Properties

Label 7448.2.a.x.1.2
Level $7448$
Weight $2$
Character 7448.1
Self dual yes
Analytic conductor $59.473$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1064)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7448.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.585786 q^{3} +1.00000 q^{5} -2.65685 q^{9} +O(q^{10})\) \(q-0.585786 q^{3} +1.00000 q^{5} -2.65685 q^{9} +5.24264 q^{11} +0.585786 q^{13} -0.585786 q^{15} +1.17157 q^{17} +1.00000 q^{19} +1.58579 q^{23} -4.00000 q^{25} +3.31371 q^{27} -6.58579 q^{29} -7.41421 q^{31} -3.07107 q^{33} -9.41421 q^{37} -0.343146 q^{39} -5.65685 q^{41} +5.24264 q^{43} -2.65685 q^{45} -7.24264 q^{47} -0.686292 q^{51} -2.58579 q^{53} +5.24264 q^{55} -0.585786 q^{57} -4.82843 q^{59} +5.00000 q^{61} +0.585786 q^{65} +8.82843 q^{67} -0.928932 q^{69} -16.2426 q^{71} +7.00000 q^{73} +2.34315 q^{75} -15.8995 q^{79} +6.02944 q^{81} +4.89949 q^{83} +1.17157 q^{85} +3.85786 q^{87} +17.5563 q^{89} +4.34315 q^{93} +1.00000 q^{95} -5.65685 q^{97} -13.9289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 6 q^{23} - 8 q^{25} - 16 q^{27} - 16 q^{29} - 12 q^{31} + 8 q^{33} - 16 q^{37} - 12 q^{39} + 2 q^{43} + 6 q^{45} - 6 q^{47} - 24 q^{51} - 8 q^{53} + 2 q^{55} - 4 q^{57} - 4 q^{59} + 10 q^{61} + 4 q^{65} + 12 q^{67} - 16 q^{69} - 24 q^{71} + 14 q^{73} + 16 q^{75} - 12 q^{79} + 46 q^{81} - 10 q^{83} + 8 q^{85} + 36 q^{87} + 4 q^{89} + 20 q^{93} + 2 q^{95} - 42 q^{99} + O(q^{100}) \)

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
\(3\) −0.585786 −0.338204 −0.169102 0.985599i \(-0.554087\pi\)
−0.169102 + 0.985599i \(0.554087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.65685 −0.885618
\(10\) 0 0
\(11\) 5.24264 1.58072 0.790358 0.612646i \(-0.209895\pi\)
0.790358 + 0.612646i \(0.209895\pi\)
\(12\) 0 0
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.58579 0.330659 0.165330 0.986238i \(-0.447131\pi\)
0.165330 + 0.986238i \(0.447131\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 3.31371 0.637723
\(28\) 0 0
\(29\) −6.58579 −1.22295 −0.611475 0.791264i \(-0.709423\pi\)
−0.611475 + 0.791264i \(0.709423\pi\)
\(30\) 0 0
\(31\) −7.41421 −1.33163 −0.665816 0.746116i \(-0.731916\pi\)
−0.665816 + 0.746116i \(0.731916\pi\)
\(32\) 0 0
\(33\) −3.07107 −0.534604
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.41421 −1.54769 −0.773844 0.633377i \(-0.781668\pi\)
−0.773844 + 0.633377i \(0.781668\pi\)
\(38\) 0 0
\(39\) −0.343146 −0.0549473
\(40\) 0 0
\(41\) −5.65685 −0.883452 −0.441726 0.897150i \(-0.645634\pi\)
−0.441726 + 0.897150i \(0.645634\pi\)
\(42\) 0 0
\(43\) 5.24264 0.799495 0.399748 0.916625i \(-0.369098\pi\)
0.399748 + 0.916625i \(0.369098\pi\)
\(44\) 0 0
\(45\) −2.65685 −0.396060
\(46\) 0 0
\(47\) −7.24264 −1.05645 −0.528224 0.849105i \(-0.677142\pi\)
−0.528224 + 0.849105i \(0.677142\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.686292 −0.0961000
\(52\) 0 0
\(53\) −2.58579 −0.355185 −0.177593 0.984104i \(-0.556831\pi\)
−0.177593 + 0.984104i \(0.556831\pi\)
\(54\) 0 0
\(55\) 5.24264 0.706918
\(56\) 0 0
\(57\) −0.585786 −0.0775893
\(58\) 0 0
\(59\) −4.82843 −0.628608 −0.314304 0.949322i \(-0.601771\pi\)
−0.314304 + 0.949322i \(0.601771\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.585786 0.0726579
\(66\) 0 0
\(67\) 8.82843 1.07856 0.539282 0.842125i \(-0.318696\pi\)
0.539282 + 0.842125i \(0.318696\pi\)
\(68\) 0 0
\(69\) −0.928932 −0.111830
\(70\) 0 0
\(71\) −16.2426 −1.92765 −0.963823 0.266542i \(-0.914119\pi\)
−0.963823 + 0.266542i \(0.914119\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 2.34315 0.270563
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −15.8995 −1.78883 −0.894416 0.447235i \(-0.852409\pi\)
−0.894416 + 0.447235i \(0.852409\pi\)
\(80\) 0 0
\(81\) 6.02944 0.669937
\(82\) 0 0
\(83\) 4.89949 0.537789 0.268895 0.963170i \(-0.413342\pi\)
0.268895 + 0.963170i \(0.413342\pi\)
\(84\) 0 0
\(85\) 1.17157 0.127075
