Properties

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

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Newspace parameters

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

Newform invariants

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

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -4.00000i q^{7} +1.00000i q^{8} -4.00000 q^{11} -1.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -7.00000 q^{19} +4.00000i q^{22} +4.00000i q^{23} -1.00000 q^{26} +4.00000i q^{28} +5.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +4.00000 q^{34} +9.00000i q^{37} +7.00000i q^{38} +5.00000 q^{41} +10.0000i q^{43} +4.00000 q^{44} +4.00000 q^{46} -3.00000i q^{47} -9.00000 q^{49} +1.00000i q^{52} +9.00000i q^{53} +4.00000 q^{56} -5.00000i q^{58} -6.00000 q^{59} +4.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -7.00000i q^{67} -4.00000i q^{68} +15.0000 q^{71} -12.0000i q^{73} +9.00000 q^{74} +7.00000 q^{76} +16.0000i q^{77} -7.00000 q^{79} -5.00000i q^{82} +6.00000i q^{83} +10.0000 q^{86} -4.00000i q^{88} +14.0000 q^{89} -4.00000 q^{91} -4.00000i q^{92} -3.00000 q^{94} -16.0000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 8q^{11} - 8q^{14} + 2q^{16} - 14q^{19} - 2q^{26} + 10q^{29} + 8q^{31} + 8q^{34} + 10q^{41} + 8q^{44} + 8q^{46} - 18q^{49} + 8q^{56} - 12q^{59} + 8q^{61} - 2q^{64} + 30q^{71} + 18q^{74} + 14q^{76} - 14q^{79} + 20q^{86} + 28q^{89} - 8q^{91} - 6q^{94} + O(q^{100}) \)

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
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 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 \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 4.00000i 0.755929i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 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
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) 9.00000i 1.47959i 0.672832 + 0.739795i \(0.265078\pi\)
−0.672832 + 0.739795i \(0.734922\pi\)
\(38\) 7.00000i 1.13555i
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) 9.00000i 1.23625i 0.786082 + 0.618123i \(0.212106\pi\)
−0.786082 + 0.618123i \(0.787894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) − 5.00000i − 0.656532i
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 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
\(66\) 0 0
\(67\) − 7.00000i − 0.855186i −0.903971 0.427593i \(-0.859362\pi\)
0.903971 0.427593i \(-0.140638\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) 15.0000 1.78017 0.890086 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 0 0
\(73\) − 12.0000i − 1.40449i −0.711934 0.702247i \(-0.752180\pi\)
0.711934 0.702247i \(-0.247820\pi\)
\(74\) 9.00000 1.04623
\(75\) 0 0
\(76\) 7.00000 0.802955
\(77\) 16.0000i 1.82337i
\(78\) 0 0
\(79\) −7.00000 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 5.00000i − 0.552158i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) − 4.00000i − 0.426401i
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) − 4.00000i − 0.417029i
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) 0 0
\(97\) − 16.0000i − 1.62455i −0.583272 0.812277i \(-0.698228\pi\)
0.583272 0.812277i \(-0.301772\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) − 5.00000i − 0.483368i −0.970355 0.241684i \(-0.922300\pi\)
0.970355 0.241684i \(-0.0776998\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 4.00000i − 0.377964i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −5.00000 −0.464238
\(117\) 0 0
\(118\) 6.00000i 0.552345i
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 4.00000i − 0.362143i
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) − 5.00000i − 0.443678i −0.975083 0.221839i \(-0.928794\pi\)
