Properties

Label 1728.3.b.f.1567.3
Level $1728$
Weight $3$
Character 1728.1567
Analytic conductor $47.085$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.3
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1567
Dual form 1728.3.b.f.1567.2

$q$-expansion

\(f(q)\) \(=\) \(q+6.00000i q^{5} +5.00000i q^{7} +O(q^{10})\) \(q+6.00000i q^{5} +5.00000i q^{7} +6.00000 q^{11} -5.19615i q^{13} +31.1769 q^{17} +25.9808 q^{19} -31.1769i q^{23} -11.0000 q^{25} -36.0000i q^{29} -26.0000i q^{31} -30.0000 q^{35} +25.9808i q^{37} +20.7846 q^{43} -31.1769i q^{47} +24.0000 q^{49} +60.0000i q^{53} +36.0000i q^{55} +18.0000 q^{59} +77.9423i q^{61} +31.1769 q^{65} +77.9423 q^{67} +25.0000 q^{73} +30.0000i q^{77} +31.0000i q^{79} +120.000 q^{83} +187.061i q^{85} -155.885 q^{89} +25.9808 q^{91} +155.885i q^{95} -85.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{11} - 44 q^{25} - 120 q^{35} + 96 q^{49} + 72 q^{59} + 100 q^{73} + 480 q^{83} - 340 q^{97}+O(q^{100}) \) 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
\(5\) 6.00000i 1.20000i 0.800000 + 0.600000i \(0.204833\pi\)
−0.800000 + 0.600000i \(0.795167\pi\)
\(6\) 0 0
\(7\) 5.00000i 0.714286i 0.934050 + 0.357143i \(0.116249\pi\)
−0.934050 + 0.357143i \(0.883751\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 0.545455 0.272727 0.962091i \(-0.412074\pi\)
0.272727 + 0.962091i \(0.412074\pi\)
\(12\) 0 0
\(13\) − 5.19615i − 0.399704i −0.979826 0.199852i \(-0.935954\pi\)
0.979826 0.199852i \(-0.0640461\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 31.1769 1.83394 0.916968 0.398961i \(-0.130629\pi\)
0.916968 + 0.398961i \(0.130629\pi\)
\(18\) 0 0
\(19\) 25.9808 1.36741 0.683704 0.729759i \(-0.260368\pi\)
0.683704 + 0.729759i \(0.260368\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 31.1769i − 1.35552i −0.735284 0.677759i \(-0.762951\pi\)
0.735284 0.677759i \(-0.237049\pi\)
\(24\) 0 0
\(25\) −11.0000 −0.440000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 36.0000i − 1.24138i −0.784056 0.620690i \(-0.786853\pi\)
0.784056 0.620690i \(-0.213147\pi\)
\(30\) 0 0
\(31\) − 26.0000i − 0.838710i −0.907822 0.419355i \(-0.862256\pi\)
0.907822 0.419355i \(-0.137744\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −30.0000 −0.857143
\(36\) 0 0
\(37\) 25.9808i 0.702183i 0.936341 + 0.351091i \(0.114189\pi\)
−0.936341 + 0.351091i \(0.885811\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 20.7846 0.483363 0.241682 0.970356i \(-0.422301\pi\)
0.241682 + 0.970356i \(0.422301\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 31.1769i − 0.663339i −0.943396 0.331669i \(-0.892388\pi\)
0.943396 0.331669i \(-0.107612\pi\)
\(48\) 0 0
\(49\) 24.0000 0.489796
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 60.0000i 1.13208i 0.824379 + 0.566038i \(0.191524\pi\)
−0.824379 + 0.566038i \(0.808476\pi\)
\(54\) 0 0
\(55\) 36.0000i 0.654545i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 18.0000 0.305085 0.152542 0.988297i \(-0.451254\pi\)
0.152542 + 0.988297i \(0.451254\pi\)
\(60\) 0 0
\(61\) 77.9423i 1.27774i 0.769314 + 0.638871i \(0.220598\pi\)
−0.769314 + 0.638871i \(0.779402\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 31.1769 0.479645
\(66\) 0 0
\(67\) 77.9423 1.16332 0.581659 0.813433i \(-0.302404\pi\)
0.581659 + 0.813433i \(0.302404\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 25.0000 0.342466 0.171233 0.985231i \(-0.445225\pi\)
0.171233 + 0.985231i \(0.445225\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 30.0000i 0.389610i
\(78\) 0 0
\(79\) 31.0000i 0.392405i 0.980563 + 0.196203i \(0.0628610\pi\)
−0.980563 + 0.196203i \(0.937139\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 120.000 1.44578 0.722892 0.690961i \(-0.242813\pi\)
0.722892 + 0.690961i \(0.242813\pi\)
\(84\) 0 0
\(85\) 187.061i 2.20072i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −155.885 −1.75151 −0.875756 0.482754i \(-0.839637\pi\)
−0.875756 + 0.482754i \(0.839637\pi\)
