Properties

Label 5700.2.f.n.3649.3
Level $5700$
Weight $2$
Character 5700.3649
Analytic conductor $45.515$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5700.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(45.5147291521\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.3
Root \(-1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 5700.3649
Dual form 5700.2.f.n.3649.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.60555i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -2.60555i q^{7} -1.00000 q^{9} -4.60555 q^{11} -4.60555i q^{13} +2.00000i q^{17} +1.00000 q^{19} +2.60555 q^{21} +2.00000i q^{23} -1.00000i q^{27} -2.60555 q^{29} +4.00000 q^{31} -4.60555i q^{33} +3.39445i q^{37} +4.60555 q^{39} +6.60555 q^{41} -10.6056i q^{43} +6.00000i q^{47} +0.211103 q^{49} -2.00000 q^{51} +1.00000i q^{57} -5.21110 q^{59} -7.21110 q^{61} +2.60555i q^{63} +4.00000i q^{67} -2.00000 q^{69} -9.21110 q^{71} -6.00000i q^{73} +12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +11.2111i q^{83} -2.60555i q^{87} -6.60555 q^{89} -12.0000 q^{91} +4.00000i q^{93} +16.6056i q^{97} +4.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 4 q^{11} + 4 q^{19} - 4 q^{21} + 4 q^{29} + 16 q^{31} + 4 q^{39} + 12 q^{41} - 28 q^{49} - 8 q^{51} + 8 q^{59} - 8 q^{69} - 8 q^{71} - 32 q^{79} + 4 q^{81} - 12 q^{89} - 48 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5700\mathbb{Z}\right)^\times\).

\(n\) \(1901\) \(2851\) \(3877\) \(4201\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.60555i − 0.984806i −0.870367 0.492403i \(-0.836119\pi\)
0.870367 0.492403i \(-0.163881\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.60555 −1.38863 −0.694313 0.719673i \(-0.744292\pi\)
−0.694313 + 0.719673i \(0.744292\pi\)
\(12\) 0 0
\(13\) − 4.60555i − 1.27735i −0.769477 0.638675i \(-0.779483\pi\)
0.769477 0.638675i \(-0.220517\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.60555 0.568578
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −2.60555 −0.483839 −0.241919 0.970296i \(-0.577777\pi\)
−0.241919 + 0.970296i \(0.577777\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) − 4.60555i − 0.801724i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.39445i 0.558044i 0.960285 + 0.279022i \(0.0900102\pi\)
−0.960285 + 0.279022i \(0.909990\pi\)
\(38\) 0 0
\(39\) 4.60555 0.737478
\(40\) 0 0
\(41\) 6.60555 1.03161 0.515807 0.856705i \(-0.327492\pi\)
0.515807 + 0.856705i \(0.327492\pi\)
\(42\) 0 0
\(43\) − 10.6056i − 1.61733i −0.588268 0.808666i \(-0.700190\pi\)
0.588268 0.808666i \(-0.299810\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) 0.211103 0.0301575
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 0 0
\(59\) −5.21110 −0.678428 −0.339214 0.940709i \(-0.610161\pi\)
−0.339214 + 0.940709i \(0.610161\pi\)
\(60\) 0 0
\(61\) −7.21110 −0.923287 −0.461644 0.887066i \(-0.652740\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 0 0
\(63\) 2.60555i 0.328269i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −9.21110 −1.09316 −0.546578 0.837408i \(-0.684070\pi\)
−0.546578 + 0.837408i \(0.684070\pi\)
\(72\) 0 0
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000i 1.36753i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.2111i 1.23058i 0.788301 + 0.615289i \(0.210961\pi\)
−0.788301 + 0.615289i \(0.789039\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.60555i − 0.279344i
\(88\) 0 0
\(89\) −6.60555 −0.700187 −0.350094 0.936715i \(-0.613850\pi\)
−0.350094 + 0.936715i \(0.613850\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.6056i 1.68604i 0.537884 + 0.843019i \(0.319224\pi\)
−0.537884 + 0.843019i \(0.680776\pi\)
\(98\) 0 0
\(99\) 4.60555 0.462875
\(100\) 0 0
\(101\) −7.21110 −0.717532 −0.358766 0.933428i \(-0.616802\pi\)
