Properties

Label 2151.2.a.k.1.7
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
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: \(0\)
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 1.7
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 \(-0.605577\pi\)
−0.325632 + 0.945497i \(0.605577\pi\)
\(3\) 0 0
\(4\) −1.15171 −0.575856
\(5\) 1.07676 0.481542 0.240771 0.970582i \(-0.422600\pi\)
0.240771 + 0.970582i \(0.422600\pi\)
\(6\) 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 \(-0.302417\pi\)
0.581626 + 0.813456i \(0.302417\pi\)
\(12\) 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 \(-0.293507\pi\)
0.604165 + 0.796859i \(0.293507\pi\)
\(18\) 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 \(-0.345090\pi\)
0.467680 + 0.883898i \(0.345090\pi\)
\(24\) 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 \(-0.513475\pi\)
−0.0423189 + 0.999104i \(0.513475\pi\)
\(30\) 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 \(-0.402029\pi\)
0.302949 + 0.953007i \(0.402029\pi\)
\(42\) 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 \(-0.500505\pi\)
−0.00158722 + 0.999999i \(0.500505\pi\)
\(48\) 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 \(-0.686633\pi\)
−0.553303 + 0.832980i \(0.686633\pi\)
\(54\) 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 \(-0.118247\pi\)
0.931790 + 0.362998i \(0.118247\pi\)
\(60\) 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 \(-0.466140\pi\)
0.106173 + 0.994348i \(0.466140\pi\)
\(72\) 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 \(-0.559980\pi\)
−0.187320 + 0.982299i \(0.559980\pi\)
\(84\) 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 \(-0.630020\pi\)
−0.397206 + 0.917729i \(0.630020\pi\)
\(90\) 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 \(-0.817350\pi\)
−0.839838 + 0.542837i \(0.817350\pi\)
\(102\) 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 \(-0.150119\pi\)
0.890837 + 0.454323i \(0.150119\pi\)
\(108\) 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 \(-0.283065\pi\)
0.629976 + 0.776614i \(0.283065\pi\)
\(114\) 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 \(-0.601675\pi\)
−0.314017 + 0.949417i \(0.601675\pi\)
\(132\) 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 \(-0.632380\pi\)
−0.403999 + 0.914759i \(0.632380\pi\)
\(138\) 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 \(-0.366751\pi\)
0.406494 + 0.913653i \(0.366751\pi\)
\(150\) 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 \(-0.534474\pi\)
−0.108092 + 0.994141i \(0.534474\pi\)
\(168\) 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 \(-0.283614\pi\)
0.628634 + 0.777701i \(0.283614\pi\)
\(174\) 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 \(-0.396564\pi\)
0.319266 + 0.947665i \(0.396564\pi\)
\(180\) 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 \(-0.246177\pi\)
0.715547 + 0.698564i \(0.246177\pi\)
\(192\) 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 \(-0.732294\pi\)
−0.666702 + 0.745325i \(0.732294\pi\)
\(198\) 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 \(-0.863784\pi\)
−0.909825 + 0.414993i \(0.863784\pi\)
\(228\) 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 \(-0.420082\pi\)
0.248441 + 0.968647i \(0.420082\pi\)
\(234\) 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 \(-0.752091\pi\)
−0.711737 + 0.702446i \(0.752091\pi\)
\(252\) 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 \(-0.654021\pi\)
−0.465210 + 0.885200i \(0.654021\pi\)
\(258\) 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 \(-0.578689\pi\)
−0.244698 + 0.969599i \(0.578689\pi\)
\(264\) 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 \(-0.839484\pi\)
−0.875524 + 0.483174i \(0.839484\pi\)
\(270\) 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 \(-0.341379\pi\)
0.477952 + 0.878386i \(0.341379\pi\)
\(282\) 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 \(-0.384555\pi\)
0.354784 + 0.934948i \(0.384555\pi\)
\(294\) 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 \(-0.906913\pi\)
−0.957543 + 0.288292i \(0.906913\pi\)
\(312\) 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 \(-0.543795\pi\)
−0.137154 + 0.990550i \(0.543795\pi\)
\(318\) 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 \(-0.332398\pi\)
0.502542 + 0.864553i \(0.332398\pi\)
\(348\) 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 \(-0.686333\pi\)
−0.552519 + 0.833500i \(0.686333\pi\)
\(354\) 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 \(-0.466634\pi\)
0.104630 + 0.994511i \(0.466634\pi\)
\(360\) 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 \(-0.561576\pi\)
−0.192244 + 0.981347i \(0.561576\pi\)
\(384\) 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 \(-0.779551\pi\)
−0.769613 + 0.638510i \(0.779551\pi\)
\(390\) 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 \(-0.354235\pi\)
0.442097 + 0.896967i \(0.354235\pi\)
\(402\) 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 \(-0.685428\pi\)
−0.550146 + 0.835069i \(0.685428\pi\)
\(420\) 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 \(-0.441260\pi\)
0.183491 + 0.983021i \(0.441260\pi\)
\(432\) 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 \(-0.447625\pi\)
0.163800 + 0.986494i \(0.447625\pi\)
