Properties

Label 4012.2.b.b.237.13
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.13
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.34

$q$-expansion

\(f(q)\) \(=\) \(q-1.58213i q^{3} -3.58424i q^{5} -1.89303i q^{7} +0.496852 q^{9} +O(q^{10})\) \(q-1.58213i q^{3} -3.58424i q^{5} -1.89303i q^{7} +0.496852 q^{9} +1.56283i q^{11} +4.45085 q^{13} -5.67074 q^{15} +(4.09490 + 0.481483i) q^{17} -5.20490 q^{19} -2.99502 q^{21} +3.19080i q^{23} -7.84674 q^{25} -5.53249i q^{27} -5.05381i q^{29} -3.34568i q^{31} +2.47260 q^{33} -6.78505 q^{35} -10.4521i q^{37} -7.04183i q^{39} +9.35341i q^{41} -7.48993 q^{43} -1.78083i q^{45} -8.79104 q^{47} +3.41645 q^{49} +(0.761771 - 6.47867i) q^{51} -2.62104 q^{53} +5.60154 q^{55} +8.23485i q^{57} +1.00000 q^{59} -12.4686i q^{61} -0.940554i q^{63} -15.9529i q^{65} +1.37777 q^{67} +5.04827 q^{69} -9.38491i q^{71} +0.343768i q^{73} +12.4146i q^{75} +2.95847 q^{77} -0.0938078i q^{79} -7.26258 q^{81} -5.61501 q^{83} +(1.72575 - 14.6771i) q^{85} -7.99581 q^{87} +7.62341 q^{89} -8.42557i q^{91} -5.29331 q^{93} +18.6556i q^{95} +17.3041i q^{97} +0.776494i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.58213i 0.913446i −0.889609 0.456723i \(-0.849023\pi\)
0.889609 0.456723i \(-0.150977\pi\)
\(4\) 0 0
\(5\) 3.58424i 1.60292i −0.598049 0.801459i \(-0.704057\pi\)
0.598049 0.801459i \(-0.295943\pi\)
\(6\) 0 0
\(7\) 1.89303i 0.715497i −0.933818 0.357748i \(-0.883545\pi\)
0.933818 0.357748i \(-0.116455\pi\)
\(8\) 0 0
\(9\) 0.496852 0.165617
\(10\) 0 0
\(11\) 1.56283i 0.471210i 0.971849 + 0.235605i \(0.0757072\pi\)
−0.971849 + 0.235605i \(0.924293\pi\)
\(12\) 0 0
\(13\) 4.45085 1.23444 0.617221 0.786790i \(-0.288258\pi\)
0.617221 + 0.786790i \(0.288258\pi\)
\(14\) 0 0
\(15\) −5.67074 −1.46418
\(16\) 0 0
\(17\) 4.09490 + 0.481483i 0.993158 + 0.116777i
\(18\) 0 0
\(19\) −5.20490 −1.19409 −0.597043 0.802209i \(-0.703658\pi\)
−0.597043 + 0.802209i \(0.703658\pi\)
\(20\) 0 0
\(21\) −2.99502 −0.653567
\(22\) 0 0
\(23\) 3.19080i 0.665327i 0.943046 + 0.332663i \(0.107947\pi\)
−0.943046 + 0.332663i \(0.892053\pi\)
\(24\) 0 0
\(25\) −7.84674 −1.56935
\(26\) 0 0
\(27\) 5.53249i 1.06473i
\(28\) 0 0
\(29\) 5.05381i 0.938469i −0.883073 0.469235i \(-0.844530\pi\)
0.883073 0.469235i \(-0.155470\pi\)
\(30\) 0 0
\(31\) 3.34568i 0.600901i −0.953797 0.300451i \(-0.902863\pi\)
0.953797 0.300451i \(-0.0971371\pi\)
\(32\) 0 0
\(33\) 2.47260 0.430425
\(34\) 0 0
\(35\) −6.78505 −1.14688
\(36\) 0 0
\(37\) 10.4521i 1.71832i −0.511708 0.859159i \(-0.670987\pi\)
0.511708 0.859159i \(-0.329013\pi\)
\(38\) 0 0
\(39\) 7.04183i 1.12760i
\(40\) 0 0
\(41\) 9.35341i 1.46076i 0.683042 + 0.730379i \(0.260656\pi\)
−0.683042 + 0.730379i \(0.739344\pi\)
\(42\) 0 0
\(43\) −7.48993 −1.14220 −0.571102 0.820879i \(-0.693484\pi\)
−0.571102 + 0.820879i \(0.693484\pi\)
\(44\) 0 0
\(45\) 1.78083i 0.265471i
\(46\) 0 0
\(47\) −8.79104 −1.28230 −0.641152 0.767413i \(-0.721543\pi\)
−0.641152 + 0.767413i \(0.721543\pi\)
\(48\) 0 0
\(49\) 3.41645 0.488064
\(50\) 0 0
\(51\) 0.761771 6.47867i 0.106669 0.907196i
\(52\) 0 0
\(53\) −2.62104 −0.360027 −0.180014 0.983664i \(-0.557614\pi\)
−0.180014 + 0.983664i \(0.557614\pi\)
\(54\) 0 0
\(55\) 5.60154 0.755312
\(56\) 0 0
\(57\) 8.23485i 1.09073i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 12.4686i 1.59644i −0.602365 0.798221i \(-0.705775\pi\)
