Properties

Label 140.2.e.a.29.1
Level $140$
Weight $2$
Character 140.29
Analytic conductor $1.118$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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 29.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 140.29
Dual form 140.2.e.a.29.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -6.00000 q^{9} +3.00000 q^{11} -1.00000i q^{13} +(-3.00000 + 6.00000i) q^{15} -5.00000i q^{17} +8.00000 q^{19} +3.00000 q^{21} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +9.00000i q^{27} +1.00000 q^{29} -2.00000 q^{31} -9.00000i q^{33} +(1.00000 - 2.00000i) q^{35} +10.0000i q^{37} -3.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(12.0000 + 6.00000i) q^{45} +11.0000i q^{47} -1.00000 q^{49} -15.0000 q^{51} -6.00000i q^{53} +(-6.00000 - 3.00000i) q^{55} -24.0000i q^{57} +10.0000 q^{59} -6.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} -10.0000i q^{67} -6.00000 q^{69} +10.0000i q^{73} +(12.0000 - 9.00000i) q^{75} +3.00000i q^{77} +7.00000 q^{79} +9.00000 q^{81} -12.0000i q^{83} +(-5.00000 + 10.0000i) q^{85} -3.00000i q^{87} -8.00000 q^{89} +1.00000 q^{91} +6.00000i q^{93} +(-16.0000 - 8.00000i) q^{95} +3.00000i q^{97} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 12 q^{9} + 6 q^{11} - 6 q^{15} + 16 q^{19} + 6 q^{21} + 6 q^{25} + 2 q^{29} - 4 q^{31} + 2 q^{35} - 6 q^{39} - 12 q^{41} + 24 q^{45} - 2 q^{49} - 30 q^{51} - 12 q^{55} + 20 q^{59} - 2 q^{65} - 12 q^{69} + 24 q^{75} + 14 q^{79} + 18 q^{81} - 10 q^{85} - 16 q^{89} + 2 q^{91} - 32 q^{95} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 3.00000i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) 0 0
\(5\) −2.00000 1.00000i −0.894427 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) −3.00000 + 6.00000i −0.774597 + 1.54919i
\(16\) 0 0
\(17\) 5.00000i 1.21268i −0.795206 0.606339i \(-0.792637\pi\)
0.795206 0.606339i \(-0.207363\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 2.00000i 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 9.00000i 1.56670i
\(34\) 0 0
\(35\) 1.00000 2.00000i 0.169031 0.338062i
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 12.0000 + 6.00000i 1.78885 + 0.894427i
\(46\) 0 0
\(47\) 11.0000i 1.60451i 0.596978 + 0.802257i \(0.296368\pi\)
−0.596978 + 0.802257i \(0.703632\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −15.0000 −2.10042
\(52\) 0 0
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) −6.00000 3.00000i −0.809040 0.404520i
\(56\) 0 0
\(57\) 24.0000i 3.17888i
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 6.00000i 0.755929i
\(64\) 0 0
\(65\) −1.00000 + 2.00000i −0.124035 + 0.248069i
\(66\) 0 0
\(67\) 10.0000i 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 12.0000 9.00000i 1.38564 1.03923i
\(76\) 0 0
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 12.0000i 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) −5.00000 + 10.0000i −0.542326 + 1.08465i
\(86\) 0 0
\(87\) 3.00000i 0.321634i
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 6.00000i 0.622171i
\(94\) 0 0
\(95\) −16.0000 8.00000i −1.64157 0.820783i
\(96\) 0 0
\(97\) 3.00000i 0.304604i 0.988334 + 0.152302i \(0.0486686\pi\)
−0.988334 + 0.152302i \(0.951331\pi\)
\(98\) 0 0
\(99\) −18.0000 −1.80907
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 5.00000i 0.492665i 0.969185 + 0.246332i \(0.0792255\pi\)
