Properties

Label 5520.2.be.b.1471.9
Level $5520$
Weight $2$
Character 5520.1471
Analytic conductor $44.077$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.be (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.0774219157\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.9
Root \(1.49885 - 1.49885i\) of defining polynomial
Character \(\chi\) \(=\) 5520.1471
Dual form 5520.2.be.b.1471.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{5} -4.84428 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{5} -4.84428 q^{7} -1.00000 q^{9} -3.09487 q^{11} -4.53183 q^{13} -1.00000 q^{15} +1.04944i q^{17} -3.05893 q^{19} -4.84428i q^{21} +(-4.79485 - 0.0971617i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -2.44565 q^{29} -3.08441i q^{31} -3.09487i q^{33} -4.84428i q^{35} +3.78545i q^{37} -4.53183i q^{39} +4.04608 q^{41} -2.53119 q^{43} -1.00000i q^{45} +3.83718i q^{47} +16.4671 q^{49} -1.04944 q^{51} +8.14281i q^{53} -3.09487i q^{55} -3.05893i q^{57} -4.64825i q^{59} -11.3650i q^{61} +4.84428 q^{63} -4.53183i q^{65} -3.77906 q^{67} +(0.0971617 - 4.79485i) q^{69} +13.2737i q^{71} +2.53642 q^{73} -1.00000i q^{75} +14.9924 q^{77} -1.04304 q^{79} +1.00000 q^{81} +4.24974 q^{83} -1.04944 q^{85} -2.44565i q^{87} +3.89654i q^{89} +21.9535 q^{91} +3.08441 q^{93} -3.05893i q^{95} -5.72909i q^{97} +3.09487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{7} - 16q^{9} + O(q^{10}) \) \( 16q + 8q^{7} - 16q^{9} - 8q^{11} + 8q^{13} - 16q^{15} - 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} + 4q^{51} - 8q^{63} + 16q^{67} + 40q^{73} + 24q^{77} - 32q^{79} + 16q^{81} + 4q^{85} + 48q^{91} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1841\) \(4417\) \(4831\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(-1\)

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
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −4.84428 −1.83097 −0.915484 0.402355i \(-0.868192\pi\)
−0.915484 + 0.402355i \(0.868192\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.09487 −0.933138 −0.466569 0.884485i \(-0.654510\pi\)
−0.466569 + 0.884485i \(0.654510\pi\)
\(12\) 0 0
\(13\) −4.53183 −1.25690 −0.628452 0.777848i \(-0.716311\pi\)
−0.628452 + 0.777848i \(0.716311\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 1.04944i 0.254526i 0.991869 + 0.127263i \(0.0406192\pi\)
−0.991869 + 0.127263i \(0.959381\pi\)
\(18\) 0 0
\(19\) −3.05893 −0.701766 −0.350883 0.936419i \(-0.614119\pi\)
−0.350883 + 0.936419i \(0.614119\pi\)
\(20\) 0 0
\(21\) 4.84428i 1.05711i
\(22\) 0 0
\(23\) −4.79485 0.0971617i −0.999795 0.0202596i
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.44565 −0.454146 −0.227073 0.973878i \(-0.572916\pi\)
−0.227073 + 0.973878i \(0.572916\pi\)
\(30\) 0 0
\(31\) 3.08441i 0.553977i −0.960873 0.276988i \(-0.910664\pi\)
0.960873 0.276988i \(-0.0893363\pi\)
\(32\) 0 0
\(33\) 3.09487i 0.538748i
\(34\) 0 0
\(35\) 4.84428i 0.818833i
\(36\) 0 0
\(37\) 3.78545i 0.622325i 0.950357 + 0.311162i \(0.100718\pi\)
−0.950357 + 0.311162i \(0.899282\pi\)
\(38\) 0 0
\(39\) 4.53183i 0.725674i
\(40\) 0 0
\(41\) 4.04608 0.631891 0.315946 0.948777i \(-0.397678\pi\)
0.315946 + 0.948777i \(0.397678\pi\)
\(42\) 0 0
\(43\) −2.53119 −0.386002 −0.193001 0.981199i \(-0.561822\pi\)
−0.193001 + 0.981199i \(0.561822\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 3.83718i 0.559711i 0.960042 + 0.279855i \(0.0902865\pi\)
−0.960042 + 0.279855i \(0.909713\pi\)
\(48\) 0 0
\(49\) 16.4671 2.35244
