Properties

Label 896.2.b.f.449.2
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 449.2
Root \(-1.84776i\) of defining polynomial
Character \(\chi\) \(=\) 896.449
Dual form 896.2.b.f.449.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.08239i q^{3} +1.08239i q^{5} -1.00000 q^{7} +1.82843 q^{9} +O(q^{10})\) \(q-1.08239i q^{3} +1.08239i q^{5} -1.00000 q^{7} +1.82843 q^{9} +5.22625i q^{11} -6.30864i q^{13} +1.17157 q^{15} +3.65685 q^{17} +4.14386i q^{19} +1.08239i q^{21} -1.17157 q^{23} +3.82843 q^{25} -5.22625i q^{27} +8.28772i q^{29} +5.65685 q^{31} +5.65685 q^{33} -1.08239i q^{35} +2.16478i q^{37} -6.82843 q^{39} +7.65685 q^{41} -5.22625i q^{43} +1.97908i q^{45} +8.00000 q^{47} +1.00000 q^{49} -3.95815i q^{51} -4.32957i q^{53} -5.65685 q^{55} +4.48528 q^{57} +6.30864i q^{59} -7.20533i q^{61} -1.82843 q^{63} +6.82843 q^{65} -7.39104i q^{67} +1.26810i q^{69} -13.6569 q^{71} +0.343146 q^{73} -4.14386i q^{75} -5.22625i q^{77} +13.6569 q^{79} -0.171573 q^{81} +5.41196i q^{83} +3.95815i q^{85} +8.97056 q^{87} -2.00000 q^{89} +6.30864i q^{91} -6.12293i q^{93} -4.48528 q^{95} -10.0000 q^{97} +9.55582i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{7} - 4 q^{9} + 16 q^{15} - 8 q^{17} - 16 q^{23} + 4 q^{25} - 16 q^{39} + 8 q^{41} + 32 q^{47} + 4 q^{49} - 16 q^{57} + 4 q^{63} + 16 q^{65} - 32 q^{71} + 24 q^{73} + 32 q^{79} - 12 q^{81} - 32 q^{87} - 8 q^{89} + 16 q^{95} - 40 q^{97} + O(q^{100}) \)

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
\(3\) − 1.08239i − 0.624919i −0.949931 0.312460i \(-0.898847\pi\)
0.949931 0.312460i \(-0.101153\pi\)
\(4\) 0 0
\(5\) 1.08239i 0.484061i 0.970269 + 0.242030i \(0.0778133\pi\)
−0.970269 + 0.242030i \(0.922187\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.82843 0.609476
\(10\) 0 0
\(11\) 5.22625i 1.57577i 0.615820 + 0.787887i \(0.288825\pi\)
−0.615820 + 0.787887i \(0.711175\pi\)
\(12\) 0 0
\(13\) − 6.30864i − 1.74970i −0.484391 0.874852i \(-0.660959\pi\)
0.484391 0.874852i \(-0.339041\pi\)
\(14\) 0 0
\(15\) 1.17157 0.302499
\(16\) 0 0
\(17\) 3.65685 0.886917 0.443459 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 0 0
\(19\) 4.14386i 0.950667i 0.879806 + 0.475333i \(0.157673\pi\)
−0.879806 + 0.475333i \(0.842327\pi\)
\(20\) 0 0
\(21\) 1.08239i 0.236197i
\(22\) 0 0
\(23\) −1.17157 −0.244290 −0.122145 0.992512i \(-0.538977\pi\)
−0.122145 + 0.992512i \(0.538977\pi\)
\(24\) 0 0
\(25\) 3.82843 0.765685
\(26\) 0 0
\(27\) − 5.22625i − 1.00579i
\(28\) 0 0
\(29\) 8.28772i 1.53899i 0.638652 + 0.769495i \(0.279492\pi\)
−0.638652 + 0.769495i \(0.720508\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) 5.65685 0.984732
\(34\) 0 0
\(35\) − 1.08239i − 0.182958i
\(36\) 0 0
\(37\) 2.16478i 0.355888i 0.984041 + 0.177944i \(0.0569447\pi\)
−0.984041 + 0.177944i \(0.943055\pi\)
\(38\) 0 0
\(39\) −6.82843 −1.09342
\(40\) 0 0
\(41\) 7.65685 1.19580 0.597900 0.801571i \(-0.296002\pi\)
0.597900 + 0.801571i \(0.296002\pi\)
\(42\) 0 0
\(43\) − 5.22625i − 0.796996i −0.917169 0.398498i \(-0.869532\pi\)
0.917169 0.398498i \(-0.130468\pi\)
\(44\) 0 0
\(45\) 1.97908i 0.295023i
\(46\) 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
\(51\) − 3.95815i − 0.554252i
\(52\) 0 0
\(53\) − 4.32957i − 0.594712i −0.954767 0.297356i \(-0.903895\pi\)
0.954767 0.297356i \(-0.0961048\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 4.48528 0.594090
\(58\) 0 0
\(59\) 6.30864i 0.821315i 0.911790 + 0.410658i \(0.134701\pi\)
−0.911790 + 0.410658i \(0.865299\pi\)
\(60\) 0 0
\(61\) − 7.20533i − 0.922548i −0.887258 0.461274i \(-0.847393\pi\)
0.887258 0.461274i \(-0.152607\pi\)
\(62\) 0 0
\(63\) −1.82843 −0.230360
\(64\) 0 0
\(65\) 6.82843 0.846962
\(66\) 0 0
\(67\) − 7.39104i − 0.902959i −0.892282 0.451479i \(-0.850896\pi\)
0.892282 0.451479i \(-0.149104\pi\)
\(68\) 0 0
\(69\) 1.26810i 0.152661i
\(70\) 0 0
\(71\) −13.6569 −1.62077 −0.810385 0.585897i \(-0.800742\pi\)
−0.810385 + 0.585897i \(0.800742\pi\)
\(72\) 0 0
\(73\) 0.343146 0.0401622 0.0200811 0.999798i \(-0.493608\pi\)
0.0200811 + 0.999798i \(0.493608\pi\)
\(74\) 0 0
\(75\) − 4.14386i − 0.478492i
\(76\) 0 0
\(77\) − 5.22625i − 0.595587i
