Properties

Label 5850.2.e.be.5149.2
Level $5850$
Weight $2$
Character 5850.5149
Analytic conductor $46.712$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5850.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.7124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1950)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5149.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5850.5149
Dual form 5850.2.e.be.5149.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +5.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +5.00000i q^{17} +5.00000i q^{22} +1.00000 q^{26} -1.00000i q^{28} -7.00000 q^{29} -9.00000 q^{31} +1.00000i q^{32} -5.00000 q^{34} +8.00000i q^{37} +2.00000 q^{41} +8.00000i q^{43} -5.00000 q^{44} -9.00000i q^{47} +6.00000 q^{49} +1.00000i q^{52} -11.0000i q^{53} +1.00000 q^{56} -7.00000i q^{58} +1.00000 q^{59} -7.00000 q^{61} -9.00000i q^{62} -1.00000 q^{64} +15.0000i q^{67} -5.00000i q^{68} +8.00000 q^{71} +4.00000i q^{73} -8.00000 q^{74} +5.00000i q^{77} +4.00000 q^{79} +2.00000i q^{82} +9.00000i q^{83} -8.00000 q^{86} -5.00000i q^{88} +16.0000 q^{89} +1.00000 q^{91} +9.00000 q^{94} -2.00000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 10q^{11} - 2q^{14} + 2q^{16} + 2q^{26} - 14q^{29} - 18q^{31} - 10q^{34} + 4q^{41} - 10q^{44} + 12q^{49} + 2q^{56} + 2q^{59} - 14q^{61} - 2q^{64} + 16q^{71} - 16q^{74} + 8q^{79} - 16q^{86} + 32q^{89} + 2q^{91} + 18q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times\).

\(n\) \(2251\) \(3251\) \(3277\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.00000i 1.21268i 0.795206 + 0.606339i \(0.207363\pi\)
−0.795206 + 0.606339i \(0.792637\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.00000i 1.06600i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) − 1.00000i − 0.188982i
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 0 0
\(47\) − 9.00000i − 1.31278i −0.754420 0.656392i \(-0.772082\pi\)
0.754420 0.656392i \(-0.227918\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) − 11.0000i − 1.51097i −0.655168 0.755483i \(-0.727402\pi\)
0.655168 0.755483i \(-0.272598\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) − 7.00000i − 0.919145i
\(59\) 1.00000 0.130189 0.0650945 0.997879i \(-0.479265\pi\)
0.0650945 + 0.997879i \(0.479265\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) − 9.00000i − 1.14300i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 15.0000i 1.83254i 0.400559 + 0.916271i \(0.368816\pi\)
−0.400559 + 0.916271i \(0.631184\pi\)
\(68\) − 5.00000i − 0.606339i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) 5.00000i 0.569803i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.00000i 0.220863i
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) − 5.00000i − 0.533002i
\(89\) 16.0000 1.69600 0.847998 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 9.00000 0.928279
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 6.00000i 0.606092i
\(99\) 0 0
\(100\) 0 0
\(101\) 7.00000 0.696526 0.348263 0.937397i \(-0.386772\pi\)
0.348263 + 0.937397i \(0.386772\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i 0.955312 + 0.295599i \(0.0955191\pi\)
−0.955312 + 0.295599i \(0.904481\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 7.00000 0.649934
\(117\) 0 0
\(118\) 1.00000i 0.0920575i
\(119\) −5.00000 −0.458349
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) − 7.00000i − 0.633750i
\(123\) 0 0
\(124\) 9.00000 0.808224
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00000 0.174741 0.0873704 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −15.0000 −1.29580
\(135\) 0 0
\(136\) 5.00000 0.428746
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) − 5.00000i − 0.418121i
\(144\) 0 0
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) − 8.00000i − 0.657596i
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −21.0000 −1.70896 −0.854478 0.519488i \(-0.826123\pi\)
−0.854478 + 0.519488i \(0.826123\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −5.00000 −0.402911
\(155\) 0 0
\(156\) 0 0