−0.969185 + 0.246332i \(0.920775\pi\)
\(104\) 0 0
\(105\) −6.00000 3.00000i −0.585540 0.292770i
\(106\) 0 0
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 30.0000 2.84747
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 0 0
\(115\) −2.00000 + 4.00000i −0.186501 + 0.373002i
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 18.0000i 1.62301i
\(124\) 0 0
\(125\) −2.00000 11.0000i −0.178885 0.983870i
\(126\) 0 0
\(127\) 2.00000i 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 2.00000 0.174741 0.0873704 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) 9.00000 18.0000i 0.774597 1.54919i
\(136\) 0 0
\(137\) 4.00000i 0.341743i 0.985293 + 0.170872i \(0.0546583\pi\)
−0.985293 + 0.170872i \(0.945342\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 33.0000 2.77910
\(142\) 0 0
\(143\) 3.00000i 0.250873i
\(144\) 0 0
\(145\) −2.00000 1.00000i −0.166091 0.0830455i
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 9.00000 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(152\) 0 0
\(153\) 30.0000i 2.42536i
\(154\) 0 0
\(155\) 4.00000 + 2.00000i 0.321288 + 0.160644i
\(156\) 0 0
\(157\) 18.0000i 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 0 0
\(159\) −18.0000 −1.42749
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 6.00000i 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 0 0
\(165\) −9.00000 + 18.0000i −0.700649 + 1.40130i
\(166\) 0 0
\(167\) 3.00000i 0.232147i 0.993241 + 0.116073i \(0.0370308\pi\)
−0.993241 + 0.116073i \(0.962969\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −48.0000 −3.67065
\(172\) 0 0
\(173\) 9.00000i 0.684257i 0.939653 + 0.342129i \(0.111148\pi\)
−0.939653 + 0.342129i \(0.888852\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) 0 0
\(177\) 30.0000i 2.25494i
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.0000 20.0000i 0.735215 1.47043i
\(186\) 0 0
\(187\) 15.0000i 1.09691i
\(188\) 0 0
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) 7.00000 0.506502 0.253251 0.967401i \(-0.418500\pi\)
0.253251 + 0.967401i \(0.418500\pi\)
\(192\) 0 0
\(193\) 8.00000i 0.575853i −0.957653 0.287926i \(-0.907034\pi\)
0.957653 0.287926i \(-0.0929658\pi\)
\(194\) 0 0
\(195\) 6.00000 + 3.00000i 0.429669 + 0.214834i
\(196\) 0 0
\(197\) 10.0000i 0.712470i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) −30.0000 −2.11604
\(202\) 0 0
\(203\) 1.00000i 0.0701862i
\(204\) 0 0
\(205\) 12.0000 + 6.00000i 0.838116 + 0.419058i
\(206\) 0 0
\(207\) 12.0000i 0.834058i
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 8.00000i 0.272798 0.545595i
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 30.0000 2.02721
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 0 0
\(223\) 19.0000i 1.27233i −0.771551 0.636167i \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) 0 0
\(225\) −18.0000 24.0000i −1.20000 1.60000i
\(226\) 0 0
\(227\) 27.0000i 1.79205i 0.444001 + 0.896026i \(0.353559\pi\)
−0.444001 + 0.896026i \(0.646441\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 9.00000 0.592157
\(232\) 0 0
\(233\) 16.0000i 1.04819i −0.851658 0.524097i \(-0.824403\pi\)
0.851658 0.524097i \(-0.175597\pi\)
\(234\) 0 0
\(235\) 11.0000 22.0000i 0.717561 1.43512i
\(236\) 0 0
\(237\) 21.0000i 1.36410i
\(238\) 0 0
\(239\) −5.00000 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000 + 1.00000i 0.127775 + 0.0638877i
\(246\) 0 0
