Properties

Label 864.3.e.e.161.2
Level $864$
Weight $3$
Character 864.161
Analytic conductor $23.542$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 864.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.5422948407\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2441150464.4
Defining polynomial: \( x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.2
Root \(-1.52833 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 864.161
Dual form 864.3.e.e.161.8

$q$-expansion

\(f(q)\) \(=\) \(q-7.37942i q^{5} +1.26611 q^{7} +O(q^{10})\) \(q-7.37942i q^{5} +1.26611 q^{7} -5.82843i q^{11} -10.4853 q^{13} -18.3399i q^{17} -20.8722 q^{19} +20.4853i q^{23} -29.4558 q^{25} -11.1777i q^{29} +61.3503 q^{31} -9.34315i q^{35} -38.4264 q^{37} +33.0988i q^{41} -49.3410 q^{43} -21.5980i q^{47} -47.3970 q^{49} +77.5925i q^{53} -43.0104 q^{55} -25.7157i q^{59} -55.8823 q^{61} +77.3753i q^{65} +91.0853 q^{67} +114.770i q^{71} -120.338 q^{73} -7.37942i q^{77} -8.21107 q^{79} -150.338i q^{83} -135.338 q^{85} -118.505i q^{89} -13.2755 q^{91} +154.024i q^{95} -72.8823 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{13} - 32 q^{25} + 32 q^{37} + 96 q^{49} + 96 q^{61} - 216 q^{73} - 336 q^{85} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 7.37942i − 1.47588i −0.674864 0.737942i \(-0.735798\pi\)
0.674864 0.737942i \(-0.264202\pi\)
\(6\) 0 0
\(7\) 1.26611 0.180873 0.0904363 0.995902i \(-0.471174\pi\)
0.0904363 + 0.995902i \(0.471174\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.82843i − 0.529857i −0.964268 0.264929i \(-0.914652\pi\)
0.964268 0.264929i \(-0.0853484\pi\)
\(12\) 0 0
\(13\) −10.4853 −0.806560 −0.403280 0.915077i \(-0.632130\pi\)
−0.403280 + 0.915077i \(0.632130\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 18.3399i − 1.07882i −0.842043 0.539410i \(-0.818647\pi\)
0.842043 0.539410i \(-0.181353\pi\)
\(18\) 0 0
\(19\) −20.8722 −1.09853 −0.549267 0.835647i \(-0.685093\pi\)
−0.549267 + 0.835647i \(0.685093\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 20.4853i 0.890664i 0.895365 + 0.445332i \(0.146914\pi\)
−0.895365 + 0.445332i \(0.853086\pi\)
\(24\) 0 0
\(25\) −29.4558 −1.17823
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 11.1777i − 0.385439i −0.981254 0.192720i \(-0.938269\pi\)
0.981254 0.192720i \(-0.0617308\pi\)
\(30\) 0 0
\(31\) 61.3503 1.97904 0.989522 0.144384i \(-0.0461200\pi\)
0.989522 + 0.144384i \(0.0461200\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 9.34315i − 0.266947i
\(36\) 0 0
\(37\) −38.4264 −1.03855 −0.519276 0.854607i \(-0.673798\pi\)
−0.519276 + 0.854607i \(0.673798\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 33.0988i 0.807287i 0.914916 + 0.403644i \(0.132256\pi\)
−0.914916 + 0.403644i \(0.867744\pi\)
\(42\) 0 0
\(43\) −49.3410 −1.14746 −0.573732 0.819043i \(-0.694505\pi\)
−0.573732 + 0.819043i \(0.694505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 21.5980i − 0.459531i −0.973246 0.229766i \(-0.926204\pi\)
0.973246 0.229766i \(-0.0737960\pi\)
\(48\) 0 0
\(49\) −47.3970 −0.967285
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 77.5925i 1.46401i 0.681299 + 0.732005i \(0.261415\pi\)
−0.681299 + 0.732005i \(0.738585\pi\)
\(54\) 0 0
\(55\) −43.0104 −0.782008
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 25.7157i − 0.435860i −0.975964 0.217930i \(-0.930070\pi\)
0.975964 0.217930i \(-0.0699304\pi\)
\(60\) 0 0
\(61\) −55.8823 −0.916102 −0.458051 0.888926i \(-0.651452\pi\)
−0.458051 + 0.888926i \(0.651452\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 77.3753i 1.19039i
\(66\) 0 0
\(67\) 91.0853 1.35948 0.679741 0.733452i \(-0.262092\pi\)
0.679741 + 0.733452i \(0.262092\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 114.770i 1.61647i 0.588858 + 0.808236i \(0.299578\pi\)
−0.588858 + 0.808236i \(0.700422\pi\)
\(72\) 0 0
\(73\) −120.338 −1.64847 −0.824234 0.566250i \(-0.808394\pi\)
−0.824234 + 0.566250i \(0.808394\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 7.37942i − 0.0958366i
\(78\) 0 0
\(79\) −8.21107 −0.103938 −0.0519688 0.998649i \(-0.516550\pi\)
