Properties

Label 547.2.a.b.1.1
Level $547$
Weight $2$
Character 547.1
Self dual yes
Analytic conductor $4.368$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.36781699056\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.72204\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.72204 q^{2} -1.76734 q^{3} +5.40952 q^{4} +0.469688 q^{5} +4.81078 q^{6} +1.03831 q^{7} -9.28087 q^{8} +0.123492 q^{9} +O(q^{10})\) \(q-2.72204 q^{2} -1.76734 q^{3} +5.40952 q^{4} +0.469688 q^{5} +4.81078 q^{6} +1.03831 q^{7} -9.28087 q^{8} +0.123492 q^{9} -1.27851 q^{10} -1.84194 q^{11} -9.56047 q^{12} -0.700934 q^{13} -2.82633 q^{14} -0.830099 q^{15} +14.4439 q^{16} +0.793975 q^{17} -0.336150 q^{18} +3.65845 q^{19} +2.54079 q^{20} -1.83505 q^{21} +5.01384 q^{22} -3.65159 q^{23} +16.4025 q^{24} -4.77939 q^{25} +1.90797 q^{26} +5.08377 q^{27} +5.61677 q^{28} +5.52848 q^{29} +2.25957 q^{30} +6.13605 q^{31} -20.7552 q^{32} +3.25534 q^{33} -2.16124 q^{34} +0.487683 q^{35} +0.668032 q^{36} -2.73114 q^{37} -9.95846 q^{38} +1.23879 q^{39} -4.35911 q^{40} -9.69220 q^{41} +4.99509 q^{42} -4.25340 q^{43} -9.96402 q^{44} +0.0580027 q^{45} +9.93978 q^{46} -5.21084 q^{47} -25.5273 q^{48} -5.92191 q^{49} +13.0097 q^{50} -1.40322 q^{51} -3.79172 q^{52} -7.01107 q^{53} -13.8382 q^{54} -0.865138 q^{55} -9.63643 q^{56} -6.46572 q^{57} -15.0488 q^{58} +6.98691 q^{59} -4.49044 q^{60} -3.87686 q^{61} -16.7026 q^{62} +0.128223 q^{63} +27.6087 q^{64} -0.329220 q^{65} -8.86117 q^{66} -5.10753 q^{67} +4.29503 q^{68} +6.45360 q^{69} -1.32749 q^{70} +1.33192 q^{71} -1.14611 q^{72} -7.83377 q^{73} +7.43429 q^{74} +8.44681 q^{75} +19.7905 q^{76} -1.91251 q^{77} -3.37204 q^{78} +4.36166 q^{79} +6.78412 q^{80} -9.35523 q^{81} +26.3826 q^{82} -13.3348 q^{83} -9.92674 q^{84} +0.372921 q^{85} +11.5779 q^{86} -9.77071 q^{87} +17.0948 q^{88} -0.208861 q^{89} -0.157886 q^{90} -0.727788 q^{91} -19.7533 q^{92} -10.8445 q^{93} +14.1841 q^{94} +1.71833 q^{95} +36.6814 q^{96} +3.08809 q^{97} +16.1197 q^{98} -0.227465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.72204 −1.92478 −0.962388 0.271679i \(-0.912421\pi\)
−0.962388 + 0.271679i \(0.912421\pi\)
\(3\) −1.76734 −1.02037 −0.510187 0.860063i \(-0.670424\pi\)
−0.510187 + 0.860063i \(0.670424\pi\)
\(4\) 5.40952 2.70476
\(5\) 0.469688 0.210051 0.105025 0.994470i \(-0.466508\pi\)
0.105025 + 0.994470i \(0.466508\pi\)
\(6\) 4.81078 1.96399
\(7\) 1.03831 0.392445 0.196222 0.980559i \(-0.437133\pi\)
0.196222 + 0.980559i \(0.437133\pi\)
\(8\) −9.28087 −3.28128
\(9\) 0.123492 0.0411640
\(10\) −1.27851 −0.404301
\(11\) −1.84194 −0.555366 −0.277683 0.960673i \(-0.589566\pi\)
−0.277683 + 0.960673i \(0.589566\pi\)
\(12\) −9.56047 −2.75987
\(13\) −0.700934 −0.194404 −0.0972021 0.995265i \(-0.530989\pi\)
−0.0972021 + 0.995265i \(0.530989\pi\)
\(14\) −2.82633 −0.755368
\(15\) −0.830099 −0.214331
\(16\) 14.4439 3.61097
\(17\) 0.793975 0.192567 0.0962837 0.995354i \(-0.469304\pi\)
0.0962837 + 0.995354i \(0.469304\pi\)
\(18\) −0.336150 −0.0792314
\(19\) 3.65845 0.839306 0.419653 0.907685i \(-0.362152\pi\)
0.419653 + 0.907685i \(0.362152\pi\)
\(20\) 2.54079 0.568138
\(21\) −1.83505 −0.400441
\(22\) 5.01384 1.06895
\(23\) −3.65159 −0.761409 −0.380704 0.924697i \(-0.624318\pi\)
−0.380704 + 0.924697i \(0.624318\pi\)
\(24\) 16.4025 3.34814
\(25\) −4.77939 −0.955879
\(26\) 1.90797 0.374184
\(27\) 5.08377 0.978372
\(28\) 5.61677 1.06147
\(29\) 5.52848 1.02661 0.513307 0.858205i \(-0.328420\pi\)
0.513307 + 0.858205i \(0.328420\pi\)
\(30\) 2.25957 0.412538
\(31\) 6.13605 1.10207 0.551034 0.834483i \(-0.314234\pi\)
0.551034 + 0.834483i \(0.314234\pi\)
\(32\) −20.7552 −3.66903
\(33\) 3.25534 0.566681
\(34\) −2.16124 −0.370649
\(35\) 0.487683 0.0824334
\(36\) 0.668032 0.111339
\(37\) −2.73114 −0.448997 −0.224498 0.974474i \(-0.572074\pi\)
−0.224498 + 0.974474i \(0.572074\pi\)
\(38\) −9.95846 −1.61547
\(39\) 1.23879 0.198365
\(40\) −4.35911 −0.689237
\(41\) −9.69220 −1.51367 −0.756833 0.653608i \(-0.773255\pi\)
−0.756833 + 0.653608i \(0.773255\pi\)
\(42\) 4.99509 0.770758
\(43\) −4.25340 −0.648638 −0.324319 0.945948i \(-0.605135\pi\)
−0.324319 + 0.945948i \(0.605135\pi\)
\(44\) −9.96402 −1.50213
\(45\) 0.0580027 0.00864653
\(46\) 9.93978 1.46554
\(47\) −5.21084 −0.760078 −0.380039 0.924970i \(-0.624090\pi\)
−0.380039 + 0.924970i \(0.624090\pi\)
\(48\) −25.5273 −3.68454
\(49\) −5.92191 −0.845987
\(50\) 13.0097 1.83985
\(51\) −1.40322 −0.196491
\(52\) −3.79172 −0.525817
\(53\) −7.01107 −0.963044 −0.481522 0.876434i \(-0.659916\pi\)
−0.481522 + 0.876434i \(0.659916\pi\)
\(54\) −13.8382 −1.88315
\(55\) −0.865138 −0.116655
\(56\) −9.63643 −1.28772
\(57\) −6.46572 −0.856406
\(58\) −15.0488 −1.97600
\(59\) 6.98691 0.909619 0.454809 0.890589i \(-0.349707\pi\)
0.454809 + 0.890589i \(0.349707\pi\)
\(60\) −4.49044 −0.579713
\(61\) −3.87686 −0.496381 −0.248191 0.968711i \(-0.579836\pi\)
−0.248191 + 0.968711i \(0.579836\pi\)
\(62\) −16.7026 −2.12123
\(63\) 0.128223 0.0161546
\(64\) 27.6087 3.45109
\(65\) −0.329220 −0.0408348
\(66\) −8.86117 −1.09073
\(67\) −5.10753 −0.623984 −0.311992 0.950085i \(-0.600996\pi\)
