Properties

Label 7623.2.a.d.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} -1.00000 q^{7} -6.00000 q^{10} +2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} -3.00000 q^{17} -4.00000 q^{19} +6.00000 q^{20} -2.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} -2.00000 q^{28} -8.00000 q^{29} +2.00000 q^{31} +8.00000 q^{32} +6.00000 q^{34} -3.00000 q^{35} +10.0000 q^{37} +8.00000 q^{38} -2.00000 q^{41} +9.00000 q^{43} +4.00000 q^{46} -9.00000 q^{47} +1.00000 q^{49} -8.00000 q^{50} +4.00000 q^{52} +8.00000 q^{53} +16.0000 q^{58} +3.00000 q^{59} -10.0000 q^{61} -4.00000 q^{62} -8.00000 q^{64} +6.00000 q^{65} +13.0000 q^{67} -6.00000 q^{68} +6.00000 q^{70} -14.0000 q^{71} -10.0000 q^{73} -20.0000 q^{74} -8.00000 q^{76} -4.00000 q^{79} -12.0000 q^{80} +4.00000 q^{82} -1.00000 q^{83} -9.00000 q^{85} -18.0000 q^{86} +9.00000 q^{89} -2.00000 q^{91} -4.00000 q^{92} +18.0000 q^{94} -12.0000 q^{95} -16.0000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) −6.00000 −1.89737
\(11\) 0 0
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −8.00000 −1.13137
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 16.0000 2.10090
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 6.00000 0.717137
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −20.0000 −2.32495
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −12.0000 −1.34164
\(81\) 0 0
\(82\) 4.00000 0.441726
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) −18.0000 −1.94099
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 18.0000 1.85656
\(95\) −12.0000 −1.23117
\(96\) 0 0
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) −2.00000 −0.202031
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) −17.0000 −1.69156 −0.845782 0.533529i \(-0.820865\pi\)
−0.845782 + 0.533529i \(0.820865\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −16.0000 −1.55406
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) −16.0000 −1.48556
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 0 0
\(122\) 20.0000 1.81071
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) −13.0000 −1.13582 −0.567908 0.823092i \(-0.692247\pi\)
−0.567908 + 0.823092i \(0.692247\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −26.0000 −2.24606
\(135\) 0 0
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) 28.0000 2.34971
\(143\) 0 0
\(144\) 0 0
\(145\) −24.0000 −1.99309
\(146\) 20.0000 1.65521
\(147\) 0 0
\(148\) 20.0000 1.64399
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 24.0000 1.89737
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 18.0000 1.38054
\(171\) 0 0
\(172\) 18.0000 1.37249
\(173\) −25.0000 −1.90071 −0.950357 0.311160i \(-0.899282\pi\)
−0.950357 + 0.311160i \(0.899282\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) 30.0000 2.20564
\(186\) 0 0
\(187\) 0 0
\(188\) −18.0000 −1.31278
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) −23.0000 −1.65558 −0.827788 0.561041i \(-0.810401\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 32.0000 2.29747
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 34.0000 2.39223
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −28.0000 −1.95085
\(207\) 0 0
\(208\) −8.00000 −0.554700
\(209\) 0 0
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) 16.0000 1.09888
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) 27.0000 1.84138
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) −26.0000 −1.76094
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −8.00000 −0.534522
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) −7.00000 −0.464606 −0.232303 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) −27.0000 −1.76129
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) −6.00000 −0.388922
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 28.0000 1.80364 0.901819 0.432113i \(-0.142232\pi\)
0.901819 + 0.432113i \(0.142232\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −20.0000 −1.28037
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) 0 0
\(250\) 6.00000 0.379473
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −26.0000 −1.63139
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 12.0000 0.744208
\(261\) 0 0
\(262\) 26.0000 1.60629
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) 26.0000 1.58820
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) 0 0
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −28.0000 −1.66149
\(285\) 0 0
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 48.0000 2.81866
\(291\) 0 0
\(292\) −20.0000 −1.17041
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −9.00000 −0.518751
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 16.0000 0.917663
\(305\) −30.0000 −1.71780
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −12.0000 −0.681554
\(311\) 25.0000 1.41762 0.708810 0.705399i \(-0.249232\pi\)
0.708810 + 0.705399i \(0.249232\pi\)
\(312\) 0 0
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 16.0000 0.902932
