Properties

Label 280.2.g.b.169.3
Level $280$
Weight $2$
Character 280.169
Analytic conductor $2.236$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 280.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.23581125660\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 169.3
Root \(-1.75233 + 1.75233i\) of defining polynomial
Character \(\chi\) \(=\) 280.169
Dual form 280.2.g.b.169.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.363328i q^{3} +(-1.75233 + 1.38900i) q^{5} +1.00000i q^{7} +2.86799 q^{9} +O(q^{10})\) \(q-0.363328i q^{3} +(-1.75233 + 1.38900i) q^{5} +1.00000i q^{7} +2.86799 q^{9} +5.14134 q^{11} +4.64600i q^{13} +(0.504664 + 0.636672i) q^{15} +3.86799i q^{17} +0.778008 q^{19} +0.363328 q^{21} -5.00933i q^{23} +(1.14134 - 4.86799i) q^{25} -2.13201i q^{27} -9.42401 q^{29} +4.72666 q^{31} -1.86799i q^{33} +(-1.38900 - 1.75233i) q^{35} +6.00000i q^{37} +1.68802 q^{39} -1.00933 q^{41} +7.00933i q^{43} +(-5.02568 + 3.98365i) q^{45} -11.4240i q^{47} -1.00000 q^{49} +1.40535 q^{51} -7.55602i q^{53} +(-9.00933 + 7.14134i) q^{55} -0.282672i q^{57} -12.5140 q^{59} +11.5047 q^{61} +2.86799i q^{63} +(-6.45331 - 8.14134i) q^{65} -11.7360i q^{67} -1.82003 q^{69} +2.72666 q^{71} -5.00933i q^{73} +(-1.76868 - 0.414680i) q^{75} +5.14134i q^{77} -5.68802 q^{79} +7.82936 q^{81} +4.67531i q^{83} +(-5.37266 - 6.77801i) q^{85} +3.42401i q^{87} +2.82936 q^{89} -4.64600 q^{91} -1.71733i q^{93} +(-1.36333 + 1.08066i) q^{95} -1.58532i q^{97} +14.7453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 8q^{9} + O(q^{10}) \) \( 6q - 8q^{9} + 14q^{11} - 18q^{15} - 8q^{19} - 2q^{21} - 10q^{25} - 6q^{29} + 20q^{31} - 2q^{35} + 10q^{39} + 36q^{41} - 28q^{45} - 6q^{49} + 42q^{51} - 12q^{55} - 12q^{59} + 48q^{61} - 22q^{65} - 36q^{69} + 8q^{71} - 40q^{75} - 34q^{79} + 30q^{81} + 14q^{85} + 10q^{91} - 4q^{95} - 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(141\) \(241\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 0.363328i 0.209768i −0.994484 0.104884i \(-0.966553\pi\)
0.994484 0.104884i \(-0.0334471\pi\)
\(4\) 0 0
\(5\) −1.75233 + 1.38900i −0.783667 + 0.621181i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.86799 0.955998
\(10\) 0 0
\(11\) 5.14134 1.55017 0.775086 0.631856i \(-0.217707\pi\)
0.775086 + 0.631856i \(0.217707\pi\)
\(12\) 0 0
\(13\) 4.64600i 1.28857i 0.764786 + 0.644284i \(0.222845\pi\)
−0.764786 + 0.644284i \(0.777155\pi\)
\(14\) 0 0
\(15\) 0.504664 + 0.636672i 0.130304 + 0.164388i
\(16\) 0 0
\(17\) 3.86799i 0.938126i 0.883165 + 0.469063i \(0.155408\pi\)
−0.883165 + 0.469063i \(0.844592\pi\)
\(18\) 0 0
\(19\) 0.778008 0.178487 0.0892436 0.996010i \(-0.471555\pi\)
0.0892436 + 0.996010i \(0.471555\pi\)
\(20\) 0 0
\(21\) 0.363328 0.0792847
\(22\) 0 0
\(23\) 5.00933i 1.04452i −0.852787 0.522259i \(-0.825090\pi\)
0.852787 0.522259i \(-0.174910\pi\)
\(24\) 0 0
\(25\) 1.14134 4.86799i 0.228267 0.973599i
\(26\) 0 0
\(27\) 2.13201i 0.410305i
\(28\) 0 0
\(29\) −9.42401 −1.74999 −0.874997 0.484128i \(-0.839137\pi\)
−0.874997 + 0.484128i \(0.839137\pi\)
\(30\) 0 0
\(31\) 4.72666 0.848933 0.424466 0.905444i \(-0.360462\pi\)
0.424466 + 0.905444i \(0.360462\pi\)
\(32\) 0 0
\(33\) 1.86799i 0.325176i
\(34\) 0 0
\(35\) −1.38900 1.75233i −0.234785 0.296198i
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 1.68802 0.270300
\(40\) 0 0
\(41\) −1.00933 −0.157631 −0.0788153 0.996889i \(-0.525114\pi\)
−0.0788153 + 0.996889i \(0.525114\pi\)
\(42\) 0 0
\(43\) 7.00933i 1.06891i 0.845196 + 0.534456i \(0.179484\pi\)
−0.845196 + 0.534456i \(0.820516\pi\)
\(44\) 0 0
\(45\) −5.02568 + 3.98365i −0.749184 + 0.593848i
\(46\) 0 0
