Properties

Label 1008.2.cs.q.703.2
Level 1008
Weight 2
Character 1008.703
Analytic conductor 8.049
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.cs (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 703.2
Root \(-1.32288 - 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 1008.703
Dual form 1008.2.cs.q.271.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{5} +2.64575 q^{7} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{5} +2.64575 q^{7} +(3.96863 + 2.29129i) q^{11} +3.46410i q^{13} +(-4.50000 - 2.59808i) q^{17} +(1.32288 + 2.29129i) q^{19} +(3.96863 - 2.29129i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(-1.32288 + 2.29129i) q^{31} +(3.96863 - 2.29129i) q^{35} +(-3.50000 - 6.06218i) q^{37} +3.46410i q^{41} -9.16515i q^{43} +(3.96863 + 6.87386i) q^{47} +7.00000 q^{49} +(1.50000 - 2.59808i) q^{53} +7.93725 q^{55} +(3.96863 - 6.87386i) q^{59} +(1.50000 - 0.866025i) q^{61} +(3.00000 + 5.19615i) q^{65} +(3.96863 + 2.29129i) q^{67} +9.16515i q^{71} +(-4.50000 - 2.59808i) q^{73} +(10.5000 + 6.06218i) q^{77} +(-3.96863 + 2.29129i) q^{79} -9.00000 q^{85} +(-1.50000 + 0.866025i) q^{89} +9.16515i q^{91} +(3.96863 + 2.29129i) q^{95} -3.46410i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{5} + O(q^{10}) \) \( 4q + 6q^{5} - 18q^{17} - 4q^{25} - 14q^{37} + 28q^{49} + 6q^{53} + 6q^{61} + 12q^{65} - 18q^{73} + 42q^{77} - 36q^{85} - 6q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 1.50000 0.866025i 0.670820 0.387298i −0.125567 0.992085i \(-0.540075\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.96863 + 2.29129i 1.19659 + 0.690849i 0.959792 0.280711i \(-0.0905701\pi\)
0.236794 + 0.971560i \(0.423903\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.50000 2.59808i −1.09141 0.630126i −0.157459 0.987526i \(-0.550330\pi\)
−0.933952 + 0.357400i \(0.883663\pi\)
\(18\) 0 0
\(19\) 1.32288 + 2.29129i 0.303488 + 0.525657i 0.976924 0.213589i \(-0.0685153\pi\)
−0.673435 + 0.739246i \(0.735182\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.96863 2.29129i 0.827516 0.477767i −0.0254855 0.999675i \(-0.508113\pi\)
0.853001 + 0.521909i \(0.174780\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −1.32288 + 2.29129i −0.237595 + 0.411527i −0.960024 0.279918i \(-0.909693\pi\)
0.722428 + 0.691446i \(0.243026\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.96863 2.29129i 0.670820 0.387298i
\(36\) 0 0
\(37\) −3.50000 6.06218i −0.575396 0.996616i −0.995998 0.0893706i \(-0.971514\pi\)
0.420602 0.907245i \(-0.361819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410i 0.541002i 0.962720 + 0.270501i \(0.0871893\pi\)
−0.962720 + 0.270501i \(0.912811\pi\)
\(42\) 0 0
\(43\) 9.16515i 1.39767i −0.715282 0.698836i \(-0.753702\pi\)
0.715282 0.698836i \(-0.246298\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.96863 + 6.87386i 0.578884 + 1.00266i 0.995608 + 0.0936230i \(0.0298448\pi\)
−0.416724 + 0.909033i \(0.636822\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.50000 2.59808i 0.206041 0.356873i −0.744423 0.667708i \(-0.767275\pi\)
0.950464 + 0.310835i \(0.100609\pi\)
\(54\) 0 0
\(55\) 7.93725 1.07026
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.96863 6.87386i 0.516671 0.894901i −0.483141 0.875542i \(-0.660504\pi\)
0.999813 0.0193585i \(-0.00616237\pi\)
\(60\) 0 0
\(61\) 1.50000 0.866025i 0.192055 0.110883i −0.400889 0.916127i \(-0.631299\pi\)
