Properties

Label 2151.2.a.k.1.18
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Root \(2.41299\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41299 q^{2} +3.82251 q^{4} -0.666388 q^{5} +0.549662 q^{7} +4.39769 q^{8} +O(q^{10})\) \(q+2.41299 q^{2} +3.82251 q^{4} -0.666388 q^{5} +0.549662 q^{7} +4.39769 q^{8} -1.60799 q^{10} +0.262501 q^{11} +3.05335 q^{13} +1.32633 q^{14} +2.96655 q^{16} +2.00829 q^{17} +1.22323 q^{19} -2.54727 q^{20} +0.633411 q^{22} +6.88399 q^{23} -4.55593 q^{25} +7.36769 q^{26} +2.10109 q^{28} +5.64739 q^{29} +2.28051 q^{31} -1.63713 q^{32} +4.84598 q^{34} -0.366288 q^{35} +8.46115 q^{37} +2.95163 q^{38} -2.93057 q^{40} +1.90913 q^{41} -7.39552 q^{43} +1.00341 q^{44} +16.6110 q^{46} +1.67407 q^{47} -6.69787 q^{49} -10.9934 q^{50} +11.6714 q^{52} +7.14980 q^{53} -0.174927 q^{55} +2.41724 q^{56} +13.6271 q^{58} +3.67831 q^{59} -13.7223 q^{61} +5.50285 q^{62} -9.88347 q^{64} -2.03472 q^{65} -2.80719 q^{67} +7.67670 q^{68} -0.883849 q^{70} -4.78655 q^{71} +2.16854 q^{73} +20.4166 q^{74} +4.67580 q^{76} +0.144287 q^{77} -5.81088 q^{79} -1.97687 q^{80} +4.60671 q^{82} -3.57961 q^{83} -1.33830 q^{85} -17.8453 q^{86} +1.15440 q^{88} -9.71189 q^{89} +1.67831 q^{91} +26.3141 q^{92} +4.03951 q^{94} -0.815145 q^{95} +7.62712 q^{97} -16.1619 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41299 1.70624 0.853120 0.521715i \(-0.174708\pi\)
0.853120 + 0.521715i \(0.174708\pi\)
\(3\) 0 0
\(4\) 3.82251 1.91125
\(5\) −0.666388 −0.298018 −0.149009 0.988836i \(-0.547608\pi\)
−0.149009 + 0.988836i \(0.547608\pi\)
\(6\) 0 0
\(7\) 0.549662 0.207753 0.103876 0.994590i \(-0.466875\pi\)
0.103876 + 0.994590i \(0.466875\pi\)
\(8\) 4.39769 1.55482
\(9\) 0 0
\(10\) −1.60799 −0.508490
\(11\) 0.262501 0.0791470 0.0395735 0.999217i \(-0.487400\pi\)
0.0395735 + 0.999217i \(0.487400\pi\)
\(12\) 0 0
\(13\) 3.05335 0.846846 0.423423 0.905932i \(-0.360828\pi\)
0.423423 + 0.905932i \(0.360828\pi\)
\(14\) 1.32633 0.354476
\(15\) 0 0
\(16\) 2.96655 0.741637
\(17\) 2.00829 0.487082 0.243541 0.969891i \(-0.421691\pi\)
0.243541 + 0.969891i \(0.421691\pi\)
\(18\) 0 0
\(19\) 1.22323 0.280628 0.140314 0.990107i \(-0.455189\pi\)
0.140314 + 0.990107i \(0.455189\pi\)
\(20\) −2.54727 −0.569588
\(21\) 0 0
\(22\) 0.633411 0.135044
\(23\) 6.88399 1.43541 0.717706 0.696347i \(-0.245192\pi\)
0.717706 + 0.696347i \(0.245192\pi\)
\(24\) 0 0
\(25\) −4.55593 −0.911185
\(26\) 7.36769 1.44492
\(27\) 0 0
\(28\) 2.10109 0.397068
\(29\) 5.64739 1.04869 0.524347 0.851505i \(-0.324310\pi\)
0.524347 + 0.851505i \(0.324310\pi\)
\(30\) 0 0
\(31\) 2.28051 0.409592 0.204796 0.978805i \(-0.434347\pi\)
0.204796 + 0.978805i \(0.434347\pi\)
\(32\) −1.63713 −0.289407
\(33\) 0 0
\(34\) 4.84598 0.831079
\(35\) −0.366288 −0.0619140
\(36\) 0 0
\(37\) 8.46115 1.39100 0.695502 0.718524i \(-0.255182\pi\)
0.695502 + 0.718524i \(0.255182\pi\)
\(38\) 2.95163 0.478818
\(39\) 0 0
\(40\) −2.93057 −0.463363
\(41\) 1.90913 0.298156 0.149078 0.988825i \(-0.452369\pi\)
0.149078 + 0.988825i \(0.452369\pi\)
\(42\) 0 0
\(43\) −7.39552 −1.12781 −0.563903 0.825841i \(-0.690701\pi\)
−0.563903 + 0.825841i \(0.690701\pi\)
\(44\) 1.00341 0.151270
\(45\) 0 0
\(46\) 16.6110 2.44916
\(47\) 1.67407 0.244188 0.122094 0.992519i \(-0.461039\pi\)
0.122094 + 0.992519i \(0.461039\pi\)
\(48\) 0 0
\(49\) −6.69787 −0.956839
\(50\) −10.9934 −1.55470
\(51\) 0 0
\(52\) 11.6714 1.61854
\(53\) 7.14980 0.982100 0.491050 0.871131i \(-0.336613\pi\)
0.491050 + 0.871131i \(0.336613\pi\)
\(54\) 0 0
\(55\) −0.174927 −0.0235872
\(56\) 2.41724 0.323017
\(57\) 0 0
\(58\) 13.6271 1.78932
\(59\) 3.67831 0.478875 0.239437 0.970912i \(-0.423037\pi\)
0.239437 + 0.970912i \(0.423037\pi\)
\(60\) 0 0
\(61\) −13.7223 −1.75697 −0.878483 0.477773i \(-0.841444\pi\)
−0.878483 + 0.477773i \(0.841444\pi\)
\(62\) 5.50285 0.698862
\(63\) 0 0
\(64\) −9.88347 −1.23543
\(65\) −2.03472 −0.252375
\(66\) 0 0
\(67\) −2.80719 −0.342953 −0.171477 0.985188i \(-0.554854\pi\)
−0.171477 + 0.985188i \(0.554854\pi\)
\(68\) 7.67670 0.930937
\(69\) 0 0
\(70\) −0.883849 −0.105640
\(71\) −4.78655 −0.568059 −0.284030 0.958816i \(-0.591671\pi\)
−0.284030 + 0.958816i \(0.591671\pi\)
\(72\) 0 0
\(73\) 2.16854 0.253808 0.126904 0.991915i \(-0.459496\pi\)
0.126904 + 0.991915i \(0.459496\pi\)
\(74\) 20.4166 2.37339
\(75\) 0 0
\(76\) 4.67580 0.536351
\(77\) 0.144287 0.0164430
\(78\) 0 0
\(79\) −5.81088 −0.653775 −0.326888 0.945063i \(-0.606000\pi\)
