Properties

Label 2151.2.a.j.1.14
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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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: \(1\)
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 1.14
Root \(-0.921026\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.921026 q^{2} -1.15171 q^{4} -1.07676 q^{5} +3.99910 q^{7} -2.90281 q^{8} +O(q^{10})\) \(q+0.921026 q^{2} -1.15171 q^{4} -1.07676 q^{5} +3.99910 q^{7} -2.90281 q^{8} -0.991725 q^{10} -3.85807 q^{11} +1.10344 q^{13} +3.68328 q^{14} -0.370136 q^{16} -4.98207 q^{17} +6.77486 q^{19} +1.24012 q^{20} -3.55338 q^{22} -4.48583 q^{23} -3.84058 q^{25} +1.01629 q^{26} -4.60582 q^{28} +0.455789 q^{29} -0.141534 q^{31} +5.46471 q^{32} -4.58862 q^{34} -4.30608 q^{35} -5.62791 q^{37} +6.23982 q^{38} +3.12563 q^{40} -3.87964 q^{41} -5.21280 q^{43} +4.44339 q^{44} -4.13156 q^{46} +0.0217628 q^{47} +8.99284 q^{49} -3.53728 q^{50} -1.27084 q^{52} +8.05621 q^{53} +4.15422 q^{55} -11.6086 q^{56} +0.419793 q^{58} -14.3144 q^{59} -13.6860 q^{61} -0.130356 q^{62} +5.77341 q^{64} -1.18814 q^{65} -8.40462 q^{67} +5.73791 q^{68} -3.96601 q^{70} -1.78926 q^{71} +8.35852 q^{73} -5.18345 q^{74} -7.80269 q^{76} -15.4288 q^{77} -16.7789 q^{79} +0.398548 q^{80} -3.57325 q^{82} +3.41314 q^{83} +5.36450 q^{85} -4.80112 q^{86} +11.1992 q^{88} +7.49447 q^{89} +4.41276 q^{91} +5.16638 q^{92} +0.0200441 q^{94} -7.29491 q^{95} -9.46397 q^{97} +8.28263 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.921026 0.651263 0.325632 0.945497i \(-0.394423\pi\)
0.325632 + 0.945497i \(0.394423\pi\)
\(3\) 0 0
\(4\) −1.15171 −0.575856
\(5\) −1.07676 −0.481542 −0.240771 0.970582i \(-0.577400\pi\)
−0.240771 + 0.970582i \(0.577400\pi\)
\(6\) 0 0
\(7\) 3.99910 1.51152 0.755760 0.654849i \(-0.227268\pi\)
0.755760 + 0.654849i \(0.227268\pi\)
\(8\) −2.90281 −1.02630
\(9\) 0 0
\(10\) −0.991725 −0.313611
\(11\) −3.85807 −1.16325 −0.581626 0.813456i \(-0.697583\pi\)
−0.581626 + 0.813456i \(0.697583\pi\)
\(12\) 0 0
\(13\) 1.10344 0.306038 0.153019 0.988223i \(-0.451100\pi\)
0.153019 + 0.988223i \(0.451100\pi\)
\(14\) 3.68328 0.984397
\(15\) 0 0
\(16\) −0.370136 −0.0925340
\(17\) −4.98207 −1.20833 −0.604165 0.796859i \(-0.706493\pi\)
−0.604165 + 0.796859i \(0.706493\pi\)
\(18\) 0 0
\(19\) 6.77486 1.55426 0.777130 0.629340i \(-0.216675\pi\)
0.777130 + 0.629340i \(0.216675\pi\)
\(20\) 1.24012 0.277299
\(21\) 0 0
\(22\) −3.55338 −0.757584
\(23\) −4.48583 −0.935359 −0.467680 0.883898i \(-0.654910\pi\)
−0.467680 + 0.883898i \(0.654910\pi\)
\(24\) 0 0
\(25\) −3.84058 −0.768117
\(26\) 1.01629 0.199311
\(27\) 0 0
\(28\) −4.60582 −0.870417
\(29\) 0.455789 0.0846379 0.0423189 0.999104i \(-0.486525\pi\)
0.0423189 + 0.999104i \(0.486525\pi\)
\(30\) 0 0
\(31\) −0.141534 −0.0254202 −0.0127101 0.999919i \(-0.504046\pi\)
−0.0127101 + 0.999919i \(0.504046\pi\)
\(32\) 5.46471 0.966033
\(33\) 0 0
\(34\) −4.58862 −0.786941
\(35\) −4.30608 −0.727861
\(36\) 0 0
\(37\) −5.62791 −0.925224 −0.462612 0.886561i \(-0.653088\pi\)
−0.462612 + 0.886561i \(0.653088\pi\)
\(38\) 6.23982 1.01223
\(39\) 0 0
\(40\) 3.12563 0.494206
\(41\) −3.87964 −0.605898 −0.302949 0.953007i \(-0.597971\pi\)
−0.302949 + 0.953007i \(0.597971\pi\)
\(42\) 0 0
\(43\) −5.21280 −0.794945 −0.397472 0.917614i \(-0.630113\pi\)
−0.397472 + 0.917614i \(0.630113\pi\)
\(44\) 4.44339 0.669866
\(45\) 0 0
\(46\) −4.13156 −0.609165
\(47\) 0.0217628 0.00317443 0.00158722 0.999999i \(-0.499495\pi\)
0.00158722 + 0.999999i \(0.499495\pi\)
\(48\) 0 0
\(49\) 8.99284 1.28469
\(50\) −3.53728 −0.500246
\(51\) 0 0
\(52\) −1.27084 −0.176234
\(53\) 8.05621 1.10661 0.553303 0.832980i \(-0.313367\pi\)
0.553303 + 0.832980i \(0.313367\pi\)
\(54\) 0 0
\(55\) 4.15422 0.560155
\(56\) −11.6086 −1.55127
\(57\) 0 0
\(58\) 0.419793 0.0551215
\(59\) −14.3144 −1.86358 −0.931790 0.362998i \(-0.881753\pi\)
−0.931790 + 0.362998i \(0.881753\pi\)
\(60\) 0 0
\(61\) −13.6860 −1.75232 −0.876159 0.482023i \(-0.839902\pi\)
−0.876159 + 0.482023i \(0.839902\pi\)
\(62\) −0.130356 −0.0165553
\(63\) 0 0
\(64\) 5.77341 0.721676
\(65\) −1.18814 −0.147370
\(66\) 0 0
\(67\) −8.40462 −1.02679 −0.513394 0.858153i \(-0.671612\pi\)
−0.513394 + 0.858153i \(0.671612\pi\)
\(68\) 5.73791 0.695824
\(69\) 0 0
\(70\) −3.96601 −0.474029
\(71\) −1.78926 −0.212346 −0.106173 0.994348i \(-0.533860\pi\)
−0.106173 + 0.994348i \(0.533860\pi\)
\(72\) 0 0
\(73\) 8.35852 0.978291 0.489146 0.872202i \(-0.337309\pi\)
