Properties

Label 3360.2.t.a.2689.2
Level $3360$
Weight $2$
Character 3360.2689
Analytic conductor $26.830$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.t (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.8297350792\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3360.2689
Dual form 3360.2.t.a.2689.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} -2.00000i q^{13} +(1.00000 - 2.00000i) q^{15} -6.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000i q^{27} +6.00000 q^{29} +2.00000 q^{31} -4.00000i q^{33} +(1.00000 - 2.00000i) q^{35} +8.00000i q^{37} +2.00000 q^{39} +12.0000 q^{41} +4.00000i q^{43} +(2.00000 + 1.00000i) q^{45} +8.00000i q^{47} -1.00000 q^{49} +6.00000 q^{51} +2.00000i q^{53} +(8.00000 + 4.00000i) q^{55} -6.00000i q^{57} +12.0000 q^{59} -10.0000 q^{61} -1.00000i q^{63} +(-2.00000 + 4.00000i) q^{65} +4.00000i q^{67} +2.00000 q^{69} -8.00000 q^{71} -2.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -4.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +(-6.00000 + 12.0000i) q^{85} +6.00000i q^{87} +12.0000 q^{89} +2.00000 q^{91} +2.00000i q^{93} +(12.0000 + 6.00000i) q^{95} -10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{5} - 2 q^{9} - 8 q^{11} + 2 q^{15} - 12 q^{19} - 2 q^{21} + 6 q^{25} + 12 q^{29} + 4 q^{31} + 2 q^{35} + 4 q^{39} + 24 q^{41} + 4 q^{45} - 2 q^{49} + 12 q^{51} + 16 q^{55} + 24 q^{59} - 20 q^{61} - 4 q^{65} + 4 q^{69} - 16 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{85} + 24 q^{89} + 4 q^{91} + 24 q^{95} + 8 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) −2.00000 1.00000i −0.894427 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 1.00000 2.00000i 0.258199 0.516398i
\(16\) 0 0
\(17\) 6.00000i 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 2.00000i 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 4.00000i 0.696311i
\(34\) 0 0
\(35\) 1.00000 2.00000i 0.169031 0.338062i
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 2.00000 + 1.00000i 0.298142 + 0.149071i
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 8.00000 + 4.00000i 1.07872 + 0.539360i
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) −2.00000 + 4.00000i −0.248069 + 0.496139i
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) −4.00000 + 3.00000i −0.461880 + 0.346410i
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −6.00000 + 12.0000i −0.650791 + 1.30158i
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 2.00000i 0.207390i
\(94\) 0 0
\(95\) 12.0000 + 6.00000i 1.23117 + 0.615587i
\(96\) 0 0
\(97\) 10.0000i 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 0 0
\(105\) 2.00000 + 1.00000i 0.195180 + 0.0975900i
\(106\) 0 0
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) −2.00000 + 4.00000i −0.186501 + 0.373002i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) −2.00000 11.0000i −0.178885 0.983870i
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 6.00000i 0.520266i
\(134\) 0 0
\(135\) −1.00000 + 2.00000i −0.0860663 + 0.172133i
\(136\) 0 0
\(137\) 14.0000i 1.19610i −0.801459 0.598050i \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 8.00000i 0.668994i
\(144\) 0 0
\(145\) −12.0000 6.00000i −0.996546 0.498273i
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 6.00000i 0.485071i
\(154\) 0 0
\(155\) −4.00000 2.00000i −0.321288 0.160644i
\(156\) 0 0
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 16.0000i 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 0 0
\(165\) −4.00000 + 8.00000i −0.311400 + 0.622799i
\(166\) 0 0
\(167\) 20.0000i 1.54765i 0.633402 + 0.773823i \(0.281658\pi\)
−0.633402 + 0.773823i \(0.718342\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 0 0
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 10.0000i 0.739221i
\(184\) 0 0
\(185\) 8.00000 16.0000i 0.588172 1.17634i
