Properties

Label 4600.2.e.w.4049.5
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 4049.5
Root \(-0.794805i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.w.4049.6

$q$-expansion

\(f(q)\) \(=\) \(q-0.794805i q^{3} -2.47193i q^{7} +2.36829 q^{9} +O(q^{10})\) \(q-0.794805i q^{3} -2.47193i q^{7} +2.36829 q^{9} -2.29993 q^{11} -3.84022i q^{13} +7.74682i q^{17} +2.29993 q^{19} -1.96471 q^{21} -1.00000i q^{23} -4.26674i q^{27} -5.28380 q^{29} -6.40148 q^{31} +1.82800i q^{33} -8.56457i q^{37} -3.05223 q^{39} +4.27699 q^{41} -1.88954i q^{43} -12.3432i q^{47} +0.889540 q^{49} +6.15721 q^{51} -7.57482i q^{53} -1.82800i q^{57} -6.07180 q^{59} -0.635155 q^{61} -5.85425i q^{63} -11.1333i q^{67} -0.794805 q^{69} +8.58163 q^{71} +16.5849i q^{73} +5.68528i q^{77} -0.335225 q^{79} +3.71363 q^{81} +15.1937i q^{83} +4.19959i q^{87} -5.55735 q^{89} -9.49277 q^{91} +5.08792i q^{93} -6.42786i q^{97} -5.44689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{9} + O(q^{10}) \) \( 10 q - 6 q^{9} - 24 q^{29} - 36 q^{31} - 18 q^{39} - 12 q^{41} - 30 q^{49} - 12 q^{51} + 2 q^{59} + 20 q^{61} + 16 q^{71} - 54 q^{81} - 28 q^{89} - 92 q^{91} + 12 q^{99} + O(q^{100}) \)

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
\(3\) − 0.794805i − 0.458881i −0.973323 0.229440i \(-0.926310\pi\)
0.973323 0.229440i \(-0.0736896\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.47193i − 0.934303i −0.884177 0.467152i \(-0.845280\pi\)
0.884177 0.467152i \(-0.154720\pi\)
\(8\) 0 0
\(9\) 2.36829 0.789428
\(10\) 0 0
\(11\) −2.29993 −0.693455 −0.346728 0.937966i \(-0.612707\pi\)
−0.346728 + 0.937966i \(0.612707\pi\)
\(12\) 0 0
\(13\) − 3.84022i − 1.06509i −0.846403 0.532543i \(-0.821237\pi\)
0.846403 0.532543i \(-0.178763\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.74682i 1.87888i 0.342713 + 0.939440i \(0.388654\pi\)
−0.342713 + 0.939440i \(0.611346\pi\)
\(18\) 0 0
\(19\) 2.29993 0.527640 0.263820 0.964572i \(-0.415017\pi\)
0.263820 + 0.964572i \(0.415017\pi\)
\(20\) 0 0
\(21\) −1.96471 −0.428734
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.26674i − 0.821134i
\(28\) 0 0
\(29\) −5.28380 −0.981177 −0.490589 0.871391i \(-0.663218\pi\)
−0.490589 + 0.871391i \(0.663218\pi\)
\(30\) 0 0
\(31\) −6.40148 −1.14974 −0.574870 0.818245i \(-0.694947\pi\)
−0.574870 + 0.818245i \(0.694947\pi\)
\(32\) 0 0
\(33\) 1.82800i 0.318213i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.56457i − 1.40801i −0.710197 0.704003i \(-0.751394\pi\)
0.710197 0.704003i \(-0.248606\pi\)
\(38\) 0 0
\(39\) −3.05223 −0.488747
\(40\) 0 0
\(41\) 4.27699 0.667954 0.333977 0.942581i \(-0.391609\pi\)
0.333977 + 0.942581i \(0.391609\pi\)
\(42\) 0 0
\(43\) − 1.88954i − 0.288152i −0.989567 0.144076i \(-0.953979\pi\)
0.989567 0.144076i \(-0.0460210\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 12.3432i − 1.80045i −0.435428 0.900223i \(-0.643403\pi\)
0.435428 0.900223i \(-0.356597\pi\)
\(48\) 0 0
\(49\) 0.889540 0.127077
\(50\) 0 0
\(51\) 6.15721 0.862182
\(52\) 0 0
\(53\) − 7.57482i − 1.04048i −0.854020 0.520241i \(-0.825842\pi\)
0.854020 0.520241i \(-0.174158\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.82800i − 0.242124i
\(58\) 0 0
\(59\) −6.07180 −0.790480 −0.395240 0.918578i \(-0.629339\pi\)
−0.395240 + 0.918578i \(0.629339\pi\)
\(60\) 0 0
\(61\) −0.635155 −0.0813233 −0.0406616 0.999173i \(-0.512947\pi\)
−0.0406616 + 0.999173i \(0.512947\pi\)
\(62\) 0 0
\(63\) − 5.85425i − 0.737566i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.1333i − 1.36015i −0.733141 0.680077i \(-0.761946\pi\)
0.733141 0.680077i \(-0.238054\pi\)
\(68\) 0 0
\(69\) −0.794805 −0.0956833
\(70\) 0 0
\(71\) 8.58163 1.01845 0.509226 0.860633i \(-0.329932\pi\)
0.509226 + 0.860633i \(0.329932\pi\)
\(72\) 0 0
\(73\) 16.5849i 1.94112i 0.240859 + 0.970560i \(0.422571\pi\)
−0.240859 + 0.970560i \(0.577429\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.68528i 0.647897i
\(78\) 0 0
\(79\) −0.335225 −0.0377157 −0.0188579 0.999822i \(-0.506003\pi\)
−0.0188579 + 0.999822i \(0.506003\pi\)
\(80\) 0 0
\(81\) 3.71363 0.412626
\(82\) 0 0
