Properties

Label 8034.2.a.bb.1.12
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-2.36221\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.36221 q^{5} -1.00000 q^{6} -4.26027 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.36221 q^{5} -1.00000 q^{6} -4.26027 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.36221 q^{10} -1.66695 q^{11} +1.00000 q^{12} -1.00000 q^{13} +4.26027 q^{14} +2.36221 q^{15} +1.00000 q^{16} +1.26660 q^{17} -1.00000 q^{18} -4.50257 q^{19} +2.36221 q^{20} -4.26027 q^{21} +1.66695 q^{22} +8.71852 q^{23} -1.00000 q^{24} +0.580037 q^{25} +1.00000 q^{26} +1.00000 q^{27} -4.26027 q^{28} +7.70479 q^{29} -2.36221 q^{30} -7.02642 q^{31} -1.00000 q^{32} -1.66695 q^{33} -1.26660 q^{34} -10.0637 q^{35} +1.00000 q^{36} +1.42734 q^{37} +4.50257 q^{38} -1.00000 q^{39} -2.36221 q^{40} -2.88901 q^{41} +4.26027 q^{42} -2.87221 q^{43} -1.66695 q^{44} +2.36221 q^{45} -8.71852 q^{46} +2.32318 q^{47} +1.00000 q^{48} +11.1499 q^{49} -0.580037 q^{50} +1.26660 q^{51} -1.00000 q^{52} -3.14222 q^{53} -1.00000 q^{54} -3.93768 q^{55} +4.26027 q^{56} -4.50257 q^{57} -7.70479 q^{58} +13.9271 q^{59} +2.36221 q^{60} -4.27701 q^{61} +7.02642 q^{62} -4.26027 q^{63} +1.00000 q^{64} -2.36221 q^{65} +1.66695 q^{66} -9.62761 q^{67} +1.26660 q^{68} +8.71852 q^{69} +10.0637 q^{70} -7.37836 q^{71} -1.00000 q^{72} -5.92878 q^{73} -1.42734 q^{74} +0.580037 q^{75} -4.50257 q^{76} +7.10165 q^{77} +1.00000 q^{78} +5.42560 q^{79} +2.36221 q^{80} +1.00000 q^{81} +2.88901 q^{82} -13.7743 q^{83} -4.26027 q^{84} +2.99198 q^{85} +2.87221 q^{86} +7.70479 q^{87} +1.66695 q^{88} -11.4042 q^{89} -2.36221 q^{90} +4.26027 q^{91} +8.71852 q^{92} -7.02642 q^{93} -2.32318 q^{94} -10.6360 q^{95} -1.00000 q^{96} +8.92335 q^{97} -11.1499 q^{98} -1.66695 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.36221 1.05641 0.528206 0.849116i \(-0.322865\pi\)
0.528206 + 0.849116i \(0.322865\pi\)
\(6\) −1.00000 −0.408248
\(7\) −4.26027 −1.61023 −0.805115 0.593118i \(-0.797897\pi\)
−0.805115 + 0.593118i \(0.797897\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.36221 −0.746996
\(11\) −1.66695 −0.502604 −0.251302 0.967909i \(-0.580859\pi\)
−0.251302 + 0.967909i \(0.580859\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 4.26027 1.13860
\(15\) 2.36221 0.609920
\(16\) 1.00000 0.250000
\(17\) 1.26660 0.307197 0.153598 0.988133i \(-0.450914\pi\)
0.153598 + 0.988133i \(0.450914\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.50257 −1.03296 −0.516480 0.856299i \(-0.672758\pi\)
−0.516480 + 0.856299i \(0.672758\pi\)
\(20\) 2.36221 0.528206
\(21\) −4.26027 −0.929667
\(22\) 1.66695 0.355395
\(23\) 8.71852 1.81794 0.908968 0.416865i \(-0.136871\pi\)
0.908968 + 0.416865i \(0.136871\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0.580037 0.116007
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −4.26027 −0.805115
\(29\) 7.70479 1.43074 0.715372 0.698744i \(-0.246257\pi\)
0.715372 + 0.698744i \(0.246257\pi\)
\(30\) −2.36221 −0.431279
\(31\) −7.02642 −1.26198 −0.630991 0.775790i \(-0.717351\pi\)
−0.630991 + 0.775790i \(0.717351\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.66695 −0.290179
\(34\) −1.26660 −0.217221
\(35\) −10.0637 −1.70107
\(36\) 1.00000 0.166667
\(37\) 1.42734 0.234653 0.117327 0.993093i \(-0.462568\pi\)
0.117327 + 0.993093i \(0.462568\pi\)
\(38\) 4.50257 0.730414
\(39\) −1.00000 −0.160128
\(40\) −2.36221 −0.373498
\(41\) −2.88901 −0.451187 −0.225594 0.974222i \(-0.572432\pi\)
−0.225594 + 0.974222i \(0.572432\pi\)
\(42\) 4.26027 0.657374
\(43\) −2.87221 −0.438007 −0.219004 0.975724i \(-0.570281\pi\)
−0.219004 + 0.975724i \(0.570281\pi\)
\(44\) −1.66695 −0.251302
\(45\) 2.36221 0.352137
\(46\) −8.71852 −1.28548
\(47\) 2.32318 0.338871 0.169435 0.985541i \(-0.445806\pi\)
0.169435 + 0.985541i \(0.445806\pi\)
\(48\) 1.00000 0.144338
\(49\) 11.1499 1.59284
\(50\) −0.580037 −0.0820296
\(51\) 1.26660 0.177360
\(52\) −1.00000 −0.138675
\(53\) −3.14222 −0.431617 −0.215809 0.976436i \(-0.569239\pi\)
−0.215809 + 0.976436i \(0.569239\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.93768 −0.530957
\(56\) 4.26027 0.569302
\(57\) −4.50257 −0.596380
\(58\) −7.70479 −1.01169
\(59\) 13.9271 1.81315 0.906575 0.422046i \(-0.138688\pi\)
0.906575 + 0.422046i \(0.138688\pi\)
\(60\) 2.36221 0.304960
\(61\) −4.27701 −0.547615 −0.273807 0.961785i \(-0.588283\pi\)
−0.273807 + 0.961785i \(0.588283\pi\)
\(62\) 7.02642 0.892356
\(63\) −4.26027 −0.536743
\(64\) 1.00000 0.125000
\(65\) −2.36221 −0.292996
\(66\) 1.66695 0.205187
\(67\) −9.62761 −1.17620 −0.588100 0.808788i \(-0.700124\pi\)
−0.588100 + 0.808788i \(0.700124\pi\)
\(68\) 1.26660 0.153598
\(69\) 8.71852 1.04959
\(70\) 10.0637 1.20284
\(71\) −7.37836 −0.875650 −0.437825 0.899060i \(-0.644251\pi\)
−0.437825 + 0.899060i \(0.644251\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.92878 −0.693912 −0.346956 0.937881i \(-0.612785\pi\)
−0.346956 + 0.937881i \(0.612785\pi\)
\(74\) −1.42734 −0.165925
\(75\) 0.580037 0.0669769
\(76\) −4.50257 −0.516480
\(77\) 7.10165 0.809309
