Properties

Label 8001.2.a.z.1.24
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $1$
Dimension $32$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(32\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.24
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56027 q^{2} +0.434445 q^{4} -1.69832 q^{5} -1.00000 q^{7} -2.44269 q^{8} +O(q^{10})\) \(q+1.56027 q^{2} +0.434445 q^{4} -1.69832 q^{5} -1.00000 q^{7} -2.44269 q^{8} -2.64985 q^{10} -1.57750 q^{11} +5.37691 q^{13} -1.56027 q^{14} -4.68015 q^{16} +7.38180 q^{17} -0.683565 q^{19} -0.737828 q^{20} -2.46133 q^{22} -8.53565 q^{23} -2.11570 q^{25} +8.38943 q^{26} -0.434445 q^{28} +5.81934 q^{29} +7.97047 q^{31} -2.41692 q^{32} +11.5176 q^{34} +1.69832 q^{35} -11.4098 q^{37} -1.06655 q^{38} +4.14848 q^{40} +5.44263 q^{41} +4.36687 q^{43} -0.685336 q^{44} -13.3179 q^{46} -3.11843 q^{47} +1.00000 q^{49} -3.30106 q^{50} +2.33597 q^{52} +7.03105 q^{53} +2.67911 q^{55} +2.44269 q^{56} +9.07974 q^{58} -5.82826 q^{59} -6.57944 q^{61} +12.4361 q^{62} +5.58925 q^{64} -9.13173 q^{65} -6.68183 q^{67} +3.20698 q^{68} +2.64985 q^{70} +4.90031 q^{71} -7.91813 q^{73} -17.8024 q^{74} -0.296971 q^{76} +1.57750 q^{77} -15.7765 q^{79} +7.94841 q^{80} +8.49197 q^{82} +6.39375 q^{83} -12.5367 q^{85} +6.81350 q^{86} +3.85334 q^{88} +2.66271 q^{89} -5.37691 q^{91} -3.70827 q^{92} -4.86560 q^{94} +1.16092 q^{95} +5.37034 q^{97} +1.56027 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \) \( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56027 1.10328 0.551639 0.834083i \(-0.314003\pi\)
0.551639 + 0.834083i \(0.314003\pi\)
\(3\) 0 0
\(4\) 0.434445 0.217222
\(5\) −1.69832 −0.759514 −0.379757 0.925086i \(-0.623992\pi\)
−0.379757 + 0.925086i \(0.623992\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.44269 −0.863621
\(9\) 0 0
\(10\) −2.64985 −0.837955
\(11\) −1.57750 −0.475634 −0.237817 0.971310i \(-0.576432\pi\)
−0.237817 + 0.971310i \(0.576432\pi\)
\(12\) 0 0
\(13\) 5.37691 1.49129 0.745643 0.666346i \(-0.232143\pi\)
0.745643 + 0.666346i \(0.232143\pi\)
\(14\) −1.56027 −0.417000
\(15\) 0 0
\(16\) −4.68015 −1.17004
\(17\) 7.38180 1.79035 0.895174 0.445717i \(-0.147051\pi\)
0.895174 + 0.445717i \(0.147051\pi\)
\(18\) 0 0
\(19\) −0.683565 −0.156821 −0.0784103 0.996921i \(-0.524984\pi\)
−0.0784103 + 0.996921i \(0.524984\pi\)
\(20\) −0.737828 −0.164983
\(21\) 0 0
\(22\) −2.46133 −0.524757
\(23\) −8.53565 −1.77981 −0.889903 0.456149i \(-0.849228\pi\)
−0.889903 + 0.456149i \(0.849228\pi\)
\(24\) 0 0
\(25\) −2.11570 −0.423139
\(26\) 8.38943 1.64530
\(27\) 0 0
\(28\) −0.434445 −0.0821023
\(29\) 5.81934 1.08062 0.540312 0.841465i \(-0.318306\pi\)
0.540312 + 0.841465i \(0.318306\pi\)
\(30\) 0 0
\(31\) 7.97047 1.43154 0.715770 0.698336i \(-0.246076\pi\)
0.715770 + 0.698336i \(0.246076\pi\)
\(32\) −2.41692 −0.427254
\(33\) 0 0
\(34\) 11.5176 1.97525
\(35\) 1.69832 0.287069
\(36\) 0 0
\(37\) −11.4098 −1.87576 −0.937881 0.346956i \(-0.887215\pi\)
−0.937881 + 0.346956i \(0.887215\pi\)
\(38\) −1.06655 −0.173017
\(39\) 0 0
\(40\) 4.14848 0.655932
\(41\) 5.44263 0.849995 0.424998 0.905194i \(-0.360275\pi\)
0.424998 + 0.905194i \(0.360275\pi\)
\(42\) 0 0
\(43\) 4.36687 0.665942 0.332971 0.942937i \(-0.391949\pi\)
0.332971 + 0.942937i \(0.391949\pi\)
\(44\) −0.685336 −0.103318
\(45\) 0 0
\(46\) −13.3179 −1.96362
\(47\) −3.11843 −0.454870 −0.227435 0.973793i \(-0.573034\pi\)
−0.227435 + 0.973793i \(0.573034\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.30106 −0.466840
\(51\) 0 0
\(52\) 2.33597 0.323941
\(53\) 7.03105 0.965789 0.482894 0.875679i \(-0.339585\pi\)
0.482894 + 0.875679i \(0.339585\pi\)
\(54\) 0 0
\(55\) 2.67911 0.361250
\(56\) 2.44269 0.326418
\(57\) 0 0
\(58\) 9.07974 1.19223
\(59\) −5.82826 −0.758775 −0.379388 0.925238i \(-0.623865\pi\)
−0.379388 + 0.925238i \(0.623865\pi\)
\(60\) 0 0
\(61\) −6.57944 −0.842411 −0.421206 0.906965i \(-0.638393\pi\)
−0.421206 + 0.906965i \(0.638393\pi\)
\(62\) 12.4361 1.57939
\(63\) 0 0
\(64\) 5.58925 0.698656
\(65\) −9.13173 −1.13265
\(66\) 0 0
\(67\) −6.68183 −0.816316 −0.408158 0.912911i \(-0.633829\pi\)
−0.408158 + 0.912911i \(0.633829\pi\)
\(68\) 3.20698 0.388904
\(69\) 0 0
\(70\) 2.64985 0.316717
\(71\) 4.90031 0.581559 0.290780 0.956790i \(-0.406085\pi\)
0.290780 + 0.956790i \(0.406085\pi\)
\(72\) 0 0
\(73\) −7.91813 −0.926747 −0.463373 0.886163i \(-0.653361\pi\)
−0.463373 + 0.886163i \(0.653361\pi\)
\(74\) −17.8024 −2.06949
