Properties

Label 177.4.a.d.1.6
Level $177$
Weight $4$
Character 177.1
Self dual yes
Analytic conductor $10.443$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 45 x^{6} + 47 x^{5} + 654 x^{4} - 157 x^{3} - 2898 x^{2} + 96 x + 2432\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-3.06139\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.06139 q^{2} +3.00000 q^{3} +8.49485 q^{4} +16.2722 q^{5} +12.1842 q^{6} +6.77038 q^{7} +2.00979 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.06139 q^{2} +3.00000 q^{3} +8.49485 q^{4} +16.2722 q^{5} +12.1842 q^{6} +6.77038 q^{7} +2.00979 q^{8} +9.00000 q^{9} +66.0875 q^{10} -16.5845 q^{11} +25.4846 q^{12} -82.3410 q^{13} +27.4971 q^{14} +48.8165 q^{15} -59.7963 q^{16} +6.48327 q^{17} +36.5525 q^{18} +151.026 q^{19} +138.230 q^{20} +20.3111 q^{21} -67.3561 q^{22} -154.283 q^{23} +6.02938 q^{24} +139.783 q^{25} -334.419 q^{26} +27.0000 q^{27} +57.5134 q^{28} +220.887 q^{29} +198.263 q^{30} -79.4162 q^{31} -258.934 q^{32} -49.7535 q^{33} +26.3311 q^{34} +110.169 q^{35} +76.4537 q^{36} +414.092 q^{37} +613.375 q^{38} -247.023 q^{39} +32.7037 q^{40} -521.903 q^{41} +82.4914 q^{42} -265.009 q^{43} -140.883 q^{44} +146.449 q^{45} -626.603 q^{46} +31.6419 q^{47} -179.389 q^{48} -297.162 q^{49} +567.714 q^{50} +19.4498 q^{51} -699.475 q^{52} -256.788 q^{53} +109.657 q^{54} -269.866 q^{55} +13.6071 q^{56} +453.078 q^{57} +897.107 q^{58} +59.0000 q^{59} +414.689 q^{60} -68.7236 q^{61} -322.540 q^{62} +60.9334 q^{63} -573.261 q^{64} -1339.87 q^{65} -202.068 q^{66} +198.971 q^{67} +55.0745 q^{68} -462.849 q^{69} +447.438 q^{70} +725.821 q^{71} +18.0882 q^{72} +571.563 q^{73} +1681.79 q^{74} +419.350 q^{75} +1282.94 q^{76} -112.283 q^{77} -1003.26 q^{78} -304.888 q^{79} -973.015 q^{80} +81.0000 q^{81} -2119.65 q^{82} +731.213 q^{83} +172.540 q^{84} +105.497 q^{85} -1076.30 q^{86} +662.661 q^{87} -33.3315 q^{88} -694.016 q^{89} +594.788 q^{90} -557.480 q^{91} -1310.61 q^{92} -238.249 q^{93} +128.510 q^{94} +2457.52 q^{95} -776.802 q^{96} -597.160 q^{97} -1206.89 q^{98} -149.261 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + O(q^{10}) \) \( 8q + 6q^{2} + 24q^{3} + 34q^{4} + 42q^{5} + 18q^{6} + 53q^{7} + 51q^{8} + 72q^{9} + 21q^{10} + 67q^{11} + 102q^{12} + 33q^{13} + 79q^{14} + 126q^{15} - 30q^{16} + 139q^{17} + 54q^{18} + 64q^{19} + 117q^{20} + 159q^{21} - 84q^{22} + 226q^{23} + 153q^{24} + 96q^{25} + 24q^{26} + 216q^{27} + 34q^{28} + 456q^{29} + 63q^{30} + 124q^{31} + 174q^{32} + 201q^{33} - 114q^{34} + 556q^{35} + 306q^{36} + 127q^{37} + 237q^{38} + 99q^{39} - 188q^{40} + 425q^{41} + 237q^{42} - 115q^{43} + 510q^{44} + 378q^{45} - 711q^{46} + 420q^{47} - 90q^{48} + 171q^{49} - 137q^{50} + 417q^{51} - 922q^{52} + 98q^{53} + 162q^{54} - 616q^{55} - 412q^{56} + 192q^{57} - 1548q^{58} + 472q^{59} + 351q^{60} - 1254q^{61} - 766q^{62} + 477q^{63} - 2019q^{64} - 734q^{65} - 252q^{66} - 1010q^{67} - 503q^{68} + 678q^{69} - 2956q^{70} - 17q^{71} + 459q^{72} - 1180q^{73} - 1228q^{74} + 288q^{75} - 2008q^{76} + 441q^{77} + 72q^{78} - 873q^{79} - 865q^{80} + 648q^{81} - 3645q^{82} + 759q^{83} + 102q^{84} - 850q^{85} - 1226q^{86} + 1368q^{87} - 3047q^{88} + 988q^{89} + 189q^{90} - 2111q^{91} - 1062q^{92} + 372q^{93} - 2240q^{94} + 1822q^{95} + 522q^{96} - 668q^{97} - 1368q^{98} + 603q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.06139 1.43592 0.717958 0.696086i \(-0.245077\pi\)
0.717958 + 0.696086i \(0.245077\pi\)
\(3\) 3.00000 0.577350
\(4\) 8.49485 1.06186
\(5\) 16.2722 1.45543 0.727713 0.685882i \(-0.240583\pi\)
0.727713 + 0.685882i \(0.240583\pi\)
\(6\) 12.1842 0.829027
\(7\) 6.77038 0.365566 0.182783 0.983153i \(-0.441489\pi\)
0.182783 + 0.983153i \(0.441489\pi\)
\(8\) 2.00979 0.0888212
\(9\) 9.00000 0.333333
\(10\) 66.0875 2.08987
\(11\) −16.5845 −0.454583 −0.227292 0.973827i \(-0.572987\pi\)
−0.227292 + 0.973827i \(0.572987\pi\)
\(12\) 25.4846 0.613063
\(13\) −82.3410 −1.75672 −0.878358 0.478004i \(-0.841360\pi\)
−0.878358 + 0.478004i \(0.841360\pi\)
\(14\) 27.4971 0.524923
\(15\) 48.8165 0.840291
\(16\) −59.7963 −0.934317
\(17\) 6.48327 0.0924956 0.0462478 0.998930i \(-0.485274\pi\)
0.0462478 + 0.998930i \(0.485274\pi\)
\(18\) 36.5525 0.478639
\(19\) 151.026 1.82356 0.911782 0.410674i \(-0.134706\pi\)
0.911782 + 0.410674i \(0.134706\pi\)
\(20\) 138.230 1.54545
\(21\) 20.3111 0.211060
\(22\) −67.3561 −0.652744
\(23\) −154.283 −1.39871 −0.699353 0.714777i \(-0.746528\pi\)
−0.699353 + 0.714777i \(0.746528\pi\)
\(24\) 6.02938 0.0512809
\(25\) 139.783 1.11827
\(26\) −334.419 −2.52250
\(27\) 27.0000 0.192450
\(28\) 57.5134 0.388179
\(29\) 220.887 1.41440 0.707201 0.707012i \(-0.249958\pi\)
0.707201 + 0.707012i \(0.249958\pi\)
\(30\) 198.263 1.20659
\(31\) −79.4162 −0.460115 −0.230058 0.973177i \(-0.573891\pi\)
−0.230058 + 0.973177i \(0.573891\pi\)
\(32\) −258.934 −1.43042
\(33\) −49.7535 −0.262454
\(34\) 26.3311 0.132816
\(35\) 110.169 0.532055
\(36\) 76.4537 0.353952
\(37\) 414.092 1.83990 0.919949 0.392038i \(-0.128230\pi\)
0.919949 + 0.392038i \(0.128230\pi\)
\(38\) 613.375 2.61849
\(39\) −247.023 −1.01424
\(40\) 32.7037 0.129273
\(41\) −521.903 −1.98799 −0.993995 0.109423i \(-0.965100\pi\)
−0.993995 + 0.109423i \(0.965100\pi\)
\(42\) 82.4914 0.303064
