Properties

Label 8001.2.a.k.1.1
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.44949 q^{2} +4.00000 q^{4} -2.44949 q^{5} +1.00000 q^{7} -4.89898 q^{8} +O(q^{10})\) \(q-2.44949 q^{2} +4.00000 q^{4} -2.44949 q^{5} +1.00000 q^{7} -4.89898 q^{8} +6.00000 q^{10} -2.89898 q^{13} -2.44949 q^{14} +4.00000 q^{16} +5.44949 q^{17} +4.44949 q^{19} -9.79796 q^{20} -6.00000 q^{23} +1.00000 q^{25} +7.10102 q^{26} +4.00000 q^{28} +7.89898 q^{29} +8.00000 q^{31} -13.3485 q^{34} -2.44949 q^{35} -7.00000 q^{37} -10.8990 q^{38} +12.0000 q^{40} -5.44949 q^{41} +2.00000 q^{43} +14.6969 q^{46} +12.0000 q^{47} +1.00000 q^{49} -2.44949 q^{50} -11.5959 q^{52} -1.89898 q^{53} -4.89898 q^{56} -19.3485 q^{58} +2.44949 q^{59} +4.44949 q^{61} -19.5959 q^{62} -8.00000 q^{64} +7.10102 q^{65} -10.2474 q^{67} +21.7980 q^{68} +6.00000 q^{70} -3.55051 q^{71} +3.34847 q^{73} +17.1464 q^{74} +17.7980 q^{76} +3.89898 q^{79} -9.79796 q^{80} +13.3485 q^{82} -6.00000 q^{83} -13.3485 q^{85} -4.89898 q^{86} +9.79796 q^{89} -2.89898 q^{91} -24.0000 q^{92} -29.3939 q^{94} -10.8990 q^{95} -2.34847 q^{97} -2.44949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} + 2q^{7} + O(q^{10}) \) \( 2q + 8q^{4} + 2q^{7} + 12q^{10} + 4q^{13} + 8q^{16} + 6q^{17} + 4q^{19} - 12q^{23} + 2q^{25} + 24q^{26} + 8q^{28} + 6q^{29} + 16q^{31} - 12q^{34} - 14q^{37} - 12q^{38} + 24q^{40} - 6q^{41} + 4q^{43} + 24q^{47} + 2q^{49} + 16q^{52} + 6q^{53} - 24q^{58} + 4q^{61} - 16q^{64} + 24q^{65} + 4q^{67} + 24q^{68} + 12q^{70} - 12q^{71} - 8q^{73} + 16q^{76} - 2q^{79} + 12q^{82} - 12q^{83} - 12q^{85} + 4q^{91} - 48q^{92} - 12q^{95} + 10q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44949 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0 0
\(4\) 4.00000 2.00000
\(5\) −2.44949 −1.09545 −0.547723 0.836660i \(-0.684505\pi\)
−0.547723 + 0.836660i \(0.684505\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −4.89898 −1.73205
\(9\) 0 0
\(10\) 6.00000 1.89737
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.89898 −0.804032 −0.402016 0.915633i \(-0.631690\pi\)
−0.402016 + 0.915633i \(0.631690\pi\)
\(14\) −2.44949 −0.654654
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 5.44949 1.32170 0.660848 0.750520i \(-0.270197\pi\)
0.660848 + 0.750520i \(0.270197\pi\)
\(18\) 0 0
\(19\) 4.44949 1.02078 0.510391 0.859942i \(-0.329501\pi\)
0.510391 + 0.859942i \(0.329501\pi\)
\(20\) −9.79796 −2.19089
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 7.10102 1.39262
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) 7.89898 1.46680 0.733402 0.679795i \(-0.237931\pi\)
0.733402 + 0.679795i \(0.237931\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) −13.3485 −2.28924
\(35\) −2.44949 −0.414039
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −10.8990 −1.76805
\(39\) 0 0
\(40\) 12.0000 1.89737
\(41\) −5.44949 −0.851067 −0.425534 0.904943i \(-0.639914\pi\)
−0.425534 + 0.904943i \(0.639914\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 14.6969 2.16695
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.44949 −0.346410
\(51\) 0 0
\(52\) −11.5959 −1.60806
\(53\) −1.89898 −0.260845 −0.130422 0.991459i \(-0.541633\pi\)
−0.130422 + 0.991459i \(0.541633\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.89898 −0.654654
\(57\) 0 0
\(58\) −19.3485 −2.54058
\(59\) 2.44949 0.318896 0.159448 0.987206i \(-0.449029\pi\)
0.159448 + 0.987206i \(0.449029\pi\)
\(60\) 0 0
\(61\) 4.44949 0.569699 0.284849 0.958572i \(-0.408056\pi\)
0.284849 + 0.958572i \(0.408056\pi\)
\(62\) −19.5959 −2.48868
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 7.10102 0.880773
\(66\) 0 0
\(67\) −10.2474 −1.25193 −0.625963 0.779853i \(-0.715294\pi\)
−0.625963 + 0.779853i \(0.715294\pi\)
\(68\) 21.7980 2.64339
\(69\) 0 0
\(70\) 6.00000 0.717137
\(71\) −3.55051 −0.421368 −0.210684 0.977554i \(-0.567569\pi\)
−0.210684 + 0.977554i \(0.567569\pi\)
\(72\) 0 0
\(73\) 3.34847 0.391909 0.195954 0.980613i \(-0.437220\pi\)
0.195954 + 0.980613i \(0.437220\pi\)
\(74\) 17.1464 1.99323
\(75\) 0 0
\(76\) 17.7980 2.04157
\(77\) 0 0
\(78\) 0 0
\(79\) 3.89898 0.438669 0.219335 0.975650i \(-0.429611\pi\)
0.219335 + 0.975650i \(0.429611\pi\)
\(80\) −9.79796 −1.09545
