Properties

Label 1600.2.f.h.1249.1
Level $1600$
Weight $2$
Character 1600.1249
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1249
Dual form 1600.2.f.h.1249.2

$q$-expansion

\(f(q)\) \(=\) \(q-0.732051 q^{3} -1.26795i q^{7} -2.46410 q^{9} +O(q^{10})\) \(q-0.732051 q^{3} -1.26795i q^{7} -2.46410 q^{9} +3.46410i q^{11} +3.46410 q^{13} -3.46410i q^{17} -2.00000i q^{19} +0.928203i q^{21} +8.19615i q^{23} +4.00000 q^{27} -9.46410 q^{31} -2.53590i q^{33} +6.00000 q^{37} -2.53590 q^{39} +2.53590 q^{41} +10.1962 q^{43} +8.19615i q^{47} +5.39230 q^{49} +2.53590i q^{51} +10.3923 q^{53} +1.46410i q^{57} +6.00000i q^{59} +12.9282i q^{61} +3.12436i q^{63} -10.1962 q^{67} -6.00000i q^{69} -4.39230 q^{71} -14.3923i q^{73} +4.39230 q^{77} +12.0000 q^{79} +4.46410 q^{81} +4.73205 q^{83} -0.928203 q^{89} -4.39230i q^{91} +6.92820 q^{93} +6.39230i q^{97} -8.53590i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{9} + 16q^{27} - 24q^{31} + 24q^{37} - 24q^{39} + 24q^{41} + 20q^{43} - 20q^{49} - 20q^{67} + 24q^{71} - 24q^{77} + 48q^{79} + 4q^{81} + 12q^{83} + 24q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −0.732051 −0.422650 −0.211325 0.977416i \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.26795i − 0.479240i −0.970867 0.239620i \(-0.922977\pi\)
0.970867 0.239620i \(-0.0770228\pi\)
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.46410i − 0.840168i −0.907485 0.420084i \(-0.862001\pi\)
0.907485 0.420084i \(-0.137999\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0.928203i 0.202551i
\(22\) 0 0
\(23\) 8.19615i 1.70902i 0.519438 + 0.854508i \(0.326141\pi\)
−0.519438 + 0.854508i \(0.673859\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −9.46410 −1.69980 −0.849901 0.526942i \(-0.823339\pi\)
−0.849901 + 0.526942i \(0.823339\pi\)
\(32\) 0 0
\(33\) − 2.53590i − 0.441443i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −2.53590 −0.406069
\(40\) 0 0
\(41\) 2.53590 0.396041 0.198020 0.980198i \(-0.436549\pi\)
0.198020 + 0.980198i \(0.436549\pi\)
\(42\) 0 0
\(43\) 10.1962 1.55490 0.777449 0.628946i \(-0.216513\pi\)
0.777449 + 0.628946i \(0.216513\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.19615i 1.19553i 0.801671 + 0.597766i \(0.203945\pi\)
−0.801671 + 0.597766i \(0.796055\pi\)
\(48\) 0 0
\(49\) 5.39230 0.770329
\(50\) 0 0
\(51\) 2.53590i 0.355097i
\(52\) 0 0
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.46410i 0.193925i
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 12.9282i 1.65529i 0.561254 + 0.827643i \(0.310319\pi\)
−0.561254 + 0.827643i \(0.689681\pi\)
\(62\) 0 0
\(63\) 3.12436i 0.393632i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.1962 −1.24566 −0.622829 0.782358i \(-0.714017\pi\)
−0.622829 + 0.782358i \(0.714017\pi\)
\(68\) 0 0
\(69\) − 6.00000i − 0.722315i
\(70\) 0 0
\(71\) −4.39230 −0.521271 −0.260635 0.965437i \(-0.583932\pi\)
−0.260635 + 0.965437i \(0.583932\pi\)
\(72\) 0 0
\(73\) − 14.3923i − 1.68449i −0.539093 0.842246i \(-0.681233\pi\)
0.539093 0.842246i \(-0.318767\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.39230 0.500550
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) 4.73205 0.519410 0.259705 0.965688i \(-0.416375\pi\)
0.259705 + 0.965688i \(0.416375\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 0 0
\(91\) − 4.39230i − 0.460439i
\(92\) 0 0
\(93\) 6.92820 0.718421
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.39230i 0.649040i 0.945879 + 0.324520i \(0.105203\pi\)
−0.945879 + 0.324520i \(0.894797\pi\)
\(98\) 0 0
\(99\) − 8.53590i − 0.857890i
\(100\) 0 0
\(101\) 12.0000i 1.19404i 0.802225 + 0.597022i \(0.203650\pi\)
−0.802225 + 0.597022i \(0.796350\pi\)
\(102\) 0 0
\(103\) 8.19615i 0.807591i 0.914849 + 0.403795i \(0.132309\pi\)
−0.914849 + 0.403795i \(0.867691\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.7321 1.61755 0.808774 0.588119i \(-0.200131\pi\)
0.808774 + 0.588119i \(0.200131\pi\)
\(108\) 0 0
\(109\) − 0.928203i − 0.0889057i −0.999011 0.0444529i \(-0.985846\pi\)
