Properties

Label 3200.2.f.f.449.2
Level $3200$
Weight $2$
Character 3200.449
Analytic conductor $25.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3200.449
Dual form 3200.2.f.f.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.00000i q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +4.00000i q^{7} -2.00000 q^{9} +3.00000i q^{11} +1.00000i q^{17} +7.00000i q^{19} +4.00000i q^{21} -4.00000i q^{23} -5.00000 q^{27} -8.00000i q^{29} +4.00000 q^{31} +3.00000i q^{33} -4.00000 q^{37} +3.00000 q^{41} -8.00000 q^{43} -9.00000 q^{49} +1.00000i q^{51} -12.0000 q^{53} +7.00000i q^{57} -8.00000i q^{59} -4.00000i q^{61} -8.00000i q^{63} +9.00000 q^{67} -4.00000i q^{69} -16.0000 q^{71} +11.0000i q^{73} -12.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -1.00000 q^{83} -8.00000i q^{87} -13.0000 q^{89} +4.00000 q^{93} +14.0000i q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 4 q^{9} - 10 q^{27} + 8 q^{31} - 8 q^{37} + 6 q^{41} - 16 q^{43} - 18 q^{49} - 24 q^{53} + 18 q^{67} - 32 q^{71} - 24 q^{77} - 8 q^{79} + 2 q^{81} - 2 q^{83} - 26 q^{89} + 8 q^{93} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000i 0.242536i 0.992620 + 0.121268i \(0.0386960\pi\)
−0.992620 + 0.121268i \(0.961304\pi\)
\(18\) 0 0
\(19\) 7.00000i 1.60591i 0.596040 + 0.802955i \(0.296740\pi\)
−0.596040 + 0.802955i \(0.703260\pi\)
\(20\) 0 0
\(21\) 4.00000i 0.872872i
\(22\) 0 0
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) − 8.00000i − 1.48556i −0.669534 0.742781i \(-0.733506\pi\)
0.669534 0.742781i \(-0.266494\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 3.00000i 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 1.00000i 0.140028i
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.00000i 0.927173i
\(58\) 0 0
\(59\) − 8.00000i − 1.04151i −0.853706 0.520756i \(-0.825650\pi\)
0.853706 0.520756i \(-0.174350\pi\)
\(60\) 0 0
\(61\) − 4.00000i − 0.512148i −0.966657 0.256074i \(-0.917571\pi\)
0.966657 0.256074i \(-0.0824290\pi\)
\(62\) 0 0
\(63\) − 8.00000i − 1.00791i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.00000 1.09952 0.549762 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) 0 0
\(69\) − 4.00000i − 0.481543i
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 11.0000i 1.28745i 0.765256 + 0.643726i \(0.222612\pi\)
−0.765256 + 0.643726i \(0.777388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 8.00000i − 0.857690i
\(88\) 0 0
\(89\) −13.0000 −1.37800 −0.688999 0.724763i \(-0.741949\pi\)
−0.688999 + 0.724763i \(0.741949\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 0 0
\(99\) − 6.00000i − 0.603023i
\(100\) 0 0
\(101\) 12.0000i 1.19404i 0.802225 + 0.597022i \(0.203650\pi\)
−0.802225 + 0.597022i \(0.796350\pi\)
\(102\) 0 0
\(103\) 16.0000i 1.57653i 0.615338 + 0.788263i \(0.289020\pi\)
−0.615338 + 0.788263i \(0.710980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.0000 1.25676 0.628379 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) 0 0
\(109\) − 20.0000i − 1.91565i −0.287348 0.957826i \(-0.592774\pi\)
0.287348 0.957826i \(-0.407226\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 9.00000i 0.846649i 0.905978 + 0.423324i \(0.139137\pi\)
−0.905978 + 0.423324i \(0.860863\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) 3.00000 0.270501
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) − 8.00000i − 0.698963i −0.936943 0.349482i \(-0.886358\pi\)
0.936943 0.349482i \(-0.113642\pi\)
\(132\) 0 0
\(133\) −28.0000 −2.42791
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000i 0.256307i 0.991754 + 0.128154i \(0.0409051\pi\)
−0.991754 + 0.128154i \(0.959095\pi\)
\(138\) 0 0
\(139\) 13.0000i 1.10265i 0.834292 + 0.551323i \(0.185877\pi\)
−0.834292 + 0.551323i \(0.814123\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −9.00000 −0.742307
\(148\) 0 0
\(149\) − 8.00000i − 0.655386i −0.944784 0.327693i \(-0.893729\pi\)
