Properties

Label 4608.2.k.bi.1153.2
Level $4608$
Weight $2$
Character 4608.1153
Analytic conductor $36.795$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1153.2
Root \(-0.923880 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 4608.1153
Dual form 4608.2.k.bi.3457.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.414214 + 0.414214i) q^{5} +3.69552i q^{7} +O(q^{10})\) \(q+(-0.414214 + 0.414214i) q^{5} +3.69552i q^{7} +(-2.93015 + 2.93015i) q^{11} +(2.41421 + 2.41421i) q^{13} +2.82843 q^{17} +(4.46088 + 4.46088i) q^{19} -6.75699i q^{23} +4.65685i q^{25} +(5.24264 + 5.24264i) q^{29} -3.06147 q^{31} +(-1.53073 - 1.53073i) q^{35} +(6.41421 - 6.41421i) q^{37} +4.00000i q^{41} +(-0.765367 + 0.765367i) q^{43} -3.06147 q^{47} -6.65685 q^{49} +(3.24264 - 3.24264i) q^{53} -2.42742i q^{55} +(0.765367 - 0.765367i) q^{59} +(0.757359 + 0.757359i) q^{61} -2.00000 q^{65} +(1.39942 + 1.39942i) q^{67} +8.02509i q^{71} -6.48528i q^{73} +(-10.8284 - 10.8284i) q^{77} -14.7821 q^{79} +(9.68714 + 9.68714i) q^{83} +(-1.17157 + 1.17157i) q^{85} +4.82843i q^{89} +(-8.92177 + 8.92177i) q^{91} -3.69552 q^{95} +5.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + O(q^{10}) \) \( 8 q + 8 q^{5} + 8 q^{13} + 8 q^{29} + 40 q^{37} - 8 q^{49} - 8 q^{53} + 40 q^{61} - 16 q^{65} - 64 q^{77} - 32 q^{85} + 64 q^{97} + O(q^{100}) \)

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
\(5\) −0.414214 + 0.414214i −0.185242 + 0.185242i −0.793635 0.608394i \(-0.791814\pi\)
0.608394 + 0.793635i \(0.291814\pi\)
\(6\) 0 0
\(7\) 3.69552i 1.39677i 0.715720 + 0.698387i \(0.246099\pi\)
−0.715720 + 0.698387i \(0.753901\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.93015 + 2.93015i −0.883474 + 0.883474i −0.993886 0.110412i \(-0.964783\pi\)
0.110412 + 0.993886i \(0.464783\pi\)
\(12\) 0 0
\(13\) 2.41421 + 2.41421i 0.669582 + 0.669582i 0.957619 0.288037i \(-0.0930026\pi\)
−0.288037 + 0.957619i \(0.593003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 4.46088 + 4.46088i 1.02340 + 1.02340i 0.999720 + 0.0236776i \(0.00753751\pi\)
0.0236776 + 0.999720i \(0.492462\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.75699i 1.40893i −0.709739 0.704464i \(-0.751187\pi\)
0.709739 0.704464i \(-0.248813\pi\)
\(24\) 0 0
\(25\) 4.65685i 0.931371i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.24264 + 5.24264i 0.973534 + 0.973534i 0.999659 0.0261248i \(-0.00831671\pi\)
−0.0261248 + 0.999659i \(0.508317\pi\)
\(30\) 0 0
\(31\) −3.06147 −0.549856 −0.274928 0.961465i \(-0.588654\pi\)
−0.274928 + 0.961465i \(0.588654\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.53073 1.53073i −0.258741 0.258741i
\(36\) 0 0
\(37\) 6.41421 6.41421i 1.05449 1.05449i 0.0560630 0.998427i \(-0.482145\pi\)
0.998427 0.0560630i \(-0.0178548\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000i 0.624695i 0.949968 + 0.312348i \(0.101115\pi\)
−0.949968 + 0.312348i \(0.898885\pi\)
\(42\) 0 0
\(43\) −0.765367 + 0.765367i −0.116717 + 0.116717i −0.763053 0.646336i \(-0.776301\pi\)
0.646336 + 0.763053i \(0.276301\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.06147 −0.446561 −0.223280 0.974754i \(-0.571677\pi\)
−0.223280 + 0.974754i \(0.571677\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.24264 3.24264i 0.445411 0.445411i −0.448415 0.893826i \(-0.648011\pi\)
0.893826 + 0.448415i \(0.148011\pi\)
\(54\) 0 0
\(55\) 2.42742i 0.327313i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.765367 0.765367i 0.0996423 0.0996423i −0.655528 0.755171i \(-0.727554\pi\)