0.975083 0.221839i \(-0.0712060\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −17.0000 −1.48530 −0.742648 0.669681i \(-0.766431\pi\)
−0.742648 + 0.669681i \(0.766431\pi\)
\(132\) 0 0
\(133\) 28.0000i 2.42791i
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 19.0000i 1.62328i 0.584158 + 0.811640i \(0.301425\pi\)
−0.584158 + 0.811640i \(0.698575\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 15.0000i − 1.25877i
\(143\) 4.00000i 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) − 9.00000i − 0.739795i
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) − 7.00000i − 0.567775i
\(153\) 0 0
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 7.00000i 0.556890i
\(159\) 0 0
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) 24.0000i 1.87983i 0.341415 + 0.939913i \(0.389094\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) − 9.00000i − 0.696441i −0.937413 0.348220i \(-0.886786\pi\)
0.937413 0.348220i \(-0.113214\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) − 10.0000i − 0.762493i
\(173\) 13.0000i 0.988372i 0.869356 + 0.494186i \(0.164534\pi\)
−0.869356 + 0.494186i \(0.835466\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) − 14.0000i − 1.04934i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −4.00000 −0.297318 −0.148659 0.988889i \(-0.547496\pi\)
−0.148659 + 0.988889i \(0.547496\pi\)
\(182\) 4.00000i 0.296500i
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) − 16.0000i − 1.17004i
\(188\) 3.00000i 0.218797i
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 12.0000i 0.863779i 0.901927 + 0.431889i \(0.142153\pi\)
−0.901927 + 0.431889i \(0.857847\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 16.0000i 1.13995i 0.821661 + 0.569976i \(0.193048\pi\)
−0.821661 + 0.569976i \(0.806952\pi\)
\(198\) 0 0
\(199\) 3.00000 0.212664 0.106332 0.994331i \(-0.466089\pi\)
0.106332 + 0.994331i \(0.466089\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 10.0000i − 0.703598i
\(203\) − 20.0000i − 1.40372i
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) − 1.00000i − 0.0693375i
\(209\) 28.0000 1.93680
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) − 9.00000i − 0.618123i
\(213\) 0 0
\(214\) −5.00000 −0.341793
\(215\) 0 0
\(216\) 0 0
\(217\) − 16.0000i − 1.08615i
\(218\) − 11.0000i − 0.745014i
\(219\) 0 0
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) − 6.00000i − 0.398234i −0.979976 0.199117i \(-0.936193\pi\)
0.979976 0.199117i \(-0.0638074\pi\)
\(228\) 0 0
\(229\) 17.0000 1.12339 0.561696 0.827344i \(-0.310149\pi\)
0.561696 + 0.827344i \(0.310149\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.00000i 0.328266i
\(233\) 8.00000i 0.524097i 0.965055 + 0.262049i \(0.0843981\pi\)
−0.965055 + 0.262049i \(0.915602\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) − 16.0000i − 1.03713i
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 7.00000i 0.445399i
\(248\) 4.00000i 0.254000i
\(249\) 0 0
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) − 16.0000i − 1.00591i
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 8.00000i − 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) 0 0
\(259\) 36.0000 2.23693
\(260\) 0 0
\(261\) 0 0
\(262\) 17.0000i 1.05026i
\(263\) 30.0000i 1.84988i 0.380114 + 0.924940i \(0.375885\pi\)
−0.380114 + 0.924940i \(0.624115\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 28.0000 1.71679
\(267\) 0 0
\(268\) 7.00000i 0.427593i
\(269\) 5.00000 0.304855 0.152428 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 0 0
\(274\) 19.0000 1.14783
\(275\) 0 0
\(276\) 0 0
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) −17.0000 −1.01413 −0.507067 0.861906i \(-0.669271\pi\)
