Properties

Label 4608.2.k.bd.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 \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 4608.1153
Dual form 4608.2.k.bd.3457.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.41421 + 2.41421i) q^{5} +1.53073i q^{7} +O(q^{10})\) \(q+(-2.41421 + 2.41421i) q^{5} +1.53073i q^{7} +(3.37849 - 3.37849i) q^{11} +(0.414214 + 0.414214i) q^{13} -2.82843 q^{17} +(0.317025 + 0.317025i) q^{19} +5.86030i q^{23} -6.65685i q^{25} +(3.24264 + 3.24264i) q^{29} +7.39104 q^{31} +(-3.69552 - 3.69552i) q^{35} +(-3.58579 + 3.58579i) q^{37} +4.00000i q^{41} +(-1.84776 + 1.84776i) q^{43} +7.39104 q^{47} +4.65685 q^{49} +(5.24264 - 5.24264i) q^{53} +16.3128i q^{55} +(1.84776 - 1.84776i) q^{59} +(-9.24264 - 9.24264i) q^{61} -2.00000 q^{65} +(-7.07401 - 7.07401i) q^{67} +11.9832i q^{71} +10.4853i q^{73} +(5.17157 + 5.17157i) q^{77} -6.12293 q^{79} +(2.48181 + 2.48181i) q^{83} +(6.82843 - 6.82843i) q^{85} -0.828427i q^{89} +(-0.634051 + 0.634051i) q^{91} -1.53073 q^{95} +10.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{5} + O(q^{10}) \) \( 8q - 8q^{5} - 8q^{13} - 8q^{29} - 40q^{37} - 8q^{49} + 8q^{53} - 40q^{61} - 16q^{65} + 64q^{77} + 32q^{85} + 64q^{97} + O(q^{100}) \)

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
\(5\) −2.41421 + 2.41421i −1.07967 + 1.07967i −0.0831305 + 0.996539i \(0.526492\pi\)
−0.996539 + 0.0831305i \(0.973508\pi\)
\(6\) 0 0
\(7\) 1.53073i 0.578563i 0.957244 + 0.289281i \(0.0934164\pi\)
−0.957244 + 0.289281i \(0.906584\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.37849 3.37849i 1.01865 1.01865i 0.0188312 0.999823i \(-0.494005\pi\)
0.999823 0.0188312i \(-0.00599452\pi\)
\(12\) 0 0
\(13\) 0.414214 + 0.414214i 0.114882 + 0.114882i 0.762211 0.647329i \(-0.224114\pi\)
−0.647329 + 0.762211i \(0.724114\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) 0.317025 + 0.317025i 0.0727306 + 0.0727306i 0.742536 0.669806i \(-0.233623\pi\)
−0.669806 + 0.742536i \(0.733623\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.86030i 1.22196i 0.791647 + 0.610979i \(0.209224\pi\)
−0.791647 + 0.610979i \(0.790776\pi\)
\(24\) 0 0
\(25\) 6.65685i 1.33137i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.24264 + 3.24264i 0.602143 + 0.602143i 0.940881 0.338738i \(-0.110000\pi\)
−0.338738 + 0.940881i \(0.610000\pi\)
\(30\) 0 0
\(31\) 7.39104 1.32747 0.663735 0.747968i \(-0.268970\pi\)
0.663735 + 0.747968i \(0.268970\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.69552 3.69552i −0.624657 0.624657i
\(36\) 0 0
\(37\) −3.58579 + 3.58579i −0.589500 + 0.589500i −0.937496 0.347996i \(-0.886862\pi\)
0.347996 + 0.937496i \(0.386862\pi\)
\(38\) 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
\(43\) −1.84776 + 1.84776i −0.281781 + 0.281781i −0.833819 0.552038i \(-0.813850\pi\)
0.552038 + 0.833819i \(0.313850\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.39104 1.07809 0.539047 0.842276i \(-0.318785\pi\)
0.539047 + 0.842276i \(0.318785\pi\)
\(48\) 0 0
\(49\) 4.65685 0.665265
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.24264 5.24264i 0.720132 0.720132i −0.248500 0.968632i \(-0.579938\pi\)
0.968632 + 0.248500i \(0.0799375\pi\)
\(54\) 0 0
\(55\) 16.3128i 2.19962i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.84776 1.84776i 0.240558 0.240558i −0.576523 0.817081i \(-0.695591\pi\)
0.817081 + 0.576523i \(0.195591\pi\)
\(60\) 0 0
\(61\) −9.24264 9.24264i −1.18340 1.18340i −0.978858 0.204541i \(-0.934430\pi\)
