Properties

Label 1728.3.b.i.1567.12
Level $1728$
Weight $3$
Character 1728.1567
Analytic conductor $47.085$
Analytic rank $0$
Dimension $12$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.116304318664704.2
Defining polynomial: \( x^{12} - 2x^{11} + x^{10} + 6x^{9} - 9x^{8} - 2x^{7} + 18x^{6} - 4x^{5} - 36x^{4} + 48x^{3} + 16x^{2} - 64x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.12
Root \(-1.41362 + 0.0408194i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1567
Dual form 1728.3.b.i.1567.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.36964i q^{5} +2.43910i q^{7} +O(q^{10})\) \(q+8.36964i q^{5} +2.43910i q^{7} +2.41295 q^{11} +20.8843i q^{13} -13.1793 q^{17} +32.7976 q^{19} +33.8139i q^{23} -45.0508 q^{25} -34.7407i q^{29} +33.7335i q^{31} -20.4143 q^{35} +30.8546i q^{37} +35.1659 q^{41} +27.9121 q^{43} -21.9865i q^{47} +43.0508 q^{49} -26.3710i q^{53} +20.1955i q^{55} -93.3081 q^{59} +27.7128i q^{61} -174.794 q^{65} -9.97035 q^{67} +64.5448i q^{71} -45.5636 q^{73} +5.88541i q^{77} -82.8072i q^{79} +59.4956 q^{83} -110.306i q^{85} -139.628 q^{89} -50.9387 q^{91} +274.504i q^{95} +150.716 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{17} - 72 q^{25} - 336 q^{41} + 48 q^{49} - 912 q^{65} + 60 q^{73} - 1248 q^{89} - 204 q^{97}+O(q^{100}) \) 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
\(5\) 8.36964i 1.67393i 0.547259 + 0.836964i \(0.315671\pi\)
−0.547259 + 0.836964i \(0.684329\pi\)
\(6\) 0 0
\(7\) 2.43910i 0.348442i 0.984707 + 0.174221i \(0.0557408\pi\)
−0.984707 + 0.174221i \(0.944259\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.41295 0.219359 0.109679 0.993967i \(-0.465018\pi\)
0.109679 + 0.993967i \(0.465018\pi\)
\(12\) 0 0
\(13\) 20.8843i 1.60648i 0.595654 + 0.803241i \(0.296893\pi\)
−0.595654 + 0.803241i \(0.703107\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −13.1793 −0.775256 −0.387628 0.921816i \(-0.626705\pi\)
−0.387628 + 0.921816i \(0.626705\pi\)
\(18\) 0 0
\(19\) 32.7976 1.72619 0.863096 0.505040i \(-0.168522\pi\)
0.863096 + 0.505040i \(0.168522\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 33.8139i 1.47017i 0.677975 + 0.735085i \(0.262858\pi\)
−0.677975 + 0.735085i \(0.737142\pi\)
\(24\) 0 0
\(25\) −45.0508 −1.80203
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 34.7407i − 1.19795i −0.800766 0.598977i \(-0.795574\pi\)
0.800766 0.598977i \(-0.204426\pi\)
\(30\) 0 0
\(31\) 33.7335i 1.08818i 0.839027 + 0.544089i \(0.183125\pi\)
−0.839027 + 0.544089i \(0.816875\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −20.4143 −0.583267
\(36\) 0 0
\(37\) 30.8546i 0.833909i 0.908927 + 0.416954i \(0.136903\pi\)
−0.908927 + 0.416954i \(0.863097\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 35.1659 0.857705 0.428852 0.903375i \(-0.358918\pi\)
0.428852 + 0.903375i \(0.358918\pi\)
\(42\) 0 0
\(43\) 27.9121 0.649119 0.324560 0.945865i \(-0.394784\pi\)
0.324560 + 0.945865i \(0.394784\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 21.9865i − 0.467799i −0.972261 0.233899i \(-0.924851\pi\)
0.972261 0.233899i \(-0.0751486\pi\)
\(48\) 0 0
\(49\) 43.0508 0.878588
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 26.3710i − 0.497567i −0.968559 0.248783i \(-0.919969\pi\)
0.968559 0.248783i \(-0.0800307\pi\)
\(54\) 0 0
\(55\) 20.1955i 0.367191i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −93.3081 −1.58149 −0.790747 0.612143i \(-0.790307\pi\)
−0.790747 + 0.612143i \(0.790307\pi\)
\(60\) 0 0
\(61\) 27.7128i 0.454308i 0.973859 + 0.227154i \(0.0729421\pi\)
−0.973859 + 0.227154i \(0.927058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −174.794 −2.68913
\(66\) 0 0
\(67\) −9.97035 −0.148811 −0.0744056 0.997228i \(-0.523706\pi\)
−0.0744056 + 0.997228i \(0.523706\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 64.5448i 0.909081i 0.890726 + 0.454541i \(0.150197\pi\)
−0.890726 + 0.454541i \(0.849803\pi\)
\(72\) 0 0
\(73\) −45.5636 −0.624159 −0.312079 0.950056i \(-0.601025\pi\)
−0.312079 + 0.950056i \(0.601025\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.88541i 0.0764339i
