Properties

Label 1950.2.e.h.1249.2
Level $1950$
Weight $2$
Character 1950.1249
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1249
Dual form 1950.2.e.h.1249.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} -1.00000i q^{13} +1.00000 q^{16} -1.00000i q^{18} -1.00000 q^{19} +4.00000i q^{22} +4.00000i q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +3.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +1.00000 q^{36} +5.00000i q^{37} -1.00000i q^{38} +1.00000 q^{39} +9.00000 q^{41} +2.00000i q^{43} -4.00000 q^{44} -4.00000 q^{46} +3.00000i q^{47} +1.00000i q^{48} +7.00000 q^{49} +1.00000i q^{52} +1.00000i q^{53} +1.00000 q^{54} -1.00000i q^{57} +3.00000i q^{58} -10.0000 q^{59} +4.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} +9.00000i q^{67} -4.00000 q^{69} +7.00000 q^{71} +1.00000i q^{72} +4.00000i q^{73} -5.00000 q^{74} +1.00000 q^{76} +1.00000i q^{78} -11.0000 q^{79} +1.00000 q^{81} +9.00000i q^{82} +6.00000i q^{83} -2.00000 q^{86} +3.00000i q^{87} -4.00000i q^{88} -10.0000 q^{89} -4.00000i q^{92} +4.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} +12.0000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + 8q^{11} + 2q^{16} - 2q^{19} + 2q^{24} + 2q^{26} + 6q^{29} + 8q^{31} + 2q^{36} + 2q^{39} + 18q^{41} - 8q^{44} - 8q^{46} + 14q^{49} + 2q^{54} - 20q^{59} + 8q^{61} - 2q^{64} - 8q^{66} - 8q^{69} + 14q^{71} - 10q^{74} + 2q^{76} - 22q^{79} + 2q^{81} - 4q^{86} - 20q^{89} - 6q^{94} - 2q^{96} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 1.00000i − 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.00000i 0.821995i 0.911636 + 0.410997i \(0.134819\pi\)
−0.911636 + 0.410997i \(0.865181\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) 1.00000i 0.137361i 0.997639 + 0.0686803i \(0.0218788\pi\)
−0.997639 + 0.0686803i \(0.978121\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 0.132453i
\(58\) 3.00000i 0.393919i
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 9.00000i 1.09952i 0.835321 + 0.549762i \(0.185282\pi\)
−0.835321 + 0.549762i \(0.814718\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −5.00000 −0.581238
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 1.00000i 0.113228i
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.00000i 0.993884i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 3.00000i 0.321634i
\(88\) − 4.00000i − 0.426401i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 4.00000i − 0.417029i
\(93\) 4.00000i 0.414781i
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) 7.00000i 0.707107i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) − 9.00000i − 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) −5.00000 −0.474579
\(112\) 0 0
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 1.00000i 0.0924500i
\(118\) − 10.0000i − 0.920575i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 4.00000i 0.362143i
\(123\) 9.00000i 0.811503i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 17.0000i 1.50851i 0.656584 + 0.754253i \(0.272001\pi\)
−0.656584 + 0.754253i \(0.727999\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) 0 0
\(134\) −9.00000 −0.777482
\(135\) 0 0
\(136\) 0 0
\(137\) − 23.0000i − 1.96502i −0.186203 0.982511i \(-0.559618\pi\)
0.186203 0.982511i \(-0.440382\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 7.00000i 0.587427i
\(143\) − 4.00000i − 0.334497i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 7.00000i 0.577350i
\(148\) − 5.00000i − 0.410997i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) − 11.0000i − 0.875113i
\(159\) −1.00000 −0.0793052
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) − 7.00000i − 0.541676i −0.962625 0.270838i \(-0.912699\pi\)
