Properties

Label 2009.2.a.n.1.1
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0419457661\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.633117.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 4 x^{2} + 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.45719\) of defining polynomial
Character \(\chi\) \(=\) 2009.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.45719 q^{2} +1.45719 q^{3} +4.03778 q^{4} +2.26685 q^{5} -3.58059 q^{6} -5.00722 q^{8} -0.876597 q^{9} +O(q^{10})\) \(q-2.45719 q^{2} +1.45719 q^{3} +4.03778 q^{4} +2.26685 q^{5} -3.58059 q^{6} -5.00722 q^{8} -0.876597 q^{9} -5.57007 q^{10} -5.41988 q^{11} +5.88382 q^{12} -3.23628 q^{13} +3.30322 q^{15} +4.22813 q^{16} -2.83206 q^{17} +2.15397 q^{18} +4.32097 q^{19} +9.15303 q^{20} +13.3177 q^{22} +6.99907 q^{23} -7.29647 q^{24} +0.138589 q^{25} +7.95216 q^{26} -5.64894 q^{27} +8.06741 q^{29} -8.11665 q^{30} -9.18123 q^{31} -0.374872 q^{32} -7.89779 q^{33} +6.95892 q^{34} -3.53951 q^{36} -0.0469023 q^{37} -10.6174 q^{38} -4.71588 q^{39} -11.3506 q^{40} +1.00000 q^{41} -6.31281 q^{43} -21.8843 q^{44} -1.98711 q^{45} -17.1980 q^{46} -5.26448 q^{47} +6.16119 q^{48} -0.340538 q^{50} -4.12685 q^{51} -13.0674 q^{52} +6.43622 q^{53} +13.8805 q^{54} -12.2860 q^{55} +6.29647 q^{57} -19.8232 q^{58} -2.45253 q^{59} +13.3377 q^{60} -5.28319 q^{61} +22.5600 q^{62} -7.53512 q^{64} -7.33615 q^{65} +19.4064 q^{66} -8.78423 q^{67} -11.4353 q^{68} +10.1990 q^{69} -12.1364 q^{71} +4.38931 q^{72} -2.42993 q^{73} +0.115248 q^{74} +0.201950 q^{75} +17.4471 q^{76} +11.5878 q^{78} +4.92882 q^{79} +9.58451 q^{80} -5.60179 q^{81} -2.45719 q^{82} +1.63593 q^{83} -6.41985 q^{85} +15.5118 q^{86} +11.7558 q^{87} +27.1385 q^{88} -1.68625 q^{89} +4.88271 q^{90} +28.2607 q^{92} -13.3788 q^{93} +12.9358 q^{94} +9.79497 q^{95} -0.546260 q^{96} -18.8278 q^{97} +4.75105 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + 2q^{11} + 2q^{12} - 5q^{13} - 5q^{15} - q^{16} - 13q^{17} + 21q^{18} + 23q^{20} + q^{22} + 2q^{23} - 2q^{24} + 22q^{25} - 10q^{27} - 5q^{29} - 33q^{30} - 17q^{31} - 12q^{32} - 3q^{33} + 8q^{34} + 15q^{36} - 7q^{37} + 3q^{38} + 5q^{39} - 7q^{40} + 5q^{41} + q^{43} - 47q^{44} + 23q^{45} - 24q^{46} - 9q^{47} + 19q^{48} + 2q^{50} + 5q^{51} - 20q^{52} + 5q^{53} - 2q^{54} - 33q^{55} - 3q^{57} - 27q^{58} - 7q^{59} - 16q^{60} - 22q^{61} + 28q^{62} - 3q^{64} - 31q^{65} + 42q^{66} - 3q^{67} - 17q^{68} + 22q^{69} - 24q^{71} - 12q^{72} - 40q^{73} - 5q^{74} - 24q^{75} + 19q^{76} + 30q^{78} - 42q^{79} - 24q^{80} + 9q^{81} - q^{82} + 12q^{83} - 23q^{85} + 16q^{86} + 32q^{87} + 26q^{88} - 8q^{89} + 59q^{90} + 12q^{92} - 11q^{93} + 23q^{94} - 17q^{95} + 17q^{96} - 16q^{97} - 45q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.45719 −1.73750 −0.868748 0.495255i \(-0.835075\pi\)
−0.868748 + 0.495255i \(0.835075\pi\)
\(3\) 1.45719 0.841309 0.420655 0.907221i \(-0.361800\pi\)
0.420655 + 0.907221i \(0.361800\pi\)
\(4\) 4.03778 2.01889
\(5\) 2.26685 1.01376 0.506882 0.862015i \(-0.330798\pi\)
0.506882 + 0.862015i \(0.330798\pi\)
\(6\) −3.58059 −1.46177
\(7\) 0 0
\(8\) −5.00722 −1.77032
\(9\) −0.876597 −0.292199
\(10\) −5.57007 −1.76141
\(11\) −5.41988 −1.63415 −0.817077 0.576528i \(-0.804407\pi\)
−0.817077 + 0.576528i \(0.804407\pi\)
\(12\) 5.88382 1.69851
\(13\) −3.23628 −0.897584 −0.448792 0.893636i \(-0.648146\pi\)
−0.448792 + 0.893636i \(0.648146\pi\)
\(14\) 0 0
\(15\) 3.30322 0.852889
\(16\) 4.22813 1.05703
\(17\) −2.83206 −0.686876 −0.343438 0.939175i \(-0.611592\pi\)
−0.343438 + 0.939175i \(0.611592\pi\)
\(18\) 2.15397 0.507694
\(19\) 4.32097 0.991298 0.495649 0.868523i \(-0.334930\pi\)
0.495649 + 0.868523i \(0.334930\pi\)
\(20\) 9.15303 2.04668
\(21\) 0 0
\(22\) 13.3177 2.83934
\(23\) 6.99907 1.45941 0.729703 0.683764i \(-0.239658\pi\)
0.729703 + 0.683764i \(0.239658\pi\)
\(24\) −7.29647 −1.48939
\(25\) 0.138589 0.0277177
\(26\) 7.95216 1.55955
\(27\) −5.64894 −1.08714
\(28\) 0 0
\(29\) 8.06741 1.49808 0.749040 0.662524i \(-0.230515\pi\)
0.749040 + 0.662524i \(0.230515\pi\)
\(30\) −8.11665 −1.48189
\(31\) −9.18123 −1.64900 −0.824498 0.565864i \(-0.808543\pi\)
−0.824498 + 0.565864i \(0.808543\pi\)
\(32\) −0.374872 −0.0662686
\(33\) −7.89779 −1.37483
\(34\) 6.95892 1.19344
\(35\) 0 0
\(36\) −3.53951 −0.589918
\(37\) −0.0469023 −0.00771068 −0.00385534 0.999993i \(-0.501227\pi\)
−0.00385534 + 0.999993i \(0.501227\pi\)
\(38\) −10.6174 −1.72238
\(39\) −4.71588 −0.755145
\(40\) −11.3506 −1.79469
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −6.31281 −0.962695 −0.481348 0.876530i \(-0.659852\pi\)
−0.481348 + 0.876530i \(0.659852\pi\)
\(44\) −21.8843 −3.29918
\(45\) −1.98711 −0.296221
\(46\) −17.1980 −2.53571
\(47\) −5.26448 −0.767903 −0.383952 0.923353i \(-0.625437\pi\)
−0.383952 + 0.923353i \(0.625437\pi\)
\(48\) 6.16119 0.889291
\(49\) 0 0
\(50\) −0.340538 −0.0481594
\(51\) −4.12685 −0.577875
\(52\) −13.0674 −1.81212
