Properties

Label 547.2.a.b.1.3
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 1.3
Root \(2.35947\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.35947 q^{2} -3.09007 q^{3} +3.56710 q^{4} -4.18174 q^{5} +7.29093 q^{6} -2.82979 q^{7} -3.69754 q^{8} +6.54852 q^{9} +O(q^{10})\) \(q-2.35947 q^{2} -3.09007 q^{3} +3.56710 q^{4} -4.18174 q^{5} +7.29093 q^{6} -2.82979 q^{7} -3.69754 q^{8} +6.54852 q^{9} +9.86669 q^{10} +2.48715 q^{11} -11.0226 q^{12} +4.60177 q^{13} +6.67680 q^{14} +12.9219 q^{15} +1.59003 q^{16} -5.70168 q^{17} -15.4510 q^{18} +0.542873 q^{19} -14.9167 q^{20} +8.74424 q^{21} -5.86836 q^{22} +6.30681 q^{23} +11.4256 q^{24} +12.4869 q^{25} -10.8577 q^{26} -10.9652 q^{27} -10.0941 q^{28} -4.16997 q^{29} -30.4887 q^{30} +5.03942 q^{31} +3.64346 q^{32} -7.68547 q^{33} +13.4529 q^{34} +11.8334 q^{35} +23.3593 q^{36} -4.02373 q^{37} -1.28089 q^{38} -14.2198 q^{39} +15.4621 q^{40} -2.04830 q^{41} -20.6318 q^{42} +7.11938 q^{43} +8.87193 q^{44} -27.3842 q^{45} -14.8807 q^{46} -4.23012 q^{47} -4.91329 q^{48} +1.00770 q^{49} -29.4625 q^{50} +17.6186 q^{51} +16.4150 q^{52} -8.60588 q^{53} +25.8720 q^{54} -10.4006 q^{55} +10.4632 q^{56} -1.67751 q^{57} +9.83893 q^{58} +2.15472 q^{59} +46.0936 q^{60} -8.53383 q^{61} -11.8904 q^{62} -18.5309 q^{63} -11.7767 q^{64} -19.2434 q^{65} +18.1336 q^{66} -3.63078 q^{67} -20.3385 q^{68} -19.4885 q^{69} -27.9206 q^{70} +7.43976 q^{71} -24.2134 q^{72} +10.8830 q^{73} +9.49386 q^{74} -38.5854 q^{75} +1.93648 q^{76} -7.03811 q^{77} +33.5511 q^{78} +4.82054 q^{79} -6.64907 q^{80} +14.2376 q^{81} +4.83290 q^{82} +11.0018 q^{83} +31.1916 q^{84} +23.8429 q^{85} -16.7980 q^{86} +12.8855 q^{87} -9.19633 q^{88} +2.44024 q^{89} +64.6122 q^{90} -13.0220 q^{91} +22.4970 q^{92} -15.5721 q^{93} +9.98085 q^{94} -2.27015 q^{95} -11.2585 q^{96} +5.35684 q^{97} -2.37763 q^{98} +16.2872 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.35947 −1.66840 −0.834199 0.551463i \(-0.814070\pi\)
−0.834199 + 0.551463i \(0.814070\pi\)
\(3\) −3.09007 −1.78405 −0.892026 0.451984i \(-0.850716\pi\)
−0.892026 + 0.451984i \(0.850716\pi\)
\(4\) 3.56710 1.78355
\(5\) −4.18174 −1.87013 −0.935065 0.354477i \(-0.884659\pi\)
−0.935065 + 0.354477i \(0.884659\pi\)
\(6\) 7.29093 2.97651
\(7\) −2.82979 −1.06956 −0.534779 0.844992i \(-0.679605\pi\)
−0.534779 + 0.844992i \(0.679605\pi\)
\(8\) −3.69754 −1.30728
\(9\) 6.54852 2.18284
\(10\) 9.86669 3.12012
\(11\) 2.48715 0.749904 0.374952 0.927044i \(-0.377659\pi\)
0.374952 + 0.927044i \(0.377659\pi\)
\(12\) −11.0226 −3.18195
\(13\) 4.60177 1.27630 0.638150 0.769912i \(-0.279700\pi\)
0.638150 + 0.769912i \(0.279700\pi\)
\(14\) 6.67680 1.78445
\(15\) 12.9219 3.33641
\(16\) 1.59003 0.397506
\(17\) −5.70168 −1.38286 −0.691430 0.722444i \(-0.743019\pi\)
−0.691430 + 0.722444i \(0.743019\pi\)
\(18\) −15.4510 −3.64185
\(19\) 0.542873 0.124544 0.0622718 0.998059i \(-0.480165\pi\)
0.0622718 + 0.998059i \(0.480165\pi\)
\(20\) −14.9167 −3.33547
\(21\) 8.74424 1.90815
\(22\) −5.86836 −1.25114
\(23\) 6.30681 1.31506 0.657530 0.753428i \(-0.271601\pi\)
0.657530 + 0.753428i \(0.271601\pi\)
\(24\) 11.4256 2.33225
\(25\) 12.4869 2.49738
\(26\) −10.8577 −2.12938
\(27\) −10.9652 −2.11025
\(28\) −10.0941 −1.90761
\(29\) −4.16997 −0.774344 −0.387172 0.922007i \(-0.626548\pi\)
−0.387172 + 0.922007i \(0.626548\pi\)
\(30\) −30.4887 −5.56646
\(31\) 5.03942 0.905106 0.452553 0.891737i \(-0.350513\pi\)
0.452553 + 0.891737i \(0.350513\pi\)
\(32\) 3.64346 0.644078
\(33\) −7.68547 −1.33787
\(34\) 13.4529 2.30716
\(35\) 11.8334 2.00021
\(36\) 23.3593 3.89321
\(37\) −4.02373 −0.661496 −0.330748 0.943719i \(-0.607301\pi\)
−0.330748 + 0.943719i \(0.607301\pi\)
\(38\) −1.28089 −0.207788
\(39\) −14.2198 −2.27699
\(40\) 15.4621 2.44478
\(41\) −2.04830 −0.319890 −0.159945 0.987126i \(-0.551132\pi\)
−0.159945 + 0.987126i \(0.551132\pi\)
\(42\) −20.6318 −3.18355
\(43\) 7.11938 1.08570 0.542848 0.839831i \(-0.317346\pi\)
0.542848 + 0.839831i \(0.317346\pi\)
\(44\) 8.87193 1.33749
\(45\) −27.3842 −4.08219
\(46\) −14.8807 −2.19404
\(47\) −4.23012 −0.617027 −0.308513 0.951220i \(-0.599831\pi\)
−0.308513 + 0.951220i \(0.599831\pi\)
\(48\) −4.91329 −0.709172
\(49\) 1.00770 0.143956
\(50\) −29.4625 −4.16663
\(51\) 17.6186 2.46709
\(52\) 16.4150 2.27635
\(53\) −8.60588 −1.18211 −0.591054 0.806632i \(-0.701288\pi\)
−0.591054 + 0.806632i \(0.701288\pi\)
\(54\) 25.8720 3.52073
\(55\) −10.4006 −1.40242
\(56\) 10.4632 1.39821
\(57\) −1.67751 −0.222192
\(58\) 9.83893 1.29191
\(59\) 2.15472 0.280520 0.140260 0.990115i \(-0.455206\pi\)
0.140260 + 0.990115i \(0.455206\pi\)
\(60\) 46.0936 5.95066
\(61\) −8.53383 −1.09264 −0.546322 0.837575i \(-0.683973\pi\)
−0.546322 + 0.837575i \(0.683973\pi\)
\(62\) −11.8904 −1.51008
\(63\) −18.5309 −2.33468
\(64\) −11.7767 −1.47209
\(65\) −19.2434 −2.38685
\(66\) 18.1336 2.23210
\(67\) −3.63078 −0.443570 −0.221785 0.975096i \(-0.571188\pi\)
−0.221785 + 0.975096i \(0.571188\pi\)
\(68\) −20.3385 −2.46640
\(69\) −19.4885 −2.34614
\(70\) −27.9206 −3.33715
