Properties

Label 8001.2.a.k.1.2
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 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} +1.00000 q^{7} +4.89898 q^{8} +O(q^{10})\) \(q+2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} +1.00000 q^{7} +4.89898 q^{8} +6.00000 q^{10} +6.89898 q^{13} +2.44949 q^{14} +4.00000 q^{16} +0.550510 q^{17} -0.449490 q^{19} +9.79796 q^{20} -6.00000 q^{23} +1.00000 q^{25} +16.8990 q^{26} +4.00000 q^{28} -1.89898 q^{29} +8.00000 q^{31} +1.34847 q^{34} +2.44949 q^{35} -7.00000 q^{37} -1.10102 q^{38} +12.0000 q^{40} -0.550510 q^{41} +2.00000 q^{43} -14.6969 q^{46} +12.0000 q^{47} +1.00000 q^{49} +2.44949 q^{50} +27.5959 q^{52} +7.89898 q^{53} +4.89898 q^{56} -4.65153 q^{58} -2.44949 q^{59} -0.449490 q^{61} +19.5959 q^{62} -8.00000 q^{64} +16.8990 q^{65} +14.2474 q^{67} +2.20204 q^{68} +6.00000 q^{70} -8.44949 q^{71} -11.3485 q^{73} -17.1464 q^{74} -1.79796 q^{76} -5.89898 q^{79} +9.79796 q^{80} -1.34847 q^{82} -6.00000 q^{83} +1.34847 q^{85} +4.89898 q^{86} -9.79796 q^{89} +6.89898 q^{91} -24.0000 q^{92} +29.3939 q^{94} -1.10102 q^{95} +12.3485 q^{97} +2.44949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} + 2q^{7} + O(q^{10}) \) \( 2q + 8q^{4} + 2q^{7} + 12q^{10} + 4q^{13} + 8q^{16} + 6q^{17} + 4q^{19} - 12q^{23} + 2q^{25} + 24q^{26} + 8q^{28} + 6q^{29} + 16q^{31} - 12q^{34} - 14q^{37} - 12q^{38} + 24q^{40} - 6q^{41} + 4q^{43} + 24q^{47} + 2q^{49} + 16q^{52} + 6q^{53} - 24q^{58} + 4q^{61} - 16q^{64} + 24q^{65} + 4q^{67} + 24q^{68} + 12q^{70} - 12q^{71} - 8q^{73} + 16q^{76} - 2q^{79} + 12q^{82} - 12q^{83} - 12q^{85} + 4q^{91} - 48q^{92} - 12q^{95} + 10q^{97} + O(q^{100}) \)

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 \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) 0 0
\(4\) 4.00000 2.00000
\(5\) 2.44949 1.09545 0.547723 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 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
\(13\) 6.89898 1.91343 0.956716 0.291022i \(-0.0939953\pi\)
0.956716 + 0.291022i \(0.0939953\pi\)
\(14\) 2.44949 0.654654
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0.550510 0.133518 0.0667592 0.997769i \(-0.478734\pi\)
0.0667592 + 0.997769i \(0.478734\pi\)
\(18\) 0 0
\(19\) −0.449490 −0.103120 −0.0515600 0.998670i \(-0.516419\pi\)
−0.0515600 + 0.998670i \(0.516419\pi\)
\(20\) 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
\(26\) 16.8990 3.31416
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) −1.89898 −0.352632 −0.176316 0.984334i \(-0.556418\pi\)
−0.176316 + 0.984334i \(0.556418\pi\)
\(30\) 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
\(34\) 1.34847 0.231261
\(35\) 2.44949 0.414039
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −1.10102 −0.178609
\(39\) 0 0
\(40\) 12.0000 1.89737
\(41\) −0.550510 −0.0859753 −0.0429876 0.999076i \(-0.513688\pi\)
−0.0429876 + 0.999076i \(0.513688\pi\)
\(42\) 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
\(52\) 27.5959 3.82687
\(53\) 7.89898 1.08501 0.542504 0.840053i \(-0.317476\pi\)
0.542504 + 0.840053i \(0.317476\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.89898 0.654654
\(57\) 0 0
\(58\) −4.65153 −0.610776
\(59\) −2.44949 −0.318896 −0.159448 0.987206i \(-0.550971\pi\)
−0.159448 + 0.987206i \(0.550971\pi\)
\(60\) 0 0
\(61\) −0.449490 −0.0575513 −0.0287756 0.999586i \(-0.509161\pi\)
−0.0287756 + 0.999586i \(0.509161\pi\)
\(62\) 19.5959 2.48868
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 16.8990 2.09606
\(66\) 0 0
\(67\) 14.2474 1.74060 0.870301 0.492519i \(-0.163924\pi\)
0.870301 + 0.492519i \(0.163924\pi\)
\(68\) 2.20204 0.267037
\(69\) 0 0
\(70\) 6.00000 0.717137
\(71\) −8.44949 −1.00277 −0.501385 0.865224i \(-0.667176\pi\)
−0.501385 + 0.865224i \(0.667176\pi\)
\(72\) 0 0
\(73\) −11.3485 −1.32824 −0.664119 0.747627i \(-0.731193\pi\)
−0.664119 + 0.747627i \(0.731193\pi\)
\(74\) −17.1464 −1.99323
\(75\) 0 0
\(76\) −1.79796 −0.206240
\(77\) 0 0
\(78\) 0 0
\(79\) −5.89898 −0.663687 −0.331844 0.943334i \(-0.607671\pi\)
−0.331844 + 0.943334i \(0.607671\pi\)
\(80\) 9.79796 1.09545
\(81\) 0 0
