Properties

Label 2240.2.g.b.449.1
Level $2240$
Weight $2$
Character 2240.449
Analytic conductor $17.886$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2240.449
Dual form 2240.2.g.b.449.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(-2.00000 + 1.00000i) q^{5} +1.00000i q^{7} +2.00000 q^{9} +1.00000 q^{11} +1.00000i q^{13} +(1.00000 + 2.00000i) q^{15} +3.00000i q^{17} -4.00000 q^{19} +1.00000 q^{21} +2.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -5.00000i q^{27} -1.00000 q^{29} -6.00000 q^{31} -1.00000i q^{33} +(-1.00000 - 2.00000i) q^{35} +2.00000i q^{37} +1.00000 q^{39} -10.0000 q^{41} +(-4.00000 + 2.00000i) q^{45} -9.00000i q^{47} -1.00000 q^{49} +3.00000 q^{51} +14.0000i q^{53} +(-2.00000 + 1.00000i) q^{55} +4.00000i q^{57} +6.00000 q^{59} +4.00000 q^{61} +2.00000i q^{63} +(-1.00000 - 2.00000i) q^{65} +10.0000i q^{67} +2.00000 q^{69} -16.0000 q^{71} +10.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +1.00000i q^{77} +11.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(-3.00000 - 6.00000i) q^{85} +1.00000i q^{87} -12.0000 q^{89} -1.00000 q^{91} +6.00000i q^{93} +(8.00000 - 4.00000i) q^{95} +19.0000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + 4q^{9} + O(q^{10}) \) \( 2q - 4q^{5} + 4q^{9} + 2q^{11} + 2q^{15} - 8q^{19} + 2q^{21} + 6q^{25} - 2q^{29} - 12q^{31} - 2q^{35} + 2q^{39} - 20q^{41} - 8q^{45} - 2q^{49} + 6q^{51} - 4q^{55} + 12q^{59} + 8q^{61} - 2q^{65} + 4q^{69} - 32q^{71} - 8q^{75} + 22q^{79} + 2q^{81} - 6q^{85} - 24q^{89} - 2q^{91} + 16q^{95} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i −0.957427 0.288675i \(-0.906785\pi\)
0.957427 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) −2.00000 + 1.00000i −0.894427 + 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 1.00000 + 2.00000i 0.258199 + 0.516398i
\(16\) 0 0
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 1.00000i 0.174078i
\(34\) 0 0
\(35\) −1.00000 2.00000i −0.169031 0.338062i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −4.00000 + 2.00000i −0.596285 + 0.298142i
\(46\) 0 0
\(47\) 9.00000i 1.31278i −0.754420 0.656392i \(-0.772082\pi\)
0.754420 0.656392i \(-0.227918\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 14.0000i 1.92305i 0.274721 + 0.961524i \(0.411414\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −2.00000 + 1.00000i −0.269680 + 0.134840i
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) −1.00000 2.00000i −0.124035 0.248069i
\(66\) 0 0
\(67\) 10.0000i 1.22169i 0.791748 + 0.610847i \(0.209171\pi\)
−0.791748 + 0.610847i \(0.790829\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) −4.00000 3.00000i −0.461880 0.346410i
\(76\) 0 0
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) −3.00000 6.00000i −0.325396 0.650791i
\(86\) 0 0
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 6.00000i 0.622171i
\(94\) 0 0
\(95\) 8.00000 4.00000i 0.820783 0.410391i
\(96\) 0 0
\(97\) 19.0000i 1.92916i 0.263795 + 0.964579i \(0.415026\pi\)
−0.263795 + 0.964579i \(0.584974\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1.00000i 0.0985329i 0.998786 + 0.0492665i \(0.0156884\pi\)
−0.998786 + 0.0492665i \(0.984312\pi\)
\(104\) 0 0
\(105\) −2.00000 + 1.00000i −0.195180 + 0.0975900i
\(106\) 0 0
\(107\) 16.0000i 1.54678i 0.633932 + 0.773389i \(0.281440\pi\)