\(78\) 0 0
\(79\) − 82.8072i − 1.04819i −0.851659 0.524096i \(-0.824403\pi\)
0.851659 0.524096i \(-0.175597\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 59.4956 0.716815 0.358407 0.933565i \(-0.383320\pi\)
0.358407 + 0.933565i \(0.383320\pi\)
\(84\) 0 0
\(85\) − 110.306i − 1.29772i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −139.628 −1.56885 −0.784426 0.620222i \(-0.787042\pi\)
−0.784426 + 0.620222i \(0.787042\pi\)
\(90\) 0 0
\(91\) −50.9387 −0.559766
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 274.504i 2.88952i
\(96\) 0 0
\(97\) 150.716 1.55377 0.776887 0.629641i \(-0.216798\pi\)
0.776887 + 0.629641i \(0.216798\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 74.0246i − 0.732916i −0.930435 0.366458i \(-0.880570\pi\)
0.930435 0.366458i \(-0.119430\pi\)
\(102\) 0 0
\(103\) − 107.152i − 1.04031i −0.854070 0.520157i \(-0.825873\pi\)
0.854070 0.520157i \(-0.174127\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 34.6694 0.324013 0.162007 0.986790i \(-0.448203\pi\)
0.162007 + 0.986790i \(0.448203\pi\)
\(108\) 0 0
\(109\) − 197.530i − 1.81220i −0.423060 0.906102i \(-0.639044\pi\)
0.423060 0.906102i \(-0.360956\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 52.7803 0.467082 0.233541 0.972347i \(-0.424969\pi\)
0.233541 + 0.972347i \(0.424969\pi\)
\(114\) 0 0
\(115\) −283.010 −2.46096
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 32.1457i − 0.270132i
\(120\) 0 0
\(121\) −115.178 −0.951882
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 167.818i − 1.34254i
\(126\) 0 0
\(127\) 74.4391i 0.586135i 0.956092 + 0.293067i \(0.0946760\pi\)
−0.956092 + 0.293067i \(0.905324\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −159.377 −1.21662 −0.608309 0.793700i \(-0.708152\pi\)
−0.608309 + 0.793700i \(0.708152\pi\)
\(132\) 0 0
\(133\) 79.9966i 0.601478i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −218.704 −1.59638 −0.798189 0.602406i \(-0.794209\pi\)
−0.798189 + 0.602406i \(0.794209\pi\)
\(138\) 0 0
\(139\) −37.8825 −0.272536 −0.136268 0.990672i \(-0.543511\pi\)
−0.136268 + 0.990672i \(0.543511\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 50.3926i 0.352396i
\(144\) 0 0
\(145\) 290.767 2.00529
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 52.1967i 0.350314i 0.984541 + 0.175157i \(0.0560432\pi\)
−0.984541 + 0.175157i \(0.943957\pi\)
\(150\) 0 0
\(151\) 38.9569i 0.257993i 0.991645 + 0.128996i \(0.0411756\pi\)
−0.991645 + 0.128996i \(0.958824\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −282.337 −1.82153
\(156\) 0 0
\(157\) − 21.8278i − 0.139031i −0.997581 0.0695154i \(-0.977855\pi\)
0.997581 0.0695154i \(-0.0221453\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −82.4754 −0.512270
\(162\) 0 0
\(163\) 262.870 1.61270 0.806350 0.591438i \(-0.201440\pi\)
0.806350 + 0.591438i \(0.201440\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 233.614i 1.39889i 0.714687 + 0.699444i \(0.246569\pi\)
−0.714687 + 0.699444i \(0.753431\pi\)
\(168\) 0 0
\(169\) −267.153 −1.58079
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 293.868i − 1.69866i −0.527864 0.849329i \(-0.677007\pi\)
0.527864 0.849329i \(-0.322993\pi\)
\(174\) 0 0
\(175\) − 109.883i − 0.627904i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −305.165 −1.70483 −0.852415 0.522866i \(-0.824863\pi\)
−0.852415 + 0.522866i \(0.824863\pi\)
\(180\) 0 0
\(181\) − 55.0270i − 0.304017i −0.988379 0.152008i \(-0.951426\pi\)
0.988379 0.152008i \(-0.0485740\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −258.242 −1.39590
\(186\) 0 0
\(187\) −31.8011 −0.170059
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 252.518i 1.32208i 0.750349 + 0.661041i \(0.229885\pi\)
−0.750349 + 0.661041i \(0.770115\pi\)
\(192\) 0 0
\(193\) 66.1013 0.342494 0.171247 0.985228i \(-0.445220\pi\)
0.171247 + 0.985228i \(0.445220\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 152.421i 0.773710i 0.922141 + 0.386855i \(0.126439\pi\)
−0.922141 + 0.386855i \(0.873561\pi\)
\(198\) 0 0
\(199\) − 36.9774i − 0.185816i −0.995675 0.0929081i \(-0.970384\pi\)
