Properties

Label 1680.2.f.b.881.1
Level $1680$
Weight $2$
Character 1680.881
Analytic conductor $13.415$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1680.881
Dual form 1680.2.f.b.881.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} -1.00000 q^{5} +(-2.00000 + 1.73205i) q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} -1.00000 q^{5} +(-2.00000 + 1.73205i) q^{7} -3.00000 q^{9} +3.46410i q^{11} +1.73205i q^{15} +6.00000 q^{17} -3.46410i q^{19} +(3.00000 + 3.46410i) q^{21} -3.46410i q^{23} +1.00000 q^{25} +5.19615i q^{27} -6.92820i q^{29} +3.46410i q^{31} +6.00000 q^{33} +(2.00000 - 1.73205i) q^{35} -2.00000 q^{37} +6.00000 q^{41} +8.00000 q^{43} +3.00000 q^{45} +12.0000 q^{47} +(1.00000 - 6.92820i) q^{49} -10.3923i q^{51} -3.46410i q^{55} -6.00000 q^{57} +12.0000 q^{59} -6.92820i q^{61} +(6.00000 - 5.19615i) q^{63} -8.00000 q^{67} -6.00000 q^{69} -3.46410i q^{71} +6.92820i q^{73} -1.73205i q^{75} +(-6.00000 - 6.92820i) q^{77} -8.00000 q^{79} +9.00000 q^{81} -6.00000 q^{85} -12.0000 q^{87} +6.00000 q^{89} +6.00000 q^{93} +3.46410i q^{95} +6.92820i q^{97} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 2q^{5} - 4q^{7} - 6q^{9} + 12q^{17} + 6q^{21} + 2q^{25} + 12q^{33} + 4q^{35} - 4q^{37} + 12q^{41} + 16q^{43} + 6q^{45} + 24q^{47} + 2q^{49} - 12q^{57} + 24q^{59} + 12q^{63} - 16q^{67} - 12q^{69} - 12q^{77} - 16q^{79} + 18q^{81} - 12q^{85} - 24q^{87} + 12q^{89} + 12q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 + 1.73205i −0.755929 + 0.654654i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 1.73205i 0.447214i
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 3.00000 + 3.46410i 0.654654 + 0.755929i
\(22\) 0 0
\(23\) 3.46410i 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 6.92820i 1.28654i −0.765641 0.643268i \(-0.777578\pi\)
0.765641 0.643268i \(-0.222422\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 2.00000 1.73205i 0.338062 0.292770i
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 10.3923i 1.45521i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 3.46410i 0.467099i
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i −0.896258 0.443533i \(-0.853725\pi\)
0.896258 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) 6.00000 5.19615i 0.755929 0.654654i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 3.46410i 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 0 0
\(75\) 1.73205i 0.200000i
\(76\) 0 0
\(77\) −6.00000 6.92820i −0.683763 0.789542i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) 3.46410i 0.355409i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 10.3923i 1.04447i
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 3.46410i 0.341328i 0.985329 + 0.170664i \(0.0545913\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) 0 0
\(105\) −3.00000 3.46410i −0.292770 0.338062i
\(106\) 0 0
\(107\) 10.3923i 1.00466i −0.864675 0.502331i \(-0.832476\pi\)
0.864675 0.502331i \(-0.167524\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 3.46410i 0.328798i
\(112\) 0 0
\(113\) 6.92820i 0.651751i −0.945413 0.325875i \(-0.894341\pi\)
0.945413 0.325875i \(-0.105659\pi\)
\(114\) 0 0
\(115\) 3.46410i 0.323029i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 + 10.3923i −1.10004 + 0.952661i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 10.3923i 0.937043i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 13.8564i 1.21999i
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 6.00000 + 6.92820i 0.520266 + 0.600751i
\(134\) 0 0
\(135\) 5.19615i 0.447214i
\(136\) 0 0
\(137\) 20.7846i 1.77575i 0.460086 + 0.887875i \(0.347819\pi\)
−0.460086 + 0.887875i \(0.652181\pi\)
\(138\) 0 0
\(139\) 17.3205i 1.46911i −0.678551 0.734553i \(-0.737392\pi\)
