Properties

Label 1150.2.b.i.599.1
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.1
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.i.599.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -0.618034i q^{3} -1.00000 q^{4} -0.618034 q^{6} -1.61803i q^{7} +1.00000i q^{8} +2.61803 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -0.618034i q^{3} -1.00000 q^{4} -0.618034 q^{6} -1.61803i q^{7} +1.00000i q^{8} +2.61803 q^{9} +3.85410 q^{11} +0.618034i q^{12} +4.09017i q^{13} -1.61803 q^{14} +1.00000 q^{16} +5.09017i q^{17} -2.61803i q^{18} +4.85410 q^{19} -1.00000 q^{21} -3.85410i q^{22} +1.00000i q^{23} +0.618034 q^{24} +4.09017 q^{26} -3.47214i q^{27} +1.61803i q^{28} +4.76393 q^{29} -2.09017 q^{31} -1.00000i q^{32} -2.38197i q^{33} +5.09017 q^{34} -2.61803 q^{36} +2.47214i q^{37} -4.85410i q^{38} +2.52786 q^{39} -12.3262 q^{41} +1.00000i q^{42} -3.85410 q^{44} +1.00000 q^{46} -9.70820i q^{47} -0.618034i q^{48} +4.38197 q^{49} +3.14590 q^{51} -4.09017i q^{52} -8.47214i q^{53} -3.47214 q^{54} +1.61803 q^{56} -3.00000i q^{57} -4.76393i q^{58} +11.7082 q^{59} +6.32624 q^{61} +2.09017i q^{62} -4.23607i q^{63} -1.00000 q^{64} -2.38197 q^{66} -5.52786i q^{67} -5.09017i q^{68} +0.618034 q^{69} +7.09017 q^{71} +2.61803i q^{72} -1.23607i q^{73} +2.47214 q^{74} -4.85410 q^{76} -6.23607i q^{77} -2.52786i q^{78} -10.4721 q^{79} +5.70820 q^{81} +12.3262i q^{82} +10.9443i q^{83} +1.00000 q^{84} -2.94427i q^{87} +3.85410i q^{88} +1.52786 q^{89} +6.61803 q^{91} -1.00000i q^{92} +1.29180i q^{93} -9.70820 q^{94} -0.618034 q^{96} -14.6180i q^{97} -4.38197i q^{98} +10.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{6} + 6 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} + 2 q^{6} + 6 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 4 q^{21} - 2 q^{24} - 6 q^{26} + 28 q^{29} + 14 q^{31} - 2 q^{34} - 6 q^{36} + 28 q^{39} - 18 q^{41} - 2 q^{44} + 4 q^{46} + 22 q^{49} + 26 q^{51} + 4 q^{54} + 2 q^{56} + 20 q^{59} - 6 q^{61} - 4 q^{64} - 14 q^{66} - 2 q^{69} + 6 q^{71} - 8 q^{74} - 6 q^{76} - 24 q^{79} - 4 q^{81} + 4 q^{84} + 24 q^{89} + 22 q^{91} - 12 q^{94} + 2 q^{96} + 18 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) − 0.618034i − 0.356822i −0.983956 0.178411i \(-0.942904\pi\)
0.983956 0.178411i \(-0.0570957\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −0.618034 −0.252311
\(7\) − 1.61803i − 0.611559i −0.952102 0.305780i \(-0.901083\pi\)
0.952102 0.305780i \(-0.0989171\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.61803 0.872678
\(10\) 0 0
\(11\) 3.85410 1.16206 0.581028 0.813884i \(-0.302651\pi\)
0.581028 + 0.813884i \(0.302651\pi\)
\(12\) 0.618034i 0.178411i
\(13\) 4.09017i 1.13441i 0.823577 + 0.567205i \(0.191975\pi\)
−0.823577 + 0.567205i \(0.808025\pi\)
\(14\) −1.61803 −0.432438
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.09017i 1.23455i 0.786748 + 0.617274i \(0.211763\pi\)
−0.786748 + 0.617274i \(0.788237\pi\)
\(18\) − 2.61803i − 0.617077i
\(19\) 4.85410 1.11361 0.556804 0.830644i \(-0.312028\pi\)
0.556804 + 0.830644i \(0.312028\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) − 3.85410i − 0.821697i
\(23\) 1.00000i 0.208514i
\(24\) 0.618034 0.126156
\(25\) 0 0
\(26\) 4.09017 0.802148
\(27\) − 3.47214i − 0.668213i
\(28\) 1.61803i 0.305780i
\(29\) 4.76393 0.884640 0.442320 0.896857i \(-0.354156\pi\)
0.442320 + 0.896857i \(0.354156\pi\)
\(30\) 0 0
\(31\) −2.09017 −0.375406 −0.187703 0.982226i \(-0.560104\pi\)
−0.187703 + 0.982226i \(0.560104\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.38197i − 0.414647i
\(34\) 5.09017 0.872957
\(35\) 0 0
\(36\) −2.61803 −0.436339
\(37\) 2.47214i 0.406417i 0.979136 + 0.203208i \(0.0651369\pi\)
−0.979136 + 0.203208i \(0.934863\pi\)
\(38\) − 4.85410i − 0.787439i
\(39\) 2.52786 0.404782
\(40\) 0 0
\(41\) −12.3262 −1.92503 −0.962517 0.271220i \(-0.912573\pi\)
−0.962517 + 0.271220i \(0.912573\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −3.85410 −0.581028
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 9.70820i − 1.41609i −0.706169 0.708044i \(-0.749578\pi\)
0.706169 0.708044i \(-0.250422\pi\)
\(48\) − 0.618034i − 0.0892055i
\(49\) 4.38197 0.625995
\(50\) 0 0
\(51\) 3.14590 0.440514
\(52\) − 4.09017i − 0.567205i
\(53\) − 8.47214i − 1.16374i −0.813283 0.581869i \(-0.802322\pi\)
0.813283 0.581869i \(-0.197678\pi\)
\(54\) −3.47214 −0.472498
\(55\) 0 0
\(56\) 1.61803 0.216219
\(57\) − 3.00000i − 0.397360i
\(58\) − 4.76393i − 0.625535i
\(59\) 11.7082 1.52428 0.762139 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) 6.32624 0.809992 0.404996 0.914319i \(-0.367273\pi\)
0.404996 + 0.914319i \(0.367273\pi\)
\(62\) 2.09017i 0.265452i
\(63\) − 4.23607i − 0.533694i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.38197 −0.293200
\(67\) − 5.52786i − 0.675336i −0.941265 0.337668i \(-0.890362\pi\)
0.941265 0.337668i \(-0.109638\pi\)