\(71\) 7.43976 0.882938 0.441469 0.897277i \(-0.354458\pi\)
0.441469 + 0.897277i \(0.354458\pi\)
\(72\) −24.2134 −2.85358
\(73\) 10.8830 1.27376 0.636881 0.770962i \(-0.280224\pi\)
0.636881 + 0.770962i \(0.280224\pi\)
\(74\) 9.49386 1.10364
\(75\) −38.5854 −4.45546
\(76\) 1.93648 0.222130
\(77\) −7.03811 −0.802067
\(78\) 33.5511 3.79892
\(79\) 4.82054 0.542353 0.271177 0.962530i \(-0.412587\pi\)
0.271177 + 0.962530i \(0.412587\pi\)
\(80\) −6.64907 −0.743388
\(81\) 14.2376 1.58195
\(82\) 4.83290 0.533704
\(83\) 11.0018 1.20760 0.603801 0.797135i \(-0.293652\pi\)
0.603801 + 0.797135i \(0.293652\pi\)
\(84\) 31.1916 3.40328
\(85\) 23.8429 2.58613
\(86\) −16.7980 −1.81137
\(87\) 12.8855 1.38147
\(88\) −9.19633 −0.980333
\(89\) 2.44024 0.258665 0.129332 0.991601i \(-0.458717\pi\)
0.129332 + 0.991601i \(0.458717\pi\)
\(90\) 64.6122 6.81072
\(91\) −13.0220 −1.36508
\(92\) 22.4970 2.34548
\(93\) −15.5721 −1.61476
\(94\) 9.98085 1.02945
\(95\) −2.27015 −0.232913
\(96\) −11.2585 −1.14907
\(97\) 5.35684 0.543904 0.271952 0.962311i \(-0.412331\pi\)
0.271952 + 0.962311i \(0.412331\pi\)
\(98\) −2.37763 −0.240177
\(99\) 16.2872 1.63692
\(100\) 44.5421 4.45421
\(101\) 16.3478 1.62666 0.813332 0.581800i \(-0.197651\pi\)
0.813332 + 0.581800i \(0.197651\pi\)
\(102\) −41.5705 −4.11609
\(103\) −11.8061 −1.16329 −0.581645 0.813443i \(-0.697591\pi\)
−0.581645 + 0.813443i \(0.697591\pi\)
\(104\) −17.0152 −1.66848
\(105\) −36.5661 −3.56848
\(106\) 20.3053 1.97223
\(107\) −12.0397 −1.16392 −0.581959 0.813218i \(-0.697714\pi\)
−0.581959 + 0.813218i \(0.697714\pi\)
\(108\) −39.1139 −3.76374
\(109\) 1.39459 0.133577 0.0667886 0.997767i \(-0.478725\pi\)
0.0667886 + 0.997767i \(0.478725\pi\)
\(110\) 24.5399 2.33979
\(111\) 12.4336 1.18014
\(112\) −4.49943 −0.425156
\(113\) 4.16686 0.391985 0.195992 0.980605i \(-0.437207\pi\)
0.195992 + 0.980605i \(0.437207\pi\)
\(114\) 3.95805 0.370705
\(115\) −26.3734 −2.45933
\(116\) −14.8747 −1.38108
\(117\) 30.1348 2.78596
\(118\) −5.08399 −0.468019
\(119\) 16.1345 1.47905
\(120\) −47.7790 −4.36161
\(121\) −4.81408 −0.437644
\(122\) 20.1353 1.82297
\(123\) 6.32938 0.570701
\(124\) 17.9761 1.61430
\(125\) −31.3083 −2.80030
\(126\) 43.7232 3.89517
\(127\) 8.57110 0.760562 0.380281 0.924871i \(-0.375827\pi\)
0.380281 + 0.924871i \(0.375827\pi\)
\(128\) 20.4998 1.81195
\(129\) −21.9994 −1.93694
\(130\) 45.4042 3.98221
\(131\) −19.4849 −1.70241 −0.851204 0.524836i \(-0.824127\pi\)
−0.851204 + 0.524836i \(0.824127\pi\)
\(132\) −27.4149 −2.38616
\(133\) −1.53621 −0.133207
\(134\) 8.56672 0.740052
\(135\) 45.8535 3.94644
\(136\) 21.0822 1.80778
\(137\) 16.6524 1.42271 0.711357 0.702830i \(-0.248081\pi\)
0.711357 + 0.702830i \(0.248081\pi\)
\(138\) 45.9825 3.91429
\(139\) −7.41300 −0.628763 −0.314381 0.949297i \(-0.601797\pi\)
−0.314381 + 0.949297i \(0.601797\pi\)
\(140\) 42.2111 3.56749
\(141\) 13.0714 1.10081
\(142\) −17.5539 −1.47309
\(143\) 11.4453 0.957103
\(144\) 10.4123 0.867693
\(145\) 17.4377 1.44812
\(146\) −25.6782 −2.12514
\(147\) −3.11385 −0.256826
\(148\) −14.3530 −1.17981
\(149\) −20.4476 −1.67513 −0.837566 0.546335i \(-0.816022\pi\)
−0.837566 + 0.546335i \(0.816022\pi\)
\(150\) 91.0412 7.43348
\(151\) 15.4794 1.25970 0.629849 0.776717i \(-0.283117\pi\)
0.629849 + 0.776717i \(0.283117\pi\)
\(152\) −2.00729 −0.162813
\(153\) −37.3375 −3.01856
\(154\) 16.6062 1.33817
\(155\) −21.0735 −1.69267
\(156\) −50.7234 −4.06112
\(157\) −16.5044 −1.31719 −0.658596 0.752497i \(-0.728849\pi\)
−0.658596 + 0.752497i \(0.728849\pi\)
\(158\) −11.3739 −0.904861
\(159\) 26.5928 2.10894
\(160\) −15.2360 −1.20451
\(161\) −17.8469 −1.40653
\(162\) −33.5931 −2.63933
\(163\) 2.84622 0.222933 0.111467 0.993768i \(-0.464445\pi\)
0.111467 + 0.993768i \(0.464445\pi\)
\(164\) −7.30649 −0.570541
\(165\) 32.1386 2.50199
\(166\) −25.9584 −2.01476
\(167\) 11.3203 0.875995 0.437997 0.898976i \(-0.355688\pi\)
0.437997 + 0.898976i \(0.355688\pi\)
\(168\) −32.3321 −2.49448
\(169\) 8.17626 0.628943
\(170\) −56.2567 −4.31469
\(171\) 3.55501 0.271859
\(172\) 25.3956 1.93640
\(173\) −14.3600 −1.09177 −0.545884 0.837861i \(-0.683806\pi\)
−0.545884 + 0.837861i \(0.683806\pi\)
\(174\) −30.4030 −2.30484
\(175\) −35.3353 −2.67110
\(176\) 3.95463 0.298092
\(177\) −6.65822 −0.500462
\(178\) −5.75768 −0.431556
\(179\) 9.00995 0.673435 0.336718 0.941606i \(-0.390683\pi\)
0.336718 + 0.941606i \(0.390683\pi\)
\(180\) −97.6823 −7.28081
\(181\) −24.6614 −1.83306 −0.916532 0.399961i \(-0.869024\pi\)
−0.916532 + 0.399961i \(0.869024\pi\)
\(182\) 30.7251 2.27749
\(183\) 26.3701 1.94933
\(184\) −23.3197 −1.71915
\(185\) 16.8262 1.23708
\(186\) 36.7420 2.69406
\(187\) −14.1809 −1.03701
\(188\) −15.0893 −1.10050
\(189\) 31.0291 2.25704
\(190\) 5.35636 0.388591
\(191\) 0.990564 0.0716747 0.0358374 0.999358i \(-0.488590\pi\)
0.0358374 + 0.999358i \(0.488590\pi\)
\(192\) 36.3908 2.62628
\(193\) −22.7042 −1.63428 −0.817141 0.576438i \(-0.804442\pi\)
−0.817141 + 0.576438i \(0.804442\pi\)
\(194\) −12.6393 −0.907449
\(195\) 59.4634 4.25826
