Properties

Label 1840.2.e.c.369.1
Level $1840$
Weight $2$
Character 1840.369
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 369.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1840.369
Dual form 1840.2.e.c.369.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803i q^{3} -2.23607 q^{5} -1.85410i q^{7} +0.381966 q^{9} +O(q^{10})\) \(q-1.61803i q^{3} -2.23607 q^{5} -1.85410i q^{7} +0.381966 q^{9} +5.61803 q^{11} +2.61803i q^{13} +3.61803i q^{15} -0.854102i q^{17} -0.145898 q^{19} -3.00000 q^{21} -1.00000i q^{23} +5.00000 q^{25} -5.47214i q^{27} +9.70820 q^{29} +2.14590 q^{31} -9.09017i q^{33} +4.14590i q^{35} +9.70820i q^{37} +4.23607 q^{39} -5.61803 q^{41} -11.2361i q^{43} -0.854102 q^{45} +1.70820i q^{47} +3.56231 q^{49} -1.38197 q^{51} -2.00000i q^{53} -12.5623 q^{55} +0.236068i q^{57} -6.00000 q^{59} +2.85410 q^{61} -0.708204i q^{63} -5.85410i q^{65} +5.23607i q^{67} -1.61803 q^{69} -0.381966 q^{71} -16.4721i q^{73} -8.09017i q^{75} -10.4164i q^{77} -7.70820 q^{79} -7.70820 q^{81} -7.70820i q^{83} +1.90983i q^{85} -15.7082i q^{87} -3.70820 q^{89} +4.85410 q^{91} -3.47214i q^{93} +0.326238 q^{95} -13.0344i q^{97} +2.14590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{9} + O(q^{10}) \) \( 4q + 6q^{9} + 18q^{11} - 14q^{19} - 12q^{21} + 20q^{25} + 12q^{29} + 22q^{31} + 8q^{39} - 18q^{41} + 10q^{45} - 26q^{49} - 10q^{51} - 10q^{55} - 24q^{59} - 2q^{61} - 2q^{69} - 6q^{71} - 4q^{79} - 4q^{81} + 12q^{89} + 6q^{91} - 30q^{95} + 22q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.61803i − 0.934172i −0.884212 0.467086i \(-0.845304\pi\)
0.884212 0.467086i \(-0.154696\pi\)
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) − 1.85410i − 0.700785i −0.936603 0.350392i \(-0.886048\pi\)
0.936603 0.350392i \(-0.113952\pi\)
\(8\) 0 0
\(9\) 0.381966 0.127322
\(10\) 0 0
\(11\) 5.61803 1.69390 0.846950 0.531672i \(-0.178436\pi\)
0.846950 + 0.531672i \(0.178436\pi\)
\(12\) 0 0
\(13\) 2.61803i 0.726112i 0.931767 + 0.363056i \(0.118267\pi\)
−0.931767 + 0.363056i \(0.881733\pi\)
\(14\) 0 0
\(15\) 3.61803i 0.934172i
\(16\) 0 0
\(17\) − 0.854102i − 0.207150i −0.994622 0.103575i \(-0.966972\pi\)
0.994622 0.103575i \(-0.0330282\pi\)
\(18\) 0 0
\(19\) −0.145898 −0.0334713 −0.0167357 0.999860i \(-0.505327\pi\)
−0.0167357 + 0.999860i \(0.505327\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 5.47214i − 1.05311i
\(28\) 0 0
\(29\) 9.70820 1.80277 0.901384 0.433020i \(-0.142552\pi\)
0.901384 + 0.433020i \(0.142552\pi\)
\(30\) 0 0
\(31\) 2.14590 0.385415 0.192707 0.981256i \(-0.438273\pi\)
0.192707 + 0.981256i \(0.438273\pi\)
\(32\) 0 0
\(33\) − 9.09017i − 1.58240i
\(34\) 0 0
\(35\) 4.14590i 0.700785i
\(36\) 0 0
\(37\) 9.70820i 1.59602i 0.602645 + 0.798009i \(0.294114\pi\)
−0.602645 + 0.798009i \(0.705886\pi\)
\(38\) 0 0
\(39\) 4.23607 0.678314
\(40\) 0 0
\(41\) −5.61803 −0.877390 −0.438695 0.898636i \(-0.644559\pi\)
−0.438695 + 0.898636i \(0.644559\pi\)
\(42\) 0 0
\(43\) − 11.2361i − 1.71348i −0.515745 0.856742i \(-0.672485\pi\)
0.515745 0.856742i \(-0.327515\pi\)
\(44\) 0 0
\(45\) −0.854102 −0.127322
\(46\) 0 0
\(47\) 1.70820i 0.249167i 0.992209 + 0.124584i \(0.0397595\pi\)
−0.992209 + 0.124584i \(0.960241\pi\)
\(48\) 0 0
\(49\) 3.56231 0.508901
\(50\) 0 0
\(51\) −1.38197 −0.193514
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) −12.5623 −1.69390
\(56\) 0 0
\(57\) 0.236068i 0.0312680i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 2.85410 0.365430 0.182715 0.983166i \(-0.441511\pi\)
0.182715 + 0.983166i \(0.441511\pi\)
\(62\) 0 0
\(63\) − 0.708204i − 0.0892253i
\(64\) 0 0
\(65\) − 5.85410i − 0.726112i
\(66\) 0 0
\(67\) 5.23607i 0.639688i 0.947470 + 0.319844i \(0.103630\pi\)
−0.947470 + 0.319844i \(0.896370\pi\)
\(68\) 0 0
\(69\) −1.61803 −0.194788
\(70\) 0 0
\(71\) −0.381966 −0.0453310 −0.0226655 0.999743i \(-0.507215\pi\)
−0.0226655 + 0.999743i \(0.507215\pi\)
\(72\) 0 0
\(73\) − 16.4721i − 1.92792i −0.266051 0.963959i \(-0.585719\pi\)
0.266051 0.963959i \(-0.414281\pi\)
\(74\) 0 0
\(75\) − 8.09017i − 0.934172i
\(76\) 0 0
\(77\) − 10.4164i − 1.18706i
\(78\) 0 0
\(79\) −7.70820 −0.867241 −0.433620 0.901096i \(-0.642764\pi\)
−0.433620 + 0.901096i \(0.642764\pi\)
\(80\) 0 0
