Properties

Label 3969.2.a.y.1.3
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) 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 1.3
Root \(0.874032\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} -0.874032 q^{5} -2.23607 q^{8} +O(q^{10})\) \(q+1.61803 q^{2} +0.618034 q^{4} -0.874032 q^{5} -2.23607 q^{8} -1.41421 q^{10} +1.00000 q^{11} -1.74806 q^{13} -4.85410 q^{16} +3.16228 q^{17} +5.45052 q^{19} -0.540182 q^{20} +1.61803 q^{22} -2.76393 q^{23} -4.23607 q^{25} -2.82843 q^{26} -3.23607 q^{29} +0.333851 q^{31} -3.38197 q^{32} +5.11667 q^{34} +2.70820 q^{37} +8.81913 q^{38} +1.95440 q^{40} +10.0270 q^{41} -11.9443 q^{43} +0.618034 q^{44} -4.47214 q^{46} -12.5216 q^{47} -6.85410 q^{50} -1.08036 q^{52} -2.70820 q^{53} -0.874032 q^{55} -5.23607 q^{58} -8.15143 q^{59} -13.1893 q^{61} +0.540182 q^{62} +4.23607 q^{64} +1.52786 q^{65} -7.47214 q^{67} +1.95440 q^{68} -3.47214 q^{71} -13.0618 q^{73} +4.38197 q^{74} +3.36861 q^{76} -12.2361 q^{79} +4.24264 q^{80} +16.2241 q^{82} +7.19859 q^{83} -2.76393 q^{85} -19.3262 q^{86} -2.23607 q^{88} -6.32456 q^{89} -1.70820 q^{92} -20.2604 q^{94} -4.76393 q^{95} +9.35931 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{11} - 6 q^{16} + 2 q^{22} - 20 q^{23} - 8 q^{25} - 4 q^{29} - 18 q^{32} - 16 q^{37} - 12 q^{43} - 2 q^{44} - 14 q^{50} + 16 q^{53} - 12 q^{58} + 8 q^{64} + 24 q^{65} - 12 q^{67} + 4 q^{71} + 22 q^{74} - 40 q^{79} - 20 q^{85} - 46 q^{86} + 20 q^{92} - 28 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) −0.874032 −0.390879 −0.195440 0.980716i \(-0.562613\pi\)
−0.195440 + 0.980716i \(0.562613\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) −1.41421 −0.447214
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) −1.74806 −0.484826 −0.242413 0.970173i \(-0.577939\pi\)
−0.242413 + 0.970173i \(0.577939\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 3.16228 0.766965 0.383482 0.923548i \(-0.374725\pi\)
0.383482 + 0.923548i \(0.374725\pi\)
\(18\) 0 0
\(19\) 5.45052 1.25044 0.625218 0.780450i \(-0.285010\pi\)
0.625218 + 0.780450i \(0.285010\pi\)
\(20\) −0.540182 −0.120788
\(21\) 0 0
\(22\) 1.61803 0.344966
\(23\) −2.76393 −0.576320 −0.288160 0.957582i \(-0.593043\pi\)
−0.288160 + 0.957582i \(0.593043\pi\)
\(24\) 0 0
\(25\) −4.23607 −0.847214
\(26\) −2.82843 −0.554700
\(27\) 0 0
\(28\) 0 0
\(29\) −3.23607 −0.600923 −0.300461 0.953794i \(-0.597141\pi\)
−0.300461 + 0.953794i \(0.597141\pi\)
\(30\) 0 0
\(31\) 0.333851 0.0599613 0.0299807 0.999550i \(-0.490455\pi\)
0.0299807 + 0.999550i \(0.490455\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) 5.11667 0.877502
\(35\) 0 0
\(36\) 0 0
\(37\) 2.70820 0.445226 0.222613 0.974907i \(-0.428541\pi\)
0.222613 + 0.974907i \(0.428541\pi\)
\(38\) 8.81913 1.43065
\(39\) 0 0
\(40\) 1.95440 0.309017
\(41\) 10.0270 1.56596 0.782978 0.622049i \(-0.213700\pi\)
0.782978 + 0.622049i \(0.213700\pi\)
\(42\) 0 0
\(43\) −11.9443 −1.82148 −0.910742 0.412975i \(-0.864490\pi\)
−0.910742 + 0.412975i \(0.864490\pi\)
\(44\) 0.618034 0.0931721
\(45\) 0 0
\(46\) −4.47214 −0.659380
\(47\) −12.5216 −1.82646 −0.913231 0.407442i \(-0.866421\pi\)
−0.913231 + 0.407442i \(0.866421\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −6.85410 −0.969316
\(51\) 0 0
\(52\) −1.08036 −0.149819
\(53\) −2.70820 −0.372000 −0.186000 0.982550i \(-0.559553\pi\)
−0.186000 + 0.982550i \(0.559553\pi\)
\(54\) 0 0
\(55\) −0.874032 −0.117854
\(56\) 0 0
\(57\) 0 0
\(58\) −5.23607 −0.687529
\(59\) −8.15143 −1.06123 −0.530613 0.847614i \(-0.678038\pi\)
−0.530613 + 0.847614i \(0.678038\pi\)
\(60\) 0 0
\(61\) −13.1893 −1.68872 −0.844358 0.535780i \(-0.820018\pi\)
−0.844358 + 0.535780i \(0.820018\pi\)
\(62\) 0.540182 0.0686031
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) 1.52786 0.189508
\(66\) 0 0
\(67\) −7.47214 −0.912867 −0.456433 0.889758i \(-0.650873\pi\)
−0.456433 + 0.889758i \(0.650873\pi\)
\(68\) 1.95440 0.237005
\(69\) 0 0
\(70\) 0 0
\(71\) −3.47214 −0.412067 −0.206033 0.978545i \(-0.566056\pi\)
−0.206033 + 0.978545i \(0.566056\pi\)
\(72\) 0 0
\(73\) −13.0618 −1.52876 −0.764382 0.644763i \(-0.776956\pi\)
−0.764382 + 0.644763i \(0.776956\pi\)
\(74\) 4.38197 0.509393
\(75\) 0 0
\(76\) 3.36861 0.386406
\(77\) 0 0
\(78\) 0 0
\(79\) −12.2361 −1.37667 −0.688333 0.725395i \(-0.741657\pi\)