−0.633932 + 0.773389i \(0.718560\pi\)
\(108\) 0 0
\(109\) −15.0000 −1.43674 −0.718370 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −2.00000 4.00000i −0.186501 0.373002i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) −2.00000 + 11.0000i −0.178885 + 0.983870i
\(126\) 0 0
\(127\) 18.0000i 1.59724i −0.601834 0.798621i \(-0.705563\pi\)
0.601834 0.798621i \(-0.294437\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) 5.00000 + 10.0000i 0.430331 + 0.860663i
\(136\) 0 0
\(137\) 8.00000i 0.683486i 0.939793 + 0.341743i \(0.111017\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) 0 0
\(143\) 1.00000i 0.0836242i
\(144\) 0 0
\(145\) 2.00000 1.00000i 0.166091 0.0830455i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 13.0000 1.05792 0.528962 0.848645i \(-0.322581\pi\)
0.528962 + 0.848645i \(0.322581\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) 12.0000 6.00000i 0.963863 0.481932i
\(156\) 0 0
\(157\) 14.0000i 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 14.0000 1.11027
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 10.0000i 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 0 0
\(165\) 1.00000 + 2.00000i 0.0778499 + 0.155700i
\(166\) 0 0
\(167\) 15.0000i 1.16073i 0.814355 + 0.580367i \(0.197091\pi\)
−0.814355 + 0.580367i \(0.802909\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 0 0
\(173\) 15.0000i 1.14043i 0.821496 + 0.570214i \(0.193140\pi\)
−0.821496 + 0.570214i \(0.806860\pi\)
\(174\) 0 0
\(175\) 4.00000 + 3.00000i 0.302372 + 0.226779i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 4.00000i 0.295689i
\(184\) 0 0
\(185\) −2.00000 4.00000i −0.147043 0.294086i
\(186\) 0 0
\(187\) 3.00000i 0.219382i
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) 0 0
\(195\) −2.00000 + 1.00000i −0.143223 + 0.0716115i
\(196\) 0 0
\(197\) 6.00000i 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 10.0000 0.705346
\(202\) 0 0
\(203\) 1.00000i 0.0701862i
\(204\) 0 0
\(205\) 20.0000 10.0000i 1.39686 0.698430i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 0 0
\(213\) 16.0000i 1.09630i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000i 0.407307i
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) 15.0000i 1.00447i −0.864730 0.502237i \(-0.832510\pi\)
0.864730 0.502237i \(-0.167490\pi\)
\(224\) 0 0
\(225\) 6.00000 8.00000i 0.400000 0.533333i
\(226\) 0 0
\(227\) 7.00000i 0.464606i −0.972643 0.232303i \(-0.925374\pi\)
0.972643 0.232303i \(-0.0746261\pi\)
\(228\) 0 0
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 9.00000 + 18.0000i 0.587095 + 1.17419i
\(236\) 0 0
\(237\) 11.0000i 0.714527i
\(238\) 0 0
\(239\) −17.0000 −1.09964 −0.549819 0.835284i \(-0.685303\pi\)
−0.549819 + 0.835284i \(0.685303\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 2.00000 1.00000i 0.127775 0.0638877i
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) −6.00000 + 3.00000i −0.375735 + 0.187867i
\(256\) 0 0
\(257\) 18.0000i 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 24.0000i 1.47990i 0.672660 + 0.739952i \(0.265152\pi\)
−0.672660 + 0.739952i \(0.734848\pi\)
\(264\) 0 0
\(265\) −14.0000 28.0000i −0.860013 1.72003i
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) 1.00000i 0.0605228i
\(274\) 0 0
\(275\) 3.00000 4.00000i 0.180907 0.241209i
