Properties

Label 4560.2.a.br.1.3
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: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.3
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.32088 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.32088 q^{7} +1.00000 q^{9} -3.70739 q^{11} +3.32088 q^{13} -1.00000 q^{15} +1.61350 q^{17} +1.00000 q^{19} -1.32088 q^{21} +7.67004 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.70739 q^{29} -9.67004 q^{31} +3.70739 q^{33} +1.32088 q^{35} +5.96265 q^{37} -3.32088 q^{39} +2.29261 q^{41} +8.73566 q^{43} +1.00000 q^{45} -11.6700 q^{47} -5.25526 q^{49} -1.61350 q^{51} +10.4431 q^{53} -3.70739 q^{55} -1.00000 q^{57} -12.6983 q^{59} +9.02827 q^{61} +1.32088 q^{63} +3.32088 q^{65} +13.2835 q^{67} -7.67004 q^{69} +8.69832 q^{71} -12.0565 q^{73} -1.00000 q^{75} -4.89703 q^{77} +14.0565 q^{79} +1.00000 q^{81} -5.02827 q^{83} +1.61350 q^{85} +1.70739 q^{87} -1.70739 q^{89} +4.38650 q^{91} +9.67004 q^{93} +1.00000 q^{95} -0.679116 q^{97} -3.70739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} + 3 q^{19} + 4 q^{21} - 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{33} - 4 q^{35} - 6 q^{37} - 2 q^{39} + 12 q^{41} + 8 q^{43} + 3 q^{45} - 6 q^{47} + 3 q^{49} - 2 q^{51} + 8 q^{53} - 6 q^{55} - 3 q^{57} + 4 q^{59} + 14 q^{61} - 4 q^{63} + 2 q^{65} + 8 q^{67} + 6 q^{69} - 16 q^{71} - 10 q^{73} - 3 q^{75} + 20 q^{77} + 16 q^{79} + 3 q^{81} - 2 q^{83} + 2 q^{85} + 16 q^{91} + 3 q^{95} - 10 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(7\) 1.32088 0.499247 0.249624 0.968343i \(-0.419693\pi\)
0.249624 + 0.968343i \(0.419693\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.70739 −1.11782 −0.558910 0.829228i \(-0.688780\pi\)
−0.558910 + 0.829228i \(0.688780\pi\)
\(12\) 0 0
\(13\) 3.32088 0.921048 0.460524 0.887647i \(-0.347662\pi\)
0.460524 + 0.887647i \(0.347662\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.61350 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.32088 −0.288241
\(22\) 0 0
\(23\) 7.67004 1.59931 0.799657 0.600457i \(-0.205015\pi\)
0.799657 + 0.600457i \(0.205015\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.70739 −0.317054 −0.158527 0.987355i \(-0.550675\pi\)
−0.158527 + 0.987355i \(0.550675\pi\)
\(30\) 0 0
\(31\) −9.67004 −1.73679 −0.868395 0.495872i \(-0.834848\pi\)
−0.868395 + 0.495872i \(0.834848\pi\)
\(32\) 0 0
\(33\) 3.70739 0.645374
\(34\) 0 0
\(35\) 1.32088 0.223270
\(36\) 0 0
\(37\) 5.96265 0.980254 0.490127 0.871651i \(-0.336950\pi\)
0.490127 + 0.871651i \(0.336950\pi\)
\(38\) 0 0
\(39\) −3.32088 −0.531767
\(40\) 0 0
\(41\) 2.29261 0.358046 0.179023 0.983845i \(-0.442706\pi\)
0.179023 + 0.983845i \(0.442706\pi\)
\(42\) 0 0
\(43\) 8.73566 1.33218 0.666088 0.745873i \(-0.267967\pi\)
0.666088 + 0.745873i \(0.267967\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −11.6700 −1.70225 −0.851125 0.524963i \(-0.824079\pi\)
−0.851125 + 0.524963i \(0.824079\pi\)
\(48\) 0 0
\(49\) −5.25526 −0.750752
\(50\) 0 0
\(51\) −1.61350 −0.225935
\(52\) 0 0
\(53\) 10.4431 1.43446 0.717232 0.696835i \(-0.245409\pi\)
0.717232 + 0.696835i \(0.245409\pi\)
\(54\) 0 0
\(55\) −3.70739 −0.499904
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −12.6983 −1.65318 −0.826590 0.562805i \(-0.809722\pi\)
−0.826590 + 0.562805i \(0.809722\pi\)
\(60\) 0 0
\(61\) 9.02827 1.15595 0.577976 0.816054i \(-0.303843\pi\)
0.577976 + 0.816054i \(0.303843\pi\)
\(62\) 0 0
\(63\) 1.32088 0.166416
\(64\) 0 0
\(65\) 3.32088 0.411905
\(66\) 0 0
\(67\) 13.2835 1.62284 0.811421 0.584462i \(-0.198694\pi\)
0.811421 + 0.584462i \(0.198694\pi\)
\(68\) 0 0
\(69\) −7.67004 −0.923365
\(70\) 0 0
\(71\) 8.69832 1.03230 0.516150 0.856498i \(-0.327365\pi\)
0.516150 + 0.856498i \(0.327365\pi\)
\(72\) 0 0
\(73\) −12.0565 −1.41111 −0.705556 0.708654i \(-0.749303\pi\)
−0.705556 + 0.708654i \(0.749303\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −4.89703 −0.558069
\(78\) 0 0
\(79\) 14.0565 1.58149 0.790743 0.612149i \(-0.209695\pi\)
