Properties

Label 5520.2.a.bv.1.3
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.73549\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +3.73549 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +3.73549 q^{7} +1.00000 q^{9} -4.84469 q^{11} -2.84469 q^{13} +1.00000 q^{15} +0.890804 q^{17} -6.84469 q^{19} -3.73549 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -0.890804 q^{29} -7.73549 q^{31} +4.84469 q^{33} -3.73549 q^{35} -1.95388 q^{37} +2.84469 q^{39} +12.3618 q^{41} +3.47098 q^{43} -1.00000 q^{45} +6.62629 q^{47} +6.95388 q^{49} -0.890804 q^{51} +12.3618 q^{53} +4.84469 q^{55} +6.84469 q^{57} -0.890804 q^{59} +8.62629 q^{61} +3.73549 q^{63} +2.84469 q^{65} -7.73549 q^{67} +1.00000 q^{69} +12.3618 q^{71} +16.5341 q^{73} -1.00000 q^{75} -18.0973 q^{77} -13.4710 q^{79} +1.00000 q^{81} +4.58018 q^{83} -0.890804 q^{85} +0.890804 q^{87} -15.1604 q^{89} -10.6263 q^{91} +7.73549 q^{93} +6.84469 q^{95} -17.2526 q^{97} -4.84469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 4 q^{11} + 2 q^{13} + 3 q^{15} - 10 q^{19} + 2 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 10 q^{31} + 4 q^{33} + 2 q^{35} + 2 q^{37} - 2 q^{39} + 8 q^{41} - 16 q^{43} - 3 q^{45} + 4 q^{47} + 13 q^{49} + 8 q^{53} + 4 q^{55} + 10 q^{57} + 10 q^{61} - 2 q^{63} - 2 q^{65} - 10 q^{67} + 3 q^{69} + 8 q^{71} + 18 q^{73} - 3 q^{75} - 12 q^{77} - 14 q^{79} + 3 q^{81} - 10 q^{83} + 2 q^{89} - 16 q^{91} + 10 q^{93} + 10 q^{95} - 20 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(7\) 3.73549 1.41188 0.705941 0.708270i \(-0.250524\pi\)
0.705941 + 0.708270i \(0.250524\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.84469 −1.46073 −0.730364 0.683058i \(-0.760649\pi\)
−0.730364 + 0.683058i \(0.760649\pi\)
\(12\) 0 0
\(13\) −2.84469 −0.788974 −0.394487 0.918902i \(-0.629078\pi\)
−0.394487 + 0.918902i \(0.629078\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0.890804 0.216052 0.108026 0.994148i \(-0.465547\pi\)
0.108026 + 0.994148i \(0.465547\pi\)
\(18\) 0 0
\(19\) −6.84469 −1.57028 −0.785139 0.619319i \(-0.787409\pi\)
−0.785139 + 0.619319i \(0.787409\pi\)
\(20\) 0 0
\(21\) −3.73549 −0.815151
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.890804 −0.165418 −0.0827091 0.996574i \(-0.526357\pi\)
−0.0827091 + 0.996574i \(0.526357\pi\)
\(30\) 0 0
\(31\) −7.73549 −1.38933 −0.694667 0.719331i \(-0.744448\pi\)
−0.694667 + 0.719331i \(0.744448\pi\)
\(32\) 0 0
\(33\) 4.84469 0.843352
\(34\) 0 0
\(35\) −3.73549 −0.631413
\(36\) 0 0
\(37\) −1.95388 −0.321216 −0.160608 0.987018i \(-0.551346\pi\)
−0.160608 + 0.987018i \(0.551346\pi\)
\(38\) 0 0
\(39\) 2.84469 0.455514
\(40\) 0 0
\(41\) 12.3618 1.93059 0.965293 0.261169i \(-0.0841080\pi\)
0.965293 + 0.261169i \(0.0841080\pi\)
\(42\) 0 0
\(43\) 3.47098 0.529319 0.264660 0.964342i \(-0.414740\pi\)
0.264660 + 0.964342i \(0.414740\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.62629 0.966544 0.483272 0.875470i \(-0.339448\pi\)
0.483272 + 0.875470i \(0.339448\pi\)
\(48\) 0 0
\(49\) 6.95388 0.993412
\(50\) 0 0
\(51\) −0.890804 −0.124737
\(52\) 0 0
\(53\) 12.3618 1.69802 0.849011 0.528376i \(-0.177199\pi\)
0.849011 + 0.528376i \(0.177199\pi\)
\(54\) 0 0
\(55\) 4.84469 0.653257
\(56\) 0 0
\(57\) 6.84469 0.906601
\(58\) 0 0
\(59\) −0.890804 −0.115973 −0.0579864 0.998317i \(-0.518468\pi\)
−0.0579864 + 0.998317i \(0.518468\pi\)
\(60\) 0 0
\(61\) 8.62629 1.10448 0.552242 0.833684i \(-0.313773\pi\)
0.552242 + 0.833684i \(0.313773\pi\)
\(62\) 0 0
\(63\) 3.73549 0.470627
\(64\) 0 0
\(65\) 2.84469 0.352840
\(66\) 0 0
\(67\) −7.73549 −0.945040 −0.472520 0.881320i \(-0.656656\pi\)
−0.472520 + 0.881320i \(0.656656\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 12.3618 1.46707 0.733537 0.679650i \(-0.237868\pi\)
0.733537 + 0.679650i \(0.237868\pi\)
\(72\) 0 0
\(73\) 16.5341 1.93516 0.967582 0.252555i \(-0.0812709\pi\)
0.967582 + 0.252555i \(0.0812709\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −18.0973 −2.06238
\(78\) 0 0
\(79\) −13.4710 −1.51560 −0.757802 0.652485i \(-0.773727\pi\)