0.755171 + 0.655528i \(0.227554\pi\)
\(60\) 0 0
\(61\) 0.757359 + 0.757359i 0.0969699 + 0.0969699i 0.753928 0.656958i \(-0.228157\pi\)
−0.656958 + 0.753928i \(0.728157\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 1.39942 + 1.39942i 0.170966 + 0.170966i 0.787404 0.616438i \(-0.211425\pi\)
−0.616438 + 0.787404i \(0.711425\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.02509i 0.952403i 0.879336 + 0.476201i \(0.157987\pi\)
−0.879336 + 0.476201i \(0.842013\pi\)
\(72\) 0 0
\(73\) 6.48528i 0.759045i −0.925183 0.379522i \(-0.876088\pi\)
0.925183 0.379522i \(-0.123912\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.8284 10.8284i −1.23401 1.23401i
\(78\) 0 0
\(79\) −14.7821 −1.66311 −0.831557 0.555440i \(-0.812550\pi\)
−0.831557 + 0.555440i \(0.812550\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.68714 + 9.68714i 1.06330 + 1.06330i 0.997856 + 0.0654452i \(0.0208468\pi\)
0.0654452 + 0.997856i \(0.479153\pi\)
\(84\) 0 0
\(85\) −1.17157 + 1.17157i −0.127075 + 0.127075i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.82843i 0.511812i 0.966702 + 0.255906i \(0.0823738\pi\)
−0.966702 + 0.255906i \(0.917626\pi\)
\(90\) 0 0
\(91\) −8.92177 + 8.92177i −0.935256 + 0.935256i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.69552 −0.379152
\(96\) 0 0
\(97\) 5.17157 0.525094 0.262547 0.964919i \(-0.415438\pi\)
0.262547 + 0.964919i \(0.415438\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0711 + 12.0711i −1.20112 + 1.20112i −0.227289 + 0.973827i \(0.572986\pi\)
−0.973827 + 0.227289i \(0.927014\pi\)
\(102\) 0 0
\(103\) 14.1480i 1.39405i −0.717049 0.697023i \(-0.754508\pi\)
0.717049 0.697023i \(-0.245492\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.39942 1.39942i 0.135287 0.135287i −0.636220 0.771507i \(-0.719503\pi\)
0.771507 + 0.636220i \(0.219503\pi\)
\(108\) 0 0
\(109\) −9.24264 9.24264i −0.885284 0.885284i 0.108781 0.994066i \(-0.465305\pi\)
−0.994066 + 0.108781i \(0.965305\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.65685 −0.720296 −0.360148 0.932895i \(-0.617274\pi\)
−0.360148 + 0.932895i \(0.617274\pi\)
\(114\) 0 0
\(115\) 2.79884 + 2.79884i 0.260993 + 0.260993i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 10.4525i 0.958179i
\(120\) 0 0
\(121\) 6.17157i 0.561052i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.00000 4.00000i −0.357771 0.357771i
\(126\) 0 0
\(127\) −11.7206 −1.04004 −0.520018 0.854155i \(-0.674075\pi\)
−0.520018 + 0.854155i \(0.674075\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.3212 + 10.3212i 0.901766 + 0.901766i 0.995589 0.0938226i \(-0.0299086\pi\)
−0.0938226 + 0.995589i \(0.529909\pi\)
\(132\) 0 0
\(133\) −16.4853 + 16.4853i −1.42946 + 1.42946i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.9706i 1.10815i −0.832467 0.554075i \(-0.813072\pi\)
0.832467 0.554075i \(-0.186928\pi\)
\(138\) 0 0
\(139\) −14.2793 + 14.2793i −1.21116 + 1.21116i −0.240511 + 0.970646i \(0.577315\pi\)
−0.970646 + 0.240511i \(0.922685\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −14.1480 −1.18312
\(144\) 0 0
\(145\) −4.34315 −0.360679
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.24264 1.24264i 0.101801 0.101801i −0.654372 0.756173i \(-0.727067\pi\)
0.756173 + 0.654372i \(0.227067\pi\)
\(150\) 0 0
\(151\) 0.634051i 0.0515983i 0.999667 + 0.0257992i \(0.00821304\pi\)
−0.999667 + 0.0257992i \(0.991787\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.26810 1.26810i 0.101856 0.101856i
\(156\) 0 0
\(157\) −1.58579 1.58579i −0.126560 0.126560i 0.640990 0.767549i \(-0.278524\pi\)
