Properties

Label 3840.2.d.p.2689.2
Level $3840$
Weight $2$
Character 3840.2689
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.6625543762\)
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 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3840.2689
Dual form 3840.2.d.p.2689.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +(2.00000 + 1.00000i) q^{5} +4.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +(2.00000 + 1.00000i) q^{5} +4.00000i q^{7} +1.00000 q^{9} +4.00000 q^{13} +(-2.00000 - 1.00000i) q^{15} +8.00000i q^{19} -4.00000i q^{21} +4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000 q^{27} -6.00000i q^{29} +8.00000 q^{31} +(-4.00000 + 8.00000i) q^{35} -4.00000 q^{37} -4.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} +(2.00000 + 1.00000i) q^{45} -4.00000i q^{47} -9.00000 q^{49} +12.0000 q^{53} -8.00000i q^{57} +6.00000i q^{61} +4.00000i q^{63} +(8.00000 + 4.00000i) q^{65} +12.0000 q^{67} -4.00000i q^{69} -16.0000 q^{71} +(-3.00000 - 4.00000i) q^{75} -8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +6.00000i q^{87} -10.0000 q^{89} +16.0000i q^{91} -8.00000 q^{93} +(-8.00000 + 16.0000i) q^{95} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + 8 q^{13} - 4 q^{15} + 6 q^{25} - 2 q^{27} + 16 q^{31} - 8 q^{35} - 8 q^{37} - 8 q^{39} - 12 q^{41} + 8 q^{43} + 4 q^{45} - 18 q^{49} + 24 q^{53} + 16 q^{65} + 24 q^{67} - 32 q^{71} - 6 q^{75} - 16 q^{79} + 2 q^{81} + 24 q^{83} - 20 q^{89} - 16 q^{93} - 16 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −2.00000 1.00000i −0.516398 0.258199i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) 4.00000i 0.872872i
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 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
\(35\) −4.00000 + 8.00000i −0.676123 + 1.35225i
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 2.00000 + 1.00000i 0.298142 + 0.149071i
\(46\) 0 0
\(47\) 4.00000i 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 6.00000i 0.768221i 0.923287 + 0.384111i \(0.125492\pi\)
−0.923287 + 0.384111i \(0.874508\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) 8.00000 + 4.00000i 0.992278 + 0.496139i
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −3.00000 4.00000i −0.346410 0.461880i
\(76\) 0 0
\(77\) 0 0
\(78\) 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
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 16.0000i 1.67726i
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −8.00000 + 16.0000i −0.820783 + 1.64157i
\(96\) 0 0
\(97\) 8.00000i 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0000i 1.39305i 0.717532 + 0.696526i \(0.245272\pi\)
−0.717532 + 0.696526i \(0.754728\pi\)
\(102\) 0 0
\(103\) 12.0000i 1.18240i −0.806527 0.591198i \(-0.798655\pi\)
0.806527 0.591198i \(-0.201345\pi\)
\(104\) 0 0
\(105\) 4.00000 8.00000i 0.390360 0.780720i
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.957826i −0.877862 0.478913i \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 8.00000i 0.752577i −0.926503 0.376288i \(-0.877200\pi\)
0.926503 0.376288i \(-0.122800\pi\)
\(114\) 0 0
\(115\) −4.00000 + 8.00000i −0.373002 + 0.746004i
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) 4.00000i 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 16.0000i 1.39793i −0.715158 0.698963i \(-0.753645\pi\)
0.715158 0.698963i \(-0.246355\pi\)
\(132\) 0 0
\(133\) −32.0000 −2.77475
\(134\) 0 0
\(135\) −2.00000 1.00000i −0.172133 0.0860663i
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 8.00000i 0.678551i 0.940687 + 0.339276i \(0.110182\pi\)
−0.940687 + 0.339276i \(0.889818\pi\)
\(140\) 0 0
\(141\) 4.00000i 0.336861i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 12.0000i 0.498273 0.996546i
\(146\) 0 0
\(147\) 9.00000 0.742307
\(148\) 0 0
\(149\) 14.0000i 1.14692i −0.819232 0.573462i \(-0.805600\pi\)
0.819232 0.573462i \(-0.194400\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.0000 + 8.00000i 1.28515 + 0.642575i
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 8.00000i 0.611775i
\(172\) 0 0
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) −16.0000 + 12.0000i −1.20949 + 0.907115i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.0000i 1.19590i 0.801535 + 0.597948i \(0.204017\pi\)
−0.801535 + 0.597948i \(0.795983\pi\)
\(180\) 0 0
\(181\) 14.0000i 1.04061i 0.853980 + 0.520306i \(0.174182\pi\)
