Properties

Label 6027.2.a.bc.1.4
Level 6027
Weight 2
Character 6027.1
Self dual yes
Analytic conductor 48.126
Analytic rank 0
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.7457527933.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.978012\)
Character \(\chi\) = 6027.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.13860 q^{2} +1.00000 q^{3} -0.703600 q^{4} +2.27123 q^{5} -1.13860 q^{6} +3.07831 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.13860 q^{2} +1.00000 q^{3} -0.703600 q^{4} +2.27123 q^{5} -1.13860 q^{6} +3.07831 q^{8} +1.00000 q^{9} -2.58601 q^{10} +4.11356 q^{11} -0.703600 q^{12} -1.02235 q^{13} +2.27123 q^{15} -2.09775 q^{16} +1.01936 q^{17} -1.13860 q^{18} +0.361035 q^{19} -1.59804 q^{20} -4.68368 q^{22} -4.61988 q^{23} +3.07831 q^{24} +0.158481 q^{25} +1.16404 q^{26} +1.00000 q^{27} -1.50587 q^{29} -2.58601 q^{30} +3.87299 q^{31} -3.76813 q^{32} +4.11356 q^{33} -1.16064 q^{34} -0.703600 q^{36} +9.23719 q^{37} -0.411073 q^{38} -1.02235 q^{39} +6.99154 q^{40} +1.00000 q^{41} -2.19077 q^{43} -2.89430 q^{44} +2.27123 q^{45} +5.26018 q^{46} -1.32485 q^{47} -2.09775 q^{48} -0.180446 q^{50} +1.01936 q^{51} +0.719323 q^{52} +7.53796 q^{53} -1.13860 q^{54} +9.34284 q^{55} +0.361035 q^{57} +1.71458 q^{58} -1.63497 q^{59} -1.59804 q^{60} +14.8701 q^{61} -4.40977 q^{62} +8.48587 q^{64} -2.32198 q^{65} -4.68368 q^{66} +5.38619 q^{67} -0.717224 q^{68} -4.61988 q^{69} -1.61302 q^{71} +3.07831 q^{72} +2.22290 q^{73} -10.5174 q^{74} +0.158481 q^{75} -0.254024 q^{76} +1.16404 q^{78} -4.15827 q^{79} -4.76446 q^{80} +1.00000 q^{81} -1.13860 q^{82} +1.53726 q^{83} +2.31521 q^{85} +2.49440 q^{86} -1.50587 q^{87} +12.6628 q^{88} +1.06059 q^{89} -2.58601 q^{90} +3.25055 q^{92} +3.87299 q^{93} +1.50847 q^{94} +0.819993 q^{95} -3.76813 q^{96} +0.418003 q^{97} +4.11356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{2} + 8q^{3} + 13q^{4} + 7q^{5} + q^{6} + 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + q^{2} + 8q^{3} + 13q^{4} + 7q^{5} + q^{6} + 6q^{8} + 8q^{9} + 8q^{10} + 11q^{11} + 13q^{12} + 10q^{13} + 7q^{15} - 17q^{16} + 3q^{17} + q^{18} + 6q^{19} + 11q^{20} + 15q^{22} + 14q^{23} + 6q^{24} + 25q^{25} + 24q^{26} + 8q^{27} + 2q^{29} + 8q^{30} + 16q^{31} + 3q^{32} + 11q^{33} - 4q^{34} + 13q^{36} - 20q^{37} + 10q^{38} + 10q^{39} - 3q^{40} + 8q^{41} + 7q^{43} + 7q^{45} - 5q^{46} + 14q^{47} - 17q^{48} - 5q^{50} + 3q^{51} + 23q^{52} + 7q^{53} + q^{54} + 48q^{55} + 6q^{57} - 20q^{58} + 22q^{59} + 11q^{60} - 33q^{62} - 10q^{64} - 14q^{65} + 15q^{66} + 12q^{67} - 27q^{68} + 14q^{69} - 5q^{71} + 6q^{72} + 2q^{73} + 6q^{74} + 25q^{75} + 43q^{76} + 24q^{78} - 15q^{79} - 7q^{80} + 8q^{81} + q^{82} + 15q^{83} - 43q^{85} + 31q^{86} + 2q^{87} + 17q^{88} + 29q^{89} + 8q^{90} + 19q^{92} + 16q^{93} + 20q^{94} + 14q^{95} + 3q^{96} + 19q^{97} + 11q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13860 −0.805109 −0.402554 0.915396i \(-0.631878\pi\)
−0.402554 + 0.915396i \(0.631878\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.703600 −0.351800
\(5\) 2.27123 1.01572 0.507862 0.861438i \(-0.330436\pi\)
0.507862 + 0.861438i \(0.330436\pi\)
\(6\) −1.13860 −0.464830
\(7\) 0 0
\(8\) 3.07831 1.08835
\(9\) 1.00000 0.333333
\(10\) −2.58601 −0.817769
\(11\) 4.11356 1.24029 0.620143 0.784489i \(-0.287075\pi\)
0.620143 + 0.784489i \(0.287075\pi\)
\(12\) −0.703600 −0.203112
\(13\) −1.02235 −0.283548 −0.141774 0.989899i \(-0.545281\pi\)
−0.141774 + 0.989899i \(0.545281\pi\)
\(14\) 0 0
\(15\) 2.27123 0.586429
\(16\) −2.09775 −0.524437
\(17\) 1.01936 0.247232 0.123616 0.992330i \(-0.460551\pi\)
0.123616 + 0.992330i \(0.460551\pi\)
\(18\) −1.13860 −0.268370
\(19\) 0.361035 0.0828271 0.0414135 0.999142i \(-0.486814\pi\)
0.0414135 + 0.999142i \(0.486814\pi\)
\(20\) −1.59804 −0.357332
\(21\) 0 0
\(22\) −4.68368 −0.998564
\(23\) −4.61988 −0.963312 −0.481656 0.876360i \(-0.659965\pi\)
−0.481656 + 0.876360i \(0.659965\pi\)
\(24\) 3.07831 0.628357
\(25\) 0.158481 0.0316963
\(26\) 1.16404 0.228287
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.50587 −0.279633 −0.139816 0.990177i \(-0.544651\pi\)
−0.139816 + 0.990177i \(0.544651\pi\)
\(30\) −2.58601 −0.472139
\(31\) 3.87299 0.695610 0.347805 0.937567i \(-0.386927\pi\)
0.347805 + 0.937567i \(0.386927\pi\)
\(32\) −3.76813 −0.666117
\(33\) 4.11356 0.716079
\(34\) −1.16064 −0.199049
\(35\) 0 0
\(36\) −0.703600 −0.117267
\(37\) 9.23719 1.51858 0.759292 0.650750i \(-0.225545\pi\)
0.759292 + 0.650750i \(0.225545\pi\)
\(38\) −0.411073 −0.0666848
\(39\) −1.02235 −0.163706
\(40\) 6.99154 1.10546
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −2.19077 −0.334089 −0.167045 0.985949i \(-0.553422\pi\)
−0.167045 + 0.985949i \(0.553422\pi\)
\(44\) −2.89430 −0.436332
\(45\) 2.27123 0.338575
\(46\) 5.26018 0.775571
\(47\) −1.32485 −0.193249 −0.0966245 0.995321i \(-0.530805\pi\)
−0.0966245 + 0.995321i \(0.530805\pi\)
\(48\) −2.09775 −0.302784
\(49\) 0 0
\(50\) −0.180446 −0.0255190
\(51\) 1.01936 0.142739