\(276\) 0 0
\(277\) 18.0000i 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 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
\(283\) 21.0000i 1.24832i 0.781296 + 0.624160i \(0.214559\pi\)
−0.781296 + 0.624160i \(0.785441\pi\)
\(284\) 0 0
\(285\) −4.00000 8.00000i −0.236940 0.473879i
\(286\) 0 0
\(287\) 10.0000i 0.590281i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 19.0000 1.11380
\(292\) 0 0
\(293\) 31.0000i 1.81104i 0.424304 + 0.905520i \(0.360519\pi\)
−0.424304 + 0.905520i \(0.639481\pi\)
\(294\) 0 0
\(295\) −12.0000 + 6.00000i −0.698667 + 0.349334i
\(296\) 0 0
\(297\) 5.00000i 0.290129i
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.00000 + 4.00000i −0.458079 + 0.229039i
\(306\) 0 0
\(307\) 25.0000i 1.42683i −0.700744 0.713413i \(-0.747149\pi\)
0.700744 0.713413i \(-0.252851\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 7.00000i 0.395663i 0.980236 + 0.197832i \(0.0633900\pi\)
−0.980236 + 0.197832i \(0.936610\pi\)
\(314\) 0 0
\(315\) −2.00000 4.00000i −0.112687 0.225374i
\(316\) 0 0
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) −1.00000 −0.0559893
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 4.00000 + 3.00000i 0.221880 + 0.166410i
\(326\) 0 0
\(327\) 15.0000i 0.829502i
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) −10.0000 20.0000i −0.546358 1.09272i
\(336\) 0 0
\(337\) 34.0000i 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) 0 0
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −4.00000 + 2.00000i −0.215353 + 0.107676i
\(346\) 0 0
\(347\) 22.0000i 1.18102i −0.807030 0.590511i \(-0.798926\pi\)
0.807030 0.590511i \(-0.201074\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) 9.00000i 0.479022i 0.970894 + 0.239511i \(0.0769871\pi\)
−0.970894 + 0.239511i \(0.923013\pi\)
\(354\) 0 0
\(355\) 32.0000 16.0000i 1.69838 0.849192i
\(356\) 0 0
\(357\) 3.00000i 0.158777i
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) −10.0000 20.0000i −0.523424 1.04685i
\(366\) 0 0
\(367\) 17.0000i 0.887393i −0.896177 0.443696i \(-0.853667\pi\)
0.896177 0.443696i \(-0.146333\pi\)
\(368\) 0 0
\(369\) −20.0000 −1.04116
\(370\) 0 0
\(371\) −14.0000 −0.726844
\(372\) 0 0
\(373\) 20.0000i 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 0 0
\(375\) 11.0000 + 2.00000i 0.568038 + 0.103280i
\(376\) 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
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 0 0
\(385\) −1.00000 2.00000i −0.0509647 0.101929i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 6.00000i 0.302660i
\(394\) 0 0
\(395\) −22.0000 + 11.0000i −1.10694 + 0.553470i
\(396\) 0 0
\(397\) 15.0000i 0.752828i −0.926451 0.376414i \(-0.877157\pi\)
0.926451 0.376414i \(-0.122843\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 13.0000 0.649189 0.324595 0.945853i \(-0.394772\pi\)
0.324595 + 0.945853i \(0.394772\pi\)
\(402\) 0 0
\(403\) 6.00000i 0.298881i
\(404\) 0 0
\(405\) −2.00000 + 1.00000i −0.0993808 + 0.0496904i
\(406\) 0 0
\(407\) 2.00000i 0.0991363i
\(408\) 0 0
\(409\) 28.0000 1.38451 0.692255 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) 0 0
\(411\) 8.00000 0.394611
\(412\) 0 0
\(413\) 6.00000i 0.295241i
\(414\) 0 0
\(415\) 4.00000 + 8.00000i 0.196352 + 0.392705i
\(416\) 0 0
\(417\) 14.0000i 0.685583i