\(82\) −1.34847 −0.148914
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 1.34847 0.146262
\(86\) 4.89898 0.528271
\(87\) 0 0
\(88\) 0 0
\(89\) −9.79796 −1.03858 −0.519291 0.854598i \(-0.673804\pi\)
−0.519291 + 0.854598i \(0.673804\pi\)
\(90\) 0 0
\(91\) 6.89898 0.723210
\(92\) −24.0000 −2.50217
\(93\) 0 0
\(94\) 29.3939 3.03175
\(95\) −1.10102 −0.112962
\(96\) 0 0
\(97\) 12.3485 1.25380 0.626899 0.779101i \(-0.284324\pi\)
0.626899 + 0.779101i \(0.284324\pi\)
\(98\) 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 \(0.563141\pi\)
\(104\) 33.7980 3.31416
\(105\) 0 0
\(106\) 19.3485 1.87929
\(107\) 7.34847 0.710403 0.355202 0.934790i \(-0.384412\pi\)
0.355202 + 0.934790i \(0.384412\pi\)
\(108\) 0 0
\(109\) 9.34847 0.895421 0.447710 0.894179i \(-0.352240\pi\)
0.447710 + 0.894179i \(0.352240\pi\)
\(110\) 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
\(116\) −7.59592 −0.705263
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 0.550510 0.0504652
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −1.10102 −0.0996817
\(123\) 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
\(130\) 41.3939 3.63048
\(131\) −10.3485 −0.904150 −0.452075 0.891980i \(-0.649316\pi\)
−0.452075 + 0.891980i \(0.649316\pi\)
\(132\) 0 0
\(133\) −0.449490 −0.0389757
\(134\) 34.8990 3.01481
\(135\) 0 0
\(136\) 2.69694 0.231261
\(137\) −21.7980 −1.86233 −0.931163 0.364604i \(-0.881204\pi\)
−0.931163 + 0.364604i \(0.881204\pi\)
\(138\) 0 0
\(139\) −20.3485 −1.72593 −0.862967 0.505260i \(-0.831397\pi\)
−0.862967 + 0.505260i \(0.831397\pi\)
\(140\) 9.79796 0.828079
\(141\) 0 0
\(142\) −20.6969 −1.73685
\(143\) 0 0
\(144\) 0 0
\(145\) −4.65153 −0.386289
\(146\) −27.7980 −2.30058
\(147\) 0 0
\(148\) −28.0000 −2.30159
\(149\) −8.69694 −0.712481 −0.356240 0.934394i \(-0.615942\pi\)
−0.356240 + 0.934394i \(0.615942\pi\)
\(150\) 0 0
\(151\) −5.34847 −0.435252 −0.217626 0.976032i \(-0.569831\pi\)
−0.217626 + 0.976032i \(0.569831\pi\)
\(152\) −2.20204 −0.178609
\(153\) 0 0
\(154\) 0 0
\(155\) 19.5959 1.57398
\(156\) 0 0
\(157\) −6.44949 −0.514725 −0.257363 0.966315i \(-0.582853\pi\)
−0.257363 + 0.966315i \(0.582853\pi\)
\(158\) −14.4495 −1.14954
\(159\) 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 \(0.268103\pi\)
\(164\) −2.20204 −0.171951
\(165\) 0 0
\(166\) −14.6969 −1.14070
\(167\) 11.1464 0.862537 0.431268 0.902224i \(-0.358066\pi\)
0.431268 + 0.902224i \(0.358066\pi\)
\(168\) 0 0
\(169\) 34.5959 2.66122
\(170\) 3.30306 0.253333
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 6.24745 0.474985 0.237492 0.971389i \(-0.423675\pi\)
0.237492 + 0.971389i \(0.423675\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) −24.0000 −1.79888
\(179\) 3.79796 0.283873 0.141936 0.989876i \(-0.454667\pi\)
0.141936 + 0.989876i \(0.454667\pi\)
\(180\) 0 0
\(181\) −24.1464 −1.79479 −0.897395 0.441228i \(-0.854543\pi\)
−0.897395 + 0.441228i \(0.854543\pi\)
\(182\) 16.8990 1.25264
\(183\) 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
\(190\) −2.69694 −0.195656
\(191\) −1.34847 −0.0975718 −0.0487859 0.998809i \(-0.515535\pi\)
−0.0487859 + 0.998809i \(0.515535\pi\)
\(192\) 0 0
\(193\) 10.4495 0.752171 0.376085 0.926585i \(-0.377270\pi\)
0.376085 + 0.926585i \(0.377270\pi\)
\(194\) 30.2474 2.17164
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) −3.55051 −0.252963 −0.126482 0.991969i \(-0.540368\pi\)
−0.126482 + 0.991969i \(0.540368\pi\)
\(198\) 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
\(203\) −1.89898 −0.133282
\(204\) 0 0
\(205\) −1.34847 −0.0941812
\(206\) −9.79796 −0.682656
\(207\) 0 0
\(208\) 27.5959 1.91343
\(209\) 0 0
\(210\) 0 0
\(211\) 4.69694 0.323351 0.161675 0.986844i \(-0.448310\pi\)
0.161675 + 0.986844i \(0.448310\pi\)
\(212\) 31.5959 2.17002
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 4.89898 0.334108
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 22.8990 1.55091
\(219\) 0 0
\(220\) 0 0
\(221\) 3.79796 0.255478
\(222\) 0 0
\(223\) 7.44949 0.498855 0.249427 0.968394i \(-0.419758\pi\)
0.249427 + 0.968394i \(0.419758\pi\)
\(224\) 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
