Properties

Label 360.2.s.b.233.2
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 233.2
Root \(0.437016 - 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 360.233
Dual form 360.2.s.b.17.2

$q$-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 \(-0.915113\pi\)
0.964651 0.263531i \(-0.0848870\pi\)
\(12\) 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 \(-0.962734\pi\)
−0.116808 0.993155i \(-0.537266\pi\)
\(18\) 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 −0.347801 0.937568i \(-0.613071\pi\)
0.937568 + 0.347801i \(0.113071\pi\)
\(24\) 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 \(-0.509868\pi\)
−0.0309972 + 0.999519i \(0.509868\pi\)
\(30\) 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 \(0.186195\pi\)
−0.833740 + 0.552158i \(0.813805\pi\)
\(42\) 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 0.957148 0.289598i \(-0.0935217\pi\)
−0.289598 + 0.957148i \(0.593522\pi\)
\(48\) 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 \(-0.339904\pi\)
0.482019 + 0.876161i \(0.339904\pi\)
\(60\) 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 \(0.270229\pi\)
−0.660772 + 0.750587i \(0.729771\pi\)
\(72\) 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 −0.172257 0.985052i \(-0.555106\pi\)
0.985052 + 0.172257i \(0.0551059\pi\)
\(84\) 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 \(-0.248710\pi\)
0.709967 + 0.704235i \(0.248710\pi\)
\(90\) 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 \(0.330116\pi\)
−0.508729 + 0.860927i \(0.669884\pi\)
\(102\) 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 0.870336 0.492459i \(-0.163902\pi\)
−0.492459 + 0.870336i \(0.663902\pi\)
\(108\) 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 −0.840027 0.542545i \(-0.817461\pi\)
0.542545 + 0.840027i \(0.317461\pi\)
\(114\) 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 \(-0.742608\pi\)
0.690497 0.723335i \(-0.257392\pi\)
\(132\) 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 0.592468 0.805594i \(-0.298154\pi\)
−0.805594 + 0.592468i \(0.798154\pi\)
\(138\) 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 \(-0.391637\pi\)
0.333896 + 0.942610i \(0.391637\pi\)
\(150\) 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 0.418711 0.908120i \(-0.362482\pi\)
−0.908120 + 0.418711i \(0.862482\pi\)
\(168\) 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 −0.694302 0.719684i \(-0.744287\pi\)
0.719684 + 0.694302i \(0.244287\pi\)
\(174\) 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 \(-0.562790\pi\)
−0.195985 + 0.980607i \(0.562790\pi\)
\(180\) 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 \(-0.918592\pi\)
0.967474 0.252971i \(-0.0814077\pi\)
\(192\) 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 \(-0.973761\pi\)
−0.0823386 0.996604i \(-0.526239\pi\)
\(198\) 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 0.917230 0.398359i \(-0.130420\pi\)
−0.398359 + 0.917230i \(0.630420\pi\)
\(228\) 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 −0.623534 0.781796i \(-0.714304\pi\)
0.781796 + 0.623534i \(0.214304\pi\)
\(234\) 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 \(-0.634158\pi\)
−0.409101 + 0.912489i \(0.634158\pi\)
\(240\) 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 \(-0.879330\pi\)
0.929000 0.370081i \(-0.120670\pi\)
\(252\) 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 0.601566 0.798823i \(-0.294544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(258\) 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 −0.120653 0.992695i \(-0.538499\pi\)
0.992695 + 0.120653i \(0.0384989\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.65841 0.405970 0.202985 0.979182i \(-0.434936\pi\)
0.202985 + 0.979182i \(0.434936\pi\)
\(270\) 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 \(0.208076\pi\)
−0.793846 + 0.608119i \(0.791924\pi\)
\(282\) 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 −0.989593 0.143897i \(-0.954037\pi\)
0.143897 + 0.989593i \(0.454037\pi\)
\(294\) 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 \(-0.980488\pi\)
0.998122 0.0612618i \(-0.0195124\pi\)
\(312\) 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 \(0.0520776\pi\)
0.162878 + 0.986646i \(0.447922\pi\)
\(318\) 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 0.999853 0.0171375i \(-0.00545529\pi\)
−0.0171375 + 0.999853i \(0.505455\pi\)