\(157\) 5.00000i 0.399043i 0.979893 + 0.199522i \(0.0639388\pi\)
−0.979893 + 0.199522i \(0.936061\pi\)
\(158\) 4.00000i 0.318223i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) − 4.00000i − 0.309529i −0.987951 0.154765i \(-0.950538\pi\)
0.987951 0.154765i \(-0.0494619\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) − 8.00000i − 0.609994i
\(173\) 11.0000i 0.836315i 0.908375 + 0.418157i \(0.137324\pi\)
−0.908375 + 0.418157i \(0.862676\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) 16.0000i 1.19925i
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) −11.0000 −0.817624 −0.408812 0.912619i \(-0.634057\pi\)
−0.408812 + 0.912619i \(0.634057\pi\)
\(182\) 1.00000i 0.0741249i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 25.0000i 1.82818i
\(188\) 9.00000i 0.656392i
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7.00000i 0.492518i
\(203\) − 7.00000i − 0.491304i
\(204\) 0 0
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) 0 0
\(208\) − 1.00000i − 0.0693375i
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 11.0000i 0.755483i
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) 0 0
\(217\) − 9.00000i − 0.610960i
\(218\) − 6.00000i − 0.406371i
\(219\) 0 0
\(220\) 0 0
\(221\) 5.00000 0.336336
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 29.0000i 1.92480i 0.271640 + 0.962399i \(0.412434\pi\)
−0.271640 + 0.962399i \(0.587566\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7.00000i 0.459573i
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.00000 −0.0650945
\(237\) 0 0
\(238\) − 5.00000i − 0.324102i
\(239\) 1.00000 0.0646846 0.0323423 0.999477i \(-0.489703\pi\)
0.0323423 + 0.999477i \(0.489703\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 14.0000i 0.899954i
\(243\) 0 0
\(244\) 7.00000 0.448129
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 9.00000i 0.571501i
\(249\) 0 0
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.0000i 1.68421i 0.539311 + 0.842107i \(0.318685\pi\)
−0.539311 + 0.842107i \(0.681315\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) 2.00000i 0.123560i
\(263\) − 26.0000i − 1.60323i −0.597841 0.801614i \(-0.703975\pi\)
0.597841 0.801614i \(-0.296025\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) − 15.0000i − 0.916271i
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) 5.00000i 0.303170i
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) − 26.0000i − 1.56219i −0.624413 0.781094i \(-0.714662\pi\)
0.624413 0.781094i \(-0.285338\pi\)
\(278\) 2.00000i 0.119952i
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) − 22.0000i − 1.30776i −0.756596 0.653882i \(-0.773139\pi\)
0.756596 0.653882i \(-0.226861\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) 2.00000i 0.118056i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) − 4.00000i − 0.234082i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) − 22.0000i − 1.27443i
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) − 21.0000i − 1.20841i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) − 5.00000i − 0.284901i
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 21.0000i 1.18699i 0.804838 + 0.593495i \(0.202252\pi\)
−0.804838 + 0.593495i \(0.797748\pi\)
\(314\) −5.00000 −0.282166
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −35.0000 −1.95962
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) − 2.00000i − 0.110432i
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) − 9.00000i − 0.493939i
\(333\) 0 0
\(334\) 4.00000 0.218870
\(335\) 0 0
\(336\) 0 0
\(337\) 13.0000i 0.708155i 0.935216 + 0.354078i \(0.115205\pi\)
−0.935216 + 0.354078i \(0.884795\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) −45.0000 −2.43689
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −11.0000 −0.591364
\(347\) − 6.00000i − 0.322097i −0.986947 0.161048i \(-0.948512\pi\)
0.986947 0.161048i \(-0.0514875\pi\)
\(348\) 0 0
\(349\) −32.0000 −1.71292 −0.856460 0.516213i \(-0.827341\pi\)
−0.856460 + 0.516213i \(0.827341\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000i 0.266501i
\(353\) 2.00000i 0.106449i 0.998583 + 0.0532246i \(0.0169499\pi\)