−0.757802 + 0.652485i \(0.773727\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.58018 0.502740 0.251370 0.967891i \(-0.419119\pi\)
0.251370 + 0.967891i \(0.419119\pi\)
\(84\) 0 0
\(85\) −0.890804 −0.0966212
\(86\) 0 0
\(87\) 0.890804 0.0955042
\(88\) 0 0
\(89\) −15.1604 −1.60699 −0.803497 0.595309i \(-0.797030\pi\)
−0.803497 + 0.595309i \(0.797030\pi\)
\(90\) 0 0
\(91\) −10.6263 −1.11394
\(92\) 0 0
\(93\) 7.73549 0.802133
\(94\) 0 0
\(95\) 6.84469 0.702250
\(96\) 0 0
\(97\) −17.2526 −1.75173 −0.875867 0.482552i \(-0.839710\pi\)
−0.875867 + 0.482552i \(0.839710\pi\)
\(98\) 0 0
\(99\) −4.84469 −0.486909
\(100\) 0 0
\(101\) −2.67241 −0.265915 −0.132957 0.991122i \(-0.542447\pi\)
−0.132957 + 0.991122i \(0.542447\pi\)
\(102\) 0 0
\(103\) −6.21839 −0.612716 −0.306358 0.951916i \(-0.599111\pi\)
−0.306358 + 0.951916i \(0.599111\pi\)
\(104\) 0 0
\(105\) 3.73549 0.364546
\(106\) 0 0
\(107\) 14.7986 1.43063 0.715316 0.698801i \(-0.246283\pi\)
0.715316 + 0.698801i \(0.246283\pi\)
\(108\) 0 0
\(109\) −12.5341 −1.20054 −0.600272 0.799796i \(-0.704941\pi\)
−0.600272 + 0.799796i \(0.704941\pi\)
\(110\) 0 0
\(111\) 1.95388 0.185454
\(112\) 0 0
\(113\) 12.3618 1.16290 0.581449 0.813583i \(-0.302486\pi\)
0.581449 + 0.813583i \(0.302486\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −2.84469 −0.262991
\(118\) 0 0
\(119\) 3.32759 0.305040
\(120\) 0 0
\(121\) 12.4710 1.13373
\(122\) 0 0
\(123\) −12.3618 −1.11462
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −11.7866 −1.04590 −0.522948 0.852365i \(-0.675168\pi\)
−0.522948 + 0.852365i \(0.675168\pi\)
\(128\) 0 0
\(129\) −3.47098 −0.305603
\(130\) 0 0
\(131\) 17.4710 1.52645 0.763223 0.646135i \(-0.223616\pi\)
0.763223 + 0.646135i \(0.223616\pi\)
\(132\) 0 0
\(133\) −25.5683 −2.21705
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 15.1604 1.29524 0.647618 0.761965i \(-0.275765\pi\)
0.647618 + 0.761965i \(0.275765\pi\)
\(138\) 0 0
\(139\) 5.51710 0.467954 0.233977 0.972242i \(-0.424826\pi\)
0.233977 + 0.972242i \(0.424826\pi\)
\(140\) 0 0
\(141\) −6.62629 −0.558035
\(142\) 0 0
\(143\) 13.7816 1.15248
\(144\) 0 0
\(145\) 0.890804 0.0739772
\(146\) 0 0
\(147\) −6.95388 −0.573547
\(148\) 0 0
\(149\) 1.15531 0.0946470 0.0473235 0.998880i \(-0.484931\pi\)
0.0473235 + 0.998880i \(0.484931\pi\)
\(150\) 0 0
\(151\) 15.4710 1.25901 0.629505 0.776996i \(-0.283258\pi\)
0.629505 + 0.776996i \(0.283258\pi\)
\(152\) 0 0
\(153\) 0.890804 0.0720172
\(154\) 0 0
\(155\) 7.73549 0.621329
\(156\) 0 0
\(157\) 18.6774 1.49062 0.745311 0.666717i \(-0.232301\pi\)
0.745311 + 0.666717i \(0.232301\pi\)
\(158\) 0 0
\(159\) −12.3618 −0.980353
\(160\) 0 0
\(161\) −3.73549 −0.294398
\(162\) 0 0
\(163\) −0.0922364 −0.00722451 −0.00361226 0.999993i \(-0.501150\pi\)
−0.00361226 + 0.999993i \(0.501150\pi\)
\(164\) 0 0
\(165\) −4.84469 −0.377158
\(166\) 0 0
\(167\) 5.37371 0.415830 0.207915 0.978147i \(-0.433332\pi\)
0.207915 + 0.978147i \(0.433332\pi\)
\(168\) 0 0
\(169\) −4.90776 −0.377520
\(170\) 0 0
\(171\) −6.84469 −0.523426
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 3.73549 0.282376
\(176\) 0 0
\(177\) 0.890804 0.0669569
\(178\) 0 0
\(179\) 9.56322 0.714788 0.357394 0.933954i \(-0.383665\pi\)
0.357394 + 0.933954i \(0.383665\pi\)
\(180\) 0 0
\(181\) −7.16035 −0.532225 −0.266112 0.963942i \(-0.585739\pi\)
−0.266112 + 0.963942i \(0.585739\pi\)
\(182\) 0 0
\(183\) −8.62629 −0.637674
\(184\) 0 0
\(185\) 1.95388 0.143652
\(186\) 0 0
\(187\) −4.31566 −0.315593
\(188\) 0 0
\(189\) −3.73549 −0.271717
\(190\) 0 0
\(191\) −14.0050 −1.01337 −0.506684 0.862132i \(-0.669129\pi\)
−0.506684 + 0.862132i \(0.669129\pi\)
\(192\) 0 0
\(193\) 26.7236 1.92360 0.961802 0.273745i \(-0.0882626\pi\)
0.961802 + 0.273745i \(0.0882626\pi\)
\(194\) 0 0
\(195\) −2.84469 −0.203712
\(196\) 0 0
\(197\) 9.16035 0.652648 0.326324 0.945258i \(-0.394190\pi\)
0.326324 + 0.945258i \(0.394190\pi\)
\(198\) 0 0
\(199\) 0.310629 0.0220199 0.0110099 0.999939i \(-0.496495\pi\)
0.0110099 + 0.999939i \(0.496495\pi\)
\(200\) 0 0
\(201\) 7.73549 0.545619
\(202\) 0 0
\(203\) −3.32759 −0.233551
\(204\) 0 0
\(205\) −12.3618 −0.863384