0.592944 + 0.805243i \(0.297965\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 3.96863 + 2.29129i 0.484845 + 0.279925i 0.722433 0.691441i \(-0.243024\pi\)
−0.237588 + 0.971366i \(0.576357\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.16515i 1.08770i 0.839181 + 0.543852i \(0.183035\pi\)
−0.839181 + 0.543852i \(0.816965\pi\)
\(72\) 0 0
\(73\) −4.50000 2.59808i −0.526685 0.304082i 0.212980 0.977056i \(-0.431683\pi\)
−0.739666 + 0.672975i \(0.765016\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.5000 + 6.06218i 1.19659 + 0.690849i
\(78\) 0 0
\(79\) −3.96863 + 2.29129i −0.446505 + 0.257790i −0.706353 0.707860i \(-0.749661\pi\)
0.259848 + 0.965650i \(0.416328\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.50000 + 0.866025i −0.159000 + 0.0917985i −0.577389 0.816469i \(-0.695928\pi\)
0.418389 + 0.908268i \(0.362595\pi\)
\(90\) 0 0
\(91\) 9.16515i 0.960769i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.96863 + 2.29129i 0.407173 + 0.235081i
\(96\) 0 0
\(97\) 3.46410i 0.351726i −0.984415 0.175863i \(-0.943728\pi\)
0.984415 0.175863i \(-0.0562716\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.50000 2.59808i −0.447767 0.258518i 0.259120 0.965845i \(-0.416568\pi\)
−0.706887 + 0.707327i \(0.749901\pi\)
\(102\) 0 0
\(103\) −1.32288 2.29129i −0.130347 0.225767i 0.793463 0.608618i \(-0.208276\pi\)
−0.923810 + 0.382851i \(0.874942\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.9059 6.87386i 1.15098 0.664521i 0.201858 0.979415i \(-0.435302\pi\)
0.949127 + 0.314893i \(0.101969\pi\)
\(108\) 0 0
\(109\) 3.50000 6.06218i 0.335239 0.580651i −0.648292 0.761392i \(-0.724516\pi\)
0.983531 + 0.180741i \(0.0578495\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 3.96863 6.87386i 0.370076 0.640991i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.9059 6.87386i −1.09141 0.630126i
\(120\) 0 0
\(121\) 5.00000 + 8.66025i 0.454545 + 0.787296i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 9.16515i 0.813276i 0.913589 + 0.406638i \(0.133299\pi\)
−0.913589 + 0.406638i \(0.866701\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.96863 6.87386i −0.346741 0.600572i 0.638928 0.769267i \(-0.279378\pi\)
−0.985668 + 0.168694i \(0.946045\pi\)
\(132\) 0 0
\(133\) 3.50000 + 6.06218i 0.303488 + 0.525657i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.5000 18.1865i 0.897076 1.55378i 0.0658609 0.997829i \(-0.479021\pi\)
0.831215 0.555952i \(-0.187646\pi\)
\(138\) 0 0
\(139\) 10.5830 0.897639 0.448819 0.893622i \(-0.351845\pi\)
0.448819 + 0.893622i \(0.351845\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −7.93725 + 13.7477i −0.663747 + 1.14964i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.50000 12.9904i −0.614424 1.06421i −0.990485 0.137619i \(-0.956055\pi\)
0.376061 0.926595i \(-0.377278\pi\)
\(150\) 0 0
\(151\) −19.8431 11.4564i −1.61481 0.932312i −0.988233 0.152953i \(-0.951122\pi\)
−0.626578 0.779359i \(1.28446\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.58258i 0.368081i
\(156\) 0 0
\(157\) −4.50000 2.59808i −0.359139 0.207349i 0.309564 0.950879i \(-0.399817\pi\)
−0.668703 + 0.743530i \(0.733150\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.5000 6.06218i 0.827516 0.477767i
\(162\) 0 0
\(163\) −3.96863 + 2.29129i −0.310847 + 0.179468i −0.647305 0.762231i \(-0.724104\pi\)
0.336459 + 0.941698i \(0.390771\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.8745 −1.22841 −0.614203 0.789148i \(-0.710522\pi\)