0.790743 + 0.612149i \(0.209695\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.02827 −0.551925 −0.275962 0.961168i \(-0.588997\pi\)
−0.275962 + 0.961168i \(0.588997\pi\)
\(84\) 0 0
\(85\) 1.61350 0.175008
\(86\) 0 0
\(87\) 1.70739 0.183051
\(88\) 0 0
\(89\) −1.70739 −0.180983 −0.0904915 0.995897i \(-0.528844\pi\)
−0.0904915 + 0.995897i \(0.528844\pi\)
\(90\) 0 0
\(91\) 4.38650 0.459831
\(92\) 0 0
\(93\) 9.67004 1.00274
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −0.679116 −0.0689537 −0.0344769 0.999405i \(-0.510977\pi\)
−0.0344769 + 0.999405i \(0.510977\pi\)
\(98\) 0 0
\(99\) −3.70739 −0.372607
\(100\) 0 0
\(101\) 12.6418 1.25790 0.628952 0.777445i \(-0.283484\pi\)
0.628952 + 0.777445i \(0.283484\pi\)
\(102\) 0 0
\(103\) −6.05655 −0.596769 −0.298385 0.954446i \(-0.596448\pi\)
−0.298385 + 0.954446i \(0.596448\pi\)
\(104\) 0 0
\(105\) −1.32088 −0.128905
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 10.6983 1.02471 0.512356 0.858773i \(-0.328773\pi\)
0.512356 + 0.858773i \(0.328773\pi\)
\(110\) 0 0
\(111\) −5.96265 −0.565950
\(112\) 0 0
\(113\) −16.3118 −1.53449 −0.767243 0.641356i \(-0.778372\pi\)
−0.767243 + 0.641356i \(0.778372\pi\)
\(114\) 0 0
\(115\) 7.67004 0.715235
\(116\) 0 0
\(117\) 3.32088 0.307016
\(118\) 0 0
\(119\) 2.13124 0.195371
\(120\) 0 0
\(121\) 2.74474 0.249521
\(122\) 0 0
\(123\) −2.29261 −0.206718
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) −8.73566 −0.769132
\(130\) 0 0
\(131\) −4.29261 −0.375047 −0.187524 0.982260i \(-0.560046\pi\)
−0.187524 + 0.982260i \(0.560046\pi\)
\(132\) 0 0
\(133\) 1.32088 0.114535
\(134\) 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
\(139\) 8.58522 0.728189 0.364094 0.931362i \(-0.381379\pi\)
0.364094 + 0.931362i \(0.381379\pi\)
\(140\) 0 0
\(141\) 11.6700 0.982795
\(142\) 0 0
\(143\) −12.3118 −1.02957
\(144\) 0 0
\(145\) −1.70739 −0.141791
\(146\) 0 0
\(147\) 5.25526 0.433447
\(148\) 0 0
\(149\) 12.0565 0.987711 0.493855 0.869544i \(-0.335587\pi\)
0.493855 + 0.869544i \(0.335587\pi\)
\(150\) 0 0
\(151\) 22.9536 1.86794 0.933968 0.357357i \(-0.116322\pi\)
0.933968 + 0.357357i \(0.116322\pi\)
\(152\) 0 0
\(153\) 1.61350 0.130443
\(154\) 0 0
\(155\) −9.67004 −0.776717
\(156\) 0 0
\(157\) −11.8688 −0.947230 −0.473615 0.880732i \(-0.657051\pi\)
−0.473615 + 0.880732i \(0.657051\pi\)
\(158\) 0 0
\(159\) −10.4431 −0.828188
\(160\) 0 0
\(161\) 10.1312 0.798454
\(162\) 0 0
\(163\) 0.0373465 0.00292520 0.00146260 0.999999i \(-0.499534\pi\)
0.00146260 + 0.999999i \(0.499534\pi\)
\(164\) 0 0
\(165\) 3.70739 0.288620
\(166\) 0 0
\(167\) 11.0283 0.853393 0.426697 0.904395i \(-0.359677\pi\)
0.426697 + 0.904395i \(0.359677\pi\)
\(168\) 0 0
\(169\) −1.97173 −0.151671
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −17.8578 −1.35771 −0.678853 0.734274i \(-0.737523\pi\)
−0.678853 + 0.734274i \(0.737523\pi\)
\(174\) 0 0
\(175\) 1.32088 0.0998495
\(176\) 0 0
\(177\) 12.6983 0.954464
\(178\) 0 0
\(179\) 19.9253 1.48929 0.744644 0.667462i \(-0.232619\pi\)
0.744644 + 0.667462i \(0.232619\pi\)
\(180\) 0 0
\(181\) −15.4713 −1.14997 −0.574987 0.818162i \(-0.694993\pi\)
−0.574987 + 0.818162i \(0.694993\pi\)
\(182\) 0 0
\(183\) −9.02827 −0.667389
\(184\) 0 0
\(185\) 5.96265 0.438383
\(186\) 0 0
\(187\) −5.98185 −0.437437
\(188\) 0 0
\(189\) −1.32088 −0.0960802
\(190\) 0 0
\(191\) −18.3492 −1.32770 −0.663849 0.747866i \(-0.731078\pi\)
−0.663849 + 0.747866i \(0.731078\pi\)
\(192\) 0 0
\(193\) 12.0192 0.865161 0.432581 0.901595i \(-0.357603\pi\)
0.432581 + 0.901595i \(0.357603\pi\)
\(194\) 0 0
\(195\) −3.32088 −0.237813
\(196\) 0 0
\(197\) 4.84049 0.344870 0.172435 0.985021i \(-0.444836\pi\)
0.172435 + 0.985021i \(0.444836\pi\)
\(198\) 0 0
\(199\) 20.6983 1.46726 0.733632 0.679547i \(-0.237823\pi\)
0.733632 + 0.679547i \(0.237823\pi\)
\(200\) 0 0
\(201\) −13.2835 −0.936949
\(202\) 0 0
\(203\) −2.25526 −0.158289
\(204\) 0 0
\(205\) 2.29261 0.160123
\(206\) 0 0
\(207\) 7.67004 0.533105
\(208\) 0 0
\(209\) −3.70739 −0.256445
\(210\) 0 0
\(211\) −2.05655 −0.141579 −0.0707893 0.997491i \(-0.522552\pi\)