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 33.1604 2.29375
\(210\) 0 0
\(211\) −5.95388 −0.409882 −0.204941 0.978774i \(-0.565700\pi\)
−0.204941 + 0.978774i \(0.565700\pi\)
\(212\) 0 0
\(213\) −12.3618 −0.847015
\(214\) 0 0
\(215\) −3.47098 −0.236719
\(216\) 0 0
\(217\) −28.8958 −1.96158
\(218\) 0 0
\(219\) −16.5341 −1.11727
\(220\) 0 0
\(221\) −2.53406 −0.170459
\(222\) 0 0
\(223\) 11.9078 0.797403 0.398701 0.917081i \(-0.369461\pi\)
0.398701 + 0.917081i \(0.369461\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −7.37874 −0.489744 −0.244872 0.969555i \(-0.578746\pi\)
−0.244872 + 0.969555i \(0.578746\pi\)
\(228\) 0 0
\(229\) 23.1604 1.53048 0.765240 0.643746i \(-0.222621\pi\)
0.765240 + 0.643746i \(0.222621\pi\)
\(230\) 0 0
\(231\) 18.0973 1.19071
\(232\) 0 0
\(233\) −22.9420 −1.50298 −0.751489 0.659746i \(-0.770664\pi\)
−0.751489 + 0.659746i \(0.770664\pi\)
\(234\) 0 0
\(235\) −6.62629 −0.432252
\(236\) 0 0
\(237\) 13.4710 0.875034
\(238\) 0 0
\(239\) −0.361783 −0.0234018 −0.0117009 0.999932i \(-0.503725\pi\)
−0.0117009 + 0.999932i \(0.503725\pi\)
\(240\) 0 0
\(241\) 28.5341 1.83804 0.919020 0.394211i \(-0.128982\pi\)
0.919020 + 0.394211i \(0.128982\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.95388 −0.444267
\(246\) 0 0
\(247\) 19.4710 1.23891
\(248\) 0 0
\(249\) −4.58018 −0.290257
\(250\) 0 0
\(251\) 30.8497 1.94722 0.973609 0.228224i \(-0.0732920\pi\)
0.973609 + 0.228224i \(0.0732920\pi\)
\(252\) 0 0
\(253\) 4.84469 0.304583
\(254\) 0 0
\(255\) 0.890804 0.0557843
\(256\) 0 0
\(257\) −12.7524 −0.795476 −0.397738 0.917499i \(-0.630205\pi\)
−0.397738 + 0.917499i \(0.630205\pi\)
\(258\) 0 0
\(259\) −7.29870 −0.453519
\(260\) 0 0
\(261\) −0.890804 −0.0551394
\(262\) 0 0
\(263\) −6.36178 −0.392284 −0.196142 0.980575i \(-0.562841\pi\)
−0.196142 + 0.980575i \(0.562841\pi\)
\(264\) 0 0
\(265\) −12.3618 −0.759378
\(266\) 0 0
\(267\) 15.1604 0.927798
\(268\) 0 0
\(269\) 10.5802 0.645085 0.322542 0.946555i \(-0.395463\pi\)
0.322542 + 0.946555i \(0.395463\pi\)
\(270\) 0 0
\(271\) 4.26451 0.259051 0.129525 0.991576i \(-0.458655\pi\)
0.129525 + 0.991576i \(0.458655\pi\)
\(272\) 0 0
\(273\) 10.6263 0.643133
\(274\) 0 0
\(275\) −4.84469 −0.292146
\(276\) 0 0
\(277\) −29.3787 −1.76520 −0.882599 0.470127i \(-0.844208\pi\)
−0.882599 + 0.470127i \(0.844208\pi\)
\(278\) 0 0
\(279\) −7.73549 −0.463112
\(280\) 0 0
\(281\) −18.7524 −1.11868 −0.559339 0.828939i \(-0.688945\pi\)
−0.559339 + 0.828939i \(0.688945\pi\)
\(282\) 0 0
\(283\) 12.8958 0.766578 0.383289 0.923628i \(-0.374791\pi\)
0.383289 + 0.923628i \(0.374791\pi\)
\(284\) 0 0
\(285\) −6.84469 −0.405444
\(286\) 0 0
\(287\) 46.1773 2.72576
\(288\) 0 0
\(289\) −16.2065 −0.953322
\(290\) 0 0
\(291\) 17.2526 1.01136
\(292\) 0 0
\(293\) 8.79857 0.514018 0.257009 0.966409i \(-0.417263\pi\)
0.257009 + 0.966409i \(0.417263\pi\)
\(294\) 0 0
\(295\) 0.890804 0.0518646
\(296\) 0 0
\(297\) 4.84469 0.281117
\(298\) 0 0
\(299\) 2.84469 0.164512
\(300\) 0 0
\(301\) 12.9658 0.747337
\(302\) 0 0
\(303\) 2.67241 0.153526
\(304\) 0 0
\(305\) −8.62629 −0.493940
\(306\) 0 0
\(307\) −10.0050 −0.571018 −0.285509 0.958376i \(-0.592163\pi\)
−0.285509 + 0.958376i \(0.592163\pi\)
\(308\) 0 0
\(309\) 6.21839 0.353752
\(310\) 0 0
\(311\) 18.5290 1.05068 0.525342 0.850891i \(-0.323937\pi\)
0.525342 + 0.850891i \(0.323937\pi\)
\(312\) 0 0
\(313\) 2.39067 0.135128 0.0675642 0.997715i \(-0.478477\pi\)
0.0675642 + 0.997715i \(0.478477\pi\)
\(314\) 0 0
\(315\) −3.73549 −0.210471
\(316\) 0 0
\(317\) −16.3157 −0.916379 −0.458190 0.888855i \(-0.651502\pi\)
−0.458190 + 0.888855i \(0.651502\pi\)
\(318\) 0 0
\(319\) 4.31566 0.241631
\(320\) 0 0
\(321\) −14.7986 −0.825975
\(322\) 0 0
\(323\) −6.09727 −0.339261
\(324\) 0 0
\(325\) −2.84469 −0.157795
\(326\) 0 0
\(327\) 12.5341 0.693135
\(328\) 0 0
\(329\) 24.7524 1.36465
\(330\) 0 0
\(331\) 24.8958 1.36840 0.684200 0.729295i \(-0.260152\pi\)
0.684200 + 0.729295i \(0.260152\pi\)
\(332\) 0 0
\(333\) −1.95388 −0.107072
\(334\) 0 0
\(335\) 7.73549 0.422635
\(336\) 0 0
\(337\) −27.4710 −1.49644 −0.748220 0.663451i \(-0.769091\pi\)
−0.748220 + 0.663451i \(0.769091\pi\)