\(47\) 11.4240i 1.66636i −0.553000 0.833181i \(-0.686517\pi\)
0.553000 0.833181i \(-0.313483\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 1.40535 0.196788
\(52\) 0 0
\(53\) 7.55602i 1.03790i −0.854805 0.518949i \(-0.826323\pi\)
0.854805 0.518949i \(-0.173677\pi\)
\(54\) 0 0
\(55\) −9.00933 + 7.14134i −1.21482 + 0.962938i
\(56\) 0 0
\(57\) 0.282672i 0.0374409i
\(58\) 0 0
\(59\) −12.5140 −1.62918 −0.814592 0.580035i \(-0.803039\pi\)
−0.814592 + 0.580035i \(0.803039\pi\)
\(60\) 0 0
\(61\) 11.5047 1.47302 0.736511 0.676426i \(-0.236472\pi\)
0.736511 + 0.676426i \(0.236472\pi\)
\(62\) 0 0
\(63\) 2.86799i 0.361333i
\(64\) 0 0
\(65\) −6.45331 8.14134i −0.800435 1.00981i
\(66\) 0 0
\(67\) 11.7360i 1.43378i −0.697187 0.716889i \(-0.745565\pi\)
0.697187 0.716889i \(-0.254435\pi\)
\(68\) 0 0
\(69\) −1.82003 −0.219106
\(70\) 0 0
\(71\) 2.72666 0.323595 0.161797 0.986824i \(-0.448271\pi\)
0.161797 + 0.986824i \(0.448271\pi\)
\(72\) 0 0
\(73\) 5.00933i 0.586298i −0.956067 0.293149i \(-0.905297\pi\)
0.956067 0.293149i \(-0.0947031\pi\)
\(74\) 0 0
\(75\) −1.76868 0.414680i −0.204229 0.0478831i
\(76\) 0 0
\(77\) 5.14134i 0.585910i
\(78\) 0 0
\(79\) −5.68802 −0.639953 −0.319976 0.947426i \(-0.603675\pi\)
−0.319976 + 0.947426i \(0.603675\pi\)
\(80\) 0 0
\(81\) 7.82936 0.869929
\(82\) 0 0
\(83\) 4.67531i 0.513181i 0.966520 + 0.256591i \(0.0825992\pi\)
−0.966520 + 0.256591i \(0.917401\pi\)
\(84\) 0 0
\(85\) −5.37266 6.77801i −0.582746 0.735178i
\(86\) 0 0
\(87\) 3.42401i 0.367092i
\(88\) 0 0
\(89\) 2.82936 0.299911 0.149956 0.988693i \(-0.452087\pi\)
0.149956 + 0.988693i \(0.452087\pi\)
\(90\) 0 0
\(91\) −4.64600 −0.487033
\(92\) 0 0
\(93\) 1.71733i 0.178079i
\(94\) 0 0
\(95\) −1.36333 + 1.08066i −0.139875 + 0.110873i
\(96\) 0 0
\(97\) 1.58532i 0.160965i −0.996756 0.0804824i \(-0.974354\pi\)
0.996756 0.0804824i \(-0.0256461\pi\)
\(98\) 0 0
\(99\) 14.7453 1.48196
\(100\) 0 0
\(101\) −9.06068 −0.901571 −0.450786 0.892632i \(-0.648856\pi\)
−0.450786 + 0.892632i \(0.648856\pi\)
\(102\) 0 0
\(103\) 5.14134i 0.506591i −0.967389 0.253295i \(-0.918486\pi\)
0.967389 0.253295i \(-0.0815145\pi\)
\(104\) 0 0
\(105\) −0.636672 + 0.504664i −0.0621328 + 0.0492502i
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 3.97070 0.380324 0.190162 0.981753i \(-0.439099\pi\)
0.190162 + 0.981753i \(0.439099\pi\)
\(110\) 0 0
\(111\) 2.17997 0.206914
\(112\) 0 0
\(113\) 4.54669i 0.427716i −0.976865 0.213858i \(-0.931397\pi\)
0.976865 0.213858i \(-0.0686030\pi\)
\(114\) 0 0
\(115\) 6.95798 + 8.77801i 0.648835 + 0.818553i
\(116\) 0 0
\(117\) 13.3247i 1.23187i
\(118\) 0 0
\(119\) −3.86799 −0.354578
\(120\) 0 0
\(121\) 15.4333 1.40303
\(122\) 0 0
\(123\) 0.366718i 0.0330658i
\(124\) 0 0
\(125\) 4.76166 + 10.1157i 0.425896 + 0.904772i
\(126\) 0 0
\(127\) 16.1214i 1.43054i 0.698849 + 0.715270i \(0.253696\pi\)
−0.698849 + 0.715270i \(0.746304\pi\)
\(128\) 0 0
\(129\) 2.54669 0.224223
\(130\) 0 0
\(131\) 0.0513514 0.00448659 0.00224330 0.999997i \(-0.499286\pi\)
0.00224330 + 0.999997i \(0.499286\pi\)
\(132\) 0 0
\(133\) 0.778008i 0.0674618i
\(134\) 0 0
\(135\) 2.96137 + 3.73599i 0.254874 + 0.321542i
\(136\) 0 0
\(137\) 14.0187i 1.19769i −0.800863 0.598847i \(-0.795626\pi\)
0.800863 0.598847i \(-0.204374\pi\)
\(138\) 0 0
\(139\) −12.6167 −1.07013 −0.535067 0.844810i \(-0.679714\pi\)
−0.535067 + 0.844810i \(0.679714\pi\)
\(140\) 0 0
\(141\) −4.15066 −0.349549
\(142\) 0 0
\(143\) 23.8867i 1.99750i
\(144\) 0 0
\(145\) 16.5140 13.0900i 1.37141 1.08706i
\(146\) 0 0
\(147\) 0.363328i 0.0299668i
\(148\) 0 0
\(149\) 11.4533 0.938292 0.469146 0.883121i \(-0.344562\pi\)
0.469146 + 0.883121i \(0.344562\pi\)
