Properties

Label 4608.2.k.m.1153.1
Level $4608$
Weight $2$
Character 4608.1153
Analytic conductor $36.795$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1153.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4608.1153
Dual form 4608.2.k.m.3457.1

$q$-expansion

\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{5} -4.00000i q^{7} +O(q^{10})\) \(q+(1.00000 - 1.00000i) q^{5} -4.00000i q^{7} +(-4.00000 + 4.00000i) q^{11} +(-3.00000 - 3.00000i) q^{13} -6.00000 q^{17} +(4.00000 + 4.00000i) q^{19} -8.00000i q^{23} +3.00000i q^{25} +(3.00000 + 3.00000i) q^{29} +4.00000 q^{31} +(-4.00000 - 4.00000i) q^{35} +(-1.00000 + 1.00000i) q^{37} +2.00000i q^{41} +(4.00000 - 4.00000i) q^{43} -8.00000 q^{47} -9.00000 q^{49} +(-7.00000 + 7.00000i) q^{53} +8.00000i q^{55} +(3.00000 + 3.00000i) q^{61} -6.00000 q^{65} +(-8.00000 - 8.00000i) q^{67} +10.0000i q^{73} +(16.0000 + 16.0000i) q^{77} -12.0000 q^{79} +(4.00000 + 4.00000i) q^{83} +(-6.00000 + 6.00000i) q^{85} +16.0000i q^{89} +(-12.0000 + 12.0000i) q^{91} +8.00000 q^{95} +8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 8 q^{11} - 6 q^{13} - 12 q^{17} + 8 q^{19} + 6 q^{29} + 8 q^{31} - 8 q^{35} - 2 q^{37} + 8 q^{43} - 16 q^{47} - 18 q^{49} - 14 q^{53} + 6 q^{61} - 12 q^{65} - 16 q^{67} + 32 q^{77} - 24 q^{79} + 8 q^{83} - 12 q^{85} - 24 q^{91} + 16 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 1.00000i 0.447214 0.447214i −0.447214 0.894427i \(-0.647584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 0 0
\(7\) 4.00000i 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 + 4.00000i −1.20605 + 1.20605i −0.233748 + 0.972297i \(0.575099\pi\)
−0.972297 + 0.233748i \(0.924901\pi\)
\(12\) 0 0
\(13\) −3.00000 3.00000i −0.832050 0.832050i 0.155747 0.987797i \(-0.450222\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000 + 4.00000i 0.917663 + 0.917663i 0.996859 0.0791961i \(-0.0252353\pi\)
−0.0791961 + 0.996859i \(0.525235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000i 1.66812i −0.551677 0.834058i \(-0.686012\pi\)
0.551677 0.834058i \(-0.313988\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 3.00000i 0.557086 + 0.557086i 0.928477 0.371391i \(-0.121119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 4.00000i −0.676123 0.676123i
\(36\) 0 0
\(37\) −1.00000 + 1.00000i −0.164399 + 0.164399i −0.784512 0.620113i \(-0.787087\pi\)
0.620113 + 0.784512i \(0.287087\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000i 0.312348i 0.987730 + 0.156174i \(0.0499160\pi\)
−0.987730 + 0.156174i \(0.950084\pi\)
\(42\) 0 0
\(43\) 4.00000 4.00000i 0.609994 0.609994i −0.332950 0.942944i \(-0.608044\pi\)
0.942944 + 0.332950i \(0.108044\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.00000 + 7.00000i −0.961524 + 0.961524i −0.999287 0.0377628i \(-0.987977\pi\)
0.0377628 + 0.999287i \(0.487977\pi\)
\(54\) 0 0
\(55\) 8.00000i 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) 3.00000 + 3.00000i 0.384111 + 0.384111i 0.872581 0.488470i \(-0.162445\pi\)
−0.488470 + 0.872581i \(0.662445\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −8.00000 8.00000i −0.977356 0.977356i 0.0223937 0.999749i \(-0.492871\pi\)
−0.999749 + 0.0223937i \(0.992871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.0000 + 16.0000i 1.82337 + 1.82337i
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 + 4.00000i 0.439057 + 0.439057i 0.891695 0.452638i \(-0.149517\pi\)
−0.452638 + 0.891695i \(0.649517\pi\)
\(84\) 0 0