0.962625 0.270838i \(-0.0873008\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 2.00000i − 0.152499i
\(173\) − 3.00000i − 0.228086i −0.993476 0.114043i \(-0.963620\pi\)
0.993476 0.114043i \(-0.0363801\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) − 10.0000i − 0.751646i
\(178\) − 10.0000i − 0.749532i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 0 0
\(183\) 4.00000i 0.295689i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) − 3.00000i − 0.218797i
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 20.0000i 1.43963i 0.694165 + 0.719816i \(0.255774\pi\)
−0.694165 + 0.719816i \(0.744226\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 12.0000i − 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) −9.00000 −0.634811
\(202\) − 2.00000i − 0.140720i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 4.00000i − 0.278019i
\(208\) − 1.00000i − 0.0693375i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) − 1.00000i − 0.0686803i
\(213\) 7.00000i 0.479632i
\(214\) 9.00000 0.615227
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 1.00000i 0.0677285i
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) − 5.00000i − 0.335578i
\(223\) − 14.0000i − 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 10.0000i 0.663723i 0.943328 + 0.331862i \(0.107677\pi\)
−0.943328 + 0.331862i \(0.892323\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 3.00000i − 0.196960i
\(233\) 8.00000i 0.524097i 0.965055 + 0.262049i \(0.0843981\pi\)
−0.965055 + 0.262049i \(0.915602\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 10.0000 0.650945
\(237\) − 11.0000i − 0.714527i
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) −9.00000 −0.573819
\(247\) 1.00000i 0.0636285i
\(248\) − 4.00000i − 0.254000i
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 23.0000 1.45175 0.725874 0.687828i \(-0.241436\pi\)
0.725874 + 0.687828i \(0.241436\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) −17.0000 −1.06667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 24.0000i − 1.49708i −0.663090 0.748539i \(-0.730755\pi\)
0.663090 0.748539i \(-0.269245\pi\)
\(258\) − 2.00000i − 0.124515i
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) − 3.00000i − 0.185341i
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) − 10.0000i − 0.611990i
\(268\) − 9.00000i − 0.549762i
\(269\) 11.0000 0.670682 0.335341 0.942097i \(-0.391148\pi\)
0.335341 + 0.942097i \(0.391148\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 23.0000 1.38948
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 12.0000i 0.719712i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −5.00000 −0.298275 −0.149137 0.988816i \(-0.547650\pi\)
−0.149137 + 0.988816i \(0.547650\pi\)
\(282\) − 3.00000i − 0.178647i
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) −7.00000 −0.415374
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) − 4.00000i − 0.234082i
\(293\) − 16.0000i − 0.934730i −0.884064 0.467365i \(-0.845203\pi\)
0.884064 0.467365i \(-0.154797\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) 5.00000 0.290619
\(297\) − 4.00000i − 0.232104i
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) 12.0000i 0.690522i
\(303\) − 2.00000i − 0.114897i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) − 3.00000i − 0.171219i −0.996329 0.0856095i \(-0.972716\pi\)
0.996329 0.0856095i \(-0.0272838\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) − 1.00000i − 0.0566139i
\(313\) 1.00000i 0.0565233i 0.999601 + 0.0282617i \(0.00899717\pi\)
−0.999601 + 0.0282617i \(0.991003\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 11.0000 0.618798
