Properties

Label 4608.2.k.y.3457.2
Level $4608$
Weight $2$
Character 4608.3457
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 3457.2
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4608.3457
Dual form 4608.2.k.y.1153.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{5} +2.82843i q^{7} +O(q^{10})\) \(q+(-1.00000 - 1.00000i) q^{5} +2.82843i q^{7} +(-3.00000 + 3.00000i) q^{13} +4.00000 q^{17} +(-5.65685 + 5.65685i) q^{19} -5.65685i q^{23} -3.00000i q^{25} +(-1.00000 + 1.00000i) q^{29} -2.82843 q^{31} +(2.82843 - 2.82843i) q^{35} +(-3.00000 - 3.00000i) q^{37} -4.00000i q^{41} -11.3137 q^{47} -1.00000 q^{49} +(5.00000 + 5.00000i) q^{53} +(8.48528 + 8.48528i) q^{59} +(1.00000 - 1.00000i) q^{61} +6.00000 q^{65} +(-2.82843 + 2.82843i) q^{67} -11.3137i q^{71} -14.0000i q^{73} +8.48528 q^{79} +(11.3137 - 11.3137i) q^{83} +(-4.00000 - 4.00000i) q^{85} -6.00000i q^{89} +(-8.48528 - 8.48528i) q^{91} +11.3137 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + O(q^{10}) \) \( 4q - 4q^{5} - 12q^{13} + 16q^{17} - 4q^{29} - 12q^{37} - 4q^{49} + 20q^{53} + 4q^{61} + 24q^{65} - 16q^{85} + 64q^{97} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 1.00000i −0.447214 0.447214i 0.447214 0.894427i \(-0.352416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 0 0
\(13\) −3.00000 + 3.00000i −0.832050 + 0.832050i −0.987797 0.155747i \(-0.950222\pi\)
0.155747 + 0.987797i \(0.450222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −5.65685 + 5.65685i −1.29777 + 1.29777i −0.367910 + 0.929861i \(0.619927\pi\)
−0.929861 + 0.367910i \(0.880073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.65685i 1.17954i −0.807573 0.589768i \(-0.799219\pi\)
0.807573 0.589768i \(-0.200781\pi\)
\(24\) 0 0
\(25\) 3.00000i 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 + 1.00000i −0.185695 + 0.185695i −0.793832 0.608137i \(-0.791917\pi\)
0.608137 + 0.793832i \(0.291917\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843 2.82843i 0.478091 0.478091i
\(36\) 0 0
\(37\) −3.00000 3.00000i −0.493197 0.493197i 0.416115 0.909312i \(-0.363391\pi\)
−0.909312 + 0.416115i \(0.863391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000i 0.624695i −0.949968 0.312348i \(-0.898885\pi\)
0.949968 0.312348i \(-0.101115\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.3137 −1.65027 −0.825137 0.564933i \(-0.808902\pi\)
−0.825137 + 0.564933i \(0.808902\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.00000 + 5.00000i 0.686803 + 0.686803i 0.961524 0.274721i \(-0.0885855\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.48528 + 8.48528i 1.10469 + 1.10469i 0.993837 + 0.110853i \(0.0353582\pi\)
0.110853 + 0.993837i \(0.464642\pi\)
\(60\) 0 0
\(61\) 1.00000 1.00000i 0.128037 0.128037i −0.640184 0.768221i \(-0.721142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −2.82843 + 2.82843i −0.345547 + 0.345547i −0.858448 0.512901i \(-0.828571\pi\)
0.512901 + 0.858448i \(0.328571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3137i 1.34269i −0.741145 0.671345i \(-0.765717\pi\)
0.741145 0.671345i \(-0.234283\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.3137 11.3137i 1.24184 1.24184i 0.282604 0.959237i \(-0.408802\pi\)
0.959237 0.282604i \(-0.0911983\pi\)
\(84\) 0 0
\(85\) −4.00000 4.00000i −0.433861 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000i 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) −8.48528 8.48528i −0.889499 0.889499i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.3137 1.16076
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.0000 11.0000i −1.09454 1.09454i −0.995037 0.0995037i \(-0.968274\pi\)
−0.0995037 0.995037i \(-0.531726\pi\)
\(102\) 0 0
\(103\) 19.7990i 1.95085i 0.220326 + 0.975426i \(0.429288\pi\)
−0.220326 + 0.975426i \(0.570712\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.48528 8.48528i −0.820303 0.820303i 0.165848 0.986151i \(-0.446964\pi\)
−0.986151 + 0.165848i \(0.946964\pi\)
