Properties

Label 4600.2.e.w.4049.2
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 18 x^{8} + 117 x^{6} + 333 x^{4} + 396 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.2
Root \(-2.29544i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.9

$q$-expansion

\(f(q)\) \(=\) \(q-2.29544i q^{3} +1.61758i q^{7} -2.26905 q^{9} +O(q^{10})\) \(q-2.29544i q^{3} +1.61758i q^{7} -2.26905 q^{9} -2.79255 q^{11} +4.88663i q^{13} -3.54388i q^{17} +2.79255 q^{19} +3.71306 q^{21} -1.00000i q^{23} -1.67786i q^{27} +3.36282 q^{29} -4.84564 q^{31} +6.41013i q^{33} -3.87204i q^{37} +11.2170 q^{39} -0.327923 q^{41} -5.38343i q^{43} -0.0121087i q^{47} +4.38343 q^{49} -8.13476 q^{51} -0.866254i q^{53} -6.41013i q^{57} -2.96752 q^{59} -7.29816 q^{61} -3.67037i q^{63} -6.94082i q^{67} -2.29544 q^{69} -2.16864 q^{71} -6.02888i q^{73} -4.51718i q^{77} -6.50561 q^{79} -10.6586 q^{81} -7.88030i q^{83} -7.71915i q^{87} +9.71986 q^{89} -7.90452 q^{91} +11.1229i q^{93} -11.9952i q^{97} +6.33643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} + O(q^{10}) \) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 2.29544i − 1.32527i −0.748941 0.662637i \(-0.769437\pi\)
0.748941 0.662637i \(-0.230563\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.61758i 0.611388i 0.952130 + 0.305694i \(0.0988885\pi\)
−0.952130 + 0.305694i \(0.901111\pi\)
\(8\) 0 0
\(9\) −2.26905 −0.756349
\(10\) 0 0
\(11\) −2.79255 −0.841985 −0.420993 0.907064i \(-0.638318\pi\)
−0.420993 + 0.907064i \(0.638318\pi\)
\(12\) 0 0
\(13\) 4.88663i 1.35531i 0.735381 + 0.677653i \(0.237003\pi\)
−0.735381 + 0.677653i \(0.762997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.54388i − 0.859516i −0.902944 0.429758i \(-0.858599\pi\)
0.902944 0.429758i \(-0.141401\pi\)
\(18\) 0 0
\(19\) 2.79255 0.640655 0.320327 0.947307i \(-0.396207\pi\)
0.320327 + 0.947307i \(0.396207\pi\)
\(20\) 0 0
\(21\) 3.71306 0.810257
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.67786i − 0.322904i
\(28\) 0 0
\(29\) 3.36282 0.624460 0.312230 0.950007i \(-0.398924\pi\)
0.312230 + 0.950007i \(0.398924\pi\)
\(30\) 0 0
\(31\) −4.84564 −0.870303 −0.435152 0.900357i \(-0.643305\pi\)
−0.435152 + 0.900357i \(0.643305\pi\)
\(32\) 0 0
\(33\) 6.41013i 1.11586i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.87204i − 0.636559i −0.947997 0.318279i \(-0.896895\pi\)
0.947997 0.318279i \(-0.103105\pi\)
\(38\) 0 0
\(39\) 11.2170 1.79615
\(40\) 0 0
\(41\) −0.327923 −0.0512130 −0.0256065 0.999672i \(-0.508152\pi\)
−0.0256065 + 0.999672i \(0.508152\pi\)
\(42\) 0 0
\(43\) − 5.38343i − 0.820965i −0.911868 0.410483i \(-0.865360\pi\)
0.911868 0.410483i \(-0.134640\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.0121087i − 0.00176623i −1.00000 0.000883117i \(-0.999719\pi\)
1.00000 0.000883117i \(-0.000281105\pi\)
\(48\) 0 0
\(49\) 4.38343 0.626204
\(50\) 0 0
\(51\) −8.13476 −1.13909
\(52\) 0 0
\(53\) − 0.866254i − 0.118989i −0.998229 0.0594946i \(-0.981051\pi\)
0.998229 0.0594946i \(-0.0189489\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.41013i − 0.849043i
\(58\) 0 0
\(59\) −2.96752 −0.386338 −0.193169 0.981166i \(-0.561877\pi\)
−0.193169 + 0.981166i \(0.561877\pi\)
\(60\) 0 0
\(61\) −7.29816 −0.934434 −0.467217 0.884143i \(-0.654743\pi\)
−0.467217 + 0.884143i \(0.654743\pi\)
\(62\) 0 0
\(63\) − 3.67037i − 0.462423i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.94082i − 0.847956i −0.905673 0.423978i \(-0.860633\pi\)
0.905673 0.423978i \(-0.139367\pi\)
\(68\) 0 0
\(69\) −2.29544 −0.276339
\(70\) 0 0
\(71\) −2.16864 −0.257370 −0.128685 0.991685i \(-0.541076\pi\)
−0.128685 + 0.991685i \(0.541076\pi\)
\(72\) 0 0
\(73\) − 6.02888i − 0.705627i −0.935694 0.352813i \(-0.885225\pi\)
0.935694 0.352813i \(-0.114775\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.51718i − 0.514780i
\(78\) 0 0
\(79\) −6.50561 −0.731938 −0.365969 0.930627i \(-0.619262\pi\)
−0.365969 + 0.930627i \(0.619262\pi\)
\(80\) 0 0
\(81\) −10.6586 −1.18429
\(82\) 0 0
\(83\) − 7.88030i − 0.864976i −0.901640 0.432488i \(-0.857636\pi\)