\(247\) 8.00000i 0.509028i
\(248\) 0 0
\(249\) −36.0000 −2.28141
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 6.00000i 0.377217i
\(254\) 0 0
\(255\) 30.0000 + 15.0000i 1.87867 + 0.939336i
\(256\) 0 0
\(257\) 6.00000i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) −6.00000 + 12.0000i −0.368577 + 0.737154i
\(266\) 0 0
\(267\) 24.0000i 1.46878i
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 3.00000i 0.181568i
\(274\) 0 0
\(275\) 9.00000 + 12.0000i 0.542720 + 0.723627i
\(276\) 0 0
\(277\) 14.0000i 0.841178i −0.907251 0.420589i \(-0.861823\pi\)
0.907251 0.420589i \(-0.138177\pi\)
\(278\) 0 0
\(279\) 12.0000 0.718421
\(280\) 0 0
\(281\) 15.0000 0.894825 0.447412 0.894328i \(-0.352346\pi\)
0.447412 + 0.894328i \(0.352346\pi\)
\(282\) 0 0
\(283\) 7.00000i 0.416107i 0.978117 + 0.208053i \(0.0667128\pi\)
−0.978117 + 0.208053i \(0.933287\pi\)
\(284\) 0 0
\(285\) −24.0000 + 48.0000i −1.42164 + 2.84327i
\(286\) 0 0
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 9.00000 0.527589
\(292\) 0 0
\(293\) 15.0000i 0.876309i −0.898900 0.438155i \(-0.855632\pi\)
0.898900 0.438155i \(-0.144368\pi\)
\(294\) 0 0
\(295\) −20.0000 10.0000i −1.16445 0.582223i
\(296\) 0 0
\(297\) 27.0000i 1.56670i
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 36.0000i 2.06815i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 19.0000i 1.08439i −0.840254 0.542194i \(-0.817594\pi\)
0.840254 0.542194i \(-0.182406\pi\)
\(308\) 0 0
\(309\) 15.0000 0.853320
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 7.00000i 0.395663i 0.980236 + 0.197832i \(0.0633900\pi\)
−0.980236 + 0.197832i \(0.936610\pi\)
\(314\) 0 0
\(315\) −6.00000 + 12.0000i −0.338062 + 0.676123i
\(316\) 0 0
\(317\) 28.0000i 1.57264i 0.617822 + 0.786318i \(0.288015\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 40.0000i 2.22566i
\(324\) 0 0
\(325\) 4.00000 3.00000i 0.221880 0.166410i
\(326\) 0 0
\(327\) 21.0000i 1.16130i
\(328\) 0 0
\(329\) −11.0000 −0.606450
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 60.0000i 3.28798i
\(334\) 0 0
\(335\) −10.0000 + 20.0000i −0.546358 + 1.09272i
\(336\) 0 0
\(337\) 22.0000i 1.19842i 0.800593 + 0.599208i \(0.204518\pi\)
−0.800593 + 0.599208i \(0.795482\pi\)
\(338\) 0 0
\(339\) 30.0000 1.62938
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 12.0000 + 6.00000i 0.646058 + 0.323029i
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) −36.0000 −1.92704 −0.963518 0.267644i \(-0.913755\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) 9.00000i 0.479022i 0.970894 + 0.239511i \(0.0769871\pi\)
−0.970894 + 0.239511i \(0.923013\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 15.0000i 0.793884i
\(358\) 0 0
\(359\) −28.0000 −1.47778 −0.738892 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 6.00000i 0.314918i
\(364\) 0 0
\(365\) 10.0000 20.0000i 0.523424 1.04685i
\(366\) 0 0
\(367\) 19.0000i 0.991792i 0.868382 + 0.495896i \(0.165160\pi\)
−0.868382 + 0.495896i \(0.834840\pi\)
\(368\) 0 0
\(369\) 36.0000 1.87409
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 32.0000i 1.65690i −0.560065 0.828449i \(-0.689224\pi\)
0.560065 0.828449i \(-0.310776\pi\)
\(374\) 0 0
\(375\) −33.0000 + 6.00000i −1.70411 + 0.309839i
\(376\) 0 0
\(377\) 1.00000i 0.0515026i
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 3.00000 6.00000i 0.152894 0.305788i
\(386\) 0 0