\(85\) −6.00000 + 6.00000i −0.650791 + 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.0000i 1.69600i 0.529999 + 0.847998i \(0.322192\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −12.0000 + 12.0000i −1.25794 + 1.25794i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.00000 + 7.00000i −0.696526 + 0.696526i −0.963660 0.267133i \(-0.913924\pi\)
0.267133 + 0.963660i \(0.413924\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 8.00000i 0.773389 0.773389i −0.205308 0.978697i \(-0.565820\pi\)
0.978697 + 0.205308i \(0.0658197\pi\)
\(108\) 0 0
\(109\) −5.00000 5.00000i −0.478913 0.478913i 0.425871 0.904784i \(-0.359968\pi\)
−0.904784 + 0.425871i \(0.859968\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 0 0
\(115\) −8.00000 8.00000i −0.746004 0.746004i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.0000i 2.20008i
\(120\) 0 0
\(121\) 21.0000i 1.90909i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.00000 + 8.00000i 0.715542 + 0.715542i
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 + 8.00000i 0.698963 + 0.698963i 0.964187 0.265224i \(-0.0854458\pi\)
−0.265224 + 0.964187i \(0.585446\pi\)
\(132\) 0 0
\(133\) 16.0000 16.0000i 1.38738 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000i 0.170872i −0.996344 0.0854358i \(-0.972772\pi\)
0.996344 0.0854358i \(-0.0272282\pi\)
\(138\) 0 0
\(139\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.00000 1.00000i 0.0819232 0.0819232i −0.664958 0.746881i \(-0.731550\pi\)
0.746881 + 0.664958i \(0.231550\pi\)
\(150\) 0 0
\(151\) 4.00000i 0.325515i −0.986666 0.162758i \(-0.947961\pi\)
0.986666 0.162758i \(-0.0520389\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 4.00000i 0.321288 0.321288i
\(156\) 0 0
\(157\) 3.00000 + 3.00000i 0.239426 + 0.239426i 0.816612 0.577186i \(-0.195849\pi\)
−0.577186 + 0.816612i \(0.695849\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −32.0000 −2.52195
\(162\) 0 0
\(163\) 12.0000 + 12.0000i 0.939913 + 0.939913i 0.998294 0.0583818i \(-0.0185941\pi\)
−0.0583818 + 0.998294i \(0.518594\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.0000 + 11.0000i 0.836315 + 0.836315i 0.988372 0.152057i \(-0.0485898\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) 12.0000 0.907115
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.00000 8.00000i −0.597948 0.597948i 0.341818 0.939766i \(-0.388957\pi\)
−0.939766 + 0.341818i \(0.888957\pi\)
\(180\) 0 0
\(181\) −9.00000 + 9.00000i −0.668965 + 0.668965i −0.957476 0.288512i \(-0.906840\pi\)
0.288512 + 0.957476i \(0.406840\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.00000i 0.147043i
\(186\) 0 0
\(187\) 24.0000 24.0000i 1.75505 1.75505i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.00000 + 1.00000i −0.0712470 + 0.0712470i −0.741832 0.670585i \(-0.766043\pi\)
0.670585 + 0.741832i \(0.266043\pi\)
\(198\) 0 0
\(199\) 4.00000i 0.283552i −0.989899 0.141776i \(-0.954719\pi\)
0.989899 0.141776i \(-0.0452813\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000 12.0000i 0.842235 0.842235i
\(204\) 0 0
\(205\) 2.00000 + 2.00000i 0.139686 + 0.139686i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 16.0000 + 16.0000i 1.10149 + 1.10149i 0.994232 + 0.107254i \(0.0342057\pi\)
0.107254 + 0.994232i \(0.465794\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000i 0.545595i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 18.0000 + 18.0000i 1.21081 + 1.21081i
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 + 12.0000i 0.796468 + 0.796468i 0.982537 0.186069i \(-0.0595747\pi\)
−0.186069 + 0.982537i \(0.559575\pi\)
\(228\) 0 0
\(229\) 7.00000 7.00000i 0.462573 0.462573i −0.436925 0.899498i \(-0.643932\pi\)
