Properties

Label 637.2.a.k.1.1
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.746052.1
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.00852\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00852 q^{2} -1.75906 q^{3} +2.03417 q^{4} -0.905722 q^{5} +3.53311 q^{6} -0.0686323 q^{8} +0.0942784 q^{9} +O(q^{10})\) \(q-2.00852 q^{2} -1.75906 q^{3} +2.03417 q^{4} -0.905722 q^{5} +3.53311 q^{6} -0.0686323 q^{8} +0.0942784 q^{9} +1.81916 q^{10} +0.716361 q^{11} -3.57822 q^{12} -1.00000 q^{13} +1.59322 q^{15} -3.93049 q^{16} -2.35227 q^{17} -0.189360 q^{18} -6.63591 q^{19} -1.84239 q^{20} -1.43883 q^{22} +3.75906 q^{23} +0.120728 q^{24} -4.17967 q^{25} +2.00852 q^{26} +5.11133 q^{27} +3.25799 q^{29} -3.20001 q^{30} -1.57050 q^{31} +8.03175 q^{32} -1.26012 q^{33} +4.72459 q^{34} +0.191778 q^{36} +5.20883 q^{37} +13.3284 q^{38} +1.75906 q^{39} +0.0621618 q^{40} -4.92168 q^{41} -9.43766 q^{43} +1.45720 q^{44} -0.0853900 q^{45} -7.55016 q^{46} +8.31986 q^{47} +6.91395 q^{48} +8.39497 q^{50} +4.13778 q^{51} -2.03417 q^{52} +14.0833 q^{53} -10.2662 q^{54} -0.648824 q^{55} +11.6729 q^{57} -6.54376 q^{58} -0.716361 q^{59} +3.24087 q^{60} +11.6527 q^{61} +3.15439 q^{62} -8.27099 q^{64} +0.905722 q^{65} +2.53098 q^{66} +9.39174 q^{67} -4.78492 q^{68} -6.61239 q^{69} +10.9914 q^{71} -0.00647055 q^{72} +3.47300 q^{73} -10.4621 q^{74} +7.35227 q^{75} -13.4986 q^{76} -3.53311 q^{78} +13.0082 q^{79} +3.55993 q^{80} -9.27395 q^{81} +9.88531 q^{82} -3.54083 q^{83} +2.13050 q^{85} +18.9558 q^{86} -5.73099 q^{87} -0.0491655 q^{88} -12.0501 q^{89} +0.171508 q^{90} +7.64656 q^{92} +2.76260 q^{93} -16.7106 q^{94} +6.01029 q^{95} -14.1283 q^{96} -7.43766 q^{97} +0.0675374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} + 9 q^{8} + 3 q^{9} + 5 q^{10} + 11 q^{11} - 5 q^{12} - 5 q^{13} + 10 q^{16} + 5 q^{17} + 9 q^{18} - 9 q^{19} - q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} - 4 q^{26} - 3 q^{29} - 13 q^{30} + 6 q^{31} + 22 q^{32} - 8 q^{33} + 22 q^{34} + 7 q^{36} + 4 q^{37} + 10 q^{38} - 28 q^{40} - 14 q^{41} + 2 q^{43} + 32 q^{45} + 3 q^{46} - q^{47} + 23 q^{48} + 9 q^{50} - 8 q^{51} - 8 q^{52} + 17 q^{53} - 23 q^{54} - 16 q^{57} - 27 q^{58} - 11 q^{59} - 29 q^{60} + 11 q^{61} + 23 q^{62} + 9 q^{64} + 2 q^{65} - 21 q^{66} + 13 q^{67} + 32 q^{68} - 18 q^{69} + 15 q^{71} - 19 q^{72} - 33 q^{74} + 20 q^{75} - 8 q^{76} - 5 q^{78} + 2 q^{79} - 55 q^{80} - 19 q^{81} - 34 q^{82} - 6 q^{83} - 22 q^{85} + 28 q^{86} + 8 q^{87} - 3 q^{88} + 4 q^{89} + 34 q^{90} + 21 q^{92} + 18 q^{93} - 20 q^{94} - 12 q^{95} + 37 q^{96} + 12 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00852 −1.42024 −0.710121 0.704080i \(-0.751360\pi\)
−0.710121 + 0.704080i \(0.751360\pi\)
\(3\) −1.75906 −1.01559 −0.507796 0.861477i \(-0.669540\pi\)
−0.507796 + 0.861477i \(0.669540\pi\)
\(4\) 2.03417 1.01709
\(5\) −0.905722 −0.405051 −0.202526 0.979277i \(-0.564915\pi\)
−0.202526 + 0.979277i \(0.564915\pi\)
\(6\) 3.53311 1.44238
\(7\) 0 0
\(8\) −0.0686323 −0.0242652
\(9\) 0.0942784 0.0314261
\(10\) 1.81916 0.575270
\(11\) 0.716361 0.215991 0.107996 0.994151i \(-0.465557\pi\)
0.107996 + 0.994151i \(0.465557\pi\)
\(12\) −3.57822 −1.03294
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.59322 0.411366
\(16\) −3.93049 −0.982623
\(17\) −2.35227 −0.570510 −0.285255 0.958452i \(-0.592078\pi\)
−0.285255 + 0.958452i \(0.592078\pi\)
\(18\) −0.189360 −0.0446327
\(19\) −6.63591 −1.52238 −0.761191 0.648528i \(-0.775385\pi\)
−0.761191 + 0.648528i \(0.775385\pi\)
\(20\) −1.84239 −0.411971
\(21\) 0 0
\(22\) −1.43883 −0.306759
\(23\) 3.75906 0.783817 0.391909 0.920004i \(-0.371815\pi\)
0.391909 + 0.920004i \(0.371815\pi\)
\(24\) 0.120728 0.0246435
\(25\) −4.17967 −0.835934
\(26\) 2.00852 0.393904
\(27\) 5.11133 0.983675
\(28\) 0 0
\(29\) 3.25799 0.604994 0.302497 0.953150i \(-0.402180\pi\)
0.302497 + 0.953150i \(0.402180\pi\)
\(30\) −3.20001 −0.584239
\(31\) −1.57050 −0.282070 −0.141035 0.990005i \(-0.545043\pi\)
−0.141035 + 0.990005i \(0.545043\pi\)
\(32\) 8.03175 1.41983
\(33\) −1.26012 −0.219359
\(34\) 4.72459 0.810261
\(35\) 0 0
\(36\) 0.191778 0.0319631
\(37\) 5.20883 0.856326 0.428163 0.903702i \(-0.359161\pi\)
0.428163 + 0.903702i \(0.359161\pi\)
\(38\) 13.3284 2.16215
\(39\) 1.75906 0.281674
\(40\) 0.0621618 0.00982864
\(41\) −4.92168 −0.768637 −0.384318 0.923201i \(-0.625563\pi\)
−0.384318 + 0.923201i \(0.625563\pi\)
\(42\) 0 0
\(43\) −9.43766 −1.43923 −0.719615 0.694373i \(-0.755682\pi\)
−0.719615 + 0.694373i \(0.755682\pi\)
\(44\) 1.45720 0.219681
\(45\) −0.0853900 −0.0127292
\(46\) −7.55016 −1.11321
\(47\) 8.31986 1.21358 0.606788 0.794863i \(-0.292458\pi\)
0.606788 + 0.794863i \(0.292458\pi\)
\(48\) 6.91395 0.997943
\(49\) 0 0
\(50\) 8.39497 1.18723
\(51\) 4.13778 0.579405
\(52\) −2.03417 −0.282089
\(53\) 14.0833 1.93449 0.967243 0.253854i \(-0.0816983\pi\)
0.967243 + 0.253854i \(0.0816983\pi\)
\(54\) −10.2662 −1.39706
\(55\) −0.648824 −0.0874874
\(56\) 0 0
\(57\) 11.6729 1.54612
\(58\) −6.54376 −0.859238