\(68\) − 5.09017i − 0.617274i
\(69\) 0.618034 0.0744025
\(70\) 0 0
\(71\) 7.09017 0.841448 0.420724 0.907189i \(-0.361776\pi\)
0.420724 + 0.907189i \(0.361776\pi\)
\(72\) 2.61803i 0.308538i
\(73\) − 1.23607i − 0.144671i −0.997380 0.0723354i \(-0.976955\pi\)
0.997380 0.0723354i \(-0.0230452\pi\)
\(74\) 2.47214 0.287380
\(75\) 0 0
\(76\) −4.85410 −0.556804
\(77\) − 6.23607i − 0.710666i
\(78\) − 2.52786i − 0.286224i
\(79\) −10.4721 −1.17821 −0.589104 0.808057i \(-0.700519\pi\)
−0.589104 + 0.808057i \(0.700519\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 12.3262i 1.36121i
\(83\) 10.9443i 1.20129i 0.799516 + 0.600645i \(0.205089\pi\)
−0.799516 + 0.600645i \(0.794911\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.94427i − 0.315659i
\(88\) 3.85410i 0.410849i
\(89\) 1.52786 0.161953 0.0809766 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 0 0
\(91\) 6.61803 0.693758
\(92\) − 1.00000i − 0.104257i
\(93\) 1.29180i 0.133953i
\(94\) −9.70820 −1.00132
\(95\) 0 0
\(96\) −0.618034 −0.0630778
\(97\) − 14.6180i − 1.48424i −0.670269 0.742118i \(-0.733821\pi\)
0.670269 0.742118i \(-0.266179\pi\)
\(98\) − 4.38197i − 0.442645i
\(99\) 10.0902 1.01410
\(100\) 0 0
\(101\) −13.7082 −1.36402 −0.682009 0.731344i \(-0.738893\pi\)
−0.682009 + 0.731344i \(0.738893\pi\)
\(102\) − 3.14590i − 0.311490i
\(103\) − 3.56231i − 0.351004i −0.984479 0.175502i \(-0.943845\pi\)
0.984479 0.175502i \(-0.0561549\pi\)
\(104\) −4.09017 −0.401074
\(105\) 0 0
\(106\) −8.47214 −0.822887
\(107\) 4.18034i 0.404129i 0.979372 + 0.202064i \(0.0647650\pi\)
−0.979372 + 0.202064i \(0.935235\pi\)
\(108\) 3.47214i 0.334106i
\(109\) −8.56231 −0.820120 −0.410060 0.912059i \(-0.634492\pi\)
−0.410060 + 0.912059i \(0.634492\pi\)
\(110\) 0 0
\(111\) 1.52786 0.145018
\(112\) − 1.61803i − 0.152890i
\(113\) 18.9443i 1.78213i 0.453878 + 0.891064i \(0.350040\pi\)
−0.453878 + 0.891064i \(0.649960\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) −4.76393 −0.442320
\(117\) 10.7082i 0.989974i
\(118\) − 11.7082i − 1.07783i
\(119\) 8.23607 0.754999
\(120\) 0 0
\(121\) 3.85410 0.350373
\(122\) − 6.32624i − 0.572751i
\(123\) 7.61803i 0.686895i
\(124\) 2.09017 0.187703
\(125\) 0 0
\(126\) −4.23607 −0.377379
\(127\) 6.18034i 0.548416i 0.961670 + 0.274208i \(0.0884158\pi\)
−0.961670 + 0.274208i \(0.911584\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −14.9443 −1.30569 −0.652844 0.757493i \(-0.726424\pi\)
−0.652844 + 0.757493i \(0.726424\pi\)
\(132\) 2.38197i 0.207324i
\(133\) − 7.85410i − 0.681037i
\(134\) −5.52786 −0.477535
\(135\) 0 0
\(136\) −5.09017 −0.436478
\(137\) − 5.32624i − 0.455051i −0.973772 0.227526i \(-0.926936\pi\)
0.973772 0.227526i \(-0.0730635\pi\)
\(138\) − 0.618034i − 0.0526105i
\(139\) −17.2361 −1.46194 −0.730972 0.682407i \(-0.760933\pi\)
−0.730972 + 0.682407i \(0.760933\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) − 7.09017i − 0.594994i
\(143\) 15.7639i 1.31825i
\(144\) 2.61803 0.218169
\(145\) 0 0
\(146\) −1.23607 −0.102298
\(147\) − 2.70820i − 0.223369i
\(148\) − 2.47214i − 0.203208i
\(149\) 1.14590 0.0938756 0.0469378 0.998898i \(-0.485054\pi\)
0.0469378 + 0.998898i \(0.485054\pi\)
\(150\) 0 0
\(151\) 17.5623 1.42920 0.714600 0.699533i \(-0.246609\pi\)
0.714600 + 0.699533i \(0.246609\pi\)
\(152\) 4.85410i 0.393720i
\(153\) 13.3262i 1.07736i
\(154\) −6.23607 −0.502517
\(155\) 0 0
\(156\) −2.52786 −0.202391
\(157\) − 9.70820i − 0.774799i −0.921912 0.387400i \(-0.873373\pi\)
0.921912 0.387400i \(-0.126627\pi\)
\(158\) 10.4721i 0.833118i
\(159\) −5.23607 −0.415247
\(160\) 0 0
\(161\) 1.61803 0.127519
\(162\) − 5.70820i − 0.448479i
\(163\) 3.61803i 0.283386i 0.989911 + 0.141693i \(0.0452546\pi\)
−0.989911 + 0.141693i \(0.954745\pi\)
\(164\) 12.3262 0.962517
\(165\) 0 0
\(166\) 10.9443 0.849440
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) −3.72949 −0.286884
\(170\) 0 0
\(171\) 12.7082 0.971821
\(172\) 0 0
\(173\) − 21.5623i − 1.63935i −0.572828 0.819676i \(-0.694154\pi\)
0.572828 0.819676i \(-0.305846\pi\)
\(174\) −2.94427 −0.223205
\(175\) 0 0
\(176\) 3.85410 0.290514
\(177\) − 7.23607i − 0.543896i
\(178\) − 1.52786i − 0.114518i
\(179\) 20.1803 1.50835 0.754175 0.656674i \(-0.228037\pi\)
0.754175 + 0.656674i \(0.228037\pi\)
\(180\) 0 0
\(181\) −18.8541 −1.40141 −0.700707 0.713449i \(-0.747132\pi\)
−0.700707 + 0.713449i \(0.747132\pi\)
\(182\) − 6.61803i − 0.490561i
\(183\) − 3.90983i − 0.289023i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 1.29180 0.0947191
\(187\) 19.6180i 1.43461i
\(188\) 9.70820i 0.708044i
\(189\) −5.61803 −0.408652
\(190\) 0 0
\(191\) −0.291796 −0.0211136 −0.0105568 0.999944i \(-0.503360\pi\)
−0.0105568 + 0.999944i \(0.503360\pi\)
\(192\) 0.618034i 0.0446028i
\(193\) 5.23607i 0.376900i 0.982083 + 0.188450i \(0.0603464\pi\)
−0.982083 + 0.188450i \(0.939654\pi\)
\(194\) −14.6180 −1.04951