0.999011 0.0444529i \(-0.0141545\pi\)
\(110\) 0 0
\(111\) −4.39230 −0.416899
\(112\) 0 0
\(113\) 0.928203i 0.0873180i 0.999046 + 0.0436590i \(0.0139015\pi\)
−0.999046 + 0.0436590i \(0.986098\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −8.53590 −0.789144
\(118\) 0 0
\(119\) −4.39230 −0.402642
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) −1.85641 −0.167387
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.80385i 0.337537i 0.985656 + 0.168768i \(0.0539790\pi\)
−0.985656 + 0.168768i \(0.946021\pi\)
\(128\) 0 0
\(129\) −7.46410 −0.657178
\(130\) 0 0
\(131\) 10.3923i 0.907980i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(132\) 0 0
\(133\) −2.53590 −0.219890
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.9282i − 1.10453i −0.833668 0.552265i \(-0.813763\pi\)
0.833668 0.552265i \(-0.186237\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i 0.905618 + 0.424094i \(0.139408\pi\)
−0.905618 + 0.424094i \(0.860592\pi\)
\(140\) 0 0
\(141\) − 6.00000i − 0.505291i
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.94744 −0.325579
\(148\) 0 0
\(149\) 18.0000i 1.47462i 0.675556 + 0.737309i \(0.263904\pi\)
−0.675556 + 0.737309i \(0.736096\pi\)
\(150\) 0 0
\(151\) 2.53590 0.206368 0.103184 0.994662i \(-0.467097\pi\)
0.103184 + 0.994662i \(0.467097\pi\)
\(152\) 0 0
\(153\) 8.53590i 0.690086i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.928203 −0.0740787 −0.0370393 0.999314i \(-0.511793\pi\)
−0.0370393 + 0.999314i \(0.511793\pi\)
\(158\) 0 0
\(159\) −7.60770 −0.603329
\(160\) 0 0
\(161\) 10.3923 0.819028
\(162\) 0 0
\(163\) −5.80385 −0.454592 −0.227296 0.973826i \(-0.572989\pi\)
−0.227296 + 0.973826i \(0.572989\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.19615i 0.634237i 0.948386 + 0.317119i \(0.102715\pi\)
−0.948386 + 0.317119i \(0.897285\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 4.92820i 0.376869i
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4.39230i − 0.330146i
\(178\) 0 0
\(179\) − 19.8564i − 1.48414i −0.670324 0.742069i \(-0.733845\pi\)
0.670324 0.742069i \(-0.266155\pi\)
\(180\) 0 0
\(181\) − 6.92820i − 0.514969i −0.966282 0.257485i \(-0.917106\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) − 9.46410i − 0.699607i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) − 5.07180i − 0.368919i
\(190\) 0 0
\(191\) 16.3923 1.18611 0.593053 0.805164i \(-0.297923\pi\)
0.593053 + 0.805164i \(0.297923\pi\)
\(192\) 0 0
\(193\) 6.39230i 0.460128i 0.973175 + 0.230064i \(0.0738936\pi\)
−0.973175 + 0.230064i \(0.926106\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.3923 0.740421 0.370211 0.928948i \(-0.379286\pi\)
0.370211 + 0.928948i \(0.379286\pi\)
\(198\) 0 0
\(199\) 6.92820 0.491127 0.245564 0.969380i \(-0.421027\pi\)
0.245564 + 0.969380i \(0.421027\pi\)
\(200\) 0 0
\(201\) 7.46410 0.526477
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 20.1962i − 1.40373i
\(208\) 0 0
\(209\) 6.92820 0.479234
\(210\) 0 0
\(211\) − 6.39230i − 0.440064i −0.975493 0.220032i \(-0.929384\pi\)
0.975493 0.220032i \(-0.0706162\pi\)
\(212\) 0 0
\(213\) 3.21539 0.220315
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000i 0.814613i
\(218\) 0 0
\(219\) 10.5359i 0.711950i
\(220\) 0 0
\(221\) − 12.0000i − 0.807207i
\(222\) 0 0
\(223\) − 15.1244i − 1.01280i −0.862298 0.506401i \(-0.830976\pi\)
0.862298 0.506401i \(-0.169024\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.5885 −1.23376 −0.616880 0.787058i \(-0.711603\pi\)
−0.616880 + 0.787058i \(0.711603\pi\)
\(228\) 0 0
\(229\) − 18.9282i − 1.25081i −0.780300 0.625405i \(-0.784934\pi\)
0.780300 0.625405i \(-0.215066\pi\)
\(230\) 0 0
\(231\) −3.21539 −0.211557
\(232\) 0 0
\(233\) − 1.60770i − 0.105324i −0.998612 0.0526618i \(-0.983229\pi\)
0.998612 0.0526618i \(-0.0167705\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.78461 −0.570622
\(238\) 0 0
\(239\) −20.7846 −1.34444 −0.672222 0.740349i \(-0.734660\pi\)
−0.672222 + 0.740349i \(0.734660\pi\)
\(240\) 0 0