\(53\) 6.43622 0.884082 0.442041 0.896995i \(-0.354255\pi\)
0.442041 + 0.896995i \(0.354255\pi\)
\(54\) 13.8805 1.88890
\(55\) −12.2860 −1.65665
\(56\) 0 0
\(57\) 6.29647 0.833988
\(58\) −19.8232 −2.60291
\(59\) −2.45253 −0.319292 −0.159646 0.987174i \(-0.551035\pi\)
−0.159646 + 0.987174i \(0.551035\pi\)
\(60\) 13.3377 1.72189
\(61\) −5.28319 −0.676443 −0.338221 0.941067i \(-0.609825\pi\)
−0.338221 + 0.941067i \(0.609825\pi\)
\(62\) 22.5600 2.86513
\(63\) 0 0
\(64\) −7.53512 −0.941891
\(65\) −7.33615 −0.909938
\(66\) 19.4064 2.38876
\(67\) −8.78423 −1.07316 −0.536582 0.843848i \(-0.680285\pi\)
−0.536582 + 0.843848i \(0.680285\pi\)
\(68\) −11.4353 −1.38673
\(69\) 10.1990 1.22781
\(70\) 0 0
\(71\) −12.1364 −1.44033 −0.720165 0.693802i \(-0.755934\pi\)
−0.720165 + 0.693802i \(0.755934\pi\)
\(72\) 4.38931 0.517286
\(73\) −2.42993 −0.284402 −0.142201 0.989838i \(-0.545418\pi\)
−0.142201 + 0.989838i \(0.545418\pi\)
\(74\) 0.115248 0.0133973
\(75\) 0.201950 0.0233192
\(76\) 17.4471 2.00132
\(77\) 0 0
\(78\) 11.5878 1.31206
\(79\) 4.92882 0.554536 0.277268 0.960793i \(-0.410571\pi\)
0.277268 + 0.960793i \(0.410571\pi\)
\(80\) 9.58451 1.07158
\(81\) −5.60179 −0.622421
\(82\) −2.45719 −0.271351
\(83\) 1.63593 0.179567 0.0897833 0.995961i \(-0.471383\pi\)
0.0897833 + 0.995961i \(0.471383\pi\)
\(84\) 0 0
\(85\) −6.41985 −0.696330
\(86\) 15.5118 1.67268
\(87\) 11.7558 1.26035
\(88\) 27.1385 2.89298
\(89\) −1.68625 −0.178742 −0.0893712 0.995998i \(-0.528486\pi\)
−0.0893712 + 0.995998i \(0.528486\pi\)
\(90\) 4.88271 0.514682
\(91\) 0 0
\(92\) 28.2607 2.94638
\(93\) −13.3788 −1.38732
\(94\) 12.9358 1.33423
\(95\) 9.79497 1.00494
\(96\) −0.546260 −0.0557524
\(97\) −18.8278 −1.91168 −0.955838 0.293894i \(-0.905049\pi\)
−0.955838 + 0.293894i \(0.905049\pi\)
\(98\) 0 0
\(99\) 4.75105 0.477498
\(100\) 0.559591 0.0559591
\(101\) −7.44903 −0.741207 −0.370603 0.928791i \(-0.620849\pi\)
−0.370603 + 0.928791i \(0.620849\pi\)
\(102\) 10.1405 1.00406
\(103\) −3.68672 −0.363264 −0.181632 0.983367i \(-0.558138\pi\)
−0.181632 + 0.983367i \(0.558138\pi\)
\(104\) 16.2048 1.58901
\(105\) 0 0
\(106\) −15.8150 −1.53609
\(107\) −8.58781 −0.830215 −0.415108 0.909772i \(-0.636256\pi\)
−0.415108 + 0.909772i \(0.636256\pi\)
\(108\) −22.8092 −2.19482
\(109\) −2.91912 −0.279601 −0.139800 0.990180i \(-0.544646\pi\)
−0.139800 + 0.990180i \(0.544646\pi\)
\(110\) 30.1891 2.87842
\(111\) −0.0683455 −0.00648707
\(112\) 0 0
\(113\) −7.03990 −0.662258 −0.331129 0.943585i \(-0.607430\pi\)
−0.331129 + 0.943585i \(0.607430\pi\)
\(114\) −15.4716 −1.44905
\(115\) 15.8658 1.47949
\(116\) 32.5745 3.02446
\(117\) 2.83692 0.262273
\(118\) 6.02633 0.554768
\(119\) 0 0
\(120\) −16.5400 −1.50989
\(121\) 18.3751 1.67046
\(122\) 12.9818 1.17532
\(123\) 1.45719 0.131390
\(124\) −37.0718 −3.32915
\(125\) −11.0201 −0.985665
\(126\) 0 0
\(127\) 19.1099 1.69573 0.847863 0.530216i \(-0.177889\pi\)
0.847863 + 0.530216i \(0.177889\pi\)
\(128\) 19.2650 1.70280
\(129\) −9.19897 −0.809924
\(130\) 18.0263 1.58101
\(131\) −9.99440 −0.873215 −0.436608 0.899652i \(-0.643820\pi\)
−0.436608 + 0.899652i \(0.643820\pi\)
\(132\) −31.8896 −2.77563
\(133\) 0 0
\(134\) 21.5845 1.86462
\(135\) −12.8053 −1.10210
\(136\) 14.1808 1.21599
\(137\) 4.66385 0.398459 0.199230 0.979953i \(-0.436156\pi\)
0.199230 + 0.979953i \(0.436156\pi\)
\(138\) −25.0608 −2.13332
\(139\) 10.3954 0.881725 0.440862 0.897575i \(-0.354673\pi\)
0.440862 + 0.897575i \(0.354673\pi\)
\(140\) 0 0
\(141\) −7.67135 −0.646044
\(142\) 29.8215 2.50257
\(143\) 17.5403 1.46679
\(144\) −3.70636 −0.308864
\(145\) 18.2876 1.51870
\(146\) 5.97080 0.494147
\(147\) 0 0
\(148\) −0.189381 −0.0155670
\(149\) 6.23050 0.510422 0.255211 0.966885i \(-0.417855\pi\)
0.255211 + 0.966885i \(0.417855\pi\)
\(150\) −0.496229 −0.0405169
\(151\) −21.1548 −1.72156 −0.860778 0.508981i \(-0.830022\pi\)
−0.860778 + 0.508981i \(0.830022\pi\)
\(152\) −21.6360 −1.75492
\(153\) 2.48258 0.200704
\(154\) 0 0
\(155\) −20.8124 −1.67169
\(156\) −19.0417 −1.52456
\(157\) −20.4912 −1.63538 −0.817688 0.575662i \(-0.804745\pi\)
−0.817688 + 0.575662i \(0.804745\pi\)
\(158\) −12.1111 −0.963504
\(159\) 9.37879 0.743787
\(160\) −0.849777 −0.0671808
\(161\) 0 0
\(162\) 13.7647 1.08145
\(163\) −1.63497 −0.128060 −0.0640302 0.997948i \(-0.520395\pi\)
−0.0640302 + 0.997948i \(0.520395\pi\)
\(164\) 4.03778 0.315298
\(165\) −17.9031 −1.39375
\(166\) −4.01979 −0.311996
\(167\) 19.8243 1.53405 0.767026 0.641616i \(-0.221736\pi\)
0.767026 + 0.641616i \(0.221736\pi\)
\(168\) 0 0
\(169\) −2.52647 −0.194344
\(170\) 15.7748 1.20987
\(171\) −3.78775 −0.289656
\(172\) −25.4898 −1.94358
\(173\) 9.31394 0.708126 0.354063 0.935222i \(-0.384800\pi\)
0.354063 + 0.935222i \(0.384800\pi\)
\(174\) −28.8861 −2.18985
\(175\) 0 0
\(176\) −22.9159 −1.72735
\(177\) −3.57380 −0.268623
\(178\) 4.14344 0.310564
\(179\) −17.3459 −1.29649 −0.648245 0.761432i \(-0.724497\pi\)
−0.648245 + 0.761432i \(0.724497\pi\)