\(78\) 1.00000 0.113228
\(79\) 5.42560 0.610427 0.305214 0.952284i \(-0.401272\pi\)
0.305214 + 0.952284i \(0.401272\pi\)
\(80\) 2.36221 0.264103
\(81\) 1.00000 0.111111
\(82\) 2.88901 0.319037
\(83\) −13.7743 −1.51192 −0.755962 0.654616i \(-0.772830\pi\)
−0.755962 + 0.654616i \(0.772830\pi\)
\(84\) −4.26027 −0.464833
\(85\) 2.99198 0.324526
\(86\) 2.87221 0.309718
\(87\) 7.70479 0.826040
\(88\) 1.66695 0.177697
\(89\) −11.4042 −1.20884 −0.604420 0.796666i \(-0.706595\pi\)
−0.604420 + 0.796666i \(0.706595\pi\)
\(90\) −2.36221 −0.248999
\(91\) 4.26027 0.446598
\(92\) 8.71852 0.908968
\(93\) −7.02642 −0.728605
\(94\) −2.32318 −0.239618
\(95\) −10.6360 −1.09123
\(96\) −1.00000 −0.102062
\(97\) 8.92335 0.906029 0.453014 0.891503i \(-0.350349\pi\)
0.453014 + 0.891503i \(0.350349\pi\)
\(98\) −11.1499 −1.12631
\(99\) −1.66695 −0.167535
\(100\) 0.580037 0.0580037
\(101\) 8.26469 0.822367 0.411184 0.911553i \(-0.365116\pi\)
0.411184 + 0.911553i \(0.365116\pi\)
\(102\) −1.26660 −0.125412
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −10.0637 −0.982112
\(106\) 3.14222 0.305199
\(107\) 1.47218 0.142321 0.0711606 0.997465i \(-0.477330\pi\)
0.0711606 + 0.997465i \(0.477330\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.0292 −1.15218 −0.576092 0.817385i \(-0.695423\pi\)
−0.576092 + 0.817385i \(0.695423\pi\)
\(110\) 3.93768 0.375444
\(111\) 1.42734 0.135477
\(112\) −4.26027 −0.402558
\(113\) 15.0560 1.41635 0.708175 0.706037i \(-0.249519\pi\)
0.708175 + 0.706037i \(0.249519\pi\)
\(114\) 4.50257 0.421704
\(115\) 20.5950 1.92049
\(116\) 7.70479 0.715372
\(117\) −1.00000 −0.0924500
\(118\) −13.9271 −1.28209
\(119\) −5.39607 −0.494657
\(120\) −2.36221 −0.215639
\(121\) −8.22128 −0.747389
\(122\) 4.27701 0.387222
\(123\) −2.88901 −0.260493
\(124\) −7.02642 −0.630991
\(125\) −10.4409 −0.933861
\(126\) 4.26027 0.379535
\(127\) −5.21000 −0.462313 −0.231157 0.972917i \(-0.574251\pi\)
−0.231157 + 0.972917i \(0.574251\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.87221 −0.252884
\(130\) 2.36221 0.207180
\(131\) 1.34822 0.117795 0.0588974 0.998264i \(-0.481242\pi\)
0.0588974 + 0.998264i \(0.481242\pi\)
\(132\) −1.66695 −0.145089
\(133\) 19.1822 1.66330
\(134\) 9.62761 0.831699
\(135\) 2.36221 0.203307
\(136\) −1.26660 −0.108610
\(137\) 9.34712 0.798579 0.399289 0.916825i \(-0.369257\pi\)
0.399289 + 0.916825i \(0.369257\pi\)
\(138\) −8.71852 −0.742170
\(139\) −19.8168 −1.68084 −0.840420 0.541935i \(-0.817692\pi\)
−0.840420 + 0.541935i \(0.817692\pi\)
\(140\) −10.0637 −0.850534
\(141\) 2.32318 0.195647
\(142\) 7.37836 0.619178
\(143\) 1.66695 0.139397
\(144\) 1.00000 0.0833333
\(145\) 18.2003 1.51146
\(146\) 5.92878 0.490670
\(147\) 11.1499 0.919628
\(148\) 1.42734 0.117327
\(149\) 22.6402 1.85476 0.927379 0.374123i \(-0.122056\pi\)
0.927379 + 0.374123i \(0.122056\pi\)
\(150\) −0.580037 −0.0473598
\(151\) 5.85954 0.476843 0.238421 0.971162i \(-0.423370\pi\)
0.238421 + 0.971162i \(0.423370\pi\)
\(152\) 4.50257 0.365207
\(153\) 1.26660 0.102399
\(154\) −7.10165 −0.572268
\(155\) −16.5979 −1.33317
\(156\) −1.00000 −0.0800641
\(157\) −17.7129 −1.41365 −0.706823 0.707390i \(-0.749872\pi\)
−0.706823 + 0.707390i \(0.749872\pi\)
\(158\) −5.42560 −0.431637
\(159\) −3.14222 −0.249194
\(160\) −2.36221 −0.186749
\(161\) −37.1432 −2.92730
\(162\) −1.00000 −0.0785674
\(163\) −18.1900 −1.42475 −0.712375 0.701799i \(-0.752380\pi\)
−0.712375 + 0.701799i \(0.752380\pi\)
\(164\) −2.88901 −0.225594
\(165\) −3.93768 −0.306548
\(166\) 13.7743 1.06909
\(167\) −13.9975 −1.08316 −0.541580 0.840649i \(-0.682174\pi\)
−0.541580 + 0.840649i \(0.682174\pi\)
\(168\) 4.26027 0.328687
\(169\) 1.00000 0.0769231
\(170\) −2.99198 −0.229475
\(171\) −4.50257 −0.344320
\(172\) −2.87221 −0.219004
\(173\) −12.2515 −0.931465 −0.465733 0.884925i \(-0.654209\pi\)
−0.465733 + 0.884925i \(0.654209\pi\)
\(174\) −7.70479 −0.584099
\(175\) −2.47111 −0.186799
\(176\) −1.66695 −0.125651
\(177\) 13.9271 1.04682
\(178\) 11.4042 0.854778
\(179\) −13.6911 −1.02332 −0.511661 0.859188i \(-0.670970\pi\)
−0.511661 + 0.859188i \(0.670970\pi\)
\(180\) 2.36221 0.176069
\(181\) 0.681068 0.0506234 0.0253117 0.999680i \(-0.491942\pi\)
0.0253117 + 0.999680i \(0.491942\pi\)
\(182\) −4.26027 −0.315792
\(183\) −4.27701 −0.316165
\(184\) −8.71852 −0.642738
\(185\) 3.37168 0.247891
\(186\) 7.02642 0.515202
\(187\) −2.11136 −0.154398
\(188\) 2.32318 0.169435
\(189\) −4.26027 −0.309889
\(190\) 10.6360 0.771618
\(191\) −24.7389 −1.79004 −0.895022 0.446021i \(-0.852840\pi\)
−0.895022 + 0.446021i \(0.852840\pi\)
\(192\) 1.00000 0.0721688
\(193\) −16.3916 −1.17989 −0.589947 0.807442i \(-0.700851\pi\)
−0.589947 + 0.807442i \(0.700851\pi\)
\(194\) −8.92335 −0.640659
\(195\) −2.36221 −0.169161
\(196\) 11.1499 0.796421
\(197\) 3.50420 0.249664 0.124832 0.992178i \(-0.460161\pi\)
0.124832 + 0.992178i \(0.460161\pi\)
\(198\) 1.66695 0.118465
\(199\) 0.914258 0.0648100 0.0324050 0.999475i \(-0.489683\pi\)
0.0324050 + 0.999475i \(0.489683\pi\)
\(200\) −0.580037 −0.0410148
\(201\) −9.62761 −0.679079
\(202\) −8.26469 −0.581501
