Properties

Label 354.4.a.b.1.2
Level 354
Weight 4
Character 354.1
Self dual yes
Analytic conductor 20.887
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 354.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.8866761420\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{51}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(7.14143\)
Character \(\chi\) = 354.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +20.1414 q^{5} -6.00000 q^{6} -28.5657 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +20.1414 q^{5} -6.00000 q^{6} -28.5657 q^{7} -8.00000 q^{8} +9.00000 q^{9} -40.2829 q^{10} +54.5657 q^{11} +12.0000 q^{12} -16.9900 q^{13} +57.1314 q^{14} +60.4243 q^{15} +16.0000 q^{16} +10.8486 q^{17} -18.0000 q^{18} +34.0000 q^{19} +80.5657 q^{20} -85.6971 q^{21} -109.131 q^{22} +100.849 q^{23} -24.0000 q^{24} +280.677 q^{25} +33.9800 q^{26} +27.0000 q^{27} -114.263 q^{28} +110.727 q^{29} -120.849 q^{30} +0.404283 q^{31} -32.0000 q^{32} +163.697 q^{33} -21.6971 q^{34} -575.354 q^{35} +36.0000 q^{36} -368.990 q^{37} -68.0000 q^{38} -50.9700 q^{39} -161.131 q^{40} -436.546 q^{41} +171.394 q^{42} +326.869 q^{43} +218.263 q^{44} +181.273 q^{45} -201.697 q^{46} +293.374 q^{47} +48.0000 q^{48} +473.000 q^{49} -561.354 q^{50} +32.5457 q^{51} -67.9600 q^{52} +109.273 q^{53} -54.0000 q^{54} +1099.03 q^{55} +228.526 q^{56} +102.000 q^{57} -221.454 q^{58} +59.0000 q^{59} +241.697 q^{60} +460.747 q^{61} -0.808567 q^{62} -257.091 q^{63} +64.0000 q^{64} -342.203 q^{65} -327.394 q^{66} +630.020 q^{67} +43.3943 q^{68} +302.546 q^{69} +1150.71 q^{70} +987.961 q^{71} -72.0000 q^{72} -629.274 q^{73} +737.980 q^{74} +842.031 q^{75} +136.000 q^{76} -1558.71 q^{77} +101.940 q^{78} +715.234 q^{79} +322.263 q^{80} +81.0000 q^{81} +873.091 q^{82} -497.577 q^{83} -342.789 q^{84} +218.506 q^{85} -653.737 q^{86} +332.181 q^{87} -436.526 q^{88} +396.729 q^{89} -362.546 q^{90} +485.331 q^{91} +403.394 q^{92} +1.21285 q^{93} -586.749 q^{94} +684.809 q^{95} -96.0000 q^{96} -1472.39 q^{97} -946.000 q^{98} +491.091 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 6q^{3} + 8q^{4} + 26q^{5} - 12q^{6} - 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 6q^{3} + 8q^{4} + 26q^{5} - 12q^{6} - 16q^{8} + 18q^{9} - 52q^{10} + 52q^{11} + 24q^{12} + 66q^{13} + 78q^{15} + 32q^{16} - 64q^{17} - 36q^{18} + 68q^{19} + 104q^{20} - 104q^{22} + 116q^{23} - 48q^{24} + 190q^{25} - 132q^{26} + 54q^{27} + 350q^{29} - 156q^{30} - 242q^{31} - 64q^{32} + 156q^{33} + 128q^{34} - 408q^{35} + 72q^{36} - 638q^{37} - 136q^{38} + 198q^{39} - 208q^{40} - 616q^{41} + 768q^{43} + 208q^{44} + 234q^{45} - 232q^{46} + 44q^{47} + 96q^{48} + 946q^{49} - 380q^{50} - 192q^{51} + 264q^{52} + 90q^{53} - 108q^{54} + 1084q^{55} + 204q^{57} - 700q^{58} + 118q^{59} + 312q^{60} + 1250q^{61} + 484q^{62} + 128q^{64} + 144q^{65} - 312q^{66} + 1460q^{67} - 256q^{68} + 348q^{69} + 816q^{70} + 162q^{71} - 144q^{72} + 284q^{73} + 1276q^{74} + 570q^{75} + 272q^{76} - 1632q^{77} - 396q^{78} - 512q^{79} + 416q^{80} + 162q^{81} + 1232q^{82} + 376q^{83} - 220q^{85} - 1536q^{86} + 1050q^{87} - 416q^{88} - 492q^{89} - 468q^{90} + 2856q^{91} + 464q^{92} - 726q^{93} - 88q^{94} + 884q^{95} - 192q^{96} - 1088q^{97} - 1892q^{98} + 468q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 20.1414 1.80150 0.900752 0.434334i \(-0.143016\pi\)
0.900752 + 0.434334i \(0.143016\pi\)
\(6\) −6.00000 −0.408248
\(7\) −28.5657 −1.54240 −0.771202 0.636591i \(-0.780344\pi\)
−0.771202 + 0.636591i \(0.780344\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −40.2829 −1.27386
\(11\) 54.5657 1.49565 0.747826 0.663894i \(-0.231098\pi\)
0.747826 + 0.663894i \(0.231098\pi\)
\(12\) 12.0000 0.288675
\(13\) −16.9900 −0.362475 −0.181238 0.983439i \(-0.558010\pi\)
−0.181238 + 0.983439i \(0.558010\pi\)
\(14\) 57.1314 1.09064
\(15\) 60.4243 1.04010
\(16\) 16.0000 0.250000
\(17\) 10.8486 0.154774 0.0773872 0.997001i \(-0.475342\pi\)
0.0773872 + 0.997001i \(0.475342\pi\)
\(18\) −18.0000 −0.235702
\(19\) 34.0000 0.410533 0.205267 0.978706i \(-0.434194\pi\)
0.205267 + 0.978706i \(0.434194\pi\)
\(20\) 80.5657 0.900752
\(21\) −85.6971 −0.890507
\(22\) −109.131 −1.05759
\(23\) 100.849 0.914277 0.457139 0.889395i \(-0.348874\pi\)
0.457139 + 0.889395i \(0.348874\pi\)
\(24\) −24.0000 −0.204124
\(25\) 280.677 2.24542
\(26\) 33.9800 0.256309
\(27\) 27.0000 0.192450
\(28\) −114.263 −0.771202
\(29\) 110.727 0.709018 0.354509 0.935053i \(-0.384648\pi\)
0.354509 + 0.935053i \(0.384648\pi\)
\(30\) −120.849 −0.735461
\(31\) 0.404283 0.00234230 0.00117115 0.999999i \(-0.499627\pi\)
0.00117115 + 0.999999i \(0.499627\pi\)
\(32\) −32.0000 −0.176777
\(33\) 163.697 0.863516
\(34\) −21.6971 −0.109442
\(35\) −575.354 −2.77865
\(36\) 36.0000 0.166667
\(37\) −368.990 −1.63950 −0.819751 0.572720i \(-0.805888\pi\)
−0.819751 + 0.572720i \(0.805888\pi\)
\(38\) −68.0000 −0.290291
\(39\) −50.9700 −0.209275
\(40\) −161.131 −0.636928
\(41\) −436.546 −1.66285 −0.831427 0.555635i \(-0.812475\pi\)
−0.831427 + 0.555635i \(0.812475\pi\)
\(42\) 171.394 0.629684
\(43\) 326.869 1.15923 0.579616 0.814890i \(-0.303202\pi\)
0.579616 + 0.814890i \(0.303202\pi\)
\(44\) 218.263 0.747826
\(45\) 181.273 0.600501
\(46\) −201.697 −0.646492
