Properties

Label 4560.2.a.bu.1.1
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.140435\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -4.98028 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -4.98028 q^{7} +1.00000 q^{9} +5.26115 q^{11} +6.98028 q^{13} +1.00000 q^{15} +0.280871 q^{17} -1.00000 q^{19} -4.98028 q^{21} -0.280871 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.26115 q^{29} -2.28087 q^{31} +5.26115 q^{33} -4.98028 q^{35} -2.98028 q^{37} +6.98028 q^{39} +0.738851 q^{41} +5.54202 q^{43} +1.00000 q^{45} +4.28087 q^{47} +17.8032 q^{49} +0.280871 q^{51} +5.71913 q^{53} +5.26115 q^{55} -1.00000 q^{57} -10.5223 q^{59} -2.24143 q^{61} -4.98028 q^{63} +6.98028 q^{65} -0.280871 q^{69} -14.5223 q^{71} +5.43826 q^{73} +1.00000 q^{75} -26.2020 q^{77} +16.4829 q^{79} +1.00000 q^{81} -10.8032 q^{83} +0.280871 q^{85} -3.26115 q^{87} +16.6600 q^{89} -34.7637 q^{91} -2.28087 q^{93} -1.00000 q^{95} +2.98028 q^{97} +5.26115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} + 6 q^{39} + 16 q^{41} + 4 q^{43} + 3 q^{45} + 14 q^{47} + 27 q^{49} + 2 q^{51} + 16 q^{53} + 2 q^{55} - 3 q^{57} - 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} - 16 q^{71} + 14 q^{73} + 3 q^{75} - 20 q^{77} - 8 q^{79} + 3 q^{81} - 6 q^{83} + 2 q^{85} + 4 q^{87} + 4 q^{89} - 48 q^{91} - 8 q^{93} - 3 q^{95} - 6 q^{97} + 2 q^{99} + O(q^{100}) \)

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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.98028 −1.88237 −0.941184 0.337894i \(-0.890285\pi\)
−0.941184 + 0.337894i \(0.890285\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.26115 1.58630 0.793148 0.609029i \(-0.208441\pi\)
0.793148 + 0.609029i \(0.208441\pi\)
\(12\) 0 0
\(13\) 6.98028 1.93598 0.967990 0.250987i \(-0.0807552\pi\)
0.967990 + 0.250987i \(0.0807552\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0.280871 0.0681212 0.0340606 0.999420i \(-0.489156\pi\)
0.0340606 + 0.999420i \(0.489156\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.98028 −1.08679
\(22\) 0 0
\(23\) −0.280871 −0.0585656 −0.0292828 0.999571i \(-0.509322\pi\)
−0.0292828 + 0.999571i \(0.509322\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.26115 −0.605580 −0.302790 0.953057i \(-0.597918\pi\)
−0.302790 + 0.953057i \(0.597918\pi\)
\(30\) 0 0
\(31\) −2.28087 −0.409656 −0.204828 0.978798i \(-0.565664\pi\)
−0.204828 + 0.978798i \(0.565664\pi\)
\(32\) 0 0
\(33\) 5.26115 0.915848
\(34\) 0 0
\(35\) −4.98028 −0.841821
\(36\) 0 0
\(37\) −2.98028 −0.489955 −0.244977 0.969529i \(-0.578781\pi\)
−0.244977 + 0.969529i \(0.578781\pi\)
\(38\) 0 0
\(39\) 6.98028 1.11774
\(40\) 0 0
\(41\) 0.738851 0.115389 0.0576946 0.998334i \(-0.481625\pi\)
0.0576946 + 0.998334i \(0.481625\pi\)
\(42\) 0 0
\(43\) 5.54202 0.845150 0.422575 0.906328i \(-0.361126\pi\)
0.422575 + 0.906328i \(0.361126\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 4.28087 0.624429 0.312215 0.950012i \(-0.398929\pi\)
0.312215 + 0.950012i \(0.398929\pi\)
\(48\) 0 0
\(49\) 17.8032 2.54331
\(50\) 0 0
\(51\) 0.280871 0.0393298
\(52\) 0 0
\(53\) 5.71913 0.785583 0.392791 0.919628i \(-0.371509\pi\)
0.392791 + 0.919628i \(0.371509\pi\)
\(54\) 0 0
\(55\) 5.26115 0.709413
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −10.5223 −1.36989 −0.684943 0.728596i \(-0.740173\pi\)
−0.684943 + 0.728596i \(0.740173\pi\)
\(60\) 0 0
\(61\) −2.24143 −0.286985 −0.143493 0.989651i \(-0.545833\pi\)
−0.143493 + 0.989651i \(0.545833\pi\)
\(62\) 0 0
\(63\) −4.98028 −0.627456
\(64\) 0 0
\(65\) 6.98028 0.865797
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −0.280871 −0.0338129
\(70\) 0 0
\(71\) −14.5223 −1.72348 −0.861740 0.507350i \(-0.830625\pi\)
−0.861740 + 0.507350i \(0.830625\pi\)
\(72\) 0 0
\(73\) 5.43826 0.636500 0.318250 0.948007i \(-0.396905\pi\)
0.318250 + 0.948007i \(0.396905\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −26.2020 −2.98599
\(78\) 0 0
\(79\) 16.4829 1.85447 0.927233 0.374485i \(-0.122181\pi\)
0.927233 + 0.374485i \(0.122181\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.8032 −1.18580 −0.592901 0.805275i \(-0.702017\pi\)