−0.326888 + 0.945063i \(0.606000\pi\)
\(80\) −1.97687 −0.221021
\(81\) 0 0
\(82\) 4.60671 0.508726
\(83\) −3.57961 −0.392914 −0.196457 0.980512i \(-0.562944\pi\)
−0.196457 + 0.980512i \(0.562944\pi\)
\(84\) 0 0
\(85\) −1.33830 −0.145159
\(86\) −17.8453 −1.92431
\(87\) 0 0
\(88\) 1.15440 0.123059
\(89\) −9.71189 −1.02946 −0.514729 0.857353i \(-0.672108\pi\)
−0.514729 + 0.857353i \(0.672108\pi\)
\(90\) 0 0
\(91\) 1.67831 0.175935
\(92\) 26.3141 2.74344
\(93\) 0 0
\(94\) 4.03951 0.416644
\(95\) −0.815145 −0.0836321
\(96\) 0 0
\(97\) 7.62712 0.774417 0.387209 0.921992i \(-0.373439\pi\)
0.387209 + 0.921992i \(0.373439\pi\)
\(98\) −16.1619 −1.63260
\(99\) 0 0
\(100\) −17.4151 −1.74151
\(101\) −1.34774 −0.134106 −0.0670528 0.997749i \(-0.521360\pi\)
−0.0670528 + 0.997749i \(0.521360\pi\)
\(102\) 0 0
\(103\) 18.5433 1.82712 0.913561 0.406702i \(-0.133321\pi\)
0.913561 + 0.406702i \(0.133321\pi\)
\(104\) 13.4277 1.31669
\(105\) 0 0
\(106\) 17.2524 1.67570
\(107\) −15.2283 −1.47217 −0.736086 0.676888i \(-0.763328\pi\)
−0.736086 + 0.676888i \(0.763328\pi\)
\(108\) 0 0
\(109\) −2.17137 −0.207979 −0.103990 0.994578i \(-0.533161\pi\)
−0.103990 + 0.994578i \(0.533161\pi\)
\(110\) −0.422098 −0.0402454
\(111\) 0 0
\(112\) 1.63060 0.154077
\(113\) −0.665124 −0.0625696 −0.0312848 0.999511i \(-0.509960\pi\)
−0.0312848 + 0.999511i \(0.509960\pi\)
\(114\) 0 0
\(115\) −4.58741 −0.427778
\(116\) 21.5872 2.00432
\(117\) 0 0
\(118\) 8.87571 0.817075
\(119\) 1.10388 0.101193
\(120\) 0 0
\(121\) −10.9311 −0.993736
\(122\) −33.1118 −2.99781
\(123\) 0 0
\(124\) 8.71727 0.782834
\(125\) 6.36796 0.569567
\(126\) 0 0
\(127\) −9.90489 −0.878917 −0.439459 0.898263i \(-0.644830\pi\)
−0.439459 + 0.898263i \(0.644830\pi\)
\(128\) −20.5744 −1.81854
\(129\) 0 0
\(130\) −4.90974 −0.430613
\(131\) 4.12041 0.360002 0.180001 0.983666i \(-0.442390\pi\)
0.180001 + 0.983666i \(0.442390\pi\)
\(132\) 0 0
\(133\) 0.672362 0.0583012
\(134\) −6.77372 −0.585160
\(135\) 0 0
\(136\) 8.83183 0.757323
\(137\) 11.0196 0.941472 0.470736 0.882274i \(-0.343988\pi\)
0.470736 + 0.882274i \(0.343988\pi\)
\(138\) 0 0
\(139\) −1.99415 −0.169142 −0.0845708 0.996417i \(-0.526952\pi\)
−0.0845708 + 0.996417i \(0.526952\pi\)
\(140\) −1.40014 −0.118333
\(141\) 0 0
\(142\) −11.5499 −0.969245
\(143\) 0.801506 0.0670253
\(144\) 0 0
\(145\) −3.76335 −0.312529
\(146\) 5.23265 0.433057
\(147\) 0 0
\(148\) 32.3428 2.65856
\(149\) −14.7199 −1.20590 −0.602952 0.797778i \(-0.706009\pi\)
−0.602952 + 0.797778i \(0.706009\pi\)
\(150\) 0 0
\(151\) −1.49585 −0.121730 −0.0608652 0.998146i \(-0.519386\pi\)
−0.0608652 + 0.998146i \(0.519386\pi\)
\(152\) 5.37938 0.436325
\(153\) 0 0
\(154\) 0.348162 0.0280557
\(155\) −1.51971 −0.122066
\(156\) 0 0
\(157\) −1.34647 −0.107460 −0.0537300 0.998555i \(-0.517111\pi\)
−0.0537300 + 0.998555i \(0.517111\pi\)
\(158\) −14.0216 −1.11550
\(159\) 0 0
\(160\) 1.09097 0.0862484
\(161\) 3.78387 0.298211
\(162\) 0 0
\(163\) −10.0209 −0.784899 −0.392450 0.919773i \(-0.628372\pi\)
−0.392450 + 0.919773i \(0.628372\pi\)
\(164\) 7.29767 0.569853
\(165\) 0 0
\(166\) −8.63756 −0.670405
\(167\) −3.14438 −0.243319 −0.121660 0.992572i \(-0.538822\pi\)
−0.121660 + 0.992572i \(0.538822\pi\)
\(168\) 0 0
\(169\) −3.67706 −0.282851
\(170\) −3.22930 −0.247676
\(171\) 0 0
\(172\) −28.2694 −2.15553
\(173\) 7.87801 0.598954 0.299477 0.954104i \(-0.403188\pi\)
0.299477 + 0.954104i \(0.403188\pi\)
\(174\) 0 0
\(175\) −2.50422 −0.189301
\(176\) 0.778721 0.0586983
\(177\) 0 0
\(178\) −23.4347 −1.75650
\(179\) −1.92732 −0.144055 −0.0720275 0.997403i \(-0.522947\pi\)
−0.0720275 + 0.997403i \(0.522947\pi\)
\(180\) 0 0
\(181\) −9.05976 −0.673406 −0.336703 0.941611i \(-0.609312\pi\)
−0.336703 + 0.941611i \(0.609312\pi\)
\(182\) 4.04974 0.300187
\(183\) 0 0
\(184\) 30.2736 2.23180
\(185\) −5.63841 −0.414544
\(186\) 0 0
\(187\) 0.527178 0.0385511
\(188\) 6.39915 0.466706
\(189\) 0 0
\(190\) −1.96693 −0.142696
\(191\) 19.2051 1.38963 0.694816 0.719188i \(-0.255486\pi\)
0.694816 + 0.719188i \(0.255486\pi\)
\(192\) 0 0
\(193\) −15.4298 −1.11066 −0.555332 0.831629i \(-0.687409\pi\)
−0.555332 + 0.831629i \(0.687409\pi\)
\(194\) 18.4042 1.32134
\(195\) 0 0
\(196\) −25.6027 −1.82876
\(197\) 17.3806 1.23831 0.619157 0.785267i \(-0.287474\pi\)
0.619157 + 0.785267i \(0.287474\pi\)
\(198\) 0 0
\(199\) −10.3156 −0.731252 −0.365626 0.930762i \(-0.619145\pi\)