\(241\) −20.3923 −1.31358 −0.656792 0.754072i \(-0.728087\pi\)
−0.656792 + 0.754072i \(0.728087\pi\)
\(242\) 0 0
\(243\) −15.2679 −0.979439
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 6.92820i − 0.440831i
\(248\) 0 0
\(249\) −3.46410 −0.219529
\(250\) 0 0
\(251\) − 15.4641i − 0.976085i −0.872820 0.488043i \(-0.837711\pi\)
0.872820 0.488043i \(-0.162289\pi\)
\(252\) 0 0
\(253\) −28.3923 −1.78501
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) − 7.60770i − 0.472719i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 20.1962i − 1.24535i −0.782481 0.622674i \(-0.786046\pi\)
0.782481 0.622674i \(-0.213954\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.679492 0.0415842
\(268\) 0 0
\(269\) 14.7846i 0.901434i 0.892667 + 0.450717i \(0.148832\pi\)
−0.892667 + 0.450717i \(0.851168\pi\)
\(270\) 0 0
\(271\) 4.39230 0.266814 0.133407 0.991061i \(-0.457408\pi\)
0.133407 + 0.991061i \(0.457408\pi\)
\(272\) 0 0
\(273\) 3.21539i 0.194604i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −19.8564 −1.19306 −0.596528 0.802592i \(-0.703454\pi\)
−0.596528 + 0.802592i \(0.703454\pi\)
\(278\) 0 0
\(279\) 23.3205 1.39616
\(280\) 0 0
\(281\) −28.3923 −1.69374 −0.846871 0.531798i \(-0.821517\pi\)
−0.846871 + 0.531798i \(0.821517\pi\)
\(282\) 0 0
\(283\) 10.5885 0.629418 0.314709 0.949188i \(-0.398093\pi\)
0.314709 + 0.949188i \(0.398093\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 3.21539i − 0.189798i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) − 4.67949i − 0.274317i
\(292\) 0 0
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 13.8564i 0.804030i
\(298\) 0 0
\(299\) 28.3923i 1.64197i
\(300\) 0 0
\(301\) − 12.9282i − 0.745169i
\(302\) 0 0
\(303\) − 8.78461i − 0.504663i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −34.1962 −1.95168 −0.975839 0.218492i \(-0.929886\pi\)
−0.975839 + 0.218492i \(0.929886\pi\)
\(308\) 0 0
\(309\) − 6.00000i − 0.341328i
\(310\) 0 0
\(311\) −7.60770 −0.431393 −0.215696 0.976460i \(-0.569202\pi\)
−0.215696 + 0.976460i \(0.569202\pi\)
\(312\) 0 0
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.3923 1.25768 0.628839 0.777536i \(-0.283531\pi\)
0.628839 + 0.777536i \(0.283531\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.2487 −0.683656
\(322\) 0 0
\(323\) −6.92820 −0.385496
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0.679492i 0.0375760i
\(328\) 0 0
\(329\) 10.3923 0.572946
\(330\) 0 0
\(331\) − 5.60770i − 0.308227i −0.988053 0.154113i \(-0.950748\pi\)
0.988053 0.154113i \(-0.0492521\pi\)
\(332\) 0 0
\(333\) −14.7846 −0.810192
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 0 0
\(339\) − 0.679492i − 0.0369049i
\(340\) 0 0
\(341\) − 32.7846i − 1.77539i
\(342\) 0 0
\(343\) − 15.7128i − 0.848412i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0526 1.50594 0.752970 0.658055i \(-0.228620\pi\)
0.752970 + 0.658055i \(0.228620\pi\)
\(348\) 0 0
\(349\) − 8.78461i − 0.470229i −0.971968 0.235115i \(-0.924453\pi\)
0.971968 0.235115i \(-0.0755466\pi\)
\(350\) 0 0
\(351\) 13.8564 0.739600
\(352\) 0 0
\(353\) − 14.7846i − 0.786905i −0.919345 0.393453i \(-0.871281\pi\)
0.919345 0.393453i \(-0.128719\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.21539 0.170177
\(358\) 0 0
\(359\) −8.78461 −0.463634 −0.231817 0.972759i \(-0.574467\pi\)
−0.231817 + 0.972759i \(0.574467\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 0.732051 0.0384227
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 10.0526i − 0.524739i −0.964967 0.262370i \(-0.915496\pi\)
0.964967 0.262370i \(-0.0845040\pi\)
\(368\) 0 0
\(369\) −6.24871 −0.325295
\(370\) 0 0
\(371\) − 13.1769i − 0.684111i
\(372\) 0 0
\(373\) −7.85641 −0.406789 −0.203395 0.979097i \(-0.565197\pi\)
−0.203395 + 0.979097i \(0.565197\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.00000i 0.102733i 0.998680 + 0.0513665i \(0.0163577\pi\)
−0.998680 + 0.0513665i \(0.983642\pi\)
\(380\) 0 0
\(381\) − 2.78461i − 0.142660i
\(382\) 0 0
\(383\) 3.80385i 0.194368i 0.995266 + 0.0971838i \(0.0309835\pi\)