\(229\) 2.55051 0.168542 0.0842712 0.996443i \(-0.473144\pi\)
0.0842712 + 0.996443i \(0.473144\pi\)
\(230\) −36.0000 −2.37377
\(231\) 0 0
\(232\) −9.30306 −0.610776
\(233\) −1.89898 −0.124406 −0.0622031 0.998064i \(-0.519813\pi\)
−0.0622031 + 0.998064i \(0.519813\pi\)
\(234\) 0 0
\(235\) 29.3939 1.91745
\(236\) −9.79796 −0.637793
\(237\) 0 0
\(238\) 1.34847 0.0874083
\(239\) −3.00000 −0.194054 −0.0970269 0.995282i \(-0.530933\pi\)
−0.0970269 + 0.995282i \(0.530933\pi\)
\(240\) 0 0
\(241\) −19.2474 −1.23984 −0.619919 0.784666i \(-0.712834\pi\)
−0.619919 + 0.784666i \(0.712834\pi\)
\(242\) −26.9444 −1.73205
\(243\) 0 0
\(244\) −1.79796 −0.115103
\(245\) 2.44949 0.156492
\(246\) 0 0
\(247\) −3.10102 −0.197313
\(248\) 39.1918 2.48868
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) 10.3485 0.653190 0.326595 0.945164i \(-0.394099\pi\)
0.326595 + 0.945164i \(0.394099\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 2.44949 0.153695
\(255\) 0 0
\(256\) −32.0000 −2.00000
\(257\) −23.1464 −1.44383 −0.721917 0.691979i \(-0.756739\pi\)
−0.721917 + 0.691979i \(0.756739\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 67.5959 4.19212
\(261\) 0 0
\(262\) −25.3485 −1.56603
\(263\) −11.1464 −0.687318 −0.343659 0.939094i \(-0.611666\pi\)
−0.343659 + 0.939094i \(0.611666\pi\)
\(264\) 0 0
\(265\) 19.3485 1.18857
\(266\) −1.10102 −0.0675079
\(267\) 0 0
\(268\) 56.9898 3.48121
\(269\) −2.75255 −0.167826 −0.0839130 0.996473i \(-0.526742\pi\)
−0.0839130 + 0.996473i \(0.526742\pi\)
\(270\) 0 0
\(271\) 22.9444 1.39377 0.696886 0.717182i \(-0.254568\pi\)
0.696886 + 0.717182i \(0.254568\pi\)
\(272\) 2.20204 0.133518
\(273\) 0 0
\(274\) −53.3939 −3.22564
\(275\) 0 0
\(276\) 0 0
\(277\) 29.7980 1.79039 0.895193 0.445679i \(-0.147038\pi\)
0.895193 + 0.445679i \(0.147038\pi\)
\(278\) −49.8434 −2.98941
\(279\) 0 0
\(280\) 12.0000 0.717137
\(281\) 24.7980 1.47932 0.739661 0.672980i \(-0.234986\pi\)
0.739661 + 0.672980i \(0.234986\pi\)
\(282\) 0 0
\(283\) 9.65153 0.573724 0.286862 0.957972i \(-0.407388\pi\)
0.286862 + 0.957972i \(0.407388\pi\)
\(284\) −33.7980 −2.00554
\(285\) 0 0
\(286\) 0 0
\(287\) −0.550510 −0.0324956
\(288\) 0 0
\(289\) −16.6969 −0.982173
\(290\) −11.3939 −0.669071
\(291\) 0 0
\(292\) −45.3939 −2.65648
\(293\) −21.5505 −1.25899 −0.629497 0.777003i \(-0.716739\pi\)
−0.629497 + 0.777003i \(0.716739\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) −34.2929 −1.99323
\(297\) 0 0
\(298\) −21.3031 −1.23405
\(299\) −41.3939 −2.39387
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) −13.1010 −0.753879
\(303\) 0 0
\(304\) −1.79796 −0.103120
\(305\) −1.10102 −0.0630443
\(306\) 0 0
\(307\) 25.4495 1.45248 0.726240 0.687442i \(-0.241266\pi\)
0.726240 + 0.687442i \(0.241266\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 48.0000 2.72622
\(311\) −17.1464 −0.972285 −0.486142 0.873880i \(-0.661596\pi\)
−0.486142 + 0.873880i \(0.661596\pi\)
\(312\) 0 0
\(313\) 20.5505 1.16158 0.580792 0.814052i \(-0.302743\pi\)
0.580792 + 0.814052i \(0.302743\pi\)
\(314\) −15.7980 −0.891530
\(315\) 0 0
\(316\) −23.5959 −1.32737
\(317\) 28.8990 1.62313 0.811564 0.584263i \(-0.198616\pi\)
0.811564 + 0.584263i \(0.198616\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −19.5959 −1.09545
\(321\) 0 0
\(322\) −14.6969 −0.819028
\(323\) −0.247449 −0.0137684
\(324\) 0 0
\(325\) 6.89898 0.382687
\(326\) 41.6413 2.30630
\(327\) 0 0
\(328\) −2.69694 −0.148914
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −13.5505 −0.744803 −0.372402 0.928072i \(-0.621466\pi\)
−0.372402 + 0.928072i \(0.621466\pi\)
\(332\) −24.0000 −1.31717
\(333\) 0 0
\(334\) 27.3031 1.49396
\(335\) 34.8990 1.90673
\(336\) 0 0
\(337\) −8.65153 −0.471279 −0.235639 0.971841i \(-0.575718\pi\)
−0.235639 + 0.971841i \(0.575718\pi\)
\(338\) 84.7423 4.60938
\(339\) 0 0
\(340\) 5.39388 0.292524
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 9.79796 0.528271
\(345\) 0 0
\(346\) 15.3031 0.822698
\(347\) −11.6969 −0.627925 −0.313962 0.949435i \(-0.601657\pi\)
−0.313962 + 0.949435i \(0.601657\pi\)
\(348\) 0 0
