Properties

Label 1728.2.c.g.1727.4
Level $1728$
Weight $2$
Character 1728.1727
Analytic conductor $13.798$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.4
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.g.1727.5

$q$-expansion

\(f(q)\) \(=\) \(q-0.317837i q^{5} +1.44949i q^{7} +O(q^{10})\) \(q-0.317837i q^{5} +1.44949i q^{7} -1.09638 q^{11} -2.89898 q^{13} -3.46410i q^{17} -4.89898i q^{19} -2.82843 q^{23} +4.89898 q^{25} -9.12096i q^{29} +7.44949i q^{31} +0.460702 q^{35} -4.89898 q^{37} -9.75663i q^{41} -6.89898i q^{43} +9.12096 q^{47} +4.89898 q^{49} -4.41761i q^{53} +0.348469i q^{55} +9.12096 q^{59} -4.00000 q^{61} +0.921404i q^{65} +5.10102i q^{67} -7.56388 q^{71} +1.89898 q^{73} -1.58919i q^{77} -2.00000i q^{79} -15.8742 q^{83} -1.10102 q^{85} -11.9494i q^{89} -4.20204i q^{91} -1.55708 q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 16q^{13} - 32q^{61} - 24q^{73} - 48q^{85} - 40q^{97} + O(q^{100}) \)

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
\(5\) − 0.317837i − 0.142141i −0.997471 0.0710706i \(-0.977358\pi\)
0.997471 0.0710706i \(-0.0226416\pi\)
\(6\) 0 0
\(7\) 1.44949i 0.547856i 0.961750 + 0.273928i \(0.0883229\pi\)
−0.961750 + 0.273928i \(0.911677\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.09638 −0.330570 −0.165285 0.986246i \(-0.552854\pi\)
−0.165285 + 0.986246i \(0.552854\pi\)
\(12\) 0 0
\(13\) −2.89898 −0.804032 −0.402016 0.915633i \(-0.631690\pi\)
−0.402016 + 0.915633i \(0.631690\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.46410i − 0.840168i −0.907485 0.420084i \(-0.862001\pi\)
0.907485 0.420084i \(-0.137999\pi\)
\(18\) 0 0
\(19\) − 4.89898i − 1.12390i −0.827170 0.561951i \(-0.810051\pi\)
0.827170 0.561951i \(-0.189949\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 4.89898 0.979796
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 9.12096i − 1.69372i −0.531817 0.846859i \(-0.678490\pi\)
0.531817 0.846859i \(-0.321510\pi\)
\(30\) 0 0
\(31\) 7.44949i 1.33797i 0.743277 + 0.668984i \(0.233271\pi\)
−0.743277 + 0.668984i \(0.766729\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.460702 0.0778728
\(36\) 0 0
\(37\) −4.89898 −0.805387 −0.402694 0.915335i \(-0.631926\pi\)
−0.402694 + 0.915335i \(0.631926\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.75663i − 1.52373i −0.647736 0.761865i \(-0.724284\pi\)
0.647736 0.761865i \(-0.275716\pi\)
\(42\) 0 0
\(43\) − 6.89898i − 1.05208i −0.850458 0.526042i \(-0.823675\pi\)
0.850458 0.526042i \(-0.176325\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.12096 1.33043 0.665214 0.746653i \(-0.268340\pi\)
0.665214 + 0.746653i \(0.268340\pi\)
\(48\) 0 0
\(49\) 4.89898 0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 4.41761i − 0.606806i −0.952862 0.303403i \(-0.901877\pi\)
0.952862 0.303403i \(-0.0981228\pi\)
\(54\) 0 0
\(55\) 0.348469i 0.0469876i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.12096 1.18745 0.593724 0.804669i \(-0.297657\pi\)
0.593724 + 0.804669i \(0.297657\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.921404i 0.114286i
\(66\) 0 0
\(67\) 5.10102i 0.623189i 0.950215 + 0.311594i \(0.100863\pi\)
−0.950215 + 0.311594i \(0.899137\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.56388 −0.897667 −0.448834 0.893615i \(-0.648160\pi\)
−0.448834 + 0.893615i \(0.648160\pi\)
\(72\) 0 0
\(73\) 1.89898 0.222259 0.111129 0.993806i \(-0.464553\pi\)
0.111129 + 0.993806i \(0.464553\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.58919i − 0.181105i
\(78\) 0 0
\(79\) − 2.00000i − 0.225018i −0.993651 0.112509i \(-0.964111\pi\)
0.993651 0.112509i \(-0.0358886\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.8742 −1.74242 −0.871209 0.490912i \(-0.836664\pi\)
−0.871209 + 0.490912i \(0.836664\pi\)
\(84\) 0 0
\(85\) −1.10102 −0.119422
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 11.9494i − 1.26663i −0.773893 0.633316i \(-0.781693\pi\)
0.773893 0.633316i \(-0.218307\pi\)