0.678551 0.734553i \(-0.262608\pi\)
\(140\) 0 0
\(141\) 20.7846i 1.75038i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.92820i 0.575356i
\(146\) 0 0
\(147\) −12.0000 1.73205i −0.989743 0.142857i
\(148\) 0 0
\(149\) 6.92820i 0.567581i 0.958886 + 0.283790i \(0.0915919\pi\)
−0.958886 + 0.283790i \(0.908408\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −18.0000 −1.45521
\(154\) 0 0
\(155\) 3.46410i 0.278243i
\(156\) 0 0
\(157\) 13.8564i 1.10586i 0.833227 + 0.552931i \(0.186491\pi\)
−0.833227 + 0.552931i \(0.813509\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 + 6.92820i 0.472866 + 0.546019i
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 10.3923i 0.794719i
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −2.00000 + 1.73205i −0.151186 + 0.130931i
\(176\) 0 0
\(177\) 20.7846i 1.56227i
\(178\) 0 0
\(179\) 10.3923i 0.776757i −0.921500 0.388379i \(-0.873035\pi\)
0.921500 0.388379i \(-0.126965\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i −0.635071 0.772454i \(-0.719029\pi\)
0.635071 0.772454i \(-0.280971\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 20.7846i 1.51992i
\(188\) 0 0
\(189\) −9.00000 10.3923i −0.654654 0.755929i
\(190\) 0 0
\(191\) 10.3923i 0.751961i 0.926628 + 0.375980i \(0.122694\pi\)
−0.926628 + 0.375980i \(0.877306\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.8564i 0.987228i −0.869681 0.493614i \(-0.835676\pi\)
0.869681 0.493614i \(-0.164324\pi\)
\(198\) 0 0
\(199\) 10.3923i 0.736691i −0.929689 0.368345i \(-0.879924\pi\)
0.929689 0.368345i \(-0.120076\pi\)
\(200\) 0 0
\(201\) 13.8564i 0.977356i
\(202\) 0 0
\(203\) 12.0000 + 13.8564i 0.842235 + 0.972529i
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 10.3923i 0.722315i
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −6.00000 6.92820i −0.407307 0.470317i
\(218\) 0 0
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.3205i 1.15987i 0.814664 + 0.579934i \(0.196921\pi\)
−0.814664 + 0.579934i \(0.803079\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i −0.973447 0.228914i \(-0.926482\pi\)
0.973447 0.228914i \(-0.0735176\pi\)
\(230\) 0 0
\(231\) −12.0000 + 10.3923i −0.789542 + 0.683763i
\(232\) 0 0
\(233\) 6.92820i 0.453882i 0.973909 + 0.226941i \(0.0728724\pi\)
−0.973909 + 0.226941i \(0.927128\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 13.8564i 0.900070i
\(238\) 0 0
\(239\) 10.3923i 0.672222i 0.941822 + 0.336111i \(0.109112\pi\)
−0.941822 + 0.336111i \(0.890888\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 15.5885i 1.00000i
\(244\) 0 0
\(245\) −1.00000 + 6.92820i −0.0638877 + 0.442627i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 10.3923i 0.650791i
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 4.00000 3.46410i 0.248548 0.215249i
\(260\) 0 0
\(261\) 20.7846i 1.28654i
\(262\) 0 0
\(263\) 24.2487i 1.49524i 0.664127 + 0.747620i \(0.268803\pi\)
−0.664127 + 0.747620i \(0.731197\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 10.3923i 0.635999i
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 24.2487i 1.47300i −0.676435 0.736502i \(-0.736476\pi\)
0.676435 0.736502i \(-0.263524\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.46410i 0.208893i
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 0 0
\(279\) 10.3923i 0.622171i
\(280\) 0 0
\(281\) 13.8564i 0.826604i 0.910594 + 0.413302i \(0.135625\pi\)
−0.910594 + 0.413302i \(0.864375\pi\)
\(282\) 0 0
\(283\) 17.3205i 1.02960i −0.857311 0.514799i \(-0.827867\pi\)
0.857311 0.514799i \(-0.172133\pi\)
\(284\) 0 0
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) −12.0000 + 10.3923i −0.708338 + 0.613438i
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) −18.0000 −1.04447
