Properties

Label 896.2.b.f.449.4
Level $896$
Weight $2$
Character 896.449
Analytic conductor $7.155$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.4
Root \(0.765367i\) of defining polynomial
Character \(\chi\) \(=\) 896.449
Dual form 896.2.b.f.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.61313i q^{3} -2.61313i q^{5} -1.00000 q^{7} -3.82843 q^{9} +O(q^{10})\) \(q+2.61313i q^{3} -2.61313i q^{5} -1.00000 q^{7} -3.82843 q^{9} +2.16478i q^{11} +0.448342i q^{13} +6.82843 q^{15} -7.65685 q^{17} +4.77791i q^{19} -2.61313i q^{21} -6.82843 q^{23} -1.82843 q^{25} -2.16478i q^{27} +9.55582i q^{29} -5.65685 q^{31} -5.65685 q^{33} +2.61313i q^{35} -5.22625i q^{37} -1.17157 q^{39} -3.65685 q^{41} -2.16478i q^{43} +10.0042i q^{45} +8.00000 q^{47} +1.00000 q^{49} -20.0083i q^{51} +10.4525i q^{53} +5.65685 q^{55} -12.4853 q^{57} -0.448342i q^{59} -12.1689i q^{61} +3.82843 q^{63} +1.17157 q^{65} +3.06147i q^{67} -17.8435i q^{69} -2.34315 q^{71} +11.6569 q^{73} -4.77791i q^{75} -2.16478i q^{77} +2.34315 q^{79} -5.82843 q^{81} -13.0656i q^{83} +20.0083i q^{85} -24.9706 q^{87} -2.00000 q^{89} -0.448342i q^{91} -14.7821i q^{93} +12.4853 q^{95} -10.0000 q^{97} -8.28772i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{7} - 4q^{9} + 16q^{15} - 8q^{17} - 16q^{23} + 4q^{25} - 16q^{39} + 8q^{41} + 32q^{47} + 4q^{49} - 16q^{57} + 4q^{63} + 16q^{65} - 32q^{71} + 24q^{73} + 32q^{79} - 12q^{81} - 32q^{87} - 8q^{89} + 16q^{95} - 40q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.61313i 1.50869i 0.656479 + 0.754344i \(0.272045\pi\)
−0.656479 + 0.754344i \(0.727955\pi\)
\(4\) 0 0
\(5\) − 2.61313i − 1.16863i −0.811529 0.584313i \(-0.801364\pi\)
0.811529 0.584313i \(-0.198636\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −3.82843 −1.27614
\(10\) 0 0
\(11\) 2.16478i 0.652707i 0.945248 + 0.326354i \(0.105820\pi\)
−0.945248 + 0.326354i \(0.894180\pi\)
\(12\) 0 0
\(13\) 0.448342i 0.124348i 0.998065 + 0.0621738i \(0.0198033\pi\)
−0.998065 + 0.0621738i \(0.980197\pi\)
\(14\) 0 0
\(15\) 6.82843 1.76309
\(16\) 0 0
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) 4.77791i 1.09613i 0.836436 + 0.548064i \(0.184635\pi\)
−0.836436 + 0.548064i \(0.815365\pi\)
\(20\) 0 0
\(21\) − 2.61313i − 0.570231i
\(22\) 0 0
\(23\) −6.82843 −1.42383 −0.711913 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) −1.82843 −0.365685
\(26\) 0 0
\(27\) − 2.16478i − 0.416613i
\(28\) 0 0
\(29\) 9.55582i 1.77447i 0.461316 + 0.887236i \(0.347377\pi\)
−0.461316 + 0.887236i \(0.652623\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) −5.65685 −0.984732
\(34\) 0 0
\(35\) 2.61313i 0.441699i
\(36\) 0 0
\(37\) − 5.22625i − 0.859191i −0.903022 0.429595i \(-0.858656\pi\)
0.903022 0.429595i \(-0.141344\pi\)
\(38\) 0 0
\(39\) −1.17157 −0.187602
\(40\) 0 0
\(41\) −3.65685 −0.571105 −0.285552 0.958363i \(-0.592177\pi\)
−0.285552 + 0.958363i \(0.592177\pi\)
\(42\) 0 0
\(43\) − 2.16478i − 0.330127i −0.986283 0.165063i \(-0.947217\pi\)
0.986283 0.165063i \(-0.0527828\pi\)
\(44\) 0 0
\(45\) 10.0042i 1.49133i
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 20.0083i − 2.80173i
\(52\) 0 0
\(53\) 10.4525i 1.43576i 0.696166 + 0.717881i \(0.254888\pi\)
−0.696166 + 0.717881i \(0.745112\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) −12.4853 −1.65372
\(58\) 0 0
\(59\) − 0.448342i − 0.0583691i −0.999574 0.0291845i \(-0.990709\pi\)
0.999574 0.0291845i \(-0.00929105\pi\)
\(60\) 0 0
\(61\) − 12.1689i − 1.55807i −0.626978 0.779037i \(-0.715708\pi\)
0.626978 0.779037i \(-0.284292\pi\)
\(62\) 0 0
\(63\) 3.82843 0.482336
\(64\) 0 0
\(65\) 1.17157 0.145316
\(66\) 0 0
\(67\) 3.06147i 0.374018i 0.982358 + 0.187009i \(0.0598793\pi\)
−0.982358 + 0.187009i \(0.940121\pi\)
\(68\) 0 0
\(69\) − 17.8435i − 2.14811i
\(70\) 0 0
\(71\) −2.34315 −0.278080 −0.139040 0.990287i \(-0.544402\pi\)
−0.139040 + 0.990287i \(0.544402\pi\)
\(72\) 0 0
\(73\) 11.6569 1.36433 0.682166 0.731198i \(-0.261038\pi\)
0.682166 + 0.731198i \(0.261038\pi\)
\(74\) 0 0
\(75\) − 4.77791i − 0.551706i
\(76\) 0 0
\(77\) − 2.16478i − 0.246700i
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) −5.82843 −0.647603