\(150\) 0 0
\(151\) 5.86799 0.477530 0.238765 0.971077i \(-0.423257\pi\)
0.238765 + 0.971077i \(0.423257\pi\)
\(152\) 0 0
\(153\) 11.0934i 0.896846i
\(154\) 0 0
\(155\) −8.28267 + 6.56534i −0.665280 + 0.527341i
\(156\) 0 0
\(157\) 5.78734i 0.461880i −0.972968 0.230940i \(-0.925820\pi\)
0.972968 0.230940i \(-0.0741801\pi\)
\(158\) 0 0
\(159\) −2.74531 −0.217718
\(160\) 0 0
\(161\) 5.00933 0.394790
\(162\) 0 0
\(163\) 3.27334i 0.256388i −0.991749 0.128194i \(-0.959082\pi\)
0.991749 0.128194i \(-0.0409180\pi\)
\(164\) 0 0
\(165\) 2.59465 + 3.27334i 0.201993 + 0.254829i
\(166\) 0 0
\(167\) 24.8773i 1.92506i −0.271166 0.962532i \(-0.587409\pi\)
0.271166 0.962532i \(-0.412591\pi\)
\(168\) 0 0
\(169\) −8.58532 −0.660409
\(170\) 0 0
\(171\) 2.23132 0.170633
\(172\) 0 0
\(173\) 6.62734i 0.503868i −0.967744 0.251934i \(-0.918933\pi\)
0.967744 0.251934i \(-0.0810665\pi\)
\(174\) 0 0
\(175\) 4.86799 + 1.14134i 0.367986 + 0.0862769i
\(176\) 0 0
\(177\) 4.54669i 0.341750i
\(178\) 0 0
\(179\) 10.0187 0.748830 0.374415 0.927261i \(-0.377844\pi\)
0.374415 + 0.927261i \(0.377844\pi\)
\(180\) 0 0
\(181\) 1.78734 0.132852 0.0664258 0.997791i \(-0.478840\pi\)
0.0664258 + 0.997791i \(0.478840\pi\)
\(182\) 0 0
\(183\) 4.17997i 0.308992i
\(184\) 0 0
\(185\) −8.33402 10.5140i −0.612730 0.773004i
\(186\) 0 0
\(187\) 19.8867i 1.45426i
\(188\) 0 0
\(189\) 2.13201 0.155081
\(190\) 0 0
\(191\) −8.59465 −0.621887 −0.310943 0.950428i \(-0.600645\pi\)
−0.310943 + 0.950428i \(0.600645\pi\)
\(192\) 0 0
\(193\) 5.17064i 0.372191i 0.982532 + 0.186095i \(0.0595834\pi\)
−0.982532 + 0.186095i \(0.940417\pi\)
\(194\) 0 0
\(195\) −2.95798 + 2.34467i −0.211825 + 0.167905i
\(196\) 0 0
\(197\) 23.9160i 1.70394i 0.523590 + 0.851971i \(0.324593\pi\)
−0.523590 + 0.851971i \(0.675407\pi\)
\(198\) 0 0
\(199\) 15.3107 1.08534 0.542672 0.839945i \(-0.317413\pi\)
0.542672 + 0.839945i \(0.317413\pi\)
\(200\) 0 0
\(201\) −4.26401 −0.300760
\(202\) 0 0
\(203\) 9.42401i 0.661436i
\(204\) 0 0
\(205\) 1.76868 1.40196i 0.123530 0.0979172i
\(206\) 0 0
\(207\) 14.3667i 0.998556i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −1.03863 −0.0715025 −0.0357512 0.999361i \(-0.511382\pi\)
−0.0357512 + 0.999361i \(0.511382\pi\)
\(212\) 0 0
\(213\) 0.990671i 0.0678797i
\(214\) 0 0
\(215\) −9.73599 12.2827i −0.663989 0.837671i
\(216\) 0 0
\(217\) 4.72666i 0.320866i
\(218\) 0 0
\(219\) −1.82003 −0.122986
\(220\) 0 0
\(221\) −17.9707 −1.20884
\(222\) 0 0
\(223\) 9.86799i 0.660810i 0.943839 + 0.330405i \(0.107185\pi\)
−0.943839 + 0.330405i \(0.892815\pi\)
\(224\) 0 0
\(225\) 3.27334 13.9614i 0.218223 0.930758i
\(226\) 0 0
\(227\) 21.6553i 1.43731i −0.695364 0.718657i \(-0.744757\pi\)
0.695364 0.718657i \(-0.255243\pi\)
\(228\) 0 0
\(229\) −20.2313 −1.33692 −0.668462 0.743747i \(-0.733047\pi\)
−0.668462 + 0.743747i \(0.733047\pi\)
\(230\) 0 0
\(231\) 1.86799 0.122905
\(232\) 0 0
\(233\) 5.45331i 0.357258i 0.983916 + 0.178629i \(0.0571663\pi\)
−0.983916 + 0.178629i \(0.942834\pi\)
\(234\) 0 0
\(235\) 15.8680 + 20.0187i 1.03511 + 1.30587i
\(236\) 0 0
\(237\) 2.06662i 0.134241i
\(238\) 0 0
\(239\) 11.5853 0.749392 0.374696 0.927148i \(-0.377747\pi\)
0.374696 + 0.927148i \(0.377747\pi\)
\(240\) 0 0
\(241\) −5.00933 −0.322679 −0.161340 0.986899i \(-0.551581\pi\)
−0.161340 + 0.986899i \(0.551581\pi\)
\(242\) 0 0
\(243\) 9.24065i 0.592788i
\(244\) 0 0
\(245\) 1.75233 1.38900i 0.111952 0.0887402i
\(246\) 0 0
\(247\) 3.61462i 0.229993i
\(248\) 0 0
\(249\) 1.69867 0.107649
\(250\) 0 0
\(251\) −23.4206 −1.47830 −0.739148 0.673543i \(-0.764772\pi\)
−0.739148 + 0.673543i \(0.764772\pi\)
\(252\) 0 0