\(180\) −8.02352 −0.598038
\(181\) −10.1355 −0.753364 −0.376682 0.926343i \(-0.622935\pi\)
−0.376682 + 0.926343i \(0.622935\pi\)
\(182\) 0 0
\(183\) −7.69861 −0.569097
\(184\) −35.0459 −2.58362
\(185\) −0.106320 −0.00781681
\(186\) 32.8742 2.41046
\(187\) 15.3494 1.12246
\(188\) −21.2568 −1.55031
\(189\) 0 0
\(190\) −24.0681 −1.74608
\(191\) 18.1959 1.31661 0.658306 0.752750i \(-0.271273\pi\)
0.658306 + 0.752750i \(0.271273\pi\)
\(192\) −10.9801 −0.792421
\(193\) 1.29950 0.0935398 0.0467699 0.998906i \(-0.485107\pi\)
0.0467699 + 0.998906i \(0.485107\pi\)
\(194\) 46.2635 3.32153
\(195\) −10.6902 −0.765539
\(196\) 0 0
\(197\) 5.64679 0.402317 0.201159 0.979559i \(-0.435529\pi\)
0.201159 + 0.979559i \(0.435529\pi\)
\(198\) −11.6742 −0.829651
\(199\) 11.5069 0.815699 0.407850 0.913049i \(-0.366279\pi\)
0.407850 + 0.913049i \(0.366279\pi\)
\(200\) −0.693943 −0.0490692
\(201\) −12.8003 −0.902863
\(202\) 18.3037 1.28784
\(203\) 0 0
\(204\) −16.6633 −1.16667
\(205\) 2.26685 0.158323
\(206\) 9.05898 0.631169
\(207\) −6.13536 −0.426437
\(208\) −13.6834 −0.948775
\(209\) −23.4191 −1.61993
\(210\) 0 0
\(211\) −9.92599 −0.683333 −0.341667 0.939821i \(-0.610991\pi\)
−0.341667 + 0.939821i \(0.610991\pi\)
\(212\) 25.9880 1.78487
\(213\) −17.6851 −1.21176
\(214\) 21.1019 1.44250
\(215\) −14.3102 −0.975946
\(216\) 28.2855 1.92458
\(217\) 0 0
\(218\) 7.17282 0.485805
\(219\) −3.54087 −0.239270
\(220\) −49.6083 −3.34459
\(221\) 9.16536 0.616529
\(222\) 0.167938 0.0112713
\(223\) 11.8232 0.791738 0.395869 0.918307i \(-0.370443\pi\)
0.395869 + 0.918307i \(0.370443\pi\)
\(224\) 0 0
\(225\) −0.121486 −0.00809909
\(226\) 17.2984 1.15067
\(227\) −1.60275 −0.106378 −0.0531892 0.998584i \(-0.516939\pi\)
−0.0531892 + 0.998584i \(0.516939\pi\)
\(228\) 25.4238 1.68373
\(229\) 9.93349 0.656423 0.328212 0.944604i \(-0.393554\pi\)
0.328212 + 0.944604i \(0.393554\pi\)
\(230\) −38.9853 −2.57061
\(231\) 0 0
\(232\) −40.3953 −2.65208
\(233\) −7.91248 −0.518364 −0.259182 0.965829i \(-0.583453\pi\)
−0.259182 + 0.965829i \(0.583453\pi\)
\(234\) −6.97084 −0.455698
\(235\) −11.9338 −0.778473
\(236\) −9.90278 −0.644616
\(237\) 7.18223 0.466536
\(238\) 0 0
\(239\) 27.1893 1.75873 0.879365 0.476148i \(-0.157967\pi\)
0.879365 + 0.476148i \(0.157967\pi\)
\(240\) 13.9665 0.901531
\(241\) −19.4014 −1.24975 −0.624877 0.780723i \(-0.714851\pi\)
−0.624877 + 0.780723i \(0.714851\pi\)
\(242\) −45.1510 −2.90242
\(243\) 8.78395 0.563490
\(244\) −21.3324 −1.36566
\(245\) 0 0
\(246\) −3.58059 −0.228290
\(247\) −13.9839 −0.889773
\(248\) 45.9724 2.91925
\(249\) 2.38386 0.151071
\(250\) 27.0784 1.71259
\(251\) −27.4697 −1.73387 −0.866935 0.498421i \(-0.833913\pi\)
−0.866935 + 0.498421i \(0.833913\pi\)
\(252\) 0 0
\(253\) −37.9341 −2.38489
\(254\) −46.9566 −2.94632
\(255\) −9.35494 −0.585829
\(256\) −32.2675 −2.01672
\(257\) −21.6328 −1.34942 −0.674710 0.738083i \(-0.735731\pi\)
−0.674710 + 0.738083i \(0.735731\pi\)
\(258\) 22.6036 1.40724
\(259\) 0 0
\(260\) −29.6218 −1.83707
\(261\) −7.07187 −0.437738
\(262\) 24.5581 1.51721
\(263\) −9.84482 −0.607058 −0.303529 0.952822i \(-0.598165\pi\)
−0.303529 + 0.952822i \(0.598165\pi\)
\(264\) 39.5460 2.43389
\(265\) 14.5899 0.896251
\(266\) 0 0
\(267\) −2.45719 −0.150378
\(268\) −35.4688 −2.16660
\(269\) 26.0627 1.58907 0.794535 0.607219i \(-0.207715\pi\)
0.794535 + 0.607219i \(0.207715\pi\)
\(270\) 31.4650 1.91490
\(271\) −25.4506 −1.54602 −0.773008 0.634396i \(-0.781249\pi\)
−0.773008 + 0.634396i \(0.781249\pi\)
\(272\) −11.9743 −0.726050
\(273\) 0 0
\(274\) −11.4600 −0.692321
\(275\) −0.751133 −0.0452950
\(276\) 41.1812 2.47882
\(277\) 29.3678 1.76454 0.882270 0.470743i \(-0.156014\pi\)
0.882270 + 0.470743i \(0.156014\pi\)
\(278\) −25.5434 −1.53199
\(279\) 8.04823 0.481835
\(280\) 0 0
\(281\) 24.2120 1.44437 0.722184 0.691701i \(-0.243139\pi\)
0.722184 + 0.691701i \(0.243139\pi\)
\(282\) 18.8500 1.12250
\(283\) −8.91251 −0.529794 −0.264897 0.964277i \(-0.585338\pi\)
−0.264897 + 0.964277i \(0.585338\pi\)
\(284\) −49.0043 −2.90787
\(285\) 14.2731 0.845467
\(286\) −43.0997 −2.54854
\(287\) 0 0
\(288\) 0.328612 0.0193636
\(289\) −8.97942 −0.528201
\(290\) −44.9361 −2.63874
\(291\) −27.4357 −1.60831
\(292\) −9.81153 −0.574176
\(293\) 8.79238 0.513656 0.256828 0.966457i \(-0.417323\pi\)
0.256828 + 0.966457i \(0.417323\pi\)
\(294\) 0 0
\(295\) −5.55950 −0.323687
\(296\) 0.234850 0.0136504
\(297\) 30.6166 1.77655
\(298\) −15.3095 −0.886856
\(299\) −22.6510 −1.30994
\(300\) 0.815430 0.0470789
\(301\) 0 0
\(302\) 51.9814 2.99120
\(303\) −10.8547 −0.623584
\(304\) 18.2696 1.04783
\(305\) −11.9762 −0.685753
\(306\) −6.10016 −0.348723
\(307\) −25.4509 −1.45256 −0.726281 0.687398i \(-0.758753\pi\)
−0.726281 + 0.687398i \(0.758753\pi\)
\(308\) 0 0
\(309\) −5.37226 −0.305617
\(310\) 51.1401 2.90456
\(311\) 8.20774 0.465418 0.232709 0.972546i \(-0.425241\pi\)
0.232709 + 0.972546i \(0.425241\pi\)
\(312\) 23.6135 1.33685