−0.0707893 + 0.997491i \(0.522552\pi\)
\(212\) 0 0
\(213\) −8.69832 −0.595999
\(214\) 0 0
\(215\) 8.73566 0.595767
\(216\) 0 0
\(217\) −12.7730 −0.867088
\(218\) 0 0
\(219\) 12.0565 0.814706
\(220\) 0 0
\(221\) 5.35823 0.360434
\(222\) 0 0
\(223\) −6.05655 −0.405576 −0.202788 0.979223i \(-0.565000\pi\)
−0.202788 + 0.979223i \(0.565000\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 28.3118 1.87912 0.939560 0.342383i \(-0.111234\pi\)
0.939560 + 0.342383i \(0.111234\pi\)
\(228\) 0 0
\(229\) −1.80128 −0.119032 −0.0595161 0.998227i \(-0.518956\pi\)
−0.0595161 + 0.998227i \(0.518956\pi\)
\(230\) 0 0
\(231\) 4.89703 0.322201
\(232\) 0 0
\(233\) −3.94345 −0.258344 −0.129172 0.991622i \(-0.541232\pi\)
−0.129172 + 0.991622i \(0.541232\pi\)
\(234\) 0 0
\(235\) −11.6700 −0.761270
\(236\) 0 0
\(237\) −14.0565 −0.913071
\(238\) 0 0
\(239\) 16.8031 1.08690 0.543452 0.839440i \(-0.317117\pi\)
0.543452 + 0.839440i \(0.317117\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −5.25526 −0.335747
\(246\) 0 0
\(247\) 3.32088 0.211303
\(248\) 0 0
\(249\) 5.02827 0.318654
\(250\) 0 0
\(251\) −24.9909 −1.57741 −0.788707 0.614770i \(-0.789249\pi\)
−0.788707 + 0.614770i \(0.789249\pi\)
\(252\) 0 0
\(253\) −28.4358 −1.78775
\(254\) 0 0
\(255\) −1.61350 −0.101041
\(256\) 0 0
\(257\) −2.91518 −0.181844 −0.0909219 0.995858i \(-0.528981\pi\)
−0.0909219 + 0.995858i \(0.528981\pi\)
\(258\) 0 0
\(259\) 7.87598 0.489389
\(260\) 0 0
\(261\) −1.70739 −0.105685
\(262\) 0 0
\(263\) 22.3118 1.37581 0.687903 0.725803i \(-0.258532\pi\)
0.687903 + 0.725803i \(0.258532\pi\)
\(264\) 0 0
\(265\) 10.4431 0.641512
\(266\) 0 0
\(267\) 1.70739 0.104491
\(268\) 0 0
\(269\) 15.0656 0.918567 0.459284 0.888290i \(-0.348106\pi\)
0.459284 + 0.888290i \(0.348106\pi\)
\(270\) 0 0
\(271\) −5.47133 −0.332359 −0.166180 0.986095i \(-0.553143\pi\)
−0.166180 + 0.986095i \(0.553143\pi\)
\(272\) 0 0
\(273\) −4.38650 −0.265483
\(274\) 0 0
\(275\) −3.70739 −0.223564
\(276\) 0 0
\(277\) −10.7730 −0.647287 −0.323644 0.946179i \(-0.604908\pi\)
−0.323644 + 0.946179i \(0.604908\pi\)
\(278\) 0 0
\(279\) −9.67004 −0.578930
\(280\) 0 0
\(281\) 17.8205 1.06308 0.531540 0.847033i \(-0.321613\pi\)
0.531540 + 0.847033i \(0.321613\pi\)
\(282\) 0 0
\(283\) 30.2070 1.79562 0.897810 0.440384i \(-0.145158\pi\)
0.897810 + 0.440384i \(0.145158\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 3.02827 0.178753
\(288\) 0 0
\(289\) −14.3966 −0.846861
\(290\) 0 0
\(291\) 0.679116 0.0398105
\(292\) 0 0
\(293\) −18.2553 −1.06648 −0.533242 0.845963i \(-0.679026\pi\)
−0.533242 + 0.845963i \(0.679026\pi\)
\(294\) 0 0
\(295\) −12.6983 −0.739325
\(296\) 0 0
\(297\) 3.70739 0.215125
\(298\) 0 0
\(299\) 25.4713 1.47304
\(300\) 0 0
\(301\) 11.5388 0.665085
\(302\) 0 0
\(303\) −12.6418 −0.726251
\(304\) 0 0
\(305\) 9.02827 0.516957
\(306\) 0 0
\(307\) 16.5852 0.946569 0.473284 0.880910i \(-0.343068\pi\)
0.473284 + 0.880910i \(0.343068\pi\)
\(308\) 0 0
\(309\) 6.05655 0.344545
\(310\) 0 0
\(311\) −11.5196 −0.653217 −0.326608 0.945160i \(-0.605906\pi\)
−0.326608 + 0.945160i \(0.605906\pi\)
\(312\) 0 0
\(313\) 32.7549 1.85141 0.925707 0.378241i \(-0.123471\pi\)
0.925707 + 0.378241i \(0.123471\pi\)
\(314\) 0 0
\(315\) 1.32088 0.0744234
\(316\) 0 0
\(317\) 15.2161 0.854619 0.427310 0.904105i \(-0.359461\pi\)
0.427310 + 0.904105i \(0.359461\pi\)
\(318\) 0 0
\(319\) 6.32996 0.354410
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.61350 0.0897773
\(324\) 0 0
\(325\) 3.32088 0.184210
\(326\) 0 0
\(327\) −10.6983 −0.591618
\(328\) 0 0
\(329\) −15.4148 −0.849844
\(330\) 0 0
\(331\) 26.9536 1.48150 0.740751 0.671779i \(-0.234470\pi\)
0.740751 + 0.671779i \(0.234470\pi\)
\(332\) 0 0
\(333\) 5.96265 0.326751
\(334\) 0 0
\(335\) 13.2835 0.725757
\(336\) 0 0
\(337\) −17.5652 −0.956839 −0.478419 0.878132i \(-0.658790\pi\)
−0.478419 + 0.878132i \(0.658790\pi\)
\(338\) 0 0
\(339\) 16.3118 0.885936
\(340\) 0 0
\(341\) 35.8506 1.94142
\(342\) 0 0
\(343\) −16.1878 −0.874058
\(344\) 0 0
\(345\) −7.67004 −0.412941