\(75\) 0 0
\(76\) −0.296971 −0.0340650
\(77\) 1.57750 0.179773
\(78\) 0 0
\(79\) −15.7765 −1.77499 −0.887496 0.460815i \(-0.847557\pi\)
−0.887496 + 0.460815i \(0.847557\pi\)
\(80\) 7.94841 0.888659
\(81\) 0 0
\(82\) 8.49197 0.937781
\(83\) 6.39375 0.701806 0.350903 0.936412i \(-0.385875\pi\)
0.350903 + 0.936412i \(0.385875\pi\)
\(84\) 0 0
\(85\) −12.5367 −1.35979
\(86\) 6.81350 0.734719
\(87\) 0 0
\(88\) 3.85334 0.410768
\(89\) 2.66271 0.282247 0.141123 0.989992i \(-0.454929\pi\)
0.141123 + 0.989992i \(0.454929\pi\)
\(90\) 0 0
\(91\) −5.37691 −0.563653
\(92\) −3.70827 −0.386614
\(93\) 0 0
\(94\) −4.86560 −0.501848
\(95\) 1.16092 0.119107
\(96\) 0 0
\(97\) 5.37034 0.545276 0.272638 0.962117i \(-0.412104\pi\)
0.272638 + 0.962117i \(0.412104\pi\)
\(98\) 1.56027 0.157611
\(99\) 0 0
\(100\) −0.919153 −0.0919153
\(101\) −18.9810 −1.88868 −0.944342 0.328965i \(-0.893300\pi\)
−0.944342 + 0.328965i \(0.893300\pi\)
\(102\) 0 0
\(103\) −6.36094 −0.626762 −0.313381 0.949627i \(-0.601462\pi\)
−0.313381 + 0.949627i \(0.601462\pi\)
\(104\) −13.1341 −1.28791
\(105\) 0 0
\(106\) 10.9703 1.06553
\(107\) 1.79748 0.173769 0.0868844 0.996218i \(-0.472309\pi\)
0.0868844 + 0.996218i \(0.472309\pi\)
\(108\) 0 0
\(109\) −16.5252 −1.58283 −0.791413 0.611281i \(-0.790654\pi\)
−0.791413 + 0.611281i \(0.790654\pi\)
\(110\) 4.18013 0.398560
\(111\) 0 0
\(112\) 4.68015 0.442232
\(113\) −14.5665 −1.37030 −0.685151 0.728401i \(-0.740264\pi\)
−0.685151 + 0.728401i \(0.740264\pi\)
\(114\) 0 0
\(115\) 14.4963 1.35179
\(116\) 2.52818 0.234736
\(117\) 0 0
\(118\) −9.09367 −0.837140
\(119\) −7.38180 −0.676688
\(120\) 0 0
\(121\) −8.51150 −0.773772
\(122\) −10.2657 −0.929414
\(123\) 0 0
\(124\) 3.46273 0.310962
\(125\) 12.0848 1.08089
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 13.5546 1.19807
\(129\) 0 0
\(130\) −14.2480 −1.24963
\(131\) 13.0083 1.13654 0.568271 0.822841i \(-0.307612\pi\)
0.568271 + 0.822841i \(0.307612\pi\)
\(132\) 0 0
\(133\) 0.683565 0.0592726
\(134\) −10.4255 −0.900624
\(135\) 0 0
\(136\) −18.0314 −1.54618
\(137\) 11.0957 0.947969 0.473984 0.880533i \(-0.342815\pi\)
0.473984 + 0.880533i \(0.342815\pi\)
\(138\) 0 0
\(139\) −4.43688 −0.376331 −0.188165 0.982137i \(-0.560254\pi\)
−0.188165 + 0.982137i \(0.560254\pi\)
\(140\) 0.737828 0.0623578
\(141\) 0 0
\(142\) 7.64580 0.641622
\(143\) −8.48207 −0.709306
\(144\) 0 0
\(145\) −9.88312 −0.820748
\(146\) −12.3544 −1.02246
\(147\) 0 0
\(148\) −4.95694 −0.407458
\(149\) 0.577502 0.0473108 0.0236554 0.999720i \(-0.492470\pi\)
0.0236554 + 0.999720i \(0.492470\pi\)
\(150\) 0 0
\(151\) −7.20366 −0.586225 −0.293113 0.956078i \(-0.594691\pi\)
−0.293113 + 0.956078i \(0.594691\pi\)
\(152\) 1.66974 0.135434
\(153\) 0 0
\(154\) 2.46133 0.198339
\(155\) −13.5364 −1.08727
\(156\) 0 0
\(157\) −4.25760 −0.339793 −0.169897 0.985462i \(-0.554343\pi\)
−0.169897 + 0.985462i \(0.554343\pi\)
\(158\) −24.6156 −1.95831
\(159\) 0 0
\(160\) 4.10471 0.324506
\(161\) 8.53565 0.672704
\(162\) 0 0
\(163\) −13.2923 −1.04113 −0.520567 0.853821i \(-0.674279\pi\)
−0.520567 + 0.853821i \(0.674279\pi\)
\(164\) 2.36452 0.184638
\(165\) 0 0
\(166\) 9.97598 0.774287
\(167\) −7.11696 −0.550727 −0.275363 0.961340i \(-0.588798\pi\)
−0.275363 + 0.961340i \(0.588798\pi\)
\(168\) 0 0
\(169\) 15.9111 1.22393
\(170\) −19.5606 −1.50023
\(171\) 0 0
\(172\) 1.89716 0.144657
\(173\) −13.8378 −1.05207 −0.526035 0.850463i \(-0.676322\pi\)
−0.526035 + 0.850463i \(0.676322\pi\)
\(174\) 0 0
\(175\) 2.11570 0.159932
\(176\) 7.38293 0.556509
\(177\) 0 0
\(178\) 4.15455 0.311397
\(179\) −17.3579 −1.29739 −0.648697 0.761047i \(-0.724686\pi\)
−0.648697 + 0.761047i \(0.724686\pi\)
\(180\) 0 0
\(181\) 20.9700 1.55869 0.779345 0.626595i \(-0.215552\pi\)
0.779345 + 0.626595i \(0.215552\pi\)
\(182\) −8.38943 −0.621866
\(183\) 0 0
\(184\) 20.8500 1.53708
\(185\) 19.3776 1.42467
\(186\) 0 0
\(187\) −11.6448 −0.851551
\(188\) −1.35479 −0.0988080
\(189\) 0 0
\(190\) 1.81134 0.131409
\(191\) −12.2635 −0.887358 −0.443679 0.896186i \(-0.646327\pi\)
−0.443679 + 0.896186i \(0.646327\pi\)
\(192\) 0 0
\(193\) −17.6311 −1.26912 −0.634558 0.772875i \(-0.718818\pi\)
−0.634558 + 0.772875i \(0.718818\pi\)
\(194\) 8.37919 0.601591
\(195\) 0 0
\(196\) 0.434445 0.0310318
\(197\) −19.0024 −1.35387 −0.676933 0.736044i \(-0.736691\pi\)
−0.676933 + 0.736044i \(0.736691\pi\)
\(198\) 0 0
\(199\) 8.01293 0.568021 0.284011 0.958821i \(-0.408335\pi\)