\(50\) 0 0
\(51\) −1.04944 −0.146950
\(52\) 0 0
\(53\) 8.14281i 1.11850i 0.828999 + 0.559251i \(0.188911\pi\)
−0.828999 + 0.559251i \(0.811089\pi\)
\(54\) 0 0
\(55\) 3.09487i 0.417312i
\(56\) 0 0
\(57\) 3.05893i 0.405165i
\(58\) 0 0
\(59\) 4.64825i 0.605151i −0.953125 0.302575i \(-0.902154\pi\)
0.953125 0.302575i \(-0.0978464\pi\)
\(60\) 0 0
\(61\) 11.3650i 1.45514i −0.686033 0.727570i \(-0.740649\pi\)
0.686033 0.727570i \(-0.259351\pi\)
\(62\) 0 0
\(63\) 4.84428 0.610322
\(64\) 0 0
\(65\) 4.53183i 0.562105i
\(66\) 0 0
\(67\) −3.77906 −0.461686 −0.230843 0.972991i \(-0.574148\pi\)
−0.230843 + 0.972991i \(0.574148\pi\)
\(68\) 0 0
\(69\) 0.0971617 4.79485i 0.0116969 0.577232i
\(70\) 0 0
\(71\) 13.2737i 1.57530i 0.616122 + 0.787651i \(0.288703\pi\)
−0.616122 + 0.787651i \(0.711297\pi\)
\(72\) 0 0
\(73\) 2.53642 0.296865 0.148433 0.988923i \(-0.452577\pi\)
0.148433 + 0.988923i \(0.452577\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 14.9924 1.70855
\(78\) 0 0
\(79\) −1.04304 −0.117351 −0.0586754 0.998277i \(-0.518688\pi\)
−0.0586754 + 0.998277i \(0.518688\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.24974 0.466470 0.233235 0.972420i \(-0.425069\pi\)
0.233235 + 0.972420i \(0.425069\pi\)
\(84\) 0 0
\(85\) −1.04944 −0.113827
\(86\) 0 0
\(87\) 2.44565i 0.262202i
\(88\) 0 0
\(89\) 3.89654i 0.413033i 0.978443 + 0.206516i \(0.0662127\pi\)
−0.978443 + 0.206516i \(0.933787\pi\)
\(90\) 0 0
\(91\) 21.9535 2.30135
\(92\) 0 0
\(93\) 3.08441 0.319839
\(94\) 0 0
\(95\) 3.05893i 0.313839i
\(96\) 0 0
\(97\) 5.72909i 0.581701i −0.956768 0.290851i \(-0.906062\pi\)
0.956768 0.290851i \(-0.0939383\pi\)
\(98\) 0 0
\(99\) 3.09487 0.311046
\(100\) 0 0
\(101\) 3.97419 0.395447 0.197724 0.980258i \(-0.436645\pi\)
0.197724 + 0.980258i \(0.436645\pi\)
\(102\) 0 0
\(103\) −13.3249 −1.31294 −0.656471 0.754351i \(-0.727952\pi\)
−0.656471 + 0.754351i \(0.727952\pi\)
\(104\) 0 0
\(105\) 4.84428 0.472754
\(106\) 0 0
\(107\) 11.4779 1.10961 0.554806 0.831980i \(-0.312793\pi\)
0.554806 + 0.831980i \(0.312793\pi\)
\(108\) 0 0
\(109\) 13.3604i 1.27970i 0.768501 + 0.639848i \(0.221003\pi\)
−0.768501 + 0.639848i \(0.778997\pi\)
\(110\) 0 0
\(111\) −3.78545 −0.359299
\(112\) 0 0
\(113\) 12.7468i 1.19912i 0.800331 + 0.599559i \(0.204657\pi\)
−0.800331 + 0.599559i \(0.795343\pi\)
\(114\) 0 0
\(115\) 0.0971617 4.79485i 0.00906038 0.447122i
\(116\) 0 0
\(117\) 4.53183 0.418968
\(118\) 0 0
\(119\) 5.08377i 0.466028i
\(120\) 0 0
\(121\) −1.42178 −0.129253
\(122\) 0 0
\(123\) 4.04608i 0.364823i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 1.34087i 0.118983i −0.998229 0.0594913i \(-0.981052\pi\)
0.998229 0.0594913i \(-0.0189479\pi\)
\(128\) 0 0
\(129\) 2.53119i 0.222859i
\(130\) 0 0
\(131\) 12.8225i 1.12030i −0.828390 0.560152i \(-0.810743\pi\)
0.828390 0.560152i \(-0.189257\pi\)
\(132\) 0 0
\(133\) 14.8183 1.28491
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 3.67251i 0.313763i 0.987617 + 0.156882i \(0.0501441\pi\)
−0.987617 + 0.156882i \(0.949856\pi\)
\(138\) 0 0
\(139\) 11.5659i 0.981009i −0.871439 0.490504i \(-0.836813\pi\)
0.871439 0.490504i \(-0.163187\pi\)
\(140\) 0 0
\(141\) −3.83718 −0.323149
\(142\) 0 0
\(143\) 14.0254 1.17287
\(144\) 0 0
\(145\) 2.44565i 0.203100i
\(146\) 0 0
\(147\) 16.4671i 1.35818i
\(148\) 0 0
\(149\) 12.2359i 1.00240i 0.865330 + 0.501202i \(0.167109\pi\)
−0.865330 + 0.501202i \(0.832891\pi\)
\(150\) 0 0
\(151\) 11.3471i 0.923415i 0.887032 + 0.461708i \(0.152763\pi\)