\(317\) − 32.0000i − 1.79730i −0.438667 0.898650i \(-0.644549\pi\)
0.438667 0.898650i \(-0.355451\pi\)
\(318\) − 1.00000i − 0.0560772i
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) 1.00000i 0.0553001i
\(328\) − 9.00000i − 0.496942i
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) − 5.00000i − 0.273998i
\(334\) 7.00000 0.383023
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.0000i − 1.19842i −0.800593 0.599208i \(-0.795482\pi\)
0.800593 0.599208i \(-0.204518\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 1.00000i 0.0540738i
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) 17.0000i 0.912608i 0.889824 + 0.456304i \(0.150827\pi\)
−0.889824 + 0.456304i \(0.849173\pi\)
\(348\) − 3.00000i − 0.160817i
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 4.00000i 0.213201i
\(353\) − 15.0000i − 0.798369i −0.916871 0.399185i \(-0.869293\pi\)
0.916871 0.399185i \(-0.130707\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) 17.0000 0.897226 0.448613 0.893726i \(-0.351918\pi\)
0.448613 + 0.893726i \(0.351918\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) − 12.0000i − 0.630706i
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 21.0000i 1.09619i 0.836416 + 0.548096i \(0.184647\pi\)
−0.836416 + 0.548096i \(0.815353\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 0 0
\(372\) − 4.00000i − 0.207390i
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) − 3.00000i − 0.154508i
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) −17.0000 −0.870936
\(382\) − 18.0000i − 0.920960i
\(383\) − 5.00000i − 0.255488i −0.991807 0.127744i \(-0.959226\pi\)
0.991807 0.127744i \(-0.0407736\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −20.0000 −1.01797
\(387\) − 2.00000i − 0.101666i
\(388\) − 12.0000i − 0.609208i
\(389\) 19.0000 0.963338 0.481669 0.876353i \(-0.340031\pi\)
0.481669 + 0.876353i \(0.340031\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 7.00000i − 0.353553i
\(393\) − 3.00000i − 0.151330i
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) − 3.00000i − 0.150566i −0.997162 0.0752828i \(-0.976014\pi\)
0.997162 0.0752828i \(-0.0239860\pi\)
\(398\) − 1.00000i − 0.0501255i
\(399\) 0 0
\(400\) 0 0
\(401\) −26.0000 −1.29838 −0.649189 0.760627i \(-0.724892\pi\)
−0.649189 + 0.760627i \(0.724892\pi\)
\(402\) − 9.00000i − 0.448879i
\(403\) − 4.00000i − 0.199254i
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000i 0.991363i
\(408\) 0 0
\(409\) 40.0000 1.97787 0.988936 0.148340i \(-0.0473931\pi\)
0.988936 + 0.148340i \(0.0473931\pi\)
\(410\) 0 0
\(411\) 23.0000 1.13451
\(412\) 0 0
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 12.0000i 0.587643i
\(418\) − 4.00000i − 0.195646i
\(419\) 37.0000 1.80757 0.903784 0.427989i \(-0.140778\pi\)
0.903784 + 0.427989i \(0.140778\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 24.0000i 1.16830i
\(423\) − 3.00000i − 0.145865i
\(424\) 1.00000 0.0485643
\(425\) 0 0
\(426\) −7.00000 −0.339151
\(427\) 0 0
\(428\) 9.00000i 0.435031i
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 19.0000i − 0.913082i −0.889702 0.456541i \(-0.849088\pi\)
0.889702 0.456541i \(-0.150912\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.00000 −0.0478913
\(437\) − 4.00000i − 0.191346i
\(438\) − 4.00000i − 0.191127i
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) − 3.00000i − 0.142534i −0.997457 0.0712672i \(-0.977296\pi\)
0.997457 0.0712672i \(-0.0227043\pi\)
\(444\) 5.00000 0.237289
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) − 2.00000i − 0.0940721i
\(453\) 12.0000i 0.563809i
\(454\) −10.0000 −0.469323