0.489146 + 0.872202i \(0.337309\pi\)
\(74\) −5.18345 −0.602564
\(75\) 0 0
\(76\) −7.80269 −0.895030
\(77\) −15.4288 −1.75828
\(78\) 0 0
\(79\) −16.7789 −1.88778 −0.943889 0.330264i \(-0.892862\pi\)
−0.943889 + 0.330264i \(0.892862\pi\)
\(80\) 0.398548 0.0445591
\(81\) 0 0
\(82\) −3.57325 −0.394599
\(83\) 3.41314 0.374641 0.187320 0.982299i \(-0.440020\pi\)
0.187320 + 0.982299i \(0.440020\pi\)
\(84\) 0 0
\(85\) 5.36450 0.581862
\(86\) −4.80112 −0.517719
\(87\) 0 0
\(88\) 11.1992 1.19384
\(89\) 7.49447 0.794413 0.397206 0.917729i \(-0.369980\pi\)
0.397206 + 0.917729i \(0.369980\pi\)
\(90\) 0 0
\(91\) 4.41276 0.462583
\(92\) 5.16638 0.538632
\(93\) 0 0
\(94\) 0.0200441 0.00206739
\(95\) −7.29491 −0.748442
\(96\) 0 0
\(97\) −9.46397 −0.960920 −0.480460 0.877017i \(-0.659530\pi\)
−0.480460 + 0.877017i \(0.659530\pi\)
\(98\) 8.28263 0.836672
\(99\) 0 0
\(100\) 4.42325 0.442325
\(101\) 16.8805 1.67968 0.839838 0.542837i \(-0.182650\pi\)
0.839838 + 0.542837i \(0.182650\pi\)
\(102\) 0 0
\(103\) −0.103493 −0.0101975 −0.00509875 0.999987i \(-0.501623\pi\)
−0.00509875 + 0.999987i \(0.501623\pi\)
\(104\) −3.20306 −0.314086
\(105\) 0 0
\(106\) 7.41997 0.720691
\(107\) −18.4298 −1.78167 −0.890837 0.454323i \(-0.849881\pi\)
−0.890837 + 0.454323i \(0.849881\pi\)
\(108\) 0 0
\(109\) −3.54270 −0.339329 −0.169664 0.985502i \(-0.554268\pi\)
−0.169664 + 0.985502i \(0.554268\pi\)
\(110\) 3.82615 0.364809
\(111\) 0 0
\(112\) −1.48021 −0.139867
\(113\) −13.3935 −1.25995 −0.629976 0.776614i \(-0.716935\pi\)
−0.629976 + 0.776614i \(0.716935\pi\)
\(114\) 0 0
\(115\) 4.83016 0.450415
\(116\) −0.524937 −0.0487392
\(117\) 0 0
\(118\) −13.1840 −1.21368
\(119\) −19.9238 −1.82641
\(120\) 0 0
\(121\) 3.88472 0.353157
\(122\) −12.6052 −1.14122
\(123\) 0 0
\(124\) 0.163006 0.0146384
\(125\) 9.51920 0.851423
\(126\) 0 0
\(127\) −2.82374 −0.250566 −0.125283 0.992121i \(-0.539984\pi\)
−0.125283 + 0.992121i \(0.539984\pi\)
\(128\) −5.61196 −0.496032
\(129\) 0 0
\(130\) −1.09431 −0.0959769
\(131\) 7.18818 0.628034 0.314017 0.949417i \(-0.398325\pi\)
0.314017 + 0.949417i \(0.398325\pi\)
\(132\) 0 0
\(133\) 27.0934 2.34929
\(134\) −7.74087 −0.668709
\(135\) 0 0
\(136\) 14.4620 1.24011
\(137\) 9.45738 0.807999 0.403999 0.914759i \(-0.367620\pi\)
0.403999 + 0.914759i \(0.367620\pi\)
\(138\) 0 0
\(139\) 7.46510 0.633182 0.316591 0.948562i \(-0.397462\pi\)
0.316591 + 0.948562i \(0.397462\pi\)
\(140\) 4.95937 0.419143
\(141\) 0 0
\(142\) −1.64795 −0.138293
\(143\) −4.25714 −0.356000
\(144\) 0 0
\(145\) −0.490776 −0.0407567
\(146\) 7.69841 0.637125
\(147\) 0 0
\(148\) 6.48174 0.532795
\(149\) −9.92379 −0.812988 −0.406494 0.913653i \(-0.633249\pi\)
−0.406494 + 0.913653i \(0.633249\pi\)
\(150\) 0 0
\(151\) 9.70932 0.790133 0.395067 0.918652i \(-0.370722\pi\)
0.395067 + 0.918652i \(0.370722\pi\)
\(152\) −19.6661 −1.59513
\(153\) 0 0
\(154\) −14.2104 −1.14510
\(155\) 0.152398 0.0122409
\(156\) 0 0
\(157\) −16.4632 −1.31391 −0.656955 0.753930i \(-0.728156\pi\)
−0.656955 + 0.753930i \(0.728156\pi\)
\(158\) −15.4538 −1.22944
\(159\) 0 0
\(160\) −5.88419 −0.465186
\(161\) −17.9393 −1.41381
\(162\) 0 0
\(163\) 20.4963 1.60540 0.802698 0.596386i \(-0.203397\pi\)
0.802698 + 0.596386i \(0.203397\pi\)
\(164\) 4.46823 0.348910
\(165\) 0 0
\(166\) 3.14359 0.243990
\(167\) 2.79371 0.216184 0.108092 0.994141i \(-0.465526\pi\)
0.108092 + 0.994141i \(0.465526\pi\)
\(168\) 0 0
\(169\) −11.7824 −0.906341
\(170\) 4.94085 0.378946
\(171\) 0 0
\(172\) 6.00365 0.457774
\(173\) −16.5368 −1.25727 −0.628634 0.777701i \(-0.716386\pi\)
−0.628634 + 0.777701i \(0.716386\pi\)
\(174\) 0 0
\(175\) −15.3589 −1.16102
\(176\) 1.42801 0.107640
\(177\) 0 0
\(178\) 6.90260 0.517372
\(179\) −8.54297 −0.638531 −0.319266 0.947665i \(-0.603436\pi\)
−0.319266 + 0.947665i \(0.603436\pi\)
\(180\) 0 0
\(181\) 0.983405 0.0730959 0.0365480 0.999332i \(-0.488364\pi\)
0.0365480 + 0.999332i \(0.488364\pi\)
\(182\) 4.06426 0.301263
\(183\) 0 0
\(184\) 13.0215 0.959957
\(185\) 6.05992 0.445534
\(186\) 0 0
\(187\) 19.2212 1.40559
\(188\) −0.0250645 −0.00182802
\(189\) 0 0
\(190\) −6.71880 −0.487433
\(191\) −19.7781 −1.43109 −0.715547 0.698564i \(-0.753823\pi\)
−0.715547 + 0.698564i \(0.753823\pi\)
\(192\) 0 0
\(193\) 21.0252 1.51342 0.756712 0.653748i \(-0.226805\pi\)
0.756712 + 0.653748i \(0.226805\pi\)
\(194\) −8.71656 −0.625812
\(195\) 0 0
\(196\) −10.3572 −0.739797
\(197\) 18.7152 1.33340 0.666702 0.745325i \(-0.267706\pi\)