\(349\) 27.0454 1.44771 0.723854 0.689953i \(-0.242369\pi\)
0.723854 + 0.689953i \(0.242369\pi\)
\(350\) 2.44949 0.130931
\(351\) 0 0
\(352\) 0 0
\(353\) −7.10102 −0.377949 −0.188975 0.981982i \(-0.560516\pi\)
−0.188975 + 0.981982i \(0.560516\pi\)
\(354\) 0 0
\(355\) −20.6969 −1.09848
\(356\) −39.1918 −2.07716
\(357\) 0 0
\(358\) 9.30306 0.491682
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −18.7980 −0.989366
\(362\) −59.1464 −3.10867
\(363\) 0 0
\(364\) 27.5959 1.44642
\(365\) −27.7980 −1.45501
\(366\) 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
\(371\) 7.89898 0.410095
\(372\) 0 0
\(373\) 24.8990 1.28922 0.644610 0.764511i \(-0.277020\pi\)
0.644610 + 0.764511i \(0.277020\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 58.7878 3.03175
\(377\) −13.1010 −0.674737
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) −4.40408 −0.225925
\(381\) 0 0
\(382\) −3.30306 −0.168999
\(383\) −3.24745 −0.165937 −0.0829684 0.996552i \(-0.526440\pi\)
−0.0829684 + 0.996552i \(0.526440\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 25.5959 1.30280
\(387\) 0 0
\(388\) 49.3939 2.50759
\(389\) −10.0454 −0.509322 −0.254661 0.967030i \(-0.581964\pi\)
−0.254661 + 0.967030i \(0.581964\pi\)
\(390\) 0 0
\(391\) −3.30306 −0.167043
\(392\) 4.89898 0.247436
\(393\) 0 0
\(394\) −8.69694 −0.438145
\(395\) −14.4495 −0.727033
\(396\) 0 0
\(397\) 22.6969 1.13913 0.569563 0.821947i \(-0.307112\pi\)
0.569563 + 0.821947i \(0.307112\pi\)
\(398\) 48.9898 2.45564
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 29.6969 1.48299 0.741497 0.670956i \(-0.234116\pi\)
0.741497 + 0.670956i \(0.234116\pi\)
\(402\) 0 0
\(403\) 55.1918 2.74930
\(404\) 0 0
\(405\) 0 0
\(406\) −4.65153 −0.230852
\(407\) 0 0
\(408\) 0 0
\(409\) −12.6969 −0.627823 −0.313912 0.949452i \(-0.601640\pi\)
−0.313912 + 0.949452i \(0.601640\pi\)
\(410\) −3.30306 −0.163127
\(411\) 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
\(419\) −7.59592 −0.371085 −0.185542 0.982636i \(-0.559404\pi\)
−0.185542 + 0.982636i \(0.559404\pi\)
\(420\) 0 0
\(421\) −38.0454 −1.85422 −0.927110 0.374790i \(-0.877715\pi\)
−0.927110 + 0.374790i \(0.877715\pi\)
\(422\) 11.5051 0.560060
\(423\) 0 0
\(424\) 38.6969 1.87929
\(425\) 0.550510 0.0267037
\(426\) 0 0
\(427\) −0.449490 −0.0217523
\(428\) 29.3939 1.42081
\(429\) 0 0
\(430\) 12.0000 0.578691
\(431\) 13.3485 0.642973 0.321487 0.946914i \(-0.395817\pi\)
0.321487 + 0.946914i \(0.395817\pi\)
\(432\) 0 0
\(433\) 15.3485 0.737600 0.368800 0.929509i \(-0.379769\pi\)
0.368800 + 0.929509i \(0.379769\pi\)
\(434\) 19.5959 0.940634
\(435\) 0 0
\(436\) 37.3939 1.79084
\(437\) 2.69694 0.129012
\(438\) 0 0
\(439\) 0.348469 0.0166315 0.00831576 0.999965i \(-0.497353\pi\)
0.00831576 + 0.999965i \(0.497353\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 9.30306 0.442502
\(443\) −26.4495 −1.25665 −0.628327 0.777950i \(-0.716260\pi\)
−0.628327 + 0.777950i \(0.716260\pi\)
\(444\) 0 0
\(445\) −24.0000 −1.13771
\(446\) 18.2474 0.864042
\(447\) 0 0
\(448\) −8.00000 −0.377964
\(449\) −28.8990 −1.36383 −0.681914 0.731433i \(-0.738852\pi\)
−0.681914 + 0.731433i \(0.738852\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 24.0000 1.12887
\(453\) 0 0
\(454\) −29.3939 −1.37952
\(455\) 16.8990 0.792236
\(456\) 0 0
\(457\) −19.4949 −0.911933 −0.455966 0.889997i \(-0.650706\pi\)
−0.455966 + 0.889997i \(0.650706\pi\)
\(458\) 6.24745 0.291924
\(459\) 0 0
\(460\) −58.7878 −2.74099
\(461\) −29.1464 −1.35748 −0.678742 0.734377i \(-0.737475\pi\)
−0.678742 + 0.734377i \(0.737475\pi\)
\(462\) 0 0
\(463\) −19.7980 −0.920089 −0.460045 0.887896i \(-0.652167\pi\)
−0.460045 + 0.887896i \(0.652167\pi\)
\(464\) −7.59592 −0.352632
\(465\) 0 0
\(466\) −4.65153 −0.215478
\(467\) 7.59592 0.351497 0.175749 0.984435i \(-0.443765\pi\)
0.175749 + 0.984435i \(0.443765\pi\)
\(468\) 0 0
\(469\) 14.2474 0.657886
\(470\) 72.0000 3.32111
\(471\) 0 0
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) −0.449490 −0.0206240
\(476\) 2.20204 0.100930
\(477\) 0 0
\(478\) −7.34847 −0.336111
\(479\) 21.2474 0.970821 0.485410 0.874286i \(-0.338670\pi\)