−0.614203 + 0.789148i \(0.710522\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −19.5000 + 11.2583i −1.48256 + 0.855955i −0.999804 0.0198012i \(-0.993697\pi\)
−0.482754 + 0.875756i \(0.660363\pi\)
\(174\) 0 0
\(175\) −2.64575 + 4.58258i −0.200000 + 0.346410i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −11.9059 6.87386i −0.889887 0.513777i −0.0159817 0.999872i \(-0.505087\pi\)
−0.873906 + 0.486096i \(0.838421\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i 0.635071 + 0.772454i \(0.280971\pi\)
−0.635071 + 0.772454i \(0.719029\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.5000 6.06218i −0.771975 0.445700i
\(186\) 0 0
\(187\) −11.9059 20.6216i −0.870644 1.50800i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.8431 + 11.4564i −1.43580 + 0.828959i −0.997554 0.0698969i \(-0.977733\pi\)
−0.438245 + 0.898856i \(0.644400\pi\)
\(192\) 0 0
\(193\) 5.50000 9.52628i 0.395899 0.685717i −0.597317 0.802005i \(-0.703766\pi\)
0.993215 + 0.116289i \(0.0370998\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −6.61438 + 11.4564i −0.468881 + 0.812125i −0.999367 0.0355680i \(-0.988676\pi\)
0.530486 + 0.847693i \(0.322009\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 + 5.19615i 0.209529 + 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.1244i 0.838659i
\(210\) 0 0
\(211\) 9.16515i 0.630955i −0.948933 0.315478i \(-0.897835\pi\)
0.948933 0.315478i \(-0.102165\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.93725 13.7477i −0.541316 0.937587i
\(216\) 0 0
\(217\) −3.50000 + 6.06218i −0.237595 + 0.411527i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.00000 15.5885i 0.605406 1.04859i
\(222\) 0 0
\(223\) −10.5830 −0.708690 −0.354345 0.935115i \(-0.615296\pi\)
−0.354345 + 0.935115i \(0.615296\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.9059 20.6216i 0.790221 1.36870i −0.135609 0.990762i \(-0.543299\pi\)
0.925830 0.377941i \(-0.123368\pi\)
\(228\) 0 0
\(229\) −1.50000 + 0.866025i −0.0991228 + 0.0572286i −0.548742 0.835992i \(-0.684893\pi\)
0.449619 + 0.893220i \(0.351560\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.5000 + 18.1865i 0.687878 + 1.19144i 0.972523 + 0.232806i \(0.0747909\pi\)
−0.284645 + 0.958633i \(0.591876\pi\)
\(234\) 0 0
\(235\) 11.9059 + 6.87386i 0.776654 + 0.448401i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −16.5000 9.52628i −1.06286 0.613642i −0.136637 0.990621i \(-0.543629\pi\)
−0.926222 + 0.376980i \(0.876963\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 10.5000 6.06218i 0.670820 0.387298i
\(246\) 0 0
\(247\) −7.93725 + 4.58258i −0.505035 + 0.291582i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −15.8745 −1.00199 −0.500995 0.865450i \(-0.667033\pi\)
−0.500995 + 0.865450i \(0.667033\pi\)
\(252\) 0 0
\(253\) 21.0000 1.32026
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.50000 0.866025i 0.0935674 0.0540212i −0.452486 0.891771i \(-0.649463\pi\)
0.546054 + 0.837750i \(0.316129\pi\)
\(258\) 0 0
\(259\) −9.26013 16.0390i −0.575396 0.996616i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.96863 2.29129i −0.244716 0.141287i 0.372626 0.927981i \(-0.378457\pi\)
−0.617342 + 0.786695i \(0.711791\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −25.5000 14.7224i −1.55476 0.897643i −0.997743 0.0671428i \(-0.978612\pi\)
−0.557019 0.830500i \(-0.688055\pi\)