\(253\) 25.7546i 1.61918i
\(254\) 0 0
\(255\) −2.46264 + 1.95204i −0.154217 + 0.122241i
\(256\) 0 0
\(257\) 13.0093i 0.811500i 0.913984 + 0.405750i \(0.132990\pi\)
−0.913984 + 0.405750i \(0.867010\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) −27.0280 −1.67299
\(262\) 0 0
\(263\) 6.01866i 0.371126i −0.982632 0.185563i \(-0.940589\pi\)
0.982632 0.185563i \(-0.0594109\pi\)
\(264\) 0 0
\(265\) 10.4953 + 13.2406i 0.644723 + 0.813367i
\(266\) 0 0
\(267\) 1.02799i 0.0629117i
\(268\) 0 0
\(269\) −3.24065 −0.197586 −0.0987929 0.995108i \(-0.531498\pi\)
−0.0987929 + 0.995108i \(0.531498\pi\)
\(270\) 0 0
\(271\) −29.1307 −1.76956 −0.884782 0.466006i \(-0.845693\pi\)
−0.884782 + 0.466006i \(0.845693\pi\)
\(272\) 0 0
\(273\) 1.68802i 0.102164i
\(274\) 0 0
\(275\) 5.86799 25.0280i 0.353853 1.50924i
\(276\) 0 0
\(277\) 4.44398i 0.267013i −0.991048 0.133507i \(-0.957376\pi\)
0.991048 0.133507i \(-0.0426237\pi\)
\(278\) 0 0
\(279\) 13.5560 0.811577
\(280\) 0 0
\(281\) −24.6974 −1.47332 −0.736660 0.676263i \(-0.763598\pi\)
−0.736660 + 0.676263i \(0.763598\pi\)
\(282\) 0 0
\(283\) 7.73937i 0.460058i −0.973184 0.230029i \(-0.926118\pi\)
0.973184 0.230029i \(-0.0738821\pi\)
\(284\) 0 0
\(285\) 0.392633 + 0.495336i 0.0232576 + 0.0293412i
\(286\) 0 0
\(287\) 1.00933i 0.0595788i
\(288\) 0 0
\(289\) 2.03863 0.119920
\(290\) 0 0
\(291\) −0.575992 −0.0337652
\(292\) 0 0
\(293\) 25.1086i 1.46686i −0.679764 0.733431i \(-0.737918\pi\)
0.679764 0.733431i \(-0.262082\pi\)
\(294\) 0 0
\(295\) 21.9287 17.3820i 1.27674 1.01202i
\(296\) 0 0
\(297\) 10.9614i 0.636043i
\(298\) 0 0
\(299\) 23.2733 1.34593
\(300\) 0 0
\(301\) −7.00933 −0.404011
\(302\) 0 0
\(303\) 3.29200i 0.189121i
\(304\) 0 0
\(305\) −20.1600 + 15.9800i −1.15436 + 0.915014i
\(306\) 0 0
\(307\) 33.9193i 1.93588i −0.251183 0.967940i \(-0.580820\pi\)
0.251183 0.967940i \(-0.419180\pi\)
\(308\) 0 0
\(309\) −1.86799 −0.106266
\(310\) 0 0
\(311\) 0.565344 0.0320577 0.0160289 0.999872i \(-0.494898\pi\)
0.0160289 + 0.999872i \(0.494898\pi\)
\(312\) 0 0
\(313\) 27.3400i 1.54535i 0.634804 + 0.772673i \(0.281081\pi\)
−0.634804 + 0.772673i \(0.718919\pi\)
\(314\) 0 0
\(315\) −3.98365 5.02568i −0.224453 0.283165i
\(316\) 0 0
\(317\) 22.7267i 1.27646i 0.769847 + 0.638228i \(0.220332\pi\)
−0.769847 + 0.638228i \(0.779668\pi\)
\(318\) 0 0
\(319\) −48.4520 −2.71279
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00933i 0.167444i
\(324\) 0 0
\(325\) 22.6167 + 5.30265i 1.25455 + 0.294138i
\(326\) 0 0
\(327\) 1.44267i 0.0797796i
\(328\) 0 0
\(329\) 11.4240 0.629826
\(330\) 0 0
\(331\) 0.462642 0.0254291 0.0127145 0.999919i \(-0.495953\pi\)
0.0127145 + 0.999919i \(0.495953\pi\)
\(332\) 0 0
\(333\) 17.2080i 0.942990i
\(334\) 0 0
\(335\) 16.3013 + 20.5653i 0.890637 + 1.12360i
\(336\) 0 0
\(337\) 10.5653i 0.575531i −0.957701 0.287765i \(-0.907088\pi\)
0.957701 0.287765i \(-0.0929124\pi\)
\(338\) 0 0
\(339\) −1.65194 −0.0897211
\(340\) 0 0
\(341\) 24.3013 1.31599
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 3.18930 2.52803i 0.171706 0.136105i
\(346\) 0 0
\(347\) 31.8760i 1.71119i 0.517643 + 0.855597i \(0.326810\pi\)
−0.517643 + 0.855597i \(0.673190\pi\)
\(348\) 0 0
\(349\) 9.62602 0.515269 0.257635 0.966242i \(-0.417057\pi\)
0.257635 + 0.966242i \(0.417057\pi\)
\(350\) 0 0
\(351\) 9.90531 0.528706
\(352\) 0 0
\(353\) 26.2534i 1.39733i 0.715451 + 0.698663i \(0.246221\pi\)
−0.715451 + 0.698663i \(0.753779\pi\)
\(354\) 0 0
\(355\) −4.77801 + 3.78734i −0.253590 + 0.201011i
\(356\) 0 0
\(357\) 1.40535i 0.0743791i
\(358\) 0 0
\(359\) 4.56534 0.240950 0.120475 0.992716i \(-0.461558\pi\)
0.120475 + 0.992716i \(0.461558\pi\)