\(90\) 0 0
\(91\) − 4.20204i − 0.440494i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.55708 −0.159753
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 15.7313i − 1.56533i −0.622446 0.782663i \(-0.713861\pi\)
0.622446 0.782663i \(-0.286139\pi\)
\(102\) 0 0
\(103\) 10.0000i 0.985329i 0.870219 + 0.492665i \(0.163977\pi\)
−0.870219 + 0.492665i \(0.836023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.66025 0.837218 0.418609 0.908166i \(-0.362518\pi\)
0.418609 + 0.908166i \(0.362518\pi\)
\(108\) 0 0
\(109\) 4.89898 0.469237 0.234619 0.972088i \(-0.424616\pi\)
0.234619 + 0.972088i \(0.424616\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 14.1421i − 1.33038i −0.746674 0.665190i \(-0.768350\pi\)
0.746674 0.665190i \(-0.231650\pi\)
\(114\) 0 0
\(115\) 0.898979i 0.0838303i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.02118 0.460291
\(120\) 0 0
\(121\) −9.79796 −0.890724
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 3.14626i − 0.281410i
\(126\) 0 0
\(127\) − 15.2474i − 1.35299i −0.736446 0.676496i \(-0.763498\pi\)
0.736446 0.676496i \(-0.236502\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.00340 −0.262408 −0.131204 0.991355i \(-0.541884\pi\)
−0.131204 + 0.991355i \(0.541884\pi\)
\(132\) 0 0
\(133\) 7.10102 0.615737
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.6780i 0.912286i 0.889906 + 0.456143i \(0.150769\pi\)
−0.889906 + 0.456143i \(0.849231\pi\)
\(138\) 0 0
\(139\) 20.0000i 1.69638i 0.529694 + 0.848189i \(0.322307\pi\)
−0.529694 + 0.848189i \(0.677693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.17837 0.265789
\(144\) 0 0
\(145\) −2.89898 −0.240747
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.78194i 0.309829i 0.987928 + 0.154914i \(0.0495101\pi\)
−0.987928 + 0.154914i \(0.950490\pi\)
\(150\) 0 0
\(151\) − 17.2474i − 1.40358i −0.712385 0.701789i \(-0.752385\pi\)
0.712385 0.701789i \(-0.247615\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.36773 0.190180
\(156\) 0 0
\(157\) 13.7980 1.10120 0.550599 0.834770i \(-0.314400\pi\)
0.550599 + 0.834770i \(0.314400\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 4.09978i − 0.323108i
\(162\) 0 0
\(163\) − 6.00000i − 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.9774 −1.77804 −0.889021 0.457867i \(-0.848614\pi\)
−0.889021 + 0.457867i \(0.848614\pi\)
\(168\) 0 0
\(169\) −4.59592 −0.353532
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 5.33902i − 0.405918i −0.979187 0.202959i \(-0.934944\pi\)
0.979187 0.202959i \(-0.0650558\pi\)
\(174\) 0 0
\(175\) 7.10102i 0.536787i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.66025 −0.647298 −0.323649 0.946177i \(-0.604910\pi\)
−0.323649 + 0.946177i \(0.604910\pi\)
\(180\) 0 0
\(181\) 9.79796 0.728277 0.364138 0.931345i \(-0.381364\pi\)
0.364138 + 0.931345i \(0.381364\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.55708i 0.114479i
\(186\) 0 0
\(187\) 3.79796i 0.277734i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.6062 1.27394 0.636971 0.770888i \(-0.280187\pi\)
0.636971 + 0.770888i \(0.280187\pi\)
\(192\) 0 0
\(193\) 2.10102 0.151235 0.0756174 0.997137i \(-0.475907\pi\)
0.0756174 + 0.997137i \(0.475907\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.2672i 0.874003i 0.899461 + 0.437002i \(0.143960\pi\)
−0.899461 + 0.437002i \(0.856040\pi\)
\(198\) 0 0
\(199\) − 20.3485i − 1.44246i −0.692693 0.721232i \(-0.743576\pi\)
0.692693 0.721232i \(-0.256424\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 13.2207 0.927913
\(204\) 0 0
\(205\) −3.10102 −0.216585
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.37113i 0.371528i
\(210\) 0 0
\(211\) 28.4949i 1.96167i 0.194841 + 0.980835i \(0.437581\pi\)
−0.194841 + 0.980835i \(0.562419\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.19275 −0.149544
\(216\) 0 0
\(217\) −10.7980 −0.733013
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0424i 0.675522i
\(222\) 0 0