−0.365626 + 0.930762i \(0.619145\pi\)
\(200\) −20.0355 −1.41673
\(201\) 0 0
\(202\) −3.25209 −0.228816
\(203\) 3.10415 0.217869
\(204\) 0 0
\(205\) −1.27222 −0.0888560
\(206\) 44.7447 3.11751
\(207\) 0 0
\(208\) 9.05790 0.628053
\(209\) 0.321098 0.0222108
\(210\) 0 0
\(211\) 5.32226 0.366400 0.183200 0.983076i \(-0.441354\pi\)
0.183200 + 0.983076i \(0.441354\pi\)
\(212\) 27.3301 1.87704
\(213\) 0 0
\(214\) −36.7456 −2.51188
\(215\) 4.92829 0.336107
\(216\) 0 0
\(217\) 1.25351 0.0850938
\(218\) −5.23948 −0.354862
\(219\) 0 0
\(220\) −0.668661 −0.0450811
\(221\) 6.13201 0.412484
\(222\) 0 0
\(223\) −7.17603 −0.480542 −0.240271 0.970706i \(-0.577236\pi\)
−0.240271 + 0.970706i \(0.577236\pi\)
\(224\) −0.899869 −0.0601250
\(225\) 0 0
\(226\) −1.60494 −0.106759
\(227\) −26.6721 −1.77029 −0.885144 0.465318i \(-0.845940\pi\)
−0.885144 + 0.465318i \(0.845940\pi\)
\(228\) 0 0
\(229\) 4.07227 0.269103 0.134552 0.990907i \(-0.457041\pi\)
0.134552 + 0.990907i \(0.457041\pi\)
\(230\) −11.0694 −0.729892
\(231\) 0 0
\(232\) 24.8354 1.63053
\(233\) −14.5101 −0.950590 −0.475295 0.879827i \(-0.657659\pi\)
−0.475295 + 0.879827i \(0.657659\pi\)
\(234\) 0 0
\(235\) −1.11558 −0.0727725
\(236\) 14.0604 0.915252
\(237\) 0 0
\(238\) 2.66365 0.172659
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −10.1087 −0.651160 −0.325580 0.945515i \(-0.605559\pi\)
−0.325580 + 0.945515i \(0.605559\pi\)
\(242\) −26.3766 −1.69555
\(243\) 0 0
\(244\) −52.4538 −3.35801
\(245\) 4.46338 0.285155
\(246\) 0 0
\(247\) 3.73494 0.237649
\(248\) 10.0290 0.636841
\(249\) 0 0
\(250\) 15.3658 0.971818
\(251\) 22.5850 1.42555 0.712775 0.701393i \(-0.247438\pi\)
0.712775 + 0.701393i \(0.247438\pi\)
\(252\) 0 0
\(253\) 1.80705 0.113608
\(254\) −23.9004 −1.49964
\(255\) 0 0
\(256\) −29.8789 −1.86743
\(257\) 6.41138 0.399931 0.199966 0.979803i \(-0.435917\pi\)
0.199966 + 0.979803i \(0.435917\pi\)
\(258\) 0 0
\(259\) 4.65077 0.288985
\(260\) −7.77771 −0.482353
\(261\) 0 0
\(262\) 9.94250 0.614250
\(263\) −3.66746 −0.226145 −0.113073 0.993587i \(-0.536069\pi\)
−0.113073 + 0.993587i \(0.536069\pi\)
\(264\) 0 0
\(265\) −4.76454 −0.292683
\(266\) 1.62240 0.0994758
\(267\) 0 0
\(268\) −10.7305 −0.655471
\(269\) 19.7784 1.20591 0.602954 0.797776i \(-0.293990\pi\)
0.602954 + 0.797776i \(0.293990\pi\)
\(270\) 0 0
\(271\) −1.14335 −0.0694534 −0.0347267 0.999397i \(-0.511056\pi\)
−0.0347267 + 0.999397i \(0.511056\pi\)
\(272\) 5.95769 0.361238
\(273\) 0 0
\(274\) 26.5903 1.60638
\(275\) −1.19593 −0.0721176
\(276\) 0 0
\(277\) −10.1579 −0.610331 −0.305165 0.952299i \(-0.598712\pi\)
−0.305165 + 0.952299i \(0.598712\pi\)
\(278\) −4.81186 −0.288596
\(279\) 0 0
\(280\) −1.61082 −0.0962650
\(281\) −12.7932 −0.763176 −0.381588 0.924333i \(-0.624623\pi\)
−0.381588 + 0.924333i \(0.624623\pi\)
\(282\) 0 0
\(283\) −5.32245 −0.316387 −0.158193 0.987408i \(-0.550567\pi\)
−0.158193 + 0.987408i \(0.550567\pi\)
\(284\) −18.2966 −1.08571
\(285\) 0 0
\(286\) 1.93402 0.114361
\(287\) 1.04938 0.0619428
\(288\) 0 0
\(289\) −12.9668 −0.762751
\(290\) −9.08092 −0.533250
\(291\) 0 0
\(292\) 8.28924 0.485091
\(293\) 7.15519 0.418011 0.209005 0.977915i \(-0.432977\pi\)
0.209005 + 0.977915i \(0.432977\pi\)
\(294\) 0 0
\(295\) −2.45118 −0.142713
\(296\) 37.2095 2.16276
\(297\) 0 0
\(298\) −35.5190 −2.05756
\(299\) 21.0192 1.21557
\(300\) 0 0
\(301\) −4.06504 −0.234305
\(302\) −3.60946 −0.207701
\(303\) 0 0
\(304\) 3.62877 0.208124
\(305\) 9.14441 0.523607
\(306\) 0 0
\(307\) −20.5867 −1.17494 −0.587472 0.809244i \(-0.699877\pi\)
−0.587472 + 0.809244i \(0.699877\pi\)
\(308\) 0.551537 0.0314267
\(309\) 0 0
\(310\) −3.66703 −0.208273
\(311\) −20.1654 −1.14347 −0.571736 0.820437i \(-0.693730\pi\)
−0.571736 + 0.820437i \(0.693730\pi\)
\(312\) 0 0
\(313\) 17.7852 1.00528 0.502639 0.864496i \(-0.332362\pi\)
0.502639 + 0.864496i \(0.332362\pi\)
\(314\) −3.24902 −0.183353
\(315\) 0 0
\(316\) −22.2121 −1.24953
\(317\) 32.3858 1.81897 0.909485 0.415737i \(-0.136476\pi\)
0.909485 + 0.415737i \(0.136476\pi\)
\(318\) 0 0
\(319\) 1.48244 0.0830009
\(320\) 6.58623 0.368181
\(321\) 0 0
\(322\) 9.13043 0.508819
\(323\) 2.45660 0.136689
\(324\) 0 0
\(325\) −13.9108 −0.771634
\(326\) −24.1804 −1.33923
\(327\) 0 0
\(328\) 8.39577 0.463579
\(329\) 0.920173 0.0507308
\(330\) 0 0
\(331\) −4.21115 −0.231466 −0.115733 0.993280i \(-0.536922\pi\)
−0.115733 + 0.993280i \(0.536922\pi\)
\(332\) −13.6831 −0.750958
\(333\) 0 0
