Properties

Label 547.2.a.b.1.13
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 1.13
Root \(-0.957552\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.957552 q^{2} +0.564538 q^{3} -1.08309 q^{4} -4.10274 q^{5} +0.540575 q^{6} +4.97202 q^{7} -2.95222 q^{8} -2.68130 q^{9} +O(q^{10})\) \(q+0.957552 q^{2} +0.564538 q^{3} -1.08309 q^{4} -4.10274 q^{5} +0.540575 q^{6} +4.97202 q^{7} -2.95222 q^{8} -2.68130 q^{9} -3.92859 q^{10} +0.0769657 q^{11} -0.611448 q^{12} -3.33082 q^{13} +4.76097 q^{14} -2.31615 q^{15} -0.660720 q^{16} -5.96711 q^{17} -2.56748 q^{18} -7.28632 q^{19} +4.44365 q^{20} +2.80689 q^{21} +0.0736987 q^{22} +2.10873 q^{23} -1.66664 q^{24} +11.8325 q^{25} -3.18943 q^{26} -3.20731 q^{27} -5.38516 q^{28} +3.24142 q^{29} -2.21784 q^{30} -6.20019 q^{31} +5.27177 q^{32} +0.0434501 q^{33} -5.71382 q^{34} -20.3989 q^{35} +2.90410 q^{36} +2.16514 q^{37} -6.97703 q^{38} -1.88037 q^{39} +12.1122 q^{40} -5.00098 q^{41} +2.68775 q^{42} -1.06278 q^{43} -0.0833611 q^{44} +11.0007 q^{45} +2.01922 q^{46} +5.86219 q^{47} -0.373001 q^{48} +17.7210 q^{49} +11.3302 q^{50} -3.36866 q^{51} +3.60759 q^{52} -9.01187 q^{53} -3.07116 q^{54} -0.315771 q^{55} -14.6785 q^{56} -4.11340 q^{57} +3.10383 q^{58} -4.47532 q^{59} +2.50861 q^{60} +4.58346 q^{61} -5.93701 q^{62} -13.3315 q^{63} +6.36944 q^{64} +13.6655 q^{65} +0.0416057 q^{66} +10.6474 q^{67} +6.46294 q^{68} +1.19046 q^{69} -19.5330 q^{70} +3.35886 q^{71} +7.91579 q^{72} +14.7261 q^{73} +2.07323 q^{74} +6.67989 q^{75} +7.89177 q^{76} +0.382675 q^{77} -1.80056 q^{78} -2.00500 q^{79} +2.71076 q^{80} +6.23324 q^{81} -4.78870 q^{82} -8.56498 q^{83} -3.04013 q^{84} +24.4815 q^{85} -1.01767 q^{86} +1.82990 q^{87} -0.227220 q^{88} -8.84846 q^{89} +10.5337 q^{90} -16.5609 q^{91} -2.28395 q^{92} -3.50024 q^{93} +5.61336 q^{94} +29.8939 q^{95} +2.97612 q^{96} -0.394709 q^{97} +16.9688 q^{98} -0.206368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.957552 0.677092 0.338546 0.940950i \(-0.390065\pi\)
0.338546 + 0.940950i \(0.390065\pi\)
\(3\) 0.564538 0.325936 0.162968 0.986631i \(-0.447893\pi\)
0.162968 + 0.986631i \(0.447893\pi\)
\(4\) −1.08309 −0.541547
\(5\) −4.10274 −1.83480 −0.917401 0.397964i \(-0.869717\pi\)
−0.917401 + 0.397964i \(0.869717\pi\)
\(6\) 0.540575 0.220689
\(7\) 4.97202 1.87925 0.939623 0.342210i \(-0.111176\pi\)
0.939623 + 0.342210i \(0.111176\pi\)
\(8\) −2.95222 −1.04377
\(9\) −2.68130 −0.893766
\(10\) −3.92859 −1.24233
\(11\) 0.0769657 0.0232060 0.0116030 0.999933i \(-0.496307\pi\)
0.0116030 + 0.999933i \(0.496307\pi\)
\(12\) −0.611448 −0.176510
\(13\) −3.33082 −0.923803 −0.461901 0.886931i \(-0.652833\pi\)
−0.461901 + 0.886931i \(0.652833\pi\)
\(14\) 4.76097 1.27242
\(15\) −2.31615 −0.598028
\(16\) −0.660720 −0.165180
\(17\) −5.96711 −1.44724 −0.723618 0.690200i \(-0.757522\pi\)
−0.723618 + 0.690200i \(0.757522\pi\)
\(18\) −2.56748 −0.605161
\(19\) −7.28632 −1.67160 −0.835798 0.549037i \(-0.814994\pi\)
−0.835798 + 0.549037i \(0.814994\pi\)
\(20\) 4.44365 0.993631
\(21\) 2.80689 0.612514
\(22\) 0.0736987 0.0157126
\(23\) 2.10873 0.439700 0.219850 0.975534i \(-0.429443\pi\)
0.219850 + 0.975534i \(0.429443\pi\)
\(24\) −1.66664 −0.340202
\(25\) 11.8325 2.36650
\(26\) −3.18943 −0.625499
\(27\) −3.20731 −0.617247
\(28\) −5.38516 −1.01770
\(29\) 3.24142 0.601916 0.300958 0.953637i \(-0.402694\pi\)
0.300958 + 0.953637i \(0.402694\pi\)
\(30\) −2.21784 −0.404920
\(31\) −6.20019 −1.11359 −0.556794 0.830651i \(-0.687969\pi\)
−0.556794 + 0.830651i \(0.687969\pi\)
\(32\) 5.27177 0.931927
\(33\) 0.0434501 0.00756369
\(34\) −5.71382 −0.979912
\(35\) −20.3989 −3.44805
\(36\) 2.90410 0.484016
\(37\) 2.16514 0.355946 0.177973 0.984035i \(-0.443046\pi\)
0.177973 + 0.984035i \(0.443046\pi\)
\(38\) −6.97703 −1.13182
\(39\) −1.88037 −0.301101
\(40\) 12.1122 1.91511
\(41\) −5.00098 −0.781021 −0.390511 0.920598i \(-0.627702\pi\)
−0.390511 + 0.920598i \(0.627702\pi\)
\(42\) 2.68775 0.414728
\(43\) −1.06278 −0.162072 −0.0810362 0.996711i \(-0.525823\pi\)
−0.0810362 + 0.996711i \(0.525823\pi\)
\(44\) −0.0833611 −0.0125672
\(45\) 11.0007 1.63988
\(46\) 2.01922 0.297717
\(47\) 5.86219 0.855089 0.427544 0.903994i \(-0.359379\pi\)
0.427544 + 0.903994i \(0.359379\pi\)
\(48\) −0.373001 −0.0538381
\(49\) 17.7210 2.53157
\(50\) 11.3302 1.60234
\(51\) −3.36866 −0.471707
\(52\) 3.60759 0.500283
\(53\) −9.01187 −1.23788 −0.618938 0.785440i \(-0.712437\pi\)
−0.618938 + 0.785440i \(0.712437\pi\)
\(54\) −3.07116 −0.417933
\(55\) −0.315771 −0.0425785
\(56\) −14.6785 −1.96150
\(57\) −4.11340 −0.544834
\(58\) 3.10383 0.407552
\(59\) −4.47532 −0.582637 −0.291319 0.956626i \(-0.594094\pi\)
−0.291319 + 0.956626i \(0.594094\pi\)
\(60\) 2.50861 0.323860
\(61\) 4.58346 0.586851 0.293426 0.955982i \(-0.405205\pi\)
0.293426 + 0.955982i \(0.405205\pi\)
\(62\) −5.93701 −0.754001
\(63\) −13.3315 −1.67961
\(64\) 6.36944 0.796180
\(65\) 13.6655 1.69500
\(66\) 0.0416057 0.00512131
\(67\) 10.6474 1.30078 0.650391 0.759599i \(-0.274605\pi\)
0.650391 + 0.759599i \(0.274605\pi\)
\(68\) 6.46294 0.783747
\(69\) 1.19046 0.143314
\(70\) −19.5330 −2.33464