−0.204541 0.978858i \(-0.565570\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −7.07401 7.07401i −0.864228 0.864228i 0.127598 0.991826i \(-0.459273\pi\)
−0.991826 + 0.127598i \(0.959273\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.9832i 1.42215i 0.703117 + 0.711074i \(0.251791\pi\)
−0.703117 + 0.711074i \(0.748209\pi\)
\(72\) 0 0
\(73\) 10.4853i 1.22721i 0.789613 + 0.613605i \(0.210281\pi\)
−0.789613 + 0.613605i \(0.789719\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.17157 + 5.17157i 0.589355 + 0.589355i
\(78\) 0 0
\(79\) −6.12293 −0.688884 −0.344442 0.938808i \(-0.611932\pi\)
−0.344442 + 0.938808i \(0.611932\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.48181 + 2.48181i 0.272414 + 0.272414i 0.830071 0.557657i \(-0.188300\pi\)
−0.557657 + 0.830071i \(0.688300\pi\)
\(84\) 0 0
\(85\) 6.82843 6.82843i 0.740647 0.740647i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.828427i 0.0878131i −0.999036 0.0439065i \(-0.986020\pi\)
0.999036 0.0439065i \(-0.0139804\pi\)
\(90\) 0 0
\(91\) −0.634051 + 0.634051i −0.0664666 + 0.0664666i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.53073 −0.157050
\(96\) 0 0
\(97\) 10.8284 1.09946 0.549730 0.835342i \(-0.314731\pi\)
0.549730 + 0.835342i \(0.314731\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.07107 + 2.07107i −0.206079 + 0.206079i −0.802599 0.596520i \(-0.796550\pi\)
0.596520 + 0.802599i \(0.296550\pi\)
\(102\) 0 0
\(103\) 2.79884i 0.275777i 0.990448 + 0.137889i \(0.0440316\pi\)
−0.990448 + 0.137889i \(0.955968\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.07401 + 7.07401i −0.683870 + 0.683870i −0.960870 0.277000i \(-0.910660\pi\)
0.277000 + 0.960870i \(0.410660\pi\)
\(108\) 0 0
\(109\) 0.757359 + 0.757359i 0.0725419 + 0.0725419i 0.742447 0.669905i \(-0.233665\pi\)
−0.669905 + 0.742447i \(0.733665\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.65685 0.344008 0.172004 0.985096i \(-0.444976\pi\)
0.172004 + 0.985096i \(0.444976\pi\)
\(114\) 0 0
\(115\) −14.1480 14.1480i −1.31931 1.31931i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.32957i 0.396891i
\(120\) 0 0
\(121\) 11.8284i 1.07531i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.00000 + 4.00000i 0.357771 + 0.357771i
\(126\) 0 0
\(127\) −13.5140 −1.19917 −0.599586 0.800311i \(-0.704668\pi\)
−0.599586 + 0.800311i \(0.704668\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.43996 6.43996i −0.562662 0.562662i 0.367401 0.930063i \(-0.380248\pi\)
−0.930063 + 0.367401i \(0.880248\pi\)
\(132\) 0 0
\(133\) −0.485281 + 0.485281i −0.0420792 + 0.0420792i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.9706i 1.79164i 0.444421 + 0.895818i \(0.353409\pi\)
−0.444421 + 0.895818i \(0.646591\pi\)
\(138\) 0 0
\(139\) −13.5684 + 13.5684i −1.15085 + 1.15085i −0.164472 + 0.986382i \(0.552592\pi\)
−0.986382 + 0.164472i \(0.947408\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.79884 0.234050
\(144\) 0 0
\(145\) −15.6569 −1.30023
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.24264 7.24264i 0.593340 0.593340i −0.345192 0.938532i \(-0.612186\pi\)
0.938532 + 0.345192i \(0.112186\pi\)
\(150\) 0 0
\(151\) 8.92177i 0.726043i 0.931781 + 0.363022i \(0.118255\pi\)
−0.931781 + 0.363022i \(0.881745\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −17.8435 + 17.8435i −1.43323 + 1.43323i
\(156\) 0 0
\(157\) 4.41421 + 4.41421i 0.352293 + 0.352293i 0.860962 0.508669i \(-0.169862\pi\)
−0.508669 + 0.860962i \(0.669862\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −8.97056 −0.706979