−0.592901 + 0.805275i \(0.702017\pi\)
\(84\) 0 0
\(85\) 0.280871 0.0304647
\(86\) 0 0
\(87\) −3.26115 −0.349632
\(88\) 0 0
\(89\) 16.6600 1.76595 0.882976 0.469418i \(-0.155536\pi\)
0.882976 + 0.469418i \(0.155536\pi\)
\(90\) 0 0
\(91\) −34.7637 −3.64423
\(92\) 0 0
\(93\) −2.28087 −0.236515
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 2.98028 0.302601 0.151301 0.988488i \(-0.451654\pi\)
0.151301 + 0.988488i \(0.451654\pi\)
\(98\) 0 0
\(99\) 5.26115 0.528765
\(100\) 0 0
\(101\) 13.0840 1.30191 0.650955 0.759116i \(-0.274369\pi\)
0.650955 + 0.759116i \(0.274369\pi\)
\(102\) 0 0
\(103\) −4.56174 −0.449482 −0.224741 0.974419i \(-0.572154\pi\)
−0.224741 + 0.974419i \(0.572154\pi\)
\(104\) 0 0
\(105\) −4.98028 −0.486025
\(106\) 0 0
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) −4.52230 −0.433158 −0.216579 0.976265i \(-0.569490\pi\)
−0.216579 + 0.976265i \(0.569490\pi\)
\(110\) 0 0
\(111\) −2.98028 −0.282875
\(112\) 0 0
\(113\) −0.241427 −0.0227115 −0.0113557 0.999936i \(-0.503615\pi\)
−0.0113557 + 0.999936i \(0.503615\pi\)
\(114\) 0 0
\(115\) −0.280871 −0.0261913
\(116\) 0 0
\(117\) 6.98028 0.645327
\(118\) 0 0
\(119\) −1.39881 −0.128229
\(120\) 0 0
\(121\) 16.6797 1.51634
\(122\) 0 0
\(123\) 0.738851 0.0666200
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 5.54202 0.487948
\(130\) 0 0
\(131\) 15.7834 1.37901 0.689503 0.724283i \(-0.257829\pi\)
0.689503 + 0.724283i \(0.257829\pi\)
\(132\) 0 0
\(133\) 4.98028 0.431845
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 10.5223 0.892490 0.446245 0.894911i \(-0.352761\pi\)
0.446245 + 0.894911i \(0.352761\pi\)
\(140\) 0 0
\(141\) 4.28087 0.360514
\(142\) 0 0
\(143\) 36.7243 3.07104
\(144\) 0 0
\(145\) −3.26115 −0.270824
\(146\) 0 0
\(147\) 17.8032 1.46838
\(148\) 0 0
\(149\) −5.43826 −0.445519 −0.222760 0.974873i \(-0.571507\pi\)
−0.222760 + 0.974873i \(0.571507\pi\)
\(150\) 0 0
\(151\) 14.2020 1.15574 0.577870 0.816128i \(-0.303884\pi\)
0.577870 + 0.816128i \(0.303884\pi\)
\(152\) 0 0
\(153\) 0.280871 0.0227071
\(154\) 0 0
\(155\) −2.28087 −0.183204
\(156\) 0 0
\(157\) −15.3988 −1.22896 −0.614480 0.788933i \(-0.710634\pi\)
−0.614480 + 0.788933i \(0.710634\pi\)
\(158\) 0 0
\(159\) 5.71913 0.453556
\(160\) 0 0
\(161\) 1.39881 0.110242
\(162\) 0 0
\(163\) 18.9408 1.48356 0.741780 0.670643i \(-0.233982\pi\)
0.741780 + 0.670643i \(0.233982\pi\)
\(164\) 0 0
\(165\) 5.26115 0.409580
\(166\) 0 0
\(167\) −11.6797 −0.903801 −0.451901 0.892068i \(-0.649254\pi\)
−0.451901 + 0.892068i \(0.649254\pi\)
\(168\) 0 0
\(169\) 35.7243 2.74802
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −16.2414 −1.23481 −0.617406 0.786644i \(-0.711817\pi\)
−0.617406 + 0.786644i \(0.711817\pi\)
\(174\) 0 0
\(175\) −4.98028 −0.376474
\(176\) 0 0
\(177\) −10.5223 −0.790904
\(178\) 0 0
\(179\) 21.9606 1.64141 0.820705 0.571353i \(-0.193581\pi\)
0.820705 + 0.571353i \(0.193581\pi\)
\(180\) 0 0
\(181\) 19.9606 1.48366 0.741828 0.670590i \(-0.233959\pi\)
0.741828 + 0.670590i \(0.233959\pi\)
\(182\) 0 0
\(183\) −2.24143 −0.165691
\(184\) 0 0
\(185\) −2.98028 −0.219114
\(186\) 0 0
\(187\) 1.47770 0.108060
\(188\) 0 0
\(189\) −4.98028 −0.362262
\(190\) 0 0
\(191\) 19.2217 1.39083 0.695417 0.718607i \(-0.255220\pi\)
0.695417 + 0.718607i \(0.255220\pi\)
\(192\) 0 0
\(193\) −6.41854 −0.462016 −0.231008 0.972952i \(-0.574202\pi\)
−0.231008 + 0.972952i \(0.574202\pi\)
\(194\) 0 0
\(195\) 6.98028 0.499868
\(196\) 0 0
\(197\) −16.2809 −1.15996 −0.579982 0.814629i \(-0.696940\pi\)
−0.579982 + 0.814629i \(0.696940\pi\)
\(198\) 0 0
\(199\) 25.3988 1.80047 0.900237 0.435400i \(-0.143393\pi\)
0.900237 + 0.435400i \(0.143393\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 16.2414 1.13992
\(204\) 0 0
\(205\) 0.738851 0.0516036
\(206\) 0 0
\(207\) −0.280871 −0.0195219
\(208\) 0 0
\(209\) −5.26115 −0.363921
\(210\) 0 0
\(211\) −27.3594 −1.88350 −0.941748 0.336318i \(-0.890818\pi\)
−0.941748 + 0.336318i \(0.890818\pi\)
\(212\) 0 0
