Properties

Label 2640.2.a.x.1.1
Level $2640$
Weight $2$
Character 2640.1
Self dual yes
Analytic conductor $21.081$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.0805061336\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2640.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.46410 q^{13} +1.00000 q^{15} +1.46410 q^{19} +2.00000 q^{21} +6.92820 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.46410 q^{29} -2.92820 q^{31} -1.00000 q^{33} +2.00000 q^{35} +8.92820 q^{37} +1.46410 q^{39} -3.46410 q^{41} -8.92820 q^{43} -1.00000 q^{45} -6.92820 q^{47} -3.00000 q^{49} -12.9282 q^{53} -1.00000 q^{55} -1.46410 q^{57} -6.92820 q^{59} +2.00000 q^{61} -2.00000 q^{63} +1.46410 q^{65} -8.00000 q^{67} -6.92820 q^{69} +13.8564 q^{71} +12.3923 q^{73} -1.00000 q^{75} -2.00000 q^{77} +13.4641 q^{79} +1.00000 q^{81} -15.4641 q^{83} -3.46410 q^{87} -12.9282 q^{89} +2.92820 q^{91} +2.92820 q^{93} -1.46410 q^{95} -10.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{5} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{5} - 4q^{7} + 2q^{9} + 2q^{11} + 4q^{13} + 2q^{15} - 4q^{19} + 4q^{21} + 2q^{25} - 2q^{27} + 8q^{31} - 2q^{33} + 4q^{35} + 4q^{37} - 4q^{39} - 4q^{43} - 2q^{45} - 6q^{49} - 12q^{53} - 2q^{55} + 4q^{57} + 4q^{61} - 4q^{63} - 4q^{65} - 16q^{67} + 4q^{73} - 2q^{75} - 4q^{77} + 20q^{79} + 2q^{81} - 24q^{83} - 12q^{89} - 8q^{91} - 8q^{93} + 4q^{95} - 20q^{97} + 2q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.46410 −0.406069 −0.203034 0.979172i \(-0.565080\pi\)
−0.203034 + 0.979172i \(0.565080\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.46410 0.335888 0.167944 0.985797i \(-0.446287\pi\)
0.167944 + 0.985797i \(0.446287\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.46410 0.643268 0.321634 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) −2.92820 −0.525921 −0.262960 0.964807i \(-0.584699\pi\)
−0.262960 + 0.964807i \(0.584699\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.92820 1.46779 0.733894 0.679264i \(-0.237701\pi\)
0.733894 + 0.679264i \(0.237701\pi\)
\(38\) 0 0
\(39\) 1.46410 0.234444
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) −8.92820 −1.36154 −0.680769 0.732498i \(-0.738354\pi\)
−0.680769 + 0.732498i \(0.738354\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −6.92820 −1.01058 −0.505291 0.862949i \(-0.668615\pi\)
−0.505291 + 0.862949i \(0.668615\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.9282 −1.77583 −0.887913 0.460012i \(-0.847845\pi\)
−0.887913 + 0.460012i \(0.847845\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −1.46410 −0.193925
\(58\) 0 0
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 1.46410 0.181599
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −6.92820 −0.834058
\(70\) 0 0
\(71\) 13.8564 1.64445 0.822226 0.569160i \(-0.192732\pi\)
0.822226 + 0.569160i \(0.192732\pi\)
\(72\) 0 0
\(73\) 12.3923 1.45041 0.725205 0.688533i \(-0.241745\pi\)
0.725205 + 0.688533i \(0.241745\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 13.4641 1.51483 0.757415 0.652934i \(-0.226462\pi\)
0.757415 + 0.652934i \(0.226462\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.4641 −1.69741 −0.848703 0.528870i \(-0.822616\pi\)
−0.848703 + 0.528870i \(0.822616\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.46410 −0.371391
\(88\) 0 0
\(89\) −12.9282 −1.37039 −0.685193 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) 2.92820 0.306959
\(92\) 0 0
\(93\) 2.92820 0.303641
\(94\) 0 0
\(95\) −1.46410 −0.150214
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) −15.4641 −1.49497 −0.747486 0.664278i \(-0.768739\pi\)