\(195\) 0 0
\(196\) −4.38197 −0.312998
\(197\) 2.43769i 0.173679i 0.996222 + 0.0868393i \(0.0276767\pi\)
−0.996222 + 0.0868393i \(0.972323\pi\)
\(198\) − 10.0902i − 0.717077i
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) −3.41641 −0.240975
\(202\) 13.7082i 0.964506i
\(203\) − 7.70820i − 0.541010i
\(204\) −3.14590 −0.220257
\(205\) 0 0
\(206\) −3.56231 −0.248198
\(207\) 2.61803i 0.181966i
\(208\) 4.09017i 0.283602i
\(209\) 18.7082 1.29407
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 8.47214i 0.581869i
\(213\) − 4.38197i − 0.300247i
\(214\) 4.18034 0.285762
\(215\) 0 0
\(216\) 3.47214 0.236249
\(217\) 3.38197i 0.229583i
\(218\) 8.56231i 0.579913i
\(219\) −0.763932 −0.0516217
\(220\) 0 0
\(221\) −20.8197 −1.40048
\(222\) − 1.52786i − 0.102544i
\(223\) − 3.05573i − 0.204627i −0.994752 0.102313i \(-0.967376\pi\)
0.994752 0.102313i \(-0.0326244\pi\)
\(224\) −1.61803 −0.108109
\(225\) 0 0
\(226\) 18.9443 1.26015
\(227\) 23.2361i 1.54223i 0.636695 + 0.771116i \(0.280301\pi\)
−0.636695 + 0.771116i \(0.719699\pi\)
\(228\) 3.00000i 0.198680i
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −3.85410 −0.253581
\(232\) 4.76393i 0.312767i
\(233\) 19.7082i 1.29113i 0.763706 + 0.645564i \(0.223378\pi\)
−0.763706 + 0.645564i \(0.776622\pi\)
\(234\) 10.7082 0.700017
\(235\) 0 0
\(236\) −11.7082 −0.762139
\(237\) 6.47214i 0.420410i
\(238\) − 8.23607i − 0.533865i
\(239\) −24.3607 −1.57576 −0.787881 0.615828i \(-0.788822\pi\)
−0.787881 + 0.615828i \(0.788822\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) − 3.85410i − 0.247751i
\(243\) − 13.9443i − 0.894525i
\(244\) −6.32624 −0.404996
\(245\) 0 0
\(246\) 7.61803 0.485708
\(247\) 19.8541i 1.26329i
\(248\) − 2.09017i − 0.132726i
\(249\) 6.76393 0.428647
\(250\) 0 0
\(251\) 12.8541 0.811344 0.405672 0.914019i \(-0.367038\pi\)
0.405672 + 0.914019i \(0.367038\pi\)
\(252\) 4.23607i 0.266847i
\(253\) 3.85410i 0.242305i
\(254\) 6.18034 0.387789
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 30.1803i 1.88260i 0.337574 + 0.941299i \(0.390394\pi\)
−0.337574 + 0.941299i \(0.609606\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 12.4721 0.772006
\(262\) 14.9443i 0.923260i
\(263\) − 21.7426i − 1.34071i −0.742041 0.670354i \(-0.766142\pi\)
0.742041 0.670354i \(-0.233858\pi\)
\(264\) 2.38197 0.146600
\(265\) 0 0
\(266\) −7.85410 −0.481566
\(267\) − 0.944272i − 0.0577885i
\(268\) 5.52786i 0.337668i
\(269\) −8.18034 −0.498764 −0.249382 0.968405i \(-0.580227\pi\)
−0.249382 + 0.968405i \(0.580227\pi\)
\(270\) 0 0
\(271\) −14.6738 −0.891368 −0.445684 0.895190i \(-0.647039\pi\)
−0.445684 + 0.895190i \(0.647039\pi\)
\(272\) 5.09017i 0.308637i
\(273\) − 4.09017i − 0.247548i
\(274\) −5.32624 −0.321770
\(275\) 0 0
\(276\) −0.618034 −0.0372013
\(277\) − 2.58359i − 0.155233i −0.996983 0.0776165i \(-0.975269\pi\)
0.996983 0.0776165i \(-0.0247310\pi\)
\(278\) 17.2361i 1.03375i
\(279\) −5.47214 −0.327608
\(280\) 0 0
\(281\) 27.2361 1.62477 0.812384 0.583123i \(-0.198169\pi\)
0.812384 + 0.583123i \(0.198169\pi\)
\(282\) 6.00000i 0.357295i
\(283\) − 9.05573i − 0.538307i −0.963097 0.269154i \(-0.913256\pi\)
0.963097 0.269154i \(-0.0867439\pi\)
\(284\) −7.09017 −0.420724
\(285\) 0 0
\(286\) 15.7639 0.932141
\(287\) 19.9443i 1.17727i
\(288\) − 2.61803i − 0.154269i
\(289\) −8.90983 −0.524108
\(290\) 0 0
\(291\) −9.03444 −0.529608
\(292\) 1.23607i 0.0723354i
\(293\) 15.8885i 0.928219i 0.885778 + 0.464109i \(0.153626\pi\)
−0.885778 + 0.464109i \(0.846374\pi\)
\(294\) −2.70820 −0.157946
\(295\) 0 0
\(296\) −2.47214 −0.143690
\(297\) − 13.3820i − 0.776500i
\(298\) − 1.14590i − 0.0663801i
\(299\) −4.09017 −0.236541
\(300\) 0 0
\(301\) 0 0
\(302\) − 17.5623i − 1.01060i
\(303\) 8.47214i 0.486711i
\(304\) 4.85410 0.278402
\(305\) 0 0
\(306\) 13.3262 0.761810
\(307\) − 27.4508i − 1.56670i −0.621579 0.783351i \(-0.713509\pi\)
0.621579 0.783351i \(-0.286491\pi\)
\(308\) 6.23607i 0.355333i
\(309\) −2.20163 −0.125246
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 2.52786i 0.143112i
\(313\) − 11.7984i − 0.666884i −0.942771 0.333442i \(-0.891790\pi\)
0.942771 0.333442i \(-0.108210\pi\)
\(314\) −9.70820 −0.547866
\(315\) 0 0
\(316\) 10.4721 0.589104
\(317\) − 0.0901699i − 0.00506445i −0.999997 0.00253222i \(-0.999194\pi\)
0.999997 0.00253222i \(-0.000806033\pi\)
\(318\) 5.23607i 0.293624i
\(319\) 18.3607 1.02800
\(320\) 0 0
\(321\) 2.58359 0.144202
\(322\) − 1.61803i − 0.0901695i
\(323\) 24.7082i 1.37480i
\(324\) −5.70820 −0.317122
\(325\) 0 0
\(326\) 3.61803 0.200384
\(327\) 5.29180i 0.292637i
\(328\) − 12.3262i − 0.680603i
\(329\) −15.7082 −0.866021
\(330\) 0 0
\(331\) 14.7639 0.811499 0.405750 0.913984i \(-0.367011\pi\)
0.405750 + 0.913984i \(0.367011\pi\)
\(332\) − 10.9443i − 0.600645i
\(333\) 6.47214i 0.354671i
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) − 29.3262i − 1.59750i −0.601662 0.798751i \(-0.705494\pi\)