\(86\) 0 0
\(87\) 3.85786 0.413606
\(88\) 0 0
\(89\) 17.5563 1.86097 0.930485 0.366331i \(-0.119386\pi\)
0.930485 + 0.366331i \(0.119386\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.34315 0.450363
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −5.65685 −0.574367 −0.287183 0.957876i \(-0.592719\pi\)
−0.287183 + 0.957876i \(0.592719\pi\)
\(98\) 0 0
\(99\) −13.9289 −1.39991
\(100\) 0 0
\(101\) 15.4853 1.54084 0.770422 0.637535i \(-0.220046\pi\)
0.770422 + 0.637535i \(0.220046\pi\)
\(102\) 0 0
\(103\) −13.6569 −1.34565 −0.672825 0.739802i \(-0.734919\pi\)
−0.672825 + 0.739802i \(0.734919\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.92893 0.283151 0.141575 0.989927i \(-0.454783\pi\)
0.141575 + 0.989927i \(0.454783\pi\)
\(108\) 0 0
\(109\) 3.31371 0.317396 0.158698 0.987327i \(-0.449270\pi\)
0.158698 + 0.987327i \(0.449270\pi\)
\(110\) 0 0
\(111\) 5.51472 0.523434
\(112\) 0 0
\(113\) −3.07107 −0.288902 −0.144451 0.989512i \(-0.546142\pi\)
−0.144451 + 0.989512i \(0.546142\pi\)
\(114\) 0 0
\(115\) 1.58579 0.147875
\(116\) 0 0
\(117\) −1.55635 −0.143885
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 16.4853 1.49866
\(122\) 0 0
\(123\) 3.31371 0.298787
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 11.5563 1.02546 0.512730 0.858550i \(-0.328634\pi\)
0.512730 + 0.858550i \(0.328634\pi\)
\(128\) 0 0
\(129\) −3.07107 −0.270392
\(130\) 0 0
\(131\) 9.65685 0.843723 0.421862 0.906660i \(-0.361377\pi\)
0.421862 + 0.906660i \(0.361377\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.31371 0.285199
\(136\) 0 0
\(137\) −10.6569 −0.910477 −0.455238 0.890370i \(-0.650446\pi\)
−0.455238 + 0.890370i \(0.650446\pi\)
\(138\) 0 0
\(139\) −12.5563 −1.06502 −0.532508 0.846425i \(-0.678750\pi\)
−0.532508 + 0.846425i \(0.678750\pi\)
\(140\) 0 0
\(141\) 4.24264 0.357295
\(142\) 0 0
\(143\) 3.07107 0.256816
\(144\) 0 0
\(145\) −6.58579 −0.546920
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.65685 0.381504 0.190752 0.981638i \(-0.438907\pi\)
0.190752 + 0.981638i \(0.438907\pi\)
\(150\) 0 0
\(151\) −5.65685 −0.460348 −0.230174 0.973149i \(-0.573930\pi\)
−0.230174 + 0.973149i \(0.573930\pi\)
\(152\) 0 0
\(153\) −3.11270 −0.251647
\(154\) 0 0
\(155\) −7.41421 −0.595524
\(156\) 0 0
\(157\) 24.3137 1.94045 0.970223 0.242215i \(-0.0778739\pi\)
0.970223 + 0.242215i \(0.0778739\pi\)
\(158\) 0 0
\(159\) 1.51472 0.120125
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.0710678 −0.00556646 −0.00278323 0.999996i \(-0.500886\pi\)
−0.00278323 + 0.999996i \(0.500886\pi\)
\(164\) 0 0
\(165\) −3.07107 −0.239082
\(166\) 0 0
\(167\) −21.4142 −1.65708 −0.828541 0.559929i \(-0.810829\pi\)
−0.828541 + 0.559929i \(0.810829\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) −2.65685 −0.203175
\(172\) 0 0
\(173\) −1.65685 −0.125968 −0.0629841 0.998015i \(-0.520062\pi\)
−0.0629841 + 0.998015i \(0.520062\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.82843 0.212598
\(178\) 0 0
\(179\) −10.4853 −0.783707 −0.391853 0.920028i \(-0.628166\pi\)
−0.391853 + 0.920028i \(0.628166\pi\)
\(180\) 0 0
\(181\) −10.7279 −0.797400 −0.398700 0.917081i \(-0.630539\pi\)
−0.398700 + 0.917081i \(0.630539\pi\)
\(182\) 0 0
\(183\) −2.92893 −0.216513
\(184\) 0 0
\(185\) −9.41421 −0.692147
\(186\) 0 0
\(187\) 6.14214 0.449157
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.757359 0.0548006 0.0274003 0.999625i \(-0.491277\pi\)
0.0274003 + 0.999625i \(0.491277\pi\)
\(192\) 0 0
\(193\) −6.24264 −0.449355 −0.224678 0.974433i \(-0.572133\pi\)
−0.224678 + 0.974433i \(0.572133\pi\)
\(194\) 0 0
\(195\) −0.343146 −0.0245732
\(196\) 0 0
\(197\) −17.4853 −1.24577 −0.622887 0.782312i \(-0.714041\pi\)
−0.622887 + 0.782312i \(0.714041\pi\)
\(198\) 0 0
\(199\) 12.0711 0.855695 0.427848 0.903851i \(-0.359272\pi\)
0.427848 + 0.903851i \(0.359272\pi\)
\(200\) 0 0
\(201\) −5.17157 −0.364775
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −5.65685 −0.395092