0.666702 + 0.745325i \(0.267706\pi\)
\(198\) 0 0
\(199\) −4.36925 −0.309728 −0.154864 0.987936i \(-0.549494\pi\)
−0.154864 + 0.987936i \(0.549494\pi\)
\(200\) 11.1485 0.788316
\(201\) 0 0
\(202\) 15.5474 1.09391
\(203\) 1.82275 0.127932
\(204\) 0 0
\(205\) 4.17745 0.291766
\(206\) −0.0953200 −0.00664126
\(207\) 0 0
\(208\) −0.408422 −0.0283189
\(209\) −26.1379 −1.80800
\(210\) 0 0
\(211\) −14.9305 −1.02786 −0.513928 0.857833i \(-0.671810\pi\)
−0.513928 + 0.857833i \(0.671810\pi\)
\(212\) −9.27843 −0.637245
\(213\) 0 0
\(214\) −16.9743 −1.16034
\(215\) 5.61295 0.382800
\(216\) 0 0
\(217\) −0.566008 −0.0384231
\(218\) −3.26291 −0.220992
\(219\) 0 0
\(220\) −4.78447 −0.322569
\(221\) −5.49740 −0.369795
\(222\) 0 0
\(223\) −3.01452 −0.201867 −0.100934 0.994893i \(-0.532183\pi\)
−0.100934 + 0.994893i \(0.532183\pi\)
\(224\) 21.8539 1.46018
\(225\) 0 0
\(226\) −12.3357 −0.820561
\(227\) 27.4158 1.81965 0.909825 0.414993i \(-0.136216\pi\)
0.909825 + 0.414993i \(0.136216\pi\)
\(228\) 0 0
\(229\) −23.6259 −1.56124 −0.780621 0.625005i \(-0.785097\pi\)
−0.780621 + 0.625005i \(0.785097\pi\)
\(230\) 4.44870 0.293339
\(231\) 0 0
\(232\) −1.32307 −0.0868636
\(233\) −7.58457 −0.496882 −0.248441 0.968647i \(-0.579918\pi\)
−0.248441 + 0.968647i \(0.579918\pi\)
\(234\) 0 0
\(235\) −0.0234334 −0.00152862
\(236\) 16.4861 1.07315
\(237\) 0 0
\(238\) −18.3504 −1.18948
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) 29.6913 1.91259 0.956293 0.292410i \(-0.0944570\pi\)
0.956293 + 0.292410i \(0.0944570\pi\)
\(242\) 3.57793 0.229998
\(243\) 0 0
\(244\) 15.7624 1.00908
\(245\) −9.68314 −0.618633
\(246\) 0 0
\(247\) 7.47563 0.475663
\(248\) 0.410845 0.0260887
\(249\) 0 0
\(250\) 8.76743 0.554501
\(251\) 22.5521 1.42347 0.711737 0.702446i \(-0.247909\pi\)
0.711737 + 0.702446i \(0.247909\pi\)
\(252\) 0 0
\(253\) 17.3066 1.08806
\(254\) −2.60074 −0.163185
\(255\) 0 0
\(256\) −16.7156 −1.04472
\(257\) 14.9158 0.930420 0.465210 0.885200i \(-0.345979\pi\)
0.465210 + 0.885200i \(0.345979\pi\)
\(258\) 0 0
\(259\) −22.5066 −1.39849
\(260\) 1.36839 0.0848641
\(261\) 0 0
\(262\) 6.62050 0.409016
\(263\) 7.93666 0.489395 0.244698 0.969599i \(-0.421311\pi\)
0.244698 + 0.969599i \(0.421311\pi\)
\(264\) 0 0
\(265\) −8.67461 −0.532877
\(266\) 24.9537 1.53001
\(267\) 0 0
\(268\) 9.67970 0.591282
\(269\) 28.7193 1.75105 0.875524 0.483174i \(-0.160516\pi\)
0.875524 + 0.483174i \(0.160516\pi\)
\(270\) 0 0
\(271\) −1.47810 −0.0897884 −0.0448942 0.998992i \(-0.514295\pi\)
−0.0448942 + 0.998992i \(0.514295\pi\)
\(272\) 1.84404 0.111812
\(273\) 0 0
\(274\) 8.71049 0.526220
\(275\) 14.8173 0.893514
\(276\) 0 0
\(277\) 29.4540 1.76972 0.884859 0.465859i \(-0.154255\pi\)
0.884859 + 0.465859i \(0.154255\pi\)
\(278\) 6.87555 0.412368
\(279\) 0 0
\(280\) 12.4997 0.747001
\(281\) −16.0239 −0.955904 −0.477952 0.878386i \(-0.658621\pi\)
−0.477952 + 0.878386i \(0.658621\pi\)
\(282\) 0 0
\(283\) 8.00350 0.475759 0.237879 0.971295i \(-0.423548\pi\)
0.237879 + 0.971295i \(0.423548\pi\)
\(284\) 2.06071 0.122281
\(285\) 0 0
\(286\) −3.92093 −0.231850
\(287\) −15.5151 −0.915827
\(288\) 0 0
\(289\) 7.82105 0.460062
\(290\) −0.452017 −0.0265434
\(291\) 0 0
\(292\) −9.62661 −0.563355
\(293\) −12.1458 −0.709567 −0.354784 0.934948i \(-0.615445\pi\)
−0.354784 + 0.934948i \(0.615445\pi\)
\(294\) 0 0
\(295\) 15.4132 0.897393
\(296\) 16.3368 0.949554
\(297\) 0 0
\(298\) −9.14006 −0.529470
\(299\) −4.94982 −0.286256
\(300\) 0 0
\(301\) −20.8465 −1.20157
\(302\) 8.94253 0.514585
\(303\) 0 0
\(304\) −2.50762 −0.143822
\(305\) 14.7366 0.843815
\(306\) 0 0
\(307\) 18.4095 1.05069 0.525343 0.850891i \(-0.323937\pi\)
0.525343 + 0.850891i \(0.323937\pi\)
\(308\) 17.7696 1.01252
\(309\) 0 0
\(310\) 0.140363 0.00797206
\(311\) 33.7729 1.91509 0.957543 0.288292i \(-0.0930874\pi\)
0.957543 + 0.288292i \(0.0930874\pi\)
\(312\) 0 0
\(313\) −13.7399 −0.776625 −0.388312 0.921528i \(-0.626942\pi\)
−0.388312 + 0.921528i \(0.626942\pi\)
\(314\) −15.1631 −0.855702
\(315\) 0 0
\(316\) 19.3245 1.08709
\(317\) 4.88391 0.274308 0.137154 0.990550i \(-0.456205\pi\)
0.137154 + 0.990550i \(0.456205\pi\)
\(318\) 0 0
\(319\) −1.75847 −0.0984552
\(320\) −6.21659 −0.347518
\(321\) 0 0
\(322\) −16.5225 −0.920765
\(323\) −33.7528 −1.87806
\(324\) 0 0
\(325\) −4.23784 −0.235073
\(326\) 18.8776 1.04554
\(327\) 0 0
\(328\) 11.2618 0.621832
\(329\) 0.0870317 0.00479822
\(330\) 0 0
\(331\) 6.51734 0.358225 0.179113 0.983829i \(-0.442677\pi\)