\(387\) 24.0000i 1.21999i
\(388\) 0 0
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) −14.0000 7.00000i −0.704416 0.352208i
\(396\) 0 0
\(397\) 17.0000i 0.853206i −0.904439 0.426603i \(-0.859710\pi\)
0.904439 0.426603i \(-0.140290\pi\)
\(398\) 0 0
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) 2.00000i 0.0996271i
\(404\) 0 0
\(405\) −18.0000 9.00000i −0.894427 0.447214i
\(406\) 0 0
\(407\) 30.0000i 1.48704i
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 10.0000i 0.492068i
\(414\) 0 0
\(415\) −12.0000 + 24.0000i −0.589057 + 1.17811i
\(416\) 0 0
\(417\) 30.0000i 1.46911i
\(418\) 0 0
\(419\) −2.00000 −0.0977064 −0.0488532 0.998806i \(-0.515557\pi\)
−0.0488532 + 0.998806i \(0.515557\pi\)
\(420\) 0 0
\(421\) −23.0000 −1.12095 −0.560476 0.828171i \(-0.689382\pi\)
−0.560476 + 0.828171i \(0.689382\pi\)
\(422\) 0 0
\(423\) 66.0000i 3.20903i
\(424\) 0 0
\(425\) 20.0000 15.0000i 0.970143 0.727607i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −9.00000 −0.434524
\(430\) 0 0
\(431\) −37.0000 −1.78223 −0.891114 0.453780i \(-0.850075\pi\)
−0.891114 + 0.453780i \(0.850075\pi\)
\(432\) 0 0
\(433\) 38.0000i 1.82616i 0.407777 + 0.913082i \(0.366304\pi\)
−0.407777 + 0.913082i \(0.633696\pi\)
\(434\) 0 0
\(435\) −3.00000 + 6.00000i −0.143839 + 0.287678i
\(436\) 0 0
\(437\) 16.0000i 0.765384i
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 12.0000i 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) 16.0000 + 8.00000i 0.758473 + 0.379236i
\(446\) 0 0
\(447\) 18.0000i 0.851371i
\(448\) 0 0
\(449\) −11.0000 −0.519122 −0.259561 0.965727i \(-0.583578\pi\)
−0.259561 + 0.965727i \(0.583578\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 27.0000i 1.26857i
\(454\) 0 0
\(455\) −2.00000 1.00000i −0.0937614 0.0468807i
\(456\) 0 0
\(457\) 22.0000i 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) 0 0
\(459\) 45.0000 2.10042
\(460\) 0 0
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) 0 0
\(465\) 6.00000 12.0000i 0.278243 0.556487i
\(466\) 0 0
\(467\) 23.0000i 1.06431i −0.846646 0.532157i \(-0.821382\pi\)
0.846646 0.532157i \(-0.178618\pi\)
\(468\) 0 0
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) −54.0000 −2.48819
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 24.0000 + 32.0000i 1.10120 + 1.46826i
\(476\) 0 0
\(477\) 36.0000i 1.64833i
\(478\) 0 0
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 3.00000 6.00000i 0.136223 0.272446i
\(486\) 0 0
\(487\) 26.0000i 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) 0 0
\(489\) −18.0000 −0.813988
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 0 0
\(493\) 5.00000i 0.225189i
\(494\) 0 0
\(495\) 36.0000 + 18.0000i 1.61808 + 0.809040i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 29.0000 1.29822 0.649109 0.760695i \(-0.275142\pi\)
0.649109 + 0.760695i \(0.275142\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 0 0
\(503\) 1.00000i 0.0445878i −0.999751 0.0222939i \(-0.992903\pi\)
0.999751 0.0222939i \(-0.00709696\pi\)
\(504\) 0 0
\(505\) 24.0000 + 12.0000i 1.06799 + 0.533993i
\(506\) 0 0
\(507\) 36.0000i 1.59882i
\(508\) 0 0
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 72.0000i 3.17888i
\(514\) 0 0
\(515\) 5.00000 10.0000i 0.220326 0.440653i
\(516\) 0 0
\(517\) 33.0000i 1.45134i
\(518\) 0 0
\(519\) 27.0000 1.18517
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 0 0