\(52\) 0.719323 0.0997522
\(53\) 7.53796 1.03542 0.517709 0.855557i \(-0.326785\pi\)
0.517709 + 0.855557i \(0.326785\pi\)
\(54\) −1.13860 −0.154943
\(55\) 9.34284 1.25979
\(56\) 0 0
\(57\) 0.361035 0.0478202
\(58\) 1.71458 0.225135
\(59\) −1.63497 −0.212855 −0.106427 0.994320i \(-0.533941\pi\)
−0.106427 + 0.994320i \(0.533941\pi\)
\(60\) −1.59804 −0.206306
\(61\) 14.8701 1.90392 0.951961 0.306218i \(-0.0990638\pi\)
0.951961 + 0.306218i \(0.0990638\pi\)
\(62\) −4.40977 −0.560042
\(63\) 0 0
\(64\) 8.48587 1.06073
\(65\) −2.32198 −0.288007
\(66\) −4.68368 −0.576521
\(67\) 5.38619 0.658027 0.329014 0.944325i \(-0.393284\pi\)
0.329014 + 0.944325i \(0.393284\pi\)
\(68\) −0.717224 −0.0869762
\(69\) −4.61988 −0.556169
\(70\) 0 0
\(71\) −1.61302 −0.191430 −0.0957149 0.995409i \(-0.530514\pi\)
−0.0957149 + 0.995409i \(0.530514\pi\)
\(72\) 3.07831 0.362782
\(73\) 2.22290 0.260171 0.130086 0.991503i \(-0.458475\pi\)
0.130086 + 0.991503i \(0.458475\pi\)
\(74\) −10.5174 −1.22263
\(75\) 0.158481 0.0182999
\(76\) −0.254024 −0.0291386
\(77\) 0 0
\(78\) 1.16404 0.131801
\(79\) −4.15827 −0.467842 −0.233921 0.972256i \(-0.575156\pi\)
−0.233921 + 0.972256i \(0.575156\pi\)
\(80\) −4.76446 −0.532683
\(81\) 1.00000 0.111111
\(82\) −1.13860 −0.125737
\(83\) 1.53726 0.168736 0.0843679 0.996435i \(-0.473113\pi\)
0.0843679 + 0.996435i \(0.473113\pi\)
\(84\) 0 0
\(85\) 2.31521 0.251119
\(86\) 2.49440 0.268978
\(87\) −1.50587 −0.161446
\(88\) 12.6628 1.34986
\(89\) 1.06059 0.112422 0.0562111 0.998419i \(-0.482098\pi\)
0.0562111 + 0.998419i \(0.482098\pi\)
\(90\) −2.58601 −0.272590
\(91\) 0 0
\(92\) 3.25055 0.338893
\(93\) 3.87299 0.401611
\(94\) 1.50847 0.155586
\(95\) 0.819993 0.0841295
\(96\) −3.76813 −0.384583
\(97\) 0.418003 0.0424418 0.0212209 0.999775i \(-0.493245\pi\)
0.0212209 + 0.999775i \(0.493245\pi\)
\(98\) 0 0
\(99\) 4.11356 0.413428
\(100\) −0.111508 −0.0111508
\(101\) 3.97091 0.395121 0.197560 0.980291i \(-0.436698\pi\)
0.197560 + 0.980291i \(0.436698\pi\)
\(102\) −1.16064 −0.114921
\(103\) −10.9705 −1.08096 −0.540479 0.841358i \(-0.681757\pi\)
−0.540479 + 0.841358i \(0.681757\pi\)
\(104\) −3.14710 −0.308598
\(105\) 0 0
\(106\) −8.58268 −0.833624
\(107\) 19.5498 1.88995 0.944974 0.327146i \(-0.106087\pi\)
0.944974 + 0.327146i \(0.106087\pi\)
\(108\) −0.703600 −0.0677039
\(109\) 17.4207 1.66860 0.834299 0.551313i \(-0.185873\pi\)
0.834299 + 0.551313i \(0.185873\pi\)
\(110\) −10.6377 −1.01427
\(111\) 9.23719 0.876755
\(112\) 0 0
\(113\) −9.78704 −0.920687 −0.460344 0.887741i \(-0.652274\pi\)
−0.460344 + 0.887741i \(0.652274\pi\)
\(114\) −0.411073 −0.0385005
\(115\) −10.4928 −0.978460
\(116\) 1.05953 0.0983748
\(117\) −1.02235 −0.0945160
\(118\) 1.86157 0.171371
\(119\) 0 0
\(120\) 6.99154 0.638237
\(121\) 5.92138 0.538307
\(122\) −16.9310 −1.53286
\(123\) 1.00000 0.0901670
\(124\) −2.72504 −0.244716
\(125\) −10.9962 −0.983530
\(126\) 0 0
\(127\) −16.9152 −1.50098 −0.750489 0.660883i \(-0.770182\pi\)
−0.750489 + 0.660883i \(0.770182\pi\)
\(128\) −2.12572 −0.187888
\(129\) −2.19077 −0.192887
\(130\) 2.64380 0.231877
\(131\) −20.7104 −1.80948 −0.904738 0.425969i \(-0.859933\pi\)
−0.904738 + 0.425969i \(0.859933\pi\)
\(132\) −2.89430 −0.251917
\(133\) 0 0
\(134\) −6.13269 −0.529784
\(135\) 2.27123 0.195476
\(136\) 3.13791 0.269074
\(137\) 6.27873 0.536428 0.268214 0.963359i \(-0.413567\pi\)
0.268214 + 0.963359i \(0.413567\pi\)
\(138\) 5.26018 0.447776
\(139\) −3.27698 −0.277950 −0.138975 0.990296i \(-0.544381\pi\)
−0.138975 + 0.990296i \(0.544381\pi\)
\(140\) 0 0
\(141\) −1.32485 −0.111572
\(142\) 1.83657 0.154122
\(143\) −4.20548 −0.351680
\(144\) −2.09775 −0.174812
\(145\) −3.42017 −0.284030
\(146\) −2.53099 −0.209466
\(147\) 0 0
\(148\) −6.49929 −0.534238
\(149\) −12.5636 −1.02925 −0.514624 0.857416i \(-0.672068\pi\)
−0.514624 + 0.857416i \(0.672068\pi\)
\(150\) −0.180446 −0.0147334
\(151\) 9.45156 0.769157 0.384578 0.923092i \(-0.374347\pi\)
0.384578 + 0.923092i \(0.374347\pi\)
\(152\) 1.11138 0.0901445
\(153\) 1.01936 0.0824106
\(154\) 0 0
\(155\) 8.79646 0.706548
\(156\) 0.719323 0.0575919
\(157\) 11.1900 0.893061 0.446530 0.894768i \(-0.352659\pi\)
0.446530 + 0.894768i \(0.352659\pi\)
\(158\) 4.73459 0.376664
\(159\) 7.53796 0.597799
\(160\) −8.55828 −0.676592
\(161\) 0 0
\(162\) −1.13860 −0.0894565
\(163\) −5.24847 −0.411092 −0.205546 0.978647i \(-0.565897\pi\)
−0.205546 + 0.978647i \(0.565897\pi\)
\(164\) −0.703600 −0.0549419
\(165\) 9.34284 0.727339
\(166\) −1.75031 −0.135851
\(167\) 13.1078 1.01432 0.507158 0.861853i \(-0.330696\pi\)
0.507158 + 0.861853i \(0.330696\pi\)
\(168\) 0 0
\(169\) −11.9548 −0.919601
\(170\) −2.63608 −0.202178
\(171\) 0.361035 0.0276090
\(172\) 1.54143 0.117533
\(173\) 2.61929 0.199141 0.0995704 0.995031i \(-0.468253\pi\)
0.0995704 + 0.995031i \(0.468253\pi\)
\(174\) 1.71458 0.129982
\(175\) 0 0
\(176\) −8.62921 −0.650451
\(177\) −1.63497 −0.122892
\(178\) −1.20758 −0.0905121
\(179\) 20.0966 1.50209 0.751044 0.660252i \(-0.229550\pi\)