0.601662 0.798751i \(-0.294506\pi\)
\(338\) 3.72949i 0.202858i
\(339\) 11.7082 0.635902
\(340\) 0 0
\(341\) −8.05573 −0.436242
\(342\) − 12.7082i − 0.687181i
\(343\) − 18.4164i − 0.994393i
\(344\) 0 0
\(345\) 0 0
\(346\) −21.5623 −1.15920
\(347\) 8.61803i 0.462640i 0.972878 + 0.231320i \(0.0743045\pi\)
−0.972878 + 0.231320i \(0.925696\pi\)
\(348\) 2.94427i 0.157830i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 14.2016 0.758027
\(352\) − 3.85410i − 0.205424i
\(353\) 24.0000i 1.27739i 0.769460 + 0.638696i \(0.220526\pi\)
−0.769460 + 0.638696i \(0.779474\pi\)
\(354\) −7.23607 −0.384593
\(355\) 0 0
\(356\) −1.52786 −0.0809766
\(357\) − 5.09017i − 0.269400i
\(358\) − 20.1803i − 1.06656i
\(359\) 18.3607 0.969040 0.484520 0.874780i \(-0.338994\pi\)
0.484520 + 0.874780i \(0.338994\pi\)
\(360\) 0 0
\(361\) 4.56231 0.240121
\(362\) 18.8541i 0.990950i
\(363\) − 2.38197i − 0.125021i
\(364\) −6.61803 −0.346879
\(365\) 0 0
\(366\) −3.90983 −0.204370
\(367\) 2.47214i 0.129044i 0.997916 + 0.0645222i \(0.0205523\pi\)
−0.997916 + 0.0645222i \(0.979448\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) −32.2705 −1.67994
\(370\) 0 0
\(371\) −13.7082 −0.711694
\(372\) − 1.29180i − 0.0669765i
\(373\) − 2.18034i − 0.112894i −0.998406 0.0564469i \(-0.982023\pi\)
0.998406 0.0564469i \(-0.0179771\pi\)
\(374\) 19.6180 1.01442
\(375\) 0 0
\(376\) 9.70820 0.500662
\(377\) 19.4853i 1.00354i
\(378\) 5.61803i 0.288960i
\(379\) −33.4508 −1.71825 −0.859127 0.511762i \(-0.828993\pi\)
−0.859127 + 0.511762i \(0.828993\pi\)
\(380\) 0 0
\(381\) 3.81966 0.195687
\(382\) 0.291796i 0.0149296i
\(383\) 17.8885i 0.914062i 0.889451 + 0.457031i \(0.151087\pi\)
−0.889451 + 0.457031i \(0.848913\pi\)
\(384\) 0.618034 0.0315389
\(385\) 0 0
\(386\) 5.23607 0.266509
\(387\) 0 0
\(388\) 14.6180i 0.742118i
\(389\) −5.67376 −0.287671 −0.143836 0.989602i \(-0.545944\pi\)
−0.143836 + 0.989602i \(0.545944\pi\)
\(390\) 0 0
\(391\) −5.09017 −0.257421
\(392\) 4.38197i 0.221323i
\(393\) 9.23607i 0.465898i
\(394\) 2.43769 0.122809
\(395\) 0 0
\(396\) −10.0902 −0.507050
\(397\) 8.32624i 0.417882i 0.977928 + 0.208941i \(0.0670016\pi\)
−0.977928 + 0.208941i \(0.932998\pi\)
\(398\) 2.00000i 0.100251i
\(399\) −4.85410 −0.243009
\(400\) 0 0
\(401\) 11.7082 0.584680 0.292340 0.956314i \(-0.405566\pi\)
0.292340 + 0.956314i \(0.405566\pi\)
\(402\) 3.41641i 0.170395i
\(403\) − 8.54915i − 0.425864i
\(404\) 13.7082 0.682009
\(405\) 0 0
\(406\) −7.70820 −0.382552
\(407\) 9.52786i 0.472279i
\(408\) 3.14590i 0.155745i
\(409\) −21.2148 −1.04900 −0.524502 0.851409i \(-0.675748\pi\)
−0.524502 + 0.851409i \(0.675748\pi\)
\(410\) 0 0
\(411\) −3.29180 −0.162372
\(412\) 3.56231i 0.175502i
\(413\) − 18.9443i − 0.932187i
\(414\) 2.61803 0.128669
\(415\) 0 0
\(416\) 4.09017 0.200537
\(417\) 10.6525i 0.521654i
\(418\) − 18.7082i − 0.915048i
\(419\) −5.52786 −0.270054 −0.135027 0.990842i \(-0.543112\pi\)
−0.135027 + 0.990842i \(0.543112\pi\)
\(420\) 0 0
\(421\) −28.7426 −1.40083 −0.700415 0.713735i \(-0.747002\pi\)
−0.700415 + 0.713735i \(0.747002\pi\)
\(422\) − 14.0000i − 0.681509i
\(423\) − 25.4164i − 1.23579i
\(424\) 8.47214 0.411443
\(425\) 0 0
\(426\) −4.38197 −0.212307
\(427\) − 10.2361i − 0.495358i
\(428\) − 4.18034i − 0.202064i
\(429\) 9.74265 0.470379
\(430\) 0 0
\(431\) 34.6525 1.66915 0.834576 0.550894i \(-0.185713\pi\)
0.834576 + 0.550894i \(0.185713\pi\)
\(432\) − 3.47214i − 0.167053i
\(433\) − 29.5066i − 1.41800i −0.705211 0.708998i \(-0.749148\pi\)
0.705211 0.708998i \(-0.250852\pi\)
\(434\) 3.38197 0.162340
\(435\) 0 0
\(436\) 8.56231 0.410060
\(437\) 4.85410i 0.232203i
\(438\) 0.763932i 0.0365021i
\(439\) 15.6180 0.745408 0.372704 0.927950i \(-0.378431\pi\)
0.372704 + 0.927950i \(0.378431\pi\)
\(440\) 0 0
\(441\) 11.4721 0.546292
\(442\) 20.8197i 0.990290i
\(443\) 13.9098i 0.660876i 0.943828 + 0.330438i \(0.107196\pi\)
−0.943828 + 0.330438i \(0.892804\pi\)
\(444\) −1.52786 −0.0725092
\(445\) 0 0
\(446\) −3.05573 −0.144693
\(447\) − 0.708204i − 0.0334969i
\(448\) 1.61803i 0.0764449i
\(449\) −18.5623 −0.876009 −0.438005 0.898973i \(-0.644315\pi\)
−0.438005 + 0.898973i \(0.644315\pi\)
\(450\) 0 0
\(451\) −47.5066 −2.23700
\(452\) − 18.9443i − 0.891064i
\(453\) − 10.8541i − 0.509970i
\(454\) 23.2361 1.09052
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) − 33.7771i − 1.58003i −0.613090 0.790013i \(-0.710074\pi\)
0.613090 0.790013i \(-0.289926\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 17.6738 0.824941
\(460\) 0 0
\(461\) −34.7639 −1.61912 −0.809559 0.587039i \(-0.800294\pi\)
−0.809559 + 0.587039i \(0.800294\pi\)
\(462\) 3.85410i 0.179309i
\(463\) 2.00000i 0.0929479i 0.998920 + 0.0464739i \(0.0147984\pi\)
−0.998920 + 0.0464739i \(0.985202\pi\)
\(464\) 4.76393 0.221160
\(465\) 0 0
\(466\) 19.7082 0.912965