\(206\) 0 0
\(207\) −4.21320 −0.292838
\(208\) 0 0
\(209\) 5.24264 0.362641
\(210\) 0 0
\(211\) −16.1421 −1.11127 −0.555635 0.831426i \(-0.687525\pi\)
−0.555635 + 0.831426i \(0.687525\pi\)
\(212\) 0 0
\(213\) 9.51472 0.651938
\(214\) 0 0
\(215\) 5.24264 0.357545
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.10051 −0.277086
\(220\) 0 0
\(221\) 0.686292 0.0461650
\(222\) 0 0
\(223\) 7.75736 0.519471 0.259736 0.965680i \(-0.416365\pi\)
0.259736 + 0.965680i \(0.416365\pi\)
\(224\) 0 0
\(225\) 10.6274 0.708494
\(226\) 0 0
\(227\) 1.41421 0.0938647 0.0469323 0.998898i \(-0.485055\pi\)
0.0469323 + 0.998898i \(0.485055\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −17.6569 −1.15674 −0.578369 0.815775i \(-0.696311\pi\)
−0.578369 + 0.815775i \(0.696311\pi\)
\(234\) 0 0
\(235\) −7.24264 −0.472458
\(236\) 0 0
\(237\) 9.31371 0.604990
\(238\) 0 0
\(239\) −15.6569 −1.01276 −0.506379 0.862311i \(-0.669016\pi\)
−0.506379 + 0.862311i \(0.669016\pi\)
\(240\) 0 0
\(241\) 16.7279 1.07754 0.538770 0.842453i \(-0.318889\pi\)
0.538770 + 0.842453i \(0.318889\pi\)
\(242\) 0 0
\(243\) −13.4731 −0.864299
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.585786 0.0372727
\(248\) 0 0
\(249\) −2.87006 −0.181883
\(250\) 0 0
\(251\) −20.2132 −1.27585 −0.637923 0.770100i \(-0.720206\pi\)
−0.637923 + 0.770100i \(0.720206\pi\)
\(252\) 0 0
\(253\) 8.31371 0.522678
\(254\) 0 0
\(255\) −0.686292 −0.0429772
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 17.4975 1.08307
\(262\) 0 0
\(263\) −26.8284 −1.65431 −0.827156 0.561973i \(-0.810043\pi\)
−0.827156 + 0.561973i \(0.810043\pi\)
\(264\) 0 0
\(265\) −2.58579 −0.158844
\(266\) 0 0
\(267\) −10.2843 −0.629387
\(268\) 0 0
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) −27.7279 −1.68435 −0.842176 0.539203i \(-0.818725\pi\)
−0.842176 + 0.539203i \(0.818725\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −20.9706 −1.26457
\(276\) 0 0
\(277\) 5.97056 0.358736 0.179368 0.983782i \(-0.442595\pi\)
0.179368 + 0.983782i \(0.442595\pi\)
\(278\) 0 0
\(279\) 19.6985 1.17932
\(280\) 0 0
\(281\) 1.89949 0.113314 0.0566572 0.998394i \(-0.481956\pi\)
0.0566572 + 0.998394i \(0.481956\pi\)
\(282\) 0 0
\(283\) 16.5563 0.984173 0.492086 0.870546i \(-0.336234\pi\)
0.492086 + 0.870546i \(0.336234\pi\)
\(284\) 0 0
\(285\) −0.585786 −0.0346990
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) 3.31371 0.194253
\(292\) 0 0
\(293\) 1.65685 0.0967945 0.0483972 0.998828i \(-0.484589\pi\)
0.0483972 + 0.998828i \(0.484589\pi\)
\(294\) 0 0
\(295\) −4.82843 −0.281122
\(296\) 0 0
\(297\) 17.3726 1.00806
\(298\) 0 0
\(299\) 0.928932 0.0537215
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −9.07107 −0.521119
\(304\) 0 0
\(305\) 5.00000 0.286299
\(306\) 0 0
\(307\) 26.6274 1.51971 0.759853 0.650094i \(-0.225271\pi\)
0.759853 + 0.650094i \(0.225271\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 27.1421 1.53416 0.767082 0.641549i \(-0.221708\pi\)
0.767082 + 0.641549i \(0.221708\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.75736 −0.0987031 −0.0493516 0.998781i \(-0.515715\pi\)
−0.0493516 + 0.998781i \(0.515715\pi\)
\(318\) 0 0
\(319\) −34.5269 −1.93314
\(320\) 0 0
\(321\) −1.71573 −0.0957626
\(322\) 0 0
\(323\) 1.17157 0.0651881
\(324\) 0 0
\(325\) −2.34315 −0.129974
\(326\) 0 0
\(327\) −1.94113 −0.107344
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.51472 −0.0832565 −0.0416282 0.999133i \(-0.513255\pi\)
−0.0416282 + 0.999133i \(0.513255\pi\)
\(332\) 0 0
\(333\) 25.0122 1.37066
\(334\) 0 0
\(335\) 8.82843 0.482349
\(336\) 0 0
\(337\) −9.07107 −0.494133 −0.247066 0.968999i \(-0.579467\pi\)
−0.247066 + 0.968999i \(0.579467\pi\)
\(338\) 0 0
\(339\) 1.79899 0.0977077
\(340\) 0 0
\(341\) −38.8701 −2.10493
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.928932 −0.0500120
\(346\) 0 0
\(347\) 9.58579 0.514592 0.257296 0.966333i \(-0.417168\pi\)