\(213\) −14.5223 −0.995051
\(214\) 0 0
\(215\) 5.54202 0.377963
\(216\) 0 0
\(217\) 11.3594 0.771124
\(218\) 0 0
\(219\) 5.43826 0.367483
\(220\) 0 0
\(221\) 1.96056 0.131881
\(222\) 0 0
\(223\) −4.56174 −0.305477 −0.152738 0.988267i \(-0.548809\pi\)
−0.152738 + 0.988267i \(0.548809\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −20.7243 −1.37552 −0.687759 0.725939i \(-0.741406\pi\)
−0.687759 + 0.725939i \(0.741406\pi\)
\(228\) 0 0
\(229\) −2.32031 −0.153331 −0.0766654 0.997057i \(-0.524427\pi\)
−0.0766654 + 0.997057i \(0.524427\pi\)
\(230\) 0 0
\(231\) −26.2020 −1.72396
\(232\) 0 0
\(233\) 23.6063 1.54650 0.773251 0.634100i \(-0.218629\pi\)
0.773251 + 0.634100i \(0.218629\pi\)
\(234\) 0 0
\(235\) 4.28087 0.279253
\(236\) 0 0
\(237\) 16.4829 1.07068
\(238\) 0 0
\(239\) 16.6994 1.08019 0.540097 0.841603i \(-0.318387\pi\)
0.540097 + 0.841603i \(0.318387\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 17.8032 1.13740
\(246\) 0 0
\(247\) −6.98028 −0.444144
\(248\) 0 0
\(249\) −10.8032 −0.684623
\(250\) 0 0
\(251\) 12.1377 0.766123 0.383061 0.923723i \(-0.374870\pi\)
0.383061 + 0.923723i \(0.374870\pi\)
\(252\) 0 0
\(253\) −1.47770 −0.0929024
\(254\) 0 0
\(255\) 0.280871 0.0175888
\(256\) 0 0
\(257\) 8.80317 0.549127 0.274563 0.961569i \(-0.411467\pi\)
0.274563 + 0.961569i \(0.411467\pi\)
\(258\) 0 0
\(259\) 14.8426 0.922275
\(260\) 0 0
\(261\) −3.26115 −0.201860
\(262\) 0 0
\(263\) −13.1179 −0.808887 −0.404444 0.914563i \(-0.632535\pi\)
−0.404444 + 0.914563i \(0.632535\pi\)
\(264\) 0 0
\(265\) 5.71913 0.351323
\(266\) 0 0
\(267\) 16.6600 1.01957
\(268\) 0 0
\(269\) 13.2217 0.806142 0.403071 0.915169i \(-0.367943\pi\)
0.403071 + 0.915169i \(0.367943\pi\)
\(270\) 0 0
\(271\) −21.9606 −1.33401 −0.667004 0.745054i \(-0.732424\pi\)
−0.667004 + 0.745054i \(0.732424\pi\)
\(272\) 0 0
\(273\) −34.7637 −2.10400
\(274\) 0 0
\(275\) 5.26115 0.317259
\(276\) 0 0
\(277\) −19.6063 −1.17803 −0.589015 0.808122i \(-0.700484\pi\)
−0.589015 + 0.808122i \(0.700484\pi\)
\(278\) 0 0
\(279\) −2.28087 −0.136552
\(280\) 0 0
\(281\) −5.78345 −0.345011 −0.172506 0.985009i \(-0.555186\pi\)
−0.172506 + 0.985009i \(0.555186\pi\)
\(282\) 0 0
\(283\) −0.418536 −0.0248794 −0.0124397 0.999923i \(-0.503960\pi\)
−0.0124397 + 0.999923i \(0.503960\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) −3.67969 −0.217205
\(288\) 0 0
\(289\) −16.9211 −0.995360
\(290\) 0 0
\(291\) 2.98028 0.174707
\(292\) 0 0
\(293\) 24.2414 1.41620 0.708100 0.706113i \(-0.249553\pi\)
0.708100 + 0.706113i \(0.249553\pi\)
\(294\) 0 0
\(295\) −10.5223 −0.612632
\(296\) 0 0
\(297\) 5.26115 0.305283
\(298\) 0 0
\(299\) −1.96056 −0.113382
\(300\) 0 0
\(301\) −27.6008 −1.59088
\(302\) 0 0
\(303\) 13.0840 0.751658
\(304\) 0 0
\(305\) −2.24143 −0.128344
\(306\) 0 0
\(307\) 9.39881 0.536419 0.268209 0.963361i \(-0.413568\pi\)
0.268209 + 0.963361i \(0.413568\pi\)
\(308\) 0 0
\(309\) −4.56174 −0.259508
\(310\) 0 0
\(311\) 19.2217 1.08996 0.544981 0.838448i \(-0.316537\pi\)
0.544981 + 0.838448i \(0.316537\pi\)
\(312\) 0 0
\(313\) −31.9606 −1.80652 −0.903259 0.429096i \(-0.858832\pi\)
−0.903259 + 0.429096i \(0.858832\pi\)
\(314\) 0 0
\(315\) −4.98028 −0.280607
\(316\) 0 0
\(317\) −9.71913 −0.545881 −0.272940 0.962031i \(-0.587996\pi\)
−0.272940 + 0.962031i \(0.587996\pi\)
\(318\) 0 0
\(319\) −17.1574 −0.960629
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) −0.280871 −0.0156281
\(324\) 0 0
\(325\) 6.98028 0.387196
\(326\) 0 0
\(327\) −4.52230 −0.250084
\(328\) 0 0
\(329\) −21.3199 −1.17541
\(330\) 0 0
\(331\) 18.2020 1.00047 0.500236 0.865889i \(-0.333247\pi\)
0.500236 + 0.865889i \(0.333247\pi\)
\(332\) 0 0
\(333\) −2.98028 −0.163318
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.37909 −0.456438 −0.228219 0.973610i \(-0.573290\pi\)
−0.228219 + 0.973610i \(0.573290\pi\)
\(338\) 0 0
\(339\) −0.241427 −0.0131125
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −53.8028 −2.90508
\(344\) 0 0
\(345\) −0.280871 −0.0151216
\(346\) 0 0
\(347\) 29.6008 1.58905 0.794527 0.607229i \(-0.207719\pi\)