\(223\) − 0.202041i − 0.0135297i −0.999977 0.00676483i \(-0.997847\pi\)
0.999977 0.00676483i \(-0.00215333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.6637 −0.774144 −0.387072 0.922050i \(-0.626513\pi\)
−0.387072 + 0.922050i \(0.626513\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.2207i 0.866119i 0.901365 + 0.433059i \(0.142566\pi\)
−0.901365 + 0.433059i \(0.857434\pi\)
\(234\) 0 0
\(235\) − 2.89898i − 0.189109i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.0703 1.36293 0.681463 0.731852i \(-0.261344\pi\)
0.681463 + 0.731852i \(0.261344\pi\)
\(240\) 0 0
\(241\) −19.7980 −1.27530 −0.637649 0.770327i \(-0.720093\pi\)
−0.637649 + 0.770327i \(0.720093\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 1.55708i − 0.0994781i
\(246\) 0 0
\(247\) 14.2020i 0.903654i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3205 1.09326 0.546630 0.837374i \(-0.315910\pi\)
0.546630 + 0.837374i \(0.315910\pi\)
\(252\) 0 0
\(253\) 3.10102 0.194959
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.6637i 0.727559i 0.931485 + 0.363779i \(0.118514\pi\)
−0.931485 + 0.363779i \(0.881486\pi\)
\(258\) 0 0
\(259\) − 7.10102i − 0.441236i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.8844 −1.53444 −0.767218 0.641386i \(-0.778360\pi\)
−0.767218 + 0.641386i \(0.778360\pi\)
\(264\) 0 0
\(265\) −1.40408 −0.0862521
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.3923i 0.633630i 0.948487 + 0.316815i \(0.102613\pi\)
−0.948487 + 0.316815i \(0.897387\pi\)
\(270\) 0 0
\(271\) − 12.5505i − 0.762389i −0.924495 0.381195i \(-0.875513\pi\)
0.924495 0.381195i \(-0.124487\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.37113 −0.323891
\(276\) 0 0
\(277\) 13.7980 0.829039 0.414520 0.910040i \(-0.363950\pi\)
0.414520 + 0.910040i \(0.363950\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 0.921404i − 0.0549663i −0.999622 0.0274832i \(-0.991251\pi\)
0.999622 0.0274832i \(-0.00874927\pi\)
\(282\) 0 0
\(283\) 3.10102i 0.184337i 0.995743 + 0.0921683i \(0.0293798\pi\)
−0.995743 + 0.0921683i \(0.970620\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.1421 0.834784
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.00680i 0.350921i 0.984486 + 0.175460i \(0.0561414\pi\)
−0.984486 + 0.175460i \(0.943859\pi\)
\(294\) 0 0
\(295\) − 2.89898i − 0.168785i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.19955 0.474192
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.27135i 0.0727972i
\(306\) 0 0
\(307\) − 18.0000i − 1.02731i −0.857996 0.513657i \(-0.828290\pi\)
0.857996 0.513657i \(-0.171710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.09978 0.232477 0.116238 0.993221i \(-0.462916\pi\)
0.116238 + 0.993221i \(0.462916\pi\)
\(312\) 0 0
\(313\) 29.4949 1.66715 0.833575 0.552406i \(-0.186290\pi\)
0.833575 + 0.552406i \(0.186290\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.5877i 1.66181i 0.556413 + 0.830906i \(0.312177\pi\)
−0.556413 + 0.830906i \(0.687823\pi\)
\(318\) 0 0
\(319\) 10.0000i 0.559893i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −16.9706 −0.944267
\(324\) 0 0
\(325\) −14.2020 −0.787787
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 13.2207i 0.728883i
\(330\) 0 0
\(331\) − 0.696938i − 0.0383072i −0.999817 0.0191536i \(-0.993903\pi\)
0.999817 0.0191536i \(-0.00609715\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.62129 0.0885808
\(336\) 0 0
\(337\) 4.20204 0.228900 0.114450 0.993429i \(-0.463489\pi\)
0.114450 + 0.993429i \(0.463489\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 8.16744i − 0.442292i
\(342\) 0 0
\(343\) 17.2474i 0.931275i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.48188 −0.294283 −0.147141 0.989115i \(-0.547007\pi\)
−0.147141 + 0.989115i \(0.547007\pi\)
\(348\) 0 0
\(349\) −24.8990 −1.33281 −0.666406 0.745589i \(-0.732168\pi\)
−0.666406 + 0.745589i \(0.732168\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.1421i 0.752710i 0.926476 + 0.376355i \(0.122823\pi\)