0.901640 0.432488i \(-0.142364\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 7.71915i − 0.827580i
\(88\) 0 0
\(89\) 9.71986 1.03030 0.515151 0.857099i \(-0.327736\pi\)
0.515151 + 0.857099i \(0.327736\pi\)
\(90\) 0 0
\(91\) −7.90452 −0.828619
\(92\) 0 0
\(93\) 11.1229i 1.15339i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 11.9952i − 1.21793i −0.793197 0.608966i \(-0.791585\pi\)
0.793197 0.608966i \(-0.208415\pi\)
\(98\) 0 0
\(99\) 6.33643 0.636835
\(100\) 0 0
\(101\) 11.4712 1.14143 0.570713 0.821150i \(-0.306667\pi\)
0.570713 + 0.821150i \(0.306667\pi\)
\(102\) 0 0
\(103\) − 4.02030i − 0.396132i −0.980189 0.198066i \(-0.936534\pi\)
0.980189 0.198066i \(-0.0634661\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 14.8318i − 1.43385i −0.697152 0.716923i \(-0.745550\pi\)
0.697152 0.716923i \(-0.254450\pi\)
\(108\) 0 0
\(109\) −12.3573 −1.18362 −0.591809 0.806078i \(-0.701586\pi\)
−0.591809 + 0.806078i \(0.701586\pi\)
\(110\) 0 0
\(111\) −8.88803 −0.843614
\(112\) 0 0
\(113\) − 5.48632i − 0.516109i −0.966130 0.258055i \(-0.916919\pi\)
0.966130 0.258055i \(-0.0830815\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 11.0880i − 1.02509i
\(118\) 0 0
\(119\) 5.73251 0.525498
\(120\) 0 0
\(121\) −3.20167 −0.291061
\(122\) 0 0
\(123\) 0.752728i 0.0678712i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.14959i 0.279481i 0.990188 + 0.139740i \(0.0446268\pi\)
−0.990188 + 0.139740i \(0.955373\pi\)
\(128\) 0 0
\(129\) −12.3573 −1.08800
\(130\) 0 0
\(131\) −2.93231 −0.256197 −0.128099 0.991761i \(-0.540887\pi\)
−0.128099 + 0.991761i \(0.540887\pi\)
\(132\) 0 0
\(133\) 4.51718i 0.391689i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 14.4054i − 1.23073i −0.788241 0.615366i \(-0.789008\pi\)
0.788241 0.615366i \(-0.210992\pi\)
\(138\) 0 0
\(139\) −18.2241 −1.54575 −0.772873 0.634561i \(-0.781181\pi\)
−0.772873 + 0.634561i \(0.781181\pi\)
\(140\) 0 0
\(141\) −0.0277948 −0.00234074
\(142\) 0 0
\(143\) − 13.6462i − 1.14115i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 10.0619i − 0.829892i
\(148\) 0 0
\(149\) 5.71986 0.468589 0.234294 0.972166i \(-0.424722\pi\)
0.234294 + 0.972166i \(0.424722\pi\)
\(150\) 0 0
\(151\) −5.43082 −0.441954 −0.220977 0.975279i \(-0.570925\pi\)
−0.220977 + 0.975279i \(0.570925\pi\)
\(152\) 0 0
\(153\) 8.04122i 0.650094i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 24.2004i 1.93140i 0.259657 + 0.965701i \(0.416390\pi\)
−0.259657 + 0.965701i \(0.583610\pi\)
\(158\) 0 0
\(159\) −1.98844 −0.157693
\(160\) 0 0
\(161\) 1.61758 0.127483
\(162\) 0 0
\(163\) − 11.2970i − 0.884849i −0.896806 0.442425i \(-0.854118\pi\)
0.896806 0.442425i \(-0.145882\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 21.6478i − 1.67516i −0.546318 0.837578i \(-0.683971\pi\)
0.546318 0.837578i \(-0.316029\pi\)
\(168\) 0 0
\(169\) −10.8791 −0.836857
\(170\) 0 0
\(171\) −6.33643 −0.484559
\(172\) 0 0
\(173\) 1.39593i 0.106130i 0.998591 + 0.0530652i \(0.0168991\pi\)
−0.998591 + 0.0530652i \(0.983101\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.81176i 0.512003i
\(178\) 0 0
\(179\) −23.5665 −1.76144 −0.880722 0.473634i \(-0.842942\pi\)
−0.880722 + 0.473634i \(0.842942\pi\)
\(180\) 0 0
\(181\) 8.33495 0.619532 0.309766 0.950813i \(-0.399749\pi\)
0.309766 + 0.950813i \(0.399749\pi\)
\(182\) 0 0
\(183\) 16.7525i 1.23838i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.89645i 0.723700i
\(188\) 0 0
\(189\) 2.71407 0.197420
\(190\) 0 0
\(191\) 18.3074 1.32468 0.662340 0.749204i \(-0.269564\pi\)
0.662340 + 0.749204i \(0.269564\pi\)
\(192\) 0 0
\(193\) 15.4434i 1.11164i 0.831303 + 0.555820i \(0.187596\pi\)
−0.831303 + 0.555820i \(0.812404\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.876228i 0.0624287i 0.999513 + 0.0312143i \(0.00993745\pi\)
−0.999513 + 0.0312143i \(0.990063\pi\)
\(198\) 0 0
\(199\) 13.5818 0.962788 0.481394 0.876504i \(-0.340131\pi\)
0.481394 + 0.876504i \(0.340131\pi\)
\(200\) 0 0
\(201\) −15.9322 −1.12377
\(202\) 0 0
\(203\) 5.43963i 0.381787i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.26905i 0.157710i
\(208\) 0 0
\(209\) −7.79833 −0.539422
\(210\) 0 0