\(196\) 3.59455 0.256754
\(197\) 10.6866 0.761389 0.380694 0.924701i \(-0.375685\pi\)
0.380694 + 0.924701i \(0.375685\pi\)
\(198\) −38.4291 −2.73104
\(199\) −3.22512 −0.228623 −0.114311 0.993445i \(-0.536466\pi\)
−0.114311 + 0.993445i \(0.536466\pi\)
\(200\) −46.1709 −3.26477
\(201\) 11.2194 0.791352
\(202\) −38.5721 −2.71392
\(203\) 11.8001 0.828207
\(204\) 62.8473 4.40019
\(205\) 8.56544 0.598236
\(206\) 27.8562 1.94083
\(207\) 41.3003 2.87057
\(208\) 7.31693 0.507337
\(209\) 1.35021 0.0933958
\(210\) 86.2766 5.95365
\(211\) −5.91177 −0.406983 −0.203491 0.979077i \(-0.565229\pi\)
−0.203491 + 0.979077i \(0.565229\pi\)
\(212\) −30.6981 −2.10835
\(213\) −22.9894 −1.57521
\(214\) 28.4072 1.94188
\(215\) −29.7714 −2.03039
\(216\) 40.5441 2.75868
\(217\) −14.2605 −0.968065
\(218\) −3.29049 −0.222860
\(219\) −33.6293 −2.27246
\(220\) −37.1001 −2.50129
\(221\) −26.2378 −1.76494
\(222\) −29.3367 −1.96895
\(223\) −12.2358 −0.819367 −0.409684 0.912228i \(-0.634361\pi\)
−0.409684 + 0.912228i \(0.634361\pi\)
\(224\) −10.3102 −0.688880
\(225\) 81.7709 5.45139
\(226\) −9.83157 −0.653987
\(227\) 0.447253 0.0296852 0.0148426 0.999890i \(-0.495275\pi\)
0.0148426 + 0.999890i \(0.495275\pi\)
\(228\) −5.98387 −0.396291
\(229\) −11.3140 −0.747652 −0.373826 0.927499i \(-0.621954\pi\)
−0.373826 + 0.927499i \(0.621954\pi\)
\(230\) 62.2273 4.10315
\(231\) 21.7482 1.43093
\(232\) 15.4186 1.01228
\(233\) 19.0486 1.24792 0.623958 0.781458i \(-0.285524\pi\)
0.623958 + 0.781458i \(0.285524\pi\)
\(234\) −71.1021 −4.64809
\(235\) 17.6893 1.15392
\(236\) 7.68609 0.500322
\(237\) −14.8958 −0.967586
\(238\) −38.0690 −2.46764
\(239\) 4.17495 0.270055 0.135027 0.990842i \(-0.456888\pi\)
0.135027 + 0.990842i \(0.456888\pi\)
\(240\) 20.5461 1.32624
\(241\) 4.77764 0.307755 0.153878 0.988090i \(-0.450824\pi\)
0.153878 + 0.988090i \(0.450824\pi\)
\(242\) 11.3587 0.730164
\(243\) −11.0995 −0.712036
\(244\) −30.4411 −1.94879
\(245\) −4.21392 −0.269217
\(246\) −14.9340 −0.952156
\(247\) 2.49817 0.158955
\(248\) −18.6334 −1.18322
\(249\) −33.9962 −2.15442
\(250\) 73.8711 4.67202
\(251\) 4.20802 0.265608 0.132804 0.991142i \(-0.457602\pi\)
0.132804 + 0.991142i \(0.457602\pi\)
\(252\) −66.1017 −4.16402
\(253\) 15.6860 0.986169
\(254\) −20.2233 −1.26892
\(255\) −73.6762 −4.61378
\(256\) −24.8154 −1.55096
\(257\) 19.5268 1.21805 0.609023 0.793152i \(-0.291562\pi\)
0.609023 + 0.793152i \(0.291562\pi\)
\(258\) 51.9069 3.23158
\(259\) 11.3863 0.707509
\(260\) −68.6431 −4.25707
\(261\) −27.3071 −1.69027
\(262\) 45.9742 2.84029
\(263\) 5.05834 0.311910 0.155955 0.987764i \(-0.450154\pi\)
0.155955 + 0.987764i \(0.450154\pi\)
\(264\) 28.4173 1.74896
\(265\) 35.9875 2.21070
\(266\) 3.62465 0.222242
\(267\) −7.54051 −0.461472
\(268\) −12.9514 −0.791131
\(269\) −16.9375 −1.03270 −0.516349 0.856378i \(-0.672709\pi\)
−0.516349 + 0.856378i \(0.672709\pi\)
\(270\) −108.190 −6.58423
\(271\) −11.5325 −0.700547 −0.350274 0.936647i \(-0.613911\pi\)
−0.350274 + 0.936647i \(0.613911\pi\)
\(272\) −9.06581 −0.549695
\(273\) 40.2389 2.43537
\(274\) −39.2910 −2.37365
\(275\) 31.0569 1.87280
\(276\) −69.5174 −4.18446
\(277\) −14.9882 −0.900556 −0.450278 0.892889i \(-0.648675\pi\)
−0.450278 + 0.892889i \(0.648675\pi\)
\(278\) 17.4908 1.04903
\(279\) 33.0007 1.97570
\(280\) −43.7545 −2.61483
\(281\) −25.9673 −1.54908 −0.774541 0.632524i \(-0.782019\pi\)
−0.774541 + 0.632524i \(0.782019\pi\)
\(282\) −30.8415 −1.83659
\(283\) −27.6595 −1.64419 −0.822094 0.569351i \(-0.807194\pi\)
−0.822094 + 0.569351i \(0.807194\pi\)
\(284\) 26.5384 1.57477
\(285\) 7.01492 0.415528
\(286\) −27.0048 −1.59683
\(287\) 5.79625 0.342142
\(288\) 23.8593 1.40592
\(289\) 15.5091 0.912300
\(290\) −41.1438 −2.41605
\(291\) −16.5530 −0.970353
\(292\) 38.8209 2.27182
\(293\) 0.415058 0.0242480 0.0121240 0.999927i \(-0.496141\pi\)
0.0121240 + 0.999927i \(0.496141\pi\)
\(294\) 7.34703 0.428488
\(295\) −9.01045 −0.524609
\(296\) 14.8779 0.864759
\(297\) −27.2720 −1.58248
\(298\) 48.2455 2.79479
\(299\) 29.0225 1.67841
\(300\) −137.638 −7.94655
\(301\) −20.1463 −1.16122
\(302\) −36.5233 −2.10168
\(303\) −50.5157 −2.90205
\(304\) 0.863182 0.0495069
\(305\) 35.6862 2.04339
\(306\) 88.0969 5.03616
\(307\) −5.03206 −0.287195 −0.143597 0.989636i \(-0.545867\pi\)
−0.143597 + 0.989636i \(0.545867\pi\)
\(308\) −25.1057 −1.43053
\(309\) 36.4817 2.07537
\(310\) 49.7224 2.82404
\(311\) −7.79527 −0.442029 −0.221015 0.975270i \(-0.570937\pi\)
−0.221015 + 0.975270i \(0.570937\pi\)
\(312\) 52.5782 2.97665
\(313\) −13.4305 −0.759134 −0.379567 0.925164i \(-0.623927\pi\)
−0.379567 + 0.925164i \(0.623927\pi\)
\(314\) 38.9416 2.19760
\(315\) 77.4914 4.36615
\(316\) 17.1954 0.967315
\(317\) −9.15464 −0.514176 −0.257088 0.966388i \(-0.582763\pi\)
−0.257088 + 0.966388i \(0.582763\pi\)
\(318\) −62.7448 −3.51856
\(319\) −10.3713 −0.580684
\(320\) 49.2470 2.75299
\(321\) 37.2034 2.07649
\(322\) 42.1093 2.34666
\(323\) −3.09529 −0.172226
\(324\) 50.7869 2.82149
\(325\) 57.4619 3.18741
\(326\) −6.71557 −0.371941
\(327\) −4.30937 −0.238309