−0.747486 + 0.664278i \(0.768739\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −8.92820 −0.847428
\(112\) 0 0
\(113\) 0.928203 0.0873180 0.0436590 0.999046i \(-0.486098\pi\)
0.0436590 + 0.999046i \(0.486098\pi\)
\(114\) 0 0
\(115\) −6.92820 −0.646058
\(116\) 0 0
\(117\) −1.46410 −0.135356
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 3.46410 0.312348
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.92820 0.437307 0.218654 0.975803i \(-0.429834\pi\)
0.218654 + 0.975803i \(0.429834\pi\)
\(128\) 0 0
\(129\) 8.92820 0.786084
\(130\) 0 0
\(131\) −5.07180 −0.443125 −0.221562 0.975146i \(-0.571116\pi\)
−0.221562 + 0.975146i \(0.571116\pi\)
\(132\) 0 0
\(133\) −2.92820 −0.253907
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 8.39230 0.711826 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(140\) 0 0
\(141\) 6.92820 0.583460
\(142\) 0 0
\(143\) −1.46410 −0.122434
\(144\) 0 0
\(145\) −3.46410 −0.287678
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −8.53590 −0.699288 −0.349644 0.936883i \(-0.613697\pi\)
−0.349644 + 0.936883i \(0.613697\pi\)
\(150\) 0 0
\(151\) −0.392305 −0.0319253 −0.0159627 0.999873i \(-0.505081\pi\)
−0.0159627 + 0.999873i \(0.505081\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.92820 0.235199
\(156\) 0 0
\(157\) −16.9282 −1.35102 −0.675509 0.737352i \(-0.736076\pi\)
−0.675509 + 0.737352i \(0.736076\pi\)
\(158\) 0 0
\(159\) 12.9282 1.02527
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) 0 0
\(163\) 17.8564 1.39862 0.699311 0.714818i \(-0.253490\pi\)
0.699311 + 0.714818i \(0.253490\pi\)
\(164\) 0 0
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) −10.3923 −0.804181 −0.402090 0.915600i \(-0.631716\pi\)
−0.402090 + 0.915600i \(0.631716\pi\)
\(168\) 0 0
\(169\) −10.8564 −0.835108
\(170\) 0 0
\(171\) 1.46410 0.111963
\(172\) 0 0
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) 6.92820 0.520756
\(178\) 0 0
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) −11.8564 −0.881280 −0.440640 0.897684i \(-0.645248\pi\)
−0.440640 + 0.897684i \(0.645248\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −8.92820 −0.656415
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 5.07180 0.366982 0.183491 0.983021i \(-0.441260\pi\)
0.183491 + 0.983021i \(0.441260\pi\)
\(192\) 0 0
\(193\) 3.60770 0.259688 0.129844 0.991534i \(-0.458552\pi\)
0.129844 + 0.991534i \(0.458552\pi\)
\(194\) 0 0
\(195\) −1.46410 −0.104846
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −16.7846 −1.18983 −0.594915 0.803789i \(-0.702814\pi\)
−0.594915 + 0.803789i \(0.702814\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) −6.92820 −0.486265
\(204\) 0 0
\(205\) 3.46410 0.241943
\(206\) 0 0
\(207\) 6.92820 0.481543
\(208\) 0 0
\(209\) 1.46410 0.101274
\(210\) 0 0
\(211\) −12.3923 −0.853121 −0.426561 0.904459i \(-0.640275\pi\)
−0.426561 + 0.904459i \(0.640275\pi\)
\(212\) 0 0
\(213\) −13.8564 −0.949425
\(214\) 0 0
\(215\) 8.92820 0.608898
\(216\) 0 0
\(217\) 5.85641 0.397559
\(218\) 0 0
\(219\) −12.3923 −0.837394
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.8564 1.19575 0.597877 0.801588i \(-0.296011\pi\)
0.597877 + 0.801588i \(0.296011\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −8.53590 −0.566547 −0.283274 0.959039i \(-0.591420\pi\)
−0.283274 + 0.959039i \(0.591420\pi\)
\(228\) 0 0
\(229\) 3.85641 0.254839 0.127419 0.991849i \(-0.459331\pi\)
0.127419 + 0.991849i \(0.459331\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) 0 0
\(237\) −13.4641 −0.874587
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 27.8564 1.79439 0.897194 0.441636i \(-0.145602\pi\)