−0.688333 + 0.725395i \(0.741657\pi\)
\(80\) 4.24264 0.474342
\(81\) 0 0
\(82\) 16.2241 1.79165
\(83\) 7.19859 0.790148 0.395074 0.918649i \(-0.370719\pi\)
0.395074 + 0.918649i \(0.370719\pi\)
\(84\) 0 0
\(85\) −2.76393 −0.299791
\(86\) −19.3262 −2.08400
\(87\) 0 0
\(88\) −2.23607 −0.238366
\(89\) −6.32456 −0.670402 −0.335201 0.942147i \(-0.608804\pi\)
−0.335201 + 0.942147i \(0.608804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.70820 −0.178093
\(93\) 0 0
\(94\) −20.2604 −2.08970
\(95\) −4.76393 −0.488769
\(96\) 0 0
\(97\) 9.35931 0.950294 0.475147 0.879906i \(-0.342395\pi\)
0.475147 + 0.879906i \(0.342395\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.61803 −0.261803
\(101\) 12.3941 1.23326 0.616628 0.787255i \(-0.288498\pi\)
0.616628 + 0.787255i \(0.288498\pi\)
\(102\) 0 0
\(103\) 10.0270 0.987991 0.493996 0.869464i \(-0.335536\pi\)
0.493996 + 0.869464i \(0.335536\pi\)
\(104\) 3.90879 0.383288
\(105\) 0 0
\(106\) −4.38197 −0.425614
\(107\) −5.94427 −0.574654 −0.287327 0.957832i \(-0.592767\pi\)
−0.287327 + 0.957832i \(0.592767\pi\)
\(108\) 0 0
\(109\) 2.94427 0.282010 0.141005 0.990009i \(-0.454967\pi\)
0.141005 + 0.990009i \(0.454967\pi\)
\(110\) −1.41421 −0.134840
\(111\) 0 0
\(112\) 0 0
\(113\) −8.23607 −0.774784 −0.387392 0.921915i \(-0.626624\pi\)
−0.387392 + 0.921915i \(0.626624\pi\)
\(114\) 0 0
\(115\) 2.41577 0.225271
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −13.1893 −1.21417
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −21.3407 −1.93210
\(123\) 0 0
\(124\) 0.206331 0.0185291
\(125\) 8.07262 0.722037
\(126\) 0 0
\(127\) 13.6525 1.21146 0.605731 0.795670i \(-0.292881\pi\)
0.605731 + 0.795670i \(0.292881\pi\)
\(128\) 13.6180 1.20368
\(129\) 0 0
\(130\) 2.47214 0.216821
\(131\) −1.87558 −0.163871 −0.0819353 0.996638i \(-0.526110\pi\)
−0.0819353 + 0.996638i \(0.526110\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −12.0902 −1.04443
\(135\) 0 0
\(136\) −7.07107 −0.606339
\(137\) 3.76393 0.321574 0.160787 0.986989i \(-0.448597\pi\)
0.160787 + 0.986989i \(0.448597\pi\)
\(138\) 0 0
\(139\) −0.206331 −0.0175008 −0.00875038 0.999962i \(-0.502785\pi\)
−0.00875038 + 0.999962i \(0.502785\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.61803 −0.471455
\(143\) −1.74806 −0.146180
\(144\) 0 0
\(145\) 2.82843 0.234888
\(146\) −21.1344 −1.74909
\(147\) 0 0
\(148\) 1.67376 0.137582
\(149\) −19.4721 −1.59522 −0.797610 0.603174i \(-0.793903\pi\)
−0.797610 + 0.603174i \(0.793903\pi\)
\(150\) 0 0
\(151\) −5.29180 −0.430640 −0.215320 0.976544i \(-0.569079\pi\)
−0.215320 + 0.976544i \(0.569079\pi\)
\(152\) −12.1877 −0.988556
\(153\) 0 0
\(154\) 0 0
\(155\) −0.291796 −0.0234376
\(156\) 0 0
\(157\) −2.16073 −0.172445 −0.0862224 0.996276i \(-0.527480\pi\)
−0.0862224 + 0.996276i \(0.527480\pi\)
\(158\) −19.7984 −1.57507
\(159\) 0 0
\(160\) 2.95595 0.233688
\(161\) 0 0
\(162\) 0 0
\(163\) −8.23607 −0.645099 −0.322549 0.946553i \(-0.604540\pi\)
−0.322549 + 0.946553i \(0.604540\pi\)
\(164\) 6.19704 0.483907
\(165\) 0 0
\(166\) 11.6476 0.904026
\(167\) 8.69161 0.672577 0.336289 0.941759i \(-0.390828\pi\)
0.336289 + 0.941759i \(0.390828\pi\)
\(168\) 0 0
\(169\) −9.94427 −0.764944
\(170\) −4.47214 −0.342997
\(171\) 0 0
\(172\) −7.38197 −0.562870
\(173\) −19.2588 −1.46422 −0.732110 0.681186i \(-0.761464\pi\)
−0.732110 + 0.681186i \(0.761464\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.85410 −0.365892
\(177\) 0 0
\(178\) −10.2333 −0.767022
\(179\) 20.3607 1.52183 0.760914 0.648852i \(-0.224751\pi\)
0.760914 + 0.648852i \(0.224751\pi\)
\(180\) 0 0
\(181\) 22.9613 1.70670 0.853349 0.521340i \(-0.174568\pi\)
0.853349 + 0.521340i \(0.174568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.18034 0.455621
\(185\) −2.36706 −0.174029
\(186\) 0 0
\(187\) 3.16228 0.231249
\(188\) −7.73877 −0.564408
\(189\) 0 0
\(190\) −7.70820 −0.559212
\(191\) −4.41641 −0.319560 −0.159780 0.987153i \(-0.551078\pi\)
−0.159780 + 0.987153i \(0.551078\pi\)
\(192\) 0 0
\(193\) −16.4721 −1.18569 −0.592845 0.805316i \(-0.701995\pi\)
−0.592845 + 0.805316i \(0.701995\pi\)
\(194\) 15.1437 1.08725
\(195\) 0 0
\(196\) 0 0
\(197\) 18.4164 1.31211 0.656057 0.754711i \(-0.272223\pi\)
0.656057 + 0.754711i \(0.272223\pi\)
\(198\) 0 0
\(199\) −11.4412 −0.811047 −0.405524 0.914085i \(-0.632911\pi\)