\(81\) −7.70820 −0.856467
\(82\) 0 0
\(83\) − 7.70820i − 0.846085i −0.906110 0.423043i \(-0.860962\pi\)
0.906110 0.423043i \(-0.139038\pi\)
\(84\) 0 0
\(85\) 1.90983i 0.207150i
\(86\) 0 0
\(87\) − 15.7082i − 1.68410i
\(88\) 0 0
\(89\) −3.70820 −0.393069 −0.196534 0.980497i \(-0.562969\pi\)
−0.196534 + 0.980497i \(0.562969\pi\)
\(90\) 0 0
\(91\) 4.85410 0.508848
\(92\) 0 0
\(93\) − 3.47214i − 0.360044i
\(94\) 0 0
\(95\) 0.326238 0.0334713
\(96\) 0 0
\(97\) − 13.0344i − 1.32345i −0.749748 0.661724i \(-0.769825\pi\)
0.749748 0.661724i \(-0.230175\pi\)
\(98\) 0 0
\(99\) 2.14590 0.215671
\(100\) 0 0
\(101\) −1.52786 −0.152028 −0.0760141 0.997107i \(-0.524219\pi\)
−0.0760141 + 0.997107i \(0.524219\pi\)
\(102\) 0 0
\(103\) − 10.8541i − 1.06949i −0.845015 0.534743i \(-0.820408\pi\)
0.845015 0.534743i \(-0.179592\pi\)
\(104\) 0 0
\(105\) 6.70820 0.654654
\(106\) 0 0
\(107\) 11.7082i 1.13187i 0.824448 + 0.565937i \(0.191486\pi\)
−0.824448 + 0.565937i \(0.808514\pi\)
\(108\) 0 0
\(109\) −7.56231 −0.724338 −0.362169 0.932113i \(-0.617964\pi\)
−0.362169 + 0.932113i \(0.617964\pi\)
\(110\) 0 0
\(111\) 15.7082 1.49096
\(112\) 0 0
\(113\) − 13.4164i − 1.26211i −0.775738 0.631055i \(-0.782622\pi\)
0.775738 0.631055i \(-0.217378\pi\)
\(114\) 0 0
\(115\) 2.23607i 0.208514i
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) −1.58359 −0.145168
\(120\) 0 0
\(121\) 20.5623 1.86930
\(122\) 0 0
\(123\) 9.09017i 0.819633i
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) − 9.70820i − 0.861464i −0.902480 0.430732i \(-0.858255\pi\)
0.902480 0.430732i \(-0.141745\pi\)
\(128\) 0 0
\(129\) −18.1803 −1.60069
\(130\) 0 0
\(131\) 14.1803 1.23894 0.619471 0.785020i \(-0.287347\pi\)
0.619471 + 0.785020i \(0.287347\pi\)
\(132\) 0 0
\(133\) 0.270510i 0.0234562i
\(134\) 0 0
\(135\) 12.2361i 1.05311i
\(136\) 0 0
\(137\) 13.8541i 1.18364i 0.806072 + 0.591818i \(0.201590\pi\)
−0.806072 + 0.591818i \(0.798410\pi\)
\(138\) 0 0
\(139\) 4.29180 0.364025 0.182013 0.983296i \(-0.441739\pi\)
0.182013 + 0.983296i \(0.441739\pi\)
\(140\) 0 0
\(141\) 2.76393 0.232765
\(142\) 0 0
\(143\) 14.7082i 1.22996i
\(144\) 0 0
\(145\) −21.7082 −1.80277
\(146\) 0 0
\(147\) − 5.76393i − 0.475401i
\(148\) 0 0
\(149\) 2.61803 0.214478 0.107239 0.994233i \(-0.465799\pi\)
0.107239 + 0.994233i \(0.465799\pi\)
\(150\) 0 0
\(151\) −14.2705 −1.16132 −0.580659 0.814147i \(-0.697205\pi\)
−0.580659 + 0.814147i \(0.697205\pi\)
\(152\) 0 0
\(153\) − 0.326238i − 0.0263748i
\(154\) 0 0
\(155\) −4.79837 −0.385415
\(156\) 0 0
\(157\) − 2.29180i − 0.182905i −0.995809 0.0914526i \(-0.970849\pi\)
0.995809 0.0914526i \(-0.0291510\pi\)
\(158\) 0 0
\(159\) −3.23607 −0.256637
\(160\) 0 0
\(161\) −1.85410 −0.146124
\(162\) 0 0
\(163\) 22.0344i 1.72587i 0.505314 + 0.862935i \(0.331377\pi\)
−0.505314 + 0.862935i \(0.668623\pi\)
\(164\) 0 0
\(165\) 20.3262i 1.58240i
\(166\) 0 0
\(167\) − 9.70820i − 0.751243i −0.926773 0.375622i \(-0.877429\pi\)
0.926773 0.375622i \(-0.122571\pi\)
\(168\) 0 0
\(169\) 6.14590 0.472761
\(170\) 0 0
\(171\) −0.0557281 −0.00426163
\(172\) 0 0
\(173\) 16.5623i 1.25921i 0.776916 + 0.629604i \(0.216783\pi\)
−0.776916 + 0.629604i \(0.783217\pi\)
\(174\) 0 0
\(175\) − 9.27051i − 0.700785i
\(176\) 0 0
\(177\) 9.70820i 0.729713i
\(178\) 0 0
\(179\) 7.52786 0.562659 0.281329 0.959611i \(-0.409225\pi\)
0.281329 + 0.959611i \(0.409225\pi\)
\(180\) 0 0
\(181\) 15.5623 1.15674 0.578369 0.815776i \(-0.303690\pi\)
0.578369 + 0.815776i \(0.303690\pi\)
\(182\) 0 0
\(183\) − 4.61803i − 0.341375i
\(184\) 0 0
\(185\) − 21.7082i − 1.59602i
\(186\) 0 0
\(187\) − 4.79837i − 0.350892i
\(188\) 0 0
\(189\) −10.1459 −0.738005
\(190\) 0 0
\(191\) −25.4164 −1.83907 −0.919533 0.393012i \(-0.871433\pi\)
−0.919533 + 0.393012i \(0.871433\pi\)
\(192\) 0 0
\(193\) − 15.7082i − 1.13070i −0.824851 0.565351i \(-0.808741\pi\)
0.824851 0.565351i \(-0.191259\pi\)
\(194\) 0 0
\(195\) −9.47214 −0.678314
\(196\) 0 0
\(197\) 20.5623i 1.46500i 0.680765 + 0.732502i \(0.261647\pi\)
−0.680765 + 0.732502i \(0.738353\pi\)
\(198\) 0 0
\(199\) 11.4164 0.809288 0.404644 0.914474i \(-0.367395\pi\)
0.404644 + 0.914474i \(0.367395\pi\)
\(200\) 0 0
\(201\) 8.47214 0.597578
\(202\) 0 0
\(203\) − 18.0000i − 1.26335i