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) − 32.0000i − 1.49690i −0.663193 0.748448i \(-0.730799\pi\)
0.663193 0.748448i \(-0.269201\pi\)
\(458\) − 5.00000i − 0.233635i
\(459\) 0 0
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) − 14.0000i − 0.650635i −0.945605 0.325318i \(-0.894529\pi\)
0.945605 0.325318i \(-0.105471\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −8.00000 −0.370593
\(467\) − 15.0000i − 0.694117i −0.937843 0.347059i \(-0.887180\pi\)
0.937843 0.347059i \(-0.112820\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 10.0000i 0.460287i
\(473\) 8.00000i 0.367840i
\(474\) 11.0000 0.505247
\(475\) 0 0
\(476\) 0 0
\(477\) − 1.00000i − 0.0457869i
\(478\) 8.00000i 0.365911i
\(479\) −25.0000 −1.14228 −0.571140 0.820853i \(-0.693499\pi\)
−0.571140 + 0.820853i \(0.693499\pi\)
\(480\) 0 0
\(481\) 5.00000 0.227980
\(482\) − 12.0000i − 0.546585i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 38.0000i 1.72194i 0.508652 + 0.860972i \(0.330144\pi\)
−0.508652 + 0.860972i \(0.669856\pi\)
\(488\) − 4.00000i − 0.181071i
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) − 9.00000i − 0.405751i
\(493\) 0 0
\(494\) −1.00000 −0.0449921
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) − 6.00000i − 0.268866i
\(499\) 1.00000 0.0447661 0.0223831 0.999749i \(-0.492875\pi\)
0.0223831 + 0.999749i \(0.492875\pi\)
\(500\) 0 0
\(501\) 7.00000 0.312737
\(502\) 23.0000i 1.02654i
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) − 1.00000i − 0.0444116i
\(508\) − 17.0000i − 0.754253i
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) 24.0000 1.05859
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 12.0000i 0.527759i
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) − 3.00000i − 0.131306i
\(523\) − 30.0000i − 1.31181i −0.754844 0.655904i \(-0.772288\pi\)
0.754844 0.655904i \(-0.227712\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) 4.00000i 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) − 9.00000i − 0.389833i
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 9.00000 0.388741
\(537\) 12.0000i 0.517838i
\(538\) 11.0000i 0.474244i
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 24.0000i 1.03089i
\(543\) − 12.0000i − 0.514969i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 22.0000i 0.940652i 0.882493 + 0.470326i \(0.155864\pi\)
−0.882493 + 0.470326i \(0.844136\pi\)
\(548\) 23.0000i 0.982511i
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) 4.00000i 0.170251i
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) − 42.0000i − 1.77960i −0.456354 0.889799i \(-0.650845\pi\)
0.456354 0.889799i \(-0.349155\pi\)
\(558\) − 4.00000i − 0.169334i
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) − 5.00000i − 0.210912i
\(563\) − 21.0000i − 0.885044i −0.896758 0.442522i \(-0.854084\pi\)
0.896758 0.442522i \(-0.145916\pi\)
\(564\) 3.00000 0.126323
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) − 7.00000i − 0.293713i
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 4.00000i 0.167248i
\(573\) − 18.0000i − 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 12.0000i − 0.499567i −0.968302 0.249783i \(-0.919641\pi\)
0.968302 0.249783i \(-0.0803594\pi\)
\(578\) 17.0000i 0.707107i
\(579\) −20.0000 −0.831172
\(580\) 0 0
\(581\) 0 0
\(582\) − 12.0000i − 0.497416i
\(583\) 4.00000i 0.165663i
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 16.0000 0.660954
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 5.00000i 0.205499i
\(593\) − 1.00000i − 0.0410651i −0.999789 0.0205325i \(-0.993464\pi\)
0.999789 0.0205325i \(-0.00653617\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 0 0
\(597\) − 1.00000i − 0.0409273i
\(598\) 4.00000i 0.163572i