\(90\) 0 0
\(91\) 25.9808 0.285503
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 155.885i 1.64089i
\(96\) 0 0
\(97\) −85.0000 −0.876289 −0.438144 0.898905i \(-0.644364\pi\)
−0.438144 + 0.898905i \(0.644364\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 156.000i − 1.54455i −0.635286 0.772277i \(-0.719118\pi\)
0.635286 0.772277i \(-0.280882\pi\)
\(102\) 0 0
\(103\) − 125.000i − 1.21359i −0.794858 0.606796i \(-0.792454\pi\)
0.794858 0.606796i \(-0.207546\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −150.000 −1.40187 −0.700935 0.713226i \(-0.747234\pi\)
−0.700935 + 0.713226i \(0.747234\pi\)
\(108\) 0 0
\(109\) 103.923i 0.953422i 0.879060 + 0.476711i \(0.158171\pi\)
−0.879060 + 0.476711i \(0.841829\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 31.1769 0.275902 0.137951 0.990439i \(-0.455948\pi\)
0.137951 + 0.990439i \(0.455948\pi\)
\(114\) 0 0
\(115\) 187.061 1.62662
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 155.885i 1.30995i
\(120\) 0 0
\(121\) −85.0000 −0.702479
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 84.0000i 0.672000i
\(126\) 0 0
\(127\) 110.000i 0.866142i 0.901360 + 0.433071i \(0.142570\pi\)
−0.901360 + 0.433071i \(0.857430\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −156.000 −1.19084 −0.595420 0.803415i \(-0.703014\pi\)
−0.595420 + 0.803415i \(0.703014\pi\)
\(132\) 0 0
\(133\) 129.904i 0.976720i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −155.885 −1.13784 −0.568922 0.822392i \(-0.692639\pi\)
−0.568922 + 0.822392i \(0.692639\pi\)
\(138\) 0 0
\(139\) 181.865 1.30838 0.654192 0.756329i \(-0.273009\pi\)
0.654192 + 0.756329i \(0.273009\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 31.1769i − 0.218020i
\(144\) 0 0
\(145\) 216.000 1.48966
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 24.0000i − 0.161074i −0.996752 0.0805369i \(-0.974337\pi\)
0.996752 0.0805369i \(-0.0256635\pi\)
\(150\) 0 0
\(151\) − 31.0000i − 0.205298i −0.994718 0.102649i \(-0.967268\pi\)
0.994718 0.102649i \(-0.0327318\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 156.000 1.00645
\(156\) 0 0
\(157\) − 20.7846i − 0.132386i −0.997807 0.0661930i \(-0.978915\pi\)
0.997807 0.0661930i \(-0.0210853\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 155.885 0.968227
\(162\) 0 0
\(163\) 306.573 1.88082 0.940408 0.340048i \(-0.110443\pi\)
0.940408 + 0.340048i \(0.110443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 155.885i − 0.933441i −0.884405 0.466720i \(-0.845435\pi\)
0.884405 0.466720i \(-0.154565\pi\)
\(168\) 0 0
\(169\) 142.000 0.840237
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 120.000i 0.693642i 0.937931 + 0.346821i \(0.112739\pi\)
−0.937931 + 0.346821i \(0.887261\pi\)
\(174\) 0 0
\(175\) − 55.0000i − 0.314286i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 192.000 1.07263 0.536313 0.844019i \(-0.319817\pi\)
0.536313 + 0.844019i \(0.319817\pi\)
\(180\) 0 0
\(181\) 337.750i 1.86602i 0.359848 + 0.933011i \(0.382828\pi\)
−0.359848 + 0.933011i \(0.617172\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −155.885 −0.842619
\(186\) 0 0
\(187\) 187.061 1.00033
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 155.885i − 0.816150i −0.912948 0.408075i \(-0.866200\pi\)
0.912948 0.408075i \(-0.133800\pi\)
\(192\) 0 0
\(193\) −325.000 −1.68394 −0.841969 0.539526i \(-0.818603\pi\)
−0.841969 + 0.539526i \(0.818603\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 30.0000i 0.152284i 0.997097 + 0.0761421i \(0.0242603\pi\)
−0.997097 + 0.0761421i \(0.975740\pi\)
\(198\) 0 0
\(199\) 259.000i 1.30151i 0.759289 + 0.650754i \(0.225547\pi\)
−0.759289 + 0.650754i \(0.774453\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 180.000 0.886700
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 155.885 0.745859
\(210\) 0 0
\(211\) 25.9808 0.123132 0.0615658 0.998103i \(-0.480391\pi\)