\(82\) 0 0
\(83\) − 13.0656i − 1.43414i −0.697002 0.717070i \(-0.745483\pi\)
0.697002 0.717070i \(-0.254517\pi\)
\(84\) 0 0
\(85\) 20.0083i 2.17021i
\(86\) 0 0
\(87\) −24.9706 −2.67713
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) − 0.448342i − 0.0469990i
\(92\) 0 0
\(93\) − 14.7821i − 1.53283i
\(94\) 0 0
\(95\) 12.4853 1.28096
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) − 8.28772i − 0.832947i
\(100\) 0 0
\(101\) 4.77791i 0.475420i 0.971336 + 0.237710i \(0.0763968\pi\)
−0.971336 + 0.237710i \(0.923603\pi\)
\(102\) 0 0
\(103\) −13.6569 −1.34565 −0.672825 0.739802i \(-0.734919\pi\)
−0.672825 + 0.739802i \(0.734919\pi\)
\(104\) 0 0
\(105\) −6.82843 −0.666386
\(106\) 0 0
\(107\) 13.5140i 1.30644i 0.757166 + 0.653222i \(0.226583\pi\)
−0.757166 + 0.653222i \(0.773417\pi\)
\(108\) 0 0
\(109\) − 11.3492i − 1.08705i −0.839391 0.543527i \(-0.817088\pi\)
0.839391 0.543527i \(-0.182912\pi\)
\(110\) 0 0
\(111\) 13.6569 1.29625
\(112\) 0 0
\(113\) 8.82843 0.830509 0.415254 0.909705i \(-0.363693\pi\)
0.415254 + 0.909705i \(0.363693\pi\)
\(114\) 0 0
\(115\) 17.8435i 1.66392i
\(116\) 0 0
\(117\) − 1.71644i − 0.158685i
\(118\) 0 0
\(119\) 7.65685 0.701903
\(120\) 0 0
\(121\) 6.31371 0.573973
\(122\) 0 0
\(123\) − 9.55582i − 0.861619i
\(124\) 0 0
\(125\) − 8.28772i − 0.741276i
\(126\) 0 0
\(127\) 4.48528 0.398004 0.199002 0.979999i \(-0.436230\pi\)
0.199002 + 0.979999i \(0.436230\pi\)
\(128\) 0 0
\(129\) 5.65685 0.498058
\(130\) 0 0
\(131\) 18.2919i 1.59817i 0.601219 + 0.799085i \(0.294682\pi\)
−0.601219 + 0.799085i \(0.705318\pi\)
\(132\) 0 0
\(133\) − 4.77791i − 0.414297i
\(134\) 0 0
\(135\) −5.65685 −0.486864
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 19.5600i 1.65906i 0.558465 + 0.829528i \(0.311390\pi\)
−0.558465 + 0.829528i \(0.688610\pi\)
\(140\) 0 0
\(141\) 20.9050i 1.76052i
\(142\) 0 0
\(143\) −0.970563 −0.0811625
\(144\) 0 0
\(145\) 24.9706 2.07369
\(146\) 0 0
\(147\) 2.61313i 0.215527i
\(148\) 0 0
\(149\) 16.5754i 1.35791i 0.734179 + 0.678956i \(0.237567\pi\)
−0.734179 + 0.678956i \(0.762433\pi\)
\(150\) 0 0
\(151\) 18.1421 1.47639 0.738193 0.674590i \(-0.235679\pi\)
0.738193 + 0.674590i \(0.235679\pi\)
\(152\) 0 0
\(153\) 29.3137 2.36987
\(154\) 0 0
\(155\) 14.7821i 1.18732i
\(156\) 0 0
\(157\) 4.77791i 0.381319i 0.981656 + 0.190659i \(0.0610626\pi\)
−0.981656 + 0.190659i \(0.938937\pi\)
\(158\) 0 0
\(159\) −27.3137 −2.16612
\(160\) 0 0
\(161\) 6.82843 0.538155
\(162\) 0 0
\(163\) − 3.95815i − 0.310026i −0.987912 0.155013i \(-0.950458\pi\)
0.987912 0.155013i \(-0.0495420\pi\)
\(164\) 0 0
\(165\) 14.7821i 1.15078i
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 12.7990 0.984538
\(170\) 0 0
\(171\) − 18.2919i − 1.39882i
\(172\) 0 0
\(173\) − 16.1271i − 1.22612i −0.790036 0.613060i \(-0.789938\pi\)
0.790036 0.613060i \(-0.210062\pi\)
\(174\) 0 0
\(175\) 1.82843 0.138216
\(176\) 0 0
\(177\) 1.17157 0.0880608
\(178\) 0 0
\(179\) 22.1731i 1.65730i 0.559770 + 0.828648i \(0.310889\pi\)
−0.559770 + 0.828648i \(0.689111\pi\)
\(180\) 0 0
\(181\) 2.61313i 0.194232i 0.995273 + 0.0971161i \(0.0309618\pi\)
−0.995273 + 0.0971161i \(0.969038\pi\)
\(182\) 0 0
\(183\) 31.7990 2.35065
\(184\) 0 0
\(185\) −13.6569 −1.00407
\(186\) 0 0
\(187\) − 16.5754i − 1.21212i
\(188\) 0 0
\(189\) 2.16478i 0.157465i
\(190\) 0 0
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) 0 0
\(193\) 13.7990 0.993273 0.496637 0.867959i \(-0.334568\pi\)
0.496637 + 0.867959i \(0.334568\pi\)
\(194\) 0 0
\(195\) 3.06147i 0.219236i
\(196\) 0 0
\(197\) − 6.12293i − 0.436241i −0.975922 0.218121i \(-0.930007\pi\)
0.975922 0.218121i \(-0.0699926\pi\)
\(198\) 0 0
\(199\) −8.97056 −0.635906 −0.317953 0.948106i \(-0.602995\pi\)
−0.317953 + 0.948106i \(0.602995\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) − 9.55582i − 0.670687i
\(204\) 0 0
\(205\) 9.55582i 0.667407i
\(206\) 0 0
\(207\) 26.1421 1.81700
\(208\) 0 0
\(209\) −10.3431 −0.715450
\(210\) 0 0
\(211\) 9.18440i 0.632280i 0.948712 + 0.316140i \(0.102387\pi\)
−0.948712 + 0.316140i \(0.897613\pi\)
\(212\) 0 0
\(213\) − 6.12293i − 0.419537i
\(214\) 0 0
