Properties

Label 360.2.s.b.233.4
Level $360$
Weight $2$
Character 360.233
Analytic conductor $2.875$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 233.4
Root \(-0.437016 + 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 360.233
Dual form 360.2.s.b.17.4

$q$-expansion

\(f(q)\) \(=\) \(q+(1.58114 + 1.58114i) q^{5} +(1.23607 - 1.23607i) q^{7} +O(q^{10})\) \(q+(1.58114 + 1.58114i) q^{5} +(1.23607 - 1.23607i) q^{7} +1.74806i q^{11} +(0.236068 + 0.236068i) q^{13} +(4.57649 + 4.57649i) q^{17} -6.47214i q^{19} +(-2.82843 + 2.82843i) q^{23} +5.00000i q^{25} +0.333851 q^{29} +10.4721 q^{31} +3.90879 q^{35} +(-2.23607 + 2.23607i) q^{37} -7.07107i q^{41} +(-6.47214 - 6.47214i) q^{43} +(-4.57649 - 4.57649i) q^{47} +3.94427i q^{49} +(-2.76393 + 2.76393i) q^{55} -7.40492 q^{59} +1.52786 q^{61} +0.746512i q^{65} +(-10.4721 + 10.4721i) q^{67} -12.6491i q^{71} +(-9.47214 - 9.47214i) q^{73} +(2.16073 + 2.16073i) q^{77} -5.52786i q^{79} +(-7.40492 + 7.40492i) q^{83} +14.4721i q^{85} -13.3956 q^{89} +0.583592 q^{91} +(10.2333 - 10.2333i) q^{95} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} + O(q^{10}) \) \( 8q - 8q^{7} - 16q^{13} + 48q^{31} - 16q^{43} - 40q^{55} + 48q^{61} - 48q^{67} - 40q^{73} + 112q^{91} + 8q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(-1\)

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
\(5\) 1.58114 + 1.58114i 0.707107 + 0.707107i
\(6\) 0 0
\(7\) 1.23607 1.23607i 0.467190 0.467190i −0.433813 0.901003i \(-0.642832\pi\)
0.901003 + 0.433813i \(0.142832\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.74806i 0.527061i 0.964651 + 0.263531i \(0.0848870\pi\)
−0.964651 + 0.263531i \(0.915113\pi\)
\(12\) 0 0
\(13\) 0.236068 + 0.236068i 0.0654735 + 0.0654735i 0.739085 0.673612i \(-0.235258\pi\)
−0.673612 + 0.739085i \(0.735258\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.57649 + 4.57649i 1.10996 + 1.10996i 0.993155 + 0.116808i \(0.0372661\pi\)
0.116808 + 0.993155i \(0.462734\pi\)
\(18\) 0 0
\(19\) 6.47214i 1.48481i −0.669951 0.742405i \(-0.733685\pi\)
0.669951 0.742405i \(-0.266315\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 + 2.82843i −0.589768 + 0.589768i −0.937568 0.347801i \(-0.886929\pi\)
0.347801 + 0.937568i \(0.386929\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.333851 0.0619945 0.0309972 0.999519i \(-0.490132\pi\)
0.0309972 + 0.999519i \(0.490132\pi\)
\(30\) 0 0
\(31\) 10.4721 1.88085 0.940426 0.340000i \(-0.110427\pi\)
0.940426 + 0.340000i \(0.110427\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.90879 0.660706
\(36\) 0 0
\(37\) −2.23607 + 2.23607i −0.367607 + 0.367607i −0.866604 0.498997i \(-0.833702\pi\)
0.498997 + 0.866604i \(0.333702\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.07107i 1.10432i −0.833740 0.552158i \(-0.813805\pi\)
0.833740 0.552158i \(-0.186195\pi\)
\(42\) 0 0
\(43\) −6.47214 6.47214i −0.986991 0.986991i 0.0129250 0.999916i \(-0.495886\pi\)
−0.999916 + 0.0129250i \(0.995886\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.57649 4.57649i −0.667550 0.667550i 0.289598 0.957148i \(-0.406478\pi\)
−0.957148 + 0.289598i \(0.906478\pi\)
\(48\) 0 0
\(49\) 3.94427i 0.563467i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) −2.76393 + 2.76393i −0.372689 + 0.372689i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.40492 −0.964038 −0.482019 0.876161i \(-0.660096\pi\)
−0.482019 + 0.876161i \(0.660096\pi\)
\(60\) 0 0
\(61\) 1.52786 0.195623 0.0978115 0.995205i \(-0.468816\pi\)