\(360\) 0 0
\(361\) −18.3947 −0.968142
\(362\) 0 0
\(363\) 5.60737i 0.294310i
\(364\) 0 0
\(365\) 6.95798 + 8.77801i 0.364197 + 0.459462i
\(366\) 0 0
\(367\) 3.79073i 0.197874i 0.995094 + 0.0989371i \(0.0315443\pi\)
−0.995094 + 0.0989371i \(0.968456\pi\)
\(368\) 0 0
\(369\) −2.89475 −0.150695
\(370\) 0 0
\(371\) 7.55602 0.392289
\(372\) 0 0
\(373\) 20.7453i 1.07415i −0.843534 0.537076i \(-0.819529\pi\)
0.843534 0.537076i \(-0.180471\pi\)
\(374\) 0 0
\(375\) 3.67531 1.73005i 0.189792 0.0893392i
\(376\) 0 0
\(377\) 43.7839i 2.25499i
\(378\) 0 0
\(379\) −3.00933 −0.154579 −0.0772894 0.997009i \(-0.524627\pi\)
−0.0772894 + 0.997009i \(0.524627\pi\)
\(380\) 0 0
\(381\) 5.85735 0.300081
\(382\) 0 0
\(383\) 7.00933i 0.358160i −0.983835 0.179080i \(-0.942688\pi\)
0.983835 0.179080i \(-0.0573121\pi\)
\(384\) 0 0
\(385\) −7.14134 9.00933i −0.363956 0.459158i
\(386\) 0 0
\(387\) 20.1027i 1.02188i
\(388\) 0 0
\(389\) 19.7653 1.00214 0.501070 0.865407i \(-0.332940\pi\)
0.501070 + 0.865407i \(0.332940\pi\)
\(390\) 0 0
\(391\) 19.3760 0.979889
\(392\) 0 0
\(393\) 0.0186574i 0.000941142i
\(394\) 0 0
\(395\) 9.96731 7.90069i 0.501510 0.397527i
\(396\) 0 0
\(397\) 9.37266i 0.470400i −0.971947 0.235200i \(-0.924425\pi\)
0.971947 0.235200i \(-0.0755745\pi\)
\(398\) 0 0
\(399\) 0.282672 0.0141513
\(400\) 0 0
\(401\) −16.1507 −0.806526 −0.403263 0.915084i \(-0.632124\pi\)
−0.403263 + 0.915084i \(0.632124\pi\)
\(402\) 0 0
\(403\) 21.9600i 1.09391i
\(404\) 0 0
\(405\) −13.7196 + 10.8750i −0.681734 + 0.540384i
\(406\) 0 0
\(407\) 30.8480i 1.52908i
\(408\) 0 0
\(409\) −15.2920 −0.756141 −0.378070 0.925777i \(-0.623412\pi\)
−0.378070 + 0.925777i \(0.623412\pi\)
\(410\) 0 0
\(411\) −5.09337 −0.251238
\(412\) 0 0
\(413\) 12.5140i 0.615773i
\(414\) 0 0
\(415\) −6.49402 8.19269i −0.318779 0.402163i
\(416\) 0 0
\(417\) 4.58400i 0.224480i
\(418\) 0 0
\(419\) 33.1820 1.62105 0.810524 0.585705i \(-0.199182\pi\)
0.810524 + 0.585705i \(0.199182\pi\)
\(420\) 0 0
\(421\) 27.8094 1.35535 0.677673 0.735363i \(-0.262988\pi\)
0.677673 + 0.735363i \(0.262988\pi\)
\(422\) 0 0
\(423\) 32.7640i 1.59304i
\(424\) 0 0
\(425\) 18.8294 + 4.41468i 0.913358 + 0.214143i
\(426\) 0 0
\(427\) 11.5047i 0.556750i
\(428\) 0 0
\(429\) 8.67869 0.419011
\(430\) 0 0
\(431\) −21.5454 −1.03780 −0.518902 0.854834i \(-0.673659\pi\)
−0.518902 + 0.854834i \(0.673659\pi\)
\(432\) 0 0
\(433\) 36.0187i 1.73095i 0.500955 + 0.865473i \(0.332982\pi\)
−0.500955 + 0.865473i \(0.667018\pi\)
\(434\) 0 0
\(435\) −4.75596 6.00000i −0.228031 0.287678i
\(436\) 0 0
\(437\) 3.89730i 0.186433i
\(438\) 0 0
\(439\) 21.1893 1.01131 0.505655 0.862736i \(-0.331251\pi\)
0.505655 + 0.862736i \(0.331251\pi\)
\(440\) 0 0
\(441\) −2.86799 −0.136571
\(442\) 0 0
\(443\) 11.1120i 0.527949i 0.964530 + 0.263974i \(0.0850334\pi\)
−0.964530 + 0.263974i \(0.914967\pi\)
\(444\) 0 0
\(445\) −4.95798 + 3.92999i −0.235031 + 0.186299i
\(446\) 0 0
\(447\) 4.16131i 0.196823i
\(448\) 0 0
\(449\) −0.961367 −0.0453697 −0.0226848 0.999743i \(-0.507221\pi\)
−0.0226848 + 0.999743i \(0.507221\pi\)
\(450\) 0 0
\(451\) −5.18930 −0.244355
\(452\) 0 0
\(453\) 2.13201i 0.100170i
\(454\) 0 0
\(455\) 8.14134 6.45331i 0.381672 0.302536i
\(456\) 0 0
\(457\) 28.2241i 1.32027i 0.751149 + 0.660133i \(0.229500\pi\)
−0.751149 + 0.660133i \(0.770500\pi\)
\(458\) 0 0
\(459\) 8.24659 0.384918
\(460\) 0 0
\(461\) 26.0887 1.21507 0.607535 0.794293i \(-0.292158\pi\)
0.607535 + 0.794293i \(0.292158\pi\)
\(462\) 0 0
\(463\) 2.90663i 0.135082i 0.997716 + 0.0675412i \(0.0215154\pi\)
−0.997716 + 0.0675412i \(0.978485\pi\)
\(464\) 0 0
\(465\) 2.38538 + 3.00933i 0.110619 + 0.139554i