\(313\) −7.84025 −0.443157 −0.221578 0.975143i \(-0.571121\pi\)
−0.221578 + 0.975143i \(0.571121\pi\)
\(314\) 50.3508 2.84146
\(315\) 0 0
\(316\) 19.9015 1.11955
\(317\) 32.4359 1.82178 0.910891 0.412647i \(-0.135396\pi\)
0.910891 + 0.412647i \(0.135396\pi\)
\(318\) −23.0455 −1.29233
\(319\) −43.7244 −2.44810
\(320\) −17.0810 −0.954855
\(321\) −12.5141 −0.698468
\(322\) 0 0
\(323\) −12.2373 −0.680899
\(324\) −22.6188 −1.25660
\(325\) −0.448512 −0.0248790
\(326\) 4.01742 0.222504
\(327\) −4.25371 −0.235230
\(328\) −5.00722 −0.276478
\(329\) 0 0
\(330\) 43.9913 2.42164
\(331\) 13.8576 0.761685 0.380843 0.924640i \(-0.375634\pi\)
0.380843 + 0.924640i \(0.375634\pi\)
\(332\) 6.60553 0.362526
\(333\) 0.0411144 0.00225305
\(334\) −48.7121 −2.66541
\(335\) −19.9125 −1.08794
\(336\) 0 0
\(337\) 18.6080 1.01364 0.506822 0.862050i \(-0.330820\pi\)
0.506822 + 0.862050i \(0.330820\pi\)
\(338\) 6.20802 0.337672
\(339\) −10.2585 −0.557164
\(340\) −25.9220 −1.40582
\(341\) 49.7611 2.69472
\(342\) 9.30722 0.503277
\(343\) 0 0
\(344\) 31.6097 1.70428
\(345\) 23.1195 1.24471
\(346\) −22.8861 −1.23037
\(347\) 23.3306 1.25245 0.626225 0.779642i \(-0.284599\pi\)
0.626225 + 0.779642i \(0.284599\pi\)
\(348\) 47.4672 2.54451
\(349\) 5.59571 0.299531 0.149766 0.988722i \(-0.452148\pi\)
0.149766 + 0.988722i \(0.452148\pi\)
\(350\) 0 0
\(351\) 18.2816 0.975798
\(352\) 2.03176 0.108293
\(353\) 19.4280 1.03405 0.517023 0.855971i \(-0.327040\pi\)
0.517023 + 0.855971i \(0.327040\pi\)
\(354\) 8.78150 0.466732
\(355\) −27.5114 −1.46016
\(356\) −6.80872 −0.360862
\(357\) 0 0
\(358\) 42.6221 2.25265
\(359\) 14.3127 0.755398 0.377699 0.925929i \(-0.376715\pi\)
0.377699 + 0.925929i \(0.376715\pi\)
\(360\) 9.94990 0.524406
\(361\) −0.329226 −0.0173277
\(362\) 24.9048 1.30897
\(363\) 26.7760 1.40537
\(364\) 0 0
\(365\) −5.50827 −0.288316
\(366\) 18.9169 0.988804
\(367\) 21.2458 1.10902 0.554512 0.832176i \(-0.312905\pi\)
0.554512 + 0.832176i \(0.312905\pi\)
\(368\) 29.5929 1.54264
\(369\) −0.876597 −0.0456338
\(370\) 0.261249 0.0135817
\(371\) 0 0
\(372\) −54.0207 −2.80084
\(373\) 28.2272 1.46155 0.730774 0.682620i \(-0.239160\pi\)
0.730774 + 0.682620i \(0.239160\pi\)
\(374\) −37.7165 −1.95027
\(375\) −16.0583 −0.829249
\(376\) 26.3604 1.35943
\(377\) −26.1084 −1.34465
\(378\) 0 0
\(379\) −8.57635 −0.440538 −0.220269 0.975439i \(-0.570693\pi\)
−0.220269 + 0.975439i \(0.570693\pi\)
\(380\) 39.5500 2.02887
\(381\) 27.8467 1.42663
\(382\) −44.7109 −2.28761
\(383\) −2.63393 −0.134588 −0.0672938 0.997733i \(-0.521436\pi\)
−0.0672938 + 0.997733i \(0.521436\pi\)
\(384\) 28.0727 1.43258
\(385\) 0 0
\(386\) −3.19311 −0.162525
\(387\) 5.53379 0.281298
\(388\) −76.0227 −3.85947
\(389\) −13.2656 −0.672595 −0.336298 0.941756i \(-0.609175\pi\)
−0.336298 + 0.941756i \(0.609175\pi\)
\(390\) 26.2678 1.33012
\(391\) −19.8218 −1.00243
\(392\) 0 0
\(393\) −14.5637 −0.734644
\(394\) −13.8752 −0.699025
\(395\) 11.1729 0.562169
\(396\) 19.1837 0.964017
\(397\) −2.28602 −0.114732 −0.0573661 0.998353i \(-0.518270\pi\)
−0.0573661 + 0.998353i \(0.518270\pi\)
\(398\) −28.2745 −1.41727
\(399\) 0 0
\(400\) 0.585970 0.0292985
\(401\) 2.79191 0.139421 0.0697107 0.997567i \(-0.477792\pi\)
0.0697107 + 0.997567i \(0.477792\pi\)
\(402\) 31.4527 1.56872
\(403\) 29.7130 1.48011
\(404\) −30.0776 −1.49642
\(405\) −12.6984 −0.630988
\(406\) 0 0
\(407\) 0.254204 0.0126004
\(408\) 20.6641 1.02302
\(409\) 10.8769 0.537826 0.268913 0.963164i \(-0.413336\pi\)
0.268913 + 0.963164i \(0.413336\pi\)
\(410\) −5.57007 −0.275086
\(411\) 6.79611 0.335227
\(412\) −14.8862 −0.733390
\(413\) 0 0
\(414\) 15.0757 0.740932
\(415\) 3.70840 0.182038
\(416\) 1.21319 0.0594816
\(417\) 15.1480 0.741803
\(418\) 57.5452 2.81463
\(419\) 24.9047 1.21668 0.608338 0.793678i \(-0.291836\pi\)
0.608338 + 0.793678i \(0.291836\pi\)
\(420\) 0 0
\(421\) −7.16405 −0.349155 −0.174577 0.984643i \(-0.555856\pi\)
−0.174577 + 0.984643i \(0.555856\pi\)
\(422\) 24.3900 1.18729
\(423\) 4.61483 0.224380
\(424\) −32.2276 −1.56511
\(425\) −0.392491 −0.0190386
\(426\) 43.4557 2.10543
\(427\) 0 0
\(428\) −34.6757 −1.67611
\(429\) 25.5595 1.23402
\(430\) 35.1628 1.69570
\(431\) −14.6538 −0.705850 −0.352925 0.935652i \(-0.614813\pi\)
−0.352925 + 0.935652i \(0.614813\pi\)
\(432\) −23.8844 −1.14914
\(433\) 29.4173 1.41370 0.706851 0.707362i \(-0.250115\pi\)
0.706851 + 0.707362i \(0.250115\pi\)
\(434\) 0 0
\(435\) 26.6485 1.27770
\(436\) −11.7868 −0.564483
\(437\) 30.2427 1.44671
\(438\) 8.70059 0.415730
\(439\) −26.5607 −1.26767 −0.633837 0.773466i \(-0.718521\pi\)
−0.633837 + 0.773466i \(0.718521\pi\)
\(440\) 61.5188 2.93280
\(441\) 0 0
\(442\) −22.5210 −1.07122
\(443\) −12.2373 −0.581409 −0.290705 0.956813i \(-0.593890\pi\)
−0.290705 + 0.956813i \(0.593890\pi\)
\(444\) −0.275964 −0.0130967
\(445\) −3.82247 −0.181203
\(446\) −29.0518 −1.37564
\(447\) 9.07902 0.429423
\(448\) 0 0
\(449\) −23.4334 −1.10589 −0.552944 0.833218i \(-0.686496\pi\)