0.0978115 + 0.995205i \(0.468816\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.746512i 0.0925935i
\(66\) 0 0
\(67\) −10.4721 + 10.4721i −1.27938 + 1.27938i −0.338357 + 0.941018i \(0.609871\pi\)
−0.941018 + 0.338357i \(0.890129\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.6491i 1.50117i −0.660772 0.750587i \(-0.729771\pi\)
0.660772 0.750587i \(-0.270229\pi\)
\(72\) 0 0
\(73\) −9.47214 9.47214i −1.10863 1.10863i −0.993331 0.115299i \(-0.963217\pi\)
−0.115299 0.993331i \(-0.536783\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.16073 + 2.16073i 0.246238 + 0.246238i
\(78\) 0 0
\(79\) 5.52786i 0.621933i −0.950421 0.310967i \(-0.899347\pi\)
0.950421 0.310967i \(-0.100653\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.40492 + 7.40492i −0.812795 + 0.812795i −0.985052 0.172257i \(-0.944894\pi\)
0.172257 + 0.985052i \(0.444894\pi\)
\(84\) 0 0
\(85\) 14.4721i 1.56972i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.3956 −1.41993 −0.709967 0.704235i \(-0.751290\pi\)
−0.709967 + 0.704235i \(0.751290\pi\)
\(90\) 0 0
\(91\) 0.583592 0.0611771
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.2333 10.2333i 1.04992 1.04992i
\(96\) 0 0
\(97\) 1.00000 1.00000i 0.101535 0.101535i −0.654515 0.756049i \(-0.727127\pi\)
0.756049 + 0.654515i \(0.227127\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.3044i 1.72185i −0.508729 0.860927i \(-0.669884\pi\)
0.508729 0.860927i \(-0.330116\pi\)
\(102\) 0 0
\(103\) 10.1803 + 10.1803i 1.00310 + 1.00310i 0.999995 + 0.00310351i \(0.000987880\pi\)
0.00310351 + 0.999995i \(0.499012\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.90879 3.90879i −0.377877 0.377877i 0.492459 0.870336i \(-0.336098\pi\)
−0.870336 + 0.492459i \(0.836098\pi\)
\(108\) 0 0
\(109\) 11.4164i 1.09349i −0.837298 0.546747i \(-0.815866\pi\)
0.837298 0.546747i \(-0.184134\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.16228 3.16228i 0.297482 0.297482i −0.542545 0.840027i \(-0.682539\pi\)
0.840027 + 0.542545i \(0.182539\pi\)
\(114\) 0 0
\(115\) −8.94427 −0.834058
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.3137 1.03713
\(120\) 0 0
\(121\) 7.94427 0.722207
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.90569 + 7.90569i −0.707107 + 0.707107i
\(126\) 0 0
\(127\) −2.76393 + 2.76393i −0.245259 + 0.245259i −0.819022 0.573762i \(-0.805483\pi\)
0.573762 + 0.819022i \(0.305483\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.5579i 1.44667i 0.690497 + 0.723335i \(0.257392\pi\)
−0.690497 + 0.723335i \(0.742608\pi\)
\(132\) 0 0
\(133\) −8.00000 8.00000i −0.693688 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.49458 + 2.49458i 0.213126 + 0.213126i 0.805594 0.592468i \(-0.201846\pi\)
−0.592468 + 0.805594i \(0.701846\pi\)
\(138\) 0 0
\(139\) 8.94427i 0.758643i 0.925265 + 0.379322i \(0.123843\pi\)
−0.925265 + 0.379322i \(0.876157\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.412662 + 0.412662i −0.0345085 + 0.0345085i
\(144\) 0 0
\(145\) 0.527864 + 0.527864i 0.0438367 + 0.0438367i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.15143 −0.667791 −0.333896 0.942610i \(-0.608363\pi\)
−0.333896 + 0.942610i \(0.608363\pi\)
\(150\) 0 0
\(151\) 12.9443 1.05339 0.526695 0.850054i \(-0.323431\pi\)
0.526695 + 0.850054i \(0.323431\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.5579 + 16.5579i 1.32996 + 1.32996i
\(156\) 0 0
\(157\) 12.7082 12.7082i 1.01423 1.01423i 0.0143277 0.999897i \(-0.495439\pi\)
