Properties

Label 3360.2.j.d.1009.2
Level $3360$
Weight $2$
Character 3360.1009
Analytic conductor $26.830$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.j (of order \(2\), degree \(1\), not 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3360.1009
Dual form 3360.2.j.d.1009.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +(2.00000 + 1.00000i) q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +(2.00000 + 1.00000i) q^{5} +1.00000i q^{7} +1.00000 q^{9} +6.00000 q^{13} +(2.00000 + 1.00000i) q^{15} +2.00000i q^{17} -4.00000i q^{19} +1.00000i q^{21} -4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +1.00000 q^{27} -6.00000i q^{29} +8.00000 q^{31} +(-1.00000 + 2.00000i) q^{35} +2.00000 q^{37} +6.00000 q^{39} -8.00000 q^{41} -6.00000 q^{43} +(2.00000 + 1.00000i) q^{45} -2.00000i q^{47} -1.00000 q^{49} +2.00000i q^{51} +6.00000 q^{53} -4.00000i q^{57} +6.00000i q^{59} +10.0000i q^{61} +1.00000i q^{63} +(12.0000 + 6.00000i) q^{65} -2.00000 q^{67} -4.00000i q^{69} +8.00000 q^{71} -6.00000i q^{73} +(3.00000 + 4.00000i) q^{75} -10.0000 q^{79} +1.00000 q^{81} +4.00000 q^{83} +(-2.00000 + 4.00000i) q^{85} -6.00000i q^{87} +6.00000i q^{91} +8.00000 q^{93} +(4.00000 - 8.00000i) q^{95} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 4 q^{5} + 2 q^{9} + 12 q^{13} + 4 q^{15} + 6 q^{25} + 2 q^{27} + 16 q^{31} - 2 q^{35} + 4 q^{37} + 12 q^{39} - 16 q^{41} - 12 q^{43} + 4 q^{45} - 2 q^{49} + 12 q^{53} + 24 q^{65} - 4 q^{67} + 16 q^{71} + 6 q^{75} - 20 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} + 16 q^{93} + 8 q^{95} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 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
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 2.00000 + 1.00000i 0.516398 + 0.258199i
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000i 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 + 2.00000i −0.169031 + 0.338062i
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 2.00000 + 1.00000i 0.298142 + 0.149071i
\(46\) 0 0
\(47\) 2.00000i 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 10.0000i 1.28037i 0.768221 + 0.640184i \(0.221142\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 12.0000 + 6.00000i 1.48842 + 0.744208i
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 0 0
\(75\) 3.00000 + 4.00000i 0.346410 + 0.461880i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −2.00000 + 4.00000i −0.216930 + 0.433861i
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 6.00000i 0.628971i
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 4.00000 8.00000i 0.410391 0.820783i
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000i 0.995037i 0.867453 + 0.497519i \(0.165755\pi\)
−0.867453 + 0.497519i \(0.834245\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) −1.00000 + 2.00000i −0.0975900 + 0.195180i
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 16.0000i 1.53252i −0.642529 0.766261i \(-0.722115\pi\)
0.642529 0.766261i \(-0.277885\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 6.00000i 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 4.00000 8.00000i 0.373002 0.746004i
\(116\) 0 0
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 10.0000i 0.873704i 0.899533 + 0.436852i \(0.143907\pi\)
−0.899533 + 0.436852i \(0.856093\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 2.00000 + 1.00000i 0.172133 + 0.0860663i
\(136\) 0 0
\(137\) 18.0000i 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(140\) 0 0
\(141\) 2.00000i 0.168430i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 12.0000i 0.498273 0.996546i
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 14.0000i 1.14692i 0.819232 + 0.573462i \(0.194400\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 16.0000 + 8.00000i 1.28515 + 0.642575i
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.0000i 1.70241i −0.524832 0.851206i \(-0.675872\pi\)
0.524832 0.851206i \(-0.324128\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 0 0
\(179\) 16.0000i 1.19590i 0.801535 + 0.597948i \(0.204017\pi\)
−0.801535 + 0.597948i \(0.795983\pi\)
\(180\) 0 0
\(181\) 10.0000i 0.743294i −0.928374 0.371647i \(-0.878793\pi\)
0.928374 0.371647i \(-0.121207\pi\)
\(182\) 0 0
\(183\) 10.0000i 0.739221i
\(184\) 0 0