\(270\) 0 0
\(271\) 14.5516 + 25.2042i 0.883949 + 1.53104i 0.846914 + 0.531730i \(0.178458\pi\)
0.0370348 + 0.999314i \(0.488209\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.93725 + 4.58258i −0.478634 + 0.276340i
\(276\) 0 0
\(277\) 8.50000 14.7224i 0.510716 0.884585i −0.489207 0.872167i \(-0.662714\pi\)
0.999923 0.0124177i \(-0.00395278\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −6.61438 + 11.4564i −0.393184 + 0.681015i −0.992868 0.119223i \(-0.961960\pi\)
0.599684 + 0.800237i \(0.295293\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.16515i 0.541002i
\(288\) 0 0
\(289\) 5.00000 + 8.66025i 0.294118 + 0.509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.7846i 1.21425i 0.794606 + 0.607125i \(0.207677\pi\)
−0.794606 + 0.607125i \(0.792323\pi\)
\(294\) 0 0
\(295\) 13.7477i 0.800424i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.93725 + 13.7477i 0.459023 + 0.795052i
\(300\) 0 0
\(301\) 24.2487i 1.39767i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.50000 2.59808i 0.0858898 0.148765i
\(306\) 0 0
\(307\) −10.5830 −0.604004 −0.302002 0.953307i \(-0.597655\pi\)
−0.302002 + 0.953307i \(0.597655\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.9059 + 20.6216i −0.675121 + 1.16934i 0.301313 + 0.953525i \(0.402575\pi\)
−0.976434 + 0.215818i \(0.930758\pi\)
\(312\) 0 0
\(313\) −22.5000 + 12.9904i −1.27178 + 0.734260i −0.975322 0.220788i \(-0.929137\pi\)
−0.296453 + 0.955047i \(0.595804\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.50000 + 12.9904i 0.421242 + 0.729612i 0.996061 0.0886679i \(-0.0282610\pi\)
−0.574819 + 0.818280i \(0.694928\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.7477i 0.764944i
\(324\) 0 0
\(325\) −6.00000 3.46410i −0.332820 0.192154i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.5000 + 18.1865i 0.578884 + 1.00266i
\(330\) 0 0
\(331\) 11.9059 6.87386i 0.654406 0.377822i −0.135736 0.990745i \(-0.543340\pi\)
0.790142 + 0.612923i \(0.210007\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.93725 0.433659
\(336\) 0 0
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.5000 + 6.06218i −0.568607 + 0.328285i
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.96863 2.29129i −0.213047 0.123003i 0.389680 0.920950i \(-0.372586\pi\)
−0.602727 + 0.797948i \(0.705919\pi\)
\(348\) 0 0
\(349\) 3.46410i 0.185429i −0.995693 0.0927146i \(-0.970446\pi\)
0.995693 0.0927146i \(-0.0295544\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −16.5000 9.52628i −0.878206 0.507033i −0.00813978 0.999967i \(-0.502591\pi\)
−0.870067 + 0.492934i \(0.835924\pi\)
\(354\) 0 0
\(355\) 7.93725 + 13.7477i 0.421266 + 0.729654i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.9059 6.87386i 0.628368 0.362789i −0.151752 0.988419i \(-0.548491\pi\)
0.780120 + 0.625630i \(0.215158\pi\)
\(360\) 0 0
\(361\) 6.00000 10.3923i 0.315789 0.546963i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.00000 −0.471082
\(366\) 0 0
\(367\) 17.1974 29.7867i 0.897696 1.55486i 0.0672642 0.997735i \(-0.478573\pi\)
0.830432 0.557120i \(-0.188094\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.96863 6.87386i 0.206041 0.356873i
\(372\) 0 0
\(373\) 0.500000 + 0.866025i 0.0258890 + 0.0448411i 0.878680 0.477412i \(-0.158425\pi\)
−0.852791 + 0.522253i \(0.825092\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 18.3303i 0.941564i 0.882249 + 0.470782i \(0.156028\pi\)
