Properties

Label 3200.2.f.r.449.1
Level $3200$
Weight $2$
Character 3200.449
Analytic conductor $25.552$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-0.437016 - 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 3200.449
Dual form 3200.2.f.r.449.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607 q^{3} -2.82843i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q-2.23607 q^{3} -2.82843i q^{7} +2.00000 q^{9} -2.23607i q^{11} +6.32456 q^{13} -5.00000i q^{17} -2.23607i q^{19} +6.32456i q^{21} -5.65685i q^{23} +2.23607 q^{27} +6.32456i q^{29} +5.00000i q^{33} -6.32456 q^{37} -14.1421 q^{39} +3.00000 q^{41} +8.94427 q^{43} -2.82843i q^{47} -1.00000 q^{49} +11.1803i q^{51} +12.6491 q^{53} +5.00000i q^{57} +8.94427i q^{59} +6.32456i q^{61} -5.65685i q^{63} -11.1803 q^{67} +12.6491i q^{69} +14.1421 q^{71} -15.0000i q^{73} -6.32456 q^{77} -14.1421 q^{79} -11.0000 q^{81} -6.70820 q^{83} -14.1421i q^{87} -1.00000 q^{89} -17.8885i q^{91} +10.0000i q^{97} -4.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} + 24 q^{41} - 8 q^{49} - 88 q^{81} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −2.23607 −1.29099 −0.645497 0.763763i \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) − 2.23607i − 0.674200i −0.941469 0.337100i \(-0.890554\pi\)
0.941469 0.337100i \(-0.109446\pi\)
\(12\) 0 0
\(13\) 6.32456 1.75412 0.877058 0.480384i \(-0.159503\pi\)
0.877058 + 0.480384i \(0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.00000i − 1.21268i −0.795206 0.606339i \(-0.792637\pi\)
0.795206 0.606339i \(-0.207363\pi\)
\(18\) 0 0
\(19\) − 2.23607i − 0.512989i −0.966546 0.256495i \(-0.917432\pi\)
0.966546 0.256495i \(-0.0825676\pi\)
\(20\) 0 0
\(21\) 6.32456i 1.38013i
\(22\) 0 0
\(23\) − 5.65685i − 1.17954i −0.807573 0.589768i \(-0.799219\pi\)
0.807573 0.589768i \(-0.200781\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.23607 0.430331
\(28\) 0 0
\(29\) 6.32456i 1.17444i 0.809427 + 0.587220i \(0.199778\pi\)
−0.809427 + 0.587220i \(0.800222\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 5.00000i 0.870388i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.32456 −1.03975 −0.519875 0.854242i \(-0.674022\pi\)
−0.519875 + 0.854242i \(0.674022\pi\)
\(38\) 0 0
\(39\) −14.1421 −2.26455
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 8.94427 1.36399 0.681994 0.731357i \(-0.261113\pi\)
0.681994 + 0.731357i \(0.261113\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.82843i − 0.412568i −0.978492 0.206284i \(-0.933863\pi\)
0.978492 0.206284i \(-0.0661372\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 11.1803i 1.56556i
\(52\) 0 0
\(53\) 12.6491 1.73749 0.868744 0.495261i \(-0.164927\pi\)
0.868744 + 0.495261i \(0.164927\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.00000i 0.662266i
\(58\) 0 0
\(59\) 8.94427i 1.16445i 0.813029 + 0.582223i \(0.197817\pi\)
−0.813029 + 0.582223i \(0.802183\pi\)
\(60\) 0 0
\(61\) 6.32456i 0.809776i 0.914366 + 0.404888i \(0.132690\pi\)
−0.914366 + 0.404888i \(0.867310\pi\)
\(62\) 0 0
\(63\) − 5.65685i − 0.712697i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.1803 −1.36590 −0.682948 0.730467i \(-0.739302\pi\)
−0.682948 + 0.730467i \(0.739302\pi\)
\(68\) 0 0
\(69\) 12.6491i 1.52277i
\(70\) 0 0
\(71\) 14.1421 1.67836 0.839181 0.543852i \(-0.183035\pi\)
0.839181 + 0.543852i \(0.183035\pi\)
\(72\) 0 0
\(73\) − 15.0000i − 1.75562i −0.479012 0.877809i \(-0.659005\pi\)
0.479012 0.877809i \(-0.340995\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.32456 −0.720750
\(78\) 0 0
\(79\) −14.1421 −1.59111 −0.795557 0.605878i \(-0.792822\pi\)
−0.795557 + 0.605878i \(0.792822\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −6.70820 −0.736321 −0.368161 0.929762i \(-0.620012\pi\)
−0.368161 + 0.929762i \(0.620012\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 14.1421i − 1.51620i
\(88\) 0 0