0.602365 0.798221i \(-0.294225\pi\)
\(62\) 0 0
\(63\) 0.940554i 0.118499i
\(64\) 0 0
\(65\) 15.9529i 1.97871i
\(66\) 0 0
\(67\) 1.37777 0.168322 0.0841610 0.996452i \(-0.473179\pi\)
0.0841610 + 0.996452i \(0.473179\pi\)
\(68\) 0 0
\(69\) 5.04827 0.607740
\(70\) 0 0
\(71\) 9.38491i 1.11378i −0.830585 0.556892i \(-0.811994\pi\)
0.830585 0.556892i \(-0.188006\pi\)
\(72\) 0 0
\(73\) 0.343768i 0.0402350i 0.999798 + 0.0201175i \(0.00640403\pi\)
−0.999798 + 0.0201175i \(0.993596\pi\)
\(74\) 0 0
\(75\) 12.4146i 1.43351i
\(76\) 0 0
\(77\) 2.95847 0.337149
\(78\) 0 0
\(79\) 0.0938078i 0.0105542i −0.999986 0.00527710i \(-0.998320\pi\)
0.999986 0.00527710i \(-0.00167976\pi\)
\(80\) 0 0
\(81\) −7.26258 −0.806954
\(82\) 0 0
\(83\) −5.61501 −0.616327 −0.308164 0.951333i \(-0.599714\pi\)
−0.308164 + 0.951333i \(0.599714\pi\)
\(84\) 0 0
\(85\) 1.72575 14.6771i 0.187184 1.59195i
\(86\) 0 0
\(87\) −7.99581 −0.857241
\(88\) 0 0
\(89\) 7.62341 0.808080 0.404040 0.914741i \(-0.367606\pi\)
0.404040 + 0.914741i \(0.367606\pi\)
\(90\) 0 0
\(91\) 8.42557i 0.883240i
\(92\) 0 0
\(93\) −5.29331 −0.548891
\(94\) 0 0
\(95\) 18.6556i 1.91402i
\(96\) 0 0
\(97\) 17.3041i 1.75696i 0.477778 + 0.878481i \(0.341442\pi\)
−0.477778 + 0.878481i \(0.658558\pi\)
\(98\) 0 0
\(99\) 0.776494i 0.0780406i
\(100\) 0 0
\(101\) −6.25828 −0.622722 −0.311361 0.950292i \(-0.600785\pi\)
−0.311361 + 0.950292i \(0.600785\pi\)
\(102\) 0 0
\(103\) −5.23906 −0.516220 −0.258110 0.966116i \(-0.583100\pi\)
−0.258110 + 0.966116i \(0.583100\pi\)
\(104\) 0 0
\(105\) 10.7349i 1.04762i
\(106\) 0 0
\(107\) 15.5877i 1.50692i −0.657493 0.753461i \(-0.728383\pi\)
0.657493 0.753461i \(-0.271617\pi\)
\(108\) 0 0
\(109\) 10.3432i 0.990702i 0.868693 + 0.495351i \(0.164960\pi\)
−0.868693 + 0.495351i \(0.835040\pi\)
\(110\) 0 0
\(111\) −16.5367 −1.56959
\(112\) 0 0
\(113\) 2.39401i 0.225209i 0.993640 + 0.112605i \(0.0359193\pi\)
−0.993640 + 0.112605i \(0.964081\pi\)
\(114\) 0 0
\(115\) 11.4366 1.06646
\(116\) 0 0
\(117\) 2.21141 0.204445
\(118\) 0 0
\(119\) 0.911460 7.75175i 0.0835534 0.710602i
\(120\) 0 0
\(121\) 8.55757 0.777961
\(122\) 0 0
\(123\) 14.7984 1.33432
\(124\) 0 0
\(125\) 10.2034i 0.912619i
\(126\) 0 0
\(127\) 5.00038 0.443712 0.221856 0.975079i \(-0.428789\pi\)
0.221856 + 0.975079i \(0.428789\pi\)
\(128\) 0 0
\(129\) 11.8501i 1.04334i
\(130\) 0 0
\(131\) 12.3002i 1.07467i 0.843368 + 0.537336i \(0.180569\pi\)
−0.843368 + 0.537336i \(0.819431\pi\)
\(132\) 0 0
\(133\) 9.85302i 0.854365i
\(134\) 0 0
\(135\) −19.8297 −1.70667
\(136\) 0 0
\(137\) 10.8131 0.923826 0.461913 0.886925i \(-0.347163\pi\)
0.461913 + 0.886925i \(0.347163\pi\)
\(138\) 0 0
\(139\) 12.0120i 1.01884i 0.860517 + 0.509422i \(0.170141\pi\)
−0.860517 + 0.509422i \(0.829859\pi\)
\(140\) 0 0
\(141\) 13.9086i 1.17132i
\(142\) 0 0
\(143\) 6.95590i 0.581682i
\(144\) 0 0
\(145\) −18.1141 −1.50429
\(146\) 0 0
\(147\) 5.40528i 0.445820i
\(148\) 0 0
\(149\) −5.55621 −0.455183 −0.227591 0.973757i \(-0.573085\pi\)
−0.227591 + 0.973757i \(0.573085\pi\)
\(150\) 0 0
\(151\) 1.65649 0.134803 0.0674017 0.997726i \(-0.478529\pi\)
0.0674017 + 0.997726i \(0.478529\pi\)
\(152\) 0 0
\(153\) 2.03456 + 0.239226i 0.164484 + 0.0193403i
\(154\) 0 0
\(155\) −11.9917 −0.963196
\(156\) 0 0
\(157\) −23.3905 −1.86677 −0.933384 0.358879i \(-0.883159\pi\)
−0.933384 + 0.358879i \(0.883159\pi\)
\(158\) 0 0