\(338\) 0 0
\(339\) −12.3618 −0.671400
\(340\) 0 0
\(341\) 37.4760 2.02944
\(342\) 0 0
\(343\) −0.172274 −0.00930193
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 1.25259 0.0672424 0.0336212 0.999435i \(-0.489296\pi\)
0.0336212 + 0.999435i \(0.489296\pi\)
\(348\) 0 0
\(349\) 9.64325 0.516192 0.258096 0.966119i \(-0.416905\pi\)
0.258096 + 0.966119i \(0.416905\pi\)
\(350\) 0 0
\(351\) 2.84469 0.151838
\(352\) 0 0
\(353\) −12.7524 −0.678744 −0.339372 0.940652i \(-0.610215\pi\)
−0.339372 + 0.940652i \(0.610215\pi\)
\(354\) 0 0
\(355\) −12.3618 −0.656095
\(356\) 0 0
\(357\) −3.32759 −0.176115
\(358\) 0 0
\(359\) 6.62629 0.349722 0.174861 0.984593i \(-0.444052\pi\)
0.174861 + 0.984593i \(0.444052\pi\)
\(360\) 0 0
\(361\) 27.8497 1.46577
\(362\) 0 0
\(363\) −12.4710 −0.654557
\(364\) 0 0
\(365\) −16.5341 −0.865432
\(366\) 0 0
\(367\) 9.60933 0.501603 0.250802 0.968039i \(-0.419306\pi\)
0.250802 + 0.968039i \(0.419306\pi\)
\(368\) 0 0
\(369\) 12.3618 0.643529
\(370\) 0 0
\(371\) 46.1773 2.39741
\(372\) 0 0
\(373\) 29.8839 1.54733 0.773665 0.633595i \(-0.218421\pi\)
0.773665 + 0.633595i \(0.218421\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 2.53406 0.130511
\(378\) 0 0
\(379\) −2.52902 −0.129907 −0.0649535 0.997888i \(-0.520690\pi\)
−0.0649535 + 0.997888i \(0.520690\pi\)
\(380\) 0 0
\(381\) 11.7866 0.603848
\(382\) 0 0
\(383\) −25.2115 −1.28825 −0.644124 0.764921i \(-0.722778\pi\)
−0.644124 + 0.764921i \(0.722778\pi\)
\(384\) 0 0
\(385\) 18.0973 0.922322
\(386\) 0 0
\(387\) 3.47098 0.176440
\(388\) 0 0
\(389\) 35.0681 1.77802 0.889012 0.457884i \(-0.151392\pi\)
0.889012 + 0.457884i \(0.151392\pi\)
\(390\) 0 0
\(391\) −0.890804 −0.0450499
\(392\) 0 0
\(393\) −17.4710 −0.881294
\(394\) 0 0
\(395\) 13.4710 0.677799
\(396\) 0 0
\(397\) 8.84972 0.444155 0.222077 0.975029i \(-0.428716\pi\)
0.222077 + 0.975029i \(0.428716\pi\)
\(398\) 0 0
\(399\) 25.5683 1.28001
\(400\) 0 0
\(401\) −23.0681 −1.15197 −0.575983 0.817461i \(-0.695381\pi\)
−0.575983 + 0.817461i \(0.695381\pi\)
\(402\) 0 0
\(403\) 22.0050 1.09615
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 9.46594 0.469209
\(408\) 0 0
\(409\) 5.29870 0.262004 0.131002 0.991382i \(-0.458181\pi\)
0.131002 + 0.991382i \(0.458181\pi\)
\(410\) 0 0
\(411\) −15.1604 −0.747805
\(412\) 0 0
\(413\) −3.32759 −0.163740
\(414\) 0 0
\(415\) −4.58018 −0.224832
\(416\) 0 0
\(417\) −5.51710 −0.270173
\(418\) 0 0
\(419\) −2.53406 −0.123797 −0.0618984 0.998082i \(-0.519715\pi\)
−0.0618984 + 0.998082i \(0.519715\pi\)
\(420\) 0 0
\(421\) 22.4079 1.09209 0.546047 0.837754i \(-0.316132\pi\)
0.546047 + 0.837754i \(0.316132\pi\)
\(422\) 0 0
\(423\) 6.62629 0.322181
\(424\) 0 0
\(425\) 0.890804 0.0432103
\(426\) 0 0
\(427\) 32.2234 1.55940
\(428\) 0 0
\(429\) −13.7816 −0.665382
\(430\) 0 0
\(431\) −31.5733 −1.52083 −0.760416 0.649436i \(-0.775005\pi\)
−0.760416 + 0.649436i \(0.775005\pi\)
\(432\) 0 0
\(433\) 7.20647 0.346321 0.173160 0.984894i \(-0.444602\pi\)
0.173160 + 0.984894i \(0.444602\pi\)
\(434\) 0 0
\(435\) −0.890804 −0.0427108
\(436\) 0 0
\(437\) 6.84469 0.327426
\(438\) 0 0
\(439\) 4.72357 0.225443 0.112722 0.993627i \(-0.464043\pi\)
0.112722 + 0.993627i \(0.464043\pi\)
\(440\) 0 0
\(441\) 6.95388 0.331137
\(442\) 0 0
\(443\) 7.87888 0.374337 0.187168 0.982328i \(-0.440069\pi\)
0.187168 + 0.982328i \(0.440069\pi\)
\(444\) 0 0
\(445\) 15.1604 0.718670
\(446\) 0 0
\(447\) −1.15531 −0.0546445
\(448\) 0 0
\(449\) −4.70633 −0.222105 −0.111053 0.993815i \(-0.535422\pi\)
−0.111053 + 0.993815i \(0.535422\pi\)
\(450\) 0 0
\(451\) −59.8890 −2.82006
\(452\) 0 0
\(453\) −15.4710 −0.726890
\(454\) 0 0
\(455\) 10.6263 0.498168
\(456\) 0 0
\(457\) −6.04612 −0.282825 −0.141413 0.989951i \(-0.545164\pi\)
−0.141413 + 0.989951i \(0.545164\pi\)
\(458\) 0 0
\(459\) −0.890804 −0.0415792
\(460\) 0 0
\(461\) −19.2526 −0.896682 −0.448341 0.893863i \(-0.647985\pi\)
−0.448341 + 0.893863i \(0.647985\pi\)
\(462\) 0 0
\(463\) 26.4418 1.22886 0.614428 0.788973i \(-0.289387\pi\)
0.614428 + 0.788973i \(0.289387\pi\)
\(464\) 0 0
\(465\) −7.73549 −0.358725