0.899498 + 0.436925i \(0.143932\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −8.00000 + 8.00000i −0.521862 + 0.521862i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −24.0000 −1.54598 −0.772988 0.634421i \(-0.781239\pi\)
−0.772988 + 0.634421i \(0.781239\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.00000 + 9.00000i −0.574989 + 0.574989i
\(246\) 0 0
\(247\) 24.0000i 1.52708i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.00000 + 4.00000i −0.252478 + 0.252478i −0.821986 0.569508i \(-0.807134\pi\)
0.569508 + 0.821986i \(0.307134\pi\)
\(252\) 0 0
\(253\) 32.0000 + 32.0000i 2.01182 + 2.01182i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 4.00000 + 4.00000i 0.248548 + 0.248548i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.0000i 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 0 0
\(265\) 14.0000i 0.860013i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.00000 + 3.00000i 0.182913 + 0.182913i 0.792624 0.609711i \(-0.208714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 12.0000i −0.723627 0.723627i
\(276\) 0 0
\(277\) −15.0000 + 15.0000i −0.901263 + 0.901263i −0.995545 0.0942828i \(-0.969944\pi\)
0.0942828 + 0.995545i \(0.469944\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −8.00000 + 8.00000i −0.475551 + 0.475551i −0.903705 0.428155i \(-0.859164\pi\)
0.428155 + 0.903705i \(0.359164\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.00000 + 1.00000i −0.0584206 + 0.0584206i −0.735714 0.677293i \(-0.763153\pi\)
0.677293 + 0.735714i \(0.263153\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −24.0000 + 24.0000i −1.38796 + 1.38796i
\(300\) 0 0
\(301\) −16.0000 16.0000i −0.922225 0.922225i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 0 0
\(307\) −24.0000 24.0000i −1.36975 1.36975i −0.860787 0.508965i \(-0.830028\pi\)
−0.508965 0.860787i \(-0.669972\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000i 0.907277i −0.891186 0.453638i \(-0.850126\pi\)
0.891186 0.453638i \(-0.149874\pi\)
\(312\) 0 0
\(313\) 8.00000i 0.452187i −0.974106 0.226093i \(-0.927405\pi\)
0.974106 0.226093i \(-0.0725954\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.0000 + 19.0000i 1.06715 + 1.06715i 0.997577 + 0.0695692i \(0.0221625\pi\)
0.0695692 + 0.997577i \(0.477838\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 24.0000i −1.33540 1.33540i
\(324\) 0 0
\(325\) 9.00000 9.00000i 0.499230 0.499230i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 32.0000i 1.76422i
\(330\) 0 0
\(331\) −16.0000 + 16.0000i −0.879440 + 0.879440i −0.993477 0.114037i \(-0.963622\pi\)
0.114037 + 0.993477i \(0.463622\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 + 16.0000i −0.866449 + 0.866449i
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.0000 + 20.0000i −1.07366 + 1.07366i −0.0765939 + 0.997062i \(0.524404\pi\)
−0.997062 + 0.0765939i \(0.975596\pi\)
\(348\) 0 0
\(349\) −11.0000 11.0000i −0.588817 0.588817i 0.348494 0.937311i \(-0.386693\pi\)
−0.937311 + 0.348494i \(0.886693\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −16.0000 −0.851594 −0.425797 0.904819i \(-0.640006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.0000i 0.844448i −0.906492 0.422224i \(-0.861250\pi\)
0.906492 0.422224i \(-0.138750\pi\)
\(360\) 0 0
\(361\) 13.0000i 0.684211i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.0000 + 10.0000i 0.523424 + 0.523424i
\(366\) 0 0
\(367\) −20.0000 −1.04399 −0.521996 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 28.0000 + 28.0000i 1.45369 + 1.45369i
\(372\) 0 0
\(373\) 15.0000 15.0000i 0.776671 0.776671i −0.202593 0.979263i \(-0.564937\pi\)