\(215\) −5.65685 −0.385794
\(216\) 0 0
\(217\) 5.65685 0.384012
\(218\) 0 0
\(219\) 30.4608i 2.05835i
\(220\) 0 0
\(221\) − 3.43289i − 0.230921i
\(222\) 0 0
\(223\) 8.97056 0.600713 0.300357 0.953827i \(-0.402894\pi\)
0.300357 + 0.953827i \(0.402894\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) − 14.3337i − 0.951363i −0.879618 0.475682i \(-0.842201\pi\)
0.879618 0.475682i \(-0.157799\pi\)
\(228\) 0 0
\(229\) 5.67459i 0.374988i 0.982266 + 0.187494i \(0.0600365\pi\)
−0.982266 + 0.187494i \(0.939964\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) − 20.9050i − 1.36369i
\(236\) 0 0
\(237\) 6.12293i 0.397727i
\(238\) 0 0
\(239\) 18.1421 1.17352 0.586759 0.809762i \(-0.300404\pi\)
0.586759 + 0.809762i \(0.300404\pi\)
\(240\) 0 0
\(241\) 1.31371 0.0846234 0.0423117 0.999104i \(-0.486528\pi\)
0.0423117 + 0.999104i \(0.486528\pi\)
\(242\) 0 0
\(243\) − 21.7248i − 1.39364i
\(244\) 0 0
\(245\) − 2.61313i − 0.166946i
\(246\) 0 0
\(247\) −2.14214 −0.136301
\(248\) 0 0
\(249\) 34.1421 2.16367
\(250\) 0 0
\(251\) − 0.819760i − 0.0517428i −0.999665 0.0258714i \(-0.991764\pi\)
0.999665 0.0258714i \(-0.00823604\pi\)
\(252\) 0 0
\(253\) − 14.7821i − 0.929341i
\(254\) 0 0
\(255\) −52.2843 −3.27417
\(256\) 0 0
\(257\) −9.31371 −0.580973 −0.290487 0.956879i \(-0.593817\pi\)
−0.290487 + 0.956879i \(0.593817\pi\)
\(258\) 0 0
\(259\) 5.22625i 0.324743i
\(260\) 0 0
\(261\) − 36.5838i − 2.26448i
\(262\) 0 0
\(263\) −2.34315 −0.144485 −0.0722423 0.997387i \(-0.523015\pi\)
−0.0722423 + 0.997387i \(0.523015\pi\)
\(264\) 0 0
\(265\) 27.3137 1.67787
\(266\) 0 0
\(267\) − 5.22625i − 0.319841i
\(268\) 0 0
\(269\) − 8.21080i − 0.500621i −0.968166 0.250311i \(-0.919467\pi\)
0.968166 0.250311i \(-0.0805327\pi\)
\(270\) 0 0
\(271\) 2.34315 0.142336 0.0711680 0.997464i \(-0.477327\pi\)
0.0711680 + 0.997464i \(0.477327\pi\)
\(272\) 0 0
\(273\) 1.17157 0.0709068
\(274\) 0 0
\(275\) − 3.95815i − 0.238685i
\(276\) 0 0
\(277\) − 14.7821i − 0.888169i −0.895985 0.444084i \(-0.853529\pi\)
0.895985 0.444084i \(-0.146471\pi\)
\(278\) 0 0
\(279\) 21.6569 1.29656
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) − 20.8281i − 1.23810i −0.785351 0.619051i \(-0.787518\pi\)
0.785351 0.619051i \(-0.212482\pi\)
\(284\) 0 0
\(285\) 32.6256i 1.93257i
\(286\) 0 0
\(287\) 3.65685 0.215857
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) − 26.1313i − 1.53184i
\(292\) 0 0
\(293\) 10.9008i 0.636834i 0.947951 + 0.318417i \(0.103151\pi\)
−0.947951 + 0.318417i \(0.896849\pi\)
\(294\) 0 0
\(295\) −1.17157 −0.0682116
\(296\) 0 0
\(297\) 4.68629 0.271926
\(298\) 0 0
\(299\) − 3.06147i − 0.177049i
\(300\) 0 0
\(301\) 2.16478i 0.124776i
\(302\) 0 0
\(303\) −12.4853 −0.717261
\(304\) 0 0
\(305\) −31.7990 −1.82080
\(306\) 0 0
\(307\) − 8.73606i − 0.498593i −0.968427 0.249297i \(-0.919801\pi\)
0.968427 0.249297i \(-0.0801994\pi\)
\(308\) 0 0
\(309\) − 35.6871i − 2.03017i
\(310\) 0 0
\(311\) −14.6274 −0.829445 −0.414722 0.909948i \(-0.636121\pi\)
−0.414722 + 0.909948i \(0.636121\pi\)
\(312\) 0 0
\(313\) −8.34315 −0.471582 −0.235791 0.971804i \(-0.575768\pi\)
−0.235791 + 0.971804i \(0.575768\pi\)
\(314\) 0 0
\(315\) − 10.0042i − 0.563671i
\(316\) 0 0
\(317\) 8.65914i 0.486346i 0.969983 + 0.243173i \(0.0781882\pi\)
−0.969983 + 0.243173i \(0.921812\pi\)
\(318\) 0 0
\(319\) −20.6863 −1.15821
\(320\) 0 0
\(321\) −35.3137 −1.97102
\(322\) 0 0
\(323\) − 36.5838i − 2.03558i
\(324\) 0 0
\(325\) − 0.819760i − 0.0454721i
\(326\) 0 0
\(327\) 29.6569 1.64003
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 18.7402i 1.03006i 0.857173 + 0.515028i \(0.172218\pi\)
−0.857173 + 0.515028i \(0.827782\pi\)
\(332\) 0 0
\(333\) 20.0083i 1.09645i
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 11.1716 0.608554 0.304277 0.952584i \(-0.401585\pi\)
0.304277 + 0.952584i \(0.401585\pi\)
\(338\) 0 0
\(339\) 23.0698i 1.25298i
\(340\) 0 0
\(341\) − 12.2459i − 0.663151i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −46.6274 −2.51034
\(346\) 0 0
\(347\) 12.6173i 0.677332i 0.940907 + 0.338666i \(0.109976\pi\)
−0.940907 + 0.338666i \(0.890024\pi\)
\(348\) 0 0