\(43\) −265.009 −0.939848 −0.469924 0.882707i \(-0.655719\pi\)
−0.469924 + 0.882707i \(0.655719\pi\)
\(44\) −140.883 −0.482702
\(45\) 146.449 0.485142
\(46\) −626.603 −2.00843
\(47\) 31.6419 0.0982011 0.0491005 0.998794i \(-0.484365\pi\)
0.0491005 + 0.998794i \(0.484365\pi\)
\(48\) −179.389 −0.539428
\(49\) −297.162 −0.866361
\(50\) 567.714 1.60574
\(51\) 19.4498 0.0534023
\(52\) −699.475 −1.86538
\(53\) −256.788 −0.665519 −0.332760 0.943012i \(-0.607980\pi\)
−0.332760 + 0.943012i \(0.607980\pi\)
\(54\) 109.657 0.276342
\(55\) −269.866 −0.661613
\(56\) 13.6071 0.0324700
\(57\) 453.078 1.05284
\(58\) 897.107 2.03096
\(59\) 59.0000 0.130189
\(60\) 414.689 0.892268
\(61\) −68.7236 −0.144248 −0.0721242 0.997396i \(-0.522978\pi\)
−0.0721242 + 0.997396i \(0.522978\pi\)
\(62\) −322.540 −0.660687
\(63\) 60.9334 0.121855
\(64\) −573.261 −1.11965
\(65\) −1339.87 −2.55677
\(66\) −202.068 −0.376862
\(67\) 198.971 0.362808 0.181404 0.983409i \(-0.441936\pi\)
0.181404 + 0.983409i \(0.441936\pi\)
\(68\) 55.0745 0.0982170
\(69\) −462.849 −0.807543
\(70\) 447.438 0.763986
\(71\) 725.821 1.21323 0.606613 0.794997i \(-0.292528\pi\)
0.606613 + 0.794997i \(0.292528\pi\)
\(72\) 18.0882 0.0296071
\(73\) 571.563 0.916388 0.458194 0.888852i \(-0.348496\pi\)
0.458194 + 0.888852i \(0.348496\pi\)
\(74\) 1681.79 2.64194
\(75\) 419.350 0.645631
\(76\) 1282.94 1.93636
\(77\) −112.283 −0.166180
\(78\) −1003.26 −1.45636
\(79\) −304.888 −0.434209 −0.217105 0.976148i \(-0.569661\pi\)
−0.217105 + 0.976148i \(0.569661\pi\)
\(80\) −973.015 −1.35983
\(81\) 81.0000 0.111111
\(82\) −2119.65 −2.85459
\(83\) 731.213 0.967001 0.483500 0.875344i \(-0.339365\pi\)
0.483500 + 0.875344i \(0.339365\pi\)
\(84\) 172.540 0.224115
\(85\) 105.497 0.134620
\(86\) −1076.30 −1.34954
\(87\) 662.661 0.816606
\(88\) −33.3315 −0.0403766
\(89\) −694.016 −0.826580 −0.413290 0.910600i \(-0.635620\pi\)
−0.413290 + 0.910600i \(0.635620\pi\)
\(90\) 594.788 0.696624
\(91\) −557.480 −0.642196
\(92\) −1310.61 −1.48523
\(93\) −238.249 −0.265648
\(94\) 128.510 0.141009
\(95\) 2457.52 2.65406
\(96\) −776.802 −0.825855
\(97\) −597.160 −0.625076 −0.312538 0.949905i \(-0.601179\pi\)
−0.312538 + 0.949905i \(0.601179\pi\)
\(98\) −1206.89 −1.24402
\(99\) −149.261 −0.151528
\(100\) 1187.44 1.18744
\(101\) 1186.90 1.16932 0.584659 0.811279i \(-0.301228\pi\)
0.584659 + 0.811279i \(0.301228\pi\)
\(102\) 78.9932 0.0766813
\(103\) −1037.17 −0.992190 −0.496095 0.868268i \(-0.665233\pi\)
−0.496095 + 0.868268i \(0.665233\pi\)
\(104\) −165.489 −0.156034
\(105\) 330.506 0.307182
\(106\) −1042.91 −0.955630
\(107\) 1786.99 1.61453 0.807266 0.590188i \(-0.200946\pi\)
0.807266 + 0.590188i \(0.200946\pi\)
\(108\) 229.361 0.204354
\(109\) −1548.23 −1.36050 −0.680248 0.732982i \(-0.738128\pi\)
−0.680248 + 0.732982i \(0.738128\pi\)
\(110\) −1096.03 −0.950021
\(111\) 1242.27 1.06227
\(112\) −404.844 −0.341555
\(113\) 1414.84 1.17785 0.588924 0.808189i \(-0.299552\pi\)
0.588924 + 0.808189i \(0.299552\pi\)
\(114\) 1840.12 1.51178
\(115\) −2510.52 −2.03571
\(116\) 1876.40 1.50189
\(117\) −741.069 −0.585572
\(118\) 239.622 0.186940
\(119\) 43.8942 0.0338133
\(120\) 98.1111 0.0746356
\(121\) −1055.95 −0.793354
\(122\) −279.113 −0.207129
\(123\) −1565.71 −1.14777
\(124\) −674.629 −0.488576
\(125\) 240.555 0.172127
\(126\) 247.474 0.174974
\(127\) 1389.63 0.970942 0.485471 0.874253i \(-0.338648\pi\)
0.485471 + 0.874253i \(0.338648\pi\)
\(128\) −256.762 −0.177303
\(129\) −795.026 −0.542621
\(130\) −5441.72 −3.67131
\(131\) −398.137 −0.265537 −0.132769 0.991147i \(-0.542387\pi\)
−0.132769 + 0.991147i \(0.542387\pi\)
\(132\) −422.649 −0.278688
\(133\) 1022.50 0.666634
\(134\) 808.096 0.520962
\(135\) 439.348 0.280097
\(136\) 13.0300 0.00821557
\(137\) −42.8475 −0.0267205 −0.0133603 0.999911i \(-0.504253\pi\)
−0.0133603 + 0.999911i \(0.504253\pi\)
\(138\) −1879.81 −1.15956
\(139\) −245.792 −0.149984 −0.0749922 0.997184i \(-0.523893\pi\)
−0.0749922 + 0.997184i \(0.523893\pi\)
\(140\) 935.868 0.564966
\(141\) 94.9258 0.0566964
\(142\) 2947.84 1.74209
\(143\) 1365.59 0.798574
\(144\) −538.167 −0.311439
\(145\) 3594.31 2.05856
\(146\) 2321.34 1.31586
\(147\) −891.486 −0.500194
\(148\) 3517.65 1.95371
\(149\) 1492.30 0.820495 0.410247 0.911974i \(-0.365442\pi\)
0.410247 + 0.911974i \(0.365442\pi\)
\(150\) 1703.14 0.927072
\(151\) −917.443 −0.494440 −0.247220 0.968959i \(-0.579517\pi\)
−0.247220 + 0.968959i \(0.579517\pi\)
\(152\) 303.531 0.161971
\(153\) 58.3495 0.0308319
\(154\) −456.027 −0.238621
\(155\) −1292.27 −0.669664
\(156\) −2098.43 −1.07698
\(157\) 2338.93 1.18896 0.594479 0.804111i \(-0.297358\pi\)
0.594479 + 0.804111i \(0.297358\pi\)
\(158\) −1238.27 −0.623489
\(159\) −770.364 −0.384238
\(160\) −4213.42 −2.08187
\(161\) −1044.56 −0.511320
\(162\) 328.972 0.159546
\(163\) 1425.51 0.684995 0.342498 0.939519i \(-0.388727\pi\)
0.342498 + 0.939519i \(0.388727\pi\)
\(164\) −4433.49 −2.11096
\(165\) −809.597 −0.381982
\(166\) 2969.74 1.38853
\(167\) 1937.97 0.897993 0.448996 0.893534i \(-0.351782\pi\)
0.448996 + 0.893534i \(0.351782\pi\)
\(168\) 40.8212 0.0187466
\(169\) 4583.05 2.08605
\(170\) 428.463 0.193304
\(171\) 1359.23 0.607855
\(172\) −2251.21 −0.997984
\(173\) 925.049 0.406533 0.203266 0.979123i \(-0.434844\pi\)