\(162\) 0 0
\(163\) 9.23880 + 9.23880i 0.723638 + 0.723638i 0.969344 0.245706i \(-0.0790198\pi\)
−0.245706 + 0.969344i \(0.579020\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.53073i 0.118452i −0.998245 0.0592259i \(-0.981137\pi\)
0.998245 0.0592259i \(-0.0188632\pi\)
\(168\) 0 0
\(169\) 12.6569i 0.973604i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −16.4142 16.4142i −1.24795 1.24795i −0.956624 0.291326i \(-0.905904\pi\)
−0.291326 0.956624i \(-0.594096\pi\)
\(174\) 0 0
\(175\) 10.1899 0.770282
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.21371 + 1.21371i 0.0907168 + 0.0907168i 0.751009 0.660292i \(-0.229568\pi\)
−0.660292 + 0.751009i \(0.729568\pi\)
\(180\) 0 0
\(181\) 4.07107 4.07107i 0.302600 0.302600i −0.539430 0.842030i \(-0.681360\pi\)
0.842030 + 0.539430i \(0.181360\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.3137i 1.27293i
\(186\) 0 0
\(187\) −9.55582 + 9.55582i −0.698791 + 0.698791i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.5140 −0.977837 −0.488918 0.872330i \(-0.662608\pi\)
−0.488918 + 0.872330i \(0.662608\pi\)
\(192\) 0 0
\(193\) −16.4853 −1.18664 −0.593318 0.804968i \(-0.702182\pi\)
−0.593318 + 0.804968i \(0.702182\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.4142 + 10.4142i −0.741982 + 0.741982i −0.972959 0.230977i \(-0.925808\pi\)
0.230977 + 0.972959i \(0.425808\pi\)
\(198\) 0 0
\(199\) 22.4357i 1.59043i 0.606329 + 0.795214i \(0.292641\pi\)
−0.606329 + 0.795214i \(0.707359\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.96362 + 4.96362i −0.348378 + 0.348378i
\(204\) 0 0
\(205\) −9.65685 9.65685i −0.674464 0.674464i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.14214 0.148175
\(210\) 0 0
\(211\) −19.0572 19.0572i −1.31196 1.31196i −0.919973 0.391982i \(-0.871789\pi\)
−0.391982 0.919973i \(-0.628211\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.92177i 0.608460i
\(216\) 0 0
\(217\) 11.3137i 0.768025i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.17157 1.17157i −0.0788085 0.0788085i
\(222\) 0 0
\(223\) 7.39104 0.494940 0.247470 0.968896i \(-0.420401\pi\)
0.247470 + 0.968896i \(0.420401\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.84776 1.84776i −0.122640 0.122640i 0.643123 0.765763i \(-0.277638\pi\)
−0.765763 + 0.643123i \(0.777638\pi\)
\(228\) 0 0
\(229\) −11.2426 + 11.2426i −0.742935 + 0.742935i −0.973142 0.230207i \(-0.926060\pi\)
0.230207 + 0.973142i \(0.426060\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.1716i 0.993923i −0.867773 0.496961i \(-0.834449\pi\)
0.867773 0.496961i \(-0.165551\pi\)
\(234\) 0 0
\(235\) −17.8435 + 17.8435i −1.16398 + 1.16398i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −29.5641 −1.91235 −0.956173 0.292803i \(-0.905412\pi\)
−0.956173 + 0.292803i \(0.905412\pi\)
\(240\) 0 0
\(241\) 18.8284 1.21285 0.606423 0.795142i \(-0.292604\pi\)
0.606423 + 0.795142i \(0.292604\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −11.2426 + 11.2426i −0.718266 + 0.718266i
\(246\) 0 0
\(247\) 0.262632i 0.0167109i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.87285 + 9.87285i −0.623169 + 0.623169i −0.946340 0.323172i \(-0.895251\pi\)
0.323172 + 0.946340i \(0.395251\pi\)
\(252\) 0 0
\(253\) 19.7990 + 19.7990i 1.24475 + 1.24475i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.9706 −1.18335 −0.591676 0.806176i \(-0.701533\pi\)
−0.591676 + 0.806176i \(0.701533\pi\)
\(258\) 0 0