\(466\) 0 0
\(467\) 12.2020i 0.564642i 0.959320 + 0.282321i \(0.0911043\pi\)
−0.959320 + 0.282321i \(0.908896\pi\)
\(468\) 0 0
\(469\) 11.7360 0.541917
\(470\) 0 0
\(471\) −2.10270 −0.0968874
\(472\) 0 0
\(473\) 36.0373i 1.65700i
\(474\) 0 0
\(475\) 0.887968 3.78734i 0.0407428 0.173775i
\(476\) 0 0
\(477\) 21.6706i 0.992228i
\(478\) 0 0
\(479\) 17.6519 0.806538 0.403269 0.915082i \(-0.367874\pi\)
0.403269 + 0.915082i \(0.367874\pi\)
\(480\) 0 0
\(481\) −27.8760 −1.27104
\(482\) 0 0
\(483\) 1.82003i 0.0828143i
\(484\) 0 0
\(485\) 2.20202 + 2.77801i 0.0999884 + 0.126143i
\(486\) 0 0
\(487\) 1.57467i 0.0713553i −0.999363 0.0356776i \(-0.988641\pi\)
0.999363 0.0356776i \(-0.0113590\pi\)
\(488\) 0 0
\(489\) −1.18930 −0.0537819
\(490\) 0 0
\(491\) −3.26270 −0.147243 −0.0736217 0.997286i \(-0.523456\pi\)
−0.0736217 + 0.997286i \(0.523456\pi\)
\(492\) 0 0
\(493\) 36.4520i 1.64172i
\(494\) 0 0
\(495\) −25.8387 + 20.4813i −1.16136 + 0.920566i
\(496\) 0 0
\(497\) 2.72666i 0.122307i
\(498\) 0 0
\(499\) −42.5293 −1.90387 −0.951936 0.306298i \(-0.900910\pi\)
−0.951936 + 0.306298i \(0.900910\pi\)
\(500\) 0 0
\(501\) −9.03863 −0.403816
\(502\) 0 0
\(503\) 18.6974i 0.833674i 0.908981 + 0.416837i \(0.136861\pi\)
−0.908981 + 0.416837i \(0.863139\pi\)
\(504\) 0 0
\(505\) 15.8773 12.5853i 0.706532 0.560039i
\(506\) 0 0
\(507\) 3.11929i 0.138533i
\(508\) 0 0
\(509\) −20.2313 −0.896738 −0.448369 0.893849i \(-0.647995\pi\)
−0.448369 + 0.893849i \(0.647995\pi\)
\(510\) 0 0
\(511\) 5.00933 0.221600
\(512\) 0 0
\(513\) 1.65872i 0.0732342i
\(514\) 0 0
\(515\) 7.14134 + 9.00933i 0.314685 + 0.396998i
\(516\) 0 0
\(517\) 58.7347i 2.58315i
\(518\) 0 0
\(519\) −2.40790 −0.105695
\(520\) 0 0
\(521\) 21.8387 0.956770 0.478385 0.878150i \(-0.341222\pi\)
0.478385 + 0.878150i \(0.341222\pi\)
\(522\) 0 0
\(523\) 20.4554i 0.894451i 0.894421 + 0.447226i \(0.147588\pi\)
−0.894421 + 0.447226i \(0.852412\pi\)
\(524\) 0 0
\(525\) 0.414680 1.76868i 0.0180981 0.0771915i
\(526\) 0 0
\(527\) 18.2827i 0.796406i
\(528\) 0 0
\(529\) −2.09337 −0.0910163
\(530\) 0 0
\(531\) −35.8900 −1.55750
\(532\) 0 0
\(533\) 4.68934i 0.203118i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 3.64006i 0.157080i
\(538\) 0 0
\(539\) −5.14134 −0.221453
\(540\) 0 0
\(541\) 27.3400 1.17544 0.587718 0.809066i \(-0.300026\pi\)
0.587718 + 0.809066i \(0.300026\pi\)
\(542\) 0 0
\(543\) 0.649390i 0.0278680i
\(544\) 0 0
\(545\) −6.95798 + 5.51531i −0.298047 + 0.236250i
\(546\) 0 0
\(547\) 22.6867i 0.970013i −0.874510 0.485007i \(-0.838817\pi\)
0.874510 0.485007i \(-0.161183\pi\)
\(548\) 0 0
\(549\) 32.9953 1.40820
\(550\) 0 0
\(551\) −7.33195 −0.312352
\(552\) 0 0
\(553\) 5.68802i 0.241879i
\(554\) 0 0
\(555\) −3.82003 + 3.02799i −0.162151 + 0.128531i
\(556\) 0 0
\(557\) 12.7453i 0.540036i −0.962855 0.270018i \(-0.912970\pi\)
0.962855 0.270018i \(-0.0870297\pi\)
\(558\) 0 0
\(559\) −32.5653 −1.37737
\(560\) 0 0
\(561\) 7.22538 0.305056
\(562\) 0 0
\(563\) 5.20333i 0.219294i 0.993971 + 0.109647i \(0.0349721\pi\)
−0.993971 + 0.109647i \(0.965028\pi\)
\(564\) 0 0
\(565\) 6.31537 + 7.96731i 0.265689 + 0.335187i
\(566\) 0 0
\(567\) 7.82936i 0.328802i
\(568\) 0 0
\(569\) 7.37605 0.309220 0.154610 0.987976i \(-0.450588\pi\)
0.154610 + 0.987976i \(0.450588\pi\)
\(570\) 0 0
\(571\) 15.8973 0.665281 0.332641 0.943054i \(-0.392060\pi\)
0.332641 + 0.943054i \(0.392060\pi\)
\(572\) 0 0
\(573\) 3.12268i 0.130452i
\(574\) 0 0
\(575\) −24.3854 5.71733i −1.01694 0.238429i
\(576\) 0 0
\(577\) 19.9707i 0.831391i −0.909504 0.415695i \(-0.863538\pi\)
0.909504 0.415695i \(-0.136462\pi\)
\(578\) 0 0
\(579\) 1.87864 0.0780736