−0.405524 + 0.914085i \(0.632911\pi\)
\(200\) 9.47214 0.669781
\(201\) 0 0
\(202\) 20.0540 1.41100
\(203\) 0 0
\(204\) 0 0
\(205\) −8.76393 −0.612100
\(206\) 16.2241 1.13038
\(207\) 0 0
\(208\) 8.48528 0.588348
\(209\) 5.45052 0.377021
\(210\) 0 0
\(211\) −3.29180 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(212\) −1.67376 −0.114954
\(213\) 0 0
\(214\) −9.61803 −0.657475
\(215\) 10.4397 0.711980
\(216\) 0 0
\(217\) 0 0
\(218\) 4.76393 0.322654
\(219\) 0 0
\(220\) −0.540182 −0.0364190
\(221\) −5.52786 −0.371844
\(222\) 0 0
\(223\) 11.6476 0.779978 0.389989 0.920819i \(-0.372479\pi\)
0.389989 + 0.920819i \(0.372479\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −13.3262 −0.886448
\(227\) 23.0888 1.53246 0.766228 0.642568i \(-0.222131\pi\)
0.766228 + 0.642568i \(0.222131\pi\)
\(228\) 0 0
\(229\) −10.0270 −0.662604 −0.331302 0.943525i \(-0.607488\pi\)
−0.331302 + 0.943525i \(0.607488\pi\)
\(230\) 3.90879 0.257738
\(231\) 0 0
\(232\) 7.23607 0.475071
\(233\) 22.9443 1.50313 0.751565 0.659659i \(-0.229299\pi\)
0.751565 + 0.659659i \(0.229299\pi\)
\(234\) 0 0
\(235\) 10.9443 0.713926
\(236\) −5.03786 −0.327937
\(237\) 0 0
\(238\) 0 0
\(239\) 13.9443 0.901980 0.450990 0.892529i \(-0.351071\pi\)
0.450990 + 0.892529i \(0.351071\pi\)
\(240\) 0 0
\(241\) −12.1089 −0.780005 −0.390002 0.920814i \(-0.627526\pi\)
−0.390002 + 0.920814i \(0.627526\pi\)
\(242\) −16.1803 −1.04011
\(243\) 0 0
\(244\) −8.15143 −0.521842
\(245\) 0 0
\(246\) 0 0
\(247\) −9.52786 −0.606243
\(248\) −0.746512 −0.0474036
\(249\) 0 0
\(250\) 13.0618 0.826099
\(251\) −2.70091 −0.170480 −0.0852399 0.996360i \(-0.527166\pi\)
−0.0852399 + 0.996360i \(0.527166\pi\)
\(252\) 0 0
\(253\) −2.76393 −0.173767
\(254\) 22.0902 1.38606
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 15.8902 0.991203 0.495602 0.868550i \(-0.334948\pi\)
0.495602 + 0.868550i \(0.334948\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.944272 0.0585613
\(261\) 0 0
\(262\) −3.03476 −0.187488
\(263\) 14.2361 0.877834 0.438917 0.898528i \(-0.355362\pi\)
0.438917 + 0.898528i \(0.355362\pi\)
\(264\) 0 0
\(265\) 2.36706 0.145407
\(266\) 0 0
\(267\) 0 0
\(268\) −4.61803 −0.282091
\(269\) −21.1344 −1.28859 −0.644293 0.764778i \(-0.722848\pi\)
−0.644293 + 0.764778i \(0.722848\pi\)
\(270\) 0 0
\(271\) −27.5865 −1.67576 −0.837879 0.545856i \(-0.816205\pi\)
−0.837879 + 0.545856i \(0.816205\pi\)
\(272\) −15.3500 −0.930732
\(273\) 0 0
\(274\) 6.09017 0.367921
\(275\) −4.23607 −0.255445
\(276\) 0 0
\(277\) 5.76393 0.346321 0.173161 0.984894i \(-0.444602\pi\)
0.173161 + 0.984894i \(0.444602\pi\)
\(278\) −0.333851 −0.0200230
\(279\) 0 0
\(280\) 0 0
\(281\) −5.00000 −0.298275 −0.149137 0.988816i \(-0.547650\pi\)
−0.149137 + 0.988816i \(0.547650\pi\)
\(282\) 0 0
\(283\) 12.1089 0.719801 0.359901 0.932991i \(-0.382811\pi\)
0.359901 + 0.932991i \(0.382811\pi\)
\(284\) −2.14590 −0.127336
\(285\) 0 0
\(286\) −2.82843 −0.167248
\(287\) 0 0
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 4.57649 0.268741
\(291\) 0 0
\(292\) −8.07262 −0.472414
\(293\) −14.9374 −0.872650 −0.436325 0.899789i \(-0.643720\pi\)
−0.436325 + 0.899789i \(0.643720\pi\)
\(294\) 0 0
\(295\) 7.12461 0.414811
\(296\) −6.05573 −0.351982
\(297\) 0 0
\(298\) −31.5066 −1.82513
\(299\) 4.83153 0.279415
\(300\) 0 0
\(301\) 0 0
\(302\) −8.56231 −0.492705
\(303\) 0 0
\(304\) −26.4574 −1.51744
\(305\) 11.5279 0.660084
\(306\) 0 0
\(307\) 18.6398 1.06383 0.531915 0.846798i \(-0.321472\pi\)
0.531915 + 0.846798i \(0.321472\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.472136 −0.0268155
\(311\) −28.4118 −1.61108 −0.805542 0.592538i \(-0.798126\pi\)
−0.805542 + 0.592538i \(0.798126\pi\)
\(312\) 0 0
\(313\) 13.3956 0.757165 0.378583 0.925567i \(-0.376412\pi\)
0.378583 + 0.925567i \(0.376412\pi\)
\(314\) −3.49613 −0.197298
\(315\) 0 0
\(316\) −7.56231 −0.425413
\(317\) 18.8328 1.05776 0.528878 0.848698i \(-0.322613\pi\)
0.528878 + 0.848698i \(0.322613\pi\)
\(318\) 0 0
\(319\) −3.23607 −0.181185
\(320\) −3.70246 −0.206974
\(321\) 0 0
\(322\) 0 0
\(323\) 17.2361 0.959040
\(324\) 0 0
\(325\) 7.40492 0.410751
\(326\) −13.3262 −0.738072
\(327\) 0 0
\(328\) −22.4211 −1.23800
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 4.44897 0.244169
\(333\) 0 0
\(334\) 14.0633 0.769511