\(83\) 15.1937i 1.66773i 0.551971 + 0.833863i \(0.313876\pi\)
−0.551971 + 0.833863i \(0.686124\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.19959i 0.450243i
\(88\) 0 0
\(89\) −5.55735 −0.589078 −0.294539 0.955639i \(-0.595166\pi\)
−0.294539 + 0.955639i \(0.595166\pi\)
\(90\) 0 0
\(91\) −9.49277 −0.995113
\(92\) 0 0
\(93\) 5.08792i 0.527593i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.42786i − 0.652650i −0.945258 0.326325i \(-0.894190\pi\)
0.945258 0.326325i \(-0.105810\pi\)
\(98\) 0 0
\(99\) −5.44689 −0.547433
\(100\) 0 0
\(101\) −14.6041 −1.45316 −0.726581 0.687081i \(-0.758892\pi\)
−0.726581 + 0.687081i \(0.758892\pi\)
\(102\) 0 0
\(103\) − 13.3980i − 1.32014i −0.751203 0.660071i \(-0.770526\pi\)
0.751203 0.660071i \(-0.229474\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.63549i − 0.158109i −0.996870 0.0790544i \(-0.974810\pi\)
0.996870 0.0790544i \(-0.0251901\pi\)
\(108\) 0 0
\(109\) −1.50182 −0.143848 −0.0719239 0.997410i \(-0.522914\pi\)
−0.0719239 + 0.997410i \(0.522914\pi\)
\(110\) 0 0
\(111\) −6.80716 −0.646107
\(112\) 0 0
\(113\) 17.6451i 1.65992i 0.557826 + 0.829958i \(0.311636\pi\)
−0.557826 + 0.829958i \(0.688364\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 9.09474i − 0.840809i
\(118\) 0 0
\(119\) 19.1496 1.75544
\(120\) 0 0
\(121\) −5.71032 −0.519120
\(122\) 0 0
\(123\) − 3.39937i − 0.306511i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 3.97362i − 0.352602i −0.984336 0.176301i \(-0.943587\pi\)
0.984336 0.176301i \(-0.0564132\pi\)
\(128\) 0 0
\(129\) −1.50182 −0.132228
\(130\) 0 0
\(131\) −14.3032 −1.24968 −0.624840 0.780753i \(-0.714836\pi\)
−0.624840 + 0.780753i \(0.714836\pi\)
\(132\) 0 0
\(133\) − 5.68528i − 0.492976i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.25585i − 0.363602i −0.983335 0.181801i \(-0.941807\pi\)
0.983335 0.181801i \(-0.0581927\pi\)
\(138\) 0 0
\(139\) 9.86950 0.837120 0.418560 0.908189i \(-0.362535\pi\)
0.418560 + 0.908189i \(0.362535\pi\)
\(140\) 0 0
\(141\) −9.81047 −0.826191
\(142\) 0 0
\(143\) 8.83224i 0.738589i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 0.707011i − 0.0583133i
\(148\) 0 0
\(149\) −9.55735 −0.782969 −0.391484 0.920185i \(-0.628038\pi\)
−0.391484 + 0.920185i \(0.628038\pi\)
\(150\) 0 0
\(151\) −24.4557 −1.99018 −0.995090 0.0989730i \(-0.968444\pi\)
−0.995090 + 0.0989730i \(0.968444\pi\)
\(152\) 0 0
\(153\) 18.3467i 1.48324i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 12.1922i − 0.973047i −0.873667 0.486524i \(-0.838265\pi\)
0.873667 0.486524i \(-0.161735\pi\)
\(158\) 0 0
\(159\) −6.02050 −0.477457
\(160\) 0 0
\(161\) −2.47193 −0.194816
\(162\) 0 0
\(163\) − 18.5762i − 1.45500i −0.686109 0.727498i \(-0.740683\pi\)
0.686109 0.727498i \(-0.259317\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.8639i 1.30497i 0.757803 + 0.652484i \(0.226273\pi\)
−0.757803 + 0.652484i \(0.773727\pi\)
\(168\) 0 0
\(169\) −1.74729 −0.134407
\(170\) 0 0
\(171\) 5.44689 0.416534
\(172\) 0 0
\(173\) 24.4626i 1.85985i 0.367745 + 0.929927i \(0.380130\pi\)
−0.367745 + 0.929927i \(0.619870\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.82589i 0.362736i
\(178\) 0 0
\(179\) −2.26173 −0.169050 −0.0845249 0.996421i \(-0.526937\pi\)
−0.0845249 + 0.996421i \(0.526937\pi\)
\(180\) 0 0
\(181\) 14.7187 1.09404 0.547018 0.837121i \(-0.315763\pi\)
0.547018 + 0.837121i \(0.315763\pi\)
\(182\) 0 0
\(183\) 0.504824i 0.0373177i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 17.8171i − 1.30292i
\(188\) 0 0
\(189\) −10.5471 −0.767189
\(190\) 0 0
\(191\) −23.7055 −1.71527 −0.857636 0.514258i \(-0.828067\pi\)
−0.857636 + 0.514258i \(0.828067\pi\)
\(192\) 0 0
\(193\) − 20.4146i − 1.46947i −0.678353 0.734736i \(-0.737306\pi\)
0.678353 0.734736i \(-0.262694\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 17.7607i − 1.26540i −0.774398 0.632699i \(-0.781947\pi\)
0.774398 0.632699i \(-0.218053\pi\)
\(198\) 0 0
\(199\) −11.1379 −0.789546 −0.394773 0.918779i \(-0.629177\pi\)
−0.394773 + 0.918779i \(0.629177\pi\)
\(200\) 0 0
\(201\) −8.84883 −0.624149
\(202\) 0 0
\(203\) 13.0612i 0.916717i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.36829i − 0.164607i