−0.311992 + 0.950085i \(0.600996\pi\)
\(68\) 4.29503 0.520849
\(69\) 6.45360 0.776922
\(70\) −1.32749 −0.158666
\(71\) 1.33192 0.158070 0.0790350 0.996872i \(-0.474816\pi\)
0.0790350 + 0.996872i \(0.474816\pi\)
\(72\) −1.14611 −0.135071
\(73\) −7.83377 −0.916874 −0.458437 0.888727i \(-0.651591\pi\)
−0.458437 + 0.888727i \(0.651591\pi\)
\(74\) 7.43429 0.864218
\(75\) 8.44681 0.975354
\(76\) 19.7905 2.27012
\(77\) −1.91251 −0.217950
\(78\) −3.37204 −0.381808
\(79\) 4.36166 0.490726 0.245363 0.969431i \(-0.421093\pi\)
0.245363 + 0.969431i \(0.421093\pi\)
\(80\) 6.78412 0.758488
\(81\) −9.35523 −1.03947
\(82\) 26.3826 2.91347
\(83\) −13.3348 −1.46368 −0.731842 0.681475i \(-0.761339\pi\)
−0.731842 + 0.681475i \(0.761339\pi\)
\(84\) −9.92674 −1.08310
\(85\) 0.372921 0.0404489
\(86\) 11.5779 1.24848
\(87\) −9.77071 −1.04753
\(88\) 17.0948 1.82231
\(89\) −0.208861 −0.0221393 −0.0110696 0.999939i \(-0.503524\pi\)
−0.0110696 + 0.999939i \(0.503524\pi\)
\(90\) −0.157886 −0.0166426
\(91\) −0.727788 −0.0762929
\(92\) −19.7533 −2.05943
\(93\) −10.8445 −1.12452
\(94\) 14.1841 1.46298
\(95\) 1.71833 0.176297
\(96\) 36.6814 3.74378
\(97\) 3.08809 0.313549 0.156774 0.987634i \(-0.449890\pi\)
0.156774 + 0.987634i \(0.449890\pi\)
\(98\) 16.1197 1.62834
\(99\) −0.227465 −0.0228611
\(100\) −25.8542 −2.58542
\(101\) −8.00261 −0.796290 −0.398145 0.917323i \(-0.630346\pi\)
−0.398145 + 0.917323i \(0.630346\pi\)
\(102\) 3.81964 0.378201
\(103\) −3.26470 −0.321681 −0.160840 0.986980i \(-0.551420\pi\)
−0.160840 + 0.986980i \(0.551420\pi\)
\(104\) 6.50528 0.637895
\(105\) −0.861901 −0.0841129
\(106\) 19.0844 1.85364
\(107\) −18.4884 −1.78735 −0.893673 0.448719i \(-0.851880\pi\)
−0.893673 + 0.448719i \(0.851880\pi\)
\(108\) 27.5008 2.64626
\(109\) −18.6769 −1.78893 −0.894463 0.447141i \(-0.852442\pi\)
−0.894463 + 0.447141i \(0.852442\pi\)
\(110\) 2.35494 0.224535
\(111\) 4.82686 0.458145
\(112\) 14.9973 1.41711
\(113\) 7.85874 0.739288 0.369644 0.929173i \(-0.379480\pi\)
0.369644 + 0.929173i \(0.379480\pi\)
\(114\) 17.6000 1.64839
\(115\) −1.71511 −0.159935
\(116\) 29.9064 2.77674
\(117\) −0.0865597 −0.00800245
\(118\) −19.0187 −1.75081
\(119\) 0.824394 0.0755720
\(120\) 7.70404 0.703279
\(121\) −7.60726 −0.691569
\(122\) 10.5530 0.955422
\(123\) 17.1294 1.54451
\(124\) 33.1931 2.98083
\(125\) −4.59326 −0.410834
\(126\) −0.349029 −0.0310940
\(127\) 9.84052 0.873205 0.436602 0.899655i \(-0.356182\pi\)
0.436602 + 0.899655i \(0.356182\pi\)
\(128\) −33.6417 −2.97354
\(129\) 7.51721 0.661854
\(130\) 0.896152 0.0785978
\(131\) 8.02091 0.700791 0.350395 0.936602i \(-0.386047\pi\)
0.350395 + 0.936602i \(0.386047\pi\)
\(132\) 17.6098 1.53274
\(133\) 3.79861 0.329381
\(134\) 13.9029 1.20103
\(135\) 2.38779 0.205508
\(136\) −7.36878 −0.631868
\(137\) 16.8434 1.43903 0.719516 0.694476i \(-0.244364\pi\)
0.719516 + 0.694476i \(0.244364\pi\)
\(138\) −17.5670 −1.49540
\(139\) 11.2506 0.954260 0.477130 0.878833i \(-0.341677\pi\)
0.477130 + 0.878833i \(0.341677\pi\)
\(140\) 2.63813 0.222963
\(141\) 9.20932 0.775565
\(142\) −3.62555 −0.304249
\(143\) 1.29108 0.107965
\(144\) 1.78370 0.148642
\(145\) 2.59666 0.215641
\(146\) 21.3239 1.76478
\(147\) 10.4660 0.863224
\(148\) −14.7742 −1.21443
\(149\) −18.8662 −1.54558 −0.772788 0.634664i \(-0.781138\pi\)
−0.772788 + 0.634664i \(0.781138\pi\)
\(150\) −22.9926 −1.87734
\(151\) −6.40292 −0.521062 −0.260531 0.965465i \(-0.583898\pi\)
−0.260531 + 0.965465i \(0.583898\pi\)
\(152\) −33.9536 −2.75400
\(153\) 0.0980495 0.00792684
\(154\) 5.20593 0.419506
\(155\) 2.88203 0.231490
\(156\) 6.70126 0.536530
\(157\) −14.9254 −1.19118 −0.595589 0.803290i \(-0.703081\pi\)
−0.595589 + 0.803290i \(0.703081\pi\)
\(158\) −11.8726 −0.944537
\(159\) 12.3909 0.982665
\(160\) −9.74845 −0.770683
\(161\) −3.79148 −0.298811
\(162\) 25.4653 2.00075
\(163\) −20.3834 −1.59655 −0.798274 0.602294i \(-0.794253\pi\)
−0.798274 + 0.602294i \(0.794253\pi\)
\(164\) −52.4302 −4.09411
\(165\) 1.52899 0.119032
\(166\) 36.2979 2.81726
\(167\) 15.8853 1.22924 0.614622 0.788822i \(-0.289309\pi\)
0.614622 + 0.788822i \(0.289309\pi\)
\(168\) 17.0309 1.31396
\(169\) −12.5087 −0.962207
\(170\) −1.01511 −0.0778551
\(171\) 0.451789 0.0345492
\(172\) −23.0089 −1.75441
\(173\) 16.4110 1.24771 0.623854 0.781541i \(-0.285566\pi\)
0.623854 + 0.781541i \(0.285566\pi\)
\(174\) 26.5963 2.01626
\(175\) −4.96250 −0.375130
\(176\) −26.6048 −2.00541
\(177\) −12.3483 −0.928151
\(178\) 0.568530 0.0426131
\(179\) −20.4269 −1.52678 −0.763390 0.645938i \(-0.776466\pi\)
−0.763390 + 0.645938i \(0.776466\pi\)
\(180\) 0.313767 0.0233868
\(181\) −10.3683 −0.770666 −0.385333 0.922778i \(-0.625913\pi\)
−0.385333 + 0.922778i \(0.625913\pi\)
\(182\) 1.98107 0.146847
\(183\) 6.85173 0.506495
\(184\) 33.8899 2.49840
\(185\) −1.28279 −0.0943122
\(186\) 29.5192 2.16445
\(187\) −1.46246 −0.106945
\(188\) −28.1881 −2.05583
\(189\) 5.27853 0.383957
\(190\) −4.67737 −0.339332
\(191\) −7.95483 −0.575591 −0.287796 0.957692i \(-0.592922\pi\)
−0.287796 + 0.957692i \(0.592922\pi\)
\(192\) −48.7939 −3.52140