\(335\) 6.53089 0.356820
\(336\) 0 0
\(337\) −17.1803 −0.935873 −0.467936 0.883762i \(-0.655002\pi\)
−0.467936 + 0.883762i \(0.655002\pi\)
\(338\) −16.0902 −0.875190
\(339\) 0 0
\(340\) −1.70820 −0.0926404
\(341\) 0.333851 0.0180790
\(342\) 0 0
\(343\) 0 0
\(344\) 26.7082 1.44001
\(345\) 0 0
\(346\) −31.1614 −1.67525
\(347\) 21.9443 1.17803 0.589015 0.808122i \(-0.299516\pi\)
0.589015 + 0.808122i \(0.299516\pi\)
\(348\) 0 0
\(349\) −28.6668 −1.53450 −0.767250 0.641348i \(-0.778375\pi\)
−0.767250 + 0.641348i \(0.778375\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.38197 −0.180259
\(353\) −20.5154 −1.09192 −0.545962 0.837810i \(-0.683836\pi\)
−0.545962 + 0.837810i \(0.683836\pi\)
\(354\) 0 0
\(355\) 3.03476 0.161068
\(356\) −3.90879 −0.207165
\(357\) 0 0
\(358\) 32.9443 1.74116
\(359\) 17.1803 0.906744 0.453372 0.891321i \(-0.350221\pi\)
0.453372 + 0.891321i \(0.350221\pi\)
\(360\) 0 0
\(361\) 10.7082 0.563590
\(362\) 37.1521 1.95267
\(363\) 0 0
\(364\) 0 0
\(365\) 11.4164 0.597562
\(366\) 0 0
\(367\) 1.74806 0.0912482 0.0456241 0.998959i \(-0.485472\pi\)
0.0456241 + 0.998959i \(0.485472\pi\)
\(368\) 13.4164 0.699379
\(369\) 0 0
\(370\) −3.82998 −0.199111
\(371\) 0 0
\(372\) 0 0
\(373\) −9.18034 −0.475340 −0.237670 0.971346i \(-0.576384\pi\)
−0.237670 + 0.971346i \(0.576384\pi\)
\(374\) 5.11667 0.264577
\(375\) 0 0
\(376\) 27.9991 1.44394
\(377\) 5.65685 0.291343
\(378\) 0 0
\(379\) −31.6525 −1.62588 −0.812939 0.582349i \(-0.802134\pi\)
−0.812939 + 0.582349i \(0.802134\pi\)
\(380\) −2.94427 −0.151038
\(381\) 0 0
\(382\) −7.14590 −0.365616
\(383\) 13.3168 0.680457 0.340229 0.940343i \(-0.389496\pi\)
0.340229 + 0.940343i \(0.389496\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −26.6525 −1.35658
\(387\) 0 0
\(388\) 5.78437 0.293657
\(389\) 10.9443 0.554897 0.277448 0.960741i \(-0.410511\pi\)
0.277448 + 0.960741i \(0.410511\pi\)
\(390\) 0 0
\(391\) −8.74032 −0.442017
\(392\) 0 0
\(393\) 0 0
\(394\) 29.7984 1.50122
\(395\) 10.6947 0.538110
\(396\) 0 0
\(397\) −28.8245 −1.44666 −0.723329 0.690504i \(-0.757389\pi\)
−0.723329 + 0.690504i \(0.757389\pi\)
\(398\) −18.5123 −0.927938
\(399\) 0 0
\(400\) 20.5623 1.02812
\(401\) 37.0689 1.85113 0.925566 0.378587i \(-0.123590\pi\)
0.925566 + 0.378587i \(0.123590\pi\)
\(402\) 0 0
\(403\) −0.583592 −0.0290708
\(404\) 7.65996 0.381097
\(405\) 0 0
\(406\) 0 0
\(407\) 2.70820 0.134241
\(408\) 0 0
\(409\) 4.11512 0.203480 0.101740 0.994811i \(-0.467559\pi\)
0.101740 + 0.994811i \(0.467559\pi\)
\(410\) −14.1803 −0.700317
\(411\) 0 0
\(412\) 6.19704 0.305306
\(413\) 0 0
\(414\) 0 0
\(415\) −6.29180 −0.308852
\(416\) 5.91189 0.289854
\(417\) 0 0
\(418\) 8.81913 0.431358
\(419\) 24.4242 1.19320 0.596600 0.802539i \(-0.296518\pi\)
0.596600 + 0.802539i \(0.296518\pi\)
\(420\) 0 0
\(421\) 14.8885 0.725623 0.362812 0.931863i \(-0.381817\pi\)
0.362812 + 0.931863i \(0.381817\pi\)
\(422\) −5.32624 −0.259277
\(423\) 0 0
\(424\) 6.05573 0.294092
\(425\) −13.3956 −0.649783
\(426\) 0 0
\(427\) 0 0
\(428\) −3.67376 −0.177578
\(429\) 0 0
\(430\) 16.8918 0.814593
\(431\) −23.4164 −1.12793 −0.563964 0.825799i \(-0.690724\pi\)
−0.563964 + 0.825799i \(0.690724\pi\)
\(432\) 0 0
\(433\) 20.5154 0.985907 0.492954 0.870056i \(-0.335917\pi\)
0.492954 + 0.870056i \(0.335917\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.81966 0.0871459
\(437\) −15.0649 −0.720651
\(438\) 0 0
\(439\) 11.6963 0.558232 0.279116 0.960257i \(-0.409959\pi\)
0.279116 + 0.960257i \(0.409959\pi\)
\(440\) 1.95440 0.0931721
\(441\) 0 0
\(442\) −8.94427 −0.425436
\(443\) −2.29180 −0.108887 −0.0544433 0.998517i \(-0.517338\pi\)
−0.0544433 + 0.998517i \(0.517338\pi\)
\(444\) 0 0
\(445\) 5.52786 0.262046
\(446\) 18.8461 0.892391
\(447\) 0 0
\(448\) 0 0
\(449\) 31.4721 1.48526 0.742631 0.669701i \(-0.233578\pi\)
0.742631 + 0.669701i \(0.233578\pi\)
\(450\) 0 0
\(451\) 10.0270 0.472154
\(452\) −5.09017 −0.239421
\(453\) 0 0
\(454\) 37.3584 1.75332
\(455\) 0 0
\(456\) 0 0
\(457\) 23.8328 1.11485 0.557426 0.830227i \(-0.311789\pi\)
0.557426 + 0.830227i \(0.311789\pi\)
\(458\) −16.2241 −0.758100
\(459\) 0 0
\(460\) 1.49302 0.0696126
\(461\) 18.1784 0.846655 0.423327 0.905977i \(-0.360862\pi\)
0.423327 + 0.905977i \(0.360862\pi\)
\(462\) 0 0
\(463\) −36.8885 −1.71436 −0.857178 0.515020i \(-0.827784\pi\)
