0.553639\pi\)
−0.167717 + 0.985835i \(0.553639\pi\)
\(158\) 0 0
\(159\) −3.67773 −0.291663
\(160\) 0 0
\(161\) −19.7995 −1.56042
\(162\) 0 0
\(163\) −20.7509 −1.62534 −0.812668 0.582727i \(-0.801986\pi\)
−0.812668 + 0.582727i \(0.801986\pi\)
\(164\) 0 0
\(165\) 4.62237 0.359851
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −12.9865 −0.998962
\(170\) 0 0
\(171\) 3.34227 0.255590
\(172\) 0 0
\(173\) −14.4252 −1.09673 −0.548363 0.836240i \(-0.684749\pi\)
−0.548363 + 0.836240i \(0.684749\pi\)
\(174\) 0 0
\(175\) 3.79475 0.286856
\(176\) 0 0
\(177\) −1.76972 −0.133020
\(178\) 0 0
\(179\) −21.8860 −1.63583 −0.817916 0.575337i \(-0.804871\pi\)
−0.817916 + 0.575337i \(0.804871\pi\)
\(180\) 0 0
\(181\) −10.9551 −0.814284 −0.407142 0.913365i \(-0.633475\pi\)
−0.407142 + 0.913365i \(0.633475\pi\)
\(182\) 0 0
\(183\) −8.66144 −0.640272
\(184\) 0 0
\(185\) 0.425955 0.0313168
\(186\) 0 0
\(187\) −15.2963 −1.11857
\(188\) 0 0
\(189\) 2.41580 0.175723
\(190\) 0 0
\(191\) −4.40538 −0.318762 −0.159381 0.987217i \(-0.550950\pi\)
−0.159381 + 0.987217i \(0.550950\pi\)
\(192\) 0 0
\(193\) 2.81738 0.202799 0.101400 0.994846i \(-0.467668\pi\)
0.101400 + 0.994846i \(0.467668\pi\)
\(194\) 0 0
\(195\) 0.215135 0.0154061
\(196\) 0 0
\(197\) 20.7266 1.47671 0.738353 0.674414i \(-0.235604\pi\)
0.738353 + 0.674414i \(0.235604\pi\)
\(198\) 0 0
\(199\) 0.813799 0.0576887 0.0288444 0.999584i \(-0.490817\pi\)
0.0288444 + 0.999584i \(0.490817\pi\)
\(200\) 0 0
\(201\) −15.1723 −1.07017
\(202\) 0 0
\(203\) −0.226200 −0.0158762
\(204\) 0 0
\(205\) 8.32874 0.581705
\(206\) 0 0
\(207\) 8.19583 0.569650
\(208\) 0 0
\(209\) 8.34278 0.577082
\(210\) 0 0
\(211\) −27.3088 −1.88002 −0.940009 0.341151i \(-0.889183\pi\)
−0.940009 + 0.341151i \(0.889183\pi\)
\(212\) 0 0
\(213\) −8.51567 −0.583484
\(214\) 0 0
\(215\) 9.91149 0.675958
\(216\) 0 0
\(217\) 3.49903 0.237530
\(218\) 0 0
\(219\) −4.74977 −0.320960
\(220\) 0 0
\(221\) −0.711923 −0.0478891
\(222\) 0 0
\(223\) 0.297772 0.0199403 0.00997015 0.999950i \(-0.496826\pi\)
0.00997015 + 0.999950i \(0.496826\pi\)
\(224\) 0 0
\(225\) −1.57081 −0.104720
\(226\) 0 0
\(227\) 2.46412 0.163550 0.0817748 0.996651i \(-0.473941\pi\)
0.0817748 + 0.996651i \(0.473941\pi\)
\(228\) 0 0
\(229\) 7.52769 0.497444 0.248722 0.968575i \(-0.419989\pi\)
0.248722 + 0.968575i \(0.419989\pi\)
\(230\) 0 0
\(231\) 6.03017 0.396756
\(232\) 0 0
\(233\) 7.47360 0.489612 0.244806 0.969572i \(-0.421276\pi\)
0.244806 + 0.969572i \(0.421276\pi\)
\(234\) 0 0
\(235\) 12.9816 0.846824
\(236\) 0 0
\(237\) −3.24808 −0.210985
\(238\) 0 0
\(239\) 12.2797 0.794306 0.397153 0.917752i \(-0.369998\pi\)
0.397153 + 0.917752i \(0.369998\pi\)
\(240\) 0 0
\(241\) −12.7966 −0.824299 −0.412150 0.911116i \(-0.635222\pi\)
−0.412150 + 0.911116i \(0.635222\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.15535 0.137700
\(246\) 0 0
\(247\) 0.388291 0.0247064
\(248\) 0 0
\(249\) −0.437500 −0.0277254
\(250\) 0 0
\(251\) −13.7924 −0.870571 −0.435285 0.900292i \(-0.643353\pi\)
−0.435285 + 0.900292i \(0.643353\pi\)
\(252\) 0 0
\(253\) 20.4579 1.28618
\(254\) 0 0
\(255\) −11.3478 −0.710630
\(256\) 0 0
\(257\) −2.03436 −0.126900 −0.0634501 0.997985i \(-0.520210\pi\)
−0.0634501 + 0.997985i \(0.520210\pi\)
\(258\) 0 0
\(259\) 0.555684 0.0345285