\(108\) 0 0
\(109\) 3.00000 3.00000i 0.287348 0.287348i −0.548683 0.836031i \(-0.684871\pi\)
0.836031 + 0.548683i \(0.184871\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −5.65685 + 5.65685i −0.527504 + 0.527504i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 11.3137i 1.03713i
\(120\) 0 0
\(121\) 11.0000i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.00000 + 8.00000i −0.715542 + 0.715542i
\(126\) 0 0
\(127\) 8.48528 0.752947 0.376473 0.926427i \(-0.377137\pi\)
0.376473 + 0.926427i \(0.377137\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.82843 + 2.82843i −0.247121 + 0.247121i −0.819788 0.572667i \(-0.805909\pi\)
0.572667 + 0.819788i \(0.305909\pi\)
\(132\) 0 0
\(133\) −16.0000 16.0000i −1.38738 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000i 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) 2.82843 + 2.82843i 0.239904 + 0.239904i 0.816810 0.576906i \(-0.195740\pi\)
−0.576906 + 0.816810i \(0.695740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 9.00000i −0.737309 0.737309i 0.234748 0.972056i \(-0.424574\pi\)
−0.972056 + 0.234748i \(0.924574\pi\)
\(150\) 0 0
\(151\) 2.82843i 0.230174i 0.993355 + 0.115087i \(0.0367147\pi\)
−0.993355 + 0.115087i \(0.963285\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.82843 + 2.82843i 0.227185 + 0.227185i
\(156\) 0 0
\(157\) −15.0000 + 15.0000i −1.19713 + 1.19713i −0.222108 + 0.975022i \(0.571294\pi\)
−0.975022 + 0.222108i \(0.928706\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) 11.3137 11.3137i 0.886158 0.886158i −0.107994 0.994152i \(-0.534443\pi\)
0.994152 + 0.107994i \(0.0344426\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.3137i 0.875481i −0.899101 0.437741i \(-0.855779\pi\)
0.899101 0.437741i \(-0.144221\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.00000 5.00000i 0.380143 0.380143i −0.491011 0.871154i \(-0.663372\pi\)
0.871154 + 0.491011i \(0.163372\pi\)
\(174\) 0 0
\(175\) 8.48528 0.641427
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.82843 + 2.82843i −0.211407 + 0.211407i −0.804865 0.593458i \(-0.797762\pi\)
0.593458 + 0.804865i \(0.297762\pi\)
\(180\) 0 0
\(181\) −9.00000 9.00000i −0.668965 0.668965i 0.288512 0.957476i \(-0.406840\pi\)
−0.957476 + 0.288512i \(0.906840\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000i 0.441129i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.00000 5.00000i −0.356235 0.356235i 0.506188 0.862423i \(-0.331054\pi\)
−0.862423 + 0.506188i \(0.831054\pi\)
\(198\) 0 0
\(199\) 8.48528i 0.601506i −0.953702 0.300753i \(-0.902762\pi\)
0.953702 0.300753i \(-0.0972379\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.82843 2.82843i −0.198517 0.198517i
\(204\) 0 0
\(205\) −4.00000 + 4.00000i −0.279372 + 0.279372i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.48528 8.48528i 0.584151 0.584151i −0.351890 0.936041i \(-0.614461\pi\)
0.936041 + 0.351890i \(0.114461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000i 0.543075i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 + 12.0000i −0.807207 + 0.807207i
\(222\) 0 0
\(223\) −25.4558 −1.70465 −0.852325 0.523013i \(-0.824808\pi\)
−0.852325 + 0.523013i \(0.824808\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.65685 + 5.65685i −0.375459 + 0.375459i −0.869461 0.494002i \(-0.835534\pi\)
0.494002 + 0.869461i \(0.335534\pi\)
\(228\) 0 0
\(229\) −17.0000 17.0000i −1.12339 1.12339i −0.991228 0.132164i \(-0.957808\pi\)
−0.132164 0.991228i \(-0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0000i 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) 0 0
\(235\) 11.3137 + 11.3137i 0.738025 + 0.738025i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.65685 0.365911 0.182956 0.983121i \(-0.441433\pi\)
0.182956 + 0.983121i \(0.441433\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 + 1.00000i 0.0638877 + 0.0638877i
\(246\) 0 0