0.751044 + 0.660252i \(0.229550\pi\)
\(180\) −1.59804 −0.119111
\(181\) −13.9771 −1.03891 −0.519456 0.854497i \(-0.673865\pi\)
−0.519456 + 0.854497i \(0.673865\pi\)
\(182\) 0 0
\(183\) 14.8701 1.09923
\(184\) −14.2214 −1.04842
\(185\) 20.9798 1.54246
\(186\) −4.40977 −0.323340
\(187\) 4.19321 0.306638
\(188\) 0.932164 0.0679850
\(189\) 0 0
\(190\) −0.933640 −0.0677334
\(191\) 11.8333 0.856230 0.428115 0.903724i \(-0.359178\pi\)
0.428115 + 0.903724i \(0.359178\pi\)
\(192\) 8.48587 0.612415
\(193\) 9.09972 0.655012 0.327506 0.944849i \(-0.393792\pi\)
0.327506 + 0.944849i \(0.393792\pi\)
\(194\) −0.475937 −0.0341703
\(195\) −2.32198 −0.166281
\(196\) 0 0
\(197\) 0.846122 0.0602837 0.0301418 0.999546i \(-0.490404\pi\)
0.0301418 + 0.999546i \(0.490404\pi\)
\(198\) −4.68368 −0.332855
\(199\) 3.86730 0.274146 0.137073 0.990561i \(-0.456231\pi\)
0.137073 + 0.990561i \(0.456231\pi\)
\(200\) 0.487855 0.0344965
\(201\) 5.38619 0.379912
\(202\) −4.52126 −0.318115
\(203\) 0 0
\(204\) −0.717224 −0.0502157
\(205\) 2.27123 0.158630
\(206\) 12.4910 0.870288
\(207\) −4.61988 −0.321104
\(208\) 2.14462 0.148703
\(209\) 1.48514 0.102729
\(210\) 0 0
\(211\) −23.6238 −1.62633 −0.813166 0.582032i \(-0.802258\pi\)
−0.813166 + 0.582032i \(0.802258\pi\)
\(212\) −5.30371 −0.364260
\(213\) −1.61302 −0.110522
\(214\) −22.2593 −1.52161
\(215\) −4.97574 −0.339343
\(216\) 3.07831 0.209452
\(217\) 0 0
\(218\) −19.8351 −1.34340
\(219\) 2.22290 0.150210
\(220\) −6.57362 −0.443193
\(221\) −1.04214 −0.0701021
\(222\) −10.5174 −0.705883
\(223\) 14.5761 0.976089 0.488044 0.872819i \(-0.337710\pi\)
0.488044 + 0.872819i \(0.337710\pi\)
\(224\) 0 0
\(225\) 0.158481 0.0105654
\(226\) 11.1435 0.741253
\(227\) 5.42919 0.360348 0.180174 0.983635i \(-0.442334\pi\)
0.180174 + 0.983635i \(0.442334\pi\)
\(228\) −0.254024 −0.0168232
\(229\) −8.50896 −0.562288 −0.281144 0.959666i \(-0.590714\pi\)
−0.281144 + 0.959666i \(0.590714\pi\)
\(230\) 11.9471 0.787767
\(231\) 0 0
\(232\) −4.63553 −0.304337
\(233\) 18.1099 1.18642 0.593210 0.805048i \(-0.297861\pi\)
0.593210 + 0.805048i \(0.297861\pi\)
\(234\) 1.16404 0.0760956
\(235\) −3.00903 −0.196288
\(236\) 1.15036 0.0748823
\(237\) −4.15827 −0.270109
\(238\) 0 0
\(239\) 2.22639 0.144013 0.0720066 0.997404i \(-0.477060\pi\)
0.0720066 + 0.997404i \(0.477060\pi\)
\(240\) −4.76446 −0.307545
\(241\) 14.2272 0.916456 0.458228 0.888835i \(-0.348484\pi\)
0.458228 + 0.888835i \(0.348484\pi\)
\(242\) −6.74206 −0.433396
\(243\) 1.00000 0.0641500
\(244\) −10.4626 −0.669800
\(245\) 0 0
\(246\) −1.13860 −0.0725942
\(247\) −0.369103 −0.0234854
\(248\) 11.9223 0.757064
\(249\) 1.53726 0.0974197
\(250\) 12.5202 0.791848
\(251\) −6.81117 −0.429918 −0.214959 0.976623i \(-0.568962\pi\)
−0.214959 + 0.976623i \(0.568962\pi\)
\(252\) 0 0
\(253\) −19.0042 −1.19478
\(254\) 19.2595 1.20845
\(255\) 2.31521 0.144984
\(256\) −14.5514 −0.909463
\(257\) 4.87985 0.304397 0.152198 0.988350i \(-0.451365\pi\)
0.152198 + 0.988350i \(0.451365\pi\)
\(258\) 2.49440 0.155295
\(259\) 0 0
\(260\) 1.63375 0.101321
\(261\) −1.50587 −0.0932109
\(262\) 23.5808 1.45682
\(263\) −16.5774 −1.02221 −0.511104 0.859519i \(-0.670763\pi\)
−0.511104 + 0.859519i \(0.670763\pi\)
\(264\) 12.6628 0.779342
\(265\) 17.1204 1.05170
\(266\) 0 0
\(267\) 1.06059 0.0649070
\(268\) −3.78972 −0.231494
\(269\) −7.34953 −0.448109 −0.224054 0.974577i \(-0.571929\pi\)
−0.224054 + 0.974577i \(0.571929\pi\)
\(270\) −2.58601 −0.157380
\(271\) 14.2752 0.867158 0.433579 0.901116i \(-0.357250\pi\)
0.433579 + 0.901116i \(0.357250\pi\)
\(272\) −2.13837 −0.129657
\(273\) 0 0
\(274\) −7.14893 −0.431883
\(275\) 0.651923 0.0393124
\(276\) 3.25055 0.195660
\(277\) −11.6467 −0.699781 −0.349891 0.936791i \(-0.613781\pi\)
−0.349891 + 0.936791i \(0.613781\pi\)
\(278\) 3.73116 0.223780
\(279\) 3.87299 0.231870
\(280\) 0 0
\(281\) 17.4871 1.04319 0.521595 0.853193i \(-0.325337\pi\)
0.521595 + 0.853193i \(0.325337\pi\)
\(282\) 1.50847 0.0898279
\(283\) 15.6673 0.931327 0.465663 0.884962i \(-0.345816\pi\)
0.465663 + 0.884962i \(0.345816\pi\)
\(284\) 1.13492 0.0673450
\(285\) 0.819993 0.0485722
\(286\) 4.78835 0.283141
\(287\) 0 0
\(288\) −3.76813 −0.222039
\(289\) −15.9609 −0.938876
\(290\) 3.89419 0.228675
\(291\) 0.418003 0.0245038
\(292\) −1.56404 −0.0915282
\(293\) −20.8334 −1.21710 −0.608550 0.793516i \(-0.708249\pi\)
−0.608550 + 0.793516i \(0.708249\pi\)
\(294\) 0 0
\(295\) −3.71339 −0.216202
\(296\) 28.4349 1.65274
\(297\) 4.11356 0.238693
\(298\) 14.3048 0.828656
\(299\) 4.72312 0.273145
\(300\) −0.111508 −0.00643789
\(301\) 0 0
\(302\) −10.7615 −0.619255
\(303\) 3.97091 0.228123
\(304\) −0.757360 −0.0434376
\(305\) 33.7734 1.93386
\(306\) −1.16064 −0.0663495
\(307\) 4.35015 0.248276 0.124138 0.992265i \(-0.460383\pi\)
0.124138 + 0.992265i \(0.460383\pi\)
\(308\) 0 0
\(309\) −10.9705 −0.624091
\(310\) −10.0156 −0.568848
\(311\) 17.4860 0.991540 0.495770 0.868454i \(-0.334886\pi\)
0.495770 + 0.868454i \(0.334886\pi\)
\(312\) −3.14710 −0.178169