\(78\) 0 0
\(79\) 13.6569 1.53652 0.768258 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) 0 0
\(81\) −0.171573 −0.0190637
\(82\) 0 0
\(83\) 5.41196i 0.594040i 0.954871 + 0.297020i \(0.0959928\pi\)
−0.954871 + 0.297020i \(0.904007\pi\)
\(84\) 0 0
\(85\) 3.95815i 0.429322i
\(86\) 0 0
\(87\) 8.97056 0.961745
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 6.30864i 0.661326i
\(92\) 0 0
\(93\) − 6.12293i − 0.634919i
\(94\) 0 0
\(95\) −4.48528 −0.460180
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 9.55582i 0.960396i
\(100\) 0 0
\(101\) 4.14386i 0.412329i 0.978517 + 0.206165i \(0.0660983\pi\)
−0.978517 + 0.206165i \(0.933902\pi\)
\(102\) 0 0
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) 0 0
\(105\) −1.17157 −0.114334
\(106\) 0 0
\(107\) − 11.7206i − 1.13307i −0.824036 0.566537i \(-0.808283\pi\)
0.824036 0.566537i \(-0.191717\pi\)
\(108\) 0 0
\(109\) 16.9469i 1.62321i 0.584203 + 0.811607i \(0.301407\pi\)
−0.584203 + 0.811607i \(0.698593\pi\)
\(110\) 0 0
\(111\) 2.34315 0.222402
\(112\) 0 0
\(113\) 3.17157 0.298356 0.149178 0.988810i \(-0.452337\pi\)
0.149178 + 0.988810i \(0.452337\pi\)
\(114\) 0 0
\(115\) − 1.26810i − 0.118251i
\(116\) 0 0
\(117\) − 11.5349i − 1.06640i
\(118\) 0 0
\(119\) −3.65685 −0.335223
\(120\) 0 0
\(121\) −16.3137 −1.48306
\(122\) 0 0
\(123\) − 8.28772i − 0.747278i
\(124\) 0 0
\(125\) 9.55582i 0.854699i
\(126\) 0 0
\(127\) −12.4853 −1.10789 −0.553945 0.832553i \(-0.686878\pi\)
−0.553945 + 0.832553i \(0.686878\pi\)
\(128\) 0 0
\(129\) −5.65685 −0.498058
\(130\) 0 0
\(131\) − 7.57675i − 0.661983i −0.943634 0.330992i \(-0.892617\pi\)
0.943634 0.330992i \(-0.107383\pi\)
\(132\) 0 0
\(133\) − 4.14386i − 0.359318i
\(134\) 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
\(139\) 10.2668i 0.870818i 0.900233 + 0.435409i \(0.143396\pi\)
−0.900233 + 0.435409i \(0.856604\pi\)
\(140\) 0 0
\(141\) − 8.65914i − 0.729231i
\(142\) 0 0
\(143\) 32.9706 2.75714
\(144\) 0 0
\(145\) −8.97056 −0.744965
\(146\) 0 0
\(147\) − 1.08239i − 0.0892742i
\(148\) 0 0
\(149\) − 19.1116i − 1.56569i −0.622219 0.782843i \(-0.713769\pi\)
0.622219 0.782843i \(-0.286231\pi\)
\(150\) 0 0
\(151\) −10.1421 −0.825355 −0.412678 0.910877i \(-0.635406\pi\)
−0.412678 + 0.910877i \(0.635406\pi\)
\(152\) 0 0
\(153\) 6.68629 0.540555
\(154\) 0 0
\(155\) 6.12293i 0.491806i
\(156\) 0 0
\(157\) 4.14386i 0.330716i 0.986234 + 0.165358i \(0.0528780\pi\)
−0.986234 + 0.165358i \(0.947122\pi\)
\(158\) 0 0
\(159\) −4.68629 −0.371647
\(160\) 0 0
\(161\) 1.17157 0.0923329
\(162\) 0 0
\(163\) 20.0083i 1.56717i 0.621283 + 0.783586i \(0.286612\pi\)
−0.621283 + 0.783586i \(0.713388\pi\)
\(164\) 0 0
\(165\) 6.12293i 0.476670i
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −26.7990 −2.06146
\(170\) 0 0
\(171\) 7.57675i 0.579408i
\(172\) 0 0
\(173\) 12.8030i 0.973394i 0.873571 + 0.486697i \(0.161798\pi\)
−0.873571 + 0.486697i \(0.838202\pi\)
\(174\) 0 0
\(175\) −3.82843 −0.289402
\(176\) 0 0
\(177\) 6.82843 0.513256
\(178\) 0 0
\(179\) 9.18440i 0.686474i 0.939249 + 0.343237i \(0.111523\pi\)
−0.939249 + 0.343237i \(0.888477\pi\)
\(180\) 0 0
\(181\) − 1.08239i − 0.0804536i −0.999191 0.0402268i \(-0.987192\pi\)
0.999191 0.0402268i \(-0.0128080\pi\)
\(182\) 0 0
\(183\) −7.79899 −0.576518
\(184\) 0 0
\(185\) −2.34315 −0.172272
\(186\) 0 0
\(187\) 19.1116i 1.39758i
\(188\) 0 0
\(189\) 5.22625i 0.380154i
\(190\) 0 0
\(191\) 3.31371 0.239772 0.119886 0.992788i \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) 0 0
\(193\) −25.7990 −1.85705 −0.928526 0.371267i \(-0.878923\pi\)
−0.928526 + 0.371267i \(0.878923\pi\)
\(194\) 0 0
\(195\) − 7.39104i − 0.529283i
\(196\) 0 0
\(197\) 14.7821i 1.05318i 0.850120 + 0.526590i \(0.176530\pi\)
−0.850120 + 0.526590i \(0.823470\pi\)
\(198\) 0 0
\(199\) 24.9706 1.77012 0.885058 0.465481i \(-0.154118\pi\)
0.885058 + 0.465481i \(0.154118\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) − 8.28772i − 0.581684i
\(204\) 0 0
\(205\) 8.28772i 0.578839i
\(206\) 0 0
\(207\) −2.14214 −0.148889
\(208\) 0 0
\(209\) −21.6569 −1.49804
\(210\) 0 0
\(211\) − 22.1731i − 1.52646i −0.646127 0.763230i \(-0.723612\pi\)