0.257296 + 0.966333i \(0.417168\pi\)
\(348\) 0 0
\(349\) 5.65685 0.302804 0.151402 0.988472i \(-0.451621\pi\)
0.151402 + 0.988472i \(0.451621\pi\)
\(350\) 0 0
\(351\) 1.94113 0.103610
\(352\) 0 0
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) −16.2426 −0.862070
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −22.6985 −1.19798 −0.598990 0.800756i \(-0.704431\pi\)
−0.598990 + 0.800756i \(0.704431\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −9.65685 −0.506853
\(364\) 0 0
\(365\) 7.00000 0.366397
\(366\) 0 0
\(367\) 28.6274 1.49434 0.747170 0.664634i \(-0.231412\pi\)
0.747170 + 0.664634i \(0.231412\pi\)
\(368\) 0 0
\(369\) 15.0294 0.782401
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −17.6569 −0.914237 −0.457119 0.889406i \(-0.651119\pi\)
−0.457119 + 0.889406i \(0.651119\pi\)
\(374\) 0 0
\(375\) 5.27208 0.272249
\(376\) 0 0
\(377\) −3.85786 −0.198690
\(378\) 0 0
\(379\) 15.4142 0.791775 0.395887 0.918299i \(-0.370437\pi\)
0.395887 + 0.918299i \(0.370437\pi\)
\(380\) 0 0
\(381\) −6.76955 −0.346815
\(382\) 0 0
\(383\) 12.2426 0.625570 0.312785 0.949824i \(-0.398738\pi\)
0.312785 + 0.949824i \(0.398738\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −13.9289 −0.708047
\(388\) 0 0
\(389\) 7.31371 0.370820 0.185410 0.982661i \(-0.440639\pi\)
0.185410 + 0.982661i \(0.440639\pi\)
\(390\) 0 0
\(391\) 1.85786 0.0939562
\(392\) 0 0
\(393\) −5.65685 −0.285351
\(394\) 0 0
\(395\) −15.8995 −0.799990
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 36.6274 1.82909 0.914543 0.404489i \(-0.132551\pi\)
0.914543 + 0.404489i \(0.132551\pi\)
\(402\) 0 0
\(403\) −4.34315 −0.216347
\(404\) 0 0
\(405\) 6.02944 0.299605
\(406\) 0 0
\(407\) −49.3553 −2.44645
\(408\) 0 0
\(409\) 6.68629 0.330616 0.165308 0.986242i \(-0.447138\pi\)
0.165308 + 0.986242i \(0.447138\pi\)
\(410\) 0 0
\(411\) 6.24264 0.307927
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.89949 0.240507
\(416\) 0 0
\(417\) 7.35534 0.360193
\(418\) 0 0
\(419\) −28.6985 −1.40201 −0.701006 0.713155i \(-0.747266\pi\)
−0.701006 + 0.713155i \(0.747266\pi\)
\(420\) 0 0
\(421\) −27.4142 −1.33609 −0.668044 0.744122i \(-0.732868\pi\)
−0.668044 + 0.744122i \(0.732868\pi\)
\(422\) 0 0
\(423\) 19.2426 0.935609
\(424\) 0 0
\(425\) −4.68629 −0.227319
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.79899 −0.0868560
\(430\) 0 0
\(431\) 5.89949 0.284169 0.142084 0.989855i \(-0.454620\pi\)
0.142084 + 0.989855i \(0.454620\pi\)
\(432\) 0 0
\(433\) −12.0000 −0.576683 −0.288342 0.957528i \(-0.593104\pi\)
−0.288342 + 0.957528i \(0.593104\pi\)
\(434\) 0 0
\(435\) 3.85786 0.184970
\(436\) 0 0
\(437\) 1.58579 0.0758585
\(438\) 0 0
\(439\) −9.51472 −0.454113 −0.227056 0.973882i \(-0.572910\pi\)
−0.227056 + 0.973882i \(0.572910\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −39.3137 −1.86785 −0.933925 0.357468i \(-0.883640\pi\)
−0.933925 + 0.357468i \(0.883640\pi\)
\(444\) 0 0
\(445\) 17.5563 0.832251
\(446\) 0 0
\(447\) −2.72792 −0.129026
\(448\) 0 0
\(449\) −14.6274 −0.690310 −0.345155 0.938546i \(-0.612174\pi\)
−0.345155 + 0.938546i \(0.612174\pi\)
\(450\) 0 0
\(451\) −29.6569 −1.39649
\(452\) 0 0
\(453\) 3.31371 0.155692
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.171573 0.00802584 0.00401292 0.999992i \(-0.498723\pi\)
0.00401292 + 0.999992i \(0.498723\pi\)
\(458\) 0 0
\(459\) 3.88225 0.181208
\(460\) 0 0
\(461\) 28.7990 1.34130 0.670651 0.741773i \(-0.266015\pi\)
0.670651 + 0.741773i \(0.266015\pi\)
\(462\) 0 0
\(463\) −2.27208 −0.105592 −0.0527962 0.998605i \(-0.516813\pi\)
−0.0527962 + 0.998605i \(0.516813\pi\)
\(464\) 0 0
\(465\) 4.34315 0.201409
\(466\) 0 0
\(467\) −17.7279 −0.820350 −0.410175 0.912007i \(-0.634532\pi\)
−0.410175 + 0.912007i \(0.634532\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −14.2426 −0.656266
\(472\) 0 0
\(473\) 27.4853 1.26377
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 6.87006 0.314558
\(478\) 0 0