−0.552944 + 0.833218i \(0.686496\pi\)
\(450\) 0.298515 0.0140721
\(451\) −5.41988 −0.255212
\(452\) −28.4256 −1.33703
\(453\) −30.8266 −1.44836
\(454\) 3.93826 0.184832
\(455\) 0 0
\(456\) −31.5278 −1.47643
\(457\) −8.25626 −0.386212 −0.193106 0.981178i \(-0.561856\pi\)
−0.193106 + 0.981178i \(0.561856\pi\)
\(458\) −24.4085 −1.14053
\(459\) 15.9981 0.746729
\(460\) 64.0627 2.98694
\(461\) 8.79238 0.409502 0.204751 0.978814i \(-0.434362\pi\)
0.204751 + 0.978814i \(0.434362\pi\)
\(462\) 0 0
\(463\) 36.8884 1.71435 0.857175 0.515025i \(-0.172217\pi\)
0.857175 + 0.515025i \(0.172217\pi\)
\(464\) 34.1100 1.58352
\(465\) −30.3277 −1.40641
\(466\) 19.4425 0.900655
\(467\) 21.8803 1.01250 0.506250 0.862387i \(-0.331031\pi\)
0.506250 + 0.862387i \(0.331031\pi\)
\(468\) 11.4549 0.529501
\(469\) 0 0
\(470\) 29.3235 1.35259
\(471\) −29.8596 −1.37586
\(472\) 12.2803 0.565249
\(473\) 34.2147 1.57319
\(474\) −17.6481 −0.810605
\(475\) 0.598837 0.0274765
\(476\) 0 0
\(477\) −5.64197 −0.258328
\(478\) −66.8093 −3.05579
\(479\) −34.3678 −1.57030 −0.785152 0.619303i \(-0.787415\pi\)
−0.785152 + 0.619303i \(0.787415\pi\)
\(480\) −1.23829 −0.0565198
\(481\) 0.151789 0.00692098
\(482\) 47.6729 2.17144
\(483\) 0 0
\(484\) 74.1945 3.37248
\(485\) −42.6798 −1.93799
\(486\) −21.5838 −0.979062
\(487\) 15.9585 0.723146 0.361573 0.932344i \(-0.382240\pi\)
0.361573 + 0.932344i \(0.382240\pi\)
\(488\) 26.4541 1.19752
\(489\) −2.38246 −0.107738
\(490\) 0 0
\(491\) −1.69863 −0.0766581 −0.0383290 0.999265i \(-0.512204\pi\)
−0.0383290 + 0.999265i \(0.512204\pi\)
\(492\) 5.88382 0.265263
\(493\) −22.8474 −1.02900
\(494\) 34.3611 1.54598
\(495\) 10.7699 0.484071
\(496\) −38.8194 −1.74304
\(497\) 0 0
\(498\) −5.85760 −0.262485
\(499\) −23.0520 −1.03195 −0.515975 0.856604i \(-0.672570\pi\)
−0.515975 + 0.856604i \(0.672570\pi\)
\(500\) −44.4967 −1.98995
\(501\) 28.8878 1.29061
\(502\) 67.4982 3.01259
\(503\) −34.6937 −1.54692 −0.773458 0.633848i \(-0.781474\pi\)
−0.773458 + 0.633848i \(0.781474\pi\)
\(504\) 0 0
\(505\) −16.8858 −0.751409
\(506\) 93.2112 4.14374
\(507\) −3.68155 −0.163503
\(508\) 77.1615 3.42349
\(509\) 7.41125 0.328498 0.164249 0.986419i \(-0.447480\pi\)
0.164249 + 0.986419i \(0.447480\pi\)
\(510\) 22.9869 1.01788
\(511\) 0 0
\(512\) 40.7573 1.80124
\(513\) −24.4089 −1.07768
\(514\) 53.1560 2.34461
\(515\) −8.35723 −0.368264
\(516\) −37.1434 −1.63515
\(517\) 28.5328 1.25487
\(518\) 0 0
\(519\) 13.5722 0.595753
\(520\) 36.7337 1.61088
\(521\) 24.2278 1.06144 0.530719 0.847548i \(-0.321922\pi\)
0.530719 + 0.847548i \(0.321922\pi\)
\(522\) 17.3769 0.760567
\(523\) 43.3265 1.89454 0.947268 0.320442i \(-0.103831\pi\)
0.947268 + 0.320442i \(0.103831\pi\)
\(524\) −40.3552 −1.76293
\(525\) 0 0
\(526\) 24.1906 1.05476
\(527\) 26.0018 1.13266
\(528\) −33.3929 −1.45324
\(529\) 25.9869 1.12987
\(530\) −35.8502 −1.55723
\(531\) 2.14988 0.0932968
\(532\) 0 0
\(533\) −3.23628 −0.140179
\(534\) 6.03778 0.261280
\(535\) −19.4672 −0.841643
\(536\) 43.9846 1.89984
\(537\) −25.2762 −1.09075
\(538\) −64.0410 −2.76100
\(539\) 0 0
\(540\) −51.7049 −2.22503
\(541\) −12.7180 −0.546788 −0.273394 0.961902i \(-0.588146\pi\)
−0.273394 + 0.961902i \(0.588146\pi\)
\(542\) 62.5370 2.68620
\(543\) −14.7693 −0.633812
\(544\) 1.06166 0.0455183
\(545\) −6.61718 −0.283449
\(546\) 0 0
\(547\) −4.71489 −0.201594 −0.100797 0.994907i \(-0.532139\pi\)
−0.100797 + 0.994907i \(0.532139\pi\)
\(548\) 18.8316 0.804446
\(549\) 4.63122 0.197656
\(550\) 1.84568 0.0786999
\(551\) 34.8590 1.48504
\(552\) −51.0685 −2.17362
\(553\) 0 0
\(554\) −72.1623 −3.06588
\(555\) −0.154929 −0.00657636
\(556\) 41.9743 1.78011
\(557\) −11.0756 −0.469288 −0.234644 0.972081i \(-0.575392\pi\)
−0.234644 + 0.972081i \(0.575392\pi\)
\(558\) −19.7760 −0.837187
\(559\) 20.4301 0.864099
\(560\) 0 0
\(561\) 22.3670 0.944337
\(562\) −59.4935 −2.50958
\(563\) −24.9735 −1.05251 −0.526255 0.850327i \(-0.676404\pi\)
−0.526255 + 0.850327i \(0.676404\pi\)
\(564\) −30.9752 −1.30429
\(565\) −15.9584 −0.671374
\(566\) 21.8997 0.920515
\(567\) 0 0
\(568\) 60.7698 2.54985
\(569\) 3.14116 0.131684 0.0658422 0.997830i \(-0.479027\pi\)
0.0658422 + 0.997830i \(0.479027\pi\)
\(570\) −35.0718 −1.46900
\(571\) 7.13418 0.298556 0.149278 0.988795i \(-0.452305\pi\)
0.149278 + 0.988795i \(0.452305\pi\)
\(572\) 70.8238 2.96129
\(573\) 26.5150 1.10768
\(574\) 0 0
\(575\) 0.969990 0.0404514
\(576\) 6.60527 0.275219
\(577\) −4.41995 −0.184005 −0.0920024 0.995759i \(-0.529327\pi\)
−0.0920024 + 0.995759i \(0.529327\pi\)
\(578\) 22.0642 0.917748
\(579\) 1.89361 0.0786959
\(580\) 73.8413 3.06609
\(581\) 0 0
\(582\) 67.4148 2.79443
\(583\) −34.8835 −1.44473
\(584\) 12.1672 0.503482
\(585\) 6.43085 0.265883
\(586\) −21.6046 −0.892476
\(587\) −29.7123 −1.22636 −0.613178 0.789944i \(-0.710109\pi\)
−0.613178 + 0.789944i \(0.710109\pi\)
\(588\) 0 0
\(589\) −39.6718 −1.63465
\(590\) 13.6608 0.562404
\(591\) 8.22845 0.338473