0.944784 0.327693i \(-0.106271\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) −9.00000 −0.704934 −0.352467 0.935824i \(-0.614657\pi\)
−0.352467 + 0.935824i \(0.614657\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000i 1.23812i 0.785345 + 0.619059i \(0.212486\pi\)
−0.785345 + 0.619059i \(0.787514\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) − 14.0000i − 1.07061i
\(172\) 0 0
\(173\) 16.0000 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 8.00000i − 0.601317i
\(178\) 0 0
\(179\) − 9.00000i − 0.672692i −0.941739 0.336346i \(-0.890809\pi\)
0.941739 0.336346i \(-0.109191\pi\)
\(180\) 0 0
\(181\) 12.0000i 0.891953i 0.895045 + 0.445976i \(0.147144\pi\)
−0.895045 + 0.445976i \(0.852856\pi\)
\(182\) 0 0
\(183\) − 4.00000i − 0.295689i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 0 0
\(189\) − 20.0000i − 1.45479i
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 7.00000i 0.503871i 0.967744 + 0.251936i \(0.0810671\pi\)
−0.967744 + 0.251936i \(0.918933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 0 0
\(203\) 32.0000 2.24596
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.00000i 0.556038i
\(208\) 0 0
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) 15.0000i 1.03264i 0.856395 + 0.516321i \(0.172699\pi\)
−0.856395 + 0.516321i \(0.827301\pi\)
\(212\) 0 0
\(213\) −16.0000 −1.09630
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 11.0000i 0.743311i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) 8.00000i 0.528655i 0.964433 + 0.264327i \(0.0851500\pi\)
−0.964433 + 0.264327i \(0.914850\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −15.0000 −0.966235 −0.483117 0.875556i \(-0.660496\pi\)
−0.483117 + 0.875556i \(0.660496\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.00000 −0.0633724
\(250\) 0 0
\(251\) 13.0000i 0.820553i 0.911961 + 0.410276i \(0.134568\pi\)
−0.911961 + 0.410276i \(0.865432\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) − 16.0000i − 0.994192i
\(260\) 0 0
\(261\) 16.0000i 0.990375i
\(262\) 0 0
\(263\) − 4.00000i − 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −13.0000 −0.795587
\(268\) 0 0
\(269\) − 16.0000i − 0.975537i −0.872973 0.487769i \(-0.837811\pi\)
0.872973 0.487769i \(-0.162189\pi\)
\(270\) 0 0
\(271\) 4.00000 0.242983 0.121491 0.992592i \(-0.461232\pi\)
0.121491 + 0.992592i \(0.461232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −19.0000 −1.12943 −0.564716 0.825285i \(-0.691014\pi\)
−0.564716 + 0.825285i \(0.691014\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) 14.0000i 0.820695i
\(292\) 0 0
\(293\) −20.0000 −1.16841 −0.584206 0.811605i \(-0.698594\pi\)
−0.584206 + 0.811605i \(0.698594\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 15.0000i − 0.870388i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 32.0000i − 1.84445i
\(302\) 0 0
\(303\) 12.0000i 0.689382i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) 16.0000i 0.910208i
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.00000 −0.449325 −0.224662 0.974437i \(-0.572128\pi\)
−0.224662 + 0.974437i \(0.572128\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 13.0000 0.725589
\(322\) 0 0
\(323\) −7.00000 −0.389490
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 20.0000i − 1.10600i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 13.0000i − 0.714545i −0.934000 0.357272i \(-0.883707\pi\)
0.934000 0.357272i \(-0.116293\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 9.00000i − 0.490261i −0.969490 0.245131i \(-0.921169\pi\)
0.969490 0.245131i \(-0.0788309\pi\)
\(338\) 0 0
\(339\) 9.00000i 0.488813i
\(340\) 0 0
\(341\) 12.0000i 0.649836i
\(342\) 0 0
\(343\) − 8.00000i − 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.00000 0.268414 0.134207 0.990953i \(-0.457151\pi\)
0.134207 + 0.990953i \(0.457151\pi\)
\(348\) 0 0
\(349\) − 4.00000i − 0.214115i −0.994253 0.107058i \(-0.965857\pi\)