0.0615658 + 0.998103i \(0.480391\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 124.708i 0.580036i
\(216\) 0 0
\(217\) 130.000 0.599078
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 162.000i − 0.733032i
\(222\) 0 0
\(223\) − 310.000i − 1.39013i −0.718945 0.695067i \(-0.755375\pi\)
0.718945 0.695067i \(-0.244625\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −360.000 −1.58590 −0.792952 0.609285i \(-0.791457\pi\)
−0.792952 + 0.609285i \(0.791457\pi\)
\(228\) 0 0
\(229\) 207.846i 0.907625i 0.891097 + 0.453812i \(0.149936\pi\)
−0.891097 + 0.453812i \(0.850064\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 311.769 1.33807 0.669033 0.743233i \(-0.266709\pi\)
0.669033 + 0.743233i \(0.266709\pi\)
\(234\) 0 0
\(235\) 187.061 0.796006
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 311.769i − 1.30447i −0.758015 0.652237i \(-0.773831\pi\)
0.758015 0.652237i \(-0.226169\pi\)
\(240\) 0 0
\(241\) −131.000 −0.543568 −0.271784 0.962358i \(-0.587614\pi\)
−0.271784 + 0.962358i \(0.587614\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 144.000i 0.587755i
\(246\) 0 0
\(247\) − 135.000i − 0.546559i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −216.000 −0.860558 −0.430279 0.902696i \(-0.641585\pi\)
−0.430279 + 0.902696i \(0.641585\pi\)
\(252\) 0 0
\(253\) − 187.061i − 0.739373i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 187.061 0.727866 0.363933 0.931425i \(-0.381434\pi\)
0.363933 + 0.931425i \(0.381434\pi\)
\(258\) 0 0
\(259\) −129.904 −0.501559
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 311.769i − 1.18543i −0.805411 0.592717i \(-0.798055\pi\)
0.805411 0.592717i \(-0.201945\pi\)
\(264\) 0 0
\(265\) −360.000 −1.35849
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 414.000i − 1.53903i −0.638627 0.769517i \(-0.720497\pi\)
0.638627 0.769517i \(-0.279503\pi\)
\(270\) 0 0
\(271\) 317.000i 1.16974i 0.811126 + 0.584871i \(0.198855\pi\)
−0.811126 + 0.584871i \(0.801145\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −66.0000 −0.240000
\(276\) 0 0
\(277\) 519.615i 1.87587i 0.346815 + 0.937934i \(0.387263\pi\)
−0.346815 + 0.937934i \(0.612737\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 311.769 1.10950 0.554749 0.832018i \(-0.312814\pi\)
0.554749 + 0.832018i \(0.312814\pi\)
\(282\) 0 0
\(283\) 103.923 0.367219 0.183610 0.982999i \(-0.441222\pi\)
0.183610 + 0.982999i \(0.441222\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 683.000 2.36332
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 270.000i 0.921502i 0.887530 + 0.460751i \(0.152420\pi\)
−0.887530 + 0.460751i \(0.847580\pi\)
\(294\) 0 0
\(295\) 108.000i 0.366102i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −162.000 −0.541806
\(300\) 0 0
\(301\) 103.923i 0.345259i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −467.654 −1.53329
\(306\) 0 0
\(307\) −436.477 −1.42175 −0.710874 0.703319i \(-0.751701\pi\)
−0.710874 + 0.703319i \(0.751701\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 467.654i 1.50371i 0.659329 + 0.751855i \(0.270841\pi\)
−0.659329 + 0.751855i \(0.729159\pi\)
\(312\) 0 0
\(313\) 85.0000 0.271565 0.135783 0.990739i \(-0.456645\pi\)
0.135783 + 0.990739i \(0.456645\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 120.000i 0.378549i 0.981924 + 0.189274i \(0.0606136\pi\)
−0.981924 + 0.189274i \(0.939386\pi\)
\(318\) 0 0
\(319\) − 216.000i − 0.677116i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 810.000 2.50774
\(324\) 0 0
\(325\) 57.1577i 0.175870i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 155.885 0.473813
\(330\) 0 0
\(331\) −25.9808 −0.0784917 −0.0392459 0.999230i \(-0.512496\pi\)
−0.0392459 + 0.999230i \(0.512496\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 467.654i 1.39598i
\(336\) 0 0
\(337\) −325.000 −0.964392 −0.482196 0.876063i \(-0.660161\pi\)
−0.482196 + 0.876063i \(0.660161\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 156.000i − 0.457478i
\(342\) 0 0