\(592\) −0.198309 −0.00815044
\(593\) 14.3020 0.587315 0.293657 0.955911i \(-0.405128\pi\)
0.293657 + 0.955911i \(0.405128\pi\)
\(594\) −75.2307 −3.08675
\(595\) 0 0
\(596\) 25.1574 1.03049
\(597\) 16.7677 0.686255
\(598\) 55.6577 2.27601
\(599\) 38.6159 1.57780 0.788902 0.614518i \(-0.210650\pi\)
0.788902 + 0.614518i \(0.210650\pi\)
\(600\) −1.01121 −0.0412824
\(601\) −2.04839 −0.0835557 −0.0417779 0.999127i \(-0.513302\pi\)
−0.0417779 + 0.999127i \(0.513302\pi\)
\(602\) 0 0
\(603\) 7.70022 0.313577
\(604\) −85.4186 −3.47563
\(605\) 41.6534 1.69345
\(606\) 26.6720 1.08347
\(607\) −5.79770 −0.235321 −0.117661 0.993054i \(-0.537540\pi\)
−0.117661 + 0.993054i \(0.537540\pi\)
\(608\) −1.61981 −0.0656920
\(609\) 0 0
\(610\) 29.4277 1.19149
\(611\) 17.0373 0.689257
\(612\) 10.0241 0.405201
\(613\) 45.3061 1.82989 0.914947 0.403573i \(-0.132232\pi\)
0.914947 + 0.403573i \(0.132232\pi\)
\(614\) 62.5378 2.52382
\(615\) 3.30322 0.133199
\(616\) 0 0
\(617\) −22.8349 −0.919300 −0.459650 0.888100i \(-0.652025\pi\)
−0.459650 + 0.888100i \(0.652025\pi\)
\(618\) 13.2007 0.531008
\(619\) −15.2514 −0.613007 −0.306503 0.951870i \(-0.599159\pi\)
−0.306503 + 0.951870i \(0.599159\pi\)
\(620\) −84.0360 −3.37497
\(621\) −39.5373 −1.58658
\(622\) −20.1680 −0.808662
\(623\) 0 0
\(624\) −19.9393 −0.798213
\(625\) −25.6737 −1.02695
\(626\) 19.2650 0.769983
\(627\) −34.1261 −1.36287
\(628\) −82.7390 −3.30165
\(629\) 0.132830 0.00529628
\(630\) 0 0
\(631\) −40.2426 −1.60203 −0.801016 0.598643i \(-0.795707\pi\)
−0.801016 + 0.598643i \(0.795707\pi\)
\(632\) −24.6797 −0.981706
\(633\) −14.4640 −0.574894
\(634\) −79.7012 −3.16534
\(635\) 43.3191 1.71907
\(636\) 37.8695 1.50162
\(637\) 0 0
\(638\) 107.439 4.25355
\(639\) 10.6388 0.420863
\(640\) 43.6707 1.72624
\(641\) 8.41075 0.332205 0.166102 0.986109i \(-0.446882\pi\)
0.166102 + 0.986109i \(0.446882\pi\)
\(642\) 30.7495 1.21358
\(643\) 23.9146 0.943101 0.471550 0.881839i \(-0.343695\pi\)
0.471550 + 0.881839i \(0.343695\pi\)
\(644\) 0 0
\(645\) −20.8526 −0.821072
\(646\) 30.0693 1.18306
\(647\) −10.9669 −0.431152 −0.215576 0.976487i \(-0.569163\pi\)
−0.215576 + 0.976487i \(0.569163\pi\)
\(648\) 28.0494 1.10188
\(649\) 13.2924 0.521772
\(650\) 1.10208 0.0432271
\(651\) 0 0
\(652\) −6.60164 −0.258540
\(653\) 29.3523 1.14865 0.574323 0.818629i \(-0.305265\pi\)
0.574323 + 0.818629i \(0.305265\pi\)
\(654\) 10.4522 0.408712
\(655\) −22.6558 −0.885234
\(656\) 4.22813 0.165081
\(657\) 2.13007 0.0831019
\(658\) 0 0
\(659\) −12.7905 −0.498245 −0.249123 0.968472i \(-0.580142\pi\)
−0.249123 + 0.968472i \(0.580142\pi\)
\(660\) −72.2887 −2.81383
\(661\) −15.1159 −0.587942 −0.293971 0.955814i \(-0.594977\pi\)
−0.293971 + 0.955814i \(0.594977\pi\)
\(662\) −34.0509 −1.32342
\(663\) 13.3557 0.518691
\(664\) −8.19146 −0.317890
\(665\) 0 0
\(666\) −0.101026 −0.00391467
\(667\) 56.4643 2.18631
\(668\) 80.0463 3.09708
\(669\) 17.2286 0.666096
\(670\) 48.9288 1.89028
\(671\) 28.6342 1.10541
\(672\) 0 0
\(673\) 10.3101 0.397425 0.198713 0.980058i \(-0.436324\pi\)
0.198713 + 0.980058i \(0.436324\pi\)
\(674\) −45.7235 −1.76120
\(675\) −0.782878 −0.0301330
\(676\) −10.2013 −0.392359
\(677\) 1.97105 0.0757535 0.0378768 0.999282i \(-0.487941\pi\)
0.0378768 + 0.999282i \(0.487941\pi\)
\(678\) 25.2070 0.968070
\(679\) 0 0
\(680\) 32.1456 1.23273
\(681\) −2.33551 −0.0894970
\(682\) −122.273 −4.68206
\(683\) −9.53970 −0.365026 −0.182513 0.983203i \(-0.558423\pi\)
−0.182513 + 0.983203i \(0.558423\pi\)
\(684\) −15.2941 −0.584785
\(685\) 10.5722 0.403944
\(686\) 0 0
\(687\) 14.4750 0.552255
\(688\) −26.6914 −1.01760
\(689\) −20.8294 −0.793538
\(690\) −56.8090 −2.16268
\(691\) −16.3915 −0.623562 −0.311781 0.950154i \(-0.600925\pi\)
−0.311781 + 0.950154i \(0.600925\pi\)
\(692\) 37.6077 1.42963
\(693\) 0 0
\(694\) −57.3276 −2.17613
\(695\) 23.5647 0.893861
\(696\) −58.8637 −2.23122
\(697\) −2.83206 −0.107272
\(698\) −13.7497 −0.520434
\(699\) −11.5300 −0.436104
\(700\) 0 0
\(701\) 12.1383 0.458457 0.229229 0.973373i \(-0.426380\pi\)
0.229229 + 0.973373i \(0.426380\pi\)
\(702\) −44.9213 −1.69544
\(703\) −0.202663 −0.00764359
\(704\) 40.8394 1.53919
\(705\) −17.3898 −0.654936
\(706\) −47.7382 −1.79665
\(707\) 0 0
\(708\) −14.4302 −0.542321
\(709\) −29.6150 −1.11222 −0.556108 0.831110i \(-0.687706\pi\)
−0.556108 + 0.831110i \(0.687706\pi\)
\(710\) 67.6008 2.53701
\(711\) −4.32059 −0.162035
\(712\) 8.44344 0.316431
\(713\) −64.2600 −2.40656
\(714\) 0 0
\(715\) 39.7611 1.48698
\(716\) −70.0388 −2.61747
\(717\) 39.6200 1.47964
\(718\) −35.1691 −1.31250
\(719\) −42.0273 −1.56735 −0.783676 0.621169i \(-0.786658\pi\)
−0.783676 + 0.621169i \(0.786658\pi\)
\(720\) −8.40175 −0.313115
\(721\) 0 0
\(722\) 0.808971 0.0301068
\(723\) −28.2715 −1.05143
\(724\) −40.9249 −1.52096
\(725\) 1.11805 0.0415234
\(726\) −65.7936 −2.44183
\(727\) 41.0718 1.52327 0.761635 0.648007i \(-0.224397\pi\)
0.761635 + 0.648007i \(0.224397\pi\)