\(59\) −0.716361 −0.0932623 −0.0466311 0.998912i \(-0.514849\pi\)
−0.0466311 + 0.998912i \(0.514849\pi\)
\(60\) 3.24087 0.418395
\(61\) 11.6527 1.49197 0.745986 0.665962i \(-0.231979\pi\)
0.745986 + 0.665962i \(0.231979\pi\)
\(62\) 3.15439 0.400607
\(63\) 0 0
\(64\) −8.27099 −1.03387
\(65\) 0.905722 0.112341
\(66\) 2.53098 0.311542
\(67\) 9.39174 1.14738 0.573692 0.819071i \(-0.305511\pi\)
0.573692 + 0.819071i \(0.305511\pi\)
\(68\) −4.78492 −0.580257
\(69\) −6.61239 −0.796038
\(70\) 0 0
\(71\) 10.9914 1.30444 0.652220 0.758030i \(-0.273838\pi\)
0.652220 + 0.758030i \(0.273838\pi\)
\(72\) −0.00647055 −0.000762561 0
\(73\) 3.47300 0.406484 0.203242 0.979129i \(-0.434852\pi\)
0.203242 + 0.979129i \(0.434852\pi\)
\(74\) −10.4621 −1.21619
\(75\) 7.35227 0.848967
\(76\) −13.4986 −1.54839
\(77\) 0 0
\(78\) −3.53311 −0.400046
\(79\) 13.0082 1.46353 0.731766 0.681556i \(-0.238696\pi\)
0.731766 + 0.681556i \(0.238696\pi\)
\(80\) 3.55993 0.398012
\(81\) −9.27395 −1.03044
\(82\) 9.88531 1.09165
\(83\) −3.54083 −0.388656 −0.194328 0.980937i \(-0.562253\pi\)
−0.194328 + 0.980937i \(0.562253\pi\)
\(84\) 0 0
\(85\) 2.13050 0.231085
\(86\) 18.9558 2.04405
\(87\) −5.73099 −0.614427
\(88\) −0.0491655 −0.00524106
\(89\) −12.0501 −1.27730 −0.638651 0.769496i \(-0.720507\pi\)
−0.638651 + 0.769496i \(0.720507\pi\)
\(90\) 0.171508 0.0180785
\(91\) 0 0
\(92\) 7.64656 0.797209
\(93\) 2.76260 0.286468
\(94\) −16.7106 −1.72357
\(95\) 6.01029 0.616642
\(96\) −14.1283 −1.44196
\(97\) −7.43766 −0.755180 −0.377590 0.925973i \(-0.623247\pi\)
−0.377590 + 0.925973i \(0.623247\pi\)
\(98\) 0 0
\(99\) 0.0675374 0.00678776
\(100\) −8.50216 −0.850216
\(101\) 1.19905 0.119310 0.0596551 0.998219i \(-0.481000\pi\)
0.0596551 + 0.998219i \(0.481000\pi\)
\(102\) −8.31083 −0.822894
\(103\) 14.4123 1.42009 0.710043 0.704158i \(-0.248675\pi\)
0.710043 + 0.704158i \(0.248675\pi\)
\(104\) 0.0686323 0.00672995
\(105\) 0 0
\(106\) −28.2866 −2.74744
\(107\) 13.5932 1.31411 0.657053 0.753845i \(-0.271803\pi\)
0.657053 + 0.753845i \(0.271803\pi\)
\(108\) 10.3973 1.00048
\(109\) −13.7248 −1.31460 −0.657299 0.753630i \(-0.728301\pi\)
−0.657299 + 0.753630i \(0.728301\pi\)
\(110\) 1.30318 0.124253
\(111\) −9.16262 −0.869677
\(112\) 0 0
\(113\) −3.25799 −0.306486 −0.153243 0.988189i \(-0.548972\pi\)
−0.153243 + 0.988189i \(0.548972\pi\)
\(114\) −23.4454 −2.19586
\(115\) −3.40466 −0.317486
\(116\) 6.62731 0.615331
\(117\) −0.0942784 −0.00871604
\(118\) 1.43883 0.132455
\(119\) 0 0
\(120\) −0.109346 −0.00998189
\(121\) −10.4868 −0.953348
\(122\) −23.4047 −2.11896
\(123\) 8.65750 0.780621
\(124\) −3.19466 −0.286889
\(125\) 8.31422 0.743647
\(126\) 0 0
\(127\) −0.950834 −0.0843729 −0.0421865 0.999110i \(-0.513432\pi\)
−0.0421865 + 0.999110i \(0.513432\pi\)
\(128\) 0.548979 0.0485233
\(129\) 16.6014 1.46167
\(130\) −1.81916 −0.159551
\(131\) 18.8196 1.64428 0.822138 0.569288i \(-0.192781\pi\)
0.822138 + 0.569288i \(0.192781\pi\)
\(132\) −2.56330 −0.223106
\(133\) 0 0
\(134\) −18.8635 −1.62956
\(135\) −4.62944 −0.398439
\(136\) 0.161442 0.0138435
\(137\) 6.18179 0.528146 0.264073 0.964503i \(-0.414934\pi\)
0.264073 + 0.964503i \(0.414934\pi\)
\(138\) 13.2811 1.13057
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −14.6351 −1.23250
\(142\) −22.0765 −1.85262
\(143\) −0.716361 −0.0599051
\(144\) −0.370560 −0.0308800
\(145\) −2.95083 −0.245053
\(146\) −6.97560 −0.577305
\(147\) 0 0
\(148\) 10.5956 0.870956
\(149\) −21.0771 −1.72670 −0.863351 0.504604i \(-0.831639\pi\)
−0.863351 + 0.504604i \(0.831639\pi\)
\(150\) −14.7672 −1.20574
\(151\) −15.7234 −1.27955 −0.639777 0.768560i \(-0.720973\pi\)
−0.639777 + 0.768560i \(0.720973\pi\)
\(152\) 0.455438 0.0369409
\(153\) −0.221768 −0.0179289
\(154\) 0 0
\(155\) 1.42244 0.114253
\(156\) 3.57822 0.286487
\(157\) 7.78499 0.621310 0.310655 0.950523i \(-0.399452\pi\)
0.310655 + 0.950523i \(0.399452\pi\)
\(158\) −26.1272 −2.07857
\(159\) −24.7733 −1.96465
\(160\) −7.27453 −0.575102
\(161\) 0 0
\(162\) 18.6269 1.46347
\(163\) 1.68991 0.132364 0.0661820 0.997808i \(-0.478918\pi\)
0.0661820 + 0.997808i \(0.478918\pi\)
\(164\) −10.0115 −0.781769
\(165\) 1.14132 0.0888514
\(166\) 7.11184 0.551986
\(167\) 21.8667 1.69210 0.846049 0.533105i \(-0.178975\pi\)
0.846049 + 0.533105i \(0.178975\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −4.27917 −0.328197
\(171\) −0.625623 −0.0478426
\(172\) −19.1978 −1.46382
\(173\) −5.84122 −0.444100 −0.222050 0.975035i \(-0.571275\pi\)
−0.222050 + 0.975035i \(0.571275\pi\)
\(174\) 11.5108 0.872634
\(175\) 0 0
\(176\) −2.81565 −0.212238
\(177\) 1.26012 0.0947164
\(178\) 24.2028 1.81408
\(179\) 2.53427 0.189421 0.0947103 0.995505i \(-0.469808\pi\)
0.0947103 + 0.995505i \(0.469808\pi\)
\(180\) −0.173698 −0.0129467
\(181\) 10.7248 0.797169 0.398585 0.917132i \(-0.369502\pi\)
0.398585 + 0.917132i \(0.369502\pi\)
\(182\) 0 0
\(183\) −20.4977 −1.51523
\(184\) −0.257993 −0.0190195
\(185\) −4.71775 −0.346856
\(186\) −5.54874 −0.406854
\(187\) −1.68508 −0.123225
\(188\) 16.9240 1.23431
\(189\) 0 0