\(479\) 24.0711 1.09984 0.549918 0.835219i \(-0.314659\pi\)
0.549918 + 0.835219i \(0.314659\pi\)
\(480\) 0 0
\(481\) −5.51472 −0.251450
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5.65685 −0.256865
\(486\) 0 0
\(487\) −40.1421 −1.81901 −0.909507 0.415689i \(-0.863541\pi\)
−0.909507 + 0.415689i \(0.863541\pi\)
\(488\) 0 0
\(489\) 0.0416306 0.00188260
\(490\) 0 0
\(491\) 40.6985 1.83670 0.918348 0.395773i \(-0.129523\pi\)
0.918348 + 0.395773i \(0.129523\pi\)
\(492\) 0 0
\(493\) −7.71573 −0.347499
\(494\) 0 0
\(495\) −13.9289 −0.626059
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9.24264 −0.413757 −0.206879 0.978367i \(-0.566331\pi\)
−0.206879 + 0.978367i \(0.566331\pi\)
\(500\) 0 0
\(501\) 12.5442 0.560432
\(502\) 0 0
\(503\) 3.72792 0.166220 0.0831099 0.996540i \(-0.473515\pi\)
0.0831099 + 0.996540i \(0.473515\pi\)
\(504\) 0 0
\(505\) 15.4853 0.689086
\(506\) 0 0
\(507\) 7.41421 0.329277
\(508\) 0 0
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.31371 0.146304
\(514\) 0 0
\(515\) −13.6569 −0.601793
\(516\) 0 0
\(517\) −37.9706 −1.66994
\(518\) 0 0
\(519\) 0.970563 0.0426030
\(520\) 0 0
\(521\) 2.10051 0.0920248 0.0460124 0.998941i \(-0.485349\pi\)
0.0460124 + 0.998941i \(0.485349\pi\)
\(522\) 0 0
\(523\) 15.3137 0.669622 0.334811 0.942285i \(-0.391328\pi\)
0.334811 + 0.942285i \(0.391328\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.68629 −0.378381
\(528\) 0 0
\(529\) −20.4853 −0.890664
\(530\) 0 0
\(531\) 12.8284 0.556706
\(532\) 0 0
\(533\) −3.31371 −0.143533
\(534\) 0 0
\(535\) 2.92893 0.126629
\(536\) 0 0
\(537\) 6.14214 0.265053
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.857864 −0.0368825 −0.0184412 0.999830i \(-0.505870\pi\)
−0.0184412 + 0.999830i \(0.505870\pi\)
\(542\) 0 0
\(543\) 6.28427 0.269684
\(544\) 0 0
\(545\) 3.31371 0.141944
\(546\) 0 0
\(547\) 9.55635 0.408600 0.204300 0.978908i \(-0.434508\pi\)
0.204300 + 0.978908i \(0.434508\pi\)
\(548\) 0 0
\(549\) −13.2843 −0.566959
\(550\) 0 0
\(551\) −6.58579 −0.280564
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5.51472 0.234087
\(556\) 0 0
\(557\) 18.5147 0.784494 0.392247 0.919860i \(-0.371698\pi\)
0.392247 + 0.919860i \(0.371698\pi\)
\(558\) 0 0
\(559\) 3.07107 0.129892
\(560\) 0 0
\(561\) −3.59798 −0.151907
\(562\) 0 0
\(563\) 10.8701 0.458118 0.229059 0.973413i \(-0.426435\pi\)
0.229059 + 0.973413i \(0.426435\pi\)
\(564\) 0 0
\(565\) −3.07107 −0.129201
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.3137 0.725828 0.362914 0.931823i \(-0.381782\pi\)
0.362914 + 0.931823i \(0.381782\pi\)
\(570\) 0 0
\(571\) 16.6985 0.698810 0.349405 0.936972i \(-0.386384\pi\)
0.349405 + 0.936972i \(0.386384\pi\)
\(572\) 0 0
\(573\) −0.443651 −0.0185338
\(574\) 0 0
\(575\) −6.34315 −0.264527
\(576\) 0 0
\(577\) −33.8284 −1.40830 −0.704148 0.710053i \(-0.748671\pi\)
−0.704148 + 0.710053i \(0.748671\pi\)
\(578\) 0 0
\(579\) 3.65685 0.151974
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −13.5563 −0.561447
\(584\) 0 0
\(585\) −1.55635 −0.0643471
\(586\) 0 0
\(587\) 34.0000 1.40333 0.701665 0.712507i \(-0.252440\pi\)
0.701665 + 0.712507i \(0.252440\pi\)
\(588\) 0 0
\(589\) −7.41421 −0.305497
\(590\) 0 0
\(591\) 10.2426 0.421326
\(592\) 0 0
\(593\) 18.3137 0.752054 0.376027 0.926609i \(-0.377290\pi\)
0.376027 + 0.926609i \(0.377290\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.07107 −0.289400
\(598\) 0 0
\(599\) −21.2132 −0.866748 −0.433374 0.901214i \(-0.642677\pi\)
−0.433374 + 0.901214i \(0.642677\pi\)
\(600\) 0 0
\(601\) −48.5269 −1.97945 −0.989727 0.142970i \(-0.954335\pi\)
−0.989727 + 0.142970i \(0.954335\pi\)
\(602\) 0 0
\(603\) −23.4558 −0.955196
\(604\) 0 0
\(605\) 16.4853 0.670222
\(606\) 0 0
\(607\) 3.89949 0.158276 0.0791378 0.996864i \(-0.474783\pi\)
0.0791378 + 0.996864i \(0.474783\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.24264 −0.171639
\(612\) 0 0
\(613\) −15.5147 −0.626634 −0.313317 0.949649i \(-0.601440\pi\)