0.179113 + 0.983829i \(0.442677\pi\)
\(332\) −3.93096 −0.215739
\(333\) 0 0
\(334\) 2.57308 0.140793
\(335\) 9.04977 0.494442
\(336\) 0 0
\(337\) 2.02175 0.110132 0.0550659 0.998483i \(-0.482463\pi\)
0.0550659 + 0.998483i \(0.482463\pi\)
\(338\) −10.8519 −0.590267
\(339\) 0 0
\(340\) −6.17836 −0.335069
\(341\) 0.546048 0.0295701
\(342\) 0 0
\(343\) 7.96956 0.430316
\(344\) 15.1318 0.815850
\(345\) 0 0
\(346\) −15.2308 −0.818813
\(347\) −18.7227 −1.00508 −0.502542 0.864553i \(-0.667602\pi\)
−0.502542 + 0.864553i \(0.667602\pi\)
\(348\) 0 0
\(349\) 14.1099 0.755283 0.377642 0.925952i \(-0.376735\pi\)
0.377642 + 0.925952i \(0.376735\pi\)
\(350\) −14.1459 −0.756132
\(351\) 0 0
\(352\) −21.0832 −1.12374
\(353\) 20.7618 1.10504 0.552519 0.833500i \(-0.313667\pi\)
0.552519 + 0.833500i \(0.313667\pi\)
\(354\) 0 0
\(355\) 1.92661 0.102254
\(356\) −8.63147 −0.457467
\(357\) 0 0
\(358\) −7.86829 −0.415852
\(359\) −3.96492 −0.209261 −0.104630 0.994511i \(-0.533366\pi\)
−0.104630 + 0.994511i \(0.533366\pi\)
\(360\) 0 0
\(361\) 26.8987 1.41572
\(362\) 0.905742 0.0476047
\(363\) 0 0
\(364\) −5.08222 −0.266381
\(365\) −9.00014 −0.471089
\(366\) 0 0
\(367\) −19.4443 −1.01499 −0.507493 0.861656i \(-0.669428\pi\)
−0.507493 + 0.861656i \(0.669428\pi\)
\(368\) 1.66037 0.0865525
\(369\) 0 0
\(370\) 5.58134 0.290160
\(371\) 32.2176 1.67266
\(372\) 0 0
\(373\) −3.77156 −0.195284 −0.0976419 0.995222i \(-0.531130\pi\)
−0.0976419 + 0.995222i \(0.531130\pi\)
\(374\) 17.7032 0.915411
\(375\) 0 0
\(376\) −0.0631732 −0.00325791
\(377\) 0.502934 0.0259024
\(378\) 0 0
\(379\) 15.1994 0.780740 0.390370 0.920658i \(-0.372347\pi\)
0.390370 + 0.920658i \(0.372347\pi\)
\(380\) 8.40163 0.430995
\(381\) 0 0
\(382\) −18.2162 −0.932019
\(383\) 7.52457 0.384488 0.192244 0.981347i \(-0.438424\pi\)
0.192244 + 0.981347i \(0.438424\pi\)
\(384\) 0 0
\(385\) 16.6132 0.846686
\(386\) 19.3647 0.985638
\(387\) 0 0
\(388\) 10.8998 0.553352
\(389\) 30.3583 1.53923 0.769613 0.638510i \(-0.220449\pi\)
0.769613 + 0.638510i \(0.220449\pi\)
\(390\) 0 0
\(391\) 22.3487 1.13022
\(392\) −26.1045 −1.31847
\(393\) 0 0
\(394\) 17.2372 0.868397
\(395\) 18.0669 0.909045
\(396\) 0 0
\(397\) 4.62399 0.232071 0.116036 0.993245i \(-0.462981\pi\)
0.116036 + 0.993245i \(0.462981\pi\)
\(398\) −4.02419 −0.201715
\(399\) 0 0
\(400\) 1.42154 0.0710769
\(401\) −17.7060 −0.884194 −0.442097 0.896967i \(-0.645765\pi\)
−0.442097 + 0.896967i \(0.645765\pi\)
\(402\) 0 0
\(403\) −0.156173 −0.00777956
\(404\) −19.4415 −0.967251
\(405\) 0 0
\(406\) 1.67880 0.0833173
\(407\) 21.7129 1.07627
\(408\) 0 0
\(409\) 4.47676 0.221362 0.110681 0.993856i \(-0.464697\pi\)
0.110681 + 0.993856i \(0.464697\pi\)
\(410\) 3.84754 0.190016
\(411\) 0 0
\(412\) 0.119194 0.00587229
\(413\) −57.2449 −2.81684
\(414\) 0 0
\(415\) −3.67514 −0.180405
\(416\) 6.02996 0.295643
\(417\) 0 0
\(418\) −24.0737 −1.17748
\(419\) 22.5224 1.10029 0.550146 0.835069i \(-0.314572\pi\)
0.550146 + 0.835069i \(0.314572\pi\)
\(420\) 0 0
\(421\) −31.4367 −1.53213 −0.766067 0.642761i \(-0.777789\pi\)
−0.766067 + 0.642761i \(0.777789\pi\)
\(422\) −13.7514 −0.669405
\(423\) 0 0
\(424\) −23.3856 −1.13571
\(425\) 19.1341 0.928139
\(426\) 0 0
\(427\) −54.7319 −2.64866
\(428\) 21.2258 1.02599
\(429\) 0 0
\(430\) 5.16967 0.249303
\(431\) −7.61874 −0.366982 −0.183491 0.983021i \(-0.558740\pi\)
−0.183491 + 0.983021i \(0.558740\pi\)
\(432\) 0 0
\(433\) −6.89352 −0.331281 −0.165641 0.986186i \(-0.552969\pi\)
−0.165641 + 0.986186i \(0.552969\pi\)
\(434\) −0.521308 −0.0250236
\(435\) 0 0
\(436\) 4.08016 0.195404
\(437\) −30.3908 −1.45379
\(438\) 0 0
\(439\) 19.2553 0.919003 0.459502 0.888177i \(-0.348028\pi\)
0.459502 + 0.888177i \(0.348028\pi\)
\(440\) −12.0589 −0.574886
\(441\) 0 0
\(442\) −5.06325 −0.240834
\(443\) −6.89520 −0.327601 −0.163800 0.986494i \(-0.552375\pi\)
−0.163800 + 0.986494i \(0.552375\pi\)
\(444\) 0 0
\(445\) −8.06976 −0.382543
\(446\) −2.77645 −0.131469
\(447\) 0 0
\(448\) 23.0885 1.09083
\(449\) 6.70630 0.316490 0.158245 0.987400i \(-0.449416\pi\)
0.158245 + 0.987400i \(0.449416\pi\)
\(450\) 0 0
\(451\) 14.9679 0.704812
\(452\) 15.4254 0.725551
\(453\) 0 0
\(454\) 25.2506 1.18507
\(455\) −4.75149 −0.222753
\(456\) 0 0
\(457\) −30.4544 −1.42460 −0.712298 0.701877i \(-0.752345\pi\)
−0.712298 + 0.701877i \(0.752345\pi\)
\(458\) −21.7600 −1.01678
\(459\) 0 0
\(460\) −5.56296 −0.259374
\(461\) −4.43558 −0.206585 −0.103293 0.994651i \(-0.532938\pi\)