0.203266 + 0.979123i \(0.434844\pi\)
\(174\) 2691.32 1.17258
\(175\) 946.386 0.408800
\(176\) 991.692 0.424725
\(177\) 177.000 0.0751646
\(178\) −2818.67 −1.18690
\(179\) 2245.15 0.937490 0.468745 0.883334i \(-0.344706\pi\)
0.468745 + 0.883334i \(0.344706\pi\)
\(180\) 1244.07 0.515151
\(181\) −1772.98 −0.728090 −0.364045 0.931381i \(-0.618605\pi\)
−0.364045 + 0.931381i \(0.618605\pi\)
\(182\) −2264.14 −0.922140
\(183\) −206.171 −0.0832819
\(184\) −310.077 −0.124235
\(185\) 6738.17 2.67784
\(186\) −967.619 −0.381448
\(187\) −107.522 −0.0420469
\(188\) 268.794 0.104276
\(189\) 182.800 0.0703533
\(190\) 9980.93 3.81102
\(191\) −620.457 −0.235051 −0.117525 0.993070i \(-0.537496\pi\)
−0.117525 + 0.993070i \(0.537496\pi\)
\(192\) −1719.78 −0.646431
\(193\) −2381.92 −0.888366 −0.444183 0.895936i \(-0.646506\pi\)
−0.444183 + 0.895936i \(0.646506\pi\)
\(194\) −2425.30 −0.897557
\(195\) −4019.60 −1.47615
\(196\) −2524.35 −0.919952
\(197\) 2044.21 0.739311 0.369655 0.929169i \(-0.379476\pi\)
0.369655 + 0.929169i \(0.379476\pi\)
\(198\) −606.205 −0.217581
\(199\) 986.827 0.351529 0.175765 0.984432i \(-0.443760\pi\)
0.175765 + 0.984432i \(0.443760\pi\)
\(200\) 280.936 0.0993257
\(201\) 596.912 0.209467
\(202\) 4820.46 1.67904
\(203\) 1495.49 0.517058
\(204\) 165.223 0.0567056
\(205\) −8492.50 −2.89337
\(206\) −4212.35 −1.42470
\(207\) −1388.55 −0.466235
\(208\) 4923.69 1.64133
\(209\) −2504.69 −0.828962
\(210\) 1342.31 0.441088
\(211\) 2539.95 0.828706 0.414353 0.910116i \(-0.364008\pi\)
0.414353 + 0.910116i \(0.364008\pi\)
\(212\) −2181.38 −0.706686
\(213\) 2177.46 0.700457
\(214\) 7257.66 2.31833
\(215\) −4312.27 −1.36788
\(216\) 54.2645 0.0170936
\(217\) −537.678 −0.168203
\(218\) −6287.98 −1.95356
\(219\) 1714.69 0.529077
\(220\) −2292.47 −0.702538
\(221\) −533.839 −0.162488
\(222\) 5045.36 1.52533
\(223\) 5064.96 1.52096 0.760482 0.649359i \(-0.224963\pi\)
0.760482 + 0.649359i \(0.224963\pi\)
\(224\) −1753.08 −0.522914
\(225\) 1258.05 0.372755
\(226\) 5746.20 1.69129
\(227\) −875.416 −0.255962 −0.127981 0.991777i \(-0.540850\pi\)
−0.127981 + 0.991777i \(0.540850\pi\)
\(228\) 3848.83 1.11796
\(229\) −4519.88 −1.30429 −0.652145 0.758095i \(-0.726131\pi\)
−0.652145 + 0.758095i \(0.726131\pi\)
\(230\) −10196.2 −2.92311
\(231\) −336.850 −0.0959443
\(232\) 443.937 0.125629
\(233\) −3300.46 −0.927983 −0.463992 0.885840i \(-0.653583\pi\)
−0.463992 + 0.885840i \(0.653583\pi\)
\(234\) −3009.77 −0.840832
\(235\) 514.883 0.142924
\(236\) 501.196 0.138242
\(237\) −914.663 −0.250691
\(238\) 178.271 0.0485530
\(239\) 2036.15 0.551079 0.275539 0.961290i \(-0.411144\pi\)
0.275539 + 0.961290i \(0.411144\pi\)
\(240\) −2919.04 −0.785098
\(241\) 1135.42 0.303480 0.151740 0.988420i \(-0.451512\pi\)
0.151740 + 0.988420i \(0.451512\pi\)
\(242\) −4288.64 −1.13919
\(243\) 243.000 0.0641500
\(244\) −583.797 −0.153171
\(245\) −4835.47 −1.26092
\(246\) −6358.95 −1.64810
\(247\) −12435.6 −3.20348
\(248\) −159.610 −0.0408680
\(249\) 2193.64 0.558298
\(250\) 976.987 0.247160
\(251\) 6147.42 1.54590 0.772952 0.634465i \(-0.218780\pi\)
0.772952 + 0.634465i \(0.218780\pi\)
\(252\) 517.621 0.129393
\(253\) 2558.71 0.635828
\(254\) 5643.82 1.39419
\(255\) 316.491 0.0777232
\(256\) 3543.28 0.865059
\(257\) 866.093 0.210216 0.105108 0.994461i \(-0.466481\pi\)
0.105108 + 0.994461i \(0.466481\pi\)
\(258\) −3228.91 −0.779159
\(259\) 2803.56 0.672605
\(260\) −11382.0 −2.71492
\(261\) 1987.98 0.471467
\(262\) −1616.99 −0.381289
\(263\) −4743.67 −1.11220 −0.556098 0.831117i \(-0.687702\pi\)
−0.556098 + 0.831117i \(0.687702\pi\)
\(264\) −99.9944 −0.0233115
\(265\) −4178.49 −0.968615
\(266\) 4152.78 0.957231
\(267\) −2082.05 −0.477226
\(268\) 1690.23 0.385250
\(269\) −5845.97 −1.32504 −0.662519 0.749045i \(-0.730513\pi\)
−0.662519 + 0.749045i \(0.730513\pi\)
\(270\) 1784.36 0.402196
\(271\) −4680.05 −1.04905 −0.524525 0.851395i \(-0.675757\pi\)
−0.524525 + 0.851395i \(0.675757\pi\)
\(272\) −387.676 −0.0864202
\(273\) −1672.44 −0.370772
\(274\) −174.020 −0.0383684
\(275\) −2318.24 −0.508345
\(276\) −3931.84 −0.857495
\(277\) −3885.21 −0.842742 −0.421371 0.906888i \(-0.638451\pi\)
−0.421371 + 0.906888i \(0.638451\pi\)
\(278\) −998.257 −0.215365
\(279\) −714.746 −0.153372
\(280\) 221.417 0.0472578
\(281\) −7580.94 −1.60940 −0.804700 0.593682i \(-0.797674\pi\)
−0.804700 + 0.593682i \(0.797674\pi\)
\(282\) 385.530 0.0814113
\(283\) 947.742 0.199072 0.0995361 0.995034i \(-0.468264\pi\)
0.0995361 + 0.995034i \(0.468264\pi\)
\(284\) 6165.74 1.28827
\(285\) 7372.56 1.53232
\(286\) 5546.17 1.14669
\(287\) −3533.49 −0.726742
\(288\) −2330.41 −0.476808
\(289\) −4870.97 −0.991445
\(290\) 14597.9 2.95592
\(291\) −1791.48 −0.360888
\(292\) 4855.34 0.973073
\(293\) −2612.95 −0.520990 −0.260495 0.965475i \(-0.583886\pi\)
−0.260495 + 0.965475i \(0.583886\pi\)
\(294\) −3620.67 −0.718237
\(295\) 960.058 0.189480
\(296\) 832.239 0.163422
\(297\) −447.782 −0.0874846
\(298\) 6060.79 1.17816
\(299\) 12703.8 2.45713
\(300\) 3562.31 0.685568
\(301\) −1794.21 −0.343577
\(302\) −3726.09 −0.709975
\(303\) 3560.70 0.675106
\(304\) −9030.79 −1.70379
\(305\) −1118.28 −0.209943
\(306\) 236.980 0.0442720
\(307\) 6751.14 1.25507 0.627537 0.778587i \(-0.284063\pi\)