\(159\) 4.14683i 0.328865i
\(160\) 0 0
\(161\) 6.04026 0.476039
\(162\) 0 0
\(163\) 16.0466i 1.25687i 0.777863 + 0.628434i \(0.216304\pi\)
−0.777863 + 0.628434i \(0.783696\pi\)
\(164\) 0 0
\(165\) 8.86239i 0.689936i
\(166\) 0 0
\(167\) 15.2908i 1.18324i −0.806218 0.591619i \(-0.798489\pi\)
0.806218 0.591619i \(-0.201511\pi\)
\(168\) 0 0
\(169\) 6.81003 0.523848
\(170\) 0 0
\(171\) −2.58607 −0.197761
\(172\) 0 0
\(173\) 18.0988i 1.37602i −0.725699 0.688012i \(-0.758484\pi\)
0.725699 0.688012i \(-0.241516\pi\)
\(174\) 0 0
\(175\) 14.8541i 1.12286i
\(176\) 0 0
\(177\) 1.58213i 0.118920i
\(178\) 0 0
\(179\) −3.81731 −0.285319 −0.142660 0.989772i \(-0.545565\pi\)
−0.142660 + 0.989772i \(0.545565\pi\)
\(180\) 0 0
\(181\) 14.6812i 1.09125i −0.838030 0.545623i \(-0.816293\pi\)
0.838030 0.545623i \(-0.183707\pi\)
\(182\) 0 0
\(183\) −19.7270 −1.45826
\(184\) 0 0
\(185\) −37.4629 −2.75433
\(186\) 0 0
\(187\) −0.752475 + 6.39961i −0.0550264 + 0.467986i
\(188\) 0 0
\(189\) −10.4731 −0.761810
\(190\) 0 0
\(191\) 24.0400 1.73947 0.869736 0.493516i \(-0.164289\pi\)
0.869736 + 0.493516i \(0.164289\pi\)
\(192\) 0 0
\(193\) 7.17007i 0.516113i −0.966130 0.258056i \(-0.916918\pi\)
0.966130 0.258056i \(-0.0830820\pi\)
\(194\) 0 0
\(195\) −25.2396 −1.80744
\(196\) 0 0
\(197\) 26.2724i 1.87183i 0.352223 + 0.935916i \(0.385426\pi\)
−0.352223 + 0.935916i \(0.614574\pi\)
\(198\) 0 0
\(199\) 18.6922i 1.32505i 0.749039 + 0.662526i \(0.230516\pi\)
−0.749039 + 0.662526i \(0.769484\pi\)
\(200\) 0 0
\(201\) 2.17982i 0.153753i
\(202\) 0 0
\(203\) −9.56700 −0.671472
\(204\) 0 0
\(205\) 33.5248 2.34148
\(206\) 0 0
\(207\) 1.58535i 0.110190i
\(208\) 0 0
\(209\) 8.13436i 0.562666i
\(210\) 0 0
\(211\) 3.31656i 0.228321i −0.993462 0.114161i \(-0.963582\pi\)
0.993462 0.114161i \(-0.0364179\pi\)
\(212\) 0 0
\(213\) −14.8482 −1.01738
\(214\) 0 0
\(215\) 26.8457i 1.83086i
\(216\) 0 0
\(217\) −6.33346 −0.429943
\(218\) 0 0
\(219\) 0.543887 0.0367525
\(220\) 0 0
\(221\) 18.2258 + 2.14301i 1.22600 + 0.144154i
\(222\) 0 0
\(223\) 23.9721 1.60529 0.802646 0.596455i \(-0.203425\pi\)
0.802646 + 0.596455i \(0.203425\pi\)
\(224\) 0 0
\(225\) −3.89867 −0.259911
\(226\) 0 0
\(227\) 5.60333i 0.371906i −0.982559 0.185953i \(-0.940463\pi\)
0.982559 0.185953i \(-0.0595373\pi\)
\(228\) 0 0
\(229\) −22.4728 −1.48504 −0.742522 0.669822i \(-0.766371\pi\)
−0.742522 + 0.669822i \(0.766371\pi\)
\(230\) 0 0
\(231\) 4.68070i 0.307968i
\(232\) 0 0
\(233\) 1.42422i 0.0933040i −0.998911 0.0466520i \(-0.985145\pi\)
0.998911 0.0466520i \(-0.0148552\pi\)
\(234\) 0 0
\(235\) 31.5092i 2.05543i
\(236\) 0 0
\(237\) −0.148416 −0.00964068
\(238\) 0 0
\(239\) 2.15838 0.139614 0.0698069 0.997561i \(-0.477762\pi\)
0.0698069 + 0.997561i \(0.477762\pi\)
\(240\) 0 0
\(241\) 3.58435i 0.230888i 0.993314 + 0.115444i \(0.0368292\pi\)
−0.993314 + 0.115444i \(0.963171\pi\)
\(242\) 0 0
\(243\) 5.10709i 0.327620i
\(244\) 0 0
\(245\) 12.2454i 0.782327i
\(246\) 0 0
\(247\) −23.1662 −1.47403
\(248\) 0 0
\(249\) 8.88369i 0.562981i
\(250\) 0 0
\(251\) 27.1042 1.71080 0.855402 0.517965i \(-0.173310\pi\)
0.855402 + 0.517965i \(0.173310\pi\)
\(252\) 0 0
\(253\) −4.98666 −0.313509
\(254\) 0 0
\(255\) −23.2211 2.73037i −1.45416 0.170982i
\(256\) 0 0
\(257\) 0.0133424 0.000832275 0.000416138 1.00000i \(-0.499868\pi\)
0.000416138 1.00000i \(0.499868\pi\)
\(258\) 0 0
\(259\) −19.7862 −1.22945