−0.857178 + 0.515020i \(0.827784\pi\)
\(464\) 15.7082 0.729235
\(465\) 0 0
\(466\) 37.1246 1.71976
\(467\) 40.3144 1.86553 0.932764 0.360488i \(-0.117390\pi\)
0.932764 + 0.360488i \(0.117390\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 17.7082 0.816819
\(471\) 0 0
\(472\) 18.2272 0.838973
\(473\) −11.9443 −0.549198
\(474\) 0 0
\(475\) −23.0888 −1.05939
\(476\) 0 0
\(477\) 0 0
\(478\) 22.5623 1.03198
\(479\) −11.2349 −0.513336 −0.256668 0.966500i \(-0.582625\pi\)
−0.256668 + 0.966500i \(0.582625\pi\)
\(480\) 0 0
\(481\) −4.73411 −0.215857
\(482\) −19.5927 −0.892421
\(483\) 0 0
\(484\) −6.18034 −0.280925
\(485\) −8.18034 −0.371450
\(486\) 0 0
\(487\) −13.9443 −0.631875 −0.315938 0.948780i \(-0.602319\pi\)
−0.315938 + 0.948780i \(0.602319\pi\)
\(488\) 29.4922 1.33505
\(489\) 0 0
\(490\) 0 0
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 0 0
\(493\) −10.2333 −0.460887
\(494\) −15.4164 −0.693617
\(495\) 0 0
\(496\) −1.62054 −0.0727646
\(497\) 0 0
\(498\) 0 0
\(499\) −13.8885 −0.621737 −0.310868 0.950453i \(-0.600620\pi\)
−0.310868 + 0.950453i \(0.600620\pi\)
\(500\) 4.98915 0.223122
\(501\) 0 0
\(502\) −4.37016 −0.195050
\(503\) −44.4295 −1.98101 −0.990507 0.137463i \(-0.956105\pi\)
−0.990507 + 0.137463i \(0.956105\pi\)
\(504\) 0 0
\(505\) −10.8328 −0.482054
\(506\) −4.47214 −0.198811
\(507\) 0 0
\(508\) 8.43769 0.374362
\(509\) −10.1545 −0.450092 −0.225046 0.974348i \(-0.572253\pi\)
−0.225046 + 0.974348i \(0.572253\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) 25.7109 1.13406
\(515\) −8.76393 −0.386185
\(516\) 0 0
\(517\) −12.5216 −0.550699
\(518\) 0 0
\(519\) 0 0
\(520\) −3.41641 −0.149819
\(521\) −21.8809 −0.958620 −0.479310 0.877646i \(-0.659113\pi\)
−0.479310 + 0.877646i \(0.659113\pi\)
\(522\) 0 0
\(523\) −17.0981 −0.747647 −0.373823 0.927500i \(-0.621953\pi\)
−0.373823 + 0.927500i \(0.621953\pi\)
\(524\) −1.15917 −0.0506388
\(525\) 0 0
\(526\) 23.0344 1.00435
\(527\) 1.05573 0.0459882
\(528\) 0 0
\(529\) −15.3607 −0.667856
\(530\) 3.82998 0.166364
\(531\) 0 0
\(532\) 0 0
\(533\) −17.5279 −0.759216
\(534\) 0 0
\(535\) 5.19548 0.224620
\(536\) 16.7082 0.721684
\(537\) 0 0
\(538\) −34.1962 −1.47430
\(539\) 0 0
\(540\) 0 0
\(541\) −18.5967 −0.799537 −0.399768 0.916616i \(-0.630909\pi\)
−0.399768 + 0.916616i \(0.630909\pi\)
\(542\) −44.6358 −1.91727
\(543\) 0 0
\(544\) −10.6947 −0.458532
\(545\) −2.57339 −0.110232
\(546\) 0 0
\(547\) −2.70820 −0.115794 −0.0578972 0.998323i \(-0.518440\pi\)
−0.0578972 + 0.998323i \(0.518440\pi\)
\(548\) 2.32624 0.0993720
\(549\) 0 0
\(550\) −6.85410 −0.292260
\(551\) −17.6383 −0.751415
\(552\) 0 0
\(553\) 0 0
\(554\) 9.32624 0.396234
\(555\) 0 0
\(556\) −0.127520 −0.00540803
\(557\) 20.7082 0.877435 0.438717 0.898625i \(-0.355433\pi\)
0.438717 + 0.898625i \(0.355433\pi\)
\(558\) 0 0
\(559\) 20.8794 0.883103
\(560\) 0 0
\(561\) 0 0
\(562\) −8.09017 −0.341263
\(563\) 34.2750 1.44452 0.722259 0.691623i \(-0.243104\pi\)
0.722259 + 0.691623i \(0.243104\pi\)
\(564\) 0 0
\(565\) 7.19859 0.302847
\(566\) 19.5927 0.823541
\(567\) 0 0
\(568\) 7.76393 0.325767
\(569\) −42.4164 −1.77819 −0.889094 0.457724i \(-0.848665\pi\)
−0.889094 + 0.457724i \(0.848665\pi\)
\(570\) 0 0
\(571\) −40.1803 −1.68149 −0.840747 0.541427i \(-0.817884\pi\)
−0.840747 + 0.541427i \(0.817884\pi\)
\(572\) −1.08036 −0.0451722
\(573\) 0 0
\(574\) 0 0
\(575\) 11.7082 0.488266
\(576\) 0 0
\(577\) 12.5216 0.521281 0.260640 0.965436i \(-0.416066\pi\)
0.260640 + 0.965436i \(0.416066\pi\)
\(578\) −11.3262 −0.471109
\(579\) 0 0
\(580\) 1.74806 0.0725844
\(581\) 0 0
\(582\) 0 0
\(583\) −2.70820 −0.112162
\(584\) 29.2070 1.20859
\(585\) 0 0
\(586\) −24.1692 −0.998418
\(587\) 2.90724 0.119995 0.0599973 0.998199i \(-0.480891\pi\)
0.0599973 + 0.998199i \(0.480891\pi\)
\(588\) 0 0
\(589\) 1.81966 0.0749778
\(590\) 11.5279 0.474595
\(591\) 0 0
\(592\) −13.1459 −0.540293
\(593\) 27.9504 1.14779 0.573893 0.818930i \(-0.305432\pi\)
0.573893 + 0.818930i \(0.305432\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.0344 −0.492950
\(597\) 0 0
\(598\) 7.81758 0.319685
\(599\) 39.8328 1.62752 0.813762 0.581198i \(-0.197416\pi\)
0.813762 + 0.581198i \(0.197416\pi\)
\(600\) 0 0
\(601\) 31.9867 1.30477 0.652383 0.757889i \(-0.273769\pi\)