\(343\) 365.000i 1.06414i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 300.000 0.864553 0.432277 0.901741i \(-0.357710\pi\)
0.432277 + 0.901741i \(0.357710\pi\)
\(348\) 0 0
\(349\) − 597.558i − 1.71220i −0.516811 0.856100i \(-0.672881\pi\)
0.516811 0.856100i \(-0.327119\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −436.477 −1.23648 −0.618239 0.785990i \(-0.712154\pi\)
−0.618239 + 0.785990i \(0.712154\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 467.654i − 1.30266i −0.758796 0.651328i \(-0.774212\pi\)
0.758796 0.651328i \(-0.225788\pi\)
\(360\) 0 0
\(361\) 314.000 0.869806
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 150.000i 0.410959i
\(366\) 0 0
\(367\) − 185.000i − 0.504087i −0.967716 0.252044i \(-0.918897\pi\)
0.967716 0.252044i \(-0.0811026\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −300.000 −0.808625
\(372\) 0 0
\(373\) − 57.1577i − 0.153238i −0.997060 0.0766189i \(-0.975588\pi\)
0.997060 0.0766189i \(-0.0244125\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −187.061 −0.496184
\(378\) 0 0
\(379\) −129.904 −0.342754 −0.171377 0.985206i \(-0.554822\pi\)
−0.171377 + 0.985206i \(0.554822\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 187.061i 0.488411i 0.969723 + 0.244206i \(0.0785272\pi\)
−0.969723 + 0.244206i \(0.921473\pi\)
\(384\) 0 0
\(385\) −180.000 −0.467532
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 354.000i 0.910026i 0.890485 + 0.455013i \(0.150365\pi\)
−0.890485 + 0.455013i \(0.849635\pi\)
\(390\) 0 0
\(391\) − 972.000i − 2.48593i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −186.000 −0.470886
\(396\) 0 0
\(397\) − 644.323i − 1.62298i −0.584367 0.811490i \(-0.698657\pi\)
0.584367 0.811490i \(-0.301343\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −311.769 −0.777479 −0.388740 0.921348i \(-0.627089\pi\)
−0.388740 + 0.921348i \(0.627089\pi\)
\(402\) 0 0
\(403\) −135.100 −0.335236
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 155.885i 0.383009i
\(408\) 0 0
\(409\) −539.000 −1.31785 −0.658924 0.752209i \(-0.728988\pi\)
−0.658924 + 0.752209i \(0.728988\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 90.0000i 0.217918i
\(414\) 0 0
\(415\) 720.000i 1.73494i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −306.000 −0.730310 −0.365155 0.930947i \(-0.618984\pi\)
−0.365155 + 0.930947i \(0.618984\pi\)
\(420\) 0 0
\(421\) − 25.9808i − 0.0617120i −0.999524 0.0308560i \(-0.990177\pi\)
0.999524 0.0308560i \(-0.00982333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −342.946 −0.806932
\(426\) 0 0
\(427\) −389.711 −0.912673
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 155.885i 0.361681i 0.983512 + 0.180841i \(0.0578818\pi\)
−0.983512 + 0.180841i \(0.942118\pi\)
\(432\) 0 0
\(433\) 250.000 0.577367 0.288684 0.957425i \(-0.406782\pi\)
0.288684 + 0.957425i \(0.406782\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 810.000i − 1.85355i
\(438\) 0 0
\(439\) 22.0000i 0.0501139i 0.999686 + 0.0250569i \(0.00797671\pi\)
−0.999686 + 0.0250569i \(0.992023\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −540.000 −1.21896 −0.609481 0.792801i \(-0.708622\pi\)
−0.609481 + 0.792801i \(0.708622\pi\)
\(444\) 0 0
\(445\) − 935.307i − 2.10181i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −155.885 −0.347182 −0.173591 0.984818i \(-0.555537\pi\)
−0.173591 + 0.984818i \(0.555537\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 155.885i 0.342603i
\(456\) 0 0
\(457\) −230.000 −0.503282 −0.251641 0.967821i \(-0.580970\pi\)
−0.251641 + 0.967821i \(0.580970\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 78.0000i 0.169197i 0.996415 + 0.0845987i \(0.0269608\pi\)
−0.996415 + 0.0845987i \(0.973039\pi\)
\(462\) 0 0
\(463\) − 55.0000i − 0.118790i −0.998235 0.0593952i \(-0.981083\pi\)
0.998235 0.0593952i \(-0.0189172\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −150.000 −0.321199 −0.160600 0.987020i \(-0.551343\pi\)