\(204\) 0 0
\(205\) 12.5623 0.877390
\(206\) 0 0
\(207\) − 0.381966i − 0.0265485i
\(208\) 0 0
\(209\) −0.819660 −0.0566971
\(210\) 0 0
\(211\) 7.70820 0.530655 0.265327 0.964158i \(-0.414520\pi\)
0.265327 + 0.964158i \(0.414520\pi\)
\(212\) 0 0
\(213\) 0.618034i 0.0423470i
\(214\) 0 0
\(215\) 25.1246i 1.71348i
\(216\) 0 0
\(217\) − 3.97871i − 0.270093i
\(218\) 0 0
\(219\) −26.6525 −1.80101
\(220\) 0 0
\(221\) 2.23607 0.150414
\(222\) 0 0
\(223\) − 22.3607i − 1.49738i −0.662919 0.748691i \(-0.730683\pi\)
0.662919 0.748691i \(-0.269317\pi\)
\(224\) 0 0
\(225\) 1.90983 0.127322
\(226\) 0 0
\(227\) 10.2918i 0.683090i 0.939865 + 0.341545i \(0.110950\pi\)
−0.939865 + 0.341545i \(0.889050\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −16.8541 −1.10892
\(232\) 0 0
\(233\) 21.1246i 1.38392i 0.721936 + 0.691960i \(0.243252\pi\)
−0.721936 + 0.691960i \(0.756748\pi\)
\(234\) 0 0
\(235\) − 3.81966i − 0.249167i
\(236\) 0 0
\(237\) 12.4721i 0.810152i
\(238\) 0 0
\(239\) −10.4721 −0.677386 −0.338693 0.940897i \(-0.609985\pi\)
−0.338693 + 0.940897i \(0.609985\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) − 3.94427i − 0.253025i
\(244\) 0 0
\(245\) −7.96556 −0.508901
\(246\) 0 0
\(247\) − 0.381966i − 0.0243039i
\(248\) 0 0
\(249\) −12.4721 −0.790390
\(250\) 0 0
\(251\) 16.7984 1.06030 0.530152 0.847903i \(-0.322135\pi\)
0.530152 + 0.847903i \(0.322135\pi\)
\(252\) 0 0
\(253\) − 5.61803i − 0.353203i
\(254\) 0 0
\(255\) 3.09017 0.193514
\(256\) 0 0
\(257\) 11.4164i 0.712136i 0.934460 + 0.356068i \(0.115883\pi\)
−0.934460 + 0.356068i \(0.884117\pi\)
\(258\) 0 0
\(259\) 18.0000 1.11847
\(260\) 0 0
\(261\) 3.70820 0.229532
\(262\) 0 0
\(263\) 18.2705i 1.12661i 0.826250 + 0.563304i \(0.190470\pi\)
−0.826250 + 0.563304i \(0.809530\pi\)
\(264\) 0 0
\(265\) 4.47214i 0.274721i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) 1.52786 0.0931555 0.0465778 0.998915i \(-0.485168\pi\)
0.0465778 + 0.998915i \(0.485168\pi\)
\(270\) 0 0
\(271\) −25.8541 −1.57052 −0.785262 0.619163i \(-0.787472\pi\)
−0.785262 + 0.619163i \(0.787472\pi\)
\(272\) 0 0
\(273\) − 7.85410i − 0.475352i
\(274\) 0 0
\(275\) 28.0902 1.69390
\(276\) 0 0
\(277\) − 7.52786i − 0.452306i −0.974092 0.226153i \(-0.927385\pi\)
0.974092 0.226153i \(-0.0726149\pi\)
\(278\) 0 0
\(279\) 0.819660 0.0490718
\(280\) 0 0
\(281\) 30.6525 1.82857 0.914287 0.405068i \(-0.132752\pi\)
0.914287 + 0.405068i \(0.132752\pi\)
\(282\) 0 0
\(283\) − 20.9443i − 1.24501i −0.782617 0.622504i \(-0.786115\pi\)
0.782617 0.622504i \(-0.213885\pi\)
\(284\) 0 0
\(285\) − 0.527864i − 0.0312680i
\(286\) 0 0
\(287\) 10.4164i 0.614861i
\(288\) 0 0
\(289\) 16.2705 0.957089
\(290\) 0 0
\(291\) −21.0902 −1.23633
\(292\) 0 0
\(293\) − 14.2918i − 0.834936i −0.908692 0.417468i \(-0.862918\pi\)
0.908692 0.417468i \(-0.137082\pi\)
\(294\) 0 0
\(295\) 13.4164 0.781133
\(296\) 0 0
\(297\) − 30.7426i − 1.78387i
\(298\) 0 0
\(299\) 2.61803 0.151405
\(300\) 0 0
\(301\) −20.8328 −1.20078
\(302\) 0 0
\(303\) 2.47214i 0.142020i
\(304\) 0 0
\(305\) −6.38197 −0.365430
\(306\) 0 0
\(307\) 16.8541i 0.961914i 0.876744 + 0.480957i \(0.159711\pi\)
−0.876744 + 0.480957i \(0.840289\pi\)
\(308\) 0 0
\(309\) −17.5623 −0.999085
\(310\) 0 0
\(311\) 22.4721 1.27428 0.637139 0.770749i \(-0.280118\pi\)
0.637139 + 0.770749i \(0.280118\pi\)
\(312\) 0 0
\(313\) 25.8541i 1.46136i 0.682720 + 0.730680i \(0.260797\pi\)
−0.682720 + 0.730680i \(0.739203\pi\)
\(314\) 0 0
\(315\) 1.58359i 0.0892253i
\(316\) 0 0
\(317\) 7.27051i 0.408353i 0.978934 + 0.204176i \(0.0654516\pi\)
−0.978934 + 0.204176i \(0.934548\pi\)
\(318\) 0 0
\(319\) 54.5410 3.05371
\(320\) 0 0
\(321\) 18.9443 1.05737
\(322\) 0 0
\(323\) 0.124612i 0.00693359i
\(324\) 0 0
\(325\) 13.0902i 0.726112i
\(326\) 0 0
\(327\) 12.2361i 0.676656i
\(328\) 0 0
\(329\) 3.16718 0.174613
\(330\) 0 0
\(331\) 25.1246 1.38097 0.690487 0.723345i \(-0.257396\pi\)
0.690487 + 0.723345i \(0.257396\pi\)
\(332\) 0 0
\(333\) 3.70820i 0.203208i
\(334\) 0 0
\(335\) − 11.7082i − 0.639688i
\(336\) 0 0
\(337\) 3.38197i 0.184227i 0.995748 + 0.0921137i \(0.0293623\pi\)
−0.995748 + 0.0921137i \(0.970638\pi\)
\(338\) 0 0
\(339\) −21.7082 −1.17903
\(340\) 0 0
\(341\) 12.0557 0.652854