\(211\) 12.9100 0.888758 0.444379 0.895839i \(-0.353424\pi\)
0.444379 + 0.895839i \(0.353424\pi\)
\(212\) 0 0
\(213\) 4.97799i 0.341086i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 7.83822i − 0.532093i
\(218\) 0 0
\(219\) −13.8389 −0.935148
\(220\) 0 0
\(221\) 17.3176 1.16491
\(222\) 0 0
\(223\) − 8.05527i − 0.539421i −0.962941 0.269710i \(-0.913072\pi\)
0.962941 0.269710i \(-0.0869280\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 4.41490i − 0.293027i −0.989209 0.146514i \(-0.953195\pi\)
0.989209 0.146514i \(-0.0468052\pi\)
\(228\) 0 0
\(229\) −17.5197 −1.15773 −0.578866 0.815423i \(-0.696505\pi\)
−0.578866 + 0.815423i \(0.696505\pi\)
\(230\) 0 0
\(231\) −10.3689 −0.682224
\(232\) 0 0
\(233\) 15.8080i 1.03562i 0.855497 + 0.517808i \(0.173252\pi\)
−0.855497 + 0.517808i \(0.826748\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 14.9332i 0.970018i
\(238\) 0 0
\(239\) 8.93972 0.578263 0.289131 0.957289i \(-0.406634\pi\)
0.289131 + 0.957289i \(0.406634\pi\)
\(240\) 0 0
\(241\) 3.50992 0.226094 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(242\) 0 0
\(243\) 19.4325i 1.24660i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.6462i 0.868284i
\(248\) 0 0
\(249\) −18.0888 −1.14633
\(250\) 0 0
\(251\) 4.33837 0.273836 0.136918 0.990582i \(-0.456280\pi\)
0.136918 + 0.990582i \(0.456280\pi\)
\(252\) 0 0
\(253\) 2.79255i 0.175566i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.6252i 1.03705i 0.855063 + 0.518524i \(0.173518\pi\)
−0.855063 + 0.518524i \(0.826482\pi\)
\(258\) 0 0
\(259\) 6.26333 0.389185
\(260\) 0 0
\(261\) −7.63039 −0.472310
\(262\) 0 0
\(263\) 13.0084i 0.802134i 0.916049 + 0.401067i \(0.131360\pi\)
−0.916049 + 0.401067i \(0.868640\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 22.3114i − 1.36543i
\(268\) 0 0
\(269\) 8.30995 0.506667 0.253333 0.967379i \(-0.418473\pi\)
0.253333 + 0.967379i \(0.418473\pi\)
\(270\) 0 0
\(271\) −8.35009 −0.507232 −0.253616 0.967305i \(-0.581620\pi\)
−0.253616 + 0.967305i \(0.581620\pi\)
\(272\) 0 0
\(273\) 18.1444i 1.09815i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 6.23195i − 0.374441i −0.982318 0.187221i \(-0.940052\pi\)
0.982318 0.187221i \(-0.0599479\pi\)
\(278\) 0 0
\(279\) 10.9950 0.658253
\(280\) 0 0
\(281\) 8.01836 0.478335 0.239168 0.970978i \(-0.423125\pi\)
0.239168 + 0.970978i \(0.423125\pi\)
\(282\) 0 0
\(283\) − 9.11741i − 0.541974i −0.962583 0.270987i \(-0.912650\pi\)
0.962583 0.270987i \(-0.0873500\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 0.530443i − 0.0313110i
\(288\) 0 0
\(289\) 4.44094 0.261232
\(290\) 0 0
\(291\) −27.5343 −1.61409
\(292\) 0 0
\(293\) 17.5323i 1.02425i 0.858911 + 0.512124i \(0.171141\pi\)
−0.858911 + 0.512124i \(0.828859\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.68550i 0.271881i
\(298\) 0 0
\(299\) 4.88663 0.282601
\(300\) 0 0
\(301\) 8.70814 0.501929
\(302\) 0 0
\(303\) − 26.3314i − 1.51270i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 30.3404i − 1.73162i −0.500375 0.865809i \(-0.666804\pi\)
0.500375 0.865809i \(-0.333196\pi\)
\(308\) 0 0
\(309\) −9.22837 −0.524984
\(310\) 0 0
\(311\) −13.7703 −0.780840 −0.390420 0.920637i \(-0.627670\pi\)
−0.390420 + 0.920637i \(0.627670\pi\)
\(312\) 0 0
\(313\) 14.3715i 0.812328i 0.913800 + 0.406164i \(0.133134\pi\)
−0.913800 + 0.406164i \(0.866866\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 26.5726i − 1.49247i −0.665685 0.746233i \(-0.731860\pi\)
0.665685 0.746233i \(-0.268140\pi\)
\(318\) 0 0
\(319\) −9.39084 −0.525786
\(320\) 0 0
\(321\) −34.0456 −1.90024
\(322\) 0 0
\(323\) − 9.89645i − 0.550653i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 28.3655i 1.56862i
\(328\) 0 0
\(329\) 0.0195868 0.00107986
\(330\) 0 0
\(331\) −12.2379 −0.672655 −0.336327 0.941745i \(-0.609185\pi\)
−0.336327 + 0.941745i \(0.609185\pi\)
\(332\) 0 0
\(333\) 8.78583i 0.481461i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 16.7474i − 0.912290i −0.889906 0.456145i \(-0.849230\pi\)
0.889906 0.456145i \(-0.150770\pi\)
\(338\) 0 0
\(339\) −12.5935 −0.683986
\(340\) 0 0
\(341\) 13.5317 0.732783