−0.926476 + 0.376355i \(0.877177\pi\)
\(354\) 0 0
\(355\) 2.40408i 0.127595i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −34.2911 −1.80981 −0.904907 0.425610i \(-0.860060\pi\)
−0.904907 + 0.425610i \(0.860060\pi\)
\(360\) 0 0
\(361\) −5.00000 −0.263158
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 0.603566i − 0.0315921i
\(366\) 0 0
\(367\) − 8.55051i − 0.446333i −0.974780 0.223167i \(-0.928361\pi\)
0.974780 0.223167i \(-0.0716394\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.40329 0.332442
\(372\) 0 0
\(373\) −34.4949 −1.78608 −0.893039 0.449979i \(-0.851431\pi\)
−0.893039 + 0.449979i \(0.851431\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 26.4415i 1.36180i
\(378\) 0 0
\(379\) − 8.00000i − 0.410932i −0.978664 0.205466i \(-0.934129\pi\)
0.978664 0.205466i \(-0.0658711\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.74983 −0.191607 −0.0958037 0.995400i \(-0.530542\pi\)
−0.0958037 + 0.995400i \(0.530542\pi\)
\(384\) 0 0
\(385\) −0.505103 −0.0257424
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 32.0662i 1.62582i 0.582388 + 0.812911i \(0.302118\pi\)
−0.582388 + 0.812911i \(0.697882\pi\)
\(390\) 0 0
\(391\) 9.79796i 0.495504i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.635674 −0.0319843
\(396\) 0 0
\(397\) −8.69694 −0.436487 −0.218243 0.975894i \(-0.570033\pi\)
−0.218243 + 0.975894i \(0.570033\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.6274i 1.12996i 0.825105 + 0.564980i \(0.191116\pi\)
−0.825105 + 0.564980i \(0.808884\pi\)
\(402\) 0 0
\(403\) − 21.5959i − 1.07577i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.37113 0.266237
\(408\) 0 0
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 13.2207i 0.650550i
\(414\) 0 0
\(415\) 5.04541i 0.247669i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −26.0915 −1.27465 −0.637327 0.770593i \(-0.719960\pi\)
−0.637327 + 0.770593i \(0.719960\pi\)
\(420\) 0 0
\(421\) 29.7980 1.45226 0.726132 0.687556i \(-0.241316\pi\)
0.726132 + 0.687556i \(0.241316\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 16.9706i − 0.823193i
\(426\) 0 0
\(427\) − 5.79796i − 0.280583i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 27.3629 1.31802 0.659011 0.752133i \(-0.270975\pi\)
0.659011 + 0.752133i \(0.270975\pi\)
\(432\) 0 0
\(433\) 32.3939 1.55675 0.778375 0.627799i \(-0.216044\pi\)
0.778375 + 0.627799i \(0.216044\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.8564i 0.662842i
\(438\) 0 0
\(439\) 19.2474i 0.918631i 0.888273 + 0.459315i \(0.151905\pi\)
−0.888273 + 0.459315i \(0.848095\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.6342 1.36045 0.680226 0.733002i \(-0.261881\pi\)
0.680226 + 0.733002i \(0.261881\pi\)
\(444\) 0 0
\(445\) −3.79796 −0.180041
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.3417i 1.05437i 0.849751 + 0.527185i \(0.176752\pi\)
−0.849751 + 0.527185i \(0.823248\pi\)
\(450\) 0 0
\(451\) 10.6969i 0.503699i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.33557 −0.0626123
\(456\) 0 0
\(457\) 25.0000 1.16945 0.584725 0.811231i \(-0.301202\pi\)
0.584725 + 0.811231i \(0.301202\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.87492i 0.0873235i 0.999046 + 0.0436618i \(0.0139024\pi\)
−0.999046 + 0.0436618i \(0.986098\pi\)
\(462\) 0 0
\(463\) 27.0454i 1.25691i 0.777847 + 0.628453i \(0.216312\pi\)
−0.777847 + 0.628453i \(0.783688\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −0.174973 −0.00809677 −0.00404838 0.999992i \(-0.501289\pi\)
−0.00404838 + 0.999992i \(0.501289\pi\)
\(468\) 0 0
\(469\) −7.39388 −0.341418
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.56388i 0.347787i
\(474\) 0 0
\(475\) − 24.0000i − 1.10120i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −26.0915 −1.19215 −0.596076 0.802928i \(-0.703274\pi\)
−0.596076 + 0.802928i \(0.703274\pi\)
\(480\) 0 0
\(481\) 14.2020 0.647557
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.58919i 0.0721612i