\(89\) −1.00000 −0.106000 −0.0529999 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) − 17.8885i − 1.87523i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) − 4.47214i − 0.449467i
\(100\) 0 0
\(101\) − 6.32456i − 0.629317i −0.949205 0.314658i \(-0.898110\pi\)
0.949205 0.314658i \(-0.101890\pi\)
\(102\) 0 0
\(103\) − 8.48528i − 0.836080i −0.908429 0.418040i \(-0.862717\pi\)
0.908429 0.418040i \(-0.137283\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.6525 1.51318 0.756591 0.653888i \(-0.226863\pi\)
0.756591 + 0.653888i \(0.226863\pi\)
\(108\) 0 0
\(109\) − 6.32456i − 0.605783i −0.953025 0.302891i \(-0.902048\pi\)
0.953025 0.302891i \(-0.0979519\pi\)
\(110\) 0 0
\(111\) 14.1421 1.34231
\(112\) 0 0
\(113\) 15.0000i 1.41108i 0.708669 + 0.705541i \(0.249296\pi\)
−0.708669 + 0.705541i \(0.750704\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.6491 1.16941
\(118\) 0 0
\(119\) −14.1421 −1.29641
\(120\) 0 0
\(121\) 6.00000 0.545455
\(122\) 0 0
\(123\) −6.70820 −0.604858
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.9706i − 1.50589i −0.658081 0.752947i \(-0.728632\pi\)
0.658081 0.752947i \(-0.271368\pi\)
\(128\) 0 0
\(129\) −20.0000 −1.76090
\(130\) 0 0
\(131\) 8.94427i 0.781465i 0.920504 + 0.390732i \(0.127778\pi\)
−0.920504 + 0.390732i \(0.872222\pi\)
\(132\) 0 0
\(133\) −6.32456 −0.548408
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.00000i 0.427179i 0.976924 + 0.213589i \(0.0685155\pi\)
−0.976924 + 0.213589i \(0.931485\pi\)
\(138\) 0 0
\(139\) − 15.6525i − 1.32763i −0.747899 0.663813i \(-0.768937\pi\)
0.747899 0.663813i \(-0.231063\pi\)
\(140\) 0 0
\(141\) 6.32456i 0.532624i
\(142\) 0 0
\(143\) − 14.1421i − 1.18262i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.23607 0.184428
\(148\) 0 0
\(149\) − 18.9737i − 1.55438i −0.629264 0.777192i \(-0.716644\pi\)
0.629264 0.777192i \(-0.283356\pi\)
\(150\) 0 0
\(151\) −14.1421 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) 0 0
\(153\) − 10.0000i − 0.808452i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) −28.2843 −2.24309
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −6.70820 −0.525427 −0.262714 0.964874i \(-0.584617\pi\)
−0.262714 + 0.964874i \(0.584617\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.82843i 0.218870i 0.993994 + 0.109435i \(0.0349042\pi\)
−0.993994 + 0.109435i \(0.965096\pi\)
\(168\) 0 0
\(169\) 27.0000 2.07692
\(170\) 0 0
\(171\) − 4.47214i − 0.341993i
\(172\) 0 0
\(173\) −12.6491 −0.961694 −0.480847 0.876804i \(-0.659671\pi\)
−0.480847 + 0.876804i \(0.659671\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 20.0000i − 1.50329i
\(178\) 0 0
\(179\) − 2.23607i − 0.167132i −0.996502 0.0835658i \(-0.973369\pi\)
0.996502 0.0835658i \(-0.0266309\pi\)
\(180\) 0 0
\(181\) − 12.6491i − 0.940201i −0.882613 0.470100i \(-0.844218\pi\)
0.882613 0.470100i \(-0.155782\pi\)
\(182\) 0 0
\(183\) − 14.1421i − 1.04542i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −11.1803 −0.817587
\(188\) 0 0
\(189\) − 6.32456i − 0.460044i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 5.00000i 0.359908i 0.983675 + 0.179954i \(0.0575949\pi\)
−0.983675 + 0.179954i \(0.942405\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 25.0000 1.76336
\(202\) 0 0
\(203\) 17.8885 1.25553
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 11.3137i − 0.786357i
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 6.70820i 0.461812i 0.972976 + 0.230906i \(0.0741690\pi\)
−0.972976 + 0.230906i \(0.925831\pi\)
\(212\) 0 0
\(213\) −31.6228 −2.16676
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 33.5410i 2.26649i
\(220\) 0 0
\(221\) − 31.6228i − 2.12718i
\(222\) 0 0
\(223\) − 22.6274i − 1.51524i −0.652694 0.757622i \(-0.726361\pi\)
0.652694 0.757622i \(-0.273639\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.94427 −0.593652 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(228\) 0 0
\(229\) − 12.6491i − 0.835877i −0.908475 0.417938i \(-0.862753\pi\)
0.908475 0.417938i \(-0.137247\pi\)