\(349\) 22.6215i 1.21090i 0.795884 + 0.605449i \(0.207007\pi\)
−0.795884 + 0.605449i \(0.792993\pi\)
\(350\) 0 0
\(351\) 0.970563 0.0518048
\(352\) 0 0
\(353\) 15.6569 0.833330 0.416665 0.909060i \(-0.363199\pi\)
0.416665 + 0.909060i \(0.363199\pi\)
\(354\) 0 0
\(355\) 6.12293i 0.324972i
\(356\) 0 0
\(357\) 20.0083i 1.05895i
\(358\) 0 0
\(359\) −6.82843 −0.360391 −0.180195 0.983631i \(-0.557673\pi\)
−0.180195 + 0.983631i \(0.557673\pi\)
\(360\) 0 0
\(361\) −3.82843 −0.201496
\(362\) 0 0
\(363\) 16.4985i 0.865947i
\(364\) 0 0
\(365\) − 30.4608i − 1.59439i
\(366\) 0 0
\(367\) −24.9706 −1.30345 −0.651726 0.758454i \(-0.725955\pi\)
−0.651726 + 0.758454i \(0.725955\pi\)
\(368\) 0 0
\(369\) 14.0000 0.728811
\(370\) 0 0
\(371\) − 10.4525i − 0.542667i
\(372\) 0 0
\(373\) 33.8937i 1.75495i 0.479622 + 0.877475i \(0.340774\pi\)
−0.479622 + 0.877475i \(0.659226\pi\)
\(374\) 0 0
\(375\) 21.6569 1.11836
\(376\) 0 0
\(377\) −4.28427 −0.220651
\(378\) 0 0
\(379\) 23.0698i 1.18502i 0.805565 + 0.592508i \(0.201862\pi\)
−0.805565 + 0.592508i \(0.798138\pi\)
\(380\) 0 0
\(381\) 11.7206i 0.600465i
\(382\) 0 0
\(383\) 19.3137 0.986884 0.493442 0.869779i \(-0.335738\pi\)
0.493442 + 0.869779i \(0.335738\pi\)
\(384\) 0 0
\(385\) −5.65685 −0.288300
\(386\) 0 0
\(387\) 8.28772i 0.421288i
\(388\) 0 0
\(389\) − 20.0083i − 1.01446i −0.861810 0.507231i \(-0.830669\pi\)
0.861810 0.507231i \(-0.169331\pi\)
\(390\) 0 0
\(391\) 52.2843 2.64413
\(392\) 0 0
\(393\) −47.7990 −2.41114
\(394\) 0 0
\(395\) − 6.12293i − 0.308078i
\(396\) 0 0
\(397\) 18.2919i 0.918043i 0.888425 + 0.459022i \(0.151800\pi\)
−0.888425 + 0.459022i \(0.848200\pi\)
\(398\) 0 0
\(399\) 12.4853 0.625046
\(400\) 0 0
\(401\) 9.51472 0.475142 0.237571 0.971370i \(-0.423649\pi\)
0.237571 + 0.971370i \(0.423649\pi\)
\(402\) 0 0
\(403\) − 2.53620i − 0.126337i
\(404\) 0 0
\(405\) 15.2304i 0.756805i
\(406\) 0 0
\(407\) 11.3137 0.560800
\(408\) 0 0
\(409\) 5.31371 0.262746 0.131373 0.991333i \(-0.458061\pi\)
0.131373 + 0.991333i \(0.458061\pi\)
\(410\) 0 0
\(411\) 5.22625i 0.257792i
\(412\) 0 0
\(413\) 0.448342i 0.0220614i
\(414\) 0 0
\(415\) −34.1421 −1.67597
\(416\) 0 0
\(417\) −51.1127 −2.50300
\(418\) 0 0
\(419\) 13.0656i 0.638298i 0.947705 + 0.319149i \(0.103397\pi\)
−0.947705 + 0.319149i \(0.896603\pi\)
\(420\) 0 0
\(421\) − 2.53620i − 0.123607i −0.998088 0.0618035i \(-0.980315\pi\)
0.998088 0.0618035i \(-0.0196852\pi\)
\(422\) 0 0
\(423\) −30.6274 −1.48916
\(424\) 0 0
\(425\) 14.0000 0.679100
\(426\) 0 0
\(427\) 12.1689i 0.588897i
\(428\) 0 0
\(429\) − 2.53620i − 0.122449i
\(430\) 0 0
\(431\) −6.82843 −0.328914 −0.164457 0.986384i \(-0.552587\pi\)
−0.164457 + 0.986384i \(0.552587\pi\)
\(432\) 0 0
\(433\) −12.3431 −0.593174 −0.296587 0.955006i \(-0.595848\pi\)
−0.296587 + 0.955006i \(0.595848\pi\)
\(434\) 0 0
\(435\) 65.2512i 3.12856i
\(436\) 0 0
\(437\) − 32.6256i − 1.56069i
\(438\) 0 0
\(439\) −0.970563 −0.0463224 −0.0231612 0.999732i \(-0.507373\pi\)
−0.0231612 + 0.999732i \(0.507373\pi\)
\(440\) 0 0
\(441\) −3.82843 −0.182306
\(442\) 0 0
\(443\) − 9.18440i − 0.436364i −0.975908 0.218182i \(-0.929987\pi\)
0.975908 0.218182i \(-0.0700127\pi\)
\(444\) 0 0
\(445\) 5.22625i 0.247748i
\(446\) 0 0
\(447\) −43.3137 −2.04867
\(448\) 0 0
\(449\) −24.6274 −1.16224 −0.581120 0.813818i \(-0.697385\pi\)
−0.581120 + 0.813818i \(0.697385\pi\)
\(450\) 0 0
\(451\) − 7.91630i − 0.372764i
\(452\) 0 0
\(453\) 47.4077i 2.22741i
\(454\) 0 0
\(455\) −1.17157 −0.0549242
\(456\) 0 0
\(457\) −10.4853 −0.490481 −0.245240 0.969462i \(-0.578867\pi\)
−0.245240 + 0.969462i \(0.578867\pi\)
\(458\) 0 0
\(459\) 16.5754i 0.773675i
\(460\) 0 0
\(461\) 10.3756i 0.483239i 0.970371 + 0.241619i \(0.0776786\pi\)
−0.970371 + 0.241619i \(0.922321\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) −38.6274 −1.79130
\(466\) 0 0
\(467\) − 1.34502i − 0.0622403i −0.999516 0.0311202i \(-0.990093\pi\)
0.999516 0.0311202i \(-0.00990745\pi\)
\(468\) 0 0
\(469\) − 3.06147i − 0.141365i
\(470\) 0 0
\(471\) −12.4853 −0.575291
\(472\) 0 0
\(473\) 4.68629 0.215476
\(474\) 0 0
\(475\) − 8.73606i − 0.400838i
\(476\) 0 0
\(477\) − 40.0166i − 1.83224i
\(478\) 0 0