\(208\) 0 0
\(209\) −5.28968 −0.365895
\(210\) 0 0
\(211\) 4.17347 0.287314 0.143657 0.989628i \(-0.454114\pi\)
0.143657 + 0.989628i \(0.454114\pi\)
\(212\) 0 0
\(213\) − 6.82072i − 0.467348i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 15.8240i 1.07421i
\(218\) 0 0
\(219\) 13.1818 0.890743
\(220\) 0 0
\(221\) 29.7495 2.00117
\(222\) 0 0
\(223\) 11.4218i 0.764863i 0.923984 + 0.382432i \(0.124913\pi\)
−0.923984 + 0.382432i \(0.875087\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 5.40014i − 0.358420i −0.983811 0.179210i \(-0.942646\pi\)
0.983811 0.179210i \(-0.0573541\pi\)
\(228\) 0 0
\(229\) 18.4333 1.21811 0.609054 0.793129i \(-0.291549\pi\)
0.609054 + 0.793129i \(0.291549\pi\)
\(230\) 0 0
\(231\) 4.51869 0.297308
\(232\) 0 0
\(233\) − 7.82122i − 0.512385i −0.966626 0.256193i \(-0.917532\pi\)
0.966626 0.256193i \(-0.0824681\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.266438i 0.0173070i
\(238\) 0 0
\(239\) 2.26133 0.146273 0.0731365 0.997322i \(-0.476699\pi\)
0.0731365 + 0.997322i \(0.476699\pi\)
\(240\) 0 0
\(241\) 13.4906 0.869006 0.434503 0.900670i \(-0.356924\pi\)
0.434503 + 0.900670i \(0.356924\pi\)
\(242\) 0 0
\(243\) − 15.7518i − 1.01048i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 8.83224i − 0.561982i
\(248\) 0 0
\(249\) 12.0760 0.765288
\(250\) 0 0
\(251\) 15.5642 0.982406 0.491203 0.871045i \(-0.336557\pi\)
0.491203 + 0.871045i \(0.336557\pi\)
\(252\) 0 0
\(253\) 2.29993i 0.144595i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 25.2354i − 1.57414i −0.616865 0.787069i \(-0.711598\pi\)
0.616865 0.787069i \(-0.288402\pi\)
\(258\) 0 0
\(259\) −21.1710 −1.31550
\(260\) 0 0
\(261\) −12.5135 −0.774569
\(262\) 0 0
\(263\) − 12.6243i − 0.778448i −0.921143 0.389224i \(-0.872743\pi\)
0.921143 0.389224i \(-0.127257\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.41701i 0.270317i
\(268\) 0 0
\(269\) −25.0644 −1.52820 −0.764101 0.645096i \(-0.776817\pi\)
−0.764101 + 0.645096i \(0.776817\pi\)
\(270\) 0 0
\(271\) −17.6777 −1.07384 −0.536922 0.843632i \(-0.680413\pi\)
−0.536922 + 0.843632i \(0.680413\pi\)
\(272\) 0 0
\(273\) 7.54490i 0.456638i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.22724i 0.434243i 0.976145 + 0.217121i \(0.0696667\pi\)
−0.976145 + 0.217121i \(0.930333\pi\)
\(278\) 0 0
\(279\) −15.1605 −0.907637
\(280\) 0 0
\(281\) −5.61315 −0.334852 −0.167426 0.985885i \(-0.553546\pi\)
−0.167426 + 0.985885i \(0.553546\pi\)
\(282\) 0 0
\(283\) − 0.873541i − 0.0519266i −0.999663 0.0259633i \(-0.991735\pi\)
0.999663 0.0259633i \(-0.00826531\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 10.5724i − 0.624071i
\(288\) 0 0
\(289\) −43.0132 −2.53019
\(290\) 0 0
\(291\) −5.10889 −0.299489
\(292\) 0 0
\(293\) 10.2737i 0.600195i 0.953909 + 0.300097i \(0.0970192\pi\)
−0.953909 + 0.300097i \(0.902981\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.81320i 0.569420i
\(298\) 0 0
\(299\) −3.84022 −0.222086
\(300\) 0 0
\(301\) −4.67082 −0.269222
\(302\) 0 0
\(303\) 11.6074i 0.666828i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 26.3994i − 1.50670i −0.657622 0.753348i \(-0.728438\pi\)
0.657622 0.753348i \(-0.271562\pi\)
\(308\) 0 0
\(309\) −10.6488 −0.605788
\(310\) 0 0
\(311\) 31.6429 1.79430 0.897151 0.441725i \(-0.145633\pi\)
0.897151 + 0.441725i \(0.145633\pi\)
\(312\) 0 0
\(313\) − 24.1327i − 1.36406i −0.731323 0.682032i \(-0.761097\pi\)
0.731323 0.682032i \(-0.238903\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.973154i 0.0546578i 0.999626 + 0.0273289i \(0.00870014\pi\)
−0.999626 + 0.0273289i \(0.991300\pi\)
\(318\) 0 0
\(319\) 12.1524 0.680402
\(320\) 0 0
\(321\) −1.29990 −0.0725531
\(322\) 0 0
\(323\) 17.8171i 0.991373i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.19365i 0.0660090i
\(328\) 0 0
\(329\) −30.5117 −1.68216
\(330\) 0 0
\(331\) 1.10352 0.0606549 0.0303274 0.999540i \(-0.490345\pi\)
0.0303274 + 0.999540i \(0.490345\pi\)
\(332\) 0 0
\(333\) − 20.2833i − 1.11152i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.33526i 0.508524i 0.967135 + 0.254262i \(0.0818326\pi\)