−0.882249 + 0.470782i \(0.843972\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.96863 + 6.87386i 0.202787 + 0.351238i 0.949425 0.313992i \(-0.101667\pi\)
−0.746638 + 0.665230i \(0.768333\pi\)
\(384\) 0 0
\(385\) 21.0000 1.07026
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.5000 18.1865i 0.532371 0.922094i −0.466915 0.884302i \(-0.654634\pi\)
0.999286 0.0377914i \(-0.0120322\pi\)
\(390\) 0 0
\(391\) −23.8118 −1.20421
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.96863 + 6.87386i −0.199683 + 0.345862i
\(396\) 0 0
\(397\) −19.5000 + 11.2583i −0.978677 + 0.565039i −0.901870 0.432007i \(-0.857806\pi\)
−0.0768065 + 0.997046i \(0.524472\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.5000 + 18.1865i 0.524345 + 0.908192i 0.999598 + 0.0283431i \(0.00902310\pi\)
−0.475253 + 0.879849i \(0.657644\pi\)
\(402\) 0 0
\(403\) −7.93725 4.58258i −0.395383 0.228274i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 32.0780i 1.59005i
\(408\) 0 0
\(409\) 16.5000 + 9.52628i 0.815872 + 0.471044i 0.848991 0.528407i \(-0.177211\pi\)
−0.0331186 + 0.999451i \(0.510544\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.5000 18.1865i 0.516671 0.894901i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 31.7490 1.55104 0.775520 0.631322i \(-0.217488\pi\)
0.775520 + 0.631322i \(0.217488\pi\)
\(420\) 0 0
\(421\) 16.0000 0.779792 0.389896 0.920859i \(-0.372511\pi\)
0.389896 + 0.920859i \(0.372511\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 9.00000 5.19615i 0.436564 0.252050i
\(426\) 0 0
\(427\) 3.96863 2.29129i 0.192055 0.110883i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.8431 + 11.4564i 0.955810 + 0.551837i 0.894881 0.446305i \(-0.147260\pi\)
0.0609292 + 0.998142i \(0.480594\pi\)
\(432\) 0 0
\(433\) 3.46410i 0.166474i 0.996530 + 0.0832370i \(0.0265259\pi\)
−0.996530 + 0.0832370i \(0.973474\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.5000 + 6.06218i 0.502283 + 0.289993i
\(438\) 0 0
\(439\) −14.5516 25.2042i −0.694512 1.20293i −0.970345 0.241724i \(-0.922287\pi\)
0.275834 0.961205i \(1.58895\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.96863 + 2.29129i −0.188555 + 0.108862i −0.591306 0.806447i \(-0.701387\pi\)
0.402751 + 0.915310i \(0.368054\pi\)
\(444\) 0 0
\(445\) −1.50000 + 2.59808i −0.0711068 + 0.123161i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) −7.93725 + 13.7477i −0.373751 + 0.647355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.93725 + 13.7477i 0.372104 + 0.644503i
\(456\) 0 0
\(457\) 0.500000 + 0.866025i 0.0233890 + 0.0405110i 0.877483 0.479608i \(-0.159221\pi\)
−0.854094 + 0.520119i \(0.825888\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.46410i 0.161339i 0.996741 + 0.0806696i \(0.0257059\pi\)
−0.996741 + 0.0806696i \(0.974294\pi\)
\(462\) 0 0
\(463\) 27.4955i 1.27782i 0.769281 + 0.638911i \(0.220615\pi\)
−0.769281 + 0.638911i \(0.779385\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.96863 6.87386i −0.183646 0.318084i 0.759473 0.650538i \(-0.225457\pi\)
−0.943119 + 0.332454i \(0.892123\pi\)
\(468\) 0 0
\(469\) 10.5000 + 6.06218i 0.484845 + 0.279925i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.0000 36.3731i 0.965581 1.67244i
\(474\) 0 0
\(475\) −5.29150 −0.242791
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.96863 6.87386i 0.181331 0.314075i −0.761003 0.648748i \(-0.775293\pi\)