−0.887032 + 0.461708i \(0.847237\pi\)
\(152\) 0 0
\(153\) 1.04944i 0.0848419i
\(154\) 0 0
\(155\) 3.08441 0.247746
\(156\) 0 0
\(157\) 10.3681i 0.827461i −0.910399 0.413730i \(-0.864226\pi\)
0.910399 0.413730i \(-0.135774\pi\)
\(158\) 0 0
\(159\) −8.14281 −0.645767
\(160\) 0 0
\(161\) 23.2276 + 0.470679i 1.83059 + 0.0370947i
\(162\) 0 0
\(163\) 11.7920i 0.923623i 0.886978 + 0.461811i \(0.152800\pi\)
−0.886978 + 0.461811i \(0.847200\pi\)
\(164\) 0 0
\(165\) 3.09487 0.240935
\(166\) 0 0
\(167\) 15.3234i 1.18576i −0.805290 0.592881i \(-0.797990\pi\)
0.805290 0.592881i \(-0.202010\pi\)
\(168\) 0 0
\(169\) 7.53751 0.579808
\(170\) 0 0
\(171\) 3.05893 0.233922
\(172\) 0 0
\(173\) 3.12803 0.237820 0.118910 0.992905i \(-0.462060\pi\)
0.118910 + 0.992905i \(0.462060\pi\)
\(174\) 0 0
\(175\) 4.84428 0.366193
\(176\) 0 0
\(177\) 4.64825 0.349384
\(178\) 0 0
\(179\) 25.9800i 1.94184i −0.239404 0.970920i \(-0.576952\pi\)
0.239404 0.970920i \(-0.423048\pi\)
\(180\) 0 0
\(181\) 15.3758i 1.14288i −0.820645 0.571439i \(-0.806385\pi\)
0.820645 0.571439i \(-0.193615\pi\)
\(182\) 0 0
\(183\) 11.3650 0.840126
\(184\) 0 0
\(185\) −3.78545 −0.278312
\(186\) 0 0
\(187\) 3.24787i 0.237508i
\(188\) 0 0
\(189\) 4.84428i 0.352370i
\(190\) 0 0
\(191\) −0.659707 −0.0477347 −0.0238674 0.999715i \(-0.507598\pi\)
−0.0238674 + 0.999715i \(0.507598\pi\)
\(192\) 0 0
\(193\) 13.2828 0.956120 0.478060 0.878327i \(-0.341340\pi\)
0.478060 + 0.878327i \(0.341340\pi\)
\(194\) 0 0
\(195\) 4.53183 0.324531
\(196\) 0 0
\(197\) −1.33165 −0.0948761 −0.0474380 0.998874i \(-0.515106\pi\)
−0.0474380 + 0.998874i \(0.515106\pi\)
\(198\) 0 0
\(199\) 3.42710 0.242941 0.121470 0.992595i \(-0.461239\pi\)
0.121470 + 0.992595i \(0.461239\pi\)
\(200\) 0 0
\(201\) 3.77906i 0.266554i
\(202\) 0 0
\(203\) 11.8474 0.831527
\(204\) 0 0
\(205\) 4.04608i 0.282590i
\(206\) 0 0
\(207\) 4.79485 + 0.0971617i 0.333265 + 0.00675321i
\(208\) 0 0
\(209\) 9.46698 0.654845
\(210\) 0 0
\(211\) 22.7762i 1.56798i 0.620774 + 0.783989i \(0.286818\pi\)
−0.620774 + 0.783989i \(0.713182\pi\)
\(212\) 0 0
\(213\) −13.2737 −0.909501
\(214\) 0 0
\(215\) 2.53119i 0.172626i
\(216\) 0 0
\(217\) 14.9418i 1.01431i
\(218\) 0 0
\(219\) 2.53642i 0.171395i
\(220\) 0 0
\(221\) 4.75587i 0.319914i
\(222\) 0 0
\(223\) 12.5342i 0.839353i 0.907674 + 0.419677i \(0.137856\pi\)
−0.907674 + 0.419677i \(0.862144\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −24.4186 −1.62072 −0.810359 0.585933i \(-0.800728\pi\)
−0.810359 + 0.585933i \(0.800728\pi\)
\(228\) 0 0
\(229\) 25.3713i 1.67658i −0.545225 0.838290i \(-0.683556\pi\)
0.545225 0.838290i \(-0.316444\pi\)
\(230\) 0 0
\(231\) 14.9924i 0.986429i
\(232\) 0 0
\(233\) 0.675614 0.0442609 0.0221305 0.999755i \(-0.492955\pi\)
0.0221305 + 0.999755i \(0.492955\pi\)
\(234\) 0 0
\(235\) −3.83718 −0.250310
\(236\) 0 0
\(237\) 1.04304i 0.0677526i
\(238\) 0 0
\(239\) 9.80756i 0.634398i −0.948359 0.317199i \(-0.897258\pi\)
0.948359 0.317199i \(-0.102742\pi\)
\(240\) 0 0
\(241\) 10.2922i 0.662978i 0.943459 + 0.331489i \(0.107551\pi\)
−0.943459 + 0.331489i \(0.892449\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 16.4671i 1.05204i
\(246\) 0 0
\(247\) 13.8625 0.882053
\(248\) 0 0
\(249\) 4.24974i 0.269316i
\(250\) 0 0
\(251\) 8.41799 0.531339 0.265669 0.964064i \(-0.414407\pi\)
0.265669 + 0.964064i \(0.414407\pi\)
\(252\) 0 0
\(253\) 14.8394 + 0.300703i 0.932947 + 0.0189050i
\(254\) 0 0
\(255\) 1.04944i 0.0657182i