\(334\) −7.58735 −0.415161
\(335\) 1.87068 0.102206
\(336\) 0 0
\(337\) 7.22450 0.393543 0.196772 0.980449i \(-0.436954\pi\)
0.196772 + 0.980449i \(0.436954\pi\)
\(338\) −8.87271 −0.482612
\(339\) 0 0
\(340\) −5.11567 −0.277436
\(341\) 0.598636 0.0324180
\(342\) 0 0
\(343\) −7.52920 −0.406539
\(344\) −32.5232 −1.75353
\(345\) 0 0
\(346\) 19.0095 1.02196
\(347\) 12.1793 0.653820 0.326910 0.945055i \(-0.393993\pi\)
0.326910 + 0.945055i \(0.393993\pi\)
\(348\) 0 0
\(349\) 7.63405 0.408641 0.204321 0.978904i \(-0.434501\pi\)
0.204321 + 0.978904i \(0.434501\pi\)
\(350\) −6.04265 −0.322993
\(351\) 0 0
\(352\) −0.429748 −0.0229057
\(353\) 7.39637 0.393669 0.196835 0.980437i \(-0.436934\pi\)
0.196835 + 0.980437i \(0.436934\pi\)
\(354\) 0 0
\(355\) 3.18970 0.169292
\(356\) −37.1238 −1.96756
\(357\) 0 0
\(358\) −4.65061 −0.245792
\(359\) −9.14233 −0.482514 −0.241257 0.970461i \(-0.577560\pi\)
−0.241257 + 0.970461i \(0.577560\pi\)
\(360\) 0 0
\(361\) −17.5037 −0.921248
\(362\) −21.8611 −1.14899
\(363\) 0 0
\(364\) 6.41535 0.336256
\(365\) −1.44509 −0.0756393
\(366\) 0 0
\(367\) 19.7352 1.03017 0.515086 0.857139i \(-0.327760\pi\)
0.515086 + 0.857139i \(0.327760\pi\)
\(368\) 20.4217 1.06455
\(369\) 0 0
\(370\) −13.6054 −0.707311
\(371\) 3.92997 0.204034
\(372\) 0 0
\(373\) −0.398821 −0.0206502 −0.0103251 0.999947i \(-0.503287\pi\)
−0.0103251 + 0.999947i \(0.503287\pi\)
\(374\) 1.27207 0.0657774
\(375\) 0 0
\(376\) 7.36204 0.379668
\(377\) 17.2434 0.888082
\(378\) 0 0
\(379\) −26.9887 −1.38632 −0.693158 0.720785i \(-0.743781\pi\)
−0.693158 + 0.720785i \(0.743781\pi\)
\(380\) −3.11590 −0.159842
\(381\) 0 0
\(382\) 46.3416 2.37104
\(383\) −0.514040 −0.0262662 −0.0131331 0.999914i \(-0.504181\pi\)
−0.0131331 + 0.999914i \(0.504181\pi\)
\(384\) 0 0
\(385\) −0.0961510 −0.00490031
\(386\) −37.2320 −1.89506
\(387\) 0 0
\(388\) 29.1547 1.48011
\(389\) 10.7557 0.545337 0.272669 0.962108i \(-0.412094\pi\)
0.272669 + 0.962108i \(0.412094\pi\)
\(390\) 0 0
\(391\) 13.8251 0.699163
\(392\) −29.4551 −1.48771
\(393\) 0 0
\(394\) 41.9391 2.11286
\(395\) 3.87230 0.194837
\(396\) 0 0
\(397\) 16.2650 0.816316 0.408158 0.912911i \(-0.366171\pi\)
0.408158 + 0.912911i \(0.366171\pi\)
\(398\) −24.8914 −1.24769
\(399\) 0 0
\(400\) −13.5154 −0.675769
\(401\) 8.64206 0.431564 0.215782 0.976442i \(-0.430770\pi\)
0.215782 + 0.976442i \(0.430770\pi\)
\(402\) 0 0
\(403\) 6.96320 0.346862
\(404\) −5.15177 −0.256310
\(405\) 0 0
\(406\) 7.49028 0.371736
\(407\) 2.22106 0.110094
\(408\) 0 0
\(409\) 9.97211 0.493089 0.246544 0.969131i \(-0.420705\pi\)
0.246544 + 0.969131i \(0.420705\pi\)
\(410\) −3.06986 −0.151610
\(411\) 0 0
\(412\) 70.8818 3.49209
\(413\) 2.02183 0.0994876
\(414\) 0 0
\(415\) 2.38541 0.117095
\(416\) −4.99873 −0.245083
\(417\) 0 0
\(418\) 0.774806 0.0378970
\(419\) 3.70284 0.180895 0.0904477 0.995901i \(-0.471170\pi\)
0.0904477 + 0.995901i \(0.471170\pi\)
\(420\) 0 0
\(421\) −29.7692 −1.45086 −0.725432 0.688294i \(-0.758360\pi\)
−0.725432 + 0.688294i \(0.758360\pi\)
\(422\) 12.8426 0.625166
\(423\) 0 0
\(424\) 31.4426 1.52699
\(425\) −9.14962 −0.443822
\(426\) 0 0
\(427\) −7.54265 −0.365015
\(428\) −58.2101 −2.81369
\(429\) 0 0
\(430\) 11.8919 0.573478
\(431\) 27.9651 1.34703 0.673516 0.739172i \(-0.264783\pi\)
0.673516 + 0.739172i \(0.264783\pi\)
\(432\) 0 0
\(433\) 36.2370 1.74144 0.870720 0.491779i \(-0.163653\pi\)
0.870720 + 0.491779i \(0.163653\pi\)
\(434\) 3.02471 0.145190
\(435\) 0 0
\(436\) −8.30006 −0.397501
\(437\) 8.42069 0.402816
\(438\) 0 0
\(439\) −2.01152 −0.0960048 −0.0480024 0.998847i \(-0.515286\pi\)
−0.0480024 + 0.998847i \(0.515286\pi\)
\(440\) −0.769276 −0.0366738
\(441\) 0 0
\(442\) 14.7965 0.703796
\(443\) −25.4185 −1.20767 −0.603834 0.797110i \(-0.706361\pi\)
−0.603834 + 0.797110i \(0.706361\pi\)
\(444\) 0 0
\(445\) 6.47189 0.306797
\(446\) −17.3157 −0.819920
\(447\) 0 0
\(448\) −5.43257 −0.256665
\(449\) −15.7520 −0.743381 −0.371690 0.928357i \(-0.621222\pi\)
−0.371690 + 0.928357i \(0.621222\pi\)
\(450\) 0 0
\(451\) 0.501149 0.0235982
\(452\) −2.54244 −0.119586
\(453\) 0 0
\(454\) −64.3593 −3.02053
\(455\) −1.11841 −0.0524317
\(456\) 0 0
\(457\) 4.73035 0.221277 0.110638 0.993861i \(-0.464710\pi\)
0.110638 + 0.993861i \(0.464710\pi\)
\(458\) 9.82634 0.459155
\(459\) 0 0
\(460\) −17.5354 −0.817593
\(461\) 26.6833 1.24277 0.621383 0.783507i \(-0.286571\pi\)
0.621383 + 0.783507i \(0.286571\pi\)
\(462\) 0 0