\(81\) 0 0
\(82\) 13.3485 1.47409
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −13.3485 −1.44784
\(86\) −4.89898 −0.528271
\(87\) 0 0
\(88\) 0 0
\(89\) 9.79796 1.03858 0.519291 0.854598i \(-0.326196\pi\)
0.519291 + 0.854598i \(0.326196\pi\)
\(90\) 0 0
\(91\) −2.89898 −0.303896
\(92\) −24.0000 −2.50217
\(93\) 0 0
\(94\) −29.3939 −3.03175
\(95\) −10.8990 −1.11821
\(96\) 0 0
\(97\) −2.34847 −0.238451 −0.119225 0.992867i \(-0.538041\pi\)
−0.119225 + 0.992867i \(0.538041\pi\)
\(98\) −2.44949 −0.247436
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 14.2020 1.39262
\(105\) 0 0
\(106\) 4.65153 0.451797
\(107\) −7.34847 −0.710403 −0.355202 0.934790i \(-0.615588\pi\)
−0.355202 + 0.934790i \(0.615588\pi\)
\(108\) 0 0
\(109\) −5.34847 −0.512290 −0.256145 0.966638i \(-0.582453\pi\)
−0.256145 + 0.966638i \(0.582453\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 14.6969 1.37050
\(116\) 31.5959 2.93361
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 5.44949 0.499554
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −10.8990 −0.986747
\(123\) 0 0
\(124\) 32.0000 2.87368
\(125\) 9.79796 0.876356
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 19.5959 1.73205
\(129\) 0 0
\(130\) −17.3939 −1.52554
\(131\) 4.34847 0.379928 0.189964 0.981791i \(-0.439163\pi\)
0.189964 + 0.981791i \(0.439163\pi\)
\(132\) 0 0
\(133\) 4.44949 0.385820
\(134\) 25.1010 2.16840
\(135\) 0 0
\(136\) −26.6969 −2.28924
\(137\) −2.20204 −0.188133 −0.0940665 0.995566i \(-0.529987\pi\)
−0.0940665 + 0.995566i \(0.529987\pi\)
\(138\) 0 0
\(139\) −5.65153 −0.479357 −0.239678 0.970852i \(-0.577042\pi\)
−0.239678 + 0.970852i \(0.577042\pi\)
\(140\) −9.79796 −0.828079
\(141\) 0 0
\(142\) 8.69694 0.729831
\(143\) 0 0
\(144\) 0 0
\(145\) −19.3485 −1.60680
\(146\) −8.20204 −0.678806
\(147\) 0 0
\(148\) −28.0000 −2.30159
\(149\) 20.6969 1.69556 0.847780 0.530349i \(-0.177939\pi\)
0.847780 + 0.530349i \(0.177939\pi\)
\(150\) 0 0
\(151\) 9.34847 0.760768 0.380384 0.924829i \(-0.375792\pi\)
0.380384 + 0.924829i \(0.375792\pi\)
\(152\) −21.7980 −1.76805
\(153\) 0 0
\(154\) 0 0
\(155\) −19.5959 −1.57398
\(156\) 0 0
\(157\) −1.55051 −0.123744 −0.0618721 0.998084i \(-0.519707\pi\)
−0.0618721 + 0.998084i \(0.519707\pi\)
\(158\) −9.55051 −0.759798
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) 17.0000 1.33154 0.665771 0.746156i \(-0.268103\pi\)
0.665771 + 0.746156i \(0.268103\pi\)
\(164\) −21.7980 −1.70213
\(165\) 0 0
\(166\) 14.6969 1.14070
\(167\) −23.1464 −1.79112 −0.895562 0.444936i \(-0.853226\pi\)
−0.895562 + 0.444936i \(0.853226\pi\)
\(168\) 0 0
\(169\) −4.59592 −0.353532
\(170\) 32.6969 2.50774
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −18.2474 −1.38733 −0.693664 0.720299i \(-0.744005\pi\)
−0.693664 + 0.720299i \(0.744005\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) −24.0000 −1.79888
\(179\) −15.7980 −1.18079 −0.590397 0.807113i \(-0.701029\pi\)
−0.590397 + 0.807113i \(0.701029\pi\)
\(180\) 0 0
\(181\) 10.1464 0.754178 0.377089 0.926177i \(-0.376925\pi\)
0.377089 + 0.926177i \(0.376925\pi\)
\(182\) 7.10102 0.526363
\(183\) 0 0
\(184\) 29.3939 2.16695
\(185\) 17.1464 1.26063
\(186\) 0 0
\(187\) 0 0
\(188\) 48.0000 3.50076
\(189\) 0 0
\(190\) 26.6969 1.93680
\(191\) 13.3485 0.965861 0.482931 0.875659i \(-0.339572\pi\)
0.482931 + 0.875659i \(0.339572\pi\)
\(192\) 0 0
\(193\) 5.55051 0.399534 0.199767 0.979843i \(-0.435981\pi\)
0.199767 + 0.979843i \(0.435981\pi\)
\(194\) 5.75255 0.413009
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) −8.44949 −0.602001 −0.301001 0.953624i \(-0.597321\pi\)
−0.301001 + 0.953624i \(0.597321\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −4.89898 −0.346410
\(201\) 0 0
\(202\) 0 0
\(203\) 7.89898 0.554400
\(204\) 0 0
\(205\) 13.3485 0.932298
\(206\) 9.79796 0.682656
\(207\) 0 0
\(208\) −11.5959 −0.804032
\(209\) 0 0
\(210\) 0 0
\(211\) −24.6969 −1.70021 −0.850104 0.526615i \(-0.823461\pi\)
−0.850104 + 0.526615i \(0.823461\pi\)
\(212\) −7.59592 −0.521690
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) −4.89898 −0.334108
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 13.1010 0.887313
\(219\) 0 0
\(220\) 0 0