0.897194 + 0.441636i \(0.145602\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −2.14359 −0.136394
\(248\) 0 0
\(249\) 15.4641 0.979998
\(250\) 0 0
\(251\) −25.8564 −1.63204 −0.816021 0.578022i \(-0.803825\pi\)
−0.816021 + 0.578022i \(0.803825\pi\)
\(252\) 0 0
\(253\) 6.92820 0.435572
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.85641 0.490069 0.245035 0.969514i \(-0.421201\pi\)
0.245035 + 0.969514i \(0.421201\pi\)
\(258\) 0 0
\(259\) −17.8564 −1.10954
\(260\) 0 0
\(261\) 3.46410 0.214423
\(262\) 0 0
\(263\) −27.4641 −1.69351 −0.846755 0.531984i \(-0.821447\pi\)
−0.846755 + 0.531984i \(0.821447\pi\)
\(264\) 0 0
\(265\) 12.9282 0.794173
\(266\) 0 0
\(267\) 12.9282 0.791193
\(268\) 0 0
\(269\) −7.85641 −0.479014 −0.239507 0.970895i \(-0.576986\pi\)
−0.239507 + 0.970895i \(0.576986\pi\)
\(270\) 0 0
\(271\) 32.3923 1.96769 0.983846 0.179016i \(-0.0572913\pi\)
0.983846 + 0.179016i \(0.0572913\pi\)
\(272\) 0 0
\(273\) −2.92820 −0.177223
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 22.5359 1.35405 0.677025 0.735960i \(-0.263269\pi\)
0.677025 + 0.735960i \(0.263269\pi\)
\(278\) 0 0
\(279\) −2.92820 −0.175307
\(280\) 0 0
\(281\) −3.46410 −0.206651 −0.103325 0.994648i \(-0.532948\pi\)
−0.103325 + 0.994648i \(0.532948\pi\)
\(282\) 0 0
\(283\) −8.92820 −0.530727 −0.265363 0.964148i \(-0.585492\pi\)
−0.265363 + 0.964148i \(0.585492\pi\)
\(284\) 0 0
\(285\) 1.46410 0.0867259
\(286\) 0 0
\(287\) 6.92820 0.408959
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) 13.8564 0.809500 0.404750 0.914427i \(-0.367359\pi\)
0.404750 + 0.914427i \(0.367359\pi\)
\(294\) 0 0
\(295\) 6.92820 0.403376
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −10.1436 −0.586619
\(300\) 0 0
\(301\) 17.8564 1.02923
\(302\) 0 0
\(303\) −10.3923 −0.597022
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −18.9282 −1.07332 −0.536660 0.843799i \(-0.680314\pi\)
−0.536660 + 0.843799i \(0.680314\pi\)
\(312\) 0 0
\(313\) 7.07180 0.399722 0.199861 0.979824i \(-0.435951\pi\)
0.199861 + 0.979824i \(0.435951\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) −11.0718 −0.621854 −0.310927 0.950434i \(-0.600639\pi\)
−0.310927 + 0.950434i \(0.600639\pi\)
\(318\) 0 0
\(319\) 3.46410 0.193952
\(320\) 0 0
\(321\) 15.4641 0.863122
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.46410 −0.0812137
\(326\) 0 0
\(327\) 10.0000 0.553001
\(328\) 0 0
\(329\) 13.8564 0.763928
\(330\) 0 0
\(331\) 17.8564 0.981477 0.490738 0.871307i \(-0.336727\pi\)
0.490738 + 0.871307i \(0.336727\pi\)
\(332\) 0 0
\(333\) 8.92820 0.489263
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −29.1769 −1.58937 −0.794684 0.607023i \(-0.792363\pi\)
−0.794684 + 0.607023i \(0.792363\pi\)
\(338\) 0 0
\(339\) −0.928203 −0.0504131
\(340\) 0 0
\(341\) −2.92820 −0.158571
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 6.92820 0.373002
\(346\) 0 0
\(347\) −1.60770 −0.0863056 −0.0431528 0.999068i \(-0.513740\pi\)
−0.0431528 + 0.999068i \(0.513740\pi\)
\(348\) 0 0
\(349\) −35.8564 −1.91935 −0.959675 0.281113i \(-0.909296\pi\)
−0.959675 + 0.281113i \(0.909296\pi\)
\(350\) 0 0
\(351\) 1.46410 0.0781480
\(352\) 0 0
\(353\) −0.928203 −0.0494033 −0.0247016 0.999695i \(-0.507864\pi\)
−0.0247016 + 0.999695i \(0.507864\pi\)
\(354\) 0 0
\(355\) −13.8564 −0.735422
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.7846 −1.09697 −0.548485 0.836160i \(-0.684795\pi\)
−0.548485 + 0.836160i \(0.684795\pi\)
\(360\) 0 0
\(361\) −16.8564 −0.887179
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −12.3923 −0.648643