0.979263 + 0.202593i \(0.0649367\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0000i 0.927047i
\(378\) 0 0
\(379\) 12.0000 12.0000i 0.616399 0.616399i −0.328207 0.944606i \(-0.606444\pi\)
0.944606 + 0.328207i \(0.106444\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 32.0000 1.63087
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −25.0000 + 25.0000i −1.26755 + 1.26755i −0.320201 + 0.947350i \(0.603750\pi\)
−0.947350 + 0.320201i \(0.896250\pi\)
\(390\) 0 0
\(391\) 48.0000i 2.42746i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.0000 + 12.0000i −0.603786 + 0.603786i
\(396\) 0 0
\(397\) −27.0000 27.0000i −1.35509 1.35509i −0.879862 0.475229i \(-0.842365\pi\)
−0.475229 0.879862i \(-0.657635\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) −12.0000 12.0000i −0.597763 0.597763i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000i 0.396545i
\(408\) 0 0
\(409\) 8.00000i 0.395575i −0.980245 0.197787i \(-0.936624\pi\)
0.980245 0.197787i \(-0.0633755\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 12.0000i −0.586238 0.586238i 0.350372 0.936611i \(-0.386055\pi\)
−0.936611 + 0.350372i \(0.886055\pi\)
\(420\) 0 0
\(421\) −9.00000 + 9.00000i −0.438633 + 0.438633i −0.891552 0.452919i \(-0.850383\pi\)
0.452919 + 0.891552i \(0.350383\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.0000i 0.873128i
\(426\) 0 0
\(427\) 12.0000 12.0000i 0.580721 0.580721i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 32.0000 32.0000i 1.53077 1.53077i
\(438\) 0 0
\(439\) 20.0000i 0.954548i 0.878755 + 0.477274i \(0.158375\pi\)
−0.878755 + 0.477274i \(0.841625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 12.0000i 0.570137 0.570137i −0.362029 0.932167i \(-0.617916\pi\)
0.932167 + 0.362029i \(0.117916\pi\)
\(444\) 0 0
\(445\) 16.0000 + 16.0000i 0.758473 + 0.758473i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −8.00000 8.00000i −0.376705 0.376705i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 24.0000i 1.12514i
\(456\) 0 0
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −19.0000 19.0000i −0.884918 0.884918i 0.109111 0.994030i \(-0.465200\pi\)
−0.994030 + 0.109111i \(0.965200\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.00000 + 4.00000i 0.185098 + 0.185098i 0.793573 0.608475i \(-0.208218\pi\)
−0.608475 + 0.793573i \(0.708218\pi\)
\(468\) 0 0
\(469\) −32.0000 + 32.0000i −1.47762 + 1.47762i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 32.0000i 1.47136i
\(474\) 0 0
\(475\) −12.0000 + 12.0000i −0.550598 + 0.550598i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.00000 8.00000i 0.363261 0.363261i
\(486\) 0 0
\(487\) 12.0000i 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.00000 8.00000i 0.361035 0.361035i −0.503159 0.864194i \(-0.667829\pi\)
0.864194 + 0.503159i \(0.167829\pi\)
\(492\) 0 0
\(493\) −18.0000 18.0000i −0.810679 0.810679i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 8.00000 + 8.00000i 0.358129 + 0.358129i 0.863123 0.504994i \(-0.168505\pi\)
−0.504994 + 0.863123i \(0.668505\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.0000i 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 0 0
\(505\) 14.0000i 0.622992i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11.0000 11.0000i −0.487566 0.487566i 0.419971 0.907537i \(-0.362040\pi\)
−0.907537 + 0.419971i \(0.862040\pi\)
\(510\) 0 0
\(511\) 40.0000 1.76950
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.00000 + 4.00000i 0.176261 + 0.176261i
\(516\) 0 0