\(190\) −12.0718 −0.875781
\(191\) −1.67861 −0.121460 −0.0607298 0.998154i \(-0.519343\pi\)
−0.0607298 + 0.998154i \(0.519343\pi\)
\(192\) 14.5491 1.04999
\(193\) −6.44816 −0.464148 −0.232074 0.972698i \(-0.574551\pi\)
−0.232074 + 0.972698i \(0.574551\pi\)
\(194\) 14.9387 1.07254
\(195\) −1.59322 −0.114093
\(196\) 0 0
\(197\) 1.87251 0.133411 0.0667054 0.997773i \(-0.478751\pi\)
0.0667054 + 0.997773i \(0.478751\pi\)
\(198\) −0.135650 −0.00964026
\(199\) 11.3967 0.807888 0.403944 0.914784i \(-0.367639\pi\)
0.403944 + 0.914784i \(0.367639\pi\)
\(200\) 0.286860 0.0202841
\(201\) −16.5206 −1.16527
\(202\) −2.40833 −0.169449
\(203\) 0 0
\(204\) 8.41694 0.589304
\(205\) 4.45767 0.311337
\(206\) −28.9475 −2.01687
\(207\) 0.354398 0.0246324
\(208\) 3.93049 0.272531
\(209\) −4.75371 −0.328821
\(210\) 0 0
\(211\) 7.53599 0.518799 0.259400 0.965770i \(-0.416475\pi\)
0.259400 + 0.965770i \(0.416475\pi\)
\(212\) 28.6478 1.96754
\(213\) −19.3345 −1.32478
\(214\) −27.3023 −1.86635
\(215\) 8.54789 0.582962
\(216\) −0.350802 −0.0238691
\(217\) 0 0
\(218\) 27.5666 1.86705
\(219\) −6.10920 −0.412822
\(220\) −1.31982 −0.0889821
\(221\) 2.35227 0.158231
\(222\) 18.4033 1.23515
\(223\) −17.6349 −1.18092 −0.590459 0.807067i \(-0.701053\pi\)
−0.590459 + 0.807067i \(0.701053\pi\)
\(224\) 0 0
\(225\) −0.394052 −0.0262702
\(226\) 6.54376 0.435284
\(227\) 5.32904 0.353701 0.176851 0.984238i \(-0.443409\pi\)
0.176851 + 0.984238i \(0.443409\pi\)
\(228\) 23.7447 1.57253
\(229\) 8.51900 0.562951 0.281476 0.959568i \(-0.409176\pi\)
0.281476 + 0.959568i \(0.409176\pi\)
\(230\) 6.83834 0.450907
\(231\) 0 0
\(232\) −0.223604 −0.0146803
\(233\) 4.75371 0.311426 0.155713 0.987802i \(-0.450233\pi\)
0.155713 + 0.987802i \(0.450233\pi\)
\(234\) 0.189360 0.0123789
\(235\) −7.53548 −0.491561
\(236\) −1.45720 −0.0948557
\(237\) −22.8821 −1.48635
\(238\) 0 0
\(239\) 14.8314 0.959365 0.479682 0.877442i \(-0.340752\pi\)
0.479682 + 0.877442i \(0.340752\pi\)
\(240\) −6.26212 −0.404218
\(241\) 6.12131 0.394308 0.197154 0.980373i \(-0.436830\pi\)
0.197154 + 0.980373i \(0.436830\pi\)
\(242\) 21.0630 1.35398
\(243\) 0.979411 0.0628292
\(244\) 23.7035 1.51746
\(245\) 0 0
\(246\) −17.3888 −1.10867
\(247\) 6.63591 0.422233
\(248\) 0.107787 0.00684448
\(249\) 6.22852 0.394716
\(250\) −16.6993 −1.05616
\(251\) 13.9708 0.881832 0.440916 0.897548i \(-0.354654\pi\)
0.440916 + 0.897548i \(0.354654\pi\)
\(252\) 0 0
\(253\) 2.69284 0.169298
\(254\) 1.90977 0.119830
\(255\) −3.74767 −0.234688
\(256\) 15.4393 0.964959
\(257\) −17.2651 −1.07696 −0.538482 0.842637i \(-0.681002\pi\)
−0.538482 + 0.842637i \(0.681002\pi\)
\(258\) −33.3443 −2.07592
\(259\) 0 0
\(260\) 1.84239 0.114260
\(261\) 0.307158 0.0190126
\(262\) −37.7996 −2.33527
\(263\) −2.60672 −0.160737 −0.0803687 0.996765i \(-0.525610\pi\)
−0.0803687 + 0.996765i \(0.525610\pi\)
\(264\) 0.0864849 0.00532278
\(265\) −12.7555 −0.783565
\(266\) 0 0
\(267\) 21.1967 1.29722
\(268\) 19.1044 1.16699
\(269\) 14.4895 0.883443 0.441721 0.897152i \(-0.354368\pi\)
0.441721 + 0.897152i \(0.354368\pi\)
\(270\) 9.29834 0.565879
\(271\) 8.63591 0.524594 0.262297 0.964987i \(-0.415520\pi\)
0.262297 + 0.964987i \(0.415520\pi\)
\(272\) 9.24558 0.560596
\(273\) 0 0
\(274\) −12.4163 −0.750095
\(275\) −2.99415 −0.180554
\(276\) −13.4507 −0.809639
\(277\) 12.2270 0.734647 0.367324 0.930093i \(-0.380274\pi\)
0.367324 + 0.930093i \(0.380274\pi\)
\(278\) −8.03410 −0.481853
\(279\) −0.148064 −0.00886437
\(280\) 0 0
\(281\) −24.1822 −1.44259 −0.721293 0.692630i \(-0.756452\pi\)
−0.721293 + 0.692630i \(0.756452\pi\)
\(282\) 29.3950 1.75044
\(283\) 30.7683 1.82899 0.914493 0.404601i \(-0.132590\pi\)
0.914493 + 0.404601i \(0.132590\pi\)
\(284\) 22.3584 1.32673
\(285\) −10.5724 −0.626257
\(286\) 1.43883 0.0850797
\(287\) 0 0
\(288\) 0.757221 0.0446197
\(289\) −11.4668 −0.674519
\(290\) 5.92682 0.348035
\(291\) 13.0833 0.766954
\(292\) 7.06467 0.413429
\(293\) 31.8295 1.85950 0.929749 0.368193i \(-0.120024\pi\)
0.929749 + 0.368193i \(0.120024\pi\)
\(294\) 0 0
\(295\) 0.648824 0.0377760
\(296\) −0.357494 −0.0207789
\(297\) 3.66156 0.212465
\(298\) 42.3338 2.45233
\(299\) −3.75906 −0.217392
\(300\) 14.9558 0.863472
\(301\) 0 0
\(302\) 31.5809 1.81728
\(303\) −2.10920 −0.121170
\(304\) 26.0824 1.49593
\(305\) −10.5541 −0.604324
\(306\) 0.445427 0.0254634
\(307\) −28.7884 −1.64304 −0.821520 0.570179i \(-0.806874\pi\)
−0.821520 + 0.570179i \(0.806874\pi\)
\(308\) 0 0
\(309\) −25.3521 −1.44223
\(310\) −2.85700 −0.162266
\(311\) −5.51862 −0.312932 −0.156466 0.987683i \(-0.550010\pi\)
−0.156466 + 0.987683i \(0.550010\pi\)
\(312\) −0.120728 −0.00683488
\(313\) 4.84799 0.274024 0.137012 0.990569i \(-0.456250\pi\)
0.137012 + 0.990569i \(0.456250\pi\)
\(314\) −15.6363 −0.882410
\(315\) 0 0
\(316\) 26.4608 1.48854
\(317\) −7.65511 −0.429954 −0.214977 0.976619i \(-0.568968\pi\)
−0.214977 + 0.976619i \(0.568968\pi\)
\(318\) 49.7577 2.79027
\(319\) 2.33390 0.130673
\(320\) 7.49121 0.418772
\(321\) −23.9112 −1.33459
\(322\) 0 0
\(323\) 15.6095 0.868534