\(463\) 26.2745 1.22108 0.610539 0.791986i \(-0.290953\pi\)
0.610539 + 0.791986i \(0.290953\pi\)
\(464\) 16.7532 0.777750
\(465\) 0 0
\(466\) −35.0127 −1.62193
\(467\) −18.8007 −0.869994 −0.434997 0.900432i \(-0.643251\pi\)
−0.434997 + 0.900432i \(0.643251\pi\)
\(468\) 0 0
\(469\) −1.54301 −0.0712494
\(470\) −2.69188 −0.124167
\(471\) 0 0
\(472\) 16.1760 0.744563
\(473\) −1.94133 −0.0892625
\(474\) 0 0
\(475\) −5.57294 −0.255704
\(476\) 4.21959 0.193405
\(477\) 0 0
\(478\) 2.41299 0.110367
\(479\) −32.0039 −1.46230 −0.731148 0.682218i \(-0.761015\pi\)
−0.731148 + 0.682218i \(0.761015\pi\)
\(480\) 0 0
\(481\) 25.8348 1.17797
\(482\) −24.3922 −1.11103
\(483\) 0 0
\(484\) −41.7842 −1.89928
\(485\) −5.08263 −0.230790
\(486\) 0 0
\(487\) 4.76961 0.216132 0.108066 0.994144i \(-0.465534\pi\)
0.108066 + 0.994144i \(0.465534\pi\)
\(488\) −60.3466 −2.73176
\(489\) 0 0
\(490\) 10.7701 0.486543
\(491\) 9.99992 0.451290 0.225645 0.974210i \(-0.427551\pi\)
0.225645 + 0.974210i \(0.427551\pi\)
\(492\) 0 0
\(493\) 11.3416 0.510800
\(494\) 9.01237 0.405486
\(495\) 0 0
\(496\) 6.76525 0.303769
\(497\) −2.63099 −0.118016
\(498\) 0 0
\(499\) 29.8371 1.33569 0.667847 0.744299i \(-0.267216\pi\)
0.667847 + 0.744299i \(0.267216\pi\)
\(500\) 24.3416 1.08859
\(501\) 0 0
\(502\) 54.4972 2.43233
\(503\) −10.1386 −0.452057 −0.226029 0.974121i \(-0.572574\pi\)
−0.226029 + 0.974121i \(0.572574\pi\)
\(504\) 0 0
\(505\) 0.898121 0.0399659
\(506\) 4.36040 0.193843
\(507\) 0 0
\(508\) −37.8615 −1.67983
\(509\) −16.9917 −0.753145 −0.376572 0.926387i \(-0.622897\pi\)
−0.376572 + 0.926387i \(0.622897\pi\)
\(510\) 0 0
\(511\) 1.19196 0.0527293
\(512\) −30.9485 −1.36774
\(513\) 0 0
\(514\) 15.4706 0.682378
\(515\) −12.3570 −0.544515
\(516\) 0 0
\(517\) 0.439445 0.0193268
\(518\) 11.2222 0.493077
\(519\) 0 0
\(520\) −8.94804 −0.392398
\(521\) −4.77975 −0.209405 −0.104702 0.994504i \(-0.533389\pi\)
−0.104702 + 0.994504i \(0.533389\pi\)
\(522\) 0 0
\(523\) 13.7212 0.599985 0.299992 0.953942i \(-0.403016\pi\)
0.299992 + 0.953942i \(0.403016\pi\)
\(524\) 15.7503 0.688055
\(525\) 0 0
\(526\) −8.84954 −0.385858
\(527\) 4.57993 0.199505
\(528\) 0 0
\(529\) 24.3893 1.06041
\(530\) −11.4968 −0.499388
\(531\) 0 0
\(532\) 2.57011 0.111428
\(533\) 5.82925 0.252493
\(534\) 0 0
\(535\) 10.1479 0.438733
\(536\) −12.3452 −0.533229
\(537\) 0 0
\(538\) 47.7249 2.05757
\(539\) −1.75820 −0.0757309
\(540\) 0 0
\(541\) −14.2257 −0.611612 −0.305806 0.952094i \(-0.598926\pi\)
−0.305806 + 0.952094i \(0.598926\pi\)
\(542\) −2.75888 −0.118504
\(543\) 0 0
\(544\) −3.28784 −0.140965
\(545\) 1.44697 0.0619815
\(546\) 0 0
\(547\) −26.3222 −1.12546 −0.562728 0.826642i \(-0.690248\pi\)
−0.562728 + 0.826642i \(0.690248\pi\)
\(548\) 42.1227 1.79939
\(549\) 0 0
\(550\) −2.88577 −0.123050
\(551\) 6.90804 0.294293
\(552\) 0 0
\(553\) −3.19402 −0.135824
\(554\) −24.5110 −1.04137
\(555\) 0 0
\(556\) −7.62265 −0.323273
\(557\) −34.8025 −1.47463 −0.737315 0.675549i \(-0.763907\pi\)
−0.737315 + 0.675549i \(0.763907\pi\)
\(558\) 0 0
\(559\) −22.5811 −0.955079
\(560\) −1.08661 −0.0459177
\(561\) 0 0
\(562\) −30.8697 −1.30216
\(563\) −40.1796 −1.69337 −0.846685 0.532095i \(-0.821405\pi\)
−0.846685 + 0.532095i \(0.821405\pi\)
\(564\) 0 0
\(565\) 0.443231 0.0186469
\(566\) −12.8430 −0.539832
\(567\) 0 0
\(568\) −21.0498 −0.883228
\(569\) 29.3863 1.23194 0.615968 0.787771i \(-0.288765\pi\)
0.615968 + 0.787771i \(0.288765\pi\)
\(570\) 0 0
\(571\) −12.7435 −0.533300 −0.266650 0.963793i \(-0.585917\pi\)
−0.266650 + 0.963793i \(0.585917\pi\)
\(572\) 3.06376 0.128102
\(573\) 0 0
\(574\) 2.53214 0.105689
\(575\) −31.3630 −1.30793
\(576\) 0 0
\(577\) −32.1954 −1.34031 −0.670157 0.742219i \(-0.733773\pi\)
−0.670157 + 0.742219i \(0.733773\pi\)
\(578\) −31.2886 −1.30144
\(579\) 0 0
\(580\) −14.3854 −0.597323
\(581\) −1.96758 −0.0816289
\(582\) 0 0
\(583\) 1.87683 0.0777302
\(584\) 9.53654 0.394625
\(585\) 0 0
\(586\) 17.2654 0.713226
\(587\) 10.6677 0.440303 0.220152 0.975466i \(-0.429345\pi\)
0.220152 + 0.975466i \(0.429345\pi\)
\(588\) 0 0
\(589\) 2.78959 0.114943
\(590\) −5.91467 −0.243503
\(591\) 0 0
\(592\) 25.1004 1.03162
\(593\) −23.8504 −0.979418 −0.489709 0.871886i \(-0.662897\pi\)
−0.489709 + 0.871886i \(0.662897\pi\)
\(594\) 0 0
\(595\) −0.735613 −0.0301572
\(596\) −56.2670 −2.30479
\(597\) 0 0
\(598\) 50.7191 2.07406
\(599\) 17.6222 0.720023 0.360012 0.932948i \(-0.382773\pi\)