−0.358766 + 0.933428i \(0.616802\pi\)
\(102\) 0 0
\(103\) − 18.4222i − 1.81519i −0.419843 0.907597i \(-0.637915\pi\)
0.419843 0.907597i \(-0.362085\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.4222i 1.78094i 0.455040 + 0.890471i \(0.349625\pi\)
−0.455040 + 0.890471i \(0.650375\pi\)
\(108\) 0 0
\(109\) 11.2111 1.07383 0.536914 0.843637i \(-0.319590\pi\)
0.536914 + 0.843637i \(0.319590\pi\)
\(110\) 0 0
\(111\) −3.39445 −0.322187
\(112\) 0 0
\(113\) 1.21110i 0.113931i 0.998376 + 0.0569655i \(0.0181425\pi\)
−0.998376 + 0.0569655i \(0.981858\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.60555i 0.425783i
\(118\) 0 0
\(119\) 5.21110 0.477701
\(120\) 0 0
\(121\) 10.2111 0.928282
\(122\) 0 0
\(123\) 6.60555i 0.595603i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 18.4222i 1.63471i 0.576137 + 0.817353i \(0.304559\pi\)
−0.576137 + 0.817353i \(0.695441\pi\)
\(128\) 0 0
\(129\) 10.6056 0.933767
\(130\) 0 0
\(131\) −15.3944 −1.34502 −0.672510 0.740088i \(-0.734784\pi\)
−0.672510 + 0.740088i \(0.734784\pi\)
\(132\) 0 0
\(133\) − 2.60555i − 0.225930i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.4222i − 1.06130i −0.847591 0.530650i \(-0.821948\pi\)
0.847591 0.530650i \(-0.178052\pi\)
\(138\) 0 0
\(139\) −9.21110 −0.781276 −0.390638 0.920544i \(-0.627745\pi\)
−0.390638 + 0.920544i \(0.627745\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 21.2111i 1.77376i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0.211103i 0.0174114i
\(148\) 0 0
\(149\) −16.4222 −1.34536 −0.672680 0.739934i \(-0.734857\pi\)
−0.672680 + 0.739934i \(0.734857\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.2111i 1.21398i 0.794710 + 0.606989i \(0.207623\pi\)
−0.794710 + 0.606989i \(0.792377\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.21110 0.410692
\(162\) 0 0
\(163\) 17.0278i 1.33372i 0.745184 + 0.666858i \(0.232361\pi\)
−0.745184 + 0.666858i \(0.767639\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 6.78890i − 0.525341i −0.964886 0.262670i \(-0.915397\pi\)
0.964886 0.262670i \(-0.0846032\pi\)
\(168\) 0 0
\(169\) −8.21110 −0.631623
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 9.21110i 0.700307i 0.936692 + 0.350154i \(0.113871\pi\)
−0.936692 + 0.350154i \(0.886129\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 5.21110i − 0.391690i
\(178\) 0 0
\(179\) 21.2111 1.58539 0.792696 0.609617i \(-0.208677\pi\)
0.792696 + 0.609617i \(0.208677\pi\)
\(180\) 0 0
\(181\) −0.788897 −0.0586383 −0.0293191 0.999570i \(-0.509334\pi\)
−0.0293191 + 0.999570i \(0.509334\pi\)
\(182\) 0 0
\(183\) − 7.21110i − 0.533060i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 9.21110i − 0.673583i
\(188\) 0 0
\(189\) −2.60555 −0.189526
\(190\) 0 0
\(191\) −5.81665 −0.420878 −0.210439 0.977607i \(-0.567489\pi\)
−0.210439 + 0.977607i \(0.567489\pi\)
\(192\) 0 0
\(193\) 0.605551i 0.0435885i 0.999762 + 0.0217943i \(0.00693788\pi\)
−0.999762 + 0.0217943i \(0.993062\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) 0 0
\(199\) −17.2111 −1.22006 −0.610031 0.792377i \(-0.708843\pi\)
−0.610031 + 0.792377i \(0.708843\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 6.78890i 0.476487i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.00000i − 0.139010i
\(208\) 0 0
\(209\) −4.60555 −0.318573
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) − 9.21110i − 0.631134i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 10.4222i − 0.707505i
\(218\) 0 0
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 9.21110 0.619606
\(222\) 0 0
\(223\) 10.4222i 0.697922i 0.937137 + 0.348961i \(0.113466\pi\)
−0.937137 + 0.348961i \(0.886534\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 13.2111i − 0.876852i −0.898767 0.438426i \(-0.855536\pi\)