\(193\) 23.9566 1.72443 0.862217 0.506539i \(-0.169075\pi\)
0.862217 + 0.506539i \(0.169075\pi\)
\(194\) −8.40593 −0.603511
\(195\) 0.581845 0.0416668
\(196\) −32.0347 −2.28819
\(197\) 5.15127 0.367013 0.183506 0.983019i \(-0.441255\pi\)
0.183506 + 0.983019i \(0.441255\pi\)
\(198\) 0.619169 0.0440024
\(199\) 1.54311 0.109388 0.0546940 0.998503i \(-0.482582\pi\)
0.0546940 + 0.998503i \(0.482582\pi\)
\(200\) 44.3569 3.13651
\(201\) 9.02674 0.636697
\(202\) 21.7835 1.53268
\(203\) 5.74028 0.402889
\(204\) −7.59078 −0.531461
\(205\) −4.55231 −0.317947
\(206\) 8.88666 0.619163
\(207\) −0.450942 −0.0313426
\(208\) −10.1242 −0.701988
\(209\) −6.73864 −0.466122
\(210\) 2.34613 0.161899
\(211\) 8.81178 0.606628 0.303314 0.952891i \(-0.401907\pi\)
0.303314 + 0.952891i \(0.401907\pi\)
\(212\) −37.9265 −2.60480
\(213\) −2.35396 −0.161291
\(214\) 50.3264 3.44024
\(215\) −1.99777 −0.136247
\(216\) −47.1818 −3.21031
\(217\) 6.37113 0.432501
\(218\) 50.8395 3.44328
\(219\) 13.8449 0.935555
\(220\) −4.67998 −0.315524
\(221\) −0.556524 −0.0374359
\(222\) −13.1389 −0.881826
\(223\) 24.2050 1.62089 0.810445 0.585815i \(-0.199226\pi\)
0.810445 + 0.585815i \(0.199226\pi\)
\(224\) −21.5503 −1.43989
\(225\) −0.590216 −0.0393478
\(226\) −21.3918 −1.42296
\(227\) −3.55162 −0.235729 −0.117865 0.993030i \(-0.537605\pi\)
−0.117865 + 0.993030i \(0.537605\pi\)
\(228\) −34.9765 −2.31637
\(229\) −18.8164 −1.24342 −0.621711 0.783247i \(-0.713562\pi\)
−0.621711 + 0.783247i \(0.713562\pi\)
\(230\) 4.66860 0.307838
\(231\) 3.38005 0.222391
\(232\) −51.3091 −3.36861
\(233\) 12.1897 0.798575 0.399288 0.916826i \(-0.369258\pi\)
0.399288 + 0.916826i \(0.369258\pi\)
\(234\) 0.235619 0.0154029
\(235\) −2.44747 −0.159655
\(236\) 37.7959 2.46030
\(237\) −7.70855 −0.500724
\(238\) −2.24404 −0.145459
\(239\) 30.6423 1.98208 0.991042 0.133553i \(-0.0426385\pi\)
0.991042 + 0.133553i \(0.0426385\pi\)
\(240\) −11.9899 −0.773942
\(241\) −7.26294 −0.467847 −0.233924 0.972255i \(-0.575157\pi\)
−0.233924 + 0.972255i \(0.575157\pi\)
\(242\) 20.7073 1.33111
\(243\) 1.28256 0.0822763
\(244\) −20.9720 −1.34259
\(245\) −2.78145 −0.177700
\(246\) −46.6270 −2.97283
\(247\) −2.56433 −0.163164
\(248\) −56.9479 −3.61620
\(249\) 23.5671 1.49350
\(250\) 12.5031 0.790764
\(251\) −25.2813 −1.59574 −0.797871 0.602828i \(-0.794041\pi\)
−0.797871 + 0.602828i \(0.794041\pi\)
\(252\) 0.693626 0.0436943
\(253\) 6.72601 0.422860
\(254\) −26.7863 −1.68072
\(255\) −0.659078 −0.0412731
\(256\) 36.3569 2.27230
\(257\) −26.5017 −1.65313 −0.826564 0.562842i \(-0.809708\pi\)
−0.826564 + 0.562842i \(0.809708\pi\)
\(258\) −20.4622 −1.27392
\(259\) −2.83578 −0.176207
\(260\) −1.78093 −0.110448
\(261\) 0.682723 0.0422595
\(262\) −21.8333 −1.34886
\(263\) 12.5983 0.776845 0.388422 0.921481i \(-0.373020\pi\)
0.388422 + 0.921481i \(0.373020\pi\)
\(264\) −30.2123 −1.85944
\(265\) −3.29301 −0.202288
\(266\) −10.3400 −0.633985
\(267\) 0.369129 0.0225903
\(268\) −27.6293 −1.68773
\(269\) 1.45914 0.0889652 0.0444826 0.999010i \(-0.485836\pi\)
0.0444826 + 0.999010i \(0.485836\pi\)
\(270\) −6.49966 −0.395557
\(271\) 5.16205 0.313572 0.156786 0.987633i \(-0.449887\pi\)
0.156786 + 0.987633i \(0.449887\pi\)
\(272\) 11.4681 0.695355
\(273\) 1.28625 0.0778473
\(274\) −45.8486 −2.76981
\(275\) 8.80336 0.530862
\(276\) 34.9109 2.10139
\(277\) 14.0110 0.841837 0.420918 0.907098i \(-0.361708\pi\)
0.420918 + 0.907098i \(0.361708\pi\)
\(278\) −30.6245 −1.83674
\(279\) 0.757753 0.0453655
\(280\) −4.52612 −0.270487
\(281\) 16.3617 0.976059 0.488030 0.872827i \(-0.337716\pi\)
0.488030 + 0.872827i \(0.337716\pi\)
\(282\) −25.0682 −1.49279
\(283\) 18.5182 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(284\) 7.20506 0.427542
\(285\) −3.03687 −0.179889
\(286\) −3.51437 −0.207809
\(287\) −10.0635 −0.594031
\(288\) −2.56310 −0.151032
\(289\) −16.3696 −0.962918
\(290\) −7.06823 −0.415061
\(291\) −5.45771 −0.319937
\(292\) −42.3770 −2.47993
\(293\) 30.5239 1.78323 0.891613 0.452798i \(-0.149574\pi\)
0.891613 + 0.452798i \(0.149574\pi\)
\(294\) −28.4890 −1.66151
\(295\) 3.28167 0.191066
\(296\) 25.3474 1.47329
\(297\) −9.36400 −0.543354
\(298\) 51.3545 2.97489
\(299\) 2.55952 0.148021
\(300\) 45.6932 2.63810
\(301\) −4.41636 −0.254555
\(302\) 17.4290 1.00293
\(303\) 14.1433 0.812514
\(304\) 52.8422 3.03071
\(305\) −1.82092 −0.104265
\(306\) −0.266895 −0.0152574
\(307\) 3.36035 0.191785 0.0958926 0.995392i \(-0.469429\pi\)
0.0958926 + 0.995392i \(0.469429\pi\)
\(308\) −10.3458 −0.589504
\(309\) 5.76984 0.328235
\(310\) −7.84501 −0.445567
\(311\) 26.0631 1.47791 0.738953 0.673757i \(-0.235321\pi\)
0.738953 + 0.673757i \(0.235321\pi\)
\(312\) −11.4970 −0.650892
\(313\) −28.4960 −1.61069 −0.805345 0.592807i \(-0.798020\pi\)
−0.805345 + 0.592807i \(0.798020\pi\)
\(314\) 40.6276 2.29275
\(315\) 0.0602249 0.00339329
\(316\) 23.5945 1.32730
\(317\) −8.26530 −0.464225 −0.232113 0.972689i \(-0.574564\pi\)
−0.232113 + 0.972689i \(0.574564\pi\)
\(318\) −33.7287 −1.89141
\(319\) −10.1831 −0.570146
\(320\) 12.9675 0.724904
\(321\) 32.6754 1.82376
\(322\) 10.3206 0.575144
\(323\) 2.90472 0.161623