−0.313317 + 0.949649i \(0.601440\pi\)
\(614\) 0 0
\(615\) 3.31371 0.133622
\(616\) 0 0
\(617\) −29.2843 −1.17894 −0.589470 0.807790i \(-0.700663\pi\)
−0.589470 + 0.807790i \(0.700663\pi\)
\(618\) 0 0
\(619\) −28.8995 −1.16157 −0.580784 0.814057i \(-0.697254\pi\)
−0.580784 + 0.814057i \(0.697254\pi\)
\(620\) 0 0
\(621\) 5.25483 0.210869
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −3.07107 −0.122647
\(628\) 0 0
\(629\) −11.0294 −0.439772
\(630\) 0 0
\(631\) 19.3848 0.771696 0.385848 0.922562i \(-0.373909\pi\)
0.385848 + 0.922562i \(0.373909\pi\)
\(632\) 0 0
\(633\) 9.45584 0.375836
\(634\) 0 0
\(635\) 11.5563 0.458600
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 43.1543 1.70716
\(640\) 0 0
\(641\) 33.3137 1.31581 0.657906 0.753100i \(-0.271442\pi\)
0.657906 + 0.753100i \(0.271442\pi\)
\(642\) 0 0
\(643\) −44.4853 −1.75433 −0.877164 0.480191i \(-0.840567\pi\)
−0.877164 + 0.480191i \(0.840567\pi\)
\(644\) 0 0
\(645\) −3.07107 −0.120923
\(646\) 0 0
\(647\) −42.0711 −1.65398 −0.826992 0.562213i \(-0.809950\pi\)
−0.826992 + 0.562213i \(0.809950\pi\)
\(648\) 0 0
\(649\) −25.3137 −0.993650
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.85786 −0.0727039 −0.0363519 0.999339i \(-0.511574\pi\)
−0.0363519 + 0.999339i \(0.511574\pi\)
\(654\) 0 0
\(655\) 9.65685 0.377325
\(656\) 0 0
\(657\) −18.5980 −0.725576
\(658\) 0 0
\(659\) −12.3848 −0.482442 −0.241221 0.970470i \(-0.577548\pi\)
−0.241221 + 0.970470i \(0.577548\pi\)
\(660\) 0 0
\(661\) 44.8701 1.74524 0.872621 0.488397i \(-0.162418\pi\)
0.872621 + 0.488397i \(0.162418\pi\)
\(662\) 0 0
\(663\) −0.402020 −0.0156132
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10.4437 −0.404380
\(668\) 0 0
\(669\) −4.54416 −0.175687
\(670\) 0 0
\(671\) 26.2132 1.01195
\(672\) 0 0
\(673\) 4.62742 0.178374 0.0891869 0.996015i \(-0.471573\pi\)
0.0891869 + 0.996015i \(0.471573\pi\)
\(674\) 0 0
\(675\) −13.2548 −0.510179
\(676\) 0 0
\(677\) 11.3137 0.434821 0.217411 0.976080i \(-0.430239\pi\)
0.217411 + 0.976080i \(0.430239\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −0.828427 −0.0317454
\(682\) 0 0
\(683\) 4.14214 0.158494 0.0792472 0.996855i \(-0.474748\pi\)
0.0792472 + 0.996855i \(0.474748\pi\)
\(684\) 0 0
\(685\) −10.6569 −0.407177
\(686\) 0 0
\(687\) 9.37258 0.357586
\(688\) 0 0
\(689\) −1.51472 −0.0577062
\(690\) 0 0
\(691\) −28.6274 −1.08904 −0.544519 0.838748i \(-0.683288\pi\)
−0.544519 + 0.838748i \(0.683288\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.5563 −0.476289
\(696\) 0 0
\(697\) −6.62742 −0.251031
\(698\) 0 0
\(699\) 10.3431 0.391214
\(700\) 0 0
\(701\) 18.7990 0.710028 0.355014 0.934861i \(-0.384476\pi\)
0.355014 + 0.934861i \(0.384476\pi\)
\(702\) 0 0
\(703\) −9.41421 −0.355064
\(704\) 0 0
\(705\) 4.24264 0.159787
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −26.5980 −0.998908 −0.499454 0.866340i \(-0.666466\pi\)
−0.499454 + 0.866340i \(0.666466\pi\)
\(710\) 0 0
\(711\) 42.2426 1.58422
\(712\) 0 0
\(713\) −11.7574 −0.440317
\(714\) 0 0
\(715\) 3.07107 0.114851
\(716\) 0 0
\(717\) 9.17157 0.342519
\(718\) 0 0
\(719\) 19.3137 0.720280 0.360140 0.932898i \(-0.382729\pi\)
0.360140 + 0.932898i \(0.382729\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −9.79899 −0.364428
\(724\) 0 0
\(725\) 26.3431 0.978360
\(726\) 0 0
\(727\) 5.24264 0.194439 0.0972194 0.995263i \(-0.469005\pi\)
0.0972194 + 0.995263i \(0.469005\pi\)
\(728\) 0 0
\(729\) −10.1960 −0.377628
\(730\) 0 0
\(731\) 6.14214 0.227175
\(732\) 0 0
\(733\) 28.6863 1.05955 0.529776 0.848137i \(-0.322276\pi\)
0.529776 + 0.848137i \(0.322276\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 46.2843 1.70490
\(738\) 0 0
\(739\) −0.627417 −0.0230799 −0.0115400 0.999933i \(-0.503673\pi\)
−0.0115400 + 0.999933i \(0.503673\pi\)
\(740\) 0 0
\(741\) −0.343146 −0.0126058
\(742\) 0 0
\(743\) 46.7279 1.71428 0.857141 0.515083i \(-0.172239\pi\)
0.857141 + 0.515083i \(0.172239\pi\)
