Properties

Label 3120.2.a.bi.1.1
Level $3120$
Weight $2$
Character 3120.1
Self dual yes
Analytic conductor $24.913$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.940.1
Defining polynomial: \( x^{3} - 7x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.602705\) of defining polynomial
Character \(\chi\) \(=\) 3120.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.43134 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.43134 q^{7} +1.00000 q^{9} +2.63675 q^{11} -1.00000 q^{13} -1.00000 q^{15} +7.84216 q^{17} -0.794590 q^{19} +3.43134 q^{21} +3.43134 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.06808 q^{29} -9.27349 q^{31} -2.63675 q^{33} -3.43134 q^{35} -0.636747 q^{37} +1.00000 q^{39} -4.63675 q^{41} -2.41082 q^{43} +1.00000 q^{45} -5.27349 q^{47} +4.77407 q^{49} -7.84216 q^{51} +11.4994 q^{53} +2.63675 q^{55} +0.794590 q^{57} +12.1362 q^{59} -11.4994 q^{61} -3.43134 q^{63} -1.00000 q^{65} +6.41082 q^{67} -3.43134 q^{69} +10.6367 q^{71} +16.0681 q^{73} -1.00000 q^{75} -9.04757 q^{77} -10.6367 q^{79} +1.00000 q^{81} +16.1362 q^{83} +7.84216 q^{85} +4.06808 q^{87} +8.63675 q^{89} +3.43134 q^{91} +9.27349 q^{93} -0.794590 q^{95} +12.2940 q^{97} +2.63675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} - 5 q^{11} - 3 q^{13} - 3 q^{15} + 7 q^{17} - 6 q^{19} + q^{21} + q^{23} + 3 q^{25} - 3 q^{27} + 10 q^{29} - 2 q^{31} + 5 q^{33} - q^{35} + 11 q^{37} + 3 q^{39} - q^{41} + 3 q^{45} + 10 q^{47} + 20 q^{49} - 7 q^{51} + 3 q^{53} - 5 q^{55} + 6 q^{57} - 8 q^{59} - 3 q^{61} - q^{63} - 3 q^{65} + 12 q^{67} - q^{69} + 19 q^{71} + 26 q^{73} - 3 q^{75} - 7 q^{77} - 19 q^{79} + 3 q^{81} + 4 q^{83} + 7 q^{85} - 10 q^{87} + 13 q^{89} + q^{91} + 2 q^{93} - 6 q^{95} + 9 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) −3.43134 −1.29692 −0.648462 0.761247i \(-0.724587\pi\)
−0.648462 + 0.761247i \(0.724587\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.63675 0.795009 0.397505 0.917600i \(-0.369876\pi\)
0.397505 + 0.917600i \(0.369876\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 7.84216 1.90200 0.951001 0.309187i \(-0.100057\pi\)
0.951001 + 0.309187i \(0.100057\pi\)
\(18\) 0 0
\(19\) −0.794590 −0.182291 −0.0911457 0.995838i \(-0.529053\pi\)
−0.0911457 + 0.995838i \(0.529053\pi\)
\(20\) 0 0
\(21\) 3.43134 0.748779
\(22\) 0 0
\(23\) 3.43134 0.715483 0.357742 0.933821i \(-0.383547\pi\)
0.357742 + 0.933821i \(0.383547\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.06808 −0.755424 −0.377712 0.925923i \(-0.623289\pi\)
−0.377712 + 0.925923i \(0.623289\pi\)
\(30\) 0 0
\(31\) −9.27349 −1.66557 −0.832784 0.553598i \(-0.813255\pi\)
−0.832784 + 0.553598i \(0.813255\pi\)
\(32\) 0 0
\(33\) −2.63675 −0.458999
\(34\) 0 0
\(35\) −3.43134 −0.580002
\(36\) 0 0
\(37\) −0.636747 −0.104681 −0.0523403 0.998629i \(-0.516668\pi\)
−0.0523403 + 0.998629i \(0.516668\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −4.63675 −0.724138 −0.362069 0.932151i \(-0.617930\pi\)
−0.362069 + 0.932151i \(0.617930\pi\)
\(42\) 0 0
\(43\) −2.41082 −0.367647 −0.183823 0.982959i \(-0.558847\pi\)
−0.183823 + 0.982959i \(0.558847\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −5.27349 −0.769218 −0.384609 0.923080i \(-0.625664\pi\)
−0.384609 + 0.923080i \(0.625664\pi\)
\(48\) 0 0
\(49\) 4.77407 0.682010
\(50\) 0 0
\(51\) −7.84216 −1.09812
\(52\) 0 0
\(53\) 11.4994 1.57957 0.789783 0.613386i \(-0.210193\pi\)
0.789783 + 0.613386i \(0.210193\pi\)
\(54\) 0 0
\(55\) 2.63675 0.355539
\(56\) 0 0
\(57\) 0.794590 0.105246
\(58\) 0 0
\(59\) 12.1362 1.57999 0.789997 0.613110i \(-0.210082\pi\)
0.789997 + 0.613110i \(0.210082\pi\)
\(60\) 0 0
\(61\) −11.4994 −1.47235 −0.736175 0.676791i \(-0.763370\pi\)
−0.736175 + 0.676791i \(0.763370\pi\)
\(62\) 0 0
\(63\) −3.43134 −0.432308
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 6.41082 0.783206 0.391603 0.920134i \(-0.371921\pi\)
0.391603 + 0.920134i \(0.371921\pi\)
\(68\) 0 0
\(69\) −3.43134 −0.413084
\(70\) 0 0
\(71\) 10.6367 1.26235 0.631175 0.775641i \(-0.282573\pi\)
0.631175 + 0.775641i \(0.282573\pi\)
\(72\) 0 0