\(71\) 3.35886 0.398624 0.199312 0.979936i \(-0.436129\pi\)
0.199312 + 0.979936i \(0.436129\pi\)
\(72\) 7.91579 0.932884
\(73\) 14.7261 1.72356 0.861779 0.507284i \(-0.169351\pi\)
0.861779 + 0.507284i \(0.169351\pi\)
\(74\) 2.07323 0.241008
\(75\) 6.67989 0.771327
\(76\) 7.89177 0.905248
\(77\) 0.382675 0.0436099
\(78\) −1.80056 −0.203873
\(79\) −2.00500 −0.225580 −0.112790 0.993619i \(-0.535979\pi\)
−0.112790 + 0.993619i \(0.535979\pi\)
\(80\) 2.71076 0.303072
\(81\) 6.23324 0.692583
\(82\) −4.78870 −0.528823
\(83\) −8.56498 −0.940129 −0.470065 0.882632i \(-0.655769\pi\)
−0.470065 + 0.882632i \(0.655769\pi\)
\(84\) −3.04013 −0.331705
\(85\) 24.4815 2.65539
\(86\) −1.01767 −0.109738
\(87\) 1.82990 0.196186
\(88\) −0.227220 −0.0242217
\(89\) −8.84846 −0.937935 −0.468967 0.883215i \(-0.655374\pi\)
−0.468967 + 0.883215i \(0.655374\pi\)
\(90\) 10.5337 1.11035
\(91\) −16.5609 −1.73605
\(92\) −2.28395 −0.238118
\(93\) −3.50024 −0.362958
\(94\) 5.61336 0.578973
\(95\) 29.8939 3.06705
\(96\) 2.97612 0.303749
\(97\) −0.394709 −0.0400766 −0.0200383 0.999799i \(-0.506379\pi\)
−0.0200383 + 0.999799i \(0.506379\pi\)
\(98\) 16.9688 1.71410
\(99\) −0.206368 −0.0207408
\(100\) −12.8157 −1.28157
\(101\) −5.67033 −0.564219 −0.282110 0.959382i \(-0.591034\pi\)
−0.282110 + 0.959382i \(0.591034\pi\)
\(102\) −3.22567 −0.319389
\(103\) −12.2307 −1.20513 −0.602565 0.798069i \(-0.705855\pi\)
−0.602565 + 0.798069i \(0.705855\pi\)
\(104\) 9.83332 0.964236
\(105\) −11.5160 −1.12384
\(106\) −8.62934 −0.838155
\(107\) −8.31411 −0.803755 −0.401878 0.915693i \(-0.631642\pi\)
−0.401878 + 0.915693i \(0.631642\pi\)
\(108\) 3.47382 0.334268
\(109\) 0.677082 0.0648527 0.0324263 0.999474i \(-0.489677\pi\)
0.0324263 + 0.999474i \(0.489677\pi\)
\(110\) −0.302367 −0.0288295
\(111\) 1.22230 0.116016
\(112\) −3.28511 −0.310414
\(113\) −8.32143 −0.782814 −0.391407 0.920218i \(-0.628012\pi\)
−0.391407 + 0.920218i \(0.628012\pi\)
\(114\) −3.93880 −0.368902
\(115\) −8.65156 −0.806763
\(116\) −3.51076 −0.325966
\(117\) 8.93091 0.825663
\(118\) −4.28535 −0.394499
\(119\) −29.6686 −2.71972
\(120\) 6.83780 0.624203
\(121\) −10.9941 −0.999461
\(122\) 4.38890 0.397352
\(123\) −2.82324 −0.254563
\(124\) 6.71539 0.603060
\(125\) −28.0319 −2.50725
\(126\) −12.7656 −1.13725
\(127\) 9.98242 0.885796 0.442898 0.896572i \(-0.353950\pi\)
0.442898 + 0.896572i \(0.353950\pi\)
\(128\) −4.44448 −0.392840
\(129\) −0.599980 −0.0528253
\(130\) 13.0854 1.14767
\(131\) 1.15417 0.100840 0.0504201 0.998728i \(-0.483944\pi\)
0.0504201 + 0.998728i \(0.483944\pi\)
\(132\) −0.0470605 −0.00409609
\(133\) −36.2277 −3.14134
\(134\) 10.1954 0.880749
\(135\) 13.1588 1.13253
\(136\) 17.6162 1.51058
\(137\) 5.05259 0.431672 0.215836 0.976430i \(-0.430752\pi\)
0.215836 + 0.976430i \(0.430752\pi\)
\(138\) 1.13992 0.0970368
\(139\) −3.30416 −0.280256 −0.140128 0.990133i \(-0.544751\pi\)
−0.140128 + 0.990133i \(0.544751\pi\)
\(140\) 22.0939 1.86728
\(141\) 3.30943 0.278704
\(142\) 3.21629 0.269905
\(143\) −0.256359 −0.0214378
\(144\) 1.77159 0.147632
\(145\) −13.2987 −1.10440
\(146\) 14.1010 1.16701
\(147\) 10.0042 0.825130
\(148\) −2.34505 −0.192762
\(149\) −23.7638 −1.94680 −0.973402 0.229103i \(-0.926421\pi\)
−0.973402 + 0.229103i \(0.926421\pi\)
\(150\) 6.39634 0.522259
\(151\) −5.72927 −0.466242 −0.233121 0.972448i \(-0.574894\pi\)
−0.233121 + 0.972448i \(0.574894\pi\)
\(152\) 21.5108 1.74476
\(153\) 15.9996 1.29349
\(154\) 0.366431 0.0295279
\(155\) 25.4378 2.04321
\(156\) 2.03662 0.163060
\(157\) 5.46006 0.435760 0.217880 0.975976i \(-0.430086\pi\)
0.217880 + 0.975976i \(0.430086\pi\)
\(158\) −1.91989 −0.152738
\(159\) −5.08754 −0.403468
\(160\) −21.6287 −1.70990
\(161\) 10.4846 0.826305
\(162\) 5.96866 0.468942
\(163\) −4.79414 −0.375506 −0.187753 0.982216i \(-0.560120\pi\)
−0.187753 + 0.982216i \(0.560120\pi\)
\(164\) 5.41653 0.422960
\(165\) −0.178264 −0.0138779
\(166\) −8.20142 −0.636554
\(167\) 8.88965 0.687902 0.343951 0.938988i \(-0.388235\pi\)
0.343951 + 0.938988i \(0.388235\pi\)
\(168\) −8.28658 −0.639323
\(169\) −1.90565 −0.146588
\(170\) 23.4423 1.79794
\(171\) 19.5368 1.49401
\(172\) 1.15109 0.0877698
\(173\) 18.2314 1.38611 0.693055 0.720885i \(-0.256264\pi\)
0.693055 + 0.720885i \(0.256264\pi\)
\(174\) 1.75223 0.132836
\(175\) 58.8314 4.44723
\(176\) −0.0508528 −0.00383317
\(177\) −2.52649 −0.189903
\(178\) −8.47286 −0.635068
\(179\) −10.4779 −0.783152 −0.391576 0.920146i \(-0.628070\pi\)
−0.391576 + 0.920146i \(0.628070\pi\)
\(180\) −11.9148 −0.888074
\(181\) −11.8241 −0.878881 −0.439441 0.898272i \(-0.644823\pi\)
−0.439441 + 0.898272i \(0.644823\pi\)
\(182\) −15.8579 −1.17547
\(183\) 2.58753 0.191276
\(184\) −6.22543 −0.458945
\(185\) −8.88299 −0.653091
\(186\) −3.35167 −0.245756
\(187\) −0.459263 −0.0335846
\(188\) −6.34931 −0.463071
\(189\) −15.9468 −1.15996
\(190\) 28.6249 2.07667
\(191\) −19.6548 −1.42217 −0.711087 0.703104i \(-0.751797\pi\)
−0.711087 + 0.703104i \(0.751797\pi\)
\(192\) 3.59579 0.259504
\(193\) 7.35743 0.529600 0.264800 0.964303i \(-0.414694\pi\)
0.264800 + 0.964303i \(0.414694\pi\)
\(194\) −0.377955 −0.0271356
