Properties

Label 5850.2.e.bk.5149.1
Level $5850$
Weight $2$
Character 5850.5149
Analytic conductor $46.712$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(46.7124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5149.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 5850.5149
Dual form 5850.2.e.bk.5149.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -2.82843i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -2.82843i q^{7} +1.00000i q^{8} -5.65685 q^{11} +1.00000i q^{13} -2.82843 q^{14} +1.00000 q^{16} -0.828427i q^{17} -2.82843 q^{19} +5.65685i q^{22} -8.48528i q^{23} +1.00000 q^{26} +2.82843i q^{28} -8.82843 q^{29} +4.00000 q^{31} -1.00000i q^{32} -0.828427 q^{34} -11.6569i q^{37} +2.82843i q^{38} +7.65685 q^{41} -9.65685i q^{43} +5.65685 q^{44} -8.48528 q^{46} +8.00000i q^{47} -1.00000 q^{49} -1.00000i q^{52} +13.3137i q^{53} +2.82843 q^{56} +8.82843i q^{58} -2.34315 q^{59} +6.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +5.65685i q^{67} +0.828427i q^{68} +5.65685 q^{71} +14.4853i q^{73} -11.6569 q^{74} +2.82843 q^{76} +16.0000i q^{77} -2.34315 q^{79} -7.65685i q^{82} +6.34315i q^{83} -9.65685 q^{86} -5.65685i q^{88} -15.6569 q^{89} +2.82843 q^{91} +8.48528i q^{92} +8.00000 q^{94} -3.17157i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{16} + 4q^{26} - 24q^{29} + 16q^{31} + 8q^{34} + 8q^{41} - 4q^{49} - 32q^{59} + 24q^{61} - 4q^{64} - 24q^{74} - 32q^{79} - 16q^{86} - 40q^{89} + 32q^{94} + O(q^{100}) \)

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −5.65685 −1.70561 −0.852803 0.522233i \(-0.825099\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) −2.82843 −0.755929
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 0.828427i − 0.200923i −0.994941 0.100462i \(-0.967968\pi\)
0.994941 0.100462i \(-0.0320319\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.65685i 1.20605i
\(23\) − 8.48528i − 1.76930i −0.466252 0.884652i \(-0.654396\pi\)
0.466252 0.884652i \(-0.345604\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 2.82843i 0.534522i
\(29\) −8.82843 −1.63940 −0.819699 0.572795i \(-0.805859\pi\)
−0.819699 + 0.572795i \(0.805859\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −0.828427 −0.142074
\(35\) 0 0
\(36\) 0 0
\(37\) − 11.6569i − 1.91638i −0.286141 0.958188i \(-0.592373\pi\)
0.286141 0.958188i \(-0.407627\pi\)
\(38\) 2.82843i 0.458831i
\(39\) 0 0
\(40\) 0 0
\(41\) 7.65685 1.19580 0.597900 0.801571i \(-0.296002\pi\)
0.597900 + 0.801571i \(0.296002\pi\)
\(42\) 0 0
\(43\) − 9.65685i − 1.47266i −0.676625 0.736328i \(-0.736558\pi\)
0.676625 0.736328i \(-0.263442\pi\)
\(44\) 5.65685 0.852803
\(45\) 0 0
\(46\) −8.48528 −1.25109
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.00000i − 0.138675i
\(53\) 13.3137i 1.82878i 0.404836 + 0.914389i \(0.367329\pi\)
−0.404836 + 0.914389i \(0.632671\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.82843 0.377964
\(57\) 0 0
\(58\) 8.82843i 1.15923i
\(59\) −2.34315 −0.305052 −0.152526 0.988299i \(-0.548741\pi\)
−0.152526 + 0.988299i \(0.548741\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.65685i 0.691095i 0.938401 + 0.345547i \(0.112307\pi\)
−0.938401 + 0.345547i \(0.887693\pi\)
\(68\) 0.828427i 0.100462i
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 0 0
\(73\) 14.4853i 1.69537i 0.530497 + 0.847687i \(0.322005\pi\)
−0.530497 + 0.847687i \(0.677995\pi\)
\(74\) −11.6569 −1.35508
\(75\) 0 0
\(76\) 2.82843 0.324443
\(77\) 16.0000i 1.82337i
\(78\) 0 0
\(79\) −2.34315 −0.263624 −0.131812 0.991275i \(-0.542080\pi\)
−0.131812 + 0.991275i \(0.542080\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 7.65685i − 0.845558i
\(83\) 6.34315i 0.696251i 0.937448 + 0.348125i \(0.113182\pi\)
−0.937448 + 0.348125i \(0.886818\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −9.65685 −1.04133
\(87\) 0 0
\(88\) − 5.65685i − 0.603023i
\(89\) −15.6569 −1.65962 −0.829812 0.558044i \(-0.811552\pi\)
−0.829812 + 0.558044i \(0.811552\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 8.48528i 0.884652i
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) − 3.17157i − 0.322024i −0.986952 0.161012i \(-0.948524\pi\)