0.994253 0.107058i \(-0.0341429\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −28.0000 −1.47778 −0.738892 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) −30.0000 −1.57895
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) − 48.0000i − 2.49204i
\(372\) 0 0
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 29.0000i − 1.48963i −0.667271 0.744815i \(-0.732538\pi\)
0.667271 0.744815i \(-0.267462\pi\)
\(380\) 0 0
\(381\) − 12.0000i − 0.614779i
\(382\) 0 0
\(383\) − 4.00000i − 0.204390i −0.994764 0.102195i \(-0.967413\pi\)
0.994764 0.102195i \(-0.0325866\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.0000 0.813326
\(388\) 0 0
\(389\) 12.0000i 0.608424i 0.952604 + 0.304212i \(0.0983931\pi\)
−0.952604 + 0.304212i \(0.901607\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) − 8.00000i − 0.403547i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) −28.0000 −1.40175
\(400\) 0 0
\(401\) −15.0000 −0.749064 −0.374532 0.927214i \(-0.622197\pi\)
−0.374532 + 0.927214i \(0.622197\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 12.0000i − 0.594818i
\(408\) 0 0
\(409\) 21.0000 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(410\) 0 0
\(411\) 3.00000i 0.147979i
\(412\) 0 0
\(413\) 32.0000 1.57462
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 13.0000i 0.636613i
\(418\) 0 0
\(419\) 31.0000i 1.51445i 0.653155 + 0.757225i \(0.273445\pi\)
−0.653155 + 0.757225i \(0.726555\pi\)
\(420\) 0 0
\(421\) − 8.00000i − 0.389896i −0.980814 0.194948i \(-0.937546\pi\)
0.980814 0.194948i \(-0.0624538\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −28.0000 −1.34871 −0.674356 0.738406i \(-0.735579\pi\)
−0.674356 + 0.738406i \(0.735579\pi\)
\(432\) 0 0
\(433\) 31.0000i 1.48976i 0.667196 + 0.744882i \(0.267494\pi\)
−0.667196 + 0.744882i \(0.732506\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 28.0000 1.33942
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) −19.0000 −0.902717 −0.451359 0.892343i \(-0.649060\pi\)
−0.451359 + 0.892343i \(0.649060\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 8.00000i − 0.378387i
\(448\) 0 0
\(449\) 23.0000 1.08544 0.542719 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 0 0
\(451\) 9.00000i 0.423793i
\(452\) 0 0
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.00000i 0.140334i 0.997535 + 0.0701670i \(0.0223532\pi\)
−0.997535 + 0.0701670i \(0.977647\pi\)
\(458\) 0 0
\(459\) − 5.00000i − 0.233380i
\(460\) 0 0
\(461\) 8.00000i 0.372597i 0.982493 + 0.186299i \(0.0596492\pi\)
−0.982493 + 0.186299i \(0.940351\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 40.0000 1.85098 0.925490 0.378773i \(-0.123654\pi\)
0.925490 + 0.378773i \(0.123654\pi\)
\(468\) 0 0
\(469\) 36.0000i 1.66233i
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) − 24.0000i − 1.10352i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 24.0000 1.09888
\(478\) 0 0
\(479\) −28.0000 −1.27935 −0.639676 0.768644i \(-0.720932\pi\)
−0.639676 + 0.768644i \(0.720932\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000i 0.543772i 0.962329 + 0.271886i \(0.0876473\pi\)
−0.962329 + 0.271886i \(0.912353\pi\)
\(488\) 0 0
\(489\) −9.00000 −0.406994
\(490\) 0 0
\(491\) 24.0000i 1.08310i 0.840667 + 0.541552i \(0.182163\pi\)
−0.840667 + 0.541552i \(0.817837\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 64.0000i − 2.87079i
\(498\) 0 0
\(499\) 8.00000i 0.358129i 0.983837 + 0.179065i \(0.0573071\pi\)
−0.983837 + 0.179065i \(0.942693\pi\)
\(500\) 0 0
\(501\) 16.0000i 0.714827i
\(502\) 0 0
\(503\) 8.00000i 0.356702i 0.983967 + 0.178351i \(0.0570763\pi\)
−0.983967 + 0.178351i \(0.942924\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) 0 0
\(509\) 12.0000i 0.531891i 0.963988 + 0.265945i \(0.0856841\pi\)
−0.963988 + 0.265945i \(0.914316\pi\)
\(510\) 0 0
\(511\) −44.0000 −1.94645
\(512\) 0 0
\(513\) − 35.0000i − 1.54529i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 16.0000 0.702322