−0.507067 + 0.861906i \(0.669271\pi\)
\(282\) 0 0
\(283\) − 8.00000i − 0.475551i −0.971320 0.237775i \(-0.923582\pi\)
0.971320 0.237775i \(-0.0764182\pi\)
\(284\) −15.0000 −0.890086
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) − 20.0000i − 1.18056i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 12.0000i 0.702247i
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.00000 −0.523114
\(297\) 0 0
\(298\) 4.00000i 0.231714i
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 40.0000 2.30556
\(302\) 4.00000i 0.230174i
\(303\) 0 0
\(304\) −7.00000 −0.401478
\(305\) 0 0
\(306\) 0 0
\(307\) 21.0000i 1.19853i 0.800549 + 0.599267i \(0.204541\pi\)
−0.800549 + 0.599267i \(0.795459\pi\)
\(308\) − 16.0000i − 0.911685i
\(309\) 0 0
\(310\) 0 0
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) 23.0000i 1.30004i 0.759918 + 0.650018i \(0.225239\pi\)
−0.759918 + 0.650018i \(0.774761\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 7.00000 0.393781
\(317\) 28.0000i 1.57264i 0.617822 + 0.786318i \(0.288015\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 0 0
\(322\) − 16.0000i − 0.891645i
\(323\) − 28.0000i − 1.55796i
\(324\) 0 0
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) 0 0
\(328\) 5.00000i 0.276079i
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) 0 0
\(334\) −9.00000 −0.492458
\(335\) 0 0
\(336\) 0 0
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) − 11.0000i − 0.590511i −0.955418 0.295255i \(-0.904595\pi\)
0.955418 0.295255i \(-0.0954048\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000i 0.213201i
\(353\) − 21.0000i − 1.11772i −0.829263 0.558859i \(-0.811239\pi\)
0.829263 0.558859i \(-0.188761\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 4.00000i 0.210235i
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) 7.00000i 0.365397i 0.983169 + 0.182699i \(0.0584832\pi\)
−0.983169 + 0.182699i \(0.941517\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) 36.0000 1.86903
\(372\) 0 0
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) −16.0000 −0.827340
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) − 5.00000i − 0.257513i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 10.0000i 0.511645i
\(383\) 5.00000i 0.255488i 0.991807 + 0.127744i \(0.0407736\pi\)
−0.991807 + 0.127744i \(0.959226\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 12.0000 0.610784
\(387\) 0 0
\(388\) 16.0000i 0.812277i
\(389\) 5.00000 0.253510 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) − 9.00000i − 0.454569i
\(393\) 0 0
\(394\) 16.0000 0.806068
\(395\) 0 0
\(396\) 0 0
\(397\) − 15.0000i − 0.752828i −0.926451 0.376414i \(-0.877157\pi\)
0.926451 0.376414i \(-0.122843\pi\)
\(398\) − 3.00000i − 0.150376i
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) − 4.00000i − 0.199254i
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) − 36.0000i − 1.78445i
\(408\) 0 0
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) 24.0000i 1.18096i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) − 28.0000i − 1.36952i
\(419\) 23.0000 1.12362 0.561812 0.827265i \(-0.310105\pi\)
0.561812 + 0.827265i \(0.310105\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 16.0000i 0.778868i
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) − 16.0000i − 0.774294i
\(428\) 5.00000i 0.241684i
\(429\) 0 0
\(430\) 0 0
\(431\) −19.0000 −0.915198 −0.457599 0.889159i \(-0.651290\pi\)
−0.457599 + 0.889159i \(0.651290\pi\)
\(432\) 0 0
\(433\) 19.0000i 0.913082i 0.889702 + 0.456541i \(0.150912\pi\)
−0.889702 + 0.456541i \(0.849088\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −11.0000 −0.526804
\(437\) − 28.0000i − 1.33942i
\(438\) 0 0