\(346\) 0 0
\(347\) 34.3118 1.84195 0.920977 0.389616i \(-0.127392\pi\)
0.920977 + 0.389616i \(0.127392\pi\)
\(348\) 0 0
\(349\) 35.2835 1.88868 0.944342 0.328965i \(-0.106700\pi\)
0.944342 + 0.328965i \(0.106700\pi\)
\(350\) 0 0
\(351\) −3.32088 −0.177256
\(352\) 0 0
\(353\) −8.71646 −0.463930 −0.231965 0.972724i \(-0.574516\pi\)
−0.231965 + 0.972724i \(0.574516\pi\)
\(354\) 0 0
\(355\) 8.69832 0.461659
\(356\) 0 0
\(357\) −2.13124 −0.112797
\(358\) 0 0
\(359\) −30.4623 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.74474 −0.144061
\(364\) 0 0
\(365\) −12.0565 −0.631069
\(366\) 0 0
\(367\) 27.5652 1.43889 0.719446 0.694548i \(-0.244396\pi\)
0.719446 + 0.694548i \(0.244396\pi\)
\(368\) 0 0
\(369\) 2.29261 0.119349
\(370\) 0 0
\(371\) 13.7941 0.716152
\(372\) 0 0
\(373\) 16.0939 0.833310 0.416655 0.909065i \(-0.363202\pi\)
0.416655 + 0.909065i \(0.363202\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −5.67004 −0.292022
\(378\) 0 0
\(379\) −9.01013 −0.462819 −0.231410 0.972856i \(-0.574334\pi\)
−0.231410 + 0.972856i \(0.574334\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 23.3401 1.19262 0.596311 0.802753i \(-0.296632\pi\)
0.596311 + 0.802753i \(0.296632\pi\)
\(384\) 0 0
\(385\) −4.89703 −0.249576
\(386\) 0 0
\(387\) 8.73566 0.444059
\(388\) 0 0
\(389\) 20.0565 1.01691 0.508454 0.861089i \(-0.330217\pi\)
0.508454 + 0.861089i \(0.330217\pi\)
\(390\) 0 0
\(391\) 12.3756 0.625860
\(392\) 0 0
\(393\) 4.29261 0.216534
\(394\) 0 0
\(395\) 14.0565 0.707262
\(396\) 0 0
\(397\) −21.7375 −1.09097 −0.545487 0.838119i \(-0.683655\pi\)
−0.545487 + 0.838119i \(0.683655\pi\)
\(398\) 0 0
\(399\) −1.32088 −0.0661269
\(400\) 0 0
\(401\) −7.17872 −0.358488 −0.179244 0.983805i \(-0.557365\pi\)
−0.179244 + 0.983805i \(0.557365\pi\)
\(402\) 0 0
\(403\) −32.1131 −1.59967
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −22.1059 −1.09575
\(408\) 0 0
\(409\) 16.2443 0.803231 0.401615 0.915808i \(-0.368449\pi\)
0.401615 + 0.915808i \(0.368449\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) −16.7730 −0.825346
\(414\) 0 0
\(415\) −5.02827 −0.246828
\(416\) 0 0
\(417\) −8.58522 −0.420420
\(418\) 0 0
\(419\) −16.9909 −0.830061 −0.415031 0.909807i \(-0.636229\pi\)
−0.415031 + 0.909807i \(0.636229\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −11.6700 −0.567417
\(424\) 0 0
\(425\) 1.61350 0.0782660
\(426\) 0 0
\(427\) 11.9253 0.577106
\(428\) 0 0
\(429\) 12.3118 0.594420
\(430\) 0 0
\(431\) −22.6418 −1.09062 −0.545308 0.838236i \(-0.683587\pi\)
−0.545308 + 0.838236i \(0.683587\pi\)
\(432\) 0 0
\(433\) −12.2070 −0.586630 −0.293315 0.956016i \(-0.594759\pi\)
−0.293315 + 0.956016i \(0.594759\pi\)
\(434\) 0 0
\(435\) 1.70739 0.0818631
\(436\) 0 0
\(437\) 7.67004 0.366908
\(438\) 0 0
\(439\) 31.8506 1.52015 0.760073 0.649837i \(-0.225163\pi\)
0.760073 + 0.649837i \(0.225163\pi\)
\(440\) 0 0
\(441\) −5.25526 −0.250251
\(442\) 0 0
\(443\) −8.95358 −0.425397 −0.212699 0.977118i \(-0.568225\pi\)
−0.212699 + 0.977118i \(0.568225\pi\)
\(444\) 0 0
\(445\) −1.70739 −0.0809380
\(446\) 0 0
\(447\) −12.0565 −0.570255
\(448\) 0 0
\(449\) 5.51960 0.260486 0.130243 0.991482i \(-0.458424\pi\)
0.130243 + 0.991482i \(0.458424\pi\)
\(450\) 0 0
\(451\) −8.49960 −0.400231
\(452\) 0 0
\(453\) −22.9536 −1.07845
\(454\) 0 0
\(455\) 4.38650 0.205643
\(456\) 0 0
\(457\) 0.131241 0.00613918 0.00306959 0.999995i \(-0.499023\pi\)
0.00306959 + 0.999995i \(0.499023\pi\)
\(458\) 0 0
\(459\) −1.61350 −0.0753115
\(460\) 0 0
\(461\) −14.1131 −0.657312 −0.328656 0.944450i \(-0.606596\pi\)
−0.328656 + 0.944450i \(0.606596\pi\)
\(462\) 0 0
\(463\) −5.98080 −0.277951 −0.138976 0.990296i \(-0.544381\pi\)
−0.138976 + 0.990296i \(0.544381\pi\)
\(464\) 0 0
\(465\) 9.67004 0.448437
\(466\) 0 0
\(467\) 6.38650 0.295532 0.147766 0.989022i \(-0.452792\pi\)
0.147766 + 0.989022i \(0.452792\pi\)
\(468\) 0 0
\(469\) 17.5460 0.810200
\(470\) 0 0
\(471\) 11.8688 0.546884
\(472\) 0 0
\(473\) −32.3865 −1.48913
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 10.4431 0.478155