\(728\) 0 0
\(729\) 29.6052 1.09649
\(730\) 13.5349 0.500948
\(731\) 17.8783 0.661252
\(732\) −31.0853 −1.14895
\(733\) −9.51418 −0.351414 −0.175707 0.984442i \(-0.556221\pi\)
−0.175707 + 0.984442i \(0.556221\pi\)
\(734\) −52.2051 −1.92693
\(735\) 0 0
\(736\) −2.62375 −0.0967128
\(737\) 47.6094 1.75372
\(738\) 2.15397 0.0792886
\(739\) −45.4606 −1.67230 −0.836148 0.548504i \(-0.815197\pi\)
−0.836148 + 0.548504i \(0.815197\pi\)
\(740\) −0.429298 −0.0157813
\(741\) −20.3772 −0.748574
\(742\) 0 0
\(743\) −51.1920 −1.87805 −0.939027 0.343845i \(-0.888271\pi\)
−0.939027 + 0.343845i \(0.888271\pi\)
\(744\) 66.9906 2.45599
\(745\) 14.1236 0.517448
\(746\) −69.3595 −2.53943
\(747\) −1.43405 −0.0524692
\(748\) 61.9777 2.26613
\(749\) 0 0
\(750\) 39.4584 1.44082
\(751\) −32.2359 −1.17631 −0.588153 0.808750i \(-0.700145\pi\)
−0.588153 + 0.808750i \(0.700145\pi\)
\(752\) −22.2589 −0.811698
\(753\) −40.0285 −1.45872
\(754\) 64.1534 2.33633
\(755\) −47.9547 −1.74525
\(756\) 0 0
\(757\) −26.3465 −0.957581 −0.478790 0.877929i \(-0.658925\pi\)
−0.478790 + 0.877929i \(0.658925\pi\)
\(758\) 21.0737 0.765432
\(759\) −55.2772 −2.00643
\(760\) −49.0456 −1.77907
\(761\) 30.0811 1.09044 0.545219 0.838293i \(-0.316446\pi\)
0.545219 + 0.838293i \(0.316446\pi\)
\(762\) −68.4246 −2.47876
\(763\) 0 0
\(764\) 73.4713 2.65810
\(765\) 5.62762 0.203467
\(766\) 6.47207 0.233845
\(767\) 7.93707 0.286591
\(768\) −47.0198 −1.69668
\(769\) 0.608700 0.0219503 0.0109751 0.999940i \(-0.496506\pi\)
0.0109751 + 0.999940i \(0.496506\pi\)
\(770\) 0 0
\(771\) −31.5232 −1.13528
\(772\) 5.24708 0.188847
\(773\) 8.67944 0.312178 0.156089 0.987743i \(-0.450111\pi\)
0.156089 + 0.987743i \(0.450111\pi\)
\(774\) −13.5976 −0.488755
\(775\) −1.27241 −0.0457064
\(776\) 94.2751 3.38428
\(777\) 0 0
\(778\) 32.5962 1.16863
\(779\) 4.32097 0.154815
\(780\) −43.1646 −1.54554
\(781\) 65.7780 2.35372
\(782\) 48.7059 1.74172
\(783\) −45.5723 −1.62862
\(784\) 0 0
\(785\) −46.4504 −1.65789
\(786\) 35.7859 1.27644
\(787\) −34.2640 −1.22138 −0.610689 0.791870i \(-0.709108\pi\)
−0.610689 + 0.791870i \(0.709108\pi\)
\(788\) 22.8005 0.812235
\(789\) −14.3458 −0.510723
\(790\) −27.4539 −0.976766
\(791\) 0 0
\(792\) −23.7895 −0.845325
\(793\) 17.0979 0.607164
\(794\) 5.61719 0.199347
\(795\) 21.2603 0.754024
\(796\) 46.4622 1.64681
\(797\) −13.1568 −0.466036 −0.233018 0.972472i \(-0.574860\pi\)
−0.233018 + 0.972472i \(0.574860\pi\)
\(798\) 0 0
\(799\) 14.9093 0.527454
\(800\) −0.0519530 −0.00183681
\(801\) 1.47816 0.0522283
\(802\) −6.86026 −0.242244
\(803\) 13.1699 0.464756
\(804\) −51.6848 −1.82278
\(805\) 0 0
\(806\) −73.0106 −2.57169
\(807\) 37.9783 1.33690
\(808\) 37.2990 1.31217
\(809\) −2.04798 −0.0720033 −0.0360017 0.999352i \(-0.511462\pi\)
−0.0360017 + 0.999352i \(0.511462\pi\)
\(810\) 31.2023 1.09634
\(811\) −5.48050 −0.192446 −0.0962232 0.995360i \(-0.530676\pi\)
−0.0962232 + 0.995360i \(0.530676\pi\)
\(812\) 0 0
\(813\) −37.0864 −1.30068
\(814\) −0.624629 −0.0218932
\(815\) −3.70622 −0.129823
\(816\) −17.4489 −0.610832
\(817\) −27.2775 −0.954318
\(818\) −26.7265 −0.934471
\(819\) 0 0
\(820\) 9.15303 0.319638
\(821\) −1.55848 −0.0543913 −0.0271957 0.999630i \(-0.508658\pi\)
−0.0271957 + 0.999630i \(0.508658\pi\)
\(822\) −16.6993 −0.582456
\(823\) 39.3703 1.37236 0.686182 0.727430i \(-0.259286\pi\)
0.686182 + 0.727430i \(0.259286\pi\)
\(824\) 18.4602 0.643093
\(825\) −1.09454 −0.0381071
\(826\) 0 0
\(827\) −26.3888 −0.917628 −0.458814 0.888532i \(-0.651726\pi\)
−0.458814 + 0.888532i \(0.651726\pi\)
\(828\) −24.7733 −0.860930
\(829\) −45.0610 −1.56503 −0.782517 0.622629i \(-0.786065\pi\)
−0.782517 + 0.622629i \(0.786065\pi\)
\(830\) −9.11224 −0.316291
\(831\) 42.7945 1.48452
\(832\) 24.3858 0.845425
\(833\) 0 0
\(834\) −37.2216 −1.28888
\(835\) 44.9387 1.55517
\(836\) −94.5613 −3.27047
\(837\) 51.8642 1.79269
\(838\) −61.1957 −2.11397
\(839\) 42.0761 1.45263 0.726314 0.687363i \(-0.241232\pi\)
0.726314 + 0.687363i \(0.241232\pi\)
\(840\) 0 0
\(841\) 36.0831 1.24425
\(842\) 17.6034 0.606655
\(843\) 35.2815 1.21516
\(844\) −40.0790 −1.37958
\(845\) −5.72712 −0.197019
\(846\) −11.3395 −0.389860
\(847\) 0 0
\(848\) 27.2131 0.934503
\(849\) −12.9872 −0.445720
\(850\) 0.964426 0.0330795
\(851\) −0.328272 −0.0112530
\(852\) −71.4086 −2.44642
\(853\) 20.2803 0.694384 0.347192 0.937794i \(-0.387135\pi\)
0.347192 + 0.937794i \(0.387135\pi\)
\(854\) 0 0
\(855\) −8.58624 −0.293643
\(856\) 43.0011 1.46975
\(857\) 49.2604 1.68270 0.841351 0.540490i \(-0.181761\pi\)
0.841351 + 0.540490i \(0.181761\pi\)
\(858\) −62.8045 −2.14411
\(859\) 12.7326 0.434430 0.217215 0.976124i \(-0.430303\pi\)
0.217215 + 0.976124i \(0.430303\pi\)
\(860\) −57.7814 −1.97033
\(861\) 0 0
\(862\) 36.0072 1.22641
\(863\) −14.7535 −0.502216 −0.251108 0.967959i \(-0.580795\pi\)
−0.251108 + 0.967959i \(0.580795\pi\)
\(864\) 2.11763 0.0720432
\(865\) 21.1133 0.717872
\(866\) −72.2838 −2.45630