\(247\) 33.9411i 2.15962i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.65685 5.65685i −0.357057 0.357057i 0.505670 0.862727i \(-0.331245\pi\)
−0.862727 + 0.505670i \(0.831245\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 8.48528 8.48528i 0.527250 0.527250i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.2843i 1.74408i 0.489432 + 0.872041i \(0.337204\pi\)
−0.489432 + 0.872041i \(0.662796\pi\)
\(264\) 0 0
\(265\) 10.0000i 0.614295i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.0000 15.0000i 0.914566 0.914566i −0.0820612 0.996627i \(-0.526150\pi\)
0.996627 + 0.0820612i \(0.0261503\pi\)
\(270\) 0 0
\(271\) −8.48528 −0.515444 −0.257722 0.966219i \(-0.582972\pi\)
−0.257722 + 0.966219i \(0.582972\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.00000 + 1.00000i 0.0600842 + 0.0600842i 0.736510 0.676426i \(-0.236472\pi\)
−0.676426 + 0.736510i \(0.736472\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.0000i 1.31241i 0.754583 + 0.656205i \(0.227839\pi\)
−0.754583 + 0.656205i \(0.772161\pi\)
\(282\) 0 0
\(283\) −2.82843 2.82843i −0.168133 0.168133i 0.618026 0.786158i \(-0.287933\pi\)
−0.786158 + 0.618026i \(0.787933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.3137 0.667827
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −13.0000 13.0000i −0.759468 0.759468i 0.216757 0.976226i \(-0.430452\pi\)
−0.976226 + 0.216757i \(0.930452\pi\)
\(294\) 0 0
\(295\) 16.9706i 0.988064i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.9706 + 16.9706i 0.981433 + 0.981433i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −2.82843 + 2.82843i −0.161427 + 0.161427i −0.783199 0.621772i \(-0.786413\pi\)
0.621772 + 0.783199i \(0.286413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.2843i 1.60385i 0.597422 + 0.801927i \(0.296192\pi\)
−0.597422 + 0.801927i \(0.703808\pi\)
\(312\) 0 0
\(313\) 16.0000i 0.904373i −0.891923 0.452187i \(-0.850644\pi\)
0.891923 0.452187i \(-0.149356\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.0000 15.0000i 0.842484 0.842484i −0.146697 0.989181i \(-0.546864\pi\)
0.989181 + 0.146697i \(0.0468644\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −22.6274 + 22.6274i −1.25902 + 1.25902i
\(324\) 0 0
\(325\) 9.00000 + 9.00000i 0.499230 + 0.499230i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 32.0000i 1.76422i
\(330\) 0 0
\(331\) −2.82843 2.82843i −0.155464 0.155464i 0.625089 0.780553i \(-0.285063\pi\)
−0.780553 + 0.625089i \(0.785063\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.65685 0.309067
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −22.6274 22.6274i −1.21470 1.21470i −0.969462 0.245241i \(-0.921133\pi\)
−0.245241 0.969462i \(-0.578867\pi\)
\(348\) 0 0
\(349\) −9.00000 + 9.00000i −0.481759 + 0.481759i −0.905693 0.423934i \(-0.860649\pi\)
0.423934 + 0.905693i \(0.360649\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −11.3137 + 11.3137i −0.600469 + 0.600469i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.3137i 0.597115i −0.954392 0.298557i \(-0.903495\pi\)
0.954392 0.298557i \(-0.0965054\pi\)
\(360\) 0 0
\(361\) 45.0000i 2.36842i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −14.0000 + 14.0000i −0.732793 + 0.732793i
\(366\) 0 0
\(367\) 2.82843 0.147643 0.0738213 0.997271i \(-0.476481\pi\)
0.0738213 + 0.997271i \(0.476481\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −14.1421 + 14.1421i −0.734223 + 0.734223i
\(372\) 0 0
\(373\) −3.00000 3.00000i −0.155334 0.155334i 0.625161 0.780496i \(-0.285033\pi\)
−0.780496 + 0.625161i \(0.785033\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000i 0.309016i
\(378\) 0 0
\(379\) 11.3137 + 11.3137i 0.581146 + 0.581146i 0.935218 0.354072i \(-0.115203\pi\)
−0.354072 + 0.935218i \(0.615203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −28.2843 −1.44526 −0.722629 0.691236i \(-0.757067\pi\)
−0.722629 + 0.691236i \(0.757067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.00000 7.00000i −0.354914 0.354914i 0.507020 0.861934i \(-0.330747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) 22.6274i 1.14432i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.48528 8.48528i −0.426941 0.426941i