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) −13.0000 −0.568450 −0.284225 0.958758i \(-0.591736\pi\)
−0.284225 + 0.958758i \(0.591736\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 16.0000i 0.694341i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.00000i − 0.388379i
\(538\) 0 0
\(539\) − 27.0000i − 1.16297i
\(540\) 0 0
\(541\) − 32.0000i − 1.37579i −0.725811 0.687894i \(-0.758536\pi\)
0.725811 0.687894i \(-0.241464\pi\)
\(542\) 0 0
\(543\) 12.0000i 0.514969i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 0 0
\(549\) 8.00000i 0.341432i
\(550\) 0 0
\(551\) 56.0000 2.38568
\(552\) 0 0
\(553\) − 16.0000i − 0.680389i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.00000i 0.167984i
\(568\) 0 0
\(569\) 37.0000 1.55112 0.775560 0.631273i \(-0.217467\pi\)
0.775560 + 0.631273i \(0.217467\pi\)
\(570\) 0 0
\(571\) − 8.00000i − 0.334790i −0.985890 0.167395i \(-0.946465\pi\)
0.985890 0.167395i \(-0.0535355\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 31.0000i 1.29055i 0.763952 + 0.645273i \(0.223257\pi\)
−0.763952 + 0.645273i \(0.776743\pi\)
\(578\) 0 0
\(579\) 7.00000i 0.290910i
\(580\) 0 0
\(581\) − 4.00000i − 0.165948i
\(582\) 0 0
\(583\) − 36.0000i − 1.49097i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.00000 −0.206372 −0.103186 0.994662i \(-0.532904\pi\)
−0.103186 + 0.994662i \(0.532904\pi\)
\(588\) 0 0
\(589\) 28.0000i 1.15372i
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) − 23.0000i − 0.944497i −0.881466 0.472248i \(-0.843443\pi\)
0.881466 0.472248i \(-0.156557\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −19.0000 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(602\) 0 0
\(603\) −18.0000 −0.733017
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000i 1.29884i 0.760430 + 0.649420i \(0.224988\pi\)
−0.760430 + 0.649420i \(0.775012\pi\)
\(608\) 0 0
\(609\) 32.0000 1.29671
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 0 0
\(619\) 40.0000i 1.60774i 0.594808 + 0.803868i \(0.297228\pi\)
−0.594808 + 0.803868i \(0.702772\pi\)
\(620\) 0 0
\(621\) 20.0000i 0.802572i
\(622\) 0 0
\(623\) − 52.0000i − 2.08334i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −21.0000 −0.838659
\(628\) 0 0
\(629\) − 4.00000i − 0.159490i
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) 15.0000i 0.596196i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 32.0000 1.26590
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 16.0000i 0.627089i
\(652\) 0 0
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 22.0000i − 0.858302i
\(658\) 0 0
\(659\) 1.00000i 0.0389545i 0.999810 + 0.0194772i \(0.00620019\pi\)
−0.999810 + 0.0194772i \(0.993800\pi\)
\(660\) 0 0
\(661\) − 40.0000i − 1.55582i −0.628376 0.777910i \(-0.716280\pi\)
0.628376 0.777910i \(-0.283720\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −32.0000 −1.23904
\(668\) 0 0
\(669\) 8.00000i 0.309298i
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) − 14.0000i − 0.539660i −0.962908 0.269830i \(-0.913032\pi\)
0.962908 0.269830i \(-0.0869676\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) 0 0
\(679\) −56.0000 −2.14908
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) −13.0000 −0.497431 −0.248716 0.968577i \(-0.580008\pi\)
−0.248716 + 0.968577i \(0.580008\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.00000i 0.305219i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 9.00000i 0.342376i 0.985238 + 0.171188i \(0.0547606\pi\)
−0.985238 + 0.171188i \(0.945239\pi\)
\(692\) 0 0
\(693\) 24.0000 0.911685
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.00000i 0.113633i
\(698\) 0 0
\(699\) 10.0000i 0.378235i
\(700\) 0 0
\(701\) 16.0000i 0.604312i 0.953259 + 0.302156i \(0.0977063\pi\)
−0.953259 + 0.302156i \(0.902294\pi\)
\(702\) 0 0
\(703\) − 28.0000i − 1.05604i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −48.0000 −1.80523
\(708\) 0 0
\(709\) 40.0000i 1.50223i 0.660171 + 0.751116i \(0.270484\pi\)