0.794527 + 0.607229i \(0.207719\pi\)
\(348\) 0 0
\(349\) 11.0446 0.591204 0.295602 0.955311i \(-0.404480\pi\)
0.295602 + 0.955311i \(0.404480\pi\)
\(350\) 0 0
\(351\) 6.98028 0.372580
\(352\) 0 0
\(353\) −19.0446 −1.01364 −0.506821 0.862051i \(-0.669179\pi\)
−0.506821 + 0.862051i \(0.669179\pi\)
\(354\) 0 0
\(355\) −14.5223 −0.770764
\(356\) 0 0
\(357\) −1.39881 −0.0740331
\(358\) 0 0
\(359\) −29.7440 −1.56983 −0.784914 0.619604i \(-0.787293\pi\)
−0.784914 + 0.619604i \(0.787293\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 16.6797 0.875456
\(364\) 0 0
\(365\) 5.43826 0.284651
\(366\) 0 0
\(367\) −27.5026 −1.43562 −0.717811 0.696238i \(-0.754856\pi\)
−0.717811 + 0.696238i \(0.754856\pi\)
\(368\) 0 0
\(369\) 0.738851 0.0384631
\(370\) 0 0
\(371\) −28.4829 −1.47876
\(372\) 0 0
\(373\) −16.4580 −0.852162 −0.426081 0.904685i \(-0.640106\pi\)
−0.426081 + 0.904685i \(0.640106\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −22.7637 −1.17239
\(378\) 0 0
\(379\) −2.84261 −0.146015 −0.0730076 0.997331i \(-0.523260\pi\)
−0.0730076 + 0.997331i \(0.523260\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −15.4383 −0.788858 −0.394429 0.918926i \(-0.629058\pi\)
−0.394429 + 0.918926i \(0.629058\pi\)
\(384\) 0 0
\(385\) −26.2020 −1.33538
\(386\) 0 0
\(387\) 5.54202 0.281717
\(388\) 0 0
\(389\) −18.4829 −0.937118 −0.468559 0.883432i \(-0.655227\pi\)
−0.468559 + 0.883432i \(0.655227\pi\)
\(390\) 0 0
\(391\) −0.0788884 −0.00398956
\(392\) 0 0
\(393\) 15.7834 0.796170
\(394\) 0 0
\(395\) 16.4829 0.829342
\(396\) 0 0
\(397\) −34.9657 −1.75488 −0.877439 0.479688i \(-0.840750\pi\)
−0.877439 + 0.479688i \(0.840750\pi\)
\(398\) 0 0
\(399\) 4.98028 0.249326
\(400\) 0 0
\(401\) 19.8229 0.989908 0.494954 0.868919i \(-0.335185\pi\)
0.494954 + 0.868919i \(0.335185\pi\)
\(402\) 0 0
\(403\) −15.9211 −0.793087
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −15.6797 −0.777213
\(408\) 0 0
\(409\) −9.64578 −0.476953 −0.238477 0.971148i \(-0.576648\pi\)
−0.238477 + 0.971148i \(0.576648\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) 52.4040 2.57863
\(414\) 0 0
\(415\) −10.8032 −0.530307
\(416\) 0 0
\(417\) 10.5223 0.515279
\(418\) 0 0
\(419\) −6.65996 −0.325360 −0.162680 0.986679i \(-0.552014\pi\)
−0.162680 + 0.986679i \(0.552014\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 4.28087 0.208143
\(424\) 0 0
\(425\) 0.280871 0.0136242
\(426\) 0 0
\(427\) 11.1629 0.540212
\(428\) 0 0
\(429\) 36.7243 1.77306
\(430\) 0 0
\(431\) 20.8371 1.00369 0.501843 0.864959i \(-0.332655\pi\)
0.501843 + 0.864959i \(0.332655\pi\)
\(432\) 0 0
\(433\) 9.58146 0.460456 0.230228 0.973137i \(-0.426053\pi\)
0.230228 + 0.973137i \(0.426053\pi\)
\(434\) 0 0
\(435\) −3.26115 −0.156360
\(436\) 0 0
\(437\) 0.280871 0.0134359
\(438\) 0 0
\(439\) −5.04459 −0.240765 −0.120383 0.992728i \(-0.538412\pi\)
−0.120383 + 0.992728i \(0.538412\pi\)
\(440\) 0 0
\(441\) 17.8032 0.847770
\(442\) 0 0
\(443\) −31.6402 −1.50327 −0.751637 0.659577i \(-0.770735\pi\)
−0.751637 + 0.659577i \(0.770735\pi\)
\(444\) 0 0
\(445\) 16.6600 0.789758
\(446\) 0 0
\(447\) −5.43826 −0.257221
\(448\) 0 0
\(449\) −7.82289 −0.369185 −0.184593 0.982815i \(-0.559097\pi\)
−0.184593 + 0.982815i \(0.559097\pi\)
\(450\) 0 0
\(451\) 3.88721 0.183041
\(452\) 0 0
\(453\) 14.2020 0.667267
\(454\) 0 0
\(455\) −34.7637 −1.62975
\(456\) 0 0
\(457\) −15.4777 −0.724016 −0.362008 0.932175i \(-0.617909\pi\)
−0.362008 + 0.932175i \(0.617909\pi\)
\(458\) 0 0
\(459\) 0.280871 0.0131099
\(460\) 0 0
\(461\) 9.92111 0.462072 0.231036 0.972945i \(-0.425788\pi\)
0.231036 + 0.972945i \(0.425788\pi\)
\(462\) 0 0
\(463\) 21.4631 0.997476 0.498738 0.866753i \(-0.333797\pi\)
0.498738 + 0.866753i \(0.333797\pi\)
\(464\) 0 0
\(465\) −2.28087 −0.105773
\(466\) 0 0
\(467\) 32.7637 1.51612 0.758062 0.652182i \(-0.226146\pi\)
0.758062 + 0.652182i \(0.226146\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −15.3988 −0.709540
\(472\) 0 0
\(473\) 29.1574 1.34066
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 5.71913 0.261861