\(342\) 0 0
\(343\) 18.4136i 0.994242i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 20.6122i − 1.10652i −0.833008 0.553260i \(-0.813383\pi\)
0.833008 0.553260i \(-0.186617\pi\)
\(348\) 0 0
\(349\) −35.5885 −1.90501 −0.952505 0.304523i \(-0.901503\pi\)
−0.952505 + 0.304523i \(0.901503\pi\)
\(350\) 0 0
\(351\) 8.19907 0.437634
\(352\) 0 0
\(353\) − 0.574309i − 0.0305674i −0.999883 0.0152837i \(-0.995135\pi\)
0.999883 0.0152837i \(-0.00486514\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 13.1586i − 0.696429i
\(358\) 0 0
\(359\) −4.12882 −0.217911 −0.108955 0.994047i \(-0.534751\pi\)
−0.108955 + 0.994047i \(0.534751\pi\)
\(360\) 0 0
\(361\) −11.2017 −0.589561
\(362\) 0 0
\(363\) 7.34924i 0.385735i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.16724i − 0.217528i −0.994068 0.108764i \(-0.965311\pi\)
0.994068 0.108764i \(-0.0346893\pi\)
\(368\) 0 0
\(369\) 0.744073 0.0387349
\(370\) 0 0
\(371\) 1.40124 0.0727486
\(372\) 0 0
\(373\) − 25.3579i − 1.31298i −0.754334 0.656491i \(-0.772040\pi\)
0.754334 0.656491i \(-0.227960\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.4328i 0.846335i
\(378\) 0 0
\(379\) −14.2580 −0.732382 −0.366191 0.930540i \(-0.619338\pi\)
−0.366191 + 0.930540i \(0.619338\pi\)
\(380\) 0 0
\(381\) 7.22969 0.370388
\(382\) 0 0
\(383\) 16.6805i 0.852334i 0.904645 + 0.426167i \(0.140136\pi\)
−0.904645 + 0.426167i \(0.859864\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 12.2153i 0.620936i
\(388\) 0 0
\(389\) −26.3853 −1.33779 −0.668894 0.743358i \(-0.733232\pi\)
−0.668894 + 0.743358i \(0.733232\pi\)
\(390\) 0 0
\(391\) −3.54388 −0.179222
\(392\) 0 0
\(393\) 6.73095i 0.339532i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.44231i 0.323330i 0.986846 + 0.161665i \(0.0516864\pi\)
−0.986846 + 0.161665i \(0.948314\pi\)
\(398\) 0 0
\(399\) 10.3689 0.519095
\(400\) 0 0
\(401\) 21.6326 1.08028 0.540141 0.841574i \(-0.318371\pi\)
0.540141 + 0.841574i \(0.318371\pi\)
\(402\) 0 0
\(403\) − 23.6789i − 1.17953i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.8129i 0.535973i
\(408\) 0 0
\(409\) −8.07200 −0.399135 −0.199567 0.979884i \(-0.563954\pi\)
−0.199567 + 0.979884i \(0.563954\pi\)
\(410\) 0 0
\(411\) −33.0666 −1.63106
\(412\) 0 0
\(413\) − 4.80020i − 0.236202i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 41.8323i 2.04853i
\(418\) 0 0
\(419\) 25.4538 1.24350 0.621751 0.783215i \(-0.286422\pi\)
0.621751 + 0.783215i \(0.286422\pi\)
\(420\) 0 0
\(421\) 31.4111 1.53089 0.765443 0.643504i \(-0.222520\pi\)
0.765443 + 0.643504i \(0.222520\pi\)
\(422\) 0 0
\(423\) 0.0274752i 0.00133589i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 11.8054i − 0.571302i
\(428\) 0 0
\(429\) −31.3239 −1.51233
\(430\) 0 0
\(431\) 3.35572 0.161639 0.0808196 0.996729i \(-0.474246\pi\)
0.0808196 + 0.996729i \(0.474246\pi\)
\(432\) 0 0
\(433\) 18.0365i 0.866777i 0.901207 + 0.433388i \(0.142682\pi\)
−0.901207 + 0.433388i \(0.857318\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2.79255i − 0.133586i
\(438\) 0 0
\(439\) −19.8080 −0.945384 −0.472692 0.881228i \(-0.656718\pi\)
−0.472692 + 0.881228i \(0.656718\pi\)
\(440\) 0 0
\(441\) −9.94621 −0.473629
\(442\) 0 0
\(443\) 11.5628i 0.549364i 0.961535 + 0.274682i \(0.0885726\pi\)
−0.961535 + 0.274682i \(0.911427\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 13.1296i − 0.621008i
\(448\) 0 0
\(449\) 19.6506 0.927368 0.463684 0.886001i \(-0.346527\pi\)
0.463684 + 0.886001i \(0.346527\pi\)
\(450\) 0 0
\(451\) 0.915742 0.0431206
\(452\) 0 0
\(453\) 12.4661i 0.585710i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 23.0682i 1.07908i 0.841959 + 0.539542i \(0.181402\pi\)
−0.841959 + 0.539542i \(0.818598\pi\)
\(458\) 0 0
\(459\) −5.94613 −0.277541
\(460\) 0 0
\(461\) 1.86556 0.0868876 0.0434438 0.999056i \(-0.486167\pi\)
0.0434438 + 0.999056i \(0.486167\pi\)
\(462\) 0 0
\(463\) − 12.9981i − 0.604071i −0.953297 0.302035i \(-0.902334\pi\)
0.953297 0.302035i \(-0.0976661\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 27.7712i − 1.28510i −0.766245 0.642548i \(-0.777877\pi\)