0.995675 0.0929081i \(-0.0296163\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 84.7359 0.417418
\(204\) 0 0
\(205\) 294.326i 1.43573i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 79.1390 0.378655
\(210\) 0 0
\(211\) 277.237 1.31392 0.656960 0.753926i \(-0.271842\pi\)
0.656960 + 0.753926i \(0.271842\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 233.614i 1.08658i
\(216\) 0 0
\(217\) −82.2793 −0.379167
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 275.241i − 1.24543i
\(222\) 0 0
\(223\) 45.6907i 0.204891i 0.994739 + 0.102446i \(0.0326668\pi\)
−0.994739 + 0.102446i \(0.967333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 399.744 1.76099 0.880493 0.474060i \(-0.157212\pi\)
0.880493 + 0.474060i \(0.157212\pi\)
\(228\) 0 0
\(229\) 189.213i 0.826258i 0.910672 + 0.413129i \(0.135564\pi\)
−0.910672 + 0.413129i \(0.864436\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −57.0895 −0.245019 −0.122510 0.992467i \(-0.539094\pi\)
−0.122510 + 0.992467i \(0.539094\pi\)
\(234\) 0 0
\(235\) 184.019 0.783061
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 309.733i − 1.29595i −0.761660 0.647977i \(-0.775615\pi\)
0.761660 0.647977i \(-0.224385\pi\)
\(240\) 0 0
\(241\) −400.843 −1.66325 −0.831624 0.555340i \(-0.812588\pi\)
−0.831624 + 0.555340i \(0.812588\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 360.320i 1.47069i
\(246\) 0 0
\(247\) 684.955i 2.77310i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 264.779 1.05490 0.527448 0.849587i \(-0.323149\pi\)
0.527448 + 0.849587i \(0.323149\pi\)
\(252\) 0 0
\(253\) 81.5912i 0.322495i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −354.023 −1.37752 −0.688760 0.724990i \(-0.741845\pi\)
−0.688760 + 0.724990i \(0.741845\pi\)
\(258\) 0 0
\(259\) −75.2574 −0.290569
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 386.105i 1.46808i 0.679106 + 0.734041i \(0.262368\pi\)
−0.679106 + 0.734041i \(0.737632\pi\)
\(264\) 0 0
\(265\) 220.716 0.832891
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 167.021i − 0.620896i −0.950590 0.310448i \(-0.899521\pi\)
0.950590 0.310448i \(-0.100479\pi\)
\(270\) 0 0
\(271\) 254.266i 0.938253i 0.883131 + 0.469126i \(0.155431\pi\)
−0.883131 + 0.469126i \(0.844569\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −108.705 −0.395292
\(276\) 0 0
\(277\) 414.204i 1.49532i 0.664082 + 0.747660i \(0.268823\pi\)
−0.664082 + 0.747660i \(0.731177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 53.6902 0.191068 0.0955341 0.995426i \(-0.469544\pi\)
0.0955341 + 0.995426i \(0.469544\pi\)
\(282\) 0 0
\(283\) 183.530 0.648517 0.324259 0.945969i \(-0.394885\pi\)
0.324259 + 0.945969i \(0.394885\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 85.7730i 0.298861i
\(288\) 0 0
\(289\) −115.305 −0.398979
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 165.240i − 0.563959i −0.959420 0.281980i \(-0.909009\pi\)
0.959420 0.281980i \(-0.0909911\pi\)
\(294\) 0 0
\(295\) − 780.955i − 2.64730i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −706.179 −2.36180
\(300\) 0 0
\(301\) 68.0804i 0.226181i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −231.946 −0.760479
\(306\) 0 0
\(307\) 168.765 0.549723 0.274861 0.961484i \(-0.411368\pi\)
0.274861 + 0.961484i \(0.411368\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 444.989i − 1.43083i −0.698699 0.715416i \(-0.746237\pi\)
0.698699 0.715416i \(-0.253763\pi\)
\(312\) 0 0
\(313\) 20.7413 0.0662660 0.0331330 0.999451i \(-0.489452\pi\)
0.0331330 + 0.999451i \(0.489452\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 69.1362i 0.218095i 0.994037 + 0.109048i \(0.0347801\pi\)
−0.994037 + 0.109048i \(0.965220\pi\)
\(318\) 0 0
\(319\) − 83.8274i − 0.262782i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −432.252 −1.33824
\(324\) 0 0
\(325\) − 940.853i − 2.89493i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 53.6273 0.163001
\(330\) 0 0
\(331\) −386.180 −1.16671 −0.583353 0.812219i \(-0.698260\pi\)
−0.583353 + 0.812219i \(0.698260\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 83.4482i − 0.249099i
\(336\) 0 0