\(260\) 0 0
\(261\) 2.51100i 0.155427i
\(262\) 0 0
\(263\) 24.6287 1.51867 0.759336 0.650699i \(-0.225524\pi\)
0.759336 + 0.650699i \(0.225524\pi\)
\(264\) 0 0
\(265\) 9.39441i 0.577094i
\(266\) 0 0
\(267\) 12.0613i 0.738137i
\(268\) 0 0
\(269\) 20.8960i 1.27405i 0.770842 + 0.637026i \(0.219836\pi\)
−0.770842 + 0.637026i \(0.780164\pi\)
\(270\) 0 0
\(271\) 4.90014 0.297662 0.148831 0.988863i \(-0.452449\pi\)
0.148831 + 0.988863i \(0.452449\pi\)
\(272\) 0 0
\(273\) −13.3304 −0.806791
\(274\) 0 0
\(275\) 12.2631i 0.739493i
\(276\) 0 0
\(277\) 6.82078i 0.409821i 0.978781 + 0.204911i \(0.0656904\pi\)
−0.978781 + 0.204911i \(0.934310\pi\)
\(278\) 0 0
\(279\) 1.66231i 0.0995196i
\(280\) 0 0
\(281\) 13.6085 0.811813 0.405907 0.913915i \(-0.366956\pi\)
0.405907 + 0.913915i \(0.366956\pi\)
\(282\) 0 0
\(283\) 0.0157633i 0.000937032i 1.00000 0.000468516i \(0.000149133\pi\)
−1.00000 0.000468516i \(0.999851\pi\)
\(284\) 0 0
\(285\) 29.5157 1.74836
\(286\) 0 0
\(287\) 17.7063 1.04517
\(288\) 0 0
\(289\) 16.5363 + 3.94325i 0.972726 + 0.231956i
\(290\) 0 0
\(291\) 27.3774 1.60489
\(292\) 0 0
\(293\) 19.9213 1.16382 0.581909 0.813254i \(-0.302306\pi\)
0.581909 + 0.813254i \(0.302306\pi\)
\(294\) 0 0
\(295\) 3.58424i 0.208682i
\(296\) 0 0
\(297\) 8.64632 0.501711
\(298\) 0 0
\(299\) 14.2017i 0.821308i
\(300\) 0 0
\(301\) 14.1786i 0.817244i
\(302\) 0 0
\(303\) 9.90143i 0.568822i
\(304\) 0 0
\(305\) −44.6904 −2.55897
\(306\) 0 0
\(307\) −26.9494 −1.53808 −0.769041 0.639200i \(-0.779266\pi\)
−0.769041 + 0.639200i \(0.779266\pi\)
\(308\) 0 0
\(309\) 8.28890i 0.471539i
\(310\) 0 0
\(311\) 26.3066i 1.49171i −0.666107 0.745856i \(-0.732041\pi\)
0.666107 0.745856i \(-0.267959\pi\)
\(312\) 0 0
\(313\) 4.40205i 0.248819i −0.992231 0.124409i \(-0.960296\pi\)
0.992231 0.124409i \(-0.0397036\pi\)
\(314\) 0 0
\(315\) −3.37117 −0.189944
\(316\) 0 0
\(317\) 31.6804i 1.77935i −0.456596 0.889674i \(-0.650931\pi\)
0.456596 0.889674i \(-0.349069\pi\)
\(318\) 0 0
\(319\) 7.89824 0.442216
\(320\) 0 0
\(321\) −24.6619 −1.37649
\(322\) 0 0
\(323\) −21.3135 2.50607i −1.18592 0.139442i
\(324\) 0 0
\(325\) −34.9246 −1.93727
\(326\) 0 0
\(327\) 16.3644 0.904952
\(328\) 0 0
\(329\) 16.6417i 0.917485i
\(330\) 0 0
\(331\) 2.43983 0.134105 0.0670526 0.997749i \(-0.478640\pi\)
0.0670526 + 0.997749i \(0.478640\pi\)
\(332\) 0 0
\(333\) 5.19316i 0.284583i
\(334\) 0 0
\(335\) 4.93827i 0.269806i
\(336\) 0 0
\(337\) 6.74925i 0.367655i 0.982959 + 0.183827i \(0.0588488\pi\)
−0.982959 + 0.183827i \(0.941151\pi\)
\(338\) 0 0
\(339\) 3.78764 0.205716
\(340\) 0 0
\(341\) 5.22871 0.283151
\(342\) 0 0
\(343\) 19.7186i 1.06471i
\(344\) 0 0
\(345\) 18.0942i 0.974158i
\(346\) 0 0
\(347\) 9.92260i 0.532673i −0.963880 0.266336i \(-0.914187\pi\)
0.963880 0.266336i \(-0.0858132\pi\)
\(348\) 0 0
\(349\) 5.02529 0.268997 0.134499 0.990914i \(-0.457058\pi\)
0.134499 + 0.990914i \(0.457058\pi\)
\(350\) 0 0
\(351\) 24.6243i 1.31435i
\(352\) 0 0
\(353\) 17.5183 0.932404 0.466202 0.884678i \(-0.345622\pi\)
0.466202 + 0.884678i \(0.345622\pi\)
\(354\) 0 0
\(355\) −33.6377 −1.78531
\(356\) 0 0
\(357\) −12.2643 1.44205i −0.649096 0.0763215i
\(358\) 0 0
\(359\) 21.3101 1.12471 0.562353 0.826897i \(-0.309896\pi\)
0.562353 + 0.826897i \(0.309896\pi\)
\(360\) 0 0
\(361\) 8.09101 0.425843
\(362\) 0 0
\(363\) 13.5392i 0.710625i
\(364\) 0 0
\(365\) 1.23215 0.0644934