0.999897 0.0143277i \(-0.00456082\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.99226i 0.551067i
\(162\) 0 0
\(163\) −6.47214 6.47214i −0.506937 0.506937i 0.406648 0.913585i \(-0.366698\pi\)
−0.913585 + 0.406648i \(0.866698\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.32456 + 6.32456i 0.489409 + 0.489409i 0.908120 0.418711i \(-0.137518\pi\)
−0.418711 + 0.908120i \(0.637518\pi\)
\(168\) 0 0
\(169\) 12.8885i 0.991426i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.333851 + 0.333851i −0.0253822 + 0.0253822i −0.719684 0.694302i \(-0.755713\pi\)
0.694302 + 0.719684i \(0.255713\pi\)
\(174\) 0 0
\(175\) 6.18034 + 6.18034i 0.467190 + 0.467190i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.24419 0.391969 0.195985 0.980607i \(-0.437210\pi\)
0.195985 + 0.980607i \(0.437210\pi\)
\(180\) 0 0
\(181\) −1.52786 −0.113565 −0.0567826 0.998387i \(-0.518084\pi\)
−0.0567826 + 0.998387i \(0.518084\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.07107 −0.519875
\(186\) 0 0
\(187\) −8.00000 + 8.00000i −0.585018 + 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.99226i 0.505942i 0.967474 + 0.252971i \(0.0814077\pi\)
−0.967474 + 0.252971i \(0.918592\pi\)
\(192\) 0 0
\(193\) 7.47214 + 7.47214i 0.537856 + 0.537856i 0.922899 0.385043i \(-0.125813\pi\)
−0.385043 + 0.922899i \(0.625813\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.1437 + 15.1437i 1.07894 + 1.07894i 0.996604 + 0.0823386i \(0.0262389\pi\)
0.0823386 + 0.996604i \(0.473761\pi\)
\(198\) 0 0
\(199\) 8.00000i 0.567105i −0.958957 0.283552i \(-0.908487\pi\)
0.958957 0.283552i \(-0.0915130\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.412662 0.412662i 0.0289632 0.0289632i
\(204\) 0 0
\(205\) 11.1803 11.1803i 0.780869 0.780869i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.3137 0.782586
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 20.4667i 1.39582i
\(216\) 0 0
\(217\) 12.9443 12.9443i 0.878714 0.878714i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.16073i 0.145346i
\(222\) 0 0
\(223\) 14.7639 + 14.7639i 0.988666 + 0.988666i 0.999936 0.0112705i \(-0.00358758\pi\)
−0.0112705 + 0.999936i \(0.503588\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.81758 7.81758i −0.518871 0.518871i 0.398359 0.917230i \(-0.369580\pi\)
−0.917230 + 0.398359i \(0.869580\pi\)
\(228\) 0 0
\(229\) 15.8885i 1.04994i 0.851119 + 0.524972i \(0.175924\pi\)
−0.851119 + 0.524972i \(0.824076\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.41577 + 2.41577i −0.158262 + 0.158262i −0.781796 0.623534i \(-0.785696\pi\)
0.623534 + 0.781796i \(0.285696\pi\)
\(234\) 0 0
\(235\) 14.4721i 0.944058i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.6491 0.818203 0.409101 0.912489i \(-0.365842\pi\)
0.409101 + 0.912489i \(0.365842\pi\)
\(240\) 0 0
\(241\) −14.9443 −0.962645 −0.481323 0.876544i \(-0.659843\pi\)
−0.481323 + 0.876544i \(0.659843\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.23644 + 6.23644i −0.398432 + 0.398432i
\(246\) 0 0
\(247\) 1.52786 1.52786i 0.0972157 0.0972157i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.7264i 0.740162i 0.929000 + 0.370081i \(0.120670\pi\)
−0.929000 + 0.370081i \(0.879330\pi\)
\(252\) 0 0
\(253\) −4.94427 4.94427i −0.310844 0.310844i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.16228 + 3.16228i 0.197257 + 0.197257i 0.798823 0.601566i \(-0.205456\pi\)
−0.601566 + 0.798823i \(0.705456\pi\)
\(258\) 0 0
\(259\) 5.52786i 0.343485i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −14.1421 + 14.1421i −0.872041 + 0.872041i −0.992695 0.120653i \(-0.961501\pi\)