\(313\) 27.3870 1.54800 0.774001 0.633184i \(-0.218252\pi\)
0.774001 + 0.633184i \(0.218252\pi\)
\(314\) −12.7409 −0.719011
\(315\) 0 0
\(316\) 2.92576 0.164587
\(317\) −31.8861 −1.79090 −0.895451 0.445161i \(-0.853147\pi\)
−0.895451 + 0.445161i \(0.853147\pi\)
\(318\) −8.58268 −0.481293
\(319\) −6.19448 −0.346824
\(320\) 19.2734 1.07741
\(321\) 19.5498 1.09116
\(322\) 0 0
\(323\) 0.368026 0.0204775
\(324\) −0.703600 −0.0390889
\(325\) −0.162023 −0.00898742
\(326\) 5.97589 0.330974
\(327\) 17.4207 0.963365
\(328\) 3.07831 0.169971
\(329\) 0 0
\(330\) −10.6377 −0.585587
\(331\) −16.1121 −0.885601 −0.442801 0.896620i \(-0.646015\pi\)
−0.442801 + 0.896620i \(0.646015\pi\)
\(332\) −1.08161 −0.0593613
\(333\) 9.23719 0.506195
\(334\) −14.9245 −0.816634
\(335\) 12.2333 0.668375
\(336\) 0 0
\(337\) 16.6380 0.906331 0.453166 0.891426i \(-0.350295\pi\)
0.453166 + 0.891426i \(0.350295\pi\)
\(338\) 13.6117 0.740378
\(339\) −9.78704 −0.531559
\(340\) −1.62898 −0.0883438
\(341\) 15.9318 0.862755
\(342\) −0.411073 −0.0222283
\(343\) 0 0
\(344\) −6.74386 −0.363605
\(345\) −10.4928 −0.564914
\(346\) −2.98231 −0.160330
\(347\) 25.4621 1.36688 0.683440 0.730007i \(-0.260483\pi\)
0.683440 + 0.730007i \(0.260483\pi\)
\(348\) 1.05953 0.0567967
\(349\) −15.6721 −0.838911 −0.419455 0.907776i \(-0.637779\pi\)
−0.419455 + 0.907776i \(0.637779\pi\)
\(350\) 0 0
\(351\) −1.02235 −0.0545688
\(352\) −15.5004 −0.826175
\(353\) 10.9229 0.581366 0.290683 0.956819i \(-0.406117\pi\)
0.290683 + 0.956819i \(0.406117\pi\)
\(354\) 1.86157 0.0989413
\(355\) −3.66353 −0.194440
\(356\) −0.746231 −0.0395501
\(357\) 0 0
\(358\) −22.8819 −1.20934
\(359\) −1.37014 −0.0723134 −0.0361567 0.999346i \(-0.511512\pi\)
−0.0361567 + 0.999346i \(0.511512\pi\)
\(360\) 6.99154 0.368487
\(361\) −18.8697 −0.993140
\(362\) 15.9143 0.836437
\(363\) 5.92138 0.310792
\(364\) 0 0
\(365\) 5.04872 0.264262
\(366\) −16.9310 −0.885000
\(367\) 33.2866 1.73755 0.868774 0.495209i \(-0.164908\pi\)
0.868774 + 0.495209i \(0.164908\pi\)
\(368\) 9.69135 0.505196
\(369\) 1.00000 0.0520579
\(370\) −23.8875 −1.24185
\(371\) 0 0
\(372\) −2.72504 −0.141287
\(373\) −13.8615 −0.717719 −0.358859 0.933392i \(-0.616834\pi\)
−0.358859 + 0.933392i \(0.616834\pi\)
\(374\) −4.77437 −0.246877
\(375\) −10.9962 −0.567841
\(376\) −4.07829 −0.210322
\(377\) 1.53952 0.0792893
\(378\) 0 0
\(379\) 1.47300 0.0756628 0.0378314 0.999284i \(-0.487955\pi\)
0.0378314 + 0.999284i \(0.487955\pi\)
\(380\) −0.576947 −0.0295968
\(381\) −16.9152 −0.866590
\(382\) −13.4734 −0.689359
\(383\) −1.29949 −0.0664006 −0.0332003 0.999449i \(-0.510570\pi\)
−0.0332003 + 0.999449i \(0.510570\pi\)
\(384\) −2.12572 −0.108477
\(385\) 0 0
\(386\) −10.3609 −0.527356
\(387\) −2.19077 −0.111363
\(388\) −0.294107 −0.0149310
\(389\) 16.8058 0.852089 0.426045 0.904702i \(-0.359907\pi\)
0.426045 + 0.904702i \(0.359907\pi\)
\(390\) 2.64380 0.133874
\(391\) −4.70934 −0.238162
\(392\) 0 0
\(393\) −20.7104 −1.04470
\(394\) −0.963390 −0.0485349
\(395\) −9.44438 −0.475198
\(396\) −2.89430 −0.145444
\(397\) 17.7317 0.889930 0.444965 0.895548i \(-0.353216\pi\)
0.444965 + 0.895548i \(0.353216\pi\)
\(398\) −4.40329 −0.220717
\(399\) 0 0
\(400\) −0.332454 −0.0166227
\(401\) 26.9691 1.34677 0.673387 0.739290i \(-0.264839\pi\)
0.673387 + 0.739290i \(0.264839\pi\)
\(402\) −6.13269 −0.305871
\(403\) −3.95954 −0.197239
\(404\) −2.79393 −0.139003
\(405\) 2.27123 0.112858
\(406\) 0 0
\(407\) 37.9977 1.88348
\(408\) 3.13791 0.155350
\(409\) −22.8930 −1.13199 −0.565993 0.824410i \(-0.691507\pi\)
−0.565993 + 0.824410i \(0.691507\pi\)
\(410\) −2.58601 −0.127714
\(411\) 6.27873 0.309707
\(412\) 7.71886 0.380281
\(413\) 0 0
\(414\) 5.26018 0.258524
\(415\) 3.49146 0.171389
\(416\) 3.85233 0.188876
\(417\) −3.27698 −0.160474
\(418\) −1.69097 −0.0827082
\(419\) 38.2790 1.87005 0.935026 0.354579i \(-0.115376\pi\)
0.935026 + 0.354579i \(0.115376\pi\)
\(420\) 0 0
\(421\) −29.0730 −1.41693 −0.708465 0.705746i \(-0.750612\pi\)
−0.708465 + 0.705746i \(0.750612\pi\)
\(422\) 26.8980 1.30937
\(423\) −1.32485 −0.0644164
\(424\) 23.2041 1.12689
\(425\) 0.161550 0.00783633
\(426\) 1.83657 0.0889823
\(427\) 0 0
\(428\) −13.7552 −0.664884
\(429\) −4.20548 −0.203043
\(430\) 5.66536 0.273208
\(431\) −28.5861 −1.37694 −0.688472 0.725263i \(-0.741718\pi\)
−0.688472 + 0.725263i \(0.741718\pi\)
\(432\) −2.09775 −0.100928
\(433\) −25.3115 −1.21639 −0.608196 0.793787i \(-0.708106\pi\)
−0.608196 + 0.793787i \(0.708106\pi\)
\(434\) 0 0
\(435\) −3.42017 −0.163985
\(436\) −12.2572 −0.587012
\(437\) −1.66794 −0.0797884
\(438\) −2.53099 −0.120935
\(439\) 11.4408 0.546042 0.273021 0.962008i \(-0.411977\pi\)
0.273021 + 0.962008i \(0.411977\pi\)
\(440\) 28.7601 1.37109
\(441\) 0 0
\(442\) 1.18658 0.0564398
\(443\) −28.2157 −1.34057 −0.670285 0.742104i \(-0.733828\pi\)
−0.670285 + 0.742104i \(0.733828\pi\)
\(444\) −6.49929 −0.308442
\(445\) 2.40884 0.114190
\(446\) −16.5963 −0.785858
\(447\) −12.5636 −0.594237
\(448\) 0 0