−0.767549 + 0.640990i \(0.778524\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 24.9706 1.96796
\(162\) 0 0
\(163\) 3.82683 + 3.82683i 0.299741 + 0.299741i 0.840912 0.541171i \(-0.182019\pi\)
−0.541171 + 0.840912i \(0.682019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.69552i 0.285968i −0.989725 0.142984i \(-0.954330\pi\)
0.989725 0.142984i \(-0.0456697\pi\)
\(168\) 0 0
\(169\) 1.34315i 0.103319i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.5858 + 13.5858i 1.03291 + 1.03291i 0.999440 + 0.0334684i \(0.0106553\pi\)
0.0334684 + 0.999440i \(0.489345\pi\)
\(174\) 0 0
\(175\) −17.2095 −1.30092
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.15640 8.15640i −0.609638 0.609638i 0.333213 0.942851i \(-0.391867\pi\)
−0.942851 + 0.333213i \(0.891867\pi\)
\(180\) 0 0
\(181\) 10.0711 10.0711i 0.748577 0.748577i −0.225635 0.974212i \(-0.572446\pi\)
0.974212 + 0.225635i \(0.0724458\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.31371i 0.390672i
\(186\) 0 0
\(187\) −8.28772 + 8.28772i −0.606058 + 0.606058i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.7206 −0.848073 −0.424037 0.905645i \(-0.639387\pi\)
−0.424037 + 0.905645i \(0.639387\pi\)
\(192\) 0 0
\(193\) 0.485281 0.0349313 0.0174657 0.999847i \(-0.494440\pi\)
0.0174657 + 0.999847i \(0.494440\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.58579 7.58579i 0.540465 0.540465i −0.383200 0.923665i \(-0.625178\pi\)
0.923665 + 0.383200i \(0.125178\pi\)
\(198\) 0 0
\(199\) 12.3547i 0.875798i 0.899024 + 0.437899i \(0.144277\pi\)
−0.899024 + 0.437899i \(0.855723\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −19.3743 + 19.3743i −1.35981 + 1.35981i
\(204\) 0 0
\(205\) −1.65685 1.65685i −0.115720 0.115720i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −26.1421 −1.80829
\(210\) 0 0
\(211\) 9.42450 + 9.42450i 0.648810 + 0.648810i 0.952705 0.303896i \(-0.0982874\pi\)
−0.303896 + 0.952705i \(0.598287\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.634051i 0.0432419i
\(216\) 0 0
\(217\) 11.3137i 0.768025i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.82843 + 6.82843i 0.459330 + 0.459330i
\(222\) 0 0
\(223\) −3.06147 −0.205011 −0.102506 0.994732i \(-0.532686\pi\)
−0.102506 + 0.994732i \(0.532686\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.765367 0.765367i −0.0507992 0.0507992i 0.681251 0.732050i \(-0.261436\pi\)
−0.732050 + 0.681251i \(0.761436\pi\)
\(228\) 0 0
\(229\) 2.75736 2.75736i 0.182211 0.182211i −0.610107 0.792319i \(-0.708874\pi\)
0.792319 + 0.610107i \(0.208874\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.8284i 1.36452i −0.731112 0.682258i \(-0.760998\pi\)
0.731112 0.682258i \(-0.239002\pi\)
\(234\) 0 0
\(235\) 1.26810 1.26810i 0.0827218 0.0827218i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.2459 0.792119 0.396060 0.918225i \(-0.370377\pi\)
0.396060 + 0.918225i \(0.370377\pi\)
\(240\) 0 0
\(241\) 13.1716 0.848456 0.424228 0.905556i \(-0.360546\pi\)
0.424228 + 0.905556i \(0.360546\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.75736 2.75736i 0.176161 0.176161i
\(246\) 0 0
\(247\) 21.5391i 1.37050i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.7486 + 12.7486i −0.804685 + 0.804685i −0.983824 0.179139i \(-0.942669\pi\)
0.179139 + 0.983824i \(0.442669\pi\)
\(252\) 0 0
\(253\) 19.7990 + 19.7990i 1.24475 + 1.24475i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.9706 0.933838 0.466919 0.884300i \(-0.345364\pi\)
0.466919 + 0.884300i \(0.345364\pi\)