\(396\) 0 0
\(397\) 15.0000 15.0000i 0.752828 0.752828i −0.222178 0.975006i \(-0.571317\pi\)
0.975006 + 0.222178i \(0.0713165\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −20.0000 −0.998752 −0.499376 0.866385i \(-0.666437\pi\)
−0.499376 + 0.866385i \(0.666437\pi\)
\(402\) 0 0
\(403\) 8.48528 8.48528i 0.422682 0.422682i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 32.0000i 1.58230i 0.611623 + 0.791149i \(0.290517\pi\)
−0.611623 + 0.791149i \(0.709483\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −24.0000 + 24.0000i −1.18096 + 1.18096i
\(414\) 0 0
\(415\) −22.6274 −1.11074
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.65685 + 5.65685i −0.276355 + 0.276355i −0.831652 0.555297i \(-0.812605\pi\)
0.555297 + 0.831652i \(0.312605\pi\)
\(420\) 0 0
\(421\) −9.00000 9.00000i −0.438633 0.438633i 0.452919 0.891552i \(-0.350383\pi\)
−0.891552 + 0.452919i \(0.850383\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.0000i 0.582086i
\(426\) 0 0
\(427\) 2.82843 + 2.82843i 0.136877 + 0.136877i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.3137 0.544962 0.272481 0.962161i \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(432\) 0 0
\(433\) −8.00000 −0.384455 −0.192228 0.981350i \(-0.561571\pi\)
−0.192228 + 0.981350i \(0.561571\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 32.0000 + 32.0000i 1.53077 + 1.53077i
\(438\) 0 0
\(439\) 2.82843i 0.134993i −0.997719 0.0674967i \(-0.978499\pi\)
0.997719 0.0674967i \(-0.0215012\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.9706 + 16.9706i 0.806296 + 0.806296i 0.984071 0.177775i \(-0.0568900\pi\)
−0.177775 + 0.984071i \(0.556890\pi\)
\(444\) 0 0
\(445\) −6.00000 + 6.00000i −0.284427 + 0.284427i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 16.9706i 0.795592i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.0000 11.0000i 0.512321 0.512321i −0.402916 0.915237i \(-0.632003\pi\)
0.915237 + 0.402916i \(0.132003\pi\)
\(462\) 0 0
\(463\) −31.1127 −1.44593 −0.722965 0.690885i \(-0.757221\pi\)
−0.722965 + 0.690885i \(0.757221\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.65685 + 5.65685i −0.261768 + 0.261768i −0.825772 0.564004i \(-0.809260\pi\)
0.564004 + 0.825772i \(0.309260\pi\)
\(468\) 0 0
\(469\) −8.00000 8.00000i −0.369406 0.369406i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 16.9706 + 16.9706i 0.778663 + 0.778663i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 28.2843 1.29234 0.646171 0.763193i \(-0.276369\pi\)
0.646171 + 0.763193i \(0.276369\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −16.0000 16.0000i −0.726523 0.726523i
\(486\) 0 0
\(487\) 2.82843i 0.128168i −0.997944 0.0640841i \(-0.979587\pi\)
0.997944 0.0640841i \(-0.0204126\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.48528 8.48528i −0.382935 0.382935i 0.489223 0.872159i \(-0.337280\pi\)
−0.872159 + 0.489223i \(0.837280\pi\)
\(492\) 0 0
\(493\) −4.00000 + 4.00000i −0.180151 + 0.180151i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 0 0
\(499\) 19.7990 19.7990i 0.886325 0.886325i −0.107843 0.994168i \(-0.534395\pi\)
0.994168 + 0.107843i \(0.0343945\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.3137i 0.504453i −0.967668 0.252227i \(-0.918837\pi\)
0.967668 0.252227i \(-0.0811629\pi\)
\(504\) 0 0
\(505\) 22.0000i 0.978987i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −23.0000 + 23.0000i −1.01946 + 1.01946i −0.0196502 + 0.999807i \(0.506255\pi\)
−0.999807 + 0.0196502i \(0.993745\pi\)
\(510\) 0 0
\(511\) 39.5980 1.75171
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 19.7990 19.7990i 0.872448 0.872448i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.0000i 0.525730i 0.964833 + 0.262865i \(0.0846673\pi\)
−0.964833 + 0.262865i \(0.915333\pi\)
\(522\) 0 0