0.986952 0.161012i \(-0.0514759\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) −16.1421 −1.60620 −0.803101 0.595843i \(-0.796818\pi\)
−0.803101 + 0.595843i \(0.796818\pi\)
\(102\) 0 0
\(103\) 1.65685i 0.163255i 0.996663 + 0.0816274i \(0.0260117\pi\)
−0.996663 + 0.0816274i \(0.973988\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 13.3137 1.29314
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) −8.82843 −0.845610 −0.422805 0.906221i \(-0.638954\pi\)
−0.422805 + 0.906221i \(0.638954\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 2.82843i − 0.267261i
\(113\) 6.48528i 0.610084i 0.952339 + 0.305042i \(0.0986705\pi\)
−0.952339 + 0.305042i \(0.901330\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.82843 0.819699
\(117\) 0 0
\(118\) 2.34315i 0.215704i
\(119\) −2.34315 −0.214796
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) − 6.00000i − 0.543214i
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 9.65685i 0.856907i 0.903564 + 0.428454i \(0.140941\pi\)
−0.903564 + 0.428454i \(0.859059\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.14214 0.536641 0.268320 0.963330i \(-0.413531\pi\)
0.268320 + 0.963330i \(0.413531\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 5.65685 0.488678
\(135\) 0 0
\(136\) 0.828427 0.0710370
\(137\) 17.3137i 1.47921i 0.673041 + 0.739605i \(0.264988\pi\)
−0.673041 + 0.739605i \(0.735012\pi\)
\(138\) 0 0
\(139\) −6.34315 −0.538019 −0.269009 0.963138i \(-0.586696\pi\)
−0.269009 + 0.963138i \(0.586696\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 5.65685i − 0.474713i
\(143\) − 5.65685i − 0.473050i
\(144\) 0 0
\(145\) 0 0
\(146\) 14.4853 1.19881
\(147\) 0 0
\(148\) 11.6569i 0.958188i
\(149\) 3.65685 0.299581 0.149791 0.988718i \(-0.452140\pi\)
0.149791 + 0.988718i \(0.452140\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) − 2.82843i − 0.229416i
\(153\) 0 0
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) 0 0
\(157\) 5.31371i 0.424080i 0.977261 + 0.212040i \(0.0680107\pi\)
−0.977261 + 0.212040i \(0.931989\pi\)
\(158\) 2.34315i 0.186411i
\(159\) 0 0
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) − 11.3137i − 0.886158i −0.896483 0.443079i \(-0.853886\pi\)
0.896483 0.443079i \(-0.146114\pi\)
\(164\) −7.65685 −0.597900
\(165\) 0 0
\(166\) 6.34315 0.492324
\(167\) − 8.97056i − 0.694163i −0.937835 0.347081i \(-0.887173\pi\)
0.937835 0.347081i \(-0.112827\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 9.65685i 0.736328i
\(173\) − 9.31371i − 0.708108i −0.935225 0.354054i \(-0.884803\pi\)
0.935225 0.354054i \(-0.115197\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.65685 −0.426401
\(177\) 0 0
\(178\) 15.6569i 1.17353i
\(179\) 7.51472 0.561676 0.280838 0.959755i \(-0.409388\pi\)
0.280838 + 0.959755i \(0.409388\pi\)
\(180\) 0 0
\(181\) −7.65685 −0.569129 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(182\) − 2.82843i − 0.209657i
\(183\) 0 0
\(184\) 8.48528 0.625543
\(185\) 0 0
\(186\) 0 0
\(187\) 4.68629i 0.342696i
\(188\) − 8.00000i − 0.583460i
\(189\) 0 0
\(190\) 0 0
\(191\) −11.3137 −0.818631 −0.409316 0.912393i \(-0.634232\pi\)
−0.409316 + 0.912393i \(0.634232\pi\)
\(192\) 0 0
\(193\) − 2.48528i − 0.178894i −0.995992 0.0894472i \(-0.971490\pi\)
0.995992 0.0894472i \(-0.0285100\pi\)
\(194\) −3.17157 −0.227706
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 13.3137i 0.948562i 0.880373 + 0.474281i \(0.157292\pi\)
−0.880373 + 0.474281i \(0.842708\pi\)
\(198\) 0 0
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 16.1421i 1.13576i
\(203\) 24.9706i 1.75259i
\(204\) 0 0
\(205\) 0 0
\(206\) 1.65685 0.115439
\(207\) 0 0
\(208\) 1.00000i 0.0693375i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 0.686292 0.0472463 0.0236231 0.999721i \(-0.492480\pi\)
0.0236231 + 0.999721i \(0.492480\pi\)
\(212\) − 13.3137i − 0.914389i
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 0 0
\(217\) − 11.3137i − 0.768025i
\(218\) 8.82843i 0.597937i
\(219\) 0 0
\(220\) 0 0
\(221\) 0.828427 0.0557260
\(222\) 0 0
\(223\) − 10.8284i − 0.725125i −0.931959 0.362563i \(-0.881902\pi\)
0.931959 0.362563i \(-0.118098\pi\)
\(224\) −2.82843 −0.188982
\(225\) 0 0
\(226\) 6.48528 0.431394