0.360012 + 0.932948i \(0.382773\pi\)
\(600\) 0 0
\(601\) 12.5309 0.511148 0.255574 0.966790i \(-0.417736\pi\)
0.255574 + 0.966790i \(0.417736\pi\)
\(602\) −9.80889 −0.399780
\(603\) 0 0
\(604\) −5.71789 −0.232658
\(605\) 7.28435 0.296151
\(606\) 0 0
\(607\) 17.3888 0.705790 0.352895 0.935663i \(-0.385197\pi\)
0.352895 + 0.935663i \(0.385197\pi\)
\(608\) −2.00259 −0.0812156
\(609\) 0 0
\(610\) 22.0653 0.893400
\(611\) 5.11152 0.206790
\(612\) 0 0
\(613\) −29.0189 −1.17206 −0.586031 0.810288i \(-0.699310\pi\)
−0.586031 + 0.810288i \(0.699310\pi\)
\(614\) −49.6754 −2.00474
\(615\) 0 0
\(616\) 0.634528 0.0255659
\(617\) 34.7121 1.39745 0.698727 0.715388i \(-0.253750\pi\)
0.698727 + 0.715388i \(0.253750\pi\)
\(618\) 0 0
\(619\) 43.2472 1.73825 0.869125 0.494592i \(-0.164683\pi\)
0.869125 + 0.494592i \(0.164683\pi\)
\(620\) −5.80909 −0.233299
\(621\) 0 0
\(622\) −48.6588 −1.95104
\(623\) −5.33826 −0.213873
\(624\) 0 0
\(625\) 18.5361 0.741444
\(626\) 42.9154 1.71525
\(627\) 0 0
\(628\) −5.14689 −0.205383
\(629\) 16.9924 0.677533
\(630\) 0 0
\(631\) −29.9746 −1.19327 −0.596635 0.802513i \(-0.703496\pi\)
−0.596635 + 0.802513i \(0.703496\pi\)
\(632\) −25.5544 −1.01650
\(633\) 0 0
\(634\) 78.1466 3.10360
\(635\) 6.60050 0.261933
\(636\) 0 0
\(637\) −20.4509 −0.810296
\(638\) 3.57712 0.141619
\(639\) 0 0
\(640\) 13.7106 0.541957
\(641\) 9.69551 0.382949 0.191475 0.981498i \(-0.438673\pi\)
0.191475 + 0.981498i \(0.438673\pi\)
\(642\) 0 0
\(643\) 4.85463 0.191448 0.0957240 0.995408i \(-0.469483\pi\)
0.0957240 + 0.995408i \(0.469483\pi\)
\(644\) 14.4639 0.569956
\(645\) 0 0
\(646\) 5.92774 0.233224
\(647\) 16.8226 0.661366 0.330683 0.943742i \(-0.392721\pi\)
0.330683 + 0.943742i \(0.392721\pi\)
\(648\) 0 0
\(649\) 0.965559 0.0379015
\(650\) −33.5667 −1.31659
\(651\) 0 0
\(652\) −38.3051 −1.50014
\(653\) 6.22373 0.243553 0.121777 0.992558i \(-0.461141\pi\)
0.121777 + 0.992558i \(0.461141\pi\)
\(654\) 0 0
\(655\) −2.74579 −0.107287
\(656\) 5.66353 0.221124
\(657\) 0 0
\(658\) 2.22037 0.0865589
\(659\) 4.01398 0.156363 0.0781813 0.996939i \(-0.475089\pi\)
0.0781813 + 0.996939i \(0.475089\pi\)
\(660\) 0 0
\(661\) 8.75196 0.340412 0.170206 0.985409i \(-0.445557\pi\)
0.170206 + 0.985409i \(0.445557\pi\)
\(662\) −10.1615 −0.394936
\(663\) 0 0
\(664\) −15.7420 −0.610909
\(665\) −0.448054 −0.0173748
\(666\) 0 0
\(667\) 38.8766 1.50531
\(668\) −12.0194 −0.465045
\(669\) 0 0
\(670\) 4.51393 0.174388
\(671\) −3.60213 −0.139059
\(672\) 0 0
\(673\) 20.6931 0.797662 0.398831 0.917024i \(-0.369416\pi\)
0.398831 + 0.917024i \(0.369416\pi\)
\(674\) 17.4326 0.671479
\(675\) 0 0
\(676\) −14.0556 −0.540600
\(677\) −3.24148 −0.124580 −0.0622900 0.998058i \(-0.519840\pi\)
−0.0622900 + 0.998058i \(0.519840\pi\)
\(678\) 0 0
\(679\) 4.19234 0.160887
\(680\) −5.88543 −0.225696
\(681\) 0 0
\(682\) 1.44450 0.0553128
\(683\) 28.0620 1.07376 0.536881 0.843658i \(-0.319602\pi\)
0.536881 + 0.843658i \(0.319602\pi\)
\(684\) 0 0
\(685\) −7.34336 −0.280575
\(686\) −18.1679 −0.693652
\(687\) 0 0
\(688\) −21.9392 −0.836423
\(689\) 21.8308 0.831688
\(690\) 0 0
\(691\) 8.47107 0.322255 0.161127 0.986934i \(-0.448487\pi\)
0.161127 + 0.986934i \(0.448487\pi\)
\(692\) 30.1137 1.14475
\(693\) 0 0
\(694\) 29.3885 1.11557
\(695\) 1.32888 0.0504072
\(696\) 0 0
\(697\) 3.83409 0.145227
\(698\) 18.4209 0.697240
\(699\) 0 0
\(700\) −9.57240 −0.361803
\(701\) 3.89669 0.147176 0.0735879 0.997289i \(-0.476555\pi\)
0.0735879 + 0.997289i \(0.476555\pi\)
\(702\) 0 0
\(703\) 10.3499 0.390354
\(704\) −2.59442 −0.0977809
\(705\) 0 0
\(706\) 17.8473 0.671694
\(707\) −0.740804 −0.0278608
\(708\) 0 0
\(709\) −11.0463 −0.414853 −0.207426 0.978251i \(-0.566509\pi\)
−0.207426 + 0.978251i \(0.566509\pi\)
\(710\) 7.69671 0.288852
\(711\) 0 0
\(712\) −42.7099 −1.60062
\(713\) 15.6990 0.587933
\(714\) 0 0
\(715\) −0.534114 −0.0199747
\(716\) −7.36721 −0.275326
\(717\) 0 0
\(718\) −22.0603 −0.823284
\(719\) 19.2426 0.717628 0.358814 0.933409i \(-0.383181\pi\)
0.358814 + 0.933409i \(0.383181\pi\)
\(720\) 0 0
\(721\) 10.1925 0.379589
\(722\) −42.2362 −1.57187
\(723\) 0 0
\(724\) −34.6310 −1.28705
\(725\) −25.7291 −0.955554
\(726\) 0 0
\(727\) 32.7014 1.21283 0.606414 0.795149i \(-0.292607\pi\)
0.606414 + 0.795149i \(0.292607\pi\)
\(728\) 7.38068 0.273546
\(729\) 0 0
\(730\) −3.48697 −0.129059
\(731\) −14.8524 −0.549334
\(732\) 0 0
\(733\) 27.7761 1.02593 0.512966 0.858409i \(-0.328547\pi\)