−0.998583 + 0.0532246i \(0.983050\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −16.0000 −0.847998
\(357\) 0 0
\(358\) 2.00000i 0.105703i
\(359\) 27.0000 1.42501 0.712503 0.701669i \(-0.247562\pi\)
0.712503 + 0.701669i \(0.247562\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 11.0000i − 0.578147i
\(363\) 0 0
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) 0 0
\(367\) 36.0000i 1.87918i 0.342296 + 0.939592i \(0.388796\pi\)
−0.342296 + 0.939592i \(0.611204\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.0000 0.571092
\(372\) 0 0
\(373\) − 17.0000i − 0.880227i −0.897942 0.440113i \(-0.854938\pi\)
0.897942 0.440113i \(-0.145062\pi\)
\(374\) −25.0000 −1.29272
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) 7.00000i 0.360518i
\(378\) 0 0
\(379\) 15.0000 0.770498 0.385249 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 18.0000i − 0.920960i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 2.00000i 0.101535i
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 6.00000i − 0.303046i
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) − 18.0000i − 0.902258i
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 9.00000i 0.448322i
\(404\) −7.00000 −0.348263
\(405\) 0 0
\(406\) 7.00000 0.347404
\(407\) 40.0000i 1.98273i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 6.00000i − 0.295599i
\(413\) 1.00000i 0.0492068i
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) 0 0
\(419\) −2.00000 −0.0977064 −0.0488532 0.998806i \(-0.515557\pi\)
−0.0488532 + 0.998806i \(0.515557\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 16.0000i 0.778868i
\(423\) 0 0
\(424\) −11.0000 −0.534207
\(425\) 0 0
\(426\) 0 0
\(427\) − 7.00000i − 0.338754i
\(428\) − 6.00000i − 0.290021i
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 9.00000 0.432014
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 0 0
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 5.00000i 0.237826i
\(443\) 32.0000i 1.52037i 0.649709 + 0.760183i \(0.274891\pi\)
−0.649709 + 0.760183i \(0.725109\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) 10.0000 0.470882
\(452\) 2.00000i 0.0940721i
\(453\) 0 0
\(454\) −29.0000 −1.36104
\(455\) 0 0
\(456\) 0 0
\(457\) − 2.00000i − 0.0935561i −0.998905 0.0467780i \(-0.985105\pi\)
0.998905 0.0467780i \(-0.0148953\pi\)
\(458\) 2.00000i 0.0934539i
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) − 11.0000i − 0.511213i −0.966781 0.255607i \(-0.917725\pi\)
0.966781 0.255607i \(-0.0822752\pi\)
\(464\) −7.00000 −0.324967
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) − 32.0000i − 1.48078i −0.672176 0.740392i \(-0.734640\pi\)
0.672176 0.740392i \(-0.265360\pi\)
\(468\) 0 0
\(469\) −15.0000 −0.692636
\(470\) 0 0
\(471\) 0 0
\(472\) − 1.00000i − 0.0460287i
\(473\) 40.0000i 1.83920i
\(474\) 0 0
\(475\) 0 0
\(476\) 5.00000 0.229175
\(477\) 0 0
\(478\) 1.00000i 0.0457389i
\(479\) 29.0000 1.32504 0.662522 0.749043i \(-0.269486\pi\)
0.662522 + 0.749043i \(0.269486\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) − 22.0000i − 1.00207i
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) 1.00000i 0.0453143i 0.999743 + 0.0226572i \(0.00721262\pi\)
−0.999743 + 0.0226572i \(0.992787\pi\)
\(488\) 7.00000i 0.316875i
\(489\) 0 0
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) − 35.0000i − 1.57632i
\(494\) 0 0
\(495\) 0 0
\(496\) −9.00000 −0.404112
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 6.00000i 0.267793i
\(503\) 2.00000i 0.0891756i 0.999005 + 0.0445878i \(0.0141974\pi\)
−0.999005 + 0.0445878i \(0.985803\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) − 8.00000i − 0.354943i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −27.0000 −1.19092
\(515\) 0 0
\(516\) 0 0
\(517\) − 45.0000i − 1.97910i
\(518\) − 8.00000i − 0.351500i
\(519\) 0 0
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) 0 0
\(523\) − 18.0000i − 0.787085i −0.919306 0.393543i \(-0.871249\pi\)
0.919306 0.393543i \(-0.128751\pi\)
\(524\) −2.00000 −0.0873704
\(525\) 0 0
\(526\) 26.0000 1.13365