−0.0519688 + 0.998649i \(0.516550\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 150.338i − 1.81130i −0.424024 0.905651i \(-0.639383\pi\)
0.424024 0.905651i \(-0.360617\pi\)
\(84\) 0 0
\(85\) −135.338 −1.59221
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 118.505i − 1.33152i −0.746166 0.665759i \(-0.768108\pi\)
0.746166 0.665759i \(-0.231892\pi\)
\(90\) 0 0
\(91\) −13.2755 −0.145885
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 154.024i 1.62131i
\(96\) 0 0
\(97\) −72.8823 −0.751363 −0.375682 0.926749i \(-0.622591\pi\)
−0.375682 + 0.926749i \(0.622591\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 113.838i − 1.12711i −0.826079 0.563554i \(-0.809434\pi\)
0.826079 0.563554i \(-0.190566\pi\)
\(102\) 0 0
\(103\) −158.766 −1.54142 −0.770709 0.637187i \(-0.780098\pi\)
−0.770709 + 0.637187i \(0.780098\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 77.7452i − 0.726590i −0.931674 0.363295i \(-0.881652\pi\)
0.931674 0.363295i \(-0.118348\pi\)
\(108\) 0 0
\(109\) −15.3970 −0.141257 −0.0706283 0.997503i \(-0.522500\pi\)
−0.0706283 + 0.997503i \(0.522500\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 76.9408i − 0.680892i −0.940264 0.340446i \(-0.889422\pi\)
0.940264 0.340446i \(-0.110578\pi\)
\(114\) 0 0
\(115\) 151.170 1.31452
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 23.2203i − 0.195129i
\(120\) 0 0
\(121\) 87.0294 0.719252
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 32.8815i 0.263052i
\(126\) 0 0
\(127\) 72.0936 0.567666 0.283833 0.958874i \(-0.408394\pi\)
0.283833 + 0.958874i \(0.408394\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 84.4214i 0.644438i 0.946665 + 0.322219i \(0.104429\pi\)
−0.946665 + 0.322219i \(0.895571\pi\)
\(132\) 0 0
\(133\) −26.4264 −0.198695
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 29.0832i − 0.212286i −0.994351 0.106143i \(-0.966150\pi\)
0.994351 0.106143i \(-0.0338502\pi\)
\(138\) 0 0
\(139\) −130.297 −0.937391 −0.468695 0.883360i \(-0.655276\pi\)
−0.468695 + 0.883360i \(0.655276\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 61.1127i 0.427362i
\(144\) 0 0
\(145\) −82.4853 −0.568864
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 92.3514i − 0.619808i −0.950768 0.309904i \(-0.899703\pi\)
0.950768 0.309904i \(-0.100297\pi\)
\(150\) 0 0
\(151\) −44.9282 −0.297538 −0.148769 0.988872i \(-0.547531\pi\)
−0.148769 + 0.988872i \(0.547531\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 452.730i − 2.92084i
\(156\) 0 0
\(157\) −131.029 −0.834582 −0.417291 0.908773i \(-0.637020\pi\)
−0.417291 + 0.908773i \(0.637020\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 25.9366i 0.161097i
\(162\) 0 0
\(163\) 258.677 1.58697 0.793487 0.608587i \(-0.208263\pi\)
0.793487 + 0.608587i \(0.208263\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 304.108i 1.82100i 0.413505 + 0.910502i \(0.364304\pi\)
−0.413505 + 0.910502i \(0.635696\pi\)
\(168\) 0 0
\(169\) −59.0589 −0.349461
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 0.651690i − 0.00376699i −0.999998 0.00188350i \(-0.999400\pi\)
0.999998 0.00188350i \(-0.000599536\pi\)
\(174\) 0 0
\(175\) −37.2943 −0.213110
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 126.765i 0.708182i 0.935211 + 0.354091i \(0.115210\pi\)
−0.935211 + 0.354091i \(0.884790\pi\)
\(180\) 0 0
\(181\) 304.735 1.68362 0.841810 0.539775i \(-0.181491\pi\)
0.841810 + 0.539775i \(0.181491\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 283.565i 1.53278i
\(186\) 0 0
\(187\) −106.893 −0.571620
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 134.132i − 0.702262i −0.936326 0.351131i \(-0.885797\pi\)
0.936326 0.351131i \(-0.114203\pi\)
\(192\) 0 0
\(193\) 182.397 0.945062 0.472531 0.881314i \(-0.343340\pi\)
0.472531 + 0.881314i \(0.343340\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 154.533i 0.784433i 0.919873 + 0.392217i \(0.128292\pi\)
−0.919873 + 0.392217i \(0.871708\pi\)
\(198\) 0 0
\(199\) 36.7171 0.184508 0.0922541 0.995735i \(-0.470593\pi\)
0.0922541 + 0.995735i \(0.470593\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 14.1522i − 0.0697155i