\(260\) 0 0
\(261\) 0.0936338 0.00579579
\(262\) 0 0
\(263\) −22.6680 −1.39777 −0.698886 0.715233i \(-0.746320\pi\)
−0.698886 + 0.715233i \(0.746320\pi\)
\(264\) 0 0
\(265\) −6.81046 −0.418363
\(266\) 0 0
\(267\) −11.9663 −0.732325
\(268\) 0 0
\(269\) −23.5210 −1.43410 −0.717052 0.697020i \(-0.754509\pi\)
−0.717052 + 0.697020i \(0.754509\pi\)
\(270\) 0 0
\(271\) −15.5238 −0.943005 −0.471503 0.881865i \(-0.656288\pi\)
−0.471503 + 0.881865i \(0.656288\pi\)
\(272\) 0 0
\(273\) 0.280657 0.0169861
\(274\) 0 0
\(275\) −3.92095 −0.236442
\(276\) 0 0
\(277\) 9.69430 0.582474 0.291237 0.956651i \(-0.405933\pi\)
0.291237 + 0.956651i \(0.405933\pi\)
\(278\) 0 0
\(279\) −1.44840 −0.0867132
\(280\) 0 0
\(281\) −6.65022 −0.396719 −0.198359 0.980129i \(-0.563561\pi\)
−0.198359 + 0.980129i \(0.563561\pi\)
\(282\) 0 0
\(283\) −20.1057 −1.19516 −0.597580 0.801809i \(-0.703871\pi\)
−0.597580 + 0.801809i \(0.703871\pi\)
\(284\) 0 0
\(285\) 6.18925 0.366620
\(286\) 0 0
\(287\) 10.8654 0.641362
\(288\) 0 0
\(289\) 20.5521 1.20895
\(290\) 0 0
\(291\) −5.72224 −0.335443
\(292\) 0 0
\(293\) 2.72431 0.159156 0.0795781 0.996829i \(-0.474643\pi\)
0.0795781 + 0.996829i \(0.474643\pi\)
\(294\) 0 0
\(295\) −3.27718 −0.190805
\(296\) 0 0
\(297\) −2.49614 −0.144841
\(298\) 0 0
\(299\) 0.952157 0.0550646
\(300\) 0 0
\(301\) 12.9302 0.745282
\(302\) 0 0
\(303\) 18.5958 1.06830
\(304\) 0 0
\(305\) −16.0393 −0.918410
\(306\) 0 0
\(307\) 7.63757 0.435899 0.217950 0.975960i \(-0.430063\pi\)
0.217950 + 0.975960i \(0.430063\pi\)
\(308\) 0 0
\(309\) −19.3000 −1.09794
\(310\) 0 0
\(311\) −19.6655 −1.11513 −0.557565 0.830133i \(-0.688264\pi\)
−0.557565 + 0.830133i \(0.688264\pi\)
\(312\) 0 0
\(313\) −27.6680 −1.56389 −0.781944 0.623349i \(-0.785772\pi\)
−0.781944 + 0.623349i \(0.785772\pi\)
\(314\) 0 0
\(315\) 4.47360 0.252059
\(316\) 0 0
\(317\) 8.67973 0.487502 0.243751 0.969838i \(-0.421622\pi\)
0.243751 + 0.969838i \(0.421622\pi\)
\(318\) 0 0
\(319\) 0.233723 0.0130860
\(320\) 0 0
\(321\) −18.2861 −1.02063
\(322\) 0 0
\(323\) −20.4814 −1.13962
\(324\) 0 0
\(325\) −0.182490 −0.0101227
\(326\) 0 0
\(327\) 16.0245 0.886158
\(328\) 0 0
\(329\) 16.9353 0.933671
\(330\) 0 0
\(331\) −12.1719 −0.669027 −0.334513 0.942391i \(-0.608572\pi\)
−0.334513 + 0.942391i \(0.608572\pi\)
\(332\) 0 0
\(333\) −0.230021 −0.0126051
\(334\) 0 0
\(335\) −28.0962 −1.53506
\(336\) 0 0
\(337\) −14.9558 −0.814692 −0.407346 0.913274i \(-0.633546\pi\)
−0.407346 + 0.913274i \(0.633546\pi\)
\(338\) 0 0
\(339\) 9.02419 0.490127
\(340\) 0 0
\(341\) −3.61539 −0.195785
\(342\) 0 0
\(343\) 19.7224 1.06491
\(344\) 0 0
\(345\) 15.1771 0.817108
\(346\) 0 0
\(347\) 18.7269 1.00531 0.502655 0.864487i \(-0.332356\pi\)
0.502655 + 0.864487i \(0.332356\pi\)
\(348\) 0 0
\(349\) 15.2459 0.816094 0.408047 0.912961i \(-0.366210\pi\)
0.408047 + 0.912961i \(0.366210\pi\)
\(350\) 0 0
\(351\) −0.116176 −0.00620100
\(352\) 0 0
\(353\) −6.76872 −0.360262 −0.180131 0.983643i \(-0.557652\pi\)
−0.180131 + 0.983643i \(0.557652\pi\)
\(354\) 0 0
\(355\) −15.7694 −0.836952
\(356\) 0 0
\(357\) −14.8040 −0.783509
\(358\) 0 0
\(359\) 10.6775 0.563537 0.281769 0.959482i \(-0.409079\pi\)
0.281769 + 0.959482i \(0.409079\pi\)
\(360\) 0 0
\(361\) −7.82920 −0.412063
\(362\) 0 0
\(363\) 4.76930 0.250323