0.485410 + 0.874286i \(0.338670\pi\)
\(480\) 0 0
\(481\) −48.2929 −2.20196
\(482\) −47.1464 −2.14746
\(483\) 0 0
\(484\) −44.0000 −2.00000
\(485\) 30.2474 1.37347
\(486\) 0 0
\(487\) 22.9444 1.03971 0.519855 0.854255i \(-0.325986\pi\)
0.519855 + 0.854255i \(0.325986\pi\)
\(488\) −2.20204 −0.0996817
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) 22.8990 1.03342 0.516708 0.856162i \(-0.327157\pi\)
0.516708 + 0.856162i \(0.327157\pi\)
\(492\) 0 0
\(493\) −1.04541 −0.0470828
\(494\) −7.59592 −0.341757
\(495\) 0 0
\(496\) 32.0000 1.43684
\(497\) −8.44949 −0.379011
\(498\) 0 0
\(499\) −38.0454 −1.70315 −0.851573 0.524236i \(-0.824351\pi\)
−0.851573 + 0.524236i \(0.824351\pi\)
\(500\) −39.1918 −1.75271
\(501\) 0 0
\(502\) 25.3485 1.13136
\(503\) −24.5505 −1.09465 −0.547327 0.836919i \(-0.684354\pi\)
−0.547327 + 0.836919i \(0.684354\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −37.0454 −1.64201 −0.821004 0.570922i \(-0.806586\pi\)
−0.821004 + 0.570922i \(0.806586\pi\)
\(510\) 0 0
\(511\) −11.3485 −0.502027
\(512\) −39.1918 −1.73205
\(513\) 0 0
\(514\) −56.6969 −2.50079
\(515\) −9.79796 −0.431750
\(516\) 0 0
\(517\) 0 0
\(518\) −17.1464 −0.753371
\(519\) 0 0
\(520\) 82.7878 3.63048
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −3.75255 −0.164088 −0.0820438 0.996629i \(-0.526145\pi\)
−0.0820438 + 0.996629i \(0.526145\pi\)
\(524\) −41.3939 −1.80830
\(525\) 0 0
\(526\) −27.3031 −1.19047
\(527\) 4.40408 0.191845
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 47.3939 2.05866
\(531\) 0 0
\(532\) −1.79796 −0.0779514
\(533\) −3.79796 −0.164508
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 69.7980 3.01481
\(537\) 0 0
\(538\) −6.74235 −0.290683
\(539\) 0 0
\(540\) 0 0
\(541\) 0.898979 0.0386501 0.0193251 0.999813i \(-0.493848\pi\)
0.0193251 + 0.999813i \(0.493848\pi\)
\(542\) 56.2020 2.41408
\(543\) 0 0
\(544\) 0 0
\(545\) 22.8990 0.980885
\(546\) 0 0
\(547\) 17.7980 0.760986 0.380493 0.924784i \(-0.375754\pi\)
0.380493 + 0.924784i \(0.375754\pi\)
\(548\) −87.1918 −3.72465
\(549\) 0 0
\(550\) 0 0
\(551\) 0.853572 0.0363634
\(552\) 0 0
\(553\) −5.89898 −0.250850
\(554\) 72.9898 3.10104
\(555\) 0 0
\(556\) −81.3939 −3.45187
\(557\) −2.20204 −0.0933035 −0.0466517 0.998911i \(-0.514855\pi\)
−0.0466517 + 0.998911i \(0.514855\pi\)
\(558\) 0 0
\(559\) 13.7980 0.583591
\(560\) 9.79796 0.414039
\(561\) 0 0
\(562\) 60.7423 2.56226
\(563\) 34.2929 1.44527 0.722636 0.691229i \(-0.242930\pi\)
0.722636 + 0.691229i \(0.242930\pi\)
\(564\) 0 0
\(565\) 14.6969 0.618305
\(566\) 23.6413 0.993719
\(567\) 0 0
\(568\) −41.3939 −1.73685
\(569\) −18.4949 −0.775346 −0.387673 0.921797i \(-0.626721\pi\)
−0.387673 + 0.921797i \(0.626721\pi\)
\(570\) 0 0
\(571\) −10.8536 −0.454208 −0.227104 0.973871i \(-0.572926\pi\)
−0.227104 + 0.973871i \(0.572926\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.34847 −0.0562840
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −35.3485 −1.47158 −0.735788 0.677212i \(-0.763188\pi\)
−0.735788 + 0.677212i \(0.763188\pi\)
\(578\) −40.8990 −1.70117
\(579\) 0 0
\(580\) −18.6061 −0.772577
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 0 0
\(584\) −55.5959 −2.30058
\(585\) 0 0
\(586\) −52.7878 −2.18064
\(587\) 0.550510 0.0227220 0.0113610 0.999935i \(-0.496384\pi\)
0.0113610 + 0.999935i \(0.496384\pi\)
\(588\) 0 0
\(589\) −3.59592 −0.148167
\(590\) −14.6969 −0.605063
\(591\) 0 0
\(592\) −28.0000 −1.15079
\(593\) −9.79796 −0.402354 −0.201177 0.979555i \(-0.564477\pi\)
−0.201177 + 0.979555i \(0.564477\pi\)
\(594\) 0 0
\(595\) 1.34847 0.0552818
\(596\) −34.7878 −1.42496
\(597\) 0 0
\(598\) −101.394 −4.14630
\(599\) 33.0000 1.34834 0.674172 0.738575i \(-0.264501\pi\)
0.674172 + 0.738575i \(0.264501\pi\)
\(600\) 0 0
\(601\) −20.3485 −0.830031 −0.415016 0.909814i \(-0.636224\pi\)
−0.415016 + 0.909814i \(0.636224\pi\)
\(602\) 4.89898 0.199667
\(603\) 0 0
\(604\) −21.3939 −0.870505
\(605\) −26.9444 −1.09545
\(606\) 0 0
\(607\) −5.34847 −0.217088 −0.108544 0.994092i \(-0.534619\pi\)
−0.108544 + 0.994092i \(0.534619\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −2.69694 −0.109196