−0.967135 + 0.254262i \(0.918167\pi\)
\(338\) 0 0
\(339\) 14.0244 0.761703
\(340\) 0 0
\(341\) 14.7229 0.797292
\(342\) 0 0
\(343\) − 19.5024i − 1.05303i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 19.4052i − 1.04173i −0.853641 0.520863i \(-0.825610\pi\)
0.853641 0.520863i \(-0.174390\pi\)
\(348\) 0 0
\(349\) −26.0825 −1.39616 −0.698081 0.716019i \(-0.745962\pi\)
−0.698081 + 0.716019i \(0.745962\pi\)
\(350\) 0 0
\(351\) −16.3852 −0.874578
\(352\) 0 0
\(353\) 24.5504i 1.30669i 0.757062 + 0.653343i \(0.226634\pi\)
−0.757062 + 0.653343i \(0.773366\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 15.2202i − 0.805540i
\(358\) 0 0
\(359\) 10.2809 0.542607 0.271303 0.962494i \(-0.412545\pi\)
0.271303 + 0.962494i \(0.412545\pi\)
\(360\) 0 0
\(361\) −13.7103 −0.721596
\(362\) 0 0
\(363\) 4.53859i 0.238214i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 13.2290i 0.690549i 0.938502 + 0.345274i \(0.112214\pi\)
−0.938502 + 0.345274i \(0.887786\pi\)
\(368\) 0 0
\(369\) 10.1291 0.527302
\(370\) 0 0
\(371\) −18.7245 −0.972125
\(372\) 0 0
\(373\) 34.2573i 1.77378i 0.461983 + 0.886889i \(0.347138\pi\)
−0.461983 + 0.886889i \(0.652862\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.2910i 1.04504i
\(378\) 0 0
\(379\) 10.2939 0.528763 0.264382 0.964418i \(-0.414832\pi\)
0.264382 + 0.964418i \(0.414832\pi\)
\(380\) 0 0
\(381\) −3.15825 −0.161802
\(382\) 0 0
\(383\) − 26.2028i − 1.33890i −0.742857 0.669450i \(-0.766530\pi\)
0.742857 0.669450i \(-0.233470\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.47497i − 0.227476i
\(388\) 0 0
\(389\) −5.59078 −0.283464 −0.141732 0.989905i \(-0.545267\pi\)
−0.141732 + 0.989905i \(0.545267\pi\)
\(390\) 0 0
\(391\) 7.74682 0.391774
\(392\) 0 0
\(393\) 11.3683i 0.573454i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.98084i 0.149604i 0.997198 + 0.0748019i \(0.0238324\pi\)
−0.997198 + 0.0748019i \(0.976168\pi\)
\(398\) 0 0
\(399\) −4.51869 −0.226217
\(400\) 0 0
\(401\) −19.8229 −0.989906 −0.494953 0.868920i \(-0.664815\pi\)
−0.494953 + 0.868920i \(0.664815\pi\)
\(402\) 0 0
\(403\) 24.5831i 1.22457i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 19.6979i 0.976389i
\(408\) 0 0
\(409\) −12.8521 −0.635498 −0.317749 0.948175i \(-0.602927\pi\)
−0.317749 + 0.948175i \(0.602927\pi\)
\(410\) 0 0
\(411\) −3.38257 −0.166850
\(412\) 0 0
\(413\) 15.0091i 0.738549i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 7.84433i − 0.384139i
\(418\) 0 0
\(419\) 5.42665 0.265109 0.132555 0.991176i \(-0.457682\pi\)
0.132555 + 0.991176i \(0.457682\pi\)
\(420\) 0 0
\(421\) 19.2456 0.937973 0.468987 0.883205i \(-0.344619\pi\)
0.468987 + 0.883205i \(0.344619\pi\)
\(422\) 0 0
\(423\) − 29.2323i − 1.42132i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.57006i 0.0759806i
\(428\) 0 0
\(429\) 7.01991 0.338924
\(430\) 0 0
\(431\) 8.53348 0.411043 0.205522 0.978653i \(-0.434111\pi\)
0.205522 + 0.978653i \(0.434111\pi\)
\(432\) 0 0
\(433\) 22.7745i 1.09447i 0.836978 + 0.547237i \(0.184320\pi\)
−0.836978 + 0.547237i \(0.815680\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 2.29993i − 0.110021i
\(438\) 0 0
\(439\) −4.67135 −0.222952 −0.111476 0.993767i \(-0.535558\pi\)
−0.111476 + 0.993767i \(0.535558\pi\)
\(440\) 0 0
\(441\) 2.10668 0.100318
\(442\) 0 0
\(443\) 3.00378i 0.142714i 0.997451 + 0.0713568i \(0.0227329\pi\)
−0.997451 + 0.0713568i \(0.977267\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 7.59623i 0.359289i
\(448\) 0 0
\(449\) −5.56914 −0.262824 −0.131412 0.991328i \(-0.541951\pi\)
−0.131412 + 0.991328i \(0.541951\pi\)
\(450\) 0 0
\(451\) −9.83678 −0.463196
\(452\) 0 0
\(453\) 19.4375i 0.913256i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 31.0180i 1.45096i 0.688242 + 0.725481i \(0.258383\pi\)
−0.688242 + 0.725481i \(0.741617\pi\)
\(458\) 0 0
\(459\) 33.0537 1.54281
\(460\) 0 0
\(461\) −9.92290 −0.462155 −0.231078 0.972935i \(-0.574225\pi\)
−0.231078 + 0.972935i \(0.574225\pi\)
\(462\) 0 0
\(463\) 10.0111i 0.465256i 0.972566 + 0.232628i \(0.0747325\pi\)