\(204\) 0 0
\(205\) 244.250 1.19146
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 121.652i 0.582066i
\(210\) 0 0
\(211\) −359.891 −1.70564 −0.852822 0.522201i \(-0.825111\pi\)
−0.852822 + 0.522201i \(0.825111\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 364.108i 1.69352i
\(216\) 0 0
\(217\) 77.6762 0.357955
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 192.299i 0.870133i
\(222\) 0 0
\(223\) 273.870 1.22812 0.614059 0.789260i \(-0.289536\pi\)
0.614059 + 0.789260i \(0.289536\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 234.500i − 1.03304i −0.856276 0.516519i \(-0.827228\pi\)
0.856276 0.516519i \(-0.172772\pi\)
\(228\) 0 0
\(229\) 25.6468 0.111995 0.0559973 0.998431i \(-0.482166\pi\)
0.0559973 + 0.998431i \(0.482166\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.30338i 0.00559390i 0.999996 + 0.00279695i \(0.000890298\pi\)
−0.999996 + 0.00279695i \(0.999110\pi\)
\(234\) 0 0
\(235\) −159.381 −0.678215
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 75.1615i 0.314483i 0.987560 + 0.157242i \(0.0502601\pi\)
−0.987560 + 0.157242i \(0.949740\pi\)
\(240\) 0 0
\(241\) 252.558 1.04796 0.523980 0.851730i \(-0.324447\pi\)
0.523980 + 0.851730i \(0.324447\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 349.762i 1.42760i
\(246\) 0 0
\(247\) 218.850 0.886034
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 246.000i − 0.980080i −0.871700 0.490040i \(-0.836982\pi\)
0.871700 0.490040i \(-0.163018\pi\)
\(252\) 0 0
\(253\) 119.397 0.471925
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 359.097i − 1.39726i −0.715482 0.698632i \(-0.753793\pi\)
0.715482 0.698632i \(-0.246207\pi\)
\(258\) 0 0
\(259\) −48.6520 −0.187846
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 215.897i − 0.820899i −0.911883 0.410450i \(-0.865372\pi\)
0.911883 0.410450i \(-0.134628\pi\)
\(264\) 0 0
\(265\) 572.588 2.16071
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 488.885i − 1.81742i −0.417432 0.908708i \(-0.637070\pi\)
0.417432 0.908708i \(-0.362930\pi\)
\(270\) 0 0
\(271\) 161.261 0.595059 0.297530 0.954713i \(-0.403837\pi\)
0.297530 + 0.954713i \(0.403837\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 171.681i 0.624295i
\(276\) 0 0
\(277\) 248.617 0.897535 0.448768 0.893648i \(-0.351863\pi\)
0.448768 + 0.893648i \(0.351863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 329.579i − 1.17288i −0.809993 0.586439i \(-0.800529\pi\)
0.809993 0.586439i \(-0.199471\pi\)
\(282\) 0 0
\(283\) 262.438 0.927343 0.463671 0.886007i \(-0.346532\pi\)
0.463671 + 0.886007i \(0.346532\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 41.9066i 0.146016i
\(288\) 0 0
\(289\) −47.3532 −0.163852
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 335.872i − 1.14632i −0.819443 0.573161i \(-0.805717\pi\)
0.819443 0.573161i \(-0.194283\pi\)
\(294\) 0 0
\(295\) −189.767 −0.643279
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 214.794i − 0.718374i
\(300\) 0 0
\(301\) −62.4710 −0.207545
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 412.379i 1.35206i
\(306\) 0 0
\(307\) −580.045 −1.88940 −0.944698 0.327941i \(-0.893645\pi\)
−0.944698 + 0.327941i \(0.893645\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 452.132i 1.45380i 0.686743 + 0.726900i \(0.259040\pi\)
−0.686743 + 0.726900i \(0.740960\pi\)
\(312\) 0 0
\(313\) −126.632 −0.404577 −0.202288 0.979326i \(-0.564838\pi\)
−0.202288 + 0.979326i \(0.564838\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 237.767i − 0.750055i −0.927014 0.375028i \(-0.877633\pi\)
0.927014 0.375028i \(-0.122367\pi\)
\(318\) 0 0
\(319\) −65.1487 −0.204228
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 382.794i 1.18512i
\(324\) 0 0
\(325\) 308.853 0.950316
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 27.3454i − 0.0831167i
\(330\) 0 0
\(331\) 222.611 0.672542 0.336271 0.941765i \(-0.390834\pi\)
0.336271 + 0.941765i \(0.390834\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 672.156i − 2.00644i
\(336\) 0 0
\(337\) 396.794 1.17743 0.588715 0.808341i \(-0.299634\pi\)