0.898767 0.438426i \(-0.144464\pi\)
\(228\) 0 0
\(229\) 3.21110 0.212196 0.106098 0.994356i \(-0.466164\pi\)
0.106098 + 0.994356i \(0.466164\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) 16.4222i 1.07585i 0.842991 + 0.537927i \(0.180792\pi\)
−0.842991 + 0.537927i \(0.819208\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.00000i − 0.519656i
\(238\) 0 0
\(239\) −25.8167 −1.66994 −0.834970 0.550295i \(-0.814515\pi\)
−0.834970 + 0.550295i \(0.814515\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.60555i − 0.293044i
\(248\) 0 0
\(249\) −11.2111 −0.710475
\(250\) 0 0
\(251\) −6.18335 −0.390289 −0.195145 0.980774i \(-0.562518\pi\)
−0.195145 + 0.980774i \(0.562518\pi\)
\(252\) 0 0
\(253\) − 9.21110i − 0.579097i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.78890i 0.173967i 0.996210 + 0.0869833i \(0.0277227\pi\)
−0.996210 + 0.0869833i \(0.972277\pi\)
\(258\) 0 0
\(259\) 8.84441 0.549565
\(260\) 0 0
\(261\) 2.60555 0.161280
\(262\) 0 0
\(263\) − 7.21110i − 0.444656i −0.974972 0.222328i \(-0.928634\pi\)
0.974972 0.222328i \(-0.0713655\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.60555i − 0.404253i
\(268\) 0 0
\(269\) 17.3944 1.06056 0.530279 0.847823i \(-0.322087\pi\)
0.530279 + 0.847823i \(0.322087\pi\)
\(270\) 0 0
\(271\) 11.6333 0.706673 0.353337 0.935496i \(-0.385047\pi\)
0.353337 + 0.935496i \(0.385047\pi\)
\(272\) 0 0
\(273\) − 12.0000i − 0.726273i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 18.2389 1.08804 0.544020 0.839073i \(-0.316902\pi\)
0.544020 + 0.839073i \(0.316902\pi\)
\(282\) 0 0
\(283\) 18.6056i 1.10599i 0.833186 + 0.552993i \(0.186514\pi\)
−0.833186 + 0.552993i \(0.813486\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 17.2111i − 1.01594i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −16.6056 −0.973435
\(292\) 0 0
\(293\) − 19.6333i − 1.14699i −0.819209 0.573495i \(-0.805587\pi\)
0.819209 0.573495i \(-0.194413\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.60555i 0.267241i
\(298\) 0 0
\(299\) 9.21110 0.532692
\(300\) 0 0
\(301\) −27.6333 −1.59276
\(302\) 0 0
\(303\) − 7.21110i − 0.414267i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 11.6333i − 0.663948i −0.943289 0.331974i \(-0.892285\pi\)
0.943289 0.331974i \(-0.107715\pi\)
\(308\) 0 0
\(309\) 18.4222 1.04800
\(310\) 0 0
\(311\) 16.6056 0.941614 0.470807 0.882236i \(-0.343963\pi\)
0.470807 + 0.882236i \(0.343963\pi\)
\(312\) 0 0
\(313\) − 0.788897i − 0.0445911i −0.999751 0.0222956i \(-0.992903\pi\)
0.999751 0.0222956i \(-0.00709749\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.42221i − 0.360707i −0.983602 0.180353i \(-0.942276\pi\)
0.983602 0.180353i \(-0.0577242\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −18.4222 −1.02823
\(322\) 0 0
\(323\) 2.00000i 0.111283i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.2111i 0.619975i
\(328\) 0 0
\(329\) 15.6333 0.861892
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) − 3.39445i − 0.186015i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.02776i 0.164932i 0.996594 + 0.0824662i \(0.0262797\pi\)
−0.996594 + 0.0824662i \(0.973720\pi\)
\(338\) 0 0
\(339\) −1.21110 −0.0657781
\(340\) 0 0
\(341\) −18.4222 −0.997618
\(342\) 0 0
\(343\) − 18.7889i − 1.01451i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.6333i 1.16134i 0.814140 + 0.580668i \(0.197209\pi\)
−0.814140 + 0.580668i \(0.802791\pi\)
\(348\) 0 0
\(349\) −34.8444 −1.86518 −0.932589 0.360939i \(-0.882456\pi\)
−0.932589 + 0.360939i \(0.882456\pi\)
\(350\) 0 0
\(351\) −4.60555 −0.245826
\(352\) 0 0
\(353\) − 0.422205i − 0.0224717i −0.999937 0.0112359i \(-0.996423\pi\)
0.999937 0.0112359i \(-0.00357656\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 5.21110i 0.275801i