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) −3.46410 −0.180334
\(370\) 0 0
\(371\) 25.8564 1.34240
\(372\) 0 0
\(373\) 0.392305 0.0203128 0.0101564 0.999948i \(-0.496767\pi\)
0.0101564 + 0.999948i \(0.496767\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −5.07180 −0.261211
\(378\) 0 0
\(379\) −9.85641 −0.506290 −0.253145 0.967428i \(-0.581465\pi\)
−0.253145 + 0.967428i \(0.581465\pi\)
\(380\) 0 0
\(381\) −4.92820 −0.252479
\(382\) 0 0
\(383\) −13.8564 −0.708029 −0.354015 0.935240i \(-0.615184\pi\)
−0.354015 + 0.935240i \(0.615184\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) −8.92820 −0.453846
\(388\) 0 0
\(389\) 24.9282 1.26391 0.631955 0.775005i \(-0.282253\pi\)
0.631955 + 0.775005i \(0.282253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 5.07180 0.255838
\(394\) 0 0
\(395\) −13.4641 −0.677452
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 2.92820 0.146594
\(400\) 0 0
\(401\) 19.8564 0.991582 0.495791 0.868442i \(-0.334878\pi\)
0.495791 + 0.868442i \(0.334878\pi\)
\(402\) 0 0
\(403\) 4.28719 0.213560
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 8.92820 0.442555
\(408\) 0 0
\(409\) 34.7846 1.71999 0.859994 0.510304i \(-0.170467\pi\)
0.859994 + 0.510304i \(0.170467\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) 13.8564 0.681829
\(414\) 0 0
\(415\) 15.4641 0.759103
\(416\) 0 0
\(417\) −8.39230 −0.410973
\(418\) 0 0
\(419\) −17.0718 −0.834012 −0.417006 0.908904i \(-0.636921\pi\)
−0.417006 + 0.908904i \(0.636921\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) −6.92820 −0.336861
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) 1.46410 0.0706875
\(430\) 0 0
\(431\) 32.7846 1.57918 0.789590 0.613635i \(-0.210294\pi\)
0.789590 + 0.613635i \(0.210294\pi\)
\(432\) 0 0
\(433\) 27.8564 1.33869 0.669347 0.742950i \(-0.266574\pi\)
0.669347 + 0.742950i \(0.266574\pi\)
\(434\) 0 0
\(435\) 3.46410 0.166091
\(436\) 0 0
\(437\) 10.1436 0.485234
\(438\) 0 0
\(439\) 29.1769 1.39254 0.696269 0.717781i \(-0.254842\pi\)
0.696269 + 0.717781i \(0.254842\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 12.9282 0.612856
\(446\) 0 0
\(447\) 8.53590 0.403734
\(448\) 0 0
\(449\) 14.7846 0.697729 0.348864 0.937173i \(-0.386567\pi\)
0.348864 + 0.937173i \(0.386567\pi\)
\(450\) 0 0
\(451\) −3.46410 −0.163118
\(452\) 0 0
\(453\) 0.392305 0.0184321
\(454\) 0 0
\(455\) −2.92820 −0.137276
\(456\) 0 0
\(457\) −8.39230 −0.392575 −0.196288 0.980546i \(-0.562889\pi\)
−0.196288 + 0.980546i \(0.562889\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.2487 −0.570479 −0.285240 0.958456i \(-0.592073\pi\)
−0.285240 + 0.958456i \(0.592073\pi\)
\(462\) 0 0
\(463\) 28.0000 1.30127 0.650635 0.759390i \(-0.274503\pi\)
0.650635 + 0.759390i \(0.274503\pi\)
\(464\) 0 0
\(465\) −2.92820 −0.135792
\(466\) 0 0
\(467\) −18.9282 −0.875893 −0.437946 0.899001i \(-0.644294\pi\)
−0.437946 + 0.899001i \(0.644294\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 16.9282 0.780010
\(472\) 0 0
\(473\) −8.92820 −0.410519
\(474\) 0 0
\(475\) 1.46410 0.0671776
\(476\) 0 0
\(477\) −12.9282 −0.591942
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −13.0718 −0.596023
\(482\) 0 0
\(483\) 13.8564 0.630488
\(484\) 0 0
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) −23.7128 −1.07453 −0.537265 0.843413i \(-0.680543\pi\)
−0.537265 + 0.843413i \(0.680543\pi\)
\(488\) 0 0
\(489\) −17.8564 −0.807495
\(490\) 0 0
\(491\) 17.0718 0.770439 0.385220 0.922825i \(-0.374126\pi\)
0.385220 + 0.922825i \(0.374126\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) −27.7128 −1.24309
\(498\) 0 0
\(499\) 12.7846 0.572318 0.286159 0.958182i \(-0.407621\pi\)