\(47\) 293.374 0.910490 0.455245 0.890366i \(-0.349552\pi\)
0.455245 + 0.890366i \(0.349552\pi\)
\(48\) 48.0000 0.144338
\(49\) 473.000 1.37901
\(50\) −561.354 −1.58775
\(51\) 32.5457 0.0893590
\(52\) −67.9600 −0.181238
\(53\) 109.273 0.283203 0.141602 0.989924i \(-0.454775\pi\)
0.141602 + 0.989924i \(0.454775\pi\)
\(54\) −54.0000 −0.136083
\(55\) 1099.03 2.69442
\(56\) 228.526 0.545322
\(57\) 102.000 0.237022
\(58\) −221.454 −0.501351
\(59\) 59.0000 0.130189
\(60\) 241.697 0.520049
\(61\) 460.747 0.967092 0.483546 0.875319i \(-0.339349\pi\)
0.483546 + 0.875319i \(0.339349\pi\)
\(62\) −0.808567 −0.00165626
\(63\) −257.091 −0.514135
\(64\) 64.0000 0.125000
\(65\) −342.203 −0.653001
\(66\) −327.394 −0.610598
\(67\) 630.020 1.14879 0.574397 0.818577i \(-0.305237\pi\)
0.574397 + 0.818577i \(0.305237\pi\)
\(68\) 43.3943 0.0773872
\(69\) 302.546 0.527858
\(70\) 1150.71 1.96480
\(71\) 987.961 1.65140 0.825700 0.564109i \(-0.190780\pi\)
0.825700 + 0.564109i \(0.190780\pi\)
\(72\) −72.0000 −0.117851
\(73\) −629.274 −1.00892 −0.504459 0.863436i \(-0.668308\pi\)
−0.504459 + 0.863436i \(0.668308\pi\)
\(74\) 737.980 1.15930
\(75\) 842.031 1.29639
\(76\) 136.000 0.205267
\(77\) −1558.71 −2.30690
\(78\) 101.940 0.147980
\(79\) 715.234 1.01861 0.509305 0.860586i \(-0.329903\pi\)
0.509305 + 0.860586i \(0.329903\pi\)
\(80\) 322.263 0.450376
\(81\) 81.0000 0.111111
\(82\) 873.091 1.17581
\(83\) −497.577 −0.658026 −0.329013 0.944325i \(-0.606716\pi\)
−0.329013 + 0.944325i \(0.606716\pi\)
\(84\) −342.789 −0.445254
\(85\) 218.506 0.278827
\(86\) −653.737 −0.819701
\(87\) 332.181 0.409352
\(88\) −436.526 −0.528793
\(89\) 396.729 0.472507 0.236254 0.971691i \(-0.424080\pi\)
0.236254 + 0.971691i \(0.424080\pi\)
\(90\) −362.546 −0.424619
\(91\) 485.331 0.559083
\(92\) 403.394 0.457139
\(93\) 1.21285 0.00135233
\(94\) −586.749 −0.643814
\(95\) 684.809 0.739578
\(96\) −96.0000 −0.102062
\(97\) −1472.39 −1.54122 −0.770609 0.637308i \(-0.780048\pi\)
−0.770609 + 0.637308i \(0.780048\pi\)
\(98\) −946.000 −0.975106
\(99\) 491.091 0.498551
\(100\) 1122.71 1.12271
\(101\) 630.969 0.621621 0.310810 0.950472i \(-0.399400\pi\)
0.310810 + 0.950472i \(0.399400\pi\)
\(102\) −65.0914 −0.0631864
\(103\) 1285.78 1.23001 0.615007 0.788521i \(-0.289153\pi\)
0.615007 + 0.788521i \(0.289153\pi\)
\(104\) 135.920 0.128154
\(105\) −1726.06 −1.60425
\(106\) −218.546 −0.200255
\(107\) −1760.57 −1.59066 −0.795330 0.606177i \(-0.792702\pi\)
−0.795330 + 0.606177i \(0.792702\pi\)
\(108\) 108.000 0.0962250
\(109\) −1227.74 −1.07886 −0.539431 0.842030i \(-0.681361\pi\)
−0.539431 + 0.842030i \(0.681361\pi\)
\(110\) −2198.06 −1.90525
\(111\) −1106.97 −0.946567
\(112\) −457.051 −0.385601
\(113\) −1128.63 −0.939579 −0.469790 0.882778i \(-0.655670\pi\)
−0.469790 + 0.882778i \(0.655670\pi\)
\(114\) −204.000 −0.167600
\(115\) 2031.23 1.64707
\(116\) 442.909 0.354509
\(117\) −152.910 −0.120825
\(118\) −118.000 −0.0920575
\(119\) −309.897 −0.238725
\(120\) −483.394 −0.367730
\(121\) 1646.42 1.23698
\(122\) −921.494 −0.683837
\(123\) −1309.64 −0.960049
\(124\) 1.61713 0.00117115
\(125\) 3135.56 2.24362
\(126\) 514.183 0.363548
\(127\) 65.9000 0.0460447 0.0230224 0.999735i \(-0.492671\pi\)
0.0230224 + 0.999735i \(0.492671\pi\)
\(128\) −128.000 −0.0883883
\(129\) 980.606 0.669283
\(130\) 684.406 0.461741
\(131\) 204.023 0.136073 0.0680365 0.997683i \(-0.478327\pi\)
0.0680365 + 0.997683i \(0.478327\pi\)
\(132\) 654.789 0.431758
\(133\) −971.234 −0.633208
\(134\) −1260.04 −0.812320
\(135\) 543.819 0.346700
\(136\) −86.7886 −0.0547210
\(137\) −2129.80 −1.32818 −0.664091 0.747652i \(-0.731181\pi\)
−0.664091 + 0.747652i \(0.731181\pi\)
\(138\) −605.091 −0.373252
\(139\) −1297.92 −0.792001 −0.396000 0.918250i \(-0.629602\pi\)
−0.396000 + 0.918250i \(0.629602\pi\)
\(140\) −2301.42 −1.38932
\(141\) 880.123 0.525672
\(142\) −1975.92 −1.16772
\(143\) −927.071 −0.542137
\(144\) 144.000 0.0833333
\(145\) 2230.20 1.27730
\(146\) 1258.55 0.713412
\(147\) 1419.00 0.796171
\(148\) −1475.96 −0.819751
\(149\) 2180.35 1.19880 0.599400 0.800450i \(-0.295406\pi\)
0.599400 + 0.800450i \(0.295406\pi\)
\(150\) −1684.06 −0.916688
\(151\) −3106.06 −1.67396 −0.836980 0.547234i \(-0.815681\pi\)
−0.836980 + 0.547234i \(0.815681\pi\)
\(152\) −272.000 −0.145145
\(153\) 97.6371 0.0515915
\(154\) 3117.42 1.63122
\(155\) 8.14284 0.00421967
\(156\) −203.880 −0.104638
\(157\) 1846.04 0.938410 0.469205 0.883089i \(-0.344541\pi\)
0.469205 + 0.883089i \(0.344541\pi\)
\(158\) −1430.47 −0.720266
\(159\) 327.819 0.163508
\(160\) −644.526 −0.318464
\(161\) −2880.81 −1.41018
\(162\) −162.000 −0.0785674
\(163\) 349.154 0.167778 0.0838892 0.996475i \(-0.473266\pi\)
0.0838892 + 0.996475i \(0.473266\pi\)
\(164\) −1746.18 −0.831427
\(165\) 3297.09 1.55563
\(166\) 995.154 0.465295
\(167\) −67.2729 −0.0311720 −0.0155860 0.999879i \(-0.504961\pi\)
−0.0155860 + 0.999879i \(0.504961\pi\)
\(168\) 685.577 0.314842
\(169\) −1908.34 −0.868612
\(170\) −437.011 −0.197160
\(171\) 306.000 0.136844
\(172\) 1307.47 0.579616
\(173\) 3237.08 1.42260 0.711302 0.702887i \(-0.248106\pi\)
0.711302 + 0.702887i \(0.248106\pi\)
\(174\) −664.363 −0.289455
\(175\) −8017.74 −3.46334
\(176\) 873.051 0.373913
\(177\) 177.000 0.0751646