−0.972566 + 0.232628i \(0.925267\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 36.9345i − 1.70912i −0.519350 0.854562i \(-0.673826\pi\)
0.519350 0.854562i \(-0.326174\pi\)
\(468\) 0 0
\(469\) −27.5209 −1.27080
\(470\) 0 0
\(471\) −9.69046 −0.446513
\(472\) 0 0
\(473\) 4.34581i 0.199821i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 17.9393i − 0.821385i
\(478\) 0 0
\(479\) 16.7734 0.766396 0.383198 0.923666i \(-0.374823\pi\)
0.383198 + 0.923666i \(0.374823\pi\)
\(480\) 0 0
\(481\) −32.8898 −1.49965
\(482\) 0 0
\(483\) 1.96471i 0.0893972i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.47391i − 0.248046i −0.992279 0.124023i \(-0.960420\pi\)
0.992279 0.124023i \(-0.0395797\pi\)
\(488\) 0 0
\(489\) −14.7644 −0.667670
\(490\) 0 0
\(491\) 35.4536 1.60000 0.800000 0.600001i \(-0.204833\pi\)
0.800000 + 0.600001i \(0.204833\pi\)
\(492\) 0 0
\(493\) − 40.9327i − 1.84351i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 21.2132i − 0.951543i
\(498\) 0 0
\(499\) 33.4110 1.49568 0.747840 0.663879i \(-0.231091\pi\)
0.747840 + 0.663879i \(0.231091\pi\)
\(500\) 0 0
\(501\) 13.4035 0.598825
\(502\) 0 0
\(503\) − 1.97733i − 0.0881650i −0.999028 0.0440825i \(-0.985964\pi\)
0.999028 0.0440825i \(-0.0140364\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.38875i 0.0616766i
\(508\) 0 0
\(509\) 23.0984 1.02382 0.511909 0.859040i \(-0.328939\pi\)
0.511909 + 0.859040i \(0.328939\pi\)
\(510\) 0 0
\(511\) 40.9969 1.81360
\(512\) 0 0
\(513\) − 9.81320i − 0.433264i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 28.3886i 1.24853i
\(518\) 0 0
\(519\) 19.4430 0.853451
\(520\) 0 0
\(521\) 21.2867 0.932588 0.466294 0.884630i \(-0.345589\pi\)
0.466294 + 0.884630i \(0.345589\pi\)
\(522\) 0 0
\(523\) 28.1943i 1.23285i 0.787414 + 0.616425i \(0.211420\pi\)
−0.787414 + 0.616425i \(0.788580\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 49.5911i − 2.16022i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −14.3797 −0.624028
\(532\) 0 0
\(533\) − 16.4246i − 0.711428i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.79763i 0.0775737i
\(538\) 0 0
\(539\) −2.04588 −0.0881223
\(540\) 0 0
\(541\) 14.8271 0.637468 0.318734 0.947844i \(-0.396742\pi\)
0.318734 + 0.947844i \(0.396742\pi\)
\(542\) 0 0
\(543\) − 11.6985i − 0.502032i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.09950i 0.389067i 0.980896 + 0.194533i \(0.0623192\pi\)
−0.980896 + 0.194533i \(0.937681\pi\)
\(548\) 0 0
\(549\) −1.50423 −0.0641989
\(550\) 0 0
\(551\) −12.1524 −0.517709
\(552\) 0 0
\(553\) 0.828654i 0.0352379i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.9824i 1.39751i 0.715362 + 0.698754i \(0.246262\pi\)
−0.715362 + 0.698754i \(0.753738\pi\)
\(558\) 0 0
\(559\) −7.25625 −0.306907
\(560\) 0 0
\(561\) −14.1612 −0.597885
\(562\) 0 0
\(563\) − 21.9384i − 0.924595i −0.886725 0.462297i \(-0.847025\pi\)
0.886725 0.462297i \(-0.152975\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 9.17985i − 0.385517i
\(568\) 0 0
\(569\) 28.2664 1.18499 0.592495 0.805574i \(-0.298143\pi\)
0.592495 + 0.805574i \(0.298143\pi\)
\(570\) 0 0
\(571\) −4.13671 −0.173116 −0.0865580 0.996247i \(-0.527587\pi\)
−0.0865580 + 0.996247i \(0.527587\pi\)
\(572\) 0 0
\(573\) 18.8413i 0.787105i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 13.8146i − 0.575111i −0.957764 0.287556i \(-0.907157\pi\)
0.957764 0.287556i \(-0.0928426\pi\)
\(578\) 0 0
\(579\) −16.2256 −0.674313
\(580\) 0 0
\(581\) 37.5579 1.55816
\(582\) 0 0
\(583\) 17.4216i 0.721527i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.2506i 0.712010i 0.934484 + 0.356005i \(0.115861\pi\)
−0.934484 + 0.356005i \(0.884139\pi\)
\(588\) 0 0
\(589\) −14.7229 −0.606649
\(590\) 0 0
\(591\) −14.1163 −0.580667
\(592\) 0 0
\(593\) 27.0042i 1.10893i 0.832207 + 0.554466i \(0.187077\pi\)
−0.832207 + 0.554466i \(0.812923\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.85247i 0.362307i
\(598\) 0 0
\(599\) 11.5259 0.470934 0.235467 0.971882i \(-0.424338\pi\)
0.235467 + 0.971882i \(0.424338\pi\)
\(600\) 0 0
\(601\) −10.8541 −0.442750 −0.221375 0.975189i \(-0.571054\pi\)
−0.221375 + 0.975189i \(0.571054\pi\)
\(602\) 0 0