\(525\) 9.00000 + 12.0000i 0.392792 + 0.523723i
\(526\) 0 0
\(527\) 10.0000i 0.435607i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) −60.0000 −2.60378
\(532\) 0 0
\(533\) 6.00000i 0.259889i
\(534\) 0 0
\(535\) 8.00000 16.0000i 0.345870 0.691740i
\(536\) 0 0
\(537\) 12.0000i 0.517838i
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) 30.0000i 1.28742i
\(544\) 0 0
\(545\) −14.0000 7.00000i −0.599694 0.299847i
\(546\) 0 0
\(547\) 8.00000i 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 7.00000i 0.297670i
\(554\) 0 0
\(555\) −60.0000 30.0000i −2.54686 1.27343i
\(556\) 0 0
\(557\) 20.0000i 0.847427i 0.905796 + 0.423714i \(0.139274\pi\)
−0.905796 + 0.423714i \(0.860726\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −45.0000 −1.89990
\(562\) 0 0
\(563\) 16.0000i 0.674320i 0.941447 + 0.337160i \(0.109466\pi\)
−0.941447 + 0.337160i \(0.890534\pi\)
\(564\) 0 0
\(565\) 10.0000 20.0000i 0.420703 0.841406i
\(566\) 0 0
\(567\) 9.00000i 0.377964i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 21.0000i 0.877288i
\(574\) 0 0
\(575\) 8.00000 6.00000i 0.333623 0.250217i
\(576\) 0 0
\(577\) 17.0000i 0.707719i 0.935299 + 0.353860i \(0.115131\pi\)
−0.935299 + 0.353860i \(0.884869\pi\)
\(578\) 0 0
\(579\) −24.0000 −0.997406
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 18.0000i 0.745484i
\(584\) 0 0
\(585\) 6.00000 12.0000i 0.248069 0.496139i
\(586\) 0 0
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 30.0000 1.23404
\(592\) 0 0
\(593\) 3.00000i 0.123195i 0.998101 + 0.0615976i \(0.0196196\pi\)
−0.998101 + 0.0615976i \(0.980380\pi\)
\(594\) 0 0
\(595\) −10.0000 5.00000i −0.409960 0.204980i
\(596\) 0 0
\(597\) 54.0000i 2.21007i
\(598\) 0 0
\(599\) 21.0000 0.858037 0.429018 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) 60.0000i 2.44339i
\(604\) 0 0
\(605\) 4.00000 + 2.00000i 0.162623 + 0.0813116i
\(606\) 0 0
\(607\) 5.00000i 0.202944i 0.994838 + 0.101472i \(0.0323552\pi\)
−0.994838 + 0.101472i \(0.967645\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) 11.0000 0.445012
\(612\) 0 0
\(613\) 12.0000i 0.484675i 0.970192 + 0.242338i \(0.0779142\pi\)
−0.970192 + 0.242338i \(0.922086\pi\)
\(614\) 0 0
\(615\) 18.0000 36.0000i 0.725830 1.45166i
\(616\) 0 0
\(617\) 34.0000i 1.36879i −0.729112 0.684394i \(-0.760067\pi\)
0.729112 0.684394i \(-0.239933\pi\)
\(618\) 0 0
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) 18.0000 0.722315
\(622\) 0 0
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 72.0000i 2.87540i
\(628\) 0 0
\(629\) 50.0000 1.99363
\(630\) 0 0
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 0 0
\(633\) 9.00000i 0.357718i
\(634\) 0 0
\(635\) −2.00000 + 4.00000i −0.0793676 + 0.158735i
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) 5.00000i 0.197181i 0.995128 + 0.0985904i \(0.0314334\pi\)
−0.995128 + 0.0985904i \(0.968567\pi\)
\(644\) 0 0
\(645\) −24.0000 12.0000i −0.944999 0.472500i
\(646\) 0 0
\(647\) 24.0000i 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) 36.0000i 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) 0 0
\(655\) −4.00000 2.00000i −0.156293 0.0781465i
\(656\) 0 0
\(657\) 60.0000i 2.34082i
\(658\) 0 0
\(659\) −39.0000 −1.51922 −0.759612 0.650376i \(-0.774611\pi\)
−0.759612 + 0.650376i \(0.774611\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 0 0
\(663\) 15.0000i 0.582552i
\(664\) 0 0