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 + 13.8564i −0.922225 + 0.798670i
\(302\) 0 0
\(303\) 10.3923i 0.597022i
\(304\) 0 0
\(305\) 6.92820i 0.396708i
\(306\) 0 0
\(307\) 24.2487i 1.38395i 0.721923 + 0.691974i \(0.243259\pi\)
−0.721923 + 0.691974i \(0.756741\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 20.7846i 1.17482i −0.809291 0.587408i \(-0.800148\pi\)
0.809291 0.587408i \(-0.199852\pi\)
\(314\) 0 0
\(315\) −6.00000 + 5.19615i −0.338062 + 0.292770i
\(316\) 0 0
\(317\) 27.7128i 1.55651i 0.627950 + 0.778253i \(0.283894\pi\)
−0.627950 + 0.778253i \(0.716106\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 20.7846i 1.15649i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.46410i 0.191565i
\(328\) 0 0
\(329\) −24.0000 + 20.7846i −1.32316 + 1.14589i
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) 17.3205i 0.929814i 0.885360 + 0.464907i \(0.153912\pi\)
−0.885360 + 0.464907i \(0.846088\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i 0.982658 + 0.185429i \(0.0593675\pi\)
−0.982658 + 0.185429i \(0.940632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 3.46410i 0.183855i
\(356\) 0 0
\(357\) 18.0000 + 20.7846i 0.952661 + 1.10004i
\(358\) 0 0
\(359\) 3.46410i 0.182828i −0.995813 0.0914141i \(-0.970861\pi\)
0.995813 0.0914141i \(-0.0291387\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) 1.73205i 0.0909091i
\(364\) 0 0
\(365\) 6.92820i 0.362639i
\(366\) 0 0
\(367\) 10.3923i 0.542474i −0.962513 0.271237i \(-0.912567\pi\)
0.962513 0.271237i \(-0.0874327\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 1.73205i 0.0894427i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 6.92820i 0.354943i
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 6.00000 + 6.92820i 0.305788 + 0.353094i
\(386\) 0 0
\(387\) −24.0000 −1.21999
\(388\) 0 0
\(389\) 6.92820i 0.351274i 0.984455 + 0.175637i \(0.0561985\pi\)
−0.984455 + 0.175637i \(0.943802\pi\)
\(390\) 0 0
\(391\) 20.7846i 1.05112i
\(392\) 0 0
\(393\) 20.7846i 1.04844i
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 13.8564i 0.695433i 0.937600 + 0.347717i \(0.113043\pi\)
−0.937600 + 0.347717i \(0.886957\pi\)
\(398\) 0 0
\(399\) 12.0000 10.3923i 0.600751 0.520266i
\(400\) 0 0
\(401\) 27.7128i 1.38391i −0.721940 0.691956i \(-0.756749\pi\)
0.721940 0.691956i \(-0.243251\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 6.92820i 0.343418i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) 0 0
\(413\) −24.0000 + 20.7846i −1.18096 + 1.02274i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −30.0000 −1.46911
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) −36.0000 −1.75038
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 12.0000 + 13.8564i 0.580721 + 0.670559i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.3923i 0.500580i 0.968171 + 0.250290i \(0.0805259\pi\)
−0.968171 + 0.250290i \(0.919474\pi\)
\(432\) 0 0
\(433\) 6.92820i 0.332948i −0.986046 0.166474i \(-0.946762\pi\)
0.986046 0.166474i \(-0.0532382\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 0 0
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) 17.3205i 0.826663i 0.910581 + 0.413331i \(0.135635\pi\)
−0.910581 + 0.413331i \(0.864365\pi\)
\(440\) 0 0
\(441\) −3.00000 + 20.7846i −0.142857 + 0.989743i
\(442\) 0 0
\(443\) 10.3923i 0.493753i −0.969047 0.246877i \(-0.920596\pi\)
0.969047 0.246877i \(-0.0794043\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 13.8564i 0.653924i −0.945037 0.326962i \(-0.893975\pi\)
0.945037 0.326962i \(-0.106025\pi\)
\(450\) 0 0
\(451\) 20.7846i 0.978709i
\(452\) 0 0
\(453\) 13.8564i 0.651031i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 0 0
\(459\) 31.1769i 1.45521i