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) −4.14214 −0.273720 −0.136860 0.990590i \(-0.543701\pi\)
−0.136860 + 0.990590i \(0.543701\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 8.82843i − 0.579615i
\(233\) 5.51472i 0.361281i 0.983549 + 0.180641i \(0.0578171\pi\)
−0.983549 + 0.180641i \(0.942183\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2.34315 0.152526
\(237\) 0 0
\(238\) 2.34315i 0.151884i
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 5.31371 0.342286 0.171143 0.985246i \(-0.445254\pi\)
0.171143 + 0.985246i \(0.445254\pi\)
\(242\) − 21.0000i − 1.34993i
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.82843i − 0.179969i
\(248\) 4.00000i 0.254000i
\(249\) 0 0
\(250\) 0 0
\(251\) −10.8284 −0.683484 −0.341742 0.939794i \(-0.611017\pi\)
−0.341742 + 0.939794i \(0.611017\pi\)
\(252\) 0 0
\(253\) 48.0000i 3.01773i
\(254\) 9.65685 0.605925
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.82843i 0.301189i 0.988596 + 0.150595i \(0.0481188\pi\)
−0.988596 + 0.150595i \(0.951881\pi\)
\(258\) 0 0
\(259\) −32.9706 −2.04869
\(260\) 0 0
\(261\) 0 0
\(262\) − 6.14214i − 0.379462i
\(263\) 16.4853i 1.01653i 0.861202 + 0.508263i \(0.169712\pi\)
−0.861202 + 0.508263i \(0.830288\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) − 5.65685i − 0.345547i
\(269\) −14.4853 −0.883183 −0.441592 0.897216i \(-0.645586\pi\)
−0.441592 + 0.897216i \(0.645586\pi\)
\(270\) 0 0
\(271\) 7.31371 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) − 0.828427i − 0.0502308i
\(273\) 0 0
\(274\) 17.3137 1.04596
\(275\) 0 0
\(276\) 0 0
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 6.34315i 0.380437i
\(279\) 0 0
\(280\) 0 0
\(281\) −8.34315 −0.497710 −0.248855 0.968541i \(-0.580054\pi\)
−0.248855 + 0.968541i \(0.580054\pi\)
\(282\) 0 0
\(283\) 17.6569i 1.04959i 0.851228 + 0.524796i \(0.175858\pi\)
−0.851228 + 0.524796i \(0.824142\pi\)
\(284\) −5.65685 −0.335673
\(285\) 0 0
\(286\) −5.65685 −0.334497
\(287\) − 21.6569i − 1.27836i
\(288\) 0 0
\(289\) 16.3137 0.959630
\(290\) 0 0
\(291\) 0 0
\(292\) − 14.4853i − 0.847687i
\(293\) − 16.6274i − 0.971384i −0.874130 0.485692i \(-0.838568\pi\)
0.874130 0.485692i \(-0.161432\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 11.6569 0.677541
\(297\) 0 0
\(298\) − 3.65685i − 0.211836i
\(299\) 8.48528 0.490716
\(300\) 0 0
\(301\) −27.3137 −1.57434
\(302\) − 12.0000i − 0.690522i
\(303\) 0 0
\(304\) −2.82843 −0.162221
\(305\) 0 0
\(306\) 0 0
\(307\) − 21.6569i − 1.23602i −0.786169 0.618011i \(-0.787939\pi\)
0.786169 0.618011i \(-0.212061\pi\)
\(308\) − 16.0000i − 0.911685i
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 30.9706i 1.75056i 0.483617 + 0.875280i \(0.339323\pi\)
−0.483617 + 0.875280i \(0.660677\pi\)
\(314\) 5.31371 0.299870
\(315\) 0 0
\(316\) 2.34315 0.131812
\(317\) − 25.3137i − 1.42176i −0.703314 0.710880i \(-0.748297\pi\)
0.703314 0.710880i \(-0.251703\pi\)
\(318\) 0 0
\(319\) 49.9411 2.79617
\(320\) 0 0
\(321\) 0 0
\(322\) 24.0000i 1.33747i
\(323\) 2.34315i 0.130376i
\(324\) 0 0
\(325\) 0 0
\(326\) −11.3137 −0.626608
\(327\) 0 0
\(328\) 7.65685i 0.422779i
\(329\) 22.6274 1.24749
\(330\) 0 0
\(331\) −8.48528 −0.466393 −0.233197 0.972430i \(-0.574919\pi\)
−0.233197 + 0.972430i \(0.574919\pi\)
\(332\) − 6.34315i − 0.348125i
\(333\) 0 0
\(334\) −8.97056 −0.490847
\(335\) 0 0
\(336\) 0 0
\(337\) 10.9706i 0.597605i 0.954315 + 0.298802i \(0.0965871\pi\)
−0.954315 + 0.298802i \(0.903413\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) −22.6274 −1.22534
\(342\) 0 0
\(343\) − 16.9706i − 0.916324i
\(344\) 9.65685 0.520663
\(345\) 0 0
\(346\) −9.31371 −0.500708
\(347\) 9.65685i 0.518407i 0.965823 + 0.259204i \(0.0834600\pi\)
−0.965823 + 0.259204i \(0.916540\pi\)
\(348\) 0 0
\(349\) −12.1421 −0.649954 −0.324977 0.945722i \(-0.605356\pi\)
−0.324977 + 0.945722i \(0.605356\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.65685i 0.301511i
\(353\) 5.31371i 0.282820i 0.989951 + 0.141410i \(0.0451636\pi\)
−0.989951 + 0.141410i \(0.954836\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 15.6569 0.829812
\(357\) 0 0
\(358\) − 7.51472i − 0.397165i
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 7.65685i 0.402435i