\(324\) −50.6073 −2.81152
\(325\) 3.35004 0.185827
\(326\) 55.4844 3.07300
\(327\) 33.0085 1.82537
\(328\) 89.9520 4.96677
\(329\) −5.41047 −0.298289
\(330\) −4.16198 −0.229110
\(331\) 12.5162 0.687951 0.343976 0.938979i \(-0.388226\pi\)
0.343976 + 0.938979i \(0.388226\pi\)
\(332\) −72.1348 −3.95891
\(333\) −0.337274 −0.0184825
\(334\) −43.2406 −2.36602
\(335\) −2.39895 −0.131068
\(336\) −26.5053 −1.44598
\(337\) −20.0461 −1.09198 −0.545990 0.837792i \(-0.683846\pi\)
−0.545990 + 0.837792i \(0.683846\pi\)
\(338\) 34.0492 1.85203
\(339\) −13.8891 −0.754351
\(340\) 2.01732 0.109405
\(341\) −11.3022 −0.612051
\(342\) −1.22979 −0.0664994
\(343\) −13.4170 −0.724448
\(344\) 39.4753 2.12836
\(345\) 3.03118 0.163193
\(346\) −44.6716 −2.40156
\(347\) 0.748255 0.0401684 0.0200842 0.999798i \(-0.493607\pi\)
0.0200842 + 0.999798i \(0.493607\pi\)
\(348\) −52.8549 −2.83332
\(349\) 4.16941 0.223183 0.111592 0.993754i \(-0.464405\pi\)
0.111592 + 0.993754i \(0.464405\pi\)
\(350\) 13.5081 0.722040
\(351\) −3.56339 −0.190200
\(352\) 38.2298 2.03765
\(353\) 28.9002 1.53820 0.769100 0.639128i \(-0.220705\pi\)
0.769100 + 0.639128i \(0.220705\pi\)
\(354\) 33.6125 1.78648
\(355\) 0.625588 0.0332028
\(356\) −1.12984 −0.0598814
\(357\) −1.45698 −0.0771118
\(358\) 55.6030 2.93871
\(359\) 22.5238 1.18876 0.594380 0.804185i \(-0.297398\pi\)
0.594380 + 0.804185i \(0.297398\pi\)
\(360\) −0.538315 −0.0283717
\(361\) −5.61576 −0.295566
\(362\) 28.2228 1.48336
\(363\) 13.4446 0.705659
\(364\) −3.93698 −0.206354
\(365\) −3.67943 −0.192590
\(366\) −18.6507 −0.974888
\(367\) −3.07935 −0.160741 −0.0803705 0.996765i \(-0.525610\pi\)
−0.0803705 + 0.996765i \(0.525610\pi\)
\(368\) −52.7431 −2.74943
\(369\) −1.19691 −0.0623085
\(370\) 3.49180 0.181530
\(371\) −7.27967 −0.377942
\(372\) −58.6635 −3.04156
\(373\) −23.8756 −1.23623 −0.618115 0.786088i \(-0.712103\pi\)
−0.618115 + 0.786088i \(0.712103\pi\)
\(374\) 3.98087 0.205846
\(375\) 8.11786 0.419205
\(376\) 48.3611 2.49403
\(377\) −3.87510 −0.199578
\(378\) −14.3684 −0.739031
\(379\) 3.60479 0.185166 0.0925828 0.995705i \(-0.470488\pi\)
0.0925828 + 0.995705i \(0.470488\pi\)
\(380\) 9.29534 0.476841
\(381\) −17.3915 −0.890996
\(382\) 21.6534 1.10788
\(383\) −3.63594 −0.185788 −0.0928939 0.995676i \(-0.529612\pi\)
−0.0928939 + 0.995676i \(0.529612\pi\)
\(384\) 59.4564 3.03412
\(385\) −0.898282 −0.0457807
\(386\) −65.2109 −3.31915
\(387\) −0.525261 −0.0267005
\(388\) 16.7051 0.848074
\(389\) 21.6735 1.09889 0.549445 0.835530i \(-0.314839\pi\)
0.549445 + 0.835530i \(0.314839\pi\)
\(390\) −1.58381 −0.0801992
\(391\) −2.89927 −0.146622
\(392\) 54.9605 2.77592
\(393\) −14.1757 −0.715069
\(394\) −14.0220 −0.706417
\(395\) 2.04862 0.103077
\(396\) −1.23048 −0.0618337
\(397\) 6.59119 0.330802 0.165401 0.986226i \(-0.447108\pi\)
0.165401 + 0.986226i \(0.447108\pi\)
\(398\) −4.20041 −0.210547
\(399\) −6.71343 −0.336092
\(400\) −69.0330 −3.45165
\(401\) −4.35340 −0.217398 −0.108699 0.994075i \(-0.534668\pi\)
−0.108699 + 0.994075i \(0.534668\pi\)
\(402\) −24.5712 −1.22550
\(403\) −4.30097 −0.214246
\(404\) −43.2903 −2.15377
\(405\) −4.39404 −0.218342
\(406\) −15.6253 −0.775471
\(407\) 5.03060 0.249358
\(408\) 13.0231 0.644742
\(409\) 31.8412 1.57445 0.787223 0.616668i \(-0.211518\pi\)
0.787223 + 0.616668i \(0.211518\pi\)
\(410\) 12.3916 0.611977
\(411\) −29.7681 −1.46835
\(412\) −17.6605 −0.870069
\(413\) 7.25459 0.356975
\(414\) 1.22748 0.0603275
\(415\) −6.26319 −0.307448
\(416\) 14.5480 0.713274
\(417\) −19.8836 −0.973703
\(418\) 18.3429 0.897180
\(419\) −29.8225 −1.45692 −0.728462 0.685087i \(-0.759764\pi\)
−0.728462 + 0.685087i \(0.759764\pi\)
\(420\) −4.66247 −0.227505
\(421\) −7.93092 −0.386530 −0.193265 0.981147i \(-0.561908\pi\)
−0.193265 + 0.981147i \(0.561908\pi\)
\(422\) −23.9860 −1.16762
\(423\) −0.643496 −0.0312879
\(424\) 65.0688 3.16002
\(425\) −3.79472 −0.184071
\(426\) 6.40758 0.310448
\(427\) −4.02539 −0.194802
\(428\) −100.014 −4.83434
\(429\) −2.28178 −0.110165
\(430\) 5.43803 0.262245
\(431\) −24.9537 −1.20198 −0.600988 0.799258i \(-0.705226\pi\)
−0.600988 + 0.799258i \(0.705226\pi\)
\(432\) 73.4294 3.53287
\(433\) 23.1505 1.11254 0.556272 0.831000i \(-0.312231\pi\)
0.556272 + 0.831000i \(0.312231\pi\)
\(434\) −17.3425 −0.832467
\(435\) −4.58918 −0.220035
\(436\) −101.033 −4.83862
\(437\) −13.3591 −0.639054
\(438\) −37.6865 −1.80073
\(439\) 18.9654 0.905167 0.452584 0.891722i \(-0.350502\pi\)
0.452584 + 0.891722i \(0.350502\pi\)
\(440\) 8.02923 0.382778
\(441\) −0.731308 −0.0348242
\(442\) 1.51488 0.0720557
\(443\) 21.6203 1.02721 0.513606 0.858026i \(-0.328309\pi\)
0.513606 + 0.858026i \(0.328309\pi\)
\(444\) 26.1110 1.23917
\(445\) −0.0980997 −0.00465037
\(446\) −65.8872 −3.11985
\(447\) 33.3429 1.57707
\(448\) 28.6664 1.35436
\(449\) 15.1299 0.714023 0.357011 0.934100i \(-0.383796\pi\)
0.357011 + 0.934100i \(0.383796\pi\)
\(450\) 1.60660 0.0757356
\(451\) 17.8524 0.840639
\(452\) 42.5121 1.99960
\(453\) 11.3161 0.531678
\(454\) 9.66767 0.453726
\(455\) −0.341833 −0.0160254
\(456\) 60.0075 2.81011
\(457\) −7.87882 −0.368556 −0.184278 0.982874i \(-0.558995\pi\)