\(178\) −793.457 −0.334113
\(179\) 3341.94 1.39547 0.697733 0.716358i \(-0.254192\pi\)
0.697733 + 0.716358i \(0.254192\pi\)
\(180\) 725.091 0.300251
\(181\) −2063.94 −0.847576 −0.423788 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(182\) −970.663 −0.395331
\(183\) 1382.24 0.558351
\(184\) −806.789 −0.323246
\(185\) −7431.99 −2.95357
\(186\) −2.42570 −0.000956242 0
\(187\) 591.960 0.231489
\(188\) 1173.50 0.455245
\(189\) −771.274 −0.296836
\(190\) −1369.62 −0.522960
\(191\) −4106.47 −1.55567 −0.777837 0.628466i \(-0.783683\pi\)
−0.777837 + 0.628466i \(0.783683\pi\)
\(192\) 192.000 0.0721688
\(193\) −861.554 −0.321327 −0.160663 0.987009i \(-0.551363\pi\)
−0.160663 + 0.987009i \(0.551363\pi\)
\(194\) 2944.77 1.08981
\(195\) −1026.61 −0.377010
\(196\) 1892.00 0.689504
\(197\) −1209.92 −0.437581 −0.218791 0.975772i \(-0.570211\pi\)
−0.218791 + 0.975772i \(0.570211\pi\)
\(198\) −982.183 −0.352529
\(199\) 1644.16 0.585687 0.292843 0.956160i \(-0.405399\pi\)
0.292843 + 0.956160i \(0.405399\pi\)
\(200\) −2245.42 −0.793875
\(201\) 1890.06 0.663256
\(202\) −1261.94 −0.439552
\(203\) −3163.00 −1.09359
\(204\) 130.183 0.0446795
\(205\) −8792.65 −2.99564
\(206\) −2571.56 −0.869752
\(207\) 907.637 0.304759
\(208\) −271.840 −0.0906188
\(209\) 1855.23 0.614015
\(210\) 3452.13 1.13438
\(211\) −4746.03 −1.54848 −0.774241 0.632890i \(-0.781868\pi\)
−0.774241 + 0.632890i \(0.781868\pi\)
\(212\) 437.091 0.141602
\(213\) 2963.88 0.953437
\(214\) 3521.14 1.12477
\(215\) 6583.60 2.08836
\(216\) −216.000 −0.0680414
\(217\) −11.5486 −0.00361278
\(218\) 2455.48 0.762871
\(219\) −1887.82 −0.582499
\(220\) 4396.13 1.34721
\(221\) −184.317 −0.0561019
\(222\) 2213.94 0.669324
\(223\) −6515.69 −1.95660 −0.978302 0.207185i \(-0.933570\pi\)
−0.978302 + 0.207185i \(0.933570\pi\)
\(224\) 914.103 0.272661
\(225\) 2526.09 0.748472
\(226\) 2257.26 0.664383
\(227\) 565.617 0.165380 0.0826901 0.996575i \(-0.473649\pi\)
0.0826901 + 0.996575i \(0.473649\pi\)
\(228\) 408.000 0.118511
\(229\) 1260.26 0.363670 0.181835 0.983329i \(-0.441796\pi\)
0.181835 + 0.983329i \(0.441796\pi\)
\(230\) −4062.47 −1.16466
\(231\) −4676.13 −1.33189
\(232\) −885.817 −0.250676
\(233\) −2409.86 −0.677576 −0.338788 0.940863i \(-0.610017\pi\)
−0.338788 + 0.940863i \(0.610017\pi\)
\(234\) 305.820 0.0854362
\(235\) 5908.98 1.64025
\(236\) 236.000 0.0650945
\(237\) 2145.70 0.588094
\(238\) 619.794 0.168804
\(239\) −4883.81 −1.32179 −0.660894 0.750479i \(-0.729823\pi\)
−0.660894 + 0.750479i \(0.729823\pi\)
\(240\) 966.789 0.260025
\(241\) 4385.86 1.17227 0.586137 0.810212i \(-0.300648\pi\)
0.586137 + 0.810212i \(0.300648\pi\)
\(242\) −3292.83 −0.874675
\(243\) 243.000 0.0641500
\(244\) 1842.99 0.483546
\(245\) 9526.90 2.48429
\(246\) 2619.27 0.678857
\(247\) −577.660 −0.148808
\(248\) −3.23427 −0.000828129 0
\(249\) −1492.73 −0.379912
\(250\) −6271.12 −1.58648
\(251\) 420.891 0.105842 0.0529212 0.998599i \(-0.483147\pi\)
0.0529212 + 0.998599i \(0.483147\pi\)
\(252\) −1028.37 −0.257067
\(253\) 5502.87 1.36744
\(254\) −131.800 −0.0325585
\(255\) 655.517 0.160981
\(256\) 256.000 0.0625000
\(257\) −5904.43 −1.43311 −0.716553 0.697533i \(-0.754281\pi\)
−0.716553 + 0.697533i \(0.754281\pi\)
\(258\) −1961.21 −0.473255
\(259\) 10540.5 2.52877
\(260\) −1368.81 −0.326500
\(261\) 996.544 0.236339
\(262\) −408.046 −0.0962181
\(263\) −7196.30 −1.68723 −0.843617 0.536945i \(-0.819578\pi\)
−0.843617 + 0.536945i \(0.819578\pi\)
\(264\) −1309.58 −0.305299
\(265\) 2200.91 0.510192
\(266\) 1942.47 0.447746
\(267\) 1190.19 0.272802
\(268\) 2520.08 0.574397
\(269\) 5610.61 1.27169 0.635846 0.771816i \(-0.280651\pi\)
0.635846 + 0.771816i \(0.280651\pi\)
\(270\) −1087.64 −0.245154
\(271\) 5150.07 1.15441 0.577204 0.816600i \(-0.304144\pi\)
0.577204 + 0.816600i \(0.304144\pi\)
\(272\) 173.577 0.0386936
\(273\) 1455.99 0.322787
\(274\) 4259.59 0.939166
\(275\) 15315.3 3.35836
\(276\) 1210.18 0.263929
\(277\) 8215.83 1.78210 0.891049 0.453907i \(-0.149970\pi\)
0.891049 + 0.453907i \(0.149970\pi\)
\(278\) 2595.84 0.560029
\(279\) 3.63855 0.000780768 0
\(280\) 4602.83 0.982400
\(281\) −2793.35 −0.593016 −0.296508 0.955030i \(-0.595822\pi\)
−0.296508 + 0.955030i \(0.595822\pi\)
\(282\) −1760.25 −0.371706
\(283\) −133.611 −0.0280649 −0.0140325 0.999902i \(-0.504467\pi\)
−0.0140325 + 0.999902i \(0.504467\pi\)
\(284\) 3951.85 0.825700
\(285\) 2054.43 0.426995
\(286\) 1854.14 0.383349
\(287\) 12470.2 2.56479
\(288\) −288.000 −0.0589256
\(289\) −4795.31 −0.976045
\(290\) −4460.41 −0.903186
\(291\) −4417.16 −0.889822
\(292\) −2517.10 −0.504459
\(293\) −8450.25 −1.68488 −0.842439 0.538792i \(-0.818881\pi\)
−0.842439 + 0.538792i \(0.818881\pi\)
\(294\) −2838.00 −0.562978
\(295\) 1188.34 0.234536
\(296\) 2951.92 0.579652
\(297\) 1473.27 0.287839
\(298\) −4360.70 −0.847679
\(299\) −1713.42 −0.331403
\(300\) 3368.13 0.648196
\(301\) −9337.23 −1.78800
\(302\) 6212.13 1.18367
\(303\) 1892.91 0.358893
\(304\) 544.000 0.102633
\(305\) 9280.11 1.74222
\(306\) −195.274 −0.0364807
\(307\) −4184.95 −0.778006 −0.389003 0.921236i \(-0.627180\pi\)
−0.389003 + 0.921236i \(0.627180\pi\)
\(308\) −6234.83 −1.15345
\(309\) 3857.34 0.710149
\(310\) −16.2857 −0.00298376