\(230\) 0 0
\(231\) 14.1421 0.930484
\(232\) 0 0
\(233\) − 10.0000i − 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 31.6228 2.05412
\(238\) 0 0
\(239\) −14.1421 −0.914779 −0.457389 0.889267i \(-0.651215\pi\)
−0.457389 + 0.889267i \(0.651215\pi\)
\(240\) 0 0
\(241\) 13.0000 0.837404 0.418702 0.908124i \(-0.362485\pi\)
0.418702 + 0.908124i \(0.362485\pi\)
\(242\) 0 0
\(243\) 17.8885 1.14755
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 14.1421i − 0.899843i
\(248\) 0 0
\(249\) 15.0000 0.950586
\(250\) 0 0
\(251\) 11.1803i 0.705697i 0.935681 + 0.352848i \(0.114787\pi\)
−0.935681 + 0.352848i \(0.885213\pi\)
\(252\) 0 0
\(253\) −12.6491 −0.795243
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.0000i − 0.623783i −0.950118 0.311891i \(-0.899037\pi\)
0.950118 0.311891i \(-0.100963\pi\)
\(258\) 0 0
\(259\) 17.8885i 1.11154i
\(260\) 0 0
\(261\) 12.6491i 0.782960i
\(262\) 0 0
\(263\) 22.6274i 1.39527i 0.716455 + 0.697633i \(0.245763\pi\)
−0.716455 + 0.697633i \(0.754237\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.23607 0.136845
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 40.0000i 2.42091i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.9737 −1.14002 −0.570009 0.821639i \(-0.693060\pi\)
−0.570009 + 0.821639i \(0.693060\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −11.1803 −0.664602 −0.332301 0.943173i \(-0.607825\pi\)
−0.332301 + 0.943173i \(0.607825\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 8.48528i − 0.500870i
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) − 22.3607i − 1.31081i
\(292\) 0 0
\(293\) −6.32456 −0.369484 −0.184742 0.982787i \(-0.559145\pi\)
−0.184742 + 0.982787i \(0.559145\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 5.00000i − 0.290129i
\(298\) 0 0
\(299\) − 35.7771i − 2.06904i
\(300\) 0 0
\(301\) − 25.2982i − 1.45817i
\(302\) 0 0
\(303\) 14.1421i 0.812444i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.1246 1.14857 0.574286 0.818655i \(-0.305280\pi\)
0.574286 + 0.818655i \(0.305280\pi\)
\(308\) 0 0
\(309\) 18.9737i 1.07937i
\(310\) 0 0
\(311\) 14.1421 0.801927 0.400963 0.916094i \(-0.368675\pi\)
0.400963 + 0.916094i \(0.368675\pi\)
\(312\) 0 0
\(313\) − 30.0000i − 1.69570i −0.530236 0.847850i \(-0.677897\pi\)
0.530236 0.847850i \(-0.322103\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.6491 −0.710445 −0.355222 0.934782i \(-0.615595\pi\)
−0.355222 + 0.934782i \(0.615595\pi\)
\(318\) 0 0
\(319\) 14.1421 0.791808
\(320\) 0 0
\(321\) −35.0000 −1.95351
\(322\) 0 0
\(323\) −11.1803 −0.622091
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 14.1421i 0.782062i
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 33.5410i 1.84358i 0.387688 + 0.921791i \(0.373274\pi\)
−0.387688 + 0.921791i \(0.626726\pi\)
\(332\) 0 0
\(333\) −12.6491 −0.693167
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 15.0000i − 0.817102i −0.912735 0.408551i \(-0.866034\pi\)
0.912735 0.408551i \(-0.133966\pi\)
\(338\) 0 0
\(339\) − 33.5410i − 1.82170i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 16.9706i − 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.1246 −1.08035 −0.540173 0.841554i \(-0.681641\pi\)
−0.540173 + 0.841554i \(0.681641\pi\)
\(348\) 0 0
\(349\) − 18.9737i − 1.01564i −0.861464 0.507819i \(-0.830452\pi\)
0.861464 0.507819i \(-0.169548\pi\)
\(350\) 0 0
\(351\) 14.1421 0.754851
\(352\) 0 0
\(353\) 10.0000i 0.532246i 0.963939 + 0.266123i \(0.0857428\pi\)
−0.963939 + 0.266123i \(0.914257\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 31.6228 1.67365
\(358\) 0 0
\(359\) −14.1421 −0.746393 −0.373197 0.927752i \(-0.621738\pi\)
−0.373197 + 0.927752i \(0.621738\pi\)
\(360\) 0 0
\(361\) 14.0000 0.736842
\(362\) 0 0
\(363\) −13.4164 −0.704179
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.9706i 0.885856i 0.896557 + 0.442928i \(0.146060\pi\)
−0.896557 + 0.442928i \(0.853940\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) − 35.7771i − 1.85745i