−0.184278 + 0.982874i \(0.558995\pi\)
\(458\) 51.2190 2.39331
\(459\) 4.03639 0.188402
\(460\) −9.27791 −0.432585
\(461\) −33.2729 −1.54967 −0.774836 0.632162i \(-0.782168\pi\)
−0.774836 + 0.632162i \(0.782168\pi\)
\(462\) −9.20065 −0.428053
\(463\) 31.9493 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(464\) 79.8528 3.70707
\(465\) −5.09353 −0.236207
\(466\) −33.1810 −1.53708
\(467\) 6.03218 0.279136 0.139568 0.990213i \(-0.455429\pi\)
0.139568 + 0.990213i \(0.455429\pi\)
\(468\) −0.468247 −0.0216447
\(469\) −5.30320 −0.244879
\(470\) 6.66211 0.307300
\(471\) 26.3783 1.21545
\(472\) −64.8446 −2.98472
\(473\) 7.83451 0.360231
\(474\) 20.9830 0.963781
\(475\) −17.4852 −0.802274
\(476\) 4.45958 0.204404
\(477\) −0.865810 −0.0396427
\(478\) −83.4096 −3.81507
\(479\) 18.1368 0.828693 0.414347 0.910119i \(-0.364010\pi\)
0.414347 + 0.910119i \(0.364010\pi\)
\(480\) 17.2288 0.786385
\(481\) 1.91435 0.0872869
\(482\) 19.7700 0.900501
\(483\) 6.70084 0.304899
\(484\) −41.1516 −1.87053
\(485\) 1.45044 0.0658612
\(486\) −3.49119 −0.158363
\(487\) −23.3591 −1.05850 −0.529251 0.848465i \(-0.677527\pi\)
−0.529251 + 0.848465i \(0.677527\pi\)
\(488\) 35.9806 1.62877
\(489\) 36.0243 1.62908
\(490\) 7.57123 0.342033
\(491\) −4.28889 −0.193555 −0.0967775 0.995306i \(-0.530854\pi\)
−0.0967775 + 0.995306i \(0.530854\pi\)
\(492\) 92.6619 4.17752
\(493\) 4.38948 0.197692
\(494\) 6.98022 0.314055
\(495\) −0.106838 −0.00480199
\(496\) 88.6285 3.97954
\(497\) 1.38295 0.0620338
\(498\) −64.1507 −2.87466
\(499\) −12.0079 −0.537549 −0.268775 0.963203i \(-0.586619\pi\)
−0.268775 + 0.963203i \(0.586619\pi\)
\(500\) −24.8474 −1.11121
\(501\) −28.0748 −1.25429
\(502\) 68.8169 3.07145
\(503\) −22.0435 −0.982872 −0.491436 0.870914i \(-0.663528\pi\)
−0.491436 + 0.870914i \(0.663528\pi\)
\(504\) −1.19002 −0.0530078
\(505\) −3.75873 −0.167261
\(506\) −18.3085 −0.813911
\(507\) 22.1071 0.981811
\(508\) 53.2325 2.36181
\(509\) −11.9642 −0.530302 −0.265151 0.964207i \(-0.585422\pi\)
−0.265151 + 0.964207i \(0.585422\pi\)
\(510\) 1.79404 0.0794414
\(511\) −8.13390 −0.359822
\(512\) −31.6816 −1.40014
\(513\) 18.5987 0.821153
\(514\) 72.1387 3.18190
\(515\) −1.53339 −0.0675693
\(516\) 40.6645 1.79016
\(517\) 9.59805 0.422122
\(518\) 7.71911 0.339158
\(519\) −29.0039 −1.27313
\(520\) 3.05545 0.133990
\(521\) −5.47217 −0.239740 −0.119870 0.992790i \(-0.538248\pi\)
−0.119870 + 0.992790i \(0.538248\pi\)
\(522\) −1.85840 −0.0813400
\(523\) −31.1363 −1.36149 −0.680747 0.732519i \(-0.738345\pi\)
−0.680747 + 0.732519i \(0.738345\pi\)
\(524\) 43.3893 1.89547
\(525\) 8.77042 0.382773
\(526\) −34.2931 −1.49525
\(527\) 4.87187 0.212222
\(528\) 47.0197 2.04627
\(529\) −9.66591 −0.420257
\(530\) 8.96373 0.389360
\(531\) 0.862827 0.0374435
\(532\) 20.5487 0.890897
\(533\) 6.79359 0.294263
\(534\) −1.00479 −0.0434813
\(535\) −8.68380 −0.375434
\(536\) 47.4023 2.04747
\(537\) 36.1013 1.55789
\(538\) −3.97183 −0.171238
\(539\) 10.9078 0.469832
\(540\) 12.9168 0.555850
\(541\) −15.0578 −0.647387 −0.323694 0.946162i \(-0.604925\pi\)
−0.323694 + 0.946162i \(0.604925\pi\)
\(542\) −14.0513 −0.603556
\(543\) 18.3242 0.786368
\(544\) −16.4791 −0.706535
\(545\) −8.77234 −0.375766
\(546\) −3.50123 −0.149839
\(547\) −1.00000 −0.0427569
\(548\) 91.1150 3.89224
\(549\) −0.478761 −0.0204330
\(550\) −23.9631 −1.02179
\(551\) 20.2257 0.861642
\(552\) −59.8950 −2.54930
\(553\) 4.52877 0.192583
\(554\) −38.1384 −1.62035
\(555\) 2.26712 0.0962338
\(556\) 60.8602 2.58105
\(557\) 23.5488 0.997796 0.498898 0.866661i \(-0.333738\pi\)
0.498898 + 0.866661i \(0.333738\pi\)
\(558\) −2.06264 −0.0873184
\(559\) 2.98136 0.126098
\(560\) 7.04403 0.297665
\(561\) 2.58466 0.109124
\(562\) −44.5374 −1.87869
\(563\) −29.0653 −1.22496 −0.612478 0.790488i \(-0.709827\pi\)
−0.612478 + 0.790488i \(0.709827\pi\)
\(564\) 49.8180 2.09772
\(565\) 3.69116 0.155288
\(566\) −50.4074 −2.11878
\(567\) −9.71364 −0.407934
\(568\) −12.3614 −0.518673
\(569\) −0.227367 −0.00953174 −0.00476587 0.999989i \(-0.501517\pi\)
−0.00476587 + 0.999989i \(0.501517\pi\)
\(570\) 8.26650 0.346246
\(571\) 0.874566 0.0365994 0.0182997 0.999833i \(-0.494175\pi\)
0.0182997 + 0.999833i \(0.494175\pi\)
\(572\) 6.98412 0.292021
\(573\) 14.0589 0.587319
\(574\) 27.3933 1.14338
\(575\) 17.4524 0.727814
\(576\) 3.40945 0.142060
\(577\) −29.2997 −1.21976 −0.609882 0.792492i \(-0.708783\pi\)
−0.609882 + 0.792492i \(0.708783\pi\)
\(578\) 44.5588 1.85340
\(579\) −42.3395 −1.75957
\(580\) 14.0467 0.583257
\(581\) −13.8457 −0.574415
\(582\) 14.8561 0.615807
\(583\) 12.9140 0.534842
\(584\) 72.7042 3.00852
\(585\) −0.0406561 −0.00168092
\(586\) −83.0875 −3.43231
\(587\) −10.9789 −0.453149 −0.226574 0.973994i \(-0.572753\pi\)
−0.226574 + 0.973994i \(0.572753\pi\)
\(588\) 56.6162 2.33481
\(589\) 22.4484 0.924971
\(590\) −8.93285 −0.367760
\(591\) −9.10404 −0.374490
\(592\) −39.4483 −1.62132
\(593\) −42.7926 −1.75728 −0.878641 0.477483i \(-0.841549\pi\)
−0.878641 + 0.477483i \(0.841549\pi\)
\(594\) 25.4892 1.04584
\(595\) 0.387208 0.0158740
\(596\) −102.057 −4.18042