\(358\) 0 0
\(359\) 13.8167 0.729215 0.364608 0.931161i \(-0.381203\pi\)
0.364608 + 0.931161i \(0.381203\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 10.2111i 0.535944i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 25.0278i 1.30644i 0.757169 + 0.653219i \(0.226582\pi\)
−0.757169 + 0.653219i \(0.773418\pi\)
\(368\) 0 0
\(369\) −6.60555 −0.343871
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 16.6056i − 0.859803i −0.902876 0.429901i \(-0.858548\pi\)
0.902876 0.429901i \(-0.141452\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −26.4222 −1.35722 −0.678609 0.734500i \(-0.737417\pi\)
−0.678609 + 0.734500i \(0.737417\pi\)
\(380\) 0 0
\(381\) −18.4222 −0.943798
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.6056i 0.539110i
\(388\) 0 0
\(389\) −36.4222 −1.84668 −0.923340 0.383984i \(-0.874552\pi\)
−0.923340 + 0.383984i \(0.874552\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) − 15.3944i − 0.776547i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 7.57779i − 0.380319i −0.981753 0.190159i \(-0.939100\pi\)
0.981753 0.190159i \(-0.0609005\pi\)
\(398\) 0 0
\(399\) 2.60555 0.130441
\(400\) 0 0
\(401\) 25.3944 1.26814 0.634069 0.773276i \(-0.281383\pi\)
0.634069 + 0.773276i \(0.281383\pi\)
\(402\) 0 0
\(403\) − 18.4222i − 0.917675i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 15.6333i − 0.774914i
\(408\) 0 0
\(409\) 12.7889 0.632370 0.316185 0.948698i \(-0.397598\pi\)
0.316185 + 0.948698i \(0.397598\pi\)
\(410\) 0 0
\(411\) 12.4222 0.612742
\(412\) 0 0
\(413\) 13.5778i 0.668120i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 9.21110i − 0.451070i
\(418\) 0 0
\(419\) 7.02776 0.343328 0.171664 0.985156i \(-0.445086\pi\)
0.171664 + 0.985156i \(0.445086\pi\)
\(420\) 0 0
\(421\) −8.42221 −0.410473 −0.205237 0.978712i \(-0.565796\pi\)
−0.205237 + 0.978712i \(0.565796\pi\)
\(422\) 0 0
\(423\) − 6.00000i − 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.7889i 0.909258i
\(428\) 0 0
\(429\) −21.2111 −1.02408
\(430\) 0 0
\(431\) −9.21110 −0.443683 −0.221842 0.975083i \(-0.571207\pi\)
−0.221842 + 0.975083i \(0.571207\pi\)
\(432\) 0 0
\(433\) 16.6056i 0.798012i 0.916948 + 0.399006i \(0.130645\pi\)
−0.916948 + 0.399006i \(0.869355\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000i 0.0956730i
\(438\) 0 0
\(439\) −34.4222 −1.64288 −0.821441 0.570293i \(-0.806830\pi\)
−0.821441 + 0.570293i \(0.806830\pi\)
\(440\) 0 0
\(441\) −0.211103 −0.0100525
\(442\) 0 0
\(443\) − 18.8444i − 0.895325i −0.894203 0.447662i \(-0.852257\pi\)
0.894203 0.447662i \(-0.147743\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 16.4222i − 0.776744i
\(448\) 0 0
\(449\) 18.2389 0.860745 0.430372 0.902651i \(-0.358382\pi\)
0.430372 + 0.902651i \(0.358382\pi\)
\(450\) 0 0
\(451\) −30.4222 −1.43253
\(452\) 0 0
\(453\) − 12.0000i − 0.563809i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 32.0555i 1.49949i 0.661725 + 0.749747i \(0.269825\pi\)
−0.661725 + 0.749747i \(0.730175\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −22.8444 −1.06397 −0.531985 0.846754i \(-0.678554\pi\)
−0.531985 + 0.846754i \(0.678554\pi\)
\(462\) 0 0
\(463\) − 33.0278i − 1.53493i −0.641091 0.767465i \(-0.721518\pi\)
0.641091 0.767465i \(-0.278482\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.8444i 1.05711i 0.848898 + 0.528557i \(0.177267\pi\)
−0.848898 + 0.528557i \(0.822733\pi\)
\(468\) 0 0
\(469\) 10.4222 0.481253
\(470\) 0 0
\(471\) −15.2111 −0.700891
\(472\) 0 0
\(473\) 48.8444i 2.24587i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −28.6056 −1.30702 −0.653510 0.756917i \(-0.726704\pi\)
−0.653510 + 0.756917i \(0.726704\pi\)
\(480\) 0 0
\(481\) 15.6333 0.712817
\(482\) 0 0