\(372\) 0 0
\(373\) 12.6491 0.654946 0.327473 0.944861i \(-0.393803\pi\)
0.327473 + 0.944861i \(0.393803\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 40.0000i 2.06010i
\(378\) 0 0
\(379\) 6.70820i 0.344577i 0.985047 + 0.172289i \(0.0551162\pi\)
−0.985047 + 0.172289i \(0.944884\pi\)
\(380\) 0 0
\(381\) 37.9473i 1.94410i
\(382\) 0 0
\(383\) 19.7990i 1.01168i 0.862627 + 0.505841i \(0.168818\pi\)
−0.862627 + 0.505841i \(0.831182\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 17.8885 0.909326
\(388\) 0 0
\(389\) − 25.2982i − 1.28267i −0.767261 0.641335i \(-0.778381\pi\)
0.767261 0.641335i \(-0.221619\pi\)
\(390\) 0 0
\(391\) −28.2843 −1.43040
\(392\) 0 0
\(393\) − 20.0000i − 1.00887i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −25.2982 −1.26968 −0.634841 0.772643i \(-0.718934\pi\)
−0.634841 + 0.772643i \(0.718934\pi\)
\(398\) 0 0
\(399\) 14.1421 0.707992
\(400\) 0 0
\(401\) 13.0000 0.649189 0.324595 0.945853i \(-0.394772\pi\)
0.324595 + 0.945853i \(0.394772\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.1421i 0.701000i
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) − 11.1803i − 0.551485i
\(412\) 0 0
\(413\) 25.2982 1.24484
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 35.0000i 1.71396i
\(418\) 0 0
\(419\) 33.5410i 1.63859i 0.573375 + 0.819293i \(0.305634\pi\)
−0.573375 + 0.819293i \(0.694366\pi\)
\(420\) 0 0
\(421\) − 12.6491i − 0.616480i −0.951309 0.308240i \(-0.900260\pi\)
0.951309 0.308240i \(-0.0997400\pi\)
\(422\) 0 0
\(423\) − 5.65685i − 0.275046i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 17.8885 0.865687
\(428\) 0 0
\(429\) 31.6228i 1.52676i
\(430\) 0 0
\(431\) −14.1421 −0.681203 −0.340601 0.940208i \(-0.610631\pi\)
−0.340601 + 0.940208i \(0.610631\pi\)
\(432\) 0 0
\(433\) 5.00000i 0.240285i 0.992757 + 0.120142i \(0.0383351\pi\)
−0.992757 + 0.120142i \(0.961665\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.6491 −0.605089
\(438\) 0 0
\(439\) 14.1421 0.674967 0.337484 0.941331i \(-0.390424\pi\)
0.337484 + 0.941331i \(0.390424\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) −29.0689 −1.38110 −0.690552 0.723283i \(-0.742632\pi\)
−0.690552 + 0.723283i \(0.742632\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 42.4264i 2.00670i
\(448\) 0 0
\(449\) −21.0000 −0.991051 −0.495526 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) 0 0
\(451\) − 6.70820i − 0.315877i
\(452\) 0 0
\(453\) 31.6228 1.48577
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000i 0.233890i 0.993138 + 0.116945i \(0.0373101\pi\)
−0.993138 + 0.116945i \(0.962690\pi\)
\(458\) 0 0
\(459\) − 11.1803i − 0.521854i
\(460\) 0 0
\(461\) 37.9473i 1.76738i 0.468069 + 0.883692i \(0.344950\pi\)
−0.468069 + 0.883692i \(0.655050\pi\)
\(462\) 0 0
\(463\) 5.65685i 0.262896i 0.991323 + 0.131448i \(0.0419627\pi\)
−0.991323 + 0.131448i \(0.958037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.94427 −0.413892 −0.206946 0.978352i \(-0.566352\pi\)
−0.206946 + 0.978352i \(0.566352\pi\)
\(468\) 0 0
\(469\) 31.6228i 1.46020i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 20.0000i − 0.919601i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 25.2982 1.15833
\(478\) 0 0
\(479\) 28.2843 1.29234 0.646171 0.763193i \(-0.276369\pi\)
0.646171 + 0.763193i \(0.276369\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 0 0
\(483\) 35.7771 1.62791
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 39.5980i − 1.79436i −0.441669 0.897178i \(-0.645614\pi\)
0.441669 0.897178i \(-0.354386\pi\)
\(488\) 0 0
\(489\) 15.0000 0.678323
\(490\) 0 0
\(491\) 26.8328i 1.21095i 0.795865 + 0.605474i \(0.207016\pi\)
−0.795865 + 0.605474i \(0.792984\pi\)
\(492\) 0 0
\(493\) 31.6228 1.42422
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 40.0000i − 1.79425i
\(498\) 0 0
\(499\) 26.8328i 1.20120i 0.799549 + 0.600601i \(0.205072\pi\)
−0.799549 + 0.600601i \(0.794928\pi\)