0.646127 0.763230i \(-0.276388\pi\)
\(212\) 0 0
\(213\) 14.7821i 1.01285i
\(214\) 0 0
\(215\) 5.65685 0.385794
\(216\) 0 0
\(217\) −5.65685 −0.384012
\(218\) 0 0
\(219\) − 0.371418i − 0.0250981i
\(220\) 0 0
\(221\) − 23.0698i − 1.55184i
\(222\) 0 0
\(223\) −24.9706 −1.67215 −0.836076 0.548613i \(-0.815156\pi\)
−0.836076 + 0.548613i \(0.815156\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) − 12.4316i − 0.825113i −0.910932 0.412556i \(-0.864636\pi\)
0.910932 0.412556i \(-0.135364\pi\)
\(228\) 0 0
\(229\) − 8.47343i − 0.559940i −0.960009 0.279970i \(-0.909675\pi\)
0.960009 0.279970i \(-0.0903245\pi\)
\(230\) 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
\(235\) 8.65914i 0.564860i
\(236\) 0 0
\(237\) − 14.7821i − 0.960199i
\(238\) 0 0
\(239\) −10.1421 −0.656040 −0.328020 0.944671i \(-0.606381\pi\)
−0.328020 + 0.944671i \(0.606381\pi\)
\(240\) 0 0
\(241\) −21.3137 −1.37294 −0.686468 0.727160i \(-0.740840\pi\)
−0.686468 + 0.727160i \(0.740840\pi\)
\(242\) 0 0
\(243\) − 15.4930i − 0.993879i
\(244\) 0 0
\(245\) 1.08239i 0.0691515i
\(246\) 0 0
\(247\) 26.1421 1.66338
\(248\) 0 0
\(249\) 5.85786 0.371227
\(250\) 0 0
\(251\) − 24.1522i − 1.52447i −0.647299 0.762236i \(-0.724101\pi\)
0.647299 0.762236i \(-0.275899\pi\)
\(252\) 0 0
\(253\) − 6.12293i − 0.384946i
\(254\) 0 0
\(255\) 4.28427 0.268291
\(256\) 0 0
\(257\) 13.3137 0.830486 0.415243 0.909710i \(-0.363696\pi\)
0.415243 + 0.909710i \(0.363696\pi\)
\(258\) 0 0
\(259\) − 2.16478i − 0.134513i
\(260\) 0 0
\(261\) 15.1535i 0.937978i
\(262\) 0 0
\(263\) −13.6569 −0.842118 −0.421059 0.907033i \(-0.638341\pi\)
−0.421059 + 0.907033i \(0.638341\pi\)
\(264\) 0 0
\(265\) 4.68629 0.287877
\(266\) 0 0
\(267\) 2.16478i 0.132483i
\(268\) 0 0
\(269\) − 27.2137i − 1.65925i −0.558324 0.829623i \(-0.688555\pi\)
0.558324 0.829623i \(-0.311445\pi\)
\(270\) 0 0
\(271\) 13.6569 0.829595 0.414797 0.909914i \(-0.363852\pi\)
0.414797 + 0.909914i \(0.363852\pi\)
\(272\) 0 0
\(273\) 6.82843 0.413275
\(274\) 0 0
\(275\) 20.0083i 1.20655i
\(276\) 0 0
\(277\) − 6.12293i − 0.367892i −0.982936 0.183946i \(-0.941113\pi\)
0.982936 0.183946i \(-0.0588871\pi\)
\(278\) 0 0
\(279\) 10.3431 0.619228
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) − 28.1103i − 1.67099i −0.549501 0.835493i \(-0.685182\pi\)
0.549501 0.835493i \(-0.314818\pi\)
\(284\) 0 0
\(285\) 4.85483i 0.287576i
\(286\) 0 0
\(287\) −7.65685 −0.451970
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) 10.8239i 0.634510i
\(292\) 0 0
\(293\) − 10.6382i − 0.621491i −0.950493 0.310746i \(-0.899421\pi\)
0.950493 0.310746i \(-0.100579\pi\)
\(294\) 0 0
\(295\) −6.82843 −0.397566
\(296\) 0 0
\(297\) 27.3137 1.58490
\(298\) 0 0
\(299\) 7.39104i 0.427435i
\(300\) 0 0
\(301\) 5.22625i 0.301236i
\(302\) 0 0
\(303\) 4.48528 0.257673
\(304\) 0 0
\(305\) 7.79899 0.446569
\(306\) 0 0
\(307\) 15.8645i 0.905433i 0.891655 + 0.452716i \(0.149545\pi\)
−0.891655 + 0.452716i \(0.850455\pi\)
\(308\) 0 0
\(309\) 2.53620i 0.144280i
\(310\) 0 0
\(311\) 30.6274 1.73672 0.868361 0.495933i \(-0.165174\pi\)
0.868361 + 0.495933i \(0.165174\pi\)
\(312\) 0 0
\(313\) −19.6569 −1.11107 −0.555536 0.831493i \(-0.687487\pi\)
−0.555536 + 0.831493i \(0.687487\pi\)
\(314\) 0 0
\(315\) − 1.97908i − 0.111508i
\(316\) 0 0
\(317\) 20.9050i 1.17414i 0.809535 + 0.587071i \(0.199719\pi\)
−0.809535 + 0.587071i \(0.800281\pi\)
\(318\) 0 0
\(319\) −43.3137 −2.42510
\(320\) 0 0
\(321\) −12.6863 −0.708080
\(322\) 0 0
\(323\) 15.1535i 0.843163i
\(324\) 0 0
\(325\) − 24.1522i − 1.33972i
\(326\) 0 0
\(327\) 18.3431 1.01438
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) − 13.8854i − 0.763210i −0.924325 0.381605i \(-0.875371\pi\)
0.924325 0.381605i \(-0.124629\pi\)
\(332\) 0 0
\(333\) 3.95815i 0.216905i
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 16.8284 0.916703 0.458351 0.888771i \(-0.348440\pi\)
0.458351 + 0.888771i \(0.348440\pi\)
\(338\) 0 0
\(339\) − 3.43289i − 0.186449i
\(340\) 0 0
\(341\) 29.5641i 1.60099i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.37258 −0.0738974
\(346\) 0 0
\(347\) 0.896683i 0.0481365i 0.999710 + 0.0240682i \(0.00766190\pi\)