\(483\) 5.21110i 0.237113i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0.366692i 0.0166164i 0.999965 + 0.00830821i \(0.00264462\pi\)
−0.999965 + 0.00830821i \(0.997355\pi\)
\(488\) 0 0
\(489\) −17.0278 −0.770022
\(490\) 0 0
\(491\) −12.2389 −0.552332 −0.276166 0.961110i \(-0.589064\pi\)
−0.276166 + 0.961110i \(0.589064\pi\)
\(492\) 0 0
\(493\) − 5.21110i − 0.234696i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000i 1.07655i
\(498\) 0 0
\(499\) 27.6333 1.23704 0.618518 0.785770i \(-0.287733\pi\)
0.618518 + 0.785770i \(0.287733\pi\)
\(500\) 0 0
\(501\) 6.78890 0.303306
\(502\) 0 0
\(503\) − 12.7889i − 0.570229i −0.958494 0.285114i \(-0.907968\pi\)
0.958494 0.285114i \(-0.0920316\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 8.21110i − 0.364668i
\(508\) 0 0
\(509\) 35.4500 1.57129 0.785646 0.618676i \(-0.212331\pi\)
0.785646 + 0.618676i \(0.212331\pi\)
\(510\) 0 0
\(511\) −15.6333 −0.691577
\(512\) 0 0
\(513\) − 1.00000i − 0.0441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 27.6333i − 1.21531i
\(518\) 0 0
\(519\) −9.21110 −0.404323
\(520\) 0 0
\(521\) −4.18335 −0.183276 −0.0916379 0.995792i \(-0.529210\pi\)
−0.0916379 + 0.995792i \(0.529210\pi\)
\(522\) 0 0
\(523\) 25.2111i 1.10240i 0.834372 + 0.551202i \(0.185831\pi\)
−0.834372 + 0.551202i \(0.814169\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 5.21110 0.226143
\(532\) 0 0
\(533\) − 30.4222i − 1.31773i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 21.2111i 0.915327i
\(538\) 0 0
\(539\) −0.972244 −0.0418775
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) − 0.788897i − 0.0338548i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 22.7889i − 0.974383i −0.873295 0.487191i \(-0.838021\pi\)
0.873295 0.487191i \(-0.161979\pi\)
\(548\) 0 0
\(549\) 7.21110 0.307762
\(550\) 0 0
\(551\) −2.60555 −0.111000
\(552\) 0 0
\(553\) 20.8444i 0.886394i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 4.42221i − 0.187375i −0.995602 0.0936874i \(-0.970135\pi\)
0.995602 0.0936874i \(-0.0298654\pi\)
\(558\) 0 0
\(559\) −48.8444 −2.06590
\(560\) 0 0
\(561\) 9.21110 0.388893
\(562\) 0 0
\(563\) − 31.6333i − 1.33318i −0.745422 0.666592i \(-0.767752\pi\)
0.745422 0.666592i \(-0.232248\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2.60555i − 0.109423i
\(568\) 0 0
\(569\) −39.4500 −1.65383 −0.826914 0.562328i \(-0.809906\pi\)
−0.826914 + 0.562328i \(0.809906\pi\)
\(570\) 0 0
\(571\) 7.63331 0.319444 0.159722 0.987162i \(-0.448940\pi\)
0.159722 + 0.987162i \(0.448940\pi\)
\(572\) 0 0
\(573\) − 5.81665i − 0.242994i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 23.2111i − 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(578\) 0 0
\(579\) −0.605551 −0.0251659
\(580\) 0 0
\(581\) 29.2111 1.21188
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.4222i 0.512719i 0.966581 + 0.256360i \(0.0825231\pi\)
−0.966581 + 0.256360i \(0.917477\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −14.0000 −0.575883
\(592\) 0 0
\(593\) 3.57779i 0.146922i 0.997298 + 0.0734612i \(0.0234045\pi\)
−0.997298 + 0.0734612i \(0.976595\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 17.2111i − 0.704404i
\(598\) 0 0
\(599\) −36.8444 −1.50542 −0.752711 0.658351i \(-0.771254\pi\)
−0.752711 + 0.658351i \(0.771254\pi\)
\(600\) 0 0
\(601\) 4.78890 0.195343 0.0976716 0.995219i \(-0.468861\pi\)
0.0976716 + 0.995219i \(0.468861\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 23.6333i − 0.959246i −0.877475 0.479623i \(-0.840773\pi\)
0.877475 0.479623i \(-0.159227\pi\)
\(608\) 0 0
\(609\) −6.78890 −0.275100
\(610\) 0 0
\(611\) 27.6333 1.11792
\(612\) 0 0
\(613\) − 4.78890i − 0.193422i −0.995313 0.0967109i \(-0.969168\pi\)