−0.103293 + 0.994651i \(0.532938\pi\)
\(462\) 0 0
\(463\) −29.3900 −1.36587 −0.682935 0.730480i \(-0.739297\pi\)
−0.682935 + 0.730480i \(0.739297\pi\)
\(464\) −0.168704 −0.00783188
\(465\) 0 0
\(466\) −6.98558 −0.323601
\(467\) −7.49113 −0.346648 −0.173324 0.984865i \(-0.555451\pi\)
−0.173324 + 0.984865i \(0.555451\pi\)
\(468\) 0 0
\(469\) −33.6109 −1.55201
\(470\) −0.0215827 −0.000995537 0
\(471\) 0 0
\(472\) 41.5520 1.91259
\(473\) 20.1114 0.924722
\(474\) 0 0
\(475\) −26.0194 −1.19385
\(476\) 22.9465 1.05175
\(477\) 0 0
\(478\) −0.921026 −0.0421267
\(479\) 5.20065 0.237624 0.118812 0.992917i \(-0.462091\pi\)
0.118812 + 0.992917i \(0.462091\pi\)
\(480\) 0 0
\(481\) −6.21005 −0.283154
\(482\) 27.3465 1.24560
\(483\) 0 0
\(484\) −4.47408 −0.203367
\(485\) 10.1904 0.462724
\(486\) 0 0
\(487\) −26.9614 −1.22174 −0.610868 0.791732i \(-0.709180\pi\)
−0.610868 + 0.791732i \(0.709180\pi\)
\(488\) 39.7279 1.79840
\(489\) 0 0
\(490\) −8.91842 −0.402893
\(491\) 5.89472 0.266025 0.133013 0.991114i \(-0.457535\pi\)
0.133013 + 0.991114i \(0.457535\pi\)
\(492\) 0 0
\(493\) −2.27077 −0.102270
\(494\) 6.88524 0.309782
\(495\) 0 0
\(496\) 0.0523868 0.00235223
\(497\) −7.15544 −0.320965
\(498\) 0 0
\(499\) 13.9968 0.626585 0.313293 0.949657i \(-0.398568\pi\)
0.313293 + 0.949657i \(0.398568\pi\)
\(500\) −10.9634 −0.490297
\(501\) 0 0
\(502\) 20.7710 0.927056
\(503\) −21.0763 −0.939747 −0.469873 0.882734i \(-0.655700\pi\)
−0.469873 + 0.882734i \(0.655700\pi\)
\(504\) 0 0
\(505\) −18.1763 −0.808835
\(506\) 15.9399 0.708613
\(507\) 0 0
\(508\) 3.25213 0.144290
\(509\) −26.8372 −1.18954 −0.594768 0.803897i \(-0.702756\pi\)
−0.594768 + 0.803897i \(0.702756\pi\)
\(510\) 0 0
\(511\) 33.4266 1.47871
\(512\) −4.17155 −0.184358
\(513\) 0 0
\(514\) 13.7378 0.605949
\(515\) 0.111438 0.00491053
\(516\) 0 0
\(517\) −0.0839625 −0.00369267
\(518\) −20.7292 −0.910788
\(519\) 0 0
\(520\) 3.44893 0.151246
\(521\) −38.1749 −1.67247 −0.836237 0.548368i \(-0.815249\pi\)
−0.836237 + 0.548368i \(0.815249\pi\)
\(522\) 0 0
\(523\) −27.7790 −1.21469 −0.607344 0.794439i \(-0.707765\pi\)
−0.607344 + 0.794439i \(0.707765\pi\)
\(524\) −8.27871 −0.361657
\(525\) 0 0
\(526\) 7.30986 0.318725
\(527\) 0.705132 0.0307160
\(528\) 0 0
\(529\) −2.87737 −0.125103
\(530\) −7.98954 −0.347043
\(531\) 0 0
\(532\) −31.2038 −1.35285
\(533\) −4.28094 −0.185428
\(534\) 0 0
\(535\) 19.8445 0.857952
\(536\) 24.3970 1.05379
\(537\) 0 0
\(538\) 26.4512 1.14039
\(539\) −34.6950 −1.49442
\(540\) 0 0
\(541\) 27.8323 1.19660 0.598302 0.801270i \(-0.295842\pi\)
0.598302 + 0.801270i \(0.295842\pi\)
\(542\) −1.36137 −0.0584759
\(543\) 0 0
\(544\) −27.2256 −1.16729
\(545\) 3.81464 0.163401
\(546\) 0 0
\(547\) −19.9472 −0.852880 −0.426440 0.904516i \(-0.640232\pi\)
−0.426440 + 0.904516i \(0.640232\pi\)
\(548\) −10.8922 −0.465291
\(549\) 0 0
\(550\) 13.6471 0.581913
\(551\) 3.08791 0.131549
\(552\) 0 0
\(553\) −67.1007 −2.85341
\(554\) 27.1279 1.15255
\(555\) 0 0
\(556\) −8.59765 −0.364621
\(557\) −13.5520 −0.574215 −0.287107 0.957898i \(-0.592694\pi\)
−0.287107 + 0.957898i \(0.592694\pi\)
\(558\) 0 0
\(559\) −5.75200 −0.243283
\(560\) 1.59384 0.0673519
\(561\) 0 0
\(562\) −14.7584 −0.622546
\(563\) −28.1055 −1.18451 −0.592253 0.805752i \(-0.701761\pi\)
−0.592253 + 0.805752i \(0.701761\pi\)
\(564\) 0 0
\(565\) 14.4216 0.606721
\(566\) 7.37143 0.309844
\(567\) 0 0
\(568\) 5.19388 0.217930
\(569\) −5.43746 −0.227950 −0.113975 0.993484i \(-0.536358\pi\)
−0.113975 + 0.993484i \(0.536358\pi\)
\(570\) 0 0
\(571\) 14.9824 0.626993 0.313496 0.949589i \(-0.398500\pi\)
0.313496 + 0.949589i \(0.398500\pi\)
\(572\) 4.90299 0.205005
\(573\) 0 0
\(574\) −14.2898 −0.596444
\(575\) 17.2282 0.718465
\(576\) 0 0
\(577\) 21.1328 0.879768 0.439884 0.898055i \(-0.355020\pi\)
0.439884 + 0.898055i \(0.355020\pi\)
\(578\) 7.20339 0.299621
\(579\) 0 0
\(580\) 0.565232 0.0234700
\(581\) 13.6495 0.566277
\(582\) 0 0
\(583\) −31.0814 −1.28726
\(584\) −24.2632 −1.00402
\(585\) 0 0
\(586\) −11.1866 −0.462115
\(587\) 26.9642 1.11293 0.556465 0.830871i \(-0.312157\pi\)
0.556465 + 0.830871i \(0.312157\pi\)
\(588\) 0 0
\(589\) −0.958872 −0.0395096
\(590\) 14.1960 0.584439
\(591\) 0 0
\(592\) 2.08309 0.0856146
\(593\) 1.66422 0.0683415 0.0341708 0.999416i \(-0.489121\pi\)
0.0341708 + 0.999416i \(0.489121\pi\)
\(594\) 0 0
\(595\) 21.4532 0.879496
\(596\) 11.4293 0.468164
\(597\) 0 0
\(598\) −4.55891 −0.186428