\(363\) 0 0
\(364\) −2.82843 −0.148250
\(365\) 0 0
\(366\) 0 0
\(367\) 25.6569i 1.33928i 0.742687 + 0.669638i \(0.233551\pi\)
−0.742687 + 0.669638i \(0.766449\pi\)
\(368\) − 8.48528i − 0.442326i
\(369\) 0 0
\(370\) 0 0
\(371\) 37.6569 1.95505
\(372\) 0 0
\(373\) 2.68629i 0.139091i 0.997579 + 0.0695455i \(0.0221549\pi\)
−0.997579 + 0.0695455i \(0.977845\pi\)
\(374\) 4.68629 0.242322
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) − 8.82843i − 0.454687i
\(378\) 0 0
\(379\) 7.51472 0.386005 0.193003 0.981198i \(-0.438177\pi\)
0.193003 + 0.981198i \(0.438177\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 11.3137i 0.578860i
\(383\) − 29.6569i − 1.51539i −0.652606 0.757697i \(-0.726324\pi\)
0.652606 0.757697i \(-0.273676\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.48528 −0.126497
\(387\) 0 0
\(388\) 3.17157i 0.161012i
\(389\) −6.48528 −0.328817 −0.164408 0.986392i \(-0.552571\pi\)
−0.164408 + 0.986392i \(0.552571\pi\)
\(390\) 0 0
\(391\) −7.02944 −0.355494
\(392\) − 1.00000i − 0.0505076i
\(393\) 0 0
\(394\) 13.3137 0.670735
\(395\) 0 0
\(396\) 0 0
\(397\) 30.2843i 1.51992i 0.649968 + 0.759962i \(0.274782\pi\)
−0.649968 + 0.759962i \(0.725218\pi\)
\(398\) 10.3431i 0.518455i
\(399\) 0 0
\(400\) 0 0
\(401\) 26.9706 1.34685 0.673423 0.739258i \(-0.264823\pi\)
0.673423 + 0.739258i \(0.264823\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 16.1421 0.803101
\(405\) 0 0
\(406\) 24.9706 1.23927
\(407\) 65.9411i 3.26858i
\(408\) 0 0
\(409\) 3.65685 0.180820 0.0904099 0.995905i \(-0.471182\pi\)
0.0904099 + 0.995905i \(0.471182\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 1.65685i − 0.0816274i
\(413\) 6.62742i 0.326114i
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) − 16.0000i − 0.782586i
\(419\) −10.8284 −0.529003 −0.264502 0.964385i \(-0.585207\pi\)
−0.264502 + 0.964385i \(0.585207\pi\)
\(420\) 0 0
\(421\) −24.1421 −1.17662 −0.588308 0.808637i \(-0.700206\pi\)
−0.588308 + 0.808637i \(0.700206\pi\)
\(422\) − 0.686292i − 0.0334081i
\(423\) 0 0
\(424\) −13.3137 −0.646571
\(425\) 0 0
\(426\) 0 0
\(427\) − 16.9706i − 0.821263i
\(428\) − 4.00000i − 0.193347i
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 22.9706i 1.10389i 0.833879 + 0.551947i \(0.186115\pi\)
−0.833879 + 0.551947i \(0.813885\pi\)
\(434\) −11.3137 −0.543075
\(435\) 0 0
\(436\) 8.82843 0.422805
\(437\) 24.0000i 1.14808i
\(438\) 0 0
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 0.828427i − 0.0394043i
\(443\) 30.3431i 1.44165i 0.693119 + 0.720823i \(0.256236\pi\)
−0.693119 + 0.720823i \(0.743764\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −10.8284 −0.512741
\(447\) 0 0
\(448\) 2.82843i 0.133631i
\(449\) 26.2843 1.24043 0.620216 0.784431i \(-0.287045\pi\)
0.620216 + 0.784431i \(0.287045\pi\)
\(450\) 0 0
\(451\) −43.3137 −2.03956
\(452\) − 6.48528i − 0.305042i
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) 20.8284i 0.974313i 0.873315 + 0.487156i \(0.161966\pi\)
−0.873315 + 0.487156i \(0.838034\pi\)
\(458\) 4.14214i 0.193549i
\(459\) 0 0
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 3.79899i 0.176554i 0.996096 + 0.0882770i \(0.0281361\pi\)
−0.996096 + 0.0882770i \(0.971864\pi\)
\(464\) −8.82843 −0.409849
\(465\) 0 0
\(466\) 5.51472 0.255464
\(467\) − 7.31371i − 0.338438i −0.985578 0.169219i \(-0.945875\pi\)
0.985578 0.169219i \(-0.0541245\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) − 2.34315i − 0.107852i
\(473\) 54.6274i 2.51177i
\(474\) 0 0
\(475\) 0 0
\(476\) 2.34315 0.107398
\(477\) 0 0
\(478\) − 16.0000i − 0.731823i
\(479\) −11.3137 −0.516937 −0.258468 0.966020i \(-0.583218\pi\)
−0.258468 + 0.966020i \(0.583218\pi\)
\(480\) 0 0
\(481\) 11.6569 0.531507
\(482\) − 5.31371i − 0.242033i
\(483\) 0 0
\(484\) −21.0000 −0.954545
\(485\) 0 0
\(486\) 0 0
\(487\) 16.4853i 0.747019i 0.927626 + 0.373510i \(0.121846\pi\)
−0.927626 + 0.373510i \(0.878154\pi\)
\(488\) 6.00000i 0.271607i
\(489\) 0 0
\(490\) 0 0
\(491\) 38.1421 1.72133 0.860665 0.509171i \(-0.170048\pi\)
0.860665 + 0.509171i \(0.170048\pi\)
\(492\) 0 0
\(493\) 7.31371i 0.329393i
\(494\) −2.82843 −0.127257