\(73\) 16.0681 1.88063 0.940313 0.340310i \(-0.110532\pi\)
0.940313 + 0.340310i \(0.110532\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −9.04757 −1.03107
\(78\) 0 0
\(79\) −10.6367 −1.19673 −0.598364 0.801225i \(-0.704182\pi\)
−0.598364 + 0.801225i \(0.704182\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.1362 1.77117 0.885587 0.464473i \(-0.153756\pi\)
0.885587 + 0.464473i \(0.153756\pi\)
\(84\) 0 0
\(85\) 7.84216 0.850601
\(86\) 0 0
\(87\) 4.06808 0.436144
\(88\) 0 0
\(89\) 8.63675 0.915493 0.457747 0.889083i \(-0.348657\pi\)
0.457747 + 0.889083i \(0.348657\pi\)
\(90\) 0 0
\(91\) 3.43134 0.359702
\(92\) 0 0
\(93\) 9.27349 0.961616
\(94\) 0 0
\(95\) −0.794590 −0.0815232
\(96\) 0 0
\(97\) 12.2940 1.24827 0.624134 0.781317i \(-0.285452\pi\)
0.624134 + 0.781317i \(0.285452\pi\)
\(98\) 0 0
\(99\) 2.63675 0.265003
\(100\) 0 0
\(101\) 5.34158 0.531507 0.265753 0.964041i \(-0.414379\pi\)
0.265753 + 0.964041i \(0.414379\pi\)
\(102\) 0 0
\(103\) 10.4108 1.02581 0.512904 0.858446i \(-0.328570\pi\)
0.512904 + 0.858446i \(0.328570\pi\)
\(104\) 0 0
\(105\) 3.43134 0.334864
\(106\) 0 0
\(107\) −5.04757 −0.487967 −0.243983 0.969779i \(-0.578454\pi\)
−0.243983 + 0.969779i \(0.578454\pi\)
\(108\) 0 0
\(109\) 6.79459 0.650804 0.325402 0.945576i \(-0.394500\pi\)
0.325402 + 0.945576i \(0.394500\pi\)
\(110\) 0 0
\(111\) 0.636747 0.0604373
\(112\) 0 0
\(113\) −8.06808 −0.758981 −0.379491 0.925196i \(-0.623901\pi\)
−0.379491 + 0.925196i \(0.623901\pi\)
\(114\) 0 0
\(115\) 3.43134 0.319974
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −26.9091 −2.46675
\(120\) 0 0
\(121\) −4.04757 −0.367961
\(122\) 0 0
\(123\) 4.63675 0.418081
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 15.6843 1.39176 0.695879 0.718159i \(-0.255015\pi\)
0.695879 + 0.718159i \(0.255015\pi\)
\(128\) 0 0
\(129\) 2.41082 0.212261
\(130\) 0 0
\(131\) −12.9308 −1.12977 −0.564883 0.825171i \(-0.691079\pi\)
−0.564883 + 0.825171i \(0.691079\pi\)
\(132\) 0 0
\(133\) 2.72651 0.236418
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −19.9102 −1.68876 −0.844382 0.535741i \(-0.820032\pi\)
−0.844382 + 0.535741i \(0.820032\pi\)
\(140\) 0 0
\(141\) 5.27349 0.444108
\(142\) 0 0
\(143\) −2.63675 −0.220496
\(144\) 0 0
\(145\) −4.06808 −0.337836
\(146\) 0 0
\(147\) −4.77407 −0.393759
\(148\) 0 0
\(149\) 8.63675 0.707550 0.353775 0.935331i \(-0.384898\pi\)
0.353775 + 0.935331i \(0.384898\pi\)
\(150\) 0 0
\(151\) 17.7253 1.44247 0.721234 0.692691i \(-0.243575\pi\)
0.721234 + 0.692691i \(0.243575\pi\)
\(152\) 0 0
\(153\) 7.84216 0.634001
\(154\) 0 0
\(155\) −9.27349 −0.744865
\(156\) 0 0
\(157\) 20.5470 1.63983 0.819914 0.572487i \(-0.194021\pi\)
0.819914 + 0.572487i \(0.194021\pi\)
\(158\) 0 0
\(159\) −11.4994 −0.911963
\(160\) 0 0
\(161\) −11.7741 −0.927927
\(162\) 0 0
\(163\) 4.22593 0.331000 0.165500 0.986210i \(-0.447076\pi\)
0.165500 + 0.986210i \(0.447076\pi\)
\(164\) 0 0
\(165\) −2.63675 −0.205270
\(166\) 0 0
\(167\) 14.4108 1.11514 0.557571 0.830129i \(-0.311733\pi\)
0.557571 + 0.830129i \(0.311733\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −0.794590 −0.0607638
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −3.43134 −0.259385
\(176\) 0 0
\(177\) −12.1362 −0.912210
\(178\) 0 0
\(179\) 20.9308 1.56444 0.782219 0.623003i \(-0.214088\pi\)
0.782219 + 0.623003i \(0.214088\pi\)
\(180\) 0 0
\(181\) −5.08860 −0.378233 −0.189116 0.981955i \(-0.560562\pi\)
−0.189116 + 0.981955i \(0.560562\pi\)
\(182\) 0 0
\(183\) 11.4994 0.850062
\(184\) 0 0
\(185\) −0.636747 −0.0468146
\(186\) 0 0
\(187\) 20.6778 1.51211
\(188\) 0 0
\(189\) 3.43134 0.249593
\(190\) 0 0
\(191\) 13.7253 0.993131 0.496566 0.867999i \(-0.334594\pi\)
0.496566 + 0.867999i \(0.334594\pi\)
\(192\) 0 0
\(193\) −5.11565 −0.368233 −0.184116 0.982904i \(-0.558942\pi\)
−0.184116 + 0.982904i \(0.558942\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −20.9988 −1.49611 −0.748053 0.663639i \(-0.769011\pi\)
−0.748053 + 0.663639i \(0.769011\pi\)
\(198\) 0 0
\(199\) −9.58918 −0.679759 −0.339879 0.940469i \(-0.610386\pi\)
−0.339879 + 0.940469i \(0.610386\pi\)
\(200\) 0 0