\(466\) 0 0
\(467\) −9.20143 −0.425792 −0.212896 0.977075i \(-0.568289\pi\)
−0.212896 + 0.977075i \(0.568289\pi\)
\(468\) 0 0
\(469\) −28.8958 −1.33429
\(470\) 0 0
\(471\) −18.6774 −0.860611
\(472\) 0 0
\(473\) −16.8158 −0.773191
\(474\) 0 0
\(475\) −6.84469 −0.314056
\(476\) 0 0
\(477\) 12.3618 0.566007
\(478\) 0 0
\(479\) 12.7524 0.582674 0.291337 0.956620i \(-0.405900\pi\)
0.291337 + 0.956620i \(0.405900\pi\)
\(480\) 0 0
\(481\) 5.55818 0.253431
\(482\) 0 0
\(483\) 3.73549 0.169971
\(484\) 0 0
\(485\) 17.2526 0.783400
\(486\) 0 0
\(487\) 30.5341 1.38363 0.691815 0.722075i \(-0.256811\pi\)
0.691815 + 0.722075i \(0.256811\pi\)
\(488\) 0 0
\(489\) 0.0922364 0.00417107
\(490\) 0 0
\(491\) 2.67241 0.120604 0.0603021 0.998180i \(-0.480794\pi\)
0.0603021 + 0.998180i \(0.480794\pi\)
\(492\) 0 0
\(493\) −0.793532 −0.0357389
\(494\) 0 0
\(495\) 4.84469 0.217752
\(496\) 0 0
\(497\) 46.1773 2.07134
\(498\) 0 0
\(499\) 20.8036 0.931297 0.465649 0.884970i \(-0.345821\pi\)
0.465649 + 0.884970i \(0.345821\pi\)
\(500\) 0 0
\(501\) −5.37371 −0.240080
\(502\) 0 0
\(503\) −3.32759 −0.148370 −0.0741849 0.997245i \(-0.523636\pi\)
−0.0741849 + 0.997245i \(0.523636\pi\)
\(504\) 0 0
\(505\) 2.67241 0.118921
\(506\) 0 0
\(507\) 4.90776 0.217961
\(508\) 0 0
\(509\) 16.2184 0.718868 0.359434 0.933171i \(-0.382970\pi\)
0.359434 + 0.933171i \(0.382970\pi\)
\(510\) 0 0
\(511\) 61.7628 2.73223
\(512\) 0 0
\(513\) 6.84469 0.302200
\(514\) 0 0
\(515\) 6.21839 0.274015
\(516\) 0 0
\(517\) −32.1023 −1.41186
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.18923 0.139723 0.0698614 0.997557i \(-0.477744\pi\)
0.0698614 + 0.997557i \(0.477744\pi\)
\(522\) 0 0
\(523\) −23.8155 −1.04138 −0.520690 0.853746i \(-0.674325\pi\)
−0.520690 + 0.853746i \(0.674325\pi\)
\(524\) 0 0
\(525\) −3.73549 −0.163030
\(526\) 0 0
\(527\) −6.89080 −0.300168
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −0.890804 −0.0386576
\(532\) 0 0
\(533\) −35.1654 −1.52318
\(534\) 0 0
\(535\) −14.7986 −0.639798
\(536\) 0 0
\(537\) −9.56322 −0.412683
\(538\) 0 0
\(539\) −33.6894 −1.45110
\(540\) 0 0
\(541\) 15.7816 0.678504 0.339252 0.940695i \(-0.389826\pi\)
0.339252 + 0.940695i \(0.389826\pi\)
\(542\) 0 0
\(543\) 7.16035 0.307280
\(544\) 0 0
\(545\) 12.5341 0.536900
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 8.62629 0.368161
\(550\) 0 0
\(551\) 6.09727 0.259753
\(552\) 0 0
\(553\) −50.3207 −2.13985
\(554\) 0 0
\(555\) −1.95388 −0.0829377
\(556\) 0 0
\(557\) 31.7405 1.34489 0.672445 0.740147i \(-0.265244\pi\)
0.672445 + 0.740147i \(0.265244\pi\)
\(558\) 0 0
\(559\) −9.87384 −0.417619
\(560\) 0 0
\(561\) 4.31566 0.182208
\(562\) 0 0
\(563\) −21.9250 −0.924029 −0.462014 0.886872i \(-0.652873\pi\)
−0.462014 + 0.886872i \(0.652873\pi\)
\(564\) 0 0
\(565\) −12.3618 −0.520064
\(566\) 0 0
\(567\) 3.73549 0.156876
\(568\) 0 0
\(569\) −14.6313 −0.613377 −0.306689 0.951810i \(-0.599221\pi\)
−0.306689 + 0.951810i \(0.599221\pi\)
\(570\) 0 0
\(571\) −1.24755 −0.0522084 −0.0261042 0.999659i \(-0.508310\pi\)
−0.0261042 + 0.999659i \(0.508310\pi\)
\(572\) 0 0
\(573\) 14.0050 0.585069
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 18.0922 0.753190 0.376595 0.926378i \(-0.377095\pi\)
0.376595 + 0.926378i \(0.377095\pi\)
\(578\) 0 0
\(579\) −26.7236 −1.11059
\(580\) 0 0
\(581\) 17.1092 0.709809
\(582\) 0 0
\(583\) −59.8890 −2.48035
\(584\) 0 0
\(585\) 2.84469 0.117613
\(586\) 0 0
\(587\) −12.5290 −0.517128 −0.258564 0.965994i \(-0.583249\pi\)
−0.258564 + 0.965994i \(0.583249\pi\)
\(588\) 0 0
\(589\) 52.9470 2.18164
\(590\) 0 0
\(591\) −9.16035 −0.376806
\(592\) 0 0
\(593\) −24.9470 −1.02445 −0.512225 0.858851i \(-0.671179\pi\)
−0.512225 + 0.858851i \(0.671179\pi\)
\(594\) 0 0
\(595\) −3.32759 −0.136418
\(596\) 0 0
\(597\) −0.310629 −0.0127132
\(598\) 0 0
\(599\) 34.5391 1.41123 0.705615 0.708596i \(-0.250671\pi\)
0.705615 + 0.708596i \(0.250671\pi\)
\(600\) 0 0
\(601\) 37.9300 1.54720 0.773599 0.633675i \(-0.218454\pi\)
0.773599 + 0.633675i \(0.218454\pi\)
\(602\) 0 0
\(603\) −7.73549 −0.315013
\(604\) 0 0
\(605\) −12.4710 −0.507017
\(606\) 0 0