\(324\) −18.8648 −1.04804
\(325\) 4.17967 0.231846
\(326\) −3.39423 −0.187989
\(327\) 24.1427 1.33510
\(328\) 0.337786 0.0186511
\(329\) 0 0
\(330\) −2.29236 −0.126190
\(331\) 11.3432 0.623477 0.311739 0.950168i \(-0.399089\pi\)
0.311739 + 0.950168i \(0.399089\pi\)
\(332\) −7.20265 −0.395297
\(333\) 0.491080 0.0269110
\(334\) −43.9199 −2.40319
\(335\) −8.50631 −0.464749
\(336\) 0 0
\(337\) 1.74149 0.0948649 0.0474324 0.998874i \(-0.484896\pi\)
0.0474324 + 0.998874i \(0.484896\pi\)
\(338\) −2.00852 −0.109249
\(339\) 5.73099 0.311265
\(340\) 4.33381 0.235034
\(341\) −1.12504 −0.0609246
\(342\) 1.25658 0.0679480
\(343\) 0 0
\(344\) 0.647729 0.0349232
\(345\) 5.98898 0.322436
\(346\) 11.7322 0.630729
\(347\) 21.0503 1.13004 0.565019 0.825078i \(-0.308869\pi\)
0.565019 + 0.825078i \(0.308869\pi\)
\(348\) −11.6578 −0.624925
\(349\) 8.35601 0.447287 0.223643 0.974671i \(-0.428205\pi\)
0.223643 + 0.974671i \(0.428205\pi\)
\(350\) 0 0
\(351\) −5.11133 −0.272822
\(352\) 5.75364 0.306670
\(353\) 8.53355 0.454195 0.227097 0.973872i \(-0.427076\pi\)
0.227097 + 0.973872i \(0.427076\pi\)
\(354\) −2.53098 −0.134520
\(355\) −9.95515 −0.528365
\(356\) −24.5119 −1.29913
\(357\) 0 0
\(358\) −5.09015 −0.269023
\(359\) −16.1713 −0.853488 −0.426744 0.904372i \(-0.640339\pi\)
−0.426744 + 0.904372i \(0.640339\pi\)
\(360\) 0.00586051 0.000308876 0
\(361\) 25.0353 1.31765
\(362\) −21.5410 −1.13217
\(363\) 18.4469 0.968212
\(364\) 0 0
\(365\) −3.14557 −0.164647
\(366\) 41.1701 2.15200
\(367\) −28.1540 −1.46963 −0.734813 0.678269i \(-0.762730\pi\)
−0.734813 + 0.678269i \(0.762730\pi\)
\(368\) −14.7749 −0.770197
\(369\) −0.464008 −0.0241553
\(370\) 9.47571 0.492619
\(371\) 0 0
\(372\) 5.61959 0.291362
\(373\) −28.5037 −1.47586 −0.737932 0.674875i \(-0.764197\pi\)
−0.737932 + 0.674875i \(0.764197\pi\)
\(374\) 3.38452 0.175009
\(375\) −14.6252 −0.755241
\(376\) −0.571012 −0.0294477
\(377\) −3.25799 −0.167795
\(378\) 0 0
\(379\) −7.26263 −0.373056 −0.186528 0.982450i \(-0.559724\pi\)
−0.186528 + 0.982450i \(0.559724\pi\)
\(380\) 12.2259 0.627178
\(381\) 1.67257 0.0856884
\(382\) 3.37152 0.172502
\(383\) −12.9325 −0.660822 −0.330411 0.943837i \(-0.607187\pi\)
−0.330411 + 0.943837i \(0.607187\pi\)
\(384\) −0.965684 −0.0492799
\(385\) 0 0
\(386\) 12.9513 0.659203
\(387\) −0.889768 −0.0452294
\(388\) −15.1295 −0.768083
\(389\) −21.1357 −1.07162 −0.535811 0.844338i \(-0.679994\pi\)
−0.535811 + 0.844338i \(0.679994\pi\)
\(390\) 3.20001 0.162039
\(391\) −8.84232 −0.447175
\(392\) 0 0
\(393\) −33.1047 −1.66991
\(394\) −3.76098 −0.189476
\(395\) −11.7818 −0.592805
\(396\) 0.137383 0.00690373
\(397\) −19.2073 −0.963988 −0.481994 0.876175i \(-0.660087\pi\)
−0.481994 + 0.876175i \(0.660087\pi\)
\(398\) −22.8905 −1.14740
\(399\) 0 0
\(400\) 16.4282 0.821408
\(401\) 16.6692 0.832420 0.416210 0.909268i \(-0.363358\pi\)
0.416210 + 0.909268i \(0.363358\pi\)
\(402\) 33.1820 1.65497
\(403\) 1.57050 0.0782321
\(404\) 2.43908 0.121349
\(405\) 8.39961 0.417380
\(406\) 0 0
\(407\) 3.73140 0.184959
\(408\) −0.283985 −0.0140594
\(409\) −12.3483 −0.610585 −0.305293 0.952259i \(-0.598754\pi\)
−0.305293 + 0.952259i \(0.598754\pi\)
\(410\) −8.95333 −0.442174
\(411\) −10.8741 −0.536381
\(412\) 29.3171 1.44435
\(413\) 0 0
\(414\) −0.711817 −0.0349839
\(415\) 3.20700 0.157426
\(416\) −8.03175 −0.393789
\(417\) −7.03622 −0.344565
\(418\) 9.54794 0.467005
\(419\) −4.35934 −0.212968 −0.106484 0.994314i \(-0.533959\pi\)
−0.106484 + 0.994314i \(0.533959\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −15.1362 −0.736820
\(423\) 0.784383 0.0381380
\(424\) −0.966567 −0.0469407
\(425\) 9.83171 0.476908
\(426\) 38.8338 1.88150
\(427\) 0 0
\(428\) 27.6509 1.33656
\(429\) 1.26012 0.0608391
\(430\) −17.1687 −0.827946
\(431\) −23.3626 −1.12533 −0.562667 0.826683i \(-0.690225\pi\)
−0.562667 + 0.826683i \(0.690225\pi\)
\(432\) −20.0900 −0.966582
\(433\) 2.71285 0.130371 0.0651856 0.997873i \(-0.479236\pi\)
0.0651856 + 0.997873i \(0.479236\pi\)
\(434\) 0 0
\(435\) 5.19068 0.248874
\(436\) −27.9186 −1.33706
\(437\) −24.9448 −1.19327
\(438\) 12.2705 0.586306
\(439\) 8.83519 0.421681 0.210840 0.977521i \(-0.432380\pi\)
0.210840 + 0.977521i \(0.432380\pi\)
\(440\) 0.0445303 0.00212290
\(441\) 0 0
\(442\) −4.72459 −0.224726
\(443\) −2.90558 −0.138048 −0.0690240 0.997615i \(-0.521989\pi\)
−0.0690240 + 0.997615i \(0.521989\pi\)
\(444\) −18.6383 −0.884536
\(445\) 10.9140 0.517373
\(446\) 35.4201 1.67719
\(447\) 37.0758 1.75362
\(448\) 0 0
\(449\) −15.2777 −0.720998 −0.360499 0.932760i \(-0.617394\pi\)
−0.360499 + 0.932760i \(0.617394\pi\)
\(450\) 0.791464 0.0373100
\(451\) −3.52570 −0.166019
\(452\) −6.62731 −0.311723
\(453\) 27.6584 1.29950
\(454\) −10.7035 −0.502341
\(455\) 0 0
\(456\) −0.801141 −0.0375169
\(457\) 23.6600 1.10677 0.553384 0.832926i \(-0.313336\pi\)
0.553384 + 0.832926i \(0.313336\pi\)
\(458\) −17.1106 −0.799527
\(459\) −12.0232 −0.561196
\(460\) −6.92566 −0.322910
\(461\) 26.6170 1.23968 0.619839 0.784729i \(-0.287198\pi\)
0.619839 + 0.784729i \(0.287198\pi\)