0.284011 + 0.958821i \(0.408335\pi\)
\(200\) 5.16799 0.365432
\(201\) 0 0
\(202\) −29.6156 −2.08374
\(203\) −5.81934 −0.408437
\(204\) 0 0
\(205\) −9.24334 −0.645583
\(206\) −9.92479 −0.691493
\(207\) 0 0
\(208\) −25.1647 −1.74486
\(209\) 1.07832 0.0745892
\(210\) 0 0
\(211\) 4.07668 0.280650 0.140325 0.990105i \(-0.455185\pi\)
0.140325 + 0.990105i \(0.455185\pi\)
\(212\) 3.05460 0.209791
\(213\) 0 0
\(214\) 2.80455 0.191715
\(215\) −7.41636 −0.505792
\(216\) 0 0
\(217\) −7.97047 −0.541071
\(218\) −25.7838 −1.74630
\(219\) 0 0
\(220\) 1.16392 0.0784717
\(221\) 39.6912 2.66992
\(222\) 0 0
\(223\) 5.02212 0.336306 0.168153 0.985761i \(-0.446220\pi\)
0.168153 + 0.985761i \(0.446220\pi\)
\(224\) 2.41692 0.161487
\(225\) 0 0
\(226\) −22.7277 −1.51183
\(227\) 5.67133 0.376420 0.188210 0.982129i \(-0.439732\pi\)
0.188210 + 0.982129i \(0.439732\pi\)
\(228\) 0 0
\(229\) 11.8322 0.781892 0.390946 0.920414i \(-0.372148\pi\)
0.390946 + 0.920414i \(0.372148\pi\)
\(230\) 22.6182 1.49140
\(231\) 0 0
\(232\) −14.2148 −0.933250
\(233\) 22.7974 1.49351 0.746755 0.665099i \(-0.231611\pi\)
0.746755 + 0.665099i \(0.231611\pi\)
\(234\) 0 0
\(235\) 5.29611 0.345480
\(236\) −2.53206 −0.164823
\(237\) 0 0
\(238\) −11.5176 −0.746575
\(239\) −10.1782 −0.658374 −0.329187 0.944265i \(-0.606775\pi\)
−0.329187 + 0.944265i \(0.606775\pi\)
\(240\) 0 0
\(241\) −26.7920 −1.72583 −0.862913 0.505353i \(-0.831362\pi\)
−0.862913 + 0.505353i \(0.831362\pi\)
\(242\) −13.2802 −0.853686
\(243\) 0 0
\(244\) −2.85840 −0.182991
\(245\) −1.69832 −0.108502
\(246\) 0 0
\(247\) −3.67547 −0.233864
\(248\) −19.4694 −1.23631
\(249\) 0 0
\(250\) 18.8555 1.19253
\(251\) −20.6400 −1.30279 −0.651393 0.758741i \(-0.725815\pi\)
−0.651393 + 0.758741i \(0.725815\pi\)
\(252\) 0 0
\(253\) 13.4650 0.846537
\(254\) −1.56027 −0.0979001
\(255\) 0 0
\(256\) 9.97031 0.623144
\(257\) −12.6267 −0.787632 −0.393816 0.919189i \(-0.628845\pi\)
−0.393816 + 0.919189i \(0.628845\pi\)
\(258\) 0 0
\(259\) 11.4098 0.708972
\(260\) −3.96723 −0.246037
\(261\) 0 0
\(262\) 20.2965 1.25392
\(263\) 1.56670 0.0966067 0.0483034 0.998833i \(-0.484619\pi\)
0.0483034 + 0.998833i \(0.484619\pi\)
\(264\) 0 0
\(265\) −11.9410 −0.733530
\(266\) 1.06655 0.0653942
\(267\) 0 0
\(268\) −2.90289 −0.177322
\(269\) 21.7358 1.32526 0.662629 0.748948i \(-0.269441\pi\)
0.662629 + 0.748948i \(0.269441\pi\)
\(270\) 0 0
\(271\) −15.5957 −0.947370 −0.473685 0.880694i \(-0.657076\pi\)
−0.473685 + 0.880694i \(0.657076\pi\)
\(272\) −34.5479 −2.09477
\(273\) 0 0
\(274\) 17.3123 1.04587
\(275\) 3.33751 0.201259
\(276\) 0 0
\(277\) 9.01303 0.541541 0.270770 0.962644i \(-0.412722\pi\)
0.270770 + 0.962644i \(0.412722\pi\)
\(278\) −6.92273 −0.415198
\(279\) 0 0
\(280\) −4.14848 −0.247919
\(281\) 5.00890 0.298806 0.149403 0.988776i \(-0.452265\pi\)
0.149403 + 0.988776i \(0.452265\pi\)
\(282\) 0 0
\(283\) 1.98588 0.118048 0.0590242 0.998257i \(-0.481201\pi\)
0.0590242 + 0.998257i \(0.481201\pi\)
\(284\) 2.12891 0.126328
\(285\) 0 0
\(286\) −13.2343 −0.782562
\(287\) −5.44263 −0.321268
\(288\) 0 0
\(289\) 37.4909 2.20535
\(290\) −15.4203 −0.905514
\(291\) 0 0
\(292\) −3.43999 −0.201310
\(293\) −13.9204 −0.813239 −0.406620 0.913598i \(-0.633293\pi\)
−0.406620 + 0.913598i \(0.633293\pi\)
\(294\) 0 0
\(295\) 9.89828 0.576300
\(296\) 27.8707 1.61995
\(297\) 0 0
\(298\) 0.901059 0.0521970
\(299\) −45.8954 −2.65420
\(300\) 0 0
\(301\) −4.36687 −0.251702
\(302\) −11.2397 −0.646770
\(303\) 0 0
\(304\) 3.19919 0.183486
\(305\) 11.1740 0.639823
\(306\) 0 0
\(307\) −28.6836 −1.63706 −0.818529 0.574465i \(-0.805210\pi\)
−0.818529 + 0.574465i \(0.805210\pi\)
\(308\) 0.685336 0.0390507
\(309\) 0 0
\(310\) −21.1205 −1.19957
\(311\) 10.1277 0.574288 0.287144 0.957887i \(-0.407294\pi\)
0.287144 + 0.957887i \(0.407294\pi\)
\(312\) 0 0
\(313\) −4.26249 −0.240930 −0.120465 0.992718i \(-0.538439\pi\)
−0.120465 + 0.992718i \(0.538439\pi\)
\(314\) −6.64300 −0.374886
\(315\) 0 0
\(316\) −6.85401 −0.385568
\(317\) 10.8781 0.610973 0.305487 0.952196i \(-0.401181\pi\)
0.305487 + 0.952196i \(0.401181\pi\)
\(318\) 0 0
\(319\) −9.18000 −0.513981
\(320\) −9.49236 −0.530639
\(321\) 0 0
\(322\) 13.3179 0.742179
\(323\) −5.04594 −0.280764
\(324\) 0 0
\(325\) −11.3759 −0.631021
\(326\) −20.7396 −1.14866
\(327\) 0 0
\(328\) −13.2946 −0.734074
\(329\) 3.11843 0.171925
\(330\) 0 0
\(331\) −14.9176 −0.819946 −0.409973 0.912098i \(-0.634462\pi\)