\(597\) −2.72720 −0.111617
\(598\) −6.96713 −0.284907
\(599\) 0.841737 0.0343925 0.0171962 0.999852i \(-0.494526\pi\)
0.0171962 + 0.999852i \(0.494526\pi\)
\(600\) −78.3938 −3.20041
\(601\) −19.7479 −0.805536 −0.402768 0.915302i \(-0.631952\pi\)
−0.402768 + 0.915302i \(0.631952\pi\)
\(602\) 12.0215 0.489960
\(603\) −0.630738 −0.0256857
\(604\) −34.6367 −1.40935
\(605\) −3.57304 −0.145265
\(606\) −38.4988 −1.56391
\(607\) −0.928802 −0.0376989 −0.0188495 0.999822i \(-0.506000\pi\)
−0.0188495 + 0.999822i \(0.506000\pi\)
\(608\) −75.9317 −3.07944
\(609\) −10.1450 −0.411098
\(610\) 4.95661 0.200687
\(611\) 3.65245 0.147762
\(612\) 0.530401 0.0214402
\(613\) 20.1162 0.812485 0.406242 0.913765i \(-0.366839\pi\)
0.406242 + 0.913765i \(0.366839\pi\)
\(614\) −9.14702 −0.369144
\(615\) 8.04548 0.324425
\(616\) 17.7497 0.715157
\(617\) −13.2513 −0.533479 −0.266739 0.963769i \(-0.585946\pi\)
−0.266739 + 0.963769i \(0.585946\pi\)
\(618\) −15.7058 −0.631778
\(619\) −29.2870 −1.17714 −0.588572 0.808445i \(-0.700310\pi\)
−0.588572 + 0.808445i \(0.700310\pi\)
\(620\) 15.5904 0.626126
\(621\) −18.5638 −0.744941
\(622\) −70.9450 −2.84464
\(623\) −0.216863 −0.00868844
\(624\) 17.8929 0.716291
\(625\) 21.7396 0.869583
\(626\) 77.5674 3.10022
\(627\) 11.9095 0.475619
\(628\) −80.7393 −3.22185
\(629\) −2.16846 −0.0864621
\(630\) −0.163935 −0.00653131
\(631\) −46.5766 −1.85418 −0.927092 0.374834i \(-0.877700\pi\)
−0.927092 + 0.374834i \(0.877700\pi\)
\(632\) −40.4800 −1.61021
\(633\) −15.5734 −0.618987
\(634\) 22.4985 0.893530
\(635\) 4.62197 0.183417
\(636\) 67.0291 2.65788
\(637\) 4.15087 0.164463
\(638\) 27.7189 1.09740
\(639\) 0.164482 0.00650679
\(640\) −15.8011 −0.624594
\(641\) −5.11165 −0.201898 −0.100949 0.994892i \(-0.532188\pi\)
−0.100949 + 0.994892i \(0.532188\pi\)
\(642\) −88.9438 −3.51033
\(643\) −4.32994 −0.170756 −0.0853781 0.996349i \(-0.527210\pi\)
−0.0853781 + 0.996349i \(0.527210\pi\)
\(644\) −20.5101 −0.808212
\(645\) 3.53074 0.139023
\(646\) −7.90677 −0.311088
\(647\) −12.0401 −0.473343 −0.236672 0.971590i \(-0.576057\pi\)
−0.236672 + 0.971590i \(0.576057\pi\)
\(648\) 86.8246 3.41079
\(649\) −12.8695 −0.505171
\(650\) −9.11895 −0.357675
\(651\) −11.2600 −0.441313
\(652\) −110.264 −4.31828
\(653\) 39.4366 1.54327 0.771636 0.636064i \(-0.219439\pi\)
0.771636 + 0.636064i \(0.219439\pi\)
\(654\) −89.8506 −3.51344
\(655\) 3.76733 0.147202
\(656\) −139.993 −5.46581
\(657\) −0.967408 −0.0377422
\(658\) 14.7275 0.574139
\(659\) 43.8625 1.70864 0.854320 0.519748i \(-0.173974\pi\)
0.854320 + 0.519748i \(0.173974\pi\)
\(660\) 8.27112 0.321953
\(661\) −5.18941 −0.201845 −0.100922 0.994894i \(-0.532179\pi\)
−0.100922 + 0.994894i \(0.532179\pi\)
\(662\) −34.0696 −1.32415
\(663\) 0.983568 0.0381986
\(664\) 123.758 4.80276
\(665\) 1.78416 0.0691868
\(666\) 0.918075 0.0355747
\(667\) −20.1877 −0.781672
\(668\) 85.9321 3.32481
\(669\) −42.7785 −1.65391
\(670\) 6.53003 0.252277
\(671\) 7.14094 0.275673
\(672\) 38.0868 1.46923
\(673\) 37.7606 1.45557 0.727783 0.685808i \(-0.240551\pi\)
0.727783 + 0.685808i \(0.240551\pi\)
\(674\) 54.5663 2.10182
\(675\) −24.2973 −0.935205
\(676\) −67.6661 −2.60254
\(677\) −36.6368 −1.40806 −0.704032 0.710168i \(-0.748619\pi\)
−0.704032 + 0.710168i \(0.748619\pi\)
\(678\) 37.8067 1.45196
\(679\) 3.20640 0.123050
\(680\) −3.46103 −0.132724
\(681\) 6.27692 0.240532
\(682\) 30.7652 1.17806
\(683\) 31.5096 1.20568 0.602841 0.797861i \(-0.294035\pi\)
0.602841 + 0.797861i \(0.294035\pi\)
\(684\) 2.44396 0.0934472
\(685\) 7.91116 0.302270
\(686\) 36.5216 1.39440
\(687\) 33.2549 1.26876
\(688\) −61.4357 −2.34221
\(689\) 4.91430 0.187220
\(690\) −8.25100 −0.314110
\(691\) 18.9407 0.720540 0.360270 0.932848i \(-0.382685\pi\)
0.360270 + 0.932848i \(0.382685\pi\)
\(692\) 88.7759 3.37475
\(693\) −0.236179 −0.00897171
\(694\) −2.03678 −0.0773153
\(695\) 5.28426 0.200443
\(696\) 90.6807 3.43724
\(697\) −7.69537 −0.291483
\(698\) −11.3493 −0.429578
\(699\) −21.5434 −0.814846
\(700\) −26.8447 −1.01464
\(701\) 42.8730 1.61929 0.809646 0.586919i \(-0.199659\pi\)
0.809646 + 0.586919i \(0.199659\pi\)
\(702\) 9.69970 0.366091
\(703\) −9.99174 −0.376846
\(704\) −50.8535 −1.91662
\(705\) 4.32551 0.162908
\(706\) −78.6675 −2.96069
\(707\) −8.30921 −0.312500
\(708\) −66.7982 −2.51043
\(709\) 35.4811 1.33252 0.666260 0.745719i \(-0.267894\pi\)
0.666260 + 0.745719i \(0.267894\pi\)
\(710\) −1.70288 −0.0639079
\(711\) 0.538630 0.0202002
\(712\) 1.93842 0.0726452
\(713\) −22.4063 −0.839124
\(714\) 3.96597 0.148423
\(715\) 0.606404 0.0226782
\(716\) −110.500 −4.12957
\(717\) −54.1553 −2.02247
\(718\) −61.3107 −2.28810
\(719\) −24.4790 −0.912913 −0.456456 0.889746i \(-0.650882\pi\)
−0.456456 + 0.889746i \(0.650882\pi\)
\(720\) 0.837785 0.0312224
\(721\) −3.38978 −0.126242
\(722\) 15.2863 0.568899
\(723\) 12.8361 0.477379
\(724\) −56.0873 −2.08447
\(725\) −26.4228 −0.981317
\(726\) −36.5968 −1.35824
\(727\) −28.7796 −1.06737 −0.533687 0.845682i \(-0.679194\pi\)
−0.533687 + 0.845682i \(0.679194\pi\)
\(728\) 6.75450 0.250339
\(729\) 25.7990 0.955517
\(730\) 10.0156 0.370693