0.766245 0.642548i \(-0.222123\pi\)
\(468\) 0 0
\(469\) 11.2273 0.518430
\(470\) 0 0
\(471\) 55.5506 2.55963
\(472\) 0 0
\(473\) 15.0335i 0.691241i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.96557i 0.0899974i
\(478\) 0 0
\(479\) 5.25003 0.239880 0.119940 0.992781i \(-0.461730\pi\)
0.119940 + 0.992781i \(0.461730\pi\)
\(480\) 0 0
\(481\) 18.9212 0.862733
\(482\) 0 0
\(483\) − 3.71306i − 0.168950i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9.90856i 0.449000i 0.974474 + 0.224500i \(0.0720748\pi\)
−0.974474 + 0.224500i \(0.927925\pi\)
\(488\) 0 0
\(489\) −25.9316 −1.17267
\(490\) 0 0
\(491\) −24.2168 −1.09289 −0.546444 0.837495i \(-0.684019\pi\)
−0.546444 + 0.837495i \(0.684019\pi\)
\(492\) 0 0
\(493\) − 11.9174i − 0.536733i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 3.50795i − 0.157353i
\(498\) 0 0
\(499\) 26.6722 1.19401 0.597007 0.802236i \(-0.296357\pi\)
0.597007 + 0.802236i \(0.296357\pi\)
\(500\) 0 0
\(501\) −49.6912 −2.22004
\(502\) 0 0
\(503\) 2.38303i 0.106254i 0.998588 + 0.0531271i \(0.0169188\pi\)
−0.998588 + 0.0531271i \(0.983081\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 24.9724i 1.10906i
\(508\) 0 0
\(509\) −36.2807 −1.60811 −0.804056 0.594554i \(-0.797329\pi\)
−0.804056 + 0.594554i \(0.797329\pi\)
\(510\) 0 0
\(511\) 9.75220 0.431412
\(512\) 0 0
\(513\) − 4.68550i − 0.206870i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.0338141i 0.00148714i
\(518\) 0 0
\(519\) 3.20427 0.140652
\(520\) 0 0
\(521\) −13.8498 −0.606773 −0.303386 0.952868i \(-0.598117\pi\)
−0.303386 + 0.952868i \(0.598117\pi\)
\(522\) 0 0
\(523\) − 11.3982i − 0.498411i −0.968451 0.249205i \(-0.919831\pi\)
0.968451 0.249205i \(-0.0801694\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.1724i 0.748040i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 6.73344 0.292206
\(532\) 0 0
\(533\) − 1.60244i − 0.0694094i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 54.0955i 2.33439i
\(538\) 0 0
\(539\) −12.2409 −0.527255
\(540\) 0 0
\(541\) −15.6397 −0.672402 −0.336201 0.941790i \(-0.609142\pi\)
−0.336201 + 0.941790i \(0.609142\pi\)
\(542\) 0 0
\(543\) − 19.1324i − 0.821050i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 45.7911i − 1.95789i −0.204131 0.978944i \(-0.565437\pi\)
0.204131 0.978944i \(-0.434563\pi\)
\(548\) 0 0
\(549\) 16.5599 0.706758
\(550\) 0 0
\(551\) 9.39084 0.400063
\(552\) 0 0
\(553\) − 10.5234i − 0.447499i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.8143i 1.30564i 0.757512 + 0.652821i \(0.226415\pi\)
−0.757512 + 0.652821i \(0.773585\pi\)
\(558\) 0 0
\(559\) 26.3068 1.11266
\(560\) 0 0
\(561\) 22.7167 0.959100
\(562\) 0 0
\(563\) 27.8036i 1.17178i 0.810390 + 0.585891i \(0.199255\pi\)
−0.810390 + 0.585891i \(0.800745\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 17.2411i − 0.724058i
\(568\) 0 0
\(569\) −7.89706 −0.331062 −0.165531 0.986205i \(-0.552934\pi\)
−0.165531 + 0.986205i \(0.552934\pi\)
\(570\) 0 0
\(571\) 6.12319 0.256248 0.128124 0.991758i \(-0.459105\pi\)
0.128124 + 0.991758i \(0.459105\pi\)
\(572\) 0 0
\(573\) − 42.0236i − 1.75556i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 27.2174i − 1.13307i −0.824036 0.566537i \(-0.808283\pi\)
0.824036 0.566537i \(-0.191717\pi\)
\(578\) 0 0
\(579\) 35.4494 1.47323
\(580\) 0 0
\(581\) 12.7470 0.528836
\(582\) 0 0
\(583\) 2.41906i 0.100187i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 26.7305i − 1.10329i −0.834080 0.551644i \(-0.814001\pi\)
0.834080 0.551644i \(-0.185999\pi\)
\(588\) 0 0
\(589\) −13.5317 −0.557564
\(590\) 0 0
\(591\) 2.01133 0.0827350
\(592\) 0 0
\(593\) − 0.0562817i − 0.00231121i −0.999999 0.00115561i \(-0.999632\pi\)
0.999999 0.00115561i \(-0.000367841\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 31.1762i − 1.27596i
\(598\) 0 0
\(599\) 39.4760 1.61295 0.806473 0.591271i \(-0.201374\pi\)
0.806473 + 0.591271i \(0.201374\pi\)
\(600\) 0 0
\(601\) 23.2118 0.946829 0.473415 0.880840i \(-0.343021\pi\)
0.473415 + 0.880840i \(0.343021\pi\)
\(602\) 0 0
\(603\) 15.7490i 0.641350i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 19.7212i 0.800459i 0.916415 + 0.400230i \(0.131070\pi\)