\(221\) −15.7980 −1.06269
\(222\) 0 0
\(223\) 2.55051 0.170795 0.0853974 0.996347i \(-0.472784\pi\)
0.0853974 + 0.996347i \(0.472784\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −14.6969 −0.977626
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 7.44949 0.492276 0.246138 0.969235i \(-0.420838\pi\)
0.246138 + 0.969235i \(0.420838\pi\)
\(230\) −36.0000 −2.37377
\(231\) 0 0
\(232\) −38.6969 −2.54058
\(233\) 7.89898 0.517479 0.258740 0.965947i \(-0.416693\pi\)
0.258740 + 0.965947i \(0.416693\pi\)
\(234\) 0 0
\(235\) −29.3939 −1.91745
\(236\) 9.79796 0.637793
\(237\) 0 0
\(238\) −13.3485 −0.865253
\(239\) −3.00000 −0.194054 −0.0970269 0.995282i \(-0.530933\pi\)
−0.0970269 + 0.995282i \(0.530933\pi\)
\(240\) 0 0
\(241\) 5.24745 0.338018 0.169009 0.985615i \(-0.445943\pi\)
0.169009 + 0.985615i \(0.445943\pi\)
\(242\) 26.9444 1.73205
\(243\) 0 0
\(244\) 17.7980 1.13940
\(245\) −2.44949 −0.156492
\(246\) 0 0
\(247\) −12.8990 −0.820742
\(248\) −39.1918 −2.48868
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) −4.34847 −0.274473 −0.137236 0.990538i \(-0.543822\pi\)
−0.137236 + 0.990538i \(0.543822\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.44949 −0.153695
\(255\) 0 0
\(256\) −32.0000 −2.00000
\(257\) 11.1464 0.695295 0.347648 0.937625i \(-0.386981\pi\)
0.347648 + 0.937625i \(0.386981\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 28.4041 1.76155
\(261\) 0 0
\(262\) −10.6515 −0.658054
\(263\) 23.1464 1.42727 0.713635 0.700518i \(-0.247048\pi\)
0.713635 + 0.700518i \(0.247048\pi\)
\(264\) 0 0
\(265\) 4.65153 0.285741
\(266\) −10.8990 −0.668259
\(267\) 0 0
\(268\) −40.9898 −2.50385
\(269\) −27.2474 −1.66131 −0.830653 0.556790i \(-0.812033\pi\)
−0.830653 + 0.556790i \(0.812033\pi\)
\(270\) 0 0
\(271\) −30.9444 −1.87974 −0.939869 0.341536i \(-0.889053\pi\)
−0.939869 + 0.341536i \(0.889053\pi\)
\(272\) 21.7980 1.32170
\(273\) 0 0
\(274\) 5.39388 0.325856
\(275\) 0 0
\(276\) 0 0
\(277\) 10.2020 0.612981 0.306491 0.951874i \(-0.400845\pi\)
0.306491 + 0.951874i \(0.400845\pi\)
\(278\) 13.8434 0.830270
\(279\) 0 0
\(280\) 12.0000 0.717137
\(281\) 5.20204 0.310328 0.155164 0.987889i \(-0.450409\pi\)
0.155164 + 0.987889i \(0.450409\pi\)
\(282\) 0 0
\(283\) 24.3485 1.44737 0.723683 0.690132i \(-0.242448\pi\)
0.723683 + 0.690132i \(0.242448\pi\)
\(284\) −14.2020 −0.842736
\(285\) 0 0
\(286\) 0 0
\(287\) −5.44949 −0.321673
\(288\) 0 0
\(289\) 12.6969 0.746879
\(290\) 47.3939 2.78306
\(291\) 0 0
\(292\) 13.3939 0.783817
\(293\) −26.4495 −1.54520 −0.772598 0.634896i \(-0.781043\pi\)
−0.772598 + 0.634896i \(0.781043\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 34.2929 1.99323
\(297\) 0 0
\(298\) −50.6969 −2.93679
\(299\) 17.3939 1.00591
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) −22.8990 −1.31769
\(303\) 0 0
\(304\) 17.7980 1.02078
\(305\) −10.8990 −0.624074
\(306\) 0 0
\(307\) 20.5505 1.17288 0.586440 0.809993i \(-0.300529\pi\)
0.586440 + 0.809993i \(0.300529\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 48.0000 2.72622
\(311\) 17.1464 0.972285 0.486142 0.873880i \(-0.338404\pi\)
0.486142 + 0.873880i \(0.338404\pi\)
\(312\) 0 0
\(313\) 25.4495 1.43849 0.719245 0.694756i \(-0.244488\pi\)
0.719245 + 0.694756i \(0.244488\pi\)
\(314\) 3.79796 0.214331
\(315\) 0 0
\(316\) 15.5959 0.877339
\(317\) 19.1010 1.07282 0.536410 0.843957i \(-0.319780\pi\)
0.536410 + 0.843957i \(0.319780\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 19.5959 1.09545
\(321\) 0 0
\(322\) 14.6969 0.819028
\(323\) 24.2474 1.34916
\(324\) 0 0
\(325\) −2.89898 −0.160806
\(326\) −41.6413 −2.30630
\(327\) 0 0
\(328\) 26.6969 1.47409
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −18.4495 −1.01408 −0.507038 0.861924i \(-0.669260\pi\)
−0.507038 + 0.861924i \(0.669260\pi\)
\(332\) −24.0000 −1.31717
\(333\) 0 0
\(334\) 56.6969 3.10232
\(335\) 25.1010 1.37142
\(336\) 0 0
\(337\) −23.3485 −1.27187 −0.635936 0.771742i \(-0.719386\pi\)
−0.635936 + 0.771742i \(0.719386\pi\)
\(338\) 11.2577 0.612336
\(339\) 0 0
\(340\) −53.3939 −2.89569
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −9.79796 −0.528271
\(345\) 0 0
\(346\) 44.6969 2.40292