−0.660171 + 0.751116i \(0.729516\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) − 16.0000i − 0.599205i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) −64.0000 −2.38348
\(722\) 0 0
\(723\) −15.0000 −0.557856
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 40.0000i 1.48352i 0.670667 + 0.741759i \(0.266008\pi\)
−0.670667 + 0.741759i \(0.733992\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) − 8.00000i − 0.295891i
\(732\) 0 0
\(733\) −32.0000 −1.18195 −0.590973 0.806691i \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27.0000i 0.994558i
\(738\) 0 0
\(739\) − 24.0000i − 0.882854i −0.897297 0.441427i \(-0.854472\pi\)
0.897297 0.441427i \(-0.145528\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) 52.0000i 1.90004i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 13.0000i 0.473746i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −32.0000 −1.16306 −0.581530 0.813525i \(-0.697546\pi\)
−0.581530 + 0.813525i \(0.697546\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 0 0
\(763\) 80.0000 2.89619
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 39.0000 1.40638 0.703188 0.711004i \(-0.251759\pi\)
0.703188 + 0.711004i \(0.251759\pi\)
\(770\) 0 0
\(771\) 18.0000i 0.648254i
\(772\) 0 0
\(773\) 8.00000 0.287740 0.143870 0.989597i \(-0.454045\pi\)
0.143870 + 0.989597i \(0.454045\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 16.0000i − 0.573997i
\(778\) 0 0
\(779\) 21.0000i 0.752403i
\(780\) 0 0
\(781\) − 48.0000i − 1.71758i
\(782\) 0 0
\(783\) 40.0000i 1.42948i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 0 0
\(789\) − 4.00000i − 0.142404i
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 26.0000 0.918665
\(802\) 0 0
\(803\) −33.0000 −1.16454
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 16.0000i − 0.563227i
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) − 8.00000i − 0.280918i −0.990086 0.140459i \(-0.955142\pi\)
0.990086 0.140459i \(-0.0448578\pi\)
\(812\) 0 0
\(813\) 4.00000 0.140286
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 56.0000i − 1.95919i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 36.0000i − 1.25641i −0.778048 0.628204i \(-0.783790\pi\)
0.778048 0.628204i \(-0.216210\pi\)
\(822\) 0 0
\(823\) 24.0000i 0.836587i 0.908312 + 0.418294i \(0.137372\pi\)
−0.908312 + 0.418294i \(0.862628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 51.0000 1.77344 0.886722 0.462303i \(-0.152977\pi\)
0.886722 + 0.462303i \(0.152977\pi\)
\(828\) 0 0
\(829\) 32.0000i 1.11141i 0.831381 + 0.555703i \(0.187551\pi\)
−0.831381 + 0.555703i \(0.812449\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) 0 0
\(833\) − 9.00000i − 0.311832i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −20.0000 −0.691301
\(838\) 0 0
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) −35.0000 −1.20690
\(842\) 0 0
\(843\) 6.00000 0.206651
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 8.00000i 0.274883i
\(848\) 0 0
\(849\) −19.0000 −0.652078
\(850\) 0 0
\(851\) 16.0000i 0.548473i
\(852\) 0 0
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.0000i 0.922302i 0.887322 + 0.461151i \(0.152563\pi\)
−0.887322 + 0.461151i \(0.847437\pi\)
\(858\) 0 0
\(859\) 5.00000i 0.170598i 0.996355 + 0.0852989i \(0.0271845\pi\)
−0.996355 + 0.0852989i \(0.972815\pi\)
\(860\) 0 0
\(861\) 12.0000i 0.408959i
\(862\) 0 0
\(863\) 56.0000i 1.90626i 0.302558 + 0.953131i \(0.402160\pi\)
−0.302558 + 0.953131i \(0.597840\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) − 12.0000i − 0.407072i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 28.0000i − 0.947656i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.00000 0.270141 0.135070 0.990836i \(-0.456874\pi\)
0.135070 + 0.990836i \(0.456874\pi\)
\(878\) 0 0
\(879\) −20.0000 −0.674583
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 0 0
\(883\) 25.0000 0.841317 0.420658 0.907219i \(-0.361799\pi\)