\(203\) −32.8245 −2.30383
\(204\) 1.26660 0.0886800
\(205\) −6.82444 −0.476640
\(206\) −1.00000 −0.0696733
\(207\) 8.71852 0.605979
\(208\) −1.00000 −0.0693375
\(209\) 7.50556 0.519170
\(210\) 10.0637 0.694458
\(211\) 8.56299 0.589501 0.294750 0.955574i \(-0.404763\pi\)
0.294750 + 0.955574i \(0.404763\pi\)
\(212\) −3.14222 −0.215809
\(213\) −7.37836 −0.505557
\(214\) −1.47218 −0.100636
\(215\) −6.78475 −0.462716
\(216\) −1.00000 −0.0680414
\(217\) 29.9344 2.03208
\(218\) 12.0292 0.814717
\(219\) −5.92878 −0.400630
\(220\) −3.93768 −0.265479
\(221\) −1.26660 −0.0852010
\(222\) −1.42734 −0.0957968
\(223\) 2.60556 0.174481 0.0872407 0.996187i \(-0.472195\pi\)
0.0872407 + 0.996187i \(0.472195\pi\)
\(224\) 4.26027 0.284651
\(225\) 0.580037 0.0386691
\(226\) −15.0560 −1.00151
\(227\) −3.21031 −0.213076 −0.106538 0.994309i \(-0.533977\pi\)
−0.106538 + 0.994309i \(0.533977\pi\)
\(228\) −4.50257 −0.298190
\(229\) −14.5877 −0.963985 −0.481992 0.876175i \(-0.660087\pi\)
−0.481992 + 0.876175i \(0.660087\pi\)
\(230\) −20.5950 −1.35799
\(231\) 7.10165 0.467254
\(232\) −7.70479 −0.505844
\(233\) −17.0098 −1.11435 −0.557175 0.830395i \(-0.688115\pi\)
−0.557175 + 0.830395i \(0.688115\pi\)
\(234\) 1.00000 0.0653720
\(235\) 5.48784 0.357987
\(236\) 13.9271 0.906575
\(237\) 5.42560 0.352430
\(238\) 5.39607 0.349775
\(239\) −8.89537 −0.575393 −0.287697 0.957722i \(-0.592890\pi\)
−0.287697 + 0.957722i \(0.592890\pi\)
\(240\) 2.36221 0.152480
\(241\) −15.3927 −0.991529 −0.495765 0.868457i \(-0.665112\pi\)
−0.495765 + 0.868457i \(0.665112\pi\)
\(242\) 8.22128 0.528484
\(243\) 1.00000 0.0641500
\(244\) −4.27701 −0.273807
\(245\) 26.3384 1.68270
\(246\) 2.88901 0.184196
\(247\) 4.50257 0.286492
\(248\) 7.02642 0.446178
\(249\) −13.7743 −0.872909
\(250\) 10.4409 0.660339
\(251\) −6.09295 −0.384584 −0.192292 0.981338i \(-0.561592\pi\)
−0.192292 + 0.981338i \(0.561592\pi\)
\(252\) −4.26027 −0.268372
\(253\) −14.5333 −0.913703
\(254\) 5.21000 0.326905
\(255\) 2.99198 0.187365
\(256\) 1.00000 0.0625000
\(257\) 1.88357 0.117494 0.0587469 0.998273i \(-0.481290\pi\)
0.0587469 + 0.998273i \(0.481290\pi\)
\(258\) 2.87221 0.178816
\(259\) −6.08085 −0.377846
\(260\) −2.36221 −0.146498
\(261\) 7.70479 0.476915
\(262\) −1.34822 −0.0832935
\(263\) −8.86701 −0.546763 −0.273382 0.961906i \(-0.588142\pi\)
−0.273382 + 0.961906i \(0.588142\pi\)
\(264\) 1.66695 0.102594
\(265\) −7.42259 −0.455966
\(266\) −19.1822 −1.17613
\(267\) −11.4042 −0.697924
\(268\) −9.62761 −0.588100
\(269\) 4.60253 0.280621 0.140311 0.990108i \(-0.455190\pi\)
0.140311 + 0.990108i \(0.455190\pi\)
\(270\) −2.36221 −0.143760
\(271\) −6.60696 −0.401344 −0.200672 0.979658i \(-0.564313\pi\)
−0.200672 + 0.979658i \(0.564313\pi\)
\(272\) 1.26660 0.0767991
\(273\) 4.26027 0.257843
\(274\) −9.34712 −0.564680
\(275\) −0.966892 −0.0583058
\(276\) 8.71852 0.524793
\(277\) −8.44048 −0.507140 −0.253570 0.967317i \(-0.581605\pi\)
−0.253570 + 0.967317i \(0.581605\pi\)
\(278\) 19.8168 1.18853
\(279\) −7.02642 −0.420660
\(280\) 10.0637 0.601418
\(281\) −1.72109 −0.102672 −0.0513359 0.998681i \(-0.516348\pi\)
−0.0513359 + 0.998681i \(0.516348\pi\)
\(282\) −2.32318 −0.138343
\(283\) 5.31556 0.315977 0.157988 0.987441i \(-0.449499\pi\)
0.157988 + 0.987441i \(0.449499\pi\)
\(284\) −7.37836 −0.437825
\(285\) −10.6360 −0.630023
\(286\) −1.66695 −0.0985688
\(287\) 12.3079 0.726515
\(288\) −1.00000 −0.0589256
\(289\) −15.3957 −0.905630
\(290\) −18.2003 −1.06876
\(291\) 8.92335 0.523096
\(292\) −5.92878 −0.346956
\(293\) −0.477341 −0.0278866 −0.0139433 0.999903i \(-0.504438\pi\)
−0.0139433 + 0.999903i \(0.504438\pi\)
\(294\) −11.1499 −0.650275
\(295\) 32.8987 1.91543
\(296\) −1.42734 −0.0829624
\(297\) −1.66695 −0.0967262
\(298\) −22.6402 −1.31151
\(299\) −8.71852 −0.504205
\(300\) 0.580037 0.0334884
\(301\) 12.2364 0.705293
\(302\) −5.85954 −0.337179
\(303\) 8.26469 0.474794
\(304\) −4.50257 −0.258240
\(305\) −10.1032 −0.578507
\(306\) −1.26660 −0.0724069
\(307\) 10.0577 0.574020 0.287010 0.957928i \(-0.407339\pi\)
0.287010 + 0.957928i \(0.407339\pi\)
\(308\) 7.10165 0.404654
\(309\) 1.00000 0.0568880
\(310\) 16.5979 0.942696
\(311\) −14.0208 −0.795044 −0.397522 0.917593i \(-0.630130\pi\)
−0.397522 + 0.917593i \(0.630130\pi\)
\(312\) 1.00000 0.0566139
\(313\) −1.47426 −0.0833301 −0.0416651 0.999132i \(-0.513266\pi\)
−0.0416651 + 0.999132i \(0.513266\pi\)
\(314\) 17.7129 0.999599
\(315\) −10.0637 −0.567022
\(316\) 5.42560 0.305214
\(317\) 13.3249 0.748401 0.374201 0.927348i \(-0.377917\pi\)
0.374201 + 0.927348i \(0.377917\pi\)
\(318\) 3.14222 0.176207
\(319\) −12.8435 −0.719098
\(320\) 2.36221 0.132052
\(321\) 1.47218 0.0821692
\(322\) 37.1432 2.06991
\(323\) −5.70297 −0.317322
\(324\) 1.00000 0.0555556
\(325\) −0.580037 −0.0321746
\(326\) 18.1900 1.00745
\(327\) −12.0292 −0.665214
\(328\) 2.88901 0.159519
\(329\) −9.89738 −0.545660
\(330\) 3.93768 0.216762
\(331\) 11.1195 0.611182 0.305591 0.952163i \(-0.401146\pi\)
0.305591 + 0.952163i \(0.401146\pi\)
\(332\) −13.7743 −0.755962
\(333\) 1.42734 0.0782177
\(334\) 13.9975 0.765910