0.942334 + 0.334674i \(0.108626\pi\)
\(480\) 0 0
\(481\) 21.0000 12.1244i 0.957518 0.552823i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.00000 5.19615i −0.136223 0.235945i
\(486\) 0 0
\(487\) 35.7176 + 20.6216i 1.61852 + 0.934453i 0.987303 + 0.158848i \(0.0507779\pi\)
0.631218 + 0.775606i \(0.282555\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 18.3303i 0.827235i 0.910451 + 0.413617i \(0.135735\pi\)
−0.910451 + 0.413617i \(0.864265\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.2487i 1.08770i
\(498\) 0 0
\(499\) 27.7804 16.0390i 1.24362 0.718005i 0.273791 0.961789i \(-0.411722\pi\)
0.969830 + 0.243784i \(0.0783888\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.7490 1.41562 0.707809 0.706404i \(-0.249684\pi\)
0.707809 + 0.706404i \(0.249684\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22.5000 + 12.9904i −0.997295 + 0.575789i −0.907447 0.420167i \(-0.861972\pi\)
−0.0898481 + 0.995955i \(0.528638\pi\)
\(510\) 0 0
\(511\) −11.9059 6.87386i −0.526685 0.304082i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.96863 2.29129i −0.174879 0.100966i
\(516\) 0 0
\(517\) 36.3731i 1.59969i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −37.5000 21.6506i −1.64290 0.948532i −0.979794 0.200011i \(-0.935902\pi\)
−0.663111 0.748521i \(-0.730764\pi\)
\(522\) 0 0
\(523\) 6.61438 + 11.4564i 0.289227 + 0.500955i 0.973625 0.228153i \(-0.0732686\pi\)
−0.684399 + 0.729108i \(0.739935\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.9059 6.87386i 0.518628 0.299430i
\(528\) 0 0
\(529\) −1.00000 + 1.73205i −0.0434783 + 0.0753066i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 11.9059 20.6216i 0.514736 0.891549i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 27.7804 + 16.0390i 1.19659 + 0.690849i
\(540\) 0 0
\(541\) −6.50000 11.2583i −0.279457 0.484033i 0.691793 0.722096i \(-0.256821\pi\)
−0.971250 + 0.238062i \(0.923488\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.1244i 0.519350i
\(546\) 0 0
\(547\) 18.3303i 0.783747i −0.920019 0.391874i \(-0.871827\pi\)
0.920019 0.391874i \(-0.128173\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −10.5000 + 6.06218i −0.446505 + 0.257790i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.50000 + 2.59808i −0.0635570 + 0.110084i −0.896053 0.443947i \(-0.853578\pi\)
0.832496 + 0.554031i \(0.186911\pi\)
\(558\) 0 0
\(559\) 31.7490 1.34284
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.96863 6.87386i 0.167258 0.289699i −0.770197 0.637806i \(-0.779842\pi\)
0.937455 + 0.348107i \(0.113175\pi\)
\(564\) 0 0
\(565\) −18.0000 + 10.3923i −0.757266 + 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.5000 18.1865i −0.440183 0.762419i 0.557520 0.830164i \(-0.311753\pi\)
−0.997703 + 0.0677445i \(0.978420\pi\)
\(570\) 0 0
\(571\) 11.9059 + 6.87386i 0.498246 + 0.287662i 0.727989 0.685589i \(-0.240455\pi\)
−0.229743 + 0.973251i \(0.573789\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.16515i 0.382213i
\(576\) 0 0
\(577\) −16.5000 9.52628i −0.686904 0.396584i 0.115547 0.993302i \(-0.463138\pi\)
−0.802451 + 0.596718i \(0.796471\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.9059 6.87386i 0.493091 0.284686i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.7490 1.31042 0.655211 0.755446i \(-0.272580\pi\)
0.655211 + 0.755446i \(0.272580\pi\)
\(588\) 0 0