\(607\) −26.0973 −1.05926 −0.529628 0.848230i \(-0.677668\pi\)
−0.529628 + 0.848230i \(0.677668\pi\)
\(608\) 0 0
\(609\) 3.32759 0.134841
\(610\) 0 0
\(611\) −18.8497 −0.762578
\(612\) 0 0
\(613\) 2.84972 0.115099 0.0575496 0.998343i \(-0.481671\pi\)
0.0575496 + 0.998343i \(0.481671\pi\)
\(614\) 0 0
\(615\) 12.3618 0.498475
\(616\) 0 0
\(617\) −9.52213 −0.383347 −0.191673 0.981459i \(-0.561391\pi\)
−0.191673 + 0.981459i \(0.561391\pi\)
\(618\) 0 0
\(619\) 23.9762 0.963683 0.481841 0.876258i \(-0.339968\pi\)
0.481841 + 0.876258i \(0.339968\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −56.6313 −2.26889
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −33.1604 −1.32430
\(628\) 0 0
\(629\) −1.74053 −0.0693993
\(630\) 0 0
\(631\) 17.8789 0.711747 0.355873 0.934534i \(-0.384183\pi\)
0.355873 + 0.934534i \(0.384183\pi\)
\(632\) 0 0
\(633\) 5.95388 0.236646
\(634\) 0 0
\(635\) 11.7866 0.467739
\(636\) 0 0
\(637\) −19.7816 −0.783776
\(638\) 0 0
\(639\) 12.3618 0.489025
\(640\) 0 0
\(641\) 2.93692 0.116001 0.0580007 0.998317i \(-0.481527\pi\)
0.0580007 + 0.998317i \(0.481527\pi\)
\(642\) 0 0
\(643\) 0.895840 0.0353285 0.0176642 0.999844i \(-0.494377\pi\)
0.0176642 + 0.999844i \(0.494377\pi\)
\(644\) 0 0
\(645\) 3.47098 0.136670
\(646\) 0 0
\(647\) 21.4659 0.843913 0.421957 0.906616i \(-0.361343\pi\)
0.421957 + 0.906616i \(0.361343\pi\)
\(648\) 0 0
\(649\) 4.31566 0.169405
\(650\) 0 0
\(651\) 28.8958 1.13252
\(652\) 0 0
\(653\) −4.84469 −0.189587 −0.0947936 0.995497i \(-0.530219\pi\)
−0.0947936 + 0.995497i \(0.530219\pi\)
\(654\) 0 0
\(655\) −17.4710 −0.682648
\(656\) 0 0
\(657\) 16.5341 0.645055
\(658\) 0 0
\(659\) 18.0973 0.704970 0.352485 0.935818i \(-0.385337\pi\)
0.352485 + 0.935818i \(0.385337\pi\)
\(660\) 0 0
\(661\) 0.747413 0.0290710 0.0145355 0.999894i \(-0.495373\pi\)
0.0145355 + 0.999894i \(0.495373\pi\)
\(662\) 0 0
\(663\) 2.53406 0.0984146
\(664\) 0 0
\(665\) 25.5683 0.991494
\(666\) 0 0
\(667\) 0.890804 0.0344921
\(668\) 0 0
\(669\) −11.9078 −0.460381
\(670\) 0 0
\(671\) −41.7917 −1.61335
\(672\) 0 0
\(673\) −19.1892 −0.739691 −0.369845 0.929093i \(-0.620589\pi\)
−0.369845 + 0.929093i \(0.620589\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −4.17731 −0.160547 −0.0802735 0.996773i \(-0.525579\pi\)
−0.0802735 + 0.996773i \(0.525579\pi\)
\(678\) 0 0
\(679\) −64.4469 −2.47324
\(680\) 0 0
\(681\) 7.37874 0.282754
\(682\) 0 0
\(683\) 26.0050 0.995055 0.497528 0.867448i \(-0.334241\pi\)
0.497528 + 0.867448i \(0.334241\pi\)
\(684\) 0 0
\(685\) −15.1604 −0.579247
\(686\) 0 0
\(687\) −23.1604 −0.883622
\(688\) 0 0
\(689\) −35.1654 −1.33969
\(690\) 0 0
\(691\) −26.8497 −1.02141 −0.510706 0.859756i \(-0.670616\pi\)
−0.510706 + 0.859756i \(0.670616\pi\)
\(692\) 0 0
\(693\) −18.0973 −0.687459
\(694\) 0 0
\(695\) −5.51710 −0.209275
\(696\) 0 0
\(697\) 11.0119 0.417106
\(698\) 0 0
\(699\) 22.9420 0.867745
\(700\) 0 0
\(701\) 11.9027 0.449560 0.224780 0.974410i \(-0.427834\pi\)
0.224780 + 0.974410i \(0.427834\pi\)
\(702\) 0 0
\(703\) 13.3737 0.504399
\(704\) 0 0
\(705\) 6.62629 0.249561
\(706\) 0 0
\(707\) −9.98277 −0.375441
\(708\) 0 0
\(709\) −39.3737 −1.47871 −0.739355 0.673315i \(-0.764870\pi\)
−0.739355 + 0.673315i \(0.764870\pi\)
\(710\) 0 0
\(711\) −13.4710 −0.505201
\(712\) 0 0
\(713\) 7.73549 0.289696
\(714\) 0 0
\(715\) −13.7816 −0.515403
\(716\) 0 0
\(717\) 0.361783 0.0135110
\(718\) 0 0
\(719\) −23.6382 −0.881557 −0.440778 0.897616i \(-0.645298\pi\)
−0.440778 + 0.897616i \(0.645298\pi\)
\(720\) 0 0
\(721\) −23.2287 −0.865083
\(722\) 0 0
\(723\) −28.5341 −1.06119
\(724\) 0 0
\(725\) −0.890804 −0.0330836
\(726\) 0 0
\(727\) −12.5513 −0.465502 −0.232751 0.972536i \(-0.574773\pi\)
−0.232751 + 0.972536i \(0.574773\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.09196 0.114360
\(732\) 0 0
\(733\) 39.1142 1.44472 0.722359 0.691519i \(-0.243058\pi\)
0.722359 + 0.691519i \(0.243058\pi\)
\(734\) 0 0
\(735\) 6.95388 0.256498
\(736\) 0 0
\(737\) 37.4760 1.38045
\(738\) 0 0
\(739\) −35.7456 −1.31492 −0.657461 0.753489i \(-0.728370\pi\)
−0.657461 + 0.753489i \(0.728370\pi\)