\(462\) 0 0
\(463\) −1.44250 −0.0670385 −0.0335193 0.999438i \(-0.510672\pi\)
−0.0335193 + 0.999438i \(0.510672\pi\)
\(464\) −12.8055 −0.594481
\(465\) −2.50214 −0.116034
\(466\) −9.54794 −0.442300
\(467\) 8.38959 0.388224 0.194112 0.980979i \(-0.437817\pi\)
0.194112 + 0.980979i \(0.437817\pi\)
\(468\) −0.191778 −0.00886496
\(469\) 0 0
\(470\) 15.1352 0.698135
\(471\) −13.6942 −0.630997
\(472\) 0.0491655 0.00226303
\(473\) −6.76077 −0.310861
\(474\) 45.9592 2.11098
\(475\) 27.7359 1.27261
\(476\) 0 0
\(477\) 1.32775 0.0607934
\(478\) −29.7893 −1.36253
\(479\) −12.6122 −0.576265 −0.288132 0.957591i \(-0.593034\pi\)
−0.288132 + 0.957591i \(0.593034\pi\)
\(480\) 12.7963 0.584069
\(481\) −5.20883 −0.237502
\(482\) −12.2948 −0.560013
\(483\) 0 0
\(484\) −21.3320 −0.969636
\(485\) 6.73645 0.305886
\(486\) −1.96717 −0.0892327
\(487\) 21.5680 0.977341 0.488671 0.872468i \(-0.337482\pi\)
0.488671 + 0.872468i \(0.337482\pi\)
\(488\) −0.799750 −0.0362030
\(489\) −2.97265 −0.134428
\(490\) 0 0
\(491\) 39.2347 1.77064 0.885318 0.464987i \(-0.153941\pi\)
0.885318 + 0.464987i \(0.153941\pi\)
\(492\) 17.6108 0.793958
\(493\) −7.66368 −0.345155
\(494\) −13.3284 −0.599673
\(495\) −0.0611701 −0.00274939
\(496\) 6.17283 0.277168
\(497\) 0 0
\(498\) −12.5101 −0.560592
\(499\) 9.16814 0.410422 0.205211 0.978718i \(-0.434212\pi\)
0.205211 + 0.978718i \(0.434212\pi\)
\(500\) 16.9125 0.756352
\(501\) −38.4648 −1.71848
\(502\) −28.0608 −1.25241
\(503\) −24.9370 −1.11188 −0.555942 0.831221i \(-0.687642\pi\)
−0.555942 + 0.831221i \(0.687642\pi\)
\(504\) 0 0
\(505\) −1.08601 −0.0483267
\(506\) −5.40864 −0.240443
\(507\) −1.75906 −0.0781224
\(508\) −1.93416 −0.0858144
\(509\) −5.89807 −0.261428 −0.130714 0.991420i \(-0.541727\pi\)
−0.130714 + 0.991420i \(0.541727\pi\)
\(510\) 7.52730 0.333314
\(511\) 0 0
\(512\) −32.1083 −1.41900
\(513\) −33.9183 −1.49753
\(514\) 34.6773 1.52955
\(515\) −13.0535 −0.575208
\(516\) 33.7700 1.48664
\(517\) 5.96003 0.262122
\(518\) 0 0
\(519\) 10.2750 0.451024
\(520\) −0.0621618 −0.00272597
\(521\) 37.1895 1.62930 0.814652 0.579950i \(-0.196928\pi\)
0.814652 + 0.579950i \(0.196928\pi\)
\(522\) −0.616935 −0.0270025
\(523\) 5.09080 0.222605 0.111303 0.993787i \(-0.464498\pi\)
0.111303 + 0.993787i \(0.464498\pi\)
\(524\) 38.2823 1.67237
\(525\) 0 0
\(526\) 5.23567 0.228286
\(527\) 3.69424 0.160924
\(528\) 4.95289 0.215547
\(529\) −8.86950 −0.385630
\(530\) 25.6198 1.11285
\(531\) −0.0675374 −0.00293087
\(532\) 0 0
\(533\) 4.92168 0.213181
\(534\) −42.5741 −1.84236
\(535\) −12.3117 −0.532280
\(536\) −0.644577 −0.0278415
\(537\) −4.45793 −0.192374
\(538\) −29.1026 −1.25470
\(539\) 0 0
\(540\) −9.41707 −0.405246
\(541\) −0.766850 −0.0329694 −0.0164847 0.999864i \(-0.505247\pi\)
−0.0164847 + 0.999864i \(0.505247\pi\)
\(542\) −17.3454 −0.745050
\(543\) −18.8655 −0.809598
\(544\) −18.8929 −0.810025
\(545\) 12.4309 0.532480
\(546\) 0 0
\(547\) 14.1428 0.604702 0.302351 0.953197i \(-0.402229\pi\)
0.302351 + 0.953197i \(0.402229\pi\)
\(548\) 12.5748 0.537170
\(549\) 1.09859 0.0468869
\(550\) 6.01383 0.256430
\(551\) −21.6197 −0.921032
\(552\) 0.453824 0.0193160
\(553\) 0 0
\(554\) −24.5582 −1.04338
\(555\) 8.29878 0.352264
\(556\) 8.13668 0.345072
\(557\) −24.8627 −1.05347 −0.526733 0.850031i \(-0.676583\pi\)
−0.526733 + 0.850031i \(0.676583\pi\)
\(558\) 0.297391 0.0125895
\(559\) 9.43766 0.399171
\(560\) 0 0
\(561\) 2.96414 0.125146
\(562\) 48.5705 2.04882
\(563\) −44.0094 −1.85478 −0.927388 0.374101i \(-0.877951\pi\)
−0.927388 + 0.374101i \(0.877951\pi\)
\(564\) −29.7703 −1.25356
\(565\) 2.95083 0.124143
\(566\) −61.7989 −2.59760
\(567\) 0 0
\(568\) −0.754366 −0.0316525
\(569\) −33.2616 −1.39440 −0.697199 0.716877i \(-0.745571\pi\)
−0.697199 + 0.716877i \(0.745571\pi\)
\(570\) 21.2350 0.889436
\(571\) −12.3540 −0.516998 −0.258499 0.966011i \(-0.583228\pi\)
−0.258499 + 0.966011i \(0.583228\pi\)
\(572\) −1.45720 −0.0609286
\(573\) 2.95276 0.123353
\(574\) 0 0
\(575\) −15.7116 −0.655219
\(576\) −0.779776 −0.0324907
\(577\) 25.9659 1.08097 0.540486 0.841353i \(-0.318240\pi\)
0.540486 + 0.841353i \(0.318240\pi\)
\(578\) 23.0314 0.957979
\(579\) 11.3427 0.471385
\(580\) −6.00250 −0.249240
\(581\) 0 0
\(582\) −26.2781 −1.08926
\(583\) 10.0887 0.417831
\(584\) −0.238360 −0.00986341
\(585\) 0.0853900 0.00353044
\(586\) −63.9303 −2.64094
\(587\) 23.9747 0.989543 0.494771 0.869023i \(-0.335252\pi\)
0.494771 + 0.869023i \(0.335252\pi\)
\(588\) 0 0
\(589\) 10.4217 0.429418
\(590\) −1.30318 −0.0536510
\(591\) −3.29385 −0.135491
\(592\) −20.4733 −0.841445
\(593\) −47.0480 −1.93203 −0.966015 0.258484i \(-0.916777\pi\)
−0.966015 + 0.258484i \(0.916777\pi\)
\(594\) −7.35433 −0.301752
\(595\) 0 0
\(596\) −42.8744 −1.75620
\(597\) −20.0474 −0.820484
\(598\) 7.55016 0.308749
\(599\) 20.1736 0.824271 0.412135 0.911123i \(-0.364783\pi\)
0.412135 + 0.911123i \(0.364783\pi\)
\(600\) −0.504604 −0.0206004
\(601\) −29.5773 −1.20648 −0.603242 0.797558i \(-0.706125\pi\)
−0.603242 + 0.797558i \(0.706125\pi\)
\(602\) 0 0