−0.853980 + 0.520306i \(0.825818\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) −8.00000 4.00000i −0.588172 0.294086i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000i 0.290957i
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 24.0000i 1.72756i 0.503871 + 0.863779i \(0.331909\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) −8.00000 4.00000i −0.572892 0.286446i
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) −12.0000 6.00000i −0.838116 0.419058i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000i 0.550743i 0.961338 + 0.275371i \(0.0888008\pi\)
−0.961338 + 0.275371i \(0.911199\pi\)
\(212\) 0 0
\(213\) 16.0000 1.09630
\(214\) 0 0
\(215\) 8.00000 + 4.00000i 0.545595 + 0.272798i
\(216\) 0 0
\(217\) 32.0000i 2.17230i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 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
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 2.00000i 0.132164i −0.997814 0.0660819i \(-0.978950\pi\)
0.997814 0.0660819i \(-0.0210498\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.00000i 0.524097i −0.965055 0.262049i \(-0.915602\pi\)
0.965055 0.262049i \(-0.0843981\pi\)
\(234\) 0 0
\(235\) 4.00000 8.00000i 0.260931 0.521862i
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −18.0000 9.00000i −1.14998 0.574989i
\(246\) 0 0
\(247\) 32.0000i 2.03611i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 16.0000i 1.00991i 0.863145 + 0.504956i \(0.168491\pi\)
−0.863145 + 0.504956i \(0.831509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0000i 1.49708i 0.663090 + 0.748539i \(0.269245\pi\)
−0.663090 + 0.748539i \(0.730755\pi\)
\(258\) 0 0
\(259\) 16.0000i 0.994192i
\(260\) 0 0
\(261\) 6.00000i 0.371391i
\(262\) 0 0
\(263\) 12.0000i 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 0 0
\(265\) 24.0000 + 12.0000i 1.47431 + 0.737154i
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) 6.00000i 0.365826i 0.983129 + 0.182913i \(0.0585527\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 16.0000i 0.968364i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 8.00000 16.0000i 0.473879 0.947758i
\(286\) 0 0
\(287\) 24.0000i 1.41668i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 8.00000i 0.468968i
\(292\) 0 0
\(293\) 20.0000 1.16841 0.584206 0.811605i \(-0.301406\pi\)
0.584206 + 0.811605i \(0.301406\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.0000i 0.925304i
\(300\) 0 0
\(301\) 16.0000i 0.922225i
\(302\) 0 0
\(303\) 14.0000i 0.804279i
\(304\) 0 0
\(305\) −6.00000 + 12.0000i −0.343559 + 0.687118i
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 12.0000i 0.682656i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 8.00000i 0.452187i 0.974106 + 0.226093i \(0.0725954\pi\)
−0.974106 + 0.226093i \(0.927405\pi\)
\(314\) 0 0
\(315\) −4.00000 + 8.00000i −0.225374 + 0.450749i
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 12.0000 + 16.0000i 0.665640 + 0.887520i
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 8.00000i 0.439720i −0.975531 0.219860i \(-0.929440\pi\)
0.975531 0.219860i \(-0.0705600\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 24.0000 + 12.0000i 1.31126 + 0.655630i
\(336\) 0 0
\(337\) 16.0000i 0.871576i 0.900049 + 0.435788i \(0.143530\pi\)
−0.900049 + 0.435788i \(0.856470\pi\)
\(338\) 0 0
\(339\) 8.00000i 0.434500i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 4.00000 8.00000i 0.215353 0.430706i
\(346\) 0 0
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) 10.0000i 0.535288i 0.963518 + 0.267644i \(0.0862451\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 24.0000i 1.27739i −0.769460 0.638696i \(-0.779474\pi\)
0.769460 0.638696i \(-0.220526\pi\)
\(354\) 0 0
\(355\) −32.0000 16.0000i −1.69838 0.849192i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 28.0000i 1.46159i 0.682598 + 0.730794i \(0.260850\pi\)
−0.682598 + 0.730794i \(0.739150\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 48.0000i 2.49204i
\(372\) 0 0
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 0 0
\(375\) −2.00000 11.0000i −0.103280 0.568038i
\(376\) 0 0
\(377\) 24.0000i 1.23606i
\(378\) 0 0
\(379\) 8.00000i 0.410932i −0.978664 0.205466i \(-0.934129\pi\)
0.978664 0.205466i \(-0.0658711\pi\)
\(380\) 0 0
\(381\) 4.00000i 0.204926i
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 30.0000i 1.52106i −0.649303 0.760530i \(-0.724939\pi\)