\(258\) 0 0
\(259\) 23.7038 + 23.7038i 1.47289 + 1.47289i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.8799i 0.794210i −0.917773 0.397105i \(-0.870015\pi\)
0.917773 0.397105i \(-0.129985\pi\)
\(264\) 0 0
\(265\) 2.68629i 0.165018i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.0711 18.0711i −1.10181 1.10181i −0.994192 0.107620i \(-0.965677\pi\)
−0.107620 0.994192i \(-0.534323\pi\)
\(270\) 0 0
\(271\) −5.59767 −0.340034 −0.170017 0.985441i \(-0.554382\pi\)
−0.170017 + 0.985441i \(0.554382\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −13.6453 13.6453i −0.822842 0.822842i
\(276\) 0 0
\(277\) 2.41421 2.41421i 0.145056 0.145056i −0.630849 0.775905i \(-0.717293\pi\)
0.775905 + 0.630849i \(0.217293\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.7990i 1.06180i 0.847435 + 0.530899i \(0.178146\pi\)
−0.847435 + 0.530899i \(0.821854\pi\)
\(282\) 0 0
\(283\) 10.3212 10.3212i 0.613531 0.613531i −0.330333 0.943864i \(-0.607161\pi\)
0.943864 + 0.330333i \(0.107161\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.7821 −0.872558
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.414214 + 0.414214i −0.0241986 + 0.0241986i −0.719103 0.694904i \(-0.755447\pi\)
0.694904 + 0.719103i \(0.255447\pi\)
\(294\) 0 0
\(295\) 0.634051i 0.0369159i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.3128 16.3128i 0.943394 0.943394i
\(300\) 0 0
\(301\) −2.82843 2.82843i −0.163028 0.163028i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.627417 −0.0359258
\(306\) 0 0
\(307\) 3.19278 + 3.19278i 0.182222 + 0.182222i 0.792323 0.610101i \(-0.208871\pi\)
−0.610101 + 0.792323i \(0.708871\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.6005i 1.39497i −0.716600 0.697484i \(-0.754303\pi\)
0.716600 0.697484i \(-0.245697\pi\)
\(312\) 0 0
\(313\) 31.3137i 1.76996i 0.465633 + 0.884978i \(0.345827\pi\)
−0.465633 + 0.884978i \(0.654173\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.8995 + 10.8995i 0.612177 + 0.612177i 0.943513 0.331336i \(-0.107499\pi\)
−0.331336 + 0.943513i \(0.607499\pi\)
\(318\) 0 0
\(319\) −30.7235 −1.72018
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.6173 + 12.6173i 0.702045 + 0.702045i
\(324\) 0 0
\(325\) −11.2426 + 11.2426i −0.623629 + 0.623629i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.3137i 0.623745i
\(330\) 0 0
\(331\) 5.99162 5.99162i 0.329329 0.329329i −0.523002 0.852331i \(-0.675188\pi\)
0.852331 + 0.523002i \(0.175188\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.15932 −0.0633402
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.97056 8.97056i 0.485783 0.485783i
\(342\) 0 0
\(343\) 1.26810i 0.0684710i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10.9552 + 10.9552i −0.588108 + 0.588108i −0.937119 0.349011i \(-0.886518\pi\)
0.349011 + 0.937119i \(0.386518\pi\)
\(348\) 0 0
\(349\) −1.92893 1.92893i −0.103253 0.103253i 0.653593 0.756846i \(-0.273261\pi\)
−0.756846 + 0.653593i \(0.773261\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −22.9706 −1.22260 −0.611300 0.791399i \(-0.709353\pi\)
−0.611300 + 0.791399i \(0.709353\pi\)
\(354\) 0 0
\(355\) −3.32410 3.32410i −0.176425 0.176425i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.634051i 0.0334639i −0.999860 0.0167320i \(-0.994674\pi\)
0.999860 0.0167320i \(-0.00532620\pi\)
\(360\) 0 0
\(361\) 20.7990i 1.09468i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.68629 + 2.68629i 0.140607 + 0.140607i
\(366\) 0 0
\(367\) −9.18440 −0.479422 −0.239711 0.970844i \(-0.577053\pi\)