\(366\) 0 0
\(367\) 11.3182i 0.590808i 0.955372 + 0.295404i \(0.0954542\pi\)
−0.955372 + 0.295404i \(0.904546\pi\)
\(368\) 0 0
\(369\) 4.64726i 0.241927i
\(370\) 0 0
\(371\) 4.96169i 0.257598i
\(372\) 0 0
\(373\) −27.2356 −1.41021 −0.705104 0.709104i \(-0.749100\pi\)
−0.705104 + 0.709104i \(0.749100\pi\)
\(374\) 0 0
\(375\) 16.1431 0.833628
\(376\) 0 0
\(377\) 22.4937i 1.15849i
\(378\) 0 0
\(379\) 15.8756i 0.815473i 0.913100 + 0.407737i \(0.133682\pi\)
−0.913100 + 0.407737i \(0.866318\pi\)
\(380\) 0 0
\(381\) 7.91127i 0.405307i
\(382\) 0 0
\(383\) −31.5223 −1.61072 −0.805358 0.592788i \(-0.798027\pi\)
−0.805358 + 0.592788i \(0.798027\pi\)
\(384\) 0 0
\(385\) 10.6039i 0.540423i
\(386\) 0 0
\(387\) −3.72139 −0.189169
\(388\) 0 0
\(389\) 0.593284 0.0300807 0.0150403 0.999887i \(-0.495212\pi\)
0.0150403 + 0.999887i \(0.495212\pi\)
\(390\) 0 0
\(391\) −1.53631 + 13.0660i −0.0776947 + 0.660775i
\(392\) 0 0
\(393\) 19.4605 0.981654
\(394\) 0 0
\(395\) −0.336229 −0.0169175
\(396\) 0 0
\(397\) 23.0032i 1.15450i −0.816568 0.577249i \(-0.804126\pi\)
0.816568 0.577249i \(-0.195874\pi\)
\(398\) 0 0
\(399\) 15.5888 0.780416
\(400\) 0 0
\(401\) 11.7157i 0.585052i 0.956258 + 0.292526i \(0.0944959\pi\)
−0.956258 + 0.292526i \(0.905504\pi\)
\(402\) 0 0
\(403\) 14.8911i 0.741778i
\(404\) 0 0
\(405\) 26.0308i 1.29348i
\(406\) 0 0
\(407\) 16.3349 0.809689
\(408\) 0 0
\(409\) 19.8877 0.983383 0.491692 0.870769i \(-0.336379\pi\)
0.491692 + 0.870769i \(0.336379\pi\)
\(410\) 0 0
\(411\) 17.1078i 0.843865i
\(412\) 0 0
\(413\) 1.89303i 0.0931498i
\(414\) 0 0
\(415\) 20.1255i 0.987922i
\(416\) 0 0
\(417\) 19.0046 0.930658
\(418\) 0 0
\(419\) 13.1372i 0.641796i −0.947114 0.320898i \(-0.896015\pi\)
0.947114 0.320898i \(-0.103985\pi\)
\(420\) 0 0
\(421\) −16.9753 −0.827326 −0.413663 0.910430i \(-0.635751\pi\)
−0.413663 + 0.910430i \(0.635751\pi\)
\(422\) 0 0
\(423\) −4.36785 −0.212372
\(424\) 0 0
\(425\) −32.1316 3.77807i −1.55861 0.183263i
\(426\) 0 0
\(427\) −23.6034 −1.14225
\(428\) 0 0
\(429\) 11.0052 0.531335
\(430\) 0 0
\(431\) 25.5419i 1.23031i 0.788407 + 0.615155i \(0.210906\pi\)
−0.788407 + 0.615155i \(0.789094\pi\)
\(432\) 0 0
\(433\) −9.83130 −0.472462 −0.236231 0.971697i \(-0.575912\pi\)
−0.236231 + 0.971697i \(0.575912\pi\)
\(434\) 0 0
\(435\) 28.6589i 1.37409i
\(436\) 0 0
\(437\) 16.6078i 0.794458i
\(438\) 0 0
\(439\) 2.76931i 0.132172i −0.997814 0.0660859i \(-0.978949\pi\)
0.997814 0.0660859i \(-0.0210511\pi\)
\(440\) 0 0
\(441\) 1.69747 0.0808319
\(442\) 0 0
\(443\) 12.6505 0.601046 0.300523 0.953775i \(-0.402839\pi\)
0.300523 + 0.953775i \(0.402839\pi\)
\(444\) 0 0
\(445\) 27.3241i 1.29529i
\(446\) 0 0
\(447\) 8.79067i 0.415784i
\(448\) 0 0
\(449\) 27.6353i 1.30419i −0.758136 0.652096i \(-0.773890\pi\)
0.758136 0.652096i \(-0.226110\pi\)
\(450\) 0 0
\(451\) −14.6178 −0.688324
\(452\) 0 0
\(453\) 2.62079i 0.123136i
\(454\) 0 0
\(455\) −30.1992 −1.41576
\(456\) 0 0
\(457\) 6.73621 0.315106 0.157553 0.987510i \(-0.449639\pi\)
0.157553 + 0.987510i \(0.449639\pi\)
\(458\) 0 0
\(459\) 2.66380 22.6550i 0.124335 1.05744i
\(460\) 0 0
\(461\) 35.4289 1.65009 0.825044 0.565068i \(-0.191150\pi\)
0.825044 + 0.565068i \(0.191150\pi\)
\(462\) 0 0
\(463\) −30.8991 −1.43600 −0.718001 0.696042i \(-0.754943\pi\)
−0.718001 + 0.696042i \(0.754943\pi\)
\(464\) 0 0
\(465\) 18.9725i 0.879827i
\(466\) 0 0