\(523\) −5.65685 5.65685i −0.247357 0.247357i 0.572528 0.819885i \(-0.305963\pi\)
−0.819885 + 0.572528i \(0.805963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.3137 −0.492833
\(528\) 0 0
\(529\) −9.00000 −0.391304
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 + 12.0000i 0.519778 + 0.519778i
\(534\) 0 0
\(535\) 16.9706i 0.733701i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.00000 + 3.00000i −0.128980 + 0.128980i −0.768650 0.639670i \(-0.779071\pi\)
0.639670 + 0.768650i \(0.279071\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 5.65685 5.65685i 0.241870 0.241870i −0.575754 0.817623i \(-0.695291\pi\)
0.817623 + 0.575754i \(0.195291\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.3137i 0.481980i
\(552\) 0 0
\(553\) 24.0000i 1.02058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 21.0000 21.0000i 0.889799 0.889799i −0.104705 0.994503i \(-0.533390\pi\)
0.994503 + 0.104705i \(0.0333898\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.65685 5.65685i 0.238408 0.238408i −0.577783 0.816191i \(-0.696082\pi\)
0.816191 + 0.577783i \(0.196082\pi\)
\(564\) 0 0
\(565\) 18.0000 + 18.0000i 0.757266 + 0.757266i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.0000i 0.838444i 0.907884 + 0.419222i \(0.137697\pi\)
−0.907884 + 0.419222i \(0.862303\pi\)
\(570\) 0 0
\(571\) −14.1421 14.1421i −0.591830 0.591830i 0.346296 0.938125i \(-0.387439\pi\)
−0.938125 + 0.346296i \(0.887439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −16.9706 −0.707721
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 32.0000 + 32.0000i 1.32758 + 1.32758i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.82843 + 2.82843i 0.116742 + 0.116742i 0.763064 0.646323i \(-0.223694\pi\)
−0.646323 + 0.763064i \(0.723694\pi\)
\(588\) 0 0
\(589\) 16.0000 16.0000i 0.659269 0.659269i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 11.3137 11.3137i 0.463817 0.463817i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 39.5980i 1.61793i −0.587857 0.808965i \(-0.700028\pi\)
0.587857 0.808965i \(-0.299972\pi\)
\(600\) 0 0
\(601\) 14.0000i 0.571072i 0.958368 + 0.285536i \(0.0921716\pi\)
−0.958368 + 0.285536i \(0.907828\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.0000 + 11.0000i −0.447214 + 0.447214i
\(606\) 0 0
\(607\) 42.4264 1.72203 0.861017 0.508576i \(-0.169828\pi\)
0.861017 + 0.508576i \(0.169828\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 33.9411 33.9411i 1.37311 1.37311i
\(612\) 0 0
\(613\) 11.0000 + 11.0000i 0.444286 + 0.444286i 0.893449 0.449164i \(-0.148278\pi\)
−0.449164 + 0.893449i \(0.648278\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000i 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 0 0
\(619\) −31.1127 31.1127i −1.25052 1.25052i −0.955485 0.295040i \(-0.904667\pi\)
−0.295040 0.955485i \(-0.595333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 16.9706 0.679911
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 12.0000i −0.478471 0.478471i
\(630\) 0 0
\(631\) 8.48528i 0.337794i −0.985634 0.168897i \(-0.945980\pi\)
0.985634 0.168897i \(-0.0540205\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.48528 8.48528i −0.336728 0.336728i
\(636\) 0 0
\(637\) 3.00000 3.00000i 0.118864 0.118864i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) −11.3137 + 11.3137i −0.446169 + 0.446169i −0.894079 0.447910i \(-0.852169\pi\)
0.447910 + 0.894079i \(0.352169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 39.5980i 1.55676i 0.627795 + 0.778379i \(0.283958\pi\)
−0.627795 + 0.778379i \(0.716042\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.00000 3.00000i 0.117399 0.117399i −0.645967 0.763366i \(-0.723545\pi\)
0.763366 + 0.645967i \(0.223545\pi\)
\(654\) 0 0
\(655\) 5.65685 0.221032
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −14.1421 + 14.1421i −0.550899 + 0.550899i −0.926700 0.375801i \(-0.877368\pi\)
0.375801 + 0.926700i \(0.377368\pi\)
\(660\) 0 0
