Properties

Label 4864.2.a.r.1.2
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \(x^{2} - 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{3} +3.31662 q^{5} -3.31662 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +3.31662 q^{5} -3.31662 q^{7} +1.00000 q^{9} +5.00000 q^{11} -6.63325 q^{15} +5.00000 q^{17} +1.00000 q^{19} +6.63325 q^{21} -6.63325 q^{23} +6.00000 q^{25} +4.00000 q^{27} +6.63325 q^{29} -10.0000 q^{33} -11.0000 q^{35} -6.63325 q^{37} +6.00000 q^{41} -1.00000 q^{43} +3.31662 q^{45} +9.94987 q^{47} +4.00000 q^{49} -10.0000 q^{51} -13.2665 q^{53} +16.5831 q^{55} -2.00000 q^{57} -6.00000 q^{59} +9.94987 q^{61} -3.31662 q^{63} -8.00000 q^{67} +13.2665 q^{69} +6.63325 q^{71} +9.00000 q^{73} -12.0000 q^{75} -16.5831 q^{77} -13.2665 q^{79} -11.0000 q^{81} +4.00000 q^{83} +16.5831 q^{85} -13.2665 q^{87} +4.00000 q^{89} +3.31662 q^{95} -12.0000 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} + 2q^{9} + 10q^{11} + 10q^{17} + 2q^{19} + 12q^{25} + 8q^{27} - 20q^{33} - 22q^{35} + 12q^{41} - 2q^{43} + 8q^{49} - 20q^{51} - 4q^{57} - 12q^{59} - 16q^{67} + 18q^{73} - 24q^{75} - 22q^{81} + 8q^{83} + 8q^{89} - 24q^{97} + 10q^{99} + O(q^{100}) \)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 3.31662 1.48324 0.741620 0.670820i \(-0.234058\pi\)
0.741620 + 0.670820i \(0.234058\pi\)
\(6\) 0 0
\(7\) −3.31662 −1.25357 −0.626783 0.779194i \(-0.715629\pi\)
−0.626783 + 0.779194i \(0.715629\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −6.63325 −1.71270
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 6.63325 1.44749
\(22\) 0 0
\(23\) −6.63325 −1.38313 −0.691564 0.722315i \(-0.743078\pi\)
−0.691564 + 0.722315i \(0.743078\pi\)
\(24\) 0 0
\(25\) 6.00000 1.20000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.63325 1.23176 0.615882 0.787839i \(-0.288800\pi\)
0.615882 + 0.787839i \(0.288800\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −10.0000 −1.74078
\(34\) 0 0
\(35\) −11.0000 −1.85934
\(36\) 0 0
\(37\) −6.63325 −1.09050 −0.545250 0.838274i \(-0.683565\pi\)
−0.545250 + 0.838274i \(0.683565\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 3.31662 0.494413
\(46\) 0 0
\(47\) 9.94987 1.45134 0.725669 0.688044i \(-0.241530\pi\)
0.725669 + 0.688044i \(0.241530\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) −10.0000 −1.40028
\(52\) 0 0
\(53\) −13.2665 −1.82229 −0.911147 0.412082i \(-0.864802\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 16.5831 2.23607
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 9.94987 1.27395 0.636975 0.770884i \(-0.280185\pi\)
0.636975 + 0.770884i \(0.280185\pi\)
\(62\) 0 0
\(63\) −3.31662 −0.417855
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 13.2665 1.59710
\(70\) 0 0
\(71\) 6.63325 0.787222 0.393611 0.919277i \(-0.371226\pi\)
0.393611 + 0.919277i \(0.371226\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) −12.0000 −1.38564
\(76\) 0 0
\(77\) −16.5831 −1.88982
\(78\) 0 0
\(79\) −13.2665 −1.49260 −0.746299 0.665611i \(-0.768171\pi\)
−0.746299 + 0.665611i \(0.768171\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 16.5831 1.79869
\(86\) 0 0
\(87\) −13.2665 −1.42232
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.31662 0.340279