\(527\) − 45.0000i − 1.96023i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 2.00000i − 0.0866296i
\(534\) 0 0
\(535\) 0 0
\(536\) 15.0000 0.647901
\(537\) 0 0
\(538\) 9.00000i 0.388018i
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) 28.0000 1.20381 0.601907 0.798566i \(-0.294408\pi\)
0.601907 + 0.798566i \(0.294408\pi\)
\(542\) 7.00000i 0.300676i
\(543\) 0 0
\(544\) −5.00000 −0.214373
\(545\) 0 0
\(546\) 0 0
\(547\) 2.00000i 0.0855138i 0.999086 + 0.0427569i \(0.0136141\pi\)
−0.999086 + 0.0427569i \(0.986386\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000i 0.170097i
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) − 30.0000i − 1.26547i
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 22.0000 0.924729
\(567\) 0 0
\(568\) − 8.00000i − 0.335673i
\(569\) −9.00000 −0.377300 −0.188650 0.982044i \(-0.560411\pi\)
−0.188650 + 0.982044i \(0.560411\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 5.00000i 0.209061i
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) 0 0
\(577\) 42.0000i 1.74848i 0.485491 + 0.874241i \(0.338641\pi\)
−0.485491 + 0.874241i \(0.661359\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 0 0
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) 0 0
\(583\) − 55.0000i − 2.27787i
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) − 15.0000i − 0.619116i −0.950881 0.309558i \(-0.899819\pi\)
0.950881 0.309558i \(-0.100181\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000i 0.328798i
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 0 0
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) −35.0000 −1.42768 −0.713840 0.700309i \(-0.753046\pi\)
−0.713840 + 0.700309i \(0.753046\pi\)
\(602\) − 8.00000i − 0.326056i
\(603\) 0 0
\(604\) 21.0000 0.854478
\(605\) 0 0
\(606\) 0 0
\(607\) − 2.00000i − 0.0811775i −0.999176 0.0405887i \(-0.987077\pi\)
0.999176 0.0405887i \(-0.0129233\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 −0.364101
\(612\) 0 0
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 18.0000i 0.721734i
\(623\) 16.0000i 0.641026i
\(624\) 0 0
\(625\) 0 0
\(626\) −21.0000 −0.839329
\(627\) 0 0
\(628\) − 5.00000i − 0.199522i
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) −36.0000 −1.43314 −0.716569 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) − 4.00000i − 0.159111i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 6.00000i − 0.237729i
\(638\) − 35.0000i − 1.38566i
\(639\) 0 0
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) − 28.0000i − 1.10421i −0.833774 0.552106i \(-0.813824\pi\)
0.833774 0.552106i \(-0.186176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 48.0000i − 1.88707i −0.331266 0.943537i \(-0.607476\pi\)
0.331266 0.943537i \(-0.392524\pi\)
\(648\) 0 0
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) 27.0000i 1.05659i 0.849060 + 0.528296i \(0.177169\pi\)
−0.849060 + 0.528296i \(0.822831\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 9.00000i 0.350857i
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) − 20.0000i − 0.777322i
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 4.00000i 0.154765i
\(669\) 0 0
\(670\) 0 0
\(671\) −35.0000 −1.35116
\(672\) 0 0
\(673\) 11.0000i 0.424019i 0.977268 + 0.212009i \(0.0680008\pi\)
−0.977268 + 0.212009i \(0.931999\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) − 45.0000i − 1.72314i
\(683\) − 19.0000i − 0.727015i −0.931591 0.363507i \(-0.881579\pi\)
0.931591 0.363507i \(-0.118421\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) 8.00000i 0.304997i
\(689\) −11.0000 −0.419067
\(690\) 0 0
\(691\) 7.00000 0.266293 0.133146 0.991096i \(-0.457492\pi\)
0.133146 + 0.991096i \(0.457492\pi\)
\(692\) − 11.0000i − 0.418157i
\(693\) 0 0
\(694\) 6.00000 0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) 10.0000i 0.378777i
\(698\) − 32.0000i − 1.21122i
\(699\) 0 0
\(700\) 0 0
\(701\) 25.0000 0.944237 0.472118 0.881535i \(-0.343489\pi\)
0.472118 + 0.881535i \(0.343489\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 7.00000i 0.263262i