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) 0 0
\(195\) −4.00000 2.00000i −0.286446 0.143223i
\(196\) 0 0
\(197\) 22.0000i 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) −24.0000 12.0000i −1.67623 0.838116i
\(206\) 0 0
\(207\) 2.00000i 0.139010i
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 8.00000i 0.548151i
\(214\) 0 0
\(215\) 4.00000 8.00000i 0.272798 0.545595i
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −3.00000 4.00000i −0.200000 0.266667i
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 22.0000i 1.44127i −0.693316 0.720634i \(-0.743851\pi\)
0.693316 0.720634i \(-0.256149\pi\)
\(234\) 0 0
\(235\) 8.00000 16.0000i 0.521862 1.04372i
\(236\) 0 0
\(237\) 8.00000i 0.519656i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 2.00000 + 1.00000i 0.127775 + 0.0638877i
\(246\) 0 0
\(247\) 12.0000i 0.763542i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) 0 0
\(255\) −12.0000 6.00000i −0.751469 0.375735i
\(256\) 0 0
\(257\) 30.0000i 1.87135i 0.352865 + 0.935674i \(0.385208\pi\)
−0.352865 + 0.935674i \(0.614792\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 2.00000i 0.123325i 0.998097 + 0.0616626i \(0.0196403\pi\)
−0.998097 + 0.0616626i \(0.980360\pi\)
\(264\) 0 0
\(265\) 2.00000 4.00000i 0.122859 0.245718i
\(266\) 0 0
\(267\) 12.0000i 0.734388i
\(268\) 0 0
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 0 0
\(275\) −12.0000 16.0000i −0.723627 0.964836i
\(276\) 0 0
\(277\) 16.0000i 0.961347i 0.876900 + 0.480673i \(0.159608\pi\)
−0.876900 + 0.480673i \(0.840392\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) −6.00000 + 12.0000i −0.355409 + 0.710819i
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) 10.0000i 0.584206i −0.956387 0.292103i \(-0.905645\pi\)
0.956387 0.292103i \(-0.0943550\pi\)
\(294\) 0 0
\(295\) −24.0000 12.0000i −1.39733 0.698667i
\(296\) 0 0
\(297\) 4.00000i 0.232104i
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 20.0000 + 10.0000i 1.14520 + 0.572598i
\(306\) 0 0
\(307\) 20.0000i 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 0 0
\(315\) −1.00000 + 2.00000i −0.0563436 + 0.112687i
\(316\) 0 0
\(317\) 14.0000i 0.786318i 0.919470 + 0.393159i \(0.128618\pi\)
−0.919470 + 0.393159i \(0.871382\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 36.0000i 2.00309i
\(324\) 0 0
\(325\) 8.00000 6.00000i 0.443760 0.332820i
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 8.00000i 0.438397i
\(334\) 0 0
\(335\) 4.00000 8.00000i 0.218543 0.437087i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −4.00000 2.00000i −0.215353 0.107676i
\(346\) 0 0
\(347\) 6.00000i 0.322097i 0.986947 + 0.161048i \(0.0514875\pi\)
−0.986947 + 0.161048i \(0.948512\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 14.0000i 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 0 0
\(355\) 16.0000 + 8.00000i 0.849192 + 0.424596i
\(356\) 0 0
\(357\) 6.00000i 0.317554i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) −2.00000 + 4.00000i −0.104685 + 0.209370i
\(366\) 0 0
\(367\) 24.0000i 1.25279i 0.779506 + 0.626395i \(0.215470\pi\)
−0.779506 + 0.626395i \(0.784530\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 12.0000i 0.621336i 0.950518 + 0.310668i \(0.100553\pi\)
−0.950518 + 0.310668i \(0.899447\pi\)
\(374\) 0 0
\(375\) 11.0000 2.00000i 0.568038 0.103280i
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 24.0000i 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) −4.00000 + 8.00000i −0.203859 + 0.407718i
\(386\) 0 0
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 4.00000i 0.201773i
\(394\) 0 0
\(395\) 16.0000 + 8.00000i 0.805047 + 0.402524i
\(396\) 0 0
\(397\) 2.00000i 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 0 0
\(405\) −2.00000 1.00000i −0.0993808 0.0496904i
\(406\) 0 0
\(407\) 32.0000i 1.58618i
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 0 0