0.286159 + 0.958182i \(0.407621\pi\)
\(500\) 0 0
\(501\) 10.3923 0.464294
\(502\) 0 0
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) 0 0
\(505\) −10.3923 −0.462451
\(506\) 0 0
\(507\) 10.8564 0.482150
\(508\) 0 0
\(509\) −7.85641 −0.348229 −0.174115 0.984725i \(-0.555706\pi\)
−0.174115 + 0.984725i \(0.555706\pi\)
\(510\) 0 0
\(511\) −24.7846 −1.09641
\(512\) 0 0
\(513\) −1.46410 −0.0646417
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −6.92820 −0.304702
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) −6.92820 −0.300658
\(532\) 0 0
\(533\) 5.07180 0.219684
\(534\) 0 0
\(535\) 15.4641 0.668571
\(536\) 0 0
\(537\) −6.92820 −0.298974
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 0.143594 0.00617357 0.00308678 0.999995i \(-0.499017\pi\)
0.00308678 + 0.999995i \(0.499017\pi\)
\(542\) 0 0
\(543\) 11.8564 0.508807
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 5.07180 0.216066
\(552\) 0 0
\(553\) −26.9282 −1.14510
\(554\) 0 0
\(555\) 8.92820 0.378981
\(556\) 0 0
\(557\) 44.7846 1.89758 0.948792 0.315900i \(-0.102306\pi\)
0.948792 + 0.315900i \(0.102306\pi\)
\(558\) 0 0
\(559\) 13.0718 0.552878
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.3923 0.437983 0.218992 0.975727i \(-0.429723\pi\)
0.218992 + 0.975727i \(0.429723\pi\)
\(564\) 0 0
\(565\) −0.928203 −0.0390498
\(566\) 0 0
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −29.3205 −1.22918 −0.614590 0.788847i \(-0.710678\pi\)
−0.614590 + 0.788847i \(0.710678\pi\)
\(570\) 0 0
\(571\) −24.3923 −1.02079 −0.510393 0.859941i \(-0.670500\pi\)
−0.510393 + 0.859941i \(0.670500\pi\)
\(572\) 0 0
\(573\) −5.07180 −0.211877
\(574\) 0 0
\(575\) 6.92820 0.288926
\(576\) 0 0
\(577\) 22.7846 0.948536 0.474268 0.880381i \(-0.342713\pi\)
0.474268 + 0.880381i \(0.342713\pi\)
\(578\) 0 0
\(579\) −3.60770 −0.149931
\(580\) 0 0
\(581\) 30.9282 1.28312
\(582\) 0 0
\(583\) −12.9282 −0.535431
\(584\) 0 0
\(585\) 1.46410 0.0605332
\(586\) 0 0
\(587\) 5.07180 0.209335 0.104668 0.994507i \(-0.466622\pi\)
0.104668 + 0.994507i \(0.466622\pi\)
\(588\) 0 0
\(589\) −4.28719 −0.176650
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) −32.7846 −1.34630 −0.673151 0.739505i \(-0.735060\pi\)
−0.673151 + 0.739505i \(0.735060\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.7846 0.686948
\(598\) 0 0
\(599\) 10.1436 0.414456 0.207228 0.978293i \(-0.433556\pi\)
0.207228 + 0.978293i \(0.433556\pi\)
\(600\) 0 0
\(601\) 36.6410 1.49462 0.747309 0.664477i \(-0.231345\pi\)
0.747309 + 0.664477i \(0.231345\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −22.7846 −0.924799 −0.462399 0.886672i \(-0.653011\pi\)
−0.462399 + 0.886672i \(0.653011\pi\)
\(608\) 0 0
\(609\) 6.92820 0.280745
\(610\) 0 0
\(611\) 10.1436 0.410366
\(612\) 0 0
\(613\) 0.392305 0.0158450 0.00792252 0.999969i \(-0.497478\pi\)
0.00792252 + 0.999969i \(0.497478\pi\)
\(614\) 0 0
\(615\) −3.46410 −0.139686
\(616\) 0 0
\(617\) −23.0718 −0.928836 −0.464418 0.885616i \(-0.653736\pi\)
−0.464418 + 0.885616i \(0.653736\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −6.92820 −0.278019
\(622\) 0 0
\(623\) 25.8564 1.03592
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.46410 −0.0584706
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 34.9282 1.39047 0.695235 0.718783i \(-0.255300\pi\)
0.695235 + 0.718783i \(0.255300\pi\)
\(632\) 0 0
\(633\) 12.3923 0.492550
\(634\) 0 0
\(635\) −4.92820 −0.195570
\(636\) 0 0
\(637\) 4.39230 0.174029
\(638\) 0 0
\(639\) 13.8564 0.548151
\(640\) 0 0
\(641\) 0.928203 0.0366618 0.0183309 0.999832i \(-0.494165\pi\)