0.588715 + 0.808341i \(0.299634\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 357.576i − 1.04861i
\(342\) 0 0
\(343\) −122.049 −0.355828
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 218.387i 0.629357i 0.949198 + 0.314678i \(0.101897\pi\)
−0.949198 + 0.314678i \(0.898103\pi\)
\(348\) 0 0
\(349\) 499.161 1.43026 0.715131 0.698990i \(-0.246367\pi\)
0.715131 + 0.698990i \(0.246367\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 423.991i − 1.20111i −0.799585 0.600554i \(-0.794947\pi\)
0.799585 0.600554i \(-0.205053\pi\)
\(354\) 0 0
\(355\) 846.933 2.38573
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 32.2254i − 0.0897643i −0.998992 0.0448822i \(-0.985709\pi\)
0.998992 0.0448822i \(-0.0142912\pi\)
\(360\) 0 0
\(361\) 74.6468 0.206778
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 888.025i 2.43295i
\(366\) 0 0
\(367\) −449.710 −1.22537 −0.612684 0.790328i \(-0.709910\pi\)
−0.612684 + 0.790328i \(0.709910\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 98.2405i 0.264799i
\(372\) 0 0
\(373\) −386.368 −1.03584 −0.517919 0.855430i \(-0.673293\pi\)
−0.517919 + 0.855430i \(0.673293\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 117.202i 0.310880i
\(378\) 0 0
\(379\) 508.603 1.34196 0.670980 0.741475i \(-0.265874\pi\)
0.670980 + 0.741475i \(0.265874\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 308.142i − 0.804549i −0.915519 0.402274i \(-0.868220\pi\)
0.915519 0.402274i \(-0.131780\pi\)
\(384\) 0 0
\(385\) −54.4558 −0.141444
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 356.707i 0.916985i 0.888698 + 0.458492i \(0.151610\pi\)
−0.888698 + 0.458492i \(0.848390\pi\)
\(390\) 0 0
\(391\) 375.699 0.960866
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 60.5929i 0.153400i
\(396\) 0 0
\(397\) 524.191 1.32038 0.660190 0.751099i \(-0.270476\pi\)
0.660190 + 0.751099i \(0.270476\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 236.141i − 0.588881i −0.955670 0.294441i \(-0.904867\pi\)
0.955670 0.294441i \(-0.0951333\pi\)
\(402\) 0 0
\(403\) −643.276 −1.59622
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 223.966i 0.550284i
\(408\) 0 0
\(409\) 261.235 0.638718 0.319359 0.947634i \(-0.396532\pi\)
0.319359 + 0.947634i \(0.396532\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 32.5589i − 0.0788351i
\(414\) 0 0
\(415\) −1109.41 −2.67327
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 392.745i 0.937339i 0.883374 + 0.468670i \(0.155267\pi\)
−0.883374 + 0.468670i \(0.844733\pi\)
\(420\) 0 0
\(421\) −266.794 −0.633715 −0.316857 0.948473i \(-0.602628\pi\)
−0.316857 + 0.948473i \(0.602628\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 540.218i 1.27110i
\(426\) 0 0
\(427\) −70.7530 −0.165698
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 601.029i − 1.39450i −0.716828 0.697250i \(-0.754407\pi\)
0.716828 0.697250i \(-0.245593\pi\)
\(432\) 0 0
\(433\) −605.500 −1.39838 −0.699191 0.714935i \(-0.746456\pi\)
−0.699191 + 0.714935i \(0.746456\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 427.572i − 0.978425i
\(438\) 0 0
\(439\) 333.992 0.760801 0.380401 0.924822i \(-0.375786\pi\)
0.380401 + 0.924822i \(0.375786\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 203.823i − 0.460098i −0.973179 0.230049i \(-0.926111\pi\)
0.973179 0.230049i \(-0.0738886\pi\)
\(444\) 0 0
\(445\) −874.500 −1.96517
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 733.492i 1.63361i 0.576912 + 0.816806i \(0.304258\pi\)
−0.576912 + 0.816806i \(0.695742\pi\)
\(450\) 0 0
\(451\) 192.914 0.427747
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 97.9655i 0.215309i
\(456\) 0 0
\(457\) −514.765 −1.12640 −0.563200 0.826321i \(-0.690430\pi\)
−0.563200 + 0.826321i \(0.690430\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 232.778i 0.504941i 0.967605 + 0.252470i \(0.0812430\pi\)
−0.967605 + 0.252470i \(0.918757\pi\)
\(462\) 0 0
\(463\) −72.7826 −0.157198 −0.0785989 0.996906i \(-0.525045\pi\)
−0.0785989 + 0.996906i \(0.525045\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 304.643i 0.652340i 0.945311 + 0.326170i \(0.105758\pi\)