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 6.63325 0.653594 0.326797 0.945095i \(-0.394031\pi\)
0.326797 + 0.945095i \(0.394031\pi\)
\(104\) 0 0
\(105\) 22.0000 2.14698
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 13.2665 1.27070 0.635350 0.772224i \(-0.280856\pi\)
0.635350 + 0.772224i \(0.280856\pi\)
\(110\) 0 0
\(111\) 13.2665 1.25920
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) −22.0000 −2.05151
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −16.5831 −1.52017
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 3.31662 0.296648
\(126\) 0 0
\(127\) 6.63325 0.588606 0.294303 0.955712i \(-0.404913\pi\)
0.294303 + 0.955712i \(0.404913\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) −3.31662 −0.287588
\(134\) 0 0
\(135\) 13.2665 1.14180
\(136\) 0 0
\(137\) −5.00000 −0.427179 −0.213589 0.976924i \(-0.568515\pi\)
−0.213589 + 0.976924i \(0.568515\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) −19.8997 −1.67586
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 22.0000 1.82700
\(146\) 0 0
\(147\) −8.00000 −0.659829
\(148\) 0 0
\(149\) 3.31662 0.271708 0.135854 0.990729i \(-0.456622\pi\)
0.135854 + 0.990729i \(0.456622\pi\)
\(150\) 0 0
\(151\) −6.63325 −0.539806 −0.269903 0.962887i \(-0.586992\pi\)
−0.269903 + 0.962887i \(0.586992\pi\)
\(152\) 0 0
\(153\) 5.00000 0.404226
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 26.5330 2.10420
\(160\) 0 0
\(161\) 22.0000 1.73384
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) −33.1662 −2.58199
\(166\) 0 0
\(167\) 6.63325 0.513296 0.256648 0.966505i \(-0.417382\pi\)
0.256648 + 0.966505i \(0.417382\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −19.8997 −1.51295 −0.756475 0.654023i \(-0.773080\pi\)
−0.756475 + 0.654023i \(0.773080\pi\)
\(174\) 0 0
\(175\) −19.8997 −1.50428
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 19.8997 1.47914 0.739568 0.673081i \(-0.235030\pi\)
0.739568 + 0.673081i \(0.235030\pi\)
\(182\) 0 0
\(183\) −19.8997 −1.47103
\(184\) 0 0
\(185\) −22.0000 −1.61747
\(186\) 0 0
\(187\) 25.0000 1.82818
\(188\) 0 0
\(189\) −13.2665 −0.964996
\(190\) 0 0
\(191\) −16.5831 −1.19991 −0.599956 0.800033i \(-0.704815\pi\)
−0.599956 + 0.800033i \(0.704815\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.5330 −1.89040 −0.945199 0.326495i \(-0.894132\pi\)
−0.945199 + 0.326495i \(0.894132\pi\)
\(198\) 0 0
\(199\) 16.5831 1.17555 0.587773 0.809026i \(-0.300005\pi\)
0.587773 + 0.809026i \(0.300005\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) −22.0000 −1.54410
\(204\) 0 0
\(205\) 19.8997 1.38986
\(206\) 0 0
\(207\) −6.63325 −0.461043
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0 0
\(213\) −13.2665 −0.909006
\(214\) 0 0
\(215\) −3.31662 −0.226192
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −18.0000 −1.21633
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.8997 1.33259 0.666293 0.745690i \(-0.267880\pi\)
0.666293 + 0.745690i \(0.267880\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) −23.2164 −1.53418 −0.767091 0.641539i \(-0.778296\pi\)
−0.767091 + 0.641539i \(0.778296\pi\)
\(230\) 0 0
\(231\) 33.1662 2.18218
\(232\) 0 0
\(233\) −3.00000 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 0 0
\(235\) 33.0000 2.15268
\(236\) 0 0
\(237\) 26.5330 1.72350