−0.999710 + 0.0240682i \(0.992338\pi\)
\(348\) 0 0
\(349\) 2.87576i 0.153936i 0.997034 + 0.0769679i \(0.0245239\pi\)
−0.997034 + 0.0769679i \(0.975476\pi\)
\(350\) 0 0
\(351\) −32.9706 −1.75984
\(352\) 0 0
\(353\) 4.34315 0.231162 0.115581 0.993298i \(-0.463127\pi\)
0.115581 + 0.993298i \(0.463127\pi\)
\(354\) 0 0
\(355\) − 14.7821i − 0.784551i
\(356\) 0 0
\(357\) 3.95815i 0.209488i
\(358\) 0 0
\(359\) −1.17157 −0.0618333 −0.0309166 0.999522i \(-0.509843\pi\)
−0.0309166 + 0.999522i \(0.509843\pi\)
\(360\) 0 0
\(361\) 1.82843 0.0962330
\(362\) 0 0
\(363\) 17.6578i 0.926796i
\(364\) 0 0
\(365\) 0.371418i 0.0194409i
\(366\) 0 0
\(367\) 8.97056 0.468260 0.234130 0.972205i \(-0.424776\pi\)
0.234130 + 0.972205i \(0.424776\pi\)
\(368\) 0 0
\(369\) 14.0000 0.728811
\(370\) 0 0
\(371\) 4.32957i 0.224780i
\(372\) 0 0
\(373\) 22.6984i 1.17528i 0.809124 + 0.587639i \(0.199942\pi\)
−0.809124 + 0.587639i \(0.800058\pi\)
\(374\) 0 0
\(375\) 10.3431 0.534118
\(376\) 0 0
\(377\) 52.2843 2.69278
\(378\) 0 0
\(379\) − 3.43289i − 0.176335i −0.996106 0.0881677i \(-0.971899\pi\)
0.996106 0.0881677i \(-0.0281012\pi\)
\(380\) 0 0
\(381\) 13.5140i 0.692342i
\(382\) 0 0
\(383\) −3.31371 −0.169323 −0.0846613 0.996410i \(-0.526981\pi\)
−0.0846613 + 0.996410i \(0.526981\pi\)
\(384\) 0 0
\(385\) 5.65685 0.288300
\(386\) 0 0
\(387\) − 9.55582i − 0.485750i
\(388\) 0 0
\(389\) − 3.95815i − 0.200686i −0.994953 0.100343i \(-0.968006\pi\)
0.994953 0.100343i \(-0.0319940\pi\)
\(390\) 0 0
\(391\) −4.28427 −0.216665
\(392\) 0 0
\(393\) −8.20101 −0.413686
\(394\) 0 0
\(395\) 14.7821i 0.743767i
\(396\) 0 0
\(397\) − 7.57675i − 0.380266i −0.981758 0.190133i \(-0.939108\pi\)
0.981758 0.190133i \(-0.0608919\pi\)
\(398\) 0 0
\(399\) −4.48528 −0.224545
\(400\) 0 0
\(401\) 26.4853 1.32261 0.661306 0.750116i \(-0.270003\pi\)
0.661306 + 0.750116i \(0.270003\pi\)
\(402\) 0 0
\(403\) − 35.6871i − 1.77770i
\(404\) 0 0
\(405\) − 0.185709i − 0.00922796i
\(406\) 0 0
\(407\) −11.3137 −0.560800
\(408\) 0 0
\(409\) −17.3137 −0.856108 −0.428054 0.903753i \(-0.640801\pi\)
−0.428054 + 0.903753i \(0.640801\pi\)
\(410\) 0 0
\(411\) − 2.16478i − 0.106781i
\(412\) 0 0
\(413\) − 6.30864i − 0.310428i
\(414\) 0 0
\(415\) −5.85786 −0.287551
\(416\) 0 0
\(417\) 11.1127 0.544191
\(418\) 0 0
\(419\) − 5.41196i − 0.264392i −0.991224 0.132196i \(-0.957797\pi\)
0.991224 0.132196i \(-0.0422028\pi\)
\(420\) 0 0
\(421\) − 35.6871i − 1.73928i −0.493685 0.869641i \(-0.664350\pi\)
0.493685 0.869641i \(-0.335650\pi\)
\(422\) 0 0
\(423\) 14.6274 0.711209
\(424\) 0 0
\(425\) 14.0000 0.679100
\(426\) 0 0
\(427\) 7.20533i 0.348690i
\(428\) 0 0
\(429\) − 35.6871i − 1.72299i
\(430\) 0 0
\(431\) −1.17157 −0.0564327 −0.0282163 0.999602i \(-0.508983\pi\)
−0.0282163 + 0.999602i \(0.508983\pi\)
\(432\) 0 0
\(433\) −23.6569 −1.13688 −0.568438 0.822726i \(-0.692452\pi\)
−0.568438 + 0.822726i \(0.692452\pi\)
\(434\) 0 0
\(435\) 9.70967i 0.465543i
\(436\) 0 0
\(437\) − 4.85483i − 0.232238i
\(438\) 0 0
\(439\) 32.9706 1.57360 0.786800 0.617209i \(-0.211737\pi\)
0.786800 + 0.617209i \(0.211737\pi\)
\(440\) 0 0
\(441\) 1.82843 0.0870680
\(442\) 0 0
\(443\) 22.1731i 1.05348i 0.850028 + 0.526738i \(0.176585\pi\)
−0.850028 + 0.526738i \(0.823415\pi\)
\(444\) 0 0
\(445\) − 2.16478i − 0.102621i
\(446\) 0 0
\(447\) −20.6863 −0.978428
\(448\) 0 0
\(449\) 20.6274 0.973468 0.486734 0.873550i \(-0.338188\pi\)
0.486734 + 0.873550i \(0.338188\pi\)
\(450\) 0 0
\(451\) 40.0166i 1.88431i
\(452\) 0 0
\(453\) 10.9778i 0.515781i
\(454\) 0 0
\(455\) −6.82843 −0.320122
\(456\) 0 0
\(457\) 6.48528 0.303369 0.151684 0.988429i \(-0.451530\pi\)
0.151684 + 0.988429i \(0.451530\pi\)
\(458\) 0 0
\(459\) − 19.1116i − 0.892055i
\(460\) 0 0
\(461\) 32.4399i 1.51088i 0.655220 + 0.755438i \(0.272576\pi\)
−0.655220 + 0.755438i \(0.727424\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 6.62742 0.307339
\(466\) 0 0
\(467\) 18.9259i 0.875788i 0.899027 + 0.437894i \(0.144275\pi\)
−0.899027 + 0.437894i \(0.855725\pi\)
\(468\) 0 0
\(469\) 7.39104i 0.341286i
\(470\) 0 0
\(471\) 4.48528 0.206671
\(472\) 0 0
\(473\) 27.3137 1.25589
\(474\) 0 0