0.0183309 + 0.999832i \(0.494165\pi\)
\(642\) 0 0
\(643\) 45.5692 1.79707 0.898537 0.438897i \(-0.144631\pi\)
0.898537 + 0.438897i \(0.144631\pi\)
\(644\) 0 0
\(645\) −8.92820 −0.351548
\(646\) 0 0
\(647\) 27.7128 1.08950 0.544752 0.838597i \(-0.316624\pi\)
0.544752 + 0.838597i \(0.316624\pi\)
\(648\) 0 0
\(649\) −6.92820 −0.271956
\(650\) 0 0
\(651\) −5.85641 −0.229531
\(652\) 0 0
\(653\) −7.85641 −0.307445 −0.153722 0.988114i \(-0.549126\pi\)
−0.153722 + 0.988114i \(0.549126\pi\)
\(654\) 0 0
\(655\) 5.07180 0.198171
\(656\) 0 0
\(657\) 12.3923 0.483470
\(658\) 0 0
\(659\) −39.7128 −1.54699 −0.773496 0.633801i \(-0.781494\pi\)
−0.773496 + 0.633801i \(0.781494\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.92820 0.113551
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) −17.8564 −0.690369
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 31.3205 1.20732 0.603658 0.797243i \(-0.293709\pi\)
0.603658 + 0.797243i \(0.293709\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −32.7846 −1.26001 −0.630007 0.776589i \(-0.716948\pi\)
−0.630007 + 0.776589i \(0.716948\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 8.53590 0.327096
\(682\) 0 0
\(683\) −8.78461 −0.336134 −0.168067 0.985776i \(-0.553752\pi\)
−0.168067 + 0.985776i \(0.553752\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −3.85641 −0.147131
\(688\) 0 0
\(689\) 18.9282 0.721107
\(690\) 0 0
\(691\) 7.71281 0.293409 0.146705 0.989180i \(-0.453133\pi\)
0.146705 + 0.989180i \(0.453133\pi\)
\(692\) 0 0
\(693\) −2.00000 −0.0759737
\(694\) 0 0
\(695\) −8.39230 −0.318338
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 32.5359 1.22886 0.614432 0.788970i \(-0.289385\pi\)
0.614432 + 0.788970i \(0.289385\pi\)
\(702\) 0 0
\(703\) 13.0718 0.493012
\(704\) 0 0
\(705\) −6.92820 −0.260931
\(706\) 0 0
\(707\) −20.7846 −0.781686
\(708\) 0 0
\(709\) 15.8564 0.595500 0.297750 0.954644i \(-0.403764\pi\)
0.297750 + 0.954644i \(0.403764\pi\)
\(710\) 0 0
\(711\) 13.4641 0.504943
\(712\) 0 0
\(713\) −20.2872 −0.759761
\(714\) 0 0
\(715\) 1.46410 0.0547543
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 18.9282 0.705903 0.352951 0.935642i \(-0.385178\pi\)
0.352951 + 0.935642i \(0.385178\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) −27.8564 −1.03599
\(724\) 0 0
\(725\) 3.46410 0.128654
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −49.9615 −1.84537 −0.922686 0.385553i \(-0.874011\pi\)
−0.922686 + 0.385553i \(0.874011\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −10.5359 −0.387569 −0.193785 0.981044i \(-0.562076\pi\)
−0.193785 + 0.981044i \(0.562076\pi\)
\(740\) 0 0
\(741\) 2.14359 0.0787469
\(742\) 0 0
\(743\) −46.3923 −1.70197 −0.850984 0.525191i \(-0.823994\pi\)
−0.850984 + 0.525191i \(0.823994\pi\)
\(744\) 0 0
\(745\) 8.53590 0.312731
\(746\) 0 0
\(747\) −15.4641 −0.565802
\(748\) 0 0
\(749\) 30.9282 1.13009
\(750\) 0 0
\(751\) −13.0718 −0.476997 −0.238498 0.971143i \(-0.576655\pi\)
−0.238498 + 0.971143i \(0.576655\pi\)
\(752\) 0 0
\(753\) 25.8564 0.942260
\(754\) 0 0
\(755\) 0.392305 0.0142774
\(756\) 0 0
\(757\) −6.78461 −0.246591 −0.123295 0.992370i \(-0.539346\pi\)
−0.123295 + 0.992370i \(0.539346\pi\)
\(758\) 0 0
\(759\) −6.92820 −0.251478
\(760\) 0 0
\(761\) −39.4641 −1.43057 −0.715286 0.698832i \(-0.753704\pi\)
−0.715286 + 0.698832i \(0.753704\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.1436 0.366264
\(768\) 0 0
\(769\) −46.4974 −1.67674 −0.838370 0.545102i \(-0.816491\pi\)
−0.838370 + 0.545102i \(0.816491\pi\)
\(770\) 0 0
\(771\) −7.85641 −0.282942
\(772\) 0 0
\(773\) −31.8564 −1.14580 −0.572898 0.819627i \(-0.694181\pi\)