\(342\) 0 0
\(343\) − 19.5836i − 1.05741i
\(344\) 0 0
\(345\) 3.61803 0.194788
\(346\) 0 0
\(347\) 14.5623i 0.781746i 0.920445 + 0.390873i \(0.127827\pi\)
−0.920445 + 0.390873i \(0.872173\pi\)
\(348\) 0 0
\(349\) −27.7082 −1.48319 −0.741593 0.670850i \(-0.765929\pi\)
−0.741593 + 0.670850i \(0.765929\pi\)
\(350\) 0 0
\(351\) 14.3262 0.764678
\(352\) 0 0
\(353\) 8.00000i 0.425797i 0.977074 + 0.212899i \(0.0682904\pi\)
−0.977074 + 0.212899i \(0.931710\pi\)
\(354\) 0 0
\(355\) 0.854102 0.0453310
\(356\) 0 0
\(357\) 2.56231i 0.135612i
\(358\) 0 0
\(359\) −13.4164 −0.708091 −0.354045 0.935228i \(-0.615194\pi\)
−0.354045 + 0.935228i \(0.615194\pi\)
\(360\) 0 0
\(361\) −18.9787 −0.998880
\(362\) 0 0
\(363\) − 33.2705i − 1.74625i
\(364\) 0 0
\(365\) 36.8328i 1.92792i
\(366\) 0 0
\(367\) 12.0000i 0.626395i 0.949688 + 0.313197i \(0.101400\pi\)
−0.949688 + 0.313197i \(0.898600\pi\)
\(368\) 0 0
\(369\) −2.14590 −0.111711
\(370\) 0 0
\(371\) −3.70820 −0.192520
\(372\) 0 0
\(373\) − 20.9443i − 1.08445i −0.840232 0.542227i \(-0.817581\pi\)
0.840232 0.542227i \(-0.182419\pi\)
\(374\) 0 0
\(375\) 18.0902i 0.934172i
\(376\) 0 0
\(377\) 25.4164i 1.30901i
\(378\) 0 0
\(379\) −24.2705 −1.24669 −0.623346 0.781946i \(-0.714227\pi\)
−0.623346 + 0.781946i \(0.714227\pi\)
\(380\) 0 0
\(381\) −15.7082 −0.804756
\(382\) 0 0
\(383\) − 0.583592i − 0.0298202i −0.999889 0.0149101i \(-0.995254\pi\)
0.999889 0.0149101i \(-0.00474620\pi\)
\(384\) 0 0
\(385\) 23.2918i 1.18706i
\(386\) 0 0
\(387\) − 4.29180i − 0.218164i
\(388\) 0 0
\(389\) 21.3262 1.08128 0.540642 0.841253i \(-0.318182\pi\)
0.540642 + 0.841253i \(0.318182\pi\)
\(390\) 0 0
\(391\) −0.854102 −0.0431938
\(392\) 0 0
\(393\) − 22.9443i − 1.15739i
\(394\) 0 0
\(395\) 17.2361 0.867241
\(396\) 0 0
\(397\) − 15.4377i − 0.774796i −0.921913 0.387398i \(-0.873374\pi\)
0.921913 0.387398i \(-0.126626\pi\)
\(398\) 0 0
\(399\) 0.437694 0.0219121
\(400\) 0 0
\(401\) −25.5279 −1.27480 −0.637400 0.770533i \(-0.719990\pi\)
−0.637400 + 0.770533i \(0.719990\pi\)
\(402\) 0 0
\(403\) 5.61803i 0.279854i
\(404\) 0 0
\(405\) 17.2361 0.856467
\(406\) 0 0
\(407\) 54.5410i 2.70350i
\(408\) 0 0
\(409\) −13.5623 −0.670613 −0.335306 0.942109i \(-0.608840\pi\)
−0.335306 + 0.942109i \(0.608840\pi\)
\(410\) 0 0
\(411\) 22.4164 1.10572
\(412\) 0 0
\(413\) 11.1246i 0.547406i
\(414\) 0 0
\(415\) 17.2361i 0.846085i
\(416\) 0 0
\(417\) − 6.94427i − 0.340062i
\(418\) 0 0
\(419\) −29.8885 −1.46015 −0.730075 0.683367i \(-0.760515\pi\)
−0.730075 + 0.683367i \(0.760515\pi\)
\(420\) 0 0
\(421\) 0.145898 0.00711064 0.00355532 0.999994i \(-0.498868\pi\)
0.00355532 + 0.999994i \(0.498868\pi\)
\(422\) 0 0
\(423\) 0.652476i 0.0317245i
\(424\) 0 0
\(425\) − 4.27051i − 0.207150i
\(426\) 0 0
\(427\) − 5.29180i − 0.256088i
\(428\) 0 0
\(429\) 23.7984 1.14900
\(430\) 0 0
\(431\) 11.2361 0.541222 0.270611 0.962689i \(-0.412774\pi\)
0.270611 + 0.962689i \(0.412774\pi\)
\(432\) 0 0
\(433\) − 27.3820i − 1.31589i −0.753065 0.657947i \(-0.771425\pi\)
0.753065 0.657947i \(-0.228575\pi\)
\(434\) 0 0
\(435\) 35.1246i 1.68410i
\(436\) 0 0
\(437\) 0.145898i 0.00697925i
\(438\) 0 0
\(439\) 33.2705 1.58791 0.793957 0.607973i \(-0.208017\pi\)
0.793957 + 0.607973i \(0.208017\pi\)
\(440\) 0 0
\(441\) 1.36068 0.0647943
\(442\) 0 0
\(443\) − 0.145898i − 0.00693182i −0.999994 0.00346591i \(-0.998897\pi\)
0.999994 0.00346591i \(-0.00110324\pi\)
\(444\) 0 0
\(445\) 8.29180 0.393069
\(446\) 0 0
\(447\) − 4.23607i − 0.200359i
\(448\) 0 0
\(449\) 17.5623 0.828816 0.414408 0.910091i \(-0.363989\pi\)
0.414408 + 0.910091i \(0.363989\pi\)
\(450\) 0 0
\(451\) −31.5623 −1.48621
\(452\) 0 0
\(453\) 23.0902i 1.08487i
\(454\) 0 0
\(455\) −10.8541 −0.508848
\(456\) 0 0
\(457\) − 1.41641i − 0.0662568i −0.999451 0.0331284i \(-0.989453\pi\)
0.999451 0.0331284i \(-0.0105470\pi\)
\(458\) 0 0
\(459\) −4.67376 −0.218153
\(460\) 0 0
\(461\) 37.3050 1.73746 0.868732 0.495282i \(-0.164935\pi\)
0.868732 + 0.495282i \(0.164935\pi\)
\(462\) 0 0
\(463\) 15.7082i 0.730022i 0.931003 + 0.365011i \(0.118935\pi\)
−0.931003 + 0.365011i \(0.881065\pi\)
\(464\) 0 0
\(465\) 7.76393i 0.360044i
\(466\) 0 0
\(467\) 23.1246i 1.07008i 0.844827 + 0.535040i \(0.179703\pi\)