\(603\) − 26.3669i − 1.07374i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 5.16930i 0.209816i 0.994482 + 0.104908i \(0.0334547\pi\)
−0.994482 + 0.104908i \(0.966545\pi\)
\(608\) 0 0
\(609\) 10.3811 0.420664
\(610\) 0 0
\(611\) −47.4008 −1.91763
\(612\) 0 0
\(613\) − 27.3190i − 1.10340i −0.834042 0.551701i \(-0.813979\pi\)
0.834042 0.551701i \(-0.186021\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 5.08325i − 0.204644i −0.994751 0.102322i \(-0.967373\pi\)
0.994751 0.102322i \(-0.0326272\pi\)
\(618\) 0 0
\(619\) 0.207299 0.00833203 0.00416602 0.999991i \(-0.498674\pi\)
0.00416602 + 0.999991i \(0.498674\pi\)
\(620\) 0 0
\(621\) −4.26674 −0.171218
\(622\) 0 0
\(623\) 13.7374i 0.550378i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4.20426i 0.167902i
\(628\) 0 0
\(629\) 66.3482 2.64547
\(630\) 0 0
\(631\) 40.7410 1.62187 0.810937 0.585133i \(-0.198958\pi\)
0.810937 + 0.585133i \(0.198958\pi\)
\(632\) 0 0
\(633\) − 3.31710i − 0.131843i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.41603i − 0.135348i
\(638\) 0 0
\(639\) 20.3237 0.803995
\(640\) 0 0
\(641\) −29.7426 −1.17476 −0.587381 0.809310i \(-0.699841\pi\)
−0.587381 + 0.809310i \(0.699841\pi\)
\(642\) 0 0
\(643\) − 34.3329i − 1.35396i −0.736003 0.676978i \(-0.763289\pi\)
0.736003 0.676978i \(-0.236711\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.0002i 0.589717i 0.955541 + 0.294859i \(0.0952726\pi\)
−0.955541 + 0.294859i \(0.904727\pi\)
\(648\) 0 0
\(649\) 13.9647 0.548163
\(650\) 0 0
\(651\) 12.5770 0.492932
\(652\) 0 0
\(653\) − 14.2038i − 0.555837i −0.960605 0.277918i \(-0.910356\pi\)
0.960605 0.277918i \(-0.0896444\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 39.2779i 1.53238i
\(658\) 0 0
\(659\) −47.9718 −1.86872 −0.934359 0.356334i \(-0.884027\pi\)
−0.934359 + 0.356334i \(0.884027\pi\)
\(660\) 0 0
\(661\) 43.6518 1.69786 0.848928 0.528508i \(-0.177248\pi\)
0.848928 + 0.528508i \(0.177248\pi\)
\(662\) 0 0
\(663\) − 23.6450i − 0.918297i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.28380i 0.204590i
\(668\) 0 0
\(669\) 9.07814 0.350981
\(670\) 0 0
\(671\) 1.46081 0.0563940
\(672\) 0 0
\(673\) − 11.9330i − 0.459983i −0.973193 0.229991i \(-0.926130\pi\)
0.973193 0.229991i \(-0.0738698\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.7598i 0.720999i 0.932759 + 0.360500i \(0.117394\pi\)
−0.932759 + 0.360500i \(0.882606\pi\)
\(678\) 0 0
\(679\) −15.8892 −0.609773
\(680\) 0 0
\(681\) −4.29206 −0.164472
\(682\) 0 0
\(683\) 10.4542i 0.400020i 0.979794 + 0.200010i \(0.0640975\pi\)
−0.979794 + 0.200010i \(0.935902\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 14.6509i − 0.558966i
\(688\) 0 0
\(689\) −29.0890 −1.10820
\(690\) 0 0
\(691\) −20.8518 −0.793240 −0.396620 0.917983i \(-0.629817\pi\)
−0.396620 + 0.917983i \(0.629817\pi\)
\(692\) 0 0
\(693\) 13.4644i 0.511469i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 33.1331i 1.25500i
\(698\) 0 0
\(699\) −6.21634 −0.235124
\(700\) 0 0
\(701\) 22.1231 0.835578 0.417789 0.908544i \(-0.362805\pi\)
0.417789 + 0.908544i \(0.362805\pi\)
\(702\) 0 0
\(703\) − 19.6979i − 0.742921i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.1004i 1.35769i
\(708\) 0 0
\(709\) −14.1669 −0.532049 −0.266024 0.963966i \(-0.585710\pi\)
−0.266024 + 0.963966i \(0.585710\pi\)
\(710\) 0 0
\(711\) −0.793908 −0.0297739
\(712\) 0 0
\(713\) 6.40148i 0.239737i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 1.79731i − 0.0671219i
\(718\) 0 0
\(719\) 6.86389 0.255980 0.127990 0.991775i \(-0.459147\pi\)
0.127990 + 0.991775i \(0.459147\pi\)
\(720\) 0 0
\(721\) −33.1189 −1.23341
\(722\) 0 0
\(723\) − 10.7224i − 0.398770i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.72298i 0.0639016i 0.999489 + 0.0319508i \(0.0101720\pi\)
−0.999489 + 0.0319508i \(0.989828\pi\)
\(728\) 0 0
\(729\) −1.37874 −0.0510645
\(730\) 0 0
\(731\) 14.6379 0.541403
\(732\) 0 0
\(733\) 13.2855i 0.490712i 0.969433 + 0.245356i \(0.0789049\pi\)
−0.969433 + 0.245356i \(0.921095\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.6059i 0.943206i
\(738\) 0 0
\(739\) 28.8043 1.05958 0.529791 0.848128i \(-0.322270\pi\)