\(185\) 4.00000 + 2.00000i 0.294086 + 0.147043i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000i 0.0727393i
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i −0.817554 0.575853i \(-0.804670\pi\)
0.817554 0.575853i \(-0.195330\pi\)
\(194\) 0 0
\(195\) 12.0000 + 6.00000i 0.859338 + 0.429669i
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −16.0000 8.00000i −1.11749 0.558744i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000i 1.37686i 0.725304 + 0.688428i \(0.241699\pi\)
−0.725304 + 0.688428i \(0.758301\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) −12.0000 6.00000i −0.818393 0.409197i
\(216\) 0 0
\(217\) 8.00000i 0.543075i
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 4.00000i 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) 0 0
\(225\) 3.00000 + 4.00000i 0.200000 + 0.266667i
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000i 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 2.00000 4.00000i 0.130466 0.260931i
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.00000 1.00000i −0.127775 0.0638877i
\(246\) 0 0
\(247\) 24.0000i 1.52708i
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 10.0000i 0.631194i −0.948893 0.315597i \(-0.897795\pi\)
0.948893 0.315597i \(-0.102205\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −2.00000 + 4.00000i −0.125245 + 0.250490i
\(256\) 0 0
\(257\) 18.0000i 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 2.00000i 0.124274i
\(260\) 0 0
\(261\) 6.00000i 0.371391i
\(262\) 0 0
\(263\) 24.0000i 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) 0 0
\(265\) 12.0000 + 6.00000i 0.737154 + 0.368577i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.0000i 0.853595i 0.904347 + 0.426798i \(0.140358\pi\)
−0.904347 + 0.426798i \(0.859642\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 6.00000i 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 4.00000 8.00000i 0.236940 0.473879i
\(286\) 0 0
\(287\) 8.00000i 0.472225i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) 0 0
\(293\) 16.0000 0.934730 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 0 0
\(295\) −6.00000 + 12.0000i −0.349334 + 0.698667i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000i 1.38796i
\(300\) 0 0
\(301\) 6.00000i 0.345834i
\(302\) 0 0
\(303\) 10.0000i 0.574485i
\(304\) 0 0
\(305\) −10.0000 + 20.0000i −0.572598 + 1.14520i
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) 16.0000i 0.910208i
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 14.0000i 0.791327i 0.918396 + 0.395663i \(0.129485\pi\)
−0.918396 + 0.395663i \(0.870515\pi\)
\(314\) 0 0
\(315\) −1.00000 + 2.00000i −0.0563436 + 0.112687i
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 18.0000 + 24.0000i 0.998460 + 1.33128i
\(326\) 0 0
\(327\) 16.0000i 0.884802i
\(328\) 0 0
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) 20.0000i 1.09930i −0.835395 0.549650i \(-0.814761\pi\)
0.835395 0.549650i \(-0.185239\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −4.00000 2.00000i −0.218543 0.109272i
\(336\) 0 0
\(337\) 8.00000i 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 4.00000 8.00000i 0.215353 0.430706i
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 6.00000i 0.321173i −0.987022 0.160586i \(-0.948662\pi\)
0.987022 0.160586i \(-0.0513385\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 0 0
\(355\) 16.0000 + 8.00000i 0.849192 + 0.424596i
\(356\) 0 0
\(357\) −2.00000 −0.105851
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 6.00000 12.0000i 0.314054 0.628109i
\(366\) 0 0
\(367\) 28.0000i 1.46159i 0.682598 + 0.730794i \(0.260850\pi\)
−0.682598 + 0.730794i \(0.739150\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 6.00000i 0.311504i
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 2.00000 + 11.0000i 0.103280 + 0.568038i
\(376\) 0 0
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) 4.00000i 0.205466i −0.994709 0.102733i \(-0.967241\pi\)
0.994709 0.102733i \(-0.0327588\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 0 0
\(383\) 26.0000i 1.32854i 0.747494 + 0.664269i \(0.231257\pi\)
−0.747494 + 0.664269i \(0.768743\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) 26.0000i 1.31825i −0.752032 0.659126i \(-0.770926\pi\)
0.752032 0.659126i \(-0.229074\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 10.0000i 0.504433i