−0.844827 + 0.535040i \(0.820297\pi\)
\(468\) 0 0
\(469\) 9.70820 0.448283
\(470\) 0 0
\(471\) −3.70820 −0.170865
\(472\) 0 0
\(473\) − 63.1246i − 2.90247i
\(474\) 0 0
\(475\) −0.729490 −0.0334713
\(476\) 0 0
\(477\) − 0.763932i − 0.0349780i
\(478\) 0 0
\(479\) −28.4721 −1.30093 −0.650463 0.759538i \(-0.725425\pi\)
−0.650463 + 0.759538i \(0.725425\pi\)
\(480\) 0 0
\(481\) −25.4164 −1.15889
\(482\) 0 0
\(483\) 3.00000i 0.136505i
\(484\) 0 0
\(485\) 29.1459i 1.32345i
\(486\) 0 0
\(487\) − 8.29180i − 0.375737i −0.982194 0.187869i \(-0.939842\pi\)
0.982194 0.187869i \(-0.0601579\pi\)
\(488\) 0 0
\(489\) 35.6525 1.61226
\(490\) 0 0
\(491\) −26.8328 −1.21095 −0.605474 0.795865i \(-0.707016\pi\)
−0.605474 + 0.795865i \(0.707016\pi\)
\(492\) 0 0
\(493\) − 8.29180i − 0.373444i
\(494\) 0 0
\(495\) −4.79837 −0.215671
\(496\) 0 0
\(497\) 0.708204i 0.0317673i
\(498\) 0 0
\(499\) −15.4164 −0.690133 −0.345067 0.938578i \(-0.612144\pi\)
−0.345067 + 0.938578i \(0.612144\pi\)
\(500\) 0 0
\(501\) −15.7082 −0.701791
\(502\) 0 0
\(503\) − 10.1459i − 0.452383i −0.974083 0.226192i \(-0.927372\pi\)
0.974083 0.226192i \(-0.0726276\pi\)
\(504\) 0 0
\(505\) 3.41641 0.152028
\(506\) 0 0
\(507\) − 9.94427i − 0.441641i
\(508\) 0 0
\(509\) −6.65248 −0.294866 −0.147433 0.989072i \(-0.547101\pi\)
−0.147433 + 0.989072i \(0.547101\pi\)
\(510\) 0 0
\(511\) −30.5410 −1.35106
\(512\) 0 0
\(513\) 0.798374i 0.0352491i
\(514\) 0 0
\(515\) 24.2705i 1.06949i
\(516\) 0 0
\(517\) 9.59675i 0.422064i
\(518\) 0 0
\(519\) 26.7984 1.17632
\(520\) 0 0
\(521\) −38.0689 −1.66783 −0.833914 0.551894i \(-0.813905\pi\)
−0.833914 + 0.551894i \(0.813905\pi\)
\(522\) 0 0
\(523\) − 4.36068i − 0.190679i −0.995445 0.0953396i \(-0.969606\pi\)
0.995445 0.0953396i \(-0.0303937\pi\)
\(524\) 0 0
\(525\) −15.0000 −0.654654
\(526\) 0 0
\(527\) − 1.83282i − 0.0798387i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −2.29180 −0.0994555
\(532\) 0 0
\(533\) − 14.7082i − 0.637083i
\(534\) 0 0
\(535\) − 26.1803i − 1.13187i
\(536\) 0 0
\(537\) − 12.1803i − 0.525620i
\(538\) 0 0
\(539\) 20.0132 0.862028
\(540\) 0 0
\(541\) −0.583592 −0.0250906 −0.0125453 0.999921i \(-0.503993\pi\)
−0.0125453 + 0.999921i \(0.503993\pi\)
\(542\) 0 0
\(543\) − 25.1803i − 1.08059i
\(544\) 0 0
\(545\) 16.9098 0.724338
\(546\) 0 0
\(547\) − 7.85410i − 0.335817i −0.985803 0.167909i \(-0.946299\pi\)
0.985803 0.167909i \(-0.0537013\pi\)
\(548\) 0 0
\(549\) 1.09017 0.0465273
\(550\) 0 0
\(551\) −1.41641 −0.0603410
\(552\) 0 0
\(553\) 14.2918i 0.607749i
\(554\) 0 0
\(555\) −35.1246 −1.49096
\(556\) 0 0
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) 29.4164 1.24418
\(560\) 0 0
\(561\) −7.76393 −0.327793
\(562\) 0 0
\(563\) 15.7082i 0.662022i 0.943627 + 0.331011i \(0.107390\pi\)
−0.943627 + 0.331011i \(0.892610\pi\)
\(564\) 0 0
\(565\) 30.0000i 1.26211i
\(566\) 0 0
\(567\) 14.2918i 0.600199i
\(568\) 0 0
\(569\) 16.3607 0.685875 0.342938 0.939358i \(-0.388578\pi\)
0.342938 + 0.939358i \(0.388578\pi\)
\(570\) 0 0
\(571\) −8.27051 −0.346110 −0.173055 0.984912i \(-0.555364\pi\)
−0.173055 + 0.984912i \(0.555364\pi\)
\(572\) 0 0
\(573\) 41.1246i 1.71801i
\(574\) 0 0
\(575\) − 5.00000i − 0.208514i
\(576\) 0 0
\(577\) 39.7082i 1.65307i 0.562882 + 0.826537i \(0.309692\pi\)
−0.562882 + 0.826537i \(0.690308\pi\)
\(578\) 0 0
\(579\) −25.4164 −1.05627
\(580\) 0 0
\(581\) −14.2918 −0.592924
\(582\) 0 0
\(583\) − 11.2361i − 0.465350i
\(584\) 0 0
\(585\) − 2.23607i − 0.0924500i
\(586\) 0 0
\(587\) − 8.85410i − 0.365448i −0.983164 0.182724i \(-0.941509\pi\)
0.983164 0.182724i \(-0.0584915\pi\)
\(588\) 0 0
\(589\) −0.313082 −0.0129003
\(590\) 0 0
\(591\) 33.2705 1.36857
\(592\) 0 0
\(593\) 6.58359i 0.270356i 0.990821 + 0.135178i \(0.0431606\pi\)
−0.990821 + 0.135178i \(0.956839\pi\)
\(594\) 0 0
\(595\) 3.54102 0.145168
\(596\) 0 0
\(597\) − 18.4721i − 0.756014i
\(598\) 0 0
\(599\) 23.6180 0.965007 0.482503 0.875894i \(-0.339728\pi\)
0.482503 + 0.875894i \(0.339728\pi\)
\(600\) 0 0
\(601\) −9.85410 −0.401957 −0.200979 0.979596i \(-0.564412\pi\)
−0.200979 + 0.979596i \(0.564412\pi\)
\(602\) 0 0
\(603\) 2.00000i 0.0814463i
\(604\) 0 0
\(605\) −45.9787 −1.86930
\(606\) 0 0