\(335\) −22.7424 −1.24255
\(336\) −4.26027 −0.232417
\(337\) 13.0367 0.710152 0.355076 0.934837i \(-0.384455\pi\)
0.355076 + 0.934837i \(0.384455\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 15.0560 0.817730
\(340\) 2.99198 0.162263
\(341\) 11.7127 0.634277
\(342\) 4.50257 0.243471
\(343\) −17.6797 −0.954612
\(344\) 2.87221 0.154859
\(345\) 20.5950 1.10880
\(346\) 12.2515 0.658645
\(347\) 0.928402 0.0498392 0.0249196 0.999689i \(-0.492067\pi\)
0.0249196 + 0.999689i \(0.492067\pi\)
\(348\) 7.70479 0.413020
\(349\) −9.85850 −0.527713 −0.263857 0.964562i \(-0.584995\pi\)
−0.263857 + 0.964562i \(0.584995\pi\)
\(350\) 2.47111 0.132087
\(351\) −1.00000 −0.0533761
\(352\) 1.66695 0.0888487
\(353\) −0.710646 −0.0378239 −0.0189119 0.999821i \(-0.506020\pi\)
−0.0189119 + 0.999821i \(0.506020\pi\)
\(354\) −13.9271 −0.740215
\(355\) −17.4292 −0.925048
\(356\) −11.4042 −0.604420
\(357\) −5.39607 −0.285590
\(358\) 13.6911 0.723598
\(359\) −7.75035 −0.409048 −0.204524 0.978862i \(-0.565565\pi\)
−0.204524 + 0.978862i \(0.565565\pi\)
\(360\) −2.36221 −0.124499
\(361\) 1.27315 0.0670080
\(362\) −0.681068 −0.0357961
\(363\) −8.22128 −0.431505
\(364\) 4.26027 0.223299
\(365\) −14.0050 −0.733057
\(366\) 4.27701 0.223563
\(367\) 4.64274 0.242349 0.121175 0.992631i \(-0.461334\pi\)
0.121175 + 0.992631i \(0.461334\pi\)
\(368\) 8.71852 0.454484
\(369\) −2.88901 −0.150396
\(370\) −3.37168 −0.175285
\(371\) 13.3867 0.695003
\(372\) −7.02642 −0.364303
\(373\) 10.6033 0.549020 0.274510 0.961584i \(-0.411484\pi\)
0.274510 + 0.961584i \(0.411484\pi\)
\(374\) 2.11136 0.109176
\(375\) −10.4409 −0.539165
\(376\) −2.32318 −0.119809
\(377\) −7.70479 −0.396817
\(378\) 4.26027 0.219125
\(379\) 14.6685 0.753468 0.376734 0.926322i \(-0.377047\pi\)
0.376734 + 0.926322i \(0.377047\pi\)
\(380\) −10.6360 −0.545616
\(381\) −5.21000 −0.266917
\(382\) 24.7389 1.26575
\(383\) −12.9463 −0.661525 −0.330762 0.943714i \(-0.607306\pi\)
−0.330762 + 0.943714i \(0.607306\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 16.7756 0.854964
\(386\) 16.3916 0.834311
\(387\) −2.87221 −0.146002
\(388\) 8.92335 0.453014
\(389\) 6.44531 0.326790 0.163395 0.986561i \(-0.447755\pi\)
0.163395 + 0.986561i \(0.447755\pi\)
\(390\) 2.36221 0.119615
\(391\) 11.0429 0.558464
\(392\) −11.1499 −0.563155
\(393\) 1.34822 0.0680089
\(394\) −3.50420 −0.176539
\(395\) 12.8164 0.644863
\(396\) −1.66695 −0.0837674
\(397\) 15.9602 0.801019 0.400510 0.916293i \(-0.368833\pi\)
0.400510 + 0.916293i \(0.368833\pi\)
\(398\) −0.914258 −0.0458276
\(399\) 19.1822 0.960309
\(400\) 0.580037 0.0290018
\(401\) −15.7998 −0.789003 −0.394502 0.918895i \(-0.629083\pi\)
−0.394502 + 0.918895i \(0.629083\pi\)
\(402\) 9.62761 0.480182
\(403\) 7.02642 0.350011
\(404\) 8.26469 0.411184
\(405\) 2.36221 0.117379
\(406\) 32.8245 1.62905
\(407\) −2.37930 −0.117938
\(408\) −1.26660 −0.0627062
\(409\) 33.5454 1.65871 0.829357 0.558718i \(-0.188707\pi\)
0.829357 + 0.558718i \(0.188707\pi\)
\(410\) 6.82444 0.337035
\(411\) 9.34712 0.461060
\(412\) 1.00000 0.0492665
\(413\) −59.3330 −2.91959
\(414\) −8.71852 −0.428492
\(415\) −32.5377 −1.59721
\(416\) 1.00000 0.0490290
\(417\) −19.8168 −0.970434
\(418\) −7.50556 −0.367109
\(419\) −2.61430 −0.127717 −0.0638585 0.997959i \(-0.520341\pi\)
−0.0638585 + 0.997959i \(0.520341\pi\)
\(420\) −10.0637 −0.491056
\(421\) −2.75049 −0.134050 −0.0670252 0.997751i \(-0.521351\pi\)
−0.0670252 + 0.997751i \(0.521351\pi\)
\(422\) −8.56299 −0.416840
\(423\) 2.32318 0.112957
\(424\) 3.14222 0.152600
\(425\) 0.734677 0.0356371
\(426\) 7.37836 0.357483
\(427\) 18.2212 0.881786
\(428\) 1.47218 0.0711606
\(429\) 1.66695 0.0804811
\(430\) 6.78475 0.327190
\(431\) 29.1265 1.40297 0.701486 0.712683i \(-0.252520\pi\)
0.701486 + 0.712683i \(0.252520\pi\)
\(432\) 1.00000 0.0481125
\(433\) 0.469219 0.0225492 0.0112746 0.999936i \(-0.496411\pi\)
0.0112746 + 0.999936i \(0.496411\pi\)
\(434\) −29.9344 −1.43690
\(435\) 18.2003 0.872639
\(436\) −12.0292 −0.576092
\(437\) −39.2558 −1.87786
\(438\) 5.92878 0.283288
\(439\) −36.5321 −1.74358 −0.871792 0.489876i \(-0.837042\pi\)
−0.871792 + 0.489876i \(0.837042\pi\)
\(440\) 3.93768 0.187722
\(441\) 11.1499 0.530947
\(442\) 1.26660 0.0602462
\(443\) −35.0074 −1.66325 −0.831626 0.555336i \(-0.812590\pi\)
−0.831626 + 0.555336i \(0.812590\pi\)
\(444\) 1.42734 0.0677385
\(445\) −26.9390 −1.27703
\(446\) −2.60556 −0.123377
\(447\) 22.6402 1.07085
\(448\) −4.26027 −0.201279
\(449\) −13.6421 −0.643812 −0.321906 0.946772i \(-0.604323\pi\)
−0.321906 + 0.946772i \(0.604323\pi\)
\(450\) −0.580037 −0.0273432
\(451\) 4.81583 0.226769
\(452\) 15.0560 0.708175
\(453\) 5.85954 0.275305
\(454\) 3.21031 0.150667
\(455\) 10.0637 0.471791
\(456\) 4.50257 0.210852
\(457\) −2.07818 −0.0972134 −0.0486067 0.998818i \(-0.515478\pi\)
−0.0486067 + 0.998818i \(0.515478\pi\)
\(458\) 14.5877 0.681640
\(459\) 1.26660 0.0591200
\(460\) 20.5950 0.960246
\(461\) 35.1380 1.63654 0.818270 0.574834i \(-0.194933\pi\)
0.818270 + 0.574834i \(0.194933\pi\)
\(462\) −7.10165 −0.330399