−0.945311 + 0.326170i \(0.894242\pi\)
\(468\) 0 0
\(469\) 115.324 0.245893
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 287.580i 0.607992i
\(474\) 0 0
\(475\) 614.807 1.29433
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 310.118i − 0.647427i −0.946155 0.323714i \(-0.895069\pi\)
0.946155 0.323714i \(-0.104931\pi\)
\(480\) 0 0
\(481\) 402.912 0.837654
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 537.829i 1.10893i
\(486\) 0 0
\(487\) 744.415 1.52857 0.764287 0.644877i \(-0.223091\pi\)
0.764287 + 0.644877i \(0.223091\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 480.754i − 0.979133i −0.871966 0.489567i \(-0.837155\pi\)
0.871966 0.489567i \(-0.162845\pi\)
\(492\) 0 0
\(493\) −204.999 −0.415820
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 145.311i 0.292376i
\(498\) 0 0
\(499\) −694.534 −1.39185 −0.695926 0.718113i \(-0.745006\pi\)
−0.695926 + 0.718113i \(0.745006\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 266.309i 0.529441i 0.964325 + 0.264720i \(0.0852796\pi\)
−0.964325 + 0.264720i \(0.914720\pi\)
\(504\) 0 0
\(505\) −840.058 −1.66348
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 205.538i − 0.403807i −0.979405 0.201903i \(-0.935287\pi\)
0.979405 0.201903i \(-0.0647127\pi\)
\(510\) 0 0
\(511\) −152.361 −0.298163
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1171.60i 2.27496i
\(516\) 0 0
\(517\) −125.882 −0.243486
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 940.550i − 1.80528i −0.430398 0.902639i \(-0.641627\pi\)
0.430398 0.902639i \(-0.358373\pi\)
\(522\) 0 0
\(523\) −94.3064 −0.180318 −0.0901591 0.995927i \(-0.528738\pi\)
−0.0901591 + 0.995927i \(0.528738\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1125.16i − 2.13503i
\(528\) 0 0
\(529\) 109.353 0.206717
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 347.050i − 0.651126i
\(534\) 0 0
\(535\) −573.714 −1.07236
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 276.250i 0.512523i
\(540\) 0 0
\(541\) −35.7645 −0.0661081 −0.0330541 0.999454i \(-0.510523\pi\)
−0.0330541 + 0.999454i \(0.510523\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 113.621i 0.208478i
\(546\) 0 0
\(547\) 443.305 0.810430 0.405215 0.914221i \(-0.367197\pi\)
0.405215 + 0.914221i \(0.367197\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 233.304i 0.423419i
\(552\) 0 0
\(553\) −10.3961 −0.0187995
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 438.098i − 0.786531i −0.919425 0.393266i \(-0.871345\pi\)
0.919425 0.393266i \(-0.128655\pi\)
\(558\) 0 0
\(559\) 517.354 0.925499
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1065.07i 1.89178i 0.324485 + 0.945891i \(0.394809\pi\)
−0.324485 + 0.945891i \(0.605191\pi\)
\(564\) 0 0
\(565\) −567.779 −1.00492
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 939.141i − 1.65051i −0.564759 0.825256i \(-0.691031\pi\)
0.564759 0.825256i \(-0.308969\pi\)
\(570\) 0 0
\(571\) −294.128 −0.515110 −0.257555 0.966264i \(-0.582917\pi\)
−0.257555 + 0.966264i \(0.582917\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 603.411i − 1.04941i
\(576\) 0 0
\(577\) −546.500 −0.947140 −0.473570 0.880756i \(-0.657035\pi\)
−0.473570 + 0.880756i \(0.657035\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 190.344i − 0.327615i
\(582\) 0 0
\(583\) 452.242 0.775716
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 43.7939i 0.0746064i 0.999304 + 0.0373032i \(0.0118767\pi\)
−0.999304 + 0.0373032i \(0.988123\pi\)
\(588\) 0 0
\(589\) −1280.51 −2.17405
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 823.454i − 1.38862i −0.719674 0.694312i \(-0.755709\pi\)
0.719674 0.694312i \(-0.244291\pi\)
\(594\) 0 0
\(595\) −171.353 −0.287988
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 826.244i − 1.37937i −0.724109 0.689686i \(-0.757749\pi\)
0.724109 0.689686i \(-0.242251\pi\)
\(600\) 0 0
\(601\) −378.632 −0.630004 −0.315002 0.949091i \(-0.602005\pi\)
−0.315002 + 0.949091i \(0.602005\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 642.227i − 1.06153i
\(606\) 0 0