\(238\) 0 0
\(239\) 3.31662 0.214535 0.107267 0.994230i \(-0.465790\pi\)
0.107267 + 0.994230i \(0.465790\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 13.2665 0.847566
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 5.00000 0.315597 0.157799 0.987471i \(-0.449560\pi\)
0.157799 + 0.987471i \(0.449560\pi\)
\(252\) 0 0
\(253\) −33.1662 −2.08514
\(254\) 0 0
\(255\) −33.1662 −2.07695
\(256\) 0 0
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) 22.0000 1.36701
\(260\) 0 0
\(261\) 6.63325 0.410588
\(262\) 0 0
\(263\) 3.31662 0.204512 0.102256 0.994758i \(-0.467394\pi\)
0.102256 + 0.994758i \(0.467394\pi\)
\(264\) 0 0
\(265\) −44.0000 −2.70290
\(266\) 0 0
\(267\) −8.00000 −0.489592
\(268\) 0 0
\(269\) −13.2665 −0.808873 −0.404436 0.914566i \(-0.632532\pi\)
−0.404436 + 0.914566i \(0.632532\pi\)
\(270\) 0 0
\(271\) 19.8997 1.20882 0.604412 0.796672i \(-0.293408\pi\)
0.604412 + 0.796672i \(0.293408\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 30.0000 1.80907
\(276\) 0 0
\(277\) −3.31662 −0.199277 −0.0996383 0.995024i \(-0.531769\pi\)
−0.0996383 + 0.995024i \(0.531769\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 11.0000 0.653882 0.326941 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(284\) 0 0
\(285\) −6.63325 −0.392920
\(286\) 0 0
\(287\) −19.8997 −1.17465
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 24.0000 1.40690
\(292\) 0 0
\(293\) 26.5330 1.55007 0.775037 0.631916i \(-0.217731\pi\)
0.775037 + 0.631916i \(0.217731\pi\)
\(294\) 0 0
\(295\) −19.8997 −1.15861
\(296\) 0 0
\(297\) 20.0000 1.16052
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 3.31662 0.191167
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 33.0000 1.88957
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) −13.2665 −0.754705
\(310\) 0 0
\(311\) 16.5831 0.940343 0.470171 0.882575i \(-0.344192\pi\)
0.470171 + 0.882575i \(0.344192\pi\)
\(312\) 0 0
\(313\) 34.0000 1.92179 0.960897 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) 0 0
\(315\) −11.0000 −0.619780
\(316\) 0 0
\(317\) 6.63325 0.372560 0.186280 0.982497i \(-0.440357\pi\)
0.186280 + 0.982497i \(0.440357\pi\)
\(318\) 0 0
\(319\) 33.1662 1.85695
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −26.5330 −1.46728
\(328\) 0 0
\(329\) −33.0000 −1.81935
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −6.63325 −0.363500
\(334\) 0 0
\(335\) −26.5330 −1.44965
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) −36.0000 −1.95525
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 9.94987 0.537243
\(344\) 0 0
\(345\) 44.0000 2.36888
\(346\) 0 0
\(347\) 27.0000 1.44944 0.724718 0.689046i \(-0.241970\pi\)
0.724718 + 0.689046i \(0.241970\pi\)
\(348\) 0 0
\(349\) 9.94987 0.532605 0.266302 0.963890i \(-0.414198\pi\)
0.266302 + 0.963890i \(0.414198\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 22.0000 1.16764
\(356\) 0 0
\(357\) 33.1662 1.75534
\(358\) 0 0
\(359\) −23.2164 −1.22531 −0.612657 0.790349i \(-0.709899\pi\)
−0.612657 + 0.790349i \(0.709899\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −28.0000 −1.46962
\(364\) 0 0
\(365\) 29.8496 1.56240
\(366\) 0 0
\(367\) 6.63325 0.346253 0.173126 0.984900i \(-0.444613\pi\)
0.173126 + 0.984900i \(0.444613\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 44.0000 2.28437