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −24.0000 −1.34164
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 8.00000 0.443760
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 3.00000 0.164895 0.0824475 0.996595i \(-0.473726\pi\)
0.0824475 + 0.996595i \(0.473726\pi\)
\(332\) −2.00000 −0.109764
\(333\) 0 0
\(334\) −6.00000 −0.328305
\(335\) 39.0000 2.13080
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 18.0000 0.979071
\(339\) 0 0
\(340\) −18.0000 −0.976187
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 50.0000 2.68802
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) −42.0000 −2.22913
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 40.0000 2.11407
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 32.0000 1.68188
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) −30.0000 −1.57027
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) −60.0000 −3.11925
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) 21.0000 1.08734 0.543669 0.839299i \(-0.317035\pi\)
0.543669 + 0.839299i \(0.317035\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.0000 −0.824042
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) −24.0000 −1.23117
\(381\) 0 0
\(382\) −40.0000 −2.04658
\(383\) −19.0000 −0.970855 −0.485427 0.874277i \(-0.661336\pi\)
−0.485427 + 0.874277i \(0.661336\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 46.0000 2.34134
\(387\) 0 0
\(388\) −32.0000 −1.62455
\(389\) −4.00000 −0.202808 −0.101404 0.994845i \(-0.532333\pi\)
−0.101404 + 0.994845i \(0.532333\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) 0 0
\(394\) 44.0000 2.21669
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 28.0000 1.40351
\(399\) 0 0
\(400\) −16.0000 −0.800000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −34.0000 −1.69156
\(405\) 0 0
\(406\) −16.0000 −0.794067
\(407\) 0 0
\(408\) 0 0
\(409\) 28.0000 1.38451 0.692255 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) 12.0000 0.592638
\(411\) 0 0
\(412\) 28.0000 1.37946
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) −3.00000 −0.147264
\(416\) 16.0000 0.784465
\(417\) 0 0
\(418\) 0 0
\(419\) −19.0000 −0.928211 −0.464105 0.885780i \(-0.653624\pi\)
−0.464105 + 0.885780i \(0.653624\pi\)
\(420\) 0 0
\(421\) 33.0000 1.60832 0.804161 0.594412i \(-0.202615\pi\)
0.804161 + 0.594412i \(0.202615\pi\)
\(422\) −14.0000 −0.681509
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 20.0000 0.966736
\(429\) 0 0
\(430\) −54.0000 −2.60411
\(431\) 4.00000 0.192673 0.0963366 0.995349i \(-0.469287\pi\)
0.0963366 + 0.995349i \(0.469287\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 26.0000 1.24517
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0000 0.570782
\(443\) 38.0000 1.80543 0.902717 0.430234i \(-0.141569\pi\)
0.902717 + 0.430234i \(0.141569\pi\)
\(444\) 0 0
\(445\) 27.0000 1.27992
\(446\) 48.0000 2.27287
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −12.0000 −0.564433
\(453\) 0 0
\(454\) 14.0000 0.657053
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) 8.00000 0.373815
\(459\) 0 0
\(460\) −12.0000 −0.559503
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 32.0000 1.48556
\(465\) 0 0
\(466\) −8.00000 −0.370593
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 54.0000 2.49083
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −29.0000 −1.32504 −0.662522 0.749043i \(-0.730514\pi\)
−0.662522 + 0.749043i \(0.730514\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) −56.0000 −2.55073
\(483\) 0 0
\(484\) 0 0
\(485\) −48.0000 −2.17957
\(486\) 0 0
\(487\) 37.0000 1.67663 0.838315 0.545186i \(-0.183541\pi\)
0.838315 + 0.545186i \(0.183541\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −6.00000 −0.271052
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 14.0000 0.627986
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) −6.00000 −0.268328
\(501\) 0 0
\(502\) 24.0000 1.07117
\(503\) −11.0000 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(504\) 0 0
\(505\) −51.0000 −2.26947
\(506\) 0 0
\(507\) 0 0
\(508\) 26.0000 1.15356
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) 42.0000 1.85074
\(516\) 0 0
\(517\) 0 0
\(518\) 20.0000 0.878750
\(519\) 0 0
\(520\) 0 0
\(521\) −1.00000 −0.0438108 −0.0219054 0.999760i \(-0.506973\pi\)
−0.0219054 + 0.999760i \(0.506973\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) −26.0000 −1.13582
\(525\) 0 0
\(526\) −36.0000 −1.56967
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −48.0000 −2.08499
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) 30.0000 1.29701
\(536\) 0 0
\(537\) 0 0
\(538\) −28.0000 −1.20717
\(539\) 0 0
\(540\) 0 0
\(541\) −33.0000 −1.41878 −0.709390 0.704816i \(-0.751030\pi\)
−0.709390 + 0.704816i \(0.751030\pi\)
\(542\) −40.0000 −1.71815
\(543\) 0 0
\(544\) −24.0000 −1.02899
\(545\) 39.0000 1.67058
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −16.0000 −0.683486
\(549\) 0 0
\(550\) 0 0