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 0 0
\(465\) −6.00000 −0.278243
\(466\) 0 0
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 16.0000 13.8564i 0.738811 0.639829i
\(470\) 0 0
\(471\) 24.0000 1.10586
\(472\) 0 0
\(473\) 27.7128i 1.27424i
\(474\) 0 0
\(475\) 3.46410i 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 12.0000 10.3923i 0.546019 0.472866i
\(484\) 0 0
\(485\) 6.92820i 0.314594i
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 0 0
\(489\) 27.7128i 1.25322i
\(490\) 0 0
\(491\) 24.2487i 1.09433i −0.837025 0.547165i \(-0.815707\pi\)
0.837025 0.547165i \(-0.184293\pi\)
\(492\) 0 0
\(493\) 41.5692i 1.87218i
\(494\) 0 0
\(495\) 10.3923i 0.467099i
\(496\) 0 0
\(497\) 6.00000 + 6.92820i 0.269137 + 0.310772i
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 20.7846i 0.928588i
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 22.5167i 1.00000i
\(508\) 0 0
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) −12.0000 13.8564i −0.530849 0.612971i
\(512\) 0 0
\(513\) 18.0000 0.794719
\(514\) 0 0
\(515\) 3.46410i 0.152647i
\(516\) 0 0
\(517\) 41.5692i 1.82821i
\(518\) 0 0
\(519\) 31.1769i 1.36851i
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 17.3205i 0.757373i −0.925525 0.378686i \(-0.876376\pi\)
0.925525 0.378686i \(-0.123624\pi\)
\(524\) 0 0
\(525\) 3.00000 + 3.46410i 0.130931 + 0.151186i
\(526\) 0 0
\(527\) 20.7846i 0.905392i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) −36.0000 −1.56227
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.3923i 0.449299i
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) 24.0000 + 3.46410i 1.03375 + 0.149209i
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) −36.0000 −1.54491
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 20.7846i 0.887066i
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 16.0000 13.8564i 0.680389 0.589234i
\(554\) 0 0
\(555\) 3.46410i 0.147043i
\(556\) 0 0
\(557\) 27.7128i 1.17423i −0.809504 0.587115i \(-0.800264\pi\)
0.809504 0.587115i \(-0.199736\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 6.92820i 0.291472i
\(566\) 0 0
\(567\) −18.0000 + 15.5885i −0.755929 + 0.654654i
\(568\) 0 0
\(569\) 27.7128i 1.16178i −0.813982 0.580891i \(-0.802704\pi\)
0.813982 0.580891i \(-0.197296\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) 3.46410i 0.144463i
\(576\) 0 0
\(577\) 34.6410i 1.44212i 0.692870 + 0.721062i \(0.256346\pi\)
−0.692870 + 0.721062i \(0.743654\pi\)
\(578\) 0 0
\(579\) 24.2487i 1.00774i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 12.0000 10.3923i 0.491952 0.426043i
\(596\) 0 0
\(597\) −18.0000 −0.736691
\(598\) 0 0
\(599\) 24.2487i 0.990775i 0.868672 + 0.495388i \(0.164974\pi\)
−0.868672 + 0.495388i \(0.835026\pi\)
\(600\) 0 0
\(601\) 13.8564i 0.565215i 0.959236 + 0.282607i \(0.0911993\pi\)
−0.959236 + 0.282607i \(0.908801\pi\)
\(602\) 0 0
\(603\) 24.0000 0.977356
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 17.3205i 0.703018i 0.936185 + 0.351509i \(0.114331\pi\)
−0.936185 + 0.351509i \(0.885669\pi\)
\(608\) 0 0
\(609\) 24.0000 20.7846i 0.972529 0.842235i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 10.3923i 0.419058i
\(616\) 0 0
\(617\) 6.92820i 0.278919i −0.990228 0.139459i \(-0.955464\pi\)
0.990228 0.139459i \(-0.0445365\pi\)
\(618\) 0 0
\(619\) 17.3205i 0.696170i −0.937463 0.348085i \(-0.886832\pi\)
0.937463 0.348085i \(-0.113168\pi\)
\(620\) 0 0
\(621\) 18.0000 0.722315
\(622\) 0 0
\(623\) −12.0000 + 10.3923i −0.480770 + 0.416359i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 20.7846i 0.830057i
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 6.92820i 0.275371i
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 10.3923i 0.411113i