\(467\) − 23.1246i − 1.07008i −0.844827 0.535040i \(-0.820297\pi\)
0.844827 0.535040i \(-0.179703\pi\)
\(468\) − 10.7082i − 0.494987i
\(469\) −8.94427 −0.413008
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 11.7082i 0.538914i
\(473\) 0 0
\(474\) 6.47214 0.297275
\(475\) 0 0
\(476\) −8.23607 −0.377500
\(477\) − 22.1803i − 1.01557i
\(478\) 24.3607i 1.11423i
\(479\) −3.88854 −0.177672 −0.0888361 0.996046i \(-0.528315\pi\)
−0.0888361 + 0.996046i \(0.528315\pi\)
\(480\) 0 0
\(481\) −10.1115 −0.461043
\(482\) 0 0
\(483\) − 1.00000i − 0.0455016i
\(484\) −3.85410 −0.175186
\(485\) 0 0
\(486\) −13.9443 −0.632525
\(487\) 42.1803i 1.91137i 0.294385 + 0.955687i \(0.404885\pi\)
−0.294385 + 0.955687i \(0.595115\pi\)
\(488\) 6.32624i 0.286375i
\(489\) 2.23607 0.101118
\(490\) 0 0
\(491\) −16.1803 −0.730209 −0.365104 0.930967i \(-0.618967\pi\)
−0.365104 + 0.930967i \(0.618967\pi\)
\(492\) − 7.61803i − 0.343447i
\(493\) 24.2492i 1.09213i
\(494\) 19.8541 0.893278
\(495\) 0 0
\(496\) −2.09017 −0.0938514
\(497\) − 11.4721i − 0.514596i
\(498\) − 6.76393i − 0.303099i
\(499\) 32.3607 1.44866 0.724331 0.689452i \(-0.242149\pi\)
0.724331 + 0.689452i \(0.242149\pi\)
\(500\) 0 0
\(501\) 4.94427 0.220894
\(502\) − 12.8541i − 0.573707i
\(503\) 20.6738i 0.921797i 0.887453 + 0.460899i \(0.152473\pi\)
−0.887453 + 0.460899i \(0.847527\pi\)
\(504\) 4.23607 0.188689
\(505\) 0 0
\(506\) 3.85410 0.171336
\(507\) 2.30495i 0.102366i
\(508\) − 6.18034i − 0.274208i
\(509\) −5.34752 −0.237025 −0.118512 0.992953i \(-0.537813\pi\)
−0.118512 + 0.992953i \(0.537813\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) − 1.00000i − 0.0441942i
\(513\) − 16.8541i − 0.744127i
\(514\) 30.1803 1.33120
\(515\) 0 0
\(516\) 0 0
\(517\) − 37.4164i − 1.64557i
\(518\) − 4.00000i − 0.175750i
\(519\) −13.3262 −0.584957
\(520\) 0 0
\(521\) −24.4721 −1.07214 −0.536072 0.844172i \(-0.680092\pi\)
−0.536072 + 0.844172i \(0.680092\pi\)
\(522\) − 12.4721i − 0.545891i
\(523\) − 26.0000i − 1.13690i −0.822718 0.568450i \(-0.807543\pi\)
0.822718 0.568450i \(-0.192457\pi\)
\(524\) 14.9443 0.652844
\(525\) 0 0
\(526\) −21.7426 −0.948024
\(527\) − 10.6393i − 0.463456i
\(528\) − 2.38197i − 0.103662i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 30.6525 1.33020
\(532\) 7.85410i 0.340519i
\(533\) − 50.4164i − 2.18378i
\(534\) −0.944272 −0.0408626
\(535\) 0 0
\(536\) 5.52786 0.238767
\(537\) − 12.4721i − 0.538212i
\(538\) 8.18034i 0.352679i
\(539\) 16.8885 0.727441
\(540\) 0 0
\(541\) −30.8328 −1.32561 −0.662803 0.748794i \(-0.730633\pi\)
−0.662803 + 0.748794i \(0.730633\pi\)
\(542\) 14.6738i 0.630292i
\(543\) 11.6525i 0.500056i
\(544\) 5.09017 0.218239
\(545\) 0 0
\(546\) −4.09017 −0.175043
\(547\) 36.9230i 1.57871i 0.613935 + 0.789356i \(0.289586\pi\)
−0.613935 + 0.789356i \(0.710414\pi\)
\(548\) 5.32624i 0.227526i
\(549\) 16.5623 0.706862
\(550\) 0 0
\(551\) 23.1246 0.985142
\(552\) 0.618034i 0.0263053i
\(553\) 16.9443i 0.720544i
\(554\) −2.58359 −0.109766
\(555\) 0 0
\(556\) 17.2361 0.730972
\(557\) − 30.8328i − 1.30643i −0.757173 0.653214i \(-0.773420\pi\)
0.757173 0.653214i \(-0.226580\pi\)
\(558\) 5.47214i 0.231654i
\(559\) 0 0
\(560\) 0 0
\(561\) 12.1246 0.511902
\(562\) − 27.2361i − 1.14888i
\(563\) 21.8885i 0.922492i 0.887272 + 0.461246i \(0.152597\pi\)
−0.887272 + 0.461246i \(0.847403\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) −9.05573 −0.380641
\(567\) − 9.23607i − 0.387878i
\(568\) 7.09017i 0.297497i
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) −30.9787 −1.29642 −0.648209 0.761462i \(-0.724482\pi\)
−0.648209 + 0.761462i \(0.724482\pi\)
\(572\) − 15.7639i − 0.659123i
\(573\) 0.180340i 0.00753381i
\(574\) 19.9443 0.832458
\(575\) 0 0
\(576\) −2.61803 −0.109085
\(577\) 12.4721i 0.519222i 0.965713 + 0.259611i \(0.0835943\pi\)
−0.965713 + 0.259611i \(0.916406\pi\)
\(578\) 8.90983i 0.370600i
\(579\) 3.23607 0.134486
\(580\) 0 0
\(581\) 17.7082 0.734660
\(582\) 9.03444i 0.374490i
\(583\) − 32.6525i − 1.35233i
\(584\) 1.23607 0.0511489
\(585\) 0 0
\(586\) 15.8885 0.656350
\(587\) − 11.3820i − 0.469784i −0.972021 0.234892i \(-0.924526\pi\)
0.972021 0.234892i \(-0.0754737\pi\)
\(588\) 2.70820i 0.111684i
\(589\) −10.1459 −0.418054
\(590\) 0 0
\(591\) 1.50658 0.0619723
\(592\) 2.47214i 0.101604i
\(593\) 34.7639i 1.42758i 0.700358 + 0.713792i \(0.253024\pi\)
−0.700358 + 0.713792i \(0.746976\pi\)
\(594\) −13.3820 −0.549069
\(595\) 0 0
\(596\) −1.14590 −0.0469378
\(597\) 1.23607i 0.0505889i
\(598\) 4.09017i 0.167259i
\(599\) 20.6180 0.842430 0.421215 0.906961i \(-0.361604\pi\)
0.421215 + 0.906961i \(0.361604\pi\)
\(600\) 0 0
\(601\) 0.270510 0.0110343 0.00551716 0.999985i \(-0.498244\pi\)
0.00551716 + 0.999985i \(0.498244\pi\)
\(602\) 0 0
\(603\) − 14.4721i − 0.589351i
\(604\) −17.5623 −0.714600
\(605\) 0 0
\(606\) 8.47214 0.344157
\(607\) − 17.5279i − 0.711434i −0.934594 0.355717i \(-0.884237\pi\)