\(603\) 0.885439 0.0360579
\(604\) −31.9841 −1.30142
\(605\) 9.49815 0.386155
\(606\) 4.23638 0.172091
\(607\) 15.4420 0.626771 0.313385 0.949626i \(-0.398537\pi\)
0.313385 + 0.949626i \(0.398537\pi\)
\(608\) −53.2980 −2.16152
\(609\) 0 0
\(610\) 21.1981 0.858287
\(611\) −8.31986 −0.336586
\(612\) −0.451115 −0.0182352
\(613\) 1.99485 0.0805711 0.0402855 0.999188i \(-0.487173\pi\)
0.0402855 + 0.999188i \(0.487173\pi\)
\(614\) 57.8222 2.33351
\(615\) −7.84129 −0.316191
\(616\) 0 0
\(617\) −2.85584 −0.114972 −0.0574858 0.998346i \(-0.518308\pi\)
−0.0574858 + 0.998346i \(0.518308\pi\)
\(618\) 50.9202 2.04831
\(619\) −31.9823 −1.28548 −0.642738 0.766086i \(-0.722202\pi\)
−0.642738 + 0.766086i \(0.722202\pi\)
\(620\) 2.89348 0.116205
\(621\) 19.2138 0.771022
\(622\) 11.0843 0.444440
\(623\) 0 0
\(624\) −6.91395 −0.276780
\(625\) 13.3680 0.534719
\(626\) −9.73730 −0.389181
\(627\) 8.36204 0.333948
\(628\) 15.8360 0.631925
\(629\) −12.2526 −0.488542
\(630\) 0 0
\(631\) 32.1115 1.27834 0.639169 0.769066i \(-0.279278\pi\)
0.639169 + 0.769066i \(0.279278\pi\)
\(632\) −0.892781 −0.0355129
\(633\) −13.2562 −0.526888
\(634\) 15.3755 0.610638
\(635\) 0.861191 0.0341753
\(636\) −50.3930 −1.99821
\(637\) 0 0
\(638\) −4.68769 −0.185588
\(639\) 1.03625 0.0409935
\(640\) −0.497222 −0.0196544
\(641\) 33.0248 1.30440 0.652200 0.758047i \(-0.273846\pi\)
0.652200 + 0.758047i \(0.273846\pi\)
\(642\) 48.0263 1.89545
\(643\) −15.7942 −0.622863 −0.311432 0.950269i \(-0.600808\pi\)
−0.311432 + 0.950269i \(0.600808\pi\)
\(644\) 0 0
\(645\) −15.0362 −0.592051
\(646\) −31.3520 −1.23353
\(647\) 4.64072 0.182445 0.0912227 0.995831i \(-0.470922\pi\)
0.0912227 + 0.995831i \(0.470922\pi\)
\(648\) 0.636493 0.0250038
\(649\) −0.513173 −0.0201438
\(650\) −8.39497 −0.329278
\(651\) 0 0
\(652\) 3.43757 0.134626
\(653\) 26.8285 1.04988 0.524941 0.851139i \(-0.324087\pi\)
0.524941 + 0.851139i \(0.324087\pi\)
\(654\) −48.4912 −1.89616
\(655\) −17.0453 −0.666016
\(656\) 19.3446 0.755280
\(657\) 0.327429 0.0127742
\(658\) 0 0
\(659\) −42.9889 −1.67461 −0.837306 0.546735i \(-0.815871\pi\)
−0.837306 + 0.546735i \(0.815871\pi\)
\(660\) 2.32163 0.0903695
\(661\) 29.4698 1.14624 0.573122 0.819470i \(-0.305732\pi\)
0.573122 + 0.819470i \(0.305732\pi\)
\(662\) −22.7830 −0.885488
\(663\) −4.13778 −0.160698
\(664\) 0.243015 0.00943082
\(665\) 0 0
\(666\) −0.986346 −0.0382201
\(667\) 12.2470 0.474205
\(668\) 44.4806 1.72101
\(669\) 31.0207 1.19933
\(670\) 17.0851 0.660056
\(671\) 8.34752 0.322252
\(672\) 0 0
\(673\) −20.1702 −0.777504 −0.388752 0.921342i \(-0.627094\pi\)
−0.388752 + 0.921342i \(0.627094\pi\)
\(674\) −3.49782 −0.134731
\(675\) −21.3637 −0.822287
\(676\) 2.03417 0.0782373
\(677\) 6.20481 0.238470 0.119235 0.992866i \(-0.461956\pi\)
0.119235 + 0.992866i \(0.461956\pi\)
\(678\) −11.5108 −0.442071
\(679\) 0 0
\(680\) −0.146221 −0.00560733
\(681\) −9.37409 −0.359216
\(682\) 2.25968 0.0865276
\(683\) −1.76952 −0.0677087 −0.0338543 0.999427i \(-0.510778\pi\)
−0.0338543 + 0.999427i \(0.510778\pi\)
\(684\) −1.27262 −0.0486600
\(685\) −5.59899 −0.213926
\(686\) 0 0
\(687\) −14.9854 −0.571729
\(688\) 37.0946 1.41422
\(689\) −14.0833 −0.536530
\(690\) −12.0290 −0.457937
\(691\) −44.9317 −1.70928 −0.854641 0.519219i \(-0.826223\pi\)
−0.854641 + 0.519219i \(0.826223\pi\)
\(692\) −11.8820 −0.451688
\(693\) 0 0
\(694\) −42.2800 −1.60493
\(695\) −3.62289 −0.137424
\(696\) 0.393331 0.0149092
\(697\) 11.5771 0.438515
\(698\) −16.7832 −0.635255
\(699\) −8.36204 −0.316281
\(700\) 0 0
\(701\) 38.5707 1.45679 0.728397 0.685156i \(-0.240266\pi\)
0.728397 + 0.685156i \(0.240266\pi\)
\(702\) 10.2662 0.387474
\(703\) −34.5653 −1.30366
\(704\) −5.92501 −0.223307
\(705\) 13.2553 0.499225
\(706\) −17.1398 −0.645066
\(707\) 0 0
\(708\) 2.56330 0.0963346
\(709\) 8.77731 0.329639 0.164819 0.986324i \(-0.447296\pi\)
0.164819 + 0.986324i \(0.447296\pi\)
\(710\) 19.9952 0.750405
\(711\) 1.22639 0.0459932
\(712\) 0.827023 0.0309940
\(713\) −5.90360 −0.221091
\(714\) 0 0
\(715\) 0.648824 0.0242646
\(716\) 5.15515 0.192657
\(717\) −26.0893 −0.974323
\(718\) 32.4804 1.21216
\(719\) −4.20437 −0.156796 −0.0783982 0.996922i \(-0.524981\pi\)
−0.0783982 + 0.996922i \(0.524981\pi\)
\(720\) 0.335625 0.0125080
\(721\) 0 0
\(722\) −50.2840 −1.87138
\(723\) −10.7677 −0.400456
\(724\) 21.8161 0.810789
\(725\) −13.6173 −0.505735
\(726\) −37.0511 −1.37509
\(727\) 28.9856 1.07502 0.537509 0.843258i \(-0.319366\pi\)
0.537509 + 0.843258i \(0.319366\pi\)
\(728\) 0 0
\(729\) 26.0990 0.966630
\(730\) 6.31796 0.233838
\(731\) 22.1999 0.821094
\(732\) −41.6958 −1.54112
\(733\) 24.0345 0.887733 0.443867 0.896093i \(-0.353606\pi\)
0.443867 + 0.896093i \(0.353606\pi\)
\(734\) 56.5480 2.08722
\(735\) 0 0
\(736\) 30.1918 1.11288
\(737\) 6.72788 0.247825
\(738\) 0.931971 0.0343063
\(739\) 11.8055 0.434273 0.217136 0.976141i \(-0.430328\pi\)
0.217136 + 0.976141i \(0.430328\pi\)
\(740\) −9.59670 −0.352782
\(741\) −11.6729 −0.428816
\(742\) 0 0
\(743\) 47.2786 1.73448 0.867241 0.497888i \(-0.165891\pi\)