\(201\) −6.41082 −0.452184
\(202\) 0 0
\(203\) 13.9590 0.979727
\(204\) 0 0
\(205\) −4.63675 −0.323844
\(206\) 0 0
\(207\) 3.43134 0.238494
\(208\) 0 0
\(209\) −2.09513 −0.144923
\(210\) 0 0
\(211\) 8.82164 0.607307 0.303653 0.952783i \(-0.401794\pi\)
0.303653 + 0.952783i \(0.401794\pi\)
\(212\) 0 0
\(213\) −10.6367 −0.728818
\(214\) 0 0
\(215\) −2.41082 −0.164417
\(216\) 0 0
\(217\) 31.8205 2.16011
\(218\) 0 0
\(219\) −16.0681 −1.08578
\(220\) 0 0
\(221\) −7.84216 −0.527521
\(222\) 0 0
\(223\) 9.93192 0.665090 0.332545 0.943087i \(-0.392093\pi\)
0.332545 + 0.943087i \(0.392093\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −9.27349 −0.615503 −0.307752 0.951467i \(-0.599577\pi\)
−0.307752 + 0.951467i \(0.599577\pi\)
\(228\) 0 0
\(229\) 12.0681 0.797481 0.398741 0.917064i \(-0.369447\pi\)
0.398741 + 0.917064i \(0.369447\pi\)
\(230\) 0 0
\(231\) 9.04757 0.595286
\(232\) 0 0
\(233\) 13.1157 0.859235 0.429617 0.903011i \(-0.358648\pi\)
0.429617 + 0.903011i \(0.358648\pi\)
\(234\) 0 0
\(235\) −5.27349 −0.344005
\(236\) 0 0
\(237\) 10.6367 0.690931
\(238\) 0 0
\(239\) 22.7729 1.47306 0.736529 0.676406i \(-0.236464\pi\)
0.736529 + 0.676406i \(0.236464\pi\)
\(240\) 0 0
\(241\) 6.82164 0.439420 0.219710 0.975565i \(-0.429489\pi\)
0.219710 + 0.975565i \(0.429489\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 4.77407 0.305004
\(246\) 0 0
\(247\) 0.794590 0.0505586
\(248\) 0 0
\(249\) −16.1362 −1.02259
\(250\) 0 0
\(251\) −9.24644 −0.583630 −0.291815 0.956475i \(-0.594259\pi\)
−0.291815 + 0.956475i \(0.594259\pi\)
\(252\) 0 0
\(253\) 9.04757 0.568816
\(254\) 0 0
\(255\) −7.84216 −0.491095
\(256\) 0 0
\(257\) −11.2464 −0.701534 −0.350767 0.936463i \(-0.614079\pi\)
−0.350767 + 0.936463i \(0.614079\pi\)
\(258\) 0 0
\(259\) 2.18489 0.135763
\(260\) 0 0
\(261\) −4.06808 −0.251808
\(262\) 0 0
\(263\) 22.0681 1.36078 0.680388 0.732852i \(-0.261811\pi\)
0.680388 + 0.732852i \(0.261811\pi\)
\(264\) 0 0
\(265\) 11.4994 0.706404
\(266\) 0 0
\(267\) −8.63675 −0.528560
\(268\) 0 0
\(269\) −13.2054 −0.805148 −0.402574 0.915387i \(-0.631884\pi\)
−0.402574 + 0.915387i \(0.631884\pi\)
\(270\) 0 0
\(271\) −14.0951 −0.856218 −0.428109 0.903727i \(-0.640820\pi\)
−0.428109 + 0.903727i \(0.640820\pi\)
\(272\) 0 0
\(273\) −3.43134 −0.207674
\(274\) 0 0
\(275\) 2.63675 0.159002
\(276\) 0 0
\(277\) −30.2723 −1.81889 −0.909444 0.415826i \(-0.863493\pi\)
−0.909444 + 0.415826i \(0.863493\pi\)
\(278\) 0 0
\(279\) −9.27349 −0.555190
\(280\) 0 0
\(281\) 2.13617 0.127433 0.0637165 0.997968i \(-0.479705\pi\)
0.0637165 + 0.997968i \(0.479705\pi\)
\(282\) 0 0
\(283\) −0.136167 −0.00809431 −0.00404715 0.999992i \(-0.501288\pi\)
−0.00404715 + 0.999992i \(0.501288\pi\)
\(284\) 0 0
\(285\) 0.794590 0.0470675
\(286\) 0 0
\(287\) 15.9102 0.939152
\(288\) 0 0
\(289\) 44.4994 2.61761
\(290\) 0 0
\(291\) −12.2940 −0.720688
\(292\) 0 0
\(293\) −15.2735 −0.892287 −0.446144 0.894961i \(-0.647203\pi\)
−0.446144 + 0.894961i \(0.647203\pi\)
\(294\) 0 0
\(295\) 12.1362 0.706595
\(296\) 0 0
\(297\) −2.63675 −0.153000
\(298\) 0 0
\(299\) −3.43134 −0.198439
\(300\) 0 0
\(301\) 8.27233 0.476809
\(302\) 0 0
\(303\) −5.34158 −0.306866
\(304\) 0 0
\(305\) −11.4994 −0.658455
\(306\) 0 0
\(307\) −7.91024 −0.451461 −0.225731 0.974190i \(-0.572477\pi\)
−0.225731 + 0.974190i \(0.572477\pi\)
\(308\) 0 0
\(309\) −10.4108 −0.592251
\(310\) 0 0
\(311\) −2.54699 −0.144426 −0.0722132 0.997389i \(-0.523006\pi\)
−0.0722132 + 0.997389i \(0.523006\pi\)
\(312\) 0 0
\(313\) 19.5892 1.10725 0.553623 0.832767i \(-0.313245\pi\)
0.553623 + 0.832767i \(0.313245\pi\)
\(314\) 0 0
\(315\) −3.43134 −0.193334
\(316\) 0 0
\(317\) −23.2735 −1.30717 −0.653585 0.756853i \(-0.726736\pi\)
−0.653585 + 0.756853i \(0.726736\pi\)
\(318\) 0 0
\(319\) −10.7265 −0.600569
\(320\) 0 0
\(321\) 5.04757 0.281728
\(322\) 0 0
\(323\) −6.23130 −0.346719
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −6.79459 −0.375742
\(328\) 0 0
\(329\) 18.0951 0.997617
\(330\) 0 0
\(331\) 20.4789 1.12562 0.562811 0.826586i \(-0.309720\pi\)
0.562811 + 0.826586i \(0.309720\pi\)
\(332\) 0 0