0.652383 + 0.757889i \(0.273769\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −3.27051 −0.133075
\(605\) 8.74032 0.355345
\(606\) 0 0
\(607\) −26.2511 −1.06550 −0.532749 0.846273i \(-0.678841\pi\)
−0.532749 + 0.846273i \(0.678841\pi\)
\(608\) −18.4335 −0.747577
\(609\) 0 0
\(610\) 18.6525 0.755217
\(611\) 21.8885 0.885516
\(612\) 0 0
\(613\) 39.1803 1.58248 0.791240 0.611506i \(-0.209436\pi\)
0.791240 + 0.611506i \(0.209436\pi\)
\(614\) 30.1599 1.21715
\(615\) 0 0
\(616\) 0 0
\(617\) 21.4164 0.862192 0.431096 0.902306i \(-0.358127\pi\)
0.431096 + 0.902306i \(0.358127\pi\)
\(618\) 0 0
\(619\) 35.5617 1.42934 0.714672 0.699460i \(-0.246576\pi\)
0.714672 + 0.699460i \(0.246576\pi\)
\(620\) −0.180340 −0.00724262
\(621\) 0 0
\(622\) −45.9712 −1.84328
\(623\) 0 0
\(624\) 0 0
\(625\) 14.1246 0.564984
\(626\) 21.6746 0.866290
\(627\) 0 0
\(628\) −1.33540 −0.0532883
\(629\) 8.56409 0.341473
\(630\) 0 0
\(631\) −11.3475 −0.451738 −0.225869 0.974158i \(-0.572522\pi\)
−0.225869 + 0.974158i \(0.572522\pi\)
\(632\) 27.3607 1.08835
\(633\) 0 0
\(634\) 30.4721 1.21020
\(635\) −11.9327 −0.473535
\(636\) 0 0
\(637\) 0 0
\(638\) −5.23607 −0.207298
\(639\) 0 0
\(640\) −11.9026 −0.470492
\(641\) −12.5279 −0.494821 −0.247410 0.968911i \(-0.579580\pi\)
−0.247410 + 0.968911i \(0.579580\pi\)
\(642\) 0 0
\(643\) −28.5393 −1.12548 −0.562740 0.826634i \(-0.690253\pi\)
−0.562740 + 0.826634i \(0.690253\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 27.8885 1.09726
\(647\) −29.6684 −1.16638 −0.583192 0.812334i \(-0.698197\pi\)
−0.583192 + 0.812334i \(0.698197\pi\)
\(648\) 0 0
\(649\) −8.15143 −0.319972
\(650\) 11.9814 0.469950
\(651\) 0 0
\(652\) −5.09017 −0.199346
\(653\) 3.29180 0.128818 0.0644090 0.997924i \(-0.479484\pi\)
0.0644090 + 0.997924i \(0.479484\pi\)
\(654\) 0 0
\(655\) 1.63932 0.0640535
\(656\) −48.6722 −1.90033
\(657\) 0 0
\(658\) 0 0
\(659\) −11.7639 −0.458258 −0.229129 0.973396i \(-0.573588\pi\)
−0.229129 + 0.973396i \(0.573588\pi\)
\(660\) 0 0
\(661\) 17.6383 0.686049 0.343024 0.939326i \(-0.388549\pi\)
0.343024 + 0.939326i \(0.388549\pi\)
\(662\) 3.23607 0.125773
\(663\) 0 0
\(664\) −16.0965 −0.624667
\(665\) 0 0
\(666\) 0 0
\(667\) 8.94427 0.346324
\(668\) 5.37171 0.207838
\(669\) 0 0
\(670\) 10.5672 0.408246
\(671\) −13.1893 −0.509167
\(672\) 0 0
\(673\) 9.65248 0.372076 0.186038 0.982543i \(-0.440435\pi\)
0.186038 + 0.982543i \(0.440435\pi\)
\(674\) −27.7984 −1.07075
\(675\) 0 0
\(676\) −6.14590 −0.236381
\(677\) 7.07107 0.271763 0.135882 0.990725i \(-0.456613\pi\)
0.135882 + 0.990725i \(0.456613\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6.18034 0.237005
\(681\) 0 0
\(682\) 0.540182 0.0206846
\(683\) −17.7639 −0.679718 −0.339859 0.940476i \(-0.610379\pi\)
−0.339859 + 0.940476i \(0.610379\pi\)
\(684\) 0 0
\(685\) −3.28980 −0.125697
\(686\) 0 0
\(687\) 0 0
\(688\) 57.9787 2.21042
\(689\) 4.73411 0.180355
\(690\) 0 0
\(691\) −18.7974 −0.715088 −0.357544 0.933896i \(-0.616386\pi\)
−0.357544 + 0.933896i \(0.616386\pi\)
\(692\) −11.9026 −0.452469
\(693\) 0 0
\(694\) 35.5066 1.34781
\(695\) 0.180340 0.00684068
\(696\) 0 0
\(697\) 31.7082 1.20103
\(698\) −46.3839 −1.75566
\(699\) 0 0
\(700\) 0 0
\(701\) −13.1803 −0.497815 −0.248907 0.968527i \(-0.580071\pi\)
−0.248907 + 0.968527i \(0.580071\pi\)
\(702\) 0 0
\(703\) 14.7611 0.556727
\(704\) 4.23607 0.159653
\(705\) 0 0
\(706\) −33.1946 −1.24930
\(707\) 0 0
\(708\) 0 0
\(709\) −5.65248 −0.212283 −0.106142 0.994351i \(-0.533850\pi\)
−0.106142 + 0.994351i \(0.533850\pi\)
\(710\) 4.91034 0.184282
\(711\) 0 0
\(712\) 14.1421 0.529999
\(713\) −0.922740 −0.0345569
\(714\) 0 0
\(715\) 1.52786 0.0571389
\(716\) 12.5836 0.470271
\(717\) 0 0
\(718\) 27.7984 1.03743
\(719\) −17.8145 −0.664368 −0.332184 0.943215i \(-0.607786\pi\)
−0.332184 + 0.943215i \(0.607786\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.3262 0.644816
\(723\) 0 0
\(724\) 14.1908 0.527399
\(725\) 13.7082 0.509110
\(726\) 0 0
\(727\) −7.02236 −0.260445 −0.130222 0.991485i \(-0.541569\pi\)
−0.130222 + 0.991485i \(0.541569\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 18.4721 0.683684
\(731\) −37.7711 −1.39701
\(732\) 0 0
\(733\) 10.0270 0.370356 0.185178 0.982705i \(-0.440714\pi\)
0.185178 + 0.982705i \(0.440714\pi\)
\(734\) 2.82843 0.104399
\(735\) 0 0
\(736\) 9.34752 0.344554
\(737\) −7.47214 −0.275240