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 4.00000i − 0.190261i
\(443\) 25.0000i 1.18779i 0.804544 + 0.593893i \(0.202410\pi\)
−0.804544 + 0.593893i \(0.797590\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) 0 0
\(448\) 4.00000i 0.188982i
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) 14.0000i 0.658505i
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 4.00000i 0.187112i 0.995614 + 0.0935561i \(0.0298234\pi\)
−0.995614 + 0.0935561i \(0.970177\pi\)
\(458\) − 17.0000i − 0.794358i
\(459\) 0 0
\(460\) 0 0
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) 0 0
\(463\) 26.0000i 1.20832i 0.796862 + 0.604161i \(0.206492\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) − 3.00000i − 0.138823i −0.997588 0.0694117i \(-0.977888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 0 0
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) − 6.00000i − 0.276172i
\(473\) − 40.0000i − 1.83920i
\(474\) 0 0
\(475\) 0 0
\(476\) −16.0000 −0.733359
\(477\) 0 0
\(478\) − 16.0000i − 0.731823i
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) 9.00000 0.410365
\(482\) − 8.00000i − 0.364390i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 22.0000i 0.996915i 0.866914 + 0.498458i \(0.166100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(488\) 4.00000i 0.181071i
\(489\) 0 0
\(490\) 0 0
\(491\) 40.0000 1.80517 0.902587 0.430507i \(-0.141665\pi\)
0.902587 + 0.430507i \(0.141665\pi\)
\(492\) 0 0
\(493\) 20.0000i 0.900755i
\(494\) 7.00000 0.314945
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) − 60.0000i − 2.69137i
\(498\) 0 0
\(499\) 23.0000 1.02962 0.514811 0.857304i \(-0.327862\pi\)
0.514811 + 0.857304i \(0.327862\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 5.00000i − 0.223161i
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) 5.00000i 0.221839i
\(509\) −4.00000 −0.177297 −0.0886484 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 0 0
\(511\) −48.0000 −2.12339
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −8.00000 −0.352865
\(515\) 0 0
\(516\) 0 0
\(517\) 12.0000i 0.527759i
\(518\) − 36.0000i − 1.58175i
\(519\) 0 0
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 0 0
\(523\) 14.0000i 0.612177i 0.952003 + 0.306089i \(0.0990204\pi\)
−0.952003 + 0.306089i \(0.900980\pi\)
\(524\) 17.0000 0.742648
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) − 28.0000i − 1.21395i
\(533\) − 5.00000i − 0.216574i
\(534\) 0 0
\(535\) 0 0
\(536\) 7.00000 0.302354
\(537\) 0 0
\(538\) − 5.00000i − 0.215565i
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) − 32.0000i − 1.37452i
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 6.00000i 0.256541i 0.991739 + 0.128271i \(0.0409426\pi\)
−0.991739 + 0.128271i \(0.959057\pi\)
\(548\) − 19.0000i − 0.811640i
\(549\) 0 0
\(550\) 0 0
\(551\) −35.0000 −1.49105
\(552\) 0 0
\(553\) 28.0000i 1.19068i
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 38.0000i 1.61011i 0.593199 + 0.805056i \(0.297865\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) 17.0000i 0.717102i
\(563\) 39.0000i 1.64365i 0.569737 + 0.821827i \(0.307045\pi\)
−0.569737 + 0.821827i \(0.692955\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) 15.0000i 0.629386i
\(569\) 8.00000 0.335377 0.167689 0.985840i \(-0.446370\pi\)
0.167689 + 0.985840i \(0.446370\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) − 4.00000i − 0.167248i
\(573\) 0 0
\(574\) −20.0000 −0.834784
\(575\) 0 0
\(576\) 0 0
\(577\) − 8.00000i − 0.333044i −0.986038 0.166522i \(-0.946746\pi\)
0.986038 0.166522i \(-0.0532537\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) − 36.0000i − 1.49097i
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) −28.0000 −1.15372