\(640\) 0 0
\(641\) 27.7128i 1.09459i 0.836940 + 0.547295i \(0.184342\pi\)
−0.836940 + 0.547295i \(0.815658\pi\)
\(642\) 0 0
\(643\) 31.1769i 1.22950i −0.788723 0.614749i \(-0.789257\pi\)
0.788723 0.614749i \(-0.210743\pi\)
\(644\) 0 0
\(645\) 13.8564i 0.545595i
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 41.5692i 1.63173i
\(650\) 0 0
\(651\) −12.0000 + 10.3923i −0.470317 + 0.407307i
\(652\) 0 0
\(653\) 41.5692i 1.62673i −0.581754 0.813365i \(-0.697633\pi\)
0.581754 0.813365i \(-0.302367\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) 20.7846i 0.810885i
\(658\) 0 0
\(659\) 10.3923i 0.404827i −0.979300 0.202413i \(-0.935122\pi\)
0.979300 0.202413i \(-0.0648785\pi\)
\(660\) 0 0
\(661\) 48.4974i 1.88633i −0.332323 0.943166i \(-0.607832\pi\)
0.332323 0.943166i \(-0.392168\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.00000 6.92820i −0.232670 0.268664i
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 30.0000 1.15987
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 5.19615i 0.200000i
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −12.0000 13.8564i −0.460518 0.531760i
\(680\) 0 0
\(681\) 41.5692i 1.59294i
\(682\) 0 0
\(683\) 17.3205i 0.662751i 0.943499 + 0.331375i \(0.107513\pi\)
−0.943499 + 0.331375i \(0.892487\pi\)
\(684\) 0 0
\(685\) 20.7846i 0.794139i
\(686\) 0 0
\(687\) −12.0000 −0.457829
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 31.1769i 1.18603i −0.805193 0.593013i \(-0.797938\pi\)
0.805193 0.593013i \(-0.202062\pi\)
\(692\) 0 0
\(693\) 18.0000 + 20.7846i 0.683763 + 0.789542i
\(694\) 0 0
\(695\) 17.3205i 0.657004i
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 20.7846i 0.785024i −0.919747 0.392512i \(-0.871606\pi\)
0.919747 0.392512i \(-0.128394\pi\)
\(702\) 0 0
\(703\) 6.92820i 0.261302i
\(704\) 0 0
\(705\) 20.7846i 0.782794i
\(706\) 0 0
\(707\) 12.0000 10.3923i 0.451306 0.390843i
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 18.0000 0.672222
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −6.00000 6.92820i −0.223452 0.258020i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.92820i 0.257307i
\(726\) 0 0
\(727\) 3.46410i 0.128476i 0.997935 + 0.0642382i \(0.0204617\pi\)
−0.997935 + 0.0642382i \(0.979538\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 12.0000 + 1.73205i 0.442627 + 0.0638877i
\(736\) 0 0
\(737\) 27.7128i 1.02081i
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.1769i 1.14377i −0.820334 0.571885i \(-0.806212\pi\)
0.820334 0.571885i \(-0.193788\pi\)
\(744\) 0 0
\(745\) 6.92820i 0.253830i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18.0000 + 20.7846i 0.657706 + 0.759453i
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 20.7846i 0.757433i
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 20.7846i 0.754434i
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 4.00000 3.46410i 0.144810 0.125409i
\(764\) 0 0
\(765\) 18.0000 0.650791
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 41.5692i 1.49902i −0.661991 0.749512i \(-0.730288\pi\)
0.661991 0.749512i \(-0.269712\pi\)
\(770\) 0 0
\(771\) 10.3923i 0.374270i
\(772\) 0 0
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 3.46410i 0.124434i
\(776\) 0 0
\(777\) −6.00000 6.92820i −0.215249 0.248548i
\(778\) 0 0
\(779\) 20.7846i 0.744686i
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 36.0000 1.28654
\(784\) 0 0
\(785\) 13.8564i 0.494556i
\(786\) 0 0
\(787\) 24.2487i 0.864373i 0.901784 + 0.432187i \(0.142258\pi\)
−0.901784 + 0.432187i \(0.857742\pi\)
\(788\) 0 0
\(789\) 42.0000 1.49524
\(790\) 0 0
\(791\) 12.0000 + 13.8564i 0.426671 + 0.492677i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 0 0
\(803\) −24.0000 −0.846942
\(804\) 0