\(256\) 0 0
\(257\) 4.93222 0.307663 0.153832 0.988097i \(-0.450839\pi\)
0.153832 + 0.988097i \(0.450839\pi\)
\(258\) 0 0
\(259\) 18.3378i 1.13946i
\(260\) 0 0
\(261\) 2.44565 0.151382
\(262\) 0 0
\(263\) −12.3726 −0.762924 −0.381462 0.924384i \(-0.624579\pi\)
−0.381462 + 0.924384i \(0.624579\pi\)
\(264\) 0 0
\(265\) −8.14281 −0.500209
\(266\) 0 0
\(267\) −3.89654 −0.238465
\(268\) 0 0
\(269\) −29.2075 −1.78081 −0.890406 0.455166i \(-0.849580\pi\)
−0.890406 + 0.455166i \(0.849580\pi\)
\(270\) 0 0
\(271\) 16.1560i 0.981409i 0.871326 + 0.490704i \(0.163260\pi\)
−0.871326 + 0.490704i \(0.836740\pi\)
\(272\) 0 0
\(273\) 21.9535i 1.32869i
\(274\) 0 0
\(275\) 3.09487 0.186628
\(276\) 0 0
\(277\) −24.8559 −1.49345 −0.746723 0.665135i \(-0.768374\pi\)
−0.746723 + 0.665135i \(0.768374\pi\)
\(278\) 0 0
\(279\) 3.08441i 0.184659i
\(280\) 0 0
\(281\) 18.4409i 1.10009i −0.835134 0.550046i \(-0.814610\pi\)
0.835134 0.550046i \(-0.185390\pi\)
\(282\) 0 0
\(283\) −0.823367 −0.0489441 −0.0244720 0.999701i \(-0.507790\pi\)
−0.0244720 + 0.999701i \(0.507790\pi\)
\(284\) 0 0
\(285\) 3.05893 0.181195
\(286\) 0 0
\(287\) −19.6004 −1.15697
\(288\) 0 0
\(289\) 15.8987 0.935217
\(290\) 0 0
\(291\) 5.72909 0.335845
\(292\) 0 0
\(293\) 6.91225i 0.403818i −0.979404 0.201909i \(-0.935285\pi\)
0.979404 0.201909i \(-0.0647145\pi\)
\(294\) 0 0
\(295\) 4.64825 0.270632
\(296\) 0 0
\(297\) 3.09487i 0.179583i
\(298\) 0 0
\(299\) 21.7294 + 0.440321i 1.25665 + 0.0254644i
\(300\) 0 0
\(301\) 12.2618 0.706758
\(302\) 0 0
\(303\) 3.97419i 0.228312i
\(304\) 0 0
\(305\) 11.3650 0.650759
\(306\) 0 0
\(307\) 30.2075i 1.72403i 0.506879 + 0.862017i \(0.330799\pi\)
−0.506879 + 0.862017i \(0.669201\pi\)
\(308\) 0 0
\(309\) 13.3249i 0.758028i
\(310\) 0 0
\(311\) 16.6281i 0.942896i 0.881894 + 0.471448i \(0.156268\pi\)
−0.881894 + 0.471448i \(0.843732\pi\)
\(312\) 0 0
\(313\) 31.9251i 1.80451i 0.431202 + 0.902255i \(0.358090\pi\)
−0.431202 + 0.902255i \(0.641910\pi\)
\(314\) 0 0
\(315\) 4.84428i 0.272944i
\(316\) 0 0
\(317\) 21.2873 1.19561 0.597806 0.801641i \(-0.296039\pi\)
0.597806 + 0.801641i \(0.296039\pi\)
\(318\) 0 0
\(319\) 7.56898 0.423781
\(320\) 0 0
\(321\) 11.4779i 0.640635i
\(322\) 0 0
\(323\) 3.21015i 0.178617i
\(324\) 0 0
\(325\) 4.53183 0.251381
\(326\) 0 0
\(327\) −13.3604 −0.738833
\(328\) 0 0
\(329\) 18.5884i 1.02481i
\(330\) 0 0
\(331\) 19.0651i 1.04791i −0.851745 0.523956i \(-0.824456\pi\)
0.851745 0.523956i \(-0.175544\pi\)
\(332\) 0 0
\(333\) 3.78545i 0.207442i
\(334\) 0 0
\(335\) 3.77906i 0.206472i
\(336\) 0 0
\(337\) 18.3374i 0.998901i −0.866342 0.499451i \(-0.833535\pi\)
0.866342 0.499451i \(-0.166465\pi\)
\(338\) 0 0
\(339\) −12.7468 −0.692311
\(340\) 0 0
\(341\) 9.54585i 0.516937i
\(342\) 0 0
\(343\) −45.8612 −2.47627
\(344\) 0 0
\(345\) 4.79485 + 0.0971617i 0.258146 + 0.00523101i
\(346\) 0 0
\(347\) 13.6634i 0.733489i −0.930322 0.366744i \(-0.880472\pi\)
0.930322 0.366744i \(-0.119528\pi\)
\(348\) 0 0
\(349\) 31.1112 1.66534 0.832671 0.553768i \(-0.186810\pi\)
0.832671 + 0.553768i \(0.186810\pi\)
\(350\) 0 0
\(351\) 4.53183i 0.241891i
\(352\) 0 0
\(353\) 31.5128 1.67726 0.838630 0.544702i \(-0.183357\pi\)
0.838630 + 0.544702i \(0.183357\pi\)
\(354\) 0 0
\(355\) −13.2737 −0.704496
\(356\) 0 0
\(357\) 5.08377 0.269061
\(358\) 0 0
\(359\) −34.1464 −1.80218 −0.901088 0.433636i \(-0.857230\pi\)
−0.901088 + 0.433636i \(0.857230\pi\)
\(360\) 0 0
\(361\) −9.64296 −0.507524
\(362\) 0 0