\(479\) −5.65685 −0.258468 −0.129234 0.991614i \(-0.541252\pi\)
−0.129234 + 0.991614i \(0.541252\pi\)
\(480\) 0 0
\(481\) 2.34315 0.106838
\(482\) 0 0
\(483\) 17.8435i 0.811909i
\(484\) 0 0
\(485\) 26.1313i 1.18656i
\(486\) 0 0
\(487\) 31.7990 1.44095 0.720475 0.693481i \(-0.243924\pi\)
0.720475 + 0.693481i \(0.243924\pi\)
\(488\) 0 0
\(489\) 10.3431 0.467733
\(490\) 0 0
\(491\) − 32.6256i − 1.47237i −0.676779 0.736187i \(-0.736625\pi\)
0.676779 0.736187i \(-0.263375\pi\)
\(492\) 0 0
\(493\) − 73.1675i − 3.29530i
\(494\) 0 0
\(495\) −21.6569 −0.973403
\(496\) 0 0
\(497\) 2.34315 0.105104
\(498\) 0 0
\(499\) − 11.7206i − 0.524686i −0.964975 0.262343i \(-0.915505\pi\)
0.964975 0.262343i \(-0.0844952\pi\)
\(500\) 0 0
\(501\) − 41.8100i − 1.86793i
\(502\) 0 0
\(503\) −5.65685 −0.252227 −0.126113 0.992016i \(-0.540250\pi\)
−0.126113 + 0.992016i \(0.540250\pi\)
\(504\) 0 0
\(505\) 12.4853 0.555588
\(506\) 0 0
\(507\) 33.4454i 1.48536i
\(508\) 0 0
\(509\) 3.50981i 0.155570i 0.996970 + 0.0777848i \(0.0247847\pi\)
−0.996970 + 0.0777848i \(0.975215\pi\)
\(510\) 0 0
\(511\) −11.6569 −0.515669
\(512\) 0 0
\(513\) 10.3431 0.456661
\(514\) 0 0
\(515\) 35.6871i 1.57256i
\(516\) 0 0
\(517\) 17.3183i 0.761657i
\(518\) 0 0
\(519\) 42.1421 1.84983
\(520\) 0 0
\(521\) 14.6863 0.643418 0.321709 0.946839i \(-0.395743\pi\)
0.321709 + 0.946839i \(0.395743\pi\)
\(522\) 0 0
\(523\) 7.46796i 0.326551i 0.986581 + 0.163276i \(0.0522060\pi\)
−0.986581 + 0.163276i \(0.947794\pi\)
\(524\) 0 0
\(525\) 4.77791i 0.208525i
\(526\) 0 0
\(527\) 43.3137 1.88677
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) 0 0
\(531\) 1.71644i 0.0744873i
\(532\) 0 0
\(533\) − 1.63952i − 0.0710155i
\(534\) 0 0
\(535\) 35.3137 1.52674
\(536\) 0 0
\(537\) −57.9411 −2.50034
\(538\) 0 0
\(539\) 2.16478i 0.0932439i
\(540\) 0 0
\(541\) 31.3575i 1.34816i 0.738656 + 0.674082i \(0.235461\pi\)
−0.738656 + 0.674082i \(0.764539\pi\)
\(542\) 0 0
\(543\) −6.82843 −0.293036
\(544\) 0 0
\(545\) −29.6569 −1.27036
\(546\) 0 0
\(547\) − 43.9748i − 1.88023i −0.340862 0.940113i \(-0.610719\pi\)
0.340862 0.940113i \(-0.389281\pi\)
\(548\) 0 0
\(549\) 46.5879i 1.98832i
\(550\) 0 0
\(551\) −45.6569 −1.94505
\(552\) 0 0
\(553\) −2.34315 −0.0996407
\(554\) 0 0
\(555\) − 35.6871i − 1.51483i
\(556\) 0 0
\(557\) − 14.7821i − 0.626337i −0.949698 0.313168i \(-0.898610\pi\)
0.949698 0.313168i \(-0.101390\pi\)
\(558\) 0 0
\(559\) 0.970563 0.0410504
\(560\) 0 0
\(561\) 43.3137 1.82871
\(562\) 0 0
\(563\) − 17.7666i − 0.748774i −0.927273 0.374387i \(-0.877853\pi\)
0.927273 0.374387i \(-0.122147\pi\)
\(564\) 0 0
\(565\) − 23.0698i − 0.970553i
\(566\) 0 0
\(567\) 5.82843 0.244771
\(568\) 0 0
\(569\) −17.5147 −0.734255 −0.367128 0.930171i \(-0.619659\pi\)
−0.367128 + 0.930171i \(0.619659\pi\)
\(570\) 0 0
\(571\) − 18.7402i − 0.784254i −0.919911 0.392127i \(-0.871739\pi\)
0.919911 0.392127i \(-0.128261\pi\)
\(572\) 0 0
\(573\) − 50.4692i − 2.10838i
\(574\) 0 0
\(575\) 12.4853 0.520672
\(576\) 0 0
\(577\) −22.9706 −0.956277 −0.478139 0.878284i \(-0.658688\pi\)
−0.478139 + 0.878284i \(0.658688\pi\)
\(578\) 0 0
\(579\) 36.0585i 1.49854i
\(580\) 0 0
\(581\) 13.0656i 0.542054i
\(582\) 0 0
\(583\) −22.6274 −0.937132
\(584\) 0 0
\(585\) −4.48528 −0.185444
\(586\) 0 0
\(587\) − 33.4454i − 1.38044i −0.723600 0.690219i \(-0.757514\pi\)
0.723600 0.690219i \(-0.242486\pi\)
\(588\) 0 0
\(589\) − 27.0279i − 1.11367i
\(590\) 0 0
\(591\) 16.0000 0.658152
\(592\) 0 0
\(593\) −25.3137 −1.03951 −0.519755 0.854316i \(-0.673977\pi\)
−0.519755 + 0.854316i \(0.673977\pi\)
\(594\) 0 0
\(595\) − 20.0083i − 0.820261i
\(596\) 0 0
\(597\) − 23.4412i − 0.959385i
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 41.3137 1.68522 0.842611 0.538523i \(-0.181018\pi\)
0.842611 + 0.538523i \(0.181018\pi\)
\(602\) 0 0
\(603\) − 11.7206i − 0.477300i
\(604\) 0 0
\(605\) − 16.4985i − 0.670760i
\(606\) 0 0
\(607\) −27.3137 −1.10863 −0.554315 0.832307i \(-0.687020\pi\)
−0.554315 + 0.832307i \(0.687020\pi\)
\(608\) 0 0
\(609\) 24.9706 1.01186
\(610\) 0 0
\(611\) 3.58673i 0.145104i
\(612\) 0 0
\(613\) 26.1313i 1.05543i 0.849421 + 0.527716i \(0.176951\pi\)