\(259\) −5.48888 5.48888i −0.341063 0.341063i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.6424i 1.27286i 0.771333 + 0.636432i \(0.219590\pi\)
−0.771333 + 0.636432i \(0.780410\pi\)
\(264\) 0 0
\(265\) 25.3137i 1.55501i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.92893 + 3.92893i 0.239551 + 0.239551i 0.816664 0.577113i \(-0.195821\pi\)
−0.577113 + 0.816664i \(0.695821\pi\)
\(270\) 0 0
\(271\) −28.2960 −1.71886 −0.859431 0.511252i \(-0.829182\pi\)
−0.859431 + 0.511252i \(0.829182\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −22.4901 22.4901i −1.35621 1.35621i
\(276\) 0 0
\(277\) 0.414214 0.414214i 0.0248877 0.0248877i −0.694553 0.719441i \(-0.744398\pi\)
0.719441 + 0.694553i \(0.244398\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 21.7990i 1.30042i −0.759755 0.650209i \(-0.774681\pi\)
0.759755 0.650209i \(-0.225319\pi\)
\(282\) 0 0
\(283\) −6.43996 + 6.43996i −0.382816 + 0.382816i −0.872116 0.489300i \(-0.837252\pi\)
0.489300 + 0.872116i \(0.337252\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.12293 −0.361425
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.41421 + 2.41421i −0.141040 + 0.141040i −0.774101 0.633062i \(-0.781798\pi\)
0.633062 + 0.774101i \(0.281798\pi\)
\(294\) 0 0
\(295\) 8.92177i 0.519446i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.42742 + 2.42742i −0.140381 + 0.140381i
\(300\) 0 0
\(301\) −2.82843 2.82843i −0.163028 0.163028i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 44.6274 2.55536
\(306\) 0 0
\(307\) 18.1606 + 18.1606i 1.03648 + 1.03648i 0.999309 + 0.0371692i \(0.0118341\pi\)
0.0371692 + 0.999309i \(0.488166\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.12840i 0.404215i 0.979363 + 0.202107i \(0.0647790\pi\)
−0.979363 + 0.202107i \(0.935221\pi\)
\(312\) 0 0
\(313\) 8.68629i 0.490978i 0.969399 + 0.245489i \(0.0789486\pi\)
−0.969399 + 0.245489i \(0.921051\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.89949 + 8.89949i 0.499845 + 0.499845i 0.911390 0.411544i \(-0.135010\pi\)
−0.411544 + 0.911390i \(0.635010\pi\)
\(318\) 0 0
\(319\) 21.9105 1.22675
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −0.896683 0.896683i −0.0498928 0.0498928i
\(324\) 0 0
\(325\) 2.75736 2.75736i 0.152951 0.152951i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.3137i 0.623745i
\(330\) 0 0
\(331\) 4.01254 4.01254i 0.220549 0.220549i −0.588180 0.808730i \(-0.700156\pi\)
0.808730 + 0.588180i \(0.200156\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 34.1563 1.86616
\(336\) 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
\(341\) 24.9706 24.9706i 1.35223 1.35223i
\(342\) 0 0
\(343\) 17.8435i 0.963461i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.3617 15.3617i 0.824661 0.824661i −0.162112 0.986772i \(-0.551830\pi\)
0.986772 + 0.162112i \(0.0518304\pi\)
\(348\) 0 0
\(349\) 16.0711 + 16.0711i 0.860265 + 0.860265i 0.991369 0.131104i \(-0.0418522\pi\)
−0.131104 + 0.991369i \(0.541852\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.9706 0.583904 0.291952 0.956433i \(-0.405695\pi\)
0.291952 + 0.956433i \(0.405695\pi\)
\(354\) 0 0
\(355\) −28.9301 28.9301i −1.53545 1.53545i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.92177i 0.470873i −0.971890 0.235437i \(-0.924348\pi\)
0.971890 0.235437i \(-0.0756520\pi\)
\(360\) 0 0
\(361\) 18.7990i 0.989421i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −25.3137 25.3137i −1.32498 1.32498i
\(366\) 0 0
\(367\) 22.1731 1.15743 0.578713 0.815531i \(-0.303555\pi\)