\(311\) −7845.63 −1.43050 −0.715249 0.698870i \(-0.753687\pi\)
−0.715249 + 0.698870i \(0.753687\pi\)
\(312\) 407.760 0.0739899
\(313\) −3180.87 −0.574420 −0.287210 0.957868i \(-0.592728\pi\)
−0.287210 + 0.957868i \(0.592728\pi\)
\(314\) −3692.09 −0.663556
\(315\) −5178.19 −0.926215
\(316\) 2860.94 0.509305
\(317\) −5932.91 −1.05118 −0.525592 0.850737i \(-0.676156\pi\)
−0.525592 + 0.850737i \(0.676156\pi\)
\(318\) −655.637 −0.115617
\(319\) 6041.91 1.06044
\(320\) 1289.05 0.225188
\(321\) −5281.71 −0.918368
\(322\) 5761.62 0.997151
\(323\) 368.851 0.0635401
\(324\) 324.000 0.0555556
\(325\) −4768.70 −0.813908
\(326\) −698.309 −0.118637
\(327\) −3683.22 −0.622882
\(328\) 3492.37 0.587907
\(329\) −8380.45 −1.40434
\(330\) −6594.19 −1.09999
\(331\) −6091.14 −1.01148 −0.505739 0.862686i \(-0.668780\pi\)
−0.505739 + 0.862686i \(0.668780\pi\)
\(332\) −1990.31 −0.329013
\(333\) −3320.91 −0.546501
\(334\) 134.546 0.0220420
\(335\) 12689.5 2.06956
\(336\) −1371.15 −0.222627
\(337\) −8467.33 −1.36868 −0.684340 0.729163i \(-0.739909\pi\)
−0.684340 + 0.729163i \(0.739909\pi\)
\(338\) 3816.68 0.614201
\(339\) −3385.89 −0.542466
\(340\) 874.023 0.139413
\(341\) 22.0600 0.00350327
\(342\) −612.000 −0.0967637
\(343\) −3713.54 −0.584584
\(344\) −2614.95 −0.409851
\(345\) 6093.70 0.950939
\(346\) −6474.15 −1.00593
\(347\) −941.766 −0.145696 −0.0728482 0.997343i \(-0.523209\pi\)
−0.0728482 + 0.997343i \(0.523209\pi\)
\(348\) 1328.73 0.204676
\(349\) −3556.31 −0.545458 −0.272729 0.962091i \(-0.587926\pi\)
−0.272729 + 0.962091i \(0.587926\pi\)
\(350\) 16035.5 2.44895
\(351\) −458.730 −0.0697584
\(352\) −1746.10 −0.264397
\(353\) 4004.32 0.603763 0.301882 0.953345i \(-0.402385\pi\)
0.301882 + 0.953345i \(0.402385\pi\)
\(354\) −354.000 −0.0531494
\(355\) 19899.0 2.97501
\(356\) 1586.91 0.236254
\(357\) −929.691 −0.137828
\(358\) −6683.89 −0.986744
\(359\) 1206.03 0.177303 0.0886514 0.996063i \(-0.471744\pi\)
0.0886514 + 0.996063i \(0.471744\pi\)
\(360\) −1450.18 −0.212309
\(361\) −5703.00 −0.831462
\(362\) 4127.87 0.599327
\(363\) 4939.25 0.714169
\(364\) 1941.33 0.279542
\(365\) −12674.5 −1.81757
\(366\) −2764.48 −0.394814
\(367\) −5836.89 −0.830199 −0.415099 0.909776i \(-0.636253\pi\)
−0.415099 + 0.909776i \(0.636253\pi\)
\(368\) 1613.58 0.228569
\(369\) −3928.91 −0.554284
\(370\) 14864.0 2.08849
\(371\) −3121.46 −0.436814
\(372\) 4.85140 0.000676165 0
\(373\) 12644.3 1.75522 0.877611 0.479373i \(-0.159136\pi\)
0.877611 + 0.479373i \(0.159136\pi\)
\(374\) −1183.92 −0.163687
\(375\) 9406.68 1.29536
\(376\) −2346.99 −0.321907
\(377\) −1881.25 −0.257001
\(378\) 1542.55 0.209895
\(379\) 7096.13 0.961751 0.480875 0.876789i \(-0.340319\pi\)
0.480875 + 0.876789i \(0.340319\pi\)
\(380\) 2739.23 0.369789
\(381\) 197.700 0.0265839
\(382\) 8212.94 1.10003
\(383\) 6841.83 0.912797 0.456399 0.889775i \(-0.349139\pi\)
0.456399 + 0.889775i \(0.349139\pi\)
\(384\) −384.000 −0.0510310
\(385\) −31394.6 −4.15589
\(386\) 1723.11 0.227212
\(387\) 2941.82 0.386411
\(388\) −5889.54 −0.770609
\(389\) −6292.35 −0.820141 −0.410070 0.912054i \(-0.634496\pi\)
−0.410070 + 0.912054i \(0.634496\pi\)
\(390\) 2053.22 0.266586
\(391\) 1094.06 0.141507
\(392\) −3784.00 −0.487553
\(393\) 612.068 0.0785618
\(394\) 2419.85 0.309417
\(395\) 14405.8 1.83503
\(396\) 1964.37 0.249275
\(397\) −92.7015 −0.0117193 −0.00585964 0.999983i \(-0.501865\pi\)
−0.00585964 + 0.999983i \(0.501865\pi\)
\(398\) −3288.33 −0.414143
\(399\) −2913.70 −0.365583
\(400\) 4490.83 0.561354
\(401\) −6882.75 −0.857128 −0.428564 0.903512i \(-0.640980\pi\)
−0.428564 + 0.903512i \(0.640980\pi\)
\(402\) −3780.12 −0.468993
\(403\) −6.86877 −0.000849027 0
\(404\) 2523.87 0.310810
\(405\) 1631.46 0.200167
\(406\) 6326.00 0.773286
\(407\) −20134.2 −2.45213
\(408\) −260.366 −0.0315932
\(409\) 4425.96 0.535085 0.267543 0.963546i \(-0.413788\pi\)
0.267543 + 0.963546i \(0.413788\pi\)
\(410\) 17585.3 2.11824
\(411\) −6389.39 −0.766826
\(412\) 5143.11 0.615007
\(413\) −1685.38 −0.200804
\(414\) −1815.27 −0.215497
\(415\) −10021.9 −1.18544
\(416\) 543.680 0.0640772
\(417\) −3893.76 −0.457262
\(418\) −3710.47 −0.434175
\(419\) 14575.7 1.69945 0.849726 0.527224i \(-0.176767\pi\)
0.849726 + 0.527224i \(0.176767\pi\)
\(420\) −6904.25 −0.802126
\(421\) −10203.0 −1.18115 −0.590575 0.806983i \(-0.701099\pi\)
−0.590575 + 0.806983i \(0.701099\pi\)
\(422\) 9492.05 1.09494
\(423\) 2640.37 0.303497
\(424\) −874.183 −0.100128
\(425\) 3044.95 0.347533
\(426\) −5927.77 −0.674182
\(427\) −13161.6 −1.49165
\(428\) −7042.27 −0.795330
\(429\) −2781.21 −0.313003
\(430\) −13167.2 −1.47669
\(431\) −5149.80 −0.575538 −0.287769 0.957700i \(-0.592914\pi\)
−0.287769 + 0.957700i \(0.592914\pi\)
\(432\) 432.000 0.0481125
\(433\) −11095.8 −1.23148 −0.615742 0.787948i \(-0.711143\pi\)
−0.615742 + 0.787948i \(0.711143\pi\)
\(434\) 23.0973 0.00255462
\(435\) 6690.61 0.737449
\(436\) −4910.95 −0.539431
\(437\) 3428.85 0.375341
\(438\) 3775.65 0.411889
\(439\) 6504.20 0.707126 0.353563 0.935411i \(-0.384970\pi\)
0.353563 + 0.935411i \(0.384970\pi\)
\(440\) −8792.25 −0.952623
\(441\) 4257.00 0.459670
\(442\) 368.634 0.0396700
\(443\) 12954.2 1.38933 0.694666 0.719332i \(-0.255552\pi\)
0.694666 + 0.719332i \(0.255552\pi\)
\(444\) −4427.88 −0.473284