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) −6.63325 −0.342540
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) −13.2665 −0.679663
\(382\) 0 0
\(383\) 13.2665 0.677886 0.338943 0.940807i \(-0.389931\pi\)
0.338943 + 0.940807i \(0.389931\pi\)
\(384\) 0 0
\(385\) −55.0000 −2.80306
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) 16.5831 0.840798 0.420399 0.907339i \(-0.361890\pi\)
0.420399 + 0.907339i \(0.361890\pi\)
\(390\) 0 0
\(391\) −33.1662 −1.67729
\(392\) 0 0
\(393\) −30.0000 −1.51330
\(394\) 0 0
\(395\) −44.0000 −2.21388
\(396\) 0 0
\(397\) −9.94987 −0.499370 −0.249685 0.968327i \(-0.580327\pi\)
−0.249685 + 0.968327i \(0.580327\pi\)
\(398\) 0 0
\(399\) 6.63325 0.332078
\(400\) 0 0
\(401\) 16.0000 0.799002 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −36.4829 −1.81285
\(406\) 0 0
\(407\) −33.1662 −1.64399
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 0 0
\(413\) 19.8997 0.979203
\(414\) 0 0
\(415\) 13.2665 0.651227
\(416\) 0 0
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 13.2665 0.646570 0.323285 0.946302i \(-0.395213\pi\)
0.323285 + 0.946302i \(0.395213\pi\)
\(422\) 0 0
\(423\) 9.94987 0.483779
\(424\) 0 0
\(425\) 30.0000 1.45521
\(426\) 0 0
\(427\) −33.0000 −1.59698
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −26.5330 −1.27805 −0.639025 0.769186i \(-0.720662\pi\)
−0.639025 + 0.769186i \(0.720662\pi\)
\(432\) 0 0
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 0 0
\(435\) −44.0000 −2.10964
\(436\) 0 0
\(437\) −6.63325 −0.317311
\(438\) 0 0
\(439\) 19.8997 0.949763 0.474882 0.880050i \(-0.342491\pi\)
0.474882 + 0.880050i \(0.342491\pi\)
\(440\) 0 0
\(441\) 4.00000 0.190476
\(442\) 0 0
\(443\) −13.0000 −0.617649 −0.308824 0.951119i \(-0.599936\pi\)
−0.308824 + 0.951119i \(0.599936\pi\)
\(444\) 0 0
\(445\) 13.2665 0.628892
\(446\) 0 0
\(447\) −6.63325 −0.313742
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 30.0000 1.41264
\(452\) 0 0
\(453\) 13.2665 0.623315
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.0000 1.35656 0.678281 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 0 0
\(459\) 20.0000 0.933520
\(460\) 0 0
\(461\) −16.5831 −0.772353 −0.386177 0.922425i \(-0.626204\pi\)
−0.386177 + 0.922425i \(0.626204\pi\)
\(462\) 0 0
\(463\) −23.2164 −1.07896 −0.539478 0.842000i \(-0.681378\pi\)
−0.539478 + 0.842000i \(0.681378\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) 26.5330 1.22518
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.00000 −0.229900
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) −13.2665 −0.607431
\(478\) 0 0
\(479\) 6.63325 0.303081 0.151540 0.988451i \(-0.451577\pi\)
0.151540 + 0.988451i \(0.451577\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −44.0000 −2.00207
\(484\) 0 0
\(485\) −39.7995 −1.80720
\(486\) 0 0
\(487\) 33.1662 1.50291 0.751453 0.659787i \(-0.229353\pi\)
0.751453 + 0.659787i \(0.229353\pi\)
\(488\) 0 0
\(489\) −40.0000 −1.80886
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 33.1662 1.49373
\(494\) 0 0
\(495\) 16.5831 0.745356
\(496\) 0 0
\(497\) −22.0000 −0.986835
\(498\) 0 0
\(499\) 3.00000 0.134298 0.0671492 0.997743i \(-0.478610\pi\)
0.0671492 + 0.997743i \(0.478610\pi\)