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 16.0000i − 0.599625i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 0 0
\(718\) 27.0000i 1.00763i
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) − 19.0000i − 0.707107i
\(723\) 0 0
\(724\) 11.0000 0.408812
\(725\) 0 0
\(726\) 0 0
\(727\) 38.0000i 1.40934i 0.709534 + 0.704671i \(0.248905\pi\)
−0.709534 + 0.704671i \(0.751095\pi\)
\(728\) − 1.00000i − 0.0370625i
\(729\) 0 0
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 0 0
\(733\) 20.0000i 0.738717i 0.929287 + 0.369358i \(0.120423\pi\)
−0.929287 + 0.369358i \(0.879577\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) 0 0
\(737\) 75.0000i 2.76266i
\(738\) 0 0
\(739\) −3.00000 −0.110357 −0.0551784 0.998477i \(-0.517573\pi\)
−0.0551784 + 0.998477i \(0.517573\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 11.0000i 0.403823i
\(743\) − 39.0000i − 1.43077i −0.698730 0.715386i \(-0.746251\pi\)
0.698730 0.715386i \(-0.253749\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 17.0000 0.622414
\(747\) 0 0
\(748\) − 25.0000i − 0.914091i
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) − 9.00000i − 0.328196i
\(753\) 0 0
\(754\) −7.00000 −0.254925
\(755\) 0 0
\(756\) 0 0
\(757\) − 33.0000i − 1.19941i −0.800223 0.599703i \(-0.795286\pi\)
0.800223 0.599703i \(-0.204714\pi\)
\(758\) 15.0000i 0.544825i
\(759\) 0 0
\(760\) 0 0
\(761\) 8.00000 0.290000 0.145000 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) 0 0
\(763\) − 6.00000i − 0.217215i
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 0 0
\(767\) − 1.00000i − 0.0361079i
\(768\) 0 0
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000i 0.359908i
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 2.00000i 0.0717035i
\(779\) 0 0
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) 0 0
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) 5.00000i 0.178231i 0.996021 + 0.0891154i \(0.0284040\pi\)
−0.996021 + 0.0891154i \(0.971596\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 0 0
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 7.00000i 0.248577i
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) − 13.0000i − 0.460484i −0.973133 0.230242i \(-0.926048\pi\)
0.973133 0.230242i \(-0.0739517\pi\)
\(798\) 0 0
\(799\) 45.0000 1.59199
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.0000i 0.705785i
\(804\) 0 0
\(805\) 0 0
\(806\) −9.00000 −0.317011
\(807\) 0 0
\(808\) − 7.00000i − 0.246259i
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) −29.0000 −1.01833 −0.509164 0.860670i \(-0.670045\pi\)
−0.509164 + 0.860670i \(0.670045\pi\)
\(812\) 7.00000i 0.245652i
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 14.0000i 0.489499i
\(819\) 0 0
\(820\) 0 0
\(821\) 16.0000 0.558404 0.279202 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(822\) 0 0
\(823\) − 28.0000i − 0.976019i −0.872838 0.488009i \(-0.837723\pi\)
0.872838 0.488009i \(-0.162277\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −1.00000 −0.0347945
\(827\) − 21.0000i − 0.730242i −0.930960 0.365121i \(-0.881028\pi\)
0.930960 0.365121i \(-0.118972\pi\)
\(828\) 0 0
\(829\) 37.0000 1.28506 0.642532 0.766259i \(-0.277884\pi\)
0.642532 + 0.766259i \(0.277884\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000i 0.0346688i
\(833\) 30.0000i 1.03944i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 2.00000i − 0.0690889i
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 28.0000i 0.964944i
\(843\) 0 0
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) − 11.0000i − 0.377742i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 8.00000i − 0.273915i −0.990577 0.136957i \(-0.956268\pi\)
0.990577 0.136957i \(-0.0437323\pi\)
\(854\) 7.00000 0.239535
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) − 26.0000i − 0.888143i −0.895991 0.444072i \(-0.853534\pi\)
0.895991 0.444072i \(-0.146466\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 12.0000i − 0.408722i
\(863\) − 27.0000i − 0.919091i −0.888154 0.459545i \(-0.848012\pi\)
0.888154 0.459545i \(-0.151988\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) 9.00000i 0.305480i