\(580\) 0 0
\(581\) −4.67531 −0.193964
\(582\) 0 0
\(583\) 38.8480i 1.60892i
\(584\) 0 0
\(585\) −18.5081 23.3493i −0.765214 0.965374i
\(586\) 0 0
\(587\) 13.8060i 0.569834i −0.958552 0.284917i \(-0.908034\pi\)
0.958552 0.284917i \(-0.0919661\pi\)
\(588\) 0 0
\(589\) 3.67738 0.151524
\(590\) 0 0
\(591\) 8.68934 0.357432
\(592\) 0 0
\(593\) 20.0666i 0.824037i −0.911175 0.412019i \(-0.864824\pi\)
0.911175 0.412019i \(-0.135176\pi\)
\(594\) 0 0
\(595\) 6.77801 5.37266i 0.277871 0.220257i
\(596\) 0 0
\(597\) 5.56279i 0.227670i
\(598\) 0 0
\(599\) −46.0267 −1.88060 −0.940299 0.340349i \(-0.889455\pi\)
−0.940299 + 0.340349i \(0.889455\pi\)
\(600\) 0 0
\(601\) −1.37605 −0.0561301 −0.0280651 0.999606i \(-0.508935\pi\)
−0.0280651 + 0.999606i \(0.508935\pi\)
\(602\) 0 0
\(603\) 33.6587i 1.37069i
\(604\) 0 0
\(605\) −27.0443 + 21.4370i −1.09951 + 0.871537i
\(606\) 0 0
\(607\) 18.7933i 0.762796i 0.924411 + 0.381398i \(0.124557\pi\)
−0.924411 + 0.381398i \(0.875443\pi\)
\(608\) 0 0
\(609\) −3.42401 −0.138748
\(610\) 0 0
\(611\) 53.0759 2.14722
\(612\) 0 0
\(613\) 24.8480i 1.00360i −0.864983 0.501801i \(-0.832671\pi\)
0.864983 0.501801i \(-0.167329\pi\)
\(614\) 0 0
\(615\) −0.509372 0.642611i −0.0205399 0.0259126i
\(616\) 0 0
\(617\) 43.1680i 1.73788i 0.494919 + 0.868939i \(0.335198\pi\)
−0.494919 + 0.868939i \(0.664802\pi\)
\(618\) 0 0
\(619\) −31.9486 −1.28412 −0.642062 0.766652i \(-0.721921\pi\)
−0.642062 + 0.766652i \(0.721921\pi\)
\(620\) 0 0
\(621\) −10.6799 −0.428571
\(622\) 0 0
\(623\) 2.82936i 0.113356i
\(624\) 0 0
\(625\) −22.3947 11.1120i −0.895788 0.444481i
\(626\) 0 0
\(627\) 1.45331i 0.0580397i
\(628\) 0 0
\(629\) −23.2080 −0.925362
\(630\) 0 0
\(631\) −8.59465 −0.342148 −0.171074 0.985258i \(-0.554724\pi\)
−0.171074 + 0.985258i \(0.554724\pi\)
\(632\) 0 0
\(633\) 0.377365i 0.0149989i
\(634\) 0 0
\(635\) −22.3926 28.2500i −0.888625 1.12107i
\(636\) 0 0
\(637\) 4.64600i 0.184081i
\(638\) 0 0
\(639\) 7.82003 0.309356
\(640\) 0 0
\(641\) 33.2266 1.31237 0.656186 0.754599i \(-0.272169\pi\)
0.656186 + 0.754599i \(0.272169\pi\)
\(642\) 0 0
\(643\) 17.4940i 0.689897i −0.938622 0.344948i \(-0.887896\pi\)
0.938622 0.344948i \(-0.112104\pi\)
\(644\) 0 0
\(645\) −4.46264 + 3.53736i −0.175716 + 0.139283i
\(646\) 0 0
\(647\) 9.91595i 0.389836i 0.980820 + 0.194918i \(0.0624441\pi\)
−0.980820 + 0.194918i \(0.937556\pi\)
\(648\) 0 0
\(649\) −64.3386 −2.52551
\(650\) 0 0
\(651\) 1.71733 0.0673074
\(652\) 0 0
\(653\) 14.6240i 0.572280i −0.958188 0.286140i \(-0.907628\pi\)
0.958188 0.286140i \(-0.0923722\pi\)
\(654\) 0 0
\(655\) −0.0899847 + 0.0713273i −0.00351599 + 0.00278699i
\(656\) 0 0
\(657\) 14.3667i 0.560499i
\(658\) 0 0
\(659\) −20.2534 −0.788959 −0.394480 0.918905i \(-0.629075\pi\)
−0.394480 + 0.918905i \(0.629075\pi\)
\(660\) 0 0
\(661\) −12.4299 −0.483469 −0.241734 0.970342i \(-0.577716\pi\)
−0.241734 + 0.970342i \(0.577716\pi\)
\(662\) 0 0
\(663\) 6.52926i 0.253575i
\(664\) 0 0
\(665\) −1.08066 1.36333i −0.0419060 0.0528676i
\(666\) 0 0
\(667\) 47.2080i 1.82790i
\(668\) 0 0
\(669\) 3.58532 0.138616
\(670\) 0 0
\(671\) 59.1493 2.28344
\(672\) 0 0
\(673\) 47.6774i 1.83783i 0.394458 + 0.918914i \(0.370932\pi\)
−0.394458 + 0.918914i \(0.629068\pi\)
\(674\) 0 0
\(675\) −10.3786 2.43334i −0.399472 0.0936592i
\(676\) 0 0
\(677\) 6.00339i 0.230729i 0.993323 + 0.115364i \(0.0368036\pi\)
−0.993323 + 0.115364i \(0.963196\pi\)
\(678\) 0 0
\(679\) 1.58532 0.0608390
\(680\) 0 0
\(681\) −7.86799 −0.301502
\(682\) 0 0
\(683\) 28.5653i 1.09302i 0.837452 + 0.546511i \(0.184044\pi\)
−0.837452 + 0.546511i \(0.815956\pi\)
\(684\) 0 0