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 34.0000 1.44452
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 18.0000 0.761319
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) 24.0000 1.01238
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) 0 0
\(569\) 40.0000 1.67689 0.838444 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 16.0000 0.665512
\(579\) 0 0
\(580\) −48.0000 −1.99309
\(581\) 1.00000 0.0414870
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −39.0000 −1.60970 −0.804851 0.593477i \(-0.797755\pi\)
−0.804851 + 0.593477i \(0.797755\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) −18.0000 −0.741048
\(591\) 0 0
\(592\) −40.0000 −1.64399
\(593\) −11.0000 −0.451716 −0.225858 0.974160i \(-0.572519\pi\)
−0.225858 + 0.974160i \(0.572519\pi\)
\(594\) 0 0
\(595\) 9.00000 0.368964
\(596\) 24.0000 0.983078
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) −34.0000 −1.38920 −0.694601 0.719395i \(-0.744419\pi\)
−0.694601 + 0.719395i \(0.744419\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 18.0000 0.733625
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −32.0000 −1.29777
\(609\) 0 0
\(610\) 60.0000 2.42933
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) −29.0000 −1.17130 −0.585649 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) −24.0000 −0.968561
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −18.0000 −0.723481 −0.361741 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) 12.0000 0.481932
\(621\) 0 0
\(622\) −50.0000 −2.00482
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 48.0000 1.91847
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) −30.0000 −1.19618
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) 39.0000 1.54767
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −3.00000 −0.117942 −0.0589711 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −16.0000 −0.627572
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) −39.0000 −1.52386
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) −18.0000 −0.701713
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) −6.00000 −0.233197
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 6.00000 0.232147
\(669\) 0 0
\(670\) −78.0000 −3.01340
\(671\) 0 0
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −4.00000 −0.154074
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) −9.00000 −0.345898 −0.172949 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.0000 1.30097 0.650487 0.759517i \(-0.274565\pi\)
0.650487 + 0.759517i \(0.274565\pi\)
\(684\) 0 0
\(685\) −24.0000 −0.916993
\(686\) 2.00000 0.0763604
\(687\) 0 0
\(688\) −36.0000 −1.37249
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −50.0000 −1.90071
\(693\) 0 0
\(694\) 32.0000 1.21470
\(695\) −6.00000 −0.227593
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) −12.0000 −0.454207
\(699\) 0 0
\(700\) −8.00000 −0.302372
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 0 0
\(703\) −40.0000 −1.50863
\(704\) 0 0
\(705\) 0 0
\(706\) 60.0000 2.25813
\(707\) 17.0000 0.639351
\(708\) 0 0
\(709\) −9.00000 −0.338002 −0.169001 0.985616i \(-0.554054\pi\)
−0.169001 + 0.985616i \(0.554054\pi\)
\(710\) 84.0000 3.15246
\(711\) 0 0
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 0 0
\(716\) −40.0000 −1.49487
\(717\) 0 0
\(718\) −36.0000 −1.34351
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 6.00000 0.223297
\(723\) 0 0
\(724\) −32.0000 −1.18927
\(725\) −32.0000 −1.18845
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 60.0000 2.22070
\(731\) −27.0000 −0.998631
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −16.0000 −0.589768
\(737\) 0 0
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 60.0000 2.20564
\(741\) 0 0
\(742\) 16.0000 0.587378
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 36.0000 1.31894
\(746\) −42.0000 −1.53773
\(747\) 0 0
\(748\) 0 0
\(749\) −10.0000 −0.365392
\(750\) 0 0
\(751\) −37.0000 −1.35015 −0.675075 0.737749i \(-0.735889\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(752\) 36.0000 1.31278
\(753\) 0 0
\(754\) 32.0000 1.16537
\(755\) −15.0000 −0.545906
\(756\) 0 0
\(757\) −9.00000 −0.327111 −0.163555 0.986534i \(-0.552296\pi\)
−0.163555 + 0.986534i \(0.552296\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) 0 0
\(761\) −45.0000 −1.63125 −0.815624 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) 0 0
\(763\) −13.0000 −0.470632
\(764\) 40.0000 1.44715
\(765\) 0 0
\(766\) 38.0000 1.37300
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) 24.0000 0.865462 0.432731 0.901523i \(-0.357550\pi\)
0.432731 + 0.901523i \(0.357550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −46.0000 −1.65558
\(773\) −15.0000 −0.539513 −0.269756 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) −12.0000 −0.429119
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) −44.0000 −1.56744
\(789\) 0 0
\(790\) 24.0000 0.853882
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 36.0000 1.27759