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) − 16.0000i − 0.717698i
\(498\) 0 0
\(499\) −0.485281 −0.0217242 −0.0108621 0.999941i \(-0.503458\pi\)
−0.0108621 + 0.999941i \(0.503458\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 10.8284i 0.483296i
\(503\) − 23.5147i − 1.04847i −0.851574 0.524235i \(-0.824351\pi\)
0.851574 0.524235i \(-0.175649\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 48.0000 2.13386
\(507\) 0 0
\(508\) − 9.65685i − 0.428454i
\(509\) −37.3137 −1.65390 −0.826951 0.562275i \(-0.809926\pi\)
−0.826951 + 0.562275i \(0.809926\pi\)
\(510\) 0 0
\(511\) 40.9706 1.81243
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) 4.82843 0.212973
\(515\) 0 0
\(516\) 0 0
\(517\) − 45.2548i − 1.99031i
\(518\) 32.9706i 1.44864i
\(519\) 0 0
\(520\) 0 0
\(521\) −26.9706 −1.18160 −0.590801 0.806817i \(-0.701188\pi\)
−0.590801 + 0.806817i \(0.701188\pi\)
\(522\) 0 0
\(523\) 10.6274i 0.464704i 0.972632 + 0.232352i \(0.0746422\pi\)
−0.972632 + 0.232352i \(0.925358\pi\)
\(524\) −6.14214 −0.268320
\(525\) 0 0
\(526\) 16.4853 0.718792
\(527\) − 3.31371i − 0.144347i
\(528\) 0 0
\(529\) −49.0000 −2.13043
\(530\) 0 0
\(531\) 0 0
\(532\) − 8.00000i − 0.346844i
\(533\) 7.65685i 0.331655i
\(534\) 0 0
\(535\) 0 0
\(536\) −5.65685 −0.244339
\(537\) 0 0
\(538\) 14.4853i 0.624505i
\(539\) 5.65685 0.243658
\(540\) 0 0
\(541\) 14.4853 0.622771 0.311385 0.950284i \(-0.399207\pi\)
0.311385 + 0.950284i \(0.399207\pi\)
\(542\) − 7.31371i − 0.314151i
\(543\) 0 0
\(544\) −0.828427 −0.0355185
\(545\) 0 0
\(546\) 0 0
\(547\) 0.686292i 0.0293437i 0.999892 + 0.0146719i \(0.00467036\pi\)
−0.999892 + 0.0146719i \(0.995330\pi\)
\(548\) − 17.3137i − 0.739605i
\(549\) 0 0
\(550\) 0 0
\(551\) 24.9706 1.06378
\(552\) 0 0
\(553\) 6.62742i 0.281826i
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) 6.34315 0.269009
\(557\) − 10.6863i − 0.452793i −0.974035 0.226396i \(-0.927306\pi\)
0.974035 0.226396i \(-0.0726945\pi\)
\(558\) 0 0
\(559\) 9.65685 0.408441
\(560\) 0 0
\(561\) 0 0
\(562\) 8.34315i 0.351934i
\(563\) − 30.3431i − 1.27881i −0.768870 0.639406i \(-0.779181\pi\)
0.768870 0.639406i \(-0.220819\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 17.6569 0.742173
\(567\) 0 0
\(568\) 5.65685i 0.237356i
\(569\) 31.6569 1.32712 0.663562 0.748121i \(-0.269044\pi\)
0.663562 + 0.748121i \(0.269044\pi\)
\(570\) 0 0
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) 5.65685i 0.236525i
\(573\) 0 0
\(574\) −21.6569 −0.903940
\(575\) 0 0
\(576\) 0 0
\(577\) − 23.4558i − 0.976480i −0.872710 0.488240i \(-0.837639\pi\)
0.872710 0.488240i \(-0.162361\pi\)
\(578\) − 16.3137i − 0.678561i
\(579\) 0 0
\(580\) 0 0
\(581\) 17.9411 0.744323
\(582\) 0 0
\(583\) − 75.3137i − 3.11918i
\(584\) −14.4853 −0.599405
\(585\) 0 0
\(586\) −16.6274 −0.686872
\(587\) − 2.62742i − 0.108445i −0.998529 0.0542226i \(-0.982732\pi\)
0.998529 0.0542226i \(-0.0172680\pi\)
\(588\) 0 0
\(589\) −11.3137 −0.466173
\(590\) 0 0
\(591\) 0 0
\(592\) − 11.6569i − 0.479094i
\(593\) − 0.343146i − 0.0140913i −0.999975 0.00704565i \(-0.997757\pi\)
0.999975 0.00704565i \(-0.00224272\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.65685 −0.149791
\(597\) 0 0
\(598\) − 8.48528i − 0.346989i
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) 29.3137 1.19573 0.597866 0.801596i \(-0.296016\pi\)
0.597866 + 0.801596i \(0.296016\pi\)
\(602\) 27.3137i 1.11322i
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) 0 0
\(606\) 0 0
\(607\) 28.9706i 1.17588i 0.808905 + 0.587939i \(0.200061\pi\)
−0.808905 + 0.587939i \(0.799939\pi\)
\(608\) 2.82843i 0.114708i
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) − 22.2843i − 0.900053i −0.893015 0.450027i \(-0.851414\pi\)
0.893015 0.450027i \(-0.148586\pi\)
\(614\) −21.6569 −0.874000
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) − 2.00000i − 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) 0 0
\(619\) 34.8284 1.39987 0.699936 0.714205i \(-0.253212\pi\)
0.699936 + 0.714205i \(0.253212\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 24.0000i − 0.962312i
\(623\) 44.2843i 1.77421i
\(624\) 0 0
\(625\) 0 0
\(626\) 30.9706 1.23783
\(627\) 0 0
\(628\) − 5.31371i − 0.212040i