\(517\) 32.0000 32.0000i 1.40736 1.40736i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000i 1.31432i −0.753749 0.657162i \(-0.771757\pi\)
0.753749 0.657162i \(-0.228243\pi\)
\(522\) 0 0
\(523\) 12.0000 12.0000i 0.524723 0.524723i −0.394271 0.918994i \(-0.629003\pi\)
0.918994 + 0.394271i \(0.129003\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 6.00000i 0.259889 0.259889i
\(534\) 0 0
\(535\) 16.0000i 0.691740i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.0000 36.0000i 1.55063 1.55063i
\(540\) 0 0
\(541\) −3.00000 3.00000i −0.128980 0.128980i 0.639670 0.768650i \(-0.279071\pi\)
−0.768650 + 0.639670i \(0.779071\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 12.0000 + 12.0000i 0.513083 + 0.513083i 0.915470 0.402387i \(-0.131819\pi\)
−0.402387 + 0.915470i \(0.631819\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 24.0000i 1.02243i
\(552\) 0 0
\(553\) 48.0000i 2.04117i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.0000 + 11.0000i 0.466085 + 0.466085i 0.900644 0.434559i \(-0.143096\pi\)
−0.434559 + 0.900644i \(0.643096\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −20.0000 20.0000i −0.842900 0.842900i 0.146336 0.989235i \(-0.453252\pi\)
−0.989235 + 0.146336i \(0.953252\pi\)
\(564\) 0 0
\(565\) −16.0000 + 16.0000i −0.673125 + 0.673125i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000i 0.754599i −0.926091 0.377300i \(-0.876853\pi\)
0.926091 0.377300i \(-0.123147\pi\)
\(570\) 0 0
\(571\) −24.0000 + 24.0000i −1.00437 + 1.00437i −0.00437833 + 0.999990i \(0.501394\pi\)
−0.999990 + 0.00437833i \(0.998606\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 16.0000 16.0000i 0.663792 0.663792i
\(582\) 0 0
\(583\) 56.0000i 2.31928i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000 16.0000i 0.660391 0.660391i −0.295081 0.955472i \(-0.595347\pi\)
0.955472 + 0.295081i \(0.0953468\pi\)
\(588\) 0 0
\(589\) 16.0000 + 16.0000i 0.659269 + 0.659269i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16.0000 0.657041 0.328521 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 0 0
\(595\) 24.0000 + 24.0000i 0.983904 + 0.983904i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 40.0000i 1.63436i 0.576386 + 0.817178i \(0.304463\pi\)
−0.576386 + 0.817178i \(0.695537\pi\)
\(600\) 0 0
\(601\) 38.0000i 1.55005i −0.631929 0.775026i \(-0.717737\pi\)
0.631929 0.775026i \(-0.282263\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −21.0000 21.0000i −0.853771 0.853771i
\(606\) 0 0
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 + 24.0000i 0.970936 + 0.970936i
\(612\) 0 0
\(613\) −7.00000 + 7.00000i −0.282727 + 0.282727i −0.834196 0.551468i \(-0.814068\pi\)
0.551468 + 0.834196i \(0.314068\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.0000i 1.28827i 0.764911 + 0.644136i \(0.222783\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 64.0000 2.56411
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.00000 6.00000i 0.239236 0.239236i
\(630\) 0 0
\(631\) 20.0000i 0.796187i −0.917345 0.398094i \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.00000 4.00000i 0.158735 0.158735i
\(636\) 0 0
\(637\) 27.0000 + 27.0000i 1.06978 + 1.06978i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) −12.0000 12.0000i −0.473234 0.473234i 0.429726 0.902959i \(-0.358610\pi\)
−0.902959 + 0.429726i \(0.858610\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.0000 19.0000i −0.743527 0.743527i 0.229728 0.973255i \(-0.426216\pi\)
−0.973255 + 0.229728i \(0.926216\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.00000 + 8.00000i 0.311636 + 0.311636i 0.845543 0.533907i \(-0.179277\pi\)