\(347\) 17.6969 0.950021 0.475011 0.879980i \(-0.342444\pi\)
0.475011 + 0.879980i \(0.342444\pi\)
\(348\) 0 0
\(349\) −17.0454 −0.912420 −0.456210 0.889872i \(-0.650793\pi\)
−0.456210 + 0.889872i \(0.650793\pi\)
\(350\) −2.44949 −0.130931
\(351\) 0 0
\(352\) 0 0
\(353\) −16.8990 −0.899442 −0.449721 0.893169i \(-0.648477\pi\)
−0.449721 + 0.893169i \(0.648477\pi\)
\(354\) 0 0
\(355\) 8.69694 0.461586
\(356\) 39.1918 2.07716
\(357\) 0 0
\(358\) 38.6969 2.04520
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) 0.797959 0.0419978
\(362\) −24.8536 −1.30627
\(363\) 0 0
\(364\) −11.5959 −0.607791
\(365\) −8.20204 −0.429314
\(366\) 0 0
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) −24.0000 −1.25109
\(369\) 0 0
\(370\) −42.0000 −2.18348
\(371\) −1.89898 −0.0985901
\(372\) 0 0
\(373\) 15.1010 0.781901 0.390951 0.920412i \(-0.372146\pi\)
0.390951 + 0.920412i \(0.372146\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −58.7878 −3.03175
\(377\) −22.8990 −1.17936
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −43.5959 −2.23642
\(381\) 0 0
\(382\) −32.6969 −1.67292
\(383\) 21.2474 1.08569 0.542847 0.839832i \(-0.317346\pi\)
0.542847 + 0.839832i \(0.317346\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −13.5959 −0.692014
\(387\) 0 0
\(388\) −9.39388 −0.476902
\(389\) 34.0454 1.72617 0.863085 0.505058i \(-0.168529\pi\)
0.863085 + 0.505058i \(0.168529\pi\)
\(390\) 0 0
\(391\) −32.6969 −1.65356
\(392\) −4.89898 −0.247436
\(393\) 0 0
\(394\) 20.6969 1.04270
\(395\) −9.55051 −0.480538
\(396\) 0 0
\(397\) −6.69694 −0.336110 −0.168055 0.985778i \(-0.553749\pi\)
−0.168055 + 0.985778i \(0.553749\pi\)
\(398\) −48.9898 −2.45564
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 0.303062 0.0151342 0.00756709 0.999971i \(-0.497591\pi\)
0.00756709 + 0.999971i \(0.497591\pi\)
\(402\) 0 0
\(403\) −23.1918 −1.15527
\(404\) 0 0
\(405\) 0 0
\(406\) −19.3485 −0.960248
\(407\) 0 0
\(408\) 0 0
\(409\) 16.6969 0.825610 0.412805 0.910819i \(-0.364549\pi\)
0.412805 + 0.910819i \(0.364549\pi\)
\(410\) −32.6969 −1.61479
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) 2.44949 0.120532
\(414\) 0 0
\(415\) 14.6969 0.721444
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 31.5959 1.54356 0.771781 0.635889i \(-0.219366\pi\)
0.771781 + 0.635889i \(0.219366\pi\)
\(420\) 0 0
\(421\) 6.04541 0.294635 0.147318 0.989089i \(-0.452936\pi\)
0.147318 + 0.989089i \(0.452936\pi\)
\(422\) 60.4949 2.94485
\(423\) 0 0
\(424\) 9.30306 0.451797
\(425\) 5.44949 0.264339
\(426\) 0 0
\(427\) 4.44949 0.215326
\(428\) −29.3939 −1.42081
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) −1.34847 −0.0649535 −0.0324767 0.999472i \(-0.510339\pi\)
−0.0324767 + 0.999472i \(0.510339\pi\)
\(432\) 0 0
\(433\) 0.651531 0.0313106 0.0156553 0.999877i \(-0.495017\pi\)
0.0156553 + 0.999877i \(0.495017\pi\)
\(434\) −19.5959 −0.940634
\(435\) 0 0
\(436\) −21.3939 −1.02458
\(437\) −26.6969 −1.27709
\(438\) 0 0
\(439\) −14.3485 −0.684815 −0.342408 0.939552i \(-0.611242\pi\)
−0.342408 + 0.939552i \(0.611242\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 38.6969 1.84063
\(443\) −21.5505 −1.02390 −0.511948 0.859017i \(-0.671076\pi\)
−0.511948 + 0.859017i \(0.671076\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) −6.24745 −0.295825
\(447\) 0 0
\(448\) −8.00000 −0.377964
\(449\) −19.1010 −0.901433 −0.450716 0.892667i \(-0.648831\pi\)
−0.450716 + 0.892667i \(0.648831\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 24.0000 1.12887
\(453\) 0 0
\(454\) 29.3939 1.37952
\(455\) 7.10102 0.332901
\(456\) 0 0
\(457\) 29.4949 1.37971 0.689857 0.723946i \(-0.257674\pi\)
0.689857 + 0.723946i \(0.257674\pi\)
\(458\) −18.2474 −0.852647
\(459\) 0 0
\(460\) 58.7878 2.74099
\(461\) 5.14643 0.239693 0.119847 0.992792i \(-0.461760\pi\)
0.119847 + 0.992792i \(0.461760\pi\)
\(462\) 0 0
\(463\) −0.202041 −0.00938964 −0.00469482 0.999989i \(-0.501494\pi\)
−0.00469482 + 0.999989i \(0.501494\pi\)
\(464\) 31.5959 1.46680
\(465\) 0 0
\(466\) −19.3485 −0.896301
\(467\) −31.5959 −1.46208 −0.731042 0.682332i \(-0.760966\pi\)
−0.731042 + 0.682332i \(0.760966\pi\)
\(468\) 0 0
\(469\) −10.2474 −0.473183
\(470\) 72.0000 3.32111