−0.239711 + 0.970844i \(0.577053\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.9832 + 11.9832i 0.622139 + 0.622139i
\(372\) 0 0
\(373\) 4.75736 4.75736i 0.246327 0.246327i −0.573135 0.819461i \(-0.694273\pi\)
0.819461 + 0.573135i \(0.194273\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.3137i 1.30372i
\(378\) 0 0
\(379\) −18.6089 + 18.6089i −0.955875 + 0.955875i −0.999067 0.0431915i \(-0.986247\pi\)
0.0431915 + 0.999067i \(0.486247\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −20.9050 −1.06820 −0.534098 0.845423i \(-0.679349\pi\)
−0.534098 + 0.845423i \(0.679349\pi\)
\(384\) 0 0
\(385\) 8.97056 0.457182
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.4142 + 14.4142i −0.730830 + 0.730830i −0.970784 0.239954i \(-0.922867\pi\)
0.239954 + 0.970784i \(0.422867\pi\)
\(390\) 0 0
\(391\) 19.1116i 0.966517i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.12293 6.12293i 0.308078 0.308078i
\(396\) 0 0
\(397\) −3.58579 3.58579i −0.179965 0.179965i 0.611375 0.791341i \(-0.290617\pi\)
−0.791341 + 0.611375i \(0.790617\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.51472 0.375267 0.187634 0.982239i \(-0.439918\pi\)
0.187634 + 0.982239i \(0.439918\pi\)
\(402\) 0 0
\(403\) −7.39104 7.39104i −0.368174 0.368174i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 37.5892i 1.86323i
\(408\) 0 0
\(409\) 31.3137i 1.54836i 0.632964 + 0.774182i \(0.281838\pi\)
−0.632964 + 0.774182i \(0.718162\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.82843 + 2.82843i 0.139178 + 0.139178i
\(414\) 0 0
\(415\) −8.02509 −0.393936
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8.79045 8.79045i −0.429442 0.429442i 0.458996 0.888438i \(-0.348209\pi\)
−0.888438 + 0.458996i \(0.848209\pi\)
\(420\) 0 0
\(421\) 6.07107 6.07107i 0.295886 0.295886i −0.543514 0.839400i \(-0.682907\pi\)
0.839400 + 0.543514i \(0.182907\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.1716i 0.638915i
\(426\) 0 0
\(427\) −2.79884 + 2.79884i −0.135445 + 0.135445i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −25.4558 −1.22333 −0.611665 0.791117i \(-0.709500\pi\)
−0.611665 + 0.791117i \(0.709500\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.1421 30.1421i 1.44189 1.44189i
\(438\) 0 0
\(439\) 38.1145i 1.81911i −0.415588 0.909553i \(-0.636424\pi\)
0.415588 0.909553i \(-0.363576\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.39942 1.39942i 0.0664883 0.0664883i −0.673081 0.739569i \(-0.735029\pi\)
0.739569 + 0.673081i \(0.235029\pi\)
\(444\) 0 0
\(445\) −2.00000 2.00000i −0.0948091 0.0948091i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.14214 0.289865 0.144933 0.989442i \(-0.453703\pi\)
0.144933 + 0.989442i \(0.453703\pi\)
\(450\) 0 0
\(451\) −11.7206 11.7206i −0.551902 0.551902i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.39104i 0.346497i
\(456\) 0 0
\(457\) 26.6274i 1.24558i −0.782390 0.622789i \(-0.785999\pi\)
0.782390 0.622789i \(-0.214001\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −27.0416 27.0416i −1.25945 1.25945i −0.951354 0.308100i \(-0.900307\pi\)
−0.308100 0.951354i \(-0.599693\pi\)
\(462\) 0 0
\(463\) 35.6871 1.65852 0.829260 0.558864i \(-0.188762\pi\)
0.829260 + 0.558864i \(0.188762\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.8966 + 26.8966i 1.24463 + 1.24463i 0.958060 + 0.286567i \(0.0925142\pi\)
0.286567 + 0.958060i \(0.407486\pi\)
\(468\) 0 0
\(469\) −5.17157 + 5.17157i −0.238801 + 0.238801i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.48528i 0.206233i
\(474\) 0 0