−0.160600 + 0.987020i \(0.551343\pi\)
\(468\) 0 0
\(469\) 389.711i 0.830941i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 124.708 0.263653
\(474\) 0 0
\(475\) −285.788 −0.601660
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 311.769i 0.650875i 0.945564 + 0.325438i \(0.105512\pi\)
−0.945564 + 0.325438i \(0.894488\pi\)
\(480\) 0 0
\(481\) 135.000 0.280665
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 510.000i − 1.05155i
\(486\) 0 0
\(487\) 785.000i 1.61191i 0.591977 + 0.805955i \(0.298348\pi\)
−0.591977 + 0.805955i \(0.701652\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 618.000 1.25866 0.629328 0.777140i \(-0.283330\pi\)
0.629328 + 0.777140i \(0.283330\pi\)
\(492\) 0 0
\(493\) − 1122.37i − 2.27661i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −623.538 −1.24958 −0.624788 0.780795i \(-0.714815\pi\)
−0.624788 + 0.780795i \(0.714815\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 654.715i 1.30162i 0.759240 + 0.650810i \(0.225571\pi\)
−0.759240 + 0.650810i \(0.774429\pi\)
\(504\) 0 0
\(505\) 936.000 1.85347
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 366.000i 0.719057i 0.933134 + 0.359528i \(0.117062\pi\)
−0.933134 + 0.359528i \(0.882938\pi\)
\(510\) 0 0
\(511\) 125.000i 0.244618i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 750.000 1.45631
\(516\) 0 0
\(517\) − 187.061i − 0.361821i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 779.423 1.49601 0.748007 0.663691i \(-0.231011\pi\)
0.748007 + 0.663691i \(0.231011\pi\)
\(522\) 0 0
\(523\) −109.119 −0.208641 −0.104320 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 810.600i − 1.53814i
\(528\) 0 0
\(529\) −443.000 −0.837429
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 900.000i − 1.68224i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 144.000 0.267161
\(540\) 0 0
\(541\) − 649.519i − 1.20059i −0.799779 0.600295i \(-0.795050\pi\)
0.799779 0.600295i \(-0.204950\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −623.538 −1.14411
\(546\) 0 0
\(547\) −420.888 −0.769449 −0.384724 0.923032i \(-0.625703\pi\)
−0.384724 + 0.923032i \(0.625703\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 935.307i − 1.69747i
\(552\) 0 0
\(553\) −155.000 −0.280289
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 690.000i 1.23878i 0.785084 + 0.619390i \(0.212620\pi\)
−0.785084 + 0.619390i \(0.787380\pi\)
\(558\) 0 0
\(559\) − 108.000i − 0.193202i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −840.000 −1.49201 −0.746004 0.665942i \(-0.768030\pi\)
−0.746004 + 0.665942i \(0.768030\pi\)
\(564\) 0 0
\(565\) 187.061i 0.331082i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1091.19 1.91774 0.958868 0.283852i \(-0.0916123\pi\)
0.958868 + 0.283852i \(0.0916123\pi\)
\(570\) 0 0
\(571\) −805.404 −1.41051 −0.705257 0.708952i \(-0.749168\pi\)
−0.705257 + 0.708952i \(0.749168\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 342.946i 0.596428i
\(576\) 0 0
\(577\) 385.000 0.667244 0.333622 0.942707i \(-0.391729\pi\)
0.333622 + 0.942707i \(0.391729\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 600.000i 1.03270i
\(582\) 0 0
\(583\) 360.000i 0.617496i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 930.000 1.58433 0.792164 0.610309i \(-0.208955\pi\)
0.792164 + 0.610309i \(0.208955\pi\)
\(588\) 0 0
\(589\) − 675.500i − 1.14686i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 498.831 0.841198 0.420599 0.907247i \(-0.361820\pi\)
0.420599 + 0.907247i \(0.361820\pi\)
\(594\) 0 0
\(595\) −935.307 −1.57195
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 311.769i − 0.520483i −0.965544 0.260241i \(-0.916198\pi\)
0.965544 0.260241i \(-0.0838021\pi\)
\(600\) 0 0
\(601\) −346.000 −0.575707 −0.287854 0.957674i \(-0.592942\pi\)
−0.287854 + 0.957674i \(0.592942\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 510.000i − 0.842975i
\(606\) 0 0
\(607\) − 595.000i − 0.980231i −0.871658 0.490115i \(-0.836955\pi\)