\(364\) 0 0
\(365\) −8.79566 −0.460386
\(366\) 0 0
\(367\) 4.63745 0.242073 0.121036 0.992648i \(-0.461378\pi\)
0.121036 + 0.992648i \(0.461378\pi\)
\(368\) 0 0
\(369\) −4.49763 −0.234137
\(370\) 0 0
\(371\) −8.88466 −0.461269
\(372\) 0 0
\(373\) 11.0186 0.570521 0.285260 0.958450i \(-0.407920\pi\)
0.285260 + 0.958450i \(0.407920\pi\)
\(374\) 0 0
\(375\) −12.1679 −0.628346
\(376\) 0 0
\(377\) 0.0108780 0.000560244 0
\(378\) 0 0
\(379\) −23.2728 −1.19544 −0.597721 0.801705i \(-0.703927\pi\)
−0.597721 + 0.801705i \(0.703927\pi\)
\(380\) 0 0
\(381\) −0.925332 −0.0474062
\(382\) 0 0
\(383\) 33.6147 1.71763 0.858816 0.512284i \(-0.171201\pi\)
0.858816 + 0.512284i \(0.171201\pi\)
\(384\) 0 0
\(385\) 11.1667 0.569108
\(386\) 0 0
\(387\) −5.35233 −0.272074
\(388\) 0 0
\(389\) −22.6735 −1.14959 −0.574795 0.818298i \(-0.694918\pi\)
−0.574795 + 0.818298i \(0.694918\pi\)
\(390\) 0 0
\(391\) −50.2239 −2.53993
\(392\) 0 0
\(393\) 2.88590 0.145574
\(394\) 0 0
\(395\) −6.01482 −0.302638
\(396\) 0 0
\(397\) −32.1120 −1.61166 −0.805828 0.592149i \(-0.798280\pi\)
−0.805828 + 0.592149i \(0.798280\pi\)
\(398\) 0 0
\(399\) 8.07426 0.404219
\(400\) 0 0
\(401\) −5.31642 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(402\) 0 0
\(403\) −0.168268 −0.00838205
\(404\) 0 0
\(405\) −1.85181 −0.0920171
\(406\) 0 0
\(407\) −0.574164 −0.0284603
\(408\) 0 0
\(409\) −28.0684 −1.38789 −0.693947 0.720026i \(-0.744130\pi\)
−0.693947 + 0.720026i \(0.744130\pi\)
\(410\) 0 0
\(411\) −18.1131 −0.893452
\(412\) 0 0
\(413\) −4.27529 −0.210373
\(414\) 0 0
\(415\) −0.810166 −0.0397695
\(416\) 0 0
\(417\) 14.8150 0.725492
\(418\) 0 0
\(419\) −5.57807 −0.272506 −0.136253 0.990674i \(-0.543506\pi\)
−0.136253 + 0.990674i \(0.543506\pi\)
\(420\) 0 0
\(421\) 6.45061 0.314383 0.157192 0.987568i \(-0.449756\pi\)
0.157192 + 0.987568i \(0.449756\pi\)
\(422\) 0 0
\(423\) −7.01021 −0.340848
\(424\) 0 0
\(425\) 9.62587 0.466923
\(426\) 0 0
\(427\) −20.9243 −1.01260
\(428\) 0 0
\(429\) −0.289991 −0.0140009
\(430\) 0 0
\(431\) 31.4574 1.51525 0.757624 0.652692i \(-0.226360\pi\)
0.757624 + 0.652692i \(0.226360\pi\)
\(432\) 0 0
\(433\) 9.89285 0.475420 0.237710 0.971336i \(-0.423603\pi\)
0.237710 + 0.971336i \(0.423603\pi\)
\(434\) 0 0
\(435\) 0.173392 0.00831350
\(436\) 0 0
\(437\) 27.3927 1.31037
\(438\) 0 0
\(439\) −15.9324 −0.760412 −0.380206 0.924902i \(-0.624147\pi\)
−0.380206 + 0.924902i \(0.624147\pi\)
\(440\) 0 0
\(441\) −1.16392 −0.0554247
\(442\) 0 0
\(443\) −34.7291 −1.65003 −0.825014 0.565112i \(-0.808833\pi\)
−0.825014 + 0.565112i \(0.808833\pi\)
\(444\) 0 0
\(445\) −22.1593 −1.05045
\(446\) 0 0
\(447\) 0.577645 0.0273217
\(448\) 0 0
\(449\) −22.3182 −1.05326 −0.526631 0.850094i \(-0.676545\pi\)
−0.526631 + 0.850094i \(0.676545\pi\)
\(450\) 0 0
\(451\) −11.2267 −0.528645
\(452\) 0 0
\(453\) −1.65587 −0.0777998
\(454\) 0 0
\(455\) 0.519723 0.0243650
\(456\) 0 0
\(457\) 3.44665 0.161227 0.0806137 0.996745i \(-0.474312\pi\)
0.0806137 + 0.996745i \(0.474312\pi\)
\(458\) 0 0
\(459\) 6.12798 0.286030
\(460\) 0 0
\(461\) −9.03541 −0.420821 −0.210411 0.977613i \(-0.567480\pi\)
−0.210411 + 0.977613i \(0.567480\pi\)
\(462\) 0 0
\(463\) 12.9438 0.601551 0.300776 0.953695i \(-0.402755\pi\)
0.300776 + 0.953695i \(0.402755\pi\)
\(464\) 0 0