\(607\) 21.0557i 0.854626i 0.904104 + 0.427313i \(0.140540\pi\)
−0.904104 + 0.427313i \(0.859460\pi\)
\(608\) 0 0
\(609\) −29.1246 −1.18019
\(610\) 0 0
\(611\) −4.47214 −0.180923
\(612\) 0 0
\(613\) − 0.763932i − 0.0308549i −0.999881 0.0154275i \(-0.995089\pi\)
0.999881 0.0154275i \(-0.00491091\pi\)
\(614\) 0 0
\(615\) − 20.3262i − 0.819633i
\(616\) 0 0
\(617\) 2.56231i 0.103155i 0.998669 + 0.0515773i \(0.0164248\pi\)
−0.998669 + 0.0515773i \(0.983575\pi\)
\(618\) 0 0
\(619\) −27.8541 −1.11955 −0.559775 0.828644i \(-0.689113\pi\)
−0.559775 + 0.828644i \(0.689113\pi\)
\(620\) 0 0
\(621\) −5.47214 −0.219589
\(622\) 0 0
\(623\) 6.87539i 0.275457i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 1.32624i 0.0529648i
\(628\) 0 0
\(629\) 8.29180 0.330616
\(630\) 0 0
\(631\) 23.4164 0.932192 0.466096 0.884734i \(-0.345660\pi\)
0.466096 + 0.884734i \(0.345660\pi\)
\(632\) 0 0
\(633\) − 12.4721i − 0.495723i
\(634\) 0 0
\(635\) 21.7082i 0.861464i
\(636\) 0 0
\(637\) 9.32624i 0.369519i
\(638\) 0 0
\(639\) −0.145898 −0.00577164
\(640\) 0 0
\(641\) −0.652476 −0.0257712 −0.0128856 0.999917i \(-0.504102\pi\)
−0.0128856 + 0.999917i \(0.504102\pi\)
\(642\) 0 0
\(643\) 41.0132i 1.61740i 0.588221 + 0.808700i \(0.299829\pi\)
−0.588221 + 0.808700i \(0.700171\pi\)
\(644\) 0 0
\(645\) 40.6525 1.60069
\(646\) 0 0
\(647\) − 13.7082i − 0.538925i −0.963011 0.269463i \(-0.913154\pi\)
0.963011 0.269463i \(-0.0868460\pi\)
\(648\) 0 0
\(649\) −33.7082 −1.32316
\(650\) 0 0
\(651\) −6.43769 −0.252313
\(652\) 0 0
\(653\) 34.9787i 1.36882i 0.729096 + 0.684411i \(0.239941\pi\)
−0.729096 + 0.684411i \(0.760059\pi\)
\(654\) 0 0
\(655\) −31.7082 −1.23894
\(656\) 0 0
\(657\) − 6.29180i − 0.245466i
\(658\) 0 0
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) −39.3951 −1.53229 −0.766146 0.642666i \(-0.777828\pi\)
−0.766146 + 0.642666i \(0.777828\pi\)
\(662\) 0 0
\(663\) − 3.61803i − 0.140513i
\(664\) 0 0
\(665\) − 0.604878i − 0.0234562i
\(666\) 0 0
\(667\) − 9.70820i − 0.375903i
\(668\) 0 0
\(669\) −36.1803 −1.39881
\(670\) 0 0
\(671\) 16.0344 0.619003
\(672\) 0 0
\(673\) − 27.7082i − 1.06807i −0.845461 0.534036i \(-0.820675\pi\)
0.845461 0.534036i \(-0.179325\pi\)
\(674\) 0 0
\(675\) − 27.3607i − 1.05311i
\(676\) 0 0
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) 0 0
\(679\) −24.1672 −0.927451
\(680\) 0 0
\(681\) 16.6525 0.638124
\(682\) 0 0
\(683\) 10.9787i 0.420089i 0.977692 + 0.210044i \(0.0673609\pi\)
−0.977692 + 0.210044i \(0.932639\pi\)
\(684\) 0 0
\(685\) − 30.9787i − 1.18364i
\(686\) 0 0
\(687\) − 16.1803i − 0.617318i
\(688\) 0 0
\(689\) 5.23607 0.199478
\(690\) 0 0
\(691\) 21.1246 0.803618 0.401809 0.915723i \(-0.368382\pi\)
0.401809 + 0.915723i \(0.368382\pi\)
\(692\) 0 0
\(693\) − 3.97871i − 0.151139i
\(694\) 0 0
\(695\) −9.59675 −0.364025
\(696\) 0 0
\(697\) 4.79837i 0.181751i
\(698\) 0 0
\(699\) 34.1803 1.29282
\(700\) 0 0
\(701\) −27.9230 −1.05464 −0.527318 0.849668i \(-0.676802\pi\)
−0.527318 + 0.849668i \(0.676802\pi\)
\(702\) 0 0
\(703\) − 1.41641i − 0.0534208i
\(704\) 0 0
\(705\) −6.18034 −0.232765
\(706\) 0 0
\(707\) 2.83282i 0.106539i
\(708\) 0 0
\(709\) −8.56231 −0.321564 −0.160782 0.986990i \(-0.551402\pi\)
−0.160782 + 0.986990i \(0.551402\pi\)
\(710\) 0 0
\(711\) −2.94427 −0.110419
\(712\) 0 0
\(713\) − 2.14590i − 0.0803645i
\(714\) 0 0
\(715\) − 32.8885i − 1.22996i
\(716\) 0 0
\(717\) 16.9443i 0.632795i
\(718\) 0 0
\(719\) 23.4508 0.874569 0.437285 0.899323i \(-0.355940\pi\)
0.437285 + 0.899323i \(0.355940\pi\)
\(720\) 0 0
\(721\) −20.1246 −0.749480
\(722\) 0 0
\(723\) 3.23607i 0.120351i
\(724\) 0 0
\(725\) 48.5410 1.80277
\(726\) 0 0
\(727\) 40.7426i 1.51106i 0.655113 + 0.755531i \(0.272621\pi\)
−0.655113 + 0.755531i \(0.727379\pi\)
\(728\) 0 0
\(729\) −29.5066 −1.09284
\(730\) 0 0
\(731\) −9.59675 −0.354949
\(732\) 0 0
\(733\) 0.111456i 0.00411673i 0.999998 + 0.00205836i \(0.000655198\pi\)
−0.999998 + 0.00205836i \(0.999345\pi\)
\(734\) 0 0
\(735\) 12.8885i 0.475401i
\(736\) 0 0
\(737\) 29.4164i 1.08357i
\(738\) 0 0
\(739\) 34.5410 1.27061 0.635306 0.772261i \(-0.280874\pi\)
0.635306 + 0.772261i \(0.280874\pi\)
\(740\) 0 0
\(741\) −0.618034 −0.0227040
\(742\) 0 0
\(743\) − 36.9787i − 1.35662i −0.734777 0.678309i \(-0.762713\pi\)