\(463\) 38.1923 1.77495 0.887473 0.460861i \(-0.152459\pi\)
0.887473 + 0.460861i \(0.152459\pi\)
\(464\) 7.70479 0.357686
\(465\) −16.5979 −0.769708
\(466\) 17.0098 0.787965
\(467\) 4.09872 0.189666 0.0948331 0.995493i \(-0.469768\pi\)
0.0948331 + 0.995493i \(0.469768\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 41.0162 1.89395
\(470\) −5.48784 −0.253135
\(471\) −17.7129 −0.816169
\(472\) −13.9271 −0.641045
\(473\) 4.78782 0.220144
\(474\) −5.42560 −0.249206
\(475\) −2.61166 −0.119831
\(476\) −5.39607 −0.247329
\(477\) −3.14222 −0.143872
\(478\) 8.89537 0.406865
\(479\) −25.1514 −1.14920 −0.574598 0.818436i \(-0.694842\pi\)
−0.574598 + 0.818436i \(0.694842\pi\)
\(480\) −2.36221 −0.107820
\(481\) −1.42734 −0.0650811
\(482\) 15.3927 0.701117
\(483\) −37.1432 −1.69008
\(484\) −8.22128 −0.373695
\(485\) 21.0788 0.957140
\(486\) −1.00000 −0.0453609
\(487\) −36.4381 −1.65117 −0.825584 0.564280i \(-0.809154\pi\)
−0.825584 + 0.564280i \(0.809154\pi\)
\(488\) 4.27701 0.193611
\(489\) −18.1900 −0.822579
\(490\) −26.3384 −1.18985
\(491\) −1.60528 −0.0724451 −0.0362226 0.999344i \(-0.511533\pi\)
−0.0362226 + 0.999344i \(0.511533\pi\)
\(492\) −2.88901 −0.130247
\(493\) 9.75892 0.439520
\(494\) −4.50257 −0.202580
\(495\) −3.93768 −0.176986
\(496\) −7.02642 −0.315495
\(497\) 31.4338 1.41000
\(498\) 13.7743 0.617240
\(499\) 1.43114 0.0640667 0.0320333 0.999487i \(-0.489802\pi\)
0.0320333 + 0.999487i \(0.489802\pi\)
\(500\) −10.4409 −0.466930
\(501\) −13.9975 −0.625363
\(502\) 6.09295 0.271942
\(503\) 39.5014 1.76128 0.880641 0.473785i \(-0.157113\pi\)
0.880641 + 0.473785i \(0.157113\pi\)
\(504\) 4.26027 0.189767
\(505\) 19.5229 0.868759
\(506\) 14.5333 0.646085
\(507\) 1.00000 0.0444116
\(508\) −5.21000 −0.231157
\(509\) 31.2303 1.38426 0.692130 0.721773i \(-0.256672\pi\)
0.692130 + 0.721773i \(0.256672\pi\)
\(510\) −2.99198 −0.132487
\(511\) 25.2582 1.11736
\(512\) −1.00000 −0.0441942
\(513\) −4.50257 −0.198793
\(514\) −1.88357 −0.0830806
\(515\) 2.36221 0.104091
\(516\) −2.87221 −0.126442
\(517\) −3.87263 −0.170318
\(518\) 6.08085 0.267177
\(519\) −12.2515 −0.537782
\(520\) 2.36221 0.103590
\(521\) 42.5234 1.86298 0.931492 0.363761i \(-0.118508\pi\)
0.931492 + 0.363761i \(0.118508\pi\)
\(522\) −7.70479 −0.337230
\(523\) −27.2989 −1.19370 −0.596850 0.802353i \(-0.703581\pi\)
−0.596850 + 0.802353i \(0.703581\pi\)
\(524\) 1.34822 0.0588974
\(525\) −2.47111 −0.107848
\(526\) 8.86701 0.386620
\(527\) −8.89969 −0.387676
\(528\) −1.66695 −0.0725447
\(529\) 53.0126 2.30489
\(530\) 7.42259 0.322417
\(531\) 13.9271 0.604383
\(532\) 19.1822 0.831652
\(533\) 2.88901 0.125137
\(534\) 11.4042 0.493507
\(535\) 3.47761 0.150350
\(536\) 9.62761 0.415849
\(537\) −13.6911 −0.590815
\(538\) −4.60253 −0.198429
\(539\) −18.5863 −0.800569
\(540\) 2.36221 0.101653
\(541\) −40.3793 −1.73604 −0.868021 0.496527i \(-0.834608\pi\)
−0.868021 + 0.496527i \(0.834608\pi\)
\(542\) 6.60696 0.283793
\(543\) 0.681068 0.0292274
\(544\) −1.26660 −0.0543052
\(545\) −28.4154 −1.21718
\(546\) −4.26027 −0.182323
\(547\) −23.7025 −1.01344 −0.506722 0.862109i \(-0.669143\pi\)
−0.506722 + 0.862109i \(0.669143\pi\)
\(548\) 9.34712 0.399289
\(549\) −4.27701 −0.182538
\(550\) 0.966892 0.0412284
\(551\) −34.6914 −1.47790
\(552\) −8.71852 −0.371085
\(553\) −23.1145 −0.982929
\(554\) 8.44048 0.358602
\(555\) 3.37168 0.143120
\(556\) −19.8168 −0.840420
\(557\) 1.20823 0.0511942 0.0255971 0.999672i \(-0.491851\pi\)
0.0255971 + 0.999672i \(0.491851\pi\)
\(558\) 7.02642 0.297452
\(559\) 2.87221 0.121481
\(560\) −10.0637 −0.425267
\(561\) −2.11136 −0.0891419
\(562\) 1.72109 0.0726000
\(563\) −44.5681 −1.87832 −0.939160 0.343480i \(-0.888394\pi\)
−0.939160 + 0.343480i \(0.888394\pi\)
\(564\) 2.32318 0.0978236
\(565\) 35.5654 1.49625
\(566\) −5.31556 −0.223429
\(567\) −4.26027 −0.178914
\(568\) 7.37836 0.309589
\(569\) −8.12797 −0.340742 −0.170371 0.985380i \(-0.554497\pi\)
−0.170371 + 0.985380i \(0.554497\pi\)
\(570\) 10.6360 0.445494
\(571\) −41.1313 −1.72129 −0.860645 0.509205i \(-0.829939\pi\)
−0.860645 + 0.509205i \(0.829939\pi\)
\(572\) 1.66695 0.0696987
\(573\) −24.7389 −1.03348
\(574\) −12.3079 −0.513724
\(575\) 5.05706 0.210894
\(576\) 1.00000 0.0416667
\(577\) 0.0968216 0.00403074 0.00201537 0.999998i \(-0.499358\pi\)
0.00201537 + 0.999998i \(0.499358\pi\)
\(578\) 15.3957 0.640377
\(579\) −16.3916 −0.681212
\(580\) 18.2003 0.755728
\(581\) 58.6821 2.43454
\(582\) −8.92335 −0.369885
\(583\) 5.23792 0.216933
\(584\) 5.92878 0.245335
\(585\) −2.36221 −0.0976654
\(586\) 0.477341 0.0197188
\(587\) 16.4362 0.678394 0.339197 0.940715i \(-0.389845\pi\)
0.339197 + 0.940715i \(0.389845\pi\)
\(588\) 11.1499 0.459814
\(589\) 31.6369 1.30358
\(590\) −32.8987 −1.35442
\(591\) 3.50420 0.144143
\(592\) 1.42734 0.0586633
\(593\) −15.3817 −0.631652 −0.315826 0.948817i \(-0.602282\pi\)
−0.315826 + 0.948817i \(0.602282\pi\)
\(594\) 1.66695 0.0683958
\(595\) −12.7467 −0.522562
\(596\) 22.6402 0.927379
\(597\) 0.914258 0.0374181
\(598\) 8.71852 0.356527
\(599\) 0.343825 0.0140483 0.00702416 0.999975i \(-0.497764\pi\)