\(465\) −2.68215 −0.124382
\(466\) 0 0
\(467\) 15.5010 0.717303 0.358651 0.933472i \(-0.383237\pi\)
0.358651 + 0.933472i \(0.383237\pi\)
\(468\) 0 0
\(469\) −36.6532 −1.69249
\(470\) 0 0
\(471\) 4.20297 0.193663
\(472\) 0 0
\(473\) −13.3602 −0.614300
\(474\) 0 0
\(475\) −5.25007 −0.240890
\(476\) 0 0
\(477\) 3.67773 0.168392
\(478\) 0 0
\(479\) 12.8336 0.586381 0.293190 0.956054i \(-0.405283\pi\)
0.293190 + 0.956054i \(0.405283\pi\)
\(480\) 0 0
\(481\) −0.0267229 −0.00121846
\(482\) 0 0
\(483\) 19.7995 0.900908
\(484\) 0 0
\(485\) −10.5965 −0.481162
\(486\) 0 0
\(487\) −4.39313 −0.199072 −0.0995359 0.995034i \(-0.531736\pi\)
−0.0995359 + 0.995034i \(0.531736\pi\)
\(488\) 0 0
\(489\) 20.7509 0.938388
\(490\) 0 0
\(491\) −16.0337 −0.723593 −0.361796 0.932257i \(-0.617836\pi\)
−0.361796 + 0.932257i \(0.617836\pi\)
\(492\) 0 0
\(493\) −0.573786 −0.0258420
\(494\) 0 0
\(495\) −4.62237 −0.207760
\(496\) 0 0
\(497\) −20.5721 −0.922786
\(498\) 0 0
\(499\) −32.1588 −1.43963 −0.719813 0.694168i \(-0.755772\pi\)
−0.719813 + 0.694168i \(0.755772\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −7.88618 −0.351627 −0.175814 0.984423i \(-0.556256\pi\)
−0.175814 + 0.984423i \(0.556256\pi\)
\(504\) 0 0
\(505\) 34.4359 1.53238
\(506\) 0 0
\(507\) 12.9865 0.576751
\(508\) 0 0
\(509\) 24.9425 1.10556 0.552779 0.833328i \(-0.313568\pi\)
0.552779 + 0.833328i \(0.313568\pi\)
\(510\) 0 0
\(511\) −11.4745 −0.507601
\(512\) 0 0
\(513\) −3.34227 −0.147565
\(514\) 0 0
\(515\) −35.7399 −1.57489
\(516\) 0 0
\(517\) −17.4985 −0.769581
\(518\) 0 0
\(519\) 14.4252 0.633196
\(520\) 0 0
\(521\) 3.76759 0.165061 0.0825305 0.996589i \(-0.473700\pi\)
0.0825305 + 0.996589i \(0.473700\pi\)
\(522\) 0 0
\(523\) −22.7409 −0.994389 −0.497195 0.867639i \(-0.665636\pi\)
−0.497195 + 0.867639i \(0.665636\pi\)
\(524\) 0 0
\(525\) −3.79475 −0.165616
\(526\) 0 0
\(527\) 8.87574 0.386633
\(528\) 0 0
\(529\) 44.1717 1.92051
\(530\) 0 0
\(531\) 1.76972 0.0767993
\(532\) 0 0
\(533\) −0.522515 −0.0226326
\(534\) 0 0
\(535\) −33.8624 −1.46400
\(536\) 0 0
\(537\) 21.8860 0.944449
\(538\) 0 0
\(539\) −2.90530 −0.125140
\(540\) 0 0
\(541\) 10.6993 0.460000 0.230000 0.973191i \(-0.426127\pi\)
0.230000 + 0.973191i \(0.426127\pi\)
\(542\) 0 0
\(543\) 10.9551 0.470127
\(544\) 0 0
\(545\) 29.6743 1.27111
\(546\) 0 0
\(547\) 20.8098 0.889763 0.444881 0.895590i \(-0.353246\pi\)
0.444881 + 0.895590i \(0.353246\pi\)
\(548\) 0 0
\(549\) 8.66144 0.369661
\(550\) 0 0
\(551\) 0.312950 0.0133321
\(552\) 0 0
\(553\) −7.84670 −0.333676
\(554\) 0 0
\(555\) −0.425955 −0.0180808
\(556\) 0 0
\(557\) −21.7669 −0.922292 −0.461146 0.887324i \(-0.652562\pi\)
−0.461146 + 0.887324i \(0.652562\pi\)
\(558\) 0 0
\(559\) −0.621811 −0.0262998
\(560\) 0 0
\(561\) 15.2963 0.645810
\(562\) 0 0
\(563\) 22.6695 0.955404 0.477702 0.878522i \(-0.341470\pi\)
0.477702 + 0.878522i \(0.341470\pi\)
\(564\) 0 0
\(565\) 16.7111 0.703040
\(566\) 0 0
\(567\) −2.41580 −0.101454
\(568\) 0 0
\(569\) 20.9889 0.879898 0.439949 0.898023i \(-0.354996\pi\)
0.439949 + 0.898023i \(0.354996\pi\)
\(570\) 0 0
\(571\) −31.4599 −1.31656 −0.658278 0.752775i \(-0.728715\pi\)
−0.658278 + 0.752775i \(0.728715\pi\)
\(572\) 0 0
\(573\) 4.40538 0.184038
\(574\) 0 0
\(575\) −12.8741 −0.536885
\(576\) 0 0