\(500\) 0 0
\(501\) − 6.32456i − 0.282560i
\(502\) 0 0
\(503\) 33.9411i 1.51336i 0.653785 + 0.756680i \(0.273180\pi\)
−0.653785 + 0.756680i \(0.726820\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −60.3738 −2.68130
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −42.4264 −1.87683
\(512\) 0 0
\(513\) − 5.00000i − 0.220755i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.32456 −0.278154
\(518\) 0 0
\(519\) 28.2843 1.24154
\(520\) 0 0
\(521\) −13.0000 −0.569540 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(522\) 0 0
\(523\) 20.1246 0.879988 0.439994 0.898001i \(-0.354981\pi\)
0.439994 + 0.898001i \(0.354981\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −9.00000 −0.391304
\(530\) 0 0
\(531\) 17.8885i 0.776297i
\(532\) 0 0
\(533\) 18.9737 0.821841
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 5.00000i 0.215766i
\(538\) 0 0
\(539\) 2.23607i 0.0963143i
\(540\) 0 0
\(541\) 12.6491i 0.543828i 0.962322 + 0.271914i \(0.0876566\pi\)
−0.962322 + 0.271914i \(0.912343\pi\)
\(542\) 0 0
\(543\) 28.2843i 1.21379i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.1246 0.860466 0.430233 0.902718i \(-0.358431\pi\)
0.430233 + 0.902718i \(0.358431\pi\)
\(548\) 0 0
\(549\) 12.6491i 0.539851i
\(550\) 0 0
\(551\) 14.1421 0.602475
\(552\) 0 0
\(553\) 40.0000i 1.70097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −44.2719 −1.87586 −0.937930 0.346825i \(-0.887260\pi\)
−0.937930 + 0.346825i \(0.887260\pi\)
\(558\) 0 0
\(559\) 56.5685 2.39259
\(560\) 0 0
\(561\) 25.0000 1.05550
\(562\) 0 0
\(563\) 44.7214 1.88478 0.942390 0.334515i \(-0.108573\pi\)
0.942390 + 0.334515i \(0.108573\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 31.1127i 1.30661i
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) 0 0
\(571\) 26.8328i 1.12292i 0.827504 + 0.561459i \(0.189760\pi\)
−0.827504 + 0.561459i \(0.810240\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 25.0000i 1.04076i 0.853934 + 0.520382i \(0.174210\pi\)
−0.853934 + 0.520382i \(0.825790\pi\)
\(578\) 0 0
\(579\) − 11.1803i − 0.464639i
\(580\) 0 0
\(581\) 18.9737i 0.787160i
\(582\) 0 0
\(583\) − 28.2843i − 1.17141i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.1803 0.461462 0.230731 0.973018i \(-0.425888\pi\)
0.230731 + 0.973018i \(0.425888\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.0000i 0.615976i 0.951390 + 0.307988i \(0.0996557\pi\)
−0.951390 + 0.307988i \(0.900344\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.1421 0.577832 0.288916 0.957354i \(-0.406705\pi\)
0.288916 + 0.957354i \(0.406705\pi\)
\(600\) 0 0
\(601\) 17.0000 0.693444 0.346722 0.937968i \(-0.387295\pi\)
0.346722 + 0.937968i \(0.387295\pi\)
\(602\) 0 0
\(603\) −22.3607 −0.910597
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 16.9706i − 0.688814i −0.938820 0.344407i \(-0.888080\pi\)
0.938820 0.344407i \(-0.111920\pi\)
\(608\) 0 0
\(609\) −40.0000 −1.62088
\(610\) 0 0
\(611\) − 17.8885i − 0.723693i
\(612\) 0 0
\(613\) 25.2982 1.02179 0.510893 0.859644i \(-0.329315\pi\)
0.510893 + 0.859644i \(0.329315\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 30.0000i − 1.20775i −0.797077 0.603877i \(-0.793622\pi\)
0.797077 0.603877i \(-0.206378\pi\)
\(618\) 0 0
\(619\) − 26.8328i − 1.07850i −0.842145 0.539251i \(-0.818707\pi\)
0.842145 0.539251i \(-0.181293\pi\)
\(620\) 0 0
\(621\) − 12.6491i − 0.507591i
\(622\) 0 0
\(623\) 2.82843i 0.113319i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 11.1803 0.446500
\(628\) 0 0
\(629\) 31.6228i 1.26088i
\(630\) 0 0
\(631\) −28.2843 −1.12598 −0.562990 0.826464i \(-0.690349\pi\)
−0.562990 + 0.826464i \(0.690349\pi\)
\(632\) 0 0
\(633\) − 15.0000i − 0.596196i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.32456 −0.250588
\(638\) 0 0
\(639\) 28.2843 1.11891
\(640\) 0 0
\(641\) 22.0000 0.868948 0.434474 0.900684i \(-0.356934\pi\)
0.434474 + 0.900684i \(0.356934\pi\)
\(642\) 0 0