\(413\) 12.0000i 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.0000i 0.685583i
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) 24.0000 18.0000i 1.16417 0.873128i
\(426\) 0 0
\(427\) 10.0000i 0.483934i
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 34.0000i 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 0 0
\(435\) 6.00000 12.0000i 0.287678 0.575356i
\(436\) 0 0
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 14.0000i 0.665160i −0.943075 0.332580i \(-0.892081\pi\)
0.943075 0.332580i \(-0.107919\pi\)
\(444\) 0 0
\(445\) −24.0000 12.0000i −1.13771 0.568855i
\(446\) 0 0
\(447\) 2.00000i 0.0945968i
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) −48.0000 −2.26023
\(452\) 0 0
\(453\) 16.0000i 0.751746i
\(454\) 0 0
\(455\) −4.00000 2.00000i −0.187523 0.0937614i
\(456\) 0 0
\(457\) 4.00000i 0.187112i −0.995614 0.0935561i \(-0.970177\pi\)
0.995614 0.0935561i \(-0.0298234\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 40.0000 1.86299 0.931493 0.363760i \(-0.118507\pi\)
0.931493 + 0.363760i \(0.118507\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) 0 0
\(465\) 2.00000 4.00000i 0.0927478 0.185496i
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) −18.0000 24.0000i −0.825897 1.10120i
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 2.00000i 0.0910032i
\(484\) 0 0
\(485\) −10.0000 + 20.0000i −0.454077 + 0.908153i
\(486\) 0 0
\(487\) 36.0000i 1.63132i 0.578535 + 0.815658i \(0.303625\pi\)
−0.578535 + 0.815658i \(0.696375\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) 36.0000i 1.62136i
\(494\) 0 0
\(495\) −8.00000 4.00000i −0.359573 0.179787i
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) 40.0000i 1.78351i −0.452517 0.891756i \(-0.649474\pi\)
0.452517 0.891756i \(-0.350526\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) 6.00000i 0.264906i
\(514\) 0 0
\(515\) 8.00000 16.0000i 0.352522 0.705044i
\(516\) 0 0
\(517\) 32.0000i 1.40736i
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) −3.00000 4.00000i −0.130931 0.174574i
\(526\) 0 0
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 24.0000i 1.03956i
\(534\) 0 0
\(535\) 6.00000 12.0000i 0.259403 0.518805i
\(536\) 0 0
\(537\) 4.00000i 0.172613i
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) 6.00000i 0.257485i
\(544\) 0 0
\(545\) 20.0000 + 10.0000i 0.856706 + 0.428353i
\(546\) 0 0
\(547\) 4.00000i 0.171028i 0.996337 + 0.0855138i \(0.0272532\pi\)
−0.996337 + 0.0855138i \(0.972747\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 0 0
\(555\) 16.0000 + 8.00000i 0.679162 + 0.339581i
\(556\) 0 0
\(557\) 10.0000i 0.423714i −0.977301 0.211857i \(-0.932049\pi\)
0.977301 0.211857i \(-0.0679510\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 0 0
\(563\) 16.0000i 0.674320i −0.941447 0.337160i \(-0.890534\pi\)
0.941447 0.337160i \(-0.109466\pi\)
\(564\) 0 0
\(565\) 6.00000 12.0000i 0.252422 0.504844i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 24.0000i 1.00261i
\(574\) 0 0
\(575\) 8.00000 6.00000i 0.333623 0.250217i
\(576\) 0 0
\(577\) 26.0000i 1.08239i 0.840896 + 0.541197i \(0.182029\pi\)
−0.840896 + 0.541197i \(0.817971\pi\)
\(578\) 0 0
\(579\) −16.0000 −0.664937
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) 0 0
\(585\) 2.00000 4.00000i 0.0826898 0.165380i
\(586\) 0 0
\(587\) 48.0000i 1.98117i 0.136892 + 0.990586i \(0.456289\pi\)
−0.136892 + 0.990586i \(0.543711\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 0 0
\(593\) 34.0000i 1.39621i −0.715994 0.698106i \(-0.754026\pi\)
0.715994 0.698106i \(-0.245974\pi\)
\(594\) 0 0
\(595\) −12.0000 6.00000i −0.491952 0.245976i
\(596\) 0 0
\(597\) 10.0000i 0.409273i
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) −10.0000 5.00000i −0.406558 0.203279i
\(606\) 0 0
\(607\) 24.0000i 0.974130i 0.873366 + 0.487065i \(0.161933\pi\)