\(486\) 0 0
\(487\) − 3.79796i − 0.172102i −0.996291 0.0860510i \(-0.972575\pi\)
0.996291 0.0860510i \(-0.0274248\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.3171 0.646122 0.323061 0.946378i \(-0.395288\pi\)
0.323061 + 0.946378i \(0.395288\pi\)
\(492\) 0 0
\(493\) −31.5959 −1.42301
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 10.9638i − 0.491792i
\(498\) 0 0
\(499\) − 14.8990i − 0.666970i −0.942755 0.333485i \(-0.891775\pi\)
0.942755 0.333485i \(-0.108225\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.1127 1.38725 0.693623 0.720338i \(-0.256013\pi\)
0.693623 + 0.720338i \(0.256013\pi\)
\(504\) 0 0
\(505\) −5.00000 −0.222497
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 26.6951i − 1.18324i −0.806217 0.591619i \(-0.798489\pi\)
0.806217 0.591619i \(-0.201511\pi\)
\(510\) 0 0
\(511\) 2.75255i 0.121766i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.17837 0.140056
\(516\) 0 0
\(517\) −10.0000 −0.439799
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 21.7060i − 0.950958i −0.879727 0.475479i \(-0.842275\pi\)
0.879727 0.475479i \(-0.157725\pi\)
\(522\) 0 0
\(523\) − 27.3939i − 1.19785i −0.800805 0.598925i \(-0.795595\pi\)
0.800805 0.598925i \(-0.204405\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25.8058 1.12412
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.2843i 1.22513i
\(534\) 0 0
\(535\) − 2.75255i − 0.119003i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.37113 −0.231351
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 1.55708i − 0.0666979i
\(546\) 0 0
\(547\) 23.7980i 1.01753i 0.860906 + 0.508764i \(0.169897\pi\)
−0.860906 + 0.508764i \(0.830103\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −44.6834 −1.90358
\(552\) 0 0
\(553\) 2.89898 0.123277
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 12.5529i − 0.531886i −0.963989 0.265943i \(-0.914317\pi\)
0.963989 0.265943i \(-0.0856832\pi\)
\(558\) 0 0
\(559\) 20.0000i 0.845910i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −22.8024 −0.961006 −0.480503 0.876993i \(-0.659546\pi\)
−0.480503 + 0.876993i \(0.659546\pi\)
\(564\) 0 0
\(565\) −4.49490 −0.189102
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.19955i 0.343743i 0.985119 + 0.171872i \(0.0549814\pi\)
−0.985119 + 0.171872i \(0.945019\pi\)
\(570\) 0 0
\(571\) 19.7980i 0.828519i 0.910159 + 0.414259i \(0.135959\pi\)
−0.910159 + 0.414259i \(0.864041\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −13.8564 −0.577852
\(576\) 0 0
\(577\) −3.79796 −0.158111 −0.0790556 0.996870i \(-0.525190\pi\)
−0.0790556 + 0.996870i \(0.525190\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 23.0095i − 0.954594i
\(582\) 0 0
\(583\) 4.84337i 0.200592i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.0881 0.952949 0.476474 0.879188i \(-0.341915\pi\)
0.476474 + 0.879188i \(0.341915\pi\)
\(588\) 0 0
\(589\) 36.4949 1.50375
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.13534i 0.334078i 0.985950 + 0.167039i \(0.0534206\pi\)
−0.985950 + 0.167039i \(0.946579\pi\)
\(594\) 0 0
\(595\) − 1.59592i − 0.0654263i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 25.5201 1.04272 0.521361 0.853336i \(-0.325425\pi\)
0.521361 + 0.853336i \(0.325425\pi\)
\(600\) 0 0
\(601\) 10.3031 0.420271 0.210135 0.977672i \(-0.432610\pi\)
0.210135 + 0.977672i \(0.432610\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.11416i 0.126608i
\(606\) 0 0
\(607\) − 23.7980i − 0.965929i −0.875640 0.482965i \(-0.839560\pi\)
0.875640 0.482965i \(-0.160440\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −26.4415 −1.06971
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 27.9985i 1.12718i 0.826055 + 0.563589i \(0.190580\pi\)
−0.826055 + 0.563589i \(0.809420\pi\)
\(618\) 0 0
\(619\) 21.5959i 0.868013i 0.900910 + 0.434007i \(0.142901\pi\)
−0.900910 + 0.434007i \(0.857099\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.3205 0.693932
\(624\) 0 0
\(625\) 23.4949 0.939796
