Properties

Label 1728.3.e.v.1025.8
Level $1728$
Weight $3$
Character 1728.1025
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
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: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.8
Root \(1.52833 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1025
Dual form 1728.3.e.v.1025.2

$q$-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 \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(0.264202\pi\)
−0.674864 + 0.737942i \(0.735798\pi\)
\(6\) 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 \(0.0853484\pi\)
−0.964268 + 0.264929i \(0.914652\pi\)
\(12\) 0 0
\(13\) 10.4853 0.806560 0.403280 0.915077i \(-0.367870\pi\)
0.403280 + 0.915077i \(0.367870\pi\)
\(14\) 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 \(-0.314907\pi\)
0.549267 + 0.835647i \(0.314907\pi\)
\(20\) 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 \(0.0617308\pi\)
−0.981254 + 0.192720i \(0.938269\pi\)
\(30\) 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 \(-0.326202\pi\)
0.519276 + 0.854607i \(0.326202\pi\)
\(38\) 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 \(-0.305495\pi\)
0.573732 + 0.819043i \(0.305495\pi\)
\(44\) 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 \(-0.738585\pi\)
0.681299 0.732005i \(-0.261415\pi\)
\(54\) 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 \(0.0699304\pi\)
−0.975964 + 0.217930i \(0.930070\pi\)
\(60\) 0 0
\(61\) 55.8823 0.916102 0.458051 0.888926i \(-0.348548\pi\)
0.458051 + 0.888926i \(0.348548\pi\)
\(62\) 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 \(-0.737908\pi\)
−0.679741 + 0.733452i \(0.737908\pi\)
\(68\) 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 \(0.360617\pi\)
−0.424024 + 0.905651i \(0.639383\pi\)
\(84\) 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 \(0.190566\pi\)
−0.826079 + 0.563554i \(0.809434\pi\)
\(102\) 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 \(0.118348\pi\)
−0.931674 + 0.363295i \(0.881652\pi\)
\(108\) 0 0
\(109\) 15.3970 0.141257 0.0706283 0.997503i \(-0.477500\pi\)
0.0706283 + 0.997503i \(0.477500\pi\)
\(110\) 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 \(-0.895571\pi\)
0.946665 0.322219i \(-0.104429\pi\)
\(132\) 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 \(-0.344724\pi\)
0.468695 + 0.883360i \(0.344724\pi\)
\(140\) 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 \(0.100297\pi\)
−0.950768 + 0.309904i \(0.899703\pi\)
\(150\) 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 \(-0.362980\pi\)
0.417291 + 0.908773i \(0.362980\pi\)
\(158\) 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 \(-0.791737\pi\)
−0.793487 + 0.608587i \(0.791737\pi\)
\(164\) 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 \(0.000599536\pi\)
−0.999998 + 0.00188350i \(0.999400\pi\)
\(174\) 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 \(-0.884790\pi\)
0.935211 0.354091i \(-0.115210\pi\)
\(180\) 0 0
\(181\) −304.735 −1.68362 −0.841810 0.539775i \(-0.818509\pi\)
−0.841810 + 0.539775i \(0.818509\pi\)
\(182\) 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 \(-0.871708\pi\)
0.919873 0.392217i \(-0.128292\pi\)
\(198\) 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 \(-0.174889\pi\)
0.852822 + 0.522201i \(0.174889\pi\)
\(212\) 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 \(0.172772\pi\)
−0.856276 + 0.516519i \(0.827228\pi\)
\(228\) 0 0
\(229\) −25.6468 −0.111995 −0.0559973 0.998431i \(-0.517834\pi\)
−0.0559973 + 0.998431i \(0.517834\pi\)
\(230\) 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 \(0.163018\pi\)
−0.871700 + 0.490040i \(0.836982\pi\)
\(252\) 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 \(0.362930\pi\)