\(500\) 0 0
\(501\) −13.2665 −0.592703
\(502\) 0 0
\(503\) 19.8997 0.887286 0.443643 0.896204i \(-0.353686\pi\)
0.443643 + 0.896204i \(0.353686\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 26.0000 1.15470
\(508\) 0 0
\(509\) −26.5330 −1.17605 −0.588027 0.808841i \(-0.700095\pi\)
−0.588027 + 0.808841i \(0.700095\pi\)
\(510\) 0 0
\(511\) −29.8496 −1.32047
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 22.0000 0.969436
\(516\) 0 0
\(517\) 49.7494 2.18797
\(518\) 0 0
\(519\) 39.7995 1.74700
\(520\) 0 0
\(521\) −16.0000 −0.700973 −0.350486 0.936568i \(-0.613984\pi\)
−0.350486 + 0.936568i \(0.613984\pi\)
\(522\) 0 0
\(523\) 6.00000 0.262362 0.131181 0.991358i \(-0.458123\pi\)
0.131181 + 0.991358i \(0.458123\pi\)
\(524\) 0 0
\(525\) 39.7995 1.73699
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.0000 0.913043
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −19.8997 −0.860341
\(536\) 0 0
\(537\) 36.0000 1.55351
\(538\) 0 0
\(539\) 20.0000 0.861461
\(540\) 0 0
\(541\) 9.94987 0.427779 0.213889 0.976858i \(-0.431387\pi\)
0.213889 + 0.976858i \(0.431387\pi\)
\(542\) 0 0
\(543\) −39.7995 −1.70796
\(544\) 0 0
\(545\) 44.0000 1.88475
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 9.94987 0.424650
\(550\) 0 0
\(551\) 6.63325 0.282586
\(552\) 0 0
\(553\) 44.0000 1.87107
\(554\) 0 0
\(555\) 44.0000 1.86770
\(556\) 0 0
\(557\) 3.31662 0.140530 0.0702650 0.997528i \(-0.477616\pi\)
0.0702650 + 0.997528i \(0.477616\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −50.0000 −2.11100
\(562\) 0 0
\(563\) −14.0000 −0.590030 −0.295015 0.955493i \(-0.595325\pi\)
−0.295015 + 0.955493i \(0.595325\pi\)
\(564\) 0 0
\(565\) 59.6992 2.51157
\(566\) 0 0
\(567\) 36.4829 1.53214
\(568\) 0 0
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 33.1662 1.38554
\(574\) 0 0
\(575\) −39.7995 −1.65975
\(576\) 0 0
\(577\) −43.0000 −1.79011 −0.895057 0.445952i \(-0.852865\pi\)
−0.895057 + 0.445952i \(0.852865\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −13.2665 −0.550387
\(582\) 0 0
\(583\) −66.3325 −2.74721
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.00000 0.206372 0.103186 0.994662i \(-0.467096\pi\)
0.103186 + 0.994662i \(0.467096\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 53.0660 2.18284
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −55.0000 −2.25478
\(596\) 0 0
\(597\) −33.1662 −1.35740
\(598\) 0 0
\(599\) 13.2665 0.542054 0.271027 0.962572i \(-0.412637\pi\)
0.271027 + 0.962572i \(0.412637\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) 46.4327 1.88776
\(606\) 0 0
\(607\) 26.5330 1.07694 0.538471 0.842644i \(-0.319002\pi\)
0.538471 + 0.842644i \(0.319002\pi\)
\(608\) 0 0
\(609\) 44.0000 1.78297
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3.31662 −0.133957 −0.0669786 0.997754i \(-0.521336\pi\)
−0.0669786 + 0.997754i \(0.521336\pi\)
\(614\) 0 0
\(615\) −39.7995 −1.60487
\(616\) 0 0
\(617\) 41.0000 1.65060 0.825299 0.564696i \(-0.191007\pi\)
0.825299 + 0.564696i \(0.191007\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −26.5330 −1.06473
\(622\) 0 0
\(623\) −13.2665 −0.531511
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −10.0000 −0.399362
\(628\) 0 0
\(629\) −33.1662 −1.32242
\(630\) 0 0