\(337\) −211.767 −0.628389 −0.314194 0.949359i \(-0.601734\pi\)
−0.314194 + 0.949359i \(0.601734\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 81.3972i 0.238701i
\(342\) 0 0
\(343\) 224.521i 0.654580i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 540.291 1.55704 0.778518 0.627623i \(-0.215972\pi\)
0.778518 + 0.627623i \(0.215972\pi\)
\(348\) 0 0
\(349\) 183.767i 0.526553i 0.964720 + 0.263276i \(0.0848031\pi\)
−0.964720 + 0.263276i \(0.915197\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 549.830 1.55759 0.778796 0.627277i \(-0.215831\pi\)
0.778796 + 0.627277i \(0.215831\pi\)
\(354\) 0 0
\(355\) −540.216 −1.52174
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 406.677i − 1.13281i −0.824129 0.566403i \(-0.808335\pi\)
0.824129 0.566403i \(-0.191665\pi\)
\(360\) 0 0
\(361\) 714.686 1.97974
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 381.351i − 1.04480i
\(366\) 0 0
\(367\) − 443.688i − 1.20896i −0.796620 0.604480i \(-0.793381\pi\)
0.796620 0.604480i \(-0.206619\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 64.3215 0.173373
\(372\) 0 0
\(373\) − 98.5364i − 0.264173i −0.991238 0.132086i \(-0.957832\pi\)
0.991238 0.132086i \(-0.0421676\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 725.534 1.92449
\(378\) 0 0
\(379\) 431.635 1.13888 0.569439 0.822033i \(-0.307160\pi\)
0.569439 + 0.822033i \(0.307160\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 260.504i − 0.680167i −0.940395 0.340083i \(-0.889545\pi\)
0.940395 0.340083i \(-0.110455\pi\)
\(384\) 0 0
\(385\) −49.2587 −0.127945
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 360.413i 0.926512i 0.886224 + 0.463256i \(0.153319\pi\)
−0.886224 + 0.463256i \(0.846681\pi\)
\(390\) 0 0
\(391\) − 445.645i − 1.13976i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 693.066 1.75460
\(396\) 0 0
\(397\) − 524.685i − 1.32162i −0.750551 0.660812i \(-0.770212\pi\)
0.750551 0.660812i \(-0.229788\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −469.300 −1.17032 −0.585162 0.810916i \(-0.698969\pi\)
−0.585162 + 0.810916i \(0.698969\pi\)
\(402\) 0 0
\(403\) −704.500 −1.74814
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 74.4505i 0.182925i
\(408\) 0 0
\(409\) 493.509 1.20662 0.603311 0.797506i \(-0.293848\pi\)
0.603311 + 0.797506i \(0.293848\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 227.587i − 0.551059i
\(414\) 0 0
\(415\) 497.957i 1.19990i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 527.341 1.25857 0.629285 0.777175i \(-0.283348\pi\)
0.629285 + 0.777175i \(0.283348\pi\)
\(420\) 0 0
\(421\) 417.579i 0.991875i 0.868358 + 0.495937i \(0.165175\pi\)
−0.868358 + 0.495937i \(0.834825\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 593.740 1.39704
\(426\) 0 0
\(427\) −67.5942 −0.158300
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 462.224i 1.07245i 0.844076 + 0.536223i \(0.180149\pi\)
−0.844076 + 0.536223i \(0.819851\pi\)
\(432\) 0 0
\(433\) 58.2079 0.134429 0.0672147 0.997739i \(-0.478589\pi\)
0.0672147 + 0.997739i \(0.478589\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1109.02i 2.53780i
\(438\) 0 0
\(439\) 621.745i 1.41627i 0.706075 + 0.708137i \(0.250464\pi\)
−0.706075 + 0.708137i \(0.749536\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −127.815 −0.288521 −0.144261 0.989540i \(-0.546080\pi\)
−0.144261 + 0.989540i \(0.546080\pi\)
\(444\) 0 0
\(445\) − 1168.63i − 2.62614i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 335.372 0.746932 0.373466 0.927644i \(-0.378169\pi\)
0.373466 + 0.927644i \(0.378169\pi\)
\(450\) 0 0
\(451\) 84.8534 0.188145
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 426.339i − 0.937008i
\(456\) 0 0
\(457\) −121.182 −0.265169 −0.132585 0.991172i \(-0.542328\pi\)
−0.132585 + 0.991172i \(0.542328\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 144.396i − 0.313224i −0.987660 0.156612i \(-0.949943\pi\)
0.987660 0.156612i \(-0.0500573\pi\)
\(462\) 0 0
\(463\) − 487.501i − 1.05292i −0.850201 0.526459i \(-0.823520\pi\)