−0.849421 + 0.527716i \(0.823049\pi\)
\(614\) 0 0
\(615\) −24.9706 −1.00691
\(616\) 0 0
\(617\) −41.7990 −1.68276 −0.841382 0.540441i \(-0.818257\pi\)
−0.841382 + 0.540441i \(0.818257\pi\)
\(618\) 0 0
\(619\) 33.9706i 1.36540i 0.730701 + 0.682698i \(0.239193\pi\)
−0.730701 + 0.682698i \(0.760807\pi\)
\(620\) 0 0
\(621\) 14.7821i 0.593184i
\(622\) 0 0
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) −30.7990 −1.23196
\(626\) 0 0
\(627\) − 27.0279i − 1.07939i
\(628\) 0 0
\(629\) 40.0166i 1.59557i
\(630\) 0 0
\(631\) −30.6274 −1.21926 −0.609629 0.792687i \(-0.708682\pi\)
−0.609629 + 0.792687i \(0.708682\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) − 11.7206i − 0.465118i
\(636\) 0 0
\(637\) 0.448342i 0.0177639i
\(638\) 0 0
\(639\) 8.97056 0.354870
\(640\) 0 0
\(641\) 27.4558 1.08444 0.542220 0.840236i \(-0.317584\pi\)
0.542220 + 0.840236i \(0.317584\pi\)
\(642\) 0 0
\(643\) − 2.98454i − 0.117699i −0.998267 0.0588495i \(-0.981257\pi\)
0.998267 0.0588495i \(-0.0187432\pi\)
\(644\) 0 0
\(645\) − 14.7821i − 0.582044i
\(646\) 0 0
\(647\) −29.6569 −1.16593 −0.582966 0.812497i \(-0.698108\pi\)
−0.582966 + 0.812497i \(0.698108\pi\)
\(648\) 0 0
\(649\) 0.970563 0.0380979
\(650\) 0 0
\(651\) 14.7821i 0.579355i
\(652\) 0 0
\(653\) 22.5445i 0.882236i 0.897449 + 0.441118i \(0.145418\pi\)
−0.897449 + 0.441118i \(0.854582\pi\)
\(654\) 0 0
\(655\) 47.7990 1.86766
\(656\) 0 0
\(657\) −44.6274 −1.74108
\(658\) 0 0
\(659\) 10.0811i 0.392703i 0.980534 + 0.196352i \(0.0629094\pi\)
−0.980534 + 0.196352i \(0.937091\pi\)
\(660\) 0 0
\(661\) 0.448342i 0.0174385i 0.999962 + 0.00871923i \(0.00277545\pi\)
−0.999962 + 0.00871923i \(0.997225\pi\)
\(662\) 0 0
\(663\) 8.97056 0.348388
\(664\) 0 0
\(665\) −12.4853 −0.484158
\(666\) 0 0
\(667\) − 65.2512i − 2.52654i
\(668\) 0 0
\(669\) 23.4412i 0.906290i
\(670\) 0 0
\(671\) 26.3431 1.01697
\(672\) 0 0
\(673\) 28.6274 1.10351 0.551753 0.834008i \(-0.313959\pi\)
0.551753 + 0.834008i \(0.313959\pi\)
\(674\) 0 0
\(675\) 3.95815i 0.152349i
\(676\) 0 0
\(677\) 8.73606i 0.335754i 0.985808 + 0.167877i \(0.0536912\pi\)
−0.985808 + 0.167877i \(0.946309\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 37.4558 1.43531
\(682\) 0 0
\(683\) 5.59767i 0.214189i 0.994249 + 0.107094i \(0.0341547\pi\)
−0.994249 + 0.107094i \(0.965845\pi\)
\(684\) 0 0
\(685\) − 5.22625i − 0.199685i
\(686\) 0 0
\(687\) −14.8284 −0.565740
\(688\) 0 0
\(689\) −4.68629 −0.178533
\(690\) 0 0
\(691\) − 35.7640i − 1.36053i −0.732968 0.680263i \(-0.761865\pi\)
0.732968 0.680263i \(-0.238135\pi\)
\(692\) 0 0
\(693\) 8.28772i 0.314824i
\(694\) 0 0
\(695\) 51.1127 1.93882
\(696\) 0 0
\(697\) 28.0000 1.06058
\(698\) 0 0
\(699\) − 26.1313i − 0.988375i
\(700\) 0 0
\(701\) 40.9133i 1.54528i 0.634847 + 0.772638i \(0.281063\pi\)
−0.634847 + 0.772638i \(0.718937\pi\)
\(702\) 0 0
\(703\) 24.9706 0.941783
\(704\) 0 0
\(705\) 54.6274 2.05739
\(706\) 0 0
\(707\) − 4.77791i − 0.179692i
\(708\) 0 0
\(709\) − 17.4721i − 0.656179i −0.944647 0.328090i \(-0.893595\pi\)
0.944647 0.328090i \(-0.106405\pi\)
\(710\) 0 0
\(711\) −8.97056 −0.336422
\(712\) 0 0
\(713\) 38.6274 1.44661
\(714\) 0 0
\(715\) 2.53620i 0.0948486i
\(716\) 0 0
\(717\) 47.4077i 1.77047i
\(718\) 0 0
\(719\) −3.31371 −0.123580 −0.0617902 0.998089i \(-0.519681\pi\)
−0.0617902 + 0.998089i \(0.519681\pi\)
\(720\) 0 0
\(721\) 13.6569 0.508608
\(722\) 0 0
\(723\) 3.43289i 0.127670i
\(724\) 0 0
\(725\) − 17.4721i − 0.648898i
\(726\) 0 0
\(727\) −20.6863 −0.767212 −0.383606 0.923497i \(-0.625318\pi\)
−0.383606 + 0.923497i \(0.625318\pi\)
\(728\) 0 0
\(729\) 39.2843 1.45497
\(730\) 0 0
\(731\) 16.5754i 0.613065i
\(732\) 0 0
\(733\) 21.7248i 0.802423i 0.915986 + 0.401211i \(0.131411\pi\)
−0.915986 + 0.401211i \(0.868589\pi\)
\(734\) 0 0
\(735\) 6.82843 0.251870
\(736\) 0 0
\(737\) −6.62742 −0.244124
\(738\) 0 0
\(739\) − 24.8632i − 0.914606i −0.889311 0.457303i \(-0.848815\pi\)
0.889311 0.457303i \(-0.151185\pi\)
\(740\) 0 0
\(741\) − 5.59767i − 0.205636i
\(742\) 0 0
\(743\) −27.5147 −1.00942 −0.504709 0.863290i \(-0.668400\pi\)
−0.504709 + 0.863290i \(0.668400\pi\)