\(738\) 0 0
\(739\) −33.3607 −1.22719 −0.613596 0.789620i \(-0.710278\pi\)
−0.613596 + 0.789620i \(0.710278\pi\)
\(740\) −1.46292 −0.0537781
\(741\) 0 0
\(742\) 0 0
\(743\) −41.2492 −1.51329 −0.756644 0.653828i \(-0.773162\pi\)
−0.756644 + 0.653828i \(0.773162\pi\)
\(744\) 0 0
\(745\) 17.0193 0.623538
\(746\) −14.8541 −0.543847
\(747\) 0 0
\(748\) 1.95440 0.0714598
\(749\) 0 0
\(750\) 0 0
\(751\) 30.3050 1.10584 0.552922 0.833233i \(-0.313513\pi\)
0.552922 + 0.833233i \(0.313513\pi\)
\(752\) 60.7811 2.21646
\(753\) 0 0
\(754\) 9.15298 0.333332
\(755\) 4.62520 0.168328
\(756\) 0 0
\(757\) 16.0689 0.584034 0.292017 0.956413i \(-0.405674\pi\)
0.292017 + 0.956413i \(0.405674\pi\)
\(758\) −51.2148 −1.86020
\(759\) 0 0
\(760\) 10.6525 0.386406
\(761\) −1.87558 −0.0679899 −0.0339949 0.999422i \(-0.510823\pi\)
−0.0339949 + 0.999422i \(0.510823\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −2.72949 −0.0987495
\(765\) 0 0
\(766\) 21.5471 0.778527
\(767\) 14.2492 0.514510
\(768\) 0 0
\(769\) 52.2958 1.88583 0.942917 0.333027i \(-0.108070\pi\)
0.942917 + 0.333027i \(0.108070\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.1803 −0.366398
\(773\) 14.3184 0.514996 0.257498 0.966279i \(-0.417102\pi\)
0.257498 + 0.966279i \(0.417102\pi\)
\(774\) 0 0
\(775\) −1.41421 −0.0508001
\(776\) −20.9281 −0.751274
\(777\) 0 0
\(778\) 17.7082 0.634870
\(779\) 54.6525 1.95813
\(780\) 0 0
\(781\) −3.47214 −0.124243
\(782\) −14.1421 −0.505722
\(783\) 0 0
\(784\) 0 0
\(785\) 1.88854 0.0674050
\(786\) 0 0
\(787\) 34.1475 1.21723 0.608613 0.793467i \(-0.291726\pi\)
0.608613 + 0.793467i \(0.291726\pi\)
\(788\) 11.3820 0.405466
\(789\) 0 0
\(790\) 17.3044 0.615663
\(791\) 0 0
\(792\) 0 0
\(793\) 23.0557 0.818733
\(794\) −46.6389 −1.65515
\(795\) 0 0
\(796\) −7.07107 −0.250627
\(797\) −27.2039 −0.963612 −0.481806 0.876278i \(-0.660019\pi\)
−0.481806 + 0.876278i \(0.660019\pi\)
\(798\) 0 0
\(799\) −39.5967 −1.40083
\(800\) 14.3262 0.506509
\(801\) 0 0
\(802\) 59.9787 2.11792
\(803\) −13.0618 −0.460940
\(804\) 0 0
\(805\) 0 0
\(806\) −0.944272 −0.0332606
\(807\) 0 0
\(808\) −27.7140 −0.974975
\(809\) 43.5410 1.53082 0.765410 0.643543i \(-0.222536\pi\)
0.765410 + 0.643543i \(0.222536\pi\)
\(810\) 0 0
\(811\) −31.5254 −1.10701 −0.553503 0.832847i \(-0.686709\pi\)
−0.553503 + 0.832847i \(0.686709\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.38197 0.153588
\(815\) 7.19859 0.252156
\(816\) 0 0
\(817\) −65.1025 −2.27765
\(818\) 6.65841 0.232806
\(819\) 0 0
\(820\) −5.41641 −0.189149
\(821\) −33.1803 −1.15800 −0.579001 0.815327i \(-0.696557\pi\)
−0.579001 + 0.815327i \(0.696557\pi\)
\(822\) 0 0
\(823\) −9.41641 −0.328235 −0.164118 0.986441i \(-0.552478\pi\)
−0.164118 + 0.986441i \(0.552478\pi\)
\(824\) −22.4211 −0.781076
\(825\) 0 0
\(826\) 0 0
\(827\) −21.6525 −0.752930 −0.376465 0.926431i \(-0.622861\pi\)
−0.376465 + 0.926431i \(0.622861\pi\)
\(828\) 0 0
\(829\) −22.2148 −0.771550 −0.385775 0.922593i \(-0.626066\pi\)
−0.385775 + 0.922593i \(0.626066\pi\)
\(830\) −10.1803 −0.353365
\(831\) 0 0
\(832\) −7.40492 −0.256719
\(833\) 0 0
\(834\) 0 0
\(835\) −7.59675 −0.262896
\(836\) 3.36861 0.116506
\(837\) 0 0
\(838\) 39.5192 1.36517
\(839\) 47.4643 1.63865 0.819324 0.573330i \(-0.194349\pi\)
0.819324 + 0.573330i \(0.194349\pi\)
\(840\) 0 0
\(841\) −18.5279 −0.638892
\(842\) 24.0902 0.830202
\(843\) 0 0
\(844\) −2.03444 −0.0700284
\(845\) 8.69161 0.299001
\(846\) 0 0
\(847\) 0 0
\(848\) 13.1459 0.451432
\(849\) 0 0
\(850\) −21.6746 −0.743432
\(851\) −7.48529 −0.256592
\(852\) 0 0
\(853\) 47.6405 1.63118 0.815590 0.578631i \(-0.196413\pi\)
0.815590 + 0.578631i \(0.196413\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 13.2918 0.454304
\(857\) 26.9675 0.921191 0.460596 0.887610i \(-0.347636\pi\)
0.460596 + 0.887610i \(0.347636\pi\)
\(858\) 0 0
\(859\) −35.0215 −1.19492 −0.597459 0.801900i \(-0.703823\pi\)
−0.597459 + 0.801900i \(0.703823\pi\)
\(860\) 6.45207 0.220014
\(861\) 0 0
\(862\) −37.8885 −1.29049
\(863\) −22.3050 −0.759269 −0.379635 0.925136i \(-0.623950\pi\)
−0.379635 + 0.925136i \(0.623950\pi\)
\(864\) 0 0
\(865\) 16.8328 0.572333
\(866\) 33.1946 1.12800
\(867\) 0 0
\(868\) 0 0
\(869\) −12.2361 −0.415080
\(870\) 0 0
\(871\) 13.0618 0.442581
\(872\) −6.58359 −0.222949
\(873\) 0 0