\(475\) 15.8645i 0.727912i
\(476\) 0 0
\(477\) − 7.91630i − 0.362463i
\(478\) 0 0
\(479\) 5.65685 0.258468 0.129234 0.991614i \(-0.458748\pi\)
0.129234 + 0.991614i \(0.458748\pi\)
\(480\) 0 0
\(481\) 13.6569 0.622699
\(482\) 0 0
\(483\) − 1.26810i − 0.0577006i
\(484\) 0 0
\(485\) − 10.8239i − 0.491489i
\(486\) 0 0
\(487\) −7.79899 −0.353406 −0.176703 0.984264i \(-0.556543\pi\)
−0.176703 + 0.984264i \(0.556543\pi\)
\(488\) 0 0
\(489\) 21.6569 0.979357
\(490\) 0 0
\(491\) − 4.85483i − 0.219096i −0.993982 0.109548i \(-0.965060\pi\)
0.993982 0.109548i \(-0.0349403\pi\)
\(492\) 0 0
\(493\) 30.3070i 1.36496i
\(494\) 0 0
\(495\) −10.3431 −0.464890
\(496\) 0 0
\(497\) 13.6569 0.612594
\(498\) 0 0
\(499\) − 13.5140i − 0.604968i −0.953154 0.302484i \(-0.902184\pi\)
0.953154 0.302484i \(-0.0978160\pi\)
\(500\) 0 0
\(501\) 17.3183i 0.773723i
\(502\) 0 0
\(503\) 5.65685 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(504\) 0 0
\(505\) −4.48528 −0.199592
\(506\) 0 0
\(507\) 29.0070i 1.28825i
\(508\) 0 0
\(509\) − 13.6997i − 0.607228i −0.952795 0.303614i \(-0.901807\pi\)
0.952795 0.303614i \(-0.0981933\pi\)
\(510\) 0 0
\(511\) −0.343146 −0.0151799
\(512\) 0 0
\(513\) 21.6569 0.956173
\(514\) 0 0
\(515\) − 2.53620i − 0.111758i
\(516\) 0 0
\(517\) 41.8100i 1.83880i
\(518\) 0 0
\(519\) 13.8579 0.608293
\(520\) 0 0
\(521\) 37.3137 1.63474 0.817372 0.576111i \(-0.195430\pi\)
0.817372 + 0.576111i \(0.195430\pi\)
\(522\) 0 0
\(523\) − 33.7080i − 1.47395i −0.675921 0.736974i \(-0.736254\pi\)
0.675921 0.736974i \(-0.263746\pi\)
\(524\) 0 0
\(525\) 4.14386i 0.180853i
\(526\) 0 0
\(527\) 20.6863 0.901109
\(528\) 0 0
\(529\) −21.6274 −0.940322
\(530\) 0 0
\(531\) 11.5349i 0.500572i
\(532\) 0 0
\(533\) − 48.3044i − 2.09229i
\(534\) 0 0
\(535\) 12.6863 0.548476
\(536\) 0 0
\(537\) 9.94113 0.428991
\(538\) 0 0
\(539\) 5.22625i 0.225111i
\(540\) 0 0
\(541\) − 12.9887i − 0.558428i −0.960229 0.279214i \(-0.909926\pi\)
0.960229 0.279214i \(-0.0900739\pi\)
\(542\) 0 0
\(543\) −1.17157 −0.0502770
\(544\) 0 0
\(545\) −18.3431 −0.785734
\(546\) 0 0
\(547\) 12.0920i 0.517018i 0.966009 + 0.258509i \(0.0832311\pi\)
−0.966009 + 0.258509i \(0.916769\pi\)
\(548\) 0 0
\(549\) − 13.1744i − 0.562270i
\(550\) 0 0
\(551\) −34.3431 −1.46307
\(552\) 0 0
\(553\) −13.6569 −0.580749
\(554\) 0 0
\(555\) 2.53620i 0.107656i
\(556\) 0 0
\(557\) − 6.12293i − 0.259437i −0.991551 0.129719i \(-0.958593\pi\)
0.991551 0.129719i \(-0.0414074\pi\)
\(558\) 0 0
\(559\) −32.9706 −1.39451
\(560\) 0 0
\(561\) 20.6863 0.873376
\(562\) 0 0
\(563\) − 35.5014i − 1.49620i −0.663584 0.748102i \(-0.730965\pi\)
0.663584 0.748102i \(-0.269035\pi\)
\(564\) 0 0
\(565\) 3.43289i 0.144423i
\(566\) 0 0
\(567\) 0.171573 0.00720538
\(568\) 0 0
\(569\) −34.4853 −1.44570 −0.722849 0.691006i \(-0.757168\pi\)
−0.722849 + 0.691006i \(0.757168\pi\)
\(570\) 0 0
\(571\) 13.8854i 0.581085i 0.956862 + 0.290543i \(0.0938358\pi\)
−0.956862 + 0.290543i \(0.906164\pi\)
\(572\) 0 0
\(573\) − 3.58673i − 0.149838i
\(574\) 0 0
\(575\) −4.48528 −0.187049
\(576\) 0 0
\(577\) 10.9706 0.456711 0.228355 0.973578i \(-0.426665\pi\)
0.228355 + 0.973578i \(0.426665\pi\)
\(578\) 0 0
\(579\) 27.9246i 1.16051i
\(580\) 0 0
\(581\) − 5.41196i − 0.224526i
\(582\) 0 0
\(583\) 22.6274 0.937132
\(584\) 0 0
\(585\) 12.4853 0.516203
\(586\) 0 0
\(587\) − 29.0070i − 1.19725i −0.801030 0.598624i \(-0.795714\pi\)
0.801030 0.598624i \(-0.204286\pi\)
\(588\) 0 0
\(589\) 23.4412i 0.965878i
\(590\) 0 0
\(591\) 16.0000 0.658152
\(592\) 0 0
\(593\) −2.68629 −0.110313 −0.0551564 0.998478i \(-0.517566\pi\)
−0.0551564 + 0.998478i \(0.517566\pi\)
\(594\) 0 0
\(595\) − 3.95815i − 0.162268i
\(596\) 0 0
\(597\) − 27.0279i − 1.10618i
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) 18.6863 0.762230 0.381115 0.924528i \(-0.375540\pi\)
0.381115 + 0.924528i \(0.375540\pi\)
\(602\) 0 0
\(603\) − 13.5140i − 0.550331i
\(604\) 0 0
\(605\) − 17.6578i − 0.717893i
\(606\) 0 0
\(607\) −4.68629 −0.190211 −0.0951054 0.995467i \(-0.530319\pi\)
−0.0951054 + 0.995467i \(0.530319\pi\)
\(608\) 0 0
\(609\) −8.97056 −0.363506
\(610\) 0 0
\(611\) − 50.4692i − 2.04176i