0.850201 0.526459i \(-0.176480\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −298.591 −0.639382 −0.319691 0.947522i \(-0.603579\pi\)
−0.319691 + 0.947522i \(0.603579\pi\)
\(468\) 0 0
\(469\) − 24.3187i − 0.0518521i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 67.3505 0.142390
\(474\) 0 0
\(475\) −1477.56 −3.11065
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 86.7847i − 0.181179i −0.995888 0.0905894i \(-0.971125\pi\)
0.995888 0.0905894i \(-0.0288751\pi\)
\(480\) 0 0
\(481\) −644.376 −1.33966
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1261.44i 2.60090i
\(486\) 0 0
\(487\) − 781.377i − 1.60447i −0.597009 0.802235i \(-0.703644\pi\)
0.597009 0.802235i \(-0.296356\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −158.487 −0.322783 −0.161392 0.986890i \(-0.551598\pi\)
−0.161392 + 0.986890i \(0.551598\pi\)
\(492\) 0 0
\(493\) 457.859i 0.928721i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −157.431 −0.316762
\(498\) 0 0
\(499\) 780.549 1.56423 0.782113 0.623137i \(-0.214142\pi\)
0.782113 + 0.623137i \(0.214142\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 517.773i − 1.02937i −0.857379 0.514685i \(-0.827909\pi\)
0.857379 0.514685i \(-0.172091\pi\)
\(504\) 0 0
\(505\) 619.559 1.22685
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 653.045i − 1.28300i −0.767124 0.641498i \(-0.778313\pi\)
0.767124 0.641498i \(-0.221687\pi\)
\(510\) 0 0
\(511\) − 111.134i − 0.217483i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 896.827 1.74141
\(516\) 0 0
\(517\) − 53.0523i − 0.102616i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −310.175 −0.595346 −0.297673 0.954668i \(-0.596210\pi\)
−0.297673 + 0.954668i \(0.596210\pi\)
\(522\) 0 0
\(523\) 779.063 1.48960 0.744802 0.667285i \(-0.232544\pi\)
0.744802 + 0.667285i \(0.232544\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 444.586i − 0.843616i
\(528\) 0 0
\(529\) −614.381 −1.16140
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 734.414i 1.37789i
\(534\) 0 0
\(535\) 290.170i 0.542374i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 103.879 0.192726
\(540\) 0 0
\(541\) 410.019i 0.757892i 0.925419 + 0.378946i \(0.123713\pi\)
−0.925419 + 0.378946i \(0.876287\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1653.26 3.03350
\(546\) 0 0
\(547\) 561.353 1.02624 0.513120 0.858317i \(-0.328490\pi\)
0.513120 + 0.858317i \(0.328490\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 1139.41i − 2.06790i
\(552\) 0 0
\(553\) 201.975 0.365235
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 363.521i − 0.652640i −0.945259 0.326320i \(-0.894191\pi\)
0.945259 0.326320i \(-0.105809\pi\)
\(558\) 0 0
\(559\) 582.924i 1.04280i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −287.769 −0.511134 −0.255567 0.966791i \(-0.582262\pi\)
−0.255567 + 0.966791i \(0.582262\pi\)
\(564\) 0 0
\(565\) 441.752i 0.781861i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 414.386 0.728270 0.364135 0.931346i \(-0.381365\pi\)
0.364135 + 0.931346i \(0.381365\pi\)
\(570\) 0 0
\(571\) 529.424 0.927188 0.463594 0.886048i \(-0.346560\pi\)
0.463594 + 0.886048i \(0.346560\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 1523.34i − 2.64930i
\(576\) 0 0
\(577\) −409.148 −0.709095 −0.354548 0.935038i \(-0.615365\pi\)
−0.354548 + 0.935038i \(0.615365\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 145.116i 0.249769i
\(582\) 0 0
\(583\) − 63.6319i − 0.109146i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 368.026 0.626961 0.313481 0.949595i \(-0.398505\pi\)
0.313481 + 0.949595i \(0.398505\pi\)
\(588\) 0 0
\(589\) 1106.38i 1.87840i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1055.08 1.77922 0.889612 0.456717i \(-0.150975\pi\)
0.889612 + 0.456717i \(0.150975\pi\)
\(594\) 0 0
\(595\) 269.048 0.452181
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 863.896i 1.44223i 0.692815 + 0.721116i \(0.256370\pi\)
−0.692815 + 0.721116i \(0.743630\pi\)
\(600\) 0 0
\(601\) −2.43642 −0.00405394 −0.00202697 0.999998i \(-0.500645\pi\)
−0.00202697 + 0.999998i \(0.500645\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 963.995i − 1.59338i
\(606\) 0 0