0.995313 0.0967109i \(-0.0308322\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000i 1.36879i 0.729112 + 0.684394i \(0.239933\pi\)
−0.729112 + 0.684394i \(0.760067\pi\)
\(618\) 0 0
\(619\) 4.36669 0.175512 0.0877561 0.996142i \(-0.472030\pi\)
0.0877561 + 0.996142i \(0.472030\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) 0 0
\(623\) 17.2111i 0.689548i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 4.60555i − 0.183928i
\(628\) 0 0
\(629\) −6.78890 −0.270691
\(630\) 0 0
\(631\) 36.8444 1.46675 0.733376 0.679823i \(-0.237943\pi\)
0.733376 + 0.679823i \(0.237943\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 0.972244i − 0.0385217i
\(638\) 0 0
\(639\) 9.21110 0.364386
\(640\) 0 0
\(641\) 35.4500 1.40019 0.700095 0.714050i \(-0.253141\pi\)
0.700095 + 0.714050i \(0.253141\pi\)
\(642\) 0 0
\(643\) − 5.39445i − 0.212736i −0.994327 0.106368i \(-0.966078\pi\)
0.994327 0.106368i \(-0.0339222\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 2.36669i − 0.0930443i −0.998917 0.0465221i \(-0.985186\pi\)
0.998917 0.0465221i \(-0.0148138\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 10.4222 0.408478
\(652\) 0 0
\(653\) − 42.0000i − 1.64359i −0.569785 0.821794i \(-0.692974\pi\)
0.569785 0.821794i \(-0.307026\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −31.2111 −1.21397 −0.606986 0.794713i \(-0.707621\pi\)
−0.606986 + 0.794713i \(0.707621\pi\)
\(662\) 0 0
\(663\) 9.21110i 0.357730i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 5.21110i − 0.201775i
\(668\) 0 0
\(669\) −10.4222 −0.402946
\(670\) 0 0
\(671\) 33.2111 1.28210
\(672\) 0 0
\(673\) − 36.6056i − 1.41104i −0.708690 0.705520i \(-0.750713\pi\)
0.708690 0.705520i \(-0.249287\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.4222i 0.400558i 0.979739 + 0.200279i \(0.0641848\pi\)
−0.979739 + 0.200279i \(0.935815\pi\)
\(678\) 0 0
\(679\) 43.2666 1.66042
\(680\) 0 0
\(681\) 13.2111 0.506251
\(682\) 0 0
\(683\) 48.0000i 1.83667i 0.395805 + 0.918334i \(0.370466\pi\)
−0.395805 + 0.918334i \(0.629534\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 3.21110i 0.122511i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −42.0555 −1.59987 −0.799934 0.600089i \(-0.795132\pi\)
−0.799934 + 0.600089i \(0.795132\pi\)
\(692\) 0 0
\(693\) − 12.0000i − 0.455842i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13.2111i 0.500406i
\(698\) 0 0
\(699\) −16.4222 −0.621145
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 3.39445i 0.128024i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.7889i 0.706629i
\(708\) 0 0
\(709\) 25.6333 0.962679 0.481340 0.876534i \(-0.340150\pi\)
0.481340 + 0.876534i \(0.340150\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 25.8167i − 0.964141i
\(718\) 0 0
\(719\) −16.2389 −0.605607 −0.302804 0.953053i \(-0.597923\pi\)
−0.302804 + 0.953053i \(0.597923\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 0 0
\(723\) − 10.0000i − 0.371904i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 17.3944i − 0.645124i −0.946548 0.322562i \(-0.895456\pi\)
0.946548 0.322562i \(-0.104544\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 21.2111 0.784521
\(732\) 0 0
\(733\) 24.4222i 0.902055i 0.892510 + 0.451027i \(0.148942\pi\)
−0.892510 + 0.451027i \(0.851058\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 18.4222i − 0.678591i
\(738\) 0 0
\(739\) −38.4222 −1.41338 −0.706692 0.707521i \(-0.749813\pi\)
−0.706692 + 0.707521i \(0.749813\pi\)
\(740\) 0 0
\(741\) 4.60555 0.169189
\(742\) 0 0
\(743\) 20.0000i 0.733729i 0.930274 + 0.366864i \(0.119569\pi\)
−0.930274 + 0.366864i \(0.880431\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 11.2111i − 0.410193i
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 40.8444 1.49043 0.745217 0.666822i \(-0.232346\pi\)