\(612\) 0 0
\(613\) − 10.8239i − 0.437174i −0.975817 0.218587i \(-0.929855\pi\)
0.975817 0.218587i \(-0.0701448\pi\)
\(614\) 0 0
\(615\) 8.97056 0.361728
\(616\) 0 0
\(617\) −2.20101 −0.0886093 −0.0443047 0.999018i \(-0.514107\pi\)
−0.0443047 + 0.999018i \(0.514107\pi\)
\(618\) 0 0
\(619\) − 14.0711i − 0.565565i −0.959184 0.282783i \(-0.908743\pi\)
0.959184 0.282783i \(-0.0912575\pi\)
\(620\) 0 0
\(621\) 6.12293i 0.245705i
\(622\) 0 0
\(623\) 2.00000 0.0801283
\(624\) 0 0
\(625\) 8.79899 0.351960
\(626\) 0 0
\(627\) 23.4412i 0.936152i
\(628\) 0 0
\(629\) 7.91630i 0.315644i
\(630\) 0 0
\(631\) 14.6274 0.582308 0.291154 0.956676i \(-0.405961\pi\)
0.291154 + 0.956676i \(0.405961\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) − 13.5140i − 0.536286i
\(636\) 0 0
\(637\) − 6.30864i − 0.249958i
\(638\) 0 0
\(639\) −24.9706 −0.987820
\(640\) 0 0
\(641\) −23.4558 −0.926450 −0.463225 0.886241i \(-0.653308\pi\)
−0.463225 + 0.886241i \(0.653308\pi\)
\(642\) 0 0
\(643\) − 29.3784i − 1.15857i −0.815124 0.579286i \(-0.803331\pi\)
0.815124 0.579286i \(-0.196669\pi\)
\(644\) 0 0
\(645\) − 6.12293i − 0.241090i
\(646\) 0 0
\(647\) −18.3431 −0.721143 −0.360572 0.932731i \(-0.617418\pi\)
−0.360572 + 0.932731i \(0.617418\pi\)
\(648\) 0 0
\(649\) −32.9706 −1.29421
\(650\) 0 0
\(651\) 6.12293i 0.239977i
\(652\) 0 0
\(653\) 39.6452i 1.55144i 0.631079 + 0.775719i \(0.282613\pi\)
−0.631079 + 0.775719i \(0.717387\pi\)
\(654\) 0 0
\(655\) 8.20101 0.320440
\(656\) 0 0
\(657\) 0.627417 0.0244779
\(658\) 0 0
\(659\) − 34.7904i − 1.35524i −0.735412 0.677621i \(-0.763011\pi\)
0.735412 0.677621i \(-0.236989\pi\)
\(660\) 0 0
\(661\) − 6.30864i − 0.245378i −0.992445 0.122689i \(-0.960848\pi\)
0.992445 0.122689i \(-0.0391517\pi\)
\(662\) 0 0
\(663\) −24.9706 −0.969776
\(664\) 0 0
\(665\) 4.48528 0.173932
\(666\) 0 0
\(667\) − 9.70967i − 0.375960i
\(668\) 0 0
\(669\) 27.0279i 1.04496i
\(670\) 0 0
\(671\) 37.6569 1.45373
\(672\) 0 0
\(673\) −16.6274 −0.640940 −0.320470 0.947259i \(-0.603841\pi\)
−0.320470 + 0.947259i \(0.603841\pi\)
\(674\) 0 0
\(675\) − 20.0083i − 0.770121i
\(676\) 0 0
\(677\) − 15.8645i − 0.609721i −0.952397 0.304860i \(-0.901390\pi\)
0.952397 0.304860i \(-0.0986098\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −13.4558 −0.515629
\(682\) 0 0
\(683\) 28.2960i 1.08272i 0.840792 + 0.541359i \(0.182090\pi\)
−0.840792 + 0.541359i \(0.817910\pi\)
\(684\) 0 0
\(685\) 2.16478i 0.0827122i
\(686\) 0 0
\(687\) −9.17157 −0.349917
\(688\) 0 0
\(689\) −27.3137 −1.04057
\(690\) 0 0
\(691\) 39.3057i 1.49526i 0.664116 + 0.747629i \(0.268808\pi\)
−0.664116 + 0.747629i \(0.731192\pi\)
\(692\) 0 0
\(693\) − 9.55582i − 0.362996i
\(694\) 0 0
\(695\) −11.1127 −0.421529
\(696\) 0 0
\(697\) 28.0000 1.06058
\(698\) 0 0
\(699\) 10.8239i 0.409398i
\(700\) 0 0
\(701\) − 4.70099i − 0.177554i −0.996052 0.0887769i \(-0.971704\pi\)
0.996052 0.0887769i \(-0.0282958\pi\)
\(702\) 0 0
\(703\) −8.97056 −0.338331
\(704\) 0 0
\(705\) 9.37258 0.352992
\(706\) 0 0
\(707\) − 4.14386i − 0.155846i
\(708\) 0 0
\(709\) 31.7289i 1.19160i 0.803131 + 0.595802i \(0.203166\pi\)
−0.803131 + 0.595802i \(0.796834\pi\)
\(710\) 0 0
\(711\) 24.9706 0.936469
\(712\) 0 0
\(713\) −6.62742 −0.248199
\(714\) 0 0
\(715\) 35.6871i 1.33462i
\(716\) 0 0
\(717\) 10.9778i 0.409972i
\(718\) 0 0
\(719\) 19.3137 0.720280 0.360140 0.932898i \(-0.382729\pi\)
0.360140 + 0.932898i \(0.382729\pi\)
\(720\) 0 0
\(721\) 2.34315 0.0872633
\(722\) 0 0
\(723\) 23.0698i 0.857975i
\(724\) 0 0
\(725\) 31.7289i 1.17838i
\(726\) 0 0
\(727\) −43.3137 −1.60642 −0.803208 0.595698i \(-0.796875\pi\)
−0.803208 + 0.595698i \(0.796875\pi\)
\(728\) 0 0
\(729\) −17.2843 −0.640158
\(730\) 0 0
\(731\) − 19.1116i − 0.706870i
\(732\) 0 0
\(733\) 15.4930i 0.572249i 0.958192 + 0.286124i \(0.0923670\pi\)
−0.958192 + 0.286124i \(0.907633\pi\)
\(734\) 0 0
\(735\) 1.17157 0.0432141
\(736\) 0 0
\(737\) 38.6274 1.42286
\(738\) 0 0
\(739\) 28.6675i 1.05455i 0.849695 + 0.527275i \(0.176786\pi\)
−0.849695 + 0.527275i \(0.823214\pi\)
\(740\) 0 0
\(741\) − 28.2960i − 1.03948i
\(742\) 0 0
\(743\) −44.4853 −1.63201 −0.816003 0.578047i \(-0.803815\pi\)