0.512966 + 0.858409i \(0.328547\pi\)
\(734\) 47.6209 1.75772
\(735\) 0 0
\(736\) −11.2700 −0.415418
\(737\) −0.736890 −0.0271437
\(738\) 0 0
\(739\) −15.1638 −0.557811 −0.278905 0.960319i \(-0.589972\pi\)
−0.278905 + 0.960319i \(0.589972\pi\)
\(740\) −21.5529 −0.792299
\(741\) 0 0
\(742\) 9.48297 0.348131
\(743\) −21.1140 −0.774599 −0.387299 0.921954i \(-0.626592\pi\)
−0.387299 + 0.921954i \(0.626592\pi\)
\(744\) 0 0
\(745\) 9.80918 0.359381
\(746\) −0.962350 −0.0352341
\(747\) 0 0
\(748\) 2.01514 0.0736809
\(749\) −8.37039 −0.305848
\(750\) 0 0
\(751\) 3.25042 0.118609 0.0593047 0.998240i \(-0.481112\pi\)
0.0593047 + 0.998240i \(0.481112\pi\)
\(752\) 4.96621 0.181099
\(753\) 0 0
\(754\) 41.6082 1.51528
\(755\) 0.996816 0.0362778
\(756\) 0 0
\(757\) 44.0840 1.60226 0.801130 0.598491i \(-0.204233\pi\)
0.801130 + 0.598491i \(0.204233\pi\)
\(758\) −65.1234 −2.36539
\(759\) 0 0
\(760\) −3.58475 −0.130033
\(761\) 38.8537 1.40845 0.704223 0.709979i \(-0.251296\pi\)
0.704223 + 0.709979i \(0.251296\pi\)
\(762\) 0 0
\(763\) −1.19352 −0.0432082
\(764\) 73.4116 2.65594
\(765\) 0 0
\(766\) −1.24037 −0.0448164
\(767\) 11.2312 0.405534
\(768\) 0 0
\(769\) 4.35090 0.156897 0.0784487 0.996918i \(-0.475003\pi\)
0.0784487 + 0.996918i \(0.475003\pi\)
\(770\) −0.232011 −0.00836110
\(771\) 0 0
\(772\) −58.9806 −2.12276
\(773\) −20.0826 −0.722321 −0.361161 0.932504i \(-0.617619\pi\)
−0.361161 + 0.932504i \(0.617619\pi\)
\(774\) 0 0
\(775\) −10.3898 −0.373214
\(776\) 33.5417 1.20408
\(777\) 0 0
\(778\) 25.9535 0.930476
\(779\) 2.33531 0.0836710
\(780\) 0 0
\(781\) −1.25647 −0.0449602
\(782\) 33.3597 1.19294
\(783\) 0 0
\(784\) −19.8696 −0.709627
\(785\) 0.897272 0.0320250
\(786\) 0 0
\(787\) 45.6772 1.62822 0.814109 0.580712i \(-0.197226\pi\)
0.814109 + 0.580712i \(0.197226\pi\)
\(788\) 66.4374 2.36673
\(789\) 0 0
\(790\) 9.34382 0.332438
\(791\) −0.365594 −0.0129990
\(792\) 0 0
\(793\) −41.8991 −1.48788
\(794\) 39.2472 1.39283
\(795\) 0 0
\(796\) −39.4314 −1.39761
\(797\) 42.3136 1.49883 0.749413 0.662103i \(-0.230336\pi\)
0.749413 + 0.662103i \(0.230336\pi\)
\(798\) 0 0
\(799\) 3.36202 0.118940
\(800\) 7.45865 0.263703
\(801\) 0 0
\(802\) 20.8532 0.736352
\(803\) 0.569242 0.0200881
\(804\) 0 0
\(805\) −2.52152 −0.0888721
\(806\) 16.8021 0.591829
\(807\) 0 0
\(808\) −5.92696 −0.208510
\(809\) −2.33468 −0.0820831 −0.0410416 0.999157i \(-0.513068\pi\)
−0.0410416 + 0.999157i \(0.513068\pi\)
\(810\) 0 0
\(811\) −21.1927 −0.744175 −0.372088 0.928198i \(-0.621358\pi\)
−0.372088 + 0.928198i \(0.621358\pi\)
\(812\) 11.8656 0.416403
\(813\) 0 0
\(814\) 5.35938 0.187846
\(815\) 6.67783 0.233914
\(816\) 0 0
\(817\) −9.04642 −0.316494
\(818\) 24.0626 0.841328
\(819\) 0 0
\(820\) −4.86308 −0.169826
\(821\) −50.6599 −1.76804 −0.884022 0.467446i \(-0.845174\pi\)
−0.884022 + 0.467446i \(0.845174\pi\)
\(822\) 0 0
\(823\) −12.9252 −0.450544 −0.225272 0.974296i \(-0.572327\pi\)
−0.225272 + 0.974296i \(0.572327\pi\)
\(824\) 81.5475 2.84084
\(825\) 0 0
\(826\) 4.87864 0.169750
\(827\) −5.83428 −0.202878 −0.101439 0.994842i \(-0.532345\pi\)
−0.101439 + 0.994842i \(0.532345\pi\)
\(828\) 0 0
\(829\) 25.8365 0.897340 0.448670 0.893697i \(-0.351898\pi\)
0.448670 + 0.893697i \(0.351898\pi\)
\(830\) 5.75597 0.199793
\(831\) 0 0
\(832\) −30.1777 −1.04622
\(833\) −13.4513 −0.466059
\(834\) 0 0
\(835\) 2.09538 0.0725135
\(836\) 1.22740 0.0424506
\(837\) 0 0
\(838\) 8.93490 0.308651
\(839\) 20.5187 0.708386 0.354193 0.935172i \(-0.384756\pi\)
0.354193 + 0.935172i \(0.384756\pi\)
\(840\) 0 0
\(841\) 2.89298 0.0997578
\(842\) −71.8328 −2.47552
\(843\) 0 0
\(844\) 20.3444 0.700283
\(845\) 2.45035 0.0842947
\(846\) 0 0
\(847\) −6.00841 −0.206451
\(848\) 21.2102 0.728362
\(849\) 0 0
\(850\) −22.0779 −0.757267
\(851\) 58.2465 1.99666
\(852\) 0 0
\(853\) 12.0881 0.413888 0.206944 0.978353i \(-0.433648\pi\)
0.206944 + 0.978353i \(0.433648\pi\)
\(854\) −18.2003 −0.622802
\(855\) 0 0
\(856\) −66.9691 −2.28896
\(857\) 47.5584 1.62457 0.812283 0.583264i \(-0.198225\pi\)
0.812283 + 0.583264i \(0.198225\pi\)
\(858\) 0 0
\(859\) 21.0304 0.717547 0.358774 0.933425i \(-0.383195\pi\)
0.358774 + 0.933425i \(0.383195\pi\)
\(860\) 18.8384 0.642385
\(861\) 0 0
\(862\) 67.4795 2.29836
\(863\) −55.2047 −1.87919 −0.939595 0.342288i \(-0.888798\pi\)
−0.939595 + 0.342288i \(0.888798\pi\)
\(864\) 0 0
\(865\) −5.24981 −0.178499
\(866\) 87.4395 2.97131
\(867\) 0 0