\(478\) 0 0
\(479\) −18.8597 −0.861721 −0.430861 0.902419i \(-0.641790\pi\)
−0.430861 + 0.902419i \(0.641790\pi\)
\(480\) 0 0
\(481\) 19.8013 0.902861
\(482\) 0 0
\(483\) −10.1312 −0.460987
\(484\) 0 0
\(485\) −0.679116 −0.0308370
\(486\) 0 0
\(487\) −10.1312 −0.459090 −0.229545 0.973298i \(-0.573724\pi\)
−0.229545 + 0.973298i \(0.573724\pi\)
\(488\) 0 0
\(489\) −0.0373465 −0.00168887
\(490\) 0 0
\(491\) −1.06562 −0.0480908 −0.0240454 0.999711i \(-0.507655\pi\)
−0.0240454 + 0.999711i \(0.507655\pi\)
\(492\) 0 0
\(493\) −2.75486 −0.124073
\(494\) 0 0
\(495\) −3.70739 −0.166635
\(496\) 0 0
\(497\) 11.4895 0.515373
\(498\) 0 0
\(499\) 14.2443 0.637664 0.318832 0.947811i \(-0.396709\pi\)
0.318832 + 0.947811i \(0.396709\pi\)
\(500\) 0 0
\(501\) −11.0283 −0.492707
\(502\) 0 0
\(503\) −30.9717 −1.38096 −0.690481 0.723351i \(-0.742601\pi\)
−0.690481 + 0.723351i \(0.742601\pi\)
\(504\) 0 0
\(505\) 12.6418 0.562551
\(506\) 0 0
\(507\) 1.97173 0.0875674
\(508\) 0 0
\(509\) −30.4057 −1.34771 −0.673855 0.738864i \(-0.735363\pi\)
−0.673855 + 0.738864i \(0.735363\pi\)
\(510\) 0 0
\(511\) −15.9253 −0.704494
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −6.05655 −0.266883
\(516\) 0 0
\(517\) 43.2654 1.90281
\(518\) 0 0
\(519\) 17.8578 0.783872
\(520\) 0 0
\(521\) −4.34916 −0.190540 −0.0952700 0.995451i \(-0.530371\pi\)
−0.0952700 + 0.995451i \(0.530371\pi\)
\(522\) 0 0
\(523\) 8.69832 0.380351 0.190175 0.981750i \(-0.439094\pi\)
0.190175 + 0.981750i \(0.439094\pi\)
\(524\) 0 0
\(525\) −1.32088 −0.0576481
\(526\) 0 0
\(527\) −15.6026 −0.679659
\(528\) 0 0
\(529\) 35.8296 1.55781
\(530\) 0 0
\(531\) −12.6983 −0.551060
\(532\) 0 0
\(533\) 7.61350 0.329777
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −19.9253 −0.859840
\(538\) 0 0
\(539\) 19.4833 0.839206
\(540\) 0 0
\(541\) 8.45398 0.363465 0.181733 0.983348i \(-0.441830\pi\)
0.181733 + 0.983348i \(0.441830\pi\)
\(542\) 0 0
\(543\) 15.4713 0.663938
\(544\) 0 0
\(545\) 10.6983 0.458266
\(546\) 0 0
\(547\) −11.5279 −0.492896 −0.246448 0.969156i \(-0.579264\pi\)
−0.246448 + 0.969156i \(0.579264\pi\)
\(548\) 0 0
\(549\) 9.02827 0.385317
\(550\) 0 0
\(551\) −1.70739 −0.0727372
\(552\) 0 0
\(553\) 18.5671 0.789552
\(554\) 0 0
\(555\) −5.96265 −0.253101
\(556\) 0 0
\(557\) 1.22699 0.0519892 0.0259946 0.999662i \(-0.491725\pi\)
0.0259946 + 0.999662i \(0.491725\pi\)
\(558\) 0 0
\(559\) 29.0101 1.22700
\(560\) 0 0
\(561\) 5.98185 0.252554
\(562\) 0 0
\(563\) −19.9144 −0.839291 −0.419646 0.907688i \(-0.637846\pi\)
−0.419646 + 0.907688i \(0.637846\pi\)
\(564\) 0 0
\(565\) −16.3118 −0.686243
\(566\) 0 0
\(567\) 1.32088 0.0554719
\(568\) 0 0
\(569\) −27.5015 −1.15292 −0.576460 0.817125i \(-0.695567\pi\)
−0.576460 + 0.817125i \(0.695567\pi\)
\(570\) 0 0
\(571\) 9.47133 0.396363 0.198181 0.980165i \(-0.436497\pi\)
0.198181 + 0.980165i \(0.436497\pi\)
\(572\) 0 0
\(573\) 18.3492 0.766547
\(574\) 0 0
\(575\) 7.67004 0.319863
\(576\) 0 0
\(577\) 36.6418 1.52542 0.762708 0.646743i \(-0.223869\pi\)
0.762708 + 0.646743i \(0.223869\pi\)
\(578\) 0 0
\(579\) −12.0192 −0.499501
\(580\) 0 0
\(581\) −6.64177 −0.275547
\(582\) 0 0
\(583\) −38.7165 −1.60347
\(584\) 0 0
\(585\) 3.32088 0.137302
\(586\) 0 0
\(587\) 3.67004 0.151479 0.0757394 0.997128i \(-0.475868\pi\)
0.0757394 + 0.997128i \(0.475868\pi\)
\(588\) 0 0
\(589\) −9.67004 −0.398447
\(590\) 0 0
\(591\) −4.84049 −0.199111
\(592\) 0 0
\(593\) −17.2270 −0.707428 −0.353714 0.935354i \(-0.615081\pi\)
−0.353714 + 0.935354i \(0.615081\pi\)
\(594\) 0 0
\(595\) 2.13124 0.0873724
\(596\) 0 0
\(597\) −20.6983 −0.847126
\(598\) 0 0
\(599\) 8.11310 0.331492 0.165746 0.986168i \(-0.446997\pi\)
0.165746 + 0.986168i \(0.446997\pi\)
\(600\) 0 0
\(601\) 7.35823 0.300149 0.150074 0.988675i \(-0.452049\pi\)
0.150074 + 0.988675i \(0.452049\pi\)
\(602\) 0 0
\(603\) 13.2835 0.540947
\(604\) 0 0
\(605\) 2.74474 0.111589
\(606\) 0 0
\(607\) −19.9253 −0.808743 −0.404372 0.914595i \(-0.632510\pi\)
−0.404372 + 0.914595i \(0.632510\pi\)
\(608\) 0 0
\(609\) 2.25526 0.0913879