\(740\) 0 0
\(741\) −19.4710 −0.715284
\(742\) 0 0
\(743\) −20.4368 −0.749753 −0.374876 0.927075i \(-0.622315\pi\)
−0.374876 + 0.927075i \(0.622315\pi\)
\(744\) 0 0
\(745\) −1.15531 −0.0423274
\(746\) 0 0
\(747\) 4.58018 0.167580
\(748\) 0 0
\(749\) 55.2799 2.01988
\(750\) 0 0
\(751\) 0.534057 0.0194880 0.00974401 0.999953i \(-0.496898\pi\)
0.00974401 + 0.999953i \(0.496898\pi\)
\(752\) 0 0
\(753\) −30.8497 −1.12423
\(754\) 0 0
\(755\) −15.4710 −0.563047
\(756\) 0 0
\(757\) −29.5171 −1.07282 −0.536409 0.843958i \(-0.680219\pi\)
−0.536409 + 0.843958i \(0.680219\pi\)
\(758\) 0 0
\(759\) −4.84469 −0.175851
\(760\) 0 0
\(761\) 38.1434 1.38270 0.691348 0.722522i \(-0.257017\pi\)
0.691348 + 0.722522i \(0.257017\pi\)
\(762\) 0 0
\(763\) −46.8208 −1.69503
\(764\) 0 0
\(765\) −0.890804 −0.0322071
\(766\) 0 0
\(767\) 2.53406 0.0914995
\(768\) 0 0
\(769\) 20.8786 0.752902 0.376451 0.926437i \(-0.377144\pi\)
0.376451 + 0.926437i \(0.377144\pi\)
\(770\) 0 0
\(771\) 12.7524 0.459268
\(772\) 0 0
\(773\) 8.31063 0.298913 0.149456 0.988768i \(-0.452248\pi\)
0.149456 + 0.988768i \(0.452248\pi\)
\(774\) 0 0
\(775\) −7.73549 −0.277867
\(776\) 0 0
\(777\) 7.29870 0.261840
\(778\) 0 0
\(779\) −84.6125 −3.03156
\(780\) 0 0
\(781\) −59.8890 −2.14300
\(782\) 0 0
\(783\) 0.890804 0.0318347
\(784\) 0 0
\(785\) −18.6774 −0.666627
\(786\) 0 0
\(787\) 1.95388 0.0696484 0.0348242 0.999393i \(-0.488913\pi\)
0.0348242 + 0.999393i \(0.488913\pi\)
\(788\) 0 0
\(789\) 6.36178 0.226485
\(790\) 0 0
\(791\) 46.1773 1.64188
\(792\) 0 0
\(793\) −24.5391 −0.871409
\(794\) 0 0
\(795\) 12.3618 0.438427
\(796\) 0 0
\(797\) −16.1773 −0.573030 −0.286515 0.958076i \(-0.592497\pi\)
−0.286515 + 0.958076i \(0.592497\pi\)
\(798\) 0 0
\(799\) 5.90273 0.208823
\(800\) 0 0
\(801\) −15.1604 −0.535665
\(802\) 0 0
\(803\) −80.1023 −2.82675
\(804\) 0 0
\(805\) 3.73549 0.131659
\(806\) 0 0
\(807\) −10.5802 −0.372440
\(808\) 0 0
\(809\) −26.8670 −0.944592 −0.472296 0.881440i \(-0.656575\pi\)
−0.472296 + 0.881440i \(0.656575\pi\)
\(810\) 0 0
\(811\) −2.13835 −0.0750878 −0.0375439 0.999295i \(-0.511953\pi\)
−0.0375439 + 0.999295i \(0.511953\pi\)
\(812\) 0 0
\(813\) −4.26451 −0.149563
\(814\) 0 0
\(815\) 0.0922364 0.00323090
\(816\) 0 0
\(817\) −23.7578 −0.831179
\(818\) 0 0
\(819\) −10.6263 −0.371313
\(820\) 0 0
\(821\) 15.1604 0.529100 0.264550 0.964372i \(-0.414777\pi\)
0.264550 + 0.964372i \(0.414777\pi\)
\(822\) 0 0
\(823\) 23.1264 0.806137 0.403068 0.915170i \(-0.367944\pi\)
0.403068 + 0.915170i \(0.367944\pi\)
\(824\) 0 0
\(825\) 4.84469 0.168670
\(826\) 0 0
\(827\) −47.4299 −1.64930 −0.824650 0.565644i \(-0.808628\pi\)
−0.824650 + 0.565644i \(0.808628\pi\)
\(828\) 0 0
\(829\) −30.1723 −1.04793 −0.523963 0.851741i \(-0.675547\pi\)
−0.523963 + 0.851741i \(0.675547\pi\)
\(830\) 0 0
\(831\) 29.3787 1.01914
\(832\) 0 0
\(833\) 6.19454 0.214628
\(834\) 0 0
\(835\) −5.37371 −0.185965
\(836\) 0 0
\(837\) 7.73549 0.267378
\(838\) 0 0
\(839\) −18.8497 −0.650765 −0.325382 0.945583i \(-0.605493\pi\)
−0.325382 + 0.945583i \(0.605493\pi\)
\(840\) 0 0
\(841\) −28.2065 −0.972637
\(842\) 0 0
\(843\) 18.7524 0.645869
\(844\) 0 0
\(845\) 4.90776 0.168832
\(846\) 0 0
\(847\) 46.5852 1.60069
\(848\) 0 0
\(849\) −12.8958 −0.442584
\(850\) 0 0
\(851\) 1.95388 0.0669782
\(852\) 0 0
\(853\) 29.7578 1.01889 0.509443 0.860504i \(-0.329851\pi\)
0.509443 + 0.860504i \(0.329851\pi\)
\(854\) 0 0
\(855\) 6.84469 0.234083
\(856\) 0 0
\(857\) −15.2865 −0.522177 −0.261089 0.965315i \(-0.584081\pi\)
−0.261089 + 0.965315i \(0.584081\pi\)
\(858\) 0 0
\(859\) −33.5171 −1.14359 −0.571794 0.820397i \(-0.693752\pi\)
−0.571794 + 0.820397i \(0.693752\pi\)
\(860\) 0 0
\(861\) −46.1773 −1.57372
\(862\) 0 0
\(863\) 31.0443 1.05676 0.528380 0.849008i \(-0.322800\pi\)
0.528380 + 0.849008i \(0.322800\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 16.2065 0.550401
\(868\) 0 0
\(869\) 65.2627 2.21388
\(870\) 0 0
\(871\) 22.0050 0.745612
\(872\) 0 0
\(873\) −17.2526 −0.583912
\(874\) 0 0
\(875\) −3.73549 −0.126283
\(876\) 0 0
\(877\) 10.6313 0.358994 0.179497 0.983758i \(-0.442553\pi\)