−0.816003 + 0.578047i \(0.803815\pi\)
\(744\) 0 0
\(745\) 20.6863 0.757887
\(746\) 0 0
\(747\) 9.89538i 0.362053i
\(748\) 0 0
\(749\) 11.7206i 0.428262i
\(750\) 0 0
\(751\) 5.45584 0.199087 0.0995433 0.995033i \(-0.468262\pi\)
0.0995433 + 0.995033i \(0.468262\pi\)
\(752\) 0 0
\(753\) −26.1421 −0.952672
\(754\) 0 0
\(755\) − 10.9778i − 0.399522i
\(756\) 0 0
\(757\) 39.6452i 1.44093i 0.693491 + 0.720465i \(0.256072\pi\)
−0.693491 + 0.720465i \(0.743928\pi\)
\(758\) 0 0
\(759\) −6.62742 −0.240560
\(760\) 0 0
\(761\) −14.9706 −0.542682 −0.271341 0.962483i \(-0.587467\pi\)
−0.271341 + 0.962483i \(0.587467\pi\)
\(762\) 0 0
\(763\) − 16.9469i − 0.613517i
\(764\) 0 0
\(765\) 7.23719i 0.261661i
\(766\) 0 0
\(767\) 39.7990 1.43706
\(768\) 0 0
\(769\) −55.2548 −1.99254 −0.996270 0.0862891i \(-0.972499\pi\)
−0.996270 + 0.0862891i \(0.972499\pi\)
\(770\) 0 0
\(771\) − 14.4107i − 0.518987i
\(772\) 0 0
\(773\) 15.8645i 0.570605i 0.958438 + 0.285303i \(0.0920941\pi\)
−0.958438 + 0.285303i \(0.907906\pi\)
\(774\) 0 0
\(775\) 21.6569 0.777937
\(776\) 0 0
\(777\) −2.34315 −0.0840599
\(778\) 0 0
\(779\) 31.7289i 1.13681i
\(780\) 0 0
\(781\) − 71.3742i − 2.55397i
\(782\) 0 0
\(783\) 43.3137 1.54791
\(784\) 0 0
\(785\) −4.48528 −0.160087
\(786\) 0 0
\(787\) 33.7080i 1.20156i 0.799414 + 0.600780i \(0.205143\pi\)
−0.799414 + 0.600780i \(0.794857\pi\)
\(788\) 0 0
\(789\) 14.7821i 0.526256i
\(790\) 0 0
\(791\) −3.17157 −0.112768
\(792\) 0 0
\(793\) −45.4558 −1.61418
\(794\) 0 0
\(795\) − 5.07241i − 0.179900i
\(796\) 0 0
\(797\) − 29.7499i − 1.05379i −0.849929 0.526897i \(-0.823355\pi\)
0.849929 0.526897i \(-0.176645\pi\)
\(798\) 0 0
\(799\) 29.2548 1.03496
\(800\) 0 0
\(801\) −3.65685 −0.129209
\(802\) 0 0
\(803\) 1.79337i 0.0632865i
\(804\) 0 0
\(805\) 1.26810i 0.0446947i
\(806\) 0 0
\(807\) −29.4558 −1.03689
\(808\) 0 0
\(809\) −36.8284 −1.29482 −0.647409 0.762143i \(-0.724148\pi\)
−0.647409 + 0.762143i \(0.724148\pi\)
\(810\) 0 0
\(811\) − 45.4286i − 1.59521i −0.603177 0.797607i \(-0.706099\pi\)
0.603177 0.797607i \(-0.293901\pi\)
\(812\) 0 0
\(813\) − 14.7821i − 0.518430i
\(814\) 0 0
\(815\) −21.6569 −0.758607
\(816\) 0 0
\(817\) 21.6569 0.757677
\(818\) 0 0
\(819\) 11.5349i 0.403062i
\(820\) 0 0
\(821\) 20.9050i 0.729590i 0.931088 + 0.364795i \(0.118861\pi\)
−0.931088 + 0.364795i \(0.881139\pi\)
\(822\) 0 0
\(823\) −7.02944 −0.245031 −0.122515 0.992467i \(-0.539096\pi\)
−0.122515 + 0.992467i \(0.539096\pi\)
\(824\) 0 0
\(825\) 21.6569 0.753995
\(826\) 0 0
\(827\) − 47.4077i − 1.64853i −0.566207 0.824263i \(-0.691590\pi\)
0.566207 0.824263i \(-0.308410\pi\)
\(828\) 0 0
\(829\) − 51.5515i − 1.79046i −0.445605 0.895230i \(-0.647011\pi\)
0.445605 0.895230i \(-0.352989\pi\)
\(830\) 0 0
\(831\) −6.62742 −0.229903
\(832\) 0 0
\(833\) 3.65685 0.126702
\(834\) 0 0
\(835\) − 17.3183i − 0.599324i
\(836\) 0 0
\(837\) − 29.5641i − 1.02189i
\(838\) 0 0
\(839\) 20.2843 0.700291 0.350145 0.936695i \(-0.386132\pi\)
0.350145 + 0.936695i \(0.386132\pi\)
\(840\) 0 0
\(841\) −39.6863 −1.36849
\(842\) 0 0
\(843\) 2.16478i 0.0745591i
\(844\) 0 0
\(845\) − 29.0070i − 0.997872i
\(846\) 0 0
\(847\) 16.3137 0.560546
\(848\) 0 0
\(849\) −30.4264 −1.04423
\(850\) 0 0
\(851\) − 2.53620i − 0.0869399i
\(852\) 0 0
\(853\) − 47.7472i − 1.63483i −0.576046 0.817417i \(-0.695405\pi\)
0.576046 0.817417i \(-0.304595\pi\)
\(854\) 0 0
\(855\) −8.20101 −0.280469
\(856\) 0 0
\(857\) −44.6274 −1.52444 −0.762222 0.647316i \(-0.775891\pi\)
−0.762222 + 0.647316i \(0.775891\pi\)
\(858\) 0 0
\(859\) − 13.6997i − 0.467427i −0.972306 0.233714i \(-0.924912\pi\)
0.972306 0.233714i \(-0.0750878\pi\)
\(860\) 0 0
\(861\) 8.28772i 0.282445i
\(862\) 0 0
\(863\) 28.6863 0.976493 0.488246 0.872706i \(-0.337637\pi\)
0.488246 + 0.872706i \(0.337637\pi\)
\(864\) 0 0
\(865\) −13.8579 −0.471182
\(866\) 0 0
\(867\) 3.92629i 0.133344i
\(868\) 0 0
\(869\) 71.3742i 2.42120i
\(870\) 0 0
\(871\) −46.6274 −1.57991
\(872\) 0 0
\(873\) −18.2843 −0.618829
\(874\) 0 0
\(875\) − 9.55582i − 0.323046i
\(876\) 0 0
\(877\) − 25.6060i − 0.864653i −0.901717 0.432326i \(-0.857693\pi\)
0.901717 0.432326i \(-0.142307\pi\)