0.627537 + 0.778587i \(0.284063\pi\)
\(308\) −953.832 −0.176460
\(309\) −3111.51 −0.572841
\(310\) −5248.42 −0.961581
\(311\) 1176.93 0.214591 0.107295 0.994227i \(-0.465781\pi\)
0.107295 + 0.994227i \(0.465781\pi\)
\(312\) −496.466 −0.0900860
\(313\) 2059.81 0.371972 0.185986 0.982552i \(-0.440452\pi\)
0.185986 + 0.982552i \(0.440452\pi\)
\(314\) 9499.28 1.70725
\(315\) 991.519 0.177352
\(316\) −2589.98 −0.461068
\(317\) 5322.47 0.943027 0.471513 0.881859i \(-0.343708\pi\)
0.471513 + 0.881859i \(0.343708\pi\)
\(318\) −3128.74 −0.551734
\(319\) −3663.30 −0.642964
\(320\) −9328.20 −1.62957
\(321\) 5360.97 0.932151
\(322\) −4242.34 −0.734213
\(323\) 979.142 0.168672
\(324\) 688.083 0.117984
\(325\) −11509.9 −1.96447
\(326\) 5789.53 0.983596
\(327\) −4644.70 −0.785482
\(328\) −1048.92 −0.176576
\(329\) 214.228 0.0358990
\(330\) −3288.09 −0.548495
\(331\) −10946.3 −1.81772 −0.908860 0.417101i \(-0.863046\pi\)
−0.908860 + 0.417101i \(0.863046\pi\)
\(332\) 6211.55 1.02682
\(333\) 3726.82 0.613299
\(334\) 7870.85 1.28944
\(335\) 3237.68 0.528040
\(336\) −1214.53 −0.197197
\(337\) 333.243 0.0538661 0.0269331 0.999637i \(-0.491426\pi\)
0.0269331 + 0.999637i \(0.491426\pi\)
\(338\) 18613.5 2.99539
\(339\) 4244.51 0.680030
\(340\) 896.180 0.142948
\(341\) 1317.08 0.209161
\(342\) 5520.37 0.872829
\(343\) −4334.14 −0.682279
\(344\) −532.613 −0.0834784
\(345\) −7531.55 −1.17532
\(346\) 3756.98 0.583747
\(347\) −2054.05 −0.317773 −0.158886 0.987297i \(-0.550790\pi\)
−0.158886 + 0.987297i \(0.550790\pi\)
\(348\) 5629.21 0.867118
\(349\) 6861.41 1.05239 0.526193 0.850365i \(-0.323619\pi\)
0.526193 + 0.850365i \(0.323619\pi\)
\(350\) 3843.64 0.587003
\(351\) −2223.21 −0.338080
\(352\) 4294.30 0.650246
\(353\) −3523.31 −0.531237 −0.265618 0.964078i \(-0.585576\pi\)
−0.265618 + 0.964078i \(0.585576\pi\)
\(354\) 718.865 0.107930
\(355\) 11810.7 1.76576
\(356\) −5895.57 −0.877709
\(357\) 131.683 0.0195221
\(358\) 9118.44 1.34616
\(359\) 1692.91 0.248882 0.124441 0.992227i \(-0.460286\pi\)
0.124441 + 0.992227i \(0.460286\pi\)
\(360\) 294.333 0.0430909
\(361\) 15949.8 2.32539
\(362\) −7200.74 −1.04548
\(363\) −3167.86 −0.458043
\(364\) −4735.72 −0.681920
\(365\) 9300.56 1.33374
\(366\) −837.339 −0.119586
\(367\) −3865.33 −0.549778 −0.274889 0.961476i \(-0.588641\pi\)
−0.274889 + 0.961476i \(0.588641\pi\)
\(368\) 9225.55 1.30683
\(369\) −4697.13 −0.662664
\(370\) 27366.3 3.84515
\(371\) −1738.55 −0.243292
\(372\) −2023.89 −0.282080
\(373\) −6121.11 −0.849703 −0.424851 0.905263i \(-0.639674\pi\)
−0.424851 + 0.905263i \(0.639674\pi\)
\(374\) −436.688 −0.0603759
\(375\) 721.665 0.0993777
\(376\) 63.5938 0.00872234
\(377\) −18188.1 −2.48470
\(378\) 742.423 0.101021
\(379\) −2939.20 −0.398356 −0.199178 0.979963i \(-0.563827\pi\)
−0.199178 + 0.979963i \(0.563827\pi\)
\(380\) 20876.3 2.81824
\(381\) 4168.89 0.560573
\(382\) −2519.92 −0.337513
\(383\) 8175.22 1.09069 0.545345 0.838212i \(-0.316399\pi\)
0.545345 + 0.838212i \(0.316399\pi\)
\(384\) −770.285 −0.102366
\(385\) −1827.09 −0.241863
\(386\) −9673.91 −1.27562
\(387\) −2385.08 −0.313283
\(388\) −5072.78 −0.663741
\(389\) 7355.93 0.958768 0.479384 0.877605i \(-0.340860\pi\)
0.479384 + 0.877605i \(0.340860\pi\)
\(390\) −16325.1 −2.11963
\(391\) −1000.26 −0.129374
\(392\) −597.234 −0.0769513
\(393\) −1194.41 −0.153308
\(394\) 8302.34 1.06159
\(395\) −4961.18 −0.631960
\(396\) −1267.95 −0.160901
\(397\) −7487.16 −0.946523 −0.473261 0.880922i \(-0.656923\pi\)
−0.473261 + 0.880922i \(0.656923\pi\)
\(398\) 4007.89 0.504767
\(399\) 3067.51 0.384881
\(400\) −8358.52 −1.04481
\(401\) −9700.71 −1.20806 −0.604028 0.796963i \(-0.706438\pi\)
−0.604028 + 0.796963i \(0.706438\pi\)
\(402\) 2424.29 0.300777
\(403\) 6539.21 0.808291
\(404\) 10082.6 1.24165
\(405\) 1318.05 0.161714
\(406\) 6073.76 0.742452
\(407\) −6867.51 −0.836387
\(408\) 39.0901 0.00474326
\(409\) −1378.30 −0.166632 −0.0833160 0.996523i \(-0.526551\pi\)
−0.0833160 + 0.996523i \(0.526551\pi\)
\(410\) −34491.3 −4.15464
\(411\) −128.543 −0.0154271
\(412\) −8810.62 −1.05356
\(413\) 399.453 0.0475927
\(414\) −5639.43 −0.669475
\(415\) 11898.4 1.40740
\(416\) 21320.9 2.51285
\(417\) −737.377 −0.0865935
\(418\) −10172.5 −1.19032
\(419\) −6367.90 −0.742463 −0.371232 0.928540i \(-0.621064\pi\)
−0.371232 + 0.928540i \(0.621064\pi\)
\(420\) 2807.60 0.326183
\(421\) 4730.02 0.547570 0.273785 0.961791i \(-0.411724\pi\)
0.273785 + 0.961791i \(0.411724\pi\)
\(422\) 10315.7 1.18995
\(423\) 284.777 0.0327337
\(424\) −516.091 −0.0591122
\(425\) 906.253 0.103435
\(426\) 8843.52 1.00580
\(427\) −465.285 −0.0527324
\(428\) 15180.2 1.71440
\(429\) 4096.76 0.461057
\(430\) −17513.8 −1.96416
\(431\) 12035.5 1.34509 0.672543 0.740058i \(-0.265202\pi\)
0.672543 + 0.740058i \(0.265202\pi\)
\(432\) −1614.50 −0.179809
\(433\) −16014.7 −1.77741 −0.888705 0.458479i \(-0.848395\pi\)
−0.888705 + 0.458479i \(0.848395\pi\)
\(434\) −2183.72 −0.241525
\(435\) 10782.9 1.18851
\(436\) −13152.0 −1.44465
\(437\) −23300.7 −2.55063
\(438\) 6964.01 0.759711
\(439\) 3032.82 0.329724 0.164862 0.986317i \(-0.447282\pi\)
0.164862 + 0.986317i \(0.447282\pi\)
\(440\) −542.375 −0.0587652
\(441\) −2674.46 −0.288787
\(442\) −2168.13 −0.233320