\(607\) −139.737 −0.230210 −0.115105 0.993353i \(-0.536720\pi\)
−0.115105 + 0.993353i \(0.536720\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 226.461i 0.370640i
\(612\) 0 0
\(613\) −508.587 −0.829669 −0.414834 0.909897i \(-0.636161\pi\)
−0.414834 + 0.909897i \(0.636161\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 429.415i 0.695973i 0.937500 + 0.347986i \(0.113134\pi\)
−0.937500 + 0.347986i \(0.886866\pi\)
\(618\) 0 0
\(619\) −720.471 −1.16393 −0.581964 0.813215i \(-0.697715\pi\)
−0.581964 + 0.813215i \(0.697715\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 150.040i − 0.240835i
\(624\) 0 0
\(625\) −493.749 −0.789999
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 704.738i 1.12041i
\(630\) 0 0
\(631\) 479.482 0.759877 0.379938 0.925012i \(-0.375945\pi\)
0.379938 + 0.925012i \(0.375945\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 532.009i − 0.837810i
\(636\) 0 0
\(637\) 496.971 0.780174
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 562.679i 0.877815i 0.898532 + 0.438907i \(0.144634\pi\)
−0.898532 + 0.438907i \(0.855366\pi\)
\(642\) 0 0
\(643\) −785.471 −1.22157 −0.610786 0.791796i \(-0.709146\pi\)
−0.610786 + 0.791796i \(0.709146\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 571.529i − 0.883352i −0.897175 0.441676i \(-0.854384\pi\)
0.897175 0.441676i \(-0.145616\pi\)
\(648\) 0 0
\(649\) −149.882 −0.230943
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 325.346i − 0.498233i −0.968474 0.249117i \(-0.919860\pi\)
0.968474 0.249117i \(-0.0801402\pi\)
\(654\) 0 0
\(655\) 622.981 0.951116
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 902.294i 1.36919i 0.728926 + 0.684593i \(0.240020\pi\)
−0.728926 + 0.684593i \(0.759980\pi\)
\(660\) 0 0
\(661\) −59.6325 −0.0902155 −0.0451078 0.998982i \(-0.514363\pi\)
−0.0451078 + 0.998982i \(0.514363\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 195.012i 0.293250i
\(666\) 0 0
\(667\) 228.979 0.343297
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 325.706i 0.485403i
\(672\) 0 0
\(673\) −1005.17 −1.49357 −0.746787 0.665064i \(-0.768404\pi\)
−0.746787 + 0.665064i \(0.768404\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 175.697i 0.259523i 0.991545 + 0.129762i \(0.0414212\pi\)
−0.991545 + 0.129762i \(0.958579\pi\)
\(678\) 0 0
\(679\) −92.2768 −0.135901
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 768.823i − 1.12566i −0.826574 0.562828i \(-0.809713\pi\)
0.826574 0.562828i \(-0.190287\pi\)
\(684\) 0 0
\(685\) −214.617 −0.313310
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 813.579i − 1.18081i
\(690\) 0 0
\(691\) −590.714 −0.854868 −0.427434 0.904047i \(-0.640582\pi\)
−0.427434 + 0.904047i \(0.640582\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 961.519i 1.38348i
\(696\) 0 0
\(697\) 607.029 0.870917
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 967.139i 1.37966i 0.723974 + 0.689828i \(0.242314\pi\)
−0.723974 + 0.689828i \(0.757686\pi\)
\(702\) 0 0
\(703\) 802.042 1.14088
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 144.131i − 0.203863i
\(708\) 0 0
\(709\) 147.647 0.208246 0.104123 0.994564i \(-0.466796\pi\)
0.104123 + 0.994564i \(0.466796\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1256.78i 1.76266i
\(714\) 0 0
\(715\) 450.976 0.630736
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 627.098i − 0.872181i −0.899903 0.436091i \(-0.856363\pi\)
0.899903 0.436091i \(-0.143637\pi\)
\(720\) 0 0
\(721\) −201.015 −0.278800
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 329.250i 0.454138i
\(726\) 0 0
\(727\) −150.518 −0.207040 −0.103520 0.994627i \(-0.533011\pi\)
−0.103520 + 0.994627i \(0.533011\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 904.910i 1.23791i
\(732\) 0 0
\(733\) −430.073 −0.586730 −0.293365 0.956000i \(-0.594775\pi\)
−0.293365 + 0.956000i \(0.594775\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 530.884i − 0.720331i
\(738\) 0 0
\(739\) 347.230 0.469865 0.234932 0.972012i \(-0.424513\pi\)
0.234932 + 0.972012i \(0.424513\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 48.7797i 0.0656523i 0.999461 + 0.0328261i \(0.0104508\pi\)