\(195\) 7.71469 0.552460
\(196\) −19.1935 −1.37096
\(197\) 0.768399 0.0547462 0.0273731 0.999625i \(-0.491286\pi\)
0.0273731 + 0.999625i \(0.491286\pi\)
\(198\) −0.197608 −0.0140434
\(199\) 21.6917 1.53768 0.768841 0.639440i \(-0.220834\pi\)
0.768841 + 0.639440i \(0.220834\pi\)
\(200\) −34.9321 −2.47008
\(201\) 6.01084 0.423972
\(202\) −5.42964 −0.382028
\(203\) 16.1164 1.13115
\(204\) 3.64858 0.255451
\(205\) 20.5177 1.43302
\(206\) −11.7116 −0.815984
\(207\) −5.65412 −0.392989
\(208\) 2.20074 0.152594
\(209\) −0.560797 −0.0387911
\(210\) −11.0271 −0.760944
\(211\) −10.4907 −0.722208 −0.361104 0.932526i \(-0.617600\pi\)
−0.361104 + 0.932526i \(0.617600\pi\)
\(212\) 9.76070 0.670368
\(213\) 1.89621 0.129926
\(214\) −7.96119 −0.544216
\(215\) 4.36031 0.297371
\(216\) 9.46869 0.644263
\(217\) −30.8275 −2.09271
\(218\) 0.648341 0.0439112
\(219\) 8.31343 0.561770
\(220\) 0.342009 0.0230583
\(221\) 19.8754 1.33696
\(222\) 1.17042 0.0785533
\(223\) −18.6471 −1.24870 −0.624352 0.781143i \(-0.714637\pi\)
−0.624352 + 0.781143i \(0.714637\pi\)
\(224\) 26.2114 1.75132
\(225\) −31.7264 −2.11509
\(226\) −7.96820 −0.530037
\(227\) 27.9582 1.85565 0.927826 0.373014i \(-0.121676\pi\)
0.927826 + 0.373014i \(0.121676\pi\)
\(228\) 4.45520 0.295053
\(229\) −9.73578 −0.643358 −0.321679 0.946849i \(-0.604247\pi\)
−0.321679 + 0.946849i \(0.604247\pi\)
\(230\) −8.28432 −0.546252
\(231\) 0.216035 0.0142140
\(232\) −9.56939 −0.628261
\(233\) 19.0910 1.25069 0.625347 0.780347i \(-0.284957\pi\)
0.625347 + 0.780347i \(0.284957\pi\)
\(234\) 8.55181 0.559050
\(235\) −24.0511 −1.56892
\(236\) 4.84719 0.315525
\(237\) −1.13190 −0.0735246
\(238\) −28.4092 −1.84150
\(239\) −26.9662 −1.74430 −0.872149 0.489241i \(-0.837274\pi\)
−0.872149 + 0.489241i \(0.837274\pi\)
\(240\) 1.53033 0.0987822
\(241\) −6.54182 −0.421396 −0.210698 0.977551i \(-0.567574\pi\)
−0.210698 + 0.977551i \(0.567574\pi\)
\(242\) −10.5274 −0.676727
\(243\) 13.1408 0.842984
\(244\) −4.96431 −0.317808
\(245\) −72.7046 −4.64493
\(246\) −2.70340 −0.172363
\(247\) 24.2694 1.54422
\(248\) 18.3044 1.16233
\(249\) −4.83526 −0.306422
\(250\) −26.8420 −1.69764
\(251\) −22.2725 −1.40583 −0.702914 0.711274i \(-0.748118\pi\)
−0.702914 + 0.711274i \(0.748118\pi\)
\(252\) 14.4392 0.909586
\(253\) 0.162300 0.0102037
\(254\) 9.55868 0.599765
\(255\) 13.8207 0.865489
\(256\) −16.9947 −1.06217
\(257\) −10.9352 −0.682119 −0.341060 0.940042i \(-0.610786\pi\)
−0.341060 + 0.940042i \(0.610786\pi\)
\(258\) −0.574512 −0.0357675
\(259\) 10.7651 0.668911
\(260\) −14.8010 −0.917920
\(261\) −8.69120 −0.537972
\(262\) 1.10518 0.0682781
\(263\) 10.3267 0.636774 0.318387 0.947961i \(-0.396859\pi\)
0.318387 + 0.947961i \(0.396859\pi\)
\(264\) −0.128274 −0.00789474
\(265\) 36.9734 2.27126
\(266\) −34.6899 −2.12698
\(267\) −4.99529 −0.305707
\(268\) −11.5321 −0.704435
\(269\) 6.36422 0.388033 0.194017 0.980998i \(-0.437848\pi\)
0.194017 + 0.980998i \(0.437848\pi\)
\(270\) 12.6002 0.766823
\(271\) −29.8982 −1.81619 −0.908095 0.418765i \(-0.862463\pi\)
−0.908095 + 0.418765i \(0.862463\pi\)
\(272\) 3.94259 0.239054
\(273\) −9.34925 −0.565843
\(274\) 4.83812 0.292281
\(275\) 0.910696 0.0549171
\(276\) −1.28938 −0.0776114
\(277\) 7.02599 0.422151 0.211076 0.977470i \(-0.432303\pi\)
0.211076 + 0.977470i \(0.432303\pi\)
\(278\) −3.16391 −0.189759
\(279\) 16.6246 0.995286
\(280\) 60.2221 3.59896
\(281\) 19.7180 1.17628 0.588139 0.808760i \(-0.299861\pi\)
0.588139 + 0.808760i \(0.299861\pi\)
\(282\) 3.16895 0.188708
\(283\) −3.99962 −0.237753 −0.118876 0.992909i \(-0.537929\pi\)
−0.118876 + 0.992909i \(0.537929\pi\)
\(284\) −3.63796 −0.215873
\(285\) 16.8762 0.999662
\(286\) −0.245477 −0.0145154
\(287\) −24.8650 −1.46773
\(288\) −14.1352 −0.832924
\(289\) 18.6064 1.09449
\(290\) −12.7342 −0.747778
\(291\) −0.222828 −0.0130624
\(292\) −15.9497 −0.933387
\(293\) −24.6884 −1.44231 −0.721156 0.692772i \(-0.756389\pi\)
−0.721156 + 0.692772i \(0.756389\pi\)
\(294\) 9.57951 0.558688
\(295\) 18.3611 1.06902
\(296\) −6.39196 −0.371525
\(297\) −0.246853 −0.0143239
\(298\) −22.7551 −1.31816
\(299\) −7.02379 −0.406196
\(300\) −7.23495 −0.417710
\(301\) −5.28416 −0.304574
\(302\) −5.48608 −0.315688
\(303\) −3.20112 −0.183899
\(304\) 4.81421 0.276114
\(305\) −18.8047 −1.07676
\(306\) 15.3204 0.875812
\(307\) 5.06946 0.289329 0.144665 0.989481i \(-0.453790\pi\)
0.144665 + 0.989481i \(0.453790\pi\)
\(308\) −0.414473 −0.0236168
\(309\) −6.90472 −0.392796
\(310\) 24.3580 1.38344
\(311\) 3.96686 0.224940 0.112470 0.993655i \(-0.464124\pi\)
0.112470 + 0.993655i \(0.464124\pi\)
\(312\) 5.55128 0.314279
\(313\) 19.7218 1.11474 0.557371 0.830263i \(-0.311810\pi\)
0.557371 + 0.830263i \(0.311810\pi\)
\(314\) 5.22829 0.295049
\(315\) 54.6955 3.08174
\(316\) 2.17160 0.122162
\(317\) −3.35701 −0.188548 −0.0942742 0.995546i \(-0.530053\pi\)
−0.0942742 + 0.995546i \(0.530053\pi\)
\(318\) −4.87159 −0.273185
\(319\) 0.249478 0.0139681
\(320\) −26.1322 −1.46083
\(321\) −4.69363 −0.261973
\(322\) 10.0396 0.559484
\(323\) 43.4783 2.41919
\(324\) −6.75119 −0.375066
\(325\) −39.4119 −2.18618