\(363\) 1.42178i 0.0746242i
\(364\) 0 0
\(365\) 2.53642i 0.132762i
\(366\) 0 0
\(367\) −9.17921 −0.479151 −0.239576 0.970878i \(-0.577008\pi\)
−0.239576 + 0.970878i \(0.577008\pi\)
\(368\) 0 0
\(369\) −4.04608 −0.210630
\(370\) 0 0
\(371\) 39.4461i 2.04794i
\(372\) 0 0
\(373\) 15.7943i 0.817795i −0.912580 0.408898i \(-0.865913\pi\)
0.912580 0.408898i \(-0.134087\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 11.0833 0.570819
\(378\) 0 0
\(379\) 11.0170 0.565905 0.282952 0.959134i \(-0.408686\pi\)
0.282952 + 0.959134i \(0.408686\pi\)
\(380\) 0 0
\(381\) 1.34087 0.0686947
\(382\) 0 0
\(383\) −0.892024 −0.0455803 −0.0227901 0.999740i \(-0.507255\pi\)
−0.0227901 + 0.999740i \(0.507255\pi\)
\(384\) 0 0
\(385\) 14.9924i 0.764085i
\(386\) 0 0
\(387\) 2.53119 0.128667
\(388\) 0 0
\(389\) 0.970666i 0.0492147i −0.999697 0.0246074i \(-0.992166\pi\)
0.999697 0.0246074i \(-0.00783356\pi\)
\(390\) 0 0
\(391\) 0.101965 5.03189i 0.00515659 0.254473i
\(392\) 0 0
\(393\) 12.8225 0.646808
\(394\) 0 0
\(395\) 1.04304i 0.0524809i
\(396\) 0 0
\(397\) −9.57904 −0.480758 −0.240379 0.970679i \(-0.577272\pi\)
−0.240379 + 0.970679i \(0.577272\pi\)
\(398\) 0 0
\(399\) 14.8183i 0.741844i
\(400\) 0 0
\(401\) 20.6070i 1.02906i −0.857471 0.514532i \(-0.827966\pi\)
0.857471 0.514532i \(-0.172034\pi\)
\(402\) 0 0
\(403\) 13.9780i 0.696296i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) 11.7155i 0.580715i
\(408\) 0 0
\(409\) 15.4878 0.765822 0.382911 0.923785i \(-0.374922\pi\)
0.382911 + 0.923785i \(0.374922\pi\)
\(410\) 0 0
\(411\) −3.67251 −0.181151
\(412\) 0 0
\(413\) 22.5174i 1.10801i
\(414\) 0 0
\(415\) 4.24974i 0.208612i
\(416\) 0 0
\(417\) 11.5659 0.566386
\(418\) 0 0
\(419\) −35.3366 −1.72631 −0.863154 0.504941i \(-0.831514\pi\)
−0.863154 + 0.504941i \(0.831514\pi\)
\(420\) 0 0
\(421\) 9.54836i 0.465359i 0.972554 + 0.232679i \(0.0747492\pi\)
−0.972554 + 0.232679i \(0.925251\pi\)
\(422\) 0 0
\(423\) 3.83718i 0.186570i
\(424\) 0 0
\(425\) 1.04944i 0.0509051i
\(426\) 0 0
\(427\) 55.0553i 2.66431i
\(428\) 0 0
\(429\) 14.0254i 0.677154i
\(430\) 0 0
\(431\) 24.9390 1.20127 0.600634 0.799524i \(-0.294915\pi\)
0.600634 + 0.799524i \(0.294915\pi\)
\(432\) 0 0
\(433\) 38.3664i 1.84377i 0.387464 + 0.921885i \(0.373351\pi\)
−0.387464 + 0.921885i \(0.626649\pi\)
\(434\) 0 0
\(435\) 2.44565 0.117260
\(436\) 0 0
\(437\) 14.6671 + 0.297211i 0.701622 + 0.0142175i
\(438\) 0 0
\(439\) 2.96764i 0.141638i 0.997489 + 0.0708189i \(0.0225612\pi\)
−0.997489 + 0.0708189i \(0.977439\pi\)
\(440\) 0 0
\(441\) −16.4671 −0.784147
\(442\) 0 0
\(443\) 0.100101i 0.00475595i −0.999997 0.00237797i \(-0.999243\pi\)
0.999997 0.00237797i \(-0.000756933\pi\)
\(444\) 0 0
\(445\) −3.89654 −0.184714
\(446\) 0 0
\(447\) −12.2359 −0.578739
\(448\) 0 0
\(449\) −32.2186 −1.52049 −0.760245 0.649637i \(-0.774921\pi\)
−0.760245 + 0.649637i \(0.774921\pi\)
\(450\) 0 0
\(451\) −12.5221 −0.589642
\(452\) 0 0
\(453\) −11.3471 −0.533134
\(454\) 0 0
\(455\) 21.9535i 1.02920i
\(456\) 0 0
\(457\) 17.8778i 0.836288i −0.908381 0.418144i \(-0.862681\pi\)
0.908381 0.418144i \(-0.137319\pi\)
\(458\) 0 0
\(459\) 1.04944 0.0489835
\(460\) 0 0
\(461\) 5.34372 0.248882 0.124441 0.992227i \(-0.460286\pi\)
0.124441 + 0.992227i \(0.460286\pi\)
\(462\) 0 0
\(463\) 25.9996i 1.20830i 0.796870 + 0.604151i \(0.206488\pi\)
−0.796870 + 0.604151i \(0.793512\pi\)
\(464\) 0 0
\(465\) 3.08441i 0.143036i
\(466\) 0 0
\(467\) 3.57869 0.165602 0.0828010 0.996566i \(-0.473613\pi\)