−0.533907 + 0.845543i \(0.679277\pi\)
\(660\) 0 0
\(661\) −7.00000 + 7.00000i −0.272268 + 0.272268i −0.830013 0.557744i \(-0.811667\pi\)
0.557744 + 0.830013i \(0.311667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 32.0000i 1.24091i
\(666\) 0 0
\(667\) 24.0000 24.0000i 0.929284 0.929284i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.00000 + 9.00000i −0.345898 + 0.345898i −0.858579 0.512681i \(-0.828652\pi\)
0.512681 + 0.858579i \(0.328652\pi\)
\(678\) 0 0
\(679\) 32.0000i 1.22805i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 12.0000i 0.459167 0.459167i −0.439215 0.898382i \(-0.644743\pi\)
0.898382 + 0.439215i \(0.144743\pi\)
\(684\) 0 0
\(685\) −2.00000 2.00000i −0.0764161 0.0764161i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 42.0000 1.60007
\(690\) 0 0
\(691\) 36.0000 + 36.0000i 1.36950 + 1.36950i 0.861145 + 0.508360i \(0.169748\pi\)
0.508360 + 0.861145i \(0.330252\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 12.0000i 0.454532i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −11.0000 11.0000i −0.415464 0.415464i 0.468173 0.883637i \(-0.344913\pi\)
−0.883637 + 0.468173i \(0.844913\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 28.0000 + 28.0000i 1.05305 + 1.05305i
\(708\) 0 0
\(709\) 33.0000 33.0000i 1.23934 1.23934i 0.279070 0.960271i \(-0.409974\pi\)
0.960271 0.279070i \(-0.0900263\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000i 1.19841i
\(714\) 0 0
\(715\) 24.0000 24.0000i 0.897549 0.897549i
\(716\) 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
\(723\) 0 0
\(724\) 0 0
\(725\) −9.00000 + 9.00000i −0.334252 + 0.334252i
\(726\) 0 0
\(727\) 52.0000i 1.92857i 0.264861 + 0.964287i \(0.414674\pi\)
−0.264861 + 0.964287i \(0.585326\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −24.0000 + 24.0000i −0.887672 + 0.887672i
\(732\) 0 0
\(733\) −37.0000 37.0000i −1.36663 1.36663i −0.865201 0.501425i \(-0.832809\pi\)
−0.501425 0.865201i \(-0.667191\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 64.0000 2.35747
\(738\) 0 0
\(739\) −24.0000 24.0000i −0.882854 0.882854i 0.110970 0.993824i \(-0.464604\pi\)
−0.993824 + 0.110970i \(0.964604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000i 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 2.00000i 0.0732743i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −32.0000 32.0000i −1.16925 1.16925i
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.00000 4.00000i −0.145575 0.145575i
\(756\) 0 0
\(757\) 7.00000 7.00000i 0.254419 0.254419i −0.568360 0.822780i \(-0.692422\pi\)
0.822780 + 0.568360i \(0.192422\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000i 0.652499i 0.945284 + 0.326250i \(0.105785\pi\)
−0.945284 + 0.326250i \(0.894215\pi\)
\(762\) 0 0
\(763\) −20.0000 + 20.0000i −0.724049 + 0.724049i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.00000 9.00000i 0.323708 0.323708i −0.526480 0.850188i \(-0.676489\pi\)
0.850188 + 0.526480i \(0.176489\pi\)
\(774\) 0 0
\(775\) 12.0000i 0.431053i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.00000 + 8.00000i −0.286630 + 0.286630i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) −12.0000 12.0000i −0.427754 0.427754i 0.460109 0.887863i \(-0.347810\pi\)
−0.887863 + 0.460109i \(0.847810\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 64.0000i 2.27558i
\(792\) 0 0
\(793\) 18.0000i 0.639199i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.00000 5.00000i −0.177109 0.177109i 0.612985 0.790094i \(-0.289968\pi\)
−0.790094 + 0.612985i \(0.789968\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\)