\(731\) −3.37710 −0.124906
\(732\) 37.0646 1.36995
\(733\) 23.0494 0.851350 0.425675 0.904876i \(-0.360037\pi\)
0.425675 + 0.904876i \(0.360037\pi\)
\(734\) 8.38213 0.309390
\(735\) 4.91577 0.181321
\(736\) 75.7893 2.79363
\(737\) 9.40776 0.346539
\(738\) 3.25804 0.119930
\(739\) −9.02906 −0.332139 −0.166070 0.986114i \(-0.553108\pi\)
−0.166070 + 0.986114i \(0.553108\pi\)
\(740\) −6.93925 −0.255092
\(741\) 4.53205 0.166489
\(742\) 19.8156 0.727453
\(743\) −47.5028 −1.74271 −0.871355 0.490653i \(-0.836758\pi\)
−0.871355 + 0.490653i \(0.836758\pi\)
\(744\) 100.646 3.68987
\(745\) −8.86121 −0.324650
\(746\) 64.9903 2.37947
\(747\) −1.64674 −0.0602510
\(748\) −7.91118 −0.289262
\(749\) −19.1968 −0.701434
\(750\) −22.0972 −0.806875
\(751\) −20.5388 −0.749473 −0.374737 0.927131i \(-0.622267\pi\)
−0.374737 + 0.927131i \(0.622267\pi\)
\(752\) −75.2647 −2.74462
\(753\) 44.6807 1.62825
\(754\) 10.5482 0.384143
\(755\) −3.00737 −0.109450
\(756\) 28.5544 1.03851
\(757\) 36.3077 1.31962 0.659812 0.751430i \(-0.270636\pi\)
0.659812 + 0.751430i \(0.270636\pi\)
\(758\) −9.81240 −0.356402
\(759\) −11.8871 −0.431476
\(760\) −15.9476 −0.578480
\(761\) 23.0597 0.835915 0.417957 0.908467i \(-0.362746\pi\)
0.417957 + 0.908467i \(0.362746\pi\)
\(762\) 47.3405 1.71497
\(763\) −19.3925 −0.702055
\(764\) −43.0318 −1.55684
\(765\) 0.0460527 0.00166504
\(766\) 9.89719 0.357600
\(767\) −4.89737 −0.176834
\(768\) −64.2550 −2.31860
\(769\) 4.40403 0.158813 0.0794066 0.996842i \(-0.474697\pi\)
0.0794066 + 0.996842i \(0.474697\pi\)
\(770\) 2.44516 0.0881176
\(771\) 46.8375 1.68681
\(772\) 129.594 4.66418
\(773\) −25.6259 −0.921699 −0.460849 0.887478i \(-0.652455\pi\)
−0.460849 + 0.887478i \(0.652455\pi\)
\(774\) 1.42978 0.0513925
\(775\) −29.3266 −1.05344
\(776\) −28.6602 −1.02884
\(777\) 5.01178 0.179797
\(778\) −58.9962 −2.11512
\(779\) −35.4584 −1.27043
\(780\) 3.14750 0.112699
\(781\) −2.45332 −0.0877867
\(782\) 7.89194 0.282215
\(783\) 28.1055 1.00441
\(784\) −85.5354 −3.05484
\(785\) −7.01029 −0.250208
\(786\) 38.5868 1.37635
\(787\) 46.4038 1.65412 0.827059 0.562115i \(-0.190012\pi\)
0.827059 + 0.562115i \(0.190012\pi\)
\(788\) 27.8659 0.992681
\(789\) −22.2655 −0.792673
\(790\) −5.57644 −0.198401
\(791\) 8.15982 0.290130
\(792\) 2.11107 0.0750136
\(793\) 2.71742 0.0964985
\(794\) −17.9415 −0.636720
\(795\) 5.81988 0.206410
\(796\) 8.34748 0.295868
\(797\) −33.6692 −1.19262 −0.596312 0.802753i \(-0.703368\pi\)
−0.596312 + 0.802753i \(0.703368\pi\)
\(798\) 18.2743 0.646902
\(799\) −4.13727 −0.146366
\(800\) 99.1971 3.50715
\(801\) −0.0257927 −0.000911340 0
\(802\) 11.8501 0.418443
\(803\) 14.4293 0.509200
\(804\) 48.8304 1.72211
\(805\) −1.78082 −0.0627655
\(806\) 11.7074 0.412376
\(807\) −2.57879 −0.0907778
\(808\) 74.2712 2.61285
\(809\) 32.2939 1.13539 0.567696 0.823238i \(-0.307835\pi\)
0.567696 + 0.823238i \(0.307835\pi\)
\(810\) 11.9608 0.420258
\(811\) 27.4278 0.963119 0.481559 0.876413i \(-0.340071\pi\)
0.481559 + 0.876413i \(0.340071\pi\)
\(812\) 31.0522 1.08972
\(813\) −9.12310 −0.319961
\(814\) −13.6935 −0.479957
\(815\) −9.57382 −0.335356
\(816\) −20.2680 −0.709523
\(817\) −15.5609 −0.544405
\(818\) −86.6732 −3.03046
\(819\) −0.0898759 −0.00314052
\(820\) −24.6258 −0.859971
\(821\) −17.5646 −0.613008 −0.306504 0.951869i \(-0.599159\pi\)
−0.306504 + 0.951869i \(0.599159\pi\)
\(822\) 81.0301 2.82625
\(823\) −7.94705 −0.277017 −0.138508 0.990361i \(-0.544231\pi\)
−0.138508 + 0.990361i \(0.544231\pi\)
\(824\) 30.2993 1.05552
\(825\) −15.5585 −0.541678
\(826\) −19.7473 −0.687097
\(827\) −5.16367 −0.179558 −0.0897792 0.995962i \(-0.528616\pi\)
−0.0897792 + 0.995962i \(0.528616\pi\)
\(828\) −2.43938 −0.0847743
\(829\) −20.0606 −0.696734 −0.348367 0.937358i \(-0.613264\pi\)
−0.348367 + 0.937358i \(0.613264\pi\)
\(830\) 17.0487 0.591768
\(831\) −24.7621 −0.858989
\(832\) −19.3519 −0.670905
\(833\) −4.70185 −0.162909
\(834\) 54.1240 1.87416
\(835\) 7.46115 0.258204
\(836\) −36.4528 −1.26075
\(837\) 31.1943 1.07823
\(838\) 81.1781 2.80425
\(839\) 39.3019 1.35685 0.678426 0.734669i \(-0.262662\pi\)
0.678426 + 0.734669i \(0.262662\pi\)
\(840\) 7.99919 0.275998
\(841\) 1.56409 0.0539343
\(842\) 21.5883 0.743983
\(843\) −28.9168 −0.995946
\(844\) 47.6675 1.64078
\(845\) −5.87518 −0.202112
\(846\) 1.75162 0.0602221
\(847\) −7.89870 −0.271403
\(848\) −101.267 −3.47753
\(849\) −32.7280 −1.12322
\(850\) 10.3294 0.354295
\(851\) 9.97300 0.341870
\(852\) −12.7338 −0.436253
\(853\) 41.2623 1.41280 0.706398 0.707815i \(-0.250319\pi\)
0.706398 + 0.707815i \(0.250319\pi\)
\(854\) 10.9573 0.374950
\(855\) 0.212200 0.00725708
\(856\) 171.589 5.86479
\(857\) 4.93880 0.168706 0.0843532 0.996436i \(-0.473118\pi\)
0.0843532 + 0.996436i \(0.473118\pi\)
\(858\) 6.21109 0.212043
\(859\) −38.1044 −1.30011 −0.650053 0.759889i \(-0.725253\pi\)
−0.650053 + 0.759889i \(0.725253\pi\)
\(860\) −10.8070 −0.368516
\(861\) 17.7857 0.606134
\(862\) 67.9250 2.31353
\(863\) −1.71273 −0.0583019 −0.0291510 0.999575i \(-0.509280\pi\)
−0.0291510 + 0.999575i \(0.509280\pi\)
\(864\) −105.514 −3.58967
\(865\) 7.70808 0.262082