\(475\) −20.7737 + 20.7737i −0.953162 + 0.953162i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 35.1618 1.60658 0.803292 0.595585i \(-0.203080\pi\)
0.803292 + 0.595585i \(0.203080\pi\)
\(480\) 0 0
\(481\) 30.9706 1.41214
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.14214 + 2.14214i −0.0972694 + 0.0972694i
\(486\) 0 0
\(487\) 6.23172i 0.282386i 0.989982 + 0.141193i \(0.0450938\pi\)
−0.989982 + 0.141193i \(0.954906\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.56420 + 3.56420i −0.160850 + 0.160850i −0.782943 0.622093i \(-0.786283\pi\)
0.622093 + 0.782943i \(0.286283\pi\)
\(492\) 0 0
\(493\) 14.8284 + 14.8284i 0.667839 + 0.667839i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −29.6569 −1.33029
\(498\) 0 0
\(499\) 12.4860 + 12.4860i 0.558949 + 0.558949i 0.929008 0.370059i \(-0.120663\pi\)
−0.370059 + 0.929008i \(0.620663\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.8072i 1.01692i 0.861085 + 0.508460i \(0.169785\pi\)
−0.861085 + 0.508460i \(0.830215\pi\)
\(504\) 0 0
\(505\) 10.0000i 0.444994i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.5563 + 18.5563i 0.822496 + 0.822496i 0.986465 0.163970i \(-0.0524299\pi\)
−0.163970 + 0.986465i \(0.552430\pi\)
\(510\) 0 0
\(511\) 23.9665 1.06021
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.86030 + 5.86030i 0.258236 + 0.258236i
\(516\) 0 0
\(517\) 8.97056 8.97056i 0.394525 0.394525i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.34315i 0.277898i −0.990300 0.138949i \(-0.955628\pi\)
0.990300 0.138949i \(-0.0443725\pi\)
\(522\) 0 0
\(523\) −10.0586 + 10.0586i −0.439830 + 0.439830i −0.891955 0.452125i \(-0.850666\pi\)
0.452125 + 0.891955i \(0.350666\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.65914 −0.377198
\(528\) 0 0
\(529\) −22.6569 −0.985081
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −9.65685 + 9.65685i −0.418285 + 0.418285i
\(534\) 0 0
\(535\) 1.15932i 0.0501216i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 19.5056 19.5056i 0.840165 0.840165i
\(540\) 0 0
\(541\) −7.24264 7.24264i −0.311385 0.311385i 0.534061 0.845446i \(-0.320665\pi\)
−0.845446 + 0.534061i \(0.820665\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.65685 0.327984
\(546\) 0 0
\(547\) −13.3827 13.3827i −0.572201 0.572201i 0.360542 0.932743i \(-0.382592\pi\)
−0.932743 + 0.360542i \(0.882592\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 46.7736i 1.99262i
\(552\) 0 0
\(553\) 54.6274i 2.32299i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.07107 8.07107i −0.341982 0.341982i 0.515130 0.857112i \(-0.327744\pi\)
−0.857112 + 0.515130i \(0.827744\pi\)
\(558\) 0 0
\(559\) −3.69552 −0.156304
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.5838 10.5838i −0.446055 0.446055i 0.447986 0.894041i \(-0.352141\pi\)
−0.894041 + 0.447986i \(0.852141\pi\)
\(564\) 0 0
\(565\) 3.17157 3.17157i 0.133429 0.133429i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.6569i 1.07559i −0.843075 0.537796i \(-0.819257\pi\)
0.843075 0.537796i \(-0.180743\pi\)
\(570\) 0 0
\(571\) 19.5056 19.5056i 0.816284 0.816284i −0.169284 0.985567i \(-0.554145\pi\)
0.985567 + 0.169284i \(0.0541455\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 31.4663 1.31224
\(576\) 0 0
\(577\) 17.3137 0.720779 0.360390 0.932802i \(-0.382644\pi\)
0.360390 + 0.932802i \(0.382644\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −35.7990 + 35.7990i −1.48519 + 1.48519i
\(582\) 0 0
\(583\) 19.0029i 0.787018i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.1200 13.1200i 0.541521 0.541521i −0.382453 0.923975i \(-0.624921\pi\)