0.649303 0.760530i \(-0.275061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 16.0000i 0.807093i
\(394\) 0 0
\(395\) −16.0000 8.00000i −0.805047 0.402524i
\(396\) 0 0
\(397\) −28.0000 −1.40528 −0.702640 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) 0 0
\(399\) 32.0000 1.60200
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) 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
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 24.0000 + 12.0000i 1.17811 + 0.589057i
\(416\) 0 0
\(417\) 8.00000i 0.391762i
\(418\) 0 0
\(419\) 16.0000i 0.781651i −0.920465 0.390826i \(-0.872190\pi\)
0.920465 0.390826i \(-0.127810\pi\)
\(420\) 0 0
\(421\) 14.0000i 0.682318i −0.940006 0.341159i \(-0.889181\pi\)
0.940006 0.341159i \(-0.110819\pi\)
\(422\) 0 0
\(423\) 4.00000i 0.194487i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −24.0000 −1.16144
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 40.0000i 1.92228i 0.276066 + 0.961139i \(0.410969\pi\)
−0.276066 + 0.961139i \(0.589031\pi\)
\(434\) 0 0
\(435\) −6.00000 + 12.0000i −0.287678 + 0.575356i
\(436\) 0 0
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) −20.0000 10.0000i −0.948091 0.474045i
\(446\) 0 0
\(447\) 14.0000i 0.662177i
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) −16.0000 + 32.0000i −0.750092 + 1.50018i
\(456\) 0 0
\(457\) 8.00000i 0.374224i −0.982339 0.187112i \(-0.940087\pi\)
0.982339 0.187112i \(-0.0599128\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.0000i 1.02464i −0.858794 0.512321i \(-0.828786\pi\)
0.858794 0.512321i \(-0.171214\pi\)
\(462\) 0 0
\(463\) 20.0000i 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) 0 0
\(465\) −16.0000 8.00000i −0.741982 0.370991i
\(466\) 0 0
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) 48.0000i 2.21643i
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −32.0000 + 24.0000i −1.46826 + 1.10120i
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) 8.00000 16.0000i 0.363261 0.726523i
\(486\) 0 0
\(487\) 4.00000i 0.181257i 0.995885 + 0.0906287i \(0.0288876\pi\)
−0.995885 + 0.0906287i \(0.971112\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 32.0000i 1.44414i −0.691820 0.722070i \(-0.743191\pi\)
0.691820 0.722070i \(-0.256809\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 64.0000i 2.87079i
\(498\) 0 0
\(499\) 24.0000i 1.07439i 0.843459 + 0.537194i \(0.180516\pi\)
−0.843459 + 0.537194i \(0.819484\pi\)
\(500\) 0 0
\(501\) 4.00000i 0.178707i
\(502\) 0 0
\(503\) 28.0000i 1.24846i −0.781241 0.624229i \(-0.785413\pi\)
0.781241 0.624229i \(-0.214587\pi\)
\(504\) 0 0
\(505\) −14.0000 + 28.0000i −0.622992 + 1.24598i
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 0 0
\(509\) 6.00000i 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 8.00000i 0.353209i
\(514\) 0 0
\(515\) 12.0000 24.0000i 0.528783 1.05757i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) 0 0
\(525\) 16.0000 12.0000i 0.698297 0.523723i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) 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
\(541\) 10.0000i 0.429934i 0.976621 + 0.214967i \(0.0689643\pi\)
−0.976621 + 0.214967i \(0.931036\pi\)
\(542\) 0 0
\(543\) 14.0000i 0.600798i
\(544\) 0 0
\(545\) 10.0000 20.0000i 0.428353 0.856706i
\(546\) 0 0
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 0 0
\(549\) 6.00000i 0.256074i
\(550\) 0 0
\(551\) 48.0000 2.04487
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) 8.00000 + 4.00000i 0.339581 + 0.169791i
\(556\) 0 0
\(557\) 4.00000 0.169485 0.0847427 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 28.0000 1.18006 0.590030 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) 0 0
\(565\) 8.00000 16.0000i 0.336563 0.673125i
\(566\) 0 0
\(567\) 4.00000i 0.167984i
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 40.0000i 1.67395i −0.547243 0.836974i \(-0.684323\pi\)
0.547243 0.836974i \(-0.315677\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) −16.0000 + 12.0000i −0.667246 + 0.500435i
\(576\) 0 0
\(577\) 16.0000i 0.666089i 0.942911 + 0.333044i \(0.108076\pi\)
−0.942911 + 0.333044i \(0.891924\pi\)
\(578\) 0 0
\(579\) 24.0000i 0.997406i
\(580\) 0 0
\(581\) 48.0000i 1.99138i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.00000 + 4.00000i 0.330759 + 0.165380i
\(586\) 0 0
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 64.0000i 2.63707i
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 0 0