0.578713 + 0.815531i \(0.303555\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.02509 + 8.02509i 0.416642 + 0.416642i
\(372\) 0 0
\(373\) −13.2426 + 13.2426i −0.685678 + 0.685678i −0.961274 0.275596i \(-0.911125\pi\)
0.275596 + 0.961274i \(0.411125\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.68629i 0.138351i
\(378\) 0 0
\(379\) −3.11586 + 3.11586i −0.160051 + 0.160051i −0.782589 0.622538i \(-0.786101\pi\)
0.622538 + 0.782589i \(0.286101\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.65914 0.442461 0.221231 0.975222i \(-0.428993\pi\)
0.221231 + 0.975222i \(0.428993\pi\)
\(384\) 0 0
\(385\) −24.9706 −1.27262
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.5858 11.5858i 0.587423 0.587423i −0.349510 0.936933i \(-0.613652\pi\)
0.936933 + 0.349510i \(0.113652\pi\)
\(390\) 0 0
\(391\) 16.5754i 0.838256i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.7821 14.7821i 0.743767 0.743767i
\(396\) 0 0
\(397\) 6.41421 + 6.41421i 0.321920 + 0.321920i 0.849503 0.527583i \(-0.176902\pi\)
−0.527583 + 0.849503i \(0.676902\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.4853 1.22274 0.611368 0.791346i \(-0.290619\pi\)
0.611368 + 0.791346i \(0.290619\pi\)
\(402\) 0 0
\(403\) 3.06147 + 3.06147i 0.152503 + 0.152503i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.2291i 1.20099i
\(408\) 0 0
\(409\) 8.68629i 0.429509i 0.976668 + 0.214755i \(0.0688952\pi\)
−0.976668 + 0.214755i \(0.931105\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.82843 + 2.82843i 0.139178 + 0.139178i
\(414\) 0 0
\(415\) −11.9832 −0.588234
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.1355 + 10.1355i 0.495151 + 0.495151i 0.909924 0.414774i \(-0.136139\pi\)
−0.414774 + 0.909924i \(0.636139\pi\)
\(420\) 0 0
\(421\) 8.07107 8.07107i 0.393360 0.393360i −0.482523 0.875883i \(-0.660280\pi\)
0.875883 + 0.482523i \(0.160280\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.8284i 0.913313i
\(426\) 0 0
\(427\) 14.1480 14.1480i 0.684671 0.684671i
\(428\) 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 \(-0.290500\pi\)
0.611665 + 0.791117i \(0.290500\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.85786 + 1.85786i −0.0888737 + 0.0888737i
\(438\) 0 0
\(439\) 18.8490i 0.899614i 0.893126 + 0.449807i \(0.148507\pi\)
−0.893126 + 0.449807i \(0.851493\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.07401 + 7.07401i −0.336096 + 0.336096i −0.854896 0.518800i \(-0.826379\pi\)
0.518800 + 0.854896i \(0.326379\pi\)
\(444\) 0 0
\(445\) 2.00000 + 2.00000i 0.0948091 + 0.0948091i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −22.1421 −1.04495 −0.522476 0.852654i \(-0.674992\pi\)
−0.522476 + 0.852654i \(0.674992\pi\)
\(450\) 0 0
\(451\) 13.5140 + 13.5140i 0.636348 + 0.636348i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.06147i 0.143524i
\(456\) 0 0
\(457\) 18.6274i 0.871354i 0.900103 + 0.435677i \(0.143491\pi\)
−0.900103 + 0.435677i \(0.856509\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −21.0416 21.0416i −0.980006 0.980006i 0.0197976 0.999804i \(-0.493698\pi\)
−0.999804 + 0.0197976i \(0.993698\pi\)
\(462\) 0 0
\(463\) −2.53620 −0.117867 −0.0589337 0.998262i \(-0.518770\pi\)
−0.0589337 + 0.998262i \(0.518770\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.6717 + 12.6717i 0.586375 + 0.586375i 0.936648 0.350272i \(-0.113911\pi\)
−0.350272 + 0.936648i \(0.613911\pi\)
\(468\) 0 0
\(469\) 10.8284 10.8284i 0.500010 0.500010i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.4853i 0.574074i
\(474\) 0 0