0.745217 + 0.666822i \(0.232346\pi\)
\(752\) 0 0
\(753\) − 6.18335i − 0.225334i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 36.0555i − 1.31046i −0.755429 0.655230i \(-0.772572\pi\)
0.755429 0.655230i \(-0.227428\pi\)
\(758\) 0 0
\(759\) 9.21110 0.334342
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) − 29.2111i − 1.05751i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) 20.4222 0.736444 0.368222 0.929738i \(-0.379967\pi\)
0.368222 + 0.929738i \(0.379967\pi\)
\(770\) 0 0
\(771\) −2.78890 −0.100440
\(772\) 0 0
\(773\) 43.2666i 1.55619i 0.628145 + 0.778096i \(0.283814\pi\)
−0.628145 + 0.778096i \(0.716186\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 8.84441i 0.317291i
\(778\) 0 0
\(779\) 6.60555 0.236668
\(780\) 0 0
\(781\) 42.4222 1.51799
\(782\) 0 0
\(783\) 2.60555i 0.0931148i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 19.6333i − 0.699852i −0.936777 0.349926i \(-0.886207\pi\)
0.936777 0.349926i \(-0.113793\pi\)
\(788\) 0 0
\(789\) 7.21110 0.256722
\(790\) 0 0
\(791\) 3.15559 0.112200
\(792\) 0 0
\(793\) 33.2111i 1.17936i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.2111i 0.609649i 0.952409 + 0.304824i \(0.0985977\pi\)
−0.952409 + 0.304824i \(0.901402\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 6.60555 0.233396
\(802\) 0 0
\(803\) 27.6333i 0.975158i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 17.3944i 0.612314i
\(808\) 0 0
\(809\) 48.4222 1.70243 0.851217 0.524814i \(-0.175865\pi\)
0.851217 + 0.524814i \(0.175865\pi\)
\(810\) 0 0
\(811\) −40.8444 −1.43424 −0.717121 0.696949i \(-0.754540\pi\)
−0.717121 + 0.696949i \(0.754540\pi\)
\(812\) 0 0
\(813\) 11.6333i 0.407998i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 10.6056i − 0.371041i
\(818\) 0 0
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) 43.2111 1.50808 0.754039 0.656830i \(-0.228103\pi\)
0.754039 + 0.656830i \(0.228103\pi\)
\(822\) 0 0
\(823\) 47.8167i 1.66678i 0.552683 + 0.833392i \(0.313604\pi\)
−0.552683 + 0.833392i \(0.686396\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 27.6333i − 0.960904i −0.877021 0.480452i \(-0.840473\pi\)
0.877021 0.480452i \(-0.159527\pi\)
\(828\) 0 0
\(829\) −15.2111 −0.528303 −0.264152 0.964481i \(-0.585092\pi\)
−0.264152 + 0.964481i \(0.585092\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 0.422205i 0.0146285i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 4.00000i − 0.138260i
\(838\) 0 0
\(839\) −27.6333 −0.954008 −0.477004 0.878901i \(-0.658277\pi\)
−0.477004 + 0.878901i \(0.658277\pi\)
\(840\) 0 0
\(841\) −22.2111 −0.765900
\(842\) 0 0
\(843\) 18.2389i 0.628180i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 26.6056i − 0.914178i
\(848\) 0 0
\(849\) −18.6056 −0.638541
\(850\) 0 0
\(851\) −6.78890 −0.232720
\(852\) 0 0
\(853\) − 29.6333i − 1.01463i −0.861762 0.507313i \(-0.830639\pi\)
0.861762 0.507313i \(-0.169361\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.42221i 0.219378i 0.993966 + 0.109689i \(0.0349855\pi\)
−0.993966 + 0.109689i \(0.965014\pi\)
\(858\) 0 0
\(859\) −8.84441 −0.301767 −0.150884 0.988552i \(-0.548212\pi\)
−0.150884 + 0.988552i \(0.548212\pi\)
\(860\) 0 0
\(861\) 17.2111 0.586553
\(862\) 0 0
\(863\) − 6.42221i − 0.218614i −0.994008 0.109307i \(-0.965137\pi\)
0.994008 0.109307i \(-0.0348632\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 36.8444 1.24986
\(870\) 0 0
\(871\) 18.4222 0.624213
\(872\) 0 0
\(873\) − 16.6056i − 0.562013i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 27.0278i 0.912662i 0.889810 + 0.456331i \(0.150837\pi\)
−0.889810 + 0.456331i \(0.849163\pi\)
\(878\) 0 0
\(879\) 19.6333 0.662215
\(880\) 0 0
\(881\) 59.2111 1.99487 0.997436 0.0715590i \(-0.0227974\pi\)
0.997436 + 0.0715590i \(0.0227974\pi\)