\(449\) 37.0482 1.74841 0.874207 0.485553i \(-0.161382\pi\)
0.874207 + 0.485553i \(0.161382\pi\)
\(450\) −0.180446 −0.00850632
\(451\) 4.11356 0.193700
\(452\) 6.88616 0.323898
\(453\) 9.45156 0.444073
\(454\) −6.18165 −0.290119
\(455\) 0 0
\(456\) 1.11138 0.0520450
\(457\) −3.13021 −0.146425 −0.0732126 0.997316i \(-0.523325\pi\)
−0.0732126 + 0.997316i \(0.523325\pi\)
\(458\) 9.68826 0.452703
\(459\) 1.01936 0.0475798
\(460\) 7.38274 0.344222
\(461\) 32.8502 1.52998 0.764992 0.644040i \(-0.222743\pi\)
0.764992 + 0.644040i \(0.222743\pi\)
\(462\) 0 0
\(463\) 15.8140 0.734937 0.367468 0.930036i \(-0.380225\pi\)
0.367468 + 0.930036i \(0.380225\pi\)
\(464\) 3.15893 0.146650
\(465\) 8.79646 0.407926
\(466\) −20.6199 −0.955197
\(467\) 36.7438 1.70030 0.850151 0.526539i \(-0.176511\pi\)
0.850151 + 0.526539i \(0.176511\pi\)
\(468\) 0.719323 0.0332507
\(469\) 0 0
\(470\) 3.42607 0.158033
\(471\) 11.1900 0.515609
\(472\) −5.03294 −0.231660
\(473\) −9.01186 −0.414366
\(474\) 4.73459 0.217467
\(475\) 0.0572173 0.00262531
\(476\) 0 0
\(477\) 7.53796 0.345139
\(478\) −2.53496 −0.115946
\(479\) −34.6662 −1.58394 −0.791969 0.610561i \(-0.790944\pi\)
−0.791969 + 0.610561i \(0.790944\pi\)
\(480\) −8.55828 −0.390630
\(481\) −9.44361 −0.430591
\(482\) −16.1991 −0.737847
\(483\) 0 0
\(484\) −4.16628 −0.189376
\(485\) 0.949381 0.0431092
\(486\) −1.13860 −0.0516477
\(487\) −24.3473 −1.10328 −0.551640 0.834082i \(-0.685998\pi\)
−0.551640 + 0.834082i \(0.685998\pi\)
\(488\) 45.7748 2.07213
\(489\) −5.24847 −0.237344
\(490\) 0 0
\(491\) −23.6572 −1.06764 −0.533818 0.845600i \(-0.679243\pi\)
−0.533818 + 0.845600i \(0.679243\pi\)
\(492\) −0.703600 −0.0317207
\(493\) −1.53503 −0.0691341
\(494\) 0.420259 0.0189083
\(495\) 9.34284 0.419929
\(496\) −8.12456 −0.364804
\(497\) 0 0
\(498\) −1.75031 −0.0784335
\(499\) 31.4749 1.40901 0.704505 0.709699i \(-0.251169\pi\)
0.704505 + 0.709699i \(0.251169\pi\)
\(500\) 7.73692 0.346006
\(501\) 13.1078 0.585615
\(502\) 7.75517 0.346130
\(503\) −17.3605 −0.774067 −0.387034 0.922066i \(-0.626500\pi\)
−0.387034 + 0.922066i \(0.626500\pi\)
\(504\) 0 0
\(505\) 9.01885 0.401334
\(506\) 21.6381 0.961929
\(507\) −11.9548 −0.530932
\(508\) 11.9015 0.528044
\(509\) −2.68874 −0.119176 −0.0595881 0.998223i \(-0.518979\pi\)
−0.0595881 + 0.998223i \(0.518979\pi\)
\(510\) −2.63608 −0.116728
\(511\) 0 0
\(512\) 20.8196 0.920105
\(513\) 0.361035 0.0159401
\(514\) −5.55618 −0.245072
\(515\) −24.9166 −1.09795
\(516\) 1.54143 0.0678575
\(517\) −5.44985 −0.239684
\(518\) 0 0
\(519\) 2.61929 0.114974
\(520\) −7.14778 −0.313451
\(521\) −39.2979 −1.72167 −0.860835 0.508884i \(-0.830058\pi\)
−0.860835 + 0.508884i \(0.830058\pi\)
\(522\) 1.71458 0.0750449
\(523\) 6.63116 0.289961 0.144980 0.989435i \(-0.453688\pi\)
0.144980 + 0.989435i \(0.453688\pi\)
\(524\) 14.5718 0.636573
\(525\) 0 0
\(526\) 18.8750 0.822988
\(527\) 3.94799 0.171977
\(528\) −8.62921 −0.375538
\(529\) −1.65667 −0.0720292
\(530\) −19.4932 −0.846732
\(531\) −1.63497 −0.0709516
\(532\) 0 0
\(533\) −1.02235 −0.0442827
\(534\) −1.20758 −0.0522572
\(535\) 44.4020 1.91967
\(536\) 16.5803 0.716162
\(537\) 20.0966 0.867231
\(538\) 8.36814 0.360776
\(539\) 0 0
\(540\) −1.59804 −0.0687686
\(541\) 32.2640 1.38714 0.693568 0.720391i \(-0.256038\pi\)
0.693568 + 0.720391i \(0.256038\pi\)
\(542\) −16.2537 −0.698157
\(543\) −13.9771 −0.599816
\(544\) −3.84109 −0.164685
\(545\) 39.5663 1.69484
\(546\) 0 0
\(547\) −7.23994 −0.309558 −0.154779 0.987949i \(-0.549466\pi\)
−0.154779 + 0.987949i \(0.549466\pi\)
\(548\) −4.41771 −0.188715
\(549\) 14.8701 0.634641
\(550\) −0.742277 −0.0316508
\(551\) −0.543671 −0.0231612
\(552\) −14.2214 −0.605304
\(553\) 0 0
\(554\) 13.2609 0.563400
\(555\) 20.9798 0.890542
\(556\) 2.30568 0.0977828
\(557\) −0.594345 −0.0251832 −0.0125916 0.999921i \(-0.504008\pi\)
−0.0125916 + 0.999921i \(0.504008\pi\)
\(558\) −4.40977 −0.186681
\(559\) 2.23973 0.0947303
\(560\) 0 0
\(561\) 4.19321 0.177038
\(562\) −19.9107 −0.839882
\(563\) 5.25348 0.221408 0.110704 0.993853i \(-0.464690\pi\)
0.110704 + 0.993853i \(0.464690\pi\)
\(564\) 0.932164 0.0392512
\(565\) −22.2286 −0.935165
\(566\) −17.8388 −0.749819
\(567\) 0 0
\(568\) −4.96536 −0.208342
\(569\) −19.6362 −0.823194 −0.411597 0.911366i \(-0.635029\pi\)
−0.411597 + 0.911366i \(0.635029\pi\)
\(570\) −0.933640 −0.0391059
\(571\) −35.2958 −1.47708 −0.738542 0.674207i \(-0.764485\pi\)
−0.738542 + 0.674207i \(0.764485\pi\)
\(572\) 2.95898 0.123721
\(573\) 11.8333 0.494345
\(574\) 0 0
\(575\) −0.732166 −0.0305334
\(576\) 8.48587 0.353578
\(577\) 34.3114 1.42840 0.714200 0.699941i \(-0.246790\pi\)
0.714200 + 0.699941i \(0.246790\pi\)
\(578\) 18.1730 0.755898
\(579\) 9.09972 0.378172
\(580\) 2.40643 0.0999217
\(581\) 0 0
\(582\) −0.475937 −0.0197282
\(583\) 31.0078 1.28421
\(584\) 6.84278 0.283156
\(585\) −2.32198 −0.0960022
\(586\) 23.7208 0.979898
\(587\) −18.7338 −0.773228 −0.386614 0.922242i \(-0.626355\pi\)
−0.386614 + 0.922242i \(0.626355\pi\)
\(588\) 0 0
\(589\) 1.39829 0.0576154