\(744\) 0 0
\(745\) 43.3137 1.58689
\(746\) 0 0
\(747\) 50.0208i 1.83017i
\(748\) 0 0
\(749\) − 13.5140i − 0.493790i
\(750\) 0 0
\(751\) −45.4558 −1.65871 −0.829354 0.558724i \(-0.811291\pi\)
−0.829354 + 0.558724i \(0.811291\pi\)
\(752\) 0 0
\(753\) 2.14214 0.0780638
\(754\) 0 0
\(755\) − 47.4077i − 1.72534i
\(756\) 0 0
\(757\) 22.5445i 0.819395i 0.912221 + 0.409697i \(0.134366\pi\)
−0.912221 + 0.409697i \(0.865634\pi\)
\(758\) 0 0
\(759\) 38.6274 1.40209
\(760\) 0 0
\(761\) 18.9706 0.687682 0.343841 0.939028i \(-0.388272\pi\)
0.343841 + 0.939028i \(0.388272\pi\)
\(762\) 0 0
\(763\) 11.3492i 0.410868i
\(764\) 0 0
\(765\) − 76.6004i − 2.76949i
\(766\) 0 0
\(767\) 0.201010 0.00725806
\(768\) 0 0
\(769\) 35.2548 1.27132 0.635661 0.771968i \(-0.280728\pi\)
0.635661 + 0.771968i \(0.280728\pi\)
\(770\) 0 0
\(771\) − 24.3379i − 0.876508i
\(772\) 0 0
\(773\) − 8.73606i − 0.314214i −0.987582 0.157107i \(-0.949783\pi\)
0.987582 0.157107i \(-0.0502168\pi\)
\(774\) 0 0
\(775\) 10.3431 0.371537
\(776\) 0 0
\(777\) −13.6569 −0.489937
\(778\) 0 0
\(779\) − 17.4721i − 0.626004i
\(780\) 0 0
\(781\) − 5.07241i − 0.181505i
\(782\) 0 0
\(783\) 20.6863 0.739268
\(784\) 0 0
\(785\) 12.4853 0.445619
\(786\) 0 0
\(787\) − 7.46796i − 0.266204i −0.991102 0.133102i \(-0.957506\pi\)
0.991102 0.133102i \(-0.0424938\pi\)
\(788\) 0 0
\(789\) − 6.12293i − 0.217982i
\(790\) 0 0
\(791\) −8.82843 −0.313903
\(792\) 0 0
\(793\) 5.45584 0.193743
\(794\) 0 0
\(795\) 71.3742i 2.53138i
\(796\) 0 0
\(797\) 27.4763i 0.973260i 0.873608 + 0.486630i \(0.161774\pi\)
−0.873608 + 0.486630i \(0.838226\pi\)
\(798\) 0 0
\(799\) −61.2548 −2.16704
\(800\) 0 0
\(801\) 7.65685 0.270542
\(802\) 0 0
\(803\) 25.2346i 0.890509i
\(804\) 0 0
\(805\) − 17.8435i − 0.628902i
\(806\) 0 0
\(807\) 21.4558 0.755281
\(808\) 0 0
\(809\) −31.1716 −1.09593 −0.547967 0.836500i \(-0.684598\pi\)
−0.547967 + 0.836500i \(0.684598\pi\)
\(810\) 0 0
\(811\) 20.9819i 0.736775i 0.929672 + 0.368388i \(0.120090\pi\)
−0.929672 + 0.368388i \(0.879910\pi\)
\(812\) 0 0
\(813\) 6.12293i 0.214741i
\(814\) 0 0
\(815\) −10.3431 −0.362305
\(816\) 0 0
\(817\) 10.3431 0.361861
\(818\) 0 0
\(819\) 1.71644i 0.0599774i
\(820\) 0 0
\(821\) 8.65914i 0.302206i 0.988518 + 0.151103i \(0.0482825\pi\)
−0.988518 + 0.151103i \(0.951717\pi\)
\(822\) 0 0
\(823\) −40.9706 −1.42814 −0.714072 0.700072i \(-0.753151\pi\)
−0.714072 + 0.700072i \(0.753151\pi\)
\(824\) 0 0
\(825\) 10.3431 0.360102
\(826\) 0 0
\(827\) 10.9778i 0.381734i 0.981616 + 0.190867i \(0.0611300\pi\)
−0.981616 + 0.190867i \(0.938870\pi\)
\(828\) 0 0
\(829\) 6.19986i 0.215330i 0.994187 + 0.107665i \(0.0343374\pi\)
−0.994187 + 0.107665i \(0.965663\pi\)
\(830\) 0 0
\(831\) 38.6274 1.33997
\(832\) 0 0
\(833\) −7.65685 −0.265294
\(834\) 0 0
\(835\) 41.8100i 1.44690i
\(836\) 0 0
\(837\) 12.2459i 0.423279i
\(838\) 0 0
\(839\) −36.2843 −1.25267 −0.626336 0.779553i \(-0.715446\pi\)
−0.626336 + 0.779553i \(0.715446\pi\)
\(840\) 0 0
\(841\) −62.3137 −2.14875
\(842\) 0 0
\(843\) − 5.22625i − 0.180002i
\(844\) 0 0
\(845\) − 33.4454i − 1.15056i
\(846\) 0 0
\(847\) −6.31371 −0.216942
\(848\) 0 0
\(849\) 54.4264 1.86791
\(850\) 0 0
\(851\) 35.6871i 1.22334i
\(852\) 0 0
\(853\) − 47.3308i − 1.62057i −0.586033 0.810287i \(-0.699311\pi\)
0.586033 0.810287i \(-0.300689\pi\)
\(854\) 0 0
\(855\) −47.7990 −1.63469
\(856\) 0 0
\(857\) 0.627417 0.0214322 0.0107161 0.999943i \(-0.496589\pi\)
0.0107161 + 0.999943i \(0.496589\pi\)
\(858\) 0 0
\(859\) 3.50981i 0.119753i 0.998206 + 0.0598766i \(0.0190707\pi\)
−0.998206 + 0.0598766i \(0.980929\pi\)
\(860\) 0 0
\(861\) 9.55582i 0.325661i
\(862\) 0 0
\(863\) 51.3137 1.74674 0.873369 0.487058i \(-0.161930\pi\)
0.873369 + 0.487058i \(0.161930\pi\)
\(864\) 0 0
\(865\) −42.1421 −1.43288
\(866\) 0 0
\(867\) 108.778i 3.69428i
\(868\) 0 0
\(869\) 5.07241i 0.172070i
\(870\) 0 0
\(871\) −1.37258 −0.0465082
\(872\) 0 0
\(873\) 38.2843 1.29573
\(874\) 0 0
\(875\) 8.28772i 0.280176i
\(876\) 0 0
\(877\) 32.2542i 1.08915i 0.838713 + 0.544573i \(0.183308\pi\)
−0.838713 + 0.544573i \(0.816692\pi\)
\(878\) 0 0
\(879\) −28.4853 −0.960785
\(880\) 0 0