−0.409973 + 0.912098i \(0.634462\pi\)
\(332\) 2.77773 0.152448
\(333\) 0 0
\(334\) −11.1044 −0.607605
\(335\) 11.3479 0.620003
\(336\) 0 0
\(337\) 9.00961 0.490785 0.245392 0.969424i \(-0.421083\pi\)
0.245392 + 0.969424i \(0.421083\pi\)
\(338\) 24.8257 1.35034
\(339\) 0 0
\(340\) −5.44649 −0.295378
\(341\) −12.5734 −0.680889
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −10.6669 −0.575122
\(345\) 0 0
\(346\) −21.5907 −1.16073
\(347\) −2.34030 −0.125634 −0.0628169 0.998025i \(-0.520008\pi\)
−0.0628169 + 0.998025i \(0.520008\pi\)
\(348\) 0 0
\(349\) −16.1455 −0.864249 −0.432125 0.901814i \(-0.642236\pi\)
−0.432125 + 0.901814i \(0.642236\pi\)
\(350\) 3.30106 0.176449
\(351\) 0 0
\(352\) 3.81268 0.203217
\(353\) 15.4517 0.822412 0.411206 0.911542i \(-0.365108\pi\)
0.411206 + 0.911542i \(0.365108\pi\)
\(354\) 0 0
\(355\) −8.32231 −0.441702
\(356\) 1.15680 0.0613103
\(357\) 0 0
\(358\) −27.0831 −1.43139
\(359\) −22.3811 −1.18123 −0.590616 0.806953i \(-0.701115\pi\)
−0.590616 + 0.806953i \(0.701115\pi\)
\(360\) 0 0
\(361\) −18.5327 −0.975407
\(362\) 32.7189 1.71967
\(363\) 0 0
\(364\) −2.33597 −0.122438
\(365\) 13.4475 0.703877
\(366\) 0 0
\(367\) −23.4592 −1.22456 −0.612280 0.790641i \(-0.709747\pi\)
−0.612280 + 0.790641i \(0.709747\pi\)
\(368\) 39.9481 2.08244
\(369\) 0 0
\(370\) 30.2343 1.57180
\(371\) −7.03105 −0.365034
\(372\) 0 0
\(373\) 26.8125 1.38830 0.694150 0.719830i \(-0.255780\pi\)
0.694150 + 0.719830i \(0.255780\pi\)
\(374\) −18.1690 −0.939497
\(375\) 0 0
\(376\) 7.61737 0.392836
\(377\) 31.2900 1.61152
\(378\) 0 0
\(379\) 19.2840 0.990551 0.495275 0.868736i \(-0.335067\pi\)
0.495275 + 0.868736i \(0.335067\pi\)
\(380\) 0.504354 0.0258728
\(381\) 0 0
\(382\) −19.1344 −0.979002
\(383\) −28.5258 −1.45760 −0.728799 0.684727i \(-0.759921\pi\)
−0.728799 + 0.684727i \(0.759921\pi\)
\(384\) 0 0
\(385\) −2.67911 −0.136540
\(386\) −27.5093 −1.40019
\(387\) 0 0
\(388\) 2.33312 0.118446
\(389\) −27.3537 −1.38689 −0.693443 0.720511i \(-0.743907\pi\)
−0.693443 + 0.720511i \(0.743907\pi\)
\(390\) 0 0
\(391\) −63.0084 −3.18647
\(392\) −2.44269 −0.123374
\(393\) 0 0
\(394\) −29.6489 −1.49369
\(395\) 26.7936 1.34813
\(396\) 0 0
\(397\) −16.9610 −0.851250 −0.425625 0.904900i \(-0.639946\pi\)
−0.425625 + 0.904900i \(0.639946\pi\)
\(398\) 12.5023 0.626685
\(399\) 0 0
\(400\) 9.90177 0.495088
\(401\) 17.7166 0.884723 0.442362 0.896837i \(-0.354141\pi\)
0.442362 + 0.896837i \(0.354141\pi\)
\(402\) 0 0
\(403\) 42.8565 2.13483
\(404\) −8.24621 −0.410264
\(405\) 0 0
\(406\) −9.07974 −0.450620
\(407\) 17.9990 0.892177
\(408\) 0 0
\(409\) −1.61511 −0.0798619 −0.0399310 0.999202i \(-0.512714\pi\)
−0.0399310 + 0.999202i \(0.512714\pi\)
\(410\) −14.4221 −0.712257
\(411\) 0 0
\(412\) −2.76348 −0.136147
\(413\) 5.82826 0.286790
\(414\) 0 0
\(415\) −10.8587 −0.533031
\(416\) −12.9955 −0.637158
\(417\) 0 0
\(418\) 1.68248 0.0822927
\(419\) 15.8383 0.773753 0.386876 0.922132i \(-0.373554\pi\)
0.386876 + 0.922132i \(0.373554\pi\)
\(420\) 0 0
\(421\) −15.1541 −0.738564 −0.369282 0.929317i \(-0.620396\pi\)
−0.369282 + 0.929317i \(0.620396\pi\)
\(422\) 6.36072 0.309635
\(423\) 0 0
\(424\) −17.1747 −0.834076
\(425\) −15.6176 −0.757567
\(426\) 0 0
\(427\) 6.57944 0.318401
\(428\) 0.780905 0.0377465
\(429\) 0 0
\(430\) −11.5715 −0.558029
\(431\) −28.5503 −1.37522 −0.687610 0.726080i \(-0.741340\pi\)
−0.687610 + 0.726080i \(0.741340\pi\)
\(432\) 0 0
\(433\) 32.9912 1.58546 0.792729 0.609575i \(-0.208660\pi\)
0.792729 + 0.609575i \(0.208660\pi\)
\(434\) −12.4361 −0.596952
\(435\) 0 0
\(436\) −7.17928 −0.343825
\(437\) 5.83468 0.279110
\(438\) 0 0
\(439\) −20.1904 −0.963636 −0.481818 0.876271i \(-0.660023\pi\)
−0.481818 + 0.876271i \(0.660023\pi\)
\(440\) −6.54422 −0.311984
\(441\) 0 0
\(442\) 61.9290 2.94566
\(443\) 40.8990 1.94317 0.971584 0.236694i \(-0.0760640\pi\)
0.971584 + 0.236694i \(0.0760640\pi\)
\(444\) 0 0
\(445\) −4.52215 −0.214370
\(446\) 7.83587 0.371039
\(447\) 0 0
\(448\) −5.58925 −0.264067
\(449\) −18.4836 −0.872293 −0.436146 0.899876i \(-0.643657\pi\)
−0.436146 + 0.899876i \(0.643657\pi\)
\(450\) 0 0
\(451\) −8.58574 −0.404287
\(452\) −6.32835 −0.297660
\(453\) 0 0
\(454\) 8.84882 0.415296
\(455\) 9.13173 0.428102
\(456\) 0 0
\(457\) 4.17547 0.195320 0.0976601 0.995220i \(-0.468864\pi\)
0.0976601 + 0.995220i \(0.468864\pi\)
\(458\) 18.4614 0.862645
\(459\) 0 0
\(460\) 6.29784 0.293638
\(461\) 27.4411 1.27806 0.639029 0.769182i \(-0.279336\pi\)