0.867241 + 0.497888i \(0.165891\pi\)
\(744\) −0.189603 −0.00695120
\(745\) 19.0900 0.699402
\(746\) 57.2503 2.09608
\(747\) −0.333824 −0.0122140
\(748\) −3.42773 −0.125330
\(749\) 0 0
\(750\) 29.3750 1.07262
\(751\) 5.47700 0.199859 0.0999294 0.994995i \(-0.468138\pi\)
0.0999294 + 0.994995i \(0.468138\pi\)
\(752\) −32.7012 −1.19249
\(753\) −24.5755 −0.895581
\(754\) 6.54376 0.238310
\(755\) 14.2410 0.518285
\(756\) 0 0
\(757\) 10.7453 0.390546 0.195273 0.980749i \(-0.437441\pi\)
0.195273 + 0.980749i \(0.437441\pi\)
\(758\) 14.5872 0.529830
\(759\) −4.73686 −0.171937
\(760\) −0.412500 −0.0149630
\(761\) 33.0399 1.19770 0.598848 0.800863i \(-0.295625\pi\)
0.598848 + 0.800863i \(0.295625\pi\)
\(762\) −3.35940 −0.121698
\(763\) 0 0
\(764\) −3.41457 −0.123535
\(765\) 0.200860 0.00726212
\(766\) 25.9753 0.938527
\(767\) 0.716361 0.0258663
\(768\) −27.1587 −0.980004
\(769\) −2.98332 −0.107581 −0.0537907 0.998552i \(-0.517130\pi\)
−0.0537907 + 0.998552i \(0.517130\pi\)
\(770\) 0 0
\(771\) 30.3702 1.09376
\(772\) −13.1167 −0.472079
\(773\) 21.9085 0.787995 0.393998 0.919111i \(-0.371092\pi\)
0.393998 + 0.919111i \(0.371092\pi\)
\(774\) 1.78712 0.0642367
\(775\) 6.56417 0.235792
\(776\) 0.510464 0.0183246
\(777\) 0 0
\(778\) 42.4516 1.52196
\(779\) 32.6598 1.17016
\(780\) −3.24087 −0.116042
\(781\) 7.87381 0.281747
\(782\) 17.7600 0.635097
\(783\) 16.6527 0.595118
\(784\) 0 0
\(785\) −7.05104 −0.251662
\(786\) 66.4917 2.37168
\(787\) 13.3632 0.476347 0.238174 0.971223i \(-0.423451\pi\)
0.238174 + 0.971223i \(0.423451\pi\)
\(788\) 3.80900 0.135690
\(789\) 4.58537 0.163244
\(790\) 23.6640 0.841927
\(791\) 0 0
\(792\) −0.00463525 −0.000164706 0
\(793\) −11.6527 −0.413798
\(794\) 38.5784 1.36910
\(795\) 22.4377 0.795782
\(796\) 23.1827 0.821691
\(797\) −32.5388 −1.15258 −0.576292 0.817244i \(-0.695501\pi\)
−0.576292 + 0.817244i \(0.695501\pi\)
\(798\) 0 0
\(799\) −19.5706 −0.692357
\(800\) −33.5701 −1.18688
\(801\) −1.13606 −0.0401407
\(802\) −33.4805 −1.18224
\(803\) 2.48792 0.0877969
\(804\) −33.6057 −1.18518
\(805\) 0 0
\(806\) −3.15439 −0.111109
\(807\) −25.4879 −0.897217
\(808\) −0.0822938 −0.00289509
\(809\) 7.68827 0.270305 0.135153 0.990825i \(-0.456848\pi\)
0.135153 + 0.990825i \(0.456848\pi\)
\(810\) −16.8708 −0.592781
\(811\) 48.3178 1.69667 0.848334 0.529461i \(-0.177606\pi\)
0.848334 + 0.529461i \(0.177606\pi\)
\(812\) 0 0
\(813\) −15.1911 −0.532773
\(814\) −7.49461 −0.262686
\(815\) −1.53059 −0.0536142
\(816\) −16.2635 −0.569336
\(817\) 62.6275 2.19106
\(818\) 24.8019 0.867178
\(819\) 0 0
\(820\) 9.06766 0.316656
\(821\) −3.73442 −0.130332 −0.0651661 0.997874i \(-0.520758\pi\)
−0.0651661 + 0.997874i \(0.520758\pi\)
\(822\) 21.8409 0.761790
\(823\) 14.2318 0.496089 0.248045 0.968749i \(-0.420212\pi\)
0.248045 + 0.968749i \(0.420212\pi\)
\(824\) −0.989150 −0.0344587
\(825\) 5.26688 0.183369
\(826\) 0 0
\(827\) 48.3016 1.67961 0.839805 0.542888i \(-0.182669\pi\)
0.839805 + 0.542888i \(0.182669\pi\)
\(828\) 0.720906 0.0250532
\(829\) −11.5101 −0.399763 −0.199882 0.979820i \(-0.564056\pi\)
−0.199882 + 0.979820i \(0.564056\pi\)
\(830\) −6.44135 −0.223582
\(831\) −21.5079 −0.746102
\(832\) 8.27099 0.286745
\(833\) 0 0
\(834\) 14.1324 0.489366
\(835\) −19.8052 −0.685386
\(836\) −9.66985 −0.334439
\(837\) −8.02734 −0.277465
\(838\) 8.75583 0.302465
\(839\) −13.1103 −0.452616 −0.226308 0.974056i \(-0.572666\pi\)
−0.226308 + 0.974056i \(0.572666\pi\)
\(840\) 0 0
\(841\) −18.3855 −0.633982
\(842\) 20.0852 0.692183
\(843\) 42.5378 1.46508
\(844\) 15.3295 0.527663
\(845\) −0.905722 −0.0311578
\(846\) −1.57545 −0.0541652
\(847\) 0 0
\(848\) −55.3541 −1.90087
\(849\) −54.1232 −1.85750
\(850\) −19.7472 −0.677325
\(851\) 19.5803 0.671203
\(852\) −39.3297 −1.34741
\(853\) −8.80346 −0.301425 −0.150712 0.988578i \(-0.548157\pi\)
−0.150712 + 0.988578i \(0.548157\pi\)
\(854\) 0 0
\(855\) 0.566640 0.0193787
\(856\) −0.932934 −0.0318870
\(857\) −16.9651 −0.579516 −0.289758 0.957100i \(-0.593575\pi\)
−0.289758 + 0.957100i \(0.593575\pi\)
\(858\) −2.53098 −0.0864063
\(859\) −14.5410 −0.496132 −0.248066 0.968743i \(-0.579795\pi\)
−0.248066 + 0.968743i \(0.579795\pi\)
\(860\) 17.3879 0.592922
\(861\) 0 0
\(862\) 46.9243 1.59825
\(863\) 39.0444 1.32909 0.664544 0.747249i \(-0.268626\pi\)
0.664544 + 0.747249i \(0.268626\pi\)
\(864\) 41.0529 1.39665
\(865\) 5.29052 0.179883
\(866\) −5.44882 −0.185159
\(867\) 20.1708 0.685036
\(868\) 0 0
\(869\) 9.31854 0.316110
\(870\) −10.4256 −0.353461
\(871\) −9.39174 −0.318227
\(872\) 0.941966 0.0318990
\(873\) −0.701211 −0.0237324
\(874\) 50.1022 1.69473
\(875\) 0 0
\(876\) −12.4272 −0.419875
\(877\) 32.5941 1.10062 0.550312 0.834959i \(-0.314509\pi\)
0.550312 + 0.834959i \(0.314509\pi\)
\(878\) −17.7457 −0.598888
\(879\) −55.9899 −1.88849
\(880\) 2.55020 0.0859671
\(881\) 43.4141 1.46266 0.731330 0.682024i \(-0.238900\pi\)
0.731330 + 0.682024i \(0.238900\pi\)
\(882\) 0 0
\(883\) 28.2902 0.952040 0.476020 0.879434i \(-0.342079\pi\)
0.476020 + 0.879434i \(0.342079\pi\)