0.923975 + 0.382453i \(0.124921\pi\)
\(588\) 0 0
\(589\) −13.6569 13.6569i −0.562721 0.562721i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.31371 0.382468 0.191234 0.981544i \(-0.438751\pi\)
0.191234 + 0.981544i \(0.438751\pi\)
\(594\) 0 0
\(595\) −4.32957 4.32957i −0.177495 0.177495i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 45.5055i 1.85931i −0.368437 0.929653i \(-0.620107\pi\)
0.368437 0.929653i \(-0.379893\pi\)
\(600\) 0 0
\(601\) 20.8284i 0.849609i −0.905285 0.424805i \(-0.860343\pi\)
0.905285 0.424805i \(-0.139657\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.55635 + 2.55635i 0.103930 + 0.103930i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.39104 7.39104i −0.299009 0.299009i
\(612\) 0 0
\(613\) −27.8701 + 27.8701i −1.12566 + 1.12566i −0.134786 + 0.990875i \(0.543035\pi\)
−0.990875 + 0.134786i \(0.956965\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.4853i 1.54936i −0.632354 0.774680i \(-0.717911\pi\)
0.632354 0.774680i \(-0.282089\pi\)
\(618\) 0 0
\(619\) 2.93015 2.93015i 0.117773 0.117773i −0.645764 0.763537i \(-0.723461\pi\)
0.763537 + 0.645764i \(0.223461\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −17.8435 −0.714886
\(624\) 0 0
\(625\) −19.9706 −0.798823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18.1421 18.1421i 0.723374 0.723374i
\(630\) 0 0
\(631\) 14.1480i 0.563224i −0.959528 0.281612i \(-0.909131\pi\)
0.959528 0.281612i \(-0.0908691\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.85483 4.85483i 0.192658 0.192658i
\(636\) 0 0
\(637\) −16.0711 16.0711i −0.636759 0.636759i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 38.1421 1.50652 0.753262 0.657721i \(-0.228479\pi\)
0.753262 + 0.657721i \(0.228479\pi\)
\(642\) 0 0
\(643\) −13.3827 13.3827i −0.527760 0.527760i 0.392144 0.919904i \(-0.371734\pi\)
−0.919904 + 0.392144i \(0.871734\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.5391i 0.846788i −0.905946 0.423394i \(-0.860839\pi\)
0.905946 0.423394i \(-0.139161\pi\)
\(648\) 0 0
\(649\) 4.48528i 0.176063i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.55635 + 4.55635i 0.178304 + 0.178304i 0.790616 0.612312i \(-0.209760\pi\)
−0.612312 + 0.790616i \(0.709760\pi\)
\(654\) 0 0
\(655\) −8.55035 −0.334090
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.02800 + 1.02800i 0.0400452 + 0.0400452i 0.726846 0.686801i \(-0.240985\pi\)
−0.686801 + 0.726846i \(0.740985\pi\)
\(660\) 0 0
\(661\) −20.5563 + 20.5563i −0.799549 + 0.799549i −0.983024 0.183475i \(-0.941265\pi\)
0.183475 + 0.983024i \(0.441265\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 13.6569i 0.529590i
\(666\) 0 0
\(667\) 35.4244 35.4244i 1.37164 1.37164i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.43835 −0.171341
\(672\) 0 0
\(673\) −26.8284 −1.03416 −0.517080 0.855937i \(-0.672981\pi\)
−0.517080 + 0.855937i \(0.672981\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.58579 9.58579i 0.368412 0.368412i −0.498486 0.866898i \(-0.666110\pi\)
0.866898 + 0.498486i \(0.166110\pi\)
\(678\) 0 0
\(679\) 19.1116i 0.733437i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −23.2011 + 23.2011i −0.887766 + 0.887766i −0.994308 0.106542i \(-0.966022\pi\)
0.106542 + 0.994308i \(0.466022\pi\)
\(684\) 0 0
\(685\) 5.37258 + 5.37258i 0.205276 + 0.205276i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.6569 0.596479
\(690\) 0 0
\(691\) −28.7988 28.7988i −1.09556 1.09556i −0.994924 0.100634i \(-0.967913\pi\)