\(478\) 0 0
\(479\) 27.7834 1.26946 0.634729 0.772735i \(-0.281112\pi\)
0.634729 + 0.772735i \(0.281112\pi\)
\(480\) 0 0
\(481\) −20.8032 −0.948543
\(482\) 0 0
\(483\) 1.39881 0.0636483
\(484\) 0 0
\(485\) 2.98028 0.135327
\(486\) 0 0
\(487\) 29.3199 1.32861 0.664306 0.747460i \(-0.268727\pi\)
0.664306 + 0.747460i \(0.268727\pi\)
\(488\) 0 0
\(489\) 18.9408 0.856534
\(490\) 0 0
\(491\) −4.69941 −0.212081 −0.106041 0.994362i \(-0.533817\pi\)
−0.106041 + 0.994362i \(0.533817\pi\)
\(492\) 0 0
\(493\) −0.915961 −0.0412528
\(494\) 0 0
\(495\) 5.26115 0.236471
\(496\) 0 0
\(497\) 72.3251 3.24422
\(498\) 0 0
\(499\) −30.4434 −1.36283 −0.681417 0.731895i \(-0.738636\pi\)
−0.681417 + 0.731895i \(0.738636\pi\)
\(500\) 0 0
\(501\) −11.6797 −0.521810
\(502\) 0 0
\(503\) 13.6797 0.609947 0.304974 0.952361i \(-0.401352\pi\)
0.304974 + 0.952361i \(0.401352\pi\)
\(504\) 0 0
\(505\) 13.0840 0.582232
\(506\) 0 0
\(507\) 35.7243 1.58657
\(508\) 0 0
\(509\) −17.7046 −0.784741 −0.392370 0.919807i \(-0.628345\pi\)
−0.392370 + 0.919807i \(0.628345\pi\)
\(510\) 0 0
\(511\) −27.0840 −1.19813
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −4.56174 −0.201014
\(516\) 0 0
\(517\) 22.5223 0.990530
\(518\) 0 0
\(519\) −16.2414 −0.712919
\(520\) 0 0
\(521\) −9.30059 −0.407466 −0.203733 0.979026i \(-0.565308\pi\)
−0.203733 + 0.979026i \(0.565308\pi\)
\(522\) 0 0
\(523\) −7.64578 −0.334327 −0.167163 0.985929i \(-0.553461\pi\)
−0.167163 + 0.985929i \(0.553461\pi\)
\(524\) 0 0
\(525\) −4.98028 −0.217357
\(526\) 0 0
\(527\) −0.640630 −0.0279063
\(528\) 0 0
\(529\) −22.9211 −0.996570
\(530\) 0 0
\(531\) −10.5223 −0.456629
\(532\) 0 0
\(533\) 5.15739 0.223391
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 0 0
\(537\) 21.9606 0.947668
\(538\) 0 0
\(539\) 93.6651 4.03444
\(540\) 0 0
\(541\) −12.8765 −0.553605 −0.276802 0.960927i \(-0.589275\pi\)
−0.276802 + 0.960927i \(0.589275\pi\)
\(542\) 0 0
\(543\) 19.9606 0.856589
\(544\) 0 0
\(545\) −4.52230 −0.193714
\(546\) 0 0
\(547\) −13.4777 −0.576265 −0.288132 0.957591i \(-0.593034\pi\)
−0.288132 + 0.957591i \(0.593034\pi\)
\(548\) 0 0
\(549\) −2.24143 −0.0956618
\(550\) 0 0
\(551\) 3.26115 0.138930
\(552\) 0 0
\(553\) −82.0892 −3.49079
\(554\) 0 0
\(555\) −2.98028 −0.126506
\(556\) 0 0
\(557\) −3.68522 −0.156148 −0.0780740 0.996948i \(-0.524877\pi\)
−0.0780740 + 0.996948i \(0.524877\pi\)
\(558\) 0 0
\(559\) 38.6848 1.63619
\(560\) 0 0
\(561\) 1.47770 0.0623887
\(562\) 0 0
\(563\) −12.7243 −0.536264 −0.268132 0.963382i \(-0.586406\pi\)
−0.268132 + 0.963382i \(0.586406\pi\)
\(564\) 0 0
\(565\) −0.241427 −0.0101569
\(566\) 0 0
\(567\) −4.98028 −0.209152
\(568\) 0 0
\(569\) 25.3006 1.06066 0.530328 0.847793i \(-0.322069\pi\)
0.530328 + 0.847793i \(0.322069\pi\)
\(570\) 0 0
\(571\) −23.0052 −0.962736 −0.481368 0.876519i \(-0.659860\pi\)
−0.481368 + 0.876519i \(0.659860\pi\)
\(572\) 0 0
\(573\) 19.2217 0.802998
\(574\) 0 0
\(575\) −0.280871 −0.0117131
\(576\) 0 0
\(577\) −25.0840 −1.04426 −0.522131 0.852865i \(-0.674863\pi\)
−0.522131 + 0.852865i \(0.674863\pi\)
\(578\) 0 0
\(579\) −6.41854 −0.266745
\(580\) 0 0
\(581\) 53.8028 2.23212
\(582\) 0 0
\(583\) 30.0892 1.24617
\(584\) 0 0
\(585\) 6.98028 0.288599
\(586\) 0 0
\(587\) 28.6848 1.18395 0.591975 0.805956i \(-0.298348\pi\)
0.591975 + 0.805956i \(0.298348\pi\)
\(588\) 0 0
\(589\) 2.28087 0.0939816
\(590\) 0 0
\(591\) −16.2809 −0.669706
\(592\) 0 0
\(593\) 2.56174 0.105198 0.0525991 0.998616i \(-0.483249\pi\)
0.0525991 + 0.998616i \(0.483249\pi\)
\(594\) 0 0
\(595\) −1.39881 −0.0573458
\(596\) 0 0
\(597\) 25.3988 1.03950
\(598\) 0 0
\(599\) −33.0446 −1.35017 −0.675083 0.737742i \(-0.735892\pi\)
−0.675083 + 0.737742i \(0.735892\pi\)
\(600\) 0 0
\(601\) −22.8371 −0.931544 −0.465772 0.884905i \(-0.654223\pi\)
−0.465772 + 0.884905i \(0.654223\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 16.6797 0.678126
\(606\) 0 0
\(607\) 28.1286 1.14171 0.570853 0.821052i \(-0.306613\pi\)
0.570853 + 0.821052i \(0.306613\pi\)
\(608\) 0 0
\(609\) 16.2414 0.658136