\(577\) −25.6636 −1.06839 −0.534196 0.845361i \(-0.679385\pi\)
−0.534196 + 0.845361i \(0.679385\pi\)
\(578\) 0 0
\(579\) −2.81738 −0.117086
\(580\) 0 0
\(581\) −1.05691 −0.0438481
\(582\) 0 0
\(583\) 9.18013 0.380202
\(584\) 0 0
\(585\) −0.215135 −0.00889474
\(586\) 0 0
\(587\) −27.5089 −1.13542 −0.567708 0.823230i \(-0.692170\pi\)
−0.567708 + 0.823230i \(0.692170\pi\)
\(588\) 0 0
\(589\) −4.84094 −0.199467
\(590\) 0 0
\(591\) −20.7266 −0.852577
\(592\) 0 0
\(593\) −27.4088 −1.12554 −0.562772 0.826612i \(-0.690265\pi\)
−0.562772 + 0.826612i \(0.690265\pi\)
\(594\) 0 0
\(595\) −27.4141 −1.12387
\(596\) 0 0
\(597\) −0.813799 −0.0333066
\(598\) 0 0
\(599\) −35.6007 −1.45460 −0.727302 0.686317i \(-0.759226\pi\)
−0.727302 + 0.686317i \(0.759226\pi\)
\(600\) 0 0
\(601\) −4.45143 −0.181578 −0.0907888 0.995870i \(-0.528939\pi\)
−0.0907888 + 0.995870i \(0.528939\pi\)
\(602\) 0 0
\(603\) 15.1723 0.617863
\(604\) 0 0
\(605\) 8.83182 0.359065
\(606\) 0 0
\(607\) 39.6466 1.60921 0.804604 0.593812i \(-0.202378\pi\)
0.804604 + 0.593812i \(0.202378\pi\)
\(608\) 0 0
\(609\) 0.226200 0.00916610
\(610\) 0 0
\(611\) −0.814417 −0.0329478
\(612\) 0 0
\(613\) 37.0509 1.49647 0.748237 0.663432i \(-0.230901\pi\)
0.748237 + 0.663432i \(0.230901\pi\)
\(614\) 0 0
\(615\) −8.32874 −0.335847
\(616\) 0 0
\(617\) 35.9525 1.44739 0.723697 0.690118i \(-0.242441\pi\)
0.723697 + 0.690118i \(0.242441\pi\)
\(618\) 0 0
\(619\) −41.1996 −1.65595 −0.827976 0.560763i \(-0.810508\pi\)
−0.827976 + 0.560763i \(0.810508\pi\)
\(620\) 0 0
\(621\) −8.19583 −0.328887
\(622\) 0 0
\(623\) −28.9081 −1.15818
\(624\) 0 0
\(625\) −14.6785 −0.587142
\(626\) 0 0
\(627\) −8.34278 −0.333178
\(628\) 0 0
\(629\) 1.40956 0.0562030
\(630\) 0 0
\(631\) 25.7109 1.02354 0.511768 0.859124i \(-0.328991\pi\)
0.511768 + 0.859124i \(0.328991\pi\)
\(632\) 0 0
\(633\) 27.3088 1.08543
\(634\) 0 0
\(635\) −1.71354 −0.0679996
\(636\) 0 0
\(637\) −0.135219 −0.00535757
\(638\) 0 0
\(639\) 8.51567 0.336875
\(640\) 0 0
\(641\) 2.43782 0.0962879 0.0481440 0.998840i \(-0.484669\pi\)
0.0481440 + 0.998840i \(0.484669\pi\)
\(642\) 0 0
\(643\) −30.5238 −1.20374 −0.601870 0.798594i \(-0.705577\pi\)
−0.601870 + 0.798594i \(0.705577\pi\)
\(644\) 0 0
\(645\) −9.91149 −0.390265
\(646\) 0 0
\(647\) 3.26910 0.128521 0.0642607 0.997933i \(-0.479531\pi\)
0.0642607 + 0.997933i \(0.479531\pi\)
\(648\) 0 0
\(649\) 4.41746 0.173401
\(650\) 0 0
\(651\) −3.49903 −0.137138
\(652\) 0 0
\(653\) 5.24631 0.205304 0.102652 0.994717i \(-0.467267\pi\)
0.102652 + 0.994717i \(0.467267\pi\)
\(654\) 0 0
\(655\) 5.34413 0.208812
\(656\) 0 0
\(657\) 4.74977 0.185306
\(658\) 0 0
\(659\) 24.5139 0.954925 0.477462 0.878652i \(-0.341557\pi\)
0.477462 + 0.878652i \(0.341557\pi\)
\(660\) 0 0
\(661\) −45.1848 −1.75749 −0.878743 0.477295i \(-0.841617\pi\)
−0.878743 + 0.477295i \(0.841617\pi\)
\(662\) 0 0
\(663\) 0.711923 0.0276488
\(664\) 0 0
\(665\) 14.9520 0.579813
\(666\) 0 0
\(667\) 0.767407 0.0297141
\(668\) 0 0
\(669\) −0.297772 −0.0115125
\(670\) 0 0
\(671\) 21.6202 0.834637
\(672\) 0 0
\(673\) 26.7834 1.03243 0.516213 0.856460i \(-0.327341\pi\)
0.516213 + 0.856460i \(0.327341\pi\)
\(674\) 0 0
\(675\) 1.57081 0.0604603
\(676\) 0 0
\(677\) 43.3493 1.66605 0.833025 0.553235i \(-0.186607\pi\)
0.833025 + 0.553235i \(0.186607\pi\)