−0.572898 + 0.819627i \(0.694181\pi\)
\(774\) 0 0
\(775\) −2.92820 −0.105184
\(776\) 0 0
\(777\) 17.8564 0.640595
\(778\) 0 0
\(779\) −5.07180 −0.181716
\(780\) 0 0
\(781\) 13.8564 0.495821
\(782\) 0 0
\(783\) −3.46410 −0.123797
\(784\) 0 0
\(785\) 16.9282 0.604193
\(786\) 0 0
\(787\) 18.7846 0.669599 0.334800 0.942289i \(-0.391331\pi\)
0.334800 + 0.942289i \(0.391331\pi\)
\(788\) 0 0
\(789\) 27.4641 0.977748
\(790\) 0 0
\(791\) −1.85641 −0.0660062
\(792\) 0 0
\(793\) −2.92820 −0.103984
\(794\) 0 0
\(795\) −12.9282 −0.458516
\(796\) 0 0
\(797\) 16.6410 0.589455 0.294728 0.955581i \(-0.404771\pi\)
0.294728 + 0.955581i \(0.404771\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −12.9282 −0.456796
\(802\) 0 0
\(803\) 12.3923 0.437315
\(804\) 0 0
\(805\) 13.8564 0.488374
\(806\) 0 0
\(807\) 7.85641 0.276559
\(808\) 0 0
\(809\) −8.53590 −0.300106 −0.150053 0.988678i \(-0.547944\pi\)
−0.150053 + 0.988678i \(0.547944\pi\)
\(810\) 0 0
\(811\) 8.39230 0.294694 0.147347 0.989085i \(-0.452927\pi\)
0.147347 + 0.989085i \(0.452927\pi\)
\(812\) 0 0
\(813\) −32.3923 −1.13605
\(814\) 0 0
\(815\) −17.8564 −0.625483
\(816\) 0 0
\(817\) −13.0718 −0.457324
\(818\) 0 0
\(819\) 2.92820 0.102320
\(820\) 0 0
\(821\) 27.4641 0.958504 0.479252 0.877677i \(-0.340908\pi\)
0.479252 + 0.877677i \(0.340908\pi\)
\(822\) 0 0
\(823\) −49.5692 −1.72787 −0.863937 0.503600i \(-0.832009\pi\)
−0.863937 + 0.503600i \(0.832009\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) −1.60770 −0.0559050 −0.0279525 0.999609i \(-0.508899\pi\)
−0.0279525 + 0.999609i \(0.508899\pi\)
\(828\) 0 0
\(829\) −25.7128 −0.893043 −0.446521 0.894773i \(-0.647337\pi\)
−0.446521 + 0.894773i \(0.647337\pi\)
\(830\) 0 0
\(831\) −22.5359 −0.781762
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 10.3923 0.359641
\(836\) 0 0
\(837\) 2.92820 0.101214
\(838\) 0 0
\(839\) −15.2154 −0.525294 −0.262647 0.964892i \(-0.584595\pi\)
−0.262647 + 0.964892i \(0.584595\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 3.46410 0.119310
\(844\) 0 0
\(845\) 10.8564 0.373472
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 8.92820 0.306415
\(850\) 0 0
\(851\) 61.8564 2.12041
\(852\) 0 0
\(853\) 24.3923 0.835177 0.417588 0.908636i \(-0.362875\pi\)
0.417588 + 0.908636i \(0.362875\pi\)
\(854\) 0 0
\(855\) −1.46410 −0.0500712
\(856\) 0 0
\(857\) −10.1436 −0.346499 −0.173249 0.984878i \(-0.555427\pi\)
−0.173249 + 0.984878i \(0.555427\pi\)
\(858\) 0 0
\(859\) −47.7128 −1.62794 −0.813970 0.580907i \(-0.802698\pi\)
−0.813970 + 0.580907i \(0.802698\pi\)
\(860\) 0 0
\(861\) −6.92820 −0.236113
\(862\) 0 0
\(863\) 10.1436 0.345292 0.172646 0.984984i \(-0.444768\pi\)
0.172646 + 0.984984i \(0.444768\pi\)
\(864\) 0 0
\(865\) 12.0000 0.408012
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 13.4641 0.456738
\(870\) 0 0
\(871\) 11.7128 0.396874
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 14.2487 0.481145 0.240572 0.970631i \(-0.422665\pi\)
0.240572 + 0.970631i \(0.422665\pi\)
\(878\) 0 0
\(879\) −13.8564 −0.467365
\(880\) 0 0
\(881\) 12.9282 0.435562 0.217781 0.975998i \(-0.430118\pi\)
0.217781 + 0.975998i \(0.430118\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) −6.92820 −0.232889
\(886\) 0 0
\(887\) 36.2487 1.21711 0.608556 0.793511i \(-0.291749\pi\)
0.608556 + 0.793511i \(0.291749\pi\)
\(888\) 0 0
\(889\) −9.85641 −0.330573
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −10.1436 −0.339442
\(894\) 0 0
\(895\) −6.92820 −0.231584
\(896\) 0 0
\(897\) 10.1436 0.338685
\(898\) 0 0
\(899\) −10.1436 −0.338308
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −17.8564 −0.594224