0.639029 + 0.769182i \(0.279336\pi\)
\(462\) 0 0
\(463\) −28.4925 −1.32416 −0.662078 0.749435i \(-0.730325\pi\)
−0.662078 + 0.749435i \(0.730325\pi\)
\(464\) −27.2354 −1.26437
\(465\) 0 0
\(466\) 35.5702 1.64776
\(467\) −31.9965 −1.48062 −0.740311 0.672265i \(-0.765322\pi\)
−0.740311 + 0.672265i \(0.765322\pi\)
\(468\) 0 0
\(469\) 6.68183 0.308538
\(470\) 8.26336 0.381161
\(471\) 0 0
\(472\) 14.2366 0.655295
\(473\) −6.88874 −0.316745
\(474\) 0 0
\(475\) 1.44622 0.0663570
\(476\) −3.20698 −0.146992
\(477\) 0 0
\(478\) −15.8808 −0.726369
\(479\) 0.125524 0.00573535 0.00286767 0.999996i \(-0.499087\pi\)
0.00286767 + 0.999996i \(0.499087\pi\)
\(480\) 0 0
\(481\) −61.3495 −2.79730
\(482\) −41.8028 −1.90406
\(483\) 0 0
\(484\) −3.69777 −0.168081
\(485\) −9.12058 −0.414144
\(486\) 0 0
\(487\) 23.4115 1.06088 0.530439 0.847723i \(-0.322027\pi\)
0.530439 + 0.847723i \(0.322027\pi\)
\(488\) 16.0715 0.727524
\(489\) 0 0
\(490\) −2.64985 −0.119708
\(491\) 20.9037 0.943369 0.471685 0.881767i \(-0.343646\pi\)
0.471685 + 0.881767i \(0.343646\pi\)
\(492\) 0 0
\(493\) 42.9572 1.93469
\(494\) −5.73472 −0.258017
\(495\) 0 0
\(496\) −37.3030 −1.67495
\(497\) −4.90031 −0.219809
\(498\) 0 0
\(499\) 5.86107 0.262377 0.131189 0.991357i \(-0.458121\pi\)
0.131189 + 0.991357i \(0.458121\pi\)
\(500\) 5.25016 0.234794
\(501\) 0 0
\(502\) −32.2040 −1.43733
\(503\) 30.4405 1.35728 0.678638 0.734473i \(-0.262571\pi\)
0.678638 + 0.734473i \(0.262571\pi\)
\(504\) 0 0
\(505\) 32.2360 1.43448
\(506\) 21.0090 0.933965
\(507\) 0 0
\(508\) −0.434445 −0.0192754
\(509\) −10.0065 −0.443531 −0.221766 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(510\) 0 0
\(511\) 7.91813 0.350277
\(512\) −11.5528 −0.510565
\(513\) 0 0
\(514\) −19.7011 −0.868977
\(515\) 10.8029 0.476034
\(516\) 0 0
\(517\) 4.91933 0.216352
\(518\) 17.8024 0.782193
\(519\) 0 0
\(520\) 22.3060 0.978182
\(521\) 9.37377 0.410672 0.205336 0.978692i \(-0.434171\pi\)
0.205336 + 0.978692i \(0.434171\pi\)
\(522\) 0 0
\(523\) 42.7780 1.87055 0.935277 0.353918i \(-0.115151\pi\)
0.935277 + 0.353918i \(0.115151\pi\)
\(524\) 5.65140 0.246882
\(525\) 0 0
\(526\) 2.44447 0.106584
\(527\) 58.8364 2.56295
\(528\) 0 0
\(529\) 49.8574 2.16771
\(530\) −18.6312 −0.809287
\(531\) 0 0
\(532\) 0.296971 0.0128753
\(533\) 29.2645 1.26759
\(534\) 0 0
\(535\) −3.05270 −0.131980
\(536\) 16.3217 0.704988
\(537\) 0 0
\(538\) 33.9138 1.46213
\(539\) −1.57750 −0.0679477
\(540\) 0 0
\(541\) 31.6416 1.36038 0.680189 0.733037i \(-0.261898\pi\)
0.680189 + 0.733037i \(0.261898\pi\)
\(542\) −24.3335 −1.04521
\(543\) 0 0
\(544\) −17.8412 −0.764934
\(545\) 28.0651 1.20218
\(546\) 0 0
\(547\) −23.6839 −1.01265 −0.506326 0.862342i \(-0.668997\pi\)
−0.506326 + 0.862342i \(0.668997\pi\)
\(548\) 4.82046 0.205920
\(549\) 0 0
\(550\) 5.20742 0.222045
\(551\) −3.97790 −0.169464
\(552\) 0 0
\(553\) 15.7765 0.670884
\(554\) 14.0628 0.597470
\(555\) 0 0
\(556\) −1.92758 −0.0817475
\(557\) 20.7846 0.880671 0.440336 0.897833i \(-0.354859\pi\)
0.440336 + 0.897833i \(0.354859\pi\)
\(558\) 0 0
\(559\) 23.4803 0.993109
\(560\) −7.94841 −0.335881
\(561\) 0 0
\(562\) 7.81523 0.329666
\(563\) 12.9904 0.547479 0.273739 0.961804i \(-0.411739\pi\)
0.273739 + 0.961804i \(0.411739\pi\)
\(564\) 0 0
\(565\) 24.7387 1.04076
\(566\) 3.09851 0.130240
\(567\) 0 0
\(568\) −11.9699 −0.502247
\(569\) 29.1174 1.22067 0.610333 0.792145i \(-0.291036\pi\)
0.610333 + 0.792145i \(0.291036\pi\)
\(570\) 0 0
\(571\) −21.9240 −0.917491 −0.458745 0.888568i \(-0.651701\pi\)
−0.458745 + 0.888568i \(0.651701\pi\)
\(572\) −3.68499 −0.154077
\(573\) 0 0
\(574\) −8.49197 −0.354448
\(575\) 18.0588 0.753106
\(576\) 0 0
\(577\) 15.9248 0.662958 0.331479 0.943463i \(-0.392452\pi\)
0.331479 + 0.943463i \(0.392452\pi\)
\(578\) 58.4960 2.43311
\(579\) 0 0
\(580\) −4.29367 −0.178285
\(581\) −6.39375 −0.265258
\(582\) 0 0
\(583\) −11.0915 −0.459362
\(584\) 19.3415 0.800358
\(585\) 0 0
\(586\) −21.7196 −0.897229
\(587\) 1.21000 0.0499420 0.0249710 0.999688i \(-0.492051\pi\)
0.0249710 + 0.999688i \(0.492051\pi\)
\(588\) 0 0
\(589\) −5.44834 −0.224495
\(590\) 15.4440 0.635819
\(591\) 0 0
\(592\) 53.3996 2.19471
\(593\) −18.1805 −0.746585 −0.373292 0.927714i \(-0.621771\pi\)
−0.373292 + 0.927714i \(0.621771\pi\)
\(594\) 0 0
\(595\) 12.5367 0.513954
\(596\) 0.250893 0.0102770
\(597\) 0 0
\(598\) −71.6093 −2.92832
\(599\) −3.41903 −0.139698 −0.0698489 0.997558i \(-0.522252\pi\)