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) 4.44949 0.204157
\(476\) 21.7980 0.999108
\(477\) 0 0
\(478\) 7.34847 0.336111
\(479\) −3.24745 −0.148380 −0.0741899 0.997244i \(-0.523637\pi\)
−0.0741899 + 0.997244i \(0.523637\pi\)
\(480\) 0 0
\(481\) 20.2929 0.925275
\(482\) −12.8536 −0.585464
\(483\) 0 0
\(484\) −44.0000 −2.00000
\(485\) 5.75255 0.261210
\(486\) 0 0
\(487\) −30.9444 −1.40222 −0.701112 0.713051i \(-0.747313\pi\)
−0.701112 + 0.713051i \(0.747313\pi\)
\(488\) −21.7980 −0.986747
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) 13.1010 0.591241 0.295620 0.955305i \(-0.404474\pi\)
0.295620 + 0.955305i \(0.404474\pi\)
\(492\) 0 0
\(493\) 43.0454 1.93867
\(494\) 31.5959 1.42157
\(495\) 0 0
\(496\) 32.0000 1.43684
\(497\) −3.55051 −0.159262
\(498\) 0 0
\(499\) 6.04541 0.270630 0.135315 0.990803i \(-0.456795\pi\)
0.135315 + 0.990803i \(0.456795\pi\)
\(500\) 39.1918 1.75271
\(501\) 0 0
\(502\) 10.6515 0.475401
\(503\) −29.4495 −1.31309 −0.656544 0.754288i \(-0.727982\pi\)
−0.656544 + 0.754288i \(0.727982\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) 7.04541 0.312282 0.156141 0.987735i \(-0.450095\pi\)
0.156141 + 0.987735i \(0.450095\pi\)
\(510\) 0 0
\(511\) 3.34847 0.148128
\(512\) 39.1918 1.73205
\(513\) 0 0
\(514\) −27.3031 −1.20429
\(515\) 9.79796 0.431750
\(516\) 0 0
\(517\) 0 0
\(518\) 17.1464 0.753371
\(519\) 0 0
\(520\) −34.7878 −1.52554
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −28.2474 −1.23517 −0.617587 0.786502i \(-0.711890\pi\)
−0.617587 + 0.786502i \(0.711890\pi\)
\(524\) 17.3939 0.759855
\(525\) 0 0
\(526\) −56.6969 −2.47210
\(527\) 43.5959 1.89907
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −11.3939 −0.494918
\(531\) 0 0
\(532\) 17.7980 0.771639
\(533\) 15.7980 0.684286
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 50.2020 2.16840
\(537\) 0 0
\(538\) 66.7423 2.87747
\(539\) 0 0
\(540\) 0 0
\(541\) −8.89898 −0.382597 −0.191299 0.981532i \(-0.561270\pi\)
−0.191299 + 0.981532i \(0.561270\pi\)
\(542\) 75.7980 3.25580
\(543\) 0 0
\(544\) 0 0
\(545\) 13.1010 0.561186
\(546\) 0 0
\(547\) −1.79796 −0.0768752 −0.0384376 0.999261i \(-0.512238\pi\)
−0.0384376 + 0.999261i \(0.512238\pi\)
\(548\) −8.80816 −0.376266
\(549\) 0 0
\(550\) 0 0
\(551\) 35.1464 1.49729
\(552\) 0 0
\(553\) 3.89898 0.165801
\(554\) −24.9898 −1.06171
\(555\) 0 0
\(556\) −22.6061 −0.958713
\(557\) −21.7980 −0.923609 −0.461805 0.886982i \(-0.652798\pi\)
−0.461805 + 0.886982i \(0.652798\pi\)
\(558\) 0 0
\(559\) −5.79796 −0.245228
\(560\) −9.79796 −0.414039
\(561\) 0 0
\(562\) −12.7423 −0.537503
\(563\) −34.2929 −1.44527 −0.722636 0.691229i \(-0.757070\pi\)
−0.722636 + 0.691229i \(0.757070\pi\)
\(564\) 0 0
\(565\) −14.6969 −0.618305
\(566\) −59.6413 −2.50691
\(567\) 0 0
\(568\) 17.3939 0.729831
\(569\) 30.4949 1.27841 0.639206 0.769035i \(-0.279263\pi\)
0.639206 + 0.769035i \(0.279263\pi\)
\(570\) 0 0
\(571\) −45.1464 −1.88932 −0.944660 0.328052i \(-0.893608\pi\)
−0.944660 + 0.328052i \(0.893608\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 13.3485 0.557154
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −20.6515 −0.859734 −0.429867 0.902892i \(-0.641440\pi\)
−0.429867 + 0.902892i \(0.641440\pi\)
\(578\) −31.1010 −1.29363
\(579\) 0 0
\(580\) −77.3939 −3.21361
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) −16.4041 −0.678806
\(585\) 0 0
\(586\) 64.7878 2.67636
\(587\) 5.44949 0.224925 0.112462 0.993656i \(-0.464126\pi\)
0.112462 + 0.993656i \(0.464126\pi\)
\(588\) 0 0
\(589\) 35.5959 1.46670
\(590\) 14.6969 0.605063
\(591\) 0 0
\(592\) −28.0000 −1.15079
\(593\) 9.79796 0.402354 0.201177 0.979555i \(-0.435523\pi\)
0.201177 + 0.979555i \(0.435523\pi\)
\(594\) 0 0
\(595\) −13.3485 −0.547234
\(596\) 82.7878 3.39112
\(597\) 0 0
\(598\) −42.6061 −1.74229
\(599\) 33.0000 1.34834 0.674172 0.738575i \(-0.264501\pi\)
0.674172 + 0.738575i \(0.264501\pi\)
\(600\) 0 0
\(601\) −5.65153 −0.230531 −0.115265 0.993335i \(-0.536772\pi\)
−0.115265 + 0.993335i \(0.536772\pi\)
\(602\) −4.89898 −0.199667
\(603\) 0 0
\(604\) 37.3939 1.52154
\(605\) 26.9444 1.09545
\(606\) 0 0
\(607\) 9.34847 0.379443 0.189721 0.981838i \(-0.439242\pi\)