0.734777 0.678309i \(-0.237287\pi\)
\(744\) 0 0
\(745\) −5.85410 −0.214478
\(746\) 0 0
\(747\) − 2.94427i − 0.107725i
\(748\) 0 0
\(749\) 21.7082 0.793201
\(750\) 0 0
\(751\) 0.875388 0.0319434 0.0159717 0.999872i \(-0.494916\pi\)
0.0159717 + 0.999872i \(0.494916\pi\)
\(752\) 0 0
\(753\) − 27.1803i − 0.990507i
\(754\) 0 0
\(755\) 31.9098 1.16132
\(756\) 0 0
\(757\) − 29.0132i − 1.05450i −0.849710 0.527251i \(-0.823223\pi\)
0.849710 0.527251i \(-0.176777\pi\)
\(758\) 0 0
\(759\) −9.09017 −0.329952
\(760\) 0 0
\(761\) 26.5066 0.960863 0.480431 0.877032i \(-0.340480\pi\)
0.480431 + 0.877032i \(0.340480\pi\)
\(762\) 0 0
\(763\) 14.0213i 0.507605i
\(764\) 0 0
\(765\) 0.729490i 0.0263748i
\(766\) 0 0
\(767\) − 15.7082i − 0.567190i
\(768\) 0 0
\(769\) 47.7082 1.72040 0.860201 0.509955i \(-0.170338\pi\)
0.860201 + 0.509955i \(0.170338\pi\)
\(770\) 0 0
\(771\) 18.4721 0.665258
\(772\) 0 0
\(773\) 14.0000i 0.503545i 0.967786 + 0.251773i \(0.0810135\pi\)
−0.967786 + 0.251773i \(0.918987\pi\)
\(774\) 0 0
\(775\) 10.7295 0.385415
\(776\) 0 0
\(777\) − 29.1246i − 1.04484i
\(778\) 0 0
\(779\) 0.819660 0.0293674
\(780\) 0 0
\(781\) −2.14590 −0.0767863
\(782\) 0 0
\(783\) − 53.1246i − 1.89852i
\(784\) 0 0
\(785\) 5.12461i 0.182905i
\(786\) 0 0
\(787\) 4.58359i 0.163387i 0.996657 + 0.0816937i \(0.0260329\pi\)
−0.996657 + 0.0816937i \(0.973967\pi\)
\(788\) 0 0
\(789\) 29.5623 1.05245
\(790\) 0 0
\(791\) −24.8754 −0.884467
\(792\) 0 0
\(793\) 7.47214i 0.265343i
\(794\) 0 0
\(795\) 7.23607 0.256637
\(796\) 0 0
\(797\) − 45.4164i − 1.60873i −0.594134 0.804366i \(-0.702505\pi\)
0.594134 0.804366i \(-0.297495\pi\)
\(798\) 0 0
\(799\) 1.45898 0.0516150
\(800\) 0 0
\(801\) −1.41641 −0.0500463
\(802\) 0 0
\(803\) − 92.5410i − 3.26570i
\(804\) 0 0
\(805\) 4.14590 0.146124
\(806\) 0 0
\(807\) − 2.47214i − 0.0870233i
\(808\) 0 0
\(809\) 42.1591 1.48223 0.741117 0.671376i \(-0.234297\pi\)
0.741117 + 0.671376i \(0.234297\pi\)
\(810\) 0 0
\(811\) −5.70820 −0.200442 −0.100221 0.994965i \(-0.531955\pi\)
−0.100221 + 0.994965i \(0.531955\pi\)
\(812\) 0 0
\(813\) 41.8328i 1.46714i
\(814\) 0 0
\(815\) − 49.2705i − 1.72587i
\(816\) 0 0
\(817\) 1.63932i 0.0573526i
\(818\) 0 0
\(819\) 1.85410 0.0647876
\(820\) 0 0
\(821\) 14.9443 0.521559 0.260779 0.965398i \(-0.416020\pi\)
0.260779 + 0.965398i \(0.416020\pi\)
\(822\) 0 0
\(823\) 32.8328i 1.14448i 0.820086 + 0.572240i \(0.193925\pi\)
−0.820086 + 0.572240i \(0.806075\pi\)
\(824\) 0 0
\(825\) − 45.4508i − 1.58240i
\(826\) 0 0
\(827\) − 32.2492i − 1.12142i −0.828014 0.560708i \(-0.810529\pi\)
0.828014 0.560708i \(-0.189471\pi\)
\(828\) 0 0
\(829\) 34.5410 1.19966 0.599830 0.800128i \(-0.295235\pi\)
0.599830 + 0.800128i \(0.295235\pi\)
\(830\) 0 0
\(831\) −12.1803 −0.422531
\(832\) 0 0
\(833\) − 3.04257i − 0.105419i
\(834\) 0 0
\(835\) 21.7082i 0.751243i
\(836\) 0 0
\(837\) − 11.7426i − 0.405885i
\(838\) 0 0
\(839\) −11.2361 −0.387912 −0.193956 0.981010i \(-0.562132\pi\)
−0.193956 + 0.981010i \(0.562132\pi\)
\(840\) 0 0
\(841\) 65.2492 2.24997
\(842\) 0 0
\(843\) − 49.5967i − 1.70820i
\(844\) 0 0
\(845\) −13.7426 −0.472761
\(846\) 0 0
\(847\) − 38.1246i − 1.30998i
\(848\) 0 0
\(849\) −33.8885 −1.16305
\(850\) 0 0
\(851\) 9.70820 0.332793
\(852\) 0 0
\(853\) 0.214782i 0.00735399i 0.999993 + 0.00367699i \(0.00117043\pi\)
−0.999993 + 0.00367699i \(0.998830\pi\)
\(854\) 0 0
\(855\) 0.124612 0.00426163
\(856\) 0 0
\(857\) 36.0000i 1.22974i 0.788630 + 0.614868i \(0.210791\pi\)
−0.788630 + 0.614868i \(0.789209\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 16.8541 0.574386
\(862\) 0 0
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 0 0
\(865\) − 37.0344i − 1.25921i
\(866\) 0 0
\(867\) − 26.3262i − 0.894086i
\(868\) 0 0
\(869\) −43.3050 −1.46902
\(870\) 0 0
\(871\) −13.7082 −0.464485
\(872\) 0 0
\(873\) − 4.97871i − 0.168504i
\(874\) 0 0
\(875\) 20.7295i 0.700785i
\(876\) 0 0
\(877\) − 25.6869i − 0.867386i −0.901061 0.433693i \(-0.857210\pi\)
0.901061 0.433693i \(-0.142790\pi\)
\(878\) 0 0
\(879\) −23.1246 −0.779974
\(880\) 0 0
\(881\) −51.7082 −1.74209 −0.871047 0.491200i \(-0.836558\pi\)
−0.871047 + 0.491200i \(0.836558\pi\)
\(882\) 0 0