\(661\) 19.0000 + 19.0000i 0.739014 + 0.739014i 0.972387 0.233373i \(-0.0749763\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 32.0000i 1.24091i
\(666\) 0 0
\(667\) 5.65685 + 5.65685i 0.219034 + 0.219034i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −23.0000 23.0000i −0.883962 0.883962i 0.109973 0.993935i \(-0.464924\pi\)
−0.993935 + 0.109973i \(0.964924\pi\)
\(678\) 0 0
\(679\) 45.2548i 1.73672i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22.6274 + 22.6274i 0.865814 + 0.865814i 0.992006 0.126192i \(-0.0402755\pi\)
−0.126192 + 0.992006i \(0.540275\pi\)
\(684\) 0 0
\(685\) −12.0000 + 12.0000i −0.458496 + 0.458496i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −30.0000 −1.14291
\(690\) 0 0
\(691\) −28.2843 + 28.2843i −1.07598 + 1.07598i −0.0791192 + 0.996865i \(0.525211\pi\)
−0.996865 + 0.0791192i \(0.974789\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.65685i 0.214577i
\(696\) 0 0
\(697\) 16.0000i 0.606043i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 33.0000 33.0000i 1.24639 1.24639i 0.289091 0.957302i \(-0.406647\pi\)
0.957302 0.289091i \(-0.0933531\pi\)
\(702\) 0 0
\(703\) 33.9411 1.28011
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 31.1127 31.1127i 1.17011 1.17011i
\(708\) 0 0
\(709\) 25.0000 + 25.0000i 0.938895 + 0.938895i 0.998238 0.0593429i \(-0.0189006\pi\)
−0.0593429 + 0.998238i \(0.518901\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.0000i 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.9706 0.632895 0.316448 0.948610i \(-0.397510\pi\)
0.316448 + 0.948610i \(0.397510\pi\)
\(720\) 0 0
\(721\) −56.0000 −2.08555
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.00000 + 3.00000i 0.111417 + 0.111417i
\(726\) 0 0
\(727\) 42.4264i 1.57351i 0.617266 + 0.786754i \(0.288240\pi\)
−0.617266 + 0.786754i \(0.711760\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 3.00000 3.00000i 0.110808 0.110808i −0.649529 0.760337i \(-0.725034\pi\)
0.760337 + 0.649529i \(0.225034\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −25.4558 + 25.4558i −0.936408 + 0.936408i −0.998096 0.0616872i \(-0.980352\pi\)
0.0616872 + 0.998096i \(0.480352\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 11.3137i 0.415060i 0.978229 + 0.207530i \(0.0665424\pi\)
−0.978229 + 0.207530i \(0.933458\pi\)
\(744\) 0 0
\(745\) 18.0000i 0.659469i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.0000 24.0000i 0.876941 0.876941i
\(750\) 0 0
\(751\) −8.48528 −0.309632 −0.154816 0.987943i \(-0.549479\pi\)
−0.154816 + 0.987943i \(0.549479\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.82843 2.82843i 0.102937 0.102937i
\(756\) 0 0
\(757\) 15.0000 + 15.0000i 0.545184 + 0.545184i 0.925044 0.379860i \(-0.124028\pi\)
−0.379860 + 0.925044i \(0.624028\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36.0000i 1.30500i −0.757789 0.652499i \(-0.773720\pi\)
0.757789 0.652499i \(-0.226280\pi\)
\(762\) 0 0
\(763\) 8.48528 + 8.48528i 0.307188 + 0.307188i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −50.9117 −1.83831
\(768\) 0 0
\(769\) −32.0000 −1.15395 −0.576975 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 13.0000 + 13.0000i 0.467578 + 0.467578i 0.901129 0.433551i \(-0.142740\pi\)
−0.433551 + 0.901129i \(0.642740\pi\)
\(774\) 0 0
\(775\) 8.48528i 0.304800i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 22.6274 + 22.6274i 0.810711 + 0.810711i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) −22.6274 + 22.6274i −0.806580 + 0.806580i −0.984115 0.177534i \(-0.943188\pi\)
0.177534 + 0.984115i \(0.443188\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 50.9117i 1.81021i
\(792\) 0 0
\(793\) 6.00000i 0.213066i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −35.0000 + 35.0000i −1.23976 + 1.23976i −0.279666 + 0.960097i \(0.590224\pi\)
−0.960097 + 0.279666i \(0.909776\pi\)
\(798\) 0 0
\(799\) −45.2548 −1.60100
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −16.0000 16.0000i −0.563926