\(599\) −19.8878 −0.812592 −0.406296 0.913741i \(-0.633180\pi\)
−0.406296 + 0.913741i \(0.633180\pi\)
\(600\) 0 0
\(601\) −13.8715 −0.565832 −0.282916 0.959145i \(-0.591302\pi\)
−0.282916 + 0.959145i \(0.591302\pi\)
\(602\) −19.2002 −0.782542
\(603\) 0 0
\(604\) −11.1823 −0.455003
\(605\) −4.18292 −0.170060
\(606\) 0 0
\(607\) −21.6595 −0.879131 −0.439565 0.898211i \(-0.644868\pi\)
−0.439565 + 0.898211i \(0.644868\pi\)
\(608\) 37.0226 1.50147
\(609\) 0 0
\(610\) 13.5728 0.549546
\(611\) 0.0240139 0.000971497 0
\(612\) 0 0
\(613\) 36.5608 1.47668 0.738339 0.674430i \(-0.235610\pi\)
0.738339 + 0.674430i \(0.235610\pi\)
\(614\) 16.9556 0.684273
\(615\) 0 0
\(616\) 44.7869 1.80452
\(617\) 20.3869 0.820746 0.410373 0.911918i \(-0.365399\pi\)
0.410373 + 0.911918i \(0.365399\pi\)
\(618\) 0 0
\(619\) 10.3450 0.415802 0.207901 0.978150i \(-0.433337\pi\)
0.207901 + 0.978150i \(0.433337\pi\)
\(620\) −0.175519 −0.00704900
\(621\) 0 0
\(622\) 31.1057 1.24722
\(623\) 29.9712 1.20077
\(624\) 0 0
\(625\) 8.95301 0.358121
\(626\) −12.6548 −0.505787
\(627\) 0 0
\(628\) 18.9609 0.756623
\(629\) 28.0387 1.11798
\(630\) 0 0
\(631\) −10.0342 −0.399454 −0.199727 0.979852i \(-0.564005\pi\)
−0.199727 + 0.979852i \(0.564005\pi\)
\(632\) 48.7060 1.93742
\(633\) 0 0
\(634\) 4.49820 0.178647
\(635\) 3.04049 0.120658
\(636\) 0 0
\(637\) 9.92302 0.393164
\(638\) −1.61959 −0.0641203
\(639\) 0 0
\(640\) 6.04274 0.238860
\(641\) 32.5006 1.28370 0.641849 0.766831i \(-0.278168\pi\)
0.641849 + 0.766831i \(0.278168\pi\)
\(642\) 0 0
\(643\) −1.29949 −0.0512467 −0.0256234 0.999672i \(-0.508157\pi\)
−0.0256234 + 0.999672i \(0.508157\pi\)
\(644\) 20.6609 0.814153
\(645\) 0 0
\(646\) −31.0872 −1.22311
\(647\) 11.1792 0.439499 0.219749 0.975556i \(-0.429476\pi\)
0.219749 + 0.975556i \(0.429476\pi\)
\(648\) 0 0
\(649\) 55.2261 2.16781
\(650\) −3.90316 −0.153095
\(651\) 0 0
\(652\) −23.6059 −0.924477
\(653\) −16.3420 −0.639512 −0.319756 0.947500i \(-0.603601\pi\)
−0.319756 + 0.947500i \(0.603601\pi\)
\(654\) 0 0
\(655\) −7.73995 −0.302425
\(656\) 1.43599 0.0560662
\(657\) 0 0
\(658\) 0.0801585 0.00312490
\(659\) −40.9312 −1.59445 −0.797227 0.603680i \(-0.793701\pi\)
−0.797227 + 0.603680i \(0.793701\pi\)
\(660\) 0 0
\(661\) 24.6581 0.959089 0.479544 0.877518i \(-0.340802\pi\)
0.479544 + 0.877518i \(0.340802\pi\)
\(662\) 6.00263 0.233299
\(663\) 0 0
\(664\) −9.90769 −0.384493
\(665\) −29.1731 −1.13128
\(666\) 0 0
\(667\) −2.04459 −0.0791668
\(668\) −3.21755 −0.124491
\(669\) 0 0
\(670\) 8.33507 0.322012
\(671\) 52.8017 2.03839
\(672\) 0 0
\(673\) 24.1505 0.930933 0.465467 0.885065i \(-0.345887\pi\)
0.465467 + 0.885065i \(0.345887\pi\)
\(674\) 1.86208 0.0717248
\(675\) 0 0
\(676\) 13.5700 0.521922
\(677\) −15.9466 −0.612876 −0.306438 0.951891i \(-0.599137\pi\)
−0.306438 + 0.951891i \(0.599137\pi\)
\(678\) 0 0
\(679\) −37.8474 −1.45245
\(680\) −15.5721 −0.597164
\(681\) 0 0
\(682\) 0.502924 0.0192579
\(683\) 41.1186 1.57336 0.786680 0.617361i \(-0.211798\pi\)
0.786680 + 0.617361i \(0.211798\pi\)
\(684\) 0 0
\(685\) −10.1833 −0.389086
\(686\) 7.34017 0.280249
\(687\) 0 0
\(688\) 1.92945 0.0735595
\(689\) 8.88951 0.338663
\(690\) 0 0
\(691\) −6.74938 −0.256759 −0.128379 0.991725i \(-0.540977\pi\)
−0.128379 + 0.991725i \(0.540977\pi\)
\(692\) 19.0456 0.724006
\(693\) 0 0
\(694\) −17.2440 −0.654575
\(695\) −8.03814 −0.304904
\(696\) 0 0
\(697\) 19.3287 0.732125
\(698\) 12.9955 0.491888
\(699\) 0 0
\(700\) 17.6890 0.668582
\(701\) −22.1034 −0.834833 −0.417417 0.908715i \(-0.637064\pi\)
−0.417417 + 0.908715i \(0.637064\pi\)
\(702\) 0 0
\(703\) −38.1283 −1.43804
\(704\) −22.2742 −0.839492
\(705\) 0 0
\(706\) 19.1221 0.719671
\(707\) 67.5070 2.53886
\(708\) 0 0
\(709\) −23.1814 −0.870594 −0.435297 0.900287i \(-0.643357\pi\)
−0.435297 + 0.900287i \(0.643357\pi\)
\(710\) 1.77445 0.0665941
\(711\) 0 0
\(712\) −21.7550 −0.815304
\(713\) 0.634896 0.0237770
\(714\) 0 0
\(715\) 4.58392 0.171429
\(716\) 9.83904 0.367702
\(717\) 0 0
\(718\) −3.65179 −0.136284
\(719\) −14.6424 −0.546068 −0.273034 0.962004i \(-0.588027\pi\)
−0.273034 + 0.962004i \(0.588027\pi\)
\(720\) 0 0
\(721\) −0.413880 −0.0154137
\(722\) 24.7744 0.922008
\(723\) 0 0
\(724\) −1.13260 −0.0420927
\(725\) −1.75050 −0.0650118
\(726\) 0 0
\(727\) −9.78781 −0.363010 −0.181505 0.983390i \(-0.558097\pi\)
−0.181505 + 0.983390i \(0.558097\pi\)
\(728\) −12.8094 −0.474747
\(729\) 0 0
\(730\) −8.28936 −0.306803
\(731\) 25.9706 0.960556