0.529791 + 0.848128i \(0.322270\pi\)
\(740\) 0 0
\(741\) −7.01991 −0.257883
\(742\) 0 0
\(743\) 15.7966i 0.579521i 0.957099 + 0.289761i \(0.0935757\pi\)
−0.957099 + 0.289761i \(0.906424\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 35.9830i 1.31655i
\(748\) 0 0
\(749\) −4.04282 −0.147722
\(750\) 0 0
\(751\) −12.7384 −0.464829 −0.232415 0.972617i \(-0.574663\pi\)
−0.232415 + 0.972617i \(0.574663\pi\)
\(752\) 0 0
\(753\) − 12.3705i − 0.450807i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.07420i 0.111734i 0.998438 + 0.0558668i \(0.0177922\pi\)
−0.998438 + 0.0558668i \(0.982208\pi\)
\(758\) 0 0
\(759\) 1.82800 0.0663520
\(760\) 0 0
\(761\) 30.6818 1.11222 0.556108 0.831110i \(-0.312294\pi\)
0.556108 + 0.831110i \(0.312294\pi\)
\(762\) 0 0
\(763\) 3.71239i 0.134398i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.3170i 0.841929i
\(768\) 0 0
\(769\) 34.5153 1.24465 0.622327 0.782758i \(-0.286188\pi\)
0.622327 + 0.782758i \(0.286188\pi\)
\(770\) 0 0
\(771\) −20.0572 −0.722342
\(772\) 0 0
\(773\) − 36.6662i − 1.31879i −0.751796 0.659396i \(-0.770812\pi\)
0.751796 0.659396i \(-0.229188\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 16.8269i 0.603660i
\(778\) 0 0
\(779\) 9.83678 0.352439
\(780\) 0 0
\(781\) −19.7371 −0.706251
\(782\) 0 0
\(783\) 22.5446i 0.805678i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 17.6991i − 0.630906i −0.948941 0.315453i \(-0.897844\pi\)
0.948941 0.315453i \(-0.102156\pi\)
\(788\) 0 0
\(789\) −10.0339 −0.357215
\(790\) 0 0
\(791\) 43.6176 1.55086
\(792\) 0 0
\(793\) 2.43913i 0.0866162i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 26.6535i − 0.944117i −0.881567 0.472058i \(-0.843511\pi\)
0.881567 0.472058i \(-0.156489\pi\)
\(798\) 0 0
\(799\) 95.6209 3.38282
\(800\) 0 0
\(801\) −13.1614 −0.465035
\(802\) 0 0
\(803\) − 38.1442i − 1.34608i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.9213i 0.701263i
\(808\) 0 0
\(809\) 24.1080 0.847591 0.423795 0.905758i \(-0.360698\pi\)
0.423795 + 0.905758i \(0.360698\pi\)
\(810\) 0 0
\(811\) −11.3363 −0.398071 −0.199035 0.979992i \(-0.563781\pi\)
−0.199035 + 0.979992i \(0.563781\pi\)
\(812\) 0 0
\(813\) 14.0503i 0.492766i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 4.34581i − 0.152041i
\(818\) 0 0
\(819\) −22.4816 −0.785570
\(820\) 0 0
\(821\) −3.95739 −0.138114 −0.0690569 0.997613i \(-0.521999\pi\)
−0.0690569 + 0.997613i \(0.521999\pi\)
\(822\) 0 0
\(823\) 46.5023i 1.62097i 0.585761 + 0.810484i \(0.300796\pi\)
−0.585761 + 0.810484i \(0.699204\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 27.7619i − 0.965375i −0.875793 0.482687i \(-0.839661\pi\)
0.875793 0.482687i \(-0.160339\pi\)
\(828\) 0 0
\(829\) −37.8234 −1.31366 −0.656831 0.754038i \(-0.728104\pi\)
−0.656831 + 0.754038i \(0.728104\pi\)
\(830\) 0 0
\(831\) 5.74424 0.199266
\(832\) 0 0
\(833\) 6.89111i 0.238763i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 27.3134i 0.944090i
\(838\) 0 0
\(839\) −20.0661 −0.692758 −0.346379 0.938095i \(-0.612589\pi\)
−0.346379 + 0.938095i \(0.612589\pi\)
\(840\) 0 0
\(841\) −1.08145 −0.0372913
\(842\) 0 0
\(843\) 4.46136i 0.153657i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 14.1155i 0.485016i
\(848\) 0 0
\(849\) −0.694295 −0.0238281
\(850\) 0 0
\(851\) −8.56457 −0.293590
\(852\) 0 0
\(853\) − 8.35933i − 0.286218i −0.989707 0.143109i \(-0.954290\pi\)
0.989707 0.143109i \(-0.0457100\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17.0931i 0.583889i 0.956435 + 0.291945i \(0.0943023\pi\)
−0.956435 + 0.291945i \(0.905698\pi\)
\(858\) 0 0
\(859\) −23.0254 −0.785617 −0.392809 0.919620i \(-0.628497\pi\)
−0.392809 + 0.919620i \(0.628497\pi\)
\(860\) 0 0
\(861\) −8.40303 −0.286374
\(862\) 0 0
\(863\) 5.49197i 0.186949i 0.995622 + 0.0934745i \(0.0297974\pi\)
−0.995622 + 0.0934745i \(0.970203\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 34.1871i 1.16106i
\(868\) 0 0
\(869\) 0.770994 0.0261542
\(870\) 0 0
\(871\) −42.7545 −1.44868
\(872\) 0 0
\(873\) − 15.2230i − 0.515220i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 7.98307i − 0.269569i −0.990875 0.134785i \(-0.956966\pi\)