\(607\) 755.681i 1.24494i 0.782642 + 0.622472i \(0.213872\pi\)
−0.782642 + 0.622472i \(0.786128\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 459.173 0.751510
\(612\) 0 0
\(613\) 183.950i 0.300082i 0.988680 + 0.150041i \(0.0479406\pi\)
−0.988680 + 0.150041i \(0.952059\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 502.269 0.814050 0.407025 0.913417i \(-0.366566\pi\)
0.407025 + 0.913417i \(0.366566\pi\)
\(618\) 0 0
\(619\) −222.995 −0.360250 −0.180125 0.983644i \(-0.557650\pi\)
−0.180125 + 0.983644i \(0.557650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 340.566i − 0.546655i
\(624\) 0 0
\(625\) 278.305 0.445288
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 406.644i − 0.646492i
\(630\) 0 0
\(631\) 1058.12i 1.67689i 0.544988 + 0.838444i \(0.316534\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −623.028 −0.981147
\(636\) 0 0
\(637\) 899.085i 1.41144i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −353.113 −0.550878 −0.275439 0.961319i \(-0.588823\pi\)
−0.275439 + 0.961319i \(0.588823\pi\)
\(642\) 0 0
\(643\) −404.720 −0.629424 −0.314712 0.949187i \(-0.601908\pi\)
−0.314712 + 0.949187i \(0.601908\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 257.672i 0.398257i 0.979973 + 0.199129i \(0.0638111\pi\)
−0.979973 + 0.199129i \(0.936189\pi\)
\(648\) 0 0
\(649\) −225.147 −0.346914
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 841.878i 1.28925i 0.764500 + 0.644624i \(0.222986\pi\)
−0.764500 + 0.644624i \(0.777014\pi\)
\(654\) 0 0
\(655\) − 1333.93i − 2.03653i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 673.789 1.02244 0.511221 0.859449i \(-0.329193\pi\)
0.511221 + 0.859449i \(0.329193\pi\)
\(660\) 0 0
\(661\) − 116.970i − 0.176959i −0.996078 0.0884796i \(-0.971799\pi\)
0.996078 0.0884796i \(-0.0282008\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −669.543 −1.00683
\(666\) 0 0
\(667\) 1174.72 1.76120
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 66.8695i 0.0996565i
\(672\) 0 0
\(673\) −772.635 −1.14805 −0.574023 0.818839i \(-0.694618\pi\)
−0.574023 + 0.818839i \(0.694618\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 172.800i 0.255244i 0.991823 + 0.127622i \(0.0407344\pi\)
−0.991823 + 0.127622i \(0.959266\pi\)
\(678\) 0 0
\(679\) 367.611i 0.541400i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 808.636 1.18395 0.591974 0.805957i \(-0.298349\pi\)
0.591974 + 0.805957i \(0.298349\pi\)
\(684\) 0 0
\(685\) − 1830.47i − 2.67222i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 550.740 0.799332
\(690\) 0 0
\(691\) 315.802 0.457022 0.228511 0.973541i \(-0.426614\pi\)
0.228511 + 0.973541i \(0.426614\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 317.063i − 0.456205i
\(696\) 0 0
\(697\) −463.463 −0.664940
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 276.941i 0.395066i 0.980296 + 0.197533i \(0.0632929\pi\)
−0.980296 + 0.197533i \(0.936707\pi\)
\(702\) 0 0
\(703\) 1011.96i 1.43949i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 180.553 0.255379
\(708\) 0 0
\(709\) 512.419i 0.722734i 0.932424 + 0.361367i \(0.117690\pi\)
−0.932424 + 0.361367i \(0.882310\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1140.66 −1.59981
\(714\) 0 0
\(715\) −421.768 −0.589885
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 129.848i − 0.180595i −0.995915 0.0902975i \(-0.971218\pi\)
0.995915 0.0902975i \(-0.0287818\pi\)
\(720\) 0 0
\(721\) 261.355 0.362490
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1565.10i 2.15875i
\(726\) 0 0
\(727\) − 581.866i − 0.800366i −0.916435 0.400183i \(-0.868947\pi\)
0.916435 0.400183i \(-0.131053\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −367.864 −0.503233
\(732\) 0 0
\(733\) 452.139i 0.616834i 0.951251 + 0.308417i \(0.0997992\pi\)
−0.951251 + 0.308417i \(0.900201\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.0579 −0.0326430
\(738\) 0 0
\(739\) −42.9916 −0.0581754 −0.0290877 0.999577i \(-0.509260\pi\)
−0.0290877 + 0.999577i \(0.509260\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 51.4541i − 0.0692518i −0.999400 0.0346259i \(-0.988976\pi\)