\(328\) 7.57366 0.418185
\(329\) 11.9703 0.659947
\(330\) −75.8301 −4.17431
\(331\) 9.02794 0.496221 0.248110 0.968732i \(-0.420190\pi\)
0.248110 + 0.968732i \(0.420190\pi\)
\(332\) 39.2445 2.15382
\(333\) −26.3495 −1.44394
\(334\) −26.7100 −1.46151
\(335\) 15.1830 0.829534
\(336\) 13.9036 0.758501
\(337\) −29.1258 −1.58658 −0.793292 0.608841i \(-0.791635\pi\)
−0.793292 + 0.608841i \(0.791635\pi\)
\(338\) −19.2917 −1.04933
\(339\) −12.8759 −0.699321
\(340\) 85.0501 4.61249
\(341\) 12.5338 0.678743
\(342\) −8.38796 −0.453569
\(343\) 16.9569 0.915589
\(344\) −26.3242 −1.41931
\(345\) 81.4956 4.38758
\(346\) 33.8819 1.82150
\(347\) −2.77201 −0.148810 −0.0744048 0.997228i \(-0.523706\pi\)
−0.0744048 + 0.997228i \(0.523706\pi\)
\(348\) 45.9639 2.46392
\(349\) 1.73834 0.0930509 0.0465255 0.998917i \(-0.485185\pi\)
0.0465255 + 0.998917i \(0.485185\pi\)
\(350\) 83.3727 4.45646
\(351\) −50.4592 −2.69331
\(352\) 9.06183 0.482997
\(353\) −25.4703 −1.35565 −0.677825 0.735224i \(-0.737077\pi\)
−0.677825 + 0.735224i \(0.737077\pi\)
\(354\) 15.7099 0.834970
\(355\) −31.1111 −1.65121
\(356\) 8.70459 0.461342
\(357\) −49.8568 −2.63870
\(358\) −21.2587 −1.12356
\(359\) −32.6283 −1.72205 −0.861027 0.508559i \(-0.830178\pi\)
−0.861027 + 0.508559i \(0.830178\pi\)
\(360\) 101.254 5.33656
\(361\) −18.7053 −0.984489
\(362\) 58.1878 3.05828
\(363\) 14.8758 0.780779
\(364\) −46.4509 −2.43469
\(365\) −45.5099 −2.38210
\(366\) −62.2195 −3.25227
\(367\) 6.81294 0.355632 0.177816 0.984064i \(-0.443097\pi\)
0.177816 + 0.984064i \(0.443097\pi\)
\(368\) 10.0280 0.522745
\(369\) −13.4133 −0.698270
\(370\) −39.7008 −2.06395
\(371\) 24.3528 1.26433
\(372\) −55.5475 −2.88000
\(373\) 10.6694 0.552440 0.276220 0.961095i \(-0.410918\pi\)
0.276220 + 0.961095i \(0.410918\pi\)
\(374\) 33.4595 1.73015
\(375\) 96.7449 4.99588
\(376\) 15.6410 0.806625
\(377\) −19.1892 −0.988296
\(378\) −73.2123 −3.76563
\(379\) 22.2741 1.14415 0.572073 0.820203i \(-0.306139\pi\)
0.572073 + 0.820203i \(0.306139\pi\)
\(380\) −8.09787 −0.415412
\(381\) −26.4853 −1.35688
\(382\) −2.33721 −0.119582
\(383\) −2.85301 −0.145782 −0.0728911 0.997340i \(-0.523223\pi\)
−0.0728911 + 0.997340i \(0.523223\pi\)
\(384\) −63.3459 −3.23261
\(385\) 29.4315 1.49997
\(386\) 53.5698 2.72663
\(387\) 46.6214 2.36990
\(388\) 19.1084 0.970082
\(389\) −12.7913 −0.648545 −0.324273 0.945964i \(-0.605119\pi\)
−0.324273 + 0.945964i \(0.605119\pi\)
\(390\) −140.302 −7.10447
\(391\) −35.9594 −1.81854
\(392\) −3.72599 −0.188191
\(393\) 60.2098 3.03718
\(394\) −25.2147 −1.27030
\(395\) −20.1582 −1.01427
\(396\) 58.0980 2.91953
\(397\) 0.584024 0.0293113 0.0146557 0.999893i \(-0.495335\pi\)
0.0146557 + 0.999893i \(0.495335\pi\)
\(398\) 7.60958 0.381434
\(399\) 4.74701 0.237648
\(400\) 19.8545 0.992726
\(401\) 1.23150 0.0614983 0.0307492 0.999527i \(-0.490211\pi\)
0.0307492 + 0.999527i \(0.490211\pi\)
\(402\) −26.4717 −1.32029
\(403\) 23.1902 1.15519
\(404\) 58.3142 2.90124
\(405\) −59.5378 −2.95845
\(406\) −27.8421 −1.38178
\(407\) −10.0076 −0.496059
\(408\) −65.1453 −3.22517
\(409\) −1.18813 −0.0587492 −0.0293746 0.999568i \(-0.509352\pi\)
−0.0293746 + 0.999568i \(0.509352\pi\)
\(410\) −20.2099 −0.998096
\(411\) −51.4572 −2.53820
\(412\) −42.1136 −2.07479
\(413\) −6.09739 −0.300033
\(414\) −97.4468 −4.78925
\(415\) −46.0065 −2.25837
\(416\) 16.7663 0.822038
\(417\) 22.9067 1.12175
\(418\) −3.18577 −0.155821
\(419\) −2.74974 −0.134333 −0.0671667 0.997742i \(-0.521396\pi\)
−0.0671667 + 0.997742i \(0.521396\pi\)
\(420\) −130.435 −6.36458
\(421\) 26.4727 1.29020 0.645100 0.764098i \(-0.276816\pi\)
0.645100 + 0.764098i \(0.276816\pi\)
\(422\) 13.9486 0.679009
\(423\) −27.7011 −1.34687
\(424\) 31.8206 1.54534
\(425\) −71.1964 −3.45353
\(426\) 54.2428 2.62807
\(427\) 24.1489 1.16865
\(428\) −42.9467 −2.07591
\(429\) −35.3667 −1.70752
\(430\) 70.2447 3.38750
\(431\) −14.7022 −0.708179 −0.354090 0.935212i \(-0.615209\pi\)
−0.354090 + 0.935212i \(0.615209\pi\)
\(432\) −17.4349 −0.838837
\(433\) 34.4192 1.65408 0.827040 0.562143i \(-0.190023\pi\)
0.827040 + 0.562143i \(0.190023\pi\)
\(434\) 33.6472 1.61512
\(435\) −53.8837 −2.58353
\(436\) 4.97464 0.238242
\(437\) 3.42380 0.163782
\(438\) 79.3473 3.79136
\(439\) 17.3151 0.826403 0.413202 0.910640i \(-0.364411\pi\)
0.413202 + 0.910640i \(0.364411\pi\)
\(440\) 38.4566 1.83335
\(441\) 6.59891 0.314234
\(442\) 61.9073 2.94463
\(443\) 19.9059 0.945757 0.472878 0.881128i \(-0.343215\pi\)
0.472878 + 0.881128i \(0.343215\pi\)
\(444\) 44.3519 2.10485
\(445\) −10.2044 −0.483737
\(446\) 28.8699 1.36703
\(447\) 63.1845 2.98852
\(448\) 33.3255 1.57448
\(449\) −9.26847 −0.437406 −0.218703 0.975791i \(-0.570183\pi\)
−0.218703 + 0.975791i \(0.570183\pi\)
\(450\) −192.936 −9.09509
\(451\) −5.09443 −0.239887
\(452\) 14.8636 0.699125
\(453\) −47.8325 −2.24737
\(454\) −1.05528 −0.0495268
\(455\) 54.4547 2.55287
\(456\) 6.20267 0.290467
\(457\) −9.70114 −0.453800 −0.226900 0.973918i \(-0.572859\pi\)
−0.226900 + 0.973918i \(0.572859\pi\)
\(458\) 26.6951 1.24738