0.934594 0.355717i \(-0.115763\pi\)
\(608\) − 4.85410i − 0.196860i
\(609\) −4.76393 −0.193044
\(610\) 0 0
\(611\) 39.7082 1.60642
\(612\) − 13.3262i − 0.538681i
\(613\) − 43.3050i − 1.74907i −0.484962 0.874535i \(-0.661167\pi\)
0.484962 0.874535i \(-0.338833\pi\)
\(614\) −27.4508 −1.10783
\(615\) 0 0
\(616\) 6.23607 0.251258
\(617\) − 22.9098i − 0.922315i −0.887318 0.461158i \(-0.847434\pi\)
0.887318 0.461158i \(-0.152566\pi\)
\(618\) 2.20163i 0.0885624i
\(619\) −21.7984 −0.876151 −0.438075 0.898938i \(-0.644340\pi\)
−0.438075 + 0.898938i \(0.644340\pi\)
\(620\) 0 0
\(621\) 3.47214 0.139332
\(622\) − 4.00000i − 0.160385i
\(623\) − 2.47214i − 0.0990440i
\(624\) 2.52786 0.101196
\(625\) 0 0
\(626\) −11.7984 −0.471558
\(627\) − 11.5623i − 0.461754i
\(628\) 9.70820i 0.387400i
\(629\) −12.5836 −0.501741
\(630\) 0 0
\(631\) −16.0689 −0.639692 −0.319846 0.947470i \(-0.603631\pi\)
−0.319846 + 0.947470i \(0.603631\pi\)
\(632\) − 10.4721i − 0.416559i
\(633\) − 8.65248i − 0.343905i
\(634\) −0.0901699 −0.00358111
\(635\) 0 0
\(636\) 5.23607 0.207624
\(637\) 17.9230i 0.710135i
\(638\) − 18.3607i − 0.726906i
\(639\) 18.5623 0.734313
\(640\) 0 0
\(641\) −44.3607 −1.75214 −0.876071 0.482183i \(-0.839844\pi\)
−0.876071 + 0.482183i \(0.839844\pi\)
\(642\) − 2.58359i − 0.101966i
\(643\) − 21.7082i − 0.856088i −0.903758 0.428044i \(-0.859203\pi\)
0.903758 0.428044i \(-0.140797\pi\)
\(644\) −1.61803 −0.0637595
\(645\) 0 0
\(646\) 24.7082 0.972131
\(647\) − 44.2492i − 1.73962i −0.493390 0.869808i \(-0.664242\pi\)
0.493390 0.869808i \(-0.335758\pi\)
\(648\) 5.70820i 0.224239i
\(649\) 45.1246 1.77130
\(650\) 0 0
\(651\) 2.09017 0.0819202
\(652\) − 3.61803i − 0.141693i
\(653\) 21.0344i 0.823141i 0.911378 + 0.411571i \(0.135020\pi\)
−0.911378 + 0.411571i \(0.864980\pi\)
\(654\) 5.29180 0.206926
\(655\) 0 0
\(656\) −12.3262 −0.481259
\(657\) − 3.23607i − 0.126251i
\(658\) 15.7082i 0.612370i
\(659\) −34.2492 −1.33416 −0.667080 0.744986i \(-0.732456\pi\)
−0.667080 + 0.744986i \(0.732456\pi\)
\(660\) 0 0
\(661\) 34.3262 1.33514 0.667568 0.744549i \(-0.267335\pi\)
0.667568 + 0.744549i \(0.267335\pi\)
\(662\) − 14.7639i − 0.573817i
\(663\) 12.8673i 0.499723i
\(664\) −10.9443 −0.424720
\(665\) 0 0
\(666\) 6.47214 0.250790
\(667\) 4.76393i 0.184460i
\(668\) − 8.00000i − 0.309529i
\(669\) −1.88854 −0.0730153
\(670\) 0 0
\(671\) 24.3820 0.941255
\(672\) 1.00000i 0.0385758i
\(673\) 6.94427i 0.267682i 0.991003 + 0.133841i \(0.0427311\pi\)
−0.991003 + 0.133841i \(0.957269\pi\)
\(674\) −29.3262 −1.12960
\(675\) 0 0
\(676\) 3.72949 0.143442
\(677\) 33.0557i 1.27043i 0.772333 + 0.635217i \(0.219090\pi\)
−0.772333 + 0.635217i \(0.780910\pi\)
\(678\) − 11.7082i − 0.449651i
\(679\) −23.6525 −0.907699
\(680\) 0 0
\(681\) 14.3607 0.550302
\(682\) 8.05573i 0.308470i
\(683\) − 11.4377i − 0.437651i −0.975764 0.218826i \(-0.929777\pi\)
0.975764 0.218826i \(-0.0702226\pi\)
\(684\) −12.7082 −0.485910
\(685\) 0 0
\(686\) −18.4164 −0.703142
\(687\) 6.18034i 0.235795i
\(688\) 0 0
\(689\) 34.6525 1.32015
\(690\) 0 0
\(691\) −24.7639 −0.942064 −0.471032 0.882116i \(-0.656118\pi\)
−0.471032 + 0.882116i \(0.656118\pi\)
\(692\) 21.5623i 0.819676i
\(693\) − 16.3262i − 0.620182i
\(694\) 8.61803 0.327136
\(695\) 0 0
\(696\) 2.94427 0.111602
\(697\) − 62.7426i − 2.37655i
\(698\) − 2.00000i − 0.0757011i
\(699\) 12.1803 0.460703
\(700\) 0 0
\(701\) −48.3394 −1.82575 −0.912877 0.408235i \(-0.866144\pi\)
−0.912877 + 0.408235i \(0.866144\pi\)
\(702\) − 14.2016i − 0.536006i
\(703\) 12.0000i 0.452589i
\(704\) −3.85410 −0.145257
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 22.1803i 0.834178i
\(708\) 7.23607i 0.271948i
\(709\) 14.9098 0.559950 0.279975 0.960007i \(-0.409674\pi\)
0.279975 + 0.960007i \(0.409674\pi\)
\(710\) 0 0
\(711\) −27.4164 −1.02820
\(712\) 1.52786i 0.0572591i
\(713\) − 2.09017i − 0.0782775i
\(714\) −5.09017 −0.190495
\(715\) 0 0
\(716\) −20.1803 −0.754175
\(717\) 15.0557i 0.562266i
\(718\) − 18.3607i − 0.685214i
\(719\) −1.72949 −0.0644991 −0.0322495 0.999480i \(-0.510267\pi\)
−0.0322495 + 0.999480i \(0.510267\pi\)
\(720\) 0 0
\(721\) −5.76393 −0.214660
\(722\) − 4.56231i − 0.169791i
\(723\) 0 0
\(724\) 18.8541 0.700707
\(725\) 0 0
\(726\) −2.38197 −0.0884031
\(727\) − 52.7984i − 1.95818i −0.203420 0.979092i \(-0.565206\pi\)
0.203420 0.979092i \(-0.434794\pi\)
\(728\) 6.61803i 0.245281i
\(729\) 8.50658 0.315058
\(730\) 0 0
\(731\) 0 0
\(732\) 3.90983i 0.144511i
\(733\) − 2.58359i − 0.0954272i −0.998861 0.0477136i \(-0.984807\pi\)
0.998861 0.0477136i \(-0.0151935\pi\)
\(734\) 2.47214 0.0912482
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) − 21.3050i − 0.784778i
\(738\) 32.2705i 1.18789i
\(739\) −21.8885 −0.805183 −0.402592 0.915380i \(-0.631890\pi\)
−0.402592 + 0.915380i \(0.631890\pi\)
\(740\) 0 0
\(741\) 12.2705 0.450768