\(444\) 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 \(-0.550584\pi\)
−0.158245 + 0.987400i \(0.550584\pi\)
\(450\) 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 \(-0.467062\pi\)
0.103293 + 0.994651i \(0.467062\pi\)
\(462\) 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 \(-0.444549\pi\)
0.173324 + 0.984865i \(0.444549\pi\)
\(468\) 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 \(-0.537909\pi\)
−0.118812 + 0.992917i \(0.537909\pi\)
\(480\) 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 \(-0.542465\pi\)
−0.133013 + 0.991114i \(0.542465\pi\)
\(492\) 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 \(-0.344300\pi\)
0.469873 + 0.882734i \(0.344300\pi\)
\(504\) 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 \(-0.297244\pi\)
0.594768 + 0.803897i \(0.297244\pi\)
\(510\) 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 \(-0.184751\pi\)
0.836237 + 0.548368i \(0.184751\pi\)
\(522\) 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 \(-0.407306\pi\)
0.287107 + 0.957898i \(0.407306\pi\)
\(558\) 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 \(-0.298239\pi\)
0.592253 + 0.805752i \(0.298239\pi\)
\(564\) 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 \(-0.463642\pi\)
0.113975 + 0.993484i \(0.463642\pi\)
\(570\) 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 \(-0.687843\pi\)
−0.556465 + 0.830871i \(0.687843\pi\)
\(588\) 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 \(-0.510879\pi\)
−0.0341708 + 0.999416i \(0.510879\pi\)
\(594\) 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 \(-0.366820\pi\)
0.406296 + 0.913741i \(0.366820\pi\)
\(600\) 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 \(-0.634601\pi\)
−0.410373 + 0.911918i \(0.634601\pi\)
\(618\) 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 \(-0.721832\pi\)
−0.641849 + 0.766831i \(0.721832\pi\)
\(642\) 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 \(-0.570524\pi\)
−0.219749 + 0.975556i \(0.570524\pi\)
\(648\) 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 \(-0.396399\pi\)
0.319756 + 0.947500i \(0.396399\pi\)
\(654\) 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 \(-0.206299\pi\)
0.797227 + 0.603680i \(0.206299\pi\)
\(660\) 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 \(-0.400863\pi\)
0.306438 + 0.951891i \(0.400863\pi\)
\(678\) 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 \(-0.788202\pi\)
−0.786680 + 0.617361i \(0.788202\pi\)
\(684\) 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 \(-0.362936\pi\)
0.417417 + 0.908715i \(0.362936\pi\)
\(702\) 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 \(-0.411973\pi\)
0.273034 + 0.962004i \(0.411973\pi\)
\(720\) 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 \(-0.526032\pi\)
−0.0816906 + 0.996658i \(0.526032\pi\)
\(744\) 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 \(-0.621390\pi\)
−0.372182 + 0.928160i \(0.621390\pi\)
\(762\) 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 \(-0.237783\pi\)
0.733720 + 0.679452i \(0.237783\pi\)
\(774\) 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 \(-0.412726\pi\)
0.270756 + 0.962648i \(0.412726\pi\)
\(798\) 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 \(-0.301553\pi\)
0.583832 + 0.811874i \(0.301553\pi\)
\(810\) 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 \(-0.508096\pi\)
−0.0254327 + 0.999677i \(0.508096\pi\)
\(822\) 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 \(-0.671534\pi\)
−0.513184 + 0.858278i \(0.671534\pi\)
\(828\) 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 \(-0.564722\pi\)
−0.201931 + 0.979400i \(0.564722\pi\)
\(840\) 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 \(-0.278634\pi\)
0.640724 + 0.767771i \(0.278634\pi\)
\(858\) 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 \(-0.416278\pi\)
0.260000 + 0.965609i \(0.416278\pi\)
\(864\) 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 \(-0.272103\pi\)
0.656341 + 0.754465i \(0.272103\pi\)
\(882\) 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 \(-0.152369\pi\)
0.887603 + 0.460610i \(0.152369\pi\)
\(888\) 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 \(-0.514891\pi\)
−0.0467639 + 0.998906i \(0.514891\pi\)
\(912\) 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 \(-0.258254\pi\)
0.688535 + 0.725203i \(0.258254\pi\)
\(930\) 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 \(-0.599705\pi\)
−0.308135 + 0.951343i \(0.599705\pi\)
\(942\) 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 \(-0.928196\pi\)
−0.974665 + 0.223670i \(0.928196\pi\)
\(948\) 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 \(-0.320744\pi\)
0.533852 + 0.845578i \(0.320744\pi\)
\(954\) 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 \(-0.193599\pi\)
0.820673 + 0.571398i \(0.193599\pi\)
\(972\) 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 \(-0.193546\pi\)
0.820768 + 0.571262i \(0.193546\pi\)
\(978\) 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 \(-0.239811\pi\)
0.729375 + 0.684114i \(0.239811\pi\)
\(984\) 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.k.1.7 yes 20
3.2 odd 2 2151.2.a.j.1.14 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.14 20 3.2 odd 2
2151.2.a.k.1.7 yes 20 1.1 even 1 trivial