\(445\) 7990.68 0.851224
\(446\) 13031.4 1.38353
\(447\) 6541.05 0.692127
\(448\) −1828.21 −0.192800
\(449\) 4722.80 0.496398 0.248199 0.968709i \(-0.420161\pi\)
0.248199 + 0.968709i \(0.420161\pi\)
\(450\) −5052.19 −0.529250
\(451\) −23820.4 −2.48705
\(452\) −4514.51 −0.469790
\(453\) −9318.19 −0.966461
\(454\) −1131.23 −0.116942
\(455\) 9775.27 1.00719
\(456\) −816.000 −0.0837998
\(457\) 13763.6 1.40883 0.704416 0.709788i \(-0.251209\pi\)
0.704416 + 0.709788i \(0.251209\pi\)
\(458\) −2520.52 −0.257153
\(459\) 292.911 0.0297863
\(460\) 8124.94 0.823537
\(461\) 4564.32 0.461131 0.230565 0.973057i \(-0.425942\pi\)
0.230565 + 0.973057i \(0.425942\pi\)
\(462\) 9352.25 0.941788
\(463\) 627.841 0.0630200 0.0315100 0.999503i \(-0.489968\pi\)
0.0315100 + 0.999503i \(0.489968\pi\)
\(464\) 1771.63 0.177254
\(465\) 24.4285 0.00243623
\(466\) 4819.72 0.479118
\(467\) 6716.18 0.665499 0.332749 0.943015i \(-0.392024\pi\)
0.332749 + 0.943015i \(0.392024\pi\)
\(468\) −611.640 −0.0604125
\(469\) −17997.0 −1.77190
\(470\) −11818.0 −1.15983
\(471\) 5538.13 0.541791
\(472\) −472.000 −0.0460287
\(473\) 17835.8 1.73381
\(474\) −4291.41 −0.415846
\(475\) 9543.02 0.921819
\(476\) −1239.59 −0.119362
\(477\) 983.456 0.0944011
\(478\) 9767.61 0.934645
\(479\) −10237.9 −0.976583 −0.488291 0.872681i \(-0.662380\pi\)
−0.488291 + 0.872681i \(0.662380\pi\)
\(480\) −1933.58 −0.183865
\(481\) 6269.14 0.594279
\(482\) −8771.72 −0.828923
\(483\) −8642.43 −0.814171
\(484\) 6585.67 0.618489
\(485\) −29656.0 −2.77651
\(486\) −486.000 −0.0453609
\(487\) 10027.8 0.933069 0.466535 0.884503i \(-0.345502\pi\)
0.466535 + 0.884503i \(0.345502\pi\)
\(488\) −3685.98 −0.341919
\(489\) 1047.46 0.0968669
\(490\) −19053.8 −1.75666
\(491\) −9590.92 −0.881532 −0.440766 0.897622i \(-0.645293\pi\)
−0.440766 + 0.897622i \(0.645293\pi\)
\(492\) −5238.55 −0.480024
\(493\) 1201.23 0.109738
\(494\) 1155.32 0.105223
\(495\) 9891.28 0.898142
\(496\) 6.46853 0.000585576 0
\(497\) −28221.8 −2.54713
\(498\) 2985.46 0.268638
\(499\) −1320.95 −0.118505 −0.0592523 0.998243i \(-0.518872\pi\)
−0.0592523 + 0.998243i \(0.518872\pi\)
\(500\) 12542.2 1.12181
\(501\) −201.819 −0.0179972
\(502\) −841.783 −0.0748418
\(503\) 16075.0 1.42495 0.712475 0.701697i \(-0.247574\pi\)
0.712475 + 0.701697i \(0.247574\pi\)
\(504\) 2056.73 0.181774
\(505\) 12708.6 1.11985
\(506\) −11005.7 −0.966927
\(507\) −5725.02 −0.501493
\(508\) 263.600 0.0230224
\(509\) 10444.5 0.909519 0.454760 0.890614i \(-0.349725\pi\)
0.454760 + 0.890614i \(0.349725\pi\)
\(510\) −1311.03 −0.113831
\(511\) 17975.7 1.55616
\(512\) −512.000 −0.0441942
\(513\) 918.000 0.0790072
\(514\) 11808.9 1.01336
\(515\) 25897.4 2.21588
\(516\) 3922.42 0.334642
\(517\) 16008.2 1.36178
\(518\) −21080.9 −1.78811
\(519\) 9711.23 0.821341
\(520\) 2737.62 0.230871
\(521\) −5162.05 −0.434076 −0.217038 0.976163i \(-0.569640\pi\)
−0.217038 + 0.976163i \(0.569640\pi\)
\(522\) −1993.09 −0.167117
\(523\) −11606.8 −0.970423 −0.485212 0.874397i \(-0.661257\pi\)
−0.485212 + 0.874397i \(0.661257\pi\)
\(524\) 816.091 0.0680365
\(525\) −24053.2 −1.99956
\(526\) 14392.6 1.19305
\(527\) 4.38590 0.000362529 0
\(528\) 2619.15 0.215879
\(529\) −1996.57 −0.164097
\(530\) −4401.82 −0.360760
\(531\) 531.000 0.0433963
\(532\) −3884.94 −0.316604
\(533\) 7416.91 0.602743
\(534\) −2380.37 −0.192900
\(535\) −35460.4 −2.86558
\(536\) −5040.16 −0.406160
\(537\) 10025.8 0.805673
\(538\) −11221.2 −0.899222
\(539\) 25809.6 2.06252
\(540\) 2175.27 0.173350
\(541\) 12568.2 0.998794 0.499397 0.866373i \(-0.333555\pi\)
0.499397 + 0.866373i \(0.333555\pi\)
\(542\) −10300.1 −0.816289
\(543\) −6191.81 −0.489348
\(544\) −347.154 −0.0273605
\(545\) −24728.4 −1.94358
\(546\) −2911.99 −0.228245
\(547\) 13248.3 1.03557 0.517785 0.855511i \(-0.326757\pi\)
0.517785 + 0.855511i \(0.326757\pi\)
\(548\) −8519.19 −0.664091
\(549\) 4146.72 0.322364
\(550\) −30630.7 −2.37472
\(551\) 3764.72 0.291075
\(552\) −2420.37 −0.186626
\(553\) −20431.2 −1.57111
\(554\) −16431.7 −1.26013
\(555\) −22296.0 −1.70524
\(556\) −5191.68 −0.396000
\(557\) 18075.5 1.37502 0.687508 0.726177i \(-0.258705\pi\)
0.687508 + 0.726177i \(0.258705\pi\)
\(558\) −7.27710 −0.000552086 0
\(559\) −5553.50 −0.420193
\(560\) −9205.67 −0.694662
\(561\) 1775.88 0.133650
\(562\) 5586.71 0.419326
\(563\) −8401.40 −0.628911 −0.314455 0.949272i \(-0.601822\pi\)
−0.314455 + 0.949272i \(0.601822\pi\)
\(564\) 3520.49 0.262836
\(565\) −22732.2 −1.69266
\(566\) 267.223 0.0198449
\(567\) −2313.82 −0.171378
\(568\) −7903.69 −0.583858
\(569\) 7993.77 0.588956 0.294478 0.955658i \(-0.404854\pi\)
0.294478 + 0.955658i \(0.404854\pi\)
\(570\) −4108.85 −0.301931
\(571\) 15831.5 1.16030 0.580149 0.814511i \(-0.302994\pi\)
0.580149 + 0.814511i \(0.302994\pi\)
\(572\) −3708.29 −0.271069
\(573\) −12319.4 −0.898169
\(574\) −24940.5 −1.81358
\(575\) 28305.9 2.05293
\(576\) 576.000 0.0416667
\(577\) 11060.2 0.797996 0.398998 0.916952i \(-0.369358\pi\)
0.398998 + 0.916952i \(0.369358\pi\)
\(578\) 9590.62 0.690168
\(579\) −2584.66 −0.185518
\(580\) 8920.81 0.638649
\(581\) 14213.6 1.01494
\(582\) 8834.31 0.629199
\(583\) 5962.55 0.423574
\(584\) 5034.19 0.356706
\(585\) −3079.83 −0.217667
\(586\) 16900.5 1.19139