\(610\) 0 0
\(611\) 29.8817 1.20888
\(612\) 0 0
\(613\) 15.9606 0.644641 0.322320 0.946631i \(-0.395537\pi\)
0.322320 + 0.946631i \(0.395537\pi\)
\(614\) 0 0
\(615\) 0.738851 0.0297934
\(616\) 0 0
\(617\) −6.59565 −0.265531 −0.132765 0.991147i \(-0.542386\pi\)
−0.132765 + 0.991147i \(0.542386\pi\)
\(618\) 0 0
\(619\) −33.8817 −1.36182 −0.680910 0.732367i \(-0.738415\pi\)
−0.680910 + 0.732367i \(0.738415\pi\)
\(620\) 0 0
\(621\) −0.280871 −0.0112710
\(622\) 0 0
\(623\) −82.9712 −3.32417
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −5.26115 −0.210110
\(628\) 0 0
\(629\) −0.837073 −0.0333763
\(630\) 0 0
\(631\) −26.7976 −1.06680 −0.533398 0.845864i \(-0.679085\pi\)
−0.533398 + 0.845864i \(0.679085\pi\)
\(632\) 0 0
\(633\) −27.3594 −1.08744
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) 124.271 4.92380
\(638\) 0 0
\(639\) −14.5223 −0.574493
\(640\) 0 0
\(641\) 25.7834 1.01838 0.509192 0.860653i \(-0.329944\pi\)
0.509192 + 0.860653i \(0.329944\pi\)
\(642\) 0 0
\(643\) 19.4237 0.765995 0.382998 0.923749i \(-0.374892\pi\)
0.382998 + 0.923749i \(0.374892\pi\)
\(644\) 0 0
\(645\) 5.54202 0.218217
\(646\) 0 0
\(647\) 8.55620 0.336379 0.168190 0.985755i \(-0.446208\pi\)
0.168190 + 0.985755i \(0.446208\pi\)
\(648\) 0 0
\(649\) −55.3594 −2.17305
\(650\) 0 0
\(651\) 11.3594 0.445209
\(652\) 0 0
\(653\) −0.842612 −0.0329740 −0.0164870 0.999864i \(-0.505248\pi\)
−0.0164870 + 0.999864i \(0.505248\pi\)
\(654\) 0 0
\(655\) 15.7834 0.616710
\(656\) 0 0
\(657\) 5.43826 0.212167
\(658\) 0 0
\(659\) 3.43826 0.133936 0.0669678 0.997755i \(-0.478668\pi\)
0.0669678 + 0.997755i \(0.478668\pi\)
\(660\) 0 0
\(661\) 9.08404 0.353328 0.176664 0.984271i \(-0.443469\pi\)
0.176664 + 0.984271i \(0.443469\pi\)
\(662\) 0 0
\(663\) 1.96056 0.0761417
\(664\) 0 0
\(665\) 4.98028 0.193127
\(666\) 0 0
\(667\) 0.915961 0.0354662
\(668\) 0 0
\(669\) −4.56174 −0.176367
\(670\) 0 0
\(671\) −11.7925 −0.455244
\(672\) 0 0
\(673\) 2.98028 0.114881 0.0574406 0.998349i \(-0.481706\pi\)
0.0574406 + 0.998349i \(0.481706\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −48.9996 −1.88321 −0.941604 0.336722i \(-0.890682\pi\)
−0.941604 + 0.336722i \(0.890682\pi\)
\(678\) 0 0
\(679\) −14.8426 −0.569607
\(680\) 0 0
\(681\) −20.7243 −0.794156
\(682\) 0 0
\(683\) −27.9211 −1.06837 −0.534186 0.845367i \(-0.679382\pi\)
−0.534186 + 0.845367i \(0.679382\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) −2.32031 −0.0885255
\(688\) 0 0
\(689\) 39.9211 1.52087
\(690\) 0 0
\(691\) 9.96056 0.378917 0.189459 0.981889i \(-0.439327\pi\)
0.189459 + 0.981889i \(0.439327\pi\)
\(692\) 0 0
\(693\) −26.2020 −0.995331
\(694\) 0 0
\(695\) 10.5223 0.399133
\(696\) 0 0
\(697\) 0.207522 0.00786045
\(698\) 0 0
\(699\) 23.6063 0.892874
\(700\) 0 0
\(701\) −10.4829 −0.395932 −0.197966 0.980209i \(-0.563434\pi\)
−0.197966 + 0.980209i \(0.563434\pi\)
\(702\) 0 0
\(703\) 2.98028 0.112403
\(704\) 0 0
\(705\) 4.28087 0.161227
\(706\) 0 0
\(707\) −65.1621 −2.45067
\(708\) 0 0
\(709\) 16.5562 0.621781 0.310891 0.950446i \(-0.399373\pi\)
0.310891 + 0.950446i \(0.399373\pi\)
\(710\) 0 0
\(711\) 16.4829 0.618155
\(712\) 0 0
\(713\) 0.640630 0.0239918
\(714\) 0 0
\(715\) 36.7243 1.37341
\(716\) 0 0
\(717\) 16.6994 0.623651
\(718\) 0 0
\(719\) 4.21655 0.157251 0.0786255 0.996904i \(-0.474947\pi\)
0.0786255 + 0.996904i \(0.474947\pi\)
\(720\) 0 0
\(721\) 22.7187 0.846090
\(722\) 0 0
\(723\) −2.00000 −0.0743808
\(724\) 0 0
\(725\) −3.26115 −0.121116
\(726\) 0 0
\(727\) −26.4580 −0.981272 −0.490636 0.871365i \(-0.663236\pi\)
−0.490636 + 0.871365i \(0.663236\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.55659 0.0575726
\(732\) 0 0
\(733\) −6.56174 −0.242363 −0.121182 0.992630i \(-0.538668\pi\)
−0.121182 + 0.992630i \(0.538668\pi\)
\(734\) 0 0
\(735\) 17.8032 0.656680
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −44.9657 −1.65409 −0.827045 0.562136i \(-0.809980\pi\)
−0.827045 + 0.562136i \(0.809980\pi\)
\(740\) 0 0
\(741\) −6.98028 −0.256427
\(742\) 0 0
\(743\) 24.4040 0.895295 0.447647 0.894210i \(-0.352262\pi\)