\(348\) 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 −0.944436 0.328694i \(-0.893391\pi\)
0.328694 + 0.944436i \(0.393391\pi\)
\(354\) 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 \(-0.459305\pi\)
0.127499 + 0.991839i \(0.459305\pi\)
\(360\) 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 −0.395546 0.918446i \(-0.629445\pi\)
0.918446 + 0.395546i \(0.129445\pi\)
\(384\) 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 \(-0.906905\pi\)
−0.957536 + 0.288315i \(0.906905\pi\)
\(390\) 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 \(-0.878757\pi\)
0.928332 0.371752i \(-0.121243\pi\)
\(402\) 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 \(-0.140330\pi\)
0.904385 + 0.426718i \(0.140330\pi\)
\(420\) 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 \(0.0435020\pi\)
−0.990676 + 0.136241i \(0.956498\pi\)
\(432\) 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 −0.867996 0.496571i \(-0.834592\pi\)
0.496571 + 0.867996i \(0.334592\pi\)
\(444\) 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 \(-0.696687\pi\)
−0.579333 + 0.815091i \(0.696687\pi\)
\(450\) 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 \(-0.929093\pi\)
0.975291 0.220923i \(-0.0709069\pi\)
\(462\) 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 0.746395 0.665504i \(-0.231783\pi\)
−0.665504 + 0.746395i \(0.731783\pi\)
\(468\) 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 \(-0.859859\pi\)
−0.904639 + 0.426179i \(0.859859\pi\)
\(480\) 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 \(-0.914759\pi\)
0.964358 0.264602i \(-0.0852405\pi\)
\(492\) 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 −0.286400 0.958110i \(-0.592459\pi\)
0.958110 + 0.286400i \(0.0924588\pi\)
\(504\) 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 \(-0.364447\pi\)
0.413098 + 0.910687i \(0.364447\pi\)
\(510\) 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 \(-0.755654\pi\)
0.719555 0.694436i \(-0.244346\pi\)
\(522\) 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 0.932388 0.361458i \(-0.117721\pi\)
−0.361458 + 0.932388i \(0.617721\pi\)
\(558\) 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 \(0.503171\pi\)
−0.999950 + 0.00996204i \(0.996829\pi\)
\(564\) 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 \(-0.621112\pi\)
−0.371369 + 0.928485i \(0.621112\pi\)
\(570\) 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 0.503338 0.864090i \(-0.332105\pi\)
−0.864090 + 0.503338i \(0.832105\pi\)
\(588\) 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 −0.0871785 0.996193i \(-0.527785\pi\)
0.996193 + 0.0871785i \(0.0277850\pi\)
\(594\) 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 \(-0.583200\pi\)
−0.258414 + 0.966034i \(0.583200\pi\)
\(600\) 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 0.754060 0.656805i \(-0.228093\pi\)
−0.656805 + 0.754060i \(0.728093\pi\)
\(618\) 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 \(0.127335\pi\)
−0.921048 + 0.389450i \(0.872665\pi\)
\(642\) 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 0.256807 0.966463i \(-0.417329\pi\)
−0.966463 + 0.256807i \(0.917329\pi\)
\(648\) 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 −0.450192 0.892932i \(-0.648644\pi\)
0.892932 + 0.450192i \(0.148644\pi\)
\(654\) 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 \(-0.636781\pi\)
−0.416607 + 0.909087i \(0.636781\pi\)
\(660\) 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 0.828590 0.559856i \(-0.189143\pi\)
−0.559856 + 0.828590i \(0.689143\pi\)
\(678\) 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 \(0.540047\pi\)
−0.992096 + 0.125479i \(0.959953\pi\)
\(684\) 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 \(0.0829940\pi\)
−0.966201 + 0.257789i \(0.917006\pi\)
\(702\) 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 \(-0.313745\pi\)
0.552314 + 0.833636i \(0.313745\pi\)
\(720\) 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 −0.927212 0.374536i \(-0.877802\pi\)
0.374536 + 0.927212i \(0.377802\pi\)
\(744\) 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 \(0.0692145\pi\)
−0.976452 + 0.215734i \(0.930785\pi\)
\(762\) 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 −0.971828 0.235692i \(-0.924264\pi\)
0.235692 + 0.971828i \(0.424264\pi\)
\(774\) 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 \(0.0200650\pi\)
0.0629944 + 0.998014i \(0.479935\pi\)
\(798\) 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 +