\(467\) −42.9151 −1.98588 −0.992938 0.118635i \(-0.962148\pi\)
−0.992938 + 0.118635i \(0.962148\pi\)
\(468\) 0 0
\(469\) 2.60816i 0.120434i
\(470\) 0 0
\(471\) 37.0070i 1.70519i
\(472\) 0 0
\(473\) 11.7055i 0.538218i
\(474\) 0 0
\(475\) 40.8415 1.87394
\(476\) 0 0
\(477\) −1.30227 −0.0596267
\(478\) 0 0
\(479\) 19.0325i 0.869619i −0.900522 0.434810i \(-0.856816\pi\)
0.900522 0.434810i \(-0.143184\pi\)
\(480\) 0 0
\(481\) 46.5208i 2.12117i
\(482\) 0 0
\(483\) 9.55650i 0.434836i
\(484\) 0 0
\(485\) 62.0218 2.81627
\(486\) 0 0
\(487\) 12.8292i 0.581346i −0.956823 0.290673i \(-0.906121\pi\)
0.956823 0.290673i \(-0.0938791\pi\)
\(488\) 0 0
\(489\) 25.3879 1.14808
\(490\) 0 0
\(491\) 41.6194 1.87826 0.939128 0.343568i \(-0.111636\pi\)
0.939128 + 0.343568i \(0.111636\pi\)
\(492\) 0 0
\(493\) 2.43332 20.6948i 0.109591 0.932049i
\(494\) 0 0
\(495\) 2.78314 0.125093
\(496\) 0 0
\(497\) −17.7659 −0.796909
\(498\) 0 0
\(499\) 13.1730i 0.589704i 0.955543 + 0.294852i \(0.0952703\pi\)
−0.955543 + 0.294852i \(0.904730\pi\)
\(500\) 0 0
\(501\) −24.1921 −1.08082
\(502\) 0 0
\(503\) 19.1004i 0.851645i −0.904807 0.425822i \(-0.859985\pi\)
0.904807 0.425822i \(-0.140015\pi\)
\(504\) 0 0
\(505\) 22.4311i 0.998172i
\(506\) 0 0
\(507\) 10.7744i 0.478507i
\(508\) 0 0
\(509\) 7.61049 0.337329 0.168665 0.985674i \(-0.446055\pi\)
0.168665 + 0.985674i \(0.446055\pi\)
\(510\) 0 0
\(511\) 0.650762 0.0287880
\(512\) 0 0
\(513\) 28.7961i 1.27138i
\(514\) 0 0
\(515\) 18.7780i 0.827459i
\(516\) 0 0
\(517\) 13.7389i 0.604235i
\(518\) 0 0
\(519\) −28.6347 −1.25692
\(520\) 0 0
\(521\) 12.2618i 0.537200i 0.963252 + 0.268600i \(0.0865609\pi\)
−0.963252 + 0.268600i \(0.913439\pi\)
\(522\) 0 0
\(523\) 12.8885 0.563576 0.281788 0.959477i \(-0.409073\pi\)
0.281788 + 0.959477i \(0.409073\pi\)
\(524\) 0 0
\(525\) 23.5012 1.02567
\(526\) 0 0
\(527\) 1.61089 13.7002i 0.0701713 0.596790i
\(528\) 0 0
\(529\) 12.8188 0.557340
\(530\) 0 0
\(531\) 0.496852 0.0215615
\(532\) 0 0
\(533\) 41.6306i 1.80322i
\(534\) 0 0
\(535\) −55.8700 −2.41547
\(536\) 0 0
\(537\) 6.03949i 0.260623i
\(538\) 0 0
\(539\) 5.33932i 0.229981i
\(540\) 0 0
\(541\) 3.06145i 0.131622i 0.997832 + 0.0658110i \(0.0209634\pi\)
−0.997832 + 0.0658110i \(0.979037\pi\)
\(542\) 0 0
\(543\) −23.2277 −0.996795
\(544\) 0 0
\(545\) 37.0726 1.58801
\(546\) 0 0
\(547\) 11.3138i 0.483742i 0.970308 + 0.241871i \(0.0777611\pi\)
−0.970308 + 0.241871i \(0.922239\pi\)
\(548\) 0 0
\(549\) 6.19505i 0.264398i
\(550\) 0 0
\(551\) 26.3046i 1.12061i
\(552\) 0 0
\(553\) −0.177581 −0.00755150
\(554\) 0 0
\(555\) 59.2713i 2.51593i
\(556\) 0 0
\(557\) 16.2845 0.689997 0.344998 0.938603i \(-0.387879\pi\)
0.344998 + 0.938603i \(0.387879\pi\)
\(558\) 0 0
\(559\) −33.3365 −1.40999
\(560\) 0 0
\(561\) 10.1250 + 1.19052i 0.427480 + 0.0502636i
\(562\) 0 0
\(563\) 10.8913 0.459015 0.229507 0.973307i \(-0.426288\pi\)
0.229507 + 0.973307i \(0.426288\pi\)
\(564\) 0 0
\(565\) 8.58068 0.360992
\(566\) 0 0
\(567\) 13.7483i 0.577373i
\(568\) 0 0
\(569\) −5.94516 −0.249234 −0.124617 0.992205i \(-0.539770\pi\)
−0.124617 + 0.992205i \(0.539770\pi\)
\(570\) 0 0
\(571\) 33.2531i 1.39160i 0.718236 + 0.695800i \(0.244950\pi\)
−0.718236 + 0.695800i \(0.755050\pi\)
\(572\) 0 0
\(573\) 38.0345i 1.58891i
\(574\) 0 0
\(575\) 25.0373i 1.04413i
\(576\) 0 0
\(577\) 33.8801 1.41045 0.705223 0.708986i \(-0.250847\pi\)