\(744\) 0 0
\(745\) 4.65685 0.170614
\(746\) 0 0
\(747\) −13.0172 −0.476276
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 22.4853 0.820500 0.410250 0.911973i \(-0.365442\pi\)
0.410250 + 0.911973i \(0.365442\pi\)
\(752\) 0 0
\(753\) 11.8406 0.431496
\(754\) 0 0
\(755\) −5.65685 −0.205874
\(756\) 0 0
\(757\) −54.3137 −1.97407 −0.987033 0.160520i \(-0.948683\pi\)
−0.987033 + 0.160520i \(0.948683\pi\)
\(758\) 0 0
\(759\) −4.87006 −0.176772
\(760\) 0 0
\(761\) −15.1421 −0.548902 −0.274451 0.961601i \(-0.588496\pi\)
−0.274451 + 0.961601i \(0.588496\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3.11270 −0.112540
\(766\) 0 0
\(767\) −2.82843 −0.102129
\(768\) 0 0
\(769\) 54.4558 1.96373 0.981864 0.189587i \(-0.0607148\pi\)
0.981864 + 0.189587i \(0.0607148\pi\)
\(770\) 0 0
\(771\) 8.20101 0.295352
\(772\) 0 0
\(773\) 15.0711 0.542069 0.271034 0.962570i \(-0.412634\pi\)
0.271034 + 0.962570i \(0.412634\pi\)
\(774\) 0 0
\(775\) 29.6569 1.06531
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.65685 −0.202678
\(780\) 0 0
\(781\) −85.1543 −3.04706
\(782\) 0 0
\(783\) −21.8234 −0.779904
\(784\) 0 0
\(785\) 24.3137 0.867793
\(786\) 0 0
\(787\) −20.2426 −0.721572 −0.360786 0.932649i \(-0.617492\pi\)
−0.360786 + 0.932649i \(0.617492\pi\)
\(788\) 0 0
\(789\) 15.7157 0.559495
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.92893 0.104009
\(794\) 0 0
\(795\) 1.51472 0.0537215
\(796\) 0 0
\(797\) −53.4975 −1.89498 −0.947489 0.319789i \(-0.896388\pi\)
−0.947489 + 0.319789i \(0.896388\pi\)
\(798\) 0 0
\(799\) −8.48528 −0.300188
\(800\) 0 0
\(801\) −46.6447 −1.64811
\(802\) 0 0
\(803\) 36.6985 1.29506
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9.37258 −0.329931
\(808\) 0 0
\(809\) 5.97056 0.209914 0.104957 0.994477i \(-0.466530\pi\)
0.104957 + 0.994477i \(0.466530\pi\)
\(810\) 0 0
\(811\) 52.4264 1.84094 0.920470 0.390813i \(-0.127806\pi\)
0.920470 + 0.390813i \(0.127806\pi\)
\(812\) 0 0
\(813\) 16.2426 0.569654
\(814\) 0 0
\(815\) −0.0710678 −0.00248940
\(816\) 0 0
\(817\) 5.24264 0.183417
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 41.9706 1.46478 0.732391 0.680884i \(-0.238404\pi\)
0.732391 + 0.680884i \(0.238404\pi\)
\(822\) 0 0
\(823\) −15.7279 −0.548241 −0.274120 0.961695i \(-0.588387\pi\)
−0.274120 + 0.961695i \(0.588387\pi\)
\(824\) 0 0
\(825\) 12.2843 0.427683
\(826\) 0 0
\(827\) −23.3137 −0.810697 −0.405349 0.914162i \(-0.632850\pi\)
−0.405349 + 0.914162i \(0.632850\pi\)
\(828\) 0 0
\(829\) −16.6863 −0.579539 −0.289769 0.957096i \(-0.593579\pi\)
−0.289769 + 0.957096i \(0.593579\pi\)
\(830\) 0 0
\(831\) −3.49747 −0.121326
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −21.4142 −0.741069
\(836\) 0 0
\(837\) −24.5685 −0.849213
\(838\) 0 0
\(839\) −13.5147 −0.466580 −0.233290 0.972407i \(-0.574949\pi\)
−0.233290 + 0.972407i \(0.574949\pi\)
\(840\) 0 0
\(841\) 14.3726 0.495606
\(842\) 0 0
\(843\) −1.11270 −0.0383234
\(844\) 0 0
\(845\) −12.6569 −0.435409
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −9.69848 −0.332851
\(850\) 0 0
\(851\) −14.9289 −0.511757
\(852\) 0 0
\(853\) 1.54416 0.0528709 0.0264354 0.999651i \(-0.491584\pi\)
0.0264354 + 0.999651i \(0.491584\pi\)
\(854\) 0 0
\(855\) −2.65685 −0.0908625
\(856\) 0 0
\(857\) −40.9706 −1.39953 −0.699764 0.714374i \(-0.746711\pi\)
−0.699764 + 0.714374i \(0.746711\pi\)
\(858\) 0 0
\(859\) −49.1838 −1.67813 −0.839064 0.544032i \(-0.816897\pi\)
−0.839064 + 0.544032i \(0.816897\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28.8701 0.982748 0.491374 0.870949i \(-0.336495\pi\)
0.491374 + 0.870949i \(0.336495\pi\)
\(864\) 0 0
\(865\) −1.65685 −0.0563347
\(866\) 0 0
\(867\) 9.15433 0.310897
\(868\) 0 0
\(869\) −83.3553 −2.82764
\(870\) 0 0
\(871\) 5.17157 0.175232
\(872\) 0 0
\(873\) 15.0294 0.508669
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −16.8701 −0.569661 −0.284831 0.958578i \(-0.591937\pi\)