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 45.0000 1.83559 0.917794 0.397057i \(-0.129968\pi\)
0.917794 + 0.397057i \(0.129968\pi\)
\(602\) 0 0
\(603\) − 9.00000i − 0.366508i
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 7.00000i 0.284121i 0.989858 + 0.142061i \(0.0453728\pi\)
−0.989858 + 0.142061i \(0.954627\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 0 0
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 3.00000 0.121070
\(615\) 0 0
\(616\) 0 0
\(617\) 27.0000i 1.08698i 0.839416 + 0.543490i \(0.182897\pi\)
−0.839416 + 0.543490i \(0.817103\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) − 30.0000i − 1.20289i
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −1.00000 −0.0399680
\(627\) − 4.00000i − 0.159745i
\(628\) 2.00000i 0.0798087i
\(629\) 0 0
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 11.0000i 0.437557i
\(633\) 24.0000i 0.953914i
\(634\) 32.0000 1.27088
\(635\) 0 0
\(636\) 1.00000 0.0396526
\(637\) − 7.00000i − 0.277350i
\(638\) 12.0000i 0.475085i
\(639\) −7.00000 −0.276916
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 9.00000i 0.355202i
\(643\) − 19.0000i − 0.749287i −0.927169 0.374643i \(-0.877765\pi\)
0.927169 0.374643i \(-0.122235\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 0 0
\(652\) − 8.00000i − 0.313304i
\(653\) − 26.0000i − 1.01746i −0.860927 0.508729i \(-0.830115\pi\)
0.860927 0.508729i \(-0.169885\pi\)
\(654\) −1.00000 −0.0391031
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) − 4.00000i − 0.156055i
\(658\) 0 0
\(659\) −13.0000 −0.506408 −0.253204 0.967413i \(-0.581484\pi\)
−0.253204 + 0.967413i \(0.581484\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 4.00000i 0.155464i
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 5.00000 0.193746
\(667\) 12.0000i 0.464642i
\(668\) 7.00000i 0.270838i
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 16.0000 0.617673
\(672\) 0 0
\(673\) 11.0000i 0.424019i 0.977268 + 0.212009i \(0.0680008\pi\)
−0.977268 + 0.212009i \(0.931999\pi\)
\(674\) 22.0000 0.847408
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) − 2.00000i − 0.0768662i −0.999261 0.0384331i \(-0.987763\pi\)
0.999261 0.0384331i \(-0.0122367\pi\)
\(678\) − 2.00000i − 0.0768095i
\(679\) 0 0
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) 16.0000i 0.612672i
\(683\) − 28.0000i − 1.07139i −0.844411 0.535695i \(-0.820050\pi\)
0.844411 0.535695i \(-0.179950\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) − 5.00000i − 0.190762i
\(688\) 2.00000i 0.0762493i
\(689\) 1.00000 0.0380970
\(690\) 0 0
\(691\) −3.00000 −0.114125 −0.0570627 0.998371i \(-0.518173\pi\)
−0.0570627 + 0.998371i \(0.518173\pi\)
\(692\) 3.00000i 0.114043i
\(693\) 0 0
\(694\) −17.0000 −0.645311
\(695\) 0 0
\(696\) 3.00000 0.113715
\(697\) 0 0
\(698\) − 18.0000i − 0.681310i
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) − 1.00000i − 0.0377426i
\(703\) − 5.00000i − 0.188579i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 15.0000 0.564532
\(707\) 0 0
\(708\) 10.0000i 0.375823i
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 11.0000 0.412532
\(712\) 10.0000i 0.374766i
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 8.00000i 0.298765i
\(718\) 17.0000i 0.634434i
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 18.0000i − 0.669891i
\(723\) − 12.0000i − 0.446285i
\(724\) 12.0000 0.445976
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) − 52.0000i − 1.92857i −0.264861 0.964287i \(-0.585326\pi\)
0.264861 0.964287i \(-0.414674\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) − 4.00000i − 0.147844i
\(733\) − 49.0000i − 1.80986i −0.425564 0.904928i \(-0.639924\pi\)