−0.995266 + 0.0971838i \(0.969017\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −25.1244 −1.27714
\(388\) 0 0
\(389\) 26.7846i 1.35803i 0.734123 + 0.679017i \(0.237594\pi\)
−0.734123 + 0.679017i \(0.762406\pi\)
\(390\) 0 0
\(391\) 28.3923 1.43586
\(392\) 0 0
\(393\) − 7.60770i − 0.383757i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −27.4641 −1.37838 −0.689192 0.724579i \(-0.742034\pi\)
−0.689192 + 0.724579i \(0.742034\pi\)
\(398\) 0 0
\(399\) 1.85641 0.0929366
\(400\) 0 0
\(401\) −4.14359 −0.206921 −0.103461 0.994634i \(-0.532992\pi\)
−0.103461 + 0.994634i \(0.532992\pi\)
\(402\) 0 0
\(403\) −32.7846 −1.63312
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.7846i 1.03025i
\(408\) 0 0
\(409\) −3.60770 −0.178389 −0.0891945 0.996014i \(-0.528429\pi\)
−0.0891945 + 0.996014i \(0.528429\pi\)
\(410\) 0 0
\(411\) 9.46410i 0.466830i
\(412\) 0 0
\(413\) 7.60770 0.374350
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 7.32051i − 0.358487i
\(418\) 0 0
\(419\) − 0.928203i − 0.0453457i −0.999743 0.0226728i \(-0.992782\pi\)
0.999743 0.0226728i \(-0.00721761\pi\)
\(420\) 0 0
\(421\) 6.00000i 0.292422i 0.989253 + 0.146211i \(0.0467079\pi\)
−0.989253 + 0.146211i \(0.953292\pi\)
\(422\) 0 0
\(423\) − 20.1962i − 0.981971i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 16.3923 0.793279
\(428\) 0 0
\(429\) − 8.78461i − 0.424125i
\(430\) 0 0
\(431\) 28.3923 1.36761 0.683805 0.729665i \(-0.260324\pi\)
0.683805 + 0.729665i \(0.260324\pi\)
\(432\) 0 0
\(433\) 26.3923i 1.26833i 0.773196 + 0.634167i \(0.218657\pi\)
−0.773196 + 0.634167i \(0.781343\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.3923 0.784150
\(438\) 0 0
\(439\) −18.9282 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(440\) 0 0
\(441\) −13.2872 −0.632723
\(442\) 0 0
\(443\) −0.339746 −0.0161418 −0.00807091 0.999967i \(-0.502569\pi\)
−0.00807091 + 0.999967i \(0.502569\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 13.1769i − 0.623247i
\(448\) 0 0
\(449\) 2.53590 0.119676 0.0598382 0.998208i \(-0.480942\pi\)
0.0598382 + 0.998208i \(0.480942\pi\)
\(450\) 0 0
\(451\) 8.78461i 0.413651i
\(452\) 0 0
\(453\) −1.85641 −0.0872216
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 22.7846i − 1.06582i −0.846172 0.532910i \(-0.821099\pi\)
0.846172 0.532910i \(-0.178901\pi\)
\(458\) 0 0
\(459\) − 13.8564i − 0.646762i
\(460\) 0 0
\(461\) 12.0000i 0.558896i 0.960161 + 0.279448i \(0.0901514\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(462\) 0 0
\(463\) − 15.8038i − 0.734467i −0.930129 0.367234i \(-0.880305\pi\)
0.930129 0.367234i \(-0.119695\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.9808 1.06342 0.531711 0.846926i \(-0.321549\pi\)
0.531711 + 0.846926i \(0.321549\pi\)
\(468\) 0 0
\(469\) 12.9282i 0.596969i
\(470\) 0 0
\(471\) 0.679492 0.0313093
\(472\) 0 0
\(473\) 35.3205i 1.62404i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −25.6077 −1.17250
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 20.7846 0.947697
\(482\) 0 0
\(483\) −7.60770 −0.346162
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 39.1244i − 1.77289i −0.462830 0.886447i \(-0.653166\pi\)
0.462830 0.886447i \(-0.346834\pi\)
\(488\) 0 0
\(489\) 4.24871 0.192133
\(490\) 0 0
\(491\) − 22.3923i − 1.01055i −0.862958 0.505275i \(-0.831391\pi\)
0.862958 0.505275i \(-0.168609\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.56922i 0.249814i
\(498\) 0 0
\(499\) − 39.5692i − 1.77136i −0.464295 0.885681i \(-0.653692\pi\)
0.464295 0.885681i \(-0.346308\pi\)
\(500\) 0 0
\(501\) − 6.00000i − 0.268060i
\(502\) 0 0
\(503\) 8.19615i 0.365448i 0.983164 + 0.182724i \(0.0584915\pi\)
−0.983164 + 0.182724i \(0.941508\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.732051 0.0325115
\(508\) 0 0
\(509\) 8.78461i 0.389371i 0.980866 + 0.194685i \(0.0623686\pi\)
−0.980866 + 0.194685i \(0.937631\pi\)
\(510\) 0 0
\(511\) −18.2487 −0.807275
\(512\) 0 0
\(513\) − 8.00000i − 0.353209i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −28.3923 −1.24869