0.990875 0.134785i \(-0.0430342\pi\)
\(878\) 0 0
\(879\) 8.16557 0.275418
\(880\) 0 0
\(881\) 44.5925 1.50236 0.751180 0.660098i \(-0.229485\pi\)
0.751180 + 0.660098i \(0.229485\pi\)
\(882\) 0 0
\(883\) 45.9134i 1.54511i 0.634949 + 0.772554i \(0.281021\pi\)
−0.634949 + 0.772554i \(0.718979\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 53.6109i − 1.80008i −0.435810 0.900039i \(-0.643538\pi\)
0.435810 0.900039i \(-0.356462\pi\)
\(888\) 0 0
\(889\) −9.82253 −0.329437
\(890\) 0 0
\(891\) −8.54109 −0.286137
\(892\) 0 0
\(893\) − 28.3886i − 0.949988i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.05223i 0.101911i
\(898\) 0 0
\(899\) 33.8241 1.12810
\(900\) 0 0
\(901\) 58.6808 1.95494
\(902\) 0 0
\(903\) 3.71239i 0.123541i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 45.1277i − 1.49844i −0.662322 0.749220i \(-0.730429\pi\)
0.662322 0.749220i \(-0.269571\pi\)
\(908\) 0 0
\(909\) −34.5867 −1.14717
\(910\) 0 0
\(911\) 6.06248 0.200859 0.100429 0.994944i \(-0.467978\pi\)
0.100429 + 0.994944i \(0.467978\pi\)
\(912\) 0 0
\(913\) − 34.9445i − 1.15649i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 35.3567i 1.16758i
\(918\) 0 0
\(919\) −38.8692 −1.28218 −0.641089 0.767467i \(-0.721517\pi\)
−0.641089 + 0.767467i \(0.721517\pi\)
\(920\) 0 0
\(921\) −20.9824 −0.691394
\(922\) 0 0
\(923\) − 32.9553i − 1.08474i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 31.7302i − 1.04216i
\(928\) 0 0
\(929\) 43.4632 1.42598 0.712991 0.701173i \(-0.247340\pi\)
0.712991 + 0.701173i \(0.247340\pi\)
\(930\) 0 0
\(931\) 2.04588 0.0670510
\(932\) 0 0
\(933\) − 25.1499i − 0.823371i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 19.7812i − 0.646223i −0.946361 0.323112i \(-0.895271\pi\)
0.946361 0.323112i \(-0.104729\pi\)
\(938\) 0 0
\(939\) −19.1808 −0.625942
\(940\) 0 0
\(941\) 31.1805 1.01645 0.508227 0.861223i \(-0.330301\pi\)
0.508227 + 0.861223i \(0.330301\pi\)
\(942\) 0 0
\(943\) − 4.27699i − 0.139278i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.65195i − 0.118673i −0.998238 0.0593363i \(-0.981102\pi\)
0.998238 0.0593363i \(-0.0188984\pi\)
\(948\) 0 0
\(949\) 63.6898 2.06746
\(950\) 0 0
\(951\) 0.773468 0.0250814
\(952\) 0 0
\(953\) 4.81987i 0.156131i 0.996948 + 0.0780655i \(0.0248743\pi\)
−0.996948 + 0.0780655i \(0.975126\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 9.65877i − 0.312224i
\(958\) 0 0
\(959\) −10.5202 −0.339715
\(960\) 0 0
\(961\) 9.97890 0.321900
\(962\) 0 0
\(963\) − 3.87331i − 0.124816i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 28.9718i 0.931670i 0.884872 + 0.465835i \(0.154246\pi\)
−0.884872 + 0.465835i \(0.845754\pi\)
\(968\) 0 0
\(969\) 14.1612 0.454922
\(970\) 0 0
\(971\) 32.8376 1.05381 0.526905 0.849924i \(-0.323352\pi\)
0.526905 + 0.849924i \(0.323352\pi\)
\(972\) 0 0
\(973\) − 24.3968i − 0.782125i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 58.0091i − 1.85587i −0.372737 0.927937i \(-0.621581\pi\)
0.372737 0.927937i \(-0.378419\pi\)
\(978\) 0 0
\(979\) 12.7815 0.408499
\(980\) 0 0
\(981\) −3.55673 −0.113558
\(982\) 0 0
\(983\) 4.41757i 0.140899i 0.997515 + 0.0704493i \(0.0224433\pi\)
−0.997515 + 0.0704493i \(0.977557\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 24.2508i 0.771913i
\(988\) 0 0
\(989\) −1.88954 −0.0600839
\(990\) 0 0
\(991\) 36.1617 1.14871 0.574357 0.818605i \(-0.305252\pi\)
0.574357 + 0.818605i \(0.305252\pi\)
\(992\) 0 0
\(993\) − 0.877082i − 0.0278334i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 11.6357i − 0.368506i −0.982879 0.184253i \(-0.941013\pi\)
0.982879 0.184253i \(-0.0589866\pi\)
\(998\) 0 0
\(999\) −36.5428 −1.15616
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.4049.5 10
5.2 odd 4 4600.2.a.bf.1.3 yes 5
5.3 odd 4 4600.2.a.bd.1.3 5
5.4 even 2 inner 4600.2.e.w.4049.6 10
20.3 even 4 9200.2.a.cv.1.3 5
20.7 even 4 9200.2.a.ct.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.3 5 5.3 odd 4
4600.2.a.bf.1.3 yes 5 5.2 odd 4
4600.2.e.w.4049.5 10 1.1 even 1 trivial
4600.2.e.w.4049.6 10 5.4 even 2 inner
9200.2.a.ct.1.3 5 20.7 even 4
9200.2.a.cv.1.3 5 20.3 even 4