0.999400 0.0346259i \(-0.0110240\pi\)
\(744\) 0 0
\(745\) −436.868 −0.586399
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 84.5620i 0.112900i
\(750\) 0 0
\(751\) − 390.652i − 0.520175i −0.965585 0.260088i \(-0.916249\pi\)
0.965585 0.260088i \(-0.0837514\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −326.055 −0.431862
\(756\) 0 0
\(757\) − 310.240i − 0.409828i −0.978780 0.204914i \(-0.934309\pi\)
0.978780 0.204914i \(-0.0656915\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 859.524 1.12947 0.564733 0.825273i \(-0.308979\pi\)
0.564733 + 0.825273i \(0.308979\pi\)
\(762\) 0 0
\(763\) 481.795 0.631448
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1948.67i − 2.54064i
\(768\) 0 0
\(769\) 1408.40 1.83147 0.915733 0.401787i \(-0.131611\pi\)
0.915733 + 0.401787i \(0.131611\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1297.40i 1.67839i 0.543828 + 0.839197i \(0.316975\pi\)
−0.543828 + 0.839197i \(0.683025\pi\)
\(774\) 0 0
\(775\) − 1519.72i − 1.96093i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1153.36 1.48056
\(780\) 0 0
\(781\) 155.743i 0.199415i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 182.691 0.232727
\(786\) 0 0
\(787\) −390.371 −0.496024 −0.248012 0.968757i \(-0.579777\pi\)
−0.248012 + 0.968757i \(0.579777\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 128.736i 0.162751i
\(792\) 0 0
\(793\) −578.762 −0.729838
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 480.484i − 0.602866i −0.953487 0.301433i \(-0.902535\pi\)
0.953487 0.301433i \(-0.0974649\pi\)
\(798\) 0 0
\(799\) 289.768i 0.362664i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −109.942 −0.136915
\(804\) 0 0
\(805\) − 690.289i − 0.857502i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1074.34 1.32798 0.663991 0.747741i \(-0.268861\pi\)
0.663991 + 0.747741i \(0.268861\pi\)
\(810\) 0 0
\(811\) −740.795 −0.913434 −0.456717 0.889612i \(-0.650975\pi\)
−0.456717 + 0.889612i \(0.650975\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2200.13i 2.69954i
\(816\) 0 0
\(817\) 915.452 1.12050
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 216.891i − 0.264180i −0.991238 0.132090i \(-0.957831\pi\)
0.991238 0.132090i \(-0.0421687\pi\)
\(822\) 0 0
\(823\) 1368.30i 1.66257i 0.555844 + 0.831287i \(0.312395\pi\)
−0.555844 + 0.831287i \(0.687605\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −936.451 −1.13235 −0.566174 0.824286i \(-0.691577\pi\)
−0.566174 + 0.824286i \(0.691577\pi\)
\(828\) 0 0
\(829\) 17.5189i 0.0211325i 0.999944 + 0.0105663i \(0.00336341\pi\)
−0.999944 + 0.0105663i \(0.996637\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −567.381 −0.681130
\(834\) 0 0
\(835\) −1955.27 −2.34164
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 115.442i 0.137595i 0.997631 + 0.0687976i \(0.0219163\pi\)
−0.997631 + 0.0687976i \(0.978084\pi\)
\(840\) 0 0
\(841\) −365.915 −0.435095
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 2235.97i − 2.64612i
\(846\) 0 0
\(847\) − 280.929i − 0.331676i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1043.32 −1.22599
\(852\) 0 0
\(853\) − 1273.82i − 1.49335i −0.665191 0.746673i \(-0.731650\pi\)
0.665191 0.746673i \(-0.268350\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 517.583 0.603947 0.301973 0.953316i \(-0.402355\pi\)
0.301973 + 0.953316i \(0.402355\pi\)
\(858\) 0 0
\(859\) −354.668 −0.412885 −0.206443 0.978459i \(-0.566189\pi\)
−0.206443 + 0.978459i \(0.566189\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1069.96i 1.23982i 0.784673 + 0.619910i \(0.212831\pi\)
−0.784673 + 0.619910i \(0.787169\pi\)
\(864\) 0 0
\(865\) 2459.57 2.84343
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 199.809i − 0.229930i
\(870\) 0 0
\(871\) − 208.224i − 0.239063i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 409.324 0.467799
\(876\) 0 0
\(877\) − 101.572i − 0.115818i −0.998322 0.0579088i \(-0.981557\pi\)
0.998322 0.0579088i \(-0.0184433\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −858.174 −0.974091 −0.487045 0.873377i \(-0.661925\pi\)