−0.0698489 + 0.997558i \(0.522252\pi\)
\(600\) 0 0
\(601\) 20.4048 0.832328 0.416164 0.909290i \(-0.363374\pi\)
0.416164 + 0.909290i \(0.363374\pi\)
\(602\) −6.81350 −0.277698
\(603\) 0 0
\(604\) −3.12959 −0.127341
\(605\) 14.4553 0.587691
\(606\) 0 0
\(607\) 10.0628 0.408435 0.204217 0.978926i \(-0.434535\pi\)
0.204217 + 0.978926i \(0.434535\pi\)
\(608\) 1.65212 0.0670023
\(609\) 0 0
\(610\) 17.4345 0.705902
\(611\) −16.7675 −0.678341
\(612\) 0 0
\(613\) 0.173649 0.00701361 0.00350680 0.999994i \(-0.498884\pi\)
0.00350680 + 0.999994i \(0.498884\pi\)
\(614\) −44.7542 −1.80613
\(615\) 0 0
\(616\) −3.85334 −0.155256
\(617\) −19.8820 −0.800421 −0.400210 0.916423i \(-0.631063\pi\)
−0.400210 + 0.916423i \(0.631063\pi\)
\(618\) 0 0
\(619\) −41.5677 −1.67075 −0.835373 0.549684i \(-0.814748\pi\)
−0.835373 + 0.549684i \(0.814748\pi\)
\(620\) −5.88084 −0.236180
\(621\) 0 0
\(622\) 15.8019 0.633599
\(623\) −2.66271 −0.106679
\(624\) 0 0
\(625\) −9.94535 −0.397814
\(626\) −6.65064 −0.265813
\(627\) 0 0
\(628\) −1.84969 −0.0738107
\(629\) −84.2250 −3.35827
\(630\) 0 0
\(631\) 21.7909 0.867483 0.433742 0.901037i \(-0.357193\pi\)
0.433742 + 0.901037i \(0.357193\pi\)
\(632\) 38.5370 1.53292
\(633\) 0 0
\(634\) 16.9727 0.674073
\(635\) 1.69832 0.0673959
\(636\) 0 0
\(637\) 5.37691 0.213041
\(638\) −14.3233 −0.567064
\(639\) 0 0
\(640\) −23.0201 −0.909948
\(641\) 14.0620 0.555415 0.277707 0.960666i \(-0.410425\pi\)
0.277707 + 0.960666i \(0.410425\pi\)
\(642\) 0 0
\(643\) −3.13529 −0.123644 −0.0618219 0.998087i \(-0.519691\pi\)
−0.0618219 + 0.998087i \(0.519691\pi\)
\(644\) 3.70827 0.146126
\(645\) 0 0
\(646\) −7.87303 −0.309760
\(647\) 31.1920 1.22628 0.613142 0.789973i \(-0.289906\pi\)
0.613142 + 0.789973i \(0.289906\pi\)
\(648\) 0 0
\(649\) 9.19408 0.360899
\(650\) −17.7495 −0.696192
\(651\) 0 0
\(652\) −5.77477 −0.226157
\(653\) 1.25769 0.0492171 0.0246085 0.999697i \(-0.492166\pi\)
0.0246085 + 0.999697i \(0.492166\pi\)
\(654\) 0 0
\(655\) −22.0923 −0.863219
\(656\) −25.4723 −0.994526
\(657\) 0 0
\(658\) 4.86560 0.189681
\(659\) 25.2125 0.982140 0.491070 0.871120i \(-0.336606\pi\)
0.491070 + 0.871120i \(0.336606\pi\)
\(660\) 0 0
\(661\) 42.0623 1.63603 0.818017 0.575193i \(-0.195073\pi\)
0.818017 + 0.575193i \(0.195073\pi\)
\(662\) −23.2755 −0.904628
\(663\) 0 0
\(664\) −15.6180 −0.606094
\(665\) −1.16092 −0.0450184
\(666\) 0 0
\(667\) −49.6719 −1.92330
\(668\) −3.09193 −0.119630
\(669\) 0 0
\(670\) 17.7058 0.684036
\(671\) 10.3791 0.400679
\(672\) 0 0
\(673\) −25.1327 −0.968793 −0.484396 0.874849i \(-0.660961\pi\)
−0.484396 + 0.874849i \(0.660961\pi\)
\(674\) 14.0574 0.541472
\(675\) 0 0
\(676\) 6.91250 0.265865
\(677\) −18.1849 −0.698903 −0.349452 0.936954i \(-0.613632\pi\)
−0.349452 + 0.936954i \(0.613632\pi\)
\(678\) 0 0
\(679\) −5.37034 −0.206095
\(680\) 30.6232 1.17435
\(681\) 0 0
\(682\) −19.6179 −0.751210
\(683\) −1.19845 −0.0458573 −0.0229286 0.999737i \(-0.507299\pi\)
−0.0229286 + 0.999737i \(0.507299\pi\)
\(684\) 0 0
\(685\) −18.8441 −0.719995
\(686\) −1.56027 −0.0595714
\(687\) 0 0
\(688\) −20.4376 −0.779176
\(689\) 37.8053 1.44027
\(690\) 0 0
\(691\) −35.3781 −1.34585 −0.672924 0.739712i \(-0.734962\pi\)
−0.672924 + 0.739712i \(0.734962\pi\)
\(692\) −6.01177 −0.228533
\(693\) 0 0
\(694\) −3.65150 −0.138609
\(695\) 7.53525 0.285828
\(696\) 0 0
\(697\) 40.1763 1.52179
\(698\) −25.1914 −0.953507
\(699\) 0 0
\(700\) 0.919153 0.0347407
\(701\) 21.6244 0.816740 0.408370 0.912816i \(-0.366097\pi\)
0.408370 + 0.912816i \(0.366097\pi\)
\(702\) 0 0
\(703\) 7.79936 0.294158
\(704\) −8.81704 −0.332305
\(705\) 0 0
\(706\) 24.1089 0.907349
\(707\) 18.9810 0.713855
\(708\) 0 0
\(709\) −20.0892 −0.754467 −0.377233 0.926118i \(-0.623125\pi\)
−0.377233 + 0.926118i \(0.623125\pi\)
\(710\) −12.9850 −0.487320
\(711\) 0 0
\(712\) −6.50418 −0.243754
\(713\) −68.0332 −2.54786
\(714\) 0 0
\(715\) 14.4053 0.538728
\(716\) −7.54107 −0.281823
\(717\) 0 0
\(718\) −34.9206 −1.30323
\(719\) 1.69180 0.0630936 0.0315468 0.999502i \(-0.489957\pi\)
0.0315468 + 0.999502i \(0.489957\pi\)
\(720\) 0 0
\(721\) 6.36094 0.236894
\(722\) −28.9161 −1.07615
\(723\) 0 0
\(724\) 9.11032 0.338582
\(725\) −12.3119 −0.457254
\(726\) 0 0
\(727\) −19.4327 −0.720718 −0.360359 0.932814i \(-0.617346\pi\)
−0.360359 + 0.932814i \(0.617346\pi\)
\(728\) 13.1341 0.486783
\(729\) 0 0
\(730\) 20.9818 0.776572
\(731\) 32.2354 1.19227
\(732\) 0 0