0.447647 + 0.894210i \(0.352262\pi\)
\(744\) 0 0
\(745\) −5.43826 −0.199242
\(746\) 0 0
\(747\) −10.8032 −0.395267
\(748\) 0 0
\(749\) −79.6844 −2.91161
\(750\) 0 0
\(751\) −5.23628 −0.191074 −0.0955372 0.995426i \(-0.530457\pi\)
−0.0955372 + 0.995426i \(0.530457\pi\)
\(752\) 0 0
\(753\) 12.1377 0.442321
\(754\) 0 0
\(755\) 14.2020 0.516863
\(756\) 0 0
\(757\) 20.5223 0.745896 0.372948 0.927852i \(-0.378347\pi\)
0.372948 + 0.927852i \(0.378347\pi\)
\(758\) 0 0
\(759\) −1.47770 −0.0536372
\(760\) 0 0
\(761\) −13.5171 −0.489996 −0.244998 0.969524i \(-0.578787\pi\)
−0.244998 + 0.969524i \(0.578787\pi\)
\(762\) 0 0
\(763\) 22.5223 0.815362
\(764\) 0 0
\(765\) 0.280871 0.0101549
\(766\) 0 0
\(767\) −73.4486 −2.65207
\(768\) 0 0
\(769\) 33.9211 1.22323 0.611613 0.791157i \(-0.290521\pi\)
0.611613 + 0.791157i \(0.290521\pi\)
\(770\) 0 0
\(771\) 8.80317 0.317038
\(772\) 0 0
\(773\) −5.79802 −0.208540 −0.104270 0.994549i \(-0.533251\pi\)
−0.104270 + 0.994549i \(0.533251\pi\)
\(774\) 0 0
\(775\) −2.28087 −0.0819313
\(776\) 0 0
\(777\) 14.8426 0.532476
\(778\) 0 0
\(779\) −0.738851 −0.0264721
\(780\) 0 0
\(781\) −76.4040 −2.73395
\(782\) 0 0
\(783\) −3.26115 −0.116544
\(784\) 0 0
\(785\) −15.3988 −0.549607
\(786\) 0 0
\(787\) −11.5669 −0.412315 −0.206158 0.978519i \(-0.566096\pi\)
−0.206158 + 0.978519i \(0.566096\pi\)
\(788\) 0 0
\(789\) −13.1179 −0.467011
\(790\) 0 0
\(791\) 1.20237 0.0427514
\(792\) 0 0
\(793\) −15.6458 −0.555598
\(794\) 0 0
\(795\) 5.71913 0.202837
\(796\) 0 0
\(797\) −17.3649 −0.615097 −0.307548 0.951532i \(-0.599509\pi\)
−0.307548 + 0.951532i \(0.599509\pi\)
\(798\) 0 0
\(799\) 1.20237 0.0425368
\(800\) 0 0
\(801\) 16.6600 0.588651
\(802\) 0 0
\(803\) 28.6115 1.00968
\(804\) 0 0
\(805\) 1.39881 0.0493017
\(806\) 0 0
\(807\) 13.2217 0.465426
\(808\) 0 0
\(809\) 26.4829 0.931088 0.465544 0.885025i \(-0.345859\pi\)
0.465544 + 0.885025i \(0.345859\pi\)
\(810\) 0 0
\(811\) −35.8422 −1.25859 −0.629295 0.777166i \(-0.716656\pi\)
−0.629295 + 0.777166i \(0.716656\pi\)
\(812\) 0 0
\(813\) −21.9606 −0.770190
\(814\) 0 0
\(815\) 18.9408 0.663468
\(816\) 0 0
\(817\) −5.54202 −0.193891
\(818\) 0 0
\(819\) −34.7637 −1.21474
\(820\) 0 0
\(821\) −12.5223 −0.437031 −0.218516 0.975833i \(-0.570121\pi\)
−0.218516 + 0.975833i \(0.570121\pi\)
\(822\) 0 0
\(823\) −4.06432 −0.141673 −0.0708366 0.997488i \(-0.522567\pi\)
−0.0708366 + 0.997488i \(0.522567\pi\)
\(824\) 0 0
\(825\) 5.26115 0.183170
\(826\) 0 0
\(827\) 33.8478 1.17700 0.588501 0.808496i \(-0.299718\pi\)
0.588501 + 0.808496i \(0.299718\pi\)
\(828\) 0 0
\(829\) −9.08404 −0.315502 −0.157751 0.987479i \(-0.550424\pi\)
−0.157751 + 0.987479i \(0.550424\pi\)
\(830\) 0 0
\(831\) −19.6063 −0.680136
\(832\) 0 0
\(833\) 5.00039 0.173253
\(834\) 0 0
\(835\) −11.6797 −0.404192
\(836\) 0 0
\(837\) −2.28087 −0.0788384
\(838\) 0 0
\(839\) −12.2753 −0.423792 −0.211896 0.977292i \(-0.567964\pi\)
−0.211896 + 0.977292i \(0.567964\pi\)
\(840\) 0 0
\(841\) −18.3649 −0.633273
\(842\) 0 0
\(843\) −5.78345 −0.199192
\(844\) 0 0
\(845\) 35.7243 1.22895
\(846\) 0 0
\(847\) −83.0695 −2.85430
\(848\) 0 0
\(849\) −0.418536 −0.0143641
\(850\) 0 0
\(851\) 0.837073 0.0286945
\(852\) 0 0
\(853\) −11.4777 −0.392989 −0.196495 0.980505i \(-0.562956\pi\)
−0.196495 + 0.980505i \(0.562956\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) −46.7637 −1.59742 −0.798709 0.601717i \(-0.794483\pi\)
−0.798709 + 0.601717i \(0.794483\pi\)
\(858\) 0 0
\(859\) −30.7298 −1.04849 −0.524244 0.851568i \(-0.675652\pi\)
−0.524244 + 0.851568i \(0.675652\pi\)
\(860\) 0 0
\(861\) −3.67969 −0.125403
\(862\) 0 0
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) −16.2414 −0.552225
\(866\) 0 0
\(867\) −16.9211 −0.574671
\(868\) 0 0
\(869\) 86.7187 2.94173
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.98028 0.100867
\(874\) 0 0
\(875\) −4.98028 −0.168364
\(876\) 0 0
\(877\) −31.1878 −1.05314 −0.526569 0.850133i \(-0.676522\pi\)
−0.526569 + 0.850133i \(0.676522\pi\)
\(878\) 0 0
\(879\) 24.2414 0.817643