\(326\) −4.59064 −0.254252
\(327\) 0.382238 0.0211378
\(328\) 14.7640 0.815205
\(329\) 29.1469 1.60692
\(330\) −0.170698 −0.00939659
\(331\) 20.3450 1.11826 0.559130 0.829080i \(-0.311135\pi\)
0.559130 + 0.829080i \(0.311135\pi\)
\(332\) 9.27668 0.509124
\(333\) −5.80537 −0.318132
\(334\) 8.51231 0.465773
\(335\) −43.6834 −2.38668
\(336\) −1.85457 −0.101175
\(337\) −36.6342 −1.99559 −0.997795 0.0663680i \(-0.978859\pi\)
−0.997795 + 0.0663680i \(0.978859\pi\)
\(338\) −1.82476 −0.0992537
\(339\) −4.69776 −0.255147
\(340\) −26.5158 −1.43802
\(341\) −0.477202 −0.0258420
\(342\) 18.7075 1.01158
\(343\) 53.3049 2.87820
\(344\) 3.13756 0.169166
\(345\) −4.88414 −0.262953
\(346\) 17.4575 0.938523
\(347\) −11.6505 −0.625434 −0.312717 0.949846i \(-0.601239\pi\)
−0.312717 + 0.949846i \(0.601239\pi\)
\(348\) −1.98196 −0.106244
\(349\) 34.6972 1.85730 0.928650 0.370958i \(-0.120970\pi\)
0.928650 + 0.370958i \(0.120970\pi\)
\(350\) 56.3341 3.01118
\(351\) 10.6830 0.570214
\(352\) 0.405746 0.0216263
\(353\) 8.18793 0.435800 0.217900 0.975971i \(-0.430079\pi\)
0.217900 + 0.975971i \(0.430079\pi\)
\(354\) −2.41924 −0.128581
\(355\) −13.7805 −0.731395
\(356\) 9.58371 0.507936
\(357\) −16.7490 −0.886454
\(358\) −10.0331 −0.530266
\(359\) −23.7939 −1.25579 −0.627896 0.778297i \(-0.716084\pi\)
−0.627896 + 0.778297i \(0.716084\pi\)
\(360\) −32.4764 −1.71166
\(361\) 34.0904 1.79423
\(362\) −11.3222 −0.595083
\(363\) −6.20657 −0.325761
\(364\) 17.9370 0.940155
\(365\) −60.4173 −3.16239
\(366\) 2.47770 0.129511
\(367\) 19.8468 1.03600 0.517998 0.855382i \(-0.326678\pi\)
0.517998 + 0.855382i \(0.326678\pi\)
\(368\) −1.39328 −0.0726296
\(369\) 13.4091 0.698050
\(370\) −8.50593 −0.442202
\(371\) −44.8072 −2.32627
\(372\) 3.79109 0.196559
\(373\) −26.5360 −1.37398 −0.686991 0.726666i \(-0.741069\pi\)
−0.686991 + 0.726666i \(0.741069\pi\)
\(374\) −0.439768 −0.0227399
\(375\) −15.8251 −0.817204
\(376\) −17.3065 −0.892515
\(377\) −10.7966 −0.556052
\(378\) −15.2699 −0.785398
\(379\) 1.81485 0.0932226 0.0466113 0.998913i \(-0.485158\pi\)
0.0466113 + 0.998913i \(0.485158\pi\)
\(380\) −32.3779 −1.66095
\(381\) 5.63545 0.288713
\(382\) −18.8205 −0.962942
\(383\) 20.8012 1.06289 0.531447 0.847092i \(-0.321649\pi\)
0.531447 + 0.847092i \(0.321649\pi\)
\(384\) −2.50908 −0.128041
\(385\) −1.57002 −0.0800155
\(386\) 7.04513 0.358588
\(387\) 2.84963 0.144855
\(388\) 0.427507 0.0217034
\(389\) −9.93221 −0.503583 −0.251792 0.967782i \(-0.581020\pi\)
−0.251792 + 0.967782i \(0.581020\pi\)
\(390\) 7.38721 0.374066
\(391\) −12.5830 −0.636350
\(392\) −52.3163 −2.64237
\(393\) 0.651573 0.0328675
\(394\) 0.735782 0.0370682
\(395\) 8.22598 0.413894
\(396\) 0.223516 0.0112321
\(397\) −27.7376 −1.39211 −0.696055 0.717989i \(-0.745063\pi\)
−0.696055 + 0.717989i \(0.745063\pi\)
\(398\) 20.7709 1.04115
\(399\) −20.4519 −1.02388
\(400\) −7.81796 −0.390898
\(401\) 37.8599 1.89063 0.945316 0.326157i \(-0.105754\pi\)
0.945316 + 0.326157i \(0.105754\pi\)
\(402\) 5.75569 0.287068
\(403\) 20.6517 1.02874
\(404\) 6.14150 0.305551
\(405\) −25.5734 −1.27075
\(406\) 15.4323 0.765891
\(407\) 0.166641 0.00826010
\(408\) 9.94504 0.492353
\(409\) 3.63895 0.179934 0.0899672 0.995945i \(-0.471324\pi\)
0.0899672 + 0.995945i \(0.471324\pi\)
\(410\) 19.6468 0.970285
\(411\) 2.85238 0.140697
\(412\) 13.2470 0.652635
\(413\) −22.2514 −1.09492
\(414\) −5.41412 −0.266089
\(415\) 35.1399 1.72495
\(416\) −17.5593 −0.860916
\(417\) −1.86533 −0.0913454
\(418\) −0.536992 −0.0262651
\(419\) 7.47377 0.365118 0.182559 0.983195i \(-0.441562\pi\)
0.182559 + 0.983195i \(0.441562\pi\)
\(420\) 12.4729 0.608614
\(421\) 23.1339 1.12748 0.563740 0.825952i \(-0.309362\pi\)
0.563740 + 0.825952i \(0.309362\pi\)
\(422\) −10.0454 −0.489001
\(423\) −15.7183 −0.764249
\(424\) 26.6051 1.29206
\(425\) −70.6058 −3.42488
\(426\) 1.81572 0.0879717
\(427\) 22.7890 1.10284
\(428\) 9.00496 0.435271
\(429\) −0.144724 −0.00698736
\(430\) 4.17523 0.201347
\(431\) 6.19805 0.298549 0.149275 0.988796i \(-0.452306\pi\)
0.149275 + 0.988796i \(0.452306\pi\)
\(432\) 2.11913 0.101957
\(433\) 2.84409 0.136678 0.0683390 0.997662i \(-0.478230\pi\)
0.0683390 + 0.997662i \(0.478230\pi\)
\(434\) −29.5189 −1.41695
\(435\) −7.50762 −0.359963
\(436\) −0.733343 −0.0351208
\(437\) −15.3649 −0.735001
\(438\) 7.96055 0.380370
\(439\) 5.09779 0.243304 0.121652 0.992573i \(-0.461181\pi\)
0.121652 + 0.992573i \(0.461181\pi\)
\(440\) 0.932225 0.0444421
\(441\) −47.5152 −2.26263
\(442\) 19.0317 0.905245
\(443\) 13.7065 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(444\) −1.32387 −0.0628280
\(445\) 36.3029 1.72092
\(446\) −17.8556 −0.845487
\(447\) −13.4156 −0.634534
\(448\) 31.6690 1.49622
\(449\) 30.5992 1.44407 0.722034 0.691858i \(-0.243207\pi\)
0.722034 + 0.691858i \(0.243207\pi\)
\(450\) −30.3797 −1.43211
\(451\) −0.384904 −0.0181244
\(452\) 9.01289 0.423931
\(453\) −3.23439 −0.151965
\(454\) 26.7714 1.25645
\(455\) 67.9451 3.18531
\(456\) 12.1437 0.568680
\(457\) 17.9572 0.840001 0.420001 0.907524i \(-0.362030\pi\)
0.420001 + 0.907524i \(0.362030\pi\)
\(458\) −9.32251 −0.435612