\(732\) 0 0
\(733\) −51.7138 −1.91009 −0.955047 0.296456i \(-0.904195\pi\)
−0.955047 + 0.296456i \(0.904195\pi\)
\(734\) −17.9087 −0.661024
\(735\) 0 0
\(736\) −24.5137 −0.903588
\(737\) 32.4256 1.19441
\(738\) 0 0
\(739\) −40.5262 −1.49078 −0.745390 0.666628i \(-0.767737\pi\)
−0.745390 + 0.666628i \(0.767737\pi\)
\(740\) −6.97928 −0.256564
\(741\) 0 0
\(742\) 29.6732 1.08934
\(743\) 4.45345 0.163381 0.0816906 0.996658i \(-0.473968\pi\)
0.0816906 + 0.996658i \(0.473968\pi\)
\(744\) 0 0
\(745\) 10.6856 0.391488
\(746\) −3.47370 −0.127181
\(747\) 0 0
\(748\) −22.1373 −0.809419
\(749\) −73.7026 −2.69304
\(750\) 0 0
\(751\) 53.0922 1.93736 0.968681 0.248309i \(-0.0798747\pi\)
0.968681 + 0.248309i \(0.0798747\pi\)
\(752\) −0.00805520 −0.000293743 0
\(753\) 0 0
\(754\) 0.463215 0.0168693
\(755\) −10.4546 −0.380483
\(756\) 0 0
\(757\) −8.84449 −0.321458 −0.160729 0.986999i \(-0.551385\pi\)
−0.160729 + 0.986999i \(0.551385\pi\)
\(758\) 13.9990 0.508467
\(759\) 0 0
\(760\) 21.1757 0.768124
\(761\) 20.5342 0.744364 0.372182 0.928160i \(-0.378610\pi\)
0.372182 + 0.928160i \(0.378610\pi\)
\(762\) 0 0
\(763\) −14.1676 −0.512902
\(764\) 22.7787 0.824104
\(765\) 0 0
\(766\) 6.93032 0.250403
\(767\) −15.7951 −0.570327
\(768\) 0 0
\(769\) −11.3547 −0.409462 −0.204731 0.978818i \(-0.565632\pi\)
−0.204731 + 0.978818i \(0.565632\pi\)
\(770\) 15.3012 0.551416
\(771\) 0 0
\(772\) −24.2149 −0.871514
\(773\) −40.7990 −1.46744 −0.733720 0.679452i \(-0.762217\pi\)
−0.733720 + 0.679452i \(0.762217\pi\)
\(774\) 0 0
\(775\) 0.543572 0.0195257
\(776\) 27.4721 0.986190
\(777\) 0 0
\(778\) 27.9608 1.00244
\(779\) −26.2840 −0.941723
\(780\) 0 0
\(781\) 6.90310 0.247012
\(782\) 20.5837 0.736073
\(783\) 0 0
\(784\) −3.32857 −0.118878
\(785\) 17.7270 0.632703
\(786\) 0 0
\(787\) 2.97511 0.106051 0.0530257 0.998593i \(-0.483113\pi\)
0.0530257 + 0.998593i \(0.483113\pi\)
\(788\) −21.5545 −0.767848
\(789\) 0 0
\(790\) 16.6401 0.592028
\(791\) −53.5619 −1.90444
\(792\) 0 0
\(793\) −15.1017 −0.536276
\(794\) 4.25881 0.151140
\(795\) 0 0
\(796\) 5.03212 0.178359
\(797\) −15.2875 −0.541512 −0.270756 0.962648i \(-0.587274\pi\)
−0.270756 + 0.962648i \(0.587274\pi\)
\(798\) 0 0
\(799\) −0.108424 −0.00383576
\(800\) −20.9877 −0.742027
\(801\) 0 0
\(802\) −16.3076 −0.575843
\(803\) −32.2478 −1.13800
\(804\) 0 0
\(805\) 19.3163 0.680811
\(806\) −0.143840 −0.00506654
\(807\) 0 0
\(808\) −49.0009 −1.72385
\(809\) −33.2118 −1.16766 −0.583832 0.811874i \(-0.698447\pi\)
−0.583832 + 0.811874i \(0.698447\pi\)
\(810\) 0 0
\(811\) 41.9154 1.47185 0.735925 0.677063i \(-0.236748\pi\)
0.735925 + 0.677063i \(0.236748\pi\)
\(812\) −2.09928 −0.0736703
\(813\) 0 0
\(814\) 19.9981 0.700934
\(815\) −22.0697 −0.773066
\(816\) 0 0
\(817\) −35.3160 −1.23555
\(818\) 4.12321 0.144165
\(819\) 0 0
\(820\) −4.81122 −0.168015
\(821\) 1.45745 0.0508654 0.0254327 0.999677i \(-0.491904\pi\)
0.0254327 + 0.999677i \(0.491904\pi\)
\(822\) 0 0
\(823\) 20.2247 0.704990 0.352495 0.935814i \(-0.385333\pi\)
0.352495 + 0.935814i \(0.385333\pi\)
\(824\) 0.300421 0.0104657
\(825\) 0 0
\(826\) −52.7240 −1.83450
\(827\) 29.5159 1.02637 0.513184 0.858278i \(-0.328466\pi\)
0.513184 + 0.858278i \(0.328466\pi\)
\(828\) 0 0
\(829\) 6.24184 0.216788 0.108394 0.994108i \(-0.465429\pi\)
0.108394 + 0.994108i \(0.465429\pi\)
\(830\) −3.38490 −0.117491
\(831\) 0 0
\(832\) 6.37059 0.220860
\(833\) −44.8030 −1.55233
\(834\) 0 0
\(835\) −3.00816 −0.104102
\(836\) 30.1033 1.04115
\(837\) 0 0
\(838\) 20.7437 0.716579
\(839\) 11.6981 0.403863 0.201931 0.979400i \(-0.435278\pi\)
0.201931 + 0.979400i \(0.435278\pi\)
\(840\) 0 0
\(841\) −28.7923 −0.992836
\(842\) −28.9540 −0.997822
\(843\) 0 0
\(844\) 17.1956 0.591897
\(845\) 12.6869 0.436441
\(846\) 0 0
\(847\) 15.5354 0.533803
\(848\) −2.98189 −0.102399
\(849\) 0 0
\(850\) 17.6230 0.604463
\(851\) 25.2458 0.865416
\(852\) 0 0
\(853\) 24.8809 0.851905 0.425953 0.904745i \(-0.359939\pi\)
0.425953 + 0.904745i \(0.359939\pi\)
\(854\) −50.4095 −1.72498
\(855\) 0 0
\(856\) 53.4981 1.82853
\(857\) −37.5138 −1.28145 −0.640724 0.767771i \(-0.721366\pi\)
−0.640724 + 0.767771i \(0.721366\pi\)
\(858\) 0 0
\(859\) −28.9804 −0.988799 −0.494399 0.869235i \(-0.664612\pi\)
−0.494399 + 0.869235i \(0.664612\pi\)
\(860\) −6.46450 −0.220437
\(861\) 0 0
\(862\) −7.01705 −0.239002
\(863\) −15.2759 −0.519999 −0.260000 0.965609i \(-0.583722\pi\)
−0.260000 + 0.965609i \(0.583722\pi\)
\(864\) 0 0
\(865\) 17.8062 0.605428