0.420658 + 0.907219i \(0.361799\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) 0 0
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) 3.00000i 0.100504i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 32.0000i − 1.06726i
\(900\) 0 0
\(901\) − 12.0000i − 0.399778i
\(902\) 0 0
\(903\) − 32.0000i − 1.06489i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 0 0
\(909\) − 24.0000i − 0.796030i
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) − 3.00000i − 0.0992855i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 32.0000 1.05673
\(918\) 0 0
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) 7.00000 0.230658
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 32.0000i − 1.05102i
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) − 63.0000i − 2.06474i
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.00000i 0.163343i 0.996659 + 0.0816714i \(0.0260258\pi\)
−0.996659 + 0.0816714i \(0.973974\pi\)
\(938\) 0 0
\(939\) 6.00000i 0.195803i
\(940\) 0 0
\(941\) 48.0000i 1.56476i 0.622804 + 0.782378i \(0.285993\pi\)
−0.622804 + 0.782378i \(0.714007\pi\)
\(942\) 0 0
\(943\) − 12.0000i − 0.390774i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −8.00000 −0.259418
\(952\) 0 0
\(953\) − 27.0000i − 0.874616i −0.899312 0.437308i \(-0.855932\pi\)
0.899312 0.437308i \(-0.144068\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 24.0000 0.775810
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −26.0000 −0.837838
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 0 0
\(969\) −7.00000 −0.224872
\(970\) 0 0
\(971\) − 45.0000i − 1.44412i −0.691831 0.722059i \(-0.743196\pi\)
0.691831 0.722059i \(-0.256804\pi\)
\(972\) 0 0
\(973\) −52.0000 −1.66704
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 31.0000i − 0.991778i −0.868386 0.495889i \(-0.834842\pi\)
0.868386 0.495889i \(-0.165158\pi\)
\(978\) 0 0
\(979\) − 39.0000i − 1.24645i
\(980\) 0 0
\(981\) 40.0000i 1.27710i
\(982\) 0 0
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000i 1.01754i
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 0 0
\(993\) − 13.0000i − 0.412543i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −24.0000 −0.760088 −0.380044 0.924968i \(-0.624091\pi\)
−0.380044 + 0.924968i \(0.624091\pi\)
\(998\) 0 0
\(999\) 20.0000 0.632772
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.f.449.2 2
4.3 odd 2 3200.2.f.a.449.1 2
5.2 odd 4 3200.2.d.a.1601.1 2
5.3 odd 4 3200.2.d.h.1601.2 yes 2
5.4 even 2 3200.2.f.b.449.1 2
8.3 odd 2 3200.2.f.e.449.1 2
8.5 even 2 3200.2.f.b.449.2 2
20.3 even 4 3200.2.d.b.1601.1 yes 2
20.7 even 4 3200.2.d.g.1601.2 yes 2
20.19 odd 2 3200.2.f.e.449.2 2
40.3 even 4 3200.2.d.b.1601.2 yes 2
40.13 odd 4 3200.2.d.h.1601.1 yes 2
40.19 odd 2 3200.2.f.a.449.2 2
40.27 even 4 3200.2.d.g.1601.1 yes 2
40.29 even 2 inner 3200.2.f.f.449.1 2
40.37 odd 4 3200.2.d.a.1601.2 yes 2
80.3 even 4 6400.2.a.v.1.1 1
80.13 odd 4 6400.2.a.c.1.1 1
80.27 even 4 6400.2.a.q.1.1 1
80.37 odd 4 6400.2.a.h.1.1 1
80.43 even 4 6400.2.a.i.1.1 1
80.53 odd 4 6400.2.a.p.1.1 1
80.67 even 4 6400.2.a.b.1.1 1
80.77 odd 4 6400.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.a.1601.1 2 5.2 odd 4
3200.2.d.a.1601.2 yes 2 40.37 odd 4
3200.2.d.b.1601.1 yes 2 20.3 even 4
3200.2.d.b.1601.2 yes 2 40.3 even 4
3200.2.d.g.1601.1 yes 2 40.27 even 4
3200.2.d.g.1601.2 yes 2 20.7 even 4
3200.2.d.h.1601.1 yes 2 40.13 odd 4
3200.2.d.h.1601.2 yes 2 5.3 odd 4
3200.2.f.a.449.1 2 4.3 odd 2
3200.2.f.a.449.2 2 40.19 odd 2
3200.2.f.b.449.1 2 5.4 even 2
3200.2.f.b.449.2 2 8.5 even 2
3200.2.f.e.449.1 2 8.3 odd 2
3200.2.f.e.449.2 2 20.19 odd 2
3200.2.f.f.449.1 2 40.29 even 2 inner
3200.2.f.f.449.2 2 1.1 even 1 trivial
6400.2.a.b.1.1 1 80.67 even 4
6400.2.a.c.1.1 1 80.13 odd 4
6400.2.a.h.1.1 1 80.37 odd 4
6400.2.a.i.1.1 1 80.43 even 4
6400.2.a.p.1.1 1 80.53 odd 4
6400.2.a.q.1.1 1 80.27 even 4
6400.2.a.v.1.1 1 80.3 even 4
6400.2.a.w.1.1 1 80.77 odd 4