\(742\) 13.7082i 0.503244i
\(743\) − 44.6312i − 1.63736i −0.574250 0.818680i \(-0.694706\pi\)
0.574250 0.818680i \(-0.305294\pi\)
\(744\) −1.29180 −0.0473595
\(745\) 0 0
\(746\) −2.18034 −0.0798279
\(747\) 28.6525i 1.04834i
\(748\) − 19.6180i − 0.717306i
\(749\) 6.76393 0.247149
\(750\) 0 0
\(751\) 29.0132 1.05871 0.529353 0.848402i \(-0.322435\pi\)
0.529353 + 0.848402i \(0.322435\pi\)
\(752\) − 9.70820i − 0.354022i
\(753\) − 7.94427i − 0.289505i
\(754\) 19.4853 0.709612
\(755\) 0 0
\(756\) 5.61803 0.204326
\(757\) − 17.8885i − 0.650170i −0.945685 0.325085i \(-0.894607\pi\)
0.945685 0.325085i \(-0.105393\pi\)
\(758\) 33.4508i 1.21499i
\(759\) 2.38197 0.0864599
\(760\) 0 0
\(761\) −35.8673 −1.30019 −0.650094 0.759854i \(-0.725270\pi\)
−0.650094 + 0.759854i \(0.725270\pi\)
\(762\) − 3.81966i − 0.138372i
\(763\) 13.8541i 0.501552i
\(764\) 0.291796 0.0105568
\(765\) 0 0
\(766\) 17.8885 0.646339
\(767\) 47.8885i 1.72916i
\(768\) − 0.618034i − 0.0223014i
\(769\) 33.4164 1.20503 0.602513 0.798109i \(-0.294166\pi\)
0.602513 + 0.798109i \(0.294166\pi\)
\(770\) 0 0
\(771\) 18.6525 0.671753
\(772\) − 5.23607i − 0.188450i
\(773\) 11.0557i 0.397647i 0.980035 + 0.198823i \(0.0637120\pi\)
−0.980035 + 0.198823i \(0.936288\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 14.6180 0.524757
\(777\) − 2.47214i − 0.0886874i
\(778\) 5.67376i 0.203414i
\(779\) −59.8328 −2.14373
\(780\) 0 0
\(781\) 27.3262 0.977810
\(782\) 5.09017i 0.182024i
\(783\) − 16.5410i − 0.591128i
\(784\) 4.38197 0.156499
\(785\) 0 0
\(786\) 9.23607 0.329440
\(787\) 43.1246i 1.53723i 0.639714 + 0.768613i \(0.279053\pi\)
−0.639714 + 0.768613i \(0.720947\pi\)
\(788\) − 2.43769i − 0.0868393i
\(789\) −13.4377 −0.478395
\(790\) 0 0
\(791\) 30.6525 1.08988
\(792\) 10.0902i 0.358539i
\(793\) 25.8754i 0.918862i
\(794\) 8.32624 0.295487
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) − 0.291796i − 0.0103359i −0.999987 0.00516797i \(-0.998355\pi\)
0.999987 0.00516797i \(-0.00164502\pi\)
\(798\) 4.85410i 0.171833i
\(799\) 49.4164 1.74823
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) − 11.7082i − 0.413431i
\(803\) − 4.76393i − 0.168116i
\(804\) 3.41641 0.120487
\(805\) 0 0
\(806\) −8.54915 −0.301131
\(807\) 5.05573i 0.177970i
\(808\) − 13.7082i − 0.482253i
\(809\) 4.25735 0.149681 0.0748403 0.997196i \(-0.476155\pi\)
0.0748403 + 0.997196i \(0.476155\pi\)
\(810\) 0 0
\(811\) 44.1803 1.55138 0.775691 0.631113i \(-0.217402\pi\)
0.775691 + 0.631113i \(0.217402\pi\)
\(812\) 7.70820i 0.270505i
\(813\) 9.06888i 0.318060i
\(814\) 9.52786 0.333951
\(815\) 0 0
\(816\) 3.14590 0.110128
\(817\) 0 0
\(818\) 21.2148i 0.741757i
\(819\) 17.3262 0.605428
\(820\) 0 0
\(821\) −50.9443 −1.77797 −0.888984 0.457939i \(-0.848588\pi\)
−0.888984 + 0.457939i \(0.848588\pi\)
\(822\) 3.29180i 0.114815i
\(823\) − 1.41641i − 0.0493729i −0.999695 0.0246864i \(-0.992141\pi\)
0.999695 0.0246864i \(-0.00785874\pi\)
\(824\) 3.56231 0.124099
\(825\) 0 0
\(826\) −18.9443 −0.659156
\(827\) − 8.29180i − 0.288334i −0.989553 0.144167i \(-0.953950\pi\)
0.989553 0.144167i \(-0.0460502\pi\)
\(828\) − 2.61803i − 0.0909830i
\(829\) −1.05573 −0.0366670 −0.0183335 0.999832i \(-0.505836\pi\)
−0.0183335 + 0.999832i \(0.505836\pi\)
\(830\) 0 0
\(831\) −1.59675 −0.0553906
\(832\) − 4.09017i − 0.141801i
\(833\) 22.3050i 0.772821i
\(834\) 10.6525 0.368865
\(835\) 0 0
\(836\) −18.7082 −0.647037
\(837\) 7.25735i 0.250851i
\(838\) 5.52786i 0.190957i
\(839\) 43.0132 1.48498 0.742490 0.669858i \(-0.233645\pi\)
0.742490 + 0.669858i \(0.233645\pi\)
\(840\) 0 0
\(841\) −6.30495 −0.217412
\(842\) 28.7426i 0.990537i
\(843\) − 16.8328i − 0.579753i
\(844\) −14.0000 −0.481900
\(845\) 0 0
\(846\) −25.4164 −0.873834
\(847\) − 6.23607i − 0.214274i
\(848\) − 8.47214i − 0.290934i
\(849\) −5.59675 −0.192080
\(850\) 0 0
\(851\) −2.47214 −0.0847437
\(852\) 4.38197i 0.150124i
\(853\) 13.7984i 0.472447i 0.971699 + 0.236224i \(0.0759098\pi\)
−0.971699 + 0.236224i \(0.924090\pi\)
\(854\) −10.2361 −0.350271
\(855\) 0 0
\(856\) −4.18034 −0.142881
\(857\) 33.4164i 1.14148i 0.821130 + 0.570741i \(0.193344\pi\)
−0.821130 + 0.570741i \(0.806656\pi\)
\(858\) − 9.74265i − 0.332608i
\(859\) −34.0689 −1.16242 −0.581208 0.813755i \(-0.697420\pi\)
−0.581208 + 0.813755i \(0.697420\pi\)
\(860\) 0 0
\(861\) 12.3262 0.420077
\(862\) − 34.6525i − 1.18027i
\(863\) 37.2361i 1.26753i 0.773525 + 0.633765i \(0.218491\pi\)
−0.773525 + 0.633765i \(0.781509\pi\)
\(864\) −3.47214 −0.118124
\(865\) 0 0
\(866\) −29.5066 −1.00267
\(867\) 5.50658i 0.187013i
\(868\) − 3.38197i − 0.114791i
\(869\) −40.3607 −1.36914
\(870\) 0 0
\(871\) 22.6099 0.766107
\(872\) − 8.56231i − 0.289956i
\(873\) − 38.2705i − 1.29526i
\(874\) 4.85410 0.164192
\(875\) 0 0
\(876\) 0.763932 0.0258109
\(877\) 23.7426i 0.801732i 0.916137 + 0.400866i \(0.131291\pi\)
−0.916137 + 0.400866i \(0.868709\pi\)