\(866\) −6.34911 −0.215751
\(867\) 0 0
\(868\) 0.651878 0.0221262
\(869\) 64.7343 2.19596
\(870\) 0 0
\(871\) −9.27396 −0.314236
\(872\) 10.2838 0.348252
\(873\) 0 0
\(874\) −27.9907 −0.946801
\(875\) 38.0683 1.28694
\(876\) 0 0
\(877\) 27.6463 0.933550 0.466775 0.884376i \(-0.345416\pi\)
0.466775 + 0.884376i \(0.345416\pi\)
\(878\) 17.7346 0.598513
\(879\) 0 0
\(880\) −1.53763 −0.0518334
\(881\) −38.9625 −1.31268 −0.656341 0.754465i \(-0.727897\pi\)
−0.656341 + 0.754465i \(0.727897\pi\)
\(882\) 0 0
\(883\) −3.02423 −0.101773 −0.0508867 0.998704i \(-0.516205\pi\)
−0.0508867 + 0.998704i \(0.516205\pi\)
\(884\) 6.33142 0.212949
\(885\) 0 0
\(886\) −6.35065 −0.213354
\(887\) −52.8701 −1.77521 −0.887603 0.460610i \(-0.847631\pi\)
−0.887603 + 0.460610i \(0.847631\pi\)
\(888\) 0 0
\(889\) −11.2924 −0.378736
\(890\) −7.43246 −0.249136
\(891\) 0 0
\(892\) 3.47186 0.116247
\(893\) 0.147440 0.00493389
\(894\) 0 0
\(895\) 9.19874 0.307480
\(896\) −22.4428 −0.749762
\(897\) 0 0
\(898\) 6.17667 0.206118
\(899\) −0.0645095 −0.00215151
\(900\) 0 0
\(901\) −40.1366 −1.33714
\(902\) 13.7858 0.459019
\(903\) 0 0
\(904\) 38.8787 1.29309
\(905\) −1.05889 −0.0351988
\(906\) 0 0
\(907\) −8.11733 −0.269531 −0.134766 0.990877i \(-0.543028\pi\)
−0.134766 + 0.990877i \(0.543028\pi\)
\(908\) −31.5751 −1.04786
\(909\) 0 0
\(910\) −4.37624 −0.145071
\(911\) 2.82293 0.0935278 0.0467639 0.998906i \(-0.485109\pi\)
0.0467639 + 0.998906i \(0.485109\pi\)
\(912\) 0 0
\(913\) −13.1681 −0.435802
\(914\) −28.0493 −0.927787
\(915\) 0 0
\(916\) 27.2102 0.899050
\(917\) 28.7463 0.949286
\(918\) 0 0
\(919\) 36.0591 1.18948 0.594739 0.803919i \(-0.297255\pi\)
0.594739 + 0.803919i \(0.297255\pi\)
\(920\) −14.0210 −0.462260
\(921\) 0 0
\(922\) −4.08528 −0.134542
\(923\) −1.97434 −0.0649860
\(924\) 0 0
\(925\) 21.6145 0.710680
\(926\) −27.0689 −0.889541
\(927\) 0 0
\(928\) 2.49075 0.0817630
\(929\) −41.9724 −1.37707 −0.688535 0.725203i \(-0.741746\pi\)
−0.688535 + 0.725203i \(0.741746\pi\)
\(930\) 0 0
\(931\) 60.9252 1.99674
\(932\) 8.73524 0.286132
\(933\) 0 0
\(934\) −6.89953 −0.225759
\(935\) −20.6966 −0.676853
\(936\) 0 0
\(937\) −9.17845 −0.299847 −0.149923 0.988698i \(-0.547903\pi\)
−0.149923 + 0.988698i \(0.547903\pi\)
\(938\) −30.9565 −1.01077
\(939\) 0 0
\(940\) 0.0269885 0.000880267 0
\(941\) 18.9045 0.616270 0.308135 0.951343i \(-0.400295\pi\)
0.308135 + 0.951343i \(0.400295\pi\)
\(942\) 0 0
\(943\) 17.4034 0.566732
\(944\) 5.29829 0.172445
\(945\) 0 0
\(946\) 18.5231 0.602238
\(947\) 59.9874 1.94933 0.974665 0.223670i \(-0.0718038\pi\)
0.974665 + 0.223670i \(0.0718038\pi\)
\(948\) 0 0
\(949\) 9.22310 0.299394
\(950\) −23.9646 −0.777513
\(951\) 0 0
\(952\) 57.8350 1.87444
\(953\) −32.9608 −1.06770 −0.533852 0.845578i \(-0.679256\pi\)
−0.533852 + 0.845578i \(0.679256\pi\)
\(954\) 0 0
\(955\) 21.2963 0.689133
\(956\) 1.15171 0.0372490
\(957\) 0 0
\(958\) 4.78993 0.154756
\(959\) 37.8211 1.22131
\(960\) 0 0
\(961\) −30.9800 −0.999354
\(962\) −5.71961 −0.184408
\(963\) 0 0
\(964\) −34.1959 −1.10137
\(965\) −22.6391 −0.728778
\(966\) 0 0
\(967\) −12.7497 −0.410001 −0.205001 0.978762i \(-0.565720\pi\)
−0.205001 + 0.978762i \(0.565720\pi\)
\(968\) −11.2766 −0.362444
\(969\) 0 0
\(970\) 9.38565 0.301355
\(971\) −51.1458 −1.64135 −0.820673 0.571398i \(-0.806401\pi\)
−0.820673 + 0.571398i \(0.806401\pi\)
\(972\) 0 0
\(973\) 29.8537 0.957066
\(974\) −24.8321 −0.795673
\(975\) 0 0
\(976\) 5.06570 0.162149
\(977\) −51.3095 −1.64154 −0.820768 0.571262i \(-0.806454\pi\)
−0.820768 + 0.571262i \(0.806454\pi\)
\(978\) 0 0
\(979\) −28.9142 −0.924102
\(980\) 11.1522 0.356244
\(981\) 0 0
\(982\) 5.42919 0.173252
\(983\) −45.7359 −1.45875 −0.729375 0.684114i \(-0.760189\pi\)
−0.729375 + 0.684114i \(0.760189\pi\)
\(984\) 0 0
\(985\) −20.1518 −0.642090
\(986\) −2.09144 −0.0666050
\(987\) 0 0
\(988\) −8.60977 −0.273913
\(989\) 23.3837 0.743559
\(990\) 0 0
\(991\) 36.0372 1.14476 0.572379 0.819989i \(-0.306021\pi\)
0.572379 + 0.819989i \(0.306021\pi\)
\(992\) −0.773441 −0.0245568
\(993\) 0 0
\(994\) −6.59034 −0.209033
\(995\) 4.70464 0.149147
\(996\) 0 0
\(997\) 33.6291 1.06504 0.532522 0.846416i \(-0.321244\pi\)
0.532522 + 0.846416i \(0.321244\pi\)
\(998\) 12.8915 0.408072
\(999\) 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.j.1.14 20
3.2 odd 2 2151.2.a.k.1.7 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.14 20 1.1 even 1 trivial
2151.2.a.k.1.7 yes 20 3.2 odd 2