0.189721 + 0.981838i \(0.439242\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 26.6969 1.08093
\(611\) −34.7878 −1.40736
\(612\) 0 0
\(613\) 38.4949 1.55479 0.777397 0.629010i \(-0.216540\pi\)
0.777397 + 0.629010i \(0.216540\pi\)
\(614\) −50.3383 −2.03149
\(615\) 0 0
\(616\) 0 0
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 0 0
\(619\) 18.3485 0.737487 0.368744 0.929531i \(-0.379788\pi\)
0.368744 + 0.929531i \(0.379788\pi\)
\(620\) −78.3837 −3.14796
\(621\) 0 0
\(622\) −42.0000 −1.68405
\(623\) 9.79796 0.392547
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) −62.3383 −2.49154
\(627\) 0 0
\(628\) −6.20204 −0.247488
\(629\) −38.1464 −1.52100
\(630\) 0 0
\(631\) 47.7980 1.90281 0.951403 0.307948i \(-0.0996421\pi\)
0.951403 + 0.307948i \(0.0996421\pi\)
\(632\) −19.1010 −0.759798
\(633\) 0 0
\(634\) −46.7878 −1.85818
\(635\) −2.44949 −0.0972050
\(636\) 0 0
\(637\) −2.89898 −0.114862
\(638\) 0 0
\(639\) 0 0
\(640\) −48.0000 −1.89737
\(641\) −1.89898 −0.0750052 −0.0375026 0.999297i \(-0.511940\pi\)
−0.0375026 + 0.999297i \(0.511940\pi\)
\(642\) 0 0
\(643\) 9.34847 0.368668 0.184334 0.982864i \(-0.440987\pi\)
0.184334 + 0.982864i \(0.440987\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) −59.3939 −2.33682
\(647\) −44.6969 −1.75722 −0.878609 0.477542i \(-0.841528\pi\)
−0.878609 + 0.477542i \(0.841528\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 7.10102 0.278525
\(651\) 0 0
\(652\) 68.0000 2.66309
\(653\) −26.4495 −1.03505 −0.517524 0.855669i \(-0.673146\pi\)
−0.517524 + 0.855669i \(0.673146\pi\)
\(654\) 0 0
\(655\) −10.6515 −0.416190
\(656\) −21.7980 −0.851067
\(657\) 0 0
\(658\) −29.3939 −1.14589
\(659\) 13.1010 0.510343 0.255172 0.966896i \(-0.417868\pi\)
0.255172 + 0.966896i \(0.417868\pi\)
\(660\) 0 0
\(661\) −20.6515 −0.803251 −0.401626 0.915804i \(-0.631555\pi\)
−0.401626 + 0.915804i \(0.631555\pi\)
\(662\) 45.1918 1.75643
\(663\) 0 0
\(664\) 29.3939 1.14070
\(665\) −10.8990 −0.422644
\(666\) 0 0
\(667\) −47.3939 −1.83510
\(668\) −92.5857 −3.58225
\(669\) 0 0
\(670\) −61.4847 −2.37536
\(671\) 0 0
\(672\) 0 0
\(673\) 43.6969 1.68439 0.842197 0.539171i \(-0.181262\pi\)
0.842197 + 0.539171i \(0.181262\pi\)
\(674\) 57.1918 2.20295
\(675\) 0 0
\(676\) −18.3837 −0.707064
\(677\) 23.4495 0.901237 0.450619 0.892717i \(-0.351203\pi\)
0.450619 + 0.892717i \(0.351203\pi\)
\(678\) 0 0
\(679\) −2.34847 −0.0901260
\(680\) 65.3939 2.50774
\(681\) 0 0
\(682\) 0 0
\(683\) 25.5959 0.979401 0.489700 0.871891i \(-0.337106\pi\)
0.489700 + 0.871891i \(0.337106\pi\)
\(684\) 0 0
\(685\) 5.39388 0.206089
\(686\) −2.44949 −0.0935220
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 5.50510 0.209728
\(690\) 0 0
\(691\) 32.5505 1.23828 0.619140 0.785281i \(-0.287481\pi\)
0.619140 + 0.785281i \(0.287481\pi\)
\(692\) −72.9898 −2.77466
\(693\) 0 0
\(694\) −43.3485 −1.64549
\(695\) 13.8434 0.525109
\(696\) 0 0
\(697\) −29.6969 −1.12485
\(698\) 41.7526 1.58036
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 24.4949 0.925160 0.462580 0.886578i \(-0.346924\pi\)
0.462580 + 0.886578i \(0.346924\pi\)
\(702\) 0 0
\(703\) −31.1464 −1.17471
\(704\) 0 0
\(705\) 0 0
\(706\) 41.3939 1.55788
\(707\) 0 0
\(708\) 0 0
\(709\) 16.3939 0.615685 0.307842 0.951437i \(-0.400393\pi\)
0.307842 + 0.951437i \(0.400393\pi\)
\(710\) −21.3031 −0.799490
\(711\) 0 0
\(712\) −48.0000 −1.79888
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) 0 0
\(716\) −63.1918 −2.36159
\(717\) 0 0
\(718\) 36.7423 1.37121
\(719\) 38.1464 1.42262 0.711311 0.702878i \(-0.248102\pi\)
0.711311 + 0.702878i \(0.248102\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −1.95459 −0.0727424
\(723\) 0 0
\(724\) 40.5857 1.50836
\(725\) 7.89898 0.293361
\(726\) 0 0
\(727\) 52.6413 1.95236 0.976179 0.216965i \(-0.0696158\pi\)
0.976179 + 0.216965i \(0.0696158\pi\)
\(728\) 14.2020 0.526363
\(729\) 0 0
\(730\) 20.0908 0.743594
\(731\) 10.8990 0.403113
\(732\) 0 0
\(733\) −29.3485 −1.08401 −0.542005 0.840375i \(-0.682335\pi\)
−0.542005 + 0.840375i \(0.682335\pi\)
\(734\) −4.89898 −0.180825
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −4.30306 −0.158291 −0.0791453 0.996863i \(-0.525219\pi\)
−0.0791453 + 0.996863i \(0.525219\pi\)