\(611\) 82.7878 3.34923
\(612\) 0 0
\(613\) −10.4949 −0.423885 −0.211942 0.977282i \(-0.567979\pi\)
−0.211942 + 0.977282i \(0.567979\pi\)
\(614\) 62.3383 2.51577
\(615\) 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
\(619\) 3.65153 0.146767 0.0733837 0.997304i \(-0.476620\pi\)
0.0733837 + 0.997304i \(0.476620\pi\)
\(620\) 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
\(626\) 50.3383 2.01192
\(627\) 0 0
\(628\) −25.7980 −1.02945
\(629\) −3.85357 −0.153652
\(630\) 0 0
\(631\) 28.2020 1.12271 0.561353 0.827577i \(-0.310281\pi\)
0.561353 + 0.827577i \(0.310281\pi\)
\(632\) −28.8990 −1.14954
\(633\) 0 0
\(634\) 70.7878 2.81134
\(635\) 2.44949 0.0972050
\(636\) 0 0
\(637\) 6.89898 0.273348
\(638\) 0 0
\(639\) 0 0
\(640\) −48.0000 −1.89737
\(641\) 7.89898 0.311991 0.155995 0.987758i \(-0.450141\pi\)
0.155995 + 0.987758i \(0.450141\pi\)
\(642\) 0 0
\(643\) −5.34847 −0.210923 −0.105462 0.994423i \(-0.533632\pi\)
−0.105462 + 0.994423i \(0.533632\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) −0.606123 −0.0238476
\(647\) −15.3031 −0.601625 −0.300813 0.953683i \(-0.597258\pi\)
−0.300813 + 0.953683i \(0.597258\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 16.8990 0.662833
\(651\) 0 0
\(652\) 68.0000 2.66309
\(653\) −21.5505 −0.843337 −0.421668 0.906750i \(-0.638555\pi\)
−0.421668 + 0.906750i \(0.638555\pi\)
\(654\) 0 0
\(655\) −25.3485 −0.990447
\(656\) −2.20204 −0.0859753
\(657\) 0 0
\(658\) 29.3939 1.14589
\(659\) 22.8990 0.892018 0.446009 0.895029i \(-0.352845\pi\)
0.446009 + 0.895029i \(0.352845\pi\)
\(660\) 0 0
\(661\) −35.3485 −1.37490 −0.687448 0.726234i \(-0.741269\pi\)
−0.687448 + 0.726234i \(0.741269\pi\)
\(662\) −33.1918 −1.29004
\(663\) 0 0
\(664\) −29.3939 −1.14070
\(665\) −1.10102 −0.0426957
\(666\) 0 0
\(667\) 11.3939 0.441173
\(668\) 44.5857 1.72507
\(669\) 0 0
\(670\) 85.4847 3.30256
\(671\) 0 0
\(672\) 0 0
\(673\) 14.3031 0.551343 0.275671 0.961252i \(-0.411100\pi\)
0.275671 + 0.961252i \(0.411100\pi\)
\(674\) −21.1918 −0.816279
\(675\) 0 0
\(676\) 138.384 5.32245
\(677\) 18.5505 0.712954 0.356477 0.934304i \(-0.383978\pi\)
0.356477 + 0.934304i \(0.383978\pi\)
\(678\) 0 0
\(679\) 12.3485 0.473891
\(680\) 6.60612 0.253333
\(681\) 0 0
\(682\) 0 0
\(683\) −13.5959 −0.520233 −0.260117 0.965577i \(-0.583761\pi\)
−0.260117 + 0.965577i \(0.583761\pi\)
\(684\) 0 0
\(685\) −53.3939 −2.04008
\(686\) 2.44949 0.0935220
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 54.4949 2.07609
\(690\) 0 0
\(691\) 37.4495 1.42465 0.712323 0.701852i \(-0.247643\pi\)
0.712323 + 0.701852i \(0.247643\pi\)
\(692\) 24.9898 0.949969
\(693\) 0 0
\(694\) −28.6515 −1.08760
\(695\) −49.8434 −1.89067
\(696\) 0 0
\(697\) −0.303062 −0.0114793
\(698\) 66.2474 2.50750
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) −24.4949 −0.925160 −0.462580 0.886578i \(-0.653076\pi\)
−0.462580 + 0.886578i \(0.653076\pi\)
\(702\) 0 0
\(703\) 3.14643 0.118670
\(704\) 0 0
\(705\) 0 0
\(706\) −17.3939 −0.654627
\(707\) 0 0
\(708\) 0 0
\(709\) −42.3939 −1.59214 −0.796068 0.605208i \(-0.793090\pi\)
−0.796068 + 0.605208i \(0.793090\pi\)
\(710\) −50.6969 −1.90262
\(711\) 0 0
\(712\) −48.0000 −1.79888
\(713\) −48.0000 −1.79761
\(714\) 0 0
\(715\) 0 0
\(716\) 15.1918 0.567746
\(717\) 0 0
\(718\) −36.7423 −1.37121
\(719\) 3.85357 0.143714 0.0718570 0.997415i \(-0.477107\pi\)
0.0718570 + 0.997415i \(0.477107\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −46.0454 −1.71363
\(723\) 0 0
\(724\) −96.5857 −3.58958
\(725\) −1.89898 −0.0705263
\(726\) 0 0
\(727\) −30.6413 −1.13642 −0.568212 0.822882i \(-0.692365\pi\)
−0.568212 + 0.822882i \(0.692365\pi\)
\(728\) 33.7980 1.25264
\(729\) 0 0
\(730\) −68.0908 −2.52015
\(731\) 1.10102 0.0407227
\(732\) 0 0
\(733\) −14.6515 −0.541167 −0.270583 0.962697i \(-0.587217\pi\)
−0.270583 + 0.962697i \(0.587217\pi\)
\(734\) 4.89898 0.180825
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −33.6969 −1.23956 −0.619781 0.784775i \(-0.712779\pi\)
−0.619781 + 0.784775i \(0.712779\pi\)
\(740\) −68.5857 −2.52126
\(741\) 0 0
\(742\) 19.3485 0.710305