\(880\) 0 0
\(881\) 1.56689 0.0527899 0.0263950 0.999652i \(-0.491597\pi\)
0.0263950 + 0.999652i \(0.491597\pi\)
\(882\) 0 0
\(883\) 36.0643 1.21366 0.606830 0.794831i \(-0.292441\pi\)
0.606830 + 0.794831i \(0.292441\pi\)
\(884\) 0 0
\(885\) −10.5223 −0.353703
\(886\) 0 0
\(887\) 9.12348 0.306337 0.153168 0.988200i \(-0.451052\pi\)
0.153168 + 0.988200i \(0.451052\pi\)
\(888\) 0 0
\(889\) 19.9211 0.668133
\(890\) 0 0
\(891\) 5.26115 0.176255
\(892\) 0 0
\(893\) −4.28087 −0.143254
\(894\) 0 0
\(895\) 21.9606 0.734060
\(896\) 0 0
\(897\) −1.96056 −0.0654611
\(898\) 0 0
\(899\) 7.43826 0.248080
\(900\) 0 0
\(901\) 1.60634 0.0535148
\(902\) 0 0
\(903\) −27.6008 −0.918497
\(904\) 0 0
\(905\) 19.9606 0.663511
\(906\) 0 0
\(907\) 15.7136 0.521761 0.260881 0.965371i \(-0.415987\pi\)
0.260881 + 0.965371i \(0.415987\pi\)
\(908\) 0 0
\(909\) 13.0840 0.433970
\(910\) 0 0
\(911\) 47.3594 1.56909 0.784543 0.620074i \(-0.212898\pi\)
0.784543 + 0.620074i \(0.212898\pi\)
\(912\) 0 0
\(913\) −56.8371 −1.88103
\(914\) 0 0
\(915\) −2.24143 −0.0740993
\(916\) 0 0
\(917\) −78.6059 −2.59580
\(918\) 0 0
\(919\) −24.4829 −0.807615 −0.403807 0.914844i \(-0.632313\pi\)
−0.403807 + 0.914844i \(0.632313\pi\)
\(920\) 0 0
\(921\) 9.39881 0.309701
\(922\) 0 0
\(923\) −101.370 −3.33662
\(924\) 0 0
\(925\) −2.98028 −0.0979909
\(926\) 0 0
\(927\) −4.56174 −0.149827
\(928\) 0 0
\(929\) 51.3988 1.68634 0.843170 0.537647i \(-0.180687\pi\)
0.843170 + 0.537647i \(0.180687\pi\)
\(930\) 0 0
\(931\) −17.8032 −0.583475
\(932\) 0 0
\(933\) 19.2217 0.629290
\(934\) 0 0
\(935\) 1.47770 0.0483260
\(936\) 0 0
\(937\) 21.5669 0.704560 0.352280 0.935895i \(-0.385407\pi\)
0.352280 + 0.935895i \(0.385407\pi\)
\(938\) 0 0
\(939\) −31.9606 −1.04299
\(940\) 0 0
\(941\) −28.8675 −0.941053 −0.470527 0.882386i \(-0.655936\pi\)
−0.470527 + 0.882386i \(0.655936\pi\)
\(942\) 0 0
\(943\) −0.207522 −0.00675784
\(944\) 0 0
\(945\) −4.98028 −0.162008
\(946\) 0 0
\(947\) −26.2414 −0.852732 −0.426366 0.904551i \(-0.640206\pi\)
−0.426366 + 0.904551i \(0.640206\pi\)
\(948\) 0 0
\(949\) 37.9606 1.23225
\(950\) 0 0
\(951\) −9.71913 −0.315164
\(952\) 0 0
\(953\) −47.0391 −1.52374 −0.761872 0.647727i \(-0.775720\pi\)
−0.761872 + 0.647727i \(0.775720\pi\)
\(954\) 0 0
\(955\) 19.2217 0.622000
\(956\) 0 0
\(957\) −17.1574 −0.554620
\(958\) 0 0
\(959\) 29.8817 0.964929
\(960\) 0 0
\(961\) −25.7976 −0.832182
\(962\) 0 0
\(963\) 16.0000 0.515593
\(964\) 0 0
\(965\) −6.41854 −0.206620
\(966\) 0 0
\(967\) 53.9460 1.73479 0.867393 0.497624i \(-0.165794\pi\)
0.867393 + 0.497624i \(0.165794\pi\)
\(968\) 0 0
\(969\) −0.280871 −0.00902287
\(970\) 0 0
\(971\) 13.6852 0.439180 0.219590 0.975592i \(-0.429528\pi\)
0.219590 + 0.975592i \(0.429528\pi\)
\(972\) 0 0
\(973\) −52.4040 −1.67999
\(974\) 0 0
\(975\) 6.98028 0.223548
\(976\) 0 0
\(977\) 50.6848 1.62155 0.810776 0.585357i \(-0.199046\pi\)
0.810776 + 0.585357i \(0.199046\pi\)
\(978\) 0 0
\(979\) 87.6505 2.80132
\(980\) 0 0
\(981\) −4.52230 −0.144386
\(982\) 0 0
\(983\) −3.19683 −0.101963 −0.0509816 0.998700i \(-0.516235\pi\)
−0.0509816 + 0.998700i \(0.516235\pi\)
\(984\) 0 0
\(985\) −16.2809 −0.518752
\(986\) 0 0
\(987\) −21.3199 −0.678621
\(988\) 0 0
\(989\) −1.55659 −0.0494967
\(990\) 0 0
\(991\) −5.04459 −0.160247 −0.0801234 0.996785i \(-0.525531\pi\)
−0.0801234 + 0.996785i \(0.525531\pi\)
\(992\) 0 0
\(993\) 18.2020 0.577622
\(994\) 0 0
\(995\) 25.3988 0.805197
\(996\) 0 0
\(997\) −1.64578 −0.0521224 −0.0260612 0.999660i \(-0.508296\pi\)
−0.0260612 + 0.999660i \(0.508296\pi\)
\(998\) 0 0
\(999\) −2.98028 −0.0942918
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 4560.2.a.bu.1.1 3
4.3 odd 2 1140.2.a.f.1.3 3
12.11 even 2 3420.2.a.k.1.3 3
20.3 even 4 5700.2.f.p.3649.1 6
20.7 even 4 5700.2.f.p.3649.6 6
20.19 odd 2 5700.2.a.z.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.f.1.3 3 4.3 odd 2
3420.2.a.k.1.3 3 12.11 even 2
4560.2.a.bu.1.1 3 1.1 even 1 trivial
5700.2.a.z.1.1 3 20.19 odd 2
5700.2.f.p.3649.1 6 20.3 even 4
5700.2.f.p.3649.6 6 20.7 even 4