\(590\) 4.22805 0.174066
\(591\) 0.846122 0.0348048
\(592\) −19.3773 −0.796401
\(593\) −5.35369 −0.219850 −0.109925 0.993940i \(-0.535061\pi\)
−0.109925 + 0.993940i \(0.535061\pi\)
\(594\) −4.68368 −0.192174
\(595\) 0 0
\(596\) 8.83973 0.362089
\(597\) 3.86730 0.158278
\(598\) −5.37773 −0.219912
\(599\) 38.4103 1.56940 0.784701 0.619874i \(-0.212816\pi\)
0.784701 + 0.619874i \(0.212816\pi\)
\(600\) 0.487855 0.0199166
\(601\) −2.61417 −0.106634 −0.0533172 0.998578i \(-0.516979\pi\)
−0.0533172 + 0.998578i \(0.516979\pi\)
\(602\) 0 0
\(603\) 5.38619 0.219342
\(604\) −6.65011 −0.270589
\(605\) 13.4488 0.546772
\(606\) −4.52126 −0.183664
\(607\) −34.7661 −1.41111 −0.705556 0.708654i \(-0.749303\pi\)
−0.705556 + 0.708654i \(0.749303\pi\)
\(608\) −1.36043 −0.0551726
\(609\) 0 0
\(610\) −38.4543 −1.55697
\(611\) 1.35445 0.0547954
\(612\) −0.717224 −0.0289921
\(613\) −7.23499 −0.292219 −0.146109 0.989268i \(-0.546675\pi\)
−0.146109 + 0.989268i \(0.546675\pi\)
\(614\) −4.95306 −0.199889
\(615\) 2.27123 0.0915848
\(616\) 0 0
\(617\) 3.70944 0.149337 0.0746683 0.997208i \(-0.476210\pi\)
0.0746683 + 0.997208i \(0.476210\pi\)
\(618\) 12.4910 0.502461
\(619\) −21.8562 −0.878473 −0.439237 0.898371i \(-0.644751\pi\)
−0.439237 + 0.898371i \(0.644751\pi\)
\(620\) −6.18919 −0.248564
\(621\) −4.61988 −0.185390
\(622\) −19.9095 −0.798298
\(623\) 0 0
\(624\) 2.14462 0.0858537
\(625\) −25.7673 −1.03069
\(626\) −31.1827 −1.24631
\(627\) 1.48514 0.0593107
\(628\) −7.87330 −0.314179
\(629\) 9.41605 0.375442
\(630\) 0 0
\(631\) 21.7931 0.867569 0.433784 0.901017i \(-0.357178\pi\)
0.433784 + 0.901017i \(0.357178\pi\)
\(632\) −12.8004 −0.509174
\(633\) −23.6238 −0.938963
\(634\) 36.3054 1.44187
\(635\) −38.4182 −1.52458
\(636\) −5.30371 −0.210306
\(637\) 0 0
\(638\) 7.05301 0.279231
\(639\) −1.61302 −0.0638099
\(640\) −4.82799 −0.190843
\(641\) −27.4415 −1.08388 −0.541938 0.840419i \(-0.682309\pi\)
−0.541938 + 0.840419i \(0.682309\pi\)
\(642\) −22.2593 −0.878504
\(643\) 0.581325 0.0229252 0.0114626 0.999934i \(-0.496351\pi\)
0.0114626 + 0.999934i \(0.496351\pi\)
\(644\) 0 0
\(645\) −4.97574 −0.195920
\(646\) −0.419032 −0.0164866
\(647\) 29.9596 1.17783 0.588917 0.808194i \(-0.299555\pi\)
0.588917 + 0.808194i \(0.299555\pi\)
\(648\) 3.07831 0.120927
\(649\) −6.72554 −0.264001
\(650\) 0.184479 0.00723585
\(651\) 0 0
\(652\) 3.69283 0.144622
\(653\) 8.27365 0.323773 0.161886 0.986809i \(-0.448242\pi\)
0.161886 + 0.986809i \(0.448242\pi\)
\(654\) −19.8351 −0.775614
\(655\) −47.0380 −1.83793
\(656\) −2.09775 −0.0819033
\(657\) 2.22290 0.0867237
\(658\) 0 0
\(659\) 35.3248 1.37606 0.688030 0.725682i \(-0.258476\pi\)
0.688030 + 0.725682i \(0.258476\pi\)
\(660\) −6.57362 −0.255878
\(661\) 32.2603 1.25478 0.627390 0.778705i \(-0.284123\pi\)
0.627390 + 0.778705i \(0.284123\pi\)
\(662\) 18.3452 0.713005
\(663\) −1.04214 −0.0404735
\(664\) 4.73215 0.183643
\(665\) 0 0
\(666\) −10.5174 −0.407542
\(667\) 6.95694 0.269374
\(668\) −9.22268 −0.356836
\(669\) 14.5761 0.563545
\(670\) −13.9287 −0.538114
\(671\) 61.1691 2.36141
\(672\) 0 0
\(673\) −5.72480 −0.220675 −0.110337 0.993894i \(-0.535193\pi\)
−0.110337 + 0.993894i \(0.535193\pi\)
\(674\) −18.9440 −0.729695
\(675\) 0.158481 0.00609995
\(676\) 8.41140 0.323515
\(677\) 51.3825 1.97479 0.987396 0.158270i \(-0.0505918\pi\)
0.987396 + 0.158270i \(0.0505918\pi\)
\(678\) 11.1435 0.427963
\(679\) 0 0
\(680\) 7.12692 0.273305
\(681\) 5.42919 0.208047
\(682\) −18.1399 −0.694611
\(683\) −12.9225 −0.494465 −0.247232 0.968956i \(-0.579521\pi\)
−0.247232 + 0.968956i \(0.579521\pi\)
\(684\) −0.254024 −0.00971286
\(685\) 14.2604 0.544863
\(686\) 0 0
\(687\) −8.50896 −0.324637
\(688\) 4.59568 0.175209
\(689\) −7.70640 −0.293591
\(690\) 11.9471 0.454817
\(691\) 7.98422 0.303734 0.151867 0.988401i \(-0.451471\pi\)
0.151867 + 0.988401i \(0.451471\pi\)
\(692\) −1.84293 −0.0700577
\(693\) 0 0
\(694\) −28.9911 −1.10049
\(695\) −7.44278 −0.282321
\(696\) −4.63553 −0.175709
\(697\) 1.01936 0.0386111
\(698\) 17.8442 0.675414
\(699\) 18.1099 0.684980
\(700\) 0 0
\(701\) −4.14530 −0.156566 −0.0782829 0.996931i \(-0.524944\pi\)
−0.0782829 + 0.996931i \(0.524944\pi\)
\(702\) 1.16404 0.0439338
\(703\) 3.33495 0.125780
\(704\) 34.9071 1.31561
\(705\) −3.00903 −0.113327
\(706\) −12.4367 −0.468063
\(707\) 0 0
\(708\) 1.15036 0.0432333
\(709\) 22.3009 0.837527 0.418764 0.908095i \(-0.362464\pi\)
0.418764 + 0.908095i \(0.362464\pi\)
\(710\) 4.17128 0.156545
\(711\) −4.15827 −0.155947
\(712\) 3.26482 0.122354
\(713\) −17.8928 −0.670090
\(714\) 0 0
\(715\) −9.55162 −0.357210
\(716\) −14.1400 −0.528435
\(717\) 2.22639 0.0831461
\(718\) 1.56004 0.0582201
\(719\) −11.8846 −0.443221 −0.221610 0.975135i \(-0.571131\pi\)
−0.221610 + 0.975135i \(0.571131\pi\)
\(720\) −4.76446 −0.177561
\(721\) 0 0
\(722\) 21.4849 0.799585
\(723\) 14.2272 0.529116
\(724\) 9.83431 0.365489
\(725\) −0.238652 −0.00886332
\(726\) −6.74206 −0.250221
\(727\) 52.7921 1.95795 0.978976 0.203977i \(-0.0653867\pi\)