\(590\) 0 0
\(591\) 0 0
\(592\) 9.00000i 0.369898i
\(593\) 29.0000i 1.19089i 0.803397 + 0.595444i \(0.203024\pi\)
−0.803397 + 0.595444i \(0.796976\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 0 0
\(598\) − 4.00000i − 0.163572i
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) − 40.0000i − 1.63028i
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) − 3.00000i − 0.121766i −0.998145 0.0608831i \(-0.980608\pi\)
0.998145 0.0608831i \(-0.0193917\pi\)
\(608\) 7.00000i 0.283887i
\(609\) 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 \(-0.0128599\pi\)
\(614\) 21.0000 0.847491
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) − 7.00000i − 0.281809i −0.990023 0.140905i \(-0.954999\pi\)
0.990023 0.140905i \(-0.0450011\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2.00000i 0.0801927i
\(623\) − 56.0000i − 2.24359i
\(624\) 0 0
\(625\) 0 0
\(626\) 23.0000 0.919265
\(627\) 0 0
\(628\) 14.0000i 0.558661i
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) − 7.00000i − 0.278445i
\(633\) 0 0
\(634\) 28.0000 1.11202
\(635\) 0 0
\(636\) 0 0
\(637\) 9.00000i 0.356593i
\(638\) 20.0000i 0.791808i
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 5.00000i 0.197181i 0.995128 + 0.0985904i \(0.0314334\pi\)
−0.995128 + 0.0985904i \(0.968567\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −28.0000 −1.10165
\(647\) 36.0000i 1.41531i 0.706560 + 0.707653i \(0.250246\pi\)
−0.706560 + 0.707653i \(0.749754\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) − 24.0000i − 0.939913i
\(653\) − 10.0000i − 0.391330i −0.980671 0.195665i \(-0.937313\pi\)
0.980671 0.195665i \(-0.0626866\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.00000 0.195217
\(657\) 0 0
\(658\) 12.0000i 0.467809i
\(659\) 33.0000 1.28550 0.642749 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) −7.00000 −0.272268 −0.136134 0.990690i \(-0.543468\pi\)
−0.136134 + 0.990690i \(0.543468\pi\)
\(662\) 28.0000i 1.08825i
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 0 0
\(667\) 20.0000i 0.774403i
\(668\) 9.00000i 0.348220i
\(669\) 0 0
\(670\) 0 0
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) − 19.0000i − 0.732396i −0.930537 0.366198i \(-0.880659\pi\)
0.930537 0.366198i \(-0.119341\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 14.0000i 0.538064i 0.963131 + 0.269032i \(0.0867037\pi\)
−0.963131 + 0.269032i \(0.913296\pi\)
\(678\) 0 0
\(679\) −64.0000 −2.45609
\(680\) 0 0
\(681\) 0 0
\(682\) 16.0000i 0.612672i
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) 10.0000i 0.381246i
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) 3.00000 0.114125 0.0570627 0.998371i \(-0.481827\pi\)
0.0570627 + 0.998371i \(0.481827\pi\)
\(692\) − 13.0000i − 0.494186i
\(693\) 0 0
\(694\) −11.0000 −0.417554
\(695\) 0 0
\(696\) 0 0
\(697\) 20.0000i 0.757554i
\(698\) − 34.0000i − 1.28692i
\(699\) 0 0
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 0 0
\(703\) − 63.0000i − 2.37609i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −21.0000 −0.790345
\(707\) − 40.0000i − 1.50435i
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.0000i 0.524672i
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 15.0000i 0.559795i
\(719\) −12.0000 −0.447524 −0.223762 0.974644i \(-0.571834\pi\)
−0.223762 + 0.974644i \(0.571834\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) − 30.0000i − 1.11648i
\(723\) 0 0
\(724\) 4.00000 0.148659
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) − 4.00000i − 0.148250i
\(729\) 0 0
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 0 0
\(733\) 3.00000i 0.110808i 0.998464 + 0.0554038i \(0.0176446\pi\)
−0.998464 + 0.0554038i \(0.982355\pi\)