\(685\) 19.4720 + 24.5653i 0.743986 + 0.938594i
\(686\) 0 0
\(687\) 7.35061i 0.280443i
\(688\) 0 0
\(689\) 35.1053 1.33740
\(690\) 0 0
\(691\) 47.8247 1.81934 0.909668 0.415337i \(-0.136336\pi\)
0.909668 + 0.415337i \(0.136336\pi\)
\(692\) 0 0
\(693\) 14.7453i 0.560128i
\(694\) 0 0
\(695\) 22.1086 17.5246i 0.838629 0.664747i
\(696\) 0 0
\(697\) 3.90408i 0.147877i
\(698\) 0 0
\(699\) 1.98134 0.0749413
\(700\) 0 0
\(701\) −4.80005 −0.181296 −0.0906478 0.995883i \(-0.528894\pi\)
−0.0906478 + 0.995883i \(0.528894\pi\)
\(702\) 0 0
\(703\) 4.66805i 0.176059i
\(704\) 0 0
\(705\) 7.27334 5.76529i 0.273930 0.217133i
\(706\) 0 0
\(707\) 9.06068i 0.340762i
\(708\) 0 0
\(709\) −11.7653 −0.441855 −0.220927 0.975290i \(-0.570908\pi\)
−0.220927 + 0.975290i \(0.570908\pi\)
\(710\) 0 0
\(711\) −16.3132 −0.611793
\(712\) 0 0
\(713\) 23.6774i 0.886725i
\(714\) 0 0
\(715\) −33.1787 41.8573i −1.24081 1.56538i
\(716\) 0 0
\(717\) 4.20927i 0.157198i
\(718\) 0 0
\(719\) 40.7640 1.52024 0.760120 0.649783i \(-0.225140\pi\)
0.760120 + 0.649783i \(0.225140\pi\)
\(720\) 0 0
\(721\) 5.14134 0.191473
\(722\) 0 0
\(723\) 1.82003i 0.0676877i
\(724\) 0 0
\(725\) −10.7560 + 45.8760i −0.399466 + 1.70379i
\(726\) 0 0
\(727\) 22.1214i 0.820436i 0.911988 + 0.410218i \(0.134547\pi\)
−0.911988 + 0.410218i \(0.865453\pi\)
\(728\) 0 0
\(729\) 20.1307 0.745581
\(730\) 0 0
\(731\) −27.1120 −1.00277
\(732\) 0 0
\(733\) 13.0500i 0.482014i −0.970523 0.241007i \(-0.922522\pi\)
0.970523 0.241007i \(-0.0774777\pi\)
\(734\) 0 0
\(735\) −0.504664 0.636672i −0.0186148 0.0234840i
\(736\) 0 0
\(737\) 60.3386i 2.22260i
\(738\) 0 0
\(739\) 34.5734 1.27180 0.635901 0.771771i \(-0.280629\pi\)
0.635901 + 0.771771i \(0.280629\pi\)
\(740\) 0 0
\(741\) 1.31330 0.0482451
\(742\) 0 0
\(743\) 3.55602i 0.130458i 0.997870 + 0.0652288i \(0.0207777\pi\)
−0.997870 + 0.0652288i \(0.979222\pi\)
\(744\) 0 0
\(745\) −20.0700 + 15.9087i −0.735308 + 0.582850i
\(746\) 0 0
\(747\) 13.4087i 0.490600i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23.6226 −0.862002 −0.431001 0.902351i \(-0.641839\pi\)
−0.431001 + 0.902351i \(0.641839\pi\)
\(752\) 0 0
\(753\) 8.50937i 0.310099i
\(754\) 0 0
\(755\) −10.2827 + 8.15066i −0.374225 + 0.296633i
\(756\) 0 0
\(757\) 15.1893i 0.552064i 0.961148 + 0.276032i \(0.0890196\pi\)
−0.961148 + 0.276032i \(0.910980\pi\)
\(758\) 0 0
\(759\) −9.35739 −0.339652
\(760\) 0 0
\(761\) 34.4626 1.24927 0.624635 0.780917i \(-0.285248\pi\)
0.624635 + 0.780917i \(0.285248\pi\)
\(762\) 0 0
\(763\) 3.97070i 0.143749i
\(764\) 0 0
\(765\) −15.4087 19.4393i −0.557104 0.702829i
\(766\) 0 0
\(767\) 58.1400i 2.09931i
\(768\) 0 0
\(769\) −40.3854 −1.45633 −0.728167 0.685400i \(-0.759627\pi\)
−0.728167 + 0.685400i \(0.759627\pi\)
\(770\) 0 0
\(771\) 4.72666 0.170226
\(772\) 0 0
\(773\) 10.7674i 0.387275i 0.981073 + 0.193638i \(0.0620286\pi\)
−0.981073 + 0.193638i \(0.937971\pi\)
\(774\) 0 0
\(775\) 5.39470 23.0093i 0.193783 0.826519i
\(776\) 0 0
\(777\) 2.17997i 0.0782060i
\(778\) 0 0
\(779\) −0.785266 −0.0281351
\(780\) 0 0
\(781\) 14.0187 0.501627
\(782\) 0 0
\(783\) 20.0921i 0.718031i
\(784\) 0 0
\(785\) 8.03863 + 10.1413i 0.286911 + 0.361960i
\(786\) 0 0
\(787\) 9.19269i 0.327684i −0.986487 0.163842i \(-0.947611\pi\)
0.986487 0.163842i \(-0.0523887\pi\)
\(788\) 0 0
\(789\) −2.18675 −0.0778503
\(790\) 0 0
\(791\) 4.54669 0.161662
\(792\) 0 0
\(793\) 53.4507i 1.89809i
\(794\) 0 0
\(795\) 4.81070 3.81325i 0.170618 0.135242i
\(796\) 0 0
\(797\) 21.1086i 0.747706i 0.927488 + 0.373853i \(0.121964\pi\)
−0.927488 + 0.373853i \(0.878036\pi\)
\(798\) 0 0
\(799\) 44.1880 1.56326
\(800\) 0 0
\(801\) 8.11458 0.286715
\(802\) 0 0