\(333\) −0.636747 −0.0348935
\(334\) 0 0
\(335\) 6.41082 0.350260
\(336\) 0 0
\(337\) −25.6843 −1.39911 −0.699557 0.714577i \(-0.746619\pi\)
−0.699557 + 0.714577i \(0.746619\pi\)
\(338\) 0 0
\(339\) 8.06808 0.438198
\(340\) 0 0
\(341\) −24.4519 −1.32414
\(342\) 0 0
\(343\) 7.63791 0.412408
\(344\) 0 0
\(345\) −3.43134 −0.184737
\(346\) 0 0
\(347\) −9.81511 −0.526903 −0.263451 0.964673i \(-0.584861\pi\)
−0.263451 + 0.964673i \(0.584861\pi\)
\(348\) 0 0
\(349\) 12.5199 0.670177 0.335088 0.942187i \(-0.391234\pi\)
0.335088 + 0.942187i \(0.391234\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 10.6367 0.564540
\(356\) 0 0
\(357\) 26.9091 1.42418
\(358\) 0 0
\(359\) 28.1362 1.48497 0.742485 0.669863i \(-0.233647\pi\)
0.742485 + 0.669863i \(0.233647\pi\)
\(360\) 0 0
\(361\) −18.3686 −0.966770
\(362\) 0 0
\(363\) 4.04757 0.212442
\(364\) 0 0
\(365\) 16.0681 0.841042
\(366\) 0 0
\(367\) −2.41082 −0.125844 −0.0629219 0.998018i \(-0.520042\pi\)
−0.0629219 + 0.998018i \(0.520042\pi\)
\(368\) 0 0
\(369\) −4.63675 −0.241379
\(370\) 0 0
\(371\) −39.4584 −2.04858
\(372\) 0 0
\(373\) −5.54815 −0.287272 −0.143636 0.989631i \(-0.545879\pi\)
−0.143636 + 0.989631i \(0.545879\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 4.06808 0.209517
\(378\) 0 0
\(379\) 14.5199 0.745839 0.372920 0.927864i \(-0.378357\pi\)
0.372920 + 0.927864i \(0.378357\pi\)
\(380\) 0 0
\(381\) −15.6843 −0.803532
\(382\) 0 0
\(383\) 22.8627 1.16823 0.584114 0.811672i \(-0.301442\pi\)
0.584114 + 0.811672i \(0.301442\pi\)
\(384\) 0 0
\(385\) −9.04757 −0.461107
\(386\) 0 0
\(387\) −2.41082 −0.122549
\(388\) 0 0
\(389\) 5.34158 0.270829 0.135414 0.990789i \(-0.456763\pi\)
0.135414 + 0.990789i \(0.456763\pi\)
\(390\) 0 0
\(391\) 26.9091 1.36085
\(392\) 0 0
\(393\) 12.9308 0.652270
\(394\) 0 0
\(395\) −10.6367 −0.535193
\(396\) 0 0
\(397\) −21.4584 −1.07697 −0.538483 0.842637i \(-0.681002\pi\)
−0.538483 + 0.842637i \(0.681002\pi\)
\(398\) 0 0
\(399\) −2.72651 −0.136496
\(400\) 0 0
\(401\) −17.3145 −0.864646 −0.432323 0.901719i \(-0.642306\pi\)
−0.432323 + 0.901719i \(0.642306\pi\)
\(402\) 0 0
\(403\) 9.27349 0.461946
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −1.67894 −0.0832220
\(408\) 0 0
\(409\) 16.4108 0.811463 0.405731 0.913992i \(-0.367017\pi\)
0.405731 + 0.913992i \(0.367017\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) −41.6433 −2.04913
\(414\) 0 0
\(415\) 16.1362 0.792093
\(416\) 0 0
\(417\) 19.9102 0.975009
\(418\) 0 0
\(419\) −25.7524 −1.25809 −0.629043 0.777370i \(-0.716553\pi\)
−0.629043 + 0.777370i \(0.716553\pi\)
\(420\) 0 0
\(421\) 13.4777 0.656865 0.328433 0.944527i \(-0.393480\pi\)
0.328433 + 0.944527i \(0.393480\pi\)
\(422\) 0 0
\(423\) −5.27349 −0.256406
\(424\) 0 0
\(425\) 7.84216 0.380400
\(426\) 0 0
\(427\) 39.4584 1.90953
\(428\) 0 0
\(429\) 2.63675 0.127303
\(430\) 0 0
\(431\) 24.2723 1.16916 0.584579 0.811337i \(-0.301260\pi\)
0.584579 + 0.811337i \(0.301260\pi\)
\(432\) 0 0
\(433\) 3.13733 0.150770 0.0753851 0.997154i \(-0.475981\pi\)
0.0753851 + 0.997154i \(0.475981\pi\)
\(434\) 0 0
\(435\) 4.06808 0.195050
\(436\) 0 0
\(437\) −2.72651 −0.130426
\(438\) 0 0
\(439\) −8.59571 −0.410251 −0.205125 0.978736i \(-0.565760\pi\)
−0.205125 + 0.978736i \(0.565760\pi\)
\(440\) 0 0
\(441\) 4.77407 0.227337
\(442\) 0 0
\(443\) −0.911399 −0.0433019 −0.0216509 0.999766i \(-0.506892\pi\)
−0.0216509 + 0.999766i \(0.506892\pi\)
\(444\) 0 0
\(445\) 8.63675 0.409421
\(446\) 0 0
\(447\) −8.63675 −0.408504
\(448\) 0 0
\(449\) 38.8139 1.83174 0.915872 0.401471i \(-0.131501\pi\)
0.915872 + 0.401471i \(0.131501\pi\)
\(450\) 0 0
\(451\) −12.2259 −0.575696
\(452\) 0 0
\(453\) −17.7253 −0.832809
\(454\) 0 0
\(455\) 3.43134 0.160864
\(456\) 0 0
\(457\) −6.25298 −0.292502 −0.146251 0.989248i \(-0.546721\pi\)
−0.146251 + 0.989248i \(0.546721\pi\)
\(458\) 0 0
\(459\) −7.84216 −0.366041
\(460\) 0 0
\(461\) −27.9513 −1.30182 −0.650910 0.759155i \(-0.725613\pi\)
−0.650910 + 0.759155i \(0.725613\pi\)
\(462\) 0 0
\(463\) 11.8832 0.552259 0.276129 0.961120i \(-0.410948\pi\)
0.276129 + 0.961120i \(0.410948\pi\)