\(593\) 8.00000i 0.328521i −0.986417 0.164260i \(-0.947476\pi\)
0.986417 0.164260i \(-0.0525237\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 22.0000 + 11.0000i 0.894427 + 0.447214i
\(606\) 0 0
\(607\) 20.0000i 0.811775i −0.913923 0.405887i \(-0.866962\pi\)
0.913923 0.405887i \(-0.133038\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 16.0000i 0.647291i
\(612\) 0 0
\(613\) 20.0000 0.807792 0.403896 0.914805i \(-0.367656\pi\)
0.403896 + 0.914805i \(0.367656\pi\)
\(614\) 0 0
\(615\) 12.0000 + 6.00000i 0.483887 + 0.241943i
\(616\) 0 0
\(617\) 8.00000i 0.322068i −0.986949 0.161034i \(-0.948517\pi\)
0.986949 0.161034i \(-0.0514829\pi\)
\(618\) 0 0
\(619\) 24.0000i 0.964641i −0.875995 0.482321i \(-0.839794\pi\)
0.875995 0.482321i \(-0.160206\pi\)
\(620\) 0 0
\(621\) 4.00000i 0.160514i
\(622\) 0 0
\(623\) 40.0000i 1.60257i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 8.00000i 0.317971i
\(634\) 0 0
\(635\) 4.00000 8.00000i 0.158735 0.317470i
\(636\) 0 0
\(637\) −36.0000 −1.42637
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) −8.00000 4.00000i −0.315000 0.157500i
\(646\) 0 0
\(647\) 12.0000i 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 32.0000i 1.25418i
\(652\) 0 0
\(653\) −20.0000 −0.782660 −0.391330 0.920250i \(-0.627985\pi\)
−0.391330 + 0.920250i \(0.627985\pi\)
\(654\) 0 0
\(655\) 16.0000 32.0000i 0.625172 1.25034i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 30.0000i 1.16686i −0.812162 0.583432i \(-0.801709\pi\)
0.812162 0.583432i \(-0.198291\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −64.0000 32.0000i −2.48181 1.24091i
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 4.00000i 0.154649i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.00000i 0.308377i 0.988041 + 0.154189i \(0.0492764\pi\)
−0.988041 + 0.154189i \(0.950724\pi\)
\(674\) 0 0
\(675\) −3.00000 4.00000i −0.115470 0.153960i
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.00000i 0.0763048i
\(688\) 0 0
\(689\) 48.0000 1.82865
\(690\) 0 0
\(691\) 8.00000i 0.304334i 0.988355 + 0.152167i \(0.0486252\pi\)
−0.988355 + 0.152167i \(0.951375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.00000 + 16.0000i −0.303457 + 0.606915i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 8.00000i 0.302588i
\(700\) 0 0
\(701\) 42.0000i 1.58632i −0.609015 0.793159i \(-0.708435\pi\)
0.609015 0.793159i \(-0.291565\pi\)
\(702\) 0 0
\(703\) 32.0000i 1.20690i
\(704\) 0 0
\(705\) −4.00000 + 8.00000i −0.150649 + 0.301297i
\(706\) 0 0
\(707\) −56.0000 −2.10610
\(708\) 0 0
\(709\) 2.00000i 0.0751116i −0.999295 0.0375558i \(-0.988043\pi\)
0.999295 0.0375558i \(-0.0119572\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 32.0000i 1.19841i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) −14.0000 −0.520666
\(724\) 0 0
\(725\) 24.0000 18.0000i 0.891338 0.668503i
\(726\) 0 0
\(727\) 28.0000i 1.03846i −0.854634 0.519231i \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 18.0000 + 9.00000i 0.663940 + 0.331970i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 40.0000i 1.47142i −0.677295 0.735712i \(-0.736848\pi\)
0.677295 0.735712i \(-0.263152\pi\)
\(740\) 0 0
\(741\) 32.0000i 1.17555i
\(742\) 0 0
\(743\) 12.0000i 0.440237i −0.975473 0.220119i \(-0.929356\pi\)
0.975473 0.220119i \(-0.0706445\pi\)
\(744\) 0 0
\(745\) 14.0000 28.0000i 0.512920 1.02584i
\(746\) 0 0
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 48.0000i 1.75388i
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 16.0000i 0.583072i
\(754\) 0 0
\(755\) −16.0000 8.00000i −0.582300 0.291150i
\(756\) 0 0
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 40.0000 1.44810
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 24.0000i 0.864339i
\(772\) 0 0
\(773\) 28.0000 1.00709 0.503545 0.863969i \(-0.332029\pi\)
0.503545 + 0.863969i \(0.332029\pi\)
\(774\) 0 0
\(775\) 24.0000 + 32.0000i 0.862105 + 1.14947i
\(776\) 0 0
\(777\) 16.0000i 0.573997i
\(778\) 0 0
\(779\) 48.0000i 1.71978i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) −8.00000 4.00000i −0.285532 0.142766i
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 0 0
\(789\) 12.0000i 0.427211i
\(790\) 0 0
\(791\) 32.0000 1.13779
\(792\) 0 0
\(793\) 24.0000i 0.852265i
\(794\) 0 0
\(795\) −24.0000 12.0000i −0.851192 0.425596i
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\) 0 0
\(804\)