−0.916415 + 0.400230i \(0.868930\pi\)
\(608\) 0 0
\(609\) 12.4864 0.505973
\(610\) 0 0
\(611\) 0.0591707 0.00239379
\(612\) 0 0
\(613\) − 10.4609i − 0.422512i −0.977431 0.211256i \(-0.932245\pi\)
0.977431 0.211256i \(-0.0677553\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.4969i 0.583622i 0.956476 + 0.291811i \(0.0942578\pi\)
−0.956476 + 0.291811i \(0.905742\pi\)
\(618\) 0 0
\(619\) 1.30293 0.0523692 0.0261846 0.999657i \(-0.491664\pi\)
0.0261846 + 0.999657i \(0.491664\pi\)
\(620\) 0 0
\(621\) −1.67786 −0.0673302
\(622\) 0 0
\(623\) 15.7227i 0.629915i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 17.9006i 0.714881i
\(628\) 0 0
\(629\) −13.7220 −0.547133
\(630\) 0 0
\(631\) −0.653161 −0.0260019 −0.0130010 0.999915i \(-0.504138\pi\)
−0.0130010 + 0.999915i \(0.504138\pi\)
\(632\) 0 0
\(633\) − 29.6340i − 1.17785i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 21.4202i 0.848699i
\(638\) 0 0
\(639\) 4.92075 0.194662
\(640\) 0 0
\(641\) −13.2594 −0.523714 −0.261857 0.965107i \(-0.584335\pi\)
−0.261857 + 0.965107i \(0.584335\pi\)
\(642\) 0 0
\(643\) 45.9135i 1.81065i 0.424716 + 0.905326i \(0.360374\pi\)
−0.424716 + 0.905326i \(0.639626\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 3.36368i − 0.132240i −0.997812 0.0661199i \(-0.978938\pi\)
0.997812 0.0661199i \(-0.0210620\pi\)
\(648\) 0 0
\(649\) 8.28694 0.325291
\(650\) 0 0
\(651\) −17.9922 −0.705169
\(652\) 0 0
\(653\) − 25.4704i − 0.996734i −0.866966 0.498367i \(-0.833933\pi\)
0.866966 0.498367i \(-0.166067\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 13.6798i 0.533700i
\(658\) 0 0
\(659\) −9.03335 −0.351890 −0.175945 0.984400i \(-0.556298\pi\)
−0.175945 + 0.984400i \(0.556298\pi\)
\(660\) 0 0
\(661\) −3.20679 −0.124730 −0.0623648 0.998053i \(-0.519864\pi\)
−0.0623648 + 0.998053i \(0.519864\pi\)
\(662\) 0 0
\(663\) − 39.7515i − 1.54382i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 3.36282i − 0.130209i
\(668\) 0 0
\(669\) −18.4904 −0.714880
\(670\) 0 0
\(671\) 20.3805 0.786779
\(672\) 0 0
\(673\) − 16.4923i − 0.635733i −0.948135 0.317866i \(-0.897034\pi\)
0.948135 0.317866i \(-0.102966\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 34.9260i − 1.34232i −0.741314 0.671158i \(-0.765797\pi\)
0.741314 0.671158i \(-0.234203\pi\)
\(678\) 0 0
\(679\) 19.4033 0.744629
\(680\) 0 0
\(681\) −10.1341 −0.388341
\(682\) 0 0
\(683\) 23.1448i 0.885612i 0.896618 + 0.442806i \(0.146017\pi\)
−0.896618 + 0.442806i \(0.853983\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 40.2153i 1.53431i
\(688\) 0 0
\(689\) 4.23306 0.161267
\(690\) 0 0
\(691\) 3.24273 0.123359 0.0616795 0.998096i \(-0.480354\pi\)
0.0616795 + 0.998096i \(0.480354\pi\)
\(692\) 0 0
\(693\) 10.2497i 0.389353i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.16212i 0.0440184i
\(698\) 0 0
\(699\) 36.2863 1.37247
\(700\) 0 0
\(701\) 19.9466 0.753373 0.376686 0.926341i \(-0.377063\pi\)
0.376686 + 0.926341i \(0.377063\pi\)
\(702\) 0 0
\(703\) − 10.8129i − 0.407814i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.5556i 0.697854i
\(708\) 0 0
\(709\) −14.3774 −0.539955 −0.269977 0.962867i \(-0.587016\pi\)
−0.269977 + 0.962867i \(0.587016\pi\)
\(710\) 0 0
\(711\) 14.7615 0.553601
\(712\) 0 0
\(713\) 4.84564i 0.181471i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 20.5206i − 0.766356i
\(718\) 0 0
\(719\) 51.4356 1.91822 0.959111 0.283029i \(-0.0913394\pi\)
0.959111 + 0.283029i \(0.0913394\pi\)
\(720\) 0 0
\(721\) 6.50317 0.242191
\(722\) 0 0
\(723\) − 8.05682i − 0.299636i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 17.4611i − 0.647597i −0.946126 0.323799i \(-0.895040\pi\)
0.946126 0.323799i \(-0.104960\pi\)
\(728\) 0 0
\(729\) 12.6305 0.467797
\(730\) 0 0
\(731\) −19.0782 −0.705633
\(732\) 0 0
\(733\) − 39.6443i − 1.46429i −0.681147 0.732147i \(-0.738518\pi\)
0.681147 0.732147i \(-0.261482\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.3826i 0.713966i
\(738\) 0 0
\(739\) 43.1149 1.58601 0.793003 0.609218i \(-0.208517\pi\)
0.793003 + 0.609218i \(0.208517\pi\)