\(740\) 68.5857 2.52126
\(741\) 0 0
\(742\) 4.65153 0.170763
\(743\) 23.6969 0.869356 0.434678 0.900586i \(-0.356862\pi\)
0.434678 + 0.900586i \(0.356862\pi\)
\(744\) 0 0
\(745\) −50.6969 −1.85739
\(746\) −36.9898 −1.35429
\(747\) 0 0
\(748\) 0 0
\(749\) −7.34847 −0.268507
\(750\) 0 0
\(751\) 43.6413 1.59249 0.796247 0.604971i \(-0.206815\pi\)
0.796247 + 0.604971i \(0.206815\pi\)
\(752\) 48.0000 1.75038
\(753\) 0 0
\(754\) 56.0908 2.04271
\(755\) −22.8990 −0.833379
\(756\) 0 0
\(757\) 43.0908 1.56616 0.783081 0.621920i \(-0.213647\pi\)
0.783081 + 0.621920i \(0.213647\pi\)
\(758\) −48.9898 −1.77939
\(759\) 0 0
\(760\) 53.3939 1.93680
\(761\) 53.6413 1.94450 0.972248 0.233952i \(-0.0751657\pi\)
0.972248 + 0.233952i \(0.0751657\pi\)
\(762\) 0 0
\(763\) −5.34847 −0.193628
\(764\) 53.3939 1.93172
\(765\) 0 0
\(766\) −52.0454 −1.88048
\(767\) −7.10102 −0.256403
\(768\) 0 0
\(769\) 32.5505 1.17380 0.586901 0.809659i \(-0.300348\pi\)
0.586901 + 0.809659i \(0.300348\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 22.2020 0.799069
\(773\) 51.2474 1.84324 0.921621 0.388090i \(-0.126865\pi\)
0.921621 + 0.388090i \(0.126865\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 11.5051 0.413009
\(777\) 0 0
\(778\) −83.3939 −2.98982
\(779\) −24.2474 −0.868755
\(780\) 0 0
\(781\) 0 0
\(782\) 80.0908 2.86404
\(783\) 0 0
\(784\) 4.00000 0.142857
\(785\) 3.79796 0.135555
\(786\) 0 0
\(787\) 55.1464 1.96576 0.982879 0.184253i \(-0.0589864\pi\)
0.982879 + 0.184253i \(0.0589864\pi\)
\(788\) −33.7980 −1.20400
\(789\) 0 0
\(790\) 23.3939 0.832317
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −12.8990 −0.458056
\(794\) 16.4041 0.582159
\(795\) 0 0
\(796\) 80.0000 2.83552
\(797\) −8.20204 −0.290531 −0.145266 0.989393i \(-0.546404\pi\)
−0.145266 + 0.989393i \(0.546404\pi\)
\(798\) 0 0
\(799\) 65.3939 2.31347
\(800\) 0 0
\(801\) 0 0
\(802\) −0.742346 −0.0262132
\(803\) 0 0
\(804\) 0 0
\(805\) 14.6969 0.517999
\(806\) 56.8082 2.00098
\(807\) 0 0
\(808\) 0 0
\(809\) −11.1464 −0.391888 −0.195944 0.980615i \(-0.562777\pi\)
−0.195944 + 0.980615i \(0.562777\pi\)
\(810\) 0 0
\(811\) 44.4949 1.56243 0.781214 0.624264i \(-0.214601\pi\)
0.781214 + 0.624264i \(0.214601\pi\)
\(812\) 31.5959 1.10880
\(813\) 0 0
\(814\) 0 0
\(815\) −41.6413 −1.45863
\(816\) 0 0
\(817\) 8.89898 0.311336
\(818\) −40.8990 −1.43000
\(819\) 0 0
\(820\) 53.3939 1.86460
\(821\) −6.79796 −0.237250 −0.118625 0.992939i \(-0.537849\pi\)
−0.118625 + 0.992939i \(0.537849\pi\)
\(822\) 0 0
\(823\) −19.0000 −0.662298 −0.331149 0.943578i \(-0.607436\pi\)
−0.331149 + 0.943578i \(0.607436\pi\)
\(824\) 19.5959 0.682656
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 24.7980 0.862310 0.431155 0.902278i \(-0.358106\pi\)
0.431155 + 0.902278i \(0.358106\pi\)
\(828\) 0 0
\(829\) 32.4949 1.12859 0.564297 0.825572i \(-0.309147\pi\)
0.564297 + 0.825572i \(0.309147\pi\)
\(830\) −36.0000 −1.24958
\(831\) 0 0
\(832\) 23.1918 0.804032
\(833\) 5.44949 0.188814
\(834\) 0 0
\(835\) 56.6969 1.96208
\(836\) 0 0
\(837\) 0 0
\(838\) −77.3939 −2.67353
\(839\) 37.8434 1.30650 0.653249 0.757143i \(-0.273405\pi\)
0.653249 + 0.757143i \(0.273405\pi\)
\(840\) 0 0
\(841\) 33.3939 1.15151
\(842\) −14.8082 −0.510323
\(843\) 0 0
\(844\) −98.7878 −3.40041
\(845\) 11.2577 0.387275
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) −7.59592 −0.260845
\(849\) 0 0
\(850\) −13.3485 −0.457849
\(851\) 42.0000 1.43974
\(852\) 0 0
\(853\) 17.7423 0.607486 0.303743 0.952754i \(-0.401764\pi\)
0.303743 + 0.952754i \(0.401764\pi\)
\(854\) −10.8990 −0.372955
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) −32.6969 −1.11691 −0.558453 0.829536i \(-0.688605\pi\)
−0.558453 + 0.829536i \(0.688605\pi\)
\(858\) 0 0
\(859\) −36.1464 −1.23330 −0.616650 0.787237i \(-0.711511\pi\)
−0.616650 + 0.787237i \(0.711511\pi\)
\(860\) −19.5959 −0.668215
\(861\) 0 0
\(862\) 3.30306 0.112503
\(863\) 37.8990 1.29010 0.645048 0.764142i \(-0.276837\pi\)
0.645048 + 0.764142i \(0.276837\pi\)
\(864\) 0 0
\(865\) 44.6969 1.51974
\(866\) −1.59592 −0.0542315
\(867\) 0 0
\(868\) 32.0000 1.08615
\(869\) 0 0
\(870\) 0 0
\(871\) 29.7071 1.00659
\(872\) 26.2020 0.887313
\(873\) 0 0