\(868\) 4.79155 0.162636
\(869\) −1.52536 −0.0517443
\(870\) 0 0
\(871\) −8.57134 −0.290429
\(872\) −9.54899 −0.323369
\(873\) 0 0
\(874\) 20.3190 0.687301
\(875\) 3.50022 0.118329
\(876\) 0 0
\(877\) −4.74754 −0.160313 −0.0801565 0.996782i \(-0.525542\pi\)
−0.0801565 + 0.996782i \(0.525542\pi\)
\(878\) −4.85378 −0.163807
\(879\) 0 0
\(880\) −0.518931 −0.0174931
\(881\) 17.3193 0.583501 0.291750 0.956494i \(-0.405762\pi\)
0.291750 + 0.956494i \(0.405762\pi\)
\(882\) 0 0
\(883\) −19.1558 −0.644644 −0.322322 0.946630i \(-0.604463\pi\)
−0.322322 + 0.946630i \(0.604463\pi\)
\(884\) 23.4397 0.788361
\(885\) 0 0
\(886\) −61.3345 −2.06057
\(887\) 11.6930 0.392614 0.196307 0.980542i \(-0.437105\pi\)
0.196307 + 0.980542i \(0.437105\pi\)
\(888\) 0 0
\(889\) −5.44434 −0.182597
\(890\) 15.6166 0.523469
\(891\) 0 0
\(892\) −27.4304 −0.918438
\(893\) 2.04777 0.0685261
\(894\) 0 0
\(895\) 1.28435 0.0429309
\(896\) −11.3090 −0.377807
\(897\) 0 0
\(898\) −38.0093 −1.26839
\(899\) 12.8789 0.429536
\(900\) 0 0
\(901\) 14.3589 0.478363
\(902\) 1.20927 0.0402642
\(903\) 0 0
\(904\) −2.92501 −0.0972844
\(905\) 6.03731 0.200687
\(906\) 0 0
\(907\) 1.98748 0.0659932 0.0329966 0.999455i \(-0.489495\pi\)
0.0329966 + 0.999455i \(0.489495\pi\)
\(908\) −101.954 −3.38347
\(909\) 0 0
\(910\) −2.69870 −0.0894610
\(911\) 17.2834 0.572625 0.286312 0.958136i \(-0.407570\pi\)
0.286312 + 0.958136i \(0.407570\pi\)
\(912\) 0 0
\(913\) −0.939651 −0.0310979
\(914\) 11.4143 0.377551
\(915\) 0 0
\(916\) 15.5663 0.514324
\(917\) 2.26483 0.0747914
\(918\) 0 0
\(919\) 33.1032 1.09197 0.545987 0.837794i \(-0.316155\pi\)
0.545987 + 0.837794i \(0.316155\pi\)
\(920\) −20.1740 −0.665117
\(921\) 0 0
\(922\) 64.3865 2.12046
\(923\) −14.6150 −0.481059
\(924\) 0 0
\(925\) −38.5484 −1.26746
\(926\) 63.3999 2.08345
\(927\) 0 0
\(928\) −9.24552 −0.303499
\(929\) −35.7910 −1.17427 −0.587133 0.809490i \(-0.699743\pi\)
−0.587133 + 0.809490i \(0.699743\pi\)
\(930\) 0 0
\(931\) −8.19303 −0.268516
\(932\) −55.4651 −1.81682
\(933\) 0 0
\(934\) −45.3659 −1.48442
\(935\) −0.351305 −0.0114889
\(936\) 0 0
\(937\) 39.1301 1.27832 0.639162 0.769072i \(-0.279281\pi\)
0.639162 + 0.769072i \(0.279281\pi\)
\(938\) −3.72326 −0.121569
\(939\) 0 0
\(940\) −4.26432 −0.139087
\(941\) −40.8856 −1.33283 −0.666417 0.745579i \(-0.732173\pi\)
−0.666417 + 0.745579i \(0.732173\pi\)
\(942\) 0 0
\(943\) 13.1425 0.427977
\(944\) 10.9119 0.355151
\(945\) 0 0
\(946\) −4.68441 −0.152303
\(947\) −3.52599 −0.114579 −0.0572896 0.998358i \(-0.518246\pi\)
−0.0572896 + 0.998358i \(0.518246\pi\)
\(948\) 0 0
\(949\) 6.62129 0.214936
\(950\) −13.4474 −0.436292
\(951\) 0 0
\(952\) 4.85452 0.157336
\(953\) −9.66364 −0.313036 −0.156518 0.987675i \(-0.550027\pi\)
−0.156518 + 0.987675i \(0.550027\pi\)
\(954\) 0 0
\(955\) −12.7980 −0.414135
\(956\) 3.82251 0.123629
\(957\) 0 0
\(958\) −77.2251 −2.49503
\(959\) 6.05708 0.195593
\(960\) 0 0
\(961\) −25.7993 −0.832234
\(962\) 62.3391 2.00989
\(963\) 0 0
\(964\) −38.6406 −1.24453
\(965\) 10.2823 0.330997
\(966\) 0 0
\(967\) 39.4687 1.26923 0.634615 0.772829i \(-0.281159\pi\)
0.634615 + 0.772829i \(0.281159\pi\)
\(968\) −48.0715 −1.54508
\(969\) 0 0
\(970\) −12.2643 −0.393783
\(971\) 14.0810 0.451882 0.225941 0.974141i \(-0.427454\pi\)
0.225941 + 0.974141i \(0.427454\pi\)
\(972\) 0 0
\(973\) −1.09611 −0.0351396
\(974\) 11.5090 0.368772
\(975\) 0 0
\(976\) −40.7080 −1.30303
\(977\) 24.7863 0.792983 0.396491 0.918038i \(-0.370228\pi\)
0.396491 + 0.918038i \(0.370228\pi\)
\(978\) 0 0
\(979\) −2.54938 −0.0814785
\(980\) 17.0613 0.545004
\(981\) 0 0
\(982\) 24.1297 0.770009
\(983\) −17.2441 −0.550000 −0.275000 0.961444i \(-0.588678\pi\)
−0.275000 + 0.961444i \(0.588678\pi\)
\(984\) 0 0
\(985\) −11.5822 −0.369040
\(986\) 27.3671 0.871547
\(987\) 0 0
\(988\) 14.2768 0.454207
\(989\) −50.9107 −1.61887
\(990\) 0 0
\(991\) −56.9132 −1.80791 −0.903953 0.427631i \(-0.859348\pi\)
−0.903953 + 0.427631i \(0.859348\pi\)
\(992\) −3.73350 −0.118539
\(993\) 0 0
\(994\) −6.34853 −0.201363
\(995\) 6.87418 0.217926
\(996\) 0 0
\(997\) −15.5192 −0.491498 −0.245749 0.969333i \(-0.579034\pi\)
−0.245749 + 0.969333i \(0.579034\pi\)
\(998\) 71.9966 2.27901
\(999\) 0 0
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 2151.2.a.k.1.18 yes 20
3.2 odd 2 2151.2.a.j.1.3 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.3 20 3.2 odd 2
2151.2.a.k.1.18 yes 20 1.1 even 1 trivial