0.0828010 + 0.996566i \(0.473613\pi\)
\(468\) 0 0
\(469\) 18.3068 0.845331
\(470\) 0 0
\(471\) 10.3681 0.477735
\(472\) 0 0
\(473\) 7.83370 0.360194
\(474\) 0 0
\(475\) 3.05893 0.140353
\(476\) 0 0
\(477\) 8.14281i 0.372834i
\(478\) 0 0
\(479\) 29.8270 1.36283 0.681416 0.731896i \(-0.261365\pi\)
0.681416 + 0.731896i \(0.261365\pi\)
\(480\) 0 0
\(481\) 17.1550i 0.782203i
\(482\) 0 0
\(483\) −0.470679 + 23.2276i −0.0214166 + 1.05689i
\(484\) 0 0
\(485\) 5.72909 0.260145
\(486\) 0 0
\(487\) 36.6321i 1.65996i −0.557793 0.829980i \(-0.688352\pi\)
0.557793 0.829980i \(-0.311648\pi\)
\(488\) 0 0
\(489\) −11.7920 −0.533254
\(490\) 0 0
\(491\) 28.1982i 1.27257i −0.771455 0.636284i \(-0.780471\pi\)
0.771455 0.636284i \(-0.219529\pi\)
\(492\) 0 0
\(493\) 2.56656i 0.115592i
\(494\) 0 0
\(495\) 3.09487i 0.139104i
\(496\) 0 0
\(497\) 64.3017i 2.88432i
\(498\) 0 0
\(499\) 25.4668i 1.14005i 0.821627 + 0.570025i \(0.193066\pi\)
−0.821627 + 0.570025i \(0.806934\pi\)
\(500\) 0 0
\(501\) 15.3234 0.684600
\(502\) 0 0
\(503\) 21.4532 0.956552 0.478276 0.878210i \(-0.341262\pi\)
0.478276 + 0.878210i \(0.341262\pi\)
\(504\) 0 0
\(505\) 3.97419i 0.176849i
\(506\) 0 0
\(507\) 7.53751i 0.334752i
\(508\) 0 0
\(509\) 9.58966 0.425054 0.212527 0.977155i \(-0.431831\pi\)
0.212527 + 0.977155i \(0.431831\pi\)
\(510\) 0 0
\(511\) −12.2871 −0.543550
\(512\) 0 0
\(513\) 3.05893i 0.135055i
\(514\) 0 0
\(515\) 13.3249i 0.587166i
\(516\) 0 0
\(517\) 11.8756i 0.522287i
\(518\) 0 0
\(519\) 3.12803i 0.137305i
\(520\) 0 0
\(521\) 13.0956i 0.573727i −0.957971 0.286863i \(-0.907387\pi\)
0.957971 0.286863i \(-0.0926126\pi\)
\(522\) 0 0
\(523\) −6.40732 −0.280173 −0.140086 0.990139i \(-0.544738\pi\)
−0.140086 + 0.990139i \(0.544738\pi\)
\(524\) 0 0
\(525\) 4.84428i 0.211422i
\(526\) 0 0
\(527\) 3.23689 0.141001
\(528\) 0 0
\(529\) 22.9811 + 0.931751i 0.999179 + 0.0405109i
\(530\) 0 0
\(531\) 4.64825i 0.201717i
\(532\) 0 0
\(533\) −18.3361 −0.794227
\(534\) 0 0
\(535\) 11.4779i 0.496233i
\(536\) 0 0
\(537\) 25.9800 1.12112
\(538\) 0 0
\(539\) −50.9635 −2.19515
\(540\) 0 0
\(541\) −30.4405 −1.30874 −0.654370 0.756174i \(-0.727066\pi\)
−0.654370 + 0.756174i \(0.727066\pi\)
\(542\) 0 0
\(543\) 15.3758 0.659841
\(544\) 0 0
\(545\) −13.3604 −0.572298
\(546\) 0 0
\(547\) 28.9045i 1.23587i −0.786230 0.617934i \(-0.787970\pi\)
0.786230 0.617934i \(-0.212030\pi\)
\(548\) 0 0
\(549\) 11.3650i 0.485047i
\(550\) 0 0
\(551\) 7.48108 0.318705
\(552\) 0 0
\(553\) 5.05277 0.214866
\(554\) 0 0
\(555\) 3.78545i 0.160684i
\(556\) 0 0
\(557\) 15.2748i 0.647212i −0.946192 0.323606i \(-0.895105\pi\)
0.946192 0.323606i \(-0.104895\pi\)
\(558\) 0 0
\(559\) 11.4709 0.485168
\(560\) 0 0
\(561\) 3.24787 0.137125
\(562\) 0 0
\(563\) −2.19626 −0.0925612 −0.0462806 0.998928i \(-0.514737\pi\)
−0.0462806 + 0.998928i \(0.514737\pi\)
\(564\) 0 0
\(565\) −12.7468 −0.536262
\(566\) 0 0
\(567\) −4.84428 −0.203441
\(568\) 0 0
\(569\) 27.6847i 1.16060i −0.814401 0.580302i \(-0.802934\pi\)
0.814401 0.580302i \(-0.197066\pi\)
\(570\) 0 0
\(571\) 28.6755 1.20003 0.600016 0.799988i \(-0.295161\pi\)
0.600016 + 0.799988i \(0.295161\pi\)
\(572\) 0 0
\(573\) 0.659707i 0.0275597i
\(574\) 0 0
\(575\) 4.79485 + 0.0971617i 0.199959 + 0.00405192i
\(576\) 0 0
\(577\) −2.26705 −0.0943785 −0.0471892 0.998886i \(-0.515026\pi\)
−0.0471892 + 0.998886i \(0.515026\pi\)
\(578\) 0 0
\(579\) 13.2828i 0.552016i
\(580\) 0 0
\(581\) −20.5869 −0.854090
\(582\) 0 0