\(874\) −24.3755 −0.824513
\(875\) 0 0
\(876\) 0 0
\(877\) 9.76393 0.329705 0.164852 0.986318i \(-0.447285\pi\)
0.164852 + 0.986318i \(0.447285\pi\)
\(878\) 18.9250 0.638686
\(879\) 0 0
\(880\) 4.24264 0.143019
\(881\) 21.7534 0.732890 0.366445 0.930440i \(-0.380575\pi\)
0.366445 + 0.930440i \(0.380575\pi\)
\(882\) 0 0
\(883\) 41.6525 1.40172 0.700859 0.713300i \(-0.252800\pi\)
0.700859 + 0.713300i \(0.252800\pi\)
\(884\) −3.41641 −0.114906
\(885\) 0 0
\(886\) −3.70820 −0.124580
\(887\) −14.1421 −0.474846 −0.237423 0.971406i \(-0.576303\pi\)
−0.237423 + 0.971406i \(0.576303\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 8.94427 0.299813
\(891\) 0 0
\(892\) 7.19859 0.241027
\(893\) −68.2492 −2.28387
\(894\) 0 0
\(895\) −17.7959 −0.594851
\(896\) 0 0
\(897\) 0 0
\(898\) 50.9230 1.69932
\(899\) −1.08036 −0.0360321
\(900\) 0 0
\(901\) −8.56409 −0.285311
\(902\) 16.2241 0.540202
\(903\) 0 0
\(904\) 18.4164 0.612521
\(905\) −20.0689 −0.667112
\(906\) 0 0
\(907\) 11.5410 0.383213 0.191607 0.981472i \(-0.438630\pi\)
0.191607 + 0.981472i \(0.438630\pi\)
\(908\) 14.2697 0.473555
\(909\) 0 0
\(910\) 0 0
\(911\) −40.2492 −1.33352 −0.666758 0.745274i \(-0.732319\pi\)
−0.666758 + 0.745274i \(0.732319\pi\)
\(912\) 0 0
\(913\) 7.19859 0.238238
\(914\) 38.5623 1.27553
\(915\) 0 0
\(916\) −6.19704 −0.204756
\(917\) 0 0
\(918\) 0 0
\(919\) 19.1803 0.632701 0.316351 0.948642i \(-0.397542\pi\)
0.316351 + 0.948642i \(0.397542\pi\)
\(920\) −5.40182 −0.178093
\(921\) 0 0
\(922\) 29.4133 0.968677
\(923\) 6.06952 0.199781
\(924\) 0 0
\(925\) −11.4721 −0.377202
\(926\) −59.6869 −1.96143
\(927\) 0 0
\(928\) 10.9443 0.359263
\(929\) −53.2974 −1.74863 −0.874315 0.485360i \(-0.838689\pi\)
−0.874315 + 0.485360i \(0.838689\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.1803 0.464492
\(933\) 0 0
\(934\) 65.2301 2.13439
\(935\) −2.76393 −0.0903902
\(936\) 0 0
\(937\) 19.5440 0.638473 0.319237 0.947675i \(-0.396574\pi\)
0.319237 + 0.947675i \(0.396574\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 6.76393 0.220615
\(941\) 46.5902 1.51880 0.759399 0.650625i \(-0.225493\pi\)
0.759399 + 0.650625i \(0.225493\pi\)
\(942\) 0 0
\(943\) −27.7140 −0.902492
\(944\) 39.5679 1.28782
\(945\) 0 0
\(946\) −19.3262 −0.628350
\(947\) −47.2361 −1.53497 −0.767483 0.641069i \(-0.778491\pi\)
−0.767483 + 0.641069i \(0.778491\pi\)
\(948\) 0 0
\(949\) 22.8328 0.741185
\(950\) −37.3584 −1.21207
\(951\) 0 0
\(952\) 0 0
\(953\) 30.5836 0.990700 0.495350 0.868694i \(-0.335040\pi\)
0.495350 + 0.868694i \(0.335040\pi\)
\(954\) 0 0
\(955\) 3.86008 0.124909
\(956\) 8.61803 0.278727
\(957\) 0 0
\(958\) −18.1784 −0.587319
\(959\) 0 0
\(960\) 0 0
\(961\) −30.8885 −0.996405
\(962\) −7.65996 −0.246967
\(963\) 0 0
\(964\) −7.48373 −0.241035
\(965\) 14.3972 0.463461
\(966\) 0 0
\(967\) 8.47214 0.272446 0.136223 0.990678i \(-0.456504\pi\)
0.136223 + 0.990678i \(0.456504\pi\)
\(968\) 22.3607 0.718699
\(969\) 0 0
\(970\) −13.2361 −0.424985
\(971\) 0.333851 0.0107138 0.00535689 0.999986i \(-0.498295\pi\)
0.00535689 + 0.999986i \(0.498295\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −22.5623 −0.722943
\(975\) 0 0
\(976\) 64.0222 2.04930
\(977\) −31.3050 −1.00153 −0.500767 0.865582i \(-0.666949\pi\)
−0.500767 + 0.865582i \(0.666949\pi\)
\(978\) 0 0
\(979\) −6.32456 −0.202134
\(980\) 0 0
\(981\) 0 0
\(982\) 35.5967 1.13594
\(983\) 24.5030 0.781524 0.390762 0.920492i \(-0.372211\pi\)
0.390762 + 0.920492i \(0.372211\pi\)
\(984\) 0 0
\(985\) −16.0965 −0.512878
\(986\) −16.5579 −0.527311
\(987\) 0 0
\(988\) −5.88854 −0.187340
\(989\) 33.0132 1.04976
\(990\) 0 0
\(991\) −6.41641 −0.203824 −0.101912 0.994793i \(-0.532496\pi\)
−0.101912 + 0.994793i \(0.532496\pi\)
\(992\) −1.12907 −0.0358480
\(993\) 0 0
\(994\) 0 0
\(995\) 10.0000 0.317021
\(996\) 0 0
\(997\) 45.1760 1.43074 0.715369 0.698746i \(-0.246258\pi\)
0.715369 + 0.698746i \(0.246258\pi\)
\(998\) −22.4721 −0.711343
\(999\) 0 0
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 3969.2.a.y.1.3 yes 4
3.2 odd 2 3969.2.a.r.1.2 yes 4
7.6 odd 2 inner 3969.2.a.y.1.4 yes 4
21.20 even 2 3969.2.a.r.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3969.2.a.r.1.1 4 21.20 even 2
3969.2.a.r.1.2 yes 4 3.2 odd 2
3969.2.a.y.1.3 yes 4 1.1 even 1 trivial
3969.2.a.y.1.4 yes 4 7.6 odd 2 inner