0.978976 + 0.203977i \(0.0653867\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −5.74846 −0.212760
\(731\) −2.23319 −0.0825975
\(732\) −10.4626 −0.386709
\(733\) −8.77886 −0.324255 −0.162127 0.986770i \(-0.551836\pi\)
−0.162127 + 0.986770i \(0.551836\pi\)
\(734\) −37.9000 −1.39892
\(735\) 0 0
\(736\) 17.4083 0.641679
\(737\) 22.1564 0.816142
\(738\) −1.13860 −0.0419123
\(739\) −49.4832 −1.82027 −0.910135 0.414311i \(-0.864023\pi\)
−0.910135 + 0.414311i \(0.864023\pi\)
\(740\) −14.7614 −0.542639
\(741\) −0.369103 −0.0135593
\(742\) 0 0
\(743\) 26.5917 0.975554 0.487777 0.872968i \(-0.337808\pi\)
0.487777 + 0.872968i \(0.337808\pi\)
\(744\) 11.9223 0.437091
\(745\) −28.5348 −1.04543
\(746\) 15.7826 0.577842
\(747\) 1.53726 0.0562453
\(748\) −2.95034 −0.107875
\(749\) 0 0
\(750\) 12.5202 0.457174
\(751\) 7.81219 0.285071 0.142535 0.989790i \(-0.454475\pi\)
0.142535 + 0.989790i \(0.454475\pi\)
\(752\) 2.77920 0.101347
\(753\) −6.81117 −0.248213
\(754\) −1.75289 −0.0638365
\(755\) 21.4666 0.781251
\(756\) 0 0
\(757\) 29.6870 1.07899 0.539497 0.841988i \(-0.318614\pi\)
0.539497 + 0.841988i \(0.318614\pi\)
\(758\) −1.67715 −0.0609167
\(759\) −19.0042 −0.689808
\(760\) 2.52419 0.0915620
\(761\) 12.7647 0.462720 0.231360 0.972868i \(-0.425683\pi\)
0.231360 + 0.972868i \(0.425683\pi\)
\(762\) 19.2595 0.697699
\(763\) 0 0
\(764\) −8.32594 −0.301222
\(765\) 2.31521 0.0837065
\(766\) 1.47959 0.0534597
\(767\) 1.67151 0.0603546
\(768\) −14.5514 −0.525079
\(769\) −26.8186 −0.967104 −0.483552 0.875316i \(-0.660654\pi\)
−0.483552 + 0.875316i \(0.660654\pi\)
\(770\) 0 0
\(771\) 4.87985 0.175744
\(772\) −6.40257 −0.230433
\(773\) −19.7548 −0.710531 −0.355265 0.934766i \(-0.615609\pi\)
−0.355265 + 0.934766i \(0.615609\pi\)
\(774\) 2.49440 0.0896594
\(775\) 0.613798 0.0220483
\(776\) 1.28674 0.0461914
\(777\) 0 0
\(778\) −19.1350 −0.686024
\(779\) 0.361035 0.0129354
\(780\) 1.63375 0.0584975
\(781\) −6.63524 −0.237428
\(782\) 5.36203 0.191746
\(783\) −1.50587 −0.0538153
\(784\) 0 0
\(785\) 25.4151 0.907104
\(786\) 23.5808 0.841098
\(787\) 38.0595 1.35667 0.678337 0.734751i \(-0.262701\pi\)
0.678337 + 0.734751i \(0.262701\pi\)
\(788\) −0.595331 −0.0212078
\(789\) −16.5774 −0.590172
\(790\) 10.7533 0.382586
\(791\) 0 0
\(792\) 12.6628 0.449953
\(793\) −15.2024 −0.539853
\(794\) −20.1893 −0.716490
\(795\) 17.1204 0.607199
\(796\) −2.72103 −0.0964444
\(797\) 42.5173 1.50604 0.753019 0.657998i \(-0.228597\pi\)
0.753019 + 0.657998i \(0.228597\pi\)
\(798\) 0 0
\(799\) −1.35050 −0.0477773
\(800\) −0.597179 −0.0211135
\(801\) 1.06059 0.0374741
\(802\) −30.7069 −1.08430
\(803\) 9.14405 0.322687
\(804\) −3.78972 −0.133653
\(805\) 0 0
\(806\) 4.50832 0.158799
\(807\) −7.34953 −0.258716
\(808\) 12.2237 0.430028
\(809\) −38.2867 −1.34609 −0.673044 0.739603i \(-0.735013\pi\)
−0.673044 + 0.739603i \(0.735013\pi\)
\(810\) −2.58601 −0.0908632
\(811\) −35.9502 −1.26238 −0.631191 0.775628i \(-0.717434\pi\)
−0.631191 + 0.775628i \(0.717434\pi\)
\(812\) 0 0
\(813\) 14.2752 0.500654
\(814\) −43.2640 −1.51640
\(815\) −11.9205 −0.417556
\(816\) −2.13837 −0.0748578
\(817\) −0.790944 −0.0276716
\(818\) 26.0659 0.911372
\(819\) 0 0
\(820\) −1.59804 −0.0558059
\(821\) −18.3527 −0.640514 −0.320257 0.947331i \(-0.603769\pi\)
−0.320257 + 0.947331i \(0.603769\pi\)
\(822\) −7.14893 −0.249348
\(823\) −41.0862 −1.43218 −0.716088 0.698010i \(-0.754069\pi\)
−0.716088 + 0.698010i \(0.754069\pi\)
\(824\) −33.7706 −1.17646
\(825\) 0.651923 0.0226970
\(826\) 0 0
\(827\) 23.0152 0.800316 0.400158 0.916446i \(-0.368955\pi\)
0.400158 + 0.916446i \(0.368955\pi\)
\(828\) 3.25055 0.112964
\(829\) 24.1424 0.838499 0.419249 0.907871i \(-0.362293\pi\)
0.419249 + 0.907871i \(0.362293\pi\)
\(830\) −3.97536 −0.137987
\(831\) −11.6467 −0.404019
\(832\) −8.67550 −0.300769
\(833\) 0 0
\(834\) 3.73116 0.129199
\(835\) 29.7709 1.03027
\(836\) −1.04494 −0.0361401
\(837\) 3.87299 0.133870
\(838\) −43.5843 −1.50559
\(839\) −15.6154 −0.539105 −0.269552 0.962986i \(-0.586876\pi\)
−0.269552 + 0.962986i \(0.586876\pi\)
\(840\) 0 0
\(841\) −26.7324 −0.921806
\(842\) 33.1024 1.14078
\(843\) 17.4871 0.602286
\(844\) 16.6217 0.572144
\(845\) −27.1521 −0.934061
\(846\) 1.50847 0.0518622
\(847\) 0 0
\(848\) −15.8127 −0.543011
\(849\) 15.6673 0.537702
\(850\) −0.183940 −0.00630910
\(851\) −42.6747 −1.46287
\(852\) 1.13492 0.0388817
\(853\) −16.3069 −0.558338 −0.279169 0.960242i \(-0.590059\pi\)
−0.279169 + 0.960242i \(0.590059\pi\)
\(854\) 0 0
\(855\) 0.819993 0.0280432
\(856\) 60.1802 2.05692
\(857\) 10.4701 0.357653 0.178827 0.983881i \(-0.442770\pi\)
0.178827 + 0.983881i \(0.442770\pi\)
\(858\) 4.78835 0.163471
\(859\) 29.4060 1.00332 0.501659 0.865065i \(-0.332723\pi\)
0.501659 + 0.865065i \(0.332723\pi\)
\(860\) 3.50093 0.119381
\(861\) 0 0
\(862\) 32.5480 1.10859
\(863\) −11.8231 −0.402463 −0.201231 0.979544i \(-0.564494\pi\)
−0.201231 + 0.979544i \(0.564494\pi\)
\(864\) −3.76813 −0.128194
\(865\) 5.94900 0.202272