\(734\) 7.00000 0.258375
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 28.0000i 1.03139i
\(738\) 0 0
\(739\) 37.0000 1.36107 0.680534 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 36.0000i − 1.32160i
\(743\) − 21.0000i − 0.770415i −0.922830 0.385208i \(-0.874130\pi\)
0.922830 0.385208i \(-0.125870\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 0 0
\(748\) 16.0000i 0.585018i
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) − 3.00000i − 0.109399i
\(753\) 0 0
\(754\) −5.00000 −0.182089
\(755\) 0 0
\(756\) 0 0
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) −35.0000 −1.26875 −0.634375 0.773026i \(-0.718742\pi\)
−0.634375 + 0.773026i \(0.718742\pi\)
\(762\) 0 0
\(763\) − 44.0000i − 1.59291i
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) 5.00000 0.180657
\(767\) 6.00000i 0.216647i
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 12.0000i − 0.431889i
\(773\) − 8.00000i − 0.287740i −0.989597 0.143870i \(-0.954045\pi\)
0.989597 0.143870i \(-0.0459547\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 16.0000 0.574367
\(777\) 0 0
\(778\) − 5.00000i − 0.179259i
\(779\) −35.0000 −1.25401
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) 16.0000i 0.572159i
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) − 16.0000i − 0.569976i
\(789\) 0 0
\(790\) 0 0
\(791\) −56.0000 −1.99113
\(792\) 0 0
\(793\) − 4.00000i − 0.142044i
\(794\) −15.0000 −0.532330
\(795\) 0 0
\(796\) −3.00000 −0.106332
\(797\) 30.0000i 1.06265i 0.847167 + 0.531327i \(0.178307\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) − 30.0000i − 1.05934i
\(803\) 48.0000i 1.69388i
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 10.0000i 0.351799i
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 20.0000i 0.701862i
\(813\) 0 0
\(814\) −36.0000 −1.26180
\(815\) 0 0
\(816\) 0 0
\(817\) − 70.0000i − 2.44899i
\(818\) − 24.0000i − 0.839140i
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 19.0000i 0.662298i 0.943578 + 0.331149i \(0.107436\pi\)
−0.943578 + 0.331149i \(0.892564\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 4.00000i 0.139094i 0.997579 + 0.0695468i \(0.0221553\pi\)
−0.997579 + 0.0695468i \(0.977845\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000i 0.0346688i
\(833\) − 36.0000i − 1.24733i
\(834\) 0 0
\(835\) 0 0
\(836\) −28.0000 −0.968400
\(837\) 0 0
\(838\) − 23.0000i − 0.794522i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) − 14.0000i − 0.482472i
\(843\) 0 0
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) 0 0
\(847\) − 20.0000i − 0.687208i
\(848\) 9.00000i 0.309061i
\(849\) 0 0
\(850\) 0 0
\(851\) −36.0000 −1.23406
\(852\) 0 0
\(853\) − 19.0000i − 0.650548i −0.945620 0.325274i \(-0.894544\pi\)
0.945620 0.325274i \(-0.105456\pi\)
\(854\) −16.0000 −0.547509
\(855\) 0 0
\(856\) 5.00000 0.170896
\(857\) 50.0000i 1.70797i 0.520300 + 0.853984i \(0.325820\pi\)
−0.520300 + 0.853984i \(0.674180\pi\)
\(858\) 0 0
\(859\) 2.00000 0.0682391 0.0341196 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 19.0000i 0.647143i
\(863\) 39.0000i 1.32758i 0.747921 + 0.663788i \(0.231052\pi\)
−0.747921 + 0.663788i \(0.768948\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 19.0000 0.645646
\(867\) 0 0
\(868\) 16.0000i 0.543075i
\(869\) 28.0000 0.949835
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) 11.0000i 0.372507i
\(873\) 0 0
\(874\) −28.0000 −0.947114
\(875\) 0 0
\(876\) 0 0
\(877\) 43.0000i 1.45201i 0.687691 + 0.726003i \(0.258624\pi\)
−0.687691 + 0.726003i \(0.741376\pi\)
\(878\) − 7.00000i − 0.236239i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) − 40.0000i − 1.34611i −0.739594 0.673054i \(-0.764982\pi\)