\(631\) −9.94987 −0.396098 −0.198049 0.980192i \(-0.563461\pi\)
−0.198049 + 0.980192i \(0.563461\pi\)
\(632\) 0 0
\(633\) −28.0000 −1.11290
\(634\) 0 0
\(635\) 22.0000 0.873043
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.63325 0.262407
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) 0 0
\(645\) 6.63325 0.261184
\(646\) 0 0
\(647\) 16.5831 0.651950 0.325975 0.945378i \(-0.394307\pi\)
0.325975 + 0.945378i \(0.394307\pi\)
\(648\) 0 0
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −36.4829 −1.42769 −0.713843 0.700306i \(-0.753047\pi\)
−0.713843 + 0.700306i \(0.753047\pi\)
\(654\) 0 0
\(655\) 49.7494 1.94387
\(656\) 0 0
\(657\) 9.00000 0.351123
\(658\) 0 0
\(659\) 26.0000 1.01282 0.506408 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) 0 0
\(661\) −39.7995 −1.54802 −0.774011 0.633173i \(-0.781752\pi\)
−0.774011 + 0.633173i \(0.781752\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.0000 −0.426562
\(666\) 0 0
\(667\) −44.0000 −1.70369
\(668\) 0 0
\(669\) −39.7995 −1.53874
\(670\) 0 0
\(671\) 49.7494 1.92055
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 24.0000 0.923760
\(676\) 0 0
\(677\) 19.8997 0.764809 0.382405 0.923995i \(-0.375096\pi\)
0.382405 + 0.923995i \(0.375096\pi\)
\(678\) 0 0
\(679\) 39.7995 1.52736
\(680\) 0 0
\(681\) −16.0000 −0.613121
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −16.5831 −0.633609
\(686\) 0 0
\(687\) 46.4327 1.77152
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −31.0000 −1.17930 −0.589648 0.807661i \(-0.700733\pi\)
−0.589648 + 0.807661i \(0.700733\pi\)
\(692\) 0 0
\(693\) −16.5831 −0.629941
\(694\) 0 0
\(695\) 36.4829 1.38387
\(696\) 0 0
\(697\) 30.0000 1.13633
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −6.63325 −0.250178
\(704\) 0 0
\(705\) −66.0000 −2.48570
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 13.2665 0.498234 0.249117 0.968473i \(-0.419860\pi\)
0.249117 + 0.968473i \(0.419860\pi\)
\(710\) 0 0
\(711\) −13.2665 −0.497533
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.63325 −0.247723
\(718\) 0 0
\(719\) −3.31662 −0.123689 −0.0618446 0.998086i \(-0.519698\pi\)
−0.0618446 + 0.998086i \(0.519698\pi\)
\(720\) 0 0
\(721\) −22.0000 −0.819323
\(722\) 0 0
\(723\) −52.0000 −1.93390
\(724\) 0 0
\(725\) 39.7995 1.47812
\(726\) 0 0
\(727\) 16.5831 0.615034 0.307517 0.951543i \(-0.400502\pi\)
0.307517 + 0.951543i \(0.400502\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −5.00000 −0.184932
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) −26.5330 −0.978684
\(736\) 0 0
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) −45.0000 −1.65535 −0.827676 0.561206i \(-0.810337\pi\)
−0.827676 + 0.561206i \(0.810337\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −53.0660 −1.94680 −0.973401 0.229107i \(-0.926420\pi\)
−0.973401 + 0.229107i \(0.926420\pi\)
\(744\) 0 0
\(745\) 11.0000 0.403009
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 19.8997 0.727121
\(750\) 0 0
\(751\) −39.7995 −1.45230 −0.726152 0.687534i \(-0.758693\pi\)
−0.726152 + 0.687534i \(0.758693\pi\)
\(752\) 0 0
\(753\) −10.0000 −0.364420
\(754\) 0 0
\(755\) −22.0000 −0.800662
\(756\) 0 0
\(757\) 9.94987 0.361634 0.180817 0.983517i \(-0.442126\pi\)
0.180817 + 0.983517i \(0.442126\pi\)
\(758\) 0 0