\(678\) 0 0
\(679\) −13.8238 −0.530508
\(680\) 0 0
\(681\) −2.46412 −0.0944253
\(682\) 0 0
\(683\) 10.0366 0.384040 0.192020 0.981391i \(-0.438496\pi\)
0.192020 + 0.981391i \(0.438496\pi\)
\(684\) 0 0
\(685\) −33.5420 −1.28157
\(686\) 0 0
\(687\) −7.52769 −0.287199
\(688\) 0 0
\(689\) 0.427263 0.0162774
\(690\) 0 0
\(691\) 17.9872 0.684266 0.342133 0.939652i \(-0.388851\pi\)
0.342133 + 0.939652i \(0.388851\pi\)
\(692\) 0 0
\(693\) −6.03017 −0.229067
\(694\) 0 0
\(695\) 27.4345 1.04065
\(696\) 0 0
\(697\) 27.5614 1.04396
\(698\) 0 0
\(699\) −7.47360 −0.282677
\(700\) 0 0
\(701\) 30.7815 1.16260 0.581301 0.813689i \(-0.302544\pi\)
0.581301 + 0.813689i \(0.302544\pi\)
\(702\) 0 0
\(703\) −0.768793 −0.0289956
\(704\) 0 0
\(705\) −12.9816 −0.488914
\(706\) 0 0
\(707\) 44.9237 1.68953
\(708\) 0 0
\(709\) −6.09943 −0.229069 −0.114534 0.993419i \(-0.536538\pi\)
−0.114534 + 0.993419i \(0.536538\pi\)
\(710\) 0 0
\(711\) 3.24808 0.121812
\(712\) 0 0
\(713\) −11.8708 −0.444565
\(714\) 0 0
\(715\) −0.537007 −0.0200829
\(716\) 0 0
\(717\) −12.2797 −0.458593
\(718\) 0 0
\(719\) 24.6860 0.920632 0.460316 0.887755i \(-0.347736\pi\)
0.460316 + 0.887755i \(0.347736\pi\)
\(720\) 0 0
\(721\) −46.6249 −1.73640
\(722\) 0 0
\(723\) 12.7966 0.475909
\(724\) 0 0
\(725\) −0.147081 −0.00546243
\(726\) 0 0
\(727\) −36.0712 −1.33781 −0.668903 0.743349i \(-0.733236\pi\)
−0.668903 + 0.743349i \(0.733236\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 32.7990 1.21311
\(732\) 0 0
\(733\) −30.1781 −1.11465 −0.557327 0.830293i \(-0.688173\pi\)
−0.557327 + 0.830293i \(0.688173\pi\)
\(734\) 0 0
\(735\) −2.15535 −0.0795014
\(736\) 0 0
\(737\) 37.8721 1.39504
\(738\) 0 0
\(739\) 11.5288 0.424093 0.212047 0.977260i \(-0.431987\pi\)
0.212047 + 0.977260i \(0.431987\pi\)
\(740\) 0 0
\(741\) −0.388291 −0.0142642
\(742\) 0 0
\(743\) 18.5601 0.680904 0.340452 0.940262i \(-0.389420\pi\)
0.340452 + 0.940262i \(0.389420\pi\)
\(744\) 0 0
\(745\) 1.06969 0.0391903
\(746\) 0 0
\(747\) 0.437500 0.0160073
\(748\) 0 0
\(749\) −44.1756 −1.61414
\(750\) 0 0
\(751\) 2.26069 0.0824937 0.0412469 0.999149i \(-0.486867\pi\)
0.0412469 + 0.999149i \(0.486867\pi\)
\(752\) 0 0
\(753\) 13.7924 0.502624
\(754\) 0 0
\(755\) −3.06636 −0.111596
\(756\) 0 0
\(757\) −45.7056 −1.66120 −0.830598 0.556872i \(-0.812001\pi\)
−0.830598 + 0.556872i \(0.812001\pi\)
\(758\) 0 0
\(759\) −20.4579 −0.742576
\(760\) 0 0
\(761\) −21.2344 −0.769745 −0.384873 0.922970i \(-0.625755\pi\)
−0.384873 + 0.922970i \(0.625755\pi\)
\(762\) 0 0
\(763\) 38.7120 1.40147
\(764\) 0 0
\(765\) 11.3478 0.410282
\(766\) 0 0
\(767\) 0.205598 0.00742373
\(768\) 0 0
\(769\) 24.5451 0.885118 0.442559 0.896739i \(-0.354071\pi\)
0.442559 + 0.896739i \(0.354071\pi\)
\(770\) 0 0
\(771\) 2.03436 0.0732658
\(772\) 0 0
\(773\) −29.6966 −1.06811 −0.534056 0.845449i \(-0.679333\pi\)
−0.534056 + 0.845449i \(0.679333\pi\)
\(774\) 0 0
\(775\) 2.27515 0.0817257
\(776\) 0 0
\(777\) −0.555684 −0.0199351
\(778\) 0 0
\(779\) −15.0323 −0.538588
\(780\) 0 0
\(781\) 21.2563 0.760609
\(782\) 0 0
\(783\) −0.0936338 −0.00334620
\(784\) 0 0
\(785\) 7.78310 0.277791
\(786\) 0 0
\(787\) 1.16617 0.0415694 0.0207847 0.999784i \(-0.493384\pi\)
0.0207847 + 0.999784i \(0.493384\pi\)
\(788\) 0 0