\(733\) 1.18340 0.0437098 0.0218549 0.999761i \(-0.493043\pi\)
0.0218549 + 0.999761i \(0.493043\pi\)
\(734\) −36.6027 −1.35103
\(735\) 0 0
\(736\) 20.6300 0.760430
\(737\) 10.5406 0.388268
\(738\) 0 0
\(739\) −13.1398 −0.483355 −0.241677 0.970357i \(-0.577698\pi\)
−0.241677 + 0.970357i \(0.577698\pi\)
\(740\) 8.41848 0.309470
\(741\) 0 0
\(742\) −10.9703 −0.402734
\(743\) 49.1903 1.80462 0.902308 0.431091i \(-0.141871\pi\)
0.902308 + 0.431091i \(0.141871\pi\)
\(744\) 0 0
\(745\) −0.980785 −0.0359332
\(746\) 41.8348 1.53168
\(747\) 0 0
\(748\) −5.05901 −0.184976
\(749\) −1.79748 −0.0656785
\(750\) 0 0
\(751\) 3.63243 0.132549 0.0662746 0.997801i \(-0.478889\pi\)
0.0662746 + 0.997801i \(0.478889\pi\)
\(752\) 14.5947 0.532215
\(753\) 0 0
\(754\) 48.8209 1.77795
\(755\) 12.2341 0.445246
\(756\) 0 0
\(757\) −46.0208 −1.67265 −0.836327 0.548231i \(-0.815301\pi\)
−0.836327 + 0.548231i \(0.815301\pi\)
\(758\) 30.0882 1.09285
\(759\) 0 0
\(760\) −2.83576 −0.102864
\(761\) 6.59912 0.239218 0.119609 0.992821i \(-0.461836\pi\)
0.119609 + 0.992821i \(0.461836\pi\)
\(762\) 0 0
\(763\) 16.5252 0.598252
\(764\) −5.32782 −0.192754
\(765\) 0 0
\(766\) −44.5079 −1.60814
\(767\) −31.3380 −1.13155
\(768\) 0 0
\(769\) 18.9717 0.684138 0.342069 0.939675i \(-0.388872\pi\)
0.342069 + 0.939675i \(0.388872\pi\)
\(770\) −4.18013 −0.150641
\(771\) 0 0
\(772\) −7.65975 −0.275681
\(773\) 21.5176 0.773934 0.386967 0.922094i \(-0.373523\pi\)
0.386967 + 0.922094i \(0.373523\pi\)
\(774\) 0 0
\(775\) −16.8631 −0.605741
\(776\) −13.1181 −0.470912
\(777\) 0 0
\(778\) −42.6792 −1.53012
\(779\) −3.72039 −0.133297
\(780\) 0 0
\(781\) −7.73023 −0.276609
\(782\) −98.3102 −3.51557
\(783\) 0 0
\(784\) −4.68015 −0.167148
\(785\) 7.23078 0.258078
\(786\) 0 0
\(787\) −19.9237 −0.710205 −0.355102 0.934827i \(-0.615554\pi\)
−0.355102 + 0.934827i \(0.615554\pi\)
\(788\) −8.25550 −0.294090
\(789\) 0 0
\(790\) 41.8052 1.48736
\(791\) 14.5665 0.517926
\(792\) 0 0
\(793\) −35.3770 −1.25628
\(794\) −26.4638 −0.939166
\(795\) 0 0
\(796\) 3.48117 0.123387
\(797\) 8.77923 0.310976 0.155488 0.987838i \(-0.450305\pi\)
0.155488 + 0.987838i \(0.450305\pi\)
\(798\) 0 0
\(799\) −23.0196 −0.814376
\(800\) 5.11346 0.180788
\(801\) 0 0
\(802\) 27.6426 0.976096
\(803\) 12.4908 0.440792
\(804\) 0 0
\(805\) −14.4963 −0.510928
\(806\) 66.8677 2.35532
\(807\) 0 0
\(808\) 46.3648 1.63111
\(809\) −10.0995 −0.355080 −0.177540 0.984114i \(-0.556814\pi\)
−0.177540 + 0.984114i \(0.556814\pi\)
\(810\) 0 0
\(811\) −17.2459 −0.605584 −0.302792 0.953057i \(-0.597919\pi\)
−0.302792 + 0.953057i \(0.597919\pi\)
\(812\) −2.52818 −0.0887217
\(813\) 0 0
\(814\) 28.0833 0.984319
\(815\) 22.5746 0.790755
\(816\) 0 0
\(817\) −2.98504 −0.104433
\(818\) −2.52000 −0.0881099
\(819\) 0 0
\(820\) −4.01572 −0.140235
\(821\) −2.22336 −0.0775958 −0.0387979 0.999247i \(-0.512353\pi\)
−0.0387979 + 0.999247i \(0.512353\pi\)
\(822\) 0 0
\(823\) −49.5127 −1.72590 −0.862951 0.505287i \(-0.831387\pi\)
−0.862951 + 0.505287i \(0.831387\pi\)
\(824\) 15.5378 0.541285
\(825\) 0 0
\(826\) 9.09367 0.316409
\(827\) −33.2176 −1.15509 −0.577544 0.816360i \(-0.695989\pi\)
−0.577544 + 0.816360i \(0.695989\pi\)
\(828\) 0 0
\(829\) −56.8197 −1.97343 −0.986715 0.162464i \(-0.948056\pi\)
−0.986715 + 0.162464i \(0.948056\pi\)
\(830\) −16.9425 −0.588081
\(831\) 0 0
\(832\) 30.0529 1.04190
\(833\) 7.38180 0.255764
\(834\) 0 0
\(835\) 12.0869 0.418285
\(836\) 0.468472 0.0162025
\(837\) 0 0
\(838\) 24.7121 0.853664
\(839\) −11.2187 −0.387312 −0.193656 0.981070i \(-0.562035\pi\)
−0.193656 + 0.981070i \(0.562035\pi\)
\(840\) 0 0
\(841\) 4.86469 0.167748
\(842\) −23.6445 −0.814842
\(843\) 0 0
\(844\) 1.77109 0.0609635
\(845\) −27.0222 −0.929593
\(846\) 0 0
\(847\) 8.51150 0.292458
\(848\) −32.9063 −1.13001
\(849\) 0 0
\(850\) −24.3677 −0.835807
\(851\) 97.3903 3.33850
\(852\) 0 0
\(853\) −15.1859 −0.519956 −0.259978 0.965614i \(-0.583715\pi\)
−0.259978 + 0.965614i \(0.583715\pi\)
\(854\) 10.2657 0.351285
\(855\) 0 0
\(856\) −4.39068 −0.150071
\(857\) −7.84113 −0.267848 −0.133924 0.990992i \(-0.542758\pi\)
−0.133924 + 0.990992i \(0.542758\pi\)
\(858\) 0 0
\(859\) 5.04032 0.171973 0.0859867 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(860\) −3.22200 −0.109869
\(861\) 0 0
\(862\) −44.5462 −1.51725
\(863\) 22.2053 0.755876 0.377938 0.925831i \(-0.376633\pi\)
0.377938 + 0.925831i \(0.376633\pi\)
\(864\) 0 0
\(865\) 23.5011 0.799061