−0.417432 + 0.908708i \(0.637070\pi\)
\(270\) 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 \(-0.648137\pi\)
−0.448768 + 0.893648i \(0.648137\pi\)
\(278\) 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 \(-0.653468\pi\)
−0.463671 + 0.886007i \(0.653468\pi\)
\(284\) 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 \(0.194283\pi\)
−0.819443 + 0.573161i \(0.805717\pi\)
\(294\) 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 \(-0.106355\pi\)
0.944698 + 0.327941i \(0.106355\pi\)
\(308\) 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 \(0.122367\pi\)
−0.927014 + 0.375028i \(0.877633\pi\)
\(318\) 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 \(-0.609166\pi\)
−0.336271 + 0.941765i \(0.609166\pi\)
\(332\) 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 \(-0.898103\pi\)
0.949198 0.314678i \(-0.101897\pi\)
\(348\) 0 0
\(349\) −499.161 −1.43026 −0.715131 0.698990i \(-0.753633\pi\)
−0.715131 + 0.698990i \(0.753633\pi\)
\(350\) 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 \(-0.326707\pi\)
0.517919 + 0.855430i \(0.326707\pi\)
\(374\) 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 \(-0.734126\pi\)
−0.670980 + 0.741475i \(0.734126\pi\)
\(380\) 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 \(-0.848390\pi\)
0.888698 0.458492i \(-0.151610\pi\)
\(390\) 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 \(-0.729524\pi\)
−0.660190 + 0.751099i \(0.729524\pi\)
\(398\) 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 \(-0.844733\pi\)
0.883374 0.468670i \(-0.155267\pi\)
\(420\) 0 0
\(421\) 266.794 0.633715 0.316857 0.948473i \(-0.397372\pi\)
0.316857 + 0.948473i \(0.397372\pi\)
\(422\) 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 \(0.0738886\pi\)
−0.973179 + 0.230049i \(0.926111\pi\)
\(444\) 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 \(-0.918757\pi\)
0.967605 0.252470i \(-0.0812430\pi\)
\(462\) 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 \(-0.894242\pi\)
0.945311 0.326170i \(-0.105758\pi\)
\(468\) 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 \(0.162845\pi\)
−0.871966 + 0.489567i \(0.837155\pi\)
\(492\) 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 \(-0.254994\pi\)
0.695926 + 0.718113i \(0.254994\pi\)
\(500\) 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 \(0.0647127\pi\)
−0.979405 + 0.201903i \(0.935287\pi\)
\(510\) 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 \(-0.471262\pi\)
0.0901591 + 0.995927i \(0.471262\pi\)
\(524\) 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 \(-0.489477\pi\)
0.0330541 + 0.999454i \(0.489477\pi\)
\(542\) 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 \(-0.632803\pi\)
−0.405215 + 0.914221i \(0.632803\pi\)
\(548\) 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 \(0.128655\pi\)
−0.919425 + 0.393266i \(0.871345\pi\)
\(558\) 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 \(-0.605191\pi\)
0.324485 0.945891i \(-0.394809\pi\)
\(564\) 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 \(-0.417083\pi\)
0.257555 + 0.966264i \(0.417083\pi\)
\(572\) 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 \(-0.988123\pi\)
0.999304 0.0373032i \(-0.0118767\pi\)
\(588\) 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 \(-0.363839\pi\)
0.414834 + 0.909897i \(0.363839\pi\)
\(614\) 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 \(-0.302285\pi\)
0.581964 + 0.813215i \(0.302285\pi\)
\(620\) 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 \(-0.290854\pi\)
0.610786 + 0.791796i \(0.290854\pi\)
\(644\) 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 \(0.0801402\pi\)
−0.968474 + 0.249117i \(0.919860\pi\)