\(418\) 0 0
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) 18.0000i 0.875190i
\(424\) 0 0
\(425\) 12.0000 + 9.00000i 0.582086 + 0.436564i
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 31.0000 1.49322 0.746609 0.665263i \(-0.231681\pi\)
0.746609 + 0.665263i \(0.231681\pi\)
\(432\) 0 0
\(433\) 6.00000i 0.288342i 0.989553 + 0.144171i \(0.0460515\pi\)
−0.989553 + 0.144171i \(0.953949\pi\)
\(434\) 0 0
\(435\) −1.00000 2.00000i −0.0479463 0.0958927i
\(436\) 0 0
\(437\) 8.00000i 0.382692i
\(438\) 0 0
\(439\) 30.0000 1.43182 0.715911 0.698192i \(-0.246012\pi\)
0.715911 + 0.698192i \(0.246012\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 24.0000 12.0000i 1.13771 0.568855i
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 0 0
\(453\) 13.0000i 0.610793i
\(454\) 0 0
\(455\) 2.00000 1.00000i 0.0937614 0.0468807i
\(456\) 0 0
\(457\) 18.0000i 0.842004i −0.907060 0.421002i \(-0.861678\pi\)
0.907060 0.421002i \(-0.138322\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) −6.00000 12.0000i −0.278243 0.556487i
\(466\) 0 0
\(467\) 27.0000i 1.24941i 0.780860 + 0.624705i \(0.214781\pi\)
−0.780860 + 0.624705i \(0.785219\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −12.0000 + 16.0000i −0.550598 + 0.734130i
\(476\) 0 0
\(477\) 28.0000i 1.28203i
\(478\) 0 0
\(479\) −2.00000 −0.0913823 −0.0456912 0.998956i \(-0.514549\pi\)
−0.0456912 + 0.998956i \(0.514549\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 2.00000i 0.0910032i
\(484\) 0 0
\(485\) −19.0000 38.0000i −0.862746 1.72549i
\(486\) 0 0
\(487\) 38.0000i 1.72194i 0.508652 + 0.860972i \(0.330144\pi\)
−0.508652 + 0.860972i \(0.669856\pi\)
\(488\) 0 0
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −5.00000 −0.225647 −0.112823 0.993615i \(-0.535989\pi\)
−0.112823 + 0.993615i \(0.535989\pi\)
\(492\) 0 0
\(493\) 3.00000i 0.135113i
\(494\) 0 0
\(495\) −4.00000 + 2.00000i −0.179787 + 0.0898933i
\(496\) 0 0
\(497\) 16.0000i 0.717698i
\(498\) 0 0
\(499\) 23.0000 1.02962 0.514811 0.857304i \(-0.327862\pi\)
0.514811 + 0.857304i \(0.327862\pi\)
\(500\) 0 0
\(501\) 15.0000 0.670151
\(502\) 0 0
\(503\) 29.0000i 1.29305i −0.762894 0.646523i \(-0.776222\pi\)
0.762894 0.646523i \(-0.223778\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000i 0.532939i
\(508\) 0 0
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 20.0000i 0.883022i
\(514\) 0 0
\(515\) −1.00000 2.00000i −0.0440653 0.0881305i
\(516\) 0 0
\(517\) 9.00000i 0.395820i
\(518\) 0 0
\(519\) 15.0000 0.658427
\(520\) 0 0
\(521\) −8.00000 −0.350486 −0.175243 0.984525i \(-0.556071\pi\)
−0.175243 + 0.984525i \(0.556071\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) 3.00000 4.00000i 0.130931 0.174574i
\(526\) 0 0
\(527\) 18.0000i 0.784092i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 10.0000i 0.433148i
\(534\) 0 0
\(535\) −16.0000 32.0000i −0.691740 1.38348i
\(536\) 0 0
\(537\) 20.0000i 0.863064i
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 41.0000 1.76273 0.881364 0.472438i \(-0.156626\pi\)
0.881364 + 0.472438i \(0.156626\pi\)
\(542\) 0 0
\(543\) 2.00000i 0.0858282i
\(544\) 0 0
\(545\) 30.0000 15.0000i 1.28506 0.642529i
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 0 0
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) 0 0
\(553\) 11.0000i 0.467768i