\(583\) 25.2009i 1.04372i
\(584\) 0 0
\(585\) 4.53183i 0.187368i
\(586\) 0 0
\(587\) 10.1170i 0.417574i −0.977961 0.208787i \(-0.933048\pi\)
0.977961 0.208787i \(-0.0669515\pi\)
\(588\) 0 0
\(589\) 9.43499i 0.388762i
\(590\) 0 0
\(591\) 1.33165i 0.0547767i
\(592\) 0 0
\(593\) 32.3649 1.32907 0.664533 0.747259i \(-0.268630\pi\)
0.664533 + 0.747259i \(0.268630\pi\)
\(594\) 0 0
\(595\) 5.08377 0.208414
\(596\) 0 0
\(597\) 3.42710i 0.140262i
\(598\) 0 0
\(599\) 3.27236i 0.133705i 0.997763 + 0.0668526i \(0.0212957\pi\)
−0.997763 + 0.0668526i \(0.978704\pi\)
\(600\) 0 0
\(601\) 35.2903 1.43952 0.719761 0.694222i \(-0.244251\pi\)
0.719761 + 0.694222i \(0.244251\pi\)
\(602\) 0 0
\(603\) 3.77906 0.153895
\(604\) 0 0
\(605\) 1.42178i 0.0578036i
\(606\) 0 0
\(607\) 5.72385i 0.232324i 0.993230 + 0.116162i \(0.0370591\pi\)
−0.993230 + 0.116162i \(0.962941\pi\)
\(608\) 0 0
\(609\) 11.8474i 0.480082i
\(610\) 0 0
\(611\) 17.3895i 0.703503i
\(612\) 0 0
\(613\) 11.1031i 0.448449i 0.974538 + 0.224224i \(0.0719848\pi\)
−0.974538 + 0.224224i \(0.928015\pi\)
\(614\) 0 0
\(615\) −4.04608 −0.163154
\(616\) 0 0
\(617\) 0.915087i 0.0368400i −0.999830 0.0184200i \(-0.994136\pi\)
0.999830 0.0184200i \(-0.00586360\pi\)
\(618\) 0 0
\(619\) 32.2320 1.29552 0.647758 0.761847i \(-0.275707\pi\)
0.647758 + 0.761847i \(0.275707\pi\)
\(620\) 0 0
\(621\) −0.0971617 + 4.79485i −0.00389896 + 0.192411i
\(622\) 0 0
\(623\) 18.8760i 0.756249i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 9.46698i 0.378075i
\(628\) 0 0
\(629\) −3.97259 −0.158398
\(630\) 0 0
\(631\) −0.724543 −0.0288436 −0.0144218 0.999896i \(-0.504591\pi\)
−0.0144218 + 0.999896i \(0.504591\pi\)
\(632\) 0 0
\(633\) −22.7762 −0.905273
\(634\) 0 0
\(635\) 1.34087 0.0532107
\(636\) 0 0
\(637\) −74.6261 −2.95679
\(638\) 0 0
\(639\) 13.2737i 0.525100i
\(640\) 0 0
\(641\) 9.49638i 0.375084i 0.982257 + 0.187542i \(0.0600521\pi\)
−0.982257 + 0.187542i \(0.939948\pi\)
\(642\) 0 0
\(643\) 7.33887 0.289417 0.144708 0.989474i \(-0.453776\pi\)
0.144708 + 0.989474i \(0.453776\pi\)
\(644\) 0 0
\(645\) 2.53119 0.0996654
\(646\) 0 0
\(647\) 8.89013i 0.349507i −0.984612 0.174753i \(-0.944087\pi\)
0.984612 0.174753i \(-0.0559128\pi\)
\(648\) 0 0
\(649\) 14.3857i 0.564689i
\(650\) 0 0
\(651\) −14.9418 −0.585614
\(652\) 0 0
\(653\) −42.3035 −1.65546 −0.827732 0.561124i \(-0.810369\pi\)
−0.827732 + 0.561124i \(0.810369\pi\)
\(654\) 0 0
\(655\) 12.8225 0.501015
\(656\) 0 0
\(657\) −2.53642 −0.0989550
\(658\) 0 0
\(659\) 2.34063 0.0911781 0.0455891 0.998960i \(-0.485484\pi\)
0.0455891 + 0.998960i \(0.485484\pi\)
\(660\) 0 0
\(661\) 6.00716i 0.233652i −0.993152 0.116826i \(-0.962728\pi\)
0.993152 0.116826i \(-0.0372719\pi\)
\(662\) 0 0
\(663\) 4.75587 0.184703
\(664\) 0 0
\(665\) 14.8183i 0.574630i
\(666\) 0 0
\(667\) 11.7265 + 0.237624i 0.454053 + 0.00920083i
\(668\) 0 0
\(669\) −12.5342 −0.484601
\(670\) 0 0
\(671\) 35.1732i 1.35785i
\(672\) 0 0
\(673\) −39.2984 −1.51484 −0.757420 0.652928i \(-0.773540\pi\)
−0.757420 + 0.652928i \(0.773540\pi\)
\(674\) 0 0
\(675\) 1.00000i 0.0384900i
\(676\) 0 0
\(677\) 10.1936i 0.391772i 0.980627 + 0.195886i \(0.0627583\pi\)
−0.980627 + 0.195886i \(0.937242\pi\)
\(678\) 0 0
\(679\) 27.7533i 1.06508i
\(680\) 0 0
\(681\) 24.4186i 0.935722i
\(682\) 0 0
\(683\) 4.94854i 0.189350i −0.995508 0.0946752i \(-0.969819\pi\)
0.995508 0.0946752i \(-0.0301813\pi\)
\(684\) 0 0
\(685\) −3.67251 −0.140319
\(686\) 0 0
\(687\) 25.3713 0.967974