\(881\) 10.9706 0.369608 0.184804 0.982775i \(-0.440835\pi\)
0.184804 + 0.982775i \(0.440835\pi\)
\(882\) 0 0
\(883\) 51.7373i 1.74110i 0.492082 + 0.870549i \(0.336236\pi\)
−0.492082 + 0.870549i \(0.663764\pi\)
\(884\) 0 0
\(885\) − 3.06147i − 0.102910i
\(886\) 0 0
\(887\) 22.6274 0.759754 0.379877 0.925037i \(-0.375966\pi\)
0.379877 + 0.925037i \(0.375966\pi\)
\(888\) 0 0
\(889\) −4.48528 −0.150432
\(890\) 0 0
\(891\) − 12.6173i − 0.422695i
\(892\) 0 0
\(893\) 38.2233i 1.27909i
\(894\) 0 0
\(895\) 57.9411 1.93676
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) − 54.0559i − 1.80286i
\(900\) 0 0
\(901\) − 80.0333i − 2.66630i
\(902\) 0 0
\(903\) −5.65685 −0.188248
\(904\) 0 0
\(905\) 6.82843 0.226985
\(906\) 0 0
\(907\) − 1.26810i − 0.0421066i −0.999778 0.0210533i \(-0.993298\pi\)
0.999778 0.0210533i \(-0.00670197\pi\)
\(908\) 0 0
\(909\) − 18.2919i − 0.606703i
\(910\) 0 0
\(911\) 38.8284 1.28644 0.643222 0.765680i \(-0.277597\pi\)
0.643222 + 0.765680i \(0.277597\pi\)
\(912\) 0 0
\(913\) 28.2843 0.936073
\(914\) 0 0
\(915\) − 83.0948i − 2.74703i
\(916\) 0 0
\(917\) − 18.2919i − 0.604051i
\(918\) 0 0
\(919\) −2.34315 −0.0772932 −0.0386466 0.999253i \(-0.512305\pi\)
−0.0386466 + 0.999253i \(0.512305\pi\)
\(920\) 0 0
\(921\) 22.8284 0.752222
\(922\) 0 0
\(923\) − 1.05053i − 0.0345786i
\(924\) 0 0
\(925\) 9.55582i 0.314193i
\(926\) 0 0
\(927\) 52.2843 1.71724
\(928\) 0 0
\(929\) 10.2843 0.337416 0.168708 0.985666i \(-0.446041\pi\)
0.168708 + 0.985666i \(0.446041\pi\)
\(930\) 0 0
\(931\) 4.77791i 0.156590i
\(932\) 0 0
\(933\) − 38.2233i − 1.25137i
\(934\) 0 0
\(935\) −43.3137 −1.41651
\(936\) 0 0
\(937\) −38.2843 −1.25069 −0.625346 0.780347i \(-0.715042\pi\)
−0.625346 + 0.780347i \(0.715042\pi\)
\(938\) 0 0
\(939\) − 21.8017i − 0.711471i
\(940\) 0 0
\(941\) 37.7749i 1.23143i 0.787970 + 0.615714i \(0.211132\pi\)
−0.787970 + 0.615714i \(0.788868\pi\)
\(942\) 0 0
\(943\) 24.9706 0.813153
\(944\) 0 0
\(945\) 5.65685 0.184017
\(946\) 0 0
\(947\) 48.3044i 1.56968i 0.619698 + 0.784841i \(0.287255\pi\)
−0.619698 + 0.784841i \(0.712745\pi\)
\(948\) 0 0
\(949\) 5.22625i 0.169651i
\(950\) 0 0
\(951\) −22.6274 −0.733744
\(952\) 0 0
\(953\) 37.3137 1.20871 0.604355 0.796715i \(-0.293431\pi\)
0.604355 + 0.796715i \(0.293431\pi\)
\(954\) 0 0
\(955\) 50.4692i 1.63314i
\(956\) 0 0
\(957\) − 54.0559i − 1.74738i
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) − 51.7373i − 1.66721i
\(964\) 0 0
\(965\) − 36.0585i − 1.16076i
\(966\) 0 0
\(967\) 29.8579 0.960164 0.480082 0.877224i \(-0.340607\pi\)
0.480082 + 0.877224i \(0.340607\pi\)
\(968\) 0 0
\(969\) 95.5980 3.07105
\(970\) 0 0
\(971\) 52.5570i 1.68663i 0.537416 + 0.843317i \(0.319401\pi\)
−0.537416 + 0.843317i \(0.680599\pi\)
\(972\) 0 0
\(973\) − 19.5600i − 0.627064i
\(974\) 0 0
\(975\) 2.14214 0.0686032
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) − 4.32957i − 0.138374i
\(980\) 0 0
\(981\) 43.4495i 1.38724i
\(982\) 0 0
\(983\) −23.0294 −0.734525 −0.367262 0.930117i \(-0.619705\pi\)
−0.367262 + 0.930117i \(0.619705\pi\)
\(984\) 0 0
\(985\) −16.0000 −0.509802
\(986\) 0 0
\(987\) − 20.9050i − 0.665414i
\(988\) 0 0
\(989\) 14.7821i 0.470043i
\(990\) 0 0
\(991\) 29.6569 0.942081 0.471041 0.882112i \(-0.343879\pi\)
0.471041 + 0.882112i \(0.343879\pi\)
\(992\) 0 0
\(993\) −48.9706 −1.55403
\(994\) 0 0
\(995\) 23.4412i 0.743136i
\(996\) 0 0
\(997\) − 0.0769232i − 0.00243618i −0.999999 0.00121809i \(-0.999612\pi\)
0.999999 0.00121809i \(-0.000387731\pi\)
\(998\) 0 0
\(999\) −11.3137 −0.357950
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 896.2.b.f.449.4 yes 4
4.3 odd 2 896.2.b.h.449.1 yes 4
8.3 odd 2 896.2.b.h.449.4 yes 4
8.5 even 2 inner 896.2.b.f.449.1 4
16.3 odd 4 1792.2.a.u.1.4 4
16.5 even 4 1792.2.a.w.1.4 4
16.11 odd 4 1792.2.a.u.1.1 4
16.13 even 4 1792.2.a.w.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.b.f.449.1 4 8.5 even 2 inner
896.2.b.f.449.4 yes 4 1.1 even 1 trivial
896.2.b.h.449.1 yes 4 4.3 odd 2
896.2.b.h.449.4 yes 4 8.3 odd 2
1792.2.a.u.1.1 4 16.11 odd 4
1792.2.a.u.1.4 4 16.3 odd 4
1792.2.a.w.1.1 4 16.13 even 4
1792.2.a.w.1.4 4 16.5 even 4