\(878\) − 15.6180i − 0.527083i
\(879\) 9.81966 0.331209
\(880\) 0 0
\(881\) 35.4164 1.19321 0.596605 0.802535i \(-0.296516\pi\)
0.596605 + 0.802535i \(0.296516\pi\)
\(882\) − 11.4721i − 0.386287i
\(883\) − 4.56231i − 0.153534i −0.997049 0.0767669i \(-0.975540\pi\)
0.997049 0.0767669i \(-0.0244597\pi\)
\(884\) 20.8197 0.700241
\(885\) 0 0
\(886\) 13.9098 0.467310
\(887\) 58.8328i 1.97541i 0.156321 + 0.987706i \(0.450037\pi\)
−0.156321 + 0.987706i \(0.549963\pi\)
\(888\) 1.52786i 0.0512718i
\(889\) 10.0000 0.335389
\(890\) 0 0
\(891\) 22.0000 0.737028
\(892\) 3.05573i 0.102313i
\(893\) − 47.1246i − 1.57697i
\(894\) −0.708204 −0.0236859
\(895\) 0 0
\(896\) 1.61803 0.0540547
\(897\) 2.52786i 0.0844029i
\(898\) 18.5623i 0.619432i
\(899\) −9.95743 −0.332099
\(900\) 0 0
\(901\) 43.1246 1.43669
\(902\) 47.5066i 1.58180i
\(903\) 0 0
\(904\) −18.9443 −0.630077
\(905\) 0 0
\(906\) −10.8541 −0.360603
\(907\) 33.1246i 1.09988i 0.835203 + 0.549942i \(0.185350\pi\)
−0.835203 + 0.549942i \(0.814650\pi\)
\(908\) − 23.2361i − 0.771116i
\(909\) −35.8885 −1.19035
\(910\) 0 0
\(911\) −22.0689 −0.731175 −0.365587 0.930777i \(-0.619132\pi\)
−0.365587 + 0.930777i \(0.619132\pi\)
\(912\) − 3.00000i − 0.0993399i
\(913\) 42.1803i 1.39597i
\(914\) −33.7771 −1.11725
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 24.1803i 0.798505i
\(918\) − 17.6738i − 0.583321i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −16.9656 −0.559034
\(922\) 34.7639i 1.14489i
\(923\) 29.0000i 0.954547i
\(924\) 3.85410 0.126791
\(925\) 0 0
\(926\) 2.00000 0.0657241
\(927\) − 9.32624i − 0.306314i
\(928\) − 4.76393i − 0.156384i
\(929\) 12.4721 0.409198 0.204599 0.978846i \(-0.434411\pi\)
0.204599 + 0.978846i \(0.434411\pi\)
\(930\) 0 0
\(931\) 21.2705 0.697113
\(932\) − 19.7082i − 0.645564i
\(933\) − 2.47214i − 0.0809341i
\(934\) −23.1246 −0.756660
\(935\) 0 0
\(936\) −10.7082 −0.350009
\(937\) 12.2016i 0.398610i 0.979938 + 0.199305i \(0.0638684\pi\)
−0.979938 + 0.199305i \(0.936132\pi\)
\(938\) 8.94427i 0.292041i
\(939\) −7.29180 −0.237959
\(940\) 0 0
\(941\) −60.5066 −1.97246 −0.986229 0.165385i \(-0.947113\pi\)
−0.986229 + 0.165385i \(0.947113\pi\)
\(942\) 6.00000i 0.195491i
\(943\) − 12.3262i − 0.401398i
\(944\) 11.7082 0.381070
\(945\) 0 0
\(946\) 0 0
\(947\) − 5.68692i − 0.184800i −0.995722 0.0924000i \(-0.970546\pi\)
0.995722 0.0924000i \(-0.0294538\pi\)
\(948\) − 6.47214i − 0.210205i
\(949\) 5.05573 0.164116
\(950\) 0 0
\(951\) −0.0557281 −0.00180711
\(952\) 8.23607i 0.266932i
\(953\) − 20.7984i − 0.673725i −0.941554 0.336863i \(-0.890634\pi\)
0.941554 0.336863i \(-0.109366\pi\)
\(954\) −22.1803 −0.718115
\(955\) 0 0
\(956\) 24.3607 0.787881
\(957\) − 11.3475i − 0.366813i
\(958\) 3.88854i 0.125633i
\(959\) −8.61803 −0.278291
\(960\) 0 0
\(961\) −26.6312 −0.859071
\(962\) 10.1115i 0.326006i
\(963\) 10.9443i 0.352674i
\(964\) 0 0
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) − 50.5410i − 1.62529i −0.582759 0.812645i \(-0.698027\pi\)
0.582759 0.812645i \(-0.301973\pi\)
\(968\) 3.85410i 0.123876i
\(969\) 15.2705 0.490559
\(970\) 0 0
\(971\) 0.729490 0.0234105 0.0117052 0.999931i \(-0.496274\pi\)
0.0117052 + 0.999931i \(0.496274\pi\)
\(972\) 13.9443i 0.447263i
\(973\) 27.8885i 0.894066i
\(974\) 42.1803 1.35155
\(975\) 0 0
\(976\) 6.32624 0.202498
\(977\) 3.43769i 0.109982i 0.998487 + 0.0549908i \(0.0175129\pi\)
−0.998487 + 0.0549908i \(0.982487\pi\)
\(978\) − 2.23607i − 0.0715016i
\(979\) 5.88854 0.188199
\(980\) 0 0
\(981\) −22.4164 −0.715701
\(982\) 16.1803i 0.516335i
\(983\) 19.2705i 0.614634i 0.951607 + 0.307317i \(0.0994312\pi\)
−0.951607 + 0.307317i \(0.900569\pi\)
\(984\) −7.61803 −0.242854
\(985\) 0 0
\(986\) 24.2492 0.772253
\(987\) 9.70820i 0.309016i
\(988\) − 19.8541i − 0.631643i
\(989\) 0 0
\(990\) 0 0
\(991\) −10.5066 −0.333752 −0.166876 0.985978i \(-0.553368\pi\)
−0.166876 + 0.985978i \(0.553368\pi\)
\(992\) 2.09017i 0.0663630i
\(993\) − 9.12461i − 0.289561i
\(994\) −11.4721 −0.363874
\(995\) 0 0
\(996\) −6.76393 −0.214323
\(997\) 41.1935i 1.30461i 0.757956 + 0.652306i \(0.226198\pi\)
−0.757956 + 0.652306i \(0.773802\pi\)
\(998\) − 32.3607i − 1.02436i
\(999\) 8.58359 0.271573
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 1150.2.b.i.599.1 4
5.2 odd 4 230.2.a.c.1.1 2
5.3 odd 4 1150.2.a.j.1.2 2
5.4 even 2 inner 1150.2.b.i.599.4 4
15.2 even 4 2070.2.a.u.1.2 2
20.3 even 4 9200.2.a.bu.1.1 2
20.7 even 4 1840.2.a.l.1.2 2
40.27 even 4 7360.2.a.bn.1.1 2
40.37 odd 4 7360.2.a.bh.1.2 2
115.22 even 4 5290.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.1 2 5.2 odd 4
1150.2.a.j.1.2 2 5.3 odd 4
1150.2.b.i.599.1 4 1.1 even 1 trivial
1150.2.b.i.599.4 4 5.4 even 2 inner
1840.2.a.l.1.2 2 20.7 even 4
2070.2.a.u.1.2 2 15.2 even 4
5290.2.a.o.1.1 2 115.22 even 4
7360.2.a.bh.1.2 2 40.37 odd 4
7360.2.a.bn.1.1 2 40.27 even 4
9200.2.a.bu.1.1 2 20.3 even 4