\(629\) −9.65685 −0.385044
\(630\) 0 0
\(631\) −33.6569 −1.33986 −0.669929 0.742425i \(-0.733676\pi\)
−0.669929 + 0.742425i \(0.733676\pi\)
\(632\) − 2.34315i − 0.0932053i
\(633\) 0 0
\(634\) −25.3137 −1.00534
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.00000i − 0.0396214i
\(638\) − 49.9411i − 1.97719i
\(639\) 0 0
\(640\) 0 0
\(641\) 4.62742 0.182772 0.0913860 0.995816i \(-0.470870\pi\)
0.0913860 + 0.995816i \(0.470870\pi\)
\(642\) 0 0
\(643\) − 39.5980i − 1.56159i −0.624786 0.780796i \(-0.714814\pi\)
0.624786 0.780796i \(-0.285186\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) 2.34315 0.0921898
\(647\) − 8.48528i − 0.333591i −0.985992 0.166795i \(-0.946658\pi\)
0.985992 0.166795i \(-0.0533419\pi\)
\(648\) 0 0
\(649\) 13.2548 0.520298
\(650\) 0 0
\(651\) 0 0
\(652\) 11.3137i 0.443079i
\(653\) − 42.2843i − 1.65471i −0.561678 0.827356i \(-0.689844\pi\)
0.561678 0.827356i \(-0.310156\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 7.65685 0.298950
\(657\) 0 0
\(658\) − 22.6274i − 0.882109i
\(659\) −7.51472 −0.292732 −0.146366 0.989231i \(-0.546758\pi\)
−0.146366 + 0.989231i \(0.546758\pi\)
\(660\) 0 0
\(661\) −8.14214 −0.316692 −0.158346 0.987384i \(-0.550616\pi\)
−0.158346 + 0.987384i \(0.550616\pi\)
\(662\) 8.48528i 0.329790i
\(663\) 0 0
\(664\) −6.34315 −0.246162
\(665\) 0 0
\(666\) 0 0
\(667\) 74.9117i 2.90059i
\(668\) 8.97056i 0.347081i
\(669\) 0 0
\(670\) 0 0
\(671\) −33.9411 −1.31028
\(672\) 0 0
\(673\) − 32.6274i − 1.25769i −0.777529 0.628847i \(-0.783527\pi\)
0.777529 0.628847i \(-0.216473\pi\)
\(674\) 10.9706 0.422570
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) − 12.3431i − 0.474386i −0.971463 0.237193i \(-0.923773\pi\)
0.971463 0.237193i \(-0.0762273\pi\)
\(678\) 0 0
\(679\) −8.97056 −0.344259
\(680\) 0 0
\(681\) 0 0
\(682\) 22.6274i 0.866449i
\(683\) 33.6569i 1.28784i 0.765091 + 0.643922i \(0.222694\pi\)
−0.765091 + 0.643922i \(0.777306\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −16.9706 −0.647939
\(687\) 0 0
\(688\) − 9.65685i − 0.368164i
\(689\) −13.3137 −0.507212
\(690\) 0 0
\(691\) −27.7990 −1.05752 −0.528762 0.848770i \(-0.677343\pi\)
−0.528762 + 0.848770i \(0.677343\pi\)
\(692\) 9.31371i 0.354054i
\(693\) 0 0
\(694\) 9.65685 0.366569
\(695\) 0 0
\(696\) 0 0
\(697\) − 6.34315i − 0.240264i
\(698\) 12.1421i 0.459587i
\(699\) 0 0
\(700\) 0 0
\(701\) −0.142136 −0.00536839 −0.00268419 0.999996i \(-0.500854\pi\)
−0.00268419 + 0.999996i \(0.500854\pi\)
\(702\) 0 0
\(703\) 32.9706i 1.24351i
\(704\) 5.65685 0.213201
\(705\) 0 0
\(706\) 5.31371 0.199984
\(707\) 45.6569i 1.71710i
\(708\) 0 0
\(709\) 7.17157 0.269334 0.134667 0.990891i \(-0.457004\pi\)
0.134667 + 0.990891i \(0.457004\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 15.6569i − 0.586765i
\(713\) − 33.9411i − 1.27111i
\(714\) 0 0
\(715\) 0 0
\(716\) −7.51472 −0.280838
\(717\) 0 0
\(718\) 28.2843i 1.05556i
\(719\) −29.6569 −1.10601 −0.553007 0.833177i \(-0.686520\pi\)
−0.553007 + 0.833177i \(0.686520\pi\)
\(720\) 0 0
\(721\) 4.68629 0.174527
\(722\) 11.0000i 0.409378i
\(723\) 0 0
\(724\) 7.65685 0.284565
\(725\) 0 0
\(726\) 0 0
\(727\) − 45.9411i − 1.70386i −0.523654 0.851931i \(-0.675432\pi\)
0.523654 0.851931i \(-0.324568\pi\)
\(728\) 2.82843i 0.104828i
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 0.343146i 0.0126744i 0.999980 + 0.00633719i \(0.00201720\pi\)
−0.999980 + 0.00633719i \(0.997983\pi\)
\(734\) 25.6569 0.947012
\(735\) 0 0
\(736\) −8.48528 −0.312772
\(737\) − 32.0000i − 1.17874i
\(738\) 0 0
\(739\) 14.1421 0.520227 0.260113 0.965578i \(-0.416240\pi\)
0.260113 + 0.965578i \(0.416240\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 37.6569i − 1.38243i
\(743\) − 36.2843i − 1.33114i −0.746335 0.665570i \(-0.768188\pi\)
0.746335 0.665570i \(-0.231812\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.68629 0.0983521
\(747\) 0 0
\(748\) − 4.68629i − 0.171348i
\(749\) 11.3137 0.413394
\(750\) 0 0
\(751\) 11.3137 0.412843 0.206422 0.978463i \(-0.433818\pi\)
0.206422 + 0.978463i \(0.433818\pi\)
\(752\) 8.00000i 0.291730i
\(753\) 0 0
\(754\) −8.82843 −0.321512
\(755\) 0 0
\(756\) 0 0
\(757\) 19.9411i 0.724773i 0.932028 + 0.362386i \(0.118038\pi\)