\(464\) 0 0
\(465\) 9.27349 0.430048
\(466\) 0 0
\(467\) 28.8680 1.33585 0.667927 0.744227i \(-0.267182\pi\)
0.667927 + 0.744227i \(0.267182\pi\)
\(468\) 0 0
\(469\) −21.9977 −1.01576
\(470\) 0 0
\(471\) −20.5470 −0.946755
\(472\) 0 0
\(473\) −6.35672 −0.292282
\(474\) 0 0
\(475\) −0.794590 −0.0364583
\(476\) 0 0
\(477\) 11.4994 0.526522
\(478\) 0 0
\(479\) 12.0464 0.550414 0.275207 0.961385i \(-0.411254\pi\)
0.275207 + 0.961385i \(0.411254\pi\)
\(480\) 0 0
\(481\) 0.636747 0.0290332
\(482\) 0 0
\(483\) 11.7741 0.535739
\(484\) 0 0
\(485\) 12.2940 0.558242
\(486\) 0 0
\(487\) −11.2518 −0.509869 −0.254934 0.966958i \(-0.582054\pi\)
−0.254934 + 0.966958i \(0.582054\pi\)
\(488\) 0 0
\(489\) −4.22593 −0.191103
\(490\) 0 0
\(491\) −31.0259 −1.40018 −0.700089 0.714055i \(-0.746857\pi\)
−0.700089 + 0.714055i \(0.746857\pi\)
\(492\) 0 0
\(493\) −31.9025 −1.43682
\(494\) 0 0
\(495\) 2.63675 0.118513
\(496\) 0 0
\(497\) −36.4983 −1.63717
\(498\) 0 0
\(499\) 5.61623 0.251417 0.125708 0.992067i \(-0.459880\pi\)
0.125708 + 0.992067i \(0.459880\pi\)
\(500\) 0 0
\(501\) −14.4108 −0.643828
\(502\) 0 0
\(503\) 3.52110 0.156998 0.0784990 0.996914i \(-0.474987\pi\)
0.0784990 + 0.996914i \(0.474987\pi\)
\(504\) 0 0
\(505\) 5.34158 0.237697
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −13.7741 −0.610525 −0.305263 0.952268i \(-0.598744\pi\)
−0.305263 + 0.952268i \(0.598744\pi\)
\(510\) 0 0
\(511\) −55.1350 −2.43903
\(512\) 0 0
\(513\) 0.794590 0.0350820
\(514\) 0 0
\(515\) 10.4108 0.458756
\(516\) 0 0
\(517\) −13.9049 −0.611535
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 25.3145 1.10905 0.554525 0.832167i \(-0.312900\pi\)
0.554525 + 0.832167i \(0.312900\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 3.43134 0.149756
\(526\) 0 0
\(527\) −72.7242 −3.16792
\(528\) 0 0
\(529\) −11.2259 −0.488084
\(530\) 0 0
\(531\) 12.1362 0.526665
\(532\) 0 0
\(533\) 4.63675 0.200840
\(534\) 0 0
\(535\) −5.04757 −0.218225
\(536\) 0 0
\(537\) −20.9308 −0.903229
\(538\) 0 0
\(539\) 12.5880 0.542204
\(540\) 0 0
\(541\) −26.6151 −1.14427 −0.572136 0.820159i \(-0.693885\pi\)
−0.572136 + 0.820159i \(0.693885\pi\)
\(542\) 0 0
\(543\) 5.08860 0.218373
\(544\) 0 0
\(545\) 6.79459 0.291048
\(546\) 0 0
\(547\) 17.7253 0.757881 0.378941 0.925421i \(-0.376288\pi\)
0.378941 + 0.925421i \(0.376288\pi\)
\(548\) 0 0
\(549\) −11.4994 −0.490783
\(550\) 0 0
\(551\) 3.23246 0.137707
\(552\) 0 0
\(553\) 36.4983 1.55206
\(554\) 0 0
\(555\) 0.636747 0.0270284
\(556\) 0 0
\(557\) 30.2723 1.28268 0.641340 0.767257i \(-0.278379\pi\)
0.641340 + 0.767257i \(0.278379\pi\)
\(558\) 0 0
\(559\) 2.41082 0.101967
\(560\) 0 0
\(561\) −20.6778 −0.873017
\(562\) 0 0
\(563\) −22.6367 −0.954025 −0.477013 0.878896i \(-0.658280\pi\)
−0.477013 + 0.878896i \(0.658280\pi\)
\(564\) 0 0
\(565\) −8.06808 −0.339427
\(566\) 0 0
\(567\) −3.43134 −0.144103
\(568\) 0 0
\(569\) 20.0410 0.840164 0.420082 0.907486i \(-0.362001\pi\)
0.420082 + 0.907486i \(0.362001\pi\)
\(570\) 0 0
\(571\) −20.3621 −0.852127 −0.426064 0.904693i \(-0.640100\pi\)
−0.426064 + 0.904693i \(0.640100\pi\)
\(572\) 0 0
\(573\) −13.7253 −0.573385
\(574\) 0 0
\(575\) 3.43134 0.143097
\(576\) 0 0
\(577\) 2.70483 0.112604 0.0563018 0.998414i \(-0.482069\pi\)
0.0563018 + 0.998414i \(0.482069\pi\)
\(578\) 0 0
\(579\) 5.11565 0.212599
\(580\) 0 0
\(581\) −55.3686 −2.29708
\(582\) 0 0
\(583\) 30.3211 1.25577
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −33.2735 −1.37334 −0.686672 0.726967i \(-0.740929\pi\)
−0.686672 + 0.726967i \(0.740929\pi\)
\(588\) 0 0
\(589\) 7.36863 0.303619
\(590\) 0 0
\(591\) 20.9988 0.863777
\(592\) 0 0
\(593\) −28.2313 −1.15932 −0.579660 0.814858i \(-0.696815\pi\)
−0.579660 + 0.814858i \(0.696815\pi\)
\(594\) 0 0
\(595\) −26.9091 −1.10316
\(596\) 0 0
\(597\) 9.58918 0.392459
\(598\) 0 0
\(599\) −29.0940 −1.18875 −0.594374 0.804189i \(-0.702600\pi\)
−0.594374 + 0.804189i \(0.702600\pi\)
\(600\) 0 0
\(601\) 46.0464 1.87827 0.939136 0.343545i \(-0.111628\pi\)
0.939136 + 0.343545i \(0.111628\pi\)
\(602\) 0 0
\(603\) 6.41082 0.261069
\(604\) 0 0