\(518\) 0 0
\(519\) −4.39230 −0.192801
\(520\) 0 0
\(521\) 31.8564 1.39565 0.697827 0.716266i \(-0.254150\pi\)
0.697827 + 0.716266i \(0.254150\pi\)
\(522\) 0 0
\(523\) −14.9808 −0.655063 −0.327531 0.944840i \(-0.606217\pi\)
−0.327531 + 0.944840i \(0.606217\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 32.7846i 1.42812i
\(528\) 0 0
\(529\) −44.1769 −1.92074
\(530\) 0 0
\(531\) − 14.7846i − 0.641597i
\(532\) 0 0
\(533\) 8.78461 0.380504
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 14.5359i 0.627270i
\(538\) 0 0
\(539\) 18.6795i 0.804583i
\(540\) 0 0
\(541\) − 15.7128i − 0.675547i −0.941227 0.337773i \(-0.890326\pi\)
0.941227 0.337773i \(-0.109674\pi\)
\(542\) 0 0
\(543\) 5.07180i 0.217652i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.80385 −0.0771270 −0.0385635 0.999256i \(-0.512278\pi\)
−0.0385635 + 0.999256i \(0.512278\pi\)
\(548\) 0 0
\(549\) − 31.8564i − 1.35960i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 15.2154i − 0.647024i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.7846 0.626444 0.313222 0.949680i \(-0.398592\pi\)
0.313222 + 0.949680i \(0.398592\pi\)
\(558\) 0 0
\(559\) 35.3205 1.49390
\(560\) 0 0
\(561\) −8.78461 −0.370887
\(562\) 0 0
\(563\) 26.1962 1.10404 0.552018 0.833832i \(-0.313858\pi\)
0.552018 + 0.833832i \(0.313858\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 5.66025i − 0.237708i
\(568\) 0 0
\(569\) 42.2487 1.77116 0.885579 0.464489i \(-0.153762\pi\)
0.885579 + 0.464489i \(0.153762\pi\)
\(570\) 0 0
\(571\) 30.3923i 1.27188i 0.771739 + 0.635939i \(0.219387\pi\)
−0.771739 + 0.635939i \(0.780613\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 0 0
\(579\) − 4.67949i − 0.194473i
\(580\) 0 0
\(581\) − 6.00000i − 0.248922i
\(582\) 0 0
\(583\) 36.0000i 1.49097i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.5167 −0.557892 −0.278946 0.960307i \(-0.589985\pi\)
−0.278946 + 0.960307i \(0.589985\pi\)
\(588\) 0 0
\(589\) 18.9282i 0.779923i
\(590\) 0 0
\(591\) −7.60770 −0.312939
\(592\) 0 0
\(593\) − 0.928203i − 0.0381167i −0.999818 0.0190584i \(-0.993933\pi\)
0.999818 0.0190584i \(-0.00606683\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.07180 −0.207575
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −20.3923 −0.831819 −0.415910 0.909406i \(-0.636537\pi\)
−0.415910 + 0.909406i \(0.636537\pi\)
\(602\) 0 0
\(603\) 25.1244 1.02314
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.19615i 0.332672i 0.986069 + 0.166336i \(0.0531936\pi\)
−0.986069 + 0.166336i \(0.946806\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 28.3923i 1.14863i
\(612\) 0 0
\(613\) 13.6077 0.549610 0.274805 0.961500i \(-0.411387\pi\)
0.274805 + 0.961500i \(0.411387\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 13.6077i − 0.547825i −0.961755 0.273913i \(-0.911682\pi\)
0.961755 0.273913i \(-0.0883179\pi\)
\(618\) 0 0
\(619\) 6.78461i 0.272696i 0.990661 + 0.136348i \(0.0435366\pi\)
−0.990661 + 0.136348i \(0.956463\pi\)
\(620\) 0 0
\(621\) 32.7846i 1.31560i
\(622\) 0 0
\(623\) 1.17691i 0.0471521i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.07180 −0.202548
\(628\) 0 0
\(629\) − 20.7846i − 0.828737i
\(630\) 0 0
\(631\) 21.4641 0.854472 0.427236 0.904140i \(-0.359487\pi\)
0.427236 + 0.904140i \(0.359487\pi\)
\(632\) 0 0
\(633\) 4.67949i 0.185993i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 18.6795 0.740108
\(638\) 0 0
\(639\) 10.8231 0.428155
\(640\) 0 0
\(641\) −4.39230 −0.173486 −0.0867428 0.996231i \(-0.527646\pi\)
−0.0867428 + 0.996231i \(0.527646\pi\)
\(642\) 0 0
\(643\) 10.5885 0.417568 0.208784 0.977962i \(-0.433049\pi\)
0.208784 + 0.977962i \(0.433049\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.5885i 1.43844i 0.694782 + 0.719220i \(0.255501\pi\)
−0.694782 + 0.719220i \(0.744499\pi\)
\(648\) 0 0
\(649\) −20.7846 −0.815867
\(650\) 0 0
\(651\) − 8.78461i − 0.344296i
\(652\) 0 0
\(653\) 19.1769 0.750451 0.375225 0.926934i \(-0.377565\pi\)