\(759\) 66.3325 2.40772
\(760\) 0 0
\(761\) −1.00000 −0.0362500 −0.0181250 0.999836i \(-0.505770\pi\)
−0.0181250 + 0.999836i \(0.505770\pi\)
\(762\) 0 0
\(763\) −44.0000 −1.59291
\(764\) 0 0
\(765\) 16.5831 0.599564
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 33.0000 1.19001 0.595005 0.803722i \(-0.297150\pi\)
0.595005 + 0.803722i \(0.297150\pi\)
\(770\) 0 0
\(771\) 16.0000 0.576226
\(772\) 0 0
\(773\) 6.63325 0.238581 0.119291 0.992859i \(-0.461938\pi\)
0.119291 + 0.992859i \(0.461938\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −44.0000 −1.57849
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 33.1662 1.18678
\(782\) 0 0
\(783\) 26.5330 0.948212
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 0 0
\(789\) −6.63325 −0.236150
\(790\) 0 0
\(791\) −59.6992 −2.12266
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 88.0000 3.12104
\(796\) 0 0
\(797\) −26.5330 −0.939847 −0.469924 0.882707i \(-0.655719\pi\)
−0.469924 + 0.882707i \(0.655719\pi\)
\(798\) 0 0
\(799\) 49.7494 1.76001
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) 45.0000 1.58802
\(804\) 0 0
\(805\) 72.9657 2.57170
\(806\) 0 0
\(807\) 26.5330 0.934006
\(808\) 0 0
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) −39.7995 −1.39583
\(814\) 0 0
\(815\) 66.3325 2.32353
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.94987 −0.347253 −0.173627 0.984812i \(-0.555549\pi\)
−0.173627 + 0.984812i \(0.555549\pi\)
\(822\) 0 0
\(823\) 29.8496 1.04049 0.520246 0.854016i \(-0.325840\pi\)
0.520246 + 0.854016i \(0.325840\pi\)
\(824\) 0 0
\(825\) −60.0000 −2.08893
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 6.63325 0.230105
\(832\) 0 0
\(833\) 20.0000 0.692959
\(834\) 0 0
\(835\) 22.0000 0.761341
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −46.4327 −1.60304 −0.801518 0.597970i \(-0.795974\pi\)
−0.801518 + 0.597970i \(0.795974\pi\)
\(840\) 0 0
\(841\) 15.0000 0.517241
\(842\) 0 0
\(843\) −52.0000 −1.79098
\(844\) 0 0
\(845\) −43.1161 −1.48324
\(846\) 0 0
\(847\) −46.4327 −1.59545
\(848\) 0 0
\(849\) −22.0000 −0.755038
\(850\) 0 0
\(851\) 44.0000 1.50830
\(852\) 0 0
\(853\) −53.0660 −1.81695 −0.908473 0.417945i \(-0.862751\pi\)
−0.908473 + 0.417945i \(0.862751\pi\)
\(854\) 0 0
\(855\) 3.31662 0.113426
\(856\) 0 0
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) 0 0
\(859\) −41.0000 −1.39890 −0.699451 0.714681i \(-0.746572\pi\)
−0.699451 + 0.714681i \(0.746572\pi\)
\(860\) 0 0
\(861\) 39.7995 1.35636
\(862\) 0 0
\(863\) −33.1662 −1.12899 −0.564496 0.825436i \(-0.690929\pi\)
−0.564496 + 0.825436i \(0.690929\pi\)
\(864\) 0 0
\(865\) −66.0000 −2.24407
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) −66.3325 −2.25018
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −12.0000 −0.406138
\(874\) 0 0
\(875\) −11.0000 −0.371868
\(876\) 0 0
\(877\) 6.63325 0.223989 0.111994 0.993709i \(-0.464276\pi\)
0.111994 + 0.993709i \(0.464276\pi\)
\(878\) 0 0
\(879\) −53.0660 −1.78987
\(880\) 0 0
\(881\) 51.0000 1.71823 0.859117 0.511780i \(-0.171014\pi\)
0.859117 + 0.511780i \(0.171014\pi\)
\(882\) 0 0
\(883\) 31.0000 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(884\) 0 0
\(885\) 39.7995 1.33785
\(886\) 0 0
\(887\) −19.8997 −0.668168 −0.334084 0.942543i \(-0.608427\pi\)