\(866\) −63.0168 −2.14140
\(867\) 28.9307 0.982537
\(868\) 34.4648 1.16981
\(869\) −8.03393 −0.272532
\(870\) 12.4920 0.423517
\(871\) 3.58004 0.121305
\(872\) 173.338 5.86997
\(873\) 0.381355 0.0129069
\(874\) 36.3642 1.23004
\(875\) −4.76924 −0.161230
\(876\) 74.8945 2.53045
\(877\) 19.2114 0.648724 0.324362 0.945933i \(-0.394850\pi\)
0.324362 + 0.945933i \(0.394850\pi\)
\(878\) −51.6245 −1.74224
\(879\) −53.9462 −1.81956
\(880\) −12.4960 −0.421238
\(881\) 0.0458679 0.00154533 0.000772664 1.00000i \(-0.499754\pi\)
0.000772664 1.00000i \(0.499754\pi\)
\(882\) 1.99065 0.0670288
\(883\) 42.1763 1.41935 0.709674 0.704531i \(-0.248842\pi\)
0.709674 + 0.704531i \(0.248842\pi\)
\(884\) −3.01053 −0.101255
\(885\) −5.79983 −0.194959
\(886\) −58.8515 −1.97715
\(887\) 36.1371 1.21337 0.606683 0.794944i \(-0.292500\pi\)
0.606683 + 0.794944i \(0.292500\pi\)
\(888\) −44.7974 −1.50330
\(889\) 10.2175 0.342685
\(890\) 0.267032 0.00895093
\(891\) 17.2318 0.577286
\(892\) 130.938 4.38412
\(893\) −19.0636 −0.637938
\(894\) −90.7609 −3.03550
\(895\) −9.59428 −0.320701
\(896\) −34.9306 −1.16695
\(897\) −4.52355 −0.151037
\(898\) −41.1842 −1.37433
\(899\) 33.9230 1.13140
\(900\) −3.19279 −0.106426
\(901\) −5.56661 −0.185451
\(902\) −48.5951 −1.61804
\(903\) 7.80520 0.259741
\(904\) −72.9360 −2.42581
\(905\) −4.86985 −0.161879
\(906\) −30.8030 −1.02336
\(907\) −21.2700 −0.706259 −0.353129 0.935574i \(-0.614882\pi\)
−0.353129 + 0.935574i \(0.614882\pi\)
\(908\) −19.2126 −0.637592
\(909\) −0.988258 −0.0327785
\(910\) 0.930485 0.0308453
\(911\) −10.2529 −0.339694 −0.169847 0.985470i \(-0.554327\pi\)
−0.169847 + 0.985470i \(0.554327\pi\)
\(912\) −93.3902 −3.09246
\(913\) 24.5619 0.812880
\(914\) 21.4465 0.709387
\(915\) 3.21818 0.106390
\(916\) −101.788 −3.36316
\(917\) 8.32821 0.275022
\(918\) −10.9872 −0.362632
\(919\) −39.3681 −1.29864 −0.649318 0.760517i \(-0.724945\pi\)
−0.649318 + 0.760517i \(0.724945\pi\)
\(920\) 15.9177 0.524791
\(921\) −5.93888 −0.195693
\(922\) 90.5703 2.98277
\(923\) −0.933589 −0.0307295
\(924\) 18.2845 0.601515
\(925\) 13.0532 0.429187
\(926\) −86.9674 −2.85793
\(927\) −0.403164 −0.0132416
\(928\) −114.745 −3.76667
\(929\) 48.6956 1.59765 0.798826 0.601562i \(-0.205455\pi\)
0.798826 + 0.601562i \(0.205455\pi\)
\(930\) 13.8648 0.454645
\(931\) −21.6650 −0.710042
\(932\) 65.9406 2.15996
\(933\) −46.0625 −1.50802
\(934\) −16.4198 −0.537274
\(935\) −0.686898 −0.0224640
\(936\) 0.803349 0.0262583
\(937\) −12.4806 −0.407723 −0.203862 0.979000i \(-0.565349\pi\)
−0.203862 + 0.979000i \(0.565349\pi\)
\(938\) 14.4356 0.471338
\(939\) 50.3621 1.64351
\(940\) −13.2396 −0.431829
\(941\) −44.4531 −1.44913 −0.724565 0.689207i \(-0.757959\pi\)
−0.724565 + 0.689207i \(0.757959\pi\)
\(942\) −71.8028 −2.33946
\(943\) 35.3919 1.15252
\(944\) 100.918 3.28461
\(945\) 2.47927 0.0806505
\(946\) −21.3259 −0.693365
\(947\) −7.49160 −0.243444 −0.121722 0.992564i \(-0.538842\pi\)
−0.121722 + 0.992564i \(0.538842\pi\)
\(948\) −41.6996 −1.35434
\(949\) 5.49096 0.178244
\(950\) 47.5954 1.54420
\(951\) 14.6076 0.473684
\(952\) −7.65109 −0.247973
\(953\) −14.6341 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(954\) 2.35677 0.0763033
\(955\) −3.73629 −0.120903
\(956\) 165.760 5.36106
\(957\) 17.9971 0.581762
\(958\) −49.3693 −1.59505
\(959\) 17.4887 0.564741
\(960\) −22.9179 −0.739673
\(961\) 6.65114 0.214553
\(962\) −5.21095 −0.168008
\(963\) −2.28317 −0.0735743
\(964\) −39.2891 −1.26542
\(965\) 11.2521 0.362219
\(966\) −18.2400 −0.586862
\(967\) 1.24602 0.0400692 0.0200346 0.999799i \(-0.493622\pi\)
0.0200346 + 0.999799i \(0.493622\pi\)
\(968\) 70.6020 2.26923
\(969\) −5.13362 −0.164916
\(970\) −3.94817 −0.126768
\(971\) 15.4693 0.496434 0.248217 0.968704i \(-0.420155\pi\)
0.248217 + 0.968704i \(0.420155\pi\)
\(972\) 6.93804 0.222538
\(973\) 11.6816 0.374494
\(974\) 63.5846 2.03738
\(975\) −5.92066 −0.189613
\(976\) −55.9969 −1.79242
\(977\) 29.2177 0.934757 0.467379 0.884057i \(-0.345199\pi\)
0.467379 + 0.884057i \(0.345199\pi\)
\(978\) −98.0598 −3.13561
\(979\) 0.384710 0.0122954
\(980\) −15.0463 −0.480637
\(981\) −2.30645 −0.0736393
\(982\) 11.6745 0.372550
\(983\) −2.39930 −0.0765259 −0.0382629 0.999268i \(-0.512182\pi\)
−0.0382629 + 0.999268i \(0.512182\pi\)
\(984\) −158.976 −5.06796
\(985\) 2.41949 0.0770913
\(986\) −11.9483 −0.380513
\(987\) 9.56214 0.304366
\(988\) −13.8718 −0.441321
\(989\) 15.5317 0.493878
\(990\) 0.290816 0.00924275
\(991\) 51.4773 1.63523 0.817614 0.575766i \(-0.195296\pi\)
0.817614 + 0.575766i \(0.195296\pi\)
\(992\) −127.355 −4.04352
\(993\) −22.1203 −0.701968
\(994\) −3.76445 −0.119401
\(995\) 0.724779 0.0229771
\(996\) 127.487 4.03957
\(997\) −41.2171 −1.30536 −0.652680 0.757634i \(-0.726355\pi\)
−0.652680 + 0.757634i \(0.726355\pi\)
\(998\) 32.6862 1.03466
\(999\) −13.8845 −0.439286
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 547.2.a.b.1.1 18
3.2 odd 2 4923.2.a.l.1.18 18
4.3 odd 2 8752.2.a.s.1.13 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.1 18 1.1 even 1 trivial
4923.2.a.l.1.18 18 3.2 odd 2
8752.2.a.s.1.13 18 4.3 odd 2