0.375225 + 0.926934i \(0.377565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 35.4641i 1.38359i
\(658\) 0 0
\(659\) 40.6410i 1.58315i 0.611073 + 0.791575i \(0.290738\pi\)
−0.611073 + 0.791575i \(0.709262\pi\)
\(660\) 0 0
\(661\) 35.5692i 1.38348i 0.722146 + 0.691741i \(0.243156\pi\)
−0.722146 + 0.691741i \(0.756844\pi\)
\(662\) 0 0
\(663\) 8.78461i 0.341166i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 11.0718i 0.428060i
\(670\) 0 0
\(671\) −44.7846 −1.72889
\(672\) 0 0
\(673\) − 39.1769i − 1.51016i −0.655633 0.755080i \(-0.727598\pi\)
0.655633 0.755080i \(-0.272402\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 43.1769 1.65942 0.829712 0.558192i \(-0.188505\pi\)
0.829712 + 0.558192i \(0.188505\pi\)
\(678\) 0 0
\(679\) 8.10512 0.311046
\(680\) 0 0
\(681\) 13.6077 0.521448
\(682\) 0 0
\(683\) −11.6603 −0.446167 −0.223084 0.974799i \(-0.571612\pi\)
−0.223084 + 0.974799i \(0.571612\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.8564i 0.528655i
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) − 5.60770i − 0.213327i −0.994295 0.106663i \(-0.965983\pi\)
0.994295 0.106663i \(-0.0340167\pi\)
\(692\) 0 0
\(693\) −10.8231 −0.411135
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 8.78461i − 0.332741i
\(698\) 0 0
\(699\) 1.17691i 0.0445150i
\(700\) 0 0
\(701\) 14.7846i 0.558407i 0.960232 + 0.279204i \(0.0900704\pi\)
−0.960232 + 0.279204i \(0.909930\pi\)
\(702\) 0 0
\(703\) − 12.0000i − 0.452589i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.2154 0.572234
\(708\) 0 0
\(709\) 10.1436i 0.380951i 0.981692 + 0.190475i \(0.0610029\pi\)
−0.981692 + 0.190475i \(0.938997\pi\)
\(710\) 0 0
\(711\) −29.5692 −1.10893
\(712\) 0 0
\(713\) − 77.5692i − 2.90499i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 15.2154 0.568229
\(718\) 0 0
\(719\) −44.7846 −1.67018 −0.835092 0.550110i \(-0.814586\pi\)
−0.835092 + 0.550110i \(0.814586\pi\)
\(720\) 0 0
\(721\) 10.3923 0.387030
\(722\) 0 0
\(723\) 14.9282 0.555186
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 34.0526i − 1.26294i −0.775401 0.631470i \(-0.782452\pi\)
0.775401 0.631470i \(-0.217548\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) − 35.3205i − 1.30638i
\(732\) 0 0
\(733\) 38.7846 1.43254 0.716271 0.697822i \(-0.245847\pi\)
0.716271 + 0.697822i \(0.245847\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 35.3205i − 1.30105i
\(738\) 0 0
\(739\) 2.00000i 0.0735712i 0.999323 + 0.0367856i \(0.0117119\pi\)
−0.999323 + 0.0367856i \(0.988288\pi\)
\(740\) 0 0
\(741\) 5.07180i 0.186317i
\(742\) 0 0
\(743\) 36.5885i 1.34230i 0.741321 + 0.671150i \(0.234199\pi\)
−0.741321 + 0.671150i \(0.765801\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −11.6603 −0.426626
\(748\) 0 0
\(749\) − 21.2154i − 0.775193i
\(750\) 0 0
\(751\) −40.3923 −1.47394 −0.736968 0.675928i \(-0.763743\pi\)
−0.736968 + 0.675928i \(0.763743\pi\)
\(752\) 0 0
\(753\) 11.3205i 0.412542i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0718 0.838559 0.419279 0.907857i \(-0.362283\pi\)
0.419279 + 0.907857i \(0.362283\pi\)
\(758\) 0 0
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) −21.7128 −0.787089 −0.393544 0.919306i \(-0.628751\pi\)
−0.393544 + 0.919306i \(0.628751\pi\)
\(762\) 0 0
\(763\) −1.17691 −0.0426072
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.7846i 0.750489i
\(768\) 0 0
\(769\) 6.78461 0.244659 0.122330 0.992490i \(-0.460963\pi\)
0.122330 + 0.992490i \(0.460963\pi\)
\(770\) 0 0
\(771\) 4.39230i 0.158185i
\(772\) 0 0
\(773\) 46.3923 1.66862 0.834308 0.551299i \(-0.185868\pi\)
0.834308 + 0.551299i \(0.185868\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 5.56922i 0.199795i
\(778\) 0 0
\(779\) − 5.07180i − 0.181716i
\(780\) 0 0
\(781\) − 15.2154i − 0.544449i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.80385 0.206885 0.103442 0.994635i \(-0.467014\pi\)
0.103442 + 0.994635i \(0.467014\pi\)
\(788\) 0 0
\(789\) 14.7846i 0.526346i
\(790\) 0 0
\(791\) 1.17691 0.0418463
\(792\) 0 0