\(554\) 0 0
\(555\) −4.00000 + 2.00000i −0.169791 + 0.0848953i
\(556\) 0 0
\(557\) 36.0000i 1.52537i −0.646771 0.762684i \(-0.723881\pi\)
0.646771 0.762684i \(-0.276119\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 3.00000 0.126660
\(562\) 0 0
\(563\) 24.0000i 1.01148i −0.862686 0.505740i \(-0.831220\pi\)
0.862686 0.505740i \(-0.168780\pi\)
\(564\) 0 0
\(565\) −14.0000 28.0000i −0.588984 1.17797i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 3.00000i 0.125327i
\(574\) 0 0
\(575\) 8.00000 + 6.00000i 0.333623 + 0.250217i
\(576\) 0 0
\(577\) 31.0000i 1.29055i −0.763952 0.645273i \(-0.776743\pi\)
0.763952 0.645273i \(-0.223257\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 14.0000i 0.579821i
\(584\) 0 0
\(585\) −2.00000 4.00000i −0.0826898 0.165380i
\(586\) 0 0
\(587\) 20.0000i 0.825488i −0.910847 0.412744i \(-0.864570\pi\)
0.910847 0.412744i \(-0.135430\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 19.0000i 0.780236i 0.920765 + 0.390118i \(0.127566\pi\)
−0.920765 + 0.390118i \(0.872434\pi\)
\(594\) 0 0
\(595\) 6.00000 3.00000i 0.245976 0.122988i
\(596\) 0 0
\(597\) 10.0000i 0.409273i
\(598\) 0 0
\(599\) −7.00000 −0.286012 −0.143006 0.989722i \(-0.545677\pi\)
−0.143006 + 0.989722i \(0.545677\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 20.0000i 0.814463i
\(604\) 0 0
\(605\) 20.0000 10.0000i 0.813116 0.406558i
\(606\) 0 0
\(607\) 1.00000i 0.0405887i 0.999794 + 0.0202944i \(0.00646034\pi\)
−0.999794 + 0.0202944i \(0.993540\pi\)
\(608\) 0 0
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 4.00000i 0.161558i 0.996732 + 0.0807792i \(0.0257409\pi\)
−0.996732 + 0.0807792i \(0.974259\pi\)
\(614\) 0 0
\(615\) −10.0000 20.0000i −0.403239 0.806478i
\(616\) 0 0
\(617\) 2.00000i 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) 0 0
\(619\) −38.0000 −1.52735 −0.763674 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) 0 0
\(621\) 10.0000 0.401286
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 4.00000i 0.159745i
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) 0 0
\(633\) 1.00000i 0.0397464i
\(634\) 0 0
\(635\) 18.0000 + 36.0000i 0.714308 + 1.42862i
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 0 0
\(639\) −32.0000 −1.26590
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 15.0000i 0.591542i 0.955259 + 0.295771i \(0.0955766\pi\)
−0.955259 + 0.295771i \(0.904423\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.0000i 1.25805i 0.777385 + 0.629025i \(0.216546\pi\)
−0.777385 + 0.629025i \(0.783454\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) 24.0000i 0.939193i 0.882881 + 0.469596i \(0.155601\pi\)
−0.882881 + 0.469596i \(0.844399\pi\)
\(654\) 0 0
\(655\) 12.0000 6.00000i 0.468879 0.234439i
\(656\) 0 0
\(657\) 20.0000i 0.780274i
\(658\) 0 0
\(659\) −21.0000 −0.818044 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 0 0
\(663\) 3.00000i 0.116510i
\(664\) 0 0
\(665\) 4.00000 + 8.00000i 0.155113 + 0.310227i
\(666\) 0 0
\(667\) 2.00000i 0.0774403i
\(668\) 0 0
\(669\) −15.0000 −0.579934
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 20.0000i 0.770943i −0.922720 0.385472i \(-0.874039\pi\)
0.922720 0.385472i \(-0.125961\pi\)
\(674\) 0 0
\(675\) −20.0000 15.0000i −0.769800 0.577350i
\(676\) 0 0