0.120653 + 0.992695i \(0.461501\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.65841 −0.405970 −0.202985 0.979182i \(-0.565064\pi\)
−0.202985 + 0.979182i \(0.565064\pi\)
\(270\) 0 0
\(271\) −11.0557 −0.671588 −0.335794 0.941936i \(-0.609005\pi\)
−0.335794 + 0.941936i \(0.609005\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.74032 −0.527061
\(276\) 0 0
\(277\) −7.18034 + 7.18034i −0.431425 + 0.431425i −0.889113 0.457688i \(-0.848678\pi\)
0.457688 + 0.889113i \(0.348678\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.3879i 1.21624i −0.793846 0.608119i \(-0.791924\pi\)
0.793846 0.608119i \(-0.208076\pi\)
\(282\) 0 0
\(283\) −8.00000 8.00000i −0.475551 0.475551i 0.428155 0.903705i \(-0.359164\pi\)
−0.903705 + 0.428155i \(0.859164\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.74032 8.74032i −0.515925 0.515925i
\(288\) 0 0
\(289\) 24.8885i 1.46403i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.4760 14.4760i 0.845696 0.845696i −0.143897 0.989593i \(-0.545963\pi\)
0.989593 + 0.143897i \(0.0459633\pi\)
\(294\) 0 0
\(295\) −11.7082 11.7082i −0.681678 0.681678i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.33540 −0.0772283
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.41577 + 2.41577i 0.138326 + 0.138326i
\(306\) 0 0
\(307\) 1.52786 1.52786i 0.0871998 0.0871998i −0.662161 0.749361i \(-0.730361\pi\)
0.749361 + 0.662161i \(0.230361\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.16073i 0.122524i 0.998122 + 0.0612618i \(0.0195124\pi\)
−0.998122 + 0.0612618i \(0.980488\pi\)
\(312\) 0 0
\(313\) 1.47214 + 1.47214i 0.0832100 + 0.0832100i 0.747487 0.664277i \(-0.231260\pi\)
−0.664277 + 0.747487i \(0.731260\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.4667 20.4667i −1.14952 1.14952i −0.986646 0.162878i \(-0.947922\pi\)
−0.162878 0.986646i \(-0.552078\pi\)
\(318\) 0 0
\(319\) 0.583592i 0.0326749i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 29.6197 29.6197i 1.64808 1.64808i
\(324\) 0 0
\(325\) −1.18034 + 1.18034i −0.0654735 + 0.0654735i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.3137 −0.623745
\(330\) 0 0
\(331\) −14.4721 −0.795461 −0.397730 0.917502i \(-0.630202\pi\)
−0.397730 + 0.917502i \(0.630202\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −33.1158 −1.80931
\(336\) 0 0
\(337\) −12.4164 + 12.4164i −0.676365 + 0.676365i −0.959176 0.282811i \(-0.908733\pi\)
0.282811 + 0.959176i \(0.408733\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 18.3060i 0.991324i
\(342\) 0 0
\(343\) 13.5279 + 13.5279i 0.730436 + 0.730436i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.3060 18.3060i −0.982716 0.982716i 0.0171375 0.999853i \(-0.494545\pi\)
−0.999853 + 0.0171375i \(0.994545\pi\)
\(348\) 0 0
\(349\) 14.4721i 0.774676i 0.921938 + 0.387338i \(0.126605\pi\)
−0.921938 + 0.387338i \(0.873395\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.5687 11.5687i 0.615742 0.615742i −0.328694 0.944436i \(-0.606609\pi\)
0.944436 + 0.328694i \(0.106609\pi\)
\(354\) 0 0
\(355\) 20.0000 20.0000i 1.06149 1.06149i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.83153 −0.254998 −0.127499 0.991839i \(-0.540695\pi\)
−0.127499 + 0.991839i \(0.540695\pi\)
\(360\) 0 0
\(361\) −22.8885 −1.20466
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 29.9535i 1.56784i
\(366\) 0 0
\(367\) 15.7082 15.7082i 0.819962 0.819962i −0.166141 0.986102i \(-0.553131\pi\)