\(795\) 0 0
\(796\) −28.0000 −0.992434
\(797\) −11.0000 −0.389640 −0.194820 0.980839i \(-0.562412\pi\)
−0.194820 + 0.980839i \(0.562412\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 32.0000 1.13137
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 6.00000 0.211472
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 52.0000 1.82597 0.912983 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 16.0000 0.561490
\(813\) 0 0
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) −56.0000 −1.95799
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) −38.0000 −1.32139 −0.660695 0.750655i \(-0.729738\pi\)
−0.660695 + 0.750655i \(0.729738\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 6.00000 0.208263
\(831\) 0 0
\(832\) −16.0000 −0.554700
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 9.00000 0.311458
\(836\) 0 0
\(837\) 0 0
\(838\) 38.0000 1.31269
\(839\) −51.0000 −1.76072 −0.880358 0.474310i \(-0.842698\pi\)
−0.880358 + 0.474310i \(0.842698\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −66.0000 −2.27451
\(843\) 0 0
\(844\) 14.0000 0.481900
\(845\) −27.0000 −0.928828
\(846\) 0 0
\(847\) 0 0
\(848\) −32.0000 −1.09888
\(849\) 0 0
\(850\) 24.0000 0.823193
\(851\) −20.0000 −0.685591
\(852\) 0 0
\(853\) −32.0000 −1.09566 −0.547830 0.836590i \(-0.684546\pi\)
−0.547830 + 0.836590i \(0.684546\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) 0 0
\(857\) 31.0000 1.05894 0.529470 0.848329i \(-0.322391\pi\)
0.529470 + 0.848329i \(0.322391\pi\)
\(858\) 0 0
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) 54.0000 1.84138
\(861\) 0 0
\(862\) −8.00000 −0.272481
\(863\) 26.0000 0.885050 0.442525 0.896756i \(-0.354083\pi\)
0.442525 + 0.896756i \(0.354083\pi\)
\(864\) 0 0
\(865\) −75.0000 −2.55008
\(866\) −8.00000 −0.271851
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) 26.0000 0.880976
\(872\) 0 0
\(873\) 0 0
\(874\) −16.0000 −0.541208
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −52.0000 −1.75491
\(879\) 0 0
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) −21.0000 −0.706706 −0.353353 0.935490i \(-0.614959\pi\)
−0.353353 + 0.935490i \(0.614959\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −76.0000 −2.55327
\(887\) 55.0000 1.84672 0.923360 0.383936i \(-0.125432\pi\)
0.923360 + 0.383936i \(0.125432\pi\)
\(888\) 0 0
\(889\) −13.0000 −0.436006
\(890\) −54.0000 −1.81008
\(891\) 0 0
\(892\) −48.0000 −1.60716
\(893\) 36.0000 1.20469
\(894\) 0 0
\(895\) −60.0000 −2.00558
\(896\) 0 0
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −48.0000 −1.59557
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −14.0000 −0.464606
\(909\) 0 0
\(910\) 12.0000 0.397796
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) −8.00000 −0.264327
\(917\) 13.0000 0.429298
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) −28.0000 −0.921631
\(924\) 0 0
\(925\) 40.0000 1.31519
\(926\) −32.0000 −1.05159
\(927\) 0 0
\(928\) −64.0000 −2.10090
\(929\) −3.00000 −0.0984268 −0.0492134 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 8.00000 0.262049
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 26.0000 0.848930
\(939\) 0 0
\(940\) −54.0000 −1.76129
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 30.0000 0.974869 0.487435 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 32.0000 1.03822
\(951\) 0 0
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) 60.0000 1.94155
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 58.0000 1.87389
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −40.0000 −1.28965
\(963\) 0 0
\(964\) 56.0000 1.80364
\(965\) −69.0000 −2.22119
\(966\) 0 0
\(967\) 1.00000 0.0321578 0.0160789 0.999871i \(-0.494882\pi\)
0.0160789 + 0.999871i \(0.494882\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 96.0000 3.08237
\(971\) −37.0000 −1.18739 −0.593693 0.804691i \(-0.702331\pi\)
−0.593693 + 0.804691i \(0.702331\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) −74.0000 −2.37111
\(975\) 0 0
\(976\) 40.0000 1.28037
\(977\) 14.0000 0.447900 0.223950 0.974601i \(-0.428105\pi\)
0.223950 + 0.974601i \(0.428105\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 84.0000 2.68055
\(983\) 8.00000 0.255160 0.127580 0.991828i \(-0.459279\pi\)
0.127580 + 0.991828i \(0.459279\pi\)
\(984\) 0 0
\(985\) −66.0000 −2.10293
\(986\) −48.0000 −1.52863
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) −18.0000 −0.572367
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 16.0000 0.508001
\(993\) 0 0
\(994\) −28.0000 −0.888106
\(995\) −42.0000 −1.33149
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 10.0000 0.316544
\(999\) 0 0
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 7623.2.a.d.1.1 1
3.2 odd 2 2541.2.a.k.1.1 yes 1
11.10 odd 2 7623.2.a.s.1.1 1
33.32 even 2 2541.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.a.1.1 1 33.32 even 2
2541.2.a.k.1.1 yes 1 3.2 odd 2
7623.2.a.d.1.1 1 1.1 even 1 trivial
7623.2.a.s.1.1 1 11.10 odd 2