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.9706i 0.676661i
\(630\) 0 0
\(631\) − 45.7423i − 1.82097i −0.413538 0.910487i \(-0.635707\pi\)
0.413538 0.910487i \(-0.364293\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.84621 −0.192316
\(636\) 0 0
\(637\) −14.2020 −0.562705
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 48.7832i − 1.92682i −0.268034 0.963409i \(-0.586374\pi\)
0.268034 0.963409i \(-0.413626\pi\)
\(642\) 0 0
\(643\) − 37.3939i − 1.47467i −0.675527 0.737335i \(-0.736084\pi\)
0.675527 0.737335i \(-0.263916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.7846 −0.817127 −0.408564 0.912730i \(-0.633970\pi\)
−0.408564 + 0.912730i \(0.633970\pi\)
\(648\) 0 0
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.5597i 0.726299i 0.931731 + 0.363150i \(0.118299\pi\)
−0.931731 + 0.363150i \(0.881701\pi\)
\(654\) 0 0
\(655\) 0.954592i 0.0372990i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27.8236 1.08385 0.541926 0.840426i \(-0.317695\pi\)
0.541926 + 0.840426i \(0.317695\pi\)
\(660\) 0 0
\(661\) 4.49490 0.174831 0.0874156 0.996172i \(-0.472139\pi\)
0.0874156 + 0.996172i \(0.472139\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 2.25697i − 0.0875215i
\(666\) 0 0
\(667\) 25.7980i 0.998901i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.38551 0.169301
\(672\) 0 0
\(673\) −36.5959 −1.41067 −0.705334 0.708875i \(-0.749203\pi\)
−0.705334 + 0.708875i \(0.749203\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 10.9638i − 0.421372i −0.977554 0.210686i \(-0.932430\pi\)
0.977554 0.210686i \(-0.0675697\pi\)
\(678\) 0 0
\(679\) − 7.24745i − 0.278132i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.3923 0.397650 0.198825 0.980035i \(-0.436287\pi\)
0.198825 + 0.980035i \(0.436287\pi\)
\(684\) 0 0
\(685\) 3.39388 0.129673
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.8066i 0.487891i
\(690\) 0 0
\(691\) 4.00000i 0.152167i 0.997101 + 0.0760836i \(0.0242416\pi\)
−0.997101 + 0.0760836i \(0.975758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.35674 0.241125
\(696\) 0 0
\(697\) −33.7980 −1.28019
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 38.7087i − 1.46201i −0.682374 0.731003i \(-0.739052\pi\)
0.682374 0.731003i \(-0.260948\pi\)
\(702\) 0 0
\(703\) 24.0000i 0.905177i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 22.8024 0.857572
\(708\) 0 0
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 21.0703i − 0.789090i
\(714\) 0 0
\(715\) − 1.01021i − 0.0377795i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.92820 0.258378 0.129189 0.991620i \(-0.458763\pi\)
0.129189 + 0.991620i \(0.458763\pi\)
\(720\) 0 0
\(721\) −14.4949 −0.539818
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 44.6834i − 1.65950i
\(726\) 0 0
\(727\) 7.24745i 0.268793i 0.990928 + 0.134396i \(0.0429096\pi\)
−0.990928 + 0.134396i \(0.957090\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −23.8988 −0.883928
\(732\) 0 0
\(733\) −4.49490 −0.166023 −0.0830114 0.996549i \(-0.526454\pi\)
−0.0830114 + 0.996549i \(0.526454\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 5.59264i − 0.206007i
\(738\) 0 0
\(739\) 19.3939i 0.713415i 0.934216 + 0.356708i \(0.116101\pi\)
−0.934216 + 0.356708i \(0.883899\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.4626 −1.15425 −0.577126 0.816655i \(-0.695826\pi\)
−0.577126 + 0.816655i \(0.695826\pi\)
\(744\) 0 0
\(745\) 1.20204 0.0440394
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.5529i 0.458675i
\(750\) 0 0
\(751\) − 47.7423i − 1.74214i −0.491156 0.871071i \(-0.663426\pi\)
0.491156 0.871071i \(-0.336574\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.48188 −0.199506
\(756\) 0 0
\(757\) 39.3939 1.43179 0.715897 0.698205i \(-0.246018\pi\)
0.715897 + 0.698205i \(0.246018\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 52.8829i − 1.91700i −0.285085 0.958502i \(-0.592022\pi\)
0.285085 0.958502i \(-0.407978\pi\)
\(762\) 0 0