−0.334084 + 0.942543i \(0.608427\pi\)
\(888\) 0 0
\(889\) −22.0000 −0.737856
\(890\) 0 0
\(891\) −55.0000 −1.84257
\(892\) 0 0
\(893\) 9.94987 0.332960
\(894\) 0 0
\(895\) −59.6992 −1.99553
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −66.3325 −2.20986
\(902\) 0 0
\(903\) −6.63325 −0.220741
\(904\) 0 0
\(905\) 66.0000 2.19391
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −19.8997 −0.659308 −0.329654 0.944102i \(-0.606932\pi\)
−0.329654 + 0.944102i \(0.606932\pi\)
\(912\) 0 0
\(913\) 20.0000 0.661903
\(914\) 0 0
\(915\) −66.0000 −2.18189
\(916\) 0 0
\(917\) −49.7494 −1.64287
\(918\) 0 0
\(919\) 33.1662 1.09405 0.547027 0.837115i \(-0.315760\pi\)
0.547027 + 0.837115i \(0.315760\pi\)
\(920\) 0 0
\(921\) 24.0000 0.790827
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −39.7995 −1.30860
\(926\) 0 0
\(927\) 6.63325 0.217865
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) −33.1662 −1.08581
\(934\) 0 0
\(935\) 82.9156 2.71163
\(936\) 0 0
\(937\) −1.00000 −0.0326686 −0.0163343 0.999867i \(-0.505200\pi\)
−0.0163343 + 0.999867i \(0.505200\pi\)
\(938\) 0 0
\(939\) −68.0000 −2.21910
\(940\) 0 0
\(941\) −46.4327 −1.51366 −0.756832 0.653609i \(-0.773254\pi\)
−0.756832 + 0.653609i \(0.773254\pi\)
\(942\) 0 0
\(943\) −39.7995 −1.29605
\(944\) 0 0
\(945\) −44.0000 −1.43132
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −13.2665 −0.430196
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) −55.0000 −1.77976
\(956\) 0 0
\(957\) −66.3325 −2.14423
\(958\) 0 0
\(959\) 16.5831 0.535497
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −33.1662 −1.06655 −0.533277 0.845940i \(-0.679040\pi\)
−0.533277 + 0.845940i \(0.679040\pi\)
\(968\) 0 0
\(969\) −10.0000 −0.321246
\(970\) 0 0
\(971\) −32.0000 −1.02693 −0.513464 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) 0 0
\(973\) −36.4829 −1.16959
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.0000 0.511885 0.255943 0.966692i \(-0.417614\pi\)
0.255943 + 0.966692i \(0.417614\pi\)
\(978\) 0 0
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 13.2665 0.423567
\(982\) 0 0
\(983\) 13.2665 0.423136 0.211568 0.977363i \(-0.432143\pi\)
0.211568 + 0.977363i \(0.432143\pi\)
\(984\) 0 0
\(985\) −88.0000 −2.80391
\(986\) 0 0
\(987\) 66.0000 2.10080
\(988\) 0 0
\(989\) 6.63325 0.210925
\(990\) 0 0
\(991\) −33.1662 −1.05356 −0.526780 0.850001i \(-0.676601\pi\)
−0.526780 + 0.850001i \(0.676601\pi\)
\(992\) 0 0
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 55.0000 1.74362
\(996\) 0 0
\(997\) −43.1161 −1.36550 −0.682751 0.730651i \(-0.739216\pi\)
−0.682751 + 0.730651i \(0.739216\pi\)
\(998\) 0 0
\(999\) −26.5330 −0.839467
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 4864.2.a.r.1.2 2
4.3 odd 2 4864.2.a.y.1.2 2
8.3 odd 2 inner 4864.2.a.r.1.1 2
8.5 even 2 4864.2.a.y.1.1 2
16.3 odd 4 1216.2.c.f.609.3 yes 4
16.5 even 4 1216.2.c.f.609.4 yes 4
16.11 odd 4 1216.2.c.f.609.2 yes 4
16.13 even 4 1216.2.c.f.609.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.f.609.1 4 16.13 even 4
1216.2.c.f.609.2 yes 4 16.11 odd 4
1216.2.c.f.609.3 yes 4 16.3 odd 4
1216.2.c.f.609.4 yes 4 16.5 even 4
4864.2.a.r.1.1 2 8.3 odd 2 inner
4864.2.a.r.1.2 2 1.1 even 1 trivial
4864.2.a.y.1.1 2 8.5 even 2
4864.2.a.y.1.2 2 4.3 odd 2