0.179497 + 0.983758i \(0.442553\pi\)
\(878\) 0 0
\(879\) −8.79857 −0.296768
\(880\) 0 0
\(881\) 20.5341 0.691810 0.345905 0.938270i \(-0.387572\pi\)
0.345905 + 0.938270i \(0.387572\pi\)
\(882\) 0 0
\(883\) −36.0390 −1.21281 −0.606404 0.795157i \(-0.707388\pi\)
−0.606404 + 0.795157i \(0.707388\pi\)
\(884\) 0 0
\(885\) −0.890804 −0.0299440
\(886\) 0 0
\(887\) 24.9759 0.838608 0.419304 0.907846i \(-0.362274\pi\)
0.419304 + 0.907846i \(0.362274\pi\)
\(888\) 0 0
\(889\) −44.0289 −1.47668
\(890\) 0 0
\(891\) −4.84469 −0.162303
\(892\) 0 0
\(893\) −45.3549 −1.51774
\(894\) 0 0
\(895\) −9.56322 −0.319663
\(896\) 0 0
\(897\) −2.84469 −0.0949813
\(898\) 0 0
\(899\) 6.89080 0.229821
\(900\) 0 0
\(901\) 11.0119 0.366860
\(902\) 0 0
\(903\) −12.9658 −0.431475
\(904\) 0 0
\(905\) 7.16035 0.238018
\(906\) 0 0
\(907\) −41.6194 −1.38195 −0.690975 0.722879i \(-0.742818\pi\)
−0.690975 + 0.722879i \(0.742818\pi\)
\(908\) 0 0
\(909\) −2.67241 −0.0886383
\(910\) 0 0
\(911\) −3.81553 −0.126414 −0.0632070 0.998000i \(-0.520133\pi\)
−0.0632070 + 0.998000i \(0.520133\pi\)
\(912\) 0 0
\(913\) −22.1895 −0.734366
\(914\) 0 0
\(915\) 8.62629 0.285176
\(916\) 0 0
\(917\) 65.2627 2.15516
\(918\) 0 0
\(919\) 39.7917 1.31261 0.656303 0.754497i \(-0.272119\pi\)
0.656303 + 0.754497i \(0.272119\pi\)
\(920\) 0 0
\(921\) 10.0050 0.329677
\(922\) 0 0
\(923\) −35.1654 −1.15748
\(924\) 0 0
\(925\) −1.95388 −0.0642432
\(926\) 0 0
\(927\) −6.21839 −0.204239
\(928\) 0 0
\(929\) 12.6141 0.413855 0.206928 0.978356i \(-0.433654\pi\)
0.206928 + 0.978356i \(0.433654\pi\)
\(930\) 0 0
\(931\) −47.5971 −1.55993
\(932\) 0 0
\(933\) −18.5290 −0.606613
\(934\) 0 0
\(935\) 4.31566 0.141137
\(936\) 0 0
\(937\) −22.1262 −0.722830 −0.361415 0.932405i \(-0.617706\pi\)
−0.361415 + 0.932405i \(0.617706\pi\)
\(938\) 0 0
\(939\) −2.39067 −0.0780164
\(940\) 0 0
\(941\) −11.0392 −0.359869 −0.179934 0.983679i \(-0.557589\pi\)
−0.179934 + 0.983679i \(0.557589\pi\)
\(942\) 0 0
\(943\) −12.3618 −0.402555
\(944\) 0 0
\(945\) 3.73549 0.121515
\(946\) 0 0
\(947\) −9.68937 −0.314862 −0.157431 0.987530i \(-0.550321\pi\)
−0.157431 + 0.987530i \(0.550321\pi\)
\(948\) 0 0
\(949\) −47.0342 −1.52679
\(950\) 0 0
\(951\) 16.3157 0.529072
\(952\) 0 0
\(953\) −14.1845 −0.459480 −0.229740 0.973252i \(-0.573788\pi\)
−0.229740 + 0.973252i \(0.573788\pi\)
\(954\) 0 0
\(955\) 14.0050 0.453192
\(956\) 0 0
\(957\) −4.31566 −0.139506
\(958\) 0 0
\(959\) 56.6313 1.82872
\(960\) 0 0
\(961\) 28.8378 0.930252
\(962\) 0 0
\(963\) 14.7986 0.476877
\(964\) 0 0
\(965\) −26.7236 −0.860262
\(966\) 0 0
\(967\) −12.8447 −0.413057 −0.206529 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(968\) 0 0
\(969\) 6.09727 0.195873
\(970\) 0 0
\(971\) 24.1945 0.776440 0.388220 0.921567i \(-0.373090\pi\)
0.388220 + 0.921567i \(0.373090\pi\)
\(972\) 0 0
\(973\) 20.6091 0.660696
\(974\) 0 0
\(975\) 2.84469 0.0911029
\(976\) 0 0
\(977\) 18.4879 0.591482 0.295741 0.955268i \(-0.404434\pi\)
0.295741 + 0.955268i \(0.404434\pi\)
\(978\) 0 0
\(979\) 73.4471 2.34738
\(980\) 0 0
\(981\) −12.5341 −0.400182
\(982\) 0 0
\(983\) 28.8325 0.919614 0.459807 0.888019i \(-0.347919\pi\)
0.459807 + 0.888019i \(0.347919\pi\)
\(984\) 0 0
\(985\) −9.16035 −0.291873
\(986\) 0 0
\(987\) −24.7524 −0.787879
\(988\) 0 0
\(989\) −3.47098 −0.110371
\(990\) 0 0
\(991\) −40.8958 −1.29910 −0.649550 0.760319i \(-0.725043\pi\)
−0.649550 + 0.760319i \(0.725043\pi\)
\(992\) 0 0
\(993\) −24.8958 −0.790046
\(994\) 0 0
\(995\) −0.310629 −0.00984759
\(996\) 0 0
\(997\) −4.65518 −0.147431 −0.0737155 0.997279i \(-0.523486\pi\)
−0.0737155 + 0.997279i \(0.523486\pi\)
\(998\) 0 0
\(999\) 1.95388 0.0618181
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 5520.2.a.bv.1.3 3
4.3 odd 2 1380.2.a.j.1.1 3
12.11 even 2 4140.2.a.s.1.1 3
20.3 even 4 6900.2.f.r.6349.6 6
20.7 even 4 6900.2.f.r.6349.1 6
20.19 odd 2 6900.2.a.x.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.j.1.1 3 4.3 odd 2
4140.2.a.s.1.1 3 12.11 even 2
5520.2.a.bv.1.3 3 1.1 even 1 trivial
6900.2.a.x.1.3 3 20.19 odd 2
6900.2.f.r.6349.1 6 20.7 even 4
6900.2.f.r.6349.6 6 20.3 even 4