\(475\) 2.11039 2.11039i 0.0968314 0.0968314i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 40.5419 1.85241 0.926204 0.377024i \(-0.123052\pi\)
0.926204 + 0.377024i \(0.123052\pi\)
\(480\) 0 0
\(481\) −2.97056 −0.135446
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −26.1421 + 26.1421i −1.18705 + 1.18705i
\(486\) 0 0
\(487\) 37.2178i 1.68650i 0.537521 + 0.843250i \(0.319361\pi\)
−0.537521 + 0.843250i \(0.680639\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.3003 12.3003i 0.555103 0.555103i −0.372806 0.927909i \(-0.621604\pi\)
0.927909 + 0.372806i \(0.121604\pi\)
\(492\) 0 0
\(493\) −9.17157 9.17157i −0.413067 0.413067i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −18.3431 −0.822803
\(498\) 0 0
\(499\) −11.6662 11.6662i −0.522251 0.522251i 0.395999 0.918251i \(-0.370398\pi\)
−0.918251 + 0.395999i \(0.870398\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.1062i 0.807314i 0.914910 + 0.403657i \(0.132261\pi\)
−0.914910 + 0.403657i \(0.867739\pi\)
\(504\) 0 0
\(505\) 10.0000i 0.444994i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.5563 + 12.5563i 0.556550 + 0.556550i 0.928324 0.371773i \(-0.121250\pi\)
−0.371773 + 0.928324i \(0.621250\pi\)
\(510\) 0 0
\(511\) −16.0502 −0.710018
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.75699 6.75699i −0.297748 0.297748i
\(516\) 0 0
\(517\) 24.9706 24.9706i 1.09820 1.09820i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.6569i 0.773561i −0.922172 0.386780i \(-0.873587\pi\)
0.922172 0.386780i \(-0.126413\pi\)
\(522\) 0 0
\(523\) 27.9790 27.9790i 1.22344 1.22344i 0.257035 0.966402i \(-0.417254\pi\)
0.966402 0.257035i \(-0.0827455\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.9050 −0.910636
\(528\) 0 0
\(529\) −11.3431 −0.493180
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.65685 + 1.65685i −0.0717663 + 0.0717663i
\(534\) 0 0
\(535\) 34.1563i 1.47671i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 15.7331 15.7331i 0.677675 0.677675i
\(540\) 0 0
\(541\) −1.24264 1.24264i −0.0534253 0.0534253i 0.679889 0.733315i \(-0.262028\pi\)
−0.733315 + 0.679889i \(0.762028\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.65685 −0.156642
\(546\) 0 0
\(547\) −0.951076 0.951076i −0.0406651 0.0406651i 0.686482 0.727147i \(-0.259154\pi\)
−0.727147 + 0.686482i \(0.759154\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.05600i 0.0875885i
\(552\) 0 0
\(553\) 9.37258i 0.398563i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.07107 6.07107i −0.257239 0.257239i 0.566691 0.823930i \(-0.308223\pi\)
−0.823930 + 0.566691i \(0.808223\pi\)
\(558\) 0 0
\(559\) −1.53073 −0.0647431
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.0991 15.0991i −0.636351 0.636351i 0.313302 0.949653i \(-0.398565\pi\)
−0.949653 + 0.313302i \(0.898565\pi\)
\(564\) 0 0
\(565\) −8.82843 + 8.82843i −0.371415 + 0.371415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14.3431i 0.601296i −0.953735 0.300648i \(-0.902797\pi\)
0.953735 0.300648i \(-0.0972029\pi\)
\(570\) 0 0
\(571\) 15.7331 15.7331i 0.658412 0.658412i −0.296592 0.955004i \(-0.595850\pi\)
0.955004 + 0.296592i \(0.0958503\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 39.0112 1.62688
\(576\) 0 0
\(577\) −5.31371 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.79899 + 3.79899i −0.157609 + 0.157609i
\(582\) 0 0
\(583\) 35.4244i 1.46713i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.5880 + 20.5880i −0.849757 + 0.849757i −0.990103 0.140346i \(-0.955179\pi\)