\(793\) 44.7846i 1.59035i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.60770 −0.0569475 −0.0284737 0.999595i \(-0.509065\pi\)
−0.0284737 + 0.999595i \(0.509065\pi\)
\(798\) 0 0
\(799\) 28.3923 1.00445
\(800\) 0 0
\(801\) 2.28719 0.0808138
\(802\) 0 0
\(803\) 49.8564 1.75939
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 10.8231i − 0.380991i
\(808\) 0 0
\(809\) −35.5692 −1.25055 −0.625274 0.780406i \(-0.715013\pi\)
−0.625274 + 0.780406i \(0.715013\pi\)
\(810\) 0 0
\(811\) 38.3923i 1.34814i 0.738669 + 0.674068i \(0.235455\pi\)
−0.738669 + 0.674068i \(0.764545\pi\)
\(812\) 0 0
\(813\) −3.21539 −0.112769
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 20.3923i − 0.713436i
\(818\) 0 0
\(819\) 10.8231i 0.378189i
\(820\) 0 0
\(821\) 50.7846i 1.77240i 0.463308 + 0.886198i \(0.346663\pi\)
−0.463308 + 0.886198i \(0.653337\pi\)
\(822\) 0 0
\(823\) 30.8372i 1.07492i 0.843290 + 0.537458i \(0.180615\pi\)
−0.843290 + 0.537458i \(0.819385\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4.73205 −0.164550 −0.0822748 0.996610i \(-0.526219\pi\)
−0.0822748 + 0.996610i \(0.526219\pi\)
\(828\) 0 0
\(829\) − 50.7846i − 1.76382i −0.471416 0.881911i \(-0.656257\pi\)
0.471416 0.881911i \(-0.343743\pi\)
\(830\) 0 0
\(831\) 14.5359 0.504245
\(832\) 0 0
\(833\) − 18.6795i − 0.647206i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −37.8564 −1.30851
\(838\) 0 0
\(839\) −8.78461 −0.303278 −0.151639 0.988436i \(-0.548455\pi\)
−0.151639 + 0.988436i \(0.548455\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 20.7846 0.715860
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.26795i 0.0435672i
\(848\) 0 0
\(849\) −7.75129 −0.266024
\(850\) 0 0
\(851\) 49.1769i 1.68576i
\(852\) 0 0
\(853\) 3.46410 0.118609 0.0593043 0.998240i \(-0.481112\pi\)
0.0593043 + 0.998240i \(0.481112\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.1436i 0.551455i 0.961236 + 0.275727i \(0.0889187\pi\)
−0.961236 + 0.275727i \(0.911081\pi\)
\(858\) 0 0
\(859\) − 51.5692i − 1.75952i −0.475419 0.879760i \(-0.657704\pi\)
0.475419 0.879760i \(-0.342296\pi\)
\(860\) 0 0
\(861\) 2.35383i 0.0802183i
\(862\) 0 0
\(863\) 40.9808i 1.39500i 0.716584 + 0.697501i \(0.245705\pi\)
−0.716584 + 0.697501i \(0.754295\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −3.66025 −0.124309
\(868\) 0 0
\(869\) 41.5692i 1.41014i
\(870\) 0 0
\(871\) −35.3205 −1.19679
\(872\) 0 0
\(873\) − 15.7513i − 0.533100i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.85641 0.265292 0.132646 0.991163i \(-0.457653\pi\)
0.132646 + 0.991163i \(0.457653\pi\)
\(878\) 0 0
\(879\) 21.9615 0.740744
\(880\) 0 0
\(881\) −7.60770 −0.256310 −0.128155 0.991754i \(-0.540905\pi\)
−0.128155 + 0.991754i \(0.540905\pi\)
\(882\) 0 0
\(883\) −5.80385 −0.195315 −0.0976575 0.995220i \(-0.531135\pi\)
−0.0976575 + 0.995220i \(0.531135\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21.3731i 0.717637i 0.933407 + 0.358819i \(0.116820\pi\)
−0.933407 + 0.358819i \(0.883180\pi\)
\(888\) 0 0
\(889\) 4.82309 0.161761
\(890\) 0 0
\(891\) 15.4641i 0.518067i
\(892\) 0 0
\(893\) 16.3923 0.548548
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 20.7846i − 0.693978i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) − 36.0000i − 1.19933i
\(902\) 0 0
\(903\) 9.46410i 0.314946i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.41154 −0.0468695 −0.0234348 0.999725i \(-0.507460\pi\)
−0.0234348 + 0.999725i \(0.507460\pi\)
\(908\) 0 0
\(909\) − 29.5692i − 0.980749i
\(910\) 0 0
\(911\) −37.1769 −1.23173 −0.615863 0.787853i \(-0.711193\pi\)
−0.615863 + 0.787853i \(0.711193\pi\)
\(912\) 0 0
\(913\) 16.3923i 0.542506i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.1769 0.435140
\(918\) 0 0
\(919\) −39.7128 −1.31000 −0.655002 0.755627i \(-0.727332\pi\)
−0.655002 + 0.755627i \(0.727332\pi\)
\(920\) 0 0
\(921\) 25.0333 0.824876
\(922\) 0 0
\(923\) −15.2154 −0.500821
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 20.1962i − 0.663329i
\(928\) 0 0