−0.284831 + 0.958578i \(0.591937\pi\)
\(878\) 0 0
\(879\) −0.970563 −0.0327363
\(880\) 0 0
\(881\) 15.3137 0.515932 0.257966 0.966154i \(-0.416948\pi\)
0.257966 + 0.966154i \(0.416948\pi\)
\(882\) 0 0
\(883\) −30.8284 −1.03746 −0.518730 0.854938i \(-0.673595\pi\)
−0.518730 + 0.854938i \(0.673595\pi\)
\(884\) 0 0
\(885\) 2.82843 0.0950765
\(886\) 0 0
\(887\) −28.2843 −0.949693 −0.474846 0.880069i \(-0.657496\pi\)
−0.474846 + 0.880069i \(0.657496\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 31.6102 1.05898
\(892\) 0 0
\(893\) −7.24264 −0.242366
\(894\) 0 0
\(895\) −10.4853 −0.350484
\(896\) 0 0
\(897\) −0.544156 −0.0181688
\(898\) 0 0
\(899\) 48.8284 1.62852
\(900\) 0 0
\(901\) −3.02944 −0.100925
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.7279 −0.356608
\(906\) 0 0
\(907\) −18.4437 −0.612411 −0.306206 0.951965i \(-0.599060\pi\)
−0.306206 + 0.951965i \(0.599060\pi\)
\(908\) 0 0
\(909\) −41.1421 −1.36460
\(910\) 0 0
\(911\) 5.55635 0.184090 0.0920450 0.995755i \(-0.470660\pi\)
0.0920450 + 0.995755i \(0.470660\pi\)
\(912\) 0 0
\(913\) 25.6863 0.850092
\(914\) 0 0
\(915\) −2.92893 −0.0968275
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.41421 0.277559 0.138780 0.990323i \(-0.455682\pi\)
0.138780 + 0.990323i \(0.455682\pi\)
\(920\) 0 0
\(921\) −15.5980 −0.513971
\(922\) 0 0
\(923\) −9.51472 −0.313181
\(924\) 0 0
\(925\) 37.6569 1.23815
\(926\) 0 0
\(927\) 36.2843 1.19173
\(928\) 0 0
\(929\) −9.62742 −0.315865 −0.157933 0.987450i \(-0.550483\pi\)
−0.157933 + 0.987450i \(0.550483\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 4.68629 0.153422
\(934\) 0 0
\(935\) 6.14214 0.200869
\(936\) 0 0
\(937\) −47.4853 −1.55128 −0.775638 0.631178i \(-0.782572\pi\)
−0.775638 + 0.631178i \(0.782572\pi\)
\(938\) 0 0
\(939\) −15.8995 −0.518860
\(940\) 0 0
\(941\) −24.5858 −0.801474 −0.400737 0.916193i \(-0.631246\pi\)
−0.400737 + 0.916193i \(0.631246\pi\)
\(942\) 0 0
\(943\) −8.97056 −0.292122
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −26.0000 −0.844886 −0.422443 0.906389i \(-0.638827\pi\)
−0.422443 + 0.906389i \(0.638827\pi\)
\(948\) 0 0
\(949\) 4.10051 0.133108
\(950\) 0 0
\(951\) 1.02944 0.0333818
\(952\) 0 0
\(953\) 51.1543 1.65705 0.828526 0.559951i \(-0.189180\pi\)
0.828526 + 0.559951i \(0.189180\pi\)
\(954\) 0 0
\(955\) 0.757359 0.0245076
\(956\) 0 0
\(957\) 20.2254 0.653794
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 23.9706 0.773244
\(962\) 0 0
\(963\) −7.78175 −0.250763
\(964\) 0 0
\(965\) −6.24264 −0.200958
\(966\) 0 0
\(967\) −16.9706 −0.545737 −0.272868 0.962051i \(-0.587972\pi\)
−0.272868 + 0.962051i \(0.587972\pi\)
\(968\) 0 0
\(969\) −0.686292 −0.0220469
\(970\) 0 0
\(971\) 4.87006 0.156288 0.0781438 0.996942i \(-0.475101\pi\)
0.0781438 + 0.996942i \(0.475101\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.37258 0.0439578
\(976\) 0 0
\(977\) −5.31371 −0.170001 −0.0850003 0.996381i \(-0.527089\pi\)
−0.0850003 + 0.996381i \(0.527089\pi\)
\(978\) 0 0
\(979\) 92.0416 2.94166
\(980\) 0 0
\(981\) −8.80404 −0.281091
\(982\) 0 0
\(983\) −17.9411 −0.572233 −0.286117 0.958195i \(-0.592364\pi\)
−0.286117 + 0.958195i \(0.592364\pi\)
\(984\) 0 0
\(985\) −17.4853 −0.557127
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.31371 0.264361
\(990\) 0 0
\(991\) 7.45584 0.236843 0.118421 0.992963i \(-0.462217\pi\)
0.118421 + 0.992963i \(0.462217\pi\)
\(992\) 0 0
\(993\) 0.887302 0.0281577
\(994\) 0 0
\(995\) 12.0711 0.382679
\(996\) 0 0
\(997\) 59.5980 1.88749 0.943743 0.330678i \(-0.107278\pi\)
0.943743 + 0.330678i \(0.107278\pi\)
\(998\) 0 0
\(999\) −31.1960 −0.986996
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 7448.2.a.x.1.2 2
7.3 odd 6 1064.2.q.k.457.2 yes 4
7.5 odd 6 1064.2.q.k.305.2 4
7.6 odd 2 7448.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.k.305.2 4 7.5 odd 6
1064.2.q.k.457.2 yes 4 7.3 odd 6
7448.2.a.x.1.2 2 1.1 even 1 trivial
7448.2.a.bd.1.1 2 7.6 odd 2