\(459\) 19.1384 0.893302
\(460\) 9.37046 0.436900
\(461\) −18.3934 −0.856668 −0.428334 0.903620i \(-0.640899\pi\)
−0.428334 + 0.903620i \(0.640899\pi\)
\(462\) 0.206864 0.00962421
\(463\) −34.4958 −1.60316 −0.801579 0.597889i \(-0.796006\pi\)
−0.801579 + 0.597889i \(0.796006\pi\)
\(464\) −2.14167 −0.0994244
\(465\) 14.3606 0.665957
\(466\) 18.2806 0.846835
\(467\) −17.7648 −0.822056 −0.411028 0.911623i \(-0.634830\pi\)
−0.411028 + 0.911623i \(0.634830\pi\)
\(468\) −9.67302 −0.447135
\(469\) 52.9389 2.44449
\(470\) −23.0301 −1.06230
\(471\) 3.08241 0.142030
\(472\) 13.2121 0.608138
\(473\) −0.0817977 −0.00376106
\(474\) −1.08385 −0.0497829
\(475\) −86.2153 −3.95583
\(476\) 32.1339 1.47285
\(477\) 24.1635 1.10637
\(478\) −25.8215 −1.18105
\(479\) 24.5625 1.12229 0.561145 0.827717i \(-0.310361\pi\)
0.561145 + 0.827717i \(0.310361\pi\)
\(480\) −12.2102 −0.557318
\(481\) −7.21167 −0.328824
\(482\) −6.26414 −0.285324
\(483\) 5.91897 0.269323
\(484\) 11.9076 0.541255
\(485\) 1.61939 0.0735327
\(486\) 12.5830 0.570778
\(487\) −35.4671 −1.60717 −0.803584 0.595191i \(-0.797076\pi\)
−0.803584 + 0.595191i \(0.797076\pi\)
\(488\) −13.5314 −0.612537
\(489\) −2.70648 −0.122391
\(490\) −69.6184 −3.14504
\(491\) 27.8265 1.25579 0.627897 0.778296i \(-0.283916\pi\)
0.627897 + 0.778296i \(0.283916\pi\)
\(492\) 3.05784 0.137858
\(493\) −19.3419 −0.871115
\(494\) 23.2392 1.04558
\(495\) 0.846675 0.0380552
\(496\) 4.09659 0.183942
\(497\) 16.7003 0.749112
\(498\) −4.63001 −0.207476
\(499\) −32.7456 −1.46590 −0.732948 0.680285i \(-0.761856\pi\)
−0.732948 + 0.680285i \(0.761856\pi\)
\(500\) 30.3612 1.35780
\(501\) 5.01855 0.224212
\(502\) −21.3271 −0.951875
\(503\) −8.43515 −0.376105 −0.188052 0.982159i \(-0.560217\pi\)
−0.188052 + 0.982159i \(0.560217\pi\)
\(504\) 39.3574 1.75312
\(505\) 23.2639 1.03523
\(506\) 0.155411 0.00690884
\(507\) −1.07581 −0.0477784
\(508\) −10.8119 −0.479700
\(509\) −11.6347 −0.515698 −0.257849 0.966185i \(-0.583014\pi\)
−0.257849 + 0.966185i \(0.583014\pi\)
\(510\) 13.2341 0.586015
\(511\) 73.2184 3.23899
\(512\) −7.38435 −0.326345
\(513\) 23.3695 1.03179
\(514\) −10.4710 −0.461857
\(515\) 50.1796 2.21118
\(516\) 0.649834 0.0286074
\(517\) 0.451188 0.0198432
\(518\) 10.3081 0.452914
\(519\) 10.2923 0.451783
\(520\) −40.3436 −1.76918
\(521\) −6.18104 −0.270796 −0.135398 0.990791i \(-0.543231\pi\)
−0.135398 + 0.990791i \(0.543231\pi\)
\(522\) −8.32228 −0.364256
\(523\) −4.44108 −0.194195 −0.0970974 0.995275i \(-0.530956\pi\)
−0.0970974 + 0.995275i \(0.530956\pi\)
\(524\) −1.25007 −0.0546098
\(525\) 33.2125 1.44951
\(526\) 9.88839 0.431154
\(527\) 36.9972 1.61162
\(528\) −0.0287083 −0.00124937
\(529\) −18.5533 −0.806664
\(530\) 35.4039 1.53785
\(531\) 11.9997 0.520741
\(532\) 39.2380 1.70118
\(533\) 16.6573 0.721510
\(534\) −4.78325 −0.206992
\(535\) 34.1106 1.47473
\(536\) −31.4334 −1.35772
\(537\) −5.91515 −0.255258
\(538\) 6.09407 0.262734
\(539\) 1.36391 0.0587477
\(540\) −14.2522 −0.613316
\(541\) 5.89672 0.253520 0.126760 0.991933i \(-0.459542\pi\)
0.126760 + 0.991933i \(0.459542\pi\)
\(542\) −28.6291 −1.22973
\(543\) −6.67517 −0.286459
\(544\) −31.4573 −1.34872
\(545\) −2.77789 −0.118992
\(546\) −8.95240 −0.383127
\(547\) −1.00000 −0.0427569
\(548\) −5.47243 −0.233771
\(549\) −12.2896 −0.524508
\(550\) 0.872039 0.0371839
\(551\) −23.6180 −1.00616
\(552\) −3.51449 −0.149587
\(553\) −9.96888 −0.423920
\(554\) 6.72776 0.285835
\(555\) −5.01479 −0.212866
\(556\) 3.57872 0.151772
\(557\) −1.66388 −0.0705009 −0.0352504 0.999379i \(-0.511223\pi\)
−0.0352504 + 0.999379i \(0.511223\pi\)
\(558\) 15.9189 0.673900
\(559\) 3.53993 0.149723
\(560\) 13.4780 0.569548
\(561\) −0.259271 −0.0109465
\(562\) 18.8810 0.796448
\(563\) −5.94258 −0.250450 −0.125225 0.992128i \(-0.539965\pi\)
−0.125225 + 0.992128i \(0.539965\pi\)
\(564\) −3.58442 −0.150931
\(565\) 34.1407 1.43631
\(566\) −3.82985 −0.160980
\(567\) 30.9918 1.30153
\(568\) −9.91611 −0.416071
\(569\) 16.7090 0.700476 0.350238 0.936661i \(-0.386101\pi\)
0.350238 + 0.936661i \(0.386101\pi\)
\(570\) 16.1599 0.676862
\(571\) −12.6323 −0.528644 −0.264322 0.964434i \(-0.585148\pi\)
−0.264322 + 0.964434i \(0.585148\pi\)
\(572\) 0.277661 0.0116096
\(573\) −11.0959 −0.463538
\(574\) −23.8095 −0.993789
\(575\) 24.9515 1.04055
\(576\) −17.0783 −0.711598
\(577\) −28.5904 −1.19023 −0.595116 0.803640i \(-0.702894\pi\)
−0.595116 + 0.803640i \(0.702894\pi\)
\(578\) 17.8166 0.741073
\(579\) 4.15355 0.172616
\(580\) 14.4037 0.598083
\(581\) −42.5853 −1.76673
\(582\) −0.213370 −0.00884446
\(583\) −0.693605 −0.0287262
\(584\) −43.4747 −1.79900
\(585\) −36.6412 −1.51493
\(586\) −23.6404 −0.976578
\(587\) −22.0036 −0.908187 −0.454094 0.890954i \(-0.650037\pi\)
−0.454094 + 0.890954i \(0.650037\pi\)
\(588\) −10.8355 −0.446847
\(589\) 45.1766 1.86147
\(590\) 17.5817 0.723827
\(591\) 0.433791 0.0178438
\(592\) −1.43055 −0.0587951
\(593\) 21.0528 0.864537 0.432268 0.901745i \(-0.357713\pi\)
0.432268 + 0.901745i \(0.357713\pi\)
\(594\) −0.236374 −0.00969856
\(595\) 121.723 4.99014
\(596\) 25.7384 1.05429
\(597\) 12.2458 0.501186
\(598\) −6.72564 −0.275032