\(610\) 0 0
\(611\) −38.7549 −1.56785
\(612\) 0 0
\(613\) 39.3219 1.58820 0.794099 0.607788i \(-0.207943\pi\)
0.794099 + 0.607788i \(0.207943\pi\)
\(614\) 0 0
\(615\) −2.29261 −0.0924470
\(616\) 0 0
\(617\) 0.0674757 0.00271647 0.00135823 0.999999i \(-0.499568\pi\)
0.00135823 + 0.999999i \(0.499568\pi\)
\(618\) 0 0
\(619\) −38.8680 −1.56224 −0.781118 0.624384i \(-0.785350\pi\)
−0.781118 + 0.624384i \(0.785350\pi\)
\(620\) 0 0
\(621\) −7.67004 −0.307788
\(622\) 0 0
\(623\) −2.25526 −0.0903552
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.70739 0.148059
\(628\) 0 0
\(629\) 9.62071 0.383603
\(630\) 0 0
\(631\) −29.2835 −1.16576 −0.582880 0.812559i \(-0.698074\pi\)
−0.582880 + 0.812559i \(0.698074\pi\)
\(632\) 0 0
\(633\) 2.05655 0.0817404
\(634\) 0 0
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) −17.4521 −0.691478
\(638\) 0 0
\(639\) 8.69832 0.344100
\(640\) 0 0
\(641\) 10.4057 0.411001 0.205500 0.978657i \(-0.434118\pi\)
0.205500 + 0.978657i \(0.434118\pi\)
\(642\) 0 0
\(643\) 11.3774 0.448682 0.224341 0.974511i \(-0.427977\pi\)
0.224341 + 0.974511i \(0.427977\pi\)
\(644\) 0 0
\(645\) −8.73566 −0.343966
\(646\) 0 0
\(647\) −45.5388 −1.79032 −0.895158 0.445750i \(-0.852937\pi\)
−0.895158 + 0.445750i \(0.852937\pi\)
\(648\) 0 0
\(649\) 47.0776 1.84796
\(650\) 0 0
\(651\) 12.7730 0.500614
\(652\) 0 0
\(653\) −2.27341 −0.0889654 −0.0444827 0.999010i \(-0.514164\pi\)
−0.0444827 + 0.999010i \(0.514164\pi\)
\(654\) 0 0
\(655\) −4.29261 −0.167726
\(656\) 0 0
\(657\) −12.0565 −0.470371
\(658\) 0 0
\(659\) 8.77301 0.341748 0.170874 0.985293i \(-0.445341\pi\)
0.170874 + 0.985293i \(0.445341\pi\)
\(660\) 0 0
\(661\) −44.4924 −1.73055 −0.865277 0.501295i \(-0.832857\pi\)
−0.865277 + 0.501295i \(0.832857\pi\)
\(662\) 0 0
\(663\) −5.35823 −0.208096
\(664\) 0 0
\(665\) 1.32088 0.0512217
\(666\) 0 0
\(667\) −13.0957 −0.507069
\(668\) 0 0
\(669\) 6.05655 0.234160
\(670\) 0 0
\(671\) −33.4713 −1.29215
\(672\) 0 0
\(673\) −5.96265 −0.229843 −0.114922 0.993375i \(-0.536662\pi\)
−0.114922 + 0.993375i \(0.536662\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 6.32996 0.243280 0.121640 0.992574i \(-0.461185\pi\)
0.121640 + 0.992574i \(0.461185\pi\)
\(678\) 0 0
\(679\) −0.897033 −0.0344250
\(680\) 0 0
\(681\) −28.3118 −1.08491
\(682\) 0 0
\(683\) −39.8506 −1.52484 −0.762421 0.647082i \(-0.775989\pi\)
−0.762421 + 0.647082i \(0.775989\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 1.80128 0.0687233
\(688\) 0 0
\(689\) 34.6802 1.32121
\(690\) 0 0
\(691\) −8.07469 −0.307176 −0.153588 0.988135i \(-0.549083\pi\)
−0.153588 + 0.988135i \(0.549083\pi\)
\(692\) 0 0
\(693\) −4.89703 −0.186023
\(694\) 0 0
\(695\) 8.58522 0.325656
\(696\) 0 0
\(697\) 3.69912 0.140114
\(698\) 0 0
\(699\) 3.94345 0.149155
\(700\) 0 0
\(701\) −2.77301 −0.104735 −0.0523676 0.998628i \(-0.516677\pi\)
−0.0523676 + 0.998628i \(0.516677\pi\)
\(702\) 0 0
\(703\) 5.96265 0.224886
\(704\) 0 0
\(705\) 11.6700 0.439519
\(706\) 0 0
\(707\) 16.6983 0.628005
\(708\) 0 0
\(709\) 10.1987 0.383021 0.191510 0.981491i \(-0.438661\pi\)
0.191510 + 0.981491i \(0.438661\pi\)
\(710\) 0 0
\(711\) 14.0565 0.527162
\(712\) 0 0
\(713\) −74.1696 −2.77767
\(714\) 0 0
\(715\) −12.3118 −0.460436
\(716\) 0 0
\(717\) −16.8031 −0.627525
\(718\) 0 0
\(719\) −34.2745 −1.27822 −0.639111 0.769115i \(-0.720698\pi\)
−0.639111 + 0.769115i \(0.720698\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) −14.0000 −0.520666
\(724\) 0 0
\(725\) −1.70739 −0.0634108
\(726\) 0 0
\(727\) −22.0939 −0.819417 −0.409709 0.912216i \(-0.634370\pi\)
−0.409709 + 0.912216i \(0.634370\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 14.0950 0.521321
\(732\) 0 0
\(733\) −13.3401 −0.492727 −0.246364 0.969177i \(-0.579236\pi\)
−0.246364 + 0.969177i \(0.579236\pi\)
\(734\) 0 0
\(735\) 5.25526 0.193843
\(736\) 0 0
\(737\) −49.2472 −1.81405
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) −3.32088 −0.121996
\(742\) 0 0
\(743\) −32.8861 −1.20647 −0.603237 0.797562i \(-0.706123\pi\)