\(688\) 0 0
\(689\) 36.9019i 1.40585i
\(690\) 0 0
\(691\) 30.0509i 1.14319i −0.820536 0.571595i \(-0.806325\pi\)
0.820536 0.571595i \(-0.193675\pi\)
\(692\) 0 0
\(693\) −14.9924 −0.569515
\(694\) 0 0
\(695\) 11.5659 0.438720
\(696\) 0 0
\(697\) 4.24610i 0.160833i
\(698\) 0 0
\(699\) 0.675614i 0.0255541i
\(700\) 0 0
\(701\) 18.9951i 0.717434i −0.933446 0.358717i \(-0.883214\pi\)
0.933446 0.358717i \(-0.116786\pi\)
\(702\) 0 0
\(703\) 11.5794i 0.436727i
\(704\) 0 0
\(705\) 3.83718i 0.144517i
\(706\) 0 0
\(707\) −19.2521 −0.724051
\(708\) 0 0
\(709\) 10.3208i 0.387605i −0.981041 0.193802i \(-0.937918\pi\)
0.981041 0.193802i \(-0.0620820\pi\)
\(710\) 0 0
\(711\) 1.04304 0.0391170
\(712\) 0 0
\(713\) −0.299687 + 14.7893i −0.0112234 + 0.553863i
\(714\) 0 0
\(715\) 14.0254i 0.524521i
\(716\) 0 0
\(717\) 9.80756 0.366270
\(718\) 0 0
\(719\) 28.4693i 1.06172i 0.847458 + 0.530862i \(0.178132\pi\)
−0.847458 + 0.530862i \(0.821868\pi\)
\(720\) 0 0
\(721\) 64.5497 2.40395
\(722\) 0 0
\(723\) −10.2922 −0.382770
\(724\) 0 0
\(725\) 2.44565 0.0908293
\(726\) 0 0
\(727\) 3.62136 0.134309 0.0671543 0.997743i \(-0.478608\pi\)
0.0671543 + 0.997743i \(0.478608\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 2.65632i 0.0982475i
\(732\) 0 0
\(733\) 3.39563i 0.125420i 0.998032 + 0.0627102i \(0.0199744\pi\)
−0.998032 + 0.0627102i \(0.980026\pi\)
\(734\) 0 0
\(735\) −16.4671 −0.607397
\(736\) 0 0
\(737\) 11.6957 0.430817
\(738\) 0 0
\(739\) 45.6318i 1.67859i 0.543675 + 0.839296i \(0.317032\pi\)
−0.543675 + 0.839296i \(0.682968\pi\)
\(740\) 0 0
\(741\) 13.8625i 0.509253i
\(742\) 0 0
\(743\) 8.29853 0.304444 0.152222 0.988346i \(-0.451357\pi\)
0.152222 + 0.988346i \(0.451357\pi\)
\(744\) 0 0
\(745\) −12.2359 −0.448289
\(746\) 0 0
\(747\) −4.24974 −0.155490
\(748\) 0 0
\(749\) −55.6023 −2.03166
\(750\) 0 0
\(751\) 39.1817 1.42976 0.714880 0.699247i \(-0.246481\pi\)
0.714880 + 0.699247i \(0.246481\pi\)
\(752\) 0 0
\(753\) 8.41799i 0.306769i
\(754\) 0 0
\(755\) −11.3471 −0.412964
\(756\) 0 0
\(757\) 37.3641i 1.35802i 0.734128 + 0.679011i \(0.237591\pi\)
−0.734128 + 0.679011i \(0.762409\pi\)
\(758\) 0 0
\(759\) −0.300703 + 14.8394i −0.0109148 + 0.538637i
\(760\) 0 0
\(761\) 13.2385 0.479896 0.239948 0.970786i \(-0.422870\pi\)
0.239948 + 0.970786i \(0.422870\pi\)
\(762\) 0 0
\(763\) 64.7217i 2.34308i
\(764\) 0 0
\(765\) 1.04944 0.0379424
\(766\) 0 0
\(767\) 21.0651i 0.760616i
\(768\) 0 0
\(769\) 33.6263i 1.21260i −0.795237 0.606298i \(-0.792654\pi\)
0.795237 0.606298i \(-0.207346\pi\)
\(770\) 0 0
\(771\) 4.93222i 0.177630i
\(772\) 0 0
\(773\) 17.5764i 0.632177i 0.948730 + 0.316089i \(0.102370\pi\)
−0.948730 + 0.316089i \(0.897630\pi\)
\(774\) 0 0
\(775\) 3.08441i 0.110795i
\(776\) 0 0
\(777\) 18.3378 0.657865
\(778\) 0 0
\(779\) −12.3767 −0.443440
\(780\) 0 0
\(781\) 41.0804i 1.46997i
\(782\) 0 0
\(783\) 2.44565i 0.0874005i
\(784\) 0 0
\(785\) 10.3681 0.370052
\(786\) 0 0
\(787\) −34.3182 −1.22331 −0.611656 0.791124i \(-0.709496\pi\)
−0.611656 + 0.791124i \(0.709496\pi\)
\(788\) 0 0
\(789\) 12.3726i 0.440475i
\(790\) 0 0
\(791\) 61.7491i 2.19555i
\(792\) 0 0
\(793\) 51.5043i 1.82897i
\(794\) 0 0
\(795\) 8.14281i 0.288796i
\(796\) 0 0
\(797\) 0.808358i 0.0286335i −0.999898 0.0143168i \(-0.995443\pi\)
0.999898 0.0143168i \(-0.00455732\pi\)
\(798\) 0 0
\(799\) −4.02688 −0.142461
\(800\) 0 0
\(801\) 3.89654i 0.137678i
\(802\) 0 0
\(803\) −7.84988 −0.277016
\(804\) 0 0
\(805\) −0.470679 + 23.2276i <