−0.487045 + 0.873377i \(0.661925\pi\)
\(882\) 0 0
\(883\) −437.517 −0.495489 −0.247745 0.968825i \(-0.579689\pi\)
−0.247745 + 0.968825i \(0.579689\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 938.095i − 1.05760i −0.848745 0.528802i \(-0.822641\pi\)
0.848745 0.528802i \(-0.177359\pi\)
\(888\) 0 0
\(889\) −181.564 −0.204234
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 721.107i − 0.807510i
\(894\) 0 0
\(895\) − 2554.12i − 2.85376i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1171.93 1.30359
\(900\) 0 0
\(901\) 347.553i 0.385741i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 460.556 0.508902
\(906\) 0 0
\(907\) −409.225 −0.451185 −0.225593 0.974222i \(-0.572432\pi\)
−0.225593 + 0.974222i \(0.572432\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1544.02i 1.69486i 0.530907 + 0.847430i \(0.321852\pi\)
−0.530907 + 0.847430i \(0.678148\pi\)
\(912\) 0 0
\(913\) 143.560 0.157240
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 388.736i − 0.423921i
\(918\) 0 0
\(919\) − 1204.58i − 1.31075i −0.755305 0.655374i \(-0.772511\pi\)
0.755305 0.655374i \(-0.227489\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1347.97 −1.46042
\(924\) 0 0
\(925\) − 1390.03i − 1.50273i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1317.76 −1.41847 −0.709234 0.704973i \(-0.750959\pi\)
−0.709234 + 0.704973i \(0.750959\pi\)
\(930\) 0 0
\(931\) 1411.97 1.51661
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 266.163i − 0.284667i
\(936\) 0 0
\(937\) −649.172 −0.692820 −0.346410 0.938083i \(-0.612599\pi\)
−0.346410 + 0.938083i \(0.612599\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1352.72i − 1.43753i −0.695252 0.718766i \(-0.744707\pi\)
0.695252 0.718766i \(-0.255293\pi\)
\(942\) 0 0
\(943\) 1189.10i 1.26097i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1194.59 −1.26145 −0.630723 0.776008i \(-0.717242\pi\)
−0.630723 + 0.776008i \(0.717242\pi\)
\(948\) 0 0
\(949\) − 951.562i − 1.00270i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −681.561 −0.715174 −0.357587 0.933880i \(-0.616400\pi\)
−0.357587 + 0.933880i \(0.616400\pi\)
\(954\) 0 0
\(955\) −2113.48 −2.21307
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 533.440i − 0.556246i
\(960\) 0 0
\(961\) −176.950 −0.184131
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 553.244i 0.573310i
\(966\) 0 0
\(967\) 1050.38i 1.08623i 0.839658 + 0.543115i \(0.182755\pi\)
−0.839658 + 0.543115i \(0.817245\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1307.27 −1.34631 −0.673157 0.739500i \(-0.735062\pi\)
−0.673157 + 0.739500i \(0.735062\pi\)
\(972\) 0 0
\(973\) − 92.3990i − 0.0949630i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.7045 0.0170977 0.00854886 0.999963i \(-0.497279\pi\)
0.00854886 + 0.999963i \(0.497279\pi\)
\(978\) 0 0
\(979\) −336.914 −0.344141
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 373.268i 0.379723i 0.981811 + 0.189862i \(0.0608040\pi\)
−0.981811 + 0.189862i \(0.939196\pi\)
\(984\) 0 0
\(985\) −1275.71 −1.29513
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 943.819i 0.954316i
\(990\) 0 0
\(991\) 1028.67i 1.03801i 0.854772 + 0.519004i \(0.173697\pi\)
−0.854772 + 0.519004i \(0.826303\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 309.487 0.311043
\(996\) 0 0
\(997\) 968.595i 0.971509i 0.874095 + 0.485755i \(0.161455\pi\)
−0.874095 + 0.485755i \(0.838545\pi\)
\(998\) 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.b.i.1567.12 yes 12
3.2 odd 2 1728.3.b.j.1567.2 yes 12
4.3 odd 2 inner 1728.3.b.i.1567.11 yes 12
8.3 odd 2 inner 1728.3.b.i.1567.1 12
8.5 even 2 inner 1728.3.b.i.1567.2 yes 12
12.11 even 2 1728.3.b.j.1567.1 yes 12
24.5 odd 2 1728.3.b.j.1567.12 yes 12
24.11 even 2 1728.3.b.j.1567.11 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.i.1567.1 12 8.3 odd 2 inner
1728.3.b.i.1567.2 yes 12 8.5 even 2 inner
1728.3.b.i.1567.11 yes 12 4.3 odd 2 inner
1728.3.b.i.1567.12 yes 12 1.1 even 1 trivial
1728.3.b.j.1567.1 yes 12 12.11 even 2
1728.3.b.j.1567.2 yes 12 3.2 odd 2
1728.3.b.j.1567.11 yes 12 24.11 even 2
1728.3.b.j.1567.12 yes 12 24.5 odd 2