\(459\) 62.5199 2.91818
\(460\) −94.0767 −4.38635
\(461\) 25.5277 1.18895 0.594473 0.804116i \(-0.297361\pi\)
0.594473 + 0.804116i \(0.297361\pi\)
\(462\) −51.3143 −2.38736
\(463\) −38.9438 −1.80987 −0.904937 0.425546i \(-0.860082\pi\)
−0.904937 + 0.425546i \(0.860082\pi\)
\(464\) −6.63036 −0.307807
\(465\) 65.1186 3.01980
\(466\) −44.9447 −2.08202
\(467\) −14.9435 −0.691501 −0.345751 0.938326i \(-0.612376\pi\)
−0.345751 + 0.938326i \(0.612376\pi\)
\(468\) 107.494 4.96891
\(469\) 10.2743 0.474425
\(470\) −41.7373 −1.92520
\(471\) 50.9996 2.34994
\(472\) −7.96714 −0.366717
\(473\) 17.7070 0.814168
\(474\) 35.1462 1.61432
\(475\) 6.77881 0.311033
\(476\) 57.5535 2.63796
\(477\) −56.3558 −2.58035
\(478\) −9.85067 −0.450559
\(479\) 3.22823 0.147502 0.0737508 0.997277i \(-0.476503\pi\)
0.0737508 + 0.997277i \(0.476503\pi\)
\(480\) 47.0802 2.14891
\(481\) −18.5162 −0.844268
\(482\) −11.2727 −0.513458
\(483\) 55.1482 2.50933
\(484\) −17.1723 −0.780560
\(485\) −22.4009 −1.01717
\(486\) 26.1890 1.18796
\(487\) −7.84073 −0.355297 −0.177649 0.984094i \(-0.556849\pi\)
−0.177649 + 0.984094i \(0.556849\pi\)
\(488\) 31.5541 1.42839
\(489\) −8.79501 −0.397724
\(490\) 9.94261 0.449161
\(491\) 18.3532 0.828269 0.414134 0.910216i \(-0.364084\pi\)
0.414134 + 0.910216i \(0.364084\pi\)
\(492\) 22.5776 1.01787
\(493\) 23.7758 1.07081
\(494\) −5.89437 −0.265200
\(495\) −68.1086 −3.06125
\(496\) 8.01280 0.359785
\(497\) −21.0529 −0.944354
\(498\) 80.2131 3.59444
\(499\) 26.3880 1.18129 0.590644 0.806932i \(-0.298874\pi\)
0.590644 + 0.806932i \(0.298874\pi\)
\(500\) −111.680 −4.99448
\(501\) −34.9806 −1.56282
\(502\) −9.92869 −0.443139
\(503\) −5.59404 −0.249426 −0.124713 0.992193i \(-0.539801\pi\)
−0.124713 + 0.992193i \(0.539801\pi\)
\(504\) 68.5188 3.05207
\(505\) −68.3620 −3.04207
\(506\) −37.0106 −1.64532
\(507\) −25.2652 −1.12207
\(508\) 30.5740 1.35650
\(509\) 38.7081 1.71571 0.857853 0.513895i \(-0.171798\pi\)
0.857853 + 0.513895i \(0.171798\pi\)
\(510\) 173.837 7.69763
\(511\) −30.7966 −1.36236
\(512\) 17.5515 0.775676
\(513\) −5.95270 −0.262818
\(514\) −46.0729 −2.03219
\(515\) 49.3700 2.17550
\(516\) −78.4741 −3.45463
\(517\) −10.5210 −0.462711
\(518\) −26.8656 −1.18041
\(519\) 44.3733 1.94777
\(520\) 71.1531 3.12027
\(521\) 5.16331 0.226209 0.113104 0.993583i \(-0.463921\pi\)
0.113104 + 0.993583i \(0.463921\pi\)
\(522\) 64.4304 2.82004
\(523\) 32.8624 1.43697 0.718487 0.695541i \(-0.244835\pi\)
0.718487 + 0.695541i \(0.244835\pi\)
\(524\) −69.5048 −3.03633
\(525\) 109.189 4.76538
\(526\) −11.9350 −0.520391
\(527\) −28.7331 −1.25163
\(528\) −12.2201 −0.531811
\(529\) 16.7758 0.729384
\(530\) −84.9115 −3.68832
\(531\) 14.1102 0.612330
\(532\) −5.47984 −0.237581
\(533\) −9.42579 −0.408276
\(534\) 17.7916 0.769918
\(535\) 50.3467 2.17668
\(536\) 13.4249 0.579869
\(537\) −27.8413 −1.20144
\(538\) 39.9636 1.72295
\(539\) 2.50629 0.107954
\(540\) 163.564 7.03868
\(541\) 2.08381 0.0895898 0.0447949 0.998996i \(-0.485737\pi\)
0.0447949 + 0.998996i \(0.485737\pi\)
\(542\) 27.2105 1.16879
\(543\) 76.2053 3.27028
\(544\) −20.7738 −0.890670
\(545\) −5.83180 −0.249807
\(546\) −94.9426 −4.06317
\(547\) −1.00000 −0.0427569
\(548\) 59.4010 2.53749
\(549\) −55.8839 −2.38507
\(550\) −73.2778 −3.12457
\(551\) −2.26376 −0.0964396
\(552\) 72.0593 3.06705
\(553\) −13.6411 −0.580079
\(554\) 35.3643 1.50249
\(555\) −51.9940 −2.20702
\(556\) −26.4430 −1.12143
\(557\) −29.8940 −1.26665 −0.633324 0.773887i \(-0.718310\pi\)
−0.633324 + 0.773887i \(0.718310\pi\)
\(558\) −77.8643 −3.29626
\(559\) 32.7617 1.38567
\(560\) 18.8154 0.795098
\(561\) 43.8200 1.85008
\(562\) 61.2692 2.58449
\(563\) 41.3819 1.74404 0.872020 0.489470i \(-0.162810\pi\)
0.872020 + 0.489470i \(0.162810\pi\)
\(564\) 46.6269 1.96335
\(565\) −17.4247 −0.733062
\(566\) 65.2619 2.74316
\(567\) −40.2893 −1.69199
\(568\) −27.5088 −1.15424
\(569\) 34.2607 1.43628 0.718141 0.695898i \(-0.244993\pi\)
0.718141 + 0.695898i \(0.244993\pi\)
\(570\) −16.5515 −0.693266
\(571\) −22.9912 −0.962151 −0.481075 0.876679i \(-0.659754\pi\)
−0.481075 + 0.876679i \(0.659754\pi\)
\(572\) 40.8265 1.70704
\(573\) −3.06091 −0.127871
\(574\) −13.6761 −0.570828
\(575\) 78.7526 3.28421
\(576\) −77.1199 −3.21333
\(577\) 13.6464 0.568109 0.284054 0.958808i \(-0.408320\pi\)
0.284054 + 0.958808i \(0.408320\pi\)
\(578\) −36.5933 −1.52208
\(579\) 70.1575 2.91564
\(580\) 62.2022 2.58280
\(581\) −31.1327 −1.29160
\(582\) 39.0563 1.61894
\(583\) −21.4041 −0.886468
\(584\) −40.2404 −1.66516
\(585\) −126.016 −5.21011
\(586\) −0.979318 −0.0404553
\(587\) −7.33098 −0.302582 −0.151291 0.988489i \(-0.548343\pi\)
−0.151291 + 0.988489i \(0.548343\pi\)
\(588\) −11.1074 −0.458062
\(589\) 2.73576 0.112725
\(590\) 21.2599 0.875256
\(591\) −33.0223 −1.35836
\(592\) −6.39782 −0.262949
\(593\) 15.0502 0.618037 0.309018 0.951056i \(-0.400000\pi\)
0.309018 + 0.951056i \(0.400000\pi\)
\(594\) 64.3476 2.64021
\(595\) −67.4703 −2.76601
\(596\) −72.9387 −2.98769
\(597\) 9.96584 0.407875
\(598\) −68.4777 −2.80026