\(394\) 0 0
\(395\) −20.0000 10.0000i −1.00631 0.503155i
\(396\) 0 0
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 48.0000 2.39105
\(404\) 0 0
\(405\) 2.00000 + 1.00000i 0.0993808 + 0.0496904i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 18.0000i 0.887875i
\(412\) 0 0
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 8.00000 + 4.00000i 0.392705 + 0.196352i
\(416\) 0 0
\(417\) 16.0000i 0.783523i
\(418\) 0 0
\(419\) 34.0000i 1.66101i −0.557012 0.830504i \(-0.688052\pi\)
0.557012 0.830504i \(-0.311948\pi\)
\(420\) 0 0
\(421\) 20.0000i 0.974740i 0.873195 + 0.487370i \(0.162044\pi\)
−0.873195 + 0.487370i \(0.837956\pi\)
\(422\) 0 0
\(423\) 2.00000i 0.0972433i
\(424\) 0 0
\(425\) −8.00000 + 6.00000i −0.388057 + 0.291043i
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 6.00000i 0.288342i −0.989553 0.144171i \(-0.953949\pi\)
0.989553 0.144171i \(-0.0460515\pi\)
\(434\) 0 0
\(435\) 6.00000 12.0000i 0.287678 0.575356i
\(436\) 0 0
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 14.0000i 0.662177i
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.00000 −0.0939682
\(454\) 0 0
\(455\) −6.00000 + 12.0000i −0.281284 + 0.562569i
\(456\) 0 0
\(457\) 12.0000i 0.561336i 0.959805 + 0.280668i \(0.0905560\pi\)
−0.959805 + 0.280668i \(0.909444\pi\)
\(458\) 0 0
\(459\) 2.00000i 0.0933520i
\(460\) 0 0
\(461\) 30.0000i 1.39724i −0.715493 0.698620i \(-0.753798\pi\)
0.715493 0.698620i \(-0.246202\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) 16.0000 + 8.00000i 0.741982 + 0.370991i
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 2.00000i 0.0923514i
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 16.0000 12.0000i 0.734130 0.550598i
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) −2.00000 + 4.00000i −0.0908153 + 0.181631i
\(486\) 0 0
\(487\) 32.0000i 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 0 0
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) 20.0000i 0.902587i 0.892375 + 0.451294i \(0.149037\pi\)
−0.892375 + 0.451294i \(0.850963\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) 4.00000i 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) 22.0000i 0.982888i
\(502\) 0 0
\(503\) 6.00000i 0.267527i 0.991013 + 0.133763i \(0.0427062\pi\)
−0.991013 + 0.133763i \(0.957294\pi\)
\(504\) 0 0
\(505\) −10.0000 + 20.0000i −0.444994 + 0.889988i
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) 0 0
\(509\) 34.0000i 1.50702i 0.657434 + 0.753512i \(0.271642\pi\)
−0.657434 + 0.753512i \(0.728358\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) −16.0000 + 32.0000i −0.705044 + 1.41009i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) −4.00000 + 3.00000i −0.174574 + 0.130931i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) 0 0
\(533\) −48.0000 −2.07911
\(534\) 0 0
\(535\) −24.0000 12.0000i −1.03761 0.518805i
\(536\) 0 0
\(537\) 16.0000i 0.690451i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000i 0.859867i 0.902861 + 0.429934i \(0.141463\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 10.0000i 0.429141i
\(544\) 0 0
\(545\) 16.0000 32.0000i 0.685365 1.37073i
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 0 0
\(549\) 10.0000i 0.426790i
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 10.0000i 0.425243i
\(554\) 0 0
\(555\) 4.00000 + 2.00000i 0.169791 + 0.0848953i
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 6.00000 12.0000i 0.252422 0.504844i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 16.0000 12.0000i 0.667246 0.500435i
\(576\) 0 0
\(577\) 38.0000i 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) 0 0
\(579\) 16.0000i 0.664937i
\(580\) 0 0
\(581\) 4.00000i 0.165948i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 12.0000 + 6.00000i 0.496139 + 0.248069i
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) 14.0000i 0.574911i 0.957794 + 0.287456i \(0.0928094\pi\)
−0.957794 + 0.287456i \(0.907191\pi\)
\(594\) 0 0
\(595\) −4.00000 2.00000i −0.163984 0.0819920i
\(596\) 0 0
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) 22.0000 + 11.0000i 0.894427 + 0.447214i
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 12.0000i 0.485468i
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 0 0