\(443\) −1755.79 −0.188308 −0.0941538 0.995558i \(-0.530015\pi\)
−0.0941538 + 0.995558i \(0.530015\pi\)
\(444\) 10552.9 1.12797
\(445\) −11293.1 −1.20303
\(446\) 20570.8 2.18398
\(447\) 4476.89 0.473713
\(448\) −3881.20 −0.409307
\(449\) −11467.7 −1.20533 −0.602666 0.797994i \(-0.705895\pi\)
−0.602666 + 0.797994i \(0.705895\pi\)
\(450\) 5109.42 0.535246
\(451\) 8655.51 0.903708
\(452\) 12018.8 1.25071
\(453\) −2752.33 −0.285465
\(454\) −3555.40 −0.367540
\(455\) −9071.41 −0.934669
\(456\) 910.593 0.0935141
\(457\) 920.816 0.0942537 0.0471268 0.998889i \(-0.484993\pi\)
0.0471268 + 0.998889i \(0.484993\pi\)
\(458\) −18357.0 −1.87285
\(459\) 175.048 0.0178008
\(460\) −21326.5 −2.16164
\(461\) 18808.7 1.90023 0.950117 0.311895i \(-0.100964\pi\)
0.950117 + 0.311895i \(0.100964\pi\)
\(462\) −1368.08 −0.137768
\(463\) −4494.12 −0.451101 −0.225550 0.974232i \(-0.572418\pi\)
−0.225550 + 0.974232i \(0.572418\pi\)
\(464\) −13208.2 −1.32150
\(465\) −3876.82 −0.386630
\(466\) −13404.4 −1.33251
\(467\) 608.207 0.0602665 0.0301332 0.999546i \(-0.490407\pi\)
0.0301332 + 0.999546i \(0.490407\pi\)
\(468\) −6295.28 −0.621793
\(469\) 1347.11 0.132630
\(470\) 2091.14 0.205228
\(471\) 7016.78 0.686446
\(472\) 118.578 0.0115635
\(473\) 4395.04 0.427239
\(474\) −3714.80 −0.359971
\(475\) 21110.9 2.03923
\(476\) 372.875 0.0359048
\(477\) −2311.09 −0.221840
\(478\) 8269.60 0.791303
\(479\) −4182.01 −0.398916 −0.199458 0.979906i \(-0.563918\pi\)
−0.199458 + 0.979906i \(0.563918\pi\)
\(480\) −12640.3 −1.20197
\(481\) −34096.7 −3.23218
\(482\) 4611.37 0.435772
\(483\) −3133.67 −0.295211
\(484\) −8970.18 −0.842428
\(485\) −9717.08 −0.909752
\(486\) 986.917 0.0921141
\(487\) −18578.4 −1.72868 −0.864340 0.502908i \(-0.832264\pi\)
−0.864340 + 0.502908i \(0.832264\pi\)
\(488\) −138.120 −0.0128123
\(489\) 4276.52 0.395482
\(490\) −19638.7 −1.81058
\(491\) 14967.0 1.37567 0.687833 0.725869i \(-0.258562\pi\)
0.687833 + 0.725869i \(0.258562\pi\)
\(492\) −13300.5 −1.21876
\(493\) 1432.07 0.130826
\(494\) −50505.9 −4.59994
\(495\) −2428.79 −0.220538
\(496\) 4748.79 0.429893
\(497\) 4914.09 0.443515
\(498\) 8909.22 0.801670
\(499\) −14608.3 −1.31053 −0.655266 0.755398i \(-0.727444\pi\)
−0.655266 + 0.755398i \(0.727444\pi\)
\(500\) 2043.48 0.182774
\(501\) 5813.92 0.518456
\(502\) 24967.0 2.21979
\(503\) 11306.5 1.00225 0.501127 0.865374i \(-0.332919\pi\)
0.501127 + 0.865374i \(0.332919\pi\)
\(504\) 122.464 0.0108233
\(505\) 19313.4 1.70186
\(506\) 10391.9 0.912997
\(507\) 13749.1 1.20438
\(508\) 11804.7 1.03100
\(509\) 6037.40 0.525743 0.262871 0.964831i \(-0.415330\pi\)
0.262871 + 0.964831i \(0.415330\pi\)
\(510\) 1285.39 0.111604
\(511\) 3869.70 0.335001
\(512\) 16444.7 1.41946
\(513\) 4077.70 0.350945
\(514\) 3517.54 0.301852
\(515\) −16877.0 −1.44406
\(516\) −6753.63 −0.576186
\(517\) −524.766 −0.0446406
\(518\) 11386.3 0.965805
\(519\) 2775.15 0.234712
\(520\) −2692.86 −0.227095
\(521\) 12456.9 1.04750 0.523750 0.851872i \(-0.324532\pi\)
0.523750 + 0.851872i \(0.324532\pi\)
\(522\) 8073.96 0.676988
\(523\) 17565.0 1.46858 0.734288 0.678838i \(-0.237516\pi\)
0.734288 + 0.678838i \(0.237516\pi\)
\(524\) −3382.11 −0.281962
\(525\) 2839.16 0.236021
\(526\) −19265.9 −1.59702
\(527\) −514.877 −0.0425586
\(528\) 2975.08 0.245215
\(529\) 11636.3 0.956378
\(530\) −16970.5 −1.39085
\(531\) 531.000 0.0433963
\(532\) 8686.02 0.707870
\(533\) 42974.1 3.49233
\(534\) −8456.00 −0.685257
\(535\) 29078.2 2.34983
\(536\) 399.890 0.0322250
\(537\) 6735.46 0.541260
\(538\) −23742.8 −1.90264
\(539\) 4928.28 0.393833
\(540\) 3732.20 0.297423
\(541\) −12600.3 −1.00135 −0.500674 0.865636i \(-0.666914\pi\)
−0.500674 + 0.865636i \(0.666914\pi\)
\(542\) −19007.5 −1.50635
\(543\) −5318.93 −0.420363
\(544\) −1678.74 −0.132308
\(545\) −25193.1 −1.98010
\(546\) −6792.43 −0.532398
\(547\) 18529.7 1.44840 0.724198 0.689592i \(-0.242210\pi\)
0.724198 + 0.689592i \(0.242210\pi\)
\(548\) −363.983 −0.0283734
\(549\) −618.512 −0.0480828
\(550\) −9415.25 −0.729941
\(551\) 33359.7 2.57925
\(552\) −930.232 −0.0717270
\(553\) −2064.21 −0.158732
\(554\) −15779.3 −1.21011
\(555\) 20214.5 1.54605
\(556\) −2087.97 −0.159262
\(557\) −17686.0 −1.34538 −0.672691 0.739923i \(-0.734862\pi\)
−0.672691 + 0.739923i \(0.734862\pi\)
\(558\) −2902.86 −0.220229
\(559\) 21821.1 1.65105
\(560\) −6587.68 −0.497108
\(561\) −322.566 −0.0242758
\(562\) −30789.1 −2.31096
\(563\) 1719.56 0.128723 0.0643614 0.997927i \(-0.479499\pi\)
0.0643614 + 0.997927i \(0.479499\pi\)
\(564\) 806.381 0.0602035
\(565\) 23022.5 1.71427
\(566\) 3849.15 0.285851
\(567\) 548.401 0.0406185
\(568\) 1458.75 0.107760
\(569\) 4196.95 0.309218 0.154609 0.987976i \(-0.450588\pi\)
0.154609 + 0.987976i \(0.450588\pi\)
\(570\) 29942.8 2.20029
\(571\) −16717.6 −1.22523 −0.612616 0.790380i \(-0.709883\pi\)
−0.612616 + 0.790380i \(0.709883\pi\)
\(572\) 11600.5 0.847971
\(573\) −1861.37 −0.135707
\(574\) −14350.8 −1.04354
\(575\) −21566.2 −1.56412
\(576\) −5159.35 −0.373217
\(577\) 4687.41 0.338196 0.169098 0.985599i \(-0.445915\pi\)
0.169098 + 0.985599i \(0.445915\pi\)
\(578\) −19782.9 −1.42363
\(579\) −7145.77 −0.512898
\(580\) 30533.1 2.18589
\(581\) 4950.59 0.353503
\(582\) −7275.89 −0.518205
\(583\) 4258.70 0.302534
\(584\) 1148.72 0.0813947