\(743\) −5.69694 −0.209000 −0.104500 0.994525i \(-0.533324\pi\)
−0.104500 + 0.994525i \(0.533324\pi\)
\(744\) 0 0
\(745\) −21.3031 −0.780484
\(746\) 60.9898 2.23300
\(747\) 0 0
\(748\) 0 0
\(749\) 7.34847 0.268507
\(750\) 0 0
\(751\) −39.6413 −1.44653 −0.723266 0.690569i \(-0.757360\pi\)
−0.723266 + 0.690569i \(0.757360\pi\)
\(752\) 48.0000 1.75038
\(753\) 0 0
\(754\) −32.0908 −1.16868
\(755\) −13.1010 −0.476795
\(756\) 0 0
\(757\) −45.0908 −1.63885 −0.819427 0.573184i \(-0.805708\pi\)
−0.819427 + 0.573184i \(0.805708\pi\)
\(758\) 48.9898 1.77939
\(759\) 0 0
\(760\) −5.39388 −0.195656
\(761\) −29.6413 −1.07450 −0.537249 0.843424i \(-0.680536\pi\)
−0.537249 + 0.843424i \(0.680536\pi\)
\(762\) 0 0
\(763\) 9.34847 0.338437
\(764\) −5.39388 −0.195144
\(765\) 0 0
\(766\) −7.95459 −0.287411
\(767\) −16.8990 −0.610187
\(768\) 0 0
\(769\) 37.4495 1.35046 0.675232 0.737606i \(-0.264044\pi\)
0.675232 + 0.737606i \(0.264044\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 41.7980 1.50434
\(773\) 26.7526 0.962222 0.481111 0.876660i \(-0.340233\pi\)
0.481111 + 0.876660i \(0.340233\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 60.4949 2.17164
\(777\) 0 0
\(778\) −24.6061 −0.882172
\(779\) 0.247449 0.00886577
\(780\) 0 0
\(781\) 0 0
\(782\) −8.09082 −0.289327
\(783\) 0 0
\(784\) 4.00000 0.142857
\(785\) −15.7980 −0.563853
\(786\) 0 0
\(787\) 20.8536 0.743350 0.371675 0.928363i \(-0.378784\pi\)
0.371675 + 0.928363i \(0.378784\pi\)
\(788\) −14.2020 −0.505927
\(789\) 0 0
\(790\) −35.3939 −1.25926
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −3.10102 −0.110120
\(794\) 55.5959 1.97303
\(795\) 0 0
\(796\) 80.0000 2.83552
\(797\) −27.7980 −0.984654 −0.492327 0.870410i \(-0.663854\pi\)
−0.492327 + 0.870410i \(0.663854\pi\)
\(798\) 0 0
\(799\) 6.60612 0.233708
\(800\) 0 0
\(801\) 0 0
\(802\) 72.7423 2.56862
\(803\) 0 0
\(804\) 0 0
\(805\) −14.6969 −0.517999
\(806\) 135.192 4.76193
\(807\) 0 0
\(808\) 0 0
\(809\) 23.1464 0.813785 0.406893 0.913476i \(-0.366612\pi\)
0.406893 + 0.913476i \(0.366612\pi\)
\(810\) 0 0
\(811\) −4.49490 −0.157837 −0.0789186 0.996881i \(-0.525147\pi\)
−0.0789186 + 0.996881i \(0.525147\pi\)
\(812\) −7.59592 −0.266564
\(813\) 0 0
\(814\) 0 0
\(815\) 41.6413 1.45863
\(816\) 0 0
\(817\) −0.898979 −0.0314513
\(818\) −31.1010 −1.08742
\(819\) 0 0
\(820\) −5.39388 −0.188362
\(821\) 12.7980 0.446652 0.223326 0.974744i \(-0.428309\pi\)
0.223326 + 0.974744i \(0.428309\pi\)
\(822\) 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
\(827\) 5.20204 0.180893 0.0904463 0.995901i \(-0.471171\pi\)
0.0904463 + 0.995901i \(0.471171\pi\)
\(828\) 0 0
\(829\) −16.4949 −0.572891 −0.286446 0.958096i \(-0.592474\pi\)
−0.286446 + 0.958096i \(0.592474\pi\)
\(830\) −36.0000 −1.24958
\(831\) 0 0
\(832\) −55.1918 −1.91343
\(833\) 0.550510 0.0190740
\(834\) 0 0
\(835\) 27.3031 0.944861
\(836\) 0 0
\(837\) 0 0
\(838\) −18.6061 −0.642738
\(839\) −25.8434 −0.892212 −0.446106 0.894980i \(-0.647190\pi\)
−0.446106 + 0.894980i \(0.647190\pi\)
\(840\) 0 0
\(841\) −25.3939 −0.875651
\(842\) −93.1918 −3.21160
\(843\) 0 0
\(844\) 18.7878 0.646701
\(845\) 84.7423 2.91523
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 31.5959 1.08501
\(849\) 0 0
\(850\) 1.34847 0.0462521
\(851\) 42.0000 1.43974
\(852\) 0 0
\(853\) −55.7423 −1.90858 −0.954291 0.298880i \(-0.903387\pi\)
−0.954291 + 0.298880i \(0.903387\pi\)
\(854\) −1.10102 −0.0376761
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) −3.30306 −0.112830 −0.0564152 0.998407i \(-0.517967\pi\)
−0.0564152 + 0.998407i \(0.517967\pi\)
\(858\) 0 0
\(859\) −1.85357 −0.0632431 −0.0316215 0.999500i \(-0.510067\pi\)
−0.0316215 + 0.999500i \(0.510067\pi\)
\(860\) 19.5959 0.668215
\(861\) 0 0
\(862\) 32.6969 1.11366
\(863\) 28.1010 0.956570 0.478285 0.878205i \(-0.341259\pi\)
0.478285 + 0.878205i \(0.341259\pi\)
\(864\) 0 0
\(865\) 15.3031 0.520320
\(866\) 37.5959 1.27756
\(867\) 0 0
\(868\) 32.0000 1.08615
\(869\) 0 0
\(870\) 0 0
\(871\) 98.2929 3.33053
\(872\) 45.7980 1.55091
\(873\) 0 0
\(874\) 6.60612 0.223455
\(875\) −9.79796 −0.331231
\(876\) 0 0