\(763\) 7.10102i 0.257074i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −26.4415 −0.954746
\(768\) 0 0
\(769\) 11.4949 0.414517 0.207258 0.978286i \(-0.433546\pi\)
0.207258 + 0.978286i \(0.433546\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 34.2911i − 1.23336i −0.787212 0.616682i \(-0.788476\pi\)
0.787212 0.616682i \(-0.211524\pi\)
\(774\) 0 0
\(775\) 36.4949i 1.31094i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −47.7975 −1.71252
\(780\) 0 0
\(781\) 8.29286 0.296742
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 4.38551i − 0.156525i
\(786\) 0 0
\(787\) 35.1918i 1.25445i 0.778837 + 0.627227i \(0.215810\pi\)
−0.778837 + 0.627227i \(0.784190\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 20.4989 0.728856
\(792\) 0 0
\(793\) 11.5959 0.411783
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.24604i 0.256668i 0.991731 + 0.128334i \(0.0409629\pi\)
−0.991731 + 0.128334i \(0.959037\pi\)
\(798\) 0 0
\(799\) − 31.5959i − 1.11778i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.08200 −0.0734720
\(804\) 0 0
\(805\) −1.30306 −0.0459269
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 11.9494i − 0.420118i −0.977689 0.210059i \(-0.932634\pi\)
0.977689 0.210059i \(-0.0673656\pi\)
\(810\) 0 0
\(811\) 24.4949i 0.860132i 0.902797 + 0.430066i \(0.141510\pi\)
−0.902797 + 0.430066i \(0.858490\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.90702 −0.0668001
\(816\) 0 0
\(817\) −33.7980 −1.18244
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 22.9774i − 0.801915i −0.916097 0.400958i \(-0.868677\pi\)
0.916097 0.400958i \(-0.131323\pi\)
\(822\) 0 0
\(823\) − 29.6515i − 1.03359i −0.856110 0.516794i \(-0.827125\pi\)
0.856110 0.516794i \(-0.172875\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −16.6206 −0.577955 −0.288978 0.957336i \(-0.593315\pi\)
−0.288978 + 0.957336i \(0.593315\pi\)
\(828\) 0 0
\(829\) 37.5959 1.30576 0.652880 0.757461i \(-0.273561\pi\)
0.652880 + 0.757461i \(0.273561\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 16.9706i − 0.587995i
\(834\) 0 0
\(835\) 7.30306i 0.252733i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.17837 −0.109730 −0.0548648 0.998494i \(-0.517473\pi\)
−0.0548648 + 0.998494i \(0.517473\pi\)
\(840\) 0 0
\(841\) −54.1918 −1.86868
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.46075i 0.0502515i
\(846\) 0 0
\(847\) − 14.2020i − 0.487988i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.8564 0.474991
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 11.6637i − 0.398423i −0.979957 0.199211i \(-0.936162\pi\)
0.979957 0.199211i \(-0.0638380\pi\)
\(858\) 0 0
\(859\) − 39.5959i − 1.35100i −0.737362 0.675498i \(-0.763929\pi\)
0.737362 0.675498i \(-0.236071\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.92820 −0.235839 −0.117919 0.993023i \(-0.537622\pi\)
−0.117919 + 0.993023i \(0.537622\pi\)
\(864\) 0 0
\(865\) −1.69694 −0.0576976
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.19275i 0.0743840i
\(870\) 0 0
\(871\) − 14.7878i − 0.501064i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.56048 0.154172
\(876\) 0 0
\(877\) 16.2020 0.547104 0.273552 0.961857i \(-0.411801\pi\)
0.273552 + 0.961857i \(0.411801\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 7.27815i − 0.245207i −0.992456 0.122604i \(-0.960876\pi\)
0.992456 0.122604i \(-0.0391244\pi\)
\(882\) 0 0
\(883\) 52.8990i 1.78019i 0.455773 + 0.890096i \(0.349363\pi\)
−0.455773 + 0.890096i \(0.650637\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 45.3191 1.52166 0.760832 0.648948i \(-0.224791\pi\)
0.760832 + 0.648948i \(0.224791\pi\)
\(888\) 0 0
\(889\) 22.1010 0.741244
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 44.6834i − 1.49527i
\(894\) 0 0
\(895\) 2.75255i 0.0920076i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 67.9465 2.26614
\(900\) 0 0
\(901\) −15.3031 −0.509819
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 3.11416i − 0.103518i