\(866\) 28.8195 0.979327
\(867\) −15.9609 −0.542061
\(868\) 0 0
\(869\) −17.1053 −0.580257
\(870\) 3.89419 0.132026
\(871\) −5.50655 −0.186582
\(872\) 53.6262 1.81601
\(873\) 0.418003 0.0141473
\(874\) 1.89911 0.0642383
\(875\) 0 0
\(876\) −1.56404 −0.0528439
\(877\) −35.9145 −1.21275 −0.606373 0.795181i \(-0.707376\pi\)
−0.606373 + 0.795181i \(0.707376\pi\)
\(878\) −13.0265 −0.439623
\(879\) −20.8334 −0.702693
\(880\) −19.5989 −0.660679
\(881\) 41.3275 1.39236 0.696180 0.717868i \(-0.254882\pi\)
0.696180 + 0.717868i \(0.254882\pi\)
\(882\) 0 0
\(883\) −29.3608 −0.988068 −0.494034 0.869443i \(-0.664478\pi\)
−0.494034 + 0.869443i \(0.664478\pi\)
\(884\) 0.733251 0.0246619
\(885\) −3.71339 −0.124824
\(886\) 32.1263 1.07930
\(887\) −25.4418 −0.854253 −0.427126 0.904192i \(-0.640474\pi\)
−0.427126 + 0.904192i \(0.640474\pi\)
\(888\) 28.4349 0.954213
\(889\) 0 0
\(890\) −2.74270 −0.0919354
\(891\) 4.11356 0.137809
\(892\) −10.2558 −0.343388
\(893\) −0.478317 −0.0160063
\(894\) 14.3048 0.478425
\(895\) 45.6439 1.52571
\(896\) 0 0
\(897\) 4.72312 0.157700
\(898\) −42.1830 −1.40766
\(899\) −5.83222 −0.194515
\(900\) −0.111508 −0.00371692
\(901\) 7.68391 0.255988
\(902\) −4.68368 −0.155950
\(903\) 0 0
\(904\) −30.1275 −1.00203
\(905\) −31.7453 −1.05525
\(906\) −10.7615 −0.357527
\(907\) −21.3485 −0.708864 −0.354432 0.935082i \(-0.615326\pi\)
−0.354432 + 0.935082i \(0.615326\pi\)
\(908\) −3.81998 −0.126770
\(909\) 3.97091 0.131707
\(910\) 0 0
\(911\) −27.8400 −0.922381 −0.461190 0.887301i \(-0.652577\pi\)
−0.461190 + 0.887301i \(0.652577\pi\)
\(912\) −0.757360 −0.0250787
\(913\) 6.32360 0.209281
\(914\) 3.56405 0.117888
\(915\) 33.7734 1.11652
\(916\) 5.98690 0.197813
\(917\) 0 0
\(918\) −1.16064 −0.0383069
\(919\) −0.843163 −0.0278134 −0.0139067 0.999903i \(-0.504427\pi\)
−0.0139067 + 0.999903i \(0.504427\pi\)
\(920\) −32.3001 −1.06490
\(921\) 4.35015 0.143342
\(922\) −37.4030 −1.23180
\(923\) 1.64906 0.0542795
\(924\) 0 0
\(925\) 1.46392 0.0481335
\(926\) −18.0057 −0.591704
\(927\) −10.9705 −0.360319
\(928\) 5.67431 0.186268
\(929\) −3.50232 −0.114907 −0.0574537 0.998348i \(-0.518298\pi\)
−0.0574537 + 0.998348i \(0.518298\pi\)
\(930\) −10.0156 −0.328425
\(931\) 0 0
\(932\) −12.7421 −0.417382
\(933\) 17.4860 0.572466
\(934\) −41.8364 −1.36893
\(935\) 9.52375 0.311460
\(936\) −3.14710 −0.102866
\(937\) −31.5255 −1.02989 −0.514946 0.857222i \(-0.672188\pi\)
−0.514946 + 0.857222i \(0.672188\pi\)
\(938\) 0 0
\(939\) 27.3870 0.893740
\(940\) 2.11716 0.0690540
\(941\) 38.5977 1.25825 0.629124 0.777305i \(-0.283414\pi\)
0.629124 + 0.777305i \(0.283414\pi\)
\(942\) −12.7409 −0.415121
\(943\) −4.61988 −0.150444
\(944\) 3.42975 0.111629
\(945\) 0 0
\(946\) 10.2609 0.333610
\(947\) −23.0815 −0.750047 −0.375023 0.927015i \(-0.622365\pi\)
−0.375023 + 0.927015i \(0.622365\pi\)
\(948\) 2.92576 0.0950242
\(949\) −2.27258 −0.0737710
\(950\) −0.0651474 −0.00211366
\(951\) −31.8861 −1.03398
\(952\) 0 0
\(953\) −0.506172 −0.0163965 −0.00819826 0.999966i \(-0.502610\pi\)
−0.00819826 + 0.999966i \(0.502610\pi\)
\(954\) −8.58268 −0.277875
\(955\) 26.8762 0.869694
\(956\) −1.56649 −0.0506638
\(957\) −6.19448 −0.200239
\(958\) 39.4707 1.27524
\(959\) 0 0
\(960\) 19.2734 0.622045
\(961\) −15.9999 −0.516127
\(962\) 10.7524 0.346673
\(963\) 19.5498 0.629983
\(964\) −10.0103 −0.322409
\(965\) 20.6676 0.665312
\(966\) 0 0
\(967\) −48.6196 −1.56350 −0.781750 0.623592i \(-0.785673\pi\)
−0.781750 + 0.623592i \(0.785673\pi\)
\(968\) 18.2278 0.585864
\(969\) 0.368026 0.0118227
\(970\) −1.08096 −0.0347076
\(971\) −3.29543 −0.105755 −0.0528777 0.998601i \(-0.516839\pi\)
−0.0528777 + 0.998601i \(0.516839\pi\)
\(972\) −0.703600 −0.0225680
\(973\) 0 0
\(974\) 27.7217 0.888260
\(975\) −0.162023 −0.00518889
\(976\) −31.1937 −0.998487
\(977\) −3.50798 −0.112230 −0.0561151 0.998424i \(-0.517871\pi\)
−0.0561151 + 0.998424i \(0.517871\pi\)
\(978\) 5.97589 0.191088
\(979\) 4.36280 0.139436
\(980\) 0 0
\(981\) 17.4207 0.556199
\(982\) 26.9360 0.859562
\(983\) −22.3364 −0.712422 −0.356211 0.934406i \(-0.615932\pi\)
−0.356211 + 0.934406i \(0.615932\pi\)
\(984\) 3.07831 0.0981328
\(985\) 1.92174 0.0612316
\(986\) 1.74777 0.0556605
\(987\) 0 0
\(988\) 0.259701 0.00826218
\(989\) 10.1211 0.321832
\(990\) −10.6377 −0.338089
\(991\) 53.4713 1.69857 0.849286 0.527933i \(-0.177033\pi\)
0.849286 + 0.527933i \(0.177033\pi\)
\(992\) −14.5939 −0.463358
\(993\) −16.1121 −0.511302
\(994\) 0 0
\(995\) 8.78352 0.278456
\(996\) −1.08161 −0.0342723
\(997\) 41.4921 1.31407 0.657033 0.753862i \(-0.271811\pi\)
0.657033 + 0.753862i \(0.271811\pi\)
\(998\) −35.8372 −1.13441
\(999\) 9.23719 0.292252
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 6027.2.a.bc.1.4 8
7.3 odd 6 861.2.i.d.247.5 16
7.5 odd 6 861.2.i.d.739.5 yes 16
7.6 odd 2 6027.2.a.bb.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
861.2.i.d.247.5 16 7.3 odd 6
861.2.i.d.739.5 yes 16 7.5 odd 6
6027.2.a.bb.1.4 8 7.6 odd 2
6027.2.a.bc.1.4 8 1.1 even 1 trivial