0.00702416 + 0.999975i \(0.497764\pi\)
\(600\) −0.580037 −0.0236799
\(601\) 5.46049 0.222738 0.111369 0.993779i \(-0.464476\pi\)
0.111369 + 0.993779i \(0.464476\pi\)
\(602\) −12.2364 −0.498717
\(603\) −9.62761 −0.392067
\(604\) 5.85954 0.238421
\(605\) −19.4204 −0.789551
\(606\) −8.26469 −0.335730
\(607\) 23.3753 0.948772 0.474386 0.880317i \(-0.342670\pi\)
0.474386 + 0.880317i \(0.342670\pi\)
\(608\) 4.50257 0.182603
\(609\) −32.8245 −1.33012
\(610\) 10.1032 0.409066
\(611\) −2.32318 −0.0939859
\(612\) 1.26660 0.0511994
\(613\) 27.5513 1.11278 0.556392 0.830920i \(-0.312185\pi\)
0.556392 + 0.830920i \(0.312185\pi\)
\(614\) −10.0577 −0.405894
\(615\) −6.82444 −0.275188
\(616\) −7.10165 −0.286134
\(617\) 35.9168 1.44596 0.722978 0.690871i \(-0.242773\pi\)
0.722978 + 0.690871i \(0.242773\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 10.0906 0.405577 0.202788 0.979223i \(-0.435000\pi\)
0.202788 + 0.979223i \(0.435000\pi\)
\(620\) −16.5979 −0.666586
\(621\) 8.71852 0.349862
\(622\) 14.0208 0.562181
\(623\) 48.5848 1.94651
\(624\) −1.00000 −0.0400320
\(625\) −27.5637 −1.10255
\(626\) 1.47426 0.0589233
\(627\) 7.50556 0.299743
\(628\) −17.7129 −0.706823
\(629\) 1.80787 0.0720847
\(630\) 10.0637 0.400945
\(631\) −12.1634 −0.484217 −0.242108 0.970249i \(-0.577839\pi\)
−0.242108 + 0.970249i \(0.577839\pi\)
\(632\) −5.42560 −0.215819
\(633\) 8.56299 0.340348
\(634\) −13.3249 −0.529199
\(635\) −12.3071 −0.488393
\(636\) −3.14222 −0.124597
\(637\) −11.1499 −0.441775
\(638\) 12.8435 0.508479
\(639\) −7.37836 −0.291883
\(640\) −2.36221 −0.0933746
\(641\) 29.9449 1.18275 0.591376 0.806396i \(-0.298585\pi\)
0.591376 + 0.806396i \(0.298585\pi\)
\(642\) −1.47218 −0.0581024
\(643\) 43.7484 1.72527 0.862634 0.505829i \(-0.168813\pi\)
0.862634 + 0.505829i \(0.168813\pi\)
\(644\) −37.1432 −1.46365
\(645\) −6.78475 −0.267149
\(646\) 5.70297 0.224381
\(647\) −7.60258 −0.298888 −0.149444 0.988770i \(-0.547748\pi\)
−0.149444 + 0.988770i \(0.547748\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −23.2157 −0.911296
\(650\) 0.580037 0.0227509
\(651\) 29.9344 1.17322
\(652\) −18.1900 −0.712375
\(653\) 44.9286 1.75819 0.879095 0.476646i \(-0.158148\pi\)
0.879095 + 0.476646i \(0.158148\pi\)
\(654\) 12.0292 0.470377
\(655\) 3.18479 0.124440
\(656\) −2.88901 −0.112797
\(657\) −5.92878 −0.231304
\(658\) 9.89738 0.385840
\(659\) −26.8676 −1.04661 −0.523306 0.852145i \(-0.675302\pi\)
−0.523306 + 0.852145i \(0.675302\pi\)
\(660\) −3.93768 −0.153274
\(661\) 29.9859 1.16632 0.583159 0.812358i \(-0.301816\pi\)
0.583159 + 0.812358i \(0.301816\pi\)
\(662\) −11.1195 −0.432171
\(663\) −1.26660 −0.0491908
\(664\) 13.7743 0.534546
\(665\) 45.3123 1.75714
\(666\) −1.42734 −0.0553083
\(667\) 67.1744 2.60100
\(668\) −13.9975 −0.541580
\(669\) 2.60556 0.100737
\(670\) 22.7424 0.878617
\(671\) 7.12956 0.275233
\(672\) 4.26027 0.164343
\(673\) −16.9433 −0.653116 −0.326558 0.945177i \(-0.605889\pi\)
−0.326558 + 0.945177i \(0.605889\pi\)
\(674\) −13.0367 −0.502153
\(675\) 0.580037 0.0223256
\(676\) 1.00000 0.0384615
\(677\) 33.3594 1.28211 0.641053 0.767496i \(-0.278498\pi\)
0.641053 + 0.767496i \(0.278498\pi\)
\(678\) −15.0560 −0.578222
\(679\) −38.0159 −1.45891
\(680\) −2.99198 −0.114737
\(681\) −3.21031 −0.123019
\(682\) −11.7127 −0.448502
\(683\) 27.1261 1.03795 0.518975 0.854789i \(-0.326314\pi\)
0.518975 + 0.854789i \(0.326314\pi\)
\(684\) −4.50257 −0.172160
\(685\) 22.0799 0.843629
\(686\) 17.6797 0.675013
\(687\) −14.5877 −0.556557
\(688\) −2.87221 −0.109502
\(689\) 3.14222 0.119709
\(690\) −20.5950 −0.784037
\(691\) −16.5747 −0.630530 −0.315265 0.949004i \(-0.602093\pi\)
−0.315265 + 0.949004i \(0.602093\pi\)
\(692\) −12.2515 −0.465733
\(693\) 7.10165 0.269770
\(694\) −0.928402 −0.0352417
\(695\) −46.8115 −1.77566
\(696\) −7.70479 −0.292049
\(697\) −3.65923 −0.138603
\(698\) 9.85850 0.373150
\(699\) −17.0098 −0.643370
\(700\) −2.47111 −0.0933993
\(701\) 8.50111 0.321082 0.160541 0.987029i \(-0.448676\pi\)
0.160541 + 0.987029i \(0.448676\pi\)
\(702\) 1.00000 0.0377426
\(703\) −6.42670 −0.242388
\(704\) −1.66695 −0.0628255
\(705\) 5.48784 0.206684
\(706\) 0.710646 0.0267455
\(707\) −35.2098 −1.32420
\(708\) 13.9271 0.523411
\(709\) 30.6892 1.15256 0.576279 0.817253i \(-0.304504\pi\)
0.576279 + 0.817253i \(0.304504\pi\)
\(710\) 17.4292 0.654108
\(711\) 5.42560 0.203476
\(712\) 11.4042 0.427389
\(713\) −61.2599 −2.29420
\(714\) 5.39607 0.201943
\(715\) 3.93768 0.147261
\(716\) −13.6911 −0.511661
\(717\) −8.89537 −0.332204
\(718\) 7.75035 0.289241
\(719\) 1.03188 0.0384825 0.0192412 0.999815i \(-0.493875\pi\)
0.0192412 + 0.999815i \(0.493875\pi\)
\(720\) 2.36221 0.0880344
\(721\) −4.26027 −0.158661
\(722\) −1.27315 −0.0473818
\(723\) −15.3927 −0.572460
\(724\) 0.681068 0.0253117
\(725\) 4.46906 0.165977
\(726\) 8.22128 0.305120
\(727\) 8.03359 0.297949 0.148975 0.988841i \(-0.452403\pi\)
0.148975 + 0.988841i \(0.452403\pi\)
\(728\) −4.26027 −0.157896
\(729\) 1.00000 0.0370370
\(730\) 14.0050 0.518349
\(731\) −3.63795 −0.134554
\(732\) −4.27701 −0.158083