−0.603237 + 0.797562i \(0.706123\pi\)
\(744\) 0 0
\(745\) 12.0565 0.441718
\(746\) 0 0
\(747\) −5.02827 −0.183975
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16.4996 −0.602079 −0.301039 0.953612i \(-0.597334\pi\)
−0.301039 + 0.953612i \(0.597334\pi\)
\(752\) 0 0
\(753\) 24.9909 0.910720
\(754\) 0 0
\(755\) 22.9536 0.835366
\(756\) 0 0
\(757\) −42.5489 −1.54647 −0.773234 0.634121i \(-0.781362\pi\)
−0.773234 + 0.634121i \(0.781362\pi\)
\(758\) 0 0
\(759\) 28.4358 1.03216
\(760\) 0 0
\(761\) 5.34009 0.193578 0.0967890 0.995305i \(-0.469143\pi\)
0.0967890 + 0.995305i \(0.469143\pi\)
\(762\) 0 0
\(763\) 14.1312 0.511585
\(764\) 0 0
\(765\) 1.61350 0.0583360
\(766\) 0 0
\(767\) −42.1696 −1.52266
\(768\) 0 0
\(769\) −7.28354 −0.262651 −0.131326 0.991339i \(-0.541923\pi\)
−0.131326 + 0.991339i \(0.541923\pi\)
\(770\) 0 0
\(771\) 2.91518 0.104988
\(772\) 0 0
\(773\) −18.1806 −0.653910 −0.326955 0.945040i \(-0.606023\pi\)
−0.326955 + 0.945040i \(0.606023\pi\)
\(774\) 0 0
\(775\) −9.67004 −0.347358
\(776\) 0 0
\(777\) −7.87598 −0.282549
\(778\) 0 0
\(779\) 2.29261 0.0821413
\(780\) 0 0
\(781\) −32.2480 −1.15393
\(782\) 0 0
\(783\) 1.70739 0.0610171
\(784\) 0 0
\(785\) −11.8688 −0.423614
\(786\) 0 0
\(787\) −20.8114 −0.741847 −0.370923 0.928663i \(-0.620959\pi\)
−0.370923 + 0.928663i \(0.620959\pi\)
\(788\) 0 0
\(789\) −22.3118 −0.794322
\(790\) 0 0
\(791\) −21.5460 −0.766088
\(792\) 0 0
\(793\) 29.9819 1.06469
\(794\) 0 0
\(795\) −10.4431 −0.370377
\(796\) 0 0
\(797\) 40.9354 1.45001 0.725004 0.688745i \(-0.241838\pi\)
0.725004 + 0.688745i \(0.241838\pi\)
\(798\) 0 0
\(799\) −18.8296 −0.666142
\(800\) 0 0
\(801\) −1.70739 −0.0603276
\(802\) 0 0
\(803\) 44.6983 1.57737
\(804\) 0 0
\(805\) 10.1312 0.357079
\(806\) 0 0
\(807\) −15.0656 −0.530335
\(808\) 0 0
\(809\) −1.48947 −0.0523670 −0.0261835 0.999657i \(-0.508335\pi\)
−0.0261835 + 0.999657i \(0.508335\pi\)
\(810\) 0 0
\(811\) −21.5460 −0.756583 −0.378292 0.925687i \(-0.623488\pi\)
−0.378292 + 0.925687i \(0.623488\pi\)
\(812\) 0 0
\(813\) 5.47133 0.191888
\(814\) 0 0
\(815\) 0.0373465 0.00130819
\(816\) 0 0
\(817\) 8.73566 0.305622
\(818\) 0 0
\(819\) 4.38650 0.153277
\(820\) 0 0
\(821\) −35.9819 −1.25578 −0.627888 0.778304i \(-0.716080\pi\)
−0.627888 + 0.778304i \(0.716080\pi\)
\(822\) 0 0
\(823\) −15.8880 −0.553819 −0.276910 0.960896i \(-0.589310\pi\)
−0.276910 + 0.960896i \(0.589310\pi\)
\(824\) 0 0
\(825\) 3.70739 0.129075
\(826\) 0 0
\(827\) −26.8789 −0.934671 −0.467335 0.884080i \(-0.654786\pi\)
−0.467335 + 0.884080i \(0.654786\pi\)
\(828\) 0 0
\(829\) 17.9253 0.622572 0.311286 0.950316i \(-0.399240\pi\)
0.311286 + 0.950316i \(0.399240\pi\)
\(830\) 0 0
\(831\) 10.7730 0.373712
\(832\) 0 0
\(833\) −8.47934 −0.293792
\(834\) 0 0
\(835\) 11.0283 0.381649
\(836\) 0 0
\(837\) 9.67004 0.334246
\(838\) 0 0
\(839\) −7.30168 −0.252082 −0.126041 0.992025i \(-0.540227\pi\)
−0.126041 + 0.992025i \(0.540227\pi\)
\(840\) 0 0
\(841\) −26.0848 −0.899477
\(842\) 0 0
\(843\) −17.8205 −0.613770
\(844\) 0 0
\(845\) −1.97173 −0.0678294
\(846\) 0 0
\(847\) 3.62548 0.124573
\(848\) 0 0
\(849\) −30.2070 −1.03670
\(850\) 0 0
\(851\) 45.7338 1.56773
\(852\) 0 0
\(853\) −37.4148 −1.28106 −0.640529 0.767934i \(-0.721285\pi\)
−0.640529 + 0.767934i \(0.721285\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −10.1806 −0.347762 −0.173881 0.984767i \(-0.555631\pi\)
−0.173881 + 0.984767i \(0.555631\pi\)
\(858\) 0 0
\(859\) 40.7367 1.38992 0.694959 0.719049i \(-0.255422\pi\)
0.694959 + 0.719049i \(0.255422\pi\)
\(860\) 0 0
\(861\) −3.02827 −0.103203
\(862\) 0 0
\(863\) 8.11310 0.276173 0.138086 0.990420i \(-0.455905\pi\)
0.138086 + 0.990420i \(0.455905\pi\)
\(864\) 0 0
\(865\) −17.8578 −0.607184
\(866\) 0 0
\(867\) 14.3966 0.488935
\(868\) 0 0
\(869\) −52.1131 −1.76782
\(870\) 0 0
\(871\) 44.1131 1.49472
\(872\) 0 0
\(873\) −0.679116 −0.0229846
\(874\) 0 0
\(875\) 1.32088 0.0446540
\(876\) 0 0
\(877\) 29.8880 1.00924 0.504622 0.863340i \(-0.331632\pi\)
0.504622 + 0.863340i \(0.331632\pi\)
\(878\) 0 0