0.425564 0.904928i \(-0.360076\pi\)
\(734\) −21.0000 −0.775124
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 36.0000i 1.32608i
\(738\) − 9.00000i − 0.331295i
\(739\) −53.0000 −1.94964 −0.974818 0.223001i \(-0.928415\pi\)
−0.974818 + 0.223001i \(0.928415\pi\)
\(740\) 0 0
\(741\) −1.00000 −0.0367359
\(742\) 0 0
\(743\) − 27.0000i − 0.990534i −0.868741 0.495267i \(-0.835070\pi\)
0.868741 0.495267i \(-0.164930\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −43.0000 −1.56909 −0.784546 0.620070i \(-0.787104\pi\)
−0.784546 + 0.620070i \(0.787104\pi\)
\(752\) 3.00000i 0.109399i
\(753\) 23.0000i 0.838167i
\(754\) 3.00000 0.109254
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 28.0000i 1.01701i
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) − 17.0000i − 0.615845i
\(763\) 0 0
\(764\) 18.0000 0.651217
\(765\) 0 0
\(766\) 5.00000 0.180657
\(767\) 10.0000i 0.361079i
\(768\) 1.00000i 0.0360844i
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) − 20.0000i − 0.719816i
\(773\) − 32.0000i − 1.15096i −0.817816 0.575480i \(-0.804815\pi\)
0.817816 0.575480i \(-0.195185\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) 12.0000 0.430775
\(777\) 0 0
\(778\) 19.0000i 0.681183i
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) 28.0000 1.00192
\(782\) 0 0
\(783\) − 3.00000i − 0.107211i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 3.00000 0.107006
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 12.0000i 0.427482i
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 0 0
\(792\) 4.00000i 0.142134i
\(793\) − 4.00000i − 0.142044i
\(794\) 3.00000 0.106466
\(795\) 0 0
\(796\) 1.00000 0.0354441
\(797\) 22.0000i 0.779280i 0.920967 + 0.389640i \(0.127401\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) − 26.0000i − 0.918092i
\(803\) 16.0000i 0.564628i
\(804\) 9.00000 0.317406
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 11.0000i 0.387218i
\(808\) 2.00000i 0.0703598i
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) −20.0000 −0.701000
\(815\) 0 0
\(816\) 0 0
\(817\) − 2.00000i − 0.0699711i
\(818\) 40.0000i 1.39857i
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 23.0000i 0.802217i
\(823\) − 31.0000i − 1.08059i −0.841475 0.540296i \(-0.818312\pi\)
0.841475 0.540296i \(-0.181688\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 1.00000i 0.0346688i
\(833\) 0 0
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) − 4.00000i − 0.138260i
\(838\) 37.0000i 1.27814i
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 2.00000i 0.0689246i
\(843\) − 5.00000i − 0.172209i
\(844\) −24.0000 −0.826114
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) 0 0
\(848\) 1.00000i 0.0343401i
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −20.0000 −0.685591
\(852\) − 7.00000i − 0.239816i
\(853\) 41.0000i 1.40381i 0.712269 + 0.701907i \(0.247668\pi\)
−0.712269 + 0.701907i \(0.752332\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 4.00000i 0.136558i
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 3.00000i − 0.102180i
\(863\) 57.0000i 1.94030i 0.242500 + 0.970151i \(0.422032\pi\)
−0.242500 + 0.970151i \(0.577968\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 19.0000 0.645646
\(867\) 17.0000i 0.577350i
\(868\) 0 0
\(869\) −44.0000 −1.49260
\(870\) 0 0
\(871\) 9.00000 0.304953
\(872\) − 1.00000i − 0.0338643i
\(873\) − 12.0000i − 0.406138i
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) 23.0000i 0.776655i 0.921521 + 0.388327i \(0.126947\pi\)
−0.921521 + 0.388327i \(0.873053\pi\)
\(878\) 19.0000i 0.641219i
\(879\) 16.0000 0.539667
\(880\) 0 0
\(881\) 56.0000 1.88669 0.943344 0.331816i \(-0.107661\pi\)