\(605\) −4.04757 −0.164557
\(606\) 0 0
\(607\) −24.1362 −0.979657 −0.489828 0.871819i \(-0.662941\pi\)
−0.489828 + 0.871819i \(0.662941\pi\)
\(608\) 0 0
\(609\) −13.9590 −0.565646
\(610\) 0 0
\(611\) 5.27349 0.213343
\(612\) 0 0
\(613\) 3.73304 0.150776 0.0753880 0.997154i \(-0.475980\pi\)
0.0753880 + 0.997154i \(0.475980\pi\)
\(614\) 0 0
\(615\) 4.63675 0.186972
\(616\) 0 0
\(617\) −13.3686 −0.538201 −0.269100 0.963112i \(-0.586726\pi\)
−0.269100 + 0.963112i \(0.586726\pi\)
\(618\) 0 0
\(619\) 21.8886 0.879776 0.439888 0.898053i \(-0.355018\pi\)
0.439888 + 0.898053i \(0.355018\pi\)
\(620\) 0 0
\(621\) −3.43134 −0.137695
\(622\) 0 0
\(623\) −29.6356 −1.18732
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.09513 0.0836716
\(628\) 0 0
\(629\) −4.99347 −0.199103
\(630\) 0 0
\(631\) −21.5892 −0.859452 −0.429726 0.902959i \(-0.641390\pi\)
−0.429726 + 0.902959i \(0.641390\pi\)
\(632\) 0 0
\(633\) −8.82164 −0.350629
\(634\) 0 0
\(635\) 15.6843 0.622413
\(636\) 0 0
\(637\) −4.77407 −0.189156
\(638\) 0 0
\(639\) 10.6367 0.420783
\(640\) 0 0
\(641\) −5.54815 −0.219139 −0.109569 0.993979i \(-0.534947\pi\)
−0.109569 + 0.993979i \(0.534947\pi\)
\(642\) 0 0
\(643\) 34.6367 1.36594 0.682970 0.730446i \(-0.260688\pi\)
0.682970 + 0.730446i \(0.260688\pi\)
\(644\) 0 0
\(645\) 2.41082 0.0949259
\(646\) 0 0
\(647\) −29.9783 −1.17857 −0.589285 0.807925i \(-0.700590\pi\)
−0.589285 + 0.807925i \(0.700590\pi\)
\(648\) 0 0
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) −31.8205 −1.24714
\(652\) 0 0
\(653\) 32.6832 1.27899 0.639495 0.768795i \(-0.279143\pi\)
0.639495 + 0.768795i \(0.279143\pi\)
\(654\) 0 0
\(655\) −12.9308 −0.505247
\(656\) 0 0
\(657\) 16.0681 0.626876
\(658\) 0 0
\(659\) −23.9296 −0.932165 −0.466082 0.884741i \(-0.654335\pi\)
−0.466082 + 0.884741i \(0.654335\pi\)
\(660\) 0 0
\(661\) −34.1632 −1.32880 −0.664398 0.747379i \(-0.731312\pi\)
−0.664398 + 0.747379i \(0.731312\pi\)
\(662\) 0 0
\(663\) 7.84216 0.304564
\(664\) 0 0
\(665\) 2.72651 0.105729
\(666\) 0 0
\(667\) −13.9590 −0.540493
\(668\) 0 0
\(669\) −9.93192 −0.383990
\(670\) 0 0
\(671\) −30.3211 −1.17053
\(672\) 0 0
\(673\) −6.90371 −0.266118 −0.133059 0.991108i \(-0.542480\pi\)
−0.133059 + 0.991108i \(0.542480\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 49.2789 1.89394 0.946970 0.321321i \(-0.104127\pi\)
0.946970 + 0.321321i \(0.104127\pi\)
\(678\) 0 0
\(679\) −42.1849 −1.61891
\(680\) 0 0
\(681\) 9.27349 0.355361
\(682\) 0 0
\(683\) −3.04219 −0.116406 −0.0582031 0.998305i \(-0.518537\pi\)
−0.0582031 + 0.998305i \(0.518537\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) −12.0681 −0.460426
\(688\) 0 0
\(689\) −11.4994 −0.438093
\(690\) 0 0
\(691\) −13.8886 −0.528346 −0.264173 0.964475i \(-0.585099\pi\)
−0.264173 + 0.964475i \(0.585099\pi\)
\(692\) 0 0
\(693\) −9.04757 −0.343689
\(694\) 0 0
\(695\) −19.9102 −0.755238
\(696\) 0 0
\(697\) −36.3621 −1.37731
\(698\) 0 0
\(699\) −13.1157 −0.496079
\(700\) 0 0
\(701\) −5.20541 −0.196606 −0.0983028 0.995157i \(-0.531341\pi\)
−0.0983028 + 0.995157i \(0.531341\pi\)
\(702\) 0 0
\(703\) 0.505953 0.0190824
\(704\) 0 0
\(705\) 5.27349 0.198611
\(706\) 0 0
\(707\) −18.3288 −0.689324
\(708\) 0 0
\(709\) −19.7524 −0.741817 −0.370908 0.928669i \(-0.620954\pi\)
−0.370908 + 0.928669i \(0.620954\pi\)
\(710\) 0 0
\(711\) −10.6367 −0.398909
\(712\) 0 0
\(713\) −31.8205 −1.19169
\(714\) 0 0
\(715\) −2.63675 −0.0986087
\(716\) 0 0
\(717\) −22.7729 −0.850470
\(718\) 0 0
\(719\) 4.13617 0.154253 0.0771265 0.997021i \(-0.475425\pi\)
0.0771265 + 0.997021i \(0.475425\pi\)
\(720\) 0 0
\(721\) −35.7230 −1.33040
\(722\) 0 0
\(723\) −6.82164 −0.253700
\(724\) 0 0
\(725\) −4.06808 −0.151085
\(726\) 0 0
\(727\) −6.09513 −0.226056 −0.113028 0.993592i \(-0.536055\pi\)
−0.113028 + 0.993592i \(0.536055\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −18.9060 −0.699265
\(732\) 0 0
\(733\) −40.0054 −1.47763 −0.738816 0.673907i \(-0.764615\pi\)
−0.738816 + 0.673907i \(0.764615\pi\)
\(734\) 0 0
\(735\) −4.77407 −0.176094
\(736\) 0 0
\(737\) 16.9037 0.622656
\(738\) 0 0