−0.932028 + 0.362386i \(0.881962\pi\)
\(758\) − 7.51472i − 0.272947i
\(759\) 0 0
\(760\) 0 0
\(761\) −27.6569 −1.00256 −0.501280 0.865285i \(-0.667137\pi\)
−0.501280 + 0.865285i \(0.667137\pi\)
\(762\) 0 0
\(763\) 24.9706i 0.903995i
\(764\) 11.3137 0.409316
\(765\) 0 0
\(766\) −29.6569 −1.07155
\(767\) − 2.34315i − 0.0846061i
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.48528i 0.0894472i
\(773\) − 53.3137i − 1.91756i −0.284148 0.958780i \(-0.591711\pi\)
0.284148 0.958780i \(-0.408289\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3.17157 0.113853
\(777\) 0 0
\(778\) 6.48528i 0.232509i
\(779\) −21.6569 −0.775937
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 7.02944i 0.251372i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 24.0000i − 0.855508i −0.903895 0.427754i \(-0.859305\pi\)
0.903895 0.427754i \(-0.140695\pi\)
\(788\) − 13.3137i − 0.474281i
\(789\) 0 0
\(790\) 0 0
\(791\) 18.3431 0.652207
\(792\) 0 0
\(793\) 6.00000i 0.213066i
\(794\) 30.2843 1.07475
\(795\) 0 0
\(796\) 10.3431 0.366603
\(797\) − 16.6274i − 0.588973i −0.955656 0.294487i \(-0.904851\pi\)
0.955656 0.294487i \(-0.0951487\pi\)
\(798\) 0 0
\(799\) 6.62742 0.234461
\(800\) 0 0
\(801\) 0 0
\(802\) − 26.9706i − 0.952364i
\(803\) − 81.9411i − 2.89164i
\(804\) 0 0
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) − 16.1421i − 0.567878i
\(809\) 13.3137 0.468085 0.234043 0.972226i \(-0.424804\pi\)
0.234043 + 0.972226i \(0.424804\pi\)
\(810\) 0 0
\(811\) −1.85786 −0.0652384 −0.0326192 0.999468i \(-0.510385\pi\)
−0.0326192 + 0.999468i \(0.510385\pi\)
\(812\) − 24.9706i − 0.876295i
\(813\) 0 0
\(814\) 65.9411 2.31124
\(815\) 0 0
\(816\) 0 0
\(817\) 27.3137i 0.955586i
\(818\) − 3.65685i − 0.127859i
\(819\) 0 0
\(820\) 0 0
\(821\) −34.2843 −1.19653 −0.598265 0.801299i \(-0.704143\pi\)
−0.598265 + 0.801299i \(0.704143\pi\)
\(822\) 0 0
\(823\) 52.9706i 1.84644i 0.384275 + 0.923219i \(0.374452\pi\)
−0.384275 + 0.923219i \(0.625548\pi\)
\(824\) −1.65685 −0.0577193
\(825\) 0 0
\(826\) 6.62742 0.230597
\(827\) − 9.65685i − 0.335802i −0.985804 0.167901i \(-0.946301\pi\)
0.985804 0.167901i \(-0.0536989\pi\)
\(828\) 0 0
\(829\) 53.3137 1.85166 0.925831 0.377938i \(-0.123367\pi\)
0.925831 + 0.377938i \(0.123367\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.00000i − 0.0346688i
\(833\) 0.828427i 0.0287033i
\(834\) 0 0
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 0 0
\(838\) 10.8284i 0.374062i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 48.9411 1.68763
\(842\) 24.1421i 0.831993i
\(843\) 0 0
\(844\) −0.686292 −0.0236231
\(845\) 0 0
\(846\) 0 0
\(847\) − 59.3970i − 2.04090i
\(848\) 13.3137i 0.457195i
\(849\) 0 0
\(850\) 0 0
\(851\) −98.9117 −3.39065
\(852\) 0 0
\(853\) − 38.2843i − 1.31083i −0.755270 0.655414i \(-0.772494\pi\)
0.755270 0.655414i \(-0.227506\pi\)
\(854\) −16.9706 −0.580721
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 20.8284i 0.711486i 0.934584 + 0.355743i \(0.115772\pi\)
−0.934584 + 0.355743i \(0.884228\pi\)
\(858\) 0 0
\(859\) −37.9411 −1.29453 −0.647267 0.762263i \(-0.724088\pi\)
−0.647267 + 0.762263i \(0.724088\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 16.0000i − 0.544962i
\(863\) − 28.2843i − 0.962808i −0.876499 0.481404i \(-0.840127\pi\)
0.876499 0.481404i \(-0.159873\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 22.9706 0.780571
\(867\) 0 0
\(868\) 11.3137i 0.384012i
\(869\) 13.2548 0.449639
\(870\) 0 0
\(871\) −5.65685 −0.191675
\(872\) − 8.82843i − 0.298968i
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) − 51.2548i − 1.73075i −0.501122 0.865376i \(-0.667079\pi\)
0.501122 0.865376i \(-0.332921\pi\)
\(878\) 22.6274i 0.763638i
\(879\) 0 0
\(880\) 0 0
\(881\) 10.2843 0.346486 0.173243 0.984879i \(-0.444575\pi\)
0.173243 + 0.984879i \(0.444575\pi\)
\(882\) 0 0
\(883\) − 31.3137i − 1.05379i −0.849930 0.526895i \(-0.823356\pi\)
0.849930 0.526895i \(-0.176644\pi\)
\(884\) −0.828427 −0.0278630
\(885\) 0 0
\(886\) 30.3431 1.01940
\(887\) 40.4853i 1.35936i 0.733508 + 0.679681i \(0.237882\pi\)
−0.733508 + 0.679681i \(0.762118\pi\)
\(888\) 0 0
\(889\) 27.3137 0.916072
\(890\) 0 0