\(874\) 65.3939 2.21198
\(875\) 9.79796 0.331231
\(876\) 0 0
\(877\) 14.3031 0.482980 0.241490 0.970403i \(-0.422364\pi\)
0.241490 + 0.970403i \(0.422364\pi\)
\(878\) 35.1464 1.18613
\(879\) 0 0
\(880\) 0 0
\(881\) 17.7526 0.598099 0.299049 0.954238i \(-0.403330\pi\)
0.299049 + 0.954238i \(0.403330\pi\)
\(882\) 0 0
\(883\) −0.202041 −0.00679922 −0.00339961 0.999994i \(-0.501082\pi\)
−0.00339961 + 0.999994i \(0.501082\pi\)
\(884\) −63.1918 −2.12537
\(885\) 0 0
\(886\) 52.7878 1.77344
\(887\) −4.89898 −0.164492 −0.0822458 0.996612i \(-0.526209\pi\)
−0.0822458 + 0.996612i \(0.526209\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 58.7878 1.97057
\(891\) 0 0
\(892\) 10.2020 0.341590
\(893\) 53.3939 1.78676
\(894\) 0 0
\(895\) 38.6969 1.29350
\(896\) 19.5959 0.654654
\(897\) 0 0
\(898\) 46.7878 1.56133
\(899\) 63.1918 2.10757
\(900\) 0 0
\(901\) −10.3485 −0.344757
\(902\) 0 0
\(903\) 0 0
\(904\) −29.3939 −0.977626
\(905\) −24.8536 −0.826161
\(906\) 0 0
\(907\) 32.7980 1.08904 0.544519 0.838748i \(-0.316712\pi\)
0.544519 + 0.838748i \(0.316712\pi\)
\(908\) −48.0000 −1.59294
\(909\) 0 0
\(910\) −17.3939 −0.576601
\(911\) 20.6969 0.685720 0.342860 0.939386i \(-0.388604\pi\)
0.342860 + 0.939386i \(0.388604\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −72.2474 −2.38973
\(915\) 0 0
\(916\) 29.7980 0.984552
\(917\) 4.34847 0.143599
\(918\) 0 0
\(919\) 14.7980 0.488140 0.244070 0.969758i \(-0.421517\pi\)
0.244070 + 0.969758i \(0.421517\pi\)
\(920\) −72.0000 −2.37377
\(921\) 0 0
\(922\) −12.6061 −0.415161
\(923\) 10.2929 0.338793
\(924\) 0 0
\(925\) −7.00000 −0.230159
\(926\) 0.494897 0.0162633
\(927\) 0 0
\(928\) 0 0
\(929\) −46.2929 −1.51882 −0.759410 0.650613i \(-0.774512\pi\)
−0.759410 + 0.650613i \(0.774512\pi\)
\(930\) 0 0
\(931\) 4.44949 0.145826
\(932\) 31.5959 1.03496
\(933\) 0 0
\(934\) 77.3939 2.53241
\(935\) 0 0
\(936\) 0 0
\(937\) −13.3031 −0.434592 −0.217296 0.976106i \(-0.569724\pi\)
−0.217296 + 0.976106i \(0.569724\pi\)
\(938\) 25.1010 0.819577
\(939\) 0 0
\(940\) −117.576 −3.83489
\(941\) −34.8434 −1.13586 −0.567931 0.823076i \(-0.692256\pi\)
−0.567931 + 0.823076i \(0.692256\pi\)
\(942\) 0 0
\(943\) 32.6969 1.06476
\(944\) 9.79796 0.318896
\(945\) 0 0
\(946\) 0 0
\(947\) −45.9898 −1.49447 −0.747234 0.664561i \(-0.768618\pi\)
−0.747234 + 0.664561i \(0.768618\pi\)
\(948\) 0 0
\(949\) −9.70714 −0.315107
\(950\) −10.8990 −0.353610
\(951\) 0 0
\(952\) −26.6969 −0.865253
\(953\) 1.34847 0.0436812 0.0218406 0.999761i \(-0.493047\pi\)
0.0218406 + 0.999761i \(0.493047\pi\)
\(954\) 0 0
\(955\) −32.6969 −1.05805
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 7.95459 0.257001
\(959\) −2.20204 −0.0711076
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −49.7071 −1.60262
\(963\) 0 0
\(964\) 20.9898 0.676036
\(965\) −13.5959 −0.437668
\(966\) 0 0
\(967\) 10.9444 0.351948 0.175974 0.984395i \(-0.443693\pi\)
0.175974 + 0.984395i \(0.443693\pi\)
\(968\) 53.8888 1.73205
\(969\) 0 0
\(970\) −14.0908 −0.452429
\(971\) −37.5959 −1.20651 −0.603255 0.797548i \(-0.706130\pi\)
−0.603255 + 0.797548i \(0.706130\pi\)
\(972\) 0 0
\(973\) −5.65153 −0.181180
\(974\) 75.7980 2.42872
\(975\) 0 0
\(976\) 17.7980 0.569699
\(977\) 57.7980 1.84912 0.924560 0.381036i \(-0.124433\pi\)
0.924560 + 0.381036i \(0.124433\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −9.79796 −0.312984
\(981\) 0 0
\(982\) −32.0908 −1.02406
\(983\) −11.9444 −0.380967 −0.190483 0.981690i \(-0.561006\pi\)
−0.190483 + 0.981690i \(0.561006\pi\)
\(984\) 0 0
\(985\) 20.6969 0.659459
\(986\) −105.439 −3.35787
\(987\) 0 0
\(988\) −51.5959 −1.64148
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 10.4495 0.331939 0.165969 0.986131i \(-0.446925\pi\)
0.165969 + 0.986131i \(0.446925\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 8.69694 0.275850
\(995\) −48.9898 −1.55308
\(996\) 0 0
\(997\) 7.50510 0.237689 0.118844 0.992913i \(-0.462081\pi\)
0.118844 + 0.992913i \(0.462081\pi\)
\(998\) −14.8082 −0.468744
\(999\) 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 8001.2.a.k.1.1 2
3.2 odd 2 2667.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.h.1.2 2 3.2 odd 2
8001.2.a.k.1.1 2 1.1 even 1 trivial