\(654\) 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 \(-0.759980\pi\)
0.728926 0.684593i \(-0.240020\pi\)
\(660\) 0 0
\(661\) 59.6325 0.0902155 0.0451078 0.998982i \(-0.485637\pi\)
0.0451078 + 0.998982i \(0.485637\pi\)
\(662\) 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 \(-0.958579\pi\)
0.991545 0.129762i \(-0.0414212\pi\)
\(678\) 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 \(0.190287\pi\)
−0.826574 + 0.562828i \(0.809713\pi\)
\(684\) 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 \(-0.359418\pi\)
0.427434 + 0.904047i \(0.359418\pi\)
\(692\) 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 \(-0.757686\pi\)
0.723974 0.689828i \(-0.242314\pi\)
\(702\) 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 \(-0.533204\pi\)
−0.104123 + 0.994564i \(0.533204\pi\)
\(710\) 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 \(-0.405225\pi\)
0.293365 + 0.956000i \(0.405225\pi\)
\(734\) 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 \(-0.575487\pi\)
−0.234932 + 0.972012i \(0.575487\pi\)
\(740\) 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 \(-0.596703\pi\)
−0.299149 + 0.954206i \(0.596703\pi\)
\(758\) 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 \(0.0729179\pi\)
−0.973876 + 0.227080i \(0.927082\pi\)
\(774\) 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 \(-0.569865\pi\)
−0.217729 + 0.976009i \(0.569865\pi\)
\(788\) 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 \(-0.901571\pi\)
0.952570 0.304320i \(-0.0984293\pi\)
\(798\) 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 \(-0.405367\pi\)
0.292937 + 0.956132i \(0.405367\pi\)
\(812\) 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 \(0.118666\pi\)
−0.931311 + 0.364224i \(0.881334\pi\)
\(822\) 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 \(-0.988024\pi\)
0.999292 0.0376151i \(-0.0119761\pi\)
\(828\) 0 0
\(829\) −3.70563 −0.00447000 −0.00223500 0.999998i \(-0.500711\pi\)
−0.00223500 + 0.999998i \(0.500711\pi\)
\(830\) 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 \(-0.231733\pi\)
0.746500 + 0.665385i \(0.231733\pi\)
\(854\) 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 \(-0.637577\pi\)
−0.418878 + 0.908042i \(0.637577\pi\)
\(860\) 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 \(-0.470189\pi\)
0.0935178 + 0.995618i \(0.470189\pi\)
\(878\) 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 \(-0.552351\pi\)
−0.163725 + 0.986506i \(0.552351\pi\)
\(884\) 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 \(-0.360524\pi\)
0.424290 + 0.905526i \(0.360524\pi\)
\(908\) 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 \(0.328769\pi\)
−0.512367 + 0.858766i \(0.671231\pi\)
\(942\) 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 \(-0.671561\pi\)
0.513256 0.858235i \(-0.328439\pi\)
\(948\) 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 \(0.206133\pi\)
−0.797543 + 0.603262i \(0.793867\pi\)
\(972\) 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 \(-0.528808\pi\)
−0.0903809 + 0.995907i \(0.528808\pi\)
\(998\) 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 1728.3.e.v.1025.8 8
3.2 odd 2 inner 1728.3.e.v.1025.2 8
4.3 odd 2 inner 1728.3.e.v.1025.7 8
8.3 odd 2 864.3.e.e.161.1 8
8.5 even 2 864.3.e.e.161.2 yes 8
12.11 even 2 inner 1728.3.e.v.1025.1 8
24.5 odd 2 864.3.e.e.161.8 yes 8
24.11 even 2 864.3.e.e.161.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.e.e.161.1 8 8.3 odd 2
864.3.e.e.161.2 yes 8 8.5 even 2
864.3.e.e.161.7 yes 8 24.11 even 2
864.3.e.e.161.8 yes 8 24.5 odd 2
1728.3.e.v.1025.1 8 12.11 even 2 inner
1728.3.e.v.1025.2 8 3.2 odd 2 inner
1728.3.e.v.1025.7 8 4.3 odd 2 inner
1728.3.e.v.1025.8 8 1.1 even 1 trivial