−0.999461 + 0.0328261i \(0.989549\pi\)
\(744\) 0 0
\(745\) −681.500 −0.914765
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 98.4338i − 0.131420i
\(750\) 0 0
\(751\) −323.248 −0.430424 −0.215212 0.976567i \(-0.569044\pi\)
−0.215212 + 0.976567i \(0.569044\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 331.544i 0.439131i
\(756\) 0 0
\(757\) 452.912 0.598298 0.299149 0.954206i \(-0.403297\pi\)
0.299149 + 0.954206i \(0.403297\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 824.757i 1.08378i 0.840449 + 0.541890i \(0.182291\pi\)
−0.840449 + 0.541890i \(0.817709\pi\)
\(762\) 0 0
\(763\) −19.4942 −0.0255495
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 269.637i 0.351547i
\(768\) 0 0
\(769\) 248.955 0.323739 0.161870 0.986812i \(-0.448248\pi\)
0.161870 + 0.986812i \(0.448248\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 351.066i − 0.454160i −0.973876 0.227080i \(-0.927082\pi\)
0.973876 0.227080i \(-0.0729179\pi\)
\(774\) 0 0
\(775\) −1807.13 −2.33178
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 690.843i − 0.886833i
\(780\) 0 0
\(781\) 668.926 0.856499
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 966.921i 1.23175i
\(786\) 0 0
\(787\) 342.705 0.435458 0.217729 0.976009i \(-0.430135\pi\)
0.217729 + 0.976009i \(0.430135\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 97.4154i − 0.123155i
\(792\) 0 0
\(793\) 585.941 0.738892
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 485.087i 0.608641i 0.952570 + 0.304320i \(0.0984293\pi\)
−0.952570 + 0.304320i \(0.901571\pi\)
\(798\) 0 0
\(799\) −396.106 −0.495752
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 701.382i 0.873452i
\(804\) 0 0
\(805\) 191.397 0.237760
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 179.278i − 0.221605i −0.993842 0.110802i \(-0.964658\pi\)
0.993842 0.110802i \(-0.0353421\pi\)
\(810\) 0 0
\(811\) −475.144 −0.585874 −0.292937 0.956132i \(-0.594633\pi\)
−0.292937 + 0.956132i \(0.594633\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1908.89i − 2.34219i
\(816\) 0 0
\(817\) 1029.85 1.26053
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 598.056i − 0.728448i −0.931311 0.364224i \(-0.881334\pi\)
0.931311 0.364224i \(-0.118666\pi\)
\(822\) 0 0
\(823\) 433.288 0.526474 0.263237 0.964731i \(-0.415210\pi\)
0.263237 + 0.964731i \(0.415210\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 62.2153i 0.0752301i 0.999292 + 0.0376151i \(0.0119761\pi\)
−0.999292 + 0.0376151i \(0.988024\pi\)
\(828\) 0 0
\(829\) 3.70563 0.00447000 0.00223500 0.999998i \(-0.499289\pi\)
0.00223500 + 0.999998i \(0.499289\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 869.257i 1.04353i
\(834\) 0 0
\(835\) 2244.14 2.68759
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1382.59i 1.64790i 0.566660 + 0.823952i \(0.308235\pi\)
−0.566660 + 0.823952i \(0.691765\pi\)
\(840\) 0 0
\(841\) 716.058 0.851436
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 435.820i 0.515764i
\(846\) 0 0
\(847\) 110.189 0.130093
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 787.176i − 0.925001i
\(852\) 0 0
\(853\) −1273.53 −1.49300 −0.746500 0.665385i \(-0.768267\pi\)
−0.746500 + 0.665385i \(0.768267\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1009.57i − 1.17802i −0.808125 0.589011i \(-0.799517\pi\)
0.808125 0.589011i \(-0.200483\pi\)
\(858\) 0 0
\(859\) 719.633 0.837757 0.418878 0.908042i \(-0.362423\pi\)
0.418878 + 0.908042i \(0.362423\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 429.608i 0.497808i 0.968528 + 0.248904i \(0.0800703\pi\)
−0.968528 + 0.248904i \(0.919930\pi\)
\(864\) 0 0
\(865\) −4.80909 −0.00555964
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 47.8576i 0.0550721i
\(870\) 0 0
\(871\) −955.055 −1.09650
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 41.6316i 0.0475790i
\(876\) 0 0
\(877\) −164.030 −0.187036 −0.0935178 0.995618i \(-0.529811\pi\)
−0.0935178 + 0.995618i \(0.529811\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 333.265i 0.378281i 0.981950 + 0.189140i \(0.0605701\pi\)