0.986102 + 0.166141i \(0.0531305\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 24.7082 + 24.7082i 1.27934 + 1.27934i 0.941036 + 0.338306i \(0.109854\pi\)
0.338306 + 0.941036i \(0.390146\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.0788114 + 0.0788114i 0.00405899 + 0.00405899i
\(378\) 0 0
\(379\) 0.944272i 0.0485040i −0.999706 0.0242520i \(-0.992280\pi\)
0.999706 0.0242520i \(-0.00772041\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.2333 + 10.2333i −0.522900 + 0.522900i −0.918446 0.395546i \(-0.870555\pi\)
0.395546 + 0.918446i \(0.370555\pi\)
\(384\) 0 0
\(385\) 6.83282i 0.348233i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 37.7711 1.91507 0.957536 0.288315i \(-0.0930952\pi\)
0.957536 + 0.288315i \(0.0930952\pi\)
\(390\) 0 0
\(391\) −25.8885 −1.30924
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.74032 8.74032i 0.439773 0.439773i
\(396\) 0 0
\(397\) −1.76393 + 1.76393i −0.0885292 + 0.0885292i −0.749985 0.661455i \(-0.769939\pi\)
0.661455 + 0.749985i \(0.269939\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.8886i 0.743504i 0.928332 + 0.371752i \(0.121243\pi\)
−0.928332 + 0.371752i \(0.878757\pi\)
\(402\) 0 0
\(403\) 2.47214 + 2.47214i 0.123146 + 0.123146i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.90879 3.90879i −0.193752 0.193752i
\(408\) 0 0
\(409\) 9.88854i 0.488957i −0.969655 0.244479i \(-0.921383\pi\)
0.969655 0.244479i \(-0.0786168\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.15298 + 9.15298i −0.450389 + 0.450389i
\(414\) 0 0
\(415\) −23.4164 −1.14947
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −37.0246 −1.80877 −0.904385 0.426718i \(-0.859670\pi\)
−0.904385 + 0.426718i \(0.859670\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −22.8825 + 22.8825i −1.10996 + 1.10996i
\(426\) 0 0
\(427\) 1.88854 1.88854i 0.0913930 0.0913930i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65685i 0.272481i −0.990676 0.136241i \(-0.956498\pi\)
0.990676 0.136241i \(-0.0435020\pi\)
\(432\) 0 0
\(433\) 10.4164 + 10.4164i 0.500581 + 0.500581i 0.911618 0.411038i \(-0.134834\pi\)
−0.411038 + 0.911618i \(0.634834\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 18.3060 + 18.3060i 0.875693 + 0.875693i
\(438\) 0 0
\(439\) 3.05573i 0.145842i −0.997338 0.0729210i \(-0.976768\pi\)
0.997338 0.0729210i \(-0.0232321\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.81758 7.81758i 0.371424 0.371424i −0.496571 0.867996i \(-0.665408\pi\)
0.867996 + 0.496571i \(0.165408\pi\)
\(444\) 0 0
\(445\) −21.1803 21.1803i −1.00404 1.00404i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.5517 1.15867 0.579333 0.815091i \(-0.303313\pi\)
0.579333 + 0.815091i \(0.303313\pi\)
\(450\) 0 0
\(451\) 12.3607 0.582042
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.922740 + 0.922740i 0.0432587 + 0.0432587i
\(456\) 0 0
\(457\) 13.0000 13.0000i 0.608114 0.608114i −0.334339 0.942453i \(-0.608513\pi\)
0.942453 + 0.334339i \(0.108513\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.48683i 0.441846i 0.975291 + 0.220923i \(0.0709069\pi\)
−0.975291 + 0.220923i \(0.929093\pi\)
\(462\) 0 0
\(463\) −20.6525 20.6525i −0.959802 0.959802i 0.0394208 0.999223i \(-0.487449\pi\)
−0.999223 + 0.0394208i \(0.987449\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.74806 1.74806i −0.0808908 0.0808908i 0.665504 0.746395i \(-0.268217\pi\)
−0.746395 + 0.665504i \(0.768217\pi\)
\(468\) 0 0
\(469\) 25.8885i 1.19542i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.3137 11.3137i 0.520205 0.520205i