\(599\) 26.1936 1.07024 0.535122 0.844775i \(-0.320266\pi\)
0.535122 + 0.844775i \(0.320266\pi\)
\(600\) 142.671 5.82452
\(601\) −11.7023 −0.477345 −0.238673 0.971100i \(-0.576712\pi\)
−0.238673 + 0.971100i \(0.576712\pi\)
\(602\) 47.5347 1.93737
\(603\) −23.7762 −0.968243
\(604\) 55.2168 2.24674
\(605\) 20.1312 0.818450
\(606\) 119.190 4.84178
\(607\) 5.53127 0.224507 0.112254 0.993680i \(-0.464193\pi\)
0.112254 + 0.993680i \(0.464193\pi\)
\(608\) 1.97793 0.0802158
\(609\) −36.4632 −1.47756
\(610\) −84.2006 −3.40918
\(611\) −19.4660 −0.787512
\(612\) −133.187 −5.38376
\(613\) 23.0113 0.929416 0.464708 0.885464i \(-0.346159\pi\)
0.464708 + 0.885464i \(0.346159\pi\)
\(614\) 11.8730 0.479155
\(615\) −26.4678 −1.06728
\(616\) 26.0237 1.04852
\(617\) −18.8683 −0.759610 −0.379805 0.925067i \(-0.624009\pi\)
−0.379805 + 0.925067i \(0.624009\pi\)
\(618\) −86.0774 −3.46254
\(619\) −6.14818 −0.247116 −0.123558 0.992337i \(-0.539431\pi\)
−0.123558 + 0.992337i \(0.539431\pi\)
\(620\) −75.1714 −3.01896
\(621\) −69.1552 −2.77510
\(622\) 18.3927 0.737481
\(623\) −6.90536 −0.276657
\(624\) −22.6098 −0.905116
\(625\) 68.4886 2.73954
\(626\) 31.6888 1.26654
\(627\) −4.17223 −0.166623
\(628\) −58.8728 −2.34928
\(629\) 22.9420 0.914757
\(630\) −182.839 −7.28447
\(631\) −32.0198 −1.27469 −0.637344 0.770580i \(-0.719967\pi\)
−0.637344 + 0.770580i \(0.719967\pi\)
\(632\) −17.8241 −0.709006
\(633\) 18.2678 0.726078
\(634\) 21.6001 0.857850
\(635\) −35.8421 −1.42235
\(636\) 94.8591 3.76141
\(637\) 4.63718 0.183732
\(638\) 24.4709 0.968812
\(639\) 48.7195 1.92731
\(640\) −85.7249 −3.38857
\(641\) −10.9848 −0.433874 −0.216937 0.976186i \(-0.569607\pi\)
−0.216937 + 0.976186i \(0.569607\pi\)
\(642\) −87.7803 −3.46441
\(643\) 30.7503 1.21267 0.606336 0.795209i \(-0.292639\pi\)
0.606336 + 0.795209i \(0.292639\pi\)
\(644\) −63.6618 −2.50863
\(645\) 91.9956 3.62232
\(646\) 7.30324 0.287342
\(647\) −26.5562 −1.04403 −0.522016 0.852936i \(-0.674820\pi\)
−0.522016 + 0.852936i \(0.674820\pi\)
\(648\) −52.6439 −2.06805
\(649\) 5.35910 0.210363
\(650\) −135.580 −5.31787
\(651\) 44.0659 1.72708
\(652\) 10.1528 0.397613
\(653\) −7.83426 −0.306578 −0.153289 0.988181i \(-0.548987\pi\)
−0.153289 + 0.988181i \(0.548987\pi\)
\(654\) 10.1678 0.397594
\(655\) 81.4809 3.18372
\(656\) −3.25684 −0.127158
\(657\) 71.2677 2.78042
\(658\) −28.2437 −1.10105
\(659\) −36.0459 −1.40415 −0.702074 0.712104i \(-0.747742\pi\)
−0.702074 + 0.712104i \(0.747742\pi\)
\(660\) 114.642 4.46242
\(661\) −6.74525 −0.262360 −0.131180 0.991359i \(-0.541877\pi\)
−0.131180 + 0.991359i \(0.541877\pi\)
\(662\) −21.3012 −0.827893
\(663\) 81.0765 3.14875
\(664\) −40.6795 −1.57867
\(665\) 6.42405 0.249114
\(666\) 62.1708 2.40907
\(667\) −26.2992 −1.01831
\(668\) 40.3808 1.56238
\(669\) 37.8093 1.46179
\(670\) −35.8238 −1.38399
\(671\) −21.2249 −0.819379
\(672\) 31.8592 1.22900
\(673\) 32.6652 1.25915 0.629576 0.776939i \(-0.283229\pi\)
0.629576 + 0.776939i \(0.283229\pi\)
\(674\) 68.7216 2.64706
\(675\) −136.921 −5.27010
\(676\) 29.1656 1.12175
\(677\) −28.5071 −1.09562 −0.547809 0.836603i \(-0.684538\pi\)
−0.547809 + 0.836603i \(0.684538\pi\)
\(678\) 30.3802 1.16675
\(679\) −15.1587 −0.581738
\(680\) −88.1600 −3.38078
\(681\) −1.38204 −0.0529600
\(682\) −29.5731 −1.13241
\(683\) −27.5926 −1.05580 −0.527900 0.849306i \(-0.677020\pi\)
−0.527900 + 0.849306i \(0.677020\pi\)
\(684\) 12.6811 0.484874
\(685\) −69.6362 −2.66066
\(686\) −40.0094 −1.52757
\(687\) 34.9611 1.33385
\(688\) 11.3200 0.431571
\(689\) −39.6023 −1.50873
\(690\) −192.287 −7.32023
\(691\) −4.93442 −0.187714 −0.0938571 0.995586i \(-0.529920\pi\)
−0.0938571 + 0.995586i \(0.529920\pi\)
\(692\) −51.2235 −1.94723
\(693\) −46.0892 −1.75078
\(694\) 6.54049 0.248273
\(695\) 30.9992 1.17587
\(696\) −47.6446 −1.80596
\(697\) 11.6787 0.442363
\(698\) −4.10155 −0.155246
\(699\) −58.8615 −2.22635
\(700\) −126.045 −4.76405
\(701\) −16.8337 −0.635799 −0.317900 0.948124i \(-0.602977\pi\)
−0.317900 + 0.948124i \(0.602977\pi\)
\(702\) 119.057 4.49352
\(703\) −2.18437 −0.0823851
\(704\) −29.2904 −1.10392
\(705\) −54.6610 −2.05865
\(706\) 60.0965 2.26176
\(707\) −46.2607 −1.73981
\(708\) −23.7506 −0.892601
\(709\) −1.51372 −0.0568489 −0.0284244 0.999596i \(-0.509049\pi\)
−0.0284244 + 0.999596i \(0.509049\pi\)
\(710\) 73.4058 2.75487
\(711\) 31.5674 1.18387
\(712\) −9.02288 −0.338147
\(713\) 31.7826 1.19027
\(714\) 117.636 4.40240
\(715\) −47.8612 −1.78991
\(716\) 32.1394 1.20111
\(717\) −12.9009 −0.481792
\(718\) 76.9855 2.87307
\(719\) −48.0138 −1.79061 −0.895306 0.445453i \(-0.853043\pi\)
−0.895306 + 0.445453i \(0.853043\pi\)
\(720\) −43.5416 −1.62270
\(721\) 33.4087 1.24421
\(722\) 44.1346 1.64252
\(723\) −14.7632 −0.549051
\(724\) −87.9697 −3.26937
\(725\) −52.0701 −1.93383
\(726\) −35.0991 −1.30265
\(727\) −52.7641 −1.95691 −0.978456 0.206456i \(-0.933807\pi\)
−0.978456 + 0.206456i \(0.933807\pi\)
\(728\) 48.1494 1.78454
\(729\) −8.41438 −0.311644
\(730\) 107.379 3.97429
\(731\) −40.5924 −1.50136
\(732\) 94.0649 3.47674