\(866\) 51.4752 1.74920
\(867\) 0 0
\(868\) −3.46273 −0.117533
\(869\) 24.8874 0.844247
\(870\) 0 0
\(871\) −35.9276 −1.21736
\(872\) 40.3659 1.36696
\(873\) 0 0
\(874\) 9.10368 0.307936
\(875\) −12.0848 −0.408539
\(876\) 0 0
\(877\) 41.0971 1.38775 0.693875 0.720095i \(-0.255902\pi\)
0.693875 + 0.720095i \(0.255902\pi\)
\(878\) −31.5025 −1.06316
\(879\) 0 0
\(880\) −12.5386 −0.422676
\(881\) 41.5727 1.40062 0.700309 0.713840i \(-0.253046\pi\)
0.700309 + 0.713840i \(0.253046\pi\)
\(882\) 0 0
\(883\) 29.2046 0.982812 0.491406 0.870931i \(-0.336483\pi\)
0.491406 + 0.870931i \(0.336483\pi\)
\(884\) 17.2436 0.579966
\(885\) 0 0
\(886\) 63.8135 2.14385
\(887\) 24.1854 0.812067 0.406033 0.913858i \(-0.366912\pi\)
0.406033 + 0.913858i \(0.366912\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −7.05577 −0.236510
\(891\) 0 0
\(892\) 2.18183 0.0730532
\(893\) 2.13165 0.0713330
\(894\) 0 0
\(895\) 29.4794 0.985388
\(896\) −13.5546 −0.452827
\(897\) 0 0
\(898\) −28.8393 −0.962381
\(899\) 46.3829 1.54696
\(900\) 0 0
\(901\) 51.9018 1.72910
\(902\) −13.3961 −0.446041
\(903\) 0 0
\(904\) 35.5815 1.18342
\(905\) −35.6139 −1.18385
\(906\) 0 0
\(907\) −49.8040 −1.65371 −0.826857 0.562412i \(-0.809874\pi\)
−0.826857 + 0.562412i \(0.809874\pi\)
\(908\) 2.46388 0.0817668
\(909\) 0 0
\(910\) 14.2480 0.472316
\(911\) −54.6229 −1.80974 −0.904869 0.425690i \(-0.860031\pi\)
−0.904869 + 0.425690i \(0.860031\pi\)
\(912\) 0 0
\(913\) −10.0861 −0.333803
\(914\) 6.51486 0.215492
\(915\) 0 0
\(916\) 5.14043 0.169845
\(917\) −13.0083 −0.429573
\(918\) 0 0
\(919\) −27.0540 −0.892428 −0.446214 0.894926i \(-0.647228\pi\)
−0.446214 + 0.894926i \(0.647228\pi\)
\(920\) −35.4100 −1.16743
\(921\) 0 0
\(922\) 42.8155 1.41005
\(923\) 26.3485 0.867271
\(924\) 0 0
\(925\) 24.1397 0.793709
\(926\) −44.4559 −1.46091
\(927\) 0 0
\(928\) −14.0649 −0.461701
\(929\) −17.5180 −0.574748 −0.287374 0.957818i \(-0.592782\pi\)
−0.287374 + 0.957818i \(0.592782\pi\)
\(930\) 0 0
\(931\) −0.683565 −0.0224030
\(932\) 9.90423 0.324424
\(933\) 0 0
\(934\) −49.9232 −1.63354
\(935\) 19.7766 0.646764
\(936\) 0 0
\(937\) −52.0412 −1.70011 −0.850056 0.526692i \(-0.823432\pi\)
−0.850056 + 0.526692i \(0.823432\pi\)
\(938\) 10.4255 0.340404
\(939\) 0 0
\(940\) 2.30087 0.0750460
\(941\) 10.3014 0.335816 0.167908 0.985803i \(-0.446299\pi\)
0.167908 + 0.985803i \(0.446299\pi\)
\(942\) 0 0
\(943\) −46.4564 −1.51283
\(944\) 27.2771 0.887795
\(945\) 0 0
\(946\) −10.7483 −0.349457
\(947\) 31.0642 1.00945 0.504726 0.863280i \(-0.331594\pi\)
0.504726 + 0.863280i \(0.331594\pi\)
\(948\) 0 0
\(949\) −42.5750 −1.38204
\(950\) 2.25649 0.0732102
\(951\) 0 0
\(952\) 18.0314 0.584402
\(953\) −28.8878 −0.935768 −0.467884 0.883790i \(-0.654984\pi\)
−0.467884 + 0.883790i \(0.654984\pi\)
\(954\) 0 0
\(955\) 20.8274 0.673960
\(956\) −4.42187 −0.143013
\(957\) 0 0
\(958\) 0.195852 0.00632768
\(959\) −11.0957 −0.358299
\(960\) 0 0
\(961\) 32.5285 1.04931
\(962\) −95.7219 −3.08620
\(963\) 0 0
\(964\) −11.6396 −0.374888
\(965\) 29.9434 0.963911
\(966\) 0 0
\(967\) 33.6661 1.08263 0.541315 0.840820i \(-0.317927\pi\)
0.541315 + 0.840820i \(0.317927\pi\)
\(968\) 20.7909 0.668246
\(969\) 0 0
\(970\) −14.2306 −0.456916
\(971\) 4.51372 0.144852 0.0724260 0.997374i \(-0.476926\pi\)
0.0724260 + 0.997374i \(0.476926\pi\)
\(972\) 0 0
\(973\) 4.43688 0.142240
\(974\) 36.5283 1.17044
\(975\) 0 0
\(976\) 30.7928 0.985652
\(977\) 49.7681 1.59222 0.796111 0.605151i \(-0.206887\pi\)
0.796111 + 0.605151i \(0.206887\pi\)
\(978\) 0 0
\(979\) −4.20043 −0.134246
\(980\) −0.737828 −0.0235690
\(981\) 0 0
\(982\) 32.6154 1.04080
\(983\) −28.8892 −0.921422 −0.460711 0.887550i \(-0.652406\pi\)
−0.460711 + 0.887550i \(0.652406\pi\)
\(984\) 0 0
\(985\) 32.2723 1.02828
\(986\) 67.0248 2.13450
\(987\) 0 0
\(988\) −1.59679 −0.0508006
\(989\) −37.2741 −1.18525
\(990\) 0 0
\(991\) −16.6458 −0.528770 −0.264385 0.964417i \(-0.585169\pi\)
−0.264385 + 0.964417i \(0.585169\pi\)
\(992\) −19.2640 −0.611632
\(993\) 0 0
\(994\) −7.64580 −0.242510
\(995\) −13.6085 −0.431420
\(996\) 0 0
\(997\) 9.55174 0.302507 0.151253 0.988495i \(-0.451669\pi\)
0.151253 + 0.988495i \(0.451669\pi\)
\(998\) 9.14485 0.289475
\(999\) 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 8001.2.a.z.1.24 yes 32
3.2 odd 2 inner 8001.2.a.z.1.9 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.z.1.9 32 3.2 odd 2 inner
8001.2.a.z.1.24 yes 32 1.1 even 1 trivial