\(877\) 43.6969 1.47554 0.737770 0.675052i \(-0.235879\pi\)
0.737770 + 0.675052i \(0.235879\pi\)
\(878\) 0.853572 0.0288067
\(879\) 0 0
\(880\) 0 0
\(881\) 42.2474 1.42335 0.711676 0.702507i \(-0.247936\pi\)
0.711676 + 0.702507i \(0.247936\pi\)
\(882\) 0 0
\(883\) −19.7980 −0.666254 −0.333127 0.942882i \(-0.608104\pi\)
−0.333127 + 0.942882i \(0.608104\pi\)
\(884\) 15.1918 0.510957
\(885\) 0 0
\(886\) −64.7878 −2.17659
\(887\) 4.89898 0.164492 0.0822458 0.996612i \(-0.473791\pi\)
0.0822458 + 0.996612i \(0.473791\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −58.7878 −1.97057
\(891\) 0 0
\(892\) 29.7980 0.997709
\(893\) −5.39388 −0.180499
\(894\) 0 0
\(895\) 9.30306 0.310967
\(896\) −19.5959 −0.654654
\(897\) 0 0
\(898\) −70.7878 −2.36222
\(899\) −15.1918 −0.506676
\(900\) 0 0
\(901\) 4.34847 0.144869
\(902\) 0 0
\(903\) 0 0
\(904\) 29.3939 0.977626
\(905\) −59.1464 −1.96609
\(906\) 0 0
\(907\) 13.2020 0.438367 0.219183 0.975684i \(-0.429661\pi\)
0.219183 + 0.975684i \(0.429661\pi\)
\(908\) −48.0000 −1.59294
\(909\) 0 0
\(910\) 41.3939 1.37219
\(911\) −8.69694 −0.288142 −0.144071 0.989567i \(-0.546019\pi\)
−0.144071 + 0.989567i \(0.546019\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −47.7526 −1.57951
\(915\) 0 0
\(916\) 10.2020 0.337085
\(917\) −10.3485 −0.341737
\(918\) 0 0
\(919\) −4.79796 −0.158270 −0.0791350 0.996864i \(-0.525216\pi\)
−0.0791350 + 0.996864i \(0.525216\pi\)
\(920\) −72.0000 −2.37377
\(921\) 0 0
\(922\) −71.3939 −2.35123
\(923\) −58.2929 −1.91873
\(924\) 0 0
\(925\) −7.00000 −0.230159
\(926\) −48.4949 −1.59364
\(927\) 0 0
\(928\) 0 0
\(929\) 22.2929 0.731405 0.365702 0.930732i \(-0.380829\pi\)
0.365702 + 0.930732i \(0.380829\pi\)
\(930\) 0 0
\(931\) −0.449490 −0.0147314
\(932\) −7.59592 −0.248813
\(933\) 0 0
\(934\) 18.6061 0.608811
\(935\) 0 0
\(936\) 0 0
\(937\) −42.6969 −1.39485 −0.697424 0.716659i \(-0.745671\pi\)
−0.697424 + 0.716659i \(0.745671\pi\)
\(938\) 34.8990 1.13949
\(939\) 0 0
\(940\) 117.576 3.83489
\(941\) 28.8434 0.940267 0.470133 0.882595i \(-0.344206\pi\)
0.470133 + 0.882595i \(0.344206\pi\)
\(942\) 0 0
\(943\) 3.30306 0.107562
\(944\) −9.79796 −0.318896
\(945\) 0 0
\(946\) 0 0
\(947\) 51.9898 1.68944 0.844721 0.535207i \(-0.179767\pi\)
0.844721 + 0.535207i \(0.179767\pi\)
\(948\) 0 0
\(949\) −78.2929 −2.54149
\(950\) −1.10102 −0.0357218
\(951\) 0 0
\(952\) 2.69694 0.0874083
\(953\) −13.3485 −0.432399 −0.216200 0.976349i \(-0.569366\pi\)
−0.216200 + 0.976349i \(0.569366\pi\)
\(954\) 0 0
\(955\) −3.30306 −0.106885
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 52.0454 1.68151
\(959\) −21.7980 −0.703893
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −118.293 −3.81391
\(963\) 0 0
\(964\) −76.9898 −2.47967
\(965\) 25.5959 0.823962
\(966\) 0 0
\(967\) −42.9444 −1.38100 −0.690499 0.723333i \(-0.742609\pi\)
−0.690499 + 0.723333i \(0.742609\pi\)
\(968\) −53.8888 −1.73205
\(969\) 0 0
\(970\) 74.0908 2.37891
\(971\) 1.59592 0.0512154 0.0256077 0.999672i \(-0.491848\pi\)
0.0256077 + 0.999672i \(0.491848\pi\)
\(972\) 0 0
\(973\) −20.3485 −0.652342
\(974\) 56.2020 1.80083
\(975\) 0 0
\(976\) −1.79796 −0.0575513
\(977\) 38.2020 1.22219 0.611096 0.791557i \(-0.290729\pi\)
0.611096 + 0.791557i \(0.290729\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 9.79796 0.312984
\(981\) 0 0
\(982\) 56.0908 1.78993
\(983\) 41.9444 1.33782 0.668909 0.743344i \(-0.266762\pi\)
0.668909 + 0.743344i \(0.266762\pi\)
\(984\) 0 0
\(985\) −8.69694 −0.277108
\(986\) −2.56072 −0.0815498
\(987\) 0 0
\(988\) −12.4041 −0.394626
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 5.55051 0.176318 0.0881589 0.996106i \(-0.471902\pi\)
0.0881589 + 0.996106i \(0.471902\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −20.6969 −0.656467
\(995\) 48.9898 1.55308
\(996\) 0 0
\(997\) 56.4949 1.78921 0.894606 0.446856i \(-0.147457\pi\)
0.894606 + 0.446856i \(0.147457\pi\)
\(998\) −93.1918 −2.94994
\(999\) 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.1.2 2
3.2 odd 2 2667.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.h.1.1 2 3.2 odd 2
8001.2.a.k.1.2 2 1.1 even 1 trivial