\(599\) 10.0460 0.410468 0.205234 0.978713i \(-0.434204\pi\)
0.205234 + 0.978713i \(0.434204\pi\)
\(600\) −19.7205 −0.805087
\(601\) −24.1270 −0.984159 −0.492079 0.870550i \(-0.663763\pi\)
−0.492079 + 0.870550i \(0.663763\pi\)
\(602\) −5.05986 −0.206225
\(603\) −28.5487 −1.16259
\(604\) 6.20534 0.252492
\(605\) 45.1059 1.83381
\(606\) −3.06524 −0.124517
\(607\) −2.94817 −0.119662 −0.0598312 0.998209i \(-0.519056\pi\)
−0.0598312 + 0.998209i \(0.519056\pi\)
\(608\) −38.4118 −1.55780
\(609\) 9.09832 0.368682
\(610\) −18.0065 −0.729062
\(611\) −19.5259 −0.789933
\(612\) −17.3291 −0.700486
\(613\) −6.60126 −0.266622 −0.133311 0.991074i \(-0.542561\pi\)
−0.133311 + 0.991074i \(0.542561\pi\)
\(614\) 4.85427 0.195902
\(615\) 11.5830 0.467073
\(616\) −1.12974 −0.0455186
\(617\) 17.3067 0.696742 0.348371 0.937357i \(-0.386735\pi\)
0.348371 + 0.937357i \(0.386735\pi\)
\(618\) −6.61163 −0.265959
\(619\) 15.8845 0.638451 0.319225 0.947679i \(-0.396577\pi\)
0.319225 + 0.947679i \(0.396577\pi\)
\(620\) −27.5515 −1.10650
\(621\) −6.76334 −0.271403
\(622\) 3.79847 0.152305
\(623\) −43.9947 −1.76261
\(624\) 1.24240 0.0497358
\(625\) 55.8453 2.23381
\(626\) 18.8847 0.754783
\(627\) −0.316591 −0.0126434
\(628\) −5.91375 −0.235984
\(629\) −12.9196 −0.515138
\(630\) 52.3738 2.08662
\(631\) −17.6658 −0.703263 −0.351631 0.936139i \(-0.614373\pi\)
−0.351631 + 0.936139i \(0.614373\pi\)
\(632\) 5.91920 0.235453
\(633\) −5.92238 −0.235394
\(634\) −3.21451 −0.127665
\(635\) −40.9553 −1.62526
\(636\) 5.51029 0.218497
\(637\) −59.0254 −2.33867
\(638\) 0.238888 0.00945768
\(639\) −9.00611 −0.356276
\(640\) 18.2345 0.720784
\(641\) 49.6629 1.96157 0.980783 0.195102i \(-0.0625037\pi\)
0.980783 + 0.195102i \(0.0625037\pi\)
\(642\) −4.49439 −0.177380
\(643\) −18.7162 −0.738096 −0.369048 0.929410i \(-0.620316\pi\)
−0.369048 + 0.929410i \(0.620316\pi\)
\(644\) −11.3558 −0.447483
\(645\) 2.46156 0.0969239
\(646\) 41.6327 1.63802
\(647\) −0.632028 −0.0248476 −0.0124238 0.999923i \(-0.503955\pi\)
−0.0124238 + 0.999923i \(0.503955\pi\)
\(648\) −18.4019 −0.722896
\(649\) −0.344446 −0.0135207
\(650\) −37.7389 −1.48024
\(651\) −17.4033 −0.682088
\(652\) 5.19251 0.203354
\(653\) 12.2114 0.477869 0.238934 0.971036i \(-0.423202\pi\)
0.238934 + 0.971036i \(0.423202\pi\)
\(654\) 0.366013 0.0143122
\(655\) −4.73526 −0.185022
\(656\) 3.30424 0.129009
\(657\) −39.4850 −1.54046
\(658\) 27.9097 1.08803
\(659\) −8.51707 −0.331778 −0.165889 0.986144i \(-0.553049\pi\)
−0.165889 + 0.986144i \(0.553049\pi\)
\(660\) 0.193077 0.00751552
\(661\) −5.76560 −0.224256 −0.112128 0.993694i \(-0.535767\pi\)
−0.112128 + 0.993694i \(0.535767\pi\)
\(662\) 19.4814 0.757165
\(663\) 11.2204 0.435764
\(664\) 25.2857 0.981277
\(665\) 148.633 5.76374
\(666\) −5.55895 −0.215405
\(667\) 6.83527 0.264663
\(668\) −9.62833 −0.372531
\(669\) −10.5270 −0.406998
\(670\) −41.8291 −1.61600
\(671\) 0.352769 0.0136185
\(672\) 14.7973 0.570819
\(673\) −18.9482 −0.730400 −0.365200 0.930929i \(-0.618999\pi\)
−0.365200 + 0.930929i \(0.618999\pi\)
\(674\) −35.0791 −1.35120
\(675\) −37.9504 −1.46071
\(676\) 2.06400 0.0793845
\(677\) 11.6427 0.447466 0.223733 0.974650i \(-0.428176\pi\)
0.223733 + 0.974650i \(0.428176\pi\)
\(678\) −4.49835 −0.172758
\(679\) −1.96250 −0.0753139
\(680\) −72.2749 −2.77162
\(681\) 15.7835 0.604824
\(682\) −0.456946 −0.0174974
\(683\) 32.4010 1.23979 0.619895 0.784685i \(-0.287175\pi\)
0.619895 + 0.784685i \(0.287175\pi\)
\(684\) −21.1602 −0.809079
\(685\) −20.7295 −0.792032
\(686\) 51.0422 1.94880
\(687\) −5.49622 −0.209694
\(688\) 0.702199 0.0267711
\(689\) 30.0169 1.14355
\(690\) −4.67682 −0.178043
\(691\) 33.5061 1.27463 0.637317 0.770602i \(-0.280044\pi\)
0.637317 + 0.770602i \(0.280044\pi\)
\(692\) −19.7463 −0.750643
\(693\) −1.02607 −0.0389770
\(694\) −11.1560 −0.423476
\(695\) 13.5561 0.514213
\(696\) −5.40228 −0.204773
\(697\) 29.8414 1.13032
\(698\) 33.2244 1.25756
\(699\) 10.7776 0.407647
\(700\) −63.7199 −2.40839
\(701\) −18.1131 −0.684122 −0.342061 0.939678i \(-0.611125\pi\)
−0.342061 + 0.939678i \(0.611125\pi\)
\(702\) 10.2295 0.386087
\(703\) −15.7759 −0.594998
\(704\) 0.490228 0.0184762
\(705\) −13.5777 −0.511367
\(706\) 7.84037 0.295076
\(707\) −28.1930 −1.06031
\(708\) 2.73642 0.102841
\(709\) −25.2354 −0.947736 −0.473868 0.880596i \(-0.657143\pi\)
−0.473868 + 0.880596i \(0.657143\pi\)
\(710\) −13.1956 −0.495222
\(711\) 5.37599 0.201615
\(712\) 26.1226 0.978987
\(713\) −13.0745 −0.489645
\(714\) −16.0381 −0.600210
\(715\) 1.05177 0.0393341
\(716\) 11.3485 0.424114
\(717\) −15.2234 −0.568530
\(718\) −22.7839 −0.850287
\(719\) 1.53980 0.0574250 0.0287125 0.999588i \(-0.490859\pi\)
0.0287125 + 0.999588i \(0.490859\pi\)
\(720\) −7.26836 −0.270876
\(721\) −60.8115 −2.26474
\(722\) 32.6433 1.21486
\(723\) −3.69311 −0.137348
\(724\) 12.8067 0.475955
\(725\) 38.3540 1.42443
\(726\) −5.94312 −0.220570
\(727\) −29.6777 −1.10068 −0.550342 0.834939i \(-0.685503\pi\)
−0.550342 + 0.834939i \(0.685503\pi\)
\(728\) 48.8915 1.81204
\(729\) −11.2812 −0.417823
\(730\) −57.8527 −2.14123
\(731\) 6.34173 0.234557