\(587\) −19130.0 −1.34511 −0.672554 0.740048i \(-0.734803\pi\)
−0.672554 + 0.740048i \(0.734803\pi\)
\(588\) 5676.00 0.398086
\(589\) 13.7456 0.000961594 0
\(590\) −2376.69 −0.165842
\(591\) −3629.77 −0.252638
\(592\) −5903.84 −0.409876
\(593\) 8882.23 0.615092 0.307546 0.951533i \(-0.400492\pi\)
0.307546 + 0.951533i \(0.400492\pi\)
\(594\) −2946.55 −0.203533
\(595\) −6241.77 −0.430063
\(596\) 8721.39 0.599400
\(597\) 4932.49 0.338146
\(598\) 3426.83 0.234337
\(599\) −18682.7 −1.27438 −0.637192 0.770705i \(-0.719904\pi\)
−0.637192 + 0.770705i \(0.719904\pi\)
\(600\) −6736.25 −0.458344
\(601\) 15263.7 1.03597 0.517987 0.855388i \(-0.326681\pi\)
0.517987 + 0.855388i \(0.326681\pi\)
\(602\) 18674.5 1.26431
\(603\) 5670.18 0.382931
\(604\) −12424.3 −0.836980
\(605\) 33161.2 2.22842
\(606\) −3785.81 −0.253776
\(607\) −354.528 −0.0237065 −0.0118533 0.999930i \(-0.503773\pi\)
−0.0118533 + 0.999930i \(0.503773\pi\)
\(608\) −1088.00 −0.0725727
\(609\) −9489.00 −0.631385
\(610\) −18560.2 −1.23194
\(611\) −4984.43 −0.330030
\(612\) 390.549 0.0257957
\(613\) −16873.7 −1.11178 −0.555892 0.831254i \(-0.687623\pi\)
−0.555892 + 0.831254i \(0.687623\pi\)
\(614\) 8369.91 0.550134
\(615\) −26378.0 −1.72953
\(616\) 12469.7 0.815612
\(617\) 17512.0 1.14264 0.571318 0.820729i \(-0.306432\pi\)
0.571318 + 0.820729i \(0.306432\pi\)
\(618\) −7714.67 −0.502151
\(619\) −1787.91 −0.116094 −0.0580471 0.998314i \(-0.518487\pi\)
−0.0580471 + 0.998314i \(0.518487\pi\)
\(620\) 32.5714 0.00210984
\(621\) 2722.91 0.175953
\(622\) 15691.3 1.01151
\(623\) −11332.8 −0.728797
\(624\) −815.520 −0.0523188
\(625\) 28070.0 1.79648
\(626\) 6361.74 0.406176
\(627\) 5565.70 0.354502
\(628\) 7384.18 0.469205
\(629\) −4003.01 −0.253753
\(630\) 10356.4 0.654933
\(631\) 16319.4 1.02958 0.514789 0.857317i \(-0.327870\pi\)
0.514789 + 0.857317i \(0.327870\pi\)
\(632\) −5721.87 −0.360133
\(633\) −14238.1 −0.894017
\(634\) 11865.8 0.743300
\(635\) 1327.32 0.0829497
\(636\) 1311.27 0.0817538
\(637\) −8036.27 −0.499857
\(638\) −12083.8 −0.749847
\(639\) 8891.65 0.550467
\(640\) −2578.10 −0.159232
\(641\) 7001.57 0.431428 0.215714 0.976457i \(-0.430792\pi\)
0.215714 + 0.976457i \(0.430792\pi\)
\(642\) 10563.4 0.649384
\(643\) 6756.97 0.414415 0.207207 0.978297i \(-0.433563\pi\)
0.207207 + 0.978297i \(0.433563\pi\)
\(644\) −11523.2 −0.705092
\(645\) 19750.8 1.20572
\(646\) −737.703 −0.0449296
\(647\) −24709.5 −1.50144 −0.750720 0.660621i \(-0.770293\pi\)
−0.750720 + 0.660621i \(0.770293\pi\)
\(648\) −648.000 −0.0392837
\(649\) 3219.38 0.194717
\(650\) 9537.41 0.575520
\(651\) −34.6459 −0.00208584
\(652\) 1396.62 0.0838892
\(653\) 7674.59 0.459923 0.229962 0.973200i \(-0.426140\pi\)
0.229962 + 0.973200i \(0.426140\pi\)
\(654\) 7366.43 0.440444
\(655\) 4109.31 0.245136
\(656\) −6984.73 −0.415713
\(657\) −5663.47 −0.336306
\(658\) 16760.9 0.993021
\(659\) −600.920 −0.0355213 −0.0177606 0.999842i \(-0.505654\pi\)
−0.0177606 + 0.999842i \(0.505654\pi\)
\(660\) 13188.4 0.777813
\(661\) −16815.7 −0.989495 −0.494748 0.869037i \(-0.664740\pi\)
−0.494748 + 0.869037i \(0.664740\pi\)
\(662\) 12182.3 0.715224
\(663\) −552.952 −0.0323904
\(664\) 3980.62 0.232647
\(665\) −19562.0 −1.14073
\(666\) 6641.82 0.386434
\(667\) 11166.7 0.648239
\(668\) −269.091 −0.0155860
\(669\) −19547.1 −1.12965
\(670\) −25379.0 −1.46340
\(671\) 25141.0 1.44643
\(672\) 2742.31 0.157421
\(673\) −10123.9 −0.579865 −0.289933 0.957047i \(-0.593633\pi\)
−0.289933 + 0.957047i \(0.593633\pi\)
\(674\) 16934.7 0.967802
\(675\) 7578.28 0.432131
\(676\) −7633.36 −0.434306
\(677\) −34456.7 −1.95610 −0.978049 0.208377i \(-0.933182\pi\)
−0.978049 + 0.208377i \(0.933182\pi\)
\(678\) 6771.77 0.383582
\(679\) 42059.7 2.37718
\(680\) −1748.05 −0.0985801
\(681\) 1696.85 0.0954824
\(682\) −44.1200 −0.00247719
\(683\) −2080.93 −0.116581 −0.0582904 0.998300i \(-0.518565\pi\)
−0.0582904 + 0.998300i \(0.518565\pi\)
\(684\) 1224.00 0.0684222
\(685\) −42897.2 −2.39272
\(686\) 7427.09 0.413364
\(687\) 3780.78 0.209965
\(688\) 5229.90 0.289808
\(689\) −1856.55 −0.102654
\(690\) −12187.4 −0.672415
\(691\) 32006.1 1.76204 0.881019 0.473081i \(-0.156858\pi\)
0.881019 + 0.473081i \(0.156858\pi\)
\(692\) 12948.3 0.711302
\(693\) −14028.4 −0.768967
\(694\) 1883.53 0.103023
\(695\) −26142.0 −1.42679
\(696\) −2657.45 −0.144728
\(697\) −4735.90 −0.257367
\(698\) 7112.62 0.385697
\(699\) −7229.58 −0.391199
\(700\) −32071.0 −1.73167
\(701\) 17360.4 0.935369 0.467685 0.883895i \(-0.345088\pi\)
0.467685 + 0.883895i \(0.345088\pi\)
\(702\) 917.460 0.0493266
\(703\) −12545.7 −0.673070
\(704\) 3492.21 0.186957
\(705\) 17726.9 0.947000
\(706\) −8008.64 −0.426925
\(707\) −18024.1 −0.958790
\(708\) 708.000 0.0375823
\(709\) −2143.27 −0.113529 −0.0567646 0.998388i \(-0.518078\pi\)
−0.0567646 + 0.998388i \(0.518078\pi\)
\(710\) −39797.9 −2.10365
\(711\) 6437.11 0.339536
\(712\) −3173.83 −0.167057
\(713\) 40.7714 0.00214152
\(714\) 1859.38 0.0974589
\(715\) −18672.5 −0.976662
\(716\) 13367.8 0.697733
\(717\) −14651.4 −0.763134
\(718\) −2412.05 −0.125372
\(719\) −7603.20 −0.394369 −0.197185 0.980366i \(-0.563180\pi\)
−0.197185 + 0.980366i \(0.563180\pi\)
\(720\) 2900.37 0.150125
\(721\) −36729.2 −1.89718
\(722\) 11406.0 0.587933
\(723\) 13157.6 0.676813