\(733\) −17.8293 −0.658541 −0.329271 0.944236i \(-0.606803\pi\)
−0.329271 + 0.944236i \(0.606803\pi\)
\(734\) −16.0749 −0.593336
\(735\) 13.0213 0.480297
\(736\) 22.9786 0.847002
\(737\) −9.03030 −0.332635
\(738\) 31.6483 1.16499
\(739\) −36.1098 −1.32832 −0.664160 0.747591i \(-0.731211\pi\)
−0.664160 + 0.747591i \(0.731211\pi\)
\(740\) 60.0207 2.20640
\(741\) −7.71953 −0.283584
\(742\) −57.4598 −2.10941
\(743\) −0.881568 −0.0323416 −0.0161708 0.999869i \(-0.505148\pi\)
−0.0161708 + 0.999869i \(0.505148\pi\)
\(744\) 57.5786 2.11093
\(745\) 85.5065 3.13272
\(746\) −25.1741 −0.921689
\(747\) 72.0453 2.63600
\(748\) −50.5849 −1.84957
\(749\) 34.0697 1.24488
\(750\) −228.267 −8.33512
\(751\) −13.6390 −0.497694 −0.248847 0.968543i \(-0.580052\pi\)
−0.248847 + 0.968543i \(0.580052\pi\)
\(752\) −6.72600 −0.245272
\(753\) −13.0031 −0.473858
\(754\) 45.2764 1.64887
\(755\) −64.7309 −2.35580
\(756\) 110.684 4.02554
\(757\) −19.9724 −0.725910 −0.362955 0.931807i \(-0.618232\pi\)
−0.362955 + 0.931807i \(0.618232\pi\)
\(758\) −52.5552 −1.90889
\(759\) −48.4708 −1.75938
\(760\) 8.39397 0.304481
\(761\) 44.3290 1.60693 0.803463 0.595355i \(-0.202989\pi\)
0.803463 + 0.595355i \(0.202989\pi\)
\(762\) 62.4912 2.26382
\(763\) −3.94639 −0.142869
\(764\) 3.53345 0.127836
\(765\) 156.136 5.64510
\(766\) 6.73161 0.243223
\(767\) 9.91550 0.358028
\(768\) 76.6813 2.76700
\(769\) −35.4396 −1.27799 −0.638993 0.769213i \(-0.720649\pi\)
−0.638993 + 0.769213i \(0.720649\pi\)
\(770\) −69.4428 −2.50255
\(771\) −60.3391 −2.17306
\(772\) −80.9882 −2.91483
\(773\) −35.9274 −1.29222 −0.646110 0.763244i \(-0.723605\pi\)
−0.646110 + 0.763244i \(0.723605\pi\)
\(774\) −110.002 −3.95394
\(775\) 62.9268 2.26040
\(776\) −19.8071 −0.711033
\(777\) −35.1844 −1.26223
\(778\) 30.1807 1.08203
\(779\) −1.11197 −0.0398403
\(780\) 212.112 7.59483
\(781\) 18.5038 0.662119
\(782\) 84.8451 3.03405
\(783\) 45.7244 1.63406
\(784\) 1.60226 0.0572236
\(785\) 69.0169 2.46332
\(786\) −142.063 −5.06723
\(787\) 4.68054 0.166843 0.0834216 0.996514i \(-0.473415\pi\)
0.0834216 + 0.996514i \(0.473415\pi\)
\(788\) 38.1202 1.35798
\(789\) −15.6306 −0.556464
\(790\) 47.5628 1.69221
\(791\) −11.7913 −0.419251
\(792\) −60.2224 −2.13991
\(793\) −39.2707 −1.39454
\(794\) −1.37799 −0.0489030
\(795\) −111.204 −3.94400
\(796\) −11.5043 −0.407761
\(797\) 51.2468 1.81526 0.907628 0.419776i \(-0.137891\pi\)
0.907628 + 0.419776i \(0.137891\pi\)
\(798\) −11.2004 −0.396491
\(799\) 24.1188 0.853262
\(800\) 45.4956 1.60851
\(801\) 15.9800 0.564624
\(802\) −2.90570 −0.102604
\(803\) 27.0677 0.955199
\(804\) 40.0206 1.41142
\(805\) 74.6311 2.63040
\(806\) −54.7167 −1.92731
\(807\) 52.3381 1.84239
\(808\) −60.4465 −2.12650
\(809\) −4.96718 −0.174637 −0.0873183 0.996180i \(-0.527830\pi\)
−0.0873183 + 0.996180i \(0.527830\pi\)
\(810\) 140.478 4.93588
\(811\) 5.27601 0.185266 0.0926328 0.995700i \(-0.470472\pi\)
0.0926328 + 0.995700i \(0.470472\pi\)
\(812\) 42.0923 1.47715
\(813\) 35.6361 1.24981
\(814\) 23.6127 0.827624
\(815\) −11.9021 −0.416914
\(816\) 28.0140 0.980685
\(817\) 3.86492 0.135216
\(818\) 2.80336 0.0980170
\(819\) −85.2750 −2.97975
\(820\) 30.5538 1.06699
\(821\) 32.6165 1.13832 0.569162 0.822225i \(-0.307268\pi\)
0.569162 + 0.822225i \(0.307268\pi\)
\(822\) 121.412 4.23472
\(823\) −41.0761 −1.43182 −0.715912 0.698191i \(-0.753989\pi\)
−0.715912 + 0.698191i \(0.753989\pi\)
\(824\) 43.6535 1.52074
\(825\) −95.9678 −3.34117
\(826\) 14.3866 0.500574
\(827\) −17.2551 −0.600019 −0.300010 0.953936i \(-0.596990\pi\)
−0.300010 + 0.953936i \(0.596990\pi\)
\(828\) 147.322 5.11981
\(829\) −7.35445 −0.255430 −0.127715 0.991811i \(-0.540764\pi\)
−0.127715 + 0.991811i \(0.540764\pi\)
\(830\) 108.551 3.76786
\(831\) 46.3147 1.60664
\(832\) −54.1936 −1.87882
\(833\) −5.74555 −0.199072
\(834\) −54.0477 −1.87152
\(835\) −47.3387 −1.63822
\(836\) 4.81633 0.166576
\(837\) −55.2581 −1.91000
\(838\) 6.48793 0.224122
\(839\) −7.34393 −0.253541 −0.126770 0.991932i \(-0.540461\pi\)
−0.126770 + 0.991932i \(0.540461\pi\)
\(840\) 135.204 4.66500
\(841\) −11.6113 −0.400391
\(842\) −62.4615 −2.15257
\(843\) 80.2409 2.76364
\(844\) −21.0879 −0.725875
\(845\) −34.1910 −1.17621
\(846\) 65.3598 2.24712
\(847\) 13.6228 0.468086
\(848\) −13.6836 −0.469896
\(849\) 85.4698 2.93332
\(850\) 167.986 5.76187
\(851\) −25.3769 −0.869908
\(852\) −82.0055 −2.80946
\(853\) 19.4141 0.664725 0.332362 0.943152i \(-0.392154\pi\)
0.332362 + 0.943152i \(0.392154\pi\)
\(854\) −56.9787 −1.94977
\(855\) −14.8661 −0.508411
\(856\) 44.5171 1.52156
\(857\) 11.7234 0.400462 0.200231 0.979749i \(-0.435831\pi\)
0.200231 + 0.979749i \(0.435831\pi\)
\(858\) 83.4468 2.84883
\(859\) 33.3903 1.13926 0.569632 0.821900i \(-0.307086\pi\)
0.569632 + 0.821900i \(0.307086\pi\)
\(860\) −106.198 −3.62131
\(861\) −17.9108 −0.610398
\(862\) 34.6894 1.18152
\(863\) −34.6843 −1.18067 −0.590334 0.807159i \(-0.701004\pi\)
−0.590334 + 0.807159i \(0.701004\pi\)
\(864\) −39.9511 −1.35917
\(865\) 60.0496 2.04175
\(866\) −81.2110 −2.75966