\(739\) −12.0270 −0.442422 −0.221211 0.975226i \(-0.571001\pi\)
−0.221211 + 0.975226i \(0.571001\pi\)
\(740\) 0 0
\(741\) −0.794590 −0.0291900
\(742\) 0 0
\(743\) −10.0410 −0.368370 −0.184185 0.982892i \(-0.558965\pi\)
−0.184185 + 0.982892i \(0.558965\pi\)
\(744\) 0 0
\(745\) 8.63675 0.316426
\(746\) 0 0
\(747\) 16.1362 0.590391
\(748\) 0 0
\(749\) 17.3199 0.632855
\(750\) 0 0
\(751\) −12.6778 −0.462619 −0.231309 0.972880i \(-0.574301\pi\)
−0.231309 + 0.972880i \(0.574301\pi\)
\(752\) 0 0
\(753\) 9.24644 0.336959
\(754\) 0 0
\(755\) 17.7253 0.645091
\(756\) 0 0
\(757\) 13.6302 0.495399 0.247699 0.968837i \(-0.420326\pi\)
0.247699 + 0.968837i \(0.420326\pi\)
\(758\) 0 0
\(759\) −9.04757 −0.328406
\(760\) 0 0
\(761\) −11.8615 −0.429980 −0.214990 0.976616i \(-0.568972\pi\)
−0.214990 + 0.976616i \(0.568972\pi\)
\(762\) 0 0
\(763\) −23.3145 −0.844043
\(764\) 0 0
\(765\) 7.84216 0.283534
\(766\) 0 0
\(767\) −12.1362 −0.438212
\(768\) 0 0
\(769\) 20.9988 0.757238 0.378619 0.925553i \(-0.376399\pi\)
0.378619 + 0.925553i \(0.376399\pi\)
\(770\) 0 0
\(771\) 11.2464 0.405031
\(772\) 0 0
\(773\) 14.4519 0.519797 0.259899 0.965636i \(-0.416311\pi\)
0.259899 + 0.965636i \(0.416311\pi\)
\(774\) 0 0
\(775\) −9.27349 −0.333114
\(776\) 0 0
\(777\) −2.18489 −0.0783826
\(778\) 0 0
\(779\) 3.68431 0.132004
\(780\) 0 0
\(781\) 28.0464 1.00358
\(782\) 0 0
\(783\) 4.06808 0.145381
\(784\) 0 0
\(785\) 20.5470 0.733353
\(786\) 0 0
\(787\) 10.5470 0.375959 0.187980 0.982173i \(-0.439806\pi\)
0.187980 + 0.982173i \(0.439806\pi\)
\(788\) 0 0
\(789\) −22.0681 −0.785645
\(790\) 0 0
\(791\) 27.6843 0.984341
\(792\) 0 0
\(793\) 11.4994 0.408356
\(794\) 0 0
\(795\) −11.4994 −0.407842
\(796\) 0 0
\(797\) 40.7729 1.44425 0.722125 0.691762i \(-0.243165\pi\)
0.722125 + 0.691762i \(0.243165\pi\)
\(798\) 0 0
\(799\) −41.3556 −1.46305
\(800\) 0 0
\(801\) 8.63675 0.305164
\(802\) 0 0
\(803\) 42.3675 1.49512
\(804\) 0 0
\(805\) −11.7741 −0.414982
\(806\) 0 0
\(807\) 13.2054 0.464852
\(808\) 0 0
\(809\) 20.2747 0.712819 0.356409 0.934330i \(-0.384001\pi\)
0.356409 + 0.934330i \(0.384001\pi\)
\(810\) 0 0
\(811\) 15.0259 0.527630 0.263815 0.964573i \(-0.415019\pi\)
0.263815 + 0.964573i \(0.415019\pi\)
\(812\) 0 0
\(813\) 14.0951 0.494338
\(814\) 0 0
\(815\) 4.22593 0.148028
\(816\) 0 0
\(817\) 1.91561 0.0670188
\(818\) 0 0
\(819\) 3.43134 0.119901
\(820\) 0 0
\(821\) −2.36209 −0.0824377 −0.0412188 0.999150i \(-0.513124\pi\)
−0.0412188 + 0.999150i \(0.513124\pi\)
\(822\) 0 0
\(823\) 16.1362 0.562471 0.281236 0.959639i \(-0.409256\pi\)
0.281236 + 0.959639i \(0.409256\pi\)
\(824\) 0 0
\(825\) −2.63675 −0.0917997
\(826\) 0 0
\(827\) −24.5880 −0.855009 −0.427505 0.904013i \(-0.640607\pi\)
−0.427505 + 0.904013i \(0.640607\pi\)
\(828\) 0 0
\(829\) −22.8216 −0.792628 −0.396314 0.918115i \(-0.629711\pi\)
−0.396314 + 0.918115i \(0.629711\pi\)
\(830\) 0 0
\(831\) 30.2723 1.05014
\(832\) 0 0
\(833\) 37.4390 1.29719
\(834\) 0 0
\(835\) 14.4108 0.498707
\(836\) 0 0
\(837\) 9.27349 0.320539
\(838\) 0 0
\(839\) −46.3211 −1.59918 −0.799590 0.600546i \(-0.794950\pi\)
−0.799590 + 0.600546i \(0.794950\pi\)
\(840\) 0 0
\(841\) −12.4507 −0.429334
\(842\) 0 0
\(843\) −2.13617 −0.0735735
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 13.8886 0.477217
\(848\) 0 0
\(849\) 0.136167 0.00467325
\(850\) 0 0
\(851\) −2.18489 −0.0748972
\(852\) 0 0
\(853\) 47.1837 1.61554 0.807770 0.589498i \(-0.200674\pi\)
0.807770 + 0.589498i \(0.200674\pi\)
\(854\) 0 0
\(855\) −0.794590 −0.0271744
\(856\) 0 0
\(857\) −11.8422 −0.404520 −0.202260 0.979332i \(-0.564829\pi\)
−0.202260 + 0.979332i \(0.564829\pi\)
\(858\) 0 0
\(859\) −10.3211 −0.352150 −0.176075 0.984377i \(-0.556340\pi\)
−0.176075 + 0.984377i \(0.556340\pi\)
\(860\) 0 0
\(861\) −15.9102 −0.542220
\(862\) 0 0
\(863\) −1.13733 −0.0387150 −0.0193575 0.999813i \(-0.506162\pi\)
−0.0193575 + 0.999813i \(0.506162\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −44.4994 −1.51128
\(868\) 0 0
\(869\) −28.0464 −0.951409
\(870\) 0 0
\(871\) −6.41082 −0.217222
\(872\) 0 0
\(873\) 12.2940 0.416089
\(874\) 0 0
\(875\) −3.43134 −0.116000