\(643\) −26.8328 −1.05818 −0.529091 0.848565i \(-0.677467\pi\)
−0.529091 + 0.848565i \(0.677467\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 45.2548i 1.77915i 0.456788 + 0.889576i \(0.349000\pi\)
−0.456788 + 0.889576i \(0.651000\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.32456 0.247499 0.123749 0.992313i \(-0.460508\pi\)
0.123749 + 0.992313i \(0.460508\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 30.0000i − 1.17041i
\(658\) 0 0
\(659\) − 6.70820i − 0.261315i −0.991428 0.130657i \(-0.958291\pi\)
0.991428 0.130657i \(-0.0417087\pi\)
\(660\) 0 0
\(661\) 44.2719i 1.72198i 0.508625 + 0.860988i \(0.330154\pi\)
−0.508625 + 0.860988i \(0.669846\pi\)
\(662\) 0 0
\(663\) 70.7107i 2.74618i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 35.7771 1.38529
\(668\) 0 0
\(669\) 50.5964i 1.95617i
\(670\) 0 0
\(671\) 14.1421 0.545951
\(672\) 0 0
\(673\) − 10.0000i − 0.385472i −0.981251 0.192736i \(-0.938264\pi\)
0.981251 0.192736i \(-0.0617360\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −37.9473 −1.45843 −0.729217 0.684282i \(-0.760116\pi\)
−0.729217 + 0.684282i \(0.760116\pi\)
\(678\) 0 0
\(679\) 28.2843 1.08545
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) 2.23607 0.0855608 0.0427804 0.999085i \(-0.486378\pi\)
0.0427804 + 0.999085i \(0.486378\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 28.2843i 1.07911i
\(688\) 0 0
\(689\) 80.0000 3.04776
\(690\) 0 0
\(691\) − 33.5410i − 1.27596i −0.770053 0.637980i \(-0.779770\pi\)
0.770053 0.637980i \(-0.220230\pi\)
\(692\) 0 0
\(693\) −12.6491 −0.480500
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 15.0000i − 0.568166i
\(698\) 0 0
\(699\) 22.3607i 0.845759i
\(700\) 0 0
\(701\) − 31.6228i − 1.19438i −0.802101 0.597188i \(-0.796285\pi\)
0.802101 0.597188i \(-0.203715\pi\)
\(702\) 0 0
\(703\) 14.1421i 0.533381i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.8885 −0.672768
\(708\) 0 0
\(709\) − 12.6491i − 0.475047i −0.971382 0.237524i \(-0.923664\pi\)
0.971382 0.237524i \(-0.0763357\pi\)
\(710\) 0 0
\(711\) −28.2843 −1.06074
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 31.6228 1.18097
\(718\) 0 0
\(719\) −14.1421 −0.527413 −0.263706 0.964603i \(-0.584945\pi\)
−0.263706 + 0.964603i \(0.584945\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) −29.0689 −1.08108
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 25.4558i 0.944105i 0.881570 + 0.472052i \(0.156487\pi\)
−0.881570 + 0.472052i \(0.843513\pi\)
\(728\) 0 0
\(729\) −7.00000 −0.259259
\(730\) 0 0
\(731\) − 44.7214i − 1.65408i
\(732\) 0 0
\(733\) 6.32456 0.233603 0.116801 0.993155i \(-0.462736\pi\)
0.116801 + 0.993155i \(0.462736\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.0000i 0.920887i
\(738\) 0 0
\(739\) 8.94427i 0.329020i 0.986375 + 0.164510i \(0.0526043\pi\)
−0.986375 + 0.164510i \(0.947396\pi\)
\(740\) 0 0
\(741\) 31.6228i 1.16169i
\(742\) 0 0
\(743\) 5.65685i 0.207530i 0.994602 + 0.103765i \(0.0330890\pi\)
−0.994602 + 0.103765i \(0.966911\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −13.4164 −0.490881
\(748\) 0 0
\(749\) − 44.2719i − 1.61766i
\(750\) 0 0
\(751\) −42.4264 −1.54816 −0.774081 0.633087i \(-0.781788\pi\)
−0.774081 + 0.633087i \(0.781788\pi\)
\(752\) 0 0
\(753\) − 25.0000i − 0.911051i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12.6491 0.459740 0.229870 0.973221i \(-0.426170\pi\)
0.229870 + 0.973221i \(0.426170\pi\)
\(758\) 0 0
\(759\) 28.2843 1.02665
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) −17.8885 −0.647609
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 56.5685i 2.04257i
\(768\) 0 0
\(769\) −21.0000 −0.757279 −0.378640 0.925544i \(-0.623608\pi\)
−0.378640 + 0.925544i \(0.623608\pi\)
\(770\) 0 0
\(771\) 22.3607i 0.805300i
\(772\) 0 0
\(773\) −31.6228 −1.13739 −0.568696 0.822548i \(-0.692552\pi\)
−0.568696 + 0.822548i \(0.692552\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 40.0000i − 1.43499i
\(778\) 0 0