0.943344 + 0.331816i \(0.107661\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 3.00000 0.100787
\(887\) 44.0000i 1.47738i 0.674048 + 0.738688i \(0.264554\pi\)
−0.674048 + 0.738688i \(0.735446\pi\)
\(888\) 5.00000i 0.167789i
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 14.0000i 0.468755i
\(893\) − 3.00000i − 0.100391i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000i 0.133556i
\(898\) 9.00000i 0.300334i
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 0 0
\(902\) 36.0000i 1.19867i
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) −12.0000 −0.398673
\(907\) 30.0000i 0.996134i 0.867139 + 0.498067i \(0.165957\pi\)
−0.867139 + 0.498067i \(0.834043\pi\)
\(908\) − 10.0000i − 0.331862i
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) 24.0000i 0.794284i
\(914\) 32.0000 1.05847
\(915\) 0 0
\(916\) 5.00000 0.165205
\(917\) 0 0
\(918\) 0 0
\(919\) −1.00000 −0.0329870 −0.0164935 0.999864i \(-0.505250\pi\)
−0.0164935 + 0.999864i \(0.505250\pi\)
\(920\) 0 0
\(921\) 3.00000 0.0988534
\(922\) − 24.0000i − 0.790398i
\(923\) − 7.00000i − 0.230408i
\(924\) 0 0
\(925\) 0 0
\(926\) 14.0000 0.460069
\(927\) 0 0
\(928\) 3.00000i 0.0984798i
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) − 8.00000i − 0.262049i
\(933\) − 30.0000i − 0.982156i
\(934\) 15.0000 0.490815
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) − 34.0000i − 1.11073i −0.831606 0.555366i \(-0.812578\pi\)
0.831606 0.555366i \(-0.187422\pi\)
\(938\) 0 0
\(939\) −1.00000 −0.0326338
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) 36.0000i 1.17232i
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) 11.0000i 0.357263i
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 32.0000 1.03767
\(952\) 0 0
\(953\) 56.0000i 1.81402i 0.421111 + 0.907009i \(0.361640\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 1.00000 0.0323762
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 12.0000i 0.387905i
\(958\) − 25.0000i − 0.807713i
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 5.00000i 0.161206i
\(963\) 9.00000i 0.290021i
\(964\) 12.0000 0.386494
\(965\) 0 0
\(966\) 0 0
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) −38.0000 −1.21760
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) − 30.0000i − 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) − 8.00000i − 0.255812i
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) −1.00000 −0.0319275
\(982\) − 32.0000i − 1.02116i
\(983\) − 32.0000i − 1.02064i −0.859984 0.510321i \(-0.829527\pi\)
0.859984 0.510321i \(-0.170473\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) − 1.00000i − 0.0318142i
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −7.00000 −0.222362 −0.111181 0.993800i \(-0.535463\pi\)
−0.111181 + 0.993800i \(0.535463\pi\)
\(992\) 4.00000i 0.127000i
\(993\) 4.00000i 0.126936i
\(994\) 0 0
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) 14.0000i 0.443384i 0.975117 + 0.221692i \(0.0711580\pi\)
−0.975117 + 0.221692i \(0.928842\pi\)
\(998\) 1.00000i 0.0316544i
\(999\) 5.00000 0.158193
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 1950.2.e.h.1249.2 2
3.2 odd 2 5850.2.e.f.5149.1 2
5.2 odd 4 1950.2.a.j.1.1 1
5.3 odd 4 1950.2.a.s.1.1 yes 1
5.4 even 2 inner 1950.2.e.h.1249.1 2
15.2 even 4 5850.2.a.bp.1.1 1
15.8 even 4 5850.2.a.l.1.1 1
15.14 odd 2 5850.2.e.f.5149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.j.1.1 1 5.2 odd 4
1950.2.a.s.1.1 yes 1 5.3 odd 4
1950.2.e.h.1249.1 2 5.4 even 2 inner
1950.2.e.h.1249.2 2 1.1 even 1 trivial
5850.2.a.l.1.1 1 15.8 even 4
5850.2.a.bp.1.1 1 15.2 even 4
5850.2.e.f.5149.1 2 3.2 odd 2
5850.2.e.f.5149.2 2 15.14 odd 2