\(740\) 0 0
\(741\) 31.3239 1.15071
\(742\) 0 0
\(743\) − 18.8929i − 0.693112i −0.938029 0.346556i \(-0.887351\pi\)
0.938029 0.346556i \(-0.112649\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 17.8808i 0.654223i
\(748\) 0 0
\(749\) 23.9917 0.876637
\(750\) 0 0
\(751\) 27.5146 1.00402 0.502012 0.864861i \(-0.332593\pi\)
0.502012 + 0.864861i \(0.332593\pi\)
\(752\) 0 0
\(753\) − 9.95847i − 0.362907i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 14.6895i − 0.533899i −0.963711 0.266949i \(-0.913984\pi\)
0.963711 0.266949i \(-0.0860157\pi\)
\(758\) 0 0
\(759\) 6.41013 0.232673
\(760\) 0 0
\(761\) −37.5884 −1.36258 −0.681289 0.732015i \(-0.738580\pi\)
−0.681289 + 0.732015i \(0.738580\pi\)
\(762\) 0 0
\(763\) − 19.9890i − 0.723651i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 14.5012i − 0.523606i
\(768\) 0 0
\(769\) 43.6879 1.57543 0.787713 0.616042i \(-0.211265\pi\)
0.787713 + 0.616042i \(0.211265\pi\)
\(770\) 0 0
\(771\) 38.1620 1.37437
\(772\) 0 0
\(773\) 7.01583i 0.252342i 0.992009 + 0.126171i \(0.0402688\pi\)
−0.992009 + 0.126171i \(0.959731\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 14.3771i − 0.515776i
\(778\) 0 0
\(779\) −0.915742 −0.0328099
\(780\) 0 0
\(781\) 6.05604 0.216702
\(782\) 0 0
\(783\) − 5.64234i − 0.201641i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 23.6386i 0.842624i 0.906916 + 0.421312i \(0.138430\pi\)
−0.906916 + 0.421312i \(0.861570\pi\)
\(788\) 0 0
\(789\) 29.8601 1.06305
\(790\) 0 0
\(791\) 8.87457 0.315543
\(792\) 0 0
\(793\) − 35.6634i − 1.26644i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 31.9000i − 1.12996i −0.825106 0.564978i \(-0.808885\pi\)
0.825106 0.564978i \(-0.191115\pi\)
\(798\) 0 0
\(799\) −0.0429117 −0.00151811
\(800\) 0 0
\(801\) −22.0548 −0.779268
\(802\) 0 0
\(803\) 16.8359i 0.594127i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 19.0750i − 0.671472i
\(808\) 0 0
\(809\) 5.68831 0.199990 0.0999951 0.994988i \(-0.468117\pi\)
0.0999951 + 0.994988i \(0.468117\pi\)
\(810\) 0 0
\(811\) −16.1989 −0.568821 −0.284410 0.958703i \(-0.591798\pi\)
−0.284410 + 0.958703i \(0.591798\pi\)
\(812\) 0 0
\(813\) 19.1671i 0.672221i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 15.0335i − 0.525955i
\(818\) 0 0
\(819\) 17.9357 0.626725
\(820\) 0 0
\(821\) −26.9329 −0.939965 −0.469982 0.882676i \(-0.655740\pi\)
−0.469982 + 0.882676i \(0.655740\pi\)
\(822\) 0 0
\(823\) 42.0280i 1.46500i 0.680765 + 0.732502i \(0.261648\pi\)
−0.680765 + 0.732502i \(0.738352\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.1311i 0.387065i 0.981094 + 0.193532i \(0.0619945\pi\)
−0.981094 + 0.193532i \(0.938006\pi\)
\(828\) 0 0
\(829\) 20.1506 0.699859 0.349929 0.936776i \(-0.386206\pi\)
0.349929 + 0.936776i \(0.386206\pi\)
\(830\) 0 0
\(831\) −14.3051 −0.496237
\(832\) 0 0
\(833\) − 15.5343i − 0.538233i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.13031i 0.281024i
\(838\) 0 0
\(839\) 12.5573 0.433526 0.216763 0.976224i \(-0.430450\pi\)
0.216763 + 0.976224i \(0.430450\pi\)
\(840\) 0 0
\(841\) −17.6914 −0.610050
\(842\) 0 0
\(843\) − 18.4057i − 0.633925i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 5.17896i − 0.177951i
\(848\) 0 0
\(849\) −20.9285 −0.718263
\(850\) 0 0
\(851\) −3.87204 −0.132732
\(852\) 0 0
\(853\) − 50.2016i − 1.71887i −0.511246 0.859434i \(-0.670816\pi\)
0.511246 0.859434i \(-0.329184\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 36.9126i 1.26091i 0.776225 + 0.630456i \(0.217132\pi\)
−0.776225 + 0.630456i \(0.782868\pi\)
\(858\) 0 0
\(859\) −7.29638 −0.248949 −0.124475 0.992223i \(-0.539725\pi\)
−0.124475 + 0.992223i \(0.539725\pi\)
\(860\) 0 0
\(861\) −1.21760 −0.0414957
\(862\) 0 0
\(863\) − 52.3903i − 1.78339i −0.452640 0.891694i \(-0.649518\pi\)
0.452640 0.891694i \(-0.350482\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 10.1939i − 0.346203i
\(868\) 0 0
\(869\) 18.1672 0.616281
\(870\) 0 0
\(871\) 33.9172 1.14924
\(872\) 0 0
\(873\) 27.2177i 0.921181i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28.1120i 0.949276i 0.880181 + 0.474638i \(0.157421\pi\)