\(883\) 51.2705i 1.72539i 0.505725 + 0.862695i \(0.331225\pi\)
−0.505725 + 0.862695i \(0.668775\pi\)
\(884\) 0 0
\(885\) − 21.7082i − 0.729713i
\(886\) 0 0
\(887\) 30.5410i 1.02547i 0.858548 + 0.512734i \(0.171367\pi\)
−0.858548 + 0.512734i \(0.828633\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) −43.3050 −1.45077
\(892\) 0 0
\(893\) − 0.249224i − 0.00833995i
\(894\) 0 0
\(895\) −16.8328 −0.562659
\(896\) 0 0
\(897\) − 4.23607i − 0.141438i
\(898\) 0 0
\(899\) 20.8328 0.694813
\(900\) 0 0
\(901\) −1.70820 −0.0569085
\(902\) 0 0
\(903\) 33.7082i 1.12174i
\(904\) 0 0
\(905\) −34.7984 −1.15674
\(906\) 0 0
\(907\) − 36.5410i − 1.21332i −0.794960 0.606662i \(-0.792508\pi\)
0.794960 0.606662i \(-0.207492\pi\)
\(908\) 0 0
\(909\) −0.583592 −0.0193565
\(910\) 0 0
\(911\) 22.3607 0.740842 0.370421 0.928864i \(-0.379213\pi\)
0.370421 + 0.928864i \(0.379213\pi\)
\(912\) 0 0
\(913\) − 43.3050i − 1.43318i
\(914\) 0 0
\(915\) 10.3262i 0.341375i
\(916\) 0 0
\(917\) − 26.2918i − 0.868232i
\(918\) 0 0
\(919\) −41.4164 −1.36620 −0.683101 0.730324i \(-0.739369\pi\)
−0.683101 + 0.730324i \(0.739369\pi\)
\(920\) 0 0
\(921\) 27.2705 0.898594
\(922\) 0 0
\(923\) − 1.00000i − 0.0329154i
\(924\) 0 0
\(925\) 48.5410i 1.59602i
\(926\) 0 0
\(927\) − 4.14590i − 0.136169i
\(928\) 0 0
\(929\) 43.3050 1.42079 0.710395 0.703804i \(-0.248516\pi\)
0.710395 + 0.703804i \(0.248516\pi\)
\(930\) 0 0
\(931\) −0.519733 −0.0170336
\(932\) 0 0
\(933\) − 36.3607i − 1.19040i
\(934\) 0 0
\(935\) 10.7295i 0.350892i
\(936\) 0 0
\(937\) 24.3262i 0.794704i 0.917666 + 0.397352i \(0.130071\pi\)
−0.917666 + 0.397352i \(0.869929\pi\)
\(938\) 0 0
\(939\) 41.8328 1.36516
\(940\) 0 0
\(941\) 27.2148 0.887177 0.443588 0.896231i \(-0.353705\pi\)
0.443588 + 0.896231i \(0.353705\pi\)
\(942\) 0 0
\(943\) 5.61803i 0.182948i
\(944\) 0 0
\(945\) 22.6869 0.738005
\(946\) 0 0
\(947\) 26.3951i 0.857726i 0.903369 + 0.428863i \(0.141086\pi\)
−0.903369 + 0.428863i \(0.858914\pi\)
\(948\) 0 0
\(949\) 43.1246 1.39988
\(950\) 0 0
\(951\) 11.7639 0.381472
\(952\) 0 0
\(953\) 37.3951i 1.21135i 0.795713 + 0.605673i \(0.207096\pi\)
−0.795713 + 0.605673i \(0.792904\pi\)
\(954\) 0 0
\(955\) 56.8328 1.83907
\(956\) 0 0
\(957\) − 88.2492i − 2.85269i
\(958\) 0 0
\(959\) 25.6869 0.829474
\(960\) 0 0
\(961\) −26.3951 −0.851456
\(962\) 0 0
\(963\) 4.47214i 0.144113i
\(964\) 0 0
\(965\) 35.1246i 1.13070i
\(966\) 0 0
\(967\) 29.2361i 0.940169i 0.882622 + 0.470084i \(0.155776\pi\)
−0.882622 + 0.470084i \(0.844224\pi\)
\(968\) 0 0
\(969\) 0.201626 0.00647716
\(970\) 0 0
\(971\) −19.1459 −0.614421 −0.307211 0.951642i \(-0.599396\pi\)
−0.307211 + 0.951642i \(0.599396\pi\)
\(972\) 0 0
\(973\) − 7.95743i − 0.255103i
\(974\) 0 0
\(975\) 21.1803 0.678314
\(976\) 0 0
\(977\) − 1.27051i − 0.0406472i −0.999793 0.0203236i \(-0.993530\pi\)
0.999793 0.0203236i \(-0.00646965\pi\)
\(978\) 0 0
\(979\) −20.8328 −0.665820
\(980\) 0 0
\(981\) −2.88854 −0.0922241
\(982\) 0 0
\(983\) 47.3951i 1.51167i 0.654762 + 0.755835i \(0.272769\pi\)
−0.654762 + 0.755835i \(0.727231\pi\)
\(984\) 0 0
\(985\) − 45.9787i − 1.46500i
\(986\) 0 0
\(987\) − 5.12461i − 0.163118i
\(988\) 0 0
\(989\) −11.2361 −0.357286
\(990\) 0 0
\(991\) −17.7295 −0.563196 −0.281598 0.959532i \(-0.590864\pi\)
−0.281598 + 0.959532i \(0.590864\pi\)
\(992\) 0 0
\(993\) − 40.6525i − 1.29007i
\(994\) 0 0
\(995\) −25.5279 −0.809288
\(996\) 0 0
\(997\) 62.7214i 1.98641i 0.116398 + 0.993203i \(0.462865\pi\)
−0.116398 + 0.993203i \(0.537135\pi\)
\(998\) 0 0
\(999\) 53.1246 1.68079
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 1840.2.e.c.369.1 4
4.3 odd 2 230.2.b.a.139.4 yes 4
5.2 odd 4 9200.2.a.bo.1.1 2
5.3 odd 4 9200.2.a.by.1.2 2
5.4 even 2 inner 1840.2.e.c.369.4 4
12.11 even 2 2070.2.d.c.829.2 4
20.3 even 4 1150.2.a.n.1.1 2
20.7 even 4 1150.2.a.l.1.2 2
20.19 odd 2 230.2.b.a.139.1 4
60.59 even 2 2070.2.d.c.829.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.1 4 20.19 odd 2
230.2.b.a.139.4 yes 4 4.3 odd 2
1150.2.a.l.1.2 2 20.7 even 4
1150.2.a.n.1.1 2 20.3 even 4
1840.2.e.c.369.1 4 1.1 even 1 trivial
1840.2.e.c.369.4 4 5.4 even 2 inner
2070.2.d.c.829.2 4 12.11 even 2
2070.2.d.c.829.4 4 60.59 even 2
9200.2.a.bo.1.1 2 5.2 odd 4
9200.2.a.by.1.2 2 5.3 odd 4