\(665\) 8.00000 16.0000i 0.310227 0.620453i
\(666\) 0 0
\(667\) 2.00000i 0.0774403i
\(668\) 0 0
\(669\) −57.0000 −2.20375
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 16.0000i 0.616755i 0.951264 + 0.308377i \(0.0997859\pi\)
−0.951264 + 0.308377i \(0.900214\pi\)
\(674\) 0 0
\(675\) −36.0000 + 27.0000i −1.38564 + 1.03923i
\(676\) 0 0
\(677\) 11.0000i 0.422764i −0.977403 0.211382i \(-0.932204\pi\)
0.977403 0.211382i \(-0.0677965\pi\)
\(678\) 0 0
\(679\) −3.00000 −0.115129
\(680\) 0 0
\(681\) 81.0000 3.10393
\(682\) 0 0
\(683\) 40.0000i 1.53056i −0.643699 0.765279i \(-0.722601\pi\)
0.643699 0.765279i \(-0.277399\pi\)
\(684\) 0 0
\(685\) 4.00000 8.00000i 0.152832 0.305664i
\(686\) 0 0
\(687\) 78.0000i 2.97589i
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) 0 0
\(693\) 18.0000i 0.683763i
\(694\) 0 0
\(695\) 20.0000 + 10.0000i 0.758643 + 0.379322i
\(696\) 0 0
\(697\) 30.0000i 1.13633i
\(698\) 0 0
\(699\) −48.0000 −1.81553
\(700\) 0 0
\(701\) −25.0000 −0.944237 −0.472118 0.881535i \(-0.656511\pi\)
−0.472118 + 0.881535i \(0.656511\pi\)
\(702\) 0 0
\(703\) 80.0000i 3.01726i
\(704\) 0 0
\(705\) −66.0000 33.0000i −2.48570 1.24285i
\(706\) 0 0
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 0 0
\(711\) −42.0000 −1.57512
\(712\) 0 0
\(713\) 4.00000i 0.149801i
\(714\) 0 0
\(715\) −3.00000 + 6.00000i −0.112194 + 0.224387i
\(716\) 0 0
\(717\) 15.0000i 0.560185i
\(718\) 0 0
\(719\) −2.00000 −0.0745874 −0.0372937 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 0 0
\(721\) −5.00000 −0.186210
\(722\) 0 0
\(723\) 54.0000i 2.00828i
\(724\) 0 0
\(725\) 3.00000 + 4.00000i 0.111417 + 0.148556i
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) 0 0
\(733\) 41.0000i 1.51437i −0.653201 0.757185i \(-0.726574\pi\)
0.653201 0.757185i \(-0.273426\pi\)
\(734\) 0 0
\(735\) 3.00000 6.00000i 0.110657 0.221313i
\(736\) 0 0
\(737\) 30.0000i 1.10506i
\(738\) 0 0
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) 30.0000i 1.10059i −0.834969 0.550297i \(-0.814515\pi\)
0.834969 0.550297i \(-0.185485\pi\)
\(744\) 0 0
\(745\) −12.0000 6.00000i −0.439646 0.219823i
\(746\) 0 0
\(747\) 72.0000i 2.63434i
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) 0 0
\(753\) 6.00000i 0.218652i
\(754\) 0 0
\(755\) −18.0000 9.00000i −0.655087 0.327544i
\(756\) 0 0
\(757\) 48.0000i 1.74459i −0.488980 0.872295i \(-0.662631\pi\)
0.488980 0.872295i \(-0.337369\pi\)
\(758\) 0 0
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 7.00000i 0.253417i
\(764\) 0 0
\(765\) 30.0000 60.0000i 1.08465 2.16930i
\(766\) 0 0
\(767\) 10.0000i 0.361079i
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 27.0000i 0.971123i 0.874203 + 0.485561i \(0.161385\pi\)
−0.874203 + 0.485561i \(0.838615\pi\)
\(774\) 0 0
\(775\) −6.00000 8.00000i −0.215526 0.287368i
\(776\) 0 0
\(777\) 30.0000i 1.07624i
\(778\) 0 0
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 9.00000i 0.321634i
\(784\) 0 0
\(785\) −18.0000 + 36.0000i −0.642448 + 1.28490i
\(786\) 0 0
\(787\) 3.00000i 0.106938i 0.998569 + 0.0534692i \(0.0170279\pi\)
−0.998569 + 0.0534692i \(0.982972\pi\)
\(788\) 0 0
\(789\) 72.0000 2.56327
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 36.0000 + 18.0000i 1.27679 + 0.638394i
\(796\) 0 0
\(797\) 43.0000i 1.52314i 0.648084 + 0.761569i \(0.275571\pi\)
−0.648084 + 0.761569i \(0.724429\pi\)
\(798\) 0 0
\(799\) 55.0000