\(789\) 22.6680 0.807004
\(790\) 0 0
\(791\) 21.8006 0.775141
\(792\) 0 0
\(793\) 1.00625 0.0357330
\(794\) 0 0
\(795\) 6.81046 0.241542
\(796\) 0 0
\(797\) −3.69459 −0.130869 −0.0654345 0.997857i \(-0.520843\pi\)
−0.0654345 + 0.997857i \(0.520843\pi\)
\(798\) 0 0
\(799\) 42.9584 1.51976
\(800\) 0 0
\(801\) 11.9663 0.422808
\(802\) 0 0
\(803\) 11.8561 0.418392
\(804\) 0 0
\(805\) 36.6648 1.29227
\(806\) 0 0
\(807\) 23.5210 0.827980
\(808\) 0 0
\(809\) 7.83155 0.275343 0.137671 0.990478i \(-0.456038\pi\)
0.137671 + 0.990478i \(0.456038\pi\)
\(810\) 0 0
\(811\) 19.6708 0.690734 0.345367 0.938468i \(-0.387754\pi\)
0.345367 + 0.938468i \(0.387754\pi\)
\(812\) 0 0
\(813\) 15.5238 0.544444
\(814\) 0 0
\(815\) 38.4267 1.34603
\(816\) 0 0
\(817\) −17.8890 −0.625855
\(818\) 0 0
\(819\) −0.280657 −0.00980695
\(820\) 0 0
\(821\) 29.8119 1.04044 0.520222 0.854031i \(-0.325849\pi\)
0.520222 + 0.854031i \(0.325849\pi\)
\(822\) 0 0
\(823\) −26.6385 −0.928559 −0.464280 0.885689i \(-0.653687\pi\)
−0.464280 + 0.885689i \(0.653687\pi\)
\(824\) 0 0
\(825\) 3.92095 0.136510
\(826\) 0 0
\(827\) 30.1035 1.04680 0.523401 0.852086i \(-0.324663\pi\)
0.523401 + 0.852086i \(0.324663\pi\)
\(828\) 0 0
\(829\) −25.4482 −0.883852 −0.441926 0.897051i \(-0.645705\pi\)
−0.441926 + 0.897051i \(0.645705\pi\)
\(830\) 0 0
\(831\) −9.69430 −0.336292
\(832\) 0 0
\(833\) 7.13247 0.247125
\(834\) 0 0
\(835\) −1.85181 −0.0640845
\(836\) 0 0
\(837\) 1.44840 0.0500639
\(838\) 0 0
\(839\) −1.40852 −0.0486275 −0.0243138 0.999704i \(-0.507740\pi\)
−0.0243138 + 0.999704i \(0.507740\pi\)
\(840\) 0 0
\(841\) −28.9912 −0.999698
\(842\) 0 0
\(843\) 6.65022 0.229046
\(844\) 0 0
\(845\) 24.0485 0.827294
\(846\) 0 0
\(847\) 11.5217 0.395889
\(848\) 0 0
\(849\) 20.1057 0.690026
\(850\) 0 0
\(851\) −1.88521 −0.0646243
\(852\) 0 0
\(853\) −27.2693 −0.933682 −0.466841 0.884341i \(-0.654608\pi\)
−0.466841 + 0.884341i \(0.654608\pi\)
\(854\) 0 0
\(855\) −6.18925 −0.211668
\(856\) 0 0
\(857\) 8.80951 0.300927 0.150464 0.988616i \(-0.451923\pi\)
0.150464 + 0.988616i \(0.451923\pi\)
\(858\) 0 0
\(859\) 28.9321 0.987152 0.493576 0.869703i \(-0.335690\pi\)
0.493576 + 0.869703i \(0.335690\pi\)
\(860\) 0 0
\(861\) −10.8654 −0.370291
\(862\) 0 0
\(863\) 1.47181 0.0501008 0.0250504 0.999686i \(-0.492025\pi\)
0.0250504 + 0.999686i \(0.492025\pi\)
\(864\) 0 0
\(865\) 26.7127 0.908259
\(866\) 0 0
\(867\) −20.5521 −0.697987
\(868\) 0 0
\(869\) 8.10765 0.275033
\(870\) 0 0
\(871\) 1.76265 0.0597251
\(872\) 0 0
\(873\) 5.72224 0.193668
\(874\) 0 0
\(875\) −29.3951 −0.993737
\(876\) 0 0
\(877\) −37.7379 −1.27432 −0.637159 0.770732i \(-0.719891\pi\)
−0.637159 + 0.770732i \(0.719891\pi\)
\(878\) 0 0
\(879\) −2.72431 −0.0918888
\(880\) 0 0
\(881\) −24.7811 −0.834897 −0.417449 0.908700i \(-0.637076\pi\)
−0.417449 + 0.908700i \(0.637076\pi\)
\(882\) 0 0
\(883\) −36.9451 −1.24330 −0.621651 0.783295i \(-0.713538\pi\)
−0.621651 + 0.783295i \(0.713538\pi\)
\(884\) 0 0
\(885\) 3.27718 0.110161
\(886\) 0 0
\(887\) −46.6482 −1.56629 −0.783146 0.621837i \(-0.786386\pi\)
−0.783146 + 0.621837i \(0.786386\pi\)
\(888\) 0 0
\(889\) −2.23542 −0.0749734
\(890\) 0 0
\(891\) 2.49614 0.0836238
\(892\) 0 0
\(893\) −23.4301 −0.784057
\(894\) 0 0
\(895\) 40.5286 1.35472
\(896\) 0 0
\(897\) −0.952157 −0.0317916