−0.880181 + 0.474638i \(0.842579\pi\)
\(878\) 0 0
\(879\) 40.2444 1.35741
\(880\) 0 0
\(881\) 41.3936 1.39459 0.697293 0.716787i \(-0.254388\pi\)
0.697293 + 0.716787i \(0.254388\pi\)
\(882\) 0 0
\(883\) − 36.7906i − 1.23810i −0.785350 0.619052i \(-0.787517\pi\)
0.785350 0.619052i \(-0.212483\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 43.7836i − 1.47011i −0.678007 0.735055i \(-0.737156\pi\)
0.678007 0.735055i \(-0.262844\pi\)
\(888\) 0 0
\(889\) −5.09471 −0.170871
\(890\) 0 0
\(891\) 29.7646 0.997151
\(892\) 0 0
\(893\) − 0.0338141i − 0.00113155i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 11.2170i − 0.374524i
\(898\) 0 0
\(899\) −16.2950 −0.543469
\(900\) 0 0
\(901\) −3.06990 −0.102273
\(902\) 0 0
\(903\) − 19.9890i − 0.665193i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.8370i 0.691882i 0.938256 + 0.345941i \(0.112440\pi\)
−0.938256 + 0.345941i \(0.887560\pi\)
\(908\) 0 0
\(909\) −26.0287 −0.863316
\(910\) 0 0
\(911\) 45.8193 1.51806 0.759031 0.651054i \(-0.225673\pi\)
0.759031 + 0.651054i \(0.225673\pi\)
\(912\) 0 0
\(913\) 22.0061i 0.728297i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.74326i − 0.156636i
\(918\) 0 0
\(919\) −30.3282 −1.00044 −0.500218 0.865900i \(-0.666747\pi\)
−0.500218 + 0.865900i \(0.666747\pi\)
\(920\) 0 0
\(921\) −69.6446 −2.29487
\(922\) 0 0
\(923\) − 10.5973i − 0.348816i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 9.12226i 0.299614i
\(928\) 0 0
\(929\) 35.7602 1.17325 0.586626 0.809858i \(-0.300456\pi\)
0.586626 + 0.809858i \(0.300456\pi\)
\(930\) 0 0
\(931\) 12.2409 0.401181
\(932\) 0 0
\(933\) 31.6088i 1.03483i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26.6768i 0.871494i 0.900069 + 0.435747i \(0.143516\pi\)
−0.900069 + 0.435747i \(0.856484\pi\)
\(938\) 0 0
\(939\) 32.9890 1.07656
\(940\) 0 0
\(941\) −9.52987 −0.310665 −0.155332 0.987862i \(-0.549645\pi\)
−0.155332 + 0.987862i \(0.549645\pi\)
\(942\) 0 0
\(943\) 0.327923i 0.0106787i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 32.5209i − 1.05679i −0.849000 0.528393i \(-0.822795\pi\)
0.849000 0.528393i \(-0.177205\pi\)
\(948\) 0 0
\(949\) 29.4609 0.956341
\(950\) 0 0
\(951\) −60.9958 −1.97793
\(952\) 0 0
\(953\) 8.80252i 0.285141i 0.989785 + 0.142571i \(0.0455368\pi\)
−0.989785 + 0.142571i \(0.954463\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 21.5561i 0.696810i
\(958\) 0 0
\(959\) 23.3018 0.752456
\(960\) 0 0
\(961\) −7.51974 −0.242572
\(962\) 0 0
\(963\) 33.6541i 1.08449i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 47.0307i 1.51241i 0.654337 + 0.756203i \(0.272948\pi\)
−0.654337 + 0.756203i \(0.727052\pi\)
\(968\) 0 0
\(969\) −22.7167 −0.729766
\(970\) 0 0
\(971\) 33.3450 1.07009 0.535046 0.844823i \(-0.320294\pi\)
0.535046 + 0.844823i \(0.320294\pi\)
\(972\) 0 0
\(973\) − 29.4789i − 0.945050i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 55.8915i 1.78813i 0.447939 + 0.894064i \(0.352158\pi\)
−0.447939 + 0.894064i \(0.647842\pi\)
\(978\) 0 0
\(979\) −27.1432 −0.867500
\(980\) 0 0
\(981\) 28.0394 0.895229
\(982\) 0 0
\(983\) 44.2540i 1.41148i 0.708469 + 0.705742i \(0.249386\pi\)
−0.708469 + 0.705742i \(0.750614\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 0.0449603i − 0.00143110i
\(988\) 0 0
\(989\) −5.38343 −0.171183
\(990\) 0 0
\(991\) −28.2426 −0.897157 −0.448579 0.893743i \(-0.648070\pi\)
−0.448579 + 0.893743i \(0.648070\pi\)
\(992\) 0 0
\(993\) 28.0913i 0.891451i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 19.7730i − 0.626218i −0.949717 0.313109i \(-0.898629\pi\)
0.949717 0.313109i \(-0.101371\pi\)
\(998\) 0 0
\(999\) −6.49673 −0.205547
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 4600.2.e.w.4049.2 10
5.2 odd 4 4600.2.a.bf.1.1 yes 5
5.3 odd 4 4600.2.a.bd.1.5 5
5.4 even 2 inner 4600.2.e.w.4049.9 10
20.3 even 4 9200.2.a.cv.1.1 5
20.7 even 4 9200.2.a.ct.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.5 5 5.3 odd 4
4600.2.a.bf.1.1 yes 5 5.2 odd 4
4600.2.e.w.4049.2 10 1.1 even 1 trivial
4600.2.e.w.4049.9 10 5.4 even 2 inner
9200.2.a.ct.1.5 5 20.7 even 4
9200.2.a.cv.1.1 5 20.3 even 4