\(884\) 4.78492 0.160934
\(885\) −1.14132 −0.0383650
\(886\) 5.83592 0.196062
\(887\) 50.2650 1.68773 0.843866 0.536554i \(-0.180274\pi\)
0.843866 + 0.536554i \(0.180274\pi\)
\(888\) 0.628852 0.0211029
\(889\) 0 0
\(890\) −21.9210 −0.734794
\(891\) −6.64349 −0.222565
\(892\) −35.8724 −1.20110
\(893\) −55.2099 −1.84753
\(894\) −74.4676 −2.49057
\(895\) −2.29535 −0.0767250
\(896\) 0 0
\(897\) 6.61239 0.220781
\(898\) 30.6856 1.02399
\(899\) −5.11668 −0.170651
\(900\) −0.801570 −0.0267190
\(901\) −33.1277 −1.10364
\(902\) 7.08145 0.235786
\(903\) 0 0
\(904\) 0.223604 0.00743695
\(905\) −9.71369 −0.322894
\(906\) −55.5526 −1.84561
\(907\) 26.8277 0.890798 0.445399 0.895332i \(-0.353062\pi\)
0.445399 + 0.895332i \(0.353062\pi\)
\(908\) 10.8402 0.359744
\(909\) 0.113045 0.00374946
\(910\) 0 0
\(911\) −22.3560 −0.740687 −0.370344 0.928895i \(-0.620760\pi\)
−0.370344 + 0.928895i \(0.620760\pi\)
\(912\) −45.8804 −1.51925
\(913\) −2.53651 −0.0839463
\(914\) −47.5217 −1.57188
\(915\) 18.5652 0.613747
\(916\) 17.3291 0.572569
\(917\) 0 0
\(918\) 24.1489 0.797034
\(919\) −8.62244 −0.284428 −0.142214 0.989836i \(-0.545422\pi\)
−0.142214 + 0.989836i \(0.545422\pi\)
\(920\) 0.233670 0.00770386
\(921\) 50.6404 1.66866
\(922\) −53.4609 −1.76064
\(923\) −10.9914 −0.361786
\(924\) 0 0
\(925\) −21.7712 −0.715832
\(926\) 2.89729 0.0952109
\(927\) 1.35877 0.0446278
\(928\) 26.1674 0.858987
\(929\) −41.3861 −1.35783 −0.678916 0.734216i \(-0.737550\pi\)
−0.678916 + 0.734216i \(0.737550\pi\)
\(930\) 5.02562 0.164796
\(931\) 0 0
\(932\) 9.66985 0.316747
\(933\) 9.70757 0.317812
\(934\) −16.8507 −0.551372
\(935\) 1.52621 0.0499124
\(936\) 0.00647055 0.000211496 0
\(937\) −21.3818 −0.698514 −0.349257 0.937027i \(-0.613566\pi\)
−0.349257 + 0.937027i \(0.613566\pi\)
\(938\) 0 0
\(939\) −8.52788 −0.278297
\(940\) −15.3285 −0.499959
\(941\) 53.1480 1.73258 0.866288 0.499545i \(-0.166499\pi\)
0.866288 + 0.499545i \(0.166499\pi\)
\(942\) 27.5052 0.896168
\(943\) −18.5009 −0.602471
\(944\) 2.81565 0.0916416
\(945\) 0 0
\(946\) 13.5792 0.441497
\(947\) −8.86936 −0.288216 −0.144108 0.989562i \(-0.546031\pi\)
−0.144108 + 0.989562i \(0.546031\pi\)
\(948\) −46.5461 −1.51175
\(949\) −3.47300 −0.112738
\(950\) −55.7082 −1.80741
\(951\) 13.4658 0.436658
\(952\) 0 0
\(953\) −39.8167 −1.28979 −0.644894 0.764272i \(-0.723099\pi\)
−0.644894 + 0.764272i \(0.723099\pi\)
\(954\) −2.66681 −0.0863413
\(955\) 1.52035 0.0491973
\(956\) 30.1696 0.975756
\(957\) −4.10546 −0.132711
\(958\) 25.3319 0.818435
\(959\) 0 0
\(960\) −13.1775 −0.425301
\(961\) −28.5335 −0.920437
\(962\) 10.4621 0.337310
\(963\) 1.28155 0.0412973
\(964\) 12.4518 0.401045
\(965\) 5.84024 0.188004
\(966\) 0 0
\(967\) −22.1611 −0.712652 −0.356326 0.934362i \(-0.615971\pi\)
−0.356326 + 0.934362i \(0.615971\pi\)
\(968\) 0.719735 0.0231332
\(969\) −27.4579 −0.882075
\(970\) −13.5303 −0.434433
\(971\) −36.0423 −1.15665 −0.578327 0.815805i \(-0.696294\pi\)
−0.578327 + 0.815805i \(0.696294\pi\)
\(972\) 1.99229 0.0639027
\(973\) 0 0
\(974\) −43.3199 −1.38806
\(975\) −7.35227 −0.235461
\(976\) −45.8007 −1.46604
\(977\) 32.9416 1.05389 0.526947 0.849898i \(-0.323336\pi\)
0.526947 + 0.849898i \(0.323336\pi\)
\(978\) 5.97064 0.190920
\(979\) −8.63219 −0.275886
\(980\) 0 0
\(981\) −1.29395 −0.0413128
\(982\) −78.8038 −2.51473
\(983\) 4.19945 0.133942 0.0669709 0.997755i \(-0.478667\pi\)
0.0669709 + 0.997755i \(0.478667\pi\)
\(984\) −0.594185 −0.0189419
\(985\) −1.69597 −0.0540382
\(986\) 15.3927 0.490203
\(987\) 0 0
\(988\) 13.4986 0.429447
\(989\) −35.4767 −1.12809
\(990\) 0.122862 0.00390480
\(991\) −13.4139 −0.426105 −0.213053 0.977041i \(-0.568341\pi\)
−0.213053 + 0.977041i \(0.568341\pi\)
\(992\) −12.6139 −0.400491
\(993\) −19.9533 −0.633198
\(994\) 0 0
\(995\) −10.3222 −0.327236
\(996\) 12.6699 0.401460
\(997\) 47.8868 1.51659 0.758295 0.651911i \(-0.226033\pi\)
0.758295 + 0.651911i \(0.226033\pi\)
\(998\) −18.4144 −0.582899
\(999\) 26.6240 0.842347
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 637.2.a.k.1.1 5
3.2 odd 2 5733.2.a.bm.1.5 5
7.2 even 3 637.2.e.m.508.5 10
7.3 odd 6 91.2.e.c.79.5 yes 10
7.4 even 3 637.2.e.m.79.5 10
7.5 odd 6 91.2.e.c.53.5 10
7.6 odd 2 637.2.a.l.1.1 5
13.12 even 2 8281.2.a.bx.1.5 5
21.5 even 6 819.2.j.h.235.1 10
21.17 even 6 819.2.j.h.352.1 10
21.20 even 2 5733.2.a.bl.1.5 5
28.3 even 6 1456.2.r.p.625.4 10
28.19 even 6 1456.2.r.p.417.4 10
91.12 odd 6 1183.2.e.f.508.1 10
91.38 odd 6 1183.2.e.f.170.1 10
91.90 odd 2 8281.2.a.bw.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.5 10 7.5 odd 6
91.2.e.c.79.5 yes 10 7.3 odd 6
637.2.a.k.1.1 5 1.1 even 1 trivial
637.2.a.l.1.1 5 7.6 odd 2
637.2.e.m.79.5 10 7.4 even 3
637.2.e.m.508.5 10 7.2 even 3
819.2.j.h.235.1 10 21.5 even 6
819.2.j.h.352.1 10 21.17 even 6
1183.2.e.f.170.1 10 91.38 odd 6
1183.2.e.f.508.1 10 91.12 odd 6
1456.2.r.p.417.4 10 28.19 even 6
1456.2.r.p.625.4 10 28.3 even 6
5733.2.a.bl.1.5 5 21.20 even 2
5733.2.a.bm.1.5 5 3.2 odd 2
8281.2.a.bw.1.5 5 91.90 odd 2
8281.2.a.bx.1.5 5 13.12 even 2