\(733\) 2.09541 0.0773957 0.0386979 0.999251i \(-0.487679\pi\)
0.0386979 + 0.999251i \(0.487679\pi\)
\(734\) −4.64274 −0.171367
\(735\) 26.3384 0.971506
\(736\) −8.71852 −0.321369
\(737\) 16.0487 0.591163
\(738\) 2.88901 0.106346
\(739\) 7.01750 0.258143 0.129071 0.991635i \(-0.458800\pi\)
0.129071 + 0.991635i \(0.458800\pi\)
\(740\) 3.37168 0.123945
\(741\) 4.50257 0.165406
\(742\) −13.3867 −0.491441
\(743\) 15.9286 0.584365 0.292182 0.956363i \(-0.405619\pi\)
0.292182 + 0.956363i \(0.405619\pi\)
\(744\) 7.02642 0.257601
\(745\) 53.4809 1.95939
\(746\) −10.6033 −0.388216
\(747\) −13.7743 −0.503974
\(748\) −2.11136 −0.0771991
\(749\) −6.27190 −0.229170
\(750\) 10.4409 0.381247
\(751\) −35.9437 −1.31161 −0.655803 0.754932i \(-0.727670\pi\)
−0.655803 + 0.754932i \(0.727670\pi\)
\(752\) 2.32318 0.0847177
\(753\) −6.09295 −0.222039
\(754\) 7.70479 0.280592
\(755\) 13.8415 0.503742
\(756\) −4.26027 −0.154944
\(757\) −35.6824 −1.29690 −0.648449 0.761258i \(-0.724582\pi\)
−0.648449 + 0.761258i \(0.724582\pi\)
\(758\) −14.6685 −0.532782
\(759\) −14.5333 −0.527526
\(760\) 10.6360 0.385809
\(761\) −49.2518 −1.78538 −0.892688 0.450675i \(-0.851184\pi\)
−0.892688 + 0.450675i \(0.851184\pi\)
\(762\) 5.21000 0.188739
\(763\) 51.2474 1.85528
\(764\) −24.7389 −0.895022
\(765\) 2.99198 0.108175
\(766\) 12.9463 0.467769
\(767\) −13.9271 −0.502877
\(768\) 1.00000 0.0360844
\(769\) −31.7831 −1.14613 −0.573065 0.819510i \(-0.694246\pi\)
−0.573065 + 0.819510i \(0.694246\pi\)
\(770\) −16.7756 −0.604551
\(771\) 1.88357 0.0678351
\(772\) −16.3916 −0.589947
\(773\) 26.1389 0.940152 0.470076 0.882626i \(-0.344226\pi\)
0.470076 + 0.882626i \(0.344226\pi\)
\(774\) 2.87221 0.103239
\(775\) −4.07558 −0.146399
\(776\) −8.92335 −0.320329
\(777\) −6.08085 −0.218149
\(778\) −6.44531 −0.231076
\(779\) 13.0080 0.466059
\(780\) −2.36221 −0.0845807
\(781\) 12.2994 0.440105
\(782\) −11.0429 −0.394894
\(783\) 7.70479 0.275347
\(784\) 11.1499 0.398210
\(785\) −41.8417 −1.49339
\(786\) −1.34822 −0.0480895
\(787\) 18.5019 0.659520 0.329760 0.944065i \(-0.393032\pi\)
0.329760 + 0.944065i \(0.393032\pi\)
\(788\) 3.50420 0.124832
\(789\) −8.86701 −0.315674
\(790\) −12.8164 −0.455987
\(791\) −64.1426 −2.28065
\(792\) 1.66695 0.0592325
\(793\) 4.27701 0.151881
\(794\) −15.9602 −0.566406
\(795\) −7.42259 −0.263252
\(796\) 0.914258 0.0324050
\(797\) −48.7167 −1.72563 −0.862816 0.505517i \(-0.831302\pi\)
−0.862816 + 0.505517i \(0.831302\pi\)
\(798\) −19.1822 −0.679041
\(799\) 2.94255 0.104100
\(800\) −0.580037 −0.0205074
\(801\) −11.4042 −0.402946
\(802\) 15.7998 0.557910
\(803\) 9.88298 0.348763
\(804\) −9.62761 −0.339540
\(805\) −87.7401 −3.09243
\(806\) −7.02642 −0.247495
\(807\) 4.60253 0.162017
\(808\) −8.26469 −0.290751
\(809\) −42.0086 −1.47694 −0.738472 0.674285i \(-0.764452\pi\)
−0.738472 + 0.674285i \(0.764452\pi\)
\(810\) −2.36221 −0.0829996
\(811\) 34.5480 1.21315 0.606573 0.795028i \(-0.292544\pi\)
0.606573 + 0.795028i \(0.292544\pi\)
\(812\) −32.8245 −1.15191
\(813\) −6.60696 −0.231716
\(814\) 2.37930 0.0833945
\(815\) −42.9686 −1.50512
\(816\) 1.26660 0.0443400
\(817\) 12.9323 0.452444
\(818\) −33.5454 −1.17289
\(819\) 4.26027 0.148866
\(820\) −6.82444 −0.238320
\(821\) −15.3923 −0.537195 −0.268598 0.963252i \(-0.586560\pi\)
−0.268598 + 0.963252i \(0.586560\pi\)
\(822\) −9.34712 −0.326018
\(823\) −29.9132 −1.04271 −0.521355 0.853340i \(-0.674573\pi\)
−0.521355 + 0.853340i \(0.674573\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −0.966892 −0.0336629
\(826\) 59.3330 2.06446
\(827\) −45.1536 −1.57015 −0.785073 0.619404i \(-0.787374\pi\)
−0.785073 + 0.619404i \(0.787374\pi\)
\(828\) 8.71852 0.302989
\(829\) 40.4980 1.40655 0.703276 0.710917i \(-0.251720\pi\)
0.703276 + 0.710917i \(0.251720\pi\)
\(830\) 32.5377 1.12940
\(831\) −8.44048 −0.292797
\(832\) −1.00000 −0.0346688
\(833\) 14.1225 0.489316
\(834\) 19.8168 0.686200
\(835\) −33.0651 −1.14426
\(836\) 7.50556 0.259585
\(837\) −7.02642 −0.242868
\(838\) 2.61430 0.0903096
\(839\) 41.1163 1.41949 0.709746 0.704457i \(-0.248810\pi\)
0.709746 + 0.704457i \(0.248810\pi\)
\(840\) 10.0637 0.347229
\(841\) 30.3638 1.04703
\(842\) 2.75049 0.0947880
\(843\) −1.72109 −0.0592776
\(844\) 8.56299 0.294750
\(845\) 2.36221 0.0812625
\(846\) −2.32318 −0.0798726
\(847\) 35.0249 1.20347
\(848\) −3.14222 −0.107904
\(849\) 5.31556 0.182429
\(850\) −0.734677 −0.0251992
\(851\) 12.4443 0.426585
\(852\) −7.37836 −0.252778
\(853\) 22.8673 0.782963 0.391481 0.920186i \(-0.371963\pi\)
0.391481 + 0.920186i \(0.371963\pi\)
\(854\) −18.2212 −0.623517
\(855\) −10.6360 −0.363744
\(856\) −1.47218 −0.0503182
\(857\) 4.48293 0.153134 0.0765671 0.997064i \(-0.475604\pi\)
0.0765671 + 0.997064i \(0.475604\pi\)
\(858\) −1.66695 −0.0569087
\(859\) −9.76511 −0.333181 −0.166591 0.986026i \(-0.553276\pi\)
−0.166591 + 0.986026i \(0.553276\pi\)
\(860\) −6.78475 −0.231358
\(861\) 12.3079 0.419454
\(862\) −29.1265 −0.992051
\(863\) −41.9300 −1.42732 −0.713658 0.700495i \(-0.752963\pi\)
−0.713658 + 0.700495i \(0.752963\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −28.9406 −0.984012