\(615\) −16.0000 8.00000i −0.645182 0.322591i
\(616\) 0 0
\(617\) 18.0000i 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) 44.0000i 1.76851i −0.467005 0.884255i \(-0.654667\pi\)
0.467005 0.884255i \(-0.345333\pi\)
\(620\) 0 0
\(621\) 4.00000i 0.160514i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.00000i 0.159490i
\(630\) 0 0
\(631\) 18.0000 0.716569 0.358284 0.933613i \(-0.383362\pi\)
0.358284 + 0.933613i \(0.383362\pi\)
\(632\) 0 0
\(633\) 20.0000i 0.794929i
\(634\) 0 0
\(635\) −8.00000 + 16.0000i −0.317470 + 0.634941i
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) −12.0000 6.00000i −0.472500 0.236250i
\(646\) 0 0
\(647\) 42.0000i 1.65119i −0.564263 0.825595i \(-0.690840\pi\)
0.564263 0.825595i \(-0.309160\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000i 0.313545i
\(652\) 0 0
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) −10.0000 + 20.0000i −0.390732 + 0.781465i
\(656\) 0 0
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) 36.0000i 1.40236i 0.712984 + 0.701180i \(0.247343\pi\)
−0.712984 + 0.701180i \(0.752657\pi\)
\(660\) 0 0
\(661\) 30.0000i 1.16686i 0.812162 + 0.583432i \(0.198291\pi\)
−0.812162 + 0.583432i \(0.801709\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) 8.00000 + 4.00000i 0.310227 + 0.155113i
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 4.00000i 0.154649i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 44.0000i 1.69608i 0.529936 + 0.848038i \(0.322216\pi\)
−0.529936 + 0.848038i \(0.677784\pi\)
\(674\) 0 0
\(675\) 3.00000 + 4.00000i 0.115470 + 0.153960i
\(676\) 0 0
\(677\) 32.0000 1.22986 0.614930 0.788582i \(-0.289184\pi\)
0.614930 + 0.788582i \(0.289184\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) 18.0000 36.0000i 0.687745 1.37549i
\(686\) 0 0
\(687\) 14.0000i 0.534133i
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.0000 + 32.0000i −0.606915 + 1.21383i
\(696\) 0 0
\(697\) 16.0000i 0.606043i
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) 50.0000i 1.88847i −0.329267 0.944237i \(-0.606802\pi\)
0.329267 0.944237i \(-0.393198\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 2.00000 4.00000i 0.0753244 0.150649i
\(706\) 0 0
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) 16.0000i 0.600893i −0.953799 0.300446i \(-0.902864\pi\)
0.953799 0.300446i \(-0.0971356\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) 32.0000i 1.19841i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) −18.0000 −0.669427
\(724\) 0 0
\(725\) 24.0000 18.0000i 0.891338 0.668503i
\(726\) 0 0
\(727\) 52.0000i 1.92857i −0.264861 0.964287i \(-0.585326\pi\)
0.264861 0.964287i \(-0.414674\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000i 0.443836i
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) −2.00000 1.00000i −0.0737711 0.0368856i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 4.00000i 0.147142i −0.997290 0.0735712i \(-0.976560\pi\)
0.997290 0.0735712i \(-0.0234396\pi\)
\(740\) 0 0
\(741\) 24.0000i 0.881662i
\(742\) 0 0
\(743\) 24.0000i 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) −14.0000 + 28.0000i −0.512920 + 1.02584i
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 38.0000 1.38664 0.693320 0.720630i \(-0.256147\pi\)
0.693320 + 0.720630i \(0.256147\pi\)
\(752\) 0 0
\(753\) 10.0000i 0.364420i
\(754\) 0 0
\(755\) −4.00000 2.00000i −0.145575 0.0727875i
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.00000 −0.290000 −0.145000 0.989432i \(-0.546318\pi\)
−0.145000 + 0.989432i \(0.546318\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) −2.00000 + 4.00000i −0.0723102 + 0.144620i
\(766\) 0 0
\(767\) 36.0000i 1.29988i
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 18.0000i 0.648254i
\(772\) 0 0
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 24.0000 + 32.0000i 0.862105 + 1.14947i
\(776\) 0 0
\(777\) 2.00000i 0.0717496i
\(778\) 0 0
\(779\) 32.0000i 1.14652i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) 44.0000 + 22.0000i 1.57043 + 0.785214i
\(786\) 0 0
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 0 0
\(789\) 24.0000i 0.854423i
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 60.0000i 2.13066i
\(794\) 0 0
\(795\) 12.0000 + 6.00000i 0.425596 + 0.212798i
\(796\) 0 0
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\)