\(474\) 0 0
\(475\) 32.3607 1.48481
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 39.5980 1.80928 0.904639 0.426179i \(-0.140141\pi\)
0.904639 + 0.426179i \(0.140141\pi\)
\(480\) 0 0
\(481\) −1.05573 −0.0481371
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.16228 0.143592
\(486\) 0 0
\(487\) 13.2361 13.2361i 0.599783 0.599783i −0.340471 0.940255i \(-0.610587\pi\)
0.940255 + 0.340471i \(0.110587\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.7264i 0.529204i 0.964358 + 0.264602i \(0.0852405\pi\)
−0.964358 + 0.264602i \(0.914759\pi\)
\(492\) 0 0
\(493\) 1.52786 + 1.52786i 0.0688115 + 0.0688115i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.6352 15.6352i −0.701333 0.701333i
\(498\) 0 0
\(499\) 17.5279i 0.784655i 0.919826 + 0.392327i \(0.128330\pi\)
−0.919826 + 0.392327i \(0.871670\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15.0649 + 15.0649i −0.671710 + 0.671710i −0.958110 0.286400i \(-0.907541\pi\)
0.286400 + 0.958110i \(0.407541\pi\)
\(504\) 0 0
\(505\) 27.3607 27.3607i 1.21753 1.21753i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.6398 −0.826195 −0.413098 0.910687i \(-0.635553\pi\)
−0.413098 + 0.910687i \(0.635553\pi\)
\(510\) 0 0
\(511\) −23.4164 −1.03588
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 32.1931i 1.41860i
\(516\) 0 0
\(517\) 8.00000 8.00000i 0.351840 0.351840i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 31.7016i 1.38887i 0.719555 + 0.694436i \(0.244346\pi\)
−0.719555 + 0.694436i \(0.755654\pi\)
\(522\) 0 0
\(523\) 25.8885 + 25.8885i 1.13203 + 1.13203i 0.989840 + 0.142187i \(0.0454135\pi\)
0.142187 + 0.989840i \(0.454587\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 47.9256 + 47.9256i 2.08767 + 2.08767i
\(528\) 0 0
\(529\) 7.00000i 0.304348i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.66925 1.66925i 0.0723034 0.0723034i
\(534\) 0 0
\(535\) 12.3607i 0.534399i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.89484 −0.296982
\(540\) 0 0
\(541\) 37.3050 1.60387 0.801933 0.597415i \(-0.203805\pi\)
0.801933 + 0.597415i \(0.203805\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 18.0509 18.0509i 0.773217 0.773217i
\(546\) 0 0
\(547\) 12.0000 12.0000i 0.513083 0.513083i −0.402387 0.915470i \(-0.631819\pi\)
0.915470 + 0.402387i \(0.131819\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.16073i 0.0920500i
\(552\) 0 0
\(553\) −6.83282 6.83282i −0.290561 0.290561i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.4744 13.4744i −0.570930 0.570930i 0.361458 0.932388i \(-0.382279\pi\)
−0.932388 + 0.361458i \(0.882279\pi\)
\(558\) 0 0
\(559\) 3.05573i 0.129244i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 23.9628 23.9628i 1.00991 1.00991i 0.00996204 0.999950i \(-0.496829\pi\)
0.999950 0.00996204i \(-0.00317107\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.7171 0.742738 0.371369 0.928485i \(-0.378888\pi\)
0.371369 + 0.928485i \(0.378888\pi\)
\(570\) 0 0
\(571\) 19.4164 0.812551 0.406276 0.913751i \(-0.366827\pi\)
0.406276 + 0.913751i \(0.366827\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −14.1421 14.1421i −0.589768 0.589768i
\(576\) 0 0
\(577\) −29.9443 + 29.9443i −1.24660 + 1.24660i −0.289383 + 0.957213i \(0.593450\pi\)
−0.957213 + 0.289383i \(0.906550\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 18.3060i 0.759459i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.74032 + 8.74032i 0.360752 + 0.360752i 0.864090 0.503338i \(-0.167895\pi\)
−0.503338 + 0.864090i \(0.667895\pi\)