−0.873366 + 0.487065i \(0.838067\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) 8.00000i 0.323117i 0.986863 + 0.161558i \(0.0516520\pi\)
−0.986863 + 0.161558i \(0.948348\pi\)
\(614\) 0 0
\(615\) 12.0000 24.0000i 0.483887 0.967773i
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 0 0
\(621\) −2.00000 −0.0802572
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 24.0000i 0.958468i
\(628\) 0 0
\(629\) 48.0000 1.91389
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 20.0000i 0.794929i
\(634\) 0 0
\(635\) 16.0000 32.0000i 0.634941 1.26988i
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 8.00000 + 4.00000i 0.315000 + 0.157500i
\(646\) 0 0
\(647\) 28.0000i 1.10079i 0.834903 + 0.550397i \(0.185524\pi\)
−0.834903 + 0.550397i \(0.814476\pi\)
\(648\) 0 0
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 0 0
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) 0 0
\(655\) −8.00000 4.00000i −0.312586 0.156293i
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) −6.00000 + 12.0000i −0.232670 + 0.465340i
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 36.0000i 1.38770i 0.720121 + 0.693849i \(0.244086\pi\)
−0.720121 + 0.693849i \(0.755914\pi\)
\(674\) 0 0
\(675\) 4.00000 3.00000i 0.153960 0.115470i
\(676\) 0 0
\(677\) 26.0000i 0.999261i −0.866239 0.499631i \(-0.833469\pi\)
0.866239 0.499631i \(-0.166531\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 46.0000i 1.76014i −0.474843 0.880071i \(-0.657495\pi\)
0.474843 0.880071i \(-0.342505\pi\)
\(684\) 0 0
\(685\) −14.0000 + 28.0000i −0.534913 + 1.06983i
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 38.0000 1.44559 0.722794 0.691063i \(-0.242858\pi\)
0.722794 + 0.691063i \(0.242858\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) −28.0000 14.0000i −1.06210 0.531050i
\(696\) 0 0
\(697\) 72.0000i 2.72719i
\(698\) 0 0
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 48.0000i 1.81035i
\(704\) 0 0
\(705\) 16.0000 + 8.00000i 0.602595 + 0.301297i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 4.00000i 0.149801i
\(714\) 0 0
\(715\) 8.00000 16.0000i 0.299183 0.598366i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) 18.0000 + 24.0000i 0.668503 + 0.891338i
\(726\) 0 0
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 46.0000i 1.69905i 0.527549 + 0.849524i \(0.323111\pi\)
−0.527549 + 0.849524i \(0.676889\pi\)
\(734\) 0 0
\(735\) −1.00000 + 2.00000i −0.0368856 + 0.0737711i
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 18.0000i 0.660356i −0.943919 0.330178i \(-0.892891\pi\)
0.943919 0.330178i \(-0.107109\pi\)
\(744\) 0 0
\(745\) 4.00000 + 2.00000i 0.146549 + 0.0732743i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 28.0000i 1.02038i
\(754\) 0 0
\(755\) −32.0000 16.0000i −1.16460 0.582300i
\(756\) 0 0
\(757\) 28.0000i 1.01768i 0.860862 + 0.508839i \(0.169925\pi\)
−0.860862 + 0.508839i \(0.830075\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 20.0000 0.724999 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) 0 0
\(763\) 10.0000i 0.362024i
\(764\) 0 0
\(765\) 6.00000 12.0000i 0.216930 0.433861i
\(766\) 0 0
\(767\) 24.0000i 0.866590i
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 0 0
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 0 0
\(775\) 6.00000 + 8.00000i 0.215526 + 0.287368i
\(776\) 0 0
\(777\) 8.00000i 0.286998i
\(778\) 0 0
\(779\) −72.0000 −2.57967
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) 18.0000 36.0000i 0.642448 1.28490i
\(786\) 0 0
\(787\) 28.0000i 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 0 0
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) 20.0000i 0.710221i
\(794\) 0 0
\(795\) 4.00000 + 2.00000i 0.141865 + 0.0709327i
\(796\) 0 0
\(797\) 18.0000i 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) −12.0000 −0.423999