\(585\) −12058.8 −0.852257
\(586\) −10612.2 −0.748098
\(587\) 15750.6 1.10749 0.553745 0.832686i \(-0.313198\pi\)
0.553745 + 0.832686i \(0.313198\pi\)
\(588\) −7573.04 −0.531134
\(589\) −11993.9 −0.839050
\(590\) 3899.16 0.272078
\(591\) 6132.64 0.426841
\(592\) −24761.1 −1.71905
\(593\) −3706.55 −0.256677 −0.128339 0.991730i \(-0.540964\pi\)
−0.128339 + 0.991730i \(0.540964\pi\)
\(594\) −1818.61 −0.125621
\(595\) 714.254 0.0492127
\(596\) 12676.8 0.871248
\(597\) 2960.48 0.202956
\(598\) 51595.1 3.52823
\(599\) 263.566 0.0179783 0.00898917 0.999960i \(-0.497139\pi\)
0.00898917 + 0.999960i \(0.497139\pi\)
\(600\) 842.807 0.0573457
\(601\) −11714.8 −0.795101 −0.397550 0.917580i \(-0.630140\pi\)
−0.397550 + 0.917580i \(0.630140\pi\)
\(602\) −7286.98 −0.493348
\(603\) 1790.74 0.120936
\(604\) −7793.55 −0.525025
\(605\) −17182.7 −1.15467
\(606\) 14461.4 0.969396
\(607\) 7579.45 0.506821 0.253410 0.967359i \(-0.418448\pi\)
0.253410 + 0.967359i \(0.418448\pi\)
\(608\) −39105.8 −2.60847
\(609\) 4486.47 0.298523
\(610\) −4541.77 −0.301461
\(611\) −2605.43 −0.172511
\(612\) 495.670 0.0327390
\(613\) −15090.5 −0.994291 −0.497146 0.867667i \(-0.665619\pi\)
−0.497146 + 0.867667i \(0.665619\pi\)
\(614\) 27419.0 1.80218
\(615\) −25477.5 −1.67049
\(616\) −225.667 −0.0147603
\(617\) −19397.1 −1.26564 −0.632819 0.774300i \(-0.718102\pi\)
−0.632819 + 0.774300i \(0.718102\pi\)
\(618\) −12637.1 −0.822552
\(619\) −14955.2 −0.971082 −0.485541 0.874214i \(-0.661377\pi\)
−0.485541 + 0.874214i \(0.661377\pi\)
\(620\) −10977.7 −0.711087
\(621\) −4165.64 −0.269181
\(622\) 4779.97 0.308134
\(623\) −4698.76 −0.302170
\(624\) 14771.1 0.947622
\(625\) −13558.6 −0.867747
\(626\) 8365.68 0.534121
\(627\) −7514.07 −0.478602
\(628\) 19868.8 1.26250
\(629\) 2684.67 0.170182
\(630\) 4026.94 0.254662
\(631\) −2751.59 −0.173596 −0.0867979 0.996226i \(-0.527663\pi\)
−0.0867979 + 0.996226i \(0.527663\pi\)
\(632\) −612.762 −0.0385670
\(633\) 7619.84 0.478454
\(634\) 21616.6 1.35411
\(635\) 22612.3 1.41313
\(636\) −6544.13 −0.408006
\(637\) 24468.6 1.52195
\(638\) −14878.1 −0.923242
\(639\) 6532.39 0.404409
\(640\) −4178.07 −0.258051
\(641\) −16315.9 −1.00536 −0.502682 0.864471i \(-0.667653\pi\)
−0.502682 + 0.864471i \(0.667653\pi\)
\(642\) 21773.0 1.33849
\(643\) 24947.6 1.53007 0.765037 0.643987i \(-0.222721\pi\)
0.765037 + 0.643987i \(0.222721\pi\)
\(644\) −8873.34 −0.542948
\(645\) −12936.8 −0.789745
\(646\) 3976.68 0.242198
\(647\) −9286.05 −0.564254 −0.282127 0.959377i \(-0.591040\pi\)
−0.282127 + 0.959377i \(0.591040\pi\)
\(648\) 162.793 0.00986902
\(649\) −978.486 −0.0591817
\(650\) −46746.1 −2.82082
\(651\) −1613.03 −0.0971118
\(652\) 12109.5 0.727367
\(653\) −7806.02 −0.467800 −0.233900 0.972261i \(-0.575149\pi\)
−0.233900 + 0.972261i \(0.575149\pi\)
\(654\) −18863.9 −1.12789
\(655\) −6478.54 −0.386470
\(656\) 31207.9 1.85741
\(657\) 5144.06 0.305463
\(658\) 870.063 0.0515480
\(659\) −30652.4 −1.81191 −0.905955 0.423374i \(-0.860846\pi\)
−0.905955 + 0.423374i \(0.860846\pi\)
\(660\) −6877.41 −0.405610
\(661\) −1119.99 −0.0659041 −0.0329520 0.999457i \(-0.510491\pi\)
−0.0329520 + 0.999457i \(0.510491\pi\)
\(662\) −44457.3 −2.61010
\(663\) −1601.52 −0.0938127
\(664\) 1469.59 0.0858902
\(665\) 16638.3 0.970237
\(666\) 15136.1 0.880647
\(667\) −34079.1 −1.97833
\(668\) 16462.8 0.953540
\(669\) 15194.9 0.878129
\(670\) 13149.5 0.758222
\(671\) 1139.75 0.0655730
\(672\) −5259.25 −0.301905
\(673\) −20636.6 −1.18199 −0.590996 0.806675i \(-0.701265\pi\)
−0.590996 + 0.806675i \(0.701265\pi\)
\(674\) 1353.43 0.0773473
\(675\) 3774.15 0.215210
\(676\) 38932.3 2.21508
\(677\) −24871.6 −1.41195 −0.705976 0.708236i \(-0.749491\pi\)
−0.705976 + 0.708236i \(0.749491\pi\)
\(678\) 17238.6 0.976467
\(679\) −4043.00 −0.228507
\(680\) 212.027 0.0119572
\(681\) −2626.25 −0.147780
\(682\) 5349.16 0.300337
\(683\) 17124.9 0.959392 0.479696 0.877435i \(-0.340747\pi\)
0.479696 + 0.877435i \(0.340747\pi\)
\(684\) 11546.5 0.645455
\(685\) −697.222 −0.0388897
\(686\) −17602.6 −0.979696
\(687\) −13559.6 −0.753032
\(688\) 15846.5 0.878116
\(689\) 21144.2 1.16913
\(690\) −30588.5 −1.68766
\(691\) −15624.6 −0.860183 −0.430092 0.902785i \(-0.641519\pi\)
−0.430092 + 0.902785i \(0.641519\pi\)
\(692\) 7858.16 0.431680
\(693\) −1010.55 −0.0553935
\(694\) −8342.29 −0.456295
\(695\) −3999.57 −0.218291
\(696\) 1331.81 0.0725319
\(697\) −3383.64 −0.183880
\(698\) 27866.8 1.51114
\(699\) −9901.37 −0.535771
\(700\) 8039.41 0.434087
\(701\) 21519.2 1.15944 0.579722 0.814814i \(-0.303161\pi\)
0.579722 + 0.814814i \(0.303161\pi\)
\(702\) −9029.31 −0.485455
\(703\) 62538.6 3.35517
\(704\) 9507.25 0.508975
\(705\) 1544.65 0.0825175
\(706\) −14309.5 −0.762812
\(707\) 8035.77 0.427463
\(708\) 1503.59 0.0798140
\(709\) −28820.4 −1.52662 −0.763309 0.646034i \(-0.776426\pi\)
−0.763309 + 0.646034i \(0.776426\pi\)
\(710\) 47967.7 2.53549
\(711\) −2743.99 −0.144736
\(712\) −1394.83 −0.0734178
\(713\) 12252.6 0.643566
\(714\) 534.814 0.0280321
\(715\) 22221.0 1.16227
\(716\) 19072.3 0.995480
\(717\) 6108.46 0.318165
\(718\) 6875.57 0.357373
\(719\) −4082.36 −0.211747 −0.105874 0.994380i \(-0.533764\pi\)
−0.105874 + 0.994380i \(0.533764\pi\)
\(720\) −8757.13 −0.453276
\(721\) −7022.05 −0.362711