\(866\) −0.469219 −0.0159447
\(867\) −15.3957 −0.522866
\(868\) 29.9344 1.01604
\(869\) −9.04420 −0.306803
\(870\) −18.2003 −0.617049
\(871\) 9.62761 0.326219
\(872\) 12.0292 0.407359
\(873\) 8.92335 0.302010
\(874\) 39.2558 1.32785
\(875\) 44.4810 1.50373
\(876\) −5.92878 −0.200315
\(877\) 2.02273 0.0683027 0.0341513 0.999417i \(-0.489127\pi\)
0.0341513 + 0.999417i \(0.489127\pi\)
\(878\) 36.5321 1.23290
\(879\) −0.477341 −0.0161003
\(880\) −3.93768 −0.132739
\(881\) −26.0273 −0.876881 −0.438440 0.898760i \(-0.644469\pi\)
−0.438440 + 0.898760i \(0.644469\pi\)
\(882\) −11.1499 −0.375436
\(883\) 9.85073 0.331503 0.165752 0.986168i \(-0.446995\pi\)
0.165752 + 0.986168i \(0.446995\pi\)
\(884\) −1.26660 −0.0426005
\(885\) 32.8987 1.10588
\(886\) 35.0074 1.17610
\(887\) −5.47297 −0.183764 −0.0918822 0.995770i \(-0.529288\pi\)
−0.0918822 + 0.995770i \(0.529288\pi\)
\(888\) −1.42734 −0.0478984
\(889\) 22.1960 0.744431
\(890\) 26.9390 0.902999
\(891\) −1.66695 −0.0558449
\(892\) 2.60556 0.0872407
\(893\) −10.4603 −0.350040
\(894\) −22.6402 −0.757202
\(895\) −32.3413 −1.08105
\(896\) 4.26027 0.142326
\(897\) −8.71852 −0.291103
\(898\) 13.6421 0.455244
\(899\) −54.1371 −1.80557
\(900\) 0.580037 0.0193346
\(901\) −3.97995 −0.132591
\(902\) −4.81583 −0.160350
\(903\) 12.2364 0.407201
\(904\) −15.0560 −0.500755
\(905\) 1.60883 0.0534792
\(906\) −5.85954 −0.194670
\(907\) −3.72175 −0.123579 −0.0617893 0.998089i \(-0.519681\pi\)
−0.0617893 + 0.998089i \(0.519681\pi\)
\(908\) −3.21031 −0.106538
\(909\) 8.26469 0.274122
\(910\) −10.0637 −0.333607
\(911\) −37.9160 −1.25621 −0.628106 0.778127i \(-0.716170\pi\)
−0.628106 + 0.778127i \(0.716170\pi\)
\(912\) −4.50257 −0.149095
\(913\) 22.9610 0.759899
\(914\) 2.07818 0.0687402
\(915\) −10.1032 −0.334001
\(916\) −14.5877 −0.481992
\(917\) −5.74380 −0.189677
\(918\) −1.26660 −0.0418042
\(919\) −55.1846 −1.82037 −0.910186 0.414199i \(-0.864062\pi\)
−0.910186 + 0.414199i \(0.864062\pi\)
\(920\) −20.5950 −0.678996
\(921\) 10.0577 0.331411
\(922\) −35.1380 −1.15721
\(923\) 7.37836 0.242862
\(924\) 7.10165 0.233627
\(925\) 0.827909 0.0272215
\(926\) −38.1923 −1.25508
\(927\) 1.00000 0.0328443
\(928\) −7.70479 −0.252922
\(929\) 15.0471 0.493681 0.246840 0.969056i \(-0.420608\pi\)
0.246840 + 0.969056i \(0.420608\pi\)
\(930\) 16.5979 0.544266
\(931\) −50.2032 −1.64534
\(932\) −17.0098 −0.557175
\(933\) −14.0208 −0.459019
\(934\) −4.09872 −0.134114
\(935\) −4.98749 −0.163108
\(936\) 1.00000 0.0326860
\(937\) 40.2496 1.31490 0.657448 0.753500i \(-0.271636\pi\)
0.657448 + 0.753500i \(0.271636\pi\)
\(938\) −41.0162 −1.33923
\(939\) −1.47426 −0.0481107
\(940\) 5.48784 0.178994
\(941\) −47.4710 −1.54751 −0.773756 0.633484i \(-0.781624\pi\)
−0.773756 + 0.633484i \(0.781624\pi\)
\(942\) 17.7129 0.577119
\(943\) −25.1879 −0.820230
\(944\) 13.9271 0.453287
\(945\) −10.0637 −0.327371
\(946\) −4.78782 −0.155666
\(947\) −47.7648 −1.55215 −0.776074 0.630642i \(-0.782792\pi\)
−0.776074 + 0.630642i \(0.782792\pi\)
\(948\) 5.42560 0.176215
\(949\) 5.92878 0.192456
\(950\) 2.61166 0.0847333
\(951\) 13.3249 0.432090
\(952\) 5.39607 0.174888
\(953\) −16.4522 −0.532939 −0.266469 0.963843i \(-0.585857\pi\)
−0.266469 + 0.963843i \(0.585857\pi\)
\(954\) 3.14222 0.101733
\(955\) −58.4385 −1.89103
\(956\) −8.89537 −0.287697
\(957\) −12.8435 −0.415171
\(958\) 25.1514 0.812604
\(959\) −39.8213 −1.28590
\(960\) 2.36221 0.0762400
\(961\) 18.3705 0.592597
\(962\) 1.42734 0.0460193
\(963\) 1.47218 0.0474404
\(964\) −15.3927 −0.495765
\(965\) −38.7204 −1.24645
\(966\) 37.1432 1.19506
\(967\) −5.40543 −0.173827 −0.0869134 0.996216i \(-0.527700\pi\)
−0.0869134 + 0.996216i \(0.527700\pi\)
\(968\) 8.22128 0.264242
\(969\) −5.70297 −0.183206
\(970\) −21.0788 −0.676800
\(971\) 34.3399 1.10202 0.551009 0.834499i \(-0.314243\pi\)
0.551009 + 0.834499i \(0.314243\pi\)
\(972\) 1.00000 0.0320750
\(973\) 84.4250 2.70654
\(974\) 36.4381 1.16755
\(975\) −0.580037 −0.0185760
\(976\) −4.27701 −0.136904
\(977\) −6.98418 −0.223444 −0.111722 0.993740i \(-0.535637\pi\)
−0.111722 + 0.993740i \(0.535637\pi\)
\(978\) 18.1900 0.581651
\(979\) 19.0102 0.607568
\(980\) 26.3384 0.841349
\(981\) −12.0292 −0.384061
\(982\) 1.60528 0.0512264
\(983\) 11.9398 0.380822 0.190411 0.981705i \(-0.439018\pi\)
0.190411 + 0.981705i \(0.439018\pi\)
\(984\) 2.88901 0.0920982
\(985\) 8.27765 0.263748
\(986\) −9.75892 −0.310787
\(987\) −9.89738 −0.315037
\(988\) 4.50257 0.143246
\(989\) −25.0414 −0.796269
\(990\) 3.93768 0.125148
\(991\) −25.6204 −0.813859 −0.406929 0.913460i \(-0.633401\pi\)
−0.406929 + 0.913460i \(0.633401\pi\)
\(992\) 7.02642 0.223089
\(993\) 11.1195 0.352866
\(994\) −31.4338 −0.997020
\(995\) 2.15967 0.0684661
\(996\) −13.7743 −0.436455
\(997\) −23.1463 −0.733051 −0.366525 0.930408i \(-0.619453\pi\)
−0.366525 + 0.930408i \(0.619453\pi\)
\(998\) −1.43114 −0.0453020
\(999\) 1.42734 0.0451590
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 8034.2.a.bb.1.12 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.12 14 1.1 even 1 trivial