\(724\) −8255.75 −0.423788
\(725\) 31078.6 1.59204
\(726\) −9878.50 −0.504994
\(727\) −2483.49 −0.126695 −0.0633477 0.997992i \(-0.520178\pi\)
−0.0633477 + 0.997992i \(0.520178\pi\)
\(728\) −3882.65 −0.197666
\(729\) 729.000 0.0370370
\(730\) 25349.0 1.28522
\(731\) 3546.06 0.179419
\(732\) 5528.97 0.279175
\(733\) −16382.6 −0.825520 −0.412760 0.910840i \(-0.635435\pi\)
−0.412760 + 0.910840i \(0.635435\pi\)
\(734\) 11673.8 0.587039
\(735\) 28580.7 1.43431
\(736\) −3227.15 −0.161623
\(737\) 34377.5 1.71820
\(738\) 7857.82 0.391938
\(739\) 35353.3 1.75980 0.879901 0.475157i \(-0.157609\pi\)
0.879901 + 0.475157i \(0.157609\pi\)
\(740\) −29727.9 −1.47679
\(741\) −1732.98 −0.0859144
\(742\) 6242.91 0.308874
\(743\) −34651.2 −1.71094 −0.855470 0.517853i \(-0.826732\pi\)
−0.855470 + 0.517853i \(0.826732\pi\)
\(744\) −9.70280 −0.000478121 0
\(745\) 43915.3 2.15964
\(746\) −25288.6 −1.24113
\(747\) −4478.19 −0.219342
\(748\) 2367.84 0.115744
\(749\) 50291.9 2.45344
\(750\) −18813.4 −0.915956
\(751\) 28892.0 1.40384 0.701919 0.712256i \(-0.252327\pi\)
0.701919 + 0.712256i \(0.252327\pi\)
\(752\) 4693.99 0.227623
\(753\) 1262.67 0.0611081
\(754\) 3762.51 0.181727
\(755\) −62560.6 −3.01565
\(756\) −3085.10 −0.148418
\(757\) 7659.14 0.367736 0.183868 0.982951i \(-0.441138\pi\)
0.183868 + 0.982951i \(0.441138\pi\)
\(758\) −14192.3 −0.680060
\(759\) 16508.6 0.789493
\(760\) −5478.47 −0.261480
\(761\) 8765.23 0.417528 0.208764 0.977966i \(-0.433056\pi\)
0.208764 + 0.977966i \(0.433056\pi\)
\(762\) −395.400 −0.0187977
\(763\) 35071.2 1.66404
\(764\) −16425.9 −0.777837
\(765\) 1966.55 0.0929422
\(766\) −13683.7 −0.645445
\(767\) −1002.41 −0.0471903
\(768\) 768.000 0.0360844
\(769\) −21094.2 −0.989177 −0.494588 0.869127i \(-0.664681\pi\)
−0.494588 + 0.869127i \(0.664681\pi\)
\(770\) 62789.2 2.93866
\(771\) −17713.3 −0.827404
\(772\) −3446.22 −0.160663
\(773\) −2971.39 −0.138258 −0.0691291 0.997608i \(-0.522022\pi\)
−0.0691291 + 0.997608i \(0.522022\pi\)
\(774\) −5883.63 −0.273234
\(775\) 113.473 0.00525945
\(776\) 11779.1 0.544903
\(777\) 31621.4 1.45999
\(778\) 12584.7 0.579927
\(779\) −14842.6 −0.682657
\(780\) −4106.43 −0.188505
\(781\) 53908.8 2.46992
\(782\) −2188.13 −0.100060
\(783\) 2989.63 0.136451
\(784\) 7568.00 0.344752
\(785\) 37182.0 1.69055
\(786\) −1224.14 −0.0555516
\(787\) 16024.0 0.725785 0.362893 0.931831i \(-0.381789\pi\)
0.362893 + 0.931831i \(0.381789\pi\)
\(788\) −4839.70 −0.218791
\(789\) −21588.9 −0.974125
\(790\) −28811.7 −1.29756
\(791\) 32240.1 1.44921
\(792\) −3928.73 −0.176264
\(793\) −7828.09 −0.350547
\(794\) 185.403 0.00828678
\(795\) 6602.73 0.294560
\(796\) 6576.65 0.292843
\(797\) 43476.9 1.93228 0.966142 0.258011i \(-0.0830669\pi\)
0.966142 + 0.258011i \(0.0830669\pi\)
\(798\) 5827.41 0.258506
\(799\) 3182.69 0.140921
\(800\) −8981.67 −0.396937
\(801\) 3570.56 0.157502
\(802\) 13765.5 0.606081
\(803\) −34336.8 −1.50899
\(804\) 7560.24 0.331628
\(805\) −58023.7 −2.54045
\(806\) 13.7375 0.000600353 0
\(807\) 16831.8 0.734211
\(808\) −5047.75 −0.219776
\(809\) −20829.5 −0.905223 −0.452611 0.891708i \(-0.649508\pi\)
−0.452611 + 0.891708i \(0.649508\pi\)
\(810\) −3262.91 −0.141540
\(811\) −37858.8 −1.63921 −0.819607 0.572926i \(-0.805808\pi\)
−0.819607 + 0.572926i \(0.805808\pi\)
\(812\) −12652.0 −0.546796
\(813\) 15450.2 0.666497
\(814\) 40268.4 1.73392
\(815\) 7032.47 0.302253
\(816\) 520.731 0.0223398
\(817\) 11113.5 0.475904
\(818\) −8851.93 −0.378362
\(819\) 4367.98 0.186361
\(820\) −35170.6 −1.49782
\(821\) −27071.8 −1.15081 −0.575403 0.817870i \(-0.695155\pi\)
−0.575403 + 0.817870i \(0.695155\pi\)
\(822\) 12778.8 0.542228
\(823\) −9426.19 −0.399242 −0.199621 0.979873i \(-0.563971\pi\)
−0.199621 + 0.979873i \(0.563971\pi\)
\(824\) −10286.2 −0.434876
\(825\) 45946.0 1.93895
\(826\) 3370.75 0.141990
\(827\) 2097.55 0.0881970 0.0440985 0.999027i \(-0.485958\pi\)
0.0440985 + 0.999027i \(0.485958\pi\)
\(828\) 3630.55 0.152380
\(829\) 1011.67 0.0423847 0.0211923 0.999775i \(-0.493254\pi\)
0.0211923 + 0.999775i \(0.493254\pi\)
\(830\) 20043.8 0.838231
\(831\) 24647.5 1.02889
\(832\) −1087.36 −0.0453094
\(833\) 5131.37 0.213435
\(834\) 7787.52 0.323333
\(835\) −1354.97 −0.0561566
\(836\) 7420.94 0.307008
\(837\) 10.9156 0.000450777 0
\(838\) −29151.4 −1.20169
\(839\) 28850.7 1.18717 0.593586 0.804771i \(-0.297712\pi\)
0.593586 + 0.804771i \(0.297712\pi\)
\(840\) 13808.5 0.567189
\(841\) −12128.5 −0.497294
\(842\) 20406.0 0.835199
\(843\) −8380.06 −0.342378
\(844\) −18984.1 −0.774241
\(845\) −38436.7 −1.56481
\(846\) −5280.74 −0.214605
\(847\) −47031.1 −1.90792
\(848\) 1748.37 0.0708009
\(849\) −400.834 −0.0162033
\(850\) −6089.89 −0.245743
\(851\) −37212.1 −1.49896
\(852\) 11855.5 0.476718
\(853\) 44788.4 1.79780 0.898901 0.438151i \(-0.144366\pi\)
0.898901 + 0.438151i \(0.144366\pi\)
\(854\) 26323.1 1.05475
\(855\) 6163.28 0.246526
\(856\) 14084.5 0.562383
\(857\) −3403.85 −0.135675 −0.0678374 0.997696i \(-0.521610\pi\)
−0.0678374 + 0.997696i \(0.521610\pi\)
\(858\) 5562.43 0.221327
\(859\) −20755.4 −0.824407 −0.412203 0.911092i \(-0.635241\pi\)
−0.412203 + 0.911092i \(0.635241\pi\)
\(860\) 26334.4 1.04418
\(861\) 37410.7 1.48078
\(862\) 10299.6 0.406967