0.705223 + 0.708986i \(0.250847\pi\)
\(578\) 0 0
\(579\) −11.3440 −0.471441
\(580\) 0 0
\(581\) 10.6294i 0.440980i
\(582\) 0 0
\(583\) 4.09623i 0.169648i
\(584\) 0 0
\(585\) 7.92622i 0.327709i
\(586\) 0 0
\(587\) −8.40852 −0.347057 −0.173528 0.984829i \(-0.555517\pi\)
−0.173528 + 0.984829i \(0.555517\pi\)
\(588\) 0 0
\(589\) 17.4139i 0.717528i
\(590\) 0 0
\(591\) 41.5665 1.70982
\(592\) 0 0
\(593\) 8.05925 0.330954 0.165477 0.986214i \(-0.447084\pi\)
0.165477 + 0.986214i \(0.447084\pi\)
\(594\) 0 0
\(595\) −27.7841 3.26689i −1.13904 0.133929i
\(596\) 0 0
\(597\) 29.5735 1.21036
\(598\) 0 0
\(599\) −12.7884 −0.522521 −0.261261 0.965268i \(-0.584138\pi\)
−0.261261 + 0.965268i \(0.584138\pi\)
\(600\) 0 0
\(601\) 27.7382i 1.13147i −0.824588 0.565733i \(-0.808593\pi\)
0.824588 0.565733i \(-0.191407\pi\)
\(602\) 0 0
\(603\) 0.684550 0.0278770
\(604\) 0 0
\(605\) 30.6723i 1.24701i
\(606\) 0 0
\(607\) 21.8395i 0.886437i 0.896414 + 0.443218i \(0.146163\pi\)
−0.896414 + 0.443218i \(0.853837\pi\)
\(608\) 0 0
\(609\) 15.1363i 0.613353i
\(610\) 0 0
\(611\) −39.1276 −1.58293
\(612\) 0 0
\(613\) −4.60258 −0.185896 −0.0929482 0.995671i \(-0.529629\pi\)
−0.0929482 + 0.995671i \(0.529629\pi\)
\(614\) 0 0
\(615\) 53.0408i 2.13881i
\(616\) 0 0
\(617\) 20.2849i 0.816638i 0.912839 + 0.408319i \(0.133885\pi\)
−0.912839 + 0.408319i \(0.866115\pi\)
\(618\) 0 0
\(619\) 13.1774i 0.529643i −0.964297 0.264822i \(-0.914687\pi\)
0.964297 0.264822i \(-0.0853131\pi\)
\(620\) 0 0
\(621\) 17.6530 0.708392
\(622\) 0 0
\(623\) 14.4313i 0.578179i
\(624\) 0 0
\(625\) −2.66235 −0.106494
\(626\) 0 0
\(627\) −12.8697 −0.513964
\(628\) 0 0
\(629\) 5.03252 42.8004i 0.200660 1.70656i
\(630\) 0 0
\(631\) −29.2829 −1.16573 −0.582867 0.812567i \(-0.698069\pi\)
−0.582867 + 0.812567i \(0.698069\pi\)
\(632\) 0 0
\(633\) −5.24724 −0.208559
\(634\) 0 0
\(635\) 17.9225i 0.711234i
\(636\) 0 0
\(637\) 15.2061 0.602487
\(638\) 0 0
\(639\) 4.66291i 0.184462i
\(640\) 0 0
\(641\) 4.14950i 0.163895i 0.996637 + 0.0819477i \(0.0261141\pi\)
−0.996637 + 0.0819477i \(0.973886\pi\)
\(642\) 0 0
\(643\) 0.693712i 0.0273573i −0.999906 0.0136787i \(-0.995646\pi\)
0.999906 0.0136787i \(-0.00435419\pi\)
\(644\) 0 0
\(645\) 42.4735 1.67239
\(646\) 0 0
\(647\) −42.9107 −1.68699 −0.843497 0.537133i \(-0.819507\pi\)
−0.843497 + 0.537133i \(0.819507\pi\)
\(648\) 0 0
\(649\) 1.56283i 0.0613463i
\(650\) 0 0
\(651\) 10.0204i 0.392729i
\(652\) 0 0
\(653\) 44.3141i 1.73415i 0.498182 + 0.867073i \(0.334001\pi\)
−0.498182 + 0.867073i \(0.665999\pi\)
\(654\) 0 0
\(655\) 44.0867 1.72261
\(656\) 0 0
\(657\) 0.170802i 0.00666361i
\(658\) 0 0
\(659\) −25.3683 −0.988208 −0.494104 0.869403i \(-0.664504\pi\)
−0.494104 + 0.869403i \(0.664504\pi\)
\(660\) 0 0
\(661\) 26.6011 1.03466 0.517331 0.855785i \(-0.326926\pi\)
0.517331 + 0.855785i \(0.326926\pi\)
\(662\) 0 0
\(663\) 3.39052 28.8356i 0.131677 1.11988i
\(664\) 0 0
\(665\) 35.3155 1.36948
\(666\) 0 0
\(667\) 16.1257 0.624389
\(668\) 0 0
\(669\) 37.9271i 1.46635i
\(670\) 0 0
\(671\) 19.4863 0.752260
\(672\) 0 0
\(673\) 23.3769i 0.901114i −0.892748 0.450557i \(-0.851225\pi\)
0.892748 0.450557i \(-0.148775\pi\)
\(674\) 0 0
\(675\) 43.4120i 1.67093i
\(676\) 0 0
\(677\) 5.91913i 0.227491i 0.993510 + 0.113745i \(0.0362848\pi\)
−0.993510 + 0.113745i \(0.963715\pi\)
\(678\) 0 0
\(679\) 32.7571 1.25710
\(680\) 0 0