\(891\) 0 0
\(892\) 10.8284i 0.362563i
\(893\) − 22.6274i − 0.757198i
\(894\) 0 0
\(895\) 0 0
\(896\) 2.82843 0.0944911
\(897\) 0 0
\(898\) − 26.2843i − 0.877117i
\(899\) −35.3137 −1.17778
\(900\) 0 0
\(901\) 11.0294 0.367444
\(902\) 43.3137i 1.44219i
\(903\) 0 0
\(904\) −6.48528 −0.215697
\(905\) 0 0
\(906\) 0 0
\(907\) 8.28427i 0.275075i 0.990497 + 0.137537i \(0.0439187\pi\)
−0.990497 + 0.137537i \(0.956081\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 0 0
\(910\) 0 0
\(911\) −24.9706 −0.827312 −0.413656 0.910433i \(-0.635748\pi\)
−0.413656 + 0.910433i \(0.635748\pi\)
\(912\) 0 0
\(913\) − 35.8823i − 1.18753i
\(914\) 20.8284 0.688943
\(915\) 0 0
\(916\) 4.14214 0.136860
\(917\) − 17.3726i − 0.573693i
\(918\) 0 0
\(919\) −41.9411 −1.38351 −0.691755 0.722132i \(-0.743162\pi\)
−0.691755 + 0.722132i \(0.743162\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 14.0000i 0.461065i
\(923\) 5.65685i 0.186198i
\(924\) 0 0
\(925\) 0 0
\(926\) 3.79899 0.124843
\(927\) 0 0
\(928\) 8.82843i 0.289807i
\(929\) −33.5980 −1.10231 −0.551157 0.834402i \(-0.685813\pi\)
−0.551157 + 0.834402i \(0.685813\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) − 5.51472i − 0.180641i
\(933\) 0 0
\(934\) −7.31371 −0.239312
\(935\) 0 0
\(936\) 0 0
\(937\) 16.6274i 0.543194i 0.962411 + 0.271597i \(0.0875518\pi\)
−0.962411 + 0.271597i \(0.912448\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) 0 0
\(940\) 0 0
\(941\) −54.9706 −1.79199 −0.895995 0.444065i \(-0.853536\pi\)
−0.895995 + 0.444065i \(0.853536\pi\)
\(942\) 0 0
\(943\) − 64.9706i − 2.11573i
\(944\) −2.34315 −0.0762629
\(945\) 0 0
\(946\) 54.6274 1.77609
\(947\) − 30.3431i − 0.986020i −0.870024 0.493010i \(-0.835897\pi\)
0.870024 0.493010i \(-0.164103\pi\)
\(948\) 0 0
\(949\) −14.4853 −0.470212
\(950\) 0 0
\(951\) 0 0
\(952\) − 2.34315i − 0.0759418i
\(953\) − 27.8579i − 0.902405i −0.892422 0.451202i \(-0.850995\pi\)
0.892422 0.451202i \(-0.149005\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 11.3137i 0.365529i
\(959\) 48.9706 1.58134
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 11.6569i − 0.375832i
\(963\) 0 0
\(964\) −5.31371 −0.171143
\(965\) 0 0
\(966\) 0 0
\(967\) − 7.51472i − 0.241657i −0.992673 0.120829i \(-0.961445\pi\)
0.992673 0.120829i \(-0.0385551\pi\)
\(968\) 21.0000i 0.674966i
\(969\) 0 0
\(970\) 0 0
\(971\) −15.5147 −0.497891 −0.248946 0.968517i \(-0.580084\pi\)
−0.248946 + 0.968517i \(0.580084\pi\)
\(972\) 0 0
\(973\) 17.9411i 0.575166i
\(974\) 16.4853 0.528222
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 8.34315i 0.266921i 0.991054 + 0.133460i \(0.0426089\pi\)
−0.991054 + 0.133460i \(0.957391\pi\)
\(978\) 0 0
\(979\) 88.5685 2.83066
\(980\) 0 0
\(981\) 0 0
\(982\) − 38.1421i − 1.21716i
\(983\) − 2.34315i − 0.0747347i −0.999302 0.0373674i \(-0.988103\pi\)
0.999302 0.0373674i \(-0.0118972\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 7.31371 0.232916
\(987\) 0 0
\(988\) 2.82843i 0.0899843i
\(989\) −81.9411 −2.60558
\(990\) 0 0
\(991\) −42.9117 −1.36313 −0.681567 0.731755i \(-0.738701\pi\)
−0.681567 + 0.731755i \(0.738701\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) 61.3137i 1.94182i 0.239435 + 0.970912i \(0.423038\pi\)
−0.239435 + 0.970912i \(0.576962\pi\)
\(998\) 0.485281i 0.0153613i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5850.2.e.bk.5149.1 4
3.2 odd 2 1950.2.e.o.1249.3 4
5.2 odd 4 5850.2.a.cl.1.2 2
5.3 odd 4 1170.2.a.o.1.1 2
5.4 even 2 inner 5850.2.e.bk.5149.4 4
15.2 even 4 1950.2.a.bd.1.2 2
15.8 even 4 390.2.a.h.1.1 2
15.14 odd 2 1950.2.e.o.1249.2 4
20.3 even 4 9360.2.a.ch.1.2 2
60.23 odd 4 3120.2.a.bc.1.2 2
195.8 odd 4 5070.2.b.q.1351.4 4
195.38 even 4 5070.2.a.bc.1.2 2
195.83 odd 4 5070.2.b.q.1351.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 15.8 even 4
1170.2.a.o.1.1 2 5.3 odd 4
1950.2.a.bd.1.2 2 15.2 even 4
1950.2.e.o.1249.2 4 15.14 odd 2
1950.2.e.o.1249.3 4 3.2 odd 2
3120.2.a.bc.1.2 2 60.23 odd 4
5070.2.a.bc.1.2 2 195.38 even 4
5070.2.b.q.1351.1 4 195.83 odd 4
5070.2.b.q.1351.4 4 195.8 odd 4
5850.2.a.cl.1.2 2 5.2 odd 4
5850.2.e.bk.5149.1 4 1.1 even 1 trivial
5850.2.e.bk.5149.4 4 5.4 even 2 inner
9360.2.a.ch.1.2 2 20.3 even 4