\(779\) − 6.70820i − 0.240346i
\(780\) 0 0
\(781\) − 31.6228i − 1.13155i
\(782\) 0 0
\(783\) 14.1421i 0.505399i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −26.8328 −0.956487 −0.478243 0.878227i \(-0.658726\pi\)
−0.478243 + 0.878227i \(0.658726\pi\)
\(788\) 0 0
\(789\) − 50.5964i − 1.80128i
\(790\) 0 0
\(791\) 42.4264 1.50851
\(792\) 0 0
\(793\) 40.0000i 1.42044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.2982 0.896109 0.448054 0.894006i \(-0.352117\pi\)
0.448054 + 0.894006i \(0.352117\pi\)
\(798\) 0 0
\(799\) −14.1421 −0.500313
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 0 0
\(803\) −33.5410 −1.18364
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) − 8.94427i − 0.314076i −0.987593 0.157038i \(-0.949806\pi\)
0.987593 0.157038i \(-0.0501945\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 20.0000i − 0.699711i
\(818\) 0 0
\(819\) − 35.7771i − 1.25015i
\(820\) 0 0
\(821\) − 37.9473i − 1.32437i −0.749340 0.662186i \(-0.769629\pi\)
0.749340 0.662186i \(-0.230371\pi\)
\(822\) 0 0
\(823\) − 5.65685i − 0.197186i −0.995128 0.0985928i \(-0.968566\pi\)
0.995128 0.0985928i \(-0.0314341\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11.1803 0.388779 0.194389 0.980924i \(-0.437728\pi\)
0.194389 + 0.980924i \(0.437728\pi\)
\(828\) 0 0
\(829\) 6.32456i 0.219661i 0.993950 + 0.109830i \(0.0350308\pi\)
−0.993950 + 0.109830i \(0.964969\pi\)
\(830\) 0 0
\(831\) 42.4264 1.47176
\(832\) 0 0
\(833\) 5.00000i 0.173240i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 56.5685 1.95296 0.976481 0.215601i \(-0.0691711\pi\)
0.976481 + 0.215601i \(0.0691711\pi\)
\(840\) 0 0
\(841\) −11.0000 −0.379310
\(842\) 0 0
\(843\) 4.47214 0.154029
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 16.9706i − 0.583115i
\(848\) 0 0
\(849\) 25.0000 0.857998
\(850\) 0 0
\(851\) 35.7771i 1.22642i
\(852\) 0 0
\(853\) −12.6491 −0.433097 −0.216549 0.976272i \(-0.569480\pi\)
−0.216549 + 0.976272i \(0.569480\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 45.0000i 1.53717i 0.639747 + 0.768585i \(0.279039\pi\)
−0.639747 + 0.768585i \(0.720961\pi\)
\(858\) 0 0
\(859\) 2.23607i 0.0762937i 0.999272 + 0.0381468i \(0.0121455\pi\)
−0.999272 + 0.0381468i \(0.987855\pi\)
\(860\) 0 0
\(861\) 18.9737i 0.646621i
\(862\) 0 0
\(863\) 8.48528i 0.288842i 0.989516 + 0.144421i \(0.0461320\pi\)
−0.989516 + 0.144421i \(0.953868\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 17.8885 0.607527
\(868\) 0 0
\(869\) 31.6228i 1.07273i
\(870\) 0 0
\(871\) −70.7107 −2.39594
\(872\) 0 0
\(873\) 20.0000i 0.676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12.6491 0.427130 0.213565 0.976929i \(-0.431492\pi\)
0.213565 + 0.976929i \(0.431492\pi\)
\(878\) 0 0
\(879\) 14.1421 0.477002
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) 15.6525 0.526748 0.263374 0.964694i \(-0.415165\pi\)
0.263374 + 0.964694i \(0.415165\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11.3137i 0.379877i 0.981796 + 0.189939i \(0.0608289\pi\)
−0.981796 + 0.189939i \(0.939171\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 0 0
\(891\) 24.5967i 0.824022i
\(892\) 0 0
\(893\) −6.32456 −0.211643
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 80.0000i 2.67112i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) − 63.2456i − 2.10701i
\(902\) 0 0
\(903\) 56.5685i 1.88248i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −8.94427 −0.296990 −0.148495 0.988913i \(-0.547443\pi\)
−0.148495 + 0.988913i \(0.547443\pi\)
\(908\) 0 0
\(909\) − 12.6491i − 0.419545i
\(910\) 0 0
\(911\) 28.2843 0.937100 0.468550 0.883437i \(-0.344777\pi\)
0.468550 + 0.883437i \(0.344777\pi\)
\(912\) 0 0
\(913\) 15.0000i 0.496428i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 25.2982 0.835421
\(918\) 0 0
\(919\) 28.2843 0.933012 0.466506 0.884518i \(-0.345513\pi\)
0.466506 + 0.884518i \(0.345513\pi\)
\(920\) 0 0
\(921\) −45.0000 −1.48280
\(922\) 0 0