\(588\) 0 0
\(589\) 67.7771i 2.79271i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22.1359 + 22.1359i −0.909014 + 0.909014i −0.996193 0.0871785i \(-0.972215\pi\)
0.0871785 + 0.996193i \(0.472215\pi\)
\(594\) 0 0
\(595\) 17.8885 + 17.8885i 0.733359 + 0.733359i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.6491 0.516829 0.258414 0.966034i \(-0.416800\pi\)
0.258414 + 0.966034i \(0.416800\pi\)
\(600\) 0 0
\(601\) 22.8328 0.931370 0.465685 0.884951i \(-0.345808\pi\)
0.465685 + 0.884951i \(0.345808\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.5610 + 12.5610i 0.510677 + 0.510677i
\(606\) 0 0
\(607\) −0.652476 + 0.652476i −0.0264832 + 0.0264832i −0.720224 0.693741i \(-0.755961\pi\)
0.693741 + 0.720224i \(0.255961\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.16073i 0.0874136i
\(612\) 0 0
\(613\) −13.2918 13.2918i −0.536851 0.536851i 0.385752 0.922603i \(-0.373942\pi\)
−0.922603 + 0.385752i \(0.873942\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.41577 2.41577i −0.0972550 0.0972550i 0.656805 0.754060i \(-0.271907\pi\)
−0.754060 + 0.656805i \(0.771907\pi\)
\(618\) 0 0
\(619\) 29.8885i 1.20132i 0.799504 + 0.600661i \(0.205096\pi\)
−0.799504 + 0.600661i \(0.794904\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −16.5579 + 16.5579i −0.663378 + 0.663378i
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −20.4667 −0.816060
\(630\) 0 0
\(631\) −36.3607 −1.44750 −0.723748 0.690064i \(-0.757582\pi\)
−0.723748 + 0.690064i \(0.757582\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.74032 −0.346849
\(636\) 0 0
\(637\) −0.931116 + 0.931116i −0.0368922 + 0.0368922i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 19.7202i 0.778900i −0.921048 0.389450i \(-0.872665\pi\)
0.921048 0.389450i \(-0.127335\pi\)
\(642\) 0 0
\(643\) −20.0000 20.0000i −0.788723 0.788723i 0.192562 0.981285i \(-0.438320\pi\)
−0.981285 + 0.192562i \(0.938320\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0509 + 18.0509i 0.709655 + 0.709655i 0.966463 0.256807i \(-0.0826706\pi\)
−0.256807 + 0.966463i \(0.582671\pi\)
\(648\) 0 0
\(649\) 12.9443i 0.508107i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.3137 + 11.3137i −0.442740 + 0.442740i −0.892932 0.450192i \(-0.851356\pi\)
0.450192 + 0.892932i \(0.351356\pi\)
\(654\) 0 0
\(655\) −26.1803 + 26.1803i −1.02295 + 1.02295i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 21.3894 0.833214 0.416607 0.909087i \(-0.363219\pi\)
0.416607 + 0.909087i \(0.363219\pi\)
\(660\) 0 0
\(661\) −21.3050 −0.828667 −0.414333 0.910125i \(-0.635985\pi\)
−0.414333 + 0.910125i \(0.635985\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 25.2982i 0.981023i
\(666\) 0 0
\(667\) −0.944272 + 0.944272i −0.0365624 + 0.0365624i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.67080i 0.103105i
\(672\) 0 0
\(673\) 10.5279 + 10.5279i 0.405819 + 0.405819i 0.880278 0.474459i \(-0.157356\pi\)
−0.474459 + 0.880278i \(0.657356\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.99226 6.99226i −0.268734 0.268734i 0.559856 0.828590i \(-0.310857\pi\)
−0.828590 + 0.559856i \(0.810857\pi\)
\(678\) 0 0
\(679\) 2.47214i 0.0948719i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.2070 29.2070i 1.11758 1.11758i 0.125479 0.992096i \(-0.459953\pi\)
0.992096 0.125479i \(-0.0400468\pi\)
\(684\) 0 0
\(685\) 7.88854i 0.301406i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 24.3607 0.926724 0.463362 0.886169i \(-0.346643\pi\)