\(906\) 0 0
\(907\) 4.49490i 0.149251i 0.997212 + 0.0746253i \(0.0237761\pi\)
−0.997212 + 0.0746253i \(0.976224\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.921404 0.0305275 0.0152637 0.999884i \(-0.495141\pi\)
0.0152637 + 0.999884i \(0.495141\pi\)
\(912\) 0 0
\(913\) 17.4041 0.575991
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.35340i − 0.143762i
\(918\) 0 0
\(919\) 2.75255i 0.0907983i 0.998969 + 0.0453991i \(0.0144560\pi\)
−0.998969 + 0.0453991i \(0.985544\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 21.9275 0.721753
\(924\) 0 0
\(925\) −24.0000 −0.789115
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 15.1278i − 0.496326i −0.968718 0.248163i \(-0.920173\pi\)
0.968718 0.248163i \(-0.0798268\pi\)
\(930\) 0 0
\(931\) − 24.0000i − 0.786568i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.20713 0.0394775
\(936\) 0 0
\(937\) −15.0000 −0.490029 −0.245014 0.969519i \(-0.578793\pi\)
−0.245014 + 0.969519i \(0.578793\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 41.5371i − 1.35407i −0.735950 0.677036i \(-0.763264\pi\)
0.735950 0.677036i \(-0.236736\pi\)
\(942\) 0 0
\(943\) 27.5959i 0.898647i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.0881 −0.750263 −0.375132 0.926972i \(-0.622402\pi\)
−0.375132 + 0.926972i \(0.622402\pi\)
\(948\) 0 0
\(949\) −5.50510 −0.178703
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 10.7423i − 0.347976i −0.984748 0.173988i \(-0.944335\pi\)
0.984748 0.173988i \(-0.0556653\pi\)
\(954\) 0 0
\(955\) − 5.59592i − 0.181080i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15.4777 −0.499801
\(960\) 0 0
\(961\) −24.4949 −0.790158
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 0.667783i − 0.0214967i
\(966\) 0 0
\(967\) − 49.2474i − 1.58369i −0.610721 0.791846i \(-0.709120\pi\)
0.610721 0.791846i \(-0.290880\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19.3383 −0.620595 −0.310298 0.950639i \(-0.600429\pi\)
−0.310298 + 0.950639i \(0.600429\pi\)
\(972\) 0 0
\(973\) −28.9898 −0.929370
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 31.4626i − 1.00658i −0.864118 0.503290i \(-0.832123\pi\)
0.864118 0.503290i \(-0.167877\pi\)
\(978\) 0 0
\(979\) 13.1010i 0.418710i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 52.5330 1.67554 0.837771 0.546022i \(-0.183858\pi\)
0.837771 + 0.546022i \(0.183858\pi\)
\(984\) 0 0
\(985\) 3.89898 0.124232
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 19.5133i 0.620486i
\(990\) 0 0
\(991\) 8.75255i 0.278034i 0.990290 + 0.139017i \(0.0443943\pi\)
−0.990290 + 0.139017i \(0.955606\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.46750 −0.205034
\(996\) 0 0
\(997\) 60.6969 1.92229 0.961146 0.276042i \(-0.0890228\pi\)
0.961146 + 0.276042i \(0.0890228\pi\)
\(998\) 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 1728.2.c.g.1727.4 8
3.2 odd 2 inner 1728.2.c.g.1727.6 8
4.3 odd 2 inner 1728.2.c.g.1727.3 8
8.3 odd 2 864.2.c.a.863.5 yes 8
8.5 even 2 864.2.c.a.863.6 yes 8
12.11 even 2 inner 1728.2.c.g.1727.5 8
24.5 odd 2 864.2.c.a.863.4 yes 8
24.11 even 2 864.2.c.a.863.3 8
72.5 odd 6 2592.2.s.a.1727.3 8
72.11 even 6 2592.2.s.h.863.1 8
72.13 even 6 2592.2.s.h.1727.1 8
72.29 odd 6 2592.2.s.h.863.2 8
72.43 odd 6 2592.2.s.a.863.3 8
72.59 even 6 2592.2.s.a.1727.4 8
72.61 even 6 2592.2.s.a.863.4 8
72.67 odd 6 2592.2.s.h.1727.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.c.a.863.3 8 24.11 even 2
864.2.c.a.863.4 yes 8 24.5 odd 2
864.2.c.a.863.5 yes 8 8.3 odd 2
864.2.c.a.863.6 yes 8 8.5 even 2
1728.2.c.g.1727.3 8 4.3 odd 2 inner
1728.2.c.g.1727.4 8 1.1 even 1 trivial
1728.2.c.g.1727.5 8 12.11 even 2 inner
1728.2.c.g.1727.6 8 3.2 odd 2 inner
2592.2.s.a.863.3 8 72.43 odd 6
2592.2.s.a.863.4 8 72.61 even 6
2592.2.s.a.1727.3 8 72.5 odd 6
2592.2.s.a.1727.4 8 72.59 even 6
2592.2.s.h.863.1 8 72.11 even 6
2592.2.s.h.863.2 8 72.29 odd 6
2592.2.s.h.1727.1 8 72.13 even 6
2592.2.s.h.1727.2 8 72.67 odd 6