\(681\) −8.86522 −0.339716
\(682\) 0 0
\(683\) 39.2331i 1.50121i −0.660749 0.750607i \(-0.729761\pi\)
0.660749 0.750607i \(-0.270239\pi\)
\(684\) 0 0
\(685\) 38.7567i 1.48082i
\(686\) 0 0
\(687\) 35.5550i 1.35651i
\(688\) 0 0
\(689\) −11.6658 −0.444433
\(690\) 0 0
\(691\) 0.796860i 0.0303140i 0.999885 + 0.0151570i \(0.00482481\pi\)
−0.999885 + 0.0151570i \(0.995175\pi\)
\(692\) 0 0
\(693\) 1.46992 0.0558378
\(694\) 0 0
\(695\) 43.0538 1.63312
\(696\) 0 0
\(697\) −4.50351 + 38.3013i −0.170583 + 1.45076i
\(698\) 0 0
\(699\) −2.25331 −0.0852281
\(700\) 0 0
\(701\) −51.4629 −1.94373 −0.971863 0.235547i \(-0.924312\pi\)
−0.971863 + 0.235547i \(0.924312\pi\)
\(702\) 0 0
\(703\) 54.4023i 2.05182i
\(704\) 0 0
\(705\) 49.8517 1.87752
\(706\) 0 0
\(707\) 11.8471i 0.445555i
\(708\) 0 0
\(709\) 9.03296i 0.339240i −0.985510 0.169620i \(-0.945746\pi\)
0.985510 0.169620i \(-0.0542540\pi\)
\(710\) 0 0
\(711\) 0.0466086i 0.00174796i
\(712\) 0 0
\(713\) 10.6754 0.399796
\(714\) 0 0
\(715\) 24.9316 0.932389
\(716\) 0 0
\(717\) 3.41484i 0.127530i
\(718\) 0 0
\(719\) 32.4318i 1.20950i −0.796414 0.604751i \(-0.793273\pi\)
0.796414 0.604751i \(-0.206727\pi\)
\(720\) 0 0
\(721\) 9.91768i 0.369354i
\(722\) 0 0
\(723\) 5.67093 0.210904
\(724\) 0 0
\(725\) 39.6560i 1.47279i
\(726\) 0 0
\(727\) −2.10920 −0.0782260 −0.0391130 0.999235i \(-0.512453\pi\)
−0.0391130 + 0.999235i \(0.512453\pi\)
\(728\) 0 0
\(729\) −29.8678 −1.10622
\(730\) 0 0
\(731\) −30.6705 3.60628i −1.13439 0.133383i
\(732\) 0 0
\(733\) 43.0998 1.59193 0.795965 0.605343i \(-0.206964\pi\)
0.795965 + 0.605343i \(0.206964\pi\)
\(734\) 0 0
\(735\) −19.3738 −0.714613
\(736\) 0 0
\(737\) 2.15322i 0.0793150i
\(738\) 0 0
\(739\) 20.3126 0.747213 0.373606 0.927587i \(-0.378121\pi\)
0.373606 + 0.927587i \(0.378121\pi\)
\(740\) 0 0
\(741\) 36.6521i 1.34645i
\(742\) 0 0
\(743\) 17.0484i 0.625444i −0.949845 0.312722i \(-0.898759\pi\)
0.949845 0.312722i \(-0.101241\pi\)
\(744\) 0 0
\(745\) 19.9148i 0.729621i
\(746\) 0 0
\(747\) −2.78983 −0.102074
\(748\) 0 0
\(749\) −29.5080 −1.07820
\(750\) 0 0
\(751\) 12.4080i 0.452774i −0.974037 0.226387i \(-0.927309\pi\)
0.974037 0.226387i \(-0.0726914\pi\)
\(752\) 0 0
\(753\) 42.8825i 1.56273i
\(754\) 0 0
\(755\) 5.93726i 0.216079i
\(756\) 0 0
\(757\) 44.1856 1.60595 0.802977 0.596010i \(-0.203248\pi\)
0.802977 + 0.596010i \(0.203248\pi\)
\(758\) 0 0
\(759\) 7.88957i 0.286373i
\(760\) 0 0
\(761\) −43.6006 −1.58052 −0.790260 0.612771i \(-0.790055\pi\)
−0.790260 + 0.612771i \(0.790055\pi\)
\(762\) 0 0
\(763\) 19.5800 0.708844
\(764\) 0 0
\(765\) 0.857441 7.29233i 0.0310009 0.263655i
\(766\) 0 0
\(767\) 4.45085 0.160711
\(768\) 0 0
\(769\) −34.3200 −1.23761 −0.618805 0.785545i \(-0.712383\pi\)
−0.618805 + 0.785545i \(0.712383\pi\)
\(770\) 0 0
\(771\) 0.0211094i 0.000760238i
\(772\) 0 0
\(773\) 17.7230 0.637453 0.318727 0.947847i \(-0.396745\pi\)
0.318727 + 0.947847i \(0.396745\pi\)
\(774\) 0 0
\(775\) 26.2527i 0.943023i
\(776\) 0 0
\(777\) 31.3043i 1.12304i
\(778\) 0 0
\(779\) 48.6836i 1.74427i
\(780\) 0 0
\(781\) 14.6670 0.524826
\(782\) 0 0
\(783\) −27.9602 −0.999215
\(784\) 0 0
\(785\) 83.8372i 2.99228i
\(786\) 0 0
\(787\) 11.9869i 0.427286i −0.976912 0.213643i \(-0.931467\pi\)
0.976912 0.213643i \(-0.0685330\pi\)
\(788\) 0 0
\(789\) 38.9659i 1.38722i
\(790\) 0 0
\(791\) 4.53192 0.161137
\(792\) 0 0
\(793\) 55.4959i 1.97072i