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 15.0000 0.508256
\(872\) 6.00000i 0.203186i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 40.0000i − 1.35070i −0.737496 0.675352i \(-0.763992\pi\)
0.737496 0.675352i \(-0.236008\pi\)
\(878\) − 4.00000i − 0.134993i
\(879\) 0 0
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) 32.0000i 1.07689i 0.842662 + 0.538443i \(0.180987\pi\)
−0.842662 + 0.538443i \(0.819013\pi\)
\(884\) −5.00000 −0.168168
\(885\) 0 0
\(886\) −32.0000 −1.07506
\(887\) 4.00000i 0.134307i 0.997743 + 0.0671534i \(0.0213917\pi\)
−0.997743 + 0.0671534i \(0.978608\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) − 16.0000i − 0.535720i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 36.0000i 1.20134i
\(899\) 63.0000 2.10117
\(900\) 0 0
\(901\) 55.0000 1.83232
\(902\) 10.0000i 0.332964i
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) 0 0
\(906\) 0 0
\(907\) − 40.0000i − 1.32818i −0.747653 0.664089i \(-0.768820\pi\)
0.747653 0.664089i \(-0.231180\pi\)
\(908\) − 29.0000i − 0.962399i
\(909\) 0 0
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 45.0000i 1.48928i
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 2.00000i 0.0660458i
\(918\) 0 0
\(919\) 50.0000 1.64935 0.824674 0.565608i \(-0.191359\pi\)
0.824674 + 0.565608i \(0.191359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 6.00000i 0.197599i
\(923\) − 8.00000i − 0.263323i
\(924\) 0 0
\(925\) 0 0
\(926\) 11.0000 0.361482
\(927\) 0 0
\(928\) − 7.00000i − 0.229786i
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000i 0.196537i
\(933\) 0 0
\(934\) 32.0000 1.04707
\(935\) 0 0
\(936\) 0 0
\(937\) − 51.0000i − 1.66610i −0.553200 0.833049i \(-0.686593\pi\)
0.553200 0.833049i \(-0.313407\pi\)
\(938\) − 15.0000i − 0.489767i
\(939\) 0 0
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.00000 0.0325472
\(945\) 0 0
\(946\) −40.0000 −1.30051
\(947\) − 41.0000i − 1.33232i −0.745808 0.666160i \(-0.767937\pi\)
0.745808 0.666160i \(-0.232063\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) 5.00000i 0.162051i
\(953\) − 51.0000i − 1.65205i −0.563632 0.826026i \(-0.690596\pi\)
0.563632 0.826026i \(-0.309404\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.00000 −0.0323423
\(957\) 0 0
\(958\) 29.0000i 0.936947i
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 8.00000i 0.257930i
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) 17.0000i 0.546683i 0.961917 + 0.273342i \(0.0881289\pi\)
−0.961917 + 0.273342i \(0.911871\pi\)
\(968\) − 14.0000i − 0.449977i
\(969\) 0 0
\(970\) 0 0
\(971\) 34.0000 1.09111 0.545556 0.838074i \(-0.316319\pi\)
0.545556 + 0.838074i \(0.316319\pi\)
\(972\) 0 0
\(973\) 2.00000i 0.0641171i
\(974\) −1.00000 −0.0320421
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) − 12.0000i − 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) 0 0
\(979\) 80.0000 2.55681
\(980\) 0 0
\(981\) 0 0
\(982\) − 30.0000i − 0.957338i
\(983\) 31.0000i 0.988746i 0.869250 + 0.494373i \(0.164602\pi\)
−0.869250 + 0.494373i \(0.835398\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 35.0000 1.11463
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 36.0000 1.14358 0.571789 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) − 9.00000i − 0.285750i
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) 11.0000i 0.348373i 0.984713 + 0.174187i \(0.0557296\pi\)
−0.984713 + 0.174187i \(0.944270\pi\)
\(998\) − 11.0000i − 0.348199i
\(999\) 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 5850.2.e.be.5149.2 2
3.2 odd 2 1950.2.e.a.1249.1 2
5.2 odd 4 5850.2.a.k.1.1 1
5.3 odd 4 5850.2.a.bu.1.1 1
5.4 even 2 inner 5850.2.e.be.5149.1 2
15.2 even 4 1950.2.a.p.1.1 yes 1
15.8 even 4 1950.2.a.l.1.1 1
15.14 odd 2 1950.2.e.a.1249.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.l.1.1 1 15.8 even 4
1950.2.a.p.1.1 yes 1 15.2 even 4
1950.2.e.a.1249.1 2 3.2 odd 2
1950.2.e.a.1249.2 2 15.14 odd 2
5850.2.a.k.1.1 1 5.2 odd 4
5850.2.a.bu.1.1 1 5.3 odd 4
5850.2.e.be.5149.1 2 5.4 even 2 inner
5850.2.e.be.5149.2 2 1.1 even 1 trivial