0.140346 + 0.990103i \(0.455179\pi\)
\(588\) 0 0
\(589\) 2.34315 + 2.34315i 0.0965476 + 0.0965476i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −13.3137 −0.546728 −0.273364 0.961911i \(-0.588136\pi\)
−0.273364 + 0.961911i \(0.588136\pi\)
\(594\) 0 0
\(595\) 10.4525 + 10.4525i 0.428511 + 0.428511i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.7875i 0.645061i 0.946559 + 0.322531i \(0.104534\pi\)
−0.946559 + 0.322531i \(0.895466\pi\)
\(600\) 0 0
\(601\) 15.1716i 0.618861i −0.950922 0.309431i \(-0.899862\pi\)
0.950922 0.309431i \(-0.100138\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 28.5563 + 28.5563i 1.16098 + 1.16098i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.06147 + 3.06147i 0.123854 + 0.123854i
\(612\) 0 0
\(613\) −25.8701 + 25.8701i −1.04488 + 1.04488i −0.0459375 + 0.998944i \(0.514627\pi\)
−0.998944 + 0.0459375i \(0.985373\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.5147i 0.866150i −0.901358 0.433075i \(-0.857429\pi\)
0.901358 0.433075i \(-0.142571\pi\)
\(618\) 0 0
\(619\) −3.37849 + 3.37849i −0.135793 + 0.135793i −0.771736 0.635943i \(-0.780611\pi\)
0.635943 + 0.771736i \(0.280611\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.26810 0.0508054
\(624\) 0 0
\(625\) 13.9706 0.558823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 10.1421 10.1421i 0.404393 0.404393i
\(630\) 0 0
\(631\) 2.79884i 0.111420i 0.998447 + 0.0557099i \(0.0177422\pi\)
−0.998447 + 0.0557099i \(0.982258\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 32.6256 32.6256i 1.29471 1.29471i
\(636\) 0 0
\(637\) 1.92893 + 1.92893i 0.0764271 + 0.0764271i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.85786 0.389362 0.194681 0.980867i \(-0.437633\pi\)
0.194681 + 0.980867i \(0.437633\pi\)
\(642\) 0 0
\(643\) −0.951076 0.951076i −0.0375068 0.0375068i 0.688105 0.725611i \(-0.258443\pi\)
−0.725611 + 0.688105i \(0.758443\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.262632i 0.0103251i −0.999987 0.00516257i \(-0.998357\pi\)
0.999987 0.00516257i \(-0.00164331\pi\)
\(648\) 0 0
\(649\) 12.4853i 0.490090i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.5563 + 26.5563i 1.03923 + 1.03923i 0.999198 + 0.0400318i \(0.0127459\pi\)
0.0400318 + 0.999198i \(0.487254\pi\)
\(654\) 0 0
\(655\) 31.0949 1.21498
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 23.3868 + 23.3868i 0.911021 + 0.911021i 0.996353 0.0853316i \(-0.0271950\pi\)
−0.0853316 + 0.996353i \(0.527195\pi\)
\(660\) 0 0
\(661\) −10.5563 + 10.5563i −0.410594 + 0.410594i −0.881946 0.471351i \(-0.843766\pi\)
0.471351 + 0.881946i \(0.343766\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.34315i 0.0908633i
\(666\) 0 0
\(667\) −19.0029 + 19.0029i −0.735794 + 0.735794i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −62.4524 −2.41095
\(672\) 0 0
\(673\) −21.1716 −0.816104 −0.408052 0.912959i \(-0.633792\pi\)
−0.408052 + 0.912959i \(0.633792\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.4142 + 12.4142i −0.477117 + 0.477117i −0.904208 0.427091i \(-0.859538\pi\)
0.427091 + 0.904208i \(0.359538\pi\)
\(678\) 0 0
\(679\) 16.5754i 0.636107i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −14.2024 + 14.2024i −0.543440 + 0.543440i −0.924536 0.381095i \(-0.875547\pi\)
0.381095 + 0.924536i \(0.375547\pi\)
\(684\) 0 0
\(685\) −50.6274 50.6274i −1.93437 1.93437i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.34315 0.165461
\(690\) 0 0