\(732\) −2.80254 −0.103585
\(733\) −34.6887 −1.28126 −0.640629 0.767851i \(-0.721326\pi\)
−0.640629 + 0.767851i \(0.721326\pi\)
\(734\) 19.0044 0.701464
\(735\) −41.0445 −1.51395
\(736\) 11.1167 0.409768
\(737\) 0.819482 0.0301860
\(738\) 12.8399 0.472644
\(739\) −3.96845 −0.145982 −0.0729909 0.997333i \(-0.523254\pi\)
−0.0729909 + 0.997333i \(0.523254\pi\)
\(740\) 9.62112 0.353679
\(741\) 13.7010 0.503319
\(742\) −42.9052 −1.57510
\(743\) 47.6574 1.74838 0.874190 0.485584i \(-0.161393\pi\)
0.874190 + 0.485584i \(0.161393\pi\)
\(744\) 10.3335 0.378845
\(745\) 97.4966 3.57200
\(746\) −25.4096 −0.930312
\(747\) 22.9653 0.840255
\(748\) 0.497425 0.0181877
\(749\) −41.3379 −1.51045
\(750\) −15.1534 −0.553322
\(751\) 11.3760 0.415117 0.207558 0.978223i \(-0.433448\pi\)
0.207558 + 0.978223i \(0.433448\pi\)
\(752\) −3.87327 −0.141243
\(753\) −12.5737 −0.458210
\(754\) −10.3383 −0.376498
\(755\) 23.5057 0.855461
\(756\) 17.2719 0.628172
\(757\) 35.7312 1.29867 0.649337 0.760501i \(-0.275047\pi\)
0.649337 + 0.760501i \(0.275047\pi\)
\(758\) 1.73781 0.0631202
\(759\) 0.0916244 0.00332575
\(760\) −88.2534 −3.20129
\(761\) 45.2992 1.64209 0.821047 0.570861i \(-0.193390\pi\)
0.821047 + 0.570861i \(0.193390\pi\)
\(762\) 5.39624 0.195485
\(763\) 3.36646 0.121874
\(764\) 21.2880 0.770174
\(765\) −65.6422 −2.37330
\(766\) 19.9183 0.719676
\(767\) 14.9065 0.538242
\(768\) −9.59415 −0.346199
\(769\) 20.9580 0.755766 0.377883 0.925853i \(-0.376652\pi\)
0.377883 + 0.925853i \(0.376652\pi\)
\(770\) −1.50337 −0.0541778
\(771\) −6.17334 −0.222327
\(772\) −7.96879 −0.286803
\(773\) 9.94311 0.357629 0.178814 0.983883i \(-0.442774\pi\)
0.178814 + 0.983883i \(0.442774\pi\)
\(774\) 2.72867 0.0980799
\(775\) −73.3637 −2.63530
\(776\) 1.16527 0.0418307
\(777\) 6.07731 0.218022
\(778\) −9.51061 −0.340972
\(779\) 36.4387 1.30555
\(780\) −8.35573 −0.299183
\(781\) 0.258517 0.00925048
\(782\) −12.0489 −0.430867
\(783\) −10.3962 −0.371531
\(784\) −11.7086 −0.418164
\(785\) −22.4012 −0.799533
\(786\) 0.623915 0.0222543
\(787\) 43.0795 1.53562 0.767808 0.640680i \(-0.221347\pi\)
0.767808 + 0.640680i \(0.221347\pi\)
\(788\) −0.832248 −0.0296476
\(789\) 5.82984 0.207548
\(790\) 7.87680 0.280244
\(791\) −41.3743 −1.47110
\(792\) 0.609244 0.0216486
\(793\) −15.2667 −0.542135
\(794\) −26.5602 −0.942586
\(795\) 20.8729 0.740285
\(796\) −23.4941 −0.832727
\(797\) 19.8313 0.702460 0.351230 0.936289i \(-0.385763\pi\)
0.351230 + 0.936289i \(0.385763\pi\)
\(798\) −19.5838 −0.693258
\(799\) −34.9804 −1.23752
\(800\) 62.3782 2.20540
\(801\) 23.7253 0.838294
\(802\) 36.2528 1.28013
\(803\) 1.13340 0.0399970
\(804\) −6.51031 −0.229601
\(805\) −43.0157 −1.51611
\(806\) 19.7751 0.696548
\(807\) 3.59284 0.126474
\(808\) 16.7401 0.588914
\(809\) 1.62413 0.0571014 0.0285507 0.999592i \(-0.490911\pi\)
0.0285507 + 0.999592i \(0.490911\pi\)
\(810\) −24.4879 −0.860415
\(811\) 41.7383 1.46563 0.732814 0.680428i \(-0.238206\pi\)
0.732814 + 0.680428i \(0.238206\pi\)
\(812\) −17.4556 −0.612570
\(813\) −16.8787 −0.591962
\(814\) 0.159568 0.00559285
\(815\) 19.6691 0.688979
\(816\) 2.22574 0.0779165
\(817\) 7.74375 0.270920
\(818\) 3.48448 0.121832
\(819\) 44.4047 1.55162
\(820\) −22.2226 −0.776047
\(821\) −34.5850 −1.20703 −0.603513 0.797353i \(-0.706233\pi\)
−0.603513 + 0.797353i \(0.706233\pi\)
\(822\) 2.73130 0.0952651
\(823\) −22.0897 −0.769999 −0.384999 0.922917i \(-0.625798\pi\)
−0.384999 + 0.922917i \(0.625798\pi\)
\(824\) 36.1079 1.25788
\(825\) 0.514123 0.0178995
\(826\) −21.3069 −0.741361
\(827\) 4.29638 0.149400 0.0746999 0.997206i \(-0.476200\pi\)
0.0746999 + 0.997206i \(0.476200\pi\)
\(828\) 6.12395 0.212822
\(829\) 9.89112 0.343533 0.171766 0.985138i \(-0.445053\pi\)
0.171766 + 0.985138i \(0.445053\pi\)
\(830\) 33.6483 1.16795
\(831\) 3.96644 0.137594
\(832\) −21.2154 −0.735513
\(833\) −105.743 −3.66378
\(834\) −1.78615 −0.0618492
\(835\) −36.4720 −1.26216
\(836\) 0.607396 0.0210072
\(837\) 19.8859 0.687358
\(838\) 7.15653 0.247218
\(839\) −45.2914 −1.56363 −0.781817 0.623509i \(-0.785707\pi\)
−0.781817 + 0.623509i \(0.785707\pi\)
\(840\) 33.9977 1.17303
\(841\) −18.4932 −0.637697
\(842\) 22.1520 0.763407
\(843\) 11.1316 0.383391
\(844\) 11.3624 0.391109
\(845\) 7.81838 0.268960
\(846\) −15.0511 −0.517466
\(847\) −54.6628 −1.87823
\(848\) 5.95432 0.204472
\(849\) −2.25794 −0.0774922
\(850\) −67.6087 −2.31896
\(851\) 4.56568 0.156510
\(852\) −2.05377 −0.0703610
\(853\) −54.9342 −1.88091 −0.940456 0.339916i \(-0.889601\pi\)
−0.940456 + 0.339916i \(0.889601\pi\)
\(854\) 21.8217 0.746723
\(855\) −80.1544 −2.74122
\(856\) 24.5451 0.838934
\(857\) −0.342745 −0.0117079 −0.00585397 0.999983i \(-0.501863\pi\)
−0.00585397 + 0.999983i \(0.501863\pi\)
\(858\) −0.138581 −0.00473108
\(859\) 13.8090 0.471157 0.235578 0.971855i \(-0.424302\pi\)
0.235578 + 0.971855i \(0.424302\pi\)
\(860\) −4.72263 −0.161040
\(861\) −14.0372 −0.478387
\(862\) 5.93495 0.202145
\(863\) −27.0862 −0.922025 −0.461013 0.887394i \(-0.652514\pi\)
−0.461013 + 0.887394i \(0.652514\pi\)
\(864\) −16.9082 −0.575229
\(865\) −74.7988 −2.54324
\(866\) 2.72336 0.0925435