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19.5000 11.2583i 0.800769 0.462324i −0.0429710 0.999076i \(-0.513682\pi\)
0.843740 + 0.536752i \(0.180349\pi\)
\(594\) 0 0
\(595\) −23.8118 −0.976187
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.96863 2.29129i −0.162154 0.0936195i 0.416727 0.909032i \(-0.363177\pi\)
−0.578881 + 0.815412i \(0.696510\pi\)
\(600\) 0 0
\(601\) 3.46410i 0.141304i −0.997501 0.0706518i \(-0.977492\pi\)
0.997501 0.0706518i \(-0.0225079\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15.0000 + 8.66025i 0.609837 + 0.352089i
\(606\) 0 0
\(607\) 1.32288 + 2.29129i 0.0536939 + 0.0930005i 0.891623 0.452778i \(-0.149567\pi\)
−0.837929 + 0.545779i \(0.816234\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23.8118 + 13.7477i −0.963321 + 0.556174i
\(612\) 0 0
\(613\) 3.50000 6.06218i 0.141364 0.244849i −0.786647 0.617403i \(-0.788185\pi\)
0.928010 + 0.372554i \(0.121518\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −9.26013 + 16.0390i −0.372196 + 0.644662i −0.989903 0.141746i \(-0.954728\pi\)
0.617707 + 0.786408i \(0.288062\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.96863 + 2.29129i −0.159000 + 0.0917985i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 36.3731i 1.45029i
\(630\) 0 0
\(631\) 45.8258i 1.82429i 0.409863 + 0.912147i \(0.365577\pi\)
−0.409863 + 0.912147i \(0.634423\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.93725 + 13.7477i 0.314980 + 0.545562i
\(636\) 0 0
\(637\) 24.2487i 0.960769i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.5000 + 18.1865i −0.414725 + 0.718325i −0.995400 0.0958109i \(-0.969456\pi\)
0.580674 + 0.814136i \(0.302789\pi\)
\(642\) 0 0
\(643\) 21.1660 0.834706 0.417353 0.908744i \(-0.362958\pi\)
0.417353 + 0.908744i \(0.362958\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.8431 34.3693i 0.780114 1.35120i −0.151761 0.988417i \(-0.548494\pi\)
0.931875 0.362780i \(-0.118172\pi\)
\(648\) 0 0
\(649\) 31.5000 18.1865i 1.23648 0.713884i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.5000 18.1865i −0.410897 0.711694i 0.584091 0.811688i \(-0.301451\pi\)
−0.994988 + 0.0999939i \(0.968118\pi\)
\(654\) 0 0
\(655\) −11.9059 6.87386i −0.465201 0.268584i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.16515i 0.357024i 0.983938 + 0.178512i \(0.0571283\pi\)
−0.983938 + 0.178512i \(0.942872\pi\)
\(660\) 0 0
\(661\) −4.50000 2.59808i −0.175030 0.101053i 0.409926 0.912119i \(-0.365555\pi\)
−0.584955 + 0.811065i \(0.698888\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 10.5000 + 6.06218i 0.407173 + 0.235081i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.93725 0.306414
\(672\) 0 0
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.50000 0.866025i 0.0576497 0.0332841i −0.470898 0.882188i \(-0.656070\pi\)
0.528548 + 0.848904i \(0.322737\pi\)
\(678\) 0 0
\(679\) 9.16515i 0.351726i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.96863 + 2.29129i 0.151855 + 0.0876737i 0.574002 0.818854i \(-0.305390\pi\)
−0.422147 + 0.906527i \(0.638723\pi\)
\(684\) 0 0
\(685\) 36.3731i 1.38974i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.00000 + 5.19615i 0.342873 + 0.197958i
\(690\) 0 0
\(691\) −1.32288 2.29129i −0.0503246 0.0871647i 0.839766 0.542949i \(-0.182692\pi\)
−0.890090 + 0.455784i \(0.849359\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 15.8745 9.16515i 0.602154 0.347654i
\(696\) 0 0