\(923\) 89.4427 2.94404
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 16.9706i − 0.557386i
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 2.23607i 0.0732842i
\(932\) 0 0
\(933\) −31.6228 −1.03528
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 25.0000i − 0.816714i −0.912822 0.408357i \(-0.866102\pi\)
0.912822 0.408357i \(-0.133898\pi\)
\(938\) 0 0
\(939\) 67.0820i 2.18914i
\(940\) 0 0
\(941\) 31.6228i 1.03087i 0.856928 + 0.515437i \(0.172370\pi\)
−0.856928 + 0.515437i \(0.827630\pi\)
\(942\) 0 0
\(943\) − 16.9706i − 0.552638i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.8328 0.871949 0.435975 0.899959i \(-0.356404\pi\)
0.435975 + 0.899959i \(0.356404\pi\)
\(948\) 0 0
\(949\) − 94.8683i − 3.07956i
\(950\) 0 0
\(951\) 28.2843 0.917180
\(952\) 0 0
\(953\) − 45.0000i − 1.45769i −0.684677 0.728846i \(-0.740057\pi\)
0.684677 0.728846i \(-0.259943\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −31.6228 −1.02222
\(958\) 0 0
\(959\) 14.1421 0.456673
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 31.3050 1.00879
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 16.9706i − 0.545737i −0.962051 0.272868i \(-0.912028\pi\)
0.962051 0.272868i \(-0.0879723\pi\)
\(968\) 0 0
\(969\) 25.0000 0.803116
\(970\) 0 0
\(971\) − 2.23607i − 0.0717588i −0.999356 0.0358794i \(-0.988577\pi\)
0.999356 0.0358794i \(-0.0114232\pi\)
\(972\) 0 0
\(973\) −44.2719 −1.41929
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 45.0000i − 1.43968i −0.694141 0.719839i \(-0.744216\pi\)
0.694141 0.719839i \(-0.255784\pi\)
\(978\) 0 0
\(979\) 2.23607i 0.0714650i
\(980\) 0 0
\(981\) − 12.6491i − 0.403855i
\(982\) 0 0
\(983\) − 19.7990i − 0.631490i −0.948844 0.315745i \(-0.897746\pi\)
0.948844 0.315745i \(-0.102254\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 17.8885 0.569399
\(988\) 0 0
\(989\) − 50.5964i − 1.60887i
\(990\) 0 0
\(991\) −42.4264 −1.34772 −0.673860 0.738859i \(-0.735365\pi\)
−0.673860 + 0.738859i \(0.735365\pi\)
\(992\) 0 0
\(993\) − 75.0000i − 2.38005i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.2982 0.801203 0.400601 0.916252i \(-0.368801\pi\)
0.400601 + 0.916252i \(0.368801\pi\)
\(998\) 0 0
\(999\) −14.1421 −0.447437
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3200.2.f.r.449.1 8
4.3 odd 2 inner 3200.2.f.r.449.8 8
5.2 odd 4 3200.2.d.r.1601.4 yes 4
5.3 odd 4 3200.2.d.m.1601.1 4
5.4 even 2 inner 3200.2.f.r.449.7 8
8.3 odd 2 inner 3200.2.f.r.449.3 8
8.5 even 2 inner 3200.2.f.r.449.6 8
20.3 even 4 3200.2.d.m.1601.4 yes 4
20.7 even 4 3200.2.d.r.1601.1 yes 4
20.19 odd 2 inner 3200.2.f.r.449.2 8
40.3 even 4 3200.2.d.m.1601.2 yes 4
40.13 odd 4 3200.2.d.m.1601.3 yes 4
40.19 odd 2 inner 3200.2.f.r.449.5 8
40.27 even 4 3200.2.d.r.1601.3 yes 4
40.29 even 2 inner 3200.2.f.r.449.4 8
40.37 odd 4 3200.2.d.r.1601.2 yes 4
80.3 even 4 6400.2.a.cp.1.1 4
80.13 odd 4 6400.2.a.cp.1.4 4
80.27 even 4 6400.2.a.cs.1.2 4
80.37 odd 4 6400.2.a.cs.1.3 4
80.43 even 4 6400.2.a.cp.1.3 4
80.53 odd 4 6400.2.a.cp.1.2 4
80.67 even 4 6400.2.a.cs.1.4 4
80.77 odd 4 6400.2.a.cs.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.m.1601.1 4 5.3 odd 4
3200.2.d.m.1601.2 yes 4 40.3 even 4
3200.2.d.m.1601.3 yes 4 40.13 odd 4
3200.2.d.m.1601.4 yes 4 20.3 even 4
3200.2.d.r.1601.1 yes 4 20.7 even 4
3200.2.d.r.1601.2 yes 4 40.37 odd 4
3200.2.d.r.1601.3 yes 4 40.27 even 4
3200.2.d.r.1601.4 yes 4 5.2 odd 4
3200.2.f.r.449.1 8 1.1 even 1 trivial
3200.2.f.r.449.2 8 20.19 odd 2 inner
3200.2.f.r.449.3 8 8.3 odd 2 inner
3200.2.f.r.449.4 8 40.29 even 2 inner
3200.2.f.r.449.5 8 40.19 odd 2 inner
3200.2.f.r.449.6 8 8.5 even 2 inner
3200.2.f.r.449.7 8 5.4 even 2 inner
3200.2.f.r.449.8 8 4.3 odd 2 inner
6400.2.a.cp.1.1 4 80.3 even 4
6400.2.a.cp.1.2 4 80.53 odd 4
6400.2.a.cp.1.3 4 80.43 even 4
6400.2.a.cp.1.4 4 80.13 odd 4
6400.2.a.cs.1.1 4 80.77 odd 4
6400.2.a.cs.1.2 4 80.27 even 4
6400.2.a.cs.1.3 4 80.37 odd 4
6400.2.a.cs.1.4 4 80.67 even 4