\(879\) 18.2553 0.615735
\(880\) 0 0
\(881\) 34.8114 1.17283 0.586413 0.810012i \(-0.300540\pi\)
0.586413 + 0.810012i \(0.300540\pi\)
\(882\) 0 0
\(883\) −27.8496 −0.937212 −0.468606 0.883407i \(-0.655244\pi\)
−0.468606 + 0.883407i \(0.655244\pi\)
\(884\) 0 0
\(885\) 12.6983 0.426849
\(886\) 0 0
\(887\) 50.7933 1.70547 0.852735 0.522343i \(-0.174942\pi\)
0.852735 + 0.522343i \(0.174942\pi\)
\(888\) 0 0
\(889\) 5.28354 0.177204
\(890\) 0 0
\(891\) −3.70739 −0.124202
\(892\) 0 0
\(893\) −11.6700 −0.390523
\(894\) 0 0
\(895\) 19.9253 0.666030
\(896\) 0 0
\(897\) −25.4713 −0.850463
\(898\) 0 0
\(899\) 16.5105 0.550657
\(900\) 0 0
\(901\) 16.8498 0.561349
\(902\) 0 0
\(903\) −11.5388 −0.383987
\(904\) 0 0
\(905\) −15.4713 −0.514284
\(906\) 0 0
\(907\) 32.3009 1.07253 0.536267 0.844049i \(-0.319834\pi\)
0.536267 + 0.844049i \(0.319834\pi\)
\(908\) 0 0
\(909\) 12.6418 0.419301
\(910\) 0 0
\(911\) 52.7367 1.74725 0.873623 0.486604i \(-0.161764\pi\)
0.873623 + 0.486604i \(0.161764\pi\)
\(912\) 0 0
\(913\) 18.6418 0.616953
\(914\) 0 0
\(915\) −9.02827 −0.298466
\(916\) 0 0
\(917\) −5.67004 −0.187241
\(918\) 0 0
\(919\) −52.5105 −1.73216 −0.866081 0.499903i \(-0.833369\pi\)
−0.866081 + 0.499903i \(0.833369\pi\)
\(920\) 0 0
\(921\) −16.5852 −0.546502
\(922\) 0 0
\(923\) 28.8861 0.950798
\(924\) 0 0
\(925\) 5.96265 0.196051
\(926\) 0 0
\(927\) −6.05655 −0.198923
\(928\) 0 0
\(929\) −35.8688 −1.17682 −0.588408 0.808564i \(-0.700245\pi\)
−0.588408 + 0.808564i \(0.700245\pi\)
\(930\) 0 0
\(931\) −5.25526 −0.172234
\(932\) 0 0
\(933\) 11.5196 0.377135
\(934\) 0 0
\(935\) −5.98185 −0.195628
\(936\) 0 0
\(937\) 10.6983 0.349499 0.174749 0.984613i \(-0.444088\pi\)
0.174749 + 0.984613i \(0.444088\pi\)
\(938\) 0 0
\(939\) −32.7549 −1.06891
\(940\) 0 0
\(941\) −22.3673 −0.729153 −0.364577 0.931173i \(-0.618786\pi\)
−0.364577 + 0.931173i \(0.618786\pi\)
\(942\) 0 0
\(943\) 17.5844 0.572628
\(944\) 0 0
\(945\) −1.32088 −0.0429684
\(946\) 0 0
\(947\) −34.8223 −1.13157 −0.565787 0.824551i \(-0.691428\pi\)
−0.565787 + 0.824551i \(0.691428\pi\)
\(948\) 0 0
\(949\) −40.0384 −1.29970
\(950\) 0 0
\(951\) −15.2161 −0.493415
\(952\) 0 0
\(953\) −29.4823 −0.955024 −0.477512 0.878625i \(-0.658461\pi\)
−0.477512 + 0.878625i \(0.658461\pi\)
\(954\) 0 0
\(955\) −18.3492 −0.593765
\(956\) 0 0
\(957\) −6.32996 −0.204618
\(958\) 0 0
\(959\) 2.64177 0.0853072
\(960\) 0 0
\(961\) 62.5097 2.01644
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 12.0192 0.386912
\(966\) 0 0
\(967\) −18.8669 −0.606719 −0.303359 0.952876i \(-0.598108\pi\)
−0.303359 + 0.952876i \(0.598108\pi\)
\(968\) 0 0
\(969\) −1.61350 −0.0518329
\(970\) 0 0
\(971\) −31.2270 −1.00212 −0.501061 0.865412i \(-0.667057\pi\)
−0.501061 + 0.865412i \(0.667057\pi\)
\(972\) 0 0
\(973\) 11.3401 0.363546
\(974\) 0 0
\(975\) −3.32088 −0.106353
\(976\) 0 0
\(977\) 6.44305 0.206132 0.103066 0.994675i \(-0.467135\pi\)
0.103066 + 0.994675i \(0.467135\pi\)
\(978\) 0 0
\(979\) 6.32996 0.202306
\(980\) 0 0
\(981\) 10.6983 0.341571
\(982\) 0 0
\(983\) −4.57429 −0.145897 −0.0729486 0.997336i \(-0.523241\pi\)
−0.0729486 + 0.997336i \(0.523241\pi\)
\(984\) 0 0
\(985\) 4.84049 0.154231
\(986\) 0 0
\(987\) 15.4148 0.490658
\(988\) 0 0
\(989\) 67.0029 2.13057
\(990\) 0 0
\(991\) 38.8296 1.23346 0.616731 0.787174i \(-0.288457\pi\)
0.616731 + 0.787174i \(0.288457\pi\)
\(992\) 0 0
\(993\) −26.9536 −0.855346
\(994\) 0 0
\(995\) 20.6983 0.656181
\(996\) 0 0
\(997\) 54.9245 1.73948 0.869738 0.493513i \(-0.164288\pi\)
0.869738 + 0.493513i \(0.164288\pi\)
\(998\) 0 0
\(999\) −5.96265 −0.188650
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.br.1.3 3
4.3 odd 2 1140.2.a.g.1.1 3
12.11 even 2 3420.2.a.m.1.1 3
20.3 even 4 5700.2.f.q.3649.6 6
20.7 even 4 5700.2.f.q.3649.1 6
20.19 odd 2 5700.2.a.w.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.g.1.1 3 4.3 odd 2
3420.2.a.m.1.1 3 12.11 even 2
4560.2.a.br.1.3 3 1.1 even 1 trivial
5700.2.a.w.1.3 3 20.19 odd 2
5700.2.f.q.3649.1 6 20.7 even 4
5700.2.f.q.3649.6 6 20.3 even 4