0.871658 0.490115i \(-0.163045\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −162.000 −0.265139
\(612\) 0 0
\(613\) − 109.119i − 0.178008i −0.996031 0.0890042i \(-0.971632\pi\)
0.996031 0.0890042i \(-0.0283685\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 592.361 0.960067 0.480034 0.877250i \(-0.340624\pi\)
0.480034 + 0.877250i \(0.340624\pi\)
\(618\) 0 0
\(619\) −701.481 −1.13325 −0.566624 0.823976i \(-0.691751\pi\)
−0.566624 + 0.823976i \(0.691751\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 779.423i − 1.25108i
\(624\) 0 0
\(625\) −779.000 −1.24640
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 810.000i 1.28776i
\(630\) 0 0
\(631\) 713.000i 1.12995i 0.825107 + 0.564976i \(0.191115\pi\)
−0.825107 + 0.564976i \(0.808885\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −660.000 −1.03937
\(636\) 0 0
\(637\) − 124.708i − 0.195773i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 436.477 0.678813 0.339407 0.940640i \(-0.389774\pi\)
0.339407 + 0.940640i \(0.389774\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 623.538i − 0.963738i −0.876243 0.481869i \(-0.839958\pi\)
0.876243 0.481869i \(-0.160042\pi\)
\(648\) 0 0
\(649\) 108.000 0.166410
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 960.000i − 1.47014i −0.677992 0.735069i \(-0.737150\pi\)
0.677992 0.735069i \(-0.262850\pi\)
\(654\) 0 0
\(655\) − 936.000i − 1.42901i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −132.000 −0.200303 −0.100152 0.994972i \(-0.531933\pi\)
−0.100152 + 0.994972i \(0.531933\pi\)
\(660\) 0 0
\(661\) 1013.25i 1.53290i 0.642301 + 0.766452i \(0.277980\pi\)
−0.642301 + 0.766452i \(0.722020\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −779.423 −1.17206
\(666\) 0 0
\(667\) −1122.37 −1.68271
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 467.654i 0.696950i
\(672\) 0 0
\(673\) 445.000 0.661218 0.330609 0.943768i \(-0.392746\pi\)
0.330609 + 0.943768i \(0.392746\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 990.000i − 1.46233i −0.682199 0.731167i \(-0.738976\pi\)
0.682199 0.731167i \(-0.261024\pi\)
\(678\) 0 0
\(679\) − 425.000i − 0.625920i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1260.00 −1.84480 −0.922401 0.386233i \(-0.873776\pi\)
−0.922401 + 0.386233i \(0.873776\pi\)
\(684\) 0 0
\(685\) − 935.307i − 1.36541i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 311.769 0.452495
\(690\) 0 0
\(691\) −103.923 −0.150395 −0.0751976 0.997169i \(-0.523959\pi\)
−0.0751976 + 0.997169i \(0.523959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1091.19i 1.57006i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 246.000i − 0.350927i −0.984486 0.175464i \(-0.943858\pi\)
0.984486 0.175464i \(-0.0561424\pi\)
\(702\) 0 0
\(703\) 675.000i 0.960171i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 780.000 1.10325
\(708\) 0 0
\(709\) − 129.904i − 0.183221i −0.995795 0.0916106i \(-0.970799\pi\)
0.995795 0.0916106i \(-0.0292015\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −810.600 −1.13689
\(714\) 0 0
\(715\) 187.061 0.261624
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 1247.08i − 1.73446i −0.497908 0.867230i \(-0.665898\pi\)
0.497908 0.867230i \(-0.334102\pi\)
\(720\) 0 0
\(721\) 625.000 0.866852
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 396.000i 0.546207i
\(726\) 0 0
\(727\) 770.000i 1.05915i 0.848264 + 0.529574i \(0.177648\pi\)
−0.848264 + 0.529574i \(0.822352\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 648.000 0.886457
\(732\) 0 0
\(733\) − 1143.15i − 1.55955i −0.626057 0.779777i \(-0.715332\pi\)
0.626057 0.779777i \(-0.284668\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 467.654 0.634537
\(738\) 0 0
\(739\) 727.461 0.984386 0.492193 0.870486i \(-0.336195\pi\)
0.492193 + 0.870486i \(0.336195\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 592.361i − 0.797256i −0.917113 0.398628i \(-0.869486\pi\)
0.917113 0.398628i \(-0.130514\pi\)