−0.100634 0.994924i \(-0.532087\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11.8294i 0.448714i
\(696\) 0 0
\(697\) 11.3137i 0.428537i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.5858 + 17.5858i 0.664206 + 0.664206i 0.956369 0.292163i \(-0.0943749\pi\)
−0.292163 + 0.956369i \(0.594375\pi\)
\(702\) 0 0
\(703\) 57.2261 2.15832
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −44.6088 44.6088i −1.67769 1.67769i
\(708\) 0 0
\(709\) 16.7574 16.7574i 0.629336 0.629336i −0.318565 0.947901i \(-0.603201\pi\)
0.947901 + 0.318565i \(0.103201\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.6863i 0.774708i
\(714\) 0 0
\(715\) 5.86030 5.86030i 0.219163 0.219163i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29.5641 1.10256 0.551278 0.834321i \(-0.314140\pi\)
0.551278 + 0.834321i \(0.314140\pi\)
\(720\) 0 0
\(721\) 52.2843 1.94717
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −24.4142 + 24.4142i −0.906721 + 0.906721i
\(726\) 0 0
\(727\) 22.0643i 0.818320i 0.912463 + 0.409160i \(0.134178\pi\)
−0.912463 + 0.409160i \(0.865822\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.16478 + 2.16478i −0.0800674 + 0.0800674i
\(732\) 0 0
\(733\) 13.1005 + 13.1005i 0.483878 + 0.483878i 0.906368 0.422490i \(-0.138844\pi\)
−0.422490 + 0.906368i \(0.638844\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.20101 −0.302088
\(738\) 0 0
\(739\) 35.1843 + 35.1843i 1.29428 + 1.29428i 0.932115 + 0.362162i \(0.117961\pi\)
0.362162 + 0.932115i \(0.382039\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.1367i 0.995550i −0.867306 0.497775i \(-0.834151\pi\)
0.867306 0.497775i \(-0.165849\pi\)
\(744\) 0 0
\(745\) 1.02944i 0.0377157i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.17157 + 5.17157i 0.188965 + 0.188965i
\(750\) 0 0
\(751\) 14.7821 0.539405 0.269703 0.962944i \(-0.413075\pi\)
0.269703 + 0.962944i \(0.413075\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −0.262632 0.262632i −0.00955817 0.00955817i
\(756\) 0 0
\(757\) 13.3848 13.3848i 0.486478 0.486478i −0.420715 0.907193i \(-0.638221\pi\)
0.907193 + 0.420715i \(0.138221\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.6274i 1.25524i 0.778519 + 0.627621i \(0.215971\pi\)
−0.778519 + 0.627621i \(0.784029\pi\)
\(762\) 0 0
\(763\) 34.1563 34.1563i 1.23654 1.23654i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.69552 0.133437
\(768\) 0 0
\(769\) 5.17157 0.186492 0.0932458 0.995643i \(-0.470276\pi\)
0.0932458 + 0.995643i \(0.470276\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −28.0711 + 28.0711i −1.00965 + 1.00965i −0.00969311 + 0.999953i \(0.503085\pi\)
−0.999953 + 0.00969311i \(0.996915\pi\)
\(774\) 0 0
\(775\) 14.2568i 0.512120i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17.8435 + 17.8435i −0.639311 + 0.639311i
\(780\) 0 0
\(781\) −23.5147 23.5147i −0.841423 0.841423i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.31371 0.0468883
\(786\) 0 0
\(787\) 5.62020 + 5.62020i 0.200339 + 0.200339i 0.800145 0.599807i \(-0.204756\pi\)
−0.599807 + 0.800145i \(0.704756\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 28.2960i 1.00609i
\(792\) 0 0
\(793\) 3.65685i 0.129859i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −28.0711 28.0711i −0.994328 0.994328i 0.00565577 0.999984i \(-0.498200\pi\)
−0.999984 + 0.00565577i \(0.998200\pi\)
\(798\) 0 0
\(799\) −8.65914 −0.306338
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.0029 + 19.0029i 0.670596 + 0.670596i
\(804\) 0 0
\(805\) −10.3431 + 10.3431i −0.364548