\(876\) 0 0
\(877\) −54.5447 −1.84184 −0.920921 0.389749i \(-0.872562\pi\)
−0.920921 + 0.389749i \(0.872562\pi\)
\(878\) 0 0
\(879\) 15.2735 0.515162
\(880\) 0 0
\(881\) 48.9555 1.64935 0.824676 0.565605i \(-0.191357\pi\)
0.824676 + 0.565605i \(0.191357\pi\)
\(882\) 0 0
\(883\) −27.3686 −0.921028 −0.460514 0.887653i \(-0.652335\pi\)
−0.460514 + 0.887653i \(0.652335\pi\)
\(884\) 0 0
\(885\) −12.1362 −0.407953
\(886\) 0 0
\(887\) 21.5265 0.722788 0.361394 0.932413i \(-0.382301\pi\)
0.361394 + 0.932413i \(0.382301\pi\)
\(888\) 0 0
\(889\) −53.8182 −1.80500
\(890\) 0 0
\(891\) 2.63675 0.0883343
\(892\) 0 0
\(893\) 4.19027 0.140222
\(894\) 0 0
\(895\) 20.9308 0.699638
\(896\) 0 0
\(897\) 3.43134 0.114569
\(898\) 0 0
\(899\) 37.7253 1.25821
\(900\) 0 0
\(901\) 90.1803 3.00434
\(902\) 0 0
\(903\) −8.27233 −0.275286
\(904\) 0 0
\(905\) −5.08860 −0.169151
\(906\) 0 0
\(907\) 39.6843 1.31770 0.658848 0.752276i \(-0.271044\pi\)
0.658848 + 0.752276i \(0.271044\pi\)
\(908\) 0 0
\(909\) 5.34158 0.177169
\(910\) 0 0
\(911\) 4.13617 0.137037 0.0685187 0.997650i \(-0.478173\pi\)
0.0685187 + 0.997650i \(0.478173\pi\)
\(912\) 0 0
\(913\) 42.5470 1.40810
\(914\) 0 0
\(915\) 11.4994 0.380159
\(916\) 0 0
\(917\) 44.3698 1.46522
\(918\) 0 0
\(919\) −17.6789 −0.583174 −0.291587 0.956544i \(-0.594183\pi\)
−0.291587 + 0.956544i \(0.594183\pi\)
\(920\) 0 0
\(921\) 7.91024 0.260651
\(922\) 0 0
\(923\) −10.6367 −0.350113
\(924\) 0 0
\(925\) −0.636747 −0.0209361
\(926\) 0 0
\(927\) 10.4108 0.341936
\(928\) 0 0
\(929\) 27.8151 0.912584 0.456292 0.889830i \(-0.349177\pi\)
0.456292 + 0.889830i \(0.349177\pi\)
\(930\) 0 0
\(931\) −3.79343 −0.124325
\(932\) 0 0
\(933\) 2.54699 0.0833846
\(934\) 0 0
\(935\) 20.6778 0.676236
\(936\) 0 0
\(937\) −54.5447 −1.78190 −0.890948 0.454105i \(-0.849959\pi\)
−0.890948 + 0.454105i \(0.849959\pi\)
\(938\) 0 0
\(939\) −19.5892 −0.639269
\(940\) 0 0
\(941\) −12.1849 −0.397216 −0.198608 0.980079i \(-0.563642\pi\)
−0.198608 + 0.980079i \(0.563642\pi\)
\(942\) 0 0
\(943\) −15.9102 −0.518109
\(944\) 0 0
\(945\) 3.43134 0.111621
\(946\) 0 0
\(947\) −39.5048 −1.28373 −0.641867 0.766816i \(-0.721840\pi\)
−0.641867 + 0.766816i \(0.721840\pi\)
\(948\) 0 0
\(949\) −16.0681 −0.521592
\(950\) 0 0
\(951\) 23.2735 0.754695
\(952\) 0 0
\(953\) 19.0205 0.616135 0.308067 0.951365i \(-0.400318\pi\)
0.308067 + 0.951365i \(0.400318\pi\)
\(954\) 0 0
\(955\) 13.7253 0.444142
\(956\) 0 0
\(957\) 10.7265 0.346739
\(958\) 0 0
\(959\) −6.86267 −0.221607
\(960\) 0 0
\(961\) 54.9977 1.77412
\(962\) 0 0
\(963\) −5.04757 −0.162656
\(964\) 0 0
\(965\) −5.11565 −0.164679
\(966\) 0 0
\(967\) 38.3404 1.23294 0.616472 0.787377i \(-0.288561\pi\)
0.616472 + 0.787377i \(0.288561\pi\)
\(968\) 0 0
\(969\) 6.23130 0.200178
\(970\) 0 0
\(971\) −15.7114 −0.504202 −0.252101 0.967701i \(-0.581121\pi\)
−0.252101 + 0.967701i \(0.581121\pi\)
\(972\) 0 0
\(973\) 68.3187 2.19020
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) 53.9566 1.72623 0.863113 0.505011i \(-0.168511\pi\)
0.863113 + 0.505011i \(0.168511\pi\)
\(978\) 0 0
\(979\) 22.7729 0.727825
\(980\) 0 0
\(981\) 6.79459 0.216935
\(982\) 0 0
\(983\) 45.0399 1.43655 0.718274 0.695760i \(-0.244932\pi\)
0.718274 + 0.695760i \(0.244932\pi\)
\(984\) 0 0
\(985\) −20.9988 −0.669079
\(986\) 0 0
\(987\) −18.0951 −0.575974
\(988\) 0 0
\(989\) −8.27233 −0.263045
\(990\) 0 0
\(991\) −7.63791 −0.242626 −0.121313 0.992614i \(-0.538710\pi\)
−0.121313 + 0.992614i \(0.538710\pi\)
\(992\) 0 0
\(993\) −20.4789 −0.649878
\(994\) 0 0
\(995\) −9.58918 −0.303997
\(996\) 0 0
\(997\) −16.0951 −0.509738 −0.254869 0.966976i \(-0.582032\pi\)
−0.254869 + 0.966976i \(0.582032\pi\)
\(998\) 0 0
\(999\) 0.636747 0.0201458
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 3120.2.a.bi.1.1 3
3.2 odd 2 9360.2.a.cy.1.1 3
4.3 odd 2 1560.2.a.q.1.3 3
12.11 even 2 4680.2.a.bh.1.3 3
20.19 odd 2 7800.2.a.bi.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.a.q.1.3 3 4.3 odd 2
3120.2.a.bi.1.1 3 1.1 even 1 trivial
4680.2.a.bh.1.3 3 12.11 even 2
7800.2.a.bi.1.1 3 20.19 odd 2
9360.2.a.cy.1.1 3 3.2 odd 2