\(744\) 0 0
\(745\) 144.000 0.193289
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 750.000i − 1.00134i
\(750\) 0 0
\(751\) 1099.00i 1.46338i 0.681636 + 0.731691i \(0.261269\pi\)
−0.681636 + 0.731691i \(0.738731\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 186.000 0.246358
\(756\) 0 0
\(757\) − 420.888i − 0.555995i −0.960582 0.277998i \(-0.910329\pi\)
0.960582 0.277998i \(-0.0896707\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 155.885 0.204842 0.102421 0.994741i \(-0.467341\pi\)
0.102421 + 0.994741i \(0.467341\pi\)
\(762\) 0 0
\(763\) −519.615 −0.681016
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 93.5307i − 0.121944i
\(768\) 0 0
\(769\) −1009.00 −1.31209 −0.656047 0.754720i \(-0.727773\pi\)
−0.656047 + 0.754720i \(0.727773\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 360.000i 0.465718i 0.972511 + 0.232859i \(0.0748081\pi\)
−0.972511 + 0.232859i \(0.925192\pi\)
\(774\) 0 0
\(775\) 286.000i 0.369032i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 124.708 0.158863
\(786\) 0 0
\(787\) −514.419 −0.653646 −0.326823 0.945086i \(-0.605978\pi\)
−0.326823 + 0.945086i \(0.605978\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 155.885i 0.197073i
\(792\) 0 0
\(793\) 405.000 0.510719
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 660.000i 0.828105i 0.910253 + 0.414053i \(0.135887\pi\)
−0.910253 + 0.414053i \(0.864113\pi\)
\(798\) 0 0
\(799\) − 972.000i − 1.21652i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 150.000 0.186800
\(804\) 0 0
\(805\) 935.307i 1.16187i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1247.08 −1.54150 −0.770752 0.637135i \(-0.780119\pi\)
−0.770752 + 0.637135i \(0.780119\pi\)
\(810\) 0 0
\(811\) 519.615 0.640709 0.320355 0.947298i \(-0.396198\pi\)
0.320355 + 0.947298i \(0.396198\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1839.44i 2.25698i
\(816\) 0 0
\(817\) 540.000 0.660955
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1242.00i 1.51279i 0.654116 + 0.756395i \(0.273041\pi\)
−0.654116 + 0.756395i \(0.726959\pi\)
\(822\) 0 0
\(823\) − 955.000i − 1.16039i −0.814478 0.580194i \(-0.802977\pi\)
0.814478 0.580194i \(-0.197023\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −90.0000 −0.108827 −0.0544135 0.998518i \(-0.517329\pi\)
−0.0544135 + 0.998518i \(0.517329\pi\)
\(828\) 0 0
\(829\) − 181.865i − 0.219379i −0.993966 0.109690i \(-0.965014\pi\)
0.993966 0.109690i \(-0.0349857\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 748.246 0.898254
\(834\) 0 0
\(835\) 935.307 1.12013
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1558.85i 1.85798i 0.370104 + 0.928990i \(0.379322\pi\)
−0.370104 + 0.928990i \(0.620678\pi\)
\(840\) 0 0
\(841\) −455.000 −0.541023
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 852.000i 1.00828i
\(846\) 0 0
\(847\) − 425.000i − 0.501771i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 810.000 0.951821
\(852\) 0 0
\(853\) 46.7654i 0.0548246i 0.999624 + 0.0274123i \(0.00872670\pi\)
−0.999624 + 0.0274123i \(0.991273\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −623.538 −0.727583 −0.363791 0.931480i \(-0.618518\pi\)
−0.363791 + 0.931480i \(0.618518\pi\)
\(858\) 0 0
\(859\) 129.904 0.151227 0.0756134 0.997137i \(-0.475909\pi\)
0.0756134 + 0.997137i \(0.475909\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 342.946i 0.397388i 0.980062 + 0.198694i \(0.0636700\pi\)
−0.980062 + 0.198694i \(0.936330\pi\)
\(864\) 0 0
\(865\) −720.000 −0.832370
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 186.000i 0.214039i
\(870\) 0 0
\(871\) − 405.000i − 0.464983i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −420.000 −0.480000
\(876\) 0 0
\(877\) − 1034.03i − 1.17906i −0.807747 0.589529i \(-0.799313\pi\)
0.807747 0.589529i \(-0.200687\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −467.654 −0.530821 −0.265411 0.964135i \(-0.585508\pi\)
−0.265411 + 0.964135i \(0.585508\pi\)