\(904\) 0 0
\(905\) 11.8564 0.394120
\(906\) 0 0
\(907\) −45.8564 −1.52264 −0.761318 0.648378i \(-0.775448\pi\)
−0.761318 + 0.648378i \(0.775448\pi\)
\(908\) 0 0
\(909\) 10.3923 0.344691
\(910\) 0 0
\(911\) −5.07180 −0.168036 −0.0840181 0.996464i \(-0.526775\pi\)
−0.0840181 + 0.996464i \(0.526775\pi\)
\(912\) 0 0
\(913\) −15.4641 −0.511787
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) 10.1436 0.334971
\(918\) 0 0
\(919\) 11.6077 0.382903 0.191451 0.981502i \(-0.438681\pi\)
0.191451 + 0.981502i \(0.438681\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) 0 0
\(923\) −20.2872 −0.667761
\(924\) 0 0
\(925\) 8.92820 0.293558
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −38.7846 −1.27248 −0.636241 0.771490i \(-0.719512\pi\)
−0.636241 + 0.771490i \(0.719512\pi\)
\(930\) 0 0
\(931\) −4.39230 −0.143952
\(932\) 0 0
\(933\) 18.9282 0.619682
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.392305 0.0128160 0.00640802 0.999979i \(-0.497960\pi\)
0.00640802 + 0.999979i \(0.497960\pi\)
\(938\) 0 0
\(939\) −7.07180 −0.230779
\(940\) 0 0
\(941\) 20.5359 0.669451 0.334726 0.942316i \(-0.391356\pi\)
0.334726 + 0.942316i \(0.391356\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) 5.07180 0.164811 0.0824056 0.996599i \(-0.473740\pi\)
0.0824056 + 0.996599i \(0.473740\pi\)
\(948\) 0 0
\(949\) −18.1436 −0.588966
\(950\) 0 0
\(951\) 11.0718 0.359028
\(952\) 0 0
\(953\) −44.7846 −1.45072 −0.725358 0.688372i \(-0.758326\pi\)
−0.725358 + 0.688372i \(0.758326\pi\)
\(954\) 0 0
\(955\) −5.07180 −0.164119
\(956\) 0 0
\(957\) −3.46410 −0.111979
\(958\) 0 0
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 0 0
\(963\) −15.4641 −0.498324
\(964\) 0 0
\(965\) −3.60770 −0.116136
\(966\) 0 0
\(967\) 18.7846 0.604072 0.302036 0.953296i \(-0.402334\pi\)
0.302036 + 0.953296i \(0.402334\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.85641 0.0595749 0.0297875 0.999556i \(-0.490517\pi\)
0.0297875 + 0.999556i \(0.490517\pi\)
\(972\) 0 0
\(973\) −16.7846 −0.538090
\(974\) 0 0
\(975\) 1.46410 0.0468888
\(976\) 0 0
\(977\) −35.5692 −1.13796 −0.568980 0.822351i \(-0.692662\pi\)
−0.568980 + 0.822351i \(0.692662\pi\)
\(978\) 0 0
\(979\) −12.9282 −0.413187
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) −13.8564 −0.441054
\(988\) 0 0
\(989\) −61.8564 −1.96692
\(990\) 0 0
\(991\) 48.7846 1.54969 0.774847 0.632149i \(-0.217827\pi\)
0.774847 + 0.632149i \(0.217827\pi\)
\(992\) 0 0
\(993\) −17.8564 −0.566656
\(994\) 0 0
\(995\) 16.7846 0.532108
\(996\) 0 0
\(997\) 48.3923 1.53260 0.766300 0.642483i \(-0.222096\pi\)
0.766300 + 0.642483i \(0.222096\pi\)
\(998\) 0 0
\(999\) −8.92820 −0.282476
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 2640.2.a.x.1.1 2
3.2 odd 2 7920.2.a.bz.1.1 2
4.3 odd 2 165.2.a.b.1.2 2
12.11 even 2 495.2.a.c.1.1 2
20.3 even 4 825.2.c.c.199.2 4
20.7 even 4 825.2.c.c.199.3 4
20.19 odd 2 825.2.a.e.1.1 2
28.27 even 2 8085.2.a.bd.1.2 2
44.43 even 2 1815.2.a.i.1.1 2
60.23 odd 4 2475.2.c.n.199.3 4
60.47 odd 4 2475.2.c.n.199.2 4
60.59 even 2 2475.2.a.r.1.2 2
132.131 odd 2 5445.2.a.s.1.2 2
220.219 even 2 9075.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.2 2 4.3 odd 2
495.2.a.c.1.1 2 12.11 even 2
825.2.a.e.1.1 2 20.19 odd 2
825.2.c.c.199.2 4 20.3 even 4
825.2.c.c.199.3 4 20.7 even 4
1815.2.a.i.1.1 2 44.43 even 2
2475.2.a.r.1.2 2 60.59 even 2
2475.2.c.n.199.2 4 60.47 odd 4
2475.2.c.n.199.3 4 60.23 odd 4
2640.2.a.x.1.1 2 1.1 even 1 trivial
5445.2.a.s.1.2 2 132.131 odd 2
7920.2.a.bz.1.1 2 3.2 odd 2
8085.2.a.bd.1.2 2 28.27 even 2
9075.2.a.bh.1.2 2 220.219 even 2