\(882\) 0 0
\(883\) − 27.8167i − 0.936105i −0.883701 0.468052i \(-0.844956\pi\)
0.883701 0.468052i \(-0.155044\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 20.8444i − 0.699887i −0.936771 0.349943i \(-0.886201\pi\)
0.936771 0.349943i \(-0.113799\pi\)
\(888\) 0 0
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) −4.60555 −0.154292
\(892\) 0 0
\(893\) 6.00000i 0.200782i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.21110i 0.307550i
\(898\) 0 0
\(899\) −10.4222 −0.347600
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 27.6333i − 0.919579i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 30.0555i 0.997977i 0.866609 + 0.498988i \(0.166295\pi\)
−0.866609 + 0.498988i \(0.833705\pi\)
\(908\) 0 0
\(909\) 7.21110 0.239177
\(910\) 0 0
\(911\) 10.4222 0.345303 0.172652 0.984983i \(-0.444767\pi\)
0.172652 + 0.984983i \(0.444767\pi\)
\(912\) 0 0
\(913\) − 51.6333i − 1.70881i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 40.1110i 1.32458i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 11.6333 0.383331
\(922\) 0 0
\(923\) 42.4222i 1.39634i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 18.4222i 0.605065i
\(928\) 0 0
\(929\) 45.6333 1.49718 0.748590 0.663033i \(-0.230731\pi\)
0.748590 + 0.663033i \(0.230731\pi\)
\(930\) 0 0
\(931\) 0.211103 0.00691861
\(932\) 0 0
\(933\) 16.6056i 0.543641i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 19.2111i − 0.627599i −0.949489 0.313800i \(-0.898398\pi\)
0.949489 0.313800i \(-0.101602\pi\)
\(938\) 0 0
\(939\) 0.788897 0.0257447
\(940\) 0 0
\(941\) −30.2389 −0.985759 −0.492879 0.870098i \(-0.664056\pi\)
−0.492879 + 0.870098i \(0.664056\pi\)
\(942\) 0 0
\(943\) 13.2111i 0.430213i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 7.21110i − 0.234329i −0.993113 0.117165i \(-0.962619\pi\)
0.993113 0.117165i \(-0.0373805\pi\)
\(948\) 0 0
\(949\) −27.6333 −0.897015
\(950\) 0 0
\(951\) 6.42221 0.208254
\(952\) 0 0
\(953\) − 5.21110i − 0.168804i −0.996432 0.0844021i \(-0.973102\pi\)
0.996432 0.0844021i \(-0.0268980\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 12.0000i 0.387905i
\(958\) 0 0
\(959\) −32.3667 −1.04518
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 18.4222i − 0.593647i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 26.2389i − 0.843785i −0.906646 0.421892i \(-0.861366\pi\)
0.906646 0.421892i \(-0.138634\pi\)
\(968\) 0 0
\(969\) −2.00000 −0.0642493
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 24.0000i 0.769405i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.42221i 0.205465i 0.994709 + 0.102732i \(0.0327585\pi\)
−0.994709 + 0.102732i \(0.967242\pi\)
\(978\) 0 0
\(979\) 30.4222 0.972298
\(980\) 0 0
\(981\) −11.2111 −0.357943
\(982\) 0 0
\(983\) 9.21110i 0.293789i 0.989152 + 0.146894i \(0.0469277\pi\)
−0.989152 + 0.146894i \(0.953072\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 15.6333i 0.497614i
\(988\) 0 0
\(989\) 21.2111 0.674474
\(990\) 0 0
\(991\) 50.4222 1.60171 0.800857 0.598856i \(-0.204378\pi\)
0.800857 + 0.598856i \(0.204378\pi\)
\(992\) 0 0
\(993\) − 8.00000i − 0.253872i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 61.6333i − 1.95195i −0.217891 0.975973i \(-0.569918\pi\)
0.217891 0.975973i \(-0.430082\pi\)
\(998\) 0 0
\(999\) 3.39445 0.107396
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 5700.2.f.n.3649.3 4
5.2 odd 4 5700.2.a.u.1.2 2
5.3 odd 4 1140.2.a.e.1.1 2
5.4 even 2 inner 5700.2.f.n.3649.2 4
15.8 even 4 3420.2.a.i.1.1 2
20.3 even 4 4560.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.e.1.1 2 5.3 odd 4
3420.2.a.i.1.1 2 15.8 even 4
4560.2.a.bl.1.2 2 20.3 even 4
5700.2.a.u.1.2 2 5.2 odd 4
5700.2.f.n.3649.2 4 5.4 even 2 inner
5700.2.f.n.3649.3 4 1.1 even 1 trivial