\(677\) 3.00000i 0.115299i 0.998337 + 0.0576497i \(0.0183606\pi\)
−0.998337 + 0.0576497i \(0.981639\pi\)
\(678\) 0 0
\(679\) −19.0000 −0.729153
\(680\) 0 0
\(681\) −7.00000 −0.268241
\(682\) 0 0
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 0 0
\(685\) −8.00000 16.0000i −0.305664 0.611329i
\(686\) 0 0
\(687\) 18.0000i 0.686743i
\(688\) 0 0
\(689\) −14.0000 −0.533358
\(690\) 0 0
\(691\) 48.0000 1.82601 0.913003 0.407953i \(-0.133757\pi\)
0.913003 + 0.407953i \(0.133757\pi\)
\(692\) 0 0
\(693\) 2.00000i 0.0759737i
\(694\) 0 0
\(695\) 28.0000 14.0000i 1.06210 0.531050i
\(696\) 0 0
\(697\) 30.0000i 1.13633i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 41.0000 1.54855 0.774274 0.632850i \(-0.218115\pi\)
0.774274 + 0.632850i \(0.218115\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 18.0000 9.00000i 0.677919 0.338960i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.0000 −0.638448 −0.319224 0.947679i \(-0.603422\pi\)
−0.319224 + 0.947679i \(0.603422\pi\)
\(710\) 0 0
\(711\) 22.0000 0.825064
\(712\) 0 0
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) −1.00000 2.00000i −0.0373979 0.0747958i
\(716\) 0 0
\(717\) 17.0000i 0.634877i
\(718\) 0 0
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) −3.00000 + 4.00000i −0.111417 + 0.148556i
\(726\) 0 0
\(727\) 36.0000i 1.33517i 0.744535 + 0.667583i \(0.232671\pi\)
−0.744535 + 0.667583i \(0.767329\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 9.00000i 0.332423i 0.986090 + 0.166211i \(0.0531534\pi\)
−0.986090 + 0.166211i \(0.946847\pi\)
\(734\) 0 0
\(735\) −1.00000 2.00000i −0.0368856 0.0737711i
\(736\) 0 0
\(737\) 10.0000i 0.368355i
\(738\) 0 0
\(739\) 15.0000 0.551784 0.275892 0.961189i \(-0.411027\pi\)
0.275892 + 0.961189i \(0.411027\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 42.0000i 1.54083i 0.637542 + 0.770415i \(0.279951\pi\)
−0.637542 + 0.770415i \(0.720049\pi\)
\(744\) 0 0
\(745\) 12.0000 6.00000i 0.439646 0.219823i
\(746\) 0 0
\(747\) 8.00000i 0.292705i
\(748\) 0 0
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 17.0000 0.620339 0.310169 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) 0 0
\(753\) 10.0000i 0.364420i
\(754\) 0 0
\(755\) −26.0000 + 13.0000i −0.946237 + 0.473118i
\(756\) 0 0
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) 0 0
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 15.0000i 0.543036i
\(764\) 0 0
\(765\) −6.00000 12.0000i −0.216930 0.433861i
\(766\) 0 0
\(767\) 6.00000i 0.216647i
\(768\) 0 0
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 13.0000i 0.467578i 0.972287 + 0.233789i \(0.0751124\pi\)
−0.972287 + 0.233789i \(0.924888\pi\)
\(774\) 0 0
\(775\) −18.0000 + 24.0000i −0.646579 + 0.862105i
\(776\) 0 0
\(777\) 2.00000i 0.0717496i
\(778\) 0 0
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 5.00000i 0.178685i
\(784\) 0 0
\(785\) 14.0000 + 28.0000i 0.499681 + 0.999363i
\(786\) 0 0
\(787\) 31.0000i 1.10503i −0.833503 0.552515i \(-0.813668\pi\)
0.833503 0.552515i \(-0.186332\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) 0 0
\(795\) −28.0000 + 14.0000i −0.993058 + 0.496529i
\(796\) 0 0
\(797\) 27.0000i 0.956389i −0.878254 0.478195i \(-0.841291\pi\)
0.878254 0.478195i \(-0.158709\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) −24.0000 −0.847998
\(802\) 0