\(882\) 0 0
\(883\) 285.788 0.323656 0.161828 0.986819i \(-0.448261\pi\)
0.161828 + 0.986819i \(0.448261\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 498.831i 0.562380i 0.959652 + 0.281190i \(0.0907290\pi\)
−0.959652 + 0.281190i \(0.909271\pi\)
\(888\) 0 0
\(889\) −550.000 −0.618673
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 810.000i − 0.907055i
\(894\) 0 0
\(895\) 1152.00i 1.28715i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −936.000 −1.04116
\(900\) 0 0
\(901\) 1870.61i 2.07615i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2026.50 −2.23923
\(906\) 0 0
\(907\) 1096.39 1.20881 0.604404 0.796678i \(-0.293411\pi\)
0.604404 + 0.796678i \(0.293411\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 623.538i − 0.684455i −0.939617 0.342227i \(-0.888819\pi\)
0.939617 0.342227i \(-0.111181\pi\)
\(912\) 0 0
\(913\) 720.000 0.788609
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 780.000i − 0.850600i
\(918\) 0 0
\(919\) − 1006.00i − 1.09467i −0.836914 0.547334i \(-0.815643\pi\)
0.836914 0.547334i \(-0.184357\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 285.788i − 0.308960i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 311.769 0.335596 0.167798 0.985821i \(-0.446334\pi\)
0.167798 + 0.985821i \(0.446334\pi\)
\(930\) 0 0
\(931\) 623.538 0.669751
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1122.37i 1.20039i
\(936\) 0 0
\(937\) 565.000 0.602988 0.301494 0.953468i \(-0.402515\pi\)
0.301494 + 0.953468i \(0.402515\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 294.000i 0.312434i 0.987723 + 0.156217i \(0.0499298\pi\)
−0.987723 + 0.156217i \(0.950070\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.0000 0.0316790 0.0158395 0.999875i \(-0.494958\pi\)
0.0158395 + 0.999875i \(0.494958\pi\)
\(948\) 0 0
\(949\) − 129.904i − 0.136885i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 904.131 0.948720 0.474360 0.880331i \(-0.342679\pi\)
0.474360 + 0.880331i \(0.342679\pi\)
\(954\) 0 0
\(955\) 935.307 0.979380
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 779.423i − 0.812745i
\(960\) 0 0
\(961\) 285.000 0.296566
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1950.00i − 2.02073i
\(966\) 0 0
\(967\) − 305.000i − 0.315408i −0.987486 0.157704i \(-0.949591\pi\)
0.987486 0.157704i \(-0.0504093\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1194.00 1.22966 0.614830 0.788660i \(-0.289225\pi\)
0.614830 + 0.788660i \(0.289225\pi\)
\(972\) 0 0
\(973\) 909.327i 0.934560i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 124.708 0.127643 0.0638217 0.997961i \(-0.479671\pi\)
0.0638217 + 0.997961i \(0.479671\pi\)
\(978\) 0 0
\(979\) −935.307 −0.955370
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1278.25i − 1.30036i −0.759780 0.650180i \(-0.774694\pi\)
0.759780 0.650180i \(-0.225306\pi\)
\(984\) 0 0
\(985\) −180.000 −0.182741
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 648.000i − 0.655207i
\(990\) 0 0
\(991\) 511.000i 0.515641i 0.966193 + 0.257820i \(0.0830043\pi\)
−0.966193 + 0.257820i \(0.916996\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1554.00 −1.56181
\(996\) 0 0
\(997\) 1060.02i 1.06320i 0.846994 + 0.531602i \(0.178410\pi\)
−0.846994 + 0.531602i \(0.821590\pi\)
\(998\) 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.f.1567.3 yes 4
3.2 odd 2 1728.3.b.a.1567.1 4
4.3 odd 2 1728.3.b.a.1567.3 yes 4
8.3 odd 2 inner 1728.3.b.f.1567.2 yes 4
8.5 even 2 1728.3.b.a.1567.2 yes 4
12.11 even 2 inner 1728.3.b.f.1567.1 yes 4
24.5 odd 2 inner 1728.3.b.f.1567.4 yes 4
24.11 even 2 1728.3.b.a.1567.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.a.1567.1 4 3.2 odd 2
1728.3.b.a.1567.2 yes 4 8.5 even 2
1728.3.b.a.1567.3 yes 4 4.3 odd 2
1728.3.b.a.1567.4 yes 4 24.11 even 2
1728.3.b.f.1567.1 yes 4 12.11 even 2 inner
1728.3.b.f.1567.2 yes 4 8.3 odd 2 inner
1728.3.b.f.1567.3 yes 4 1.1 even 1 trivial
1728.3.b.f.1567.4 yes 4 24.5 odd 2 inner