\(697\) 9.00000 15.5885i 0.340899 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 9.26013 16.0390i 0.349252 0.604923i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −11.9059 6.87386i −0.447767 0.258518i
\(708\) 0 0
\(709\) −17.5000 30.3109i −0.657226 1.13835i −0.981331 0.192328i \(-0.938396\pi\)
0.324104 0.946021i \(-0.394937\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.1244i 0.454061i
\(714\) 0 0
\(715\) 27.4955i 1.02827i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.96863 6.87386i −0.148005 0.256352i 0.782485 0.622669i \(-0.213952\pi\)
−0.930490 + 0.366317i \(0.880618\pi\)
\(720\) 0 0
\(721\) −3.50000 6.06218i −0.130347 0.225767i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.1660 −0.785004 −0.392502 0.919751i \(-0.628390\pi\)
−0.392502 + 0.919751i \(0.628390\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −23.8118 + 41.2432i −0.880710 + 1.52543i
\(732\) 0 0
\(733\) 19.5000 11.2583i 0.720249 0.415836i −0.0945954 0.995516i \(-0.530156\pi\)
0.814844 + 0.579680i \(0.196822\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.5000 + 18.1865i 0.386772 + 0.669910i
\(738\) 0 0
\(739\) −19.8431 11.4564i −0.729942 0.421432i 0.0884593 0.996080i \(-0.471806\pi\)
−0.818401 + 0.574648i \(0.805139\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.8258i 1.68118i −0.541669 0.840592i \(-0.682207\pi\)
0.541669 0.840592i \(-0.317793\pi\)
\(744\) 0 0
\(745\) −22.5000 12.9904i −0.824336 0.475931i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 31.5000 18.1865i 1.15098 0.664521i
\(750\) 0 0
\(751\) −11.9059 + 6.87386i −0.434452 + 0.250831i −0.701241 0.712924i \(-0.747370\pi\)
0.266790 + 0.963755i \(0.414037\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −39.6863 −1.44433
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.5000 + 12.9904i −0.815624 + 0.470901i −0.848905 0.528545i \(-0.822738\pi\)
0.0332809 + 0.999446i \(0.489404\pi\)
\(762\) 0 0
\(763\) 9.26013 16.0390i 0.335239 0.580651i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.8118 + 13.7477i 0.859793 + 0.496402i
\(768\) 0 0
\(769\) 51.9615i 1.87378i 0.349624 + 0.936890i \(0.386309\pi\)
−0.349624 + 0.936890i \(0.613691\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.5000 + 14.7224i 0.917171 + 0.529529i 0.882732 0.469878i \(-0.155702\pi\)
0.0344397 + 0.999407i \(0.489035\pi\)
\(774\) 0 0
\(775\) −2.64575 4.58258i −0.0950382 0.164611i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −7.93725 + 4.58258i −0.284382 + 0.164188i
\(780\) 0 0
\(781\) −21.0000 + 36.3731i −0.751439 + 1.30153i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.00000 −0.321224
\(786\) 0 0
\(787\) 6.61438 11.4564i 0.235777 0.408378i −0.723721 0.690093i \(-0.757570\pi\)
0.959498 + 0.281715i \(0.0909031\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −31.7490 −1.12887
\(792\) 0 0
\(793\) 3.00000 + 5.19615i 0.106533 + 0.184521i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.46410i 0.122705i −0.998116 0.0613524i \(-0.980459\pi\)
0.998116 0.0613524i \(-0.0195413\pi\)
\(798\) 0 0
\(799\) 41.2432i 1.45908i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.9059 20.6216i −0.420149 0.727720i
\(804\) 0 0
\(805\) 10.5000 18.1865i 0.370076 0.640991i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −22.5000 + 38.9711i −0.791058 + 1.37015i 0.134255 + 0.990947i \(0.457136\pi\)
−0.925312 + 0.379206i \(0.876197\pi\)
\(810\) 0 0
\(811\) −21.1660 −0.743239 −0.371620