\(867\) −13.0847 −0.444381
\(868\) 0 0
\(869\) −26.7136 −0.906197
\(870\) −65.4804 −2.21999
\(871\) 28.4282 0.963254
\(872\) 14.6167 0.494982
\(873\) 16.5044 0.558590
\(874\) −74.3122 −2.51365
\(875\) 0 0
\(876\) −14.2973 −0.483060
\(877\) −20.7432 −0.700450 −0.350225 0.936666i \(-0.613895\pi\)
−0.350225 + 0.936666i \(0.613895\pi\)
\(878\) 65.2648 2.20258
\(879\) 12.8122 0.432144
\(880\) −51.9469 −1.75113
\(881\) 19.5041 0.657110 0.328555 0.944485i \(-0.393438\pi\)
0.328555 + 0.944485i \(0.393438\pi\)
\(882\) 0 0
\(883\) 27.7626 0.934286 0.467143 0.884182i \(-0.345283\pi\)
0.467143 + 0.884182i \(0.345283\pi\)
\(884\) 37.0077 1.24470
\(885\) −8.10125 −0.272321
\(886\) 30.0693 1.01020
\(887\) −35.3228 −1.18602 −0.593012 0.805194i \(-0.702061\pi\)
−0.593012 + 0.805194i \(0.702061\pi\)
\(888\) 0.342221 0.0114842
\(889\) 0 0
\(890\) 9.39254 0.314839
\(891\) 30.3610 1.01713
\(892\) 47.7394 1.59843
\(893\) −22.7476 −0.761221
\(894\) −22.3089 −0.746120
\(895\) −39.3204 −1.31434
\(896\) 0 0
\(897\) −33.0068 −1.10206
\(898\) 57.5802 1.92148
\(899\) −74.0687 −2.47033
\(900\) −0.490535 −0.0163512
\(901\) −18.2278 −0.607255
\(902\) 13.3177 0.443430
\(903\) 0 0
\(904\) 35.2503 1.17241
\(905\) −22.9756 −0.763734
\(906\) 75.7469 2.51652
\(907\) 10.5803 0.351312 0.175656 0.984452i \(-0.443795\pi\)
0.175656 + 0.984452i \(0.443795\pi\)
\(908\) −6.47156 −0.214766
\(909\) 6.52980 0.216580
\(910\) 0 0
\(911\) −18.0853 −0.599192 −0.299596 0.954066i \(-0.596852\pi\)
−0.299596 + 0.954066i \(0.596852\pi\)
\(912\) 26.6223 0.881552
\(913\) −8.86654 −0.293440
\(914\) 20.2872 0.671041
\(915\) −17.4516 −0.576930
\(916\) 40.1093 1.32525
\(917\) 0 0
\(918\) −39.3105 −1.29744
\(919\) 9.90775 0.326826 0.163413 0.986558i \(-0.447750\pi\)
0.163413 + 0.986558i \(0.447750\pi\)
\(920\) −79.4436 −2.61918
\(921\) −37.0869 −1.22205
\(922\) −21.6046 −0.711508
\(923\) 39.2770 1.29282
\(924\) 0 0
\(925\) −0.00650012 −0.000213722 0
\(926\) −90.6419 −2.97868
\(927\) 3.23177 0.106145
\(928\) −3.02425 −0.0992757
\(929\) 15.2665 0.500878 0.250439 0.968132i \(-0.419425\pi\)
0.250439 + 0.968132i \(0.419425\pi\)
\(930\) 74.5208 2.44363
\(931\) 0 0
\(932\) −31.9489 −1.04652
\(933\) 11.9602 0.391561
\(934\) −53.7641 −1.75921
\(935\) 34.7948 1.13791
\(936\) −14.2051 −0.464307
\(937\) −21.2126 −0.692985 −0.346492 0.938053i \(-0.612627\pi\)
−0.346492 + 0.938053i \(0.612627\pi\)
\(938\) 0 0
\(939\) −11.4247 −0.372832
\(940\) −48.1859 −1.57165
\(941\) 34.1282 1.11255 0.556274 0.830999i \(-0.312230\pi\)
0.556274 + 0.830999i \(0.312230\pi\)
\(942\) 73.3707 2.39055
\(943\) 6.99907 0.227921
\(944\) −10.3696 −0.337502
\(945\) 0 0
\(946\) −84.0720 −2.73342
\(947\) −18.3584 −0.596567 −0.298284 0.954477i \(-0.596414\pi\)
−0.298284 + 0.954477i \(0.596414\pi\)
\(948\) 29.0003 0.941886
\(949\) 7.86394 0.255274
\(950\) −1.47146 −0.0477403
\(951\) 47.2653 1.53268
\(952\) 0 0
\(953\) −16.0790 −0.520849 −0.260424 0.965494i \(-0.583862\pi\)
−0.260424 + 0.965494i \(0.583862\pi\)
\(954\) 13.8634 0.448844
\(955\) 41.2474 1.33473
\(956\) 109.785 3.55069
\(957\) −63.7147 −2.05960
\(958\) 84.4482 2.72840
\(959\) 0 0
\(960\) −24.8902 −0.803328
\(961\) 53.2949 1.71919
\(962\) −0.372974 −0.0120252
\(963\) 7.52805 0.242588
\(964\) −78.3386 −2.52312
\(965\) 2.94576 0.0948273
\(966\) 0 0
\(967\) 38.0340 1.22309 0.611546 0.791209i \(-0.290548\pi\)
0.611546 + 0.791209i \(0.290548\pi\)
\(968\) −92.0080 −2.95725
\(969\) −17.8320 −0.572847
\(970\) 104.872 3.36725
\(971\) −60.7488 −1.94952 −0.974761 0.223250i \(-0.928333\pi\)
−0.974761 + 0.223250i \(0.928333\pi\)
\(972\) 35.4677 1.13763
\(973\) 0 0
\(974\) −39.2129 −1.25646
\(975\) −0.653567 −0.0209309
\(976\) −22.3380 −0.715021
\(977\) −4.23580 −0.135515 −0.0677576 0.997702i \(-0.521584\pi\)
−0.0677576 + 0.997702i \(0.521584\pi\)
\(978\) 5.85415 0.187195
\(979\) 9.13928 0.292093
\(980\) 0 0
\(981\) 2.55889 0.0816990
\(982\) 4.17386 0.133193
\(983\) 7.21490 0.230120 0.115060 0.993359i \(-0.463294\pi\)
0.115060 + 0.993359i \(0.463294\pi\)
\(984\) −7.29647 −0.232603
\(985\) 12.8004 0.407855
\(986\) 56.1404 1.78788
\(987\) 0 0
\(988\) −56.4639 −1.79636
\(989\) −44.1838 −1.40496
\(990\) −26.4637 −0.841071
\(991\) −20.9196 −0.664533 −0.332267 0.943186i \(-0.607813\pi\)
−0.332267 + 0.943186i \(0.607813\pi\)
\(992\) 3.44178 0.109277
\(993\) 20.1932 0.640813
\(994\) 0 0
\(995\) 26.0843 0.826927
\(996\) 9.62551 0.304996
\(997\) 18.4455 0.584175 0.292088 0.956392i \(-0.405650\pi\)
0.292088 + 0.956392i \(0.405650\pi\)
\(998\) 56.6432 1.79301
\(999\) 0.264948 0.00838258
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 2009.2.a.n.1.1 5
7.6 odd 2 287.2.a.e.1.1 5
21.20 even 2 2583.2.a.r.1.5 5
28.27 even 2 4592.2.a.bb.1.5 5
35.34 odd 2 7175.2.a.n.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
287.2.a.e.1.1 5 7.6 odd 2
2009.2.a.n.1.1 5 1.1 even 1 trivial
2583.2.a.r.1.5 5 21.20 even 2
4592.2.a.bb.1.5 5 28.27 even 2
7175.2.a.n.1.5 5 35.34 odd 2