\(863\) 129.134 0.00509359 0.00254679 0.999997i \(-0.499189\pi\)
0.00254679 + 0.999997i \(0.499189\pi\)
\(864\) −864.000 −0.0340207
\(865\) 65199.4 2.56283
\(866\) 22191.7 0.870790
\(867\) −14385.9 −0.563520
\(868\) −46.1946 −0.00180639
\(869\) 39027.3 1.52349
\(870\) −13381.2 −0.521455
\(871\) −10704.0 −0.416409
\(872\) 9821.91 0.381436
\(873\) −13251.5 −0.513739
\(874\) −6857.70 −0.265406
\(875\) −89569.5 −3.46057
\(876\) −7551.29 −0.291249
\(877\) 11326.3 0.436102 0.218051 0.975937i \(-0.430030\pi\)
0.218051 + 0.975937i \(0.430030\pi\)
\(878\) −13008.4 −0.500014
\(879\) −25350.8 −0.972764
\(880\) 17584.5 0.673606
\(881\) 11445.9 0.437708 0.218854 0.975758i \(-0.429768\pi\)
0.218854 + 0.975758i \(0.429768\pi\)
\(882\) −8514.00 −0.325035
\(883\) −31.3773 −0.00119584 −0.000597922 1.00000i \(-0.500190\pi\)
−0.000597922 1.00000i \(0.500190\pi\)
\(884\) −737.269 −0.0280509
\(885\) 3565.03 0.135409
\(886\) −25908.5 −0.982406
\(887\) 23118.1 0.875117 0.437558 0.899190i \(-0.355843\pi\)
0.437558 + 0.899190i \(0.355843\pi\)
\(888\) 8855.76 0.334662
\(889\) −1882.48 −0.0710195
\(890\) −15981.4 −0.601906
\(891\) 4419.82 0.166184
\(892\) −26062.7 −0.978302
\(893\) 9974.73 0.373787
\(894\) −13082.1 −0.489408
\(895\) 67311.5 2.51394
\(896\) 3656.41 0.136331
\(897\) −5140.25 −0.191336
\(898\) −9445.59 −0.351006
\(899\) 44.7651 0.00166074
\(900\) 10104.4 0.374236
\(901\) 1185.45 0.0438326
\(902\) 47640.9 1.75861
\(903\) −28011.7 −1.03230
\(904\) 9029.03 0.332191
\(905\) −41570.6 −1.52691
\(906\) 18636.4 0.683391
\(907\) 17582.7 0.643688 0.321844 0.946793i \(-0.395697\pi\)
0.321844 + 0.946793i \(0.395697\pi\)
\(908\) 2262.47 0.0826901
\(909\) 5678.72 0.207207
\(910\) −19550.5 −0.712191
\(911\) 18975.5 0.690104 0.345052 0.938583i \(-0.387861\pi\)
0.345052 + 0.938583i \(0.387861\pi\)
\(912\) 1632.00 0.0592554
\(913\) −27150.7 −0.984179
\(914\) −27527.3 −0.996194
\(915\) 27840.3 1.00587
\(916\) 5041.05 0.181835
\(917\) −5828.06 −0.209879
\(918\) −585.823 −0.0210621
\(919\) 35075.0 1.25900 0.629498 0.777002i \(-0.283260\pi\)
0.629498 + 0.777002i \(0.283260\pi\)
\(920\) −16249.9 −0.582329
\(921\) −12554.9 −0.449182
\(922\) −9128.63 −0.326069
\(923\) −16785.5 −0.598592
\(924\) −18704.5 −0.665945
\(925\) −103567. −3.68137
\(926\) −1255.68 −0.0445619
\(927\) 11572.0 0.410005
\(928\) −3543.27 −0.125338
\(929\) −14369.6 −0.507482 −0.253741 0.967272i \(-0.581661\pi\)
−0.253741 + 0.967272i \(0.581661\pi\)
\(930\) −48.8571 −0.00172267
\(931\) 16082.0 0.566129
\(932\) −9639.44 −0.338788
\(933\) −23536.9 −0.825898
\(934\) −13432.4 −0.470579
\(935\) 11922.9 0.417028
\(936\) 1223.28 0.0427181
\(937\) 11260.5 0.392598 0.196299 0.980544i \(-0.437108\pi\)
0.196299 + 0.980544i \(0.437108\pi\)
\(938\) 35993.9 1.25293
\(939\) −9542.61 −0.331642
\(940\) 23635.9 0.820126
\(941\) 41046.7 1.42198 0.710991 0.703201i \(-0.248247\pi\)
0.710991 + 0.703201i \(0.248247\pi\)
\(942\) −11076.3 −0.383104
\(943\) −44025.0 −1.52031
\(944\) 944.000 0.0325472
\(945\) −15534.6 −0.534751
\(946\) −35671.6 −1.22599
\(947\) −13131.7 −0.450604 −0.225302 0.974289i \(-0.572337\pi\)
−0.225302 + 0.974289i \(0.572337\pi\)
\(948\) 8582.81 0.294047
\(949\) 10691.4 0.365708
\(950\) −19086.0 −0.651824
\(951\) −17798.7 −0.606902
\(952\) 2479.18 0.0844019
\(953\) −46349.4 −1.57545 −0.787725 0.616027i \(-0.788741\pi\)
−0.787725 + 0.616027i \(0.788741\pi\)
\(954\) −1966.91 −0.0667517
\(955\) −82710.2 −2.80255
\(956\) −19535.2 −0.660894
\(957\) 18125.7 0.612248
\(958\) 20475.9 0.690548
\(959\) 60839.2 2.04859
\(960\) 3867.15 0.130012
\(961\) −29790.8 −0.999995
\(962\) −12538.3 −0.420219
\(963\) −15845.1 −0.530220
\(964\) 17543.4 0.586137
\(965\) −17352.9 −0.578871
\(966\) 17284.9 0.575706
\(967\) 28322.6 0.941875 0.470937 0.882167i \(-0.343916\pi\)
0.470937 + 0.882167i \(0.343916\pi\)
\(968\) −13171.3 −0.437338
\(969\) 1106.55 0.0366849
\(970\) 59311.9 1.96329
\(971\) −11450.1 −0.378424 −0.189212 0.981936i \(-0.560593\pi\)
−0.189212 + 0.981936i \(0.560593\pi\)
\(972\) 972.000 0.0320750
\(973\) 37076.0 1.22159
\(974\) −20055.7 −0.659780
\(975\) −14306.1 −0.469910
\(976\) 7371.95 0.241773
\(977\) −28464.8 −0.932107 −0.466054 0.884756i \(-0.654325\pi\)
−0.466054 + 0.884756i \(0.654325\pi\)
\(978\) −2094.93 −0.0684952
\(979\) 21647.8 0.706707
\(980\) 38107.6 1.24214
\(981\) −11049.6 −0.359621
\(982\) 19181.8 0.623337
\(983\) −24122.9 −0.782708 −0.391354 0.920240i \(-0.627993\pi\)
−0.391354 + 0.920240i \(0.627993\pi\)
\(984\) 10477.1 0.339429
\(985\) −24369.6 −0.788305
\(986\) −2402.46 −0.0775963
\(987\) −25141.3 −0.810798
\(988\) −2310.64 −0.0744041
\(989\) 32964.2 1.05986
\(990\) −19782.6 −0.635082
\(991\) 4710.10 0.150980 0.0754901 0.997147i \(-0.475948\pi\)
0.0754901 + 0.997147i \(0.475948\pi\)
\(992\) −12.9371 −0.000414065 0
\(993\) −18273.4 −0.583978
\(994\) 56443.6 1.80109
\(995\) 33115.8 1.05512
\(996\) −5970.93 −0.189956
\(997\) 21534.7 0.684064 0.342032 0.939688i \(-0.388885\pi\)
0.342032 + 0.939688i \(0.388885\pi\)
\(998\) 2641.90 0.0837954
\(999\) −9962.73 −0.315522
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 354.4.a.b.1.2 2
3.2 odd 2 1062.4.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
354.4.a.b.1.2 2 1.1 even 1 trivial
1062.4.a.g.1.1 2 3.2 odd 2