\(878\) 0 0
\(879\) −11.5147 −0.388382
\(880\) 0 0
\(881\) −22.9706 −0.773898 −0.386949 0.922101i \(-0.626471\pi\)
−0.386949 + 0.922101i \(0.626471\pi\)
\(882\) 0 0
\(883\) 21.4303i 0.721186i 0.932723 + 0.360593i \(0.117426\pi\)
−0.932723 + 0.360593i \(0.882574\pi\)
\(884\) 0 0
\(885\) 7.39104i 0.248447i
\(886\) 0 0
\(887\) −22.6274 −0.759754 −0.379877 0.925037i \(-0.624034\pi\)
−0.379877 + 0.925037i \(0.624034\pi\)
\(888\) 0 0
\(889\) 12.4853 0.418743
\(890\) 0 0
\(891\) − 0.896683i − 0.0300400i
\(892\) 0 0
\(893\) 33.1509i 1.10935i
\(894\) 0 0
\(895\) −9.94113 −0.332295
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) 46.8824i 1.56362i
\(900\) 0 0
\(901\) − 15.8326i − 0.527460i
\(902\) 0 0
\(903\) 5.65685 0.188248
\(904\) 0 0
\(905\) 1.17157 0.0389444
\(906\) 0 0
\(907\) − 17.8435i − 0.592485i −0.955113 0.296243i \(-0.904266\pi\)
0.955113 0.296243i \(-0.0957337\pi\)
\(908\) 0 0
\(909\) 7.57675i 0.251305i
\(910\) 0 0
\(911\) 33.1716 1.09902 0.549512 0.835486i \(-0.314814\pi\)
0.549512 + 0.835486i \(0.314814\pi\)
\(912\) 0 0
\(913\) −28.2843 −0.936073
\(914\) 0 0
\(915\) − 8.44157i − 0.279070i
\(916\) 0 0
\(917\) 7.57675i 0.250206i
\(918\) 0 0
\(919\) −13.6569 −0.450498 −0.225249 0.974301i \(-0.572320\pi\)
−0.225249 + 0.974301i \(0.572320\pi\)
\(920\) 0 0
\(921\) 17.1716 0.565823
\(922\) 0 0
\(923\) 86.1562i 2.83587i
\(924\) 0 0
\(925\) 8.28772i 0.272499i
\(926\) 0 0
\(927\) −4.28427 −0.140714
\(928\) 0 0
\(929\) −46.2843 −1.51854 −0.759269 0.650777i \(-0.774443\pi\)
−0.759269 + 0.650777i \(0.774443\pi\)
\(930\) 0 0
\(931\) 4.14386i 0.135810i
\(932\) 0 0
\(933\) − 33.1509i − 1.08531i
\(934\) 0 0
\(935\) −20.6863 −0.676514
\(936\) 0 0
\(937\) 18.2843 0.597321 0.298661 0.954359i \(-0.403460\pi\)
0.298661 + 0.954359i \(0.403460\pi\)
\(938\) 0 0
\(939\) 21.2764i 0.694330i
\(940\) 0 0
\(941\) 39.4595i 1.28634i 0.765722 + 0.643172i \(0.222382\pi\)
−0.765722 + 0.643172i \(0.777618\pi\)
\(942\) 0 0
\(943\) −8.97056 −0.292122
\(944\) 0 0
\(945\) −5.65685 −0.184017
\(946\) 0 0
\(947\) − 1.63952i − 0.0532772i −0.999645 0.0266386i \(-0.991520\pi\)
0.999645 0.0266386i \(-0.00848034\pi\)
\(948\) 0 0
\(949\) − 2.16478i − 0.0702719i
\(950\) 0 0
\(951\) 22.6274 0.733744
\(952\) 0 0
\(953\) 14.6863 0.475736 0.237868 0.971298i \(-0.423551\pi\)
0.237868 + 0.971298i \(0.423551\pi\)
\(954\) 0 0
\(955\) 3.58673i 0.116064i
\(956\) 0 0
\(957\) 46.8824i 1.51549i
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) − 21.4303i − 0.690581i
\(964\) 0 0
\(965\) − 27.9246i − 0.898925i
\(966\) 0 0
\(967\) 58.1421 1.86973 0.934863 0.355010i \(-0.115523\pi\)
0.934863 + 0.355010i \(0.115523\pi\)
\(968\) 0 0
\(969\) 16.4020 0.526909
\(970\) 0 0
\(971\) 45.5825i 1.46281i 0.681943 + 0.731405i \(0.261135\pi\)
−0.681943 + 0.731405i \(0.738865\pi\)
\(972\) 0 0
\(973\) − 10.2668i − 0.329138i
\(974\) 0 0
\(975\) −26.1421 −0.837218
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) − 10.4525i − 0.334063i
\(980\) 0 0
\(981\) 30.9861i 0.989310i
\(982\) 0 0
\(983\) −56.9706 −1.81708 −0.908539 0.417799i \(-0.862802\pi\)
−0.908539 + 0.417799i \(0.862802\pi\)
\(984\) 0 0
\(985\) −16.0000 −0.509802
\(986\) 0 0
\(987\) 8.65914i 0.275623i
\(988\) 0 0
\(989\) 6.12293i 0.194698i
\(990\) 0 0
\(991\) 18.3431 0.582689 0.291345 0.956618i \(-0.405897\pi\)
0.291345 + 0.956618i \(0.405897\pi\)
\(992\) 0 0
\(993\) −15.0294 −0.476945
\(994\) 0 0
\(995\) 27.0279i 0.856843i
\(996\) 0 0
\(997\) 36.7695i 1.16450i 0.813009 + 0.582250i \(0.197828\pi\)
−0.813009 + 0.582250i \(0.802172\pi\)
\(998\) 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.449.2 4
4.3 odd 2 896.2.b.h.449.3 yes 4
8.3 odd 2 896.2.b.h.449.2 yes 4
8.5 even 2 inner 896.2.b.f.449.3 yes 4
16.3 odd 4 1792.2.a.u.1.2 4
16.5 even 4 1792.2.a.w.1.2 4
16.11 odd 4 1792.2.a.u.1.3 4
16.13 even 4 1792.2.a.w.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.b.f.449.2 4 1.1 even 1 trivial
896.2.b.f.449.3 yes 4 8.5 even 2 inner
896.2.b.h.449.2 yes 4 8.3 odd 2
896.2.b.h.449.3 yes 4 4.3 odd 2
1792.2.a.u.1.2 4 16.3 odd 4
1792.2.a.u.1.3 4 16.11 odd 4
1792.2.a.w.1.2 4 16.5 even 4
1792.2.a.w.1.3 4 16.13 even 4