\(691\) 14.0936 + 14.0936i 0.536147 + 0.536147i 0.922395 0.386248i \(-0.126229\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 65.5139i 2.48508i
\(696\) 0 0
\(697\) 11.3137i 0.428537i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −20.4142 20.4142i −0.771034 0.771034i 0.207253 0.978287i \(-0.433548\pi\)
−0.978287 + 0.207253i \(0.933548\pi\)
\(702\) 0 0
\(703\) −2.27357 −0.0857493
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.17025 3.17025i −0.119230 0.119230i
\(708\) 0 0
\(709\) −25.2426 + 25.2426i −0.948007 + 0.948007i −0.998714 0.0507063i \(-0.983853\pi\)
0.0507063 + 0.998714i \(0.483853\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 43.3137i 1.62211i
\(714\) 0 0
\(715\) −6.75699 + 6.75699i −0.252697 + 0.252697i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12.2459 0.456694 0.228347 0.973580i \(-0.426668\pi\)
0.228347 + 0.973580i \(0.426668\pi\)
\(720\) 0 0
\(721\) −4.28427 −0.159555
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21.5858 21.5858i 0.801676 0.801676i
\(726\) 0 0
\(727\) 42.8155i 1.58794i −0.607958 0.793969i \(-0.708011\pi\)
0.607958 0.793969i \(-0.291989\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.22625 5.22625i 0.193300 0.193300i
\(732\) 0 0
\(733\) −32.8995 32.8995i −1.21517 1.21517i −0.969304 0.245867i \(-0.920927\pi\)
−0.245867 0.969304i \(-0.579073\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −47.7990 −1.76070
\(738\) 0 0
\(739\) 22.2275 + 22.2275i 0.817652 + 0.817652i 0.985767 0.168115i \(-0.0537681\pi\)
−0.168115 + 0.985767i \(0.553768\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.5587i 1.04772i −0.851806 0.523858i \(-0.824492\pi\)
0.851806 0.523858i \(-0.175508\pi\)
\(744\) 0 0
\(745\) 34.9706i 1.28122i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10.8284 10.8284i −0.395662 0.395662i
\(750\) 0 0
\(751\) 6.12293 0.223429 0.111715 0.993740i \(-0.464366\pi\)
0.111715 + 0.993740i \(0.464366\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −21.5391 21.5391i −0.783887 0.783887i
\(756\) 0 0
\(757\) 23.3848 23.3848i 0.849934 0.849934i −0.140190 0.990125i \(-0.544771\pi\)
0.990125 + 0.140190i \(0.0447715\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10.6274i 0.385244i −0.981273 0.192622i \(-0.938301\pi\)
0.981273 0.192622i \(-0.0616990\pi\)
\(762\) 0 0
\(763\) −1.15932 + 1.15932i −0.0419700 + 0.0419700i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.53073 0.0552716
\(768\) 0 0
\(769\) 10.8284 0.390483 0.195242 0.980755i \(-0.437451\pi\)
0.195242 + 0.980755i \(0.437451\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 13.9289 13.9289i 0.500989 0.500989i −0.410756 0.911745i \(-0.634735\pi\)
0.911745 + 0.410756i \(0.134735\pi\)
\(774\) 0 0
\(775\) 49.2011i 1.76735i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.26810 + 1.26810i −0.0454344 + 0.0454344i
\(780\) 0 0
\(781\) 40.4853 + 40.4853i 1.44868 + 1.44868i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −21.3137 −0.760719
\(786\) 0 0
\(787\) 34.4734 + 34.4734i 1.22884 + 1.22884i 0.964402 + 0.264441i \(0.0851875\pi\)
0.264441 + 0.964402i \(0.414813\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.59767i 0.199030i
\(792\) 0 0
\(793\) 7.65685i 0.271903i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.9289 + 13.9289i 0.493388 + 0.493388i 0.909372 0.415984i \(-0.136563\pi\)
−0.415984 + 0.909372i \(0.636563\pi\)
\(798\) 0 0
\(799\) −20.9050 −0.739566
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 35.4244 + 35.4244i 1.25010 + 1.25010i
\(804\) 0 0
\(805\) 21.6569 21.6569i