\(722\) 64778.5 3.33906
\(723\) 3406.26 0.175214
\(724\) −15061.2 −0.773127
\(725\) 30876.3 1.58168
\(726\) −12865.9 −0.657712
\(727\) 34069.5 1.73806 0.869028 0.494763i \(-0.164745\pi\)
0.869028 + 0.494763i \(0.164745\pi\)
\(728\) −1120.42 −0.0570406
\(729\) 729.000 0.0370370
\(730\) 37773.2 1.91513
\(731\) −1718.12 −0.0869318
\(732\) −1751.39 −0.0884334
\(733\) 23108.8 1.16445 0.582226 0.813027i \(-0.302182\pi\)
0.582226 + 0.813027i \(0.302182\pi\)
\(734\) −15698.6 −0.789435
\(735\) −14506.4 −0.727995
\(736\) 39949.1 2.00074
\(737\) −3299.83 −0.164926
\(738\) −19076.9 −0.951530
\(739\) −22291.4 −1.10961 −0.554805 0.831981i \(-0.687207\pi\)
−0.554805 + 0.831981i \(0.687207\pi\)
\(740\) 57239.7 2.84348
\(741\) −37306.9 −1.84953
\(742\) −7060.93 −0.349346
\(743\) 2619.28 0.129330 0.0646648 0.997907i \(-0.479402\pi\)
0.0646648 + 0.997907i \(0.479402\pi\)
\(744\) −478.831 −0.0235951
\(745\) 24282.9 1.19417
\(746\) −24860.2 −1.22010
\(747\) 6580.92 0.322334
\(748\) −913.383 −0.0446478
\(749\) 12098.6 0.590219
\(750\) 2930.96 0.142698
\(751\) −16907.8 −0.821538 −0.410769 0.911739i \(-0.634740\pi\)
−0.410769 + 0.911739i \(0.634740\pi\)
\(752\) −1892.07 −0.0917509
\(753\) 18442.3 0.892528
\(754\) −73868.7 −3.56783
\(755\) −14928.8 −0.719621
\(756\) 1552.86 0.0747051
\(757\) 3540.46 0.169987 0.0849936 0.996382i \(-0.472913\pi\)
0.0849936 + 0.996382i \(0.472913\pi\)
\(758\) −11937.2 −0.572005
\(759\) 7676.12 0.367096
\(760\) 4939.11 0.235737
\(761\) −23640.9 −1.12613 −0.563063 0.826414i \(-0.690377\pi\)
−0.563063 + 0.826414i \(0.690377\pi\)
\(762\) 16931.5 0.804937
\(763\) −10482.1 −0.497351
\(764\) −5270.69 −0.249590
\(765\) 949.472 0.0448735
\(766\) 33202.7 1.56614
\(767\) −4858.12 −0.228705
\(768\) 10629.8 0.499442
\(769\) 7631.43 0.357863 0.178931 0.983862i \(-0.442736\pi\)
0.178931 + 0.983862i \(0.442736\pi\)
\(770\) −7420.54 −0.347296
\(771\) 2598.28 0.121368
\(772\) −20234.1 −0.943318
\(773\) −18739.3 −0.871936 −0.435968 0.899962i \(-0.643594\pi\)
−0.435968 + 0.899962i \(0.643594\pi\)
\(774\) −9686.73 −0.449848
\(775\) −11101.0 −0.514531
\(776\) −1200.17 −0.0555200
\(777\) 8410.68 0.388329
\(778\) 29875.3 1.37671
\(779\) −78821.0 −3.62523
\(780\) −34145.9 −1.56746
\(781\) −12037.4 −0.551513
\(782\) −4062.44 −0.185770
\(783\) 5963.95 0.272202
\(784\) 17769.2 0.809456
\(785\) 38059.4 1.73044
\(786\) −4850.96 −0.220137
\(787\) 11942.8 0.540933 0.270467 0.962729i \(-0.412822\pi\)
0.270467 + 0.962729i \(0.412822\pi\)
\(788\) 17365.3 0.785042
\(789\) −14231.0 −0.642126
\(790\) −20149.3 −0.907442
\(791\) 9578.99 0.430581
\(792\) −299.983 −0.0134589
\(793\) 5658.77 0.253403
\(794\) −30408.2 −1.35913
\(795\) −12535.5 −0.559230
\(796\) 8382.96 0.373274
\(797\) 11411.4 0.507167 0.253583 0.967314i \(-0.418391\pi\)
0.253583 + 0.967314i \(0.418391\pi\)
\(798\) 12458.3 0.552657
\(799\) 205.143 0.00908317
\(800\) −36194.6 −1.59959
\(801\) −6246.15 −0.275527
\(802\) −39398.3 −1.73467
\(803\) −9479.09 −0.416575
\(804\) 5070.68 0.222424
\(805\) −16997.2 −0.744188
\(806\) 26558.3 1.16064
\(807\) −17537.9 −0.765011
\(808\) 2385.43 0.103860
\(809\) 30559.4 1.32808 0.664038 0.747699i \(-0.268841\pi\)
0.664038 + 0.747699i \(0.268841\pi\)
\(810\) 5353.09 0.232208
\(811\) −38485.1 −1.66633 −0.833165 0.553025i \(-0.813473\pi\)
−0.833165 + 0.553025i \(0.813473\pi\)
\(812\) 12704.0 0.549041
\(813\) −14040.1 −0.605669
\(814\) −27891.6 −1.20098
\(815\) 23196.1 0.996960
\(816\) −1163.03 −0.0498947
\(817\) −40023.2 −1.71387
\(818\) −5597.80 −0.239270
\(819\) −5017.32 −0.214065
\(820\) −72142.5 −3.07235
\(821\) 34212.3 1.45434 0.727172 0.686455i \(-0.240834\pi\)
0.727172 + 0.686455i \(0.240834\pi\)
\(822\) −522.061 −0.0221520
\(823\) 24648.8 1.04399 0.521995 0.852949i \(-0.325188\pi\)
0.521995 + 0.852949i \(0.325188\pi\)
\(824\) −2084.50 −0.0881275
\(825\) −6954.71 −0.293493
\(826\) 1622.33 0.0683391
\(827\) −20204.0 −0.849532 −0.424766 0.905303i \(-0.639644\pi\)
−0.424766 + 0.905303i \(0.639644\pi\)
\(828\) −11795.5 −0.495075
\(829\) −12267.7 −0.513964 −0.256982 0.966416i \(-0.582728\pi\)
−0.256982 + 0.966416i \(0.582728\pi\)
\(830\) 48324.1 2.02091
\(831\) −11655.6 −0.486557
\(832\) 47202.9 1.96691
\(833\) −1926.58 −0.0801346
\(834\) −2994.77 −0.124341
\(835\) 31535.0 1.30696
\(836\) −21277.0 −0.880239
\(837\) −2144.24 −0.0885492
\(838\) −25862.5 −1.06612
\(839\) −8916.91 −0.366920 −0.183460 0.983027i \(-0.558730\pi\)
−0.183460 + 0.983027i \(0.558730\pi\)
\(840\) 664.250 0.0272843
\(841\) 24402.0 1.00053
\(842\) 19210.4 0.786265
\(843\) −22742.8 −0.929187
\(844\) 21576.5 0.879968
\(845\) 74576.1 3.03609
\(846\) 1156.59 0.0470029
\(847\) −7149.21 −0.290023
\(848\) 15355.0 0.621806
\(849\) 2843.23 0.114934
\(850\) 3680.64 0.148524
\(851\) −63887.3 −2.57348
\(852\) 18497.2 0.743785
\(853\) 5454.65 0.218949 0.109475 0.993990i \(-0.465083\pi\)
0.109475 + 0.993990i \(0.465083\pi\)
\(854\) −1889.70 −0.0757193
\(855\) 22117.7 0.884688
\(856\) 3591.48 0.143405
\(857\) 30739.7 1.22526 0.612630 0.790370i \(-0.290112\pi\)
0.612630 + 0.790370i \(0.290112\pi\)
\(858\) 16638.5 0.662039
\(859\) −9450.34 −0.375368 −0.187684 0.982229i \(-0.560098\pi\)
−0.187684 + 0.982229i \(0.560098\pi\)
\(860\) −36632.1 −1.45249
\(861\) −10600.5 −0.419585