\(867\) −47.9242 −1.62759
\(868\) −50.8686 −1.72659
\(869\) 11.9894 0.406713
\(870\) 127.137 4.31035
\(871\) −16.7080 −0.566129
\(872\) −5.15654 −0.174623
\(873\) 35.0793 1.18726
\(874\) −8.07835 −0.273254
\(875\) 88.5959 2.99509
\(876\) −119.959 −4.05305
\(877\) 11.7682 0.397384 0.198692 0.980062i \(-0.436331\pi\)
0.198692 + 0.980062i \(0.436331\pi\)
\(878\) −40.8544 −1.37877
\(879\) −1.28256 −0.0432596
\(880\) −16.5372 −0.557470
\(881\) −38.4286 −1.29469 −0.647346 0.762197i \(-0.724121\pi\)
−0.647346 + 0.762197i \(0.724121\pi\)
\(882\) −15.5699 −0.524267
\(883\) −43.7693 −1.47296 −0.736478 0.676462i \(-0.763512\pi\)
−0.736478 + 0.676462i \(0.763512\pi\)
\(884\) −93.5929 −3.14787
\(885\) 27.8429 0.935929
\(886\) −46.9673 −1.57790
\(887\) −20.7665 −0.697271 −0.348635 0.937258i \(-0.613355\pi\)
−0.348635 + 0.937258i \(0.613355\pi\)
\(888\) −45.9737 −1.54277
\(889\) −24.2544 −0.813466
\(890\) 24.0771 0.807066
\(891\) 35.4110 1.18631
\(892\) −43.6462 −1.46138
\(893\) −2.29642 −0.0768467
\(894\) −149.082 −4.98605
\(895\) −37.6772 −1.25941
\(896\) −58.0102 −1.93798
\(897\) −89.6814 −2.99437
\(898\) 21.8687 0.729767
\(899\) −21.0142 −0.700864
\(900\) 291.685 9.72284
\(901\) 49.0679 1.63469
\(902\) 12.0202 0.400227
\(903\) 62.2536 2.07167
\(904\) −15.4071 −0.512433
\(905\) 103.127 3.42807
\(906\) 112.859 3.74950
\(907\) 12.9518 0.430056 0.215028 0.976608i \(-0.431016\pi\)
0.215028 + 0.976608i \(0.431016\pi\)
\(908\) 1.59540 0.0529451
\(909\) 107.054 3.55075
\(910\) −128.484 −4.25921
\(911\) 5.57453 0.184692 0.0923462 0.995727i \(-0.470563\pi\)
0.0923462 + 0.995727i \(0.470563\pi\)
\(912\) −2.66729 −0.0883228
\(913\) 27.3631 0.905585
\(914\) 22.8896 0.757119
\(915\) −110.273 −3.64551
\(916\) −40.3583 −1.33348
\(917\) 55.1382 1.82082
\(918\) −147.514 −4.86868
\(919\) −24.5634 −0.810273 −0.405137 0.914256i \(-0.632776\pi\)
−0.405137 + 0.914256i \(0.632776\pi\)
\(920\) 97.5167 3.21503
\(921\) 15.5494 0.512370
\(922\) −60.2320 −1.98363
\(923\) 34.2361 1.12689
\(924\) 77.5782 2.55214
\(925\) −50.2439 −1.65201
\(926\) 91.8869 3.01959
\(927\) −77.3125 −2.53928
\(928\) −15.1931 −0.498738
\(929\) 28.8268 0.945775 0.472888 0.881123i \(-0.343212\pi\)
0.472888 + 0.881123i \(0.343212\pi\)
\(930\) −153.646 −5.03823
\(931\) 0.547050 0.0179289
\(932\) 67.9484 2.22572
\(933\) 24.0879 0.788603
\(934\) 35.2587 1.15370
\(935\) 59.3009 1.93935
\(936\) −111.424 −3.64202
\(937\) −15.5358 −0.507534 −0.253767 0.967265i \(-0.581670\pi\)
−0.253767 + 0.967265i \(0.581670\pi\)
\(938\) −24.2420 −0.791529
\(939\) 41.5010 1.35433
\(940\) 63.0994 2.05808
\(941\) −53.4497 −1.74241 −0.871205 0.490919i \(-0.836661\pi\)
−0.871205 + 0.490919i \(0.836661\pi\)
\(942\) −120.332 −3.92063
\(943\) −12.9182 −0.420675
\(944\) 3.42605 0.111508
\(945\) −129.756 −4.22095
\(946\) −41.7791 −1.35836
\(947\) 26.4923 0.860885 0.430443 0.902618i \(-0.358357\pi\)
0.430443 + 0.902618i \(0.358357\pi\)
\(948\) −53.1349 −1.72574
\(949\) 50.0811 1.62570
\(950\) −15.9944 −0.518927
\(951\) 28.2885 0.917316
\(952\) −59.6580 −1.93353
\(953\) −26.3995 −0.855162 −0.427581 0.903977i \(-0.640634\pi\)
−0.427581 + 0.903977i \(0.640634\pi\)
\(954\) 132.970 4.30506
\(955\) −4.14228 −0.134041
\(956\) 14.8925 0.481657
\(957\) 32.0482 1.03597
\(958\) −7.61691 −0.246091
\(959\) −47.1229 −1.52168
\(960\) −152.177 −4.91148
\(961\) −5.60426 −0.180783
\(962\) 43.6886 1.40858
\(963\) −78.8420 −2.54065
\(964\) 17.0424 0.548897
\(965\) 94.9429 3.05632
\(966\) −130.121 −4.18656
\(967\) 23.1774 0.745334 0.372667 0.927965i \(-0.378443\pi\)
0.372667 + 0.927965i \(0.378443\pi\)
\(968\) 17.8002 0.572121
\(969\) 9.56464 0.307261
\(970\) 52.8542 1.69705
\(971\) 9.57093 0.307146 0.153573 0.988137i \(-0.450922\pi\)
0.153573 + 0.988137i \(0.450922\pi\)
\(972\) −39.5932 −1.26995
\(973\) 20.9772 0.672499
\(974\) 18.5000 0.592777
\(975\) −177.561 −5.68651
\(976\) −13.5690 −0.434333
\(977\) 15.5814 0.498493 0.249247 0.968440i \(-0.419817\pi\)
0.249247 + 0.968440i \(0.419817\pi\)
\(978\) 20.7516 0.663562
\(979\) 6.06925 0.193974
\(980\) −15.0315 −0.480163
\(981\) 9.13249 0.291578
\(982\) −43.3039 −1.38188
\(983\) −45.6013 −1.45445 −0.727227 0.686397i \(-0.759191\pi\)
−0.727227 + 0.686397i \(0.759191\pi\)
\(984\) −23.4031 −0.746064
\(985\) −44.6885 −1.42390
\(986\) −56.0984 −1.78654
\(987\) −36.9892 −1.17738
\(988\) 8.91125 0.283505
\(989\) 44.9006 1.42776
\(990\) 160.700 5.10739
\(991\) 26.6593 0.846860 0.423430 0.905929i \(-0.360826\pi\)
0.423430 + 0.905929i \(0.360826\pi\)
\(992\) 18.3609 0.582959
\(993\) −27.8970 −0.885283
\(994\) 49.6738 1.57556
\(995\) 13.4866 0.427554
\(996\) −121.268 −3.84253
\(997\) −21.8151 −0.690891 −0.345446 0.938439i \(-0.612272\pi\)
−0.345446 + 0.938439i \(0.612272\pi\)
\(998\) −62.2616 −1.97086
\(999\) 44.1208 1.39592
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.1.3 18
3.2 odd 2 4923.2.a.l.1.16 18
4.3 odd 2 8752.2.a.s.1.18 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.3 18 1.1 even 1 trivial
4923.2.a.l.1.16 18 3.2 odd 2
8752.2.a.s.1.18 18 4.3 odd 2