\(929\) −7.60770 −0.249600 −0.124800 0.992182i \(-0.539829\pi\)
−0.124800 + 0.992182i \(0.539829\pi\)
\(930\) 0 0
\(931\) − 10.7846i − 0.353451i
\(932\) 0 0
\(933\) 5.56922 0.182328
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 9.60770i 0.313870i 0.987609 + 0.156935i \(0.0501613\pi\)
−0.987609 + 0.156935i \(0.949839\pi\)
\(938\) 0 0
\(939\) − 16.1051i − 0.525571i
\(940\) 0 0
\(941\) − 20.7846i − 0.677559i −0.940866 0.338779i \(-0.889986\pi\)
0.940866 0.338779i \(-0.110014\pi\)
\(942\) 0 0
\(943\) 20.7846i 0.676840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.8038 1.48843 0.744213 0.667943i \(-0.232825\pi\)
0.744213 + 0.667943i \(0.232825\pi\)
\(948\) 0 0
\(949\) − 49.8564i − 1.61841i
\(950\) 0 0
\(951\) −16.3923 −0.531557
\(952\) 0 0
\(953\) 24.9282i 0.807504i 0.914869 + 0.403752i \(0.132294\pi\)
−0.914869 + 0.403752i \(0.867706\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −16.3923 −0.529335
\(960\) 0 0
\(961\) 58.5692 1.88933
\(962\) 0 0
\(963\) −41.2295 −1.32860
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.19615i 0.263570i 0.991278 + 0.131785i \(0.0420709\pi\)
−0.991278 + 0.131785i \(0.957929\pi\)
\(968\) 0 0
\(969\) 5.07180 0.162930
\(970\) 0 0
\(971\) 33.0333i 1.06009i 0.847970 + 0.530045i \(0.177825\pi\)
−0.847970 + 0.530045i \(0.822175\pi\)
\(972\) 0 0
\(973\) 12.6795 0.406486
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 5.32051i − 0.170218i −0.996372 0.0851091i \(-0.972876\pi\)
0.996372 0.0851091i \(-0.0271239\pi\)
\(978\) 0 0
\(979\) − 3.21539i − 0.102764i
\(980\) 0 0
\(981\) 2.28719i 0.0730243i
\(982\) 0 0
\(983\) − 11.4115i − 0.363972i −0.983301 0.181986i \(-0.941747\pi\)
0.983301 0.181986i \(-0.0582525\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −7.60770 −0.242156
\(988\) 0 0
\(989\) 83.5692i 2.65735i
\(990\) 0 0
\(991\) 44.1051 1.40105 0.700523 0.713630i \(-0.252950\pi\)
0.700523 + 0.713630i \(0.252950\pi\)
\(992\) 0 0
\(993\) 4.10512i 0.130272i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −25.6077 −0.811004 −0.405502 0.914094i \(-0.632903\pi\)
−0.405502 + 0.914094i \(0.632903\pi\)
\(998\) 0 0
\(999\) 24.0000 0.759326
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 1600.2.f.h.1249.1 4
4.3 odd 2 1600.2.f.e.1249.4 4
5.2 odd 4 320.2.d.b.161.3 yes 4
5.3 odd 4 1600.2.d.b.801.2 4
5.4 even 2 1600.2.f.d.1249.4 4
8.3 odd 2 1600.2.f.i.1249.2 4
8.5 even 2 1600.2.f.d.1249.3 4
15.2 even 4 2880.2.k.l.1441.3 4
20.3 even 4 1600.2.d.h.801.3 4
20.7 even 4 320.2.d.a.161.2 4
20.19 odd 2 1600.2.f.i.1249.1 4
40.3 even 4 1600.2.d.h.801.2 4
40.13 odd 4 1600.2.d.b.801.3 4
40.19 odd 2 1600.2.f.e.1249.3 4
40.27 even 4 320.2.d.a.161.3 yes 4
40.29 even 2 inner 1600.2.f.h.1249.2 4
40.37 odd 4 320.2.d.b.161.2 yes 4
60.47 odd 4 2880.2.k.e.1441.4 4
80.3 even 4 6400.2.a.cd.1.1 2
80.13 odd 4 6400.2.a.bf.1.2 2
80.27 even 4 1280.2.a.p.1.1 2
80.37 odd 4 1280.2.a.c.1.2 2
80.43 even 4 6400.2.a.y.1.2 2
80.53 odd 4 6400.2.a.ck.1.1 2
80.67 even 4 1280.2.a.b.1.2 2
80.77 odd 4 1280.2.a.m.1.1 2
120.77 even 4 2880.2.k.l.1441.1 4
120.107 odd 4 2880.2.k.e.1441.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.2 4 20.7 even 4
320.2.d.a.161.3 yes 4 40.27 even 4
320.2.d.b.161.2 yes 4 40.37 odd 4
320.2.d.b.161.3 yes 4 5.2 odd 4
1280.2.a.b.1.2 2 80.67 even 4
1280.2.a.c.1.2 2 80.37 odd 4
1280.2.a.m.1.1 2 80.77 odd 4
1280.2.a.p.1.1 2 80.27 even 4
1600.2.d.b.801.2 4 5.3 odd 4
1600.2.d.b.801.3 4 40.13 odd 4
1600.2.d.h.801.2 4 40.3 even 4
1600.2.d.h.801.3 4 20.3 even 4
1600.2.f.d.1249.3 4 8.5 even 2
1600.2.f.d.1249.4 4 5.4 even 2
1600.2.f.e.1249.3 4 40.19 odd 2
1600.2.f.e.1249.4 4 4.3 odd 2
1600.2.f.h.1249.1 4 1.1 even 1 trivial
1600.2.f.h.1249.2 4 40.29 even 2 inner
1600.2.f.i.1249.1 4 20.19 odd 2
1600.2.f.i.1249.2 4 8.3 odd 2
2880.2.k.e.1441.2 4 120.107 odd 4
2880.2.k.e.1441.4 4 60.47 odd 4
2880.2.k.l.1441.1 4 120.77 even 4
2880.2.k.l.1441.3 4 15.2 even 4
6400.2.a.y.1.2 2 80.43 even 4
6400.2.a.bf.1.2 2 80.13 odd 4
6400.2.a.cd.1.1 2 80.3 even 4
6400.2.a.ck.1.1 2 80.53 odd 4