\(862\) 48881.0 1.93143
\(863\) 20494.1 0.808373 0.404186 0.914677i \(-0.367555\pi\)
0.404186 + 0.914677i \(0.367555\pi\)
\(864\) −6991.22 −0.275285
\(865\) 15052.5 0.591678
\(866\) −65042.0 −2.55221
\(867\) −14612.9 −0.572411
\(868\) −4567.50 −0.178607
\(869\) 5056.41 0.197384
\(870\) 43793.6 1.70660
\(871\) −16383.4 −0.637350
\(872\) −3111.63 −0.120841
\(873\) −5374.44 −0.208359
\(874\) −94633.3 −3.66249
\(875\) 1628.65 0.0629239
\(876\) 14566.0 0.561804
\(877\) 9745.52 0.375237 0.187618 0.982242i \(-0.439923\pi\)
0.187618 + 0.982242i \(0.439923\pi\)
\(878\) 12317.5 0.473456
\(879\) −7838.84 −0.300794
\(880\) 16137.0 0.618156
\(881\) −15386.5 −0.588405 −0.294203 0.955743i \(-0.595054\pi\)
−0.294203 + 0.955743i \(0.595054\pi\)
\(882\) −10862.0 −0.414674
\(883\) −38659.6 −1.47339 −0.736693 0.676227i \(-0.763614\pi\)
−0.736693 + 0.676227i \(0.763614\pi\)
\(884\) −4534.89 −0.172539
\(885\) 2880.17 0.109397
\(886\) −7130.96 −0.270394
\(887\) −2987.34 −0.113083 −0.0565417 0.998400i \(-0.518007\pi\)
−0.0565417 + 0.998400i \(0.518007\pi\)
\(888\) 2496.72 0.0943517
\(889\) 9408.32 0.354944
\(890\) −45865.8 −1.72744
\(891\) −1343.35 −0.0505093
\(892\) 43026.1 1.61505
\(893\) 4778.75 0.179076
\(894\) 18182.4 0.680212
\(895\) 36533.5 1.36445
\(896\) −1738.37 −0.0648158
\(897\) 38111.5 1.41862
\(898\) −46574.7 −1.73076
\(899\) −17542.0 −0.650788
\(900\) 10686.9 0.395813
\(901\) −1664.83 −0.0615576
\(902\) 35153.4 1.29765
\(903\) −5382.63 −0.198364
\(904\) 2843.53 0.104618
\(905\) −28850.2 −1.05968
\(906\) −11178.3 −0.409904
\(907\) −8601.67 −0.314899 −0.157450 0.987527i \(-0.550327\pi\)
−0.157450 + 0.987527i \(0.550327\pi\)
\(908\) −7436.53 −0.271795
\(909\) 10682.1 0.389772
\(910\) −36842.5 −1.34211
\(911\) −6239.74 −0.226928 −0.113464 0.993542i \(-0.536195\pi\)
−0.113464 + 0.993542i \(0.536195\pi\)
\(912\) −27092.4 −0.983682
\(913\) −12126.8 −0.439583
\(914\) 3739.79 0.135340
\(915\) −3354.84 −0.121211
\(916\) −38395.7 −1.38497
\(917\) −2695.54 −0.0970714
\(918\) 710.939 0.0255604
\(919\) 44905.0 1.61184 0.805918 0.592027i \(-0.201672\pi\)
0.805918 + 0.592027i \(0.201672\pi\)
\(920\) −5045.63 −0.180815
\(921\) 20253.4 0.724617
\(922\) 76389.3 2.72858
\(923\) −59764.9 −2.13129
\(924\) −2861.50 −0.101879
\(925\) 57883.1 2.05750
\(926\) −18252.4 −0.647743
\(927\) −9334.54 −0.330730
\(928\) −57195.2 −2.02319
\(929\) −20966.7 −0.740470 −0.370235 0.928938i \(-0.620723\pi\)
−0.370235 + 0.928938i \(0.620723\pi\)
\(930\) −15745.3 −0.555169
\(931\) −44879.2 −1.57987
\(932\) −28036.9 −0.985385
\(933\) 3530.80 0.123894
\(934\) 2470.16 0.0865377
\(935\) −1749.61 −0.0611962
\(936\) −1489.40 −0.0520112
\(937\) 42889.9 1.49536 0.747679 0.664060i \(-0.231168\pi\)
0.747679 + 0.664060i \(0.231168\pi\)
\(938\) 5471.12 0.190446
\(939\) 6179.43 0.214758
\(940\) 4373.85 0.151765
\(941\) 17830.3 0.617697 0.308848 0.951111i \(-0.400056\pi\)
0.308848 + 0.951111i \(0.400056\pi\)
\(942\) 28497.8 0.985679
\(943\) 80520.8 2.78061
\(944\) −3527.98 −0.121638
\(945\) 2974.56 0.102394
\(946\) 17850.0 0.613480
\(947\) −20658.3 −0.708874 −0.354437 0.935080i \(-0.615328\pi\)
−0.354437 + 0.935080i \(0.615328\pi\)
\(948\) −7769.93 −0.266198
\(949\) −47063.1 −1.60983
\(950\) 85739.5 2.92816
\(951\) 15967.4 0.544457
\(952\) 88.2184 0.00300333
\(953\) −44095.2 −1.49883 −0.749415 0.662101i \(-0.769665\pi\)
−0.749415 + 0.662101i \(0.769665\pi\)
\(954\) −9386.23 −0.318543
\(955\) −10096.2 −0.342099
\(956\) 17296.8 0.585167
\(957\) −10989.9 −0.371215
\(958\) −16984.8 −0.572810
\(959\) −290.094 −0.00976812
\(960\) −27984.6 −0.940832
\(961\) −23484.1 −0.788294
\(962\) −138480. −4.64114
\(963\) 16082.9 0.538177
\(964\) 9645.22 0.322253
\(965\) −38759.1 −1.29295
\(966\) −12727.0 −0.423898
\(967\) 45160.1 1.50181 0.750906 0.660409i \(-0.229617\pi\)
0.750906 + 0.660409i \(0.229617\pi\)
\(968\) −2122.25 −0.0704667
\(969\) 2937.43 0.0973826
\(970\) −39464.8 −1.30633
\(971\) 49299.5 1.62935 0.814673 0.579920i \(-0.196916\pi\)
0.814673 + 0.579920i \(0.196916\pi\)
\(972\) 2064.25 0.0681181
\(973\) −1664.11 −0.0548292
\(974\) −75454.0 −2.48224
\(975\) −34529.7 −1.13419
\(976\) 4109.42 0.134774
\(977\) 27045.5 0.885632 0.442816 0.896613i \(-0.353980\pi\)
0.442816 + 0.896613i \(0.353980\pi\)
\(978\) 17368.6 0.567880
\(979\) 11509.9 0.375749
\(980\) −41076.6 −1.33892
\(981\) −13934.1 −0.453498
\(982\) 60786.8 1.97534
\(983\) −23169.0 −0.751757 −0.375878 0.926669i \(-0.622659\pi\)
−0.375878 + 0.926669i \(0.622659\pi\)
\(984\) −3146.76 −0.101946
\(985\) 33263.8 1.07601
\(986\) 5816.19 0.187855
\(987\) 642.684 0.0207263
\(988\) −105639. −3.40164
\(989\) 40886.4 1.31457
\(990\) −9864.26 −0.316674
\(991\) −58402.5 −1.87206 −0.936032 0.351915i \(-0.885531\pi\)
−0.936032 + 0.351915i \(0.885531\pi\)
\(992\) 20563.6 0.658159
\(993\) −32839.0 −1.04946
\(994\) 19958.0 0.636851
\(995\) 16057.8 0.511625
\(996\) 18634.6 0.592833
\(997\) −3853.02 −0.122394 −0.0611968 0.998126i \(-0.519492\pi\)
−0.0611968 + 0.998126i \(0.519492\pi\)
\(998\) −59329.8 −1.88182
\(999\) 11180.5 0.354089
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 177.4.a.d.1.6 8
3.2 odd 2 531.4.a.e.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.4.a.d.1.6 8 1.1 even 1 trivial
531.4.a.e.1.3 8 3.2 odd 2