\(898\) 0 0
\(899\) −0.135619 −0.00452314
\(900\) 0 0
\(901\) −22.5371 −0.750819
\(902\) 0 0
\(903\) −12.9302 −0.430288
\(904\) 0 0
\(905\) 20.2867 0.674353
\(906\) 0 0
\(907\) 43.9961 1.46087 0.730433 0.682985i \(-0.239319\pi\)
0.730433 + 0.682985i \(0.239319\pi\)
\(908\) 0 0
\(909\) −18.5958 −0.616784
\(910\) 0 0
\(911\) 16.0348 0.531258 0.265629 0.964075i \(-0.414420\pi\)
0.265629 + 0.964075i \(0.414420\pi\)
\(912\) 0 0
\(913\) 1.09206 0.0361419
\(914\) 0 0
\(915\) 16.0393 0.530244
\(916\) 0 0
\(917\) 6.97174 0.230227
\(918\) 0 0
\(919\) 5.37241 0.177219 0.0886096 0.996066i \(-0.471758\pi\)
0.0886096 + 0.996066i \(0.471758\pi\)
\(920\) 0 0
\(921\) −7.63757 −0.251667
\(922\) 0 0
\(923\) 0.989314 0.0325637
\(924\) 0 0
\(925\) 0.361318 0.0118801
\(926\) 0 0
\(927\) 19.3000 0.633895
\(928\) 0 0
\(929\) −23.0096 −0.754920 −0.377460 0.926026i \(-0.623202\pi\)
−0.377460 + 0.926026i \(0.623202\pi\)
\(930\) 0 0
\(931\) −3.89013 −0.127494
\(932\) 0 0
\(933\) 19.6655 0.643820
\(934\) 0 0
\(935\) 28.3258 0.926352
\(936\) 0 0
\(937\) −50.5567 −1.65162 −0.825808 0.563951i \(-0.809281\pi\)
−0.825808 + 0.563951i \(0.809281\pi\)
\(938\) 0 0
\(939\) 27.6680 0.902911
\(940\) 0 0
\(941\) 16.0633 0.523648 0.261824 0.965116i \(-0.415676\pi\)
0.261824 + 0.965116i \(0.415676\pi\)
\(942\) 0 0
\(943\) −36.8618 −1.20039
\(944\) 0 0
\(945\) −4.47360 −0.145526
\(946\) 0 0
\(947\) −10.2063 −0.331659 −0.165830 0.986154i \(-0.553030\pi\)
−0.165830 + 0.986154i \(0.553030\pi\)
\(948\) 0 0
\(949\) 0.551808 0.0179124
\(950\) 0 0
\(951\) −8.67973 −0.281459
\(952\) 0 0
\(953\) −10.7589 −0.348515 −0.174257 0.984700i \(-0.555752\pi\)
−0.174257 + 0.984700i \(0.555752\pi\)
\(954\) 0 0
\(955\) 8.15792 0.263984
\(956\) 0 0
\(957\) −0.233723 −0.00755519
\(958\) 0 0
\(959\) −43.7575 −1.41300
\(960\) 0 0
\(961\) −28.9022 −0.932327
\(962\) 0 0
\(963\) 18.2861 0.589262
\(964\) 0 0
\(965\) −5.21724 −0.167949
\(966\) 0 0
\(967\) 32.7046 1.05171 0.525855 0.850574i \(-0.323745\pi\)
0.525855 + 0.850574i \(0.323745\pi\)
\(968\) 0 0
\(969\) 20.4814 0.657957
\(970\) 0 0
\(971\) 5.93206 0.190369 0.0951845 0.995460i \(-0.469656\pi\)
0.0951845 + 0.995460i \(0.469656\pi\)
\(972\) 0 0
\(973\) 35.7900 1.14737
\(974\) 0 0
\(975\) 0.182490 0.00584434
\(976\) 0 0
\(977\) 36.2266 1.15899 0.579496 0.814975i \(-0.303249\pi\)
0.579496 + 0.814975i \(0.303249\pi\)
\(978\) 0 0
\(979\) 29.8695 0.954634
\(980\) 0 0
\(981\) −16.0245 −0.511623
\(982\) 0 0
\(983\) −6.41022 −0.204454 −0.102227 0.994761i \(-0.532597\pi\)
−0.102227 + 0.994761i \(0.532597\pi\)
\(984\) 0 0
\(985\) −38.3816 −1.22294
\(986\) 0 0
\(987\) −16.9353 −0.539055
\(988\) 0 0
\(989\) −43.8668 −1.39488
\(990\) 0 0
\(991\) 11.8990 0.377986 0.188993 0.981978i \(-0.439478\pi\)
0.188993 + 0.981978i \(0.439478\pi\)
\(992\) 0 0
\(993\) 12.1719 0.386263
\(994\) 0 0
\(995\) −1.50700 −0.0477751
\(996\) 0 0
\(997\) −12.7858 −0.404930 −0.202465 0.979290i \(-0.564895\pi\)
−0.202465 + 0.979290i \(0.564895\pi\)
\(998\) 0 0
\(999\) 0.230021 0.00727754
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 8016.2.a.q.1.2 5
4.3 odd 2 2004.2.a.b.1.2 5
12.11 even 2 6012.2.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.b.1.2 5 4.3 odd 2
6012.2.a.f.1.4 5 12.11 even 2
8016.2.a.q.1.2 5 1.1 even 1 trivial