\(867\) 10.5040 0.356735
\(868\) 33.3891 1.13330
\(869\) −0.154316 −0.00523481
\(870\) −7.18894 −0.243728
\(871\) −35.4644 −1.20167
\(872\) −1.99890 −0.0676912
\(873\) 1.05833 0.0358191
\(874\) −14.7127 −0.497663
\(875\) −139.375 −4.71175
\(876\) −9.00423 −0.304225
\(877\) 37.0243 1.25022 0.625111 0.780536i \(-0.285054\pi\)
0.625111 + 0.780536i \(0.285054\pi\)
\(878\) 4.88140 0.164739
\(879\) −13.9376 −0.470102
\(880\) 0.208636 0.00703311
\(881\) 6.89398 0.232264 0.116132 0.993234i \(-0.462950\pi\)
0.116132 + 0.993234i \(0.462950\pi\)
\(882\) −45.4983 −1.53201
\(883\) 9.37818 0.315601 0.157801 0.987471i \(-0.449560\pi\)
0.157801 + 0.987471i \(0.449560\pi\)
\(884\) −21.5269 −0.724027
\(885\) 10.3655 0.348434
\(886\) 13.1247 0.440934
\(887\) 33.7032 1.13164 0.565822 0.824528i \(-0.308559\pi\)
0.565822 + 0.824528i \(0.308559\pi\)
\(888\) −3.60851 −0.121094
\(889\) 49.6328 1.66463
\(890\) 34.7620 1.16522
\(891\) 0.479746 0.0160721
\(892\) 20.1966 0.676232
\(893\) −42.7138 −1.42936
\(894\) −12.8461 −0.429638
\(895\) 42.9879 1.43693
\(896\) −22.0980 −0.738244
\(897\) −3.96520 −0.132394
\(898\) 29.3004 0.977766
\(899\) −20.0974 −0.670286
\(900\) 34.3627 1.14542
\(901\) 53.7748 1.79150
\(902\) −0.368566 −0.0122719
\(903\) −2.98311 −0.0992717
\(904\) 24.5667 0.817076
\(905\) 48.5114 1.61257
\(906\) −3.09710 −0.102894
\(907\) −25.0104 −0.830455 −0.415228 0.909718i \(-0.636298\pi\)
−0.415228 + 0.909718i \(0.636298\pi\)
\(908\) −30.2814 −1.00492
\(909\) 15.2038 0.504280
\(910\) 65.0609 2.15675
\(911\) −40.4013 −1.33855 −0.669277 0.743013i \(-0.733396\pi\)
−0.669277 + 0.743013i \(0.733396\pi\)
\(912\) 2.71781 0.0899955
\(913\) −0.659210 −0.0218167
\(914\) 17.1949 0.568758
\(915\) −10.6160 −0.350954
\(916\) 10.5448 0.348409
\(917\) 5.73856 0.189504
\(918\) 18.3260 0.604847
\(919\) 8.97289 0.295988 0.147994 0.988988i \(-0.452718\pi\)
0.147994 + 0.988988i \(0.452718\pi\)
\(920\) 25.5413 0.842073
\(921\) 2.86190 0.0943028
\(922\) −17.6127 −0.580043
\(923\) −11.1878 −0.368250
\(924\) −0.233986 −0.00769757
\(925\) 25.6189 0.842346
\(926\) −33.0316 −1.08548
\(927\) 32.7943 1.07710
\(928\) 17.0880 0.560942
\(929\) −1.61228 −0.0528973 −0.0264487 0.999650i \(-0.508420\pi\)
−0.0264487 + 0.999650i \(0.508420\pi\)
\(930\) 13.7510 0.450914
\(931\) −129.121 −4.23176
\(932\) −20.6774 −0.677310
\(933\) 2.23944 0.0733161
\(934\) −17.0107 −0.556607
\(935\) 1.88424 0.0616212
\(936\) −26.3660 −0.861801
\(937\) −12.9284 −0.422351 −0.211176 0.977448i \(-0.567729\pi\)
−0.211176 + 0.977448i \(0.567729\pi\)
\(938\) 50.6918 1.65514
\(939\) 11.1337 0.363335
\(940\) 26.0496 0.849643
\(941\) −18.4517 −0.601510 −0.300755 0.953701i \(-0.597239\pi\)
−0.300755 + 0.953701i \(0.597239\pi\)
\(942\) 2.95157 0.0961672
\(943\) −10.5457 −0.343415
\(944\) 2.95693 0.0962399
\(945\) 65.4256 2.12829
\(946\) −0.0783255 −0.00254658
\(947\) 12.2973 0.399607 0.199803 0.979836i \(-0.435970\pi\)
0.199803 + 0.979836i \(0.435970\pi\)
\(948\) 1.22595 0.0398170
\(949\) −49.0499 −1.59223
\(950\) −82.5556 −2.67846
\(951\) −1.89516 −0.0614548
\(952\) 87.5883 2.83875
\(953\) −2.51896 −0.0815970 −0.0407985 0.999167i \(-0.512990\pi\)
−0.0407985 + 0.999167i \(0.512990\pi\)
\(954\) 23.1378 0.749114
\(955\) 80.6387 2.60941
\(956\) 29.2069 0.944619
\(957\) 0.140840 0.00455271
\(958\) 23.5199 0.759894
\(959\) 25.1216 0.811218
\(960\) −14.7526 −0.476138
\(961\) 7.44239 0.240077
\(962\) −6.90555 −0.222644
\(963\) 22.2926 0.718369
\(964\) 7.08541 0.228206
\(965\) −30.1857 −0.971711
\(966\) 5.66773 0.182356
\(967\) −45.0973 −1.45023 −0.725116 0.688627i \(-0.758214\pi\)
−0.725116 + 0.688627i \(0.758214\pi\)
\(968\) 32.4570 1.04321
\(969\) 24.5451 0.788503
\(970\) 1.55065 0.0497884
\(971\) 35.4672 1.13820 0.569098 0.822270i \(-0.307293\pi\)
0.569098 + 0.822270i \(0.307293\pi\)
\(972\) −14.2328 −0.456516
\(973\) −16.4284 −0.526669
\(974\) −33.9616 −1.08820
\(975\) −22.2495 −0.712554
\(976\) −3.02838 −0.0969360
\(977\) 8.98700 0.287520 0.143760 0.989613i \(-0.454081\pi\)
0.143760 + 0.989613i \(0.454081\pi\)
\(978\) −2.59159 −0.0828699
\(979\) −0.681028 −0.0217658
\(980\) 78.7459 2.51545
\(981\) −1.81546 −0.0579631
\(982\) 26.6454 0.850287
\(983\) −8.91145 −0.284231 −0.142116 0.989850i \(-0.545390\pi\)
−0.142116 + 0.989850i \(0.545390\pi\)
\(984\) 8.33484 0.265705
\(985\) −3.15254 −0.100448
\(986\) −18.5209 −0.589825
\(987\) 16.4546 0.523754
\(988\) −26.2860 −0.836270
\(989\) −2.24111 −0.0712633
\(990\) 0.810735 0.0257669
\(991\) 24.7481 0.786149 0.393074 0.919507i \(-0.371411\pi\)
0.393074 + 0.919507i \(0.371411\pi\)
\(992\) −32.6860 −1.03778
\(993\) 11.4855 0.364481
\(994\) 15.9914 0.507218
\(995\) −88.9954 −2.82134
\(996\) 5.23704 0.165942
\(997\) −39.8404 −1.26176 −0.630878 0.775882i \(-0.717305\pi\)
−0.630878 + 0.775882i \(0.717305\pi\)
\(998\) −31.3557 −0.992546
\(999\) −6.94426 −0.219707
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.1.13 18
3.2 odd 2 4923.2.a.l.1.6 18
4.3 odd 2 8752.2.a.s.1.5 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.13 18 1.1 even 1 trivial
4923.2.a.l.1.6 18 3.2 odd 2
8752.2.a.s.1.5 18 4.3 odd 2