−0.981950 + 0.189140i \(0.939430\pi\)
\(882\) 0 0
\(883\) 289.138 0.327450 0.163725 0.986506i \(-0.447649\pi\)
0.163725 + 0.986506i \(0.447649\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 483.338i 0.544913i 0.962168 + 0.272457i \(0.0878361\pi\)
−0.962168 + 0.272457i \(0.912164\pi\)
\(888\) 0 0
\(889\) 91.2784 0.102675
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 450.796i 0.504811i
\(894\) 0 0
\(895\) 935.449 1.04519
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 685.759i − 0.762802i
\(900\) 0 0
\(901\) 1423.04 1.57940
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 2248.77i − 2.48483i
\(906\) 0 0
\(907\) −769.663 −0.848581 −0.424290 0.905526i \(-0.639476\pi\)
−0.424290 + 0.905526i \(0.639476\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 332.059i − 0.364499i −0.983252 0.182250i \(-0.941662\pi\)
0.983252 0.182250i \(-0.0583379\pi\)
\(912\) 0 0
\(913\) −876.235 −0.959731
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 106.887i 0.116561i
\(918\) 0 0
\(919\) −1339.15 −1.45718 −0.728591 0.684949i \(-0.759825\pi\)
−0.728591 + 0.684949i \(0.759825\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 1203.39i − 1.30378i
\(924\) 0 0
\(925\) 1131.88 1.22366
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1194.16i − 1.28543i −0.766106 0.642714i \(-0.777808\pi\)
0.766106 0.642714i \(-0.222192\pi\)
\(930\) 0 0
\(931\) 989.277 1.06260
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 788.808i 0.843645i
\(936\) 0 0
\(937\) 631.705 0.674178 0.337089 0.941473i \(-0.390558\pi\)
0.337089 + 0.941473i \(0.390558\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1616.20i − 1.71753i −0.512367 0.858766i \(-0.671231\pi\)
0.512367 0.858766i \(-0.328769\pi\)
\(942\) 0 0
\(943\) −678.038 −0.719022
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1625.50i 1.71647i 0.513256 + 0.858235i \(0.328439\pi\)
−0.513256 + 0.858235i \(0.671561\pi\)
\(948\) 0 0
\(949\) 1261.78 1.32959
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 143.033i 0.150087i 0.997180 + 0.0750435i \(0.0239096\pi\)
−0.997180 + 0.0750435i \(0.976090\pi\)
\(954\) 0 0
\(955\) −989.817 −1.03646
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 36.8225i − 0.0383968i
\(960\) 0 0
\(961\) 2802.87 2.91661
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1345.98i − 1.39480i
\(966\) 0 0
\(967\) 132.178 0.136689 0.0683443 0.997662i \(-0.478228\pi\)
0.0683443 + 0.997662i \(0.478228\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1171.53i − 1.20652i −0.797543 0.603262i \(-0.793867\pi\)
0.797543 0.603262i \(-0.206133\pi\)
\(972\) 0 0
\(973\) −164.971 −0.169548
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 797.623i 0.816400i 0.912893 + 0.408200i \(0.133843\pi\)
−0.912893 + 0.408200i \(0.866157\pi\)
\(978\) 0 0
\(979\) −690.699 −0.705515
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 960.676i 0.977290i 0.872483 + 0.488645i \(0.162509\pi\)
−0.872483 + 0.488645i \(0.837491\pi\)
\(984\) 0 0
\(985\) 1140.37 1.15773
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1010.76i − 1.02201i
\(990\) 0 0
\(991\) 474.492 0.478802 0.239401 0.970921i \(-0.423049\pi\)
0.239401 + 0.970921i \(0.423049\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 270.951i − 0.272313i
\(996\) 0 0
\(997\) 180.219 0.180762 0.0903809 0.995907i \(-0.471192\pi\)
0.0903809 + 0.995907i \(0.471192\pi\)
\(998\) 0 0
\(999\) 0 0
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 864.3.e.e.161.2 yes 8
3.2 odd 2 inner 864.3.e.e.161.8 yes 8
4.3 odd 2 inner 864.3.e.e.161.1 8
8.3 odd 2 1728.3.e.v.1025.7 8
8.5 even 2 1728.3.e.v.1025.8 8
12.11 even 2 inner 864.3.e.e.161.7 yes 8
24.5 odd 2 1728.3.e.v.1025.2 8
24.11 even 2 1728.3.e.v.1025.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.e.e.161.1 8 4.3 odd 2 inner
864.3.e.e.161.2 yes 8 1.1 even 1 trivial
864.3.e.e.161.7 yes 8 12.11 even 2 inner
864.3.e.e.161.8 yes 8 3.2 odd 2 inner
1728.3.e.v.1025.1 8 24.11 even 2
1728.3.e.v.1025.2 8 24.5 odd 2
1728.3.e.v.1025.7 8 8.3 odd 2
1728.3.e.v.1025.8 8 8.5 even 2