0.739594 0.673054i \(-0.235018\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 25.0000 0.839891
\(887\) 36.0000i 1.20876i 0.796696 + 0.604381i \(0.206579\pi\)
−0.796696 + 0.604381i \(0.793421\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) 0 0
\(892\) − 2.00000i − 0.0669650i
\(893\) 21.0000i 0.702738i
\(894\) 0 0
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 27.0000i 0.901002i
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 20.0000i 0.665927i
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) 50.0000i 1.66022i 0.557598 + 0.830111i \(0.311723\pi\)
−0.557598 + 0.830111i \(0.688277\pi\)
\(908\) 6.00000i 0.199117i
\(909\) 0 0
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) − 24.0000i − 0.794284i
\(914\) 4.00000 0.132308
\(915\) 0 0
\(916\) −17.0000 −0.561696
\(917\) 68.0000i 2.24556i
\(918\) 0 0
\(919\) −29.0000 −0.956622 −0.478311 0.878191i \(-0.658751\pi\)
−0.478311 + 0.878191i \(0.658751\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 24.0000i − 0.790398i
\(923\) − 15.0000i − 0.493731i
\(924\) 0 0
\(925\) 0 0
\(926\) 26.0000 0.854413
\(927\) 0 0
\(928\) − 5.00000i − 0.164133i
\(929\) 11.0000 0.360898 0.180449 0.983584i \(-0.442245\pi\)
0.180449 + 0.983584i \(0.442245\pi\)
\(930\) 0 0
\(931\) 63.0000 2.06474
\(932\) − 8.00000i − 0.262049i
\(933\) 0 0
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 0 0
\(937\) − 22.0000i − 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) 28.0000i 0.914232i
\(939\) 0 0
\(940\) 0 0
\(941\) −52.0000 −1.69515 −0.847576 0.530674i \(-0.821939\pi\)
−0.847576 + 0.530674i \(0.821939\pi\)
\(942\) 0 0
\(943\) 20.0000i 0.651290i
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −40.0000 −1.30051
\(947\) 8.00000i 0.259965i 0.991516 + 0.129983i \(0.0414921\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 16.0000i 0.518563i
\(953\) 36.0000i 1.16615i 0.812417 + 0.583077i \(0.198151\pi\)
−0.812417 + 0.583077i \(0.801849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) − 15.0000i − 0.484628i
\(959\) 76.0000 2.45417
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) − 9.00000i − 0.290172i
\(963\) 0 0
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) 0 0
\(967\) − 44.0000i − 1.41494i −0.706741 0.707472i \(-0.749835\pi\)
0.706741 0.707472i \(-0.250165\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) 21.0000 0.673922 0.336961 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 0 0
\(973\) − 16.0000i − 0.512936i
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) 0 0
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) 0 0
\(982\) − 40.0000i − 1.27645i
\(983\) − 48.0000i − 1.53096i −0.643458 0.765481i \(-0.722501\pi\)
0.643458 0.765481i \(-0.277499\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 20.0000 0.636930
\(987\) 0 0
\(988\) − 7.00000i − 0.222700i
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) 13.0000 0.412959 0.206479 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) − 4.00000i − 0.127000i
\(993\) 0 0
\(994\) −60.0000 −1.90308
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) − 23.0000i − 0.728052i
\(999\) 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.c.5149.1 2
3.2 odd 2 1950.2.e.n.1249.2 2
5.2 odd 4 5850.2.a.by.1.1 1
5.3 odd 4 5850.2.a.a.1.1 1
5.4 even 2 inner 5850.2.e.c.5149.2 2
15.2 even 4 1950.2.a.e.1.1 1
15.8 even 4 1950.2.a.x.1.1 yes 1
15.14 odd 2 1950.2.e.n.1249.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.e.1.1 1 15.2 even 4
1950.2.a.x.1.1 yes 1 15.8 even 4
1950.2.e.n.1249.1 2 15.14 odd 2
1950.2.e.n.1249.2 2 3.2 odd 2
5850.2.a.a.1.1 1 5.3 odd 4
5850.2.a.by.1.1 1 5.2 odd 4
5850.2.e.c.5149.1 2 1.1 even 1 trivial
5850.2.e.c.5149.2 2 5.4 even 2 inner