0.463362 + 0.886169i \(0.346643\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −14.1421 + 14.1421i −0.536442 + 0.536442i
\(696\) 0 0
\(697\) 32.3607 32.3607i 1.22575 1.22575i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.6507i 0.515578i −0.966201 0.257789i \(-0.917006\pi\)
0.966201 0.257789i \(-0.0829940\pi\)
\(702\) 0 0
\(703\) 14.4721 + 14.4721i 0.545827 + 0.545827i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −21.3894 21.3894i −0.804432 0.804432i
\(708\) 0 0
\(709\) 19.8885i 0.746930i 0.927644 + 0.373465i \(0.121830\pi\)
−0.927644 + 0.373465i \(0.878170\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −29.6197 + 29.6197i −1.10927 + 1.10927i
\(714\) 0 0
\(715\) −1.30495 −0.0488024
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −29.6197 −1.10463 −0.552314 0.833636i \(-0.686255\pi\)
−0.552314 + 0.833636i \(0.686255\pi\)
\(720\) 0 0
\(721\) 25.1672 0.937275
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.66925i 0.0619945i
\(726\) 0 0
\(727\) −31.1246 + 31.1246i −1.15435 + 1.15435i −0.168676 + 0.985671i \(0.553949\pi\)
−0.985671 + 0.168676i \(0.946051\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 59.2393i 2.19105i
\(732\) 0 0
\(733\) −11.1803 11.1803i −0.412955 0.412955i 0.469811 0.882767i \(-0.344322\pi\)
−0.882767 + 0.469811i \(0.844322\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.3060 18.3060i −0.674309 0.674309i
\(738\) 0 0
\(739\) 32.3607i 1.19041i 0.803575 + 0.595203i \(0.202929\pi\)
−0.803575 + 0.595203i \(0.797071\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 15.0649 15.0649i 0.552677 0.552677i −0.374536 0.927212i \(-0.622198\pi\)
0.927212 + 0.374536i \(0.122198\pi\)
\(744\) 0 0
\(745\) −12.8885 12.8885i −0.472200 0.472200i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.66306 −0.353081
\(750\) 0 0
\(751\) −12.3607 −0.451048 −0.225524 0.974238i \(-0.572409\pi\)
−0.225524 + 0.974238i \(0.572409\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 20.4667 + 20.4667i 0.744859 + 0.744859i
\(756\) 0 0
\(757\) −2.70820 + 2.70820i −0.0984313 + 0.0984313i −0.754608 0.656176i \(-0.772173\pi\)
0.656176 + 0.754608i \(0.272173\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11.9026i 0.431469i −0.976452 0.215734i \(-0.930785\pi\)
0.976452 0.215734i \(-0.0692145\pi\)
\(762\) 0 0
\(763\) −14.1115 14.1115i −0.510869 0.510869i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.74806 1.74806i −0.0631189 0.0631189i
\(768\) 0 0
\(769\) 32.0000i 1.15395i −0.816762 0.576975i \(-0.804233\pi\)
0.816762 0.576975i \(-0.195767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20.4667 20.4667i 0.736136 0.736136i −0.235692 0.971828i \(-0.575736\pi\)
0.971828 + 0.235692i \(0.0757357\pi\)
\(774\) 0 0
\(775\) 52.3607i 1.88085i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −45.7649 −1.63970
\(780\) 0 0
\(781\) 22.1115 0.791210
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 40.1869 1.43433
\(786\) 0 0
\(787\) −21.5279 + 21.5279i −0.767385 + 0.767385i −0.977645 0.210260i \(-0.932569\pi\)
0.210260 + 0.977645i \(0.432569\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7.81758i 0.277961i
\(792\) 0 0
\(793\) 0.360680 + 0.360680i 0.0128081 + 0.0128081i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.9535 29.9535i −1.06101 1.06101i −0.998014 0.0629944i \(-0.979935\pi\)
−0.0629944 0.998014i \(-0.520065\pi\)
\(798\) 0 0
\(799\) 41.8885i 1.48191i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16.5579 16.5579i 0.584316 0.584316i
\(804\) 0 0
\(805\) −11.0557 +